# Cylinder Radial Heat Conduction (Q=2πkL·ΔT/ln(r₂/r₁))

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.

> Canonical page: https://elysiatools.com/en/tools/cylinder-radial-conduction

- **Category:** Math & Numbers

- **Keywords:** cylinder conduction, radial conduction, cylindrical wall, Fourier's law, thermal conductivity, heat flow rate, heat flux, thermal resistance, steady state, radial heat transfer, pipe insulation, ln(r2/r1), hollow cylinder

## 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.

## Inputs

- **Thermal Conductivity k (W/(m·K))** (number): Thermal conductivity k of the cylinder-wall material in W/(m·K).
- **Cylinder Length L** (number): Cylinder length L, in the selected Length Unit.
- **Length Unit** (select)
- **Inner Radius r₁** (number): Inner radius r₁ of the cylinder wall, in the selected Radius Unit.
- **Outer Radius r₂** (number): Outer radius r₂ of the cylinder wall, in the selected Radius Unit.
- **Radius Unit** (select)
- **Temperature Difference ΔT** (number): Temperature difference ΔT across the cylinder wall, in the selected Temperature Unit. May be negative to indicate the reverse heat-flow direction.
- **Temperature Unit** (select): Unit of the temperature difference. This is a difference, not an absolute temperature: Δ°C = ΔK and Δ°F ×5/9 = ΔK.
- **Decimal Places** (number)

## 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

- Enter thermal conductivity k, cylinder length L, inner radius r₁, outer radius r₂, and temperature difference ΔT.
- Select length and radius units from meters, centimeters, or millimeters, and select K, °C, or °F for the temperature difference.
- The calculator converts inputs to consistent units and applies Q = 2πkL·ΔT/ln(r₂/r₁).
- The 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.

## Frequently asked questions

### 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.

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