A theoretical analysis demonstrates the advantages of cycled mass transfer operations in liquid extraction, in which only one phase jlows at any given time and where the phases use same interstage jlow passages during their respectiue jlow periods ontrolled cycling, which should not, for reasons to become obvious later, be confused with pulsation, is a relatively new concept, having been advanced by M. R. Cannon about 15 years ago (2, 3). Basically, controlled cycling can he described as a mode of operating a countercurrent stagewise mass transfer apparatus such that only one phase flows at any given time and the phases use the same interstage flow passages during their respective flow periods. For example, a cycled distillation or gas absorption column has an operating cycle consisting of two parts: a vapor flow period, when vapor flows upward through the column and liquid remains stationary on each plate, and a liquid flow period, when no vapor flows and liquid drains from plate to plate. Times for each flow period are on the order of 1 to 10 seconds, which illustrates the basic difference between pulsation and cycling. Further differences will be apparent later. The cycle sequence involved in liquid extraction is more complex, as there are coalescing periods following each phase flow period to allow phase separation. Previous studies have amply demonstrated that the performance of cycled apparatus routinely exceeds that of dimensionally similar conventional equipment. Studies illustrating these results are available in the literature for distillation (7,6-8,72-75) and liquid-liquid extraction (77, 79-27). However, in none of these studies was a rational explanation of the enhanced performance offered. It is the purpose of this paper to present a theoretical analysis which will demonstrate the advantages of cycled mass transfer operations. Several recent papers have dealt with controlled cycling in a reasonably fundamental manner. McWhirter (73, 7 5 ) first recognized that, while over a number of cycles a cycled column appears to have a steady-state mode of operation, for any single cycle the operation is definitely unsteady-state. For instance, during the vapor flow period in a distillation column, the composition of the liquid on any tray varies with time, and thus the composition of vapor leaving this stage and contacting liquid on the stage above also varies with time. McWhirter applied these ideas in developing computer simulations of cycled and conventional columns, solving the unsteady-state material balance equations which describe compositions as a function of time by finite difference techniques. This analysis demonstrated that the enhanced performance of cycled apparatus was a direct result of the timevariant composition gradients existing on each plate, and
C
22
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
ANALYSIS OF CONTROLLED CYCLING MASS TRANSFER OPERATIONS R. G. ROBINSON AND A. J. ENGEL provided the first available method for predicting the performance of cycled distillation columns. Recently, analogous simulations have been made by Sommerfeld, Schrodt, Parisot, and Chien (4,78), who also presented an analytic solution of the unsteadystate material balance equations obtained by matrix methods. This solution could be used to predict column performance from the basic operating parameters of the system, although evaluation of the matrices becomes difficult for more than a few stages. T h e most important concept generated by these studies was the idea that there existed cyclic variations in composition within the liquid on any tray, which in turn caused similar variations in the composition of
vapors leaving that stage. These time-axis composition profiles, which \vex repeated exactly during each cycle of pseudo steady-state operation, were the basic distinctions between cycled and continuous apparatus. Composition Profiles and Stage Performance
All the studies previously mentioned had shown that, in cycled distillation columns, the overall stage efficiency was always higher than the point efficiency. I t is instructive to note that such behavior can also occur in conventional equipment. \Vhen there is imperfect liquid mixing on a crossflow tray, a transverse composition profile can occur between the liquid inlet and outlet. I n such a case, the overall stage efficiency, defined as
Ffl - Fff-1
E, = -
Y,*-
-
Y,-1
will always exceed the point efficiency, defined as
since, with a composition gradient, Fff may exceed Yfl* but Y,, can never exceed Ynl*. The effect of the liquid phase composition profile was first discussed by Lewis (70), who, in a classic paper, derived the mathematical expressions applicable to operation under these conditions and presented predicted values of overall plate efficiencies for various plate configurations as functions of point efficiency and the ratio of the slopes of the equilibrium and operating lines. I n his derivations, Lewis assumed no lateral liquid mixing on a tray and no lateral vapor mixing between trays. He considered three cases: Case I.
Vapor flowing to the tray, liquid flows a t uniform conipositioii at all points Case 11. Vapor laterally unmixed flowing between stages, liquid flows in same lateral direction on all stages Case 111. Vapor laterally unmixed flowing between stages, liquid flows in opposite directions on alternate stages The highest attainable plate efficiencies were those for Case 11, with Case I11 providing the least enhancement. Most commercial types of conventional distillation columns correspond to a Case I11 configuration, although lateral vapor mixing would tend to produce a Case I situation and actually improve performance.
The effect of these composition gradients was greatest at high point efficiencies, since the magnitude of the gradients themselves depended directly on this point efficiency. Other studies, such as that of Gautreaux and O'Connell ( 9 ) ,have pointed out the effect of partial lateral liquid mixing, which decreases the overall stage efficiency. T o prevent such liquid mixing, plates consisting of successive contacting segments separated by weirs have been developed, and the advantage of sieve trays, in which lateral liquid mixing is minimized, over bubble-cap plates has been recognized (5). Application to Controlled Cycling
The basic analogy between cycled operation and conventional operation with lateral concentration gradients has been discussed previously, having been developed independently and at about the same time by the present authors and the group of Parisot, Sommerfeld, Chien, Schrodt, and Robinson (76, 77). It may be described as follows. [$'hen a specific liquid subvolume passes through a conventional distillation column in which no lateral liquid mixing occurs on the trays, its concentration-time history will appear as in Figure l a . No composition changes occur while the subvolume is transferring between stages in a dowmconier, and the concentration decreases as liquid crosses a plate and contacts vapor. NOW consider a cycled column in which all of the liquid on plate n flows, with no mixing, to plate (n - 1) during the liquid flow period. For this case, the concentration-time history of any particular liquid subvolume Trill be, as shown in Figure l b , identical to that of the conventional column, provided the vapor flow period is identical to the time required to flow across the conventional plate and provided the liquid flow period corresponds to the residence time of liquid in downcoini:rs. I n effect, then, cycled operation has replaced the distance-axis composition profiles found in a conventional column with a composition profile along the time axis. The analogy is precise only when all of the liquid on any plate drops, in plug flow, to the plate below during the liquid flow period. Also, it must be assumed that mass transfer is negligible during the liquid flow period in a cycled column, which corresponds to neglecting transfer in downcorners of a conventional column. Because the lateral concentration gradient has been replaced by a time-axis concentration gradient in cycled operation, it is entirely natural to expect apparent overall efficiencies greater than the point efficiency at any VOL. 5 9
NO. 3
MARCH 1967
23
MATHEMATICAL DEVELOPMENT Derivation of Vapor Flow Period Material Balance
The mathematical analysis of a conventional column developed by Lewis (70) can readily be adapted to cyclic time-variant concentration profiles. T h e shape of the plate under consideration is completely immaterial so long as the vapor is evenly distributed throughout the liquid; a binary mixture is assumed.
TIME
Figure l o . Rtpre~entation of concentmtion projles in conventional column: internal (Lagrangian) dcwpoint
I X I
VAPOR FLOW PERIOD I LlPUlO OM SlkGt n I
-3 E
,z-
I I
.
I VAPOR FLUS rtwv 1 LlQUlO ON ITA6t
I
I
II
I
I
T,
1,
n-I
I
I
I
I I
!
TIM
Figure l b . Reprermtation of concentration pr@les in a cycled column: inicrnal (Lagrangian) impoint
given time. T h e definitions of Equations 1 and 2 apply, with time averages replacing distance average compositions and the second subscripts denoting various times within a cycle rather than locations. The cycled column corresponds very well to several of the models proposed by Lewis (70). The plate directly above the reboiler is analogous to a Lewis Case I plate, with invariant (at least over the short times of one cycle) composition vapor flowing to it. Other plates in the column correspond to Lewis Case I1 plates. Since in the cycled column gradients are along a time axis, no antiparallelism of gradients corresponding to the Lewis Case 111 is possible. Lateral liquid mixing has no effect on a cycled plate; in fact, good lateral mixing is desirable. Similarly, lateral vapor mixing is unimportant, since at any given time all the liquid on a stage should have the same composition. However, axial vapor mixing will lead to a more uniform composition for the vapor flowing between stages and adversely affect performance. Also, axial mixing of the liquid during the liquid flow 1 short-circuiting period, such as liquid from Stage n through Stage n and ending up on Stage n - 1, would partially destroy the concentration difference between adjacent stages and adversely affect performance.
I , , V
H=IIOLDUP OF LIQUID PER STAG, MOLES L=OVEULL OR GROSS APPARENT LlpUlD HOW RATE, MOLES/UNIT TIME t.=TOTAL CYCLE TlMf, SUM OF LENGTHS OF LIDUlD AND VUOR HOW WlOM ?.=LENGTH OF VAPOR FLOW PERIOD V=INSTANTANEOUS VAPOR R O W RATE DURING YAFM HOW FEIIOD, MOLES/UNll IIME V’=OVERALL OR GROSS APPARENT VAMR FLOW RAIE, MOLES/UNll TIM€ X=INSTANTANEOUS LIQUID COMPOSITION IN TERMS OF MORE VOUTILE COMPONENT, MOLE R1CTION Y=INSTANTANEOUS VAPOR COMPOSITION II TERMS OF MORE VOLATILE COMPONENT, MOLE FRACTION Y*==COMPOSITION OF VAPOR IN EPUlLlBRlUM WITH LIQUID, MOLE FRAinlON MORE VOLATILE COMJONENT V+RACTlON OF H ON RATE THAT HOW TO )UTE BELOW DURING THE LlPUlD FLOW PERIOD
LE
Y’=V
(-11.
t.
11
Figure 2. Reprcstntationof a cycled stage during thc uaporflow pm’ad
2.4
+
0.2
AUTHORS Alfred
J. Engel is Associate Professor in the De-
partment of Chemical Engineering at Pennsylvania State University. Robert G. Robinson was a graduate student at Penn State and is presently an Engineer for The Upjohn Go., Kalamaroo, M i c h . 24
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
0
0.2
Figurc 3. Variation of overall X for constant Case Iplatc
+:
0.4 0.6 M I N T EFFICIENCY, E stage aflciency
0.8
I .o
with point e.ciic;mcv and
During the vapor flow period, the plate can be pictured as in Figure 2, which also includes the nomenclature involved in the following equations. A unit cross section of column is chosen as a basis. A material balance can be written on the plate in terms of the more volatile component; during the vapor flow period this is
(Y,- Y,-i)Vdt
=
HdX,
Solution for Case I
In this application, Y,- is a constant throughout the vapor flow period, and the final term of Equation 13 can be discarded. Equation 1 3 is integrated from 7 = 7 to 7 = 0 and from (Y,,), to (Y,),, that is, the vapor concentration at any time during the cycle to the concentration at the end of the vapor flow period :
(3)
H , the holdup, is related to the overall apparent liquid flow rate by
or
(4) and L' is related to the overall apparent vapor flow rate by
The average composition of vapor leaving the plate during the entire cycle is
(5) or
A new variable, defined as r = t/tL', is now introduced. This may be considered as a measure of the remaining time for vapor-liquid contact during a cycle, and thus varies from 1 to 0 during the vapor flow period. With these substitutions, the material balance can be written
By definition,
The internal reflux ratio is defined as
and, by substitution,
R = -L
V'
(7)
Over the range of compositions occurring on one plate, the equilibrium curve is assumed straight, but not necessarily passing through the origin :
Y*=KX+b
(8)
dY* K = dX
or
Further defining X, the ratio of the slopes of the operating and equilibrium lines, by A = -
K
R
the material balance becomes
The definition of point efficiency provided by Equation 2 can be rewritten as
Y,* =
Y, -
E
-1 + E7 Yn-l
and, taking the derivative of this with respect to 7 and substituting the result in Equation 11, we arrive a t :
Equation 19 now describes the apparent overall efficiency of a cycled Case I plate for a single cycle of operation. For a cycled column operating in a pseudo steady-state manner, the compositions and flows are repeated exactly every cycle, and the solution obtained by integration over a single cycle applies to all cycles, describing the overall performance. Naturally, the equation is not valid during true transient operation periods such as start-up. Some typical E , values for various E and X are shown in Figure 3. I t is easily seen that the increase in E, over point efficiency is small except for rather high E values. Systems with low point efficiencies would gain little from cycled operation. The overall efficiency is also strongly dependent on A, but in most cases of industrial interest X will be approximately 1.O. Table I illustrates the effect of variations in 4 on E,. It is seen that E , increases continuously with increasing
9. VARIATION OF E, W I T H 9 E, = 1 . 0 x = 1.0 Q 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1 . 5 E, 1 . 3 0 1 . 3 7 1 . 4 5 1 . 5 3 1 . 6 2 1 . 7 2 1 . 8 2 2 . 3 2 TABLE I.
Solution for Case II
This basic material balance equation will now be
applied to the cases of interest.
According to Lewis' development, a number of assumptions are necessary to obtain a model that can be VOL. 5 9
NO. 3
MARCH 1967
25
treated mathematically. These are :
1. Vapors rise from plate to plate with no axial mixing 2. The equilibrium relation can be represented by a straight line, Y" = K X 6,over a range of several plates 3. The operating line is straight and of slope R , over the same range 4. Plug flow of liquid from plate to plate occurs during the liquid flow period
For given 4, A, and E, values of a are calculated from Equations 28 through 30, as appropriate, and these a values are used to calculate E , from the relation
E,
+
From these assumptions, the Y us. T curves for plates n and n - 1 must have the same shape, but may differ in size. The following important variables, which are illustrated in Figure 4, are now defined :
Z, = Yn - (m), k , = ( Y J 0 - (Y,--i),
(20)
(21)
Since the Y us. T curves have the same shape for plates n and n - 1,
2, = QZ, -1
(22)
k , = Qk, -1
(23)
where a is an arbitrary constant. These variables can now be substituted into Equation 13 to yield [(Q
- 1)Zn-l
+ k n l a +'XE E-I
dZ, - 1 -~ dr
(24)
The variables, Z,-I and T , are separated and integrated between T = 1 and T = 0:
(25) Expressions defining the relationship between k , and may be derived from the definitions for E, 2,and k , and the relation between (Y,*), and (Yn-1*)1, as determined by the liquid flow period material balance. These derivations are included in the appendix; the results are:
Z,- 1 for various ranges of
0 < ' < l
11'12
+
(2,
(Zn-di
-1) 1
=
k,
=
4kn
+ (4 - l ) k , + i
a - 1
= __
A-I
This equation was first derived geometrically by Lewis, and is valid for all values of E,. Solution of the preceding equations for Q is rather difficult, bvith trial-and-error methods required. Such equations are readily solved by simple digital computer programs, however, and very accurate solutions obtained. Some typical results are shown in Figures 5 and 6. Figure 5 illustrates the variation of the overall stage efficiencl-, E,, with point efficiency and X for a given \-slue of q5. As in Case I, the overall efficiency rapidly increases with increasing point efficiency and is strongly dependent on A. Cy-cled operation would yield little improvement unless point efficiencies were high. Figure 6 compares the performance of Cases I and 11 plates for similar +, A, and E values. The superior performance associated with Case I1 is clearly evident. Effect of
+
\2'hile the variation of E, with point efficiency and X is simpll- represented, the variation of E, Lvith 4 is somewhat more difficult to explain. This is illustrated in Figure 7, Lvhich indicates that E , values increase to a maximum a t 4 = 1.0, and periodically return to the same maximum at each integral value of 4. This periodicity was noted by both MctVhirter (73) and Chien et al. ( d ) , although in both these studies the maxinia became successively lower as C$ increased. The decrease in the maxima was caused by the fact that these authors considered a finite column, \vith a small number of plates. In such a case, as 4 exceeds 1.0, liquid reflux from the condenser progressively affects the second plate below the condenser and destroys some of the difference behveen liquid compositions on the top two plates. This in part diminishes the efficiency of the top plate and the overall column efficiency falls. At = 2.0, both top plates have the same initial composition, and the column efficiency cannot equal that a t 4 = 1.0.
+
(26)
(27)
Similar expressions can be derived for other values of 4 . Substitution of Equations 26 and 27 into 25 leads to the following relations, \vhich describe cycled operation for various ranges of 4 :
l$=l
XE(Q - 1) = In aS-E-1 ( a - 1) 4XE-
11912
26
a:+E-l
=
01
In [(+ - I ) a 2
+
INDUSTRIAL A N D ENGINEERING CHEMISTRY
(Z' IT =(Y,), K,=(Y "1 0-
4Y"I, (Y*-i
Jo
Figure 4. Concentration projiles on adjacent plates: Case I I cycled plates
Parisot et al. (76) explained the periodicity of E. between integral values of 4 i n r r m s of the net result of two effects: the increasing liquid compositions, at the end of the liquid flow period, with increasing 4, and the decreasing liquid compositions at the end of the vapor flow period as this period becomes longer with increasing 4. The former effect predominates as 4 approaches integral values, while the latter takes precedence for 4 values just above 1, 2, etc. Their explanation of decreasing maxima with increasing 4 values as a result of condenser effects similar to those mentioned above is certainly valid, but the occurrence of minima between integral 4 values is in no way related to condenser effects. T h e present work, considering a plate as a part of a n infinite sequence in which no such end effects are possible, demonstrates that these minima arise naturally because of the competing effects noted above. In terms of the concentration profiles the behavior is as follows. A change in 4, for a given column geometry, requires either a change in holdup or a change in the lengths of both flow periods (to maintain constant A). Very small 4 values, corresponding to pulsed operation, would result in extremely shallow concentration gradients and negligible performance increases. The gradients increase in magnitude u p to 4 = 1.0, a t which point all compositions between top and bottom product composition exist a t some time within the column. The column then approximates a true countercurrent contactor, and the separation is a maximum. Further increases in 4 lead to a n overlap of concentration gradients on adjacent plates. This overlap is a maximum at 4 = 2.0, in which case the liquid is simply transferred two stages down the column each cycle
2.0 1.9 1.8 1.7 1.6 I
2 1.5
- 1.4
I .o
0.9 0.8
0.;
0.c
0.8
0.7
fOIwT WICltNCV, E
0.9
I
1 .o
Figure 6. Comparison of Case I and Case II stage pnformame at equal valuer of 0 and h
i
k l
0.6
2.c 1.5
-
::1.' h
.1.2'.:i I.'
O.
1. I n any event, the knowledge that performance deexceeds 1.0 is creases, for a number of reasons, as sufficient to restrict operating values of I$ to 1.0 or less in practical applications.
+
+
limitations of Analysis
The assumptions made in the development of the equations describing controlled cycling equipment performance limit their applicability, and mention should be made of these limitations. All the assumptions (and their shortcomings) usually associated with the McCabe-Thiele method are inherent in this analysis. Plug flow of both liquid and vapor is assumed, and axial mixing in either case can cause decreases in performance. The assumption of constant point efficiency on a plate may in some cases not be justified, since large concentration changes during one cycle can cause drastic variations in this parameter. Excessively long vapor flow periods may cause liquid and vapor to approach equilibrium on some plate, although common sense operation would easily avoid such a situation. The equations were developed by integration of the time-variant material balances over a single cycle, and their relevance for repetitive cyclic operation depends on the concentration gradients being reproduced exactly during each successive cycle. The equations definitely do not apply to true transient situations such as start-up or response to a change in operating conditions. Finally, the development assumes negligible mass transfer during the liquid flow period. This assumption is justified in such operations as distillation or absorption of a moderately soluble gas, where the ratio of molar holdups of liquid to gas is high and transfer during the liquid flow period would have a negligible effect on liquid compositions. This was shown experimentally by McWhirter (75). The equations developed would not describe conditions in a cycled liquidliquid extractor, where mass transfer is significant during both phase flow periods. 28
INDUSTRIAL A N D ENGINEERING CHEMISTRY
I a
TIME
I
cycled column
Recommended Design Procedures
A conventional McCabe-Thiele analysis is used to specify minimum and actual reflux ratios. LVhile in a cycled column, liquid and vapor flows are not simultaneous, the same ratio of flows is required to prevent approach to equilibrium from occurring on any plate as in a continuous column. LVhile the operating line concept is not strictly valid in cy-cled columns, overall operation can be described in terms of a pseudo operating line representing the locus of the average compositions of liquid and vapor flowing between stages. Three variables must be specified to solve the design equations: the point efficiency, E, the ratio of liquid flow per cycle to liquid holdup per stage, +, and the ratio of the slopes of equilibrium and operating lines, A. The reflux ratio will specify X for any short segment of the column. together with the equilibrium relation; + can be arbitrarily chosen and cycle times adjusted to obtain the desired value. The point efficiency must be estimated for each concentration range of interest, as in conventional columns. Design is started with the first plate above the reboiler, which is a Case I plate. Equation 19 is employed, with the values of E and X that apply in the expected concentration range for the plate, and E , calculated. IYith E , known, the actual stage separation can be plotted on the McCabe-Thiele diagram. This procedure is repeated through the stripping section. The lower two plates should be treated as Case I ; the others can be considered as Case 11. The design equations are not valid for the feed plate itself, if an intermediate feed is used, and a conservative estimate of performance must be made in this region. The enriching section is then treated in a manner analogous to the stripping section. The use of @ values greater than 1.0 is not recommended because of diminished performance in this region. The effect of liquid mixing during the liquid flow period can be moderated by specifying + less than 1.0. Optimum performance would probably be atbetween 0.6 and 0.9, depending on the tained for actual plate design. Packed plates (75) would allow less liquid mixing than sieve plates. For sieve plates, the actual plate configurations are
+
determined by standard design procedures, with hole size and total free area designed to prevent weepage. Hydrodynamic tests may be necessary to determine possible liquid flow rates and fix the liquid flow period required for a given 4. The reflux ratio then specifies the amount of vapor that must flow during the vapor flow period; this is regulated by adjusting the boilup rate. The chosen conditions must provide a suitable superficial vapor velocity. Control of a cycled column could be obtained by varying boilup rate and reflux ratio or by adjustment of cycle times. Since very precise control of cycle times is possible, extremely small changes in operating conditions can easily be made. I n this respect, controlled cycling adds an entirely new set of easily regulated variables to the conventional control parameters.
CONCLUSIONS The performance of controlled cycling equipment may be described in terms of the time-axis concentration gradients existing on each stage, and equations analogous to those developed for conventional columns with lateral concentration gradients can be used to predict this performance. The distillation data of McWhirter (75) very closely matched the predicted results calculated by the method suggested here. The cycled column has, a t best, exactly the same performance as a similar conventional column in which no later liquid mixing occurs. Controlled cycling is better than conventional operation only as it produces better concentration gradients. However, the magnitude of the concentration gradient, which in conventional apparatus depends on the plate diameter or length of the liquid path, is a function of the vapor flow period time. This may be regulated a t will, and small-diameter cycled columns can give performance equivalent to that which can be obtained only in much larger conventional columns.
APPENDIX
Since, in general, be true,
01
# 1 - E , for the above equality to
(5)
( Z , -111 = +kn An analogous derivation results in Equation 27. NOMENCLATURE
E
point efficiency overall stage efficiency H liquidholdup on one stage of a cycled column, moles interval between I-' values on successive stages at a given k, time, mole fraction K = d Y * / d X = slope of the equilibrium curve L = overall or gross apparent liquid flow rate, moleslunit overall time R = L / V ' = overall reflux ratio, moles liquid/inole vapor 1 = time 2, = total cycle time t, = length of vapor flow period V = instantaneous vapor flow rate during vapor flow period, moles/unit time V' = overall or gross apparent vapor-flow rate, moles/unit overall time X , = instantaneous liquid composition on stage n in terms of more volatile component, mole fraction = average liquid composition on stage n in terms of more volatile component, mole fraction Y , = instantaneous composition of vapor leaving stage n in terms of more volatile component, mole fraction Pn = average composition of vapor leaving stage n in terms of more volatile component, mole fraction Yn* = composition of vapor in equilibrium with liquid X,,, mole fraction more volatile component 2, = Y, - ( Y n ) o height , of vapor concentration profile at any time above concentration at end of vapor flow period, mole fraction = ratio of size of concentration profiles on adjacent stages a = L / K V ' = ratio of slopes of equilibrium and operating lines X = t / t u = reduced time variable, dimensionless T = fraction of liquid holdup on a stage that flows during a 4 cycle
Eo
= = = =
xn
SUBSCRIPTS i = value at an intermediate location on a plate or time during a cycle n = stage number = value at reduced time 7 during vapor flow period T
Derivation of Equation 26
LITERATURE C I T E D
For 4 5 1, a material balance for stage n - 1 during the liquid flow period may be written as
(1) Cannon, 3f. R., I N J . EXG. CHEM. 53, 629 (1961). (2) Cannon, ?vi. R., 021 Gar J . 51, 268 (1952). (3) Ibid., 5 5 , 68 (1956). (4) Chien, H. H., Sommerfeld, J. J., Schrodt, V.N., Parisot, P. E., to be published, 1767. ( 5 ) Finch, R . N., V a n \Vinkle, 51.,I & E C PROC.DESIGN ])EVELOP. 3, 106 (1964). (6) Gaska, R. .4.,h1.S. Thesis, T h e Pennsylvania State University, University Park, Pa., 1957. (7) Gaska, R. A , , Ph.D. Thesis, Ibid.,1 9 5 9 . (8) Gaska, R. A,, Cannon, hi. R., ISD. E N G .CHEM.53, 030 (1761). (7) Gautreaux, hi. F., O'Connell, H. E., Chern. €ne. Progr. 51, 232 (1955). (10) Lewis, \V. K., Jr., I N D .ENG.CHEM.28, 399 (1936). (1 1) Lunde, P. J., L1.S. Thesis, T h e Pennsylvania State University, University Park, Pa., 1760. (12) hfc\?'hirter, J. R., hf.S. Thesis, T h e Pennsylvania Stare University, University Park, Pa., 1961. (13) Mcb'hirtcr, J. R., Ph.D.Thesis, Zbzd., 1762. (14) McI\hirler, J. R., Cannon, 31. R.. IND.ENO.CHEDI. 53, 632 (1761). (15) LfclVhirtpr: J. R., Lloyd, I V . A , , Chern. En!. PIOQT. 59, S o . 6, 58 (June 1763). (16) Parisot, P. E., Sommerfrld, J. T., Chien, H. H., Schrodt, V. K.,to be published, 1967. (17) Robinson, R. G., Ph.D. Thesis, T h e Pennsylvania St.ite University, University Park, Pa.. 1964. (18) Sommerfeld, J. T., Schrodt, V. S., Parisot, P. E., Chien, H. H., t o be puhlished, 1967. (19) Speaker, S. X i . , Ph.D. Thesis, T h e Pennsylvania State Uniwrsity, University Park, Pa., 1957. (20) Szabo, T. T., Ph.D. Thesis, T h e Pennsylvania State University, University Park, Pa., 1958. ( 2 1 ) Szaho, T. T., Lloyd, \V, A , , Cannon, hi. R., Speaker, S. hi., Chern. En!. Progr. 60, S o . 1, 66 (Jan. 1964).
+
Since the equilibrium relation is given by I.'* = k X b for both plates, this relation can be directly transposed to (Yn-1*)1
=
+ (1 - 4 ) ( Y f l - l * ) o
4 (Yfl*),
(2)
From the definition of E a t r = 1 and the definitions for Z and k, the point efficiency can be written in terms of Z , k , and 4 as
' I
L'
t
E
El
This can be rearranged to give
I
E
\-fl-L,
VOL. 5 9
NO. 3
M A R C H 1967
29
