Influence of Inclination Angles on Convective Heat Transfer in Solar Panels

Flat plates, as exemplified by solar collectors, heat sinks, and building roofs, play a crucial role in various engineering applications through their capacity to gain or dissipate heat from their environment via natural, forced, or mixed convection phenomena. Consequently, they have garnered significant attention from researchers [1-5].

In order to elucidate the influence of different types of convection on an inclined flat plate, numerous experimental and numerical investigations have been undertaken. Pioneering work by Jürges [6] offered the first convective heat transfer coefficients derived from wind tunnel experiments involving a plate subjected to different air flow speeds. A numerical study by Yang and Yang [7] on mixed convection on an inclined heated plate located within a horizontal channel revealed that forced convection serves to enhance the heat transfer rate. It was further noted that the angle of inclination substantially affects the interaction between forced and natural convections. Siddiqa et al. [8] numerically analysed the influence of parameters such as the inclination angle, heat generation, viscosity variation and Prandtl number on laminar natural convection over an inclined flat plate.

The plate inclination angle or angle of attack has been demonstrated by various researchers to have a significant role in either augmenting or diminishing the heat transfer coefficient. Rida [9] demonstrated that thermal transmittance peaks at a tilt angle of 20°, with the maximum value beginning to decrease with an increase in panel tilt. Further experimental work by Ramirez et al. [10] examined the variation in local heat transfer coefficients across the surface of a narrow rectangular plate at different angles, leading to observations of various flow patterns. An empirical correlation expressing the average Nusselt number as a function of Reynolds number and the plate inclination angle was deduced. Other researchers have emphasized the importance of the angle of attack on flow separation for different fluids, leading to the development of several convective coefficient correlations [11-14].

Heat generation has also been recognized as a key factor. Cossali [15] analysed the effect of periodic heating on average heat transfer, finding it to be significant for relatively high-frequency fluctuations, whereas the impact of instantaneous heat transmittance oscillations at low excitation frequencies was found to be negligible. Vynnycky et al. [16] considered the impact of thermal conductivity of a heated plate on forced convection heat transfer, finding that the thickness of the plate affects the heat flux distribution in conjugate flow. For a narrow plate, Shavazi et al. [17] analysed the effect of heat flux on temperature distribution and the transition from laminar to turbulent flow. An inclination angle of 30° was found to enhance the convection rate compared to other angles. Sasaki and Ashiwake [18] added rectangular grids to an inclined plate to improve convective heat transfer for different heat fluxes, resulting in a 20% increase compared to the case without grids for inclination angles ±30°.

The majority of studies have addressed laminar natural convection or forced convection for low turbulence experimentally and higher Reynolds numbers numerically [19-22]. To simulate turbulent flow, various turbulence models are available: Reynolds Averaged Navier Stokes (RANS) and Large Eddy simulation (LED) models. Within RANS models, standard k-ε and k-ω are most commonly used. Other models such as R k-ε, RNG k-ϵ, S k-ω, SST k-ω are either based on k-ε or k-ω with certain modifications or a mixture of both models in the case of SST k-ω. Karava et al. [3] found that the SST k-ω turbulence model provided more accurate results than the R k-ε model compared to experimental data, with an error of 10%. El Shamy et al. [21] used the k-ε model because its results better corresponded with their experimental findings and correlations in literature. These models are employed to calculate the flow velocity, which significantly affects the improvement in heat transfer. The investigated range of Reynolds numbers varies between 5600 and 150745. However, a higher range for forced convection around an inclined heated plate has not been explored.

In the present paper, both experimental and CFD analyses of air flow over a heated plate at various angles of attack, α, ranging from 0° to 90°, variable heat flux, q, from 1000 to 4000 W/m², and different Reynolds numbers, Re, ranging between 67000 and 367000 are undertaken. This includes predictions about the general relationship between heat transfer and friction factor. The Nusselt number, Nu, for an inclined plate surface is also investigated across a broad range of angles of attack.

The plate of a length L=25 cm, is heated with a flux, q, and inclined with an angle of attack, α (Figure 3(a)). It is surrounded by air with a velocity similar to the experiment values and even higher. Reynolds number varies from 67000 to 650000. The angle of attack will be varying between 0° and 90°. The flow is incompressible. An inflow limit of three times the length of the plate establishes a uniform flow velocity. At the downstream limit of the plate length, the pressure was set to zero. Constant wall heat flux and no-slip condition are specified on the plate surface. The simulation is two-dimensional, and the results are obtained for steady state regime. The computed domain is given by Figure 3(b).

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Figure 3. Problem configuration

The equations describing the thermo-fluid properties of inclined plates are a set of nonlinear PDEs. Airflow is governed by the equations of mass, momentum, and energy. In general, the equations can be written as follows [24, 25]:

Mass:

$\frac{\partial \rho}{\partial t}+\operatorname{div}(\rho \boldsymbol{u})=0$      (1)

Momentum:

$\begin{gathered}\frac{\partial(\rho u)}{\partial t}+\operatorname{div}(\rho u \boldsymbol{u})=-\frac{\partial p}{\partial x}+\operatorname{div}(\mu \operatorname{grad} u)-  \left\lceil\frac{\partial \overline{\left(\rho u^{\prime 2}\right)}}{\partial x}+\frac{\partial \overline{\left(\rho u^{\prime} v^{\prime}\right)}}{\partial y}\right\rceil\end{gathered}$                 (2)

$\begin{gathered}\frac{\partial(\rho v)}{\partial t}+\operatorname{div}(\rho v \boldsymbol{u})=-\frac{\partial p}{\partial y}+\operatorname{div}(\mu \operatorname{grad} v)- \left\lceil\frac{\partial \overline{\left(\rho u^{\prime} v^{\prime}\right)}}{\partial x}+\frac{\partial\left(\rho v^{\prime 2}\right)}{\partial y}\right\rceil\end{gathered}$                  (3)

Energy:

$\frac{\partial(\rho E)}{\partial t}+\operatorname{div}(\rho E \boldsymbol{u})=\operatorname{div}(k \operatorname{grad} T)-\left[\frac{\partial \overline{\left(\rho u^{\prime} E^{\prime}\right)}}{\partial x}+\frac{\partial \overline{\left(\rho v^{\prime} E^{\prime}\right)}}{\partial y}\right]$                         (4)

where, u is the velocity vector: u=ux+vy, p is the pressure, μ the fluid viscosity (kg/(m.s)), ρ the fluid density (kg/(m³)), T the temperature (K) and E the energy (J).

As already mentioned in the introduction, there are many turbulence models. As shown by Karava et al. [3], k-ε model is adequate to describe the turbulent flow around the inclined plate. But we decided to use RNG k-ε model because it offers better performance than k-ε by increasing the accuracy and calculating the turbulent Prandtl number, which is more suitable for separation flows [26].

The equations in RNG k-ε model are written as bellow:

$\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_i}\left(\rho k u_i\right)=\frac{\partial}{\partial x_i}\left(\alpha_k \mu_{e f f} \frac{\partial k}{\partial x_i}\right)+G_k+G_b-\rho \varepsilon+S_k$               (5)

$\begin{gathered}\frac{\partial}{\partial t}(\rho \varepsilon)+\frac{\partial}{\partial x_i}\left(\rho \varepsilon u_i\right)=\frac{\partial}{\partial x_j}\left(\alpha_{\varepsilon} \mu_{e f f} \frac{\partial \varepsilon}{\partial x_j}\right)+ \\ C_{1 \varepsilon} \frac{\varepsilon}{k}\left(G_k+C_{3 \varepsilon} G_b\right)-C_{2 \varepsilon} \rho \frac{\varepsilon^2}{k}-R_{\varepsilon}+S_k\end{gathered}$                 (6)

where, $G_k=-\rho \overline{u_i^{\prime} u_j^{\prime}} \frac{\partial u_j}{\partial x_i}, R_{\varepsilon}=\frac{C_\mu \rho \xi^3\left(1-\xi / \xi_o\right)}{1+\theta \xi^3} \frac{\varepsilon^2}{k}$, $G_b=g_i \frac{\mu_t}{\rho P r_t} \frac{\partial \rho}{\partial x_i}, C_{3 \varepsilon}=\tanh \left|\frac{v}{u}\right|, S_k=\xi \varepsilon$.

The model constants; C_1ε, C_2ε, a_k, a_e, ξ_o and θ in Eqns. (5) and (6) are: C_1ε=1.42; C_2ε1.68; αε_k; ξ=4.34; and θ=0.012, G_k and G_b are turbulent kinetic energy production due to velocity and buoyancy forces, respectively. α_k and α_ε are the reciprocal of (Pr) for both k and ε respectively.

For the energy equation, it becomes as the following:

$\frac{\partial(\rho E)}{\partial t}+\frac{\partial}{\partial x_i}\left(u_i(\rho E+p)\right)=\frac{\partial}{\partial x_j}\left(k_{e f f} \frac{\partial T}{\partial x_j}+u_i\left(\tau_{i j}\right)_{e f f}\right)+S_h$               (7)

where, E is the total energy, $k_{\text {eff }}$ the effective thermal conductivity and $\left(\tau_{i j}\right)_{e f f}$ the deviatoric stress tensor defined:

$\left(\tau_{i j}\right)_{e f f}=\mu_{e f f}\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)-\frac{2}{3} \mu_{e f f} \frac{\partial u_k}{\partial x_k} \delta_{i j}$                  (8)

The Eqs. (1)-(8) will be resolved numerically using Ansys Fluent software.

In order to quantify the heat transfer, Nusselt number is determineted. Local Nusselt number is defined as [26]:

$\mathrm{Nu}_{\mathrm{x}}=\frac{q_p X_p}{\left(T_p-T_b\right) K}$                   (9)

With q_p: heat flux at point p, X_p: location of point p on the plate, T_p: plate temperature, T_b: bulk temperature of the air at point p and K: air thermal conductivity.

Average Nusselt number, Nu, is the mean of local Nusselt number calculated on the plate.

The coefficient of skin friction is also determined using the expression:

$f=\frac{\tau_w}{\rho u_b^2}$                  (10)

where, τ_w is wall shear stress at the point the plate, ρ is fluid density and u_b is bulk velocity at the location on the plate. Pressure drag force coefficient is expressed as below:

$C_d=\frac{2 \Delta p}{\rho u_0^2}$              （11）

With $\Delta p$: difference of pressure and u₀ the velocity of undistributed air flow.

This research combined experimental and numerical methods to study Reynolds number, heat flux and angle of attack influences on fluid flow and heat transfer properties of the hot plate.

In Figure 5, for q=2000 W/m², the average Nusselt number (Nu) is presented as a function of Reynolds number (Re) both experimentlly and numerically at α= 0°, 20°, 30° and 40°. For 67000≤Re≤367000, experimental measurements are in good agreement with numerical results with a maximal deviation of 5% that increases sligtly for α=40°. In general, Nu augments when Re increase regardless of α. By increasing the air velocity, the airflow contact becomes higer with the plate which enhances its heating effect. With increasing Re until 370000, Nu at α=20° becomes quite similar to the ones obtained by α=30° and α=40°. Beyond that Re value, Nu at α=20° is the maximal compared to the others.

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Figure 5. Experimental and numerical results for Nu vs. Re for q=2000 W/m² and different angles of attack

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Figure 6. Skin Friction coefficient as function of the plate’s length for Re=2.74×10⁵and different angles of attack

For α=10°, Nu is minimal, especially for very high Re, due to the high skin friction effect noticed from the leading edge of the plate which is greater than the other cases (Figure 6). While for α=20°, skin friction coefficient is low at the leading edge of the plate. Beyond the middle of the plate, the friction becomes more important with the angle increment. With increasing the angle of attack, airflow is hindered due to skin friction resistance. In fact, the drag force is the combination of the skin friction due to viscous force with pressure force. From Figure 7, it is obvious that pressure force is higher than viscous force for low angles of attack, quite similar for α=60° and they switch roles for higher angles with a maximum value of friction at α=90°. This drag force leads to hydrodynamic boundary layer with different thicknesses depending on the chosen angle of attack that generate the turbulent flow on and behind the plate translated by the creation of vortices. Actually, even at the first edge of the plate, a reciculation of air is observed at α=10° (Figure 8), which becomes weaker for high angles. Behind the plate, the airflow is slowed down with the angle increase. This affects the distribution of the temperature in the medium (Figure 9). Whenever the angle is important, the temperature decreases behind the plate. In fact, thermal boundary layer is also playing an important role in either enhancing or decreasing heat transfer. Both hydrodynamic and thermal boundary layers develop at leading edge of the plate and one overcomes the other depending on the plate angle. By increasing Re to very high values, the inclined plate becomes submerged by the turbulent flow, which reduces the skin friction (Figure 10). However, for high angles, the skin friction rises slightly due to the plate resistance over the flow.

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Figure 7. Average skin friction and average pressure coefficients for α=0 to 90° and Re=2.74×10⁵

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Figure 8. Velocity contours for Re=2.74×10⁵ and different angles of attack

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Figure 9. Isotherms for Re=2.74×10⁵ and different angles of attack

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Figure 10. Friction factor vs. Reynolds number for q=2000 W/m² and different angles of attack

Correlations of average Nusselt number and skin friction coefficient were correlated using both experimental and numerical results for the studied Re range (7.3×10⁴≤Re≤ 5.83×10⁵) and angles of attack (0° to 90°):

$N u=0.186 \operatorname{Re}^{0.664} \operatorname{Pr}^{\frac{1}{3}}\left(0.04 \sin \alpha-0.09 \sin ^2 \alpha\right)$        (12)

with maximum deviations between -9% and 12%, (Figure 11).

$f=0.1985 R e^{-0.314}(1+\sin \alpha)^{0.352}$                   (13)

with maximum deviation of ±15%.

In Figure 12, the average Nusselt number calculated and correlated are plotted as a function of the angle of attack for different Re. A good agreement between the two Nu is observed with a relative difference between the two varying between 0.1% and 12%. For all the plotted Re values, Nu is quite constant for 0°≤α≤30° with a slight increment for α=20° then decreases with α increase. This decrease is more prounounced for high Re. As already explained, the friction is more important when α is high which impacts negatively the circulation of the aiflow and therefore diminish the heat transfer. For 7.19×10⁴≤ Re≤1.5×10⁵ [21], Nu was also found to augment slightly for α≤20°, then decreases beyond α=20°.

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Figure 11. Comparison of the present correlation with the present experimental and numerical data

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Figure 12. Correlated and numerical average Nusslet number for different angles of attack and Re

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Figure 13. Effect of heat flux on Nusselt number at α=30°

By increasing the heat flux heating the plate (q) from 1000 W/m² to 4000 W/m² at α=30°. For Re≤200000, Nu is quite similar for all the heat fluxes. Beyond Re=200000, Nu increases when the heat flux is high, due to the turbulent layer developed that favorites convection circulation, as seen in Figure 13.