(SEM V) THEORY EXAMINATION 2022-23 HEAT AND MASS TRANSFER

SECTION A – Short Answer Type Questions (2 Marks each)

(a) Explain the Significance of Thermal Diffusivity.

Thermal diffusivity (α) is a physical property that indicates how quickly heat spreads through a material.

α=kρcp\alpha = \frac{k}{\rho c_p}α=ρcp​k​

Where:

kkk = thermal conductivity (W/m·K)

ρ\rhoρ = density (kg/m³)

cpc_pcp​ = specific heat (J/kg·K)

Significance:

High thermal diffusivity ⇒ material adjusts rapidly to temperature changes (e.g., metals).

Low thermal diffusivity ⇒ slow temperature equalization (e.g., wood, insulation).

It helps predict transient heat conduction behavior in unsteady-state problems like cooling or heating of solids.

Example: Metals have high α (fast heat transfer), while ceramics have low α (good insulators).

(b) Define Critical Radius of Insulation.

The critical radius of insulation (r_c) is the radius at which the heat loss from an insulated cylindrical or spherical surface is maximum.

For a cylinder:

rc=khr_c = \frac{k}{h}rc​=hk​

where

kkk = thermal conductivity of insulation material

hhh = convective heat transfer coefficient

Explanation:
Adding insulation increases conduction resistance but also increases surface area, which can enhance convection.
Initially, total heat loss increases until r=rcr = r_cr=rc​, then decreases beyond it.

Importance:
Used in designing insulation for pipes and cables to ensure efficient heat control.

 SECTION B – Long Answer Type Questions (10 Marks each)

(a) Derive the Energy Equation for the Thermal Boundary Layer Over a Flat Plate.

Assumptions:

Steady-state flow.                                        Incompressible, laminar flow.

Constant fluid properties.                            Neglect viscous dissipation.

Energy Equation (2D steady form):

u∂T∂x+v∂T∂y=α∂2T∂y2u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2}u∂x∂T​+v∂y∂T​=α∂y2∂2T​

Where:

u,vu, vu,v = velocity components

TTT = temperature

α=kρcp\alpha = \frac{k}{\rho c_p}α=ρcp​k​ = thermal diffusivity

Boundary Conditions:
At y=0y = 0y=0: T=TsT = T_sT=Ts​ (surface temperature)
At y→∞y → ∞y→∞: T=T∞T = T_∞T=T∞​ (free stream temperature)

Interpretation:

The term on LHS represents energy convection, and RHS represents energy diffusion due to conduction.

The equation governs heat transfer in boundary layers, forming the basis of correlations like Nusselt and Prandtl numbers.

(b) Explain Working Principle of Heat Pipe and Discuss Pool Boiling with Diagram.

Heat Pipe:

A heat pipe is a highly efficient device that transfers heat using the phase change of a working fluid (evaporation and condensation) inside a sealed tube.

Working Principle:

Evaporator section: Absorbs heat causing liquid to vaporize.

Adiabatic section: Vapor travels with negligible temperature loss.

Condenser section: Vapor condenses, releasing latent heat to surroundings.

Wick structure: Capillary action returns condensed liquid to the evaporator.

Advantages: High thermal conductivity, passive operation, and compact size.

Pool Boiling:

Occurs when a stationary liquid is heated from below.
Regimes of Pool Boiling (Boiling Curve):

Natural convection region: Low heat flux; no bubble formation.

Nucleate boiling: Bubbles form at heated surface; high heat transfer rate.

Transition boiling: Unstable vapor film formation.

Film boiling: Stable vapor film insulates surface; heat transfer decreases.

Applications: Heat exchangers, electronics cooling, and nuclear reactors.

SECTION C – Very Long Answer Type Questions (10 Marks each)

(a) Derive Expression for LMTD (Logarithmic Mean Temperature Difference) for a Parallel Flow Heat Exchanger.

Consider a parallel-flow heat exchanger where both fluids enter and exit in the same direction.

Let:

Th,i,Th,oT_{h,i}, T_{h,o}Th,i​,Th,o​ = inlet & outlet temperatures of hot fluid.

Tc,i,Tc,oT_{c,i}, T_{c,o}Tc,i​,Tc,o​ = inlet & outlet temperatures of cold fluid.

Differential Heat Transfer:

dQ=U dA ΔTdQ = U \, dA \, \Delta TdQ=UdAΔT

where ΔT=(Th−Tc)\Delta T = (T_h - T_c)ΔT=(Th​−Tc​).

Integrating:

Q=UA(ΔT1−ΔT2)ln⁡(ΔT1/ΔT2)Q = U A \frac{(ΔT_1 - ΔT_2)}{\ln(ΔT_1/ΔT_2)}Q=UAln(ΔT1​/ΔT2​)(ΔT1​−ΔT2​)​

Thus,

LMTD=ΔT1−ΔT2ln⁡(ΔT1/ΔT2)\text{LMTD} = \frac{ΔT_1 - ΔT_2}{\ln(ΔT_1/ΔT_2)}LMTD=ln(ΔT1​/ΔT2​)ΔT1​−ΔT2​​

where

ΔT1=Th,i−Tc,i,ΔT2=Th,o−Tc,oΔT_1 = T_{h,i} - T_{c,i}, \quad ΔT_2 = T_{h,o} - T_{c,o}ΔT1​=Th,i​−Tc,i​,ΔT2​=Th,o​−Tc,o​

Conclusion:
LMTD gives the effective average temperature difference between the two fluids, accounting for exponential decay along the heat exchanger length.

(b) Explain Radiation Laws – Kirchhoff’s, Lambert’s, Wien’s, and Planck’s Laws.

Kirchhoff’s Law:

1. EA=f(T,λ)\frac{E}{A} = f(T, λ)AE​=f(T,λ)

For a body in thermal equilibrium, emissivity (ε) = absorptivity (α).
A perfect black body has ε = α = 1.

Lambert’s Cosine Law:
The intensity of radiation emitted in a given direction varies with the cosine of the angle (θ) from the normal:

1. I=I0cos⁡θI = I_0 \cos θI=I0​cosθ

Wien’s Displacement Law:
The wavelength corresponding to maximum emissive power (λ_max) is inversely proportional to absolute temperature (T):

1. λmaxT=2897 μm⋅Kλ_{max} T = 2897 \, μm·Kλmax​T=2897μm⋅K

Planck’s Law:
Describes the spectral distribution of radiation from a black body:

1. Eλ=C1λ5(eC2/(λT)−1)E_λ = \frac{C_1}{λ^5 (e^{C_2/(λT)} - 1)}Eλ​=λ5(eC2​/(λT)−1)C1​​

where C1C_1C1​ and C2C_2C2​ are constants.

Significance:
These laws explain thermal radiation behavior, forming the foundation for radiation heat transfer in furnaces, solar collectors, and space engineering.