Process Modeling & Simulation
Chapter 3: Mathematic Models of Chemical Engineering System
Chemical Engineering System 1. Seri Series es of Iso Isoth ther erma mal, l, Cons Consta tant nt-- Hold Holdup up CSTRs 2. CS CSTR TRss wit with h Var Varia iable ble Ho Hold ldup upss 3. Two He Hea ated Ta Tan nss !. "a "ass-ph phas ase, e, #re #ress ssur uri$ i$ed ed CST CSTR R %. &o &oni niso soth ther erma mall CS CSTR TR '. Si Sin( n(le le-C -Com ompo pone nent nt Vapo Vapori ri$e $err ). *u *ult lti+ i+om ompo pone nent nt l las ash h r rum um . /at+h Re Rea a+t +tor or 0. Re Rea+ a+to torr with with *as *asss Tra Trans nsfe ferr 1.Ideal /inar istillation Column 11. /at /at+h +h istillat istillation ion Column Column
1. SERIES OF ISOTHERMAL, O!STA!T"HOL#$P STRs
A Product
k
B
-r# k$ A
B is produced and reactant A is consumed.
A first-order
reaction occurring in the liquid.
Assume
that the temperatures temperatures and holdups (volumes) (volumes) of the three tanks can be different, but both temperatures and the liquid volumes are assumed to be constant (isothermal and constant holdup).
Densit
is assumed constant throughout the sstem, !hich is a binar mi"ture of A and B.
%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'
%he
specific reaction rates k, are given b the Arrhenius equation
f
the temperatures in the reactors are different, the k/s are different. %he n refers to the stage number. %he
volumes 0, can be pulled out of the time derivatives because the are constant. %he flo!s are all equal to but can var !ith time. An
energ equation is not required because !e have assumed isothermal operation. An heat addition or heat removal required to keep the reactors at constant temperatures could be calculated from a stead state energ balance (1ero time derivatives of temperature).
%he
three first-order nonlinear ordinar differential equations given in &qs. (*.*) are the mathematical model of the sstem.
%he
parameters that must be kno!n are 0 2, 0) , 0*, kl, k), and k*, .
%he
variables that must be specified before these equations can be solved are and $ A,. 34pecified5 does not mean that the must be constant. %he can be timevaring, but the must be kno!n or given functions of time. %he are the forcing functions.
%he
initial conditions of the three concentrations (their values at time equal 1ero) must also be kno!n.
f
the throughput is constant and the holdups and temperatures are the same in all three tanks, &qs. (*.*) become
+here !ith units of minutes. %here is onl one forcing function or input variable, $A 6
EXAMPLE: $onsider a sstem !ith * $4%7s in series previousl discussed. 8iven are'
4imulate the concentration profile of this reaction sstem !ith step si1e, 9t # 6.2 min starting at t#6min to t#6.:min b using e"plicit &uler method. (4tate our calculation value to * decimal places)
Answer TIME (MIN)
CA1
CA
CA3
. .1
.! .!%
.2 .2
.1 .1
.2 .3 .! .%
.!0% .%3' .%)2 .'%
.23 .2) .213 .22
.1 .1 .1 .11
%. STRs ITH 'ARIA(LE HOL#$PS f
the previous e"ample is modified slightl to permit the volumes in each reactor to vary with time, both total and component continuity equations are required for each reactor . %o sho! the effects of higherorder kinetics, assume the reaction is now nth-order in reactant A.
;ur
mathematical model no! contains six first-order nonlinear ordinary differential equations.
Parameters
that must be kno!n are k2, k, k*, and n.
nitial
conditions for all the dependent variables that are to be integrated must be given< $A2, $A, $A*, 0, 0, and 0*.
%he et
forcin! functions $Ao(t) and o(t), must also be given.
us no! chec" the de!rees of freedom of this sstem.
%here
are six equations. But there are nine un"no#ns< $A2,$A,$A*,0, 0,0*,2, and *
$learl
this sstem is not sufficiently specified and a solution could not be obtained.
What
have we missed in our modeling?
A good
plant operator could take one look at the sstem and see !hat the problem is. +e have not specified how the flows out of the tanks are to be set. Phsicall there !ould probabl be control valves in the outlet lines to regulate the flo!s. =o! are these control valves to be set> A common configuration is to have the level in the tank controlled b the outflo!, i.e., a level controller opens the control valve on the e"it line to increase the outflo! if the level in the tank increases. Thus there must be a relationship 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.
). TO HEATE# TA!*S et
us consider a process in !hich t!o energ balances are needed to model the sstem. %he flo! rate of oil passing through t!o perfectl mi"ed tanks in series is constant at ?6 ft *min. %he densit p of the oil is constant at @6 lbmft*, and its heat capacit $ p, is 6. Btulbm.. %he volume of the first tank 02, is constant at @:6 ft *, and the volume of the second tank 0, is constant at ?6 ft *. %he temperature of the oil entering the first tank is %o, and is 2:6 at the initial stead state. %he temperatures in the t!o tanks are % 2 and %. %he are both equal to :6 at the initial stead state. A heating coil in the first tank uses steam to heat the oil. et C 2 be the heat addition rate in the first tank. %here is one energ balance for each tank.
4ince 4ince
the throughput is constant ; # 2 # # .
volumes, densities, and heat capacities are all constant, &qs. (*.26) and (*.22) can be simplified
Let’s
check the degrees of freedom of this system.
%he
parameter values that are kno!n are p, $p, 02, 0, and . %he heat input to the first tank C 2 !ould be set b the position of the control valve in the steam line.
%hus
!e are left !ith t!o dependent variables, % 2 and %, and !e have t!o equations. 4o the sstem is correctl specified.
T$TORIAL
#lease refer handout. 1.inal 4am 5+t 2! 2.62, inal 4am 7pril 211
A!SER +1 t(min)
T1 (°)
0
28
dT1!dt T"(°) dT"!dt 0.4933
28
0.4933
1
28.493 0.4265 28.493 0.4922 3 3
2
28.919 0.3686 28.985 0.4824 8 5
3
29.288 0.3187 29.467 0.4661 4 9
4
29.607 0.2755 29.934 0.4454 1 0
4. GAS-PHASE PRESSURIZED CSTR
-r $% "$&A$.' -r (% "(&)
Assume < sothermal sstem, ∴ % # constant 4stem volume, 0 constant Perfect gases appl
Pa!e *+
ρ ρ d (d V ) balance the vessel gives: V total =mass F F − F ooaround F o ρ = f ρ −o ρ f f f ρ dt
. !O!ISOTHERMAL STR
A- PERFETL PERFETL MI/E# OOLI!0 A*ET
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
537.34
537.81
0.0456
0.5215
537.21
537.81
(- PL$0 FLO OOLI!0 A*ET Plug Flow: constant velocity of flow in every part of a system.
- L$MPE# A*ET MO#EL
#- SI0!IFIA!T METAL ALL APAITA!E - Eass of metal !alls and its effects on the thermal dnamics must be considered.
Tutorial Handout9 63, inal 4am, 5+t 2 7n endothermi+ rea+tion of rea+tant : to produ+t ; is +arried out in
,EA/,AE M0EL L12-P6A,E /3AM&, M0EL
L12 A3 4AP05 /3AM&, M0EL 6E5MAL E12L)52M M0EL
%&'L*+,-+''T #-+/&0/
To describe these boiling systems rigorously, conservtion e!utions "or both the v#or nd li!uid #hses re needed.
The
bsic #roblem is $nding the rte o" v#ori%tion o" mteril "rom the li!uid #hse into the v#or #hse.
&et consider &i!ue$ed #etroleum gs &'(/ v#ori%er system. (s is drn o
The li!uid in the tn) is ssumed #er"ectly mied &'( is "ed into #ressuri%ed tn) to hold the li!uid level in the tn). *ssume &'( is #ure com#onent+
the to# o" the tn) t volumetric o rte, F v forcing functions/. -et losses nd the mss o" the lls re ssumed negligible
-et is dded t rte, Q to hold the desired #ressure in the tn) by v#ori%ing the li!uid t rte, W v mss #er time/.
#. %T#2%T#T ,+L The
sim#lest model ould neglect the dynmics o" both v#or nd li!uid #hses nd relte the gs rte, F v to the het in#ut by+ ρ v F v ( H v − h0 ) = Q
where, H v = enthalpy of vapor leaving tank(Btu/lb m or cal/g) h0 = enthalpy of liquid feed(Btu/l b m or cal/g)
3. L&45&*-6#% 2'#,&% ,+L ore relistic model is obtined i" ssume tht the volume o" the v#or #hse is smll enough to m)e its dynmics negligible. "
only "e moles o" li!uid hve to be v#ori%ed to chnge the #ressure in the v#or #hse. o,
e cn ssume tht this #ressure is lys e!ul to the v#or #ressure o" the li!uid t ny tem#erture - 7 -
1/
* totl continuity e!utions "or the li!uid #hse, #otal continuity : d"$ = ρ o F o − ρ v F v ρ dt
23/ The to controller e!utions relting #ressure to het in#ut nd the li!uid level to "eed o rte,Q=0 f lso needed. F = f !( P )
0
(V L )
4/ *n energy e!ution "or li!uid
#hse gives the "unction o" time/.
tem#erture
s
The li!uid is ssumed incom#ressible so, # v nd internl energy, U is # T. The
enthl#y o" the v#or leving the v#ori%er is ssumed to be sim#le "orm + Energy : # T :d ;(vV . T ) L C p ρ = ρ o C p F oT o − ρ v F v (C pT + λ v ) + Q dt
5/
*n e!ution o" stte "or the v#or is needed to be ble to clculte density,
MP = RT
6/
The v#or #ressure reltionshi# gives the #ressure in the v#ori%er t "apor pressure : tht tem#erture A + B ln P = T
rom the system, e hve 6 e!utions. =n)non
vribles re Q, F 0 , P, V L , ρv nd T . >egree
o" "reedom , 6?6 0.
. L&45& #' #-+/ 2'#,&% ,+L or
the cses, i" the dynmics o" the v#or #hse cannot :e neglected i" e hve lrge volume o" v#or/.
Totl continuity nd energy e!utions "or the gs in the tn) is needed.
The
e#ression "or boiling rte, W v in term o" #ressure dierentil s driving "orce W v = K MT ( P − P v ) K MT is the pseudo mass transfer coefficient&
*t some tem#erture, li!uid boils becuse it eerts v#or #ressure P greter thn the #ressure Pv in the v#or #hse bove it. *t
e!uilibrium, P = Pv . " li!uid nd v#or re in e!uilibrium, K is very
@!utions describing the system "or li;uid
@!utions describing the system "or + va
= W v − ρ v F v
nergy : d (V v ρ vU v ) dt
= W v H L − ρ v F v H v
%tate : ρv
=
MP v RT v
where, U L = internal energy of liquid at temperature # * $ = enthalpy of vapor boiling off liquid U v = internal energy of vapor at temperature #v * = enthalpy of vapor phase
The
systems hve 10 vribles. Q, F 0 ,V L ,W v , T, V v , ρv T v , P nd Pv
@!utions+
10 e!utions &i!uid #hse 3 A#or #hse 3 Boiling rte 1 ontroller e!ution 2 totl volume A & : Av/ o" tn) 1 >egree
o" "reedom 0
. T6/,#L 45&L&3/&5, ,+L or this cse, therml e!uilibrium beteen li!uid nd v#or is ssumed to hold t ll times.
The v#or nd li!uid tem#ertures re ssumed e!ul to ech other, T 7 Tv .
@limintes the need "or n energy blnce "or the v#or #hse. C sensible het o" v#or is usully smll com#red ith ltent het eects.
ρ C p
The
used,
d (V LT )
=
ρ o F oC pT o − W v (C pT + λ v ) + Q
sim#le dt enthl#y reltionshi#s cn be
Tutorial 2Final E3am, A4ril %5587 =#" is feed into
a pressuri$ed tan to hold the li>uid le8el in the tan. The pressure in the tan is maintained b 8aporisin( the li>uid at a rate ? 8 @massAtimeB, and this is done b addin( heat at a rate 6. Heat losses and the mass of the tan walls are assumed ne(li(ible. "as is drawn off the top of the tan at a 8olumetri+ flow rate, 8. The 8olume of the 8apor is lar(e and the rate of the 8apori$ation ?8 is (i8en as ? 8 DE *T @#-#8B (i8en that E *T is the pseudo mass transfer +oeffi+ient. / referrin( i(ure 1, deri8e the e>uations that des+ribed the beha8iour of this sin(le +omponent 8aporiser.
,EA/,AE M0EL
P5A&&AL M0EL 570502, M0EL
,5LT&+,-+''T 9L#%6 /5,
ystem "or v#or?li!uid ith more thn one com#onent.
f drum P G bubblepoint P of feed at %6, some of liquid !ill vapori1e. iquid stream at high % F P flashed into a drum
P6 is high enough to prevent an vapori1ation of feed at %6 and "6H
P is reduced as it flo!s through restriction (valve)
%his e"pansion is irreversible and occurs at constant enthalp
8as is dra!n off through a control valve !hose steam position is set b P controller
Adiabatic conditions are assumed (no heat losses)
iquid comes off the bottom of tank on level controller
The
"orcing "unctions in this system re "eed tem#erture, T0D "eed rte, D nd "eed com#osition, 0E. >ensity o" li!uid in the tn), < & is ssumed to be )non "unction o" tem#erture, T nd com#osition, E. >ensity o" v#or in the drum is )non "unction o"av T, com#osition, y E nd #ressure, M v P wher e M vav = average molecular ρ v = '. RT weight of gas C
M = ∑ M j y j where M j = molecular weight of av v
j =!
jth component
#. %T#2%T#T ,+L The
system hich neglects dynmics com#letely.
'ressure is ssumed constnt, nd the stedy stte T@, @ nd energy blnces re used.
A#or nd li!uid #hses re ssumed to be in e!uilibrium.
#otal ontinuity : ρ o F o
F v + ρ L F L
= ρ v
omponent ontinuity : ρ o F o av 0
M
$0 j
=
ρ v F v av v
M
y j
+
ρ L F L av L
M
$ j
"apor + liquid equilibrium : y j
=
f ( $ j ,T , P )
nergy equation : h0 ρo F o = Hρv F v + hρ L F L #hermal properties : h0
=
f #$0 j !T 0 "
h = f #$0 j !T 0 "
H = f #y j !T!P"
3. /&+/+5% ,+L %7I JJ
An equilibrium-flash calculation is made at each point in time to find vapor and liquid flo! rates and properties immediatel after the pressure letdo!n valve (using same equations as in $ase A).
These
to strems re then "ed into the v#or nd li!uid #hses. 0apor phase
%7I JJ
iquid phase
or
li!uid #hses,
Boiling rate : W v
= K MT ( P L − P v)
#otal ontinuity : d ρ LV L dt
= F L- ρ L- − F L ρ L − W v
nergy : d ( ρ LV LU L ) dt
= ρ L- F L- h L- − W v H L − ρ L F L h L
CCE : d
V L ρ L $ j %v M v dt
F - ρ = L L $ - − j %v M L -
W v %v v
M - -
y j,
−
ρ L F L %v L
M
$ j
or
v#or #hses,
#otal ontinuity : d ρ vV v dt
= F v H v + W v − F v ρ v
nergy : d ( ρ vV vU v ) dt
= ρ v- F v H v + W v H L − ρ v F v H v
CCE :
V v ρv y j d M %v v dt
F - ρv = y j + %v v
M v -
&t%te : ρ v
=
M v%v P v RT v
W v %v v
M - -
, j
y
−
ρv F v %v v
M
y j
*ddition
o" 1/multi?com#onent v#or?li!uid e!uilibrium e!ution to clculte ' & . 2/F?1 @ "or ech #hse ontroller
e!utions relting A & to & nd 'v to v . F L = f (V L ) F v = f ( P v )
. -/#T&#L ,+L or
cses tht ignore the dynmics o" the v#or #hse s in cse B+ &i!uid?#hse dynmics model/.
The
v#or is ssumed to be lys in e!uilibrium ith the li!uid.
o,
conservtion e!utions re ritten "or li!uid #hse only.
#otal ontinuity : ρ LV d L
dt
= F 0 ρ 0 − F v ρ v − F L ρ L
CCE : d
V L ρ L $ j %v M L dt
F ρ = 0 0 $ − F v ρ v y − F L ρ L $ 0 j j j %v %v %v M 0
M v
M L
nergy : d ( ρ LV L h ) dt
= ρ 0 F 0 h0 − ρ v F v H − ρ L F L h
The
F v#or?li!uid e!uilibrium e!utions, three enthl#y reltionshi#s, to density e!utions, moleculr eight e!utions, nd "eedbc) controller e!utions re ll needed. totl
number o" e!utions must e!ul to totl number o" vribles *ll
the "eed #ro#erties or "orcing "unctions re given.
