Question

Explain how the finite difference form of a heat conduction problem is obtained by the energy balance method.

Short answer

Answer: The finite difference form of the heat conduction problem derived using the energy balance method is: \begin{equation} \frac{\partial T_i}{\partial t} = \alpha \left( \frac{T_{i+1} - 2T_i + T_{i-1}}{\Delta x^2} \right) \end{equation} where \(T_i\) is the temperature at position \(x_i\) and time \(t\), \(\alpha\) is the thermal diffusivity, and \(\Delta x\) is the length of a small control volume.

Step-by-step solution

Understand the Heat Conduction Equation

The heat conduction equation, also known as the heat equation or diffusion equation, describes the heat transfer through a material. In one dimension, it is given by: \begin{equation} \frac{\partial T(x,t)}{\partial t} = \alpha \frac{\partial^2 T(x,t)}{\partial x^2} \end{equation} where \(T(x,t)\) is the temperature at position \(x\) and time \(t\), and \(\alpha\) is the thermal diffusivity of the material, which is a constant.

Apply the Energy Balance Method

The energy balance method states that the rate of change of thermal energy in a control volume is equal to the rate at which heat enters the control volume minus the rate at which heat leaves the control volume. In the context of the heat conduction problem, we consider a small control volume with a length \(\Delta x\) in our one-dimensional problem. Let the heat entering and leaving the control volume be \(Q_{in}\) and \(Q_{out}\), respectively. Then, the energy balance equation can be written as: \begin{equation} \frac{\partial T}{\partial t} \Delta x = - (Q_{out} - Q_{in}) \end{equation}

Calculate Heat Fluxes

Next, we need to determine the heat flux entering and leaving the control volume. For this, we use Fourier's law of heat conduction, which states: \begin{equation} q = -k \frac{\partial T}{\partial x} \end{equation} where \(q\) is the heat flux, and \(k\) is the thermal conductivity of the material. For the control volume, the heat fluxes can be written as: \begin{align} Q_{in} = -k \frac{\partial T}{\partial x} \Big|_{x = x_i - \frac{\Delta x}{2}} \\ Q_{out} = -k \frac{\partial T}{\partial x} \Big|_{x = x_i + \frac{\Delta x}{2}} \end{align} Here, \(x_i\) denotes the position of the center of the control volume.

Discretize the Equation using Finite Difference Approximation

To obtain the finite difference form of the heat conduction problem, we need to discretize the spatial derivatives using finite difference approximations. For the heat fluxes, we use the central difference approximation, which gives: \begin{align} \frac{\partial T}{\partial x} \Big|_{x = x_i - \frac{\Delta x}{2}} \approx \frac{T_{i-1} - T_i}{\Delta x} \\ \frac{\partial T}{\partial x} \Big|_{x = x_i + \frac{\Delta x}{2}} \approx \frac{T_i - T_{i+1}}{\Delta x} \end{align} Substituting these approximations into the expressions for \(Q_{in}\) and \(Q_{out}\), we get: \begin{align} Q_{in} = k \frac{T_{i-1} - T_i}{\Delta x} \\ Q_{out} = k \frac{T_i - T_{i+1}}{\Delta x} \end{align} Now, substituting the heat fluxes back into the energy balance equation, we have: \begin{equation} \frac{\partial T_i}{\partial t} \Delta x = - k (T_i - T_{i+1} - T_{i-1} + T_i) \end{equation} Finally, divide both sides of the equation by \(\Delta x\), and substitute the thermal diffusivity \(\alpha = k/\rho c_p\), where \(\rho\) is the material density and \(c_p\) is the specific heat capacity, to obtain the finite difference form: \begin{equation} \frac{\partial T_i}{\partial t} = \alpha \left( \frac{T_{i+1} - 2T_i + T_{i-1}}{\Delta x^2} \right) \end{equation} This is the finite difference form of the heat conduction problem derived using the energy balance method.