Calculos Em Engenharia Quimica

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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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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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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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2. S

TEADY S

TATE M

ATERIAL B

ALANCES ON A S

EPARATION

T

RAIN


Solut ion of simu ltan eous linear equat ions.


Mater ial balances on a steady st at e process with n o recycle.

2.3 Course Useage

Int roduction to Chemical Engineering.


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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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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3. VAPOR PRESSURE DATA REPRESENTATIONBY POLYNOMIALSAND EQUATIONS


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.


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.


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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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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4. REACTION EQUILIBRIUMFOR MULTIPLE GAS PHASE REACTIONS


Solution of systems of nonlinear algebraic equations.


Complex chemical equilibrium calculations involving multiple reactions.

4.3 Course Useage

Therm odynam ics or Reaction En gineering.


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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5. TERMINAL VELOCITYOF FALLING PARTICLES


Solution of a single nonlinear algebraic equation..


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.


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

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Calculos Em Engenharia Quimica