Operating lines intersection

Minimum Reilux with Pinch Zone. There are some distillations where the minimum reflux does not occur at the intersection of the upper and lower operating lines and the q line. These cases arise when the equiUbrium is skewed from positive activity coefficients and when the operating line intersects the equiUbrium line in a 2one of constant composition, a pinch 2one, which is not at the line intersection. Figure 14 illustrates such a case. An example of such a pinch 2one in an ethanol—water column is available (37). 

The reason for this simple relationship is that the concept of minimum reflux implies an infinite number of stages and thus no change in composition from stage to stage for an infinite number of stages each way from the pinch point (the point where the McCabe-Thiele operating lines intersect at the vapor curve for a well-behaved system, this is the feed zone). The liquid refluxed to the feed tray from the tray above is thus the same composition as the flash liquid. 

The top and bottom operating lines intersect the diagonal at xd and xb respectively mark these points on the diagram. 

It is seen from the example shown in Figure 11.15 in which the feed enters as liquid at its boiling point that the two operating lines intersect at a point having an X-coordinate of xj. The locus of the point of intersection of the operating lines is of considerable importance since, as will be seen, it is dependent on the temperature and physical condition of the feed. 

If the two operating lines intersect at a point with coordinates (xq, yq), then from equations 11.35 and 11.37 ... 

The ratio L/V depends on the amount of reflux in the column. At minimum reflux, the operating lines intersect each other and the q line at the point where the q line intersects the xy curve (in Fig. 8.2, at point D). 

For stripping, the entering liquid and gas concentrations are known. The fraction stripped and therefore the exit liquid concentration is also known, but the exit gas concentration is unknown. Therefore point 2 - at the bottom of the column is fixed, but point 1 - at the top of the column - is not fixed. The maximum exit and minimum gas flow rate gas concentration is obtained when the operating line intersects the equilibrium curve, as shown by the dashed line in Figure 6.16. The intersection is given by Equation 6.21.1S, which is also obtained by the simultaneous solution of a component balance and an equilibrium relation. 

The recovery of the key component is specified to calculate the exit composition of the gas stream for absorbers or the exit composition of the hquid stream for strippers from Equation 6.21.2A or 6.21.2S. For both cases, it is assumed that the operating line intersects the equilibrium cinve at one end and not at some intermediate point between the ends of the operating line. This is the case for dilute solutions when both the operating and equilibrium lines are linear. Separation of dilute solutions occurs frequently when purifying waste streams. Because both the equilibrium and operating curves are linear for dilute solutions, the equation derived by Kremser [59] can be used to calculate the number of equilibrium stages. 

C THE EXTENDED OPERATING LINE INTERSECT THE VAPOUR-LIQUID... 

It is also known that D=V, and when this expression for D is substituted in the above equation, the result is Y=X= Kj. That is, the rectifying section operating line intersects the diagonal at the distillate composition. Similarly, it may be shown that the stripping section operating line intersects the diagonal at Y = X = Xf the bottoms composition. 

Applied to the bottom stage, j-1 = N,j = N+l,B = L, Y= = 0, and X = Xg. That is, the operating line intersects the X-axis at X = Xg. This intersection point represents passing streams below the column. With open steam, the operating line crosses the diagonal and ends at the X-axis. Equilibrium stage steps are constructed in the usual manner, between the operating line and the equilibrium curve. 

Therefore, the feed mole fraction za is known, since the operating line intersects the y = x line at za- Then, the V/F ratio can be specified and the equilibrium values obtained, or vice versa. This is shown in Figure 3.18. 

Now remember that the feed plate is where the operating lines intersect each other. Solving mass balances for the top and bottom sections of the column gives ... 

Comparison of Eqs. (4.56) and (4.57) shows that the operating lines intersect at... 

If there are infinite numbers of both extracting and scrubbing stages, the equilibrium lines intersect the operating lines at the operating-line intersection, as illustrated in Fig. 4.16h ... 

To locate the point where the operating lines intersect, let and x i — j... 

The x and y values are from the operating lines (Figure 6.15). The temperatures are bubble and dew points for the liquids and vapors, respectively. The operating lines intersect at x = xF = 0.505, y = yF = 0.691, the dividing point between the rectifying and stripping sections. 

The bottoms, however, becomes leaner xu2 < xbj. As overhead is withdrawn at higher and higher rales, a point is reached where the lower end of the operating line intersects the equilibrium line at yp (Condition 3). Here xaj and yp are equilibrium compositions. The overhead is still pure low-boiler, but there are two invariant zones, one at the bottom and one at the top of the column. 

This operating line intersects the (x = y) line at x = yp. Figure 8.16, which is discussed later, shows the corresponding McCabe-Thiele construction. 

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