and has a slope of p/(p + 1). The effects of various values of p are shown in Fig. 7-4 for a given slope of the operating line above the feed. Thus, if p = 0, the mols of overflow above and below the feed are equal, and the operating lines must intersect in a horizontal line through the diagonal at zF. A value of p = —1, i.e., Vf - F/-would put the intersection on a vertical line at zf.

The value of p is best obtained by an enthalpy balance around the feed plate. However, when the molal enthalpy of the overflow from the feed plate and the plate above is essentially the same and the enthalpy of the vapor from the feed plate and the plate below is also the same, then, by Eqs. (7-6) and (7-6a), — p becomes approximately the heat necessary to vaporize 1 lb. mol of the feed divided by the latent heat of vaporization of the feed. Thus, an all-vapor feed at its boiling point would have a value of p « 0, for an all-liquid feed at its boiling point, p would equal — 1; p would be less than —1 for a cold feed, between — 1 and zero for a partially vaporized feed, and greater than zero for a superheated vapor feed.

A little study of Fig. 7-4 indicates that for a given 0/D fewer plates are required for a given separation the colder the feed. This results from the fact that the cold feed condenses vapor at the feed plate and increases the reflux ratio in the lower portion of the column. This higher reflux ratio is obtained at the expense of a higher heat consumption in the still.

Fig. 7-4. The effect of the thermal condition of the feed on the intersection of the operating lines.

1, p is greater than 0 (superheated vapor feed)

Fig. 7-4. The effect of the thermal condition of the feed on the intersection of the operating lines.

1, p is greater than 0 (superheated vapor feed)

Logarithmic Plotting. When the design involves low concentrations at the terminals of the tower, it is necessary to expand this part of the diagram in order to plot the steps satisfactorily. This may be done by redrawing these regions of the y,x diagram to a larger scale. In some cases, it may be necessary to make more than one expansion of successive portions of the diagram. Alternately, the y,x diagram may be plotted on logarithmic paper, and the steps constructed in the usual manner. On this type of plot in the low-concentration region, the equilibrium curve is generally a straight line, since, for small values of x: y « aX—— becomes y - ax; however, the operating line * 1 + (a — l)x which is of the form ym = axm+i + b is a curved line unless 6 = 0. The operating line is constructed from points calculated from the operating-line equation.

Minimum Number of Plates. The slope of the operating line above the feed is On/Fn, and as this slope approaches unity the number of theoretical plates becomes smaller. When On/Vn is equal to 1, Or/D is equal to infinity, and only an infinitesimal amount of product can be withdrawn from a finite column. Frequently it is assumed that total reflux corresponds to the addition of no feed or to the removal of no products. If such is the case, the tower is not meeting the design conditions. It is better to visualize a tower with an infinite cross section, which is separating the feed at a finite rate into the desired products. Under such conditions the column is said to operate at total reflux or with an infinite reflux ratio, and both operating lines have a slope of unity causing them to coincide with the y = x diagonal. Since a higher reflux ratio than this is not possible, the size of the steps on the y,x diagram is a maximum, and a minimum number of theoretical plates to give a given separation is obtained. This number is determined by simply using the y = x diagonal as the operating line and con-A column with the minimum number of plates serves as a reference below which no column with fewer plates can give the desired separation, but such a column would have a zero capacity per unit volume and would require infinite heat consumption per unit of product.

Minimum Reflux Ratio. In general, it is desired to keep the reflux ratio small in order to conserve heat and cooling requirements. As the reflux ratio Or/D is reduced from infinity, the slope of the operating line ~ = ,n ?j(PL decreases from unity. Thus, in Fig. 7-5 a reflux V n iy/ls) ~r 1

ratio of infinity would correspond to operating lines coinciding with the diagonal as ac6, and a lower reflux ratio would correspond to adb. It is obvious that the average size of the steps between the equilibrium curve and the line adb will be much smaller than the size of the steps

s true ting the steps from xD to xw.

between the equilibrium curve and the line acb. Thus, a reduction of the reflux ratio requires an increase in the number 6f theoretical plates to~effect a ^^^ejBMSSa- AS the reflux ratio is fur.ther decreased, the size ofthe steps between the operating lines and the equilibrium curve becomes still smaller, and still more theoretical plates are required, until the conditions represented by aeb are encountered, when the operating line just touches the equilibrium curve. In this final case, the size of the step at the point of contact would be zero, and an infinite number of plates would be required to travel a finite distance down the operating line. The reflux ratio corresponding to this case is called the minimum reflux ratio and represents the theoretical limit below Whicfr^tttr^ be re3uced and produce the desired separation even if an infinite column is employed. This reflux ratio is easily determined by laying out the operating line of the flattest slope through xD that just touches but does not cut the equilibrium curve at any point; the slope of this line™ = (0/D)^+ 1 g*ves value of O/D. Alternately, it may be calculated from the equation

D Va

where xc and yc are the coordinates of the point of contact. For mixtures having normal-shaped equilibrium curves, such as benzene-toluene, the point of contact of the operating line with the equilibrium curve will occur at the intersection of the operating lines. For cases that deviate widely from Raoult's law, the operating line may become tangent to the equilibrium curve before the intersection of the operating lines touches the equilibrium curve, and in such cases it is usually best to plot the diagram and determine the slope On/Vn.

Optimum Reflux Ratio. The choice of the proper reflux ratio is a matter of economic balance. At the minimum reflux ratio, fixed charges are infinite, because an infinite number of plates is required. At total reflux, both the operating and the fixed charges are infinite. This is due to the fact that an infinite amount of reflux and a column of infinite cross section would be required for the production of a finite amount of product. The tower cost therefore passes through a minimum as the reflux ratio is decreased above the minimum. The costs of the still and condenser both increase as the reflux ratio is increased. The heat and cooling requirements constitute the main operating costs, and the sum of these increases almost proportionally as the x c reflux ratio is increased. The total cost, the sum of operating and fixed costs, therefore passes through a minimum.

Optimum Reflux Ratio Example. The following estimates illustrate these factors for the fractional distillation of a methanol-water mixture to produce 250 gal. of methanol per hour. In making the calculations, it was assumed that the heat-transfer surface required was proportional to the vapor rate which is equal to

Z>(^ + l) and that the tower costs were proportional to the total square feet of plate area. The charges on the equipment, including maintenance, repairs, depreciation, interest, etc., were taken at 25 per cent per year, and the heating costs were based on the heat load. The costs per hour as a function of the reflux ratio are summarized in Table 7-1. Labor charges have been excluded since these should be

Table 7-1. Estimated Cost fob the Fractionation of a Methanol-Water

Mixture

Table 7-1. Estimated Cost fob the Fractionation of a Methanol-Water

Mixture

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