Multicomponent Distillation Problems SINGLE EQUILIBRIUM LINE SOLUTION HENRY J. HIBSHSIAN Purdue University and Purdue Research Foundation, Lafayette, Ind.
literature (6) and because i t illustrates several features which are often but not always encountered in such problems. The composition of the feed is 60, 30, and 10 mole per cent of benzene, toluene, and m-xylene, respectively. The distillate is to contain not over 0.5 mole per cent of toluene, and the bottom not over 0.5 mole per cent of benzene. The reflux ratio, O / D , is to be 2, and the feed is to be preheated to a temperature such that the change in moles of overflow across the feed plate will be equal to the moles of feed. The absolute pressure on the column is to be one atmosphere. The following nomenclature is used:
The method is a graphical one which uses a Cox vapor pressure chart on a transparent sheet as equilibrium lines. Any number of problems involving any or all of the same components can be solved by superposition of the chart on the appropriate operating lines. The method reduces appreciably the time required to solve these problems. The method was checked by the solution of several problems which had been solved by other methods j approximately identical solutions were obtained.
mole fraction of a component in the liquid mole fraction of a component in the vapor D distillate withdrawn as overhead product, moles per unit of time V = total vapor passing from one plate to the next, moles per unit of time n = number of plate under consideration, counting up from feed plate m = number of plate under consideration, counting up from still 7 = total absolute pressure in column P = vapor pressure of pure component in question B = benzene T = toluene X = m-xylene
z y
H E solution of multicomponent distillation problems may be accomplished by the use of one equilibrium line for each component. The method is fundamentally the same as that proposed by Sore1 (7) with the usual simplifying assumptions. Sorel's method was improved by Lewis and Matheson (4) and by Lewis and Cope ( 3 ) . Excellent reviews and criticisms of all the methods proposed for the solution of multicomponent distillation problems were given by Gilliland ( 2 ) and Robinson and Gilliland (6). A new treatment was recently proposed by Nelson (6). The single equilibrium line method is a graphical one which differs from previous graphical methods in the following ways: It requires only one equilibrium line for each component. This fact eliminates the series of y/z lines of previous methods, with the inherent necessity for interpolation between these lines. All the data are plotted on two sheets of paper which are superimposed. This superposition permits all the data to be conveniently mounted on a drawing board, and greatly simplifies and speeds u p the manipulations involved in proceeding from one plate to the next. Once the equilibrium data are plotted for a series of components, any number of problems involving several or all of these components may be solved without replotting any of the data. This is possible because the equilibrium data are on a separate, permanent, transparent sheet. This sheet need merely be superimposed on the correct operating lines in order to solve a problem. If the reflux ratio is constant, a condition permitting the use of the same operating lines, a problem can be solved for as many pressures as desired without replotting any data. The problem solved by Robinson and Gilliland ( 6 ) involving benzene, toluene, and m-xylene was chosen to illustrate the method. This problem was selected because its solution by two other methods has already appeared in the
T
= = =
The conventional methods of material balances were used to obtain the following: --DistillateMoles
O/Dfor top of column 0" = 120.2 Vm = 180.3
Mole %
-ResidueMoles
Mole )$'
=2
The equations of the operating lines below the feed plate are : B : yms = 1.221 z(* + 1)B - 0.0011 T : y m ~ = 1.221 ~ ( m+ 1 ) ~ - 0.164 X : y m x = 1.221 ~ ( m+ I)X - 0.0555
above the feed plate are: B :
T :
X
Y ~ B= Y ~ T =
: ynx
=
0.667 ~ ( +n 1 ) ~ 0.667 z ( + ~ I)T 0.667 ~ ( +n IJX
+ + 0.332 0.0017
THE vapor pressures of the components are plotted according to the method of Cox (1). If actual fugacities or reliable equilibrium data are known, they may be used with slight modification to take care of pressure in drawing up the equilibrium lines. These lines are plotted on a sheet of 988
JULY, 1940
INDUSTRIAL AND EXGINEERING CHEMISTRY
I
989
proximately 10 cm. long. Such paper gave all the accuracy which was justified by the method and permitted ease of manipulation on a drawing board. Figure 1 shows the data plotted for the problem under consideration. The continuous log scale of the log paper upon which the celluloid is superimposed is represented by the continuous scale on the left. The segment of the scale on the right, the Cox lines, and the alignment points are copied on the celluloid. The vertical log scale and the horizontal arbitrary Cox temperature scale are the vertical and horizontal coordinates, respectively. The temperature scale is placed on the celluloid but not the log scale. The Cox temperature scale is left a t an angle for ease of manipulation. The bottom-product Cox line is represented by W . The topproduct Cox line is coincident with the benzene line. The operating lines are plotted on the log paper itself. The coordinates start with 1,1, one cycle down from the top, on the left-hand side of the paper. The extra coordinate lines a t the top are of service in solving problems. The horizontal log scale represents ralues of y on the plate under consideration. The vertical log scale represents values of z on the plate above. Figure 2 shon-s these lines for the problem under consideration. Figure 3 shows the two plots superimposed for one atmosphere working pressure. Subscripts m and T refer, respectively, to operating lines below and above the feed plate.
Cox Temperature Scale- OC. FIGURE 1. ECJUILIBRIT-JI LIXES
transparent celluloid. This may be done ea& by superimposing the celluloid sheet on a piece of logarithmic paper. Alignment points are placed along the margin of the celluloid, and one complete cycle of the logarithmic scale is copied along the margin. No coordinate lines are drawn. The vapor pressures of the two products from the distillation are calculated for two temperatures a t the extremes of the temperature scale. From these the Cox lines of the products may be drawn. These product Cox lines should not be drawn permanently, so that they may be replaced by others if it is desired to repeat the solution for other pairs of end products. They are used only t o obtain rapidly the relation between the temperatures in the bottom and top of the column with respect to pressure. This relation becomes immediately apparent for all pressures as soon as these two lines are drawn. For the case where the temperature in the still or condenser is the controlling factor, the pressure a t which the column must operate can be chosen from these lines. If the pressure is set, the two end temperatures of the column may be read directly. Special inks which will adhere to the celluloid may be pui chased, or they may be made by suspension or solution of coloring matter in a solution of celluloid in acetone. An alternative method of placing lines on celluloid is to scratch the celluloid with a sharp point and fill the scratch with ordinary drawing ink. These Cox plots may be used for the solution of any number of problems, so some care is warranted to in constructing them. A sheet of celluloid about inch thick should be used; too thin a sheet will warp out of shape. If another substance is used in place of celluloid, one should be certain that i t will not change its shape with changes in room temperature and humidity; this is very disconcerting. Celluloid and paper slowly change shape but not enough to invalidate the method. It is advantageous to place the lines on the underside of the celluloid where they will not be easily rubbed off by the continued use I n the case of the solution under consideration, a logarithmic paper was used with 4 X 5 cycles. Each cycle waq ap-
yn 1.101Fraction in Vapor on Plate n FIQURE 2. OPERATISGLIXES
Fundamentally the solution of a problem resolves itself into proceeding from a known value of z for each component on a given plate to the value of 5 for the respective component on the plate above, or vice versa. This procedure is used to go u p from the bottom of the column and down from the top until the feed plate is reached. The usual simplifying assumptions are made: constant molar overflows in the top and bottom sections of the column; no heat transfer to or from the column except a t the pot, feed plate, and condenser; obedience of all components to Raoult's law unless equilibrium data are used; and negligible changes in heats of mixing, heats of vaporization, and sensible heats of the components throughout the column.
INDUSTRIAL AND ENGINEERING CHEMISTRY
VOL. 32, NO. 7
square. Then the T square is moved up the paper until one side of i t passes through the desired point on an equilibrium line. The 45" triangle is moved across the paper with its hypotenuse parallel to the 45' diagonal (through 1,l and 0.1,O.l) until its hypotenuse passes through the desired value of x on the horizontal scale. I n this position the point of intersection of the T square and hypotenuse of the triangle is the value of y, read on the horizontal scale (the same scale as that from which x was read). The value of x on the plate above is obtained on the vertical scale by dropping from the value of y just determined down to the appropriate operating line. This process is indicated for benzene in the still by the dotted lines in Figure 3. The temperature t. is first found on the temperature scale and projected vertically to the benzene Cox line, then horizontally until i t intersects the 45" line through the value X,. This value of the mol fraction of benzene in vapor (Y,) is projected vertically to the operating line B , and thus gives the value of x.cl on the vertical scale. This value of z is used to go up to the next plate in the same manner as before. Before going to the next plate, the values of z and y for all the components on the given plate are obtained such that Zs and Zy are both approximately equal to one.
% M l F r a c t i o n i n Liquid on P l a t e n y,
h@l F r a c t i o n i n Vapor
Arbitrary
COX
on P l a t e n
Temperature Scale
OC.
FIGURE 3. SUPEBIMPOSED DIAGRAMS
IT I S POSSIBLE to go u p the column in the following manner: I n the pot x is known or estimated for each component. B y Raoult's law x,P = ymx; therefore y m = x,P/n. When ym is known, zm 1 may be obtained from the operating line of the component under consideration. The Cox lines when originally drawn represent the vapor pressures, P , of the various components. Since the vapor pressure scale is logarithmic, the Cox lines will read P / r directly on the log scale when the transparent sheet is moved u p or down (depending on whether r is less or greater than 1, respectively) on the log paper an amount equivalent to r. This operation divides every point on all the Cox lines by r. The principle is t h a t of subtraction of logs. The log scale on the transparent sheet forms a continuous slide rule with the log scale on the paper. The transparent sheet is tacked in position t o read P / x directly on the vertical scale of the undersheet and remains there throughout the remainder of the solution. Since the horizontal scale on the operating line plot is logarithmic, i t is simple t o multiply P / a by x to obtain y by addition of logs. Any value on the 45" diagonal passing through the points 1,1and 0.1,O.l will read the same on both axes. If a length equivalent to x is added in a horizontal direction to a value of P/r,the value P / r will be multiplied by x. It is evident from geometrical considerations that, if a horizontal line be drawn through a point on a Cox line corresponding to a value of P / x at temperature T,and another line be drawn parallel to the 45" diagonal (through the points 1,l and 0.1,O.l) and through a given value of x on the horizontal scale (vertical value of this point = l), the intersection of these two lines when read on the horizontal scale will be the value of xP/x or y. A quick way to obtain y is t o mount the plot on a drawing board with the horizontal lines parallel t o the arm of the T
I N COMING down the column the process is reversed; xm is read on the vertical scale projected to the appropriate operating line and from there u p to the horizontal axis for the value of ym-l on the plate below. The T square is placed so that its edge passes through the appropriate equilibrium line at the desired temperature. The hypotenuse of the triangle is set to pass through the point formed by the intersection of the T square and the vertical projection of the point ym-l. If the hypotenuse of the triangle is parallel with the 45" diagonal (through the points 1,l and O.l,O.l), the intersection of the hypotenuse with the horizontal axis will be the value of xm-1.
+
/
/
/
I
/
/
/ 1
-
Liquid Line
e***
Vapor Line
P l a t e Number OF COMPONENTS FIGURE 4. VAPORAND LIQUIDDISTRIBUTION
IN
COLUMN
JULY, 1940
INDUSTRIAL AND ENGINEERING CHEMISTRY
The temperatures on each successive plate are obtained by estimation. It is possible to estimate a satisfactory temperature with some degree of perception and sagacity because the general trends in direction and magnitude of the temperature differences between plates becomes apparent upon an analytical examination of the operating and equilibrium lines. I n most cases it is not necessary to obtain the exact temperature necessary to make 8z and Z y both equal to one. In fact, i t is meaningless to estimate temperatures closer than is justified by the accuracy of the data and the method. A satisfactory method is to estimate one temperature high and the next low in such a manner that 2x and Z y are, on the average, both equal t o one. A little care in determining the accuracy needed will pay large dividends in time and energy saved. On plate 2 the concentration of toluene is critically sensitive to temperature because of the relationship of the operating and equilibrium lines. Table I contains the estimated values of z and y on each plate. The subscripts on y are the estimated temperatures; fractions were obtained by interpolation or extrapolation from other estimates.
TABLEI. ESTIMATED VALUESOF UllE
28
B T X
0.005
0.0133
0.744 0.281 6
1,000
.
0.858 0
0.997
5 AND
56
UlOO
0.280 0.672 0.066 1.018
0.491 0.491 O Z 1.002
21
1/113,8
27
B
0.0117 0.842
0.403 0.541
0.618 0.345
2
0 2 1.001
0.0299 0.920
0 3 1.006
0
1.001
T
B T X
0.0510
2/96
22
Ylll
28
0.0251 0.890 0.087 1.002
0.0601 0.902
0.521 0,428
0.0382 1.000
23
0
3
1.004
2/81
-
0.077 1.007
0
0 3 6 ,
0.012
0.998
1,008
24
VI07
210
us7
0.203 0.756 0.0282
0.710 0.280
1.010
0.987
1.006
0.863 0.138 0.004 __ 1.007
L’106
Xll
V8s
0.167 0.760
0.340 0.650
0,800 0.203
0.996
1.015
0.920 0.095 0,0012 1.016
26
B T 2
2/90
0.806 0.190
0.095 0.841
0.0184
0.069
0.025
0 ,006 __
1,009
2/83
214
0.702 0.240 0,0136 0.956
X
0.074
513
0.965
29
T
0.00030 0.978
2/02
0.604 0.338
2
0,058
s
2/109,3
B
1.020
0.979
0.114 0.856 2
0.0018
2182
0.920
0.957 0.036 0.00015 0.993
0,050 0,880
1.005
212
0.880 0.138
0.880 0.084 0.0009
B
T
y ON EACHPLATE
0:052
... 216
0:big
.. . XI6
o:Oi4
...
o:oi1
...
2184
o:oii
...
2/84
:...
0 0054
-
991
the sections of the equilibrium lines being used determine the rates of change of the various components. For total condensation, the Cox lines for the bottom and top products intersect the horizontal axis a t the temperature of the pot and condenser, respectively. Maxima are determined by the relative positions of all the operating and equilibrium lines and by the temperature and concentrations of all the components. It is helpful when working a problem to make a plot of the concentrations of the components in the liquid and vapor phases against the number of the plates. Figure 4 shows these plots for the example. It may happen in some cases that the Concentration of one or more components will remain practically constant for a considerable number of plates. I n such cases the temperature is usually constant. 4 s long a5 the concentration of a t least one component is changing, the solution should be continued. However, if the z and y values of all the components become asymptotic to some value such that the feed plate cannot be reached, the initial assumptions with respect to end concentrations are such that the assumed separation cannot be obtained. The statement that only one and not two concentrations need be changing is true because in those cases where i t is desirable to take one component down to a very low concentration, that component will be present in such low concentration that the other components will often remain a t constant concentrations, within the precision of the method, for a considerable number of plates. The concentration of the other components mill start changing when the low-concentration component reaches a value significant with respect to the components present in larger amounts. I n the present example, a t the top of the column from the thirteenth plate on, the concentration of benzene is high and the concentration of xylene insignificant; so the problem simplifies to the problem of determining the number of plates required to take the concentration of toluene down to the required value. I n this region the composition is quite insensitive to temperature and the mole fraction of benzene may be considered to be 1 minus the concentration of toluene. Here the same temperature may be assumed over and over, since i t is obvious that the only factor affecting the concentration of benzene is the concentration of the toluene. I n multicomponent problems involving feeds brought in a t other temperatures than that needed to make the change in molar overflow across the feed equal to the moles of feed, the choice of a feed plate necessitates special material and heat balances across the feed plate. I n general, the feed plate should be so chosen that when the components are all balanced correctly a t the feed plate, the column has the least possible number of plates.
Acknowledgment The intercepts and asymptotes of the operating lines are of considerable service in determining where to introduce components which do not appear in the bottoms or distillate. For example, it is obvious that a component which does not appear in the bottoms must be introduced for consideration on such a plate and at such a concentration that it will reach a value of z on the feed plate higher than the value of z determined by the asymptote of the operating line above the feed plate of the component in question. Similarly, a component not appearing in the distillate must be introduced for consideration below the feed plate so as to make its value of y on the first plate below the feed plate higher than the value of y determined b y the asymptote of the operating line below the feed plate of the component in question.
THIS method facilitates analysis of trends. The slopes and positions of the operating lines in combination with
The author wishes to acknowledge the cooperation of E. R. Gilliland and C. S.Robinson, Massachusetts Institute of Technology, in giving permission for the use of their problem as an example for this publication.
Literature Cited C O X , I N D . ENG.C H E X . , 15, 592 (1923). Gilliland, E. R., Ibid., 27, 260 (1935). Lewis and Cope, Ibid., 24, 498 (1932). Lewis and Matheson, Ibid.,24, 494 (1932). Nelson, W. L., Oil Gas J . , 37, 41 (July 2 1 ) , 44 (Aug. 4 ) , 58 (Aug. 18), 46 (Sept. 15), 56 (Oct. 6 ) , 59 (Oct. 20), 176 (Oct. 27), 44 (Nov. lo), 40 (Dec. 1. 1938). (6) Robinson, C . S., and Gilliland, E. R . , “Elements of Fractional Distillation”, Chap. XV, p. 139, New York, McGraw-Hill Book Co., 1939. (7) Sorel, “La rectification de l’aicool”, Paris, 1893.
(1) (2) (3) (4) (5)
