Question

Obtain a relation for the fin efficiency for a fin of constant cross-sectional area \(A_{c}\), perimeter \(p\), length \(L\), and thermal conductivity \(k\) exposed to convection to a medium at \(T_{\infty}\) with a heat transfer coefficient \(h\). Assume the fins are sufficiently long so that the temperature of the fin at the tip is nearly \(T_{\infty}\). Take the temperature of the fin at the base to be \(T_{b}\) and neglect heat transfer from the fin tips. Simplify the relation for \((a)\) a circular fin of diameter \(D\) and \((b)\) rectangular fins of thickness \(t\).

Short answer

Based on the step-by-step solution provided, create the following short answer: To find the fin efficiency for a fin with constant cross-sectional area, length, perimeter, and thermal conductivity exposed to convection, follow these steps: 1. Define fin efficiency and fin parameters 2. Derive the differential equation for heat conduction in the fin using Fourier's law and Newton's law of cooling 3. Solve the differential equation for temperature distribution using the given boundary conditions 4. Calculate the heat transfer rate of the fin 5. Calculate the fin efficiency 6. Simplify the fin efficiency relation for a circular fin of diameter D 7. Simplify the fin efficiency relation for a rectangular fin of thickness t The fin efficiency, denoted by η, reveals the ratio of the actual heat transfer through the fin to the maximum possible heat transfer (assuming the entire fin is at the base temperature).

Step-by-step solution

Define fin efficiency and fin parameters

Fin efficiency is defined as the actual heat transfer rate through the fin divided by the maximum possible heat transfer rate (i.e., if the entire fin were at the base temperature \(T_b\)). Let's denote the fin efficiency by \(\eta\). The fin is also characterized by the following parameters: Cross-sectional area \(A_{c}\), Perimeter \(p\), Length \(L\), and Thermal Conductivity \(k\). Convection occurs between the fin and the surrounding medium, and the heat transfer coefficient is given by \(h\).

Define the differential equation for heat conduction in the fin

We can now model the heat conduction in the fin by considering its cross-sectional area \(A_{c}\), thermal conductivity \(k\), and perimeter \(p\). Using Fourier's law of heat conduction and Newton's law of cooling, we can derive the following differential equation for the temperature distribution in the fin: \((A_c k\frac{d^2T}{dx^2}) + (hp(T-T_{\infty}))=0\).

Solve the differential equation for temperature distribution

The general solution for the above differential equation is given by the following formula: \(T(x)=T_\infty + C_1\mathrm{e}^{mx}+C_2\mathrm{e}^{-mx}\), where \(m=\sqrt{\frac{hp}{Ak}}\). We're given the boundary conditions: \(T(x=0)=T_{b}\) and \(T(x=L)\approx T_{\infty}\). Using these boundary conditions, we can solve for the constants \(C_1\) and \(C_2\).

Calculate the heat transfer rate of the fin

Using the temperature distribution in the fin, we can calculate the heat transfer rate \(q_{f}\) through the fin by applying Fourier's law at the fin's base: \(q_{f}= -kA_c\left(\frac{dT}{dx}\right)_(x=0)\).

Calculate the fin efficiency

Now that we have the heat transfer rate of the fin, the fin efficiency \(\eta\) can be found as the ratio of the actual heat transfer through the fin to the maximum possible heat transfer (i.e., if the entire fin were at the base temperature \(T_b\)): \(\eta=\frac{q_{f}}{q_{max}}\), where \(q_{max}=h\cdot p \cdot L \cdot (T_b-T_\infty)\). Calculate \(\eta\) using the expression for \(q_{f}\) obtained in Step 4.

Simplify the relation for a circular fin of diameter \(D\)

Now, let's simplify the fin efficiency relation for the case of circular fins with a diameter of \(D\). For a circular fin, \(A_c= \frac{1}{4}\pi D^2\) and \(p=\pi D\). Substitute these expressions into the fin efficiency relation derived in Step 5 and simplify the resulting expression.

Simplify the relation for a rectangular fin of thickness \(t\)

Finally, let's simplify the fin efficiency relation for the case of rectangular fins with a thickness of \(t\). For a rectangular fin, \(A_c= t\cdot w\) (where \(w\) is the width of the fin along the base) and \(p=2(t+w)\). Substitute these expressions into the fin efficiency relation derived in Step 5 and simplify the resulting expression.