## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

Implicit schemes require important sub-iterations when dealing with highly nonlinear problems such as the combined heat and moisture transfer through porous building elements. The computational cost rises significantly when the whole-building is simulated, especially when there is important coupling among the building elements themselves with neighbouring zones and with HVAC (Heating Ventilation and Air Conditioning) systems. On the other hand, the classical Euler explicit scheme is generally not used because its stability condition imposes very fine time discretisation. Hence, this paper explores the use of an improved explicit approach - the Dufort-Frankel scheme - to overcome the disadvantage of the classical explicit one and to bring benefits that cannot be obtained by implicit methods. The Dufort-Frankel approach is first compared to the classical Euler implicit and explicit schemes to compute the solution of nonlinear heat and moisture transfer through porous materials. Then, the analysis of the Dufort-Frankel unconditionally stable explicit scheme is extended to the coupled heat and moisture balances on the scale of a one- and a two-zone building models. The Dufort-Frankel scheme has the benefits of being unconditionally stable, second-order accurate in time O(dt^2) and to compute explicitly the solution at each time step, avoiding costly sub-iterations. This approach may reduce the computational cost by twenty, as well as it may enable perfect synchronism for whole-building simulation and co-simulation.

To read the full-text of this research,

you can request a copy directly from the authors.

... Thus, no costly sub-iterations are required to treat the nonlinearities, as in implicit approaches. Furthermore, as demonstrated in [16,17], it has an extended stability region, so the so-called Courant-Friedrichs-Lewy (CFL) condition [18] is not an impediment. Interested readers may consult [16,17] for further details and its applications for heat and moisture transfer in building porous materials. ...

... Furthermore, as demonstrated in [16,17], it has an extended stability region, so the so-called Courant-Friedrichs-Lewy (CFL) condition [18] is not an impediment. Interested readers may consult [16,17] for further details and its applications for heat and moisture transfer in building porous materials. Since many details are provided in [17] for a similar system of coupled partial differential equations, the demonstration of the fully discrete equations is not detailed. ...

... Interested readers may consult [16,17] for further details and its applications for heat and moisture transfer in building porous materials. Since many details are provided in [17] for a similar system of coupled partial differential equations, the demonstration of the fully discrete equations is not detailed. ...

... Another improved explicit approach, namely DU FORT-FRANKEL method is used for a critical assessment of the STS method. The method was deeper explored in the works of GASPARIN et al. [26,27]. ...

... The method was first presented in the listed works [43][44][45] almost 50 years ago. In this article, the detailed discretization of the scheme is omitted, and interested readers can refer to the recent papers [26,27]. There the method has been extended for coupled equations of heat and mass transfer, and its performance has been discussed in more details. ...

... This condition imposes certain restrictions for the application of the EULER explicit scheme into building simulation tools [27]. Secondly, compared to EULER implicit scheme, the STS methods are much easier to implement, since it has an explicit formulation. ...

... In addition, it has the advantage to explicitly computing the solution at each time step, avoiding costly sub-iterations. All these characteristics make the method to be an interesting option in co-simulation and in parallel simulations (Gasparin et al., 2018c). ...

... This approach may reduce the computational cost by a factor of twenty, as well as it enables perfect synchronism for whole-building simulation and co-simulation. These results are shown in An improved explicit scheme for whole-building hygrothermal simulation (Gasparin et al., 2018c). For the first case study, of heat and moisture transfer through a wall, the Euler implicit scheme required around 3 sub-iterations, making it three times more costly than the DuFort-Frankel scheme. ...

... For coupling the model of heat and moisture transfer with others through co-simulation approaches (Wetter, 2011), the DuFort-Frankel is a very promising approach. It is explicit and avoids extra sub-iterations due to model coupling (Gasparin et al., 2018c). For strong reduction of the computational cost with a very satisfactory accuracy of the solution, the Spectral has been shown as the best option (Gasparin et al., 2018e). ...

Building energy consumption is directly impacted by weather parameters such as temperature, solar radiation, atmospheric pressure, relative humidity and wind velocity. The knowledge of the building hygrothermal performance enables the design of energy efficient buildings and the prediction of overall durability and sustainability of envelopes. Therefore, designers and builders are interested in modeling the long-term performance of the envelopes by means of accurate, reliable and fast simulation tools.Several numerical models have been proposed in the literature to study the heat and moisture transfer in building materials. In general, this problem is solved by traditional methods, such as finite-difference and finite-volume methods, using mainly implicit schemes. Nevertheless, these methods impose costly sub-iterations to treat the nonlinearities and very fine discretization, which increase substantially the simulation computational cost. Therefore, this research has been focused on the development and analyses of numerical methods for efficiently simulate the problem of heat and mass transfer through porous materials.In the first part of this thesis, improved schemes of the traditional numerical methods have been developed to eliminate costly sub-iterations to treat nonlinearities, to improve the order of accuracy and to save computer run time. Despite the great progress with the new numerical schemes, the conclusion of the first part shows that we still have to deal with large systems of equations, particularly when treating multi-dimensional transfer problems. For this reason, to reduce even more the computational burden and the size of the system, a reduced-order model, based on spectral methods is proposed in the sequence to provide an accurate description of the physical phenomena. The degrees of freedom of the solution is strongly decreased while maintaining the model fidelity. It ensures a computational cost much lower than the complete original model.All these methods are applied to problems related to building physics, such as single and multilayer nonlinear transfer, the determination of optimum insulation thickness, the process of moisture buffer effects and transfer in one- or two-zone building models. In conclusion, we show how to build efficient numerical models, in terms of computational cost and accuracy, to investigate the heat and mass transfer in porous materials.

... It is an explicit numerical scheme with an increased stability domain. Interested readers are invited to consult [18] for the original work, [19,20] for the results on the stability analysis and [20,21] for further details and its applications for heat and moisture transfer in building porous materials. Since a complete description is provided in [21], only the main steps are recalled here. ...

... Interested readers are invited to consult [18] for the original work, [19,20] for the results on the stability analysis and [20,21] for further details and its applications for heat and moisture transfer in building porous materials. Since a complete description is provided in [21], only the main steps are recalled here. The idea of the approach is to replace the term u n j ←− ...

... If T ≡ T ′ then dT dt ≡ dT ′ dt . Thus, from equations (20) and (21), one obtain: ...

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.

... , N t }. The solution u ( x ⋆ , t ⋆ ) is obtained using the Dufort-Frankel numerical scheme [28][29][30]. It is an explicit numerical scheme with a relaxed stability condition. ...

... A simple uniform initial condition is set as u 0 = 0 to illustrate the approach. The Fourier number, the thermal conductivity, and the heat capacity are given as: 30 ] . The problem is solved by implementing the Dufort-Frankel numerical scheme. ...

... An efficient sensitivity analysis for energy performance of a building envelope: a continuous derivative based approach C Continuous function derivatives he second order partial differentiation of Eq. (30) gives us the second-order sensitivity coefficients: ...

Within the framework of building energy assessment, this article proposes to use a derivative based sensitivity analysis of heat transfer models in a building envelope. Two, global and local, estimators are obtained at low computational cost, to evaluate the influence of the parameters on the model outputs. Ranking of these estimators values allows to reduce the number of model unknown parameters by excluding non-significant parameters. A comparison with variance and regression-based methods is carried out and the results highlight the satisfactory accuracy of the continuous-based approach. Moreover, for the carried investigations the approach is $100$ times faster compared to the variance-based methods. A case study applies the method to a real-world building wall. The sensitivity of the thermal loads to local or global variations of the wall thermal is investigated. Additionally, a case study of wall with window is analyzed.

... Most simplified models of the moisture in rooms intended for people's stay [35][36][37][38] are based on the moisture balance, which includes ventilation parameters, internal sources of moisture, and the moisture buffering by the interior materials. The aim of the article is to develop a humidity model in the passenger car cabin as a function of time, the number of people, and the cubic capacity of the car cabin based on empirical research. ...

... where V (m 3 ) is the volume of the car, t (h) is the time, ω (g/m 3 ) is the absolute humidity, Q n (g/h) is the humidity supplied from the outside or discharged from the car by supply ventilation, and Q g (g/h) is the humidity generated by the driver and passengers. It should be noted that the moisture buffering by materials, which in many models has been included [35][36][37][38], is omitted here. The humidity that is supplied from the outside or removed from the car by the supply ventilation is described by the following formula: ...

This paper presents research on humidity in a passenger car cabin with the use of supply ventilation without cooling the air. Based on the tests carried out and the humidity balance in the car, a model was developed for changing the humidity in the passenger car cabin as a function of time. The study of thermohumid conditions was carried out in two passenger cars. During the tests, the heating and cooling functions were turned off. The relative humidity and temperature were measured outside the car before and after driving the car and in the supply air duct and inside the passenger car cabin while driving the car. The tests were carried out for an average temperature range from 20 to 42.9 °C. In order to develop a model of humidity changes as a function of time, a humidity balance was prepared. Human-generated humidity in the car cabin depends mainly on the temperature inside the car and the age of the person and can range from 20 to 180 g/(h × person) for an adult in the temperature range of 20–43 °C, while for a child under six years old the humidity ranges from 8 to 19.5 g/(h × person) in the temperature range 22–34 °C. A formula of humidity generated by an adult and a child aged six years old was obtained as a function of temperature inside a passenger car. Based on the experimental research and the model developed, the humidity generated by a single adult and a six-year-old child in the car was determined. The developed model can be used in the automatic airflow adjustment systems in passenger cars.

... It provides an explicit numerical scheme with an increased stability domain. Interested readers may consult [27] for the original work [28,29], for the results on the stability analysis and [29,30] for further details and its applications for heat and moisture transfer in building porous materials. Since a complete description is provided in Refs. ...

... Since a complete description is provided in Refs. [30,31] for an analogous system of coupled diffusion equations, the whole demonstration of the scheme is not detailed here. ...

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.

... Thus, efficient numerical models are worth of investigation. In this paper, an innovative numerical model based on the Du Fort-Frankel scheme is studied, which has already demonstrated a promising efficiency in [15][16][17] for the simulation of one-dimensional heat and mass transfer through porous materials in building envelopes. Here, the model is extended to simulate two-dimensional heat transfer in a building facade over one year. ...

... Interested readers may consult [15,16,25] for example of its applications for one-dimensional heat and moisture transfer in building porous materials. ...

A two-dimensional model is proposed for energy efficiency assessment through the simulation of heat transfer in building envelopes, considering the influence of the surrounding environment. The model is based on the Du Fort–Frankel approach that provides an explicit scheme with a relaxed stability condition. The model is first validated using an analytical solution and then compared to three other standard schemes. Results show that the proposed model offers a good compromise in terms of high accuracy and reduced computational efforts. Then, a more complex case study is investigated, considering non-uniform shading effects due to the neighboring buildings. In addition, the surface heat transfer coefficient varies with wind velocity and height, which imposes an addition non-uniform boundary condition. After showing the reliability of the model prediction, a comparison over almost 120 cities in France is carried out between the two- and the one-dimensional approaches of the current building simulation programs. Important discrepancies are observed for regions with high magnitudes of solar radiation and wind velocity. Last, a sensitivity analysis is carried out using a derivative-based approach. It enables to assess the variability of the solution according to the modeling of the two-dimensional boundary conditions. Moreover, the proposed model computes efficiently the solution and its sensitivity to the modeling of the urban environment.

... The optimization processes have been carried out for the simulation time = 120 h. For the sake of clarity, the parameters of both models are optimized for the time-averaging periods τ = [6,12,24,48] h only. The evolutions of the temperature for the representative simulation time with both CM and ARM with different EMs are shown in Figure 3. ...

... The total simulation time for the first part of the study is taken as = 100 days. The results are compared for four τ = [6,12,24,48] h averaging periods and different t Δ timestep sizes. Figure 6 illustrates the evolution of the global relative error and computational time. ...

The design of numerical tools to model the behavior of building materials is a challenging task. The crucial point is to save computational costs and maintain the high accuracy of predictions. There are two main limitations on the time scale choice, which places an obstacle to solving the above issues. The first one is the numerical restriction. A number of research studies are dedicated to overcome this limitation and it is shown that it can be relaxed with innovative numerical schemes. The second one is the physical restriction. It is imposed by the properties of a material, the phenomena itself, and the corresponding boundary conditions. This study is focused on the study of a methodology that enables to overcome the physical restriction on the time grid; a so-called average reduced model is suggested. It is based on smoothing the time-dependent boundary conditions. Besides this, the approximate solution is decomposed into average and fluctuating components. The primer is obtained by integrating the equations over time, whereas the latter is a user-defined EM. The methodology is investigated for both heat diffusion and coupled heat and mass transfer. It is demonstrated that the signal core of the boundary conditions is preserved and the physical restriction can be relaxed. The model proved to be reliable, accurate, and efficient also in comparison with the experimental data of 2 years. The implementation of the scarce time-step of 1 h is justified. It is shown, that by maintaining the tolerable error, it is possible to cut computational effort up to almost four times in comparison with the complete model with the same time grid.

... Thus, efficient numerical models are worth of investigation. In this article, an innovative numerical model based on the DU FORT-FRANKEL scheme is studied, which has already demonstrated a promising efficiency in [15][16][17] for the simulation of one-dimensional heat and mass transfer through porous materials in building envelopes. Here, the model is extended to simulate two-dimensional heat transfer in a building facade over one year. ...

... Furthermore, as demonstrated in next section, it has an extended stability region, so the so-called Courant-Friedrichs-Lewy (CFL) restriction [24] is relaxed. Interested readers may consult [15][16][17] for example of its applications for one-dimensional heat and moisture transfer in building porous materials. ...

A two-dimensional model is proposed for energy efficiency assessment through the simulation of heat transfer in building envelopes, considering the influence of the surrounding environment. The model is based on the Du Fort–Frankel approach that provides an explicit scheme with a relaxed stability condition. The model is first validated using an analytical solution and then compared to three other standard schemes. Results show that the proposed model offers a good compromise in terms of high accuracy and reduced computational efforts. Then, a more complex case study is investigated, considering non-uniform shading effects due to the neighboring buildings. In addition, the surface heat transfer coefficient varies with wind velocity and height, which imposes an addition non-uniform boundary condition. After showing the reliability of the model prediction, a comparison over almost 120 cities in France is carried out between the two- and the one-dimensional approaches of the current building simulation programs. Important discrepancies are observed for regions with high magnitudes of solar radiation and wind velocity. Last, a sensitivity analysis is carried out using a derivative-based approach. It enables to assess the variability of the solution according to the modeling of the two-dimensional boundary conditions. Moreover, the proposed model computes efficiently the solution and its sensitivity to the modeling of the urban environment.

... It is an explicit numerical scheme with an increased stability domain. Interested readers are invited to consult [18] for the original work, [19,20] for the results on the stability analysis and [20,21] for further details and its applications for heat and moisture transfer in building porous materials. Since a complete description is provided in [21], only the main steps are recalled here. ...

... Interested readers are invited to consult [18] for the original work, [19,20] for the results on the stability analysis and [20,21] for further details and its applications for heat and moisture transfer in building porous materials. Since a complete description is provided in [21], only the main steps are recalled here. The idea of the approach is to replace the term ←− 1 2 (︀ +1 + −1 )︀ in the forward in time central scheme to obtain the following fully discrete dynamical system: ...

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.

... According to the standard von Neumann stability analysis, the Du Fort-Frankel scheme is unconditionally stable [16,41]. Further details and examples of applications of this approach are presented in [15,16]. ...

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.

... According to the standard von Neumann stability analysis, the Du Fort-Frankel scheme is unconditionally stable [27,28]. Further details and examples of applications of this scheme may be consulted in [27,29]. ...

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.

... Recently, innovative and efficient methods of numerical simulation have been proposed. Some improved explicit schemes, enabling to overcome the stability restrictions of standard Euler explicit schemes, have been proposed in [13,14]. An accurate and fast numerical scheme based on the Scharfetter-Gummel idea has been proposed in [15] to solve the advective-diffusive moisture differential equation. ...

This paper aims at estimating the sorption isotherm coefficients of a wood fiber material using experimental data. First, the mathematical model, based on convective transport of moisture, the Optimal Experiment Design (OED) and the experimental set-up are presented. Then, measurements of relative humidity within the material are carried out, after searching the OED using the computation of the sensitivity functions and a priori values of the unknown parameters. It enables to plan the experimental conditions in terms of sensor positioning and boundary conditions out of 20 possible designs, ensuring the best accuracy for the identification method and, thus, for the estimated parameter. After the measurements, the parameter estimation problem is solved. The determined sorption isotherm coefficients calibrate the numerical model to fit better the experimental data. However, some discrepancies still appear since the hysteresis effects on the sorption capacity are not included in the model. Therefore, the latter is improved proposing an additional differential equation for the sorption capacity to consider the hysteresis effects. The OED approach is developed for the estimation of five of the coefficients involved in the hysteresis model. To conclude, the prediction of the model with hysteresis have better reliability when compared to the experimental observations.

... The solution u ( x , t ) is obtained using the Dufort-Frankel numerical scheme. This numerical model is chosen due to its explicit formulation without loss of accuracy or reliability [13][14][15]. For the non-linear case, the solution is calculated with the following expression: ...

The in situ estimation of the thermal properties of existing building wall materials is a computationally expensive procedure. Its cost is highly proportional to the duration of measurements. To decrease the computational cost a methodology using a D-optimum criterion to select an optimal experiment duration is proposed. This criterion allows to accurately estimate the thermal properties of the wall using a reduced measurement plan. The methodology is applied to estimate the thermal conductivity of the three-layer wall of a historical building in France. Three different experiment sequences (one, three and seven days) and three spatial distributions of the thermal conductivity are investigated. Then using the optimal duration of observations the thermal conductivity is estimated using the hybrid optimization method. Results show a significant reduction of computational time; and reliable simulation of physical phenomena using the estimated values.

... According to the standard von Neumann stability analysis, the Du Fort-Frankel scheme is unconditionally stable [16,41]. Further details and examples of applications of this approach are presented in [15,16]. ...

... Our idea is to use Dufort-Frankel numerical scheme to numerically solve these equations. This innovative numerical model has benefits of being unconditionally stable, second-order accurate in time and having an explicit formulation (Gasparin et al., 2018a). This approach allows us retrieving values of sensitivity coefficients with higher accuracy at a low computational cost. ...

... Therefore, no sub-iterations are required to treat the non-linearities of the problem as it is the case when using Crank-Nicolson approach for instance. This feature may reduce significantly the CPU time of the algorithm [3,11,12]. When using a fully implicit approach, as for instance in [35], a special iterative approach (e.g. the Picard one) is required to treat the non-linearities at each time iteration. ...

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 intend to explain the discrepancies by analysing the importance of hysteresis effects as well as carrying out the sensitivity analysis on the input parameters as convective transfer coefficients. This article intends to investigate the accuracy and efficiency of the coupled solution by adding advective transfer of both heat and moisture in the physical model. The efficient Scharfetter-Gummel numerical scheme is proposed to solve the system of advection-diffusion equations, which has the advantages of being well-balanced and asymptotically preserving. Moreover, the scheme is particularly efficient in terms of accuracy and reduction of computational time when using large spatial discretization parameters. Several linear and nonlinear cases are studied to validate the method and highlight its specific features. At the end, an experimental benchmark from the literature is considered. The numerical results are compared with the experimental data for a purely diffusive model and also for the proposed model. The latter presents better agreement with the experimental data. The influence of the hysteresis effects on the moisture capacity is also studied, by adding a third differential equation.

... An improved explicit model study was conducted to overcome these shortcomings. Using the improved explicit analysis method DUFORT-FRANKEL, compared to the classical EULER implicit and explicit scheme, the calculation time was significantly reduced [13,14]. ...

Several building energy simulation programs have been developed to evaluate the indoor conditions and energy performance of buildings. As a fundamental component of heating, ventilating, and air conditioning loads, each building energy modeling tool calculates the heat and moisture exchange among the outdoor environment, building envelope, and indoor environments. This paper presents a simplified heat and moisture transfer model of the building envelope, and case studies for building performance obtained by different heat and moisture transfer models are conducted to investigate the contribution of the proposed steady-state moisture flux (SSMF) method. For the analysis, three representative humid locations in the United States are considered: Miami, Atlanta, and Chicago. The results show that the SSMF model effectively complements the latent heat transfer calculation in conduction transfer function (CTF) and effective moisture penetration depth (EMPD) models during the cooling season. In addition, it is found that the ceiling part of a building largely constitutes the latent heat generated by the SSMF model.

... The solution u ( x ⋆ , t ⋆ ) is obtained using the Dufort-Frankel numerical scheme. This numerical model is chosen due to its explicit formulation without loss of accuracy or reliability [13][14][15]. For the non-linear case, the solution is calculated with the following expression: ...

The \emph{in situ} estimation of the thermal properties of existing building wall materials is a computationally expensive procedure. Its cost is highly proportional to the duration of measurements. To decrease the computational cost a methodology using a D-optimum criterion to select an optimal experiment duration is proposed. This criterion allows to accurately estimate the thermal properties of the wall using a reduced measurement plan. The methodology is applied to estimate the thermal conductivity of the three-layer wall of a historical building in France. Three different experiment sequences (one, three and seven days) and three spatial distributions of the thermal conductivity are investigated. Then using the optimal duration of observations the thermal conductivity is estimated using the hybrid optimization method. Results show a significant reduction of computational time; and reliable simulation of physical phenomena using the estimated values.

... The advective-diffusive model is solved using the Dufort-Frankel scheme. Interested readers may consult (Gasparin et al. 2017(Gasparin et al. , 2018 for further details on this scheme. Before comparing models with each other and with experimental data, each model was validated by comparing the results with a reference solution, obtained with the MATLAB package Chebfun (Driscoll et al. 2014). ...

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.

The design of numerical tools to model the behavior of building materials is a challenging task. The crucial point is to save computational cost and maintain high accuracy of predictions. There are two main limitations on the time scale choice, which put an obstacle to solve the above issues. First one is the numerical restriction. A number of research is dedicated to overcome this limitation and it is shown that it can be relaxed with innovative numerical schemes. The second one is the physical restriction. It is imposed by properties of a material, phenomena itself and corresponding boundary conditions. This work is focused on the study of a methodology that enables to overcome the physical restriction on the time grid. So-called Average Reduced Model (ARM) is suggested. It is based on smoothing the time-dependent boundary conditions. Besides, the approximate solution is decomposed into average and fluctuating components. The primer is obtained by integrating the equations over time, whereas the latter is an user-defined empirical model. The methodology is investigated for both heat diffusion and coupled heat and mass transfer. It is demonstrated that the signal core of the boundary conditions is preserved and the physical restriction can be relaxed. The model proved to be reliable, accurate and efficient also in comparison with the experimental data of two years. The implementation of the scarce time-step of $1 \, \sf{h}$ is justified. It is shown, that by maintaining the tolerable error it is possible to cut computational effort up to almost four times in comparison with the complete model with the same time grid.

One possibility to improve the accuracy of building performance simulation (BPS) tools is via co-simulation techniques, where more accurate mathematical models representing particular and complex physical phenomena are employed through data exchanging between the BPS and a specialized software where those models are available. This article performs a deeper investigation of a recently proposed co-simulation technique that presents as novelty the employment of artificial intelligence as a strategy to reduce the computational burden generally required by co-simulations. Basically, the strategy, known as intelligentco-simulation, constructs new mathematical models through a learning procedure (training period) that is performed using the input–output data generated by a standard co-simulation, where the models of specialized software are employed. Once the learning phase is complete, the specialized software is disconnected from the BPS and the simulation goes on by using the synthesized models, requiring a much lower computational cost and with a low impact on the accuracy of the results. The synthesis of accurate-and-fast models is performed through machine learning techniques and the purpose of this paper is precisely a deep investigation of two techniques – recurrent neural networks and proper orthogonal decomposition reduction method, whose main goal is to reduce the training time period and to improve the accuracy. The case study focuses on a co-simulation between Domus and CFX programs, performing a two-dimensional diffusive heat transfer problem through a building envelope. The results show that for a standard co-simulation of 14 h, the intelligent co-simulation provided a reduction of 90% in the computer run time with accuracy error at the order of .

Within the framework of building energy assessment, this article proposes to use a derivative based sensitivity analysis of heat transfer models in a building envelope. Two, global and local, estimators are obtained at low computational cost, to evaluate the influence of the parameters on the model outputs. Ranking of these estimators values allows to reduce the number of model unknown parameters by excluding non-significant parameters. A comparison with variance and regression-based methods is carried out and the results highlight the satisfactory accuracy of the continuous-based approach. Moreover, for the carried investigations the approach is 100 times faster compared to the variance-based methods. A case study applies the method to a real-world building wall. The sensitivity of the thermal loads to local or global variations of the wall thermal properties is investigated. Additionally, a case study of wall with window is analyzed.

The purpose of the paper is to analyse air humidity in classrooms with stack ventilation. Based on experimental research, new humidity models in classrooms were developed. Experimental studies were conducted in five classrooms of the Faculty of Civil and Environmental Engineering at Bialystok University of Technology in north-eastern Poland (23°10′E, 53°08′N). The tests were performed for both unventilated classrooms and those ventilated by opening windows before and between classes. A linear increase in relative humidity during classes was reported in all classrooms. Stack ventilation provides a low, constant flow of air through the room. It should be noted that stack ventilation was equal to about 0.45 air changes per hour (ACH). The increase in relative humidity in rooms of this type depends primarily on the volume of the room and the number of people in the room. The developed models can be used in engineering practice for the design of automatic control systems in ventilation systems. The results indicate a need to control humidity in educational buildings. In this study, the approximate value of the moisture emitted by one class participant was also determined by comparing the moisture balance in the classroom and experimental data.

Simulation models for moisture transfer in building materials are highly incongruent with respect to the moisture potential used. Often the relatively better numerical efficiency and accuracy of a certain moisture potential is put forward as motivation. Various claims are made in that respect, but factual evidence is typically lacking. This paper aims at providing such support by assessing simulation efficiency and accuracy for capillary pressure, relative humidity and -log(-capillary pressure). To that goal, a suite of benchmark simulations are performed with those three potentials and performances are compared, based on deviations from reference solutions and on numbers of iterations required. The study initially reveals mixed results, showing no consistent advantages for either potential. Further analysis uncovers though that -log(-capillary pressure) suffers from a strongly nonlinear moisture capacity near saturation. This finally results in a decision in favour of capillary pressure and relative humidity, at least for general-purpose moisture transfer simulation.

This paper presents the Umidus program which has been developed to model coupled heat and moisture transfer within porous media, in order to analyze higrothemal performance of building elements when subjected to any kind of climate conditions. Both diffusion and capillary regimes are taken into account, that is the transfer of water in the vapor and liquid phases through the material can be analyzed. The model predicts moisture and temperature profiles within multi-layer walls and low-slope roofs for any time step and calculates heat and mass transfer. Umidus has been built in an OOP language to be a fast and precise easy-to-use software.

A whole building hygrothermal model has been developed on the basis of an existing detailed model for thermal simula- tion of buildings. The thermal model is a well-proven transient tool for hour-by-hour simulation of the thermal conditions in multizone buildings. The model has been expanded with new capabilities for transient simulation of indoor humidity condi- tions, taking into account the moisture buffer capacity of build- ing components and furnishings and the supply of humidity from indoor activities. Also integrated in the model are tran- sient calculations of the moisture conditions in the layers of all the external building envelope components. The advantage of the new model is that both the boundary conditions for the envelope and the capacity of building mate- rials to buffer the indoor humidity are considered in the same calculation. The model considers the latent heat effect asso- ciated with the absorption or evaporation of moisture, and it takes into account the way in which moisture in the building materials affects their thermal conductivity. The paper presents the principles for the model and some applications and calculation results. The model is validated against experimental data from a full-scale test cell. In the test cell, it is possible to control the release or withdrawal of humidity from the indoor space and measure the response in humidity of the air and the moisture content of building materials in the room. A sequence of exper- iments has been conducted using different interior materials to provide source data for the effect of moisture absorption and release. Examples of comparisons between simulated and measured data are presented.

Humidity of indoor air is an important factor influencing the air quality and energy consumption of buildings as well as durability
of building components. Indoor humidity depends on several factors, such as moisture sources, air change, sorption in materials
and possible condensation. Since all these phenomena are strongly dependent on each other, numerical predictions of indoor
humidity need to be integrated into combined heat and airflow simulation tools. The purpose of a recent international collaborative
project, IEA ECBCS Annex 41, has been to advance development in modelling the integral heat, air and moisture transfer processes
that take place in “whole buildings” by considering all relevant parts of its constituents. It is believed that full understanding
of these processes for the whole building is absolutely crucial for future energy optimization of buildings, as this cannot
take place without a coherent and complete description of all hygrothermal processes. This paper will illustrate some of the
modelling work that has taken place within the project and present some of the simulation tools used.

While the transfer equations for moisture and heat in building components are currently undergoing standardisation, atmospheric boundary conditions, conservative modelling and numerical efficiency are not addressed. In a first part, this paper adds a comprehensive description of those boundary conditions, emphasising wind-driven rain and vapour exchange, the main moisture supply and removal mechanism, respectively. In the second part the numerical implementation is tackled, with specific attention to the monotony of the spatial discretisation, and to the mass and energy conservation of the temporal discretisation. Both issues are illustrated with exemplary hygrothermal simulations. Numerical efficiency is treated in two follow-up papers.

The standardised Glaser method for calculation, prediction and evaluation of moisture performance is considered as rarely applicable. The present state of knowledge, analytical as well as experimental, concerning heat, air and moisture demands updating of standards. This paper presents five numerical benchmark cases for the quality assessment of simulation models for one-dimensional heat, air and moisture (HAM) transfer. In one case, the analytical solution is known and excellent agreement between several solutions from different universities and institutes is obtained. In the remaining four cases, consensus solutions have been found, with good agreement between different HAM models. The work presented here is an outcome of the EU-initiated project for standardisation of HAM calculation methods (HAMSTAD WP2).

Implicit schemes have been extensively used in building physics to compute the solution of moisture diffusion problems in porous materials for improving stability conditions. Nevertheless, these schemes require important sub-iterations when treating nonlinear problems. To overcome this disadvantage, this paper explores the use of improved explicit schemes, such as Dufort–Frankel, Crank–Nicolson and hyperbolization approaches. A first case study has been considered with the hypothesis of linear transfer. The Dufort–Frankel, Crank–Nicolson and hyperbolization schemes were compared to the classical Euler explicit scheme and to a reference solution. Results have shown that the hyperbolization scheme has a stability condition higher than the standard Courant–Friedrichs–Lewy condition. The error of this schemes depends on the parameter τ representing the hyperbolicity magnitude added into the equation. The Dufort–Frankel scheme has the advantages of being unconditionally stable and is preferable for nonlinear transfer, which is the three others cases studies. Results have shown the error is proportional to O(dt). A modified Crank–Nicolson scheme has been also studied in order to avoid sub-iterations to treat the nonlinearities at each time step. The main advantages of the Dufort–Frankel scheme are (i) to be twice faster than the Crank–Nicolson approach; (ii) to compute explicitly the solution at each time step; (iii) to be unconditionally stable and (iv) easier to parallelize on high-performance computer systems. Although the approach is unconditionally stable, the choice of the time discretization remains an important issue to accurately represent the physical phenomena.

Innovative and efficient ways to carry out numerical simulations are worth of investigation to reduce the computational complexity of building models and make it possible to solve complex problems. This paper presents a reduced order model, based on Proper Generalised Decomposition (PGD), to assess 2-dimensional heat and moisture transfer in walls. This model is associated with the multizone model Domus using an indirect coupling method. Both models are co-simulated to perform whole-building hygrothermal simulation, considering 2D transfer in walls. The whole-building model is first validated with data from the IEA Annex 41. Then, a case study is considered taking into account a 2-zones building with an intermediary shared wall modelled in 2 dimensions to illustrate the importance of the technique to analyse the hygrothermal behaviour of the wall. It has been highlighted that the whole model enables to perform more precisely analyses such as mould growth on the internal surface. In addition, important theoretical numerical savings (90%) are observed when compared to the large original model. However, the effective numerical savings are not so important (40%) due to the limitations of the co-simulation method.

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

Water infiltration is known to play an important part in the degradation process of construction materials. Over time, microscopic and macroscopic cracks progressively develop under the effects of mechanical loading and sorption/desorption cycles: their influence is to be accounted for in long-term hygrothermal performance assessments of the building envelope. The present work aims at showing the potential consequences of cracking on the heat and moisture transfer across building facades, in order to justify the need for the identification of damage to prevent durability and thermal issues. Specific simulation cases of insulated and non-insulated building facades were defined, and submitted to atmospheric boundary conditions for simulation times of one month. Some of the simulation geometries included previous measurements of crack patterns in concrete. The comparison of fractured and non-fractured building facades showed the effects of cracks on the moisture accumulation and thermal performance of these wall configurations, thus giving an estimate of what these effects might be in real conditions. A methodology is thus proposed for the identification of renovation needs, which may be applied for the purpose of durability assessments as well.

A standard for binary floating-point arithmetic is being proposed and there is a very real possibility that it will be adopted by many manufacturers and implemented on a wide range of computers. This development matters to all of us concerned with numerical software. One of the principal motivations for the standard is to distribute more evenly the burden of portability between hardware and software. At present, any program intended to be portable must be designed for a mythical computer that enjoys no capability not supported by every computer on which the program will be run. That mythical computer is so much grubbier than almost any real computer that a portable program will frequently be denigrated as "suboptimal" and then supplanted by another program supposedly "optimal" for the real computer in question but often inferior in critical respects like reliability. A standard --- almost any reasonable standard --- will surely improve the situation. A standard environment for numerical programs will promote fair comparisons and sharing of numerical codes, thereby lowering costs and prices. Furthermore, we have chosen repeatedly to enrich that environment in order that applications programs be simpler and more reliable. Thus will the onus of portability be shared among hardware manufacturers and software producers.

The International Building Physics Toolbox (IBPT) is a software library developed originally for heat, air and moisture system analysis in building physics. The toolbox is constructed as a modular structure of standard building elements, using the graphical programming language Simulink. To enable development of the toolbox, a common modelling platform is defined: a set of unique communication signals, material database and documentation protocol. The IBPT is an open source and available on the Internet. Any user can utilize, expand and develop the contents of the toolbox. This paper presents structure and essence of the library. Potential applications of the toolbox are illustrated through examples.

This paper describes the coupling of a model for heat and moisture transport in porous materials to a commercial Computational Fluid Dynamics (CFD) package. The combination of CFD and the material model makes it possible to assess the risk of moisture related damage in valuable objects for cases with large temperature or humidity gradients in the air. To couple both models the choice was made to integrate the porous material model into the CFD package. This requires the heat and moisture transport equations in the air and the porous material to be written down in function of the same transported variables. Validation with benchmark experiments proved the good functionality of the coupled model. A simulation study of a microclimate vitrine for paintings shows that phenomena observed in these vitrines are well predicted by the model and that data generated by the model provides additional insights in the physical mechanisms behind these phenomena.

Due to the lack of building simulation programs that can simulate in details the combined heat, vapor and liquid transfer in porous elements and the HVAC systems, a generic and flexible computational algorithm has been elaborated in order to integrate models for both HVAC systems and multizone building hygrothermal model.In the algorithm, models for the primary components (chiller, cooling tower, primary pumps and condensation pumps) and for the secondary components (cooling and dehumidifying coil, humidifier, fan and mixing box) have been presented. Those mathematical models have been integrated into a whole-building simulation model, allowing to perform whole-building hygrothermal simulation combined to HVAC systems and plant simulation.The building hygrothermal model is based on the simultaneous calculation of temperature and moisture content distributions within the porous envelope taking into account both vapor diffusion and capillary migration. First, results are presented for a multizone building to show the usability of the proposed algorithm. Then, the analysis of the influence of the adsorption and desorption phenomena on the calculation of thermal loads and energy consumption is presented, showing that the disregard of moisture may oversize the HVAC system in 13% and underestimate the cooling energy consumption in 4% for the case study defined in this paper.

Most building materials are porous, composed of solid matrix and pores. The time varying indoor and outdoor climatic conditions result heat, air and moisture (HAM) transfer across building enclosures. In this paper, a transient model that solves the coupled heat, air and moisture transfer through multilayered porous media is developed and benchmarked using internationally published analytical, numerical and experimental test cases. The good agreements obtained with the respective test cases suggest that the model can be used to assess the hygrothermal performance of building envelope components as well as to simulate the dynamic moisture absorption and release of moisture buffering materials.

In order to precisely predict ground heat transfer, room air temperature and humidity, a combined model has been developed and conceived to calculate both the coupled heat and moisture transfer in soil and floor and the psychrometrics condition of indoor air. The present methodology for the soil is based on the theory of Philip and De Vries, using variable thermophysical properties for different materials. The governing equations were discretized using the finite-volume method and a three-dimensional model for describing the physical phenomena of heat and mass transfer in unsaturated moist porous soils and floor. Additionally, a lumped transient approach for a building room and a finite-volume multi-layer model for the building envelope have been developed to integrate with the soil model. Results are presented in terms of temperature, humidity and heat flux at the interface between room air and the floor, showing the importance of the approach presented and the model robustness for long-term simulations with a high time step.

A mathematical formulation applied to a numerically robust solver is presented, showing that moisture content gradients can be used as driving forces for heat and moisture transport calculation through the interface between porous materials with different pore size distribution functions. For comparison purposes, several boundary conditions are tested—in order to gradually increase the discontinuity effects—and a detailed analysis is undertaken for the temperature and moisture content distributions and sensible and latent heat fluxes, when the discontinuity on the moisture content profile is taken or not into account.

A mathematical model applied to the analysis of hygrothermal behavior of buildings is described in this paper. A lumped capacitive and transient approach to model the room air temperature and humidity is described. To evaluate the whole building performance, it was employed a multi-layer model in finite differences for the building envelope and, in this code, we can include air infiltration, conduction loads, internal gains of people, lights and equipment and short and long-wave radiation. Different numerical methods are used to integrate the differential governing equations in the air domain, and results in terms of accuracy and computer run time are discussed in the paper. In the results section, we also show the influences of simulation time step on both room air temperature and humidity and temperature profiles within the building envelope.

It is shown that the Du Fort-Frankel method is unstable for the diffusion equation, if the usual central difference approximations are made to linear boundary conditions involving first order space derivatives. This is shown to be true even when the corresponding differential equation is stable. A modified boundary condition is presented which is proved to be stable provided the differential equation is stable.

Simulation program for the calculation of coupled heat, moisture, air, pollutant, and salt transport

- Bauklimatik-Dresden

Bauklimatik-Dresden (2011). Simulation program for the calculation
of coupled heat, moisture, air, pollutant, and salt transport.
Available at http://www.bauklimatik-dresden.de/delphin/index.
php?aLa=en.

An analysis of moisture accumulation in walls subjected to hot and humid climates

- D Burch

Burch D (1993). An analysis of moisture accumulation in walls
subjected to hot and humid climates. ASHRAE Transactions,
99(2): 1013-1022.

Analysis aequationum universalis seu adaequationes algebraicas resolvendas methodus generalis, et expedita, ex nova infinitarum serierum doctrina deducta ac demonstrata. Microfilm copy: University Microfilms

- J Raphson

J. Raphson. Analysis aequationum universalis seu adaequationes algebraicas resolvendas
methodus generalis, et expedita, ex nova infinitarum serierum doctrina deducta ac demonstrata. Microfilm copy: University Microfilms, Ann Arbor(MI), 1690. 10

The MathWorks Inc. Available at https

- Matlab

E-mail address: Denys.Dutykh@univ-smb

- France Chambéry
- Lama Cnrs
- Université Savoie Mont
- Blanc

Chambéry, France and LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus
Scientifique, F-73376 Le Bourget-du-Lac Cedex, France
E-mail address: Denys.Dutykh@univ-smb.fr
URL: http://www.denys-dutykh.com/
N. Mendes: Thermal Systems Laboratory, Mechanical Engineering Graduate Program, Pontifical Catholic University of Paraná, Rua Imaculada Conceição, 1155, CEP:
80215-901, Curitiba -Paraná, Brazil
E-mail address: Nathan.Mendes@pucpr.edu.br
URL: https://www.researchgate.net/profile/Nathan_Mendes/

E-mail address: suelengasparin@hotmail

- S Gasparin

S. Gasparin: Thermal Systems Laboratory, Mechanical Engineering Graduate Program, Pontifical Catholic University of Paraná, Rua Imaculada Conceição, 1155, CEP:
80215-901, Curitiba -Paraná, Brazil
E-mail address: suelengasparin@hotmail.com
URL: https://www.researchgate.net/profile/Suelen_Gasparin/