Compute radial steady-state heat conduction through a single-layer cylindrical wall (Fourier's law in cylindrical coordinates): radial heat flow rate Q = 2π·k·L·ΔT/ln(r₂/r₁) (W), inner-surface heat flux q_inner = Q/(2π·r₁·L) (W/m²), outer-surface heat flux q_outer = Q/(2π·r₂·L) (W/m²), and cylindrical thermal resistance R = ln(r₂/r₁)/(2π·k·L) (K/W). k is the thermal conductivity (W/(m·K)); L the cylinder length; r₁ the inner radius, r₂ the outer radius (must have r₂ > r₁ > 0); ΔT the temperature difference (K; a °C difference equals a K difference, a °F difference is converted by ×5/9). ΔT may be negative (indicating reverse heat flow), but k, L, r₁ and r₂ must be positive. Length and radius in m/cm/mm.
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Key facts
Category
Math & Numbers
Input types
number, select
Output type
json
Sample coverage
4
API ready
Yes
Overview
Calculate steady-state radial heat conduction through a single-layer cylindrical wall using Fourier's law. Enter thermal conductivity, cylinder length, inner and outer radii, temperature difference, and units to obtain heat flow rate, inner and outer surface heat flux, and cylindrical thermal resistance.
When to use
Estimate radial heat transfer through a hollow cylinder or cylindrical wall.
Evaluate heat loss or transfer in pipe walls and insulation layers.
Compare heat flux and thermal resistance for different materials, dimensions, or temperature differences.
How it works
1Enter thermal conductivity k, cylinder length L, inner radius r₁, outer radius r₂, and temperature difference ΔT.
2Select length and radius units from meters, centimeters, or millimeters, and select K, °C, or °F for the temperature difference.
3The calculator converts inputs to consistent units and applies Q = 2πkL·ΔT/ln(r₂/r₁).
4The JSON result includes Q in W, inner and outer heat flux in W/m², and cylindrical thermal resistance R in K/W.
Use cases
Analyze heat transfer through steel or other cylindrical pipe walls.
Estimate the effect of a low-conductivity insulation layer around a pipe.
Check thermal resistance and surface heat flux for hollow-cylinder designs.
Examples
1. Steel steam pipe heat transfer
Thermal engineer
Background
A thermal engineer is evaluating radial conduction through a 2 m steel pipe wall with an inner radius of 0.05 m and an outer radius of 0.1 m.
Problem
Determine the heat flow rate, heat flux at both cylindrical surfaces, and thermal resistance for a 100 K temperature difference.
How to use
Enter k = 50 W/(m·K), L = 2 m, r₁ = 0.05 m, r₂ = 0.1 m, and ΔT = 100 K.
Length unit: m; radius unit: m; temperature unit: K; decimal places: 4.
Outcome
The result is Q = 90647.2028 W, q_inner = 144269.5041 W/m², q_outer = 72134.752 W/m², and R = 0.0011 K/W.
2. Cylindrical insulation layer
Pipe insulation designer
Background
A designer is checking a 1 m insulation section with inner and outer radii of 5 cm and 10 cm, respectively.
Problem
Calculate heat transfer through the insulation when its conductivity is 0.5 W/(m·K) and the temperature difference is 80 °C.
How to use
Enter k = 0.5 W/(m·K), L = 1 m, r₁ = 5 cm, r₂ = 10 cm, and ΔT = 80 °C.
FAQ
What equation does the calculator use?
It uses Q = 2π·k·L·ΔT/ln(r₂/r₁) for steady-state radial conduction through a single-layer cylindrical wall.
What values must be positive?
Thermal conductivity, length, inner radius, and outer radius must be positive, with r₂ greater than r₁.
Can the temperature difference be negative?
Yes. A negative ΔT indicates that heat flows in the reverse direction.
How are Celsius and Fahrenheit differences converted?
A temperature difference in °C equals the same difference in K. A difference in °F is multiplied by 5/9 to convert it to K.
What results are provided?
The tool returns heat flow rate Q, inner-surface heat flux, outer-surface heat flux, and cylindrical thermal resistance.