A Boulemtafes-Boukadoum

C Abid

A Benzaoui
email: abenzaoui@gmail.com

3D numerical study of the effect of aspect ratios on mixed convection air flow in upward solar air heater

Keywords: Unsteady, mixed convection, CFD, solar collector, PRB flow, heat transfer, duct height

3D Numerical study of mixed convection air flow in upward solar air heater with large spanwise aspect ratio (A = 10 to 40) was performed using CFD commercial code Fluent14.5 ( ANSYS).The main objective of this study is to investigate the channel height's effect (Aspect ratio) on flow pattern and heat transfer in upward solar air heater in the particular case of low Re and high aspect ratio. The bottom plate (absorber) was submitted to Constant Heat Flux (CHF) in the range of 200 to 1000 W/m2 and Reynolds number was varied from 50 to 1000. The simulation results of flow visualization and Nusselt number calculation show that depending on Ri*, the velocity and temperature distribution in SAH vary greatly with channel's height. Our results are in concordance with most of authors conclusions about Poiseuille-Rayleigh-Benard flows. In mixed convection, increasing heat flux enhance heat transfer unlike forced convection flows. Simulation results of flow visualization and Nusselt number calculation have shown that depending on Ri*, the velocity and temperature distribution in SAH vary greatly with the channel's height. The obtained results were different from previous studies. Indeed, our investigation of channel's height was achieved for the same heat flux but different Grashof number. For low channel's height (high aspect ratio), enhancing heat flux has not a significant effect but for higher channel's height, an augmentation of heat flux enhance buoyancy effects in the flow and causes high turbulence, Also, increasing Reynolds number in low channel's height (high A), can enhance substantially heat transfer . For higher channel's height (low A), increasing Reynolds number decrease Ri* and thus buoyancy forces. Heat transfer is reduced and so Nusselt number. Those results may be very useful for engineers in designing and testing solar collectors.

Introduction

Mixed convection is a heat transfer mode where free convection phenomena coexist along those of forced convection ones. When buoyancy is small compared to inertia, mixed convective flow is simple and steady. But when the buoyancy is dominated over the inertia, the flow can become unsteady, transitional and even turbulent. Additionally, various complex flow structures like longitudinal vortices, transverse waves and flow reversal may appear during this buoyancy induced flow transition (Lin and Lin 1996b). Mixed convective heat transfer is a complex mechanism and understanding this process helps to improve design and optimization of energy devices by reducing heat losses and enhancing efficiency. Mixed convection is prevailing in wide range of engineering applications [START_REF] Wessels | On the Impact of the Aspect Ratio on the Formation of Large-Scale Structures in Turbulent Mixed Convection in a Cuboidal Sample[END_REF]. Detailed understanding of these buoyancy induced flow is important in fundamental fluid mechanics and heat transfer studies such as cooling of microelectronic components, heat transfer in compact heat exchangers, growth of single crystal through chemical vapor deposition (Lin and Lin 1996b), chemical and nuclear power, and in renewable energies as solar collectors [START_REF] Abid | Mixed Convection in a Horizontal Rectangular Duct Heated from Below[END_REF] still relevant, many recent scientific papers are related to this heat transfer mode over the world. [START_REF] Wessels | On the Impact of the Aspect Ratio on the Formation of Large-Scale Structures in Turbulent Mixed Convection in a Cuboidal Sample[END_REF][START_REF] Chakkingal | Assisting and Opposing Mixed Convection with Conjugate Heat Transfer in a Differentially Heated Cavity Filled with Coarse-Grained Porous Media[END_REF][START_REF] Taher | Experimental Determination of Heat Transfer in a Poiseuille-Rayleigh-Bénard Flow[END_REF][START_REF] Zhu | CFD Model Evaluation in Mixed Convection with High Richardson Number[END_REF].

Mixed convection heat transfer mode has been investigated both experimentally and numerically in a wide range of scientific papers. Most of those studies were about mixed convection between parallel plates or in rectangular channel heated from below, called Poiseuille-Rayleigh-Benard flows (PRB). In PRB flows at relatively low Reynolds and Rayleigh numbers, the combination of Rayleigh-Bénard cells with a Poiseuille flow give rise to very complex flow structures ranging from time-dependant transverse rolls to steady longitudinal rolls time-dependent mixed rolls and time-oscillating rolls (wavy, snaking or varicose) [START_REF] Benzaoui | Efficient Vectorized Finite-Difference Method to Solve the Incompressible Navier-Stokes Equations for 3-D Mixed-Convection Flows in High-Aspect-Ratio Channels[END_REF]. An exhaustive bibliographical review on the PRB flows was achieved by [START_REF] Nicolas | Revue bibliographique sur les écoulements de Poiseuille-Rayleigh-Bénard : écoulements de convection mixte en conduites rectangulaires horizontales chauffées par le bas[END_REF]. The study of Mori and Uchida [START_REF] Uchida | Forced Convective Heat Transfer between Horizontal Flat Plates[END_REF]) is considered as one of the first experimental studies on mixed convection of air flow between two horizontal parallel plates, where they could observe longitudinal rolls. According to literature, the longitudinal rolls appear as soon as Ra exceeds the critical value of 1708. [START_REF] Ostrach | Heat Transfer Augmentation in Laminar Fully Developed Channel Flow by Means of Heating From Below[END_REF] have shown experimentally that these rolls become unstable as soon as Ra> 8000. Other authors have noticed that the rolls structure is destroyed beyond Ra = 18000. [START_REF] Abid | Mixed Convection in a Horizontal Rectangular Duct Heated from Below[END_REF]) observed an original structure composed of two longitudinal rolls nested in a large transverse roll in the case of water laminar flow in a rectangular pipe (A=2) during stationary regime. Based on numerical and experimental study, [START_REF] Benderradji | Characterization of Fluid Flow Patterns and Heat Transfer in Horizontal Channel Mixed Convection[END_REF]) have noted for the first time the existence of two mechanisms of roll initiation in a horizontal channel uniformly heated from below at constant heat flux for water flow. The first initiation mechanism is due to lateral wall effect and the second one combines the wall effect with the existence of high vertical temperature gradient in the boundary layer at the bottom of the channel, which simultaneously triggers pairs of rolls in the whole zone between the two lateral rolls [START_REF] Benderradji | Characterization of Fluid Flow Patterns and Heat Transfer in Horizontal Channel Mixed Convection[END_REF].

Most of those studies revealed that buoyancy forces have a great effect on increasing heat transfer in mixed convection flows. In the entry region, [START_REF] Kamotani | Convective Heat Transfer Augmentation in Thermal Entrance Regions by Means of Thermal Instability[END_REF] noted that heat transfer depends not only on Rayleigh number but also on Ri=Gr/Re 2 which expresses the ratio of buoyancy forces to inertia forces. (Lin and Lin 1996a) explored experimentally the effect of Reynolds and Grashof numbers on the transition of mixed convection flow in a bottom-heated pipe. One of the main results obtained is that the buoyancy forces tend to thicken the boundary layer in the inlet region of the channel before the onset of instability. The Nusselt number tends to fall below the asymptotic value of the case of forced convection, but heat transfer is increased after the onset of instability. (Maughan and Incropera 1990) studied numerically the increase in heat transfer for air flow between two parallel plates heated from below. Their results showed that for large values of heat flux and flow rate, there is a sensitive enhancement of heat transfer prior to the establishment of thermal instability. Experimental measurements and visualization confirmed this result. [START_REF] Gau | Secondary Flow and Enhancement of Heat Transfer in Horizontal Parallel-Plate and Convergent Channels Heating from Below[END_REF] highlighted the effect of Ri on Nusselt number and they observed that when Ri increases, the locations for both heat transfer enhancement onset and maximum Nusselt number move upstream. In the literature and for heat transfer, some stability analysis express the Nusselt number as a function of Reynolds (Re), Rayleigh (Ra), and Prandtl (Pr) numbers and also of the transverse aspect ratio A (width to height) [START_REF] Rahli | Three-Dimensional Mixed Convection Heat and Mass Transfer in a Rectangular Duct: Case of Longitudinal Rolls[END_REF]. This parameter has been found to have a great impact on this heat transfer mode. Hence, (H. K. [START_REF] Moffat | Three-Dimensional Flow Effects in Silicon CVD in Horizontal Reactors[END_REF]H. Moffat and Jensen 1986) noticed that the structure of the secondary flow due to thermal instability is very sensitive to the channel aspect ratio A. In the case of a large transverse elongation (A ≥ 6), the longitudinal rolls appear gradually in the duct: they are first initiated near the vertical walls and then spread progressively to the interior of the channel [START_REF] Rahli | Three-Dimensional Mixed Convection Heat and Mass Transfer in a Rectangular Duct: Case of Longitudinal Rolls[END_REF]. This parameter has been investigated by Combined flow visualization and temperature measurement (Chang and Lin 1998). Thus, Reynolds number was varied from 2.5 to 50, Rayleigh number from 3000 to 20,000 and the aspect ratio A from 2 to 12, covering the steady and unsteady longitudinal vortex flows. [START_REF] Gau | Secondary Flow and Enhancement of Heat Transfer in Horizontal Parallel-Plate and Convergent Channels Heating from Below[END_REF]) achieved experimentations to investigate the effect of aspect ratio A (width to height) at the entrance of a channel of A=6.67. Reynolds number ranges from 100 to 2000 and the buoyancy parameter Ri from 2.5 to 907. In his paper [START_REF] Ozsunar | Numerical Analysis of Grashof Number, Reynolds Number and Inclination Effects on Mixed Convection Heat Transfer in Rectangular Channels[END_REF] [ reports preliminary results of a numerical investigation of mixed convection heat transfer in horizontal and inclined channel (aspect ratio A=10), uniformly heated from below. Reynolds number was ranged from 50 to 2000 and Gr from 7x10 5 to 4x10 6

Since the process is very complex, the first numerical studies on mixed convection were in 2D because it was difficult to model the complex phenomenon. 3D numerical simulations of PRB (Poiseuille-Rayleigh-Benard) flows are relatively recent and not so numerous because computations of such flows necessitate large meshes and high computational costs. [START_REF] Benzaoui | Efficient Vectorized Finite-Difference Method to Solve the Incompressible Navier-Stokes Equations for 3-D Mixed-Convection Flows in High-Aspect-Ratio Channels[END_REF] have presented and validated an efficient vectorized finite difference code to solve 3D incompressible Navier-Stokes equations under Boussinesq approximation for mixed convection flows in high aspect ratio channels. They dealt with steady longitudinal rolls, unsteady transverse rolls and for the first time, wavy rolls. With the development of CFD techniques [START_REF] Bammou | A Numerical Study of the Longitudinal Thermoconvective Rolls in a Mixed Convection Flow in a Horizontal Channel with a Free Surface[END_REF]) used this powerful tool to resolve complex flow and visualize the roll development mechanism in mixed convection in details. [START_REF] Rahli | Three-Dimensional Mixed Convection Heat and Mass Transfer in a Rectangular Duct: Case of Longitudinal Rolls[END_REF] Most of those studies studied the effect of Aspect ratio in the range of A=1-12 but higher aspect ratios have not been examined. Furthermore, the range of parameters used was not extended enough. Based on this literature survey, our aim in this study is to extend the most important parameters that impact mixed convection in the particular case of upward solar air heater (SAH). SAHs are particular type of heat exchangers that convert solar radiation into thermal energy using air as working fluid. The most important applications of SAHs are space heating and cooling, and some industrial heat process like drying [START_REF] Kalogirou | Solar Thermal Collectors and Applications[END_REF]. SAHs can also be integrated with photovoltaic (PV) systems to produce both thermal energy and electricity, called PV/T collectors. SAHs may be classified according to number of air passes into single-pass and double-pass. In single-pass SAH, air flows in one way either above the absorber plate or below it. They are generally divided into two types:

! type I (Upward air pass) with air flowing between the absorber plate and the cover glazing; ! type II (Downward air pass) with air flowing between the absorber plate and the back panel; There are several parameters impacting the solar air heaters efficiency: type and shape of absorber plate, number of glass cover plate, the use of artificial roughness [START_REF] Yadav | Heat Transfer and Fluid Flow Analysis of Solar Air Heater: A Review of CFD Approach[END_REF][START_REF] Boulemtafes-Boukadoum | CFD Based Analysis of Heat Transfer Enhancement in Solar Air Heater Provided with Transverse Rectangular Ribs[END_REF] or baffles. As it is described by (Hollands, K.G.T 1979), the collector geometry is one of the most important parameters to investigate in order to reduce convective losses. To reach high thermal efficiencies in SAHs, heat has to be transferred efficiently from the absorber to the flowing air. Several investigators have attempted to design more efficient solar air heaters by proposing appropriate design of SAH's air channel geometry. By theoretical analysis, (Altfeld, Leiner, and Fiebig 1988b;1988a) demonstrated optimal air velocity corresponding to the channel depth in downward SAH (type II). Hegazy [START_REF] Hegazy | Optimum Channel Geometry for Solar Air Heaters of Conventional Design and Constant Flow Operation[END_REF]) suggested a criterion ratio of depth to length as 0.0025 for SAH type II. [START_REF] Gan | Effect of Air Gap on the Performance of Building-Integrated Photovoltaics[END_REF] performed CFD simulations to find the optimal air gap in Trombe wall and behind building integrated photovoltaic panels [35].

Most of above-mentioned design and optimization studies are limited to forced convection flows neglecting the effect of mixed convection. However, the air flow rate in some practical operating conditions of SAH is very low and vary widely due to solar radiation randomness. At low Reynolds number and under specific solar radiation, buoyancy effects are very high in upward solar air heater leading to the onset of thermo-convective instability and longitudinal rolls development. [START_REF] Sun | Influence of Channel Depth on the Performance of Solar Air Heaters[END_REF] carried out 2D CFD analysis to found out the optimum channel height of SAH type I (upward), and type III (double pass). The optimal channel depths for type I and type III SAHs with black-painted absorber are suggested to be equal to 10 mm in the range of investigated parameters of mass flow rates between 0.01 to 0.06 Kg/s, corresponding to Reynolds number of 500 to 4000. For the parameters investigated, Ri* is very low, inertia forces are higher than buoyancy forces and hence, the results obtained are comparable to that of forced convection flows. The analysis was based on first and second laws of thermodynamics and thus does not take into account the effect of mixed convection. The aim of this study is to investigate 3D unsteady mixed convection air flow in Upward solar air heater under some specific parameters to approach the real operating conditions of SAH at low Reynolds numbers (50 to 1000). Our main objective is to extend mixed convection parameters range to large channel's aspect ratios (A= 10 to 40), various bottom wall heat flux (200-1000W/m 2 ) and to consider some additional effects, such as mixed heat boundary condition on the top wall (glazing). Since channel's height (Aspect ratio) is a principal variable to be fixed in the design of solar air heaters (SAHs), we emphasize on the influence of channel height on the flow structure and heat transfer augmentation using Computational Fluid dynamics (CFD) techniques. Combined flow visualizations and heat transfer calculations are carried out to investigate all these parameters in a large range of Grashof number (6.87x10 2 to 8.8x10 5 ) and Richardson number Ri*(6.87x10 -4 -351) covering forced, natural and mixed convection of steady and unsteady air flows.

Model and Numerical procedures

As depicted in Fig. 1, 3D mixed convection air flow in upward solar air heater was numerically analyzed. The physical domain consisted in cross section rectangular channel (LxWxH) heated from below. The top wall (glazing) is subjected to external heat losses: convective heat transfer with ambient and radiative heat exchange with the sky. Conductive heat transfer into glass thickness (ev) has been taken into account using SHELL CONDUCTION option of ANSYS Fluent code. The bottom wall (absorber)is submitted to Constant Heat Flux (CHF) (as received solar radiation). The lateral walls are considered adiabatic.

9> " + 9 " : 9@ " J + gβ(T -T M )cosθ (2) 9= 9B + u 9= 9; v 9= 9> + w 9= 9@ = - G ρ 9H 9> + ν I 9 " = 9; " + 9 " = 9> " + 9 " = 9@ " J + gβ(T -T M )sinθ (
Uniform air flow at the channel's entrance (u=u0, v=0, w=0), atmospheric pressure at the outlet and non-slip condition on the walls are the hydrodynamic boundary conditions applied to the studied domain. Thermal conditions applied are: T=Ti=Tamb at the inlet mixed boundary condition was applied on the glazing (top wall) : Qg=hw(Tg-Tamb)+εσ(Tg 4 -Ts 4 ) (6) With Ts, the sky temperature calculated by Swinbank expression [37]:Ts= 0.0552 * Tamb 1.5 (7) on the absorber: Constant heat flux imposed Qabs=200 to 1000 W/m 2 lateral walls: they are considered adiabatic with zero heat flux, q=0

Simulations parameters used to investigate the effect of channel's height on flow patterns and heat transfer in mixed convection airflow are: Qabs=200 to 1000 W/m 2 Re =50 to 1000 Gr*= 6.87x10 2 to 8.8 x10 5 Ri*=Gr*/Re 2 = 6.87x10 -4 to 351 Since this is a PRB flow study and as the major scientific works in this area, Reynolds number and modified Grashof number were calculated using H as characteristic length:

𝑅𝑒 = 6* , ( 8
)
𝐺𝑟 * = '()* ! +, " (9)
To estimate heat transfer we have calculated the local Nusselt number by

𝑁𝑢(𝑥) = 2(3)Z # + (10)
𝑤𝑖𝑡ℎ ℎ(𝑥)local convective heat transfer coefficient expressed by :

ℎ(𝑥) = ) S $(&) _S ((&) (11)

3-Numerical resolution

Considering the nonlinearity in the inertia terms of fundamental equations, they were solved numerically using a CFD(Computational fluid dynamics) code FLUENT 14.5 (ANSYS), based on finite volumes method (FVM). Second order Upwind scheme was used to discretize momentum and energy equations. Pressure was interpolated using PRESTO! scheme. We used PISO algorithm to handle the pressure-velocity coupling. Time advancement was carried out using First order implicit scheme. Due to the iterative process of numerical resolution, convergence was used as the monitor of final solution achievement. Convergence was considered as being completed when residuals become less than 10 -5 for the velocities and 10 -7 for energy for most of the cases tested. Visualization and calculations of some averaged variables were performed to confirm the convergence. As depicted in Fig. 2, temperature contours on the absorber is stable after almost 2000 to 3000 iteration in the steady state. (Case H=1cm, L=1m, Q=600W/m 2 , Re=50).

Grid independence checking

Velocity and temperature gradients are very important near the walls. Hence ,the generated grids were very dense near the walls. The meshes applied to the solution domain are structured and non-uniform with a refining near the walls. For L=1m, grids size vary from 530.000 to 2 400 000 cells depending on channel's height. Several grid sizes have been tested to ensure that the results are grid-independent. The appropriate mesh size is the one who gives the best compromise between accuracy and calculation cost. The results of test grid performed for the case of L=1m, H=1cm, Re=50, Q=600W/m 2 are presented below.(G1=135000 cells, G2=880 000, G3= 1 116 000 to G4=2340 000). In Fig. 3, is depicted Tabs evolution along the channel for the four grid meshes. We can observe that from the grid of 1116000 there is no noticeable difference. Thus G3 was selected to balance the computational load and the numerical accuracy.

Validation case

The model validation was performed for the case of forced convection air flow in a rectangular channel heated from below studied by [START_REF] Maughan | Experiments on Mixed Convection Heat Transfer for Airflow in a Horizontal and Inclined Channel[END_REF] as shown in Fig. 5. As predicted by theory, Nusselt number decrease quickly to reach an asymptotic value as expected for the case of pure forced convection. We can observe that our predicted longitudinal Nusselt number variations agree well with experimental results of Maughan. In the case of mixed convection, the reliability of our numerical results was verified by comparing them to data reported in literature: numerical solution of [START_REF] Ozsunar | Numerical Analysis of Grashof Number, Reynolds Number and Inclination Effects on Mixed Convection Heat Transfer in Rectangular Channels[END_REF] and experimental data of [START_REF] Maughan | Experiments on Mixed Convection Heat Transfer for Airflow in a Horizontal and Inclined Channel[END_REF]. The present results are related to the case of Gr*=2.10 5 , Re=250, A=10.Fig. 6 shows Nusselt number longitudinal variation in the flow direction. In mixed convection Nu decrease to the minimum then it start to increase to a maximal value before reaching the asymptotic value . This mechanism will be explain in details in the following results. This comparison has showed that our numerical predictions are much closer to experimental results of Maughan than those of Oszunar. This can be explained by the fact that we have used different CFD codes with different simulations parameters. Also, Oszunar has not used Boussinesq approximation and considered that air density is dependent of temperature only. Notice that in figure 5 and6, Nusselt number was calculated using H as characteristic length to follow Maughan results.

Results and Analysis

All computations were carried out for air flow (Pr = 0.71) in a rectangular channel of large aspect ratio A=10 to A=40, with Reynolds number varying from 50 to 1000.The modified Grashof number varies from 6.87x10 2 to 8.8 x10 5 and consequently the buoyancy-to-inertia ratio variation is within a wide range from very low to very high values.

In the present three-dimensional unsteady flow simulations, a large number of results data were obtained. Only a few of these results will be presented here (L=100cm, θ°=0) for horizontal channel to mainly explain the effects of various parameters on the characteristics of mixed convective flow patterns and heat transfer in the particular case of upward solar air heater.

Preliminary investigation of flow pattern Forced convection

Fig. 7 Nusselt number variation along the channel for the case of pure forced convection (A=20, Re=1000, Q=600W/m 2 ) Fig. 7 depicts Nusselt number variation in the longitudinal direction for H=1cm (A=20), Re=1000, Q=600W/W/m 2 . In this case Gr*=3,30x10 4 and Ri*=0.032 which verifies the conditions of pure forced convection. After a quick decrease, Nu reaches the asymptotic value of 5.61 which is close to the theoretical value of 5.39 for the case of parallels plates with one plate heated for fully developed laminar flow [START_REF] Meyer | Single-Phase Mixed Convection of Developing and Fully Developed Flow in Smooth Horizontal Circular Tubes in the Laminar and Transitional Flow Regimes[END_REF]. Is also plotted the correlation of (R.K. [START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF] for simultaneously thermally and hydrodynamically developing flow. Fig. 8 shows local Nusselt number as function of the inverse of Graetz number. Local Nusselt number at different heat fluxes is presented in Fig. 9. As expected, there is no significant difference between the four heat fluxes, which confirms that is pure forced convection flow. After the onset of the instability, ascending plumes of warm fluid growth along the side walls from the heated surface. Thus, the bottom plate is cooled by fresh fluid descending from the core of the channel to the plate. This secondary flow is a strong mechanism to heat transfer enhancement. This is the region of mixed convection, where Nusselt number is highly enhanced depending on Grashof and Reynolds numbers. Toward the channel exit, the amplitude of variations in Nusselt number decrease to reach a stable shape. The mechanism of roll development is well illustrated in Fig. 11, where are plotted the contours of local temperature on the absorber (bottom heated plate) and also in Fig. 12 where are shown the contours of both temperature and velocity longitudinal component u in different cross sections along the channel. At x=0.01m, the flow is in laminar forced convection regime characterized by vertical temperature gradient and parabolic shape of velocity's contours.

As the flow moves downstream, the boundary layer at the bottom wall becomes thicker and the temperature is higher. This leads to the rise of longitudinal rolls near the side walls (x=0.15m). This is confirmed by the contours of u velocity that exhibits a distortion at the same locations. Then at x=0.3m more longitudinal rolls are induced but the fluid's core is still not destabilized yet and rolls number increase in pairs to reach the fully developed state and their maximum number (20) at x=0.45 m.

Here, the roll's height and lateral extension are higher as observed by [START_REF] Gau | Secondary Flow and Enhancement of Heat Transfer in Horizontal Parallel-Plate and Convergent Channels Heating from Below[END_REF]. This length where the rolls invade the whole cross section corresponds to the establishment (development) length and it will be noted Le. At the channel exit, the roll size is approximately equal to the channel's height [START_REF] Benderradji | Characterization of Fluid Flow Patterns and Heat Transfer in Horizontal Channel Mixed Convection[END_REF] .The rolls can be identified in both temperature and velocity contours.

3-1 Effect of heat flux supplied to the absorber (effect of Grashof number)

To examine the influence of Grashof number on rolls generation and temperature distribution, we varied the heat flux from 200 to 1000 W/m 2 . Fig. 14 shows the temperature contours as visualized by ANSYS code for the case of H=1cm, A=20, Re=50.

The main observations are that as heat flux increases, more rolls are induced and development length Le decreases as noticed by many authors [START_REF] Benderradji | Characterization of Fluid Flow Patterns and Heat Transfer in Horizontal Channel Mixed Convection[END_REF][START_REF] Gau | Secondary Flow and Enhancement of Heat Transfer in Horizontal Parallel-Plate and Convergent Channels Heating from Below[END_REF]. The longitudinal rolls are induced by the first roll initiation mechanism: rolls are primary initiated at the sidewalls of the channel and then they spread gradually to the flow's core in the downstream direction as already observed by previous authors. As heat flux increase (Q=600 w/m 2 , Ri*=13.19), buoyancy forces are higher and the boundary layer at the heated wall becomes thicker. Hence, rolls initiation occurs at a much earlier stage and more rolls are induced [START_REF] Incropera | Mixed-Convection Flow and Heat Transfer in the Entry Region of a Horizontal Rectangular Duct[END_REF]. Nevertheless, these rolls are parallel and stationary. Such, the establishment length is shifted towards the channel inlet. When Q=800W/m 2 (Ri*=17.59), the number of induced rolls is again increased and one can notice a rolls destabilization so that several dislocations are observed. The development to unsteady state is more highlighted when the heat flux is more important (Q=1000W/m 2 , Ri*=22.26).

The flow structure evolves versus the longitudinal direction from laminar forced convection, laminar mixed convection, transitional and finally turbulent.

To examine locally the effect of heat flux on the roll development for different channel's height, we have plotted temperature (Fig. 13) and u velocity contours (Fig. 14) at the same location (x=0.25 m) for five various heat flux values to highlight the effect of heat flux on the flow structure and roll development.

The temperature contours indicate clearly that for Q = 200 W/m 2 , the flow's core is still dominated by forced convection and longitudinal rolls induced by side walls effect have a mushroom shape as described by [START_REF] Gau | Secondary Flow and Enhancement of Heat Transfer in Horizontal Parallel-Plate and Convergent Channels Heating from Below[END_REF]. At Q=400 W/m 2 , more longitudinal rolls are induced in the flow's core. For higher heat fluxes (Q=600, 800, 1000 W/m 2 ), corresponding to a higher Ri*, one can observe that the longitudinal rolls in the central zone develop simultaneously at the lower wall. These central rolls, referred as thermal plumes by [START_REF] Gau | Secondary Flow and Enhancement of Heat Transfer in Horizontal Parallel-Plate and Convergent Channels Heating from Below[END_REF], are initiated by the supercritical vertical thermal gradient at the lower wall (similar to Rayleigh-Benard instability). This is the second mechanism of roll initiation highlighted by [START_REF] Benderradji | Characterization of Fluid Flow Patterns and Heat Transfer in Horizontal Channel Mixed Convection[END_REF]) . Also, one noticed that the number of rolls produced is proportional to Grashof number as reported in the literature [START_REF] Gau | Secondary Flow and Enhancement of Heat Transfer in Horizontal Parallel-Plate and Convergent Channels Heating from Below[END_REF], [START_REF] Rahli | Three-Dimensional Mixed Convection Heat and Mass Transfer in a Rectangular Duct: Case of Longitudinal Rolls[END_REF], (Incropera et al.1987). ) corresponding to Grashof number of (1.1 to 5.5x10 4 ). First, as expected, the maximum Nusselt numbers are close to the inlet where thermal boundary layer is thinner [START_REF] Meyer | Single-Phase Mixed Convection of Developing and Fully Developed Flow in Smooth Horizontal Circular Tubes in the Laminar and Transitional Flow Regimes[END_REF]. For all the curves, Nusselt number decrease near the inlet, secondary flow effects are very small and Nusselt number variation is close to that of forced convection. With the increase of heat flux and consequently the Grashof number, the point where the onset of instability occurs is moved towards the channel inlet and secondary flow effects become larger. This effect is due to an increase in the thermal boundary layer thickness and temperature gradient that accelerate development of secondary flows [START_REF] Maughan | Experiments on Mixed Convection Heat Transfer for Airflow in a Horizontal and Inclined Channel[END_REF]. The onset of instability in the flow is very fast, resulting in large fluctuations in Nusselt number values and thus higher enhancement in the heat transfer is obtained. Consequently, at the same Reynolds number, with increasing heat flux, instability occurs earlier and faster and Nusselt number is higher [START_REF] Ozsunar | Numerical Analysis of Grashof Number, Reynolds Number and Inclination Effects on Mixed Convection Heat Transfer in Rectangular Channels[END_REF] . These observations agree well with [START_REF] Meyer | Single-Phase Mixed Convection of Developing and Fully Developed Flow in Smooth Horizontal Circular Tubes in the Laminar and Transitional Flow Regimes[END_REF] who reported that for mixed convection flow in circular ducts, the entrance length decreases when thermal effects are high enough. As reported by [START_REF] Maughan | Experiments on Mixed Convection Heat Transfer for Airflow in a Horizontal and Inclined Channel[END_REF], Nusselt number for each surface heat flux appears to approach a final asymptotic value, suggesting the existence of a fully developed condition. This is confirmed experimentally by Meyer and al for mixed convection in circular ducts. The authors describe 3 regions with different transfer mechanism: the Leveque region where Nusselt number decrease due to thermal entrance effects (FCD) to reach its minimal value where free convection and entrance effects are balanced. In region2, Nu is increasing due to buoyancy forces (MCD) before reaching the In Fig. 15-b), one can notice that for larger Grashof numbers, there is a noticeable departure in Nusselt number's curve from the forced convection zone before reaching the minimum as observed by [START_REF] Maughan | Experiments on Mixed Convection Heat Transfer for Airflow in a Horizontal and Inclined Channel[END_REF]. Rapid boundary layer growth in this upstream region prevents the secondary flow from fully compensating for the effect of the thickening boundary layer. Hence, Nusselt number, although enhanced above the forced convection limit, continues to decline. An additional contribution to heat transfer enhancement in this region may come from an induced streamwise pressure gradient [START_REF] Maughan | Experiments on Mixed Convection Heat Transfer for Airflow in a Horizontal and Inclined Channel[END_REF] .

Q=200W/m 2 Q=400W/m 2 Q=600W/m 2 Q=800W/m 2 Q=1000 W/m 2

3-2 Effect of channel's height (Aspect ratio)

Results presented on Fig. 16 exhibit the effect of the channel's height on the flow pattern for typical case of Re=50 and Q=200W/m 2 . The channel' height varies from 0.5cm to 2cm corresponding to aspect ratio A= 40 to 10. In this case, even if heat flux is the same, modified Grashof number which is very sensitive to height ( α to H 4 ) changes widely from 6.87x10 2 to 1.76x10 5 . In previous works, comparison were performed for two aspects ratio channels for the same Grashof number. In this study, we have decided to keep heat flux constant accordingly to real operating SAH under solar radiation. The goal is to find the optimal height for the same solar radiation. For H=0.5cm, the flow looks to be entirely in forced convection regime because the buoyancy to inertia ratio is low (Ri*=0.27). For H=1cm, Ri* is more important (Ri*=4.39), the onset of instability occurs close to the inlet and the roll length development is approximately reached at X=0.35cm. As the value of the channel's height increases (H=1.5cm, Ri*=22.26), the point of instability onset is shifted toward the channel's inlet and the number of rolls decreases as expected. For higher value of the channel's height (H=2cm), the buoyancy to inertia ratio is very high (Ri*= 70.36) and the longitudinal steady roll structure is destroyed, the regime is turbulent. Consequently, increasing channel's height enhances rolls induction and decreases development length but beyond a certain value, the regime becomes first chaotic and then turbulent. So, reducing the duct's height can stabilize the flow. As reported by [START_REF] Rahli | Three-Dimensional Mixed Convection Heat and Mass Transfer in a Rectangular Duct: Case of Longitudinal Rolls[END_REF], the aspect ratio has an effect on the developed rolls number. This number is even despite the value of A. On Fig. 17 and Fig. 18, we have plotted the temperature and u velocity distribution in a cross section at x=0.3m. One can see that for H=0.5 cm, there is no rolls and the flow seems entirely in forced convection regime. At H=1cm, the temperature and U velocity contours show that the length roll development is not reached yet. When the channel height increases (H=1.5cm), one can notice that the cross section of the rolls increases and their number decreases. The rolls structure symmetry is destroyed for H=2cm. To highlight the effect of channel's height on heat transfer, we have plotted longitudinal spanwise-averaged Nusselt number distribution versus the longitudinal position in Fig. 19. One can observe that near the inlet, all the curves show the same decrease rate corresponding to the forced convection developing region. At low channel's height, the curve shape is analogue to that of forced convection regime. As the buoyancy forces increase, the Nusselt number raises rapidly to reach its maximum value related to heat transfer enhancement before reaching an asymptotic value (H=1cm ). For H=1.5cm and H=2cm, the modified Grashof number is very high, consequently Nusselt number increases with channel's height increase and after reaching its maximum value the Nusselt number shows large fluctuations. One can notice that the point of instability onset advances upstream as the channel's height increases.

Hence , for the same heat flux, increasing H (decreasing A) , enhances heat transfer and Nusselt number is augmented. This is an interesting result because it is different from previous studies: (Abou-Ellail and Morcos 1983), [START_REF] Ozsunar | Numerical Analysis of Grashof Number, Reynolds Number and Inclination Effects on Mixed Convection Heat Transfer in Rectangular Channels[END_REF], [START_REF] Maughan | Experiments on Mixed Convection Heat Transfer for Airflow in a Horizontal and Inclined Channel[END_REF]) performed comparative study for different aspect ratio and One can notice that for low channel's heights, increasing heat flux will enhance the onset of instability and induce the longitudinal rolls development first near the wall sides. It causes also development length Le decrease. At high channel's height, reducing surface heat flux can change the flow state from chaotic to turbulent. From Fig. 21, the main result is that for low channel's height (high aspect ratio), enhancing heat flux has not a significant effect but for higher channel's height, an augmentation of heat flux enhance buoyancy effects in the flow and causes high turbulence. This conclusion may be very useful in designing and testing solar collectors.

3-3 Effect of Reynolds number

Mixed convection flows are deeply dependent to Grashof and Reynolds numbers. In the following we will present the results obtained from investigation of Reynolds number effect on the flow structure and heat transfer enhancement in the case of channel's height of H=1cm, subjected to heat flux Q=600w/m 2 . Reynolds number is varying in the range of 50 to 1000. In Fig. 21 To investigate the effect of Re on heat transfer, we have plotted the longitudinal spanwiseaveraged Nusselt number distribution versus the longitudinal position for various Reynolds number on Fig. 22. The Grashof number is the same (Gr*=3,30x10 4) but Buoyancy to inertia ratio varies widely.

For small Reynolds number (Re=50, 100) corresponding to high buoyancy to inertia ratio (Ri*=13.19 and 3.29 respectively), the induced secondary flow enhances heat transfer and Nusselt number is increased. As Reynolds number increases, (Ri*=0.52, 0.13 and 0.032) respectively) secondary flow effects withdraw, they are almost non-existent for Re=1000 and Nusselt number variation is similar to that of forced convection. So, increasing Reynolds number, reduces fluctuations in Nusselt number and also decrease its average value. This result is in agreement with the conclusions reported by Ozsunar, Maughan and Lin. For low channel's height ( high A), increasing Reynolds number can enhance substantially heat transfer and increased Nusselt number. For higher channel's height, increasing Re decrease Ri* from very high values to low one but still in the free convection region. One can observe that onset of secondary flow is shifted towards the exit and Nusselt number shows large fluctuations. So, one can conclude that decreasing Reynolds number will reduce the roll development length, enlarge the mixed convection region and enhance substantially the heat transfer depending on buoyancy to inertia ratio (Ri*).

4-Conclusion

3D Numerical study of mixed convection air flow in upward solar air heater (Type I) with large spanwise aspect ratio (A = 10 to 40) was carried out. The analysis was achieved using CFD commercial code Fluent14.5 ( ANSYS).

The studied domain concerns an air flow into a rectangular channel uniformly heated from below with thermally insulated lateral walls. The top wall (glazing) is subjected to external heat losses: convective heat transfer with the ambiance and radiative heat exchange with the sky. The bottom plate (absorber) is submitted to heat flux in the range of 200 to 1000 W/m 2 that represents absorbed solar radiation. The air flows at low and moderate Reynolds number from 50 to 1000. Heat transfer characteristics of flow exhibiting longitudinal rolls was investigated in a wide range of parameters. The main objective was studying the influence of channel's height (spanwise) aspect ratio on the flow pattern and heat transfer. It was found that our results are in agreement with most of author's conclusions about PRB flows: H=2cm/Re=50 H=2cm/Re=250

• Such water flows, in mixed convection air flows, longitudinal rolls are initiated by first or second mechanism depending on Ri*.

• At the same Reynolds number, increasing heat flux (modified Grashof number) will faster the onset of instability and Nusselt number is higher at the difference of forced convection flows.

• When Re increases, the instability onset point is moved downstream, the roll length development Le is increased and the roll's number decrease.

• Decreasing Reynolds number will decrease the roll development length, enlarge the mixed convection region and enhance substantially the heat transfer depending on buoyancy to inertia ratio.

Simulation results of flow visualization and Nusselt number calculation have shown that depending on Ri*, the velocity and temperature distribution in SAH vary greatly with the channel's height. The obtained results were different from previous studies. Indeed, our investigation of channel's height was achieved for the same heat flux but different Grashof number. Hence,:

• At the same heat flux and Re number increasing channel's height enhances rolls induction and decreases length roll development but beyond a certain value, the regime becomes first chaotic and then turbulent. So, reducing the channel's height can stabilize the flow. For heat transfer, Nusselt number increases with channel's height and shows large fluctuations. One can notice that the point of instability onset advances upstream as the channel's height increases. • For low channel's height (high aspect ratio), enhancing heat flux has not a significant effect but for higher channel's height, an augmentation of heat flux enhance buoyancy effects in the flow and causes high turbulence. This conclusion may be very useful in designing and testing solar collectors. So, reducing surface heat flux can change the flow state from chaotic to turbulent. • For low channel's height ( high A), increasing Reynolds number can enhance substantially heat transfer . For higher channel's height (low A), increasing Reynolds number decrease Ri* and thus buoyancy forces. Heat transfer is reduced and so Nusselt number. So, one can conclude that reducing Reynolds number will reduce the roll development length, enlarge the mixed convection region and enhance substantially the heat transfer depending on buoyancy to inertia ratio (Ri*).

Fig. 1

 1 Fig.1 Schematic view of the computational domain

  Fig.2 Temperature contours for different iteration numbers

Fig

  Fig.3Local absorber temperature evolution along the channel (z=0.1) obtained with four mesh grids

Fig. 5

 5 Fig.5 Comparison of Nusselt number longitudinal distribution in the case of forced convection

Fig. 6

 6 Fig.6 Comparison of Nusselt number longitudinal distribution in the case of mixed convection (Gr*=2.10 5 , Re=250, A=10)

Fig. 8 Fig. 10

 810 Fig.8Nusselt number as function of the inverse of Graetz number for (A=20, Re=1000, Q=600W/m 2 )

Fig

  Fig.11 Temperature contours on the absorber plate (H=1cm, Q=200W/m 2 ,Re= 50)

Fig. 14

 14 Fig.14 Longitudinal roll structures induced by various heat flux supplied to the absorber plate (H=1cm, Re=50)

  Fig. 15-b) Zoom on the region of instability onset

  Fig.16 Effect of channel's height on roll's development (L=1m, Re=50, Q=200 W/m 2 ) x=0.3m

  number increases with Aspect ratio (decreasing H) but for the same Grashof number.

Fig. 19

 19 Fig. 19 Effect of channel's height on Nusselt number distribution (Re=50, Q=200 W/m 2 )

Fig. 20

 20 Fig. 20 Comparison of heat flux effect on longitudinal rolls development for different channel's height

Fig. 22

 22 Fig.22 Contours of temperature for H=1cm, Q=600 W/m 2 and Re=50-1000

Fig. 24

 24 Fig.24 exhibits a comparison of the flow pattern of different channel's height (H=0.5cm to H=2cm) for two Reynolds number flow (Re=50 and Re=250). At low channel's height, reducing Reynolds number promotes rolls appearance and reduces greatly the roll length development Le. At high channel's height, it causes the vortex flow to change from unsteady or chaotic to steady state with symmetric rolls splitting and merging.

  . Mixed convection studies are

	atm	atmospheric
	b	bulk
	g	glazing, top wall
	i	inlet
	o	outlet
	Nomenclature s sky
	Cp	constant-pressure specific heat of air
	D	hydraulique diameter
	e	glass thickness
	g	gravitational acceleration
	G	incident solar radiation intensity on the solar air heater
	H	channel's height
	h	local convective heat transfer coefficient
	hw	heat transfer coefficient between glazing and the external surrounding air
	k	air thermal conductivity
	L	channel's length
	𝑚̇	air mass flow rate
	P	pressure
	Q	heat flux imposed on the absorber
	q	useful energy gain
	S	aperture area of SAH
	T	temperature
	U	air averaged velocity in the duct
	u	longitudinal velocity component
	v	transversal velocity component
	w	azimuthal velocity component
	W	channel's width
	Dimensionless numbers/parameters
	𝐺𝑟 * =	'()* ! +, "	modified Grashoff number
	𝑁𝑢(𝑥) = 𝑅𝑒 = 6* ,	2(3)* +	local Nusselt number Reynolds number
	Ri*=Gr*/Re 2	modified Richardson number
	A=W/H		Channel's spanwise Aspect ratio
	X + =X/L		Dimensionless abscissa
	1/Gz=x/RePrD	Inverse Graetz number
	Greek letters
	ε	glazing emissivity
	θ	inclination angle
	σ	stefan-boltzmann coefficient
	𝛽	air thermal expansion coefficient
	𝜈	air kinematic viscosity
	Subscripts
	abs, p absorber, bottom wall
	amb	ambient

  , flow pattern represented by temperature contours on the absorber (bottom plate) are shown. For Re=50(Ri*=13,19), the flow is characterized by three regions: the forced convection region, the instability onset region and the steady roll development region. When Re increases, flow velocities are high, which delay the enlargement of the thermal boundary layer and inhibit the buildup of instability and[START_REF] Ozsunar | Numerical Analysis of Grashof Number, Reynolds Number and Inclination Effects on Mixed Convection Heat Transfer in Rectangular Channels[END_REF], consequently, the instability onset point is moved downstream. Moreover, the roll length development Le is increased and the roll's number decreases. This effect is more apparent for Re= 250 (Ri*=0.52), where we can observe the stretching of the forced convection region. For Re=500 to 1000 (Ri*=0.13; 0.032), inertia forces are very high, only wall sides rolls are induced and the flow reaches the exit without reaching steady state longitudinal rolls.

	50
	Nu				H=0.5cm/Q=200W/m2
					H=0.5cm/Q=600W/m2
	40				H=1cm/Q=200W/m2
					H=1cm/Q=600W/m2
					H=2cm/Q=200W/m2
	30				H=2cm/Q=600W/m2
	20
	10
	0
	0	0,2	0,4	X +	0,6	0,8	1

Absorber temperature contours with 135000 cells (G1)

Acknowledgments: The authors thank The IUSTI (AMU, France) for providing computer resources and CFD code to achieve this study.


Rachid Bessaïh
email: bessaih.rachid@gmail.com

Chérifa Abid

HEAT TRANSFER ANALYSIS FOR MIXED CONVECTION IN A HORIZONTAL CYLINDRICAL DUCT

Keywords: Mixed convection, heat transfer, horizontal duct, local and average Nusselt numbers S

This paper deals with three-dimensional, mixed convection in a cylindrical duct horizontally. This latter is partially subjected to a uniform volumetric heat generation at the solidliquid interface. The working fluid (water) with a parabolic velocity profile and a constant temperature T0 enters the tube. The study was carried out for different Richardson numbers values Ri=0.1-8 at Reynolds number Re=600. Results were conducted so that to show the influence of Richardson number Ri on the dynamic and thermal fields and local Nusselt number Nu(q,z) and peripherally averaged axial Nusselt number Nu(z). Also, analyses of the results showed that the hydrodynamic effects are manifested by the existence of secondary flow, inducing temperature gradient at a crosssection between the top and bottom of the duct. The reversed flow is observed for a low Reynolds number Re=10. A significant increase in heat exchange is observed in mixed convection compared to pure forced convection flow. Correlations for the variation of average Nusselt number Nu (Ri,z) at the entrance region with Ri and z=z¢/Di and local Nusselt Nu(Ri) number in the hydrodynamics establishment zone are proposed and compared with the present numerical results.

Introduction

Heat transfer with mixed convection in ducts is of great interest for industrial applications like heat exchangers, cooling of nuclear reactors, solar collectors, geothermal power generation, the food, and biomedical industries…etc... Mixed convection was the topic of many theoretical, experimental, and computational studies in pipes. For example, Abid et al. [START_REF] Abid | Etude de la convection mixte dans un conduit cylindrique. Approches analytique/numérique et détermination expérimentales de la température de paroi par thermographie infrarouge[END_REF] investigated mixed convection in a long heated horizontal cylindrical duct using analytical and numerical approaches. The wall temperature was experimentally determined by infrared thermography. Results show the hydrodynamic establishment of a secondary flow characterized by the presence of two counter-rotating transverse rollers. Besides, the thermal establishment is characterized by an increase in the mean temperature of the cylinder wall. K.Choi and H. Choi [START_REF] Choi | Developing mixed convection flow in a horizontal tube under circumferentially non-uniform heating[END_REF] have shown the increase in heat exchange and the appearance of natural convection rolls, related to the establishment of mixed convection inside a non-uniformly heated horizontal tube on its circumference. Hwang et al. [START_REF] Hwang | Laminar convective heat transfer in a horizontal isothermal tube for high Rayleigh numbers[END_REF] studied numerically mixed convection within a cylindrical horizontal duct. A correlation was obtained giving Nu with Ra . Ouzzane and Galanis [START_REF] Ouzzane | Effets de la conduction pariétale et de la répartition du flux thermique sur la convection mixte près de l'entrée d'une conduite inclinée[END_REF] analyzed the effects of parietal conduction and heat flow distribution on laminar mixed convection inside a conduit inclined for four different thermal boundary conditions. They illustrated the effects on conjugated transfer flow and non-uniform heating conditions. Habib et Negm [START_REF] Habib | Laminar mixed convection in horizontal concentric annuli with non-uniform circumferential heating[END_REF] presented a numerical analysis on steady laminar mixed convection in a concentrated horizontal annulus with nonuniform circumferential heating. Results suggested that the effects of buoyancy on the mixed convection are clearly sensitive to heaters. Orfi and Galanis [START_REF] Orfi | Developing laminar mixed convection with heat and ass transfer in horizontal and vertical tubes[END_REF] were interested in the effects on flow, thermal, and concentration fields of the solutal and thermal Grashof numbers in horizontal and vertical tubes. Results show the influence of the Grashof number and the inclination of the tube on the flow field. The effect of the Lewis number on laminar mixed convective heat and mass transfer in a horizontal tube with uniform heat flux and concentration was investigated numerically by Orfi and Galanis [START_REF] Orfi | Mixed convection with heat and mass transfer in horizontal tubes[END_REF]. The authors concluded that the impact of Lewis number on the number of Sherwood is greatest near tube entrance. [START_REF] Mohammed | Combined convection heat transfer for thermally developing aiding flow in an inclined circular cylinder with constant heat flux[END_REF] experimentally approached laminar heat transfer with mixed convection for airflow in a cylindrical duct tilts for different Re values, heat flux, and inclination angle. They deduced that the surface temperature along the duct, local Nusselt number distribution according to dimensionless axial distance (x/D.Re.Pr). They have also found that if the heat flow increases for the same Re, the surface temperature increases. Correlations that give Nu based on Ra for three inclinations were obtained. In Mohammed and Salman's experimental work on mixed convection in a horizontally oriented cylinder [START_REF] Mohammed | Free and forced convection heat transfer in the thermal entry region for laminar flow inside a circular cylinder horizontally oriented[END_REF], authors were able to conclude that free convection reduces heat transfer for a small Reynolds number (Re = 400) and increases heat transfer for Re = 1600. Mohammed and Salman [START_REF] Mohammed | Combined natural and forced convection heat transfer for assisting thermally developing flow in a uniformly heated vertical circular cylinder[END_REF][START_REF] Mohammed | Experimental investigation of mixed convection heat transfer for thermally developing flow in a horizontal circular cylinder[END_REF][START_REF] Mohammed | The effects of different entrance sections lengths and heating on free and forced convective heat transfer inside a horizontal circular tube[END_REF][START_REF] Mohammed | The effect of different inlet geometries on laminar flow combined heat transfer inside a horizontal circular pipe[END_REF] performed experimental work on natural and forced convective heat transfer in both vertical and horizontal circular cylinders. For example, in [START_REF] Mohammed | Combined natural and forced convection heat transfer for assisting thermally developing flow in a uniformly heated vertical circular cylinder[END_REF], the average heat transfer results were correlated with the empirical correlation, and an increase in the Nu values as the heat flux rises was observed in [START_REF] Mohammed | Experimental investigation of mixed convection heat transfer for thermally developing flow in a horizontal circular cylinder[END_REF] for all entrance sections.

Mohammed and Selman

The effect of geometric parameters on laminar mixed convection in an inclined eccentric straight annulus was investigated numerically by Shiniyan et al. [START_REF] Shiniyan | The effect of geometric parameters on mixed convection in an inclined eccentric annulus[END_REF]. Results showed that for an angle of inclination equal to 55 °, Nu has the maximum value. Testi [START_REF] Testi | A novel correlation for azimuthal and longitudinal distributions of heat transfer coefficients in developing horizontal pipe flow under transitional mixed convection[END_REF] measured heat transfer coefficients at six cross-sections along the uniformly heated horizontal cylindrical pipe under transitional mixed convection; for the entire heat transfer surface, a single Nusselt number correlation was proposed. Chae and Chung [START_REF] Chae | Laminar mixed-convection experiments in horizontal pipes and derivation of a semi-empirical buoyancy coefficient[END_REF] performed experimentally mixed convection in horizontal pipes for different values of Gr and Re.

A correlation was developed using a semi-empirical buoyancy coefficient. Schriener and El-Genk [START_REF] Schriener | Convection heat transfer of NaK-78 liquid metal in a circular tube and a trilobe channel[END_REF] performed a numerical analysis of turbulent heat transfer by convection in a uniformly heated circular tube. The average Nusselt numbers for fully developed flows are correlated with Peclet number. Colombo et al. [START_REF] Colombo | Fully developed laminar mixed convection in uniformly heated horozonatl annular ducts[END_REF] studied numerically heat transfer by mixed convection in a uniformly heated concentric annulus. Results show that mass flow reduction is affected by axial shear stress. In a horizontal duct, Ganesan et al. [START_REF] Ganesan | Mixed convection and radiation heat transfer in a horizontal duct with variable wall temperature[END_REF] conducted a numerical analysis of mixed convection heat transfer with radiation. Results show that average Nu is influenced by flow conditions and surface radiation.

The present work deals with laminar mixed convection in a long, uniformly heated, horizontal duct at the wall. The fluid employed is water. The effect of Ri on the fluid flow, thermal field, Nu(q,z), and Nu(z) have been investigated numerically.

To the literature relevant, there are very few experimental or numerical works, showing the variation of Nu(z) in the entry region and Nu(q,z) along the peripheral wall. The study focuses on heat transfer, including the variation of average and local Nusselt numbers.

This paper is structured according to the following. Section 2 provides a description of the geometry and the governing equations with boundary conditions. The numerical approach and code validation are discussed in Section 3. Section 4 describes the results and discussions.

2.Geometry description and governing equations 2.1 Geometry description

Figure 1 depicts the geometry under consideration. A horizontal cylinder having a length L=100cm, an internal diameter Di=1cm and an external diameter Do=1.04cm, is made of Inconel (thermal conductivity kS=20W/m.K). An electric current I produces a rate of energy generation per unit volume qv by Joule effect along with the thickness of the wall e=0.2 cm. At the inlet duct, the axial velocity v of the water flow has a parabolic profile and the fluid is held at 15 °C.

Governing equations

By neglecting the viscous dissipation and using Di, v0, ρv0 2 , and qvDi 2 /kS as typical scales for L z¢ r¢ q Heated section, qv Unheated section

Di

Inconel of thickness, e

Inlet fluid lengths, velocities, pressure, and temperature, the non-dimensional equations for a steady, threedimensional, incompressible laminar flow with constant thermophysical properties, only the density was related with the temperature variation in the buoyancy terms [START_REF] Bejan | Convection Heat Transfer[END_REF], can be written as follows in cylindrical coordinate: Continuity:

(1) r-Momentum:

(2) z-Momentum:

(3) q-Momentum: (4)
Energy equation: [START_REF] Habib | Laminar mixed convection in horizontal concentric annuli with non-uniform circumferential heating[END_REF] where, , , .

0 θ w r 1 z v r (ru) r 1 = ¶ ¶ + ¶ ¶ + ¶ ¶ Θ cosθ Re Gr θ w r 2 r u z u ν z θ u ν θ r 1 r u rν r r 1 Re 1 r P r w z u v θ u r w r u u 2 2 2 * * 2 * 2 + ú û ù ê ë é ¶ ¶ - - ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ¶ ¶ - = - ¶ ¶ + ¶ ¶ + ¶ ¶ ú û ù ê ë é ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ¶ ¶ - = ¶ ¶ + ¶ ¶ + ¶ ¶ z v ν z θ v ν θ r 1 r v rν r r 1 Re 1 z P z v v θ v r w r v u * * 2 * Θ sinθ Re Gr θ u r 2 r w z w ν z θ w ν θ r 1 r w rν r r 1 Re 1 θ P r 1 r uw z w v θ w r w r w 2 2 2 * * 2 * - ú û ù ê ë é ¶ ¶ - - ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ¶ ¶ - = + ¶ ¶ + ¶ ¶ + ¶ ¶ u S z Θ k z θ Θ k θ r 1 r Θ rk r r 1 RePr 1 z Θ v θ Θ r w r Θ u * * 2 * + ú û ù ê ë é ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ÷ ø ö ç è ae ¶ ¶ ¶ ¶ + ÷ ø ö ç è ae ¶ ¶ ¶ ¶ = ¶ ¶ + ¶ ¶ + ¶ ¶ ï ï î ï ï í ì ¥ ® = = zone

Boundary conditions

The dimensionless boundary conditions of unheated and heated regions along the horizontal duct are:

• Fluid region: At z=0 (inlet duct), 0£r£0.5, 0£q£2p: u=0, v=2(1-4r 2 ), w=0, Q=0.

At z=L/ Di =100 (outlet duct), 0£r£0.5, 0£q£2p:

At r=0 (axis), 0£q£2p, 0£z£100, the dynamical axial conditions are considered and u, v, w, and Q are correctly interpolated near the axis:

• Solid region: At 0.5£r£0.52, 0£q£2p,0£z£100: u=v=w=0,

• Unheated region : At 0.5£r£0.52, 0£q£2p, 0£z£25: u=v=w=0, Where, hr is the coefficient of radiative heat transfer, defined as [START_REF] Bergman | Fundamentals of Heat and Mass Transfer[END_REF] : , and h the coefficient of the convective heat transfer, h=2.7W/m 2 .K and the emissivity coefficient e=0.9.

Nusselt number

The local Nusselt number along the peripheral wall is:

(7)
Where Qw and Qm (z) are the dimensionless wall and mean temperatures, respectively. Qm (z) is defined as follows:

(8)
The Nu(z) peripherally averaged axial number is calculated by the integration of the local Nusselt number Nu(q,z) along the heated wall:

(9) ν Di v Re 0 = 2 S 5 v ν k Di q gβ Gr ÷ ÷ ø ö ç ç è ae = a n = Pr 0 z Θ k z z w z v z u * = ÷ ø ö ç è ae ¶ ¶ ¶ ¶ = ¶ ¶ = ¶ ¶ = ¶ ¶ 0 r Θ r r w r r v r r u r = ÷ ø ö ç è ae ¶ ¶ ¶ ¶ = ÷ ø ö ç è ae ¶ ¶ ¶ ¶ = ÷ ø ö ç è ae ¶ ¶ ¶ ¶ = ÷ ø ö ç è ae ¶ ¶ ¶ ¶ ( ) Θ k Di h h r Θ k f r 0.52 r * + = ¶ ¶ - = 0 r Θ = ¶ ¶ ( ) ( ) ¥ ¥ + + = T T T T h r 2 2 se ( ) (z) Θ Θ r Θ z , Nu m w 0.5 r - ¶ ¶ = = q ò ò ò ò = Q 0.5 0 2π 0 0.5 0 2π 0 m rdrdθ v rdrdθ Θ v (z) ( ) ò = 2π 0 dθ z , Nu 2π 1 Nu(z) q
3.Numerical method and validation

3.1Numerical method

The Finite-volume approach is used to solve the governing equations ( 1)-( 5). The numerical procedure known as SIMPLER [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF] is used to treat the coupling of pressure-velocity. The discretized algebraic equations are solved by the tridiagonal matrix algorithm (TDMA) line-by-line [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF]. Convergence is confirmed when the maximal relative change falls below 10 -5 for u, v, w, and Θ between two consecutive iteration levels.

3.2Analysis of grid independence

In this work, three grids are used:32×82×42, 32×162×42, and 32×202×42 nodes. In the fluid, the non-uniform mesh is adopted in the radial and axial direction; the mesh is more refined in regions where the variations in velocity and temperature are important (At the inlet duct and the solid-fluid interface). In the wall of thickness e, five nodes have been chosen in the direction r. In the azimuthal direction, a uniform mesh was adopted. The values of Nu(z) by using the grids 32×162×42 and 32×202×42 nodes are 10.98, respectively, for Re=600 and Ri=2. To maximize the processing time and cost of computations, the grid of 32×162×42 nodes has been used. For the results presented here the total processor time on the workstation, hp Z820 was around 10 h for each case.

Code validation

The FORTRAN code for mixed convection in a heated long circular duct was validated using experimental data from Abid et al. [START_REF] Abid | Etude de la convection mixte dans un conduit cylindrique. Approches analytique/numérique et détermination expérimentales de la température de paroi par thermographie infrarouge[END_REF].It should be noted that considered comparison [START_REF] Abid | Etude de la convection mixte dans un conduit cylindrique. Approches analytique/numérique et détermination expérimentales de la température de paroi par thermographie infrarouge[END_REF] for temperature profiles at Grashof number Gr=2.57×10 5 and Reynolds number Re=143.28. Figure 2 shows a good comparison of temperature at the top (q=0) and bottom (q=p) of the duct walls. Our results were compared with the experimental work of Barozzi et al. [START_REF] Barozzi | Experimental investigation of combined forced and free convection in horizontal and inclined tubes[END_REF] and numerical data of Akbari et al. [START_REF] Akbari | Fully developed mixed convection in horizontal and inclined tubes with uniform heat flux using nanofluid[END_REF] at Rayleigh number Ra=2.4×10 4 and Re=200. The obtained peripherally averaged local axial Nusselt number Nu(z) in comparison with refs. [START_REF] Barozzi | Experimental investigation of combined forced and free convection in horizontal and inclined tubes[END_REF] and [START_REF] Akbari | Fully developed mixed convection in horizontal and inclined tubes with uniform heat flux using nanofluid[END_REF] are illustrated in figure 3. As shown in figures 2-3, It is evident that our predictions are in excellent agreement with the references [START_REF] Abid | Etude de la convection mixte dans un conduit cylindrique. Approches analytique/numérique et détermination expérimentales de la température de paroi par thermographie infrarouge[END_REF], [START_REF] Barozzi | Experimental investigation of combined forced and free convection in horizontal and inclined tubes[END_REF] and [START_REF] Akbari | Fully developed mixed convection in horizontal and inclined tubes with uniform heat flux using nanofluid[END_REF].

T, C

4.Results and discussion

Results presented in this section are relative to numerical simulations carried out in mixed and forced convection in a horizontal cylindrical duct. The aspect ratio L/Di=100 (where L is the duct length equal to 100cm and Di its internal diameter Di=1cm). The Richardson number Ri= Gr/ Re 2 is ranged between 0 and 8, which corresponds to Grashof number values Gr=0 and 2.88×10 6 and the Reynolds number is equal to Re=10 and 600. All results are presented in dimensionless form.

4.1Reversed flow

Numerical simulations (see Figure 4) clearly show the influence of natural convection on the reversed flow, represented by velocity vectors in the r-z plane, at low Re and Gr numbers (Re =10 and Gr=10 5 ). For these values, forced convection is dominated by natural convection. The horizontal channel is composed of two zones: one unheated and the other heated. In the unheated zone (0≤(z¢/Di)/10≤2.5), the imposed flow at the duct inlet is of Poiseuille type, only the axial velocity exists while the other components of the velocity are zero. In the heated zone (2.5<(z¢/Di)/10≤4), we see that near the upper wall, a part of the fluid is reversed (returned); velocity vectors take a direction opposite to that of the main flow (see, also Figure 5). This can be attributed to the importance of heat transfer by natural convection versus forced convection, which generates convective rolls. The heat transfer by natural convection is carried out from the hot fluid of the heated zone to the cold fluid of the unheated zone.

Flow structure

For the case (Ri=2 and Re=600), the value of z¢/Di=36 corresponds to the hydrodynamic established zone, which agrees with the Abid et al. [START_REF] Abid | Etude de la convection mixte dans un conduit cylindrique. Approches analytique/numérique et détermination expérimentales de la température de paroi par thermographie infrarouge[END_REF] experimental data. It should be noted that in the absence of heating in the zone (0≤(z¢/Di)/10≤2.5), the flow profile is parabolic, imposed by the condition at the entrance of the duct, the maximum of which the dimensionless axial velocity v=2 is located at r=0, and the minimum at the wall (Figure 6). Upon entry of the heated zone, the flow represented by the transverse components (u-w) in the r-q plane begins to grow gradually; the flow pattern consists of two convective rolls induced by natural convection (Figure 7). The existence of the secondary flow for Ri=2 and Re=600 is characterized by relatively low transverse velocities in comparison with that of the main flow, and contributes to the modification of the axial profile (Figure 6). It is visible in Figure 7 that there is a formation of two counter-rotating and symmetrical cells in the cross-section. Near the wall, the movement is upward while in the central area of the duct; it is descending creating a counter-clockwise vortex: the less dense fluid particles will be pushed upwards. By symmetry, the flow follows the same path but in a clockwise direction. During the hydrodynamic establishment, one notices an increase in transverse velocities parallel to the dumping of the convective rolls; as z¢/Di increases, the convective rolls undergo compaction downwardly (Figure 7).

1 0 . at Ri = 2 and Re = 600.

Figure 8 presents the profiles of the axial velocity vs as function of the diameter of the tube, for varying values of z¢/Di at Ri=Gr/Re 2 =2 and Re=600. The fluid accelerates in the central part of the section near the duct entrance. Subsequently, there is a slowing of the fluid as z¢/Di increases until the duct exit. The parietal heating that accelerates the hot fluid at the wall may explain this slowing. and Re=600.

Thermal field

Figures 9a-b illustrate the thermal fields in the r-q plane for two values of Ri = 0.1 and 2 and different z¢/Di, at Re = 600. It is clear that in the unheated zone; isotherms are uniform, thus highlighting the absence of heat transfer between the fluid and the wall. On the other hand, in the heated zone, it is noted that the convective rolls disturb the isotherms along the duct. In the lower zone, the heat is transferred by the secondary flow to the upper zone of the cross-section. However, in the upper part of the transverse section, isotherms are almost horizontal (i.e. there is no thermal convection in the axial direction), evolving towards stratification of the flow. Similar results were observed in the works of Abid et al. [START_REF] Abid | Etude de la convection mixte dans un conduit cylindrique. Approches analytique/numérique et détermination expérimentales de la température de paroi par thermographie infrarouge[END_REF] and [START_REF] Abid | Thermal instabilities in a horizontal cylindrical duct: A physical approach[END_REF]. This stratification occupies an increasingly important area. In a cross-section, the coldest zone of the fluid moves downward when z¢/Di increases. The effect of Richardson number from 0.1 to 2 on the thermal field is clearly shown in Figure 9b; due to the contribution of natural convection. 1 .

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3 .

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5 .

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7 .

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9 Figure 10 displays the profiles of the temperature Q in respect of the diameter of the tube, for three values of z¢/Di at Ri=2 and Re=600. The values of these profiles show that the temperature in the upper zone increases with increasing z¢/Di. This is highlighted by the temperature contours in a crosssection as shown in Figure 9.

Heat Transfer

The various obtained results allowed us to determine heat transfer coefficient. Thus, figure 11 illustrates the variation of Nu (z) with z¢/Di for different values of Ri, at Re = 600. We can notice that the hydrodynamics establishment zone varies according to Richardson number Ri. At z¢/Di= 25, the Nu(z) values are identical to those obtained by forced convection (Ri=0). We can see in this region that the peripherally averaged local axial Nusselt number decreases rapidly for different Richardson number values. This can be attributed to low natural convection activity coupled with thermal boundary layer formation. A significant impact of mixed convection on Nusselt number values is observed. In the region of hydrodynamic establishment, Nu (z) takes an almost horizontal to the exit duct. The behavior was observed experimentally by Barozzi et al. [START_REF] Barozzi | Experimental investigation of combined forced and free convection in horizontal and inclined tubes[END_REF] and numerically by Akbari et al. [START_REF] Akbari | Fully developed mixed convection in horizontal and inclined tubes with uniform heat flux using nanofluid[END_REF]. Nu (z) values in mixed convection are greater than those obtained in forced convection (Fig. 11).

Figure 12 relates the variation of Nu (z) as a function of z¢/Di at Ri=0 (pure forced convection) and Re = 600. Here, the fluid is air and the value of our prediction Nu (z)=4.43, which is in very good agreement with that resulting from an analytical solution with constant surface heat flux (Nu (z)=4.36) [START_REF] Bergman | Fundamentals of Heat and Mass Transfer[END_REF].

Figure 13 displays the local Nusselt number Nu(q,z), at z¢/Di =80, like a function of angular coordinate q for various values of Ri (Re = 600) in the cross-section. The shape of Nu(q,z) profiles is Gaussian. The values of Nu(q,z) are maximum in q=p (180°) and minimum in q=0 (0°), for the different Richardson numbers. To summarize the discussion of this figure, it can be said that the heat transfer increases sharply as one goes from top to bottom of the cross-section.

In the inlet region, a correlation for the variation of the peripherally averaged local axial Nusselt number average Nu (Ri, z) on the heated wall based on the Richardson number and z=z¢/Di in a crosssection is proposed: Nu(Ri, z)= 2390 Ri 0.140 (z¢/Di) -1.537 Re=600, Ri=2 Q These two correlations (10) -(11) compared with the numerical results are illustrated in Figures 14 and15, respectively. Figure 14 shows a slight difference between the numerical results and correlation, and a good agreement is observed in figure 15. Figu.13. Variation of Nu(q,z), at z¢/Di =80 (in the established zone), as a function of angular coordinate q for different values of Ri , at Re = 600.

Conclusion

In this paper, a numerical analysis was conducted of mixed convection in a horizontal cylindrical duct. We found the main following results:

• Reversed flow was observed at low Re, showing a natural convection effect on flow.

• Natural convection is causing a secondary flow, characterized by convective rolls.

• Secondary flow causes temperature gradient through a cross-section between the top and bottom of the duct. • In the hydrodynamic establishment zone, the average values of the Nusselt number are higher than those obtained under forced convection. • In mixed convection, the heat exchange between the fluid and the wall is better than forced convection.

Fig. 1 .

 1 Fig. 1. Horizontal duct with boundary conditions.

  Grashof and Prandtl numbers are defined, respectively, as :

Fig. 2 .

 2 Fig. 2. Comparison of our numerical results and the Abid et al. experimental data[1].

Fig. 3 .

 3 Fig.3. Comparison of our results with those of Akbari et al.[START_REF] Akbari | Fully developed mixed convection in horizontal and inclined tubes with uniform heat flux using nanofluid[END_REF] and Barozzi et al.[START_REF] Barozzi | Experimental investigation of combined forced and free convection in horizontal and inclined tubes[END_REF] experimental data.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Velocity vector field in the r-z plane along the horizontal duct, showing the reversed flow, for Re=10, Gr=10 5 .

Fig. 6 .Fig. 7 .

 67 Fig.6. Contours of the axial velocity v for different values of z¢/Di, at Ri = 2 and Re = 600.

Fig. 8 .

 8 Fig. 8. Axial velocity profiles v with the diameter of the conduit, for different values of z¢/Di at Ri=Gr/Re 2 =2and Re=600.

Fig. 9 .

 9 Fig. 9. Temperature fields for various values of z¢/Di and Ri, at Re = 600.

Fig. 10 .

 10 Fig.10. Profiles of the temperature Q with the tube diameter, for various values of z¢/Di at Ri=2 and Re=600.

( 10 ) 11 )

 1011 And in the established zone, z¢/Di = 80, another correlation for various Richardson number values is given as follow:Nu(Ri)= 9.576 Ri 0.71 (

Fig. 11 .

 11 Fig. 11. Variation of Nu (z) with z¢/Di for various values of Ri, at Re = 600 (for water).

Fig. 12 .

 12 Fig. 12. Variation of Nu(z) with z¢/Di at Ri=0 and Re = 600 (for air).

Fig. 14 .

 14 Fig.14. Variation of the peripherally averaged local axial Nusselt number Nu (Ri,z) as a function of z¢/Di in the entry zone: comparison between numerical results and correlation.

Fig. 15 .

 15 Fig. 15. Variation of Nu (Ri) with Ri in the established zone: comparison between numerical results and correlation.