Revisiting thermo-physical property models of Al₂O₃-Water nanofluid for natural convective flow

Abstract

One of the comprehensive ways of heat transport performance augmentation of thermo-fluid systems is to use nanofluid over base fluid. This study mainly scrutinizes several existing models of thermal conduction coefficient and absolute viscosity of Al₂O₃-water nanofluid with the experimental data. A benchmark problem of natural convective flow is selected to test the performance of the available nanofluid models. The Rayleigh number varies between 10³ and 10⁹, while the solid-volume proportion (φ) changes from 0 to 4%. The governing mathematical model is numerically discretized via the Galerkin finite element procedure under appropriate auxiliary conditions. The results produced by the models are verified with the existing experimental findings based on the evaluation of the Prandtl number and average Nusselt number. It has been confirmed that the AH model (Azmi's viscosity and Ho's conductivity models) is suitable for lower nanoparticle concentration (φ = 0.01), the AM model (Azmi's viscosity and Maxwell's conductivity models) for moderate concentration (0.01 < φ < 0.04), and the NH model (Ngueyn's viscosity and Ho's conductivity models) for higher value of the solid-volume proportion (φ = 0.04).

Graphical abstract

Highlights

• •Review of dynamic viscosity and thermal conductivity models of Al₂O₃-water nanofluid.
• •Comparison of various models with existing experimental data in terms of Pr and Nu.
• •Azmi's model is preferable in finding dynamic viscosity.
• •Ho's model is a good choice for calculating thermal conductivity.
• •Joint models of Azmi's viscosity and Maxwell's conductivity show reasonable accuracy.

1. Introduction

Heat transfer augmentation demonstrates the improvement of thermal performance of any heat exchanging media, heat transferring process, heat generating components, and so on. It is presently an essential research and development topic in any thermal management system, energy and power sector, aerospace technologies, heating and cooling of electronic components and electronic devices, thermal energy storage, solar collectors, etc. One of the best techniques for augmentation is raising the thermal conduction coefficient of typical working liquids by adding nano-sized particles of higher thermal conductivity. Choi and Eastman [1] developed the name ‘nanofluid’ in 1995. They demonstrated that tiny amounts of nanoparticles, sizes between 1 and 100 nm, uniformly dispersed into a typical liquid, e.g., water, ethylene glycol, or oil, could appreciably enhance the heat transport rate. They also argued that the few experimental investigations focused on evaluating nanofluids' thermal conductivity and dynamic viscosity. Mansour et al. [2] examined how different models of nanofluid's properties affected the thermo-hydraulic behavior of pipe flow. Besides, applying Fourier sine transforms to find the exact solution of nanofluids' temperature and velocity profiles opened the door to groundbreaking research in the augmentation of heat transfer performance using nanofluids [3,4].

A significant number of computational and experimental investigations were conducted by several researchers who analyzed the effectivity of nanofluid in the field of fluid flow, energy, and thermal sciences. Kakac and Pramuanjaroenkij [5] reviewed that the thermo-fluid performance of a thermal system improved by convection with nanofluids. They suggested that a significant number of experimental studies and theoretical modeling on effective thermal diffusivity and conductivity were necessary to utilize the full potential of nanofluids for forced convection enhancement. Kasaeian et al. [6] surveyed the effect of nanofluids in porous media and confirmed that the nanofluids enhanced the thermal performance. Pinto and Fiorelli [7] reviewed the investigations of heat transfer enhancement mechanisms where nanofluid was used as a working liquid. Besides, the hydrodynamic properties of nanofluid and the forced convective flow enhancement were reviewed by Hussein et al. [8]. It was summarized that the heat transport performance of nanofluids improved because of the dispersion of nanoparticles, which improved the thermal conduction coefficient of liquids and the chaotic motion of particles, which increased the fluids’ turbulence and fluctuation. In another investigation, Ismail et al. [9] reviewed the recent applications of nanofluids, concluding that nanofluids enhanced the heat transfer performance. Guo [10] studied the thermal performance of nanofluid with an artificial neural network, which was highlighted in particular. In a recent study, Souayeh et al. [11] numerically investigated the natural convective flow around two horizontally arranged spheres at a steady temperature utilizing three separate nanofluids. They found that the thermo-hydraulic performance of the nanofluids increased with the increase of Rayleigh number.

Because of the reports of considerably improved thermal characteristics, nanofluids have gained an immense focus in recent decades. Nanofluids’ improved thermal conductivity and dynamic viscosity have widened the potential for enhanced thermal performance. Many experimentalists used their experimental data to provide the correlations of those properties for determining heat transfer rates. Einstein [12], Brinkman [13], and Batchelor [14] provided correlations for evaluating the absolute viscosity of the nanofluid, considering the solid-volume fraction of nanoparticles as one of the varying parameters. Besides, different correlations were given by many researchers to predict the thermal conduction coefficient of nanofluid. Xue [15] provided a novel thermal conductivity model of nanofluids based on average polarization and Maxwell theories. On the other hand, Jang and Choi [16], Prasher et al. [17], and Patel et al. [18] considered the effect of Brownian motion in devising the theoretical models of nanofluid thermal conductivity. Ren et al. [19] and Xie et al. [20] acknowledged the impact of interfacial nanolayers of liquid molecules in their thermal conductivity models. To suggest another correlation of thermal conductivity, Yang et al. [21] experimentally observed the heat transfer performance of nanofluids considering the effect of nanoparticle size and dispersed phase. A combined model of effective thermal conductivity, including the influences of Brownian motion, nanolayer, nanoparticle size, and surface chemistry, was proposed by Murshed et al. [22]. Among the recent studies, Xu et al. [23] investigated a novel method, called the improved steady-flow method, in determining the thermal conductivity of nanofluid and suggested that Al₂O₃-water nanofluid had great potential in the enhancement of thermal performance for a lower solid-volume fraction of nanofluid. Topal and Servantie [24] conducted a molecular dynamics simulation to calculate the thermal conductivity of nanofluids for various solid-volume fractions of nanofluids. They found enhanced thermal conductivity with a solid-volume fraction of 2%–3%. However, Raja and Sunil [25] estimated nanofluid thermal conductivity using different available models. They suggested that an extensive future investigation was necessary to determine the most reliable model for specific applications.

Despite the recent development, the above literature review confirms that a thorough investigation is still necessary to choose the best model of the thermo-physical properties of nanofluids. Thus, the current study compares the reliability of the available thermal conduction coefficient and absolute viscosity correlations of water-based Al₂O₃ nanofluid by estimating the free convection heat transfer characteristics. Firstly, the Prandtl numbers of the Al₂O₃-water nanofluid are analyzed from the evaluation of available models. Subsequently, a square-shaped differentially heated chamber filled with that nanofluid is considered to find the average Nusselt number employing different correlations of the thermal conduction coefficient and the absolute viscosity. These findings are compared to an experimental work to see which correlation produces the closest match.

2. Effective thermo-fluid properties of nanofluid

An essential task is accurately predicting the nanofluid's thermo-fluid properties to assess its thermal augmentation. A prevalent practice is gathering experimental data on nanofluid properties, mainly thermal conduction coefficient and absolute viscosity, and providing empirical correlations. The mixing theory calculates mass density, specific heat capacitance, and volumetric thermal expansion coefficient. These are the primary functions of base fluid properties, nanoparticle, and nanoparticle volume proportion. The thermo-fluid properties of water and Al₂O₃ listed in Table 1 are taken to compute the effective properties of Al₂O₃-water nanofluid.

Table 1.

Thermo-fluid properties of base fluid: water and nanoparticles: Al₂O₃ at 25°C [26].

Properties	Water	Al2O3
Dynamic viscosity, μ (Pa.s)	0.001003	–
Thermal conduction coefficient, k (Wm−1K−1)	0.613	25
Heat capacitance, Cp (Jkg−1K−1)	4179	765
Density, ρ (kgm−3)	997.1	3970
Coefficient of volumetric thermal expansion, β (K−1)	2.10 × 10−4	0.85 × 10−5

The effective density, coefficient of volumetric thermal expansion, and specific heat capacity at fixed pressure of Al₂O₃-water nanofluid can be calculated following the mixing rule theory as given below,

$${\rho }_{nf}=\left(1-\phi \right){\rho }_{bf}+\phi {\rho }_s\text{,}$$ (1)

$${\left(\rho \beta \right)}_{nf}=\left(1-\phi \right){\left(\rho \beta \right)}_{bf}+\phi {\left(\rho \beta \right)}_s\text{,}$$ (2)

$${\left(\rho C_p\right)}_{nf}=\left(1-\phi \right){\left(\rho C_p\right)}_{bf}+\phi {\left(\rho C_p\right)}_s\text{,}$$ (3)

where, the subscripts ‘bf’, ‘s’, and ‘nf’ indicate the thermo-fluid properties of base fluid, nanoparticles, and nanofluid, respectively.

2.1. Effective dynamic viscosity

To estimate the effective absolute viscosity of nanofluid, Einstein [12], Brinkman [13], and Batchelor [14] developed the correlations for colloid dispersion, which many researchers frequently used. The calculated dynamic viscosity calculated from these correlations momentously underestimates the actual values. The degree of underestimation rises as the diameter of nanoparticles reduces and the concentration of nanoparticles increases [1]. Table 2 summarizes some experimental works measuring the absolute viscosity of water-Al₂O₃ nanofluid. The earlier model of viscosity, developed by Brinkman [13], can moderately evaluate the dynamic viscosity of any nanofluid for a specific range of nanoparticle solid-volume proportions (0 ≤ φ ≤ 0.04). The model can be written as follows,

$${\mu }_{nf}=\frac{{\mu }_{bf}}{{\left(1-\phi \right)}^{2.5}}\text{.}$$ (4)

Table 2.

Some experimental works measuring the absolute viscosity of Al₂O₃-water nanofluid.

Researcher(s)	Nanoparticle Size (nm)	Measuring Technique	Remarks/Findings
Murshed et al. [22]	80	–	Derived theoretical model
Nguyen et al. [27]	36, 47	Piston-type viscometer	Proposed correlation
Ho et al. [28]	–	–	Proposed correlation
Azmi et al. [29]	–	–	Proposed correlation
Corcione [30]	25–200	–	Derived empirical model
Wang et al. [31]	28	1D steady state parallel plate	Discussed discrepancies associated with models
Putra et al. [32]	38	Rotating disk-type	Provided experimental data
Lee et al. [34]	–	Ultrasonic vibration.	Measured viscosity
Chandrasekar et al. [35]	43	Sonicator	Provided experimental data

After that, many researchers developed different effective dynamic viscosity models for different types of nanofluid. Nguyen et al. [27] formulated a dynamic viscosity model for Al₂O₃-water nanofluid (0 ≤ φ ≤ 0.094) as given below,

$${\mu }_{nf}=0.904e^{0.1483\phi }{\mu }_{bf}\text{.}$$ (5)

Later, Ho et al. [28] conducted another experimental work for Al₂O₃-water nanofluid (0 ≤ φ ≤ 0.04) and provided an absolute viscosity model based on their experimental data as follows,

$${\mu }_{nf}=\left(1+4.93\phi +222.4{\phi }^2\right){\mu }_{bf}\text{.}$$ (6)

Similarly, Azmi et al. [29] considered higher solid-volume fraction (0 ≤ φ ≤ 0.10) in their experiment and correlated the temperature-dependent dynamic viscosity by the following relation:

$${\mu }_{nf}={\mu }_{bf}{\left(1+\phi \right)}^{11.3}{\left(1+\frac{t}{70}\right)}^{-0.038}{\left(1+\frac{d_s}{170}\right)}^{-0.061}\text{,}$$ (7)

where, t refers to the nanoparticles' temperature in °C and d_s is nanoparticles diameter in nm. Corcione [30] gathered experimental data on dynamic viscosity, which was a function of temperature, sizes, and solid-volume fraction of nanofluid, and provided a distinct correlation for a wide range of nanoparticle-volume fraction (0.002 ≤ φ ≤ 0.09). According to the data analysis of Corcione [30], the magnitude of dynamic viscosity was independent of temperature between 293 and 333 K. The correlation is written as,

$${\mu }_{nf}=\frac{{\mu }_{bf}}{1-34.87{\left(d_s/d_{bf}\right)}^{-0.3}{\phi }^{1.03}},d_{bf}=0.1{\left(\frac{6M}{N\pi {\rho }_{bf}}\right)}^{1/3}\text{,}$$ (8)

where, M and N indicate the molar mass of the base fluid and Avogadro number, respectively, and d_bf is the equivalent diameter of the base fluid molecule.

2.2. Effective thermal conductivity

Experimental works measuring the thermal conduction coefficient of Al₂O₃-water nanofluid are also listed in Table 3. Maxwell [36] gave an empirical model to find the thermal conduction coefficient of nanofluids (0 ≤ φ ≤ 0.04) at room temperature:

$$k_{nf}=\frac{k_s+2k_{bf}-2\phi \left(k_{bf}-k_s\right)}{k_s+2k_{bf}+\phi \left(k_{bf}-k_s\right)}{k}_{bf}\text{.}$$ (9)

Table 3.

Some experimental works measuring the thermal conduction coefficient of Al₂O₃-water nanofluid.

Researcher(s)	Nanoparticle Size (nm)	Measuring Method	Remarks/Findings
Choi and Eastman [1]	38.4	Hot-wire transient	Measured conductivity
Murshed et al. [22]	80	Hot-wire transient	Derived theoretical model
Ho et al. [28]	–	–	Proposed correlation
Azmi et al. [29]	–	–	Proposed correlation
Corcione [30]	10–150	–	Derived empirical model
Li and Peterson [33]	36	Hot-wire transient	Proposed correlation
Lee et al. [34]	–	Ultrasonic vibration	Measured conductivity
Chandrasekar et al. [35]	43	Sonicator	Measured conductivity
Das et al. [37]	28.6	Oscillation	Measured conductivity
Chon et al. [38]	47	Hot-wire transient	Derived empirical correlation
Zang et al. [39]	–	Transient SWH	Measured conductivity
Moldoveanu et al. [40]	43	Transient	Proposed correlation
Kumar et al. [41]	–	KD2 Pro digital recorder	Measured conductivity

This model tends to fail drastically when the suspension temperature is above 20°C. Li and Peterson [33] experimentally showed the limitation of Maxwell's model and derived a temperature-dependent equation for estimating the thermal conduction coefficient of nanofluids as,

$$k_{nf}=\left(0.764\phi +0.0187t-0.462\right)k_{bf}+k_{bf}\text{.}$$ (10)

Another two thermal conductivity models of water-Al₂O₃ nanofluid were given by Ho et al. [28] and Azmi et al. [29], where they considered the nanoparticle-volume proportion within 0 ≤ φ ≤ 0.04 and 0 ≤ φ ≤ 0.10, respectively. Those models are given below:

$$k_{nf}=\left(1+2.944\phi +19.672{\phi }^2\right)k_{bf}\text{,}$$ (11)

$$k_{nf}=0.8938{\left(1+\phi \right)}^{1.37}{\left(1+\frac{t}{70}\right)}^{0.2777}{\left(1+\frac{d_s}{170}\right)}^{-0.0336}{\left(\frac{{\alpha }_s}{{\alpha }_{bf}}\right)}^{0.01737}{k}_{bf}\text{,}$$ (12)

where, α denotes thermal diffusivity. Corcione [30] also provided a model of thermal conductivity, which can be given as follows,

$$k_{nf}=k_{bf}\left[1+4.4R{e}^{0.4}P{r}^{0.66}{\left(\frac{T}{T_{fr}}\right)}^{10}{\left(\frac{k_s}{k_{bf}}\right)}^{0.03}{\phi }^{0.66}\right],Re=\frac{2{\rho }_{bf}{k}_{b}T}{\pi {\mu }_{bf}^2{d}_s},P{r}_{bf}=\frac{{\mu }_{bf}{C_p}_{,bf}}{k_{bf}}\text{,}$$ (13)

where, Pr_bf and T_fr denote the Prandtl number and the freezing point temperature of the base fluid (water), Re indicates the Brownian-motion Reynolds number, k_b is Boltzmann's constant, and T is the nanofluid's temperature in K.

3. Test problem and numerical simulation

A differentially heated chamber of equal length L filled with water-Al₂O₃ nanofluid, as shown in Fig. 1, is selected as a test configuration. The nanofluid is supposed to be incompressible, and Newtonian, and its flow regime is steady, two-dimensional, and laminar. The chamber's left and right solid surfaces are fixed at differential temperature (T_h > T_c), while the horizontal top and bottom boundaries are kept adiabatic.

Fig. 1.

Computational domain of the present investigation.

A summary of the dimensionless auxiliary conditions applied to the square cavity has been presented in Table 4. The nanoparticle-volume fraction of water-Al₂O₃ nanofluid is changed inside the chamber between 0 and 4%. The effective density, coefficient of volumetric thermal expansion, and heat capacitance of Al₂O₃-water nanofluid are calculated from (1), (2), (3), respectively. The correlations shown in (4), (5), (6), (7), (8), (9), (10), (11), (12), (13) are utilized in calculating the nanofluid's viscosity and thermal conduction coefficient. The Rayleigh number (Ra) is changed from 10³ to 10⁹ to compare the system's thermal and hydrodynamic performances under different nanofluid models. The governing models used in the test problem are continuity, momentum, and thermal energy equations. The dimensionless forms of those equations are presented as,

$$\nabla \cdot \mathit{U}=0\text{,}$$ (14)

$$\left(\mathit{U}\cdot \nabla \right)\mathit{U}=\nabla \left[-P\mathbf{I}+\frac{{\nu }_{nf}}{{\nu }_{bf}}P{r}_{bf}\left\{\nabla \mathit{U}+{\left(\nabla \mathit{U}\right)}^T\right\}\right]+\mathit{F}\text{,}$$ (15)

$$\mathit{U}\cdot \nabla \mathrm{\Theta }=\frac{{\alpha }_{nf}}{{\alpha }_{bf}}{\nabla }^2\mathrm{\Theta }\text{,}$$ (16)

where, U, P, and Θ are the non-dimensional nanofluid's velocity vector, pressure, and temperature, respectively. The body force vector is expressed as F = (0, β_nfRaPr_bfΘ/β_bf). In those governing equations, the crucial controlling parameter (Ra) is introduced in terms of base fluid's properties, which is expressed as (17):

$$Ra=\frac{g{\beta }_{bf}\left(T_h-T_c\right)L^3}{{\nu }_{bf}{\alpha }_{bf}}\text{,}$$ (17)

where, ν and g are kinematic viscosity, and gravitational acceleration, respectively. The average Nusselt number is calculated at the hot isothermal left surface to express the system's thermal performance. The following formula defines it as,

$$Nu=-\frac{k_{nf}}{k_{bf}}\underset{0}{\overset{1}{\int }}\frac{\partial \mathrm{\Theta }}{\partial X}dY\text{,}$$ (18)

where, X and Y are dimensionless Cartesian frameworks.

Table 4.

List of auxiliary conditions imposed on the square cavity.

Boundary wall	Velocity Vector	Temperature State
Left wall	U = 0	Θ = 1
Right wall		Θ = 0
Top and bottom walls		∂Θ/∂Y = 0

To solve the governing equations (14), (15), (16) alongside the auxiliary conditions, the Galerkin finite element method-based commercial software program “COMSOL Multiphysics 6.1” is used. First, the computational domain has been discretized into triangular mesh elements, and the governing equations are applied to the individual element. Then, the solution of the nonlinear governing algebraic finite element equations is obtained using the Newton-Raphson iteration procedure. When the relative percentage of error goes below a certain threshold, the solution procedure is assumed to be completed. The error limit follows this terminating criterion: γ^m+1 – γ^m < 10⁻⁵, where γ and m are the general independent variable and number of iterations, respectively. The step-by-step solution procedure is depicted in Fig. 2. Before performing parametric simulation, an optimum mesh of 16,946 elements is selected via an extensive test of grid refinement, as illustrated in Fig. 3(a). The present model is validated against a computational study by Santra et al. [42], as shown in Fig. 3(b), to support the justification of the computational strategy and numerical accuracy. The comparison of the average Nusselt number (calculated using (18)) reveals that the current model and the computations of the numerical scheme are consistent and accurate, thus providing enough confidence for further computational outcomes.

Fig. 2.

Flow diagram of the step-by-step solution algorithm using the finite element method.

Fig. 3.

(a) Grid independence test of the benchmark problem for Ra = 10⁵ and φ = 0.01, and (b) validation of the present model with the numerical simulation of Santra et al. [42] in terms Nu_bf for base fluid (water) with Pr = 7.02 (color online). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

4. Results and discussion

The effective dynamic viscosity of nanofluid is calculated using the correlations (4) to (8), ranging the solid-volume proportion of nanoparticles from 0 to 4%. The variation of the normalized dynamic viscosity is illustrated in Fig. 4(a). It can be said that Brinkman [13] and Corcione [30] models are not predicting well enough compared to the experimental results of Ho et al. [28]. It is important to note that their experimental work [28] is considered here since their evaluated geometrical configuration (square cavity), boundary conditions (differentially heated), and thermal transport mode (free convection) are identical to the current test problem.

Fig. 4.

Discrepancy of (a) the effective normalized absolute viscosity and (b) the effective normalized thermal conduction coefficient of water-Al₂O₃ nanofluid with the variation of φ (color online). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Similarly, the effective thermal conduction coefficient of Al₂O₃-water nanofluid is computed using the correlations (9) to (13) within similar relative proportions of nanoparticle volume proportion. The results from various models are presented in Fig. 4(b). It is observed that Li and Peterson's [33] model shows a poor estimation in comparison with the experimental measurements of Ho et al. [28].

Since Brinkman's [13] dynamic viscosity model and Li and Peterson's [33] thermal conduction coefficient model differ significantly from the experimental data, further investigation excludes these two models. Ten combinations of different existing models, along with the experimental results of Ho et al. [28] on thermal conduction coefficient and dynamic viscosity listed in Table 5, are taken for further analysis of the performance of water-Al₂O₃ nanofluid. For comparison, the solid-volume proportion is chosen within 0–4%.

Table 5.

Combinations of different existing models of the effective absolute viscosity and the thermal conduction coefficient of nanofluid.

Model ID	Viscosity Model	Thermal Conductivity Model	Symbol
NM	Nguyen et al. [27]	Maxwell [36]
NH	Nguyen et al. [27]	Ho et al. [28]
NA	Nguyen et al. [27]	Azmi et al. [29]
AM	Azmi et al. [29]	Maxwell [36]
AH	Azmi et al. [29]	Ho et al. [28]
AA	Azmi et al. [29]	Azmi et al. [29]
HM	Ho et al. [28]	Maxwell [36]
HH	Ho et al. [28]	Ho et al. [28]
HA	Ho et al. [28]	Azmi et al. [29]
CC	Corcione [30]	Corcione [30]
HH_exp	Ho et al. [28] experimental	Ho et al. [28] experimental

The Prandtl number of nanofluid is (Pr_nf = ν_nf/α_nf, where ν is kinematic viscosity) evaluated using the properties listed in Table 1. The normalized Prandtl numbers (Pr_nf/Pr_bf) for all eleven combinations are displayed in Fig. 5. The values of Pr_nf/Pr_bf calculated from the experimental results of Ho et al. [28] are also shown here. It can be observed that the trend of variation of the Prandtl number with the variation of φ is similar for all combinations except the Corcione [30] models. It might be due to including all the uncertainties of different experimental works in a single model since he took a wide range of data from his literature review.

Fig. 5.

Comparison of normalized Prandtl numbers of Al₂O₃-water nanofluid with the variation of φ for different combinations of models as listed in Table 5 (color online). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

For all combinations, the percentage change of Nu is plotted in Fig. 6 for solid-volume fraction 1–4% at three different Rayleigh numbers (10³, 10⁶, and 10⁹). From Fig. 6(a), it can be seen that the percentage change of Nu is almost greater than unity for the entire range of φ, which contemplates the augmentation of heat transfer performance at Ra = 10³. For Ra = 10⁶, as shown in Fig. 6(b), heat transfer performance is enhanced only for lower solid-volume proportion (φ = 0.01), and for higher values (φ > 0.01), it deteriorates. When both Ra and solid-volume fraction increase, two opposite effects occur: higher Ra increases the buoyancy effect to increase the heat transfer, and higher solid-volume fraction means higher viscosity, which ultimately reduces the impact of Ra. For a moderate value of Ra (=10⁶), the effect of increasing viscosity dominates over the effect of buoyancy, for which overall heat transfer performance deteriorates. However, superior heat transfer performance is obtained at Ra = 10⁹, as shown in Fig. 6(c), except for slight deterioration at φ = 0.04 by a few combinations. At this value of Ra, severe mixing of the nanofluid occurs due to higher buoyancy-induced fluid motion, and this phenomenon surpassed the negative effect of the increasing viscosity of nanofluids at higher solid-volume fractions. The next task is to find the closeness of the results evaluated by the nanofluid models with the experimental data to get into the final selection.

Fig. 6.

Comparison of % change of Nu of Al₂O₃-water nanofluid with the variation of φ for different combinations of models as listed in Table 5 at (a) Ra = 10³, (b) Ra = 10⁶, and (c) Ra = 10⁹ (color online). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

The mean squared error (MSE) of Nu with the experimental measurements of Ho et al. [28] for all combinations of models are illustrated in Fig. 7. It can be summarized from the figure that among all the varieties, NM, NA, AA, HH, HA, CC show very poor closeness for the entire range of solid-volume fraction. Besides, the quantitative data of the mean squared errors are presented in Table 6 to select the best combinations. Observing the MSE values confirmed that for solid-volume proportion φ = 0.01, the AH combination gives a closer result to the experimental work. However, for solid-volume proportions in the range of 0.02–0.03, it can be emphasized that the combination of AM outperforms other combinations. Finally, using the NH combination to estimate the nanofluid's thermal conduction coefficient and absolute viscosity is better at φ = 0.04.

Fig. 7.

Variation of mean squared error of Nu with respect to the experimental results of Ho et al. [28] within 10³ ≤ Ra ≤ 10⁹.

Table 6.

Mean squared error of Nu with respect to the experimental results of Ho et al. [28] within 10³ ≤ Ra ≤ 10⁹.

Combination	Solid-volume fraction (φ)
	0.01	0.02	0.03	0.04
NM	0.00734	0.01030	0.01679	0.12472
NH	0.01696	0.05541	0.01515	0.00224
NA	0.11931	0.03258	0.05275	0.38810
AM	0.00124	0.00937	0.00063	0.01417
AH	0.00018	0.05319	0.05634	0.08565
AA	0.05292	0.03088	0.01430	0.15208
HM	0.00057	0.03530	0.00074	0.03555
HH	0.00417	0.09438	0.07057	0.04603
HA	0.07848	0.06353	0.00896	0.21287
CC	0.06821	0.29319	0.29596	0.29504

Further, the variation of percentage change of Nu with the interpretation of Ra is plotted, as shown in Fig. 8(a)-(c), to compare how the best-selected combinations predict the heat transfer performance. It is apparent for all three varieties that the heat transfer rate increases with increasing solid-volume proportions at a lower Rayleigh number. In this case, the effect of enhanced thermal conduction coefficient is dominant compared to the impact of enhanced viscosity. However, a worsened heat transfer performance is observed for a wide range of Rayleigh numbers (10⁴ ≤ Ra ≤ 10⁸). The reduction of thermal performance is prominent for higher solid-volume fractions (φ ≥ 0.02) because of the impediment in natural convection by the nanoparticles, and it is justified by the finding of Nnanna [43]. Finally, the deterioration continues until turbulence (Ra > 10⁸) is introduced in the flow regime.

Fig. 8.

Variation of percentage change of Nu of water-based Al₂O₃ nanofluid against Ra for different φ during the combination of (a) AH, (b) AM, (c) NH, and (d) CC (color online). Solid and dashed lines indicate correlational and experimental data, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Now, the closeness of the models with the experimental data can be visualized in those figures. For φ = 0.01, the AH and AM models are relatively close to the experimental data according to Fig. 8(a) and (b). However, the AH model is more closely fitted than the AM model. Then, in the case of φ = 0.02 and 0.03, the AM model shows the best fit compared to any other models, as seen in Fig. 8(b). Finally, according to Fig. 8(c), the NH model can be chosen to get the best result for φ = 0.04.

In the case of statistical analysis, Corcione's proposed models [30] can be considered since he gathered extensive data from the literature and presented the correlations. Though the variation trend of Nu is similar to those of AH, AM, and NH models, as shown in Fig. 8(d), it contemplates the enhancement of heat transfer for almost the entire range of Ra. This result is contradictory to the findings of the experimental work of Ho et al. [28].

5. Conclusion

This paper comparatively studies and presents different available effective absolute viscosity and thermal conduction coefficient models. As heat transfer augmentation in various devices is highly desired, making a linkage between simulation and experimental work with the closest prediction of thermo-physical properties is compulsory.

A benchmark problem of the differentially heated cavity is selected to analyze the Al₂O₃-water nanofluid's natural convection. Ten combinations of the absolute viscosity and the thermal conduction coefficient models of the nanofluid are thoroughly investigated since these properties are vital functions of the solid-volume fraction of nanoparticles, nanoparticle size, temperature, etc. The coefficient of heat transfer also strongly depends on the absolute viscosity and thermal conduction coefficient. The rest of the required properties are estimated by using the general mixing theory. The findings can be summarized as follows:

• i.According to the experimental and simulation data agreement, AH, AM, and NH combinations seemed the most reliable. However, AH can be suggested for lower nanoparticle concentration (φ = 0.01), AM for moderate concentration (0.01 < φ < 0.04), and NH for higher value of the solid-volume proportion (φ = 0.04).
• ii.To calculate individual properties of nanofluid, viscosity models of Nguyen et al. [27] and Azmi et al. [29] and conductivity models of Ho et al. [28] and Maxwell [36] can be used with minimum uncertainty.
• iii.Moreover, nanofluids are often considered as single-phase fluids in simulation-based works. This assumption limits the effect of particle size, stability, and other physical mechanisms happening inside the nanofluid, such as thermophoresis, dispersion, clustering, Brownian motion, etc. Besides, using a complex empirical correlation associated with much detailed physics is often considered impractical in many engineering applications. Therefore, identifying the most effective correlations with a trade-off between the accuracy of the results and the computational complexity is vital. In this regard, the selected models show good concurrency.
• iv.The nanofluid properties and the heat transfer mechanism still require a lot of detailed computational and experimental investigations to make a more generalized way of solving such thermal system problems.

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Tahmidul Haque Ruvo: Writing – review & editing, Writing – original draft, Visualization, Validation, Investigation, Formal analysis, Conceptualization. Md. Shahneoug Shuvo: Writing – review & editing, Writing – original draft, Methodology, Investigation, Data curation. Sumon Saha: Writing – review & editing, Writing – original draft, Supervision, Software, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Nomenclature

C_p

Specific heat (Jkg⁻¹K⁻¹)

d

diameter (m)

F

Dimensionless body force vector

g

Gravitational acceleration (ms⁻²)

I

Identity vector

k

Thermal conductivity (Wm⁻¹K⁻¹)

k_b

Boltzmann constant (1.38066 JK⁻¹)

L

Length of the cavity (m)

M

Molar mass of base fluid

N

Avogadro number

N_E

Number of mesh elements

Nu

Average Nusselt number

P

Dimensionless pressure

Pr

Prandtl number

Ra

Rayleigh number

Re

Reynolds number

t

temperature of nanofluid (°C)

T

Temperature of nanofluid (K)

U

Dimensionless velocity vector (ms⁻¹)

X, Y

Dimensionless Cartesian coordinates

Greek Symbols

α

Thermal diffusivity (m²s⁻¹)

β

Thermal expansion coefficient (K⁻¹)

Θ

Dimensionless temperature

μ

Dynamic viscosity (Pa.s)

ν

Kinematic viscosity (m²s⁻¹)

ρ

density (kgm⁻³)

φ

Solid-volume fraction of nanoparticles

Subscripts

bf

Base fluid

c

Cold wall

fr

Freezing point of base fluid

h

Hot wall

nf

Nanofluid

s

nanoparticles