Question

Consider transient heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes 0,1 , \(2,3,4\), and 5 with a uniform nodal spacing of $\Delta x$. The wall is initially at a specified temperature. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary nodes for the case of insulation at the left boundary (node 0 ) and radiation at the right boundary (node 5) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {surr }}\)

Short answer

The explicit finite difference formulation for transient heat conduction in this scenario involves discretizing the energy balance equation using the finite difference method, with the following equations for each node: \begin{equation} \frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2 T_i^n + T_{i-1}^n}{(\Delta x)^2} + Q_i^n \end{equation} For the boundary nodes, we have: Node 0 (Left boundary, insulation): \begin{equation} T_1^n = T_{-1}^n \end{equation} Node 5 (Right boundary, radiation): \begin{equation} k \frac{T_5^n - T_4^n}{\Delta x} = \varepsilon\sigma\left({T_5^n}^4 - {T_{\text{surr}}}^4\right) \end{equation}

Step-by-step solution

Set up the energy balance equation

The energy balance equation for heat conduction in a plane wall with variable heat generation and constant thermal conductivity can be written as: \begin{equation} \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} + Q \end{equation} where \(T\) is the temperature, \(t\) is the time, \(\alpha\) is the thermal diffusivity, and \(Q\) is the heat generation term.

Discretize the equation using finite difference

We can discretize the energy balance equation using the finite difference method with uniform nodal spacing of \(\Delta x\). The explicit finite difference equation for node \(i\) is given by: \begin{equation} \frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2 T_i^n + T_{i-1}^n}{(\Delta x)^2} + Q_i^n \end{equation} where \(T_i^n\) represents the temperature at node \(i\) at time step \(n\), and \(\Delta t\) is the time step.

Apply the boundary conditions

We have insulation at the left boundary (node 0) and radiation at the right boundary (node 5) with an emissivity \(\varepsilon\) and surrounding temperature \(T_{\text{surr}}\). For the left boundary (node 0), insulation implies no heat flux, which translates to: \begin{equation} \frac{\partial T_0}{\partial x} = 0 \end{equation} For the right boundary (node 5), radiation is considered, and the boundary condition is given by: \begin{equation} -k \frac{\partial T_5}{\partial x} = \varepsilon\sigma(T_5^4 - T_{\text{surr}}^4) \end{equation} where \(k\) is the thermal conductivity and \(\sigma\) is the Stefan-Boltzmann constant.

Obtain finite difference equations for boundary nodes

Using the finite difference discretization from Step 2, we can obtain the equations for the boundary nodes as: Node 0 (Left boundary, insulation): \begin{equation} T_1^n = T_{-1}^n \end{equation} Node 5 (Right boundary, radiation): \begin{equation} k \frac{T_5^n - T_4^n}{\Delta x} = \varepsilon\sigma\left({T_5^n}^4 - {T_{\text{surr}}}^4\right) \end{equation} With the above equations, we have obtained the explicit finite difference formulation for the boundary nodes in the given transient heat conduction problem.