**3. PCM wall numerical characterization**

A numerical investigation by using specific FORTRAN program was achieved to solve the energy and the exergy mathematic relations to evaluate the PCM wall performances by determining the melting phase proprieties (velocity, isotherm, melting front evolution, etc.).

The proposed numeric investigations describe the heat transfer phenomena inside the PCM wall and evaluate its thermal behavior and effects on test cell ambiance. It allows also the appraisal of the energy and exergy stored during the charging process and to evaluate the thermal characteristics. The considered assumptions are the flow is two-dimensional and laminar, the expansion of the PCM is negligible and the phase change is isothermal. The PCM wall is subjected to an imposed temperature superior to the melting temperature of PCM-27 (**Figure 5**).

The other walls are maintained adiabatic (**Figure 5**). Considering the mentioned assumptions:

• The continuity equation is given by [1, 23]:

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial y} = \mathbf{0} \tag{1}$$

• The quantity of movement relations are given by [1, 23]:

$$
\rho\_l \frac{\partial u}{\partial t} + \rho\_l \left(\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial u}{\partial \mathbf{y}}\right) = -\frac{\partial p}{\partial \mathbf{x}} + \mu\_l \left(\frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial \mathbf{y}^2}\right) + Bu \tag{2}
$$

$$
\rho\_l \frac{\partial v}{\partial t} + \rho\_l \left(\frac{\partial uv}{\partial \mathbf{x}} + \frac{\partial vv}{\partial \mathbf{y}}\right) = -\frac{\partial p}{\partial \mathbf{y}} + \rho\_l \mathbf{g} + \mu\_l \left(\frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial \mathbf{y}^2}\right) + Bv \tag{3}
$$

In the Boussinesq approximation, the source terms *Bu* and *Bv* that appear in the momentum Eqs. (2) and (3) are used to account for this buoyancy force when the PCM is solid. The technique used to cancel the velocity introduces a Darcy term [1].

**Figure 5.** *Boundary conditions through the PCM wall.*

$$B = -C\frac{\left(\mathbf{1} - H^2\right)}{\left(H^3 + b\right)}\tag{4}$$

• In order to account for the phase change process happening when the PCM is melting, the energy balance was applied at the interface as follows

*Energy Storage in PCM Wall Used in Buildings' Application: Opportunity and Perspective*

*Ts* � *kl*∇ !

temperatures of liquid and solid at the interface is given by [1]:

• Since the phase change of pure substances occur at a single temperature, the

where the subscripts s and l stand for the solid and liquid phase, *L* is the latent heat (enthalpy) of fusion, and *X* is the position of the melting interface.

• The Plexiglas cavity was partially filled with PCM-27 heated by a lamp placed at 0.1 m. The lamp imposes the uniformity of heat and temperature,*TH*, on the directly exposed PCM wall surface. The condition of adhesion was used to express the velocity fields. The thermal and the dynamic bounder conditions in

the PCM vertical enclosure are given by the following expressions:

upper to the PCM melting temperature:

• The left vertical side of area (*x = L, y, t*) is maintained to a temperature *TH*

• The right vertical side of the PCM wall (*x = 0, y, t*) of the area is maintained

• The horizontal walls of the domain are maintained adiabatic (*x, y = 0, t*) and

*u x*ð Þ¼ , *y* ¼ *0*, *t v x*ð Þ¼ , *y* ¼ *0*, *t 0*,

*u x*ð Þ¼ , *y* ¼ *H*, *t v x*ð Þ¼ , *y* ¼ *H*, *t 0*,

• The energy stored, *ES*, in the PCM wall is given by [25]:

*Tl* <sup>¼</sup> *<sup>ρ</sup><sup>L</sup> dX*

*dt* (9)

*Tl* ¼ *Ts* ¼ *Ti* (10)

*u x*ð Þ¼ ¼ *L*, *y*, *t v x*ð Þ¼ ¼ *L*, *y*, *t 0* (11)

*u x*ð Þ¼ ¼ *0*, *y*, *t v* ¼ ð Þ¼ *x* ¼ *0*, *y*, *t 0* and ∂t*=*∂x ¼ 0 (13)

*ES* ¼ *MPCM*�<sup>27</sup>*:CPCM*�<sup>27</sup>*:*ð*TPCM*,*<sup>F</sup>* � *TPCM*,1Þ þ *MPCM*�<sup>27</sup>*:L* (16)

) is the aperture area of the PCM wall,*Tcell* (°C) is the test

where *MPCM-27* (kg) represents the PCM-27 mass, *CPCM-27* (kJ/kg°C) represents the specific heat of PCM-27,*TPCM, I* (°C) is the initial temperature of PCM, *TPCM-27, F* (°C) is the final temperature of PCM, *L* (kJ/kg) is the latent heat of

*T x*ð Þ¼ ¼ *L*, *y*, *t TH* (12)

*<sup>∂</sup>T x*ð Þ , *<sup>y</sup>* <sup>¼</sup> 0, *<sup>t</sup>*

*<sup>∂</sup>T x*ð Þ , *<sup>y</sup>* <sup>¼</sup> *<sup>H</sup>*, *<sup>t</sup>*

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> 0 (14)

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> 0 (15)

*ks*∇ !

[27, 28]:

*DOI: http://dx.doi.org/10.5772/intechopen.92557*

adiabatic:

(*x, y = H, t*):

**3.1 Energy and exergy analysis**

fusion, *APCM wall* (m<sup>2</sup>

**153**

The constant C is chosen so that it is high enough to cancel the velocities in the solid region and a low number b is introduced to avoid a division by zero in the case of a zero liquid fraction: where b = 0.001 is a small computational constant used to avoid division by zero, and C is a constant reflecting the morphology of the melting front. A value of C = 105 has been used in the literature [1]. The liquid fraction (*H*) is given by [1, 24].

$$H = \begin{cases} \mathbf{0} & \text{if} \quad T < T\_{\text{melting}} \\ \left[ \mathbf{0}, \mathbf{1} \right] & \text{if} \quad T = T\_{\text{melting}} \\ \mathbf{1} & \text{if} \quad T > T\_{\text{melting}} \end{cases} \tag{5}$$

The energy equation is written as [25, 26].

$$\text{Cp}\_{eq} \frac{\partial T}{\partial t} = \lambda\_{eq} \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial \mathbf{y}^2} \right) + \text{Cp}\_{eq} \left( \nu \frac{\partial T}{\partial \mathbf{x}} + \nu \frac{\partial T}{\partial \mathbf{y}} \right) \tag{6}$$

*Cpeq* and *λeq* that appear in Eq. (8), respectively, represent the equivalent volume capacity (*J/K*) and the equivalent thermal conductivity (*W/m K*) of the two solid and liquid phases of PCM, such as:

$$\mathbb{C}p\_{eq} = \sum \theta\_i \rho\_i (\mathbb{C}p)\_i \tag{7}$$

$$
\lambda\_{\rm eq} = \sum \theta\_i \lambda\_i \tag{8}
$$

The term *θ<sup>i</sup>* that appears in Eqs. (9) and (10) is a quantity related to the *H* value, which provides information on the state-owned PCM whether liquid or solid. The thermal and dynamic initial conditions are, *respectively, u = v = 0 (once t = 0) and T=T0 = 288.15 K.*

*Energy Storage in PCM Wall Used in Buildings' Application: Opportunity and Perspective DOI: http://dx.doi.org/10.5772/intechopen.92557*

• In order to account for the phase change process happening when the PCM is melting, the energy balance was applied at the interface as follows [27, 28]:

$$k\_s \vec{\nabla} T\_s - k\_l \vec{\nabla} T\_l = \rho L \frac{d\mathbf{X}}{dt} \tag{9}$$

• Since the phase change of pure substances occur at a single temperature, the temperatures of liquid and solid at the interface is given by [1]:

$$T\_l = T\_s = T\_i \tag{10}$$

where the subscripts s and l stand for the solid and liquid phase, *L* is the latent heat (enthalpy) of fusion, and *X* is the position of the melting interface.


$$u(\mathbf{x} = L, \mathbf{y}, t) = v(\mathbf{x} = L, \mathbf{y}, t) = \mathbf{0} \tag{11}$$

$$T(\mathbf{x} = L, \mathbf{y}, t) = T\_H \tag{12}$$

• The right vertical side of the PCM wall (*x = 0, y, t*) of the area is maintained adiabatic:

$$\mu(\mathbf{x} = \mathbf{0}, \mathbf{y}, t) = \boldsymbol{\nu} = (\mathbf{x} = \mathbf{0}, \mathbf{y}, t) = \mathbf{0} \text{ and } \partial \mathbf{t} / \partial \mathbf{x} = \mathbf{0} \tag{13}$$

• The horizontal walls of the domain are maintained adiabatic (*x, y = 0, t*) and (*x, y = H, t*):

$$\mu(\mathbf{x}, \mathbf{y} = \mathbf{0}, t) = \nu(\mathbf{x}, \mathbf{y} = \mathbf{0}, t) = \mathbf{0}, \frac{\partial T(\mathbf{x}, \mathbf{y} = \mathbf{0}, t)}{\partial \mathbf{y}} = \mathbf{0} \tag{14}$$

$$\mu(\mathbf{x}, \mathbf{y} = H, \mathbf{t}) = \nu(\mathbf{x}, \mathbf{y} = H, \mathbf{t}) = \mathbf{0}, \frac{\partial T(\mathbf{x}, \mathbf{y} = H, \mathbf{t})}{\partial \mathbf{y}} = \mathbf{0} \tag{15}$$

#### **3.1 Energy and exergy analysis**

• The energy stored, *ES*, in the PCM wall is given by [25]:

$$E\_S = M\_{\rm PCM-\mathcal{Z}\mathcal{I}} \cdot C\_{\rm PCM-\mathcal{Z}\mathcal{I}} \cdot (T\_{\rm PCM,F} - T\_{\rm PCM,1}) + M\_{\rm PCM-\mathcal{Z}\mathcal{I}} \cdot L \tag{16}$$

where *MPCM-27* (kg) represents the PCM-27 mass, *CPCM-27* (kJ/kg°C) represents the specific heat of PCM-27,*TPCM, I* (°C) is the initial temperature of PCM, *TPCM-27, F* (°C) is the final temperature of PCM, *L* (kJ/kg) is the latent heat of fusion, *APCM wall* (m<sup>2</sup> ) is the aperture area of the PCM wall,*Tcell* (°C) is the test

*B* ¼ �*C*

*H* ¼

The energy equation is written as [25, 26].

*∂T <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>λ</sup>eq*

*Cpeq*

*Boundary conditions through the PCM wall.*

*Thermodynamics and Energy Engineering*

solid and liquid phases of PCM, such as:

8 ><

>:

*∂*<sup>2</sup>*T ∂x*<sup>2</sup> þ

*∂*<sup>2</sup>*T ∂y*<sup>2</sup> � �

*Cpeq* and *λeq* that appear in Eq. (8), respectively, represent the equivalent volume capacity (*J/K*) and the equivalent thermal conductivity (*W/m K*) of the two

The term *θ<sup>i</sup>* that appears in Eqs. (9) and (10) is a quantity related to the *H* value, which provides information on the state-owned PCM whether liquid or solid. The thermal and dynamic initial conditions are, *respectively, u = v = 0 (once t = 0) and*

is given by [1, 24].

**Figure 5.**

*T=T0 = 288.15 K.*

**152**

<sup>1</sup> � *<sup>H</sup>*<sup>2</sup> � �

0 *if T* <*Tmelting* �0, 1½ *if T* ¼ *Tmelting* 1 *if T* >*Tmelting*

þ *Cpeq u*

*∂T ∂x* þ *v ∂T ∂y*

*Cpeq* <sup>¼</sup> <sup>X</sup>*θ<sup>i</sup> <sup>ρ</sup>i*ð Þ *Cp <sup>i</sup>* (7)

*<sup>λ</sup>eq* <sup>¼</sup> <sup>X</sup>*θ<sup>i</sup> <sup>λ</sup><sup>i</sup>* (8)

� �

The constant C is chosen so that it is high enough to cancel the velocities in the solid region and a low number b is introduced to avoid a division by zero in the case of a zero liquid fraction: where b = 0.001 is a small computational constant used to avoid division by zero, and C is a constant reflecting the morphology of the melting front. A value of C = 105 has been used in the literature [1]. The liquid fraction (*H*)

*<sup>H</sup>*<sup>3</sup> <sup>þ</sup> *<sup>b</sup>* � � (4)

(5)

(6)

cell temperature, and *hPCM-27* (W/m<sup>2</sup> °C) is the heat transfer coefficient of PCM-27.

Energy input during charging is given by:

$$E\_{ic} = I.A\_{PCM} \,\text{wall} \tag{17}$$

where *I* (W/m<sup>2</sup> ) is the irradiance intensity of the lamp and *APCM wall* (m<sup>2</sup> ) is the PCM wall area.

• The energy efficiency of the PCM wall during the charging and the discharging processes are given by:

$$\eta\_c = \frac{E\_0}{E\_{ic}} \text{ and } \eta\_d = \frac{E\_S}{E\_0} \tag{18}$$

where E0 is the energy transferred to the PCM wall.

• The overall exergy transferred to the PCM wall is given by [29, 30]:

$$\begin{split} \mathbf{EX}\_{0} &= \mathbf{M}\_{\text{PCM}-\text{27}} \mathbf{C}\_{\text{PCM}-\text{27}} \left( T\_{f} - T\_{i} \right) + \mathbf{M}\_{\text{PCM}-\text{27}} \mathbf{L} \cdot \left( \mathbf{1} - \frac{T\_{a}}{T\_{s}} \right) \\ &- \mathbf{M}\_{\text{PCM}-\text{27}} \mathbf{T}\_{a} \cdot \mathbf{C}\_{p} \cdot \ln \left( \frac{T\_{f}}{T\_{i}} \right) \end{split} \tag{19}$$

where *Ta(K)* is the ambient temperature and *Ts(K)* is the temperature of sun.

• Exergy input during the thermal storage is given by [30]:

$$EX\_{ic} = IA\_{PCM\,wall} \cdot \left(1 - \frac{T\_a}{T\_s}\right) \tag{20}$$

• Exergy efficiency of the PCM wall during the thermal storage and the thermal discharging are respectively given by:

$$\Psi \mathcal{C} = \frac{EX\_0}{EX\_{ic}} \text{ and } \Psi\_d = \frac{E\_S X}{EX\_0} \tag{21}$$

temperature obtained show an acceptable agreement, of about 0–5°C. It is concluded that the numerical model permits the simulation of the PCM wall thermal

*Energy Storage in PCM Wall Used in Buildings' Application: Opportunity and Perspective*

*DOI: http://dx.doi.org/10.5772/intechopen.92557*

*The simulated and the experimental temperature profile inside the PCM wall for two different times: (a) 4000*

**Figure 7** shows the variation of the energy and the exergy stored in the PCM wall during storage process. It is found that the recovered energy incessantly increases vs. charging time. It ranges between 95 and 780 W. This variation takes roughly 130 min and then the energy stored reaches a maximum, which value is due to the fact that the test cell temperature also fluctuates that is in the range of 22–24°C (**Figure 7**). On the other hand, it is found that the exergy stored grows with the charging time. However, it is seen that the exergy is lesser than the stored energy. It

**Figure 8** shows the variation of the energy and exergy efficiencies of PCM wall during the charging process. It is seen that the PCM wall performance increases gradually from 10 to 95%. **Figure 9** shows the variation of the energy and exergy efficiencies of PCM wall during the discharging process. It is seen that the PCM wall performance decreases regularly from 100 to 10%. It is found that the energy and exergy efficiencies are more important than the charging process. It is also seen that

behavior with an acceptable accuracy.

**Figure 6.**

**155**

*and (b) 6000 s.*

varies between 50 and 460 W.

**4.2 Exploitation of the numerical model**
