Library / Heat transfer and thermal systems / HTS-07
Heat transfer and thermal systems
Radial temperature in a heat-generating cylinder
Temperature in a long solid cylinder of radius R with uniform volumetric heating and a fixed surface temperature Ts.
- Ref
- HTS-07
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- T [K]
A long solid cylinder of radius R with uniform heating q‴ and surface temperature Ts has the parabolic profile T(r) = Ts + q‴(R² − r²)/(4k). The centre is the hottest point. All the heat generated, q‴πR² per metre of length, leaves through the surface, giving a surface flux of q‴R/2.
Temperature in a long solid cylinder of radius R with uniform volumetric heating and a fixed surface temperature Ts. Steady radial conduction, uniform conductivity and negligible end effects.
Calculator
- Centreline temperature, Ts + q‴R²/(4k)372.5 K
- Heat generated per metre of length, q‴πR²3,142 W/m
- Heat flux at the surface, q‴R/210,000 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 = 0.03 m the temperature is 350 K, and the centreline reaches 372.5 K.
- The cylinder generates 3,142 W per metre, all removed at the surface at 10,000 W/m², which is the load the coolant must carry.
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 | 310.1 | 100,300 | 100,000 | +10% | +1.14 % |
| R Cylinder radius | 0.03 to 0.5 m | 310 | 6,538 | 6,228 | +10% | +3.75 % |
| k Thermal conductivity | 0.1 to 500 W/(m K) | 310.3 | 1,910 | 1,600 | +10% | −1.04 % |
| Ts Surface temperature | 250 to 1,000 K | 290 | 1,040 | 750 | +10% | +8.86 % |
| r Radius of interest | 0 to 0.03 m | 350 | 372.5 | 22.5 | +10% | −1.35 % |
Sweep of Cylinder radius
| Cylinder radius (m) | Temperature at r (K) |
|---|---|
| 0.03 | 310 |
| 0.08875 | 484.4 |
| 0.1475 | 831.4 |
| 0.2062 | 1,351 |
| 0.265 | 2,043 |
| 0.3237 | 2,908 |
| 0.3825 | 3,945 |
| 0.4413 | 5,155 |
| 0.5 | 6,538 |
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 341) moves the result by +8.86 %, from 350 to 381 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 440,000) moves the result by +1.14 %, from 350 to 354 K.
Cylinder radius
The cylinder radius, in m. A larger radius raises the temperature rise with the square of R. Raising it by 10 % (to 0.055) moves the result by +3.75 %, from 350 to 363.1 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 0.033) moves the result by −1.35 %, from 350 to 345.3 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 −1.04 %, from 350 to 346.4 K.
Worked example with the values above
- Geometry term: R² − r² = 0.05² − 0.03² = 0.0016 m².
- Rise above the surface: 400,000 × 0.0016 / (4 × 4) = 40 K.
- T = 310 + 40 = 350 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