GATE Past Year Questions: Free & Forced Convection | Heat Transfer - Mechanical Engineering PDF Download

1. Grashof Number (A): The Grashof number is used in fluid dynamics to describe free convection. It relates the buoyancy forces to the viscous forces within a fluid. In simpler terms, it helps in analyzing the flow of fluid when there are temperature differences causing movement.
   Matches with: 3 (Free convection)

2. Schmidt Number (B): The Schmidt number is a dimensionless number used in fluid dynamics that describes the ratio of momentum diffusivity (viscosity) to mass diffusivity. It is important in mass transfer operations.
   Matches with: 1 (Mass diffusion)

3. Weber Number (C): The Weber number is a dimensionless number in fluid mechanics that is used to analyze the relative importance of the fluid's inertia compared to its surface tension. It is commonly used in the study of droplets, bubbles, and liquid surfaces. 
   Matches with: 5 (Surface tension)

4. Fourier Number (D): The Fourier number is a dimensionless number used in heat conduction problems. It represents the ratio of heat conduction rate to the rate of heat storage, making it crucial in transient heat conduction analysis.
   Matches with: 2 (Transient heat conduction)

The correct matching code is (b) 3, 1, 5, 2,

The ratio of momentum diffusivity (ν) to thermal diffusivity (α) is called the Prandtl number.

• Prandtl Number (Pr): This is a dimensionless number that represents the ratio of momentum diffusivity (kinematic viscosity, ν) to thermal diffusivity (α). It is used to relate the relative thickness of the momentum boundary layer to the thermal boundary layer in fluid flow.

• Nusselt Number (Nu): It represents the ratio of convective to conductive heat transfer across a boundary.

• Biot Number (Bi): It relates the internal resistance to heat conduction within a body to the external convective heat transfer.

• Lewis Number (Le): This is the ratio of thermal diffusivity to mass diffusivity.

The Grashof number signifies the ratio of buoyancy force to viscous force in a fluid.

Grashof Number (Gr): It is a dimensionless number that compares the relative significance of buoyancy forces to viscous forces in fluid flow. It is important in the study of natural convection, where fluid motion is caused by temperature-induced density differences (buoyancy).

A high Grashof number indicates that buoyancy forces dominate, leading to strong natural convection.

The highest Heat Transfer Coefficient (HTC) in pool boiling occurs in the nucleate boiling zone.

• Nucleate Boiling Zone: This is the regime where small bubbles form on the heated surface and then detach into the liquid. The heat transfer is extremely efficient in this zone because the bubbles enhance the mixing of the liquid, thus increasing the heat transfer rate significantly.

• Subcooled Boiling Zone: This occurs when the bulk liquid is at a temperature below its boiling point, so boiling happens only near the surface. The HTC here is lower compared to nucleate boiling.

• Partial Film Boiling Zone: In this zone, part of the surface is covered by a vapor film, which reduces heat transfer because vapor has lower thermal conductivity than liquid.

• Film Boiling Zone: In this zone, the surface is completely covered by a stable vapor film, which significantly lowers the heat transfer coefficient due to the insulating effect of the vapor.

The heat transfer coefficients for free convection in gases, forced convection in gases and vapours, and for boiling water lie, respectively, in the range of:

• Free convection in gases: 5-15 W/m²K
• Forced convection in gases and vapours: 20-200 W/m²K
• Boiling water: 3000-50000 W/m²K

Explanation:

• Free convection in gases: Heat transfer by natural or free convection in gases has a relatively low heat transfer coefficient because the fluid movement is primarily due to buoyancy forces.

• Forced convection in gases and vapours: In forced convection, the movement of gases is induced by external forces (e.g., a fan or pump), leading to a higher heat transfer coefficient compared to free convection.

• Boiling water: Boiling water has a significantly higher heat transfer coefficient due to the phase change from liquid to vapor, which enhances heat transfer.

Correct Answer:

A: 5-15; 20-200 and 3000-50000 W/m²K

Given data;


Volume,
V = 2×1×2=4 m³
Heat generated,
Q = q_G × V = 100 × 4 = 400 W
Convection heat transfer from face PQRS,

At the inlet of pipe
x = O
q "_wi = 2500 x 0 = 0
At the exit of pipe,
x = 1

Average heat transfer,

Net heat transfer

By solving above equation, we get
T_o = 62° C

Heat transfer through pipe wall,

Q = mC_P(T₀–T_i), amount of heat gained by the water
2355 = 0.01 × 4180 (T₀ – 20)
T₀ = 76.33°C

5000 = 1000 (Tso – 76.33)
Tso = 81°C

Forlaminar flow, Nu = 0.664 (Re)^0.5 (Pr)^0.33

So when free stream velocity increases by a factor of 2, then the average heat transfer coefficient rises by a factor of √2.

Prandtl number = 

When

(Nu)_q for constant wall heat flux and (Nu)_T at constant wall temperature for a hydrodynamically and thermally fully developed laminar flow through a circular pipe of constant cross-section is 4.36 and 3.66 respectively.

In case of uniform heat flux, bulk mean temperature varies linearly. The difference between bulk mean temperature and wall temperature is constant in thermally developed region so bulk tempeature,

Here,

So, Q and R in thermally developed region.