

ORIGINAL ARTICLE 

Year : 2016  Volume
: 1
 Issue : 2  Page : 6474 

Numerical study of natural convection and radiation exchange in an asymmetrically heated inclined channel
Shafqat Hussain
Department of Mechanical Engineering, Al Imam Mohammad Ibn Saud Islamic University, Riyadh 11432, Kingdom of Saudi Arabia
Date of Submission  22May2016 
Date of Acceptance  13Jun2017 
Date of Web Publication  21Aug2017 
Correspondence Address: Shafqat Hussain Department of Mechanical Engineering, Al Imam Mohammad Ibn Saud Islamic University, P.O. Box 5701, Riyadh 11432 Kingdom of Saudi Arabia
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/ijas.ijas_21_16
Introduction: In this paper, numerical study is carried out to investigate natural convection and radiation heat transfer in an asymmetrical heated inclined air channel with open ends. The Reynoldsaveraged Navier–Stokes equations are solved using a commercial computational fluid dynamics solver ANSYSFLUENT^{©} in conjunction with the discrete ordinate radiation model. Simulations were run considering the channel with inclination angle to horizontal in the range 18° to 45° and a wall surface emissivity of 0.27–0.95. The channel length to channel space ratio was selected in the range 44–220. A uniform heat flux in the range 100–500 W/m^{2} was applied along the upper wall of the channel while the lower wall and side walls were assumed thermally insulated. Results: Temperature profiles along the upper and lower walls of the channel were obtained with variations in the channel length to space ratios, angles of inclination, radiation emissivity, and input Ohmic heat flux. The effect of various parameters on the maximum wall temperature and heat transfer was investigated. The numerical predictions are validated by comparison with the experimental data available in literature. Conclusion: The numerical results obtained are found in good agreement with the experimental measurements. From the numerical results, a correlation of local and average Nusselt number with modified channel Rayleigh number in the range of 10^{1}–10^{5} is developed at asymmetric heat flux boundary conditions. Nu = 0.6851Ra” ^{0.2612} Keywords: Computational fluid dynamics modeling, inclined air channel, natural convection and radiation heat transfer, validation
How to cite this article: Hussain S. Numerical study of natural convection and radiation exchange in an asymmetrically heated inclined channel. Imam J Appl Sci 2016;1:6474 
How to cite this URL: Hussain S. Numerical study of natural convection and radiation exchange in an asymmetrically heated inclined channel. Imam J Appl Sci [serial online] 2016 [cited 2021 May 12];1:6474. Available from: https://www.eijas.org/text.asp?2016/1/2/64/213388 
Introduction   
In recent years, computational fluid dynamics (CFD) techniques are being widely used to model the heating and cooling processes and are becoming a valuable tool in analyzing the design and operation of various components and equipment used in thermal systems. In the beginning, these techniques were mostly employed to study the flow and heat transfer mechanisms in vertical parallelplate channels,^{[1],[2],[3],[4],[5],[6]} but later on their application was extended to the analysis of flow and heat transfer in inclined air channels.^{[7],[8],[9],[10],[11],[12],[13],[14],[15]} However, in these earlier studies, there remained some concerns about the accuracy of the presented results mainly due to the complex nature of these flows resulting from the interaction of various heat transfer modes and due to the lack of availability of experimental data for comparison. Hence, more comparative studies between CFD and experiments are required which could address a few of these concerns. Some experimental results on vertical and inclined air channels were reported in various studies,^{[16],[17],[18],[19]} but the need still exists for a more comprehensive numerical as well as experimental study to improve the design of inclined air channels.
Natural convection in closed channels is of interest in a number of engineering applications such as passive cooling of solar collectors. Detailed thermal analysis of the channel is of importance in component and system design. In the past, vertical channels formed by parallel walls had been studied extensively. In comparison to vertical channels, the study of natural convection heat transfer in inclined channels, however, received relatively less attention. Fedorov and Viskanta^{[20]} developed a model to predict induced flow and heat transfer in an asymmetrically heated, vertical parallelplate channel. The flow of air in the channel was induced by thermal buoyancydriven force. The model predictions were compared with the available experimental data. It was found that the low Reynolds number kε turbulence model predicted the heat transfer reasonably well in the thermal buoyancydriven flows. The local heat flux and Nusselt number results were presented over a wide range of conditions to develop a better understanding of the flow physics.
Cheng and Muller^{[21]} performed numerical and experimental investigations on turbulent natural convection coupled with thermal radiation in a vertical rectangular channel with oneside heated wall. They developed their own radiation model and combined it with the CFD code FLUTAN. They found the numerical method to be efficient and accurate for investigating the flow and heat transfer in the channel. It was shown that when wall emissivity was not very low, thermal radiation contributed significantly to the total heat transfer by natural convection even at low temperatures of the heated wall. Based on the experimental and numerical results, they developed a semiempirical correlation for turbulent natural convection coupled with thermal radiation in vertical rectangular channels. Yilmaz and Fraser^{[22]} investigated both experimentally and numerically turbulent natural convection in a vertical parallelplate channel. The velocity and temperature measurements were made at the channel outlet. A commercial CFD code was used to simulate heat transfer and fluid flow in the channel. The numerical method used was found to predict heat transfer and flow rate fairly accurate.
Nouanégué and Bilgen^{[23]} carried out a numerical study of a solar chimney system for heating and ventilation of dwellings. The governing conservation equations were solved using a finite differencecontrol volume method. The Nusselt number, the dimensionless volume flow rate, and radiation heat flux ratio were calculated as a function of the governing parameters. It was shown that surface radiation had pronounced effect on the flow and temperature profiles, the Nusselt number and volume flow rates, which consequently improved the ventilation performance of the chimneys. The Nusselt number was found an increasing function of the surface emissivity, the conductivity ratio and the wall thickness. Bacharoudis et al.^{[24]} investigated the buoyancydriven flow and heat transfer inside a wall solar chimney with one adiabatic wall and other wall under a heat flux. The governing elliptic equations were solved numerically using a control volume method. The flow was turbulent and six different turbulence models were tested. Comparison with experimental data showed that the performance of realizable kε model was better under strong adverse pressure gradients.
Yang and Yang^{[25]} studied the effect of inclination on natural convection in an airfilled differentially heated enclosure both experimentally and numerically. They calculated local and mean Nusselt numbers at various inclination angles, ranging between 0^{o} and 180^{o} for Rayleigh numbers between I0^{4} and l0^{6}. Comparison of the numerical predictions with experimental data was made, the heat flux at the hot and cold boundaries showed strong dependence on the angle of inclination and the Rayleigh number. Said et al.^{[26]} studied numerically the effect of the angle of inclination on the heat transfer characteristics of a parallelwalled isothermal vertical channel. They plotted the velocity vectors, the temperature contours, and the local and average Nusselt number distributions provided comparisons of the computed Nusselt numbers with published experimental and numerical results which showed good agreement. It was reported that the local and average heat transfer decreased as the angle of inclination was increased.
Onur et al.^{[27]} studied experimentally the effect of channel spacing and inclination angle on the natural convection of air between inclined parallel plates, in which the upper plate was heated isothermally whereas the lower plate was insulated. Experiments were performed for inclination angles of 0°, 30°, and 45° with respect to the vertical position. The results indicated that the heat transfer rate was influenced by both the channel spacing and inclination angle. Lin and Harrison^{[28]} investigated experimentally heat transfer by natural convection and radiation exchange in an asymmetrically heated, inclined air channel. They conducted experiments on channel with inclination angles ranging from 18° to 30° with respect to horizontal position, and wall surface emissivity ranged from 0.29 to 0.95. The channel length/space (L/S) ratio was varied in the range of 44–220. In each test, a uniform Ohmic heat flux (q”) was applied along the top wall of the channel while the bottom wall was thermally insulated. Temperature profiles along both the top and bottom walls of the channel were recorded at different q” and L/S ratios. The dependency of maximum wall temperature and heat transfer on the channel spacing (S) and surface emissivity (ε) was explored, and correlations for local and average Nusselt numbers were developed as a function of modified channel Rayleigh number.
Although natural convection in vertical parallelplate channels had been studied extensively in the past several decades covering various aspects of the problem, the literature survey showed that not enough research was conducted on the combined natural convection and radiation heat transfer in inclined parallelplate channels. The objective of this study is to develop a CFD model to investigate the combined effect of natural convection and radiation on the flow and temperature fields in the air channels proposed for passive cooling of residential solar collectors under stagnation conditions. In the present study, the inclined parallelplate air channel with asymmetric heating studied earlier experimentally by Lin and Harrison^{[28]} is selected. The Reynoldsaveraged Navier– Stokes (RANS) equations are solved using a commercial CFD solver ANSYSFLUENT^{©} in conjunction with the discrete ordinate (DO) radiation model.
The numerical results of the airflow velocity and temperature distributions are presented for different values of channel length to space ratios, angles of inclination, radiation emissivity, and input Ohmic heat flux. The CFD model predictions are validated by comparing the numerical results with the experimental data recorded by Lin and Harrison.^{[28]} The numerical results are found to be in good agreement with the experimental data. From the results of this study, a correlation for overall average Nusselt number is developed as a function of modified channel Rayleigh number.
Description of an Inclined Air Channel   
A schematic of the inclined air channel geometry^{[28]} selected for the present study is shown in [Figure 1]. It consisted of two 2.4 m long (L) and 0.34 m wide (W) plates. The top plate of the channel (ceiling) was made of 2 mm thick aluminum sheet with a surface emissivity of 0.27. The bottom plate (floor) of the channel was made of 1 mm thick galvanized steel sheet with surface emissivity of 0.37. To investigate the effect of radiation exchange on the natural heat transfer in the channel, both surfaces were coated with black paint to have an emissivity of 0.95. The channel was mounted on a steel frame with an adjustable angle of inclination (θ) that varied from 18° to 30°.  Figure 1: Side view of the air channel used in the experimental study of Lin and Harrison^{[28]}
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The inlet of the channel was located 0.3 m above the floor. The height of the channel space (s) could be adjusted by moving the top plate. Electric heating strips were bonded to the upper surface of the channel ceiling with a silicon adhesive that made good surface contact providing a high heat transfer rate. A length of approximately 0.1 m at the entrance and exit of the channel was not heated. A 75 mm foam insulation layer was applied on top of the heating strip to minimize the heat loss. The sides and bottom of the channel were also insulated with 25 mm thick foam board. Further details of the air channel can be found in study by Lin and Harrison^{[28]} and its application can be seen in an Enerworks flatplate solar collector shown in [Figure 2].
Experimental Data   
The experimental data recorded by Lin and Harrison^{[28]} are used to validate the numerical predictions of the present work and the experimental procedures followed by them are briefly described here. In the experimental tests, an AC power supply was used to regulate the power input to the heaters in the range of 50–400 W/m^{2}. In each test, the Ohmic heat flux(q) was calculated from the measured voltage and resistance across the heaters. The expected accuracy of the measurement of q was ±1.5%. The temperature distribution along the top wall of the channel was measured with copper/constantan thermocouples, starting at 0.1 m from entrance, and spaced at 0.3 m intervals. The accuracy of the temperature measurements was reported ± 0.5°C. Three rows comprising eight thermocouples each were placed along the center, left and righthand sides of the channel's top wall. On the bottom wall of the channel, eight thermocouples were placed along its centerline opposite to the ones on the top wall. Measurements were recorded using a computerbased dataacquisition system at 5min intervals until steadystate was reached. The ambient temperature was measured at a location close to the channel inlet. The tests were carried out in an airconditioned room with temperatures maintained between 18°C and 24°C throughout the testing. More details on the experimental measurements can be found in study by Lin and Harrison.^{[28]}
Mathematical Modeling   
Governing equations and assumptions
Numerical investigations were conducted along with the experiments to achieve better understanding of thermal phenomena for passive cooling of collectors under stagnation conditions. In the present study, attention has been focused on the modeling of natural convective flows and temperature distributions in the inclined air channel of the collector. The governing equations that must be solved numerically in dealing with the threedimensional flow in this situation consist of the continuity equation, the momentum equations, and the energy equation. In the numerical model, it was assumed that (i) the flow is steady, turbulent, and threedimensional, (ii) the flow is single phase, i.e., the effects of dust particles and water vapor are neglected, (iii) the air properties are constant except for the density change with temperature that gives rise to the buoyancy forces which have been treated using the Boussinesq approximation, and (v) external ambient conditions are steady. Turbulence was modeled using the RANS approach. With these assumptions, the set of governing equations for conservation of mass, momentum, and energy in Cartesian coordinates (the ydirection being in the vertical direction) in terms of eddy viscosity and turbulent diffusivity are as follows:
Mass conservation:
Momentum conservation,
xdirection:
ydirection:
zdirection:
Energy conservation:
Turbulence modeling
To describe the additional terms arose in equations^{[8]} to^{[11]} due to time averages, a turbulence model is used. The kω turbulence model allows an accurate prediction of the velocity field in the proximity of solid walls (Versteeg and Malalasekera, 1995; Zaidi et al., 2010). Keeping in view the length of the paper, a brief summary of the mathematical expressions of the turbulence model is described below.
SST kω turbulence model
It was developed by Mentor (1994) and incorporates a cross diffusion term in the ω equation and a blending function to allow proper calculation of the nearwall and the farfield areas. This makes the model more precise for a larger variety of flows. The transport equations of the model are as follows:
Where represents the generation of turbulent kinetic energy that arises due to the mean velocity gradient, G_{ω} represents the generation of specific dissipation of kinetic energy, ω and Y_{k} and Y_{ω} represent the dissipation of K and ω due to turbulence, D_{ω} is the cross diffusion term. The constants specific to the SST kω model are defined as follows:
Simulations have been conducted using the commercial CFD solver ANSYSFLUENT14^{©}. Results obtained by employing the turbulence model have been compared to experiments to validate the model used.
Radiation modeling
The radiation exchange between the walls of the channel was modeled using the DOs, radiation model. The DOmodel solve the radiation transport equation (given below) for a finite number of discrete solid angles, each associated with a vector direction fixed in the global Cartesian system (x, y, and z). The DOmodel does not perform ray tracing.
Where I is radiation intensity, is position vector, is direction vector, s is path length, a is absorption coefficient,
σ_{s} is scattering coefficient, n is refractive index, σ is Stephan–Boltzma constant, T is local temperature, ϕ is phase function, and Ω is solid angle. The details of the model are available in the (fluent 6.3_documents).
Solution procedures
To get the solution, for pressure discretization, the body forceweighted scheme was employed using the SIMPLEalgorithm for pressurevelocity coupling discretization. The secondorder upwind scheme was used to discretize the momentum and other equations in the numerical simulations. To get a converged solution, the equations for mass and momentum conservation were iteratively solved until the sum of the absolute normalized residuals for all the cells in flow domain became <10^{−4} while the energy equation was iterated until the residual fell below 10^{−6}, the solution then being considered to be converged. Underrelaxation factors 0.3, 1, 0.2, 0.8, 0.8, 1, and 0.9 for pressure, density, momentum, turbulence kinetic energy, turbulence dissipation rate, turbulent viscosity, energy respectively were used. In all the cases considered, the convergence criteria were met after about 10,000 iterations using a mesh size of approximately 900,000 cells. The mesh size was selected based on the results of the mesh independent tests.
Computational domains and spatial discretization
A modeling of the air flow phenomena involved in the air channel implies that the mesh should be used to properly define a minimum cell size to compute the air flow appropriately with the proposed geometry. Mesh density used depends on the nearwall modeling strategy determined by the y + characteristic parameter. Keeping in view the required y + values near the walls and computational capability of the available computers, cell count in the range 500,000–1,400,000 was used in the present simulations. Mesh sensitivity test was carried out to examine the mesh independence of the numerical results. Three mesh densities were investigated: Mesh 1 (544,620 elements), Mesh 2 (970,200 elements, [Figure 3], and Mesh 3 (1,378,440 elements).  Figure 3: Computational grid for the air channel, the enlarged view is shown on the right
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Boundary conditions
To solve the mathematical model, an appropriate set of boundary conditions close to the conditions used in the experimental tests has been imposed using the experimental data. The lower and side walls of the channel were assumed to be adiabatic. All boundaries of the domain, except the channel openings, and heat source, were modeled as noslip (u_{j} = 0) wall boundaries with zero heat flux. A constant relative pressure of 0 P_{a} was imposed across the channel inlet and the channel outlet. The turbulence parameters such as the turbulence intensity were specified at the inlet. At the outlet opening, it was assumed zero gradients normal to the opening for all of the solved variables. Water tubes were not considered for simplicity in the simulations. Density at the inlet plane was computed using the Boussinesq approximation. At the outlet, FLUENT uses the boundary condition ambient pressure as the static pressure of the fluid and extrapolate all other conditions from the interior of the domain.
Validation of the Model   
To validate the CFD model, the numerical results obtained are compared with the experimental measurements obtained by Lin and Harrison.^{[28]} A comparison of the predicted and measured average temperature values along the dimensionless axial positions of the channel for (L/S) ratios of 110, 73, and 44 are shown in [Figure 4].  Figure 4: Comparison of the predicted and measured^{[28]} average temperature values on the heated wall along the dimensionless axial position of the channel for (L/S) ratio of 73 with q” of 400 W/m^{2}, ε of 0.27 and θ of 18°
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From [Figure 4], it will be seen that the numerical predictions are in good agreement with the experimental measurements. The accuracy of the present simulations is within ±4% of the experimentally determined temperature values by Lin and Harrison.^{[28]}
Results and Discussions   
Air flow and thermal analysis in the channel
In this subsection, airflow and thermal analysis are performed and are presented to see the effect of channel space geometry on the velocity profiles and temperature distribution for the conditions; channel length to space (L/S) ratio = 73, angle of inclination (θ) = 18°, Ohmic heat flux (q”) = 400 W/m^{2} and emissivity (ε) = 0.27. Representative air velocity contours and vector plots in the channel space along the xy plane at Z = 0.17 m are shown in [Figure 5] and [Figure 6] with enlarged views near the inlet and outlet of the channel.  Figure 5: Velocity contours (m/s) along the xy plane of the channel at Z = 0.17 m, with enlarged views near the inlet (a) and near the outlet (b) for conditions (L/s ratio = 73, θ = 18°, q” = 400 W/m^{2} and ε = 0.27)
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 Figure 6: Velocity vectors along the xy plane at Z = 0.17 m, with enlarged views near the inlet (a) and near the outlet (b)
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As can be seen from [Figure 5] and [Figure 6], air is drawn into the channel toward the heated wall and accelerated upward in a boundarylayer like flow. Due to the heating of the air, higher velocity gradients are encountered close to the heated wall resulting in higher temperatures and higher momentum contribution toward the upper wall. At the inlet, the velocity profile is uniform. However, at the outlet due to asymmetric heating, the profile does not remain uniform anymore. The magnitude of velocities is found to be much larger close to the heated wall.
Representative temperature contours in the channel space along the xy plane at Z = 0.17 m are shown in [Figure 7] with enlarged views near the inlet and outlet of the channel. The temperature distribution on the top heated wall and the bottom wall along the inclined (xz) plane is shown in [Figure 8].  Figure 7: Temperature contours (°C) along the xy plane at Z = 0.17 m with enlarged views near the inlet (a) in the middle section (b) and near the outlet (c)
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 Figure 8: Temperature distribution on the channel top heated wall (a) and bottom wall (b)
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From the temperature contours shown in [Figure 7] and [Figure 8], it can be seen that as the flow moves toward the channel exit, heat transfer rate increases due to larger temperature gradients. It is observed that as a result of mutual interaction between temperature and velocity fields the temperature contours are in agreement with the velocity vectors.
To illustrate in detail, the air flow and thermal mechanism in the channel, twodimensional velocity and temperature profiles at four different axial positions (x = 0.50, 1.00, 1.75, and 2.00 m) from inlet are plotted as a function of the channel height (ydirection) and are shown in [Figure 9] and [Figure 10], respectively for the conditions L = 2.2 m, S = 0.03 m, q” = 400 W/m^{2} and θ = 18^{0}. The heated wall is at y = 0.04 m.  Figure 9: Velocity profiles at four different axial positions (x = 0.50, 1.00, 1.75, and 2.00 m) along the height (ydirection) of the channel
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 Figure 10: Temperature profiles at four different axial positions (x = 0.50, 1.00, 1.75, and 2.00 m) along the height (ydirection) of the channel
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It will be seen from [Figure 9] and [Figure 10] when the lower wall of the channel is heated up by radiation heat transfer; the top wall sees a drop in temperature. A hydrodynamic boundary layer is developed along both walls of the channel. Velocity near the channel walls increases with increasing axial location. The maximum velocity near the top and bottom walls is approximately 0.60 and 0.48 m/s, respectively. The temperature decreases by about 5°C of both walls near the outlet. From [Figure 10], it will be seen that the temperature of the top and bottom walls increases with increasing axial locations with maximum value reaching to approximately 118°C and 70°C, respectively at x = 0.2 m.
The temperature profiles on the top and bottom walls along the length of the channel for the pure convection and the combined radiationconvection heat transfer are shown in [Figure 11].  Figure 11: Temperature profiles on the top and bottom walls along the length of the channel for the pure convection and the combined radiationconvection heat transfer
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From [Figure 11], it will be seen that when radiation is included in the model, heat is radiated directly from the walls to the surroundings as a result the temperature of top wall decreases while temperature of bottom wall increases. The radiation effect is significant in terms of reducing the maximum temperature of the heated wall.
The local heat transfer coefficient (W/m^{2}  K) as a function of the channel length is plotted in [Figure 12].  Figure 12: Local top wall heat transfer coefficient (W/m^{2}  K) in the channel
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It is observed that heat transfer coefficient decreases along the length of the channel. It is higher near the inlet of the channel due to the fact that the flow has not been fully developed close to the inlet.
Parameteric study
Effect of variation in the channel space (S = 0.02 m, 0.03 m, and 0.05 m) on the temperature distribution
The effect of variation in the channel space, S, on the temperature distribution is examined. Simulations are carried out for conditions: dimensionless channel space L/S ratios = 110, 73 and 44 (L = 2.2 m), q” = 400 W/m^{2}, θ = 18°, ε = 0.27. The comparative effect of variation in the L/S ratio on the temperature profiles along the length of the channel is displayed in [Figure 13].  Figure 13: Comparative effect of variation in the L/S ratio on the predicted temperature profiles
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It will be seen that with the decrease in the the L/S ratio, the maximum temperature on the heated wall decreases. However, when the L/S ratio is reduced further (i.e., L/S < 44), no significant change in temperature distribution is observed.
Effect of inclination angle
The channel performance is evaluated and compared at two different angles of inclination. Angle of inclinations of θ = 18°, 30° are selected using q” = 400 W/m^{2}, ε = 0.27 and L/S ratio = 44. A series of simulations are carried out to investigate the effect of angle of inclination on the temperature distribution over the heated wall along the length of the channel. A comparison of the predicted and measured average temperature values along the dimensionless axial position of the channel for angle of inclination 18° and 30° are shown in [Figure 14].  Figure 14: Predicted and measured average temperature values on the heated wall of the channel
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From [Figure 14], it will be that for the same thermal boundary conditions, with the increase in the angle of inclination, the maximum temperature values on the heated wall decrease. This can be attributed to the fact that at higher inclinations the buoyancy effect will be higher which causes a reduction in temperature values, but this decrease is relatively small.
Effect of surface emissivity
To study the effect of surface emissivity, a series of simulations were run to investigate the effect of the surface emissivity on the temperature distribution over the heated wall along the length of the channel by assuming a coating on the channel surfaces with emissivity of 0.27 and 0.95. The numerical results obtained are compared with the experimental measurements recorded by Lin and Harrison.^{[28]} The predicted average temperature values along the dimensionless axial position of the channel for emissivity of 0.27 and 0.95 are shown in [Figure 15] for the conditions: L/S = 100, q” = 400 W/m^{2}, θ = 18°.  Figure 15: Predicted and measured average temperature values on the heated wall along the dimensionless axial position of the channel for ε =0.27 and 0.95 for the conditions L/S = 100, q = 400 W/m^{2}, θ = 18°
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From [Figure 15], it will be seen that at lower emissivity the agreement between the predicted and measured values is good but at higher emissivity the difference is a little more which can be attributed to the fact that at higher emissivity, there is more radiation exchange between the top and bottom walls of the channel and also through the side walls, wherever, in numerical simulations, the side walls are assumed to be adiabatic. It is noted that the radiation exchange between the top and bottom walls of the channel has a significant impact on wall temperature and heat transfer in the channel. By increasing the surface emissivity from 0.27 to 0.95, the top wall temperatures decrease by approximately 20°.
Effect of Ohmic heat flux
To investigate the effect of the Ohmic input heat flux (W/m^{2}) on the heat transfer from the heated wall along the length of the channel, simulations were carried out with variation in heat flux applied in the range 20–250 W/m^{2}. The numerical results obtained are compared with the experimental measurements recorded by Lin and Harrison.^{[28]} Heat transfer from the top wall consists of natural convection to the channel air (Q_{conv)}, conduction through the top insulation layer (Q_{cond}) and radiation to the bottom surface (Q_{rad}). The energy balance on the top wall can be represented as:
To calculate the radiation exchange, the channel is partitioned into seven segments along the channel length. [Figure 16] plots the numerically predicted and calculated (from measured temperatures) average heat transfer values from the top wall of the channel by convection and radiation as a function of the heat input for a channel L/S ratio of 100.  Figure 16: Predicted and measure average heat transfer values from the top wall of the channel by convection and radiation as a function of the heat input for a channel L/S ratio of 100
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From [Figure 16], it will be seen that there is a good agreement between the predicted and measured heat transfer rate values. It is observed that both radiation and convection heat transfer from the top wall of the channel increase linearly with increase in input Ohmic heat flux. It is also worth noting that approximately 45% of the heat input to the top surface is dissipated to the bottom surface by radiation where it is transferred to the air stream by convection.
Correlation of Nusselt number
It is well documented in literature that natural convection heat transfer between inclined parallel plates depends on the dimensionless Nusselt number, Nu, the Rayleigh number, Ra, the ratio of plate spacing (channel space), s, to plate length, L, (s/L) and the plate inclination, θ. For these reasons, the numerical results are often presented in the form of:
It should be noted that Ra_{s} is based on plate spacing whereas Ra_{L} based on plate length and are defined as follows:
Where β is the coefficient of thermal expansion, T_{H} is the surface temperature of heated wall, T_{o} is the ambient air temperature, ν is the kinematic viscosity of air, α is the thermal diffusivity of air, and g is the acceleration due to gravity. The thermophysical properties in the Nusselt and Rayleigh numbers are evaluated at the film temperature defined as (T_{H} + T_{o})/2. The heat transfer coefficient, h, is defined by the relation:
Where q is the heat flux of the heated plate, A is the surface area of the wall, and ΔT is the temperature difference between the heated wall and ambient air. The overall average Nusselt number Nu_{s} based on plate spacing(s) and Nu_{L} based on plate length (L) are defined as follows:
Where h is local heat transfer coefficient, s is channel spacing, and k is the thermal conductivity of air inside the channel. Since the space dimension(s) used in simulations is very small (0.01–0.04 m) relative to the channel length, the radiation losses from the channel to surroundings are considered negligible. From the calculated wall temperatures and input heat flux, the heat transfer coefficient (h) can be determined from equation (12) which is then used in equation (13) to find the overall average Nusselt number. The relationship of Nusselt number as function of the modified channel Raleigh number, Ra^{”}, is calculated and plotted in [Figure 17].  Figure 17: Correlation of Nusselt number versus channel modified Rayleigh number
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All the numerical data plotted in [Figure 17] is obtained from simulations of the air channel run at angle of inclination of 18^{o} and surface emissivity of 0.95. The L/S ratio used in the simulations ranges from 44 to 220 while the input heat flux is varied between 25 and 460 W/m^{2}. A power law curve fit is provided to the numerical data, and following correlation is obtained between Nusselt number and channel modified Raleigh number;
Nu = 0.6851Ra”^{0.2612}
Conclusions   
In this study, an inclined air channel with a onesided heated wall is studied numerically using commercial CFD code FLUENT^{©}. The air velocity and temperature patterns in the channel are obtained and a parametric study is carried out to study the effect of various parameters such as the heated wall temperature, the input heat flux, the surface emissivity, and the channel geometry which includes the lengthtospace ratio and the angle of inclination. The CFD model is validated by comparing the numerical predictions with the experimental data available in the literature. Based on the numerical results obtained, a numerical correlation for Nusselt number as a function of modified Rayleigh number is developed. The following specific conclusions may be drawn from this study:
 In general, the numerical results of this study are in good agreement with the experimental measurements reported by Lin and Harrison.^{[28]} The small quantitative differences in the numerical values may be attributed to the heat loss through the sides which were assumed adiabatic in the numerical simulations
 The CFD code ANSYSFLUENT^{©} in conjunction with and the DO radiation model is an efficient and accurate numerical tool to model the airflow and heat transfer in inclined air channels employed for passive air cooling of flat plate solar collectors under stagnation conditions
 The velocity contours along the channel indicates a developing character of the airflow and provides some indepth insight to the further understanding of airflow in inclined parallelplate channels
 It was found that by decreasing the channel L/S ratio, the maximum temperature achieved inside the channel can be reduced. However, reducing the L/S below a threshold value would not cause any significant effect on the temperature distribution
 The angle of inclination has little effect on the wall temperature as the temperature decreased only slightly when the angle of inclination is increased from 18° to 30°
 At moderate to high wall emissivity, thermal radiation contributes significantly to the total heat transfer by natural convection, even at low temperatures of the heated wall
 In the region near the channel inlet, the convective heat transfer is found to be stronger than in other parts of the channel. This entrance effect diminishes rapidly as the distance from the entrance increases and vanishes completely at about half the channel length. The radiation heat flux is lowest at the inlet, increases along the channel length, and reaches it maximum close to the outlet
 The results indicate that both radiation and convection heat transfer from the top surface increase linearly with increasing input heat flux. It is noticed that approximately 45 percent of input heat flux to the top surface is dissipated by radiation to the bottom surface where it is transferred to the air stream by convection
 The numerical results obtained are put together as a correlation for Nusselt number (equation 14) expressed as function of modified Rayleigh number. The numerically correlated values from equation (14) agree well with the experimental correlations developed by Lin and Harrison.^{[28]} Hence, the proposed numerical methodology could be used with confidence to predict the heat transfer coefficient and airflow field in an asymmetric heated air channel designed for passive air cooling of solar collectors under stagnation conditions with small inclination angles and high surface emissivity.
Acknowledgment
This work was supported by Calorimetry Laboratory, Solar Research Centre of Queen's University at Kingston, Canada.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11], [Figure 12], [Figure 13], [Figure 14], [Figure 15], [Figure 16], [Figure 17]
