A new model for simulating heat, air and moisture transport in porous building materials
Introduction
Moisture is a key factor on durability and performance of buildings. An excessive level compromises the construction quality, impacts the indoor air quality and the thermal comfort, as well as the building energy efficiency [1]. As a consequence, a number of models for predicting the impact of moisture on building energy efficiency are proposed in the literature. A primary overview may be consulted in [2]. Among the physical phenomena, the air transfer through the porous building media has a crucial impact on the amount of moisture. Diverse studies enhance these effects using both experimental and numerical results [3], [4], [5].
Several numerical models are proposed in the literature for the prediction of physical phenomena of coupled heat, air and moisture transport through porous building materials. Their physical representations are based on the mass conservation laws for the dry air, vapor and liquid water, as well as the energy conservation law as detailed in the early work of Luikov [6]. As the continuity of his work, the numerical models proposed in the literature can be divided into two main groups. The first group considers the three evolution differential equations to compute the temperature, the mass content and the air pressure in the porous media. In [7], a model is proposed for the simulation of transfer through hollow porous blocks. It is based on an implicit finite–difference numerical scheme. More recently, in [8], [9], the commercial COMSOL™ software is used to propose a numerical model for such physical problems. As mentioned by the authors, the scheme is based on an explicit in time finite–element approach. The main drawback of these numerical models is their computational cost. The implicit approach requires costly sub-iterations at each time step to handle severe nonlinearities of the problem. The explicit scheme requires very fine time steps to satisfy the so-called Courant-Friedrichs-Lewy (CFL) stability conditions. Indeed, the characteristic time of air transfer is very small compared to the ones for heat and mass transfer.
To handle this computational issue, the second group of models does not consider the transient phenomena for the air transport through the porous matrix. In other words, the evolution differential equation for air transfer is transformed into a simple steady boundary value problem which is solved at prescribed time instances. It enables somehow to relax the stability conditions and, therefore, to save computational efforts comparing to the models from the first group. Some examples can be found in [10] or [11] for one-dimensional transfer. The former references uses an explicit scheme provided by the commercial COMSOL™ software. The latter is based on an implicit scheme based on the generic ODE–solver from the SUN-DIALS solver package [12]. More recently, a numerical model is proposed in [13], [14] for the simulation of two–dimensional transfer in building structures. It is also based on an explicit scheme from COMSOL™ software. The assumption of neglecting the transient term is justified by the low velocities occurring through the porous matrix. As mentioned by authors, the numerical predictions are reliable only in the context of simulations with standard hourly climate driven boundary conditions. It should be noted that this condition is not often satisfied, particularly for the air pressure surrounding building walls as detailed in [15]. Moreover, even if the restriction on the time discretisation is relaxed, the numerical models are based on standard approaches and still have a high computational cost.
Therefore, the goal is to propose an efficient numerical model considering the three transient equations to improve the reliability of its predictions. It requires to be accurate with a reduced computational cost. To address this issue, this paper proposes to use the Scharfetter–Gummel scheme combined with a two–step Runge–Kutta approach. The Scharfetter–Gummel numerical scheme was proposed in 1969 in [16] with very recent theoretical results in [17], [18]. In the context of building porous media, it is successfully applied in [19] to water transport and then in [20] to combined heat and moisture transfer. The contributions of the present paper is two fold. First, the model proposed in [19] is extended by including the air transport equation. Then, the extension of the Scharfetter–Gummel approach to a system of three coupled advection–diffusion equations is proposed. The combination with a two-step Runge–Kutta scheme is investigated in order to relax the stability restrictions on the choice of the time discretisation.
The paper is organized as follows. Section 2 presents the demonstration of the mathematical model to describe the physical phenomena and its dimensionless formulation. Then, Section 3 presents the numerical method to solve the system of three differential advection–diffusion equations. In Section 4, one case studie is considered to validate the numerical model. The purpose is to quantify the accuracy and efficiency in terms of computational time and relaxation of the stability condition. For each case, a reference solution is proposed, computed by a numerical pseudo–spectral approach. In the last Section, the reliability of the numerical predictions is evaluated by comparing them to experimental observations. A wall composed of two layers of wood fiberboard is submitted to a controlled environment on the inner side and to climate driven boundary condition on the outer side. Three points of measurements of temperature and vapor pressure within the wall are used for comparison purposes.
Section snippets
Formulation of the physical phenomena
First, the mathematical model including the governing equations to describe the physical phenomena is presented.
Elaborating an efficient numerical model
Since the physical phenomena have been described, the second part in the elaboration of a numerical model consists in detailing the numerical method to solve the physical problem. For this, it is of major importance to define the strategy of building a numerical model that reduces the computational effort and maximize the accuracy of the solution. First, since we have a nonlinear problem, explicit scheme is preferred than implicit approaches to avoid costly subiterations to treat the
Validation of the numerical model
After presenting the numerical model, a case study with nonlinear coefficients and Robin–type boundary conditions is considered to validate its implementation. The reference solution is computed using a numerical pseudo–spectral approach obtained with the Matlab™ open source toolbox Chebfun [36].
Comparison of the numerical predictions with experimental data
Previous section aimed at validating the results of the numerical model with several reference solutions. The proposed numerical model showed a high accuracy with a relaxed stability condition compared to standard approaches. In other words, we verified that the differences due to numerical approximations of the mathematical model by the numerical one are minors. The numerical predictions of the numerical model are now compared with experimental data. The issue is to evaluate the physical
Conclusion
Within the context of predicting the impact of air transport on the combined heat and moisture transfer in porous building materials, this article proposes a new model with an efficient numerical scheme. In contrast to earlier proposed models in literature, the approach takes into account the transient effects of air convection without being constrained by the high numerical cost induced by the very small characteristic times of the physical phenomena. After describing in details the
Conflict of interest
We wish to confirm that there are no known conflicts of interest associated with this publication.
Acknowledgments
The authors acknowledge the Junior Chair Research program “Building performance assessment, evaluation and enhancement” from the University of Savoie Mont Blanc in collaboration with The French Atomic and Alternative Energy Center (CEA) and Scientific and Technical Center for Buildings (CSTB). The authors thanks the grants from the Ministry of Education and Science of the Republic of Kazakhstan and the visiting professor grants from the University of Savoie Mont Blanc. The authors also
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