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already considered. However, the mole fraction of the benzene remaining in the still was also higher than in either of the two previous cases. Because the overhead composition is constant, the integral of

can be obtained analytically. Integration of r dx

J a + bx produces l/£> In (a + bx). Using the values from above

which gives the same result as obtained by graphical integration.

Note in Figs. 5.3a to e (deliberately stopped at a reflux ratio of 13:1 for clarity of illustration) that increases in reflux ratio produce an ever-diminishing increment in reduction of the concentration of the light component in the still. Increasing reflux ratio to 20:1, for example, would only decrease the residual mole fraction of the benzene to 0.23 (from 0.25 for 13:1),

### 5.1.4 Time and boil-up requirements

Constant reflux ratio. With a constant reflux ratio and a constant vapor rate in the distillation system, the moles of vapor that must be produced during the distillation can be simply calculated from Eq. '5.4). In Example 5.3, it was calculated that 52.2 moles of distillate were produced when a reflux ratio of 1.64 was used.

### Therefore

Because the boil-up rate associated with a specific distillation system should be known from past experience with the system, this calculation of total vapor load immediately produces a time required for the new separation.

Conversely, in the design of a new system, the time available for the distillation can be divided into the vapor load to determine the boil-up rate required.

= 138 moles of vapor

Varying reflux ratio. The amount of material which must be vaporized to achieve some specified separation under conditions of varying reflux ratio may be developed from Eq. (5.3).

Differentiating dV = dL + dD

Multiplying dh by dV/dV

Substituting from Eq. (5.5)

(where R is variable). Rearranging dV(1-Rh)'dD dV = dDi{ 1 - Ri{R + 1)]

Integrating

/'v f>d t dv~ Jo 1~^R/(R + 1) From Eqs. (5.1) and (5.2)

but in batch distillation

Multiplying the third of these by xD

Dxd = WpXo - Wxw Setting equals to Dxd equal to each other.

Differentiating (remembering that W0, jc0, and xD are all constant when reflux ratio varies) and rearranging,

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