NUMERICAL STUDY OF HEAT TRANSFER FROM A WALL INCORPORATING A PHASE CHANGE MATERIAL

A numerical study of the thermal behavior of walls made up of construction materials used in Algeria and walls containing a phase change materials is presented. The model, based on the enthalpy formulation, is described by an equation of heat transfer. This equation is solved by an implicit method of finite differences and algorithm of Thomas. We analyzed the influence of the wall's thickness and its composition on the evolution during the time of the temperature of the inside face of thewall.


Introduction
The building sector in Algeria is one of the most dynamic sectors, result of a high rate of growth of the population and urbanization.The growth of the population in Algeria is remarkable, increasing from 18,8 million inhabitants in 1980 to 34,4 million in 2008.Consequently, the request for housing increases considerably and is making construction one of the main engines driving the growth of the country.
In Algeria, the building sector is the largest energy consumer among the economic sectors, with 41% from national energy and 21% of the CO2 emission [1].Most of this energy comes from heating and air-conditioning systems.It thus proves necessary to reduce the share of energy used in the building sector and thus the environmental impact of this sector by promoting concept of buildings with low energy intake.
The thermal inertia of the building plays a significant role in the improvement of thermal comfort and the reduction of energy consumptions in the building sector [2].The techniques based on thermal inertia contribute to improve thermal comfort and to allow energy savings.
Also, the integration of phase change materials (PCM) in building was the purpose of many researchers who analyzed their impacts on the energy efficiency of the envelope of a building.Maha et al. [3,4] carried out tests by incorporating PCM coupled with the use of a super insulation material VIP (Vacuum Insulation Panel) in walls made up of PVC.The concept of coupling PCM with a super insulation material proves to be a promising solution for light envelopes of low thickness having a good insulation and a significant inertia.The determination, with the software CODYMUR, of the optimal thickness of a plasterboard in which a PCM has been added, showed that a one cm thickness can double the thermal inertia of this plate [5].
Castellón et al. [6] proved the feasibility of the use of the micro PCM encapsulated (Micronal BASF) in sandwich panels to increase their thermal inertia and to reduce the energy demand of the buildings.An experimental study on two prototypes, on scale 1, of exchangers of heat PCM-air intended for natural ventilation in buildings showed that this type of exchanger can ensure the natural cooling of a house with a low thermal conductivity of the PCM [7].
This work deals with a numerical study of the thermal behavior of walls built with construction materials used in Algeria and in which PCM were added.The model, based on the enthalpy formulation, is described by an equation of heat transfer which we solved by an implicit method of finite differences and the algorithm of Thomas.We analyzed the influence of wall thickness and its composition as well as the effect of PCM materials on the evolution during the time of the temperature of the wall inner face.The results obtained from the model were confronted with the results of a similar study of Maha Ahmed et al. [3,4].Confrontation shows a good agreement.

Physical model and mathematical formulation 2. 1 Physical model
Let us consider a vertical wall with a thickness e in which a phase change material (PCM) is built-in.This wall is between the inside environment characterized by a temperature fixed at 23 °C, and the external environment which has sinusoidally varying temperature with which it heat transfers by convection (figure 1).For the homogeneous materials as the plaster, the concrete, the BTS and the stone, the drifted partial of the enthalpy is given by [8]: For a wall in plaster containing a PCM material, the equation ( 1) is written [8]: For the considered mixture (plaster 70%, GR 30%), the specific heat of this mixture varies according to the temperature [3,4], as it is reported on the figure 2.

Initial conditions and
• A coefficient of heat exchange between the outer wall and the atmosphere [9]: he= 17 [W/m²K] at x = e, Fourier-type boundaries conditions: • A coefficient of heat exchange between the interior wall and the interior air [9]: hi= 9 [W/m²K]

Numerical Methodology
In order to solve the nonlinear differential equation which governs heat transfer through a wall integrating an PCM material, the method of finite differences according to an implicit scheme was established.Discretiszation of the equation (1) leads to the following expression : 1st International Conference on Numerical Physics The equation (7) written for each point 1<i<N results in a system with N simultaneous equations and N unknown factors.We obtained a system of tridiagonal algebraic equations, which we solved with Thomas Algorithm.

Composition of the walls
We considered walls in: plaster, concrete, stabilized earth brick Brick (BTS), stone, as well as walls made up of a mixture plasters /PCM Tables 1 and 2 represent thermal conductivity values (λ), density (ρ) and the specific heat (C) of various materials studied in this article.

Results and discussion
We analyzed the influence of these walls thickness by considering a thickness ranging between 1 cm and 8 cm. the calculations were carried out for an interior temperature Ti=23°C and an outside air temperature t) 10 (7.27 sin 8

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The convection heat transfer coefficients between the wall and ambient air (he) and between the wall and the interior air (hi) are respectively equal to 17 W/m 2 .K and 9 W/m 2 .K. From Figure 3, we see the effect of PCM on the thermal stability of wall's inner face made up of plaster and PCM.The temperature of the wall's inner face, thickness equal to 1 cm, varies between 18,8 and 26,5 °C with a time lag of 3,5 h.For a thickness equal to 8 cm, the temperature is almost constant during the day; with a very small variation with time (lower than 1°C).Let us note that the temperature of the inner face of the wall is close to 23 °C, this value contributes to improve thermal comfort of a habitat whose walls would be submitted to the same climatic conditions as the wall retained in this study.We note that the variation, during the time, of the wall inner face temperature decreases with the increase its thickness and that for a thickness at least equal to 3 cm, the time lag is very high.Thus, for a 3 cm thickness, time lag is estimated at 8:00 and for a thickness equal to 5 cm it is 12 h.Figure 4 illustrates the evolution during the time of the temperature of the inner face of a plaster wall according to the thickness (e).For a thickness equal to 1 cm, the temperature is set between 19 and 28 °C with amplitude, defined by the difference between the maximum and minimal temperatures of the day, equal to 9 °C.For a 8 cm thickness, it oscillates during time between a minimal value of 21 and one maximum value of 25,5 °C, with an amplitude of 4,5 °C.We note that just like the preceding case, the amplitude of the temperature of interior surface decreases with the increase of its thickness.Figure 5 illustrates the evolution during a day of the temperature of the inner face of a concrete wall according to its thickness.For a thickness equal to 1 cm, the temperature varies from 18,8 to the 28,8 °C with an amplitude of 9 °C; it lies between 20 and 27 °C for a thickness equal to 8 cm.It follows a 2,2 hours time lag between the outside temperature and that of the inner face wall.For a wall made up of stabilized earth brick of a thickness equal to 1 cm, the variation in the temperature of its inner face lies between 18,8 and 28,8 °C with an amplitude of 10 °C (figure 6).The amplitude is reduced to 6°C for a thickness equal to 8 cm and this temperature varies between 20,5 and 26,5 °C.The temperature of the inner face of a stone wall, varies from 18,8 to 28,5 °C with an amplitude of 10 °C for a thickness equal to 1 cm and oscillates between a minimal value of 22,2 and one maximum value of 26,8 °C, with an amplitude of 4,6 °C for a thickness equal to 8 cm (figure 7).

Conclusion
By using the model based on the enthalpy formulation, we proceeded to a numerical study of the thermal behavior of a wall made up of construction materials and PCM.We showed that a wall made up of plaster and MPC of thickness equal to 8 cm can stabilize the temperature of its inner face during 24 hours, with a very low amplitude (lower than 1°C).For other construction materials, it varies between 20 and 26 °C, with a time lag compared to the outside air temperature with a maximum period equal to 3 a.m. and of the amplitudes of temperature varying between 5

Fig. 1 .
Fig.1.Diagram of the physical model 2.2 mathematical formulation 2.2.1 Assumptions -The heat transfer is unidirectional; -The thermo-physical properties of homogeneous materials are constant.-The thermo-physical properties of mixture plaster /PCM are variable.Considering the formulated assumptions above, the equation of transfer verifies the following expression [8]: ) 1 ( x T t h outer temperature varying sinusoidally according to the relation:

Fig. 6 .
Fig. 6.Indoor surface temperature variation for walls in BTS

Table 1 .
Thermo physical properties of studied materials