WRC RESEARCH REPORT NO. MATHEMATICAL IN ANALYSIS …

WRC RESEARCH REPORT NO. '194

MATHEMATICAL MODELS AND OPTIMIZATION TECHNIQUES FOR USE I N ANALYSIS AND DESIGN OF

WASTEWATER TREATMENT SYSTEMS

Chi-Chung Tang '

Department of C i v i l Engineer ing

E. Downey B r i l l , J r . Department of C i v i l Engineer ing

and I n s t i t u t e f o r Environmental S t u d i e s

John T. P f e f f e r Department of C i v i l Engineer ing

REPORT

PROJECT N O . S-092-ILL

UNIVERSITY OF ILLINOIS

WATER RESOURCES CENTER

2535 Hydrosystems Laboratory

Urbana, I l l i n o i s 61801

November, 1984

T h i s r e p o r t is based on t h e d o c t o r a l d i s s e r t a t i o n o f Chi-Chung Tang a t

t h e U n i v e r s i t y o f I l l i n o i s a t Urbana-Champaign. The a u t h o r s thank M. T.

Su idan , J . W. E h e a r t , and J . S . Liebman f o r t h e i r h e l p f u l s u g g e s t i o n s .

Also, S . A . Burns , B.D.C. Wong, P . H . H u t t o n , M . Okumura, and J . G . Uber

p rov ided v a l u a b l e a s s i s t a n c e . T h i s work was p a r t i a l l y suppor ted by t h e

Water Resources Cen te r a t t h e U n i v e r s i t y o f I l l i n o s under g r a n t number

S-092-.ILL.

v

ABSTRACT

A mathemat ica l framework is developed f o r use i n t h e d e s i g n o f a

secondary wastewater t r e a t m e n t sys tem. Mathemat ica l models p r e d i c t i n g t h e

performance o f v a r i o u s u n i t p r o c e s s e s a r e used t o c o n s t r u c t a comprehensive

sys tem model. Three e f f i c i e n t o p t i m i z a t i o n approaches t o g e n e r a t e c o s t

e f f e c t i v e sys tem d e s i g n s a r e s t u d i e d . The f i r s t approach t r a n s c r i b e s t h e

comprehensive sys tem model i n t o a n o n l i n e a r program t h a t i n c l u d e s 64

v a r i a b l e s and 58 c o n s t r a i n t s . A g e n e r a l i z e d reduced g r a d i e n t a l g o r i t h m is

a p p l i e d t o s o l v e t h i s model. The second approach u s e s a n e x i s t i n g a l g o r i t h m

f o r s o l v i n g g e n e r a l i z e d geomet r i c programs. P a r t i t i o n i n g of model v a r i a b l e s

i n t o two s e t s is n e c e s s a r y . A number of geomet r i c programming subproblems

r e s u l t i n g from t h e p a r t i t i o n i n g a r e s o l v e d . The t h i r d approach decomposes

t h e wastewater sys tem i n t o a l i q u i d and a s l u d g e subsystem. The l i q u i d

subsystem is o p t i m a l l y des igned , whi le t h e s l u d g e subsystem d e s i g n i n c l u d e s

embedded o p t i m i z a t i o n s t e p s . The o v e r a l l o p t i m a l d e s i g n is o b t a i n e d from

c o o r d i n a t i o n between t h e two subsystem d e s i g n s . The comprehensive sys tem

model can be used a s a t o o l f o r t h e a n a l y s i s of p r o c e s s performance.

Impor tan t i n s i g h t s abou t p r o c e s s d e s i g n , modeling, and i n t e g r a t i o n can be

ga ined by e x e r c i s i n g t h e model. P o t e n t i a l l y f r u i t f u l a r e a s f o r r e s e a r c h can

a l s o be i d e n t i f i e d . T h i s is i l l u s t r a t e d through t h e u s e of an example

problem.

Tang, Chi-Chung; B r i l l , E . Downey, J r . ; and P f e f f e r , John T .

MATHEMATICAL MODELS A N D OPTIMIZATION TECHNIQUES FOR USE I N ANALYSIS A N D DESIGN OF WASTEWATER TREATMENT SYSTEMS, Water Resources Cen te r Research Repor t 1 9 4 . Urbana, I L : Water Resources C e n t e r , U n i v e r s i t y of I l l i n o i s a t Urbana-Champaign.

KEYWORDS: Wastewater t r e a t m e n t , mathemat ica l models, o p t i m i z a t i o n

621 ............................................................................................................ L~cluluns 'S'I

...................................................... 8 (5 uo!lcz!u~!ldg J~J 11xo~dd~ UO!~!SO~UIO~~(J .+'I

.................................................................... 98 uo!lc'L!UI!ldg JOJ U-'~J!JO%IV J33I 'I'E

........................... f.9 uo!lcz!m!ldo JOJ mq?!~o%lv ?ua!pcJa pa3npaU paz!lcJauaa 'z.1

....................................................................................................... 19 uo!?3npo~?u1 '1.1

....................... S9 13a011 N8LSAS 3AISNSIBUdkYO3 3HL d0 NOILF'ZIkVILdO 'E

.................................................................................................... 8 I au!I?nO s!saqL 'P'T

viii

P a5e

4.2. Primary Sedimentation ...................................................................................... 135

4.3. Activated Sludge ............................................................................................ 141

4.4. Secondary Sedimentation .................................................................................. 147

4.5. Sludge Thickening ......................................................................................... 152

4.6. An aerobic Digestion ......................................................................................... 163

4.7. Vacuum Filter .................................................................................................. 165

4.8. Design Under Uncertainty: A Multi-objective Approach ................................. 167

............................................................................................................ 4.9. Summary 170

5 . SUMMARY AND FUTURE RESEARCH .................................................................. 173

5.1. Introduction ................................................................................................... 173

5.2. Comprehensive System Design Model ............................................................... 174

' 5.3. Optimization Techniques for Wastewat.er Treatment Systenr h.fodel ............. 176

5.4. Use of Model for Process Analysis ...................................................................... 178

5.5. Future Research ................................................................................................. 179

APPENDIX

A . COST FUNCTIONS O F UNIT PROCESSES .................................................... 181

I3 . ESTIMATING OPERATION COST FOR SLUDGE LANDFILL ........................ 191

C . AN.GYSIS PROGRIZM FOR DESIGN O F

..................................................... WASTEWATER TREATiLlENT SYSTEMS 102

D . GRG hlODEL FOR BASE SYSTEhI DESIGN OPTIhlIZATION ........................ 200

E . IGGP h.lODEL FOR BASE SYSTEhl DESIGN ..................................................... 217

ix

Page

F. GRG MODEL FOR LIQUID SUBSYSTEM OPTIMIZATION .............................. 224

G. SLUDGE SUBSYSTEM DESIGN .......................................................................... 231

H. GRG MODEL FOR SYSTEM WITHOUT A PRIMARY CLARIFIER ................ 235

I. G R G MODEL FOR THE SYSTEM WHERE WASTE ACTIVATED

SLUDGE IS RECIRCULATED TO PRIMARY CLARIFIER ........................... 241

x i

LIST OF TABLES

Table

2.1 - Modcls for Saspendcd Solids Rrrnovnl Efficiency in the Primary Settling

Tank ........................................................................................................................

2.2 - !'viodels for Organic matter Removal Efficiency in the Primary Settling

Tank ........ ... .... .... ... ................. ................... .............. ..................... ...... ..................... .

2.3 - Empirical Models Predicting Total Suspended Solids Concentration in Secon-

dary Clarifier Emuent .............................................................................................

2.4 - Summary of Cost Functions ....................................................................................

2.5 - Summary of Decision Variables in the htodel .........................................................

2.6 - Bounds on the Decision Variables ...........................................................................

2.7 - Base Design Conditions ...........................................................................................

2.8 - Summary of Parameters in the System Model .......................................................

2.9 - Decision Variables for Example Treatment System Design ...................................

3.1 - Computing Experience in Optimizing Wastewater Treatment Systeni

Design ............ ...... . ..... . ... ..... ............. ............. .... . .... ............... ................ ... .......,.....,.. .

3.2 - Summary of Wastewater Treatment System Designs Obtained Using Diflerent

Start ing Points ........................................................................................................

3.3 - Solution Obtained Using GRG with Dinerent Bounds on Selected Varial)les ......

3.4 - Exploring Design Space : Design No. 1 ...................................................................



3.7 - Exploring Design Spacc : Design No. 4 ...................................................................


Page

xii

Page

.................... . 3.9 Summary of Final ~ o l u t i o n s Obtained From Solving Program (3.3) 86

.......................................................... 3.10 - Summary of Cost Functions Used in IGGP 92

.......................... 3.11 - Wastewater Treatment System Designs Obtained Using IGGP 95

3.12 . Optimization of the Liquid Treatment Subsystem ...............................................

3.13 . Optimization of the Liquid Treatment Subsystem ...............................................

3.14 . Liquid Subsystem Design Optimization for Different Influent Conditions ..........

3.15 . Approximate Designs of Wastewater Treatment System :

Q,, = 1.0 3/hr .....................................................................................................


.................................................................................................... Qlo = 4.0 m3/hr


Qlo = 7.0 m3/hr .....................................................................................................

3.18 . Fine-tuning Solutions in the Decomposition Approach ........................................

3.19 . Examination of Assumptions in the Decomposition Approach ............................



4.1 . Treatment P lan t Design Optimization : Base System Without 3 Primary

Clarifier ....................................................................................................................

.................... 4.2 . Final Designs li'itli and M'ithout a Primary Clarifier in the System

4.3 . Summary of Wastewater Treatment System Design : Influent Volatile Biode-

................................................................................... gradable Solids = 200 mg/l

4.4 . Treatment Plant Design Optimization : Base System Without 3 Primary

Clarifier. Influent Volatile Biodegradable Suspended Solids Concentration =

200 mg/l ..................................................................................................................

6SI ...... " ........................... sf. ,@ a%pnl~ .CJ~LLI!JJ 'quc~ %u!111a~ .i~cw!~d 01 pauJn1

-a~ aZpn1~ palcq3y alscA\ : uo!?cz!ru!)do u%!saa rualsbg jo .i~cmmns - ~1.f

8SI .............................................................................................. no!jcJlua3uoj aSpnl~

bJcm!Jd ao!mJa$aa 03 slapojy ?uaJaU!a bq so%!saa ~crn!)do jo uos!~cdmo3 - 11.p

LSI ..................................................................................................... ~.f. a~nZ!d u! ma?

-sbs aq? puu u~a?siig asca arl? JOJ su%!saa J~)sJ%!~ .<JCII~!J J JO UOS!JL'~~LIO~ - 01.f

SSI .. " .... .............................................................................. 111C1 %u!]))~s .iJcIlI!Jd 01

pauJn3aU a%pnls pa)cA!l3y a?scl\\ : uo!lcz!u~!ld~) oS!l;aa ma]c.i~ jo L~crnmns - C, f.

............................................................................... uo!jc~g!~rlj a)aldrr~oj %II!~LI~S

-SV [)au!e?qo uZ!saa ma1s.C~ ~UJI~I~U~JL ~alc.+\alse~\\ ICIU!~~~) JO .CJCIIIIII~S - s'f-

xiv

Page

C.3 - Input Data to the Analysis Program : Decision Variables .................................... 196

D.l - Summary of hiode1 Variables: Base System ......................................................... 200

D.2 - Description of the Constraints in the GRG Model ................................................ 211

E . l - Parameters that are Unique in the IGGP Model ................................................. 217

F.l - Summary of Model Variables: Liquid Subsystem .................................................. 22-1

F.2 - Input Data to the Analysis Program: Liquid Subsystem ....................................... 225

M.1 - Summary of Model Variables: Base System Without a Primary Clarifier ........... 235

1.1 - Summary of Model Variables : Waste Activated Sludge Recirculated to the

Primary Clarifier ..................................................................................................... 241

............................................................................. 1 P ~auaq>!q~ .Cj!.ic~3 aq1 JO uZ!saa - 9.z

.................................... 6E a%pnls paJc.i!?sy aJsc.j\ puc dSpnl5 ~JVUI!JJ jo Su!puala - 9.z

.................................................................. 0'2 ssaso~d a%pnl~ pa~c.i!l>y aqJ jo n%!saa - 1.'~

...................................................................... 9Z quc~ %u!l~las .<JCU!J~ aql jo uZ!saa - g'c

.................................................. PZ ssssa>o~d q!u,l JO uS!saa ~oj uc~%c!a ~cuo!j>unj - c.-,

xvi

Page

3.6 . Using IGGP for Optimal System Design: Total System Cost vs . Mass Fraction

................................................................................................... of Primary Sludge 94

.............................................................. . 3.7 Design Obtained From IGGP: fp = 0.487 97

3.8 . Subsystem Formed By Tearing the Interactions Between Liquid and Sludge

Processing Trains .................................................................................................... 100

3.9 . Flowchart of the Decomposition Approach ............................................................ 106

3.10 . Concept of the Decomposition Approach ............................................................ 107

3.11 . Golden Section Search For the Optimal t ............................................................ 113

3.12 . Total System Cost vs . Digester Supernatant Flowrate: Q,, = 1.0 m3/hr .......... 120



................................ 3.15 . Best Design Obtained From The Decomposition Approach

........... 4.1 . Performance of Primary Clarifier as Predicted by the Voshel-Sak Model

................................ 4.2- Wastewater Treatment System Without a Primary Clarifier

4.3- Final Design for the Wastewater Treatment System Without a Primary

Clarifier ....................................................................................................................

4.4- Best Design for the Bnse System: Influent Volatile Biodegradable Suspended

.......................................................................... Solids Concentration = 200 mg/l

4.5- Best Design for the System Without 3 Primary Clarifier: Inliuent Volatile

.................................................. Biodegradable Solids Concentration = 'LOO mg/l

.................. 4.6- Best Design for the Base System: Influent Soluble BOD, = 200 mg/l

.............................. 4.7- Recirculation of Waste Activated Sludge to Primary Clarifier

4.8- Best Design for the System \Vhere Waste Activated Sludge is Returned to Pri-

.......................................................................................................... mary Clarifier

xvi i

Page

4.9- Best Design for the System Where Waste Activated Sludge is Returned to Pri-

mary Clarifier: Primary Sludge @ 4% ..................................................................

4.10- Best Design for the Base System: Improved Settling Properties of Digested

Sludge ......................................................................................................................

4.11- Vacuum Filter Surface Area Requirement vs . Solids Cake Concentration ..........

4.12- Approximation of the Noninferior Set With Flow Safety Factor and System

.............................................................................. Cost as Two Design 0bject.ives

A.l- Cost Functions for Primary Clarifier ......................................................................

A.2- Cost Functions for Primary Sludge Pumping .........................................................

.......................................................................... A.3- Cost Functions for Aeration Tank

....................................................... A.4- Cost Functions for Activated Sludge Aeration

................................................................... A.5- Cost Functions for Secondary Clarifier

............................................................. A.6- Cost Functions for Recirculation Pumping

A.7- Cost Functions for Gravity Thickener ....................................................................

................................................................... A.8- Cost Functions for Anaerobic Digester

........................................................................... A.9- Cost Functions for Vacuum Filter

H.1- Wastewater Treatment System Without a Primary Clarifier ................................

............................... 1.1- Recirculation of Waste Activated Sludge to Primary Clarifier

CHAPTER 1

INTRODUCTION

1.1. P r e l i m i n a r i e s

The objective of present wastewater treatment plant design is to provide a cost

effective processing system for a given wastewater. Such a system is relatively complex, con-

taining a series of unit processes. Generally each of the unit processes is designed to achieve

a specific goal, and only limited consideration is given during the design procedure to

interactions among the unit processes. I t would be desirable, of course, for engineers to have

design procedures tha t take into full account the tradeoffs tha t are possible among the unit

processes. For instance, minor modifications in the design of the liquid waste treatment por-

tion of a conventional plant may produce significant cost savings in the solids handling por-

tion of the plant.

Design engineers, however, are generally limited to using their past experiences and

trial and error in considering these tradeofis for a small number of options. One reason is

tha t many of the unit processes are not well understood, and therefore a complete and com-

patible set of unit process models is not available for use in comprehensive, systematic dcsign

procedures. A second reason is tha t only limited progress has been made in combining the

existing knowledge of individual unit processes to form comprehensive dcsign procedures.

One approacll t ha t researchers have identified is to connect various unit process

models within an overall system model and to apply a matl~ematical or enumerative optimi-

zation technique. T h e literature review in Section 1.3 provides an overview of the consider-

able progress tha t has been made since the first work in this area was reported by Lynn e t

al. in 1962.

T h e goals of lhis research are to extend the current capabililics in combining unit pro-

cess models within an overall optimization framework as an anaiysis and design tool, and to

highlight research needs tha t will improve the usefulness of unit process models in

comprehensive system design. It is important to stress tha t wastewater t r e a t n ~ e n t plant

design is a complex process and lha t good designs generally cannot be achieved using only a

mathematical, computerized model. T h e best system models are designed for use as tools by

designers, who ultimately have the responsibility for taking into account factors not con-

sidered in the model. System models can be very userul, however, for obtaining an optimal

solution for given input d a t a and effluent requirements based on specified assumptions. By

varying these conditions, the designer can use the model to facilitate the evaluation of

options and tradeoffs.

Research in developing comprehensive design procedures is important because the need

for wastewater treatment will clearly continue to require the commitment of significant

resources a t the national and international level. It is also important t o improve the under-

standing of complete wastewater treatment systems so that innovative regulatory

approaches to water quality management can be better evaluated. Examples of such

approaches are time varying eflluent requirements tha t change with receiving body condi-

tions (see Reheis e l al., 1982, for an illustration) and basin wide management of a particular

pollutant using transrerable discharge permits (see Joeres and David, 1983, for a discussion

of the program recently implemented by the IVisconsin Department of Natural Resources for

the Fox River). In general, as more cost effective regulatory approaches are developed it will

be even more important to understand better the options and tradeoffs ir: wastewater treat-

ment. Perrect understanding (e.g., or costs) cannot be expected, but relative performances,

costs, trends, etc. provide fundamental insights.

In the remainder or this chapter, research objectives and procedure are outlined in Sec-

tion 1.2. Section 1.3 provides a thorough literature review or past research eflorts on the

optimization of wastewater t,reatment system design. Several guidelines to improve this

research over previous studies are summarized. Section 1.4 describes the organization of the

thesis.

1.2. Research Objectives

Progress in developing comprehensive system models can be roughly divided into two

branches: 1) efforts to develop models tha t consider a wide range of unit processes and

emphasize the selection from among them (e.g., an activated sludge process or a trickling

filter) to form a treatment train, and 2) efforts tha t focus on a particular process train

(perhaps with some options) and tha t emphasize the selection of design parameters (e.g.,

basin volumes). Models of each type can be used jointly since they emphasize different stages

of the design process. Models ~f the first type could be used in selecting a general plant lay-

out , and models of the second type could be used in refining recycle flows and in selecting

design parameters for the given layout.

T h e overall objective of this thesis is t o extend research along the second branch

described above by developing a comprehensive system model of a conventional activated

sludge secondary wastewater treatment system; several variations of the base treatment sys-

tem are also considered. There are two major tasks under the objective: one is t o develop

efficient optimization techniques for solving the comprehensive system design model, and the

other is to illustrate the use of the system model for the analysis of process performance and

d ~ s i g n . T h e specific steps taken to achieve this objective are tlir followino,:

1) Evaluate current unit process models to determine their suitability for use in a

comprehensive system model and design procedure, and construct an overall wastewa-

ter system model which can be used to describe the performance of the system with

given influent and design conditions.

2 ) Develop and apply optimization approaches for the design of the wastewater treatment

system. Several approaches are examined for their applicability to optimizing the

comprehensive system model.

3) Illustrate the use of the comprehensive system model as a tool for the analysis of per-

formances, integration, and limitations of unit processes considered in the study.

Several variations of the base treatment system are modeled to verify the insights

obtained from the design optimization of the base system.

1.3. Literature Review

Pas t studies on the use of optimization models in the design and planning of wastewa-

ter treatment and sludge disposal systems can be roughly divided into two general

categories: Optimal process synthesis and optimal process design. Process synthesis studies

deal with the selection of the combination of unit processes tha t composes the least cost

treatment system. Lynn , e t al. (1962) pioneered the study of the optimal wastewater treat-

men t plant synthesis. A network linear programming model was formulated to represent

the BOD removal in a treatment plant t h a t consists only of liquid waste treatment. Many

assumptions had t o be made in order to render the optimization model a linear program.

T h e model was solved for the combination of unit processes t h a t would remove a given

amount of BOD a t the least treatment cost.

Evenson e t a / . (1969) applied dynamic programming to select the unit processes t h a t

would result in the least cost design of a plant treating cannery processing wastes. Both

liquid and sludge treatments were included in their system, with the sludge t rea tment train

being a diverging branch in the dynamic programming framework. The removal of BOD

was considered t o be the only function of the plant. T h e structure of the waste treatment

plant , with each unit processes represented 3s a lL~ tage ' l and with the absence of recycle

streams, made the application of dynamic programming possible. However, the design of

5

unit processes in this study was very simplistic.

S l ~ i l ~ and Krishnan (1969, 1973) also applied dynamic programming for the optimiza-

tion of industrial waste treatment plant design. The problem was formulated as an initial-

Gnal s ta te problem since the characteristics of the raw waste and the requirement of the

treated effluent quality represent the boundary conditions. The performance of a unit pro-

cess was considered to be its ability for removing BOD. The Decision Inversion Method pro-

posed by Aris e t al. (1964) was used to identify the least expensive liquid treatment system.

T h e same methodology of process optimization was again demonstrated on a simplified prob-

lem by Shih and DeFilippi (1970). Lack of confidence in the performances of individual unit

processes was considered by these authors a major handicap of the study.

T h e study o l Shih and Krishnan (1969) appears to have attracted attention from other

researchers. Ecker and McNamara (1971) formulated a geometric program for each of the

process trains considered by Shih and Icrishnan. T h e primal-dual relationship was used for

solving these programs. T h e flowchart tha t has the lowest treatment cost was then

identified by comparing the optimal cost of each process flowchart. Computational simpli-

city and the ease of performing sensitivity analysis for variations in effluent quality are

features of the geometric programming approach for this problem.

Adam and Panagiotakopoulos (1977) discussed the weakness of using linear program-

ming, dynamic programming, and geometric programming for wastewater treatment process

desisn optimization. They proposed a network approach as an alternative solution tech-

nique fo'r the problem stutlicd by Shih and Krishnan (1369). Advantages of the network

approach as claimed by the authors included its capability of handling multiple wastewater

parameters (other than BOD), its indifference to the types of the cost [unctions and perfor-

mance relationships, and its flexibility and ctficiency. Unfortunately, with a simple example

problem, none of these adv:intages were demonstrated by the proposed approach.

The fact t h a t various: optimization approaches have been appiied to solve the same

process optimization problcrn is indicative of the many special characteristics contained in

this problem. T h e special arrangement of the unit processes in the system or the unique

characteristics of the process performance relationships or cost functions may warrant the

application of a specific optimization technique or the deve!opment of an innovative optimi-

zation procedure.

Sterling (1976) conducted a similar study to those discussed above on the optimal pro-

cess selection and design using dynamic programming. Only BOD was included in the

analysis of process performance, and the treatment included only liquid waste.

Patterson (1977) also developed a dynamic programming model lor the optimal process

selection and design of a liquid waste treatment system. An effort was specially made to

identify those flowcharts tha t are good with respect to the total system cost, bu t diflerent in

the units being included. This allows the designer to examine different flowcharts and

tradeoffs among these systems in more detail.

Mishra et al. (1973) considered optimization of both the structure and the design of a

biological wastewater treatment system tha t included only liquid waste processing. Struc-

tural parameters, or stream splitting factors, were introduced into the model formulation t o

specify the arrangement of the unit processes. These structural parameters were continuous

variables varying between zero and one. Both BOD and total suspended solids concentra-

tions were modeled. Tho objective function was not complete because only the constrr~ction

cost of t,he system was included. T h e simples pattern search technique was employed t o

opt,imize this nonlinear programming model. Because the operation and maintenance costs

were not included in the objective function, the optimal system selected by the technique

was an activated sludge system, not a trickling filter system.

Bush and Silveston (1978) considered the optimal synthesis of the liquid processing por-

tion of a complete waste treatment system. T h e structural parameter method used by

Mishra e l al. was adopted. The constraints on the decision and state variables were

espressed in terms of penalty functions. Five wastewater paranieters were modeled. T h e

complex method by Box (1365) was selected as the optimization algorithm.

While most efforts in optimal process synthesis focused on the liquid treatment system,

Hasit e t al. (1981) studied the optimization of a sludge management system using 3 mixed

integer model. T h e design of the sludge treatment and disposal units were based on empiri-

cal loading factors to avoid nonlinearity in the model, and to make the model amenable to

efficient optimization. Since the process performances were not modeled, the tradeoffs

between performance and costs could not be evaluated. This model can be used to minimize

overall sludge handling, transportation, and disposal costs both for a single plant and for 3

group of plants with or without centralized treatment.

T h e U. S. Army Corps of Engineers (1978) developed 3 computer program ( C t V D E T )

in an effort to aid in the design of wastewater treatment facilities. T h e design procedures

for a wide range of physical, chemical, and biological unit processes were programmed (the

1980 version of CAPDET contains 79 liquid stream processes and 14 sludge stream

processes). Once the user specifies the unit processes to be considered for the design, CAP-

D E T synthesizes and designs all possible treatment flowcharts tha t can be constructed from

these unit processes using user-provided or default design criteria. Among all designs exam-

ined, the more cost-effective process trains and their detailed designs are given to the user as

outputs. T h e effectiveness or CAPDET as 3 screening device and design aid was demon-

strated by McGhee e t al. (1383). Some problenis cncountercd in the application of CAPDET

were also noted by these autliors, among them the most noticeable being the high computer

user costs because of the enumerative nature of the program.

Rossman (1979, 1980) also developed a computer-aided procedure for the synthesis and

design of wastewater treatment and-sludge disposal systems. Information requirements from

the user are similar to tha t for CAPDET. T h e computational procedure uses implicit

enumeration coupled with a heuristic penalty method that accounts for the impact of return

sidestreams from sludge processing. A unique feature: of this work is tha t planning objec-

tives other than system cost can be optimized in the program. Alternative designs tha t are

energy efficient, or low in the initial construction cost, etc. can be identified and evaluated.

The optimal design of the system is approximate in the sense tha t discrete values for the

decision variables are supplied by the user.

T o summarize, optimal process synthesis studies often deal with a variety of wastewa-

ter treatment unit processes. The mathematical models are basically used as screening dev-

ices for planning and design of wastewater treatment systems. They are used as design aids

to specify good process trains; but the system design and performance in general cannot be

predicted a t a detailed level. If the tradeofIs among unit process designs or the applicability

of unit process models for design are to be further explored, a process design optimization

model will have to be employed.

Process design optimization models usually employ fairly detailed mathematical state-

ments to describe the perfcrmance of a specified configuration (or possibly a few variations)

of unit processes. They d o not deal with the breadth of the options considered by the syn-

thesis models. T o use process synthesis models and process design models conjunctively, a

process design model could be used in evaluating more thoroughly a process train selected

using a synthesis model.

Naito e l a l . (19G9) and Fan e l a l . (1970) studied the optimal design of an activated

sludge subsyst,em consisting of aeration and final sedimentation. Various Bow regimes in the

aeration tank were considered. The simplex method of Nelder and hlead (1965) was

employed to minimize the total capital cost of the system. The objective function was not

complete since it left out the operation and maintenance costs which often play an impor-

t an t role in the design of wastewater treatment systems.

Berthouex and Polkowski (1970) investigated wastewater treatment plant dcsign under

uncertainty. Uncertainty in performance of system components was considered by appiying

the concept of propagation of variance. Only the liquid treatment train was optimized,

sludge train design and cost estimation were based on typical design criteria. Thickening of

activated sludge in the final clarifier was modeled by the limiting flux theory. T h e pattern

search technique of Hooke and Jeeves (ISGI), with modifications t o handle inequality con-

straints, was applied to solve this problem. Only a single local minimum was reported for

the problem.

Scherfig et al. (1970) attempted to optimize the design of an activated sludge system

using geometric programming. The primal problem of their model had a high degree of

difficulty and was not amenable to the classic geometric programming solution approach. As

a result, the system was decomposed into a sludge disposal system and a liquid waste t r e a t

ment system. The sludge subsystem was optimally designed using a search algorithm t o solve

the dual problem. The liquid treatment train was designed by experience. These authors did

not coordinate the designs of the two subsystems t o identify the overall optimal system

design. T h e capability of the classic geometric programming for solving the entire waste

treatment plant design was shown t o be limited by tlie high degree of dificulty and the lack

of an efficient nonlinear programming technique.

Parkin and Dague (1072) indicated tha t an overall waste treatment system made up of

individually op timized unit processes was seldom op tirnal. They assembled a design model

for a treatment system that included both liquid and sludge processing. Six decision vari-

ables were identified and 720 alternative designs formed by dillerelit combinations of the

values of the six decision variables were evaluated. This complete enumeration approach

indicated tha t more than GO% of the treatment alternatives investigated were a t least 20%

more expensive than the least cost design. The importance of the cost-eflective design of a

waste treatment system was clearly demonstrated.

Middleton and Lawrence (1074) presented 3. unique technique for optimization of the

activated sludge system. By adopting the concept of sludge age and the set of design equa-

tions proposed by Lawrence and McCarty (1970)) they observed tha t the liquid and sludge

process trains could be optimized independently for a fixed sludge age. An enumerative

graphical search technique was developed based on the fact t ha t each subsystem had only

two decision variables. This optimization technique was specially designed to solve this for-

mulation of the problem. I t would become more complicated and inefficient if recycle

streams generated in sludge processing are recycled to the liquid treatment train.

Middleton and Lawrence (1976) applied the same optimization technique to the design

of a sirnila: system where anaerobic digestion was substituted for aerobic digestion. Primary

settling and sludge dewatering by vacuum filtration were also included. Simplifying assump-

tions were made such tha t the number of decision variables in this problem remained the

same as in the previous problem even though more units were included. T h e assumptions

tha t the primary settling tank removes suspended solids a t a constant efliciency and tha t the

final settling tank performs perfect clarification are unrealistic. However, they are essential

for the solution technique to work. This is clearly a drawback of this approach. Only a sin-

gle local minimum was found for this problem. T h e overall system cost was found to be

quite insensitive to the sludge age.

Craig el al. (1078) used the complex algorithm (Box, 1065) to design the system studied

by Middleton and Lawrence (1976). It was shown tha t this nonlinear programming algorithm

was much more eRicient than the graphical enumeration technique previously used. Multi-

ple start ing points were used in solving the nonlinear programming model, bu t only one local

minimum was identified in this problem. Since the formul;ztion had been purposely res-

tricted by Middleton and Lawrence to include only five decision variables, the complex algo-

rithm worked satisractorily, outperforming the graphical enumeration significantly as far as

computing time was concerned. The same algorithm was also successtully applied to an

optimal activated sludge operation problem by the same group o l researchers (Hughey e t al. ,

1982).

Bowden e t al . (1976, 1978) reported another effort to develop a computerized pro-

cedure for wastewater treatment system design. Their model included liquid waste and

sludge treatment units and recycle streams generated from sludge processing. Because of the

presence of the recycle streams in the model, an iterative approach was used t o determine a

steady s ta te solution. The objective function value corresponding to a set of decision vari-

ables could not be determined until a steady-state design was obtained. T h e search method

by Powell (1964) was $elected as the optimization algorithm. Although the computational

experience was not explicitly reported, it is expected tha t the overall opt.imization procedure

would not be very efficient beca~ise of the time requirement for obtaining the steady-state

design by iteration.

In his study of sensitivity of the optimal wastewater treatment plant design with

respect t o s ta te variables and technological parameters, Voelkel (1978) assembled an optimi-

zation model for a complete wastcwatcr treatment system that contains recycle streams from

sludge processing. Nine degrees of freedom were identified in his model. A modified com-

plex algorithm was selected as the optimization technique. Voelkel applied the equation ord-

ering algorithm of Rudd and Watson (1968) to select the decision variables in his model.

Fixing the values of these selected decision variables permits more efficient solution for the

steady-state design than the iterative approach. Voelkel did not report any computational

experience with his optimization approach, nor did he discuss the quality of the solutions

obtained from using this search technique.

Based on the above studies tha t used search techniques for optimization of wastewater

t rea tment system design, it appears tha t these methods are not computationally efficient

because of the nature of these methods and the need for obtaining 3 feasible solution by

iteration. Although these metliods are straightforward, they are likely to be very slow in

obtaining the optimal system design for a complex arrangement of unit processes.

An optimization procedure that incorporates embedded optimization steps may serve

well for the purpose of process design optimization. Tarrer e t al. (1976) studied t,he

activated sludge design under uncertainty. In developing their solution strategy, Tarrer e t

al. assumed tha t either the effluent BOD or total suspended solids constraint would be limit-

ing if a least cost design is to be achieved. They subsequently developed a solution pro-

cedure with embedded nonlinear programming steps for the optimal design of their system.

T h e mnjor shortcoming of this work, however, is t h a t it optimized only the liquid t rea tment

process train, although the costs of sludge treatment were estimated using typical design cri-

teria (and were included in the overall objective function).

Grady (1977) outlined the steps for using discrete dynamic programming for optimiza-

tion of the activated sludge system. T h e problem formulation was similar to t h a t of Tarrer

e t al., i.e., only the liquid treatment train was considered. Grady observed t h a t the problem

could be formulated as three stages in series, each having one decision variable, provided

tha t the sludge age was fixed. T o implement the solution procedure, the sludge age was first

calculated from an assumed emuent soluble BOD requirement. Designs were then made

based on this sludge age using dynamic programming. It should be noted t h a t if the com-

plete treatment plant design is to be optimized, dynamic programming may not be an

attractive technique because of the recycle streams, branches, and additional s ta te variables

t h a t would be required in the system model.

Lauria e t al. (1977) considered optimization of an activated s h ~ d g e s ~ ~ b s y s t e m t h a t

included aeration and final settling. Through substitutions they reduced the objective to a

function with only two variables. They solved the problem by using the classical calculus

technique with Newton's method for solving systems of nonlinear equations. This approach

would become impractical for a more complete treatment system because of the extensive

computing requirements.

Hughes (1978) employed the same design equations used by Laur is e l al. and optim-

ized the system design using geometric programming. The problem had ten degrees of

difficulty and a concave objective function. A problem of this type was considered unsolv-

able by Scherfig c t (11. in 1970. Advances made by Avriel e t al. (1975), however, on the

development of a solution technique for generalized geometric programs made the problem

amenable to very efficient solution. With only two degrees of freedom in the problem,

Hughes was able to verify t h a t his solution was indeed the global minimum by mapping the

response surface.

Although their main objective was to identify the most cost-efiective sludge treatment

and management scheme, Dick e t a l . (1976, 1978, 1979, 1981) considered both the liquid and

sludge treatment trains and performed 3 sequence of very comprehensive studies on treat-

ment process selection and design optimization. The interactions between the liquid and the

sludge subsystems were considered in more detail than in previous studies. Side streams

generated throughout the sludge processing train were assumed t o be recycled to the liquid

t rea tment train. T h e authors called for the use of fundamental design equations instead of

empirical observations grounded purely on experience. Process models were complete except

tha t the authors assumed a constant emuent solids concentration from the secondary

clarifier regardless of the design condition. This assumption is unrealistic since the perfor-

mance of the secondary clarilier varies signilicantly with the design and operation of the

activated sludge subsystem. Based on their modeling work, the authors indicated t h a t the

physical properties of sludge influenced the optimal design t o such an extent t h a t more

research on this aspect would be needed (Dick e t a!., 1978). Predictive models for sludge

characteristics as functions of basic design and operational variables were subsequently

developed (Dick e t al., 1979, 1081).

Dick e t al . developed a computer program for the selection of the least cost

configuration of unit processes among alternative sludge management schemes. Because of

the presence of recycle streams in the system, this program calculates the design parameters

iteratively until a steady state design is achieved. This information is used interactively

with a nonlinear programming code. This code uses the penalty function approach with the

Davidon-Fletcher-Powell me t l~od (Davidon, 1959, and Fletcher and Powell, 1963) for the

minimization of the resulting unconstrained problem. Dick e l a l , recognized t h a t the

number of potential decision variables can be very great when complex systems are being

optimized. They discussed the factors tha t limit the number of desig~i parameters actually

needed as decision variables for the purposes of their study. In the demonstration runs

presented, the design of the liquid train was fixed, i.e., the optirnization was carried ou t for

the sludge treatment system only. The computational requirements of their approach would

be expected to increase considernbly if it is applied t o the entire wastewater t rea tment sys-

tem. Dick et al. did not report an a t tempt t o verify tha t the local optimum resulting from

their solution strategy was indeed the global optimum, nor did lhey discuss the general issue

of local optimality.

Tyteca e l al. (1977) presented a thorough review of mathematical models developed for

or used in wastewater treatment process design and optimization. Based on this work,

Tyteca formulated an optimization model for a complete activated sludge system (Tyteca,

1981). His model included quite detailed models for unit processes except t h a t he assumed

perfect clarification in the secondary clarifier. The model had eight degrees of freedom and

was unique in tha t dimensionless variables were used. The model was formulated as a

geometric program which allows eflicient computation of the analytical derivatives of the

objective function and the constraints and systematic input of model d a t a when implement-

ing the optimizing code (Tyteca and Smecrs, 1981, Smeers and Tyteca, 198-1). T h e authors

discussed one potential problem with their approach: the use of inequality constraints to

replace equations as required for the standard geometric program formulation. In view of

the size of this problem, Tyteca and Smeers decided t o employ a well-tested nonlinear pro-

grnmming algorithm based on the generalized reduced gradient (GRG) method, rather than

a special-purpose geometric programming (GP) code. A more general study of tlie use of a

GRG algorithm t o solve geometric programs was carried out by Ratner e t a l . (1978). They

reported that for many test problems GRG compared well w ~ t h special-purpose G P codes.

An interesting conclusion of the Tyteca and Smeers study was t h a t only a single local

minimum was found for their highly nonlinear model.

I<oelling (1983) used a quasi-enumerative se3rch procedure for optimization of sewage

treatment plant design. His study concentrated on the design of the activated sludge sub-

system. Sludge processing units were then sized accordingly. The model has two degrees of

freedom. An interesting feature of this work was t h a t three objective functions were con-

sidered: the total system cost, the costs incurred by the federal government, and the costs

incurred by the local municipality. It was observed t h a t the "least cost1' design changes

with different objective functions. Koelling concluded t h a t a design reached as 3 comprom-

ise of different interest gtoups seems to be more realistic than that obtained based on a sin-

gle objective.

Suidan el a l . (1983) formulated an optimization model for a simplified activated sludge

system. Waste sludge was assumed to be dewatered by vacuum filtration and incinerated.

Separate sludge thickening was not considered and recycle streams frorn the sludge process-

ing system were ignored. Consequently, it was possible to simplify the model sufficiently t o

have only two decision variables. The univariate search technique was selected t o solve this

problem. Fibonacci search was employed for minimization in one dimension. It was

reported t h a t the response surface was unimodal and very flat in the vicinity of the

optimum. These researchers applied the limiting flux theory to thickening in the primary

settling tank and obtained an unrealistically high underflow solids concentration. As a

consequence, they assumed that only a fraction of the primary clarifier area was effective for

sludge thickening.

As shown in the literature review, the s ta te or the a r t has evolved considerably over

the last twenty years in the application of optimization concepts to wastewater t rea tment

system analysis and design. There still, however, a re areas vihere additional improvements

can be made. The following summary of guidelines for future work is from the above discus-

sion. These guidelines serve as a basis for the development of the comprehensive system

model described in Chapter 2.

1) Construction, operation and maintenance costs: Both categories of costs should be

included since different unit processes have dinerent relative costs for construction,

operation and maintenance.

2) Complete treatment system: T h e sludge treatment and disposal systems should be

optimized together. Since the costs of these systenls comprise a large portion of the

total system cost, designs based on optimizing only the liquid treabment train are not

likely to be optimal for an entire treatment system consisting of both liquid and sludge

treatment units and sludge disposal.

3) Descriptive process models: Mathematical models describing the performance of units

and interactions among units should be taken into account. For example, recycle

streams from the sludge treatment system to the liquid treatment system have often

been neglected b u t should be considered. Ultimate sludge disposal costs also must be

considered.

-I) Realistic assumptions: For many unit processes, a predictive modcl lor process pcrfor-

mance is not avai1:ible due to the complexity of the process. Assumptions about pro-

cess performance are necessary lor a complete design of the treatment system. Limita-

tions of the state-of-the-art in this area are probably best exemplified by an assump-

tion made by a number o l researchers tha t 100% of the solids are captured in the final

settling tank. Because a substantial portion o l emuent BOD results lrom the

.suspended solids. it is essential t ha t a model relating the design and operational

parameters to the final etlluent suspended solids concentralion be included in the

optimization model.

5) Meaningful constraints: Several previous studies have formulated the optimization

problem with constraints on various design parameters or s ta te variables. These con-

straints are based on past experience rather than on scientific fundamentals. Such

constraints have often been used to force the model to produce "reasonable" results.

This limits the usefulness of an optimization model by forcing it to work only in the

range of variables found in conventionally designed systems. Important insights on

process research may be lost with such a restricted optimization model. However,

empirical models should only be used in the ranges within which they are developed

when they are used to construct the overall system model.

6 ) Efficient optimization technique: Many optimization methods used previously can be

applied only to a special and limited process scheme or only when simplifying assump-

tions about process designs are made. Few studies developed and presented methods

t h a t are specially tailored to take advantage of the unique structure of a complete

wastewater treatment system.

One major purpose of this thesis is to make additional progress toward developing an

efficient optimization method for use in designing a complete activated sludge t rea tment sys-

tem. Significant cost savings in water pollution control eflorts may ultimately be made pos-

sible with the aid of such n method for treatment process design. Attempts are made to

incorporate many of the internctions tha t were omitted in previous s t ~ ~ d i e s into an optimiza-

tion model to provide the designer with realistic insights about system design. Three optimi-

zation approaches tha t can be used efficiently to solve a complete mathematical model for

t,he waste treatment system design are also presented.

A treatment plant design optimization model has been perceived by a number of

researchers as a means to obtain the least cost system design. This role of an optimization

model is suggested by the very nature of the optimizstion concept. T h e planning and design

of a wastewater treatment system, however, is a complex problem. Many important issues

such as energy requirements and system reliability may not be captured in a cost-

optimization model. As a result, the optimal design obtained from solving such a model may

only be meaningful mathematically. Another view suggests tha t the most appropriate role of

this type of optimization model is as a decision-making aid. This role is more appropriate

because of the importance of unmodeled issues and the uncertainties associated with plan-

ning a waste treatment system. T h e other major purpose of this thesis is t o illustrate the

use of such an optimization model as a tool for the analysis of process performance. An

optimization model can lead to the examination of the validity of process models from the

cost-effectiveness point of view. Useful insights about process performance, integration, or

limitations are gained as valuable by-products from exercising an optimization model.

1.4. Thesis Outline

A comprehensive system design model of a wastewater treatment system is prerequisite

for this research. Chapter 2 defines the base treatment system selected for this study, and

provides a review of the representative process design models tha t describe the performances

of those unit processes included in the base treatment system. Design equations and cost

information used for the co~istruction of the comprehensive system model are also described

in this chapter. Several solution tcchniqucs examined for solving the comprcticnsive system

model arc tlescribcd in Chapter 3. together ivith Jiscussions of t l ~ ~ performances o f thcse

techniques. Chapter 4 cniphnsizes the use of thc optiniization nlotlel :IS a tool for system and

process analysis. Insights obtained from optirnizing the system design are used as examples

to illustrate the role of an optimization model. A summary, epnclusions, and future research

directions are presented in Chapter 5.

CHAPTER 2

DEVELOPMENT OF THE COMPREHENSIVE SYSTEM MODEL

2.1. Introduction

Design of unit processes in a wastewater treatment system follows two general

approaches in current practice. One approach is simplistic, and involves the use of empiri-

cally determined design parameters. It has been observed qualitatively tha t these design

parameters affect the performances of unit processes. However, quantit3tiv.e measures of

process performance cannot be obtained. As 3 result, designs based on past experience deny

the engineers the opportunity to analyze the interactions among unit processes in a waste-

water t rea tment system, which are essential to achieving a cost effective design.

The other approach for designing unit processes is to employ mathematical models

which predict the process performance under given input and design conditions. Interrela-

tionships among unit processes can be studied in detail to strive for cost-eficiency; perfor-

mances of unit processes can be predicted to insure satisfactory emuent water quality.

These process performance models may be developed from physical, chemical or biological

principles, or from empirical d a t a fitting; they m y be time-dependent or time-independent;

deterministic or stochastic. A thorough review of process performance models for unit

processes typically employed in secondary waslewater treatment was given by Tyteca e t al.

(1977). T h e revlew of proccss performnnce rr~odels in this c l~npter is intended to highlight

and updnte tha t ellort. Time-dependent or stochastic moticls are not considered in this

study. Unit process performance models were selected based on this review, and serve as

building blocks for the comprehensive system model.

Ideally, a process model suitable for design should be able to describe the unit process

performance over a wide range of operating and influent conditions. It should reflect realis-

tic process performance, and include all relevant process variables tha t aRect the process

performance. Based on these guidelines, models developed from fundamental principles were

given Erst priority for use in constructing the comprehensive system model because such

models are valid regardless of the external conditions. Models developed from plant-scale

studies were then considered, followed by models developed in laboratories. For complicated

processes for which only empirical models are currently possible, those empirical models tha t

predict process performance consistent with fundamental knowledge of treatment processes

or with observed process responses were preferred. Since the development of an empirical

model is generally specific to the system studied, the limitations of such models should be

recognized.

Assuming tha t cost efficiency is a primary objective in the design of a wastewater

treatment system, it is appropriate to write the comprehensive system model in the form of

an optimization model. T h e objective function is to minimize the total system cost, which

includes capital, operatiod and maintenance costs. T h e constraint set in the model is a col-

lection of the independent design equations for all unit processes in the system and the mass

balance relationships among the interconnected units. Restrictions on emuent water quality

are also imposed on the design of the system as constraints.

T h e base wastewater treatment system selected for this study and the definition o f the

variables in the model are described in Section 2.2. Section 2.3 presents process performance

models for units included in the base treatment system. Representative models for designing

each unit process are reviewed, followed by the mathematical expression of a particular

model tha t is incorporated into the overall system model. Section '7.4 deals with the formu-

lation of the objective function in the cost minimization model, and includes discussions

about available cost information. Section 2.5 illustrates the complete design procedure using

the comprehensive system model with a numerical example.

2.2. System Description

2.2.1. Flowchart

A typical secondary wastewater treatment plant was selected as the base system for

evaluation in this study. T h e flowsheet of the plant includes primary sedimentation of raw

wastewater, organic material stabilization by the activated sludge process, gravity thickening

of combined primary and waste activated sludge, two-stage anaerobic digestion of the thick-

ened sludge, and sludge dewatering by vacuum filter. Final sludge disposal by sanitary

landfill was assumed. Figure 2.1 depicts this study system. Supernatants generated in

sludge processing were assumed to be recirculated to the head end of the plant for BOD and

suspended solids removal.

2.2.2. Definition of System Variables

T h e complete design of the wastewater treatment system requires the specification of

three groups of variables:

1) T h e parameters are those quantities tha t remain constant in the design; examples are

the biological coefficients in the activated sludge process, pumping eficiency, cost of

energy, etc.. A complete list of the parameters used in the system design and economic

analysis is provided in Section 2.5.

2) T h e decision variables specify the dimensions or the design condition of a unit process.

More specifically, the decision variables selected in Lhis study are: overflow ra te of the

primary settling tank (L,), mean cell residence time (O,), liydrauiic retention time (O),

and sludge recycle ratio ( r ) in the activated sludge process, solids loading on the grav-

ity thickener (Lg), digestion temperature ( T d ) and solids retention time ( O J ) of the pri-

mary anaerobic digester, solids loading on the secondary digester (Ld), and Blter yield

from the vacuum filter (L,).

L A N D F l L L

I NFLIJEPIT E F F L U E N T \

f

Figure 2.1 - Act iv~ted Sludge Treatment System

0 4

S E T T L l NG A E R A T I O N

3 , - 1

S E C O N b A R Y

S E T T L l N G

8 6

7

7

9

10 GRAV l T Y T H I C K E N I N G

ANAEROB l C D l GEST l ON (FR I MARY)

13 ANAEROB l C D l G E S T l ON

(SECONDARY)

14

15

j l ~ e v

VACUUM F I L T R A T I O N

L E G E N C : -\ . -/ S L U D G E

, I lIe;JiL 7. L I W

3) T h e s ta te variables represent the wastewater characteristics a t a particular stage dur-

ing the treatment processes, and are defined a t bhe seventeen control points shown in

Figure 2.1. These s ta te variables include flowrate, concentrations of soluble BOD6,

active biomass, biodegradable and inert (with respect t o aerobic stabilization) volatile

suspended solids, fixed suspended solids, and total suspended solids. T h e following

notation and units for the s ta te variables are used in the development of the

comprehensive system model:

Qi = Flowrate a t control point j, m3/hr

Sj = Soluble BOD:, concentration a t control point j, g/m3

Maj = Active biomass concentration a t control point j in kg/m3 unless noted otherwise

Mdj = Biodegradable volatile solids concentration a t control point j in kg/m3 unless noted otherwise

Mij = Inert volatile solids concentration a t control point j in kg/m3 unless noted otherwise

M,j = Fixed, or inorganic, solids concentration a t control point j in kg/m3 unless noted otherwise

Mti = Total suspended solids concentration a t control point j in kg/m3 unless noted otherwise

j = Index of the control point, j = 0,1, ... ,16.

2.3. Problem Formulation

T h e constraints in the comprehensive system model are described in this section. In

general, t he design of a unit process can be considered using Figure 2.2. Vectors Z; and Y;

represent the input states t o unit i and outprit states from unit i, respcctively. Vector di

denotes the decisions made a t unit i. T h e input and ou tpu t states are related by a transfor-

mation function, or a tcchnological function,

Y; = T;(Z;,d;)

where Ti is a vector function tha t defines the performance of the unit process.

T h e total cost of unit i , c;, can be expressed as

Figure 2.2 - Functional Diagram for Design of Unit Processes

C; = ci(Z;,di) (2 .2 )

T h e Ti functions are described in this section, while the ci functions are described in

the next.

2.3.1. Primary Sedimentation

Primary sedimentation is provided mainly for the removal of influent settleable solids.

Organic matter in the form of suspended matter and semi-colloidal solids may also be

removed from the wastewater. Fundamental understanding of the solids removal mechan-

ism is limited to the ideal conditions of discrete spherical particles settling in laminar flow.

T h e overflow rate was shown to be the single most important parameter controlling the

solids removal efficiency according to the theory (Hazen, 1904, Camp, 1916). In practice,

however, because of the flocculant nature of :vastewnter and the disturbance in the settling

tank caused by hydraulic turbulence, density currents, scour and wind action, it is not possi-

ble to apply this basic knowledge to design.

Empirical relationships developed from plant operating d a t a to describe the suspended

solids and organic matter removal eficiencies in the primary settling tank are abundant in

the literature. Smith (1968) proposed tha t solids removal efficiency is a function of the sur-

face overflow rate. He developed a model using d a t a from the W P C F Xlanual of Practice

(1959). Berthouex and Polkowski (1970) developed a linear model with respect to the

overflow ra te based on the same data. This model is mathenlatically simple, but it is not an

adequate representation of the observed data .

Other researchers have found that the influent suspended solids concentration is also

important in predicting the solids removal efficiency. This observation seems reasonable

considering tha t sewage contains a large portion of flocculant particles. Voshel and Sak

(1968) developed two models relating the solids removal efficiency to both the influent solids

concentration and the overflow rate based on their plant-scale study performed in Michigan.

In England, two models have evolved over the past decade for the solids removal efficiency

of primary sedimentation. The model of Tebbu t t and Christoulas (1975) was developed

from a pilot-scale study and was shown to describe plant operating d a t a adequately. T h e

CIRIA (1973) model used detention time instead of overflow rate t o represent the hydraulic

features of the settling tank. This model was based on d a t a observed a t sewage works in the

London area. Dick e t al. (1978) fitted the W P C F (1959) d a t a to 3 model of the form pro-

posed by Tebbu t t and Christoulas. These models are summarized in Table 2.1.

It is noted tha t all models indicate tha t the solids removal efficiency increases with

decreasing overflow rate and with increasing influent solids concentration when it is con-

sidered. Parameters in the models represent the degree of dependence of the solids removal

on influent solids concentration and overflow rate. These parameters are related t o the

characteristics of the influent to the primary settling tank.

Figure 2.3 is used to illustrate the design of the primary settling tank. The overflow

ra te is the decision variable of this unit,. The model of Voshel and Sak is selected to describe

the removal of total suspended solids in the primary settling tank. T h e fraction of influent

suspended solids remaining in the primary emuent is calculated as

where v 1 , v 2 and v , are positive parameters, and L p is the overflow rate defined as

Table 2.1 - Models for Suspended Solids Removal Efficiency in the Primary Settling Tank

Suspended Solids Domain of Models Source of Data

Removal Efficiencv Ex~er i rnen t

Smith (1968) .82 exp(- .2112Lp) W P C F (1959) .42<Lp <3.75

Voshel and Sak (1968)

1. .139dfi:7Lc" Voshel and 70<hf t1<160 2. .340i\!;;~L,'~ Sak (1968) 1 .71<Lp<1.88 (Polymer addition)

Berthouex and .82 - .142Lp k W C F (1959) ,42<Lp <3.75 Polkowski (1970)

Tebbut t and 2 6 5 Tebbut t and lOO<,Cf,, < l o 0 0 .955 exp(-- - .050+1Lp)

Christoulas (1975) Mt 1 Christoulas (1975) 1.04< L, <6.25

Dick e t al. (1976) 4 0 .84 exp(-- - .177Lp) W P C F (1959) .42<Lp <3.75

Mt 1 M,,--230

Note -- L p : overflow rate (rn3/myhr)

Mtl : influent suspended solids concentration (g/m3 1 : detention time (hours)

Figure 2.3 - Design of the Primary Settling Tank

Note that 'titl and , I t t2 are in g/m3, and the surface area of the primary clarifier, ,4p, is in

Primary sludge concentration has been modeled by two approaches. T h e first

approach assumes tha t this concentration is controlled by the hydraulic limitations of the

sludge withdrawal mechanisms. As a result, a constant concentration is assigned to the pri-

mary sludge (see, for example, Voelkel, 1978). T h e second approach uses the diuerential

thickening technique (see, for example, Dick and Suidan, 1975) which is based on the limit-

ing flux theory (Dick, 1972) to calculate the primary sludge concentration. Thickening con-

s tants for primary sludge can be obtained from batch settling tests.

Many models have been proposed to define the batcb sludge settling velocity as a func-

tion of the initial solids concentration (Vesilind, 1979). Vesilind (1968) proposed an exponen-

tial relationship,

ui = a' exp(-b1Ci) (2.5)

where u; is the batcb settling velocity,

C; is the initial solids concentration,

and a' and b ' are empiric?~lly determined constants for the sludge.

Berthouex and Polkowski (1970) used equation (2.5) to develop a mathematical expres-

sion of the limiting flux.

GL = a'b1C; esp(-b1CU)

where GL is the limiting flux, and Cu is the underflow solids concentration.

Dick and Suidan (1075) also derived an expression, equation (2.8), for calculating the

limiting flux based on tho following batch settling velocity model proposed by Duncan and

Kawata (10G8),

where Q, is the underflow flowrate from a thickener,

A is the surface area of a thickener.

and a and n are empirically determined constants.

The underflow solids concentration from a thickener can be calculated as

Dick and Young (1972) have shown tha t equation (2.0) provides adequate prediction of

pilot plant thickening data. This equation is used to describe the sludge thickening in the

overall system model.

The thickening function of the primary settling tank is modeled, i.e., the primary

sludge concentration is calculated, from equation (2.9) as

where a, and n, are settling constants of the primary sludge obtained when the batch set-

tling velocity is expressed in meters/hr and the sludge solids concentration in kg/m3.

The flow and mass balance relationships around the primary settling tank are

Q1 = Q2 + Q8 (2.11)

~ ~ b ! , , = &,hit2 + 1 0 ~ ~ ~ ~ 1 1 , ~ (3.12)

A unit conversion Factor, lo3, is inserted in equation (2.13) since Aft, and h!,, are in

g /m3 while A!!, is in kg/m3

Empirical motlels predicting the removal of organic matter in the primary settling tank

also exist in the literature. Table 2.2 provides a sample of these models. Most of the

models were developed from actual plant d a t a except the one by Tebbut t and Christoulas

(1975) which was developed from a pilot-plant study. .4 common feature of these models is

the lack of fit of the d a t a to the pro~posed model, generally with R~ less than 0.6. Theretore

none of these models is used in this study. Instead, the total BOD in the primary elf uent is

modeled by considering the soluble and suspended portions respectively.

The soirrble BODj concentration is assumed unaffected by primary sedimentation, i.e.,

S2 = S , (2.13)

S, = S l

The concentrations of individual solids components are calculated based on the

assumption tha t the settleable portion of each solids component is t he same:

Table 2.2 - Models for Organic Matter Removal Efficiency in the Primary Settling Tank

Models BOD Removal Efficiency Source or D3ta Domain or Exper iment

Berthouex and Polkowski (1970)

WPCF (1959)

T e b b u t t and 0.311 + 0.779e T e b b u t t and 200 < ,\it, < 800 Christoulas (1975). 411 Christoulas (1975) 0.26< e <0.63 (-) + 1.09

M, ,

T e b b u t t (1979) 0.08 + 0.508e W P C F (1959) 0.2< e <0.8

T e b b u t t (1979) - 0.31 + l . 2 l l e T e b b u t t (1979) 0.6< e <0.8 Th i s model is Tor C O D removal efficiency in the primary sett l ing tank.

No te -- L p : overflow ra te (m3/rn"hr)

hi t1 : influent suspended solids concentration (g/m3)

e : suspended solids removal efficiency

M,, Mo8 = M p l -

'$1;

where hlol, M d l , M i , and iVf1 and Ma,, Md2, h1i2, M!2 are in g/m3

2.3.2. Activated Sludge

The activated sludge process consists of aerobic waste stabilization in the aeration

tank, clarification o f the aeration tank effluent and sludge concentration in the secondary

clarifier, and recycle of the thickened sludge to the aeration tank to maintain the microbial

population (Figure 2.4).

Tyteca et al. (1977) have reviewed various kinetic models proposed for the design of

biological wastewater treatment processes. Among the models proposed, the first order

models by McKinney (1962) and Eckenfelder (1966) and the Monod model by Lawrence and

McCarty (1970) are the most widely accepted design models in practice. T h e design equa-

tions developed by Lawrence and McCarty are chosen as the basis for design of the

activated sludge process. The aeration tank is assumed t o be completely mixed. All

Figure 2.4 - Design of the Activated Sludge Process

biological activities are assumed to occur in the aeration tank, and the biodegradable volatile

solids are assumed to be completely consurncd in the tank, i.e., M,, = 0. The substrate util-

ized in the process, S , is then

1.42 g BODL g BOD, S = S 2 + (

g VSS )( 1.5 g BODL lMd2 - S3

where Md2 is the volatile biodegradable solids concentration in the primary effluent, S, is the

soluble BOD, in the aeration tank effluent, and can be calculated as

where K, is the half-velocity constant, g BOD,/m3,

k is the maximum specific utilization coefficient, day-',

y is the growth yield coefficient, g cell/g BOD,,

b is the endogeneous decay coefficient, day-'

and 0, is the mean cell residence time, days.

T h e mean cell residence time, by definition, is

where V is the volume of the aeration tank, m3,

Ma, is the biomass concentration in the treated effluent, g/m3,

Ma, is the biomass concentration in the underflow from the secondary clarifier,

0 is the hydraulic retention time in days, which is dcfincd as

w is the sludge wasting ratio defined as

and 10"s a unit conversion factor,

T h e biomass concentration in the aeration t ank , A!a3, can be derived from the mass

balance relationship of the substrate as

where is a unit conversion factor.

T h e volatile inert suspended solids concentration in the mixed liquor is derived from

the mass balance relationship and the assumption tha t the solid compositions remain

unchanged through secondary sedimentation,

where f d is the fraction of microbial cells t ha t is degradable, and 10" is a unit conversion

factor.

Similarly, the conccntrations of the fixed suspended solids can be calcrllatcd and a ratio

defined,

Mass balance of biomass around the aeration tank yields

Qe . where r =- 1s the sludge recycle ratio, and 1 0 ' ~ is a unit conversion factor. Q2

The oxygen requirement for aeration is estimated using the Lawrence-L,lcCarty Model

3s

where O2 is the oxygen requirement in kg/day, and 2 4 ~ 1 0 - ~ is a unit conversion factor.

The air flow ra te is calculated as

where Qa is the air flow ra te in m3 air/min,

a and p are correction factors,

y is the weight fraction of oxygen in air,

C, is the dissolved oxygen saturation concentration a t 20°C, g/m3,

D O is the dissolved oxygen concentration maintained in the aeration tank, g/m3,

O T E is the oxygen transfer efficiency,

TL is the temperature of the aeration tank content, O C,

pa,, is the density of air, kg/m3,

and (111440) is a unit conversion factor.

A minimum requirement for mixing or the aeration tank content is imposed on the

modcl lo maintain the complete-mk Bow required in the tank. This constraini is trail-

scribed as :

Qa -27 (2.26) v where is the minimum mixing requirement in m3/m3/min, whose value is assumed to be

0.02.

T h e dissolved oxygen concentration maintained in the aeration tank is assumed to be

1.5 g/m3 so that the biological activity of a non-nitrifying activated sludge system will not

be inhibited.

2.3.3. Secondary Sedimentation

A secondary clarifier performs two functions: clarification and thickening. Mixed

liquor suspended solids (MLSS) from the aeration basin must be removed from the plant

effluent to meet the water quality standards, while the settled solids should be concentrated

for biomass recycle and further sludge processing. The surface area of the clarifier is deter-

mined from either the clarification or the thickening requirement (Dick, 1970).

Clarification efficiency of the secondary clarifier is a critical factor in determining the

efficiency of the entire waste treatment system for both BOD and suspended solids removal.

The effluent BOD from a secondary treatment plant consists of both soluble organics

remaining or produced from the activated sludge process and the biodegradable suspended

solids in the effluent. Depending on the operating conditions of the activated sludge process,

suspended solids may account for more than half of the effluent total BOD. Thus the degree

of uncertainty inherent in 3 model of the clarifier is very important. Influent solids concen-

t r i t ion to a clarifier is usually in the range of 1500 to 3000 g/m3, or possibly even greater.

Since the desired effluent concentration is in the 10 to 20 g/m3 range, removal efficiencies in

excess of 99 percent are required. A slight deviation in this efficiency can have a pronounced

impact on the quality of the effluent from the system.

The design conditions of both the aeration tank and the secondary clarifier affect the

clarification efficiency. Parker (1983) provided an excellent review of how these design con-

ditions influence thc solids removal efficiency in the secondary clarifier. Becar~se of the com-

plesity involved in modeling the performance of this unit, a predictive r~iodel describing the

clarification efficiency based on fundamental mechanisms is not currently available.

Therc are, however, a number of empirical models in the literature tha t predict the

clarification performance of the secondary sedimentation tank (Table 2.3). Villiar (1967)

developed a regression model based on results from bench scale experiments. Takarnatsu

(s~noq) auoz ~cal3 u! au!? uo!,jnasjap = 11

(u!u/,u) yucg uo!?.elae 03 a)el MO~ J!Z = '6 (u) qqdap Jag'eM ap!s = H

(~E~/SSTJC S/aoa 9) ua?sl(s azpnls pa?c~y3a aqg u! o!?c~ L~S!UESJOOJ~!~ 03 pooj = J~J

Lm) J~IJ!J'~~ X~cpuo3as jo eale a3'ejJns = Iv : WON

(~861) V?aU!a?I 0.09'8Z + ;08FZ- (~861) q?Eu!aX put uo!qs.c3 ~BGSE' - 08G'l + '01E't. - Z'8P pu'e uo!qsv~

(9~61) MaJPT pua Xqsna

and Naito (1967) considered the effects of flow conditions on clarification efficiency using a

calcium carbonate suspension. Pllanz (1069) reported results from a series of in-plant stu-

dies carried out in Germany. These experiments were carefully controlled t o simulate steady

s ta te operation. T h e efRuent solids concentration was shown to be proportional to the feed

flow ra te and solids concentration. Sludge settleability, temperature, and wind were also

shown to affect clarifier performance. Lech (1973), Busby and Andrews (1975) and Keinath

e t al. (1977) have developed regression models from Pflanz's data.

Agnew (1972) proposed two models based on in-plant operating data . One of the

models provided a satisfactory Et for short-term observations of efRuent suspended solids

concentrations. However, this model did not adequately predict the clarifier performance

under varying operating conditions over a long period of time. A second model was then

developed from d a t a representing a wide range of operating conditions and sludges with

different properties. This model included design parameters for the biological treatment unit

as well as parameters representing the hydraulic efficiency of the clarifier. Both of Agnew's

models predict t ha t the effluent solids concentration decreases as MLSS concentration

increases, which contradicts Pflanz's observations.

Tuntoolavest e l al. (1080) used a laboratory-scale pilot plant facility supplied with syn-

thetic wastewater in an a t tempt to resolve the issue over the effect of hfLSS on clarification

efficiency and to determine other design parameters t h a t are important in influencing the

clarification efficiency. Their results supported the trend predicted by Pflanz, i.e., t h a t the

emuent solids concentration increases with higher LESS concentrations. They also observed

t h a t the turbulence level in the aeration tank, as measured by the air Row rate in their

study, affected the clarification eficiency. This observation was consistent with the conclu-

sion reached by Parker e l al. (1071) t h a t tlie floc-destructing environment of the aeration

tank has a direct impact on sludge settling characteristics and the clarification efficiency.

The thickening characteristics of the sludge were not found to be significantly related to

changes of the design parameters they studied.

Dietz and ICeinath (1982) presented a model based on a laboratory-scale s tudy using

calcium carbonate as settling particles. I t was shown that the steady-state clarifier perfor-

mance was most sensitive to the clear zone detention time in the clarifier. No consideration

was given though to the issue of upstream operating conditions in an actual treatment plant.

Chapman (1983) studied the effects on clarification efficiency caused by a number of

design variables. Among them, the side water depth of the clarifier. MLSS concentration,

clarifier feed flow and underflow rates were found to have significant impacts on clarification

efficiency. T h e air flow rate, however, was not an important factor. Chapman's results were

also in agreement with Pflanz's observation that the effluent solids concentration increases

with the MLSS concentration.

Cashion and Iceinath (1983)' studied the effects of solids retention time (SRT),

hydraulic retention time (HRT), and clarifier overflow ra te on the final clarifier solids remo-

val efficiency in a laboratory-scale unit treating real wastewater. T h e S R T values in their

study ranged from two to eight days, and the H R T values ranged from four to 12 hours.

T h e effluent solids concentration was found to be insensitive to the overflow rate. High

solids removal was attained in the regions defined by low S R T values and high H R T values

or high S R T values and low H R T values. No apparent correlation was observed between

the solids concentration of the influent to the clarifier and the ef luent solids concentration.

Sludge settling characteristics were rcportcd by Bisogni and Lawrence (1371) tn be a

function of sludge age. In their stutly with synthetic fced, activated sludge flocculated and

settled better with increasing sludge age for sludge ages beyond three days. Dick e l al.

(1970) conducted similar experiments using real wastewater. They found that the correla-

tion between sludge settling properties and sludge age was not significant. They also

observed tha t influent suspended sdlids concentration exerted an effect on sludge settling

behavior. It appears t h a t currently there is no satisfactory model for predicting activated

sludge settling characteristics a s a function of operating parameters in the aeration tank.

The soluble B O D , concenl;r:ition is assumed to be unchanged through sedimentation

and sludge separation, i.e.,

S3 = S, = S, = S, = S, (2.27)

The total effluent BOD concentration includes both the soluble and the suspended por-

tions. The effluent total suspended solids concentration is assumed to follow the model

developed by Chapman for the secondary clarifier. The side water depth in Chapman's

model is assumed to be a constant of 1.94 meters because the side water depth in the origi-

nal pilot study was varied over only a small range (1.48 to 1.94 meters) and the effluent

solids concentration is not very sensitive to this depth. The resulting model for secondary

clarification becomes

Q3 Mt4 = - c1 + c2Mt3 + c3- A J

where Mt3 and 1 2 1 , , are both in g/m3,

A j is the surface area of the secondary clarifier, m2,

and c,, c, and c, a re model parameters.

The effluent water quality requirements can be formulated as

1.42 g BODL g BOD, s3 + ( g cell )( 1.5 g BODL If clu,, SBOD

Mt, S ~ s s (2.30)

where SBoD and STss represent BOD, and total suspended solids restrictions, respectively, in

the emuent, and are in g/m3

Since the volatile biodegradable suspended solids are assumed to be completely con-

sumed during the activated sludge process (Section 2.3.2),

T h e ratios between the volatile inerts and the b ion~ass and the inorganic solids and the

biornass are assumed to be unnffectcd by secondary sedimentation or slridge separation. In

other words,

Dick (1970) discussed the importance of including sludge thickening as an integral part

of the design of a secondary clarifier. T h e underflow solids concentration from the clarifier

is governed by the thickening model (equation (2.9)),

where a, and n, are constants representing thickening properties of the waste activated

sludge, and

Q5 = (r + w)Q2 (2.34)

Decision variables selected for the design of the activated sludge process are the mean

cell residence time, hydraulic retention time, and sludge recycle ratio.

2.3.4. Sludge Blending

$ Figure 2.5 - Blending of Primary Sludge and Waste Activated Sludge

Since the primary and the waste activated sludges are combined before thickening (see

Figure 2.5), 3 set of mass balance relationships is needed to calculate the characteristics of

the influent to the thickener :

Q g = Q7 + Q8 (2.35)

= Q7Mt7 + Qsltfts (2.36)

Q g S g = Q7S7 + QsSs (2 .37)

The settling characteristics of combined primary and waste activated sludge have been

studied by Dick e t al. (1978) and Suidan (1982) using plant operating data . Regression

models were developed in both studies based on limited experimental da ta to relate the set-

tling constants in equation (2 .7) to the mass fraction of either the primary or the waste

activated sludge.

The empirical relationships developed by Suidan are used to determine the thickening

constants of the combined sludge:

*

where f p is the mass fraction of the primary sludge defined as

a l , a2, and n , are constants and a,, nc are constants characterizing the thickening of the

combined primary and activated sludge.

2.3.5. Gravity Thickening

The design of the gravity thickener is illustrated by Figure 2.6. The underflow solids

concentration is again calculated from equation (2.9),

1 - 1

" nc )( A , )C Aft11 = lac(nc-111 = (- -

n~ - 1 Qi i

where A, is the surface area of the thickener in m2.

T h e solids loading on the thickener is the decision variable. By definition, it is

Combining equations (2.41) and (2.42),

T h e flow and mass balance equations are

QioMtio + Ql lh f ! l l = Q9h49 (2.45)

There is no model available t o predict t he overflow solids concentration, Aftlo. As a result,

this concentration is treated as a parameter in the model, and is given a value of 0.2 kg/m3.

T h e solids compositions in the thickener overflow and underflow are calculated from

mass balance relationships based on the assumption tha t thickening does not affect t he

solids composition. For example,

Figure 2.6 - Design of the Gravity Thickener

Similarly,

T h e soluble BOD is assumed not affected by gravity thickening, i.e.,

2.3.6. Anaerobic Digester : Primary Tank

Conventional designs of an anaerobic digester use two-stage systems. The primary

digester is generally mixed and heated to the fermentation temperature. Most sludge stabili-

zation occurs in this unit. The secondary digester is not mixed and is primarily used to

thicken the digested sludge.

T h e design of the primary digester depends on the kinetic model assumed for waste

stabilization. There are several modeling approaches for the design of the primary digester.

Lawrence and h4cCarty (1969) developed design equations based on hlonod kinetics of sob-

s t ra te utilization. The underlying assunlption for the Monod kinetics is t ha t methane fer-

mentation is the limiting step. Sewage sludge is a mix of complex organic solids, however,

and it has been reported (PfeBer, 1968) tha t except for very high loading rates, hydrolysis of

the organic solids is the rate limiting step. A second modeling approach assumes t h a t the

stabilization ra te is first order with respect to the biodegradable (under an anaerobic

environment.) volatile solids. The percent volatile solids tha t is degradable as weil as the

first order ra te coelFicient were found to be functions or digestion temperature for tempera-

tures ranging from 25 to 35 O C (Pfeffer, 1981).

Chen and Hashimoto (1979, 1980) also proposed a set of equations for predicting diges-

tion performance. T o use this model for design, the biodegradable volatile solids concentra-

tion must be determined as a function of fermentation temperature.

Gossett and Belser (1982) studied the effect of sludge retention time in the activated

sludge system on the performance of the anaerobic digester. A first order reaction was pos-

tulated for the conversion of active biomass in the digester influent into available substrate

in the digester. T h e effect of temperature on digestion ra te was not studied

Wise (1980) summarized experimental results from studies involving stabilization of

various organic residues a t different temperatures. A first order kinetic model was assumed

for total volatile solids destruction. Figure 2.7 depicts the correlation between the digestion

, ra te coefficient and the fermentation temperature. T h e mathematical expression describing

this relationship is

where K 1 is the first-order rate coefficient in day-', and Td is the fermentation temperature

in OC. This model is selected for primary digester design in this study because it covers a

wide range of digestion temperatures. Consequently one decision variable for designing the

prilrlary digester is the fermentation temperature.

T h e primary digester is modeled as a complete-mix reactor where a11 sludge stabiliza-

tion is assumed to take place (Figure 2.8). T h e solids compositions in the digester effluent

are calculated based on the following assumptions: the volatile solids in the digester efRuent

are assumed to be nondegradable in- the aerobic environment and to consist of no microor-

ganisms tha t are capable of aerobic degradation of organic material. These assumptions are

K,, day-1 P

Figure 2.8 - Design of the P r i n ~ a r y Digester

necessxy in order to calculate the solids compositions of the digester supernatant which is

recycled to the liquid processing train. T h e inorganic solids are assumed to be unaffected by

anaerobic digestion. With these assumptions, the solids compositions of the primary digester

effluent can be calculated as

Vd where e d = - 24Qi1

is the sludge age in days, which is equivalent lo the hydraulic retention time for this diges-

tion system without solids recycle, and 1/, is the volume of the primary digester in m3. T h e

sludge age is the other decision variable for the design of this unit. No model is available for

prediction of the soluble BOD6 concentration of the digested sludge (S,,) when Erst order

kinetics is used t o describe the performance of the primary digester. Therefore, it is

assumed to be a constant , 500 g/rn3, in this study.

T h e flowrate of the digester emuent is

T h e methane gas produced'during digestion is calculated as

kg B O D L m3 CII, G = (1.42

kg VS )'0'35 kg BOD,

g B O D , m3 CH, + (1.5 g B O D , )(0'35 kg BODL

) ( 1 0 - 3 k ) ~ l l s l l g

where G is the methane production ra te in m3/hr. The Erst term is the methane produced

from stabilization or the volatile sr~spended solids, while the second term represents t h a t

from the soluble organics.

T h e energy value of the methane gas, E in kWhr/yr, is estimated to be

= 87113.3 G (2.53)

T h e heat requhements for raising the influent sludge to the digestion temperature, qR

in kWhr/yr, is

= 10.22 X lo3 Qll(Td - TO)

where To is the influent sludge temperature in C .

Assuming t h a t the digester is approximately cylindrical, and all digester units are uni-

form in size, then the heat loss of the digester to the environment, q, in kkVl~r /~ r , can be

estimated as

qL = (8.76 k IVhr Wall -yr ) u v , a ( T d - T,)

where U is the average heat conduction coefficient of the digester outside surfaces, iVall/m2-0C,

a is the rat io of the digester outside surface area to its volume, m2/m3,

and T, is the average ambient temperature in OC.

47

T h e total heat requirement for the digester to maintain its operating temperature is

Q R + QL q =

where E is the heat transfer efficiency or the heat exchanger.

The net energy value of the digester gas is the energy produced by methane less the

total heat requirement, or

N = E - q (2.57)

where N is the net energy value in kWhr/yr. This net energy production from the digestion

system is given a cost credit of 2.37 dol lars l10~kJ (0.25 dollars/therm) (Rimkus e t a l . , 1980)

in the overall system economic analysis assuming the methane is used on site.

2.3.7. Anaerobic Digestion : Secondary Tank

T h e secondary digester is assumed to be unmixed and unheated, and is modeled as a

gravity thickener with no methane fermentation taking place (Figure 2.9). T h e underflow

solids concentration, rrom equation (2.43), is

where Ld = Q 14Mt 14

Ad

is the solids loading, 2nd the decision variable for this unit,

Figure 2.9 - Design or the Secondary Digester

Ad is the surface area of the secondary digester in m',

a, 2nd n i are settling properties of a fully tfigcstcd sludgt-,

and 6 is a factor to discount the settling velocity of the digested sludge. In practice, the

gas production in the secondary digester may be suficiently high to cause some turbulence

in the digester. The rising gas will reduce the settling velocity of the digested sludge.

preventing the sludge from thickening to the degree expected from thickening theory alone.

The use of the factor 6 is intended to account for this observation. Initially, the value of 6

is assumed to be 0.25. T h e sensitivity of the overall system design to this value is examined

in Chapter 4.

There is no model available to predict the suspended solids concentration in the diges-

ter supernatant, :\ft13. Therefore it is treated as a parameter in the model and is assumed to

be a typical value o f 4 kg/m3.

T h e mass and flow balances around the secondary digester are

Q l c = Q13 + Q14

QicJft i2 = Q13hft13 + Qi4-bfti4

The soluble BOD, concentration is unaffected by this unit; therefore

Sl2 = Sl3 = s14 The solids compositions are assumed to remain the same, or

2.3.8. Vacuum Filtration

T h e design of the vacuum filter is shown schematically in Figure 2.10. Coackley and

Jones (1956) compared several filtration theories and concluded t h a t the model proposed by

Carman (1033) fits experimental da ta most adequately. They developed the following equa-

tion for calculating the filter yield from Carman's analysis for given operating conditions and

a sludge with known specific resistance,

where Li is the filter yield in kg/m2/hr,

x is form time per cycle time,

P is the vacuum pressure applied in Newtons/m2,

p is the viscosity of filtrate in Newton-sec/m2,

r, is the specific resistance in m/kg,

t, is the cycle time in minutes,

and W = Q ieM: le Q 16

is the mass of solids filtered per unit volume of filtrate in kg/m3. Christensen (1983) has

summarized the values of specific resistance for various sludges t o be dewatered. The filter

yield is the decision variable of this unit.

Figure 2.10 - Design of the Vacuum Filter

T h e size of the filter is

A, = QleMtte

J

where A, is the filter area in m2.

The mass and flow balance relationships around the unit give

Qi4Mti4 = Q15Mtis + Qieh*!~, (2.68)

T h e suspended solids concentration in the filtrate (Mt1,) is assumed to be a constant of 2

kg/m3 due t o the lack of a predictive model.

T h e soluble BOD, concentration is the same throughout the process:

s 1 4 = S16 = Sib

T h e solids components are:

2.3.9. Recycle Streams

T h c side streams generatrd in sludge treatment are recyclcd back t o the hcad end of

the plant for the removal of the organics and the suspended solids (Figure 2.11). T o arrive

a t a steady-state design of the system, flow and mass balances must be met where the recy-

cle streams join the influent stream to the plant:

= Qo + Q l o + Q l 3 + Q i a (2.71)

.sar3~ u! luamar!nbar puq aql s! 7y araqm

B1'jtrg1gZ-07 x Z9.1 = 7~

'(8~67) 'ID la

93!a Lq padolaaap uo!lcnba %u!molloj aql %u!sn pa3cmysa s! luamar!nbar sarc pucl aqJ

.ma?sLs ascq aql u! pamnssc s! llypuc[ Lrcl!nss

Lq ~csods!a .sa!Sa)sqs luamaScucm a%pnls lcm!ldo JO ~namdola~ap r!aql u! (7861 '8~67)

'ID la 93!a Lq L[aa!sualr;a papnls uaaq aacq Laql puc 'alqcl!cac arz suoydo ald!llnn 'smal

-sLs luamlcarl ralnmaqsem jo lrcd ~zrSalu! uc s! aye3 aSpnls aql JO ~ssods!p alem!lln

.rol3ej uo!slaauo3 l!un s s! E-~~ pus cEm/~ u! an O/jY PUF 'O!W 'Opjv 'OVjv alaqm

(LL'Z) TljV + 1.'~~ + IPJq + IVj\l = IlJq

m3 h r W , = ( & l , T ) ( 1 . ~ 4 ~ ) ( 1 0 6 ems)(24 -)( 1 ton r c m m3 day 9 . 0 7 2 ~ lo5 g 1

2.4. Cost Information

T h e total cost of the wastewater treatment system is the sum of the costs of all unit

processes. Although cost d a t a are abundant in the literature, only those d a t a t h a t relate

costs to the capacities of the units are useful for this study. Smith (1968) developed cost

functions from cost da ta collected by Logan et al. (1962) and Swanson (1966). Patterson

and Banker (1971) presented the capital, operation and maintenance costs in graphical forms

with respect t o the sizes of the unit processes. Cost functions have been developed from this

information by Middleton and Lawrence (1975), the U. S . Army Corps of Engineers (1978),

and Rossman (1979). Dick et al. (1978) also developed a set of cost functions based on d a t a

presented by Patterson and Banker, Metcalf and Eddy, Inc. (1975), and Ettlich (1977).

These cost functions were compared lor unit processes considered in this study using

constant year (1971) dollars. The results of this comparison are summarized in Appendix A.

Considerable variations in unit process costs were observed among different sources of data:

Costs of wastewater treatment systems vary locally and depend on many factors. Therefore

the cost functions considered in this study are only meaningful in the sense tha t they

represent typical relative costs among unit processes.

Cost functions selected for use in this study are summarized in Table 2.4. They a re

based primarily on the d a t a collected by Patterson and Banker. T h e firm pumping capacity , , i I

is assumed to be two and a half times the average daily flow.

Costs for final sludge disposal by sanitary landfill a re not listed in Table 2.4. These

costs include capital and operation costs. T h e capital cost is calculated according to the

equation presented by Rossman, -

Table 2.4 - Summary of Cost Functions

Capital Operation Maintenance Material Power and Supply 1 1 9 i l $ / ~ r )

( k lVhr /yr )

Primary 8 2 4 ~ ; ~ ~ 17. 15Ape (-4,2279) 9.23Ape ( A , 2 2 7 9 ) 8.62Apie - Clarifier 92.45Ap3 ( A p <279) 106Ap14 ( A , <279)

primary 16042Qt3 3 7 4 Q i 1 166 Q t 3 385Q;Ie4 23.85 Q , H / E , ~ Sludge

Pumping

Aeration 461 v7' - Tank

Diffused 8533 Qkee 187 Qi4, 74.4 Qib6 Aeration

Secondary 824A j77 1 7 . 1 5 ~ je ( A l 2 2 7 9 ) 9 . 2 3 ~ je (.dl 2 2 7 9 ) 8 . 6 2 A f e - Clarifier 9 2 . 4 5 ~ j ~ ( A l <279) 106Aj14 ( A l <279)

Return & 2779Qib3 .333Q6+390 .2375Qb+370 300 ( Q 6 <63.2) 23.85Q, l I /~ , Waste 40.57Q;jS2 ( Q b <252) Sludge 5.97Qig7 (Qb<632) Pumping 2.5dQb (Qb>632)

Gravity 8 2 4 ~ 1 ; ~ ~ 1 7 . 1 5 ~ ; ~ ( A , 2 2 7 9 ) 9 . 2 3 ~ ; ~ ( ~ ~ 2 2 7 9 ) 8.62.4p - Thicker 92.45Ai3 ( A g <279) 106A;14 ( A g <279)

Anaerobic 2323 V f g 1.29 Vig3 ( V d 2 5 6 7 8 ) 0.83 V d 2 5 6 7 8 ) 14.4 lriee ( Vd2"39) - D~gester 14 Vib5 ( V d 21968) 8.5 Vib5 ( V d 21968) 142 V;" ( V d <2839)

192 b2 ( vd <1968) 113 b21 ( V d <1968)

vacuum 2 9 1 8 0 ~ ; ~ ~ 197.55Qit8Mifg 5 . 5 7 Q i w i (QleAI t l e2519) 230Qii1AI;:;+ - F ~ l t e r ~ O Q ~ ~ ~ A I ; ~ ~ ( Q ,,A1,,,2103) 182Qi~8L\1;~~

41.5Q;",8Al;:,8 ( Q l e i l l ~ l e <103)

Recirculation 2779 c ) ; ~ ~ ~ 0.333 Q , +390 0.2375 Q , +370 300 ( ~ ~ 2 6 3 . 2 ) 2 3 . 8 5 Q r l l / ~ ,

Pumplng 40.57 Q;6" Q , <252) 5.97 Q;87 ( Q , <632) 2.54Q, ( Q r >632)

t H is the pumping head in meters, and E , is the pumping efficiency.

C C = AL CL + 62001;; 1VP.74

where C C is the capital cost in present value (P.V.) dollnts,

CL is the unit cost of land, P.V. dollars/acre,

and F, is a factor updating the cost from 1971 dollar to the present value.

Equation (2.80) can be rewritten in terms of Q,, and ,lltl, by substituting (2.78) and

(2.79) Tor AL and 1V,, respectively,

C C = 3.62 X 10-'CL Q1,httl, + 72053F., Q:z4 (2.81)

The annual manhours lor the landfill operation is estimated using d a t a from the U. S.

Environmental Protection Agency (USEPA) Process Design hlanual (1974),

OHRS = 8024 Qp:e7 (2.82)

where OII'RS is the annual operation manhour requirement for a landfill. T h e development

of equation (2.82) is described in Appendix B.

The total annual cost in 1980 dollars is used to express the total system cost. A

3 twenty-year design life and a 7-% discount rate are assumed t o amortize the capital costs. 8

The USEPA National Average Wastewater Treatment Plant Index is used to update the

capital costs and the costs for material and supply. Annual operation and maintenance

costs are calculated by multiplying the manhour requirement by the hourly wage rates. T h e

cost for pumping is the product of the power requirement and the unit power cost.

2.5. System Design

A complete set of equnt.ions for designinq a secondary wastewater treatment system is

presented in Section 2.3. T h e design of the overall t reatment system lor specified influent

conditions and decision variables using these equations is illustrated in this section. In

Chapter 3, a comprehensive model assembled based on a subset of the design equations

described in Section 2.3 is presented. This model is optimized using a nonlinear program-

ming algorithm to generate cost-emective designs for the studied wastewater treatment

system. Description of this model is provided in Section 3.2.1.

There are nine degrees of freedom in the wastewater treatment system model. There-

fore a complete system design requires specification of nine decision variables. T h e selected

decision variables in the model are summarized in Table 2.5.

Bounds are imposed on the decision variables in the comprehensive system model.

Table 2.6 summarizes these bounds. Most of the bounds cover typical range observed in

practice for the conventional activated sludge process. T h e values of these bounds are set

arbitrarily, but are relatively reasonable to avoid 'the possible lack or efficiency of an

Table 2.5 - Summary of Decision Variables in the Model

Unit Decision Variables

Primary Sedimentation Overflow Rate (L,)

Activated Sludge (Aeration Mean Cell Residence Time (8,) + Final Sedimentation) Hydraulic Retention Time (8)

Sludge Recycle Ratio ( r )

Gravity Thickening Solids Loading (Lg)

Anaerobic Digestion -Primary Digestion Temperature ( T d )

Solids Residence Tirne (Bd) -Secondary Solids Loading (Ld)

Vacuum Filtration Filter Yield ( L , )

Table 2.6 - Bounds on the Decision Variables

Variables Lower Bound Upper Bound

Overflow Rate, Primary Clarifier (m/hr ) 0.5 6.0 Mean Cell Residence Time (days) 2.0 6.0 Hydraulic Retention Time (days) 0.1 0.5 Sludge Recycle Ratio 0.1 1 .O Solids Loading, Gravity Thickener (kg/m"hr) 0.5 2.0 Digestion Temperature ( C) 20 60 Residence Time, Pr imary Digester (days) 5 30 Solids Loading, Secondary Digester (kg/m"hr) 0.5 2.0 Filter Yield (kg/m2-hr) 5 50

optimization algorithm. Exceptions are the bounds on the activated sludge mean cell

residence time and on the digestion temperature. Bounds imposed on the mean cell

residence time are to prevent the process from failure. In addition to insure against process

failure, the bounds on the digestion temperature define the domain on which the empirical

model (equation (2.48)) is based. T h e solids concentration of the filtered cake is also con-

strained to be less than 150 kg/m3 because the process model used for vacuum filter design

does not predict a maximum cake concentration that can be obtained in practice. If the final

solution obtained from optimizing t,he system design model suggests tha t some of the deci-

sion variables are a t their imposed bounds, then the roles of these bounds are examined in

detail. This is carried out in Chapter 4.

Design of the overall system may be carried out using several approaches once the

decision variables are specified. A straightforward approach was employed in this study: unit

processes are designed sequentially according to the system flowchart. Since only a few

eqliations are solved in the design of each unit process, the computation required for one

iteration of design is not excessive. However, a steady s ta te design cannot be obtained in one

iteration because of the presence of the recycle streams in the system. Characteristics of the

recycle streams, however, a re determined a t the end of each iteration. A new set of influent

conditions to the plant is calculated by mass balance relationships between the design plant

influent and the recycle streams. A new iteration is then initiated using the newly calculated

influent conditions. This direct substitution process is continued until the fractional changes

of all influent s ta te variable values becon$ less than lo-'.

An analysis computer program was written to carry out the calculations. Figure 2.12

shows the logic on which the design of the analysis program is based. T h e listing of the pro-

gram and the instructions for using the program are given in Appendix C. More efficient

calculation schemes than direct substitution for updating the initial design conditions are

available (Westerberg e t al., 1979). However, since a typical steady-state design can be

Figure 2.12 - Flow Diagram of the Analysis Program

( Start )

I Input:

(1) Design conditions, Vo (2) Parameters (3) Decision variables

1 Influent conditions to the system, V, = Vo

1 Perform sequential design

of each unit process b

I Obtain recycle stream

characteristics, V,

1 Calculate new influent conditions, VI1,

from mass balance of Vo and V,

No 1

Let V 1 = V 1 ,

Yes

V

Calculate total system cost

achieved in less than ten iterations with computer time less than 0.3 seconds on the CDC

Cyber 175 computer, the direct substitution strategy was considered adequate for this s tudy.

An example system design obtained from using the analysis program is presented

below. T h e wastewater treatment system was assl~med to receive 3 typical domestic sewage

with characteristics listed in Table 2.7. The parameters in the model are tabulated in Table

2.8. T h e values of the nine decision variables used for the system design are summarized in

Table 2.9. Figure 2.13 (refer to Section 2.2.2 for the notation) describes the complete system

design obtained from the analysis program for the conditions listed in Tables 2.7 t o 2.9. I t is

noted tha t any arbitrarily selected values for the decision variables may lead to 3 design

tha t does not meet the effluent requirements or may result in a filtered cake more concen-

trated than 150 kg/m3. Such a design is called an infeasible design.

T h e analysis program is useful for examining the responses from the system model for

given influent and design conditions and for generating system designs tha t can be used as

I

start ing solutions in various optimization procedures. This is illustrated in more detail in

Chapter 3.

Table 2.7 - 'Base Design Conditions

Flowrate ( m y h r ) 1500 Soluble BOD, (g/m3) 100

Active Biomass Conc. (g/m3) 5 Volatile Biodegradable Suspended Solids Conc. (g/m3) 100 Volatile Inert Suspended Solids Conc. (g/m3) 45 Fixed Suspended Solids Conc. (g/ms) 50 Total Suspended Solids Conc. (g/m3) 200

Table 2.8 - Summary of Parameters in the System Model

Names (LTnits) Value

Economic Data:

Capital Recovery Factor Base (1071) Cost Index Cost Index Tor 1980 Operating/h,laintenance Wages jdollars/hr) Land Cost, CL (dollars/acre) Electricity Cost (dollars/ klVhr) Pumping Head, 11 (meters) Pumping Efficiency, E

P

Primary Sedimentation:

Constant in Voshel-Sak Model, vl Constant in Voshel-Sak hlodel, v2 Constant in Voshel-Sak Model, v,

Sludge Settling Characteristics:

Tbickening Constant, a, Thickening Constant, a , Tbickening Constant, a2 Thickening Constant, n, Thickening Constant, n l

Activated Sludge Kinetics:

Growth Yield Coefficient, y (g cell/g BOD,)

Half-Velocity Constant, K, (g BOD,/m3)

Maximum Specific Utilization Coefl., k (day-') Endogeneous Decay Coeficient, b (day-') Fraction of cells Degradable. j d Conversion (g BODL/g cell) Conversion (g 130DL/g BOD,)

Secondary Sedirrlentation

Constant in Chapman Modcl, c , Const.ant in Chaprnan Modcl, c,

Constant in Chapman hlodel, c3

Aeration:

Alpha Factor in Aeration Beta Factor in Aeration DO Concentration in Aeraton Tank, D O (g/1113)

60

Table 2.8 (continued)

Names (Units) Val~le

DO Saturation Concentration, C, (g/m3)

Temperature of Mixed Liquor, TL ( ' C ) Oxygen Transfer Efficiency, O T E Density of Air, pair (kg/m3) Weight Fraction of Oxygen in Air, y Mixing Requirement, 7 (m3 air/m3/min)

Gravity Thickening:

TSS of Thickener Supernatant, M,,, (kg/m3) 0.2

Anaerobic Digestion:

Temperature of Digester Influent, To ( C ) Methane Production (m3/kg BODL)

Average Ambient Temperature, T , ( ' C ) Efficiency of Heat Exchanger, a Heat Conduction Coefficient, U (W/m2- C ) Outside Surface Area and Volume Ratio for Digester, a Worth of Digester Gas (dollars/lOO k J ) Soluble BOD5 in Digester Supernatant, S12 (g/m3) Factor Accounting For Effect of Rising Gas

on Thickening in Secondary Digester, 8 Thickening Constant for Digested Sludge, a,

Thickening Constant for Digested Sludge, n,

TSS of Digester Supernatant, A!,,, (kg/m3) Height of Digester (m)

Vacuum Filtration:

Form Time per Cycle Time, x Pressure Applied on Vacuum Filter, P (Nt/m2) Viscosity of Filtrate, p (Nt-sec/m2) 0 Cycle Time, t , (min)

Specific Resistance of Sludge, r, (m/kg)

TSS of Filtrate, All , , (kg/n13)

Emuen t Standards:

BODs Concentration (mg/l) 30 TSS Concentration (me/]) 30

Table 2.9 - Decision Variables for Example Treatment System Design

Decision Variables (Unit) Value Primary Clarifier Overflow Rate (m/hr) 3.0 . . Mean Cell Residence Time (days) 3.0 Hydaulic Retention Time (days) 0.15 Sludge Recycle Ratio 0.15 Solids Loading on Thickener (kg/m2/hr) 1 .O Digestion Temperature (OC) 35 Retention Time in Digester (days) 15

Solids Loading on Digester (kg/m2/hr) 1 .O Filter Yield (kg/m2/hr) 8.0

B- r '6

GE'L

CHAPTER 3

OPTIMIZATION OF THE COMPREHENSIVE SYSTEM MODEL

3.1. Introduction

As described in Chapter 2, the design of a wastewater treatment system is formulated

as an optimization model in which the total system cost is to be minimized subject to the

unit process performance models and the effluent water quality requirements. This chapter

discusses the techniques tha t were used in this study for solving the comprehensive system

model. Illustrations of the use of these solution techniques are presented, and performances

of these techniques are discussed.

T h e comprehensive system model is highly nonlinear; the objective function and the

majority of the constraints are nonlinear. Most constraints are equations; exceptions are the

ones specifying effluent water quality and the mixing requirement in the aeration tank. T h e

problem is poorly scaled, usually with overflow and underflow rates (expressed in the same

unit) from a separation unit differing in magnitude by several orders of ten. T h e complex

arrangement of the units in the system appears to make it impractical to apply dynamic

programming as the solution technique even though stages and states are clearly defined by

the model. One approach to optimization examined in this study is to apply a well-tested

nonlinear programming algorithm to solve the comprehensive system model directly. T h e

ger~ernlized red uccd grad icn t (GRG) algorithm dcvcloped by Lasdon el al. (1978), named

GRG2, has been applied to many liighly nonlinear programs with success. Studies of the

computational experience with various constrained nonlinear programming methods have

shown tha t the GRG algorithm is among the most efficient ones (Warren and Lasdon, 1979).

GRG2 is well designed so tha t it competes favorably with more advanced algorithms such as

sequential quadratic programming in terms of robustness and reliability (Schittkowski,

1983). Section 3.2 describes the use of GRG2 to optimize the comprehensive system model.

Special-purpose optimization algorithms developed for efficient solution of models with

special characteristics may also be used to solve the comprehensive system model. T h e

Interactive Generalized Geometric Programming (IGGP) code designed by Burns and

Ramamurthy (1982) is an efficient algorithm with the capability of solving large-scale

geometric programs. This algorithm is based on the primal condensation method proposed

by Avriel e t al. (1975). Burns and Ramamurthy extend this algorithm to solve problems

with equality constraints. This extension allows the use of IGGP for solving tbe comprehen-

sive system model. This is illustrated in Section 3.3.

A unique optimization procedure designed to take advantage of the special structure of

the wastewater treatment system model was developed and is evaluated in Section 3.4. T o

solve the comprehensive system model by nonlinear programming directly, all equations

have to be solved simultaneously. This mathematical operation is very costly with respect

to computing requirements. A wastewater treatment system is generally composed of a

liquid processing train And a sludge processing train, each consisting of individual unit

processes provided to perform various treatment functions. By decomposing the entire treat-

ment system, 3 series of subproblems with lower dimensionality can be solved instead of a

large problem. Optimization techniques can be applied more eflectively for solving these

smaller problems, bu t coordination of the solutions is also required.

3.2. Generalized Reduced Gradient Algorithm for Optimization

T h e generalized reduced gradient algorithm is an extension of the reduced gradient

algorithm by Wolfe (10G3, 1967) to allow the solution of problems with nonlinear constraints.

T h e earliest development of the algorithm was by Abadie and Carpentier (10GO). Later

improvements of the algorithm have incorporated many strategies for solving subproblems

during the overall optimization procedure (see, for example, tIimmelblau, 1972). GRG:! was

used in this study.

GRG2 solves the following general nonlinear program:

Minimize c (X)

subject to g(X) 0

h(X) = 0

X, 5 X 5 Xu

where c is a scalar objective function, and is the total cost of the wastewater treatment sys-

tem in the comprehensive system model,

X is the vector of the variables in the model,

g is the vector of the inequality constraints,

h is the vector of the equality constraints,

and X,, and X, are vectors representing the lower and the upper bounds of the variables,

respectively.

T h e underlying concepts in developing GRG2 are described in detail by Lasdon et al.

(1078).

3.2.1. Optimization Procedure

T o make an optimization run, the user is asked t o provide two Eles: one containing the

program control parameter, initial solution to the problem, and bounds on the variables,

and another specifying the model objective function and constraints. Instructions on using

the program on the CDC Cyber computer can be found in the GRG User's Guide prepared

by the Computing Services Ofice a t the University of Illiriois (1082).

T h e optimization model solved by C R C 2 includes 6.1 variables and 55 design equations

and three inequality constraints. T h e model is constructed based on the design equations

discussed in Section 2.3. Detailed descriptions about the variables and the constraints in

this optimization model are provided in Appendix D.

The control parameters in GRG2 are critical to the likelihood of obtaining convergence

of the optimization procedure as well as to the quality of the final solution. The derivatives

of the functions were approximated by the central differencing method. An equality con-

straint , g(X) = 0, is considered to be satisfied when its value is in the (-neighborhood of

zero, i.e., Ig(x)I 5 (. The value of this tolerance, (, was initially set to be The objec-

tive function generally improved significantly as the algorithm proceeded with this tolerance

level. When the fractional change in the objective function became less than lo4 for three

consecutive iterations, the value of ( was tightened to lo4. Then a phase-I optimization,

which minimizes the sum of the constraint infeasibilities, was initiated until a11 constraints

were satisfied to this final tolerance level and a feasible solution was found. Optimization of

the true objective function was then begun until the termination criteria were met. The

final solution obtained with this strategy was generally found to be superior to tha t obtained

using a tight tolerance level throughout the optimization.

The basic variables were estimated using quadratic extrapolation. The one step ver-

sion of the Broyden-Fletcher-Shanno variable metric method (see, for example, Avriel, 1976)

was selected for generating search directions in the GRG2 runs.

Scaling of the variables as well as the constraints in the model has a direct effect on

whether the optimization will be successful or not. No general rules are available; scaling

nonlinear programs, as described by Lasdon and Beck (1981), is a "black art". Most vari-

ables in the model were scaled to have numerical values between 0.1 and 100 as suggested by

the authors of GRG2. Some constraints were also scaled by trial-and-error in an a t tempt to

achieving a balance among all constraints. Scaling factors in the optimization model solved

by GRG2 are discussed in Appendix D.

3.2.2. Performance of GRG2

The elficiency of GRG2, the quality of the soiutions obtained, and the effects on the

solution of the imposed bounds on the selected variables are discussed in this subsection.

T h e computing time required for an optimization run varies with the start ing solution

and is highly dependent on the quality of the final solution. For a11 the GRG runs made in

this s tudy, the computing time never exceeded two minutes of central processing (CP) time

on a CDC Cyber 175 computer when the program was run in batch mode with the control

parameter values specified in Section 3.2.1. A FORTRAN V compiler was used to compile

the program tha t contains the objective function and the constraints.

Based on the results from a number of test runs, it was noticed tha t varying some of

the control parameters may result in a slightly better solution or a slightly faster optimiza-

tion process for a particular starting solution and set of design conditions. However, in

order for the results t o be consistent and comparable, the control parameters used for run-

ning GRG2 were kept the same for all runs.

Computing experiences of some previous studies involving wastewater treatment sys-

tem design models are listed in Table 3.1 for comparison. Although a straight comparison of

the computing time requirements is not meaningful, this table does seem t o indicate tha t the

computing time using GRG2 for the comprehensive system model is a t least comparable

since the model solved is more complex than the others listed.

Because the model is highly nonlinear, rnultiple local optima are expected t o be

present. Different start ing solutions were used t o examine this issue. Table 3.2 summarizes

the results of using live diBerent start ing solutions. T h e final solutions have objective func-

tion values t h a t vary from 502,000 t o 584,700 dollars/year, representing improvements in

the objective function from the initial solutions from 17 (starting point No. 1) to 33% (start-

ing point No. 5). All solutions call for designs tha t produce emuents exactly meeting the

Table 3.1 - Computing Experience in Optimizing W:lstewater Treatment System Design

Optimizstion Execution Time Machine Met,hod

Comment s (seronds)

T a n g CRC:! 51-105. C D C 9 degrees of freedom, Cyber 175 58 constraints,

64 variables.

O the r Investigators :

Middleton & Lawrence Graphical 9 6 IBM 5 degrees of freedom, (1976) Enumerat ion 360165 N o recycle.

Craig e t al. (1978)

Box-Complex 1.65-2.82 C D C 5 degrees of freedom, Cyber 173 No recycle.

T y t e c a & Smeers C R C for a 124263 IBhl 8 degrees of freedom, (1981) geometric program 3701158 35 constraints,

33 variables. - -

* For the base t r ea tmen t sys tem shown in Figure 2.1.

BOD5 and total suspended solids standards. Among the Eve start ing solutions, No. 4 and

No. 5 differ only in the primary clarifier overflow rate, but the optimization results are very

different. This is due to the fact tha t the initial solutions are quite different in the values of

variables other than the decision variables. Figures 3.1 and 3.2 illustrate these two designs

(notation is defined in Section 2.2.2). T h e design obtained with the higher overflow ra te

(starting point No. 5, Figure 3.2) has a higher mixed liquor suspended solids concentration in

the aeration tank and has to waste more activated sludge for the same hydraulic retention

time, sludge age, and sludge recycle ratio. Therefore the combined primary and waste

activated sludges in the two designs exhibit quite dirrcrent charncteristics which result in

very different values of the state variables when the sludge processing train is designed using

the same design criteria. T h e importance of the choice of start ing solution when using

GRG2 t o solve the comprehensive system model is obvious from this example.

The solution obtained by GRG is directly related to the bounds on the variables. This

is best illustrated by an example. Two optimization runs start ing from the same solution

Table 9.2 - Summary of Wastewater Treatment System Designs Obtained Using DiRerent Starting Points

Variables (Units)

Primary Clarifier Overflow Ra te (m/dav) initial final

Mean Cell Residence Time (days) initial final

Hydraulic Retention Time (hr) initial final

Sludge Recycle Ratio (%) initial final

Solids Loading on Thickener (kg/m2/day) initial final

Digestion Temperature ( C) initial final

Retention Time in Digester (days) initial final ,

Solids Loading on Digester (kg/m2/day) initial final

Filter Yield (kg /mvhr ) initia.1 final

Cake Solids Concentration (kg/m3)

in i t id final

Effluent BOD, (mg/l)

initial final

Effluent 'rSS (mg/l)

initial final

Tota l System Cost ( lo3 $/yr) initial final

Computer Time ( C P seconds)

Starting Point 1 2 3 4 5

t : infeasible

Table 3.3 - Solution Obtained Using GRG with Different Rounds on Selected Variables

Variables (Units) Starting Solution With Solution With

Point Default Bounds Modified Bounds

Primary Clnrifier Overflow Rate (m!day) 36.0 80.0 43.8 Mean Cell Residence Time (days) 2.0 2.22 2.26 Hydraulic Retention Time (hr) 2.4 3.7 3.5 Sludge Recycle Ratio (%) 15.0 11.6 11.5 Solids Loading on Thickener (kg/rn2/day) 24.0 12.5 12.6 Digestion Temperature (OC) 30.0 60.0. 60.0. Retention Time in Digester (days) 15.0 14.7 14.7 Solids Loading on Digester (kg/m2/day) 12.0 38.4 36.3 Filter Yield (kg/m2/hr) 10.0 6.79 6.92 Cake Solids Concentration (kg/m3) 164.1 150.0. 150.0. EfIluent BOD5 (mg/l) 30.8 30.0 30.0 EfRuent TSS (mg/l) 23.9 30.0 30.0 Total System Cost (lo3 $/yr) 678.0 506.1 517.5 Computer Time (CP seconds) - 50.79 39.53

* These values are a t their specified bounds.

(starting point No. 2 in Table 3.2) were made with slightly diflerent bounds on the decision

variables. T h e solution shown in the second column of Table 3.3 was obtained using the

default bound set summarized in Table 2.6. In the second optimization run, the upper

bound on the primary clarifier overflow rate was changed from the default value of 144 to

240 meters/day, and the lower bound of the solids loadings on both the gravity thickener

and the secondary digester were changed from If! to 2:i kg/m2/day. These numbers have

little physical significance and were used only for this experiment. The results of this run

are summarized in the 1 s t column of Table 3.3. It is observed that the final objective func-

tion vn111es are clilTcrent by 2.3%. Note tha t none or the thrce decision varizbles for which

the bounds were modified is s t its bound in the linal solut.ion. Tlie ovrrllow rnie for the pri-

mary clarifier in the two final solutions is the variable that showed the most significant

diflerence in the two designs. This appears to be 3 weakness of GRG2 since most bounds on

the decision variables, as described in Section 2.5, were arbitrarily selected and have little

fundamental significance. DiNerent nonbasic variables could be selected in GRGS if different

bounds are specified on the variables, resulting in diflerent optimization processes and

diBerent solutions. Ideally, the optimal solution should not depend heavily on the bounds

specified for the variables (which are not limiting the solution).

T h e solution process by GRC2 may terminate due t o several reasons : a local optimum

may be found, a feasible solution may be unxiailable in the phase-I optimization, or some

numerical difficulties such as scaling may cause the solution process to stop prematurely.

Most of the optimization runs presented in this study terminated because the fractional

change in the objective function was less than the specified tolerance for a specified number

of iterations. T h e characteristics of the final solutions of this type are uncertain since they

may or may not be local optima.

In summary, the solution obtained by GRG2 is observed to be affected by the start ing

point, t he bounds on the variables, the tolerance levels of the equality constraints, the stop-

ping criteria, and the various optimization strategies tha t arc employed within the GRG2

optimization procedure. These difficulties associated with using GRG2 t o optimize the

comprehensive system model prompted the development of a strategy t o evaluate the qual-

ity of the solutions obtained and to generate alternative good solutions t h a t may be exam-

ined further from a practical perspective. The following subsection examines a strategy t h a t

is designed for this purpose.

Developing alternative procedures for optimization of the comprehensive system model

is also suggested by the difficulties of using GRG2. Sections 3.3 and 3.4 describe two alter-

native solution procedures.

3.2.3. E x p l o r a t i o n o f t h e F e a s i b l e Des ign S p a c e

Brill (1979) proposed t h a t when using an optimization model of a complex planning

problem with important unmodeled issues it may be desirable to use the m o d ~ l to explore

alternative soln tions. These alternatives can then be evaluated with respect t o the unmo-

deled issues. The first step in his Hop-Skip-Jump (HSJ) method is t o obtain an initial design

using a single or multiple objective procedure. The next step thcn is to solve the following

optimization problem :

Minimize C z k kCK

subject to fj(X) 2 Ti , W- j

where K is the index set of those variables which nre nonzero in the initial design,

fj(X) is the j t h objective function, and is a function of the solution vector, X,

Ti is the target specified for the j t h objective,

and Fd is the feasible solution space.

This formulation is designed to generate a maximally different solution from the initial solu-

tion. T h e objective function space can be explored by solving a sequence of problems in the

form of program (3.2), and alternative designs can be generated and examined

Extending this idea by a slight modification of the objective function in program (3.2),

we can explore the feasible design space of the wastewater treatment system model by solv-

ing

Optimize F (X)

subject to c(X) 5 T (3.3)

X € F,

where the objective function, F , is a function of the variables, and may be minimized or

maximized. This function may be formed a t random or using knowledge or engineering judg-

ment of the problem. The totnl system cost, c(X), which is the objective o l the original

optimization problem (program (3.1)), is constrained to be less than or equal to a target, T,

which mny be arbitrnrily determined, or which may be the same as the cost of the solution

obtained from GRG2.

If a feasible solution can be obtaiqed from solving the constrained formulation (3.3)

with T set t o the current best value of the objective function, then the new solution will be

a t least as good. The new solution may meet the target exactly, bu t it may represent a

design tha t is different from the current solution.

Table 3.2 reveals characteristics in the decision variable values t h a t result in cost

effective designs of the base wastewater treatment system. While all five final designs have

similar values for the mean cell residence time, hydraulic retention time, sludge recycle ratio,

and solids loading ra te on the gravity thickener, i t is noted that "good" designs exhibit some

special characteristics. Design No. 5 has its overflow ra te on the primary clarifier a t its

upper bound (144 meterslday); designs No. 2, 3, and 5 all have the digestion temperature a t

the specified upper bound (60 O C ) ; and designs No. 2 and 3 have the cake solids concentra-

tion a t the upper bound or 150 kg/m3. If these characteristics indeed lead to a more cost

effective design than other feasible designs, then it may be possible to improve further the

solution obtained from GRG by using program (3.3) to examine it with respect to these

characteristics.

Program (3.3) was constructed for each of the Eve designs examined in Table 3.2. T h e

objective functions and target values used to form program (3.3) as well as the results of

solving program (3.3) are summarized in Tables 3.4 through 3.8. The solution obtained from

GRG2 using start ing point No. 1 was used as the start ing point in Table 3.4 with the cake

solids concentration being the objective function t o be maximized. A different solution was

obtained, but t.he total system cost remained the same. The difference between the two

designs is primarily in the sludge processing train because the objective function chosen is

related directly t o the design of sludge treatment units. This solution was then used as the

start ing point for the next optimization where the overflow rate of the primary clarifier was

maximized. This run produced another different design with the same total system cost.

Ilowever, the major difference between this and the two previous designs is on the liquid

::::.1:1 I 't:.91 SC'L (spuo~as d3) aul!L ~alndulo3

0 COS G'ZPS 0'CI.S IVU'J

6';l : G'F.1.Y Ci';.f cJ [r!]!:~! (LC/$ ,o[) ?so,-, u1als.C~ Il:?oJ,

00:: 0 OC 0'01' 1ruy O'OC 0.0: 0.0: IT!l!U!

(l/Sm) SSL 'JU3nUJB 0'06 O'OS O'OI: ITU'J O'OC O'OC 0'0'2 I"!?!U!

([/Pm) 'sou ?uanw3

U'OSI O'OSI O'OSI ITUY 0'0'21 O'OSI G'C1.l lT!l!U!

(Lm/UQy) uo!lclquaJuoj sp!los aye3

111'9 qt.1 - A Ct 'L IT"Y PA': CF'L .. 11: LI 1 lT!l!u!

(1~1/p/Py) PI^!,^ la?l!d

"11- 6'SC C'GC [Tug

6% C'6C 6'GZ lT!?!U! (.i~p/,m/Py) la?saZ!a uo Bu!peoy sp!los

P.81 L'81 L'81 IEUY L.81 "81 L'LI [T!I!u!

(s,C~p) la~saZ!a u! >LII!L uo!luqax

0'09 8'CF C'1-t- ITUY 8'Cf. Z't-f 6't-F IT!?!U!

(3 ) a~nlc~3drnaL uo!qsa%!a

0.C I O'CI C'CI ITUY 0'Cl S.CI V'CI [V!I!U!

(.[ep/;ur/Py) lauaqJ!qL uo %u!peoy sp!log

t.01 i.'O I :'CI ITUY t.01 L'CI L'CI I"!'I!U!

(%) O!~TH a1363a~ a%pn[g

C'F Z'f S'C ITUY C'F S'C S'C I'F!?!U!

(lq) am!& uo!?ua?ax 3!1n.~pX~

61.Z GI'C OG'Z lCUY

61.Z OZ'Z OC'Z [T!?!U! (sX~p) am!& a>uap!sax lla3 wary

O'i-f 1 O'ft-l G'SI 1 ITUY O'ff 1 6'Sll 6'SI I IT!?!"!

(X~p/ur) a?rx MoUla.to J~IJ!~CI~ X1~m!1d

.ouiaj, uo!~sa;!a alxa moi11.j~~ .IALJ!.II:~;) ,~JXU!.I~I .3u03 SII!IOS ayz;)

az!ru!xzj~ ae!m!xcjy az!m!x~ly (3!~i2) alqC!leA ,I 'uo!?nlln:j a.t!i:~a(qg L.

processing train. This solution was then used for a third optimization run, and the digestion

temperature was maximized. The total system cost improved sig~iificantly f r o n ~ 542,000 to

502,000 dollars/year. This cost reduction is the result of diflerent designs in the sludge

treatment system. Thus, in this case, the modified HSJ approach ied to an improved solu-

tion in comparison t o the first solution obtained using GRG2. T h e objective function value

of the improved solution is the same as the best solution obtained using GRG2 and listed in

Table 3.2.

Table 3.5 lists two optimization runs t h a t started from the two final solutions given in

Table 3.3, the solution obtained using start ing point No. 2 and another solution obtained

using different bounds on selected decision variables. The overflow rate of the primary

clarifier was maximized in solving program (3.3). Two designs with very similar characteris-

tics in sludge processing were obtained. T h e total system costs differ only slightly due t o the

difference in the activated sludge process design; both designs have slightly better objective

function values than those obtained so Tar. This example illustrates tha t the eflect of the

bounds on the GRG:! solution can become less critical if an HSJ type approach is followed

(i.e., by solving the constrained rormulation of (3.3)). DilTcrcnt bounds on the variables or

different control parameters used in runing GRG2 aflcct the solution in 3 complex problem.

Solving the constrained formulation provides confidence to the solution quality, and gen-

erates dimerent good designs.

Final solution No. 3 in Table 3.2 was used as the start ing solution in Table 3.6. The

primary clarifier overflow ra te was first maximized. With the primary clarifier overflow ra te

a t its specified upper bound, there was one equality constraint not satisfied to the specified

tolerance level. T o continue the optimization, the decision variables in this infeasible solu-

tion were used as input to the analysis program which generated a slightly diflerent solution.

This solution satisfied all constraints in the model, but the cake solids concentration violated

its upper bound of 150 kg/m3. Program (3.2) WRS then solved r~s i r~g this new start ing point;

7 8

Table 3.5 - Esploring Design pace : Design No. 3

Objective Function. F

Variable (Unit) Maximize Max~mize Primary Clarifier Overflow Rate Primary Clarifier Overflow R3t.e

Primary Clarifier Overflow Rate (m/day) initial 80.0 43.8 final 144.0 144.0

Mean Cell Residence Time (days) initial 2.22 2.26 final 2.19 2.19

Hydaulic Retention Time (hr) initial 3.7 3.5 final 3.8 4.2

Sludge Recycle Ratio (%) initial 11.6 11.5 final 12.5 10.0



Retention Time in Digester (days) initial final


Filter Yield (kg/m2/hr) initial final

Cake Solids Concentration (kg/mT initial final


initial final

Effluent T S S (mg/l) initial final

To ta l System Cost ( lo3 $/yr) initial final


Table 3.6 - Exploring Design Space : Design No. 3

Obiective F~tnct ion. F

Variable (Unit) M:~ximize Final GRG:! Primary Clarifier Overflow Ra te Solution

Primary Clarifier Overflow Rate (m/day) initial 78.2 144.0 final 144.0 14.1.0


Hydraulic Retention Time (hr) initial 3.6 4.2 final 4.2 3.8


Solids Loading on Thickener (kg/m2/day) initial 13.9 12.0 final 12.0 12.0

Digestion Temperature ("C) initial 60.0 60.0 final 60.0 60.0

Retention Time in Digester (days) initial 12.9 13.0 final 13.0 16.2

Solids Loading on Digester (kg/m2/day) initial 41.6 37.2 final 37.2 40.3

Filter Yield (kg/m2/hr) initial 6.62 6.86 final 6.86 6.69

Cake Solids Conccntration (kg/m3) initial 150.0 155. L final 150.0 150.0

Emuen t BOD, (rng/l)

initial final

EfAuent T S S (mg/l) initial final

To ta l System Cost ( lo3 $/yr) initial

final Computer Time (CP seconds)

One constraint is violated in this solution.

the final GRG3 solution is given in Table 3.6. The total system cost of this solution is com-

parable t o the best solution obtained so far (first solution in Table 3.5). Using the analysis

program in this case helped to restart an optimization in which GRG2 failed t o find a feasi-

ble solution by solving simultaneous design equations.

Table 3.7 provides another example of using the analysis program t o restart the G R G

optimization. Final solution No. 4 in Table 3.2 was used for the first optimization run in

Table 3.7 in which the cake solids concentration was maximized. T h e primary clarifier

overflow ra te in this solution was then maximized. This resulted in an infeasible design with

all constraints satisfied t o l u 3 , but not the specified tolerance level of T h e decision

variables in this final solution, with a minor modification of the value of the solids loading

value on the secondary digester, were used as input t o the analysis program. This

modification is necessary for the analysis program to produce a feasible design of the secon-

dary digester (i.e., the underflow solids concentration is higher than or equal to the influent

solids concentration). T h e resulting design was used as the new start ing point, and primary

clarifier overflow rate was again maximized using program (3.3). A very different solution

was obtained with an improved total system cost (from 551,500 t o 523,700 dollarsjyear); it is

the third solution listed in Table 3.7. Finally, the digestion temperature was maximized.

T h e solution obtained, the last in Table 3.1, has a total system cost of 501,200 dollars/year,

which represents a 10% savings of the total system cost from the GRG2 solution

Final solution No. 5 has the best objective function value among the five designs in

I Table 3.2. blarginnl reduction of the total system cost, however, was observed when the

cake solids concentration was maximized (see Table 3.8). An alternative design with a

nearly identical total system cost was obtained using a diflerent objective [unction. This

objective function minimizes the solids loading on the gravity thickener. Consequently it

has a larger thickener which provides a digester influent with higher solids concentration.

This allows the primary digester to be smaller, yet to achieve the same solids retention time.

8 1

T a b l e 3.7 - E x p l o r i n g Design S p a c e : Design No. 4

Objective F u n c t ~ o n , F

Vzriable (Unit) Maximize Maximize Maximize Max~mize

Cake Solids Primary Clarifier Primary Clarifier Digestion Concentration Overflow Rate Overflow Ra te Temperature

Primary Clarifier Overflow Ra te (m/day) initial 17.4 16.0 18.7 144.0 final 16.0 18.7 144.0 144.0

Mean Cell Residence Time (days) initial 2.36 2.37 2.36 2.19 final 2.37 2.36 2.19 2.19

Hydaulic Retention Time (hr) initial 3.4 3.3 3.5 4.2 final 3.3 3.5 4.2 4.2

Sludge Recycle Rat io (%) initial 10.0 10.0 10.0 10.0 final 10.0 10.0 10.0 10.0

Solids Loading on Thickener (kg/ms/day) initial 13.2 13.3 13.1 12.0 final 13.3 13.1 12.0 12.0

Digestion Temperature ( C) initial 60.0 60.0 59.8 50.0 final 60.0 59.8 50.0 60.0

Retention Time in Digester (days) initial 16.3 16.3 16.3 16.2 final 16.3 16.3 16.2 16.1

Solids Loading on Digester (kg/mZ/day) initial 36.6 36.2 35.3 30.0 final 36.2 36.5 30.0 39.4


Cake Solids Concentration (kg/m3) initial final


initial final

Emuen t T S S (mg/l) initizl final

Totnl System Cost ( l ~ " / ~ r ) initial

final 560.0 55 1.5' 523.7 501.2 Computer Time ( C P seconds) 5.074 12.93 34.422 8.977

* : Solution infeasible with respect t o the contraint tolerance of lo-', b u t all satisfied to

8 2

T a b l e 3.8 - Exploring Design s p a c e : Design No. .5

Objective Function. F

Variable (Unit) hlaximize Maximize

Cake Solids Conc. Cake Solids Conc . -100~~*

Primary Clarifier Overflow Rate (m/day) initial 1*4.0 144.0 final 144.0 144.0


Hydaulic Retention Time (hr) initial 3.7 3.7 final 3.7 3.7


Solids Loading on Thickener (kg/m2/day) initial 12.6 12.6 final 12.6 12.0

Digestion Temperature (OC) initial 60.0 60.0 final 60.0 60.0

Retention Time in Digester (days) initial 13.9 13.9 final 13.9 13.9

Solids Loading on Digester (kg/m2/day) initial 40.4 40.4 final 40.6 38.2

Filter Yield (kg/m2/hr) initial 6.70 6.70 final 6.67 6.80

Cake Solids Concentration (kg/mq initial 143.7 143.7 final 150.0 150.0

Eflluent BOD6 (mg/l)

initial 30.0 30.0 final 30.0 30.0

Emuent TSS (mg/l) initial 30.0 30.0 final 30.0 30.0

Total System Cost (10"/~r) initial 502.0 502.0 final 500.1 500.5

Computer Time (CP seconds) 6.052 12.386

L is the solids loading on the gravity thickener as defined in Chapter 2.

8 3

These differences are given in Figures 3.3 and 3.4, which show the details of these two

designs. Since the design of the liquid treatment train and the total system cost of these two

designs are almost the same, the difference in the design or the thickener and the digester

irnplies t h a t there may be many possible combinations of the sizes of thickener and digester

t h a t would result in practically the same cost for sludge treatment.

These illustrations show tha t program (3.3) is potentially useful for generating altena-

tive good designs for the wastewater treatment system considered. By relaxing the target

values and forming difierent objective functions, many alternative designs can be produced

which can then be evaluated for other important issues not present in a cost minimization

model. Table 3.9 summarizes the final designs obtained from solving program (3.3) using the

five solutions listed in Table 3.2. T h e total system cost ranges from 500,384 t o 501,963

dollars/year; the differences are practically insignificant. These designs are similar because

the objective functions used in obtaining them are similar. The size of the primary digester

represents the most significant difference in the sludge processing train design, while the sizes

or the aeration tank and the final clarifier are the major diflerences in the liquid train

design. These solutions are discussed in more detail in Chapter 4.

Finally, it was observed tha t objective function values obtained using multiple start ing

points varied considerably and the best value is 502,000 dollars/year (Table 3.2). All solu-

tions obtained by solving program (3.3) achieved a better objective value than this.

Although the improvement may be small from a practical point of view for the particular

wastewater treatment system model considered here, the diflerence could be greater in other

cases. It suggests tha t this strategy may serve as a useful fine-tuning step for solving such

problems using GRG2.

0-7-

BJ CP'L

zl" LEZ L

11 1 j3NVl

IW

;JU,+.JJ : Q i alnbli :t

3cx7: :L \r I

: a~m3i

- - - - - 0

FSO' _etOV L+O'

'050' 1 c

(St'L

IL'DL - It

18E'L -

LZL'O = J

.ul g8g'Jy ;cr,L: nAmr6L'Z=za I E~I mwOS9S-n 1'

ZUC' 1

LLLC Y ibb sx

A LO!

LOLL * LG' -. ;OF

i LLFL -1 -

ICLTL , -

lN3rl23'

Table 3.9 - Summary of Final Solutions Obtained From Solving Program (3.3)

Design No. : 1 2 3 4 5 6 7 Startine; Point No. : 1 2 2 3 4 5 5

Liquid Processing:

Primary Clarifier (m2) 252 252 25 1 252 25 1 25 1 252

Aeration Tank (m3) 6273 5687 6407 5686 6408 5637 5650

Final Clarifier (m2) 658 684 653 684 653 687 686

Air Flow Rate (m3/min) 242 242 242 242 242 242 242

Effluent BOD, (g/m3) 30 30 30 30 30 30 30

Effluent TSS (g/m3) 30 30 30 30 30 30 30

Sludge Processing: Mass Fraction of Primary Sludge

Thickener (my

Thickener Supernatant (rn3/hr)

Primary Digester (m3)

Secondary ~ i g e s t e r * (my

Digester Supernatant (m3/hr)

Vacuum Filt,er (my

Filtrate (m3/hr)

Cake Flowrate (m3/hr)

Cake Concentration (kg/m3)

Total System Cost (103$/yr) 501.963 500.384 500.954 500.627 501.228 500.422 500.467

Height of the digester is assumed to be 10 m.

3.3. IGGP Algorithm for Optimization

3.3.1. Introduction

T h e Generalized Geometric Programming (GGP) algorithm for solving geometric pro-

grams was developed by Avriel el al. (1975). Thc algorithm condenses polynomials t o mono-

mials ( a posynomial is a polynomial with only positive coeflicients, and a monomial is a posy-

nomial with only a single term) a t a given point and then linearizes the monomials by loga-

rithmic transformation. A linear program is then solved in each iteration. There are a

number of computer codes tha t implement this basic idea (Dembo, 1980). Burns and

Ramamurthy (1982) have developed a code t h a t can be used interactively on the CDC

Cyber computers a t the University of Illinois. T h e original algorithm developed by Avriel e t

al. deals exclusively with inequality constraints. Equality constraints have to be converted

to inequalities in order for the optimization to proceed. Strategies for this conversion have

been proposed (See, for exampie, Blau and Wilde, 1969). Burns and Ramamurthy (1983)

discussed the deficiencies of these strategies and extended the idea of condensation of poly-

nomials to the treatment of equality constraints. Favorable results were obtained from their

algorithm when it was applied to solve generalized geometric programs with equality con-

straints. This algorithm, named Interactive Generalized Geometric Programming (IGGP),

was used to solve the comprehensive system model described in Chapter 2.

IGGP solves the following geometric program:

Minimize P: (XI) - P,(Xi)

subject to P i ) - P i ) 5 0 , k = l,...,K

P,?(Xi) - P;(Xi) = O , j = 1, ... , J

0 < XiL 5 X i

where P:, P i , P:, P;, PJ and PJ are posynomials,

XiL is the vector of the lower bounds on the model variables, N x 1,

Xi = [ zl,...., ZN ]

and N is the number of variables in the model.

Two restrictions are noted in program (3.4). The objective function and the con-

straints in the model have to be polynomials in order to apply the algorithm. The variables

in the model have to be strictly positive.

Program (3.4) can be restated as

Minimize zo

subject to p a x )

P i ( X ) + 2 0

Pk+(X) , k = 1, ..., K

P;(X)

P f (X) = 1 , j = 1, ..., J

Pj(X)

0 < X L S X

where X = [ zo, z,, ...., zN ] * is the (N+l) x 1 solution vector, and XIL is the (N+l) x 1 vec-

tor of lower bound. The denominator of each inequality constraint in program (3.5) is con-

- densed to a monomial a t a point X = X, while both the denominator and the numerators

- are condensed to monomials for each equality constraint a t X. The resulting program

becomes

Minimize zo

subject to P;(x,%) S 1 , k = 0,1, ..., K

M / ( x , ~ ) = 1 , j = 1 ,..., J

X, s X - where P i is a posynomial and A!,: is a monomial resulting from the condensation a t point X.

Program (3.6) is linearized by logarithmic transformation. A linear program (LP) is

solved, and the most violated inequality polynomial is linearized a t the LP solution and is

appended to the LP tableau as a cutting plane. Additional cutting planes are added until a11

of the inequality polynomials are satisfied within a specified tolerance. Cutting planes are

added only for the inequality constraints, the equality constraints are simply log-linearized

once in each iteration. The detailed development of this method is documented by Burns

and Ramamurthy (1983).

Convergence to a Kuhn-Tucker solution of the GGP without equality constraints was

shown by Avriel and Williams (1970). Burns and Ramamurthy did not prove their method

will converge to 3 Kuhn-Tucker solution. Nevertheless, it is an attractive approach to test

because it can solve large-scale problems efficiently by transforming the nonlinear program

t o a linear program. Also it is interesting to test the proposed strategy of Burns and

Rarnamurthp for handling equality constraints in G G P using the comprehensive system

model which includes primarily equality constraints. These tests and their results are pro-

vided in the next two subsections.


As mentioned in the previous subsection, the optimization model has to be transformed

to a G G P and the variables have to be strictly positive to apply the IGGP. Most of the

design equations in the comprehensive system model can be transformed to polynomials with

the exceptions of equations (2.39), (3.41), and (2.18). T h e requirement for the variables to be

strictly positive is not a practical problem. Although one variable became zero in final solu-

tions obtained using GRG2, most variables are strictly positive because of what they

represent in the system. Where necessary, however, a small positive number can be imposed

as the lower bound for those variables tha t otherwise may turn ou t to be zero.

Modifications of equations (2.39) and (3.41) are necessary in order to use IGGP. If the

mass fraction of the primary sludge, f,, is fixed in the model, then the thickening constants

of the combined primary and waste activated sludge can be calculated immediately from

equations (2.38) and (2.39). When these constants become known, the thickening equation

(2.41) for thc combined sludge can be transformed into a polynomial. Thus , by fixing f,

two equations were dropped from the model, and equation (3.41) was simplified to form a

poly nonlial.

Equation (3.48) calculates the first-order digestion rate coefficient as a function of the

fermentation temperature. Because this model is empirical, alternative modeling of the

experimental d a t a used to develop equation (2.48) is possible. Polynomial models tha t

satisfy the standard G G P format were used to fit the experimental da ta . It was found tha t

a third degree polynomial fits the d a t a reasonably well,

K l = 0.06457 - 5 . 1 3 5 8 ~ 1 0 - ~ T , + 1.2061 ~ I O - ~ T , " + 1 . 9 1 8 ~ 1 0 ~ T ~ (3.7)

where K , is the rate coefficient in day-', and Td is the fermentation temperature in OC. Fig-

ure 3.5 presents equation (3.7) in graphical form.

With the above modifications, the comprehensive system model can be transcribed to a

G G P which has 62 variables and 57 constraints; 54 of the constraints are equalities. T h e

design of IGGP allows the objective function to be specified only interactively. Since many

cost functions describing the costs of unit processes are composed of several piecewise seg-

ments (see Table 2.4), it is necessary to guess the capacities of these units in advance to

determine the segment of the function in which the Enal solution falls. Ideally, if the Enal

solution specifies a size of a particular unit tha t is not in the range assumed, the cost func-

tion for tha t unit should be revised in IGGP. However, knowing that the cost functions

used in this study are only approximations of the cost d a t a and involve uncertainty, and

that the differences are small (see Table 3.11 for a comparison of the total system costs cal-

culated by the complete and the simplified cost functions), this trial-and-error approach was

not performed. Consequently the objective function value obtained from the IGGP solution

may be slightly different from tha t obtained from the GRG2. The cost functions used in the

G G P model are summarized in Table 3.10. A listing of the G G P model is attached in

Appendix E.

The solution process proceeds by searching over a range of values of f, for the best

solution. A start ing point can be obtained from the analysis program. T h e value of f, is

then fixed a t a given value, and an optimal design is obtained by IGGP. Theoretically, the

initial solution does not have to be feasible since IGGP can s tar t from an infeasible solution

and perform Phase-I optimization. Any one-dimensional search technique can be used to

obtain the optimal value of f, which results in the least cost design of the system. T h e

design found in this manner will be a locally optimal design for the overall system. This

solution strategy is somet.imes referred to as partitioning, or projection in the operations

I I

I I

I

m S

ewag

e S

lud

ge

0

Mun

lolp

al

Raf

use

-

A

Du

lry

Ma

nu

re

- B

eef

Ma

nu

re

x E

lep

han

t G

rass

-

+ S

llll

age

- - Bes

t F

lt

- -

- -

- -

- -

- -

- -

- w

X

es 'P

I

I I

I I

10

20

30

40

50

60

70

Te

mp

era

ture

, 'C

Fig

ure

3.5

Mod

elin

g D

iges

tion

Rat

e C

oefi

cien

t as

a P

olyn

orn

ial

Wit

h R

esp

ect

to F

erm

enta

tion

Ter

np

crn

ture

9 2

Tablc 3.10 - Summsry of Cost Functions Used in IGGP

Capital Operation Maintenance Material Power

(1971 $) and Supply

(manhours/yr) (manhours/yr) (197l$/yr) (kl .Vhr/~r)

Primary 824A;77 92.45Ai3 106A;'" 8.62A;7e -- Clarifier

Primary l6042QiC3 374 Qi4' 166 9t3 385 Qie4 23.85 Q 8 ~ / e P t Sludge Pumping

Aeration 461 Va7' -- -- -- -- Tank

Diffused 8533 Qiee 187 Q;48 74.4 Qib6 -- -- Aeration

Secondary 824A j77 17.15Aje 9.23Aje 8.62Aj7' -- Clarifier

Return & 2779 Q t 3 .333 Qb+390 .2375 Qb+370 40.57Qe2 23.85 QsH/ep Waste Sludge Pumping

Gravity 824A;77 17.15Aje 9.23Aie 8.62Ai7' -- Thicker

Anaerobic 2323 V;16g 192 V;" 113 G2l 142 V;i37 -- Digester

Vacuum 29180A;71 197.55Qii8hl$: 20 Q;8,3,1fi~~ 230 + -- Filter 182 Q;8,8hl;f;

t H is the pumping head iri rneters, and cp is the pumping elficicncy.

$ Qr = Qlo + Q13 + Q15

research literature (Geoflrion, 1971).

3.3.3. Performance of IGGP

Solutions were obtained for the conditions listed in Tables 2.6 to 2.8. Because the

majority of the constraints in the model are equalities, IGGP essentially solves the linear

program resulting from the log-linearization of the condensed equality constraints. If the

operating point is near the final solution, the condensation and the linearization are more

accurate than if the operating point is far away. It was noticed during the test runs t h a t

when f , is specified to be very different from the value in the initial solution provided by

the analysis program, i.e., the start ing solution for optimization is infeasible, IGGP may not

be able to find a feasible start ing solution using its Phase-I optimization routine. As a result,

feasible start ing solutions were used. The initial designs were generated by the analysis pro-

gram. These designs corresponded to different values of f,, and the optimal solutions

corresponding to these f,'s were obtained by IGGP. In this approach, the values of f, can-

not be controlled directly, and an efficient one-dimensional search method could not be used

t o locate the optimal f,. For the eleven initial designs specified, the values of f, ranged

from 0.44 t o 0.61. Figure 3.6 depicts the total system cost versus f,. The computing time

for individual IGGP runs varied from 2.5 to 5.7 seconds. The results are summarized in

Table 3.11.

It is observed from Figure 3.6 thnt the system cost is very sensitive t o f , when f, is

less than about 0.47, and is relatively insensitive to f , otherwise. The solutior~s obtained

with f, less than 0.47 are characterized by a high primary clarifier overtlow rate (a t its

upper bound of 1.14 metersldny), and by emuent BOD6 and suspended solids values tha t are

below the assumed standards. The solutions obtained with f, greater than 0.49 exhibit the

opposite characteristics. This observation reveals the two extrernes of the system design

when f , is fixed. When the mass fraction of tlre primary sludge is relatively small, the pri-

Total System Cost, lo3 dollars/year

P 0

O P IU

? P P

? P 0,

? P 0,

0 UI 0

? UI IU

0 in P

O UI 0

P cn- a

0 b 0

0 0 IU

P 0 P

0 b 0,

0 b 0,

? 4'

PPPPPcncncncncntncnulUl~0, cncn~~eorm~~cnm~coeo

p0000000000000000 I I I I I I I I I I 1 I I L

- -

- -

- -

- -

- -

- -

- -

- -

-

- -

- -

- -

- -

- -

I I 1 I I I I I I I I I I I

,. . . t : 1 . -: )~ :: ,. -

I,' .:

mary settling tank is small. A large secondary clarifier is needed t o produce a highly con-

centrated waste activated slutlge for recycle to the aeration tank. This thickening require-

ment causes the plant t o produce a high quality (low suspended solids) effluent. T h e thick-

ening characteristics of the combined primary and secondary sludge a re not as good as when

f p is large. Therefore a large thickener is needed. On the other hand, when f p is large, t he

primary clarifier is large, the waste activated sludge is thickened t o a smaller degree, and the

clarification requirement of the secondary clarifier dominates the system design. Conse-

quently the effluent water quality s tandards are binding. I t appears tha t an optimal f,

value exists between 0.471 and 0.487 (see Figure 3.6) where the primary clarifier overflow

ra te is high and both the effluent BOD, and total suspended solids standards are binding.

More points may be used as start ing points to run IGGP and t o refine the curve shown

in Figure 3.6 if it is desired t o know the optimal value of f p more accurately. This was not

carried ou t in this study because: 1) the cost is relatively insensitive near the optimal f,, and

2) the trend for optimal design conditions has become obvious through the analysis. If the

Phase-I optimization in the IGGP performed more reliably for the system model, then locat-

ing the optimal system design could be done eflectively by using a one-dimensional search

technique such as Fibonacci search.

The total computer time Tor running IGGP and generating the points in Figure 3.6 was

about 50 seconds for the test problem. Therefore the total time required in this optimiza-

tion approach is comparable to tha t of GRG2. The solution obtained with f p equnl to 0.487

is shown in Figure 3.7. This dcsign is similar to desisn No. .i in Table 3.0 except for the

digestion system. T h e total system cost calculated using the complete cost functions (Table

2.4) is 500,500 dollars/year which compares well t o the solutions obtained using GRG:! (see

Table 3.9). It is noted t h a t the solutions in Table 3.9 have f, values tha t range from 0.485

to 0.487 and t h a t are within the final interval for f, (0.471 to 0.487) determined by the

IGGP solution process.

l+JJ/D ;; 1 'JU /.uJ

S' I?

LO'S 1 I*'S

L.LI

3.4. Decomposition Approach for Optimization

3.4.1. Introduction

A wastewater treatment system is very complex in nature - not only because the

design of individual unit processes may be complicated, but also because various interactions

among the unit processes are complicated. In general, however, a wastewater treatment sys-

tem can be considered to consist of a liquid treatment portion and a sludge treatment and

disposal portion. For the base system (Figure 2.1), the liquid subsystem includes the primary

settling tank and the activated sludge process, while the sludge subsystem contains the other

units in the system. The inputs to the liquid subsystem are the influent wastewater and the

recycle streams generated in the sludge treatment. T h e liquid subsystem produces primary

and secondary sludges which are inputs to the sludge subsystem.

This section presents a specially tailored approach for solving the comprehensive sys-

tem model. The overall system is decomposed into a liquid subsystem and a sludge subsys-

tem. The design of the liquid subsystem is optimized. The optimal design of the liquid sub-

system has been studied by many researchers (Section 1.3) and many alternative optimiza-

tion techniques have been shown to be applicable to this problem. T h e solution obtained

from optimizing the liquid subsystem design is then treated as input to the sludge subsys-

tem. Embedded optimization steps are used in the sludge subsystem design. The optimal

solution for the entire system is then obtained by coordinating the designs of the liquid and

sludge subsystenls. This approach may be especially useful for design engineers since alter-

native designs of each system are explicitly examined and tradeolTs betweeu the two subsys-

tems can be readily evaluated.

Formal decomposition techniques for nonlinear programs were Erst developed by

researchers in the mid 1060's (for example, the feasible decomposition method by Brosilow e l

al. (1065) and the dual-feasible method by Brosilow and Lasdon (196.5)). A large complex

system is deconlposcd into a number of small subsystems each with its goals and constraints.

Each subsystem is optimized separately, and results from the subsystem optimization are

coordinated so that an optimal solution for the overall system can be obtained. Mathemati-

cal programming basis for nonlinear decomposition is well documented in Schoemer (1970)

and Lasdon (1970).

Although the decomposition approaches have numerous advantages for solving com-

plex, interconnecting large-scale system models as discussed by Haimes (1977), the efficiency

and robustness of these methods depend strongly on the characteristics of the problem.

Westerberg (1972) discussed the use of decomposition techniques for steady-state chemical

process synthesis and design problems. Limitations or the decomposition approaches were

identiEed, and some computational experiences were reported.

While general decomposition approaches were not used to solve the comprehensive sys-

tem model, the idea of decomposing the model into smaller problems was adopted for

developing an optimization procedure that is unique for this particular problem. The pro-

cedure preserves such advantages of the decomposition approaches as conceptual

simplification of a complex system, reduction in dimensionality, and flexibility in using

dimerent techniques for optimizing diflcrent subsystems.


T h e overall wastewater treatment system was divided into two subsystems, one

represents liquid processing arid the other sludge processing. This conceptual simplification

of the system and the interactions between the two subsystems are shown in Figure 3.8. Tlie

input to the liquid subsystem is the combination of the plant influent and the recycle

streams generated from sludge processing, i.e., the output from the sludge subsystem. T h e

output from the liquid subsystem (i.e., the combined primary and waste activated sludge)

serves as input to the sludge subsystem.

l NFLUENT EFFLUENT

LANDF l L L

Figure 3.8- Subsystem Formed I3y Tearing the Interactions Between

Liquid anci Sludge Processing Trains

T h e design of the liquid system cannot be determined unless the characteristics of the

recycle streams, i.e., the s ta te variables a t control points 10, 13 and 15, are known. There

are twelve unknown s ta te variables a t these three control points tha t connect the liquid and

subsystems. These interacting variables are Qlo , Slo, iMal0 , MdlO, ly10, Ml10, Q13,

Mil , , bf, , , , Q , , , Mi l , and M I 1 5 . The soluble BOD, of the digester supernatant (s13) and the

filtrate ( S 1 5 ) have been assumed to be a constant (Section 2.3) . Because of the lack of pro-

cess models for predicting the total suspended solids concentrations of the thickener super-

na tant ( M t l o ) , digester supernatant (M, , , ) , and filtrate (Mt l , ) , these concentrations have been

assumed to be constants, or

Malo + Md10 + Milo + = Mtlo = constant

Mi13 + 12lll3 = Mt13 = constant

Mil , + Ml15 = Mt15 = constant

It is desirable to eliminate as many of the interacting variables as possible in order to

efficiently coordinate the designs of the two subsystems. The solids concentration in the

thickener supernatant is usually much less than the solids concentration in the digester

supernatant or in the filtrate for a well-operatcd gravity thickener with high solids recovery

efficiency, i.e.,

M:10 << Aft,, (3 .11)

<< Mt16 ( 3 . 13 )

Consequently the contribution of the suspendrd solids from the thickener supernatant to the

plant influent is small compared t o tha t of the solids from the digester supcrnatnnt and

filtrate i f the thickener decant, digester supernatant, and filtrate have flowrates in the same

order of magnitude. I t is assumed that the suspended solids mass in the thickener superna-

t an t can be neglected in the recycle mass balances. This additional assumption is made only

for the decomposition solution approach. This assumption allows the variables Al,,,, 1V,,,,

h i i lo , and d l l l o to be eliminated from the group of interacting variables. I t is also assunied

for the decomposition approach that the soluble BOD6 concentration of the thickener decant

(S,,) is much tess than tha t of the digester supernatant or of the filtrate in the calculation of

recycle BOD mass balance. This assumption allows the interacting variable Sl0 to be elim-

inated.

It has also been assumed that the total suspended solids in the digester supernatant

consist of only the volatile and aerobically nondegradable solids (hli13) and the inert solids

( 1 , ) Since the secondary digester is modeled as a thickener and the vacuum filter is a

physical separation unit, the solids species in the filtrate are expected to be in the same pro-

portion as in the digester supernatant, i.e.,

Once the ratio, z, is determined, the solids compositions in the digester supernatant and in

the filtrate can be calculated from equations (3.9), (3.10), and (3.13).

With the above assumptions, the recycle stream characteristics can be determined with

the specification of only four interacting variables: the flowrates of thickener decant (Qlo),

digester supernatant (QI3), and filtrate (Q16), and the ratio between the volatile inert solids

and the inorganic solids concentrations in the digested sludge ( 2 ) . T h e liquid subsystem can

be readily designed for known characteristics of the recycle stream.

T h e complete decomposition procedure is now stated as follows :

1) Assume valrles for Qlo, QI3, Q16 and z . Calculate from mass balance relationships

(equations (3.14) to (3.20) below) the influent characteristics to the liquid srlbsystenl.

M:, = nf,, + AI,, + .\I;, + nl, , (3.20)

where Ma,, 121d0, hliO and hlJo are in g/m3, and is a unit conversion lactor. T h e

magnitudes for Qlo, QI3, Q16, and z can be roughly decided from running the analysis

program using several different start ing points. The assumed value for t is not critical

in this approach. This is explained in more detail in step (4).

2) Op~ini ize the liquid subsystem design using any efficient optimization technique. GRG2

was used in this study. The model has 21 variables in 17 equations and three inequal-

ity constraints. Therefore it has lour degrees o l freedom. A listing of the GRG pro-

gram describing this model is attached in Appendix F.

3) Calculate the mass and flow characteristics of the combined primary and waste

activated sludge based on the optimal design lrom the liquid subsystem optimization.

The combined sludge is the input to the sludge processing train

4) Determine the most cost eEective sludge subsystem design for the assumed values of

Qlo, QL3, and Q16. This is an optimization problem with one degree of lreedom in the

ratio z. Except for the solids compositions of the waste activated sludge, the liquid

subsystem design is not affected by the value of z specified in step (1) because neither

the volatile inert solids nor the inorganic solids is removed in the activated sludge pro-

cess. This is illustrated by a numerical example in the next subsection where the liquid

subsystem design is optimized lor dillerent influent conditions. Start ing lrom the

optimal design lor the liquid subsystem, the solids compositions o l the waste activated

sludge can be readily calculated lor a given value o l z using equations (2.21) and (2.22):

In the above equations, hf,, and iVfl are determined by 2 , and all other variables in

the right-hand-side are known from the optimal design of the liquid subsystem, which

is obtained in step(2).

The influent characteristics of the combined primary and waste acbivated sludge can

be determined once the solids compositions of the waste activated sludge are calculated

(see equations (2.31) and (2.32)). The sludge subsystem design then proceeds as fol-

lows:

4.1) For the gravity thickener, there is one degree of freedom in the design for given

influent conditions, i.e., complete design of this unit requires one design variable to be

specified. The supernatant flowrate, Q,,, is treated as tha t variable in this approach

since its value is specified in step (1).

4.2) For the primary digester, there are two decision variables, digestion temperature and

solids retention time. Since the characteristics of the digester influent are known from

the thickener design, and tile digester elTluent is characterized by the ratio between the

two effluent solids concentrations, z , the primary digester design can be formulated as

another optirnizatiou problem. The net cost of the prirriary digest~on system is minim-

ized subject to tile elfluent characteristics as specified by z . Recall from Section 2.3.6

t h a t

hf112 = illf 11 , (3.21)

the solids ratio z can be written from equations (3.23) and (3.24) m

since the solids compositions are assu,med to be unaffected by the secondary digester.

Specification of the digestion temperature results in the determination of the digest.ion

ra te coeficient, K,. The solids retention time, 8,, can then be calculated from equa-

tion (3.25), and the primary digester design is completely defined. This is a one-

dimensional optimization problem with respect to the digestion temperature.

Fibonacci search was employed to find the optimal digestion temperature tha t is accu-

ra te to within 1 O C .

4.3) T h e design of the secondary digester is similar to tha t of the gravity thickener. T h e

decision variable is chosen to be the digester supernatant flowrate, Q13, whose value is

specified in step (1).

4.4) T h e design of the vacuum filter requires the specification of one design variable which

is chosen as the filtrate flowrate, Q,,. Its value is specified in step (1).

Repeat steps (4.1) to (4.4) for different values of t . Golden section search was used to

identify the optimal value of z for the sludge subsystem design. The computer pro-

gram designed to carry out the calculations in step (4) is attached in Appendix G.

5) Sum the costs for the liquid subsystem obtained in step (2) and for the best sludge sub-

system obtained in step ('1) and obtain t.he total cost for the entire system. This cost is

for an assumed set of interacting variables Q,,, Q13, and QI5. A complete flowchart

describing steps (1) to (5) is shown in Figure 3.0.

6) Different combinations of values for the interacting variables can be selected. T h e

total system cost can be calculated for each combination following steps (1) through

(5), and the trend for a cost-eflective design can be identified.

This proposed procedure transforms the original problem which has nine decision vari-

ables into two subproblems. The liquid subsystem design has four decision variables; and

I Assume values for QIO, Q13, QIS, z I

Calculate characteristics of the influent to the liquid subsystem from mass balance

between plant illfluent and recycle streams I Optimize liquid subsystem design and

Assume a range for z

G Calculate characteristics of the

combined primary and waste activated sludge from the output of liquid train I

design optimization and the value of z

I , Design sludge subsystem for given a

with one-dimensional optimization for the primary digester design

Reduce the range of uncertainty for z by comparing the total sludge subsystem costs

calculated for dificrent values of : , using

golden section search

Calculate total system cost

Figure 3.9 - Flowchart of the Decomposition Approach

the sludge subsystern design has two decision variables ( z and T d ) , each can be determined

optimally using embedded one-dimensional optimization. The search for the overnll optimal

system design is a problem with three decision variables (Qlo, Q13 and Q,,). This concept is

illustrated by Figure 3.10. The solutions obtained using the decomposition approach are

only approximations t o the comprehel~sive system design model described in Chapter 2

because of the additional assumptions made in developing this approach. These assumptions

neglect the soluble BOD and suspended solids concentrations in the thickener supernatant.

T h e validity of these assumptions are examined in the next subsection, so are the perfor-

mance of the decomposition procedure for optimizing the complete wastewater treatment

system design and the performance of the embedded techniques for optimizing the subsys-

tem designs are also discussed.

3.4.3. Performance of the Optimization Approach

Step (2) in the above decomposition approach is essential to the overall optimization

procedure. T o examine the objective function surface of the liquid subsystem, diflerent

design conditions and multiple start ing points were investigated. Table 3.12 summarizes

solutions obtained when the base design conditions (see Section 2.5) are treated as the

Original hlodel (9 degrees of freedom)

Liquid Subsystem (4 degrees of frec.dom)

Sludge Subsystem (2' dcgrccs of freetlom)

Coordination (3 degrees of freedom)

Figure 3.10 - Concept of the Decomposition Approach

Ta

ble

3.1

2 - O

pti

miz

atio

n o

f th

e L

iqui

d T

ren

trr~

en

t Su

l)sy

stem

Infl

uent

Con

diti

ons:

Flo

wra

te

= 1

500

m3/

hr

Solu

ble

BO

D6

=

100

g/m

3

Act

ive

Bio

mas

s =

5

g/m

3

Vol

atil

e B

iode

grad

able

Sol

ids

=

100

g/rn

3

Vol

stil

e In

ert

Sol

ids

=

45 g

/m3

Inor

gani

c So

lids

=

50

g/r

n3

Sol

utio

n O

btai

ned

Usi

ng G

RC

!! V

aria

bles

(IJ

nits

)

Pri

mar

y C

lari

fier

Ove

rflo

w R

ste

(m/d

ay)

init

ial

fina

l M

ean

Cel

l R

esid

ence

Tim

e (d

ays)

in

itia

l fi

nal

Hyd

raul

ic R

eten

tion

Tim

e (h

r)

init

ial

fina

l S

ludg

e R

ecyc

le R

atio

(%

) in

itia

l fi

nal

Em

uent

BO

I),

(mg/

l)

init

ial

fina

l E

mue

nt T

SS

(m

dl)

in

ilia

1 fi

nal

Liq

uid

Sys

tem

Cos

t (l

o3 $

/yr)

in

ilis

l fi

nal

Com

pute

r T

irr~

e (CP

sec

onds

)

inlluent to the liquill train. All GRG runs were made interactively with the same control

pnranieter values specified in Section 3.2.1. The: computing time requirement is much less

than tha t for the complete model which includes 64 variables and 58 constraints (as opposed

to 21 and 20, respectively). The solution process also appears to be robust; widely diflerent

initial solutions converge to essentially the same solution. These observations are encourag-

ing for the approach of decomposing the overall system model into smaller subsystems whose

mathematical expressions are amenable to efficient and robust solution techniques.

Table 3.13 summarizes the liquid subsystem design optimization for a diflerent set of

influent conditions which has a higher flowrate and suspended solids concentration than the

base conditions. Five start ing points were tested, and four of them converged to the same

optimal solution. T h e optimization runs with start ing point No. 4 stopped short of the

optimum, bu t the objective function value and the design are almost the same as the

optimal solution. This indicates the flatness of the objective function surface of this sub-

problem.

T h e influent conditions examined in Table 3.13 were varied one a t a time to observe

the etlect of each condition on the liquid system design. T h e results are tabulated in Table

3.14. Case 1 is the original solution from the first column of Table 3.13. A change in the

flowrate (Case 2) aRects the liquid system cost, bu t has little etlect on the system design. An

increase in the influent soluble BOD, (Case 3) increases the cost of the subsystem. A higher

biomass concentration is maintained in the aeration tank when the size of the tank remains

a t the minimum level. A large secondary clarifier is included tor thickening purposes. Thus

the effluent suspended solids concentration decreases. The elTect o t the increased volatile

suspended solids in the influent (Case 4) is similar to that caused by an increased soluble

B O D , concentration.

Increasing the influent volatile inert solids (Case 5) or the inorganic solids (Case 6) by

the same amount (5 mg/l) results in two almost identical designs with the only difference

110

Table 3.13 - Optimization of the Liquid Treatment Subsystcm

Influent Conditions :

Flowrate = 1515 m3/11r

Soluble BOD, = 100 g/m3

Active Biomass = 5 ,5/m3

Volatile Biodegradable Solids = 100 g/m3

Volatile Inert Solids = 50 g/m3

Inorganic Solids = 55 g/m3

Solution Obtained Using GRG2 Variables (Units)

Primary Clarifier Overflow Rate (m/day) initial final

Mesn Cell Residence Time (days) initial final


Sludge Recycle Ratio (%) initial final

EfRuent BODb (mg/l)

initial final

EfRuent TSS (mg/l) initial final

Liquid System Cost ( lo3 $/yr) initial final


Table 3.14 - Liquid Subsystem Design Opti~nization for Di5erent Influent Conditions

Case

1 2 3 4 5 6

Influent Conditions :

Flowrate (m3/hr) 1515 1510 1515 1515 1515 1515 Soluble BOD, (mg/l) 100 100 105 100 100 100

Active Biomass (mg/l) 5 5 5 5 5 5 Volatile Degradable Solids (mg/l) 100 100 100 105 100 100 Volatile Iner t Solids (mg/l) 50 50 50 50 55 50 Inorganic Solids (mg/l) 55 55 55 55 55 60

Final. Solutions : Primary Clarifier Overflow Ra te (m/day) Mean Cell Residence Time (days) Hydraulic Retention Time (hr) Sludge Recycle Ratio (%) Ef luen t BOD, (mg/l)

Effluent T S S (mg/l)

Liquid System Cost (lo3 $/yr) Computer Time ( C P seconds)

Note : Starting point No. 1 in Table 3.13 was used in all runs.

being the composition of the sludge produced. T h e volatile inert and inorganic solids are

not treated in the activated sludge process, and they d o not contribute to the effluent BOD.

T o avoid excessive build-up of these solids in the system, which would require a larger aera-

tion tank and a larger final clarifier, more solids have to be wasted either in the overflow or

to the sludge processing train. A low sludge age and high solids concentration in the effluent

are direct consequences of this increased solids concentration in the influent. T h e fact t ha t

the liquid system cost is not affected by the ratio between the volatile inert and inorganic

solids has important implication in the analysis of the sludge treatment subsystem design

(step (4) of the decomposition procedure). It allows the optimization of value of the ratio of

the volatile inert and the inorganic solids concentrations ( z ) in the sludge sybsystem based

on only one optimization run for the liquid subsystem design.

As mentioned above, golden section search was used in the sludge subsystem design

optimization of the value of the ratio z . A typical cost curve resulting from this search is

shown in Figure 3.11. T h e cost curves exhibited this general shape for all runs made in this

study. This shape results in fast convergence of the sludge subsystem design.

T h e search for the cost-effective overall system design was carried out by examining

various combinations of Q,,, Q13, and Q15. During the liquid subsystem design, the solution

obtained from each GRG run was saved and used as the start ing solution for the next run.

It was observed t h a t this strategy saves computing time by about 50% when compared to

the strategy of start ing from an arbitrarily chosen solution. This is because the start ing

solution is closer to the final optimal solution. As was shown in the test runs for liquid sub-

system design optimization (Tables 3.12 through 3.14), the cost surface of this problem is

flat, and convergence to a unique local optimum was often observed. These observations sup-

por t the use of a previously determined optimal solution as the start ing point for a new

optimization run.

Tables 3.15 t o 3.17 present results obtained from the proposed optimization approach.

A coarse grid enumeration was performed for various combinations of values of QI3 and Q15

for Q1, equal to 1.0, 4.0, and 7.0 m3/hr, respectively. T h e computing time required to solve

the liquid subsystem problem ranged from 1.58 t o 2.91 seconds when GRG:! was used

interactively on 3 CDC Cyber 175 computer. T h e computing time for sludge subsystem

design averaged about 0.08 seconds. Fifty-three runs altogether were made to explore any

trends exhibited by the cost-effective designs.

The followi~lg observations can be made from the results in Tables 3.15 t o 3.17. For

fised vnlues of Q,, and Q13, the total systerrl cost decreases as Q,, increases, which implies

an increasingly efficient vacuum filter for sludge dewatering. T h e total system cost keeps

decreasing until the cake concentration equals the assumed upper bound of 15%. For fixed

Q,, and Q,,, an increase in Q,, implies a larger secondary digester which produces a more

concentrated sludge for dewatering and final disposal. Therefore the total system cost

decreases. For fixed values of Q13 and Qla, increasing Q1, produces decreasing system costs.

Sludge Subsystem Cost, lo3 dollars/year R3 R3 R3 R3 R3 w W W W W w Ln 0, 4 OJ (D 0 P rU W P ul 0 0 0 0 0 0 0 0 0 0

Tab

le 3

.15

- App

roxi

mat

e D

esig

ns o

f W

aste

wat

er T

reat

men

t Sy

stem

: Q,, =

1.0

m3/

hr

-

Liq

uid

Su

bsy

stem

:

~r

ima

rj C

lari

iier

Ove

rffo

w K

ate

(mld

ay)

hlea

n C

ell

Res

iden

ce T

ime

(day

s)

Hyd

raui

ic R

eter

~ti

on

Tim

e (1

11)

Slu

dge

Rec

ycle

Rat

io (

%)

Co

st (

lo3

$/y

r)

Co

mp

ute

; T

iril

e (C

P s

econ

ds)

Slu

dge

Su

bsy

stem

:

Sol

ids

Lo

adin

g o

n T

hick

erie

r (k

g/m

2/da

y)

Dig

esti

on T

emp

erat

ure

("C

)

Ret

enti

on

Tir

ue i

n lj

iges

tcr

(day

>)

Sol

ids

Lo

adin

g o

n D

iges

ter

(kg/

rn2/

day)

Fil

ter

Yie

ld (

kg/r

n2/h

r)

Cak

e S

olid

s C

on

cen

trst

ion

(k

g/n

lg

Co

st (

lo3

$/yr

) C

om

pu

ter

Tim

e (C

P s

econ

ds)

Infe

asib

le i

f th

e b

ou

nd

s c

n t

he

Jrci

sion

var

iabl

es (

see

Tab

le 2

.6)

are

cons

ider

ed.

817.

9 77

9.1

729.

5 67

1.3

597.

0 54

2.1

I T

ota

l S

yste

rn C

ost

(lo

3 $/

yr)

838.

2 79

9.3

751.

7 69

6.6

630.

4 54

2.6

Tab

le 3

.15

(co

nti

nu

ed)

Q,, (

m3/

hr)

Q,,

(1?l

3jhr

)

Litj

did

Sut

syst

ern

:

1'ri

.nar

y C

lari

fier

Ove

rflo

t*, R

ate

(rn

lday

) h

lesn

Cel

l R

esid

ence

Tim

e (d

ays)

Il

yd

r~u

lic R

eten

tion

Tir

ne (

hr)

3iud

ge I

l~cy

cle I

iati

o (7;)

C

ost

(10,

8/y

r)

C,~

n.p

utc

r Tim

e (C

P se

cori

ds)

To

tal

Com

pute

r T

ime

: 45

.188

CP

seco

nds.

S!ud

;;e

Sub

syst

em :

Soli

ds L

oadi

iig

on T

hick

trie

r (l

~~

/m~

/da

y)

Dig

esti

on T

einp

erat

ure

('(2)

Ret

enti

or1

Tim

e ir

~ Dig

escc

r (d

:~j.

s)

Soli

tls

Loa

tlin

g on

Dig

este

r (k

g/rn

2/da

y)

Fil

ter

Yie

ld (

kg/m

2/hr

)

Cak

e So

lids

Cor

rren

trat

ion

(kg/

n13)

Cos

t (1

0"$/

r)

Co

mp

ute

r T

ime

(CP

sec

or~

ds)

Tot

i1 S

y:t.c

m

Cos

t (l

o3

$/yr

) -

* Ir

~fe

usib

le if

the

bo

ur~

ds or

1 th

e de

cisi

on v

aria

bles

(se

e T

able

2.6

) ar

e co

nsid

ered

.

50.8

' 51

.3-

51.8

. 52

.3.

59.3

59

.3

59.3

59

.3

7.04

6.

89

6.82

6.

62

70.3

. 70

.1.

70.5

. 74

.5.

14.4

8.

30

6.42

5.

45

17

24

45 '

15

1'

478.

9 42

8.6

363.

4 28

1.8

.078

,0

82

.086

.0

79

735.

7 68

6.1

621.

7 54

0.7

49.6

' 50

.3.

50.8

' 51

.3.

51.7

'

59.3

59

.3

59.3

59

.3

59.3

7.

27

7.16

7.

05

6.79

6.

74

102.

10

1.

100.

98

. 10

3.

14.5

8.

33

6.44

5.

45

4.88

.

13

18

26

5 1

15."

522.

0 47

8.3

422.

3 35

3.0

282.

3 .0

78

.081

,0

85

,077

.0

85

777.

8 73

4.8

679.

5 61

1.0

540.

8

51.9

. 52

.4*

52.8

'

59.3

59

.3

59.3

6.

81

7.17

6.

42

43.3

46

.2

48.0

11.3

8.

18

6.31

23

39

150

432.

7 37

3.9

283.

0 .0

80

.079

.0

83

-- 69

0.4

632.

4 5-

12.2

Tab

le 3

.16

- App

roxi

mat

e D

esig

ns o

f W

aste

wat

er T

reat

men

t Sy

stem

: Q,, =

4.0

m3/

hr

- -

Liq

uid

Subs

y3tt

.m :

P

rim

ary

Cla

r~fi

er O:,e

rflo

w

Rat

e (m

/day

) hl

can

Ccl

l R

rsid

ence

Tim

e (d

ays)

II

~d

rau

lic R

z~

~n

ti

~n

T

ime

(hr)

S

ludg

e R

ecyc

le R

atio

(%

) co

st (l

o3

$1

~~

) C

ornp

uter

Tll

rle

(CP

sec

onds

)

Q,,

(1n3

/hr)

0.

0

Q,,

(m3/

hr)

I 1.0

3.0

5.0

7.0

7.53

Slu

dge

Subs

y)-z

tem

:

1 .o

1.0

3.0

5.0

6.68

Soli

ds L

oadi

ng o

n T

hick

ener

(kg

/mz/

day)

D

iges

tion

Trm

per

stil

re (

'C)

Ret

enti

on T

ill~

e in D

iges

ter

(day

s)

Soli

ds L

oadi

ng o

n D

iges

ter

(kg/

m2/

day)

Fil

ter

Yie

ld j

kg/n

i2/h

r)

Cak

e So

lid5

Con

cent

rati

on (

kg/m

3)

Cos

t (l

o3

8/y

r)

Co

mp

ute

r T

iii~

e (C

P se

cond

s)

To

tal

Sys

tem

Cos

t (l

o3

$/yr

)

Infe

asib

le i

f th

e bo

unds

on

t1;e

dec

isio

n va

riab

les

(see

Tab

le 2.6)

are

cor~

side

red.

I-'

I-' a

Tab

le 3.16

(con

tin

ued

)

Liq

uid

S~

ibsy

stcm

: P

rim

ary

Cla

rifi

er O

verf

low

Rat

e (m

/day

) M

ean

Ckl

l R

esid

ence

Tin

ie (

days

) H

ydra

~ll

ic Ret

enti

on T

irne

(hr

) S

ludg

e It

ecyc

le R

atio

(%

) co

st (

103

$I~

)

Co

~n

pu

t~er

T

ime

(CP

sec

onds

)

Sl~

ldg

e Sub

syst

em :

Solit

ls L

oadi

ng o

n T

hick

ener

(kg

/m2/

day)

D

iges

tion

Trm

per

atu

re (

"C)

Ret

enti

on T

ime

in D

iges

ter

(tln

ys)

Soli

ds L

oadi

ng o

n D

iges

ter

(kg/

m2/

day)

Fil

ter

Yie

ld (

kg/m

2/hr

)

Cak

e So

lids

Con

cent

rati

on (

kg/m

3)

Cos

t (l

o3

$/yr)

Co

mp

ute

r T

ime

(CP

sec

onds

)

To

tal

Com

pute

r T

ime

: 32

.781

CP

seco

nds.

I I

I

Tot

al S

yste

m C

ost

(lo

3 $

/yr)

Infe

asib

le i

f th

e bo

unds

on

the

deci

sion

vsr

inbl

es (

see

Tab

le 2

.6)

are

cons

itle

red.

665.

3 60

7.5

520.

8 61

7.2

510.

9 .5

22.5

Tab

le 3

.17

- App

roxi

mat

e D

esig

ns o

f W

aste

wat

er T

reat

men

t Sy

stem

: Q,, =

7.0

ms/

hr

-

- p

pp

Liq

uid

Sub

syst

em :

Pri

mar

y C

lari

fier

Ove

rflo

w R

ate

[rn/

day)

M

ean

Cel

l R

esid

ence

Tim

e (d

ays)

Il

ydra

blic

Ret

enti

on T

ime

(hr)

Sl

udge

Rec

ycle

Rat

io (5

%)

. -

Cos

t (l

o3

$/yr

) C

ornr

filt

~r T

ime

(CP

seco

nds)

Slud

ge S

ubsy

stem

:

Soli

ds L

oadi

ng o

n T

hick

eric

-r (

kg/l

~~

','ds

~)

Dig

esti

on T

emp

erat

ure

(OC

)

Ret

enti

on T

ime

in D

iges

ter

(day

sj

Soli

ds L

oadi

ng o

n D

iges

ter

(kg/

m2/

day)

Fil

ter

Yie

ld (

kg/m

2/hr

)

Cak

e So

lids

Con

cent

r;it

ion

(kg/

nl?

Cos

t (1

0"//r

) C

ompu

ter

Tim

e (C

P se

cond

s)

Tut

al S

yste

m C

ost

(lo

3 $/

yr)

* In

fess

ible

if t

ile

boun

ds o

n ti

le t

lcci

sion

var

isbl

es (

xee

Tab

le 2

.6)

are

cons

ider

ed.

To

tal

Co

mp

~~

ter

Tim

e :

25.0

48 C

P s

econ

ds.

This is at tr ibuted to a larger gravity lhickcner which reduces the volume of sludge to be

processed in lhc subsequent unit processes.

Thus , the trend t h a t indicates a cost-eflective design is obvious from this analysis: for

this example problem, the cost of the liquid subsystem is not very sensitive to the recycle

flowrates, and it is the design of the sludge subsystem t h a t determines the most cost-effective

overall system design. T o make the sludge subsystem design cost efficient, the volume of the

sludge t o be processed should be minimized. T h e above analysis indicates t h a t the gravity

thickener is most cost effective for achieving this goal. Although an increased level of sludge

concentration produces higher BOD and suspended solids mass in the recycle streams t o the

liquid train, t he marginal increase in liquid subsystem cost is much less than the reduced

cost for sludge t rea tment and disposal. The best design obtained from the coarse grid

enumeration has Qlo = 7.0, Q13 = 0.0, and Q15 = 4.1 m3/hr (Table 3.17).

Figures 3.12 to 3.14 depict the cost surfaces for the diflerent combinations of superna-

t a n t flowrates. These Figures are graphical representations of the results in Tables 3.15 to

3.17. I t is obvious from these plots tha t the total system cost decreases as Q13 or Q16

increases for a fixed Q,,; the total system cost decreases more rapidly for a unit increase of

Q15 than a unit increase of Q13. T h e boundary of the feasible region outside which the cake

concentration exceeds its upper bound is also shown approximately in each case by the

hashed line. It is noted t h a t the boundary is very flat, meaning tha t many alternative

designs are availnblc a t approximately the same total system cost. These alternative designs

are diflcrent mainly in thcir designs or the sludge subsystem, although some of them may

violate other constraints set on the decision variables. For example, the design with Q l o =

4.0 m v h r , Q13 = 1.0 and QI5 = 6.68 m3/hr has a total system cost of 518,200 d ~ l l a r s / ~ e a r ;

another design with Qlo = 4.0, Q13 = 5.0 and Q15 = 3.28 m3/hr has a. total system cost of

522,500 dollars/year (see Table 3.16), and the third design with Qlo = 7.0, Q13 = 1.0 and

0

I- B

ound

ary

of F

easi

ble

Reg

ion

1 F

igur

e 3.

12-

Tot

al S

yste

m C

ost

vs. D

iges

ter

Supe

rnat

ant

Flo

wra

te- Q,, =

1.0

mS/

hr

Fig

ure

3.1

3- T

otal

Sys

tem

Cos

t vs

. D

iges

ter

Su

per

nat

ant

Flo

wrn

te-

QIo

= 4

.0 m

s/h

r

Q16 = 3.0 m3/hr has a total system cost of 518,700 dollars/year (see Table 3.17). These ,

three designs are very different in bhcir design of the gravity thickener, the secondary diges-

ter and the vacuum filter. ,411 three designs have approximately the same total system cost.

However, the solids loading on the secondary digester in the first design is 77.2 kg/m2/day

which is infeasible if the bounds on the decision variables (see Table 2.6) are considered.

If more accurate identification of the most cost efficient design is desired, a fine-tuning

step can be employed. ,4s an example, the neighborhood around the best solution described

above (given in Table 3.17) was explored based on t,he trend observed in the coarse grid

enumeration. Five runs were made, and the results are summarized in Table 3.18.

Although the second design in Table 3.18 with Qlo =7.2, Q13 = 0 and Ql6 = 3.9 m3/hr has

the lowest total system cost among the five designs, the extent of violation of its cake solids

concentration is also the greatest. Therefore, the fine-tuning process was continued. T h e

final design Qlo = 7.27, Q13 = 0 and Q15 = 3.80 m3/hr has a total system cost about

501,700 d ~ l l a r s / ~ e a r . This design is shown in detail in Figure 3.15. Compared with the

designs obtained by CRC:! (Tabel 3.9), this design is most similar to the one shown in Figure

3.4 in terms of the s ta te variables in the model. However, this design suggests a smaller

aeration tank, a smaller primary digester, and larger final settling tank, secondary digester

and vacuum filter. Also, the cake solids concentration is slightly above the upper bound used

in the original model solved by CRG2 (see Table 2.6). It is noted tha t the maximum diges-

tion temperature tli:rt can be obtained in the decomposition ;tpproacti is 59.3 O C because of

the stopping criterion spcciliccl in the Fi1)ollncci search. The a c t ~ ~ n l upper bor~ntl for this

variable in the model is 60 OC.

As mentioned above, the solutions obtained using this approach are only approxima-

tions to the comprehensive system model because the soluble BOD and the solids concentra-

tions in the thickener supernatant are neglected. T h e approximation is better when the

Table 3.18 - Fine-tuning Solutior~s in the Decomposition Approach

Q,, (m3/hr) 7.1 7.2 7.3 7.26 7.27 1 Q13 (m3/hr) 0 0 0 0 0

Q,, (m3/hr) 4.0 3.9 3.75 3.82 3.80 I /

Liquid Subsystem : Primary Clarifier Overflow Rate (rn/day) 144 144 144 144 144 i Mean Cell Residence Time (days) 2.16 2.16 2.16 2.16 2.16 Hydraulic Retention Time (hr) 3.66 3.66 3.66 3.66 3.66

i Sludge Recycle Ratio (%) 12.5 12.5 12.5 12.5 12.5 \

Cost ( l o 3 $/yr) 255.6 255.6 255.5 255.5 255.5 Computer Time ( C P seconds) 1.873 1.656 1.670 2.006 1.651 I

Sludge Subsystem :

Solids Loading on Thickener (kg/m2/day)

Digestion Temperature (OC) Retention Time in Digester (days)

Solids Loading on Digester (kg/m2/day)

Filter Yield (kg/m2/hr)

Cake Solids Concentration (kg/my

Cost (lo3 $ / ~ r ) Computer Time ( C P seconds)

Total System Cost ( lo3 $ / ~ r )

Infeasible in the optimization model solved by GRG.

Total Computer Time : 9.265 C P seconds.

thickener supernatant flowrate is small compared to tha t of the digester and filter superna-

tants. It is interesting to examine the errors associated with the designs with high thickener

supernatant flowrates. Tables 3.19 to 3.21 summarize thrce designs tha t have high thick-

ener supcrnatant flowratcs. The vnlues of t,he decision vnrinl)les obtained from the dccom-

position approach were used ns inputs to the analysis program (Section 2.5) which calculates

the exact values of the s ta te variables in the model. Important design vari:lbles calculated

from the decomposition approach as well ns using the analysis program are compared with

each other. T h e errors in Tables 3.19 t o 3.21 for thesc variables are a11 less than 1%. These

values olTer an indication of the maximum possible errors in the decomposition approach;

the errors are expected to be smaller when the thickener flowrate is smallcr. For the

---- ;'<I - -- t LL*' olnoli :t

33onis :+ :ON3337 I

,

- C;IZ' 19'LZ I SSO'

8)0' LtO' LSO' & I LZ'L -

ICO 9 I

I 9S'L

SZL'O = J '

--- C Z - -- '6Et1 i -1

v lN311-lAA3

Table 3.19 - Esamination of Assurnprions in the Decomposition Approach

P r ~ m a r y Clarifier Overflow Rate

hlean Cell Residence T ime Hydrau l~c Retention T ime

Sludge Recycle Rat io

Solids Loading on Thickener

Digestion Temperature

Retention Time in Digester

Solids Loading on Digester

Filter Yield

= 144 m j d a y = 2.17 days

= 3.65 hr

= 12.5 %

= 12 8 kg/mC/day

= 59.3 C

= 12 9 days

= 27 .3 kg/m2/day

= 13 5 kg/m:/br

A p p r o x i m a t e d Design E x a c t Design ~ r r o r * ( % )

P r i m a r y Clar if ier - Sur face A r e s ( m y

Solids R e m o v a l (5) Under f low Solids (%)

A e r a t i o n T a n k - \'olume (m3)

Biomass ( m g / l )

h f L S S (mg/ l ) F i n a l Clar if ier -

S u r f a c e A r e s (m2)

Ef f luen t B O D 5 (mg/ l )

E m u e n t TSS (mg/ l )

G r a v i t y T h i c k e n e r - S u r f a c e A r e a (m?

I n f l u e n t Sol ids (%) Under l low Solids (%) S u p e r n a t a n t (m3/hr )

P r i m a r y Diges te r - V o l u m e (mJ)

E m u e n t Sol ids (%) S e c o n d a r y Diges te r -

S u r f a c e A r e s ( m y

S u p e r n a t a n t (m3/hr )

V a c u u m Fi l t e r -

S u r f a c e A r e a (m-)

C a k e Solitla (5) S u p e r n a t a n t (m3/hr )

T o t a l S y s t e m C o s t r)

I A p p r o u i m n t e d decign v3lue - Esnct d ~ s i g n v n l ~ l e i E r r o r (%) = ' x 100

E x a c t design v s l u e

Table 3.20 - Examination of Assumptions in the Decomposition Approach

Primary ClariEer Overflow Rate Mean Cell Residence Time Hydraulic Retention Time

Sludge Recycle Ratio

Solids Loading on Thickener




Filter Y i ~ l d

= i 4 4 m/day

= 2.16 days

= 3 .68 hr

= 12.6 %

= 13.7 kg/m2/day

= 59 .3 T = 12 .5 days

= 29.0 kg/m2/day

= 7.76 kg/mYhr

Approximated Design Exact D e s ~ g n Error*(%)

Pr imary Clarifier - Surface Area (m? 251.57 251.58 .0040

Solitls Removal (%) 39.454 39.503 ,12 Underflow Solids (%) 7.7203 7.6947 .33

Aeration T a n k - Volume (m3) 5550.2 5550.4 .036

Biomass (mg/l) 713.26 714.40 .16

m-23 (mg/l) 1540.2 1542.8 .17 Final Clarifier -

Surface Area (m? 705.9 1 708.65 .39 EfIluent BOD5 (mg/l) 30.000 2S.:,G6 .11

Etfluent TSS (mg/l) 29.190 29.000 .34

Gravi ty Thickener - Surface Area ( m y 441.01 443.84 .64

Influent Solids (5%) 2.1534 2.1517 .060 Underflow Solids (%) 5.2872 5.2864 .015

Supe rna t an t (m3/hr) 7.0000 7.0372 .53 Pr imary Digester -

Volume (m3)

Eflluent Solids (%) Secondary Digcster -

Surface Area ( m 3

Supe rna t an t (m3/hr)

V s c u ~ l r n Filter - Surface Area ( m y 10.465 10.535 .66

Cake Solids (5%) 10.606 10.606 .OO

Supe rna t an t (m"/hr) 3.0000 3.0202 .67

T o t s l System Cost ( lo3 $/yr) 518.67 520.00 '.25

I Approximated decign vslue - Exact design value 1 Error (%) =

Exact design value - x I00

Table 3.21 - Examination of Assu~nptions in the Decomposition Approach

Primary Clarifier Overflow Rate

Mean Cell Rev~dence Time

Hydraulic Retention Time

Sludge Recycle Ratio

Solids Load~ng on Thickener




Filter Yield

= 144 m/day

= 2.16 days

= 3.66 hr

= 12.5 %

= 12.0 kg/m2/day

= 59.3 C = 12.7 days

= 36.5 kg/rn2/day

= 6.91 kg/m2/hr

Approximated Design Exact Design Error*(%)

Pr imary Clarifier - Surface Area (m")

Solids Removal (%) Underflow Solids (96)

Aeration T a n k - Volume ( m 3

Biomass (mg/l)

MLSS (mg/l)

Final Clarifier - Surface Area (rn?

Effluent BODB (mg/l)

Effluent TSS (mg/l)

Gravity Thickener -

Surface Area (mT)

Influent Solids (5%) Underflow Solids (%) Superna t an t (m3/hr)

Pr imary Digester - Volume (m3)

Effluent Solids (%) Secondary Digester -

Surface Area (mZ)

Supe rna t an t (m3/hr)

Vacuum Filter - Surface Area jm')

Cake Solids (9,) Superna t an t ( m y h r )

T o t a l System Cost ( l o 3 $/yr)

I Approximnted design vnlue - Exnct design vnl~le I Error ('5) = A - x 100

Exact design value

parameters a ~ ~ d design conditions considered in this example, the simplifying assumptions

appear t.o be very reasonable. It is also noted that thc objective iunction value calculated in

the decomposition approach is slightly lower than that calculated from the analysis program

in ail three cases. Because the suspended solids in the thickener supernatant are ignored in

tha decomposition approach, the cost for liquid treatment is underestimated, bu t this error

appears insignificant from a practical point of view.

There may be many modifications of the basic decomposition approach outlined in this

section. Alternative optimization techniques may be used to optimize the liquid subsystem

design. For example, IGGP can be applied to solve this subsystem design. Dynamic pro-

gramming or any other nonlinear programming techniques are also possible candidates. As

for the coordination of the subsystem designs, it may be possible to employ more efficient

optimization technique than the coarse grid enumeration to find the combination of recycle

flowrates (Qlo, QI3, and QIS) tha t results in the least total system cost. These modifications

are potentially capable of refining and improving the proposed basic approach.

3.5. Summary

The comprehensive system model described in Chapter 2 can be optimized using threc

optimization techniques. T h e first approach solves the nonlinear programming model, which

contains 6.1 variables, 55 equality constraints, and thrce incquality constraints, directly using

the generalized reduced gradient algorithm developed by Lasdon e l al . (GRG2). T h e solu-

t ~ o n obtair~ed from applytng GRG2 dcpcnds on the various control pararrictors ass~gncd, thc

initial solution, bounds on ~ r ~ o d e l v a r l a b l ~ ' ~ , and constraint and variable bcaling. Con~pu ta -

tional experience with a particular problem is helpful for obtaining "good quality" solutions.

hlultiple start ing points are necessary to ascertain the quality of the solution obtained. An

approach dcrived from the Ilop-Skip-Jump method can be used as a tool to improve and

fine-tune the solution obtained by solving the base nonlinear programming wastewater treat-

ment system model. Good bui diflerent solutions can also be obtained using this approach.

T h c computing time requirements Tor GRG:! arc comparable to those reported in the litera-

ture for solving wastewater treatmcnt syst,em models using other optimization techniques.

The comprehensive system model can also be formulated as 3 geometric program by

modifying the constraint set and by assigning a value to one variable in the model. An

efficient package for solving geometric programs (IGGP) can be employed for solving the

subproblems resulting from the partitioning process. ti one-dimensional enumeration can be

used to search for the optimal value of the fixed variable. This second level search could be

more efficient if IGGP be able to s tar t from an infeasible start ing point and to

proceed with the optimization efficiently. This is prevented by the large number of equality

constraints in the model. The computing time for solving the geometric programming sub-

problems is usually less than five seconds. Therefore IGGP would be more at tractive for

wastewater treatment systems tha t can be described completely as a geometric program.

Because of the unique structure of the wastewater treatmcnt system under study, an

approach that decomposes the wastewater treatmcnt system into two interacting subsystems

was developed for optimization of the overall system design. The liquid subsystem design

can be optimized using GRG2 for spec~fied recycle charncteristics from the sludge subsystem.

This problem contains 21 variables, 1'7 equality constraints, 2nd three incqullity constraints,

and it can be solved very cfficicntly by GRG2. The solution obtained from the liquid sub-

system optimization provides input to the sludzc subsystem. The design or the sludge sub-

system is ca r r~cd ou t sequrn tially for rach unit process. Two one-dirne~~sional optimization

searches arc embedded in the sludge subsystem design. The computing time requirement for

the sludge subsystem design is trivial. A coarse grid cnumer:ltion is employed for the second

level optimization that searches for the combination of the interacting variables tha t pro-

duces the lo~vcst total system cost. Trends for cost-ellcctive system designs can be identified

in this approach with confidence. The totnl computing time for one set or design conditions

is con~parable to that required when using GRG? for the entire model. Improvenient in the

computing time may bc poss~ble if an another optimization technique is subst i t i~ tcd for

enumeration in the second level problem. Several simplifying assumptions are necessary in

using the decomposition approach. These assumptions appear very reasonable for the exam-

ple problem. It is noted tha t if the same assumptions are applied to the original model

evaluated using GRG2, three variables and three constraints can be omitted. However, the

model is still of considerable size, and the same dificulties discussed above in using GRG2

for solving the entire system model are expected to occur.

Using the GRG2 algorithm to solve the comprehensive system model is the most

straightforward approach for optimization. Once formulated, the model can be used repeti-

tively to examine various influent and design conditions with only minor adjustments of the

input d a t a files. However, if the flowchart is modified, the system model needs to be revised

and most variables and constraints in the model need to be relabeled which may involve

extensive effort. If the size of the problem increases, however, the efficiency of the algorithm

decreases drastically. Therefore although it is useful as a tool for process analysis because it

can be applied directly, it may not be the best strategy for optimizing a complex wastewater

treatment system. T h e use of this algorithm for the analysis of wastewater treatment

processes is illustrated in more detail in Chapter 4.

IGGP is an eficient program for solving geometric programs. However, for the waste-

water treatment system model tha t contains a large number of equality constraints, t he

optimization performs better with feasible start ing solution. Therefore the second level

problem of finding the optimal value of the partitioned variable cannot be solved by elficient

optimization technique. In addition, the model has to be formulated as a geometric program

before IGGP can be applied, which may not always be possible because process design equa-

tions may be of any mathematically complicated forms.

T h e decompos~tion approach is specially deveioped to solve the compreliens~ve system

model by taking advantage of the unique structure of the waste treatment system and

reducing the dimensionality of the problem. By decomposing the overall system into

interacting subsystems, different optimization algorithms can be appiied to solve different

subsystem designs. Nonlinear programming algorithms are also more efficient for solving

problems of smaller size. This approach is also quite flexible, since design of some unit

processes is done on a modular basis. Consequently, modifications of the process flowchart

will not cause extensive revision of the system model in terms of human effort. The

identification of any trend related to cost-efficient design is especially useful since it suggests

design guidelines. Also, many solutions with good total system costs are identified in this

approach. These solutions can then be evaluated with respect to other planning issues tha t

are not captured in the cost minimization model.

CHAPTER 4

A N ILLUSTRATION OF THE USE OF THE OPTIkIIZATION MODEL FOR PROCESS ANALYSIS AND DESIGN

4.1. Introduction

An optimization model can be used to obtain cost eflcctive designs of the wastewater

treatment system defined by the selected process performance models and parameters.

Using an optimization model also enables the designer to analyze process performances sys-

tematically and effectively. Detailed design of the entire wastewater treatment system can

then be performed following the guidelines or trends suggested from the modeling study.

In this chapter the role of an optimization model is explored, and it is shown t h a t such

a model may be used for more than just identifying a least-cost system design. Specifically,

such a model can be used as a tool for the analysis of treatment process performance and of

alternative t rea tment plant configurations. Potentially important research areas or design

guidelines can also be identified from these insights.

The hypothetical wastewater treatment system described in Figure 2.1 was designed

using various optimization approaches described in Chapter 3 for the design conditions sum-

marized in Tables 2.6 to 2.8. T h e final designs obtained from using GRG2 are summarized

in Table 3.9. These designs provide the basis for the following discussion. They have

several common characteristics; the overflow rate of the primary settling tank, the digester

operating temperature, and the solids concentration of the cake from the vacuum filter are

a t their upper bounds. The implications associated with a variable being a t its specified

bound in the final solution may provide useful insights. Relaxing such a bound may imply

tha t the total system cost could be reduced. It may be necessary, however, to extrapolate

process models. Additional research may be needed to justify such extensions if bounds

imposed on the decision variables represent ranges recommended for design or limits within

which the process model is developed. On the other hand, if lhe bounds represent the limits

outside which process failure will occur, then extrapolation of a process model is inappropri-

ate. hlodification or the process flowchart may also be suggested when a variable is a t its

bound. For instance if an unusually high upper bocnd on a loading rate is approached in

the optimization solutions, then it may be desirable to eliminate tha t unit process.

Design of wastewater treatment systems is subject to uncertainties. Uncertainties

arise from parameter estimation, cost information, the prediction of influent characteristics?

possible changes in the water quality regulations, and the lack of knowledge about the per-

formance of some unit processes. While design is usually carried out by assuming steady-

s ta te conditions, an operating wastewater treatment plant is more likely to receive sewage

varying with time in quanti ty as well as in strength. There may also be other impor tant

planning issues tha t are specific for each plant; examples are energy requirements, effluent

limitation on a specific pollutant, and system reliability concerns. In light of these realistic

considerations, t he design obtained from the mathematical optimization of a comprehensive

system design model needs to be examined carefully or modified so tha t the final plant being

constructed will meet the design goals.

This chapter presents observations and discussions drawn from an examination of the

solutions obtained from the optimization or the example wastewater treatment system. T h e

discussion is on a unit-by-unit basis. Finally lhe design of wastewater treatment plant is

considered as a two-objective problem to illustrate a simplistic approach for design under

uncertainty. T h e tradeom between economic efficiency and a flow safety factor is studied.

This design approach allows the use of an optimization model as a useful preliminary design

aid.

4.2. Primary Sedimentation

Typical design guidelines for a primary settling tank generally call for the overflow

ra te to be less than or equal to 40 metersjday under the average flow conditions (see, for

example, Great Lakes- Upper Mississippi River Board of Sta te Sanitary Engineers, 1978). In

their pilot scale studies, Tebbut t and Christoulas (1975) investigated the performance of pri-

mary settling tanks for overflow rates up to 150 meters/day. Their results implied t h a t the

current practice is too conservative. As a result, an upper bound of 144 metersjday was

imposed on the overflow rate in the comprehensive system model. T h e final design showed

t h a t the overflow rate is a t this upper bound.

This solution suggests tha t the total system cost may be further reduced by relaxing

the upper bound on the overflow rate because of a negative reduced gradient associated with

this variable in the final solution. Two major questions arise:

1) Is the Voshel-Sak model a valid representation of the primary clarifier performance

when the overflow rate is as high as t h a t assumed in the comprehensive system model?

2) Is the primary clarifier a cost-effective unit in the assumed wastewater t rea tment sys-

tem?

Extrapolating the Voshel-Sak model to high overflow rates shows tha t solids removal

efficiency decreases only marginally as the loading increases substnntially. This is depicted

in Figure 4.1. It is expected tha t the solids removal efficiency will decrease sharply when the

overIlow rate reaches a critical value. Therefore t,he behavior o i the prim:~ry set,tling tank a t

high overflow rates should be an area of further investigation.

T o address the second question, the primary settling tank was eliminated from the

base system. The modified system is shown in Figure 4.2. T h e GRG:! was used to deter-

mine an optimal design under the base conditions listed in Tables 2.6 through 2.8. T h e

GRG2 model describing the system design has 51 variables, 43 equality constraints, and

1/6w 'UO!~PJ~U?~UO~ sp!los papuadsns 00s OOP OOC 002 OOt 0 -rl

00'0 7 0

OT'O -f.

03'0 o' 3

OC'O 0 - OP'O

m OS'O 2 -- 09'0 ,n OL'O

w 08'0 0 3 06'0 0

oo-r 2 -

Xap/s~aiaw MOI ~J~AO OSZ 002 OST OOt 0s 0 7

I I I 1

- - - - - -

00'0 7 u OT'O 0 3

OZ'O 0- 3

0'2'0 0 - ----------=-- - -.-- - a \ .

- - -1/Bw OOE = sp[los suenlful ---

1/6w 003 = SPIIOS iuQnlfu1 - - -1/fiw 001 = SPllOS luanl3ul -

($Pol4 Y DS-la WoA I

three inequality constraints. T h e computer program listing of this model is included in

Appendix H. Table 4.1 summarizes the results of the optimization.

Three different starting points were used for the GRG2 runs. T h e final designs are

very similar, and the total system cost without the primary clarifier is about 492,500

dollars/year, or 1.6% less than the final design with the primary clarifier. The final design

obtained from using starting point No. 1 is shown in Figure -1.3 (refer to Section 2.2.2 for the

notation). A comparison between this design and the one with the primary clarifier in the

system (design No. 6 in Table 3.9) is shown by Table 4.2. Without the primary clarifier in

the system, a larger aeration tank and final clarifier are needed to achieve the same ef luent

water quality. However, the total sludge production is less because of the absence of primary

sludge. Therefore the costs for sludge treatment and disposal are less. However, the biologi-

cal parameters used for design of the system without the primary clarifier are likely to be

different from those of the system with the primary clarifier. This is a weakness of this

analysis and further research is necessary to determine how the biological parameters are

affected by the absence of the primary clarifier. For the base design conditions, with the

assumption t h a t biological parameters are constant, provision of the primary clarifier

appears to be unjustified 3s far as the economic efficiency of the system is concerned.

T o explore further the role of the primary clarifier, the influent volatile biodegradable

suspended solids concentration was increased to 200 mg/l while the other parameters in the

model remained unchanged. Five different starting points were used for the GRG2 optimiza-

tion runs, and the results are tabulated in T:rble 4.3. In contrast to the results when the

base design corlditions were evaluated, the prinlary clarifier overflow ra te is not a t the

upper bound of 144 meters/day in any of the final solutions. This suggests tha t the pres-

ence of this unit is cost-effective for these design conditions. The final design obtained from

start ing point No. .5 is shown in Figure 4.4; this design h s the lowest total system cost

(545,000 dollars/ye3r) among the five final designs.

O'OSI S'SZI

O'OC

0.1. I

O'OSI Z'L91

O'OC

C'6C

0'0'2

I 'SZ

O'OSI I." G aPI

(spuo3as d3) am!L ~a2ndmo3

lCUY I'F!?!U!

(JL/$ cO1) 1503 ma?si~ l"10.L

l"UY

lqu!

(l/sm) SSL 3uanB3 I"UY

["!?!U!

(I/s~) 'a08 gUanw3 I"UY

l"!?!U! &m/Sy) uo!go~~ua3uo3 sp!los ayq

CO'L CO'L ZO'L I"UY

0'6 6'9 0'8 p!?!u!

(J~/~~/SX) Pla!A Ja?l!d

L'tC 9'1.C 6'f.t' IEuY

0'81 0'9'2 O'PZ [a!?!u! (Lep/,m/Sy) ~a?saS!a uo Su!p.eo~ sp!~os

6'ZI P'CI P'ZT IZUY 0'01 0'05 031 I"!?!u!

(shp) ~a?saS!a u! am!L uo!?ua?aa

09 09 09 leuY sc sc sc lqu!

(3 ) a~n?.e~aduraL uo!?sa%!a

I .ZT O'LI C'ZT IQUY 0.8 1 0.9'2 0'f.Z ["!?!u!

(L~p/~m/Sy) Jauays!qL uo Su!poo~ sp!log

S'P I 9'FI L'PI IZUY 0's I O'OC O'SZ I'E!?!U!

(%) 0!3"~ al3L3aa aoOpn~s

L'P L'P 93 I"UY

O'P 0'8 0'9 I'F!?!U! (~q) am!L uo!?ua?aH 3!lnc~pL~

80'L 80'E 80.2 lc"!l

S'Z S'S 0-2 IZ!?!~! (sXcp) am!L a3uap!saU [la3 uzalq

C " G T

E~:ID 2u!s11 pau!c:qg uc!lnlos (3!un) ~I~"!J"A

I ! I , :, !? t; i

LANDF l LL 2 9 . t i 2 3 1 t3

LEGEND :

3: SLUDGE

+: L IQUID

Figure 4 .3 - Final Design for the \Yastewater Treainlent System Without a Pri~nary Clarifier

Table 4.2 - Final Designs With and Without a Primary Clarifier in the System

Variables (Units) With Primary Clarifier Without Primary Clarifier

Primary Clarifier Surface Area (m2) 25 1 -- Mean Cell Residence Time (days) 2.10 2.08

Aeration Tank Volume ( m 3 5637 7038 Sludge Recycle Ratio (%) 12.8 14.7

Final Clarifier Surface Area (m2) 68 7 717

Thickener Influent Flowrate (m3/hr) 11.8 17.8

Thickener Influent Solids Concentration (kg/m3) 21.3 12.0

Thickener Surface Area (m2) 475 41 1

Digestion Temperature ( C) 60 60

Primary Digester Volume (my 1500 1170 Retention Time in Digester (days) 14.0 15.4

Vacuum Filter Surface Area (m2) 6.7 10.8

Cake Solids Concentration (kg/mg 150 150 Effluent BOD, (mg/l) 30.0 30.0

Effluent TSS (mg/l) 30.0 30.0

Total System Cost ( lo3 $/yr) 500.4 492.5

T h e same design conditions were then examined for a system without a primary

cIarifier. Four start ing Ijoints were tested, and the final designs were very similar (Table

4.4). The total system would cost 556,300 dollars/year, which is slightly (2%) higher than

t h a t for the base system designed lor the same conditions. This design is shown in Figure

4.5. A comparison or the two designs is shown by Table 4.5. It is not surprising to observe

t h a t the primary clarifier is cost-effective when the influent wastewater contains high con-

centration of suspended organic materials. This trend would be expected t o apply to even

higher, or lower, influent suspended solids levels than those considered here. In general,

depending on the design conditions, the observations t h a t can be drawn from a wastewater

t rea tment system optimization study may be very different.

4.3. Activated Sludge

T h e final designs for the base system are characterized by an emuent t h a t just meets

the assumed wat,er quality standards. 1-Iowever, it is possible t h a t only one of the two con-

T a b l e 4.3 - S u ~ n m a r y of \Vastewater Treatment System Design : Influent Volntilc Dioclegradnble Sol ids = 200 m % / l

Solution Obta ined Using GItC;:! Variables (Units)

1 2 3 4 5

Pr imary Clarifier Overflow Rate (m/day)

initial 36.0 24.0 32.0 36.0 24.0 final 130.0 79.9 116.6 69.2 113.3

Mean Cell Residence Time (days) initial 2.0 3.0 4.0 5.0 6.0 final 2.38 2.41 2.30 2.42 2.39

Hydraulic Retention Time (hr) initial 2.4 3.6 6.0 4.8 10.8 final 4.3 4.2 4.3 4.1 4.3

Sludge Recycle Rat io (96) initial 15.0 30.0 25.0 25.0 10.0 final 14.1 13.6 14.0 13.5 13.9

Solids Loading on Thickener (kg/m2/dzy) initial 12.0 36.0 40.0 24.0 12.0 final 12.0 13.0 12.0 12.0 12.0

Digestion Tempera tu re ( C) initial 35.0 25.0 35.0 35.0 35.0 final 60.0 60.0 60.0 60.0 60.0

Retention Time in Digester (dzys) initial 15.0 20.0 15.0 15.0 15.0

final , 14.7 17.1 14.4 16.0 15.1

Solids Loading on Digester (kg/m2/day) initial 12.0 24.0 18.0 24.0 12.0 final 48.0 48.0 48.0 48.0 48.0

Filter Yield (kg/m2/hr) initial 13.0 7.8 12.0 8.0 10.0 final 6.41 6.31 6.37 6.31 6.3 1

Cake Solids Concenlration (kg/m3) initial 70.9 186.3 52.2 142.2 164.3 final 12l.O 150.0 132.3 148.3 150.0


initial 3 1.5 35.7 3.1.9 17.8 2-L.7 final 30.0 30.0 30.0 30.0 30.0

Effluent T S S (rng/lj

initial 21.7 50.7 59.5 23.4 41.1 f ind 30.0 20.0 20.0 BO.O 30.0

Tota l System Cost ( lo3 $ / ~ r ) initial 788.4 774.3 709.9 768.7 779:l

final 55.1.6 5.17.0 550.9 5.18.1 545.0

Computer Time (CP seconds). 536 88.273 666 570 525

* Except Tor s tar t ing point No. 2, 311 computer times reported on this tnble are rccordcd when the optimization model and GIIC2 are run on 3 IIarris computer. A subroutine cxlculating the

analytical derivatives Tor all Tunctions in the model is incorporated in thrse runs.

ioot -, ILL'! , -

T a b l e 4.4 - T r e a t r n c n t P l a n t Dcsign Opt in i i za t ion : I32sc S y s t e m W i t l l o u t a P r i r n a r y Cl,\r if ier, I n l l u c n t Vola t i le G iodcgradnb le S u s p e n d e d Sol ids Concc.nt rnt ion = 200 r n g j l

Solution Obtained Using GItCi2 Variables (Units)

1 2 3 4

Mean Cell Residence Time (days) in i t id 4.0 3.0 5.0 6.0 final 2.27 2.27 2.27 2.57

Hydraulic Retention Time (hr) initial 4.8 6.0 4.8 12.0 final 5.8 5.8 5.8 5.8

Sludge Recycle Rat io (%) initial 15.0 10.0 50.0 30.0 final 17.1 17.1 17.1 17.1

Solids Loading on Thickener (kg/m"day) initial 12.0 2i.O 18.0 36.0 final 12.0 12.0 12.0 12.0

Digestion Temperature ( C) initial 35 35 35 35 final 60 60 60 60

Retention Time in Digester (days) initial 15.0 20.0 25.0 10.0 final 14.6 14.0 15.1 14.4

Solids Loading on Digester (kg /myday) initial 12.0 24.0 18.0 30.0 final 45.5 44.9 46.0 45.4

Filter Yield (kg/m"hr) initial 10.0 7.8 8.5 7.4 final 6.42 6.45 6.40 6.43

Cake Solids Concentration (kg/mg initial final

Emuen t BOD, (mg/l)

initial final


Totnl System Cost ( lo3 $/yr) initial final


Table 4.5 - Final Designs With and Without a Primary Clarifier in the System : Influent Volatile Biodegradable Suspended Solids Concentration = 200 mg/l

Variables (Units) Wi th Pr imary Clarifier Wi thou t Pr imarv Clarifier

Pr imary Clarifier Surface A r e a (m? 320 -- Mean Cell Residence T ime (days) 2.39 2.27

Aeration T a n k Volume (m3) 6500 8778 Sludge Recycle Ra t io (%) 13.9 17.1

Final clarifier Surface Area (m2) 702 752

Thickener Influent Flowrate (m3/hr) 15.2 23.7

Thickener Influent Solids Concentration (kg/m3) 53.7 11.4

Thickener Surface Area (ml) 716 533

Digestion Tempera tu re ( C) 60 60

Primary Digester Volume (m3) 2075 1639 Retention T ime in Digester (dxys) 15.1 14.0

Vacuum Filter Surface Area (m? 14.1 12.6

Cake Solids Concentration (kg/m3) 150 150 Effluent BOD5 (mg/l) 30.0 30.0

Effluent T S S (mg/l) 30.0 30.0

Tota l System Cost ( l o 3 $/yr) 545.0 556.3

straints would be binding in the final solution if a dilferent set of design conditions are con-

sidered. In the final solutions listed in Table 3.9, the sludge ages are about 2.2 days for the

desien conditions assumed in Tables 5.6 to 2.8 since no provision for nitrification is con-

sidered in the model. T h e sludge recycle ralios (10-13s) are lower than what is usually

experienced in practice because lhe emuent suspended solids concentration increases with

the recycle rat io according to Chnpmnn's model. Good slr~dge thickrning in thc final settling

tank is also suggested a t this low value of the sludge recycle ratio.

Sludge sctt l ir~g characteristics could be atrccted by the sludge age. Bisogni and

Lawrence (1071) showed t h a t sludge llocculated and setlled better with longer sludge ages.

This observation was questioned by Dick and IIasit (1081). Currently there is no consensus

on how sludge age aflccts the activated sludge settling properlics. If longer sludge ages d o

enhance sludge thickening, then the dcsign sludge age should perhaps be longer than t h a t

obtained for the base system design.

Increased organic loading to the wastewater treatment plant would be expected to

have a direct effect on the design of the activated sludge process. T h e influent soluble B O D ,

concentration was increased to 200 mg/l and the model was optimized with GRG2 using five

diBerent start ing points. T h e results are summarized in Table 4.6. Although the initial

designs are quite diflerent, with sludge ages ranging from two t o six days, the final solutions

obtained by GRG:! are very similar. T h e system design obtained with start ing point No. 1 is

shown in Figure 4.6. A comparison of this design with the final design (No. 6 in Table 3.9)

obtained for the base design conditions (influent soluble BOD:, = 100 mg/l) is shown by

Table 4.7. It is observed that the design of the primary clarifier is not affected by changing

the influent soluble BOD,. This is consistent with the assumption made in the primary

clarifier design tha t the soluble B O D is unaffected by this unit. T h e design determined for

the high influent soluble B O D , condition has a slightly higher sludge age in order to meet

the same effluent water quality requirements. T h e aeration tank is bigger, and the hLSS

concentration is higher because of the higher organic loading. T h e sludge production ra te is

high, resulting in higher costs for sludge treatment and disposal.

4.4. Secondary Sedimentation

The clarification model describing the solids removal of the final settling tank in the

activated sludge process plays a critical role in the design of wastewater treatment plants.

Most previous researchers (see, for example, Middleton and Lawrence, 1976, Tyteca, 1981)

assumrti t ha t the final clarifier is 100% eficicnt in t.hc removnl of suspended solids. If the

eflluent is assunred to be free of s u s p e ~ ~ d e d solids, then the system design model is subject

only to a restriction on the B O D 6 concentratio:.

This assumption can be expected t o have significant impact on the entire t rea tment

plant design. T h e comprehensive sys t em model was modified to examine this issue; the

water quality constraints are reduced t,o

148

T a b l e 1.6 - T r e a t m e n t P l m t Design Optimizat ion : Influent Soiuble BOD6 = '"30 rng/l

Solrltion Obt7inetl Using GRG'? Vsriables (Units)

1 5 3 4 5

Primary Clarifier OverHow Rate jm/day) initial final

hlean Cell Residence Time (days) initial final


Sludge Recycle Rstio (%) initial final



Retention Time in Digester (days) initial final

Solids Losding on Digester (kg/m"day) initial final


Cske Solids Concentration (kg/m3) initial final

Etlluent BOD, (mg/l)

initial final

Emuent TSS (mg/l) initid final

Total Systen~ Cost ( l~"/yr) initial final


Table 4.7 - Final Designs With DifIerent Influent Soluble BODS Concentration

Influent Soluble BOD, Variables (Units)

100 mq/ l 200 mg/1

Primary Clarifier Overflow Rate (mjday) Mean Cell Residence Time (days) Hydraulic Retention Time (hr) Aeration Tank Volume (m3) MLSS Concentration (mg/l) Sludge Recycle Ratio (%) Final Clarifier Surface Area (m2) Thickener Influent Flowrate (m3/hr) Thickener Influent Solids Concentration (kg/m3) Mass of Sludge Processed (kg/hr) Solids Loading on Thickener (kg/m"day) Thickener Surface Area (m2) Digestion Temperature ( C ) Retention Time in Digester (days) Primary Digester Volume (m3) Solids Loading on Digester (kg/m2/day) Filter Yield (kg/m"hr) Vacuum Filter Surface Area (m2) Cake Solids Concentration (kg/m3) E m r ~ e n t BOD, (mg/l) Emuent TSS (mg/l) Total Svstem Cost ( lo3 $/yr)

St 5 B O D , standard (4.1)

where S3 is the soluble BOD, concentration in the plant emurnt as defined in Chapter 2

Since the total suspended solids concentration of the emucnt is assumed to be zero, no con-

straint is needed lor suspended solids.

Optirnizatiori runs were lnnde with a t o t ~ l (solu1)le) DOD5 s t a~ idnrd of 15 :rncl 10 rng/l

for the bnse design conditions except thc i~ifiucnt solr11)lc DO,!), concentration was clinnged

from 100 to 200 mg/l. T h e results o l these two runs are summarized in Table 4.8. These

final designs show that the total system costs are much less than tha t obtained originally

(577,100 dollars/year in Table 4.6) even though the BOD, standards are much more

stringent (30 mg/l initially). If Chapman's model for clarification correctly calculates the

I 'EL L'86 (spuo2as ~3) am!^ ~aqndmo3

8'SCS L'9GP It UY 0'848 0'8S8 [r!'l!u!

(lJC/$ c~~) Is03 ma9sLs 1VJ.L

0'021 I'GCI IzUY 6'IC G.IC I'q!U!

(1/9m) SSL guanU13

0'1s C'SL I= UY

C'F2 C'b2 ['!'l!"!

(1/9m) Sa~u qUarlW3 L'OZI L'90 IzuY 2'ZS 2.29 [t!'I!U!

&m/9y) uo!grlgua2uo3 spqos ayq

1 t'9 99.9 1"UY

0'21 0'21 [=!?!u!

(lV/:m/9?1) PIa!A laql!d

0'8t 0'8P IrUY 0'81 0'81 IF!g!U!

(L~p/,m/By) ~aqsaS!a ho Bu!pcoq sp![os

9'91 9.91 It UY 0.91 0'9 I I"!'I!U!

(sLap) ~agsa9!a u! am!L uo!guagax

0'09 0'09 IFUY

O'SC 0.SC [yp! (3 ) a~nq~~adma~ uoysaZ!a

P'9I 2'01 IruY O'Ot O'OP ['!'I!"!

(L~p/~m/Sy) lauay2!qL uo Su!pzoq SP!IOS

8'82 8'ZC IZUY

O'SZ O'SZ ["!?!U! (%) o!qq a13L3ax a%pnls

6.01 S'L It UY 0'9 0'9 ["!g!U!

(~q) am!L uo!guaqax 2!ln.e~pL~

LOT 8L'Z IEUY

O'P O'P IE!?!U! (sLvp) am!L a2uap!sax 11" uzatq

O'PPI ow1 1"UY

O'ZS O'ZC ["!?!U! (Lzp/m) aqq n\oUlaAo lay!l~i3 L~zm!~d

elfluent suspended solids concentration, t.lien the designs shown in Table 4.8 are in fact

unacceptable because the actual B O D 5 conccntration wo~:ld be greater than 50 mg/l and the

actual total suspended solids concentration greater than 120 mg/l; these concentrations are

well beyond the water quality restrictions.

This example illustrates tha t it is important , of course, for a comprehensive system

model to include complete performance relationships for a11 unit processes in the system.

Performance relationships for some unit processes, however, may not be available or not be

reliable. In such cases, making simplifying assumptions are crucial since an "optimal design"

obtained is not likely to be optimal or even feasible when the process mechanisms are taken

into account. This example also supports the view tha t in general it is more important to

use such a systern model as a tool to identify the limitations of current process models and

future research arezs, and to analyze the trends for cost-effective process synthesis or design,

rather than to use such a model to obtain the "optimal systern design."

4.5. S l u d g e T h i c k e n i n g

Sludge thickening in a wastewater treatrnent plant is provided to reduce the volume of

sludges for processing and final disposal. Very large thickeners are specified by the solutions

to the base system model, and the digester inllrlcnts have coricentrations higher than 5';71 in

all desigris in Table 3.9. LVith these high solids concrntrations, the costs of hentin5 t h r diges-

ter influent become outweighed by the benetits tha t can be derived from the methane pro-

duction in the digester. Therefore an rfficicntly drsigncd th icke~ir r is the key to 3 cost-

cuective sludge treatrnent train. However, there should I)e n practical limit on thickener

design beyond which the lirnitirig flux theory is no longer :~pplicable for predicting underflow

solids conccntration. T h e lower bound for the solids loatling of the gravity thickener in the

model is 12 kg/m2/day which is lower than values usually observed in practice. The model-

ing study suggests tha t the limitations of the limiting flux theory be investigated. A long

detention time in the thickener may cause sludge degradation in the thickener and problems

in sludge cransport.

There are a number of possible schemes for sludge thickening other than tha t assumed

in the base flowchart. One such scheme has been analyzed using the system model. If the

limiting flux theory is valid for primary sludge thickening as assumed, then recirculation of

the waste activated sludge to the primary clarifier would appear to be very attractive. The

separate thickener could then be eliminated from the system. and yet a very concentrated

sludge could be obtained from the primary settling tank and pumped directly to the diges-

ter. This scheme is depicted in Figure 4.7. The thick sludge would have a significant impact

in reducing the cost of sludge treatment and disposal.

The GRG optimization model was modified to represent the flowchart shown in Figure

4.7. The revised optilnization model has 51 variables, 43 equations, and three inequality

constraints. A listing of the optimization model is in Appendix I.

Results of optimizing the treatment system design using GRG2 are listed in Table 4.9.

Five start ing points were used in this exercise. The final solutions have objective function

values ranging from 466,200 to 469,200 dollars/year, representing cost reductions of 6.2 to

6.8% from the cost of the base system designed for the same conditions (which has total sys-

tem cost of about 500,400 dollars/year). T h e final design obtained from start ing point No. 1

in Table 4.9 is shown in Figure 4.8.

Because of the use o i the primary clarilier as a thickener, the f ind soll~tions speciiy

tha t the size of this unit be from 400 t o 750 rn2, which are sigr~ificnr~tly larger values t11:ln

the 250 m2 obtained for the base system. In the design shown in Figure 4.8, the primary

sludge is about 7.5% (75.5 kg/m3) with 3 flowrate of 3.45 m3/hr. This sludge is 3 highly

concentrated digester influent with a high organic content, which helps to produce more

methane gas a t a moderate digester retention time (17.G days). A comparison of the pri-

Figure 1.7- Recircul:ition o f IC'nste Activated Sludge to Primary C!nritier

I N F L U E N T EFFLUE?Y'T I

?R I MARY

S E T T L l NG

SEC3NDAE ' f

S E T T L l NG A E R A T l O N -+

I I v

I

I I

7.7

ANAEROB 1 C D I G E S T I O N (PR I M A R Y )

-

/

- ANAEROB l C D l GEST I O N

(SECONDARY)

J L EGEI'IE : Ti

F I L T R A T I O N : S L U D G E

+: L I Q L I I D

L A N D F l L L

OSG'CL' 6CO'LC S91.'9P LOS'GC GC6'CZ (spuo2as d3) au!~ ~aqnduo3

O'L9F P'L91 '9 0'891 C'991 lVUY

L'CS9 F'OL9 Z f CL C CG9 I'S09 IC!I!U!

(~d/$ '01) ?so3 lwsbs IVWL

O'OC O'OC O'OC O'OC O'OC ICUY 0 Z'OZ 1'9s 1'82 9'CC IC!l!u!

(1/9u) SSL 3uanw3 O'OC O'OC O'OC O'OC O'OC I'UY

L'91 8'F I 8'82 8'LC 0'9Z IC!3!u!

(I/S~) 'a08 quanu13

O'OSI O'OSI O'OSI O'OSI O'OSI ICUY

5'891 C.CSI F'191 1'191 I'b9I [C!?!u! 6~/83) UO!3CJ'JUa2UO3 Sp!lOS ayE3

OC'L 01'8 F9'8 8G'L 69'L lzUY 0.61 S'P I 0.0; O'OC 0'01 I'!?!U!

(~q/,u/B3) Pla!A Ja?l!d

C'ZC 8'ZC 8'81 8'CZ S'9Z IcuY Z'L 8'P P'Z F'Z O'ZI IC!l!u!

(d~p/,u/Sy) ~agsaP!a uo Bu!peol sp![os

C'SI O'OC

I'CZ 9'LI O'OZ O'SZ

I'UY

[qu! (sbep) ~a3saB!a u! au!L uo!guagaa

IEUY

I"!l!U!

(3 ) a~nge~adma~ uo!gsa%!a

I"UY

lZ!?!U! (%) o!qe~ a13b3a~ aSpnls

1"UY

['!3!U! (~q) au!L uo!guagaH 2![nslpb~

I'UY

IE!?!U! (sbcp) au!L a2uap:saH Ila3 uCalq

I'UY

IE!'l!U! (icp/m) 39x8 nio~~ao ~ay!~cl3 XJC~!J~

S'l I O'OC

9'01 O'OZ

C'11 O'OC

F'ZI S'I I 0.91 O'OZ

P'C 9'C

S'C 8'F

C'C 0'9

SC'C S'P

OC'Z 0'4

0C'Z LZ'Z S'Z O'C

8'00 O'OZ 1

C'LS 0'09

18 L'OL 0'9'2 O'ZL

mary digester designs for the base system (design No. 6 in Table 3.9) and the modified sys-

tem is shown by Table 4.10.

It is recognized t h a t thickening characteristics of the combined waste activated sludge

and the raw wastewater may be different from those of the raw influent alone. Also because

the waste activated sludge contains a high concentration of microbial mass, biological stabili-

zation of soluble organics is possible in the primary clarifier. Experimental work on the use

of the primary clarifier as a thickener is necessary to verify the results from the modeling

study.

Rimary sludge concentration has been modeled by many researchers as a constant.

This modeling approach was examined by fixing the primary sludge concentration to 4% in

the optimization models. Table 4.11 summarizes the designs obtained from this approach

and from using the differential thickening technique for the base system. T h e influent solu-

ble BOD, is 200 mg/l in these runs. In general, the two solutions show the same characteris-

tics for a cost-effective design. T h e liquid subsystem designs appear to be similar regardless

of the approach selected to model the primary sludge concentration. T h e mass fractions of

the primary sludge are about the same in the two designs, which results in very similar

Table 4.10 - Comparison of Primary Digester Designs for the Base System and the System in Figure 4.7

Wasted Activated Sludge Design Conditions Base System

Returnrd to Primary Clarifier

Inlluent Flowrnte (m3/lir) Influent Volatile Solids (kg/m3) In tluent Tota l Solids (kg/rnZ) Digester Volume (m3) Digestion Temperature ("C) Solids Retention Time (days) Methane Production (m3/day) Heating Requirement (10' klVhr/~r) Net Value from Diccster Gas (10' k\Vhr/yr)

Table 4.11 - Comparison of Optimal Designs by Diflerent Models to Determine Primary Sludge Concentration

Varinbles (Units) Limitins Flux Constnnt (4%) Primary Clarifier Overflow Rate (m/day) 144 144 Primary Sludge Concentration (%) 7.7 4.0 Mean Cell Residence Time (days) 2.47 2.47 Hydraulic Retention Time (h r ) 5.1 5.1 Sludge Recycle Ratio (%) 15.8 15.7 Solids Loading on Thickener (kg/m2/day) 12.0 12.0 Thickener Surface Area (m2) 620 621 Digestion Temperature ( C ) 60.0 60.0 Primary Digester Volume (m3) 1970 1920 Retention Time in Digester (days) 14.2 13.8 Solids Loading on Digester (kg/m2/day) 48.0 48.0 Filter Yield (kg/m2/hr) 6.31 6.31 Cake Solids Concentration (kg/m3) 150.0 150.0 Effluent BOD5 (mg/l) 30.0 30.0 Effluent TSS (mg/l) 30.0 30.0 Total System Cost ( l o 3 $ / ~ r ) 577.1 581.8 Computer Time ( C P seconds) 64.4 54.4

Notes: 1) Start ing point No. 1 in Table 4.6 is used for these runs. 2) Influent Soluble BOD5 = 200 mg/l

thickener sizing. However, because of the difierence in the digester influent flowrate and

solids concentration due to the difierent modeling approaches for the primary sludge concen-

trat ion, the primary digesters are designed differently in the two solutions in Table 4.11

Therefore the total system costs in the two designs are slightly diflerent.

A similar modification of the primary sludge concentration was also rnade in the model

describing the wastewater treatment system with recirculation of the waste activated sludge

to the primary clarifier (see Figure 4.7). Solutions were obtained for the base conditions in

which the influent soluble BOD6 concentration is 100 mg/l. 'Three dillcrcnt start ing points

were used to run GRG2. Results are tabulated in Table -1.12.

It is interesting to note tha t starting points No. 1 and No. 3, although very difierent,

converge to exactly the same point in the optimization. This solution is displayed in Figure

4.9. A comparison between this design and tha t obtained by modeling the primary sludge

Table 4.12 - Summary of System Dcs~gn Optimization : Waste ~?ctivlrted Sludge Returned to Primary Settling Tank , Primary Slud%e @ 476

Solution Obtained Using CRC;? Variables (Units)

1 'l 9

Primary Clarifier Overflow Rnte (m/day) initial final

Mean Cell Residence Time (days) initial final

Hydraulic Retent ion Time (hr) initial final

Sludge Recycle Rat io (%) initial final

Digestion Temperature ( C ) initial final

Retent ion T ime in Digester (days) initial final


Fil ter Yield (kg/m2/hr) initial final

Cake Solids Concentration (kg/m3) initial final


initial final


Totxl System Cost ( l o 3 $/yr) initial Rnal

Compute r Time ( C P seconds)

concentration using the limiting flux theory (Solution No. 1 in Table 4.0) is shown in Table

4.13. The difference in total system cost is more pronounced in this case than in the prcvi-

ous example, with the limiting flux approach costing about 5% less than the approach which

assumes tha t the primary sludge concentration is independent of the primary clarifier sur-

face area. T h e major difference in the system design is, as expected, in the primary clarifier.

When the limiting flux theory is used t o calculate the primary sludge concentration, the

underlying assumption is t h a t the primary clarifier serves as a thickener as well. In this

case, this use is necessary in the most cost-efficient design since it reduces the volume of the

sludge t o be processed. This is also the reason why this thickening scheme, i.e., returning

the waste activated sludge to the primary clarifier, is potentially attractive. On the other

hand, if the primary sludge is fixed a t 4%, then the thickening function of the primary

clarifier is neglected. No matter how small the primary clarifier is, the thickened sludge

from the clarifier is always a t the same concentration of 4%. This causes the optimization

program to select the size of the primary clarifier t h a t is as small as possible.

In summary, modeling the primary sludge concentration as a constant has little effect

on the solution obtained for a cost-efficient wastewater treatment system design for the base

system; for this system, however, the thickening potential of the primary clarifier is limited

because of the sludge thickening scheme specified. In contrast , when a system flowchart is

designed specifically to take advantage of the thickening capability of the primary settling

lank, then this capability may be more irnportsnt. It is also noted t h a t if the primary sludge

concentration were modeled initially as a constant in the base system design, the final solu-

tions obtained may have suggested designs with a good total system cost, but these solutions

would not have suggested the alternative sludge thickening scbeme of returning the waste

activated sludge to the primary clarifier. This insight was directly provided by the model,

however, when the primary sludge concentration was modeled using the limiting flux theory.

Table -1.13 - comparison of Designs by Dilferent Modeling Approaches on Primary Sludge Concentration :

Waste Activated Sludgc Returned to Primary Clarifier

Variahlrs ILJnits! Limitin: Flux Constant ( 4 5 )

Primary Clarifier Surface Area (m2) 514 253 Primary Sludge Concentration (%) 7.6 4 .O h4ean Cell Residence Time (days) 2.27 2.21 Aeration Tank Volume (m3) 5115 541G Hydraulic Retention Time (hr) 3.4 3.6 Sludge Recycle Ratio (%) 11.5 11.7 Digester Influent Flowrate (mJ/hr) 3.15 6.62 Digestion Temperature ( C ) 60 60 Primary Digester Volume (m3) 1456 1861 Retention Time in Digester (days) 17.6 11.7 Secondary Digester Volume (m3) 760 450 Filter Yield (kg/m2/lrr) 7.69 6.31 Vacuum Filter Surface Area (m2) 10.1 12.9 Cake Solids Concentration (kg/m3) 150.0 150.0 Effluent BOD, (mg/l) 30.0 30.0 Effluent TSS (mg/l) 30.0 30.0 Total System Cost ( lo3 $/yr) 466.2 489.4

This example has illustrated tha t the comprehensive systcm modcl can be used to

analyze cost-efficient process integration. Results presented here arc dcperidcnt on the set-

tling properties of the primary, the activated, and the combined primary and activated

sludges, as well as the limitations of all of the unit process models. Consequently it is the

methodology of the analysis and the philosophy of using the optimizntion modcl for process

analysis tha t are important. An optimization niodel enables the design engineer to invcsti-

gate alternative flowcharts eficiently. Insights about the impact on the entire plant due to

design modification of 3 sinqltt unit process can be obtained. Such information should be

viewed as supplementing the traditional knowledge used by the design engineer (not as

replacing any of it).

4.8. Anaerobic Digestion

The final solutions obtained for the base system and a11 design conditions considered -

suggest thermophilic digestion because the digestion temperature is a t its specified upper

bound or 60 O C . This upper bound cannot be relaxed because the activities of the anaerobic

microorganisms will decrease drastically and finally stop completely when the digestion tem-

perat,ure goes higher than this temperature. Theimophilic digestion results in a high degree

of organics stabilization and high methane production. which is given a cost credit in the

model. T h e solids concentrations of the digester influent in the final solutions are all higher

than 5%. This high concentration results in low energy requirements for heating the

influent.

It is recognized tha t the unit process model used in this analysis (equation (2.48)) is

based on a number of assumptions and is developed from limited experimental da ta . I t

appears tha t fine-tuning of tha t model would be worthwhile to verify the benefits associated

with a thermophilic digest,ion system.

The final solutions in Table 3.0 also call for the elimination of the secondary digester

since the influent solids concentration to this unit is almost identical to the underflow solids

concentration a t the design loading rate. Both the secondary digester and the vacuum filter

are provided to achieve the same purpose, sludge concentration. Because of the poor set-

tling characteristics of the digested sludge, it is more economical t o concentrate the sludge

by vacuum filter than by the secondary digester. It is interesting thxt this insight, wliich

was obtained using the rnodel for the example problem, is consistent with the observations

by Lawler and Singer (1084) who suggested the elimination of the secondary digester in n

t reatment plant based on their survey of the performance of the secondary digester as a

thickener a t a number or existing plants.

Eliminating the secondary digester from the system layout may not be desirable in

practice, however, since the secondary digester provides reliability to the system. Because or

the reliability problems associated with operating an anaerobic digester, it may be desirable

to haye the secondary digester in the system. Cleaning the primary cell is also possible

without the necessity of operational modifications if secondary digesters are present. In

addition, methane production is generally observed in the secondary digester, which contri-

butes to the net energy production. Since these considerations are not captured in the

mathematical model, it would be desirable to explore the role of the secondary digester in

more detail in an actual design exercise.

This example also brings up the general question of the role of an optimization model.

Planning and design of a wastewater treatment system in general is very complicated. Using

mathematical models for design may not include all important considerations in a realistic

t rea tment system design situation. For example, the system that has the least system cost

may not satisfy other design criteria such as ease o t operation or high degree of system relia-

bility. Mathematical models should be used to generate alternative system designs tha t are

good with respect to these important design criteria. Traditional engineering design con-

cepts can then be exercised to determine the most adequate system design.

Because of the consideration given to the rising digester gas on the digested sludge set-

tling characteristics, the digested sludge settling velocity was assunied to be only one-fourth

of tha t of a fully digested sludge in the calculation of the digested sludge solids concentra-

tion (see Section 2.3.7). This factor discounts the digested sludge settling velocity fro111 what

is predicted by the limiting flus theory alone. For a tt~ermophilic digestion system, the per-

cent of organics stnbilization is very high in the primary digcstcr. Therefore the eIlect of

digester g m on sludge settling in the secondary digester beconies less signiscant, and a larger

factor is more appropriate. A factor of 0.90 was substituted for 0.25 in the secondary diges:

ter design, and one optimization run was made tor the base system and design conditions.

T h e results are summarized in Table 4.14 and depicted in Figure 4.10. Also listed in Table

4.14 tor compnrison is the solution obtained tor the base drsign conditions (design No. 6 in

165

Table 4.14 - Sensit.ivity of the System Design to the Digested Sludse Settling Characteristics

Variables (TJnits) Initial ~ i n a l * Base Design

Primary Clarifier Overflow Rate (m/(lay) 72.0 144.0 144.0 Xlean Cell Residence Tirne (days) 3.0 2.19 2.19 Hydraulic Retention Time (hr) 3.6 3.8 3.7 Sludge Recycle Ratio (%) 30.0 12.6 12.8 Solids Loading on Thickener (kg/m"day) 36.0 12.0 12.6 Digestion Temperature ( C) 25.0 60.0 60.0 Retention Time in Digester (days) 20.0 14.0 13.9 Solids Loading on Digester (kg/m"d a y ) 24.0 94.0** 40.6 Filter Yield (kg/m2/hr) 12.0 7.73 6.67

Cake Solids Concentration (kg/m3) 196.9 150.0 150.0 Eflluent BOD, (mg/l) 26.1 30.0 30.0 Effluent TSS (mg/l) 36.8 30.0 30.0 Total System Cost ( lo3 $/yr) 644.6 484.9 500.4

* Computer time for optimization : 67.034 CP seconds. ** Upper bound of digester solids loading (48 kg/m2/day) is relaxed in this run.

Table 3.9). When the digested sludge is assumed to have better settling properties, the total

system cost is lowered to 484,900 dollars/year. The solution specifies a small secondary

digester surface area (22 m2) which would concentrate the digested sludge from 2.0 to 2.4%

(20.2 to 24.5 kg/m3 in Figure 4.10). The solids loading on the secondary digester, however,

is extremely high a t 94 kg/m2/hr, and the secondary digester begins to play a role in the

overall wastewater treatment system. This suggests t h a t the settling properties of the dig-

ested sludge have a direct eflect on the arrangement of the digestion system (i.e., should a

secondary digester be included or not) i f the limiting flux theory is valid a t the high solids

loading. Since d a t a in this area are lacking, laboratory analysis of digested sludge settling

chnracteristics under various fermentation conditions sl~ould be performed to identify the

appropriate role of the secondary digester.

4.7. Vacuum Filter

The solids cake concentrations in the final solutions are a t the specified upper bound of

15% for all conditions considered. As discussed in Section 2.5, this upper bound was arbi-

trariiy set because tile model used for tlie vacuum lilter dus~gn does not prcdict a maxlmunl

cake concelr tration that cnn pract~ically be attained. Since the i ir~al tlispos:~i of ?rwntered

sludge is relatively expensive, and since the filter area requirement is insensitive to the

filtered cake concentration a t high co~~cen t ra t ion levels (see Figure 4.11), the cake concentra-

tion was driven to its upper bound in the solutions obtained.

The limitation of the vacuum filter design model appears to be that it is only applica-

ble within a limited range of design conditions. For example, the air drying mechanism is

not considered in the development of this design equation. This is an area where additional

research is needed t o refine the present model for vacuum filter design.

4.8. Design Under Uncertainty: A Multi-objective Approach

As discussed in the introduction of this chapter, the design of wastewater treatment

plants involves many uncertainties. Parameter uncertainty in the design of wastewater

t rea tment systems has been dealt with by Berthouex and Polkowski (1970), and Tarrcr e l a l .

(1976). Key parameters were assumed to follow a certain statistical distribution, and the

means and the standard deviations were taken into account in mathemnticnl ~rlodrls. Thcr r

are three major difficulties with this approach: 1) The statistical distributions of the design

paralrleters are usually unknown and have to be assumed, 2 ) the r c s u l t ~ ~ ~ g 111nt11rm:itlcnl

modcl brcomrs very complicntcd, and 3) ~lncertaintiea on process perforrnnnrr n~o(lcls, rost

information, and design conditions are not included.

An alternative approacl~ to handling uncertainty in e ~ ~ g i ~ ~ e c r i r ~ g tlc,sig~~ is ro pcrrorrn

sensitivity analysis for model para~netcrs . L'oelkcl (197s) pcrlorrl~cti w~r>itivily a11:iIysis of

the parameters in his model and recorded the sensitivity of [.he overall systern design to the

unit changes of these parameters. The major drawback o f this approach is tha t tlie optin~i-

zation procedure may terminate a t local optima because l l ~ e rliodel is nonlinear. A distinct

trend for the system cost as 3 function of the perturbed parameter may not be attained.

Cak

e C

once

ntra

tion

, %

Fig

ure

-1.1

1- V

acuu

m F

ilter

Sur

face

Are

a R

equi

rem

ent

vs.

Solit

ls C

ake

Con

cent

rati

on

l'f 88

0.0'2

O'OC O'OSI

9L.9

6.8'2 9'f.i

03

L'89L

0.0'2

0'0'2 O'OSI

Z8'9

0'8'2

L'CI

09

9 F'OOS

0.0'2 0'0'2

G'OC O'OC O'OCl O'OSI

18'9 69'9

E'8C C'OP

(;.El Z'1.i

09 09

G'Zl O.:! O'ZI O'G1 S'ZI !,i~l)/~u1/3yj 1aua)[3!rl~ uo Surpcoy sp!ln~

I'CI Z'ZI C'EI l,'E I ?'Z I (s) O!~TU a!>K>a?~ aspn~s

8'C 8.C 8'C 8'2 8'C (JY) am!& uo!qua~a~ >!lnc~pL~

61'Z 6 I I 8I'Z (sn'sp) am!& asuap!sa~ 1123 uratq

0'ff.l O'Pf 1 0'1.1.1 0'11~1 O'i-f.1 [Kcp/m) alc~ mou~aho ~ay!~z.l~ d~rw!~d

O'C q'E 0': S' I 0' 1 10 jix\,~ Xjaj~~ alclrno~~j (sl!un) s31q=!Je~

sqlnsa~ a11~ .IIO!ICZ!UI!I~O JOJ pxn se,n ~3x3 pue 'pamnss~ a~a~ SUO!?!PUOJ oZ!sap aseq arl~

palc~ado aq plnoa ?! ?cq? 0s ma?sis aqj 01 .(?!acdea aJom ap!ilo~d 03 pasn SEA\ JO?aZJ dlaj~s

aq? u! sa.~!?aafqo o.~i? aq 01 paulnssc aJaM dqajes puc qsoa 'aldmexa ail!?cJ?snl[! ue sv

-aJU!UOn ?cql su%!sap ail!?cnJa?lr: duem a~c~aua9 09 pasn aq uca lapom uo!~ez!m!~do aqL

.mqqo~d ail!laaJqo-!llnm e sc palapom aq nc3 ma3s.C~ ?uam

-?Val? ~alc.+ia?sc?n E JO u%!sap aq? '333ds31 s!?? .dl!l!qc!la~ PUT: ?so3 s,ma?sds aql uaamqaq

Boapel? e s?s!sa alaq? 'sp~om Jaqlo UI .?sea ma~sis lclo? aq? a?clcasa SJOJ~EJ i?ajns '~aila

-MO~-I .u2!sap USISAS arll o?u! ?pn(l a(! 03 ,[auepunpal puc 6?!!!q!x31] duos S.+~OIIC JOJ~CJ s!rlL

'1033CI d?a!es r: L!d(lc 01 s! .{?rl!c?~aaun u91sap tll!~i %u!leap JCJ dSa?cl?s uoulmoa y

are surnrnarized in Table 4.15. It is interesting to noie tha t the final designs exhibit similar

values for thc decision variables. Tlic tradcofi between the design safety factor and the sys-

tem cost is depicted in Figure 4.12. This curve is sligl~tly convex due to t.he economies of

scale in the design of wastewater treatment systems.

T o account for the design uncertainties mentioned above, the design engineer may

select 3 safety factor greater than one based on the design Bowrate or influent pollutant con-

centrations. This is similar to design based on the maximum daily flow except t h a t the

peaking factor becomes the second objective in the model. The design made according to

this approach is more realistic since design flows may be exceeded, and because there are

uncertainties in the model. With better knowledge about the design parameters or process

performance models, a smaller safety factor may be used.

4.9. Summary

T h e role of the comprehensive system model developed in Chapter 3 as a tool for use

in the analysis and design of secondary wastewater treatment systems is illustrated in this

chapter. Recognizing the limitations of a cost-minimization system model, the intent of this

work has not been to obtain the "least-cost design." Through tlie use of the model, poten-

tially important research arcas in treatment process design are identified from the cost-

cffect.iueness viewpoint. For example, the solids removal hrhavior of the primary clarifier a t

overflow rates higher than usually recommended in design practice should be examined.

The importance of a model describing c1arific:ntion in the nc:tivntrtl slutlge final clarilicr is

also illustrated. Sludge thickening a t low solids lo:lding is crilical to the dcsign of the sludge

processing train. Anaerobic digestion in the thcrmophilic range is another area tha t should

be investigated. T h e settling cl~aractcristics of the digcsted slutlge de te rn~ ine the role of the

secondary digester in the overall t reatment system; correlations between the digested sludge

settling properties and the degree of-organics stabilization should be studied. Refinements of

Economlc Efficiency, 1-03 doIlars/year

.~,)~)(IuI II:IIIYIJO .)II) jo IIO! ~u>!ld(lu ~ltlu~!s c .icl 1)au!clqo ,,uZ!sa~ lcrn!ldo aq?,,

u\:rl.) .)!Js!~u~J ~JOIII 3'1 I)lno.\+ IIJI:uJ(I(~\: JCI 3d.i) S!II~ uo p~scq apcm ul-;!s~p c ~cq~ pa~aadxa

S! ?I .[).))U.)S~J~ Sl ~d,iIJ.).l~(l~ O.\\J 5C .~J!I!({c![~J PUC ~U!JJP!SUO~ I~IJIS~S ?U31UlCdJj 131

-l:.\i,))\C.\\ 1: JO ll~lSd1) 311) JOJ 1131:0J(1(1l! D!]S!I(~UI!S V '.i)!l!({~!1~1 PUC ?SO3 S,UIJ~S~S 3ql U3JM13q

1103[)CJ') I: y JJJ~I~~ ~.)IIIS IIOIJI:J.)~)!:"KI~~IS~~ luc)~otlrn! uc s! uZ!sap puc 3u1~aporu ssa30~d

I]! .ijn!r:]~~.)rl,l .I).)J.)I)!suoJ .)(I 01 J.\CII 1%)~ 111a)s.i~ 3111 UCIII ~31110 s~.i!13arqo a3u!s a.i!13arqo

..ion).; Su~laporr~ jcn!lcularllcru arl? UIOJJ pau!c?qo srlo!snl3

-uo~ all] .ij!Jd.i Jdr{]Jnj ol .i~cs'sa3~u .)(I .icm y~o,\i ~I:JUJLU!J~~X~ 'sl!un ?ualu?caJi mca~?sdn

JO uS!sdp aq] o) YIIIPJOJJ~ a311cq3 .[em uY!sap U! pasn s~alamc~cd aq? 's!sarl~uis ssa3

-old JOJ paw y I~~OUI UO!~UZ!ILI!~~O UF uaq.*i ?cq? paz!uSo3a~ aq plnorls ?! 'ia~a,iio~] .s!ssq?

-u.<s sss~o~d JO quarrla~!nba~ ~c!.i!~luou ale q3!q,+\ y~o~ u%!sap puz suo!?c?ndmo3 u! s9u1,ics

lc!lua~od s~uasa~da~ s!q~, 'paz.i[zue aq uc3 s?~cq3.no~ rna?sbs ~nj%u!tteam ?sou aq? 'sassa3o~d

?!un '-;uoi~c su0!]3clslu! aql ?uno33z o~u! %u!ye? k8 .p!z s!saqluis ssa3o~d c sc lapom m?lsiCs

a.i!susqa~druo~ aq? jo ssaulnjasn aq? MO~S q3!q~i saldlucxa ale s?q%!su! asaqL ~Lllc3!uouo3a

J.~!)JCJ~~C 3q .\em ~aua!f3!ql ,i?!.icl% c sc Jag!JclJ li~clu!~d aq? jo asn aq? uaq? '~c!?ua?od

[["J SI! 01 a111as 01 pa.\iollc s! ~Zpnls L~cm!~d aq? JI .s~!~s!J~?~cJc~~ %u!l??as ~ood scq azpnls

[)alsaS!p aql uaq.*i adi!?3aUa-?so3 aq ?ou Leu ~a?sa%p li~cpuo3as aqL .q%!q LJ~A ?ou s! uo!yc~?

-uaJuon sp!los a[!lclo.i luanuu! aqj uaqM a.i!qaaja-?so3 aq ?ou Xcru J~IJ!JCI~ .i~cm!~d i? .uS!sap

3-~~;s,iioU ssa3oad dh!?CAouu! h!$uap! 01 sdlaq oslc ~apolu lua,1shs aa!suaqa~druoa aqL

CHAPTER 5

SUMMARY AND FUTURE RESEARCH

5.1. Introduction

T h e design of wastewater t rea tment systems involves many tradeoffs because of the

complex arrangement of the unit processes. With increasing understanding of the funda-

mentals of the wastewater treatment mechanisms, researchers have been developing

mathematical models t h a t can be used t o describe the levels of performance of the various

unit processes. Use of these mathematical models for design allows engineers to examine the

tradeoffs in a wastewater treatment system in detail and to strive for cost-effective system

designs.

There are other uses of a comprehensive design model for a wastewater t rea tment sys-

tem in addition t o obtaining cost-effective system designs. Limitations of process perfor-

mance models and potential research areas can be identified. Important insights about pro-

cess flowsheets can be gained from exercising such a model. Innovative water quality

management strategies for a river basin can be better evaluated using a model for wastewa-

ter treatment plants as the basis. Also, issues tha t are important in planning and design of

wastewater t rea tment systems but tha t are unmotielcd can br rvnlunted.

Efficient mathematical programmin:: techniques are essential if a comprehensive system

model is to achieve extensive use. Because a system model is very complicated mathemati-

cally, research must be done to develop efficient optimization procedures.

In this thesis, a complete model Tor use in the design of a secondary wastewater treat-

ment system is developed. This model includes state-of-the-nrt process design models t o

predict the performance of the treatment system. T h e construction of the model is described

in detail in Chapter 2, and is briefly summarized in Section 5.2.

Chapter 3 discusses the use of two existing optimizat.ion algorithms and one new

approach for solving the comprehensive system model developed in Chapter '2. Some key

observations are summarized in Section 5.3.

Recognizing the unmodeled issues and uncertainties involved in the design of wastewa-

ter treatment systems, the use of the comprehensive system model as a tool for the analysis

of process performance is illustrated in Chapter 4. Conclusions from using the model as an

analysis tool are summarized in Section 5.4.

The comprehensive system model developed in this research may serve as basis for

additional research in the area of environmental systems analysis. Several areas in treat-

ment process design and modeling were identified in Chapter 4 as potentially fruitful for

achieving more cost-effective system designs. Sectiou 5.5 provides a summary of these possi-

ble future research directions.

5.2. Comprehensive System Design Model

A typical secondary wastewater treatment system was selected for initial evaluation in

this study. This system includes primary sedimentation, aeration and secondary sedimenta-

tion (activated sludge), gravity thickening of combined primnry and waste activated sludge,

two-stage anaerobic digestion, vacuum filter dewatering, and final sludge disposal by snni-

tary landfill. S u p e r ~ l a t n ~ ~ t s gener:~ted in sludge processing are recyclrd to the Iread end ot

the plant.

Wastewater parnmeters represent the state of the wastewater or sludge during diflerent

stages of the treatmcut process. These s ta te variables include flowrate, soluble $BODS$ con-

centration, and concentrations of active biomass, volntile biodegradable suspended solids,

volatile inert susperlded solids, fixed suspended solids, and total suspended solids. Nine deci-

sion varirrbles need t o be specified in order to define the system design completely.

T h e solids removal eficiency in the prlniary clarifier was modeled using the Voslrel-Sak

(1968) equation. T h e primary sludge concentration was calculated based on the limiting flux

theory. T h e Lawrence-McCarty (1970) model was selected for the design of the activated

sludge process. Clarification of the aeration tank effluent is critical in determining the

efficiency of the overall wastewater treatment. This function of the Enal clarifier was

modeled based on an equation proposed by Chapman (1983). Thickening in the final

clarifier and in the gravity thickener was modeled using the differential thickening technique

be Dick and Suidan (1975). Sludge stabilization in the primary anaerobic digester is a func-

tion of digestion temperature and solids retention time. A mathematical model based on

limited experimental d a t a summarized by Wise (1980) was developed to describe the perfor-

mance of the p r i m x i digester. T h e secondary digester was modeled as a gravity thickener,

and the differential thickening technique was employed for design. Vacuum filter desi, gn was

based on the estimated filter yield.

T o estimate the total system cost, cost information summarized by Patterson and

Banker (1971) was used to calculate the costs of each unit process in the system. Sludge

disposal costs were estimated based on models developed by Dick e t al. (1978), Rossman

(1979), and USEPA Process Design Manual (1374). An analysis computer program was writ-

ten t o design the wastewater treatment system for specified influent and design conditions.

Unit processes were designed sequentially according t o the system flowchart. T h e steady-

s ta te design of the overall system was obtained tl~rouglr iterations because of the presence of

the recycle streams in the system. This program is useful for examining the system response

corresponding to different input and design conditions and for generating systenr designs

tha t can be used as initial solutions in an optimization procedure.

5.3. Optimization Techniques for Wastewater Treatment System Model

Three optiniization approaches were taken in this study to solve the compreh~nslve

wastewater treatment model. Because the model is more complicated than previously stu-

died ones, efficient optimization techniques are essential.

The first technique examined was to formulate the system design problem as a non-

linear program and to solve it directly using a generalized reduced gradient algorithm

(GRGZ). The resulting nonlinear program has 64 variables, 55 equations, and three inequal-

ity constraints. Computing time for this model ranged from 51 to 105 central processing

seconds on a CDC Cgber 175 computer. This performance can be considered to be a t least

comparable to previous studies tha t solved less complicated wastewater treatment system

models using other optimization techniques. Once the GRG:! model is formulated, i t can be

used repetitively to examine different influent or design conditions with minor adjustments

of the d a t a file. This allows its use as a tool for process analysis. However, extensive revi-

sion of the model is necessary i f an alternative treatment flovchart is to be examined.

The solutions obtained from using GRG:! depend on the various control parameters

specified, the bounds on the variables, the initial solutions, and the n~rmerical characteristics

of the model. A modified IISJ (Brill, 1070) approach was usrd to examine the quality of

these solutions. This strategy explores the feasible design space using objective functions

tha t are formed based on the knowledge about the problcm. Numerical examples have

shown that this strateqy I~elpc*d to irr~prove the tot:~l systrln cost of thc solution obtnincd

from solving the original rnodel directly using GRC:!. This str:ltc.qy can also b r used to itlrn-

tify designs tha t are similar in the total system cost, but are different with respect to the

sizes of the unit processes in the wastewater treatment system. This is particularly useful if

there are unmodeled issues in the design of the treatment system. For the example problems,

seven diffcrcnt systrnl designs were obtained using the proposed strategy (see Table 3.9).

The dilTcrences among these designs were not significant, however, because similar objective

functions were used to generate these designs.

Tlie system design can also be formulated as a generalized geometric program (GGP) if

one variable in the model is fixed and one equation is modified. An efficient algorithm

(IGGP) for solving G G P was used to solve the subproblems resulting from the specification

of the variable. These subproblems have 62 variables, 54 equality constraints, and three ine-

quality constraints. The computing time for solving one subproblem ranged from 2.5 to 5.7

seconds. T o obtain the optimal design for the original problem, different values of the fixed

variable have to be examined; a subproblem has to be solved for each value assumed. For

the example problem, eleven subproblems were solved for a total computer time of about 50

seconds on the Cyber 175 computer. The final solution obtained from this approach com-

pared well with tha t obtained from GRG:! as far as the characteristics of the cost-eflective

designs.

A unique approach was also developed for the identification of cost-effective designs.

This approach decomposes the overall system into a liquid subsystem and a sludge subsys-

tem. T h e liquid subsystem design was optimized using GRG:! for a specified set of recycle

stream characteristics. T h e output from the liquid subsystem, i.e., the combined primary

and waste activated sludge, was treated as input to the sludge subsystem. T h e design of the

sludge subsystem was carried out for the specified set of recycle stream characteristics. Two

one-dimensional optimizations were embedded in the sludge subsystem design. One advan-

tage of this appronch is thnt the overall system which contains nine degrees of freedom can

be reduced to two sr~hproblems with four and two degrees of freedom, respectively. Optimi-

zation techniques can be applied to solve these smaller problems more eficiently and Inore

reliably. The subsystem designs, however, must be coordinated to obtain the overall optinial

design. This coordination involved determination of the values of three interacting vari-

ables. A coarse grid enumeration technique was employed to identify the set of interacting

variables tha t results in tlie least system cost. Several assumptions were used in this

approach to reduce the number of the interacting variables so that the coordination could

he carrictl out. more efficiently. These assumplions were shown to be adeqr~nte with three

numerical examples. Total computer time of about 100 seconds was necessary for the exam-

ple problem tested. This computing time is comparable to those required in the pre,vious

two approaches. Trends for cost-eflective designs were clearly identified using this approach.

Another advantage of the decomposition approach is tha t many alternative solutions

can be obtained during the optimization process. These solutions are very different in their

designs, bu t the total system costs are similar. Therefore they can be evaluated with respect

to unmodeled issues.

5.4. Use of Modei for Process Analysis

A mathematically optimal solution is the result of optimizing the comprehensive systern

model. This mathematically least-cost design is not expected to be the best final plan to be

implemented in a realistic design situation because the design of wastewater treatment sys-

tem typically involves other importan t but unmodeled issues. IIowever, the characteristics of

this solution provide useful insights about process research and design. This use o l a

comprehensive system model as an analysis and design tool is illustrated in Chapter 4.

Several research areas in process modeling were identified by an examination of the

solutions obtained from GIZG2. T h e solids removal ellicicncy of the prirn:lry clarilier a t high

overflow rates, sludge thickening a t low solids loadings, and sludge solids stabilization by

thermophilic anaerobic digestion are examples o l these potential rese:lrch areas.

Information on process flowsheets was also obtained from the modcling study. T h e use

of the primary clarifier as a thickener was cost-eflcctive if primary sludge is allowed to

thicken to its lull potential. The role of the primary clnrifier depends on the chnracteristics

o l the influent wastewater. The role of the secondary digester depends on the settling

characteristics of the digested sludge. These rrsults of course depend heavily on the parame-

ter values used. It is recognized that parameter values used for desigu of some unit

processes may be functions of the influent characteristics to these u n ~ t s . Without such Infor-

matlion, results obtained on process synthesis from the use of the comprehensive system

model shouid be examined carefully. Experimental evaluation may be necessary to confirm

the modeling results.

Uncertainties in designing wastewater treatment systems have been dealt with by

researchers using various approaches. Traditionally, engineers have employed peaking fac-

tors to design some units in a wastewater treatment system to provide a system with relia-

bility. There is a tradeoff between the system's cost and reliability, i.e., the niore reliable the

system is, the more it costs. This problem can be considered as a two-objective problem,

and noninferior designs can be generated. These designs can be evaluated based on other

design criteria.

5.5. Future Research

Several potential research areas in process modeling and design have been suggested

from the use of the comprehensive system model. There are other areas tha t deserve future

investigation:

1) Optimization techniques: As mentioned above, one advantage of the decomposition

approach is tha t difTerent opt imi~at ion tecliniqucs can be used to solve difierent subsys-

tem designs. Alternative optimization techniques for optimizing the liquid subsystem

and for coordination could be studied to irnprove the efficiency of this approach.

There are alternative strategies for defining the subsysterns and for implementation of

the decomposition approach. The robustness of the decomposition approach, i.e., the

performance of this approach under diITerent influent conditions should be studied.

T h e applicability of the approach to other system flowcharts can also be investigated.

This proposed work is potentially capable of refining the decomposition approach and

making it a useful design and analysis tool

2 ) Sensitivity analysis: Information on the effect of a particular pararneter in the rnodcl

on the overall system design is nseful for systcm design and process analysis. Potential

research areas can be identified where the system model is very sensitive to a particu-

lar parameter. The settling characteristics of the digested sludge serves as an csamplc

t o illustrate the importance of the sensitivity analysis.

3) Reliability analysis: The reliability of the systcm designed based on the optimization of

a comprehensive model can be evaluated. Design safety factors on particular unit

processes can be determined more rationally when dinerent influent conditions nre

imposed on the system designed based on the average flow and are subject to the con-

s t ra in t t ha t the efRuent water quality standards have to be met. This information

helps to establish guidelines for practical wastewater treatment system design. Consid-

ering system cost and a flow safety factor as two objectives in wastewater t rea tment

system design is an alternative approach of analyzing the system reliability. This has

been illustrated using an example problem in Chapter 4.

4) Model verification: Realistic plant operating d a t a may be used in a given situation t o

determine the most appropriate process performance relationships. These models can

thcn be used in a realistic design condition. If the facility already esists, thcn this

information can be usctl in the comprehensive systcm model to idcntify cost-cflcctive

operation of a wnstewater treatnlent systrm.

5) Water quality rnanngolncnt: Tlic lnodcl call be uscd to gcncr:lte inforlnntlon tha t

relates the cost of a waste\vater treatment systcm to its waste removal elficicncy. Sucli

infornlation is useful in studies involving innovative water quality n~anngemcnt strn-

tegics. hlodificntions of the comprehensive system model may be necessary, however,

for specific situations (for example, if multiple pollutants are to be controlled).

APPENDIX A

COST FUNCTIONS OF UNIT PROCESSES

Five sets of cost functions representing three sources of cost information were com-

pared in this study. Table A . l summarizes the cost functions studied and the sources of the

information (see Section 2.4 for a discussion of this cost information).

In Figures A.l t o A.9, capital costs are expressed in 1971 dollars. The USEPA National

Average Wastewater Treatment Plant Index is used to convert costs to this common basis.

Middleton and Lawrence, CAPDET, and Rossman all developed their cost functions

based on the information furnished by Patterson and Banker. However, these functions

vary considerably in the degree of complexity. The function tha t is the simplest among the

three was selected for use in the study if no significant difference is observed among the

predictions of these three sets of functions. Cost functions incorporated into the comprehen-

sive system model are summarized in Table 2.4.

Table A.l - Summary or Cost Information

Cost Functions Source

Smith (1368) Logan e l a!. (1962) Swanson (1066)

Middleton k Lawrence (1975) Patterson 6. Banker (1971)

Dick e l a!. (1378) Patterson 6. Banker (1971) Metcalf 6. Eddy, Inc. (1975) Ettlich (1377)

CAPDET (1378) Pattcrson A l Brnkcr (1371)

Rossman (1.379) Patterson IC. Banker (1371) Ettlich (13771

Air Flow RaTe, m3/minute

10' '103 1 o4 Blower Capacity, rn3/minure

Figure A.4- Cost Functions for Activated Sludge Aeration

l o 2 1 O2 10" i o4

Firm Pumping C a p a c i t y ; m3/hour

1 0 3

(J?

-, -3 - - 0 - u

.3 10' 0 4

- t

C/) 0

10' - u t . - Q C, 0

1 o0

Figure A.6- Cost Functions for Iiecirculation Pumping

. : . . . , , , . , , l l . l , , , . . , . ,

Smirn - - ;\AiddieTon

Click $ 1 (71. I -I Rossm a n

/

- / :--<

_-----

r

! . . , . . * ! I I I . , l , 0 < a , . ,

10' 1 02 1 o3 1 o4 Firm Pumping C a p a c i t y , m3/hour

vCj T c0 1 a0 T . . 10 1 ....... : : ... ill,, , , ,

10 1 /-

/ 0 L1 -c -. -t-

'2 -

- ' iCT C? 0 fl i - w 0

: ?(,? L.

I Lv .

uDUJsSO& - - - -- -. a 13GdV3 0 -.

.!L> !? XalC - :-:

I. d

1 ua]alpp!v\; -- -. - -.

I: Ul'urS - u, t-

.., , . , . , . , . I.., I.. , . - vOT

Figure A.8- Cost Functions tor .,\nnerobic Digester

cn ~ m i t n c - - UiddleTon r - - - , , - 3icK P! t71. - G CAPDET -0 . . - . . - - Rossman

,? LO3 : 0

1 4

.. +- I/) C

102 : - - G +- . - E G 0

10' I I ! ! ! ! I I . . . ! , , I , . , . , . 1 O2 1 o3 1 o4 1 O5

Diges re r Vo iu rne , m3

APPENDIX B

ESTIMATING OPERATION COST FOR SLUDGE LANDFILL

T h e following developnient is based on Figure 9-1 on Page 9-4 in the USEPA Process

Design Manual - Sludge Treatment and Disposal.

Let 1V, = wet tons of sludge landfilled per day,

0"dC = annual operation cost for sludge landfill.

According t o Figure 9-1,

Since the labor ra te is 6.25 dollars/hr in Figure 9-1, the annual manhours, O H R S , can

be calculated as

Using equation (2.77),

APPENDIX C

ANALYSIS PROGRAM FOR DESIGN OF WASTEWATER TREATMENT SYSTEMS

, T h e analysis program can be used to determine a complete system design for the base

wastewater treatment system (Figure 2.1) and two variations of the base system (Figures 4.2

and 4.7). Two input d a t a Gles are necessary to run this program. T h e Grst file contains the

parameters in the model, and the second the decision variables. Specifications of these two

files are described below.

Model parameters include the influent wastewater characteristics, effluent water quality

standards, and parameters for process design and economic analysis. T h e input order of

these parameters in the d a t a file is shown in Table C.1.

Table C . l - Input Da ta to the Analysis Program : Model Parameters

Card No. Comment

1 Influent Characteristics 2 Emuen t Water Quality Standards

2 3 Parameters

T h e influent characteristics are the design flowrate (m3/hr), the soluble BOD, concen-

tration (mg/l), and the concentrations of active biomass, volatile biodegradable solids, vola-

tile inert solids, and fixed solids (a11 in mg/l). The second card specifies, in order, the

emuent BOD, and the total suspended solids standards (both in mg/l). Table C.2 lists the

parameters used for process design and cost calculations according to their input order.

Free format input is used. An example input file looks like this:

T h e decision variables selected in this study were summarized in Table 2.5. T h e values

of these variables are specified in the second input d a t a Ele to run the analysis program.

Table C.3 summarizes the information requirements of this file.

An example input file containing the decision variables is shown below:

INITIAL DESIGN FOR BASE SYSTEM 0 0 1 0 1.50000 2.00000 .150000 0.100000 1 .booooo 30.0000 15.0000 0.5000000 10.00000

Three files are produced from running the analysis program. The first file contains the

detailcd design information for the specified flowchart and the values of the decision vari-

ables. The itemized costs for the unit processes included in the flowchart are summarized in

a second output file. The third file has the values of the variables tha t are in the GRG

optimization model.

The analysis program is listed on the next few pages.

Table C.2 - Summary of Parameters in the System Model

-- Names ilrnitsl Vallle Index

Economic Data:

Capital Recovery Factor Base (1971) Cost Index Cost Index for 1980 Operating/XIaintenance Wages (dollars/hr) Land Cost, CL (dollars/acre) Electricity Cost (dollars/kWhr) Pumping Head, 1-1 (meters) Pumping Efficiency, E

P

Primary Sedimentation:

Consbant in Voshel-Sak %,lodel, o,

Constant in Voshel-Sak Pclodel, v 2 Constant in Voshel-Sak hlodel, L!,

Sludge Settling Characteristics:

ThickeningConstant, a, Thickening Constant, a, Thickening Constant, a2 Thickening Constant, n, Thickening Constant, n,

Activated Sludge Kinetics:

Growth Yield Coeficient, y (g cell/g BOD6)

Half-Velocity Constant, K, (g BOD^/^^) Maximum Specific LTt.ilization Cocfl., k (day-')

Endogencous Decny Coelficicnt, b (dny-') Fraction of cells Degradable, f d Conversion (g BODL/g cell) Conversion (g BODL/g BOD,)

Secondary Sed irncnt:ltion:

Constant in C l ~ a p r r ~ a n h l o d ~ l , c , Constant in Chapman hlodcl, c 2 Constant in Chapman l lodcl , c ,

Aeration:

Alpha Factor in Aeration Beta Factor in Aeration

DO Concentration in :\ernton Tank . DO (-g/rr13)

Table C.2 (continued)

DO Saturation Concentration, C, (g/m3)

Temperature of hlised Liquor, TL (OC) Oxygen Transfer Efficiency, O T E Density of Air, pair (kg/m3) Weight Fraction of Oxygen in Air, y Temperature Coefficient Mixing Requirement, q (m3 air/m3/min)

Gravity Thickening:

TSS of Thickener Supernatant, 1\!,,, (kg/m3) 0.2 48

Anaerobic Digestion:

Coeff. for Digestion Rate Model CoeB. for Digestion Rate Model Temperature of Digester Influent, To (OC)

Methane Production (m3/kg BODL)

Average Ambient Temperature, T , (OC) Efficiency of Heat Exchanger, e

Heat Conduction Coefficient, U (W/~"OC) Outside Surface Area and Volume Ratio for Digester, a Worth of Digester Gas (dollars/therm) Soluble BOD:, in Digester Supernatant, S12 (g/m3) Factor Accounting For Effect of Rising G3s

on Thickening in Secondary Digester, 6 Thickening Constant for Digested Sludge, ad

Thickening Constant for Digested Sludge, nd

TSS of Digester Supernatant, Mt13 (kg/m3) Height of Digester (m)

Vacuum Filtration:

Coefi. for Calculating Filtcr Yield Form Time per Cycle Time, x Pressure Applied on Vacun~n Filter, P (Nt /m2) Viscosity of Filtrate, p (Nt-sec/rn2) Cycle Time, 1 , (min)

Specific Resistance of Sludge, r, (m/kg)

TSS of Filtrate, hfl16 (kg/m3) Unit Conversion Factor

Table C.3 - Input Da ta bo the Analysis Program : Decision Variables

Card KO. ( 'ornmcnls

1 Title 2 O u t p u t pr in t level.

=O : only the final results a rc printed =1 : design of every iteration is printed

3 Process flowchart, 0. 1. 0. : Figure 1.1 1. 1. 0. : Figure 4.2 0. 0. 1. : Figure 4.7

4 Pr imary clarifier overflow rate (m/hr) , delete this card if primary clarifier is not in the system (Figure 4.2)

5 Sludge age, ec (days), Activated sludge recycle ratio 6 Hydraulic retention time (days) 7 Gravity thickener solids loading (kg/m2/hr) , delete this card

if waste activated sludge is recycled to primary settling tank (Figure 4.7)

8 Digestion temperature (OC), Solids retention time, e,, (days)

9 Secondary digester solids loading (kg/m"hr) 10 Vacuum filter filter yield (kg/m2/hr)

Sep 16 13:25 1984 DESIGN Page 1

PROGRAM MAIN (PAR. DECVAR. 0UTPUT.DETAIL. COST.GRGDATA.RCYCLE.

+ 'WE~=PAR,

TAPE~ZDECVAR.

T~PE~-DETAIL,

+ WEB- GHGDATA, TPPE9=COST,WE7=RCYCLE)

COMM

ON/STATE/'VIN,VCIIT,VSIDE,ARY1,A

RY2,ITE

COMMON/PARVAR N

. V . X

CO

MM

ON

;'C

ON

T~

?L;;

~R

I~~

T,

Bl, B2.83

COMMON.'DPST/OR,IUNITl

COMT.ON;'DAS,/SRT,HIIT,ASFtR, IllNIT2. BODSTD. TssSTD

COmlON/BLEm;AC, NC

COMMCN,/DGT.'SLGT.

IWI T3

COMMON/DPAI.U?D/SRTD,

TE.9

COMMCN/DSAND,!SLSAND. I

UNIT

^ COI4MON./DM,/Y

I ELD

REAL VIN(7) ,VOUT (7) ,VSICE (7) ,U(100) ,V(20) ,X(100)

REAL ARYl 7 ARY2 7

REAL VOLD 171

TEST 171 , INFLOW (7)

INTEGER LPRINT

REAL NC

. CHARACTER* 1 TITLE (GO)

C

READ(4,'

(INFLOW(I),I=1,6)

INFLOW (7 =INFLOW(3) +INFLOW (4) tINFLOW(5) +INFLOW(6)

READ [4, *I BODS

TD, TSSSTD

READ 4, *

(U(1). I=l, 80)

- L.

READ (5.90) TITLE

90

FORMAT (BOAl)

READ 15. *l LPRINT

READ(S;*~B~,B~,B~

IF(Bl.EQ.1.0) GOT0 11

READ 5

11

READ I5::Ig:T.ASRR

L.

WRITE (6.99)

99

FOWAT(2X.104 (

I*

'))


WRITE (6,100) TITLE

100

FORMAT (/

, 15X. 80A1, /)

WRITE (6.99)

C

ITE=O

DO 20 I=1.7

VIN (I) =o. 0

VOLD (I) =INFLOW (I)

20

CONTINUE

L. 32

CALL MB (VIN,INFLOW)

IF(ITE.EQ.0) GOT0 26

DO 21 I=1.7

TEST (I) - (VI

N (I -VOD (I)

) IN (I)

IF (A

Bs (TEsT(1) .GT.l.E-6f"TiIEN

IF(ITE.EQ.40 1 TH

EN

PRINT *

. 'MAXIMUM NUMBER OF ITERATION 1401

+ REACHED.FCR RECYCLE STREAM MASS

BA

LA

IJ~E

''

STOP

ELSE

- ITE=ITE+l

CALL M

PY (VIN, VOLD)

GO TO 27

ENDIF

ENDIF

CONTINUE

LPRINT=LPRINT+l

IF ILPRINT. EO. 11 THEN

~TE=ITE+~-

' GO TO 28

ELSE

GO TO 40

ENDIF

ITE=ITE+l

FO.PMAT (E14.6)

WRITE(~,IIO) ITE

FORMAT (//

/, ZOX, 'ITERATION', IS,///)

CALL BRANCH (Bl.VIN.ARY1)

IF(B~.EQ.I.O) THEN

CALL AS

DO 51 I=1.7

ARY1 (I) =O.O

VSIDE (I) =O. 0

CONTINUE

VOUT (1) =O .O

GOT0 52

EmIF

CALL PET

CALL KB (ARY 1, VOUT)

CALL AS

- ~p

IF (B3. EQ. 1.0 .AND. B2. EQ.0 .O) THEN

DO 6

1 I=3,7

VIN(I)=VIN(I)*l.E-3

Ssp 16 13:25 1984 DESIGN Page 3

Sol' Ib 13:25 1984 DESIGN Page 4

6 1

CONTINUE

GOT0 33

ENDIF

CALL BRANCH B2,VSIDE.VOUT)

CALL BRANM[B3,VIN,ARYl)

5a

CALL SLMIX

CALL GT

CALL UJPY (VOUT, VINj

33

CALLPAND

CALL COPY (VOUT, VIN)

CALL smn

CALL MB (ARY1, VSIDE

CALL V

F CALL MB (ARY1, VSIDE

C

DO 35 I=3,7

VIN(1) =VIN(I) *l.E3

35

CONTINUE

c~

GO TO 31

C 40 CALL OBJ

C

IF (81. EQ. 1.0. OR. B3. EQ. 1 .O) THEN

WAR-51

ELSE

WAR=64

END IF

-

---

WRITE (8,120) (K,X(K) ,K=l.NVAR)

120 FORHAT(3X,I3,4X,E20.lO)

C

STOP

END

SUBROUTINE PST

COM.ION/STATE/VIN,VOUT,VSIDE,ARY~,ARY~,ITE

COmON/PARVAR/U,V.X

COMMCN/CONTRL/LPRIIJT. 81. BZ.B3

. .

CO~ON~DPST~OR,

IUNI TI.

REAL VIN(7) ,VOUT (7) ,VSICE (7) ,ARY1(7) ,ARY2 (7) ,U(100) ,V(20) ,X(100)

REAL W

1

- L

IF (VIN (1) . EQ. 0 .O) RETURN

c L.

XPl=l.E3*V(6) /VIN(7)

C C USE NEWTON'S KETHOD TO SOLVE FOR PRIMARY SLUDGE FLOWRATE:

C

N= 1

VSIDE (l)=l.E-3*VIW(l)

21

VOUT (1) =VIN (1) -VSIDE (1)

FzVIN(1) -VOUT 1) *R1-KPl*VSIDE (1 *

* ((V(5)

-l.)/V(5))

+ * (

VOUT (1) *U (16) /OR, **

(1 ./V (5) 1

IF (ABS (F) .LE. l.E-6) COT0 22

FPRIME-R1--KP1*

(U (16) *VOUT (1) /OR/'VSIDE (1) )

** (1

. /V (5)

) +

* (1. -VIN (1) /V (5) /VOUT (1) )

VSIDE (1) =VSIDE (1) -F/FPRIME

N=N+l

IF(N.GE.50)

THEN

PRINT *,'MN(IWJM NUMBFR OF ITERATION REACIED IN PRIMAhY

+ SETTLING TANK DESIGN

STOP

22

IF(VSIDEI(l).LT.O.)

THEN

PRINT *

, FAILED TO FIND A

FYIBLE SOLUTION IN

+PRIMARY SETTLING TANK DESIGN

STOP

DO 1 I=3,6

VOUT (I) =VIN(I) *R1

VSIDE (I) =VIN(I) *RATIO

1

CONTINUE

C

IF (LPRINT.LT. 1) GOT0 100

WRITE 16.131 0R*24. .APST.R*100.

13

TOR MA

^ (ix,' **PRIMARY SETTLING y

DESIY-- I

,//;

+

OVERFLOW RATE

= ,F12.5,

M/DAY, ,/

, +

' SURFACE

AREA ='.F12.5.'

SO M

a./.

-

-----

+I

SOLIDS REMOVAL

='

.. ,

WRITE (6.14)

:S(C,/CU M) ',3X,'p;IA/C/CU M) ',3X,

,3Xa1MI(G,/CU M) ,SX,'MT(G/'CU

Ed)')

FORMAT l2X. ' IhFLUENT' .3X.F12.5.5 f lX.Fl2.51 .lX.F13.51

FORMAT 2x1 '

EFFLWNT~;~X;F~~.~;~(~X;F~~.~~

;lX;~13.51

FORMATI2X. 'UNDERFLOW',2X.Fl2.5,5(lX,F11.5), lX.Fl3.5)

X 1) =VIN (1) /60.0

X 11 VOUT (1) 60.

x I 121 1

vsIDE

Sep 16 13:25

1984 DESIGN Page 5

X 6 =VIN 6

X [PI =VIN [21

IF(B3.NE.1,)THEN

X(64) :-VSIDE (7) *l.E-.5

ELSE

x-56 =VSIDE 7 *l.E-5

XI511 =VSIDE 161 *l .E- 3

ENDIF

C 100

RETVRN

END

C

SUBROUTINE AS

COMMON/STATE/VIN,VOUT,VSIDE.ARY1,ARY2,ITE

COMMON/PARVAR/U,V,X

COMM

ON/C

ONTRL/LPRINT.Bl.B2.B3

COMMON)DAS/SR~,HRT.ASRR; IUTJIT~,

BODSTD, TSSSTD

REAL VIN (7) ,VOUT (7) ,VSIDE 7 ,ARY1(7) ,MY2 (7) ,U(100)

REAL V (20) ,X (100) . SN

(3) , F (31

REAL N

u

C C C AERATION:

- L IF (IUNIT2. EQ.0 .A

ND

. ITE . E

Q. 1) HRT=HRT/24.

C

. -

IF (AFw . LT. u (45) )

TIIEN

PRINT *,'MIXING REQUIREMENT CONTROLS OXYGEN D

EMANDn

ENDIF

C

IF (LPRINT. LT. 1)

GOT0 150

rXR*l.E2,VAT,HRTh24.,0RMT,AFR

WRITE16,lOO) SRT,A,

loo

SO~YA~(I/,~X,'**ACTIVATSD

SLUDG~: SYSTE~ DESIGN--',//,

+ SLLQGE

AGE

= .F12.5,

CAYS, ,/,

+'

SLUDGE R

ECYCLE =',F12.5,' PERCENT,',//,

+2X.'fA\. AERATION

Tl.

NK

--I.

//.

+'

RETENTION TIME =',F12.5,'

HO

UR

S;",

/, +

' OXYGEN REQUIREENT=:,F12.5.' KG/D,

,/,

t '

AIR

FLOWRATE

= .F12.5.'

CU

M/MIN.'./\

.r

,

WRITE (6,110)

110

FORMAT (15X. 'Q (N M,"HR) ' ,

4X, 'S (G/N M) ' ,3X, 'MA(G/N M) ' ,3X,

Ssp 16 13:25 1984 DESIGN P

age 6

+'lm(GjN

WRITE 6.120 M) i j3x

, ARYl(1)

'MI (G/N

,1=1,71

M ' ,3X, 'MF (G/N M) ' ,5X, 'MT(C/CU

M) '

)

WRITE 6,130 VIN (I), I=1,7)

120

FORMAT 2X, 'IMLUENT1, 3X,F12.5,5 lX,F11.5 , lX.Fl3.5

130

FOMT(2X. 'EFFLUENT', 3X.Fl2.5.5~lX.Fl2.5~ ,lX,F13.51

DO i

s5 r=i,i'

x (I) =

my1 (I)

155

CONTINUE

C

C SECONDARY SETTLING:

C 160

YY=VIN 151 /VIN 131

ZZ=VIN 6 /VIN 3

XX=VIN 7 /VIN 3

C

Q2=ARY1(1) '

~

xA2=kRY1(3)

C C SETTING UP THE

COEFFICIENTS FOR SIMULTANEOUS EQUATIONS:

NW=U 1201

ml=dRyi (3) *~.oE-3

Al2=-1.0/Xx*l.OE-3

m3=VIN(l) /XX*l.OE-3

RHSl= VIN (3) *VAT/SRT/24.O+Q2*XAZ+ASRR*Q2 *MY1 (3) ) *l

.~2l=v17)

RHS2-ARYl(7) *l.OE-3

A31=U(32) *Q2* (1. OtASRR) (-1.0)

A32~1.0

RHS3=-U(30) tU(31) *VIN (7)

- C; C PROVIDING STARTING V

ALUES FOR VARIABLES:

IAREA=l

N=3

C

CALL FUNC (N, SN,F ,NW,Al1,Al2,A13,~21,A31.~32,

RHs~,RHs~,RHs~)

- L CALL QUASI (N, SN,F,NW,~l,Al2,A13,~1,A31,A32

,lU!Sl,

tRHS2.RHS3, ITEQN)

C

IFfITEON.GE.50) THEN

IF (IAR~A.GT.

20) TXEN

PRlNT *,

'HAXII4I.N NUMBER F)F ITERATIOIJS REkCIiED IN

+ ACTIVATED SLUDGE DESIGN

STOP

ELSE

IAREA=IAREA+l

GOT0 180

ENDIF

ENDIF

C



IF(SN(1) ;LE.O.O.OR.SN(2) .LE.O.O.OR.SN(3) .LE.O.O)

THEN

PRINT *

, FAILED TO FIND,A FEASIBLE SOLUTION IN

+ACTIVATED SLUDGE DESIGN

STOP

ENDIF

- X 19 =ZZ

X 12-71 =AE'R/60.

ELSE

I.

ARYl 1 =SN 1

VOUT [7{ =SN [2{

M-SN (3)

-

C

C ACTIVATED SLUDGE SEPARATION:

C - L 700

VOUT (1) =ASRRhQ2

VIN (1) =ARY1(1) -VOUT (1)

C

L

IF (EFFBOD. GT. BODSTD) THEN

PRINT *

,'WING-BODS STANDARD VIOLATED1

EMIIF

ARYl I =

k~1

I *l.OE-3

VOUT [I{ =MY1 [I{

VIN (1) =MY1 (1)

IF(EFFTSS .?T. TSSSTD) THEN

PRINT

*, WARNING-TSS STANDARD VIOLATED'

ENDIF

WRITE(6.2001AF.OFRATE.SLFST.EFFBOD.EFFTSS

. ,

60

CON~I~E

vour (2) :=ARY1(2)

VIN (2) :MY1 (2)

C

IF (LPRINT. LT. 1) CO TO 20000

WRITE (6,800)

800

FORMAT (/

, 2X. ' (C) . ACTIVATED SLUDGE SEPARATION- - ' . j

)

WRITE (6.9001

t '

:SURFACE '

A~

EA

=',F12.5, SQ M, I

,/

,

t '

OVERFLOW RATE =',F12.5,' M/DAY,',/,

+I

SOLIDS ~ar

n~rr

; ='.~'12.5.' KGIHRISO M.'./.

. .,.

+I

EFFLUENT BODS

= : 1~

12.5;

I GI.~)CU' M:

I

,

+'

EFFLUENT TSS

= ,F12.5, ' C;I.l/CU M. ';{

goo

FORMA? (i~

x,

'9 (CU MIHR) ,4x, @SJC/CU M)

I, 2x. IIIA(~G/CU M) I

, 2x.

+ 'MD (KG/CU M)

,2X, 'MI (KG/CU M) ,2X, 'ME (KG/CU M)

,

+4X, 'MT(KG/CU M) '

) WRITE (6.9101 (ARYllIl. I=1.7)

WRITE (6,300)

FORMAT(l5X. :Q(CU M'HR) ',4X, :S(G/CU M) ',3X, 'p(G/CU M) ',3X,

t1MD'(G/C7J

M) ,3X,'MI(G/CU

t4)

,3X,'MZ(G/CU M) ,SX,'MT(G/CU M)

WRITEf6.4001 (VINfI) ,I=1,7)

WRITE 16: 9201 ~VOUT (11 ;1=1: 71

WRITE 6 500 VO'JT

I WRITE 16: 6001 (ART1 [I{ ::I;: :{

FORMAT (2X. 'INFLUENT' .3X.Fl2.5.5(1X.F12.51.1X.F13.51

IF(Bl.NE.1.0) THEN

X 21 =VIN(l)/Q2*1.E2

X 22 =>.SR9tX(21) *1.E-2

X 1 28 I

=VIN(II

X (29) =VIN (7)

GOT0 20000

ELSE

X 16 =VIN(l) /Q2*1.E2

X[l7{=ASRR+X(16)*1.E-2



CALL MATVEC(N,H,Y,HY

C

COMPUTE TRA?iSPOSE (S (l*Nj) l H(N*N)

Y (N*l) :

SHY=O. 0

20000 RETURN

END

C

DO 35 I=l,N

SIfl=SKi+S (I) *HY (I)

CONTINUE

COMPUTE TRANSPOSE (S (1*N)

UPDATE :

DO 40 I=l,N

DO 40 J=l.N

SUBROUTINE OUASI (N.X. F.NC,All.Fd2.Al3 .AZl.A31.

- A~~;~S~;RHSZ.RHS~,ITE

- 'DIMENSION F 3) ,F;Y(II)

,H (3.3) ,HY(3) ,S (31 ,SH(3)

DIMENSION UI3.3) ,X(3) ,Y(3)

REAL NC

40

CONTINUE

- TIATE H (I, J) :

DO 10 I=l.N

C CONTINUE ITERATIONS:

' EZSE '

H(I,J)=O.O

ENDIF

CONTINUE

100

RETURN

END

C C SUBROUTINE 'MATVEC' PERFORMS THE POST-MULTIPLICATION OF

C

A MATRIX (N*N) BY A VECTOR (Ntl)

C

L

C COMPUTE X' (I) (THE NEW SOLUTION) :

15

CALLMATVEC(N,H,F,S)

DO 16 I=l,N

s (I) =-S (I)

16

CONTINUE

C

- SUBROUTINE MATVEC (N,A, B, C

DIMENSION A(3.3) ,B (3) ,C(3{

c

c TEST FOR CONVERGENCE:

~d i

J=~,N

C(I)=C(I) tA(1,J) *B(J)

2

CONTINUE

1

CONTINUE

DO 20 I=l,N

IF (I. EQ. 2) THEN

IF fABS IS fII I .LE

THEN

THEN

NS ~OP-NS~OP~

1

ENDIF

ELSE

IF (ABS (S (I) ) . LE

NSTOP=NSTOP +1

EhDIF

ENDIF

RETURN

END

C C SUBROUTINE V

ENT PERFORMS THE PRE-MULTIPLICATION OF

C

A MATRIX (N*N) BY A

VECTOR (l*lJ)

c -

SUBROUTINE VEI3MAT (N,X,Y, Z

- DIMENSION X (3) ,Y (3,3) , Z (31

-- - --

CONTINUE

c

UPDATE X (II :

DO 2< i

=1,~

x (I) =x (I) ts (I)

25

CONTINUE

C

Z (J) =Z (J) +X (I) *Y (I, J)

2

CONTINUE

1

CONTINUE

RETURN

C COMPUTE Y (I) :

C.UL FUNC (N,X.FNEW,tiC,Al1.A12,Al3,A21,A31,

tA32.RHSl.RIIS2.TUIS3)

DO 3

0 I=l.N

END

C

SUI?ROUTINE FUNC~,X,F,NC,A~~,A~~,A~~,A~~,A~~,A~~,

+WISl, RHS2, RIIS3)

REAL NC

- . . -

- - . -

Y I FNEW I F (I)

F I11 :FNEwIII

- '

DIMENSION X (3) ,F (3)

c

, .

30

~~NTINUE

C C UPDATE H (I, J) :

C

COMPUTE H (I, J) *Y (J) :



L.

SUBROUTINE SLGMIX

CO~ON/'STATE/VIN.VOUT,VSIDE,ARYl,AilY2.ITE

COMMON /P.WVAR ,'U .

V . X

, .

RVJ. NC

C

DO 1 11-3.7

VOUT (I) =VOUT(I) '1

.E-3

VSIDE (I) :VSIDE (I) '1. E-3

1

CONTINUE

C

IF (VOUT (1) . EQ. 0.0) T

IIEN

DO 11 I-2,7

VOUT(I)=O.O

11

CONTIhvE

GOT0 13

ENDIF

C

IF (VIN (1) . EQ . 0 . 0) THEN

DO 12 I=2.7

VIN (I) -0.0

12

CONTINUE

ErnIF

C 13

DO

2 I=1,7

TEMP (I) -VIN (I)

2

CONTINUE

- L

C THICKENING CIiARACTERISTICS OF CCMBIMD SLUDGE:

PRISLC-VOUT (1) 'VOUT (7)

FP PRISLGI PRISLCv'IEkP (1) *TEMP (7)

) AC-U(l7l -U 1181 *FPh*U1191

L

VIN (1) --VOUT (1)

t TEMP (1)

DO 10 1~2.7

VIN (I) = (TE:.P (I) *TEMP (1) tVOUT (I) *VOUT (1) )PIN (1)

10

CONTIIvVE

C

IF (LPRINT . LT .l) C3T3 1000

WRITE (6,100)

100

FORMATI//. 2X. ' **SLIIDGE .@LENDING--', /)

WRITE (6',1io)

110

FOWiT(I5X. '?(CiJ

HsM)

',d

.X, 'S,(G/CU M) ',2X, 'MA(fC/CU M) ',2X.

+ 'MD IKG.<CU MI ,2X, ' MI (KC; CU M) ,2X, 'M

F (KC/CU M) ,4X,

t'MT KG CU M 'I

MI~E 6,120 ?OUT I ,I:1,7

L'RITE 16,130] ITEW I1{,1=1,7I

WRITE 6,140 VIN (I). 1~1.7)

120

FORMAT 2X, 'PRIMARY' ,4X,F12.5,5 (1X.Fl2.5) ,1X,F13.5)

130

FORMAT 2X,'ACTIVATED:,2X,F12.5,5 1X.Fl2.5 ,1X,F13.5

140

FORMAT 2X. 'COMBINED ,2X,F12.5,5[1X,Fl2.5~

,LX,P13.5~

C

I IF(Bl.NE.1.0)

THEN

Xf30I =VINf11

X (24) =VIN (7)

ENDIF

1000 RETURN

END

L

SUBROUTINE GT

CO~ON/STATE/VIN,VOUT,VSIDE,ARY~,ARY~,ITE

COrnON/PbRVAR/U. V. X

CO~WON)CONTRL)LPR~NT,

~1,

B2, B3

COMMON/DGT,'SLGT,IUNIT3

COMMON,/BLEND/AC, NC

REAL VIN(7) ,VOUT (7) ,VSIDE (7) ,ARY1(7) ,ARY2 (7) .U(lOC)

, +v

(20) , x (100)

REAL NC,KC

10

C

0

L

KC= (AC* (NC-1 .O) )

** (

l.O/NC) *NC,/ (NC-1 .O)

VOUT (7) =SLGT* (KC/SLCT)*

(NC/ (NC-1.0) )

- L IF (VOUT(71 .LT.VIN[7) 1 THEN

SLCT-KC::NC/VIN

(7)

&* (NC-1.0)

PRINT

*, THICKENER DESIGN INFEASIBLE--INFLUENT

+SOLIDS CONCENTRATION GREATER THAN UNDEPSLOW SOLIDS

+ CONCENTW,TION FOR THE SOLIDS LOMINC SPECIFIED. -

---

t CHANGE SLUDGE THICKfNINC SFIIEME, OR IJSE,SLUCG

tE LOADING LESS T

i . SLCT.

KC/S0 M/HR

STOP

ENDIF

-VIN (1) *ARY2 (7) )

,/ (VOUT (7) -MY2 (7)

)

R-VGUT (7) /VIN (7)

ZSTAR-ARY2 (71 /VIN (7)

. .

DO 10 1=3,6 .

10

CONTINUE

L

IF (LPRINT. LT. 1) GOT0 1000

WRITE(6.100) SLCT*24. ,AG,PSR*l.E2

100

FORMATf//.2X.'**GRAVITY THICKENER DESIGN--'.//.

nrrr bWN 000

++ ++ WHHH.

uuu-unu- -mmm- ,-mmm

u. WHHII R- -N x 11 11 r - I! I! . x. rp. ' . 1. 1- . 4 r1r.I.I- NrN-

N. x - NN In' .In . . ' In In. In .-E !J'!J' - In- W-W

N\ - - xn . . . - d 'um - mro 1. 1 rqr 6s Ez NPN A - PI. . N R r - In. In - In- ?E ?: . cr. 2.- .-. F5;F 1- 1 xF a 'A

rqr wrw - X

W. ' R

In. In -C"-

3- -0 $ggmr <<x. - rrou-N WC~Z

ZZ-H mozm

WWH 11 R- x.x;.r ?.- ILII I!>II - - ZJZJL2 9e LLk L

moo SLY mro c urn I1 + In 4\D m NY

$5 i *nz ' In-

*

; " 2 n m 0 '

iU . , C"!" InIn - 55 11 rr NN . . InIn - . . FF -1 wr WW . .

666 NNN ... PC"? - - - E25 !.$ g z*u

5 -

6 6 6. NNN . . . "C"P --- 22: x.m. . -. -? >>--

>

Inm - 9 Y

EOZ &

H v

2 F2 -mo zg C, I, 4 - <- GF i- MrU Ez < Zidz" zG=i '5%-

2 mzm Ul- Z u4 m- %J

xi mr HW 9- 3 F m u w P m I I

m XXX

3- HPPW -I=

2s* u- rn - \ 3 5 - 4 -


VOUT (I) =VIN (I) *R

VSIDE (I) -VIN (I) *RATIO

10

CONTINUE

IF (LPRINT . LT .l) GOT0 1000

WRITE (6,100) YIELD,AV,PSR*l.E2

FORMAT (

/I, 2X, '**VACUUM FILTER DFSIGN--',//.

t

FILTER YIELD

= ,F12.5, KG/SQ,M/HR,',/,

+'

SURFACE AREA

=',F12.5,' SQ M, ,/

, +'

SOLIDS RECOVERY ='.F12.5.'

PERCENT.',/)

-. .

WRITE (6,110)

FORMAT (15X. 'Q (CU M

; HR) ' ,4X, 'S (G/CU M)

' ,2X, 'MA(KG/CU M) ' , 2X,

+'MD fKG/CU M

I ' .2X. 'MI fKG/N M1 ' .2X.

FOR

MA^ 2~,'~~~~~~~';3~,~12~5,5(lX,F12.5)

,lX,F13.5)

FORMAT 2X. 'UNDERFLOW' ,2X, F12.5.5 (lX.Fl2.5) , lX, F13.5)

FORMAT I 2X.

'SUPERNATANT',F12.5,5(lX,F12.5) ,lX,F13.5,/////)

L

IF Bl.NE.1.O)THEN

IF[B3.NE.l.) THEN

X (58) =VSIDE (1)

1030 RETURN

EM)

- L SUBROUTINE BRPAM (F.A.B)

REAL A (7) , B (7)

C -

DO 1 I-2,7

B (I) =A

(I)

1

COIiTINUE

C

RETURN

EID

C


C C

SUBROUTINE MB (A, B)

REAL, A (7) . B (

7)

c -

D=A (1) +B (1)

DO 1 I=2,6

A(1) =

(A(1) *A(I) +B(1) *B (I) ) /D

1

CONTINUE

(3) +A (4) +A (5) +A(6)

rn

L

RETURN

END

SUBROUTINE COPY (A, B)

REAL A(7) .B(7)

C

DO 1 I=1,7

B(1) =A(I)

1

CONTINUE

C

RETURN

END

C

SWROUTINE OBJ

COMMON/PARVAR/'U,V,X

COMMcjN/CoNTRL/LPRINT,Bl,B2,B3

REAL U (100) , V (20) , X (100)

REAL EEIETBEN

rn

L

WRITE (9.81)

81

FORMAT(/////,2X,'SUMMARY

OF COST ESTIMATES:',///)

WRITE (9.98)

98

~0RMAf(126.('*') ,

//)

WRITE (9,10000)

10000 FORMAT (37X. 'CAPITAL', IOX, 'OPERATION', 8X. 'MAINTEW2JCE'.

1

9X,'MATERIALa.13X,'POWER'I

.

.

WRITE (9,10001)

10001 FORMAT (34.7.. 'DOLLARS/YEAR'. 5X, 4

('DOLmS/YEAR1, 7X), /;)

WRITE (9.98)

- L IF(Bl.NE.1.0) THEN

APST=l. E2*X (9)

PSF=X (12)

nn nnn nnn nnn

mmnmonm ssmrm"Ir

mmmnm -l-lmmmmm nl"I'd 'dm

I1 11.

r, !J's m a. P 4P

0, 6g g ?P *-

I1 . I1 . 11 11 11 LP UP.. N ou OHNW4

0' WW4 mm . r4wa .JW LnLn *. *P m*m* 0. . m mm In- -m Ln

Sop 16 13:25 1984 DESIGN Page 21

L

CCSAND=2323.*VSAND**.59

IF (VSAND . GE .5678. ) THEN

COSAND=1.29*VSAND**.83

ELSE

IF(VSAND.GE.1968.) THEN

COSAND=14.*VSAND**.55

ELSE

COSAt0=192. 'VSAND*'

.2

O4SAND=113. ''.'SAND**.

21

ENDIF

IF (VSAND.GE. 2839.) TKEN

CSSAND=14.4*VSAND**.66

ELSE

CSSAND=142.*VSAND'*.37

'ENDIF

- L C VACUUM FILTER:

ELSE

-

IF (Q16X16 .GE .lo3 .)

TH

EN

C?-lVF=2O.'Q16X16**.63

ELSE

- ENDIF

ENDIF

C c RECIRCULATION PUMPING:

L

CCRP=2779.*QCYCLE**.53

COW=. 333*QCYCLE

W=

.

2375*QCYCLE

IF (QCYCLE . LT .158. ) EfEN

CSW=300.

ELSE

----

IF(QCYCLE.LT.631.) THEN

CSP.P=40.57*QCYCLE'*.52

- ELSE

IF (QCYCLE . LT .1580. ) THEN

CSW=5.97*QCYCLE**.87

ELSE

CSRP=2.540'QCYCLE

EhDIF

EFDIF


L

TO~=CCPST+CCPSP+CCATtC(3DAA+CCFST+CCKSP~CCRP

t

tCC

GT

t~A

ND

tCC

SA

ND

tCC

VF

tCC

SD

t

+COPST+~ST~U)PSP+WSP~CCIDAA+U~~+~FST

t

+OFST+ CORSPt~SPtCOGTt

CM

CT

t

+COPAND+ W

AN

D+

COSEUTD tC2:SA.m

t

tCO

VF

tCM

VF

tCO

MS

DtC

OB

Pt~

P

t

tCSPSTtCSPSPtCSFST+CSRSP+CSST+CSPF~JD+CSSAND+CSW

t

tcsw


.,--.-

--

~

~~

.-

j I

WRITE 9,1009 CCVF,CCVF,U~'+~.CSVE. .

WRITE 9,1013 CCRP,C3IiP,~GP,CSRP,CFW

WRITE 9,1010 CCSD, COMSD

WRITE 9,1011 NETBEN

WRITE 9.981

FORMAT 5X,'PRIWY SETTLING TANK',lOX.FB.O,3 10X.FB.0

/ FORMAT ISX.

'PRIWY SLUDGE PUMPING1 ,9X.F8.0,4llOX,F8.0] :/I

-

- --

FORMAT 15x1 'AERATIOti TANK' ,18X, E8 .O, /)

5X. 'DIFFUSED AIR MRATION' , fOX.FB.O.2 (10X.FB .O) ,/

) 5X. ' SECOhiARY SETTLING TANK ,8X,F8 .0,3 (10X,F8 .O) ,

) 5X. *RECYCLE SLUDGE PUHPING', 9X.EB.O.I (lOX,EB.O), /(

5x. 'cmv~n

THICKENER . ~

~x,FB.o,

3 (~ox,F~.o)

.I)

SX,'PRIMARY ANAEROBIC DIGESTER',SX.FB.O,

+3(10X,FB.O) ,/)

FORMAT(SX.'SECONDAilY ANAEROBIC DIGESTER',

-.

+~x,F~.o,~(~ox,F~.o),

')

ILTER',l~X.F8.~.3(lOX,FB.O)~/)

STRm PUMPING .9X,

UDGE DISPOSAL' ,lOX,F8.O.lOX,F8.O./)

OF

tET

ENERGY FROM E

TIi

AN

E

t=

' F8.O.' DOLLAR

+s,&R.

' /

j)

1012 FOFT(//i20X,'TOTAL SYSTEM COST

+=

,F8.0, DOLLARS/-.

')

C

RETURN

END

APPENDIX D

GRG MODEL FOR BASE SYSTEM DESIGN OPTIMIZATION

T h e GRG optimization model for the base treatment system (Figure 2.1) has 64 vari-

ables and 59 functions (constraints plus objective function). The file containing all functions

in the model is listed as GCObIP8 on the followiug pages. A list of the variables in the

model is provided in Table D.1. The reader is refered to Chapter 2 for the notation used in

this table.

Table D . l - Summary of Model Variables : Base System

Varisble Index Unit Meaning

1 m3/min Q 1/60

2 d m 3 1

3 d m 3 M4 1

4 d m 3 Md 1

5 g/m3 M, 1

6 . g/m3 4 1

7 d m 3 Mt 1 8 m3/hr P

9 1 OOmZ Ap/lOO 10 -- Mt 3 J 4 1

11 m3/min QJ60 12 m3/hr Q s 13 days 0 c 14 days 0 15 1000m3 v/lO00 16 kg/m3 nia

17 d l " 3 S3

18 -- hi, 3 / l i 1 4 3

10 -- 1Cl13/1\la3 20 r 2 1 -- 100 w

22 -- r S w

23 . d m 3 hi4 4

24 kg/m3 hi, 6

25 100rn2 A1/lOO

26 g/m3 s 27 m3/sec Q J G O

Table D.1 (continued)

Vnrinblc Index Unit L,l.lrariino,

rn3/hr

rn3/hr

kg/m3

kg/m3

kg/rn3 -- "C

days

1000rn3

day-'

106kH'hr/year

kg/rn3

kg/m3

rn3/hr

1 0 ~ k l V h t / ~ e a r

kg/rn2/hr

100rn'

rn3/hr

rn3/hr

kg/rn3 --

T h e constraints in the GRG model and their corresponding equation numbers (see

Chap,tcr 2 ) are summarized in Table D.2. It is convenient to define some "secondary

vsriables" in constructing the GRG model to avoid repetitive computation. -4s shown in

Tablc D.2, several secondary variables (variable No.23, 38, 39, 40, 46, :ind 55) are defined by

the constraints which are not described bu t are derived from design equations presented in

Chapter 2 .

Table D.2 - Description of the Constraints in the GRG Model

Constraint No. in GRG Model Corresponding Equation No. from Chapter 2

definition of X(29)' 2.42 2.44 2.45 2.11

definition of ~ ( 3 9 ) '

definition of X(.IO)'

definition of ~ ( 3 8 ) ' 2.48

definition of S(.IB)* 2.58 2.49 2.52

2.54 - 2.56 2.53 & 2.57

2.60 2.59 2.58 2.61

Table D.2 (continued)

~ o n s t r m \ ! o . in CiRG kyodel Correspondiri~ Eq~lntion No. from i:hnpcer 2

40 definition of ~ ( 5 5 ) . 4 1 2.67 42 5.68 43 2.64 & 2.65 4 4 2.66 45 2.7 1 46 2.72 47 2.73 48 5.74 49 2.75 50 2.76 5 1 2.77 52 2.40 53 2.38 54 2.39 55 2.10 5 6 5.59 57 2.30 58 2.56

: S denotes the variables in the GRG model. See Table D.1.

The initial solution to the model and the control parameters for the optimizaticn (see

Section 3.2.1) are specified in another input da ta file. .4n example is of this file is also listed

in this Appenuis under the name GRGDXTA. The user's manual for GRG2 should be con-

sulted for the details.

Aug 22 16:14 1984 CCOMP8 P

age 1

PROGRAM HATN(INPUT,PAR, OUTPUT,LASTVAR, TAPE7, WEB,

+ TAPE5=INPUT, TAPE6=OUTPUT, TAPE4=LASTVAR,

+ TAPE9=PAR)

DIMENSION Zf2OOOOI

COrnON Z

DATA NCORE/20000/

CALL CRG (Z,NCORE)

END

L

C SUBROUTINE CCOMP:

C

SUBROUTINE GCOMP IG. XI

50

EPBND=l.E-6

DO 100 I

=1,64

IF (X (I) . LT . E

PBND) THEN

DO 200 5

~1.59

C(J) =1.~30.

200

CONTINUE

RETURN

ENDIF

100

CONTINUE

L

C PRIMARY SETTLING

TAN

K DESIGX:

- L

C ACTIVATED SLUDGE DESISN:

C

*X(23)+

+ +

X(10))

Aug 22 16:14 1984 CCOMP8 P

age 2

G 6 =X 14

- (l.E3*X(15))/(6O.*X(11))/24.

G 171 =

X 1171 -U (24) * (

1. +U (26) *X (13) ) / (X (13) * (

U (23) *U (25)

+ -U(26)) -1.)

G(8) =X (14) (l.E3*X(16) ) /X (13) -

U (23) *X(26) : (1. tU(26)

t

X(13)

+ +U (80) *U (281 *V (9)

+ (1. E3*X (15)

(l.E3*X(16) ) ,3

: (13)

G (16) =X (26) +X(171 -X (2) -V (8) *X (4) *X (10)

C

C SLUDGE MIXING:

- c GRAVITY THICKENER DESIGN:

C

C

C PRIMARY DIGESTER DESIGN:

C

G (29) =

X (44) -U(53) *EXP (ALOG (10 .)

*lo. /3. * (

U (54) -

+ + +

C

C SECONDARY DIGESTER DESIGN:

C

noon - PPPP PWPJP -

- - Vlm Vl aw N - - I1 I/ ,I

XXXX - mmNm WNWI- - I I *I CCX- -r NI-N. 04OM -N * 9- * mc x F; -z -m P cy :: NX * b-- X -01 -

I4 It 11 11 XXXX - mmmm 4mww -

- - - *I- mmn. oomm . . OW

Aug 22 16: 14 1984 CCOMP8 Page 5

C C RECIRCULATION PUMPING:

-.

IF(QcIcLE.L?.~~~.)

THEN

csRP=300.

ELSE

---

-

IF (QCYCLE.LT.631.) THEN

CSRP=40.57*QCYCLE**.52

- ELSE

IF (QCICLE . LT. 1580. )

THEN

CSRP=5.97*OCYCLE**.87

I.

C4%T=824.

1. E2"

(34) ) ".

77

IF((l.E2*X[34)) .GE.279.) THEN

COGT=17.15* fl.E2*X134~1**0.6

. ..

ELSE

COGT=92.45*(1.E2*X(34 )**.3

C?-IGT-106. (1 E2.X

(3%) 1 **

.I4

ENDIF

CSGT=8.62* (1 .EZ*X(34))t*. 76

C c PRIMARY DIGESTER:

- L CBANDz2323. (l.E3*X(43)

** .5

9

IF((lE3*X(43)) .GE.16781 THEN

COPAND=1.29* 11.E3'X 143) 1 *'.El3

CMPAND=.83* (i.~3*~(43)

I;* .

82

ELSE

IF((l.E3*X(43)) .GE.1968.) THEN

COPAND=14.* 1 E3*X 4

3

WAND-8.5* !l:E3*X14;!

1 :::;;

,

..

ELSE

CGPAND=192.* 1 E3*X 43

*' 2

CMPAND=113. * [

l:E3*X[431{ **:21

ENDIF

ENDIF

IF((l.E3*X(43)) .GE.2839.) THEN

CSPAND=14.4* (1. E3*X(43) ) *

* .66

ELSE

~~~%D=111142.

(l.E3'X(43))

** .3

7

ENDIF

C

C SECONDARY DIGESTER:

-

Aug 22 16:14 1984 CCOMP8 Page 6

cCsAND=2323.*vsAND**.59

IF (VSAND. GE ,5678. ) THEN

COSAND=l. 29*VSAND** .83

ELSE

IF (VSAND.GE.1968.) THEN

COsAND=14.*vsAND**.55

CMSAND=8.5*VSAND* .55

ELSE

ENDI

F IF(VSAND.GE.2839.) THEN

CSSAND=14.4*VSAND**.66

ELSE

CSSAND=142.*VSAND**.37

ENDIF

C

C VACUUM FILTER :

- L Q16X16=X (59) *X (60)

CCVF=29180. *X(57) **.71

CSVF=230.*Q16X16**.71+182.*Q16X16**.86

COVF=107.55*Q16X16**.58

IF (Q16X16. GE ,519. ) THEN

E-=5.57iQ16X16**.84

IF (Q16X16. GE .103. ) THEN

CMVF=20.*Q16X16**.63

ELSE

+

tu(1j *

l +

3.6~3/1.'0$5~~(49) '

' '

' '

* '

RETURN

END

- L C SUBROUTINE REPORT

n

L

SUEROUTINE REPORT (G.X.H.N. CON, VAR, XO)

DIMENSION

X (N) .G (M) ,CON (M) ,VAR (N) , xo (N)

COMMON/INITBK,/INIT

IF (INIT. EQ .I) RETURN

WRITE (4.10)

(I,X(I) , I=l,N)

10

FORMAT(3X. 13,4X,E20. 10)

RETURN

Aug 29 14:54 1984 CRGDATA Paga 1

Aug 29 14:54 1984 CROATA Page 2

64

59

NA

ME

EXAMPLE DATA PILE FOR GRC2

ROU

1. OGCOE - 1

END

IN1

SEPARATE

1

62

.5294457576E+02

63

.2574041692E101

64

.7736809844E100

END

ROW

E

1 55

L

56

58

END

PRI

IPR

1

END

LIM

NST

3

END

EPS

EPN

1.OOOOE-4

EPI

1.0000E-2

EPT

---

1.0000E -4

MET

FDC

::

A

MIN

END

GO

217

AFPENDIX E

IGGP MODEL FOR BASE SYSTEIi4 DESIGN

T h e IGGP model contains 62 variables, 57 constraints, and 90 parameters. Variables

No. 1 to 60 are defined the same as in the GRG model (Table D. l in Appendix D). Variable

No. 61 is the total recycle stream Bowrate in m3/hr, or

X(61) = (21, + Ql3 + Q15

Variable No. 62 is the primary sludge concentration in 100 kg/rn3 (.41t$100).

Parameters in the G G P model are the same as tha t in the GRG model. Exceptions are

listed in Table E.1.

Table E.l - Parameters tha t are Unique in the IGGP Model

Parameter Index Meaning in G G P Model

1 C, (See equation (2.2.5))

1440 y a ( P C, -DO)(OTE)p,,,

51 - 54 Paranieters in equation (3.7)

81 657.3(*)l~ (See equation (2.62)) CLr, I ,

82 Influent flowrate to plant , m3/hr 83 Inliucnt soluble BOD,, g/m3

84 Influent active biomass, f i lm9 85 Influent volatile degradable solids, g /m2

86 Influent volatile inert solids, g/nl" 87 Influent fixed solids, g/m3 88 BOD$ s tandard , g/m3

89 Total suspended solids standard, g/m3 90 hlass fraction of the primary sludqe, f,

If the value of f , is changed, then parameter No. 46 in the C G P model, which

corresponds to this f , , needs to be calculated using equations (2.38) a n J (2.39) for at and n,.

Exponents in constraint No. 25 which represents thickening of the combined primary and

activated sludge aiso have to be modified since this t-qurt~on is:

The listing of the IGGP model is on the next few pages.

Au

g 23 22: 19 1984 SUM P

aga 1

Au

g 23 22:19 1984 SUM Page 2

I G

CP

INTERACTIVE GENERALIZED GEOMETRIC PROGRAMMING

84,'08/ 04. 10.33.08.

SESSION TITLE : IGGP MODEL FOR W

TEWATER TREARiENT PLANT DESIGN

7) : EQUALITY- -AS3

G(

+ 1.P23P24--1.P25X13Xl7

- 1.P24-- 1 .P26X13X17

- 1.P24--1.X17 -

1.

- 1.P26X13

-CO

MM

AM

)-->

REPD

READING DATA FILE:

UPDATED MODEL

-CO

MM

AN

D--

, CHE OBJ

-cG

MM

AM

)-->

L ALL

G(

9) : EQUALITY- -AS5

+ 1.P27

+ 1.?26--1.X13--l.X18

+ .1E-02P26--

l.X3XlOX14^-1.X16-- l.Xl.8

- 1.

- OBJECTIVE : d

+ 20571.PlP2'-1.P3X9-

.77

+ 368.P4X9'

.3

+ '

202.P4x9- .14

r 285.4P2--l.P3X3' .76

G (

10) : EQUALITY- -AS6

+ , 1.

- l.X3X6^-1.X19

- 1

00

0.X

6--

1.X

10

--1

.X1

3"-

1.X

14

X1

6X

19

11) : EQUALITY--AS7

G(+

1.

+

1.X20

- 1.X13--1.X14

- h)

. lE-O2X3XlOX16-- 1.

- 1 .X16-- 1 .X2SX24

t-' \D

G(


+ 1.

+ 1.X18

+ 1.X19

- 1.24P22Xll'-

.421106X22--.421106X24 -1.X25'

,421106

G(


+ 1.

- 1.X2OX22--1.

- .1E-OlX21X22--1.

GI 141

: EOUALITY--AS10

-CO

MM

AN

D--

.> CHE ALL

G(

1) :

EQUALITY--PSTl

t

1.

- 1.PllX7- .27X8'-.22

- l.Xl0

G (

2) : EQUALITY- -PST2

+ 1.

- .6X8'-1

X9--1.Xll

3) : EQUALITf - -PST3

'(t

60.Xll

+ 1.X12

- 60.X1

G(

4) : EQUALITY--PST4

+

.6E-O3X7XlOXll

3 1.X12X62

- .6E-O3XlX7

C(

5) : EQUALITY- -AS1

Au

g 23 22:19 1984 SUM Page 3

Aug 23 22:19 1984 SUM Page 4

C ( 21) : EQUALITY- -MIX5

+ 1.

+

1.X18

+ 1.X19

- 1.X24'-1.X29

C (

22) : EQUALITY- -CT1

+

1.X33X34

- .lE-O1X36X37

C ( 24) : EQUALITY- -CT3

t

1.P48X35

+

1.X36X37

- 1.X30X32

C (

27) : EQUALITY- -CT6

+

1.P48X12X62

- lE-04X7X30X32X40

C ( 29) : EQUALITY- -Pm1

+

1.X44

t 1.P5iX41

- 1.P51

- 1.P53X41' 2.

- 1.P54Y.41 ' 3.

C ( 33) : EQUALITY- -PANDS

+ 1.

+

1.P28--1.P56--1.X36--1.X47X~~1.X48

- 1.X46X47'-1.

- .lE-02P28--1.P29X3lX47--1.

C (

34) : EQUALITY- -PAW6

+

1.

t

1.X45X4gA-1.

- .871133E-OU(48X49'-1.

C( 39) : EQUALITY-.

+

1.X36X38

C ( 41) : EQUALITY- -M1

+ 1.

- 1.X53'-1.X58

- 1.X53'-1.X59

C (

42) : EQUALITY- -M2

+ 1.

- l.P79X53^-1.X54'-l.X58

- l.X53'-1.X54^-1.X59X60

C (

43) : EQUALITY- -M3

+ 1.

- l.P81X56^-1.X58'-.5X59^

.5X60' .5

C (

44) : EQUALITY- -MI

+

1.

- l.X56'-1.X57^-1.X59X60

G (

45) : EQUALITY- -RECYCLE

+ 1.

- 1.X35X61'-1.

- l.X52X61'-1.

- 1.X58X61--1.

C ( 46) : EQUALITY- -Mi31

+ 60.X1

- 1.P82

- 1.X61

C(

48) : EQUALITY--ME33

+ 1.

- .166667E-01P82P84X1--1.X33-1.

- .166667E-01x1--1.X3'-1. X35X39 -

.166667E-01Xl'-1.X35X40

C ( 49) : EQUALITY- -KB4

+ 1.

- .166667E-OlP82P85Xl^-l.X4--1.

.166667E-OlX1'-1.X35X40

C ( 51) : EQUALITY- -ME36

+

1.

+ 16.6667P70Xl'-l,X6'-1.X52X55

t

Au

q 23 22:19 1984 SUM Page 5

Aug 23 22:19 1984 SUM Page 6

C (

54) : EQUALITY- -DEFN OF FP

+ 1.

- 1.P90

- .l

E-O

lP9

0X

12

^-1

.X2

8X

29

X6

2'-

1.

C (-

55) : BOD STANDARD

+ 1.X17

+

l.P27P28P29'-1.X23

- 1.P88

C(

56) : TSS STANDARD

t

1.X23

+ 1.X18X23

+

1.X19X23

- 1.P89

C (

57) : MIXING REQUIREMENT

+ 1.P45

- .6E-OlX15--1.X27

OBJECTIVE VALUE: 675263.

- -

LOWER

OPERATING PREV.

UPPER

VARIABLE

BOUND

PT.

OP.PT.

BOUND

NAME

..---.------------------.---------.------------------------

XI

'11

25.0000

25.1589 25.1508 30.0030 X(

1)

50. OOCO

.100000E-03

.400000

PARAMETER VALUE

NAME

....

....

....

....

....

....

....

....

ioo . oso

100.000

300.000

6.00000

30. OCOO

2.00000

2.00000

1.00000

5.00000

1.50000

30.0030

20.0030

100.000

200.030

10.0000

100.000

100 .ooo

100.000

100. OCO

1CO. G30

2.00033

100.050

100.000

100. OGO

nnnn -: 8 . uum~rmrrr. u. m. m. mmraww. m. . . . N. r. I-. w. . . . . . . . a. r. r. war

POOOTO. OW r. r. muwm. O. ~NOI-I-I-r. P. ~OWOPI-mm~-I-NI-. w. w. mo.. :Z m. OOOOPOOOO~WW~W. ~;D~moooootnoo~~ m. ~N~POOOO~OOWNO. I-VI I P 40000. 0. Oa0000000~ mooo. 0. 00000000000t~OWm0000NONN00040

awwrld rooooooo rooooooow~ooomooooooooooooommm~ooooa~~~~~~~~ -I. moooooooowooo~oooo. OO~OOOOOOOOOOOOOOOOOOI-~OOOOO~OOOOOOOO II IIIIU 1 moooooooowoooooooomoooooooooooooooooooooooooooooo~~~~~~~ I Ill I

m m m mmmm mmmmmm ... I

'+I + + + + #,,,++ Y '? rrrr oooorr r o '? 4I-I-4 1 t; 8 ++PI- mwwrrr I- r *oar I < B romw I 4omci. F ao4m 8

corn4 l s mmmm 1

mmmm 1 r3lO.Qlo 1

CCCC 1

CCCF : 8 HHHH l

2722; I I!

rrr c

4NN

m5ug gszz OHHVI

IS$& +I

HH Z POTU

ZEgU O'ZXO

pn H - m VIP

?J 2E

IIIIIII I,, I . PO.. . . . . . . . .

,,I,,, I I I,! I I, I I,

wr. ~~~~NL~L~WCLC'~~CI~C~WC~~~C'~~CL~L'NCW'NWI~CLW'NC~'N;D'NCWC'~C'~~ m. ~NI-OWOW~~OI-~~~I-O~W~O~WPI-PI-OPOI-PI-OP~WO~WI-~WI-~WOLI~OPL wmmm~mm1-~m1-OLO~a0~m0oo~~m~~00mm~m~0m~mmro1-~m~~m~PmO~4 ~WO~~WW~~WWO~L~~~O~I-OOON~NW~O~~~NPOW~~~~N~~~~~~~O~OW~ m~~~mmm~~~~~am~m~~~moo~~mam~omm~m~oo~m~aa~o~ommo~*~omm ~ONI-NP~P~OLO~N~~WOO~OOON~OPWON~PWOOO~~~OW~~~~PWI-~~~~O m mmmmmmmmmmmmnmmm mmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm

II,IIII I,,II,I,I I I I I I I III

&&~&~&~~&&~&&~&& &&&&&ooI-0000 0000000000000000000 ~WWPLPLWNLWPWLN~ WWWLWWPP~WLN NWWPWWLWWWWPLWWWL~P

mmmm r3r3lOlo CCCC

EEmm 2222 I I I I

I, I

:i3&i3 S4mm Z rl

mmmm lor3r3lo

Egg 2222 I4II

4, I

?ski3 AWNF

mmmm lolor3cl

&&EE 2222 I I, I II I

%is Q&WN 2 P m

mmmm Ioor3Io

EZEE caaq I,I

IIII s""" ,-ZZZ

222

mmmm lor3aIo CCCC

FCFS 8587 , I I I

VIVIww zzz2 222%

mmmmmmm r3r3r3lor3r3~ ~CCCCCC

PC~FF~F CIHHHUHU

22227y2 I,!,

mmmm r3mlOtcJ

EEE 7qq1 , , I I

nnnn HUH+? WPWN

mmmm r3r3lON

EZ? HHHH

?772

PXXX mLW

mmmm r3lolor3

sggz a::: , I ,,

iiii xxrr NrNP

mmmm IoIoIoro

E?? HHHH

2222 I I I I

ikkk I-am4 0

mmmmmm Ior3r3tOr310 CCCCCC

FFCFFg HHHHHH

22222q I,,,,.

kikkii ~~PWNP

zzz

Aug

23 22:19 1984 SUM Page 9

MAX

TRM

70

MAXIMUM NO. OF CHARACTERS PER TERM.

P

5

NUI4BER CE LP COLUMNS PRICED.

FACTOR

4.0

FACTORIZATION FREOUENCY/NVAR

CNCP

25

DEFAULT M

AX

. NO.. 6

~ CUTTING PLANES.

I N

CP

2 5

CUTTING PLANE INCREMENT.

TPACE

0

TO TERMINTL

GW

D

NO

EXTRA EEASIBILITX CHECK.

SYSTEM

0

CURRENT SUBSYSTEM

LIST

1

1 : E

G'll

i 0 : SUM ONLY

BIG= 100.

SMALL= -100.

METHOD

= SAFE

AUTOBOUND =

1

1-TRUE,O=EALSE

-Corn--> EXIT

CP TIME =

1.169 SEC.

APPENGIX F

GRG PdODEL FOR LIQUID SUBSYSTEM OPTIMIZATION

T h e liquid subsystem to be optimized is shown in Figure 3.8. T h e GRG model describ-

ing the design of this subsystem has 21 variables and 21 functions. The wastewater pararne-

ters a t control point No.1 (see Figure 3.8) are input t o the program. The variables in this

model is given in Table F . l , followed by the listing of the program t h a t defined the model.

Another program is used to generate the initial solution needed for the GRG run. This

program is listed under the name DGSIGN11. A file specifying the decision variables is

needed to run DESIGN11. Input requirements of this file are summarized in Table F.2.

Table F.1 - Summary of Model Variables: Liquid Subsystem

Variable Index Unit Mesnine;

days days

1000m3

kg/m7

g/m3

Table F.2 - Input Da ta to the Analysis Program : Liquid Subsystem

Card No. Comments

1 Title 2 Primary clarifier overflow rate (m/hr) 3 Sludge sge (days), Activated sludge recycle ratio 4 Hydrsulic retention time (days)

Aug 22 16:21 1984 GMMPll Page 1

Aug 22 16:21 1984 GMMPll Page 2

PROGRAM MAIN(INPUT, PM1, OUTPUT, LAST11, TAPE7. WEB,

t

TAPE5=INPUT, TAPE~zOUTPUT,

TAPE9=PAR1,

t

TAPE3;LASTll)

DIMENSION 2 (9000)

COMMON Z

DATA NCORE/9000/

CALL CRG (2, NCORE)

EhD

C C SUBROUTINE GCOMP:

L

SUBROUTINE G

COM

P (G,X)

COPWON,/ INITBK/INIT

C

COMMON,:BONDRY 101. S1. XAl. XD1. XI 1. El. XT1

DIMENSION ~(zii,x(21)

. REAL INF (6) , STD (2) , U (80) , V (20)

IF IINIT. EO. OI COT0 53

EPBND=l.E-6

DO 100 I=1,21

IF (X (I)

. LT. EPBND)

DO 200 J:1,21

G (J) =I. E30

W)NTINuE

RE TURN

THEN

END IF

100

CGNTINUE

C C PRIMARY SETTLING T

AN

K DESIGN:

-

C C ACTIVATED SLUDGE DESIGN:

c - C THICKENING MODEL FOR PRIMARY SEDIMENTATION:

C

G(17)=X(21)

-V(6)

* ((l.E2*X(2))

*U(16)/X(5))

** (l./V(5))

'1.E-2

C C

C EFFLUENT WATER QUALITY STANDARDS:

,. L C C MIXING REQUIREMENT IN AERATION TANK:

c -

G(20)=(U(45)

- (60.'X(20))/(1.E3*X(8)))

*l.E2

C

C OBJECTIVE FUNCTION:

L

C PRIMARY SETTLING TANK:

C

CCPST=824. ' (l.E2*X(2))

*** .

77

IF ((1. E2*X (2)

) .GE. 279

THEN

COPST=17.15* (1.EZ1X(2

b*O.6

CIIPSTi9.

ELSE

23' (l.E2'X(2)i"O.6

COPST=92.45* (l.E2*X(2

) **

.3

~sT=lO6.

ENDIF

* (l.E2'X(2)1

**.I4

-. .- . -

CSPST-8.62' (l.E2'X(2))

*V6

C C PRIMARY SLUDGE PUMPING:

C

CCPSP=16042. *X (5)

.53

COPSP=374.'Xf51

*'.41

Au

g 23 21:53 1984 DESIGN11 Page 2

A

ug

23 21:53 1984 DESIGN11 Page 1

COMMON/PARVA~/U,

V, x

COMMON,~DPST/'OR,IUNITl

REAL VIN (7) ,VOUT(7) ,VSIDE (7) ,MY1 (7) ,U (100) ,V (20) ,

X (30)

REAL KP1

COMMON DAS :SRT;HRT,ASKR, ILTIT2,BODSTD.TSSSTD

REAL VIN (7) ,VOUT (7) , VSIDE (7) ,U (100) ,V(20) ,X (30)

REAL ARYl(7) , INFLOW (7)

REAL N2

CHARACTER* 1 TITLE (ao)

m

C USE NEWTON'S METHOD TO SOLVE FOR PRIMARY SLiTDGE FLOWRATE:

C

L

READ 4

,~1NFLOW(I),I~1,6)

READ 4

* BODSTD,TSSSTD

READl4:*/ (U(1) ,1=1,80)

C

N-1

VSIDE (1) =l.E-3'VIN(l)

VOUT (1) =VIN(l) -VSIDE (1)

F=VIN (1) -VOUT (1) *R1-KPl*VSIDE (11 **((V(5) -l.)/V(5))

VSIDE (1) )

** (1. /

V (5) )

C

READ (5.90) TITLE

90

FORMAT f8OAll

+

(v~u

? (1) *U (lk) /OR)

** (1. /V (5) )

IF (ABS (F) . LE. 1. E-6) GOT0 22

FPRIME-R1-KPl* (U (16) 'VOUT (1) /OR

+ (1. -VIN I11 /V(51 NOUT (11 I

..

. N=N+l ' .

N

IE(N.GE.50) THEN

N

PRINT *

, 'MPXIMUM NUMBER OF ITERATION REACHED IN

Or,

+PRIMARY SETTLING TANK DESIGN

STOP

ENDIF

GOT0 21

IF (VSIDE (1) .LT.O.) THEN

PRINT *,'FAILED TO FIND A

FTIBLE SOLUTION IN

+PRIMARY SETTLING T

JdN

DESIGN

STOP

ENDIF

v (9

1 =U (3

9) /1410: ,'u (36) '(U

+ /U42/U43/~(44j**

V (10) =

U l2el *U 1561

VOUT (71 =VIN (71 *R1

WST-VOUT (1) OR

VSIDE (7) =V(6{

(VSIDE(l)/U(16666)/APST)

** (l./V(S))

*l.E3

VOUT (2) =VIN (2(

VOUT (I j

=VIN(I) *RI

VSIDE (I) =VIN (I) *RATIO

CONTINUE

- WUL PST

DO 88 I=1,7

ARY1 (I) zVOUT (I)

U0

CONTINUE

ULL AS

C

h'RITE (8,120) (K,X(K) , K=1,21)

120 FORMAT (3

X. 13.4X.EZO. 13)

L

STOP

END

C

RETURN

END

L

SUBROUTINE PST

Au

q 23 21:53 1984 DESIGN11 Page 3

SUBROUTINE AS

COMMON/SmTE/VIN, VOUT, VSIDE, ARYl

MM.ION./PARVAR/U,V, , , , ,X

COMMON/DAS/SRT,HRT,ASRR, IUNIT2,BODSTD, TSSSTD

REAL VIN (7) , VOUT (7) , VSIDE (7) ,ARY1(7) , U(100) , V(20)

,X (30)

REAL SN(3).F(3)

REAL hW

C C

C AERATION:

- L IF (IUNIT2. EQ. 0) HRT=HRT/'L4

c

VIN (5) = (ARYl (5) 'C+

(1.0-U (27) ) 'U

(26) WIN (3) ' SRT) /

(l.0tCA ARYl(3) 'VIN(3)

) *VIN 6 =MY1 (6) *C) 1. O+CR ARYl(31 /*IN

(3) )

) VIN 171 =VIN (3) +VIM 14) rVIN IS)

*VIN (6)

,.

ORMT=ARYl(l) 'FOOD* (U(29) -U(28) 'U(23)/(1.

tU(26) 'SRT))

'1.E-3+24.

AFR=ORMT *V (9)

AEW-AFR .*AT

IF (AEW.LT;U(45)) TiEN

PRINT *

, MIXING REZUIREMENT CONTROLS OXYGEN DEMAND'

ENDIF

C

-C SECONDARY SETTLING:

Aug 23 21:53 1984 DESIGN11 Page 4

C

CALL FUNC (N, SN, F,NW,Al1,Al2,A13,A21,A31,A32,

t

RIiSl,RHS2,RHS3)

C

CALL QUASI(N,SN,E,NW,Al1,Al2,A13,A21.A31,A32,

t

RHSl.RHSZ.RHS3,ITEQN)

- L IF 1TEQN.GE. 50) THEN

IF IIAREA. GT .5) THEN

PRINT *,'MAXIMUM MJMBER OF fTERATIONS REACHED

+ IN ACTIVATED SLUDGE DESIGN

STOP

FCSE

C

IF (SN (1) . LE .O. 0. OR. SN(2) .LE. 0.0. OR. SN (3) .LE .O. 0) TIEN

PRINT *,'FAILED TO FIND,A FEASIBLE SOLUTION IN

+ACTIVATED SLUDGE DESILT

C

)

ARYl 1 =SN 1

VOUT 171 =SA 121

- M=SN (3)

'VIN (3) -XA2) /ASRR

C C SETTING UP

THE COEFFICIENTS FCR SIMLTL'PAKEOUS EQUATIONS:

NW:u

1201

Al2=-1.0,jXk*l.0~-3

Al3=VIN(1) /XX'l

.CE-.3

RHSl= VIN (3) 'VAT/SRT,'24. O+Q2*XA2+ASRR'QZ'ARYl

(3)) '1. E-3

MI-v

171

~~~2-kkl(7)*1.OE-3

A31-U(32) 'Q2*

(1.OtASRR) ' (

-l.C)

A32=1.0

RHS3=-U (30) tU (31) 'VIN (7)

C C PROVIDING STARTING VALUES FOR VARIMLES:

. . .. STOP

ENDIF

L

C - ACTIVATED SLUDGE SEPARATION:

L

VOUT (I) =ASRR*pi

VIN (1) =MY1 (1) -VOUT (1)

Aug 23 21:53 1984 DESIGN11 Page 6

X(15)=ASRR+X(14) *1.E-2

RETURN

L

SUBROUTINE QUASI (N ,X, F , NC.Al1 ,M2, Al3, A21,A31,

A32.RHSl.RHS2.RIIS3,ITE

'DIMEN~ION F 3) .FNEh'(3) H (3.3) ,

?E (3) ,S (31 ,SH (3)

DIMENSION ~13.3) ,x(3) ,i(3)

REAL NC

C

H(I~J)=O.O

END IF

10

CONTINUE

- L C COMPUTE X' (I)

(THE NEW SOLUTION) :

15

CAtLMATVEC(N,H,F,S)

DO 16 1:l.N

--

-

-

-

s (I) =-s (I)

16

CONTINUE

C C

TEST FOR CONVERGENCE :

DO 20 I-1.N

IF (I .EQ. 2) THEN

IF~ADS~S~I))

.LE.

KS~OP-NSTOP;~

EXDIF

ELSE

IF(ABS(S(I)) .LE.

NSTOP=NSTOPtl

ENDIF

END IF

20

CONTINUE

TIEN

THEN

C

UPDATE X(1)

: DO 25 I=l,N

X (I) =X (I)

4 S (I)

25

CONTINUE

L

C COMPUTE Y (I) :

CALL F

UNC(N,X.F~TW,!IC,A~~.~~.~~,~~,~~,~~~

t

RIISl.RIiSi.RHS3)

DO 30 I=l,N

Y I =FNEW I -F (I)

F bj

-Fh=wL{

30

CONTINUE

L c UPDATE HII,J) :

c

CO~UTE H (I, J) *Y (J) :

CALL PATVEC(N,H.Y,HY

c ComuTE masPosE (s

(1*N{) l

H(N*N)

* Y (N41) :

SHY-0.0

sHY=sHY+s (I) *HY (I)

35

CONTINUE

,-

-

.

C

UPDATE:

4O

CONTINUE

L C CONTINUE ITERATIONS:

ITEyITE f1

GO TO 15

C 100

RETURN

END

C C SUBROUTINE 'MAl'VEC' PERFORMS THE POST-MULTIPLICATION OF

C

A MATRIX (N'N)

BY A VECTOR

(N*l)

c -

SUBROUTINE MATVEC (N,A, B,C

DIMENSION A(3.3)

,B(3) ,C(3{

c

DO 2 J=l,N

C(I)=C(I)

+A(I,J) *B(J)

2

CONTINUE

i CONTINUE

RETURN

END

- L C SUBROUTINE VECMAT PERFORMS THE PRE-MLKTIPLICATION OF

C

A MATRIX (N*N) BY A VECTOR

(ltN)

C

SUBROUTIhT VECMAT(N,X,Y,Z

DIMENSION X(3)

,Y (3.3) , 1(3{

C

z(J)=o.o.

DO 2 I=1,N

Z(J)-Z(J)+X(I)

'Y(1,J)

2

CONTINUE

1

CONTINUE

RETURN

END

- L SUBROUTINE FUNC(N,X,F,NC.All.A12,Al3,~1,A31,A32,

+ RHSl,RHS2,RHS3)

REAL NC

DIMENSION X(3) ,F (3)

- L =All*X(l) +A12'X

1) 'X(2) +A13'X(2)

-RHS1

=A21"NCt (X (3) /j

(I)) RHS2"NC

7P.31/Xf31 +A32*Xf2l --RHS3

. .,

. .

RET~JRN

END

APPENEIX G

SLUDGE SUBSYSTEM DESIGN

The inputs to this program are: ('11 wastewater parameters at control point No. 1 (see

Figure 3.8), [2] optimal solution for the liquid subsystem (see Appendix F), and [3] the recy-

cle tlowrates Qlo, Q13, and Q16. Only the En31 design of the sludge subsystem is printed out.

The program listing is on the next few pages.

735m gcr 3PZ m3.i "'1

mcr C

PPuPc'o-OP'u --CiZCZTemP rsnl zcl--O. cro-ozo gZ N

T rl-n u r ma* N or .. 0 0. "00.

0 -n-u - z op- 23 nn 3 G-++c *rlWW+3 rz . <TOrOaJ mzrrr

V vl mp

W 9 rs aJ 5

r ul

5

x

d -

4 XPPPUUU rlmmmppu888888 2 p~&&s52$gggggg ~Z~g:~~:?.~~ - * * .. o"Im<wmm

rc~~rooc~n - ..uu E E ~~~.mo~zmFl~xc~~ 3 U--rX. roo'-. \. H 'U r. u- co-ra~E *~-gi:,~" .Pmm GrLr ..9.na~ma~ mmar r- . r ura4 ~,.n

gr EF z?. k, - 2 EggE K- xw u - rn

:? .;= Go Gg?L - o g p .r -0 --- fig r. \ 5 EEm3

? ??YS 8 fi GfiF'ul rD rov,n O. -\ +ir r 0 t: t: LZ " P .m P.

- uu XOO -- LO;~; -aa Y

4mn mu,- NP04 r w-

rj Go tjuz r " ' 9 ."I m. - E 5:: n a ro-

W N

WW I-' 0

Ew8;?8R?4EFE$ ~GZ~~~~R~rvy

Wcl I1 OH "8888

& g ?az:z - H rn-n- 8" "88 H O ba p L sw . 0

gp $

8;2E227zu Z-Z I-OPI-I- H u- I----. H. 11 0 ,I 11 I- 11 0

,Z!-!-y? $F" 2 I OOUP "8N 0 I- 0 z-I- 0 z;g 8s H I

OE I- I-

&7uu uxnm LZGk I I-HN

5Gi Pal- * -. - * X *. HCJ anm

W - c n N - + c - P - * 3 cl H

B cl H - + C cl '2 C - W - a n N -

8? -I-

T: I-" -- 7

'? I- - * t-

3

. ~

e.= .c 38 I2 :.O !-' e e I-. I-

P I-'

ZW82 HXHH

5"02 R 8;

H m

+ I-, umn " E8 83; n PI-

; $; $:g m HH II f * F!i'28 & 8 rr\ " g rr;sg L.NN2 .* 2 m

XNn z 0 275 a, P NO* * z5 w 5; - N Y

sEg2 ;??; E QN8 Hmm" WOW *\- s r3 I-$$ ::cl N n

" i 8 - I-'

P -

rw-7

I-' P. .cl Lh: ". .

*-

m d p Z

I1 11 PI-' I-'- WN . . ;-?

, . . NN I-

-. . . - .I! Q .- >.", > I-I- 5 I-' ON X. >.$? - m

- C - V1 P - I-.

-. I-' 0 x' Zd x u . I-

C? - N X I1

:C! .- P. N -' I1 I- - N

~2 . . t-0 " m g" m I! .

€ €Z

Sep 16 23:17 1984 SLUDGE3 Page 5

Sep 16 23:17 1984 SLUDGE3 Page 6

~-

+ ,~',10X,'XF14=',F12.6,'KG/CU M',/,1OX,'XT14=',F12.6,

+ 'KG/CU M1,/)

81

FORHAT(t0X. 'AREA OF

SEU)?DARY DIGESTER=' ,F12.6, 'SQ M

' ,/, 1OX.

+ SOLIDS LOADING 014 DIGESTER:.'

,F12.6, 'KG/SQ M/HR ' , /)

C 604

016=014-015

1~16-

(~1~*~~14-~15*~

(79) ) /Q16

W - Q16'XT16.'Q15

YIELD U (73) *SQRT (WiU (74) "U (75) /U

(76) /U

(77) /U

(78) )

AV:Q16*XT16,,YIEL?

-

IF (Q16X16 .~~-519

.) TXN

W-5.57'Ql6X16'*.04

ELSE

IF (Q16X16. GE ,103. ) THEN

13MVF=20.*Q16X16".63

ELSE

~:41.5'Q16Xl6**.48

ENDIF

EhiIF

U)HSD-i 141 *

8024. *Q16* * .667

TCVFSD- U (1) 'U

(3) ./U (2) *CM+U (4)

(CDVF+CMVF) +U (3) /

U (2) * C

SVE

+ t CCSD* COI-ISD

IF(IC.EQ.0 GOTO 605

WRITE 6 90 Q16,XIl6,XF16,XT16

WRITLf6:9l/AV.YIELD

FOPMT(?OX, '

AR

EA

-O~.

VW

4 FILTER=' ,F12.6, 'SQ M1 ,/, IOX,

+ FILTER YIELD-',Fl2.6,'KG/'SQ M/HR1,/)

QCYCLE-QlOtQ13tQ15

CCRP=2779.*QCYCLE*'.53

W-;

P=

. 333'QCYCLE

W=.2375*QCYCLE

IF (QCYCLE . LT. 158. )

TiEN

CSRP- 300.

ELSE

IF (GCYCLE . LT. 631. ) TIiEISN

CSKP-40.57'0CYCLEt*.52

ELSE


CSKP 5.97*QCYCLEt*.87

ELSE

CSRP=2.540tQCYCLE

ENDIE

EIiDIF

ENDIF

cPRP=OCYCLE

TCRP=~

(1) *U (3) ~(

2)

*CCRP+U(4) * (

CORPtCMRP) +CSRP*U (3) /'U (2)

+ TU)ST=TCGT+TtPAND+TCSAND+TCVFSD+TCRP

tCPRPa23.85*U (6) *U(7) /U

(8)

IF IIC. EO. OI GOT0 606

92

FOR

MA^(

^/(:

'TOTAL COST FOR SLUDGE TR!3TMENT

AND DISPOSAL

=',

+

F1O.O,

DOLLARS/YEAR' , //

) 606

RETURN

END

L

SUBROUTINE COST (TC,TEMP.U.CHEC.Q11, S9, VSDEST)

DIMENSION U (80)

-

Ql=lO. 22E3*Q11* TEMP-u (55) )

Q2-U (59) *VCIGLU 160) (TEMP-U (57) ) '8.76

Q- 1011 021 /U (58)

~~

~T

=]

E~

E~

-Q

J

*l.E-6

CCI."AND-2323. *VDIG**.59

IF (MIG. GE .5678. ) THEN

MPEND=1.29*MIG**.83

ELSE

IF (VDIG.GE. 1968 .)

THEN

COPAND-14.*MIGk*.55

CM?ANDr8.5*MIG** .55

ELSE

ENDIF

ENDIF

IF(MIG.GE.2839.) TKE

N CSPAND=14.4*MIG**.66

ELSE

CSPAND=142.*MIG**.37

ENDIF

-

---

TClU (1) *U (3) U (2) *mANDtU(4) A (COPANDTC?PP-WJ)

3) /U (2{*CSPAND-U (61) *3.6E3/1.055*EVtIET

END

APPENDIX H

GRG MODEL FOR SYSTE?/I IYITIIOUT A PRZPJAXY CLARIFIER

T h e Howchart of the system is shown in Figure H.1. There are 51 variables and 47

equations in the GRG model describing the design of this system. Table fI.1 provides a list

of tlie variables in the model.

Table H.l - Summary of hfiodel Variables : Base System Without a Primary Clarifier

Variable Index Unit Meaning

1 m3/min QJGO

2 .dm3 5'2

3 g/m3 hf4

4 g/m3 Md2

5 g/m3 MI 2

6 g/m3 A!, 7 g/m3 M:2 8 days 0, 9 days 0 10 1000m3 V/lOOO 11 kg/m3 A f 4 3

12 g/m3 5'3

13 - .%fl J,! fa

14 - --

Aff J h f a 3

15 r 16 - 100w

17 - r + w

18 g/m3 4

19 kg/m3 A!,, 20 loom" Af/lOO

2 1 g/m3 S

32 m3/sec Q,/co 23 m3/hr Q e

24 k d m 3 A f t ,

25 kg/m2/hr L , 26 100m2 A,/lOO

- 27 m3/hr Q l o

28 m3/hr Q11 29 kg/ms hf:ll

Aug

22 16:30 1984 GCOMP7 Page 1

Aug 22 16:30 1984 CCOMP7 Page 2

PROGRAM MAIN(INPUT.PhR.OUTPUT.LASlVAR.TAPE7.TAPE8.

t

TAPES-ItITUT, TAPE6=OUTPUT, TAPE4=LASlVAR,

t

TAPE9:PAR)

DIMENSION z I200001

COMMON Z

DATA NCORE/20000/

CALL GRG (z, NC

ORE)

END


C

SUBROUTINE GCOMP (G.XI

50

EPBND-1.E-6

DO 100 I=1,51

, IF (X (I) .LT. EPBND) THEN

DO 200 J=1.47

G (J) =l. E30

200

CONT1h-m

RErnN

ENDIF

100

CONTINUE

C

C ACTIVATED SLUDGE DESIGN:

-

, , --

.--,

/xi151 /i18\) fix [ill

* '

'-

' '

' ~

~ '

C SLUDGE MIXING:

G 13 =X 23

- 60 *X(1 )

* l.E-2*X(16))

G[~~~=u[~o~*[~.~x~~~~x[~~))*~~.E~*x(~~))-x(~~)

C c GRAVITY THICKENERR DESIGN:

- C PRIMARY DIGESTER DESIGN:

- L

C SECONDARY DIGESTER DESIGN:

VACUUM FILTER DESIGN:

L

C MASS U'CE OF RECYCLE STREAM:

C

G 37 =

(60. 'X

1) ) -INFLOW (1) -X (27) -X (43) -X 49)

~1381

=U(80) 160. *X(l) ) *X(2) -U(80) *mFLo"[l) *IIIFLOW(

- *X (27) *X(12)

-$:Fix

(43) *U (66) -U (80) lU(66) *x(m

*~(39)

=U 1801 *LO. *X (1) *X (3)

-U (80) *Irm,ow(J -IIELOW(I)

- *X(27) *X(31)

'G 40 =$4g0!60.*X

1 *X 4 -U 80 'INFLOW

1 *INFLOW 4

c[411 =u[aol *Lo. *x 111 *x b1

-u [aoj *I

rm.,ov 111 *itrLow[jj

-U 8

0 'X

13 *X 27 'X

31

t

-U 1701 *X 1431 *X 1461 -X 1461 *X (49)

gg: HHHLl """'a

N

ul '2 rO 01

ON . 4NN m m. N. *a N . L WPX. * *01 *ul-* h -* n*N- I-. c.h P-OI-.

- -. M *I- mp-. 00""

OW

VI a

E TI I :98::9S:RS mmm-mcnrn-cn 8 !225!B!BE'R $z:ZEQXP$ZE HHHH~HH~H mgg~mg5~$ 5 :;gmp;:ggm5pm5$g LqGd d:Vb 'a~~~~m'a~m'a~m I:qwqH I;g A$

11115 ;i;gLr g 11 Ilm. 1111 llllm llllmtl n - ON .jmN m~ I- I-w PI- mew . PWN m m m. ;I 5 ; %& pp . P. m. WNONN - wm L P* PQ . L* m. *w rn N . P ErN,? dm ulm

WN 01. X UNXN H * *"I *W%* 8 4. % ,U)mw * FJ . . * - *10-(*: m A* - *M < < *. . *W mW n*W. !a P-NP 3

rO ro"b *r cnmr mcm2 3 ,a - rrz P-L*

mmC4 P . 1-01. !a

!-'yz .I-y -. P-- m m. m. -m .. 2 2"!3 mmY m. -I-

68E * * $g?gr - D Eb DD "Y %D"k N 3 Nm s":%"x' NM-N P m 9; mm 0z

* *- * *- x' 2 2 m m - -x ^xmh *o 6: . * *h *. N N- N-. N .- isp a&-** n nN M WWP W-01% a3 mN pNNm m.

wi%olE w ww ww LL~ ~wm- - -m -ad- L PW PL -m -*4w Y -U) K. - YI -a-

Z za *- *-. Y Y. K. **. ? **-• ~4 * *- ** **- **- . .* Z

* . . ** . . .* ..& I-&

-1 rn m b~ HgLLm: mm Z BID

Z

L, 2 !I !I C0NW OPWW . mlrw

. 01, P *lo

- m

Aug 22 16: 30 1984 GCOMP7 Page 5

ELSE

CSS~=l42.*VSAND'*.37

ErnIF

C

C VACUUM FILTER:

CSVF-230. ~~16i16~'.

717182. *Q16X16** .86

W)VF=197.55*Q16X16**.58

IF (Q16X16. GE ,519. ) TXN

CM

VF

=5

.55

55

7*0

16

X1

6*-

.84

ELSE

IF (Q16X16 .GE ,103 .)

THEN

CMVF=20.'016X16**.63

- ELSE

DNF~41.5*Q16X16**

.48

ENDIF

ENDIF

- L C (47) =V (1) (CCATtCCDM+CCFST+CCRSP 7 C

CR

P t

tCCGTtCBANDtCCSANDtCCVFt72053 .*X(50)

.74)

t +U(4)*(CODAA+UOAArCOFST

t +WSTtCORSP+aGtSP TCQCT+(~*IGT+COPAND

. ,

RETU

RN

'

Em

C C SUBROUTINE REPORT

- L W'BRO'JTINE REPORT(G,X,M, ~~.CON,VAR,XO)

DIENSION X (N) . G

(M) , CQN (N) , VAR (N) , XO (N)

COMMON :INITBK/INIT

IF (INIT.EQ. 1)' RETGIU

WRITE(4.10)

(I,X(I) ,I=l,N)

10

FORMAT (3X, I3

,4X

, E2O. 10)

RETURN

END

APPENDIX I

GRG h/lODEL FOR THE SYSTEM WHERE WASTE ACTIVATED SLUDGE IS RECIRCULATED TO PRIMARY CLARIFIER

T h e system is shown in Figure 1.1. There are 51 variables and 47 equations in the

GRG model. A list of the model variables is provided in Table 1.1.

Table 1.1 - Summary of Model Variables : Waste Activated Sludge Recirculated to the Primary Clarifier

Variable Index Unit hleaninq

m3/min

g/m3

g/m3

g/m3

g/m3

g/m3

g/m3 m3/hr

100m2 - m3/min

m3/hr days days

1000m3 kg/m3

g/m3 - - - - -

g/m3 kg/m3

100m2

g/m3 m3/sec

m3/hr

kg/m3 "C

days

Aug 22 16: 19 1984 GCOMP6 Page 1

Aug 22 16:19 1984 GCOMP6 Page 2

PROGRAM HAIN (INPUT,PAR, 0UTPUT.LASlVAR. TAPE7, WEB,

+ TAPES-INPUT, TAPE6:OUTPUT.

TAPE4=LASlVAR.

+ TAPES-PAR)

DIMENSION Z (20000)

COMMON Z

DATA NU)RE/20000,'

CALL GRC (Z,NCORE)

END


- L SUBROUTINE GCOMP (G,X)

COmON./INITBK..'INIT

DIMEIJSION C 1471

.X 1511

REAL I~TLOW~~)

ST^

(21 ,U(80) ,V(20)

l./v(5))*V(5)/(V(5)

-1.0

(I. /U (20) ) *U (20) /

(U (201 - 1.0)

+ DO 100 1~1.51

IF(X(1)

.LT.EPBND) THEN

DO 200 J--1.47

G (J) =1. E30

200

CONTINUE

RETURN

- --

END IF

L

C PRIMARY SETTLING TANK DESIGN:

- L. C ACTIVATED SLUDGE DESIGN:

L C SLUDGE MIXING:

- L. C PRIMARY DIGESTER DESIGN:

C

L. C SECONDARY DIGESTER DESIGN:

C

C

C C VACUUM FILTER DESIGN:

C c MASS BALANCE OF RECYCLE STREAMS:

-

Aug 22 16:19 1984 CCOMP6 Page 3

G(41) =U (80) "60. *X (1) 'X

(6) -U (80) *INFLOW(l) *INFLOW(6)

t

-X 19 *X 28

*X (24) -U (70) *X (41) (1.-X (44) )

+

-xSI *u I 79

I (I. -X (44) )

G (42) =X (3) +X (4) tX (5) tX (6) -X (7)

C

C THICKENING MODEL FOR PRIMARY SEDIMENTATION:

C

G(43)=X(50)-V(6)*((l.EZ*X(9))*U(16) /X(12))'*(l./V(5))*1.E-2

C

C EFFLUENT WATER QUALITY SWARDS:

C

'X 23 -STD(l)

G 44 =X 17 tU(27)

=x 1231 (I. +xi;!:]

+X 1191 ) -

sm

(2)

C

C MIXIlJG REQUIREMENT IN MRATION TA

NK:

C

G(46)-(U(45)

- (6O.*X(27))/ (l.E3*X(lS)))*l.E2

C

C OBJECTIVE FUNCTION:

C

C PRIMARY SETTLING TANK:

C

CCPST=824. (l.E2*X(9)) **

.77

IF((l.Ez"X(9)) .GE.279. THEN

CCPST-17.15* (1 .E2*X (9

*.0.6

WST=9.23* (1.E2.X

(9) 1 **

O. 6

ELSE

COPST=92.45* (l.E2*X (5

) *

* .3

rwsr=los.*(l.E2*x(e)] **I4

ENDIF

CSPST-8.62* (l.E2*X(9)) **

.76

C

C PRIMARY SLUDGE PUMPING:

C

CCPSP=16042. *X (12) *

.53

CPPSP-x ( (12)

C

C AERATION:

C

CCAT=461.* (l.E3*X(15) **.71

COAA=8533.*(60.*X(27 )**.66

CODAA-l87* [60.*xi2711 **48

0fDAA~74.4~

60.*X 27

**.55

C C FINAL SETTLING TA

NK:

C

CCFST=824.*(l.E2*X(25))**.77

IF (

(1. E2'X

(25) )

. GE .279

TIE

N

COFST=l7.15* (1.EZ1X(25 I'k0.6

WST=9.23* (1E2*X(25)i4*0.6

ELSE

COFST=92.45* (l.E2*X(25 )

'* .3

Q.FST=106. ll.E2*X[251] **

. 14

Au

g 22 16:19 1984 CCOMP6 Page 4

CORSP=.~~~*Q~

CMRSP=.2375'Q5

IF (Q5. LT. 158 .)

THEN

CSRSP=300.

ELSE

IF(Q5.LT.631.) THEN

CSRSP=40.57*Q5**.52

ELSE

IF (Q5.LT. 1580.) THEN

CSRSP=5. .97*Q5** .87

ELSE

CSRSP=2.54*05

- ENDIF

ENDIF

ENDIF

C RECIRCULATION PUMPINC:

-

CM

W=

. 2375;~CYc~~


CSW=300.

~-

~-

ELSE

IF (QCYCLE. LT. 631. )

CSW=40. 57*QCYCLEA

ELSE

IF (PCYCLE . L

T .1580.

CSRP=5.97*QCYCLE*A

ELSE

CSRP=2.54*QCYCLE

ENDIF

ENDIF

ENDIF

CPRP=QCYCLE

C

C PRIMARY DIGESTER:

C

THEN

'*.52

) THEN

'.87

IF ((1.~3*~(32)')

.~~.5678'.

1 THEN

COPAND=1.29* (l.E3*X(32 )

**

3

WAND=.83*(lE3*X(32)I**.82

ELSE

IF((l.E3*X(32)) .GE.1968.) THEN

ELSE

COPAN'D=192. l.E3*X 32

CJPAND=113. 1l.E3*Xl32II

ENDIF

ENDIF

IF((l.E3*X(32)) .GE.2839.) THEN

CSPAND=14.4* (l.E3*X (32)) **

. 66

ELSE

CSP&"D=142.* (l.E3*X(32)) **.37

ENDIF

C RETURN SLUDGE PUMPING:

C

Q5= (60. *X (11) ) *X (22)

L

C SECONDARY DIGESTER:

C

Aug 22 16:19 1984 GCOMF'6

Page 5

ELSE

IF (VSAND. GE .1968. )

TE

N

u)sAND=~~.*vs~D**.~~

CMSAW=8. 5*VSNiD**. 55

ELSE

COSAND=192. *VSAniD** .2

(2.ISAND=l13.*VSAND*'.21

E.W I F

IF (VSAND. GE. 2839. ) THEN

CSSkW=14.4*VSAKD**.

66

ELSE

C V

AC

lTU

M FILTER:

C

Ql6X16=X (48) *X

(49)

CCW=29180. *X (46) **

.71

CSVF~230.*Q16X16**.71+182.*Q16X16*=.86

COVF=197.55*Q16X16a*.5E

IF (Ql6Xl6. GE .519. ) TEN

CNW=5.57*016X16".

04

-

ELSE

IF (Ql6X16 .CE. 103 .)

THEN

U4VF-:2O.*QlbXlb**. 63

ELSE

L

G (47) =V(l) * (CCPST b

CC

PSP

+CC

AT

+C(I

DA

A*m

ST+C

(3R

SP+C

(;X

P

t

+CCPAND+CCSANDrCCVEt72053. 'X(48)

*' .74)

+

tU(4) * (COPSTrCI.FSTtCCPSPtWSP+CODAAt03DAA+COFST

t

+~STtCORS?t(SERS?tCOPk'iD+Q.PAND+COS9ND+(2.ISAh?)

t

+COVF~CMVF*CO~GTQ?FP.~O~~.*X(~~)

**.667)

+

+V (2) * (

CSP

STT

CSF

'SP

TC

SFST

+CSR

SP tCSPAND + CSSAND bCSW

+ +CSVFI

+ tV 3

* (CPPSPtCPKSPt C

?W

) +

tullj

:u(5)

(3.62E-2) 'X(48) AX(4g) -U(61)

+ 3.6E3 i1 .C55*X (38)

RETURN

E h?

) L

C SUBROUTINE REPORT

C

SUBROUTINE REPOST (C, X

.H.N

. CON,VAR,XO)

DIMENSION X (N) , G (H) ,CON (M)

,VAR (N) , XO (N)

CObPlONiINITBK/INIT

IF (INIT.EQ.l) RETURN

WRITE(4.10) (I,X(I) ,I=l,N)

10

FOWAT (3X. 13.4X. E2O. 10)

RE TURN

END

REFERENCES

Abadie, J., and Carpentier, J., "Generalization of the Wolfe Reduced Gradient Method t o the Case of Nonlinear Constraints", in Opt imiza t ion , edited by R. Fletcher, Academ- ic Press, 1069

Adams B. J., and Panagiotakopoulos D., "Network Approach t o Optimal Wastewater Treat- ment System Design", Jour. W a t e r Pol lut ion Control Federat ion, Vol. 49, No. 4, pp. 623-632, Apr. 1977

Agnew, R. W., .4 Mathemat ica l h fode l of a Final Clarifier, Water Pollution Cont.rol Series 17090 F J W 02/72, U. S. Environmental Protection Agency, Washington D. C., 1972

Aris, R., Nemhauser, G. L., and Wilde, D. J., "Optimization of h4ultistage Cyclic and Branching System by Serial Procedure", A. I. Ch. E . Jour., Vol. 10, No. 6, pp. 913- 919. Nov. 1964

Avriel, M., and Williams, A. C., "Complementary Geometric Programming", S I A M Jour. of Appl ied Mathemat ics , Vol. 19, pp. 125-141, 1970

Avriel, M., Dembo, R., and Passy, U., "Solution of Generalized Geometric Programs", In ter - nat ional Jour. fog Numer ica l h le thods i n Engineer ing , Vol. 9, pp. 149-168, 1975

Avriel, M., ~ V o n l i n e a r P r o g r a m m i n g , Analys i s and hlethods, Printice-Hail, Inc., 1976

Berthouex, P . M., and Polkowski L. B.,, "Optimum Waste Treatment Plant Design Under Uncertainty", Jour. W a t e r Pollution Control Federat ion, Vol. 42, No. 9, pp. 1589- 1613, Sept. 1970

Bisogni, J. J., and Lawrence, A. W., "Rclationship Between Biological Solids Retenlion Time and Settling Characteristics of Activated Sludge," lVater Research , Vol. 5, No. 9 , pp. 753-763, Sept. 1971

Blau, G. E., and Wilde. D. J., "Generalized Polynomial Programming," T h e Canadian Jour. of , fC'hemical Engineer ing , Vol. 47, pp. 317-326, June l!)G9

Box, M. J., "A New Method of Constrained Optimization and a Comparison with Other Methods", C o m p u t e r Jour.; Vol. 8 , No. 1, pp. 42-52, 1965

Bowden, K., Gale, R. S., and D. E. Wright, "Evaluation o f the CIRL4 Prototype hlodel for the Design of Sewage-Treatment Works", lVater Pol lut ion Control , Vol. 75, No. '7, pp. 192-203, 1076

Bowden, Ii. , and Wright, D.E., "CIRIX Ilodel for Cost-ellcctive CVastcwatcr Treat~ncnt" , l l f a t h c m a l i c a l i l l ode i s i n l Y a t e r Po l lu t i on Corlt;ol, edited by 12. Janics, John-il'iley and Sons. 1978

Brill. E. D. Jr., "The Use of Optimization Models in Public-sector P!anningU, M a n a g e m e n t S c i e n c e , Vol. 25, No. 5, pp. 413-422, 1079

Brosilow, C. B., and Lasdon, L. S., "A Two Level Optimization Technique for Recycle Processes", Proceedings, A. I. C h . E. - I. C h e m . E n g . S y m p . , Ser. 4 , 1965

Brosilow, C. B., Lasdon, L. S., and Pcarson, J . D., "Feasible Optimization hlethods for Inter- connected Systems", Proceeding, Jo in t A u t o m . C o n t r o l C ~ n j . , pp. 79-84, 1965

Burns, S., and Ramamurthy S. , "Intcrnctive Generalized Geometric P rogra~nming (1GGP)- User's hlanual", 2nd ed., Department of Civil Engineering, L'niversity of Illinois, Mar. 1983

Burns, S. A,, and Ramamurthy S., "Generalized Geometric Programming with Equality Constraints!' , submitted to J o u r . of O p t i m i ~ a t i o n T h e o r y a n d A p p l i c c t i o n s for publi- cation, 1983

Busby, J. B., and J. F. Andrews, "Dynamic Llodeling and Control Strategies for the Activat- ed Sludge Process", Jour . W a t e r P o l l u t i o n C o n t r o l F e d e r a t i o n , Vol. 47, No. 5, pp. 1055-1050, hlay 1975

Bush, h1. J., and Silveston, P . L., "Optimal Synthesis of Waste Treatment Plants", C O ? I I ~ U ~ - e r s a n d C h e m i c a l E n g i n e e r i n g , Vol. 3, No. 4 , pp. 153-159, 1978

Camp, T. R., "Sedime~~tat ion a ~ ~ d the Design of Setlling Ta~ iks ! ' , Trar tsacl ior ts , A S C E , Vol. 111, pp. 895-036, 19.16

Carman, P . C., "A Study of the klachanism of Filtration", Jotrr. of t h e S o c i e t y of C h e m i c a l I n d u s t r y , Vol. 52, P a r t 2, pp. 3SOT-2S2T, Sept. 8, 1033

Cashion. B. S., and Iieinath. T . \I., "Inflr~encc of Three Factors on Clnrificntion in the t\c- tivalcd Slutlgc I'roccss", Jorrr. Ii'atcr I'ollutiort C ~ t t t r o l F 'edcrafzon, Vol. 5 5 , No. 11, pp. 1331-1337, NOV. 1083

Chapman, D. T. , "Thc Influence of'Process Variables on Secondary Clarification", Jour . 1Ya- t e r P o l l u t i o n C o n t r o l Federatiorz, Vol. 55, No. 12, pp. 142'5-1434, Dcc. 1083

Chen, Y. R., and IIashimoto, '4: G., "Iiinetics of hiethane Fermentation", B i o t e c h n o l o g y a n d B i o e n g i n e e r i n g S y m p o s i u t n , No. 8 , pp. 260-282, 1078

Chcn, Y. R., and Hashimoto, A. C., "A Substrate Utilization Kinetics Model for Biologicai Treatment Processes", Uiotechnology and Bioengineering, Vol. 22, No. 10, pp. 2081- 2095, 1980

Christenscn, G. L.. "Units of Specific Resistance". Jorrr. Water Pollution Control Federation, Vol. ,55, No. 4, pp. 417-410, Apr. 1083

Coackley, P. , and Jones, B. R. S., "Vacuum Sludge Filtration", Sewage and Industrial Wastes, Vol. 28, No. 8, pp. 963-976, Aug. 1956

Computing Services Office, GRG User's Guide, University of Illinois, Urbana, IL 61801. Dec. 1982

Construction Industry Research and Information Association, Cost Effective Sewage Treat- ment - The Creation o f an Optimizing hlodel, Report No. 46, CIRL4, London, Eng- land, 1973

Craig, E. W., Meredith, D. D., and A. C. Middleton, "Algorithm for Optimal Activated Sludge Design", Jour. of the Environmerital Engineering Division, ASCE, Vol. 104, No. EE6, pp. 1107-1117, Dec. 1978

Davidon, W. C., "Variable Metric Methods for Minimization", AEC Research and Develop- ment Report, ANL-5900 (Rev.), NOV. 1959

Dembo, R. S., "Current Sta te of the Art of Algorithnis and Computer Software for Geometric Programming", in Advances in Geometric Programming, edited by M . Avriel, Plenum Press, 1080

Dick, R. I., "Hole of Activated Sludge Final Settling Tanks", Jour. o f the Sanitary Engineer- ing Division, ASCE, Vol. 06, NO. SA2, pp. 423-436, Apr. 1970

Dick, R. I., and Young, K. W., "Analysis of Thickening Performance of Final Settling Tanks," Proceedings, 27th Industrial Wastes Conference, Purdue University, pp. 33-54, 1072

Dick, R. I., "Sludge Treatment", C l~ap tc r 12, Physicochernical Prucesses for 1Vater Quciity Control, edited by W . J. Weber, Jr., Wilcy-Interscience, Ncw York, 1072

Dick, R. I., and M. T . Suidan, "Xiodeling and Simulation of Clarification and Thickening Processes", Mathematical IZIodeling for Water Pollution Control Processes, edited by T. M. Keinath, and M. P . Waniclista, Ann Arbor Science Publishers Inc., 1975

Dick, R. I., Simmons, D. L.. Ball, R . O., and K. Perlin, P r o c e s s S e l e c t i o n f o r O p t i m a l ~ t ! a n a g e n t e n t of R e g i o n a l IYas t ewa te r Res id r la l s , First Annual Progress Report , Department of Civil Engineering, Delaware University, Newark, Delaware, Oct. 1076

Dick, R. I.; Simmons, D. L.? Ball, R. O., Perlin. K., and hl. W. O'I~Iara, P r o c e s s S e l e c t i o n f o r O p t i m a l i t f a n a g e m e n t of R e g i o n a l l V a s t e w a t e r R e s i d u a l s , Department of Civil En- gineering, Delaware University, Newark, Delaware, Aug. 1978

Dick, R. I., Hasit, Y., Driscoll, C. T. , Moore, I(. A., and D. J. Sander, I n t e g r a t i o n of P r o c e s s e s f o r 1Yas t ewa te r R e s i d u a l s A f a n a g e m e n t , First Annual Report, School of Civil and Environmental Engineering, Cornell University, Ithaca, New York, Mar. 1979

Dick, R. I., and Y. Hasit, I n t e g r a t i o n o f P r o c e s s e s f o r l V a s t e w a t e r R e s i d u a l s ~ t i a n a g e m e n t , School of Civil and Environmental Engineering, Cornell University, Ithaca, New York, May 1981

Dietz, J. D., and T . M. Keinath. "Clarification hlechanisms for a Flocculent Slurry", presented a t the 55th Annual Conference of the Water Pollution Control Federation, St . Louis, hlissouri, Oct. 1982

Duncan, J. W. K . , and Kawata , I<., Discussion or "Evaluation of Activated Sludge Thicken- ing Theories", J o u r . of t h e S a n i t a r y E n g i n e e r i n g D i v i s i o n , A S C E , pp. 431-433. Apr. 1968

Eckenfeldcr, W. \V., I n d u s t r i a l lVa t e r P o l l u t i o n C o n t r o l , klcGraw-IIill, 1966

Ecker, J. G., and J. R. McNamara, "Geometric Programming and the Preliminary Design of Industrial Waste Treatment Plants", lVa t e r R e s o u r c e s R e s e a r c h , Vol. 7 , No. 1, pp. 18-22, Feb. 1971

Ettlich, W. F . , T r a n s p o r t o f S e w a g e S l u d g e s , EP.1 60012-77-216, U . S. Environmental Pro- tection Agency, Cincinnati, Ohio, 1977

Evenson, D. T., Orlob, G. T. , and J. R. hionser, "Preliminary Selection of Waste Treat r r~ent Systems", Jour . Li'atcr Pol lr~l ior t C'ontrol F I - d ~ r a l i o r ~ . Vol. 41, No. 11, P n r t 1, pp. 1845-1858, NOV. 1969

Fan, L. T. , Chen, K. C., Erickson,. L. E. , and 31. Naito, "EBects of Axial Dispersion on the Optimal Design of the Activated Sludge Proce.ss", ] r a t e r R e s e a r c h , Vol. 4, pp. 271-

- 284, 1970

Fletcher, R . , and 11. J. D. Powell, "A Rapidly Convergent Descent hiethod for hlinimiza- tion", C o m p u t e r Jour . , Vol. 6 , pp. 16.3-368, 19G3

Goelfrion, A. hl., "Large-scaie Linear and Nonlinear Programming", Optimization illelhods f o r Large-Scale Systems, edited by D. A. Wismer, McGraw-Hill, 1971

Gossett, J . h,l., and Belser, R. L., "Anaerobic Digestion o l Waste Activated Sludge", Jour. o f Ihe Etrvironmet~tal Erlgineering Division, ASCE, Vol. 108, No. EE6, pp. 1101-1120, Dec. 1982

Grady, C. P . L., Jr., "Simplified Optimization of Activated Sludge Process", Jour, o f the En- vironmental Engineering Division, ASCE, Vol. 103, No. EE3, pp. 413-429, June 1977

Great Lakes - Upper Mississippi River Board of Sta te Sanitary Engineers, Recommended Standards For Sewage Works, 1978

Haimes, Y. Y., Hierarchical Analysis o f IYater Resources Systems, h,lcGraw-Hill, 1977

Hasit, Y., Rajagopal, R., and P . A. Vesilind, "Sludge Management Systems: Optimal Plan- ning", Jour. o f the Environmental Engineering Division, ASCE, Vol. 107, No. EE3, pp. 493-509, June 1981

Hazen, A., "On Sedimentation", Transactions, ASCE,, Vol. 53, pp. 45-71, 1904

Himmelblau, D. M., Applied Nonlinear Programming, McGraw-Hill, 1972

Hooke, R., and T. A. Jeeves, "Direct Search Solution of Numerical and Statistical Prob- lems", Jour. o f the Association f o r Computing Machinery, Vol. 8, pp. 212-229, 1961

Hughes, h4. 3I., "The Least Cost Design of the Activated Sludge Process with Geometric Programming", Presented to the Faculty of the University of North Carolina a t Chapel Hill in Partial Fulfillment of the Requirements lor the Degree o l Master o l Science in Environmental Engineering, 1978

Hughey, P . W., Meredith. D. D., and A. C. hliddlcton, "Algorithm lor Optimal A ~ t ~ i v a t e d Sludge Operation", Jour. o f the Envirotarrrental Engineering Division, ASCE, Vol. 108, No. EE2, pp. 340-366, Apr. 1982

Joeres, E . F., and David, h4. [I., editors, Buying A Better Environment - Cost-Effective Re- gulation Through Permit Trading, Land Economics hlonographs, No. 6 , The Univer- sity of Wisconsin Press, 198'3

Keinath, T. M., Ryckman, kt. D., Dana, C. H., and D. A. Holer, "Activated Sludge- Unified System Design and Operation", Jour. oj the Environmental Engineering Division, ASCE, Vol. 103, No. EE5, pp. 829-849, Oct. 1977

Iioelling, C. P . "0ptimiz:rtlon of Wastewater Treatment Process", Presented a t the ORSA/TIXIS National J o i ~ l t hieeing, San Diego, California. Oct. 27, 1982

Lasdon, L. S., O p t i m i z a t i o n T h e o r y F o r L a r g e S y s t e m s , Machlillan Publishing Company, 19'70

Lasdon, L. S., Waren, A. D., Jain, A., and M. Ratner, "Design and Testing of a Generalized Reduced Gradient Code for Nonlinear Programming", AC, l1 T r a n s a c t i o n s o n M a t h e m a t i c a l S o f t w a r e , Vol. 4 , No. 1, pp. 34-50, Mar. 1978

Lasdon, L. S., and Beck, P . O., "Scaling Nonlinear Programs", O p e r a t i o n s R e s e a r c h L e t t e r s , Vol. 1, No. 1, pp. 6-9. Oct. 1981

Lauria, D. T. , Uunk, J. B., and J. I<. Schaefer, "Activated Sludge Process Design", J o u r . o f t h e E n v i r o n m e n t a l E n g i n e e r i n g D i v i s i o n , A S C E , Vol. 103, No. EE-I, pp. 655-645, Aug. 1977

Lawler, D. F., and Singer, P . C., "Return Flows from Sludge Treatment", J o u r . l Y a t e r P o l - l u t i o n C o n t r o l F e d e r a t i o n , Vol. 56, No.2, pp. 118-156, Feb. 1984

Lawrence, A. W., and McCarty, P . L., "Kinetics of hlethane Fermentation in Anaerobic Treatment", J o u r . W a t e r P o l l u t i o n C o n t r o l F e d e r a t i o n , Vol. 4 1 , No. 5, pp. R1-17, Feb. 1969

Lawrence, A. W., and McCarty, P . L., "A Unified Basis for Biological Treatment Design and Operation", J o u r . of t h e S a n i t a r y E n g i n e e r i n g D i v i s i o n , A S C E , Vol. 96, No. SA3, pp. 757-778, June 1970

Lech, R. F., "Applicability of Control Strategies to the Activated Sludge Wastewater Treat- ment Process", A Thesis Submitted to the Faculty of Purdue University in Part ial F~llfillrnent of the Requirements for the Degrre of Doctor of Philosophy, 1973

Logan, J. A., IIatfield, W. D., Russell, G. S., and \V. R. Lynn, "An Analysis of the Economics of \Vas~cwatcr Trratrnent", J o n r . Il'otcr P o l l u t i o n C o n t r o l F e d e r a t i o n . Yol. 31, No. 9, pp. 860-882, Sept. 1962

Lynn, W. R., Logan, J . A., and .4. Charnes, "Systems .4nalysis for Planning Wastewster Treatment Plants", J o u r . l V a t e r I Jo l l u t i on C o n t r o l F e d e r a t i o n , i'ol. 34, No. 6, pp. 565-581, June 1965

~ c ~ h e e , T . J . , hiojgani, P . , and Vicidomina, F . , "Use of EPA's C M D E T Program for Evaluation of \Vastewater Treatment Alternatives", J o u r . ll.'ater P o l l u t i o n C o n t r o l F e d e r a t i o n , Lrol. 55, No. 1, pp. 35-43, Jan. 1983

McKinncy, R. E., "Mathematics of Complete hlixing Activated Sludge", Jour. o f the Sani- tary Engineering Division, ASCE, Vol. 58, No. SA3, pp. 87-113, May 1962

h.1etcalf and Eddy, Inc., Water Pollution Abatement Technology: Capabilities and Costs- LrO1.

3, Report to National Commission on Water Quality on Assessment of Technolo~ies and Costs for Publicly-Owned Treatment Works, PB-250-690-03, National Technical Information Service, Springfield, Virginia, 1975

Middleton, A. C., and A. W. Lawrence, "Cost Optimization of Activated Sludge Systems", Biotechnology and Bioengineering, Vol. 16, No. 6 , pp. 807-826, June 1374

Middleton, A. C., and A. W. Lawrence, Least Cost Design o f Activated Sludge Wastewater Treatment Systems, E P M Technical Report No. 75-1, Department of Environmental Engineering, Cornell University, Ithaca, N. Y., 1975

Middleton, A. C., and A. W. Lawrence, "Least Cost Design of Activated Sludge Systems", Jour. Water Pollution Control Federation, Vol. 48, No. 5, pp. 889-905, May 1976

Mishra, P . N., Fan L. T., and L. E . Erickson, "Biological Wastewater Treatment System Design - P a r t I. Optimal Synthesis", The Canadian Jour. o f Chemical Engineering, Vol. 51, pp. 694-701, Dec. 1973

Naito, M., Takamatsu , T. , and L. T . Fan , "Optimization of the Activated Sludge Process- Optimum Volume Ratio of Aeration and Sedimentation Vessels", IVater Research, Vol. 3, pp. 433-443, 1969

Nelder, J. A., and R. Mead, "A Simplex hlethod for Function Minimization", Computer Jour., Vol. 7, pp. 308-313, 1965

Parker , D. S., Kaufman, W. J., and Jenkins, D., "Physical Conditioning of Activated Sludge Floc", Jour. Water Pollution Control Federation, Vol. 43, No. 9 , pp. 1817-1833, Sept. 197 1

Ps rkc r , D. S., "Assessment of Secondary Clarification Design Concepts", Jour. IVater Pollu- tion Control Federation, Vol. 55, NO. 4, pp. 349-359, Apr. 1983

Parkin , G. F., and R. R. Dague, "Optimal Design of Wastewater Treatment Systems By Enumeration". Jour. of the Sanitary Engineering Division, ASCE, Vol. 98, No. SA6, pp. 833-851, Dec. 1972

Patterson, W. L., and Banker, R. F., Estimating Costs and Manpower Requirements for Conventional lvastewater Treatment Facilities, Report No. 17090 Dan 10/71, U. S. Environmental Protection Agency, Washington D. C., Oct. 1971

Patterson, I i . E. , "Dynamo 1: A Cost Elfective Solution to Treatment Problems", W c t e r 8 1Ynstes E n g i n e e r i n g , pp. 121-126, Sept. 1977

Pfefier, J. T. , "Increased Loading on Digesters with Recycle of Digested Solids", Jour . W a t e r P o l l u t i o n C o n t r o l F e d e r a t i o n , Vol. 40, No. 11, P a r t 1, pp. 1920-1033, Nov. 1968

Pfeffer, J. T., "hlethane Fermentation - Engineering -4spects of Reactor Design", Presented a t the 31th International Conference of CEBEDEAU, Liege, Belgium, May 25-27, 1981

Pflanz. P. , "Performance of (Activated Slutlge) Secondary Sedimentation Basins", in A d - v a n c e s i n W a t e r P o l l u t i o n R e s e a r c h , Proceedings of the Fourth International Conference, pp. 569-581, edited by S. H. Jenkins, 1969

Powell, M.J. D., "An Efficient Method for Finding the Minimum of a Function of Several Variables Without Calculating Derivatives", C o m p u t e r Jour . , Vol. 7, No. 4, pp. 303- 307, 1964

Ratner, M., Lasdon, L. S., and A. Jain, "Solving Geometric Programs Using GRG: Results and Comparisons", J o u r . of O p t i n z i t a t i o n T h e o r y a n d A p p l i c a t i o n s , Vol. 26, No. 2, pp. 253-264, Oct. 1978

Reheis, H. F., Dozier, J. C., Word, D. M., and J . R. Holland, "Treatment Costs Savings Through Monthly Variable Eflluent Limits", Jour . IVater P o l l u t i o n C o n t r o l F e d e r a - t i o n , Vol. 54, No. 8, pp. 122.1-1230, .4ug. 1982

Rimkus, R. P . , Ryan, J.M., and Michuda, A. J., "Digester Gas Utilization Program", J o u r . of t h e E n v i r o n m e n t a l E n g i n e e r i n g D i v i s i o n , A S C E , Vol. 106, No. GE6, pp. 1071-1078, Dcc. 1980.

Rossman, L. ,I., Computer-Aided S y n t h e s i s of 1Vnstewater T r e a t m e n t arid S l u d g e D i sposa l S y s t e m s , EPA-GOO/?-70-158, U. S. Environmental Protection Agency, Cincinnati, Ohio, Dec. 1079

Rossman, L. A . , "Synthesis of Waste Treatment Systems by Implicit Enumeration", J o u r . 1Vater P o l l u t i o n C o n t r o l F r d e r n t i o n , Vol. 5 2 , No. 1, pp. 148-160, Jan. 1980

Rudd, D. F., and C. C. Watson, S t r a t e g y o f P r o c e s s E n g i n e e r i n g , John Wiley and Sons, 1968

Scherfig, J . , Schinzinger, R., and T . W. hlorgan, "Waste Treatment Optimization by Geometric Programming", in . .idvnnces in 1Vater P o l l r ~ t i o n R e s e a r c h , Proceedings of the Fifth International Conference Vol. 2, pp. 11-??/I- 11-22/18, edited by S. FI . Jt-n- kins. 1970

Schittkowski, I<., "The Current State of Constrained Optimization Software", in Noniinear Programming, edited by h4. J. D. Powell, Academic Press, 1982

Schoeffler, J. D., "Static h4ultilevel Systems", in Optimization Methods for Large-Scale Sys- tems, edited by D . A. Wismer, McCraw-Hill, 1971

Shih, C. S., and Krishnan, P. , "Dynamic Optimization for Industrial Waste Treatment Design", Jour. Water Pollution Control Federation, Vol. 41, No. 10, pp. 1787-1802, Oct. 1969

Shih, C. S., and DeFilippi, J. A., "System Optimization of Waste Treatment Plant Process Design", Jour. of the Sanitary Engineering Divsion, ASCE, Vol. 96, No. SA2, pp. 409-421, Apr. 1970

Shih, C. S., and Icrishnan, P . , "System Optimization for Pulp and Paper Industrial Waste- water Treatment Design", Water Research, Vol. 7, pp. 1805-1820, 1973

Smeers, Y., and Tyteca, D., "A Geometric Programming Model for the Optimal Design of Wastewater Treatment Plants", Operations Research, Vol. 32, No. 2, pp. 314-342, March-April 1984

Smith, R., Preliminary Design and Simulation of Conventional IVaste Renovation System Using the Digital* Computer, Report No. WP-20-9, U. S. Department of the Interior, Federal Water Pollution Control Administration, Cincinnati, Ohio, March 1968

Sterling, R. A., "Computerized Algorithms Simplify Meeting Federal Effluent Standards", Water d Sewage IVorks, pp. 68-72, Oct. 1972

Suidan, M. T., Department of Civil and Environmental Engineering, University of Illiuois, Urbana, Illinois, Personal Communication, 1982

Suidan, M. T., Saunders, F. M., Godfrey, C. S., and 1-1. T . Stewart, "Wastewater Treatment Optimization and Sensitivity Analysis", Jour. of the Environrrlental Engineering Division, ./ISCE7 Vol. 109, No. E E 1, pp. 120-135, Feb. 1983

Swanson, C. L,., "Unit Process Operating and klaintenance Costs for Conve~~t iona l Sewage Treatment Processes1', International FIVPCA ~\lemorandurn, Aug. 1966

Takamatsu, T. , and M. Nait.0, "EITects of Flow Conditions on the Efficiency of a Sedimenta-

- tion Vessel", IVater Research, Vol. 1, pp. 433-450, 1967

Tarrer, A. R., Grady, C. P . L., Jr., Lim, 11. C., and L. B. Koppcl, "Optimal Activated Sludge Design Under ~ n c e r t , a i b t ~ " , Jour. of the Environmental Engineering Division, .4SCG, Vol. 102, No. EE3, pp. 657-673, June 1076

Tebbut t , T . H. Y., and D. G. Christoulas, "Performance Relationships for Primary Sedimen- tation", lYater Research, Vol. 0, pp. 347-356, 1975

Tebbut t , T . H. Y., "Primary Sedimentation of Wastewater", Jour. IVater Pollution Control Federation, Vol. 51, No. 12, pp. 2858-2867, Dec. 1979

Tuntoolavest, hi., Miller, E., and C. P . L. Grady, Jr . , Characterization of IVastewater Treat- ment Plant Final Clarifier Performance, Technical lieport No. 120, Water Resources Research Center, Purdue University, West Lafayette, Indiana, June 1080

Tyteca, D., Smeers, Y., and E . J. Nyns, "Mathematical hlodeling and Economic Optimiza- tion of Wastewater Treatment Plants", C R C Critical Re~kiews in Environmental Control, Vol. 8, No. 1, pp. 1-89, Dec. 1977

Tyteca, D., "Nonlinear Programming Model of Wastewater Treatment Plant", Jour. of the Environmental Engineering Division, ASCE, Vol. 107, No. EE4, pp. 747-766, Aug. 1981

Tyteca, D., and Smeers, Y., "Nonlinear Programming Design of Wastewater Treatment Plant", Jour. of the Environmental Engineering Division, ASCE, Vol. 107, No. EE4, pp. 767-779, Aug. 1981

U. S. Environmental Protection Agency Technology Transfer, Sludge Treatment and Dispo- '

sal, E P A 62511-74-006, Oct. 1974

U. S. Army Corps of Engineers and U. S. Environmental Protection Agency, Computer- Assisted Procedure for the Design and Evaluation of IVastewater Treatment Systems- User's Guide, 43010-79-01, Office of the Chief of Engineers, Department of the Army, 1978

Vesilind, P. A., Treatment and Disposal of Il'astewater Sludges, Revised Edition, Ann Arbor Science, Ann Arbor! h.[icliigan, 1070

Vesilind, P . A., Discussion of "Evaluation of Activated Sludge Thickening Theories", Jour. of the Sanitary Engineering Ditjision, ASCE, Vol. 9.1, No. SA1, pp. 185-191, Feb. 19G8

Villier, R . V., "Removal of Organic Carbon from Domestic \VastewaterM. Thesis Presented to the University of Cincinnati, Ohio, in Partial Fulhllment of the Requirements for the Degree of l las ter of Science, 1367

Voelkel, I<. C., "Senstivily ~Znalysis of \Vastcwater Treatment Plan1 Independent State Vari- ables and Technological Parameters", A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree or Doctor or Philosophy a t the University of Wisconsin a t Madison, 1978

Voshel, D., and J . G. Sak, "Enect of Primary E m r ~ e n t Suspended Solids and BOD on Ac- tivated Sludge Production", Jour. lVater Pollution Control Federation, Vol. 40, No. 5, P a r t 2, pp. R203-212, May 1968

Warren, A. D., and L. S. Lasdon, "The Status of Nonlinear Programming Sortware", Opera- tions Research, VoI. 27, No.3, pp. 431-456, May-June 1979

Water Pollution Control Federation, Sewage Treatment Plant Design, Manual of Practice, No. 8, Washington, D. C., 1959

Westerberg, A. M., "Decomposition Methods for Solving Steady Sta te Process Design Prob- lems", in Decomposition of Large Scale Problems, edited by D. M. Himmelblau, North-Holland, 1973

Westerberg, A. W., Hutchison, H. P. , Motard, R. L., Winter, P . , Process Flowsheeting, Cam- bridge University Press, London, 1979

Wise, D. L., Dynatech R I D Co., Personal Communication, Cambridge, hlass., 1980

Wolfe, P. , "hlethods of Nonlinear Programming", in Recent Advances in Mathematical Pro- gramming, edited by Graves and Wolfe, h1cGraw-Hill, 1963

Wolre, p. , "Methods for Linear Constraints", in Nonlinear Programming, edited by J . Abadie, North-IIolland, 19G7

Date post:	14-Nov-2021
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

WRC RESEARCH REPORT NO. MATHEMATICAL IN ANALYSIS …

Documents