BATCH RECTIFICATION Effect of Fractionation and Column Holdup Equations are derived f o r both binary and complex mixtures showing the yield of distillate of given Composition obtainable by batch reciijication of a given charge, taking into account both degree of fractionation and column holdup. It is assumed that the reflux ratio is continuously increased throughout the distillation t o maintain the product composition constant. The equations for binary mixtures are exact solutions; those for complex mixtures aflord a good approximation to results obtained by plate-to-plate calculations.
R. Edgeworth-Johnstone TRINIDAD LEASEHOLDS, LTD., ' POINTE-A-PIERRE, TRINIDAD, B. W. I
R
OSE, Welshans, and Long (6) and Colburn and Stearns (6)published methods of calculating the distillation curve of a binary mixture subjected to batch rectification, where the effect of column holdup is recognized. I n both cases a constant reflux ratio is assumed so that the distillate composition varies continually. I n practice the important question usually is, how much distillate of a given composition can be obtained from a given charge? To conserve heat, common practice is to start with a low reflux ratio and increase it as the cutting point is approached; the composition of the distillate is thus maintained approximately constant at the desired value throughout the distillation. Under these conditions the yield of dis.tillate cannot readily be computed by methods which start b y assuming a constant reflux ratio. Two factors reduce the actual yield of distillate below 100 pkr cent-incomplete fractionation and column holdup. The relative importance of these factors varies with circumstances a n d governs the type of column which will be most efficient for a particular duty. For example, packed columns usually .have a much lower holdup than plate columns of equal fractionating ability; but the number of theoretical plates which can be economically provided in a packed column is limited by the well-known channeling effect, which increases with the height of the column. I n certain circumstances, however, %he holdup effect may be preponderant, so that a packed column with relatively few theoretical plates and low holdup will give far better yields than a plate column with many more theoretical plates and a higher holdup. It does not seem possible by existing methods to decide upon the most advantageous arrangement without long and tedious calculations, and then only in the case of binary mixtures. Equations are developed here which give the yield from batch rectification directly and take into account both the factors mentioned above. The basic assumption is that the reflux ratio is progressively increased throughout the distillation, so as to maintain the distillate composition constant, and reaches infinity a t the cutting point. The assumption of
constant distillate composition permits the final distribution of components to be computed from conditions at the cutting point alone, using the comparatively simple equations applicable to total reflux. The method can be extended with good approximation to complex mixtures. BINARY MIXTURES
Consider first an ideal column having N theoretical plates and no holdup. A batch of F moles of a binary mixture, in which the mole fraction of the more volatile component ( A ) is a,, is distilled with progressively increasing reflux ratio to maintain the distillate composition constant at a,. This may be done by keeping the overhead vapor a t constant temperature by automatic control; a constant rate of heat input to the still is also maintained. If temperature control is not sensitive enough, Bogart's method (1) of calculating how the reflux ratio should vary may be applied. Such conditions will henceforth be referred to as constant distillate composition (C. D. C.) conditions. At the end of the distillation the reflux ratio is infinity. For total reflux the Fenske equation (4) applies, and the composition of the bottoms is, therefore,
From material balances at the end of the distillation, P+W=F Pa, Vu, = Fur
+
Substituting for ulo and simplifying, aN+l
P = ( F Z )
- (A ('+ ) ) 1 - a, aN+l - 1
The total quantity of distillate of composition a, contained in the original charge is F(af/a,). Dividing by this quantity gives the yield fraction as shown in Equation 1. 40iI
INDUSTRIAL AND ENGINEERING CHEMISTRY
408
The correct value for plate columns is found by integrating between 0.5 and ( N 0.5) theoretical plates:
+
1 - a, ti=
-1
aN+l
This equation represents the yield from an ideal column having no holdup. It is less than unity owing to incomplete fractionation-i. e., because N is finite and a certain quantity of A must therefore remain in the still. Kow consider an actual column having a holdup of Q moles per theoretical plate with other conditions the same as for the ideal column. At the beginning of the distillation the column is assumed to be empty. The final bottoms composition, a,, is the same as for the ideal column, but there is held up in the column NQ moles of a mixture, the average composition of which lies between a, and a,. To calculate the yield fraction it is necessary to find the average composition of this mixture, deduct the total moles of A and B held up from those present in the original batch, correct F and a/ accordingly, and insert these corrected quantities in Equation 1. If QZ represents the total moles of A held up in the column, the material balance equations become: P+W=F-NQ Pa,
+ W a w = Fa, - QZ
The problem is to determine 8. Applying the Fenske equation liquid on the top plate is:
(4, the composition of the
Similarly, a2 =
ap
a2(1 - up)
Vol. 35, No. 4
+ a,
Substituting for azoin the material balance equations and simplifying,
Dividing by F(a,./ap) and putting Q/F = q,
Equation 4 gives the yield fraction from an actual batch column operated under C. D. C. conditions; both incomplete fractionation end column holdup are taken into account. It is an exact solution, provided that the appropriate equation is used to evaluate Z and that the underlying assumptions apply. These assumptions were total reflux a t the cutting point, validity of the Fenske equation, and neglect of the effect of vapor on the average composition of the material held up. According to Underwood (6),the Fenske equation may be applied to all binary mixtures in which the components can be assumed to have a constant average relative volatility throughout the distillation, even though they do not conform to Raoult's law. If the column is not empty at the beginning of the distillation, its contents must be added to the original charge; both F and a/are corrected accordingly. COMPLEX MIXTURES
The proportion of vapor in the column is assumed to be so small that the average composition of the total material held up is not sensibly different from that of the liquid held up. The total moles of component A retained in the column are then :
The summation of the series represented by Z was found by Rose, Welshans, and Long (6) to be:
For present purposes it is more convenient to have Z in terms of product composition. Substituting according to the Fenske equation:
Consider a mixture consisting of components A , B, C, etc., from which a distillate containing a relatively high proportion of A is required to be separated by batch rectification under C. D. C. conditions. A and B are the key components, and the distillate may, without serious error, be assumed to consist of these components only. Provided the key components obey Raoult's law, the Fenske equation applies in this case at infinite reflux ratio as if the charge were a binary mixture of A and B only; that is,
For an ideal column with no holdup, the material balance equations are: P + W = F Waw = Fa/ Pa, Pb, Wb, = Fb/
++
Evaluating the yield fraction from these equations,
(5)
This expression was derived by integrating between 1 and N theoretical plates, assuming a continuous change of composition. Hence it is strictly valid only for packed columns. For a plate column it gives a value of Z somewhat too low. This can be seen by putting N = 1,when for a plate column Z should be numerically equal to at = ap/[cr(l - a,) a,].
+
This is the same as Equation 1 except that for binary mixtures bf = 1 - a/. The only effect of the heavier components present is to raise the temperature a t the bottom of the column and thus slightly reduce the average value of a! compared with that applying to a binary mixture of A and B. The important difference between binary and complex mixtures lies in the effect of column holdup. Consider a complex
April, 1943
INDUSTRIAL AND ENGINEERING CHEMISTRY
mixture and a corresponding binary mixture in which the relative proportions of A and B are the same. F moles of the complex mixture contain much less of component A than F moles of the binary mixture owing to the presence of the heavier components. Yet at the end of the distillation the amount of A held up in the column from the complex mixture will be nearly as great as that from the binary mixture. Therefore the reduction in yield fraction due to holdup will be much greater in the case of the complex mixture.
409
consist principally of the key components. The holdup of A in this region is thus practically the same as that with the corresponding binary mixture. Lower down the column substantial concentrations of heavier components appear, which reduce the holdup of lighter component per theoretical plate below the value obtaining for the corresponding binary mixture. But since the holdup per plate in that region is already low, the effect on the total quantity of distillate held up is slight.
Courtesy. Foster Wheeler Corporation
S t e d m a n B a t c h Distillation Unit at Trinidad
The fact that nearly as much of component A is held up from the complex mixture as from a corresponding binary mixture is easily understood when it is remembered that the lighter component is held up chiefly in the top few plates. Here, even with the complex mixture, both liquid and vapor
Table I Mole Fraction Mole Fraction (from Underwood Curves) plste No. Hexane Heptane Octane Distillate 0.64 0.46 Nil 1 0,346 0.64 0.016 2 0.035 0.18 0.786 3 0.096 0.08 0.826 4 0.04 0.60 0.79 5 0.316 0.02 0.666 6 0.01 0.60 0.49 7 N i l 0.70 0.30 Total for 7 plates 0.690
~
~
~
Mist. 0.54 0.360 0.186 0.103 0.048 0.029 0.020 N i l 0.736
~
An illustration is the plate-to-plate computation of Underwood (6) for a mixture of hexane, heptane, octane, nonane, and decane; the column holdup of hexane may be compared with that which would occur with the corresponding binary mixture of hexane and heptane. Only the first seven plates need be considered, since below them the hexane concentration is negligible. I n this region the only components present in appreciable quantity are hexane, heptane, and octane as shown in Table I. The total column holdup of hexane for the five-component mixture is thus only 6 per cent lower than that for the corresponding $ binary ~ $ mixture. $ ~ ~ ~ ~ ~ I n view of the relatively small proportion the quantity of material held up in the column normally bears to the total quantity of distillate, a reasonable approximation in the case of complex mixtures is to assume that the value of I: is the same as that calculated for the corresponding binary mixture. This is given by Equations 2 or 3 according as the column is of the packed or plate variety. I n other words, the column is assumed to contain A and B only.
I N D U S T R I A L A N D E N G 1.N.E E R I N G C H E M I S T R Y
410
For a n actual column with appreciable holdup, the material balance equations applying t o a complex mixture are: P + W = F - NQ Pa, Wa, = Fa/ - QZ Pb, Wb, = Fbf - ( N Q - QZ)
++
(The column is again assumed to be empty a t the start of the distillation.) Combining these with the Fenske equation and solving for P ,
Dividing b y F(af/ap) and putting Q / F
= q,
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condition. The choice lies between a plate column having ten theoretical plates and a holdup of 15 moles per theoretical plate, and a packed column of five theoretical plates with a holdup of 3 moles per theoretical plate. The charge in each case is 2500 moles, and distillation is to take place under C. D. C. conditions a t atmospheric pressure. What n-illbe the respective yield fractions from the two columns? The molal average boiling points of the key components are found to be 110" and 150" F. From flash vaporization curves or other data the temperatures a t the top and bottom of the column are estimated at 120" and 240' F., respectively. Using these temperatures and the equation mentioned above, the relative volatilities of the key components at the top and bottom of the column are calculated to be 2.24 and 1.74, respectively, giving a n average relative volatility a = 1.99: bf = 0.05 a, = 0.95 For packed column: Z = 3.03 (Equation 2) For plate column: Z = 3.86 (Equation 3) a/
Equation 6 is a n approximate solution for complex mixtures, based upon the assumptions underlying Equation 4 plus the further assumption that the key components follow Raoult's law and that a t the cutting point the column contains these key components only. The last assumption brings the calculated yield fraction slightly below the true value; hut since column holdup is usually small compared with the total yield of distillate, the error is likely to lie within the limits of accuracy of the data and simplifying assumptions. A comparison will be given of the yield fraction as calculated from Equation 4 and that obtained from a plate-toplate calculation. Consider a mixture of three components A , B , and C in which at = by = 0.21, cj = 0.58, cy = p = 1.5. A thousand moles of this mixture are t o be rectified under C. D. C. conditions in a column having ten theoretical plates and a holdup of 2 moles per theoretical plate. The product is required to contain 90 mole per cent of component A . A plate-to-plate calculation gave the following figures: an = 0.897, b, = O.lOO,c, = 0.003; a, = 0.025, b, = 0.239, cw = 0.736. Moles of each component held up in the column under total reflux are: A , 8.63; B , 6.84; C, 4.53. Material balances based on these figures give' y = 0.867
Taking a, = 0.897 and applying Equation 3, Z = 4.97
substituting in Equation 5 and solving:
=
0.04
Substituting these values in Equation 6 and solving, For packed column: y = 0.560 For plate column: y = 0.415
The utility of the method presented is principally that i t offers a means of comparing the suitability of different columns for a separation which is intended to be carried to a high final reflux ratio. Whether this is a n economic procedure in any given case must be decided by other methods. ACKNOWLEDGMENT
The author is indebted to Arthur Rose for valuable criticism, and desires to thank the chairman and the board of Trinidad Leaseholds, Ltd., for permission t o publish this paper. NOMENCLATURE
A , B, C, etc. = components in order of decreasing volatility aj, a,, aw,an = mole fraction of component A in charge, distil-
late, residue, and on nth plate, counting from top of column, respectively same for component B total moles of charge, distillate, and residue, respectively number of theoretical plates in column moles held up in the column per theoretical plate Q/F
molal average boiling points on absolute scale of fractions taken as key components inocalculations relating to complex mixtures, R. or K. Y = yield fraction (1OOy = per cent yield) W , P = average relative volatility of Components A / B , B/C-i. e., mean of values at top and bottom temperatures of column relative volatility of components A / B at absolute temperature 1' O
y = 0.863
I n t,he case of complex mixtures such as petroleum, it is usual to treat comparatively narrow fractions as pure components for calculation purposes. ElseTTliere it has been shonn that the relative volatilities of such fractions at any temperature can be conveniently estimated from their molal average boiling points by the equation (3) :
ap
a,
a(1
- ap) + ap + az(1 - a,)
a, 4 1 - ap)
+ a,
.
+ ap +
ap
' ' ' '
+a ~ (1 a,)
+ ap
LITERATURE CITED
As an example of the use of Equation 6, a certain petroleum distillate contains 4.0 mole per cent of material boiling between 80" and 140" F., and 5.0 mole per cent boiling between 140" and 160" F. This distillate is to be rectified by the batch process, and it is required to separate the 80-140" F. fraction (which is the lightest material present) in a 95 per cent pure
(1) Bogart, M. J. B , Trans. Am. Inst. Chem. Engrs., 33, 139 (1937). ( 2 ) Colburn, A. P., and Steams, R. F., Ihid., 37, 291 (1941). (3) Edgeworth-Johnstone, R., J . Inst. PetToZeum Tech., 25, 558 (1939). (4) Fenske, M. R., IND.ENG.CHEIIZ., 24, 482 (1932). (6) Rose, Arthur, Welshans, L. M., and Long, H. H., Ibid., 32, 673 (1940). ( 6 ) Underwood, A. J. V., Trans. Inst. Chem. Engrs. (London), 10, 112 (1932).
