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 , and 4 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 uniform heat flux \(\dot{q}_{0}\) 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 \(T_{\infty}\). Do not simplify.

Short answer

Answer: For the boundary nodes 0 and 4, the explicit finite difference formulations are as follows: Node 0: \(T^{n+1}_0 = T^n_0 + \frac{\alpha \Delta t}{\Delta x^2}(T^n_1 - 2 T^n_0) + \frac{\Delta t q}{\rho c_p} + \frac{\dot{q}_{0} \Delta x}{k}\) Node 4: \(T^{n+1}_4 = T^n_4 + \frac{\alpha \Delta t}{\Delta x^2}(T^n_3 - 2 T^n_4) + \frac{\Delta t q}{\rho c_p} + \frac{\Delta x h (T_\infty - T^n_4)}{k}\)

Step-by-step solution

Write down the energy balance equation

For transient heat conduction with variable heat generation (given as \(q\)) and constant thermal conductivity (denoted by \(k\)), we can write the energy balance equation as: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} + \frac{q}{\rho c_p}\) where \(\alpha\) is the thermal diffusivity, T is the temperature, \(t\) is time, \(\rho\) is the density, and \(c_p\) is the heat capacity.

Apply finite difference approximations to the derivatives

To obtain the explicit finite difference formulation, we need to replace the derivatives in the energy balance equation by the finite difference approximations. For the time derivative, we use forward difference: \(\frac{T^{n+1}-T^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{\Delta x^2} + \frac{q}{\rho c_p}\) For the spatial second-order derivative, we use central difference: \(\frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{\Delta x^2} = \alpha \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{\Delta x^2} + \frac{q}{\rho c_p}\)

Obtain the explicit finite difference formulation for the boundary nodes

For the left boundary (node 0) with uniform heat flux \(\dot{q}_{0}\), we have: \(-\dot{q}_{0} = k\frac{T_1^0 - T_0^0}{\Delta x}\) Now, isolate \(T^0_0\) and use it in the finite difference equation at node 0: \(T^{n+1}_0 = T^n_0 + \frac{\alpha \Delta t}{\Delta x^2}(T^n_1 - 2 T^n_0) + \frac{\Delta t q}{\rho c_p} + \frac{\dot{q}_{0} \Delta x}{k}\) For the right boundary (node 4) with convection, we have: \(-k\frac{T_4^n - T_3^n}{\Delta x} + h(T_4^n - T_\infty) = 0\) Now, using this in the finite difference equation at node 4: \(T^{n+1}_4 = T^n_4 + \frac{\alpha \Delta t}{\Delta x^2}(T^n_3 - 2 T^n_4) + \frac{\Delta t q}{\rho c_p} + \frac{\Delta x h (T_\infty - T^n_4)}{k}\) These are the explicit finite difference formulations for the boundary nodes 0 and 4 in the given problem.