Heat and Mass Transfer for SSC JE Mechanical: Conduction, Convection, Radiation, Heat Exchangers, Fins and Formulas

Heat and mass transfer is a scoring subject in the SSC JE Mechanical paper, because the questions come from a fixed set of laws and formulas. This guide on heat and mass transfer for SSC JE covers the three modes of heat transfer, the laws, the dimensionless numbers, the heat exchangers, the fins and the transient conduction, with solved examples, practice questions and FAQs.

Modes of Heat Transfer

Conduction Formulas

• Thermal conductivity is the highest for pure metals (silver, copper, aluminium) and the lowest for gases; air has k about 0.026 W/mK.
• Alloys have lower conductivity than pure metals, and the conductivity of most liquids decreases with temperature (water is an exception at low temperatures).
• Thermal resistance in series adds up; in parallel, the reciprocals add up.
• Critical radius of insulation: r_c = k/h for a cylinder and 2k/h for a sphere; insulation increases the heat loss if the outer radius is less than the critical radius.

Dimensionless Numbers

• For fully developed laminar flow in a tube, Nu = 3.66 (constant wall temperature) and 4.36 (constant heat flux).
• Dittus–Boelter equation for turbulent flow in tubes: Nu = 0.023 Re^0.8 Prⁿ (n = 0.4 for heating and 0.3 for cooling).
• In natural convection, Nu depends on Gr and Pr; the heat transfer coefficient is the lowest for natural convection of gases and highest for boiling and condensation.

Radiation

• Stefan–Boltzmann constant σ = 5.67 × 10⁻⁸ W/m²K⁴; a black body emits E_b = σT⁴ and a real body E = εσT⁴.
• Wien’s displacement law: λ_max T = 2898 µm K; the wavelength of the maximum emission decreases as the temperature rises.
• Kirchhoff’s law: at thermal equilibrium, emissivity equals absorptivity (ε = α).
• A black body has absorptivity = emissivity = 1; a white body has a reflectivity of 1; a grey body has constant emissivity at all wavelengths.
• Radiation shields reduce the heat exchange between two surfaces.
• View factor (shape factor) F₁₂ is the fraction of the radiation leaving surface 1 that reaches surface 2; reciprocity: A₁F₁₂ = A₂F₂₁.

Heat Exchangers

• LMTD = (ΔT₁ − ΔT₂) / ln(ΔT₁/ΔT₂); the heat transfer Q = U A × LMTD.
• For the same terminal temperatures, LMTD of counter flow is more than that of parallel flow.
• Effectiveness = actual heat transfer / maximum possible heat transfer; the NTU = UA/C_min.
• Fouling factor accounts for the deposits that reduce the heat transfer.

Fins and Transient Conduction

• Fins (extended surfaces) increase the heat transfer by increasing the surface area; used in air-cooled engines and radiators.
• Fin efficiency = actual heat transfer from the fin / heat transfer if the whole fin were at the base temperature.
• Fins are useful when h is low (such as for gases) and not so much for liquids.
• Lumped system analysis: T − T_∞ = (T_i − T_∞) e^−t/τ, where the time constant τ = ρVc_p/(hA).

Mass Transfer (Basics)

Fick’s law of diffusion: the mass flux is proportional to the concentration gradient, J = −D (dC/dx). Mass transfer happens by diffusion or convection; the Schmidt number Sc = ν/D and the Sherwood number Sh = h_mL/D are the dimensionless numbers; Lewis number Le = α/D.

Solved Examples

Example 1. A wall of area 10 m², thickness 0.2 m and k = 1.2 W/mK has a temperature difference of 30°C. Find the heat flow.
Q = 1.2 × 10 × 30 / 0.2 = 1,800 W.

Example 2. Find the emissive power of a black body at 500 K.
E = 5.67 × 10⁻⁸ × 500⁴ = 3,544 W/m².

Example 3. Find the LMTD if ΔT₁ = 40°C and ΔT₂ = 20°C.
20 / ln 2 = 28.85°C.

Example 4. The sun has a surface temperature of about 5,800 K. Find the wavelength of the maximum emission.
λ_max = 2898/5800 = 0.5 µm.

Example 5. For a small sphere, h = 50 W/m²K, k = 50 W/mK and the characteristic length is 0.01 m. Can the lumped system method be used?
Bi = hL/k = 0.01, which is less than 0.1, so yes.

Quick Tips

• Memorise the formulas of conduction through a wall, a cylinder and a sphere.
• Remember the meaning of each dimensionless number.
• Remember that the counter flow exchanger is the most effective.
• Remember the lumped system condition Bi < 0.1.

Quick Revision Points

• Fourier’s law governs conduction; Newton’s law governs convection.
• E = σT⁴ for a black body.
• Nu = hL/k and Bi = hL/k (with different k).
• The critical radius is k/h for a cylinder.
• LMTD of counter flow is greater than that of parallel flow.

Practice Questions With Answers

1. What is the Stefan–Boltzmann law?
   Answer: E = σT⁴
2. What is the Biot number?
   Answer: hL/k
3. What does Wien’s law relate?
   Answer: The wavelength of maximum emission and temperature (λ_max T = constant)
4. What is the critical radius of insulation for a cylinder?
   Answer: k/h
5. In which heat exchanger is the effectiveness the highest?
   Answer: Counter flow
6. Which mode of heat transfer needs no medium?
   Answer: Radiation
7. What is the Reynolds number limit for laminar flow in a pipe?
   Answer: Re < 2300
8. What is the Prandtl number?
   Answer: μc_p/k
9. What is the emissivity of a black body?
   Answer: 1
10. Which law says ε = α at thermal equilibrium?
    Answer: Kirchhoff’s law

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Mechanical Engineering notes page and the next topic Machine Design.

Frequently Asked Questions

What is Fourier’s law?

The heat flow by conduction is proportional to the area and the temperature gradient: q = −kA dT/dx.

What is the Nusselt number?

The ratio of convective to conductive heat transfer, hL/k.

What is the Prandtl number?

The ratio of momentum diffusivity to thermal diffusivity.

What is emissivity?

The ratio of the emission of a surface to that of a black body at the same temperature.

What is LMTD?

The log mean temperature difference used for the heat exchanger calculation.

What is the purpose of fins?

To increase the surface area and the heat transfer rate.

What is the critical radius of insulation?

The radius at which the heat loss is maximum; insulation below this radius increases the heat loss.

What is the Biot number?

The ratio of the internal conduction resistance to the surface convection resistance.

Which is a better conductor: copper or aluminium?

Copper has a higher thermal conductivity than aluminium.

Is this topic asked in SSC JE?

Yes. It is a regular part of the mechanical paper.

What is Kirchhoff’s law?

Emissivity equals absorptivity at thermal equilibrium.

What is Fick’s law?

The law of diffusion in mass transfer.