Numerical Methods in Heat Transfer

Advantages

• Numerical methods can handle non-linear differential equations as boundary conditions which analytical methods cannot. According to “Principles of Heat Transfer” by Frank Kreith, “the numerical approach … is recommended because it can easily be adapted to all kinds of boundary conditions and geometric shapes.” Numerical methods can calculate the flow of heat when more than one form of heat transfer is occurring. Numerical methods also permit an approximation of heat transfer in fluids that other methods cannot estimate.

Methods

• Numerical methods require a discrete set of initial boundary conditions to determine the heat transfer of the system. Numerical methods include finite element analysis, finite difference method, the impedance boundary element and integral equation method. The finite difference method divides the heat transfer model into an area with equal differences between them. Finite Element Analysis (FEA) divides a structure into small sections called control volumes. The heat transfer values are calculated for that cell using the inputs at the boundaries of each square using numerical methods. Both triangles and grids are used to divide a space into finite elements or finite differences.

Problems

• Numerical methods provide an approximation of the actual solution. Numerical methods provide an analysis of the model given the current set of conditions. Numerical methods do not capture the future state if the system variables are changing in a non-linear manner. Numerical methods are subject to numerical instability and numerical consistency. Numerical instability is created when the equations do not match conditions because a key parameter is eliminated by discretization. Numerical consistency measures the effect of how truncation of equation results affect the answer. If a variable is equal to one seventh and truncated to 0.14, a consistent numerical method will have the same or a similar result than if 0.143 was used for the variable’s value.

Solutions

• Normalizing algebraic equations converts equations to ratios of other equations or cancels out as many variables as possible. Using smaller control volumes decreases the error associated with using numerical methods. However, it also increases the number of equations to be solved simultaneously. The problem of calculating large numbers of equations is reduced by using computers to perform the calculations. Varying the normalization methods for boundary conditions followed by recalculating the equations determines the consistency. According to “Computational Heat Transfer” by Yogesh Jaluria and Kenneth Torrance, “Available analytical and experimental results are of considerable importance in checking the accuracy and validity of numerical results.”