A straight fin of uniform circular cross section and adiabatic tip has an aspect ratio (length/diameter) of 4. If the Biot number (based on radius of the fin as the characteristic length) is 0.04, the fin efficiency is \(\underline{\hspace{1cm}}\) % (round off to nearest integer).
Show Hint
Correct Answer: 42
Solution and Explanation
\[ \eta = \frac{\tanh(mL)}{mL} \] where \(m = \sqrt{\frac{hP}{kA}}\) and \(L\) is the length of the fin. The Biot number is defined as:
\[ \text{Bi} = \frac{hL}{k} \] Given: \(\text{Bi} = 0.04\), \(L = 4D\), so:
\[ \eta \approx 42 \text{ to } 44 % \] Final answer: 42–44 %.
Top Questions on Steady and unsteady heat conduction
- Consider a solid slab of thickness 2L and uniform cross section A. The volumetric rate of heat generation within the slab is \(\dot{q}\) (W m\(^{-3}\)). The slab loses heat by convection at both the ends to air with heat transfer coefficient \(h\). Assuming steady state, one-dimensional heat transfer, the temperature profile within the slab along the thickness is given by: \[ T(x) = \frac{\dot{q}L^2}{2k} \left[1 - \left(\frac{x}{L}\right)^2 \right] + T_s \text{for} -L \leq x \leq L \] where \(k\) is the thermal conductivity of the slab and \(T_s\) is the surface temperature. If \(T_s = 350\) K, ambient air temperature \(T_\infty = 300\) K, and Biot number (based on L as the characteristic length) is 0.5, the maximum temperature in the slab is \(\underline{\hspace{1cm}}\) K (round off to nearest integer).
- GATE CH - 2021
- Heat Transfer
- Steady and unsteady heat conduction
- One side of a solid food block of 10 cm thickness is subjected to a heating medium having a film heat transfer coefficient of 70 W·(m\(^2\)·°C)\(^{-1}\). The other side of the food block is being cooled by a medium having a film heat transfer coefficient of 100 W·(m\(^2\)·°C)\(^{-1}\). The food block has a thermal conductivity of 0.2 W·(m·°C)\(^{-1}\) and the contact area of the block available for heat transfer is 1 m\(^2\). Heat transfer rate in the block at steady state is 100 J·s\(^{-1}\). The temperature difference between the two sides of the block in °C is \(\underline{\hspace{1cm}}\) (round off to 2 decimal places).
- GATE AG - 2021
- Dairy and Food Engineering
- Steady and unsteady heat conduction
Questions Asked in GATE CH exam
- Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?
An electrical wire of 2 mm diameter and 5 m length is insulated with a plastic layer of thickness 2 mm and thermal conductivity \( k = 0.1 \) W/(m·K). It is exposed to ambient air at 30°C. For a current of 5 A, the potential drop across the wire is 2 V. The air-side heat transfer coefficient is 20 W/(m²·K). Neglecting the thermal resistance of the wire, the steady-state temperature at the wire-insulation interface __________°C (rounded off to 1 decimal place).
GIVEN:
Kinematic viscosity: \( \nu = 1.0 \times 10^{-6} \, {m}^2/{s} \)
Prandtl number: \( {Pr} = 7.01 \)
Velocity boundary layer thickness: \[ \delta_H = \frac{4.91 x}{\sqrt{x \nu}} \]- GATE CH - 2025
- Fluid Mechanics
The first-order irreversible liquid phase reaction \(A \to B\) occurs inside a constant volume \(V\) isothermal CSTR with the initial steady-state conditions shown in the figure. The gain, in kmol/m³·h, of the transfer function relating the reactor effluent \(A\) concentration \(c_A\) to the inlet flow rate \(F\) is:
- GATE CH - 2025
- Fluid Mechanics
A hot plate is placed in contact with a cold plate of a different thermal conductivity as shown in the figure. The initial temperature (at time $t = 0$) of the hot plate and cold plate are $T_h$ and $T_c$, respectively. Assume perfect contact between the plates. Which one of the following is an appropriate boundary condition at the surface $S$ for solving the unsteady state, one-dimensional heat conduction equations for the hot plate and cold plate for $t>0$?
- GATE CH - 2025
- Thermodynamics
The following data is given for a ternary \(ABC\) gas mixture at 12 MPa and 308 K:
\(y_i\): mole fraction of component \(i\) in the gas mixture
\(\hat{\phi}_i\): fugacity coefficient of component \(i\) in the gas mixture at 12 MPa and 308 K
The fugacity of the gas mixture is _________ MPa (rounded off to 3 decimal places).- GATE CH - 2025
- Thermodynamics