Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v14i48.1357 RESEARECH ARTICLE Natural Convection Modeling in a Solar Tower Moctar Moctar ousmane.m2001@gmail.com 1 Ky Thierry Sikoudouin Maurice 2 Konfé Amadou 2 Dianda Boureima 3 Ouédraogo Salifou 2 Bathiébo Dieudonné Joseph 2 University of Agadez PO BOX 199 Niger Laboratory L.E.T.R.E, University Joseph KI­ZERBO Ouagadougou 03, PO BOX 7021 Burkina Faso Institute of Research in Applied Science and Technologies PO Box 7047 Burkina Faso 14 48 3475 2021 Abstract

Objectives: We present in this study a modeling of thermal laminar convection airflow in a solar tower. Methods: To formulate with precision, the boundary conditions of the solar chimney model chosen, the Cartesian equations are transformed into hyperbolic coordinates. An orthogonal grid is elaborated. It then makes it possible to draw up the diagrams of physique and calculation fields. The computer code uses the heat equation, the vorticity, and the stream function formalism as the boundary conditions for pressure are difficult to set. We use the Boussinesq approximation, which consists in considering that the density (ρ) of the fluid varies only in the term of the gravity forces, whose variations with temperature, assumed to be linear, generate natural convection. These variations are then translated into an equation of state which relates density to temperature. The system of dimensionless equations is solved by using an intégro-interpolation method referring to finite differences scheme. Findings: The solutions obtained from the dimensionless equations enabled us to determine the space evolution parameters (temperatures and velocities) in the tower according to the Rayleigh number. The fluid temperature and velocity evolution in the collector increase when one moves in the direction of radius decrease. The fluid temperature evolution in the chimney showed that the highest temperature is located at the chimney base while we obtained a parabolic profile of the transverse temperature distribution within the chimney. Finally, the evolution of the fluid velocity in the chimney showed that there was a preferred zone for turbine installation. Novelty: The use of dimensionless geometric parameters is unique and in general, the approach adopted in this paper differs from that encountered in the literature.

Keywords None
Introduction

Nowadays, the development level of each nation is measured by the rate of access to electricity and its consumption per capita. In our countries, the energy crisis, the deterioration of the environment, and the decline in agricultural yields have negative consequences on the quality of life. The population growth and development needs have led to an increase in the consumption of traditional energy resources. Indeed, in 2.day to 6.5 kWh / m2.day; and duration from 3 000 to 3 500 hours per year we immediately understand that this tremendous, "free", non-polluting, almost inexhaustible energy potential can cover a large part of the energy needs of our countries . It is with this in mind that researchers and policymakers have turned to renewable energies. Thus, the solar chimney introduced by Jörg Schlaich and Rudolf Bergermann  in 1976 is a new technology for producing electricity by transforming solar energy into mechanical energy. Figure 1 shows the description of a solar chimney power plant as consisting of a collector, a turbine, and a vertical chimney given by Isidoro Cabanyes in 1903. In 1926, Bernard Dubos  proposed to use a vertical wind blowing in a tube on the side of a mountain as a solar chimney.

Solar chimney power plant project proposed by Isidoro Cabanyes 

Figure 2, Figure 4 show respectively between 1940 and 1960, then in 1975, two solar tower projects close to each other studied by Edgard Nazaré  and Louis Michaud.

Nazaré aerothermal power plant 

Michaud type aerothermal power plant 

Finally, faced with these different original concepts, Figure 4 shows the first prototype tower built in 1981 thanks to the team of Pr J. Schlaich, in Spain. The first experimental model is that of Manzanares which operated from 1982 to 1989 (J. Schlaich) [2,5]. This tower served as an information provider .

Prototype of the solar tower prototype plant at Manzanares, Spain 

Pasumarthi and Scherif [7,8] built a conical solar tower in Florida. This study reviewed the heat transfer within the collector and the possibilities to improve it. In China, a pilot plant with an energy storage system was built [9,10]. Likewise, in Brazil [11-13], a solar tower was built. Koyun et al, Buğutekin et al [14,18], built two pilot plants. Golder et al, Akbarzadeh et al [15,16], built a hybrid system consisting of a solar power plant associated with a solar pool. Ahmed et al , built a prototype which made it possible to study the influence of the nature of the soil on the temperature profile of the chimney. Najmi et al, Kasaeian et al [19,20] built prototypes with a cone at the entrance of the chimney to improve the power output and decrease the height of the air inlet. Mehla et al, Shuia et al, Al-Dabbas [21-23] developed prototypes in which they reviewed the influence of geometric parameters and their variation with solar radiation. Chappell et al  designed a prototype solar chimney at low material cost and technically less cluttered. Figure 5 shows four experimental prototypes in which experimental results differ from one model to another, nevertheless, have the same appearance as the simulated results developed by Koonsrisuk et al .

Experimental prototypes in Thailand 

Aja et al  showed that there is a preferred direction and orientation for which the velocity and direction of the wind strongly impact on the performance of the prototype. An experimental prototype of a solar chimney was designed at the Laboratory of Renewable Thermal Energies of Ouagadougou University, (Burkina-Faso)  which revealed that at the chimney entrance the absorber’s temperature decreases slightly while that of the fluid is maintained at a maximum level. An increase in the stack height results in an intensification of convective movements within the prototypeWe also notice an increase in the fluid’s temperature at the expense of that of the absorber. The maximum velocities are obtained at the chimney entrance but decrease very quickly along the ascending axis of the latter. Flow can be improved in this area by using curved junctions. Figure 6 shows the experimental prototype of Ouagadougou’s university.

Experimental prototype of Ouagadougou’s nuiversity 

This confirms the choice of this zone, by several authors for the turbine installation. The ideal location for the drying racks is the area above the chimney entrance, characterized with relatively high temperatures and maximum flow velocities. These conditions are favorable to mass and heat transfers. This study confirms the feasibility of the system but also highlights the need to optimize the thermo-aeraulic parameters to produce electricity and allow the conservation of food products. Al-Azawie et al, Li et al, Tan et al [27-29] studied, experimentally the conversion capacity of materials then the contribution of phase change materials, and finally the effects of the ambient air velocity and the internal thermal load on the solar chimney. Shahreza et al  brought an innovation in the design of a solar tower by associating concentrators able to track the sun. The maximum velocity obtained then reached 5.12 m. s-1. This is especially important about the dimensions of the prototype. Ghalamchi et al  developed a solar chimney pilot plant and concluded that reducing the collector inlet height increases the performance of the system. During the same year, Okada et al  experimented with two types of chimneys: one in the shape of a cylinder and another in the shape of a diffuser. They proved that the diffuser-shaped chimney increases the air speed resulting in better power output than in the cylindrical case. Ky et al  presented an experimental study of a solar chimney prototype made up of 49 hemispherical concentrators, this collector works according to the theory of hot spots. The resulting temperature profile suggests that the innovation provided could replace flat absorbers, allowing air to circulate more smoothly through the manifold and thereby reducing its surface area. Ahmed et al  built two hybrid models of solar chimney designated respectively by (A) and (B). System A consists of a glass collector, a photovoltaic panel acting as an absorber, and a chimney, while system (B) comprises a photovoltaic panel acting as a collector and an absorber in plywood. They obtained the following results: system (A) recorded the highest temperature (90°C) against 67° C for system (B) at noon. The velocity level recorded by system (A) is higher than that of the system (B) because the glass collector increases the absorbed energy. Figure 7 and Figure 8 show respectively system (A) and system (B).

Hybrid gazed PV / solar chimney

Hybrid PV/solar chimney

Fadaei et al  experimentally studied the effect of the latent heat storage of a paraffin phase change material. The results showed that the maximum temperatures and velocities (72° C and 2 m / s) were recorded at the phase change material absorber against the devoid absorber which was recorded (69° C and 1.9 m / s). In the context of thermal building energy, Abdeen et al, Elghamry et al, Hamood et al [36-38], built prototypes for the realization of thermal comfort. Mahal et al , developed a hybrid solar chimney system coupled with a liquid desiccation system that could simultaneously produce electricity and freshwater using calcium chloride acting on the air of the prototype.

The study of experimental models demonstrates the enthusiasm and reliability of the system, hence the numerous numerical studies that have followed.

From the literature review, we distinguish five simple theoretical models compared with the results of the CFD calculation code. These are Chitsomboon et al, Schlaich et al, Tingzhen et al, Zhou et al, Koonsrisuk and Chitsomboon [40-44]. Nevertheless, Mullett  began the development of theoretical models of the solar chimney while expressing that the overall efficiency is proportional to the size of the chimney. Schlaich et al [46,47], gave the expression of the flux at the collector outlet and the efficiency of the chimney. Using the Boussinesq approximation, Unger et al  gave the expression of the maximum velocity reached in natural convection while Nizetic et al  showed the parabolic profile of the velocity at the chimney entrance. Padki and Sherif [50,51] developed models capable of producing significant energies in the medium and long term, Pasumarthi and Sherif  introduced two innovations which allowed them to increase the total power output of the chimney. Bernades et al  used the numerical CFD (Computational Fluid Dynamic) model in the study of convective flow in a solar chimney. They presented a solution for Navier-Stokes's equations and the energy for a steady-state laminar natural convection, using the finite volume method in generalized coordinates. This method provides a detailed view of the geometric effects and operational, optimal geometric characteristics. Kröger and Blaine , Kröger and Buys  respectively developed analytical relations to determine the pressure difference due to friction phenomena and heat transfers to develop a radial flow between the roof and the collector. Padki and Sherif  established differential equations, the power formula, and the relative efficiency of the solar chimney. Gannon and Von Backström  succeeded in predicting the performance of a large-scale power plant. Chitsomboon et al [58,59,85] proposed a model of dimensionless variables which they validated with CFD theory.

Gannon and Von Backström [60-62] with one model of turbine claimed to be able to extract 80% of the power generated by the flow, then with another type found a total efficiency between 85 and 90%. Bernardes et al  showed that the power output can be improved by increasing the height of the chimney, the collector’s surface, and its transmittance. Von Backström  obtained the average density in the chimney. First, Schlaich et al [65,66,47] gave some results of design, construction, and operation of a prototype like that of Manzanares and proposed commercial prototypes accompanied by basic technical, economic data, then they confirmed that a solar chimney is an option of low-cost electricity production. Finally, they presented a model providing information from a theoretical, practical, experimental, and economic point of view for the design of a 200 MW chimney. Pastohr et al [67[ showed that the stationary solution hypothesis does not consider the heat storage of the soil and the fluid, hence the dynamic regime solution would be of particular interest in the future. Serag-Eldin  from the k-epsilon model studied the effects of atmospheric winds on the performance of a solar chimney, from which it results in total degradation of the performance in the presence of strong wind and a significant degradation for weak wind, except for sensors with low intake height. Bilgen and Rheault  built an inclined collector field at the lower level of a mountain, thus playing the role of a chimney. They developed a mathematical model which results showed that the production of electrical energy can reach 85% of that produced by a horizontal collector solar chimney. Tingzhen et al [70-73] developed a model which studies the difference in static pressure between the inlet and the outlet of the chimney. They showed that this static pressure is negative and decreases throughout the flow in the collector, but increases inside the chimney, and in another model validated by the data of the Manzanares prototype. They proved that the output power could exceed 10 MW. In addition, they concluded that the soil storage energy, velocity, and average chimney outlet temperature increase with increasing incident solar radiation.

Mathematical formulation

We use here the formalism, the heat equation, the vorticity, and the current function by considering the assumption of Boussinesq (We suppose the density constant in all the terms except in the force of gravity). The conservation equations for momentum are then replaced by that of vorticity. The theoretical study is carried out in steady state in an interval corresponding to the minimum and maximum equilibrium temperatures of the absorber, obtained during the experimental study (320 <T <370) K1. Figure 11 shows the movement of the fluid.

Axisymmetric representation of the optimized solar tower

The following assumptions are considered in this study.

The air follows ideal gas law, permanent two-dimensional laminar regime, physical properties of air are assumed to be constant except density in buoyancy force. The radiation exchange, the viscous heat dissipation and the power density are neglected. Finally, the walls of the chimney are adiabatic with the non-slip condition .

<bold id="strong-397f0a9ca4fd4a5dabc9abf138bb8c96">Mass conservation</bold> ux+vy=0                                                                                                                    (1) <bold id="strong-8049645cb4d744ccb967b22492a933f5">Vorticity equation</bold> (uw)x+(vw)y=Tx(g.β)+ν2ωx2+2ωy2                                                       (2) <bold id="strong-ed7a635a256747738a94d17aca3a5114">Energy conservation equation</bold> uTx+vTy=λρ.Cp2Tx2+2Ty2                                                                           (3) <bold id="strong-19556d8185ee4c9aa85ead85525a8fa3">Boundary conditions</bold>

The boundary conditions are represented on figure 10 below.

Boundary conditions in axisymmetric solution

Numerical methods present shortcomings in the face of certain geometries, hence the use of the mesh generation technique. According to Hoffmann , the basic idea is to transform a complex geometry into a simple geometry by using generalized or hyperbolic coordinates. We are going to transform the cartesian equations into hyperbolic coordinates to formulate with precision the boundary conditions for the selected tower model. The relations allowing the passage from cartesian coordinates (x, y) to hyperbolic coordinates (ξ, η) is carried out using the following relations :

x=r+ξ2  ; y=r-ξ2                                                                                          (4) η=2xy ; ξ=x2-y2 ; r=ξ2+η2=x2+y2                                              (5)

Figure 11, Figure 12, and Figure 13 show the coordinate system, the physical domain, and the computation domain.

Coordinate system<inline-formula id="inline-formula-ff906bbd69b14a7eb3d6dd19f149ef09"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msup><mml:mi>ξ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>

Real physical domain

Computation domain

Equations (1), (2), (3) become:

Vξξ+Vηη=0                                                                                                         (6) VξTξ+VηTη=1hλρ.Cp2Tξ2+2Tη2                                                                   (7) Vξhwξ+Vηhwη=νh22wξ2+2wη2+gβAξ,ηTξ+Bξ,ηTη              (8)
Numerical solution

The resulting equations having been transformed into curvilinear coordinates are subsequently discretized with a finite difference scheme using the Samarsky method  exposed in Nogotov book  the solutions obtained from the system of dimensional algebraic equations make it possible to generally determine the evolution of the aeraulic parameters. We use a method of dimensional analysis like that used by Tayebi  and Heisler  but presenting a fundamental difference as to the choice of the characteristic length appearing in the Rayleigh number defined by Tahar 

d=Hc;H=h·d=Hc2r;Vξ*=Vξ·da;Vη=*Vη·da;  ψ*=ψa;w*=w·d2a ξ*=ξd2; η*=ηd2 ; T*=T-TcTheat-Tc ; t*=tad2                                                                         (10)

d2a Characteristic time

Equations (1), (2), and (3) can be summed up in the equations below, by posing:

Vξ*=1Hψ*η* ;  Vη*=-1Hψ*ξ* ; Ra=gβΔTd3νa ; Pr=νa  ;ν=μρ  ; a=λρ                                                                                     11 ξ*1HT*ξ*-Vξ*T*+ η*1HT*η*-Vη*T*=0                                    (12) ξ*PrHw*ξ*-Vξ*w*+ η*PrHw*η*-Vη*w*=-Pr.RaCξ*,η*T*ξ*  +Dξ*,η*T*η*                           (13) w*=-1H22ψ*ξ*2+2ψ*η*2                                                                                     (14)

The dimensional boundary conditions are given below:

-Outlet

T*ξ*=0  T1,j=4T*2,j-T*3,j3

-Inlet

T*NI,j=0

-The ground

T*i,j=0

-Roof

T*i,j=0

-Axisymmetric axis

T*i,1=4T*i,2-T*i,33

-Boundary conditions of the stream function

-Outlet

ψ*1,j=8ψ*2,j-ψ*3,j7

-Inlet

ψ*NI,j=4ψ*NI-1,j-ψ*NI-2,j3

-Roof

ψ*i,1=4ψ*i,2-ψ*i,33

-Axisymmetric axis

T*ξ*=0 ψ*i,1=8ψ*i,2-ψ*i,37

-Boundary conditions of the vorticity equation

-Outlet

ω*1,j=-1H27ψ*1,j-8ψ*2,j+ψ*3,j2Dξ*2

-Inlet

ω*NI,j=-1H27ψ*NI,j-8ψ*NI-2,j+ψ*NI-3,j2*Dξ*2

-Axisymmetric axis

ω*i,1=0

-Velocity boundary conditions

-Outlet

V*1,j=4V*2,j-V*3,j3 U*1,j=0

-Inlet

U*NI,j=0 V*NI,j=0

-Ground

U*i,j=0 V*i,j=0

-Roof

U*i,j=0 V*i,j=0

Axisymmetric axis

U*i,j=4V*i,2-V*i,33 U*i,1=0

For the resolution we use the so-called integro-interpolation method described by Nogotov , a system of differential equations is obtained by integration of the system constituted by equations (12) to (13). Our calculation program gives the following results.

Results and discussion 4.1 Temperature evolution in the collector

Figure 14 presents the temperature evolution in the collector.

Temperature evolution in the collector near theabsorber

From the dimensionless study, the fluid temperature in the collector increases very quickly along the x-axis, as the radius decreases. At the chimney inlet, we notice a slight decrease in temperature. This has also been observed by Ghalamchi . This is in fact the area where heat energy is transformed into kinetic energy in the chimney. The increase in the Rayleigh number generally leads to an intensification of convective movements in the ducts.

4.2 Velocity evolution in the collector

The velocity evolution shown in Figure 15 increases in the collector when one moves in the direction of radius decrease. But we notice that the slope of the curve is more pronounced near the base of the chimney. In fact, the fluid having acquired maximum energy is no longer confined and can then move by free convection in the chimney.

Fluid velocity evolution in the collector as afunction of X * for different Rayleigh numbers

Fluid temperature evolution in the chimney

Figure 16 and Figure 17 show respectively the fluid temperature evolution in the chimney and in a cross-section. The fluid temperature evolution in the chimney naturally shows an almost hyperbolic profile. In fact, the base of the chimney receives hot air when it has acquired the maximum energy, leading to the highest temperature at the base. This then decreases very quickly. In the laminar regime, for 106 < Ra <107, outlet temperatures remain close to ambient temperature as depicted in Figure 16. Figure 17 shows a parabolic profile of the transverse temperature distribution within the chimney when it is assumed that the walls are at a constant temperature.

Dimensional evolution of the temperature in the chimney

Evolution of the temperature in a cross-section of the chimney Y * = 0.45

4.4 <bold id="strong-0bdd95f8edca44e58047de6d8b3f2852">Evolution of the fluid velocity in the chimney</bold>

Figure 18 shows the velocity evolution in the chimney. In the variation interval chosen for the Rayleigh numbers [106, 107], the maximum velocities are always observed between 0.2 <X * <0.4 for the chosen geometry as in Figure 18. Heisler  and Tahar  place this zone close to 0.2 for Rayleigh's number of the order of 105. It is therefore the preferred zone for turbines installation. We can notice that the Rayleigh numbers chosen and represented in Figure 18 correspond to the temperatures that are usually noted in the vicinity of the absorbers, in our climate, during a day.

Evolution of the fluid velocity in the chimney as a function of X * for different Rayleigh numbers

Conclusion

The aeraulic parameters evolution is simulated using curves showing an acceptable precision, of the order of 4%, for temperatures, however, those concerning the velocities are less precise (> 10%) The solutions obtained from the system of dimensionless equations allowed us to determine:

The spatial evolution of temperatures and velocities in the collector for different Rayleigh numbers.

An increase in Rayleigh numbers leads to an intensification of convection movements in the collector and an increase in the fluid temperature. The maximum velocities concentration region hardly varies as a function of the Rayleigh number. The temperatures close to the entrance of the chimney are decreasing slightly using our calculation code.

Spatial evolution of temperatures and velocities in the chimney for different Rayleigh numbers.

The velocity curves in the chimney, a bell curve in the vicinity of the inlet, obtained using our calculation program are in perfect agreement with the literature data. This is the sensitive area that must be determined with precision to place a turbine or drying racks. However, this first approach, using our calculation program, makes it possible to pre-dimension a system using the calculation of its aeraulic parameters.

References

1. Moctar O, Boureima D, Sié K, Amadou K, Ky T, Bathiebo D J. Experimental study, in natural convection. Global journal of pure and applied sciences. 2015; 21, 155-169.

95. Hoseini H, Mehdipour R. Evaluation of solar-chimney power plants with multiple-angle collectors. Journal of computational and applied research in mechanical engineering. 2018; 8(1), 85-96. Doi: 10.22061/JCARME.2017.2282.1213.