Forced Convection over an Inclined Heated Plate with Varying Aspect Ratios: 3D Numerical and Experimental Investigations

Forced convection is a complex mode of energy transfer that combines fluid motion with heat transfer. Many applications related to this phenomenon exist such as wind-loaded buildings, solar panels and heat sinks [1-3]. Mostly, solar panels can be modeled as inclined surfaces. Therefore, optimizing the parameters controlling forced convection on inclined plates, as air velocity, angle of inclination or attack, heat generation and plate’s dimension, became a challenge.

Some authors analyzed the impact of air speed on heat transfer both experimentally and numerically [4-7]. By increasing the air velocity, the flow becomes turbulent. In order to simulate accurately those turbulences, the choice of a turbulence model is decisive. Reynolds Averaged Navier Stokes (RANS) models, especially standard k-ε and k-ω, are the most used for this kind of problems. Other models derived from the previously cited models exist like R k-ε, RNG k-ε, S k-ω and SST k-ω. Several researchers were interested in detecting the most adequate turbulence model. Therefore, the choice was based on testing the findings of those latter with experimental results. El-Shamy et al. [8] showed that SST k-ω was in better agreement with experiment compared to R k-ε, while, Karava et al. [9] found that k-ε was the most reliable.

Studies were also elaborated to evaluate the flow structure over a heated plate. Hourigan et al. [10] determined experimentally and numerically both the flow patterns and the thermal fields over a heated plate. They found that the heat transfer improved when the large vortices were formed. The sound was also used as a method to enhance the forced convection. Zuercher et al. [11] experimentally examined the stability of the boundary layer on the upper side of an inclined heated plate, while Griffiths [12] focused on an inelastic non-Newtonian fluid. Liu and Ren [13] simulated the transient convective flow adjacent to a heated plate and determined the parameters characterizing the thermal boundary layer flow.

In general, convective heat transfer correlations were established for laminar, mixed, and forced flows [14-17], which relate the Nusselt number to the Reynolds number and the angle of inclination. Altering the plate's inclination angle has a significant influence on heat transfer, leading to either enhancement or deterioration. According to Yang et al. [18], when the inclination angle becomes important, heat transfer is improved and flow separation also can occur. Alhendal and Touzani [19] showed that average Nusselt number increases with an angle of inclination varying from 0° to 20° approximately then decreases with higher angles (up to 90°). For a rectangular duct equipped by inclined baffles on its bottom wall, Arslan et al. [20] observed heat transfer improvement when the baffles inclination angle becomes steeper.

Another interesting parameter that can influence the flow as well as the heat transfer around the plate is the plate’s heating mode. Cossali [21] applied periodic heating condition and studied its impact on heat transfer coefficient. Shavazi et al. [22] considered a uniform flux on an inclined narrow flat plate. An increasing boundary layer thickness in the laminar region and uniform temperature distribution in turbulent region, were observed. Turgut et al. [23] examined numerically heat transfer over an inclined flat plate with unheated starting length with two boundary conditions: constant temperature and uniform flux. In order to enhance heat transfer, Sasaki and Ashiwake [24] attached rectangular grids on the inclined plate. An improvement of 20% was observed compared to without grids case.

Most researches mentioned above were interested in understanding the interaction between the air flow around the inclined plate and heat transfer, but the numerical part was mostly bidimensional. Thanks to developed CFD techniques, 3D studies became performed. Roy et al. [25] presented a three-dimensional study of an inclined surface with a heating pad in the middle, where heat transfer coefficients were determined numerically and experimentally. Moreover, heat transfer coefficient was calculated either locally or as an average on the inclined plate for a fixed plate’s dimension. In a 3D numerical study, Salhi et al. [26] showed that the presence of inclined baffles in a rectangular channel generated vortices inducing a mixing phenomenon, which leads to heat transfer rate’s improvement. The dimension of the plate is also an important parameter. Motwani et al. [27] were interested in studying the impact of two aspect ratios on heat transfer coefficient for different angle of plate’s inclination. Lately, Shademan and Naghib-Lahouti [28] investigated aspect ratio effect on airflow over an inclined flat plate using Large Eddy Simulation (LES) turbulence model.

Due to the lack in tridimensional studies as well as the effect of plate’s aspect ratio on heat transfer over inclined heated plate, the aim of this paper is to set out a comprehensive experimental and CFD analysis of this latter. This also includes predictions about the general heat transfer correlation, h, and general correlations for the average Nusselt number, Nu, with angle of inclination 0 ≤ α ≤ 90°, Reynolds number 63848 ≤ Re ≤ 319403, and aspect ratio 0.25 ≤ AR ≤ 2.

The computational field is depicted in Figure 3 as a rectangle with an inclined heated plate with an ignored thickness, length L, and width w inside of it. The plate has an angle of α and is heated using flux. The division of the plate's length by its width (AR = L/w) yields the aspect ratio AR. The turbulent, incompressible flow has an entrance velocity (V) that ranges from 4 to 32 m/s. A uniform flow velocity is defined by the flow limit, which is three times the plate's length. The pressure is null at the plate's minimum length, and no slip condition is given to the surface of the plate or any other surfaces that limit the computing field.

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Figure 3. 3D problem configuration

The equations governing the phenomenon are the three equations of conservation (mass, momentum, and energy). They can be expressed as the following:

Mass:

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

Momentum:

$\frac{\partial(\rho \vec{v})}{\partial t}+\nabla \cdot(\rho \vec{v} \vec{v})=-\nabla p+\rho \vec{g}+\mu \Delta \vec{v}$     (2)

Energy:

$\frac{\partial(\rho E)}{\partial t}+\nabla \cdot(\vec{v}(\rho E))=k \Delta T$     (3)

The conversion formula for the k-ε RNG model is:

$\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_i}\left(\rho k u_i\right)=\frac{\partial}{\partial x_j}\left(\alpha_k \mu_{e f f} \frac{\partial k}{\partial x_j}\right)+G_k+G_b-\rho \varepsilon+S_k$     (4)

$\begin{gathered}\frac{\partial}{\partial t}(\rho \varepsilon)+\frac{\partial}{\partial x_i}\left(\rho \varepsilon u_i\right) \\ =\frac{\partial}{\partial x_j}\left(\alpha_{\varepsilon} \mu_{e f f} \frac{\partial \varepsilon}{\partial x_j}\right)+C_{1 \varepsilon} \frac{\varepsilon}{k}\left(G_k+C_{3 \varepsilon} G_b\right)-C_{2 \varepsilon} \rho \frac{\varepsilon^2}{k}-R_{\varepsilon}+S_k\end{gathered}$     (5)

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

The model constants; $C_{1 \varepsilon}, C_{2 \varepsilon}, \alpha_k, \alpha_{\varepsilon}, \xi_o$ and $\theta$ in Eqs. (4) and (5) are: $C_{1 \varepsilon}=1.42$; $C_{2 \varepsilon}=1.68$; $\alpha \varepsilon_k$; $\xi=4.34$; and $\theta=0.012$, turbulent kinetic energy is represented by $G_{\xi}$ while turbulent kinetic energy produced by buoyant forces and velocity, respectively, is represented by $G_b$. For $k$ and $\varepsilon, \alpha, k$ and $\alpha_{\varepsilon}$ represent the reciprocal of (Pr).

For the energy equation, it becomes as the following:

$\frac{\partial(\rho E)}{\partial t}+\frac{\partial}{\partial x_i}\left(u_i(\rho E+p)\right)=\frac{\partial}{\partial x_j}\left(k_{e f f} \frac{\partial T}{\partial x_j}+u_i\left(\tau_{i j}\right)_{e f f}\right)+S_h$     (6)

With E, $k_{e f f}$ and $\left(\tau_{i j}\right)_{e f f}$ representing the total energy, the effective thermal conductivity and the deviatoric stress tensor respectively; which can be formulated as:

$\left(\tau_{i j}\right)_{e f f}=\mu_{e f f}\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)-\frac{2}{3} \mu_{e f f} \frac{\partial u_k}{\partial x_k} \delta_{i j}$     (7)

The Eqs. (1)-(7) will be solved numerically utilizing Ansys Fluent software [30].

Pressure drag force coefficient is expressed as below:

$C_d=\frac{2 \Delta p}{\rho u_0^2}$     (8)

With $u_0$ being the velocity of the undistributed air flow and $\Delta p$ being the pressure difference.

The Nusselt number is determined to assess the heat transfer. Expression of the local Nusselt number is:

$N u_x=\frac{q_p X_p}{\left(T_p-T_b\right) K}$     (9)

With $T_P$ representing the plate temperature, $T_b$ representing the bulk air temperature, $K$ representing air thermal conductivity, and $q_p$ representing the heat flux at point $p$ on the plate.

The average Nusselt number, $N u$, represents the average of the local Nusselt numbers computed across the plate, expressed as:

$N u=\frac{h l}{K}$     (10)

where, h for the entire plate is:

$h=\frac{q}{A\left(T_s-T_{\infty}\right)}$      (11)

where, A is changed by changing the aspect ratio, AR.

The experiment and CFD cases were carried out within an average air velocity between 2-20 m/s and 4-32 m/s respectively. The considered Reynolds number range is between 63848 and 319403. Both 2D and 3D numerical simulations and experiments were used to analyze the airflow on heated inclined flat plates for angles of inclination α varying from 0° to 90° and different aspect ratios (0.25 ≤ AR ≤ 2).

Before elaborating the 3D simulations, the CFD for AR=1 and α = 0° was compared to the correlation given by ASHRAE [29]. As it is noticed from Figure 5, the calculated heat transfer coefficient (h) is quietly similar to the correlated one with a relative difference between approximatively 2% and 14% observed at V= 8m/s and V=20 m/s respectively. The same tendency is noticed for both curves; the heat transfer increases with higher air inlet velocity.

In a previous work [19], heat transfer coefficient was found to be decreasing from α = 30°. As noticed from Figure 6, a vortex is created on the first edge of the plate helping the heat transfer improvement all over the plate. However, with the angle of inclination increment, the airflow becomes hindered due to the plate’s resistance which weakens the created vortex. In fact, when the inclination angle is high, an important skin friction occurs as a result of the created boundary layer around the plate. This latter leads to heat dissipation and therefore decreases the heat transfer coefficient on the plate.

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Figure 5. Validation with correlation from ASHRAE [29]

In order to have more realistic results, the width of the plate is taken into account in the following paragraphs.

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Figure 6. Streamlines for α = 20°, α = 30° and α = 60° for 2D case at V=20 m/s

For α = 30°, the variation of average Nusselt number with Re is presented in Figure 7 for 3D results where 0.25 ≤ AR ≤ 2 and 2D case. Independently of the plate’s dimension, Nu increases with Re augmentation as observed for 2D case. However, when Re ≤ 100000, Nu decreases slightly for AR increment. Beyond Re = 100000, Nu decreases more for higher AR. By increasing the aspect ratio, the surface of exchange becomes important but leads to more skin friction, which impacts negatively the heat transfer.

Another interesting constatation is that for AR= 1, Nu is higher than all cases, regardless of Re. This can be related to the shape of the plate. For AR = 1, the plate is a square while for the other AR it becomes a rectangle. The square shape has less surface; thus, less friction occurs with the airflow. Compared to the 2D case, the impact of improving the surface of contact is obvious. More surface offers better heat transfer. As noticed, Nu can be correlated to Re in a linear or power empirical formula. Therefore, we will present these two types of correlations to see which one is more adequate for each studied case depending on AR and α.

The following empirical relationship formula of linear and power law are suggested for both dimensional and non-dimensional forms:

h = a⸳V + b     (12)

h = c⸳V^m     (13)

Nu = d⸳V + e     (14)

Nu = f⸳Vⁿ     (15)

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Figure 7. Comparison between Nu vs. Re for different AR and 2D case

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Figure 8. Average heat transfer coefficients vs. V for different angles of inclination and AR

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Figure 9. Average Nu vs. Re for different angles of inclination and AR

Average heat transfer coefficient and Nusselt number are presented in function of inlet air velocity and Reynolds number respectively, experimentally as well as numerically, for different angles of inclination from 0⁰ to 90⁰ and aspect ratios AR= 0.34, 1 and 2 (Figures 8 and 9). Simulation results are in good agreement with experimental data for most studied angles. Table 2 shows a relative error between experimental and numerical heat coefficients for AR=1, varying between 1.67% to 11% approximatevely. This difference can be related to the measurments uncertainty and numerical errors. However, for α = 0°, α = 10°, and α = 90°, depending on AR, the two curves (experiment and CFD) become distant for higher V and Re. In experiment, when the velocity is very high therefore Re is important, the airflow submerging the plate for those particular angles, is facing hydrodynamic and thermal boundary layers that one overpowers the other depending the angle of inclination creating turbulences that can lead the plate to vibrate. Actually, McCormick et al. [31] observed the plate vibration experimentally when the velocity is high, showing that it impacts heat transfer coefficient values. For all cases, we can see that the power law is the most suitable to correlate average heat transfer coefficient and Nusselt number for all angles with R² very close to 1.

Table 2. Comparison between experimental and numerical convective coefficients for AR =1

At first, a correlation of average heat transfer coefficient as a function of air inlet velocity and the aspect ratio at α = 0° is elaborated. It was obtained by taking the average values of the constant variables’ correlations for the 3D plates with aspect ratio 0.25,0.33,0.5 at α = 0° (Figure 10). Then the found average value from this process is plotted against (V·AR^0.3), and the following correlation for h is obtained:

h = 12.33·V^0.8 ⸳AR^0.24     (16)

For 4 ≤ V ≤ 24 and 0.25 ≤ AR ≤ 1

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Figure 10. Correlation for heat transfer coefficient, h, vs. V and AR of (0.25 to 0.5)

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Figure 11. Residual normal probability plot

A general correlation relating average Nusselt number to Reynolds number, angle of inclination and aspect ratio is generated from the different data that we collected for the following ranges cover:

0 ≤ α ≤ 90° for

63848 ≤ Re ≤ 319403

and 0.25 ≤ AR ≤ 1.

the correlation expression is:

Nu = 0.1·Re^0.72(1-sin(α)^-0.02AR^-0.11)     (17)

As shown in Figure 11, this general correlation presents a good association between the considered variables. The maximum error between the normalized residues and the normalized quantities is between -18% and 15%.