# The extended cup method: Cross-effects of water vapor diffusivity and gas permeability in porous media

Patrick Perré, Joel Casalinho, Julia P Lancha, Romain Rémond

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## To cite this version:

Patrick Perré, Joel Casalinho, Julia P Lancha, Romain Rémond. The extended cup method: Cross-effects of water vapor diffusivity and gas permeability in porous media. International Journal of Heat and Mass Transfer, 2025, 239, pp.126569. 10.1016/j.ijheatmasstransfer.2024.126569. hal-05202894

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Submitted on 7 Aug 2025

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# The extended cup method: Cross-effects of water vapor diffusivity and gas permeability in porous media

Patrick Perré<sup>a,b,*</sup>, Joel Casalinho<sup>b</sup>, Julia P. Lancha<sup>b</sup>, Romain Rémond<sup>c</sup>

## ARTICLE INFO

**Keywords:**

Anisotropy

Computational model

Material characterization

Theoretical formulation

Vapor permeability

## ABSTRACT

The cup method is a standard method for measuring the mass diffusivity of building materials, a crucial parameter for assessing the energy performance of buildings. However, during this test, the respective gradients of air and vapor concentration give rise to a cross flux of these two species by binary diffusion. A gradient of the total gaseous pressure therefore exists to ensure, via a Darcy's regime, a total flux of air equal to zero. The extended cup method consists in measuring this pressure gap during the cup method. This paper proposes a large set of challenging experiments performed with an original experimental device. The results are analyzed using both a rigorous computational solution and an approached analytical solution. The theory was first confirmed by measurements performed with light concrete at various temperatures. Then, the results obtained for various porous media were plotted on a synthetic diagram, which allowed the following conclusions to be drawn:

* By choosing suitable operating conditions using the synthetic diagram, the extended cup method allows a rigorous determination of both the mass diffusivity and the permeability in the case of an inert solid phase,

* In the case of bound water migration, the extended cup method combined with conventional permeability measurement can quantify the respective contribution of vapor and bound water diffusion,

* As practical outcome, the correction due to the pressure gradient can be discarded for routine measurement of mass diffusivity.

## 1. Introduction

Increasingly demanding thermal standards of buildings require efficient simulation tools together with relevant and accurate material characterization [1–3]. It is now well established that, due to the latent heat of vaporization, accounting for the coupling between heat and mass transfer is mandatory. This is particularly needed for material having a high moisture buffering effect such as bio-based materials.

These trends gave rise to a huge demand in material characterization in relation with moisture. This includes equilibrium with the so-called sorption isotherm, but also the water vapor diffusivity. Note that this ability of moisture migration is often called moisture permeability or vapor permeability in the domain of building materials by using the water vapor pressure gradient as driving force in the Fick's law. This need concerns both new and innovative materials used in construction or traditional materials that need to be well characterized to design relevant renovation solutions.

water diffusivity, often

Even though some works pointed out the need to account for transient states, namely in summer comfort [4–6], norms still propose steady-state methods for routine characterization [7,8].

In this context, the classical cup method [9] is still a widely-spread technique to determine water vapor diffusivity in building materials [10–13]. It is also widely used to characterize polymer films or fabrics [14,15]. The cup method consists in imposing a relative humidity (RH) gap through the sample and measuring the resulting mass flux at steady-state $Q_v$ (m²/s). In practice, the sample is air-tightly mounted on the top of a cup containing a saturated saline solution. This allows the internal RH ($\phi_1$) to be set. The whole device is put in a climatic chamber set at constant temperature and RH ($\phi_2$).

Although simple, the cup method suffered from several weaknesses. Several works are devoted to a deeper analyses or improvements of the

<footnote><sup>a</sup> Université Paris-Saclay, CentraleSupélec, LGPM, Centre Européen de Biotechnologie et de Bioéconomie (CEBB), 3 rue des Rouges Terres, F-51 110 Pomacle, France</footnote>

<footnote><sup>b</sup> Université Paris-Saclay, CentraleSupélec, LGPM, 3 rue Joliot Curie, F-91 190 Gif-sur-Yvette, France</footnote>

<footnote><sup>c</sup> Université de Lorraine, INRAE, LERMAB, F-88 000 Epinal, France</footnote>

<footnote>* Corresponding author at: Université Paris-Saclay, CentraleSupélec, LGPM, 3 rue Joliot Curie, F-91 190 Gif-sur-Yvette, France. E-mail address: patrick.perre@centralesupelec.fr (P. Perré).</footnote>
cup method, such as the correction due to the diffusion resistance inside the cup [16,17] or, even better, the measurement of the actual RH inside the cup [18]. This internal resistance is particularly important for highly permeable materials [19,20]. Note that the problem of actual internal RH is more pronounced for the dry cup, due to the desiccants behavior [21]. The possible error due to air buoyancy was studied in detail [22]. This work reports corrections of the order of some mg, hence negligible compared to the mass losses of some grams obtained in our tests.

Modified cup methods have been proposed to easily assess the effect of temperature in building materials [23] and to avoid the internal overpressure when characterizing impermeable films at high temperature [24]. In the cup test, a gradient is imposed across the sample and we must keep in mind that the diffusivity is averaged over the corresponding range of MC, which needs a proper interpretation [25]. All these drawbacks of an apparently simple method explains the poor reproducibility on similar materials between different laboratories [26]. This problem of reproducibility is also raised in [27].

To complete these refined analyses of the cup method, the present work focuses on the cross fluxes due to the binary diffusion in the gaseous phase, cross fluxes that gives rise to an pressure gap over the sample thickness.

Assuming moisture diffusion to occur only as water vapor diffusion in the gaseous phase of the pores, the flux $Q_v$ (kg s⁻¹) is simply expressed using Fick's law [28]:

$$Q_v = -\rho_g D_{eff} S \frac{d\omega_v}{dx} \qquad (1)$$

Where $D_{eff}$ is the macroscopic mass diffusivity to be determined (m² s⁻¹), $\rho_g$ the density of gas (kg m⁻³), S is the surface area of the sample (m²), and $\omega_v = \rho_v/\rho_g$ the mass fraction of vapor (-).

Assuming $\rho_g$ and $D_{eff}$ to be constant throughout the sample thickness $\ell$, which stands at low temperature, a simple integration over the thickness gives:

$$Q_v = -D_{eff} S \frac{M_v P_{sv}(T)}{RT} \frac{\phi_2 - \phi_1}{\ell} \qquad (2)$$

The water vapor diffusivity, also called water vapor permeability in the building sector, can be expressed as a fraction of the binary diffusion coefficient of water vapor in air ($D_v$), either using a dimensionless factor ($f$) or its inverse, the resistance factor ($\mu$):

$$D_{eff} = f D_v = \frac{D_v}{\mu} \qquad (3)$$

This method seems to be straightforward. However, the simple formulation presented above neglects an important physical effect: in a gas, the diffusion coefficient is called binary because the kinetics theory of gases tells us that, in a presence of a concentration gradient in a mixture of gases A and B, two opposite gradients of molar (or mass) fraction coexist [28]. Consequently, a macroscopic flux of A in one direction necessarily co-exists with a flux of B in the opposite direction. This phenomenon has been underlined decades ago during drying below the boiling point of water [29,30]. More recently, the effect of total gaseous pressure was pointed out, but not fully addressed, during mass diffusivity determination using the cup method [31]. These authors claim that, due to the pressure, the water vapor permeability defined for most current building materials might be underestimated. The present work clearly addresses this point and proves that the correction is negligible.

In the case of the cup method, the binary diffusion implies that a dry air diffusive flux necessarily exists due to the imposed gradient of $\omega_v$. This seems paradoxical as the gas-liquid interface inside the cup is non-permeable to air (contrary to water vapor) and no accumulation term can exist at steady-state. The explanation lies in the distinction between the pure diffuse flux density $j_v$ and the total flux density $q_v$. This occurs for example when a liquid evaporates in a tube, which is known as the Stefan diffusion column [28]. In this case, a barycentric velocity

is induced by binary diffusion. This barycentric velocity ensures that the total flux of air (diffusion + convection) is equal to zero. As a consequence (see detail in Section 2.3), the total evaporation flux reads as:

$$q_v = \rho_v v_g + j_v = -\frac{1}{1 - \omega_v} \rho_g D_{eff} \frac{d\omega_v}{dx} \qquad (4)$$

Eq. (4) must be integrated along the sample thickness. The fields of variables $\rho_g$ and $\omega_v$ therefore depend on the mass diffusivity but also on the permeability.

This paper proposes a comprehensive analysis of this physical issue. It includes: (i) a new device, carefully developed to get relevant data, (ii) a set of data obtained over a large range of temperature levels and (iii) theoretical considerations based on the steady-state formulation integrated along the sample thickness. The theoretical analysis includes two solutions of the formulation: a full computational solution of the non-linear profiles of two coupled variables (vapor and air densities) along the sample thickness and an easy-to-use analytical solution.

Light concrete was firstly used as reference porous medium [32] to validate the approach. Expanded polystyrene, two species of wood in different directions and MDF (Medium Density Fiberboard) were further characterized to broaden the discussion. This paper has two main outcomes:

1. A dimensionless number and corresponding chart to help the reader guide experiments and rigorously analyze data using the analytical solution,

2. The potential of the extended cup method to better analyze the interaction between water vapor diffusivity and permeability.

## 2. Materials and methods

### 2.1. Experimental device

The cup method consists in imposing a relative humidity gap through the sample and measuring the resulting mass flux, $Q_v$, at steady-state. This is obtained by mounting the sample air-tightly on top of a cup containing a saturated saline solution to set the internal RH ($\phi_1$). Sodium chloride (Merck KGaA, Germany) was used in this study, which imposes $\phi_2$ very close to 75% over the temperature range 40 °C to 70 °C [33]. The whole device is put in a chamber to control the values of temperature and RH ($\phi_1$). Compared to the classical method, the extended cup method includes a pressure tap to collect the pressure inside the cup (Figs. 1 and 2). The pressure at the front face of the sample is also measured to determine accurately the pressure gap across the sample. It might seem very simple but, in practice, it is quite tricky to do this in a rigorous way:

* A perfect air-tightness should be ensured in spite of the tubing and connection of the pressure sensor,

* The tubing or wire of the pressure sensor is likely to perturb the mass measurement,

* The pressure difference can be very small, especially at low temperatures, and requires very sensitive pressure sensors.

Fig. 1 depicts the principle of the device specifically developed to fulfill these requirements. A chamber with forced convection (Memmert UFP 400) was used. As this chamber is only temperature controlled, the actual relative humidity inside the chamber ($\phi_{chamber}$) is determined from the ambient conditions, that are registered during the tests using a temperature/relative humidity sensor (Sensirion SHT75):

$$\phi_{chamber} = \frac{\phi_{room} P_{sv}(T_{room})}{P_{sv}(T_{chamber})} \qquad (5)$$

Over the range of temperature tested in this study and with room conditions equal to 20 °C and 50% RH, Eq. (5) gives a relative humidity
<figure><img src="image_5.png" /><figcaption>Fig. 1. Schematic diagram of the experimental device.</figcaption></figure>

in the chamber $\phi_1$ equal to 16%, 9%, 6% and 4% at 40 °C, 50 °C, 60 °C and 70 °C respectively. In order to be able to perform tests over a wide range of temperature levels, the laboratory balance and the pressure sensor are placed outside the chamber. The sample and its cup are placed on a custom support, itself resting on the laboratory balance. Two silicone tubes collect the internal cup pressure and the chamber pressure. The opening of the later is placed close to the upper surface of the sample (Fig. 2b). Although these silicone tubes are very flexible, an extra precaution has been taken to reduce the noise of the mass measurement. A large loop of flexible silicone tubing, held horizontally by stainless steel wires, was shaped on the tube connected to the cup. This allowed both parameters, mass loss and pressure gap, to be grabbed simultaneously with accuracy. In order to ensure reliable pressure readings, these tubes are connected to two different sensor kinds to measure the pressure gap:

* To get a comprehensive figure of physical effects, two high-precision absolute pressure sensors (Druck, DPS 8100) with 2.10⁵ Pa of full scale (FS) and an accuracy of 0.014% of FS. Its resonant silicon crystal pressure sensor technology with its integrated electronics, increases the accuracy and stability of one order of magnitude in comparison of current pressure measurement technologies. A test performed with the two sensors used in the experiments with open tubing gave a pressure difference of 0.3 Pa with a standard deviation of ca. 1 Pa. In addition to the variation of the total pressure due to the variation of the atmospheric pressure, these two sensors are also able to give the pressure gap with a good accuracy,

* To secure the measurement, a differential pressure transducer (Druck, LPX 9481) with a full scale of 50 mm H₂O, approximately 500 Pa, with an accuracy better than 0.1% of FS was also connected to the silicone tubes (Fig. 2a).

After the first test, the two absolute pressure sensors were been added to the experiment because the pressure gap observed for the oak samples was too large for the 500-Pa differential pressure transducer. For the following tests, the quality of the absolute sensors was so good that this is the information used for all tests except the samples with a very small pressure gap (EPS and oak L).

A original cup was specifically designed by computer aided design (CAD) and realized in the lab in aluminum using a lathe. This cup has a thickness of 8 mm, which allows a Swagelok connector to be screwed for

pressure measurement, hence insuring a perfect air-tightness. However, this thickness would have reduced the surface area of the internal face of the sample. This point was addressed by the conical shape at to top of the cup (Fig. 1). To test the materials, discs of 72 mm diameter were prepared using a Computer Numerical Control (CNC) machine. Their thickness depends on the material (typically in the range 5 mm to 20 mm).

Airtightness over the lateral face of the sample must be guaranteed. To that purpose, the lateral face of the sample was covered by two successive layers of epoxy resin. This creates a very smooth lateral surface, which is then coated with vacuum grease before the smaller-diameter rubber tube is fitted to ensure a perfect seal. This method was already successfully used to measure mass diffusivity [17,34]. However, as sealing is very important in this case, it has been reinforced by the application of three successive rubber tubes, which triple the pressure applied to the side face, as shown in Fig. 1.

The experimental protocol includes the following key stages:

* The chamber and electronic devices remained turned on during each characterization campaign. The allows the warming up phase to be avoided and the time to transient state to be reduced by some hours,

* Before each test, the cup was cleaned from any salt deposit, especially at the pressure tap and a new saturated salt solution higher than 2 cm was prepared in the cup,

* The sample is careful mounted on the cup, with a layer of vacuum grease and three successive rubber tubes,

* The pressure gauge tube is connected and the cup is placed in the chamber over the balance plateau. The absence of friction of the plateau rod is checked before closing the chamber door,

* The acquisition software is launched, acquiring mass and pressures every two minutes, which cumulates several thousand acquisitions per experiment. The chamber fan is automatically switched off 5 s before weighing by the acquisition software to eliminate any aeraulic disturbance, and restarted immediately after weighing.

Weighting in the absence of air flow together with the caution taken regarding the pressure tube allow the mass loss to be obtained with an exceptional quality, in both accuracy and frequency, by comparison with standard manual weighting: the water vapor flux can be obtained with an accuracy of ± 2% using the slope over one hour (see
<figure><img src="image_6.png" /><figcaption>Fig. 2. Two pictures of the experimental set-up. left) Cup and sample without the rubber tubes, right) The extended cup placed on the balance support in the chamber. (1) Silicon tube for chamber pressure measurement, (2) thermocouple for temperature control, (3) silicon tube and connector for internal pressure measurement, (4) balance support.</figcaption></figure>

## Appendix A). As the resin penetrates slightly into the porosity of the sample, an effective diameter of 70 mm was used to calculate the mass flow density from the mass loss rate.

### 2.2. Permeability measurement

An in-house device was used to measure independently the permeability of all samples tested in the extended device presented above. The measurement principle consists in imposing a pressure difference across the sample and calculating its permeability by determining the flow rate from the pressure relaxation kinetics in rigid volumes. The apparatus and the experimental protocol are described in details in [35]. The device is composed of two rigid tanks of identical volume built in series with a sample support. The pressure in the tanks is measured by the same high-accuracy pressure sensors (GE Druck, TERPS RPS/DPS 8000) as those used in the extended cup device. The pressure range of the sensors extends from about 0 to 2000 mbar with a precision of 0.2 mbar between 35 mbar and 2000 mbar at 25 °C and a resolution of 1 Pa. The device is completely air-tight and placed in a temperature-controlled chamber (accuracy ±0.5 °C, standard deviation 0.04 °C) to avoid pressure changes due to thermal dilatation. For the tests, the chamber temperature was set at 30 °C.

Two sets of tanks ($V_{small} = 0.2$ L and $V_{large} = 24$ L) can be selected to fit the expected permeability of the sample. The sample support relies on the ALU-CHA system developed by [36]. It consists of a rubber inner tube mounted coaxially in a hollow aluminum housing. The housing is connected to a vacuum pump to mount the sample, then to a compressed air supply during the measurement to ensure the lateral seal.

The measurement protocol begins by setting the two tanks at different pressures ($P_1 > P_2$). The flow valve remains close until thermal equilibrium. Then, the valve is open and the two pressures are recorded at a frequency between ca. 1 Hz to 0.01 Hz depending on the time constant of the system (sample permeability, sample thickness and selected volumes). Subsequently, the pressure values allow both the driving force (pressure gap) and gas flow (pressure relaxation in each volume) to be obtained and analyzed to calculate the sample permeability [35].

### 2.3. Theoretical analysis of the extended cup method in steady-state

The comprehensive macroscopic formulation of coupled heat and mass transfer in porous media is well-established and was already used in building materials [6,37]. This formulation assumes water to exist in three different forms: liquid, vapor and bound water. Yet, all liquid water contributions can been discarded in the present work, as the sample stays within the hygroscopic domain. Indeed, using Kelvin's

equation, the radius needed to obtain a water activity of 0.75 equals 3.4 nm at 40 °C and 2.8 nm at 70 °C, at the transition between micropores and mesopores, at the limit of the possible use of Kelvin's equation [38]. The proportion of this size of pore is only marginal in AAC [39]. In wood, it was proved that no liquid exist in the pores at 75% RH [40,41]. The water bound by capillary condensation can contribute to water migration, as explained in the discussion for wood, but all materials studied in our paper depict a large porosity that is not affected by capillary condensation in such small pores. Similarly, as light concrete was used as reference material, the transport terms of bound water can be temporary discarded. This set of macroscopic equations can be further simplified when applied to the configuration of the cup method:

* Thanks to the steady-state regime, all time derivatives can be discarded,

* A 1-D configuration can be reasonably assumed,

* Due to the huge contrast between the air density and the liquid density (a factor 900 to 1000 depending on the temperature level), the velocity of the liquid level can be neglected as stated in [28]. As the liquid surface is impermeable to dry air, the total air flow (sum of diffusive and convective flows) is zero in any cross-section of the sample,

* The coupling between heat and mass transfer can be ignored as the samples tested in this study have a quite low diffusivity and a not too low thermal conductivity [6].

The resulting set of equations reads as :

Convective velocity of gas

$$v_g = -\frac{K}{\mu_g} \frac{dP_g}{dx} \qquad (6)$$

Diffusive flux of water vapor

$$j_v = -\rho_g f D_v \frac{d\omega_v}{dx} \qquad (7)$$

Diffusive flux of air

Keeping in mind that $\omega_v + \omega_a = 1$, we obtain

$$j_a = \rho_g f D_v \frac{d\omega_v}{dx} \qquad (8)$$

Total flux of air

$$q_a = \rho_g f D_v \frac{d\omega_v}{dx} - \rho_a \frac{K}{\mu_g} \frac{dP_g}{dx} = 0 \qquad (9)$$
Total flux of water vapor

$$q_v = -\rho_g f D_v \frac{d\omega_v}{dx} - \rho_v \frac{K}{\mu_g} \frac{dP_g}{dx} \qquad (10)$$

Combining Eqs. (9) and (10) yields

$$q_v = \frac{1}{1 - \omega_v} j_v = - \frac{\rho_g}{1 - \omega_v} f D_v \frac{d\omega_v}{dx} \qquad (11)$$

Finally (see detail in Appendix B), the local pressure increment along the thickness can be relied to the local increment of vapor concentration through Eq. (9):

$$dP_g = \frac{1}{1 - \omega_v} f D_v d\omega_v \frac{\mu_g}{K} \qquad (12)$$

Eq. (12) involves the ratio $f D_v \mu_g / K$ which is of major importance in the analysis of the extended cup method.

To solve this set of equations, Eqs. (11) and (12) have been developed using the law of perfect gases to obtain two primary variables ($P_a$ and $P_v$). The two coupled and non-linear equations could have been solved using a classical Newton-Raphson method, with computation of the Jacobian matrix and its inversion to update the fields until convergence. However, the form of the physical equations allows a trick to be implemented to propose a simple, quasi-explicit, solution. This numerical solution is detailed in algorithm 1 of Appendix B and the commented R script of this numerical solution is proposed as supplementary material.

However, in most cases, such a numerical integration can be avoided. If the total pressure remains close to the atmospheric pressure throughout the sample, i.e. if the pressure gap is much smaller than the atmospheric pressure, Eqs. (11) and (12) can be integrated over the sample thickness, assuming the factor $\rho_g$ to be constant and equal to the average air density $\overline{\rho_g}$.

Diffusive flux of water vapor

$$j_v = -\overline{\rho_g} f D_v \left( \frac{\omega_{v2} - \omega_{v1}}{\ell} \right) \qquad (13)$$

Total flux of water vapor

$$q_v = -\overline{\rho_g} f D_v \ln \left( \frac{1 - \omega_{v1}}{1 - \omega_{v2}} \right) \times \frac{1}{\ell} \qquad (14)$$

Total pressure gap

$$\Delta P_g = f D_v \frac{\mu_g}{K} \ln \left( \frac{1 - \omega_{v1}}{1 - \omega_{v2}} \right) \qquad (15)$$

The averaged value $\overline{\rho_g}$ is computed from the average values of pressure:

$$\overline{P_v} = P_{sv}(T) \frac{\phi_1 + \phi_2}{2} \qquad (16)$$

$$\overline{P_a} = P_{atm} - \overline{P_v} \qquad (17)$$

In Eq. (13), the flux increases with temperature in three different ways:

* the most important one, the increase of the driving force due to the effect of temperature on the saturated vapor pressure,

* the factor $1/(1 - \omega_v)$ embedded in the $\ln$ function, which increases with temperature, with a singularity at the boiling point,

* the thermal activation of the binary diffusivity $D_v$

Note that Eqs. (13) and (14) tell us that the vapor flux is inversely proportional to sample thickness while the pressure gap is independent of sample thickness.

In the following, we will refer to two different solutions:

1. The **analytical solution** (equations Eq. (13) and Eq. (14)), which is a rigorous integration over the thickness. However, this solution assumes the gaseous pressure to be equal to the atmospheric pressure throughout the thickness,

2. The **numerical solution**: rigorous numerical integration of equations Eq. (11) and Eq. (12) along the sample thickness, able to consider for the coupling with the pressure field.

At this point, it is very important to emphasize the coupling between vapor concentration $\omega_v$ and the effect of total pressure on $\rho_g$. When the pressure difference is no longer negligible with respect to atmospheric pressure, the gas density increases significantly, reducing the concentration fraction of water vapor. As a result, the driving force for vapor diffusion is reduced. It is worth mentioning that **only the numerical solution is able to account for this coupling**.

In order to analyze the experimental data, the previous solutions must be inverted: determine the value of $f$ and $K$ capable of giving the measured values $q_v$ and $\Delta P$. This is straightforward for the analytical solution and was done by inverse analysis for the numerical solution, using in algorithm 2 of Appendix B. By using experimental data at one single temperature level, the two degrees of freedom ($f$ and $K$) allow the two experimental data to be obtained accurately. The inverse procedure is then based on a lumped objective function that sums the errors of $q_v$ and $\Delta P$.

## 3. Results

### 3.1. Theoretical solutions

In this section, the numerical solution (numerical integration of the differential expressions (11) and (12)) and the analytical solution (Eqs. (13) and (14)) were used to analyze the coupling between the water vapor flux and the pressure gap. In the proposed simulations, the following physical values were used:

* $K = 2.10^{-13}$ m²

* $f = 0.1$

* $\ell = 20$ mm

* $D_v = 2.18 \times 10^{-5} \times \frac{P_{atm}}{P} \times \left(\frac{T_K}{273}\right)^{1.75}$ m² s⁻¹, where P is the total gas pressure [42,43]

* $\mu_g = 2.10^{-5}$ Pa s

* $P_{sv} = 10^5 \exp(65.832 - 8.2 \ln(T_K) + 5.717 \times 10^{-3} T_K - 7235.46/T_K)$ Pa

Where $T_K$ is the temperature in Kelvin. The values chosen for permeability $K$ and dimensionless diffusivity $f$ are typical values for autoclaved aerated concrete (AAC), also named light concrete in the following. The dynamic viscosity of air varies little with temperature and has been considered constant here. The effect of temperature on the vapor flux was computed using a large RH gap [10% ; 90%] over a temperature interval from 20 °C to 95 °C by increment of 1 °C. This large gap is more demanding as it reinforces the non-linear effect due to the factor $1/(1 - \omega_v)$ over the sample thickness. In this case, for which the pressure gap remains negligible compared to the atmospheric pressure, the analytical and numerical solutions remain very close, even at 95 °C, near the singularity (top plot of Fig. 3). This proves that the analytical solution performs perfectly thanks to the proper integration of $d\omega_v/(1 - \omega_v)$. The small difference between the two solutions are explained by two effects:

* the pressure gap that, which, albeit small, slightly deviates the pressure inside the sample from atmospheric pressure,
<table><caption>**Table 1**
Summary of the experimental data obtained with a sample of light concrete at different temperature levels. The pressure gaps and the mass fluxes are the average of two tests.</caption><thead><tr><th>Temperature (°C)</th><th>Pressure gap (Pa)</th><th>Vapor flux (kg m<sup>-2</sup>s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>40</td><td>6.13</td><td>3.93.10<sup>-6</sup></td></tr><tr><td>50</td><td>9.78</td><td>7.62.10<sup>-6</sup></td></tr><tr><td>60</td><td>16.7</td><td>1.49.10<sup>-5</sup></td></tr><tr><td>70</td><td>37.8</td><td>3.01.10<sup>-5</sup></td></tr></tbody></table>

* the assumption made to integrate the analytical solution: the air density in factor of the flux which is properly treated in the numerical solution while assumed to be constant, equal to the average air density, in the analytical solution.

Note that pure diffusion flow is only approximate as temperature increases.

To better emphasize on the effect of total pressure, the simulations were also performed using a permeability 100 times smaller ($K = 2.10^{-15}$ m²). In this case, the pressure gap attains 20 kPa. This seriously reduces the driving force for diffusion as a higher air density reduces the vapor density fraction (bottom plot of Fig. 3). Unsurprisingly, the analytical model predicts the same values as before and is not able to account for the effect of a total pressure increasing along the sample thickness.

As shown in Fig. 3, the pressure difference also shows an exponential trend, but with a more pronounced increase with temperature than the vapor flux. The analytical model is excellent for the parameters of light concrete, but fails again when the variations of the total pressure become important.

## 3.2. Validation of the theory with light concrete experiments

Fig. 4 depicts two examples of tests performed with light concrete at 40 °C and 70 °C. The pressure gap stabilizes quite rapidly, typically within two hours whereas a longer duration is needed for an accurate determination of the vapor flux as it requires the mass loss to be derived. According to the very small characteristic time of the transient mass regime for ACC (Appendix C), it is clear that the time needed to attain the steady-state regime (about 2 h) is due to the thermal stabilization of the chamber after sample handling. The perfectly linear mass loss obtained after stabilization enables highly accurate determination of the vapor flux. The dashed thick cyan line indicates the time range over which the average slope of the mass loss is computed to obtain the water vapor flux. The noise of the pressure gap is due to both the small temperature variations in the chamber and to the temporal change in atmospheric pressure. The pressure gap was then defined as the average value over several hours after stabilization. A R script was written to process and plot the experimental data and compute the average pressure and the vapor flux density over the linear part of the curve.

Table 1 summarizes the experimental data obtained for temperature levels ranging from 40 °C to 70 °C for the same sample of light concrete. Using the numerical solution, the two key parameters of the light concrete sample, dimensionless diffusivity *f* and permeability, noted $K_{cup}$ in the following to emphasize the method of determination, were identified from the experimental results of Table 1. The system is well-conditioned as the dimensionless diffusivity acts on the vapor flux while the permeability acts on the pressure gap. The value of *f* was determined as best fit to minimize the root mean square error (RMSE) of the vapor fluxes for the 4 temperature levels. Then, the permeability *K* was determined using this value of *f* to minimize the RMSE pressure gaps:

$$ \cdot f = 0.088 $$

<table><caption>**Table 2**
Summary of the parameters measured on two samples of light concrete: Dimensionless diffusivity *f* and permeability ($K_{cup}$) values obtained with the extended cup method and by a specific measurement of the permeability ($K_{direct}$). To be consistent with all measurements presented in the following, the value of *f* = 0.98 for sample 1 was computed using only the water vapor flux obtained at 70 °C, which slightly higher than the one obtained from all temperatures, as can be observed in Fig. 5.</caption><thead><tr><th>Sample</th><th>f</th><th>K<sub>cup</sub> (m²)</th><th>K<sub>direct</sub> (m²)</th></tr></thead><tbody><tr><td>Sample n°1</td><td>0.092</td><td>2.59.10<sup>-13</sup></td><td>2.94.10<sup>-13</sup></td></tr><tr><td>Sample n°2</td><td>0.130</td><td>3.94.10<sup>-13</sup></td><td>3.36.10<sup>-13</sup></td></tr></tbody></table>

<table><caption>**Table 3**
Summary of the raw experimental results obtained for various porous media.</caption><thead><tr><th>Sample</th><th>Temp. °C</th><th>Thickness mm</th><th>Density kg m<sup>-3</sup></th><th>Vapor flux (SD) kg m<sup>-2</sup>s<sup>-1</sup> (%)</th><th>ΔP (SD) Pa (Pa)</th></tr></thead><tbody><tr><td>EPS n°1</td><td>60</td><td>39.8</td><td>47.7</td><td>3.02.10<sup>-6</sup> (1.7)</td><td>1.75 (0.96)</td></tr><tr><td>EPS n°2</td><td>60</td><td>41.8</td><td>45.8</td><td>3.33.10<sup>-6</sup> (1.6)</td><td>1.14 (0.95)</td></tr><tr><td>AAC n°1</td><td>70</td><td>16.0</td><td>624</td><td>3.01.10<sup>-5</sup> (0.7)</td><td>37.8 (2.2)</td></tr><tr><td>AAC n°2</td><td>70</td><td>16.1</td><td>648</td><td>4.24.10<sup>-5</sup> (1.0)</td><td>35.1 (1.3)</td></tr><tr><td>Pine (T)</td><td>70</td><td>9.88</td><td>669</td><td>1.06.10<sup>-6</sup> (7.9)</td><td>220 (25)</td></tr><tr><td>Pine (R)</td><td>70</td><td>9.75</td><td>671</td><td>1.27.10<sup>-5</sup> (4.9)</td><td>44.9 (1.7)</td></tr><tr><td>Oak (T1)</td><td>70</td><td>5.37</td><td>670</td><td>1.56.10<sup>-6</sup> (6.2)</td><td>508 (54)</td></tr><tr><td>Oak (T2)</td><td>70</td><td>9.94</td><td>640</td><td>1.04.10<sup>-6</sup> (20.0)</td><td>473 (58)</td></tr><tr><td>Oak (R)</td><td>70</td><td>15.7</td><td>573</td><td>3.34.10<sup>-6</sup> (9.3)</td><td>24.1 (2.8)</td></tr><tr><td>Oak (L)</td><td>70</td><td>15.4</td><td>537</td><td>3.44.10<sup>-5</sup> (0.2)</td><td>5.06 (1.6)</td></tr><tr><td>MDF</td><td>70</td><td>10.2</td><td>650</td><td>6.91.10<sup>-5</sup> (0.5)</td><td>47.6 (1.9)</td></tr></tbody></table>

$$ \cdot K_{cup} = 2.58.10^{-13} m^2 $$

Fig. 5 depicts the curves computed with the identified parameters together with the experimental points added as markers. It is worth noting that the predicted effect of temperature is very nicely confirmed by the experimental points. The increase in vapor flux with temperature is primary due to the increase in saturated vapor pressure, but the correction from the pure diffusive flux (Eq. (13)) and the thermal activation of binary diffusivity are also significant.

The permeability of this sample, measured by pressure relaxation in our in-house device (see Section 2.2) was determined at $K_{direct} = 2.94.10^{-13}$ m², a difference of 14%. Keeping in mind the small values of the pressure gap measured in the extended cup method, this good match is remarkable.

The second sample of light concrete was characterized at one single temperature level (70 °C). We obtained a pressure gap of 35.1 Pa and a mass loss flux equal to $4.24.10^{-5}$ kg m$^{-2}$ s$^{-1}$. This gives 0.130 and $3.94.10^{-13}$ m$^2$ for the dimensionless diffusivity and permeability respectively. The permeability determined by the specific method was $K_{direct} = 3.36.10^{-13}$ m$^2$, a difference of -15%. This is consistent with the uncertainties analysis proposed in Appendix A and confirms that the extended cup method could become a routine test to determine two mass transfer parameters at the same time: the dimensionless diffusivity *f* and the permeability *K* (Table 2). To that purpose, we just need to add an accurate pressure sensor to the classical cup test. For example, in the work on air buoyancy cited previously [22], a pressure sensor was added in several cups. This work demonstrates the possibility to measure the mass loss and the pressure gap simultaneously on several samples.

## 3.3. Experimental results for various porous media

Now that the consistency of experimental results with theory had been confirmed with a reference porous medium, the experimental setup was used to test various porous media. In line with the results presented in the detailed analysis of the previous section, only one temperature level was performed per sample. To optimize the accuracy of pressure gap and mass loss, high temperature levels were chosen when possible (70 °C for wood, MDF and mineral materials and 60 °C
<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Effect of temperature on the water vapor flux. This flux is computed with three methods: the analytical and numerical solutions of the complete formulation and the pure diffusive expression. A large RH gap [10% ; 90%] was chosen here to enhance the non-linear effects. Top) The simulations were done with the typical values of light concrete ($K = 2.10^{-13} \text{ m}^2$ and $f = 0.1$). Bottom) The permeability is decreased by a factor 100 to depict the effect of pressure gap on the driving force for diffusion.</figcaption></figure>

for Expanded Polystyrene, EPS). Two contrasted behaviors are shown in Fig. 6. The EPS sample was 40-mm thick, ca. 3 times thicker than the AAC sample and the vapor flux is roughly 10 times smaller. This indicates that the dimensionless diffusivity *f* is about 3 times smaller than for AAC. Yet, the pressure gap is very low (1.14 Pa as averaged over more than 10 hours, which is at the limit of the sensor accuracy). The tangential oak sample had a very small vapor flux (of the order of $10^{-6} \text{ kg m}^{-2} \text{ s}^{-1}$) and a large pressure gap (nearly 500 Pa). We will see, however, that this pressure difference is much smaller than the value expected from the sample permeability. In addition, the very low tangential permeability of oak explains the time- variations of the pressure gap. To understand these variations, the absolute pressures, in the chamber and inside the cup, were plotted in Fig. 6 (bottom). The characteristic relaxation time linked to the volume of the cup is so long that the internal pressure is affected by variations in atmospheric pressure: the pressure gap increases when atmospheric pressure decreases, and vice versa. This phenomenon also affects the mass loss, whose slope

depicts slight variations. In this case, the pressure gap and the slope were averaged over more than 50 hours.

Table 3 summarizes the sample characteristics and experimental results of all experiments, including the standard deviation of the pressure gap and the standard deviation of the mass loss curve as derived on a sliding window of 4 h. This table highlights the considerable variability of the porous media tested in the present study. The richness of this dataset will be analyzed in detail in the following section, in order to draw practical conclusions on the extended cup method.

# 4. Discussion

## 4.1. A synthetic diagram

The extended cup method could become a routine test to determine two mass transfer parameters at the same time : mass diffusivity and permeability. However, depending on the medium properties, optimal
<figure><img src="image_8.png" /><figcaption>Fig. 4. Two examples of experimental tests for light concrete at 40 °C(top) and 70 °C (bottom). Even at the lowest temperature level, the experimental device is able to determine the low mass flux and the very low pressure difference.</figcaption></figure>

conditions should be chosen for the double determination to be relevant and easy to analyze:

* The pressure difference should be sufficient for its accurate measurement,

* The pressure variation along the sample thickness should remain small enough for the linear solution to be valid.

As can be seen in Eq. (14), the pressure gap is independent of the sample thickness, and, from the analytical equation, proportional to the ratio of diffusivity and permeability. This defines the intrinsic pressure gap of the sample, $\Delta P_g^*$ (Pa):

$$ \Delta P_g^* = \frac{f D_v \mu_g}{K} \qquad (18) $$

Using the analytical solution, the actual pressure gap depends linearly on this intrinsic pressure gap, intrinsic to the sample, weighted by a coefficient defined by the test conditions:

$$ \Delta P_g = \Delta P_g^* \times \ln \left( \frac{1 - \omega_{v1}}{1 - \omega_{v2}} \right) \qquad (19) $$

To plot a complete chart of the coupled effects of these two parameters, a set of 10,000 runs (100 × 100) was performed for all combinations of f values between 0 and 1 and permeability values from 10⁻¹⁷ m² to 10⁻¹² m² using the full numerical solution. A linear progression was used for f and a geometrical one for K. The procedure was repeated for two temperature levels: 70 °C, the temperature at which most tests were performed, and 40 °C, to emphasize that the temperature level could be used to adapt the experimental conditions to a given sample. The external pressure is supposed to be equal to the atmospheric pressure. Due to the non-linear profiles inside the sample,
<figure><img src="image_9.png" /><figcaption>Fig. 5. Validation of the theoretical analysis with experimental data obtained for sample n°1 of light concrete, a reference porous medium. The best fits of the root mean square error (RMSE) over the 4 temperature levels, are obtained for f = 0.088 for vapor fluxes and K = 2.58.10⁻¹³ m² for the pressure gaps.</figcaption></figure>

it is not possible to plot a general graph that would be valid whatever the RH gap. The plot proposed here used a relative humidity of 75% in the cup, which corresponds to a saturated NaCl solution, and a vapor pressure in the chamber equals to 0.50 × P<sub>v</sub>(20 °C), which is a typical value when using a temperature-controlled oven.

As can be seen in Fig. 7, for each temperature level, all points lie on a unique master curve when plotted against the characteristic pressure difference ($\Delta P^* = f D_v \mu_g / K$). As the solution accounts for non-linear profiles of both $\omega_v$ and $P_g$, these master curves deviate from a straight lines when the pressure difference is no longer negligible compared to atmospheric pressure. This is explained by a pressure gap that exceeds 1 bar for the worse cases: high diffusivity and low permeability. For the sake of illustration, the blue dot curves represent the pressure gap as calculated by the analytical solution (Eq. (19)), which remains a straight line. Similarly, the vapor flux is also affected by these non-linear effects. The dimensionless correction represented on the second y-axis (dark red lines) is simply the ratio of the fluxes obtained by the rigorous numerical solution and the approaches analytical solution. As can be seen, the ratio is less than one: the increase in total pressure reduces the actual water vapor flux, simply because the increase in pressure reduces the vapor density fraction $\omega_v$. In this case, the analytical solution is not longer valid, as proven by the deviation between the red solid line and the blue dashed line in Fig. 7.

Two colored zones are highlighted in this graph. The blue zone represents the optimal range of pressure gap : large enough to be easily measured ($\Delta P_g > 10$ Pa) but small enough to avoid non-linear effects due to pressure variation ($\Delta P_g < 8.10^3$ Pa). The orange zone corresponds to this pressure limitation, that allows the error on the vapor flux determined by the analytical solution to remain below 4%. Depending on the sample, the degree of freedom to fulfill these constraints is the temperature level. As the pressure is much faster to determine than the mass flux, the pressure gap can be used as fast information to adjust the temperature level at which the mass rate should be measured for an unknown sample.

## 4.2. Detailed analysis of the wide range of measured properties

Table 4 summarizes the whole data set. In this table, the values of f are computed with the analytical solution, which means that the effect of the pressure gap on the water vapor flux is not considered. $K_{direct}$ is the permeability determined by the dedicated permeability device.

In the present work, a wide variety of porous media were selected to observed contrasted behaviors of the cross-effects of water vapor diffusivity and gas permeability. With one or two samples per type of material, it is not intended for material characterization, but it is always informative to compare with literature data. In spite of the variability range reported even for the same sample among different labs [26], the values of f are in rather good agreement, in particular for AAC and MDF, with those determined in another work with a completely different approach [44]. The values obtained with wooden samples and EPS are also consistent with published data [36,45], even though the values obtained here for EPS are a bit larger than those reported for this density. In this material, the pathways for mass migration (diffusivity and permeability) occur between beds and there organization might have been altered by the tests at 60 °C. However, the permeability was measured after the diffusion tests, so the cross-effects of diffusivity and permeability remain valid. Regarding permeability, it is well-known that this physical parameter presents a huge variability, even among samples of the same product. Yet, the values obtained with our device are within the literature ranges, for example for softwood [46], for AAC [47], or for MDF [48,49].

The intrinsic pressure gap $\Delta P^*$ is calculated with these values using Eq. (18). With these data, the theoretical pressure gap was computed using the full numerical solution (Theor. $\Delta P$), to be compared with the actual pressure gap measured experimentally. These value were also reported in Fig. 7 as open markers for the measured pressure gap. For the sake of clarity, one single point was plotted per material: in case of duplicate samples, the averaged values were plotted. For each material, the marker of measured pressure gap is connected to the theoretical solution (small plain markers) by vertical lines. Consistently, these theoretical points lie on the 70 °C red curve, except for EPS which was tested at 60 °C.

This comparison is very interesting. As already stated in Section 3.2, the agreement between theory and experiment is rather good for light concrete (AAC). This is also the case for MDF and longitudinal oak. The order of magnitude remains consistent for EPS. Indeed, even when averaged over several hours, the most sensitive differential pressure sensor cannot be expected to be accurate to more than one Pa, due to the drift of the sensor offset (see Appendix A. For such a material, extremely high pressure accuracy would be required to determine the permeability $K_{cup}$ with the extended cup method.
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Experimental results for two contrasted behaviors. Top) A sample of Expanded polystyrene at 60 °C with a very low pressure gap. Bottom) A sample of oak in tangential direction at 70 °C. In this case, the change rate of the atmospheric pressure partly impacts the pressure gap. For a complete information, the absolute pressure of the chamber and inside the cup were plotted instead of temperature in this case.</figcaption></figure>

<table><caption>**Table 4**

Data computed for various porous media and comparison between the theoretical and measured pressure gap.</caption><thead><tr><th>Sample</th><th>f<br>(-)</th><th>K<sub>direct</sub><br>(m<sup>2</sup>)</th><th>ΔP<sup>*</sup><br>(Pa)</th><th>Theor. ΔP<br>(Pa)</th><th>Meas. ΔP<br>(Pa)</th></tr></thead><tbody><tr><td>EPS n°1</td><td>0.040</td><td>4.1.10<sup>-12</sup></td><td>6.02</td><td>0.57</td><td>1.75</td></tr><tr><td>EPS n°2</td><td>0.046</td><td>4.4.10<sup>-12</sup></td><td>6.50</td><td>0.61</td><td>1.14</td></tr><tr><td>AAC n°1</td><td>0.092</td><td>2.9.10<sup>-13</sup></td><td>197</td><td>33.8</td><td>37.8</td></tr><tr><td>AAC n°2</td><td>0.131</td><td>3.4.10<sup>-12</sup></td><td>238</td><td>40.9</td><td>35.1</td></tr><tr><td>Pine (T)</td><td>0.0020</td><td>9.4.10<sup>-17</sup></td><td>1.32.10<sup>4</sup></td><td>2.3.10<sup>3</sup></td><td>220</td></tr><tr><td>Pine (R)</td><td>0.024</td><td>7.9.10<sup>-14</sup></td><td>185</td><td>31.9</td><td>44.9</td></tr><tr><td>Oak (T1)</td><td>0.0016</td><td>1.9.10<sup>-18</sup></td><td>5.2.10<sup>5</sup></td><td>8.9.10<sup>4</sup></td><td>508</td></tr><tr><td>Oak (T2)</td><td>0.0020</td><td>8.3.10<sup>-19</sup></td><td>1.47.10<sup>6</sup></td><td>2.5.10<sup>5</sup></td><td>473</td></tr><tr><td>Oak (R)</td><td>0.010</td><td>1.3.10<sup>-13</sup></td><td>47.7</td><td>8.21</td><td>24.1</td></tr><tr><td>Oak (L)</td><td>0.104</td><td>1.5.10<sup>-12</sup></td><td>42.9</td><td>7.38</td><td>5.06</td></tr><tr><td>MDF</td><td>0.135</td><td>2.9.10<sup>-13</sup></td><td>288</td><td>49.5</td><td>47.6</td></tr></tbody></table>

### 4.3. Discrepancy between theory and measurement for wood

On the contrary to the other porous media, a clear mismatch is observed for the wood samples in transverse direction, with two contradictory results. In tangential direction, in which wood depicts very low permeability values, the measured pressure gap is much lower than the theoretical pressure gap. For example, around 500 Pa instead of the order of 10⁵ Pa for the two samples of oak and 220 Pa instead of 2300 Pa for pine. This represents a 200- and 10-fold reduction respectively. On the contrary, in radial direction, paradoxical trends are observed: the measured pressure gap is higher than the theoretical one.

Two complementary facts are needed to explain the discrepancy between theory and measurement for wood in the transverse directions: (i) the pore morphology resulting from the cellular structure of wood and (ii) the ability of water to diffusion through the cell walls as bound water diffusion [50].
<figure><img src="image_11.png" /><figcaption>Fig. 7. Combined effects of mass diffusivity and permeability on the pressure gap and on the dimensionless vapor flux. The red lines are from the numerical simulation while the blue dashed lines are obtained from the analytical solution (Eq. (14)). The dimensionless vapor flux is the ratio of the vapor flux obtained by the numerical solution over the analytical solution. The experimental values of the different materials are put as open markers. For each point, these markers are connected to the theoretical solution by vertical lines.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 8. Schematic presentation of the path ways for water vapor and bound water in tangential and radial directions. The background images were prepared from micro-tomographic images [51]. A softwood is used here as example, but the principle would be similar for hardwood.</figcaption></figure>
The **tangential direction** offers the worse connectivity between elements [52]. This is for example attested by the huge tortuosity values, up to 400, found in oak for the diffusion of oxygen, a gas that is not able to diffuse though the cell wall [53]. Regarding Darcy's law, this is confirmed by the very low tangential permeabilities measured for oak and pine. Due to this huge tortuosity, the easiest path for moisture transport is as bound water diffusion through the cell walls in series with water vapor diffusion across the cell lumens (Fig. 8). The continuous path in the gaseous phase, which is the sole possible pathway for permeability, contributes only marginally to the macroscopic vapor flux [52,54,55]. Interestingly, thanks to the extended cup method aided by a direct measurement of permeability, we learned that the direct vapor flux through the openings, along the huge tortuosity needed to get along the cells length orthogonal to the flux direction, represents only ca. 0.5% and 10% for oak and pine respectively.

In **radial direction**, the structure resembles the organization in the tangential direction, but with localized shortcuts formed by the rays. This organization explains the much higher permeability values measured in radial direction: Table 4 reports permeability values in radial direction $10^5$ and $10^3$ higher than in tangential direction for oak and pine respectively. One part of the bound water diffusing through the cell walls connected to the rays can diffuse as water vapor in the cell length to benefit for these shortcuts (Fig. 8). As a result, and contrary to the tangential direction, the vapor flux in the gaseous phase is boosted by the bound water diffusion, resulting in an higher pressure gap.

### 4.4. The negligible effect of pressure gradient on the value of *f*

So far, the discussion has focused on the pressure gradient triggered by diffusion inside the sample. We have shown that measuring the pressure difference provides very interesting additional information compared with the classical cup method. Form a practical point of view, it is also important to analyze the effect of this pressure gap on the calculated value of *f*. To do this, the value of *f* was calculated with different assumptions:

1. The analytical solution,

2. The pure diffusion (Eq. (7)), which ignores the factor $1/(1-\omega_v)$

3. The regular numerical solution with the measured value of permeability ($K_{direct}$)

4. The numerical solution with a value of $K$ corrected to obtain the experimental pressure gap (for wood in tangential direction)

Appendix B explains in detail how the numerical solution was used to generate Table 5. To ease the reading, a representative subset of all samples was selected. This selection covers all observed behaviors. Four digits are displayed to allow even the smallest differences to be seen. The difference between the analytical diffusion and the pure diffusion simply lies in the term $1/(1-\omega_v)$. therefore, it depends on the temperature level and on the RH gap. For the range of RH used in this work, the error is ca. +5% at 60 °C (EPS in Table 5) and +9% at 70 °C (all other samples in Table 5). The comprehensive numerical solution, using the measured permeability value, always increases the value of *f*. This is consistent as the pressure gap tends to reduce the driving force by increasing the gaseous density. Yet, the correction remains completely negligible for most samples, which confirms that the analytical solution using a proper integration preforms well. The difference is only significant for oak in the tangential direction, due to the very low permeability in this direction. In this case, the value of *f* using the measured permeability almost doubles (from 0.0016 to 0.0029) for sample T1) and almost triples for sample T2 (from 0.0020 to 0.0057). However, as explained in subsection 4.3, the measured pressure gap is not consistent with the measured permeability $K_{direct}$. When the permeability is corrected to obtain the actual experimental pressure gap (last column of Table 5), the values of the numerical solution are very close to the analytical solution, even for these tangential oak samples, with a difference of less than 1%).

<caption>Table 5</caption>

Effect of the pressure gap on the value of *f*. The value of *f* was computed with different assumptions from the experimental data (see explanations in the text). The pressure gaps are also provided to show the differences between the analytical and numerical solutions. This table confirms that the pressure correction can be discarded for routine measurements of *f*.

<table><thead><tr><th></th><th>Analytical solution</th><th>ΔP Pa</th><th>Pure diffusion</th><th>Numerical K = K<sub>direct</sub></th><th>ΔP Pa</th><th>Numerical K = K<sub>corrected</sub></th></tr></thead><tbody><tr><td>EPS 1</td><td>0.03990</td><td>0.570</td><td>0.04212</td><td>0.03993</td><td>0.572</td><td>0.03993</td></tr><tr><td>AAC 1</td><td>0.09212</td><td>33.8</td><td>0.1006</td><td>0.09239</td><td>34.7</td><td>0.09239</td></tr><tr><td>Pine T</td><td>0.002003</td><td>2.3.10<sup>3</sup></td><td>0.002187</td><td>0.002063</td><td>2.3.10<sup>3</sup></td><td>0.002013</td></tr><tr><td>Oak T1</td><td>0.001602</td><td>8.9.10<sup>4</sup></td><td>0.001750</td><td>0.002951</td><td>6.8.10<sup>4</sup></td><td>0.001616</td></tr><tr><td>Oak T2</td><td>0.001977</td><td>2.5.10<sup>5</sup></td><td>0.002159</td><td>0.005750</td><td>1.5.10<sup>5</sup></td><td>0.001993</td></tr><tr><td>MDF</td><td>0.1348</td><td>49.5</td><td>0.1472</td><td>0.1352</td><td>50.8</td><td>0.1352</td></tr></tbody></table>

This is a key conclusion of this work. The pressure gap measured in the extended cup method is very informative in the characterization of a porous medium, but, contrary to what is stated in [31], the effect of pressure gap can be discarded to determine the dimensionless diffusivity *f* from the experimental water vapor flux. Besides, the convective flux triggered by diffusion (factor $1/(1-\omega_v)$ in Eq. (13)) might be important and is easy to considered. It must not be discarded.

### 5. Conclusion

The cup method is a standard method to measure mass diffusivity in steady-state regime. The extended cup method consists in the measurement of both the mass loss and the internal pressure gap inside the cup. This extended method has been investigated in detail in the present work, both theoretically and experimentally. The experimental device developed for that purpose allows an accurate measurement of the pressure gap and the mass flux versus time, over a wide range of temperature levels. Using light concrete as reference porous media, the experimental results are in excellent agreement with the theory, both in terms of mass flux and pressure gap, over the tested range of temperature (from) 40 °C to 70 °C). To validate the results, the permeability of all samples were determined with an independent method. Then a data set was collected and analyzed for a wide variety of contrasted porous media.

The key outcomes of this work are as follows:

* For classical building materials, the extended cup method allows a simultaneous determination of both mass diffusivity and permeability.

* In the case of bound water migration, an independent measurement of permeability is able to quantify the respective contributions of vapor and bound water diffusion,

* A global chart is proposed to define the domain of validity of the simple analytical solution depending on material characteristics and temperature level. The provide a simple guide for an easy implementation of the extended cup method, namely the selection of the working temperature,

* For routine measurements of mass diffusivity, the correction related to the pressure gap across the sample can be neglected to determine the dimensionless diffusivity *f* or its inverse $\mu$, the resistance factor used in the building sector.

### CRediT authorship contribution statement

**Patrick Perré:** Writing – review & editing, Writing – original draft, Visualization, Supervision, Software, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **Joel Casalinho:** Methodology, Investigation. **Julia P. Lancha:** Visualization, Software, Methodology, Investigation. **Romain Rémond:** Writing – review & editing, Formal analysis, Conceptualization.
<table><caption>**Table 6**

List symbols (Latin letters)</caption><thead><tr><th>Symbol</th><th>Name</th><th>Unit</th></tr></thead><tbody><tr><td>a<sub>w</sub></td><td>sorption isotherm (water activity)</td><td>–</td></tr><tr><td>c</td><td>molar concentration</td><td>mole m<sup>-3</sup></td></tr><tr><td>D</td><td>diffusion coefficient</td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>f</td><td>dimensionless diffusion factor</td><td>–</td></tr><tr><td>h</td><td>specific enthalpy</td><td>J kg<sup>-1</sup></td></tr><tr><td>Δh<sub>v</sub></td><td>specific enthalpy of evaporation</td><td>J kg<sup>-1</sup></td></tr><tr><td>j<sub>k</sub></td><td>diffusive flux density of component k</td><td>kg m<sup>-2</sup> s<sup>-1</sup></td></tr><tr><td>K</td><td>permeability</td><td>m<sup>2</sup></td></tr><tr><td>ℓ</td><td>thickness of the sample</td><td>m</td></tr><tr><td>m</td><td>mass</td><td>kg</td></tr><tr><td>P</td><td>pressure</td><td>Pa</td></tr><tr><td>⟨r⟩</td><td>equivalent pore radius</td><td>m</td></tr><tr><td>q<sub>k</sub></td><td>total flux density of component k</td><td>kg m<sup>-2</sup> s<sup>-1</sup></td></tr><tr><td>Q<sub>k</sub></td><td>total flux of component k</td><td>kg s<sup>-1</sup></td></tr><tr><td>R</td><td>gas constant</td><td>J kg<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>S</td><td>surface area</td><td>m<sup>2</sup></td></tr><tr><td>T</td><td>temperature</td><td>k or °C</td></tr><tr><td>ν</td><td>velocity</td><td>m s<sup>-1</sup></td></tr></tbody></table>

<table><caption>**Table 7**

List of symbols (Greek letters).</caption><thead><tr><th>Symbol</th><th>Name</th><th>Unit</th></tr></thead><tbody><tr><td>ε</td><td>void fraction</td><td>–</td></tr><tr><td>φ</td><td>relative humidity</td><td>(–)</td></tr><tr><td>μ</td><td>dynamic viscosity</td><td>kg m<sup>-1</sup> s<sup>-1</sup></td></tr><tr><td>ρ</td><td>density</td><td>kg m<sup>-3</sup></td></tr><tr><td>τ</td><td>tortuosity factor</td><td>(–)</td></tr><tr><td>ω</td><td>density fraction</td><td>(–)</td></tr></tbody></table>

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

Communauté Urbaine du Grand Reims, Département de la Marne, Région Grand Est and European Union (FEDER Grand Est 2021–2027) are acknowledged for their financial support to the Chair of Biotechnology of CentraleSupélec and the Centre Européen de Biotechnologie et de Bioéconomie (CEBB). This study was also carried out with the financial support of the Smart-Réno project (dispositif des Certificats d'Economie d'Energie), in particular through the funding of Julia Lancha's post-doc. Sébastien Gauthier, who manufactured the original sample support, is warmly acknowledged.

## Notations

The main notations are summarized in (Tables 6–8)

## Appendix A. Uncertainties analysis

The two key experimental values obtained from the experiments are the pressure gap and the water vapor flux. The water vapor flux is obtained as the time-derivative of the mass loss. Thanks to the continuous measurement of the mass (every 2 min), we get a very rich information compared to most cup experiments in which the cup is weighted once or twice a day. The vapor flux is obtained as a linear fit over a very large time interval of several days. In Table 3, the slope

<table><caption>**Table 8**

Subscripts.</caption><thead><tr><th>Subscripts</th><th>Meaning</th></tr></thead><tbody><tr><td>a</td><td>air</td></tr><tr><td>cup</td><td>determined by the cup method</td></tr><tr><td>direct</td><td>determined by a specific device</td></tr><tr><td>eff</td><td>effective property</td></tr><tr><td>g</td><td>relative to the gaseous phase</td></tr><tr><td>K</td><td>absolute temperature (kelvin)</td></tr><tr><td>ν</td><td>water vapor</td></tr><tr><td>sv</td><td>saturated water vapor</td></tr></tbody></table>

accuracy was assessed by the standard deviation of the local slope, as defined over a sliding window of 4 h. Fig. A.1 (top) proposes to go further in the analysis. In a standard test (AAC n°1 at 70 °C), the slope was also evaluated on a sliding window of one hour. In this case, the standard deviation increases only slightly, from 0.7% to 0.9%. In terms of maximum values, the interval is ± 1% for a window of 4 h to ± 2% for an interval of one hour. This means that the local mass slope determined over any window of one hour in the full interval of about 30 h gives the global slope with an accuracy of 2%. This emphasizes the exceptional quality of the raw data.

In the worse case of oak T n°2, the sample permeability is so small that the characteristic time of pressure relaxation in the internal cup volume is comparable to the variation of the atmospheric pressure. Consequently, the slope needs to be averaged over a longer time. As observed in Fig. A.1 (bottom) the 4-hour interval is too short and the standard deviation of 20% reported in Table 3 is too severe. By increasing the sliding window to half a day (12 h), the standard deviation drops to 6.7%, with peak values in the range ± 12%. Except for very particular samples, the water vapor flux accuracy is equal of better than 2%.

The accuracy of the pressure gap can be quantified by its standard deviation, together with the sensor offset. The latter was evaluated in real conditions through the pressure gap measured once the cup was empty :we obtained 0.7 Pa with a standard deviation less than 1 Pa, as averaged over several hours, for the differential pressure sensor. Surprisingly, the data were very similar for the pressure difference between the two absolute sensor (0.3 Pa with a standard deviation of 1 Pa). This is underline the excellent resonant silicon crystal technology of these sensors. As depicted in Table 3, the standard deviation of the pressure is of the order of 5%, except for large pressure gap due to the effect of atmospheric pressure, where the standard deviation is of the order of 10%. The offset is negligible, except for the EPS samples, for which both the offset value and the standard deviation are of the order of the measurement. In this case, the measurements are just able to confirm that we get the right order of magnitude. The pressure gap accuracy is therefore equal or better than ± 12%, except for very permeable samples (less than 10 Pa of pressure gap).

The accuracy of these two key experimental values can now be used to estimate the error on the physical parameters *f* and *K*. Classically with formula involving multiplications, the total relative error is obtained as the sum of the relative errors of each term of the expression. As the samples are prepared by a CNC machine, the diameter *D* is well defined but the resin penetrates slightly into the porosity of the sample, which reduces the effective radius by about 1 mm. A actual section of 70 mm was used to calculate the mass flow density with an uncertainty of ca. ± 1 mm. The thickness *ℓ* is measured with a caliper at ± 0.1 mm. With an uncertainty of ± 0.1 °C at 70 °C, the saturated vapor pressure is accurate at ± 0.9%.

Using Eqs. (13) and (14) respectively, and neglecting the uncertainties of the physical properties of the gas by comparison with the experimental uncertainties, the relative errors are therefore:

$$ \frac{\Delta f}{f} = \frac{\Delta \ell}{\ell} + 2 \frac{\Delta D}{D} + \frac{\Delta(\Delta \omega_{\nu})}{\Delta \omega_{\nu}} + \frac{\Delta Q_{\nu}}{Q_{\nu}} \quad (A.1) $$
<figure><img src="image_13.png" /><figcaption>Fig. A.1. Evaluation of the accuracy of the water vapor flux determination by deriving the mass loss evolution. Local derivative using a sliding window of different duration compared to the average value obtained as linear fit over the whole time interval Top) An example of representative case: AAC n°1 and Bottom) The worse case of the paper: oak tangential n°2.</figcaption></figure>

$$ \frac{\Delta K}{K} = \frac{\Delta f}{f} + \frac{\Delta(\Delta\omega_v)}{\Delta\omega_v} + \frac{\Delta(\Delta P_g)}{\Delta P_g} \qquad (A.2) $$

Under the above conditions, i.e. when the pressure difference is greater than 10 Pa, the relative accuracies are better than 6.8% and 19.8% for f and K respectively.

Note that Eq. (13) does not account for the transfer resistance at the face of the sample. On the external face, this resistance can be evaluated using the classical correlation to obtain the average Nusselt number for a flow over a plate [28]. The fan of the climatic chamber ensures an air flow over 2 m s⁻¹. At 60 °C and for a diameter of 70 mm, we obtain a heat transfer coefficient close to 20 W m⁻² K⁻¹, which represents a thermal boundary layer of ca. 1 mm. The mass boundary layer is of the same order of magnitude as the Schmidt number is close to the unit

for water vapor diffusion in air. This represents a negligible resistance compared to the sample resistance, even for the most diffusive material of this study (1% of equivalent sample thickness).

The internal resistance is higher if we consider the air at rest in the cup [17] and could represent up to 20% of the sample resistance in the worse cases (AAC, MDF or Oak in longitudinal direction). Our experimental device has not internal RH sensor that would allows the actual RH gap to be measured [18]. However, as the air gap increases from 2 cm to 4 cm when the cup is empty, the assumption of the air at rest would then double the resistance during the experiment. This would significantly reduce the water vapor flux during the test duration, which is not the case. Thanks to the accurate and frequent mass automatic recording, Fig. A.1 reports variations in the total flux less than ±2%, which constitutes an upper bound for the internal
resistance. The small temperature gradients due to evaporation are then enough to trigger a air circulation by natural convection inside the cup, which reduced the internal resistance. Therefore, we can affirm that the external resistance remains small, even for the worse configurations for which 4% could be added to the global accuracy of 6.8% for f. The relative external and internal resistances are much smaller, hence completely negligible, for the other samples.

## Appendix B. Numerical solution

The two solutions are derived from a combination of binary gas diffusion and Darcy's law. The gradient of the mass fraction of water vapor ($\omega_v = \rho_v/\rho_g$) is the driving force for binary diffusion. As $\omega_v+\omega_a = 1$, $\nabla\omega_v = -\nabla\omega_a$. The diffusive flux of air is then if the same magnitude and opposite to the diffusive flux of water vapor. The total fluxes of air and water vapor are obtained as the sum of diffusive and convective fluxes:

$$q_v = j_v + \rho_v v_g \qquad (B.1)$$

$$q_a = j_a + \rho_a v_g = -j_v + \rho_a v_g \qquad (B.2)$$

with

$$v_g = -\frac{K}{\mu_g} \nabla P_g \qquad (B.3)$$

$$j_v = -\rho_g f D_v \nabla \omega_v \qquad (B.4)$$

The value of $v_g$ is obtained by the condition $q_a = 0$:

$$v_g = j_v/\rho_a \qquad (B.5)$$

This allows the two key physical quantities of the coupling between diffusivity and permeability, the total water vapor flux and the pressure gradient, to be obtained:

$$q_v = -\frac{\rho_g}{1 - \omega_v} f D_v \nabla \omega_v \qquad (B.6)$$

$$\nabla P_g = \frac{1}{1 - \omega_v} f D_v \nabla \omega_v \frac{\mu_g}{K} \qquad (B.7)$$

Assuming that the gas mixture obeys the law of perfect gases, $P_g = P_v + P_a$ and the densities and mass fraction are given by:

$$\rho_v = \frac{P_v M_v}{RT_k} \qquad (B.8)$$

$$\rho_a = \frac{P_a M_a}{RT_k} \qquad (B.9)$$

$$\omega_v = \frac{\rho_v}{\rho_g} = \frac{P_v M_v}{P_a M_a + P_v M_v} \qquad (B.10)$$

Where $T_k$ is the temperature in Kelvin.

The integration of Eqs. (B.6) and (B.7), reduced to their 1D version, should account for the combined variations of $\rho_a$ and $\rho_v$ over the sample thickness. The numerical solution is obtained by integration over the sample thickness. Eqs. (B.6) and (B.7) are therefore written in an incremental form $dx$:

$$q_v = -\frac{\rho_g}{1 - \omega_v} f D_v \frac{d\omega_v}{dx} \qquad (B.11)$$

$$dP_g = \frac{1}{1 - \omega_v} f D_v \frac{\mu_g}{K} d\omega_v \qquad (B.12)$$

Due to the non-linear behavior, the resulting flux is therefore not known during the resolution. However, a simple trick allows the integration over the thickness to be treated in an explicit way. In expression (B.11), we can observe that $q_v$ is inversely proportional to $dx$ while expression (B.12) tells us that the solution $dP_g$ is proportional to $d\omega_v$ and independent of $dx$.

Instead of a classical discretization in equal space steps, we proceed here in $n$ increments of relative humidity $d\phi$. The solution can then be computed with any guess value of $q_{v,guess}$. Using this guess value, the thickness increment $dx$ is computed for each relative humidity increment, thank to Eq. (B.11):

$$dx = -\frac{\rho_g}{1 - \omega_v} \frac{D_v(T, P_g)}{q_{v,guess}} d\omega_v \qquad (B.13)$$

The integration starts from the external face, where the total pressure equals the atmospheric pressure. Note that, due to the non-linear behavior, the value of $dx$ varies over the sample thickness. Remembering that the driving force for diffusion $d\omega_g$ depends on the gas density, hence of the gaseous pressure, a loop in needed inside each space step to update the pressure gap until convergence. After summation of $dx$ over all RH increment, we obtain the virtual thickness corresponding to the guess flux. Once done, the actual water vapor flux is computed by the ratio of virtual over actual thickness. As $dx$ is not involved in Eq. (B.12), the linear correction of the thickness does not change the pressure field.

The following pseudo-code details the integration over the thickness

### Algorithm 1 Numerical solution

Set the problem configuration (K, f, T, $\phi_1$, $\phi_2$, $\ell$)

Set the number of space increments $n$

$d\phi \leftarrow (\phi_2 - \phi_1)/(n - 1)$

Set the guess value $q_{v,guess}$

Initialization:

$x[1] \leftarrow 0$

$P_v[1] \leftarrow \phi_1 P_{sv}(T)$

$P_g[1: n] \leftarrow P_{atm}$

for $i = 2$ to $n$ do

$P_v[i] \leftarrow P_v[i - 1] + d\phi P_{sv}(T)$

$\rho_{a,i-1} \leftarrow (P_g[i - 1] - P_v[i - 1])M_a/(RT_k)$

$\rho_{v,i-1} \leftarrow P_v[i - 1]M_v/(RT_k)$

$\omega_{v,i-1} \leftarrow \rho_{v,i-1}/(\rho_{a,i-1} + \rho_{v,i-1})$

error $\leftarrow 2 \times$ tolerance

$dP_{g,old} \leftarrow 0$

while error > tolerance do

$\rho_{a,i} \leftarrow (P_g[i] - P_v[i])M_a/(RT_k)$

$\rho_{v,i} \leftarrow P_v[i]M_v/(RT_k)$

$\omega_{v,i} \leftarrow \rho_{v,i}/(\rho_{a,i} + \rho_{v,i})$

$d\omega_v \leftarrow \omega_{v,i} - \omega_{v,i-1}$

$\bar{\rho}_a = (\rho_{a,i-1} + \rho_{a,i})/2$

$\bar{\rho}_v = (\rho_{v,i-1} + \rho_{v,i})/2$

$\bar{\rho}_g = \bar{\rho}_a + \bar{\rho}_v$

$\bar{P} \leftarrow (P_g[i - 1] + P_g[i])/2$

Compute $dx$ using equation (B.13) and $q_{v,guess}$

Compute $dP_g$ using equation (B.12)

$P_g[i] \leftarrow P_g[i - 1] + dP_g$

$$\text{error} \leftarrow \left| \frac{dP_g - dP_{g,old}}{dP_g} \right| \\ dP_{g,old} \leftarrow dP_g$$

end while

$x[i] \leftarrow x[i - 1] + dx$

end for

$\Delta P \leftarrow P_g[n] - P_{atm}$

$q_v \leftarrow q_{v,guess} \times \frac{x[n]}{\ell}$

In this work, the numerical solution was computed with $n = 100$), a value large enough for this first order solution to be accurate ($10^{-5}$ relative accuracy on the pressure gap, the more demanding value).

Algorithm 1 allows the values of $q_v$ and $\Delta P$ to be computed whatever the configuration (boundary conditions, temperature, sample thickness, $f$ and $K$). In order to deduce $f$ and $K$ from the experimental water vapor flux $q_v$ and the measured pressure gap $\Delta P_{exp}$, guess values
$f_{\text{guess}}$ and $K_{\text{guess}}$ are used and corrected after integration over the sample thickness. As the pressure gap affects by the values, a loop until convergence is needed (see algorithm 2).

**Algorithm 2 Analysis of experimental data**

Set the problem configuration ($q_{v,exp}$, $\Delta P_{exp}$, T, $\phi_1$, $\phi_2$, $\ell$)

Set the guess values $f_{\text{guess}}$ and $K_{\text{guess}}$
error ← 2 × tolerance

while error > tolerance do

Compute $x[n]$ and $\Delta P$ using algorithm 1 and $q_{v,exp}$ as guess value of the flux

$$ \begin{gathered} \text{error} \leftarrow \left| \frac{\Delta P - \Delta P_{exp}}{\Delta P_{exp}} \right| + \left| \frac{x[n] - \ell}{\ell} \right| \\ f \leftarrow f \times \frac{x[n]}{\ell} \\ K \leftarrow K \times \frac{\Delta P}{\Delta P_{exp}} \end{gathered} $$

end while

This algorithm was used to compute column 6 of Table 5. To compute the theoretical pressure gap using the measured permeability $K_{exp}$ (column 4 of Table 5), the initial guess of permeability is just set to this value without any update in the convergence loop that updates only the value of $f$. Consistently, the contribution of pressure gap must be removed from the error. In this case, the solution gives the pressure gap obtained with the identified value of $f$ and the measured permeability $K_{exp}$ (column 5 of Table 5).

**Appendix C. Evaluation of the characteristic time of the transient state**

To prove that the steady-state flux observed in our experiments is a true steady-state diffusion flux, let us evaluate the characteristic time of the transient period. For a diffusion process, the characteristics time is given by:

$$ \tau = \frac{\ell^2}{D} \qquad (C.1) $$

Where $\ell$ is the characteristic length (half-thickness here as we have two exchange surfaces) and $D$ the diffusion coefficient in $m^2 s^{-1}$. This is the inverse of the time factor involved in analytical solutions of the second Fick's law, the combination of Fick's law with mass conservation. This is needed as the characteristic time constant represents the balance between the migration ability and the storage capacity. This required the driving force to be consistent with the storage capacity. Let us do it in 1-D using the moisture content $X$ as driving force:

$$ \frac{\partial X}{\partial t} = \frac{\partial}{\partial x} \left( D \frac{\partial X}{\partial x} \right) \qquad (C.2) $$

The dimensionless diffusivity $f$ adopted in this paper expresses the vapor flux using the mass fraction of water vapor as driving force:

$$ j_v = -\rho_g f D_v \frac{\partial \omega_v}{\partial x} \qquad (C.3) $$

It must then be converted to have the gradient of moisture content as driving force.

$$ j_v = -\rho_0 D \frac{\partial X}{\partial x} \qquad (C.4) $$

Where $\rho_0$ is the density of the porous medium. Assuming $\rho_g$ to be nearly constant, we obtain a classical relation involving the derivative of the sorption isotherm:

$$ D = \frac{f D_v(T)}{\rho_0} \frac{M_v P_{sv}(T)}{RT} \frac{da_{av}}{dX} \qquad (C.5) $$

Let us put some orders of magnitude of physical values into this equation. For light concrete: $f \approx 0.1$, $\rho_0 \approx 620 \text{ kgm}^{-3}$ and $dX/dav \approx 2.10^{-2}$ [44]. We obtain 700 s (0.2 h) at 40 °C and 250 s (0.07 h) at 70 °C. These short characteristic time is explained by two feature of

light concrete: a very porous structure with a high value of $f$ and a very small storage capacity. For such small times, the coupling between heat and mass transfer matters, which extends the transient period [6]. However, it will remain negligible compared with the duration of the experiment. The worse of all tests is the case of oak T2, which combines a low value of $f$, a high hygroscopicity, a high density and a thickness of 10 mm. In this case, the time constant equals 7 h, which remains much smaller than the test duration. This confirms that the long period at constant mass loss rate is a true steady-state regime.

Note also that the first instants of each test are usually discarded as the pressure depicts huge variations at the beginning of the tests due to thermal dilatation of the trapped air. For example, the test oak T2 lasted 95 h in total by only 80 h were plotted.

**Appendix D. Supplementary data**

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.ijheatmasstransfer.2024.126569.

**Data availability**

Data will be made available on request.

**References**

[1] D. Crawley, L. Lawrie, F. Winkelmann, W. Buhl, Y. Huang, C. Pedersen, R. Strand, R. Liesen, D. Fisher, M. Witte, et al., EnergyPlus: creating a new-generation building energy simulation program, Energy Build. 33 (2001) 319-331.

[2] M. Woloszyn, C. Rode, Tools for performance simulation of heat, air and moisture conditions of whole buildings, in: Building Simulation, vol. 1, Springer, 2008, pp. 5-24.

[3] J. Delgado, N. Ramos, E. Barreira, V. De Freitas, A critical review of hygrothermal models used in porous building materials, J. Porous Media 13 (2010) 221-234.

[4] J. Jacques, M. Labat, M. Woloszyn, Dynamic coupling between vapour and heat transfer in wall assemblies: Analysis of measurements achieved under real climate, Build. Environ. 87 (2015) 129-141.

[5] R. Rémond, G. Almeida, Mass diffusivity of low-density fibreboard determined under steady-and unsteady-state conditions: Evidence of dual-scale mechanisms in the diffusion, Wood Mater. Sci. Eng. 6 (1-2) (2011) 23-33.

[6] P. Perré, A. Challansonnx, J. Colin, On the importance of heat and mass transfer coupling for the characterization of hygroscopic insulation materials, Int. J. Heat Mass Transfer 133 (2019) 968-975.

[7] American Society for Testing and Materials, Standard Test Methods for Water Vapor Transmission of Materials, ASTM International, 2013.

[8] ISO-12572, Hygrothermal performance of building materials and products — Determination of water vapour transmission properties — Cup method, ISO STANDARDS, 2016.

[9] P.W. Gibson, Comparison of sweating guarded hot plate and upright cup methods of measuring water vapor permeability, Technical Report, ARMY NATICK RESEARCH DEVELOPMENT AND ENGINEERING CENTER MA, 1992.

[10] G. Galbraith, J. Guo, R. McLean, The effect of temperature on the moisture permeability of building materials, Build. Res. Inform. 28 (4) (2000) 245-259.

[11] P. Mukhopadhyaya, M.K. Kumaran, J. Lackey, Use of the modified cup method to determine temperature dependency of water vapor transmission properties of building materials, J. Test. Eval. 33 (5) (2005) 316.

[12] N. Issaadi, A. Nouviaire, R. Belarbi, A. Aït-Mokhtar, Moisture characterization of cementitious material properties: assessment of water vapor sorption isotherm and permeability variation with ages, Constr. Build. Mater. 83 (2015) 237-247.

[13] K. Svennberg, L. Wadsø, A modified cup-method for lightweight and highly permeable materials, in: Research in Building Physics, CRC Press, 2020, pp. 177-182.

[14] J. Huang, X. Qian, Comparison of test methods for measuring water vapor permeability of fabrics, Text. Res. J. 78 (4) (2008) 342-352.

[15] T.-C. Codau, E. Onofrei, S. Petrusic, G. Bedek, D. Dupont, D. Soulat, Mathematical correlation of test methods for measuring water-vapor transmission through fabrics, J. Porous Media 18 (3) (2015).

[16] A. Gennadios, C.L. Weller, C.H. Gooding, Measurement errors in water vapor permeability of highly permeable, hydrophilic edible films, J. Food Eng. 21 (4) (1994) 395-409.

[17] S. Zohoun, E. Agoua, G. Degan, P. Perré, An experimental correction proposed for an accurate determination of mass diffusivity of wood in steady regime, Heat Mass Transf. 39 (2) (2003) 147-155.
[18] T. Colinart, P. Glouannec, Accuracy of water vapor permeability of building materials reassessed by measuring cup's inner relative humidity, Build. Environ. 217 (2022) 109038.

[19] Y. Hu, V. Topolkaraev, A. Hiltner, E. Baer, Measurement of water vapor transmission rate in highly permeable films, J. Appl. Polym. Sci. 81 (7) (2001) 1624-1633.

[20] R. Mustapha, A. Zoughaib, N. Ghaddar, K. Ghali, Modified upright cup method for testing water vapor permeability in porous membranes, Energy 195 (2020) 117057.

[21] K. Pietrak, M. Krakowiak, P. Łapka, Experimental verification of conditions within the cups in the cup method of water vapor permeability measurement, in: J. Phys.: Conf. Ser., 2423, (1) IOP Publishing, 2023, 012009.

[22] J. Richter, K. Staněk, P. Kopecký, J. Tywoniak, Measurement of water vapor transmission properties using the cup method-error caused by air buoyancy, J. Phys.: Conf. Ser. 2654 (1) (2023) 012038.

[23] M.K. Kumaran, An alternative procedure for the analysis of data from the cup method measurements for determination of water vapor transmission properties, J. Test. Evaluat. 26 (6) (1998) 575-581.

[24] S. Iizuka, K. Murata, M. Sekine, C. Sato, A novel cup with a pressure-adjusting mechanism for high-temperature water vapor transmission rate measurements, Polym. Test. 50 (2016) 73-78.

[25] K. Fukui, S. Takada, Potential errors and improvements in estimating the humidity dependency of vapor permeability using average humidity during cup tests, J. Build. Eng. 71 (2023) 106578.

[26] C. Hall, G.J. Lo, A. Hamilton, Water vapour permeability of inorganic construction materials, Mater. Struct. 57 (2) (2024) 39.

[27] J. Richter, K. Staněk, Water vapour permeability of wood-remarks on cup method procedure and measured data of spruce wood, J. Phys.: Conf. Ser. 2069 (1) (2021) 012009.

[28] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, second ed., John Wiley & Sons, New York, 2002.

[29] P. Perré, Le séchage convectif de bois résineux: choix, validation et utilisation d'un modèle (Ph.D. thesis), Paris 7, 1987.

[30] P. Perré, M. Moser, M. Martin, Advances in transport phenomena during convective drying with superheated steam and moist air, Int. J. Heat Mass Transfer 36 (11) (1993) 2725-2746.

[31] T. Duforestel, I. Oubrahim, R. Belarbi, H. El Hardouz, M.C. Daåge, Assessment of the water vapor permeability: Effect of the total pressure, Int. J. Heat Mass Transfer 196 (2022) 123210.

[32] O. Krischer, K. Kröll, Die Wissenschaftlichen Grundlagen Der Trocknungstechnik, Springer-Verlag, 2013.

[33] L. Greenspan, Humidity fixed points of binary saturated aqueous solutions, J. Res. Natl Bureau Stand. Sect. A Phys. Chem. 81 (1) (1977) 89.

[34] P. Perré, F. Pierre, J. Casalinho, M. Ayouz, Determination of the mass diffusion coefficient based on the relative humidity measured at the back face of the sample during unsteady regimes, Drying Technol. 33 (9) (2015) 1068-1075.

[35] W. Ai, H. Duval, F. Pierre, P. Perré, A novel device to measure gaseous permeability over a wide range of pressures: characterisation of slip flow for Norway spruce, European beech, and wood-based materials, Holzforschung 71 (2) (2017) 147-162.

[36] E. Agoua, P. Perré, Mass transfer in wood: Identification of structural parameters from diffusivity and permeability measurements, J. Porous Media 13 (11) (2010) 1017-1024.

[37] P. Perré, I.W. Turner, A 3D version of Transpore: a comprehensive heat and mass transfer computational model for simulating the drying of porous media, Int. J. Heat Mass Transf. 42 (12) (1999) 4501-4521.

[38] J. Haynes, Pore size analysis according to the Kelvin equation, Matér. Construct. 6 (1973) 209-213.

[39] L.N. Trong, S. Asamoto, K. Matsui, Sorption isotherm and length change behavior of autoclaved aerated concrete, Cem. Concr. Compos. 94 (2018) 136-144.

[40] W. Simpson, Sorption theories applied to wood, Wood Fiber Sci. (1980) 183-195.

[41] G. Almeida, S. Gagné, R.E. Hernández, A NMR study of water distribution in hardwoods at several equilibrium moisture contents, Wood Sci. Technol. 41 (4) (2007) 293-307.

[42] N.H. Chen, D.F. Othmer, New generalized equation for gas diffusion coefficient, J. Chem. Eng. Data 7 (1) (1962) 37-41.

[43] E.N. Fuller, P.D. Schettler, J.C. Giddings, New method for prediction of binary gas-phase diffusion coefficients, Indust. Eng. Chem. 58 (5) (1966) 18-27.

[44] A. Challansonnex, F. Pierre, J. Casalinho, P. Lv, P. Perré, Mass diffusivity determination of various building materials based on inverse analysis of relative humidity evolution at the back face of a sample, Constr. Build. Mater. 193 (2018) 539-546.

[45] C. Buratti, E. Belloni, F. Merli, Water vapour permeability of innovative building materials from different waste, Mater. Lett. 265 (2020) 127459.

[46] G.L. Comstock, Directional permeability of softwoods, Wood Fiber Sci. (1970) 283-289.

[47] F. Wägner, G. Schober, H. Mörtel, Measurement of the gas permeability of autoclaved aerated concrete in conjunction with its physical properties, Cem. Concr. Res. 25 (8) (1995) 1621-1626.

[48] H. Thoemen, A. Klueppel, An investigation on the permeability of different wood furnish materials, Holzforschung 62 (2008) 215-222.

[49] Y. Li, B.Y. Lattimer, S.W. Case, Measurement and modelling of thermal and physical properties of wood construction materials, Constr. Build. Mater. 284 (2021) 122780.

[50] A.J. Stamm, Bound-water diffusion into wood in the fiber direction, For. Prod. J. 9 (1959) 27-32.

[51] P. Perré, D.M. Nguyen, G. Almeida, A macroscopic washburn approach of liquid imbibition in wood derived from X-ray tomography observations, Sci. Rep. 12 (1) (2022) 1750.

[52] J. Siau, Transport Processes in Wood, Springer-Verlag, New York, 1984.

[53] C. Roussey, P. Perré, J. Casalinho, J. Colin, Inverse analysis of oxygen diffusivity in oak wood using the back-face method: application to cooperage, Wood Sci. Technol. (2022) 1-21.

[54] E.J. Carr, I.W. Turner, P. Perre, A dual-scale modeling approach for drying hygroscopic porous media, Multiscale Model. Simul. 11 (1) (2013) 362-384.

[55] M. Louërat, M. Ayouz, P. Perré, Heat and moisture diffusion in spruce and wood panels computed from 3-D morphologies using the Lattice Boltzmann method, Int. J. Therm. Sci. 130 (2018) 471-483.