The physical model deals with a simple habitat with length L and height h1 where a solar chimney is integrated into the right side of the roof top whereas the photovoltaic Trombe wall chimney integrated to the south façade of the building.

The associated chimneys are designed as a solar collector with plate separation d and height h2. The principal components elements of the solar chimneys are the front glass cover, the silcon polycristallin semi-transparent photovoltaic cells (PV), air flow, absorber plate and back adiabatic panel. The associated chimneys act as the exhaust fan by sucking the room air and venting it out during sunshine hours. In the system, the solar PV panels absorbed the incident solar radiation and transfer heat to the air in the gaps by convection and radiation phenomena. The back walls temperature rise and in turn, together with the solar PV cells, heat the air in the gaps. The right side wall is maintained at the ambient temperature whereas the upper wall and the floor of the room are assumed to be perfectly insulated and adiabatic. The physical system is sketched in Figure 1.

The non dimensional set of the governing equations (continuity, momentum and energy equations) for a two-dimensional, incompressible laminar flow are the following:

∂ U / ∂ X + ∂ V / ∂ Y = 0 (1)

∂ ω / ∂ τ + U ∂ ω / ∂ X + V ∂ ω / ∂ Y = R i ∂ θ / ∂ X + 1 / R e ( ∂ 2 ω / ∂ X 2 + ∂ 2 ω / ∂ Y 2 ) (2)

∂ θ / ∂ τ + U ∂ θ / ∂ X + V ∂ θ / ∂ Y = 1 / R e P r ( ∂ 2 θ / ∂ X 2 + ∂ 2 θ / ∂ Y 2 ) (3)

ω = − ( ∂ 2 ψ / ∂ X 2 + ∂ 2 ψ / ∂ Y 2 ) (4)

The stream function and the vorticity are related to the velocity components by the following expressions:

U = ∂ ψ / ∂ Y ; V = − ∂ ψ / ∂ X     and     ω = ( ∂ V / ∂ X − ∂ U / ∂ Y ) (5)

Where the scales are defined: X = x/d; Y = y/d; U = u/u₀; V = v/u₀

Ψ = ψ / ( u 0 d ) ;     ω = Ω d / u 0 ;     θ = λ ( T − T a ) / ϕ d ; R e = ( ρ v 0 d ) / μ ;     G r = ( g β ϕ d 4 ) / ( λ V 2 ) (6)

By introducing the non dimensional parameters into the physical boundary conditions illustrated in Figure 1, the following non dimensional boundary conditions are obtained:

θ = 0 ; U = V = ω = ψ = 0 at τ = 0

at τ > 0

The boundary conditions, associated with the problem are as follows:

X = L *     and     0 ≤ Y ≤ e * : θ = 0 ; V = 0 ; U = − 1 ; ω = 0 ; ψ = − Y (7)

X = L *     and     e * ≤ Y ≤ H 1 * : θ = 0 ; V = 0 ; U = 0 ; ω = − ∂ 2 ψ ∂ X 2 | X = L * ; ψ = 0

Y = 0     and     0 ≤ X ≤ L * : U = V = ψ = 0 ; ∂ θ ∂ Y | Y = 0 = 0 ; ω = − ∂ 2 ψ ∂ Y 2 | Y = 0

Y = H 1 *     and     d 1 * ≤ X ≤ L A * : U = V = ψ = 0 ; ∂ θ ∂ Y | Y = H 1 * = 0 ; ω = − ∂ 2 ψ ∂ Y 2 | Y = H 1 *

Y = H 1 *     and     d 2 * ≤ X ≤ L * : U = V = ψ = 0 ; ∂ θ ∂ Y | Y = H 1 * = 0 ; ω = − ∂ 2 ψ ∂ Y 2 | Y = H 1 * (8)

X = L A *     and     H 1 * ≤ Y ≤ H * : U = V = ψ = 0 ; ω = − ∂ 2 ψ ∂ X 2 | X = L A *

X = L A *     and     H 1 * ≤ Y ≤ H * : ∂ θ ∂ X | X = L A * = ( η e l − τ g l α P V ) − ( q r P V , a b s ϕ ) (9)

X = d 2 *     and     H 1 * ≤ Y ≤ H * : U = V = ψ = 0 ; ω = − ∂ 2 ψ ∂ X 2 | X = d 2 *

X = d 2 *     and     H 1 * ≤ Y ≤ H * : − ∂ θ ∂ X | X = d 2 ∗ = ( τ g l α P V α a b s ) − ( q r a b s , P V ϕ )

Y = H *     and     L A * ≤ X ≤ L B * : ∂ U ∂ Y | Y = H * = ∂ V ∂ Y | Y = H * = ∂ ψ ∂ Y | Y = H * = ∂ ω ∂ Y | Y = H * ∂ θ ∂ Y | Y = H * = 0

X = 0     and     0 ≤ Y ≤ H * : U = V = ψ = 0 ; ω = − ∂ 2 ∂ X 2 | X = 0 ; ∂ θ ∂ X | X = 0 = ( η e l − τ g l α P V ) − ( q r P V , a b s ϕ ) (10)

X = d 1 *     and     e w * ≤ Y ≤ H * : U = V = 0 ; ψ = 0 ; ω = − ∂ 2 ψ ∂ X 2 | X = d 1 *

X = d 1 *     and     e w * ≤ Y ≤ H * : − ∂ θ ∂ X | X = d 1 * = ( τ g l α P V α a b s ) − ( q r a b s , P V ϕ )

At the outlet of the chimneys:

If V ( X , H * ) ≥ 0 and V ( X , H * ) ⋅ n > 0

Y = H *     and     0 ≤ X ≤ d 1 *     or     L A * ≤ X ≤ d 2 * : U = 0 ; ∂ V ∂ Y | Y = H * = ∂ θ ∂ Y | Y = H * = ∂ ω ∂ Y | Y = H * = ∂ ψ ∂ Y | Y = H * = 0 (11)

If V ( X , H * ) ≤ 0 and V ( X , H * ) ⋅ n < 0

Y = H *     and     0 ≤ X ≤ d 1 *     or     L A * ≤ X ≤ d 2 * : U = θ = 0 ; ∂ V ∂ Y | Y = H * = ∂ ω ∂ Y | Y = H * = ∂ ψ ∂ Y | Y = H * = 0

q r a b s , P V = σ ( T P V 4 − T a b s 4 ) ( 1 ε P V + 1 ε a b s − 1 ) ;     q r P V , a b s = σ ( T a b s 4 − T P V 4 ) ( 1 ε P V + 1 ε a b s − 1 ) (12)

Heat Transfer

The most important heat transfer is evaluated through the PV cells modules plates along the chimneys. These are best represented by Nusselt number. The local Nusselt number along the solar PV cells panels of the chimneys are given by:

X = 0     and     0 ≤ Y ≤ H * : N u P V ( Y ) = ϕ d λ ( T P V ( 0 , Y ) − T a ) = 1 θ P V 1 ( 0 , Y ) (13)

X = L A *     and     H 1 * ≤ Y ≤ H * : N u a b s ( Y ) = ϕ d λ ( T P V ( L A * , Y ) − T a ) = 1 θ P V 2 ( L A * , Y ) (14)

The electrical efficiency of the solar PV cells is given as follows:

η e l = η r e f + β p v ( T P V ¯ − 298 ) + γ log ( ϕ 1000 ) (15)

T P V ¯ = 1 H ∫ 0 H T P V ( 0 , y ) d y (16)

where T P V ¯ is the average absolute temperature of the PV plate

The non linear partial differential governing Equations, (1-3), were discretized using a finite difference technique. The first and second derivatives of the diffusive terms were approached by central differences while a second order upwind scheme was used for the convective terms to avoid possible instabilities frequently encountered in mixed convection problems. The integration of Equations (2-3) was assured by Thomas algorithm. At each time step, the Poisson equation, Equation (4), was treated by using the Point Successive Under-Relaxation method (PSUR) with an optimum under-relaxation coefficient equal to 0.8 for the uniform grid (101 × 101) adopted in the present study. Convergence of iteration for stream function solution is obtained at each time step. The following criterion is employed to check for steady-state solution. Convergence of solutions is assumed when the relative error for each variable between consecutive iterations is recorded below the convergence criterion ε such that ∑ ​ | ( ϕ i , j n + 1 − ϕ i , j n ) / ϕ i , j n + 1 | < 10 − 5 where ϕ stands for ψ , θ , ω , n refers to time and i and j refer to space coordinates. The time step used in the computations is 2.5.10-5. Grid independency solutions are assured by comparing different grid meshes for the fixed Grashof and Reynolds numbers used in this work (Gr = 104 and Re = 30) (Table 1). The vorticity computational formula of Woods [33] for approximating the wall vorticity was used:

ω P = 1 2 ω P + 1 − 3 Δ η 2 ( ψ P + 1 − ψ P ) ,

where ψ P and ψ P + 1 are stream function values at the points adjacent to the boundary wall; n the normal abscise on the boundary wall. Numerical model input parameters are given in Table 2.

In order to test the computer code developed for this study, the problem of a ventilated rectangular enclosure with its left wall submitted to a constant heat flux, while the remaining walls are considered adiabatic was studied. Very good agreement is obtained between the test problem solution and the ventilated rectangular enclosure solutions according to the work of A. Raji et al.

Where the horizontal cold jet enters into the enclosure from the top of its heated wall and leaves from the top of the other vertical one with no slip boundary conditions applied to all the walls. The Reynolds number, Re was set at 200, for Raleigh number Ra set at 106. The numerical analysis predicted values of streamlines and isotherms together with ventilated rectangular enclosure results are shown in Figure 2.

Figures 3(a)-(c) indicates the shape of streamlines and isotherms at a fixed Rayleigh number (Ra = 104) for different Reynolds numbers. One can observe in Figure 3(a) that for Re = 50, the isotherms are very concentrated in the gaps of the chimneys. This tendency proves that the integration of the associated solar hybrid photovoltaic chimneys into the building promotes the passive cooling for the relative size of the habitat. The cold zone observed in the room varies increasingly when the Reynolds number increases, Figure 3(a), Figure 3(b). There are two types air flows in the integrated solar chimneys.

One can observe the manifestation of the back flow or reverse flow phenomenon in the chimney type II, whereas the steady flow is more established in the solar hybrid photovoltaic-thermal Trombe wall chimney. The open lines are very concentrated in the gap of the Trombe wall chimney integrated in the south façade of the building. The presence of the open lines in the chimney type I, indicates really the forced convection in this zone of the cavity. There are permanent closed cells in the room, where their sizes increase with the Reynolds number. The simultaneous existence of closed cells of natural convection and the open lines indicates that the manifestation of mixed convection phenomenon in the vented bioclimatic building. Then mixed convection is indicated to evacuate the heat excess in the associated chimneys integrated into a building. Then, the air flow behind the solar photovoltaic cells plates induced by buoyancy forces has reduced the module’s temperature. The same results are obtained by Brinkworth et al. [13].

The variation of the absolute temperature along the PV cells plates of the chimneys is observed in Figure 4(a), Figure 4(b). In this figure, one can observe that the temperature increases with the Reynolds number in the chimney type II, while it decreases in the Trombe wall chimney. One can conclude that the associated chimneys are playing simultaneously the antagonist role for the habitat passive cooling. The opposite phenomenon is observed the variation of heat transfer exchange along the heated solar PV plates along the chimneys, Figure 5(a), Figure 5(b).

The position of the chimney is a key controlling parameter for air cooling in the building. Then, Figures 6(a)-(c) gives more illustration of the movement of the coolant fluid in the bioclimatic building. Hence, in Figure 6(a), where the chimney type II is located near the Trombe wall channel, one can observe the simultaneous existence of the closed cells and the open lines in the bioclimatic building design. This position of the hybrid photovoltaic-thermal chimney promotes the best heat exchange in the associated chimneys. The thermal plume is clearly observed in the gaps of the chimneys, due to a large cold zone obtained for maintaining the thermal comfort in the building. This obvious situation is observed via the isothermal shape obtained in the chimneys. The isotherms bifurcations indicate the mixed convection phenomenon in the vented bioclimatic building, due to the existence of open lines and the closed cells in the room. Then , one can observe in Figure 6(b), Figure 6(c), that the size of the closed cells decrease when the location of the chimney type II, is far from the Trombe wall chimney which is constantly located at the south façade of the building. This situation prevents the existence of Rayleigh Bernard natural convection closed cells in the cavity. Hence the forced flow is dominant than the natural convection, due to a large number of open lines in the last architect cavity design. Then, Figure 6(c) the extremely location of chimney type II, is a strategy to keep the ambient temperature in the building for thermal comfort. In this way, one can reduce the air conditioned energy load.

For any location case of the chimney type II, in Figure 7(a), the local Nusselt number increases to attain the maximal value before decreasing along the remain PV cells plate in the Trombe wall chimney. But the opposite variation of local Nusselt number is observed along the PV cells plate in the hybrid photovoltaic – thermal chimney. Hence, in this situation, the Nusselt number decreases to attain the minimal value before increasing along the remain PV cells plate, Figure 7(b). These variations of the local Nusselt number prove sufficiently that the association of the cooling cavity and the Trombe wall chimney integrated into the building, give more possibility for passive cooling in the large habitat. This passive cooling technique can reduce the building energy consumption.

In Figure 8(a), Figure 8(b), the outlet velocity increases to attain the maximum value before it decreases along the solid walls which limited the separation of the chimneys. No sleep conditions are observed along the limitation solid walls along the chimneys. In particularly, one can observe that, the outlet velocity is a decreasing function of the height of the chimney at the outlet opening of the Trombe wall chimney, (chimney type I), Figure 8(a); while it becomes an increasing function at the outlet opening of the chimney type II, Figure 8(b). The existence of the two peaks in the variation of the outlet velocity in chimney type II, is a proof of the back flow phenomenon observed in the shape of the streamlines. Other where, the stabilized flow is observed in the chimney type I, due to the unique peak of the outlet velocity variation in this design chimney. In Figure 9(a), Figure 9(b), the outlet opening temperature curves have the maximum value near the chimneys limitation plates.

The minimum values are observed at the middle of the chimneys, because, the heat exchange is more important at the vicinity of the heated plates. One can observe that the outlet temperature is an increasing function of the height of the chimney, because, longer is the chimney height, higher is the flow duration along the chimney. Due to the coolant fluid to have the best heat exchange along the bioclimatic building, this practice has an important interest to keep constant the temperature of habitation space in the room at the fresh air ambient temperature to overcome to the thermal comfort.

The width of chimney is an important controlling parameter of the heat exchange in the depth of the chimney. Hence, Figure 10(a), Figure 10(b) indicates that the intensity of the heat exchange is an increasing function of the chimney width.

The local Nusselt number is an inverse relation of the dimensionless temperature. Then, smaller is the width, higher is Nusselt number along the heated PV cells plates. The peak observed in the variation of Nusselt number along the PV cells plate in the Trombe wall chimney, gives more explication that the heat exchange is weak in this entry zone of the PV cells plate perfectly vented by the fresh air, and then it decreases along the remain PV plate as the same variation observed along the heated PV plate in the chimney type II.

The chimney width size has an important effect on the variation of the solar PV cells electrical output. Then the curves of variation of the electrical efficiency of the solar PV cells along the chimneys in Figure 11(a), Figure 11(b) indicated that there is a strongly relation between the chimney size and the electrical efficiency of the solar PV module.

In Figure 11(a), the PV cells electrical efficiency decreases for the weak values of the size of the chimney while it increases for the high value of the width size. This result indicates that authors to attain the optimum value of the Trombe wall chimney width to obtain the maximum PV cells electrical efficiency. Other where, the maximum value of PV cells electrical efficiency is attained for the weak value of the chimney width value, Figure 11(b)

One can conclude that the associated hybrid photovoltaic chimneys integrated into the building are doing the complementary role in the passive cooling effect for the space habitation in the room. Then, in Figure 12, the outlet temperature in the Trombe wall chimney is a decreasing function of the width of the chimney. Consequence, larger is the chimney width, higher is the PV cells electrical efficiency, thus the PV cells plate are perfectly vented and the excess of the heat is permanent evacuated.