These equations reduce to Eqs. (7-14), (7-17), (7-27), and (7-31) for the special cases considered.
Method of Ponchon and Savarit. Ponchon (Ref. 14) and Savarit (Ref. 15) showed that Eqs. (7-13) and (7-27), or in general any equation
of this type, could be easily solved by plotting the enthalpy (or other property) of the saturated vapor and liquid vs. the mol fraction. For example, in Fig. 7-16, if the value Md for the upper section is plotted at xd, it is easily shown that any straight line drawn through the point (Md, xD) will intersect the enthalpy lines to give values of x and y that will satisfy Eq. (7-13). Thus if yn is known, a line through the vapor enthalpy curve at this composition and the point (Md, xD) will cut the liquid enthalpy at hn+i and rcn+i. When xn+i is known, 2/wH.x is obtained from the equilibrium curve and xn+z is determined by drawing a new line through (yn+i} Hn+1), (MD, xD), etc.
Similarly if Mw for the lower section is plotted at xw, then a straight line through this point intersects the two enthalpy curves at values that satisfy Eq. (7-27) and these straight lines give the relation between ym and xm+1.
A heat and material balance around the whole column (no side streams) gives
Equation (7-40) is of the same type as Eqs. (7-13) and (7-27), and similar reasoning leads to the conclusion that a straight line through (zf, Hf) and (xw, Mw) will also pass through (xD, Md). In other words, the point (zf, hF) lies on a straight line between (#/>, Md) and (xw, Mw). This line will be termed the terminal tie line.
In general the same type of information given by the constant O/V method can be obtained by the use of the Ponchon and Savarit method. For example, the cases of total reflux, minimum reflux ratio, and optimum feed-plate location can be easily solved.
Total Reflux. In the case of total reflux the values of Md and Mw are infinite, and lines drawn through them and the enthalpy curves will therefore be vertical. Thus, for this case the diagram shows that the composition of the liquid leaving the plate is equal to the composition of the vapor entering the plate, and the same number of theoretical plates will be obtained by both the constant O/V method and Ponchon-Savarit methods, regardless of the value of the enthalpies.
Minimum Reflux Ratio. The case of the minimum reflux ratio corresponds to conditions that require an infinite number of plates to obtain the desired separation. As in the case of the y,x diagram, this necessitates a region in which succeeding plates differ only differentially in composition, i.e., a pinched-in region.
The step equivalent to a theoretical plate on the enthalpy diagram involves going from the vapor below a plate to the liquid on a plate by means of the enthalpy operating line through one of the terminal enthalpy points and then proceeding from the composition of the liquid on the plate to the vapor above the plate by the equilibrium relationship, i.e., by an equilibrium tie line. If the composition of this vapor above the plate is to be equal to the composition of the vapor entering the plate, it is necessary for the enthalpy operating line to coincide with the equilibrium tie line. In the general case it is a trial-and-error procedure to determine the least value of Md that can be employed. However, if the pinched-in region occurs at the feed plate, the minimum reflux ratio can be easily determined by finding the equilibrium tie line that passes through the point Qif, Zf) and extrapolating this line until it intersects the vertical line at Xt>. The enthalpy value at this intersection will correspond to the minimum Mj>. In other cases, the equilibrium tie lines for a number of compositions above the feed plate can be extrapolated to the vertical line through xi>, and the maximum value of MD so obtained corresponds to the minimum reflux ratio for this section. A similar procedure can be used below the feed plate and the minimum value of Mw determined. Since Mn and Mw must fall on the line through the feed point, it can be determined which of the two values is the limiting one, and thus whether the pinched-in region is above or below the feed plate.
Optimum Feed-plate Location. The optimum feed-plate location again corresponds to making the total number of theoretical plates required for the operating conditions chosen a minimum, which is equivalent to making the change in composition per plate a maximum at all points. In general, it will be found that, when the enthalpy operating line is on the Xd side of the terminal tie line through (M»} xD) and (Mw, xw), larger steps will be obtained by using enthalpy operating lines drawn through (Md, xD) than those drawn through (MW) xw); on the xw side of the terminal tie line the reverse will be true. The enthalpy operating lines should be drawn through Mw for values of x less than the composition given by the intersection of the terminal tie line with the liquid enthalpy curve,* and the (Md, xD) point should be used for all operating lines corresponding to liquid mol fractions greater than this value. It should be noted that the value of the mol fraction at which the change-over is made is not equal to the composition of the feed. It will be equal to the composition of the feed when the feed enters with an enthalpy equal to that of a saturated liquid. If the enthalpy of the feed is greater than that of a saturated liquid, the change-over value will be at composition lower than that of the feed. If the enthalpy of the feed is less than that of the saturated liquid, the reverse will be true. This is similar to the conditions found for the y,x diagram.
General, It is frequently advantageous to use enthalpy diagrams to determine a series of values of yn, xn+h and ym, xm+1, which can be plotted on the y,x diagram to give the actual operating lines which are then employed in the usuaLstepwise manner. The y,x curves and the enthalpy composition values can be plotted on the same diagram, and the combined graphical procedure shown in Fig. 7-16 completes both the y,x and the enthalpy diagrams. To illustrate the procedure, (1) starting at (hw, xw•), a vertical line is drawn to the equilibrium curve giving yW) (2) this value is transposed to the H,y line by going horizontally to y = x and then vertically to the II curve giving the point (Hw, yw), (3) the line through this point and (Mw, xw) gives the value of xx on the h curve, and (4) the process is repeated. The intersection of the vertical lines with the horizontal lines on the y,x diagram corresponds to points of the operating lines, and the triangles above the lines drawn through these points are the usual steps on the y,x diagram.
Heat losses from the column can be taken into account by shifting the value of Md from plate to plate by an amount equal to the heat loss per plate divided by N. A side stream in the upper section of the column is handled by locating a point (MS) xNS) where
and drawing lines through this point.
In this case, (zF, hF) lies on a straight line through (xw, Mw) and (xns, Ms), and (xNS, Ms) lies on a straight line through (xD) Mn) and (xL, hL). In a similar way any number of side streams or feeds can be handled.
In general the Ponchon-Savarit diagram is somewhat more difficult to use than the constant 0/V diagram, but it is the exact solution for theoretical plates assuming that the enthalpy data employed are correct. This graphical procedure suffers because the absolute values of MD and Mw are frequently large, and to plot them on the diagram .requires the use of an ordinate scale such that the enthalpy curves for the liquid and vapor are crowded together, making it difficult to obtain accurate results. Likewise when low concentrations are encountered, it is impossible to obtain accurate results from the diagram unless the plot is greatly expanded. For such regions Eqs. (7-14) and (7-28) can be used algebraically, or in most cases the con-
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