Free Convection Heat Transfer and Entropy Generation in an Odd-Shaped Cavity Filled with a Cu-Al₂O₃ Hybrid Nanofluid

Abstract

The present paper aims to analyze the thermal convective heat transport and generated irreversibility of water-Cu-Al₂O₃ hybrid nanosuspension in an odd-shaped cavity. The side walls are adiabatic, and the internal and external borders of the enclosure are isothermally kept at high and low temperatures of T_h and T_c, respectively. The control equations based on conservation laws are formulated in dimensionless form and worked out employing the Galerkin finite element technique. The outcomes are demonstrated using streamlines, isothermal lines, heatlines, isolines of Bejan number, as well as the rate of generated entropy and the Nusselt number. Impacts of the Rayleigh number, the hybrid nanoparticles concentration (ϕ_hnf), the volume fraction of the Cu nanoparticles to ϕ_hnf ratio (ϕ_r), width ratio (WR) have been surveyed and discussed. The results show that, for all magnitudes of Rayleigh numbers, increasing nanoparticles concentration intensifies the rate of entropy generation. Moreover, for high Rayleigh numbers, increasing WR enhances the rate of heat transport.

1. Introduction

The interest in the field of heat transfer has increased considerably recently. While device performance and heat generation have gone up, the sizes of the components have shrunk. For instance, computer chips that are made today are at least several orders of magnitude smaller in size as compared to the past. Thus, the generated heat per surface area is growing continuously. Thus, novel designs with improved heat transfer capabilities are demanded to cool devices that produce large amounts of heat.

Heat transfer in enclosed geometries with various initial and boundary conditions have been a hot topic in the past few decades. This is because these types of geometries have been utilized extensively in real-life applications, such as the cooling of electronic devices, thermal design of buildings, lubrication and drying applications, furnace and nuclear reactors, biochemical, and food processing. Nonetheless, rectangular and square lid-driven cavities have been studied more because of their simplicity and applicability. As such, a large portion of convective heat transport in cavities’ studies has been devoted to rectangular geometries.

Among all the commonly used approaches to enhance heat transfer efficiency, including the passive and active methods, the dispersion of nano-sized particles of highly conductive material in the base working fluids seems to be both simpler and more promising. Hence, in recent years, many studies have focused on applying nanofluids in heat transfer devices, including solar collectors, electronics, nuclear reactors, food, glass, and others [1,2,3,4,5]. For instance, Abu-Nada and Chamkha [6] investigated the role of Cu nanoparticles, Rayleigh number, and aspect ratio of the cavity in free convection heat transfer of CuO-EG-Water nanoliquid. They found that the aspect ratio of the enclosure considerably impacts the rate of heat transfer. Oztop and Abu-Nada [7] surveyed the free convection heat transfer problem of nanoliquids in rectangular cavities. They [7] revealed that the average Nusselt number enhances as the volume fraction of the nanoparticles increases. Parvin and Chamkha [8] presented a thermal convective motion, energy transport, and entropy production in an odd-shaped chamber. The computational analysis on the influence of thermal convection and solid particle concentration on Nu, total entropy generation, and Be was performed in the chamber saturated with Cu-H₂O nanosuspension, which concluded horizontal and vertical parts. Furthermore, the entropy production resulting from energy transference, streamlines, and isotherms is shown for different Ra and particle concentrations.

Hybrid nanoliquids are a new type of nanoliquids. Generally, these types of fluid can be prepared by two separate approaches: (a) dispersion of two (or even more) nano-sized particles to a base fluid, and (b) by suspending the so-called hybrid (also composite) nanoparticles to a host liquid. The efficacy of hybrid nanosuspensions through their chemo-physical characteristics was tested and proved by many numerical and experimental studies. The properties and applicability of hybrid nanosuspensions were extensively reviewed in the literature [9,10,11,12,13,14]. Furthermore, the role of different hybrid nanoparticles on the natural, mixed, and forced convection heat transfer has been discussed in some published research studies on the boundary layer by Suresh et al. [15], Aly and Pop [16], Ismael et al. [17], Waini et al. [18,19], Roslan et al. [20], Khashi’ie et al. [21,22,23], Ali et al. [24], Khan et al. [25] and Liu et al. [26]. For example, according to the experimentations of Suresh et al. [15], by employing Cu–Al₂O₃ hybrid nanofluid, the rate of heat transfer could be enhanced up to around 13.5%. Ismael et al. [17] numerically surveyed the irreversibility and heat transfer of Cu–Al₂O₃ hybrid nanofluid in a lid-driven cavity. They reported that using hybrid nanofluids is promising for the improvement of heat transfer and reducing the size of heat transfer devices. Roslan et al. [20] investigated the mixed convection heat transfer of Al₂O₃-Cu-water hybrid nanofluids in a rectangular cavity. The top lid of the cavity was moving with a constant velocity, inducing a mixed convection flow. The results of Roslan et al. [20], in agreement with the outcomes of Roslan et al. [20], showed that the presence of hybrid nanoparticles improved the heat transfer. Following [20], Ali et al. [24] analyzed the mixed convection of the same hybrid nanofluid for a double lid-driven cavity, where both top and bottom lids were moving. This research also confirmed the heat transfer enhancement of using hybrid nanofluids.

With the above review, it can be clearly observed that the natural convection heat transfer of hybrid nanofluids is an important task that could be promising for reducing the size of heat transfer devices. The literature reviewed above shows that the heat transfer and energy generation of nanosuspensions have been addressed in many recent publications. However, the entropy generation and visualization of heat transfer paths (heat-lines) of hybrid nanofluids, and the shape of enclosures, are the influential aspects that have been overlooked in the literature and will be tackled in the present work.

2. Problem Physics

Free convective heat transport of Cu–H₂O hybrid nanoliquid in the region formed between a hot block and its chamber is schematically shown in Figure 1.

The vertical (or horizontal) length of the hot (or cold) surfaces is L (or L—W). The width of the cavity is W and can vary to have different width ratios (W/L). The temperature of the interior walls, shown in red in Figure 1, is isothermally kept constant at T_h, and the external walls, depicted in blue, are at the temperature of T_c. The remaining borders are assumed to be adequately insulated. It is assumed that the circulation is steady, incompressible, laminar, and the radiation has no significant effect on the process.

2.1. Governing Equations

Under the above-mentioned assumptions, the control equations for the hybrid nanoliquid, based on conservation laws, are [8,27]:

Continuity Equation:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

Momentum Equation:

$${\rho }_{hnf}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+{\mu }_{hnf}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)$$

(2)

$${\rho }_{hnf}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+{\mu }_{hnf}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+{\left(\rho \beta \right)}_{hnf}g\left(T-T_c\right)$$

(3)

Energy Equation:

$${\left(\rho c\right)}_{hnf}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=k_{hnf}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$$

(4)

where u and v are the velocity components in x and y directions, T is the temperature of the hybrid nanofluid, g is the gravity and p is the pressure. The corresponding boundary conditions are

$$\begin{array}{l}T=T_h\mathrm{at}\mathrm{the}\mathrm{inner}\mathrm{walls}\\ T=T_c\mathrm{at}\mathrm{the}\mathrm{outer}\mathrm{walls}\\ \frac{\partial T}{\partial n}=0\mathrm{at}\mathrm{the}\mathrm{rest}\mathrm{of}\mathrm{the}\mathrm{surfaces}\\ u=v=0\mathrm{at}\mathrm{all}\mathrm{solid}\mathrm{surfaces}\end{array}$$

(5)

2.2. Thermophysical Characteristics

In Equations (1)–(4), ρ_hnf, (ρc)_hnf, β_hnf and μ_hnf denote the density, thermal capacitance, heat expansion coefficient and dynamic viscosity of the nanoliquid which could be calculated from the below relations [28]:

$${\mu }_{hnf}={\mu }_f{\left(1-{\phi }_{hnf}\right)}^{-2.5}$$

(6)

$${\rho }_{hnf}={\rho }_{A{l}_2{O}_3}{\phi }_{A{l}_2{O}_3}+{\rho }_{Cu}{\phi }_{Cu}+\left(1-{\phi }_{hnf}\right){\rho }_f$$

(7)

$$\frac{k_{hnf}}{k_f}=\frac{\left\{\frac{{\phi }_{A{l}_2{O}_3}{k}_{A{l}_2{O}_3}+{\phi }_{Cu}{k}_{Cu}}{{\phi }_{hnf}}+2k_f+2\left({\phi }_{A{l}_2{O}_3}{k}_{A{l}_2{O}_3}+{\phi }_{Cu}{k}_{Cu}\right)-2k_f{\phi }_{hnf}\right\}}{\left\{\frac{{\phi }_{A{l}_2{O}_3}{k}_{A{l}_2{O}_3}+{\phi }_{Cu}{k}_{Cu}}{{\phi }_{hnf}}+2k_f-\left({\phi }_{A{l}_2{O}_3}{k}_{A{l}_2{O}_3}+{\phi }_{Cu}{k}_{Cu}\right)+k_f{\phi }_{hnf}\right\}}$$

(8)

$${\left(\rho c\right)}_{hnf}={\phi }_{A{l}_2{O}_3}{\left(\rho c\right)}_{A{l}_2{O}_3}+{\phi }_{Cu}{\left(\rho c\right)}_{Cu}+\left(1-{\phi }_{hnf}\right){\left(\rho c\right)}_f$$

(9)

$${\left(\rho \beta \right)}_{hnf}={\phi }_{A{l}_2{O}_3}{\left(\rho \beta \right)}_{A{l}_2{O}_3}+{\phi }_{Cu}{\left(\rho \beta \right)}_{Cu}+\left(1-{\phi }_{hnf}\right){\left(\rho \beta \right)}_f$$

(10)

Here, ${\phi }_{hnf}={\phi }_{A{l}_2{O}_3}+{\phi }_{Cu}$ is the volume fraction of the hybrid nanofluid. Moreover, the subscripts f, Cu, and Al₂O₃ refer to the base fluid, Cu, and Al₂O₃ nanoparticles’ properties, respectively. The physical characteristics of the hybrid nano-sized particles and water (as the host fluid) are gathered in

Table 1. Thermal characteristics of Cu and Al₂O₃ nanoparticles [29].

	ρ (kg·m−3)	c (J·kg−1·K−1)	k (W·m−1·K−1)	α (m2·s−1) × 107	β (K−1) × 106
Al2O3	3970	765	40	131.7	25.5
Cu	8933	385	400	1163.1	50.1
Host fluid (water)	997.1	4179.0	0.613	1.47	210

.

2.3. Non-Dimensional Form of the Governing Equations

Equations (1)–(4) can be non-dimensionalized by introducing the below parameters:

$$X=\frac{x}{L},Y=\frac{y}{L},U=\frac{uL}{{\nu }_f},V=\frac{vL}{{\nu }_f},P=\frac{p{L}^2}{{\rho }_f{\nu }_f^2},\theta =\frac{T-T_c}{T_h-T_c}$$

(11)

Substituting Equation (7) into Equations (1)–(4), results in the next equations

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$

(12)

$$\frac{{\rho }_{hnf}}{{\rho }_f}\left(U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}\right)=-\frac{\partial P}{\partial X}+\frac{{\mu }_{hnf}}{{\mu }_f}\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)$$

(13)

$$\frac{{\rho }_{hnf}}{{\rho }_f}\left(U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}\right)=-\frac{\partial P}{\partial Y}+\frac{{\mu }_{hnf}}{{\mu }_f}\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)+Ra\cdot Pr\frac{{\left(\rho \beta \right)}_{hnf}}{{\left(\rho \beta \right)}_f}\theta$$

(14)

$$\frac{{\left(\rho c\right)}_{hnf}}{{\left(\rho c\right)}_f}\left(U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}\right)=\frac{k_{hnf}}{k_f}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)$$

(15)

where Rayleigh and Prandtl numbers are defined as

$$\begin{array}{c}Ra=\frac{\rho g\beta \left(T_h-T_c\right)L^3}{{\alpha }_f{\nu }_f}\\ Pr=\frac{C_p{\mu }_f}{k_f}\end{array}$$

(16)

The boundary conditions (5) in non-dimensional form become

$$\begin{array}{l}\theta =1\mathrm{at}\mathrm{the}\mathrm{inner}\mathrm{walls}\\ \theta =0\mathrm{at}\mathrm{the}\mathrm{outer}\mathrm{walls}\\ \frac{\partial \theta }{\partial N}=0\mathrm{at}\mathrm{the}\mathrm{rest}\mathrm{of}\mathrm{the}\mathrm{surfaces}\\ U=V=0\mathrm{at}\mathrm{all}\mathrm{solid}\mathrm{surfaces}\end{array}$$

(17)

2.4. Rate of Heat Transfer

The parameter of interest in this research is the rate of energy transference from the hot border, which can be quantified by the average Nusselt number Nu_avg as:

$$N{u}_{avg}=-\frac{1}{1-WR}\frac{k_{hnf}}{k_f}\left\{{{\displaystyle \int }}_{Y=0}^{Y=1-WR}{\frac{\partial \theta }{\partial X}|}_{X=1-WR}dY+{{\displaystyle \int }}_{X=0}^{X=1-WR}{\frac{\partial \theta }{\partial Y}|}_{Y=1-WR}dX\right\}$$

(18)

3. Entropy Generation

The strength of generated entropy is expressed as:

$$s_{gen}=\frac{k_{hnf}}{T_0^2}\left[{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right]+\frac{{\mu }_{hnf}}{T_0}\left[2{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial v}{\partial y}\right)}^2+\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right]$$

(19)

The first term in the above equation is the energy transference component of entropy production. The second one indicates that the portion of generated entropy caused by friction of the hybrid nanofluid. The generated entropy could be transformed into the non-dimensional form using the below parameters [30,31]:

$$S=\frac{s{T}_0^2{L}^2}{k_f\left(T_h-T_c\right)}$$

(20)

Equation (15) in the non-dimensional form reads as:

$$S_{gen}=S_{th}+S_{viscous}$$

(21)

where

$$S_{th}=\frac{k_{hnf}}{k_f}\left[{\left(\frac{\partial \theta }{\partial X}\right)}^2+{\left(\frac{\partial \theta }{\partial Y}\right)}^2\right]$$

(22)

$$S_{viscous}=\chi \frac{{\mu }_{hnf}}{{\mu }_f}\left[2{\left(\frac{\partial U}{\partial X}\right)}^2+2{\left(\frac{\partial V}{\partial Y}\right)}^2+\left(\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}\right)\right]$$

(23)

Here χ is called the irreversibility function and can be defined as:

$$\chi =\frac{T_0{\mu }_f}{k_f}{\left[\frac{L}{{\nu }_f\left(T_h-T_c\right)}\right]}^2$$

(24)

The ratio of generated entropy induced by energy transference to the total entropy production is called the Bejan number, which is

$$Be=\frac{S_{th}}{S_{gen}}$$

(25)

It is apparent that for Be > 0.5 (Be < 0.5), the irreversibility of the energy transference (fluid friction) governs the total entropy production.

4. Heat Function

The x- and y-derivatives of the heat function equal the energy flux components (j_,x, j_,y) and for the hybrid nanofluid read as [32]:

$$j_x=\frac{\partial h}{\partial y}={\left(\rho c\right)}_{hnf}u\left(T-T_c\right)-k_{hnf}\frac{\partial T}{\partial x}$$

(26)

$$j_y=-\frac{\partial h}{\partial x}={\left(\rho c\right)}_{hnf}v\left(T-T_c\right)-k_{hnf}\frac{\partial T}{\partial y}$$

(27)

Using the non-dimensional heat function as $H=\frac{h}{\left[{\alpha }_f\left(T_h-T_c\right)\right]}$, the dimensionless form of the above equations is obtained as:

$$J_X=\frac{\partial H}{\partial Y}=\frac{{\left(\rho c\right)}_{hnf}}{{\left(\rho c\right)}_f}U\theta -\frac{k_{hnf}}{k_f}\frac{\partial \theta }{\partial X}$$

(28)

$$J_Y=-\frac{\partial H}{\partial X}=\frac{{\left(\rho c\right)}_{hnf}}{{\left(\rho c\right)}_f}V\theta -\frac{k_{hnf}}{k_f}\frac{\partial \theta }{\partial Y}$$

(29)

With a cross derivation, the dimensionless presentation of the heat function could be calculated as the following:

$$\frac{{\partial }^{2}H}{\partial X^2}+\frac{{\partial }^{2}H}{\partial Y^2}=\frac{{\left(\rho c\right)}_{hnf}}{{\left(\rho c\right)}_f}\left\{\frac{\partial \left(V\theta \right)}{\partial X}-\frac{\partial \left(U\theta \right)}{\partial Y}\right\}$$

(30)

The boundary conditions for the above equation can be obtained by integrating over the Equations (23) or (24) and imposing the no-slip condition (U = V = 0). In addition, the heat lines on the bottom wall can be arbitrary set to zero [32]:

on the bottom wall:

$$H\left(1-WR\le 0\le 1,0\right)=0$$

(31)

on the inner walls:

$$\begin{array}{c}H\left(1-WR,0\le Y\le 1-WR\right)=-\frac{k_{hnf}}{k_f}{{\displaystyle \int }}_0^Y\frac{\partial \theta }{\partial X}dY\mathrm{and}\\ H\left(0\le X\le 1-WR,1-WR\right)=\frac{k_{hnf}}{k_f}{{\displaystyle \int }}_0^X\frac{\partial \theta }{\partial Y}dX-\frac{k_{hnf}}{k_f}{{\displaystyle \int }}_0^{1-WR}{\frac{\partial \theta }{\partial X}|}_{1-WR,0\le Y\le 1-WR}dY\end{array}$$

(32)

on the outer walls:

$$\begin{array}{c}H\left(1,0\le Y\le 1\right)=-\frac{k_{hnf}}{k_f}{{\displaystyle \int }}_0^Y\frac{\partial \theta }{\partial X}dY\mathrm{and}\\ H\left(0\le X\le 1,1\right)=\frac{k_{hnf}}{k_f}{{\displaystyle \int }}_0^X\frac{\partial \theta }{\partial Y}dX-\frac{k_{hnf}}{k_f}{{\displaystyle \int }}_0^1{\frac{\partial \theta }{\partial X}|}_{1,0\le Y\le 1}dY\end{array}$$

(33)

on the right adiabatic wall:

$$\begin{array}{c}H\left(0,1-WR\le Y\le 1\right)=N{u}_{avg}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-\frac{k_{hnf}}{k_f}{{\displaystyle \int }}_0^{1-WR}{\frac{\partial \theta }{\partial Y}|}_{0\le X\le 1-WR,1-WR}dX\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{k_{hnf}}{k_f}{{\displaystyle \int }}_0^{1-WR}{\frac{\partial \theta }{\partial X}|}_{1-WR,0\le Y\le 1-WR}dY\hfill \end{array}$$

(34)

The employed numerical method, details of the grid check and validations are discussed in the Appendix A and Appendix B.

5. Results and Discussion

In this section, different parameters including Rayleigh number (Ra = 10³, 10⁴ and 10⁵), width ratio (WR = W/L = 0.2, 0.3 and 0.4), the concentration of the hybrid nanoparticles (0.0 ≤ ϕ_hnf ≤ 0.05) and the concentration of Cu to ϕ_hnf ratio (0.0 ≤ ϕ_r = ϕ_Cu/ϕ_hnf ≤ 1.0) have been surveyed and discussed. Moreover, the Prandtl number Pr = 6.2 is considered to be constant in all simulations.

Figure 2 shows the isotherms, heat lines, and streamlines in the cavity filled with hybrid nanosuspension for different Ra and WR values. Figure 2a represents the isotherms, which as seen, the isotherms are layered in the horizontal direction at Ra = 10³. The layered arrangements of the isotherms confirm that conduction energy transference is the dominant mode of heat transport in the chamber. It should be noted that the isotherms signify the strength of the conduction heat transfer; for instance, it can be observed from Figure 2 that the conduction energy transference has larger values near the walls (especially the hot walls). Conversely, the heat lines at Ra = 10³ and WR = 0.2 show the perpendicular lines toward isotherms, and heat lines in this figure represent the routes of the energy transference from the hot border to the cold border with the conduction mechanism. Furthermore, the first row of Figure 2c demonstrates a vortex zone in the vertical part of the chamber, which represents a slight fluid velocity in this region. By increasing the WR from 0.2 to 0.3 at the same Ra (second row), the isotherms tend to separate as a result of conduction energy transference reduction. In this new state, the maximum value of the heat line is reduced from 9 to 5.5, and this reduction shows the lower energy transport strength in the cavity. Therefore, it can be concluded that the rate of energy transport from the hot border to the cold border decreases if the wall width increases when the conduction is the dominant phenomenon.

The uniformity of isotherms is gone when the Ra number is increased to 10⁵ at the same WR = 0.3 (third row), especially in the horizontal zone of the chamber. It depicts that the conduction energy transference is mainly restricted to the walls, and Ra increment leads to increasing buoyancy force. Moreover, the streamlines show different vortices at Ra = 10⁵ and WR = 0.3, and these vortices are started to evolve in a horizontal direction, and the routes of heat transfer are mainly similar to streamlines. Therefore, it can be concluded; the free convective energy transference now is the dominant mode in the cavity. In the last line of Figure 2, the effect of WR is depicted at the high value of Ra. By increasing the WR from 0.3 to 0.4 at the Ra = 10⁵, the conduction effect on heat transfer is weakened, even in the cavity’s vertical part. Furthermore, the stronger vortexes and more extensive heat lines distribution, especially in the horizontal part, show the dominance convection heat transfer, which is boosted by increasing the wall width. It can be found from Figure 2 that the maximum value of the heat lines is raised in the high Ra when the wall width is increased. Therefore, increasing wall width has a direct influence on energy transference at the high Ra. Figure 2d shows the isentropic lines as the Be number is changed in the range between 0 and 1. It can be observed from the figures that the lines, with larger values, are close to walls. It shows in these areas the thermal entropy has mainly the same values as total entropy. In addition, some lines with lower values of Be are emerged in the middle of vertical cavity by increasing Ra and WR that show the effect of viscous entropy in these areas. Furthermore, these lines extend in horizontal part when the convective energy transport is the dominant mode in the chamber. Figure 3 shows the alteration of average Nu values against the total amount of nanoparticle volume fractions, with an equal proportion of Cu and Al₂O₃ in the water-based fluid. It can be obtained from Figure 3a, that the Nu_avg has a more considerable value in the higher Ra as a result of natural convection increment, which was discussed in Figure 2. On the other hand, Figure 3a shows that the Nu_avg rises when a higher concentration of solid particles presents in the medium. The investigated nanoparticles in this research have a high heat transfer conduction coefficient, and the presence of these materials can enhance the conduction phenomena. For this reason, the same slops are observed in Figure 3a, and this represents that the nanoparticles have the same and constant effect in different Ra values. Figure 3b shows that the Be number goes down when the Ra number is increased. For high Ra, the energy transport is enhanced. Therefore, the generated entropy due to energy transference increases, but it can be found from Figure 3b that the generated entropy caused by friction of the hybrid nanofluid has a higher value increment compared to S_th that leads to Be number reduction. However, nanoparticle volume fractions have no influence on the Be number in any values of the Ra (see Figure 3b). After increasing the nanoparticle volume fractions, heat transfer entropy and overall entropy both increased, so Be remained almost constant.

Figure 4 and Figure 5 demonstrate the influence of Cu nanoparticles concentration (in the constant value of ϕ_r) on the Nu_avg and Be numbers in different values of wall width. Figure 4a reveals that the percentage of Cu nanoparticle volume fractions has no considerable effect on the Nuavg. Both Cu and Al₂O₃ have a similar influence on hybrid nanofluid at Ra = 10³. It can be seen from this Figure that the Nu_avg reduces considerably when the wall width is increased from 0.2 to 0.4. As previously described at Ra = 10³, the conduction energy transference is the dominant regime; therefore, it is evident that the Nu_avg decreases according to the Fourier law when the WR increases. This reduction in heat transfer rate leads to entropy decrement, generated by heat transfer, and the Be reduction in Figure 4b is caused by this S_th alterations. Similar to the Nu_avg, the percentage of the Cu nanoparticle volume fractions has no considerable effect on Be number, because it does not affect the rate of energy transference at the low Ra; therefore, no entropy is generated by heat transfer and nanofluid frictions; then, Be remains constant at different values of ϕ_r.

Figure 5a shows that increasing the Cu particles in the hybrid nanofluid has a positive influence on energy transference at Ra = 10⁵, in opposite results, as obtained from Figure 4a. This phenomenon can be attributed to the larger conduction heat transfer coefficient of Cu compared to Al₂O₃ nanoparticles. By increasing the wall width from 0.2 to 0.3, Nu_avg is increased, as shown in Figure 5a. At WR = 0.3, the nanofluid can be circulated more easily in the cavity as a result of buoyancy force at Ra = 10⁵. However, energy transference strength decreases considerably when WR increases to 0.4. Figure 5b shows similar results in Figure 4b, but for different reasons. The generated entropy caused by frictions increases when the wall width changes from 0.2 to 0.3 because of the nanofluid movement in the cavity. As for the previous results, Be number more depends on S_viscous than S_th at high Ra numbers. Therefore, a considerable reduction can be observed on Be number in Figure 5b when WR reaches 0.3. However, by increasing WR to 0.4, the convection effect is decreased (see Figure 5a); therefore, the nanofluid velocity decreases that leads to entropy reduction (S_viscous). In this state, the S_viscous tends to S_th value, and because of this, the Be does not change noticeably at WR = 0.4 compared to WR = 0.3. Figure 5b shows that the Cu nanoparticle volume fractions in hybrid nanofluid have no substantial effect on Be number as a result of convection mechanisms dominance at high Ra numbers. However, Figure 5b shows that Be slightly decreases with ϕ_r increment at WR = 0.2. These slight changes can be attributed to Cu density that has a larger value compared to Al₂O₃. The greater density at high Ra with a narrow wall width leads to increase entropy caused by friction factors and decrease Be number.

Figure 6 shows the influence of the Ra and the hybrid nanoparticles’ concentration on the averaged of produced entropy. As previously mentioned, by increasing the Ra number, the buoyancy force intensifies, and hence, the average velocity and the rate of heat transfer in the enclosure increase (See Equations (22) and (23)). As a result, the velocity and temperature gradient are enhanced, and the generated entropy increases. As can be seen from Figure 6a,b, increasing the Rayleigh number has a considerable impact on both components of the produced entropy. It is evident that for low Rayleigh numbers, the friction share of entropy is almost feeble and the entropy generation is mainly controlled by the heat transfer mechanism. However, in high Rayleigh numbers, S_viscous overtakes its counterpart, S_th and the entropy generation induced by the friction of the hybrid nanofluid completely prevails. On the other hand, increasing the volume fraction of the hybrid nanoparticles, according to Equations (6) and (8) leads to the increment of the thermal conductivity and also the viscosity of the suspension. Hence, it slightly enhances the S_th (See Equation (22)) and S_viscous (See Equation (23)). Finally, Figure 6c represents the final result that the volume fraction of the hybrid nanoparticles, as well as the Ra number, have a direct influence on the total generated entropy in the cavity; however, the impact of the Rayleigh number (buoyancy force) is more substantial than the ϕ_hnf.

6. Conclusions

The thermo-gravitational convection of water-Cu-Al₂O₃ in an enclosed cavity was investigated. The enclosed cavity concluded two vertical and horizontal parts, and the heat was transferred from the left vertical border and bottom horizontal border toward other directions. The Cu and Al₂O₃ nanoparticles were chosen as composite nanoparticles. Then, the different volume fractions of hybrid nanoparticles and their proportion were investigated on the energy transference. Furthermore, the effect of Ra and wall width and their relationship with nanoparticle volume fractions were studied as two other important factors, which affect the energy transference and generated entropy in the chamber. The conclusions are:

-

Hybrid nanoparticles enhanced energy transport when the conduction mechanism was dominant. Conversely, they had no significant influence on convective transport;

-

The wall-width ratio (WR) is a parameter that can have a different influence on the energy transference rate in different conditions. Increasing the wall width led to a reduction of the energy transference rate at low Ra (10³) owing to the dominant conduction heat transfer mechanism;

-

WR had a positive influence on energy transference at high Ra (10⁵) when WR was increased from 0.2 to 0.3. However, a further increase of wall width reduced the heat transfer in the cavity when WR > 0.4, and therefore, an optimum wall width can enhance the heat transfer at high Ra;

-

At Ra = 10³, the nanoparticles of Cu and Al₂O₃ had a similar effect on nanofluid in the range of ϕ_hnf = 0–0.05, and they enhanced the strength of energy transference to be the same as each other. Conversely, Cu nanoparticles had a stronger impact on heat transfer compared to Al₂O₃ in convection heat transfer at Ra = 10⁵;

-

Ra and ϕ_hnf both could enhance generated thermal and viscous entropy, however, Ra had a more intensive influence on generated entropy in the cavity.