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Optimization of multistage feedback processes by the method of dynamic programming
Letters to the Editors 141 OLDSHLJE J. Y., in: Biological treatment of sewage and industrial wastes, 1. Ed. by MCCABE and ECKENFELDER,Reinhold, New York 1956. [51 RUSHTONJ. H. and OLDSHUE J. Y., Chem. Engng. Progr. Symp. No. 25, 181 1959. T. Academic PI HYMAND., in: Advances in Chemical Enginqering. 3 E$l. by DREW T. B., HERPESJ. W. and VERMEULEN Press, New York 1962. T., WILLIAMSC. M. and LANGLQISG. E. Chem. Engng Progr. 1955 51 85. [71 VERMEULEN PI CALDERBANKP. H. Trans. Inst. Chem. Engrs 1958 36 443. P. H. and Moo YOUNG M. B. Int. Sym. on Distillation, Inst. Chem. Engrs, London, 1960, 59. 191 CALDERBANK [lOI SULLIVAND. M. and LINDSAYE. E. Industr. Engng. Chem. (Fund.) 1962 1 87. [111 RODRIQUEZF., GROTZ L. C. and ENGLED. L. Amer. Inst. Chem. Engrs J. 19617 663. Chemical Engineering Science, 1964, Vol. 19, p. 514. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Design of agitators for gas-liquid contacting: reply to Dr. Bourne’s communication (Received 29 January 1964) The following conclusions tigations:
3. Much power can be wasted if the corresponding optimal rotational speed and corresponding optimal D/T ratio are not chosen for a certain tank (see for example Figs.7 and 8). K. R. WESTERTERP. N. V.Petrochemie AK&Amoco Postbus 135, Delfr rijl Holland
can be drawn from my inves-
1. Agitators with different D/T ratios cannot be compared as such and it is only fortuitous if the specific interfacial areas at equal power inputs are equal. 2. Agitation rate together with D/T is a more adequate but not the sole variable for scaling-up.
Chemical Engineering Science, 1964, Vol. 19, pp. 514-517. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Optimization
of multistage feedback processes by the method of dynamic programming (Received 21
February
1. INTRODUCTION RECENTLYthere has been a great interest in the multistage feedback problem. Methods to solve it have been proposed by JACKSON[l] and FAN and WANG [2]. The former used the classical variation approach, the latter Pontryagin’s maximum principle. It would be very valuable to develop the dynamic programming method such that it can be applied to recycle (or feedback) systems. Indeed, dynamic programming always leads to the extremum extremorum of the objective function, irrespective of the eventual existence of other stationary values (which are only local extrema). Moreover, such a method would have the additional advantage, to generate simultaneously the solution of a complete set of problems (for varying feed conditions). All attempts known by the author, to solve the problem using dynamic programming [3], were fallacious because they started out from a fundamental misconception [4]. Indeed, the feedforward cascade within the feedback system should have been optimized for variable entry and exit conditions, i.e. for constant feed conditions to the entire feedback
1964)
system. However, if the feedback system is optimized for constant conditions of the feed to the feedforward cascade, the policy obtained will always include the optimal feedforward cascade as an optimal sub-process, but will not necessarily be the optimal feedback policy. On the other hand, if the objective function is optimized for constant conditions of the feed to the entire feedback system, nothing guarantees that such an optimal feedforward cascade will be found as part of the optimal feedback policy. To allow for this fact, the objective function of the feedforward stages will be modified by introducing a Lagrangian multiplier 6. By doing so, the non-sequential R-stage optimization problem, will be transformed into a sequential (R -I- I)-stage optimization problem. 2. DYNAMICPROGRAMMING ALGORITHM FOR FEEDBACKSYSTEMS Consider the multi-stage cross-current extraction cascade with recycle as represented in the figure.
+q yR
Rtl
yz
Multistage process with recycle
514
w2
y1
Wl
Letters to the Editors The objective function of the open-loop cascade may be found, applying Bellman’s principle of optimality :
fr(xr+d = rnz
((xr+l - xl) -
h : + AL)
(1)
in which xr = solute concentration in the solvent stream, leaving the rth stage. y, = solute concentration in the extract; leaving the rth stage. wr = amount of wash water added at the rth stage (operating variable). 4 = solvent flow rate through the open-loop cascade h = relative cost of the operation (i.e. extraction and wash water costs). fr = objective or return function of the last r stages. Equation (1) expresses that the process return equals the amount of solute extracted by the wash water minus the extraction costs, assuming solvent and wash water to be immiscible. The material balance of the rth stage, may be written as :
It is obvious that the dynamic programming algorithm cannot be applied directly to this non-sequential system, because optimization for constant xR+r leads to an optimal value of xi and this corresponds to a certain feed state xf, which generally will not be equal to the given xf . Hence the objective function should be maximized for varying xR+l and then the best XR+~should be determined which satisfies (5). Therefore, a Lagrangian multiplier 6 may be introduced into (6), so: R 1 -max max C (xr+i - xr) 1 -/3ZR+1 1 5, [ r=l
F(xf)=
(Its-$
wry7 = q(xr+l - xl)
or
vryr = xr+i - xr
(2)
1 =
y, = !?
where
l_s2+;
&l 1
-8(xR+l-xl)]]
Xf)
fi)(XR+l-
+
4
The equilibrium relationship tions in the two phases is:
+rn~i~~~~+l-x~~(l+~-:;,il
between the solute concentra1
xr =
CNyr)
or
yr =Wx,)
(3)
=iqg+;
The open-loop optimization problem, as defined by equations (l)-(3), has already been solved by ARKS et al. PI. If a fraction /I of the solvent flow is recycled back to the entry of the Rth stage, and a flow rate q is entering and leaving the feedback system, the objective becomes:
+y;
r=l,...,R R
q max
max C (xr+l ZRfl i WI [ r=l
x7) +
+Xf-XRfl-
where vr now equals: Vr =
-
(& 1 This profit must be maximized using for each of the R stages the equilibrium relationship (3) and for the mixing stage R + 1 the feedback constraint: pm
Substituting these equations (2), (3) and (5) into (4):
-
-
xR)(l
+s
xf) +
-;,
+fR-l("(xR)])
Clearly, among these policies, this one should be selected which satisfies equation (7), thus leading to the optimal feedback policy.
W7
xn+1= (1 - fi)XY+
[(x,+1
p,
in which fR(@(xR+l) is a mathematical objective function, representing the profit obtainable from an R stage open-loop cascade with feed state XR+~and Lagrangian parameter 6. Applying equation (7), the problem is reduced to the optimization of a sequence of R + 1 decisions. Starting from the outlet and progressing in counterflow direction, thefr(@(xr+i), vr and xr are computed up to r = R. For each XR+l a value of S may be determined, which satisfies (5). Hence, for each x~+r there is an optimal R-stage policy determined by the value of 6 = '%xR+l) (8)
F’(xf) = q max max xf - XI &&~I1 ZR+1 i WI [
=
SC1-(xRfl 1 p
3.
EXAMPLE
Assume a linear equilibrium relationship : 1 Xr = CYyr= -yr
(5)
k
A material balance for stage r, reduces to: 515
(9)
Letters to the Editors kx,+l
&+I
and
y*=1
(10)
X’=1
xR+l
First one optimizes:
((x2-x1)(1++-)) (114
fi’d’(xa)=yx
=
(1
-
=
(1-Ph
+ pxl
h
R’R+l(XR+l)l,R+l (18)
+B[,(,
1
+a)
The optimal value S*, which determines the optimal feedback policy, follows from:
which requires : x2=(1
j%f
1
F(xf)=
+S)gAc or
x1 = [$&I"'
-1 -pzR+l max fR(d)(xR+l) + S(y = &fn’d*)(x~+1’)
hence
which will be maximum for:
[gy3
=
(xR+l*
+ ;
-
xf))
with S* = S(XR+~*) Clearly the value of S* depends on the number of stages over which a feedback loop is closed. Whatever S* may be, equations (12) till (17) show that equal distribution of the wash waters over the different stages will be the optimal policy for the recycle cascade. The same equations for 6 = 0, reduce to the well-known optimal policy of the feedforward cascade (1). Furthermore, it should be noticed that the equations (13) for S = 0 are completely analogous to the formulas of PIMEN and IOFFE[6], giving the optimal holding time of an isothermal feedforward cascade of CSTR, in which a first order irreversible reaction is occurring.
Then another stage is added and optimized:
X2
(xR+~- xf))
(12a)
Wb)
for which r2 = ;([”
+;)kx3]w
- 1) = u1
(13b)
4. SINGLE-STAGE RECYCLE SYSTEM
This can be continued until the R stage cascade is completed, and the optimal operating and stage variables are found from qP(Xr+1)
Consider function :
= 0. They are:
^
OXr
Fl(xf)
1 (1 +&kxr+l l/r+l~ h 1
v* = k
=
xR
xR-1
xr
xf - x1 - 1_~
01 1
which is maximal for:
(13~)
JACK.WN [4] proved that the true optimal policy should satisfy his equation (17), which becomes here:
l , . . . , R.
x3 =_=-=
x2
Xl
Xl
(1 + S)kxl
ax1
G
=-ax1
s2
h
I
au1
l-/32
(14)
zf
=-e
x 1-B
Using the Lagrangian multiplier approach optimal feedback policy, first
(15) f1(Yx2,=m$
or xR+l-n
=
x XR+l
kcl +sjxR+l
1
(22)
Ul
Hence
In particular:
x
I
WC)
Hence, it is seen that the optimal feedback policy requires an equal-size cascade. Moreover, for each value of 6 (particularly for the optimal S*): Xr+1
= mu?
with objective
v _1= v
I’ r=l
XR+l XR -rv=.~.=-=...
the one stage recycle system,
n/R+1
((x2-x1(1 +s+-J
=max ((x2 21
(16)
is calculated.
x1)(1 + 6) - hvl}
(17)
l +s -
x
Y(m)
x2-x1=
YyXl) +wx1)1~
516
(23)
The maximum is reached for :
A lRIR+l = [k&]R’R+l(xR+l)llR+l x1 = xR+1 k(1 + S)XR+I The best values S(xR+r) of the parameter 6, should be determined from the feedback constraint :
to establish the
(24)
Letters to the Editors To obtain 6 as a function of xa (or xl), this should be substituted into the feedback constraint:
fl(Xf - Xl) =
(1 -
l+
h s- Y(x1)
(25)
Y(x1)
&I = Y(xl*) (x2* - x
0 -@ -
l*)xYyxl*)
Y(xl*)
Hence
xrwIP
Finally, 6 must be chosen among these values 6(x1) such that it satisfies :
x
I
Fl(xf) = y
xf - XI - i-q
=yx
((x,-xX1)(1
and (28) becomes
1 -/I
-&),
(26)
hYyx1)
(27)
to: 1 -p
[ 1+
1+s*=(1-17)[1-&]+& +By(xl*) l - p[ l -
h -Y(xl*)
I
(28)
vl*Y(xl*)
vl*Yyxl*)
1
(29)
Equation (29) defines the critical feed concentration, under which it becomes unprofitable to extract with a cascade having a cost parameter h. Formula (28) shows that this is the only case for which the optimal feedback policy will require an optimal feedforward policy! However, (27) can be transformed, using (2) and (5), into
PI
[31 [41 151 161 L71
-
-aal
228
(32)
(33)
Using the Lagrangian multiplier method the optimal closedloop policy of feedback systems may be determined. The general feedback algorithm, contains the feedforward algorithm as a limit case (for 6 = 0). The method may be applied to any chemical system with recycle, particularly, to chemical recycle reactors, as will be shown in a forthcoming publication [7]. A. R. VAN CAUWENBERGHE Laboratorium Anorganische Technische Chemie Rijksuniversiteit Gent, Belgium
RWBRBNCBS
111
ax1
h
(1=
5. CONCLUSION
Y(xl*)
0
+
Substituting this into (32), proves that the optimal solution found by the multiplier method, indeed satisfies JACKSON'S condition, as formulated by equation (22).
0 i.e. if x1* = xf cdt, because
=
(31)
1+6
1 ax1 axz vr* = 1 + vl*~(x1*)
be equal to the optimal feedforward policy only if 1 - -
x
= _- x
but from (2):
which shows that for fi = 0 (i.e. the feedforward cascade), 6* = 0. Further, for any p the optimal feedback policyhwill
WXf,,,,)
(30)
vl*Yyxl*)
which of course is also satisfied for 6 = 6* hence (30) reduces
which determines the optimal value of xl, called xi*. The optimal 6* satisfies both (25) and (27), hence:
-= h
ul*Yyxl*)
However from (23) :
au1 z2
I - Y-(x1)=(Xf - “l)([ycx1)]2 I
l-
+
-ax1
-x
=
1 -/3+
= (1 - /Nl + ol*\T'(xl*)l
This requires that
= 1 -B
B
1+6*=(1-@+1+
v1 I
JACKSONR. Chem. Engng Sci. 19 1964 19. FAN L. T. and WANG C. S. Chem. Engng Sci. 19 1964 86. RUDD D. F. and BLUM E. D. Chem. Engng Sci. 17 1962 277. JACKKINR. Chem. Engng Sci. 1963 18 215. ARIS R., RUDD D. F. and AMUND~~NN. R. Chem. Engng Sci. 1960 12 88. PISMENL. M. and IOFFEI. I., Znt. Chem. Engng 1963 3 24. VAN CAUWENBERGHE A. R. and LAPIDUSL. Chem. Engng Sci. Submitted for publication.
517
Design of agitators for gas-liquid contacting: reply to Dr. Bourne’s communication (Received 29 January 1964) The following conclusions tigations:
3. Much power can be wasted if the corresponding optimal rotational speed and corresponding optimal D/T ratio are not chosen for a certain tank (see for example Figs.7 and 8). K. R. WESTERTERP. N. V.Petrochemie AK&Amoco Postbus 135, Delfr rijl Holland
can be drawn from my inves-
1. Agitators with different D/T ratios cannot be compared as such and it is only fortuitous if the specific interfacial areas at equal power inputs are equal. 2. Agitation rate together with D/T is a more adequate but not the sole variable for scaling-up.
Chemical Engineering Science, 1964, Vol. 19, pp. 514-517. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Optimization
of multistage feedback processes by the method of dynamic programming (Received 21
February
1. INTRODUCTION RECENTLYthere has been a great interest in the multistage feedback problem. Methods to solve it have been proposed by JACKSON[l] and FAN and WANG [2]. The former used the classical variation approach, the latter Pontryagin’s maximum principle. It would be very valuable to develop the dynamic programming method such that it can be applied to recycle (or feedback) systems. Indeed, dynamic programming always leads to the extremum extremorum of the objective function, irrespective of the eventual existence of other stationary values (which are only local extrema). Moreover, such a method would have the additional advantage, to generate simultaneously the solution of a complete set of problems (for varying feed conditions). All attempts known by the author, to solve the problem using dynamic programming [3], were fallacious because they started out from a fundamental misconception [4]. Indeed, the feedforward cascade within the feedback system should have been optimized for variable entry and exit conditions, i.e. for constant feed conditions to the entire feedback
1964)
system. However, if the feedback system is optimized for constant conditions of the feed to the feedforward cascade, the policy obtained will always include the optimal feedforward cascade as an optimal sub-process, but will not necessarily be the optimal feedback policy. On the other hand, if the objective function is optimized for constant conditions of the feed to the entire feedback system, nothing guarantees that such an optimal feedforward cascade will be found as part of the optimal feedback policy. To allow for this fact, the objective function of the feedforward stages will be modified by introducing a Lagrangian multiplier 6. By doing so, the non-sequential R-stage optimization problem, will be transformed into a sequential (R -I- I)-stage optimization problem. 2. DYNAMICPROGRAMMING ALGORITHM FOR FEEDBACKSYSTEMS Consider the multi-stage cross-current extraction cascade with recycle as represented in the figure.
+q yR
Rtl
yz
Multistage process with recycle
514
w2
y1
Wl
Letters to the Editors The objective function of the open-loop cascade may be found, applying Bellman’s principle of optimality :
fr(xr+d = rnz
((xr+l - xl) -
h : + AL)
(1)
in which xr = solute concentration in the solvent stream, leaving the rth stage. y, = solute concentration in the extract; leaving the rth stage. wr = amount of wash water added at the rth stage (operating variable). 4 = solvent flow rate through the open-loop cascade h = relative cost of the operation (i.e. extraction and wash water costs). fr = objective or return function of the last r stages. Equation (1) expresses that the process return equals the amount of solute extracted by the wash water minus the extraction costs, assuming solvent and wash water to be immiscible. The material balance of the rth stage, may be written as :
It is obvious that the dynamic programming algorithm cannot be applied directly to this non-sequential system, because optimization for constant xR+r leads to an optimal value of xi and this corresponds to a certain feed state xf, which generally will not be equal to the given xf . Hence the objective function should be maximized for varying xR+l and then the best XR+~should be determined which satisfies (5). Therefore, a Lagrangian multiplier 6 may be introduced into (6), so: R 1 -max max C (xr+i - xr) 1 -/3ZR+1 1 5, [ r=l
F(xf)=
(Its-$
wry7 = q(xr+l - xl)
or
vryr = xr+i - xr
(2)
1 =
y, = !?
where
l_s2+;
&l 1
-8(xR+l-xl)]]
Xf)
fi)(XR+l-
+
4
The equilibrium relationship tions in the two phases is:
+rn~i~~~~+l-x~~(l+~-:;,il
between the solute concentra1
xr =
CNyr)
or
yr =Wx,)
(3)
=iqg+;
The open-loop optimization problem, as defined by equations (l)-(3), has already been solved by ARKS et al. PI. If a fraction /I of the solvent flow is recycled back to the entry of the Rth stage, and a flow rate q is entering and leaving the feedback system, the objective becomes:
+y;
r=l,...,R R
q max
max C (xr+l ZRfl i WI [ r=l
x7) +
+Xf-XRfl-
where vr now equals: Vr =
-
(& 1 This profit must be maximized using for each of the R stages the equilibrium relationship (3) and for the mixing stage R + 1 the feedback constraint: pm
Substituting these equations (2), (3) and (5) into (4):
-
-
xR)(l
+s
xf) +
-;,
+fR-l("(xR)])
Clearly, among these policies, this one should be selected which satisfies equation (7), thus leading to the optimal feedback policy.
W7
xn+1= (1 - fi)XY+
[(x,+1
p,
in which fR(@(xR+l) is a mathematical objective function, representing the profit obtainable from an R stage open-loop cascade with feed state XR+~and Lagrangian parameter 6. Applying equation (7), the problem is reduced to the optimization of a sequence of R + 1 decisions. Starting from the outlet and progressing in counterflow direction, thefr(@(xr+i), vr and xr are computed up to r = R. For each XR+l a value of S may be determined, which satisfies (5). Hence, for each x~+r there is an optimal R-stage policy determined by the value of 6 = '%xR+l) (8)
F’(xf) = q max max xf - XI &&~I1 ZR+1 i WI [
=
SC1-(xRfl 1 p
3.
EXAMPLE
Assume a linear equilibrium relationship : 1 Xr = CYyr= -yr
(5)
k
A material balance for stage r, reduces to: 515
(9)
Letters to the Editors kx,+l
&+I
and
y*=1
(10)
X’=1
xR+l
First one optimizes:
((x2-x1)(1++-)) (114
fi’d’(xa)=yx
=
(1
-
=
(1-Ph
+ pxl
h
R’R+l(XR+l)l,R+l (18)
+B[,(,
1
+a)
The optimal value S*, which determines the optimal feedback policy, follows from:
which requires : x2=(1
j%f
1
F(xf)=
+S)gAc or
x1 = [$&I"'
-1 -pzR+l max fR(d)(xR+l) + S(y = &fn’d*)(x~+1’)
hence
which will be maximum for:
[gy3
=
(xR+l*
+ ;
-
xf))
with S* = S(XR+~*) Clearly the value of S* depends on the number of stages over which a feedback loop is closed. Whatever S* may be, equations (12) till (17) show that equal distribution of the wash waters over the different stages will be the optimal policy for the recycle cascade. The same equations for 6 = 0, reduce to the well-known optimal policy of the feedforward cascade (1). Furthermore, it should be noticed that the equations (13) for S = 0 are completely analogous to the formulas of PIMEN and IOFFE[6], giving the optimal holding time of an isothermal feedforward cascade of CSTR, in which a first order irreversible reaction is occurring.
Then another stage is added and optimized:
X2
(xR+~- xf))
(12a)
Wb)
for which r2 = ;([”
+;)kx3]w
- 1) = u1
(13b)
4. SINGLE-STAGE RECYCLE SYSTEM
This can be continued until the R stage cascade is completed, and the optimal operating and stage variables are found from qP(Xr+1)
Consider function :
= 0. They are:
^
OXr
Fl(xf)
1 (1 +&kxr+l l/r+l~ h 1
v* = k
=
xR
xR-1
xr
xf - x1 - 1_~
01 1
which is maximal for:
(13~)
JACK.WN [4] proved that the true optimal policy should satisfy his equation (17), which becomes here:
l , . . . , R.
x3 =_=-=
x2
Xl
Xl
(1 + S)kxl
ax1
G
=-ax1
s2
h
I
au1
l-/32
(14)
zf
=-e
x 1-B
Using the Lagrangian multiplier approach optimal feedback policy, first
(15) f1(Yx2,=m$
or xR+l-n
=
x XR+l
kcl +sjxR+l
1
(22)
Ul
Hence
In particular:
x
I
WC)
Hence, it is seen that the optimal feedback policy requires an equal-size cascade. Moreover, for each value of 6 (particularly for the optimal S*): Xr+1
= mu?
with objective
v _1= v
I’ r=l
XR+l XR -rv=.~.=-=...
the one stage recycle system,
n/R+1
((x2-x1(1 +s+-J
=max ((x2 21
(16)
is calculated.
x1)(1 + 6) - hvl}
(17)
l +s -
x
Y(m)
x2-x1=
YyXl) +wx1)1~
516
(23)
The maximum is reached for :
A lRIR+l = [k&]R’R+l(xR+l)llR+l x1 = xR+1 k(1 + S)XR+I The best values S(xR+r) of the parameter 6, should be determined from the feedback constraint :
to establish the
(24)
Letters to the Editors To obtain 6 as a function of xa (or xl), this should be substituted into the feedback constraint:
fl(Xf - Xl) =
(1 -
l+
h s- Y(x1)
(25)
Y(x1)
&I = Y(xl*) (x2* - x
0 -@ -
l*)xYyxl*)
Y(xl*)
Hence
xrwIP
Finally, 6 must be chosen among these values 6(x1) such that it satisfies :
x
I
Fl(xf) = y
xf - XI - i-q
=yx
((x,-xX1)(1
and (28) becomes
1 -/I
-&),
(26)
hYyx1)
(27)
to: 1 -p
[ 1+
1+s*=(1-17)[1-&]+& +By(xl*) l - p[ l -
h -Y(xl*)
I
(28)
vl*Y(xl*)
vl*Yyxl*)
1
(29)
Equation (29) defines the critical feed concentration, under which it becomes unprofitable to extract with a cascade having a cost parameter h. Formula (28) shows that this is the only case for which the optimal feedback policy will require an optimal feedforward policy! However, (27) can be transformed, using (2) and (5), into
PI
[31 [41 151 161 L71
-
-aal
228
(32)
(33)
Using the Lagrangian multiplier method the optimal closedloop policy of feedback systems may be determined. The general feedback algorithm, contains the feedforward algorithm as a limit case (for 6 = 0). The method may be applied to any chemical system with recycle, particularly, to chemical recycle reactors, as will be shown in a forthcoming publication [7]. A. R. VAN CAUWENBERGHE Laboratorium Anorganische Technische Chemie Rijksuniversiteit Gent, Belgium
RWBRBNCBS
111
ax1
h
(1=
5. CONCLUSION
Y(xl*)
0
+
Substituting this into (32), proves that the optimal solution found by the multiplier method, indeed satisfies JACKSON'S condition, as formulated by equation (22).
0 i.e. if x1* = xf cdt, because
=
(31)
1+6
1 ax1 axz vr* = 1 + vl*~(x1*)
be equal to the optimal feedforward policy only if 1 - -
x
= _- x
but from (2):
which shows that for fi = 0 (i.e. the feedforward cascade), 6* = 0. Further, for any p the optimal feedback policyhwill
WXf,,,,)
(30)
vl*Yyxl*)
which of course is also satisfied for 6 = 6* hence (30) reduces
which determines the optimal value of xl, called xi*. The optimal 6* satisfies both (25) and (27), hence:
-= h
ul*Yyxl*)
However from (23) :
au1 z2
I - Y-(x1)=(Xf - “l)([ycx1)]2 I
l-
+
-ax1
-x
=
1 -/3+
= (1 - /Nl + ol*\T'(xl*)l
This requires that
= 1 -B
B
1+6*=(1-@+1+
v1 I
JACKSONR. Chem. Engng Sci. 19 1964 19. FAN L. T. and WANG C. S. Chem. Engng Sci. 19 1964 86. RUDD D. F. and BLUM E. D. Chem. Engng Sci. 17 1962 277. JACKKINR. Chem. Engng Sci. 1963 18 215. ARIS R., RUDD D. F. and AMUND~~NN. R. Chem. Engng Sci. 1960 12 88. PISMENL. M. and IOFFEI. I., Znt. Chem. Engng 1963 3 24. VAN CAUWENBERGHE A. R. and LAPIDUSL. Chem. Engng Sci. Submitted for publication.
517