reactor isothermally. This \vi11 produce the greatest profit from the system. T h e cost of achieving the desired temperature level, the rate of recycle, the flo\v and composition of the fresh feed, and the value of the final product all serve to change the level of the optimal temperature, but in any problem the temperature is independent of the position along the axis of the reaction vessel. The optimal temperature level, yield. and profit can be determined from Equations 20! 26, 29. 36. and 38. the kinetic equation being directly integrable, since the temperature fiinction is now independent of 1.
and
(33) for this case. Further, the boundary condition expressed by Equation 11, since t h e f , do not depend upon yo. leads to the usual result 1'0
=
1
(34)
W e therefore find from Equation 32. using Equations 26 and 34. that
Conclusions
T h e general problem of determining the equations that govern the choice of a suitable operating policy for a chemical reaction system lvith recycle has been presented for the case involving a tubular reactor. Based upon Pontryagin's \vork. these eqilations can be sho\vn to reduce to the expressions derived in the pioneering \vork of Katz for the case of no recycle. This optimal policy involves the most suitable choice of the system operating variables. such as the reactor temperature profile, the recycle rate: and the separation factors, in order to optimize a particular objective functional which normally will be a profit or cost relation.
T h e adjoint function v ( l ) may be related to the local composition of the reacting mixture by taking the ratio of Equations 35 and 26. and then integrating the resultant expression. These steps lead to
[u(4
-
al(r
+ q)li"(l)
=
+
[40) - a ~ ( r 411
(36)
If optimum profit function is assumed to lie a t a stationary point, the optimal temperature policy will be defined by Equation 22. I t is clear that this general condition gives
[u(l) -
al(r
+ q)1
Acknowledgment
(37) O n e of the authors (R. S. S) acknowledges the support of the Universitv of Texas Research Institute.
where
Z ' = bZ bT
literature Cited
(1) Fan, I,.>\Vang. C . , Chem. Eng. Sci. 19, 86 (1964). (2) Fan, L., \Vaiig, C.: J . .4ppI. M a t h . Phys. 15, 46 (1964). (3) Jackson, K.. Chem. Eng. Sci. 18, 215 (1963). (4) Ibid., 19, 19 (1964). (5) Zbid.. in press. (6) Katz. S.,.'inn. .V.1.. Acad. Sci.84, 441 (1960). (7) Kopp. R. E.. "Optimization Techniques," G . Leitmann, ed.. Chap. 7 , pp. 255-60. Academic Press, New York. 1962. (8) Kuhn, H. \V., Tucker. A. \V., "Second Berkeley Symposium on Mathematical Statistics and Probability." J. Neyinan. ed., p. 481, Cniversity of California Press, Berkeley, 1951. (9) Pontryagin, L. S., Boltyanskiy, V. G.. Gamkrelidze, K. V.? Mischenko, Ye. F., "Mathematical Theory of Optimal Processes," Chap. I and 11. U. S. Department of Comrnerce. Joint Publications Research Service. No. 15,089 (Sept. 4: 1962). (IO) Rudd, D. F., Blum, E. D.: Chem. Eng. Sci.17, 277 (1962).
for the example under consideration. Rearranging and substituting for u(1) from Equation 36, we have
2'
+
-a(r 4) x n ( 0 ) K [ v ( O ) - a1(r
+ q)l
(38)
T h e right-hand side of the equation is independent of the position, I , along the reactor, being only a function of the inlet and outlet conditions of the reactor. This equation cannot be solved directly for the temperature, but it is obvious that if the temperature is independent of the reactor position, it must be a constant throughout the reactor. T h e optimal operating policy in this case is therefore to operate the tubular
RECEIVED for review June 5, 1964 ACCEPTED February 4, 1965
OPTIMIZATION OF INITIAL COMPOSITION Adiabatic Equilibrium Gas-Phase Reactions in the Presence of Inerts C
HE composition of the particular initial reactant mixture T w h i c h will realize the greatest equilibrium yield can be determined theoretically in many cases. T h e classic distribution of stoichiometric ratios ( 7 . 8) is optimum for isothermal perfect gas reactions. HoLvever. deviations from ideal behavior. from an isothermal process, or maximization of a function other than yield. all may require substantial mod-
260
l&EC FUNDAMENTALS
. J.
P I N G S, California Instztute of Technology, Pasadena, Calij.
ification of the simple result. I n four previous papers (4-7). Lve have presented derivation of the formulas required to predict the optimum initial feed composition under a variety of practical conditions. T h e present paper considers the simultanrous effect of inerts and an adiabatic constraint. T h e results have general utility. but should have particular significance for reactions inlrolving air as a source of oxygen.
Expressions are derived for the initial distribution of mole fractions of reactants required to obtain maximum yield in gas phase reactions proceeding adiabatically to equilibrium in the presence of inerts. The initial composition necessary to attain the maximum equilibrium adiabatic temperature i s also identified. The required corrections to the classic distribution of stoichiometric ratios involve heat capacities, the enthalpy of reaction, and the concentration of inerts. The shifts in composition are nontrivial, and the resulting increases in yield and temperature can be substantial in certain cases.
Thermodynamics
Consider inerts carried in a fixed ratio to a given reactant. As in ( 4 )and (7). the present analysis treats only ideal solutions. Products of the reaction are assumed absent from the initial reactant mixture. 'I'he energy balance equation given by Equation 3 of ( 4 ) is noiv generalized to the following, including the effect of inerts :
]=1
problem, since the unkno\vn mole fraction distribution appears in the right-hand side of the equation. Nevertheless, as in our several previous solutions (4-7) of problems of this type, there exists a simple iterative process which involves assuming an initial mole fraction distribution. followed by refinements to that assumed distribution on the basis of computations from Equation 6 . Moreover, as noted previously. the selection of a n initial starting distribution need not be entirely arbitrary, since the final solution will frequently represent only small corrections to the classic distribution of stoichiometric ratios. M hich is given by the following expression in the presence of inerts:
where
If this stoichiometric distribution is substituted into Equation 6, the f o l l o ~ i n gfirst approximation is obtained for the composition distribution required for maximum yield
and J
(Pf> =
(Cp,)-'
c
Til (CPJ
(3) (,,o)!,-l
i=1
h
1
T h e normalization and equilibrium constraints are as follo\vs : Yf
+ [v(l + r(J))](L)[
IS1
(1
+
- (1
+ r,(J))
v(L![
!SI
+ G3[
- GI
T
~ ) Y( ( ~ )
(12)
j=l
where
By using the same undetermined multiplier techniques (2, 4-8), the unique distribution \\hich \till maximize [. subject to the three constraints discussed above. may be identified
and TIS]are obtained by simultaneous solution of the following specific versions of Equations 5 and 1,
and
Z-'
=
R-'[,0(1
+
(p))(Cp)](L)
(10)
T h e L equations represented by Equation 6 may be solved simultaneously with Equations 1 and 5 to give the maximum [['I. and the (jl0)1'1 distribution. yield values of TI'], Equation 6 reduces to Equation 5 of ( 6 ) in the absence of inerts and reduces to Equation 9 of (7). if the enthalpy of reaction is zero Equation 6 is an implicit solution to the
Maximization of Temperature
T h e technique used in arriving at Equation 6 may be employed for optimization of variables other than yield, Maximization of the final equilibrium temperature in an adiVOL. 4
NO. 3
AUGUST
1965
261
abatic reaction is a practical goal in the treatment of flame temperature problems. T h e maximum is sought subject to the same three constraints as before-namely, normalization to unity of the initial mole fraction distribution, the criteria of thermodynamic equilibrium, and satisfaction of an energy balance of equation of the type of Equation 1. Application of the undetermined multiplier technique followed by tedious straightforward algebra results in the following expression for that initial mole fraction distribution which will result in maximum equilibrium adiabatic temperature.
I
I
SO3
S02+AIR (79/21)=
ADIABATIC
780
0.1I
k' 760
.09
[y,o](c
1
=
3 l-
n
+ [y(l + T(J))](L)[
y1
(I
[TI
- (I
+
+ e[[ [Ti]z~,
y , ( J ) ) u ( L ) [ [TI
+ ~ , ( ~ ) -) de[~ )
::
a
a a
h
W
a
0
[TIL,
I
W
i = 1,2
.
. .L
(19)
740
.07
a
z
where
LL
L,
=
X,([[Tl)ZR-'
(20)
and
T h e L equations represented by Equation 19 may be solved simultaneously ti-ith Equations 1 and 5 to give values of Tf*l, [ and the (y:)ITi distribution corresponding to maximum temperature. Several features of this solution might be noted. First, this is obviously a different composition distribution than required for maximum yield. Secondly, this solution does not explicitly involve the enthalpy of reaction, in contrast to the solution for the maximum yield case. This solution again is implicit, since it involves the unknown initial composition on the right-hand side of the equation. Nevertheless, in conjunction with Equations 1 and 5, it represents an exact solution. I n actual practice this set of equations is most easily solved by substitution of a trial composition distribution into the right-hand side of Equation 19, follobved by iterative refinement. The starting distribution might again correspond to stoichiometric ratios or perhaps to the composition actually found for maximum temperature. Examples. Consider the oxidation of SO2 with air (assumed 79y0 nitrogen, 2ly0 oxygen) with reactants combined a t 500' C. and 1 atm. Assume that the reaction proceeds to equilibrium adiabatically. Pertinent data are summarized below :
Soz $-
112
T o = 500" C.
0 9
=
=
y ( T
-1,
YO*
=
-112,
773.16' K.
VgOa
=
1,
g max
= - 112, y(L) = - 312%
rq,-Or =
7gl(J)
VX2 =
==
2/3
= 791'21 = 3.7619
l&EC FUNDAMENTALS
700
Figure 1 . Extent of reaction, mole fraction of SOa, and final temperature vs. original mole fraction of oxygen Curves obtained b y simultaneous iterative solutions of Equations 1 ond 5 . Arrows indicate composition for maximum yield predicted b y Equation 6 and for maximum temperature predicted b y Equation 19
and TIs] = 738' C. T h e pertinent heat capacities a t the equilibrium temperature and the average heat capacities over the temperature range 500" to 738' C. were estimated from the tables in ( 3 ) . These data were then used in Equation 12, the first approximation for the composition distribution required for maximum yield. This provided the following solution : ;Yo:
'v
0.0885
normalizing
~ ~ N0 02. 8 7 9 0 ----+ yX,O
N
0.3329
0 0681 0.6759 0.2560 1 .oooo
This normalized approximate distribution was used to solve Equations 1 and 5 simultaneously for improved values of { a n d T . These data were then substituted into Equation 6, the exact expression for the composition distribution required for maximum yield, giving the following :
0
As an exercise the initial composition distribution which Lvould result in maximum equilibrium yield for this system \vas found. (As a separate problem considered further along, a solution was also sought for the composition which \vould maximize the equilibrium temperature.) Equations 17 and 18, corresponding to an initial stoichiometric mixture. were first solved simultaneously. yielding a value of [ ['I = 0.09731 262
.03 C
1.3004
AHR(T o ) = 23,490 cal./(gram mole SO3 formed) =
720
SO3
P = 1 atm.
YSr&
.05
0.0864 0.5478 y ~ : = 0.3252 '0' = .YSO,O . =
-
normalizing
-
0 3391
0 9594 Another cycle resulted in :
YN,"
1 ,0000
-
0.0900 normalizing 0.5687 = 0 3386 0 9973
YO: = ~ ~ 8 0 2= 0
0.0901
0.5708
0.0902 0.5703 0 3395 1 .0000
T h e last cycle results in a solution which differs from the immediately preceding one by only small amounts in the fourth place after the decimal. T h e equilibrium yield and temperature for this computed distribution are { I y ] = 0.1160 and TIy] = 747' C. For the sake of comparison and for general verification of the correctness of the analysis. Equations 1 and 5 Lvere solved simultaneously for equilibrium yield and temperature for a sequence of assumed values of the initial mole fraction of oxygen. T h e yield, t h r temperature, and also the mole fraction of SO:, are shown in Figure 1 as a function of jo,O, The maximum in yield obviously occurs at an initial composition different from that corresponding to stoichiometric ratios and actually represents a 205% increase in yield or equilibrium SO3 concentration. The figure also indicates the answer obtained by the direct solution described in the preceding paragraphs. T h a t computation corrrctly predicted the locus of maximum yield. As is apparent i n preceding paragraphs, the utilization of the expressions derived in this paper will usually require several steps of iteration because of the appearance of the unknown mole fraction distribution in the correction terms in the right-hand side of Equation 6. Hoxvever, even two or three steps of such iteration represent substantially less work than required to locate the maximum by mapping out a curve as in Figure 1. T h e preceding chemical reaction was used also to check the equations for locating maximum temperature. Since Equation 19 again contains the unknown yield and composition distribution in the right-hand side of the equation, some first approximation is necessary. .4 reasonable trial distribution is the one found in the previous paragraphs for the solution of the maximum yield problem. Upon substitution of that distribution, yield, and temperature into the right-hand side of Equation 19, the following estimate was obtained for the initial composition distribution for maximization of temperature Yo: = 0.1191 0.1133 t.so,O = 0 , 4 8 3 8 -------+ 0.4604 0.4263 YN: 0 4480 ~.
,
1 0509
1 0000
niques are applicable to maximization of any function which can be characterized in terms of {, T , and the y j 0 distribution. Discussion
Although the solutions to the problems considered have been given in terms of analytical expressions, the resulting equations are implicit and are sufficiently complex so that it may not be easy to deduce from them which factors are important in the phenomena However. there is certainly nothing more involved than the effects of simultaneous variation of several parameters. each of which tends to increase or decrease the yield. For an isothermal perfect gas reaction ( 7 ) any deviation from stoichiometric ratios tends to decrease the yield. O n the other hand, decreasing the dilution by an inert will increase the effective yield if the reaction is one in which moles are decreased. Therefore, the adjustment of initial composition when inerts are carried in fixed ratio to a reactant involves two competing processes: the tendency to decrease yield as the system moves away from stoichiometric ratios, and the tendency to increase yield as the dilutant is decreased. This results in a maximum in yield and a corresponding optimum in initial composition. Similarly. if the heat capacities of the species are different, either greater yield or greater equilibrium temperature may be obtained due to the change in temperature arising from preferential increase or decrease of a given component. For the problems considered in the paper, all three of these factors are operating at the same time. I n a sense, the procedures of this and the preceding papers (4-7) might be regarded as multivariate Le Chatelier analysis. Acknowledgment
A. P. Kendig assisted in the numerical computations. Nomenclature
Cp,
= isobaric molal heat capacity of coinponent
(CP,)
=
( T - To)-' f 7 C P ( , ) do
J
TO
fugacity of pure component defined by Equation 13 enthalpy of reaction index denoting an inert index denoting reactant or product = total number of inerts = equilibrium constant
One more cycle of substitution of this refined distribution of mole fractions in Equations 19 resulted in the following estimates: 0.1112 yo; = 0 . 1 1 0 7 normalizing 0.4706 .yso,O. = 0.4688 0.4182 y X , O = 0 4166 --1 .oooo 0,9961
= = = = =
This solution is compared in Figure 1 with the location of the maximum by the numerical-graphical technique. Increases in temperature are definite but not spectacular. amounting to 4' C. higher than for stoichiometric mixture and 14' C. higher than for a mixture corresponding to maximum yield. T h e numerical results of both these examples have no general significance. of course, and are offered here only as exercises to illustrate the use of the equations and verify their correctness The magnitude of the e f t c t s in any particular case \till depend upon the parameters, including initial temperature, intrinsic equilibrium yield, enthalpy of reaction, heat capacity distribution, and concentration of inerts.
= total number of reactants = defined by Equation 20 = total number of products
-
h
=
rII(f,t/P)YI 1
Optimization of Other Variables
As discussed in reference (4):the analysis of these two papers has a generality extending beyond the maximum yield and maximum temperature problems considered here. The tech-
= =
= = = =
total number of moles M . total number of reactants plus products pressure defined by Equation universal gas constant mz,O m,O, where m,, is the number of moles of component 2 carried with component I
L
+
'
J
=
11, 1=1
= absolute temperature = defined by Equation 14 = defined by Equation 15 = = = = =
defined by Equation 16 defined by Equation 8 defined by Equation 9 mole fraction of component defined by Equation 10 VOL. 4
NO. 3
AUGUST
1965
263
GREEKLETTERS = defined by Equation 2 6, = defined by Equation 21 8 = extent of reaction or yield, moles
1
=
[J]
[TI [Y]
tima
U,
= stoichiometric coefficient of component
u(L)
=
j
L
Literature Cited (1) De Donder. Th., Van Lerberghe. G., Bull. Acad. Roy. Belg. (C1.Sc.) 12 (5), 151 (1926). (~, 2 ) Franklin. Philip. “Treatise on Advanced Calculus.” p. 353, Wiley. New York. 1940. (3) Hougeii. 0 . A . . \Vatson, K. M.. Ragatz, R. A . , “Chemical Process Principles;” Part I, “Material and Energy Balances.” 2nd ed., pp. 253. 258. IViley, New York, 1954. (4) Pings, C. J., A.I.Ch.E. J . 10, 934 (1964). ( 5 ) Pings, C. J., Chem. Eng. Progr. 59, No. 12. 90 (1963). (6) Pings, c. J., IND. ENG.C H E M . FUNDAMENTALS 2, 244 (1963). (7) Ibzd., p. 321. (8) Prigogine, I.? Defay. R., E\rerett, D. H.. “Chemical Thermodynamics,” p. 134. Longnians Green. London, 1954.
Vj
1 A;
”(N)
=
Vf 1
SUBSCRIPTS
i
= index denoting an inert = index denoting a reactant or product
I
SUPERSCRIPTS = indicates ( J ) = indicates ( L ) = indicates ( N ) = indicates
0
property under conditions of stoichiometric feed = indicates value of property under conditions of maximum temperature = indicates value of property under conditions of maximum yield = indicates value of
condition in initial reaction mixture summation over the range 1 to J summation over the range 1 to L summation over the range 1 to S
RECEIVED for review June 15, 1964 ACCEPTED February 1. 1965
CONTROL OF A CONTINUOUS-FLOW AGITATED-TANK REACTOR M A S W . W E B E R , State University o j .Vew Y o r k , Buffalo, P E T E R H A R R I 0 T T , Cornell Uniuersity, Ithaca, N . Y,
T HO
iV. Y
The stability criteria for a tank-flow reactor are reviewed for the case where there is no mixing delay and the dynamics of the cooling system are not important. Stability criteria are then developed for cases where the dynamics of a cooling coil are significant. Zero-order kinetics are assumed, so the stability criteria are conservative. A zero-order, exothermic reaction was simulated in a 2-foot tank. Hot water was used as a feed and live steam was sparged into the tank to simulate the heat of reaction. The reactor was made inherently stable or unstable by controlling the change in steam rate with temperature. The reactor temperature was controlled by the cooling water rate. For a stable reactor, the system was stable for controller gains below a certain maximum; when the reactor was inherently unstable, the system was conditionally stable. Good control was achieved with about the same controller settings for both cases because the absolute value of the largest time constant was much larger than the second largest.
design of a chemical reactor, the desired temperature A moderate conversion may be called for to minimize by-product formation or because the reactor is the first of a series. If the conversion is low or moderate (less than 70y0)and the reaction exothermic, the temperature control system deserves careful study, since the reactor may lack inherent stability Two approaches are possible. If sufficient cooling area is used to permit a very low driving force for heat removal, the reactor will be inherently stable. A controller is not essential, though some type of temperature control would probably be provided for improved response to upsets ‘The other alternative is to use less area for heat transfer and a larger temperature driving force, and operate within the unstable region with the aid of a good control system. The steady- and unsteady-state behavior of tank-flow reactors has received considerable attention in the literature. Several authors have developed stability criteria assuming a constant coolant flow rate and average coolant temperature (7-3, 7, 8) For the case of control of a reactor, in some studies the change in average coolant temperature with change in the flow rate of coolant was accounted for ( 7 , 4, 7). In the first of these articles. it was even assumed that the over-all coN THE
Iand conversion are usually fixed first.
264
l&EC FUNDAMENTALS
efficient of heat transfer varied with the eight-tenths power of the coolant flow rate. However, in none of these cases were the dynamics of a coil or jacket considered. It was the purpose of this study to examine a case where the dynamics of a cooling coil were significant. As an aid to understanding systems where coil dynamics are important, it is worthwhile to consider first the simple case where the dynamics of the heat removal system are neglected. This case has merit because the analyses of the steady and unsteady states are greatly simplified. At the same time, the mathematical results are fairly easy to interpret physically. Two other assumptions made for this case are that the contents of the tank are perfectly mixed and the over-all coefficient of heat transfer is constant. Steady-State Analysis
A steady-state analysis can be used as a preliminary check on reactor stability. Following the method of van Heerden (70), the rate of heat generation, Qo, and the rate of heat removal, Q R , are plotted against reactor temperature. The heat generation curve is sigmoidal, but Q R , the heat removed through the coil and as sensible heat of the product, is a linear function of reactor temperature. The reactor is inherently
