This paper presents a parallel solution method of large sparse systems of linear equations arising in the context of a chemical process flowsheeting application on a message passing multicomputer. To maximize the performance, the algorithm uses a novel matrix decomposition and solution method, called parallel two-phased LU decomposition, which schedules the concurrent tasks in a maximally overlapping manner, and at the same time, tries to minimize the interprocessor data dependencies and obtain optimal load balancing. The forward elimination step is performed concurrently with the parallel two-phased LU decomposition step and backward substitution is parallelized in a piecewise manner. Implementation results on an Intel iPSC/2 multicomputer are presented. With 16 processors, speedups of up to 11.3 are observed for relatively large problem sizes and close to 71% processor utilization is achieved despite the high fraction of sequential operations in the process. Although this work was motivated by the need to reduce the computational bottleneck of the DZINE flowsheeting program, it should also be applicable to a wide range of scientific and engineering problems where fast and efficient solution of large sparse systems of linear equations is required.