Gas Flow Through Microchannels:
mathematical models
M. Bergoglio, D. Mari, V. Ierardi,
A. Frezzotti, G.P. Ghiroldi
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
For the gas flow through a tube of cross-section A, the gas flow-rate is
defined as the ratio of the quantity DM to the time flowing to the
cross sectional area A
q = DM/Dt
The quantity DM of the fluid can be measured as an element of volume
or mass or number of particles.
Volum flow rate qv = dV/dt m3/s
Mass flow rate qm = dm/dt kg/s
Molar flow rate qn = dn/dt mol/s
Number of particles qN = dN/dt 1/s
d(p V ) / dt = p dV / dt = R T dn/dt
throughput
dT/dt=0
Pa.m3 /s
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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.
p1,n1
B
dJ
dA
p2, n2
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
The conductance of an infinitely thin orifice of opening area A in
the molecular flow regime is
_
C  14v A
v
is the mean thermal velocity of the gas molecules.
Molecular conductance is thus independent of inlet and outlet
pressure
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
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
C  C 0  K 0  K 1,2
d2
d3
d4
2
f
e
b
g
b
 d4  v
C 0    
 
 2  4
K 1,2  K 1  K 2  0.9999  0.0001
d5
K1: transmission probability of cylindrical step
d6
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
 d 42   1
1
  2 
K 0  d 3  Tup

 1  1 2
 Tmid Tlow

Tup: transmission probability of upper
spherical segment
Tmid: transmission
cylindrical segment
probability
of
Tlow: transmission probability of lower
spherical segment
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
Molecular flow through a tube with circular cross section
p2
p1
C  CA K  A
K 
4d
K 
3l
T
l
T
le
1
 1
3l
l
3
7r
1
3l
1 e
8r
RT
K
2M
Short tube
2
4d 1  d   2l 
K 
   ln 
3l 2  l   d 
Long tube
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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
 d4 p12  p22
q
 
128 l
2
The Hagen–Poiseuille equation can be derived from the Navier-Stokes
equations.
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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
r
r
l
Z  0.1472 
l 1  3.095 r
l
C = Cv + Z Cm
1  2.507
q pV  3.342
r
l
3
r

1  2.507

RT
l
p 1  p 2 )   0.1472 r 
M
l 1  3.095 r

l

Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013





q pV  3.342
r
l
r

1  2.507

RT
l
p 1  p 2)    0.1472 r 
M
l 1  3.095 r


l

3
C  3.342
r
l
3
r

1  2.507

RT
r
l
  0.1472 
M 
l 1  3.095 r

l

C M  3.342
r
l
3












r

1

2
.
507

r
l
RT   0.1472 
l 1  3.095 r


l

Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013






Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
2.0E-08
1.8E-08
1.6E-08
1.4E-08
1.2E-08
1.0E-08
8.0E-09
6.0E-09
4.0E-09
1.E+06
1.E+07
1.E+08
4.0E-09
3.5E-09
3.0E-09
2.5E-09
2.0E-09
1.5E-09
1.E+05
1.E+06
1.E+07
2.3E-09
2.2E-09
2.1E-09
2.0E-09
1.9E-09
1.8E-09
1.7E-09
1.6E-09
1.5E-09
1.E+03
1.E+04
1.E+05
1.E+06
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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.
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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.
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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.
External force field
r1
v1
Rebound of a molecule from the wall
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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.
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
Kn=0.001
Diameters:
14 mm
16 mm
Length:
0.74 mm
Kn<<1
Kn>>1
Kn≈1
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
Diameters:
11 mm
6 mm
Length:
0.38 mm
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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.
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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)
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
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.
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013
Temperature distribution
Metrologia del vuoto negli ambienti industriali – Torino - 27 giugno 2013