Question

Consider steady one-dimensional heat conduction in a pin fin of constant diameter \(D\) with constant thermal conductivity. The fin is losing heat by convection to the ambient air at \(T_{\infty}\) with a convection coefficient of \(h\), and by radiation to the surrounding surfaces at an average temperature of \(T_{\text {surr- }}\). The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the finite difference formulation of this problem to determine \(T_{1}\) and \(T_{2}\) for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in \({ }^{\circ} \mathrm{C}\).

Short answer

Answer: The energy balance equations for nodes 1 and 2 are: For node 1: \(\frac{kAD(T_2 - T_0)}{2\Delta x} + hP(T_1 - T_\infty) + \epsilon\sigma P(T_1^4 - T_{surr}^4) = 0\) For node 2: \(\frac{kAD(T_2 - T_1)}{2\Delta x} + hP(T_2 - T_\infty) + \epsilon\sigma P(T_2^4 - T_{surr}^4) = 0\)

Step-by-step solution

Setup energy balance formulation

Since we are using the energy balance approach, we need to set up the equations for the balance of heat conduction, convection, and radiation at each node. For steady-state conduction, the heat conducted into a node is equal to heat conducted out plus heat lost by convection and radiation. Starting with the general energy balance equation for a one-dimensional pin fin, we have: \(Q_{cond} + Q_{conv} + Q_{rad} = 0\) Here, \(Q_{cond}\) is the heat conduction, \(Q_{conv}\) is the heat convection, and \(Q_{rad}\) is the heat radiation.

Calculate the heat conduction term

The heat conduction term \(Q_{cond}\) can be calculated using Fourier's Law, given as: \(Q_{cond} = kA\frac{dT}{dx}\) where \(k\) is the thermal conductivity, \(A\) is the cross-sectional area of the fin, and \(\frac{dT}{dx}\) is the temperature gradient along the fin. Since we are using finite differences, we will approximate the temperature gradient using the central difference scheme: \(\frac{dT}{dx} \approx \frac{T_{i+1} - T_{i-1}}{2\Delta x}\) This approximation will be used to calculate the heat conduction term for each node.

Calculate the heat convection term

The heat convection term \(Q_{conv}\) can be calculated using Newton's Law of Cooling, given as: \(Q_{conv} = hP(T - T_\infty)\) where \(P\) is the perimeter of the fin and \(T_\infty\) is the ambient air temperature. Since we are only concerned about nodes 1 and 2, this term can be applied directly to each of the nodes.

Calculate the heat radiation term

The heat radiation term \(Q_{rad}\) can be calculated using Stefan-Boltzmann Law, given as: \(Q_{rad} = \epsilon\sigma P(T^4 - T_{surr}^4)\) where \(\epsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant, and \(T_{surr}\) is the surrounding surfaces temperature. This term can also be applied directly to nodes 1 and 2.

Apply boundary conditions

Now that we have expressions for \(Q_{cond}\), \(Q_{conv}\), and \(Q_{rad}\), we need to apply the boundary conditions of specified temperature at the fin base (node 0) and negligible heat transfer at the fin tip (node 2). For node 0 (fin base): The temperature is specified, so we don't need an energy balance equation for this node. For node 1 (middle): The general energy balance equation holds true. We will need to combine the expressions for \(Q_{cond}\), \(Q_{conv}\), and \(Q_{rad}\) and solve for \(T_1\). For node 2 (fin tip): The heat transfer is negligible, so we will set \(Q_{cond}\), \(Q_{conv}\), and \(Q_{rad}\) to zero and solve for \(T_2\).

Solve for temperatures \(T_1\) and \(T_2\)

Now we have to solve the energy balance equations for both nodes 1 and 2. For node 1: \(\frac{kAD(T_2 - T_0)}{2\Delta x} + hP(T_1 - T_\infty) + \epsilon\sigma P(T_1^4 - T_{surr}^4) = 0\) For node 2: \(\frac{kAD(T_2 - T_1)}{2\Delta x} + hP(T_2 - T_\infty) + \epsilon\sigma P(T_2^4 - T_{surr}^4) = 0\) These are two non-linear equations with two unknowns \(T_1\) and \(T_2\). Depending on the specific problem, there might be different methods to solve the system of equations, such as numerical methods or using a computer program. In conclusion, we have derived the finite difference formulation for the given heat conduction problem in a pin fin system. By applying the boundary conditions and solving the energy balance equations for nodes 1 and 2, we can determine the temperatures \(T_1\) and \(T_2\).