Numerical Study of Combined Natural Convection and Radiation in Three Dimensional Solar Thermal Collector: Focus on the Inclination Effect on Heat Transfer
Kaouther Ghachem^{1}, Mohamed Bechir Ben Hamida^{2, 3}, Chamseddine Maatki^{1}, Lioua Kolsi^{1, 4}, Mohamed Naceur Borjini^{1}, Habib Ben Aissia^{1}
^{1}Research Unit of Metrology and Energy Systems, National Engineering School, Energy Engineering Department, University of Monastir, Monastir city, Tunisia
^{2}High School of Sciences and Technology of Hammam Sousse (ESSTHS), Department of Physics, University of Sousse, Sousse, Tunisia
^{3}Research Unit of Ionized Backgrounds and Reagents Studies (UEMIR), Preparatory Institute for Engineering Studies of Monastir (IPEIM), University of Monastir, Monastir city, Tunisia
^{4}College of Engineering Mechanical Engineering Department, Haïl University, Saudi Arabia
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To cite this article:
Kaouther Ghachem, Mohamed Bechir Ben Hamida, Chamseddine Maatki, Lioua Kolsi, Mohamed Naceur Borjini, Habib Ben Aissia. Numerical Study of Combined Natural Convection and Radiation in Three Dimensional Solar Thermal Collector: Focus on the Inclination Effect on Heat Transfer. American Journal of Modern Energy. Vol. 1, No. 2, 2015, pp. 4451. doi: 10.11648/j.ajme.20150102.13
Keywords: Natural Convection, Radiation, Inclinaned Cavity, Solar Collector
1. Introduction
Coupling of natural convection and radiation is prevalent in many industrial applications and natural phenomena. We quote for example: solar collectors, fire research, and building construction and insulation systems.
Many researchers have had a great interest to study pure natural convection in an inclined cavity. Mention may be made of the work of Ozoe, Sayama and Churchill [11] from one side and Ozoe, Yamamoto and Churchill [12] from the other side who reported that if the cavity is seriously inclined (ϕ between 80° and 90°), complex flow pattern with 3D multiple rollcells would appear in the cavity. Wang Q.W, Wang, G., Zeng and Ozoe [14] studied the natural convection in 2D cavity with an inclined angle varying between 0° and 80°.
Recently, Henderson, Junaidi, Muneer, Grassie and Currie [8] conducted a review paper about the works in pure natural convection and pure radiation for inclined cavity.
It is worth noting that the first numerical study in literature about the coupled heat transfer problem involving both convection and radiation in a rectangular cavity seems to be that of Larson and Viskanta [9]. They found that radiation heats up the cavity surface and the gas body very quickly and thus considerably modifies the flow pattern and the corresponding convection process. Several twodimensional studies are found for the problem of combined radiation and natural convection in rectangular participating medium (Chang, Yang and Lloyd [5]; Yang [16], Yucel, Acharya and Williams [1]) but many works take only into account the effects of surfacetosurface radiation in order to reduce computational effort (Balaji and Venkateshan [4] and Velusamy, Sundararajan, Seetharamu [13]), Chang , Yang and Lloyd [5] investigated combined radiation and natural convection in twodimensional enclosures with partitions. Desreyaud and Lauriat [7] computed natural convection and radiation in rectangular enclosures using a onedimensional P1 radiation analysis. Webb and Viskanta [15] measured the natural convection induced by irradiation and compared experimental results with the results of an analysis based on a spectral onedimensional radiation model. A review paper on this subject was given by Yang [16]. A 3D numerical simulation of radiation and convection in a differentially heated cubic cavity using the discrete ordinates method is effectuated by Colomer, Costa, Cònsul and Oliva [6]. The authors detail the effect of the Planck number and the optical thickness on heat transfer and effectuate a comparison between dimensional results obtained from a twodimensional model and those obtained in the midplane of a long rectangular enclosure. Later, Borjini, Mbow and Daguenet [2] studied numerically the effect of the radiative heat transfer on the threedimensional convection in a cubic differentially heated cavity for different optical parameters of the medium, Pr = 13.6 and Ra=105. Their results show that the structure of the main flow is considerably altered by of the conduction–radiation parameter. Indeed, they found that the inner spiraling flows were very sensible in location and direction to the radiative heat transfer. However, the peripheral spiralling motion was qualitatively insensitive to these parameters. It is also found that radiation favorites the merging of the vortices near the front and back walls. Recently, Kolsi, Abidi, Maatki, Borjini, and Ben Aissia [10] studied the effect of radiative transfer and the aspect ratio on the 3D natural convection. Prandtl and Rayleigh numbers are respectively fixed at 13.6 and 105. They found that the principal flow structure is considerably modified when the radiationconduction parameter was varied. However, the peripheral spiraling motion is qualitatively insensitive to these parameters. In all these works, radiation is found to play an important and sometimes major role in heat transfer and fluid flow processes.
In the present paper, we further study the combined radiation and natural convection in an inclined 3D cavity filled with emitting, absorbing, and isotropically scattering fluid. The work will be divided into two main parts, namely: effect of Rayleigh number on heat transfer and cavity inclination effect on the flow and heat transfer.
2. Mathematic Formulation
Figure1, presents the considered physical system which is composed of a square basic parallelepipedic enclosure, with aspect ratio F=H/W equal to 2. The two vertical walls are kept respectively at constant cold temperature and hot temperature. All the other walls are kept adiabatic. In the present study, the cavity is inclined around with an angle γ. All these surfaces are gray and diffuse. Even more, the cavity is filled with a gray, emitting–absorbing and isotropically scattering fluid. The flow is supposed to be laminar and the Boussinesq approximation is used. The dimension equations describing the combined radiation and natural convection are respectively:
Continuity:
(1)
Momentum:
(2)
Energy:
(3)
As numerical method we had used the vorticityvector potential formalism . This allows, in a 3D configuration, the elimination of the pressure, which is a delicate term to treat. For this, one applies the rotational to the equation of momentum (Eq.2). The vector potential and the vorticity are, respectively, defined by the two following relations:
and (4)
The foregoing dimensionless parameters are given as follow: time, velocity, the vector potential, the vorticity, are put respectively in their dimensionless forms by , , and . The adimensional temperature is defined by: . Whereas is the mean temperature of the hot wall.
The system of equations controlling the phenomenon becomes:
(5)
(6)
(7)
Pr  is the Prandtl number 

Ra  is the Rayleigh number 

Rc  is the radiation conduction parameter 

τ  is the optical width 

ϕt  is the temperature ratio 

The initial and boundary conditions are as follows:
Temperature
At x=0, T=1 and at x=1, T=0. On the other walls,
Velocity: on all walls
Vorticity


 at x=0 and 1 


 at and 1 


 at z=0 and 1 
Vector potential
 at and 1 
 at and 1 
 at and 1 
Radiative flux
 at y=0 and y=1 
 at z=0 and z=1 
The radiative transfer equation, for a gray semitransparent medium which absorbs, emits and isotropically diffuses the radiation, can be viewed in the manuscript of Borjini, Mbow and Daguenet [2] et Kolsi, Abidi, Maatki, Borjini, and Ben Aissia [10].
3. Validation Test
The comparison of radiative and conductive fluxes on the heated wall, with the results of Colomer, Costa, Cònsul and Oliva [6] is presented, for several optical thicknesses, in Table 1. A remarkable difference is observed between the two results. This is a consequence that Colomer, Costa, Cònsul and Oliva [6] used the discrete ordinates method with suitable directions and for the classical 3D furnace case they compared their results only with the approximation P3 of the spherical harmonics.
Comparison of thermal transfer on the hot face between our results and those of the literature for Pr=0.71, Rc=1/(0.016´17) et Ft=1/17.
t = 0  t = 1  t =10  
 rc/F_{t} 
 rc/F_{t} 
 rc/F_{t}  
Ra = 103  Our results  1.06  6.49  1.70  4.61  1.65  1.25 
Colomer et al.  1.76  6.20  1.76  4.64  1.54  1.16  
Ra = 104  Our results  2.04  6.89  2.45  5.12  2.23  1.65 
Colomer et al.  2.26  6.28  2.25  4.69  2.11  1.54  
Ra = 105  Our results  4.13  7.23  4.04  5.88  4.46  2.99 
Colomer et al.  4.37  6.52  3.92  5.44  4.21  2.8 
4. Results and Discussion
To extract simulations for the present study, the parameters of simulations are respectively: a times step equal to10^{4}, a space grid of 41x81x41 for F=2 and an angular grid FT6FVM.The Rayleigh number is fixed at 10^{5} and the Prandtl number at 0.71. The optical with τ is equal to 1, Φt=0.1 and albedo equal to 1. The effects of the conductionradiation parameter, the inclination angle of cavity and the oscillatory excitation on the principal flow and the heat transfer are discussed.
A. Effect of the variation of the Ra number on heat transfer
This section is reserved to show the influence of the Ra number on the heat transfer. Figure 2 shows that the increase of the Ra number and consequently the effect of natural convection, do not alter the average radiative flux. For the rest of the results the Ra number is fixed at 10^{5}.
B. Cavity Inclination effect on the flow and heat transfer
As shown in Figure 1, the cavity is gradually inclined with a step equal to 30°around zaxis. Four angles presented here are respectively: γ=0°, γ=30°, γ=60° and γ=90°. In these cases, the effect of γ is carefully investigated with the variation of the conductionradiation parameter rc which evolved from 0 to infinite.
Figure 3 represents the projection of flow lines in the main plan XY for γ=0°. When rc=0 (Figure 3(a)), the flow is characterized by one thermal vortex slightly titled towards the cold wall and turns counter the clockwise. The streamlines are not closed and they form a spiral. When rc=10 (Figure 3(b)), the vortex moves toward the hot wall and stabilizes in the center of the cavity. For rc>∞ (Figure 3 (c)), one center vortex is shown and streamlines are closed.
(a) (b) (c)
Figure 4 represents the projection of flow lines in the main plan XY for γ=30°. For rc= 0 (Figure 4(a)), one vortex situated in the center of the cavity turns counter the clockwise. When rc= 10 (Figure 4(b)), one notes the apparition of one cell structure with three inner vortexes. The two vortexes situated in the top and in the bottom of the cavity turn counter the clockwise however the third one, situated in the center, turns in the clockwise. When rc>∞ (Figure 4(c)), these vortex disappear and one center vortex takes place. In this case, one notes the apparition of one small vortex in the top corner near the hot wall of the cavity turning in the clockwise.
(a) (b) (c)
Figure 5 represents the projection of flow lines in the main plan XY for γ=60°. In absence of radiation (Figure 5 (a)), one notes the apparition of one cell with three inner vortexes. The top and the bottom vortex are slightly titled to the cold wall and turn counter clockwise. The central vortex turns in the clockwise. For rc=10 (Figure 5(b)) one notices the coalescence of the three vortex and one vortex, slightly titled to the hot wall, takes place and turns counter the clockwise. By the increasing of rc which tends to infinite (Figure 5(c)), one center vortex appears and turns in the clockwise. For γ=90° (Figure 6) the problem of RayleigBenard convection is retrieved. Figure 6 shows the projection of the flow lines on the mid XY plane for rc=1, rc=10 and rc>∞. When rc=1 (Figure 6(a)) and rc= 10 (Figure 6(b)), two contrarotative cells take place and when rc>∞ the flow presents height rollcells. This result is earlier demonstrated by Ozoe et al. [11].
(a) (b) (c)
(a) (b) (c)
(a) (b) (c)
(a) (b) (c)
(a) (b) (c)
(a) (b) (c)
The (Figure 7, Figure 8, Figure 9 and Figure 10) present the isothermal surfaces respectively for γ=0°, γ =30°, γ =60° and γ= 90°. In absence of radiative transfer (rc=0), and by increasing the inclination angle, one notes that the distribution of the isothermal surfaces is largely modified. In fact, isothermal surfaces are distorted and a 3D aspect appears in the core of the cavity. When the effect of the radiation increases rc=10, a vertical stratification appears in the core of the cavity and the thermal gradient decreases in the bottom near the hot wall and in the top near the cold wall. When rc goes to infinity the radiative heat transfer prevails and the isotherms are parallel to isothermal walls. Furthermore, one notes the 3D distribution of the temperature for low rc number. However for higher value of rc, the temperature field becomes independent of the flow and pure radiative transfer takes place. It should be noted that the inclination of the cavity has no effect on the temperature distribution when the conductive radiative parameter goes to infinity. In fact, the isothermal surfaces shown earlier in (Fig. (8(c), 9(c) and 10(c))), are identical and equidistant expected near active sides and they are symmetrical about the middle plane.
Figure 11 shows the evolution of mean radiative flux as a function of the angle of tilt for rc=10 and Ra =10^{5}. The inclination of the cavity minimizes the average value of the radiative flux which knows its minimum value to an angle close to 60 °. An inclination greater than 80° increases the average radiant flux which reaches its highest values.
The distribution of the mean radiative flux along the Y direction shows an almost identical look for an inclination angle of the cavity, below 80°. When the configuration becomes similar to the RayleighBenard case, the distribution of mean radiative flux is characterized by two extrema (Figure 12).
The distributions of conductive and radiative heat fluxes on the hot wall are represented (Figure 13 and Figure 14) for the different inclinations angles and for rc=10. It is remarkable that for γ=0°, radiation increases conductive heat transfer at the top of the hot wall and decreases it at the bottom. Due to the temperature levels, the radiative flux is higher on the hot wall while the conductive flux is more important on the cold wall. These characteristics are qualitatively similar to those obtained in 3D by Colomer et al. [6] and 2D Tan and Howell, (1991) computations.
By increasing the inclination angle of the cavity, it is remarkable that the maximum values of the conductive heat transfer decrease (Figure 13) and those of the radiative heat transfer increases (Figure 14).
(a) (b) (c) (d)
(a) (b) (c) (d)
5. Conclusion
The present numerical results are carried for Ra=10^{5}, Pr=0.71 and the optical with τ is equal to 1, Φt=0.1 and albedo equal to 1. Many conclusions are finding: in absence of radiation and by the increasing of the inclination angle of the cavity, we find that the formation of multiroll cells is favorites.
The radiation promotes the formation of three vortex when γ=30 ° and rather tends to make the flow monocellular when γ = 60 °. For γ=90°, the flow is with two contrarotative cells for low values of rc and it is with height rollcells when rc tends to infinite.
In absence of radiative transfer (rc=0), and by increasing the inclination angle, the isothermal surfaces are distorted and a 3D aspect appears in the core of the cavity. By the introduction of radiative fluxes, one notes the 3D distribution of the temperature for low rc number. However for higher value of rc, the temperature field becomes independent of the flow and pure radiative transfer takes place. It should be noted that the inclination of the cavity has no effect on the temperature distribution when the conductive radiative parameter goes to infinity. By increasing the inclination angle of the cavity, it is remarkable that the maximum values of the conductive heat transfer decrease and those of the radiative heat transfer increases.
Nomenclature
A – Dimension amplitude of the sinusoidal excitation
F – Aspect ratio
– acceleration of gravity
H– height of the cavity
i– refractive index
I – dimensionless radiant intensity,
I^{0}– dimensionless black body intensity
L – total number of discrete solid angles
n– unit vector normal to the control volume surface
P– pressure
Pr– Prandtl number
q_{c}– dimensionless local conductive heat flux on isothermal walls
q_{r}– dimensionless local radiative heat flux on isothermal walls
Ra– Rayleigh number
Rc– radiation conduction parameter
S– distance in the direction of the intensity
t– dimensionless time
T– dimensionless temperature
Tc– colde temperature
Th– hot temperature
– velocity vector
W– cavity width
Greek symbols
α– thermal diffusivity
β– extinction coefficient
β_{t}– coefficient of thermal expansion
ΔA– area of a control volume face
ΔV– control volume
ΔΩ^{l}– control solid angle
ε– emissivity
γ – inclinaition angle
ϕ_{t}– temperature ratio
k – absorption coefficient
– dimensionless vector potential
ϑ– kinematic viscosity
σ– Stefan–Boltzmann constant
τ– optical width
– dimensionless vorticity vector
ω_{0}– scattering albedo
– unit vector in the direction of the intensity
Subscript
x, y, z – Cartesian coordinates
Superscript
^{,}– real variables
References