(16) Hiby. .J. \ Y . , \Virtz. K., Physik. Z. 41, 77 (1940). (17) Hoffman, D. T., Jr.. Ph.D. thesis in chemical engineering, Purdue University. Lafaayette, Ind., 1959. (18) Horne. F. H., Bearman. R. J., J . Chem. Phvs.37, 2842 (1962). (19) .Jones, .4.L., Hughes, E. C. (to Standard Oil Co., Ohio), U. S. Patents 2,541,069-70-71 (Feb. 13, 1951). (20) .Jones, R. C., Furry. W. H., Rem. .Wad. Phys. 18, 151 (1946). (21) Ihid., p, 177. (22) Klemm, A , Z. .VntilrJorsch. 7a, 417 (1952). (23) Korsching, H . , LVirtz, K . , Ber. Deut. Chem. Ges. 73, 249 (1940). (24) Korschinq, H.. IVirtz, K.. Naturzvissenschaften 27, 110, 367 (1939). (25) I,ud\vig, C.. @‘pin. A k a d . B e y . 20, 539 (1856). (26) Majumdar, S. D., Phys. Rer,. 81, 844 (1951). (27) Mizushina, T., Ito, T i . , Kagaku Kogaku 25, 34 (1961). (28) Moseley, H. M., Rosen, N.: “Approach to Equilibrium by a
Thermal Diffusion Column. I . Column Closed at Both Ends,” USAEC AECD-2236 (dec1:rssified Aug. 25, 1948). (29) Powers, J. E., in “New Chemical Engineering Separation Techniques.” H. M. Schoen. ed., pp. 42-6, Wiley, New York, 1962. (30) Powers, ~ J .E., “Proceedings of Joint Conference o n Thermodynamic and Transport Properties of Fluids.” London, July 1957, pp, 198-204, Institution of Mechanical Engineers, 1 Birdcage LYalk. $Vestminster. London SLYl, 1958. (31) Soret, C., Arch. S i . Ph-ys. .Vat. (Geneun) [3] 2, 48 (1879). (32) Waldmann, Id., Z. Phyrik 114, 53 (1939). RECEIVED for review Auguqt 16, 1963 ACCEPTED January 22, 1964
Part of bachelor thesis of T. C. Ruppel, University of Pittsburgh, 1963.
OPTIMIZATION B Y A GRADIENT TECHNIQUE E. STANLEY LEE Process Development Dicision, Phillips Pelroleurn Co., Bartlrsuzlle, Okla.
A numerical gradient technique, used to obtain the optimum conditions of multistage chemical engineering problems, has the advantage of being able to investigate problems with a fairly large number of state variables. The optimum temperature profile in a tubular reactor and the optimum solvent distribution in multistage cross-current extraction with both miscible and immiscible solvents are obtained b y this method.
v.
~ R I O U Smethods
can be used to obtain the optimum conditions of continuous or multistage chemical processes. T h e most promising ones are dynamic programming, maxim u m principle, and various versions of gradient methods. However, each method has its limitations and can be used only for a certain class of optimization problems. T h e various 6, 8, methods have been compared in detail in the literature (4, 9, 7 7 ) . For complicated optimization problems, a combination of these methods must be used. ‘This paper describes the use of a gradient technique in obtaining the optimum temperature profile in a tubular reactor and in determining optimum solvent distribution in crosscurrent extraction with miscible and immiscible solvents. This technique has been applied to aerospace systems by Bryson (5) and Kelley (7, 10). As has been pointed out elsewhere. both the maximum principle and dynamic programming can be applied only to problems with a rather restricted number of state variables. T h e gradient technique has the advantage of being able to handle problems with a fairly large number of state variables. Bryson has obtained the numerical solutions of variational problems involving eight state variables. However, this technique cannot conveniently handle problems with inequality constraints on the state and control variables. Convergence to a n extreme which is not the absolute maximum or minimum may occur in applying this method. The Numerical Method
Equations of the gradient technique for continuous processes were obtained by Kelley and Bryson (5, 7, 70). Later Dreyfus obtained the same results by the use of the embedding technique ( 2 , 6). T h e results of Dreyfus are outlined and some extensions of the method to optimization of a n integral equation and to stagewise processes are discussed.
Let the system be represented by the following set of differential equations :
where u is the control variable that is to be chosen optimally. Since differential equations must be reduced to difference equations for digital computation, Equation 1 will be treated in the following difference form:
T h e initial conditions are
i
= xtO,
xi(t0)
=
I, . . ., n
(3)
T h e problem is optimization of a specified function d y 1 ,
,
,
’, x,, t )
(4)
of the state variables and time t a t some unspecified future time, t , , where t l is the first time that a terminal condition $(x,,
,
.
. , x,, t )
=
0
(5)
is satisfied. Estimate a reasonable control variable sequence, u f , where ui; = u(k.4) and is a nominal sequence. Define
. xno, t o )
the value of C$ a t time t l where the starting state is ( X I O , . . . , .r,o) a t time t o and using the nominal control sequence uk T h e function Ssatisfies the following relation : S(xl0, . .
~
S(XI, . . ., xn, t )
=
=
+
S(XIf 11.4; . . . , . ~ n
fn.4, t
+ A)
(6)
This result forms our basic equation. I t is essentially saying that the value of C$ a t t , \\ith starting state x1, . . . , x, a t t VOL. 3
NO. 4
NOVEMBER
1964
373
+
equals the kalue of Q a t t~ \\ith starting state A , f1A, , fnA at t I where the f's are evaluated using the particular cominal control sequence, u k To obtain the direction of steepest ascent, Equation 6 is differentiated with respect to the control variable, u,
x,
+
+
3s z=I
1
b(u,
+ /LA)
b(xi
Since
+ fJ)
bu
1
evaluated in terms of the state and decision
variables a t time t , Equation 7 can be put into the following form by the use of Equation 2 :
where
D,(@)I f
bS = -
bU
evaluated in terms of the state and control
variables at time t
D T l ( @ I,)+,
'@
is the change in the final value of 4 due to a change
~
6x1
(7) bS
Substitute into Equation 12
tl
in x I , Equation 1 5 can be reduced to Equation 11. These equations are called influence functions or adjoint equations They are usually derived by the use of the theory of adjoint equations These equations can be solved in a backward recursive fashion, since the boundary conditions are given a t the final time, t = t I , by Equation 11. The symbol D has been used in the above equations instead of X which is urually used in the literature We have reserved the symbol X for the stagewise processes Dreyfus has pointed out that Equation 9 is the discrete analog of the multiplier rule in the calculus of variations The optimality condition is obtained when D,(p) = 0, as no more improvement would be possible. The sequence D,(@)is essentially the gradient of @ with respect to the control variable, u . If an improvement of A ~ J is asked, the greatest improvement will be obtained if
bS
evaluated in terms of the state and control 3x1 variables a t time ( t f A) T h e values of D,,(q) l t + , can be obtained by the following recurrence relation : =
-
The procedure for obtaining the solution is then :
A. Estimation of a nominal control sequence, u t . B. Integration of Equation 1. C. Determination of D,(@)and LIZ($) along the nominal
j = 1,
.. , n
which was obtained by partial differentiation of Equation 6 with respect to x I . T h e change of S with respect to time can be obtained in the same way:
trajectory with boundary conditions Equation 11. D. Determination of an improved u ( t ) from Equation 16. E. Repetition of the above Procedures B through D until
2
{ [ D u ( @ ) ] * becomes 1~) so small that possible further im-
t=n
provement is not significant. T h e method can be extended easily to problems involving additional final conditions. Assume the relationship At the final time tl. the following relationships can be obtained (2, 6 ) .
, yn, 2)
O(r1,
must be satisfied a t the final time used above get the results
=
0
tl.
(17)
T h e same arguments as
j = 1, . . . , n A similar relationship exists for the time variable,
t , a t the final time t l . Equation 11 corresponds to the transversality condition in the calculus of variations and can be obtained in the following manner. A variation in x j a t the final time t l varies the value of @ due to Equation 4 directly and due to the change in the final time determined by Equation 5 indirectly. Let 6 x , denote the variation in x j . then:
j
j
where At is the change in the final time. final time, d $ l c , = 0. Thus
374
I & E C FUNDAMENTALS
Since
=
0 a t the
= 1, , . . , n
=
1, . , , , n
T h e final condition can be made to satisfy Equation 17 in essentially the same way as that for the criterion 'function, @ (6). T h e greatest change for 8 will be determined by an equation similar to Equation 16. T h e problem of maximizing an integral of the form
can also be treated by the above method b! introduction of the (n 1)th variable
+
O n e inore diflerential relation can be added to the sbstem (23)
a n d the criterion function now becomes 9 =
An+,
(24)
For certain simple stage\vise processes. the above result, can be applied without change. Ho\vever: if the process equation cannot be put in the form of Equation 2. tht. above results cannot be directly applied. The required equations must t h r n be obtained by direct differentiation \cith thr aid of Lagrange multipliers. This point \vi11 be made clrar in the cross-current extraction problem. Optimum Temperature Profile in Tubular Reactor
'Ioillustrate the use of the optimization procedure outlined, the optimum temperature profile in a tubular reactor \vas obtained for a particular set of numerical conditions. .I'he reaction scheme ib .A B C:. \\.here B is the desired product. Both reactions are first order. This problem has been solved by various authors by other methods (3,Y). T h e kinetics of the reactions are given by -f
TI ME, M l N U T E S Figure 1 .
Temperature profiles
-f
d.x
- - = -k,n dt
(25)
T h e problem is to determine the temperature 7 ( t ) such that a t a specified time i I the value ofy is a maximum Thus 7 is the control variable a n d
$ = t - t l = O
G1 = 0.535 X 10" per minute 0.461 X per minutc 18.0(10 cal. per mole 30.000 cal. per mole 2 cal. '(mole-' K , )
G2 = E, = E? = R =
where x a n d y represent the concentrations of A a n d B. respectively, and k , a n d k? are the rate constants of the reactions
0 = ?
tion 34 is not needed in carr)-ing out the numerical calcula: tions. T h e numerical values used are
(28) (29)
T h e variational equations can be obtained eahil) from Equation 8 :
1 0 minutc, 0.9.5 mole per liter 0.0.3mole per 1irc.r yg LQ = 0.002 lI
to
= = =
This set of numerical values is the wine a.: that u d by Bilous and .Amundson (.3) in their esarnple .i. T h e p r o b k m \vas solved on a digital computer. '1 hc. approach of the coiitrol variable from thc initially rhtirnared nominal profile to the optimum one is sholvn in I'igure 1. T h e numerical Runge-Kutta integration zchvme \vas used, .After 4.5 iterations and 2 miiiutes of computing. the total yield of i \\-as 0.680mole per litrr and the final value of I \vas 0.1'mole per liter. This compared favorably \\.ith the results of Bilous and .Amundson Further improvement diould br expected if a smaller time incremenl i, used during the fir,t fr\v seconds of the reaction. A constant value of IQ \vas used during the above calculations. The rate of convcrgence to the optimum solution should be much faster if the value of -10 \Yere adjusted according to the magnitude of the gradient during the calculations. Optimum Solvent Distribution in CrossCurrent Extraction with Miscible Solvents
Thp optimum solvenr distribution for. immiscible solvents in cross-currcnt extraction \\-as studied by .Ark. Rudd. and Amundson ( 7 ) by dynamic programming tcchniciue. 'l'he same problem \vas solved by the usc of the gradient tcchnicjue. T h e process is shown in Figure 2. A mixture contairiing lIf0 solute is being extracted by a second solvent, u . T h e flo\v rate! q. of the raffirlate \vi11 be assumed constant. T h e extract leaves the extractor with a solute concentration .vlc, T h e streams q and u are expressed in total \veight and all concentrations a r r in weight fractions. VOL. 3
NO. 4
NOVEMBER 1 9 6 4
375
wI
*2
i
i
'El
X
'E2
Figure 2.
En
X~
N
Cross-current extraction with immiscible solvents
n
=
1. . .
.!
.\
The final conditions are obtained from Equations 11 i\ mass balance on the solute around the nth extractor gives Xzi(n- 1)
+
= fin
(35)
L'nxEn
Lvhere u n = re.,/'g
(36)
Follo\ving the treatment of Aris, Rudd. arid Amundson, the v
( c n x E r i-
problem is to maximize the quantity
At',)
kvhere X
1
is the cost of rhe exrract solvent divided by the cost of the solute and has the meaning of Lagrange multiplier. T h e quantity to be maximized is the total solute extracted instead of the final quantity This is analogous to the maximization of an integral in the continuous case. Introduce the variable: 2,
=
9
(cnYs,(
-
X
0,)
I
137)
n-1
=
(u,xEn -
+
xLnj
L'nXEn
- xu,
DTKiO)
.v
=
0.
Dz(P)
.L' =
1
(45)
Since the process equations are expressed in a back\vard recursive fashion. a minus sign must be placed in front of .IQ in Equation 16. The above set of equations can also be obtained by direct differentiation The problem is to maximize the quantithv
( Z ~ , ~ Y , ~ ~,
XC,~) subject to the constraint of .I' equations as
1
shown in Equation 35. Introduce the set of Lagrange multipliers Azn, n = 1. . . . .I'>and consider the folloiving equation :
(46 1
where xli is the state variable and X~ \vi11 be considered as a function' of ,tR through the equilibrium relationship. Equation 39. Differentiation of Equation 46 with respect to x K and 1' alloivs the follo\ving equation to be obtained :
1
n-1
Since z,(-1
=
( z ~ n x E 7 ~- X u , / : Equation 3- becomes:
4
z,l-
1
= 2,
-
U,XEn
+ xu,,
(38)
The problem is to choose the sequence! un: such that the quantity Q = z.\- ia maximized. The state variables are xil and z. The other variable, A,:,. can be eliminated by the use of the equilibrium relationship xb =
JbH)
(39)
T h e stopping condition is $ = n - '1- = 0. The required equations can be obtained in the same \vay as for the continuous case. However, Equations 35 and 38 are not in the same form as the difference Equation 2. Equations 35 and 38 are written in a back\vard recursive fashion. while Equation 2 is \vritten in a fonvard recursive fashion. Form the equation S(\n,?.
2,)
=
+
S ( X R ~ ~ n 2 . r ~2,. - cnx.rn
This equation can be rearranged to obtain
+h~n)
Choose A,, vanishes :
in the above equations so that the coefficient of ii\,
(40)
+
not x ~ ~ 1 . !as~ inL Equation where ,xHn L J , , Y ~ ~ is equal to nlirn6 . A similar condition exists for the term'z, - o,xEn f Xa,,. Partial differentiation of Equation 40 gives : Equation 48 is reduced to
where n = 1, 376
. . ., .Y
l&EC FUNDAMENTALS
-1
A N-3
0.6k
N=10 -N=20
-1
1
0.2
0.12-
0.08
I I
tI 7
This problem \vas solved by the gradient technique on a large digital computer. T h e results are shown in Figure 3: The initially estimated nominal control variable sequence u n , h = 1, . . . . .I. {vere 0.65, 0.2: and 0.1 for -Y = 3, 10, and 20: respectively. For the case .Y = 3 and with A@ = 0.0001, the optimum allocation as sho\vn in Figure 3 \vas obtained in approximately 0.2 minute. Most of the calculation time was spent in solving Equatioi,s 35 and 38 by a trial arid error procedure. Maximum values of z y Lvere 0.1077, 0.1143, = 3 and 0.11578 for .I- = 3. l o 3 and 20, respectively. For the calculated z.\- compared favorably with the value of 0.1075 obtained by Aris, Rudd. and Amundson. If the same amount of solvents is used and distributed equally among all the stages, the values of tS are 0,1054, 0.1133, and 0.1153 for .Y = 3. 10>20, respectively. T h e maximum (Figure 3) on the profile of the solvent dis: tribution \vas very sensitive to the total profit value, z , ~ especially for .V = 20. The maximum value of the profile for u can increase or decrease a fairly large amount without changing total profit z.\- appreciably. A\.
Optimum Solvent Distribution in CrossCurrent Extraction with Miscible Solvents
T h e flowsheet for miscible solvents is shown in Figure 4 .
.4mixture containing solvent A and solute C is being extracted by a second solvent, B. The concentrations of C and A in
0.04
I
~ ' t
N=lu
N=3
0-
5
I
I
IO
I
'
I
I
15
I
I
'20
n Figure 3. solvents
Optimum
allocation
with
immiscible
the original mixture and raffinate are repesented by x R and y R , respectively. T h e corresponding components in the extract streams are represented by .xE and y B . T h e extract stream has a n inlet flow rate, zc, of pure solvent B and an outlet rate, (r. Mass balance on the total component in each stream, the solute x? and the solvent y , respectively, gives: qn-1
Since L I Z ( @ ) = 1: for n = 0:1, , . . > .V, Equations 42 t o 4 5 are equivalent to Equations 49, 50, and 52. T h e symbols D , and D,,? in the former correspond to -A, and A, respectively. in the latter set of equations. In order to solve the above equations numerically, the equilibrium relationship. Equation 39, must be known. For the purpose of comparison. the equilibrium data of Ark, R u d d , and .\mundson \\.ere used. T o avoid numerical differentiation which is known to be very inaccurate, the d a t a which were obtained from the small plot in the above reference were correlated into the following equation : .xE = a
+ b x , + cxR2 + dxR3 + exR4 + f x R j
(53)
+
= qn
+ + +
un
( 54)
qn-lXR(n-1)
= qn.YKn
U ~ X E ~
(55)
qn-1YK'n-1:
= qnYRn
UnYEn
(56)
There are six unknowns, u,, x E n ? y E n , q n : x R n , and y R n . In addition to the above three equations, three more equations can be obtained from the equilibrium relationship. Typical equilibrium data are used and are sho\vn in Figure 5 by the use of triangular coordinates. From the saturation curve in the raffinate layer the following relationship is obtained :
T h e tie-line relationship gives
\\.here
d = -633.84 e = 3371.3 J = -5916 0
a = 0.00099
b = 1 7971 c = 35 196
By the combined use of both tie-line and saturation curve relationships the third equation is obtained :
T h e other numerical values used are X = 0 0 5 xo=02
wl
With given initial conditions a trial and error procedure must be used to solve the above set of equations.
0
20 =
I
i2 Figure 4.
Cross-current extraction with miscible solvents VOL. 3
NO. 4
NOVEMBER 1 9 6 4
377
and the original equation was reduced to : h.,
where A,,
Figure
5.
= ,A,
-
Equilibrium diagram
yR = a
Equations 54 to 56 cannot be put into the form of Equation 2. Thus the required equations for the gradient technique will be obtained by direct differentiation. Introduce the set of ,,A, and ,,A, n = 1, . . ., A' and Lagrange multipliers ,A,, consider the equation : .v =
((UnYEn
- ~wn) - ~ u n ( q n-t u n
n=l Azn[qnXRn
+
qn-1
- wn) -
- yn-lXR(n-lll -
UdEn
A@njqnYRn
-
f
UnYEn
-
qn-IYR(n-l]l
1
(61)
where u , q . and x R are considered as the state variables a n d a'is the control variable. T h e problem is to find the sequence such that the quantity @ is maximized. *The variables i E ,y E , and y R will be considered as a function of xR through Equations 57 to 59. Differentiating Equation 61 with fespect to the state and control variables, a fairly complicated equation is obtained. Setting the coefficients of dq, du, a n d dxR in this equation equal to zero as \vas done for the case of immiscible solvents, the follo\ving results \\'ere obtained: XEn
-
-
A2nxEn
n = 1, kun
+
AznxRn
+
AqnYRn
,
-
XqnYEn
(62)
=
. ., (A- - 1)
= Xu(n+I)
+
Xz(n+l)XRn
+
Xo(n+l)YRn
(63) n = 1, . . ., ( N
378
l&EC FUNDAMENTALS
-
1)
n = 1, . . . ! .V
(69)
T h e A's for the above set of equations have thr same meaning as the D's for'the continuous case. T h e A w n ' s are essentially the gradient of 0 with respect to the variables zt n: n = 1, . . . , .V. Thus the above problem can be solved by the use of Equation 16 and the procedure discussed earlier. T h e equilibrium relationships, which are usually given in tabular form, can be correlated into the following equations :
T h e quantity to be maximized is
o
A,
+ b x R + cxR2
(70)
which correspond to the Equations 57, 59, and 58. These equations can be used only within the range for which they are correlated. They cannot be used in the completely miscible region of the three components, as shown in Figure 5. This problem was solved for both the linear and nonlinear equilibrium relationships numerically by the gradient technique. Linear Equilibrium Relationship. T h e coefficients in Equations 70 a n d 72, which represent the equilibrium relationship, are assumed to be the following values : a = 1.0 e = 0.4
c = d ~ f = g = i = j = @
b = -1.4 h = 1.6
T h e other numerical values used are : X = 0.05 $70
=
1.0
T h e optimum results obtained for this problem are shown in Figure 6. T h e obtained maximum values of @ a t n = S are 0.1110, 0.1224, and 0.1251 for JV = 3, 10, a n d 20, respectively. A constant value of A ~ J= 0.0001 was used for all three cases. * For the case of immiscible solvents and linear equilibrium, Ark, R u d d , a n d Amundson ( 7 ) have shown that the optimum policy is to distribute the solvent, B, equally amoilg all the stages. This is not the case for miscible solvents. As shown in Figure 6, there is a fairly large drop in the amount of solvent used between the first and second stages and then it rises slowdy from the second to the Nth stage. T h e relatively large amount of solvent used for the first stage is due to the fact that solvent B is not present in the feed to the first stage and thus a fairly large amount of B is needed to obtain the equilibrium mixture in the raffinate froin the first stage. Nonlinear Equilibrium Relationship. T h e equilibrium diagram shown in Figure 5 was correlated into Equations 70 to 72. T h e coefficients obtained for thesevequations a r e : f = 2.6043 a = 0.9865 b = -0.8295 g = 0.0150 c = -1,2203 h = 4.1882 d = 0.009 i = -11.23 e = -0,1156 j = 8.481
T h e other numerical values used are:
0.3
;G
X = 0.05
N=3
I
r' 0 . 2 1
* -
0.1 -
-
N=IO N =20
0
qo = 1 . 0 T h e optimum results obtained for this problem are showm in Figure 7. T h e obtained maximum values of Q a t n = .Y are 0.3490 and 0.3644 for 5 = 3 a n d 10: respectively. iVith a starting value for the nominal Fontrol variable sequence us, = 0.2, n = l 3 , . . , .Y a n d for -1-= 10, approximately 1 minute computation time is needed to obtain the optimum shoivn in Figure 7. Most of this computation time !vas spent in the trial and error procedure used to obtain the solution of Equations 54 to 59. Discussion
/ I
," O.OE
Figure 6. Optimum allocation with miscible solvents Linear equilibrium
U O.I8
c Q K
N-3
t
T h e gradient technique seems to be a very powerful tool for solving multistage optimization problems with a fairly large number of state variables. By the use of this method, the two-point boundary value difficulties kvhich are generally preser.t in both the classical and maximum pri. :ciple approach can be overcome. I n addition, it can ha.!dle stagelvise processes equally as well as the coitinuous processes which have been studied by several authors in the literature. Various generalizations can be made for the problems discussed. Since temperature influences the equilibrium relationship, the temperature can also be considered as a control variable in addition to the one already discussed for crosscurrent extraction. Furthermore, a countercurrent flow arrangement for the extraction problem can be considered and the temperature instead of the solvent allocation can be considered as the control variable. In the example of cross-current extraction with miscible solvents, the'value of X was chosen so that the equilibrium relationship Equations 70 to 72 are valid. If the value of X is chosen too large o r too small, a coxtrairit on the control variable EN, for n = 1 is needed, because there is no solvent B in the feed to the first stage. Thus, if the value of ~ e is' ~too small or too large, the concentrations of the total mixture in stage one may fall outside the two-phase region, as shown in Figure 5. Although this problem with a control variable constraint cannot be handled as conveniently as Lvith dynamic programming, it can still be solved by the gradient technique ( 6 ) . T h e constraint on the control variable can be taken care of by successive approximation. T h e gradient technique may also be used to solve multistage optimization problems containing branching a n d parallel operations. 4 s an example of a branching type problem, a mixture with several components can be separated in several extraction towers. Each tower would be folloived by two towers, one to process the raffinate stream of the preceding tower a n d the second to process the extract stream of the preceding tower. This type of problem has not been investigated in detail but appears to be amenable to optimization by gradient techniques.
0.32-ll 4
L1
0.2411
Acknowledgment
T h e author is indebted to the Phillips Petroleum Co. for permission to publish this paper.
o'0 O I
3
5
7
9
n Figure 7. Optimum allocation with miscible solvents Nonlinear equilibrium
Nomenclature
xi
= ith state variable,
U
=
t ti
A
i
=
control variable = time = final time = time increment VOL.
3
NO. 4
1, . .
.n
NOVEMBER
1964
379
profit function value controlling step size terminal condition a t t l = additional terminal condition = defined by Equation 6
9 A+
4
9
S
kl.
k?
=
SUBSCRIPTS
= =
0
= =
x. y
E,. E! R
= =
xRn. ynn = x k n , )En
=
qn
= = =
u.n &I
evaluated in terms of variables a t time t reaction rate constants concentrations of A and B, respectively activation energies of reactions gas constant concentrations of solute C and solvent A in raffinate stream from nth stage, respectively concentrations of solute C and solvent A in extract stream from nth stage, respectively f l o ~rate of raffinate stream from nth stage feed rate of extracting solvent B to nth stage flow rate of extract stream from nth stage
x
= Zb,,:q = relative
zn
=
an
cost of extracting solvent B state variable defined by Equation 38 A’ = total number of stages ATn. AQn. A,,, A t n , , ,A = Lagrange multipliers a. b , c. etc = constants
tl,
= initial condition
AV
=
final condition
literature Cited
(1) Aris, R., Rudd, D. F.; Amundson, N . R., Cheni. Eng. Scz. 1 2 , 88 (1960). (2) Bellman, R.. Dreyfus, S..”Applied Dynamic Programming,” Princeton University Press, Princeton, X. J.. 1962. (3) B~lous, 0.; Amundson. N. R., Chern. Eng. Sci. 5, 81, 115 (1936). (4) Brvson, .A. E.. “Gradient Method for Optimizing Multistage Allocation Processes,” Raytheon Co.. Bedford. Mass.? Kept. BR-1818 (1962). (5) Bryson, A. E., Denham. LV, F.: J . 4,bpl. .Vlech. 29, 247 (1962). (6) Dreyfus. S., “Variational Problems iiith State Variable Inequality Constraints,” RAND Corp., Paper P-2605 (1962). (7) Kelley, H. J., A R S J . 30, 947 (1960).’ (8) Lee, E. S..Ind. En?. Chem. 5 5 , 30 (August 1963). (9) ,Lee, E. S.. “Optimization by Pontryagin’s Maximum Principle on the Analog Computer” Joint Automatic Control Conference, Minneapolis, Minn., June 1963. (10) Leitmann, G., “Optimization Techniques.” Chap. 6. by H. J. Kelley, Academic Press, New York. 1962. (11) Storey, C., Chem. Eng. Sci. 17, 45 (1962). I
RECEIVED for review August 9. 1963 ACCEPTED April 27. 1964
HEAT AND M A S S TRANSPORT
IN M U L T I P A R T I C L E SYSTEMS ROBERT
PFEFFER
The City College of The City Cntversity of ‘lere Y o r k ,
.lea
York. .I‘ 1,
The “free surface model” suggested b y Happel was combined with the Levich “thin boundary layer solution” of the diffusion equation to give an expression relating the Sherwood (or Nusselt) number to the Peclet number and the fractional void volume of a multiparticle system. The theoretical result, Nsh =
1.26
[
1 - ( 1 -
E)6/a
1
l/3
NP~”~ agrees , very closely with the available experimental mass transfer data for
fixed and fluidized beds in the low Reynolds number-high Peclet number range in which it i s applicable.
and mass transfer involving particles a t very low Reynolds numbers are important in such diverse fields as combustion of finely dispersed fuel, spray drying, meteorological studies. ion exchange, and gas chromatography. I n many chemical process applications, such as flow through packed and fluidized beds, the higher velocities involved will introduce some inertial effects. Nevertheless, ’ an analytical treatment involving viscous forces alone will usually apply to such systems a t Reynolds numbers appreciably higher than for a single particle (S),since a t higher particle concentrations, the separation phenomenon would be expected to become less pronounced. To obtain a simple analytical relationship among the rate of heat or mass transfer, the voidage of the system, and the Peclet number, the “free surface model’’ was combined with the Levich “thin boundary layer solution” of the diffusion equation. T h e free surface model developed by Happel (4)in his study EAT
380
I & E C FUNDAMENTALS
of the rate of sedimentation and pressure drop in packed beds assumes that two concentric spheres \\ill describe a typical cell in a random assemblage \vhich is considered to consist of many cells, each of which contains a panicle surrounded b>- a fluid envelope. Each cell contains the same amount of fluid as the relative volume of fluid to particle volume in the entire assemblage. and the outside surface of each cell is assumed to be frictionless or having a .’free surface.” Thus. the entire disturbance due to each particle is confined to the cell of fluid with which it is associated. For the case of heat or mass transport, the temperature or concentration of the fonvard moving fluid is assumed to be that of the cell boundary. If a solid sphere of radius e is placed a t the origin of a spherical coordinate system surrounded by a spherical envelope of fluid having a free surface a t radius b . and if the fluid envelope moves past the solid sphere with a constant velocity I- then the solution to the Navier-Stokes equations. ivith the inertial terms
