Mixed Convection in an Open Cavity with an Internal Fin Filled with Nanofluid in a Porous Medium Using Two-Phase Mixture Model

Enhancing heat transfer refers to improving the rate and efficiency of thermal transfer between two or more objects or mediums. Efficient heat transfer is crucial in various industrial, technological, and everyday applications to optimize performance, increase energy efficiency, and prevent overheating. Heat transfer can be improved when pure fluids are used instead of nanofluids. Base fluids like water, oil, or ethylene glycol are mixed with high thermally conductive metallic or non-metallic nanoparticles like Cu, Al, Al₂O₃, SiO₂, etc. [1, 2]. The problem of mixed convection in different cavity shapes with varying boundary conditions filled with nanofluid has gained significant interest over the years [3-6]. Through all these studies, they discovered that mixed convection across cavities is significantly impacted by flow direction, cavity steering, obstacle existence, and fluid characteristics on heat transfer rate. Hence, Selimefendigil and Ӧztop [7] studied the mixed convection in an enclosure containing two inner adiabatic spinning circular cylinders utilizing various types of nanoparticles. They concluded that the nanofluid provides a better heat transfer rate than other particles, equal to 4 percent when the Rayleigh number is high. In a square cavity that had a complicated fin, Shulepova et al. [8] Simulated the mixed convection of an Al₂O₃-water nanofluid. They mentioned that the heat transfer intensity was influenced by both the interior block's position and the nanoparticles' concentration. Ching-Chang-Cho [9] studied nanofluid mixed convection and entropy generation in a lid-driven cavity. According to his findings, there is a correlation between the amount of nanoparticle and an increase in the Nu_avg as well as total entropy production. Hussain et al. [10] performed a computational investigation into the mixed convection of a hybrid nanofluid within an open cavity that included an adiabatic obstacle. More studies that are comparable to this one has been carried out using a variety of nanofluids [11-13].

Because of the presence of interconnected pores that provide a large surface area for fluid-solid interaction, using porous mediums is another method for enhancing heat transfer in nanofluid thermal systems. This method helps improve the convective heat transfer. Porous mediums can be found in materials like porous media. Olak et al. [14] investigated a problem involving mixed convection conditions in a cavity driven by a lid and containing a heated porous block. They discovered that a reduction in Darcy's number could lead to an increase in the Nu_avg. In their study, Sheremet et al. [15] investigated the impact of Darcy's number and the dimensions of the inlet and outlet sections in the context of utilizing mixed convection within an open square cavity containing nanofluid. They have discovered an improvement in heat transfer after making the parameter adjustments. Laminar mixed convection was investigated by Tham et al. [16] over a circular cylinder. Siavashi et al. [17], conducted a study on the natural convection flow of a nanofluid composed of copper and water. Their investigation was carried out within a cavity featuring a set of fins attached to the heated wall. The researchers used the two-phase method for their investigation. They discovered that incorporating porous fins with high Darcy numbers improved heat transfer and that the opposite was true. Rajarathinam et al. [18] predicted Cu-water nanofluid mixed convection in an inclined porous cavity. They considered the effects of the moving wall(s) direction with three different cases and discovered that it plays a major role in flow and heat transfer. Using the two-phase mixture model, Emami et al. [19] investigated the natural convection of Cu-water nanofluid within an inclined porous cavity. They did this by observing the flow of the nanofluid. According to the results of their investigation, the positioning of the hot wall has a significant bearing on how the inclination angle is behaved. If you have a square and tilted cavity at an angle, it might be advantageous in some circumstances; however, in other circumstances, the overall heat transfer would be reduced.

According to the comprehensive review of the relevant literature that was discussed earlier, it is abundantly clear that the scenario of mixed convection utilizing the two-phase model in an open cavity that is both filled with Cu-water nanofluid and contains an internal fin that is covered with a porous medium has not been given any consideration or investigated. We discussed this topic and investigated how important factors, such as the Da number, affect the amount of heat transferred via convection.

The fluid is Newtonian, permanent, and incompressible; the flow of fluid within the cavity is laminar and three-dimensional; and the local thermal equilibrium between the fluid and the porous medium is verified.

The steady-state mixture model is governed by continuity, momentum, energy, and entropy production equations [20]:

$\nabla \cdot\left(\rho_m \mathrm{~V}_m\right)=0$                               (1)

$\begin{gathered}\nabla \cdot\left(\rho_m V_m V_m\right)=-\nabla \mathrm{P}+\nabla \cdot\left(\nabla V_m+\nabla_m^T\right) \mu_m+\nabla \sum_{k=1}^n\left(\rho_k \varphi_k V_{d r, k} V_{d r, k}\right)-F_{n p}+(\rho \beta)_m\left(T-T_i\right) \mathrm{g}\end{gathered}$                               (2)

$\begin{gathered}\nabla \cdot \sum_{k=1}^n\left(\rho_k \varphi_k V_k C_{p, k} T\right)=\nabla \cdot\left(\mathrm{K}_m \nabla T\right)\end{gathered}$                               (3)

$\begin{gathered}\nabla \cdot\left(\varphi_{n f} \rho_{n f} V_m\right)=\nabla \cdot\left(\varphi_{n f} \rho_{n f} V_{d r, n p}\right)\end{gathered}$                               (4)

where, $\mathrm{V}_m, \mathrm{~V}_{d r, k}, \mathrm{~V}_{n p, f}$ are respectively the mixture, drift, and slip velocities and represented as follows [21]:

$\begin{gathered}\mathrm{V}_m=\sum_{k=1}^n \frac{\varphi_k \rho_k \mathrm{~V}_k}{\rho_m} \end{gathered}$                  (5)

$\begin{gathered}\mathrm{~V}_{d r, k}=\mathrm{V}_k-\mathrm{V}_m \end{gathered}$                  (6)

$\begin{gathered}\mathrm{~V}_{n p, f}=\mathrm{V}_{n p}-\mathrm{V}_f\end{gathered}$                  (7)

Relationship between drift and relative velocity:

$\begin{gathered}\mathrm{V}_{d r, n p}=\mathrm{V}_{n p, f}-\sum_{k=1}^n \frac{\varphi_k \rho_k \mathrm{~V}_k}{\rho_m}\end{gathered}$                           (8)

$\begin{gathered}\mathrm{~V}_{n p, f}=\frac{\rho_p d_p^2}{\mu_f f_{d r a g}} \frac{\left(\rho_{n p}-\rho_m\right)}{\rho_{n p}} g-\left(\mathrm{V}_m . \nabla\right) \mathrm{V}_m\end{gathered}$                           (9)

$\begin{gathered}f_{\text {drag }}= \begin{cases}1+0.15 \operatorname{Re}_p^{0.687}, \operatorname{Re}_p \leq 1000 \\ 0.0183 \operatorname{Re}_p, \operatorname{Re}_p \geq 1000\end{cases}\end{gathered}$                           (10)

$\begin{gathered}\mathrm{F}_{n p}=\sum_{k=1}^2 \varphi_k\left(\frac{\mu_k}{\mathrm{~K}_k} u_k+\frac{c_{d, k} \rho_k}{\sqrt{\mathrm{K}_k}}\left|u_k\right|\right)\end{gathered}$                           (11)

where, $C_d$ and $k$ are the inertia coefficient and permeability of the porous medium and calculated as:

$\begin{aligned} C_d=\frac{1.75}{\sqrt{150} \varepsilon^{3 / 2}}\end{aligned}$                  (12)

$\begin{aligned}k=\frac{\varepsilon^3 D_{n p}^2}{150(1-\varepsilon)^2}\end{aligned}$                  (13)

The entropy is generated as a result of irreversibility source in the flow field, such as the viscous dissipation effect and heat transfer, as per the method suggested by Bejan [22], S_gen can be calculated using the formula [23]:

$S_{g e n}=S_{g e n, T}+S_{g e n, f}$                      (14)

where, $S_{\text {gen}, T}$ represents the entropy generation rate caused by the heat transfer irreversibility and is defined as:

$S_{g e n, T}=\frac{k_m}{T^2}\left[\left(\frac{\partial T}{\partial x}\right)^2+\left(\frac{\partial T}{\partial y}\right)^2+\left(\frac{\partial T}{\partial z}\right)^2\right]$                (15)

$S_{g e n, f}$ is the entropy production attributed to fluid friction given by:

For the clear region:

$\begin{gathered}S_{g e n, f}=\frac{\mu_m}{T_o}\left\{2\left[\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial y}\right)^2+\left(\frac{\partial w}{\partial z}\right)^2\right]+\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)^2\right. \\\left.+\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)^2+\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)^2\right\}\end{gathered}$                (16)

For the porous region:

$\begin{gathered}S_{g e n, f}=\frac{\mu_m}{T_o}\left\{2\left[\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial y}\right)^2+\left(\frac{\partial w}{\partial z}\right)^2\right]+\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)^2\right. \\\left.+\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)^2+\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)^2\right\} \\+\frac{\mu_m}{K T_{in}}\left|\overrightarrow{v_m}\right|^2\end{gathered}$                (17)

The dimensionless Bejan number quantifies the relative importance of heat transfer irreversibility in a system, it aids to analyzing and optimizing the efficiency of heat transfer processes, and is defined as [24]:

$B e=\frac{S_{g e n, T}}{S_{g e n}}$                        (18)

The specific heat, density, thermal conductivity, thermal expansion coefficient, and viscosity of the copper-water nanofluid are determined using the following equations:

$\begin{gathered}\rho_{n f}=(1-\varphi) \rho_f+\varphi \rho_{n p}\end{gathered}$                        (19)

$\begin{gathered}\rho_{n f} C p_{n f}=\left(1-\varphi_{n p}\right) \rho_f C p_f+\varphi_{n p} \rho_{n p} C p_{n p}\end{gathered}$                        (20)

$\begin{gathered}\frac{k_{n f}}{k_f}=\left[\frac{\left(k_{n p}+2 k_f\right)+2 \varphi\left(k_f-k_{n p}\right)}{\left(k_{n p}+2 k_f\right)+\varphi\left(k_f-k_{n p}\right)}\right]\end{gathered}$                        (21)

$\begin{gathered}\rho_{n f} \beta_{n f}=\left(1-\varphi_{n p}\right) \rho_f \beta_f+\varphi_{n p} \rho_{n p} \beta_{n p}\end{gathered}$                        (22)

$\begin{gathered}\mu_{n f}=\mu_f(1-\varphi)^{-2.5}\end{gathered}$                        (23)

The dimensionless form of variables used is given below:

$\mathrm{X}=\frac{x}{L}, \mathrm{Y}=\frac{y}{L}, \mathrm{Z}=\frac{z}{L}, \mathrm{~V}=\frac{u}{u_0}, \Theta=\frac{T-T_C}{T_h-T_c}$

The boundary conditions are taken to be:

• The bottom wall of the fin:

T=T_h

• The inlet (velocity inlet):

u=U₀; v=w=0; T=T₀

• The outlet (outflow):

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=\frac{\partial w}{\partial z}=0 ; \frac{\partial T}{\partial x}=0$

• The rest of the walls (adiabatic):

$u=v=w=0 ; \frac{\partial T}{\partial x}=\frac{\partial T}{\partial y}=\frac{\partial T}{\partial z}=0$

The research involved examining various parameters, including the range of 10^-4 to 10^-1 for Da, 0.2 to 0.4 for ε, and 0 to 0.08 for φ. This section provides a comprehensive discussion of how these parameters influence heat transfer.

5.1 Darcy number effect

The calculation for different values of the Darcy number will be done to examine its influence on the mixed convection flow (Da=10^-1, 10^-2, 10^-3, and 10^-4), where Reynolds number (Re) and porosity (ε) are kept constant (Re=600 and ε=0.4), as well as the geometrical parameters of the cavity.

In porous media, the Da number represents the relative effect of the permeability as the fluid passes through it, where high Da values denote a higher permeability and vice versa. It also indicates how dominant the inertial forces are compared to the viscous forces in determining the fluid flow behavior [29]. The effects of Da on the average Nu_avg, Be, and S_gen, respectively, can be seen in Figure 6. It is observed that when Da is increased, the Nu_avg also increases for any value of φ, and the impact of Da on the Nu_avg is evident when the value of Da is raised from 10^-4 to 10^-3. However, the increase in Nu_avg becomes almost negligible for higher values of Da. Higher Da makes the flow more dominated by inertial forces than viscous forces, which offers the nanofluid a better chance to flow faster in the porous region. Due to these higher inertial forces, the fluid can enter deeper into the porous structure and be interconnected with a larger heated surface area, promoting better mixing and more effectively carrying the heat away from the heated wall. This necessarily leads to improving convection flow and overall heat transfer, indicating higher Nu_avg. Also, at a constant value of Da, the Nu_avg increases when the φ increases, proving that adding nanoparticles signifies a higher thermal conductivity and improves heat transfer. It implies that the Nu_avg varies steadily and linearly with the φ for all cases of Da.

Figure 6(b) illustrates the variation of S_gen as a function of φ for different Da. We notice that by decreasing the Da, S_gen decreases due to increased flow resistance and decreased speed (velocity gradients). Besides, for all values of Da, φ effectively reduces S_gen.

Figure 6(c) depicts the variation of Be against φ for different values of Da. It is observed that the Be values at Da=10^-1 and 10^-2 are almost the same, while a noticeable difference is found at Da=10^-3 and 10^-4. However, all these values are higher than 0.7, which means that the total entropy generation in the system is dominated by heat transfer irreversibility. This explains why the Be increases as the Da increases. In addition, for a given value of Da, the enhancement of φ causes the Be to increase slightly.

6.jpg

Figure 6. Variation of Nu_avg (a), Be (b), and total entropy generation (c) versus φ for different Da at Re=600 and ε=0.4

The effects of Da on the streamlines and isotherms within the cavity for φ=4%, Re=600, and ε=0.4, are presented in Figure 7. As shown in the streamlines, three vortices of different sizes are created, the largest in the center and the others at the corners of the cavity. As the Da increases, the strength of the vortices increases, which means a better convection flow regime is created. Furthermore, the Da has a clear impact on the isotherms, so that the more Da increases, the cavity’s temperature distribution is improved, and the isothermal lines generated from the heated surface extend more and more over the cavity allowing the fluid to carry the heat easily at lower temperatures.

7.jpg

7-2.jpg

Figure 7. Isotherms (a) and streamlines (b) for different values of Da at Re=600, φ=4% and ε=0.4

Figure 8 illustrates the distribution of the Nu_local along the fin for different Da at Re=600, φ=4%, and ε=0.4. According to this figure, the local Nusselt number increased strongly when heading from bottom to the top of the fin due to the increase in surface area exposed to the fluid, which increases the heat transfer rate between the solid surface and the surrounding fluid, promoting a better convection.

The variation of the Nu_local versus distance along the heated surface for Re=600, ε=0.4, and φ=4% and for different Da is shown in Figure 9. It is observed that the heat transfer rate reached its maximum at the middle of the cavity due to the enhancement of convection flow, and the rate is minimum near the adiabatic walls due to boundary layer effects for all values of Da. It is also observed that Nu_local increases for high Da, which confirm what we explained before.

8.png

Figure 8. Distribution of the local Nusselt number along the fin for different Da at Re=600, φ=4% and ε=0.4

9.png

Figure 9. Distribution of the local Nusselt number along the heated surface from back to the front side for different Da at Re=600, φ=4%, and ε=0.4

5.2 The porosity effects

The effects of modifying the porosity of the medium on the Nu_avg, S_gen, and Be are shown in Figure 10. Different porosity values (ε=0.2, 0.4, 0.6, and 0.8) are investigated for different φ. It can be observed that the Nu_avg increases when a larger porosity ratio is used. Since porosity measures the empty space within the medium, the amount of void spaces increases as the porosity ratio increases. This implies more space for the fluid to flow through, which might increase convection and heat transfer.

Figure 10(b) reveals the variation of S_gen with φ for different ε. As illustrated, S_gen increases by decreasing the porosity due to the rise in temperature gradients since the convection is weak, which leads to higher rates of entropy generation.

Figure 10(c) reveals the effects of the Be on the φ for different Da. High values of ε indicate a predominance of convective heat transfer, which means higher fluid perturbation emerges. As fluid flows through the pores of the material, it forms a thermal boundary layer along the solid-fluid interface. This boundary layer is where most of the convective heat transfer take place. With higher ε, the thickness of this boundary layer may decrease due to increased fluid flow and turbulence, leading to enhance heat transfer. This results an increase in temperature gradients. Therefore, the cavity's larger heat transfer irreversibility makes the Be rise. In addition, there is a slight augmentation of Be when increasing φ.

10.jpg

Figure 10. Variation of Nu_avg (a), Be (b), and total entropy generation (c) versus φ for different values of ε at Re=600 and Da=10^-1

Figure 11 displays the impact of the ε on fluid flow and heat transfer in terms of isotherms and streamlines. it can be seen that the ε has a significant effect on isotherms since convection is considerable at high ε values and it is accompanied by decreasing viscous effects; the intensity of the temperature gradients along the fin as it is the main source of heat diffusion become less by increasing the ε values.

On the other hand, streamlines show three recirculation cells with a major vortex at the middle of the cavity, and two other vortices at the corners are generated. The size of the vortex near the porous medium increased by increasing the ε, and the fluid circulation in this area became significant.

11a.jpg

11b.jpg

Figure 11. Isotherms (a) and streamlines (b) for different values of ε at Re=600, φ=4% and Da=10^-1

5.3 Correlation

Finally, two correlations for Nu are done in terms of the pertinent parameters, Re, Da, φ and the porosity are presented in Figure 12. The correlations are expressed as follows:

$\begin{aligned} N u & =0.328 R e^{0.671} D a^{0.0380}(1+\varphi)^{1.291}\end{aligned}$                 (24)

$\begin{aligned}N u & =0.286 R e^{0.774} \varepsilon^{0.027}(1+\varphi)^{1.235}\end{aligned}$                               (25) 

12.jpg

Figure 12. Comparison between the numerical results and correlation