Nomenclature u
C
= =
C,
=
E
=
e
=
f H
=
H = n
,‘
= =
L)‘,
=
C,i = u
=
x
=
y
=
pd =
Ap =
pulse amplitude, inches tracer concentration hundredth-value stages, number of plate spacing distances required to decrease the backmixing tracer concentration CICO to 0.01, stages backmixing coefficient based upon superficial velocity, E = He, sq. ft./’hr. backmixing coefficient based upon actual velocity, sq. ft.,’hr, transition frequency. cycles,;minute phase fractional holdup in the column stage number stage height equal in distance to the plate spacing, but may be located between or across the plates, ft. continuous phase superficial velocity based on a single phase in the column. ft . ,’hr. discontinuous phase superficial velocity based on a single phaue in the column, ft. ‘hr. C’H. actual velocity based on t\vo phases uniformly distributed in the column, f t . ‘hr. column distance, ft. interfacial tersion: Ib.; sq. hr. orgacic viscosity, lb. (ft.)(hr.) phase dersity difference, Ib./cu. ft.
literature Cited
(1) .\skins, J. \I,..Chem. Efig. Progr. 47, 401 (1951). (2) Babb. A . L.. C.S. I t . Enerey Conin;. .Votebook. H W N 1901, 195;
(3) Bell. R. I... Uni\-ei.sity of Lyashington, Seattle, \Vash., unpublished data. 1963. 46, 233 14) Bernard. R. A , , \\‘ilhelm. R. H.. C h m . Ene. - Prner. . (1 950). ( 5 ) Rcyer. G. I,;Znd. Eng. Chem. Fundamentois 1, 93 (1962). (26) Smoot. L. D.; Mar. B. \V.. Rabb. A. L.. I d . Enq. Cheni. 50, 1055 (1958). (271 Swift. \V, H.. Burger, I>,L . , “Backinixin? in Pulsed Columns 11.’‘ C‘.S. At. Enerty Conini. R(,pt.. HW-29010, 1953. (28) Taylor. G. I . . Proc. Roy. Soc. (London) A219, 186 (1953). (29) Ibid.. A223. 446 11354). - 7
-.
-
i--
,.
(31) Van der Lann, E. G.. Chem. En:. Sci. 7 , 187 (1958). (12) Wilburn, N. P.. Dierks. R. D.. “Program for the Development of a Mathematical Model of a Solvent Extraction System.“ lT.S.A t . Energy Comm. Rcpt.. HW-64623, 1960. RECEIVED for review March 4. 1963 ACCEPTED March 5 . 1964
0 PT IM IZAT IO N OF BACKM IX REACTORS I N SERIES FOR A SINGLE REACTION STEPHEN SZEPE AND OCTAVE LEVENSPIEL’ Department of ChPmicai Engineerln,q, Zllinois Instilute of Technolo,fy, Chicago, Ill.
The optimum volumetric ratio in an isothermal sequence of two backmix reactors i s determined as a function o f reaction order and terminal conversion, for single reactions. This solution i s then generalized for the optimization o f multistage systems. The graphical method o f Kubota and its relationship to Denbigh’s “maximization of rectangles” i s discussed. Analysis of the volume reductions achieved b y optimization shows that the convenient equal-volume arrangement i s satisfactory for the design of such simple reaction systems. N HIS B A S I C
WORK
on continuous reactors, Denbigh has
I shoivn ( 3 ) that the total volume required by a stirred tank
reactor system can be reduced in two distinct ways: by increasing the number of reactors in series, and by correctly selecting the relative sizes of reactors. T h e second of these methods is the subject of the present note, In this respect. Denbigh has found that there exists a Present England. 214
address,
University
of
Cambridge, Cambridge,
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
volumetric ratio of reactors: ( T ’ ? VI)*. for \vhich the total volume in a two-stage system is a t a minimum. For an nth order irreversible reaction. this ratio is approximately n Actually, for first-ordcr rcactions the above result is exact; but for reaction ordcrs other than one it is restricted to high conversions. as assumed in Dcnbigh’s derivation. For low conversions the optimum might be quite far from n. as demonstrated in a specific example given by Leclerc (6). I n connection Lvith this problem the purpose of this article is twofold :
-00I
.01
I .o
.I c2 = I - x p CO
Figure 1.
Optimum intermediate concentrations in a two-reactor system
To obtain a general relationship for the optimum volumetric ratio, applicable to the entire range of conversion, for any reaction order. and To evaluate the significance of this optimum-i.e., to calculate the volume reduction achieved by using the optimum, instead of the more convenient equal-sized, arrangement.
and Equation 2 becomes, after substitution and rrarrangement : (Cl*)n+'
+ (n
-
l)rancl* -
=
nCg(.zn
(1 -
X1*)%+1
+
(72
- 1)(1 -
x2)n
(1 -
.XI*)
n(l
A-R taking place in a n isothermal sequence of two backmix reactors (continuous flow stirred tank reactors with perfect mixing). T h e total volume required to achieve some specified terminal concentration of A , c2. can be calculated as:
For a given floiv rate and initial concentration of A: CI is the only independent variable in Equation 1. The optimum volumetric ratio corresponding to a minimum total volume can be obtained in two steps-the essential first step being to find the optimum intermediate concentration, c1*. By setting
-d V= o dci
(4)
or in terms of conversions:
Optimum Volumetric Ratios in a Two-Reactor System
Consider a single re(4ction of the type:
0
- XJ?? =
(5)
0
As indicated by Equation 5. the optimum intermediate conversion is independent of initial concentration and rat? constant, depending solely on the terminal coil\ ersion and reaction order. \.\'bile it is not possible to solve Equation 5 analytically. except for n = 1 . numerical solutions for 1 , * can be obtained for various values of n and xa. Such calculations were performed on an IBM 1620 computer, tising r\'e\\.ton's mrthod (5). Subsequently. the optimum volumetric rdtio \vas calculated as:
(;)*
1 - XI*
=
- .k,*
x7
(KJ (T)
(6)
T h e results obtained are shoLvn in Figures 1 and 2. C h Figure 1: a function of the optimum intermediate coriversion, (1 - x i * ) > is plotted against fraction remaining. (1 - A ~ ) : with reaction order as a parameter. I n Figure 2. the optinrum volumetric ratio is given in a similar fashion. Kcpresriirative values of Figure 2 a r r presented in Table I .
the following equation results: Table 1.
cz=
7 -
cn
Equation 2 is a specific case of the general expression given by Kubota ~t ai. ( 4 )for ai-stage system, For irrevrrsible rractions of order n:
0 5
0 1 0 01 0 001 (0)
-r
= kcn
Optimum Volumetric Ratios, ( V 2 / V , ) * ,in a TwoReactor System A2
0, 0 85 0 58 0 36 0 26 ( 0 1)
o-5
0 0 0 0 (0
n ~
7
~
92 76 63 56
5)
-
1 00 1 00 1 00 1 00 (1 0 )
~~
2 1 16
7 7
1 08 1 21 1 36 1 44
i l 5,
1 44 1 '1 1 85 ( 2 0)
__ i
l 31 1 82 2 il
2 60 (3 0 )
(3) VOL.
3
NO. 3
JULY
1964
215
-Exact solution --- Denbigh approximation
.ooI
I
1
I
Figure 2.
I
1
1
1
1
.oI
This equation fits the values of Table I with an error in all cases less than 0.06. Graphical Method
.4 graphical method for obtaining the optimum intermediate concentration for any rate equation was given by Kubota et al. ( 4 ) . This method is a geometrical representation of the equations for the optimum condition-for a two-stage system of Equation 2. An interesting relationship exists between Kubota’s method and the “maximization of rectangles,” introduced by Denbigh (2) for another optimization problem involving backmix reactors: yield maximization in multiple reactions. T h e essential identity of these methods becomes clear when the followi’ng is recognized :
Minimization of o f )
=
I
,
,
I
I
I
l
l
l
l
I
*I
method this rectangle represents the yield resulting from the addition of a second reactor, and the curve is the instantaneous fractional yield us. conversion relationship.) O n Figure 3. the curve in question is the reciprocal rate plotted against concentration. T h e rectangle ABCD is the space time required by a single backmix reactor, and the rectangle CEFG is the space time reduction to be maximized T h e condition of Equation 2 states that the optimum is a t that intermediate concentration where the slope of the reciprocal rate curve is equal to the slope of the diagonal EG. Multistage Systems
T h e results obtained for two-reactor systems can be extended to the optimization of a multistage setup. For a ]stage system ( I - 1) intermediate conversions are to be opti-
’1
-r
(-; - 3 -‘I
Maximization of (achieved space timebyreduction addition) of second reactor to a one-reactor system
a n d \\hen it is considered that in both methods a rectangle ( I n Denbigh‘s bounded by a curve is to be maximized. 216
I
Optimum volumetric ratios in a two-reactor system
As shown in Figure 2 , for high conversions the optimum volumetric ratio approaches the result given by Denbigh, while for low conversions this ratio tends to 1. T h e difference between the approximation and the exact optimum is, of course, especially pronounced for reaction orders far from unity. I t might be pointed out that the family of curves on Figure 2 can be well represented by the simple empirical equation :
(; :lr;mcacrtixne
I
I
l&EC PROCESS DESIGN AND DEVELOPMENT
I c2
CI
CO
C
Figure 3. Graphical method for finding the optimum intermediate concentration in a two-reactor system
mized. and the following equations result for the condition of optimality :
Table II.
Per Cent Reduction in Total Volume in a TwoReactor System
-
veq
X 100 =
‘Opt
% reduction
v e9
c1= CO
2
= 1,2,
. , I - 1 (8)
Since these equations are identical in form to Equation 5, Figure 1 can be used directly for a trial and error solution. (The coordinates of Figure 1 should now read [(l - x i * ) / (1 x t - ~ ) ]and [(l - x t + 1 ) / ( 1 - xi-J], respectively.) The optimum volumetric ratio between stage? can be calculated as :
-
As might be expected. for a given terminal conversion, x j , the calculated optimum ratios tend to unity with increasing number of stages. Additional Considerations
To determine the advantages of the optimum system, the reduction in total volume as compared with the equal-volume case was also calculated in the computer program. Representative values of this calculation are shown in Table 11. These values show that the volume reduction, in general, is negligibly small. becoming appreciable only a t very high conversions. T h e calculations also showed that in a wide range of terminal conversions the equal-volume arrangement is actually better than the one with a ratio n, as suggested by Denbigh-up to about 85% conversions for second- and third-order reactions, and u p to 95yGconversions for a halforder reaction. In addition to the above, if one considers the extra cost of installing units of unequal sizes and the fact that a given volumetric ratio is only optimum for a specific conversion, it becomes doubtful whether the design for such an optimum would ever be justified If size considerations are really critical. the addition of another unit is far superior to trying to choose an optimum volumetric ratio. The above discussion shows that in designing a backmix reactor system. one should use the convenient equal-volume arrangement. This conclusion is not limited to a two-reactor system but holds even more strongly for the general ]-reactor case
n
- x2
0.1
0.5
1
7.5
2
0.5 0.1 0.01 0.001
