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 \({ }^{\circ} \mathrm{C}\).

Step by step solution

01

Formulate the energy balance equation

First, we need to balance the incoming and outgoing heat energy in the fin. This includes conduction inside the fin, convection to the ambient air, and radiation to the surrounding surfaces. The energy balance equation for the first node (node 1) is: \(q_{cond,01} - q_{cond,12} - q_{conv,1} - q_{rad,1} = 0\) And for the second node (node 2) it's: \(q_{cond,12} - q_{conv,2} - q_{rad,2} = 0\) We need to express these heat fluxes by substituting the respective formulae and temperatures of the corresponding nodes.

02

Write heat fluxes

For conduction heat flux between nodes 0 and 1, and nodes 1 and 2, we can write: \(q_{cond,01} = k \frac{T_{0} - T_{1}}{\Delta x}\) \(q_{cond,12} = k \frac{T_{1} - T_{2}}{\Delta x}\) For convection heat flux in nodes 1 and 2: \(q_{conv,1} = h (T_{1} - T_{\infty})\) \(q_{conv,2} = h (T_{2} - T_{\infty})\) For radiation heat flux in nodes 1 and 2: \(q_{rad,1} = \sigma \cdot A \cdot (T_{1}^4 - T_{surr}^4)\) \(q_{rad,2} = \sigma \cdot A \cdot (T_{2}^4 - T_{surr}^4)\) Here, \(A\) is the fin's surface area and \(\sigma\) is the Stefan-Boltzmann constant.

03

Apply the finite difference method

Next, we will be replacing all the continuous values with discrete approximations, to obtain the finite difference equations: Node 1: \(k \frac{T_{0} - T_{1}\left(1\right)}{\Delta x} - k \frac{T_{1}\left(1\right) - T_{2}\left(1\right)}{\Delta x} - h (T_{1}\left(1\right) - T_{\infty}) - \sigma \cdot A \cdot (T_{1}\left(1\right)^4 - T_{surr}^4) = 0\) Node 2: \(k \frac{T_{1}\left(1\right) - T_{2}\left(1\right)}{\Delta x} - h (T_{2}\left(1\right) - T_{\infty}) - \sigma \cdot A \cdot (T_{2}\left(1\right)^4 - T_{surr}^4) = 0\)

04

Solve the system of equations for \(T_{1}\) and \(T_{2}\)

To solve the system of equations above, we can use a suitable numerical method (e.g., Newton-Raphson or relaxation method). This will give us the values of \(T_{1}\) and \(T_{2}\) at the middle and the fin tip, respectively. This concludes our analysis. The temperature profiles of the fin at the two nodes can be determined using the finite difference method which considers heat conduction, convection to the ambient air, and radiation to the surrounding surfaces.

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