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Process Modeling &Simulation

Chapter 3:Mathematic Models of ChemicalEngineering System

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Chemical Engineering System

1. Series of Isothermal, Constant- HoldupCSTRs

2. CSTRs with Variable Holdups

3. Two Heated Tan s!. "as-phase, #ressuri$ed CSTR %. &onisothermal CSTR '. Sin(le-Component Vapori$er

). *ulti+omponent lash rum. /at+h Rea+tor

0. Rea+tor with *ass Transfer1 .Ideal /inar istillation Column11. /at+h istillation Column

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1. SERIES OF ISOTHERMAL, O!STA!T"HOL#$PSTRs

Product B is produced and reactant A is consumed.

A first-order reaction occurring in the liquid.

Assume that the temperatures and holdups (volumes) of the threetanks can be different, but both temperatures and the liquid volumes areassumed to be constant (isothermal and constant holdup).

Densit is assumed constant throughout the s stem, !hich is a binar

mi"ture of A and B.

A Bk -r# k$ A

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%hus the %$& for the first reactor is'

ike!ise total mass balances on tanks and * give

+here is defined as the throughput (m * min)

$$& for reactant A'

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%he specific reaction rates k, are given b the Arrhenius equation

f the temperatures in the reactors are different, the k/s aredifferent. %he n refers to the stage number.

%he volumes 0, can be pulled out of the time derivativesbecause the are constant. %he flo!s are all equal to butcan var !ith time.

An energ equation is not required because !e haveassumed isothermal operation. An heat addition or heatremoval required to keep the reactors at constanttemperatures could be calculated from a stead state energbalance (1ero time derivatives of temperature).

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%he three first-order nonlinear ordinar differentialequations given in &qs. (*.*) are the mathematical modelof the s stem.

%he parameters that must be kno!n are 0 2 , 0 ) , 0 * , k l, k ) ,and k * , .

%he variables that must be specified before theseequations can be solved are and $ A A ,. 34pecified5 doesnot mean that the must be constant. %he can be time-var ing, but the must be kno!n or given functions oftime. %he are the forcing functions.

%he initial conditions of the three concentrations (theirvalues at time equal 1ero) must also be kno!n.

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+here !ith units of minutes. %here is onl oneforcing function or input variable, $A 6

f the throughput is constant and the holdups and

temperatures are the same in all three tanks, &qs. (*.*)become

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EXAMPLE:

$onsider a s stem !ith * $4%7s in series previousldiscussed. 8iven are'

4imulate the concentration profile of this reaction s stem!ith step si1e, 9t # 6.2 min starting at t#6min to t#6.:min

b using e"plicit &uler method. (4tate our calculationvalue to * decimal places)

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Answer

TIME(MIN)

CA1 CA CA3

. .! .2 .1.1 .!% .2 .1.2 .!0% .2 3 .1.3 .%3' .2 ) .1.! .%)2 .213 .1.% .' % .22 .1 1

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%. STRs ITH 'ARIA(LE HOL#$PSf the previous e"ample is modified slightl to permit the volumes in

each reactor to vary with time , both total and component continuityequations are required for each reactor . %o sho! the effects of higher-order kinetics, assume the reaction is now nth-order in reactant A.

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;ur mathematical model no! contains six first-order nonlinear ordinarydifferential equations.

Parameters that must be kno!n are k 2, k , k *, and n.

nitial conditions for all the dependent variables that are to be integrated must begiven< $A 2, $A , $A *, 0 , 0 , and 0 *.

%he forcin! functions $A o(t) and o(t) , must also be given.

et us no! chec" the de!rees of freedom of this s stem.

%here are six equations . But there are nine un"no#ns < $A2,$A ,$A *,0 ,0 ,0 *, 2, and *

$learl this s stem is not sufficiently specified and a solution could not beobtained.

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What have we missed in ourmodeling?

A good plant operator could take one look at the s stem and see !hatthe problem is.+e have not specified how the flows out of the tanks are to be set.Ph sicall there !ould probabl be control valves in the outlet lines toregulate the flo!s.=o! are these control valves to be set> A common configuration is tohave the level in the tank controlled b the outflo!, i.e., a levelcontroller opens the control valve on the e"it line to increase theoutflo! if the level in the tank increases. Thus there must be arelationship between tank holdup and flow.

%he f functions !ill describe the level controller and the control valve.%hese three equations reduce the degrees of freedom to 1ero.

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). T O HEATE# TA!*Set us consider a process in !hich t!o energ balances are needed tomodel the s stem.

%he flo! rate of oil passing through t!o perfectl mi"ed tanks inseries is constant at ?6 ft * min. %he densit p of the oil is constant at@6 lbm ft*, and its heat capacit $ p, is 6. Btu lbm. . %he volume of the

first tank 0 2, is constant at @:6 ft *, and the volume of the second tank0 , is constant at ?6 ft *. %he temperature of the oil entering the firsttank is % o, and is 2:6 at the initial stead state. %he temperatures inthe t!o tanks are % 2 and % .%he are both equal to :6 at the initial stead state. A heating coilin the first tank uses steam to heat the oil. et C 2 be the heat additionrate in the first tank.%here is one energ balance for each tank.

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4ince the throughput is constant ; # 2 # # .

4ince volumes, densities, and heat capacities are all constant, &qs.(*.26) and (*.22) can be simplified

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Let’s check the degrees of freedom ofthis system.%he parameter values that are kno!n are p , $ p, 0 2, 0 , and . %he heat input tothe first tank C 2 !ould be set b the position of the control valve in the steam line.

%hus !e are left !ith t!o dependent variables, %2

and % , and !e have t!oequations. 4o the s stem is correctl specified.

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T$TORIAL

#lease refer handout.1. inal 4am 5+t 2 !2.62, inal 4am 7pril 2 11

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A!S ER +1

t(min) T1(° )

dT1!dt T"(° ) dT"!dt

0 28 0.4933 28 0.4933

1 28.4933

0.4265 28.4933

0.4922

2 28.919

8

0.3686 28.985

5

0.4824

3 29.2884

0.3187 29.4679

0.4661

4 29.6071

0.2755 29.9340

0.4454

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4. GAS-PHASE PRESSURIZED CSTR

-r $% "$&A$.'

-r ( % "( &)

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Assume < sothermal s stem, ∴ % # constant

4 stem volume, 0 constant Perfect gases appl

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Pa!e *+

V d ρ )(d ρtotal mass balance around the vessel gives:

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oo f f F F dt Vd

ρ ρ ρ −=)( oo f f F F dt d

V ρ ρ ρ −=total mass balance around the vessel gives:

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. !O!ISOTHERMAL STR

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A- PERFE TL MI/E# OOLI!0 A *ET

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Tutorial 2+ , Final E3am A4ril %5567

Refer handout (i8en..T k # T $ f(T)

530.00 0.0302 0.5308 530.00 530.59

530.59 0.0311 0.5302 530.55 531.13

537.30 0.0444 0.5222 536.74 537.34537.81 0.0456 0.5215 537.21 537.81

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(- PL$0 FLO OOLI!0 A *ETPlug Flow: constant velocity of flow in every part of a system.

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- L$MPE# A *ET MO#EL

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#- SI0!IFI A!T METAL ALL APA ITA! E

- Eass of metal !alls and its effects on the thermald namics must be considered.

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TutorialHandout963, inal 4am, 5+t 2 7n endothermi+ rea+tion of rea+tant : to produ+t ; is

+arried out in

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, EA /, A EM0 EL

L 12 A3 4AP05/3AM &, M0 EL

6E5MAL

E12 L )5 2M M0 EL

L 12 -P6A,E/3AM &, M0 EL

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%&' L * +,-+' 'T#-+/&0 /

To describe these boiling systems

rigorously , conserv tion e!u tions "orboth the v #or nd li!uid #h ses reneeded .

The b sic #roblem is $nding the r te o"v #ori% tion o" m teri l "rom the li!uid#h se into the v #or #h se .

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&'( is "ed into #ressuri%ed t n) to hold

the li!uid level in thet n). *ssume &'( is

#ure com#onent+#ro# ne

The li!uid inthe t n) is

ssumed#er"ectly

mi ed -e t is dded t

r te, Q to hold thedesired #ressure inthe t n) by

v #ori%ing theli!uid t r te, W v

m ss #er time/.

-e t losses ndthe m ss o" thells re ssumed

negligible

( s is dr n othe to# o" the t n)

t volumetrico r te, F v

forcing functions /.

&et consider &i!ue$ed #etroleum g s&'(/ v #ori%er system.

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#. %T # 2%T#T ,+ L

The sim#lest model ould neglect thedyn mics o" both v #or nd li!uid#h ses nd rel te the g s r te, F v to the

he t in#ut by+

cal/g)or bfeed(Btu/lliquidof enthalpycal/g)orlbtank(Btu/leavingvaporof enthalpywhere,

)(

m0

m

0

==

=−

h H

Qh H F

v

vvv ρ

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3. L&45& *-6#% 2'#,& %

,+ L ore re listic model is obt ined i"ssume th t the volume o" the v #or

#h se is sm ll enough to m )e itsdyn mics negligible .

" only "e moles o" li!uid h ve to bev #ori%ed to ch nge the #ressure in the

v #or #h se. o, e c n ssume th t this #ressure

is l ys e!u l to the v #or #ressure o"

the li!uid t ny tem#er ture - 7 - v

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1/ * tot l continuity e!u tions"or the li!uid #h se ,

2 3/ The t o controllere!u tions rel ting #ressure to he tin#ut nd the li!uid level to "eed

o r te, 0 lso needed.)(0)(! LV P f F f Q ==

vvoo F F ρ ρ ρ −=dt

d"

:continuity#otal

$

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4/ *n energy e!u tion "or li!uid

#h se gives the tem#er ture s "unction o" time/.

The li!uid is ssumed incom#ressibleso, # v nd intern l energy, U is # T.

The enth l#y o" the v #or le ving thev #ori%er is ssumed to be sim#le "orm +

# T : ; v.QT C F T F C

dt T V d

C

Energy

v pvvoo po L

p ++−= )()(

:

λ ρ ρ ρ

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5/ *n e!u tion o" st te "or the v #or is needed to be ble to c lcul te density,< v "rom the #ressure or tem#er ture. The#er"ect g ses is used.

6/ The v #or #ressure rel tionshi#gives the #ressure in the v #ori%er tth t tem#er ture

BT A

P +=ln

: pressure"apor

RT MP

v = ρ

:%tate

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. L&45& #' #-+/

2'#,& % ,+ L or the c ses, i" the dyn micso" the v #or #h se cannot :eneglected i" e h ve l rgevolume o" v #or /.

Tot l continuity nd energye!u tions "or the g s in the t n)is needed .

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The e #ression "or boiling r te, W v in

term o" #ressure di erenti l s driving"orce

*t some tem#er ture, li!uid boilsbec use it e erts v #or #ressure P

gre ter th n the #ressure Pv in thev #or #h se bove it.

*t e!uilibrium, P = P v . " li!uid ndv #or re in e!uilibrium, K MT is very

t&coefficientransfermass pseudotheis

)(

MT

v MT v

K

P P K W −=

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@!u tions describing the system "orli;uid

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The systems h ve 10 v ri bles.

Q, F 0 ,V L ,W v , T, V v , ρ v T v , P nd P v

@!u tions+ 10 e!u tions

&i!uid #h se 3 A #or #h se 3 Boiling r te 1 ontroller e!u tion 2 tot l volume A & : A v/ o" t n) 1

>egree o" "reedom 0

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. T6 /,#L 45&L&3/&5,,+ L or this c se, therm l e!uilibrium

bet een li!uid nd v #or is ssumed tohold t ll times.

The v #or nd li!uid tem#er tures ressumed e!u l to e ch other, T 7 T v .

@limin tes the need "or n energyb l nce "or the v #or #h se. C sensiblehe t o" v #or is usu lly sm ll com# red

ith l tent he t e ects.

The sim#le enth l#y rel tionshi#s c n beused,

QT C W T C F dt

T V d C v pvo poo

L p

++−= )()(

λ ρ ρ

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, EA /, A EM0 EL

P5A& &AL M0 EL

5 70502, M0 EL

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,5LT& +,-+' 'T 9L#%6

/5, ystem "or v #or?li!uid ith more th n

one com#onent.

f d P G b bbl i

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iquid streamat high % F Pflashed into a

drum

P is reducedas it flo!s

throughrestriction

(valve)

%his e"pansion isirreversible and occurs

at constant enthalp

f drum P G bubblepointP of feed at % 6, some of

liquid !ill vapori1e. 8as is dra!n offthrough a control

valve !hose steamposition is set b Pcontroller

iquid comesoff the bottom

of tank on levelcontroller

P 6 is highenough to

prevent anvapori1ationof feed at % 6

and " 6H

Adiabatic conditions areassumed (no heatlosses)

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The "orcing "unctions in this system re"eed tem#er ture, T 0D "eed r te, D nd "eed

com#osition, 0E. >ensity o" li!uid in the t n), < & isssumed to be )no n "unction o"

tem#er ture, T nd com#osition, E. >ensity o" v #or in the drum is )no n"unction o" T, com#osition, y E nd #ressure,

' .

componentth

of weightmolecularwhere

gasof htweig

molecularaveragee wher

!

av

avav

j

M y M M

M

RT

P M

j j

C

j jv

vv

v

==

==

∑=

ρ

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#. %T # 2%T#T ,+ L

The system hich neglects dyn micscom#letely .

'ressure is ssumed const nt , ndthe ste dy st te T @, @ nd energy

b l nces re used .

A #or nd li!uid #h ses re ssumedto be in e!uilibrium .

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!T!P"#y " !T #$ " !T #$

L Lvvoo

P T $ j

j L

L L j

v

vv j

oo

L Lvvoo

j j j

j

f H f h f h

F hρ F Hρ F ρh

f y

$ M

F y

M F

$ M

F

F F F

===

+=

=

+=

+=

00000

0

),,(

avav0av0

: properties#hermal

:equationnergy

:mequilibriuliquid+"apor

:ontinuityomponent

:ontinuity#otal

ρ ρ ρ

ρ ρ ρ

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3. /& +/+5% ,+ L

An equilibrium-flash calculation is made at each point intime to find vapor and liquid flo! rates and propertiesimmediatel after the pressure letdo!n valve (usingsame equations as in $ase A).

%7I JJ

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These t o stre ms re then "ed into

the v #or nd li!uid #h ses.0apor phase

iquid phase

%7I JJ

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j%v L

L L j%v

v

v j%v

L

L L%v

v

j L L

L L L Lv L L L L L L

v L L L L L L

v L MT v

$ M

F ρ y

M W

$ M

ρ F dt

M

$ ρV d

CCE

h F H W h F dt

U V d

W F F

dt

V d

P P K W

−−=

−−=

−−=

−=

,---

---

--

---

:

)(

:nergy

:ontinuity#otal

)( :rateBoiling

ρ ρ ρ

ρ ρ ρ

or li!uid #h ses,

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or v #or #h ses,

v

v%v

v

j%vv

vv j%v

v

v j%v

v

v%vv

jvv

vvv Lvvvvvvv

vvvvvvv

RT

P M

&t%te

y M F ρ

y M W

y M

ρ F

dt

M

y ρV d

CCE H F H W H F dt

U V d

F W H F dt

V d

v

v

=

−+=

−+=

−+=

ρ

ρ ρ ρ

ρ ρ

:

---

:

)(

:nergy

:ontinuity#otal

,--

---

--

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*ddition o"1/multi?com#onent v #or?li!uid

e!uilibrium e!u tion to c lcul te ' &.

2/F ?1 @ "or e ch #h se

ontroller e!u tions rel ting A & to & nd ' v to v .

)()( v L P vV L f F f F ==

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. -/# T& #L ,+ L

or c ses th t ignore the dyn micso" the v #or #h se s in c se B+

&i!uid?#h se dyn mics model/.

The v #or is ssumed to be l ysin e!uilibrium ith the li!uid .

o, conserv tion e!u tions reritten "or li!uid #h se only.

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h F H F h F dt

hV d

$ M

ρ F y

M F

$ M

ρ F dt

M $ ρV d

CCE

F F F dt

V d

L Lvv L L

j%v L

L L j%v

v

vv j%v

%v L

j L L

L Lvv L L

ρ ρ ρ ρ

ρ

ρ ρ ρ ρ

−−=

−−=

−−=

000

00

00

00

)(

:nergy

:

:ontinuity#otal

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The F v #or?li!uid e!uilibriume!u tions, three enth l#y

rel tionshi#s, t o density e!u tions,molecul r eight e!u tions, nd

"eedb c) controller e!u tions re llneeded.

tot l number o" e!u tions must

e!u l to tot l number o" v ri bles *ll the "eed #ro#erties or "orcing

"unctions re given.

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Tutorial 2+ , Final E3am Oct %5567

7 brine solution is to be e8aporated in an e8aporator to ield asolution of % F salt. The feed mole fra+tions of salt and waterare 4 so and 4 wo respe+ti8el with its densit , G o . The feedtemperature and pressure for the s stem are T o and # o ,respe+ti8el . / +onsiderin( a ri(orous model @li>uid and 8apor

d nami+sB for the e8aporation pro+ess and with an aid of adia(ram,aB eri8e the total +ontinuit e>uations for the 8apor and li>uid

phase. bB eri8e the +omponent +ontinuit e>uation for e8aporated

water.+B eri8e the ener( +ontinuit e>uations for the 8apor and li>uid

phase of the s stem.