LITERATURE CITED
L. =
%
;
1.3 .
t-
4
FEHROUSLWL
E'u 1.0 .
/;/f
//,,/e-.
2
8
F t H R l C I RON
2 a
b lua
-
/-'
3
r_____"'
ClRCOSdi
2XH
DLTIEX 2:
SUNUEX L3
59 5 IO 125 5 5"
-
__-. 375
30
by the nitrogen bases, as shoivn by the results wit>hlatex No. 3. The osidntion of the latex seem2 t o inlply lhnt ferrous iron accelcrates the effect of the nitroycn bases, but ferric iron does not,.
jC
60
(1) hdains, Norman G., Richardson, Dorothy 31., Anal. CItem. 23, 129-33 (1951). (2) Eby, L. T., Ibid., 25, 1057 (1953). (3) Froidel, Robert d.,Orchin, Alilton, "Ultra Violet Spectra of Aromatic C'ompounds," Wiley, K e w York, 1951. (4) Hersh, R. E., Fenske, El. R., Uatson, H. J., Koch, E. F.,Booser, E. R., Braun, W.G., Anal. Chena. 20, 434-44 (1948). (5) Johnson, Paul H., Afiller, K. L . , Jr., Benedict, B. C., IKD. EXG.CHEW47, 1578 (1955). (6) Kurtz. S. F., Jr., Nartin, C . C . , India Rubher World 126, 495 (1952). (7) Linnig, Frederick 3.. private caoinmunication. (8) Nostler, F'.S.,White, R. R I . , IND.ENG.CHEM. 41,598 (1949). (9) Taft, Mi. K., Duke, J., Larchsr, T. B., Sr., Hitzmiller, W. G.. Feldon, SI.,Ibid., 48, 1220 (1956). (10) Taft, W. K., Duke, June, Lauridrie, It. W., Snyder, A. D., Prem, D. C., hlooney, Howard, I b i d . , 46, 396-412 (11) Taft, W. K., Duke, J., Snyder R. W., R u b b e r Aye (N. Y.) 75, Xo. 1, (1954). (12) Taft, W. K., E'eldon, Milton, Duke, June,
RECEIVEDio!. review November 16, 1956. ACCEPTEDFebruary 13, 1956. l\leeting, Pa., L)ivision of Rubber Chemistry, I , ~ , . 1955. Work performed as p a r t of the government synthetic rubber
Design Equations for Continuous Stirred-Tank Reactors 1)EAIt S1R:
The authow of this article [Acton, F. S.,Lapidus, I,., Isu. ICNG. CHFM.47, 706 (1955)] ( 1 ) made t v o mistakes, which some\rh:*f \yeaken their treatment,. The basic points in the folloxing discussion are taken from some unpublished lectuw riotcs (111 liomogeneous reactors. ('onqicier the irreversible second-orc1c.r r t w c l ion
A f B (in excese) -+ products
(11
which occurs in the cont,inuous stirred-tank reactor, then the material lsalance for A4on the nth tank in the form is
- a,
=
da, dT
+ Ka,b,
(2)
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1956 where T = tq/V
1229
Thus, Equation 2 can be written as
K = kV/q
(3)
and where the symbols have the same significance as in the original article. Similarily, for B we find (4)
To find the relationship between an and b,, Equation 2 is subtracted from Equation 4, thus obtaining the stoichiometric equation (which has the same form as the “purging” equation in the absence of a chemical reaction)
da, d~ + (1 + Keofn)an + K u ~= a n - 1
wheref, = 1 for case A and is given by Equation 11 for case 13. Again, we must consider the initial and boundary conditions for an and we will discuss only two cases: Case a. a,(O) = ao(T) = a0 = constant
e, = b,
- an
(6)
is the excess of B in the nth tank. To solve the system of Equation 5, the initial and boundary conditions for the e, must be considered. Two typical case8 are considered : Case A. e,(O) = eo(T) = eo = constant
(7)
nhere the subscript 0 refers to conditions in the feed to the first tank-i.e., the initial excess of B in each tank is the same as the excess in the feed, which is constant. (This is the case actually considered by Acton and Lapidus, although their initial and boundary conditions are clearly inconsistent, being a mixkure of those for this case and for case B, considered below.) The solution of Equation 5 subjected to the conditions Equation 7 is en(T) = eo or b,(T) = a,(T)
+ eo
a,(O)
=
0, ao(T) = a0 = constant
(14)
Hence altogether there are four cases: Aa, Ab, Ba, and Bb. Of these four, case Ab (which is the one actually studied by Acton and Lapidus) and case Bb are of greatest practical importance, as there will inevitably be some loss of unreacted A as well as B in the remaining cases during starting up. We have, therefore, the problem of solving the system of nonlinear difference-differential Equation 12, subject to the eet of conditions, Equation 13 or 14. It is probably impossible to obtain the general solution in closed form in terms of tabulated functions. Thus Acton and Lapidus found an analytic solution for case Ab only for a l ( T ) and some approximate or numerical method is, in general, necessary. The practical problem thus arises as to the number of parameters-in addition to the variable T-on which the concentrations a , depend. These authors claim that there are three parameters-say, K , eo, and so-so that a complete approximate solution is impossible for practical reasons. I n fact, this claim is the basis of their article. The number of parameters is, however, only two, if the quantities are defined as
(8)
that is, the excess is constant. [This is the result which these authors state without proof, apparently believing that it applies in all cases. This result seems to have been taken from another article (W), while forgetting that it is, in general, valid only for the steady state.]
(13)
Case b.
(5) n herc
(12)
A,, = Ka,,
Eo = Keo, and A.
=
Kao
(15)
Equations 12 through 14 then can be written in the form
+ (1 + Eofn)An + Af = A,-I
(16)
A,(O) = Ao(Tj = A. = constant
(17)
A,(Oj = 0, Ao(T) = AO = constant
(18)
dT with either
Case I3
e,(O) = 0, eo(??) = eo = constant
(9) or
Here the solution of Equation 5 is e,(T) = eOf4T) or bn(T) = an(T)
+ eoSn(T)
(10)
where
Here the excess in the nth tank is not constant, but only approaches eo for large values of T . These are not the only cases which arise in practice. For example, some or all of the effluent from the last tank may be recycled to the feed during starting up to avoid loss of unreacted A in cases where a.(O) = 0, while in studies concerning the regulation of such reactors, it would be desirable to study the influence of a sinusoidal or step perturbation of eo(T). Then there is the whole field of shutting-down problems. I n all cases, however, there is no great difficulty in solving Equation 5 for e, and thus in finding the relationship between bn and a,. It should be noted that for any reaction there can be derived, by a similar procedure, a system of linear stoichiometric equations for the sums or differences of the concentrations of the reactants and products with respect to that of one of the constituents, so that the concentrations of all the other constituents can be expressed in terms of that of this base one. Similar remarks apply to systems of reactions. This approach can also easily be extended to startingup procedures in Thich the feed is run into empty tanks ( 5 ) .
Thus a complete solution, although laborious, is possible, a t least with a machine. Equation 16 has the form of a Riccatti equation and can profitably be transformed to a linear equation of the second order. Let
An(T) =
dlnu,(T)
7
(19)
when this equation is found
If we analytically or otherwise find a fundamental set of solutions of these equations-say, (on and $n-so that the general solution is Un =
+
C I P ~ Cz$n
(21)
where CI and CZare integration constants, we can express A , as
and find the value of the ratio CZ/Clfrom the value of A.(O).
_.
-
-
- -__
INDUSTRIAL AND ENGINEERING CHEMISTRY
1230
For the case .Ab (and Aa) it is convenient to introdure the quantities
D = (I
+ Eo)/2, S = D T , oCn(S) cyo =
Ao/D, and
y =
A , , ( T ) / D , C,,(S)
(1
Un(T),
+ CUO/D)'/~
(23)
The substitutions
u l ( T ) = U 1 ( S ) = SO(1 iY ) exp ( - S ) V ( S ) n-ith S = ( 2 0 - 1) esp ( - ?')
S-dS2
so that a,,(S)is given by Equation 22, but x i t h c y , , ( S ) , @,l(S), and *,,(AS)in place of A , ( T ) , &(2'), and +bn(T)> respectively. I n particular for n = I , it is easy t o see that a suitnblc fnndaincntal set of solutions is
(1
+ 1?0)/2]transform this ccjuation to
+ (1 + 2 D y
-
S)
dV - [ D ( r - 1) dS ~
+ l]V = 0
=
exp ( -8) Y
(y
cosh y S
-
sinh y S )
(26)
Thus
For q ( 0 ) = 0 (case Ab) we get Acton and Lapidus' result, and
(35
+
+
+
+
+ +
Q:(S)
1
which is the standard form of t h e confluent hypergconicti,i(. 1: I equation (4,6 ) , a solution of which is F [ D ( r - 1) 20,; SI. Here taking the exponent of S in I
