A new segregated pressure-based algorithm is developed to solve two-dimensional incompressible fluid flows with convective heat transfer and liquid to solid phase change by the finite volume method. The predictive-correction scheme is based on the SIMPLER algorithm with n added iterative cycles to increase the accuracy of pressure in satisfying the discretized continuity equation. The performance of SIMPLERnP is compared with SIMPLER and IDEAL in the solution of the laminar fluid flow in a lid driven cavity, natural heat convection inside a square cavity, convective heat transfer in the annular space between two concentric cylinders and conjugate natural heat convective solidification of a binary alloy that included the transient heat conduction in the thick mold walls. The results indicate that the proposed algorithm is highly stable, even when a value of 0.99 is used for the under-relaxation of the velocity components, and allows significant improvements in the reduction of the number of iterations and time of computation.