UPSC ESE Mechanical Engineering Heat Transfer Syllabus

Heat Transfer syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Heat Transfer in UPSC ESE Mechanical Engineering, not a summary of it.

1. 1

   Conduction

   3 topics

   • Fourier's Law
   • Steady State Conduction
   • Transient Conduction
2. 2

   Convection

   3 topics

   • Newton's Law of Cooling
   • Forced Convection
   • Natural Convection
3. 3

   Radiation

   3 topics

   • Stefan-Boltzmann Law
   • Radiative Heat Exchange
   • View Factors

Heat Transfer flashcards for UPSC ESE Mechanical Engineering

19 of 51 cards from the Heat Transfer deck — real questions with worked answers.

1. State Fourier's law of heat conduction in one dimension and define each term.

   $$Q = -kA\frac{dT}{dx}$$ where $Q$ is the heat conduction rate (W), $k$ is thermal conductivity (W/m·K), $A$ is the cross-sectional area normal to heat flow ($\text{m}^2$), and $\frac{dT}{dx}$ is the temperature gradient. The negative sign indicates heat flows in the direction of decreasing temperature.

2. What is the physical significance of the negative sign in Fourier's law $Q = -kA\frac{dT}{dx}$?

   It ensures $Q$ is positive in the direction of decreasing temperature, consistent with the second law of thermodynamics — heat flows spontaneously from a hot region to a cold region (down the temperature gradient).

3. Define thermal conductivity $k$ and give its SI units.

   Thermal conductivity is a material property representing its ability to conduct heat — the rate of heat transfer per unit area per unit temperature gradient. Its SI unit is $\text{W/m·K}$ (or $\text{W/m·°C}$).

4. What is the general three-dimensional heat conduction equation (Fourier-Biot) with internal heat generation?

   $$\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}}+\frac{\dot{q}}{k}=\frac{1}{\alpha}\frac{\partial T}{\partial t}$$ where $\dot{q}$ is the volumetric heat generation rate and $\alpha$ is thermal diffusivity.

5. Define thermal diffusivity $\alpha$ and give its formula and units.

   $$\alpha = \frac{k}{\rho c_{p}}$$ It measures how quickly heat diffuses through a material relative to its heat storage capacity. Units: $\text{m}^2/\text{s}$. A high $\alpha$ means heat propagates rapidly.

6. For steady-state 1-D conduction with no heat generation through a plane wall, what is the temperature distribution?

   Linear. The governing equation reduces to $\frac{d^{2}T}{dx^{2}}=0$, giving $T(x)=C_{1}x+C_{2}$, a straight-line temperature profile between the two surface temperatures.

7. Write the conduction thermal resistance for a plane wall of thickness $L$, area $A$, conductivity $k$.

   $$R_{\text{cond}} = \frac{L}{kA}$$ with units $\text{K/W}$. The heat rate is $Q = \frac{\Delta T}{R_{\text{cond}}}$.

8. Write the conduction thermal resistance for a hollow cylinder (length $L$, inner radius $r_1$, outer radius $r_2$).

   $$R_{\text{cyl}} = \frac{\ln\left(\frac{r_{2}}{r_{1}}\right)}{2\pi k L}$$

9. Write the conduction thermal resistance for a hollow sphere (inner radius $r_1$, outer radius $r_2$).

   $$R_{\text{sph}} = \frac{r_{2}-r_{1}}{4\pi k\, r_{1} r_{2}}$$

10. In the thermal resistance (electrical analogy) for heat conduction, what corresponds to voltage and to current?

    Temperature difference $\Delta T$ is analogous to voltage (potential), and heat transfer rate $Q$ is analogous to electric current. Thermal resistance $R$ is analogous to electrical resistance, so $Q=\frac{\Delta T}{R}$.

11. For composite walls in series, how do the thermal resistances combine?

    They add directly: $R_{\text{total}}=R_{1}+R_{2}+\cdots+R_{n}$, and $Q=\frac{\Delta T_{\text{overall}}}{R_{\text{total}}}$ is the same through each layer.

12. Define the critical radius of insulation for a cylinder and give its formula.

    The outer insulation radius at which heat loss is maximum (adding insulation up to this radius increases heat loss). For a cylinder: $$r_{c}=\frac{k}{h}$$ where $k$ is insulation conductivity and $h$ is the outer convective coefficient.

13. What is the critical radius of insulation for a sphere?

    $$r_{c}=\frac{2k}{h}$$ where $k$ is the insulation thermal conductivity and $h$ is the outside convective heat transfer coefficient.

14. Define the overall heat transfer coefficient $U$ and how it relates to total resistance.

    $U$ combines all conduction and convection resistances into a single coefficient: $$Q = U A\,\Delta T_{\text{overall}}, \qquad \frac{1}{UA}=R_{\text{total}}=\sum R_{i}$$ Units of $U$: $\text{W/m}^2\text{·K}$.

15. What is a fin (extended surface) and why is it used?

    A fin is an extended surface attached to a body to increase the surface area exposed to a fluid, thereby enhancing convective heat transfer. Used when $h$ is low (e.g., gas-side cooling) to dissipate more heat for a given temperature difference.

16. Define fin efficiency $\eta_{f}$.

    $$\eta_{f}=\frac{\text{actual heat transferred by fin}}{\text{heat that would transfer if entire fin were at base temperature}}$$ It accounts for the temperature drop along the fin due to its finite conductivity.

17. For a long (infinitely long) fin of uniform cross-section, what is the temperature distribution and fin parameter $m$?

    $$\frac{\theta}{\theta_{b}}=e^{-mx}, \qquad m=\sqrt{\frac{hP}{kA_{c}}}$$ where $\theta=T-T_{\infty}$, $P$ is perimeter, $A_c$ is cross-sectional area, $\theta_b$ is base excess temperature.

18. Define the Biot number and its physical meaning.

    $$Bi=\frac{hL_{c}}{k}$$ It is the ratio of internal conduction resistance within a body to the external convective resistance at its surface. Small $Bi$ means uniform internal temperature.

19. What is the lumped capacitance method and under what condition is it valid?

    It assumes a body has uniform (spatially constant) temperature during transient cooling/heating, treating it as a single 'lump.' Valid when $Bi=\frac{hL_{c}}{k}<0.1$, where $L_{c}=V/A_{s}$.

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