Question

A fin of length \( L \) is applied the boundary conditions associated with a very long fin, the adiabatic tip fin, and the finite convective tip fin, which of the following observation about the fin-tip temperature (\( T \)) for these three cases is correct.

• A. \( T_{\text{very long}}>T_{\text{convective tip}}>T_{\text{adiabatic tip}} \)
• B. \( T_{\text{very long}}<T_{\text{convective tip}}<T_{\text{adiabatic tip}} \)
• C. \( T_{\text{very long}}<T_{\text{adiabatic tip}}<T_{\text{convective tip}} \)
• D. \( T_{\text{convective tip}}>T_{\text{adiabatic tip}}>T_{\text{very long}} \)

Hint:

In fin heat transfer, the tip temperature increases as the tip condition restricts heat loss: very long fin (lowest), convective tip, adiabatic tip (highest).

Answer 1

The Correct Option is B

Step 1: Understand the fin boundary conditions.
A fin of length \( L \) is analyzed under three boundary conditions at the tip:
Very long fin (infinitely long): The fin is assumed to be so long that the tip temperature approaches the surrounding temperature \( T_\infty \).
Adiabatic tip: The fin tip has no heat transfer (\( q = 0 \)), meaning the temperature gradient at the tip is zero.
Finite convective tip: The fin tip loses heat to the surroundings via convection, with a heat transfer coefficient \( h \).
The base of the fin is at temperature \( T_b \), and the surrounding temperature is \( T_\infty \), where \( T_b>T_\infty \). We need to compare the tip temperatures under these conditions. Step 2: Derive the tip temperatures for each case.
The general temperature distribution in a fin is governed by the fin equation: \[ \frac{d^2 \theta}{dx^2} - m^2 \theta = 0, \] where \( \theta = T - T_\infty \), \( m = \sqrt{\frac{hP}{kA}} \), \( h \) is the convective heat transfer coefficient, \( P \) is the perimeter, \( k \) is the thermal conductivity, and \( A \) is the cross-sectional area. The boundary condition at the base (\( x = 0 \)) is \( \theta(0) = T_b - T_\infty = \theta_b \). Very long fin:
For an infinitely long fin, the temperature at the tip (\( x \to \infty \)) approaches the surrounding temperature: \[ \theta(x) = \theta_b e^{-mx}, \] \[ T(x \to \infty) = T_\infty, \] \[ T_{\text{very long}} = T_\infty. \] Adiabatic tip:
The boundary condition at the tip (\( x = L \)) is \( \frac{d\theta}{dx} = 0 \). The solution is: \[ \theta(x) = \theta_b \frac{\cosh[m(L - x)]}{\cosh(mL)}, \] At the tip (\( x = L \)): \[ \theta(L) = \theta_b \frac{\cosh(0)}{\cosh(mL)} = \theta_b \frac{1}{\cosh(mL)}, \] \[ T_{\text{adiabatic tip}} = T_\infty + (T_b - T_\infty) \frac{1}{\cosh(mL)}. \] Since \( \cosh(mL)>1 \), the tip temperature is higher than \( T_\infty \). Finite convective tip:
The boundary condition at the tip is convective heat loss: \( -k \frac{d\theta}{dx} \bigg|_{x=L} = h \theta(L) \). The solution is more complex, but the tip temperature lies between the adiabatic and very long cases: \[ \theta(x) = \theta_b \frac{\cosh[m(L - x)] + \frac{h}{mk} \sinh[m(L - x)]}{\cosh(mL) + \frac{h}{mk} \sinh(mL)}, \] At the tip (\( x = L \)): \[ \theta(L) = \theta_b \frac{1}{\cosh(mL) + \frac{h}{mk} \sinh(mL)}, \] \[ T_{\text{convective tip}} = T_\infty + (T_b - T_\infty) \frac{1}{\cosh(mL) + \frac{h}{mk} \sinh(mL)}. \] The denominator is larger than in the adiabatic case due to the additional \( \frac{h}{mk} \sinh(mL) \) term, so \( T_{\text{convective tip}}<T_{\text{adiabatic tip}} \), but still greater than \( T_\infty \). Step 3: Compare the tip temperatures.
\( T_{\text{very long}} = T_\infty \),
\( T_{\text{adiabatic tip}} = T_\infty + (T_b - T_\infty) \frac{1}{\cosh(mL)} \),
\( T_{\text{convective tip}} = T_\infty + (T_b - T_\infty) \frac{1}{\cosh(mL) + \frac{h}{mk} \sinh(mL)} \).
Since \( \cosh(mL) + \frac{h}{mk} \sinh(mL)>\cosh(mL) \), the convective tip temperature is lower than the adiabatic tip but higher than the very long fin: \[ T_{\text{very long}}<T_{\text{convective tip}}<T_{\text{adiabatic tip}}. \] Step 4: Select the correct answer.
The correct observation is \( T_{\text{very long}}<T_{\text{convective tip}}<T_{\text{adiabatic tip}} \), matching option (2).