SUBSCRIPTS e i
j m o
+ -
= = = = = = =
electrolyte as a neutral species any arbitrary species any arbitrary species membrane solvent positive ion negative ion
literature Cited
Banks, W., Sharples, A;, J . Appl. Chem. 16,28-32 (1966). Bennion, D. N., “Mass Transport of Binary Electrolyte Solutions in Membranes,” UCLA Department of Engineering, Rept. 66-17 (March 1966). Clark, W. E., Science 138, Ser. 2, 148-9 (1962). Coleman, B. D., Truesdell, Clifford, J . Chem. Phys. 33, 28-31 (1960). 5 , 281-3 Cullinan, H. T., Jr., IND.ENC. CHEM.FUNDAMENTALS (1966). Currie, D. J., Gordon, A. R., J.Phys. Chem. 64, 1751-3 (1960); Denbigh, K. G., “Principles of Chemical Equilibrium,” Cambridge University Press, Cambridge, 1957. Denbigh, K. G., “Thermodynamics of the Steady State,” Wiley, New York, 1951. Duncan, B. C., “Investigation of Membrane Transport Phenomena,” OSW Saline Water Conversion Rept., pp. 120-211 (1965): Fitts, D. D., “Nonequilibrium Thermodynamics,” McGraw-Hill, New York, 1962. George, J. H. B., “Transport Properties of Ion Exchange Membranes,” OSW Saline Water Conversion Rept. 50 (1965). Harned, H. S., Owen, B. B., “The Physical Chemistry of Electrolyte Solutions,” Reinhold, New Yorkj 1958. Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” Chap. 2, Wiley, New York, 1954. Jagur-Grodzinski, J., Kedem, O., Desalination 1, 327-41 (1966). Kedem, O., Katchalsky, A., Biochim. Biophys. Acta 27, 227-46 (1958). Kedem, O., Katchalsky, A., J . Gen. Physiol. 45, 143-79 (1961). Kedem, O., Katchalsky, A., Trans. Faraday Soc. 59, 1918-30 (1963). Kirkwood, J. G., “Ion Transport Across Membranes,” H. T. Clarke, Ed., Academic Press, New York, 1954. Klemm, Alfred, 2.h‘aturforsch. Sa, 397-400 (1953). Klemm, A,, 2.Naturforsch. 17a, 805-7 (1962). Laity, R. LV., J . Chem. Phys. 30, 682-91 (1959a). Laity, R. IV., J . Phys. Chem. 63, 80-3 (1959b). Lamm, Ole, Adoan. Chem. Phys. 6, 291-313 (1964). Loeb, Sidney, Desalination 1, 35-49 (1966a).
Loeb, S., “Preparation and Performance of High-Flux Cellulose Acetate Desalination Membranes,” Ulrich Merten, Ed., “Desalination by Reverse Osmosis,” M.I.T. Press, Cambridge, Mass., 1966b. Loeb, Sidney, “Sea Water Demineralization by Means of a Semipermeable Membrane,” UCLA Engineering Rept. 63-32 (Julv 1963). Loeb, ’Sidney, Manjikian, Serop, Ind. Eng, Chem. Process Design Deuelop. 4, 207-12 (1965). Loeb, Sidney, Rosenfeld, J., “Turbulent Region Performance of Reverse Osmosis Desalination Tubes 1. ExDerience at Coalinga Pilot Plant,” UCLA Engineering Rept. 66-62 (October 1966).Loeb, Sidney, Sourirajan, Srinivasa, Advan. Chem. Ser., No. 38, 117-32 (1963). Longsworth, L. G., J . Am. Chem. Sac. 54, 2741-58 (1932). Lonsdale, H. K., “Properties of Cellulose Acetate Membranes,” Ulrich Merten, Ed., “Desalination by Reverse Osmosis,” M.I.T. Press. Cambridge. Mass.. 1966. Lonsdale, H. K., Merte; U., Rilky, R. L., J . Appl. Polymer Sei: 9,1341-62 (1965). Manjikian, Serop, Ind. Eng. Chem. Product Research Deuelop. 6,23-32 /rn,7\ \17U/ J.
Merten, Ulrich, Desalination 1, 297-310 (1966). Merten. Ulrich. “TransDort ProDerties of Osmotic Membranes,” Ulrich Mertkn, Ed.: “Desdination by Reverse Osmosis;” M.I.T. Press, Cambridge, Mass., 1966. Newman, John, Bennion, Douplas, Tobias, C. W., Ber. Bunsengesell. 69, 608-12 (1965). Rhee, B. b‘.. “TransDort Parameters in SemiDermeable, Cellulose Acetate Membranes,” M.S. thesis, D e p a r t k n t of Engineering, University of California at Los Angeles, August 1967. Reid, C. E., Breton, E. J., J . Appl. Polymer Sci. 1, 133-43 (1959). Riley, Robert, Gardner, J. O., Merten, Ulrich, Science 2, 143, Ser. 2, 801-3 (1964). Robinson, R. A , , Stokes, R. H., “Electrolyte Solutions,” 2nd ed., Butterworth’s. London. 1959. Sherwood, T. K.,Brian, P . L. T., Fisher, R. E., IND.ENG.CHEM. FUNDAMENTALS 6, 2-12 (1967). Spiegler, K. S., Trans. Faraday Soc. 54, 1408-28 (1958). Spiegler, K. S., Kedem, O., Desalination 1, 311-26 (1966). Steel. B. 3.. Stokes. R . H.. J . Phvs. Chem. 62. 450-2 11958). Thau, G., Bloch, R., Kedem, O:, Desalinatzdn 1, 129-38 (1966). Vignes, Alain, IND.END.CHEM.FUNDAMENTALS 5 , 189-99 (1966). Washburn, E., J . Am. Chem. Soc. 31, 322-55 (1909). Wills, G. B., Lightfoot, E. N., IND.ENG.CHEM.FUNDAMENTALS 5 , 115-20 (1966). RECEIVED for review December 19, 1967 ACCEPTED August 23, 1968 Work supported by the State of California through the University of California Statewide Water Resources Center.
OPTIMUM ADIABATIC CASCADE REACTOR W I T H DIRECT INTERCOOLING D. C. DYSON AND F. J . M. HORN
Chemical Engineering Department, Rice University, Houston, Tex. 77001 An efficient algorithm is presented for the determination of the minimum mass of catalyst required by an N-stage adiabatic tubular reactor with feed preheater and direct (cold shot) intercooling by cold feed bypass for the single reversible exothermic reaction with arbitrary kinetics. The inlet and outlet conversions and the outlet flow rate are fixed and the minimization is over the inlet temperature and the catalyst and cold shot distributions. The algorithm is believed to be the best available for this problem. A numerical example is presented and the connection between this problem and the variational problem arising when N+ is established. HE problem of minimizing the volume (or mass of catalyst) Trequired for a chemical reactor for a given duty is of fundamental importance in reactor theory. I t is desirable for the design engineer to have at his disposal efficient programs for the solution of this problem for various different types of reactors. I n this paper we treat the case of a single reversible exothermic reaction taking place in a n N-stage adiabatic cascade
with main feed preheater and interstage cooling by cold feed bypass (Figure 1) and consider the minimum volume problem with exit conversion as a parameter. The classical method of expressing the volume as a function of a suitable set of free variables, differentiating, and solving the equations D = 0 for the vanishing of the derivatives is used. The equations D = 0 for the case where the adiabatic temVOL. 8
NO. 1
FEBRUARY 1969
49
JgqYI;,$-;:i$Tk Y, hnl
r-------
‘YO,
L_
hj
i
~ei
to be the free variables which determine the remainder. We can easily get the derivatives of the remaining variables with respect to the elements of S as follows: Differentiating Equation 4 we obtain
I
_ _ _ _ _ _J
Figure 1. N-stage adiabatic cascade reactor with feed preheater and cold feed bypass for interstage cooling
Using Equation 4 again perature rise coefficient is constant have been published (Horn, 196la; Konoki, 1960). We derive the equations D = 0, making no assumptions about the thermochemistry; discuss the essential features of an algorithm for finding the correct solution to these equations, when it exists; consider the degeneracy which occurs when the cold shot is unreasonably hot; show that the optimal control for the corresponding variational problem which arises in the case of perfect direct control (Dyson, 1966) is a plausible limiting case of the optimal N-stage control; and give numerical examples to demonstrate the efficiency of the method. Derivation of Equations D
< < u2
u3
. ...
Using Equation 5 again
=0
(9)
We use some convenient stoichiometric and enthalpy relations which have appeared elsewhere (Dyson, 1966; Dyson and Horn, 1967). We consider the mass flow rate through the Nth stage to be constant and take w N = 1 for simplicity. The constraints u1
Differentiating Equation 5
< < wj
W j f l
.....
> . . ,Ye,v-l; 50
.
WI,~Z.. . ~ N - I ;
l&EC FUNDAMENTALS
hN)
For minimum M we simply equate Expressions 10, 11, and 12 to zero. It turns out that the resulting equations are satisfied by the optimum reactor even if some of the inequalities 1 become equalities. I t is convenient to write these conditions as follows:
where
and
( 0 for N = 1
Algorithm for Solution of Equations D
=0
We wish to determine the relation between M * = min M and parameter y e H for various N. T h e best procedure is in essence as follows:
1. Assign some reasonable value to hl--e.g., point A in Figure 2. 2. Evaluate integrals ti and tl’ by a stepwise numerical procedure starting from y = yal and along with them Fl(y,hl). When Fl(y,h,) changes sign from to - findy,, by a suitable interpolation (point B , Figure 2). Calculate t,,, tel’, Vel. 3. Determine p l by solving Equation 14 with i = 1 (the root pl = 1 to be ignored). Determine ya2and hz from Equations 4, 5, and 14a (point C, Figure 2). 4. Proceed with stage 2 as for stage 1 in 2 above (line CD, Figure 2), determine p2, yaS, and h8 as in 3 above (point E, Figure 2), and continue until yo,,, is reached. 5. Compute the final stage, determine y e N by solving FN(y,hN) = 0 and M by 2 from stored values using the condition: w N = 1. I t has been assumed that N 2. T h e modifications required to cater for the cases N = 1 and N = 2 are elementary. Further (for given h l ) , the machine time involved in calculating optimal 1, 2, , . . , n 1 and n-stage reactors is only slightly longer than that required for the n-stage reactor itself, but some additionallogic will be required to keep track of F N ( y ) for N = 1 , 2 , 3 . . . . n - l a s w e l l a s F i C y ) f o r N = n a n d i < n.
+
>
-
Existence and Uniqueness of Solutions to Equations D
=0
Let the rootspt = 1 of Equation 14 and y e , = ya, of Equation 15 be called trivial roots. We make the following assumptions (these have always been satisfied in calculations carried out so far) :
>
ASSUMPTION A. For all N > 1; R 0. ASSUMPTION B. For i # A’; given a n h, and a y e , satisfying Equation 15 nontrivially, and such that: V ( y & , h J > 0, F$’(ye,,h,) 0 (see Equation 17), there exists a correspondingp, < 1 satisfying Equation 14. ASSUMPTION C. At all points y,h for which V ( y , h ) = 0, dV/dy # 0 and d V / b h # 0.
, QeN an optimal N-stage reactor ( N > 1) will in fact be identical to a one-stage reactor-Le., w 1 = 0 2 = w 3 . . . = w N ) . I t is conceivable that this degeneracy (cold feed to hot degeneracy) might occur for ye,,, < jl,,,, although this has not occurred in any of the examples we have computed so far. This brings us to Assumption B, which is made merely for purposes of simplification. Should B turn out to be untrue, this would mean that (for the hl and N in question) step 3 or 4 of the algorithm could not be completed-Le., we would have a degenerate case. The reader will satisfy himself that Assumption A is not essential either. If A is not satisfied, one has either a degenerate case or two (nontrivially) independent solutions to the equations D = 0. In the latter case two M,y,, relations would be obtained and hence the min M,y,, relation.
I
I
I
I
I
0.5
I
1.5
I
I
0
.C
E I
0
B
-I
-2
J
-3
-4
-5
0
Loglo [Ye,
Figure 3.
1 - ye,
I]
Results of an example with T, =
600’ K.
Minimum volume for a reactor with 1, 2, 3, co stages as a function of outlet conversion. Degenerate point A
2
l
.-C E I
0
0 0
Case N + cc
The variational problem arising when N -+ co has been treated (Dyson, 1966; Dyson and Horn, 1967). The trajectories traced out in the y,h diagram (Figure 2 ) by optimal reactors are in part lines of constant h, and in part sections of curve L1 (part of which becomes an accumulation curve for the sets of zigzagging trajectories discussed above, as one might plausibly infer). satisfying Equation 19, has the same significance in the variational case as for N finite. [The mapping which brings Equations 19 of this paper into the corresponding equations for the variational problem is given by Dyson (1 966).] Computational Details
-2
-3
Figure 4.
Results of an example with T,,, = 300’ K.
Minimum volume as a function of outlet conversion. appears for conversions of practical interest
No degenerate point
Numerical Results
There is nothing that deserves special mention except the evaluation of the integrals t and t’. I t often turns out (at any rate, for moderate hl and for N = 1, 2, 3, or 4) that the conversions a t the end of each stage are very close to conversionsyr’ a t which V = 0, so that these integrals must be evaluated for yei close to the poles of the integrands. The method given by Horn (196lb) of interpolating 1 / V by a function having a simple pole a t y = yi’, and d V / b h by a quadratic so that the contributions to t, and ti’ over a subinterval whose ends were interpolation base points could be obtained in closed form, proved very satisfactory. First yc’ was determined precisely, then the function (J, - y r ’ ) / [ B C(y, - y r ’ ) ] was used to interpolate V(y,,h,) close to yt’, and further away l/V(yr,h,) was interpolated by u(y, - yr’) b c/hr- yi‘). Provided V(y,hi) is a decreasing function ofy over the interval of integration, this interpolation can be used satisfactorily in every subinterval. Simpson’s rule quadrature was used over the earliest part of the interval (ya,,yeJ; in some cases that part of the interval in which V is a n increasing function of y. Details are given by Dyson (1966).
+ + +
52
- I
J
l&EC FUNDAMENTALS
Results of calculations are given by Dyson (1966) and Dyson et ul. (196?), where adiabatic cascades are compared with several different types of reactors. In Figure 3 the results for a reactor with reaction rate functions u = H(I
- y)e
-A
- ”ye
- _A’ T
(20)
with H = 8.7678 X 1014, H’ = 1.244 x 1020, A = 2.2748 x lo4 O K . , A’ = 2A, a constant adiabatic temperature rise coefficient of 158.5’ K. and ym = 0, and T, = 600’ K. are presented. The N = 3 and N = m curves in Figure 3 were h, the points a t the ends near B corredetermined for h1 sponding to hl = h,, but for the two-stage reactor the results were obtained for h down to the degenerate point A . I n these cases it is worthwhile to bypass feed whenever h i > E l satisfying Equation 19 and in particular when < hl < h,i.e., when the feed heat exchanger actually cools the main feed. But if any cooling is to be carried out in practice, it should presumably be the subsidiary feed which is cooled, so this latter case is not of practical importance. I n Figure 4 the
>
corresponding results are presented for a 300’ K . cold feed temperature. Even for y e N = 0.99, there is no sign of cold shot degeneracy (the N = 1 curves in Figures 3 and 4 are the same). Provided hl is far from the degeneration enthalpy, the optimal distribution of volume between the stages follows the same pattern found earlier by Kuchler (1961) for the cascade with indirect intercooling-Le., the volume increases as the stage number increases. I n the examples calculated there 1 as is roughly two to four times as much volume in stage i in stage i. The above method requires only a few times as much computer time to find a value of M * as is required to find any one value of M regardless of N . I t is therefore very efficient, especially for large N . (For N = 4 we have already a sevendimensional problem.) T o get a relative precision of 1 in 100,000 in all relevant numbers for one-, two-, and threestage reactors a t most 1 second’s computation is required on a n I C T Atlas. I t is orders of magnitude faster than the dynamic programming method described by Lee and Aris (1963), which is not surprising, since dynamic programming is a procedure which is necessarily worse than solving Equations 14 and 15 by calculating backward from the Nth stage, and this involves a one-dimensional loss of information (Dyson, 1966).
+
Nomenclature
a, b , c
A , A’ B, C F h
H, H’ M N
Pi
= = =
=
=
-
constants in interpolation formulas parameters in Equation 20 parameters in interpolation formulas function defined by Equation 16 enthalpy of a mass of mixture which contains 1 y moles of reference substance parameters in Equation 20 objective function, total volume (or catalyst mass) number of stages see Equation 14a
-
R
= = =
S t t’
=
T = u(y,T) = V(y,h) = =
Y
function defined by Equation 16a set of 2iV - 1 chosen free variables function defined by Equation 3 function defined by Equation 13 temperature, O K. reaction rate (as function of conversion and temperature) reaction rate (as function of conversion and enthalpy, fractional conversion of reference substance
SUBSCRIPTS entrance to stage i exit from stage i 2, 1, k , 1 = stage i, j, k , or 1 m = “cold shot” stream ai et
.
= =
GREEKLETTERS w 6ik
-
quantity such that ~ ( 1 y) is molal flow rate of reference reactant = Kroeneker 6-i.e., ijik = 0 for i # k, 8 i b = 1 for =
i = k
literature Cited
Dyson, D. C., Ph.D. thesis, University of London, England, January 1966. Dyson, D. C., et al., Can. J . Chem. Eng. 45, 310-18 (1967). Dyson, D. C., Graves, J. R., Chem. Eng. Sci. 23, 435-46 (1968). Dyson, D. C., Horn, F. J. M., J . Optimization Theory Appl. 1 ( l ) , 40-52 (1967). Horn, F., Chem. Eng.Sci. 14,20-1 (1961a). Horn, F., 2. Elektrochem. 6 5 , 295 (1961b). Konoki, K., Kagaku Kogaku 24, 569-71 (1960). Kuchler. L.. Chem. Ene. Sci. 14. 11-19 (1961). Lee, K.:Y.,‘ Aris, R.: h d . E&. Chem. Pro&ss Design Develop. 2, 300-6 (1963). RECEIVED for review December 21, 1967 ACCEPTED July 12, 1968 Work supported by the Systems Grant from the National Science Foundation (No. GU-1153) and by Imperial College, London.
STRATEGY FOR ESTIMATION OF RATE CONSTANTS FROM ISOTHERMAL REACTION DATA J A M E S M. E A K M A N ’
Research Laboratory, Ashland Chemical Co., Minneapolis, Minn. 55420
A two-step computational strategy for the estimation of rate constants from isothermal multiresponse timeconcentration data is proposed. Required expressions for derivatives of the integral kinetic expressions with respect to the rate constants are obtained. The method is effectively applied to a simulated homogeneous reaction problem.
NE
major problem facing the chemical engineer engaged in
0 process research is the need to develop accurate mathe-
matical descriptions for the processes with which he is working. I n many cases the most important unit in the process is the chemical reactor(s) in which the formation of products takes place. Most important to an accurate mathematical descrip1 Present address, Department of Chemical Engineering, University of Nebraska, Lincoln, Neb. 68508
tion of a reactor is a good kinetic model for the chemical reactions occurring therein. This requires knowledge of both the reaction mechanism and the associated rate constants. Here it is assumed that a plausible mechanism has been proposed and that estimates of the rate constants are required. The specific purpose of this work is to propose a computational strategy for the estimation of rate constants from multiresponse time-cqncentration data collected from isothermal homogeneous reaction systems. The speed of convergence VOL. 8
NO. 1
FEBRUARY 1969
53
