UPSC ESE Mechanical Engineering Heat Transfer Flashcards

24 sample cards from the Heat Transfer deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Define the Reynolds number and its role in convection.

   $$Re=\frac{\rho V L}{\mu}=\frac{VL}{\nu}$$ It is the ratio of inertial to viscous forces and determines whether flow is laminar or turbulent, governing forced-convection heat transfer correlations.

2. Define the Prandtl number and its physical meaning.

   $$Pr=\frac{\mu c_{p}}{k}=\frac{\nu}{\alpha}$$ It is the ratio of momentum diffusivity to thermal diffusivity, linking the velocity and thermal boundary layers. For gases $Pr\approx 0.7$; for liquid metals $Pr\ll 1$; for oils $Pr\gg 1$.

3. For forced convection, the Nusselt number is generally a function of which dimensionless groups?

   $$Nu=f(Re,\ Pr)$$ Typical correlations take the form $Nu=C\,Re^{m}Pr^{n}$, where constants depend on geometry and flow regime.

4. State the Dittus-Boelter equation for turbulent forced convection in tubes.

   $$Nu=0.023\,Re^{0.8}Pr^{n}$$ where $n=0.4$ for heating ($T_s>T_f$) and $n=0.3$ for cooling. Valid for $Re>10{,}000$, $0.6<Pr<160$, and fully developed turbulent flow.

5. For fully developed laminar flow in a circular tube with constant wall heat flux, what is the Nusselt number?

   $$Nu=4.36$$ (constant). For constant wall temperature, $Nu=3.66$. These are independent of $Re$ and $Pr$ in the fully developed laminar region.

6. Define the Grashof number and its physical meaning in natural convection.

   $$Gr=\frac{g\beta(T_{s}-T_{\infty})L^{3}}{\nu^{2}}$$ It is the ratio of buoyancy forces to viscous forces. In natural convection it plays the role $Re$ plays in forced convection. $\beta$ is the coefficient of volumetric thermal expansion.

7. For natural convection, the Nusselt number is a function of which dimensionless groups?

   $$Nu=f(Gr,\ Pr)=f(Ra)$$ where the Rayleigh number $Ra=Gr\cdot Pr$. Correlations take the form $Nu=C(Gr\,Pr)^{n}=C\,Ra^{n}$.

8. Define the Rayleigh number and its significance.

   $$Ra=Gr\cdot Pr=\frac{g\beta(T_{s}-T_{\infty})L^{3}}{\nu\alpha}$$ It governs the onset and intensity of natural convection. Above a critical value, buoyancy-driven flow (and transition to turbulence) occurs.

9. For an ideal gas, what is the coefficient of volumetric thermal expansion $\beta$?

   $$\beta=\frac{1}{T}$$ where $T$ is the absolute temperature in kelvin. This is used in the Grashof and Rayleigh numbers for gases.

10. What dimensionless number determines whether forced or natural convection dominates in a combined (mixed) convection situation?

    The ratio $\frac{Gr}{Re^{2}}$. If $\frac{Gr}{Re^{2}}\ll 1$, forced convection dominates; if $\frac{Gr}{Re^{2}}\gg 1$, natural convection dominates; if $\approx 1$, both must be considered (mixed convection).

11. Define the hydrodynamic boundary layer and the thermal boundary layer.

    The hydrodynamic (velocity) boundary layer is the thin region near a surface where velocity changes from zero (no-slip) to the free-stream value. The thermal boundary layer is the region where temperature changes from the surface value to the free-stream value. Their relative thickness is set by $Pr$.

12. State the Stefan-Boltzmann law for a black body.

    $$E_{b}=\sigma T^{4}$$ The total emissive power of a black body is proportional to the fourth power of its absolute temperature, where $\sigma=5.67\times10^{-8}\ \text{W/m}^2\text{·K}^4$ is the Stefan-Boltzmann constant.

13. What is the value of the Stefan-Boltzmann constant $\sigma$?

    $$\sigma = 5.67\times10^{-8}\ \text{W/m}^{2}\text{·K}^{4}$$

14. Define emissivity $\varepsilon$ of a surface.

    $$\varepsilon=\frac{E}{E_{b}}=\frac{\text{emissive power of the surface}}{\text{emissive power of a black body at the same temperature}}$$ It ranges from 0 to 1; $\varepsilon=1$ for a black body. For a gray body, $\varepsilon$ is constant with wavelength.

15. State Kirchhoff's law of thermal radiation.

    At thermal equilibrium, the emissivity of a surface equals its absorptivity: $$\varepsilon=\alpha$$ A good emitter is also a good absorber at the same temperature and wavelength.

16. For an opaque surface, what is the relation among absorptivity, reflectivity, and transmissivity?

    $$\alpha+\rho+\tau=1$$ For an opaque body $\tau=0$, so $\alpha+\rho=1$. For a black body $\alpha=1$, $\rho=\tau=0.

17. State Wien's displacement law.

    $$\lambda_{\max}T = 2898\ \mu\text{m·K}$$ The wavelength at which black-body spectral emissive power peaks is inversely proportional to absolute temperature. Hotter bodies emit at shorter wavelengths.

18. Write the net radiative heat exchange between two large parallel gray plates (areas equal, infinite extent).

    $$Q_{12}=\frac{\sigma A(T_{1}^{4}-T_{2}^{4})}{\frac{1}{\varepsilon_{1}}+\frac{1}{\varepsilon_{2}}-1}$$

19. Write the net radiation heat exchange between a small gray body (1) completely enclosed by a large surface (2).

    $$Q_{12}=\varepsilon_{1}\sigma A_{1}(T_{1}^{4}-T_{2}^{4})$$ When $A_1\ll A_2$, the enclosure behaves as a black body and only the small body's emissivity matters.

20. Define the radiation shape factor (view factor) $F_{12}$.

    $F_{12}$ is the fraction of radiation leaving surface 1 that is directly intercepted by surface 2. It is a purely geometric quantity depending on the size, shape, orientation, and spacing of the surfaces.

21. State the reciprocity relation for view factors.

    $$A_{1}F_{12}=A_{2}F_{21}$$ This relates the view factors between two surfaces through their areas and is fundamental for radiation network analysis.

22. State the summation (enclosure) rule for view factors.

    For any surface $i$ in an enclosure of $N$ surfaces: $$\sum_{j=1}^{N}F_{ij}=1$$ All radiation leaving surface $i$ must strike some surface of the enclosure (including itself, $F_{ii}$, if the surface is concave).

23. What is the view factor $F_{ii}$ of a flat or convex surface with respect to itself, and why?

    $$F_{ii}=0$$ A flat or convex surface cannot 'see' itself — no radiation leaving it strikes itself directly. For a concave surface, $F_{ii}>0$.

24. In a radiation network, define surface resistance and space (geometric) resistance.

    Surface resistance accounts for a surface's non-black emissivity: $$R_{\text{surface}}=\frac{1-\varepsilon}{\varepsilon A}$$ Space (geometric) resistance accounts for the view factor between surfaces: $$R_{\text{space}}=\frac{1}{A_{1}F_{12}}$$ Heat flows between black-body emissive powers $E_b=\sigma T^4$ through these resistances.

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