Cylindrical pipe conduction calculator

A cylindrical pipe conduction calculator turns thermal conductivity, axial length, inner and outer radii, and the temperature drop across the wall into the steady radial heat transfer rate predicted by Fourier’s law in cylindrical coordinates. Enter k for the pipe material, L for the length of the pipe section, ri and ro for the inner and outer radii of the wall, and ΔT for the temperature difference between the inner and outer surfaces. The tool returns Q equals two pi k L ΔT over the natural log of ro over ri, the cylindrical thermal resistance, the heat fluxes at both surfaces, a readable equation string, and substitution steps you can copy into a notebook. Homework checks, insulated-pipe estimates, steam-line heat-loss sketches, and coding-kata expected values all need that logarithmic radial model instead of the plane-wall quotient Q equals k A ΔT over L.

What radial Fourier conduction says about a pipe wall

Fourier’s law in cylindrical coordinates says that the radial heat flux is minus k times dT/dr. For steady conduction through a long hollow cylinder with constant conductivity and no generation, energy conservation forces the heat transfer rate Q to be the same at every radius, so the temperature field must vary as the logarithm of r rather than as a straight line in r.

Integrating from the inner surface at ri and Ti to the outer surface at ro and To produces the classic pipe-wall formula Q equals two pi k L times (Ti minus To) divided by the natural logarithm of ro over ri. The same result is often written with a conduction resistance R_cyl equal to ln(ro/ri) over two pi k L, so Q equals ΔT over R_cyl, which makes it easy to place the metal wall in series with inner and outer convection films in a later resistance network.

Thermal conductivity k is a material property: tens of watts per metre-kelvin for many steels, a few hundred for copper, and a fraction of one for common plastics. Length L is the axial length of the pipe section you are analysing, not the wall thickness; the thickness enters only through the two radii. Temperature difference ΔT is the inner-surface temperature minus the outer-surface temperature; a positive ΔT with positive k and L yields heat flowing radially outward.

The logarithm is defined only when both radii are positive and ro is strictly larger than ri. If the radii are equal there is no wall and the formula would divide by zero; if they are inverted the argument of the log is less than one and the sign of Q would flip for the wrong geometric reason.

Practical tips, limits, and related searches

Keep k, L, ri, ro, and ΔT inside the published absolute range so the product two pi k L ΔT and the logarithm cannot overflow into a non-finite heat rate the solver must reject. Use a coherent unit set: watts per metre-kelvin, metres, and kelvin (or Celsius difference) yield watts for Q, kelvin per watt for thermal resistance, and watts per square metre for the surface fluxes. Imperial inputs need the same coherence if you want BTU per hour.

This tool does not convert units, look up material properties from a metal name, add convection or radiation films, treat multi-layer insulation, or solve the transient radial equation. When the wall is thin relative to the radius, ln(ro/ri) is approximately (ro minus ri) over ri and the cylindrical result approaches the plane-wall formula with area equal to two pi ri L and thickness equal to ro minus ri; use that check as a sanity test, not as a substitute for the log form on thick tubes.

Prefer this capability when you already have k, L, ri, ro, and ΔT and only need Q; the sibling plane-wall conduction tool is the right page when the geometry is a flat slab.