Thermal Conduction, Convection & Radiation Worked Problems Cheat Sheet

Key Facts

• Conduction through a flat wall is modeled by $\frac{Q}{t}=\frac{kA\Delta T}{L}$, where $k$ is thermal conductivity, $A$ is area, and $L$ is thickness.
• Thermal resistance for conduction is $R_{\mathrm{cond}}=\frac{L}{kA}$, so heat transfer rate can be written as $\dot{Q}=\frac{\Delta T}{R_{\mathrm{cond}}}$.
• For layers in series, total thermal resistance is $R_{\mathrm{total}}=R_1+R_2+R_3+\cdots$ and the same heat rate passes through each layer at steady state.
• Convective heat transfer is estimated by Newton's law of cooling, $\dot{Q}=hA\left(T_s-T_{\infty}\right)$, where $h$ is the convection coefficient.
• Radiation from an object is given by the Stefan-Boltzmann law $P=\varepsilon\sigma A T^4$, where $\sigma=5.67\times10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$.
• Net radiation between an object and large surroundings is $P_{\mathrm{net}}=\varepsilon\sigma A\left(T^4-T_{\mathrm{surr}}^4\right)$, with all temperatures measured in kelvin.
• Temperatures must be converted using $T_{\mathrm{K}}=T_{^\circ\mathrm{C}}+273.15$ before using any $T^4$ radiation formula.
• When conduction, convection, and radiation act together, the total heat transfer rate is often found by adding parallel contributions, such as $\dot{Q}_{\mathrm{total}}=\dot{Q}_{\mathrm{conv}}+\dot{Q}_{\mathrm{rad}}$.

Vocabulary

Conduction

Conduction is heat transfer through direct particle interactions, usually strongest in solids with high thermal conductivity.

Convection

Convection is heat transfer between a surface and a moving fluid such as air or water.

Radiation

Radiation is heat transfer by electromagnetic waves and can occur through empty space.

Thermal Conductivity

Thermal conductivity $k$ measures how easily a material conducts heat, with larger $k$ giving a larger heat transfer rate.

Emissivity

Emissivity $\varepsilon$ is a number from $0$ to $1$ that describes how effectively a surface emits thermal radiation.

Steady State

Steady state means temperatures at each point are constant in time, even though heat continues to flow.