Heat Transfer — Convection and Radiation
Convection and radiation are the two modes of heat transfer involving energy transport by motion or electromagnetic waves. In GATE, convection problems test your ability to apply dimensional analysis and Nusselt number correlations, while radiation requires understanding of black body radiation laws and view factor algebra. Together these topics contribute 4–7 marks annually.
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
Newton’s Law of Cooling $$q = hA(T_s - T_\infty)$$
Where $h$ = convective heat transfer coefficient, $A$ = surface area, $T_s$ = surface temperature, $T_\infty$ = fluid temperature.
Dimensional Analysis — Key Correlations:
| Regime | Correlation | Application |
|---|---|---|
| Laminar (internal) | $Nu_D = 0.664 Re_D^{1/3} Pr^{1/3}$ | Entry length |
| Turbulent (internal) | $Nu_D = 0.023 Re_D^{0.8} Pr^{0.4}$ | Fully developed |
| Laminar (external) | $Nu_L = 0.664 Re_L^{1/3} Pr^{1/3}$ | Flat plate |
| Natural convection | $Nu_L = 0.59 Ra_L^{1/4}$ (laminar) | Vertical plates |
Black Body Radiation Laws:
- Stefan-Boltzmann: $E = \sigma T^4$ (W/m²)
- Wien’s Law: $\lambda_{max}T = 2898$ μm·K
- Emissivity $\varepsilon$: $E = \varepsilon \sigma T^4$
Kirchhoff’s Law: $\varepsilon = \alpha$ (emissivity equals absorptivity at thermal equilibrium)
⚡ Exam Tip: Always convert $T$ to Kelvin in radiation calculations. The $T^4$ dependence makes Kelvin vs Celsius errors devastating.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Forced Convection — External Flow
Governing Parameters
- Reynolds Number $Re_L = \rho UL/\mu = UL/\nu$ — Ratio of inertial to viscous forces
- Prandtl Number $Pr = \mu C_p/k = \nu/\alpha$ — Ratio of momentum to thermal diffusivity
- Nusselt Number $Nu_L = hL/k$ — Ratio of convective to conductive heat transfer
Flat Plate — Laminar and Turbulent
Local Nusselt number:
- Laminar ($Re_x < 5 \times 10^5$): $Nu_x = 0.332 Re_x^{1/2} Pr^{1/3}$
- Turbulent ($Re_x > 5 \times 10^5$): $Nu_x = 0.0296 Re_x^{0.8} Pr^{0.33}$
Average Nusselt number (entire plate): $$Nu_L = (0.037 Re_L^{0.8} - 850) Pr^{1/3}$$
Flat Plate — Combined Laminar-Turbulent
Use critical Reynolds $Re_c = 5 \times 10^5$: $$Nu_L = 0.037(Re_L^{0.8} - 18700) Pr^{1/3} + 0.664 Re_L^{0.5} Pr^{1/3}$$
Forced Convection — Internal Flow
Entrance Effects
Thermal entrance length: $L_t \approx 0.05 Re_D Pr \cdot D$ (laminar)
Hydrodynamic entrance: $L_h \approx 0.05 Re_D \cdot D$
Fully Developed Laminar Flow (Circular Pipe)
| Boundary Condition | $Nu_D$ |
|---|---|
| Constant wall temperature (CFT) | 3.66 |
| Constant heat flux (CHF) | 4.36 |
Turbulent Flow in Pipes (Dittus-Boelter)
Heating: $Nu_D = 0.023 Re_D^{0.8} Pr^{0.4}$ Cooling: $Nu_D = 0.023 Re_D^{0.8} Pr^{0.3}$
⚠️ Limitations: $0.7 < Pr < 160$, $Re_D > 10,000$, $L/D > 10$.
Sieder-Tate Correlation (More General)
$$Nu_D = 0.027 Re_D^{0.8} Pr^{1/3} \left(\frac{\mu}{\mu_s}\right)^{0.14}$$
Where $\mu_s$ = viscosity at surface temperature.
Natural (Free) Convection
Driven by buoyancy: $\Delta \rho/\rho = \Delta T/T$ causes fluid motion.
Rayleigh Number: $Ra_L = Gr_L \cdot Pr = \frac{g\beta\Delta T L^3}{\nu^2} \cdot Pr$
Grashof Number: $Gr_L = \frac{g\beta\Delta T L^3}{\nu^2}$ — Ratio of buoyancy to viscous forces
| Geometry | $Nu_L$ Correlation |
|---|---|
| Vertical plate (laminar) | $Nu_L = 0.59 Ra_L^{1/4}$ for $10^4 < Ra < 10^9$ |
| Vertical plate (turbulent) | $Nu_L = 0.13 Ra_L^{1/3}$ for $10^9 < Ra < 10^{13}$ |
| Horizontal plate (heated top) | $Nu_L = 0.54 Ra_L^{1/4}$ for $10^4 < Ra < 10^7$ |
| Horizontal cylinder | $Nu_D = [0.6 + \frac{0.387 Ra_D^{1/6}}{[1+(0.559/Pr)^{9/16}]^{8/27}}]^2$ |
Radiation — Black Body and Real Surfaces
Black Body Characteristics
A black body absorbs all incident radiation and emits maximum energy at any temperature.
Spectral distribution — Planck’s Law: $$E_{\lambda,b}(\lambda,T) = \frac{2\pi hc^2}{\lambda^5} \cdot \frac{1}{e^{hc/(\lambda kT)} - 1}$$
This peaks at $\lambda_{max}T = 2898$ μm·K (Wien’s displacement law).
Stefan-Boltzmann Law
$$E_b = \sigma T^4$$
Where $\sigma = 5.67 \times 10^{-8}$ W/m²·K⁴
Gray surface: $E = \varepsilon \sigma T^4$
Emissivity and Absorptivity
| Surface | $\varepsilon$ |
|---|---|
| Black body | 1.0 |
| White polished | 0.02–0.05 |
| Black paint | 0.9–0.95 |
| Concrete | 0.85–0.95 |
| Human skin | 0.95 |
Kirchhoff’s Law of Radiation: $$\varepsilon_\lambda = \alpha_\lambda \quad \text{(at each wavelength and temperature)}$$
At thermal equilibrium, good emitters is good absorber.
Radiation Heat Transfer Between Surfaces
Between two black surfaces: $$q_{12} = A_1 F_{12} \sigma (T_1^4 - T_2^4)$$
Where $F_{12}$ = view factor from surface 1 to 2.
For gray surfaces: $$q_{12} = \frac{\sigma(T_1^4 - T_2^4)}{\frac{1-\varepsilon_1}{\varepsilon_1 A_1} + \frac{1}{A_1 F_{12}} + \frac{1-\varepsilon_2}{\varepsilon_2 A_2}}$$
⚡ Common Mistake: For large enclosures ($A_2 \gg A_1$), radiative resistance simplifies to $(1-\varepsilon_1)/\varepsilon_1$.
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Thermal Boundary Layer — Detailed Theory
Velocity Boundary Layer
- δ: Thickness where $u = 0.99U_\infty$
- δ_c: Thermal boundary layer thickness (where $T = T_\infty$ at wall)
- Prandtl analogy: When $Pr \approx 1$, momentum and thermal boundary layers are similar
Energy Equation for Forced Convection
For steady, fully developed laminar flow: $$\frac{d}{dx}\left[\frac{(T_s - T)}{T_s - T_m}\right] = \text{const}$$
Colburn Analogy: $St \cdot Pr^{2/3} = f/8$ where $St = h/(\rho u C_p)$ (Stanton number).
Combined Convection and Radiation
In many engineering applications, both modes occur simultaneously:
$$q_{total} = h_{conv}A(T_s - T_\infty) + \varepsilon\sigma A(T_s^4 - T_{surr}^4)$$
⚠️ Note: $T_{surr}$ is the surrounding surface temperature, not ambient air temperature.
View Factor Algebra
View Factor Definitions
- $F_{12}$: Fraction of radiation leaving surface 1 that strikes surface 2
- Reciprocity: $A_1 F_{12} = A_2 F_{21}$
- Summation: $\sum_{j=1}^{N} F_{ij} = 1$
View Factor for Special Cases
Infinite parallel plates: $$F_{12} = 1 \quad \text{(large surfaces facing each other)}$$
Small object in large enclosure: $$F_{12} \approx 1 \quad \text{(object “sees” mostly the enclosure)}$$
Disk to coaxial parallel disk: $$F_{12} = \frac{1}{2}\left[1 - \frac{(D/2)^2 - (H/2)^2 + S^2}{S^2}\right]$$ Where $S = \sqrt{H^2 + (D/2)^2}$
Radiation Shielding
For two parallel infinite plates with $n$ radiation shields of emissivity $\varepsilon_s$: $$q_{shielded} = \frac{\sigma(T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} + 2(n)\left(\frac{2}{\varepsilon_s} - 1\right)}$$
Each shield approximately halves the radiative heat transfer.
Boiling and Condensation
Nucleate Boiling (Pool Boiling — Rohsenow Correlation)
$$q = \mu_{lg} h_{fg} \left[\frac{g(\rho_l - \rho_v)}{\sigma}\right]^{1/2} \left[\frac{C_{p,l}\Delta T}{C_s h_{fg} Pr_l^n}\right]^3$$
Critical Heat Flux (CHF)
$$q_{CHF,max} \approx 0.15 h_{fg} \rho_v^{1/2} [\sigma g(\rho_l - \rho_v)]^{1/4}$$
Condensation — Nusselt Theory
For film condensation on vertical plate: $$h = 0.943 \left[\frac{\rho_l(\rho_l - \rho_v)g h_{fg}k_l^3}{\mu_l L(T_{sat} - T_s)}\right]^{1/4}$$
Example Problem
GATE 2023 (ME) Style: Air at 300 K flows over a flat plate at 350 K with $h = 50$ W/m²·K and $\varepsilon = 0.8$. Surface emissivity is 0.8. Find the total heat transfer per unit width for a plate of length 1 m.
Solution approach:
- Convection: $q_{conv} = hA\Delta T = 50 \times 1 \times 50 = 2500$ W/m
- Radiation: Treat as gray surface in large enclosure at $T_{surr} \approx T_\infty = 300$ K $q_{rad} = \varepsilon\sigma A(T_s^4 - T_{surr}^4)$ $= 0.8 \times 5.67\times10^{-8} \times 1 \times (350^4 - 300^4)$ $= 0.8 \times 5.67\times10^{-8} \times 1 \times (1.5\times10^{10} - 8.1\times10^9)$ $\approx 305$ W/m
- Total: $q_{total} \approx 2805$ W/m
⚡ GATE Tip: At moderate temperatures, convection dominates. Radiation becomes significant above ~500 K or when convection coefficient is very low (natural convection).
Dimensionless Number Summary
| Number | Formula | Physical Meaning |
|---|---|---|
| Reynolds $Re$ | $\rho UL/\mu$ | Inertial / Viscous |
| Prandtl $Pr$ | $\mu C_p/k$ | Momentum / Thermal diffusivity |
| Nusselt $Nu$ | $hL/k$ | Convective / Conductive |
| Grashof $Gr$ | $g\beta\Delta TL^3/\nu^2$ | Buoyancy / Viscous |
| Rayleigh $Ra$ | $Gr \cdot Pr$ | Combined free convection |
| Stanton $St$ | $h/(\rho u C_p)$ | Heat transfer / Thermal capacity |
| Biot $Bi$ | $hL/k$ | Internal / External resistance |
Previous Year GATE Pattern
| Year | Topic Focus | Marks |
|---|---|---|
| 2023 | Convection correlation, radiation | 4 |
| 2022 | Natural convection, Nusselt number | 3 |
| 2021 | View factor, radiation shield | 2 |
| 2020 | Boiling, condensation | 3 |
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