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Using the 2D Finite Difference Method for Heat Transfer Analysis

Key Takeaways

  • The finite difference method provides an approximate numerical solution to the governing partial differential equation of a given problem.

  • For the 2D finite difference method, the nodes are often organized in a rectangular grid.

  • CFD tools use the right discretization and approximation schemes to define the temperature field for heat transfer solutions. 

 Heat transfer simulation

Heat transfer simulation

Most engineering problems related to fluid flow, heat transfer, or electromagnetism can be expressed in the form of a partial differential equation (PDE). These PDEs describe the behavior of a phenomenon using a mathematical function. This can be done using numerical techniques such as the finite difference method. This method usually involves:

  1. Building a grid.
  2. Performing differential approximation.
  3. Solving for boundary conditions.

Following these steps, the complex fluid flow and thermal conductivity equations can be solved. This article will focus mainly on the heat transfer aspect of the system. Using CFD, the heat transfer solution can be simplified by the use of the 1D or 2D finite difference method. Let’s take a closer look at how this works.

The 2D Finite Difference Method

The main aim of the finite difference method is to provide an approximate numerical solution to the governing partial differential equation of a given problem. The first step in the process is to define grids with the set of nodes where each PDE is assigned. In a heat transfer problem, each node represents the temperature at each point on the surface. The nodes are often organized in a rectangular grid for the 2D finite difference method, where the points can be defined as:

 (xₘ,yₙ), i.e., (mΔx, nΔy)

At each node, the numerical approximation of the assigned differential equation is done. For a function ‘u’ at point (x)i, this may look like:

Numerical approximation for a given function

The approximation allows for the identification of a discretization error, which is an important factor in deriving reasonably accurate results.

By removing the limit and employing Taylor’s series of expansion, we can derive the following finite difference formula:

 Finite difference formula

Analyzing the Heat Transfer Equation 

The discretization and approximation of the finite difference method can be used to analyze the heat transfer scheme in a given system. In general, 1D steady-state, the second-order derivative of the heat equation, can be written as:

1D steady state heat equation

This heat conduction equation can be finite difference approximated at point m:

Finite difference approximated heat conduction equation

Temperature analysis in a nodal network on 2D domain

However, as mentioned earlier, when using the 2D finite difference method, temperature relations should be analyzed along both the x and y direction within their designated domain. So, for temperature T at point (m,n), the equations can be written as:

 Temperature equations in a 2D domain

These equation can be approximated using the 2D finite difference method for the heat transfer equation as:

 2D heat conduction equation

For a uniformly spaced grid, i.e., Δx=Δy, with no internal heat generation, the above equation can be written as: 

Temperature value using 2D finite difference method

The nodal temperature is simply the average of the temperature of the surrounding nodes.

In the finite difference method, an increased number of nodes in the domain means an increase in solution accuracy. However, system designers should be aware of the increased calculations this may entail. Using CFD tools can aid designers with these complex calculations.

CFD Tools Help Designers With 2D Finite Difference Method Calculations

When solving the heat transfer equation using a 2D finite difference method, the 2D domain must be discretized in equal spacing and the heat equation must be solved at each node to identify the unknown temperature. CFD solvers can help in this process by simplifying meshing and enabling numerical analysis. With the right discretization and approximation schemes, a CFD solver can simplify heat transfer problems.

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