Heat Transfer: Conduction, Convection, and Radiation

Conduction

Fourier's Law: $\dot{Q} = -kA\frac{dT}{dx}$

For a plane wall (steady state, 1D): $\dot{Q} = \frac{kA(T_1 - T_2)}{L}$

Thermal Resistance (conduction through plane wall): $R_{cond} = \frac{L}{kA}$

Composite Walls (Series Resistance)

For walls in series: $\dot{Q} = \frac{T_1 - T_n}{\sum R_i} = \frac{\Delta T_{overall}}{R_{total}}$

Cylindrical Wall (Radial Conduction)

$\dot{Q} = \frac{2\pi kL(T_1 - T_2)}{\ln(r_2/r_1)}$

$R_{cyl} = \frac{\ln(r_2/r_1)}{2\pi kL}$

Thermal Conductivity Values

Convection

Newton's Law of Cooling: $\dot{Q} = hA(T_s - T_\infty)$

Thermal Resistance (convection): $R_{conv} = \frac{1}{hA}$

Convection Coefficient (h) Ranges

Dimensionless Numbers for Convection

Radiation

Stefan-Boltzmann Law: $\dot{Q} = \epsilon \sigma A (T_s^4 - T_{surr}^4)$

where:

• ε = emissivity (0 to 1; 1 for a blackbody)
• σ = Stefan-Boltzmann constant = 5.67 × 10⁻⁸ W/(m²·K⁴)
• T in Kelvin (absolute temperature!)

Radiation between two surfaces: $\dot{Q}_{1 \to 2} = \frac{\sigma(T_1^4 - T_2^4)}{\frac{1-\epsilon_1}{\epsilon_1 A_1} + \frac{1}{A_1 F_{12}} + \frac{1-\epsilon_2}{\epsilon_2 A_2}}$

where F₁₂ is the view factor (fraction of radiation leaving surface 1 that reaches surface 2).

Key Radiation Properties

• Emissivity (ε): Ratio of emitted radiation to blackbody radiation
• Absorptivity (α): Fraction of incident radiation absorbed
• Reflectivity (ρ): Fraction reflected
• Transmissivity (τ): Fraction transmitted
• Kirchhoff's Law: α = ε at thermal equilibrium

Overall Heat Transfer Coefficient (U)

For a composite wall with convection on both sides:

$\frac{1}{UA} = \frac{1}{h_1 A_1} + \frac{L}{kA} + \frac{1}{h_2 A_2}$

$\dot{Q} = UA \cdot \Delta T$

Heat Exchangers — LMTD Method

$\dot{Q} = UA \cdot LMTD$

Log Mean Temperature Difference: $LMTD = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}$

Counterflow heat exchangers are more effective than parallel flow for the same surface area.