Question

Consider steady one-dimensional heat conduction in a composite plane wall consisting of two layers \(A\) and \(B\) in perfect contact at the interface. The wall involves no heat generation. The nodal network of the medium consists of nodes 0,1 (at the interface), 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 insulation at the left boundary (node 0 ) and radiation at the right boundary (node 2 ) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {sarr }}\).

Short answer

Answer: The finite difference formulation can be represented by the following equations: 1. \(T_1 - T_0 = 0\) 2. \(\frac{\Delta x_A}{k_A} \frac{T_1 - T_0}{\Delta x} = \frac{\Delta x_B}{k_B} \frac{T_2 - T_1}{\Delta x}\) 3. \(k_B \frac{T_2 - T_1}{\Delta x_B} = \varepsilon \sigma(T_2^4 - T_{sarr}^4)\) These equations can be solved numerically to find the temperature distribution across the composite wall.

Step-by-step solution

Energy balance equation

In a steady-state one-dimensional heat conduction, the energy balance equation for each layer can be written as: Layer A: \(k_{A} \frac{T_1 - T_0}{\Delta x} = k_{A} \frac{T_1 - T_0}{\Delta x_A}\) Layer B: \(k_{B} \frac{T_2 - T_1}{\Delta x} = k_{B} \frac{T_2 - T_1}{\Delta x_B}\) where \(k_A\) and \(k_B\) are the thermal conductivities of layers A and B, respectively, and \(T_0\), \(T_1\), and \(T_2\) are the temperatures at nodes 0, 1, and 2, respectively.

Insulation Boundary Condition

At node 0, there is insulation, which means there is no heat flow through the left boundary. Mathematically, this condition is expressed as \(k_A \frac{dT_A}{dx}=0\). Since \(T_0\) is insulated, we can set the energy balance equation of Layer A at this node as: \(k_A \frac{T_1 - T_0}{\Delta x_A} = 0\)

Radiation Boundary Condition

At node 2, there is radiation, which means there is heat loss by radiation. The heat loss due to radiation is given by the Stefan-Boltzmann law, \(q_R = \varepsilon \sigma (T^4 - T_{surr}^4)\), where \(\varepsilon\) is emissivity and \(\sigma\) is the Stefan-Boltzmann constant. We can write the energy balance of Layer B at node 2 as: \(k_B \frac{T_2 - T_1}{\Delta x_B} = \varepsilon \sigma(T_2^4 - T_{sarr}^4)\)

Finite difference formulation

Now we have three equations with three unknowns, \(T_0\), \(T_1\), and \(T_2\). We can rearrange them in the form of finite difference equations: 1. \(T_1 - T_0 = 0\) 2. \(\frac{\Delta x_A}{k_A} \frac{T_1 - T_0}{\Delta x} = \frac{\Delta x_B}{k_B} \frac{T_2 - T_1}{\Delta x}\) 3. \(k_B \frac{T_2 - T_1}{\Delta x_B} = \varepsilon \sigma(T_2^4 - T_{sarr}^4)\) These finite difference equations can be solved numerically to obtain the temperature distribution across the composite wall.