| Complex systems of interlinked distillation columns have been shown to frequently achieve more efficient separations of multicomponent mixtures than the simpler, more conventional, sequential arrangements. It is now generally recognized that in performing simulation or design calculations for complex separation systems a simultaneous rather than iterative sequential approach to solution is preferable. That is, rather than repeatedly solve a sequence of single-column problems until the solution to the entire system is obtained, it is preferable to solve all of the columns simultaneously.; In this study, several alternatives to the Newton-Raphson method for solving the nonlinear set of equations are considered as a means of reducing the total solution time. These methods are referred to collectively as quasi-Newton methods. Also, several new routines are proposed for solving the almost-block-tridiagonal linear subproblem that arises. These methods use a block structure in the manner of the Naphtali-Sandholm approach to single column problems, but employ bordered matrix forms to maintain sparsity during the solution. These new linear equation solvers are compared with several other solution methods, including a modified Thomas algorithm and several general sparse matrix solution routines.; The various solution methods have been compared using nine example problems. The results show that quasi-Newton methods can dramatically reduce the total solution time required, reductions in solution time of 40-70% over the Newton-Raphson method being obtained for one method. The quasi-Newton methods that were found to be the most effective require the use of a linear equation solver that retains the inverse. In terms of storage required, the most efficient linear equation solvers of those that retain the inverse are the new routines that have been proposed. Reductions in storage of up to 50% are obtained using the new methods. In terms of solution time, the most efficient linear equation solvers of those that retain the inverse are again the new routines. |
