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Bottom term of Eq. (3.30) is 2.49/(1.5514 1.73162) = 0.5353

Ns = [log [( - 0.0352)/( -0.2985)]/log 0.5353 = 3.42

Total number of stages is therefore

This checks with the graphical solution of Example 2.1 which gave just above 9 stages.

3.2.9 The Jafarey, Douglas, and McAvoy equation: design and control

Jafarey et al. (61) derived a simple, approximate equation for binary distillation by simplifying the solution to Smoker's equation. Their equation is powerful for predicting the effect of disturbances on column performance. This makes their equation particularly useful in computer and microprocessor control, where it can be applied to estimate the effect of disturbances and the control action needed to compensate for them. This application is highlighted in Examples 3.8 and 3.9. The Jafarey et al. equation is

S is given by Eq. (3.3). Equation (3.40) is stated (61) to predict N within plus or minus two to three stages. The effect of disturbances on column performance can be evaluated by combining Eq. (3.40) with the column overall mass and component balance equations [Eqs. (3.1), (3.2)]. Douglas et al. (62) illustrate several applications. Example 3.8 illustrates this for binary distillation.

Extension to multicomponent separations. Douglas et al. (62) extended Eq. (3.40) to multicomponent distillation. Their procedure lumps the light key and light nonkeys into a pseudo-light-key component and the heavy key and heavy nonkeys into a pseudo-heavy-key component according to Hengstebeck's method (63). The Douglas et al. (62) procedure is complex but was shown to give accurate results. Douglas et al. (62) also describe a simple, but less accurate approach. The author proposes an alternative variation of the simple procedure, illustrated in Example 3.9.

Example 3.8 A benzene-toluene column normally operates as described in Example 2.1. The column is computer-controlled, using the Jafarey et al. algorithm. The algorithm manipulates boilup flow to control toluene purity. If the

In S

In S

toluene purity is to be temporarily increased from 90 to 95 percent, and the benzene purity to remain unaffected, what would the controller set the boilup flow rate at? Assume that any boilup changes will be compensated by reflux and distillate rate changes, such that the benzene purity remains unaffected. Also assume a relative volatility of 2.49.

solution First, determine N for normal operation from Eq. (3.40)

In S = In 1(0.95/0.05)(0.90/0.10)1 - 5.142 [Eq. (3.3)1

The denominator of Eq. (3.40) is ln[2-49V^ir

This checks with the x-y diagram (Example 2.1) and with Smoker's method (Example 3.7).

Solve the material and component balances for the new conditions:

Solving these simultaneously gives B = 122 and D = 78 lb-mole/h. Recalculate S for the new conditions

The denominator of Eq. (3.40) will be equal to (In S)/N, i.e., 5.889/8.80 = 0.6692. The term inside the square root will be equal to (exp 0.6692/2.49)2, i.e., 0.6150. Therefore,

(R + l)(0.4i? + 0.75) This gives the quadratic equation

Solving

D 0.5572 ± Vo.55722 + 4 x 0.1540 x 0,4612 R =-2 x 0.1540-- 4 314

The computer control will therefore increase the boilup rate from 234 lb-mole/h (Example 2.1) to 364 lb-mole/h.

Example 3.9 A depropanizer normally operates as described in Example 2.4. The column is computer-controlled, using the Jafarey et al. algorithm. The algorithm manipulates reflux flow to control top product purity. Distillate flow rate must remain fixed, but bottom purity is allowed to vaiy. The top product purity spec is temporarily relaxed from 0.5 mole percent to 0.9 mole percent. What would the controller set the reflux flow at?

solution First, lump all the light nonkeys into a pseudo light key (PLK), and all the heavy nonkeys into a pseudo heavy key (PHK). The material balance of Example 2.4 can now be written in terms of the pseudo nonkeys.

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