1bHeat Transfer: Conduction, Convection, Radiation
Analyze the three mechanisms of heat transfer, apply Fourier's conduction law, and use the Stefan-Boltzmann law for radiative heat exchange.
Heat transfer governs everything from building insulation and cooking to climate science and spacecraft thermal design. Understanding conduction, convection, and radiation is essential for engineering and environmental physics.
Why does a thermos keep coffee hot for hours, and why does a metal handle feel hotter than a plastic one even at the same temperature?
Lesson Overview
Heat is thermal energy in transfer from a hotter object to a cooler one. There are three mechanisms: Conduction — direct particle-to-particle energy transfer through a material; Convection — heat transfer by the bulk movement of a fluid (liquid or gas); and Radiation — energy transfer by electromagnetic waves that requires no medium. The rate of conductive heat flow is governed by Fourier's Law: Q/t = kA(ΔT/d). Radiation power follows the Stefan-Boltzmann Law: P = εσAT⁴, where σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
Key Equations
Worked Examples
A glass window (k = 0.96 W/m·K) has area A = 2 m², thickness d = 0.005 m, and temperature difference ΔT = 20 °C. Find the rate of heat conduction through the window.
A blackbody (ε = 1) sphere has surface area A = 0.04 m² and temperature T = 500 K. Find the power radiated.
Compare the conduction rates of copper (k = 385 W/m·K) and wood (k = 0.15 W/m·K) for identical slabs (A = 0.01 m², d = 0.02 m, ΔT = 30 °C).
A radiator (ε = 0.9, A = 1.5 m²) is at 350 K. The surrounding environment is at 300 K. Find the net power radiated.
Explain why a convection current forms when a pot of water is heated from below.
Guided Problems
A steel rod (k = 50 W/m·K, A = 0.002 m², d = 0.5 m) has one end at 200 °C and the other at 20 °C. Find the rate of heat conduction.
Hint: Use Q/t = kA(ΔT/d). Calculate ΔT = 200 − 20 = 180 °C, then substitute all values.
A perfect blackbody (ε = 1) has area A = 0.1 m² and is at T = 400 K. Calculate the power it radiates.
Hint: Use P = εσAT⁴. Compute T⁴ = (400)⁴ = 2.56 × 10¹⁰ K⁴, then multiply by σ = 5.67 × 10⁻⁸ W/m²K⁴.
Identify the primary mode of heat transfer in each scenario: (a) heat moving through a metal spoon in hot soup, (b) warm air rising from a heater, (c) heat from the Sun reaching Earth.
Hint: Conduction requires direct contact through a solid. Convection involves fluid movement. Radiation travels through vacuum as electromagnetic waves.
A wall has thermal conductivity k = 0.04 W/m·K, area A = 10 m², and thickness d = 0.1 m. The inside is at 20 °C and outside at −10 °C. Find the heat loss rate.
Hint: ΔT = 20 − (−10) = 30 °C. Use Q/t = kA(ΔT/d).
If the temperature of a radiating object doubles (in Kelvin), by what factor does the radiated power increase?
Hint: Use P = εσAT⁴. If T → 2T, then P → ε σ A (2T)⁴ = 16 εσAT⁴. The power increases by a factor of 16.
Key Vocabulary
Conduction
Heat transfer through direct particle-to-particle collisions within a material, without bulk movement of matter. Rate depends on thermal conductivity k, area A, temperature gradient ΔT/d.
Example: A metal spoon in hot coffee conducts heat to your hand. Metals have high k; insulators like wood have low k.
Convection
Heat transfer by the bulk movement of a fluid (liquid or gas). Warmer, less-dense fluid rises; cooler, denser fluid sinks, creating convection currents.
Example: Ocean currents and atmospheric circulation are large-scale convection systems driven by temperature differences.
Radiation
Heat transfer by electromagnetic waves (infrared radiation). Requires no medium — it can travel through a vacuum. Power follows the Stefan-Boltzmann Law: P = εσAT⁴.
Example: The Sun heats Earth through 150 million km of vacuum entirely by radiation.
Thermal Conductivity (k)
A material property measuring how readily it conducts heat [W/(m·K)]. High k = good conductor (metals); low k = good insulator (wood, air, foam).
Example: Copper k ≈ 385 W/m·K; air k ≈ 0.025 W/m·K — air is ~15,000× less conductive than copper.
Emissivity (ε)
A dimensionless factor (0 to 1) describing how efficiently a surface emits radiation compared to a perfect blackbody (ε = 1). Dark, rough surfaces have ε near 1; shiny surfaces have ε near 0.
Example: A blackbody (ε = 1) radiates maximally. A polished silver surface (ε ≈ 0.02) radiates very little — useful in thermos flasks.
Stefan-Boltzmann Law
The power radiated by an object: P = εσAT⁴, where σ = 5.67 × 10⁻⁸ W/(m²·K⁴). Power depends on the fourth power of absolute temperature.
Example: Doubling the Kelvin temperature of a radiator increases its radiated power by 2⁴ = 16 times.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
Which mode of heat transfer can occur through a vacuum?
A glass pane (k = 0.96 W/m·K, A = 1 m², d = 0.004 m) has ΔT = 10 °C. What is the conduction rate?
If the absolute temperature of a blackbody doubles, its radiated power changes by a factor of:
Which of the following is the best thermal insulator?
Warm air rising near a heater and cool air sinking is an example of:
Independent Practice
A brick wall (k = 0.72 W/m·K, A = 15 m², d = 0.20 m) separates inside (22 °C) from outside (2 °C). Calculate the rate of heat loss through the wall.
A human body (ε = 0.97, A = 1.8 m², T = 307 K) radiates heat. The room is at 293 K. Find the net power radiated by the body.
Explain why a thermos flask uses a double-walled silvered vacuum container to minimize heat transfer. Address all three modes of heat transfer.
Two rods of equal dimensions are connected in series. Rod 1 has k₁ = 200 W/m·K; Rod 2 has k₂ = 50 W/m·K. Which rod has the greater temperature drop across it? Explain using the concept of thermal resistance.
★ A solar panel (ε = 0.85, A = 2 m²) reaches a steady-state temperature T when it absorbs 800 W/m² of solar radiation and radiates heat to the environment at 300 K. Set up the energy balance equation P_absorbed = P_radiated and solve for T.
ChallengeCommon Mistakes
Saying metals feel colder because they are at a lower temperature
Metals feel colder because they conduct heat away from your hand faster (high k), not because they are at a lower temperature — both are at room temperature.
Using Celsius temperature in the Stefan-Boltzmann Law
The Stefan-Boltzmann Law requires absolute temperature in Kelvin: P = εσAT⁴. Using Celsius gives completely wrong answers.
Thinking convection occurs in solids
Convection requires bulk fluid movement. In solids, only conduction (and radiation from the surface) can transfer heat.
Assuming a shiny surface is a poor emitter but a good absorber
By Kirchhoff's Law, good absorbers are also good emitters (high ε), and poor absorbers are poor emitters (low ε). A shiny surface both reflects and emits poorly.
Math Tips
Conduction: Q/t = kA(ΔT/d). Higher k, larger area, larger ΔT, or thinner material → faster heat flow.
Stefan-Boltzmann: P = εσAT⁴. ALWAYS use Kelvin. Power scales as T⁴ — doubling T multiplies power by 16.
Net radiation: P_net = εσA(T_object⁴ − T_environment⁴). An object both emits and absorbs radiation.
Thermal resistance R = d/(kA). Higher R = better insulator. For layers in series, total R = R₁ + R₂ + …