Library / Heat transfer and thermal systems / HTS-47
Heat transfer and thermal systems
Adiabatic mixing temperature of two streams
Temperature after steady adiabatic mixing of two streams of heat-capacity rates C = ṁcp (W/K).
- Ref
- HTS-47
- Rev
- v1
- Field
- Heat transfer and thermal systems
- Result
- Tout [K]
The temperature after steady adiabatic mixing of two streams is the heat-capacity-rate-weighted average, Tout = (C₁T₁ + C₂T₂)/(C₁ + C₂), with C = ṁcp. It follows from an energy balance with constant cp, and it assumes no heat loss, reaction, phase change or enthalpy of mixing. The stream with the larger heat-capacity rate dominates.
Temperature after steady adiabatic mixing of two streams of heat-capacity rates C = ṁcp (W/K). Assumes constant cp, no heat loss, reaction, phase change or enthalpy of mixing.
Calculator
- Total heat-capacity rate, C₁ + C₂600 W/K
- Heat gained by stream 2, C₂(Tout − T₂)8,000 W
- Weight of stream 1, C₁/(C₁ + C₂)0.3333
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 two streams leave mixed at 320 K, 33.3 % of the way from stream 2 to stream 1.
- Stream 2 gains 8,000 W, exactly what stream 1 loses, which confirms the energy balance.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T2 Temperature of stream 2 | 250 to 1,000 K | 286.7 | 786.7 | 500 | +10% | +6.25 % |
| T1 Temperature of stream 1 | 250 to 1,000 K | 283.3 | 533.3 | 250 | +10% | +3.75 % |
| C1 Heat-capacity rate of stream 1 | 1 to 100,000 W/K | 300.1 | 359.8 | 59.61 | +10% | +0.403 % |
| C2 Heat-capacity rate of stream 2 | 1 to 100,000 W/K | 300.1 | 359.7 | 59.58 | +10% | −0.391 % |
Sweep of Heat-capacity rate of stream 2
| Heat-capacity rate of stream 2 (W/K) | Mixed temperature (K) |
|---|---|
| 1 | 359.7 |
| 4.217 | 358.8 |
| 17.78 | 355.1 |
| 74.99 | 343.6 |
| 316.2 | 323.2 |
| 1,334 | 307.8 |
| 5,623 | 302.1 |
| 23,710 | 300.5 |
| 100,000 | 300.1 |
Sweep one variable
Variables and how each one contributes
Mixed temperature
What the equation gives you. Everything else feeds it.
Heat-capacity rate of stream 1
The heat-capacity rate ṁcp of stream 1, in W/K. A larger rate pulls the result towards T₁. Raising it by 10 % (to 220) moves the result by +0.403 %, from 320 to 321.3 K.
Temperature of stream 1
The temperature of stream 1, in K. Raises the mixed temperature in proportion to its weight. Raising it by 10 % (to 396) moves the result by +3.75 %, from 320 to 332 K.
Heat-capacity rate of stream 2
The heat-capacity rate ṁcp of stream 2, in W/K. A larger rate pulls the result towards T₂. Raising it by 10 % (to 440) moves the result by −0.391 %, from 320 to 318.8 K.
Temperature of stream 2
The temperature of stream 2, in K. Raises the mixed temperature in proportion to its weight. Raising it by 10 % (to 330) moves the result by +6.25 %, from 320 to 340 K.
Worked example with the values above
- Weighted sum: C₁T₁ + C₂T₂ = 200 × 360 + 400 × 300 = 192,000 W.
- Total: C₁ + C₂ = 600 W/K.
- Tout = 192,000 / 600 = 320 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