Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and variable thermal conductivity. The nodal network of the medium consists of nodes 0,1 , and 2 with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the finite difference formulation of this problem for the case of specified heat flux \(\dot{q}_{0}\) to the wall and convection at the left boundary (node 0 ) with a convection coefficient of \(h\) and ambient temperature of \(T_{\infty}\), and radiation at the right boundary (node 2 ) with an emissivity of \(\varepsilon\) and surrounding surface temperature of \(T_{\text {surr }}\).

The heat conduction equation for a one-dimensional plane wall with variable thermal conductivity \(k(x)\) and heat generation \(q_{g}(x)\) can be written as: $$\frac{d}{dx}\left(k(x)\frac{dT}{dx}\right) + q_{g}(x) = 0$$ We also have to take into account the given heat flux at the left boundary (node 0) and the convection at the right boundary (node 2).

To discretize the heat conduction equation, we can use the finite difference method. We will replace the derivatives of temperature with respect to x with their finite difference approximations: $$\frac{k_{1+1/2}\left(T_1-T_0\right) - k_{1-1/2}\left(T_0-T_{-1}\right)}{\Delta x^2} + q_{g,1} = 0$$ $$\frac{k_{2+1/2}\left(T_2-T_1\right) - k_{2-1/2}\left(T_1-T_0\right)}{\Delta x^2} + q_{g,2} = 0$$ Here, the subscript refers to the node number, and \(k_{1\pm1/2}\) are thermal conductivities at the midpoints between nodes.

We are given the following boundary conditions: - At the left boundary (node 0), the heat flux \(\dot{q_{0}}\) to the wall and convection is specified with a convection coefficient \(h\) and ambient temperature \(T_\infty\). Therefore, we have: $$k_{1-1/2}\frac{T_0-T_{-1}}{\Delta x} + q_0 = h\left(T_\infty - T_0\right)$$ - At the right boundary (node 2), radiation with an emissivity \(\varepsilon\), surrounding temperature \(T_\text{surr}\), and surface temperature \(T_2\) is specified. For the radiant heat flux, we can use the Stefan-Boltzmann law: $$q_\text{rad}=\varepsilon \sigma \left(T_2^4 - T_\text{surr}^4\right)$$ Here \(\sigma\) is the Stefan-Boltzmann constant. The radiant heat flux can be related to the conductive heat flux as follows: $$k_{2+1/2}\frac{T_2-T_1}{\Delta x} = \varepsilon \sigma \left(T_2^4 - T_\text{surr}^4\right)$$

Now we will rewrite the finite difference equations for each node, incorporating the boundary conditions derived above: - Node 0: $$k_{1-1/2}\frac{T_0-T_{-1}}{\Delta x} + q_0 = h\left(T_\infty - T_0\right)$$ - Node 1: $$\frac{k_{1+1/2}\left(T_1-T_0\right) - k_{1-1/2}\left(T_0-T_{-1}\right)}{\Delta x^2} + q_{g,1} = 0$$ - Node 2: $$\frac{k_{2+1/2}\left(T_2-T_1\right) - k_{2-1/2}\left(T_1-T_0\right)}{\Delta x^2} + q_{g,2} = \varepsilon \sigma \left(T_2^4 - T_\text{surr}^4\right)$$ These are the finite difference formulations for the given problem for nodes 0, 1, and 2 considering steady one-dimensional heat conduction in a plane wall with variable heat generation and variable thermal conductivity along with the specified boundary conditions.