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Lesson 4.9 • Competency 4.9Complete Lesson
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Heat Transfer Mechanisms and Newton’s Law of Cooling

Conduction, Fourier’s law dQ/dt = -kA dT/dx, thermal conductivity k, composite rods in series/parallel, convection currents, black-body radiation, Stefan-Boltzmann law E = σeAT⁴, and Newton’s Law of Cooling.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

  • Derive rate of heat flow through uniform and composite conductors in steady state using Fourier’s law.
  • Apply the Stefan-Boltzmann law and Newton’s Law of Cooling to solve practical cooling rate problems.

Core Theory & Concepts

Heat energy spontaneously transfers from regions of higher temperature to regions of lower temperature through three distinct physical mechanisms: conduction (particle contact), convection (bulk fluid movement), and radiation (electromagnetic waves).

Thermal Conduction & Fourier’s Law

In steady state along a conductor of cross-sectional area A and length L with end temperatures θ₁ > θ₂: dQ/dt = k A (θ₁ - θ₂) / L, where k is the thermal conductivity (W m⁻¹ K⁻¹).

\frac{dQ}{dt} = k A \frac{\theta_1 - \theta_2}{L}

Stefan-Boltzmann Radiation & Net Exchange

A surface of area A, emissivity e, and absolute temperature T emits radiation at rate P = σ e A T⁴, where σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. In surroundings at temperature T₀: P_net = σ e A (T⁴ - T₀⁴).

P_{\text{net}} = \sigma e A \left(T^4 - T_0^4\right)

Common Misconceptions

Misconception

Newton’s law of cooling applies accurately at all temperature differences, even several hundred degrees Celsius.

Physical Clarification:Newton’s law of cooling dθ/dt ∝ (θ - θ₀) is a linear approximation of Stefan’s T⁴ law, valid ONLY when temperature excess (θ - θ₀) is small (typically < 30 °C). At high excess, T⁴ radiation dominates.

Syllabus Topic Outline

Competency 4.9
  • 1Mechanism of thermal conduction: free electrons and lattice vibrations (phonons)
  • 2Fourier’s law of heat conduction: dQ/dt = kA(θ₁ - θ₂)/L
  • 3Composite thermal conductors in series and interface temperature
  • 4Thermal radiation: Stefan-Boltzmann law E = σ e A T⁴, net heat exchange
  • 5Newton’s Law of Cooling: dθ/dt = -k(θ - θ₀) for small temperature excess

Practice Problems