Method of Tomich

This method (Tomich, 1970) differs from the foregoing methods mainly in that the summation statements and the energy balances (Equations 13.3 and 13.4) are solved simultaneously. The benefits of the simultaneous solution are twofold. First, distillation columns and absorbers and columns that are hybrids of both types of processes can all be solved with the same method. Second, different types of column performance specifications can be incorporated in the simultaneous solution of the equations. The method is also computationally stable and efficient because it uses Broyden s modification of the Newton-Raphson technique for solving the equations (Broyden, 1965). A brief description of the method follows  

Assume initial temperature and vapor profiles, I j and Vj (the independent variables). 

Compute the liquid profile, Lj, from a total material balance on each stage. 

Solve Equations 13.3 and 13.4 simultaneously for a new set of Tj and Vj. At this point, any performance specifications that the column may be required to meet (Equation 13.9) are expressed as functions of T, and Vj and solved simultaneously with Equations 13.3 and 13.4. 

If the convergence criteria are met, a solution has been reached and the calculations are stopped. Otherwise, the calculations are repeated beginning at step 2, using the updated values of TJ and Vj. 

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In the classical Newton-Raphson technique, the Jacobian matrix is inverted every iteration in order to compute the corrections AT] and Al]. The method of Tomich, however, uses the Broyden procedure (Broyden, 1965) in subsequent iterations for updating the inverted Jacobian matrix. 

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