I
OD1
STANLEY M. WALAS
-0
f
University of Kansas, Lawrence, Kan. --0.1
2
Use Nornographs to S o l v e .
.
3 r0.2 4 5-
70.3
678.90.102
REACTIONS IN
STIRRED TANKS
K8 - 0.4 0.5
2 0.6
3
Design calculations for the continuous stirred tank reactor battery are simplified for many types of liquid phase reactions
0.7 4
5-
0.8
67-
0.9 1.0
Figure 1 . This nomograph can solve a first-order reaction whose rate i s given by Equation 2
STIRRED
tanks are often used in series to accomplish continuous reaction, thus affording a highly flexible system adaptable to small or medium production capacities and especially suitable for relatively slow rates of reaction. For reactions of order higher than the first, the process design for a series of stirred tanks involves tedious graphical or algebraic calculations (2). To minimize this labor, generalized charts have been developed for direct solution of the principal types of reactions (7). Constant volume and complete mixing are assumed. The charts presented here eliminate the tedious solution of successive quadratic or cubic equations and are particularly time saving when the specific rate or the residence time is to be evaluated.
Theoretical The material balance on the rnth unit of a continuous stirred tank reactor (CSTR) battery is r,R
+ C,
=
Cm-l
(1)
where 7 m is the reaction rate in the mth reactor, e is the residence time or the ratio of the reactor volume to the volumetric flow rate, and C is concentration. Upon substituting for the rates in terms of concentrations, the CSTR equations are developed in those shown below for the stated reactions, in terms of x =
c/co:
Rate Equation
CSTR Equation x m = 1/(1
r = kC
+ kR)",
(29
r = kC3
I n Equation 3, C, designates the excess of the concentration of the second reactant over that of the first, and C,, is the initial concentration of the first reactant.
Examples I n the case of the first-order reaction, any one of the three variables x,,, rn, or k0 can be evaluated directly by Figure 1 when the other two are specified. Solution of higher order reactions requires a stepwise procedure. Consider Equation 3 with x, and l/keCo specified. The fractions X I , X Z , . . . are determined in succession with Figure 2. O n the example (Figure 3) the various points marked are located as follows: Point Determined
Line
PlPZ pap4 paps (P7by translation of P6) plp7
PKPS and so on.
Pa PS P6
(or XI)
Equation 4, is solved similarly with Figure 4. For this case, as shown on the example (Figure 3), point P d is not needed, but otherwise the construction is exactly the same. When x, and ni are specified and either k or e is required, the solution is accomplished by trial. Thus several positions P1on Figure 3 are tried until one is found for which x, attains the specified m. Because each graphical solution is very rapid, the over-all problem is likewise solved quickly. More complex reactions also can be handled by these charts. The revcrsible rate equation 7 = klC(C C,) kz(C0 -- C P (5) leads to the CSTR equation .", P X n = qm-1 (6) with
+ +
O n Figure 2, the lines connecting p with 4 0 , 41, . . . intersect the x, axis in XI, x2, . . . , respectively. The third-order rate equation, inwhich Cf designates the excess of the concentration of the third reactant over that of the first r = kC(C
Pa P, (or xt)
T h e third-order reaction,
-
leads to x:
+ C,)(C + C f )
+ P X 6 = qm-
VOL. 52, NO. 10
(9)
1
(10)
OCTOBER 1960
831
the form of Equation 6 applies, but with
where x,
p
= XeXf
=
-
+
x,,
xe ~
+3
xi
(11
+3 x.fP + kRCo1
(xe
>
:
(12
Figure 3. Construction procedure used with either Figure 2 or 4 when k or 0 i s required
h
h
kecao
4
where .Y = C,/Cao. To start the solution, Cal is evaluated from Figure 1. Then ;b and yo are calculated from Equations 16 and 17 and located on their respective axes on Figure 2, thus determining xl. Similarly, x2 is determined by the line passing through p and q l , and so on.
’4
Figure 4 is used to solve Equation 10 just as Figure 2 is used for Equation 6. As a final example, consider the C -+ D for two reactions A -+ B and B which the rate equations are
+
la t6
= klC,
= k?Cb(Cb
f
c,) - kiC,
I
p = - + x e
Literature Cited
(1) Lipka, J., “Graphical and Mechanical Computation,” p. 110, Wiley, New York, 1918. ( 2 ) Walas, S. M., “Reaction Kinetics for Chemical Engineers,‘’ Chap. 4,McGrawHill, Sew York, 1959.
(14) (15)
The CSTR equation corresponding to Equation 14 is solved with Figure 1. For the other reaction, an equation of
RECEIVED for review March 18, 1960 ACCEPTED July 25, 1960
2c
Figure 2. To solve a second-order reaction whose rate is given by Equations 3 or 5, determine the fractions in succession from this nomograph
19 18 17
16
Typical construction is shown in Figure 3
Figure 4. This nomograph i s used for solving a third-order reaction whose rate i s given by Equations 4 or 9 Figure 3 shows a typical construction
15 14
13 12
E:: I,”
P
7 c
i
832
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Crn/Co
