Library / Heat transfer and thermal systems / HTS-43
Heat transfer and thermal systems
Crossflow effectiveness with both streams unmixed
Engineering approximation for a single-pass crossflow exchanger with both streams unmixed.
- Ref
- HTS-43
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- ε
For a single-pass crossflow exchanger with both streams unmixed, an engineering approximation is ε ≈ 1 − exp[(NTU^0.22/Cr)(exp(−Cr NTU^0.78) − 1)]. It is not exact and should be labelled as approximate. At Cr = 0 it uses the limiting form 1 − exp(−NTU). The unmixed arrangement is more effective than the mixed one, because the temperature varies across each stream.
Engineering approximation for a single-pass crossflow exchanger with both streams unmixed. At Cr = 0 it takes the limiting form 1 − e⁻ᴺᵀᵁ. It is a correlation, not an exact solution.
Calculator
- Phase-change limit (Cr = 0), 1 − exp(−NTU)0.6321
- Mixed-stream crossflow value (entry above)0.4685
- Difference, unmixed minus mixed0
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 approximation gives 46.85 % effectiveness, and that is approximate by a few per cent compared with the exact series solution.
- The mixed-stream crossflow case gives 46.85 % at the same NTU and Cr, so the choice of which streams are mixed matters.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| NTU Number of transfer units UA/Cmin | 0 to 20 | 0 | 0.8553 | 0.8553 | +10% | +4.59 % |
| Cr Capacity ratio Cmin/Cmax | 0 to 1 | 0.4685 | 0.6321 | 0.1636 | +10% | — |
Sweep of Capacity ratio Cmin/Cmax
| Capacity ratio Cmin/Cmax | Effectiveness (approximate) |
|---|---|
| 0 | 0.6321 |
| 0.125 | 0.6094 |
| 0.25 | 0.5872 |
| 0.375 | 0.5656 |
| 0.5 | 0.5448 |
| 0.625 | 0.5246 |
| 0.75 | 0.5052 |
| 0.875 | 0.4865 |
| 1 | 0.4685 |
Sweep one variable
Variables and how each one contributes
Effectiveness (approximate)
What the equation gives you. Everything else feeds it.
Number of transfer units UA/Cmin
The number of transfer units, UA/Cmin. Raises the effectiveness, with falling returns. Raising it by 10 % (to 1.1) moves the result by +4.59 %, from 0.4685 to 0.49.
Capacity ratio Cmin/Cmax
The capacity ratio Cmin/Cmax, between 0 and 1. A larger ratio lowers the effectiveness. Raising it by 10 % (to 1.1) gives no valid result at the other default values.
Worked example with the values above
- Powers of NTU: NTU^0.22 = 1, NTU^0.78 = 1.
- Inner term: exp(−Cr NTU^0.78) − 1 = −0.63212; × NTU^0.22/Cr = −0.63212.
- ε ≈ 1 − exp(that) = 0.46854.
Assumptions and limits
- Steady operation with constant properties and overall coefficient, and an insulated exterior.
- The flow arrangement matches the formula exactly, including which fluid is mixed.
Common mistakes
- Using this approximation when one stream is mixed.
- Treating it as exact and quoting more digits than it supports.
- Using a Cr above 1 instead of the ratio of the smaller to the larger capacity rate.
Statistics and uncertainty
Effectiveness formulas are exact for their ideal arrangements, but real exchangers have bypass, leakage and non-uniform flow, which lower performance by a few per cent to 10 %. Fouling allowances are largely judgement.
Where it is used
- Rating exchangers from known UA and flow rates.
- Estimating the effect of fouling on duty.
Maths you need
- Capacity rate: C = ṁcp, in W/K
- NTU: UA divided by Cmin
- Effectiveness: actual heat transfer over the thermodynamic maximum