Ind. Eng. Chem. Res. 1987,26, 376-378
376
For a classical Rice-Herzfeld (1944) free-radical chain reaction of a pure compound A in tetralin, the long-chain and steady-state approximations (Gavalas, 1966) facilitate development of an analytical expression for the rate of reaction, rR This is shown as eq 1,for the case of initiation
impact both molecular and free-radical fragmentation of carbon-carbon and carbon-hydrogen bonds. A molecular reaction might be viewed formally as the simultaneous occurrence of the free-radical reaction steps of @-scission and hydrogen abstraction. Thus, this model served as a quantitative assignment of the relative reactivities of primary, secondary, and tertiary bonds and not as a verification of a molecular mechanism. Literature Cited
by first-order fission, chain transfer by reaction of substrate-derived 0-radicals with tetralin and also the thusderived tetralyl radicals with substrate, and second-order chain termination by all possible combinations of p - , 0-, and y-radicals (tetralyl). The rate constants for termination by two different radicals were taken to be twice those for self-termination (k,) to reflect their statistically expected values (Pryor, 1966). The subscripts in eq 1 denote the elementary steps in the free-radical mechanism as follows: ki = chain initiation, k, = p-radical decomposition (@-scission),k, = H abstraction from A by y-radicals, kH = H abstraction from A by &radicals, kD = D abstraction from tetralin-d,, by 0,and k, = chain termination. Equation 1 shows that the primary effect of R on the rate of the free-radical reaction is through reactant dilution: in the liquid phase, A will approach zero as R m. In this limit, the reaction order in A approaches 1.5. On the other hand, the order of the molecular reaction remains 1.0 at all R, and thus, at very high tetralin loadings, reactant dilution decreases the rate of the radical reaction to a greater extent than the rate of the molecular reaction. Therefore, the value of DI as 1/R 0 is less than or equal to the free-radical component at all other values of R and hence serves as a lower bound for the free-radical fraction of neat pyrolysis. Since DI 1.0 as 1/R 0, these data suggest the PDD pyrolysis mechanism is entirely freeradical at all R. The accurate predictions of product profiles by Blouri et al, (1985) suggest that the same energetic fundamentals
-
-
-
-
Blouri, B.; Hamdan, F.; Herault, D. Ind. Eng. Chem. Process Des Dev. 1985, 24, 30. Cronauer, D. C.; Jewell, D. M.; Shah, Y. T.; Modi, R. J. Ind. Eng. Chem. Fundam. 1979, 18, 153. Gavalas, G. R. Chem. Eng. Sci. 1966, 21, 133. Gilbert, K. C.; Gajewski, F. J. J. Org. Chem. 1982, 47, 4899. Klein, M. T.; Virk, P. S. Ind. Eng. Chem. Fundam. 1983, 22, 35. Kossiakoff, A,; Rice, F. 0. J . Am. Chem. Soc. 1943, 65, 520. Miller, D. B. Ind. Eng. Chem. Product Res. Deu. 1963, 2, 220. Miller, R. E.; Stein, S. E. J . Phys. Chem. 1981, 85, 580. Mushrush, G. W.; Hazlett, R. N. Ind. Eng. Chem. Fundan. 1984,23. 288. O'Malley, M. M. BChE Thesis, University of Delaware, Newark, 1985. O'Malley, M. M.; Bennett, M. A.; Simmons, M. B.; Thompson. E. D.; Klein, M. T. Fuel 1985, 64, 1027. Poutsma, M. L.; Dyer, C. W. J . Org. Chem. 1982, 47, 4903. Pryor, W. A. Free Radicals; McGraw-Hill: New York, 1966; p 14. Rebick, C. Adu. Chem. Ser. 1979, 183, 1. Rice, F. 0.;Herzfeld, K. F. J. Am. Chem. SOC. 1944, 56, 284. Virk, P. S. Fuel 1979, 58, 149. Woodward, R. B.; Hoffmann, R. The Conservation of Orbrtal Symmetry; Academic: New York, 1970.
* Author to whom correspondence should he addressed. 'Present address: Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109.
Phillip E. Savage,+Michael T. Klein* Department of Chemical Engineering and Center for Catalytic Science and Technolog3 University of Delaware Newark, Delaware 19716 Received for review December 11, 1985 Accepted September 2, 1986
Optimization of Consecutive Reactions with Recovery and Reuse of Unconverted Reactant T h e operating conditions for consecutive reactions are determined, which give the maximum yield of intermediate in terms of the reactant recovery ratio m and t h e rate constant ratio k , / k , . T h e effectiveness of CSTR and PFR reactors in the recycle system (RS) is compared. T h e difference in general yield between CSTR and PFR in the RS can be reduced by raising the value of the recovery ratio m. For consecutive reactions such as k,
A-R-S
k,
the maximum amount of intermediate obtainable from a once through operation depends only on the ratio of the rate constants k,/k,. However, when some of the unconverted reactant can be separated from the exit stream and returned to the reactor, the yield of the intermediate can be increased. For brevity, the no-recycle system is called NRS, and the recycle system is called RS, in this paper. 0888-5885/87/2626-0376$01.50/0
I. Simplified Process a n d Optimum Equation The system studied consists of a reaction unit and a recovery unit. The reaction unit contains reactors and other auxiliary equipment. Reactors can be CSTR, PFR, or nonideal flow vessels modeled as tanks in series. The recovery unit includes all equipment needed to separate and purify various components from the mixture flowing from the reactor. Figure 1 shows the system considered with all streams labeled, where nAO= initial moles of the reactant A/unit time, nA = moles of component A at the outlet of the reactor/unit time, nR = moles of component R at the outlet 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 2, 1987 377
111. First-Order Consecutive Reaction Consider the consecutive first-order reactions
T I A'
O-mnA
kl
A-R-S
Reactron Unit
A n*o
which are very simple and are often met in practice. Let the general case of nonideal flow in the reactor be represented by a series of N stirred tanks. Then we find (Butt, 1980) CAN
nR nR
Y= nAO
k2
-
- mnA
nAO
1-m-
nA
(1)
=
CAO
(1 + klt)N
where CAN = moles of component A at outlet of Nth reactor, Cm = moles of component R at outlet of Nth reactor, and t = is the residence time in each reactor. Substitute eq 6 into eq 7. This gives
nAO
Because nR/nAO= f(nA/nAo)and because m is a constant, the problem we are discussing becomes a one-dimensional optimization. So, to maximize the production of intermediate R, differentiate Y with respect to nA/nAOand put dY , ,
where
=o
(9) Substitute eq 8 into eq 3. This gives the optimum equation for RS of first-order consecutive reactions. 1. When k # 1 we find
This gives the optimum for consecutive reactions as
For first-order consecutive reactions, it is convenient to use concentration C instead of moles n. Thus, eq 2 becomes
+ k # 1,eq 10 becomes -k + [ ( l m)k]'/'
( 1 ) For CSTR, N = 1, if m
-
1-m-k if m
+ k = 1, eq 10 becomes
(2) For PFR, N
11. Relationship between the Optimum Point of RS and That of NRS If m = 0, eq 3 can be simplified into
m(1 - k)(
This is the equation which gives the optimum point of NRS. If m = 1, eq 3 becomes CR = 0
-
1 2 m,
eq 10 becomes
+ k(
$)pFR
").-' A '0
(4)
(5)
This equation tells us that if unreacted A is recovered completely, the optimum single-pass conversion would approach zero and the optimum reaction time would approach zero. In practice, m = 1is impossible and m always lies between zero and unity, and so with it comes its corresponding optimum conversion and optimum reaction time.
(11)
- 1= 0
(13)
PFR
2. When k = 1, we find
(1)For CSTR, N = 1, eq 14 becomes 1 - (1 - m)I/*
(2) For PFR, N
-
m m,
eq 14 becomes
(15)
378 Ind. Eng. Chem. Res., Vol. 26, No. 2 , 1987 Table I. Changes of Y * C S T R / Y *with ~ F Rm Value
Y*CSTR/ Y*PFR
US
-
-3
-2
-/
1
0
0.80 0.810
0.50 0.740
0.00 0.680
m
2
3
4
Ly u Figure 2. Comparison of maximum yields for consecutive first-order reaction in the CSTR and PFR in the RS.
By substitution of eq 11 and 13 into eq 1, two equations in which the optimum intermediate yield is a function of m and k can be given. for CSTR
for PFR
(5) -(") 'A0
PFR
'A0
PFR
Y*PFR
0.95 0.890
0.99 0.947
1.00 1.000
for NRS. In the RS, eq 17 and 18 should be used to calculate Y*csTR/ Y*pFR. The calculated results under k = 1 are shown in Table I. It can be seen that Y*csm/ Y*pm is a function of m. The ratio approaches unity with increasing m values; that is, the differences between CSTR and PFR become smaller with increasing m values. The maximum yield ratios are calculated by eq 17 and 18. The results obtained are shown in Figure 2. We, thus, conclude that in the RS, although the maximum yield of intermediate in CSTR is always smaller than in PFR, this difference can be reduced by raising the value of recovery ratio m. Also the difference in general yield between the PFR and CSTR is not great in the RS if the m value approaches unity.
Acknowledgment We are thankful to Prof. 0. Levenspiel of Oregon University for his guidance to this paper.
Nomenclature A = reactant C = concentration, kmol/m3 K = ratio of reaction rate constants ( K = k 2 / k l ) ki = reaction rate constants, time-' (i = 1 and 2) m = reactant recovery ratio N = number of equal-sized backmix reactors in series n = moles/unit time R = intermediate S = byproduct t = reaction time Y = yield of intermediate Superscripts * = optimum value
IV. Comparison of Effectiveness in Two Ideal Reactors In optimization of consecutive reactions, the main way discussed in the literature (Debigh and Turner, 1979) was often the comparison of PFR and CSTR. The ratio of the maximum concentrations of intermediate in CSTR and PFR in NRS was considered as the ratio of the maximum yield of intermediate of CSTR to that of PFR. The figure displaying the ratio of the maximum yields was made so that the difference between CSTR and PFR could be clearly shown. For example, we find that the yield ratio equals 0.68 for the NRS for k = 1. However, this conclusion is correct only
Literature Cited Butt, J. B. Reaction Kinetics and Reactor Design; Prentice-Hall: Englewood Cliffs, NJ, 1980; p 227. Denbigh, K. G.; Turner, J. C. R. Chemical Reactor Theory, 2nd ed.; Cambridge University Press: New York, 1979; p 130.
Liu Da-Zhuang,* Xu Hai-Sheng, Wang An-Zhong Chemical Engineering Department Zhengzhou Institute of Technology Zhengzhou, The People's Republic of China Received for review September 25, 1985 Revised manuscript received May 5 , 1986 Accepted September 23, 1986
