Starting with an energy balance on a disk volume element, derive the one- dimensional transient implicit finite difference equation for a general interior node for \(T(z, t)\) in a cylinder whose side surface is subjected to convection with a convection coefficient of \(h\) and an ambient temperature of \(T_{\infty}\) for the case of constant thermal conductivity with uniform heat generation.

We start by considering a cylindrical coordinate system for our problem. Let \(z\) represent the spatial coordinate in the axial direction and \(t\) denote time. A disk volume element has a thickness of \(\Delta z\) in the \(z\) direction. The energy balance equation for the disk volume element can be given as: $$\frac{\partial}{\partial z}\left( k\frac{\partial T}{\partial z} \right) - q_g + hP(T_{\infty} - T) = \rho c_p \frac{\partial T}{\partial t}$$ where: - \(k\) is the constant thermal conductivity, - \(T\) is the temperature, - \(z\) is the axial coordinate, - \(q_g\) is the uniform heat generation, - \(h\) is the convection coefficient, - \(T_{\infty}\) is the ambient temperature, - \(P\) is the perimeter of the disk, - \(\rho\) is the density of the material, - \(c_p\) is the specific heat capacity, and - \(t\) is time.

The equation derived in Step 1 represents the transient heat conduction governing equation with heat generation and convection heat transfer on the side surface of the cylinder. It can be further simplified to: $$\frac{\partial^2 T}{\partial z^2} = \frac{1}{\alpha}\frac{\partial T}{\partial t} + \frac{q_g - hP(T_{\infty} - T)}{k}$$ where \(\alpha = \frac{k}{\rho c_p}\) is the thermal diffusivity.

Now we need to discretize the governing equation using the finite difference method. We can replace the second derivative in space with a central difference approximation: $$\frac{\partial^2 T}{\partial z^2} \approx \frac{T_{i+1}-2T_i+T_{i-1}}{(\Delta z)^2}$$ and the time derivative with a forward difference approximation: $$\frac{\partial T}{\partial t} \approx \frac{T_i^{n+1}-T_i^n}{\Delta t}$$ where \(i\) is the index for spatial nodes and \(n\) is the index for time steps. Substitute the approximated derivatives back into the governing equation and rearrange: $$T_{i+1}^{n+1} - 2T_i^{n+1} + T_{i-1}^{n+1} = \frac{\alpha \Delta t}{(\Delta z)^2}(T_i^{n+1} - T_i^n) + \frac{\Delta t}{k}(q_g - hP(T_{\infty} - T_i^{n+1}))$$

Since the side surface of the cylinder is subjected to convection, we need to apply a convection boundary condition at the nodes adjacent to the surface. This can be done by using the Robin boundary condition: $$k\frac{\partial T}{\partial z} + h(T-T_{\infty}) = 0$$ The discretized equation derived in Step 3, along with the convection boundary condition, forms the one-dimensional transient implicit finite difference equation for a general interior node for \(T(z, t)\) in the cylinder.