A new scheme of plotting ternary distillation problems is presented which leads t o a graphical solution. The procedare outlined shows how a ternary system can be represented from the three limiting binary systems and how such a representation eliminates trial-and-error calculations of equilibrium relations.
er
ems
J
N THE design of fractionating LAWRENCE H . CORN balances and equilibrium relations columns the engineer is congraphically and to continue the procThe M . Mr. Kellogg Company, K~~~york, N y . fronted with t h e problem of esses of plate-to-plate determinadetermining the number of plates t o tions on the McCabe-Thiele diagram use and the reflux ratio required for economical pel foimance if the system resolvcr itself into B binary mixture. With the and low original cost. I n his study t o set the equipment detemperature scale so chosen for the benzene-toluene system to sign, graphical methods are often helpful in visualizing column give the resulting equilibrium curve and with the geometric performance expected and in saving time in making calculations. properties of t h e isosceles right-angle triangle, it is possible t o The use of t h e McCabe-Thiele diagram has become standard carry out the necessary steps. practice in designing distillation apparatus; and although it is primarily for a two-component system, it has been adapted t o HE bubble-point and dew-point curves will, in general, be more complex problems. Bonilla (1) presented a graphical Tcurved lines. However, when equilibrium constants are asmethod for ternary systems, and this discussion will extend the sumed t o be a function of pressure and temperature for any comgraphical applications. ponent and unaffected by the varying amounts of other compoFor t h e determination of the number of plates required for a nents present, then i t follows t h a t t h e isotherms are straight lines. given separation, the usual practice is t o calculate the number When considering hydrocarbons far removed from their critical of theoretical plates-that is, plates on which equilibrium condipoints, the use of equilibrium constants uncorrected for the tions are reached. T o set the number of plates it is necessary t o other components present is acceptable. Dourson, Sage. and arrive at, a reflux ratio t o use. The number of plates varies over a Lacey (2)give data on t h e methane-propane+-pentane system wide range ds t h e reflux ratios vary, and t h a t maximum variation which Till serve t o show limitations of equilibrium constants. is of primary interest at this time. The limiting conditions of From the binary systems the isotherms are drawn in as shown reflux ratios are total reflux, requiring a minimum of plates, and in Figure 1. T o cover the complete temperature range requires minimum reflux, requiring a n infinite number of plates. The reall three binary curves if t h e lines are drawn from two binary flux ratios are, in turn, affected by the supply of heat t o the curves. By use of the slopes of t h e lines, only one auxiliary curve column, and the condition of the feed thereby sets the relation is required. between the top or fractionating section LIT' and the bottom The slope of the liquid isotherm (bubble-point line) is or stripping section L/V. I n order t o limit the discussion, (Kt K s ) / ( K b - Kt) the feed shall be considered as all liquid at its boiling point, which gives a change in moles of liquid across the feed plate The slope of the vapor isotherm (dew-point line) is equal t o t h e moles of feed. The discussion has been limited t o a ternary system, and is now further limited t o a two-phase equilibrium system. It thus has three degrees of freedom according t o t h e phase rule. ConsiderThe diagram is now complete for the solution of ternary distiling a practical distillation problem, i t is also advisable t o work at lation problems for this particular system of benzene-tolueneonly one pressure so t h a t temperature and concentration remain xylene a t atmospheric pressure. Taking the familiar problem ai; as t h e two variables. It is now possible t o represent one temperaset up by Robinson and Gilliland (6),t h e plates required at total ture o n a plane, and i t appears as a dew-point curve and a bubblereflux and theminimum reflux ratio for infiniteplateswill be found, point curve. Thus the system can be shown as a triangular diawhich will effect the separation outlined in the following table: gram with compositions as t h e axes and any one temperature --Distillate-Bottoms-Feed---represented. Moles Mole % Moles hIole % hloles Mole % It is next desired t o show all the temperature ranges and conBenaene 60.0 60.0 59.8 99.5 0.2 0.5 Toluene 30.0 30.0 0.3 0.5 29.7 74.4 centrations on one diagram. This is usually done by a triangular 10.0 10.0 0.0 0.0 10.0 25.1 Xylene - prism in which concentration is shown in t h e X-Y plane and tem100.0 100.0 60.1 100.0 39.9 100.0 perature along the Z-axis. In such a figure the surfaces parallel For the case of total reflux t h e condition of t h e feed is immat o t h e Z-axis represent the three limiting cases of the ternary terial. Starting with La,t h e bottom product composition, it is mixture-that is, t h e three binary systems. By revolving these necessary t o find V,,the vapor in equilibrium with t h a t liquid. surfaces into the X-Y plane, it is possible t o show any temperaDraw a line through L,parallel t o t h e nearest liquid isotherm. ture in the plane of t h e triangle. By projections the limits of the From the Y intercept proceed horizontally t o the liquid line of the isotherms discussed above are located on t h e triangle. benzene-xylene boiling point curve, vertically at constant temFigure 1 is such a triangular diagram with the auxiliary curves perature t o t h e vapor line, horizontally back t o the Y-axis, and representing the benzene-toluene-xylene system a t atmospheric locate the point through which is drawn a line parallel t o the pressure. Equilibrium constants were used t o plot the equilibnearest vapor isotherm. A line parallel to the vapor line just rium curves. From such a diagram it is desired t o solve material
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378
April,
1944
INDUSTRIAL AND ENGINEERING CHEMISTRY
379
\
BOILING POINT CURVE L
'TEMPERATURE, OF.
BENZENE-XYLENE BOILING POINT CURVE
*'
\
TEMPERATURE, 'F.
21
380
Vol. 36, No. 4
INDUSTRfAL AND ENGINEERING CHEMISTRY
located and through the Y intercept of the liquid line through L. intersects a horizontal through L,. Through this point of intersection a radial line from the origin is drawn which intersects the vapor line a t Vs. This is merely a geometric construction based on similar triangles and using the equilibrium relations known when one component drops t o zero concentration. I n other words, point L,is known and V,lies on a given line at a point where the ordinate bears the same ratio t o the ordinate of L,as the ratio of the known ordinates when the toluene concentration is zero, Point L1 must coincide with V , for this hypothetical case, since there are no products and, by material balance, L, must coincide with V , in quantity and quality. Point V1 is located as was V,, and L,must coincide with it. This procedure is carried on until the xylene concentration is negligible and the steps are completed on the McCabe-Thiele diagram. T o keep the diagram as rlear as possible, the steps are not all drawn in, but between eleven and twelve plates are required to reach the desired distillate.
HE other limiting case, minimum reflux, is influenced by the Tcondition of the feed, which is arbitrarily taken t o be at its boiling point, as stated above. It is now necessary t o determine the minimum reflux ratio and then show graphically that it is the the dialimiting case. From the method suggested by Jenny (0, gram can be used t o find the limiting L/V ratios and thereby the reflux ratio. Since the temperaure of the feed tray is approximately 195" F. and since the light key component, benzene, is preqent in the bottoms in small concentration, the stripping section L/'V equals Kb a t 195' F., which is 1.33. Assuming that xylene drops t o a negligible value a t constant ratio of benzene to toluene, the temperature a t which that takes place is also readily determined. I n this case that temperature is approximately 190" F., and since the heavy key component, toluene, is present in the distillate in small concentration, the fractionating section L/V equals Kt a t 190' F., which is 0.50. These L / V ratios correspond to a reflux ratio L I D equal to 1.0. The steps up t o the resulting binary system for the problem a t minimum reflux were determined, but only the operating line is shown for the completion on the McCabe-Thiele diagram. Starting with L,, V , is lo-. cated as before and coincides with V,for total reflux. However, LI must be determined by material balance. Using the procedure similar t o t h a t shown by Bonilla (1) and otherwise known as the lever rule, Ll lies on LsVaand a t such a distance that L,V,/LILs is equal to L/V or, in this case, L1/Vawhich must be 1.33 as shown above. Having thus obtained L1, VI is found by similar procedure to that used t o locate V8. Construction lines are not shown as the procedure is the same as before. It is now easier t o locate Lz as it lies on a line through L,V1 and on a line through LI parallel to VqVl. The constant L/V requires that the line VmVm+lbe parallel to Lm+1Ln+2. This method is continued until the change in concentration for each theoreticA1 plate is very small. It is now necessary to change t o the top section L/V, and thus the feed plate is located. This point corresponds t o plate 16 as shown on Figure 1. This diagram helps to visualize the actual operation of a column and the sections shown by Gilliland (3) t o exist in a fractionating system. On the diagram for minimum reflux these divisions are definitely isolated. Varying from the previous presentation and holding t o the special ternary system under consideration the sections are:
1. Starting at the bottom of the tower is the section in which the heavy component drops to a nearly constant value. 2. Up to the lower pinch point is a section in which the light key component has increased in concentration and the heavy key component has decreased in concentration. 3. There is a lower pinched-in region where infinite plates are required to produce a finite change in composition. 4. Next is the intermediate region where the components heavier than the heavy key component drop t o a negligible value.
5 . The second pinched-in point occurs a t the start of the binary system and thus appears on the McCabe-Thiele diagram. 6. I n the final section the desired overhead concentration is reached. The occurrence of two pinched-in points is necessary, as shown by the diagram. If only one existed, it could be relieved by adjusting the feed location. Another interesting factor is that, in the intermediate region between the pinched-in points, the ratio of light key component to heavy key Component decreases in passing upward through thc column. With the limits of the operations for producing the required overhead and bottom products, the problem is t o choose a suitable reflux ratio and determine the number of plates required and the feed location. The reflux ratio was chosen equal t o 2.0, and Figure 2 was used as an aid in the algebraic calculations and as a piot t o show concentration and temperature gradients. Trialand-error calculations were entirely eliminated since temperatures can be determined for any concentration of liquid or vapor a t equilibrium bubble-point or dew-point conditions. The problem shown here is completely defined by what has been stated above and the following equations (6): The equations for the top sections are, for benzene: =
(L/V )nZn+1
=
0.667~,+1
%I
+ (DIVIJsn
+ 33.2
for toluene: yn =
0.6672,+1
+ 0.17
for xylene: yn
= 0.667~,+1
Below the feed plate, the equations are, for benzene: ym = =
(L/V)ms(m,S 1.2212,+1
- (L*/Vm)ZLa
- 0.11
for toluene: y m = 1.221~,+1 - 16.4
for xylene: y m = 1.221sm+1- 5.55
The feed plate is taken as that plate where a greater ratio of light key component t o heavy key component occurs by using the top-section L/V ratio as the calculations are made from the bottom of the tower. The temperature relations are readily determined for the feed and the feed plate from the diagram. It is believed that this method of plotting compositions and temperatures will be useful whenever ternary distillation problems are encountered. It completely eliminates trial-and-error calculations and serves very well t o illustrate temperature and concentration gradients through a column. NOMENCLATURE
Kb
Kt
=
=
K, = L = V =
F
D
= =
L, = y =
z = n m, m n,
equilibrium constants for benzene equilibrium constants for toluene equilibrium constants for xylene liquid leaving a tray in equilibrium with its vapor vapor leaving a tray in equilibrium with liquid on t h a t tray feed to column distillate bottoms product mole % of any component in vapor mole % of any component in liquid 1, etc. = subscript for plates above feed 1, etc. = subscript for plates below feed
+ +
LITERATURE CITED
(1) Bonilla, C. F., Trans. Am. Inst. Chem. Engrs., 37, 669 (1941). (2) Dourson, R. H., Sage, B. H., and Lacey, W '. N., PetroZeum Tech.,
5, T.P.1490 (1942).
Gilliland, E. R., IND.ENG.CHEM.,32, 1101 (1940). (4) Jenny, F. J., Trans. Am. I n s t . Chsm. Engrs., 35, 636 (1939). ( 5 ) Robinson, C. S., and Gilliland, E. R., "Elements of Fractional Distillation", 3rd ed., New York, McGraw-Hill Book Co., 1939. (3)
