the direct methanation process is slow. At high temperatures the carbon-steam reaction proceeds a t a rapid rate, and tlie H2) is proportionately greater than the CH4 yield of (CO yield. Consequently, there is a region of intermediate temperature where preformed niet~hane is maximized, even though the amount of carbon gasified to both methane and carbon oxides continues t o increase with temperature (Figure 10). The gasification process is quite coniples and many other variables, including the composition of the Stage 1 gas, influence the preformed met’liane.
+
Discussion
The PEDU t’ests wit’li lignite hare showii that gasification proceeds smoothly and predictably in an internally fired unit in which coiidit’ioiis closely simulating those in Stage 2 are achieved. One of the distinctions between the PEDU and an integrated t,wo-stage gasifier, however, is in the amount of nitrogen required by the PEDU for purging aiid coal feeding. Such nitrogen will not be needed in the integrated gasifier so that the partial pressure of the reacting gases, particularly hydrogeii, will be increased proportioiiately. This in turn will lead to higher met’haiie yields and higher values of preformed methane. The PEDU tests have further shoivii conditions of operation more favorable t’lian others-namely, those at interniediate t,emperatures where maximum preformed methane is achieved. Actually, the tests have provided considerable information on the effect of temperature on the process. The previous experiments in the CFR varied gas composition and rate while maintaining a nearly constant temperature; the results established the validit’y of the two-step model and correlating equation for the direct methanation process. The current PEDU tests have extended the applicability of t’he model to other temperatures and prorided quantitative iiiformation on the effects of gasification variables on yields. The yield expressions for methane and carbon oxides have been inserted into a computer program simulating tlie integrated gasifier. This program is being used in conjuiictioii with economic informat’ion and gas processing requirement’s
to project cost aiid production figures for operation of t,he process on a commercial scale. A4dditional P E D P tests have already begun with R subbituminous Wyoming coal. Tests at’ higher pressures are plaiiiied to determine the effect of this variable on gasifier performance. The information aiid experience gained froin operation of the 100 lb/lir PEDU are being usrd in the design of a fully integrated 5 ton Air pilot plant for production of pipeline gas with the two-stage concept. This plant will esplore the problems of operating a slagging stage and entrained stage 15-ithin the same vessel aiid will provide design data for a commercial scale plant. literature Cited
I h i i a t h , E. E.j GIen~i~ I?. A , , “Pipeliiie (;as from Coal by TwnStage Entraiiied Gasific.ation,” i n “Operating Section Proceedi ~ i g s S~ e’ w ~ York, Amer. Gas Assoc., pp 65, 147, 151 (1965). Field, 11. .4.,Gill, 1). W.j Morgan, B. B.,Hawksley, P. (;. W., “Combustion of Pulverized Coal,” Leatherhead, England, Brit. Coal TJtil. Ile.;. .4\soc. (1967). Glenn, It. h.,Grace, 11. J., “hn Internally-Fired Process aiid 1)evelopment Unit for C ification of Coal under Condition. Simrilating Stage 2 of the BCIi Two-Stage Super-Pressure Procew,“ Amer. (;ai A s ~ o c . ,Synthetic Pipeline Gay Synip., Pittsburgh, Pa., 1968 Glenn, I? A , , lloiiath, E. E.j Grace, 1:. J., “Gaiification of Coal iiiider Conditions Simulating Stage 2 of the BCR Siiper-
Pressure Ga.C., Amer. Chem. Soc.. D D 81-103 11967). Lacey, J. A , , “The ’eisificatioii‘of Coal i n a Slaggiiig Preisiire Gaqifier,” d i j i e r . Chem. S O C . Diu. Fuel Cheiu. P r e p . , 10 (4), 151-67 (1966). lIosele!; F., Paterwn, I)., J . I n s f . F 7 ~ l38 , 128R), 13-23 (196%). on, I]., &fd., (296), 378-91 (1965h). (111, I ) . > z’bicl., 40, 523-30 (1967). 1-(!1i Fredersdorff, C. G.j Elliut, 11. h.,“Coal Ga.dkttioii,” i n “Chemistry of Coal Utilization,’’ IViley, Kew York, N.Y., 1963, pp 892-1022.
Zshradnik,11. I,,>Glenn, I?. .4.,Fitel, 50, 77-00 (1971). R ~ x i : r v e ofor review Fehrriary IT! I W l ACCI:PTI:D,Iiigrist 13j 1971
Based on work carried oiit at Bitnmiiiou.~Coal ReAearch, Iiic.) with support from the Office of Coal Research, U.S. ljepsrtmeiit of the Interior, tirider contract So. 14-01-0001-324. Pre.Geuted at the Symposium 011 Synthetic. Hydrocarbon Fiiels from \Vestern Coals, -4ICHE, Ileiiver, Colo., duguit 30-September 2, 1970.
Complex Method for Solving Variational Problems with State-Varia ble Inequality Constraints Tomio Umeda and Akio Shindo Chiyoda Chemical Engineering R. Construction Go., I’okohama, J a p a n
Atsunobu lchikawa T o k y o Institute of Technology, Tokyo, Japan
V a r i o u s methods for solving variational problems with or without inequality constraints have been developed. The maximum principle and dynamic programniiiig have received much attention. In applying the maximum principle, one must solve two-point boundary value problems, whereas in dynamic programming the curse of dimensionality restricts its applicability to simple problems. Thus, in most cases, the solu102
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
tioii of variational problems requires application of a numerical method, such as gradient,or conjugate gradient. With nonlinear problems e>peciallj-. digital computers are necessary to solve optimization problems by a sequence of discrete optimization. T h a t state vuriahles must satkfy various inequality constrailits usually complicates solutioii of variationnl problems.
The application of the Complex method for solving variational problems with state-variable inequality constraints has been attempted. The basic concept of the extensive use of the method has been presented, and optimization problems of tubular reactor design have been solved to demonstrate its applicability. The simplicity of the Complex method has been preserved in its application to the variational problems.
With variational methods a jump coiiditjoii a t each junct,ion point on the boundary between feasible a i d iionfeasible regions must be satisfied. Gradient iiiethods require the introduction of penalty fuiictioiis to replace the constrained problems by uiicoiistrained problems. However, no clear-cut method exists for choosing the penalty function>, and the algorithm must be repeated several times for increasing values of the peiialty functions. Also, the valucs of the fuiictional between iterations are iiot coinparable since the constraints are not' satisfied. Several approaches have been investigated to eliminate or reduce the disadvantages of these complicated approaches. The Complex method (Box, 1965), one of the promising methods of direct search for finite space problems, is simple and offers easy handling of various inequalit'y constraints. Extensive use of the Complex method has been attempted in this work. Since this method is applicable only to optimization problems iii Euclidean space, a finite difference approsiiiiation must be made first. The two possible types of approsimatioii are compared by solving numerical examples. To deinoristrat,e the applicaliility of the proposed method, two variational problems of tubular reactor design are solved. Variational Problem and Discrete Approximation
Generally, the variational problem niay be represented as follows : Maximize (or minimize) the object'ive function
is replaced by a simple summation of the function values on each interval. The first type of discretization can be represented as follow: Maximize (or minimize) the sum of scalar functions K-1
JI
=
k=O
(6)
Fk[x(Ok), d(Bi,), O,]
subject to the following coiistraiiits:
- x(ed
x(o,+d q[X(8,),
= Sk[X(ex),
d(O,), e,] 2 0 X-
d ( W , e,]
('i1
0 , 1, 2 . . , K - 1
=
(8)
.JT[x(0o)]= 0 Jr[x(o,)]
=
(9)
0
(10)
where 7, aiid fk denote, respectively, the function values of F a n d f f o r t e [Oh., 8k+l]-that is, Ft[x(Od, d(ed,
=
Ft[x(Od, d(ek), 0,l
1
F[x(Ot),d(od,
e,]
.
(Oi,+i
-
oh)
S[x(oJ, d(en.), 0,l . (OX+L - 0,)
The second type of discretization can be represented as folIaximize (or minimize) the sum of scalar functions K-l J2
,E r.',[x(ek), d(6d,
=
(11)
k=O
subject to the follon iiig constraints x(Ok+J
g[x(Bk), d(O,),
subject to the following constraints: Equality constraints (system equations)
where Fr[x(&), folio!? s:
Inequality constraints s[X(t),
d(01 2 0
- ~ ( 8 , ) = .fk[x(oi,),d ( @ d0,l , 0 k
=
0,1,2, . . . K - 1
(13)
A'- [x(Oo)]
=
0
(14)
X[x(e,)]
=
0
(15)
e,] 2
e,]
aiid f,[x(Bk), d(&),
(3)
Constraints on state variables at the initial and terminal times
N[x(to),t o ]
=
0
(4)
*lI[x(t,), t,]
=
0
(5)
Since the variational problem is not used to obtain analytic solut'ioiis for general forms of iioiilinear optimization problems by applying the maximum principle or variat,ional methods, numerical solut'ioiis must be obtained 011 a digital computer. Discretized forms of problems may be defined by E q u a t'lolls 1-5. Two types of discrete approximation are considered. The first replaces the differential equations by their difference approximate forins. The state variables are iiot continuous with respect to t on [to, tl]. The second type discretizes d ( t ) with respect to t and solves the system equations (Equation 2 ) for the state variables by integrating them numerically or analytically. In both cases t'he objective function (Equation I)
(12)
r.'t[x(od, d(oi,), or1
=
fic[x(eJ, d(ed, or1
=
e,]
are defined as
L;"
F[x(l), d ( t ) , t ] d t
1:""
f[x(t), d(0, tldt
(16) (1 'i)
The second type of discretization generally gives bet'ter results than the first t,ype. The computing time by the second type, hoxever, is larger than that by the first type owing to the iiumerical integration of Equations 16 and 17. Complex Method for Variational Problems
The Complex method of Box (1965) is a direct search method for solving optimization problems in Euclidean space. Since the computational procedure of the Complex method is based on evaluation aiid compariioii of objective fuiiction values a t all vertices of R simplex, aiid the replacemelit, of the worst vertex by a new 01112 is iberatively carried out, the Complex method cannot be directly applied to solve optiniization problems in function space. Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , NO. 1 , 1972
103
Cons t mint
Figure 1 . Assignment of three different constants for onedecision variable in two-dimensional approximate problem
The finite difference approximation of the problem and the proper definition of a vertex must be made. A brief description of the Complex method is given, and the extensive use of the method for variational problems is attempted by defining a vertex in the space of decision variables. The Complex method is a constrained simplex method and has two functions: to find sequentially a new vertex with improved values of objective function and to make the vertex satisfy inequality constraints on decision and/or state variables. The initial points are provided or generated with the aid of random numbers, pseudorandom deviates rectangularly distributed in the interval (0, 1). The objective function is evaluated a t each of these points. Comparison of these function values gives the worst function associated with the worst vertex, and the worst vertex is replaced by its reflection of a times as far from the centroid of the remaining vertices. If the trial point satisfies all the constraints, the objective function is evaluated and the computation is continued. If the trial point does not satisfy some implicit constraint, or the trial point is again the worst point, a further trial point is constructed by a move halfway back toward the centroid. This process is repeated as necessary. The satisfact'ion of explicit constraints is made by resetting the trial point just inside its appropriate boundary to give a further trial point. The Complex method applies in finding an approximate solution for a discretized optimization problem if the vertex in the space of d ( t ) , t e [ B p , Ok+l] is properly defined. If a piecewise constant denoting d(&) is given to each d ( t ) , t e [&, k = 0,1,2 . . . K - 1, the state variables and thus the objective function may be calculated by Equations 6 and 7 or 11 and 12. Since a set of the piecewise constants d(&) assigned to all d(t),te[&, IC = 0,1,2, . . . K - I,may specify a unique vertex in the n K dimensional Euclidean space of d(&), a setto-point mapping of the set d(ek)into a vertex in the space of the same set may be constituted. On the other hand, any vertex in the space corresponds to a set of piecewise constant's for d(&), and thus the simplex method is applied to find an approximate solution of a discrete unconstrained optimization problem. Since the Complex method can handle any implicit form of state-variable inequality constraints in Euclidean space. its application can be extended to the variational problems with state-variable inequality constraints. To apply this method, 104 Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
Figure 2. Extensive use of complex method two-dimensional problem for one-decision variable
the number of vertices of the simplex may be specified in such a way that i t is larger t,han nK 1, where n is the number of elements of decision vector d ( t ) , t e [ & , 6'k+l], k = 0,1,2, . . , K - 1. Using t'he Complex method can schematically be shown for a one-dimensional problem (Figure 1). The interval [Bo, O K ] is decomposed in two parts, and three sets of piecewise constants for the decision variable are assigned. These three sets, associated wit'h different values of an objective function, form a simplex ABC in Figure 2. The point B' is a vertex with a better value of the objective function, and the point C" is obtained by making the point C' a halfway move toward the centroid of the remaining points so that the point satisfies an inequality constraint'. Thus, a comput,ational procedure has been developed. The basic procedure is solving variational problems wit'h statevariable inequality constraints and specified terminal time. Other types of variational problems arising owing to the combination of terminal condit,ions with a terminal time condition can be solved as described below. For problems with unspecified terminal time, the terminal time is treated as one of the independent variables, and for each assumed value of the t'ermiiial time, the whole computation for solving the variational problem with specified terminal time is iteratively carried out. For problems with specified terminal conditions expressed by implicit or explicit forms of equality constraints (Equations 5 or 10), the number of independent variables decreases since state variables a t both ends of the final interval are independently given, and the decision variatle on the interval may be computed by these given state variables. dlthough the first type of discretization requires only solution of Equation 7 wit,h respect to the decision variable, the second type of discretization requires numerical or analytic integrations involved in Equations 16 and 17. To satisfy the terminal conditiom, iterative integrations are generally necessary.
+
Applications
The proposed method is applied to find optimal temperature profiles for a t,ubular reactor where t,he first-order consecutive reactions A + B -+ C take place. As the objective function, the yield of B , which should be maximized, is chosen.
350 h
5 345 W
CL 3
ki
340
CL W
a
E
335
7 1 -
c I
M
1
0
65 0
.
T
I ,+
04
I
w
I
I
0.2
8 330
2
Physicochemical data = ' 5 . 3 5 X 10I0permin
Ai Az
= 4 . 6 1 X 1017per min
18,000 caljmol 30,000 cal/mol 1000 cal/l. "K R = 1987 cal/mol OK (- AH)1 = 10,000 cal/mol (- AH)z = 20,000 cal/mol Initial conditions Xao = 0 . 9 5 mol/l. a 0 = 0 . 0 5 mol/l. xco = 0 . 0 moljl. To = 335°K = = =
E1
E2 C,p
+--I--
0
Table 1. Numerical Constants
4 6 TIME ( m i n )
8
0.0 io
Figure 3. Optimal solution from Problem 1 X, optimum = 0.6400 (first-type discretization)
The kinetic equations are given by the following forms:
Upper and lower bounds on T , ze and UT = 345°K LT = 310°K Ue = 5.0min Le = 0 . 1 min = 10.0min KO U, = 0.1 mol/l.
e
dxu
- = -k1Xa,
at
Xc(0)
= xco
xc(&+l)
where xu, x b , aiid x c are the concentrations of A, B , and C, respectively, and k l and kz are rate constants expresed by the following Arrhenius types: kl
=
AI exp (-E1/RT)
(21)
k2
=
-42 esp (-Ez/RT)
(22)
Problem 1 is a modification of the well-known problem of finding an optimal temperature profile for a tubular reactor. Modification has been made by adding to it an inequality constraint imposed on a state variable, x , . Problem 2 is finding an optimal temperature for an adiabatic reactor. The temperature in Problem 2 is a state variable on which an inequality constraint is imposed.
= X&O)
+
XbjeO)
+
xc(O0)
- x a ( e k ) - z b ( 0 k ) (26) k=0,1,2, ... K - 1
I n the second type of discretization, the temperature is discretized and some specified value on each interval is assigned. The state variables x u , x b , and x , are obtained as continuous functioiis by integrating Equations 18, 19, and 20 with respect to t for these given values of temperature on the intervals. They are given by the following equations: = xa(Br)
exp( -A1
. exp [-E1/RT(ex+1)1. t }
(27)
Temperature Profile for Tubular Reactor (Problem 1)
The original problem without any state-variable inequality constraints has been solved by many authors. Bilous and Amundson (1956) used the method of functional differentiation, and Aris (1961) applied dynamic programming. Katz (1960) and Lee (1964a) applied variational methods. Lee (1964b) aiid Flynn and Lapidus (1969) also solved the problem by the methods of gradient and conjugate gradient, respectively. The present problem with state-variable coiistraints may be formulated in two types of discretized problems: Maximize x b ( K )
T(ek)
(23)
subject to xc(&) I U , and F ( e k ) 5 UT where in the first type of discretization, x u , x b , and z, can be obtained by the following equations:
denotes holding time on each interval. The where t e [ O r , results of computation for the numerical constarits given in Table I are shown in Figures 3 and 4,respectively, for t h e first and second types of discretization. The inequality constraints on the state x c and the decision T are satisfied in the solutions. The optimal temperature near the entrance of the reactor is located on the boundary ( U T = 345'K), and the concentration x c a t the exit is also located on the boundary ( U , = 0.1). Since the original problem does not impose any inequality constraints, the higher temperature near the entrance is allowed to obtain a better value for x b (Figure 6). With respect to both types of discretization, the continuity of the state variables with respect to time in the second type gives better results than those obtained by the first type. Figures 3 and 4 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
105
350
A
350
b.
b.
/08
10.8
I
W
LT
I
$
340
W
c
340
a" E
%
W
345
g
I
3
k!
L -
W
335
c
33 5
-
I-
.0.2
330
5
9
330
TtME ( m i n )
TIME (min)
Figure 4. Optimal solution for Problem 1 X, optimum = 0.6646 (second-type discretization)
Figure 5. Optimal solution for Problem 2 X, optimum = 0.6362
also shovc that the concentration profiles are insensitive to the temperature profile in this problem. About 6 min of IBM 360/40 machine time is required to satisfy the preset stopping criterion defined as 10-5 for the relative error on the objective function. Since Equations 27 and 28 are the analytical solutions for Equations 18 and 19, respectively, the computing time for solving the problem by these two tvpes of discretization is about the same. For solving more general problems which require numerical integration of Equations 27 and 28, much computer time is required for solution of the second type.
constraints and that with inequality constraint LT 5 T ( e k )5 C T . A s a result of optimizabion, the optimal temperatures in both cases increase monotonically with time, and the outlet temperature in the second case is located on the upper boundary, though it is higher than the boundary in the first case. Furthermore, (&+I - ex) gradually decreases in bot,h cases, though its distribution is different, in each, showing that the inequalit,y constraint imposed on the temperature affects the distribution of holding time. The numerical coilstants involved in the equality and inequality constants are specified (Table I) by taking the above investigation into account. The optimal solution for Problem 2 on the basis of Table I is given in Figure 5 which shows satisfaction of bot,h equalit,!- and inequality const'raints. The outlet temperature is located just below the upper boundary (ITT = 345), and t,he distribution of holding time is also affected by the inequality constraint as well as the - 0,) = K O . equality constraint
Temperature Profile for Adiabatic Reactor (Problem 2)
The optimization problem for an adiabatic reactor design must take into account a heat balance relationship, together with Equations 24-26. Since this may be classified as any optimization problem with specified terminal time and statevariable inequality constraints, it is taken as an example. This problem, however, has a different character from general variational problems and cannot be an optimization problem in the second type of discretization owing to its lack of freedom. The heat balance equation in the discrete form can be expressed by :
T(ekTl)=
vek)+
AH)^
. exp [ - E ~ / R T ( B ~I +. Jz a ( ~ k ++l )
k
=
0, 1 , 2 , . . K - 1
The present optimization problem may be formulated as fol10~7sin the first type of discretization: Maximize za(KB)
~ e -~ ek)i + ~ subject to inequality constraints, 2 (8k+l - 8,) 2 L ' S , and Lr I T(&) 5 1'2.and equality constraints, Equations 24-26, 30, and (Bk+l - O k ) = K O ,where k runs from 0-9. k
The independent variables are 9 in - Or), X- = 0,1,2, this problem. Before solving this problem, two cases have been investigated: that without these equality and inequality 106 Ind.
Eng. Chern. Process Der. Develop., Vol. 11,
NO. 1, 1972
k
With the same computer, 7 rnin \%asrequired to satisfy the same stopping criterion as that for Problem 1. Discussion
In the previous section, two constrained optimization problems are solved to give discrete approximate solutions. T o compare a discrete approximate solution with the original iiondiscrete solution, the previously mentioned original unconstrained problem for finding the optimal temperature profile is also solved. The optimal solutions obtained b y the method of functional differentiation (Bilous and Xmundson, 1956) and the present approach of discretization are given in Figure 6. The former method gives a better result (Line a) than that obtained by Bilous and Amundson (1956), perhaps because of the more precise computationa by a digital computer. Line b is the temperature profile obtained by the present approach by use of Equations 27-29. These lines in Figure 6 may give the degree of approximation, and the result of the present approach orving to the second type of discretization shows that t'he average deviation in temperature is about 0.66'K. By this comparison t'he proposed approach gives a well-approximated solution to the original nondiscrete problem. Since the proposed method is based on extensive use of the Complex method (Box, 1965)) monotonic decrease in tem-
h
=rl
Acknowledgment
The authors acknowledge Tsumoru Maeda's contributions to this work. Nomenclature
355
\
-08
%
E
ili = frequency factor in Arrhenius rate constaiit ki d ( B k ) = decision vector at kth discrete time or 011 kth interval ( k = 0, I , 2, d ( t ) = decision vector, (ei, I t K - 1) E t = activabion energy in Arrhenius rate constant ki k = index denoting kth discrete time or number of vertices ki = reaction rate constant in Arrhenius type espressioll K = total number of interval Kg = constant for (6k+l - B k ) k
v
3551 0
U
'
"
2
"
"
"
4 6 TIME ( m i n )
a
iJ 0.0 IO
Figure 6. Optimal solution for original problem a:
x g
optimum = 0.6801 (Bilous and Amundson, 1956)
b: X g optimum = 0.6796 (Present method)
perature might not be obtained on intervals insensitive to the objective function, provided that some inequality constraints, such as T ( & ) 2 T(Bt+l),are not imposed. I n addition, this is a direct search method and to check whether a global solution is obtained is necessary in general to solve the problems several times b y starting from different initial values of decision variables. With respect to solving large dimensional problems which require finer discrete intervals, unreasonable computing time may be necessary for this approach to yield solutions. The Complex method , however, does not require considerable increase in the computing time with increasing the dimensionality. This character has been confirmed by Box (1965). I n general, the following two procedures for decreasing the computing time are possible: starting the computations by coarser discrete intervals, stopping the computations b y a milder stopping criterion. The results b y these procedures give a rough approximation to the solution, and the computations vcith finer discrete intervals are restarted from these results to obtain a more precise solution which satisfies the more severe criterion for stopping the computations.
L T = lower boundary for temperature Le = lower boundary for holding time R = gas constant
T
=
temperature
T(Bk) = temperature a t kth discrete time t = time t , = terminal time
U, UT Ue
= upper boundary for concentration C = upper boundary for temperature = upper boundary for holding time ~ ( 0 , ) = st'ate vector at kth discrete time x(t) = state vector, (0, 5 t B k f l , k = 0, 1 , 2 , za = concentration of A r,(Bk) = concentration of d at kt'h discrete time xo = concentration of B zb(Bk)= concentration of B a t kth discrete time 2 , = concentrat'ioii of C z,(B,) = concentration of C a t ' k t h discrete time
I
GREEKLETTERS a: = reflection factor in the Complex method AH = heat of reaction = kt'h discrete time p = density
Literature Cited
Aris, R., "The Optimal Design of Chemical Reactors," pp 142-52, Academic Press, New York, N.Y., 1961. Bilous, O., Aniimdson, N. R., Chem. Eng. Sci., 5 , pp 81, 115 (1956). Box, M. J., Comput. J . , 8 (l),42 (1965). Flvnn, J. V., Jr., Lapidus, L., A I C h E J . , 15 (Z), 308 (1969). Katz, S., Ann. X.Y. Acad. Ski., 84, Art. 12, 441 (1960). Lee, E. S., A I C h E J . , 10 (3), 309 (1964a). Lee, E. S., Ind. Eng. Chem. Fundam., 3 (4), 373 (1964b). RECEIVED for review March 2, 1971 ACCEPTEDJ d y 14, 1971 The authors express their gratitude to Chiyoda Chemical Engineering Construct~ionCo. for supporting this work.
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1 , 1972
107
