a collection of 10 numerical problems in chemical engineering

8/8/2019 A Collection of 10 Numerical Problems in Chemical Engineering

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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



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*

SimultaneousODE’s 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

SimultaneousODE’s 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*

SimultaneousODE’s 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 Stiff ODE’s

10




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/




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.)




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

V 2

-------+ V b–( ) R T =

a27

64------

R2T c

2

Pc

--------------

=

b R T c

8Pc

-----------=




Reduced pressu re is defined as

(4)

and the compressibility factor is given by

(5)

Pr P

Pc

------=

Z PV

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 of Pr .?




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




3. VAPOR PRESSURE DATA REPRESENTATION BY POLYNOMIALS AND 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 of T 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 a2T 2 a3T 3 ...+an T n+ + + +=




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 th e absolut e tempera tu re in K. Both 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 C which 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 AB

T 273.15+---------------------------–=

P( )log AB

T 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.




4. REACTION EQUILIBRIUM FOR 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 C A0 and C B0. The equilibrium const an ts K C1,

K C2 and K C3 have kn own values.

A B+ C D+↔ B C X Y +↔+

A X Z ↔+

K C 1

C C C D

C A C B----------------= K C 2

C X C Y

C B C C

-----------------= K C 3

C Z

C A C X

-----------------=

C A C A 0 C D– C Z –= C B C B 0 C D– C Y –=

C C C D C Y –= C Y C X C Z +=

C A C B C C C D C X C Y ,,,,, C Z

Solve this system of equations when C A0 = C B0 = 1.5, , an d

sta rting from four sets of initial estimat es.

(a)

(b)

(c)

K C 1 1.06= K C 2 2.63= K C 3 5=

C D C X C Z 0= = =

C D C X C Z 1= = =

C D C X C Z 10= = =




5. TERMINAL VELOCITY OF 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, ρ is the fluid density in kg/m 3, is th e diameter of the spherical

par ticle in m an d C D 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 µ is th e viscosity in Pa ⋅s or k g/m ⋅s.

v t

4 g ρ p ρ–( ) D p

3C Dρ-------------------------------------=

v t ρ p D p

C D 24 R e-------= for R e 0.1<

C D24

R e------- 1 0.14 R e

0.7+( )= for 0.1 R e 1000≤≤

C D 0.44= for 1000 R e 350000≤<

C D 0.19 84×10 R e ⁄ –= for 350000 R e<

R e D p v t ρ µ ⁄ =

(a) Calculate th e terminal velocity for par ticles of coal with ρp = 1800 kg/m

3

and =0.208×10 -3 m falling in water at T = 298.15 K wher e ρ = 994.6 kg/m 3 and µ = 8.931×10−4 kg/

m ⋅s.

(b) Estimate t he terminal velocity of the coal particles in water within a centrifugal separator

where t he a cceleration is 30.0 g.

D p




6. HEAT EXCHANGE IN A SERIES OF TANKS


Solut ion of simu ltan eous first order ordinar y different ial equat ions.


Uns teady sta te ener gy balan ces, dynam ic response of well mixed heated t an ks in series.

6.3 Course Useage

Heat Transfer.


Three t an ks in series are used t o preheat a m ulticomponent oil solution before it is fed to a d istillation

column for separa tion as sh own in Figure (2). Each tank is initially filled with 1000 kg of oil at 20 °C .

Satur ated steam at a temperatur e of 250°C condenses within coils immersed in each tank. The oil is

fed into the first tank at the rate of 100 kg/min and overflows into the second and the third tanks at

th e same flow rate. The temp erat ur e of the oil fed to the first ta nk is 20°C . The ta nks are well mixed

so tha t th e temperatur e inside the tan ks is uniform, and th e outlet stream t emperature is the temper-

ature within the tank. The heat capacity, C p, of the oil is 2.0 KJ /kg. For a par ticular ta nk, th e rat e at

which heat is tra nsferred t o the oil from th e stea m coil is given by th e expression

(18)

where UA = 10 kJ/min·°C is the product of the heat transfer coefficient and the area of the coil foreach tan k, T = temper atu re of th e oil in the tan k in , an d Q = rate of heat tran sferred in kJ /min.

Energy balances can be made on each of the individual tanks. In these balances, the mass flow

rat e to each t ank will remain at the sa me fixed value. Thus W = W 1 = W 2 = W 3. The mass in each ta nk will be assumed constant as the tank volume and oil density are assumed to be constant. Thus M =

M 1 = M 2 = M 3. For t he first ta nk, the en ergy balance can be expressed by

Accumulat ion = Inpu t - Outpu t

(19)

Note that the u nsteady state m ass balance is not needed for t ank 1 or any other tan ks since the m ass

Q U A T s t eam T –( )=

°C

T0=20oC

W1=100 kg/min

Steam

T1

Steam

T2

Steam

T3

T1 T2 T3

Figure 2 Series of Tank s for Oil Heating

M C p

d T 1

d t ----------- W C p T 0 U A T s t eam T 1–( ) W C p T 1–+=




in each ta nk does not chan ge with time. The a bove different ial equat ion can be rear ra nged an d explic-

itly solved for the derivative which is th e usu al forma t for nu merical solution.

(20)

Similar ly for the second ta nk

(21)

For the th ird tank

(22)

d T 1

d t ----------- W C p T 0 T 1–( ) U A T s t e a m T 1–( )+[ ] M C p( ) ⁄ =

d T 2


d T 3


Determine the steady sta te temperat ures in all three ta nks. What t ime interval will be required

for T 3 to reach 99% of th is steady stat e value during sta rtu p?




7. DIFFUSION WITH CHEMICAL REACTION IN A ONE DIMENSIONAL SLAB


Solution of second order ordinary differential equations with two point boundary conditions.


Methods for solving second order ordinary different ial equat ions with t wo point bound ary values typ-

ically used in tr an sport phenomena an d reaction kinet ics.

7.3 Course Useage

Transport Phenomena and Reaction Engineering.


The diffusion an d simulta neous first order irreversible chemical reaction in a single pha se cont aining

only reactan t A and product B resu lts in a second order ordinary different ial equat ion given by

(23)

where C A is the concentration of reactant A (kg mol/m 3), z is the distan ce variable (m), k is the homo-

geneous r eaction r ate const an t (s-1) an d D AB is th e bina ry diffus ion coefficient (m2 /s). A typical geom -

etry for Equation (23) is that of a one dimension layer which has its surface exposed to a known

concentra tion an d a llows n o diffusion across its bottom sur face. Thus th e initial a nd boun dar y condi-tions are

(24)

(25)

where C A0 is the const an t concentra tion a t th e surface (z = 0) an d th ere is no tran sport across the bot-

tom su rface (z = L) so th e deriva tive is zero.

This different ial equation ha s an an alytical solut ion given by

(26)

z2

2

d

d C A k

D A B

------------C A=

C A C A 0 for z = 0=

zd

d C A0 for z = L=

C A C A 0

L k D A B ⁄ ( ) 1 z L ⁄ –( )[ ]cosh

L k D A B ⁄ ( )cosh

-----------------------------------------------------------------------------=




(a ) Numerica l ly solve Equ ation (23) with t he bounda ry conditions of (24) and (25) for the case

where C A0 = 0.2 kg mol/m 3, k = 10 -3 s-1, D AB = 1.2 10-9 m2 /s, and L = 10 -3 m. This solution

should ut ilized a n ODE solver with a shooting t echn ique an d employ Newton’s m ethod or

some other technique for converging on the boundary condition given by Eq ua tion (25).

(b) Compare t he concentra tion profiles over the th ickness as predicted by the num erical solu-

tion of (a) with th e an alytical solution of Equation (26).




8. BINARY BATCH DISTILLATION


Solution of a system of equations comprised of ordinary differential equations and nonlinear

algebraic equat ions.


Batch distillation of an ideal binary m ixture.

8.3 Course Useage

Separ ation P rocesses.


For a binary batch distillation process involving two components designated 1 and 2, the moles of liq-

uid remaining, L , as a fun ction of the m ole fraction of th e componen t 2, x2, can be express ed by th e fol-

lowing equation

(27)

where k 2 is the vapor liquid equilibrium ratio for component 2. If the system may be considered ideal,

th e vapor liquid equilibrium ra tio can be calculat ed from where Pi is the vapor pressur e of

component i and P is the total pressure.

A common vapor pressure model is the Antoine equation which utilizes three parameters A , B,and C for componen t i as given below where T is the tempera tur e in °C.

(28)

The tem perat ure in th e batch still follow th e bubble point curve. The bubble point t empera tur e

is defined by th e implicit algebraic equation which can be written using t he vapor liquid equilibrium

rat ios as

(29)

Consider a binary mixture of benzene (component 1) and toluene (component 2) which is to be

considered a s ideal. The Antoine equ at ion constan ts for benzene ar e A1

= 6.90565, B1

= 1211.033 and

C 1 = 220.79. For t oluen e A2 = 6.95464, B 2 = 1344.8 an d C 2 = 219.482 (Dean 1). P is the pressure in mm

Hg and T the tempera ture in °C.

d L

d x2

---------L

x2 k 2 1–( )--------------------------=

k i P i P ⁄ =

P i 10

AB

T C +---------------–

=

k 1 x1 k 2 x2+ 1=

The batch distillation of benzene (component 1) and toluene (component 2) mixture is being car-

ried out at a pressure of 1.2 atm. Initially, there are 100 moles of liquid in the still, comprised of

60% benzene and 40% toluene (mole fraction basis). Calculate the amount of liquid remaining in

the still when concentr at ion of toluene rea ches 80%.




9. REVERSIBLE, EXOTHERMIC, GAS PHASE REACTION IN A CATALYTIC REACTOR


Simulta neous ordinar y different ial equat ions with k nown initial conditions.


Design of a gas p ha se cat alytic reactor with pr essure dr op for a first order reversible gas pha se reac-

tion.

9.3 Course Useage

Reaction En gineering


The elementar y gas pha se reaction is car ried out in a packed bed reactor. There is a heat

excha nger su rroun ding the r eactor, an d th ere is a pressu re drop along the length of the reactor.

The various para meter s values for this reactor design problem are sum mar ized in Table (3).

Table 3 Parameter Values for Problem 9.

CPA = 40.0 J /g-mol.K R = 8.314 J /g-mol.K

CPC = 80.0 J /g-mol.K FA0 = 5.0 g-mol/min

= - 40,000 J /g-mol Ua = 0.8 J /kg.min .K

EA = 41,800 J /g-mol.K Ta = 500 K

k = 0.5 dm 6 /kg⋅min ⋅mol @450 K = 0.015 kg-1

KC = 25,000 dm 3 /g-mol @450 K P 0 = 10 atm

CA0 = 0.271 g-mol/dm3 yA0 = 1.0 (Pur e A feed)

T0 = 450 K

2 A C

q

q

Ta

Ta

FA0

T0

X

T

Figure 3 Packed Bed Ca ta lytic Reactor

H R∆

α




Addition Information

The notat ion u sed here a nd t he following equations an d relationships for t his par ticular problem ar e

adapted from the textbook by Fogler.2 The problem is to be worked a ssumin g plug flow with n o radial

gradients of concentrations and temperature at any location within the catalyst bed. The reactor

design will use th e conversion of A designat ed by X and t he t empera tu re T which ar e both functions of

location within th e cata lyst bed specified by th e cata lyst weight W.

The general reactor design expression for a catalytic reaction in terms of conversion is a molebalance on reactan t A given by

(30)

The simple cata lytic reaction ra te expression for th is reversible reaction is

(31)

where t he ra te const an t is based on reacta nt A and follows th e Arrh enius expression

(32)

an d th e equilibrium const an t var iation with temper atu re can be determ ined from van’t H off’s equa-

tion with

(33)

The stoichiometry for an d th e stoichiometric ta ble for a gas allow th e concentr at ions to

be expressed as a function of conversion an d t empera tur e while allowing for volumetr ic cha nges du e

to decrease in moles du ring t he r eaction. Therefore

(34)

and

(35)

(a) Plot th e conversion (X), reduced pressure (y) and t emperatur e (T ×10 -3) along the reactorfrom W = 0 kg up to W = 20 kg.

(b) Aroun d 16 kg of cat alyst you will observe a “knee” in th e conversion pr ofile. Explain why t his

knee occurs and wh at pa ram eters affect th e knee.

(c) Plot th e concentr ation profiles for reactant A and product C from W = 0 kg up t o W = 20 kg.

F A 0

d X

d W --------- r ' A–=

r ' A– k C A2

C C

K C

--------–=

k k @T =450°K ( )

E A

R--------1

450---------1

T ----–exp=

C ˜P∆ 0=

K C K C @T =450°K ( )H R∆ R

-------------1

450---------

1

T ----–exp=

2 A C

C A C A 0

1 X –

1 ε X +----------------- P

P0------

T 0

T ------ C A 0

1 X –

1 0.5 X –----------------------

y

T 0

T ------= =

yP

P0

------=

C C

0.5C A 0 X

1 0.5 X –------------------------

yT 0

T ------=




The pr essure drop can be expressed a s a different ial equat ion (see Fogler2 for details)

(36)

or

(37)

The general energy balance may be written at

(38)

which for only reactant A in th e rea ctor feed simplifies to

(39)

d P

P0

------

d W ----------------

α 1 ε X +( )–

2-----------------------------

P0

P------

T

T 0------=

d y

d W ---------

α 1 0.5 X –( )–

2 y----------------------------------

T

T 0------=

d T

d W

---------U a T a T –( ) r ' A H R∆( )+

F A 0 θiC P i X C ˜

P∆+∑( )

---------------------------------------------------------------=

d T

d W ---------

U a T a T –( ) r ' A H R∆( )+

F A 0 C P A( )---------------------------------------------------------------=




10. DYNAMICS OF A HEATED TANK WITH PI TEMPERATURE CONTROL


Solution of ordinary differential equations, generation of step functions, simulation of a proportional

integral controller.


Closed loop dynamics of a process including first order lag and dead time. Padé approximation of time

delay.

10.3 Course Useage

Pr ocess Dynam ics an d Control


A continuous pr ocess system consisting of a well-stirr ed ta nk, hea ter an d PI temper atu re cont roller is

depicted in Figure (4). The feed strea m of liquid with den sity of ρ in kg/m3 and heat capacity of C in

kJ / kg⋅°C flows into the heated tan k at a constan t ra te of W in kg/min and t emperatur e T i in °C. The

volume of th e ta nk is V in m3. I t is desired to heat t his stream t o a higher set point temperatur e T r in

°C. The outlet temper at ur e is measur ed by a ther mocouple as T m in °C, an d the required hea ter inpu t

q in kJ /min is adjusted by a P I tem perat ur e cont roller. The cont rol objective is to maint ain T 0 = T r in

the presence of a change in inlet temperature T i which differs from th e steady sta te design tempera -

tur e of T is.

V, T

Heater TC

controllerPI

Set pointT r

Feed

W , T i, ρ, C pq

T m

Thermocouple

W, T 0 , ρ , C p

Measured

Figure 4 Well Mixed Tank with H eater and Tempera tur e Contr oller




Modeling and Control Equations

An energy balan ce on t he stirr ed tan k yields

(40)

with initial condition T = T r a t t = 0 which corr esponds t o steady stat e operat ion a t t he set point t em-

perature T r ..

The th ermocouple for tem perat ur e sensing in the outlet st ream is described by a first order sys-

tem plus the dead time τd which is th e time for the output flow to reach the mea sur ement point . The

dead t ime expression is given by

(41)

The effect of dead time m ay be calculat ed for t his situ at ion by t he Pa dé appr oximation which is a firstorder different ial equa tion for the measu red temper atu re.

I. C. T 0 = T r a t t = 0 (steady st ate) (42)

The above equation is used to genera ted th e tempera tur e input to the th ermocouple, T 0.

The thermocouple shielding and electronics are modeled by a first order system for the input

temperature T 0 given by

I. C. T m = T r a t t = 0 (steady sta te) (43)

where th e therm ocouple time const an t τm is known.The energy input to the ta nk, q, as m an ipulated by t he pr oport iona l/integral (PI) controller can

be described by

(44)

where K c is th e proportiona l gain of th e cont roller, τ I is the integra l time constan t or reset t ime. The qs

in th e above equat ion is th e energy input r equired at steady sta te for th e design conditions as calcu-

lated by

(45)

The integra l in Equa tion (44) can be conveniently be calculat ed by defining a new var iable as

I. C. errsum = 0 at t = 0 (steady sta te) (46)

Thus Equ ation (44) becomes

(47)

Let us consider some of the int eresting aspects of th is system as it r esponds t o a variety of par amet er

d T

d t --------

W C p T i T –( ) q+

ρV C p---------------------------------------------=

T 0 t ( ) T t τd –( )=

d T 0

d t ----------- T T 0–

τd

2-----

d T

d t --------

–2

τd

-----=

d T m

d t ------------

T 0 T m–

τm

---------------------=

q qs K c T r T m–( )K c

τ I ------- T r T m–( ) t d

0

t ∫ + +=

q s W C p T r T i s–( )=

t d

d errsum( ) T r T m–=

q q s K c T r T m–( )K c

τ I ------- errsum( )+ +=




an d opera tiona l cha nges.The nu merical values of th e system an d contr ol par am eters in Table (4) will

be considered a s leading to baseline steady sta te operation.

Table 4 Baseline System and Control Parameters for Problem 10

ρVC p = 4000 kJ /°C WC p = 500 kJ/min ⋅°C

T is = 60 °C T r = 80 °C

τd = 1 min τm = 5 min

K c = 50 kJ/min ⋅°C τ I = 2 min

(a) Demonst ra te th e open loop performa nce (set K c = 0) of this system wh en th e system is ini-

tially operating at design steady state at a temperature of 80°C, and inlet temperature T i is sud-denly changed to 40°C at time t = 10 min. Plot the temperatures T , T 0, and T m to steady state,

and verify that Padé approximation for 1 min of dead time given in Equation (42) is working

properly.

(b) Demonst ra te the closed loop performa nce of th e system for th e conditions of par t (a) an d the

baseline param eters from Table (4). Plot temperat ures T , T 0, an d T m to steady state.

(c) Repea t pa r t (b ) wit h K c = 500 kJ/min ⋅°C.

(d) Repeat part (c) for proportional only control action by setting the term K c / τ I = 0.

(e ) I mp lement lim it s on q (as per Equ ation (47)) so that th e maximum is 2.6 times th e baseline

steady state value and the minimum is zero. Demonstrate the system response from baseline

steady st ate for a proport iona l only cont roller wh en t he set point is cha nged from 80°C to 90°C at

t = 10 min. K c = 5000 kJ/min ⋅°C. Plot q and q lim versus time to steady stat e to demonstra te th e lim-

its. Also plot t he t empera tu res T , T 0, an d T

mto steady st at e to indicate controller per forman ce




REFERENCES

1. Dean, A. (Ed.), Lan ge’s Han dbook of Chemist ry, New York : McGraw-H ill, 1973.

2. Fogler, H. S. Elements of Chemical Reaction Engineering, 2nd ed., Englewood Cliffs, NJ : Pren tice-Hall,

1992.

3. Per ry, R.H., Green , D.W., an d Malorey, J.D., Eds. Perry’s Ch emical E ngineers Ha nd book . New York:

McGraw-Hill, 1984.

4. Shacham, M., Brau ner; N., and Pozin, M. Computers Chem Engng., 20 , Supp l. pp. S1329-S1334 (1996).

a collection of 10 numerical problems in chemical engineering

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