Library / Heat Transfer / HTR-12
Heat Transfer
Log-mean temperature difference
The correct average temperature difference along a heat exchanger when the difference changes from one end to the other.
- Ref
- HTR-12
- Rev
- v1
- Field
- Heat Transfer
- Result
- ΔTlm [K]
ΔT₁ and ΔT₂ are the temperature differences between the hot and cold streams at the two ends of the exchanger. The heat transferred is Q = U A ΔTlm, where U is the overall coefficient and A the area. Which end is called 1 does not matter, since the result is symmetric in the two.
The correct average temperature difference along a heat exchanger when the difference changes from one end to the other.
Calculator
- Arithmetic mean of the two differences60 K
- Area needed for 100 kW at U = 500 W/m²·K3.466 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.
- The log mean (57.7 K) is a little below the simple average (60 K). Using the average would oversize the heat flow by about 4 % and undersize the area.
- If one end difference falls to zero, ΔTlm falls to zero and the area needed becomes unlimited, which is why exchangers are not designed with a zero approach temperature.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ΔT2 Temperature difference at end 2 | 1 to 500 K | 18.03 | 229.2 | 211.2 | +10% | +4.35 % |
| ΔT1 Temperature difference at end 1 | 1 to 500 K | 10.57 | 182.1 | 171.6 | +10% | +5.49 % |
Sweep of Temperature difference at end 2
| Temperature difference at end 2 (K) | Log-mean difference (K) |
|---|---|
| 1 | 18.03 |
| 63.38 | 71.37 |
| 125.8 | 101.2 |
| 188.1 | 126.5 |
| 250.5 | 149.4 |
| 312.9 | 170.8 |
| 375.3 | 191 |
| 437.6 | 210.4 |
| 500 | 229.2 |
Sweep one variable
Variables and how each one contributes
Log-mean difference
What the equation gives you. Everything else feeds it.
Temperature difference at end 1
The temperature difference between the two streams at end 1. Raises the mean difference; the effect is about half in proportion at the defaults. A 10 % rise in ΔT₁ raises ΔTlm by about 5.5 %.
Temperature difference at end 2
The temperature difference between the two streams at end 2. Raises the mean difference, with a smaller effect than ΔT₁ because it is the smaller one. A 10 % rise in ΔT₂ raises ΔTlm by about 4.3 %.
Worked example with the values above
- Difference of the ends: ΔT₁ − ΔT₂ = 80 − 40 = 40 K.
- Logarithm of the ratio: ln(80/40) = 0.69315.
- ΔTlm = 40 / 0.69315 = 57.71 K.
Assumptions and limits
- Steady flow with constant specific heats and a constant overall coefficient U.
- No heat loss to the surroundings and no phase change (or a constant temperature stream for boiling or condensing).
Common mistakes
- Subtracting the inlet and outlet temperatures of one stream instead of taking the difference between the streams at each end.
- Using the same formula for a cross-flow exchanger without a correction factor.
Statistics and uncertainty
For ratios ΔT₁/ΔT₂ under about 2 the average and the log mean differ by less than 4 %, so the choice hardly matters. When one end difference is small, a 1 K error in it can change ΔTlm by tens of per cent and so the required area.
Where it is used
- Sizing counter-flow and parallel-flow heat exchangers.
- Checking the performance of an existing exchanger from measured temperatures.
Maths you need
- Natural logarithm: ln of a ratio
- Heat exchanger terms: hot and cold streams