Evaluation of Thermal Effect of PCM Wallboards by Coupling Simplified Phase Change Model with Design Tool ()
1. Introduction
Phase Change Materials (PCM) have been widely studied in building applications [1] . They are able to influence the thermal comfort and the energy performance of buildings [2] -[4] . They lead to alternative and passive ways to store energy in buildings: a PCM material captures or releases heat (more specifically latent heat) when it undergoes a phase change from solid to liquid, liquid to gas or vice versa. The phase change occurs at a constant temperature (or over a limited range of temperature values), which is the temperature that corresponds to the phase transition temperature of the material. This latent heat storage is attractive since it provides a high energy storage density. Any material can potentially achieve a change phase. Concerning the industrial applications, one can distinguish inorganic PCMs (salthydrates for instance) and organic (parafines for instance). The order of magnitude of the melting latent heat of parafines is about 200 kJ/kg.
Many authors have studied the thermal effect of PCM in specific systems, for instance PCM integrated in a storage tank [5] , PCM integrated in ventilation systems [6] or in active walls [7] . This paper focuses on PCM wallboards used as internal building finishes. Such kinds of industrial products have recently been developed, but it is likely that they will be widely used in the near future [8] . This use of PCM has already been adopted by several architects as a promising solution to balancing the lack of thermal inertia in lightweight constructions. Professionals (architects and engineers) therefore need appropriate tools to evaluate the actual benefits of this material, especially in the early design phase of projects.
The integration of PCM wallboards into simulation tools has mainly been carried out in two kinds of building thermal model. Several studies deal with thermal models solved by the finite difference method [9] -[11] . However, as explained by Ibanez et al. [12] or Schranzhofer et al. [13] , it is also possible to couple a PCM model with a thermal model based on the transfer functions method.
As noted by Dutil et al. [14] , the enthalpy method and the heat capacity method are the two main approaches when modelling heat transfers with phase change. The enthalpy method has been used by Dauvergne [15] and Dauvergne and Palomo Del Barrio [16] to model plasterboards with micro-encapsulated PCM. Some authors have used the effective heat capacity method, assuming that this heat capacity is temperature dependent: Borreguero et al. [17] fitted the heat capacity model according to the experiment, and Chen et al. [18] or Ahmad et al. [19] chose a rectangular function. In the case of research by Athienitis et al. [20] , the approach is different again: the heat flux due to phase change is modelled by a triangular function, and is injected as a source term in the heat balance equation.
The aforementioned studies concentrated for the most part on the accuracy of the calculations, but did not necessarily target the application to the real needs of architects and engineers. For instance, whether or not PCM wallboards are suitable to ensure a satisfying level of comfort in summer is a question that has to be considered at the early design phase, because the answer can strongly influence the design of the building. Indeed, professionals need simulation tools which are accurate, but quick and easy to use, and able to help them during the whole building design process, especially in the early phase.
This paper proposes a consistent response to this professional requirement. The basic principle consists in coupling a straightforward model of PCM wallboards with an existing thermal model of buildings (implemented in professional software) reduced by modal analysis, the aim being to ensure both fairly good accuracy and short computation times. First, both models are described, then the coupling methodology is explained. The simulation tool is then applied to two real case studies to illustrate the ability of the methodology and the resulting coupled models simulation tool to evaluate easily the performance of PCM wallboards: a project for a timber frame family house, and an existing lightweight office building.
2. Modelling
Adding the possibility of taking PCM wallboards into account in thermal simulation tools for buildings requires a specific methodological and numerical treatment. One of the main reasons is that integrating the phase change phenomenon into the heat balance equation necessarily leads to a non-linear differential equation. It is generally assumed that heat conduction occurs in materials with isotropic, homogeneous and constant properties. With the phase change phenomenon, the last assumption must be revised as the thermo-physical properties of the PCM are temperature-dependent.
This section describes the existing building thermal design tools used in this paper (COMFIE), then the PCM wallboards model adopted, and finally the coupling of these two models to obtain a PCM wallboard-building thermal design tool.
2.1. Building Thermal Model and Calculation Algorithm
This section gives a brief description of the model and the whole calculation algorithm which has been developed and applied in the thermal simulation tool for multi-zone buildings. This simulation tool is called COMFIE and is linked to the Pleiades professional interface, widely used in France by architects and engineers, which gives fast and accurate results on heating and cooling loads and thermal comfort [21] . It focuses on the preliminary design of buildings, and uses a standard (or user-defined) library containing different physical objects (e.g. materials, walls or windows).
In this model, a wall is made of different material layers, with constant and homogeneous physical properties (e.g. conductivity or heat capacity). Surface heat transfer coefficients (convection and long wave radiation) are also assumed to be time independent. Convection heat transfer coefficients depend on the inclination of the wall and, for external surfaces, a mean building wind exposure. Long wave radiation heat transfer coefficients depend on surface emissivity.
This model is a finite volume model. It is especially useful for building designers, who have to perform simulations within a short computation time. For this reason, heat transfer in walls is assumed to be one-dimensional. The building is decomposed into small volumes, or “nodes”, assumed to be at a uniform temperature. Actually, the smaller the nodes are, the more realistic is this assumption of uniform temperature. Nevertheless, a limited number of nodes are imposed. The chosen compromise consists in placing this limited number of nodes so that temperature uniformity is maximal in each one. This strategy leads to a simulation tool which is able to perform year-long simulations within a computation time of between 10 s and few minutes, depending on the number of thermal zones and the CPU [21] .
For example, Figure 1 illustrates the discretization procedure on a wall composed of an 11.5 cm concrete wall and a 1.5 cm plasterboard placed on its internal surface. The wall is divided into n nodes, whose thicknesses follow a geometrical sequence with a common ratio r. The internal layer (finite volume) is the thinnest because this side of the wall has more influence on the indoor temperature than the external side of the wall.
The discretization method creates thermal building models with varying degrees of accuracy, depending on the values of n and r. Several values of n and r have been tested, for various types of walls and simple building configurations. Simulation results, performed on real case studies of buildings with simple geometries, showed that three nodes are sufficient to obtain discrepancies of less than 0.1˚C on zone temperatures, when compared to results obtained with twenty nodes reference simulations; furthermore, the effect of the r value on these discrepancies was very small (less than 10% of the aforementioned discrepancy). Thus three was finally chosen as a default value for both n and r.
Because the numbers and positions of nodes in a wall is a consequence of the n and r values, (see Figure 1, where n = 3 and r = 3), the model layers (bounded by dotted lines in Figure 1) do not necessarily match the material layers. As a result, the thermo-physical properties of the materials that constitute a layer are combined
Figure 1. Example of the spatial discretization of a multi-layer wall (composed of an 11.5 cm concrete wall and a 1.5 cm plasterboard placed on its internal surface). Number of nodes: n = 3; common ratio of the geometrical sequence of the node thicknesses: r = 3.
(thermal masses and thermal resistances are added together). This discretization methodology is applicable to all types of wall. The complete methodology is described by Peuportier [21] .
An energy balance is written for each node. The building is divided into thermal zones, and for each zone a matrix equation system is built:
(1)
in which T is the discrete temperature field vector, U is a vector made of the driving forces (external temperature, solar irradiation, internal heat source), Y is the output vector, C is the diagonal matrix of the capacities of the discrete elements of the node grid, A is the matrix containing all exchange terms between the nodes, E is the matrix containing the exchange terms between the nodes and the driving forces, J is the matrix relating the outputs to the nodes , and G is the matrix relating the outputs to the driving forces. A particular output within the vector Y is the zone temperature Tz of a given thermal zone, which is an equivalent temperature taking into account the air temperature and the surface temperature of the different walls [21] .
The temperature vector T is divided into a steady and a dynamic term, and a diagonalization step transforms the system into the equivalent in the eigenbase, composed of the eigenvectors:
(2)
in which X is the modal state vector, F is the diagonal matrix of the eigenvalues of the system.
A reduction step selects the preponderant eigenvectors: only the largest time constants are considered, and multiple eigenvalues are eliminated (this happens if a zone has several identical walls). Only a few eigenvectors are kept, and therefore the order of the model is reduced.
The modal models of the different zones are combined, in order to build the global reduced matrix system of the multi-zone building:
(3)
The simulation algorithm includes a controller which calculates the heating power in each zone, according to the temporal scenario concerning the set point temperatures. The time step is typically half an hour, with the possibility of reducing it to a few minutes.
The complete modelling and simulation procedure is described by Peuportier [21] . The reliability of the model and of the resulting simulation tool has been demonstrated in several validation procedures [22] [23] .
The aim of this work is to integrate a PCM wallboard model into the existing building thermal design tool described briefly in this section. As pointed out above, in order to ensure short computation times, this building thermal design tool considers restrictive assumptions, for example constant and homogeneous values for thermo-physical properties. For this reason, the PCM wallboard model proposed in the next sub-section has been developed to be associated with this existing building thermal design tool, but without producing any major modifications in this software solver. Thus, an enthalpy method has been chosen and the PCM phase change thermal effect is described as an external heat source term for the heat balance equations that takes place in the building thermal design tool.
2.2. PCM Wallboard Model
In this sub-section, a simple expression of the PCM wallboard enthalpy is proposed, assuming the usual restrictions retained in the literature: one-dimensional heat transfer, isotropic and homogeneous material properties and negligible natural convection of the PCM liquid phase [11] [12] , identical thermo-physical properties for PCM liquid and solid phases [9] [20] , no consideration of a possible hysteresis and sub-cooling phenomena for the PCM [10] [17] [18] . Note that the last assumption is in accordance with the level of precision of the usual PCM wallboard manufacturers’ commercial data, as used by architects and engineers.
The PCM is associated with a structuring material to enhance its mechanical properties (e.g. plaster). The resulting product is considered to be a homogeneous medium, namely the PCM wallboard, whose thermal conductivity, mass density and heat capacity can be written as follows:
(4)
where e is the ratio between the mass of PCM and the total mass of the PCM wallboard. In the previous equation, the subscripts “m”, “pcm”, and “w” refer respectively to “structuring material”, “Phase Change Material (PCM)” and “PCM wallboard”.
The main characteristics of the PCM are its latent heat of fusion L, and its enthalpy variation on its melting temperature range (PCM temperature range over which the phase change occurs); this range may reduce to a single temperature if the PCM is pure or eutectic. These characteristics are commonly obtained by using a DSC (Differential Scanning Calorimeter). Given these experimental data, the PCM wallboard mass enthalpy is then written as follows:
(5)
where f is the mass fraction of the melted PCM over the total PCM mass. As already mentioned above, for impure materials, the phase change takes place over a given temperature range; the melted PCM mass fraction versus PCM temperature curves usually have the same shape as the one shown in Figure 2. As suggested by Ahmad et al. [19] , for instance, these curves can be approximated by a linear function (Figure 2) and expressed as follows:
(6)
where Tlow and Tup are the limit temperatures of the PCM phase change temperature range.
2.3. Coupling the PCM Wallboard Model and the Building Thermal Model
A simple and efficient coupling methodology is used to integrate a non-linearity (the PCM wallboard enthalpy model described above) into the existing thermal simulation tool (the building thermal model described above) without modifying its intrinsic calculation algorithm. The thermal effect of the PCM wallboard enthalpy modification (due to the PCM phase change) is considered as an external heat source term in the heat balance equation. This integration strategy, using a heat source term, has already been considered by Athienitis et al. [20] . Note that a method based on the enthalpy formulation is well adapted to this context, compared to the variable
Figure 2. Mass fraction of melted PCM (paraffin) versus temperature (experimental data: square dots; linear approximation: solid line).
heat capacity formulation. In the present paper, the modelling originality lies in the fact that this heat source term is not calculated by the building thermal model, but is calculated independently and then injected into the building thermal model.
As pointed out earlier, the building thermal model is one-dimensional. Thus, the heat balance equation in a wall is written as follows:
(7)
where Δ is the Laplace operator and H is the enthalpy of the wall. If this wall is made of PCM wallboards, the enthalpy expression is that proposed in Equations (5) and (6). In this case, the heat balance is:
(8)
where q , the rate of energy generation per unit volume, expresses the heat stored or released due to PCM melting or solidification:
(9)
Thus, the heat transfer generated by the PCM phase change can be considered as a heat source in the heat balance equation. Here, the resolution of this equation is performed by coupling a PCM model that calculates the rate of energy generation 
given the local temperature T in the PCM wallboard, with the existing building thermal model which calculates the temperature T according to this local rate of energy
; this iterative procedure is illustrated in Figure 3. According to this algorithm, the interaction between the PCM phase change and the building thermal evolution is dynamically taken into account: at each time step, given the discrete temperature field vector in the building at the beginning of this time step Tn, an iterative procedure that couples the PCM model and the Building model leads to a convergence at the temperature vector at the end of this time step Tn+1. This algorithm has been implemented using an object oriented approach (in a Delphi language environment): both PCM and building thermal models are objects that exchange variables (vectors 
and Tn) during the iteration procedure. For a traditional building thermal simulation, a half hour time step is often sufficient to ensure the convergence of this iterative procedure.
3. Numerical Comparisons between the PCM Wallboard-Building Coupled Model and a Reference Model
The aim of this section is to verify the reliability of the PCM wallboard-building model resulting from the coupling presented in Figure 3. Numerical results from the PCM wallboard-building coupled model are compared with numerical results from a reference model which has recently been experimentally validated by Dauvergne [15] and Dauvergne and Palomo Del Barrio [16] . This reference model assumes unidirectional heat transfers, constant thermo-physical properties, no convection in the liquid PCM and no hysteresis or sub-cooling effect.
A simple configuration is considered: a wall made of 15 cm gypsum (the matrix), in which PCM (paraffin) is homogeneously mixed. The PCM mass fraction is ε = 0.15 and its latent heat of fusion is L = 112 kJ/kg. The phase change temperature range is defined from Tlow = 17˚C to Tup = 28˚C (see Figure 2). Other material thermo-physical parameters are given in Table 1. Newton boundary conditions are considered on each side of the wall. The air temperature on the interior side is 24˚C, and the external side is subjected to climatic driving forces corresponding to a meteorological station located in the south of France (city of Agen). The simulation time step is 1 hour, and the simulation is run for a whole year.
In order to compare numerical results from these two models, a mean wall temperature is considered. Note that the interior surface wall temperature would probably have been more appropriate for making this comparison, because this surface exchanges both with the thermal zone and with the PCM wallboard; however, as pointed out in Section 2.1 (see Figure 1), this interior wall surface temperature is not calculated directly by the building simulation tool. As a consequence, instead of extrapolating node temperatures to obtain an internal wall surface temperature, the spatial averaged wall temperature has been considered as a reasonable quantity to compare those model outputs.
Table 1. Thermo-physical properties of materials used to compare the coupled PCM-Building model with the reference model developed by Dauvergne [15] .
Figure 3. PCM model—Building model coupling procedure.
Figure 4 deals with the influence of the number of nodes retained for the simulations with the PCM wallboard-building coupled model on the reliability of the corresponding simulated results. Results obtained with three nodes appear to be quite close to those obtained with the reference model from Dauvergne and Palomo Del Barrio [16] . This number of nodes requires a compromise between accuracy and computation time, in accordance with building design objectives and with the level of precision of the usual PCM wallboard manufacturer commercial data: the mean discrepancy between the 3-node model and the reference model is 0.08˚C for the whole period (1 year), with a maximum discrepancy of 0.48˚C. This difference may be in part a consequence of the approximation, using a linear function, of the PCM enthalpy-temperature curve (see Figure 2). Simulations for the simplified model with fifteen nodes have also been performed, and the temperature results in this case are quite close to those obtained with the 3-node model [24] . As a consequence, the PCM wallboard-building coupled model, which contains the discretization procedure described in Section 2.1, is considered to be a reliable model.
One objective of this paper is to point out that using a building-PCM wallboard coupled design tool of this kind, architects and engineers will be able to evaluate the thermal performance (or non-performance) of PCM wallboards (through the professional interface Pleiades) in the early design phase of building projects. Such an evaluation is performed next using two indicators: a PCM Utilization Factor UF defined specifically in this paper, and degrees∙hours of discomfort DH. Consideration is mainly given to thermal comfort in summer, but some focuses on winter heating load are also briefly considered. With this objective in mind, the PCM wallboard-building simulation tool is applied in the next section to two real case studies to illustrate this evaluation procedure for using PCM wallboard in buildings.
4. PCM Wallboards Performance Indicators
To evaluate quantitatively the influence of the PCM on thermal comfort, two indicators are used. The first, commonly used to evaluate the summer comfort in buildings, is the degrees hours of discomfort DH: this is the sum, over a given simulation period (July and August in this paper) and for each hour h in this period, of the hourly average value of the over-temperature in a considered building zone, relative to a reference temperature Tref (26˚C in our case).
(10)
The second indicator is the PCM utilization factor UF. It is proposed and defined in this paper in order to
Figure 4. Mean wall temperature versus time. Dashed line: PCM wallboardbuilding coupled model with one node, dotted line: PCM wallboard-building coupled model with three nodes, body line: reference model [15] .
evaluate the performance of the melting/solidification process. It is the ratio between the energy stored and released by the PCM over a given simulation period (July and August in this paper), and the maximum energy which would have been stored and released by the PCM if it had completely melted and solidified each day during this period:
(11)
In the preceding equation 
is the number of days over the considered period. This indicator has been proposed to help to assess the efficiency of the PCM wallboards and their interaction with the building envelope: a unit value for the UF indicator means that the PCM melts and solidifies completely every day (24 hours), which gives a maximum thermal storage performance. If UF is lower than unity, either the sizing (for instance the area covered with PCM wallboards, or the number of layers) or the management of the building (windows opening during the night) is not appropriate.
5. Using PCM Wallboard-Building Coupled Model on Two Case Studies
Two different and complementary case studies are considered: a timber frame family house project and an existing office building. The PCM wallboard characteristics which are considered in both cases are given in Table 2. In order not to overload the present paper, the descriptions of the building envelopes are concise and focus on the main significant parameters.
5.1. Case Studies: A Short Description
The first case study is a timber frame family house project (see Figures 5 and 6). This is a two-storey building with a useful area of 84 m2, and a south oriented main facade. The external walls and the first floor are timber framed. The ground floor, made of concrete, is the main contributor to the thermal mass of the house. The external envelope is well insulated: 23 cm insulation on the external walls, 8 cm on the floor, and 35 cm on the ceiling (all insulation layers are placed in the external side of the walls). The thermal bridges are treated, and the double glazed windows are filled with argon and coated with a low emissivity surface.
This timber frame house is occupied by a single family (4 persons). The heating set point temperature is constant at 20˚C. The house is empty from 9 am to 6 pm from Monday to Friday. The air change rate acr (given by the ratio between the ventilation flow rate and the volume of the zone) during the day is fixed at 0.5 h−1 and the nominal efficiency of the heat recovery ventilation system is 80%. During summer, the air change rate is supposed to be 6 h−1 at night (when family members are sleeping), achieved by appropriately opening the windows; furthermore, it is assumed that during the day the occupants use the external manual roller blinds, that 75% of the window areas are covered by these blinds, and that these blinds are completely opaque to solar radiation
Table 2. PCM wallboard thermo-physical properties considered for both case studies.