EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 47

FJ

TJ
TJ v
T
‘J CA
FJ -
TJO
F
I
CA
T
FIGURE 3.3
k Nonisothermal CSTR.
A - B

inlet temperature of I”, . The volume of water in the jacket V, is constant. The
mass of the metal walls is assumed negligible so the “thermal inertia” of the
metal need not be considered. This is often a fairly good assumption because
the heat capacity of steel is only about 0.1 Btu/lb,“F, which is an order of mag-
nitude less than that of water.

A. PERFECTLY MIXED QlCU.ING JACKET. We assume that the temperature

e in the jacket is TJ. The heat transfer between the process at tem-
rature T and the cooling water at temperature TJ is described by an cuuxall
heat transfer coefficient.
Q = f-J4rV’ - ‘I’
where Q = heat transfer rate
U = overall heat transfer coefficient
AH = heat transfer area
In general the heat transfer area c* vary with the J&&p in the reactor if
some area was not completely-d with reaction mass liquid at all times. The
equations describing the system are:
Reactor total continuity:

Reactor component A continuity :

d(vcA)
~ = Fo CA0 - FCA - Vk(CJ
dt
Reactor energy equation :

p y = p(Foho - Fh) - LVk(CJ” - UA,,(T - TJ) (3.20)

48 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS

Jacket energy equation :

PJ v, % = F.,PJ@,, - h,) + UAAT - TJ)

where pJ = density of cooling water

h = enthalpy of process liquid
h, = enthalpy of cooling water
The-n of constant densities makes C, = C, and permits us to use en-
thalpies in the time derivatives to replace internal energies,
A hydraulic- between reactor holdup and the flow out of the
reactor is also needed. A level controller is assumed to change the outflow as the
volume in the tank rises or falls: the higher the volume, the larger the outflow.
The outflow is shut off completely when the volume drops to a minimum value
Vm i n *
F = K,(V - Vrni”) (3.22)

The level controller is a proportional-only feedback controller.

Finally, we need enthalpy data to relate the h’s to compositions and tem-
peratures. Let us assume the simple forms

h=C,T and h,=CJT, (3.23)

where C, = heat capacity of the process liquid
C, = heat capacity of the cooling water
Using Eqs. (3.23) and the Arrhenius relationship for k, the five equations
that describe the process are

dv=F -F (3.24)
dt ’

d(VCd = F,-, CA0 - FCA - V(CA)“ae-E’RT

- (3.25)
dt

4W
- = pC,(F, To - FT) - ,lV(C,)“ae-E’RT - UA,,(T - TJ) (3.26)
PC, d t

PJ YrC, $ = FJPJCGJO - G) + U&V - Trl (3.27)

F = I&@ - Vmin) (3.28)

Checking the degrees of freedom, we see that there are five equations and
five unknowns: V, F, CA, T, and TJ. We must have initial conditions for these
five dependent variables. The forcing functions are To, F, , CA,, and F, .
The parameters that must be known are n, a, E, R, p, C,, U, A,, pJ, V,,
CJ, TJo, K,, and Vmin. If the heat transfer area varies with the reactor holdup it
 EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 49

would be included as another variable, but we would also have another equation;
the relationship between area and holdup. If the reactor is a flat-bottomed verti-
cal cylinder with diameter D and if the jacket is only around the outside, not
around the bottom
A,+ (3.29)

We have assumed the overall heat transfer coefficient U is constant. It may be a

function of the coolant flow rate F, or the composition of the reaction mass,
giving one more variable but also one more equation.

B. PLUG FLOW COOLING JACKET. In the model derived above, the cooling
water inside the jacket was assumed to be perfectly mixed. In many jacketed
vessels this is not a particularly good assumption. If the water flow rate is high
enough so that the water temperature does not change much as it goes through
the jacket, the mixing pattern makes little difference. However, if the water tem-
perature rise is significant and if the flow is more like plug flow than a perfect mix
(this would certainly be the case if a cooling coil is used inside the reactor instead
of a jacket), then an average jacket temperature TJA may be used.

(3.30)
‘GO + TJexit

2
&A =

where TJ.aait is the outlet cooling-water temperature.

The average temperature is used in the heat transfer equation and to rep-
resent the enthalpy of jacket material. Equation (3.27) becomes
&A
PJGCJ -
dt = FJ PJ CJ(TJO - &exit) + UAdT - TJA) (3.31)

Equation (3.31) is integrated to obtain TJA at each instant in time, and Eq. (3.30)
is used to calculate TJexi, , also as a function of time.

_ C. $UMPEb JACKET MODEL. Another alternative is to 9 up the jacket

volume into a number of perfectly mixed “lumps” as shown in Fig. 3.4.
An energy equation is needed for&lump. Assuming four lumps of&
volume and heat transfer area, we@ fp energy equations for the jacket:
dTr,
:PJ vJ c J - = FJ PJ CJ(TJO - T,,) + iu&tT - TJI)
dt

&z
iPJ GcJ -& = F,f’, ~,(TJ, - T,,) + ~~A,XT - qz)

(3.32)
d=h
iPJ vJcJ -dt = FJ L’J ~,(TJ, - TJ& + %&IV - %)

d%
tPJ vJ c J - = FJI’JCJ(TJ, - &4) + tU&V - TJ~)
dt
 50 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS

Q4-M TJ~ *
4

Q3-- TJ~
I

Q2** 7~2 !
4 i’
t
F~
QI-- TJI -
TJO
FIGURE 3.4
Lumped jacket model.

D. S-CANT METAL WA&s In -.-- some reactors, particularly

m-pressure v& or smaller-scale equipment, the mass of the metal walls and
its effects on the thermal dynamics m be considered. To be rigorous, the
energy equation for the wall should be a partial differential equation in time and
radial position. A less rinorous but frequently used approximation is to “hunp”
the mass of the metal and assume the metal is all at one temperature I’,‘,. This
assumption is a fairly ed one when the wall is not too sand the thermal
-
conductivity of the metal is large.
Then effective i@ide and @e film coefficients h, and h, are used as
in Fig. 3.5.
P
The three energy equations for the process are: ,
PC 4v”r)
= pCkF, To - FT) - AV(CA)“ae-E’RT - h,AXT - TM)
* at

(3.33)

PJ vJ cJ dt
is- - FJ PJ CJ~%I - r,) + ho ‘%(TM - TJ)

.-
where hi - inside heat transfer film coefficient
h, = outside heat transfer film coefficient

FIGURE 3.5
Lumped metal model.
 EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 51

pM = density of metal wall

CM = heat capacity of metal wall
V, = volume of metal wall
Ai = inside heat transfer area
A, = outside heat transfer area

3.7 SINGLE-COMPONENT VAPORIZER

aoiling systems represent some of the most interesting and important operations
in chemical engineering processing and are among the most dificult to model. To
describe these systems rigorously, conservation equations must be written for
both the vapor and liquid phases. The basic problem is finding the rate of vapor-
ization of material from the liquid phase into the vapor phase. The equations
used to describe the boiling rate should be physically reasonable and mathemati-
cally convenient for solution.
Consider the vaporizer sketched in Fig. 3.6. Liquefied petroleum gas (LPG)
is fed into a pressurized tank to hold the liquid level in the tank. We will assume
that LPG is a pure component: propane. Vaporization of mixtures of com-
ponents is discussed in Sec. 3.8.
The liquid in the tank is assumed perfectly mixed. Heat is added at a rate Q
to hold the desired pressure in the tank by vaporizing the liquid at a rate W,
(mass per time). Heat losses and the mass of the tank walls are assumed negligi-
ble. Gas is drawn off the top of the tank at a volumetric flow rate F,. F, is the
forcing function or load disturbance.

A. STEADYSTATE MODEL. The simplest model would neglect the dynamics of

both vapor and liquid phases and relate the gas rate F, to the heat input by
P,F,W, - ho) = Q (3.34)
where I-Z, = enthalpy of vapor leaving tank (Btu/lb, or Cal/g)
h, = enthalpy of liquid feed @&u/lb, or Cal/g)

FIGURE 3.6
LPG vaporizer.
52 MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS

B. LIQUID-PHASE DYNAMICS MODEL. A somewhat more realistic model is

obtained if we assume that the volume of the vapor phase is small enough to
make its dynamics negligible. If only a few moles of liquid have to be vaporized
to change the pressure in the vapor phase, we can assume that this pressure is
always equal to the vapor pressure of the liquid at any temperature (P = P, and
WV = pu F,). An energy equation for the liquid phase gives the temperature (as a
function of time), and the vapor-pressure relationship gives the pressure in the
vaporizer at that temperature.
A total continuity equation for the liquid phase is also needed, plus the two
controller equations relating pressure to heat input and liquid level to feed flow
rate F, . These feedback controller relationships will be expressed here simply as
functions. In later parts of this book we will discuss these functions in detail.
Q =fim Fo =fz(YL) (3.35)
An equation of state for the vapor is needed to be able to calculate density
pv from the pressure or temperature. Knowing any one property (T, P, or p,) pins
down all the other properties since there is only one component, and two phases
are present in the tank. The perfect-gas law is used.
The liquid is assumed incompressible so that C, = C, and its internal
energy is C, T. The enthalpy of the vapor leaving the vaporizer is assumed to be
of the simple form: C, T + I,.
Total continuity :

Energy :
c p d(l/, T)
? -=PoC,FOTO-~,F,(C,T+~,)+Q
dt
State :

(3.38)

Vapor pressure :

lnP=$+B

Equations (3.35) to (3.39) give us six equations. Unknowns are Q, F,, P, V’, pv,
and T.

C. LIQUID AND VAPOR DYNAMICS MODEL. If the dynamics of the vapor

phase cannot be neglected (if we have a large volume of vapor), total continuity
and energy equations must be written for the gas in the tank. The vapor leaving
the tank, pv F, , is no longer equal, dynamically, to the rate of vaporization W, .