Gas Flow Through Microchannels : m athematical models

1. Gas Flow ThroughMicrochannels: mathematicalmodels M. Bergoglio, D. Mari, V. Ierardi, A. Frezzotti, G.P. Ghiroldi

2. For the gas flow through a tube ofcross-section A, the gas flow-rate isdefinedas the ratioof the quantityDM to the timeflowingto the cross sectional area A q = DM/Dt The quantityDMof the fluid can bemeasuredasanelementof volume or mass or numberofparticles. Volumflow rate qv= dV/dt m3/s Mass flow rate qm = dm/dt kg/s Molar flow rate qn = dn/dt mol/s NumberofparticlesqN = dN/dt 1/s d(p V ) / dt = pdV / dt = R Tdn/dtdT/dt=0 Pa.m3 /s throughput

3. Modes of flow of gas in leaks: • Turbulent flow • Choked flow occurs when the flow velocity approximates the speed of sound in the gas • Laminar flow occurs with leakage rates in the range from 10-2 to 10-4 Pa m3/s • Transition flow occurs in the gradual transition from laminar to molecular flow • Molecular flow is the most probable with leakage rate below than 10-4 Pa m3/s

4. p1,n1 B dA dJ p2, n2 Molecular flow: At low inlet pressure, gas may be considered as individual molecules which hardly interact. As a consequence, each molecule travels through the duct/volume in which the orifice is inserted without interacting with other molecules. The total gas flow is given by the independent motion of a lot of molecules. This flow is called molecular.

5. The conductance of the orifice may be calculated as the product between the conductance of an ideal aperture C0 and a factor K taking into account the transmission probability. Define molecular transmission probability, K, of a duct as the ratio of the flux of gas molecules at the exit aperture to the flux at the inlet aperture The conductance of an infinitely thin orifice of opening area A in the molecular flow regime is is the mean thermal velocity of the gas molecules. Molecular conductance is thus independent of inlet and outlet pressure

6. d2 d3 d4 f b b g e d5 d6 K1: transmission probability of cylindrical step K2: transmission probability of conical segment K0:main correction factor related to the transmission probability of the upper spherical segment, the cylindrical segment and the lower spherical segment Tup: transmission probability of upper spherical segment Tmid: transmission probability of cylindrical segment Tlow: transmission probability of lower spherical segment

7. p2 T p1 T l Molecular flow through a tube withcircular cross section Short tube Long tube

8. The Hagen–Poiseuille equation can be derived from the Navier-Stokes equations. For a compressible fluid in a tube the flow rate and the linear velocity are not constant along the tube. The flow is usually expressed at outlet pressure. As fluid is compressed or expands, work is done and the fluid is heated and cooled. For an ideal gas in the isothermal case, where the temperature of the fluid is permitted to equilibrate with its surroundings, and when the pressure difference between ends of the pipe is small, the flow rate at the duct outlet is given by

9. Transition flow occurs when the mean free path length of the gas molecules is about equal to the cross sectional dimension of the duct. Transition flow occurs under leakage conditions intermediate between those for viscous flow and those for molecular flow. For transition flow Knudsen’s law for molecular flow is modified by additional term that depends on the ratio R equal r/l that applies for the average pressure (p1+p2) / 2 existing within the leakage path. This correction term for transitional flow in leakage paths is give as factors Z C = Cv + Z Cm

13. The aim of the JRP task is to assess the transfer function of a calibration using helium in vacuum at 20 °C to a different gas species such as a refrigerant, CO2, methane and the new refrigerant gas 1234yf (automotive area) with reference to atmosphere. Starting from a specific design of capillary leak detailed analysis of the flow field within the capillary and suitable mathematical model will be developed. The aim of the model is to describe: the behaviour of the capillary conductance in a way completely independent from the gas species, to extend the calibration curve to various other gases; be of general use and predict gas flows in the range from about 10-7 Pa m³/s to about 10-3 Pa m³/s with relative uncertainty of few parts in 102. From an available and accurate design of the manufactured capillary leaks the gas flow will be modelled via DSMC and CFD tools in order to establish behaviour of the leak flow for different conditions of gas and pressures.

15. Direct Simulation Monte Carlo of gas flows Monte Carlo method is a generic numerical method for a variety of mathematical problems based on computer generation of random numbers. Direct simulation Monte Carlo (DSMC) method is the Monte Carlo method for simulation of dilute gas flows on molecular level, i.e. on the level of individual molecules. To date DSMC is the basic numerical method in the kinetic theory of gases and rarefied gas dynamics. Kinetic theory of gases is a part of statistical physics where the flow of gases are considered on a molecular level and described in terms of probability of changes of states of gas molecules in space and in time.

16. External force field r1 v1 Rebound of a molecule from the wall Basic approach of the DSMC method Gas is represented by a set of N simulated molecules X(t) = (r1(t), v1(t),…, rn(t), vn(t)) Velocities vi (and coordinates ri) of gas molecules are random variables. Gas flow is simulated as a change of X(t) in time due to – Free motion of molecules or motion under the effect of external (e.g. gravity) forces – Collisions between gas molecules – Interaction of molecules with surfaces of bodies, channel, walls, etc.

17. Computational fluid dynamics, CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. The fundamental basis of almost all CFD problems are the Navier–Stokes equations Navier–Stokes equations describe the motion of fluid substances. These equations arise from applying Newton’s second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow. A solution of the Navier–Stokes equations is called flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest (such as flow rate or drag force) may be found.

18. Kn=0.001 Diameters: 14 mm 16 mm Length: 0.74 mm Kn<<1 Kn>>1 Kn≈1

19. Diameters: 11 mm 6 mm Length: 0.38 mm

20. Solving the full non-linear Navier-Stokes equations it is possible to quantify the compressibility effects. The Mach number, Ma = u/c, where u is the gas velocity and c is the sound velocity, can be used to determine the importance of the compressibility effects. In general, the compressibility effects can be neglected for the Mach numbers lower than 0.3.

21. Modeling activity Test of a simple phenomenological model which expresses the mass flow rate as: MG = MNS [1+W (Kn; Dp=p2 , a)] Development of a DSMC code (2D, axisymmetric with reservoirs). Single species polyatomic gas or mixtures with arbitrary number of monatomic species Compressible NS+Fourier computations by a commercial code (Reservoirs + channel)

22. Long tube MG is obtained with the analytical formula by Gallis et al., in equation the diameter is set equal to the mean diameter: Dmean = (12,28+6,94)/2 = 9,61mm. Accomodation coefficient: a = 1

25. The full non-linear Navier-Stokes equations with no slip boundary condition are also used to quantify the effects of tubes with non constant cross sections.

26. Temperature distribution