A pulsed column crystallizer was developed for p-xylene purification and showed appreciably higher capacity and efficiency when compared to conventional units (6). In addition the crystallizer \\-as fairly simple and said to be easy to operate. This unit had solid and liquid phases moving countercurrently and had only a single stage. T h e application of cycling to fluid bed reactions \vas reported to have two beneficial effects (5). T h e first was that the particle size range that could be handled was extended from a ratio of about 5 to 1 to the handling of everything from fine po\vders to coarse metal chips from machining operations. In addition. flou. rates up to 10 times those available in steady flow systems were achieved. T h e pulses were timed so that the bed settled benveen pulses to the position it \could assume during steady-state f l o ~ v . .A recent report presents laboratory data for the production of butadiene by dehydrogenation (8). By operating the reactor \vith reactant butenes pulsed into a diluent stream, conversion to butadiene far in excess of equilibrium can be achieved. Furthermore, acceptable conversions Ivere achieved a t temperatures \\.here no measurable quantity of butadiene \vould be produced under conventional steady floiv conditions. T h e application of cycling to heat transfer \vas made to \velding operations ( 7 ) and to film boiling ( ~ 3 ) . \Velding is a particularly complicated heat transfer operation and as such makes a n interesting study. T h e cycling nature of the operation is some\vhat different from that described previously. 'T\vo po\ver sources \\-ere. used; one provided background current a t a low enough level not to fuse the metal while a second pulsed unit provided current in shots with which the \veld \vas made. T h e operation is especially suited to joining thin sheets of metal \\.here the problems of maintaining an arc and not burning the metal a t the same time are eliminated.
Up to 1007, increases in heat transfer rate were observed when cycling \\-asapplied to stable film boiling. There is some information available Ivhich indicates that fuel cell performance can be improved by cycling ( d ) . T h e improvement is primarily a n increase in lifetime of the elec; trodes a t high current densities. Nomenclature
H
=
holdup
n = stage number t = time V
=
vapor flow rate
X = mole fraction in liquid Y = mole fraction in vapor Literature Cited
(1) British LVelding Research Association, Steel 153, No. 8. 52
(1964).
(2) Cannon, M. K.. Znd. Eng. Chem. 53, 629 (1961). (3) DiCicco, D. A , . Schoenhels. R. J.. J . Heat Transfer 86, 457
(1964). (4) Interagency .idvancrd PoLver Group. Project Brief. Contract NAS3-2752, August 1963. (5) Levey, R. P.. Heidt. H. M.. Hamrin. C. M.. J r . . 53rd National Meeting, X.1.Ch.E.. Pittsburgh. Pa.. May 1964. (6) Marbvil. S. J., Kolner. S. J.. Chem. En!. Progr. 59, N o . 2, 60 (1963). (7) Robertson. D. C.. M. S. thesis. T h e Pennsylvania State University, University Park. Pa.. 1957. (8) Semenenko, E. I.. Rojinskii. S. Z . , Kinrtika i Katiiiz 5, No. 3. 490 (1964). (9) Szabo. T. T.. Lloyd, LV. A , Cannon, hf. R.. Speaker. S. S.. Chem. En?. Progr. 60, NO. 1, 66 (1964). (10) Ziolkewski. Z.. Filip, S., Intern. Chem. Eng. 3, 433 (1963).
\'ERLE N . SCHRODT Monsanto Co. St. Louis, .Wo. RECEIVED for review September 23, 1964 ACCEPTED December 3. 1964
CORRESPONDENCE DISCRETE MAX I M UM PR I NC I PLE SIR: T h e maximum principle of Pontryagin ( 7 7 ) is now a well known method of dealing with a wide class of extrema1 problems associated \vith the solution of ordinary differential equations with given initial conditions. In a particularly lucid exposition of this principle. Rozonoer (72) has pointed out that. although one might hypothesize a discrete analog of the maximum principle for difference equations rather than differential equations, such a result is invalid except in certain very special conditions \vhich render it almost trivial. Nevertheless. in three recent papers (8,-70), Katz has presented a proof of a discrete maximum principle around \vhich a significant amount of \vork-some already published (7.-3, 73. 71) and some still in press-is beginning to build up. HoLvever. the purpose of this note is to reaffirm, largely by means of simple counterexamples. Rozonoer's original statement that the dixrete maximum principle is invalid. and to show that the "proof" given of it is fallacious. Firstly. we will briefly recapitulate Katz's results. H e considers a system of difference equations of the form r I n = Fl"(ui:'"-l.e"): ( i = 1, 2.
,
Lvith given initial conditions 110
l&EC FUNDAMENTALS
,S); (n
=
I , 2. .
xio = a t ;
(2
=
1. 2: . . . S )
(2)
I t is customary and convenient to regard each Equation 1 as describing the relation bet\veen outputs and inputs for some physical unit. so that the complete set of equations describes the behavior of a sequential chain of units as shown in Figure 1. I t is then required to find those values of the variables 8'. Bz. . .e.' Lvhich minimize (maximize) the value of ~1.'. T h e proposed solution makes use of the solutions t i nof a set of difference equations adjoined to Equations 1-namely
(3)
.Y) (1) Figure 1 .
Sequential chain
I\
H' can then be written down by substituting into Equation 5, giving
ith the terminal conditions
zi'
= =
I ; forz = 1 0 othentise
(4)
I t i, then assrrtrd that each On should be chosen to minimize (maximize) the corresponding quantity s H" = zlnFjn (3
where
1'1
\kith the zl"regarded as Constants from the point of view of the tninimization (maximization) process. In Kaiz'.; formulation of the problem. the functions FIn are ajsurnrd to be the samis for each value of n and are written F,. I-Io\\-ever. this restriction is not vital to the argument, and Fan and LVang (1.3 ) derive the same result without any such assllmption. Indecd it has no Eearing on the validity or Some dortbt may be thrown on the correctness of the above algorithm by the ver:: simple example shown in Figure 2, \\-here the objrctive is to maximize t 2 . Direct elimination of x1 sho\cs that ';2
=
.I-
(x0
+
- (0*)2
O'j2
\vith a stationary maximum value a t 0' = - .Y 0 > O2 = 0:which also gives the largest value for any choice of 8' and 02. Hoivever, F' is linear in the adjustable variable O'? so it is not possible to determine a value for 0' by the condition that
H'
=
const. x1
should be maximized tvith respect to 01, as \\-odd be required by the computational procedure suggested by Katz ( 8 ) . Althou5h this may cast some doubt on the result, it is easy to see that 9 = 0 for the optimal policy, so that H' is actually indeprndent of 8' and is. in a sense. maximized for any value of 0'. T o obtain a completely unambiguous counterexample, therefore. one must take S = 2. corresponding to a two-dimensional system. Consider. for example, the system shown in Figure 3, \\-here the problem is to minimize x 1 2with respect to 0' and 02. By direct elimination of x l l and x p i , it is easily shown that
\\-ith a unique stationary minimum at 0' = P = 0, which also This is, therefore, the solution gives the smallest value of of the problem. LVe now pursue Katz's proposed procedure, solving the adjoined equations backward along the chain, starting from the boundary conditions. zl*
= 1,
2?2 =
0
using the value of 21' found above. I t is seen that b2H1/b(Ot)2 is negative for all values of O', so it follow that H' can never be minimized with respect to O', as would be required by Katz's principle expressed in Equation 5. Indeed the values 0' = O2 = 0 and the corresponding solutions x l l = x2' = 1, zll = 1, 22' = 2. ivhich have been shown by direct calculation (Equation 6) to minimize . x l Z 3 actually maximize H' in direct contradiction of Katz's algorithm. In this simple counterexample the functions Fin are different for different values of n. I t remains to demonstrate the truth of the assertion made above that Katz's result remains invalid even if all the functions Fin are the same, as in his original derivation. We shall d o this by showing that, from any S dimensional counterexample, it is possible to generate a n (S 1) dimensional counterexample in iyhich all the functions Ftn are the same. Consider then a n example in S dimensions with state variables
+
xin
(i
=
1: 2, . .S); ( n
=
1 2
,
. .'I.)
and recurrence relations x i n = F Z n ( x k n p 10'). . \\-ith boundary conditions x i o given, and suppose that this contradicts Katz's result in the same way as the example just quoted, and is therefore a counterexample. Let us call it Example I. Now introduce a second example (Example 11) with S 1 dimensions and recurrence relations
+
with the functions G i independent of n. T h e functions G i of any S 1 variables, El, Er. . . E S + l are defined by the following relations:
+
.v
Gi(El. .
.Es~I)
G S + I ( ~.E.s+1) I.
=
+m(ELy+l)Fim(Ei>
m=l
= Es-i
. . .Es, 0 ) ;
(i
=
1
1. , . S )
+1
(8) where the functions $m have the following properties
According to Equation 3: we then have (i) (ii) (iii)
and
1 ( x = integer = rn) (.r = integer # m and 1 0 $m(.r) arbitrary for other values of x . +m(x) = &(x)
=
< < N) x
1
(9) There are many such sets of functions-for
I Figure 2.
example,
e2
One-dimensional example
Figure 3.
Two-dimensional example
VOL. 4
NO. 1
FEBRUARY
1965
111
$n(.T)
(x -
stationary values of the objective function and of the functions
=
l ) ( x - 2) . .
,
(.r
-
m
( m - l)(n - 2) . . . (rn - m
+ l ) ( x m 1 ) . . . ( x - 2V) + l ) ( m - m - 1 ) . . .(m - .\-) -
-
T h e boundary conditions for the recurrence relations (Equation 7) are . x t o , the same as those given for Example I
(i = 1 , 2, and
X&10
=
I.
.S)
..
1
(10)
Putting E l . . . Es+l equal to x l n - l , . . . xSLln--l in Equations 8 and substituting into Equation 7 : it follows on taking account of Equation 9 that only one term in the sum over n retains a nonzero value-namely. the term m = n-so we have xin
= Fin(Xkn-l,
x&ln
=
Ys+1
n-l+
e“);
(i
=
1. 2:
,
,
.S) (11)
1
Thus the variables sin(i = 1. .S) take the same values in Example 11 as in Example I . I t follows that identical values of 81: 02. . in the t\\-o examples give identical values of XI.’’, so the same set of values of the 8’s minimizes (maximizes) xl-v in both cases. Similarly it can be shown that the adjoined variables z i n (i = 1. 2, . .S) are identical in Examples I and 11: so the function ,
,
,
,
S+I
=
C ]=I
S
zlnGJ = ]=I
+
tJnFJn
Zg+inGS+i
for Example I1 differs from the function S
HIn
tlnFln
= /=1
for Example I only by the term zS+lnGS+l,which is independent of the adjustable variables 6”, 8*,, ,e”. Thus, if a set of values of e’, . , .e‘‘ which minimizes xy” also maximizes some H n in Example I, thus contradicting Katz‘s result, the same will be true in Example 11. Example I1 is therefore a counterexample if Example I is, and furthermore Example I1 has the same recurrence relations at all stages. thus proving our assertion. Though Katz’s discrete maximum principle is false. as has no\v been demonstrated. what he refers to as his “weaker algorithm” is true. This states merely that takes a stationary value with respect to variations in e‘, P , . , e.’ if and only if each of the functions H“ takes a stationary value. and makes no comment on the relation between the natures of these stationary values. This result was earlier derived and used by the first of the present writers ( 4 ) . T h e extension of Katz’s results by Fan and LVang (2, 3) to systems topologically more complex than a simple sequential chain is also false. but once again a weaker algorithm relating stationary values is true. and was published by the second of the present writers (6,7). T h e fallacy in the proofs given by Katz and by Fan and TVang lies in the attempt to deduce the natures of stationary values from a consideration of first-order variations only. and the results obtained are simply consequences of a confusion in orders of magnitude of small quantities. T h e natures of
112
l&EC
FUNDAMENTALS
derivatives \vith respect to the variables 8’. 02. . . .9-‘. Indeed, there is no difficulty in writing do\vn the Hessian for variations of x i s and hence deducing correctly the nature of a stationary value of t1,’, as is sho\vn in more detail in another publication (5).in \\rhich we also investigate certain‘special circumstances in which a stronger result is true. \-cry briefly. \ve may enumerate these cases here. I n order that x;’ should take a stationary minimum (maximum) value with respect to variations in 8’. . .e.’. it is necessary and sufficient that each function H n should take a stationary minimum (maximum) value with respect to the same variables in the follo\ving circumstances. .4. \Vhen the functions F i n ( , t k n - l . O n ) take the special form ,
,
@”
HIIn
H“ are determined, of course. by Hessian matrices of second
\\.here the are constants. This is the case quoted by Rozonoer ( 7 2 ) . B. \\’hen S = 1. in other ri-ords \vhen there is only one xvariable at each stage [though there are exceptions in this case. as is discussed elsewhere (.5)]. The condition is also necessary. but not sufficient. if the functions F i n are linear in the variables ~ k ’ ‘ - ~ , but the coefficients may depend on the 0’s. These results refer to relations betiveen local minima (maxima) of ,11-’- and local minima (maxima) of the functions H n . Pontryagin’s principle is a more powerful result relating the absolute minimum (maximum) of ~1.’- \vith absolute minima (maxima) of the Hn. and this is valid only in the case A above. as was asserted by Rozonoer.
literature Cited
(1) Fan. L. T.. ”Optimization of Multistage Heat Exchanger System by the Discrete Maximum Principle,” Kansas State Univ., Eng. Expt. Sta.. Spec. Rept., 1964. (2) Fan, L. T.. LVang. C . S.. J . Electron. Control 16, 441 (1964). (3) Fan; L. T.. LVang. C . S.. J . Soc. Ind. Appl. .Math. 12, 226 (1 964). (4) Horn, F.. Chem. Eng. Sci. 15, 176 (1961). (5) Horn, F.. Jackson, R.: J.Electron. Control (in press). (6) Jackson. R.. Chem. Eng. Sci. 19, 19 (1964). (7) Ibin’.,p. 253. (8) Katz. s.: IND.ENG.CHEM. FUNDAMENTALS 1, 226 (1962). (9) Katz, S., J.Electron. Control 13, 179 (1962). (10) Ibid.. 16, 189 (1964). (11) Pontryagin. L. S., L‘spekhi. .Mat. .L’auk 14, 3 (1959). (12) Rozonoer. L. I., Automation 3 Remote Control 20, 1288, 1405, 1.5-1.7 11059) i - -’,. -
(13) it’ang. C . s.. Fan. L. T.. IND.EXG.CHEM.FUXDAMESTALS3, 38 (1964). (14) Zahradnik. R. L.; Archer, D. H.. Ibid., 2, 238 (1963).
F. Horn Imperial College oj Science C8 Technolog) Prince Consort R d . South Kensington. London, E n g l a n d
R Jackson L.niLersit) o j Edinburgh and Heriot- T l h t t College Chambers St Edinburgh. Scotland
