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 with the ambient air at \(T_{\infty}\left(\right.\) in ${ }^{\circ} \mathrm{C}\( ) with a convection coefficient of \)h$, and by radiation to the surrounding surfaces at an average temperature of \(T_{\text {sur }}\) (in \(\mathrm{K}\) ). 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 for the case of a specified temperature at the fin base and convection and radiation heat transfer at the fin tip.

Short answer

Question: Analyze the steady one-dimensional heat conduction in a pin fin, using an energy balance approach and obtain the finite difference formulation for the case of a specified temperature at the fin base and convection and radiation heat transfer at the fin tip. Assume a pin fin with constant diameter and thermal conductivity. Answer: The finite difference formulations for node 1 (fin middle) and node 2 (fin tip) are: Node 1: \((T_0 - T_1) + (T_2 - T_1) = \frac{hP\Delta{x}}{kA}(T_1 - T_{\infty})\) Node 2: \[\frac{kA}{\sigma \varepsilon PD\Delta{x}}(T_1 - T_2) = \frac{hP}{\sigma \varepsilon PD\Delta{x}}(T_2 - T_{\infty}) + (T_2^4 - T_{\text{sur}}^4)\]

Step-by-step solution

Energy balance at node 0 (fin base)

At node 0, there is a specified temperature (\(T_0\)). Therefore, there is no need to set up an energy balance equation since we already have the temperature information for this node.

Energy balance at node 1 (fin middle)

At node 1, we'll have heat conduction from node 0, heat conduction to node 2, and heat loss by convection from the surface of the fin. Write down the energy balance using Fourier's law and Newton's law of cooling: \[k\frac{A}{\Delta{x}}(T_0 - T_1) + k\frac{A}{\Delta{x}}(T_2 - T_1) = hP(T_1 - T_{\infty})\] Here, A is the cross-sectional area of the fin, and P is the perimeter of the fin.

Energy balance at node 2 (fin tip)

At node 2, we'll have heat conduction from node 1, heat loss by convection from the surface of the fin, and heat loss by radiation. Write down the energy balance using Fourier's law, Newton's law of cooling and Stefan-Boltzmann law: \[k\frac{A}{\Delta{x}}(T_1 - T_2) = hP(T_2 - T_{\infty}) + \sigma \varepsilon P(D\Delta{x})(T_2^4 - T_{\text {sur }}^4)\] Here, \(\sigma\) is the Stefan-Boltzmann constant, and \(\varepsilon\) is the emissivity of the fin's surface.

Rewrite the equations in finite difference form

Divide each term by \(kA/\Delta{x}\) in the equation for node 1 and by \(\sigma \varepsilon PD\Delta{x}\) in the equation for node 2 to obtain the finite difference equations. For node 1, \[(T_0 - T_1) + (T_2 - T_1) = \frac{hP\Delta{x}}{kA}(T_1 - T_{\infty})\] For node 2, \[\frac{kA}{\sigma \varepsilon PD\Delta{x}}(T_1 - T_2) = \frac{hP}{\sigma \varepsilon PD\Delta{x}}(T_2 - T_{\infty}) + (T_2^4 - T_{\text{sur}}^4)\] These are the finite difference formulations of the problem for the given case.