Impact of Temperature and Nanoparticle Concentration on Turbulent Forced Convective Heat Transfer of Nanofluids

Cieśliński, Janusz T.; Lubocki, Dawid; Smolen, Slawomir

doi:10.3390/en15207742

Open AccessArticle

Impact of Temperature and Nanoparticle Concentration on Turbulent Forced Convective Heat Transfer of Nanofluids

by

Janusz T. Cieśliński

^1,*

,

Dawid Lubocki

¹ and

Slawomir Smolen

²

¹

Faculty of Mechanical Engineering and Ship Technology, Institute of Energy, Gdańsk University of Technology, Narutowicza 11/12, 80233 Gdańsk, Poland

²

Faculty of Nature and Engineering, J.R. Mayer–Institute for Energy Engineering, City University of Applied Sciences Bremen, Neustadtswall 30, 28199 Bremen, Germany

^*

Author to whom correspondence should be addressed.

Energies 2022, 15(20), 7742; https://doi.org/10.3390/en15207742

Submission received: 19 September 2022 / Revised: 13 October 2022 / Accepted: 17 October 2022 / Published: 19 October 2022

(This article belongs to the Section D3: Nanoenergy)

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Abstract

:

Theoretical analysis of the influence of nanoparticles and temperature on the average Nusselt (Nu) number and the average heat transfer coefficient (HTC) during the turbulent flow of nanofluid in a horizontal, round tube was carried out. The Nu number is a function of the Reynolds (Re) number and the Prandtl (Pr) number, which in turn are functions of the thermophysical properties of the liquid and the flow conditions. On the other hand, the thermophysical properties of nanoliquids are primarily a function of nanoparticle concentration (NPC) and temperature. Hence, the correct determination of the value of the Nu number, and then the HTC, which is needed for engineering calculations, depends on the accuracy of determining the thermophysical properties of nanofluids. In most cases, the thermophysical properties of the nanofluids are calculated as functions of the corresponding thermophysical properties of the base liquid. Therefore, the accuracy of the calculations of the thermophysical properties of nanofluids is equally determined by the reliable correlations for the base liquids. Therefore, new correlations for the calculation of the thermophysical properties of water have been developed. The results of calculations of the thermophysical properties of the base liquid (water) and the water-Al₂O₃ nanofluids by use of carefully selected correlations is presented. It was established that even for small concentrations of nanoparticles, a significant intensification of heat transfer using nanofluids as compared to the base liquid is obtained for the tested temperature range.

Keywords:

nanofluids; thermophysical properties; forced convective heat transfer; horizontal tube

1. Introduction

Constant technological progress means that the heat fluxes that occur in devices are increasing. This presents a real challenge for heat transfer engineers and researchers. Two directions of intensification of heat transfer are being developed. The first is related to the modification of the heat exchange surface, e.g., [1,2,3]. The second is related to the search for new thermal fluids that are characterized by better thermophysical properties, such as thermal conductivity or contact angle. Among the new thermal fluids, the greatest hope is raised by a mixture of a base liquid and nanoparticles with a size below 100 nm, referred to as a nanofluid [4]. The use of nanofluids in a great number of areas is considered, such as single-phase systems, e.g., [5,6], two-phase systems, e.g., [7,8], magnetic nanofluids, e.g., [9,10], non-Newtonian nanofluids, e.g., [11,12], viscoelastic nanofluids, e.g., [13,14], chemical reaction systems, e.g., [15,16], bio-nanofluids, e.g., [17,18], nanofuels, e.g., [19,20], and others [21].

The single-phase convective heat transfer of nanofluids is of particular interest due to potential applications in many cooling/heating systems, e.g., heat exchangers, electronics, nuclear reactors, car radiators, solar devices, e.g., [22,23,24,25], medical applications, e.g., [26,27], thermal energy storage, e.g., [28,29], or even rocketry applications [30]. Although different aspects, e.g., flow regimes—laminar or turbulent, boundary conditions, nanoparticle type and concentration, of single-phase forced convection of nanofluids in tubes were studied both experimentally [31,32,33,34] and numerically [35,36,37,38,39,40,41,42] there is still controversy about the influence of nanoparticles on heat transfer efficiency and flow resistance. Experimental works show that the addition of nanoparticles can both intensify and deteriorate heat transfer. In turn, numerical works most often show the intensification of heat transfer as a result of adding nanoparticles. A similar controversy concerns flow resistance. Intensive work is still underway to determine the thermophysical properties of nanofluids and their stability [43,44].

The nanofluid, which is inherently a two-phase mixture of liquid and solid particles, can be modeled as a single-phase continuum (single phase fluid) with the thermophysical properties which take into account the effect of the presence of nanoparticles. The results of numerical simulations obtained with the single-phase approach and the two-phase model [45,46], did not differ significantly. The condition for the accuracy of the single-phase approach is the correct selection of the correlation on the thermophysical properties of nanofluids.

In the present study, a theoretical analysis of the influence of temperature and NPC during forced convection of nanofluids inside a horizontal circular tube is presented. It was assumed that the analyzed nanofluid can be treated as a homogeneous, single-phase liquid. The Nu number and average HTC were parameters of the intensity of the convective heat transfer. For forced convection, the Nu number is a function of the Re number and Pr number. The Re number and Pr number are functions of the thermophysical properties of nanofluids. The thermophysical properties of nanofluids varied first of all with temperature and NPC. Therefore, an analysis was conducted to evaluate the effects on the performance of nanofluids due to variations of thermal conductivity, viscosity, density, and specific heat, which are functions of NPC and temperature. Water-based nanofluids with dispersed alumina (Al₂O₃) nanoparticles at mass concentrations of 0.1%, 1%, and 5% within the temperature range 20–70 °C are considered. Water-Al₂O₃ nanofluids were selected because frequent application in both numerical and experimental studies. Moreover, thermophysical properties of water-Al₂O₃ nanofluids are comprehensively and thoroughly investigated [47].

2. Materials and Methods

2.1. Tested Nanofluids

In this study alumina (Al₂O₃) nanoparticles were selected while distilled, deionized water was tested as base fluid. Alumina nanoparticles were tested at the concentration of 0.1%, 1%, and 5% by weight. It was assumed that nanoparticles have a spherical form with mean diameter of d_p = 47 nm.

The properties of alumina (Al₂O₃) nanoparticles are shown in Table 1.

2.2. Correlations for Forced Convection Heat Transfer inside Horizontal Tubes

Recognized correlation equations used for the determination of an average Nu number for forced convection inside horizontal cylinders in base fluids are collected in Table 2. The Nu number for water is a benchmark to the values calculated for the analyzed nanofluids. Hence, the selection of the correct correlation is important when comparing the results for the base fluid and nanofluids.

Figure 1 shows the dependence of the Nu number on the Re number on the basis of the correlations shown in Table 1. The calculations were performed for the Pr = 5. The values of the correction coefficients were assumed to be equal to one. Noteworthy is the fact that the values of the Nu number are significantly higher from the Notter and Sleicher correlation [55] compared to the other correlations. Note, however, that the Notter and Sleicher correlations (Equations (6) and (7)) are suitable for fluids with a very low Pr number. Figure 1 shows that the correlations proposed by Dittus and Boelter, Kraußold, Sieder and Tate, Mikhejew, Hausen, and Gnieliński give similar values of the Nu number, although the difference between the Kraußold correlation and Hausen correlation for Re = 10⁵ is as much as 45%.

In Table 3 are collected experimental and numerical correlation equations used for determination of an average Nu number for forced convection inside horizontal cylinders in nanofluids.

Figure 2 shows the dependence of the Nu number on the Re number for nanofluids and correlations shown in Table 3. The calculations were performed for the Pr = 5, w = 1 m/s, φ_v = 1% and d_p = 40 nm.

As seen in Figure 2, except Equation (17), the correlations show pretty good consistency. The numerically developed correlation equation proposed by Saha and Paul, Equation (18), was selected for the analysis, as it fits the experimental data for water-Al₂O₃ nanofluids very well and contains parameters that characterize both nanoparticles and molecules of the base fluid.

2.3. Correlations for Thermophysical Properties of Nanofluids

A large number of correlations devoted to calculate viscosity of nanofluids are published in the literature, e.g., [67,68]. In Table 4 are shown correlations proposed for water-Al₂O₃ nanofluids.

The correlation proposed by Corcione, Equation (26), was selected for the analysis, as it fits the experimental data of the viscosity of water-Al₂O₃ nanofluids very well.

Equally important as reliable correlations for nanofluids are the formulas for calculating the properties of base liquids. Tested correlations for calculation viscosity of water are shown in Table 5.

Figure 3 shows the viscosity of water against temperature calculated from correlations presented in Table 5. As seen in Figure 3, predictions realized with Equation (27) deviate by about 20% for minimum temperature 20 °C. Higher temperature difference between predictions is negligible.

Figure 4 shows influence of NPC on the viscosity of the tested nanofluids against temperature, calculated by the use of Equation (26) in combination with Equation (31).

As seen in Figure 4, viscosity of the tested nanofluids decreases sharply with a temperature increase and slightly increases with NPC increase. The slight influence of nanoparticles on the viscosity of the tested nanofluids is due to the very small used NPC.

Similarly to viscosity several correlations devoted to thermal conductivity of nanofluids are published in literature, e.g., [79,80]. In Table 6 are collected correlations proposed for water-Al₂O₃ nanofluids.

The correlation proposed by Corcione, Equation (33), was selected for the analysis, as it fits the experimental data of the thermal conductivity of water-Al₂O₃ nanofluids very well, as it takes into account the influence of Brownian motion on the thermal conductivity of nanofluids.

Correlations for calculation thermal conductivity of water are shown in Table 7.

Figure 5 shows thermal conductivity of water against temperature calculated from correlations presented in Table 7.

Figure 6 shows influence of NPC on the thermal conductivity of the tested nanofluids against temperature calculated by use of Equation (33) in combination with Equation (41).

As seen in Figure 6 thermal conductivity of the tested nanofluids increases moderately with temperature increase and for given temperature increases with NPC increase.

The density of nanofluids is generally calculated by the use of the mixture model (Table 8).

The correlation proposed by Pak and Cho, Equation (43), was selected for the analysis, as it fits the experimental data of the density of water-Al₂O₃ nanofluids very well and is based on the general mixture theory.

Density of pure water can be estimated by use of correlations given in Table 9.

Figure 7 shows the density of water against temperature calculated from correlations presented in Table 9. The maximum difference between the predictions is equal to 0.4%.

Figure 8 illustrates influence of the NPC on density of the tested nanofluids as a function of temperature by use of Equations (43) and (49).

As seen in Figure 8, the density of the analysed nanofluids decreases with temperature increase and increases with NPC increase because the density of the nanoparticle material (Table 1) is higher than density of water.

Several investigators have been involved in the physics of the specific heat of nanofluids [87,88,89]. In Table 10 are collected correlations proposed for water-Al₂O₃ nanofluids.

The correlation proposed by Williams et al. [90] was selected for the analysis. As shown in [49] Equation (52) much better fits the experimental data of the specific heat of water-Al₂O₃ nanofluids than Equation (51) based on the mixture theory. The better accuracy of Equation (52) results from the assumption of thermal equilibrium between the particles and the liquid, which does not have to be present due to the Brownian motion of nanoparticles and different thermophysical properties of the nanoparticle material and the base liquid.

The specific heat of water can be estimated by use of correlations given in Table 11.

Figure 9 shows the specific heat of water against temperature calculated from correlations presented in Table 11. The difference between the individual correlations seems large, but for 70 °C it does not exceed 3.4%.

Figure 10 illustrates influence of NPC on the specific heat of the tested nanofluids as a function of temperature calculated by the use of Equation (52) in combination with Equation (60).

As seen in Figure 10, the specific heat of water-Al₂O₃ nanofluids decreases with temperature increases, and for the given temperature decreases with NPC increases because the specific heat of the nanoparticle material (Table 1) is lower than the specific heat of water. It is worthy to note the non-monotonic course of specific heat for water-based nanofluids against the temperature that results from the data for water (Figure 9).

Figure 11 illustrates the influence of NPC on thermal diffusivity of the tested nanofluids as a function of temperature. The thermal diffusivity of water was calculated by the use of present correlations, i.e., Equations (41), (49) and (60), while thermal diffusivity of nanofluids was determined by use of Equations (33), (43) and (52). As seen in Figure 11, the thermal diffusivity of water and nanofluids increases with temperature increase, however the growth rate is higher for nanofluids. Moreover, thermal diffusivity increases with NPC increase.

As seen in Figure 12, the thermal diffusivity of the tested nanofluids increases slightly with NPC, although the rate of growth increases with temperature increase.

3. Results and Discussion

3.1. Variation of Pr Number

Figure 13 shows a comparison of Pr numbers of the water-Al₂O₃ nanofluids as a function of temperature. The Pr number for water was calculated by use of present correlations, i.e., Equations (31), (41), (49) and (60), while the Pr number of nanofluids was determined by use of Equations (26), (33), (43) and (52). For the tested nanofluids, the Pr number decreases with an increase in temperature, predominantly due to the decrease in viscosity (Figure 4).

Figure 14 illustrates the influence of NPC on Pr number of the tested nanofluids at three temperatures, namely 30 °C, 50 °C, and 70 °C. As seen in Figure 12, the Pr number for water-Al₂O₃ decreases slightly with the NPC increase. Regardless of the temperature, the decrease in Pr number with the increase in NPC does not exceed 3%.

As shown in Figure 13 and Figure 14, the Pr number decreases with both temperature increase and NPC increase. However, the decrease due to the increase in NPC for a given temperature is small, and the decrease due to the increase in temperature is very large for a given NPC. The Pr number of the nanofluid for the tested range of temperature and NPC is determined by the temperature, not by the NPC. As it results from the correlations presented in Table 2, the Nu number is a function of

P r^{m}

, hence the higher the value of the Pr number, the higher the value of the Nu number.

3.2. Variation of Re Number

Figure 15 illustrates the change of Re number versus temperature for the tested nanofluids. In this case the average velocity and the diameter of the horizontal tube are held constant at the values taken from the experiment presented in [93], i.e., w = 1 m/s, D = 10 mm. The Re number for water was calculated by use of present correlations, i.e., Equations (31) and (49), while the Re number of nanofluids was determined by use of Equations (26) and (43).

As seen in Figure 15 Re number definitively increases with temperature increase, which also results in increase of the Nu number.

Figure 16 shows the change of Re number against NPC for selected temperatures. It is observed that an increase in NPC results in a gradual increase of Re number for all tested temperatures.

As shown in Figure 15 and Figure 16, Re number increases with both temperature increase and NPC increase. However, the increase due to the increase in NPC for a given temperature is small, and the increase due to the increase in temperature is substantial. It follows that the value of the Re number in the studied temperature and NPC range is primarily determined by temperature, and the influence of NPC is modest.

3.3. Variation of Nu Number

Figure 17 shows the variation of Nu number against temperature for water-Al₂O₃ nanofluids predicted from the Saha and Paul correlation, Equation (18). Results are compared to the predictions for water from the commonly accepted Gnieliński correlation, Equation (10). Calculations for water were conducted by the use of present correlations for thermophysical properties of water. As seen in Figure 17, Nu number increases with temperature increase. For the tested NPC range 0.1% ≤ φ_m ≤ 5% Nu number for nanofluids is higher than for pure water. It means that the addition of Al₂O₃ nanoparticles to water results in substantial heat transfer improvement.

The influence of nanoparticles on the Nu number is clearer in Figure 18. Figure 18 shows the variation of the Nu number against NPC for water-Al₂O₃ nanofluids and three selected temperatures, namely 30 °C, 50 °C, and 70 °C. As seen in Figure 18, the addition of even the smallest amount of nanoparticles (

φ_{m} = 0.001

) causes a sharp increase in the Nu number. However, the Nu number increase for

φ_{m} > 0.001

is negligible, and the increase in the Nu number is practically due to the increase in temperature.

3.4. Variation of Heat Transfer Coefficient

Figure 19 shows the variation of heat transfer coefficient against temperature for water-Al₂O₃ nanofluids predicted from the Saha and Paul correlation (Equation (18)). The results are compared to the predictions from the Gnieliński correlation (Equation (10)) for water. As seen in Figure 19, heat transfer coefficient increases with temperature. For the tested mass concentration range 0.1% ≤ φ_m ≤ 5% heat transfer coefficient for nanofluids is higher than for water. It means that addition of Al₂O₃ nanoparticles to water results in substantial heat transfer improvement. In the literature, several mechanisms of heat transfer improvement in nanofluids have been discussed, e.g., [44,94]. According to Buongiorno [95], two mechanisms are responsible for heat transfer improvement in nanofluids, namely Brownian motion and thermophoresis.

Figure 20 shows variation of heat transfer coefficient against NPC for water-Al₂O₃ nanofluids and three selected temperatures, namely 30 °C, 50 °C and 70 °C. As seen in Figure 20, the influence of NPC on HTC for the tested mass concentration range 0.1%

< φ_{m} \leq 5 %

is negligible for the given temperature.

As shown in Figure 19 and Figure 20, HTC increases with both temperature increase and NPC increase. However, the increase due to the increase in NPC for a given temperature is small, and the increase due to the increase in temperature is significant. It is worth noting that maximum HTC increase is higher than Nu number increase and equals 38%, while for the Nu number it is 31%.

4. Conclusions

The analysis showed that the addition of Al₂O₃ nanoparticles with a mass concentration of 0.1% ≤ φ_m ≤ 5% within the temperature range of 20–70 °C to water as a base liquid causes the intensification of heat transfer under conditions of turbulent forced convection in round tubes.

The slight influence of the studied concentration of nanoparticles on the thermophysical properties of nanofluids indicates that the intensification of heat transfer results from the transport mechanisms, and not from the improvement of the thermophysical properties of nanofluids.

Author Contributions

Conceptualization, J.T.C.; methodology, J.T.C. and S.S.; software, D.L.; validation, J.T.C.; formal analysis J.T.C. and S.S.; investigation, D.L.; data curation, D.L.; writing—original draft preparation, J.T.C.; writing—review and editing, J.T.C. and S.S.; funding acquisition, S.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors thank Paweł Dąbrowski (Gdańsk University of Technology) for developing present correlations for thermal properties of water.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

a	Thermal diffusivity	(m²/s)
c_p	Specific heat	(J/(kgK))
$d_{f}$	Base fluid molecule diameter	(m)
$d_{p}$	Particle diameter	(m)
D	Inside diameter of tube	(m)
f	Friction factor	(-)
h	Local heat transfer coefficient	(W/(m² K))
$\bar{h}$	Average heat transfer coefficient	(W/(m² K))
k	Thermal conductivity	(W/(m K))
L	Length	(m)
$N u = \frac{h D}{k}$	Local Nusselt number	(-)
$\bar{N u} = \frac{\bar{h} D}{k}$	Average Nusselt number	(-)
$P e = R e P r$	Peclet number	(-)
$P e_{p} = \frac{w d_{p}}{a_{n f}}$	Peclet number related to d_p	(-)
$P r = \frac{ν}{a}$	Prandtl number	(-)
$q$	Heat flux	(W/m²)
$R e = \frac{w D}{ν}$	Reynolds number	(-)
$t$	Temperature	[°C]
$T$	Temperature	[K]
w	Velocity	(m/s)
x	Axial coordinate	(m)
Greek symbols
µ	Dynamic viscosity	(Pas)
$ν$	Kinematic viscosity	(m²/s)
ρ	Density	(kg/m³)
$φ$	Nanoparticle concentration	(-)
Subscripts
bf	Base fluid
f	Fluid
m	Mass
nf	Nanofluid
nl	Nanolayer
p	Particle
v	Volume
w	Wall
Abbreviations
HTC	Heat transfer coefficient
NPC	Nanoparticle concentration

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Figure 1. Nu-Re relationships for water.

Figure 2. Nu-Re relationships for water-Al₂O₃ nanofluids.

Figure 3. Viscosity of water.

Figure 4. Viscosity of water-Al₂O₃ nanofluids.

Figure 5. Thermal conductivity of water.

Figure 6. Thermal conductivity of water-Al₂O₃ nanofluids.

Figure 7. Density of water.

Figure 8. Density of water-Al₂O₃ nanofluids.

Figure 9. Specific heat of water.

Figure 10. Specific heat of water-Al₂O₃ nanofluids.

Figure 11. Thermal diffusivity of water-Al₂O₃ nanofluids.

Figure 12. Thermal diffusivity of water-Al₂O₃ nanofluids.

Figure 13. Prandtl number for water-Al₂O₃ nanofluids.

Figure 14. Variation of Pr number for water-Al₂O₃ nanofluids.

Figure 15. Variation of Re number for water-Al₂O₃ nanofluids.

Figure 16. Variation of Re number for water-Al₂O₃ nanofluids.

Figure 17. Variation of Nu number for water-Al₂O₃ nanofluids.

Figure 18. Variation of Nu number for water-Al₂O₃ nanofluids.

Figure 19. Variation of heat transfer coefficient for water-Al₂O₃ nanofluids.

Figure 20. Variation of heat transfer coefficient for water-Al₂O₃ nanofluids.

Table 1. Properties of Al₂O₃ nanoparticles.

Thermal Conductivity k_p [W/(mK)]	Density ρ_p [kg/m³]	Specific Heat c_p,p [J/(kg K)]
35 *	3600 **	765 **

*—[48]; **—[49].

Table 2. Correlation equations for turbulent flow of base fluid.

Authors	Correlation	Range	Remarks	Equation
Dittus and Boelter [50]	$\bar{N u} = 0.23 R e^{0.8} P r^{n}$	Re > 10⁴ 0.7 < Pr < 100	n = 0.4—heating n = 0.3—cooling	Equation (1)
Krauβold [51]	$\bar{N u} = 0.032 R e^{0.8} P r^{n} {(\frac{L}{D})}^{- 0.054}$	Re > 10⁴	n = 0.37—heating n = 0.3—cooling	Equation (2)
Sieder and Tate [52]	$\bar{N u} = 0.027 R e^{4 / 5} P r^{1 / 3} {(\frac{μ_{f}}{μ_{w}})}^{0.14}$	Re > 10⁴ 0.7 < Pr < 16,700	T_w = const.	Equation (3)
Mikhejev [53]	$\bar{N u} = 0.021 R e^{0.8} P r_{f}^{0.43} {(\frac{P r_{f}}{P r_{w}})}^{0.25} ε_{L}$	10⁴ < Re < 5 × 10⁶ 0.6 < Pr < 2500	ε_L = f(L/D, Re)	Equation (4)
Petukhov [54]	$\bar{N u} = \frac{(f / 8) R e P r}{1.07 + 12.7 {(f / 8)}^{\frac{1}{2}} (P r^{\frac{2}{3}} - 1)}$	10⁴ < Re < 5 × 10⁶ 0.5 < Pr < 2000	$f = {(1.82 l n R e - 1.64)}^{- 2}$	Equation (5)
Notter and Sleicher [55]	$\bar{N u} = 4.8 + 0.0156 P e^{0.85} P r^{0.08}$ $\bar{N u} = 6.3 + 0.0167 P e^{0.85} P r^{0.08}$	10⁴ < Re < 10⁶ 0.004 < Pr < 0.1	T_w = const. q_w = const.	Equation (6) Equation (7)
Churchill and Ozoe [56]	$\bar{N u} = \frac{0.3387 P r^{1 / 3} R e^{1 / 2}}{{[1 + {(0.0468 / P r)}^{2 / 3}]}^{1 / 4}}$	Re > 100 10⁻⁴ < Pr → ∞	q_w = const.	Equation (8)
Hausen [57]	$N u = 0.0235 [1 + {(\frac{d}{L})}^{2 / 3}] [R e^{0.8} - 230] P r_{f}^{0.3} {(\frac{μ_{f}}{μ_{w}})}^{0.14}$	2300 < Re < 2 × 10⁶ 1.5 < Pr < 500 d/L < 1		Equation (9)
Gnieliński [58]	$\bar{N u} = \frac{(f / 8) (R e - 1000) P r}{1 + 12.7 {(f / 8)}^{0.5} (P r^{\frac{2}{3}} - 1)}$	3 × 10³ < Re < 5 × 10⁶ 0.5 < Pr < 2000	$f = {(0.79 l n R e - 1.64)}^{- 2}$	Equation (10)
Kutateladze [59]	$\bar{N u} = 1.61 {(P e \frac{D}{L})}^{1 / 3}$	Pe > 12 d/L < 12		Equation (11)

Table 3. Nu number correlation equations for water- Al₂O₃ nanofluids.

Authors	Equation	Remarks	Equation
Xuan and Li [60]	$\bar{N u} = 0.0059 (1 + 7.6286 φ_{v}^{0.6886} P e_{p}^{0.001}) R e_{}^{0.9238} P r_{}^{0.4}$	$P e_{p} = \frac{w d_{p}}{a_{n f}}$	Equation (12)
Vasu et al. [61]	$\bar{N u} = 0.0256 \cdot R e^{0.8} P r^{0.4}$	10⁴ < Re < 8 × 10⁴	Equation (13)
Hussein et al. [62]	$\bar{N u} = 0.02 \cdot R e^{0.788} P r^{0.45}$	5000 < Re < 5 × 10⁴ 6.8 < Pr < 11.97	Equation (14)
Sahin et al. [63]	$\bar{N u} = 0.106 \cdot R e^{0.588} \cdot (1 + φ_{v}^{- 0.1096}) \cdot P r^{0.258}$	4000 < Re < 20,000 5 < Pr < 7 0.5% < φ_v < 4%	Equation (15)
Chavan and Pise [64]	$\bar{N u} = 0.508358 \cdot R e^{0.7401} \cdot P r^{- 0.7026}$	6000 < Re < 14,000 0.3% < φ < 1%	Equation (16)
Durga and Gupta [65]	$\bar{N u} = 0.09589 \cdot R e^{0.8} \cdot P r^{0.4} \cdot {(1 + φ_{v})}^{2833}$	3000 < Re < 30,000 5.12 < Pr < 6.54 0% < φ_v < 0.03%	Equation (17)
Saha and Paul [66]	$\bar{N u} = 0.0126 R e_{}^{0.85589} P r_{}^{0.44709} {(\frac{d_{f}}{d_{p}})}^{- 0.00176}$	10⁴ < Re < 10⁵ 8.45 < Pr < 20.29 4% < φ_v < 6% 10 < d_p [nm] < 40	Equation (18)

Table 4. Correlations for viscosity of water- Al₂O₃ nanofluids.

Authors	Correlation								Remarks	Equation
Krieger and Dougherty [69]	$μ_{n f} = μ_{b f} (1 - {(\frac{φ_{a}}{φ_{v s}})}^{- 2.5 φ_{m}})$								$φ_{a} —$ volume fraction of aggregates $φ_{a} = φ_{v} {(d_{a} / d_{p})}^{3 - d_{f}}$ $φ_{v}$ —volume fraction of the well-dispersed individual particles, $d_{a}$ —diameter of aggregates, $d_{f}$ —fractal dimension of aggregates $φ_{v s}$ —volume fraction of densely packed spheres	Equation (19)
Palm et al. [70]	$μ_{b f} = 0.034 - 2 \cdot 10^{- 4} T + 2.9 \cdot 10^{- 7} T^{2}$ $μ_{b f} = 0.039 - 2.3 \cdot 10^{- 4} T + 3.4 \cdot 10^{- 7} T^{2}$								$φ_{v} = 1 %$ $φ_{v} = 4 %$	Equation (20) Equation (21)
Nguyen et al. [71]	$μ_{n f} = 0.904 e^{0.148 φ_{v}} μ_{b f}$ $μ_{b f} = (2.1275 - 0.0215 \cdot T + 0.0002 \cdot T^{2}) μ_{b f}$								d_p = 47 nm $φ_{v} = 4 %$	Equation (22) Equation (23)
Khanafer and Vafai [72]	$μ_{n f} = - 0.4491 + \frac{28.837}{t} + 0.574 φ_{v} - 0.1634 φ_{v}^{2} + 23.053 \frac{φ_{v}^{2}}{t^{2}} + 0.0132 φ_{v}^{3} - 2354.735 \frac{φ_{v}}{t^{3}} + 23.498 \frac{φ_{v}^{2}}{d_{p}^{2}} - 3.0185 \frac{φ_{v}^{3}}{d_{p}^{2}}$									Equation (24)
Pastoriza-Gallego et al. [73]	$μ_{n f} = e^{(A + \frac{B}{T - T_{o}})}$									Equation (25)
		ϕ_v
		0.000	0.005	0.010	0.015	0.021	0.031	0.048
	A	−3.694	−3.632	−2.381	−1.702	−3.450	−3.302	−1.379
	B [K]	999.0	999.0	689.3	534.7	999.0	999.0	518.4
	T_o [K]	145.7	145.5	169.8	185.5	146.2	145.3	189.9
Corcione [74]	$μ_{n f} = μ_{b f} (\frac{1}{1 - 34.87 {(\frac{d_{p}}{d_{f}})}^{- 0.3} φ_{v}^{1.03}})$								d_f—equivalent diameter of liquid molecule $d_{f} = 0.1 {(\frac{6 M}{N π ρ_{f o}})}^{1 / 3}$ M—molecular weight of water N—Avogadro number $ρ_{f o}$ - density of water at T_o = 293 K	Equation (26)

Table 5. Correlations for viscosity of water.

Authors	Correlation	Range	Equation
Minea et al. Equation (31) in [36]	$μ_{b f} = 1.055787 - 0.0132897 \cdot T + 0.00006309 \cdot T^{2} - 1.33730666 \cdot 10^{- 7} \cdot T^{3} - 1.0666666 \cdot 10^{- 10} \cdot T^{4}$	294 < T < 344	Equation (27)
Chon and Khim [75]	$μ_{b f} = 2.414 \cdot 10^{- 5} \cdot 10^{\frac{247.8}{T - 140}}$	294 < T < 344	Equation (28)
Purohit et al. [76]	$μ_{b f} = 999.79 + 0.068 \cdot t - 0.0107 \cdot t^{2} + 0.00082 \cdot t^{2.5} - 2.303 \cdot 10^{- 5} \cdot t^{3}$	300 < T < 350	Equation (29)
Saeed and Dulaimi [77]	$μ_{b f} = 0.414092804247831 - 4.792184560427 \cdot 10^{- 3} T + 2.0927097596 \cdot 10^{- 5} T^{2} - 4.0781184 \cdot 10^{- 8} T^{3} + 2.9885 \cdot 10^{- 11} \cdot T^{4}$		Equation (30)
Present work [78]	$μ_{b f} = 2.2551419 - 0.033948 \cdot T + 0.0002053 \cdot T^{2} - 6.229 \cdot 10^{- 7} \cdot T^{3} + 9.4741 \cdot 10^{- 10} \cdot T^{4} - 5.775 \cdot 10^{- 13} \cdot T^{5}$	283 < T < 343	Equation (31)

Table 6. Correlations for thermal conductivity of water- Al₂O₃ nanofluids.

Authors	Correlation	Remarks	Equation
Khanafer and Vafai [72]	$k_{n f} = k_{b f} (0.9843 + 0.398 φ_{v}^{0.7383} {(\frac{1}{d_{p}})}^{0.2246} {(\frac{μ_{n f}}{μ_{b f}})}^{0.0235} - 3.9517 \frac{φ_{v}}{t} + 34.034 \frac{φ_{v}^{2}}{t^{3}} + 32.509 \frac{φ_{v}}{t^{2}})$	water-Al₂O₃ $0.01 \leq φ_{v} \leq 0.09$ $20 \leq t [° C] \leq 70$ $13 nm \leq d_{p} \leq 131 nm$	Equation (32)
Corcione [74]	$k_{n f} = k_{b f} (1 + 4.4 R e_{p}^{0.4} P r^{0.66} {(\frac{T}{T_{f r}})}^{10} {(\frac{λ_{p}}{λ_{f}})}^{0.03} φ_{v}^{0.66}$	$R e_{p}$ —Re number based on nanoparticle diameter $R e_{p} = \frac{ρ_{b f} u_{B} d_{p}}{μ_{b f}}$ $u_{B}$ —Brownian velocity of the nanoparticle $u_{B} = \frac{2 k_{b} T}{π μ_{b f} d_{p}^{2}}$	Equation (33)
Chen [81]	$k_{p} = k_{n f} \frac{0.75 d_{p} / l_{p}}{0.75 d_{p} / l_{p} + 1}$	l_p—mean free path of nanoparticle	Equation (34)
Hassani et al. [82]	$k_{n f} = k_{b f} (1.04 + φ_{v}^{1.11} {(\frac{k_{p}}{k_{b f}})}^{0.33} P r^{- 1.7} [\frac{1}{P r^{- 1.7}} - \frac{262}{{(\frac{k_{p}}{k_{b f}})}^{0.33}} + (135 {(\frac{d_{r e f}}{d_{p}})}^{0.23} {(\frac{ν_{b f}}{d_{p} u_{B r}})}^{0.82}) {(\frac{c_{p}}{T^{- 1} u_{B r}^{2}})}^{- 0.1} {(\frac{T_{B}}{T})}^{- 7}])$	Various base fluids, metal and oxide nanoparticles	Equation (35)
Sawicka et al. [83]	$k_{n f} = k_{b f} (1 + 0.1046 φ_{m}^{0.2388} {(\frac{100}{d_{p}})}^{3.14 \cdot 10^{- 3}})$		Equation (36)

Table 7. Correlations for thermal conductivity of water.

Authors	Correlation	Range	Equation
Minea et al. Equation (22) in [36]	$k_{b f} = - 0.98249 \cdot 10^{- 5} \cdot T^{2} + 7.535211 \cdot 10^{- 3} \cdot T - 0.76761$	294 < T < 344	Equation (37)
Minea et al. Equation (30) in [36]	$k_{b f} = - 0.743567 + 0.077513 \cdot T - 9.9999999 \cdot 10^{- 6} \cdot T^{2} - 8.63331959 \cdot 10^{- 18} \cdot T^{3} + 7.301424 \cdot 10^{- 21} \cdot T^{4}$	294 < T < 344	Equation (38)
Purohit [76]	$k_{b f} = 0.56112 + (0.00193 \cdot t) - (2.601 \cdot 10^{- 6} \cdot t^{2}) - (6.08 \cdot 10^{- 8} \cdot t^{3})$	300 < T < 350	Equation (39)
Saeed and Dulaimi [77]	$k_{b f} = - 0.46662403 + 0.00575419 \cdot T - 7.18 \cdot 10^{- 6} \cdot T^{2}$		Equation (40)
Present work [78]	$k_{b f} = 27.689 - 0.415 \cdot T + 0.000249 \cdot T^{2} - 7.389 \cdot 10^{- 6} \cdot T^{3} + 1.0839 \cdot 10^{- 8} - 6.3292 \cdot 10^{- 12}$	283 < T < 343	Equation (41)

Table 8. Correlations for density of nanofluids.

Authors	Correlation	Remarks	Equation
Khanafer and Vafai [72]	$ρ_{n f} = 1001.064 + 2738.6191 φ_{v} - 0.2095 \cdot t$	water-Al₂O₃ $0 \leq φ_{v} \leq 0.04$ $5 \leq t [° C] \leq 40$	Equation (42)
Pak and Cho [84]	$ρ_{n f} = φ_{v} ρ_{p} + (1 - φ_{v}) ρ_{b f}$	Mixture model	Equation (43)
Sharifpur et al. [85]	$ρ_{n f, n l} = \frac{ρ_{n f}}{(1 - φ_{v}) + φ_{v} {(r_{p} + t_{v})}^{3} / r_{p}^{3}}$	$r_{p}$ —radius of nanoparticle $t_{v}$ —nanolayer thickness $t_{v} = - 0002833 r_{p}^{3} + 0.0475 r_{p} - 0.1417$	Equation (44)

Table 9. Correlations for density of water.

Autors	Correlation	Range	Equation
Minea et al. Equation (20) in [36]	$ρ_{b f} = - 2.0546 \cdot 10^{- 10} \cdot T^{5} + 4.0505 \cdot 10^{- 7} \cdot T^{4} - 3.1285 \cdot 10^{- 4} \cdot T^{3} + 0.11576 \cdot T^{2} - 20.674 \cdot T + 2446$		Equation (45)
Minea et al. Equation (28) in [36]	$ρ_{b f} = - 413.15683 + 13.24245 \cdot T - 0.040578 \cdot T^{2} + 0.00004 \cdot T^{3} - 2.27018 \cdot 10^{- 17} \cdot T^{4}$		Equation (46)
Purohit et al. [76]	$ρ_{b f} = 999.79 + 0.068 \cdot t - 0.0107 \cdot t^{2} + 0.00082 \cdot t^{2.5} - 2.303 \cdot 10^{- 5} \cdot t^{3}$	300 < T < 350	Equation (47)
Saeed and Dulaimi [77]	$ρ_{b f} = 765.33 + 1.8142 \cdot T - 0.0035 \cdot 10^{- 2} \cdot T^{2}$		Equation (48)
Present work [78]	$ρ_{b f} = - 5859.3637 + 98.96855 \cdot T + 0.574747 \cdot T^{2} + 0.0016856 \cdot T^{3} - 2.4989 \cdot 10^{- 6} \cdot T^{4} - 1.4908 \cdot 10^{- 9} \cdot T^{5}$		Equation (49)
Saha and Paul [86]	$ρ_{b f} = 330.12 + 5.92 \cdot T - 1.63 \cdot 10^{- 2} \cdot T^{2} + 1.33 \cdot 10^{- 5} \cdot T^{3}$	278 < T < 363	Equation (50)

Table 10. Correlations for specific heat of nanofluids.

Authors	Correlation	Remarks	Equation
Pak and Cho [84]	$c_{p, n f} = φ_{v} c_{p, p} + (1 - φ_{v}) c_{p, b f}$	Mixture model	Equation (51)
Williams et al. [90]	$c_{p, n f} = \frac{ρ_{b f} c_{p, b f} (1 - φ_{v}) + φ_{v} ρ_{p} c_{p, p}}{ρ_{b f}}$	water-Al₂O₃, water-ZrO₂	Equation (52)
Corcione et al. [91]	$c_{p, n f} = \frac{(1 - φ_{v}) {(ρ c_{p})}_{b f} + φ_{v} {(ρ c_{p})}_{p}}{(1 - φ_{v}) ρ_{b f} + φ_{v} ρ_{p}}$	water-Al₂O₃, water-CuO, water-TiO₂	Equation (53)
Sekhar and Sharma [92]	$c_{p, n f} = c_{p, H_{2} O} [0.8429 {(1 + \frac{t}{50})}^{- 0.3037} {(1 + \frac{d_{p}}{50})}^{0.4167} {(1 + \frac{φ_{v}}{100})}^{2.272}]$	water-Al₂O₃, water-CuO, water-TiO₂, water-SiO₂ $0.01 % \leq φ_{v} \leq 4 %$ , $20 \leq t [° C] \leq 50$ , $15 nm \leq d_{p} \leq 50 nm$	Equation (54)

Table 11. Correlations for specific heat of water.

Authors	Correlation	Range	Equation
Minea et al. Equation (21) in [36]	$c_{p, b f} = - 2.0546 \cdot 10^{- 10} \cdot T^{5} + 4.0505 \cdot 10^{- 7} \cdot T^{4} - 3.1285 \cdot 10^{- 4} \cdot T^{3} + 0.11576 \cdot T^{2} - 20.674 \cdot T + 2446$	293 < T < 313	Equation (55)
Minea et al. Equation (25) in [36]	$c_{p, b f} = 6108.94345 - 12.426 \cdot T + 0.02 \cdot T^{2} - 5.540012 \cdot 10^{- 12} \cdot T^{3} + 6.25929269 \cdot 10^{- 21} \cdot T^{4}$	293 < T < 313	Equation (56)
Saha and Paul [66]	$c_{p, b f} = 10.01 - 5.14 \cdot 10^{- 2} \cdot T + 1.49 \cdot 10^{- 4} \cdot T^{2} - 2.62 \cdot 10^{- 12} \cdot T^{3}$	278 < T < 363	Equation (57)
Purohit et al. [76]	$c_{p, b f} = 4217.4 - 5.61 \cdot T + 1.299 \cdot T^{1.52} - 0.11 \cdot T^{2} + 4149.6 \cdot 10^{- 6} \cdot T^{2.5}$	300 < T < 350	Equation (58)
Saeed and Dulaimi [77]	$c_{p, b f} = 10444.58656104 - 54.08920728 \cdot T + 0.15359377 \cdot T^{2} - 0.00014301 \cdot T^{3}$		Equation (59)
Present work [78]	$c_{p, b f} = 185614 - 2737 \cdot T + 16.54446 \cdot T^{2} - 0.5006 \cdot T^{3} + 7.58 \cdot 10^{- 5} \cdot T^{4} - 4.5942 \cdot 10^{- 8} \cdot T^{5}$	283 < T < 343	Equation (60)

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Cieśliński, J.T.; Lubocki, D.; Smolen, S. Impact of Temperature and Nanoparticle Concentration on Turbulent Forced Convective Heat Transfer of Nanofluids. Energies 2022, 15, 7742. https://doi.org/10.3390/en15207742

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Cieśliński JT, Lubocki D, Smolen S. Impact of Temperature and Nanoparticle Concentration on Turbulent Forced Convective Heat Transfer of Nanofluids. Energies. 2022; 15(20):7742. https://doi.org/10.3390/en15207742

Chicago/Turabian Style

Cieśliński, Janusz T., Dawid Lubocki, and Slawomir Smolen. 2022. "Impact of Temperature and Nanoparticle Concentration on Turbulent Forced Convective Heat Transfer of Nanofluids" Energies 15, no. 20: 7742. https://doi.org/10.3390/en15207742

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Impact of Temperature and Nanoparticle Concentration on Turbulent Forced Convective Heat Transfer of Nanofluids

Abstract

1. Introduction

2. Materials and Methods

2.1. Tested Nanofluids

2.2. Correlations for Forced Convection Heat Transfer inside Horizontal Tubes

2.3. Correlations for Thermophysical Properties of Nanofluids

3. Results and Discussion

3.1. Variation of Pr Number

3.2. Variation of Re Number

3.3. Variation of Nu Number

3.4. Variation of Heat Transfer Coefficient

4. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

Nomenclature

References

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