Engineering task and calculation objective
This module calculates heat transfer and flow in rarefied gases according to the VDI Heat Atlas (VDI-Wärmeatlas, 12th edition 2019). At very low pressures or very small gap widths, the mean free path of the gas molecules is no longer small compared with the characteristic dimension of the system: continuum mechanics loses its validity, and heat transport must be described by free molecular motion or the transition regime. The classic geometries considered are plane parallel plates, concentric cylinders and concentric spheres in the steady state.
In practice, this calculation is needed wherever a vacuum acts — or is intended to act — as thermal insulation: in vacuum-insulated vessels and piping for cryogenic media (LNG, liquid nitrogen), in Dewar flasks, vacuum panels and thermos bottles, but also when estimating heat losses in vacuum systems and in measurement devices such as thermal-conductivity vacuum gauges. To calculate the heat flux in rarefied gases, you need — besides the temperature and the individual gas constant of the gas — above all the accommodation coefficient, which describes the energy exchange when molecules collide with the wall.
Standard and calculation basis: VDI-Wärmeatlas, 12. Auflage 2019
Calculation workflow
- Define geometry and boundary conditions: First the configuration is selected: parallel plates, concentric cylinders or concentric spheres. This includes the dimensions (gap width or diameters) and the surface temperatures of the two walls.
- Provide gas data: For the filling gas, the temperature, the individual gas constant and the molecular characteristics (isentropic exponent, molecular structure) are required, since they determine the energy transported per molecular collision.
- Assess the pressure range and degree of rarefaction: Using the ratio of the mean free path to the characteristic dimension (Knudsen number), it is checked whether the continuum regime, the transition regime or free molecular flow applies. This determines which relation of the VDI Heat Atlas governs.
- Set the accommodation coefficient: The accommodation coefficient describes how completely the gas molecules adopt the wall temperature on impact. It depends on the gas species, the wall material and the surface condition, and enters the heat flux directly.
- Calculate and evaluate the heat flux: The module determines the heat flux under free molecular motion and compares it with the continuum value. From this you can read off how far the pressure must be reduced for the vacuum insulation to reach its full effectiveness.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Dynamic viscosity | η | Pa·s |
| Density | ρ | kg/m³ |
| Arithmetic mean value molecular velocity | cm | m/s |
| Temperature | T | K |
| Knudsen number | Kn | – |
| Correction factor | f | – |
| Accommodation coefficient | γ | – |
| Thermal diffusivity | a | m²/s |
| Prandtl-Zahl | Prandtl Pr | – |
| Isentropic exponent | κ | – |
| Pressure | p | Pa |
| Mean flow velocity | u | m/s |
| Plate distance | s | m |
| Heat flux | q | W/m² |
| Heat flux continuum | qkont | W/m² |
| Thermal conductivity | λ | W/(m·K) |
| Temperature 1 | T1 | K |
| Temperature 2 | T2 | K |
| Specific heat capacity constant pressure | cp | J/(kg·K) |
| Length of heating wire | L | m |
| Heat duty continuum | Qkont | W |
| Heat performance | Qh | W |
| Heat duty radiation | Qs | W |
| Arithmetic mean value molecular velocity | cm1 | m/s |
Frequently asked questions
When is a gas considered rarefied?
The governing quantity is the Knudsen number, the ratio of the molecular mean free path to the characteristic dimension (e.g. gap width). For Kn « 1 the ordinary continuum treatment with pressure-independent thermal conductivity applies; for Kn » 1 free molecular flow prevails, with the transition regime in between. Since the mean free path is inversely proportional to pressure, the rarefied regime is reached either by low pressure or by very small gaps.
What is the accommodation coefficient and how do I choose it?
The accommodation coefficient indicates what fraction of the energy required for complete thermal equilibration a molecule actually exchanges during a wall collision. It lies between 0 and 1 and depends on the gas species, the wall material, the temperature and the surface coverage. For engineering surfaces and heavy gases it is often close to 1; for helium and hydrogen on clean metal surfaces it is significantly lower. Since measured values scatter, the sensitivity of the result to this quantity should be checked.
Why does the heat flow in a vacuum not keep decreasing indefinitely as I lower the pressure?
In the free molecular flow regime, the heat flux is proportional to pressure: every further pressure reduction lowers the gaseous heat transport. In parallel, however, thermal radiation between the walls and conduction through solid connections (spacers, suspensions) remain. Below a certain pressure these contributions dominate, so that further evacuation hardly reduces the total losses any more.
Does the calculation also cover the transition regime between continuum and free molecular flow?
Yes, the VDI Heat Atlas provides interpolation relations for the transition regime that link the continuum solution and free molecular flow (temperature-jump approach). This regime is particularly important in practice, because many vacuum insulations in the rough and fine vacuum range operate precisely there.