semi-infinite extent, everywhere a t the melting temperature. T,, Lvhich has been in contact with a porous plate of temperature T, T,, sufficiently long for a steady state to have been established by the imposition of a constant suction. The dimensionless heat equation in the melt may then be written
>
0" = -BO'
T 7',T 'r, uL x
X 6
(1)
pL
u i t h boundary conditions: 0 ( 0 ) = 1 ; O(1) = 0; 0 ' ( l ) = -AB. T h e first t\vo equations represent the temperature conditions a t the melt surface. and the last the energy conservation requirement in the melting process. T h e solution is immediate:
.I
' - x0 )=' x( [
X + l
- I]
where
T h e dimensionless temperature distribution and the Peclet number are thus uniquely determined by the thermal conditions. but the melt thickness is inversely proportional to the suction velocity for a given Peclet number. and hence to the pressure difference across the porous plate. Finally, the assumption of equal phase densities has not been made. so that the analysis applies equallv \vel1 to vaporization providing the thermal variations in vapor density can be neglected. T h e same solution applies even when 7', < 7,. Nomenclature
X
0
temperature melting temperature \\.all temperature suction velocity = distance from plate = dimensionless distance, ,Y '6 = melt thickness = melt densit?. = latent heat parameter, .i,'CL(T, - T,) = latent heat of melting = ( 7 ' - 7-J # ( T W - T,) = = = =
literature Cited (1) Bankoff. S. G.. .-1.I.Ch.E. J . 7 , 485 (1961). (2) Bankoff, S. G., "Advances in Chemical Engineering." Val. V , T. B. Drew et al.. eds.. Academic Press. New York. i n press. (3) Carslaw. H. S.. .Jaeger, .J. C.. ,'Conduction of Heat in Solids." 2nd ed.. Oxford Lniv. Press, London, 1959. i..I..Bankoff, S. G.; J . Heat Transfer. to be pub(4) Frankel, i
lished. (5) Hamill. T. D.. Bankoff. S. G.. A.1.Ch.E. J . 9, '41 (1963). Bankoff. s. G.. IND. I;NG. CHEM.F U N D A \ I E N T A L S (6) Hamill. .r. n.. 3, 177 (1964). (7) .Jain. K. C.. Bankoff. S . G.. J . Heat Transfer, to be published. (8) Pai. V. K.. Rankoff, S . G.. A.1.Ch.E. J . ! to be published. (9) \Vayner. P. C.. Jr.. Bankoff, S. G.! Ibid., to be published.
T. D. HAMILL Lhpartnient of Chemical Engineering .Veert, )-ark 1 Tniwrszty .Vert. York, .V. I?.
S. G. BANKOFF Department of Chemical Engineering .\'orthwestern Cniuersity Evanston, 111.
B-1 = Peclet number, p1,CLvL6/k C L = liquid specific heat k
=
RECEIVED for review July 21. 1964 ACCEPTED October 6, 1964
liquid thermal conductivity
COM M UN I C A T I ON
CONTROLLED CYCLE OPERATIONS Considerable improvement in the performance of process equipment can b e obtained b y operating this equipment in a transient manner as opposed to the usual steady-state operation. O r d e r .of magnitude increases in throughput have been observed and 100% increases in efficiency are possible. The operation of equipment in the following categories is discussed: extraction, distillation, absorption, screening, crystallization, fluid b e d reactors, heat transfer, and electrolysis.
w
H A V E OBSERVED improved performance of process equipment \\.hen it is operated in controlled, repeating, nonsteady st3tes. The improvement involves throughput. efficiency, and control. and the magnitude of the improvement is more than just a fe\v per cent. One hundred per cent increases in efficiency are possible a n d order of magnitude increases in throughput have been observed. Control is improved since the higher throughputs result in considerably shorter delay times. T h e principle of this cyclic operation \vas first reported by Cannon ( 2 ) . Since the appearance of this report, a dozen other papers and theses have reported the effects of controlled
108
E
I&EC FUNDAMENTALS
cycle operation. These workers describe the operation of equipment in the folloiving categories: extraction. distillation, absorption, screening, crystallization. fluid bed reactors! heat transfer, and electrolysis. SVe have done work in the areas of distillation and extraction, the most widely used separation methods. Controlled cycling is more easily applied to extraction than to any other unit operation. Figure 1 sho\vs schematically an extractor for the purification of chlorinated phenols. T h e column was built of 6-inch sections of 1-inch borosilicate glass pipe with tainless steel perforated plates flanged between the sections. Air pressure
on the feed tanks \vas used to force material through the column. The control valves were electrically operated solenoid valves and the sequencing \vas completely automatic. l'his sequence \vas : heeavy phase feed, coalesce, light phase feeed. water feed. coalesce, and repeat. T h e heavy phase was the mixture of chlorinated phenols and the light phase was aqueous sodium hydroxide. Extraction is an operation where order of magnitude throughput increases and improved efficiencies have been observed ( 9 ) . T h e only type of extraction unit that approaches a cycled extractor in terins of capacity and efficiency is a rotating disk contactor. T h e cycled extractor has the higher capacity of the t\vo but has a somewhat lower efficiency. O u r particular application illustrates a further and very real advantage of controlled cycling. Sodium chlorophenate soaps tend to emulsify badly-so badly in fact. that it is virtually impossible to run a conventional extractor because of emulsion formation. The cycled extractor gives one the ability to control the agitation level and thereby also eliminate the emulsion problem. 'The next unit operation that \vas investigated in the controlled cycle mode was distillation. T h e laboratory units were made of glass pipe sections all 11;2 inches in diameter. T h e first column \vas a five-stage batch vacuum still. and again chlorinated phenols were used as a test system. With this unit a five times increase in throughput over conventional operation was demonstrated. '4packed column was also' tested; a three times increase in throughput \vas demonstrated. A 28-stage continuous fractionator .using live steam as a heat source was built and operated satisfactorily. T h e continuous column was used for drying lo\\ boiling organic solvents. T h e water content was reduced to a fraction of a per cent and, most important, there \rere no apparent hydrodynamic delays in the liquid portion of the cycle. All of the plate-type columns employed perforated trays. Csing this preliminary information as a base, a semiplant scale column was designed, built, and operated successfully. Increased throughput similar to that observed in laboratory units was obtained. Nothing was observed that would cast doubt on the usefulness of the concept. Extensive simulation of the distillation columns using both analog and digital computers has been done. T h e model
used to describe the controlled cycle column is quite simple. Figure 2 shows a column section during the vapor flow period. For this period, the material balance is
During the liquid flow period, we have
x, = x,,, This expression can be modified so that a fraction of a plate can be dropped and mixing can occur. As it stands, it is for the ideal case of plug flow and complete dumping of the trays. T h e equations have been solved using a digital computer. T h e results of the simulation show that? on the average, the mass transfer driving force is about double that for conventional operation. This is the reason for the improved efficiency of cycled operation. T h e simulation technique is well worked out, and the separation to be obtained by operating any distillation column in cyclical fashion can be predicted. Calculated performance has been shown to agree with actual operation for both plate and packed columns. Further simulation has been done to try to determine the transient pressure response of the system. No predictions have been made about the throughput. This is strictly an experimental fact a t this stage. Some recent \vork on absorption has been reported (70). This is quite similar to distillation, and the work deals mainly \vith observed separation improvements. These reports show an increase in throughput of three times with a simultaneous improvement in separation. T h e application of controlled cycling to screening utilizes an on-off air blast through the screen in place of the usual shake and hammer operation (7). T h e air blast is sufficient to fluidize the bed which drops onto the screen when the air is turned off. T h e throughput is increased two to three times, I n addition, blinding and adhesion problems are very much decreased. This is especially true for solids mixtures which have a high percentage of near mesh sizes and are difficult to screen. T h e use of an air blast is simpler than other mechanical methods of shaking.
1 " '
PHASE FEED
I
I
V
Yn
? I -
n t h PLATE
H
LIGHT PHASE FEED
WATER
u
HEAVY h
Figure 2. VOL. 4
Vapor flow period NO. 1
FEBRUARY 1965
109
A pulsed column crystallizer was developed for p-xylene purification and showed appreciably higher capacity and efficiency when compared to conventional units (6). I n 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 u p 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 \\-as applied 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. I n 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
