where A i , Bi,M i , and mi are large numbers, arbitrarily chosen. This penalty function becomes large when the yi terms approach the limits and is small otherwise. For either of these two possibilities we may treat the problem as continuous and employ differentiation, thus taking full advantage of continuity. The problem formulation recommended by Converse does not solve the problem proposed by me but rather a "continuous" approximation of it. His proposal avoids the discretization process and therefore yields a n approximation to the correct anscer. I feel that many problems are inherently discrete and for this reason methods should be developed for handling them. I appreciate Professor Converse's careful study of my paper and his constructive comments.
\Ve may give a geometric interpretation to the problem in the following manner ( 7 ) : Over the three time stages the objecti\ e function may be \\ritten as :
+ + ~ 2 '
~ 3 '
-
3s
- 2 ~ 3- ~2
+ 300
(1)
\Ye desire both to maximize and minimize J subject to the 3 % terms. taking on only the discrete values ~1
=
=
(2)
=
3.1'
f (y!
= J
+ 3s - 298.75
+ + JZ
JJ
= 18
- +)'
+
(j3-
1)* - 3s - 298.75
(5)
or as
where C = J
+ 3s - 298.75
(71
C is a constant to be determined for the maximization and minimization problems. Equation 6 is the equation of a sphere of radius d ? n . i t h theoriginatjl = 0 , j ?=
1 2 ?
-j3
=
1
O u r problem may now be interpreted as finding the point or
dc
tangency between the sphere of radius and the plane y1 y2 j 3 = 18, subject to the permissible discrete values for y i given in Equation 2. Itre may visualize the solution space as consisting of concentric spheres and tangent planes to them. For some set ofy2values \Ye desire to find the largest and smallest sphere that satisfies all the constraints. The selection of values for j l , j 2 :j 3 determines the radius of 3
yi.
the sphere and of course determines the value of
The
i=l
= sphere of minimum radius, minimum of J f o r 5' = 0 and j 1 = 6,
jvp =
6>y3 = 6.
The
l / c = d 1 3 1 . 2 5 , corresponds to
=
S = 0 and
j1
= 10,
32
=
6,
y3
2.
(3)
l y e may rearrange J by completing the squares. so J no\* appears as J
+)*+ (y8 - 1)'
sphere of maximum radius, the maximum value of J for
2, 4, 6
a n d subject to the constraint that ~i
-
de d91.25corresponds . to the
2, 4, 6, 8, 10
y2 = 2, 4, 6. 8. 10 ~3
+ (y2
+ +
Geometric Interpretation
J = yi2
ylz
(4)
.or as
literature Cited
(1) Bellman, R., "-idaptive Control Processes," pp. 106-9, Princeton L-niversity Press, Princeton, N. J., 1961. S. M . Roberts
International Business .Wachtnes Cork. Houston 7 7, Tex.
CORRESPONDENCE A MACROSCOPIC APPROACH TO IRREVERSIBLE T H E R M O D Y N A M I C S
SIR: This article by C. M. Sliepcevich and D. Finn [IND. ENG.CHEM.FUXDAMENTALS 2 , 249 (1963)]> is an attempt to derive the Onsager reciprocal relations from classical thermodynamics alone-i.e., icithout recourse to statistical or microscopic reversibility considerations. However, there appears to be an arbitrary definition in the course of their development tyhich cannot be justified without recourse to considerations outside classical thermodynamics. The first portion of the paper is concerned \cith the justification of the validity of a quadratic form in the state variables for the rate of change of lost work (Equation 16 or 1 6 n ) . This portion of the paper seems to be quite proper and convincing.
At this point in the development it is necessary to define the forces, Xi, and the fluxes, Ji,such that
Sliepcevich and Finn choose to define the J , in accordance Icith their Equations 17 and 18, but any number of other choices seem to be equally acceptable unless some reason can be given for preferring one such choice above all the rest. To cite one possible alternate choice J1 and J S could be defined as follows:
(3)
VOL. 2 NO. 4 N O V E M B E R 1 9 6 3
325
These definitions still give the necessary quadratic form for the rate of change of lost work, but the resulting matrix of phenomenological coefficients is no longer symmetric. Classical thermodynamics appears to offer no reason to prefer the choice of Sliepcevich and Finn rather than Equations 2 and 3 above. To put the same argument in physical terms, there seems to be no way of knowing, from classical thermodynamics only, whether the heat and mass fluxes are dependent upon the temperature and potential gradients in accordance with Equations 2 and 3 above or whether Equations 17 and 18 of the subject paper are more realistic.
SIR: Mixon’s principal objection seems to center around the point that the J’s, as given by Equations 17 and 18, cannot be defined uniquely without recourse to microscopic thermodynamics and Onsager’s theorem of microscopic reversibility. As clearly stated in the paper: “Although his (Onsager) theory is based upon assumptions regarding the microscopic behavior of physical systems, the result is macroscopic in that it conforms to limited experimental evidence.” Classical (macroscopic) thermodynamics is a science based on empirical observations on the macroscopic scale. In this respect, all thermodynamic functions, including the J’s, are defined such that they conform to experimental observations. There is nothing (as yet) in the theorem of microscopic reversibility which in itself affords a priori the basis for selection of macroscopic thermodynamic variables; like all of statistical thermodynamics, it must continously seek confirmation on the macroscopic level. Mixon’s argument is based on the mathematical fact that from the quadratic equation, 15 = L I ~ AT2 * f (LI,*
+ L21*) ( A T + ) + L ~ z (AM)* *
(21)
Of course, the Onsager theory provides a basis for the selection of the definitions adopted by Sliepcevich and Finn; such a basis seems to constitute the contribution of microscopic reversibility over and above classical thermodynamics to the derivation of the reciprocal relations. It is suggested that some justification is needed for the choice of definitions indicated by Equations 17 and 18. Forest 0 . Mixon Research Triangle Institute Durham, N . C.
and
so that
Indeed since lw was postulated as a continuous function of two arbitrary, independent variables, one might expect that the final form of the equation would be symmetric. One of the most important contributions of the paper is that the J’s and L’s have for the first time been defined precisely in terms of partial derivatives of well-established, macroscopic thermodynamic variables as given by Equations 16, 16a, b , and c. Furthermore-as noted briefly in the original articleif one accepts as a first approximation that Q = C‘AIT and J? = K . 4 4 ~then by comparing Equations 10 and 21, it follow^ that
it is not possible to define the J’s uniquely such that
I C = [ J I ]A T
+ [ J : ] Ap
(19)
+ L I ? *AM]
(17)
+
(18)
where [ L I I *A T
JI
=
J2
= [L2?’ Ap
and Lll*
AT]
so that
In general, Equation 21 can be expressed in matrix form, fw
=
XTL*X
=
IJX
(a)
for which there are many matrices, L , and vectors, J , which satisfy the above relations. However, the J’s as defined by Equations 17 and 18 do not merely represent an arbitrary grouping of the terms in Equation 16 or 16a.
so that they will have the same functional form as those proposed by Onsager. In fact the terms in the brackets on the right-hand side of these equations are clearly and uniquely defined by the Equations 14, 15, 17, and 18 from which it is obvious that 326
l&EC FUNDAMENTALS
and J?
=
( K A ) Ap
+ ( siT K)A
AT
(= + 2 $)
(2)
Therefore, it is misleading to continue to refer to J1 as the heat flux, ( = Q) [refer to the last paragraph preceding the conclusions in the paper] and JQ as the mass flux As a final point, the alternative definitions for the J’s as tentatively proposed by Mixon are not incorrect in the sense that when evaluated properly according to his definitions they \vi11 give the correct value for 1w. However, it does not follow that the L21, as originally defined in the paper, becomes zero. T o avoid confusion, it would be better to denote these alternative definitions of J’s and L’s by some other symbols.
(=a).
C. M . Sliepcevich Don Finn Hadi Hashemi Michael Heyrnann Cnwersitj of Oklahoma .Torman. Okla.
