Calculation of the Effect of Reflux Ratio in Batch Fractionation ARTHUR ROSE AND HARRY H. LONG The Pennsylvania State College, State College, Penna.
The reflux ratio is one of the most important variables in batch fractionation because of its relation to the sharpness of separation and to the time and expense required for collection of a given amount of product of specified purity. The effect of reflux
ratio in batch fractionation may be estimated by calculating distillation curves for the cases of interest by an extension of the methods of Young ( 1 2 ) and of Rayleigh ( 1 1 ) . The present paper summarizes the results of some calculations of this kind.
HE details of the derivations and methods of calculation were described in previous papers (6, 7, 8, 10). The procedure always depends upon the solution of equations of the following types:
Results of Calculations
T
log L =
The results of the calculations are given in the accompanying figures and a study of these leads t o the following general conclusions:
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1. Increased reflux ratio always gives sharper separation, but there are many instances where the increase is insignificant. 2. The magnitude of the effect of reflux ratio in any given case depends upon CY and n as well as R, with the order of importance generally CY, R, n. 3. The effect of increasing R is always proportionately greater at small values of R than at larger values (assuming CY and n remain the same). 4. With a given value of CY", a small value of LI results in reflux ratio changes having but a small effect on sharpness of separation at all reflux ratios, while a large value of CY results in a marked effect for small reflux ratios with little effect for higher ones. a. When CY is small, R has but relatively little effect on sharp ness of separation, regardless of the values of an,n, and R, unless n and CY*are made very lar e. 6. Increase of n causes to have more effect (at low values of R ). 7. With given values of CY and R, the effect of increasing n is much more marked when R is large than when it is small.
The solution of such equations makes it possible to construct graphs of per cent distilled (100  L ) against distillate composition (q,) such as those of Figures 1 to 4 of this paper. The shape of such curves depends upon the value of the initial still composition, the vapor pressure ratio, the equivalent number of plates, the reflux ratio, and the holdup in the column and mixture undergoing distillation. By calculating a series of curves involving different values of these factors, the effect of each on the sharpness of separation may be estimated. The calculations may be made for nonideal multicomponent systems for distillations in which holdup is appreciable and the usual simplifying assumptions (4) are not valid, but the present paper considers only the batch fractionation of ideal binary mixtures when holdup is negligible and the usual simplifying assumptions are justifid. This procedure is comparatively simple and the conclusions apply to many batch fractionations. Distillation curves calculated on the basis of these assumptions coincide (within the limits of experimental error involved in ordinary reflux ratio and efficiency determinations) with actual experimental curves (6, 20) obtained under comparable conditions. The calculations for the curves of this paper were made by using a graphical method for evaluating the integral of Equation 1, after obtaining the corresponding values of 2, and 2, by either the McCabeThiele procedure or the Smoker equation, as already described (IO). The Smoker equation (9) was used in the form: (rnC*)"
x.=k+ a"
 mc
(01

 IC)  k)
(xp
1) (xp
CY* (01

(rnC2)"
%
Calculated Distillation Curves for Typical Cases The calculated curves give a definite idea of the relative advantage of successive increases in reflux ratio. This was illustrated in an earlier paper (6) which showed the effect of reflux ratio in the distillation of an ideal binary mixture with vapor pressure ratio 1.25 in a column with the equivalent of twenty plates and negligible holdup. A reflux ratio of a t least 19 was required in this case to produce a reasonably sharp separation. Figure 1 shows the effect of reflux ratio when the vapor pressure ratio is 1.5 and there are eleven plates; Figures 2, 3, and 4 give similar sets of curves for mixtures with higher and lower vapor pressure ratios. All the curves in this paper were calculated for mixtures with initial composition of 50 mole per cent. The curves for mixtures with different initial compositions will be identical except for a horizontal displacement. Each of these sets of curves has distinct characteristics in
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gradual that i t is of considerable value to know the limiting sharpness of separation for a specific case. Thus by making the calculations for infinite reflux, i t is possible to show definitely that curve D (Figure 7) approaches a value of x, = 0.94 as a limit while curve B will approach a value of only x, = 0.7 as its l i t .
Methods of Summarizing the Effect of Reflux Ratio I n order to show clearly the effect of reflux ratio under different conditions of distillation, i t is desirable to have some method of combining and summarizing all results from studies such as those of Figures 1 to 4. This may be done in a variety of ways, some of which are illustrated in Figures 6 to 8. I n batch fractionation i t is always advantageous if the intermediate fraction between pure components is as small as possible. In Figure 5 the size of an arbitrarily chosen intermediate fraction (including the distillate with composition zp = 0.9 to 2, = 0.1) is plotted against reflux ratio for each of the curves of Figures 1 to 4, as well as for some additional similar studies. Such curves are advantageous in that they show the relation between the reflux ratio and the results directly. They suffer the disadvantage (when compared with some of the succeeding methods) that they require the righthand portion of the distillation curve to be located accurately. This is a matter of appreciable additional care in calculation when the distillation curve has a sharp break.
MOLS REMA~N~NG IN 5 7 7 ~ ~
spite of the fact that the overall fractionating factor, a",is approximately equal in each case. Thus in Figures 1 and 2 with a = 1.5 and 2, respectively, a reflux ratio of 9 is sufficient for a fair separation, and increase of R beyond about 30 results in little further improvement in fractionation. This is emphasized by comparison with the curves for the sharpest possible separation calculated on the assumption of distillation under total reflux. Figures 3 and 4 were calculated for smaller values of a (1.1 and 1.05) but larger 12 so that a" is approximately the same as for Figures 1 and 2. I n Figures 3 and 4 a reflux ratio of 9 results in poor separation, and higher reflux ratios are necessary to achieve sharp separation. Increase of R to 99 or 199 results in an appreciable improvement in these cases. I n each case as R is increased, the sharpness of separation approaches that of distillation under total reflux. When a is large (1.5 or 2.0), the limiting sharpness of separation is nearly reached a t low values of R and further increase in R can have but little effect. With the smaller values of a (1.1 or 1.05), the limiting sharpness is approached more gradually and regularly as R is increased. This approach is often so
Another method of summarizing the effect of reflux ratio is given in Figure 6 in which the slope of the steepest portion of the break in the various curves is used as a measure of the sharpness of separation and is plotted against the rcflux ratio. This method would be useful in cases where an analytical solution for Equation 1 or 2 exists, because the value of the slope at the point of inflection where it has a maximum value can be found directly by mathematical methods and without resort to the plotting of graphs. Probably the simplest and most useful criterion in batch fractionation is the purity of an arbitrarily chosen fraction collected just prior t o the break in the distillation curve. In Figure 7 the purity (as to the more volatile component) of the fraction coming over between 40 and 45 per cent distilled
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tion under total reflux at the beginning of a batch distillation, and also a comparison of operation R ith constant reflux as compared with continuous or periodic variation of reflux, is possible when the methods of calculation described here are combined with the procedure recommended by Bogart ( I ) and with the fundamental material and heat balance procedures.
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Effect of Reflux Ratio on Effect of Efficiency /O
20
8
30
40
50
60 70 80 R REFLUX RAT/O
90
is plotted against the reflux ratio. This set of curves probably shows most clearly the effect of reflux ratio, and the variation Qf this effect with a, an,and n. The larger values of xp,a t 4045 per cent distilled over, represent sharper fractionation. Curves A , B, and E show the marked effect of reflux ratio when a and anare both large and R is small; curves C and D show the relatively slight effect when CY is small even though a" is just as large as for curves A , B, 9 w A00 and E. Comparisonof curvesCand $ .90 h G shows the effect of increasing n .go when CY is small. Since the time required for distil.70 lation of a specified amount of product .50 is directly propor0 20 40 60 80 IO0 tional to the reflux 1. R € f L U ! RAD0 OR P ratio used, curves R€U77V€ 77h9€ RfQL4PED such as those of Figures 5 to 8 give a relation between the time required and the results of a distillation.
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Economic Factors 1 ~ 1Figure 8 the average composition of the first 40 per cent distilled over is plotted against the reflux ratio for various typical cases of batch fractionation. Such relations are of great practical interest since the proportion of the charge that is recovered and the purity of the product obtained in a distillation determine the return on the operation, while the reflux ratio is directly related to the time, heat, and cooling water consumed, and so to the cost of the operation. A complete study of such a question would include prices of raw material, distillate, residue, and the variations in these with composition, as well as the relation of construction costs t o the number of plates and the effect of this variable on the results. An evaluation of the conventional period of opera
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Figure 9 gives calculated curves for the distillation of an ideal binary mixture with a = 1.25 and with reflux ratio 9 in apparatus with the equivalent of ten, twenty, thiity, and forty plates. Figure 10 gives similar curves for the identical case when R = 49. Figure 11 summarizes these and a number of other cases by the method of Figure 7. The heavy lines on Figure 11 refer to Figures 9 and 10 and show the large effect of n when R is large (49) and the slight effect when €2 is small. They also show that increase of reflux ratio beyond 49 gives little improvement in separation when a = 1.25, a d that there is little to be gained by using a large number of plates when R has as low a value as 9. The other curves of Figure 11 indicate the changes in these relations when a has smaller or larger values. When a is as small as 1.1, both high reflux and a relatively large number of plates are necessary for sharp separation. The gradual slope of both curves for CY = 1.1 indicates that increasing the number of plates will have but little effect in these cases, and that with too low a reflux ratio sharp separation will be impossible with any reasonable number of plates
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StillProduct Compositions by Transfer Unit Method The procedure described by Chilton and Colburn (8) was followed for obtaining a series of values of x, and x., and these were then used with Equation 1 in order to obtain the required distillation curve. The calculations were made for the distillation of a benzenetoluene mixture with initial composition 2, = 0.484 through a column with the equivalent of 11 transfer units with reflux ratio 2.38. Figure 12 shows the calculated curves for both the transfer unit and the correwonding McCabeThiele procedure (assuming eleven perfect plates). It is not yet known over how wide a range the two methods give substantially the same results. The transfcr unit method involves considerably more labor in cases where may be considered constant and when the number of plates becomes large.
Analytical Solutions An analytical solution of Equations 1 and 2 would have numerous advantages (7) but necessitates the use of a valid relation between product composition and still compositiorl in order to eliminate either z, or z, from Equation 1 or 2. 1357 assuming total reflux, relatively simple solutions have been
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TH€OR€7/CAL /?LA TES /n)
obtained (7, 8 ) , and these have been used to reach certain conclusions concerning maximum sharpness of separation, minimum plates required, and the importance of holdup when reflux ratio is large and separation is sharp. For cases of finite reflux, an analytical solution involves use of the Smoker equation or its equivalent for elimination of 2,. By replacing both sa and x+ by a new variable (Smoker’s quantity c) Equation 1 may be separated into a number of partial fractions, all but one of which may be exactly integrated. This one may be integrated by expansion into a series but unfortunately the series is so complex that such an analytical solution has no advantage over the graphical procedure. 0. T
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CURVES
always complicates the calculations or makes them more laborious. The use of the McCabeThiele procedure instead of the Smoker equation for establishing still and product compositions makes the method applicable to nonideal mixtures in which the vapor pressure ratio undergoes appreciable changes. If the usual simplifying assumptions are not justified, suitable corrections in still and product composition relations are possible by standard methods (8). The methods may be extended to the case of mixtures with three or more components by simultaneous solution of the proper number of equations similar to Equation 1. A general equation (6)was derived to apply to cases in which holdup is appreciable. It was thus shown (8) that a relatively small holdup cannot have a large effect on the shape of the distillation curve when the reflux ratio is equivalent to total reflux, and the agreement between experimental and calculated curves (IO)makes i t appear that this is equally true for fractionation a t finite reflux. The effect of reflux ratio and other operating conditions on the height equivalent to a theoretical plate is little understood, and such evidence as exists (6, IO) indicates that there are many cases in which the effect is unimportant. The methods described in this paper should serve as a useful means of investigating the nature of such relations when they are important, and the equations can then be modified to provide for such cases. I n evaluating and using the methods and conclusions presented here, it should be recalled that in ordinary batch fractionation practice i t is not usually convenient to make accurate measurements of reflux ratio and equivalent plates; in fact, it is probably difficult to make such measurements accurately. The exact confirmation of the theory of batch fractionation is less important than an approximate knowledge of how various conditions of distillation will affect the results of a particular fractionation.
tOMPAR/SON Of TRANSFER U N h AND THEORErr/CAI PLATE M€PfODS
Nomenclature
L x,
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The best hope for a practical analytical solution would seem to necessitate an entirely different method of expressing the stillproduct composition relations. It does not seem possible to obtain a solution based on the transfer unit concept because the relation between transfer units, still, and product composition cannot be solved for either of the last two. The same situation apparently applies to the Lewis relation (3).
687
= moles remaining in still at any time
mole fraction of more volatile component in still at any time z p = mole fraction of more volatile component in distillate at any time f ~ ( z J = holdup function (8) denoting holdup of more volatile component in colqmn when still composition is xa IC1 = integration constant or = vapor pressure ratio n = total equivalent plates R = reflux ratio (moles reflux t o moles distillate) or” = overall fractionating fqctor =
c 1 or1
k:
i=
m
=
R
e
= 1
+ (a  l)k
R+1
>
quantities used for convenience in Smoker equation
Limitations and Assumptions
Literature Cited
All the results obtained by the methods described in this paper are subject to limitations because of the assumptions involved. These are: 1. The mixture distilled follows Raoult’s law. 2. The variation in the vapor pressure ratio is negligible. 2. The usual simplifying assumptions regarding heat losses from the column and equimolal overflow, and equal heats of vaporization are justified. 4. The mixture distilled has only two components. 5. The holdup is negligible. 6. The efficiency is not affected by the reflux ratio or other operating conditions.
Bogart, TTans. Am. Znst. Chem. Engrs., 33, 139 (1937). Chilton and Cdburn, IND.ENQ.CHEM.,27, 256, 904 (1935). Lewia, Zbid., 14, 492 (1922). Robinson and Gilliland, “Elements of Fractional Distillation”, 3rd. ed., Chap. XI11 (1939). (5) Rose, IND. ENO.CHEW, 32, 675 (1940). (6) Rose, J . Am. Chem. SOC., 62, 793 (1940). (7) Rose and Welshans, IND.ENQ.CHEM.,32,668 (1940). (8) Rose, Welshans, and Long, Zbid., 32, 673 (1940). (9) Smoker, Trans. Am. Znst. Chem. Engrs., 34, 165 (1938). (10) Smoker and Rose, Ibid.,36, 285 (1940). (11) Walker, Lewis, McAdams, and Gilliland, “Principles of Chemical Engineering”, p. 632 (1937). (12) Young, “Distillation Principles and Processes”, p. 117 (1922).
The procedure may be extended to include cases in which the assumptions are not justified, although such extension
PRESENTED before the Division
(1) (2) (3) (4)
’of Petroleum Chemistry at the 99th Meeting of the American Chemical Society, Cinoinnati, Ohio.