## Finite Difference Methods in Heat TransferFinite Difference Methods in Heat Transfer presents a clear, step-by-step delineation of finite difference methods for solving engineering problems governed by ordinary and partial differential equations, with emphasis on heat transfer applications. The finite difference techniques presented apply to the numerical solution of problems governed by similar differential equations encountered in many other fields. Fundamental concepts are introduced in an easy-to-follow manner. Representative examples illustrate the application of a variety of powerful and widely used finite difference techniques. The physical situations considered include the steady state and transient heat conduction, phase-change involving melting and solidification, steady and transient forced convection inside ducts, free convection over a flat plate, hyperbolic heat conduction, nonlinear diffusion, numerical grid generation techniques, and hybrid numerical-analytic solutions. |

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### Contents

Basic Relations | 1 |

Discrete Approximation of Derivatives | 19 |

Methods of Solving Sets of Algebraic Equations | 43 |

OneDimensional SteadyState Systems | 75 |

OneDimensional Parabolic Systems | 99 |

ii Onedimensional transient convection and diffusion | 168 |

Elliptic Systems | 189 |

Hyperbolic Systems | 227 |

Phase Change Problems | 275 |

Hybrid NumericalAnalytic Solutions | 359 |

Convection Inside a Parallel Plate Duct with Step Change | 369 |

377 | |

Appendices | 395 |

Program to Solve Example 101 | 401 |

409 | |

Nonlinear Diffusion | 249 |

### Common terms and phrases

algebraic equations algorithm boundary conditions boundary node boundary surface calculations central difference central difference formula central differencing computational domain Consider the following control volume convection convection boundary condition convection term convergence coordinates Crank-Nicolson method derivative determined difference approximation difference equations dimensionless discretize elliptic energy equation enthalpy exact solution example explicit method Figure finite finite-difference approximation finite-difference equations finite-difference form finite-difference representation finite-difference scheme first-order flow fluid Gauss-Seidel given by Eq grid points heat conduction equation heat conduction problem heat transfer coefficient implicit initial condition integral transform interface internal nodes iteration level n+1 mesh nonlinear obtained from Eq one-dimensional Ozisik parabolic parameter partial differential equations physical domain Poisson's equation prescribed heat flux second-order accurate simple explicit scheme solved stability criterion steady-state heat conduction step stream function Taylor series thermal conductivity Thomas algorithm transient heat conduction truncation error two-dimensional unknown node temperatures upwind values variable vector velocity components vorticity