Fourier's Law of Conduction
Jean-Baptiste Joseph Fourier published his heat equation in 1822 — one of the founding documents of mathematical physics. His law states that the heat flux through a material is proportional to the temperature gradient and the material's thermal conductivity: The thermal resistance concept — R = L/(kA) for conduction, analogous to electrical resistance R = L/(σA) — is one of the most useful in thermal engineering. It allows complex heat transfer problems involving multiple materials and modes to be solved as simple resistor networks.
Newton's Law of Cooling: Convection
Convection transfers heat between a surface and an adjacent fluid in motion. Newton's cooling law is empirical — it says the heat flux is proportional to the temperature difference between surface and fluid:
Radiation: Heat Transfer Through Empty Space
Electromagnetic radiation transfers heat even through vacuum — the mechanism by which the Sun warms the Earth. The Stefan-Boltzmann law gives the power radiated by a black body:
Thermal Resistance Networks
Complex heat transfer problems — a composite wall with multiple layers, a heat exchanger with fouling, a circuit board cooled by forced convection — can be solved by building a thermal resistance network analogous to an electrical circuit. Resistances in series add; resistances in parallel give a combined resistance via the reciprocal rule. Temperature differences are the "voltages" and heat flows are the "currents."
Extended Surfaces and Fins
When the convective heat transfer coefficient is low (natural convection in air, h ≈ 5–20 W/m²·K), increasing surface area is the most effective way to increase heat transfer. Fins — extended surfaces attached to a base surface — increase the area available for convection. The fin's effectiveness depends on the fin parameter m = √(hP/kA_c), where P is perimeter and A_c is cross-sectional area:
Combined Modes: Real Engineering Problems
Real thermal engineering problems almost always involve multiple heat transfer modes simultaneously. A gas turbine blade is heated by convection from hot combustion gases, cooled internally by convection from cooling air through internal passages, and radiates to the surrounding casing. A building wall loses heat by conduction through each layer, by natural convection on both surfaces, and by radiation between the warm interior surfaces and cool exterior. The thermal resistance network method handles these combined-mode problems systematically by placing radiation and convection resistances in parallel (since both occur at the same surface) and conduction resistances in series. Enter fluid temperatures, flow rates, and geometry. EngForge computes the Log Mean Temperature Difference, NTU-effectiveness, overall heat transfer coefficient U, and required heat exchanger area — the complete thermal design in seconds.