Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes \(0,1,2,3\), and 4 with a uniform nodal spacing of \(\Delta x\). Using the finite difference form of the first derivative (not the energy balance approach), obtain the finite difference formulation of the boundary nodes for the case of uniform heat flux \(q_{0}\) at the left boundary (node 0 ) and convection at the right boundary (node 4) with a convection coefficient of \(h\) and an ambient temperature of \(T_{\infty}\).

Consider the steady one-dimensional heat conduction with variable heat generation, the general equation can be given as $$-\kappa \frac{d^{2} T}{d x^{2}}=q_{g}(x)$$ where \(\kappa\) is the constant thermal conductivity and \(q_{g}(x)\) is the heat generation term.

Using the central difference approximation for the second derivative, the finite difference form can be written as $$ \frac{d^2 T}{d x^2} \approx \frac{T_{i+1} - 2T_{i} + T_{i-1}}{(\Delta x)^2} $$ Now, substituting the finite difference form back to the general equation, we get $$ -\kappa \frac{T_{i+1} - 2T_{i} + T_{i-1}}{(\Delta x)^2} = q_{g}(x) $$

The given boundary conditions are: 1. Uniform heat flux \(q_{0}\) at the left boundary (node 0) 2. Convection at the right boundary (node 4) with a convection coefficient \(h\) and ambient temperature \(T_{\infty}\) To incorporate Boundary condition 1, we apply the heat flux relation: $$ q_{0} = -\kappa \frac{dT}{dx} \approx -\kappa \frac{T_{1}-T_{0}}{\Delta x} $$ We want to express \(T_{0}\) in terms of the other variables, so we rearrange the equation as: $$ T_{0} = T_{1} - \frac{q_{0} \Delta x}{\kappa} $$ Now, for boundary condition 2, we apply the convection condition at node 4: $$ -\kappa \frac{dT}{dx} = h (T_{4}-T_{\infty}) $$ Approximate the first derivative at node 4 as $$ -\kappa \frac{dT}{dx} \approx -\kappa \frac{T_{4}-T_{3}}{\Delta x} $$ Therefore, $$ -\kappa \frac{T_{4}-T_{3}}{\Delta x} = h (T_{4}-T_{\infty}) $$ Rearrange the equation to express \(T_{4}\) in terms of other variables, we get $$ T_{4} = \frac{h \Delta x T_\infty + \kappa T_{3}}{h \Delta x + \kappa} $$ The finite difference formulations for the boundary nodes are: $$ T_{0} = T_{1} - \frac{q_{0} \Delta x}{\kappa} $$ $$ T_{4} = \frac{h \Delta x T_\infty + \kappa T_{3}}{h \Delta x + \kappa} $$