5
Numerical Analysis in Heat Conduction
5.1 Finite-Difference Energy Balance Method for 2-D
Steady-State Heat Conduction
The above-discussed principle of separation of variable is a powerful method
to solve the 2-D heat conduction problem. However, the solutions become
tedious for various nonhomogenous BCs. With the help of modern computers,
the heat conduction problem can be easily solved by using the finite-difference
energy balance method for complex BCs. For example, Figure 5.1 shows the
typical numerical grid distribution for 2-D heat conduction with given surface temperatures as BCs. It requires detailed mathematical procedures if we
choose the separation of variable method to solve this problem. Of course,
the accuracy of the numerical solutions depends on the number of finitedifference grids used for energy balance calculations. In general, the accuracy
improves with the increase of grid points. It should be noted that each grid
point actually represents the temperature of a small area ΔxΔy. So we obtain
the discrete temperature distribution by using the finite-difference method.
However, when Δx and Δy become very small (approaching zero), the temperature distribution predicted by the finite-difference method will be the
same as those calculated using the separation of variable method.
In general, the grid size in the x-direction is not necessarily the same as that
in the y-direction. We need to use smaller grid size (more grid points) in the
high-temperature gradient direction. The energy balance can be performed
for each grid point shown in Figure 5.1. The number of unknown temperatures
is the same as the number of energy balance equations (the number of grid
points). Therefore, the unknown temperatures can be solved. Note that we do
not need to perform energy balance on the boundary points if the boundary
temperatures are already given. But, we need to perform energy balance on
the boundary points if the boundary is exposed to heat flux or convection
in which their boundary temperatures are unknown and to be determined
using the finite-difference method. The following outlines the finite-difference
method to solve the 2-D heat conduction problem shown in Figure 5.2. We
can begin the energy balance at the interior points and then extend to energy
balance at the boundary points with various BCs.
105
Numerical Analysis in Heat Conduction
5.1 Finite-Difference Energy Balance Method for 2-D
Steady-State Heat Conduction
The above-discussed principle of separation of variable is a powerful method
to solve the 2-D heat conduction problem. However, the solutions become
tedious for various nonhomogenous BCs. With the help of modern computers,
the heat conduction problem can be easily solved by using the finite-difference
energy balance method for complex BCs. For example, Figure 5.1 shows the
typical numerical grid distribution for 2-D heat conduction with given surface temperatures as BCs. It requires detailed mathematical procedures if we
choose the separation of variable method to solve this problem. Of course,
the accuracy of the numerical solutions depends on the number of finitedifference grids used for energy balance calculations. In general, the accuracy
improves with the increase of grid points. It should be noted that each grid
point actually represents the temperature of a small area ΔxΔy. So we obtain
the discrete temperature distribution by using the finite-difference method.
However, when Δx and Δy become very small (approaching zero), the temperature distribution predicted by the finite-difference method will be the
same as those calculated using the separation of variable method.
In general, the grid size in the x-direction is not necessarily the same as that
in the y-direction. We need to use smaller grid size (more grid points) in the
high-temperature gradient direction. The energy balance can be performed
for each grid point shown in Figure 5.1. The number of unknown temperatures
is the same as the number of energy balance equations (the number of grid
points). Therefore, the unknown temperatures can be solved. Note that we do
not need to perform energy balance on the boundary points if the boundary
temperatures are already given. But, we need to perform energy balance on
the boundary points if the boundary is exposed to heat flux or convection
in which their boundary temperatures are unknown and to be determined
using the finite-difference method. The following outlines the finite-difference
method to solve the 2-D heat conduction problem shown in Figure 5.2. We
can begin the energy balance at the interior points and then extend to energy
balance at the boundary points with various BCs.
105