Derivation of Heat Equation

PDEs · Axiom Academy

LESSON Derivation of the Heat Equation Derive the fundamental heat equation from physical principles: conservation of energy and Fourier's Law Step 1: Conservation of Energy The foundation of the heat equation is the principle of conservation of energy. For any small element of material, the change in thermal energy must equal the net heat flow into the element plus any internal heat generation. Consider a small element between positions x and x + Δx in a one-dimensional rod. The energy balance states: Heat In: Heat flowing into the element from the left Heat Out: Heat flowing out of the element to the right Heat Stored: Increase in thermal energy of the element The thermal energy stored in the element depends on: ρ (rho): Density of the material (kg/m³) c: Specific heat capacity (J/(kg·K)) u(x,t): Temperature at position x and time t (K) For a small element of length Δx and cross-sectional area A: Step 2: Fourier's Law of Heat Conduction Fourier's Law describes how heat flows through materials. It states that heat naturally flows from hot regions to cold regions, and the rate of heat flow is proportional to the temperature gradient. The heat flux q (heat flow per unit area per unit time) is given by: q: Heat flux (W/m²) - energy per unit area per unit time k: Thermal conductivity (W/(m·K)) - material property ∂u/∂x: Temperature gradient - rate of temperature change with position Negative sign: Heat flows from high to low temperature (opposite to gradient)

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