Simplified Calculation of Theoretical Plates Fractionation of a Binary Mixture in Which the Concentration of One Component Is Small GEORGE W. THOMSON AND HAROLD A. BEATTY Ethyl Corporation, Detroit, Mich.
T
method of Dodge and Huffman for the treatment of this difficulty is given below. Also, the separation of a binary mixture containtheir equation is necessarily per cent Of One ing less than somewhat inexact when applied ponent. The assumptions involved are that to a plate column, owing to the operating lines are linear and that the the use of integration methods curve is also linear near the which hold strictly only when the number of plates is very ends of the x-Y diagram. The data prelarge* In general, for a plate sented show that the number Of dates column or for the comparison calculated by these methods does not differ of any separating device with a significantly from the number calculated plate column, the only exact methodisbythesummationof b; the smoker equation (10) or by finite steps. The use of any the McCabe-Thiele graphical method. integration method involves introducing the concept of the transfer unit which is foreign to the present problem, The use of Dodge and Huffman's Review of Different Methods Equation 9 gives good results when the slopes of the equilibrium curve and the operating line are about the same, but Sore1 in 1893 (11, 12) discussed the most general case, in increasingly poorer results as the slopes become more diverwhich variations in the volume of overflow and the heat congent. This relation appears to be independent of the number tents of the liquid and vapor streams a t various parts of the of plates. column were all taken into account. His method involves An exact equation was obtained by Smoker (10). It uses a heat and material balances around every plate and, although summation of finite steps and so can be applied to any probvery tedious to work out, is exact. An excellent summary in lem in the present class, the only assumptions required being modern notation is given in Robinson and Gilliland's text OD constance of molal overflow from plate to plate and condistillation (9). stancy of the relative volatility, a. The numerical work Since then many contributions have appeared which make required to solve the equation is considerable; each single use of various simplifying assumptions. The most useful is value of the number of theoretical plates, n, takes about 30 the graphical method of McCabe and Thiele (6), which asminutes with a calculating machine. sumes constancy of molal overflow. If the molal latent heats When the feed is far from pure but the product (either overof the two components are very different, a modified equihead or bottoms) is obtained in a fairly high state of purity, librium curve can be drawn and the McCabe-Thiele graphical and where the above two assumptions of constant overflow method can still be applied (8). Thus this method is of wide and constant a hold, a useful although inexact solution is utility; but when the number of plates is large, the labor of given by Underwood in his classic paper (16) in a set of two drawing the steps is considerable, especially near the ends of e"quations (numbered 37 and 39). Owing to the approxithe equilibrium curve where replotting on a larger scale or mations made in the derivation of these equations, they can even on log-log paper is necessary. be shown to give inaccurate results when applied over a Various analytical methods of computation have been denarrow range close to the ends of the x-y diagram. His vised. At infinite reflux, the problem is readily solved if the Equation 37 gives successful results when the overhead comrelative volatility, a, is assumed to be constant or if an position, ZD, is near unity, and the composition of the feed, appropriate mean value can be chosen. The more complex ZF, is not close to ZD. For the stripping column, his correproblem of operation a t any workable reflux ratio, again with sponding Equation 39 gives successful results when the a assumed to be constant, also has an exact solution. bottoms composition, ZW, is very small and ZF is not too close Dodge and Huffman (4, 6) presented a solution of this to ZW. Underwood suggested a small correction, to make his problem in their Equation 9; but in addition to being someEquation 37 more useful when ZD is not very close to unity. what laborious to solve, it becomes impractical when either Instead of the reflux ratio, R, a modified value equal to R ( 2 component is separated in a high state of purity, as small ZD) may be used. These equations are not recommended for differences between large numbers become involved. The
HE calculation of the number of theoretical needed to achieve a given separation of a binary mixture is a common problem in continuous distillation, and one which usually requires a considerableamount of numerical and graphical computation. This paper presents a simplified method of making this calculation for a separation over a narrow concentration range near one end or the other of the equilibrium curve and shows how the method can be extended to wider concentration ranges.
Simplified methods are presented for calculating the number of theoretical plates for
exact
1124
INDUSTRIAL AND ENGINEERING CHEMISTRY
September, 1942
use over small concentration ranges, where the errors may become very large; however, over suitable ranges they are valuable, since they afford a rapid solution. The decision as to whether or not they should be used in any particular case will be further discussed below. Frequently in practice, both the feed and one of the products (either overhead or bottoms) contain one of the components as an “impurity” in a low concentration. Then for the separation over the narrow concentration range between the feed and the one product it becomes permissible t o make the simplifying assumption that the liquid-vapor equilibrium curve is a straight line over this range. Dodge and Huffman (6) make use of this assumption in their Equations 11 and 12, in order to avoid the difficulties inherent in their abovementioned general Equation 9, near either end of the equilibrium curve. These new equations are inexact for the same reason as is their Equation 9-namely, the use of integration methods. I n some cases the error from this source compensates in part for the error due to the assumption of a linear equilibrium curve, but in other cases these two errors are cumulative. Underwood (16) gives the derivation of an exact equation, due to Thormann (16),obtained by making this assumption of a linear equilibrium curve. The equation given applies to the reduction in concentration of a higher boiling impurity in the rectifvine: section: Underwood mentions that “a similar equation can be derived for the exhausting column”, but he does not indicate the-utility of such Case equations other than by reference to one Type example, the rectification of aqueous No* alcohol, nor does he discuss the concentration ranges over which they apply. Rectifying I After the present work was completed, Suen (14) published a method using the absorption factor charts of Brown and associates (e,$,13) for a graphical solution Rectifying I1 near the ends of the x-y diagram and making the same basic assumption of a linear equilibrium curve. A certain amount of calculation is necessary before the charts can be used. One definite disadvantage of the method is that even the best of these ‘I1 Stripping charts (2) covers but a limited range of the variables involved and is difficult to read accurately for large values of n. The present authors have developed a set of four equations for the rapid solution IV Stripping of plate calculations in the distillation of compounds containing small amounts of impurities. One of these is identical with Underwood’s equation modified from Thormann. The two inherent assumptions are: (1) The operating line is linear (which is equivalent to the usual assumptions for the McCabeThiele diagram) ; and (2) the equilibrium line is straight. The latter assumption is true, of course, only near the ends of the diagram. The concentration ranges over which the equations may properly be applied are considered in detail below.
Derivation of Basic Equations For any linear equilibrium curve (y = a bx) and operating line (y = c dx) the number of steps, n, between any two concentrations, $0 and G,may readily be shown by a stepwise summation to be given by:
+
+
c - a
a);(
=I
xo xn
-b‘, where x,, > xo - a - cb--d
(l)
1125
This relation may be applied to either the rectifying or the stripping section, for impurities of volatility either higher or lower than the principal component. For these four general cases, the corresponding values of a, b, e, d in terms of the relative volatility, a,the reflux ratio R or R‘ ( R = L / D above the feed plate, R’ = €/W below the feed plate), and the terminal compositions are given in Table I. TABLE I. VALUESOF a, b, c, d Case
No.
Type
Volatility of Impurity
Rectifying
Lower
I1
Rectifying
Higher
I11
Stripping
Lower
Stripping
b
d
C
- - R+ 1 R+1
I
IV
a a - 1 a
1
a
R
XD
0
- -1
a-1 01
Higher
R
XD
0
cy
01
R+1 xw R’ R ’ - 1 ?R
R
-
T
xw
-
R’
Substitution of these values into Equation 1 gives the set of equations listed in Table 11.
TABLE 11. GENERAL EQUATIONS Volatility of Impurity
Equation
No.
- xo (aR - R - 1) + a (L.z\n = - xD R +.~l / l-Xz,.- . . - (a& - ti - 1) + a 1 - ID Xn -(aR - R + -1 1
Lower
(\-3 ) ,
01)
Higher
-a(R=f 1)
(
I
)
Where XF 4 xo
XD
(3)
S(aR-R+a)-l ZD
< xn Q XD
- xo
+ 1) - a (018’- R’ + 1) 1 -XK 1
Lower
(aR’- R’
(4)
01
- (aR’- R’ (W)” xw - - R’ Xm
Higher
= xw xo
(aR’
Where xw
+1 a) + 1 a)
(5)
< xo < xn C XF
When the terminal concentrations xo and sn are, respectively, equal to XF and XD (for rectification) or to ZI and XF (for stripping), the equations simplify to the forms given in Table 111. Graphical solutions of Equations 6 to 9 have been devised, but have always been found to be more time consuming and less reliable than the direct numerical solutions.
Allowance for q Line Placing X E equal t o one of the terminal concentrations in developing Equations 6 to 9 implies that the feed is wholly liquid a t its boiling point. A procedure presented below was worked out for any condition of the feed. If we define p as the difference in molal overflow above and below the feed plate, divided by the number of moles of feed
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
1126
Vol. 34, No. 9
Applicability of Equations
TABLE111. SIMPLIFIED EQUATIONS
Equations 2 to 9 do not give exact results, as does the Smoker equation, because the assumption of a linear equilibrium curve always introduces more or less error. It is desirable to estimate the magnitude of this error for (a - 1)(R 1) any given set of conditions to be sure I1 Rectifying Higher (7) that it is small enough to be neglected. ?E (aR - R + a) - 1 XD The error depends on R, a,XF, and ZD (or ZW), but owing to the complexity of (a - 1)(R’ - 1) Lower Stripping I11 = 1 - ZF the Smoker equation, it has not been (&’ R’ + 1 ) - a found possible to express it in a simple 1 - xw manner. However, a systematic study XF has been made of the magnitude of the (aR’ - E’ - 01) + 1 xw error for different values of the above IV Stripping Higher variables. This study was confined to Note that n includes the feed plate but not the still pot. the use of Equation 9, since the results can be extended to include the remaining equations. For Equation 9 the percentage error increases as: (1) Z F increases, (2) the separation, Z F / Z W , decreases, (3) the reflux (i. e., q = (L--L)/F), then p is readily obtained for any ratio decreases, and (4) a decreases. condition of the feed by a heat balance. An exactly equivaA limiting case may be made up so that the error, for all lent definition is that q is the total heat needed to convert one practical cases, will be less than the error of the limiting case. mole of feed into saturated vapor, divided by the molal latent First, an arbitrary upper limit of ZF = 0.05 is taken as reasonheat (1). It is readily shown that the line able; above this concentration the increase of the curvature of the equilibrium curve might seriously invalidate the use of the equation. Secondly, the lower limit of the reflux ratio is taken as 20 per cent greater than the minimum, since commeris the locus of the intersection of all possible operating lines. cial columns are seldom run a t reflux ratios less than this, It has been named the “g line”. The effects of varying X F / X W and a,for these limiting condiIf xi is the intersection of the operating lines and the p line, tions of XF = 0.05 and R‘ = 1.2 RImin.,are presented in a separation may be considered to be either from xw to xi or Table V. from xi to Z D , according to whether the stripping or rectifying At a reflux ratio of only 10 per cent above the minimum and sections are being considered. The value of zi is easily oba t the same XF = 0.05, the error for a! = 1.5 is 3.2 plates tained from the equations: (14.7 per cent) a t Z F / ~ W= 10, and 6.7 plates (7.7 per cent) a t Z F / $ W = 1000. From these results the general statement may safely be made that, for the majority of all practical cases, the error To apply these results, xi is substituted for Z F in the approcaused by the use of Equation 9 is less than 10 per cent for priate equation of Table 111. X F / X W = 10, less than 7 per cent for Z F / Z W = 100, and less than 5 per cent for Z P / Z W = 1000. Limiting Conditions These conclusions for Equation 9 are directly applicable to Equation 6, as may be noted from the symmetry of the Table IV presents the exact relations for the minimum equations. Although no analogous systematic study has reflux ratio and for the minimum number of plates at total been made for Equations 7 and 8, it is believed that the errors reflux, based on the use of Raoult’s law with constant a. It introduced are similar to those given above, for corresponding has been found, under the conditions where Equations 6 to 9 values of the variables involved. For example, Dodge and may properly be applied, that the limiting values which can Huffman (5) present a problem in the concentration of ordibe calculated from these equations are but little different nary water from 0.2 mole per cent HzO1*t o 2.54 mole per cent, from the values calculated from Raoult’s law. with a = 1.006, R’ = 3920 (R’/R’min. = 2), and (1 - ZW)/ (1 - Z F ) = 12.7. Smoker’s equation gives n = 523.7, and Equation 8 gives 518.8, an error of 0.94 per cent. LIMITING CONDITIONS FROM RAOULT’S LAW WITH TABLE IV. I n conclusion, the equation may be used over the conCONSTANT 01 centration ranges of 0-5 and 95-100 per cent with an error of Case Minimum No. of Plates less than 10 per cent, when using reasonable reflux ratios, No. Minimum Reflux Ratio a t Total Reflux Usually the error is much less than 10 per cent, being often of the order of a fraction of a plate. Case No.
Volatility of Impurity
Type
Equationa
+
-
XF(1
111, IV
01
-
(XW/XF)
(a
-
-
1)(1
(a
-
- XR)
UZW
- xw)
log X r p ( 1 log a
XF)
The calculated minimum number of plates includes the feed plate but not the still pot.
Choice of Proper Value of a Since no two compounds have exactly the same vapor pressure-temperature relation, it follows that a is not strictly constant, as has been assumed in the derivation of the equations, although the variation over the concentration range involved may be very small. For example, for the benzenetoluene system a t 760 mm. a decreases from 2.60 a t x = 0 to
INDUSTRIAL AND ENGINEERING CHEMISTRY
September, 1942
-
PRACTICAL ERROR FOR EQUATION ga TABLE V. MAXIMUM (ZF
1.6 3 6 12 0
16.1 5.5 3.0 1.9
15.6 5.0 2.7 1.7
9.3 8.9 8.7 8.6
P
0.05 and R’
37.0 11.7
(38:;)
34.6 11.0 5.8 3.7
1.2 R’min.)
6.6
(g:6;). 1
59.8 (18.3) (9.4) 5.8
57.1 17.5 9.0 5.6
Values in parentheses are interpolated.
2.35 at z = 1; and for the n-heptane-methylcyclohexane system a increases from 1.07. to 1.09. Although these variations may appear trivial, substitution in Equations 2 to 9 will readily show that small changes in the value of a, especially when a itself is small, cause relatively large changes in the number of plates calculated. I n practice, no significant error is introduced by assuming that a is constant over a narrow concentration range, if a proper choice is made of its value. Inspection of McCabeThiele diagrams near x = 0 and x = 1, and consideration of the fundamental derivation of Equations 2 to 5 show that the best value of a to use is that corresponding to the concentration where the change in composition per plate is least. This point is readily located from the relative slopes of the operating and equilibrium lines, since these determine where the “pinch” is. In Equations 6 and 8 the use of the value of a a t x = 1 is usually entirely adequate, while in Equations 7 and 9 the similar value a t z = 0 is equally adequate. Several authors have advocated the use of an average relative volatility or an average value of K = y/x to be applied over a section of a distillation column. Such a procedure may lead to erroneous results. For example, in this laboratory it has been found, in the rectification of ethyl chloride containing 1 per cent of hydrogen chloride to 0.001 per cent, that a varies from 12.00 to 12.56 over the concentration range Assuming a reflux ratio of 1.2, and using the terminal value of a = 12.00, Equation 9 gives 9.09 plates and the Smoker equation gives 9.26 plates for the rectifying section. A carefully drawn McCabe-Thiele diagram, which took into account the change of a with concentration, gave 9.20 plates. But taking a mcati value of a = 12.28, the Smoker equation gives onlj 9.00 plates, which is lower than any other of the values c a l d a t e d . The equilibrium constant, K = y/x, varies from 12 (at 0.001 per cent) to 10.8 (at 1 per cent). Taking an average value of K = 11.4 in place of a,Equation 9 gives 9.74 plates and the Smoker equation gives an even higher value, close to 9.9. Thus the use of average a or K values is less satisfactory than the general rule of using the value of a near the point where the maximum number of plates is concentrated.
1127
ing line lies close to the y = x diagonal (which corresponds to total reflux) over most of its length. The method assumes total reflux up t o a certain selected concentration, x’, which is somewhat below XF, and then applies Equation 4 from x’ to z”, the concentration on the plate 4.5 below the feed plate. The selection of 2’ is described below. 3.9 This method gjves good results which are always slightly less than the true number of plates. The details of computation are presented below for a typical example, and a comparison of the results with the true (Smoker equation) values is given, for various values of R and x’, in Table VI. As n’ is decreased, the error in using a straight line instead of the true equilibrium curve becomes dominant; while as x’ is increased toward x”,the error inherent in the assumption of total reflux up to x‘ is magnified. Thus, for every R there is an optimum value of x’ which gives the minimum error. EXAMPLB. Given the conditions: a = 4, X D = 0,9999, XF 0.99, and zw = 0.10:
{a:]
L=
The calculation of the number of plates, n, in the stripping section (for example, for R = 1) is as follows: y” = mole fraction of more volatile component in vapor from plate below feed plate = (RXF ~ D ) / ( R 1) = 0.99495, and 1 - y” = 0.00505. Whence,
+
1
-
4 1
=
(a
+
- ti’)
- 1)(1 - y”) + 1 = 0.01990
From a material balance, R’
=
(-)
+ (-)
R
for R = 1, R‘ = 180.7878.. . . . . . . . . . .
(11)
Arbitrarily (see below), let x’ = 0.80; 1 - Z’ = 0.20. Let n2 designate the number of plates between n’ and x”, as given by general Equation 4 for case 111. Then, 1 - 2’ [(a - 1)R’ 11 - a 1 - xw (12) 11 - a 1 - xw [(a - l)R’
+
+
whence% = 1.924
Assuming total reflux from x w to x’,and designating by nl the number of plates over this range, the equation for total reflux gives:
Divided-Range Method (In applying Equation 13 to this example, a is assumed to be When a small percentage of impurity is being removed by constant. If a varies over the range zw to z’, either a suitable distillation, it is usually required to calculate not only the mean value must be taken or for abnormally-shaped x-y number of plates needed to reduce the concentration of the diagrams such as for the alcohol-water system, a McCabeimpurity to the desired level a t one end of the column, but Thiele diagram must be drawn.) also the number of d a t e s needed t o concentrate the impurity at &e other end of the column to the point where it can economically be disTABLE VI. DIFFERENCE IN 12 BETWEEN DIVIDED-RANGE METHODAND carded. A “divided-range” method has been SMOKER EQUATION FOR VARIOUS VALUESOF x’, worked out for this specific problem. The foln (Smoker Eq.) - n (Divided-Range Method) at 3’ Value of: sm2ker lowing two cases are considered: case DR-I R Eq. 0.75 0.80 0.85 0.90 0.95 0.96 0.97 with Z F close to 1 and ZW small; case DR-I1 0.35 7.704 0.297 0.257 0.224 0.216 0.247 0.284 0.357 0.4 7.024 0.272 0.231 0.198 0.189 0.217 0.251 0,321 with Z F close to 0 and ZD large. 0.5 6.467 0.256 0.215 0.182 0.171 0.193 0.224 0.287 0.7 6.007 0.242 0.200 0.166 0.152 Case D R I only will be considered in detail 0.165 0.190 0.242 1.0 5.697 0.231 0.188 0.153 0.135 0.138 0.158 0.198 (an identical treatment can be applied to case 8.0 5.197 0.203 0.159 0.117 0.093 0.070 0.075 0,103 DR-11). Since ZF is close to 1, the lower operat-
1128
INDUSTRIAL A N D ENGINEERING CHEMISTRY
The total number of plates, below and including the feed n2 1= plate but excluding the still pot, is equal to nl 5.509, which may be compared with the Smoker equation value of 5.697. The number of plates above the feed plate may be readily obtained by the application of Equation 6. For various values of R and d , the differences between the number of plates in the stripping section by the divided-range method and the corresponding Smoker equation values are shown in Table VI. The differences are given to three decimal places to show the positions of the minima. The value of x' which gives the minimum error in the divided-range method is readily obtained by a simple trialand-error procedure. Near the minimum point the error is not very sensitive to the value of x', as seen from Table VI. Experience has shown that the minimum error occurs when the value of nl/n lies between 0.4 and 0.6. The following rule has been found t o give rapid, reliable results in the selection of a proper value of x': (1) Take a trial value of z' somewhat below ZF; (2) calculate nl and n as in the example above; (3) if nI/n lies between 0.4 and 0.6, no further calculation is necessary; (4) if nl/n lies outside these limits, take a new value of nl equal to 0.45 of the value of n obtained above, calculate the corresponding-z' using Equation 13, and proceed as before. I n the example above, nr/n is 0.47 for R = 1. A plot of the values in Table VI for R = 1 gives the minimum error a t 2' = 0.91 which corresponds to nl/n = 0.58. Since the slope of these error curves is much steeper on the feed plate side of the minimum, the lower value of nl/n is by far the safer value. A further example illustrating this principle with a much larger number of plates follows: Table VI1 presents the error in the number of plates computed by the divided-range method for a = 1.4, XF = 0.99, zry = 0.10, and R = 2.7 (near the minimum R of 2.49). The Smoker equation gives 28.852 plates, including the feed plate, for the stripping section.
+ +
Vol. 34, No. 9
TABLEVIII. RULESFOR CHOICEOF BESTCALCULATIOK METHOD Stripping Section 100 zw Nethod" 0-2 (9)
100 ZF 0-2 2-5
0-5
5-20
0-20
(9)
M-9 or
S
20-80
Rectifying Section 100 ZD Methoda
(7) Ivl or S DR-I1 (7)
0-5 5-70 70b-100 2-5 5-100 5-100 98-100 20-98 98-100 80-100
11-6 or S M-6 or S U(37) M or S U(37) M-6 or S
0-2 U(39) 2-80 M or S 80-95 0-2 U(39) 0-95 M-9 or S 95-98 0-95 M-9 .or S 95-100 (6) 95-98 (8) 98-100 0-30 b DR-I 9s-100 (6) 30-95 M or S 95-100 (8) a (6) (7) (8), and 9) refer to equations of Table 111. DR-I and DR-I1 ale oorrespbndi&gdivide6-range methods. U(37) U(39) :re Underwood's equation numbers: M signifies McCabe-T'hiele &phical method; &I-6 refera. to the use of Equation 6 to supplement the McCabe-Thiele method below Z~ 0.02; M-9 refers to the similar use of Equation 9 above zD = 0.98; S sign,fies Smoker's equation. b Theae DR limits are to be used with discretion, as noted in connection with the method.
7
tion range. We believe that when the methods indicated are employed, the error will always be less than 10 per cent of the true value, in many instances being only a fraction of a plate. Of course, if only a single value, in contradistinction to an R vs. n or an R us. concentration range study, is needed, the exact answer may be obtained by Smoker's equation with less than half an hour's computation on a calculating machine.
Use of Equations for R us. n Studies
In economic balances of continuous distillation columns, &heR us. n curve, which gives the number of theoretical plates against reflux ratio, is invaluable. Over the proper concentration ranges, Equations 6 to 9 and the divided-range IN DIVIDED-RANGE METHOD FOR VARIATION TABLE VII. ERROR method give a ready means of obtaining this curve. By means IN 7LI of suitable tabular forms, which will be illustrated below, a nl Plates Error ndn complete R us. n study may be computed with but little more 8 26,882 1.970 0.298 12 27.751 1,101 0.432 effort than that required for a single value. This ease of cal15 27.701 1.151 0,541 18 26.603 2,249 0.677 culation is mainly due to the linearity of the functions which 19 25.469 3.383 0.746 appear in the tables. The errors involved in the use of the equations for this purpose are small, as shown above, and are well within the limit of the other errors and approximations The minimum error lies a t about n~ = 13.5, with nl/n = inherent in economic balances. When R us. n is plotted, the familiar hyperbolic type of 0.49. Even in this extreme case (low reflux ratio and a) the curve results. Interpolation, which is quite difficult on this actual error is very small, 1plate out of 29. The basic assumptions inherent in this divided-range method must be kept in mind when applying it to a specific problem since, TABLEIx. CALCULATION O F PLATES IN RECTIFYING SECTION if a is very large and ZF is greater than 2 per a + cent (DR-11) or less than 98 per cent (DR-I), 1 - ZF 1) x nreot. any assumption of total reflux may be unallowTheoretical Actual R Bo % R (G)( R + 1) able, owing to the geometry of the correspond1.2 0.068 45.2 1.0309 125.2 157 88.5 1.958 51.1 63.9 315 2.136 147 1,1025 1.4 0.246 ing McCabe-Thiele diagram. 1.6 0.424 542 2.314 234 1.1631 36.1 46.1
(e)"
(OL-
Selection of Method to Use for a Given Problem
0.602 1.8 2.0 0.780 1.670 3.0 2.560 4.0 4 B =R(a 1)
The derived Equations 2 to 9, the dividedrange method, Underwood's Equations 37 and 39, and the McCabe-Thiele graphical method may be employed singly or in suitable combination to expedite the solution of all problems in the fractionation of binary mixtures, provided the operating lines are straight. Table VI11 summarizes our experience in regard to the choice of the most expedient methods to be employed for any concentra-
-
-
768 995 2130 3260 1
2.492 2.670 3.570 4.450
308 373 597 733
OB (1 TABLE X. CALCULATION
R 1.6 1.8 2.0 3.0
n(1
- 2F) + (1 - ZD) 0.02070 0.02328 0.02587 0.03880
1.2150 1.2600 1.4176 1,5120
29.4 25.6 18.3 16.0
36.8 32.0 22.9 20.0
- 2") FOR VARIOUSR VALUES R + l
R(1
- ZF) + (1 - ZD) 125.6 120.3 116.0 103.1
1 - 2" 0.01494 0.01559 0.01617 0.01817
1129
INDUSTRIAL AND ENGINEERING CHEMISTRY
September, 1942
type of curve, is simplifiedby plotting (n- wn.) TABLE XI. us. ( R R ~ ~or. (R' ) - R',~~.)on log-logpaper, The curves are readily drawn from a few points, since they have only a slight curvature. R R'
-
CALCULATION OF
nz
FOR 2' = 0.90, 10 P E R BOTTOMS~
CENT
TOLUENE IN
(E;)B-~
-
Bb ( 1 - ) ~xw - a 15.96 176.5 158.1 190.0 170.1 17.31 18.67 203.5 182.1 270.0 241.3 25.35 n1 is readily caloulated, as in Equation 13, to be:
1.6 1.8 2.0 2.0
0.777
20.54 15.68 13.01 8.28
n2 4.70 4.29 4.00 3.30
Typical Example of Use of Equations 1.104 1,435 PROBLEM. Toluene containing 1.5 weight 3.061 per cent of a high-boiling impurity is to be a purified to meet a specification calling for less log X /log 1.89 6.66 than 0.001 per cent of impurities boiling above b B ss (a - 1) R'+ 1 138" C. There is available a sixty-plate fractionating tower, with a measured plate efTABLE XII. SUMMARY OF PLATE CALCULATIONS FOR 10 PER ficiency- of 80' per cent based on benCENTTOLUENE IN BOTTOMS zene-toluene separations. I n connection with a complete economic balance on the use of the tower to purify the toluene, Stripping Rectifying Total R Theoretical Actual (Actual) (Actual) R vs. n curves are desired for the fractionation a t atmospheric pressure, with the feed preheated t o its boiling point, assuming that a bottoms concentration of 10 per cent toluene will be economical. Since this may prove t o be too high, a bottoms concentration of 2 per cent is also to be assumed in a separate calculation. SOLUTION. From Table VI11 it is seen that Equation 6 Since the column available has only sixty actual plates, it should be applied over the rectifying section and the DR-I follows from Table IX that a reflux ratio less than R = 1.6 method over the stripping section. cannot be used. Also, from Table I X it is evident that the The molecular weight of the impurity is first estimated on reflux ratio of 4.0 is higher than need be considered. the assumption that it boils a t 138" C. and is similar to The divided-range method is now applied to the stripping toluene in nature. A plot of molecular weight us. boiling section as follows: First, 1 - d',the concentration of the point for benzene, toluene, ethylbenzene, and propylbenzene impurity on the plate below the feed plate, is calculated in gives a molecular weight of 107 for the impurity. To estimate Table X, using the relation: the relativevolatility, on the safe side, the impurity is assumed CY to be ethylbenzene, boiling at 136" C. Interpolation of vapor 1 - 3'' = pressure data for ethylbenzene (7) gives 402 mm. for the R+1 (14) - + R(1 - SF) (1 - ZD) vapor pressure a t 110.6" C., the boiling point of toluene, whence (Y = 760/402 = 1.89, using the terminal value of 01 For the bottoms concentration of 10 per cent toluene, the as recommended above. relation between R and R' is as follows: Using the estimated molecular weight, the molecular concentrations are as follows:
E(-
:Fo)
9
+
99.999 weight 70:ZD 98.5 weight %: XF 10 weight yo: xw (or 2 weight %: xw
0.999989842 0.98707 0.1144 0.0232)
= = = =
Applying the above results by means of Equation 12, values of n2are obtained for the reasonable arbitrary choice of x' = 0.90. These calculations, details of which have already been given, are outlined in Table XI. The total number of theoretical and actual plates required for the stripping section, excluding the feed plate, and the actual plates required for both sections (i. e., for the whole column, in addition to the plates furnished by the total condenser and still pot), are presented in Table XII. Inspection shows that the values of nl/n are suitable.
(1 - X F ) / ( ~ - ZD) = 1273, which value is not changed by a += 5 per cent variation in the molecular weight. Limiting conditions for the rectifying section, obtained from Table IV, are:
0 99999 (0:98707) log 1.89
log 1273 nmin. =
~
11.25
CALCULATION FOR RECTIFYINQ SECTION. Table IX presents a typical tabular form for R us. n studies using Equation 6. Also, the
TABLE XIII. CALCULATION OF nz FOR x' = 0.85, 2 PERCENTTOLUENE IN BOTTOMS'
R
R'
B
1.6 1.8 2.0 3.0
195.0 209.9 224.8 299.4
174.6 187.8 201.1 267.5 0.9768 X
( L s ) B 1-xw
24.93 26.97 29.00 39.20
E)
-a
-
1
- xw 0.781 1.107 1.438 3.086
-a
( K 1 ) n 2
na
31.92 24.36 20.17 12.70
5.40 4.98 4.68 3.97
number of actual plates based on a n efficiency a ni = log (m / l o g 1.89 8.60. of 80 Der cent are given. The d a t e s indicated includk the feed plite. CALCULATION FOR STRIPPINQ SECTION.For TABLEXIV. SUMMARY OF PLATE CALCULATIONS FOR 2 PER a bottoms concentration of 10 per cent, the minimum numCENTTOLUENE IN BOTTOMS ber of plates in the stripping section, including the feed plate, Stripping Rectifying Total is calculated from Table IV to be 10 theoretical plates or 13 R Theoretical Actual (Actual) (Actual)
1130
INDUSTRIAL AND ENGINEERING CHEMISTRY
The calculations for the stripping section for 2 per cent toluene in the bottoms are as follows:
Vol. 34, No. 9
n, = number of theoretical plates a t total reflux between xw and x ‘ in divided-range method DR-I, or between ZD
and x‘ in DR-I1
nz = number of theoretical plates between
2’ and 2‘’ in divided-range method q = L ) / F , number of moles of liquid which appear on feed plate as a result of introduction of one mole of feed R = reflux ratio above feed plate, or moles of reflux divided by moles of overhead product or distillate, per unit time ( R = L / D ) R’ = reflux ratio below feed plate, or moles of reflux in lower part of column divided by moles of bottoms product or waste, per unit time (R’ = L / W ) W = moles of bottoms product removed per unit time x = mole fraction of more volatile component in liquid phase Z’ = an arbitrary value of x near XR, used in divided-range method x”,y” = corresponding values of x and y on plate below feed plate in divided-range method DR-I, or on plate above feed plate in method DR-I1 Z< = value of x at intersection of upper and lower operating lines y = mole fraction of more volatile component in vapor phase LY = relative volatility of more volatile with respect t o less volatile of two comDonents Subscripts D = distillate or overhead product F = feed W = waste or bottoms product 0, n = terminal concentrations or plate numbers
(z-
12.68 (including the feed plate) R’ = 75.6 74.6 R =
+
Selecting 2’ = 0.85 (the range being wider than before) gives the values of n2and n1listed in Table XIII. Adding the plates required for the rectifying section (Table IX) gives the total number presented in Table XIV. A comparison of Tables XI1 and XIV shows that, at all the reflux ratios studied, an additional 3.3 plates are required for the production of 2 per cent toluene in the bottoms instead of 10 per cent. The bottoms concentration between these limits is seen to have but little effect on the total number of plates required. EFFECT OF INCREASE IK IMPURITY. Assume that it has been decided to use 40 actual plates of the column for the rectifying section, operating a t a value of R = 1.8. Then, if the feed should at times contain more than 1.5 per cent of impurity, the problem arises whether a reasonable increase in the reflux ratio will compensate for this. The relation between the amount of impurity in the feed and the reflux ratio required to obtain the specified product is readily solved by the use of Equation 6, taking n = (40) X (0.80) = 32 theoretical plates in the stripping section, including the feed plate. Table X V presents a tabular calculation of this equation. T u r n . ” and “Den.” refer to the numerator and denominator of the large fraction in Equation 6A, obtained by rearranging Equation 6:
Literature Cited (1) Badger. W. L., and McCabe, W. L., “Elements of Chemical Engineering”, 2nd ed., P. 348, New York, MoGraw-Hill Book Co., 1936. (2) Brown, G. G., and Souders, M., in “Science of Petroleum”, Fig. 21 opposite p. 1665, Oxford Unir. Press, 1938. (3) Brown, G. G., Souders, M., Nyland, H. V., and Hesler, W. W., IND. ENC.CHEM..27, 387, Fig. 7 ( 1 9 3 5 ) . (4) Dodge, B. F., Trans. Am. Inst. Chem. Engrs., 3 4 , 585 (1938). (5) Dodge, B. F., and Huffman, J. R., IND.EKG.CHEM.,2 9 , 1434 (1937). (6) McCabe, W. L., and Thiele, E. W., Ibid., 17,
BETWEEN REFLUX RATIOAND FEEDCOMPOSITION TABLE XV. RELATION
R 1.7 1.8
2.0 2.1
(a
- 1)(R f 1) 2.403 2.492 2.670 2.759
--
0
Numerator
b
Denominator
1
%)l&( 262 510 1630 2700
(a
- 1)(R + 1)
(a -
1)(R
Num.a 629 12f9 4350 7470
Den.b 0,513 0.602 0.780 0.869
-
ZQ
E 1224 2110 5580 8600
Wt. ,% Impurity in Feed 1.4 2.5 6.6
(a)% a.
+ 1) - a.
Thus, the recommended reflux ratio of 1.8 is adequate to give a satisfactory product with a feed as high as 2.5 weight per cent of impurity. A slight increase of the reflux ratio above this value is evidently sufficient to take care of a large increase of the impurity in the feed.
B
Nomenclature a quantity used to simplify tabular calculations, equal t o following values: For Equation 6, B = a!R - R - 1 For Equation 7, B = aR - R + a For Equation 8, B = a!R’ - R’ 1 For Equation 9, B = a!R‘ - R’ - a! = moles of overhead product removed per unit time = moles of feed to column per unit time = equilibrium constant, defined as equal t o y/x = moles of liquid overflow above feed plate per unit time = moles of liquid overflow below feed plate per unit time = number of theoretical plates between any two points of the column under consideration =
+
D
F K L L n
..-(192.5). (7) Perry, J. H., Chemical Engineer’a Handbook, 1st ed., p. 327. New York, McGraw-Hill Book Co.. 1934. (8) Peters, W. A., J. IND. ENG.CHEM.,1 4 , 4 7 6 (1922). (9) Robinson, C. S., and Gilliland, E. R., “Ele-
ments of Fractional Distillation”, 3rd ed., pp. 81-9, New York, McGraw-Hill Book
Co., 1939. (10) Smoker, E . H., Tvans. Am. Inst. Chem. Engrs., 34, 165, 583 (1938). (11) Sorel, “Distillation et rectification industrielles”, 1R 9 9 (12) Sorel, “La rectification de l’alcool”, 1893. (13) Souders, M., and Brown, G. G., IND.*ENG. CHEY., 24, 520, Fig. 1 (1932). (14) Suen, T.-J., Ihid.,3 3 , 656-7 (1941). (15) Thormann, “Destillieren und Rektifizieren”, Leipzig, Otto Spanner, 1928. (16) Underwood, A. J. V., Trans. Inst. Chem. Engrs. (London), 10, 112 (1932).
10.0
