Conduction and the Heat Equation: From Fourier's Law to Multi-Dimensional Heat Transfer

This video explains how thermal conduction redistributes energy inside an object, from a car rotor to a brick wall. It covers the three modes of heat transfer, with emphasis on conduction, and describes how atoms vibrate and how metals rely on free electrons to move energy. Fourier's law is introduced as a practical definition of heat transfer rate, with a simple one dimensional example through a wall showing how temperature difference and material properties control energy loss. It then generalizes to two and three dimensions, discusses the need to know the temperature field to apply the law, and introduces the heat equation. The talk also highlights material properties like thermal conductivity and the concept of thermal diffusivity used to compare how fast heat spreads.

Overview of Thermal Conduction

Thermal conduction is the process by which heat flows within and between bodies due to temperature differences. At the microscopic level, energy is the random motion of atoms and molecules, and in solids these vibrations propagate through a lattice. The stiffer the bonds and the more orderly the lattice, the more effectively energy is transferred. In metals, free electrons contribute alongside lattice vibrations to conduct heat efficiently.

Conduction is contrasted with convection and radiation, but it is often the dominant mechanism in solids. Gases and non-metallic liquids conduct heat mainly through molecular collisions, which are less effective than electron- or lattice-based transfer in many solids.

Fourier's Law and Heat Flux

The rate of heat transfer, Q, through a defined area is governed by Fourier's law. It relates Q to the temperature gradient, the area, and a material property called thermal conductivity, K. In one dimension the law can be written as Q = -K A (dT/dx). The minus sign ensures the heat flows from hot to cold, yielding a positive transfer rate when the gradient is negative. When the temperature profile is linear, the gradient simplifies to (T1 - T2)/L, making it easy to quantify energy loss across a wall.

One-Dimensional Conduction Through a Wall

Consider a wall of thickness L with hot interior temperature T1 and cold exterior temperature T2, and area A. The heat loss is Q = K A (T1 - T2)/L. An example with T1 = 25°C, T2 = 5°C, a 5 cm thick steel wall, and A = 2 m² yields about 36 kW of heat transfer, illustrating how material properties, geometry, and boundary temperatures control energy loss.

From 1D to Higher Dimensions

Real problems require extending Fourier's law to two and three dimensions. The temperature field T becomes a function of x, y, and z, and heat flow is described by a heat flux vector q = -K ∇T. The gradient operator (∇) and its divergence lead to a more general form of the heat equation, which in vector form involves the Laplacian ∇² T. The heat equation links the rate of change of temperature to the curvature of the temperature field and any internal heat sources.

Material Properties and Diffusivity

Three key material properties appear in the heat equation: thermal conductivity K, density ρ, and specific heat capacity Cp. The product ρ Cp is the volumetric heat capacity, and the ratio K/(ρ Cp) defines the thermal diffusivity α, which measures how quickly temperature changes diffuse through a material. Materials with high K or low Cp have higher α and enable faster heat diffusion.

Generalized Heat Equation and Internal Sources

A complete form of the heat equation includes an internal heat generation term qGen, accounting for sources like electrical heating in cables. The equation can be written as ∂T/∂t = α ∇²T + qGen/(ρ Cp). This generalized form covers transient conduction with internal heating and can be simplified for steady state or reduced dimensionality as needed.

Assorted Analytical and Numerical Approaches

In simple steady-state one-dimensional cases with no internal generation, the equation reduces to a second-order ODE that integrates to a linear temperature distribution. In more complex situations, analytical solutions become challenging and engineers rely on numerical methods such as finite difference or finite element analysis. A practical technique to simplify layered conduction problems is the concept of thermal resistance, which can extend to conduction and convection problems as well as insulation design decisions.

Extensions and Practical Considerations

The video also discusses multidimensional conduction, where heat flows perpendicular to isotherms and not along them, and the conditions under which isothermal lines exist. Real materials vary with temperature and the temperature field can be solved using boundary and initial conditions. In engineering practice, it is common to approximate conductivity as constant over a range of temperatures for many analyses, though this approximation has limits. Materials vary across the spectrum from gases with low conductivity to metals with high conductivity, with diamonds and aerogels as notable examples of extreme conductivity and insulation properties, respectively.