Graur

M V Johansson

) M Wüest

P Perrier

I Graur

Microporous media

Low-pressure gas flow properties of sintered stainless steel microporous media

I. INTRODUCTION

In-situ calibration is typically done using standard leaks from glass, quartz, or fluoric resin. However, these types of leaks have a limit of applicability to different gases 1 , 2 because each standard leak is specific for a single gas. As the indirect pressure sensors are highly gas-dependent, several standard leaks are required to cover the desired calibration range. The capillary or orifice leaks work well with the different gases. But in the typical pressure ranges used for calibration, the gas flow is usually in the transitional regime, where an explicit analytical expression of conductance is absent. When decreasing the tube or aperture diameter to ensure free molecular regime, to have an analytical expression relating the flow parameters and the constant conductance, the flow rate becomes very low, leading to, for example, a long time for the calibration process. The authors of Ref. 3 suggested the implementation of a porous medium, namely, sintered stainless steel with small pore sizes, for sensor calibration. This kind of microporous media can ensure a sufficiently high gas flow rate and the free molecular flow regime (i.e. constant conductance) inside a porous sample for relatively high-pressure due to micro-metric pore size. The last property has several useful characteristics. The first consists of the independence of the conductance of the mean pressure when a flow inside a medium is in free molecular regime. Therefore, when the conductance has been measured at a certain pressure, it will a) Also at SINTEF Energy Research, Gas Technology department, Kolbørn Hajes vei 1, 7034 Trondheim, Norway not change as the mean pressure decreases. Secondly, the different gas species' conductance has a simple correlation by the square of the molecular mass. Furthermore, it is possible to calibrate the sensors for gas mixtures, as the interaction between the different gas molecules in a mixture is negligible in the free molecular regime. Lastly, the dependence of temperature can easily be corrected [START_REF] Yoshida | Newly developed standard conductance element for in situ calibration of high vacuum gauges[END_REF] . All mentioned above properties of the conductance of a microporous medium have allowed its successful application for in-situ calibration of high-vacuum gauges [START_REF] Yoshida | New leak element using sintered stainless steel filter for in-situ calibration of ionization gauges and quadrupole mass spectrometers[END_REF] . Furthermore, this approach can also be extended to calibrate ultrahigh vacuum (UHV) sensors [START_REF] Yoshida | Calibration of ultrahigh vacuum gauge from 10 -9 pa to 10 -9 pa by the two-stage flow-dividing system[END_REF] .

A microporous leak for calibration as a stand-alone device could be useful in practice, especially if this device will allow users to directly connect to a vacuum vessel and calibrate the sensors in place. This article is the first step in this direction and it presents the experimental methodology for the characterization of gas flowing through the porous sintered stainless steel sample. Firstly, the global methodology of the pressure measurements in time is presented, which allows the further extraction of the flow characteristics through the porous medium such as the mass flow rate, conductance, and permeability, by fitting the pressure measurements in time. Then, the conductance and permeability behaviors are analyzed in a large mean pressure range. The proposed approach also allows the calculation of the effective pore size to characterize the gas rarefaction. This is useful for determining the free molecular flow regime, for a certain gas type, pressure and temperature, at which the conductance becomes constant, whatever is the pressure. This study allows several preliminary conclusions in the development of using the porous conductance element for in situ sensor calibration.

In this section, the methodology for measuring different characteristics of a gas flow through a porous sample like the mass flow rate, conductance and permeability is presented. The constant volume technique [START_REF] Knudsen | Die Gase der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren[END_REF][START_REF] Arkilic | Gaseous slip flow in long microchannels[END_REF][START_REF] Arkilic | Mass flow and tangential momentum accommodation in silicon micromachined channels[END_REF][START_REF] Ewart | Mass flow rate measurements in gas micro flows[END_REF][START_REF] Ewart | Mass flow rate measurements in microchannel, from hydrodynamic to near free molecular regimes[END_REF][START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF] is adopted for the mass flow rate measurements through a porous sample between the two tanks of known volumes. The pressure evolutions in both tanks due to the gas flowing from the high to the low-pressure tank through a porous sample are measured. Then, the gas pressure variations in time are directly related to the mass flow rate, conductance, and permeability. This technique is similar to the Brace technique Ref. 11 (pulse-decay method), usually used to analyze the permeability of the porous samples. However, the proposed methodology could be applied not only to characterize the porous samples, but also other type of connections as tube, bend etc. Moreover, it also allows taking into consideration the effects of gas rarefaction. These effects are usually characterized by the Knudsen number, Kn, or rarefaction parameter, δ , which are defined as following

Kn = ℓ a , δ = a ℓ , (1)
and represent the ratio of two parameters, the characteristic flow dimension, a, and the equivalent molecular free path ℓ, defined as 12

ℓ = µv 0 p , v 0 = √ 2RT , (2)
where v 0 is the most probable molecular speed, µ is the gas viscosity, see A, p is the gas pressure, T is the gas temperature, R is the specific gas constant. The characteristic flow dimension, a, is initially unknown due to the complex geometry of the porous media. As shown in Ref. 13, the gas rarefaction in a porous medium can nevertheless be qualified by a single dimension, the characteristic flow dimension, which, furthermore, gives morphological information of the porous structure. FIG. 1. Principle of the pressure gradient driven flow experiment. A closed system with two tanks connected by a microporous medium, with an initial gas pressure difference causing gas to flow from the high-pressure to the low-pressure tank.

To characterize a porous sample, i.e., to measure its conductance and permeability, we use a closed system consisting of two tanks with the known volumes, V 1 , V 2 , only connected by the porous sample, see Fig. 1. To relate the mass of the gas flowing through a porous sample, generated by setting an initial gas pressure drop between the reservoirs, we use the ideal gas law, which is valid under equilibrium condition. Its use could be also justified by the pressure (around atmospheric and lower) and temperature (lower than the critical temperature) ranges for all considered here gases, see also Appendix in Ref. 13.

However, the gas pressure and mass of a gas in a tank change in time. Therefore, we assume that the evolution of the gas state in the tanks is a quasi-steady process, that is, a succession of local equilibrium. This assumption is true when a small unbalancing force exists and modifies the system slower than the system reaches a local equilibrium. According to the ideal gas law, the mass variation in the high-pressure tank reads

Ṁ1 = - dM 1 dt = - V 1 RT 1 dp 1 dt 1 - dT 1 /T 1 dp 1 /p 1 , (3)
where p 1 is the gas pressure, M 1 is the total mass of a gas, and T 1 is the gas temperature in the high-pressure side, tank 1. When a pressure gradient is applied under nearly isothermal conditions, the relative temperature change will be much lower than the relative change in pressure. Then, the quantity defined as

ε = dT 1 /T 1 dp 1 /p 1 (4)
is negligible in comparison to unity as the temperature T 1 is nearly constant in time. As the temperature difference between the tanks is negligible, that is T 1 ≈ T 2 , we denote in the following the temperature in each tank as T . Taking into account previous analysis of the temperature effects, which leads to the conclusion that parameter ε, defined by Eq. ( 4), is much smaller than 1 (ε ≪ 1), and considering an infinitesimal time interval dt, we obtain the pressure gradient driven mass flow rates for the high and low-pressure tanks as

Ṁ1 = - V 1 RT dp 1 dt , Ṁ2 = V 2 RT dp 2 dt . (5)
To calculate the mass flow rate in this system using one of Eqs. (5) we need to know the pressure change in time in highpressure or low-pressure tanks, namely, the derivative of p 1 or p 2 . Sometimes, it is also useful to use the pressure difference ∆p(t) = p 1 (t) -p 2 (t) instead of the pressure in a tank. In this case, the mass flow rate reads

Ṁ(t) = V 0 RT d(∆p(t)) dt , where V 0 = V 1 V 2 V 1 +V 2 . (6)
From the conservation of mass, the gas which leaves the highpressure tank has to go to the low-pressure tank and the gas mass flow rate between the two tanks has to be equal

Ṁ1 (t) = Ṁ2 (t) = Ṁ(t).
From the ideal gas law and admitting the mass conservation along the microporous medium at any time, we can write the following relation for two tanks maintained at the same temperature

-V 1 dp 1 = V 2 dp 2 . ( 7
)
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

It is worth underlining that in Eq. ( 7) both expressions are exact (perfect) differentials, so their integration does not depend on the form (linear or exponential) of the pressure variation in time. Let us define the time t 0 as the initial time of an experimental run and t f as the final time of this run, when the pressures in both tanks become equal, so p 1 (t f ) = p 2 (t f ) = p f . Then, by integrating Eq. ( 7) from t 0 to t f and by solving for final pressure, one obtains

p f = V 1 p 1 (t 0 ) +V 2 p 2 (t 0 ) V 1 +V 2 . ( 8
)
Equation ( 8), derived from the mass conservation along with a porous sample, will be used in the next sections.

B. Pressure evolution and mass flow rate

To measure the system's mass flow rate using the constantvolume technique, we need to know the pressure (or the pressure difference) derivative in time, as seen from Eqs. ( 5), (6). In the following, we will derive the analytical expressions for the pressure and the pressure difference evolutions in time.

As various characteristics of an element of a vacuum system, through what gas can flow, like a tube, bend or porous element etc., depend on applied pressure difference at its ends, we can assume that the mass flow rate in each tank is proportional to the applied pressure difference

Ṁ1 = - V 1 RT dp 1 dt = A(p 1 (t) -p 2 (t)), Ṁ2 = V 2 RT dp 2 dt = A(p 1 (t) -p 2 (t)), (9)
where A is a proportionality coefficient. The physical meaning of this coefficient could be associated with different gas flow properties of interest, see more details in Sections II C and VI C. By subtracting the second expression of Eqs. ( 9) from the first one, we obtain the first-order differential equation for ∆p(t)

d(∆p(t)) dt = - ART V 0 ∆p(t). (10)
By integrating this differential equation, completed by the initial condition for pressures in each tank, namely, p 1 (t 0 ) = p 01 and p 2 (t 0 ) = p 02 , we obtain the following solution of Eq. ( 10)

∆p(t) = ∆p 0 exp (-(t -t 0 )/τ) , (11)
where

∆p 0 = p 1 (t 0 )-p 2 (t 0 ) = p 01 -p 02 . In previous equation τ is equal to τ = V 0 ART , (12)
and it can be associated with the gas relaxation time. It is important to note that equation ( 10) can be integrated under the condition of τ constancy in time. As the isothermal conditions are assumed, τ is constant in time when the coefficient A is independent of time. This latter condition will be discussed in Section II C. Analogously, we can derive the expressions for the pressure evolution in each tank. Namely, for the first tank we use the first expression of (9), then, we integrate Eq. ( 7) between arbitrary time t and final time t f . As the final pressure in each tank, p f , is the same, we easily obtain the first-order differential equation for the pressure evolution in the first tank, which can be integrated using the value of initial pressure in the first tank at t 0 , p 1 (t 0 ), as initial condition. Then, the analytical expression of the pressure evolution in the first tank becomes

p 1 (t) = p f + (p 1 (t 0 ) -p f ) exp (-(t -t 0 )/τ 1 ) . ( 13
)
Obtained in the similar way the pressure evolution in the second tank reads

p 2 (t) = p f + (p 2 (t 0 ) -p f ) exp (-(t -t 0 )/τ 2 ) . (14)
It is to note that, when deriving expressions ( 11) and ( 13), (14), the hypothesis of the mass conservation between tanks is used, so τ is the same in all these formulas. However, the different τ has been used in Eqs. ( 11) and ( 13), ( 14) to simplify the explanation of experimental verification of the mass conservation. Finally, by deriving expressions for pressure variation in each tank, Eqs. ( 13), (14), and the pressure difference between the tank, Eq. ( 11); and using the expressions of the mass flow rates, Eqs. ( 5), (6), we see that by deriving expressions for p 1 (t), p 2 (t), or ∆p(t), we have the same expression for the mass flow rate

Ṁ1 (t) = Ṁ2 (t) = Ṁ(t) = V 0 RT ∆p(t) τ . (15)
Therefore, we can measure the pressure evolution in any tank (or pressure difference between them), then fit these measured data using τ as a fitting parameter, as suggested in Ref. 10, to finally obtain the mass flow rate. In addition, once we have confirmed the conservation of mass experimentally in a setup, we do not need to specify whether the relaxation time is obtained from the fitting of p 1 (t), p 2 (t), or ∆p(t) variations. In practice, if the pressure sensors have different accuracy, we simply take the relaxation time by using the pressure variations measured by the sensor with the highest accuracy.

C. Conductance

One of the important characteristics of an element in a vacuum system is its conductance. To connect the measurement of the pressure evolution in time and the conductance, let us first introduce the throughput Q, sometimes called pV -flow, which is defined as in Ref. 14

Q = d (pV ) dt . ( 16
)
The throughput between two tanks depends on the applied pressure difference and the conductance C of an element that This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

connects both tanks Q = C∆p(t). (17)
In the frame of constant-volume technique and under isothermal condition the throughput (16) in each tank is expressed as

Q 1 = - dp 1 dt V 1 , Q 2 = dp 2 dt V 2 . ( 18
)
Using relation (17) between conductance and throughput one obtains

dp 1 dt = - C V 1 (p 1 -p 2 ), dp 2 dt = C V 2 (p 1 -p 2 ). (19)
By subtracting the second expression of ( 19) from the first one the first-order differential equation for the pressure difference is obtained

d(∆p(t)) dt = - C V 0 ∆p(t). (20)
The comparison between Eqs. ( 10) and ( 20) allows us to identify A constant in Eq. (10), which is equal to

A = C RT . (21)
From Eqs. ( 12) and ( 21) we obtain the expression for the conductance via the gas relaxation time as

C = V 0 τ . (22)
As it was discussed in Section II B, the relaxation time τ can be determined either from p 1 (t), p 2 (t) or ∆p(t) measurements. This is useful in practice as we do not always have the same accuracy on the pair of pressure sensors, then we can use the one with the highest accuracy to measure the pressure evolution. Additionally, we do not strictly need to measure the pressure difference to obtain the conductance, this can be done by measuring the pressure evolution in one tank using one pressure sensor.

To integrate Eq. ( 20) we have to assume the constancy of C/V 0 (or τ) over time. However, it is well known that the conductance changes as a function of gas rarefaction depending on the gas nature, temperature, and pressure. To ensure the constant level of gas rarefaction during a time interval, we need to have constant pressure, specifically constant mean pressure, under isothermal conditions. In Section II E the conditions for the constancy of mean pressure and the conditions, where its variation remains small, are outlined. It is interesting to note that the importance of this requirement also depends on the flow regime as the intensity of the conductance variation due to a mean pressure change depends on the gas rarefaction. More specifically, in the hydrodynamic regime, the conductance varies essentially with a mean pressure, but in the free molecular regime, it is independent of mean pressure, at least for a tube, aperture, and studied here porous medium. Therefore, if we use the proposed approach in the free molecular regime when the conductance is constant, we do not have any restrictions on the variation in mean pressure, volume difference, or pressure ratios.

D. Permeability

Another important characteristic of a porous medium is its permeability, K, which can be extracted from the pressure variation in the same way as the mass flow rate and the conductance. To express the gas permeability, we write the classical Darcy law [START_REF] Do | Adsorbtion analysis: equilibria and kinetics[END_REF] in the form where the pressure difference through a sample is replaced by the pressure gradient [START_REF] Sander | Laboratory measuremant of low permeability unconventional gas reservoir rocks: A review of experimental methods[END_REF] :

Q V = - KS µ dp dx , ( 23
)
where S is the surface of porous sample. In previous equation Q V is the volumetric flow rate, which is related to the mass flow rate as

Q V = Ṁ RT p . ( 24
)
Replacing Q V in Eq. ( 23) by its expression involving Ṁ, Eq. ( 24), then by integrating the obtained equation along with the porous sample and by using the mass conservation property, we obtain the expression, analogous to Eq. ( 9),

Ṁ = K µ S L p m RT ∆p(t), (25)
where the mass flow rate is also proportional to the pressure difference applied to the sample ends. In previous relation p m is the mean pressure p m = 0.5(p 1 (t) + p 2 (t)), L is the porous sample thickness. As the mass flow rate is related to the pressure derivative in time through Eq. ( 9), we rewrite Eq. ( 25) in the following form

d(∆p) dt = - ART V 0 ∆p(t), where A = KSp m µLRT . (26)
Applying the same reasoning as in previous sections and the definition of the relaxation time τ by Eq. ( 12) we find the relation between the permeability and the relaxation time in the form

K = L S µ p m V 0 τ , (27)
where the relaxation time τ can be determined from the fitting of p 1 (t), p 2 (t), or ∆p(t) measurements. As it was pointed out in Section II B the differential equation for the pressure evolution in time, here Eq. ( 26), can be solved when the proportionality coefficient A is independent of time. Since the permeability depends on the mean pressure, the constancy of the mean pressure during an experimental run ensures the constancy of A coefficient and so the validity of expression (27).

The conditions of the mean pressure constancy are provided in Section II E. From Eqs. ( 27) and ( 22) we derive a useful relation between permeability and conductance

K = L S µ p m C. (28)
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

E. Conditions of the constancy of mean pressure

In this section, we establish the conditions of the constancy of mean pressure during the measurement procedure, which leads to the pressure relaxation time constancy, and justify the use of Eqs. ( 5), (6) for the mass flow rate, Eq. ( 22) for the conductance, and Eq. ( 27) for the permeability calculations.

In addition, as it has been pointed out in Ref. 17, in the general case, the constancy of the mean pressure is an important point, especially for high-pressure experiments, where the viscosity and compressibility factor may change as a function of pressure. In the presented, experiments the implemented pressure and temperature conditions allow us to stay under the ideal gas law assumptions, and the viscosity does not change with pressure. However, it is still important to have a constant mean pressure as for low permeable porous media, the permeability can be a function of mean pressure due to rarefaction effects.

The expression for the final pressure (8) in the system as a function of initial pressures in the tanks and the tank volumes was obtained in Section II. From this expression, the maximal variation of the mean pressure p m (t) with time, from its initial value p m (t 0 ) to its final value p m (t f ) = p f , is estimated as

p m (t f ) p m (t 0 ) = p 1 (t 0 )V 1 + p 2 (t 0 )V 2 (V 1 +V 2 )p m (t 0 ) . (29)
From the analysis of Eq. ( 29) it leads that for equal tanks volumes, the mean pressure p m (t) does not vary in time, i.e., between its initial value, p m (t 0 ), and its final value, p m (t f ), so p m (t 0 ) = p m (t f ). When the volumes are different, the maximal amplitude of mean pressure variation between its initial state p m (t 0 ) and its final state p m (t f ) for a given volume ratio,

k V = V 1 /V 2 ,
and initial pressure ratio, k p = p 1 (t 0 )/p 2 (t 0 ), is calculated from

p m (t f ) p m (t 0 ) = 2(1 + k V k p ) (1 + k V )(1 + k p ) . (30)
In our experimental conditions, two tanks volumes are related as k V = V 1 /V 2 = 1.071, so from Eq. ( 30) we can find that for the initial pressure ratio between the tanks, k p , equal to 1.5, 2, and 3, the ratio p m (t f )/p m (t 0 ) is equal 1.007, 1.011 and 1.017, respectively. Therefore, the initial pressure ratio equal to 2 leads to approximately 1% of the deviation of the mean pressure from its initial value. Under our experimental conditions, the experimentally evaluated value of p m (t f )/p m (t 0 ) was found lower than 1%.

III. EXPERIMENTAL SETUP

The sketch of the experimental setup, shown in Fig. 2, provides more details on the experiment, compared to the simplified presentation of the measurement principles given in Fig. 1. The experiments are performed within a narrow temperature range, excluding any heat source in an environment. The temperature is measured using three 4-wire RTD sensors, one precision probe measuring the ambient temperature, and two flat surface sensors attached to the reservoirs. The pressure is measured close and perpendicular to the gas flux in an ISO-KF 25 cross. For high-pressure measurements around atmospheric pressure, two additional reservoirs, V ref , were connected to increase the time of pressure relaxation and, therefore, to make it measurable easily. Both volumes V ref are the reference volumes. These volumes are also used to determine the total volume of each high and low-pressure side, by using the gas expansion method of Ref. 14 for each pair of pressure sensors, as some pressure sensors have different internal volumes. The values of high and low-pressure side tank volumes, V 1 and V 2 , respectively, see Fig. 2, used for different pressure ranges, are provided in Table I.

The method applied here requires that the tank volume is relatively large compared to the volume occupied by a gas inside a microporous medium. In our experimental setup, this ratio is larger than 10 3 . FIG. 2. Schematic of the experimental setup. It includes a diaphragm vacuum pump (DVP), a turbo molecular pump (TMP), and a gas delivery system with four gas bottles, with helium, nitrogen, argon, and krypton. The initial pressure is set using the valves v Pump and v Gas . Once equilibrium has been reached the initial pressure difference is set by an instance opening and closing the valves v Drop or v Gas . The pressure is measured using a pair of two capacitance diaphragm gauges (CDG) measuring pressure in the high-pressure and low-pressure sides, p 1 and p 2 , respectively. The temperature is measured using three 4-wire RTD sensors, one precision probe of ambient temperature, T Ambient , and two flat surface sensors, attached to measure the temperature of each reservoir.

F.S p 2 [kPa] F.S p 1 [kPa] V 2 [cm 3 ] V 1 [cm 3 ] V 0 [cm
V 0 is equal to V 1 V 2 /(V 1 +V 2 ).
The analyzed sintered stainless steel microporous medium has a cylindrical shape with thickness specified by the manu-This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS

IV. EXPERIMENTAL PROCEDURE A. Leak and degassing

The pressure relaxation experiments have been done in a closed vacuum system without pumping, so the leak and degassing need to have negligible effects. Several leakage tests were performed showing the absence of the detectable increase in pressure, measured with the lowest F.S. pressure sensor of 1.33 kPa for 30 minutes. In addition, for all realized measurements, we did not detect any linear increase of the mean pressure during a time interval equal to several times the typical experimental duration.

The main part of the degassing in our pressure range comes from water vapor absorbed in the connections and tank's walls made of stainless steel. To get rid of the water, we pumped down the system for 24 hours before starting the measurements. Moreover, the degassing process was accelerated by heating the system and flushing it with dry gas.

B. Temperature measurements and stability

To extract the mass flow rate, conductance and permeability from the pressure measurements we have to use the values of the gas temperature in each tank, T 1 and T 2 . However, the gas temperature in each tank was not measured directly. The ambient temperature T in the vicinity of the setup and the temperature of the external surfaces of both tanks were measured and their variation during an experiment time length was found typically around 0.1 K. Then, it was assumed that T 1 ≈ T 2 and denoted in the following as T . This is a reasonable assumption, as the experimental setup is in a closed room with covered windows, so any change in the room temperature occurs much slower than the typical time of one measurement. Moreover, we can expect small gradients in the system as the material of the tanks and connection tubes is stainless steel. The only source of heat in the experiments is the pressure sensors, but they do not have integrated internal heating, and only small heating comes from the electronics.

C. Measurement procedure

First, the setup is evacuated, typically overnight, with all valves open and the pressure sensors are zeroed. To ensure a high gas purity, the experiment is flushed three times with the test gas before starting the experiments. Then, the system is pressurized up to the full-scale of the pressure sensors. The gas flow is initiated by a small pressure difference set between the low-and high-pressure sides, either by rapidly opening the valve v Drop , see Fig. 2, or by adding additional gas in the high-pressure side from the gas supply. After that, the two sides are only connected by the porous medium and the pressure relaxation is recorded in both the high-pressure, p 1 , and low-pressure, p 2 , sides. A larger pressure ratio than intended for analysis is set to ensure that the system has sufficient time to reach an isothermal state and only a small pressure ratio is used for analysis. Then, several tests on the mass conservation are carried out, see details in Section V C, to be sure that all relaxation times extracted from the measurements of the pressure in each tank and the pressure difference between tanks are the same, i.e. τ 1 = τ 2 = τ. After that, from the measured pressure relaxation time and with known volumes, temperature, and gas type, the mass flow rate, conductance, and permeability are extracted.

V. DATA ANALYSIS

A. Fitting range Initially, the system is in equilibrium, and p 1 = p 2 . Then, the pressure relaxation is actuated by a rapid opening and closing of the valve connecting tank 2 (low-pressure tank), and the pump, which leads to the pressure drop in this tank and to the establishment of the pressure difference between two tanks. During the first few seconds, the pressure relaxation may be in a non-equilibrium and not yet in a quasi-equilibrium state. In the non-equilibrium state, a small and rapid temperature drop may take place in the low-pressure side due to the pressure drop, then the system reaches thermal equilibrium again, see Fig. 4.

If the porous sample is of significant size, the time to reach a quasi-equilibrium could be longer, related to filling the sample with a gas. Moreover, some time is also needed for the pressure sensors to reach equilibrium. Therefore, for an experimental run, we set a larger pressure ratio than we intend to measure.

From time t 0 , see Fig. 4, the system reaches the quasistationary state, where there is still a pressure change in time, up to the final pressure equilibrium, p 1 = p 2 , but determined by one single relaxation time. This range, from t 0 , of the quasi-stationary relaxation process is used for the data fitting. The first non-equilibrium part of the pressure relaxation process (marked with the gray color in Fig. 5) is cut away.

B. Fitting method

The pressure measurements in time, p 1 (t), p 2 (t), and ∆p(t), are fitted using Levenberg-Marquardt non-linear regression, using Eqs. ( 13), ( 14) and (11), respectively. These equations have three parameters, the initial and final pressures and the relaxation time, but two first of them can be obtained by a technique described below, and only the relaxation time is used as the fitting parameter.

The final pressure value, p f , is obtained from the mean value of p m at the end of the measurements when the mean This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227 The blue and red curves correspond to the high and low-pressure tanks, respectively. After initial pressure equilibrium, p 1 (t i ) = p 2 (t i ), a sudden pressure drop is made by a valve opening, where the system is briefly in a non-equilibrium state, then shortly afterward, the quasiequilibrium is reached again, where the pressure decay is measured. At the final stage, the system is in equilibrium again, and p 1 (t f ) = p 2 (t f ). pressure becomes constant. The initial pressure values, p 1 (t 0 ), p 2 (t 0 ), ∆p(t 0 ), are obtained from the intercept of a linear fit of a short time interval, which is typically around 1 or 2 seconds of measurements and the data acquisition frequency is typically 10 -30 Hz. This is a suitable strategy as the exponential decay is locally linear and it is important to average over a few measurements to get a reliable value.

This fitting procedure proved to be a robust technique as it is not sensitive for small variations in the data or their fitting range in contrast to using, for example, all three parameters p 1 (t 0 ), p 1 (t f ), and τ in Eq. ( 13) simultaneously as fitting parameters.

C. Different tests of measurement quality

It is essential to ensure the measurement quality and to confirm that the physical assumptions are satisfied, especially when the geometry of connection is complex as for a porous medium, where there are not any reference values. In the following, we propose different tests to guaranty the measurement quality.

Conservation of mass

To confirm conservation of mass, we verify if the mass flow rates, Ṁ1 and Ṁ2 , calculated from the pressure measurements in each tank, are equal within experimental uncertainty. The error bars are also provided for both measurements.

Logarithm of the pressure difference

Another useful method to check measurement reliability, is to see if the logarithm of the pressure difference ln (∆p(t)) = ln (∆p 0 ) -

1 τ t (31)
is linear in time. One example of the measured and fitted logarithmic pressure difference in time is shown in Fig. 7, where the linearity in time of the measured pressure difference is clearly seen. However, in practice this test is difficult to apply in the range close to the final pressure, because there the logarithmic pressure variation can deviate from a linear due to the fluctuations of the pressure measurements which become to be of the same order as the pressure difference itself.

Relaxation time

When the mass is conserved between the tanks, the relaxation times in Eqs. ( 13), ( 14) and (11) have to be the same. We can obtain an overview of the measurement quality by plotting the relaxation times τ 1 and τ 2 , obtained from the fits of pressures p 1 and p 2 , measured in the high and low-pressure tanks, respectively, as a function of mean pressure. Figure 9 shows these relaxations times τ 1 and τ 2 , divided by the corresponding tank volume, to remove the volume dependence, see more details in Section VI A. From Fig. 9 we can immediately see that the relaxation times from p 1 and p 2 measurements are equal, and thus, the mass is conserved.

Real-time final pressure check

Another way, also very useful, to check the quality of the measurements, is to plot in real-time the pressure calculated as

p f (t) = V 1 p 1 (t) +V 2 p 2 (t) V 1 +V 2 . ( 32
)
The previous expression is similar to Eq. ( 8), to calculate the final pressure as a function of initial pressures. But now, in Eq. (32) the pressures in both tanks and final pressure depend on time. The calculated final pressure from Eq. (32) p f (t), should be constant in time, and any deviation from the constant value can be used as an indicator of the quality of the measurements. Equation ( 8) and, consequently, Eq. ( 32), has a simple form, but several assumptions were made for their derivation, namely, the conservation of mass and assumption of the quasisteady state. Therefore, the latter could be checked by analyzing the constancy of the final pressure in time. In addition, Eq. ( 32) also allows verifying if the volumes, V 1 and V 2 , are correctly determined.

D. Measurement uncertainty

The uncertainty of the conductance measurements, when using Eq. ( 22), is calculated by the classical way,

δC C = δV 0 V 0 + δ τ τ . ( 33
)
The relative measurement uncertainty on the conductance is presented in Table II, and it is lower than 4.4%. The uncertainty of the permeability measurements, when using Eq. ( 27), reads

δ K K = δV 0 V 0 + δ L L + δ S S + δ µ µ + δ p m p m + δ τ τ . ( 34
)
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

The relative measurement uncertainty on the permeability is presented in Table II, the permeability uncertainty lies in the range 5.0% -6.4%.

δV0 V0 δ L L δ S S
δ µ µ δ pm pm δ τ τ uncertainty conductance 3.0% < 1.4 % <4.4% permeability 3.0% 0.5% 0.1% 1.0% < 0.6 % < 1.4 % 5.0% -6.4 % TABLE II. Measurement uncertainties of the microporous media conductance and permeability.

VI. RESULTS

In this section, we present the main results on the relaxation time, conductance, and permeability. The data on the mass flow rate are not really presented, but these data are used to determine the effective pore size in Section VI B 2 following the methodology developed in Ref. 13.

A. Pressure relaxation time

The pressure relaxation time, sometimes called relaxation time, is the most important parameter that characterizes the properties of a gas flowing through a medium, i.e. tube, bend, porous medium etc. The relaxation time is directly related to the conductance, as outlined in Ref. The pressure relaxation time is obtained from the fitting of the pressure evolution in time in each tank, see also Ref. 10. For this study, the tanks of different volumes were used, depending on the pressure range, see Table I. Figure 8 shows the relaxation time obtained for different mean pressures using three volumes and pressure sensor pairs. As it is clear from Fig. 8, the relaxation time changes when the volume changes, which leads to its discontinuity as a function of mean pressure. We can remove this dependency by dividing the pressure relaxation time by the volume V 0 , used in the experimental system, which depends on two (high-pressure (V 1 ) and lowpressure (V 2 )) volumes. Figure 9 presents the pressure relaxation times, reduced by the volume used for the specific measurement range (Table I), which makes the relaxation time a smooth function of the mean pressure.

Furthermore, Figure 9 gives the relaxation times obtained from the sensors providing the measurements of pressure p 1 and p 2 . Finally, only a small difference was detected in relaxation times, obtained from p 1 and p 2 pressure fits. From this result, we can conclude that we have conservation of mass, a fundamental property in this type of measurements.

The full-scale of the sensors used for each pressure range is not always the same at the respective high and low-pressure sides. The experimental confirmation of the conservation of mass allows us to use the pressure measurements made by the sensor with the highest accuracy for data evaluation and Three different pairs of volumes were used for different pressure ranges, see Table I. HELIUM( 1) ARGON( 1) NITROGEN( 1) KRYPTON( 1) HELIUM( 2) ARGON( 2) NITROGEN( 2) KRYPTON( 2) FIG. 9. Double logarithmic representation of the gas pressure relaxation time over volume, V 0 , as a function of mean pressure. The squares with a central dot represent the pressure relaxation time from the sensor providing the measurements of pressure p 1 , and the triangles are the measurements coming from the sensor providing the measurements of pressure p 2 . Three sets of volumes used with corresponding pressure ranges are given in Table I. relaxation time extraction. In the following, the relaxation time associated with the sensor with the highest accuracy will be used to calculate the conductance and permeability, and it will be called τ.

τ /V0 [s•m -3 ] pm [Pa]

B. Conductance

The conductance of microporous sintered stainless steel porous sample is measured in the large range of the mean pressure, varying from low vacuum conditions, p m ∼ 300 Pa to the pressure slightly higher than atmospheric pressure. Figure 10 provides the conductance variation as a function of mean pressure. This figure shows that the conductance becomes constant at a certain mean pressure value, which depends on the gas nature and increases rapidly for the high pressures. The conductance is also gas-dependent: the lighter the gas is, the higher the conductance. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS

Flow regimes and free molecular conductance

One of the motivations of this work is the development of a pressure sensor calibration device involving the porous medium [START_REF] Johansson | Porous conductance element for high-vacuum pressure sensor calibration[END_REF] . To use the sintered steel microporous sample for the calibration purpose, we need to find the flow conditions where the porous element conductance is constant, whatever is a gas nature.

It is known that the conductance of a tube is constant in the free molecular flow regime [START_REF] Jousten | Handbook of Vacuum Technology[END_REF] , i.e. when the rarefaction parameter δ , Eq. ( 1), is lower than 0.1 (δ < 0.1). However, in the case of the porous media it is much more difficult to determine a characteristic dimension of a gas flow, which is a tube radius in the case of a flow through a tube. Nevertheless, by comparing the curves of the conductance obtained for microporous element with that of a single tube, we can, by analogy, distinguish different flow regimes as it was done when the flows through a ceramic microporous media were analyzed, see Ref. 13, even if a characteristic dimension of the porous sample, a, has not yet been identified to calculate the rarefaction parameter. To do this the conductance of the sintered stainless steel microporous element is plotted on Fig. 11 in the function of the inverse molecular mean free path, instead of the mean pressure, which allows us to eliminate the gas dependency of conductance in the x-axis. In addition, as the conductance is proportional to the most probable velocity, v 0 , so it is also normalized by v 0 to eliminate its dependency on the gas nature in the y-axis. From this curve, we can identify the flow regimes visually by comparing with similar results in terms of conductance for a single capillary, see Ref. 18. This identification could be refined when the flow dimension a of the porous element is extracted, as it has been done in Ref. 10 for the ceramic microporous media using the mass flow rate data, see Section VI B 2.

By analyzing Fig. 11 we determine (visually) the limits of the gas flow regimes in term of the molecular mean free path (or inverse of it), see Table III. It is clear that the conductance becomes constant for all gases when the equivalent mean free path is less than 6.7 µm. The corresponding limits, expressed in term of pressure, p FM , for any gas, can be easily obtained FIG. 11. Conductance over most probable molecular velocity, C/v 0 , as a function of inverse equivalent mean free path ℓ. The colors indicate the gas flow regimes: pink color corresponds to the free molecular regime, green is the transitional regime, violet corresponds to the hydrodynamic regime. The outlined limits are provided in Table III.

C /v 0 [10 -8 m 2 ] -1 [m -1 ]
Regime ℓ -1 [µm -1 ] ℓ [µm] HYDRO ℓ -1 > 1500. 0.00067 < ℓ SLIP ℓ -1 > 12. 0.083 < ℓ FM ℓ -1 < 0.15 6.7 > ℓ TABLE III.
Regime identification in the function of the inverse of the molecular mean free path, ℓ -1 , second column, and of the mean free path, ℓ, third column.

from

p FM m = µv 0 ℓ FM , ( 35
)
and they are provided in Table IV, assuming that the ambient temperature is equal to T = 25 • C. Here we should note that the free molecular limit p FM m is calculated in terms of the mean pressure. Therefore, for the calibration purpose, the pressure in the test chamber, p 2 , lower pressure tank, needs to be much lower, so p m = 0.5 (p 1 + p 2 ) ≈ 0.5p 1 , so we should consider the upper-pressure limit in the gas inlet tank, p 1 , as p 1 ≈ 2p m . IV. p FM m is the upper mean pressure limit to stay in the free molecular flow regime with an equivalent mean free path of 6.7 µm. The ambient temperature is assumed to be equal to 25 • C.

The porous element conductance in the free molecular regime for various tested gases is provided in Table V, where the influence of the gas nature is eliminated by dividing on the most probable velocity, see third column of this table. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

GAS C [10 -6 m 3 • s -1 ] C/v 0 [

Porous medium model: a bundle of capillaries

The representation of a porous element as a bundle of tortuous capillaries of the same radius a is the simplest model of this type of media [START_REF] Do | Adsorbtion analysis: equilibria and kinetics[END_REF][START_REF] Carman | Fluid flow through granular beds[END_REF] . In Ref. 13 this model was used to characterize a ceramic microporous sample, namely to extract a porous medium characteristic dimension (effective pore size), which was used in Ref. 13 to establish the gas flow regime inside the porous sample.

To know the effective flow regime through the sintered stainless steel porous medium we need to know its characteristic dimension (effective pore size), denoted a in the following. When this is later known, the pressure threshold can be established, which guarantees free molecular flow regime inside the medium. In this regime, the conductance does not depend anymore on pressure and the porous element can be successfully used in this pressure range. To apply the methodology of Ref. 13 we need first make an assumption on the type of the gas-surface interaction because the effective pore size extracted from the mass flow rate by a fitting procedure is proportional to the velocity slip coefficient. In the frame of Maxwellian scattering kernel [START_REF] Cercignani | Theory and application of the Boltzmann equation[END_REF] this velocity slip coefficient depends on the accommodation coefficient, α, which takes a value between 0 and 1, that corresponds to fully specular and fully diffusive reflection, respectively.

The gas-surface interaction is a very complex process, which is evidently gas, geometry, and surface dependent. Below we resume some data on the accommodation coefficients characterizing the interaction between several gases and stainless steel surfaces to justify an assumption on this kind of interaction, which we have to do. Several types of surfaces, especially metal surfaces, namely stainless steel with low roughness, could show some particular properties. Analyzing the gas flows through a long capillary the authors of Ref. 22 have found the accommodation coefficients experimentally for different gases on stainless steel surface of a capillary tube very close to each other, but lower than one, so not completely diffusive, namely He (0.930), N 2 (0.881), Ar (0.883), which are different in 5.2%. The authors of Ref. 23 measured the tangential momentum accommodation coefficient for various gases flowing through microchannels made from engineering stainless steel (SUS304). They found the accommodation coefficients to be identical for Ar, N 2 and O 2 and equal to 0.95. Similar behaviors were observed by the authors of Ref. 24, who measured the tangential momentum transport coefficients for different gases interacting with a stainless steel sphere used in Spinning Rotor Gauges (SRG) devices. They also found for gases of interest, namely for He (1.015), Ar (1.013), N 2 (1.018) and Kr (1.025), very similar values of the accommodation coefficients, different in only 1.2%. The authors of a series of works [START_REF] Bentz | The spinning rotor gauge measurements of viscosity, velocity slip coefficients, and tangential momentum, accommodation coefficients for N2 and CH4[END_REF][START_REF] Tekasakul | The spinning rotor gauge: Measurements of viscosity, velocity slip coefficients, and tangential momentum accommodation coefficients[END_REF][START_REF] Bentz | Measurements of viscosity, velocity slip coefficients, and tangential momentum accommodation coefficients using a modified measurements of viscosity, velocity slip coefficients, and tangential momentum accommodation coefficients using a modified spinning rotor gauge[END_REF] also investigated the interaction of stainless steel sphere of the SGR devices with different gases. In their latter paper [START_REF] Bentz | Measurements of viscosity, velocity slip coefficients, and tangential momentum accommodation coefficients using a modified measurements of viscosity, velocity slip coefficients, and tangential momentum accommodation coefficients using a modified spinning rotor gauge[END_REF] they recognized some problems related to the previously used experimental setup. Therefore, we refer only to this latter data of Ref. 27, where information on He (0.8315) and Ar (0.7929) was reported. Anew the difference is found to be small, 4.6%. Therefore, this kind of surface is less sensitive to a gas type, so the values of the accommodation coefficients of the gases used in our experiments could be considered to be very close to one another.

An additional argument in favor of the similarity of gassurface interaction for the considered gases could be derived by analyzing the conductance in the free molecular regime. It is known that the pressure-driven mass flow rate through a long tube in the free molecular regime [START_REF] Sharipov | Data on internal rarefied gas flows[END_REF] is proportional to the accommodation coefficient. As we adopt here the model of porous media as a bundle of the long tubes the conductance of the porous sample in the free molecular regime could also be proportional to the accommodation coefficient as

C FM v 0 ∝ 2 -α α . (36)
Analyzing the conductance divided by the most probable velocity in the free molecular regime, shown in Table V, one can see that the maximum relative variation is less than one percent (0.5%). This negligible difference suggests that the sintered stainless steel surface of the microporous medium has probably similar properties as the stainless steel surface of the capillary tube analyzed in Ref. 22, or the stainless steel surface of the sphere of the SRG in Refs. 24, 27, i.e. the accommodation coefficient of various gases has very similar values. Another hypothesis could be made here that due to the complex geometry of the porous medium the gas-surface interaction is not only dependent on the surface type and roughness, and gas nature, but also on the geometrical configuration, as pointed out in Ref. 29, 30. The complex geometry could also be the reason for a less pronounced gas-surface interaction compared to that studied usually in the classical geometries like tubes or bends. We will provide some additional comments on this subject in Section VI C.

In the frame of a bundle of capillaries model and following the methodology developed in Ref. 13, the characteristic flow dimension of the porous element has been extracted, by fitting the data on mass flow rate in slip flow regime. The pore diameter, 2a, where a characteristic pore dimension, is found to be close to 1.5 µm, whatever a gas used for this analysis, see Table VI. The data for helium is not included in the analysis because helium flow was in the transitional and not in the free molecular flow regime in the tested pressure range. Now, by using the characteristic pore dimension, a = 0.75 µm, the conductance is plotted over the rarefaction parameter in Fig. 12, where the constant conductance is clearly seen in the free molecular flow regime, δ < 0.1. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

Comparison with orifice conductance

In the previous section, the conductance through the microporous sintered stainless steel was analyzed in the frame of the bundle of capillaries model and an effective pore size (characteristic pore dimension) was extracted. It is also of interest to compare the conductance of sintered stainless steel microporous medium with an orifice's conductance. For the latter case, an analytical model of the conductance exists in the free molecular flow regime.

In the free molecular flow regime, the mass flow rate through an orifice can be expressed in the analytical form [START_REF] Jousten | Handbook of Vacuum Technology[END_REF][START_REF] Ho | Numerical study of unsteady rarefied gas flow throuth an orifice[END_REF] as

ṀFM = R 2 orifice √ π ∆p v 0 , (37)
where R orifice is the orifice radius. By using the definition of conductance, it can be related to the mass flow rate as

Ṁ = 2C ∆p v 2 0 . (38)
From Eqs. ( 37) and (38) the orifice radius in function of the conductance reads

R orifice = 2 √ π C v 0 . ( 39
)
Using the previous expression for the orifice radius and the values of the conductance obtained for our stainless steel microporous element, we can estimate the corresponding radius of an orifice that could have the same conductance. Then the orifice radius, R orifice , would be 97.2 µm, while the characteristic dimension of the porous sample is a = 0.75 µm. Therefore, to reach the same rarefaction level (free molecular flow regime, where the conductance is constant) with an orifice flow, the working pressure has to be around 100 times lower compared to the stainless steel microporous sample. This example illustrates that with the microporous conductance element, we can combine high conductance and high rarefaction level.

C. Permeability

In Section VI C, it was shown that permeability is related to the conductance through Eq. ( 28), so in this section we will give only a brief overview of the permeability behaviors of the microporous sintered stainless steel conductance element. First, the permeability is given in Fig. 13 as a function of mean pressure, where one can see the permeability increasing with the pressure decreasing. This property is sometimes called the apparent permeability [START_REF] Klinkenberg | The permeability of porous media to liquid and gases[END_REF] , to make the difference to the intrinsic permeability, which is constant whatever the pressure is in the high-pressure range. Figure 13 shows the gas dependency of the permeability: the porous sample is more permeable for the lighter gases.

This gas dependency is eliminated by plotting the permeability as a function of the Knudsen number, defined by Eq. ( 1), see Fig. 14. To calculate the Knudsen number, the effective pore size, a, calculated in Section VI B 2 and provided in Table VI, is used. From the graph presented in Fig. 14 we can identify the flow regimes visually. We do not measure the intrinsic (hydrodynamic) permeability directly because it was out of the intended pressure measurement range. However, the permeability can be extrapolated to the hydrodynamic flow regime by fitting the measured data in the transitional and slip flow regimes, using the expression proposed in Ref. 32

K = K D 1 + b p m . ( 40
)
Doing so, we find a gas-independent hydrodynamic permeability, K D , which is provided in VII, the hydrodynamic (or intrinsic, or Darcy) permeability was found the same whatever the gas for its determination. However, for the b coefficient, its gas dependency is clearly seen, as was also the case when the microporous ceramic membranes were analyzed in Ref.

13.

We can eliminate the gas dependency of b coefficient by rewriting Eq. (40) in the form proposed in Ref. 30 as:

K = K D 1 + b ′ Kn , (41) where b
′ = b a µv m . (42)
The b ′ coefficient for all gases considered in experiments is shown in Figure 15 Unfortunately, it is difficult to compare b ′ coefficient directly with the experimental results from [START_REF] Klinkenberg | The permeability of porous media to liquid and gases[END_REF] because the characteristic pore dimension a is involved in its definition, see Eq. (42). Therefore, in Figure 16 we compare b ′ /a in function of the molecular mean free path, ℓ, to avoid the use of the characteristic dimension. Finally, we observe the same behavior of b ′ /a decreasing with the mean free path increasing (pressure decreasing).

Several other numerical approaches were developed recently [START_REF] Kawagoe | A study on pressure-driven gas transport in porous media: from nanoscale to microscale[END_REF][START_REF] Ahmadian | A dusty gas model-direct simulation Monte Carlo algorithm to simulate flow in microporous media[END_REF][START_REF] Shariati | Direct simulation Monte Carlo investigation of fluid characteristics and gas transport in porous microchannels[END_REF] , which consider the porous media as a set of random distributed spheres. However, the obtained here experimental results agree more than sufficiently with our simulations using a bundle of tubes, that we did not take into account the more involved models. These recently developed approaches [START_REF] Kawagoe | A study on pressure-driven gas transport in porous media: from nanoscale to microscale[END_REF][START_REF] Ahmadian | A dusty gas model-direct simulation Monte Carlo algorithm to simulate flow in microporous media[END_REF][START_REF] Shariati | Direct simulation Monte Carlo investigation of fluid characteristics and gas transport in porous microchannels[END_REF] will allow to go forward in the understanding of the flow properties inside the porous media and probably further comparison with experimental results.

Some comments on the conductance and permeability in the free molecular flow regime

In Section VI C, the relation between conductance and permeability is established via Eq. ( 28). This equation can be rewritten in following form

K ℓ = L S C v 0 . (43)
As the conductance in the free molecular regime is constant and the ratio C/v 0 is gas independent (except possible dependence on the gas-surface interaction), so the ratio K/ℓ is also constant and gas independent in this regime, also except pos- This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

sible influence on the gas-surface interaction. As K D , is the intrinsic permeability of a homogeneous porous medium in the hydrodynamic flow regime, the quantity L S C v 0 is similarly suitable for this purpose, but in the free molecular regime. Equation ( 28) can also be rewritten in the following form

C v 0 = K ℓ S L , (44)
from which one can see that in the free molecular regime, where the ratio C/v 0 is constant, the conductance is proportional to the surface area S of the sample. Therefore, to reduce the conductance of a sample, for example, in two times one have to reduce in two times its surface area.

VII. CONCLUSION

The global approach to calculate the mass flow rate, conductance, and permeability of the sintered stainless steel microporous element is proposed, which can also be used to determine all the three properties of any other connection elements as a tube, bend, etc. This approach is based on the pressure measurements in each tank (or pressure difference between them). The pressure evolution is then fitted using only one fitting parameter, the gas relaxation time, which is directly related to the conductance and permeability. From the conductance or permeability measurements the characteristic flow dimension is found to be 0.75 µm independent of gas type. The characteristic flow dimension then clearly defines the free molecular flow regime in terms of the mean pressure, gas type, and temperature. This information is indispensable for using the microporous element for the calibration of pressure sensors. As it concerns permeability, the proposed approach permits finding the Klinkenberg parameter to describe the apparent gas permeability of the sintered stainless steel microporous element, which is found gas dependent and the intrinsic (hydrodynamic) permeability. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS

p f = p m
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS

  DOI: 10.1063/5.0082227 facturer of L = 1.0 mm, and a diameter of D = 16 mm, fixed in an ISO-KF 16 centering ring, see Fig. 3(a). The images of the tomography analysis with 35 magnification (resolution of 4.31µm) and 700 times magnification (resolution of 0.22µm) are provided on Fig. 3(b) and (c), (d), respectively.

FIG. 3 FIG. 4 .

 34 FIG. 3. a) Picture of the conductance element, from the front and behind; b)-d) images of the tomography analysis of the porous structure: b) 35 times magnification with the resolution of 4.31 µm, c) and d) 700 times magnification with 0.22 µm resolution, in c) the colors are inverted.

FIG. 5 .

 5 FIG.5. Schema of the typical pressure evolution with time, p 1 and p 2 are the pressures in high-pressure and low-pressure sides, respectively, p m denotes initial mean pressure and p f the final mean pressure.

FIG. 6 .

 6 FIG.6. Mass flow rate of helium at mean pressure p m = 1.13 × 10 5 Pa, Ṁ1 is the mass flow rate of the gas leaving the high-pressure tank and Ṁ2 the mass flow rate of the gas entering the low-pressure tank. The error bars are also provided for both measurements.

FIG. 7 .

 7 FIG. 7. Logarithm of the measured and fitted pressure difference in time.

  5, and permeability, Ref. 11, and more recently in Refs. 18 and 10, respectively.

4 FIG. 8 .

 48 FIG.8. Pressure relaxation time as a function of mean pressure. Three different pairs of volumes were used for different pressure ranges, see TableI.

5 FIG. 10 .

 510 FIG.10. Conductance as a function of mean pressure.

GAS 2 a 8 TABLE 1 FIG. 12 .

 28112 FIG.12. Conductance over most probable velocity as a function rarefaction parameter.

K [m 2 ] 3 FIG. 13 .

 2313 FIG.13. Permeability as a function of inverse mean pressure.

  as a function of the Knudsen number. Here one observes anew that the behavior of b ′ coefficient is similar for different gases. The authors of Ref.30 have solved the linearized BGK model kinetic equation for a simplified porous medium geometry and for different values of the accommodation coefficient. The tendency of decreasing of b ′ coefficient with Kn increasing was found for the accommodation coefficient different from 1. Taking into account this finding, we can conclude that the accommodation coefficient of the analyzed here porous medium is different from 1.

FIG. 15 .FIG. 16 .

 1516 FIG. 15. The normalized Klinkenberg correction factor b ′ = ba/µ/ √ 2RT for all gases used in experiments in function of the Knudsen number.

µ 5 3 ] 5 C /v 0 [ 10 - 8 m 2 ] 7 C /v 0 [ 1 K [m 2 ] 3 K [m 2 ] 1

 535010827012321 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227 τ /V 0 [s•m -This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5ln ∆p [ln(0.133322 kPa)] t [s] ln (∆p) ln (∆p fit )

TABLE I .

 I Volume configurations for different pairs of pressure sensors of respective full scales. V 1 and V 2 are the volumes of the high and low-pressure tanks, respectively, and the effective volume

			3 ]
	133 13.3 1.33	133 133 13.3	388 ± 8 380 ± 8 192 ± 4 183 ± 4 167 ± 4 87 ± 2 183 ± 4 171 ± 4 88 ± 2

TABLE V .

 V Conductance, C, and conductance over most probable velocity, C/v 0 , in the free molecular flow regime. AVG in last line means the averaged of the values provided in previous four lines.

			10 -9 m 2 ]
	He	9.33	8.36
	N 2	3.52	8.37
	Ar	2.96	8.38
	Kr	2.06	8.40
	AVG	4.47	8.38

CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

  Table VII together with the b This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE FIG. 14. Permeability as a function of Knudsen number. coefficient. The relation between b coefficient and the velocity slip coefficient in slip flow regime is explained in detail in Ref. 13. As it is clear from Table

	10 -11
		KRYPTON
		NITROGEN
		ARGON
		HELIUM
	10 -12
	]
	[m 2
	K
	10 -13
	10 -14	10 -1	10 0	10 1
			Kn

ARTICLE AS DOI: 10.1063/5.0082227

  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE

	p 2 V 2 T 2 ,	2 p 2 p 1 > Ṁ Ṁ	1	p 1 V 1 T 1 ,	Pressure p	p 1 p 2
		Porous sample
	Low pressure			High pressure	Time t

CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE

		ISO-KF 16
		Ṁ	R	Ṁ
	DVP TMP
	v Pump						T Ambient
	v Drop	T 2	p 2	>	p 1	T 1
					Ṁ
							v Jump
	V Ref	V 2					V 1
				Porous media	v Gas
	Reference volume	CDG				CDG	V Ref
		p 2					p 1	Reference volume

CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE

	Pressure	Equilibrium Non-equilibrium
			Quasi-equilibrium
				Equilibrium
		t i	t 0	t f	Time

CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

				p 1
	p 1 (t 0 )		p 2 p m
			∆p(t * )	p f
	p/ pressure	p f p m
	p 2 (t 0 )
		t 0	t *	t/ time	f t

This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0082227

ACKNOWLEDGMENT

This work has been carried out in the framework of the Labex MEC (ANR-10-LABX-0092) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR). The authors (M.V. Johansson, P. Perrier, and I. Graur) would like to acknowledge financial support provided by the European Union network program H2020, MIGRATE project under Grant Agreement No.643095.

Appendix A: Gas viscosity

The gas viscosity coefficient, involved in Eqs. ( 2) and ( 23), depends on the gas temperature and the gas nature and it is calculated as [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] :

where ω is the gas viscosity index,