Martin Victor

V Johansson

Fabrice Testa

Pierre Perrier
email: pierre.perrier@univ-amu.fr

Jérôme Vicente
email: jerome.vicente@univ-amu.fr

Jean Philippe Bonnet

Philippe Moulin
email: philippe.moulin@univ-amu.fr

Irina Martin Graur

M V Johansson

Determination of an effective pore dimension for microporous media

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The determination of characteristics of porous media permeability like the micro and nanoporous membranes or ultra-tight shale-gas reservoirs is still a challenge up to now. The low porous media find a broad application in medicine [START_REF] Adiga | Nanoporous membranes for medical and biological applications[END_REF], biotechnology for separation and filtration [START_REF] Abedini | Application of membrane in gas separation processes: its suitability and mechanisms[END_REF]. The recent development of porous ceramic media with high thermal, chemical and structural stability and the ability to have catalytic properties has opened up new horizons for membranes applications, for example, in high-temperature gas separation and catalytic reactions [START_REF] Lin | Experimental studies on pore size change of porous ceramic membranes after modification[END_REF]. Unconventional resources, such as ultra-tight shale-gas reservoirs of very small pores (in nanoscale) play a significant role in securing hydrocarbon energy because of their potential to offset declines in conventional gas production [START_REF] Naraghi | A stochastic permeability model for the shalegas system[END_REF]. The morphology of the porous structure dominates the fluid flow through a porous medium. Therefore, it is important to characterize the geometrical properties of a porous medium quantitatively. Different methods exist for the measurements of the average pore size and pore size distribution.

The choice of the most appropriate method depends on the application of the porous solid, its chemical and physical nature and the range of pore size. The most commonly used methods are [START_REF] Lowell | Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density (Particle Technology Series)[END_REF]: mercury porosimetry, where the pores are filled with mercury under pressure. This method is suitable for many materials with pores in the appropriate diameter range from 0.003 µm to 300 µm. From mesopore to micropore size analysis, BET method [START_REF] Brunauer | Adsorption of gases in multilayers[END_REF], can be done by gas adsorption, usually nitrogen, at liquid nitrogen temperature. This method can be used for pores in the approximate diameter range from 1 nm to 0.1 µm. The pore size diameter can also be determined via direct observation methods: scanning electron microscopy (SEM), field-emission scanning electron microscopy (FE-SEM), environmental scanning electron microscopy (ESEM), and atomic force microscopy (AFM), [START_REF] Wyart | Membrane characterisation by optical methods: Ellipsometry of the scattered field[END_REF], [START_REF] Wyart | Membrane characterization by microscopic methods: multiscale structure[END_REF]. The tomography analysis of a porous structure can allow the determination of the internal structure of a sample limited by the characteristics of their spatial resolution [START_REF] Hachem | Hygromorphic characterization of softwood under high resolution x-ray tomography for hygrothermal simulation[END_REF]. All these methods require either preliminary sample preparation or lead to the complete sample destruction, furthermore, they only use a small part of the sample for analysis.

We propose here a simple approach for the non-destructive porous sample characterization by measuring the pressure variation in the inlet and outlet tanks (or just the pressure difference between them). The experimental methodology, based on the constant volume technique, was initially developed for the isothermal and non-isothermal measurements of the mass flow rate through the microchannels [START_REF] Cardenas | Thermal transpiration flow: a circular cross-section microtube submitted to a temperature gradient[END_REF] and has been recently adapted for the analysis of porous samples [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF]. The gas permeability of the porous sample can be easily obtained directly from the pressure evolution in time without calculation of the mass flow rate.

The measurements are analyzed by assuming the porous media have similar behavior as the classical bundle of capillaries model, first suggested by Kozeny [START_REF] Kozeny | Uber kapillare leitung der wasser in boden[END_REF] and then extended by Carman [START_REF] Carman | Fluid flow through granular beds[END_REF] to allow for torturous capillaries. In our analysis we assume that the capillary tubes have the same radius. This allows us to find an unique parameter (capillary's radius) to characterize the porous structure. This unique parameter helps also to determine the gas flow regime, by introducing the Knudsen number as the ratio of the molecular mean free path and the capillary radius, and then by referring on this Knudsen number to distinguish the flow regimes. Recently, the models of a bundle of capillary tubes of variable shape and size cross-section were developed, [START_REF] Cai | Generalized modeling of spontaneous imbibition based on Hagen-Poiseuille flow in tortuous capillaries with variably shaped apertures[END_REF], [START_REF] Golparvar | A comprehensive review of pore scale modeling methodologies for multiphase flow in porous media[END_REF], but all the models were used either for the liquid or for two phase flows, which physics is different from the single phase flows.

The model of a bundle of capillaries with gas flow inside was considerably improved by Klinkenberg [START_REF] Klinkenberg | The permeability of porous media to liquid and gases, Drilling and Production Practice[END_REF] taking into account the slip flow regime through the capillaries. In the present article, from the measured mass flow rate the effective pore size is estimated by using the fitting procedure via slip flow expression.

The obtained effective pore sizes are then compared to mercury porosimetry and micro-computed tomography (µCT) results. The proposed technique of the effective pore size measurement can be used as a non-destructive method for quality verification. Furthermore, this method is independent of the exterior sample geometry. When the effective pore size is known and by using the infor-mation about porosity the permeability, apparent permeability, and tortuosity coefficients as well as the surface-to-volume ratio can be easily obtained.

Experimental methodology

The experimental methodology, applied in this article, is described in details in Ref. [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF]. We present here only the summary of this technique, essential to understand the data treatment. From measurements of pressure variation over time we calculate the important characteristics of porous media such as mass flow rate and permeability, and then effective pore size dimension.

Experimental apparatus

The experimental setup is a high vacuum system capable of measuring up to 5 decades of pressure. In the presented experiment the mean pressure is varied from 75 Pa up to 131 kPa. This large pressure measurement range is achieved by using three pairings of four Capacitance Diaphragm Manometers (CDM) with full-scale (CDM 1 -CDM 2 ): 133 kPa -133 kPa, 133 kPa -13.3 kPa and 13.3 kPa -1.33 kPa. Four high purity gas bottles with test gases, Helium, Neon, Nitrogen, Argon (Air Liquide, France) are used. The pumping is performed by a twostage Diaphragm Vacuum Pump (DVP) and a Turbomolecular Pump (TMP).

Each side of the porous medium is connected to two reservoirs, of volumes V 1 and V 2 , for the high and low-pressure, respectively. Both tanks volumes, including the volumes of the valves, connecting tubes and pressure sensors, are measured accurately, and these volumes are equal to V 1 = 255.8 ± 5.5 cm 3 and V 2 = 238.8 ± 5.1 cm 3 , for the high and low pressure tanks, respectively. The reservoirs are connected only by a porous sample which is fixed with a vacuum glue.

Two microporous samples, used in the experiments and mentioned in the following as the first and second discs, have a cylindrical shape (disc) with the same radius and thickness (in main flow direction) equal to 4.75 ± 0.01 mm and L = 2.3 ± 0.01 mm, respectively. The characteristics of these microporous discs are the same as of the ceramic membranes used in micro-to-nano filtration. For such ceramic microporous media, depending on manufacturer, porosity is in the range 15% -30% with pore diameter ranging from 1 µm to 10 µm. The total volume of each porous disc is 0.14 cm 3 , so by taking 30% of porosity a gas volume inside the medium is approximately 0.042 cm 3 , which is much smaller than the volume of each tank.

The experiments are performed within a narrow temperature range (around room temperature, 29-31 • C), excluding any heat source in an environment.

The temperature is measured using the thermocouple with the accuracy of 0.6 K.

Mass flow rate measurements

The constant volume technique [START_REF] Ewart | Mass flow rate measurements in gas micro flows[END_REF], [START_REF] Rojas-Cardenas | Time-dependent experimental analysis of a thermal transpiration rarefied gas flow[END_REF] and the methodology, analogous to the pulse decay method [START_REF] Brace | Permeability of granite under high pressure[END_REF], [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF], are implemented here to measure the mass flow rate through samples of a microporous ceramic medium. The applied experimental technique allows us to deduce the mass flow rate and also the permeability from the pressure variation in time in both tanks. This approach needs to have a stable temperature during the measurements, see discussion in Ref. [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF]. Therefore, if the temperature variations during the experimental time are small compared to the pressure variations we can calculate the mass flow rate from the pressure variation in each tank as following:

Ṁ1 = - dM 1 dt = - V 1 RT dp 1 dt , Ṁ2 = dM 2 dt = V 2 RT dp 2 dt . (1)
Here Ṁi and p i , i = 1, 2 are the mass flow rate and pressure in the tank i, respectively, R is the specific gas constant, T is the gas temperature, the same in each tank, t is the time. Equations ( 1) are obtained with an assumption that the gas follows the ideal gas law. In the present study, the maximal considered pressure is slightly above atmospheric pressure (up to 131 kPa); therefore we do not consider here the real gas effects. However, the proposed approach can be generalized to take into account the real gas effects by using, for example, the van der Waals equation instead of the ideal gas law. The estimations of the compressibility factor under our experimental conditions are provided in AppendixA.

The ideal gas law is valid under equilibrium condition; however, the gas pressure and gas mass in a tank change in time. Here we assume that we have a quasi-steady process, that is, we have a succession of local equilibrium. This assumption is true when we have a small unbalancing force which modifies the system slower than the system reaches a local equilibrium, see AppendixB for a further discussion.

Very often it is convenient to express the mass flow rate in function of the pressure difference, ∆p(t) = p 1 (t)p 2 (t), between two tanks:

Ṁ (t) = - V 0 RT d(∆p(t)) dt , V 0 = V 1 V 2 V 1 + V 2 , (2)
where V 0 is the effective volume. It is clear that the mass flow rate can be calculated using expressions (1) and (2), when the pressure variation in each tank or the pressure difference between them in time is known. To measure the mass flow rate, first, the initial pressure difference is settled between the tanks, then the gas starts to flow from higher to lower pressure tank up to the same final pressure p f is reached, see Fig. 2 in [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF]. During the experiments the pressure variations over time in each tank are recorded, then, their difference is fitted by using the exponential fitting function [START_REF] Brace | Permeability of granite under high pressure[END_REF], [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF]:

∆p(t) = ∆p 0 exp (-(t -t 0 )/τ ), (3)
where τ is the pressure relaxation time, ∆p 0 is the initial pressure difference between the tanks at time t = t 0 . Similar exponential representations of the pressure evolution over time in the first, p 1 (t), and second, p 2 (t), tank can be also written in a form similar to Eq. ( 3), see Refs. [START_REF] Brace | Permeability of granite under high pressure[END_REF], [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF]. The pressure evolution in time in each tank and the pressure difference between two tanks as a function of time and their corresponding fitting functions are shown in Fig. 2 (a) and (b), respectively.

By using Eq. ( 2) for the mass flow rate, the exponential representation of the pressure difference in time, Eq. ( 3), and its analytical derivative, we can now express the mass flow rate as

Ṁ (t) = - V 0 RT d(∆p(t)) dt = V 0 RT ∆p 0 τ exp - t -t 0 τ . (4)
From Eqs.

(1) we can also express the mass flow rate using the analogous to Eq. ( 3) exponential representation of the pressure variation over time in each tank, see Refs. [START_REF] Brace | Permeability of granite under high pressure[END_REF], [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF]. To obtain the mass flow rate from the pressure variation measurements, the pressure variation in time, τ , is fitted using pressure relaxation time as a single fitting parameter, then the mass flow rate can be calculated from Eq. ( 4).

The classical uncertainty calculation technique is used to estimate the measurement uncertainty of the mass flow rate, which for our experimental conditions lies in the range 3.6 -5.1%, see Ref. [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF] for more details.

Gas permeability measurements

The Darcy law [START_REF] Do | Adsobtion analysis: equilibria and kinetics[END_REF] allows us to relate the instantaneous discharge (or volumetric) flow rate through a porous medium, Q, to the pressure drop over a given distance L, which is the thickness of a porous sample (disc):

Q = KS µ ∆p L , ( 5
)
where K is the permeability, S is the cross-section of the porous sample, µ is the viscosity, which is calculated as [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF]:

µ = µ ref T T ref ω , (6)
where ω is the gas viscosity index, µ ref is the gas viscosity at temperature T ref = 273.15 K [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF], see also Table 1.

Gas The volumetric flow rate, used in Eq. ( 5), is related to the mass flow rate and the gas density ρ as:

µ ref × 10 -5 [Pa • s] ω R [J • kg -1 • K -1 ] Molar mass M [g • mol -1 ]
Q = Ṁ ρ = Ṁ RT p , (7)
then by integrating along the porous sample and by using the mass conservation property, we obtain the expression, analogous to Eq. ( 5), which relates the mass flow rate, instead of volumetric flow rate, to the permeability

Ṁ = KS µ ∆p L p m RT . (8)
Here p m is the mean pressure between two tanks, p m = 0.5(p 1 +p 2 ). Then, using the expression of the mass flow rate via the pressure variation in tanks, Eq. ( 2), and following the technique developed in Ref. [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF] we can relate permeability to the pressure difference variations between the tanks and finally obtain the expression of the gas permeability through the fitting parameter, τ , pressure relaxation time, see Ref. [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF] for more details:

K = µ τ p m LV 0 S . (9)
It is worth to note that the previous expression is obtained under the condition of the mean pressure constancy during an experimental run. This condition is satisfied, when the tanks volumes are equal, V 1 = V 2 . For the case of different tanks volumes V 1 = V 2 , the analytical expression was derived in [START_REF] Johansson | Mass flow rate and permeability measurements in microporous media[END_REF]. This expression relates the variation of the mean pressure during the experimental time to the tanks volumes ratio, V 1 /V 2 , and the initial pressure ratio, p 1 (t 0 )/p 2 (t 0 ), between the tanks.

The uncertainty of the permeability measurements, when using Eq. ( 9), is calculated by the classical way, similar to the calculation of the uncertainty on the mass flow rate. Under our experimental conditions the measurement uncertainty on the permeability lies in the range 5.0 -6.4%.

Modeling of the porous structure

Different type of modeling can be used to characterize the flow through microporous media. One of the simplest and, in the same time, efficient models of a microporous medium is its representation as a bundle of several numbers of capillaries with the circular cross-section of the same or different diameters [START_REF] Kozeny | Uber kapillare leitung der wasser in boden[END_REF], [START_REF] Do | Adsobtion analysis: equilibria and kinetics[END_REF]. All the capillaries (pores) can be parallel and have a length L c equal to the length L (thickness) of the porous medium, see Fig. 1 (left). However, in the real samples, this capillary length can be different from the length of the porous medium because of the random orientation of the capillaries. As a result, the capillary length is generally longer than the medium thickness, see Fig. 1 (right). To account for this fact a tortuosity l τ [START_REF] Epstein | On tortuosity and tortuosity factor in flow and diffusion through porouse media[END_REF] is introduced as

l τ = L c L . (10)
The sample porosity ε is defined as

ε = V c V , (11)
where V c is the volume of void-space (such as fluids) and V is the total or bulk volume of a solid material.

If the porous medium is represented as a bundle of N capillaries of the same radius a and of the length L c , different from the membrane thickness L, Eq. ( 10), the porosity is calculated as:

ε = N πa 2 L c SL = N πa 2 l τ S . (12)
With the same set of parameters a very useful characteristic of porous medium can be calculated, the Specific Surface Area (SSA) defined as the the ratio of the internal surface to the sample volume

S A = 2πaL c N SL = 2πaN l τ S m 2 m 3 . ( 13
)
Finally the used here model of the porous medium has 4 unknown parameters.

Three among them, the parameters a, N and l τ (or ε) are defined above. The forth parameter, the velocity slip coefficient, σ p , (or the accommodation coefficient, α), are presented in Section 4.2, where their physical meaning and the typical values are given. In the following, we will show how these characteristics of a porous medium can be extracted when using the presented above model of a bundle of capillaries.

Expressions of the mass flow rate through a single capillary

Flow regimes

The microporous medium is modeled here as a bundle of capillaries, so it is worth to define first different possible flow regimes in a capillary and to present then the expressions of the mass flow rate through a capillary for these flow regimes. Usually the flow regimes could be identified through the Knudsen number, which is calculated as the ratio between the equivalent molecular mean free path and the characteristic dimension a of the capillary (its radius):

Kn = a . (14)
The equivalent molecular mean free path is defined as

= µv 0 p m , (15)
where v 0 is the most probable molecular speed

v 0 = √ 2RT . ( 16
)
It is convenient to introduce also the rarefaction parameter δ which is directly proportional to the pressure and related to the Knudsen number as

δ = a = 1 Kn . ( 17
)
We adopt here the classical definition of gas flow regimes in terms of the Knudsen number or rarefaction parameter [START_REF] Cercignani | Theory and application of the Boltzmann equation[END_REF]: continuum flow regime (Kn < 0.01 or δ > 100); slip flow regime (0.01 < Kn < 0.1 or 10 < δ < 100); transitional flow regime (0.1 < Kn < 10 or 0.1 < δ < 10); free molecular flow regime (Kn > 10 or δ < 0.1).

Mass flow rate expressions for a single capillary

In the case of the slip flow regime (10 ≤ δ ≤ 100) the mass flow rate through a tube (capillary) of a radius a can be obtained from the Stokes equation

µ r ∂ ∂r r ∂u ∂r = dp dz ( 18
)
subjected to the velocity slip boundary condition at the solid surface

u = σ p ∂u ∂r r=a . ( 19
)
In the previous relations u is the longitudinal flow velocity, z is the longitudinal flow direction, p is the local gas pressure, σ p is the velocity slip coefficient, which depends on the type of the gas-surface interaction. Using the kinetic theory the velocity slip coefficient was calculated in Ref. [START_REF] Cercignani | Theory and application of the Boltzmann equation[END_REF] to be equal to 1.018 in the case of diffuse gas-surface interaction. The accommodation coefficient, α, characterizes also the gas-surface interaction: it is equal to 1 for the case of diffuse interaction (complete accommodation) in the frame of Maxwellian scattering kernel [START_REF] Cercignani | Mathematical methods in kinetic theory[END_REF]. Both coefficients are related between them: the authors of Refs. [START_REF] Loyalka | Some numerical results for the bgk model: thermal creep and viscous slip problems with arbitrary accommodation of the surface[END_REF], [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF] suggested to use the following relation:

σ p (α) = 2 -α α (σ p (α = 1) -0.1211(1 -α)) . (20)
By integrating Eq. ( 18) with the boundary condition [START_REF] Brace | Permeability of granite under high pressure[END_REF] and the symmetry condition on the tube axis we obtain the velocity profile over the capillary crosssection. Next, by integrating this velocity profile and then, by integrating along the tube (according to z variable) from 0 to the capillary length L c and using the mass conservation property we obtain finally the mass flow rate through a single tube (capillary) for the slip flow regime:

Ṁ = Ṁ P 1 4 + σ p δ , (21)
where Ṁ P is the classical Poiseuille masse flow rate through a capillary of a radius a and the length L c in the hydrodynamic flow regime (δ ≥ 100):

Ṁ P = πa 4 L c ∆p p m µv 2 0 = πa 4 L c M S0 , (22)
where

M S0 = ∆pp m µv 2 0 kg s m -3 . (23)
In the hydrodynamic flow regime the molecule-molecule collisions dominate the molecule-surface collisions. Contrarily, in the free molecular flow regime (δ ≤ 0.1, Knudsen diffusion regime) the molecule-molecule collisions can be practically neglected and the molecule-surface collisions guide the flow. In this regime the mass flow rate is calculated from following expression [START_REF] Sharipov | Rarefied gas dynamics. Fundamentals for research and practice[END_REF]:

Ṁ FM = 2 -α α 8 3 √ π Ṁ G , (24)
with

Ṁ G = πa 3 L c ∆p v 0 = πa 3 L c M G0 , (25)
where

M G0 = ∆p v 0 kg s m -2 . ( 26
)
In the case of the transitional flow regime (0.1 ≤ δ ≤ 10) the mass flow rate can be found only numerically by the solution of the linearized Boltzmann equation (or of others model kinetic equations [START_REF] Sharipov | Data on internal rarefied gas flows[END_REF], [START_REF] Graur | Gas flow through an elliptical tube over the whole range of the gas rarefaction[END_REF]). In this flow regime the number of molecule-surface collisions is comparable to the number of intermolecular collisions.

The approximate expression of the dimensionless mass flow rate G (G = Ṁ / Ṁ G ) through a capillary of the radius a and the length L c , which covers all flow regimes, was proposed in [START_REF] Sharipov | Leak rate of water into vacuum through microtubes[END_REF] for the diffuse scattering, α = 1:

G(δ) = Ṁ / Ṁ G = 8 3 √ π 1 + 0.04 δ 0.7 ln δ 1 + 0.78 δ 0.8 + δ 4 + σ p δ 1 + δ . ( 27
)
The typical shape of the normalized mass flow rate, G function, Eq. ( 27), for a capillary is shown on Fig. 3. The function G has the finite limit in the free molecular regime, i.e. when δ → 0. In the case of diffuse scattering of the molecules from the surface, α = 1, lim δ→0 G(δ) = 8/(3 √ π), see Eqs. ( 24) and [START_REF] Sharipov | Rarefied gas dynamics. Fundamentals for research and practice[END_REF]. In the opposite limit case, δ → ∞, the hydrodynamic flow regime, the normalized mass flow rate, G function, Eq. ( 27), tends to infinity. Therefore, in the following to have a finite value of the mass flow rate in the hydrodynamic flow regime, we will use the representation of the mass flow rate in form ( 21), which limit case for δ → ∞ gives very known Poiseuille flow rate, Eq. ( 22).

Determination of porous medium characteristics from pressure measurements

In the previous Section, we introduced the complete description of the flow through a single capillary. In this Section, the model of the porous media as a bundle of capillaries is presented, and its parameters as the capillary radius, capillary number, tortuosity, and specific surface area are extracted from the measurements. The proposed geometrical model corresponds to a homogeneous porous medium with a signature of a single pore size.

Fitting range

The analytical expressions for the mass flow rate through a single capillary, provided in the previous Section, could be used to calculate the mass flow rate through a microporous sample by representing it as a bundle of capillaries. To choose the analytical expression for the mass flow rate the flow regime in a capillary must be known. However, the flow regimes depend on the rarefaction parameter (Knudsen number), which includes the characteristic dimension of a flow (capillary radius), which is a priori unknown.

To have an idea about the flow regime it is useful to note that the mass flow rate through a capillary in free molecular regime is proportional to ∆p/v 0 , this motivates our definition of the dimensionless quantity G 0 as

G 0 = Ṁ /(∆p/v 0 ) = Ṁ /M G0 m 2 . ( 28
)
By defining the dimensionless quantity G 0 in this way, we eliminate its dependence from the geometrical parameters, a, and L c , which are unknown a priori in this model. By analogy, in the hydrodynamic flow regime, the mass flow rate is proportional to ∆pp m /(µv 2 0 ), so we define dimensionless quantity S 0 as

S 0 = Ṁ / ∆pp m µv 2 0 = Ṁ /M S0 m 3 . (29)
When we plot these normalized quantities, G 0 and S 0 , as a function of inverse molecular mean free path -1 and molecular mean free path , respectively, we find similar behavior as for the mass flow rate through a tube. That is, with these normalizations we find a constant value in the respective regime and we can make the first identification of two known limits, the free molecular and hydrodynamic regimes. Once the definition of the slip flow regime has been done in a suitable way, we have to find three properties which present the additional argumentation that we have defined our regime correctly. First, the Knudsen number and rarefaction parameter are close to the theoretical values for a tube. Second, the relative difference of our linear fit in the slip flow regime and the measured values do not have a trend but is rather scattered around a constant value. Third, the intrinsic permeability is gas independent within experimental uncertainty.

Effective pore size

Let us use the analytical expression for the mass flow rate through one tube,

Eqs. ( 21) and ( 22), and write it for a bundle of N capillaries, where the capillary length L c can be different from the thickness L of the porous sample (disc). In this case the mass flow rate through a bundle of N capillaries reads:

Ṁ = N πa 4 L c ∆pp m µv 2 0 1 4 + σ p δ . (30)
In the previous expression four parameters are unknown: N , a, L c and σ p . To determine them from experimental data we can write the previous expression in the following form

S 0 = Ṁ /M S0 = N πa 4 L c 1 4 + σ p δ , (31)
where M S0 is defined in [START_REF] Cercignani | Theory and application of the Boltzmann equation[END_REF]. Then we fit previous expression according to the linear regression:

F S = A S X + B S , (32)
where

A S = σ p πa 3 N L c , B S = πa 4 N 4L c , X = , (33)
A S and B S are the fitting coefficients of the S-fit. The mass flow rate is fitted via the molecular mean free path, X = , Eq. ( 15). From the previous expressions, it is clear that the slope of the fitting curve, A S coefficient, depends on the gas nature only via the slip coefficient σ p .

As it was mentioned previously, initially the characteristic dimension a of the flow are not known, and we can refer to different flow regimes only by using the molecular mean free path. The intervals associated with different flow regimes and expressed in term of the molecular mean free path are presented in Table 2.

The S-type fit, Eq. ( 32), is realized in the hydrodynamic and slip flow regimes, so for > 0.19 µm for the first disc.

The fitting coefficients, A S and B S , for the first disc and various gases are shown in Table 3. It is worth to note that even if the measurements for four gases are carried out for the first disc only for two gases, Nitrogen and Argon, the number of the experimental points in slip regime is large enough to obtain the good fitting curves. The measured mass flow rate normalized using Eq. ( 31), and the fitting curve, according to Eq. ( 32), are shown in Fig. 5 for Argon. The quality of the fit is also tested by plotting the relative deviation between the measured and fitted quantities, (S 0 -F S )/F S , see Fig. 6. As it is clear from this last figure, the points are homogeneously distributed around 0, and they do not show any trend, which confirms the quality of fitting function and supports also our choice of the fitting range, see comments at the end of Section 5.1. The results for other gases are close to that measured for Argon.

From the fitting coefficients, A S and B S , the effective flow dimension of the porous medium a, i.e. effective pore radius, can be found as

a = 4σ p B S A S . (34)
To calculate the characteristic dimension of the porous medium, a, from the previous expression, Eq. ( 34), we need only the information on the velocity slip coefficient, σ p , which characterizes the gas-surface interaction. In addition, Eq. ( 34) is independent of the external geometrical parameters of a sample, so we are not restricted to only cylindrical shape of the porous media. In the following, we assume that all the gases interact with the wall of the porous medium diffusively and the analytical value of this coefficient (σ p = 1.018) is used for further calculations. To estimate the error, induced by this assumption, we calculated the relative (compared to Nitrogen) σ p value from following relation

σ p σ N2 p = A S /A N2 S B S /B N2 S . (35)
This relative value σ p /σ N2 p of the velocity slip coefficient for Argon and the first disc is of the order of 3%, which is less than the measurement accuracy. Therefore, the assumption on the equality of σ p coefficient for analyzed here gases is justified, so we continue to use the same value of the velocity slip coefficient (analytical value 1.018 [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF]) for all gases used in the experiments.

To obtain the characteristic flow dimension (effective pore size) we used here the experimental data in the hydrodynamic and slip flow regimes. Another possible approach to calculated the effective pore size from the measurements is presented in AppendixC. This methodology is based on the utilization of the measurements in the free molecular and hydrodynamic flow regimes, but it was not applied in this work.

Other characteristic parameters of porous sample

As it discussed in previous Section, from the mass flow rate fitting expression we can extract: the characteristic dimension of porous medium, a, and also the number N of the capillaries as

N = B S L c πa 4 . (36)
However, in Eq. ( 36) the capillary length, L c , is still unknown, so we can make two assumptions to obtain this value. One of possibilities is to assume that the capillary length is equal to the porous disc thickness, L c = L, so the tortuosity factor, l τ , Eq. ( 10), is equal to 1. However, with this assumption the sample porosity, ε = 2.2%, is much smaller than that provided by the manufacturer,

15.9%.

The second possible choice is to assume that the capillary length, L c , is equal to l τ L. However, to calculate l τ we have to introduce new additional parameter, the porosity, ε, which can be known either from the manufacturer or from the tomography analysis (see Section 8), then the tortuosity is calculated as

l τ = a 2 ε B S S L . (37)
Finally, to have the complete realistic description of a porous sample by using the model of the bundle of the tortuous capillaries we need to use additionally the information about the porosity.

The last important data, which can be extracted from the measurements, also by using additional information about the porosity, is the value of the surface-to-volume ratio, Eq. ( 13), which can be also calculated as

S A = ε 2 a . (38)
These results extracted from the measurements are compared with the results of the computer tomography analysis and with that of the mercury porosimetry in Section 9.

Permeability

In Section 2.3 we provided the definition of the permeability as it was proposed by Darcy, i.e. for the incompressible fluid, and then its expression through the mass flow rate, Eq. ( 8), more adapted for the gas flows, so the permeability is calculated as

K = Ṁ RT p m µ S L ∆p . (39)
By using the same model of the porous media as a bundle of N capillaries with length L c and replacing the mass flow rate in the previous expression by its representation provided in Section 5.2, Eqs. ( 30), [START_REF] Vicente | Characterization (2d-3d) of ceramic microfiltration membrane by synchrotron radiation: new and abraded membranes[END_REF], we expresse the permeability as

K = Ṁ /M K0 , M K0 = p m ∆p µv 2 0 2S L = M S0 2S L , (40)
or

K = πa 4 N L c L 2S 1 4 + σ p δ . ( 41
)
The last expression can be rewritten in the form analogous to Eq. ( 32) and then fitted according to the linear regression:

F K = A K X + B K , (42)
where

A K = σ p πa 3 N L c L 2S , B K = πa 4 N 4L c L 2S , X = , (43)
A K and B K are the fitting coefficients. By comparing the coefficients A K and B K of the permeability fit, Eq. ( 43), and coefficients A S and B S , Eq. ( 33), for the fit of the mass flow rate, Eq. ( 32), one can see that they differ only in the factor L/(2S) and the pore characteristic dimension can be also found from the ratio of coefficients A K and B K , as it was done in Section 5.2 for A S and B S coefficients.

From Eq. ( 41) we also obtain the well known Klinkenberg expression

K = K D 1 + 4 σ p δ = K D (1 + 4σ p Kn) , (44)
where K D is the hydrodynamic (Darcy or intrinsic) permeability

K D = πa 4 8 N L L c S = ε l 2 τ a 2 8 , (45)
which can be found from the fitting coefficient B K (or B S , obtained from the fit of the mass flow rate) as

K D = B K = B S L 2S . ( 46
)
As it is clear from Eq. ( 44), in the slip flow regime, the permeability becomes inversely proportional to the rarefaction parameter, i.e. to the mean pressure.

Therefore, we can rewrite Eq. ( 44) in the following form, proposed initially by Klinkenberg [START_REF] Klinkenberg | The permeability of porous media to liquid and gases, Drilling and Production Practice[END_REF],

K = K D 1 + b p m , ( 47
)
here b is a gas dependent coefficient. By identifying Eqs. ( 44) and (47) we can deduce the expression for b:

b = 4σ p µv 0 a [Pa]. (48)
It is clear that b coefficient depends not only from the gas nature through the gas viscosity, the most probable molecular velocity and the velocity slip coefficient, but also from the characteristic dimension a (effective pore radius) of the porous medium, therefore, expression (47) is not at all universal.

It is worth to note that the Klinkenberg expression, Eq. ( 47), is derived from the expression of the mass flow rate through a bundle of capillaries in the slip flow regime, so, theoretically, expression (47) is valid only in the slip flow regime. To find the limits of the validity of the Klinkenberg expression the model proposed in [START_REF] Sharipov | Leak rate of water into vacuum through microtubes[END_REF] for the mass flow rate through a single capillary for all flow regimes, Eq. ( 27), can be used:

K = G K D 4δ . ( 49
)
The limits of the validity of the Klinkenberg expression for the analyzed porous samples are discussed in Section 9.

Tomography analysis

To have additional information about the samples, a typical sample from the same batch was scanned with MicroXCT-400 tomograph at CEREGE, 1which uses the linear attenuation method. The focal spot size of X-ray beam was 5-7 µm. The geometrical voxel size is determined by the size and number of detector elements and the source-object-detector distances (magnification).

The microXCT-400 is also equipped with many optical lenses that lead to an additional optical magnification. In this work, an x40 optical lens was used [START_REF] Vicente | Characterization (2d-3d) of ceramic microfiltration membrane by synchrotron radiation: new and abraded membranes[END_REF]. Finally, the geometrical voxel size is fixed to 1.8 µm. The image stack corresponds to a cylinder of 1.8 mm of diameter and 1.8 mm thick composed of 1000 slices of 1000×1000 voxels size. The porous morphological analysis was perform with the iMorph software [START_REF] Vicente | Characterization (2d-3d) of ceramic microfiltration membrane by synchrotron radiation: new and abraded membranes[END_REF], [START_REF] Hachem | Hygromorphic characterization of softwood under high resolution x-ray tomography for hygrothermal simulation[END_REF] on a cubic Region Of Interest (ROI) made of 500×500×500 voxels, see Fig. 7, which represents 0.35% of the total volume of the analyzed sample.

The pore network segmentation is a crucial step consisting to binarize the reconstructed volume from X-ray computed tomography acquisitions. Because our 3D images of porous sample are under resolved and weakly contrasted, hysteresis method is well adapted for the binarization compared to classical Otsu binarization method which results in our case to a mixing of phases (i.e. the solid and poral phases). The hysteresis function [START_REF] Canny | A computational approach to edge detection[END_REF] performs a dual thresholding operation on the original grayscale image using two threshold values (lower and upper). For the specific application of membrane binarization, the lower threshold is chosen as the smaller threshold that allows the percolation of the poral network and the upper threshold is tuned to obtain the porosity given by the manufacturer [START_REF] Perrin | Morphological characterization of ceramic membranes from 3d xray computed tomography[END_REF].

The local thickness is computed for every voxel of the poral space by filling the pore space with spheres [START_REF] Remigy | Morphological characterization of a polymeric microfiltration membrane by synchrotron radiation computed microtomography[END_REF], [START_REF] Bossa | Micro-and nano-x-ray computed-tomography: A step forward in the characterization of the pore network of a leached cement paste[END_REF]. The volumetric distribution of the local thickness gives the mean pore size diameter equal to 7.26 ± 5.31 µm. Its cumulative distribution (Fig. 8)

shows that 65% of the total pore volume is composed of structures with local thickness inferior to 5 µm, and that 80% of the total pore volume is occupied by structures with local thickness inferior to 7 µm.

To quantify the size of the constrictions in the vicinity of the pores, we use a watershed pore segmentation method based on the ultimates eroded. The watershed operator [START_REF] Digabel | Iterative algorithms[END_REF] that is used to individualize each pore surrounded by constrictions, relies on iterative erosion and dilation. The Euclidean distance of the pixel to the nearest background pixel is therefore called the Euclidean distance map. The erosion process (that consist in peeling the distance map successively) needs to be constrained such that the isolated pixels (i.e. pixels surrounded by eight neighbors in each and any erosion iteration) may not be eroded. These pixels coincide with the local maxima in the Euclidean distance map and are called ultimate eroded points [START_REF] Lantuejoul | On the use of the geodesic metric in image analysis[END_REF], because further constrains erosions do not change the image of ultimate's points. The ultimate eroded points are now iteratively dilated through the watershed region growing process and following the distance map values (Fig. 9). The voxels that correspond to the meeting region coming from different ultimate's points labels are identified as throats. For every throat surface we compute the equivalent ellipsoid and we report here the distribution values of the major and the minor axis (Fig. 9).

The minor axis length gives the diameter of the maximal inscribed disk into the throat. The mean values are 7.7 µm and 18.8 µm for the minor and major axis, respectively.

We compute the penetration length of the porous sample for different particle diameters [START_REF] Perrin | Determination of molecular weight cut off by x-ray micro ct[END_REF]. The results are presented on Figure 10. From this information we can the easily deduce the geometrical cutt-off size of the membrane. From Figure 10, it is clear that only the particles with a diameter of around 3.5 µm are able to cross the porous sample.

Results and comparison

In this Section we present the results obtained with the proposed methodology on the effective pore size, tortuosity, surface-to-volume ratio and the permeability. We compare these porous sample characteristics to the data obtained from the tomographic and porosimetry analyses, when they are available.

Pore size

We start by the analysis of the pore size with one porous sample from a batch, called previously first disc. Applying the presented here experimental methodology, explained in details in Section 5.2, the effective pore dimensions are extracted from the mass flow rate measurements, see Table 4. By analyzing the data presented in Table 4, we can see that for this porous sample the effective pore diameters, 2a, calculated with different gases, are very close one to another. The uncertainty in the estimation of the characteristic pore dimension is of the order of 16% for Nitrogen and decreases up to 13.9% for Argon. The average pore dimension, estimated with two gases, 2a = 3.6 µm, is obtained with an uncertainty of 13.9%.

The uncertainty of the effective pore dimension is calculated using the square root of the summation of the fitting coefficients uncertainty, which is calculated from the limits of a 95% confidence interval of the fitting parameters.

The iMorph computer analysis of the tomographic data, see Section 8, allows to obtain the representation of the porous structure of a sample as the system of the pores which are connected by the constrictions (throats). From the analysis of the aperture map distribution, Fig. 8, it was found that 65 % of total pore volume is composed of the structures with the local thickness smaller that 5µm, which is close to the results found for the first disc. From the distribution of the throat size dimensions, shown on Fig. 9, it is clear that 25% of the throats have the dimension equal to 3.6 µm. In addition, from the iMorph analysis it was also found that only the particles with diameter of around 3.5 µm are able to cross the sample, see Fig. 10.

All this information confirms our experimental finding for the first disc. From this analysis, we can conclude that the proposed gas flow methodology allows us to estimate the effective pore size which determines the flow through a porous medium. This dimension is also correlated to the throat size dimension and to the particle cut-off dimension, obtained from the tomography analysis.

Tortuosity and surface-to-volume ratio

The information about the effective pore dimension is extracted without any additional knowledge about the analyzed porous sample. However, if we assume that the sample porosity is equal to 13.6%, the value obtained from the tomographic analysis, then additional characteristics of the porous sample are obtained: the tortuosity, Eq. ( 37), the capillaries number, N , Eq. ( 36), and surface-to-volume ratio, S A , Eq. [START_REF] Perrin | Determination of molecular weight cut off by x-ray micro ct[END_REF]. All these values are presented in Table 4 for the first disc. These characteristics can be compared with the data on the tortuosity and surface-to-volume ratio, obtained from the tomography analysis, see Table 5. The tortuosity in three directions is given in Table 5 and the gas permeation direction corresponds to the z axis. The computer tomography tortuosity in this direction is equal to 1.61, which is close to the tortuosity calculated for the first disc, 2.5, see Table 4.

The averaged over two gases value of the surface-to-volume ratio is equal to 15.0 × 10 4 m 2 /m 3 for the first disc, see Table 4. This value is of the same order of magnitude as that provided by the tomographic analysis, 6.27 × 10 4 m 2 /m 3 , see Table 5.

Both data on the tortuosity and surface-to-volume ratio, found from the proposed methodology, are close to that obtained from the tomography analysis.

Non-destructive analysis

To check the repeatability of the determination of the effective pore size dimension, we have analyzed the second disc, which was provided by the same manufacturer, so supposed to be identical to the first disc.

The same analysis was carried out for this second disc and the effective pore diameters, 2a, calculated with three gases are provided in Table 4. As for the first disc, the effective pore diameters obtained with different gases are very close one to another. The uncertainty in the estimation of the characteristic pore dimension is of the order of 16% for Argon and decreases up to 12% and 10%

for Helium and Nitrogen, respectively. The average pore dimension, estimated with three gases, 2a = 22 µm, is obtained with an uncertainty of 13.6%. The much larger effective pore size, 22 µm, found for the second disc, represents a possible sample imperfection: the largest pores are interconnected for this sample and so they determine the gas flow rate. This structural defect is also visible through the mercure porosimetry analysis of a sample from the second batch and it results in a peak between 20 and 30 microns, see Fig. 11.

This finding suggests that the proposed gas flow method could potentially be used as a method of the non-destructive analysis of a porous sample.

Permeability

The hydrodynamic (intrinsic) permeability K D , calculated from the mass flow rate measurements, is provided in Table 6, for the first and second discs, respectively. As it is clear from Table 6 the hydrodynamic permeability is gas independent within the experimental uncertainty and it is found to be much smaller for the second disc. It is also worth to underline that two parameters for description of the permeability in form of Eq. ( 44), a and K D , are obtained directly from the fit of the measured mass flow rate or permeability data without any additional information about the sample porosity.

The intrinsic permeability is also provided by iMorph computer analysis, based on the analytical relation for the channel conductivity, and it is equal to 1.3 × 10 -14 m 2 . This value is close to that obtained for the first disc, 0.9 × 10 -14 m 2 , see Table 6.

The permeability for both discs is plotted on Fig. 12 as a function of the Knudsen number. The permeability increases in more than two orders of magnitude (first disc) when the mean pressure is decreasing.

In Table 6 the b coefficient is provided for two porous discs and it is two times smaller for the first disc compared to the second one which confirms that the Klinkenberg expression is not universal and the b coefficient is gas and porous sample dependent.

The measured and calculated from Eqs. (49), [START_REF] Sharipov | Rarefied gas dynamics. Fundamentals for research and practice[END_REF] Therefore, the deviation between measured and semi-analytical data is found for the large Knudsen number range. It also confirms that the Klinkenberg expression is valid in the slip regime only.

The implementation of Eqs. (49), [START_REF] Sharipov | Rarefied gas dynamics. Fundamentals for research and practice[END_REF] shows an interesting potential of extracting intrinsic permeability from apparent permeability in free molecular and transitional regimes.

Conclusion

The classical model of the porous media presentation as a bundle of capillaries was revised. The original methodology was suggested to determine the characteristic flow dimension. The experimental procedure is developed to determine the effective pore size (characteristic flow dimension) and the number of capillaries, related to the model a bundle of capillaries. The experimentally obtained effective pore dimension is in very good agreement with the results of the mercury porosimetry and micro-computed tomography. The use of additional information on the sample porosity allows to find the tortuosity and the surface-to-volume ratio, which were close to that calculated from the tomography analysis. The Klinkenberg formula was also analyzed, and it was shown that this expression is not general and b coefficient reveals gas and porous medium dependency. In addition, the Klinkenberg expression is valid only for the slip flow regime, which was shown experimentally. Therefore, this formula has to be used with precaution in the case of low porous structures. The intrinsic permeability obtained by tomography analysis is very close to the measured permeability, which is not surprising in the case of the homogeneous porous medium. The proposed approach is the first very promising stage to evolve towards measurements of even lower permeabilities and also the characteristic dimension (pore size) of membranes used for microfiltration (> 100 nm ) and ultrafiltration (> 10 nm). where the mean free path, , and the most probable molecular velocity, v 0 , are defined by Eqs. ( 15) and ( 16), respectively. To estimate the average time for a gas molecule to travel through the porous media we introduce the Fluid Travel Time, FTT, as

F T T = τ L u , (B.2)
where τ is the tortuosity, Eq. ( 10), L is the thickness of the porous sample and u is the fluid velocity. In the hydrodynamic flow regime we have

∆p = 0.5ρu 2 , (B.3)
where ρ is the gas density. Under typical experimental conditions, for the sample thickness L = 2 mm, and assuming τ = 2, the mean free time, M F T , is five orders of magnitude less than the average fluid travel time, F T T , through the porous media, therefore, we are well within the quasi-stationary state assumption.

AppendixC. Two limits approximation

Additional possibility to obtain the porous medium characteristic dimension can be done by using two limits of the flow regimes, which can be easily dis- However, two problems are related to the realization of this approach. First, the value of the accommodation coefficient α is unknown a priori. The second curtail point is the correct determination of the flow regime, i.e. the correct choice of the pressure range. From the theoretical point of view we know that the both normalized mass flow rates ṀP /M S0 and ṀFM /M G0 have to be constant.

If this is not the case, the hydrodynamic (or free molecular) regime has not been reached yet, and the determination of the characteristic dimension can be affected by the essential error. Under our experimental conditions and for the porous samples used here, we did not arrive to reach both regimes, so this theoretically possible approach was not realized here.

  However, it is more complicated to identify the slip flow regime. The G 0 curve for the first disc is presented in Fig.4. It is clear that the analytical curve of dimensionless mass flow rate G trough a single capillary, Fig.3, has a very similar shape to the experimental curve of the normalized quantity G 0 through the microporous medium, see Fig.4. Therefore, by analogy, we can identify the slip flow regimes visually in terms of inverse molecular mean free path. In Table2different flow regimes are represented by different molecular mean free path ranges.When the flow regime is determined the corresponding analytical mass flow rate expression can be chosen and then the measured mass flow rate can be fitted to determine the characteristic flow dimension, pore radius, and the number of capillaries in the representation of the porous medium as a bundle of capillaries.Once the flow dimension has been extracted, it is useful to calculate the Knudsen number and the rarefaction parameter and compare how close the chosen Knudsen number (rarefaction parameter) range is to the classical definition of the slip regime range for a capillary. For the slip flow regime, we should find the Knudsen number Kn ≈ 0.1 and the rarefaction parameter δ ≈ 10. We can use this point as a guideline to further refine our definition of the regimes. This is done by reiterating the process of fitting and extracting the equivalent flow dimension and again calculating the Knudsen number and rarefaction parameter until become close to the theoretical values of a tube.

  permeabilities are shown on Fig. 13. Very good agreement between experimental and analytical data are found in the near hydrodynamic, slip and beginning of the transitional flow regimes, see Fig. 13. However, in the free molecular flow regime, the semianalytical expression overestimate the experimental data. It can be explained by the fact that the experimental data are fitted only in the slip flow regime.

NFigure 1 :

 1 Figure 1: Models of the flat ceramic microporous membrane: bundle of N parallel capillaries of the same radius a (left) of the same length Lc equal to the membrane thickness L; (right) of the capillary length Lc is greater than the membrane thickness L, Lc = lτ L.

Figure 2 :

 2 Figure 2: Upper figure: Pressure evolution in time, where the upstream tank pressure, p 1 , is red and the downstream tank pressure, p 2 , is blue, together with the respective fitting curves of the pressure evolution, in black, and the mean pressure pm in grey. Lower figure shows the evolution of the pressure difference and the exponential fitting of the measurements.

Figure 3 :

 3 Figure 3: Dimensionless mass flow rate G for a single circular tube, Eq.(27), as a function of rarefaction parameter and Knudsen number.

G 0 [m 2 ]

 2

Figure 4 :[ 10 -16 m 3 ]ℓ [ 1 ×Figure 5 :

 410315 Figure 4: Experimental points for the first disc in normalized form of G 0 function, G 0 = Ṁ /(∆p/v 0 ) as a function of the inverse molecular mean free path, -1 .

Figure 6 :Figure 7 :

 67 Figure 6: Difference between the experimental points for normalized mass flow rate, Eq. (31), and the fitting curve, Eq. (32), in form (S 0 -F S )/F S as a function of molecular mean free path, .

Figure 8 :

 8 Figure 8: Aperture map distribution (65% of the voxels belong to as sphere with diameter inferior or equal to 5 µm (blue voxels)).

Figure 9 :

 9 Figure 9: Throats size dimension and shape (70 % of the throats present an inner circle diameter equal to 7.6 µm, and 25 % equal to 3.6 µm).

Figure 10 :Figure 13 : 0 2SL

 10130 Figure 10: Microtomographic analysis: penetration length as a function of a particle diameter.

  estimate this time, we introduce a measure of the average time between two successive collisions, the Mean Free Time, M F T , as M F T = /v 0 , (B.1)

  tinguished. The first limit is the free molecular regime (or Knudsen diffusion regime), where the molecule-molecule collisions can be neglected because they are a few numerous comparing to molecule-surface collisions. In this regime, the mass flow rate is proportional to a 3 . For the second limit regime, hydrodynamic regime (Poiseuille flow), the opposite situation is realized: molecule-surface collisions are very few numerous compared to molecule-molecule collisions. The mass flow rate is proportional to a 4 . Therefore, we can use the ratio of the normalized mass flow rates, measured in these two regimes to find the characteristic dimension of the porous media as followingṀ P /M S0 Ṁ FM /M G0

Table 1 :

 1 Useful characteristics of the gases [21] used in present experiments

	He	1.865	0.66	2077.1	4.003
	Ne	2.976	0.66	412.02	20.18
	N 2	1.656	0.74	296.80	28.00
	Ar	2.117	0.81	208.13	39.95

Table 2 :

 2 Flow regimes identification. S [10 -9 m 2 ] B S [10 -16 m 3 ] A S [10 -9 m 2 ] B S [10 -15 m 3 ]

			1st disc	2nd disc
		Regime	[µm]	[µm]
		HYDRO 1.9×10 -3 <	0.11 <
		SLIP	0.19 <	1.1 <
		FM	19 >	110 >
		1st disc	2nd disc
	GAS A He N 2 Ar AVR	1.2 ± 0.2 1.2 ± 0.1 1.2 ± 0.1	5.5 ± 0.1 5.5 ± 0.1 5.5 ± 0.1	2.4 ± 0.2 2.9 ± 0.2 2.4 ± 0.4 2.6 ± 0.3	7.1 ± 0.1 7.2 ± 0.1 7.4 ± 0.2 7.2 ± 0.1

Table 3 :

 3 Fitting coefficients A S and B S with S-fit.

Table 4 :

 4 Estimation of the porous media characteristic dimension, a, the number of capillaries N , and the surface to volume ratio, S A , by using S-type fit and the porosity obtained from the micro-computed tomography, ε = 13.6%. The tortuosity is calculated from Eq. (37).

	τ

Table 5 :

 5 Results from iMorph analysis of the one part of the first porous disc, obtained with 1.8 µm space resolution: porosity, ε, specific surface area, S A , and tortuosity, lτ , in three directions (x, y and z). D [10 -15 m 2 ] b [kPa] K D [10 -13 m 2 ]

		1st disc		2nd disc
	GAS K b [kPa]
	He N 2 Ar AVR	8.9 ± 0.2 9.0 ± 0.2 9.0 ± 0.2	17 ± 2 18 ± 3 18 ± 3	1.16 ± 0.02 1.16 ± 0.02 1.20 ± 0.03 1.18 ± 0.02	7.3 ± 0.8 3 ± 0.2 2.6 ± 0.5 4.3 ± 0.5

Table 6 :

 6 Klinkenberg coefficients.

								p1
								p2
								p m
								p fit 1 p fit 2
	Pa]
	[10 3
	p
		0	6	12	18	24	30	36	42	48
						t [s]
		6						p1 -p2 ∆p fit
		5
	Pa]	4
	∆p [10 3	3
		2
		1
		0
			6	12	18	24	30	36	42	48
						t [s]

Centre for Research and Teaching in Environmental Geoscience, Aix Marseille University, Aix-en-Provence, France, https://www.cerege.fr

Acknowledgement

The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University -A*MIDEX, a French "Investissements d'Avenir" programme. It has been carried out in the framework of the Labex MEC. The authors (M.V. Johansson, P. Perrier, and I. Graur) would like to acknowledge the financial support provided by the European Union network program H2020, MIGRATE project under Grant Agreement No.643095.

AppendixA. Real gas effects

The ideal gas law assumes that gas molecules do not occupy any space and that there is no molecular potential for attraction and repulsion. Under high pressure, the first assumption breaks down as the volume occupied by the gas molecules cannot be neglected. When the temperature is low, the molecular potential cannot be neglected, otherwise, if there is no attraction condensation cannot occur. The ideal gas law is considered accurate when the temperature is significantly larger than the boiling point, i.e. at least two times greater than the critical temperature. Additionally, the pressure needs to be not much greater than atmospheric pressure, it has to be much lower than critical pressure.

To quantify the deviation from ideal gas law conditions, it is useful to introduce the compressibility factor [START_REF] Reid | The properties of gases and liquids, fourth edition Edition, no. 30 in McGraw-Hill chemical engineering series[END_REF] Z = pV M RT , (A.1) which assumes unity when the ideal gas law assumption is valid. In Table A [START_REF] Reid | The properties of gases and liquids, fourth edition Edition, no. 30 in McGraw-Hill chemical engineering series[END_REF]. compressibility factor close to unity. To extend the presented mass flow rate measurement method for real gases Eq. A.1 can be used to relate the gas mass and the compressiblity factor to pressure.

AppendixB. Quasi-stationary assumption

To derive the mass flow rate we have to consider the mass variation in time as a quasi-steady process. We assume that there are infinitesimal unbalanced forces which modify the state of the system slower than the system reaches its local equilibrium. In this case, we can approximate the thermodynamic processes as a succession of equilibrium states. This approximation can be considered as an accurate one when the average time needed for a gas molecule to travel through the porous medium is much greater than the time between two successive intermolecular collisions in the reservoir [START_REF] Spakovszky | Thermodynamics and propulsion[END_REF]. To quantitatively


Camille Lavayssière
email: camille.lavayssiere@univ-pau.fr

Bastien Letowski

Benoît Larroque

Franck Luthon

LABOREM -A NETWORK OF OPEN SOURCE REMOTE LABORATORIES FOR LEARNING 1

Keywords: e-learning, game-based learning, industrial engineering, labwork, student motivation, online education, virtual instruments, remote laboratory, SCADA. 1

The increase in numerical technologies development makes possible the emergence of new pedagogical tools as remote laboratories. The instrumentation of technological systems and distant interfaces offers new possibility for a distant learner. The LaboREM platform, designed with this objective, is running and evolving since 2009. In this paper, we present the evolution from a local solution to a network of remote laboratories. To deploy remote laboratories informatics and electronics development are needed. Based on these objectives, a novel approach using Python as unique programming language is used. A generic PCB (Printed Circuit Board) approach has been designed to answer robustness and adaptive needs. These developments will help to spread LaboREM to other education remote laboratories.

INTRODUCTION

Distance education is growing easier and faster with the widespread use of computers and internet. Moreover as shown in [START_REF] Araneda | Massification of higher education in Chile: challenges for chilean universities[END_REF] the "massification" of higher education leads to a diversification in post-secondary student profile. To answer this issue, it appears to be decisive to create new tools for teachers and learners as remote laboratories and game-like scenarios. The perspectives of applications with this new teaching format are various (Fig. 1).

Fig.1. Applications of LaboREM in education

From one side, training centers can share equipment through a network of distant laboratories and can offer a new educational tool. This tool is not designed to replace traditional practical works in classroom, but its objective is to offer a combined learning process: on-site courses, simulation, online courses.

On the other side, online technologies erase time and transfer constraints. This functionality gives good perspectives to learners with motor disabilities and learners in continuing education. Remote laboratories allow teachers to show live experiments in a lecture to apply directly theoretical knowledge. Another advantage is to provide to the students a new way to deal with in depth revision to prepare for an exam.

The innovative character is the set-up of a learning scenario that allows connecting in realtime to a platform with the integration of the lesson in the LMS (Learning Management System) [START_REF] Luthon | LaboREM-A Remote Laboratory for Game-Like Training in Electronics[END_REF]. The lesson adopts a game-like scenario.

Creating a network of remote laboratories can be very useful to share expensive experiments, to mutualize equipment between training institutes which have the same learning program. This kind of network was developed in the project WebLab Deusto [START_REF] Gustavsson | On Objectives of Instructional Laboratories, Individual Assessment, and Use of Collaborative Remote Laboratories[END_REF]. As, since 2009, we have developed and deployed a remote laboratory for 40 undergraduate students, we get some interesting feedback of their feeling to use this kind of lab [START_REF] Luthon | Remote Laboratory for Game-Based Distance Learning in Electronics[END_REF]. Main important things that emerge from their perception are listed below:

Feeling of immersion (video, robot) Real-time process Integration in the LMS of the university (interaction with tutor) Work on the same equipment as it was during hands-on labworks.

For this reasons, our approach is quite different from the approach chosen by the authors of [START_REF] Gustavsson | On Objectives of Instructional Laboratories, Individual Assessment, and Use of Collaborative Remote Laboratories[END_REF] who are working on remote lab network which is not linked with a game-like pedagogical scenario in a LMS.

METHODOLOGY AND ISSUES

The actual prototype of LaboREM evolving since 2009 is facing several issues to develop a network of remote laboratories sharing resources between various scientific centers and universities (Fig. 2).

Fig.2. LaboREM hardware resources on server side

To describe the set-up of a generic platform, it is necessary to list the issues and transpose them to each part of the system (Fig. 3).

Fig.3. Diagram of LaboREM 2018

The methodology we choose to create an e-lab network is a low-cost solution, easy to deploy and open source. To have an immersion of the learner, we design experiences embedded in a game-like scenario. To have high availability and easy access to the platform, we use a web HMI (human machine interface) with 24/7 or scheduling access. To carry out various equipment compatibility, we would like to develop an open community to easily share practical works.

2.1.

Server and software LaboREM uses a supervisory control and data acquisition (SCADA) system. The initial prototype of LaboREM uses a technically solution running on Windows Server operating system, and using the proprietary LabVIEW software from National Instruments© to program and create the web remote access, which allows to control the instruments (Fig. 4) and exploit data results. The cost of the software, the powerful server needed and the non-standard protocol used by LabVIEW to communicate data through the internet lead us to develop an entirely new IT system for migration to an open source and light software (Fig. 5). Similar issues exist for low and medium size industries [START_REF] Merchán | Open Source SCADA System for Advanced Monitoring of Industrial Processes[END_REF]. Linux (Debian) is used as operating system for the stability. The programming language used is Python. This language enables clear programming, in addition of the Django framework which eases the creation of complex, databasedriven websites. To communicate with the instruments, we used the Django application of PyScada [START_REF] Schröder | PyScada is a open source scada system that uses the Django framework as backend[END_REF] and other Python open source applications. One advantage of this solution is to provide an HTML5 interface compatible with every client (computer, smartphone…). Another advantage on the server side is to develop with a light and open software solution which enables to use as a server a single-board computer (SBC). These choices require additional developments in comparison with the solution with LabVIEW© but it ensures that the deployment will be robust on client-side (HMI) and on server-side (open-source applications)

Data acquisition and control

The deployment of the instrumentation and the server system needs the design of an electronic box. The "LaboREM Box" will store the electronic part designed in our laboratory in University of Pau. The approximate size for practical works in electronics is 20x15x10cm3. The box is composed of several printed circuit boards (PCB): a motherboard (Fig. 6) where we can place 16 different plugs and a robot extension board. The robot extension-board interconnects the movable elements. The robot extension will be optional to adapt the box to all the university's needs. The motherboard receives and manages all the information of the experiment that the learner wants to work on. Each plug is a device under test (DUT) and can be easily replaced by the teacher to adapt the scenario to the learners. The learners will have access to 16 DUT to achieve their work. The generic plug schematics is illustrated in Fig. 7.

Fig.7. Pin mapping of the generic plug

Each plug is supplied with different voltages in order to deal with CMOS or TTL circuits and also circuits with symmetric supply. Three input voltages and one output voltage are present. C ij interconnect the external components placed by the robot.

Instruments communication

The data acquisition system has to be compatible with several protocols (VISA, Modbus, BACnet…) and connections (USB, Serial, Ethernet). To fulfill these requirements, we take profits to use an open source solution in Python and we can easily find in the Python community usefull applications:

PyUSB, PySerial, PyVISA, PyModbus. As shown in Fig. 8, we have to develop the instrument commands part called by PyScada. The idea is to provide a list of experiments scripts available in a bank of instruments selected by PyScada after the teacher has informed the experiment he wants to set-up.

Interaction between learner and laboratory

Improving motivation and immersion of the learner is a priority to have a good learning process. To ensure these issues, it is necessary to use some digital tools to bring immersion feeling to the student. A live video coming from a simple camera and/or a drone replaces the eye of the learner (Fig. 9). It allows to focus on the experiment and move around. Image processing techniques allow to work on facial expression recognition to detect the emotion of the learner, and give him the feeling that he is not alone: this mimics the eye of the teacher [START_REF] Khattar | Using computer vision for student-centred remote lab in electronics[END_REF]. Live call will be possible between the teacher and the student. Implementation of the robotic arm on server side permits to interact directly on the experiment via some controls available on the web interface: this mimics the arm of the student.

Integration of the pedagogical scenario in the LMS

An adaptive scenario for the learner will improve his immersion and his motivation in the learning process. To interact in real-time with the scenario and the remote lab, a bidirectional communication between the laboratory (instrument drivers database, HMI) and the LMS (user database, game-like scenario, SCORM database) is required. A typical game-like scenario contains video presentations, quizzes, online lessons, remote laboratories, online simulators. The game-like scenario designed in the LMS plans some different paths depending on student's results during the sequence of the scenario, allowing teachers or students to adapt it or to redo one step as described in Fig. 10.

TECHNICAL RESULTS AND PROGRESS

The actual HMI version of LaboREM is directly accessible from the learning management system (LMS) of the University of Pau. The remote laboratory activity is embedded into a game-like scenario which gives an access to practical works in analog electronics, to the list of connected users and a "TOP 10" ranking which offers the opportunity to the students to compare their experimental results with other classmates, and to compete for best measurements

3.1.

Open source SCADA development

The SCADA system was developed and validated, the web HMI allows to enter the values to control function generator, digital multimeter and oscilloscope (Fig. 11). An algorithm which computes the frequency response of any filter was validated. Fig. 12 and Fig. 13 present a Bode plot of a simple RC low-pass filter. Based on our first results, we are currently working towards three major improvements and developments:

Improve the graphic part of the HMI to be more attractive, Create a library to permit to SCADA system to communicate with different instruments brands (Tektronix, Agilent, Rigol, etc.), Migrate to Moodle to be compliant with the most used LMS in higher education.

PCB design

A PCB design software -called Ultiboardwas used to design PCBs. Connectors and plugs are designed in PCI Express logic because this is a well-known electronic standard. It simplifies PCB design, reduces the cost thanks to widely used connector types and assures a mechanical resistance that allows the teacher to change easily the plug on the motherboard. 3D views of the designed PCB are visible in Fig. 14 and Fig. 15.

CONCLUSION

We have shown in this article all the topics which refer to the deployment of LaboREM in some universities. Development with open-source applications, standardized protocols and low-cost hardware set-up allows to design a solution easy to duplicate and favors interconnection of the remote labs to create a network. The use of opensource applications on the server-side allows to embed the remote lab into a blended learning scenario in the LMS of the university.

As generic motherboard were designed, new practical DUT can be imagine in other fields in applied sciences like mechanics, optics and renewable energies.

Further developments are currently conducted toward a full design of the "LaboREM Box" and a transfer to universities in Senegal, Burkina Faso and Germany. The impact of LaboREM for learners and teachers will be measured in all partner universities to improve the pedagogical approach, the immersion of the learner in the practical work, and the interaction with the professor.

This pedagogical tool will be a new pillar for blended learning approach and thus the frontier between theoretical and practical studies becomes more porous.

Fig. 4 .

 4 Fig.4. Diagram of the proprietary structure

Fig. 6 .

 6 Fig.6. Block schematics of the modular motherboard

Fig

  Fig.8. Instrument communication 2.4.Interaction between learner and laboratoryImproving motivation and immersion of the learner is a priority to have a good learning process. To ensure these issues, it is necessary to use some digital tools to bring immersion feeling to the student. A live video coming from a simple camera and/or a drone replaces the eye of the learner (Fig.9). It allows to focus on the experiment and move around. Image processing techniques allow to work on facial expression recognition to detect the emotion of the learner, and give him the feeling that he is not alone: this mimics the eye of the teacher[START_REF] Khattar | Using computer vision for student-centred remote lab in electronics[END_REF]. Live call will be possible between the teacher and the student.

Fig. 9 .

 9 Fig.9. Drone flying above the LaboREM measurement instruments

Fig. 10 .

 10 Fig.10. Learning map as a treasure hunt with various steps

Fig. 11 .

 11 Fig.11. Input Controls and plot information

Fig. 14 .

 14 Fig.14. 3D view of the "LaboREM Box" motherboard; grey rectangles represent connectors to interconnect plugs and robot extension board

ACKNOWLEDGMENTS

The authors would like to thank the staff of ISITE program E2S for financial support for the engineer fellowship of Camille Lavayssière.


Camille Lavayssière

Bastien Letowski

Benoît Larroque

Franck Luthon

EASY APPLIED SCIENCES LEARNING THROUGH OPEN SOURCE REMOTE LABORATORY

Keywords: e-learning, game-based learning, industrial engineering, labwork, student motivation, online education, virtual instrument, remote laboratory, SCADA

The development in numerical technologies modifies and shapes interests of today's students but it also makes possible the emergence of new pedagogical tools as remote laboratories. The instrumentation of technological systems and distant interfaces offers new possibility for a distant learner and it takes more into account the student's interest in internet, web communication and gaming. The remote laboratory platform, designed with this objective, is running and evolving since 2009. In this paper, we present the evolution from a proprietary solution to an open source solution. So, informatics and electronics developments are needed. Based on these objectives, a novel approach using Python as unique programming language is used. A generic PCB (Printed Circuit Board) approach has been designed to answer robustness and adaptive needs. These developments will help to reduce the cost of the remote laboratory and give more flexibility to use it on wide applied sciences practical works.

INTRODUCTION

Distance education is growing easier and faster with the widespread use of computers and internet. Moreover, the "massification" of higher education leads to a diversification in post-secondary student profile [START_REF] Araneda | Massification Of Higher Education In Chile: Challenges For Chilean Universities[END_REF]. To answer this issue, it appears decisive to create new tools for teachers and learners as remote laboratories and game-like scenarios [START_REF] Luthon | LaboREM: A Remote Laboratory for Game-Like Training in Electronics[END_REF]. Since 2009, we have developed and deployed a remote laboratory called LaboREM where student can build their electronic circuit with a robotic arm. The perspectives of applications with this new teaching format are various and illustrated in Figure 1.

Figure 1. Applications of LaboREM in education

Creating a network of these remote laboratories can be very useful to share expensive experiments or to mutualize equipment between engineering education training centres. We would like to develop an open community to easily share practical works and to create an open source, low-cost and easy to deploy e-lab network inspired by [START_REF] Gustavsson | The VISIR projectan Open Source Software Initiative for Distributed Online Laboratories[END_REF], [START_REF] Gustavsson | On Objectives of Instructional Laboratories, Individual Assessment, and Use of Collaborative Remote Laboratories[END_REF].

The methodology and the solution are globally presented in the second section of the article. Then, the third section describes the technical implementations in regard to important software and hardware developments. Finally, conclusions and future objectives are summarized in the last section.

METHODOLOGY

The most important points for a remote laboratory are summarized in the Table 1, comparing the goals and the proposition of LaboREM. To describe the set-up of a generic platform, it is necessary to illustrate the issues presented and to transpose them to each part of the system. LaboREM answers these objectives (Figure 2).

Figure 2. Diagram of LaboREM 2018

The development of LaboREM will allow to use the same equipment as the one available in hands-on work, where student's arms and eyes are replaced by the robot arm and the drone [START_REF] Khattar | Using Computer Vision For Student-Centred Remote Lab In Electronics[END_REF]. Students and teachers will be able to access to the platform from any device over the internet. The normalized developments for the software and hardware reduce the cost and allow us to share and reproduce LaboREM easily.

Software Methodology

Server and software

The supervisory control and data acquisition (SCADA) system was developed and validated on basic electronic filters, the web human machine interface (HMI) allows to enter the values to control function generator, digital multimeter and oscilloscope. An algorithm which computes the frequency response of any filter was validated.

Previously, the SCADA system used by LaboREM ran on Windows Server operating system. The proprietary LabVIEW software from National Instruments was used for data acquisition and robot control. The cost of the software and the proprietary protocols library used by LabVIEW lead us to develop an entirely new IT system for migration to an open source and light software (Figure 4). For instance, low and medium size industries face similar issues [START_REF] Merchan | Open Source SCADA System for Advanced Monitoring of Industrial Processes[END_REF] and other universities substitute LabVIEW by open source solution like the National University of Distance Education in Spain [START_REF] Sáenz | Open and Low-Cost Virtual and Remote Labs on Control Engineering[END_REF]. To ease the creation of complex, database-driven websites we use a Python framework called Django [START_REF] Schröder | PyScada is a open source scada system that uses the Django framework as backend[END_REF]. Communication with the instruments is done with the Django application PyScada. Running under Linux, the solution provides an HTML5 interface compatible with every client. Another advantage on the server side is to provide a solution which can be developed with a light and open software solution which enables to use as a server a single-board computer (SBC). The solution will be embedded in a Raspberry Pi like in [START_REF] Bermúdez-Ortega | Remote Web-based Control Laboratory for Mobile Devices based on EJsS, Raspberry Pi and Node.js**This work has been supported by the Spanish National Research Project DPI2013-46665-C2-1-R[END_REF].

The teacher can easily add the HMI web page in the leaning management system (LMS) and receives real time results of the practical work: he can adapt and improve the pedagogical scenario. The authentication can be managed by the central authentication service (CAS) of the university or with Django. The integration of LaboREM is unnoticeable for students and enhances their immersion.

Instruments communication

The data acquisition system must be compatible with several communication protocols (VISA, Modbus, etc) and connections (serial, USB, ethernet, GPIB, etc).

To fulfill these requirements and go beyond the proprietary solutions, we can easily find in the Python community useful applications like PyUSB or PySerial. As shown in Figure 5, we have to develop the instrument commands part called by PyScada.

Figure 5. Instruments communication with the LabView based solution and the open source proposed solution

The idea is to provide a bank of experiments scripts available for a list of instruments selected by PyScada. This bank of scripts can be filled-in by the teacher: he informs about the experiment which he wants to set-up and the instruments he has got.

Result and progress of the open source SCADA development

The SCADA system was developed and validated, the web HMI allows to sign in with the university ID and then to enter the values to control function generator, digital multimeter and oscilloscope. An algorithm which computes the frequency response of any filter was validated and presented on the Figure 6. It is a Bode plot of a Sallen-Key band-pass filter.

Hardware Methodology

Data acquisition and control

Hardware development is also conducted. A modular motherboard where 16 different devices under test (DUT) can be plugged has been designed ( Figure 7). The motherboard receives and manages all the information of the experiment that the learner wants to work on. The learners will have access to 16 DUT to achieve their work. Each DUT can be easily replaced by the teacher to adapt the scenario to the learners. The DUT is connected to the motherboard with a generic plug, the schematic is illustrated in Fig. 8. Each plug is supplied with several voltages levels in order to deal with CMOS or TTL circuits and also circuits with symmetric supply. Twelve bidirectional input/outputs are present. The Vj are mainly used for voltage signals and the Cj for other connections like robot interconnection or logical signals if need by the DUT. As plugs are generic, it will be possible to interconnect different sources, multimeters and robot controller.

PCB design

The Finally, all the electronics parts will be stored in a "LaboREM Box" designed in our laboratory in University of Pau. The approximate size for practical works in electronics is 25x25x10cm 3 . The box will be composed of the motherboard, the different plugs, a relay to turn-off/on all the elements, a power supply and the SBC.

Moreover, PCB designs of each plug will be available as open hardware permitting community to create new DUT and to share it.

CONCLUSIONS

As we shown in this article, developments with open-source applications, standardized protocols and low-cost hardware set-up allow designing a solution easy to duplicate and favors interconnection of the remote labs to create a network. The solution will transfer to universities in Senegal, Burkina Faso and Germany. The second objective is to continue developments toward a full design of the "LaboREM Box" to control an easy application to applied sciences experimentation. As modular motherboard was designed, new practical DUT can be thought of in others fields in applied sciences like automatic control [START_REF] Sáenz | Open and Low-Cost Virtual and Remote Labs on Control Engineering[END_REF], [START_REF] Bermúdez-Ortega | Remote Web-based Control Laboratory for Mobile Devices based on EJsS, Raspberry Pi and Node.js**This work has been supported by the Spanish National Research Project DPI2013-46665-C2-1-R[END_REF], mechanics, optics [START_REF] Galan | A remote laboratory for optical levitation of charged droplets[END_REF] and renewable energies.

Figure 3

 3 Figure 3 illustrates the actual prototype of LaboREM evolving since 2009 with the last updates.

Figure 3 .

 3 Figure 3. LaboREM 2018 hardware resources on server side

Figure 4 .

 4 Figure 4. Diagram of the proprietary structure versus the open source structure

Figure 6 .

 6 Figure 6. The Web HMI of Laborem developed under PyScada

Figure 7 .Figure 8 .

 78 Figure 7. Block schematic of the modular motherboard

  deployment of the new open source version of LaboREM needs the design of plugs and the motherboard. Connectors and plugs are designed in PCI Express logic because this is a well-known electronic standard. It simplifies PCB design, reduces the cost thanks to widely used connector types and assures a mechanical resistance that allows the teacher to change easily the plug on the motherboard. The modular motherboard is visible in Figure 9. A 4-wire bus and a multiplexer are used to select a plug among the 16 ones. All the different signals are isolated with switches operating by the multiplexer.

Figure 9 .

 9 Figure 9. Top view of the motherboard; PCIe connectors are visible to interconnect plugs (only two connectors and switches are soldered on the picture) Two examples of plugs are shown in the Figure 10.

Figure 10 .

 10 Figure 10. Plugs examples of an active filter remotely configurable by the robotic arm and a second test plug to check electrical continuity after components soldering

Table 1 .

 1 Remote laboratory pillars and the LaboREM answer

	Goals	Topics	LaboREM
	Student immersion	Real Time Immersion	Robotic arm, camera, drone
	Pedagogical efficiency	Real lab or simulation	Practical works
		Accessibility and
		compatibility	Standard website
	Network and	Normalized	Multi-protocols
	collaboration	development	communication
			Open source
			software (Python)
		Cost	and hardware (generic plugs)

ACKNOWLEDGEMENTS

The authors would like to thank the staff of ISITE program E2S for financial support for the engineer fellowship of Camille Lavayssière.