Heat Flow Through Cylindrical Walls

Many everyday systems—from hot-water pipes to nuclear fuel rods—feature heat moving radially through cylindrical layers. Fourier’s law for radial conduction in steady state gives the rate of heat transfer as $Q=\frac{2\pi kL\mathrm{\Delta T}}{ln}$. Here $k$ is the thermal conductivity, $L$ the cylinder length, $r{1}_{}$ and $r{2}_{}$ the inner and outer radii, and $\mathrm{\Delta T}$ the temperature difference between inner and outer surfaces. The logarithmic term accounts for the expanding area as heat travels outward. This calculator rearranges the relation to solve for any missing variable, aiding engineering estimates and classroom exercises alike.

Derivation Overview

The derivation begins with Fourier’s law in differential form $q=-kA\frac{\mathrm{dT}}{\mathrm{dr}}$, where $A=2\pi rL$ is the area of a cylindrical shell at radius $r$. Integrating from $r{1}_{}$ to $r{2}_{}$ for a steady radial gradient leads to the logarithmic expression above. Unlike planar conduction, where area remains constant, the cylindrical case requires accounting for the increasing circumference with radius, hence the logarithm of the radius ratio.

Using the Calculator

Enter known values for thermal conductivity, cylinder length, inner and outer radii, temperature difference, and heat transfer rate. Leave exactly one field blank. The script solves algebraically for conductivities, lengths, radii, or temperature differences, and computes heat flow directly when all geometry is specified. If a radius is unknown, exponential functions invert the logarithm: $r{2}_{}=r{1}_{}·exp\frac{2\pi kL\mathrm{\Delta T}}{Q}$. Inputs must remain positive, and $r{2}_{}$ must exceed $r{1}_{}$; the code checks these conditions and warns otherwise.

Sample Values

Consider a copper pipe ($k$ ≈ 385 W/m·K) 1 m long with inner radius 1 cm and outer radius 1.2 cm. If the inner surface is 80 K hotter than the outer, the heat transfer rate is about 1.02 kW. Increasing insulation thickness by doubling $r{2}_{}$ cuts the rate nearly in half, demonstrating the effectiveness of radial insulation.

k (W/m·K)	L (m)	r₁ (cm)	r₂ (cm)	ΔT (K)	Q (W)
385	1	1.0	1.2	80	1020
385	1	1.0	2.0	80	583
50	0.5	2.0	3.0	40	262

Design Considerations

Engineers commonly confront this calculation when sizing pipe insulation, evaluating geothermal wells, or analyzing fuel cladding in reactors. The formula assumes constant material properties and one-dimensional radial conduction. Real systems may involve temperature-dependent conductivities, contact resistances between layers, or additional heat transfer modes such as convection at surfaces. Still, the analytic expression offers an essential baseline, enabling quick comparisons between materials or preliminary assessments before detailed simulations.

Logarithmic Mean Radius

The denominator of the formula, $ln\frac{r}{2_{}}$, effectively introduces a logarithmic mean radius. For thin walls where $r{2}_{}$ is only slightly larger than $r{1}_{}$, the logarithm approximates $\frac{r}{2_{}}r{1}_{}$, recovering the planar formula with area equal to the inner surface. For thick insulation, the logarithm grows, reflecting the expanding area and thus reduced thermal resistance per unit thickness.

Common Pitfalls

A typical mistake is to treat $\mathrm{\Delta T}$ as the difference between fluid temperatures inside and outside the pipe without accounting for convective resistances. The formula only covers conduction through the solid wall; any convective film coefficients must be incorporated separately as thermal resistances in series. Another oversight is neglecting unit consistency. Radii should be entered in meters, and conductivities in watts per meter-kelvin to maintain SI coherence.

Historical Perspective

The logarithmic form for cylindrical conduction emerged in the nineteenth century as engineers sought to quantify heat losses in steam engines and boilers. It remains central to modern heat transfer curricula. With computers now ubiquitous, students might be tempted to rely on finite-element software for every problem, but mastering this analytic solution fosters intuition about how geometry and material properties influence heat flow.

Conclusion

Whether insulating household plumbing, designing industrial process equipment, or studying energy flows in advanced reactors, the cylindrical conduction formula is indispensable. This calculator offers a fast, offline-capable way to explore how geometry, materials, and temperature differences govern radial heat transfer.