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Mathematicians, scientists, and engineers often look for simple and robust computer programs in order to solve the Fourier differential equation for heat transfer in a numerical way. Specifically, the 2D differential equation may take various initial and boundary conditions applicable in many practical applications. Simplifications to 1D differential equations may also be applied to the numerical solution of various practical problems in heat and mass transfer. The implicit scheme for the finite difference solution of the heat transfer equation, as formulated by Patankar, is mostly deployed throughout the book. It is worth noting that the method is proven to be stable and fast for most time scales. In fact, fast convergence is attained for steady state problems that have searched for solutions within a time span of more than the universe age. Furthermore, mathematical problems that finalized phenomena within a period of a few milliseconds have proven the remarkable stability of the numerical solution. A practitioner has thirty selected cases in order to grasp the salient and advanced features of the implicit scheme. A program core developed in C++ allows the introduction of the relevant initial and boundary conditions and solves various problems with only minor changes. About two-thirds of the selected cases were based on mathematical problems for which analytical solutions exist. In this way, analytical and numerical solution programs are given in order for the practitioner to compile, run the code, and plot the desired results. Orthogonal and cylindrical coordinates are deployed according to border geometry; Python programs were developed for the analytical solutions in cylindrical symmetry. Induction heating of billets and slabs, rebar production by quench and tempering, and steel billet solidification are some of the most practical examined cases. |
