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. The temperature at the right boundary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary node 0 for the case of combined convection, radiation, and heat flux at the left boundary with an emissivity of \(\varepsilon\), convection coefficient of \(h\), ambient temperature of \(T_{\infty}\), surrounding temperature of \(T_{\text {surr }}\), and uniform heat flux of \(\dot{q}_{0}\) toward the wall. Also, obtain the finite difference formulation for the total amount of heat transfer at the right boundary for the first 20 time steps.

Short answer

2. What are the boundary conditions for node 0 (left boundary) and node 4 (right boundary)? 3. What is the explicit finite difference formulation for calculating temperatures at node 0? 4. How can the finite difference formulation for the total amount of heat transfer at the right boundary for the first 20 time steps be determined?

Step-by-step solution

Governing equation for transient heat conduction

The governing equation for transient heat conduction with variable heat generation and constant thermal conductivity can be represented as: $$\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} + q(x, t)$$ where \(T\) is the temperature, \(\alpha\) is the thermal diffusivity, and \(q(x, t)\) represents the heat generation rate per unit volume in the medium.

Define boundary conditions

The boundary conditions for our problem are as follows: At the left boundary (node 0): 1. Convection heat transfer: \(h(T_0 - T_{\infty})\) 2. Radiation heat transfer: \(\varepsilon\sigma(T_0^4 - T_\text{surr}^4)\) 3. Heat flux toward the wall: \(\dot{q}_{0}\) At the right boundary (node 4): Temperature is specified.

Develop explicit finite difference formulation for node 0

Developing an explicit finite difference formulation for node 0 using the forward difference for time and central difference for the spatial derivative, we get: $$\frac{T_0^{n+1} - T_0^n}{\Delta t} = \alpha \frac{T_1^n - 2T_0^n + T_{-1}^n}{\Delta x^2} + q(x_0, t_n)$$ Now, we can substitute the convection, radiation, and heat flux boundary conditions for node 0 in the above equation: $$\frac{T_0^{n+1} - T_0^n}{\Delta t} = \alpha \frac{T_1^n - 2T_0^n + (T_0^n - \frac{\Delta x}{k}(h(T_0^n - T_{\infty}) + \varepsilon\sigma(T_0^n)^4 - \varepsilon\sigma T_\text{surr}^4 + \dot{q}_{0}))}{\Delta x^2} + q(x_0, t_n)$$ This explicit finite difference formulation can be used to calculate temperatures at node 0.

Determine the finite difference formulation for the total amount of heat transfer at the right boundary for the first 20 time steps

For the first 20 time steps, the total amount of heat transfer at node 4 can be calculated as follows: 1. Write down the explicit finite difference equation for node 4, similar to the one in step 3. 2. Calculate the temperature at each time step using this equation. 3. Calculate the heat transfer at the right boundary by taking the difference between node 4 and node 3. The result will give us the finite difference formulation for the total amount of heat transfer at the right boundary for the first 20 time steps.