Radial Conduction Through a Spherical Wall
Calculate steady heat flow through a concentric spherical shell.
Try this formula
Calculate heat rate; positive radially outward or choose another supported quantity.
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Try this formula
Calculate heat rate; positive radially outward or choose another supported quantity.
Use decimals or scientific notation, such as 1.2e3. Calculations update as you type.
Heat rate; positive radially outward
Enter the known values to see the result.
Conduction is steady, radial, and one-dimensional, with constant positive conductivity. Use a concentric spherical shell with no heat generation and 0 < r_i < r_o. The specified temperatures are wall temperatures; external convection resistances are not included.
Difficulty
Intermediate
Read time
5 min
Prerequisites
Basic algebra, mechanics, and consistent SI units
Source
Teng-Jui Lin — CHEM E 330 — Transport Processes IReference equation: Radial transport in spherical systems. FormuLab wrote new explanations, assumptions, applications, and numerical examples. Integrated source flux over the spherical area 4πr² to obtain heat rate.
Adapted from Teng-Jui Lin, Equation Sheets — Mechanical Engineering Expansion, under CC BY 4.0.
Adapted standard equations and added original lesson text, variables, SI examples, and assumptions. Integrated source flux over the spherical area 4πr² to obtain heat rate.
Definition
The heat-flow area grows with radius squared. Integrating through a spherical wall produces a difference of inverse radii, which differs from both plane-wall and cylindrical-wall conduction.
Variables & Units
| Symbol | Meaning | Units |
|---|---|---|
| Heat rate; positive radially outward | ||
| Thermal conductivity | ||
| Inner-wall absolute temperature | ||
| Outer-wall absolute temperature | ||
| Inner radius | ||
| Outer radius |
Real-World Applications
Spherical insulation
Estimate radial conduction through a spherical layer.
Thermal resistance
Compare the effect of inner and outer radii.
Worked Example
Problem
A spherical shell has k = 0.5 W/(m·K), inner radius 0.1 m, outer radius 0.2 m, and wall temperatures 400 K and 300 K. Find the outward heat rate.
Given
Solution
Final Answer
Contents
Info
Tools
Source & review
Reference equation: Radial transport in spherical systems. FormuLab wrote new explanations, assumptions, applications, and numerical examples. Integrated source flux over the spherical area 4πr² to obtain heat rate.
Adapted from Teng-Jui Lin, Equation Sheets — Mechanical Engineering Expansion, under CC BY 4.0.
Adapted standard equations and added original lesson text, variables, SI examples, and assumptions. Integrated source flux over the spherical area 4πr² to obtain heat rate.
Assumptions
- • Conduction is steady, radial, and one-dimensional, with constant positive conductivity.
- • Use a concentric spherical shell with no heat generation and 0 < r_i < r_o.
- • The specified temperatures are wall temperatures; external convection resistances are not included.
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