Library / Heat transfer and thermal systems / HTS-05
Heat transfer and thermal systems
Temperature within a cylindrical wall
Steady radial temperature profile in a cylindrical wall with fixed temperatures Ti and To at radii ri and ro, for uniform conductivity and no generation or end effects.
- Ref
- HTS-05
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- T [K]
With fixed wall temperatures, steady radial conduction in a cylinder gives a logarithmic temperature profile: T(r) = Ti + (To − Ti) ln(r/ri)/ln(ro/ri). Most of the temperature drop occurs near the inner surface, where the area is smallest and the flux is highest, so the profile is steeper there than a straight line would be.
Steady radial temperature profile in a cylindrical wall with fixed temperatures Ti and To at radii ri and ro, for uniform conductivity and no generation or end effects. The profile is logarithmic, not linear.
Calculator
- Fraction of the total drop reached at r0.5
- Temperature at the same r if the profile were linear366.7 K
- Temperature at the geometric-mean radius √(ri ro)350 K
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 = 0.04 m the wall is at 350 K, 50 % of the way from the inner to the outer temperature.
- A straight-line profile would give 366.7 K, so the log profile is lower here, because the temperature drops fastest where the area is smallest, near the inner wall.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Ti Inner-surface temperature | 250 to 1,000 K | 275 | 650 | 375 | +10% | +5.71 % |
| To Outer-surface temperature | 250 to 1,000 K | 325 | 700 | 375 | +10% | +4.29 % |
| r Radius of interest | 0.02 to 0.08 m | 300 | 400 | 100 | +10% | −1.96 % |
| ro Outer radius | 0.08 to 0.5 m | 350 | 378.5 | 28.47 | +10% | +0.919 % |
| ri Inner radius | 0.005 to 0.02 m | 325 | 350 | 25 | +10% | +1.05 % |
Sweep of Radius of interest
| Radius of interest (m) | Temperature at r (K) |
|---|---|
| 0.02 | 400 |
| 0.0275 | 377 |
| 0.035 | 359.6 |
| 0.0425 | 345.6 |
| 0.05 | 333.9 |
| 0.0575 | 323.8 |
| 0.065 | 315 |
| 0.0725 | 307.1 |
| 0.08 | 300 |
Sweep one variable
Variables and how each one contributes
Temperature at r
What the equation gives you. Everything else feeds it.
Inner-surface temperature
The temperature at the inner radius, in K. Raises the temperature at every radius, most near the inner wall. Raising it by 10 % (to 440) moves the result by +5.71 %, from 350 to 370 K.
Outer-surface temperature
The temperature at the outer radius, in K. Raises the temperature at every radius, most near the outer wall. Raising it by 10 % (to 330) moves the result by +4.29 %, from 350 to 365 K.
Inner radius
The inner radius, in m. A larger inner radius moves the interior point closer to the inner surface in log terms. Raising it by 10 % (to 0.022) moves the result by +1.05 %, from 350 to 353.7 K.
Outer radius
The outer radius, in m. A larger outer radius spreads the log range, so the point at r sits nearer the inner temperature. Raising it by 10 % (to 0.088) moves the result by +0.919 %, from 350 to 353.2 K.
Radius of interest
The radius at which the temperature is wanted, in m. A larger radius moves the temperature from Ti towards To. Raising it by 10 % (to 0.044) moves the result by −1.96 %, from 350 to 343.1 K.
Worked example with the values above
- Log terms: ln(r/ri) = ln(0.04/0.02) = 0.69315; ln(ro/ri) = 1.3863.
- Fraction of the drop: 0.5.
- T = 400 + (300 − 400) × 0.5 = 350 K.
Assumptions and limits
- Steady state, constant properties unless the page states a temperature law, and no internal heat generation.
- One-dimensional conduction in the stated geometry, with no edge or end losses.
Common mistakes
- Using diameters where radii are needed.
- Using °C in a formula that needs an absolute temperature, or leaving a temperature difference out of the heat rate.
- Applying a plane-wall formula to a curved wall when the thickness is not small compared with the radius.
Statistics and uncertainty
Thermal conductivities are rarely known better than 5 % to 10 %, and contact resistances can vary by a factor of two between nominally identical joints. Treat results as estimates unless the inputs were measured.
Where it is used
- Insulation sizing for vessels, pipes and walls.
- Estimating interface and layer temperatures in composite constructions.
Maths you need
- Fourier’s law: q″ = −k dT/dx
- Thermal resistance: R = ΔT/Q, in K/W
- Series and parallel paths: resistances add in series; conductances add in parallel