This document compares the limits of safe operation for a continuously stirred tank reactor (CSTR) and a semi-batch reactor (SBR) for an exothermic liquid phase reaction. It outlines the stability criteria used to define the limits for a CSTR based on temperature and conversion. Equations are presented to calculate the critical coolant temperature values corresponding to the multiple steady states, limit cycles, and transition between damped and aperiodic return to the steady state for a CSTR operating at a fixed temperature. The analysis shows that for the same reaction and conditions, an SBR has a more limited range of safe operation than a CSTR.

1. Chemical Engineering Science. Vol. 41, No. 4. pp. 1081-1087. 1986. 0009-2509/86 $3.00 + 0.00
Printed in Great Britain. @ 1986. Pergamon Press Ltd.
A COMPARISON OF THE LIMITS OF SAFE OPERATION OF A SER AND A CSTR
P. Hugo* and J. Steinbach**
*Institut fur Technische Chemie, Technische Universitat Berlin,
StraRe des 17. Juni 135, 1000 Berlin 12
**Schering AG, Abt. APC, MullerstraBe 170-178, 1000 Berlin 66
Federal Republic of Germany
ABSTRACT
In the case of exothermal reactions it is necessary to calculate the limits of safe operation
from stability criteria. In their general form these criteria are often difficult to apply in
practice. _
However, under industrial conditions the temperature of a reaction is usually fixed. Unter these
conditions from the stability criteria a corresponding limit of the coolant temperature can be
calculated. For safe operation the actual coolant temperature must be chosen higher than this
limit value.
Using the well known criteria of a CSTR the principles of these calculations are demonstrated.
Using empirical stability criteria of a SBR similar calculations can be performed. Taking the
same reaction and reaction conditions the SBR is more critical than the CSTR.
KEYWORDS
CSTR; Semi-Batch-Reactor(SBR); thermal stability; safe operation conditions; critical coolant
temperature.
MODEL ASSUMPTION
The aim of this paper is to compare the limits of safe operation of a CSTR and a Semi-Batch-Reac-
tor (SBR). The problems are discussed for a special case of a liquid phase reaction of the type:
A + B = Products
with a rate law
r = k(T)CA cB k(T) = k, exp(-E/RT) (la,b)
The reaction may be carried out with or without solvent S. All products formed by the reaction
remain in solution. A stoichiometric input of A and B is assumed.
Under these conditions the density P and the heat capacity cP remain nearly constant during the
reaction.
LIMITS OF STABILITY OF A CSTR
Stability Criteria
The stability criteria of an indirectly cooled CSTR are well known; see Perlmutter (1972). They
define the limits of safe operation. It is convenient to describe these limits using dimension-
less groups as defined In Table 1. CBo is the virtual initial concentration of B, to be CalCUlat-
ed from the input rate mg by CB, = mBr/MBY.
From the discussion of the dynamic behavior of the balance equations of an indirectly cooled CSTR
three limits can be obtained:
a) Multiple stady states with ignition/extinction(denoted by the index B):
Bg = (1 + St) r (l/S) + 2/(1 - 3) ] (2)
If the actual thermal reaction number B > Ba there are multiple steady States.
1081

2. 1082 P. HLJCXJandJ. STEINBACH K-3
TABLE 1 Definitions and Dimensionless Groups
Conversion 3 = 1 - (c&c&o)
Adiabatic temperature rise ATad = (-~H)Cgo/~cp
Damkohler number Da = k CBo T
Thermal reaction number B = E ATad/RTZ
Modified Stanton number St = kw Ar/Vpcp
(Westerterp and co-workers, 1984)
b) Limit cycles of temperature and conversion (denoted by the index a):
6, = [z + St + 23/(1 - 3)] /3 (3)
A limit cycle is obtained in the range B6 > B > B,.
c) For real but stable eigen values a CSTR can return from small disturbations to the stationary
state either with damped oscillations or aperiodically.The limit between these two cases
(denoted by the index 0) is:
Bw = [-/x - 4 23/(l - 3)]'/3 (4)
If B < Bw; the return is aperiodic.
To apply these equations the conversion 3 must be calculated. For a reaction with a rate law (1)
3= [I -31 -Q']/Q Q = 2 Da/(t + 2 Da) (5a,b)
Critical Coolant Temperature
From the Eq.(2),(3),(4)and(5a,b) the conversion and temperature at the three limits can be cal-
culated. This is, however, somewhat troublesome because B and Da both depend on temperature.
This problem can be avoided. Under industrial conditions the temperature at which the reaction is
to be performed is usually fixed. So in practice the question is how to find the other reaction
conditions for a sufficiently safe operation at the prescribed temperature T.
The most important parameter for controlling the reaction temperature is the coolant temperature
Tc- For simplicity we assume that the input temperature of A, B and of the solvent S is equal to
the coolant temperature. At the stationary state the coolant temperature of a CSTR can be calcu-
lated by:
Tc = T - ATad3/(1 f St) (6a)
Using the definition of B from Table 1
T, = T [1 - (RT/E) B3/(1 + St)] (Sb)
So, inserting the critical values of B from the Eq.(2), (3) and (4) three corresponding critical
values of the coolant temperature T,, Tg, T, can be calculated. Comparing these values with the
actual coolant temperature T, the range of dynamic behavior of a CSTR can be determined as shown
in Table 2.
TABLE 2 Ranges of Dynamics of a CSTR
Range Coolant Temperature T, Type of Dynamics
AS Tc ' Tw Aperiodic stable
PS Tw > T, > Ta Periodic stable
LC Ta ' T, > TB Limit cycles
IE TB ' Tc Ignition/extinction

3. K-3 Safe operation of a SBR and a CSTR 1083
Independent Reaction and Reactor Parameters
The dimensionless groups of Table 1 contain several reaction and reactor parameters within one
group. Under the conditions described in Section 1 three reaction and reactor conditions can be
changed independently:
1) Dilution by solvent x= (i, + i,)/i (7)
2) Reactor size Y = VPcp/(kwA*,) (8)
3) Input rate z= = 1/r (9)
In these equations ~1 = 1 hour is the chosen time unit. X, Y, Z are defined in such a way that
increasing values will lead to more critical conditions.
The total input is rig = VP/~ = 6~ + r&j + 6s and the stoichiometric input is fs = mB/fi’ = MB/MA-
~A
Using these equations and Eq.(7) the virtual initial concentration CBo = mBz/MB V can be rear-
ranged as:
‘
Bo =
By Eq.(8), (9), (10) the dimensionless groups of Table 1 can be separated into the characteristic
reaction data presented in Table 3 and the variable groups X. Y, Z. The result is
--
ET
/(l + fs) (11)
3 :
B = [ (E/RT)~ fe/(i + fS)] x (12)
St = a (13)
TABLE 3 Characteristic Reaction Data
Denotation Definition Example Values
Stoichiometric factor f, = MB/MA f, = 1.25
Kinetic factor fk = kmpq,/MA In fk = 32.38
Energetic factor f, = R(-AH)/E MBcp f, = 0.0827
Calculations for an Example
Taking the data of Table 3 and E/R = 10000 K some calculations of the different coolant tempera-
tures were carried out. Fig. 1 shows the influence of changing the reactor size Y at fixed X = 1
(i.e. no solvent) and Z = 0.25 (spacetime 4 hours). Only small reactors (Y < 0.50) work in the
stable range.
Fig. 1 Limits of stable operation of a CSTR changing the dimen-
sionless reactor size Y, (X = 1. Z = 0.25); T = 350 K

4. 1084 P. HUGO and J. STEINBACH K-3
Fig. 2 shows the change of input rate Z at fixed X = 1 and Y = 1. Increasing input rate Z Ieads
to limit cycles (range LC) and at very low coolant temperatures to extinction (range IE).
If it is possible to increase the reactor temperature T this is the best way of stabilizing the
reactor. Fig. 3 shows the same conditions as Fig. 1 but at ten degrees higher reaction tempera-
ture.
In the same manner the addition of solvent (i.e. decreasing X) can be discussed. Due to the limit
space of this paper these results are not demonstrated as a figure. In the case of a CSTR the re-
suits for changing X are similar to the changeof 2. Dilution with solvent stabilizes the CSTR.
.m
1IKl
t
Fig. 2 Limits of stable operation of a CSTR vs. the dimen-
sionless input rate Z, (X = 1, Y = 1); T = 350 K
I I
Fig. 3 Limits of stable operation of a CSTR. same conditions
as Fig. 1, but T = 360 K
LIMITS OF SAFE OPERATION OF A SBR
Dynamic Behavior of a SBR
Taking the model assumptions of Section 1 the performance of the SBR is as follows: The component
B with or without solvent S is placed into the reactor at coolant temperature. A is then fed at
constant rate, at the same temperature, up to the stoichiometric ratio. The total time for addi-
tion of A is denoted as T. Due to the heat of reaction there is an increase of the reactor temper-
ature T. The dynamic behavior of this operating condition was investigated in detail by one of us
(Steinbach. 1985). The results are briefly summarized.
If the reaction rate is sufficiently high most of the added component A is immediately converted.
This is a safe operation as shown in Fig. 4. However, if the starting temperature is too low, un-
stable operation results: Fig. 5. The reaction conditions used for simulation are the same as in
Fig. 7. Fig. 4 corresponds to Z = 0.1; Fig. 5 to Z = 0.2.

5. K-3 Safe operation a SBR and a CSTR
of 1085
Fig. 4 Temperatureand molar fraction of a SBR. working
in a safe condition, see Fig. 7:.Z = 0-l
Fig. 5 Temperature and molar fraction of a SBR. working
in an unstable condition, see Fig. 7: Z = 0.2
Limits of Safe Operation of a SBR
A discontinuous process possesses no stationary state. Therefore, a stability analysis similar to
the CSTR cannot be performed. What can be done is to develop empirical correlations from numeri-
cal solutions of the balance equations. This was demonstrated in an earlier paper (Hugo, 1981),
investigatingthe thermal design of a batch-reactor.
To compare the SBR and the CSTR corresponding dimensionless groups have to be used. As the volume
and cooling area increase with time in a SBR, somewhat different definitions must be used:
a) The virtual initial concentration of B is calculated from the input mB and the final volume V:
'Bo = mB/MB V (14)
b) The increase of volume due to the input of mA leads to an additional parameter:
G = mA/m (15)
m = V/pis the total final mass.
c) As an approximation a mean value of the cooling area A is used.
All other definitions are the same for a SBR and a CSTR.
Of course, low values of the thermal reaction number B will lead to a safe operation of the SBR
in any case. From numeric calculations (Steinbach, 1985) we found that the limit of the thermal
reaction number BY can be calculated by:
B Y= [ JSt + 2 G'+ 1.2 ]' (16)
When the actual B exceeds E$ the SBR may become critical, as demonstrated in Fig. 5. To avoid
this, the initial reaction rate at the coolant temperature T, must be sufficiently high. The lim-
it value To can be calculated with sufficient accuracy by the empirical correlation (Steinbach,
1985):
Da (Tb) = St (17)
If the actual coolant temperature T, is greater then Ta the "stable" maximum temperature Tm (see
Fig. 4) can be calculated from
T, = T, ATad
(18)
[ l-G 3
This equation corresponds to Eq.(6a) of the CSTR at 5 = 1.
Desing of a SBR
In a SBR the dilution by solvent is described by

6. 1086 P. HUGO and J. STEINBACH K-3
x=
mAm mB
+ (7a)
and the stoichiometric input iS fs = mB/mA. With these equations the volume increase parameter
can be rearranged as
B=h (19)
The design principle of a SBR corresponds to that of a CSTR. From Eq.(18) one gets:
Tc=Tm Cl- [?I&1 (6~)
Inserting B at T,, the actual coolant temperature is obtained; inserting B, from Eq.(16) the cor-
responding T-,is obtained. If T, > T-,the SBR operates safely: Range ASBR.
safe operation demands Eq.(17). By the general definitions of Da and St given in Eq.
[:lpazdTli3) the corresponding coolant temperature Ta can be calculated as
Tg = (E/R)/ln [fk X Y /(I + fs) ] (17a)
In the range Tr > Tc > Ta a safe operation of the type of Fig. 4 is obtained: Range BSBR.
If T, < T6 the SBR IS unstable: Range DSBR. The last two Fig. 6 and Fig. 7 show the result of a
typical design of the SBR. The fixed data f,, fK, f, and E/R are the same as for the CSTR.
Fig. 6 Limits of safe operation of a SBR changing the dimen-
sionless reactor size Y, (X = 1, Z = 0.15); T = 350 K
Fig. 7 Limits of safe operation of a SBR changing the dimen-
sionless input rate 2. (X = 1, Y = 0.75); T = 350 K

7. K-3 Safeoperation a SBR and a CSTR
of 1087
The conditions for Fig. 6 (SBR) and Fig. 1 (CSTR) are identical with the exception of the input
rate Z. As the Iimit size of safe operation Y = 0.5 is nearly the same, the lower input rate
Z = 0.15 of the SBR (Z = 0.25, CSTR) indicates that the SBR is the more difficult to control.
The same conclusion can be taken from a comparison of Fig. 7 and Fig. 2. Note that the Z-scale
differs in the two figures. Generally, the input rate is a rather critical condition of safe oper-
ation.
So. taking the same reaction and reaction conditions the SBR is more critical than the CSTR. This
is due to the different wav of conversion of the comoonent B. In a CSTR at a stationarv dearee of
conversion the concentratibnof both components A anh B usually is small. In a SBR theFe is a high
initial concentrationof B and a small concentrationof A as shown in Fig. 4. Therefore, critical
conditions can result from a comparativelysmall increase of the feed rate of component A.
REFERENCES
Hugo, P. (1981). Start-up and Operation of Exothermic Batch-Processes.Ger. Chem. Eng..4. 161-173
Perlmutter, D. D. (1972). Stability of Chemical Reactors. Prentice-Hall, Inc., Englewooa Cliffs,
_. _
Sternbich, J. (1985). Untersuchung zur thermischen Sicherheit des indirekt gekuhlten Semibatch-
Reaktors. Dissertation.Technische Universitat Berlin.
Westerterp, K. R., W. P. M. van Swaaij and A. A. C. M_ Beenackers (1984). Chemical Reactor Design
and Operation. John Wiley & Sons, Chichester. p. 306
NOTATION
c, ATad, Da, B, St: see Table 1
fs, fk, fez see Table 3
Cooling area (mZ)
Heat capacity (J/kg K)
Concentration of component i (mol/m')
Activation enerav (J/mol)
G Volume increase-of‘a.SBR; see Eq.(15)
AH Heat of reaction (J/mol)
k Rate constant (m'/mol h)
kw Heat transfer coefficient (W/m* K)
Mi Molar mass of component i (kg/mol)
?i Total input of component i in a SBR (kg)
mi Input rate of component i in a CSTR (kg/h)
R Gas constant, 8.314 J/mol K
7 Temperature, K
V Volume of reaction phase (ma)
x. Y, z Dimensionless reaction conditions, Ea.(7), (8). (9)
_..
6 Density (kg/m')
T Space time (CSTR) or time of dosage (SBR). (h)