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ABSTRACT
Curr ent personal compu ters provide exceptiona l compu ting capabilities to engineer-
ing student s tha t can great ly improve speed and a ccur acy during sophisticated prob-
lem solving. The need t o actually creat e program s for mat hema tical problem solving
has been reduced if not eliminated by available mathematical software packages.
This paper summarizes a collection of ten typical problems from throughout thechemical engineering curriculum t ha t requires n umer ical solutions. These problems
involve most of th e stan dar d nu merical methods fam iliar to under gradu at e engineer-
ing students. Complete problem solution sets have been generated by experienced
users in six of the leading mat hema tical softwar e packages. These deta iled solut ions
including a write up and the electronic files for each package are available through
the INTERNET at www.che.utexas.edu/cache and via FTP from ftp.engr.uconn.edu/
pub/ASEE/. The written materials illustrate the differences in these mathematical
software packages. The electronic files allow hands-on experience with the packages
during execution of the actual software packages. This paper and the provided
resources should be of considera ble value dur ing ma them atical problem solving and/
or th e selection of a pa ckage for classr oom or per sonal u se.
iNTRODUCTION
Session 12 of th e Chemical En gineering Summ er School* at Snowbird, Utah on
* The Ch. E. Summer School was sponsored by the Chemical Engineering Division of the AmericanSociety for Engineering Education.
Micha el B. Cutlip, Depar tm ent of Chem ical En gineerin g, Box U-222, Un iversity
of Conn ecticut , Storr s, CT 06269-3222 (mcutlip@uconnvm .uconn .edu)John J. Hwalek, Department of Chemical Engineering, University of Maine,
Orono, ME 04469 (hwa lek@ma ine.ma ine.edu)
H. Eric Nuttall, Department of Chemical and Nuclear Engineering, University
of New Mexico, Albuqu erqu e, NM 87134-1341 (nut ta ll@un m.edu )
Mordechai Shacham, Department of Chemical Engineering, Ben-Gurion Uni-
versity of the N egev, Beer Sheva , Isra el 84105 (sha cham @bgum ail.bgu.ac.il)
Joseph Brule, John Widmann, Tae Han, and Bruce Finlayson, Department of
Chemical Engineering, University of Washington, Seattle, WA 98195-1750
([email protected] ington.edu )
Edward M. Rosen, EMR Technology Group, 13022 Musket Ct., St. Louis, MO
63146 (EMR ose@comp us er ve.com)
Ross Taylor, Department of Chemical Engineering, Clarkson University, Pots-
dam , NY 13699-5705 (taylor@sun .soe.clar kson.edu )
A COLLECTION OF TEN NUMERICAL PROBLEMS IN
CHEMICAL ENGINEERING SOLVED BY VARIOUS
MATHEMATICAL SOFTWARE PACKAGES
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Page 2 A COLLECTION OF TEN NUMERICAL PROBLEMS
August 13, 1997 was concerned with The Use of Mathematical Software in Chemical Engineering.
This session provided a ma jor overview of thr ee ma jor mat hema tical softwar e pa ckages (Math CAD,Mathematica, and POLYMATH), and a set of ten problems was distributed that utilizes the basic
numerical methods in problems that are appropriate to a variety of chemical engineering subject
area s. The problems a re titled according to th e chemical engineering principles th at ar e used, and t he
numerical methods required by the mathematical modeling effort are identified. This problem set is
summ ar ized in Table 1.
* Problem originally suggest ed by H. S. Fogler of the Un iversity of Michigan
** Problem preparation assistance by N. Brauner of Tel-Aviv University
Table 1 Problem Set for Use with Mathematical Software Packages
SUBJECT AREA PROBLEM TITLEMATHEMATICAL
MODEL PROBLEM
Introduction toCh. E.
Molar Volum e and Comp res sibility Factorfrom Van Der Waals Equ ation
Single NonlinearEquation
1
Introduction toCh. E.
Steady Stat e Material Balances on a Sep-ara tion Train*
Simultaneous Lin-ear Equa tions
2
MathematicalMethods
Vapor Pressure Data Representation byPolynomials and Equ ations
Polynomial F it-ting, Linear a ndNonlinear Regres-sion
3
Thermodynamics React ion Equil ibr ium for Mult iple GasPhase Reactions*
SimultaneousNonlinear Equa -tions
4
F lu id Dyn am ics Ter m in al Velocit y of Fa llin g P ar ticles S in gle N on lin ea rEquation
5
H ea t Tr an sfer Un st ea dy St at e H ea t E xch an ge in aSeries of Agitated Tan ks*
SimultaneousODEs with kn owninitial conditions.
6
Ma ss T ra n sfer Diffu sion wit h Ch em ica l Rea ct ion in aOne Dimensional Slab
SimultaneousODEs with splitbounda ry condi-tions.
7
SeparationProcesses
Binary Batch Dist illa t ion** Simultaneous Dif-ferential and Non-linear AlgebraicEquations
8
ReactionEngineering
Reversible, Exoth ermic, Gas Ph ase Reac-tion in a Cat alytic Reactor*
SimultaneousODEs an d Alge-braic Equations
9
Pr ocess Dynam icsand Control
Dynamics of a Heat ed Tan k with P I Tem-perature Control**
Simultaneous StiffODEs
10
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A COLLECTION OF TEN NUMERICAL PROBLEMS Page 3
ADDITIONAL CONTRIBUTED SOLUTION SETS
After th e ASEE Sum mer School, th ree more sets of solut ions wer e provided by au thors wh o hadconsiderable experience with additional ma th emat ical softwar e packages. The curr ent total is n ow six
packages, an d t he pa ckages (listed a lphabetically) and a ut hors a re given below.
Excel - Edwar d M. Rosen, EMR Techn ology Group
Maple - Ross Taylor, Clarks on Un iversity
Math CAD - John J. Hwalek, University of Maine
MATLAB - Joseph Bru le, John Widman n, Tae H an , and Br uce Finlayson, Depart ment of Chemi-
cal Engin eerin g, Un iversity of Wash ington
Math emat ica - H. Eric Nu tta ll, University of New Mexico
POLYMATH - Michael B. Cutlip, University of Connecticut and Mordechai Shacham, Ben-
Gurion Un iversity of the Negev
The complete pr oblem set ha s now been solved with the following ma them atical softwar e pack-
ages: Excel*, Maple, MathCAD, MATLAB , Mathematica#, and Polymath . As a service to the aca-
demic community, the CACHE Corporation** provides this problem set as well as the individual
package writeups and problem solution files for downloading on the WWW at http://
www.che.ut exas.edu/cache/. The pr oblem set a nd d eta ils of th e various solutions (about 300 pa ges) ar e
given in separa te docum ents as Adobe PDF files. The pr oblem solut ion files can be executed with t he
particular mathematical software package. Alternately, all of these materials can also be obtained
from an FTP site a t th e Un iversity of Conn ecticut: ftp.engr.uconn .edu/pub/ASEE /
USE OF THE PROBLEM SET
The complete problem writeups from the various packages allow potential users to examine the
detailed tr eatm ent of a variety of typical pr oblems. This met hod of present at ion s hould indicat e th e
convenience and str engths/weaknesses of each of the ma th emat ical softwar e pa ckages. The p roblem
files can be executed with t he corresponding softwar e package to obtain a sense of the pa ckage opera-
tion. Param eters can be cha nged, and the problems can be resolved. These activities should be very
helpful in t he evalua tion an d selection of appr opriate softwar e packages for p ersonal or educational
use.
Additionally attr active for engineering faculty is t ha t ind ividua l problems from th e problem set
can be easily int egrated into existing coursework. P roblem var iations or even open-ended pr oblems
can quickly be created. This problem set and the various writeups should be helpful to engineering
faculty who are continually faced with the selection of a mathematical problem solving package for* Excel is a tr adem ar k of Microsoft Corpora tion (htt p://www.microsoft.com) Maple is a t ra dema rk of Waterloo Maple, Inc. (ht tp://ma plesoft.com) MathCAD is a trademark of Mathsoft, Inc. (http://www.mathsoft.com) MATLAB is a tr adem ar k of The Ma th Works, Inc. (http ://www.mat hworks.com)
# Mathematica is a trademark of Wolfram Research, Inc. (http://www.wolfram.com)
POLYMATH is copyrighted by M. B. Cutlip an d M. Sha cham (http ://www.che.utexas /cache/)
** The CACHE Corporat ion is n on-profit educational corporation supported by m ost chemical engineering depar tmen tsand ma ny chemical corporation. CACHE stan ds for comput er a ides for chemical engineering. CACHE can be conta ctedat P. O. Box 7939, Aust in, TX 78713-7939, Ph one: (512)471-4933 Fax: (512)295-4498, E-m ail: cache@ut s.cc.utexa s.edu ,Internet: http://www.che.utexas/cache/
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A COLLECTION OF TEN NUMERICAL PROBLEMS
use in conjunction with t heir courses.
THE TEN PROBLEM SET
The complete pr oblem set is given in th e Appendix to th is paper. Each problem sta temen t car efully
identifies the n um erical met hods used, the concepts ut ilized, and t he genera l problem cont ent.
APPENDIX
(
Note to Reviewers
- The Appendix which follows can either be printed with the article or provided
by the au th ors as a Acrobat P DF file for the disk wh ich n ormally accompanies th e CAEE Jour na l. File
size for t he P DF docum ent is about 135 Kb.)
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A COLLECTION OF TEN NUMERICAL PROBLEMS
Page 5
1. M
OLAR
V
OLUME AND C
OMPRESSIBILITY
F
ACTOR
FROM V
AN
D
ER
W
AALS
E
QUATION
1.1 Numerical Methods
Solution of a single nonlinear algebraic equation.
1.2 Concepts Utilized
Use of the van der Waals equation of state to calculate molar volume and compressibility factor for a
gas.
1.3 Course Useage
Int roduction to Chemical En gineering, Thermodynamics.
1.4 Problem Statement
The ideal gas law can represen t t he pr essure-volume-temperat ur e (PVT) relationship of gases only at
low (near atmospheric) pressures. For higher pressures more complex equations of state should be
used. The calculation of the molar volume and the compressibility factor using complex equations of
sta te typically requires a nu merical solution when the pressu re an d tempera tu re ar e specified.
The van der Waa ls equation of sta te is given by
(1)
where
(2)
and
(3)
The variables are defined by
P = pressure in atm
V
= molar volume in liters/g-mol
T = temperatu re in K
R
= gas constan t (
R
= 0.08206 atm
.
liter/g-mol
.
K)
T
c
= critical tem perat ur e (405.5 K for a mmonia)
P
c
= critical pr essure (111.3 atm for a mmonia)
Pa
V2
-------+ V b( ) R T=
a27
64------
R2Tc
2
Pc--------------
=
bR Tc
8Pc-----------=
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A COLLECTION OF TEN NUMERICAL PROBLEMS
Reduced pressu re is defined as
(4)
and the compressibility factor is given by
(5)
PrP
Pc------=
ZPV
R T---------=
(a ) Calculate the molar volume and compressibility factor for gaseous ammonia at a pressure
P = 56 atm a nd a t emperature T = 450 K using th e van der Waa ls equation of sta te.
(b ) Repeat the calculations for th e following redu ced pr essures: Pr= 1, 2, 4, 10, and 20.
(c ) How does th e compr essibility factor var y as a function ofPr.?
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A COLLECTION OF TEN NUMERICAL PROBLEMS
Page 7
2. S
TEADY S
TATE M
ATERIAL B
ALANCES ON A S
EPARATION
T
RAIN
2.1 Numerical Methods
Solut ion of simu ltan eous linear equat ions.
2.2 Concepts Utilized
Mater ial balances on a steady st at e process with n o recycle.
2.3 Course Useage
Int roduction to Chemical Engineering.
2.4 Problem Statement
Xylene, styrene, toluene and benzene are t o be separa ted with t he ar ray of distillat ion column s th at is
shown below wher e F, D, B, D1, B1, D2 and B2 a re t he m olar flow rat es in m ol/min.
15% Xylene
25% Styrene
40% Toluene
20% Benzene
F=70 m ol/min
D
B
D1
B1
D2
B2
{
{
{
{
7% Xylene4% Styrene
54% Toluene35% Benzene
18% Xylene24% Styrene42% Toluene16% Benzene
15% Xylene10% Styrene54% Toluene21% Benzene
24% Xylene65% Styren e10% Toluene
1% Benzene
#1
#2
#3
Figure 1 Separation Train
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A COLLECTION OF TEN NUMERICAL PROBLEMS
Materia l balances on individual components on the overall separ at ion t ra in yield the equa tion set
(6)
Overall balances and individual component balances on colum n #2 can be u sed to determine t he
molar flow ra te a nd m ole fractions from th e equat ion of stream D from
(7)
where X
Dx
= mole fraction of Xylene, X
Ds
= mole fraction of Styrene, X
Dt
= mole fraction of Toluene,
and X
Db
= mole fraction of Benzene.
Similarly, overall balances and individual component balances on column #3 can be used to
determ ine the m olar flow rat e and mole fra ctions of str eam B from the equa tion set
(8)
Xylene: 0.07D1
0.18B1
0.15D2
0.24B2
0.15 70=+ + +
Styrene: 0.04D1
0.24B1
0.10D2
0.65B2
0.25 70=+ + +
Toluene: 0.54D1
0.42B1
0.54D2
0.10B2
0.40 70=+ + +
Benzene: 0.35D1
0.16B1
0.21D2
0.01B2
0.20 70=+ + +
Molar Flow Rat es: D = D1 + B 1
Xylene: XDx
D = 0.07D1
+ 0.18B1Styrene: XDsD = 0.04D1 + 0.24B1
Toluen e: XDt D = 0.54D1 + 0.42B1Ben zen e: XDbD = 0.35D1 + 0.16B1
Molar Flow Rat es: B = D2 + B 2
Xylene: XBxB = 0.15D2 + 0.24B2
Styrene: XBsB = 0.10D2 + 0.65B2Toluene: XBt B = 0.54D2 + 0.10B2Benzene: XBbB = 0.21D2 + 0.01B2
(a ) Calculat e the molar flow rates of strea ms D1, D2, B1 and B2.
(b ) Determine t he molar flow rates a nd compositions of str eams B and D.
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A COLLECTION OF TEN NUMERICAL PROBLEMS Page 9
3. VAPOR PRESSURE DATA REPRESENTATIONBY POLYNOMIALSAND EQUATIONS
3.1 Numerical Methods
Regression of polynomia ls of var ious degr ees. Linea r regr ession of ma th ema tical models with var iable
tr an sformat ions. Nonlinear regression.
3.2 Concepts Utilized
Use of polynomials, a modified Clausius-Clapeyron equation, and the Antoine equation to model
vapor pressure versus temperatu re data
3.3 Course Useage
Math emat ical Meth ods, Therm odyna mics.
3.4 Problem Statement
Table (2) presents data of vapor pressure versus temperature for benzene. Some design calculations
require these data to be accurately correlated by various algebraic expressions which provide P inmmH g as a function ofT in C.
A simple polynomial is often used a s an empirical modeling equa tion. This can be written in gen-
eral form for th is problem a s
(9)
where a0... an ar e the p ara meter s (coefficients) to be determined by regression a nd n is the degree of
th e polynomial. Typically the degree of th e polynomial is selected which gives th e best data r epresen -
Table 2 Vapor Pressure of Benzene (Perry3)
Temperature,T(oC)
Pressure, P(mm Hg)
-36.7 1
-19.6 5
-11.5 10
-2.6 20
+7.6 40
15.4 60
26.1 100
42.2 200
60.6 400
80.1 760
P a0 a1T a2T2 a3T
3 ...+an Tn+ + + +=
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Page 10 A COLLECTION OF TEN NUMERICAL PROBLEMS
tat ion when using a least-squares objective function.
The Clausius-Clapeyron equation which is useful for the correlation of vapor pressure data isgiven by
(10)
where P is the vapor pressure in m mHg an d T is the tempera tu re in C. Note tha t th e denominator is
just the abs olu te t em per a ture in K. Bot h A and B are the parameters of the equation which are typi-
cally determined by regression.
The Ant oine equa tion which is widely used for the r epresenta tion of vapor pressu re da ta is given
by
(11)
where typically P is the vapor pressure in mmH g and T is the temperatu re in C. Note tha t t his equa-
tion has par ameters A ,B, and Cwhich m ust be determined by n onlinear regression a s it is not possi-
ble to linear ize this equa tion. The Antoine equ at ion is equivalent t o the Clau sius-Clapeyron equat ion
when C= 273.15.
P( )log A BT 273.15+---------------------------=
P( )log A BT C+---------------=
(a) Regress the data with polynomials having the form of Equ ation (9). Determ ine th e degree of
polynomial which best repr esents t he dat a.
(b) Regress the data using linear regression on Equa tion (10), the Clausius-Clapeyron equa tion.
(c) Regress the data using nonl inear regression on Equa tion (11), the Antoine equa tion.
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A COLLECTION OF TEN NUMERICAL PROBLEMS Page 11
4. REACTION EQUILIBRIUMFOR MULTIPLE GAS PHASE REACTIONS
4.1 Numerical Methods
Solution of systems of nonlinear algebraic equations.
4.2 Concepts Utilized
Complex chemical equilibrium calculations involving multiple reactions.
4.3 Course Useage
Therm odynam ics or Reaction En gineering.
4.4 Problem Statement
The following rea ctions ar e ta king place in a const an t volume, gas-pha se bat ch reactor.
A system of algebraic equations describes the equilibrium of the above reactions. The nonlinear
equilibrium r elationsh ips utilize the t herm odyna mic equilibrium expressions, an d th e linear relation-
ships h ave been obtained from th e stoichiometry of th e rea ctions.
(12)
In th is equat ion set an d are concentra tions of th e various species at
equilibrium resulting from initial concentrations of only CA0 and CB0. The equilibrium const an ts KC1,
KC2 and KC3 have kn own values.
A B+ C D+B C X Y++A X Z+
KC1
CCCD
CA CB----------------= KC2
CXCY
CB CC
-----------------= KC3
CZ
CA CX-----------------=
CA CA 0 CD CZ= CB CB 0 CD CY=
CC CD CY= CY CX CZ+=
CA CB CC CD CX CY,,,,, CZ
Solve this system of equations when CA0 = CB0 = 1.5, , an d
sta rting from four sets of initial estimat es.
(a)
(b)
(c)
KC1 1.06= KC2 2.63= KC3 5=
CD CX CZ 0= = =
CD CX CZ 1= = =
CD CX CZ 10= = =
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Page 12 A COLLECTION OF TEN NUMERICAL PROBLEMS
5. TERMINAL VELOCITYOF FALLING PARTICLES
5.1 Numerical Methods
Solution of a single nonlinear algebraic equation..
5.2 Concepts Utilized
Calculat ion of term ina l velocity of solid par ticles falling in flu ids un der t he force of gravity.
5.3 Course Useage
Fluid dyna mics.
5.4 Problem Statement
A simple force balance on a spherical particle reaching terminal velocity in a fluid is given by
(13)
wher e is th e ter min al velocity in m/s, g is th e accelera tion of gravit y given by g = 9.80665 m/s2,
is the particles density in kg/m 3, is the fluid density in kg/m 3, is th e diameter of the sphericalpar ticle in m an d CD is a dimensionless drag coefficient.
The dr ag coefficient on a spherical part icle at term inal velocity varies with th e Reynolds n um ber
(R e) as follows (pp. 5-63, 5-64 in Perry3).
(14)
(15)
(16)
(17)
where and is th e viscosity in Pa s or k g/m s.
v t
4 g p ( )Dp3CD
-------------------------------------=
v t pDp
CD 24R e-------= for R e 0.1