Numerical Simulation of the Natural Convective Heat Transfer of Nanofluids in a Square Cavity Based on Different Predictive Models for Single-Phase and Multiphase Flow Mixtures

Among the various convective heat transfer methods, natural convective heat transfer has the lowest heat flow density. However, it is safe, economical, and noiseless; therefore, it is widely used in industrial fields such as solar collectors, building heating and ventilation, air cooling equipment, and cooling of electronic equipment [1]. For the practical application of natural convective heat transfer in engineering, conventional fluids (such as water, ethanol, and oil) are generally used as the heat transfer medium. The low thermal conductivity of these working fluids limits the strength of the natural convective heat transfer. Choi and Eastman [2] first introduced the concept of nanofluids in 1995, and many scholars have since explored various types of the nanofluid, their thermophysical properties, forced convective heat transfer, natural convective heat transfer, and boiling heat transfer [3–8]. Compared with traditional heat transfer media, nanofluids not only have a higher thermal conductivity, but are also more stable than suspensions of millimetre- and micron-sized particles; thus, nanofluids do not clog or wear out pipes and do not cause a significant increase in the system pressure drop [9].

Compared with conventional heat transfer media, nanofluids have higher thermal conductivities, and thus a large number of experimental and theoretical studies have investigated the thermal properties of nanofluids. Thong et al. [10] studied the effect of nanoparticles of different shapes on the thermal conductivity and viscosity. The experimental results showed that the shape and size of the nanoparticles were important factors affecting the thermal conductivity. This result is consistent with the findings of Main et al. [11]. Vijayta et al. [12] analyzed the effects of temperature and concentration on the density, viscosity, and sound speed of nanofluids to determine the most suitable mass concentration of a Co₃O₄ nanofluid from the perspective of its acoustic and thermophysical properties. Nikhil and Hemadri [13] compared the influence of three different nanoparticles on the thermorheological behavior of the base fluid; the results indicated that the surfactant not only affected the stability of nanofluids but also the thermal conductivity and viscosity. With further understanding of the thermophysical properties of nanofluids, a large number of studies have focused on mechanisms to enhance these thermophysical properties. Bahiraei et al. [14] studied the effect of the nanoparticle shape on the heat transfer and pressure drop; they suggested that flakes had the highest heat transfer efficiency and the smallest pressure drop, whereas oblate spheroids had the lowest efficiency. Ambreen et al. [15] and Li et al. [16] numerically studied the effect of the interfacial layer on the heat transfer and pressure drop of nanofluids using the multiphase method and molecular dynamics method, respectively. The results indicated that considering the influence of the interfacial layer could effectively improve the simulation accuracy. Elsaidy et al. [17] studied the effect of the nanoparticle agglomeration structure on the thermal conductivity. The experimental results showed that an appropriate increase in the diameter of the agglomerate structure was beneficial for improving the thermal conductivity of nanofluids without causing a sharp increase in viscosity. Esfe et al. [18] used an artificial neural network to build a prediction model for nanofluid thermophysical properties with respect to the temperature and nanoparticle volume fraction.

Many experimental and simulation studies have been conducted on the application of nanofluids in square cavities. Giwa et al. [19] studied the effect of different magnetic field sources on nanofluids and suggested that the magnetic field could effectively improve natural convection in a square cavity. Similar findings were obtained for Yan et al. [20] and Jelodari et al. [21] regarding the flow characteristics and heat transfer of magnetic nanofluids under the influence of different magnetic field sources. Giwa et al. [22] experimentally investigated the natural convective heat transfer of mixed nanofluids and single-property nanofluids. The results showed that nanofluids can effectively improve the natural convective heat transfer compared to the base liquid, and the enhanced heat transfer of mixed nanofluids was better than that of nanofluids containing a single particle species. This result was consistent with the findings of Solomon et al. [23]. Based on extensive experimental data, researchers have used numerical simulations to investigate the mechanism of enhanced heat transfer by nanofluids. Cho [24] numerically studied the effects of the Richardson number, Reynolds number, and volume fraction on the Nusselt number and provided the optimal curve of the cavity wall. Salman et al. [25] studied the enhanced heating transfer of various types of nanofluids through experiments and single-phase flow numerical models. The numerical simulation results were in good agreement with the experimental values. Kalbani et al. [26], Wang et al. [27], and Shuleprova et al. [28] also used single-phase flow models to study the flow and heat transfer of nanofluids in square cavities and obtained results that conformed to the experimental observations. At the same time, after comparing the numerical simulation results of single-phase and two-phase flow models, Lotfi et al. [29] concluded that the two-phase flow model was better than the single-phase flow model. Many subsequent studies on two-phase flow models have since been performed. Liao [30], Corcione et al. [31], and Borrelli et al. [32] used two-phase flow models to study the flow and enhanced heat transfer in nanofluids cavities with the aim of analyzing the Brownian motion and thermophoresis of nanoparticles. The enhanced heat transfer mechanism of nanofluids has been studied from various aspects, such as the nanoparticle shape.

As can be seen from the previous experimental and simulation studies, several studies have been conducted on the thermal properties of nanofluids and natural convective heat transfer. However, comparisons of the prediction accuracy of nanofluid thermophysical property prediction models based on different theories and methods in simulation studies and discussion of the simulation accuracies of the single-phase and two-phase flow models are insufficient. Therefore, in this study, the natural convective heat transfer and flow characteristics of nanofluids in a square cavity are investigated using single-phase and mixed-phase flow models, and the results of the two models are compared with available experimental data [33]. The effects of different thermal property prediction models on the single-phase flow simulation results are investigated, and the effects of the Grashof number and nanoparticle volume fraction on the Nusselt number are examined based on a two-phase flow mixing model. The velocity and temperature distributions of the nanofluid in the vertical and horizontal central cross section of the square cavity at different Grashof numbers and volume fractions are investigated to explore the applicability of using a multiphase flow mixing model to analyze the properties of nanofluids.

The model used in this study is the same as the experimental model [33] in the literature, i.e., a square cavity with a size of 40 mm × 40 mm × 90 mm. To improve the simulation efficiency, the model is simplified into two dimensions, as shown in Fig. 1. The side length of the square cavity is H = W = 40 mm, the upper and lower walls are adiabatic (heat flow density q = 0), u is the velocity in the x-direction, v is the velocity in the y-direction, the left wall is a high-temperature wall with temperature T_H, and the right wall is a low-temperature wall with temperature T_C (T_H ≥ T_C). The cavity is filled with a liquid (water or nanofluid). The selected nanofluids are consistent with those in the literature; they are water-based alumina (Al₂O₃-H₂O) nanofluids with an average nanoparticle size of 33 nm.

In single-phase flow simulations, the thermophysical parameters of nanofluids are mainly obtained through classical formulas and prediction models. The thermophysical properties of nanofluids are an important factor affecting the accuracy of single-phase flow simulations. The thermophysical parameters of the nanofluids used in the two-phase flow mixture model are the respective thermophysical parameters of the nanoparticles and base fluid. In this section, we introduce the prediction model for the nanofluid thermophysical parameters considering different factors.

It is clear and obvious from Eq. (1) that Maxwell model only considers the volume fraction of solid particles and the thermal conductivities of the basic fluid and solid particles. Thus, Eq. (1) is suitable for the thermal conductivity of dilute suspensions with large particle sizes and relatively small particle volume fractions. It can be used for rate calculations and is a homogeneous model.

where h is the thickness of the boundary layer, $rp$ is the radius of the nanoparticles, and $γ$ is the ratio of the thermal conductivity of the liquid to that of the nanoparticles.

where $κ$ is the Boltzmann constant; $T0=273 K$ is the reference temperature; $ρf$⁠, $Cp,f$⁠, and T are the density, specific heat capacity, and temperature of the base fluid, respectively; and $ρp$ and $dp$ are the density and diameter of the nanoparticles, respectively.

where M is the molar mass of the base liquid molecule, $ρf0$ is the density of the base liquid at 293 K, and N is Avogadro's constant.

These prediction models can be divided into three groups according to whether the interfacial layer and nanoparticle micromotion are considered and whether they are corrected with extensive experimental data, as summarized in Table 1.

Equations (1)–(13) are the prediction models for the thermophysical properties of nanofluids used in this study. We summarize the specific formulas in Table 2 and list the relevant major references in Table 3.

When simulating a single-phase flow, the following assumptions are made.

1. The nanofluid is an isotropic, incompressible, Newtonian fluid with laminar motion.

2. The nanoparticles have a uniform shape and size distribution, and the nanoparticles are in thermal equilibrium with the base fluid, maintaining a uniform flow rate and no mutual slip.

3. The thermophysical parameters of nanofluids are constant, except for the density, and the variation in the density of the nanofluid due to the buoyancy force is determined using the Boussinesq assumption.

4. Viscous dissipation is neglected.

Based on the above assumptions, the governing equations for the natural convective heat transfer of nanofluids in a square cavity can be obtained.

where p is the static pressure, $τ¯¯$ is the stress tensor (described below), and $ρg$ and $F$ are the gravitational body force and external body forces (for example,, that arise from interaction with the dispersed phase), respectively. $F$ also contains other model-dependent source terms such as porous-media and user-defined sources.

Mixture models can be used for two-phase or multiphase flows (fluids or particles). In the mixture model, the different phases are regarded as an interpenetrating continuum, the volume fraction of which is a continuous function of time and space; the combination of the volume fractions of each phase is equal to 1.

α_k is the volume fraction of phase, k.

The commercial software Fluent was used to perform the simulation calculations. The set of linear differential equations is solved using the Finite Volumes Methods. The Semi-Implied Method of Pressure Correlation Equation (SIMPLE) algorithm is used to deal with the coupling of the pressure and velocity fields; the second-order windward format is used for the momentum, energy, and multiphase flow volume fraction equations; and the pressure staggered option (PRESTO) algorithm is used for the pressure correction equation with a suitable subrelaxation factor.

where q is the average heat flux, and $ΔT$ is the temperature difference between the high-temperature wall and the low-temperature wall.

where L is the characteristic length (L is the side length of the square cavity), and k is the thermal conductivity.

The calculation model uses a quadrilateral uniform grid. First, de-ionized water is used as the working medium to verify the effect of the mesh number on the calculation results. The results are summarized in Table 4. When a grid number of 100 × 100 is used, the Nusselt number does not change notably with the grid number. Therefore, a grid number of 100 × 100 is selected for the subsequent simulations.

It is also crucial to verify the reliability of the numerical method before performing numerical simulations. The two-dimensional (2D) model and the three-dimensional (3D) model were used to simulate the natural convection heat transfer of de-ionized water in the square cavity, respectively, and compared with the experimental data [33]. The results are shown in Fig. 2(a). The accuracy of the two-dimensional model is comparable to that of the three-dimensional model, and the maximum errors are 4.5% and 2.1%, respectively, indicating that the two-dimensional model meets the accuracy requirements and can improve the simulation calculation speed. In addition, the single-phase model and the mixture model were used to simulate the natural convection heat transfer in the cavity, respectively, and the results are compared with those obtained experimentally [33]. It can be seen from Fig. 2(b) that the maximal error of the single-phase flow model is 4.5% and that of the mixture model is 5.3%, indicating that the calculation accuracies of the two models are satisfactory.

Figure 3 shows a comparison of the simulation results of the single-phase flow model and mixture model with the experimental results [33] for the nanofluid. As shown in Fig. 3, the trends of the values obtained using the two simulation methods are consistent with the experimental results; the maximum error of the single-phase flow model is 13.9%, and that of the mixture model is 7.2%. The results of the two models thus satisfy the simulation accuracy requirements. The accuracy of the mixture model is higher than that of the single-phase flow model. Furthermore, as the volume fraction of nanoparticles increases, the error of the single-phase flow model increases further compared to that of the mixture model. At high volume fractions, the mixture model maintains agreement with the experimental results, whereas the calculated data obtained with the single-phase model is only more accurate in the case of low volume percentage. This shows that the nanofluid cannot be simply regarded as a solid–liquid two-phase system, and the flow and heat transfer characteristics of the nanofluid cannot be accurately reflected using the classical experimental formula directly. Owing to the small size effect of nanoparticles, the particles undergo random Brownian motion within the base fluid. As the concentration of nanoparticles increases, the likelihood of particle-to-particle and particle-to-base fluid collisions increases. The mixture model can calculate the relative velocity of the second phase. This may be one of the reasons why the prediction results of the mixture model considering the relative velocities of the discrete phases are more in line with the experimental data.

In the calculation process of the single-phase flow models, nanofluids are regarded as a liquid conforming to the Navier–Stokes (N–S) equation and different prediction models for the thermophysical properties are a key factor affecting the simulation accuracy. Figure 4 shows the computational results obtained using the different prediction models introduced in Sec. 2, which are compared with the experimental data [33]. As shown in Fig. 4, the calculation accuracy of the Choi and Ward model considering the interface layer and Brownian motion is higher than that of the traditional Maxwell and Einstein formulas. On this basis, the Vajjha and Corcione correction models modified based on a large amount of experimental data have better prediction accuracies for different types of nanofluids. It is suggested that random Brownian motion and the interfacial layer of nanoparticles in the basic liquid are key factors influencing the thermophysical properties of the nanofluid. The interior of the nanofluid is composed of particles with random motion, and this motion will lead to violent collisions between particles, between particles and base liquid molecules, and between particles and heat exchange interfaces, which will affect the heat and mass transfer inside the nanofluid. Moreover, the base liquid molecules will also wrap around the nanoparticles and nanoparticle aggregates to form small micelles, which affects the flow and heat transfer inside the base liquid. This suggests that Brownian motion and interfacial layers should affect the prediction accuracy of the nanofluids properties of heat transfer and flow characteristics. However, there is still considerable uncertainty about the effects of interactions between different types of particles in the base fluid, and purely theoretical models still require a large amount of experimental data to improve the prediction accuracy. Therefore, different prediction models are key factors affecting the accuracy of single-phase flow simulations. Simultaneously, more theoretical and experimental studies are required to uncover factors that affect the heat transfer and flow characteristics of the nanofluid.

The impacts of the Grashof number and volumetric percentage on the improved heat transfer of nanofluids are explored using a mixture model since its calculation findings are more precise than those of the single-phase flow model. For natural convective heat transfer, the Rayleigh number criterion obtained by the dimensionless energy differential equation is usually used to evaluate the transformation of the heat transfer law; however, the effect is not ideal. From a theoretical perspective, the criterion to reflect the transition of the flow shape should be derived from the dimensionless form of the momentum differential equation. Yang et al. [40,41] showed that it was better to use the Grashof number than the Rayleigh number criterion as the basis for the transformation of the heat transfer law. Therefore, this study adopts the Grashof number criterion as the basis for determining the transformation of the heat transfer classification.

Figure 5 shows the variation in the convective heat transfer coefficient (h) with the Grashof number. It is clear to observe that after the addition of nanoparticles, as relative to de-ionized water, the natural convection heat transfer efficiency is much higher, and it increases with increasing Grashof number. An increase in the Grashof number indicates that the temperature difference between the two ends of the square cavity increases. The addition of nanoparticles increases the thermal conductivity of the base liquid, allowing the liquid to heat more rapidly and improving the driving force of the flow inside the square cavity. When the Grashof number is constant, an enhancement in the nanoparticle volume fraction improves convective heat transfer even further. In this case, the temperature difference is constant and the driving force is the same, which means that under the same driving force, the nanoparticles can enhance the heat and mass transfer capacity inside the liquid. While the addition of nanoparticles enhances the basic fliud's thermal conductivity directly, it reduced the mobility of the base liquid, which decreases the natural convective heat transfer to a certain extent. As a result, it is vital to think about the impact of adding nanoparticles on the enhanced heat transfer and the mobility of the base liquid in a comprehensive and balanced manner.

To further analyze the flow state of the nanofluid in the square cavity, Fig. 6 illustrates the influence of the Grashof number on the velocity distribution in the horizontal and vertical directions at the cavity center. It can be seen that at a given Grashof number, with an increase in the volume fraction of nanoparticles, the velocity of the nanofluid in both directions increases significantly. It leads to the enhancement of the natural convective heat transfer inside the square cavity, which facilitates the enhanced transfer of heat and mass. When the Grashof number increases, the extreme velocity value appears closer to the wall. Because the micromotion of nanoparticles enhances the heat and mass transfer inside the fluid, the flow in the nanofluid tends to be accelerated at the near-wall, while the flow decelerates in the central part. This suggests that nanofluids are more likely to produce natural convective heat transfer when driven by the same temperature difference. This phenomenon is the same as the flow trend for de-ionized water, indicating that the flow behavior of nanofluids is more similar to that of pure liquids than to that of millimetre- or micron-scale suspensions.

The Nusselt number varies with nanoparticles volume percentage, which can be seen in Fig. 7. It was demonstrated in Fig. 7 that the incorporation of the Al₂O₃ nanoparticle to a basic fluid can increase the Nussle number and improve the heat transfer capacity of the liquid. As nanoparticles volume fraction increases, the Nusselt number further increases. Compared with the base fluid, the Nusselt number of the nanofluid is increased by 19.7% with a volume fraction of nanoparticles of 2%. There are two main reasons for this phenomenon. On one hand, with an increase in the volume concentration, the thermal conductivity of nanofluids increases significantly (considering the basic fluid, water). On the other hand, the irregular thermal motion of Al₂O₃ nanoparticles inside the base liquid leads to more intense collisions between the Al₂O₃ nanoparticles and between particles and molecules of the basic liquid, thus intensifying the heat and mass transfer inside the liquid. Therefore, one of the most critical elements influencing the natural convective heat transfer of the nanofluid is the volume fraction of nanoparticles.

Figure 8 shows cloud diagrams of the temperature and velocity distributions of water and the nanofluid (with a concentration of 2%) in the square cavity for the same Grashof number. The solid and dashed lines represent the nanofluid and water, respectively. The liquid in the square cavity is heated on the left high-temperature wall, and its density decreases. Owing to the action of buoyancy, the liquid flows upward from the left wall, while the cooler wall on the right cools the heated liquid, thereby driving the liquid to flow in the square cavity. As shown in Fig. 8, the isotherm on the left wall is shifted to the right, which is caused by the convection of the liquid. The angle of the isotherm of the nanofluid shifted to the right exceeds that of the base fluid, demonstrating that the Al₂O₃ nanoparticles' unpredictable Brownian thermal movement and the interference between the nanoparticles with base fluid molecules improve the energy transfer inside the fluid. At the same time, the addition of nanoparticles reduces the angle between the velocity and temperature gradients. According to the principle of field synergy [42], reducing the angle between the velocity and temperature gradients can effectively strengthen convective heat transfer. This also suggests that the flow properties of nanofluids are more similar to those of pure liquids, and thus the theory of enhanced heat and mass transfer from pure liquids can be used to explain some of the flow and heat transfer properties of nanofluids. The small-scale effect of nanoparticles is the key reason why their behavior differs from that of a conventional solid–liquid two-phase flow. Therefore, when studying the enhanced heat transfer mechanism of nanofluids, it is important to consider the micromotion within the nanofluid. The use of mesoscopic or microscopic simulation tools may be a better research direction for simulation studies of nanofluids.

In this study, numerical simulations have been performed to determine the accuracy of different prediction models for the thermophysical properties of the nanofluids, and the simulation accuracies of the single-phase flow model and multiphase flow mixture model were compared. According to the principle of field synergy, the influence of the nanofluid on the flow and heat transfer within the cavity under the same Grashof number was analyzed, and the following conclusions were obtained.

1. The mixture model outperformed the single-phase flow model in terms of simulation precision. The simulation reliability of the single-phase flow model was more dependent on a high-precision thermophysical property prediction model. Thermal conductivity and viscosity prediction models that consider the Brownian motion of nanoparticles and the boundary layer had better simulation accuracy. Moreover, the models proposed by Vajjha and Corcioen, which are based on a large amount of experimental data, were suitable for most types of nanofluids.

2. The experimental and simulation results both demonstrated that the inclusion of nanoparticles could enhance the heat transfer and that this effect has risen as the nanoparticle concentration increased. When the Grashof number was constant, the volume fraction of nanoparticles increased and the heat transfer enhancement increased.

3. The addition of a nanofluid enhanced the flow strength of the fluid in the square cavity. Meanwhile, when subjected to the same Grashof number conditions, as the volume concentration of Al₂O₃ nanoparticles rises, the flow velocity in the x- and y-directions increased, the heat transfer inside the fluid increased, and the angle between the velocity and temperature gradients decreased. The nanofluid strengthened the flow state of the fluid inside the square cavity, which is conducive to strengthening the energy transmission inside the fluid, thereby improving the heat exchange.

• National Natural Science Foundation of China (Grant No. 52071107; Funder ID: 10.13039/501100001809).

• Natural Science Foundation of Heilongjiang Province of China-Outstanding Youth Foundation (Grant No. YQ2021E008; Funder ID: 10.13039/501100005046).