Library / Heat transfer and thermal systems / HTS-44
Heat transfer and thermal systems
One-shell-pass exchanger effectiveness
Effectiveness of an ideal shell-and-tube exchanger with one shell pass and two or an even number of tube passes, with equal area per pass and no bypass or leakage.
- Ref
- HTS-44
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- ε
An ideal exchanger with one shell pass and two or any even number of tube passes has ε = 2/[1 + Cr + S/tanh(NTU·S/2)], with S = √(1 + Cr²). It gives zero at NTU = 0 and approaches 2/(1 + Cr + S) at large NTU, which is below 1 for Cr above zero. The model assumes equal area per pass and no bypass or leakage.
Effectiveness of an ideal shell-and-tube exchanger with one shell pass and two or an even number of tube passes, with equal area per pass and no bypass or leakage. Returns zero when NTU = 0.
Calculator
- Parameter, S = √(1 + Cr²)1.118
- Limit at large NTU, 2/(1 + Cr + S)0.7639
- Counterflow effectiveness at the same NTU and Cr0.5647
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 shell-and-tube unit reaches 53.99 % effectiveness, and its ceiling at this capacity ratio is 76.39 %.
- A true counterflow unit would get 56.47 %, so the multi-pass arrangement gives up some performance for compactness.
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 | 0.7639 | 0.7639 | +10% | +4.58 % |
| Cr Capacity ratio Cmin/Cmax | 0 to 1 | 0.4627 | 0.6321 | 0.1694 | +10% | −1.55 % |
Sweep of Number of transfer units UA/Cmin
| Number of transfer units UA/Cmin | Effectiveness |
|---|---|
| 0 | 0 |
| 2.5 | 0.7237 |
| 5 | 0.7615 |
| 7.5 | 0.7638 |
| 10 | 0.7639 |
| 12.5 | 0.7639 |
| 15 | 0.7639 |
| 17.5 | 0.7639 |
| 20 | 0.7639 |
Sweep one variable
Variables and how each one contributes
Effectiveness
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 +4.58 %, from 0.5399 to 0.5647.
Capacity ratio Cmin/Cmax
The capacity ratio Cmin/Cmax, between 0 and 1. A larger ratio lowers the effectiveness. Raising it by 10 % (to 0.55) moves the result by −1.55 %, from 0.5399 to 0.5316.
Worked example with the values above
- S = √(1 + 0.5²) = 1.118; tanh(NTU·S/2) = tanh(0.55902) = 0.50725.
- Denominator: 1 + Cr + S/tanh = 1.5 + 2.2041 = 3.7041.
- ε = 2 / 3.7041 = 0.53994.
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 the wrong stream as Cmin when forming the capacity ratio.
- Mixing up NTU with UA.
- Using a counterflow formula for a parallel or crossflow unit.
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