An Introduction to Heat Transfer

Morteza Heydari

Numerical Methods In Heat Conduction

WHY NUMERICAL METHODS?• The ready availability of high-speed computers and easy-to-

use powerful software packages.• Engineers in the past had to rely on analytical skills to solve

significant engineering problems, and thus they had to undergo a rigorous training in mathematics.

• Today’s engineers, on the other hand, have access to a tremendous amount of computation power under their fingertips, and they mostly need to understand the physical nature of the problem and interpret the results.

• But they also need to understand how calculations are performed by the computers to develop an awareness of the processes involved and the limitations, while avoiding any possible pitfalls.

WHY NUMERICAL METHODS?•Limitations of exact solution•Better Modeling of numerical method•Flexibility of numerical method•Complications of exact solution

• The numerical methods for solving differential equations are based on replacing the differential equations by algebraic equations. In the case of the popular finite difference method, this is done by replacing the derivatives by differences.

Finite Difference Formulation of Differential Equation

Finite Difference Formulation of Differential Equation (cont)

Finite Difference Formulation of Differential Equation (example 1-D)

Finite Difference Formulation of Differential Equation (example 2-D)

One-Dimensional Steady Heat Conduction

One-Dimensional Steady Heat Conduction

Boundary Condition

• Specified temperature

• Specified Heat Flux Boundary Condition

• Convection Boundary Condition

• Radiation Boundary Condition

• Combined Convection and Radiation Boundary Condition

Boundary Condition (cont)

• Interface Boundary Condition (perfect contact)

Boundary Condition (cont)

Transient Heat Conduction

Transient Heat Conduction (cont)

Transient Heat Conduction example

explicit

Transient Heat Conduction example (cont)

implicit

Stability Criterion for Explicit Method: Limitation on Δt

•The explicit method is easy to use, but it suffers from an undesirable feature that severely restricts its utility: the explicit method is not unconditionally stable, and the largest permissible value of the time step t is limited by the stability criterion. If the time step t is not sufficiently small, the solutions obtained by the explicit method may oscillate wildly and diverge from the actual solution.