Suppose now that for this discrete process the optimal control is used at all stages except at number n, where the optimal control is changed a small amount, do,". The corresponding small change, dP, in the profit may then be written, neglecting higher order terms, (4)
where summation is implied over repeated subscripts here and throughout. The partial derivatives of P in Equation 4 may in turn be expressed as (5)
A necessary condition for a maximum of P is then that all Orn are chosen so that dP is negative for all such dern, whether the maximum occurs at a stationary point where the first derivatives vanish, or at the boundary of some 0," where the first derivatives determine the sign of dP. To determine the corresponding condition when Or may be piecewise continuous, let At + 0, all the time choosing the optimal control at all stages except the one starting at tl. Since each ern is chosen independently, it is necessary to assume that there really exists a piecewise continuous control that is optimal, to which Orn will converge as At 0. For very small A t Equation 3 may be approximated by -+
xin
xfn--l
+
f((Xkn-',
P)At
(7)
Substituting this expression for x i n in Equations 5 and 6, neglecting terms of order (At)z, and inserting the result into Equation 4, gives
a~
b P bfin 1 byin dP = -nAt de," f - - ___ At do," do," 2 bxin born d0,n bxin 387
(8)
where f i n stands for f i ( x X n P 1 ,O r n ) . dP, given by Equation 8, must be negative for arbitrary small At, and from our continuity assumption it follows that the expression
evaluated at tl must be negative. T o evaluate b P / d x i at t l , the discrete case is again treated first. Following Horn and Jackson ( 2 ) one defines
From the chain rule of calculus one then gets
Inserting Equation 7 in Equation 11 yields
Rearranging, dividing through by At, and letting At gives the differential equation
4
0
with boundary conditions
zi-i.e., b P / b x r i s evaluated along the optimal trajectory, and is thus independent of 0, and is a function of time only. Defining a function
H = rift (1 5) Equation 9 means that H must have a maximum with regard to 0, at t l . Furthermore, for Equation 8 to hold in a given neighborhood of the maximum of P, it must hold for increasing do," as At 0. For sufficiently small At, the permitted increment in Orn becomes the whole region of Or, since 0, is bounded. In other words, for P to have a maximum, Or must be chosen at all points (except at the possible discontinuities) to give N a global maximum, which is just the maximum principle. [If only first derivatives had been used throughout before letting At + 0, the result would have been the ''weak'' maximum principle (7).] That the function H is constant along the optimal path and zero if the final time is unspecified has been proved in a simple manner ( 7 ) and will not be repeated. The maximum principle may thus be considered a natural extension of ordinary differentiation, and only elementary calculus is needed to prove it. However, Pontryagin (4) has proved the maximum principle under more general conditions than those used here. -+
literature Cited
(1) Denn, M. M., Aris, R., A.Z.Ch.E. J. 11,367 (1965). (2) Jackson, R., Horn, F., IND.ENG. CHEM.FUNDAMENTALS 4, 487 (1965). (3) Jackson, R., Horn, F., Intern. J. Control 1, 389 (1965). (4) Pontryagin, L. S., et al., "Mathematical Theory of Optimal Processes," Wiley, New York, 1962.
JORGEN LOVLAND Norwegian Institute of Technology Trondheim, Norway REC~~IVED for review May 2, 1966 ACCEPTED December 10, 1966
OPTIMUM ALLOCATION OF ADSORBENT I N STAGEW I SE ADS0 R PTIO N 0 PE RAT10 NS study is concerned with the relative performance of several stagewise contacting operations for adsorption systems. The general agreement is that two-stage operation requires less allocation of adsorbent than single-stage operation to obtain the same final concentration of solute in the fluid phase. Also, countercurrent operation is believed to be better than crossflow operation for an equal number of stages, but no general quantitative results have been published to THIS
308
I&EC FUNDAMENTALS
confirm this belief for systems obeying a variety of adsorption isotherms. Further, Treybal (1955) describes two alternative methods of conducting a two-stage crossflow (referred to by him as cocurrent) operation: splitting the adsorbent in two portions, the portions being so determined as to provide a minimum total expenditure of adsorbent, or splitting the solution into two portions and treating each portion with the entire batch of adsorbent, the resulting solutions then being blended
Calculations on stagewise contacting operations for adsorption systems obeying the Freundlich and KobleCorrigan isotherms have shown that countercurrent operation is always superior to crossflow operation, and that crossflow operation in which the adsorbent i s split into two portions is always superior to the crossflow operation in which the solution is split into two portions. Hence the latter contacting operation need never be employed. The above conclusion may be valid in general and may apply to systems having equilibrium relationship other than the ones considered in this study.
to give the desired final composition. Except for a few very specific cases studied (Sanders, 1928)) these two types of crossflow contacting operations have not been compared either with each other, or with countercurrent operation. The various contacting operations studied here are shown schematically in Figure 1. Two different adsorption equilibrium curves have been considered, the Freundlich isotherm [written using the symbols of Treybal (1955) but in dimensionless form] Y = xn (1) and the Koble-Corrigan isotherm
The Koble-Corrigan isotherm was selected for study because the curve contains other equilibrium curves as limiting cases. For example, when m = 0, Equation 2 reduces to Equation 1, and when n = 1, the Langmuir isotherm results. Further, by proper choice of parameters m and n, Equation 2 can be made to take on shapes similar to that of the Brunauer-EmmettTeller isotherm. Calculations were made on the university's digital computer for the two-stage split-adsorbent crossflow case and the counter-
current case, using a trial and error technique embodying the Newton-Raphson method. The two-stage split-solution crossflow case was treated as a problem in constrained optimization. For this purpose, the modified pattern search technique of Weisman, Wood, and Rivlin (1965) was employed. Only minor changes had to be made in the published program of these authors for use on the computer. No computational difficulties were experienced. The initial dimensionless fluid-phase solute concentration, Y o ,is always 1.0 in comparing the four contacting operations. The calculations, presented herein, determined the minimum total pounds of adsorbent required per pound of solution to reduce the dimensionless fluid-phase solute concentration to a final value of one tenth the initial value-Le., single-stage: Y1 = 0.1, two-stage split adsorbent: Yz = 0.1, two-stage split solution: Y = 0.1, and countercurrent: Yz = 0.1. Theinitial dimensionless solid-phase concentration, X,, was taken to be zero in all cases. Results for systems obeying the Freundlich isotherm are shown in Figure 2 for a range of values of parameter n. Figure 3 presents some of the results for systems obeying the Koble300 -
t
2 00-
100: C
.-+0
50-
-0 3
30-
m
n -
20-
\
+
c
0)
IO
L
c
0 VI -0 0
5-
4
3-
2-
0.2
0.4
1.0
2.0
4.0
n
Figure 1. Stagewise adsorption operations considered in this study a. b. C.
d.
Single-stage Two-stage crossflow, split adsorbent feed Two-stage crossflow, split solution feed Two-stage countercurrent flow
Figure 2. Adsorbent requirement for four operations using Freundlich isotherm and final dimensionless fluid-phase solute concentration = 0.1 0 8
0
Single-stage Two-stage crossflow, split adsorbent feed Two-stage crossflow, split solution feed Two-stage countercurrent flow
VOL. 6
NO. 2
M A Y 1967
309
E
c
1
.-0 c
1
The conclusions drawn from these calculations for adsorption operations may be extended to other mass transfer operations, such as distillation and gas absorption. Acknowledgment
Financial support by the National Research Council of Canada is greatly appreciated. Nomenclature
i
-3 I '
'
"
1.3
1.4
1.5
1.6
I,? 1.8
1.9
2.0 2.1
m Figure 3. Adsorbent requirement for four operations using Koble-Corrigan isotherm and final dimensionless fluid-phase solute concentration =
0.1 Bracketed numbers. Values of parameter n Single-stage @ Two-stage counterflow, split adsorbent feed 0 Two-stage crossflow, split solution feed ( I Two-stage countercurrent flow
0
Corrigan isotherm. In the latter case, several values of the parameter n (0.2, 0.4, 0.6) and various values of the second parameter m are shown. These graphs confirm the belief that countercurrent operation is always superior to crossflow operation. Of greater interest, perhaps, is the fact that crossflow operation with the adsorbent being split is always better than the alternative crossflow operation in which the solution is split. This suggests that the latter method should never be employed. Conclusions similar to the above were obtained for other values (other than 0.1) of the final dimensionless solute concentration in the fluid phase. Although this may be difficult to prove in general, any rearrangement of the relative amount of adsorbent required for each operation for systems obeying other adsorption isotherms is doubtful.
G,
solute-free solvent, lb. solute-free adsorbent, lb. parameter in Koble-Corrigan isotherm m n parameter in Freundlich and Koble-Corrigan isotherms X = solid-phase concentration, lb. solute adsorbed/lb. adsorbent Xrer= solute concentration in solid phase which is in equilibrium with Yo,lb. solute adsorbed/lb. adsorbent X = X/Xler,dimensionless solid-phase concentration Y = concentration of solution in fluid phase, lb. solute/lb. solvent Y = Y/Y,, dimensionless fluid-phase concentration
L,
= = = =
SUBSCRIPTS 0 = initial 1 = stage 1 2 = stage 2 Literature Cited
Sanders, M. T., Ind. Eng. Chem. 20,791-4 (1928).
Trevbal. R. E., "Mass Transfer Operations."
DD.
472-6. McGraw-
ew York, 1955.
R. G. LERCH D. A. RATKOWSKY
University of British Columbia Vancouver, Canada RECEIVED for review June 21, 1966 ACCEPTED December 10, 1966 21st Annual Northwest Regional Meeting, American Chemical Society, Vancouver, B. C . , June 16-17, 1966.
LOW TEMPERATURE THERMODYNAMIC FUNCTIONS FOR n-HALOGENATED HYDROCARBONS -
-
The low temperature (298.1 6' to 100' K.) thermodynamic functions [Cpo, So, (H' Ho')/T, - (F" Ho')/ T ] are calculated for n-halogenated hydrocarbons, n-C,H2,+1X (X = F, CI, Br, I) in the ideal gas state a t 1 - a h . pressure. The functions for the three lower members (C1-3) of the series were determined by statistical mechanical means treating the restricted internal rotational contribution by the approximate LielmezsBondi method. The values for the higher members of the series were calculated by the linear incremental method using calculated low temperature methylene group contributions, The agreement between the computed results and the few available experimental values is satisfactory.
Morgan and Lielmezs (1965) presented calculated thermodynamic functions of the n-halogenated hydrocarbons in the ideal gas state a t 1-atm. pressure, from 800" to 298.16' K. As several of these hydrocarbons are used as refrigerants, especially the lower number compounds, extension of these calculations down to 100' K. appeared worthwhile. The internal rotational contribution was determined by the Lielmezs-Bondi (1958, 1965) method. The calculations, using the structural and spectroscopic data as suggested ECENTLY
310
l&EC FUNDAMENTALS
by Morgan and Lielmezs (1965) (in this calculation the rotational isomerization energy for n-propyl fluoride is assumed to be -200 cal. per gram-mole " K.), were performed by means of a digital computer program (Cross, 1966) for each successive member in the given series from methyl to n-propyl ( C I - ~ ) , inclusive (Table I). Utilizing the calculated, low temperature, incremental values of the methylene group (Gelles and Pitzer, 1953; Kennedy, 1966; Person and Pimentel, 1953; Pitzer, 1940; Thomson, 1954) (Table 111), the
