D. S. BILLINGSLEY, W. S. McLAUGHLJN,Jr.l, N. E. WELCH, and C. D. HOLLAND Department of Chemical Engineering, The Agricultural and Mechanical College of Texas, College Station, Tex.
Use of 'Computers in Reactor Design This simplification of a tedious, time-consuming problem will be especially valuable for its application to many types of industrial reactors
A flow-
The effects considered here are heat of reaction, heat capacity of the reacting mixture, over-all heat transfer coefficient, temperature of the condensing vapor, and variation of the rate of reaction, with respect to both temperature and concentration, on the reactor volume (or time) required to obtain a specified conversion. For illustration two of the possible reactors described by the equations stated below are shown. Although in each of these the reactants are contained in a single inner tube, the equations are applicable when the condensing vapor is contained in the inner tube. In
method devised for either or batch-type tubular reactors heated indirectly by condensing vapors is applicable to first-order and pseudofirst-order, endothermic, liquid phase reactions. The design method is primarily a graphical one based on several hundred solutions to the differential equations describing the reactors. I t consists of two determinations: the volume of the preheat and of the reaction sections of a flow-type reactor. The preheat section is designed by use of an analytical solution of the equations describing it, and the reaction section is designed with nomographs based on numerical solutions to its equations. In the design of the batch reactor, the same approach is used, except that in the design calculations preheat and the reaction times are determined. Qualitatively the preheat section (or preheat time) is the volume (or time) required to raise the temperature of the reacting mixture to a value such that the reaction proceeds at an appreciable rate. The solutions for the reaction section (flow) and the reaction time (batch) were obtained by use of an IBM 650 digital computer in service at the Texas Engineering Experiment Station. Little information pertaining to design of the type of reactors considered in this work has appeared in the literature. Hougen and Watson (3) and recently Smith (8) have suggested sizing reactors of this type by the numerical solution of the difference equations describing them. Murdoch and Holland (7) have shown a design method for fired tubular reactors in which the rate of heat input was constant. This method was developed for first-order, endothermic, vapor phase reactions. Gee, Linton, Maier, and Raines ( 2 ) investigated (by use of a computer) the possibility of carrying out a particular homogeneous, gas phase reaction in a nonadiabatic, nonisothermal, tubular
reactor. However, they did not attempt to obtain a general correlation.
'Present address, 922 East 12th St., Dallas, Tex.
Batch- and flow-type tubular reactors are typical of reactors described by the equations. Condensing vapor could also be contained in inner tube
DESIGN
Differential Equations Describing Reactors
When heat is supplied indirectly to a liquid phase, first-order, endothermic reaction by a condensing vapor, the equations describing either a batch or flow-type reactor are of the same form, provided that steady state exists and only radial mixing of the reactants occurs in the flow-type reactor and that uniform (very thorough) radial and longitudinal mixing occurs in the batch reactor.
'
. '. REACTANTS
. . .. !BATCH SYSTEM
CONDENS\NG
VAPOR
FLOW SYSTEM
VOL. 50, NO. 5
MAY 1958
741
Symbols in Equations 1 and 2 for Various Reactors
Table I. Type and
Arrangement Flow Reactants inside tubes
mls
S
Ua
b/kl
Reactants outside tubes
PCPSkl
v/kl
Ua Batch Reactants inside tubes
PC,Slkl
l/kl
Ua Reactants outside tubes
1/k1
pCvSk1 ~
Ua
n:AH,S,h v UQtl
klV7
nzAHrSkl v Uatl
klVP
iVj A E A k i VUatl
Jzle
hT:A ErSki VUatl
kle
V
V
mixture and the heat of reaction, respectively. The precise definitions of s, mls, c/s, and Zdepend upon the type and arrangement of the reactor. Equations 1 and 2 are stated in dimensionless form because X , 2, 6, $, m/s, c/s, and @ are dimensionless to facilitate the construction of the nomographs.
addition, any number of tubes may be employed. The equations describing this class of reactors follow :
d4 cdX + +sm- dZ + -s -d Z = @
z
c/s
pCpSik1
(2)
Design of Preheat Section or Calculation of Preheat l i m e
where klV, 2 = - (for a flow-type reactor)
Because the preheat section functions primarily as a heat exchanger, it was possible to design it analytically. This approach reduced the number of digital solutions which would have been required had the temperature, to, of the cold feed been taken as a parameter. The preheat section, more precisely, is defined as the volume of reactor required to obtain a conversion, X , of 0.01. For a batch reactor, the preheat time is the
where V, is the reactor volume swept out by the reactant stream and u is the volumetric flow rate of the stream Z = k $ (for a batch reactor) and X = fraction of the base component reacted. The remaining symbols are defined in Table I and in the nomenclature section. Parameters m / s and c i s (Table I) are proportional to the heat capacity of the
8
-I
6
2
i
i
\
I
I
I
\
I
IO
I
I
I
I
i
I
I
I
I \
time required to effect a conversion, X, of 0.01. This specification permits calculation of the temperature, tl, the rate constant, kl, and the corresponding volume (or time) required. By definition little reaction occurs in the preheat section, so essentially all of the heat transferred from the condensing vapors reappears in the form of sensible heat of the reactants. I n view of this, the term corresponding to the heat absorbed by the reaction may be neglected in Equation 2 to give
When Equations 1 and 3 are combined 1
1
d X - k d Z ki
Equation 4 may be solved for d Z to give
When this value of d Z is substituted in Equation 5 , the following relationship is obtained:
Starting with Equation 3, the following relationships are readily developed
When the term kd+/@ - r+h of Equation 7 is replaced by the relationships shown in Equation 8
where
0001
y = In k l / k ,
08
06
d = b(1
04
Because
0.2
I
x
I
1
I
I
\
I
I
I
0.0001
0.I
0.08 006 004
002
0
IO
20
30
40
50
60
70
X
Figure 1.
742
-X
Plot of the function / ( x )
INDUSTRIAL AND ENGINEERING CHEMISTRY
80
90
100°ooool
yo = In kl/ko
-
1/@)
USE OF COMPUTERS IN REACTOR DESIGN
k, = k , eb (1 - I/*) = A8 - W R T ,which is the value the rate constant would have if
was constant throughout the preheat section. The upper limit of infinity applies exactly when the vaIue of the rate constant at the inlet to the preheat section is zero. If this were not the case, the integrals in Equation 12 would have finite upper limits. I n applying Equation 12, the term rn/skl (Table I) is readily evaluated without the precise knowledge of t i . For a specified condensation temperature, k,, is evaluated as indicated previously, Although lc/p is usually close to unity, it is nevertheless a function of t1, so that Equation 12 must be solved by trial for tl. In the first trial, it is suggested that one take 1c/? = 1. Then the numerical value of the right hand side of Equation 12 may be obtained. The lower limit corresponding to this value of the integral is obtained from Figure 1. From this value of the lower limit, the temperature tl is readily calculated by use of the following relationship
Equation 12 is used to calculate the temperature tl of the reacting mixture a t the end of the preheat section. A plot of the function I(x) is shown in Figure 1. Values of the function I have been tabulated previously (7, 4). I n those cases the function designed herein as A plot I(x) is referred to as -Ei(-x). of the function 1c/ us. x with w as a parameter is shown in Figure 2. I n the derivation of Equation 12, it was assumed that m/s, which involves certain physical properties of the system,
as E/R and T are known. After ti has been evaluated, the corresponding value of b = E/Rtl may be calculated. Using the value of b ( l - l/Q) obtained in the first trial, the value of J / p for the second trial is obtained from Figure 2. This process may be repeated until a ti of the desired accuracy is obtained. After tl has been determined, the volume of the preheat section (for a flow system) or the preheat time (for a batch system) may be calculated with
Because ko is usually zero, Equation 11 reduces to Equation 12: Z[b (1
-
k)]
=
In -1
1
- xi
(12)
where
+p=1-
eb(1
e*Z(b) Z[b(l
- I/@)
- l/@)j
the reacting mixture were at the temperature of the condensing vapor
the following expression :
This expression is obtained by the 'direct integration of Equation 4. For a flow and a batch system, Equation 14 has the following forms.
-
T to (V,)p = m In T
- ti
(15)
and
-
(8)= ~ m In T T - to ti
(16)
The value of m, characteristic of the type and arrangement of the reactor, is obtained from Table I. Equation 15 may be rearranged to the form customarily used in heat transfer calculations, q = UA ATm. The simplifying assumption of negligible heat absorption by chemical reaction which led to Equation 4, upon which Equations 12 and 14-16 depend, is examined below. The error in the volume or time as calculated by Equations 15 and 16 is proportional to the ratio (c/m) and inversely proportional to (Q - 1) and the logarithmic term in Equations 15 and 16:
X
Figure 2.
Plot o f the function $ VOL. 50, NO. 5
0
MAY 1958
743
~~
Table II.
~
When the term corresponding to the heat absorbed by the chemical reaction is taken to be negligible, Equation 2 reduces to Equation 4. T o develop an inequality relating the error associated with this approximation and the physical variables of the process, Equation 2 is solved for dZ:
Range of Parameters
0.0, 0.1, 0.2, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0,7.0,8.0,9.0(provided 0 6 c/s 5 @ - 1)
a:
b
5, 10, 15, 20, 25, 35,45, 55
4 s
0.005, 0.058, 0.111
@
1.02, 1.06, 1.10, 1.14, 1.18
Values for Figure 3
Table 111.
b
-
Integrating termwise over the preheat section
~
35
45
55
F1.40 F3.40 F4.60 F5.20 F5.40
F2.05 F4.10 F5.10 F5.60 F5.85
F2.50 F4.75 F5.50 F6.00 F6 20 e
F3.00 F5.50 F5.95 F6.25 F6.40
F1.45 F2.50 F3.10 F3.50
a 0.0, m/s = 0.58 F0.60 F0.95 F1.15 F2.85 F3.15 F2.25 F3.20 F3.80 F4.20 F4.00 F4.50 F4.80 F4.50 F4.80 F5.20
F1 .SO F3.90 F4.85 F5.40 F5.80
F2.25 F4.50 F5.30 F5.75 F6.05
F2.60 F4.85 F5.60 F6.05 F6.15
F0.65 F2.00 F3.10 F3.80 F4.25
F1.65 F3.75 F4.65
F1.60 F2.00
F0.45 F1.40 F2.25 F2.90 F3.35
F5.35 F5.60
F2.15 F4.30 F5.20 F5.65 F5.95
F2.40 F4.65 F5.50 F5.90 F6.20
1.02 1.06 1.10 1.14 1.18
G1.20 G1.60 GI.80 G2.20 G2.40
G1.40 G2.05 G2.65 G3.00 G3.40
G1.60 G2.55 G3.25 G3.60 G3.85
m / s = 0.005 G1.80 G2.00 G2.80 G3.20 G3.55 G3.80 G3.80 G4.10 G4.15 G4.35
62.30 G3.65 G4.10 G4.40 G4.60
G2.60 G4.00 G4.45 G4.65 G4.85
GZ 90
1.02 1.06 1.10 1.14 1.18
G1.05 GI.50 GI.70 G1.95 G2.30
G1.40 G1.95 G2.45 G2.85 G3.20
a: = 0.1, m/s = 0.058 G1.50 G1.70 G1.90 G2.40 G2.65 G3.00 G2.95 G3.40 G3.60 G3.75 G4.10 G3.40 G4.05 G4.25 G3.70
G2.25 G3.40 G4.00 G4.30 G4.50
G2.50 G3.80 G4.25 G4.55 G4.70
G2.60 G3.90 G4 40 G4.60 G4.85
35
45
55
G1.80 G2.85 G3.40 G3.80 G4.05
G2.10 G3.25 G3.85 G4.20 G4.40
G2.45 G3.60 G4.20 G4.45 G4.60
G2.80 G3.80 G4.35 G4.60 G4.70
a: = 0.2, m/s = 0.005 A0.28 A0.32 A0.13 C0.76 C0.87 B1.04 C1.04 B1.24 B1.45 A1.34 B1.58 A1.72 C1.72 C1.86 (21.58
A0.61 B1.34 B1.72 A1.92 C2.03
A0.71 C1.58 B1.92 A2.03 C2.08
A0 .93 B1.72 B2.03 (2.14 c2.21
D0.50 D1.21 D1.58 D1.78 D1.89
D0.66 D1.45 D1.78 D1.89 D2.02
D0.79 D1.58 D1.89 D1.97 D2.11
E0.55
E0.66
E1.02 E1.47 E1.70 E1.87
El .26 El.70
E0.87 E1.47 El.76 EZ.03 E2.17
0
5
1.02 1.06 1.10 1.14 1.18
F0.05 F0.80 F1.35 F1.85 F2.30
F0.55 F1.80 F2.65 F3.40 F3.90
1.02 1.06 1.10 1.14 1.18
FO.00 F0.60 F1.20 F1.70 F2.10
FO.40
1.02
1.06
F0.05 FO .65
1.10
F1.15
1.14 1.18
10
20
15 a: =
0.0, m/s = 0.005
F0.80 F2.45 F3.60 F4.30 F4.75
a: =
25
F1.20 F3.15 F4.20 F4.65
F5.15
0.0, m/s = 0.111 F0.95 F1.20 F2.60 F3.00 F3.80 F4.00 F4.35 F4.65 F4.65 F5.10
a = 0.1,
where 1)
@
- $0
9 - 1
Z denotes the correct value of Z for the preheat section (or preheat time), thereby distinguishing it from the approximate value given by Equation 14. Although the precise value of the integral in Equation 19 is not known, it can be said
o
