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Stage compositions in the TG method are obtained by stage-to-stage calculations from both ends toward the feed stage. With reference to Fig. 13-1, the calculations work with the ratios vn/d, €n/d, vm/b, and €m/b instead of v or € directly. The working equations are derived as follows:

In the rectifying section, the equilibrium relationship for component i at any stage n can be expressed in terms of component flow rate in the distillate d = DxD and component absorption factor An = Ln/KnVn.

The general component-i balance around a section of stages from stage n to the top of the column is vn = €n + 1 + d vn /d = (€n + i/d) + 1

Increasing the subscripts in Eq. (13-47) by 1 and substituting for €n + 1/d in Eq. (13-48) gives the following combined equilibrium and material-balance relationship for component i vn /d = (vn + 1/d)An + 1 + 1

Equation (13-50) is used to calculate, from the previous stage, the (€/d) ratio on each stage in the rectifying section. The assumed temperature and phase-rate-profile assumptions conveniently fix all the An values for ideal solutions. The calculations are started by writing the equation for stage N:

A knowledge of the reflux ratio (obtained from the specified distillate and top vapor rates) permits the calculation of (^Nd) from which (€m- 1/d) is obtained, etc. Equation (13-50) is applied to each stage in succession until the ratio €m+2/d in the overflow from the stage above the feed stage is obtained. The calculations are then switched to the stripping section.

The equilibrium relationship for component i in the stripping section can be expressed in terms of component flow rate in the bottoms, b = Bxb, and Sm = KmVm /Lm as ym = Kmxm Vmym (KmVm /Lm)Lmxm vm Sm€m vm /b = (€m /b)Sm

Combination with the material balance

The bottom-up calculations are started by writing Eq. (13-55) for stage 1 as

The Sm values all are fixed by assumed temperature and phase-rate profiles. Equation (13-55) is applied to each of the stripping stages in sequence until the ratio Cm + 2/b in the liquid entering the feed stage is obtained.

The manner in which rectifying and stripping-section calculations are meshed at the feed stage depends upon the thermal condition of the feed. Figure 13-45 shows three possible ways in which fresh feed can affect the L and V rates between the feed stage and stage M + 2. The superscript bar denotes the stream rate when the stream enters a stage, while the lack of a bar denotes the rate when the stream leaves a stage.

Top-down calculations for the example problem are shown in Table 13-10 and bottom-up calculations in Table 13-11. Top-down and bottom-up calculations have provided values of Cm + 2/d and Cm + 2/b respectively. For a bubble-point feed,

and a combination of Eqs. (13-48) and (13-54) provides for each component i b Vm + 1/d Cm + 2/d + 1 d v + 1/b Cm + 2/b - 1

The b/d ratios obtained from this equation can then be used to calculate the individual b and d values as follows. Since d + b = Fxf d = -

FXf and

Calculated values of d from the first column iteration in the example problem are as follows:

d

1 - 1 b

b d

FXf

d

c

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