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where a,, = Kji/Kjb, the relative volatility of component i at the temperature of plate j. The quantity Kjb is the K value of the base component, evaluated at the temperature of plate j.

It can be shown that the xj('s and y/s defined by Eq. (2-29) form a consistent set in that they give the same value of Kjb (see Prob. 2-7). Component b represents a hypothetical base component whose K value is given by

where the constants a and b are evaluated on the basis of the values of K at the upper and lower limits of the curve fits of the midboiling component of the mixture or one just lighter. Thus, after Kjb has been computed by use of Eq. (2-30), the temperature 7} n+1 to be assumed for the next trial is calculated directly by use of Eq. (2-31).

The corrected compositions and the new temperatures are used in the enthalpy balances to determine the total flow rates to be used for the next trial through the column.

Determination of a Set of Improved Total-Flow Rates by Use of the Constant-Composition Method

In the constant-composition method, one of the total-flow rates (Vj or L}) is eliminated from the enthalpy balance given by Eq. (2-1) for each stage by use of the component-material balances for the respective stage. The restatement of the enthalpy balances given by Eq. (2-1) in the form called the constant-composition method may be initiated by first observing that

and c c

Use of relationships of this type permits the enthalpy balance to be restated in terms of the component-flow rates. For example, the enthalpy balance enclosing plate j

Vj+! Hj+! - Ljhj -DHd-Qc = 0 (i = 1, 2, ...,/- 2) (2-32)

may be restated in terms of the component-flow rates as follows i [Hj+ u ¡Vj+ u- hjJji - H M -Qc = o (2-33)

i = i where HDi = hu for a distillation column having a total condenser and, HDi = Hu for a distillation column having a partial condenser. When the component-material balance enclosing plate j [see Eq. (2-12)]

is used to eliminate vj+l from Eq. (2-33), the following result is obtained t [(Hj+, - hjlj, + (Hj+u, - H„,K] ~Qc = 0

Lj l (Hj+ u■ - hjfci + D t (Hj+U, - HDi)XDi -Qc = 0 i = i i = i

The desired expression for calculating L} is then given by

Similarly and

Qc = Lt £ (H2i - hu)xu + D X (H2i - HDi)XDi (2-36)

The flow rates in the stripping section may be determined by use of the enthalpy balances which enclose either the top or the bottom of the column and the given plate. When the reboiler is enclosed, the following formula is obtained

QR-Bt{hsi-hn)xBi

This expression is developed in a manner analogous to that demonstrated above for Eq. (2-34). The above formulas are given the name "constant-composition method" because each of the summations appearing in Eqs. (2-34) through (2-37) may be represented by a thermodynamic process which occurs at constant composition. The reboiler duty QR is found by use of the overall enthalpy balance [the last expression given by Eq. (2-1)].

The total-flow rates of the vapor and liquid streams are related by the following total material balances

After the L/s for the rectifying section and the V/s for the stripping section have been determined by use of the enthalpy balances, the remaining total-flow rates are found by use of Eq. (2-38). These most recent sets of values of the variables {Tj,n+i}, {^-.n+i}» and {Lj,n+ i} are used to make the next trial through the column. The procedure described is repeated until values of the desired accuracy have been obtained. A summary of the steps of the proposed calculational procedure follow.

### Calculational Procedure for Ideal Solutions

1. Assume a set of temperatures {7}} and a set of vapor rates {Vj}. [The set of liquid rates corresponding to the set of assumed vapor rates are found by use of the total-material balances; see Eq. (2-38)].

2. On the basis of the temperatures and flow rates assumed in step 1, compute the component-flow rates by use of Eqs. (2-18) through (2-21) [or (2-22)] for each component i.

3. Find the 6 >0 that makes g(Q) = 0; see Eqs. (2-26) through (2-28). (Newton's method4 always converges to the desired 6, provided that the first assumed value of 0 is taken to be equal to zero.)

4. Use Eq. (2-29) to compute the corrected x}- s or s for each component i and plate j.

5. Use the results of step 4 to compute the Kjb for each stage j by use of either one of the expressions given by Eq. (2-30). Use the Kjh's so obtained to compute a new set of temperatures {7},n+1} by use of Eq. (2-31).

6. Use the results of steps 4 and 5 to compute new sets of total-flow rates, {Vj,n+1} and {Lj,n+hy use of Eqs. (2-34) through (2-38).

7. If 6, the Tj\ and I^-'s are within the prescribed tolerances, convergence has been achieved; otherwise, repeat steps 2 through 6 on the basis of the most recent set of T/s and V/s.

In the above calculational procedure, it is supposed that the pressure drop from plate to plate is negligible relative to the total pressure. If this assumption is not valid, the calculational procedure is modified as described in Sec. 2-4.

The solution of the component-material balances and equilibrium relationships by use of the above recurrence formulas is demonstrated by the following numerical example.

Overhead vapor, V2 = 100

Overhead vapor, V2 = 100

Figure 2-5 Flow diagram for Example 2-1.

Example 2-1 (a) On the basis of the initial set of temperatures (7\ = T2 = T3 — T4 = 560°R) and the total-flow rates displayed in Fig. 2-5, solve Eq. (2-18) for the component-flow rates by use of the above recurrence formulas given by Eqs. (2-20) and (2-21). (b) Repeat (a) by use of the recurrence formulas given by Eqs. (2-22) and (2-21).

Component A', C, £, Specifications

1 1/3 4 x 103/Pi 4.6447 x 103 Total condenser, P = 1 atm, boiling point

2 1/3 8 x 103/P 4.6447 x 103 liquid feed (ln = FX,), N = 4, / = 3,

3 1/3 12 x 103/P 4.6447 x 103 F = 100 lb mol/h, D = Lt = L2 = 50 lb mol/h, I3 = 150 lb mol/h, V2 = V3 = V4 = 100 lb mol/h t T is in °R. \$ P is in atm.

Solution (a) Use of Eqs. (2-20) and (2-21) The correspondence of the symbols in the recurrence formulas and the elements of A, and f{ follow