Library / Heat transfer and thermal systems / HTS-08
Heat transfer and thermal systems
Radial temperature in a heat-generating sphere
Temperature in a solid sphere of radius R with uniform volumetric heating q‴ and a fixed surface temperature Ts.
- Ref
- HTS-08
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- T [K]
A solid sphere with uniform heating and a fixed surface temperature has the profile T(r) = Ts + q‴(R² − r²)/(6k), with a factor 6 where the slab has 2 and the cylinder 4. The centre has zero radial gradient. The surface flux is q‴R/3 and the total heat is q‴ times the volume of the sphere.
Temperature in a solid sphere of radius R with uniform volumetric heating q‴ and a fixed surface temperature Ts. Constant conductivity; the radial gradient is zero at the centre.
Calculator
- Centre temperature, Ts + q‴R²/(6k)322.5 K
- Total heat generated, q‴ × 4πR³/367.86 W
- Heat flux at the surface, q‴R/36,000 W/m²
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At r = 0.01 m the temperature is 320 K, and the centre reaches 322.5 K.
- The sphere generates 67.86 W in total, so for the same R, k and q‴ a sphere runs cooler at its centre than a cylinder (factor 6 against 4).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| q‴ Volumetric heat generation | 1,000 to 1e9 W/m³ | 300 | 33,630 | 33,330 | +10% | +0.625 % |
| R Sphere radius | 0.01 to 0.3 m | 300 | 2,547 | 2,247 | +10% | +1.48 % |
| k Thermal conductivity | 0.1 to 500 W/(m K) | 300.2 | 1,100 | 799.8 | +10% | −0.568 % |
| Ts Surface temperature | 250 to 1,000 K | 270 | 1,020 | 750 | +10% | +9.38 % |
| r Radius of interest | 0 to 0.01 m | 320 | 322.5 | 2.5 | +10% | −0.164 % |
Sweep of Radius of interest
| Radius of interest (m) | Temperature at r (K) |
|---|---|
| 0 | 322.5 |
| 0.00125 | 322.5 |
| 0.0025 | 322.3 |
| 0.00375 | 322.1 |
| 0.005 | 321.9 |
| 0.00625 | 321.5 |
| 0.0075 | 321.1 |
| 0.00875 | 320.6 |
| 0.01 | 320 |
Sweep one variable
Variables and how each one contributes
Temperature at r
What the equation gives you. Everything else feeds it.
Surface temperature
The surface temperature, in K. Raises the temperature everywhere by the same amount. Raising it by 10 % (to 330) moves the result by +9.38 %, from 320 to 350 K.
Volumetric heat generation
The uniform volumetric heat generation, in W/m³. Raises the temperature rise above the surface in direct proportion. Raising it by 10 % (to 660,000) moves the result by +0.625 %, from 320 to 322 K.
Sphere radius
The sphere radius, in m. A larger radius raises the temperature rise with the square of R. Raising it by 10 % (to 0.033) moves the result by +1.48 %, from 320 to 324.7 K.
Radius of interest
The radius at which the temperature is wanted, in m. Moving outwards lowers the temperature towards Ts. Raising it by 10 % (to 0.011) moves the result by −0.164 %, from 320 to 319.5 K.
Thermal conductivity
The thermal conductivity, in W/(m K). A higher conductivity lowers the temperature rise in inverse proportion. Raising it by 10 % (to 4.4) moves the result by −0.568 %, from 320 to 318.2 K.
Worked example with the values above
- Geometry term: R² − r² = 0.03² − 0.01² = 8×10−4 m².
- Rise above the surface: 600,000 × 8×10−4 / (6 × 4) = 20 K.
- T = 300 + 20 = 320 K.
Assumptions and limits
- Uniform volumetric generation and constant conductivity in a steady state.
- The stated boundary condition holds exactly, for example a fixed surface temperature or an insulated inner surface.
Common mistakes
- Mixing the factor 2, 4 or 6 between the slab, cylinder and sphere.
- Using the half-thickness or the full thickness in the wrong place.
- Forgetting that generation is per unit volume, not per unit area.
Statistics and uncertainty
Volumetric heating is often known only to 10 % to 20 %, and the peak temperature scales directly with it. Use a margin on predicted centreline or hot-spot temperatures.
Where it is used
- Hot-spot checks in busbars, fuel rods, heated plates and spherical particles.
- Finding the surface temperature a coolant must hold for a given peak.
Maths you need
- Volumetric generation: q‴ in W/m³
- Steady conduction equation: k∇²T + q‴ = 0
- Symmetry: zero gradient at the centre