yields t [(Jfi+1§ f - hjfo + (Hj+ u t - hpi)wu + (Hj+ u, - Hu)dA - Qc = 0 ¿=1
Since Iji = LjXji, wu = Wl xpi, d{ = M^, the above form of the energy balance may be solved for L} to give
- wx i (Hj+ lt, - hpi)xpi - D £ (HJ+ lB, - H 4- 6c
The complete set of enthalpy balances for the column which are obtained in a manner analogous to that demonstrated above. After the L/s have been computed by use of these enthalpy-balance expressions, the corresponding vapor rates may be computed by use of the total-material balances.
To simplify the geometrical representation of the g functions, consider the case where only one sidestream Wt is withdrawn in addition to the distillate D and the bottoms B as shown in Fig. 3-4. For this column, only two 9 multipliers exist, 90 and 9V These two multipliers are defined by the first two expressions of Eq. (3-10). If D and Wx are specified, then the corresponding g functions are ffo(0o,0i)= iiddco-D (3-21)
where
Again, (di)eo may be stated in terms of pt as demonstrated above by Eqs. (3-13)
Traces of the functions g0 in the g090 , go0i9 and the 9^9^ planes are shown in Fig. 3-5. and traces of g1 in the gx 90 , g191, and 9t 9(f planes are shown in Fig. 3-6. The desired solution is the intersection of the traces of g0 and gY in the 909x plane where g0 = gt = 0 as shown in Fig. 3-7. Examination of these graphs reveals that g0 and gt arc continuous, monotonic functions for all positive values of 90 and 9t. Such behavior is desirable in the numerical solution of problems.
The set of 0's that makes gQ = gx = 0 may be found by use of the Newton-Raphson method3 as described in App. A. The Newton-Raphson method consists of the successive solution of the equations corresponding to the linear terms
-Trace of gQ -The desired solution b f x
"Trace of g Figure 3-7 Traces of the functions g0 and fQ] gi in the 6o0i plane.
of the Taylor series expansions of the functions g0 and gt about the assumed values of the variables 60 and 0U namely
Expressions for the partial derivatives are given in Prob. 3-1. The Newton-Raphson method is illustrated by the following example.
Example 3-1 Make two trials by use of the Newton-Raphson method for the set of values x and y which make fx = f2 = 0 simultaneously
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