The complete set of component-material balances may be represented by a matrix equation of the same form as the one used to represent these balances for a conventional column, namely,
The elements of A, are the same as those displayed beneath Eq. (2-18) except that in this case, the following expressions for p,-, must be used for plates p and q from which sidestreams Wl and W2 are withdrawn
When model 2, shown in Fig. 2-2, is used for the feed plates, the column vector / contains the vapor and liquid rates (vFi and lFi) for each feed. The elements of v, and / may be displayed as follows
The elements vf-iti and vFU, vfi and lFii, v,-Ui and vF2i, vti and lF2i lie in rows / - 1,/ t - 1, i, respectively.
The component-material balances A,vf = —/, may be solved for the vapor rates and dv by use of the recurrence formulas given by Eqs. (2-20) and (2-21) in the same manner as was demonstrated in Example 2-1.
After the component-material balances have been solved for the component-flow rates di9 v2i, v3i, ..., vNi, the corresponding set of flow rates for the liquid may be calculated by use of the equilibrium relationship given by Eq. (2-10). Then the 0 method is applied for the purpose of finding a set of terminal-component flow rates which are in component-material balance and in agreement with the specified values of the total-flow rates of the terminal streams.
The formulation of the 6 method of convergence for a complex column follows that originally proposed by Lyster et al.9 First, a 0 multiplier is defined for each stream withdrawn from the column which may be specified independently. For the column shown in Fig. 3-1, any three of the four streams D, Wu W2, and B may be specified independently. For definiteness, suppose that D, Wl9 W2 are specified. Then B may be found by an overall material balance. The 0 multipliers are defined by the following equations
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