Dynamic Simulation of Multiple Effect Evaporatorsin Paper Industry Using MATLAB

Deepak Kumar', Anjana Rani Gupta' , Somesh Kumar'UDepartment of Mathematics, NIET Greater Noida, UP: India

'Department of Information Technology, NIET Greater Noida, U'P; India'dkriitr@gmail.com, 'argad76@gmail.com, 3someshkumarrajput@gmail.com

Abstract- The present study attempts to develop adynamic model for the MEE to study the transientbehavior of the system. Each effect in the process isrepresented by a number of variables which are relatedby the energy and material balance equations for thefeed, product and liquor flow. In the present studydynamic equations are being written for the MEEsystem for a Paper Industry. In this study a generalizedmodel is given which could be applied to any number ofeffects in the MEE system with simple modifications itcould also be applied for Forward Feed, Backward Feedas well as for the Mixed Feed. For such situation basicequations for an effect will be same but the equationsfor the parameters like density, boiling point elevationand specific heat etc. should be changed and then themodel can be used for the other type of evaporator also.

Keywords - Multiple effect evaporator (MEE);backward feed; Runge-Kutta method; boiling pointelevation (BPE), multiple effect distillation (MED).

I. INTRODUCTION

Mathematical Modeling is an indispensable tool toanalyze, correlate, simulate, optimize, and finallycontrol any chemical process system. Pulp and paperindustry is very capital intensive industry and consistsof many subsystems. Huge amount of raw material,chemical, energy and water are consumed in theprocess of paper manufacture with requirement oflarge labor force. Systematic mathematical modelingof each subsystem is therefore an imperative necessityto solve the above issues one after another. One suchenergy intensive subsystem called evaporator is usedto concentrate black liquor from pulp mill. MultipleEffect Evaporators (MEE) dealing with theconcentration of weak black liquor from pulp mill ofpaper industry consume a large amount of thermalenergy in form of steam (25-40%).

A wide variety of mathematical models for multipleeffect evaporators can be found in the scientificliterature. Normally the main difference among thesemathematical models is the heuristic knowledge whichis incorporated in their development.

El-Nashar [1] developed a simulation model forpredicting the transient behavior of ME stack typedistillation plants. Transient heat balance equationswere written for each plant component in terms of theunknown temperatures of each effect. The equationswere solved simultaneously to yield the time-dependent effect oftemperature as well as performanceratio and distillate production. The results of thesimulation program were compared with actual plantoperating data taken during plant start-up, andagreement was found to be reasonable.

Lambert, Joyo and Koko [6] developed a system ofnon-linear equations governing the MEE system andpresented a calculation procedure for reducing thissystem to a linear form and solved iteratively by theGaussian elimination technique. Boiling point rise andnonlinear enthalpy relationships in temperature andcomposition were included. The results of linear andnonlinear techniques were compared.

Hanbury [II] presented a steady-state solution tothe performance equations of an MED plant. Thesimulation was based on a linear decrease in boilingheat transfer coefficient.

Miranda and Simpson [10] describe aphenomenological, stationary and dynamic model of amultiple effect evaporator for simulation and controlpurposes. The model includes empirical knowledgeabout thermo physical properties that must becharacterized into a thermodynamic equilibrium. Theproperties selected evolved from an economicaloptimization because of their influence on thetemperature and concentration variations parameters.The developed model consists of differential andalgebraic equations that are validated using aparameter sensitivities method that uses data collectedin the industrial plant. The simulation results show aqualitatively acceptable behavior.

Tonelli, Romagnoli, and Porras [12] presented acomputed package for the simulation of the open-loopdynamic response of MEE for the concentration of

8 NIET Journal of Engineering & Technology, Vol. 5, 2014

liquid foods. It is based on a non-linear mathematicalmodel. An illustrative case study for a triple effectevaporator for apple juice concentrators waspresented. The response of the unit to largedisturbances in steam pressure and feed flow ratebased on the solution of the mathematical model was inexcellent agreement with the experimentallydetermined response.

Narmine and Marwan [4] developed a dynamicmodel for the MEE process to study the transientbehavior of the system. This model allowed the studyof system start up, shut down, load changes andtroubleshooting in which plant performance changedsignificantly. Each effect in the process is representedby a number of variables which are related by theenergy material balance equation for the feed, productand brine flow. These equations were solvedsimultaneously to predict the system time dependentparameters under various transients. The effect of feedflow, feed temperature, live steam flow changes onplant performance such as temperature, brine salinity,product flow rate and brine level was investigated totest the validity ofthe model.

In the present investigation the dynamic model ofMEE system of a paper industry is developed by usingenergy and material balance equations to study thetransient behavior of the system. Further theparametric equations were solved for steady state dataand the system of simultaneous ordinary differentialequations for sextuple backward feed evaporator tostudy the transient behavior of the system.

II. MATHEMATICAL MODELING

The Mathematical modeling is carried out forsextuple backward feed evaporator system. Inbackward feed evaporator system the steam input isgiven in the first effect and the feed input is in the lasteffect. The material and energy flow for ith effect isgiven in the Fig-I. Model equations are presented fOJith effect using material and energy balance equationsand equations stating the physical properties of theblack liquor. It is assumed that the vapor generated bythe process of concentration of black liquor issaturated. It is also assumed that the energy and massaccumulation in the vapor lump is neglected as it isvery small as compared to the enthalpy of the steam.

Material balance for liquor in the itheffect:

~ MI(i)(t) =WI(i +1) - W\(i) - Wv(i)dt

Energy balance for liquor in the itheffect:

~(MI(iXl) * hl(i)(l»)= WI(i + I )hl(i + I) - WI(i)hl(i)dt- W\(i)h\'(i) + W\'(i - I )h\{i - I) - Wvf i-1)hc(i -1) (2)

Material balance for solids in ith effect:

~(Ml(i)(t)*X(i)(t»)= WI(i + I )X(i + 1)-Wl(i)X(i)dt

(3)

Where MI(i) can be written as:

MI(i) = A L(i)PI(i) (4)

The PI(i) is the density of the liquor given by[3]

Pl(i) = (997 + 649X(i))[ 1008- O.237(Tl(i)1l (00)

-1.94(Tl(iYlOOoiJ (5)

Where TI(i) is in °C

The vapor and liquor in ith effect are in equilibriumand the relation for the liquor and vapor temperature isdefined in terms of boiling point elevation (BPE) is asfollows:

TI(i) = Tvti) + BPE(i)

and the boiling point elevation (BPE) is given by [3]

(6)

BPE(i) = (6. 173X(i) -7.48X(i) 15 + 32.74 7X(i)C) *(l + 0.6(Ty(i) - 3.7316)/1 00) (7)

Where Tv(i) is in OK.

hv(i-1), WV(i-1), Wv(i), Tv(i),

Steam input Vapor generatedthroughevaporationstream

Black liquorinput

WI(i+1), X(i+1),

TI(i+1), hIO+1),WI(i), XCi), TI(i),

hl(i), PIOj

Black liquor

Wv(i-1), hc(i-1)

(1)Fig-l Block Diagram with terms used for the i" effect

NIET Journal of Engineering & Technology, Vol. 5, 2014 9

Specific heat of water at constant pressure, Cp givenby [3]

Cpt i) = 4216( 1- XCi»~+ (1675 +:U 1TI(i)/lOOO) * XCi) +

(·+87 - 20TI(i)IlOOO) * (1- X(I» * X(i)'

Where TI(i) is in -x.Differentiating equation (6) with respect to time we get

d ( dTI(I)(t) = - T,· I (I)dt dl ){ I + (il~\' BPE(I) H +

) (9)(iJ . Ia .- BPE(I) - X(I)(t)ax ill

Differentiating equation (4) with respect to time and

using the value of the value of *TI(i)(I) from equation (9)

and by rearranging the terms we get the value of *MI(ii!tl

and then comparing the resultant differential equation

with equation (1) we get

CI =C(~L(J)(t) )+C3(~T\'(I)(t) )+CI(~X(l)(t) )(10)lit tit dt

Where:

Cl = WI(i + 1) - WI(i) - Wv(i)

C2 = APl(i)

C3 = AL(i) (~PI(i) Y 1 + (_a-BPE(i) )JaTI A aTv

C4 = AL(i)( ( a~l PI(i) X a~ BPE(i) )+ (a~ Pl(i) ))

Differentiating MI (i) (t)*hl (i) (t) with respect to timed

and using the value of cttTI(i)(t) from equation (9) and

the value of ~ Ml(i)(tifrom the equation which is obtained

by the differentiation of equation (4) and thencomparing the resultant differential equation withequation (2) we get

C5 = C6( :t L(i)(t)) + C7 ( :t TV(i)(t))+ C8 ( :t X(i)(t) )

(II)

Where:

C5 = WI(i + 1)hl(i + 1) - WI(i)hl(i)

- Wv(i)hv(i)+ Wv(i -I )hv(i -I) - Wv(i -1 )hc(i -1)

(8)C6 = API(i)hl(i)

C7 = AL(i{ 1+ ( a~v BPE(i)))

(Pl(i{ a~l hl(i))+ hl(i{ a~l Pl(i)))

C8 = AL(i)

Pl(i) ( ~ hl(i) Y ~ BPE(i)) +aTI A ax

PI(i) (~hl(i))+ hl(i) (~PI(i))ax aTl

(a~ BPE(i)) + hl(i) ( a~ PI(i))

Differentiating MI(i)(t)*X(i)(t) with respect to time

and using the value of i.MI(i)(t) from the equation whichdt

is obtained by the differentiation of equation (4) andrearranging the equation and representing it in the formof coefficients

.!X(i)(t) = C9 + C 10(.!L(i)(t)l+ C 11 (~TV(i)(t)1ili ili) ili )

(12)

\\"\0 + ll-'\(i +1) - \\ l(i)X(i)C9=---r------~--------------------~

. \L(i{PI(i) + X(i){( ()~I PI(i)I()~BPE(i) )+ (()~ PI(i) )}]

, PI(i)X(i)(10= .. -

L(i{ PI(i) + X(i){( a~1 PI(i) Ia~ BPE(i) )+ ( a~ PI(i) )}]

X(i)( a PI(i) ~ 1+ ( a BPE(i»)}aTI A aT\"

C11=------~----~~~--~~~--~PI(i) + XI(i)[(-4-PI(i) Y ~ BPE(i»)+ (~PI(i) )11

(HI JJ\ a:\. ~

10 NIET Journal of Engineering & Technology, Vol. 5,2014

III. MODEL VALIDATION

System of simultaneous ordinary differentialequations given by equations (LO), (II) and (12) issolved by Runge-Kutta method of 4th order. Runge-Kutta 's methods do not require the calculations of higherorder derivatives and give greater accuracy. Runge-Kutta's formulae has advantage of requiring only thefunction values at some selected points. Error of 4thorder Runge- Kutta's Method is of the order of h5. Acomputer code is developed in MATLAB for thesolution.

For the validation of the model, steady state data iscalculated and compared with the data of the paper mill.The steady state results are in good agreement with thedata ofthe paper mill.

IV. SIMULATION

A. Effect of varyingfeedflow rate:

To study the transient behavior a step change of a10% disturbance in the liquor flow rate to the last effectwas applied to the model and the response of the systemtoward this change is shown through the graphs of thevarious system variables from Fig-2 to Fig-7. Thus thefeed flow rate is decreased by 10% and the response of itis seen in following variables.

• First Effect Temperature• First Effect Liquor Level• Product Flow• Last Effect Temperature• Last Effect Liquor Level• Product Concentration

B. Effect of varying steamflow rate:

Similarly a step change of 10% in the steam flow rateto the first effect was applied and the response of thesystem toward this change is shown through the graphsof the various system variables from the Fig-8 to Fig-I3.The feed flow rate is decreased by 10% and the responseof it is seen in following variables.

• First Effect Temperature• First Effect Liquor Level• Product Flow• Last Effect Temperature• Last Effect Liquor Level• Product Concentration

These graphs are drawn with respect to time and bythe graphical study it could easily be estimated that howthe variables approach to the new conditions from theprevious one.

V. RESULTS A D DISCUSSION

A. Effect of varyingfeedflow rate

The MEE system is backward feed; hence the inputliquor enters the evaporator system from the last effect.The variation in the feed flow effects the last effectvariables more than it affects the previous effectsvariables. The steady state is reached more quickly inlast effect from other effects due to the same reason.Hence we can say that the response of disturbance infeed flow is greater in last effect than in other effects asinferred by the results shown in the graphs.

B. Effect of varying steam flow rate

In MEE with backward feed; the steam enters theevaporator system from the first effect. The variationin the steam flow effects the first effect variables morethan it affects the subsequent effects variables. Thesteady state is reached later in first effect from othereffects due to the same reason. Hence we can say thatthe response of disturbance in steam flow is greater infirst effect than in other effects as inferred by theresults shown in the graphs.

VI. CONCLUSION

A dynamic mathematical model was developed fora sextuple backward feed evaporator for concentratingthe black liquor by using material, energy balanceequations and parametric correlations. The model issuccessfully validated using the data obtained from themill. The transient behavior of liquor temperature,liquor level, and product flow & product concentrationwas studied by disturbing the liquor flow rate andsteam flow rate by 10%.

The transient study shows that the liquortemperature and product flow rate first increase thendecrease to reach a steady state with a decreasing feedflow rate, while they directly decrease before reachingsteady state with a decreasing steam flow rate. Theproduct concentration increases and reaches steadystate with decreasing feed flow rate, while it decreaseswith decreasing steam flow rate. The variation in thefeed flow affects the last effect variables more than itaffects the previous effects variables and the steadystate is reached quickly in last effect than other effects.The variation in the steam flow affects the first effectvariables more than it affects the subsequent effectsvariables. The steady state is reached quickly in firsteffect from other effects.

NIET Journal of Engineering & Technology, Vol. 5, 2014 11

Variation in First Effect Temperature

r---~-----------------~---- -----

"r--T-1-----0..---- _

'--J--

'"1.~.1--

Fig-2 Variation in first effect temperature

Variation in Product Flow

ia

. ) 3 -e 19 Z< 2 28 34 37 ' J. '.{, 49 :J2 55 50 n &l 67 7~ 7& 7~

Tlmejrrinl

-[)Y"~"rBeo""l[JJPfIMCl.!.SlOado,SltIlC

12

Fig-4 Variation in product flow

Variation in Last Effect Liquor Level

e e

., lJ" 16192:'

Fig-6 Variation in last effect liquor level

08

02

Fig-3 Variation in first effect temperature

Variation in First Effect Liquor Level

1~ 13 -e ·922 25 213 '1 34 37 .f() J1:oF:') 5:i 55 5'; 0:.' ~ q 7: p 75 '>'Q

TIme(min\

Variation In Last Effect Temperature

-o,narTlo,;&riW0.1

Po: "",Se<a.:,s:.W

051

05

'0 1!1 4. l~ l~ Jl jot 37 4(; 4~ 40 4\1 ~ 50 ~b ti1 ~ ~I lU 73 7~ nl!Inejmlnl

Fig-5 Variation in last effect temperature

Variation in product concentration

-D,lnaI'rKBcoo.o.If

Pl'evOlosSteildyStale

0.48 '---- •. ---1~ 13 te 19 U 25 28 31 34 37 <10 .3 4 49 ~ ~:) :>ti s- 64

Time (mm)

Fig-7 Variation in product concentration

A. Effect of varying feed flow rate Inferencefrom the Graphs (Fig. 2 to 7)

NIET Journal of Engineering & Technology, Vol. 5, 2014

"0

-DyrBTi[;Bsha~(U

-PrevwsStoodySt<*!

••----------------------1

71013161922 2526 3134 37 4043 46 49 52 55 56 61 64 671013 767e

T1me(miro)

Fig-8 Variation in first effect temperature

-Oyn(fTlit5et-~"oor

-P~cusSleadyStale

10 -a 16 19 22 23 ~8 a- 34 37 ~o 43 46 4" !;2 ~5 58 61 64 r:;r 70 ?3 Jt; j'<j

Iimelmin}

Fig-IO Variation in product flow

r s

-01lam~Beha\·D.J'

-Pr~llU!Stoodyltate

-DynU'lleBeh •••O\.I'

-1'"vDusSl~ySl'i"

0'0.•

• 2

.a ••• H.M •• ea •••• y ••• mnRTime (mif'l)

Fig-9 Variation in first effect liquor level

-DjwvNc8a/lavour

1~-P(""''''''I~s.IIit-i)Sbtll

••--------------------------~

"

Tlme(min)

Fig-12 Variation in first effect liquor level

- J 16 19 aa 25 2! J1 l4 37 ~o 0 ~€ '9 ;2 ~5 sa <>1 64 P 1J 7J 70 l!i

T1m.~Inl

Fig-II Variation in last effect temperature

'51

,d

.n •••• ~M ••• a ••••• M.nronn~Hone!rrin)

Fig-13 Variation in product concentration

B. Effect of varying steam flow rate Inferences from the graphs (Fig. 8 to 13)

NIET Journal of Engineering & Technology, Vol. 5, 2014

-O\f'lI/I1o<:&Il~'oI::lJr

p((I\IO()ysSt:l.1dI'8~te

13

NOMENCLATURE

A - Shell area, m'

BPE - Boilingpointelevation, °C

C - Constant

Cp - Specific heat of water at constant pressure, KJ/Kg

h - Enthalpy, KJ/Kg °C

L - Liquor level, m

M - Mass,kg

PI - Liquor density

- Time,sec

T - Temperature.X'

W - Mass flow rate, Kg/s

X - Solid content, %

Sym bols used with the above terms:c - Condensate

- Liquor

- Effect number

v - Vapor

REFERENCES

[I) A.M. El-Nashar and A. Qamhiyah, "Simulation of theperformance of M ES evaporators under unsteady state operatingconditions", Desalination, 79, pp-65, 1990.

[2) c., Cadet, Y, Toure, G., Gilles, & J.P. Chabriat, "Knowledgemodelling and non-linear predictive control of evaporators in canesugar production plants". Journal of Food Engineering, vol.40(1/2), pp 59-70, 1999.

[3) Joban Gullicbsen and Carl- Joban Fogelbolm, Chemical Pulpingvol. 6B, B 18-20.

[4) H.A., Nannine, and M.A., Marwan. "Dynamic response of Multi-effect evaporators", Desalination, vol. 114, pp 189-196, 1997.

[5) R. H., Perry, C. H., Chilton, S.D., Kirkpatrick, "ChemicalEngineers Handbook", 4th edition, McGraw Hill, New York, 1963.

[6) R.N. Lambert, D. Joyo and F.W. Koko, "Design calculations formultiple-effect evaporators I. Linear method", Ind. Eng Chern.Res., vol. 26, pp-I 00, (1987).

[7) C. H., Runyon, T. R., Rumsey and K.L., McCarthy. "Dynamicsimulation of a non-linear model of a double effect evaporator",Journal of Food Engineering, vol. 14, pp 185-20 I, 1991.

[8) c., Riverol, and v., Napolitano. " on-linear Control of anevaporator using an error trajectory technique", Journal of

Chemical Technology and Biotechnology, vol. 75, pp1047-1053,2000.

[9) Steam Table and Mollier Diagrams (S.l.Units), emchand andBrothers, Roorkee, 1984.

[10) V. Miranda, R., Simpson. "Modelling ans simulation of anindustrial multiple effect evaporator : tomato concentrate",Journal of Food Engineering, vol. 66, pp 203-21 0, 2005.

[II) W,'T, Hanbury, Proc., IDA World Congress on Desalination andWater Sciences, Abu Dhabi, UAE, vol. 4, pp 375,1995.

[12) S,Tonelli, J. A., Romagnoli, and J. A.,Porras. "Computer packagefor transient analysis of industrial multiple effect evaporators",Journal of Food Engineering, vol. 12, pp 267-281, 1990.

Deepak Kumar received hisDoctorate degree in AppliedMathematics from Indian Institute ofTechnology, Roorkee 2012. During2006-2012, he stayed in Researchduring Ph.D. program and received

Junior and senior research fellowships from Ministry ofHuman Resource Development, Govt. of India. Dr.Kumar has published many research papers In

International Journal of repute.Anjana Rani Gupta is working asProfessor and Head(Mathematics) atNIET. She completed her Ph.D degreefrom IIT Roorkee. She is having morethan 15 years of teaching experience atIIT Roorkee and various Engineering

Colleges affiliated to UPTU. Dr. Rani has publishedmany research papers in reputed journals and she is alsoa member of different Mathematical Societies.

Somesh Kumar received his M.C.Adegree from MJP RohilkhandUniversity, Bareilly in 2000, and M.E.and Ph.D. degrees from Dr. B. R.Ambedkar University, Agra in 2006and 20 II receptively. Between 2000

and 2011, he served with the SGI and Apeejay Groups.Currently he is working at NIET in the capacity ofProfessor & Head of IT department. Prof. Somesh haspublished a number of research papers in Elsevier,Springer,Inderscience, etc.

14 NIET Journal of Engineering & Technology, Vol. 5,2014

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