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# Variation of the ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{2}$$\end{document} error as a function of Δx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} (Δt=10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t\,=\,10^{-4}$$\end{document}) (a) and Δt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} (Δx=10-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x\,=\,10^{-2}$$\end{document}) (b) using an explicit Scharfetter–Gummel scheme

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Comparisons of experimental observation of heat and moisture transfer through porous building materials with numerical results have been presented in numerous studies reported in the literature. However, some discrepancies have been observed, highlighting underestimation of sorption process and overestimation of desorption process. Some studies int...

## Citations

... Then, for the heat and mass advection-diffusion Equations (2a) and (2b), the SCHARFETTER-GUMMEL numerical scheme is used. Preliminary studies 23,24 showed the efficiency of the approach to extend the stability conditions and the accuracy of the solution. As a last step of the proposed methodology, the time discretisation of these two equations, an innovative two-step RUNGE-KUTTA approach is used of the time discretisation of these two advection-diffusion equations, enabling to extend further the stability region of the numerical scheme. ...

... = (1) and we obtain Δt ≤ CΔx. 24 Thus, it is one order less restrictive than standard approach and the so-called COURANT-FRIEDRICHS-LEWY (CFL) conditions Δt ≤ CΔx 2 . Using, the SCHARFETTER-GUMMEL scheme, the spatial discretisation does not need to be extremely refined. ...

This article proposes an efficient explicit numerical model with a relaxed stability condition for the simulation of heat, air and moisture transfer in porous material. Three innovative approaches are combined to solve the system of two differential advection-diffusion equations coupled with a purely diffusive equation. First, the DuFort-Frankel scheme is used to solve the diffusion equation, providing an explicit scheme with an extended stability region. Then, the two advection-diffusion equations are solved using both the Scharfetter-Gummel numerical scheme for the space discretisation and the two-step Runge-Kutta method for the time variable. This combination enables to relax the stability condition by one order. The proposed numerical model is evaluated on three case studies. The first one considers quasi-linear coefficients. The theoretical results of the numerical schemes are confirmed by our computations. Indeed, the stability condition is relaxed by a factor of 40
compared to the standard Euler explicit approach. The second case provides an analytical solution for a weakly nonlinear problem. A very satisfactory accuracy is observed between the reference solution and the one provided by the numerical model. The last case study assumes a more realistic application with nonlinear coefficients and Robin-type boundary conditions. The computational time is reduced 10 times by using the proposed model in comparison with the explicit Euler method.

... In Refs. [3][4][5], it was demonstrated that the simulation of material moisture buffering capacity is inaccurate without hysteresis effects. In Ref. [6], the absence of hysteresis in the model yield in the prediction of erroneous indoor air comfort. ...

... Recently, the so-called Bang-Bang model, inspired by the control literature [17][18][19], has been introduced to take into account the hysteresis effects on moisture sorption capacity in a model of heat and mass transfer [5]. In Ref. [20], the model is coupled with a moisture transport equation to determine the optimal experiment design to estimate the coefficients of the moisture sorption curves. ...

... As mentioned for the previous case, the good reliability of the model depends on the value of the parameter β in Eqs. (4) and (5). Here, the value for the desorption phase is higher The computed temperature and relative humidity at x ¼ 1:5 cm are compared to experimental measurements in Fig. 11(a) and (b). ...

The reliability of mathematical models for heat and mass transfer in building porous
material is of capital importance. A reliable model permits to carry predictions of the physical phenomenon with sufficient confidence in the results. Among the physical phenomena, the hysteresis effects on moisture sorption and moisture capacity need to be integrated in the mathematical model of transfer. This article proposes to explore the use of an smooth Bang-Bang model to simulate the hysteresis effects coupled with heat and mass transfer in porous material. This model adds two supplementary differential equations to the two classical ones for heat and mass transfer. The solution of these equations ensures smooth transitions between the main sorption and desorption curves. Two parameters are required to control the
speed of transition through the intermediary curves. After the mathematical description of the model, an efficient numerical model is proposed to compute the fields with accuracy and reduced computational efforts. It is based on the DuFort-Frankel scheme for the heat and mass balance equations. For the hysteresis numerical model, an innovative implici-explicit approach is proposed. Then, the predictions of the numerical model are compared with experimental observations from literature for two case studies. The first one corresponds to a slow cycle of adsorption and desorption while the second is based on a fast cycling case
with alternative increase and decrease of moisture content. The comparisons highlight a very satisfactory agreement between the numerical predictions and the observations. In the last Section, the reliability and efficiency of the proposed model is investigated for long term simulation cases. The importance of considering hysteresis effects in the reliability of the predictions are enhanced by comparison with classical approaches from literature.

... This can be explained by the fact that advection of vapor due to gas transport is almost always neglected. But this quite common simplification has not got any consensus, since some authors have already underlined that air flows may have a significant impact on the hygroscopic behavior of porous material like concrete [20], textiles [21], or even earthen and bio-based materials [22]. ...

The hygroscopic behavior of earthen materials has been extensively studied in the past decades. However, while the air flow within their porous network may significantly affect the kinetics of vapor transfer and thus their hygroscopic performances, few studies have focused on its assessment. For that purpose, a key parameter would be the gas permeability of the material, and its evolution with the relative humidity of the air. Indeed, due to the sorption properties of earthen material, an evolution of the water content, and thus of relative permeability, are foreseeable if the humidity of in-pore air changes. To fill this gap, this paper presents the measurement of relative permeabilities of a compacted earth sample with a new experimental set-up. The air flow through the sample is induced with an air generator at controlled flow rate, temperature, and humidity. The sample geometry was chosen in order to reduce, as much as possible, its heterogeneity in water content, and the tests were realized for several flow rates. The results, which show the evolution of gas permeability with the relative humidity of the injected air and with the water content of the material, either in adsorption or in desorption, were eventually successfully compared to predictions of the well-known Corey's law.

... As discussed and thoroughly motivated in [15][16][17], it is of capital importance to obtain a dimensionless problem before elaborating a numerical model. For this, dimensionless fields are defined: ...

The fidelity of a model relies both on its accuracy to predict the physical phenomena and its capability to estimate unknown parameters using observations. This article focuses on this second aspect by analyzing the reliability of two mathematical models proposed in the literature for the simulation of heat losses through building walls. The first one, named DuFort-Frankel (DF), is the classical heat diffusion equation combined with the DuFort-Frankel numerical scheme. The second is the so-called RC lumped approach, based on a simple ordinary differential equation to compute the temperature within the wall. The reliability is evaluated following a two stages method. First, samples of observations are generated using a pseudo-spectral numerical model for the heat diffusion equation with known input parameters. The results are then modified by adding a noise to simulate experimental measurements. Then, for each sample of observation, the parameter estimation problem is solved using one of the two mathematical models. The reliability is assessed based on the accuracy of the approach to recover the unknown parameter. Three case studies are considered for the estimation of (i) the heat capacity, (ii) the thermal conductivity or (iii) the heat transfer coefficient at the interface between the wall and the ambient air. For all cases, the DF mathematical model has a very satisfactory reliability to estimate the unknown parameters without any bias. However, the RC model lacks of fidelity and reliability. The error on the estimated parameter can reach 40% for the heat capacity, 80% for the thermal conductivity and 450% for the heat transfer coefficient.

... The Scharfetter-Gummel numerical scheme was proposed in 1969 in [36] with very recent theoretical results in [18,19]. In the context of building porous media, it is successfully applied in [5] to water transport and then in [6] to combined heat and moisture transfer. The contributions of the present paper is two fold. ...

... It has been shown that the diffusion of moisture can be written using the vapor pressure P 1 , introducing the global moisture permeability k m [6]. Thus, the total moisture flux yields: ...

... The moisture and heat equations are advection-diffusion types. The Scharfetter-Gummel approach has shown great efficiency in preliminary studies for a single equation [5] and a system of two coupled equations [6]. Therefore it will be used for the spatial discretisation of the moisture and heat equations. ...

This work presents a detailed mathematical model combined with an innovative efficient numerical model to predict heat, air and moisture transfer through porous building materials. The model considers the transient effects of air transport and its impact on the heat and moisture transfer. The achievement of the mathematical model is detailed in the continuity of Luikov's work. A system composed of two advection-diffusion differential equations plus one exclusively diffusion equation is derived. The main issue to take into account the transient air transfer arises in the very small characteristic time of the transfer, implying very fine discretisation. To circumvent these difficulties, the numerical model is based on the Du Fort-Frankel explicit and unconditionally stable scheme for the exclusively diffusion equation. It is combined with a two-step Runge-Kutta scheme in time with the Scharfetter-Gummel numerical scheme in space for the coupled advection-diffusion equations. At the end, the numerical model enables to relax the stability condition, and, therefore, to save important computational efforts. A validation case is considered to evaluate the efficiency of the model for a nonlinear problem. Results highlight a very accurate solution computed about 16 times faster than standard approaches. After this numerical validation, the reliability of the mathematical model is evaluated by comparing the numerical predictions to experimental observations. The latter is measured within a multi-layered wall submitted to a sudden increase of vapor pressure on the inner side and driven climate boundary conditions on the outer side. A very satisfactory agreement is noted between the numerical predictions and experimental observations indicating an overall good reliability of the proposed model.

... It is based on the Scharfetter-Gummel numerical scheme. This approach is particularly efficient for socalled advection-diffusion equations as highlighted from a mathematical point of view in [12] and illustrated in [2,3] for the case of heat and moisture transfer in building porous materials. In our work, the proposed numerical model is compared to the standard methods in the context of capillary adsorption phenomena. ...

The goal of this study is to propose an efficient numerical model for the predictions of capillary adsorption phenomena in a porous material. The Scharfetter-Gummel numerical scheme is proposed to solve an advection-diffusion equation with gravity flux. Its advantages such as accuracy, relaxed stability condition, and reduced computational cost are discussed along with the study of linear and nonlinear cases. The reliability of the numerical model is evaluated by comparing the numerical predictions with experimental observations of liquid uptake in bricks. A parameter estimation problem is solved to adjust the uncertain coefficients of moisture diffusivity and hydraulic conductivity.

... Some studies try to explain the deviations observed between measurements and simulations, stating that modelling some additional phenomena, not taken into account in the purely diffusive model, could improve predictions (Langmans et al. 2013;Teasdale-St-Hilaire and Derome 2007;Samri 2008;Perré et al. 2015). On the one hand, taking into account the advection phenomenon seems to have an impact on the transient state (Berger et al. , 2017a. The air within the porous matrix may move between fibres or within connected pores due to a total pressure gradient. ...

Prediction of moisture transfer within material using a classic diffusive model may lack accuracy, since numerical simulations underestimate the adsorption process when a sample is submitted to variations of moisture level. Model equations are always established with assumptions. Consequently, some phenomena are neglected. This paper therefore investigates the impact of improving traditional diffusive models by taking into account additional phenomena that could occur in moisture transport within hygroscopic fibrous materials such as wood-based products. Two phenomena in the porous material are investigated: (1) non-equilibrium behaviour between water vapour and bound water, and (2) transport by air convection. The equations of each model are established by starting from averaging conservation equations for the different species considered within material (water vapour, bound water and air). In addition, the validity of assumptions currently used in the models is verified. Then the three models are compared with experimental data to highlight their capacity to predict both the vapour pressure and the mass of adsorbed water. This comparison tends to show a slight improvement in predictions with the new models. To increase our understanding of these models, the influence of the main parameters characterising phenomena (sorption coefficient, intrinsic permeability, Péclet number and Fourier number) is studied using local sensitivity analysis. The shape of the sensitivity coefficients shows that the first kinetics period is only impacted a little by the non-equilibrium. In other periods, the influence of the diffusion phenomenon represented by the Fourier number is much greater than that of the two other phenomena: advection and sorption. Nevertheless, the sensitivity study shows that these two phenomena have some influence on vapour pressure.

... Dimensionless formulation. As discussed and thoroughly motivated in [15][16][17], it is of capital importance to obtain a dimensionless problem before elaborating a numerical model. For this, dimensionless fields are defined: ...

The fidelity of a model relies both on its accuracy to predict the physical phenomena
and its capability to estimate unknown parameters using observations. This article focuses on this second aspect by analyzing the reliability of two mathematical models proposed in the literature for the simulation of heat losses through building walls. The first one, named DF, is the classical heat diffusion equation combined with the DuFort-Frankel numerical scheme. The second is the so-called RC lumped approach, based on a simple ordinary differential equation to compute the temperature within the wall. The reliability is evaluated following a two stages method. First, samples of observations are generated using a pseudospectral numerical model for the heat diffusion equation with known input parameters. The results are then modified by adding a noise to simulate experimental measurements. Then, for each sample of observation, the parameter estimation problem is solved using one of the two mathematical models. The reliability is assessed based on the accuracy of the approach to recover the unknown parameter. Three case studies are considered for the estimation of (i) the heat capacity, (ii) the thermal conductivity or (iii) the heat transfer coefficient at the interface between the wall and the ambient air. For all cases, the DF mathematical model has a very satisfactory reliability to estimate the unknown parameters without any bias. However, the RC model lacks of fidelity and reliability. The error on the estimated parameter can reach 40% for the heat capacity, 80% for the thermal conductivity and 450% for the heat transfer coefficient.

... It is based on the Scharfetter-Gummel numerical scheme. This approach is particularly efficient for so-called advection-diffusion equations as highlighted from a mathematical point of view in [16] and illustrated in [17] and [18] for the case of heat and moisture transfer in building porous materials. In our work, the proposed numerical model is compared to the standard methods in the context of capillary adsorption phenomena. ...

Rising damp is one the main causes of moisture damages in historical buildings. The
goal of this study is to propose an efficient numerical model for the predictions of capillary adsorption phenomena in porous material. The Scharfetter-Gummel numerical scheme is proposed to solve an advection-diffusion equation with gravity flux. Its advantages such as accuracy, relaxed stability condition and reduced computational cost are discussed along with the study of linear and nonlinear cases. Last, the reliability of the numerical model is evaluated by comparing the numerical predictions with experimental observations of liquid uptake in bricks. A parameter estimation problem is solved to adjust the uncertain coefficients of moisture diffusivity and the hydraulic conductivity and thus improve the predictions. The efficient numerical model enables to solve the inverse problem two times faster than standard approaches. A very satisfactory reliability of the proposed numerical model is observed.

... The Scharfetter-Gummel numerical scheme was proposed in 1969 in [36] with very recent theoretical results in [18,19]. In the context of building porous media, it is successfully applied in [5] to water transport and then in [6] to combined heat and moisture transfer. The contributions of the present paper is two fold. ...

... It has been shown that the diffusion of moisture can be written using the vapor pressure P 1 , introducing the global moisture permeability k m [6]. Thus, the total moisture flux yields: ...

... The moisture and heat equations are advection-diffusion types. The Scharfetter-Gummel approach has shown great efficiency in preliminary studies for a single equation [5] and a system of two coupled equations [6]. Therefore it will be used for the spatial discretisation of the moisture and heat equations. ...