1D Heat Conduction: Fourier's Law and the Idea of Thermal Resistance

Press a hand against an outside wall on a cold morning and you can feel heat leaving your body. It flows from warm to cold, through the wall material, at a rate set by how thick the wall is, how large it is, and what it is made of. The same process keeps a coffee cup warm, decides how much insulation a house needs, and governs how fast a heat sink sheds power from a chip. Strip it down to its simplest form — heat flowing straight through a flat slab — and you have one-dimensional steady conduction, the most useful starting point in all of thermal engineering.

This article explains Fourier's law, introduces the thermal resistance analogy that makes layered walls almost trivial, works a full example, and points out the assumptions that quietly break the simple result.

Why this calculation matters

One-dimensional conduction is the workhorse calculation behind a surprising amount of design. Building envelopes are sized with it: every U-value on an insulation datasheet is a conduction calculation in disguise. Furnace and oven walls, cold-store panels, pipe lagging, and the casing of almost any appliance are checked the same way. Even when the real geometry is two- or three-dimensional, the 1D model gives a fast first estimate that is often within engineering tolerance.

The calculation matters because heat loss costs money and heat gain causes failures. Undersize the insulation on a chilled warehouse and the refrigeration plant runs harder every hour of every year. Underestimate conduction into an electronics enclosure and components run hot and age fast. A reliable 1D conduction number, produced early, is what turns those risks into design decisions rather than surprises.

The core formula

Steady one-dimensional conduction through a flat slab is described by Fourier's law. For a wall of thermal conductivity k, face area A, and thickness L, with a temperature difference dT held across it:

Q = k * A * dT / L

Q is the heat flow rate in watts. Read the formula physically: heat flow rises with a more conductive material, rises with a larger area, rises with a bigger temperature difference, and falls as the wall gets thicker. Conductivity k is the material property that does the sorting — copper is a fast conductor, glass wool is a slow one, and they differ by a factor of thousands.

The formula becomes far more powerful when rewritten in the language of thermal resistance. Define the conductive resistance of the slab as:

R = L / (k * A)        (K/W)

Then Fourier's law takes the same shape as Ohm's law for electricity, with temperature difference playing the role of voltage and heat flow the role of current:

Q = dT / R

This analogy is the reason the resistance form is worth learning. For a wall built from several layers — say brick, insulation, and plasterboard — the layers carry the same heat flow one after another, exactly like resistors in series. Their resistances simply add:

R_total = R_1 + R_2 + R_3 + ...

Compute each layer's R, sum them, and divide the overall temperature difference by the total. A multilayer wall becomes no harder than a single slab.

A worked example

Consider steady one-dimensional conduction through a flat insulating wall: thermal conductivity k = 0.04 W/m.K, face area A = 10 m^2, thickness L = 0.10 m, with a temperature difference dT = 20 K held across it.

Step 1 — apply Fourier's law directly.

Q = k * A * dT / L
Q = 0.04 * 10 * 20 / 0.10
Q = 8 / 0.10 = 80 W

The wall passes 80 watts of heat from the warm side to the cold side.

Step 2 — solve it again through thermal resistance. First the conductive resistance of the slab:

R = L / (k * A)
R = 0.10 / (0.04 * 10)
R = 0.10 / 0.40 = 0.25 K/W

Then the heat flow:

Q = dT / R
Q = 20 / 0.25 = 80 W

Both routes give 80 W, as they must — the resistance form is just Fourier's law rearranged. The value of the second route appears as soon as the wall has more than one layer. If a second insulation layer added another 0.25 K/W, the total resistance would be 0.50 K/W and the heat flow would drop to 20/0.50 = 40 W, halving the loss. Resistances in series simply add, and the arithmetic stays this short no matter how many layers you stack.

Common mistakes

Mixing up resistance in series and in parallel. Layers stacked through the thickness of a wall are in series, so their resistances add. Parallel paths — a window set into a wall, or a metal stud bridging insulation — carry heat side by side, and there the conductances add instead. Treating a thermal bridge as a series layer badly underestimates the loss.

Ignoring the surface films. The wall does not see the room and outdoor air temperatures directly. Thin layers of nearly still air cling to each face and add their own convective resistances. For a well-insulated wall these films are small, but for a bare metal panel they can dominate. A pure conduction calculation between surface temperatures is correct; using air temperatures without surface resistances is not.

Assuming the 1D model when the geometry is curved. Fourier's law in the form Q = k*A*dT/L assumes a constant cross-sectional area. A pipe wall does not have one — the area grows with radius — so a flat-slab calculation overestimates the resistance of insulation around a pipe. Curved geometry needs the cylindrical form.

Forgetting that conductivity is not truly constant. Tabulated k values are quoted at a reference temperature. For large temperature differences, or for insulation that absorbs moisture, the real conductivity drifts and the simple linear result becomes an approximation rather than an exact answer.

Try the interactive NovaSolver calculator

Steady conduction is the long-time limit of a process that is really time-dependent, and watching it settle makes the physics click. The 1D Transient Heat Conduction Simulator on NovaSolver runs a finite-difference simulation of heat conduction through a slab — pick a material, set the initial temperature profile and the left and right boundary conditions, and watch the temperature profile T(x,t) evolve toward its steady shape, with the centre temperature, Fourier number, and diffusivity reported live. It is a direct way to see how a temperature profile relaxes into the linear steady-state gradient this article describes.

Related calculators

• 2D heat conduction — for corners, fins, and openings where heat spreads in two directions and the 1D model no longer holds.
• Cylindrical heat conduction — for pipe walls and lagging, where the area grows with radius and the resistance takes a logarithmic form.
• Transient heat conduction — for warm-up and cool-down problems, where time and thermal diffusivity drive the answer.

The full set lives in the thermal tools hub.

Closing note

One-dimensional conduction is a small calculation that anchors a large field. Keep two ideas close: Fourier's law sets the heat flow from conductivity, area, temperature difference, and thickness, and the resistance form Q = dT/R turns any layered wall into a sum of series resistances. Reach for the resistance analogy whenever a wall has more than one material, watch for parallel paths and surface films, and respect the constant-area assumption. Get those right and most building-envelope and insulation problems become a few lines of arithmetic.