Library / Heat Transfer / HTR-16
Heat Transfer
Effectiveness of a counterflow heat exchanger
Fraction of the largest possible heat transfer that an ideal counterflow exchanger achieves, from its number of transfer units and the ratio of the two heat-capacity rates.
- Ref
- HTR-16
- Rev
- v1
- Field
- Heat Transfer
- Result
- ε
The effectiveness ε of a heat exchanger is the heat actually transferred divided by the maximum possible, set by the stream with the smaller heat-capacity rate. For counterflow, ε = [1 − e−NTU(1−Cr)]/[1 − Cre−NTU(1−Cr)], with NTU = UA/Cmin and Cr = Cmin/Cmax. When Cr = 1 the formula becomes ε = NTU/(1 + NTU). The result lies between 0 and 1.
Fraction of the largest possible heat transfer that an ideal counterflow exchanger achieves, from its number of transfer units and the ratio of the two heat-capacity rates. At equal heat-capacity rates the second form applies.
Calculator
- Parallel-flow effectiveness for the same NTU and Cr0.5179
- Effectiveness when Cr = 0 (boiling or condensing)0.6321
- Effectiveness at double the NTU0.7746
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.
- An exchanger with NTU = 1 and Cr = 0.5 has an effectiveness of 0.565: it transfers 56.5 % of the thermodynamic maximum. A parallel-flow exchanger of the same size would only reach 0.518.
- Doubling the area (NTU = 2) lifts the effectiveness to only about 0.775: the second half of the area buys much less than the first.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| NTU Number of transfer units | 0.1 to 10 | 0.09301 | 0.9966 | 0.9036 | +10% | +5.28 % |
| Cr Heat-capacity-rate ratio, Cmin/Cmax | 0 to 1 | 0.5 | 0.6321 | 0.1321 | +10% | −1.18 % |
Sweep of Number of transfer units
| Number of transfer units | Effectiveness |
|---|---|
| 0.1 | 0.09301 |
| 1.338 | 0.6556 |
| 2.575 | 0.8399 |
| 3.813 | 0.9197 |
| 5.05 | 0.9583 |
| 6.287 | 0.978 |
| 7.525 | 0.9883 |
| 8.762 | 0.9937 |
| 10 | 0.9966 |
Sweep one variable
Variables and how each one contributes
Effectiveness
What the equation gives you. Everything else feeds it.
Number of transfer units
The number of transfer units, UA/Cmin: a measure of the size of the exchanger relative to the flow. Raises the effectiveness, with diminishing returns: a larger exchanger improves performance but ever more slowly. Raising it by 0.25 (to 1.25) moves the result by +12.4 %, from 0.5647 to 0.6346.
Heat-capacity-rate ratio, Cmin/Cmax
The ratio of the smaller to the larger heat-capacity rate, between 0 (one stream condensing or boiling) and 1 (balanced). Lowers the effectiveness: with more balanced streams the temperature difference shrinks along the exchanger. Raising it by 0.05 (to 0.55) moves the result by −1.18 %, from 0.5647 to 0.5581.
Worked example with the values above
- Exponential term: e−NTU(1−Cr) = e−1 × 0.5 = 0.6065.
- Numerator 1 − 0.6065 = 0.3935; denominator 1 − 0.5 × 0.6065 = 0.6967.
- ε = 0.3935 / 0.6967 = 0.5647.
Assumptions and limits
- A counterflow exchanger with constant overall coefficient and constant specific heats, and no heat loss to the surroundings.
- Steady flow of both streams, with no phase change except when Cr = 0.
Common mistakes
- Using it for a crossflow or shell-and-tube exchanger, which have different formulas.
- Taking Cr as Cmax/Cmin, which is above 1.
Statistics and uncertainty
Heat-capacity rates carry the uncertainty of flow and specific-heat measurement (about 2 to 5 %), so Cr is known to a few per cent. The effectiveness is only weakly sensitive to Cr at low NTU but more so at high NTU.
Where it is used
- Rating an exchanger when the outlet temperatures are unknown.
- Comparing counterflow and parallel-flow performance for a given size.
Maths you need
- Heat-capacity rate: mass flow times specific heat, in W/K
- Maximum heat transfer: Cmin × the inlet temperature difference