INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1950
LITERATURE CITED
Baylis, W. S., U. S. Patent 1,776,990 (Sept. 30, 1930). Davidson, R. C., Ewing, F. J., and Shute, R. S., Nutl. Petroleum
News, 35, No. 27 R,318 (1943). Ewing, F . J., U. S. Patent 2,410,436 (Nov. 5, 1946). Ewing, F. J., Secor, R.B., and Warner, J. G., Ibid., 2,391,312 (Dec. 18, 1945) Glaeser, R., Compt. rend., 222, 1241 (1946). IND.ENG* C H E & f s i 2148 (1948); 413 1485 ( 1949) Grim, R. E., J. Geol., 50, 225 (1942). Hauser, E. A., Chem. Rev., 37, 287 (1945). Hofmann, U., and Bilke, VV., KolloidZ., 77, 238 (1936). I
Grenal’l
.
401
(10) (11) (12) (13)
87 1
Hofmann, U., Endell, K., and Wilm, D., 2. Krist., 86,340 (1933). Marshall, C. E., Ibid., 91, 433 (1935). Prutzman, P.W., U. S. Patent 1,397,113 (Nov. 15, 1921). Ross, C. S., and Hendricks, S. B., U . S . Geol. Survey, Profess.
Paper 205B (1945). (14) Shankland, R. U., and Schmitkons, G. E., Proc. Am. Petroleum Inst., 27, 111, 57 (1947). (15) Thomas, C. L., IND. ENG.CHEM.,41, 2564 (1949). (16) Van Horn, L., U. S. Patent 2,391,050 (Dec. 18, 1945). (17) Whitmore, F. C., IND.ENG.CHEM.,26, 94 (1934); J . Am. Chem. Soc., 54, 3274 (1932). RECEIVED June 2, 1949.
Calculation of Minimum Reflux in Distillation Columns Three methods for calculatR . N. SHIRAS volatility rates are valid. It Shell ~~~~l~~~~~~ company, sari ~ r a n c i s c o ,calif, is also suitable as an approxiing minimum reflux rates in distillation columns fracmate method for all column tionating multicomponent D. N. HANSON AND C. H. GIBSON design problems which normixtures are presented and mally occur. The third Unicersity of Caltlfornia, Berkeley, CalV. evaluated. The first method method is an adaptation of is based on some overlooked the Thiele and Geddes plateequations originally published in 1932. It is very simple, byplate method of calculating finite plate columns. I t is but is rigorous for only a restricted class of separations. completely general, since with it variable overflow and The second method is an extension and elaboration of a variable volatility ratios can be taken into account, and method recently published by Underwood. This ‘ @ functhere are no restrictions as to the type of separation. I t is tion” method is fairly rapid and is exact for all separations recommended as a tool in distillation research but is too for which the postulates of constant overflow and constant laborious for plant design work.
T
a?
HE calculation of minimum reflux is important both as an initial step in the design of distillation columns and as a correlating property in research on distillation calculations. In binary systems, methods for the rigorous calculation of this quantity have been well defined (5, 13) although some of the more useful equations appear to have gained less attention than they deserve. For multicomponent systems, both approximate niethods and an exact method have been proposed. The available rigorous method for multicomponent systems (2, 7 ) is quite tedious and can hardly be justified even in application to unusual design problems or to research in distillation calculations Of the proposed approximate methods (1, 46, 9, 1118), the best is the method introduced by Underwood ( 1 1 , l a ) which is quite accurate and is, in addition, rapid in calculation. I t is thus adequate for most design purposes. In a recent paper Cnderwood has shown that this method is accurate not only for sharp separations between adjacent keys but also for cases where the separation is not sharp and where one or more components lie between the keys in volatility. Actually, Underwood’s equations are exact for any multicomponent system for which the assumptions of constant molal flows and constant relative volatility are valid. Further equations are presented here which facilitate the exact solution of a multicomponent system by Undei wood’s equations. When this elaborated Vnderwood method is not exact and a precise evaluation of minimum reflux is required, there is also a new rigorous method which, although timeconsuming, is less tedious than that proposed by Jenny ( 7 ) and by Brown and Holcomb ( 8 ) . This new method consists of an adaptation of the finite plate method of Thiele and Geddes (IO) to the calculation for infinite plates (minimum reflux). For the calculation of minimum reflux, all multicomponent
separations can be classified under two categories: class 1, separations such that, with infinite plates, all components of the feed are present in both the top product and the bottom product; class 2, separations such that, with infinite plates, some of the components are completely in the top product or completely in the bottom product. Failure to recognize the distinction between these two classes has led to much confused thinking about the socalled pinch points. When there are infinite plates in a rectifying or stripping section, as one proceeds away from the section terminus toward the feed plate, a point will be reached where there are only infinitesimal changes in component concentrations and molal flows for successive plates. Some writers on distillation have called this the “pinch point”; the present authors believe that a better designation is the “point of infinitude.” With binary systems the point of infinitude for both sections occurs at the feedplatc. The same is true of all class 1 separations of which a binary separation is merely a special species. With class 2 separations, for one or both of the sections, the point of infinitude occurs away from the feed plate. If the top product contains all components of the feed, then the point of infinitude for the rectifying section will be at the feed plate. If not all the components are present in the top product, then the point of infinitude will be at an intermediate point of the rectifying section. In the top product the mathematical concentrations of the vanished heavy components are infinitesimals of the second order; the infinite plates between the top of the column and the point of infinitude build the concentrations of these. heavy components up to infinitesimals of the first order; the infinite plates between the point of infinitude and the feed plate build the concentrations for these components up to the
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
872
finite values of the feed plate. Similar considerations apply in the stripping section. The compositions a t the points of infinitude of the rectifying section and the stripping section, respectively, are given by the familiar equations :
Vol. 42, No. 5
is the same as that of the feed plate vapor and the composition of the residual liquid, Lj, in equilibrium with this vapor portion of the feed is the same ac; that of the feed plate liquid, and LI = AL. In view of this identity of flashed feed compositions with feed plate compositions, which holds for class 1 separations, for such separations (X$)@ in Equation 5 may be replaced by (Xt),>the mole fraction of component i in the flashed feed liquid. E y u n tiori 5 then becomes:
and
If L, is equal to F , boiling point feed, Equation 8 becomes simply
These equations have been published by Brown and hlartin (3) and by others, and are presented here merely for convenient reference. ’From the operating equations for the two sections it also follows that:
Lm
+
(Lm d)(Kd)w d ( X i l d Lm(Xi)m = Lm 1. (L, d)(Ki),
.
This is a modified form of equation No. 60 in Underxood’s old paper (13). Fenske ( 5 ) also developed it, but only for two coinponent systems, For the special case of A V = F , dew point feed, Underwood’s equation KO,61 likewise holds true and one may write
(3)
+
Lh,
By solving Equation 3 for’(&), and then applying it to two components, A and B, Underwood (19)has shown that at the point of infinitude in the rectifying section:
If AV is equal to neither zero nor F (usually I), then the more general form, Equation 8, must be used. This entails evaluating the (X,), from the \$ell known flash equations. This may be done even though AV is negative or greater than F , in which case the values of (Xt)fconstitute a mathematical solution of the flash equation without physical counterpart other than their identity with the values of ( X % ) + .This is permissible because Equation 7 is true regardless of the sign or magnitude of AT’, and by sub= (K,)+(X,)+and (K,)+ = stituting the relationships, ( Y % ) + (at3)+(K7)+, this equation transforms to
Similarly for the point, of infinitude in the stripping section:
These equations, 1 t o 6, are valid for all cases.
(11)
MI[NINIUM REFLUX IN CLASS 1 SEPARATIONS
For separations of class 1, the liquid leaving the feed tray has %hesame composition as the liquid leaving the plat>eabove the feed tray and the liquid leaving the plate below the feed tray, since the points of infinitude for both the rectifying and the stripping sections are a t the feed tray. From this there follow some interesting relationships as t o the compositions of the liquid and vapor leaving the feed tray. Thus for any component, i :
( X i ) + I
( X i ) + = (Xi)++ 1
(Yd+ 1 = ( Y i h
=
(Yi)‘$+
1
+ V+(Yi)’$
L++l(Xi)++l f F(Xi)F f ~’Gl(~’i)+l = L i ( x i ) +
+
~ + + l ( x i F) (+ x~)F
F(xi)F =
(v+
+
~ ;  1 ( ~ i ) +=
v i  l ) ( Y i ) + t’ ( L i
F(X& = 4V(Y&
+
L ; ( x ~ ) +V+(Yi)+
 b+l)(xi)’$
+ 4L(Xi)+
(7)
where A V represents the increment of vapor flow a t the feed tray, (V, V i ,), and AL represents the increment of liquid flow a t the feed tray, (LG L++,). According to Equation 7, then, the total feed can be divided into a vapor portion of amount AV and an equilibrium liquid portion of amount F  4V or AL and the compositions of these two portions are the same, respectively, as the compositions of the vapor and liquid leaving the feed tray. In other words, if the feed is flashed to a degree of vaporization, V / equal to AV, the composition of this vapor portion of the feed



Equation I t is one of the alpha value valiants of the standard single flash equation. Equations 8, 9, and 10 are completely grneral, lioldirig for any two components of the multicompoiient mixture. If they are applied to the two key components whose specified distributions define the separation, then they give a simple, direct, and exact solution for the minimum reflux for separations which belong to class 1. Seither constancy of reflux nor constancy of alpha values Tras assumed in the derivation of these equations. The alpha value used is the alpha value for the key components at the feed plate conditions, and the minimum reflux found is the minimum reflux rate at the feed plate. By overall heat balance calculations the corresponding top internal reflux or the external reflux can then be obtained. Theoretically, at least, the method can break down in extreme cases of variable alpha, analogous to the breakdown of the graphical evaluation of a binary minimum reflux when the rectifying line crosses the equilibrium curve betxeen the feed plate composition and the top product coniposition, due to a pronounced nonideality in the binary system and consequent concavity in the equilibrium curve. Equations 9 and 10 were published by Underwood in 1932. They represent such a simple method of calculating the minimum reflux, it is surprising that they have been so completely ignored in subsequent discussions of minimum reflux. The explanation probably lies in a failure to recognize the distinction between class 1 arid class 2 separations. Shortly after the publication of UndeiJvood’s compendium of distillation calculation methods,
May 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY
1% orkers
in distillation theory became acutely aware that with the multicomponent separations normally encountered, the feed plate composition is markedly different from that of the feed and that with infinite plates in both sections, the points of infinitude were remote from the feed plate. A calculation method which presumed identity of feed plate and feed compositions was regarded accordingly as being erroneous, when actually it is merely restricted. Equation 8 also provides a means of determining the distribution of the components other than the two keys. Thus if it be applied to any such component, i, and component B, and again to component A and component B, and Lg + 1 is then eliminated, one obtains:
By this equation and the flash, if any, of the feed the d ( X & values for all components are readily found. MIKIMUM REFLUX FOR CLASS 2 SEPARATIONS
As has already been brought out, USE OF @ FUNCTION. Equation 8 can be used for the evaluation of the minimum reflux only when, because of the choice of key components or the volatility relationships of the components, the separation belongs to class 1 and all components are present in both products. Unfortunately in engineering practice the separations normally encountered belong instead to class 2, and it is for this class of separations that the various approximate methods which have appeared in the literature have been devised. Of these various methods, the best which has been published thus far is that recently presented by Underwood (11, l a ) ; it is relativelyrapid and with suitable elaboration it is in fact exact for all separations for which constancy of reflux and constancy of volatility ratios are valid presumptions. Even where these presumptions are not valid, the method usually gives good approximate values of minimum reflux adequate for design purposes. This method, in its essential features, wm contained in Underwood’s recent paper ( I I ) , wherein he also sketchily indicated the mathematical argument for its validity. When filled out in detail, this argument is completely rigorous within the limitations of the presumptions of constant reflux and constant volatility ratios. The crux of the method is the setting up of a set of parameters, Q1, for the rectifying section, and another set of parameters, in this nomenclature for the stripping section, which are defined as the roots, respectively, of the two equations:
Equation 13 has as many roots, @>, as there are components in the top product. Furthermore it can be shown that one of these is less in value than the smallest air and the remaining roots must lie successively between adjacent airvalues. This least root can be given a physical interpretation. If in Equation 1 for the rectifying point of infinitude one replaces (&), by its equivalent, a,,(K,), , and then multiplies the numerator and the denominator by a*,.,this equation becomes
ai,
_ _ V(i,),
Summing this equation componentwise gives:
The expression
L
V(K,), ~
873
is nothing more than the absorption fac
tor for the reference component at the rectifying point of infinitude and its value must be less than that of the smallest azV. Otherwise by Equation 15 for at least one component, V ( Y , ) , would be negative and would not be physically real. Since Equation 15 and Equation 13 have the same form, the value of the absorption factor for the reference component at the point of infinitude which satisfies Equation 15 is that root of Equation 13 which is smaller than the smallest aZT. I t can be shown, following the line of reasoning indicated by Underwood, that if V and VI are the vapor flow rates in the rectifying and stripping sections, respectively, under minimum reflux conditions, then some of the @1 roots for Equation 13 will be identical with certain @; roots of Equation 14, and these common roots of the two equations will be all the and @;, the values of which lie between the alpha values for those components and only those components which distribute between top and bottom products. Thus, if only the two key components defining the separation are present in both products, there will be only one common root; if three components distribute, there will be two common roots; if in addition to the keys two other coinponents distribute, there will be three common roots; etc. Furthermore, adding Equations 13 and 14 gives:
+
for all common roots of Equations 13 and 14. In other words, if Equation 16 be solved for those roots which lie between the 01~). values of those components which are present in both top and bottom products, these roots will also be common roots of Equations 13 and 14. Now substituting any one of these common roots in Equation 13, performing numerically the summation and subtracting the yield of the top product, d, will give the value of the minimum reflux:
+
Or an alternative form may be used to calculate L,i, directly:
These relationships are valid, regardless of whether a distributing component other than the keys is more volatile than either key, less volatile, or intermediate in volatility. If only the two key components distribute, Equation 16 can be solved for the Q root lying in value between the alpha values of the key components, and Equation 17 or 18 can be solved directly for Lmin, the value of the minimum reflux. If any other components distribute, the exact distribution of these components must be established, and the values of the d ( X i ) dfound for these components must be used in solving Equation 17 or 18 for L,,,. These equations, however, also provide the means for calculating these d(X& values of the distributing components other than the keys. The d(X&values for the keys are given in the problem specifications. By definition the d(Xi)dfor the light nondistributing components are the same as the F ( X , ) , and are zero for the heavy nondistributing components. Thus the only unknowns in Equation 18 are Lminand the d ( X , ) , for the nonkey distributing components. It has already been pointed out that the number of roots of Equation 16 applicable to Equation 18 are one less than the total numbel: of distributing components and hence one more than the number of nonkey distributing components. Each such appropriate GI root of Equation 16 gives rise to an independent equation of Equation 18 type. Thus the number of these simultaneous independent equations is equal to the number
INDUSTRIAL AND ENGINEERING CHEMISTRY
874
TABLE I.
COMPOSITION AND VOLATILITY
COVPONEST FEED
Component
Rlok Fiactioii 0.05 0.05 0.10 0.30 0.05 0.30 0.10 0.05 1.00

RATIOSO F EIGHTTolatility Ratio 10 5 2.05 2.0 1.5 1.0 0.9 0.1
IN TOPPRODUCT (EQCATIOX 12) T.~BLE 11. RECOVERIES
Component
.4 B C E
G H
7 Recovery in %op Product 730 330 '34 50 2  62
of unknown values of Lminand d ( X i ) dand t,he unknown quant,ities can be found by any of the standard methods for solving simultaneous linear equations. In many cases it is obvious which coinponents will distribute at minimum reflux, but in some cases with components lying relatively close to but outside t8hekeys on a volatility scale, it is not obvious. If a component is presumed to distribute at minimum reflux, which in fact does not, then the solution of the set of simultaneous equat'ions, 18, will give rise to an impossible value for d(Xi)dfor this component; it will be greater than its F(Xi), if a light component, or negative if a heavy component. At the same time the d(X,), found for other distribut,ing nonkey components will also be slightly in error and must be recalculated after dropping as a variable the component or components thus shown to be nondistributing. Since each solution of Equation 16 for an appropriate @ root is a trial and error procedure of the same nature as the trial and error solution of the standard single flash equation, it is highly desirable that no superfluous roots be evaluated. Equation 12 developed for class l separations, although it will not, give correct values for component distributions in class 2 separations, can still be used to establish which are the distributing components and thus forestall evaluating unnecessary Q roots of Equation 16. The procedure is to determine the apparent d(Xijd value through Equation 12 for each doubtful component. If the ratio of this apparent d(Xi)dvalue to the corresponding F(XijFis greater than 1.01 or less t,han 0.01 it may safely be presumed t o be nondistributing; if this ratio lies betxeen 0.99 and 0.01, t'he component probably does distribute; if the ratio lies between 1.01 and 0.99 or 0.01 and 0.01, usually there will be no serious error in assuming nondist,ribution of that component. In general, since minimum reflux is only a guide for the estimation of design reflux flows, complete rigor in its calculation is unnecessary. Similarly there is no occasion to determine the exact distribut,ion of all components a t minimum reflux. For these reasons, the use of the values of d ( X i ) dfor the distributing component,s obtained by Equation 12 gives an adequate approximation of minimum reflux. Under these circumstances, only one value of Q need be calculated from Equation 16, preferably the value lying between the (\I values of the keys; the minimum reflux is directly calculated by Equation 18. Xormally, this procedure gives a value of minimum reflux which differs by only a few per cent from the true value. The following numerical example illustrates this practical form of the @ function minimum reflux method and t.he accuracy that may be expected with it. An eightcomponent feed has the composition and volatility ratios relative to component F , the heavy key, shown in Table I. The feed is at its boiling point and constant internal molal flows
VoI. 42, No. 5
are presumed. It is desired to calculate the niinimum reflux for the specifications of 90% recovery of component D in the top product and 90al,recovery of component F i n the bottom product. Recoveries in top product for all the other components were calculated according to Equation 12, and the values found are listed in Table 11. From this it is quite evident that components A , B , and H do not distribute, but that components C, E, and G do. Component E lies between the two keys, so obviously it must distribute; components C and G, respectively, are more volatile and less volatile than either key, but their volatilities are so close to that of one or the other of the key components that they also distribute. The next step is to substitute the F(X,)r values ( F = I) and aLI values from Table I into Equation 16 with Vi equal to zero since boiling point feed has been specified, and then to solve this equation for either of the @ roots intermediate in value to 2.0 and 1.0, the of the key components. @E,? was selected and a value of 1.2477 was obtained for it by trial and error procedure, The remainder of the computation of the minimum reflux is shown in Table I11 from which it appears that the minimuin reflux is 0.5831 moles reflux per mole feed.
T.IBLE111. COMPCTATION OF ~ I I S I M REFLUX U~I O E , F = 1.2477
Eq. 18
Recovery in T o p Product, Compn.
%
air.
100 100 94 90 50 10 2
1;
2.05 2.0 1.5 1.0 0.9 0.1
0
Coniment Kondistributing Eondistributing From Table I1 Light key From Table I1 Heavy key From Table I1 Nondistributina
d(.Yi)d 0.050
0 050
0 094 0.270 0 025 0 030 0 002
1.2477 d ( X i ) d air  1 . 2 4 7 7 0.0071 0.0166 0.1462 0,4478 0.1237 0.1511 0.0072
In order to determine the accuracy of this value for the minimum reflux, the exact value was found by solving for the exact distributions of components C, E , and G through the set of four simultaneous linear equations of Equation 18 type. To do this necessitated trial and error evaluation of three more @ roots of Equation 16, namely, @c,D, * D , E , and *.F,G. The component recoveries thus found and the resultant mininiuni reflux value are shonm in Table IV.
TABLE I\. C o a r ~ o m RECOVERIES :~~ A N D 111xnin1 REFLUX I'.II,UES G C , D = 2.0375
O D . E = 1.5604
Recovery in TOP Product. Coinin.
c70
j:
100 100 94.42 90 48.36 10 2.603
c
D
E
r
c
d(Xi)d 0,0~000
0.0~000
0.09442 0.27000 0.02418 0.08000
0,51212
O E F = 1.2477 1.5604 d(.Yi)d air 1.5604

96.0= 0.9192
1.2477 d ( X i ) d O Z T  1.2477
0,00925 0,02268 0,30095 0,958SZ 0,62417 0.08353 0 00615 0,57755
0 00713 0.01RF3 0,14684 0,44783 0 1185'3 0.15110 0.00994 0,57758 Lmin = 0.3776
The simplified method t,hus gave an anwer xhich was in error by only 0..5832  0.5776 = 0.0057 or 1%. Had @D.E been evaluated and used in the simplified method instead of @ ' E , P , the minimum reflus value found would have been 0.5565 and hence in error by 0.0211 or 3.775, which is still close enough for most practical cases. The discussion of class 2 separat,ionsthus far has been restricted to the calculation of minimum reflux under the postulates of constant reflux and constant volatility ratios. I n most practical problems of class 2 separations which arise in the petroleum and chemical industries, these presumptions are not valid, either because of appreciable departures from ideal solution behavior or, commonly, because the feed embraces too broad a volatility range, and as a consequence there is a wide temperature spread in
INDUSTRIAL A N D ENGINEERING CHEMISTRY
May 1950
the column. For these, unfortunately usual, cases there is no rapid method of precise evaluation of the minimum reflux, but the 0 function method may still be used to give good approximate values. I n applying the function to class 2 separations involving variable alpha values and variable molal flows throughout the column, alpha values are found for all the components at an estimated feed plate temperature. These alpha values are then used to evaluate component distributions by Equation 12 and one @ root of Equation 16. The minimum reflux is then found by Equation 18. The value for the minimum reflux thus found is regarded as the minimum value for the internal reflux to the feed plate The corresponding top internal reflux or the external reflux can then be found by overall heat balance around the rectifying section, employing feed vapor and liquid compositions for feed plate compositions. Alternatively the true feed plate compositions may be calculated for this purpose, but since this entails evaluation of all but one each of the 0 and roots of Equations 13 and 14 and still only leads to an approximate value of the minimum reflux, the authors do not feel this extra labor to be warranted. In order to solve Equation 16, the change in vapor flow rate, V f ,in passing from the stripping section to the rectifying section must be estimated. This estimate should be made on the basis of the feed plate temperature used for obtaining the alpha values of the components. Considerable research in comparisons of the results of this approximate method with precise values obtained by laborious exact calculations for a wide variety of cases is needed to delineate the trustworthiness of the approximate @ function method. Probably such a research program would show that the biggest sources of error were poor estimates of the feed plate temperature and of the change in vapor flow rate a t the feed plate, the apparent vaporization of the feed. There are various criteria which can be used for arriving a t the feed tray temperature, pending confirmation by a research study; two of the simplest are an arithmetic mean of the top plate and reboiler temperatures weighted by the molal amounts of top and bottom product and, alternatively, for ordinary petroleum feed streams, the temperature a t which the K values of the two keg components are equally distant from unity. On the whole, the method should be as dependable or more so than the other approximate methods which have been published and which involve the same or greater uncertainties. Moreover, it has the advantage of being fairly rapid to perform. Its use is recommended for all ordinary plant design problems involving class 2 separations. Two examples of class 2 sepal ations employing variable alpha values which have been reported in the literature (2, 7 ) have been calculated by the approximate @ function method to estimate the degree of inaccuracy in the assumption of constant alpha values. Both procedures proposed above for estimating feed plate temperature and thus obtaining average alpha values were used in these calculations. For the problem given by Brown and Holcomb ( 2 ) both procedures gave the same alpha values within the accuracy to which their equilibrium constant charts could be read. The approximate minimum reflux found was 0.67 mole per mole of feed, compared to their value of 0.66. For the problem given by Jenny ( 7 ) , the weighting of top and bottom temperatures for obtaining feed plate temperature gave 158" F. and an alpha value between the keys of 2.09. The second procedure gave a temperature of 180" F. and an alpha value betueen the keys of 2.02. The respective minimum reflux values were found to be 0.54 and 0.57, compared to Jenny's value of approximately 0.57. I n both problems the accuracy of the minimum reflux calculated by the approximate @ function method with either procedure for obtaining average alpha values is as good as the accuracy of reading the charts of equilbrium constants. Moreover, since minimum reflux serves primarily as a base point for selecting a design reflux, the accuracy obtained with the 0 function method is easily sufficient.
+
*'
*
d
875
EXACT CALCULATION OF MINIMUM REFLUX WITH VARIABLE REFLUX AND EQUILIBRIUM CONST4NTS (K VALUES)
The exact calculation of minimum reflux for those .elms 2 separations for which the simplifying assumptions of constant reflux and constant volatility ratios are not valid is necessarily tedious because a certain amount of trial and error platebyplate calculation cannot be avoided. The recognized technique for such a calculation has been well outlined by Brown and Holcomb ( 2 ) for distribution of the key components only and need merely be summarized here. The approximate value of the minimum reflux is estimated or computed by one of the available methods. With this reflux, tbe compositions and magnitudes of the vapors at the two points of infinitude are computed by Equations 1 and 2 and to these are added very small amounts of the missing nondistributing components; in the rectifying section these are the components heavier than the heavy key and in the stripping section, the components lighter than the light key. Platebyplate calculations of the Lewis and Matheson ( 8 ) type are then made from both points toward the feed plate, and adjustments are made in the quantities added at the points of infinitude (or equivalently part way toward the feed plate) until the concentration of each nondistributing component in the feed plate vapor is the same when the composition of the feed plate vapor is calculated from either point of infinitude. In these computations variation in reflux flow can be taken into account by a running heat balance in the usual manner for Lewis and Matheson type calculations. After this balance for the nondistributing components has been achieved, the nature and degree of unbalance for the two key components is observed, and from this evidence a new estimate is made of the minimum reflux and the whole process is repeated until all components are in agreement. Even as outlined by Brown and Holcomb ( 2 )or in the modified version of Jenny ( 7 ) , this procedure is quite laborious. If the added complication be present, that several components other than the key components are distributing, the computation time and labor become prohibitive, since these distributions must also be established for each reflux estimate. This Brown and Holcomb method is simply an adaptation to minimum reflux determination of the method of Lewis and Matheson for calculations of finite plates and reflux. The concentration ratio platebyplate method of Thiele and Geddes (10) for calculatingwith finite plates can be similarly adapted to infinite plates minimum reflux computations. The Thiele and Geddes approach does not require preestimation of component distributions and with it the computation time does not mount impossibly with an increase in distributing components. I t will be recalled that in a Thiele and Geddes type calculation one proceeds from plate n 1 to plate n in the rectifying section according to the recursion equation:
+
and from plate n to the equation:
 1 to plate n in the stripping section according
With finite plate and reflux calculations, the starting point in which is equal to the rectifying section is the ratio (Xs)L/(Xi)d, l/(Ki)t,and for the stripping section is the ratio (Y