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Explaining the Finite Difference Method and Heat Transfer

Key Takeaways

  • FDM supports numerical analysis of heat transfer problems through discretization and approximation of the governing equations.

  • A well-defined boundary condition in the domain enables convergence or divergence analysis. 

  • In different heat flow systems, systems designers can use CFD solvers to apply the finite difference method to heat transfer equations.

Heat transfer simulation

CFD modeling of fluid flow and heat transfer in engineering systems is quite a prevalent problem. Systems engineers have relied on the finite difference method (FDM) as a mathematical approach to analyzing heat transfer problems. FDM generally uses a regular structured grid, which makes solving the equation for heat transfer much easier.

When analyzing a problem in a CFD application, the first step is to discretize the partial differential equation, then approximate and calculate heat transfer in order to derive a numerical solution. A CFD solver with the ability to implement the finite difference method is one way to analyze the equation for heat transfer in varying geometries.

Solving With Finite Difference Methods

The finite difference method is one way to solve the governing partial differential equations into numerical solutions in a heat transfer system. This is done through approximation, which replaces the partial derivatives with finite differences. This provides the value at each grid point in the domain.

The geometric domain is discretized on a spatial and temporal basis, followed by numerical approximation of the differential equation at each grid point. For a simple geometry at point (x)i and function ‘u’,  the derivative can be represented as:

 derivative for simple geometry

The idea behind replacing derivatives with differential factors is that by identifying small truncation/ discretization errors, i.e., the error in the difference between the exact and approximated solution, accurate results can be obtained.

From the above equation, if we remove the limit, a finite difference approximation can be achieved. FDM uses Taylor series expansion properties to make this approximation possible. For the function of u when ∆x>0,

Taylor series

Here, H is the higher-order term.

The above equation can be reiterated as:

Taylor series reiterated

Here, the remaining terms are truncation errors. If we eliminate the higher-order terms, the equation can be expressed as:

Equation with higher-order terms eliminated

With the above equations, at point i, the finite difference formula can be written as:  

 finite difference formula

In order to minimize the truncation error, ∆x has to be smaller. However, note that not all approximations with finite difference methods yield a precise numerical solution. For this reason, it is necessary to take stability and convergence into account. 

Applying the Finite Difference Method to the Heat Transfer Equation

In the development of a CFD model, numerical computing allows for solving complex heat transfer problems. With the finite difference method, the discretized governing equation can be presented in the form of a heat equation. By doing this, one can identify the temperature distribution within the system.

The first step is to generate the grid by replacing the object with the set of finite nodes. The second-degree heat equation for 2D steady-state heat generation can be expressed as:

steady-state heat generation equation

Note that T= temperature, k=thermal conductivity, and q=internal energy generation rate.

Now, using Taylor series expansion on the grid, at temperature T(m,n), the above heat equation can be written for grid spacing ∆x and ∆y as:

Taylor series expansion

 

 For uniform grid spacing, ∆x=∆y, 

Heat transfer equation in terms of finite difference method

For formulating the boundary conditions, if surface temperature T = T(m,n+1) = T(m,n) = T(m, n-1), on a node point (m-1,n), using Fourier’s law for heat conduction, we get the following equation,  where q’(m,n) is the heat flux.

 Fourier’s law for heat conduction

The above equation can be used to identify the temperature value at each grid point. 

Solving for Finite Difference Method Heat Transfer Problems

Numerical evaluation of finite difference method heat transfer problems can be a challenge given the different geometries and boundary conditions. However, CFD solvers can help you attain the best simulation by helping you adjust the grid point for optimized meshing. Through properly defined boundary conditions, discretization, approximation, and numerical analysis, CFD solvers can make heat transfer problems easier.

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