Rapid Solution of Multicomponent Distillation Problems
GEORGE KARNOFSKY
Purdue University, Lafayette, Ind
The methods described in this paper are a revision of the Hibshman method and a new solution requiring only one operating line, which applies to all the components.
’
m H E calculation of t h e n u m b e r of theoretical plates
the equilibrium line eorresponding to the assumed plate temperature are drawn for each component; and the plate temperature is found that makes By = 1. These methods were described in detail by Robinson and Gilliland (4, Chapter XV). They pointed out that it is not necessary to guess the plate temperature exactly, but that if By is within 10 per cent of 1,the value of y for each component can usually be corrected by dividing by By, which makes the sum of the corrected y values come out 1. This eliminates most of the trial and error from the solution. Hibshman (1) plots the operating lines on log-log paper, and superimposes a Cox type graph of vapor pressure vs. temperature.
in multicomponent
tillation has received considerable treatment in the literature. The problem has been approached in two waysby the application of exact methods based on a stepwise calculation from plate to plate, and by analytical methods involving approximations. Although the stepwise calculations are more tedious, they give accurate indications of liquid and vapor compositions, whereas the analytical methods give only limited results. The methods described here make possible a more rapid solution by the stepwise procedure; and the only assumption made is the usual one of constant reflux ratios above and below the feed plate. The new schemes depend in part on methods already described in the literature, which are reviewed below Robinson and GilliIand (4, page 139) pointed out that all the stepwise methods are based on Sorel’s method (6) of algebraic computation. In the case of binary mixtures the assumption af constant reflux ratio leads to the familiar McCabeThiele method. For multicomponent mixtures the operating lines for each component (Equations 1 and 3 below) are computed in the same way as for the components of a binary mixture, but there is n o single equilibrium line. The equilibrium compositions of liquid and vapor on each plate must be found by trial and error, either from experimental liquidvapor equilibrium data for the system, or by Raoult’s law if the components are normal. The application of the algebraic stepwise procedure was demonstrated by Lewis and Matheson (3). The equations of the operating lines (Equations 1 and 3) relate the vapor composition below any plate to the liquid composition on the plate. The equilibrium data relate the liquor composition to the vapor composition above the plate. Given the liquor composition, a -plate temperature is found by trial that makes the sum of the mole fractions of the components in the vapor come out 1. I n this way liquid and vapor composition and temperature can be found for each plate. The problem is simplified by neglecting, above the feed plate, the components that do not appear in the overhead product and, below the feed plate, the components that do not appear in the bottoms. Lewis and Cope (8) described the equivalent graphical method. For each component the operating line and a series of y = K x lines, each corresponding to a different temperature, are drawn. The steps between the operating line and
Revised Hibshman Method The basis of the Hibshman method is that of the slide ruletwo numbers can be multiplied by adding their corresponding logarithmic lengths. Thus, if y = K x , y can be found by adding on a logarithmic scale the length corresponding to K (or P/?r if Raoult’s law holds) to the length corresponding to z. If the operating line is plotted on log-log coordinates, y. can be found from znby adding the length K , expressed in the same coordinates. Several schemes can be devised for performing the addition. Hibshman plots the operating lines y. + 1 vs. x, and ym vs.xm+ 1 for the several components on log-log paper, and then superimposes a transparent Cox type chart (with the vapor pressure plotted as ordinate on the same logarithmic scale) on the plot, so that the ordinate for ?r on the chart falls over the ordinate 1on the plot. P/Tcan then be read directly from the scale for any value of P. Hibshman uses a T square and 45’ triangle to perform the addition, as described in detail in his paper. The Hibshman method can be criticized on the following grounds: 1. The cumbersome mechanics of the method makes it tedious and subject t o error. 2. The Cox chart must be plotted on transparent material which has the same coefficient of ex ansion as paper, if it is t o be accurate over a period of time. d h o u h the effort required t o make the chart is justified if a number of problems involving the components is to be solved, for a single problem the method has little advantage over the algebraic solution.
I n the revised method these defects are eliminated by substituting, for the Cox chart, an easily constructed nomograph of vapor pressure us. temperature on log-log paper, and by performing the addition of lengths with dividers. The revised method is carried out in the following manner when Raoult’s law is obeyed: Construct a vapor pressuretemperature nomograph, similar to Figure 1, on log-log paper, using the logarithmic s a l e for the vapor pressure P , and laying off the corresponding temperatures on lines parallel
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Vol. 34, No. 7
INDUSTRIAL AND ENGINEERING CHEMISTRY
840
E
0
FIGURE 1 (Above). VAPOR NOXOGRAPH FOR BENZEXE, TOLUENE,AND XYLENE F I G U R E 2 (Right). OPERATING LINESFOR BENZENETOLUENEXYLENE PROBLEM, ILLUSTRATING REVISEDHIBSHMAN METHOD PRESSURE
to the P line. This nomograph is the equivalent of a vapor pressure-temperature graph for each component. Figure 1 is the vapor pressure nomograph for benzene, toluene, and xylene. Draw the operating lines, determined by the equation for each component, pn log-log paper of the same size as that used for the nomograph. On the nomograph dram the line representing m-, as in Figure 1. Then, for any temperature the value of P / n is represented by the distance, on the line representing the component, between the temperature and the intersection with the a line. Thus, the length AB in Figure 1 represents P/a for benzene at 116' C. To find y, given x and the temperature, add the length P/ato z on the logarithmic scale, using dividers. The length is considered positive if P is greater than x , negative if P is less than a. The method is illustrated by solving the problem given by Robinson and Gilliland (4, page 139) and Hibshman (1). The feed composition is 60, 30, and 10 per cent of benzene, toluene, and xylene, respectively. The distillate is to contain not over 0.5 mole per cent of toluene, and the bottoms not over 0.5 mole per cent of benzene. The reflux ratio is 2, and the change in moles of overflow across the feed plate is equal to the feed. The still pressure is 760 mm. Rlaterial halaiices give:
B
T 1
Ilistillate Moles ZD
Residue Lloles
59,s 0.3 0.0 60.1
0.2 29.7 10.0 39.9
0.995 0.005
... - -
0,
=
V,
-
1.000
120.2; =
o,,
ZA
0.005 0.744
0.251 1.000
The equations of the operating lines below the feed plate are:
Bm : yn T m : ym X, :
ym
1 . 2 2 1 ~ m+ 1,2212, + = 1.2212, +
=
1
1
- 0.0011 - 0.164 - 0.0555
The equations of the operating lines above the f e d Idate are:
Bn : Tn :
yn + I
=
yn + I = ~n + I =
Xa :
++
0.6672, 0.332 0.667~n 0.0017 0.6672,
These equations are plotted in Figure 2. B,, T,,, and X,are the operating lines above the feed plate; B,, T,, and X , are the operating lines below the feed. Assume the still temperature is 116" C. Then the lengths AB, CD;and EF on Figure 1 represent P/T for benzene, toluene, and xylene, respectively. This length is added t o zw for each component laid out along the ordinate axis, as shown in Figure 2, to give y m . Note that for xylene P is less than m-, so that the length EF is considered negative. The values of yw read from Figure 2 are: for henzene, 0.013; toluene, 0.855; xylene, 0.132; total = 1.000. Since Zyw = 1, the assumed temperature is correct. The values of XI can now be read from the operating lines and the process repeated. Above the feed plate t'he procedure is the same.
Single Operating Line Method A single operating line, which applies to all the components, can be constructed in the following manner: The equation of the operating line above the feed plate is usually written (6) in the form: yn+1 =
= 220.2
V,,, = 180.3
1
Substituting,
&+
I%+
1
=
0 _LI Vt,
2,.
+ DE
ZD
(1)
July, 1942
INDUSTRIAL AND ENGINEERING CHEMISTRY
841
Equation 4 is used to determine the number of plates below the feed plate in the following way, when Raoult’s law is obeyed: Construct the operating line T y m + l/Pm+ 1 ym us. xwym for the ratio b,. Starting a t the still, assume a temperature and find yw such that Zy, = 1. Compute xw/yw and read nyl/Plyw from the plot. Assume the first plate temperature, and add the length Pl/a (from the vapor pressure nomograph) to xyl/Plyw on the logarithmic scale. The value of yJyW can then be ’read from the scale and y1 computed. If the temperature is chosen correctly, Zyl = 1. Values of OPERATING LINESBASEDON EQUATIONS 2 AND 4 FIQURB 3. GENERALIZED x,/yl are computed, and the mocess is reaeated. To demonstrate the method, the problem givkn under “Revised Hibshman Method” is continued. The operating lines are drawn in Figure 4, where b, = 2 and b, = 180.3/ (220.2-180.3) = 4.52. The values of y w were found in the preceding section:
This is a general equation which can be applied to any problem, since it does not depend on the state of the feed or products. If the usual assumption is made that b, is a constant, then a plot of K , .+ 1 (2%+ I/$*) vs. XD/X, gives a simple family of curves (Figure 3) on log-log coordinates. Equation 2 is used to determine the number of plates above the feed in the following way, when Raoult’s law is obeyed. Construct the operating line P, + 1 xn + l/nx,‘ us. XD/X% for the reflux ratio b,. Starting a t the top plate, find x1 for each component in the usual manner by guessing a top plate temperature that makes ZxI = 1. Compute xD/zl and read Pzx2/nzl from the plot. Assume the second plate temperature and subtract the length PZ/Tfrom Pnxs/rrxl with dividers as described under “Revised Hibshman Method”. This gives values .of zz/zl for each component, from which zz can be computed. If the temperature is correct, Zzz = 1. Using the correct values of 2 2 , compute X ~ / X Zand repeat the process. Below the feed plate an operating line similar to Equation 2 exists. The operating equation (6)is usually written: (3) TI
B
0.013
X
0.132
T
0.005 0.744 0.251
0.855
0.385 0.870 1.900
Assume the temperature of the first plate above the still
(m = 1) is 116” C. Add the lengths of Pl/n for each component, found from Figure 1, to nyt/Ply, as demonstrated for benzene in Figure 4. This gives:
B
T X
0.031 0.965
2.40 1.13 0.65
0.029 0.891
0.086 1.082
0.080 -
0 173 0 835 3.14
1.000
TABLBI. LIQUID AND VAPOR COMPOSITIONS OF PLATES Plate Number Still 1 2 3
?
6 7 8 9 (feed)
Assumed Temp., C. 116 116 110 110 1ro 105 100 95 95 90
F--Z
B
(Corrected)-T X
Below Feed 0.005 0.744 0 . 2 5 1 0.012 0.827 0 . 1 6 1 0.025 0.867 0.108 0.049 0.852 0,099 0.094 0.817 0.089 0.165 0.748 0.087 0.272 0.647 0,081 0.380 0.544 0.076 0.490 0.438 0.072 0.578 0.355 0.067
Above Feed 1 2
This equation has the same form as Equation 2 so Figure 3 can be used to represent both equations.
0.900 0.985 1.230
3 4 5 6 7
8 (feed)
80
80 80 85 85 85 85 90
0,986 0.971 0.946 0,912 0.862 0.793 0,706 0.623
0.014 0.029 0,054 0.088 0,138 0.207 0.294 0.377
._ . ., . ,,. .. ,. .. ... .,. ...
CY
B
(Corrected)T X
0.013 0,029 0.060 0,114 0.203 0.330 0.479 0,617 0.717 0.781
0.856 0.891
0.995 0.989 0.979 0.963 0,942 0.905 0.858 0.801
0.005 0.011 0.021 0.037 0.058 0.095 0.142 0.199
0.132 0.080
0,886 0.054 0.842 0.760 0.638 0.495 0.362 0,265 0.204
0.044 0.037 0.032 0,026 0.021 0.018 0,015
... . .. .. . .. .. .. . ..
... .. .
.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 34, No. 7
Equations 6 and 8 represent generalieed operating lines, applicable to any distillation problem. Since they are similar in form, both are plotted as a single family of curves for different values of b in Figure 5 . Above the feed plate the vapor composition above any plate is related to the vapor composition of the plate below it; below the feed plate the liquid composition on any plate is related to the liquor com0.51 position on the plate above I it. Working with these equations in a manner analogous to that described for Equations 2 and 4, the I computation can be carried . ... . - .. I I 1 1 1 1 1 1 a11 I I I I I i l l l I I II I , I Ill I I I 1 1 1 1 1 1 out from the feed plate up 91 .05 0.1 03 10 5 10 50 to the top and from the feed plate down to the botFIGURE4. OPERATING LIKES FOR BENZENE-TOLUENE-XYLENE PROBJJCM, ILLUSTRATING tom. However, the steep SINGLEOPERATING LINEM ~ T H O D slope of the lines of Figure 5 makes them more difficult to read accuratelv than those of Figure 3, so that Figure 3 gives better result;. Just as in all trial-and-error methods, it is not necessary t o guess the plate temperature exactly, since y, can be corrected Raoult’s Law Not Obeyed if Zy, found is not far from 1, as shown by Robinson and Gilliland (4). The corrected d u e s of y1 are used to compute For simplicity, the two methods have been demonstrated xw/yl, and the process is repeated. for the case where Raoult’s law is obeyed. However, if values of K as a function of temperature are available, the The values of z and y for each plate computed in thifi way vapor pressure vs. temperature nomograph is replaced by a K are shown in Table I. us. temperature nomograph, with K on the logarithmic scale. Using Equations 2 and 4, the calculation of the plate comThe logarithmic lengths corresponding to K are then handled position must be carried out from the top plate down to the in the same way as the lengths P/n-, measuring the lengths feed and from the still up to the feed. However, this choice fromK = 1. mas arbitrary, since operating lines can be developed which make possible the calculation in the other direction. Above the feed, Equation 1 can be 5written : L
-,.
Q5 2
Below the feed, Equation 3 can be writ ten :
FIGURE 6.
GENERALIZED OPERATINQ LINESBASEDON EQUATIONS 6 AND 8
July, 1942
INDUSTRIAL AND ENGINEERING CHEMISTRY
Acknowledgment The writer wishes to thank C. L. Lovell, professor of Chemical Engineering, Purdue University, for advice and criticism, and C. S. Robinson and E. R. Gilliland for permission to use their problem.
Nomenclature = mole fraction of a component in liquid = mole fraction of a component in vapor = number of a plate above feed counting down from top
x g
n
(contrary t o usual practice but used here because that is the direction of the computation)
6.
=
reflux ratio, 0
D
=
a43
0,
~
Vm
- On
Vm
b, =
O m - Vm P = vapor pressure of a component T
= absolute pressure in still, same unit as P
XD, xw =
K
=
mole fraction of a component in distillate and bottoms, respectively
(in equilibrium); ?Tr! (when Raoult’s law is obeyed)
B, T,X = benzene, toluene, xylene, respectively
m = number of a plate below feed, counting up from first
plate above still D = moles of distillate withdrawn as overhead product, per unit time W = moles of bottoms withdrawn per unit time Vn,V,,, = total moles of vapor passing from one plate to the next, per unit time, above and below feed plate, respectively O,,, Om = total moles of overflow from one plate t o the next, per unit time, above and below feed plate, respectively
Literature Cited (1) Hibshman, IND.EN*. CREM.,32,988 (1940). (2) Lewis and Cope, Ibid., 24, 498 (1932). (3) Lewis and Matheson, Ibid., 24, 494 (1932). (4) Robinson, C. S.. and Gilliland, E. R., “Elements of Fractional Distillation”, New York, McGraw-Hill Book Co., 1939. (6) Sorel, “La rectification de l’alcool”, 1893. (6) Walker, Lewis, McAdams, and Gilliland, “Principles of Chemical Engineering”, New York, McGraw-Hill Book Co., 1939.
Critical States of Two-Component Paraffin Svstems J
Empirical correlations of literature data are presented by means of which critical temperatures, pressures, and compositions of two-component normal paraffin hydrocarbon systems can be predicted. Values calculated by means of these correlations are compared with existing published data on ten systems. Disregarding positive and negative signs for the errors, and excluding two systems, the average error for the calculated pressures is 13 pounds per square inch or 1.3 per cent, and the average error for the calculated temperatures is 6.4’ F.
HE past decade has seen considerable compilation of data on the phase behavior of multicomponent hydrocarbon systems. From the standpoint of the engineering profession, probably the most needed of these data have been volumetric and vaporization equilibrium data on the compounds found in the mixtures encountered in oil and gas production and in petroleum refining. Knowledge of the critical states is of prime importance in such studies, particularly in work on vaporization equilibria. The purpose of this paper is to present empirical correlations by means of which the critical states of two-component normal paraffin hydrocarbon systems can be predicted. The critical state has been defined by Gibbs (W) as that state of a system at which all distinction between the two coexistent phases vanishes. A one-component system has only one critical state, whereas a multicomponent system has
T
1
Present address, The Dow Chemical Company, Midland, Mich.
F. DREW MAYFIELDl Phillips Petroleum Company, Bartlesville, Okla. an infinite number of critical states. Confining our attention to binary normal paraffin hydrocarbon systems, it is to be noted that all critical temperatures of such mixtures lie between the critical temperatures of the two pure components; the critical pressures of the mixtures may be far in excess of, or even many t i e s the critical pressure of either of the two pure components. This is readily apparent from Figure 1. For a more detailed discussion of phase behavior of multicomponent hydrocarbon systems, work such as that of Sage and Lacey (I?‘) and of Katz and Kurata (6)should be consulted. Previous means of predicting critical states of multicomponent systems have been reported by Roess (IS) and by Smith and Watson (10). These two correlations were developed for complex petroleum hydrocarbon fractions, and no attempt has been made here to compare the accuracy of these previous methods with that of this paper.
Data Selected The following systems are considered in this study: methane-propane ( I 8 ) , methane-n-butane ( I @ , methane-npentane ( $ I ) , methane-n-hexane ( I @ , ethane-n-butane (8), ethane-n-heptane (T), propane-n-butane ( I I ) , propane-npentane (167, n-butane-n-heptane (Q), n-pentane-n-heptane ( I ) . Direct experimentally determined critical states were reported for all of these systems except methane-+pentane and methane-n-hexane. Critical states for these two systems have been estimated from the published data by extrapolation of the reported bubble-point pressure-composition isotherms, the maximum pressure on such an isotherm being the critical pressure. This method of estimating critical pressures does not permit accurate estimations of critical state compositions. Furthermore, such estimated critical pressures should be accepted with some reservation until more data become available.
