# The Application of Numerical Methods in Heat Transfer Problems

### Key Takeaways

• Numerical methods solve heat transfer problems by step-wise, iterative solution methods.

• Numerical methods can help in achieving sufficiently accurate approximate solutions for complex heat transfer equations in a faster, more accurate way.

• The most widely-used numerical methods in heat transfer problems are the finite element method, finite volume method, finite difference method, and boundary element method. Numerical methods in heat transfer problems help to develop efficient thermal management techniques

Heat transfer is an area in engineering that is associated with energy utilization and conservation. The heat transfer problems in engineering are governed by mathematical equations. It is very difficult to obtain closed-form results for the heat transfer governing equations through analytical  calculations. Cheap, fast, and efficient computing devices that use numerical methods to solve heat transfer problems were developed to solve this issue.

Numerical methods solve heat transfer problems by step-wise, iterative solution methods. The numerical properties, merits, demerits, and mathematical formulations of each numerical method differ. However, the common objective of all numerical methods in heat transfer problems is to obtain the approximate solution in the shortest amount of time. Various numerical methods are used to solve heat transfer problems such as the finite difference method, finite volume method, finite element method, and the boundary element method. We will discuss the application of numerical methods in heat transfer in this article.

## Complex Heat Transfer Problems and Solutions

Heat transfer is governed by mathematical equations, which can be applied to solve heat flow problems in various applications. Depending on the heat transfer problem, the mathematical expression for heat transfer differs with different initial and boundary conditions. Generally, the heat transfer equation is in partial differential form. It can be used for determining the change in time over which the heat spreads throughout a space.

To solve simple heat transfer problems associated with simple geometries and boundary conditions, analytical methods can be employed. Analytical methods give the solution function corresponding to each point in the medium. Analytical mathematical methods such as the D-Alembert method or the separation of the variable method are used to find solutions to heat transfer equations. As the complexity of a heat transfer problem increases, the number of variables in the mathematical equation also increases. It is not practical to use analytical methods every time to obtain a heat flow solution.

Real heat transfer situations are associated with complex geometries and complex boundary conditions or variable properties. When using analytical methods to solve complicated heat transfer problems, the solutions might be unobtainable or non-existent. It is necessary to converge heat transfer problem solving in such cases to get an approximate solution. When using the analytical method, iteration and convergence become tedious and difficult. Numerical methods can help in achieving sufficiently accurate approximate solutions for complex heat transfer equations in a faster and more accurate way.

## Numerical Methods in Heat Transfer Problem Solving

Numerical methods are used to solve complex heat transfer problems involving mechanisms such as conduction, convection, radiation, or a combination of them. There are various numerical methods available such as the finite element method, finite volume method, finite difference method, and the boundary element method. All these methods vary in numerical properties, mathematical formulations, merits, and shortcomings. The common objective of any numerical method applied in heat transfer problem solving is to achieve the approximate solution of the governing equations to form a closed-form result in minimum time, which is not obtainable with analytical methods.

The steps involved in solving heat transfer problems using numerical methods are:

1. The physical problem of heat transfer is defined.

2. The differential equations are formulated with appropriate initial and boundary conditions.

3. The differential equations are discretized into a set of algebraic equations.

4. Numerical properties such as convergence, consistency, stability, and error bounds of the discretization scheme are analyzed.

5. If the discretization scheme turns out to be satisfactory, the matrix solver or iteration method is used to solve algebraic equations. If not, another suitable discretization scheme is selected.

6. The result obtained is interpreted.

## The Importance of Numerical Method Selection

In a physically complicated heat transfer problem, if you are applying an incorrect numerical method, it takes valuable time to confirm the correctness of the procedure. The incorrect numerical method will fail to explore the complexities of the physical heat transfer phenomenon. Moreover, the accuracy of the results obtained will be doubtful. The accuracy of the numerical solution depends on the quality of discretization. Therefore, it is important to choose the right numerical method to solve complex heat transfer problems.

In the electronics industry, the application of numerical methods in heat transfer problems helps engineers develop efficient thermal management techniques for semiconductor devices and integrated circuits, preventing the damage of components due to thermal hotspots. Cadence software offers you analysis tools that can analyze the heat transfer in electronic systems.