Library / Heat transfer and thermal systems / HTS-48
Heat transfer and thermal systems
Entropy generation in an insulated heat exchanger
Rate of entropy generation in a heat exchanger with no heat leakage or pressure drop, from the heat-capacity rates and the inlet and outlet temperatures of the two single-phase streams.
- Ref
- HTS-48
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- Ṡgen [W/K]
The entropy generation rate of an insulated heat exchanger with constant heat-capacity rates is Ṡgen = Ch ln(Th,out/Th,in) + Cc ln(Tc,out/Tc,in). The hot stream contributes a negative term and the cold stream a positive one; the sum must be non-negative. A negative result means the data break the second law. First check that the hot-side heat loss equals the cold-side gain.
Rate of entropy generation in a heat exchanger with no heat leakage or pressure drop, from the heat-capacity rates and the inlet and outlet temperatures of the two single-phase streams. It must be non-negative for a physically possible exchanger.
Calculator
- Heat released by the hot stream, Ch(Thi − Tho)10,000 W
- Heat gained by the cold stream, Cc(Tco − Tci)10,000 W
- Energy-balance mismatch, hot minus cold0 W
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 exchanger generates 4.124 W/K of entropy; this is physically allowed.
- The hot stream gives 10,000 W and the cold stream takes 10,000 W, a mismatch of 0 W; the formula assumes these are equal.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W/K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Cc Heat-capacity rate of the cold stream | 1 to 100,000 W/K | −26.55 | 15,390 | 15,410 | +10% | +74.8 % |
| Ch Heat-capacity rate of the hot stream | 1 to 100,000 W/K | −13,320 | 30.7 | 13,350 | +10% | −64.8 % |
| Th,in Hot-stream inlet temperature | 250 to 1,000 K | −179.1 | 98.12 | 277.3 | +10% | −462 % |
| Tc,out Cold-stream outlet temperature | 250 to 1,000 K | −63.17 | 214.1 | 277.3 | +10% | +462 % |
| Th,out Hot-stream outlet temperature | 250 to 1,000 K | −63.17 | 214.1 | 277.3 | +10% | +462 % |
| Tc,in Cold-stream inlet temperature | 250 to 1,000 K | −236.7 | 40.59 | 277.3 | +10% | −462 % |
Sweep of Hot-stream outlet temperature
| Hot-stream outlet temperature (K) | Entropy generation rate (W/K) |
|---|---|
| 250 | −63.17 |
| 343.8 | 0.5202 |
| 437.5 | 48.75 |
| 531.3 | 87.58 |
| 625 | 120.1 |
| 718.8 | 148 |
| 812.5 | 172.6 |
| 906.3 | 194.4 |
| 1,000 | 214.1 |
Sweep one variable
Variables and how each one contributes
Entropy generation rate
What the equation gives you. Everything else feeds it.
Heat-capacity rate of the hot stream
The heat-capacity rate of the hot stream, in W/K. Scales the hot-stream term, which is negative as the stream cools. Raising it by 10 % (to 220) moves the result by −64.8 %, from 4.124 to 1.453 W/K.
Hot-stream inlet temperature
The hot-stream inlet temperature, in K. A hotter inlet makes the hot stream’s entropy drop larger in magnitude. Raising it by 10 % (to 440) moves the result by −462 %, from 4.124 to −14.94 W/K.
Hot-stream outlet temperature
The hot-stream outlet temperature, in K. A hotter outlet lessens the hot stream’s entropy drop, raising the result. Raising it by 10 % (to 385) moves the result by +462 %, from 4.124 to 23.19 W/K.
Heat-capacity rate of the cold stream
The heat-capacity rate of the cold stream, in W/K. Scales the cold-stream term, which is positive as the stream warms. Raising it by 10 % (to 220) moves the result by +74.8 %, from 4.124 to 7.207 W/K.
Cold-stream inlet temperature
The cold-stream inlet temperature, in K. A warmer inlet lessens the cold stream’s entropy gain. Raising it by 10 % (to 330) moves the result by −462 %, from 4.124 to −14.94 W/K.
Cold-stream outlet temperature
The cold-stream outlet temperature, in K. A warmer outlet raises the cold stream’s entropy gain, raising the result. Raising it by 10 % (to 385) moves the result by +462 %, from 4.124 to 23.19 W/K.
Worked example with the values above
- Hot stream: Ch ln(Tho/Thi) = 200 × ln(350/400) = −26.706 W/K.
- Cold stream: Cc ln(Tco/Tci) = 200 × ln(350/300) = 30.83 W/K.
- Ṡgen = 4.1239 W/K.
Assumptions and limits
- Constant properties in each phase, a single melting temperature and no losses unless stated.
- Steady or quasi-steady balances, with absolute temperatures where logarithms or powers are used.
Common mistakes
- Forgetting the latent heat term, or counting it twice.
- Using °C in an entropy or radiation formula.
- Using these simple formulas when the Stefan number is not small.
Statistics and uncertainty
Latent heats and specific heats are tabulated to a few per cent, but real storage materials melt over a range and lose heat, so real systems typically deliver 5 % to 20 % less than the ideal values.
Where it is used
- Sizing thermal storage, ice banks and phase-change units.
- Checking exchanger data, mixing streams and cooling times.
Maths you need
- Latent heat: energy per kilogram to change phase
- Heat-capacity rate: C = ṁcp
- Second law: entropy generation cannot be negative