14. If the convergence criterion is met, compute values of L, from Eq. (13-74) and values of Q1 and QN from Eq. (13-72). Otherwise, set k = k + 1 and repeat steps 7 to 14.

Step 14 implies that if the calculations are not converged, values of Tj(k) computed in step 10 and values of Vj(k) computed in step 12 are used as values of the tear variables to initiate iteration k + 1. This is the method of successive substitution, which may require a large number of iterations and/or may result in oscillation. Alternatively, values of Tj(k) and Vj(k) can be adjusted prior to initiating iteration k + 1. Experience indicates that values of T, should be reset if they tend to move outside of specified upper and lower bounds and that negative Vj values be reset to small positive values. Also, damping can be employed to prevent values of absolute Tj and Vj from changing by more than, say, 10 percent on successive iterations. Orbach and Crowe [Can. J. C'hem. Eng., 49, 509 (1971)] show that the dominant eigenvalue method of adjusting values of T, and Vj can generally accelerate convergence and is a worthwhile improvement to the BP method.

Example 4: Calculation of the BP Method Use the BP method with the SRK equation-of-state for K values and enthalpy departures to compute stage temperatures, interstage vapor and liquid flow rates and compositions, and reboiler and condenser duties for the light-hydrocarbon distillation-column specifications shown in Fig. 13-51 with feed at 260 psia. The specifications are selected to obtain three products: a vapor distillate rich in C2 and C3, a vapor side-stream rich in n-C4, and a bottoms rich in n-C5 and n-C6.

Initial estimates provided for the tear variables were as follows compared with final converged values (after 23 iterations), where numbers in parentheses are consistent with specifications:

Stage |
T(0) |
T(23) Op |
V<0), (lb-mol)/h |
V<23), (lb-mol)/h |

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