e Ewart Æ Pierre Perrier Æ Irina Graur
J. Gilbert Me
Mass ﬂow rate measurements in gas micro ﬂows
Received: 10 June 2005 / Revised: 1 June 2006 / Accepted: 2 June 2006 / Published online: 11 July 2006
Ó Springer-Verlag 2006
Abstract The main objective of this experimental inves-
tigation on the gas ﬂow slip regime is to measure the
mass ﬂow rate in isothermal steady ﬂows through
cylindrical micro tubes. Two technical procedures de-
voted to mass ﬂow rate measurements are compared,
and the measured values are also compared with the
results yielded by diﬀerent approximated analytical
solutions of the gas dynamics continuum equations.
Satisfactory results are obtained and the way is clearly
opened to measuring mass ﬂow rates for higher Knudsen
numbers, over all the micro ﬂow transitional regime.
Since the early 1980s, mass ﬂow rates in microchannels
have mostly been measured using a liquid drop method
(Colin et al. 2004; Harley et al. 1995; Lalonde 2001;
Maurer et al. 2003; Pong et al. 1994; Zohar et al. 2002).
So far, in order to determine volumes variations and
mass ﬂow rates, the drop movement has been either
observed through a low power microscope (Harley et al.
1995) or determined visually as a meniscus of water
travelling along the marked scale of a syringe (Zohar
et al. 2002) or detected by means of optoelectronic
sensors (Colin et al. 2004). Other authors (Arkilic et al.
1997; Arkilic et al. 2001) have used a diﬀerent method
involving a sensitive dual-tank accumulation technique
based on the measurements of the pressure diﬀerences
between an accumulation reservoir and a reference tank.
Another approach used by some authors involves
the utilization of ﬂowmeters (Yao et al. 2004) or high
precision ﬂow sensors (Jang and Wereley 2004)to
measure the mass ﬂow rate. But this kind of mea-
surement is restricted to relatively high mass ﬂow rates
The main aim of the present study focusing on the gas
ﬂow regime is the validation of a mass ﬂow rate mea-
surement method based on direct pressure change
measurements. Two methods are presented:
1. The liquid drop method already used and tested in
previous years, developed here using new equipment.
2. A new method (new as far as the measurement range
is concerned) based on constant volume pressure
measurements, using up-to-date Inﬁconä pressure
gauges with a resolution of 1.9, 0.19 and 0.019 Pa.
Let us point out that the experimental methods
mentioned at the beginning of the introduction generally
concern experiments carried out in channels with rect-
angular (or trapezoidal) cross-sections, while the present
measurements were carried out in microtubes. Experi-
ments in this type of geometry are rare (Porodnov et al.
1974; Dong 1956; Tison 1993) and they were obtained in
tubes of relatively large diameter (3.64 cm in Dong
1956), or using capillary packets of 10–640 capillaries
(1974), which makes it diﬃcult to control the diameter of
Then the diﬀerent features and the respective poten-
tialities of the two experimental techniques are com-
pared. Their respective results are analyzed and
compared with theoretical results derived from contin-
uum approaches taking into account the rarefaction ef-
fects present in microﬂows.
Although the NS equations are derived from a ﬁrst
order kinetic solution, many authors (Karniadakis and
Beskok 2002; Cercignani 1964; Colin et al. 2004;
Deissler 1964; Hadjiconstantinou 2003) have suggested
to use in this framework the velocity slip conditions of
second order according to the Knudsen number, to
better take into account the rareﬁed eﬀects for the
moderately rareﬁed gas ﬂows. The implementation of
T. Ewart (&) Æ P. Perrier Æ I. Graur Æ J. Gilbert Me
partement de Me
tique - UMR CNRS 6595,
de Provence - Ecole Polytechnique Universitaire de
Marseille, 5, rue Enrico Fermi, 13453 Marseille cedex 13, France
Experiments in Fluids (2006) 41: 487–498