Entrainment in Plate Columns EDWIN M. BAKER University of Michigan, Ann Arbor, Mich.
- q,). Likewise, for any given value of Zm + I , a point on the line of equation 4a will lie above a point on the line of Equation 2a by the amount e(xm+l - xm). Figure 1shows a simple method of constructing these curved operating lines to meet the above requirements. Figure 1 is constructed for the conditions :
M
e(xn -tl
0:DERN design of plate columns frequently calls for the use of vapor velocities sufficiently highso that the effect of entrainment on column performance cannot be neglected. The method of calculating the required number of plates in columns when entrainment is an important factor has been treated by Sherwood and Jenny (4), Colburn (I), Rhodes (3)) and others. The purpose of this paper is to present a modification of t h e method of Sherwood and Jenny for binary systems. This modification seems to possess some merit from the viewpoint of simplicity, particularly as applied to the teaching of distillation calculations. When there is no entrainment, a general material balance o n the assumption of constant molal overflow results in the following equations for the upper and lower operating lines on the McCabe-Thiele .diagram (9):
If one mole of vapor entrains or carries up the column e moles of liquid, a general material balance yields the following equations for the operating lines:
Composition of feed: Composition of distillate: Composition of waste: Thermal condition of feed: Reflux ratio: Relative volatility Entrainment:
q = 0.50 0.95 xw = 0.05 q = 0.50 xd
VJD = 2.70 = 4.0
e =
0.40
1.0
0.9 0.8 0.7 0.6
Y
0.5
V and L in Equations 3 to 4a, inclusive, refer to the quantity of dry vapor and to t h e quantity of reflux resulting from it. If there is zero entrainment, the last terms of Equations 3a and 4a drop out, and these ,equations become identical with Equations l a and 2a. For a system and conditions that make la and 2a equations of straight lines, Equations 3a and 4a are equations of curved lines, since neither (2, + 1 - 2), nor ( x m + I - s m ) are linear with x due to the curvature of the equilibrium curve. Further, for any given value of x m + l , a point on the line of Equation 3a will lie above a point on the line of Equation l a by the amount
0.4
0.3 0.2
0. I 0.0
x
FIGURID 1. CONBTRUCTION OF OPERATING LINESFOR e = 0.4 717
INDUSTRIAL AND ENGINEERING CHEMISTRY
718
VOL. 31, NO. 6
sufficient so that there is no entrainment in the vapor going to the single total condenser. From this condenser, product is withdrawn and reflux 0.< a t the boiling point is returned to the column. It should be noted that these conditions call for using lines l a and 2a for the stepwise construc0.F tion a t x = xw, and at z = xd as shown in Figure 2. With zero entrainment, 6.4 theoretica1 0.i plates are required, and 8.6 theoretical plates with 0.4 entrainment. 0.6 If the entrainment in the upper and lower parts of the column is not the same, the conY struction would obviously be slightly modified to 0.5 meet this situation. Thus, if the entrainment in the lower part of the colunin were 0.5, the en0.4 trainment in the upper part remaining 0.4, the spread between line 2a and line CZ would be increased in the ratio of 5 to 4 over that shown in 0.3 Figure 1. The curved operating lines intersect the normal 0.2 operating lines a t the points where extensions of the equilibrium line below x = 0 and above x = 1.0 intersect the normal operating lines for zero 0.I entrainment. The two curved operating lines themselves intersect on the vertical through the 0.0 intersection of the p line with the normal operat,‘XYO.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ing lines. x If it is desired to apply Murphree plate efficienFIGURE 2. DETERMINING THE NCMBE R OF THEORETICAL PLATES FOR cies to the stepwise calculation to determine e = 0.4 directly the numbers of actual plates required, this is readily done by proportioning the distance between the curved operating lines and the equilibThen D x d / V nof Equation la = 0.95/2.70 = 0.352. rium curve. Line l a is then drawn through y = 0.352, x = 0, ppint A, While the method presented is not basically new, it is hoped and x = y = 0.95. This line is extended, intersecting the that it will prove of advantage. line x = 1 at point B. The q line is drawn through x = y = 0.50, with a slope of - 1. Nomenclature Line 2a is drawn through the intersection of the q line with line la, and through the point z = y = 0.05. This line is D = moles of distillate in unit time extended, intersecting the y axis at point C. These are the F = moles of feed in unit time L, = moles of reflux above feed plate in unit time, not includordinary McCabe-Thiele operating lines. The amount of ing the returned entrainment entrainment, e , is now laid off above point B , giving point D , L, = moles of reflux (L, QF)below feed p1,ate and a straight line is drawn connecting points A and D. V, = moles dry vapor above feed plate in unit time = V, Select any point, G, on the upper operating line where x = (1 - q ) F V , = moles dry vapor below feed plate zn. The corresponding value of y on the equilibrium curve, W = moles of waste (bottoms) in unit time point K , is y,. From point G on line AB draw a line, GH, e = moles of liquid entrained by vapor per mole of vapor parallel to line A D . From K on the equilibrium curve draw Q = moles of liquid resulting on feed plate, per mole of feed, from introduction of feed a horizontal line, KL, intersecting GH a t R. Then R is a 2 = mole fraction of more volatile substance in liquid point on the curved operating line (Equation 3a). The reason y = mole fraction of more volatile substance in vapor for this follows directly from the construction. Distance GS is equal t o ezn,distance U T is ez, and distance U R is thereSubscripts fore e(x, + - x,). Thus for a liquid of composition xn+ d = distillate w = waste leaving plate n 1, a column operating without entrainment m = any plate in lower part of column would require that the vapor from the plate below be of comm + 1 = plate above mth plate position U ,whereas with e = 0.4 the composition of the vapor n = any plate in upper part of column must be of composition R. Other points on the curved upper n + 1 = plate above nth plate. operating line are readily determined in like fashion. Literature Cited T o plot the curved lower operating line, erect a vertical line from the intersection of the q line with lines l a and 2a (1) Colburn, A. P., IND.ENG.CHEM.,28, 526-30 (1938). to give point Z on line A D . Draw line CZ. The same type ( 2 ) McCabe, W. L., and Thiele, E. W., I b i d . , 17, 747 (1925). of construction as used previously will yield points on the (3) Rhodes, F. H., Ibid., 26, 1333 (19341, 27, 272 (1935); Rhodes, F. H., and Slachman, P. G., I b i d . , 29, 51 (1937). curved lower operating line. This construction is shown in (4) Sherwood, T. K., and Jenny, F. J., Ibid., 27, 265-72 (1935). Figure 1; the same letters as before were used with prime mark added. Figure 2 shows the completed chart, with the curved operating lines for e = 0.4 in solid lines, and the straight operating lines for zero entrainment as dotted lines. The problem has been worked on the assumptions that there is no entrainment from the still, and that the spacing above the top plate is I.(
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