= Schmidt number = v / D = Sherwood number = k d / D = total number of purification stages excluding
NSO
Nsh
P
= osmotic pressure corresponding to solute con-
T
= quantity defined by Equation 31
centration
feed stage = operating pressure, atm.
SUBSCRIPTS
= quantity of final product, cc. or moles
f, 1,2,. . .i,. . . p
P P [PRI
= product rate, grams per hour per 7.6 sq. cm.
[PWP1
= pure water permeability, grams per hour per
of film area
S t
a0 VuC v1
v10
u:
W X
X XA1,XAz,XAs
XAl
d X A )
7.6 sq. cm. of film area = membrane surface area, sq. cm. = time, sec. = average fluid velocity at membrane entrance, cm. per sec. = quantity defined by Equation 32, cm. per sec. = quantity of solution on high pressure side of membrane at any time, cc. or moles = value of Vl at time 0 = total number of concentration stages excluding feed stage = quantity of final concentrated effluent from reverse osmosis unit, cc. or moles = longitudinal distance along length of membrane from channel entrance, cm. = quantity defined by Equation 47 = mole fraction of solute in feed solution, concentrated boundary solution, and product solution, respectively, at any time = value of X A , at time 0
2
GREEKLETTERS = constant = constant P = quantity defined by Equation 27 Y A = fraction product recovery defined by Equation 30 e = quantity defined by Equation 28 x = quantity defined by Equation 29 = kinematic viscosity, sq. cm. per sec. V cy
XA,
atm.
= feed stage, and purification stages 1,2,. . .i,
. . .p, respectively
SUPERSCRIPTS
1,2,. . .j,. . .w
= concentration stages 1,2,. . .j,.
. .w,respec-
tively Literature Cited
Gosting, L. J., Akeley, D. F., J . Am. Chem. SOC. 74, 2058 (1952). Gucker, F. T., Gage, F. W., Moser, C. E., J . A m . Chem. SOC.60, 2582 (1938). International Critical Tables, Vol. V, p. 22, McGraw-Hill, New York, 1929. Kimura, S., Sourirajan, S.,A.1.Ch.E. J . 13, 497 (1967). Kimura, S.,Sourirajan, S., IND. ENG.CHEM.PROCESS DESION DEVELOP. 7 , 197 (1968a). Kimura, S., Sourirajan, S.,IND. ENG.CHEM.PROCESS DESIGN DEVELOP. 7 , 548 (1968b). Loeb, S.,Sourirajan, S., Advan. Chem. Ser., No. 38, 117 (1963). Loeb, S., Sourirajan, S., U.S. Patent 3,133,132 (May 12, 1964). Ohya, H., Sourirajan, S., “Some General Equations for Reverse Osmosis Process Design,” A.1.Ch.E. J., in press, 1968. Scatchard, G., Hamer, W. J., Wood, S.E., J . Am. Chem. Sod. 60, 3061 (1938). Sourirajan, S.,2nd. Eng. Chem. Fundamentals 3, 206 (1964). Sourirajan, S., Govindan, T. S.,Proceedings of First International Symposium on Water Desalination, Washington, D. C., October 1965 (U.S. Dept. Interior, Office of Saline Water, Vol. I, pp. 251-74, 1967). Sourirajan, S., Kimura, S., IND. ENG.CHEM.PROCESS DESIGN DEVELOP. 6, 504 (1967). Timmermans, J., “Physicochemical Constants of Binary Systems in Concentrated Solutions,” Vol. 4, pp. 115-17, Interscience, New York, 1960. RECEIVED for review June 10, 1968 ACCEPTED September 25, 1968 Issued as N.R.C. No. 10494
CORRESPONDENCE ALGORITHM FOR OPTIMIZATION OF ADIABATIC REACTOR SEQUENCE WITH COLD-SHOTS COOLING SIR: In a recent note Malengt and Villermaux (1967) quesrioned the algorithm of Lee and Aris (1963) for the optimization of a multibed adiabatic reactor sequence with cold-shot cooling, on the basis of the incorrect use of the bypass parameter, AB. They have evolved a new algorithm wherein X1 has been explicitly included as a decision variable in the optimization function. I t appears that the inadvertent omission of an important operational feature of the last stage in a reactor sequence of this type has resulted in the rather controversial formulation of algorithms for optimal operation of a reactor sequence by application of the principles of dynamic programming. Once this fact is recognized, an analytical design scheme for the case of cold-shot cooling emerges which is as elegant as that evolved for the heat-exchanger cooling, if not as gt neral. Prior to entry to reactor 1, the mixing of the main stream from reactor 2 and the bypass stream provides the final complete feed to the first reactor and therefore the algorithm for the 142
I & E C P R O C E S S D E S I G N A N D DEVELOPMENT
optimization of the first reactor cannot distinguish between the type of cooling effected between the first and second stages. In other words, the optimal design for the first reactor is a common design for either type of interstage cooling and can be evolved by following the procedure for one-stage optimization with heat exchanger cooling, after neglecting the cooling costs. For any given feed state (XZ, T z ) ,therefore, the optimum Xz is not an independent parameter and hence does not appear in the algorithm. In Figure 1, 71 and 71 are drawn by the usual maximizing procedure for p l , the profit function for the first reactor. If the feed state (XZ, T?)is designated by D or any point D’ on the ray joining origin 0 (fresh feed) to D, the optimal inlet state will be given by B Tl),where the ray intersects the locus i l . In either case A 2 will be calculated as the ratio of segments, OB/PD or OBIPD’. When D’ coincides with B , X Z is unity, which means that there is no bypass of feed and B is a common point specifying inlet feed state to reactor 1, irrespective of whether it is reached for exchanger
(XI,
The profit function, P,for two stages may be written as
For all possible feed states (X2, T z )representing a stream from T z ) is the second reactor, the optimal one-bed policy f1 (XZ, already known, since the inlet state to stage 1 , x 1 and can be straightaway calculated and located on i l . 21
Figure
1.
One-bed optimal design, cold-shot cooling
TI
h2Tz
=
A2
XZ
'
+ (1 -
(4)
X2)Ty
T h e decision variables for a two-stage scheme are then XZ XZ) ; and T z . This choice takes care of the bypass stream, (1 p l may be rewritten as:
-
P Z l
R
XI # f (X2). Differentiating Equation 5 write X2, Figure 2.
Two-bed design
Similarly,
32 = hz (1 dx 2
f
For P (= pi
-
A)
(7)
+ p ~to) be maximum,
X
Applying the above condition we get rl(f1)
Figure 3. Rate matching and loci of exit states, cold-shot cooling
cooling by moving horizontally from C or along the ray OB produced. I n this note a new algorithm is developed for a multibed reactor sequence. A two-bed operational sequence is indicated in Figure 2. T h e profit function for the first stage may be written as:
(9)
= rz(x2)
This condition of matching of reaction rates between reactors is almost identical to that obtained for a heat exchanger cooling problem, except that the exit state from reactor 2 is located on a ray joining the feed point to the inlet state to reactor 1. This procedure is indicated in Figure 3, where 0 represents the feed state. 71 and i l represent the strategy for a one-bed optimal policy. B ( 8 1 , TI) is the feed state to reactor 1. O n joining OB and extending it to intersect the constant rate curve, r ( x l ) , at D gives the exit state from second reactor. Following other adiabatic paths in the first stage, the locus of exit states from stage 2, 7 2 , can be drawn. Since by definition, OBIOD equals X2, 7 2 can be graduated in terms of XZ. I n order to locate i 2 , the following condition may be specified :
or x2
Since the first reactor operates on the total stream, Xi = 1. T h e decision variables are then X1 and TI. T h e loci 71, and for one-stage optimal policy, are drawn as indicated by Aris (1965), after differentiating p l with respect to X1 and Ti and setting the derivatives to zero.
(br/dT) ad rZ2(x)
dx
bppl + dT2 -p
=
0
(10A)
since
VOL. 8
NO. 1
JANUARY
1969
143
T h e condition therefore for locating 7 2 becomes almost identical to that applicable for a problem with heat exchanger coolingnamely,
Since A? is already known, 72, the locus of inlet states, to stage 2 can be drawn. Figure 4 indicates a two-bed optimal design, for a feed located at point 0. The procedure is to preheat to R , then react along RS, mix with fresh cold feed along SM (A, = P M / P S ) , and react further in reactor 1 along M N .
calculated for a reversible first-order system by the simultaneous solution of the equations for rm, the loci of maximum reaction rate at constant conversion, and the appropriate rate equation, identical in magnitude to P. Specifically the two equations are :
and
This fact does not appear to have been mentioned explicitly in the literature and may be helpful in a preliminary analysis of the optimization problem. Nomenclature
t
E
f ko
X
P
= activation energy = optimal policy
= pre-exponential factor = profit function (total)
Pn
= profit function for nth stage
r
= reaction rate
R = gas constant T, = temperature of feed T,, T, = exit and inlet temperatures to bed n Top, = optimum temperature (curve)
X, X,,, X,
J-.
Figure 4. cooling
An
Two-bed optimal design, cold-shot
TS
Tm 7n
The treatment can be extended to a three-stage design where the locus 73 will be graduated in terms of A S . 73 could be obtained on the basis of an equation similar to 12, where instead of Af, A 3 would be used. In this case where preheating is done only ahead of reactor 3,72 will be drawn on the basis of the vanishing of the integral given as LHS in Equation 12. The graphical scheme enumerated above is specific for a given fresh T I ) ,as changing the feed quality alters the position of feed (X,, loci, T~ and f n , except, of course, T I and 71. In this respect the applicability of the design procedure is not so general as that possible for the case of exchanger cooling. il‘hatever the mode of interstage cooling, once parameter P, which is the cost of reactor holding time, is fixed for a given reacting system, the optimization procedure fixes an upper limit for the conversion theoretically attainable. This may be
7,
B c1
conversion x,,== equilibrium exit and inlet conversion of stage n = conversion in fresh feed = fraction of total flow through = locus of equilibrium states
n
locus of maximum reaction rate locus of optimal inlet states for nth stage locus of optimal exit states for nth stage cost of holding time, units of rate = cost function for preheater
= = = =
literature Cited
Ark, R., “Introduction to Analysis of Chemical Reactors,” p. 235, Prentice-Hall, Englewood Cliffs, N. J., 1965. Lee, K. Y . , Aris, R., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 2, 300 (1963). MalengC, J. P., Villerrnaux, J., IND.END.CHEM.PROCESS DESIGN DEVELOP. 6 , 535 (1967).
P. G. Bhandarkar G. Narsimhan‘ Indian Institute of Technology Kanpur, India 1 Present address, Monash University, Clayton, Victoria, Australia.
DYNAMIC SORPTION BY HYGROSCOPIC SALTS SIR: With reference to a recent article, “Dynamic Sorption by Hygroscopic Salts,” by Dharamvir Punwani, C. TV. Chi, DESIGN DEVELOP. and D. T. M’asan [IND.ENG.CHEM.PROCESS 7, 410 (1968)], we wish to submit the following clarification: Tl’hile throughout the paper we deal with salts coated on asbestos, the comparison of the cooling effects produced by LiCl and LiBr was made on the basis of the vapor pressure data of unsupported (not coated on asbestos) LiCl and LiBr. In Figure 11, the equilibrium curves for LiCl and LiBr are based on the data of unsupported salts and not the salts coated on asbestos. This was due to the unavailability of the solidvapor equilibrium data for the salts coated on asbestos. Since then, data have been obtained that indicate that the vapor pressure of salts coated on the support material used in 144
I&EC PROCESS DESIGN A N D DEVELOPMENT
this study is different from that of the pure salt. Therefore, the calculated cooling effects shown on page 414 apply to the pure salts and not to those coated on asbestos. Dharamvir Punwani Institute of Gas Technology Chicago, Ill.
C. TV. Chi W . R. Grace & Co. Clarksville, Md. D. T . TVasan Illinois Institute of Technology Chicago, Ill.
