#
A CFD Study on Heat Transfer Performance of SiO_{2}-TiO_{2} Nanofluids under Turbulent Flow

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{2}-P25 particles were added to water/ethylene glycol as a base fluid. The result is considered a new hybrid nanofluid (HN) for investigating heat transfer (HT). The volume concentrations were 0.5, 1.0, and 1.5%. The Reynolds number was in the range of 5000–17,000. The heat flux (HF) was 7955 W/m

^{2}, and the wall temperature was 340.15 K. The numerical experiments were performed strictly following the rules that one should follow in HT experiments. This is important because many studies related to nanofluid HT overlook these details. The empirical correlations that contain the friction factor perform better with higher Reynolds numbers than the correlations based only on Reynolds and Prandtl numbers. When temperature differences are moderate, researchers may consider using constant properties to lower computational costs, as they may give results that are similar to temperature-dependent ones. Compared with previous research, our simulation results are in agreement with the experiments in real time.

## 1. Introduction

_{2}O

_{3}/water hybrid nanofluids with 0.1 and 2 vol% via a two-step method. Nanofluids’ k and viscosity (µ) increased with an increase in concentration growth. Madhesh et al. [16] studied the HT of Cu-Ti/water hybrid nanofluids with 0.1–1.0 vol%; they observed that the convective HT coefficient was maximum, with 48.4%, at 0.7 vol%.

_{2}O

_{3}hybrid nanofluids. They observed that the enhancement of the HT coefficient was 13.56% at Re = 1730, compared with water. Mosayebidorcheh et al. [18] studied the HT under a magnetic field and turbulent flow. The HT is proportional to the concentration and Reynolds number but inverse to the Hartmann number and turbulent parameter. Abbasi et al. [19] investigated the effect of the functionalisation technique on k and the stability of carbon nanotube/Al

_{2}O

_{3}HNs. The functional groups significantly affected the k of HNs. The k enhancement was 20.68% for 0.1 vol%.

_{2}O

_{3}nanoparticles were used to determine the impact of base fluid on HT. The computational model was validated using carbon nanotubes with water nanofluids and compared with the data presented in previous research. It was found that ethylene glycol (EG) had better HT increments than water.

_{3}O

_{4}-MWCNT with a water HN under turbulent flow in a circular tube. Their experiment presented an HT increment of 31.10% with a cost of 1.18-times increase in pumping energy at 0.3 vol% and Re = 22,000 in comparison with water. The proposed correlations fit the experimental results. Using ultrasonication, Baby and Ramaprabhu [22] prepared Fe

_{3}O

_{4}/MWCNT and Fe

_{3}O

_{4}@SiO

_{2}/MWCNT nanofluids, which have k enhancements of 20% and 24.5%, respectively.

_{w}number at wall temperature ratio account for the two phenomena. This study investigates the HT properties of SiO

_{2}-TiO

_{2}HNs in water/EG under turbulent flow through a tube. Their properties are analysed using CFD simulation with commercial software. The TiO

_{2}nanoparticles in the nanofluids have different sizes. The concentrations of nanofluid are 0.5–1.5 vol%. The water/EG mixture is the generally applied HT fluid to set the water/EG ratio according to temperature limitations, for example, in a four-season climate. Freezing can be avoided, and the pumping work is less, compared with a pure EG application.

## 2. Empirical Correlations and Fluid Properties

#### 2.1. Theoretical Background

#### 2.2. Forced Flow in Circular Tubes

_{eqH}) and thermal (D

_{eqT}) equivalent diameter even in complex permanent cross-sectional structures. The definitions are as follows:

#### 2.2.1. Turbulent Flow

#### 2.3. Fluid Properties

_{2}-P25 nanoparticles and water/ethylene glycol mixture as base fluid were considered hybrid nanofluids (HNs), and the authors investigated these properties. The HNs in point were considered incompressible, Newtonian, and single-phase fluids. Table 4 illustrates the density ($\rho $), k, dynamic viscosity (µ), and specific heat (c

_{p}) of HNs obtained from the authors’ previous study [24].

^{2}values confirm. If needed, this prediction method works for other properties as well.

_{mean,f}practically is the so-called completely stirred temperature.

## 3. CFD Analysis

#### 3.1. Arrangements of the Numerical Experiment

^{2}and (b) a constant tube wall temperature of 67 °C. Figure 4 represents the arrangement, flow in a pipe with L = 1.5 m length, d = 16 mm hydraulic diameter, and wt = 2 mm wall thickness. We assumed that the flow in the tube is symmetrical and steady. The following equations were used for calculating the mass, energy, and momentum conservation with variable properties [26]:

#### 3.2. Grid Independence Test

^{5}, 8.0 × 10

^{5}, 9.0 × 10

^{5}, 1.0 × 10

^{6}, 1.1 × 10

^{6}, and 1.2 × 10

^{6}. Figure 6 shows the calculated ratio (Nu divided by Nu of the highest mesh) variation against the mesh density. We assumed that if mesh independence is proved at the end of the investigated range, the same mesh used is validated inside the Re range.

#### 3.3. Fully Developed Flow

_{e}), we used Equation (6) from [27]. However, in planning the experiment, the tube should be long enough to neglect that part. Figure 7 represents the local Nu number variation decrease from [23]. According to the investigated NFs, the Pr = 5 curve was assumed to determine that part of the length where the local Nu number varied. Practically, this means that our investigated zone started at $L=~23\xb7D=0.37$ m from the tube entrance, as shown in Table 6, which lists the calculation results based on [27]. It is also worth mentioning that in [28], the proposed equation for hydrodynamic entry length is $Le=D\xb71.359\xb7R{e}^{0.25}$ and in [29], $Le=~10\xb7D$. The smallest approximation for entry length is in [27,28,29] offers the largest one. We accepted this latter value.

#### 3.4. Velocity and Temperature Profiles

#### 3.5. Results

#### 3.5.1. Constant HF with Non-Variable Properties, Developed Flow

#### 3.5.2. Constant Wall Temperature with Non-Variable Properties and Developed Flow

- The Nu numbers of the constant wall temperature are 7–10% higher than the Nu numbers of the constant HF, and the difference increases as the Re number increases. The difference for constant HF and constant wall temperature is considered small in the literature, for example, in [23], but due to the greater temperature difference alongside the tube length, the HT is more intensive in the case of constant wall temperature. Moreover, the observed differences in the context of the accuracy of HT measurements are satisfactory;
- The DB correlation underestimates, while the ST correlation gives practically the same Nu numbers for constant HF;
- The ST correlation values are less than the numerical simulation result till Re = 14,000 and are above the simulation result only when Re = 17,000. It should also be mentioned that the ST correlation by [23] is valid only to Re = 10,000, but according to our results, it can be used at least to Re = 17,000;
- The correlation that contains the friction factor performs better at higher Re numbers. It should be mentioned that the application range for vonK starts from Re = 10,000, and for Gn, from Re = 2300, but vonK performs well below Re = 10,000 as well.

#### 3.5.3. Constant HF with Variable Properties

#### 3.5.4. Constant Wall Temperature with Variable Properties

#### 3.5.5. HT in Flow Development

^{4}, but for smaller Reynolds numbers, the flow development occurs at a somewhat greater distance. The authors advise the use of short tubes in the practical engineering application, considering flow development. A more intensive HT would occur than expected. Generally, this is not considered a problem and can be ignored.

#### 3.5.6. Friction Factors

## 4. Conclusions

_{2}-P25 nanofluids passing through a circular tube was investigated by CFD with constant HF and constant wall temperature. During the analysis, a comprehensive treatment of turbulent HT was offered by mentioning different known but often ignored details of the phenomenon. The following conclusions were drawn:

- Most of the simplifications in the context of the turbulent flow are acceptable, but one has to refer to the borders of the simplifications;
- When the real conditions do not meet the simplified circumstances, one should refer to the complex phenomenon;
- In the case of moderate temperature differences, the constant properties give similar results to the temperature-dependent case with lower computational cost;
- The Nusselt number and pressure drop increase with increasing concentrations of hybrid nanofluids and flow rate, in agreement with many experimental and theoretical studies.
- Compared with previous research, the simulated results are acceptable.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

HN | Hybrid nanofluid |

HT | Heat transfer |

HF | Heat flux |

CFD | Computer fluid dynamics |

EG | Ethylene glycol |

PD | Pressure drop |

MWCNT | Multi wal carbon nanotubes |

Re | Reynolds number |

Nu | Nusselt number |

Pr | Prandtl number |

## References

- le Ba, T.; Mahian, O.; Wongwises, S.; Szilágyi, I.M. Review on the recent progress in the preparation and stability of graphene-based nanofluids. J. Therm. Anal. Calorim.
**2020**, 142, 1–28. [Google Scholar] [CrossRef][Green Version] - Xiong, Q.; Altnji, S.; Tayebi, T.; Izadi, M.; Hajjar, A.; Sundén, B.; Li, L.K.B. A comprehensive review on the application of hybrid nanofluids in solar energy collectors. Sustain. Energy Technol. Assess.
**2021**, 47, 101341. [Google Scholar] [CrossRef] - Bianco, V.; Manca, O.; Nardini, S. Numerical investigation on nanofluids turbulent convection heat transfer inside a circular tube. Int. J. Therm. Sci.
**2021**, 50, 341–349. [Google Scholar] [CrossRef] - Wang, J.; Zhu, J.; Zhang, X.; Chen, Y. Heat transfer and pressure drop of nanofluids containing carbon nanotubes in laminar flows. Exp. Therm. Fluid Sci.
**2013**, 44, 716–721. [Google Scholar] [CrossRef] - Hussein, A.M.; Sharma, K.V.; Bakar, R.A.; Kadirgama, K. The effect of cross sectional area of tube on friction factor and heat transfer nanofluid turbulent flow. Int. Commun. Heat Mass Transf.
**2013**, 47, 49–55. [Google Scholar] [CrossRef] - Hussein, A.M.; Bakar, R.A.; Kadirgama, K. Study of forced convection nanofluid heat transfer in the automotive cooling system. Case Stud. Therm. Eng.
**2014**, 2, 50–61. [Google Scholar] [CrossRef][Green Version] - Kumar, A.; Sharma, K.; Dixit, A.R. A review of the mechanical and thermal properties of graphene and its hybrid polymer nanocomposites for structural applications. J. Mater. Sci.
**2018**, 54, 5992–6026. [Google Scholar] [CrossRef] - Kumar, A.; Sharma, K.; Dixit, A.R. Carbon nanotube- and graphene-reinforced multiphase polymeric composites: Review on their properties and applications. J. Mater. Sci.
**2019**, 55, 2682–2724. [Google Scholar] [CrossRef] - Pak, B.C.; Cho, Y.I. Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Exp. Heat Transf.
**1998**, 11, 151–170. [Google Scholar] [CrossRef] - Sharma, K.V.; Sundar, L.S.; Sarma, P.K. Estimation of heat transfer coefficient and friction factor in the transition flow with low volume concentration of Al
_{2}O_{3}nanofluid flowing in a circular tube and with twisted tape insert. Int. Commun. Heat Mass Transf.**2009**, 36, 503–507. [Google Scholar] [CrossRef] - Kim, J.-O.; Jung, J.-P.; Lee, J.-H.; Suh, J.; Kang, H.-S. Effects of laser parameters on the characteristics of a Sn-3.5 wt.%Ag solder joint. Met. Mater. Int.
**2009**, 15, 119–123. [Google Scholar] [CrossRef] - le Ba, T.; Alkurdi, A.Q.; Lukács, I.E.; Molnár, J.; Wongwises, S.; Gróf, G.; Szilágyi, I.M. A Novel Experimental Study on the Rheological Properties and Thermal Conductivity of Halloysite Nanofluids. Nanomaterials
**2020**, 10, 1834. [Google Scholar] [CrossRef] [PubMed] - Eshgarf, H.; Kalbasi, R.; Maleki, A.; Shadloo, M.S.; Karimipour, A. A review on the properties, preparation, models and stability of hybrid nanofluids to optimize energy consumption. J. Therm. Anal. Calorim.
**2021**, 144, 1959–1983. [Google Scholar] [CrossRef] - Ghalandari, M.; Maleki, A.; Haghighi, A.; Shadloo, M.S.; Nazari, M.A.; Tlili, I. Applications of nanofluids containing carbon nanotubes in solar energy systems: A review. J. Mol. Liq.
**2020**, 313, 113476. [Google Scholar] [CrossRef] - Suresh, S.; Venkitaraj, K.P.; Selvakumar, P.; Chandrasekar, M. Synthesis of Al2O3-Cu/water hybrid nanofluids using two step method and its thermo physical properties, Colloids Surfaces A Physicochem. Eng. Asp.
**2011**, 388, 41–48. [Google Scholar] [CrossRef] - Madhesh, D.; Kalaiselvam, S. Experimental Analysis of Hybrid Nanofluid as a Coolant; Elsevier Ltd.: Amsterdam, The Netherlands, 2014; pp. 1667–1675. [Google Scholar] [CrossRef][Green Version]
- Suresh, S.; Venkitaraj, K.P.; Selvakumar, P.; Chandrasekar, M. Effect of Al 2O 3-Cu/water hybrid nanofluid in heat transfer. Exp. Therm. Fluid Sci.
**2012**, 38, 54–60. [Google Scholar] [CrossRef] - Mosayebidorcheh, S.; Sheikholeslami, M.; Hatami, M.; Ganji, D.D. Analysis of turbulent MHD Couette nanofluid flow and heat transfer using hybrid DTM-FDM. Particuology
**2016**, 26, 95–101. [Google Scholar] [CrossRef] - Abbasi, S.M.; Rashidi, A.; Nemati, A.; Arzani, K. The effect of functionalisation method on the stability and the thermal conductivity of nanofluid hybrids of carbon nanotubes/gamma alumina. Ceram. Int.
**2013**, 39, 3885–3891. [Google Scholar] [CrossRef] - Labib, M.N.; Nine, M.J.; Afrianto, H.; Chung, H.; Jeong, H. Numerical investigation on effect of base fluids and hybrid nanofluid in forced convective heat transfer. Int. J. Therm. Sci.
**2013**, 71, 163–171. [Google Scholar] [CrossRef] - Sundar, L.S.; Singh, M.K.; Sousa, A.C.M. Enhanced heat transfer and friction factor of MWCNT-Fe
_{3}O_{4}/water hybrid nanofluids. Int. Commun. Heat Mass Transf.**2014**, 52, 73–83. [Google Scholar] [CrossRef] - Baby, T.T.; Sundara, R. Surfactant free magnetic nanofluids based on core-shell type nanoparticle decorated multiwalled carbon nanotubes. J. Appl. Phys.
**2011**, 110, 064325. [Google Scholar] [CrossRef] - Rohsenow, W.M.; Hartnett, J.R.; Cho, Y.I. Handbook of Heat Transfer; McGraw-Hill: New York, NY, USA, 1998. [Google Scholar]
- le Ba, T.; Várady, Z.I.; Lukács, I.E.; Molnár, J.; Balczár, I.A.; Wongwises, S.; Szilágyi, I.M. Experimental investigation of rheological properties and thermal conductivity of SiO
_{2}–P25 TiO_{2}hybrid nanofluids. J. Therm. Anal. Calorim.**2020**, 146, 1–15. [Google Scholar] [CrossRef] - Mishra, S.K.; Chandra, H.; Arora, A. Effect of velocity and rheology of nanofluid on heat transfer of laminar vibrational flow through a pipe under constant heat flux. Int. Nano Lett.
**2019**, 9, 245–256. [Google Scholar] [CrossRef][Green Version] - Saedodin, S.; Zaboli, M.; Ajarostaghi, S.S.M. Hydrothermal analysis of heat transfer and thermal performance characteristics in a parabolic trough solar collector with Turbulence-Inducing elements. Sustain. Energy Technol. Assess.
**2021**, 46, 101266. [Google Scholar] [CrossRef] - Entrance Length, Aerospace, Mech. Mechatron. Engg. 2005. Available online: http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/pipeflow/node9.html (accessed on 7 June 2021).
- Zhi-qing, W. Study on correction coefficients of liminar and turbulent entrance region effect in round pipe. Appl. Math. Mech.
**1982**, 3, 433–446. [Google Scholar] [CrossRef] - Cengel, Y.A. Heat Transfer. A Practical Approach, 2nd ed.; McGraw-Hill: New York, NY, USA, 2002. [Google Scholar]

**Figure 3.**Velocity profiles for constant and variable viscosity [25], reprinted from Springer Nature open access article under the terms of the Creative Commons CC-BY license.

**Figure 8.**Velocity and temperature profiles in fully developed flow (

**a**) constant HF (

**b**) constant temperature of wall.

**Figure 11.**Comparison of the simulation results and correlations with (

**a**) Re = 5000 (

**b**) Re = 17,000.

**Table 1.**Correlations for Nu number based only on Re and Pr numbers [23].

Correlations (a) | Application Range |
---|---|

Dittus–Boelter (1) $Nu=\{\begin{array}{c}0.023R{e}^{0.8}P{r}^{0.4}forheating\\ 0.026R{e}^{0.8}P{r}^{0.3}forcooling\end{array}$ | $0.7\le Pr\le 120$ $2500\le Re\le 1.24\xb7{10}^{5}$ $L/d>60$ |

Colburn (2) $Nu=0.023R{e}^{0.8}P{r}^{1/3}$ | $0.5\le Pr\le 3$ ${10}^{4}\le Re\le {10}^{5}$ |

Drexel–McAdams (3) $Nu=0.021R{e}^{0.8}P{r}^{0.4}$ | $Pr\le 0.7$ ${10}^{4}\le Re\le 5\xb7{10}^{5}$ |

Gnielinski (4,5) $Nu=0.0214\left(R{e}^{0.8}-100\right)P{r}^{0.4}$ $Nu=0.012\left(R{e}^{0.87}-280\right)P{r}^{0.4}$ | $0.5\le Pr\le 1.5$ ${10}^{4}\le Re\le 5\xb7{10}^{6}$ $1.5\le Pr\le 500$ $3\xb7{10}^{3}\le Re\le {10}^{6}$ |

Sieder–Tate (6) $Nu=0.027R{e}^{4/5}P{r}^{1/3}{\left(\frac{\mu}{{\mu}_{w}}\right)}^{0.14}$ | $0.7\le Pr\le 16$ $700\le Re\le {10}^{4}$ |

Hausen (7) $Nu=0.037\left(R{e}^{0.75}-180\right)P{r}^{0.42}\left[1+{\left(\frac{x}{D}\right)}^{-\frac{2}{3}}\right]$ | $0.7\le Pr\le 3$ ${10}^{4}\le Re\le {10}^{5}$ |

**Table 2.**Correlations for Nu number containing the friction factor [23].

Correlations (a) | Application Range |
---|---|

von Kármán (8) $Nu=\frac{\left(\xi /8\right)RePr}{1+5\sqrt{\xi /8}\left[\mathrm{Pr}-1+ln\left(\frac{5Pr+1}{6}\right)\right]}$ | 0.7 ≤ Pr ≤ 10 10 ^{4} ≤ Re ≤ 5 × 10^{6} |

Prandtl (9) $Nu=\frac{\left(\xi /8\right)RePr}{1+8.7\sqrt{\xi /8}\left(Pr-1\right)}$ | 0.5 ≤ Pr ≤ 5 10 ^{4} ≤ Re ≤ 5 × 10^{6} |

Friend–Metzner (10) $Nu=\frac{\left(\xi /8\right)RePr}{1.2+11.87\sqrt{\xi /8}\left(Pr-1\right)P{r}^{-1/3}}$ | 50 ≤ Pr ≤ 600 5·10 ^{4} ≤ Re ≤ 5 × 10^{6} |

Pethukov–Kirillov–Popov (11) $Nu=\frac{\left(\xi /8\right)RePr}{C+12.7\sqrt{\xi /8}\left(P{r}^{2/3}-1\right)}$ $C=1.07+\frac{900}{Re}-\frac{0.63}{1+10Pr}$ | 0.5 ≤ Pr ≤ 10^{6}4000 ≤ Re ≤ 5 × 10 ^{6} |

Webb (12) $Nu=\frac{\left(\xi /8\right)RePr}{1.07+9\sqrt{\xi /8}\left(Pr-1\right)P{r}^{1/4}}$ | 0.5 ≤ Pr ≤ 100 10 ^{4} ≤ Re ≤ 5 × 10^{6} |

Gnielinski (13) $Nu=\frac{\left(\xi /8\right)\left(Re-1000\right)Pr}{1+12.7\sqrt{\xi /8}\left(P{r}^{2/3}-1\right)}\left[1+{\left(\frac{D}{L}\right)}^{2/3}\right]{\varphi}_{T}$ | 0.5 ≤ Pr ≤ 2000 2300 ≤ Re ≤ 5 × 10 ^{6}$\xi ={\left(1.82lo{g}_{10}Re-1.64\right)}^{-2}$ ${\varphi}_{T}={\left(\frac{Pr\left(Tmean\right)}{Pr\left(Twall\right)}\right)}^{0.14}$ For the case of constant fluid properties: D/L = 0, and Φ _{T} = 1. |

Sandall et al. (14) $\frac{\left(\xi /8\right)RePr}{12.48P{r}^{2/3}-7.853P{r}^{\frac{1}{3}}+3.613\mathrm{ln}\left(Pr\right)+5.8+C}$ $C=2.78ln\left(\frac{Re\sqrt{\xi /8}}{45}\right)$ | 0.5 ≤ Pr ≤ 2000 10 ^{4} ≤ Re ≤ 5 × 10^{6} |

**Table 3.**Correlations for friction factor, [23].

Correlations | Application Range |
---|---|

Blasius $\left(1\right)\text{}\xi /4=0.0791R{e}^{-0.25}$ | 4 × 10^{3} ≤ Re ≤ 10^{5} |

Drew et al. $\left(2\right)\text{}\xi /4=0.00128+0.1143R{e}^{-0.311}$ $\left(3\right)\text{}\xi /4=0.0014+0.125R{e}^{-0.32}$ | 4 × 10^{3} ≤ Re ≤ 5 × 10^{6}4 × 10 ^{3} ≤ Re ≤ 10^{7} |

Bhatti and Shah $\left(4\right)\text{}\xi /4=0.00128+0.1143R{e}^{-0.311}$ | 4 × 10^{3} ≤ Re ≤ 10^{7} |

Prandtl–Kármán–Nikuradse $(5)\text{}\frac{2}{\sqrt{\xi}}=1.7272\mathrm{ln}\left(\frac{Re\sqrt{\xi}}{2}\right)-0.3946$ | 4 × 10^{3} ≤ Re ≤ 10^{7} |

Colebrook $(6)\text{}\frac{2}{\sqrt{\xi}}=1.5635\mathrm{ln}\left(\frac{Re}{7}\right)$ | 4 × 10^{3} ≤ Re ≤ 10^{7} |

Filonenko $(7)\text{}\frac{2}{\sqrt{\xi}}=1.58\mathrm{ln}Re-3.28\text{}$ | 10^{4} ≤ Re ≤ 10^{7} |

Techo et al. $(8)\text{}\frac{2}{\sqrt{\xi}}=1.7372\mathrm{ln}(\frac{Re}{1.964\mathrm{ln}Re-3.8215})$ | 10^{4} ≤ Re ≤ 10^{7} |

Temperature °C | Ρ g/cm ^{3} | k W/mK | µ [mPas] | c_{p}J/gK |
---|---|---|---|---|

0.50 vol% SiO_{2}-P25 | ||||

20 | 1.033 | 0.510 | 1.789 | 3848.04 |

30 | 1.030 | 0.535 | 1.470 | 3863.36 |

40 | 1.026 | 0.549 | 1.152 | 3887.89 |

50 | 1.022 | 0.561 | 0.978 | 3903.14 |

60 | 1.017 | 0.568 | 0.815 | 3925.54 |

1.00 vol% SiO_{2}-P25 | ||||

20 | 1.044 | 0.522 | 2.054 | 3799.25 |

30 | 1.041 | 0.546 | 1.594 | 3814.19 |

40 | 1.037 | 0.56 | 1.313 | 3838.14 |

50 | 1.033 | 0.571 | 1.079 | 3852.96 |

60 | 1.028 | 0.578 | 0.885 | 3874.74 |

1.50 vol% SiO_{2}-P25 | ||||

20 | 1.055 | 0.531 | 2.281 | 3733.92 |

30 | 1.052 | 0.553 | 1.825 | 3766.05 |

40 | 1.048 | 0.568 | 1.445 | 3789.45 |

50 | 1.045 | 0.579 | 1.212 | 3803.85 |

60 | 1.039 | 0.587 | 1.008 | 3825.04 |

Concentrations | ||||
---|---|---|---|---|

0.50% | 1% | 1.50% | ||

Pr Numbers | ||||

Temperature, °C | 20 | 13.50 | 14.95 | 16.04 |

30 | 10.62 | 11.14 | 12.43 | |

40 | 8.16 | 9.00 | 9.64 | |

50 | 6.80 | 7.28 | 7.96 | |

60 | 5.63 | 5.93 | 6.57 | |

70 | 5.05 | 5.57 | 6.00 | |

80 | 4.37 | 4.83 | 5.20 | |

90 | 3.86 | 4.26 | 4.59 | |

100 | 3.45 | 3.80 | 4.10 |

Re | L_{e}/D, [27] | L_{e}, m [29] | L_{e}/D, [28] |
---|---|---|---|

5000 | 18.2 | 0.29 | 11.4 |

8000 | 19.7 | 0.31 | 12.9 |

11,000 | 20.8 | 0.33 | 13.9 |

14,000 | 21.6 | 0.35 | 14.8 |

17,000 | 22.3 | 0.36 | 15.5 |

0.5% | 1.0% | 1.5% | ||||
---|---|---|---|---|---|---|

Re | Const. | Variable | Const. | Variable | Const. | Variable |

5000 | 43.4 | 43.7 | 44.1 | 44.5 | 45.6 | 45.5 |

8000 | 63.1 | 64.2 | 64.2 | 65.3 | 66.4 | 66.9 |

11,000 | 81.9 | 83.3 | 83.4 | 84.8 | 86.4 | 86.9 |

14,000 | 100.0 | 101.5 | 101.8 | 103.4 | 105.5 | 105.9 |

17,000 | 117.5 | 119.0 | 119.7 | 121.2 | 124.1 | 124.2 |

0.5% | 1.0% | 1.5% | ||||
---|---|---|---|---|---|---|

Re | Const. | Variable | Const. | Variable | Const. | Variable |

5000 | 47.2 | 47.2 | 48.2 | 48.6 | 50.2 | 50.0 |

8000 | 69.2 | 70.1 | 70.7 | 72.6 | 73.8 | 74.8 |

11,000 | 90.7 | 91.6 | 92.7 | 95.2 | 96.9 | 98.3 |

14,000 | 111.6 | 112.1 | 114.1 | 117.0 | 119.3 | 120.8 |

17,000 | 131.8 | 132.0 | 134.8 | 138.1 | 141.0 | 142.7 |

Case Identification | No. | Friction Factors | ||
---|---|---|---|---|

Re = 5000 | Re = 17,000 | |||

Numbers of Table 3 correlations | 1 | 1 | 0.037626513 | 0.027709216 |

2 | 2 | 0.037458997 | 0.027222256 | |

3 | 3 | 0.038356659 | 0.027742485 | |

4 | 4 | 0.037458997 | 0.027222256 | |

5 | 5 | 0.037777816 | 0.027226969 | |

6 | 6 | 0.037893426 | 0.026929325 | |

7 | 7 | 0.038619473 | 0.027272146 | |

8 | 8 | 0.037320168 | 0.026953634 | |

Constant HF non-variable properties | 0% | 9 | 0.036984055 | 0.027878525 |

0.5% | 10 | 0.036983276 | 0.027878481 | |

1% | 11 | 0.036983498 | 0.02787853 | |

1.5% | 12 | 0.036983328 | 0.027878432 | |

Constant wall temperature non-variable properties | 0% | 13 | 0.036984055 | 0.027878525 |

0.5% | 14 | 0.036983276 | 0.027878481 | |

1% | 15 | 0.036983498 | 0.02787853 | |

1.5% | 16 | 0.036983328 | 0.027878432 | |

Constant HF variable properties | 0% | 17 | 0.037219022 | 0.028036842 |

0.5% | 18 | 0.03727405 | 0.028011376 | |

1% | 19 | 0.037350228 | 0.028012885 | |

1.5% | 20 | 0.036843771 | 0.02796081 | |

Constant wall temperature variable properties | 0% | 21 | 0.036975466 | 0.027986751 |

0.5% | 22 | 0.036819132 | 0.027951593 | |

1% | 23 | 0.036852728 | 0.027979754 | |

1.5% | 24 | 0.036324745 | 0.027667343 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ba, T.L.; Gróf, G.; Odhiambo, V.O.; Wongwises, S.; Szilágyi, I.M.
A CFD Study on Heat Transfer Performance of SiO_{2}-TiO_{2} Nanofluids under Turbulent Flow. *Nanomaterials* **2022**, *12*, 299.
https://doi.org/10.3390/nano12030299

**AMA Style**

Ba TL, Gróf G, Odhiambo VO, Wongwises S, Szilágyi IM.
A CFD Study on Heat Transfer Performance of SiO_{2}-TiO_{2} Nanofluids under Turbulent Flow. *Nanomaterials*. 2022; 12(3):299.
https://doi.org/10.3390/nano12030299

**Chicago/Turabian Style**

Ba, Thong Le, Gyula Gróf, Vincent Otieno Odhiambo, Somchai Wongwises, and Imre Miklós Szilágyi.
2022. "A CFD Study on Heat Transfer Performance of SiO_{2}-TiO_{2} Nanofluids under Turbulent Flow" *Nanomaterials* 12, no. 3: 299.
https://doi.org/10.3390/nano12030299