# Heat Transfer and Pressure Drop of Nanofluid with Rod-like Particles in Turbulent Flows through a Curved Pipe

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{4}≤ Sc ≤ 3 × 10

^{5}, particle aspect ratio 2 ≤ λ ≤ 14, and Dean number 5 × 10

^{3}≤ De ≤ 1.5 × 10

^{4}. The momentum and energy equations of nanofluid, together with the equation of particle number density for particles, are solved numerically. Some results are validated by comparing with the experimental results. The effect of Re, Φ, Sc, λ, and De on the friction factor f and Nusselt number Nu is analyzed. The results showed that the values of f are increased with increases in Φ, Sc, and De, and with decreases in Re and λ. The heat transfer performance is enhanced with increases in Re, Φ, λ, and De, and with decreases in Sc. The ratio of energy PEC for nanofluid to base fluid is increased with increases in Re, Φ, λ, and De, and with decreases in Sc. Finally, the formula of ratio of energy PEC for nanofluid to base fluid as a function of Re, Φ, Sc, λ, and De is derived based on the numerical data.

## 1. Introduction

_{2}O

_{3}particles, the Nusselt number (Nu), i.e., heat transfer, was enhanced with increases in the Reynolds number (Re) and Prandtl number (Pr), and the pressure drop (PD) was increased with increases in particle volume concentration (Φ) in the pipe with U-bend [1] and in a U-bend heat exchanger [2]. There was an obvious enhancement of heat transfer with increasing Re and Φ in a curved pipe with triangular cross-section [3] and in a duct of square cross-section [4]. The frictional entropy generation was lower than the thermal entropy generation [5]. Both friction factor and average value of Nu were larger than that for pure water in a finned bend tube [6]. A new equation estimating the pressure loss in turbulent regime was formulated [7]. The local and average value of Nu was increased with increases in Φ, regardless of Re in a curved channel; the function of Φ on the increase in heat transfer was more remarkable at larger Re [8]. For the nanofluid with Fe

_{3}O

_{4}particles, both Nu and consumed power were increased with increasing Φ in a heat exchanger [9]. For the nanofluid with CuO particles, the increase in Nu value was about 18.6% at 0.06% of Φ compared to base fluid with a pumping penalty of 1.09 times in a heat exchanger [10].

## 2. Model and Equation

#### 2.1. Flow of Nanofluid

_{i}and u

_{i}′ are the mean and fluctuation velocity of nanofluid, respectively; P is the pressure; T is the temperature; ρ

_{nf}is the density of nanofluid; μ is the fluid viscosity; $\overline{{{u}^{\prime}}_{i}{{u}^{\prime}}_{j}}$ is the Reynolds stress; a

_{kl}and a

_{ijkl}are the mean second- and fourth-order tensors of particle orientation, respectively; ε

_{kl}is the mean rate-of-strain tensor; C

_{nf}is the thermal diffusivity coefficient of nanofluid; C

_{T}= 0.1k

^{2}/ε (k is the turbulent kinetic energy, ε is the turbulent dissipation rate) is the eddy thermal diffusivity coefficient; and μ

_{a}is the generalized viscosity coefficient to account for two-particle interactions [26]:

#### 2.2. Density and Thermal Diffusivity of Nanofluid

_{nf}and thermal diffusivity D

_{nf}of nanofluid in Equations (2) and (3) are [27]:

_{nf}is the thermal conductivity, and (ρC

_{p})

_{nf}is the heat capacitance [28]:

^{0.2}λ is the shape factor.

#### 2.3. Probability Density Function and Tensor of Particle Orientation

_{i}and p are the unit vector of principal axis and orientation vector of the particle, respectively; and ψ(p) is the mean probability density function of particle orientation and can be used to determine the likelihood of particle orientation falling within a specific range of values. ψ(p) is given by:

_{j}is the gradient operator projected onto the surface of unit sphere; ω

_{ij}= (∂U

_{j}/∂x

_{i}− ∂U

_{i}/∂x

_{j}), D

_{rI}is the rotary diffusion coefficient resulted from particle interaction, D

_{rI}= 0.01$\sqrt{2{\epsilon}_{{}_{ij}}{\epsilon}_{j}{}_{i}}$ for isotropic D

_{rI}[30]; α

_{ψx}= 1.3(5k

^{2}ν/3ε)

^{1/2}and α

_{ψp}= 0.7(4ε/15ν)

^{1/2}(ν is the fluid viscosity) are the dispersion coefficient of linear and angular displacement [31]; and D

_{rB}is the Brownian rotary diffusion coefficient [32,33]:

_{b}is the Boltzmann constant; L

_{p}is the particle length.

#### 2.4. Turbulent Model

_{T}= 0.09ρ

_{nf}k

^{2}/ε, and k-equation and ε-equation are [34]:

_{k}and S

_{ε}are the source terms resulting from the rod-like particles:

#### 2.5. Equation of Particle Number Density

_{t}= 0.09k

^{2}/ε, and D

_{tB}is the Brownian translational diffusion coefficient [32,33]:

#### 2.6. Pressure Drop and Nusselt Number

_{Sa}is the average velocity of nanofluid in the flow direction.

_{w}is the wall temperature, and T

_{m}is the mean temperature over cross-section.

## 3. Numerical Method and Parameters

#### 3.1. Main Steps

- (1)
- Solving Equations (1)–(4) and (12)–(14) with Φ = μ
_{a}= S_{k}= S_{ɛ}= 0 to obtain U_{j}, P, k, ε and $\overline{{{u}^{\prime}}_{i}{{u}^{\prime}}_{j}}$. - (2)
- Solving Equations (17)–(19) to obtain n(v) and Φ.
- (3)
- Substituting Φ into Equations (4)–(8) to obtain μ
_{a}, ρ_{nf},D_{nf}, k_{nf}and (ρC_{p})_{nf}. - (4)
- Substituting U
_{j}, k, ε and Equation (11) into Equation (10) and solving it to obtain ψ. - (5)
- Substituting ψ into Equation (9) to get a
_{ij}and a_{ijkl}. - (6)
- Substituting ρ
_{nf}, μ_{a}, a_{kl}, a_{ijkl}and D_{nf}into Equations (1)–(4) and (12)–(14) to obtain U_{j}, P, k, ε, $\overline{{{u}^{\prime}}_{i}{{u}^{\prime}}_{j}}$ and T. - (7)
- Repeating steps (2) to (6) using the new values of U
_{j}, P, k, ε, $\overline{{{u}^{\prime}}_{i}{{u}^{\prime}}_{j}}$, and T until the difference between the successive results of U_{i}, p, and T is less than a definite value. - (8)
- Calculating the friction factor f and Nusselt number using Equations (20) and (21).

#### 3.2. Numerical Method

_{s}are located at the center of the meshes, whereas U

_{r}and U

_{θ}are located at the mesh lines. The no-slip condition is applied on the walls, and the standard wall function is employed, and the distance between the first mesh center and wall is laid at y

^{+}= 30. Equation (9) is integrated by the Simpson formula. The in-house code is used in the numerical simulation.

#### 3.3. Parameters in Numerical Simulation

_{0}= 293 K. For water, ρ

_{f}= 998.3 kg/m

^{3}, C

_{p}= 4180 J/kg·K, k

_{f}= 0.602 W/m·K, and μ

_{f}= 1.005 × 10

^{−3}Pa·S. For ZnO particles, ρ

_{p}= 5606 kg/m

^{3}, C

_{p}= 520 J/kg·K, and k

_{p}= 90 W/m·K. The Boltzmann constant k

_{b}is 1.38 × 10

^{−23}J/K. We choose ZnO as the nanofluid because it is insoluble in water and has good dispersion and stability in water. The parameter values given above are actual values. The value of dimensionless parameters given in numerical simulation comes from the range of application in practical application.

_{p}is the particle diffusion coefficient, and d

_{p}is the equivalent diameter of particles.

#### 3.4. Mesh Independence Test

^{−4}.

## 4. Results and Discussion

#### 4.1. Validation

_{2}O

_{3}spherical particles was used in the experiment. In addition, the pressure drop at low Re deviates highly from the experimental data compared to that at high Re, which can be attributed to the fact that the k-ε turbulent model has higher accuracy when used in the flow with high Re.

#### 4.2. Friction Factor

#### 4.2.1. Impact of Reynolds Number

^{1/4}) is also given as a comparison. We can see that the values of f in the nanofluid are larger than those in pure water. The reason is that the rod-like particles are enforced by the fluid to align with flow direction in the nanofluid, which makes the fluid expend extra energy, resulting in an increase in pressure drop. This conclusion is also obtained in the nanofluid with carbon nanotube additives [39]. The values of f are decreased with the increase in Re for different particle volume concentration Φ, indicating that the law that f decreases with the increase in Re in pure water does not change for the nanofluid. The magnitude of decrease for f is large, in the range of Re < 20,000, because the turbulent flow has not yet reached a fully developed state. When Re > 20,000, the magnitude of decrease for f becomes small, and f gradually reaches a stable value with increasing Re, which shows that the turbulent flow has reached a fully developed state. The values of f for Blasius solution are obviously larger than numerical results with Φ = 0% in the laminar flow and transition areas (5000 ≤ Re ≤ 10,000) because the calculation accuracy is not high when the Blasius solution is applied to the laminar flow and transition areas. However, the values of f for Blasius solution and numerical results with Φ = 0% are basically consistent because the Blasius solution is suitable for the flow, which reaches a fully developed turbulent state.

#### 4.2.2. Impact of Particle Volume Concentration

#### 4.2.3. Impact of Schmidt Number

_{p}, i.e., a large Sc corresponds to a large μ or d

_{p}. The value of f is large in the flow with large μ. The particles with large d

_{p}have large inertia so are more likely to be thrown to the outer wall of the curved pipe by centrifugal force, and the accumulation of particles near the outer wall leads to an increase in f. In addition, the force of particles acting on the fluid is closely related to the particle size; the particles with large d

_{p}lead to an increase in the turbulence of the flow as well as f.

#### 4.2.4. Impact of Particle Aspect Ratio

_{a}, as shown in Equation (4), which leads to an increase in f. On the other hand, for the particles with large λ, the alignment phenomenon formed by particles under shear is more obvious, which leads to a decrease in viscosity of the nanofluids, in a manner similar to shear thinning, resulting in a decrease in f. As shown in Figure 6, the values of f increase with decreasing λ, which indicates that the effect of λ on decreasing f is larger than that on increasing f in the parameter range discussed in this paper.

#### 4.2.5. Impact of Dean Number

#### 4.3. Heat Transfer

#### 4.3.1. Impact of Reynolds Number

#### 4.3.2. Impact of Particle Volume Concentration

#### 4.3.3. Impact of Schmidt Number

#### 4.3.4. Impact of Particle Aspect Ratio

#### 4.3.5. Impact of Dean Number

#### 4.4. Energy Performance Evaluation Criterion

_{out}and T

_{in}are the temperatures at outlet and inlet, respectively.

#### 4.4.1. Impact of Re and Φ

_{nf}/PEC

_{f}, and the relationship between PEC

_{nf}/PEC

_{f}and Re for different Φ is shown in Figure 12. It can be seen that PEC

_{nf}/PEC

_{f}is less than 1 at low Re, indicating that the difference in the friction factor between the nanofluid and base fluid is larger than that in the Nusselt number. The opposite is true for the case that PEC

_{nf}/PEC

_{f}is larger than 1. In Figure 12, PEC

_{nf}/PEC

_{f}is increased with increases in Φ, and the increase rate is roughly the same at the given Φ. The point of PEC

_{nf}/PEC

_{f}= 1 shifts to low Re with increasing Φ, i.e., Re ≈ 18,000 for Φ = 0.1% to Re ≈ 12,500 for Φ = 5%. It can be inferred that it is better to apply nanofluids to enhance heat transfer at higher Re and Φ from a comprehensive point of view.

#### 4.4.2. Impact of Sc, λ and De

_{nf}/PEC

_{f}as a function of Re for different Sc, λ, and De, respectively. It can be seen that PEC

_{nf}/PEC

_{f}is increased with increasing λ and De, and with decreasing Sc. The point of PEC

_{nf}/PEC

_{f}= 1 also shifts to low Re with decreasing Sc and increasing λ and De. Thus, it is better to apply nanofluids to enhance heat transfer at low Sc and high λ and De.

#### 4.4.3. Correlation Model

_{nf}/PEC

_{f}is directly proportional to Re, Φ, λ, and De and inversely proportional to Sc. It is necessaryto build a correlation model relating PEC

_{nf}/PEC

_{f}to Re, Φ, Sc, λ, and De in order to more effectively describe the effect of these parameters on the energy performance evaluation criterion. Firstly, Re, Φ, Sc, λ, and De are combined into a dimensionless parameter:

## 5. Conclusions

^{4}≤ Sc ≤ 3 × 10

^{5}, 2 ≤ λ ≤ 14, 5 × 10

^{3}≤ De ≤ 1.5 × 10

^{4}are solved numerically. Some results are validated by comparing the present numerical results with the experimental ones. The main conclusions are summarized as follows:

- (1)
- The values of f in nanofluid are larger than that in pure water, and are increased with increases in Φ, Sc, and Re, and with decreases in Re and λ. The magnitude of decrease for f is large and small at Re < 20,000 and Re > 20,000, respectively.
- (2)
- Rod-like nanoparticles added to the base fluid can promote convective heat transfer. Heat transfer performance is enhanced with increasesinRe, Φ, λ, and De, and with decreases in Sc. The effect of Φ on the heat transfer is more obvious at low Re than that at high Re.
- (3)
- The ratios of energy PEC for the nanofluid to the base fluid are increased with increases in Re, Φ, λ, and De, and with decreases in Sc. Finally, the formula of ratio of energy PEC for nanofluid to the base fluid as a function of Re, Φ, Sc, λ, and De is derived based on the numerical data.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Choi, J.; Zhang, Y.W. Numerical simulation of laminar forced convection heat transfer of Al
_{2}O_{3}-water nanofluid in a pipe with return bend. Int. J. Therm. Sci.**2012**, 55, 90–102. [Google Scholar] [CrossRef] - Prasad, P.V.D.; Gupta, A.V.S.S.K.S.; Sreeramulu, M.; Sundar, L.S.; Singh, M.K.; Sousa, A.C.M. Experimental study of heat transfer and friction factor of Al
_{2}O_{3}nanofluid in U-tube heat exchanger with helical tape inserts. Exp. Therm. Fluid Sci.**2015**, 62, 141–150. [Google Scholar] [CrossRef] - Vasa, A.; Barik, A.K.; Nayak, B. Turbulent convection heat transfer enhancement in a 180-degree U-bend of triangular cross-section using nanofluid. In Proceedings of the International Conference on Materials, Alloys and Experimental Mechanics, Conference Series-Materials Science and Engineering, Telangana, India, 3–4 July 2017. [Google Scholar]
- Barik, A.K.; Satapathy, P.K.; Sahoo, S.S. CFD study of forced convective heat transfer enhancement in a 90 degrees bend duct of square cross section using nanofluid. Sadhana-Acad. Proc. Eng. Sci.
**2016**, 41, 795–804. [Google Scholar] - Mukherjee, A.; Rout, S.; Barik, A.K. Entropy generation analysis in a 180-degree return bend pipe using nanofluid. In Proceedings of the International Conference on Functional Materials, Characterization, Solid State Physics, Power, Thermal and Combustion Energy, Visakhapatnam, India, 7–8 April 2017. [Google Scholar]
- Yasinm, N.J.; Jehhef, K.A.; Mohsen, Z.A. Assessment the effect of nanofluid on turbulent heat transfer and pressure drop in bend finned tube. In Proceedings of the 2nd International Conference on Sustainable Engineering Techniques, Baghdad, Iraq, 6–7 March 2019. [Google Scholar]
- Özbey, M. Experimental study on pressure drop of Aluminum-Oxide/Water nanofluids. J. Thermophys. Heat Transf.
**2016**, 30, 342–349. [Google Scholar] [CrossRef] - Ma, Y.; Mohebbi, R.; Rashidi, M.M.; Yang, Z.G. Study of nanofluid forced convection heat transfer in a bent channel by means of lattice Boltzmann method. Phys. Fluids
**2018**, 30, 032001. [Google Scholar] [CrossRef] - Kumar, N.T.R.; Bhramara, P.; Addis, B.M.; Sundar, L.S.; Singh, M.K.; Sousa, A.C.M. Heat transfer, friction factor and effectiveness analysis of Fe
_{3}O_{4}/water nanofluid flow in a double pipe heat exchanger with return bend. Int. Commun. Heat Mass Transf.**2017**, 81, 155–163. [Google Scholar] [CrossRef] - Rao, V.N.; Sankar, B.R. Heat transfer and friction factor investigations of CuO nanofluid flow in a double pipe U-bend heat exchanger. Mater. Today-Proc.
**2019**, 18, 207–218. [Google Scholar] - Tillman, P.; Hill, J. Modelling the thermal conductivity of nanofluids. Solid Mech. Appl.
**2007**, 144, 105–118. [Google Scholar] - Murugesan, C.; Sivan, S. Limits for thermal conductivity of nanoflfluids. Therm. Sci.
**2010**, 14, 65–71. [Google Scholar] [CrossRef] - Yang, Y.; Zhang, Z.G.; Grulke, E.A.; Anderson, W.B.; Wu, G.F. Heat transfer properties of nanoparticle-in-fluid dispersions (nanofluids) in laminar flow. Int. J. Heat Mass Transf.
**2005**, 48, 1107–1116. [Google Scholar] [CrossRef] - Xie, H.; Yu, W.; Li, Y. Thermal performance enhancement in nanofluids containing diamond nanoparticles. J. Phys. D Appl. Phys.
**2009**, 42, 095413. [Google Scholar] [CrossRef] - Ji, Y.; Wilson, C.; Chen, H.; Ma, H. Particle shape effect on heat transfer performance in an oscillating heat pipe. Nanoscale Res. Lett.
**2011**, 6, 290–296. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jeong, J.; Li, C.; Kwon, Y.; Lee, J.; Kim, S.H.; Yun, R. Particle shape effect on the viscosity and thermal conductivity of ZnO nanofluids. Int. J. Refrig.
**2013**, 36, 2233–2241. [Google Scholar] [CrossRef] - Ekiciler, R.; Cetinkaya, M.S.A.; Arslan, K. Effect of shape of nanoparticle on heat transfer and entropy generation of nanofluid-jet impingement cooling. Int. J. Green Energy
**2020**, 17, 555–567. [Google Scholar] [CrossRef] - Anwar, T.; Kumam, P.; Shah, Z.; Sitthithakerngkiet, K. Significance of shape factor in heat transfer performance of molybdenum-disulfide nanofluid in multiple flow situations; A comparative fractional study. Molecules
**2021**, 26, 3711. [Google Scholar] - Alsarraf, J.; Shahsavar, A.; Mahani, R.B.; Talebizadehsardari, P. Turbulent forced convection and entropy production of ananoflui dinasolar collector considering various shapes for nanoparticles. Int. Commun. Heat Mass Transf.
**2020**, 117, 104804. [Google Scholar] [CrossRef] - Yu, L.; Liu, D.; Botz, F. Laminar convective heat transfer of aluminapoly alphaolef in nanofluids containing spherical and non-spherical nanoparticles. Exp. Therm. Fluid Sci.
**2012**, 37, 72–83. [Google Scholar] [CrossRef] - Elias, M.M.; Miqdad, M.; Mahbubul, I.M.; Saidur, R.; Kamalisarvestani, M.; Sohel, M.R.; Hepbasli, A.; Rahim, N.A.; Amalina, M.A. Effect of nanoparticle shape on the heat transfer and thermodynamic performance of a shell and tube heat exchanger. Int. Commun. Heat Mass Transf.
**2013**, 44, 93–99. [Google Scholar] [CrossRef] - Elias, M.M.; Shahrul, I.M.; Mahbubul, I.M.; Saidur, R.; Rahim, N.A. Effect of different nanoparticle shapes on shell and tube heat exchanger using different bafflfle angles and operated with nanoflfluid. Int. J. Heat Mass Transf.
**2014**, 70, 289–297. [Google Scholar] [CrossRef] - Ghosh, M.M.; Ghosh, S.; Pabi, S.K. Effects of particle shape and fluid temperature on heat-transfer characteristics of nanoflfluids. J. Mater. Eng. Perform.
**2013**, 22, 1525–1529. [Google Scholar] [CrossRef] - Lin, J.Z.; Xia, Y.; Ku, X.K. Friction factor and heat transfer of nanoflfluids containing cylindrical nanoparticles in laminar pipe flow. J. Appl. Phys.
**2014**, 116, 133513. [Google Scholar] [CrossRef] - Batchelor, G.K. Stress generated in anon-dilute suspension of elongated particles by pure straining motion. J. Fluid Mech.
**1971**, 46, 813–829. [Google Scholar] [CrossRef] - Mackaplow, M.B.; Shaqfeh, E.S.G. A numerical study of the rheological properties of suspensions of rigid, non-Brownian fibres. J. Fluid Mech.
**1996**, 329, 155–186. [Google Scholar] [CrossRef] - Batchelor, G.K. The effect of brownian motion on the bulk stress in a suspension of spherical particles. J. Fluid Mech.
**1977**, 83, 97–117. [Google Scholar] [CrossRef] - Zhang, X.; Gu, H.; Fujii, M. Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and rod-like nanoparticles. Exp. Therm. Fluid Sci.
**2007**, 31, 593–599. [Google Scholar] [CrossRef] - Advani, S.G.; Tucker, C.L. The use of tensors to describe and predict fiber orientation in short fiber composites. J. Rheol.
**1987**, 31, 751–784. [Google Scholar] [CrossRef] - Folgar, F.; Tucker, C.L., III. Orientation behaviour of fibres in concentrated suspensions. J. Reinf. Plast. Comp.
**1984**, 3, 98–119. [Google Scholar] [CrossRef] - Olson, J.A. The motion of fibers in turbulent flow, stochastic simulation of isotropic homogeneous turbulence. Int. J. Multiph. Flow
**2001**, 27, 2083–2103. [Google Scholar] [CrossRef] - Li, G.; Tang, J.X. Diffusion of Actin filaments within a thin layer between two walls. Phys. Rev. E
**2004**, 69, 061921. [Google Scholar] [CrossRef] - de la Torre, J.G.; Bloomfield, V.A. Hydrodynamic properties of complex, rigid, biological macromolecules: Theory and application. Quart. Rev. Biophys.
**1981**, 14, 81–139. [Google Scholar] [CrossRef] - Lin, J.Z.; Shen, S.H. A theoretical model of turbulent fiber suspension and its application to the channel flow. Sci. China Phys. Mech. Astron.
**2010**, 53, 1659–1670. [Google Scholar] [CrossRef] - Friedlander, S.K. Smoke, Dust and Haze: Fundamentals of Aerosol Behavior; Wiley: New York, NY, USA, 2000. [Google Scholar]
- Patankar, S.V.; Spalding, D.B. Calculation procedure for heat, mass and momentum-transfer in 3-dimensional parabolic flows. Int. J. Heat Mass Transf.
**1972**, 15, 1787–1806. [Google Scholar] [CrossRef] - Ferrouillat, S.; Bontemps, A.; Poncelet, O.; Soriano, O.; Gruss, J.A. Influence of nanoparticle shape factor on convective heat transfer and energetic performance of water-based SiO
_{2}and ZnO nanofluids. Appl. Therm. Eng.**2013**, 51, 839–851. [Google Scholar] [CrossRef] - Blasius, H. Grenzschichten in flüssigkeiten mit kleiner reibung. Z. Math.Phys.
**1908**, 56, 1–37. [Google Scholar] - Steele, A.; Bayer, I.S.; Loth, E. Pipe flow drag reduction effects from carbon nanotube additives. Carbon
**2014**, 77, 1183–1186. [Google Scholar] [CrossRef] - Bernstein, O.; Shapiro, M. Direct determination of the orientation distribution function of cylindrical particles immersed in laminar and turbulent shear flows. J. Aerosol Sci.
**1994**, 25, 113–136. [Google Scholar] [CrossRef]

**Figure 2.**Viscosity of nanofluid with ZnO particles (λ = 8, Φ = 0.93%). ■: numerical result; ●: experimental result [37].

**Figure 3.**Pressure drop as function of Re (λ = 1, Φ = 2%). ■: present result; ●: experimental result [7].

**Figure 4.**Relationship between f and Re for different Φ (De = 1.2 × 10

^{4}, Sc = 10

^{5}, λ = 10). ●: Φ = 0%; ◄: Blasius solution (straight pipe) simulation: ■: Φ = 0.1%, ▲: Φ = 1%, ▼: Φ = 3%, ♦: Φ = 5%.

**Figure 5.**Relationship between f and Re for different Sc (De = 1.2 × 10

^{4}, Φ = 3%, λ = 10). Simulation: ■: Sc = 10

^{4}, ▲: Sc = 5 × 10

^{4}, ▼: Sc = 10

^{5}, ♦: Sc = 3 × 10

^{5}.

**Figure 6.**Relationship between f and Re for different λ (Sc = 10

^{5}, Φ = 3%, De = 1.2 × 10

^{4}). Simulation: ■: λ = 2, ▲: λ = 6, ▼: λ = 10, ♦: λ = 14.

**Figure 7.**Relationship between f and Re for different De (Sc = 10

^{5}, Φ = 3%, λ = 10). Simulation: ■: De = 5 × 10

^{3}, ▲: De = 9 × 10

^{3}, ▼: De = 1.2 × 10

^{4}, ♦: De = 1.5 × 10

^{4}.

**Figure 8.**Relationship between Nu and Re for different Φ (De = 1.2 × 10

^{4}, Sc = 10

^{5}, λ = 10). Simulation: ●: Φ = 0%, ■: Φ = 0.1%, ▲: Φ = 1%, ▼: Φ = 3%, ♦: Φ = 5%.

**Figure 9.**Relationship between Nu and Re for different Sc (De = 1.2 × 10

^{4}, Φ = 3%, λ = 10). Simulation: ■: Sc = 10

^{4}, ▲: Sc = 5 × 10

^{4}, ▼: Sc = 10

^{5}, ♦: Sc = 3 × 10

^{5}.

**Figure 10.**Relationship between Nu and Re for different λ (Sc = 10

^{5}, Φ = 3%, De = 1.2 × 10

^{4}). Simulation: ■: λ = 2, ▲: λ = 6, ▼: λ = 10, ♦: λ = 14.

**Figure 11.**Relationship between Nu and Re for different De (Sc = 10

^{5}, Φ = 3%, λ = 10). Simulation: ■: De = 5 × 10

^{3}, ▲: De = 9 × 10

^{3}, ▼: De = 1.2 × 10

^{4}, ♦: De = 1.5 × 10

^{4}.

**Figure 12.**Relationship between PEC

_{nf}/PEC

_{f}and Re for different Φ (De = 1.2 × 10

^{4}, Sc = 10

^{5}, λ = 10). Simulation: ●: Φ = 0%, ■: Φ = 0.1%, ▲: Φ = 1%, ▼: Φ = 3%, ♦: Φ = 5%.

**Figure 13.**Relationship between PEC

_{nf}/PEC

_{f}and Re for different Sc (De = 1.2 × 10

^{4}, Φ = 3%, λ = 10). Simulation: ■: Sc = 10

^{4}, ▲: Sc = 5 × 10

^{4}, ▼: Sc =10

^{5}, ♦: Sc = 3 × 10

^{5}.

**Figure 14.**Relationship between PEC

_{nf}/PEC

_{f}and Re for different λ (Sc = 10

^{5}, Φ = 3%, De = 1.2 × 10

^{4}). Simulation: ■: λ = 2, ▲: λ = 6, ▼: λ = 10, ♦: λ = 14.

**Figure 15.**Relationship between PEC

_{nf}/PEC

_{f}and Re for different De (Sc = 10

^{5}, Φ = 3%, λ = 10). Simulation: ■: De = 5 × 10

^{3}, ▲: De = 9 × 10

^{3}, ▼: De = 1.2 × 10

^{4}, ♦: De = 1.5 × 10

^{4}.

**Figure 16.**Relationship between PEC

_{nf}/PEC

_{f}and dimensionless parameter ξ. ●: numerical data; —: Equation (25).

r × θ × S Nu | r × θ × S Nu | r × θ × S Nu |
---|---|---|

56 × 48 × 112 177.308 | 64 × 40 × 112 177.315 | 64 × 48 × 104 177.301 |

60 × 48 × 112 177.336 | 64 × 44 × 112 177.338 | 64 × 48 × 108 177.333 |

64 × 48 × 112 177.357 | 64 × 48 × 112 177.357 | 64 × 48 × 112 177.357 |

68 × 48 × 112 177.372 | 64 × 52 × 112 177.370 | 64 × 48 × 116 177.374 |

72 × 48 × 112 177.383 | 64 × 56 × 112 177.378 | 64 × 48 × 120 177.386 |

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**

Lin, W.; Shi, R.; Lin, J.
Heat Transfer and Pressure Drop of Nanofluid with Rod-like Particles in Turbulent Flows through a Curved Pipe. *Entropy* **2022**, *24*, 416.
https://doi.org/10.3390/e24030416

**AMA Style**

Lin W, Shi R, Lin J.
Heat Transfer and Pressure Drop of Nanofluid with Rod-like Particles in Turbulent Flows through a Curved Pipe. *Entropy*. 2022; 24(3):416.
https://doi.org/10.3390/e24030416

**Chicago/Turabian Style**

Lin, Wenqian, Ruifang Shi, and Jianzhong Lin.
2022. "Heat Transfer and Pressure Drop of Nanofluid with Rod-like Particles in Turbulent Flows through a Curved Pipe" *Entropy* 24, no. 3: 416.
https://doi.org/10.3390/e24030416