heat to vaporise 1 mole of feed molar latent heat of the feed
A material balance of the more volatile component over the whole column gives:
Thus, from equation 11.42:
This equation is commonly known as the equation of the q-line. If xq = x/, then yq = x/. Thus, the point of intersection of the two operating lines lies on the straight line of slope q/(q — 1) passing through the point (x/, x/). When yq = 0, xq = x//q. The line may thus be drawn through two easily determined points. From the definition of q, it follows that the slope of the q-line is governed by the nature of the feed as follows.
These various conditions are indicated in Figure 11.16.
Mole fractionC6H6 in liquid (x)
Figure 11.16. Effect of the condition of the feed on the intersection of the operating lines for a fixed reflux ratio
Mole fractionC6H6 in liquid (x)
Figure 11.16. Effect of the condition of the feed on the intersection of the operating lines for a fixed reflux ratio
Altering the slope of the q -line will alter the liquid concentration at which the two operating lines cut each other for a given reflux ratio. This will mean a slight alteration in the number of plates required for the given separation. Whilst the change in the number of plates is usually rather small, if the feed is cold, there will be an increase in reflux flow below the feed plate, and hence an increased heat consumption from the boiler per mole of distillate.
11.4.3. The importance of the reflux ratio Influence on the number of plates required
The ratio Ln/D, that is the ratio of the top overflow to the quantity of product, is denoted by R, and this enables the equation of the operating line to be expressed in another way, which is often more convenient. Thus, introducing R in equation 11.35 gives:
Any change in the reflux ratio R will therefore modify the slope of the operating line and, as may be seen from Figure 11.15, this will alter the number of plates required for a given separation. If R is known, the top line is most easily drawn by joining point A (xd, xd) to B (0, xd/(R + 1)) as shown in Figure 11.17. This method avoids the calculation of the actual flow rates Ln and Vn, when the number of plates only is to be estimated.
Figure 11.17. Influence of reflux ratio on the number of plates required for a given separation
If no product is withdrawn from the still, that is D = 0, then the column is said to operate under conditions of total reflux and, as seen from equation 11.47, the top operating line has its maximum slope of unity, and coincides with the line x = y. If the reflux ratio is reduced, the slope of the operating line is reduced and more stages are required to pass
Figure 11.17. Influence of reflux ratio on the number of plates required for a given separation
Mole fraction CSH6 in liquid (x)
Mole fraction CSH6 in liquid (x)
from x/ to xd, as shown by the line AK in Figure 11.17. Further reduction in R will eventually bring the operating line to AE, where an infinite number of stages is needed to pass from xd to x/. This arises from the fact that under these conditions the steps become very close together at liquid compositions near to x/, and no enrichment occurs from the feed plate to the plate above. These conditions are known as minimum reflux, and the reflux ratio is denoted by Rm. Any small increase in R beyond Rm will give a workable system, although a large number of plates will be required. It is important to note that any line such as AG, which is equivalent to a smaller value of R than Rm, represents an impossible condition, since it is impossible to pass beyond point G towards x/. Two important deductions may be made. Firstly that the minimum number of plates is required for a given separation at conditions of total reflux, and secondly that there is a minimum reflux ratio below which it is impossible to obtain the desired enrichment, however many plates are used.
Figure 11.17 represents conditions where the q-line is vertical, and the point E lies on the equilibrium curve and has co-ordinates (x/, y/). The slope of the line AE is then given by:
If the q-line is horizontal as shown in Figure 11.18, the enrichment line for minimum reflux is given by AC, where C has coordinates (xc, yc). Thus:
Underwood and Fenske equations
For ideal mixtures, or where over the concentration range concerned the relative volatility may be taken as constant, Rm may be obtained analytically from the physical properties of the system as discussed by Underwood(28) . Thus, if xnA and xnB are the mole fractions of two components A and B in the liquid on any plate n, then a material balance over the top portion of the column above plate n gives:
and:
Under conditions of minimum reflux, a column has to have an infinite number of plates, or alternatively the composition on plate n is equal to that on plate n + 1. Dividing equation 11.51 by equation 11.52 and using the relations X(n+1)A = xnA and X(n+1)B = xnB, then:
A JnA LnxnA + DXdA
JnB LnxnB + DxdB
Thus:
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