For a gas flow simulation, the set up of the simulation domain and the particle statistics requires to decide values for certain parameters, which are listed in the following. For simplicity we assume that all cells have a cubical shape.

*a*= Cell dimension [m]*V*= Cell volume =_{c}*a*^{3}*N*= Quotient between physical number of particles_{REAL}*N*and number of simulation particles*N*_{S}*δt*= Period of one time cycle

The cell dimension should be less than the mean free path *λ* which itself reciprocally depends on total pressure *p*^{1)}, which leads to the following considerations:

,

.

Since the travelling distance of a particle during one time cycle should be also less than the mean free path, a similar relation holds for the time cycle:

.

From the ideal gas equation, we know for the physical number *N _{c}* of particles within a cell, that:

,

provided that the cell dimension *a* is adjusted reciprocally with total pressure *p* as mentioned above. It is possible to keep the number *N _{s}* of simulation particles per cell constant by adjusting

.

Especially at higher pressure, the computational effort is almost completely dominated by the collision routine. In the so-called *No time counter method* ^{2)}, the number *N _{COLL}* of pre-selected collision pairs is determined as follows:

.

If we put in the above described relations, we find for each cell that *N _{COLL}* is independent from the total pressure

For Argon we know, that for a total pressure of 1 Pa, the mean free path is about 6 mm. Thus, in order to be on the safe side, we chose a cell size of 5 mm at *p* = 1 Pa.

For the time cycle we consider the mean thermal velocity *c* of Ar at room temperature, which is approx. 400 m/s. As mean travelling distance at *p* = 1 Pa, we chose a length of 4 mm which is well below the mean free path resulting in:

.

The physical number of particles within a cell sized 5x5x5 mm^{3} at T = 300 K (room temperature) can be estimated from the ideal gas equation as *N* = 3.02 x 10^{13}. If we would like to maintain a constant
number of simulation particles per cell of *N _{s}* = 10, the resulting value for

Based on these starting considerations and on the scaling relations shown above, we obtain the following set of simulation parameters as a function of total pressure:

Pressure [Pa] | Cell dimension a | Time cycle [s] | N for 10 simulation particles per cell_{REAL} |
---|---|---|---|

0.1 | 50 mm | 1 x 10^{-4} | 3 x 10^{14} |

1.0 | 5 mm | 1 x 10^{-5} | 3 x 10^{12} |

10.0 | 0.5 mm | 1 x 10^{-6} | 3 x 10^{10} |

100.0 | 0.05 mm | 1 x 10^{-7} | 3 x 10^{8} |

1000.0 | 5 µm | 1 x 10^{-8} | 3 x 10^{6} |

10000.0 | 0.5 µm | 1 x 10^{-9} | 3 x 10^{4} |

100000.0 (~ 1 bar) | 0.05 µm | 1 x 10^{-10} | 3 x 10^{2} |

see Mean free path

Bird, G. A.; *Monte Carlo Simulation in an engineering context*, In: Progr. Astro. Aero. **74** (1981) 239-255.

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