Hot air line, a worked cooling example
Written by Haimi Jordaan, MEng (Mechanical), University of Pretoria. Seven years in a specialist engineering analysis and design group across CFD, FEA and DEM. He wrote the solver behind Fluid Network Studio.Published . Last updated .
This worked example couples two effects that usually get treated separately, compressible gas flow and heat loss along the same pipe. It is a built-in network in Fluid Network Studio. It shows what a hot gas line does as it cools, which runs opposite to the intuition you build from isothermal lines, and why the solver marches a gas pipe segment by segment when heat transfer is on.
The scenario
Hot air enters a bare line and travels to a draw-off. The pipe is uninsulated, so the air loses heat to a cooler ambient all the way along, and by the outlet it has cooled close to ambient temperature. The interesting question is what that cooling does to the flow. On an isothermal line the gas expands and speeds up toward the outlet. Cool the gas and the story reverses, and this example is built to make that reversal visible.
The real setup
The example uses these parameters exactly as built:
- An inlet boundary at 500 kPa absolute.
- Inlet air supplied at 400 K, which is 126.85 degrees C, close enough to call it 127 degrees C air.
- A single bare pipe, 300 m long, 50 mm internal diameter, roughness 4.6e-5 m.
- A direct overall heat-transfer coefficient on the line of 3.0 W/m.K (per metre of pipe) to an ambient of 15 degrees C.
- A draw-off pulling a fixed mass flow of 0.18 kg/s.
- Working fluid set to the gas phase, air, with gas constant 287.05 J/kg.K, viscosity 2.3e-5 Pa.s and thermal conductivity 0.0257 W/m.K.
The physics and method
With heat transfer on, a gas pipe is solved by a steady one-dimensional segment march rather than a single closed relation. Each segment accounts for friction, the acceleration term and wall heat loss together, in total-energy form, so the temperature, pressure, density and velocity all move down the pipe in step. The wall heat path reuses the same film and composite-cylinder treatment used for liquid heat loss, now driven by the gas conductivity. The march is solved with a midpoint scheme and step doubling for accuracy, and an always-on energy residual is reported so the heat balance is shown, not assumed.
The solved result
The air enters at 126.9 degrees Celsius and arrives at 15.78 degrees, within a degree of the ambient, and the density and the velocity move the wrong way on the way there:
| Quantity | Inlet | Outlet |
|---|---|---|
| Temperature | 126.9 degC | 15.78 degC |
| Absolute pressure | 500.0 kPa | 395.4 kPa |
| Density | 4.355 kg/m3 | 4.768 kg/m3 |
| Velocity | 21.05 m/s | 19.23 m/s |
| Quantity | Value |
|---|---|
| Mass flow (fixed by the draw-off) | 0.18 kg/s |
| Heat lost to the 15 degree ambient | 20.20 kW |
| Overall coefficient on the line, entered directly | 3.0 W/m.K, per metre of pipe |
| Reynolds number | 199 300 |
| Marching segments used along the pipe | 160 |
| Energy-balance residual | 1.4 x 10^-8 |
This is the reversal the example exists to show. On an isothermal line the gas expands as it loses pressure and speeds up toward the outlet. Here the pressure still falls, by 104.6 kPa, and yet the density RISES from 4.355 to 4.768 kg/m3 and the velocity FALLS from 21.05 to 19.23 m/s, because cooling from 400 K down to 289 K shrinks the gas faster than the pressure drop expands it. Temperature wins over pressure along this particular line, and only a solve that marches the two together can tell you which one does.
The bare line sheds 20.20 kW over 300 m, enough to take essentially all the heat out of the stream. By the outlet the air is 0.78 K above ambient, so the last stretch of pipe is doing almost nothing, which is the practical signal that the line is far longer than the cooling duty needs.
Every number is solver output, from a 160-segment march down the pipe, not a hand calculation. Open the example, press Solve, and colour the network by temperature to watch it happen.
What the solver computes and what you learn
Solve it and the air arrives near the 15 degrees C ambient. The signature is in the density and velocity. As the gas cools, its density rises, and because the mass flow is fixed the velocity falls along the pipe, the opposite of a hot isothermal line that speeds up toward the outlet. Switch the view to Colour by, then Temperature, and you can watch the line cool from inlet to draw-off. The lesson is that cooling and compressibility pull in opposite directions, and only a coupled march captures which one wins where.
Gas and heat transfer are both part of the Advanced plan, so solving this example needs that plan.
Take it further
This example sits behind two application pages, pipe heat loss for the thermal side and compressed air system design for the gas side. The glossary explains the terms the solver uses. Fluid Network Studio supports your engineering work and does not replace a qualified engineer, and it makes no claim of compliance with any standard.
Open this example in FNS and add insulation to slow the cooling.