September 22, 2026
Fourier's Law, Written Like Ohm's Law
Joseph Fourier's 1822 treatise set down the law that still governs conduction: heat flux is proportional to the temperature gradient, q = −k dT/dx, with k the material's thermal conductivity. For a flat wall of thickness L and area A at steady state, the gradient is constant and the heat flow is simply Q = kA(T₁ − T₂)/L.
Rearranged, that becomes Q = ΔT/R, with R = L/(kA). This is exactly the form of Ohm's law, with temperature difference playing voltage, heat flow playing current, and R a thermal resistance in kelvin per watt. The analogy is not just a memory aid. It means every rule for combining electrical resistors carries straight over to heat flow, and that turns most steady conduction problems into a few lines of arithmetic.
Layers in Series, and Where the Temperature Drop Goes
A furnace wall made of firebrick, then insulation, then a steel casing passes the same heat through each layer in turn, so the resistances add: R_total = L₁/(k₁A) + L₂/(k₂A) + L₃/(k₃A). The heat flow is the overall temperature difference divided by that total. Then, just like voltage across resistors in series, the temperature drop across each layer is proportional to that layer's share of the total resistance.
Conductivities span a huge range. Carbon steel sits around 50 W/m·K, firebrick around 1, and mineral wool or similar insulation around 0.04. For layers of equal thickness, the insulation's resistance is over a thousand times the steel's, so almost the entire temperature drop happens across the insulation and the steel casing sits within a degree or so of the outer surface temperature. Plotted through the wall, the temperature profile is a set of straight lines with the steepest slope in the lowest-conductivity layer.
Convection and Contact Resistances Join the Same Circuit
The surfaces of a wall exchange heat with fluids on each side, and convection fits the same scheme with a resistance of 1/(hA), where h is the heat transfer coefficient. A wall between hot gas and outside air therefore has five resistances in series: gas film, three layers, and air film. When one fluid is a gas, its film resistance is often the largest in the circuit, which is why finned surfaces are put on the gas side.
Real interfaces add one more term. Two solid surfaces pressed together touch only at microscopic high points, with tiny gaps of air between, and that creates a thermal contact resistance at each joint. It is usually small next to insulation but can matter a great deal between metal layers, which is why thermal paste is used between a processor and its heat sink.
Cylinders and the Critical Radius of Insulation
Pipes and tubes need a slightly different resistance because the area grows with radius. For a cylindrical shell of length L the result is R = ln(r₂/r₁)/(2πkL). This has a surprising consequence. Adding insulation to a pipe adds conduction resistance, but it also increases the outer surface area, which lowers the outer convection resistance. Below a certain outer radius, the second effect wins and adding insulation actually increases heat loss.
That threshold is the critical radius of insulation, r_c = k/h for a cylinder and 2k/h for a sphere. With typical values, k = 0.05 W/m·K and h = 10 W/m²·K, the critical radius is about 5 mm. Steam pipes are far larger than that, so insulating them always helps. For thin electrical wires, though, a layer of plastic insulation can increase heat loss, and that is part of why cable insulation helps keep conductors cool.
References
- Fourier, J. (1822). Théorie analytique de la chaleur. Paris: Firmin Didot.
- Bergman, T. L., Lavine, A. S., Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer (7th ed.), Chapter 3. Hoboken, NJ: Wiley.
- Holman, J. P. (2010). Heat Transfer (10th ed.), Chapter 2. New York: McGraw-Hill.
- Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena (2nd ed.), Chapter 10. New York: Wiley.