Library / Heat transfer and thermal systems / HTS-22
Heat transfer and thermal systems
Mean temperature ratio of a convectively cooled plane wall
Volume-averaged dimensionless temperature of the same symmetric wall, from the first mode.
- Ref
- HTS-22
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- θm/θi
Averaging the first mode over the wall volume gives the mean temperature ratio θm/θi ≈ C₁ [sin λ/λ] exp(−λ²Fo), with the same λ from λ tan λ = Bi and the same C₁. The mean temperature gives the stored energy directly, since the energy released is proportional to 1 − θm/θi. Use Fo > 0.2 as a first-term guideline.
Volume-averaged dimensionless temperature of the same symmetric wall, from the first mode. Supply the Biot number Bi = hL/k with L the half-thickness; λ tan λ = Bi gives the first root. Use Fo > 0.2 as a first-term guideline.
Calculator
- First eigenvalue, λ0.7854
- Coefficient, C₁1.1
- Fraction of the initial excess energy released, 1 − θm/θi0.4655
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 Fo = 1 the wall’s mean temperature excess is 53.45 % of the initial value, so 46.55 % of the stored energy has gone.
- The mean excess is below the centre value, because the surfaces have cooled more, and the two agree only as Bi → 0 where the wall is nearly uniform.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Fo Fourier number αt/L² | 0 to 5 | 0.04533 | 0.9905 | 0.9452 | +10% | −5.98 % |
| Bi Biot number hL/k (L = half-thickness) | 0.01 to 100 | 0.07287 | 0.9901 | 0.9172 | +10% | −4.73 % |
Sweep of Fourier number αt/L²
| Fourier number αt/L² | Mean temperature ratio |
|---|---|
| 0 | 0.9905 |
| 0.625 | 0.6737 |
| 1.25 | 0.4581 |
| 1.875 | 0.3116 |
| 2.5 | 0.2119 |
| 3.125 | 0.1441 |
| 3.75 | 0.09801 |
| 4.375 | 0.06666 |
| 5 | 0.04533 |
Sweep one variable
Variables and how each one contributes
Mean temperature ratio
What the equation gives you. Everything else feeds it.
Biot number hL/k (L = half-thickness)
The Biot number hL/k, with L the half-thickness. A larger Biot number raises λ, so the mean temperature falls faster. Raising it by 10 % (to 0.8639) moves the result by −4.73 %, from 0.5345 to 0.5092.
Fourier number αt/L²
The Fourier number αt/L². More time lowers the mean ratio exponentially. Raising it by 10 % (to 1.1) moves the result by −5.98 %, from 0.5345 to 0.5026.
Worked example with the values above
- Root of λ tan λ = 0.7854: λ = 0.785398; C₁ = 1.1002; sin λ/λ = 0.90032.
- Decay: exp(−λ²Fo) = 0.53964.
- θm/θi = 1.1002 × 0.90032 × 0.53964 = 0.53454.
Assumptions and limits
- Constant properties, a uniform initial temperature and convection to a fixed ambient on both sides for the wall solutions.
- For lumped models, a body with small internal temperature differences and constant conductances.
Common mistakes
- Using the full thickness instead of the half-thickness in Bi and Fo.
- Using the first-term form at Fo below about 0.2.
- Reading the mean ratio as the centre or surface value.
Statistics and uncertainty
First-term wall solutions are accurate to within about 2 % for Fo above 0.2, and lumped formulas need a small Biot number, below about 0.1, to stay within a few per cent.
Where it is used
- Cooling and heating of slabs, plates and electronics enclosures.
- Estimating warm-up and settling times.
Maths you need
- Biot number: Bi = hL/k
- Fourier number: Fo = αt/L²
- Time constant: τ = C/G, capacitance over conductance