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Numerical Investigation of Forced Convective Heat Transfer and Performance Evaluation Criterion of Al_{2}O_{3}/Water Nanofluid Flow inside an Axisymmetric Microchannel

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## Abstract

**:**

_{2}O

_{3}/water nanofluid conjugate heat transfer inside a microchannel is studied numerically. The fluid flow is laminar and a constant heat flux is applied to the axisymmetric microchannel’s outer wall, and the two ends of the microchannel’s wall are considered adiabatic. The problem is inherently three-dimensional, however, in order to reduce the computational cost of the solution, it is rational to consider only a half portion of the axisymmetric microchannel and the domain is revolved through its axis. Hence. the problem is reduced to a two-dimensional domain, leading to less computational grid. At the centerline (r = 0), as the flow is axisymmetric, there is no radial gradient (∂u/∂r = 0, v = 0, ∂T/∂r = 0). The effects of four Reynolds numbers of 500, 1000, 1500, and 2000; particle volume fractions of 0% (pure water), 2%, 4%, and 6%; and nanoparticles diameters in the range of 10 nm, 30 nm, 50 nm, and 70 nm on forced convective heat transfer as well as performance evaluation criterion are studied. The parameter of performance evaluation criterion provides valuable information related to heat transfer augmentation together with pressure losses and pumping power needed in a system. One goal of the study is to address the expense of increased pressure loss for the increment of the heat transfer coefficient. Furthermore, it is shown that, despite the macro-scale problem, in microchannels, the viscous dissipation effect cannot be ignored and is like an energy source in the fluid, affecting temperature distribution as well as the heat transfer coefficient. In fact, it is explained that, in the micro-scale, an increase in inlet velocity leads to more viscous dissipation rates and, as the friction between the wall and fluid is considerable, the temperature of the wall grows more intensely compared with the bulk temperature of the fluid. Consequently, in microchannels, the thermal behavior of the fluid would be totally different from that of the macro-scale.

## 1. Introduction

_{2}O

_{3}/water nanofluid flowing inside a tube that is put in a uniform wall heat flux. They showed that using nanoparticles enhances the coefficient of heat transfer, and it increases with an augmentation in the fraction of particles. Akbari and Behzadmehr [29] investigated the influence of using nanofluid on heat transfer in a tube installed horizontally with uniform heat flux. They used three different Al

_{2}O

_{3}particle concentrations of 0, 2, and 4 percent. They concluded that, for a given Grashof number, the coefficient of convective heat transfer enhances with an augmentation in the particle’s concentration. Thermophysical specifications of a nanofluid such as thermal conductivity and viscosity can influence convective heat transfer [30,31]. Different models, based on various approaches such as correlations, artificial neural network, and support vector machines, are represented for modeling of nanofluids’ properties [32,33,34,35,36,37]. Koo et al. [38] and Chon et al. [39] modeled nanofluid effective thermal conductivity including effects due to both temperature and particle size. Ben Mansour et al. [40] studied nanofluid forced convective heat transfer. They showed that different modeling of nanofluids’ thermophysical features leads to contradictory results. Li et al. [41] investigated the impact of thermophysical properties of the fluid on thermal features. They showed that the thermophysical properties can considerably affect the heat transfer rate. Lelea [42] numerically studied a microchannel and investigated heat transfer and laminar flow of fluid inside it. He used three different fluids with temperature dependency. The results indicated that thermal conductivity has remarkable impact on local Nusselt number behavior in the cases of low Re numbers. As the Re increases, the thermal conductivity has a weaker effect on the local Nusselt number. Another important parameter in microtubes is viscous dissipation, which is like an energy source in the fluid flow and changes the distribution of the temperature. This energy source is actually induced by shear stresses. Many works have been done so far studying this effect. Tso and Mahulikar [43,44,45] investigated the impact of the Brinkman number on the thermal behavior of the fluid flowing in microchannels. Morini [46] studied conjugate heat transfer and the heating due to viscosity in microtubes. The results revealed that the dependency of the average Nusselt number on Re number can be described by including the viscous dissipation and the conjugate effects. In a study by Safaei et al. [47], erosion rate of a two phase fluid flow inside a 90 degree elbow was analyzed. Parameters of fluid velocity, particle diameters, and its volume fraction were investigated, and the results showed that particles’ size and volume fraction, as well as the velocity of the fluid, have a considerable effect on the erosion rate.

_{2}O

_{3}/water laminar flow in an axisymmetric microchannel, are investigated.

## 2. Numerical Model

_{i}= 0.1 mm and outer diameter of D

_{o}= 0.3 mm with L = 100 mm length and a constant heat flux applied on the outer wall q

_{o}= 5305.3 W m

^{−2}is studied. The wall thermal conductivity is equal to that of the steel, k = 16.3 W m

^{−1}K

^{−1}. It is considered that, at the microchannel inlet, the temperature and velocity are uniform. Heat flux in uniform condition is applied on the outer wall and the condition adopted at the outlet is fully developed. At the interface of solid and liquid, conjugate heat transfer procedure is considered and the continuity of heat flux and temperature are defined at the interface.

_{v}is the viscous dissipation term.

_{0}, T = T

_{0}.

_{2}O

_{3}nanoparticles are represented in the work of [54] as follows:

## 3. Results and Discussion

_{2}O

_{3}nanoparticles in water is considered as the single phase nanofluid. The nanofluid flows inside the microchannel with an inlet temperature of 293.15 K, and its temperature increases as a result of the imposed heat flux on the outer wall. This paper aims to evaluate the impact of different parameters including Reynolds number, nanoparticles diameter, and particle volume fraction on heat transfer characteristics.

^{2}K, while in nanofluid with 6% of Al

_{2}O

_{3}, this is about 25,000 W/m

^{2}K, which is an increase of 19%. The augmentation in the coefficient of heat transfer is the result of increased thermal conductivity of the nanofluid. In Figure 6, it can be concluded that PEC decreases with an augmentation in the solid phase volume fraction. This is the result of greater pressure losses, which is related to the nanoparticles. However, this decrement is not severe. More specifically, PEC is 2.05 for pure water and reduces to 1.77 for a volume fraction of 6%.

## 4. Conclusions

_{2}O

_{3}/water nanofluid inside a microchannel with an inner diameter of 0.1 mm was investigated numerically. The fluid flow was considered as laminar. A constant heat flux was applied on the microchannel’s outer wall. Four different Re numbers of 500, 1000, 1500, and 2000 and different nanoparticle diameters in the range of 10 nm to 70 nm with four various nanoparticle volume fractions of 0% (pure water), 2%, 4%, and 6% were used in the study. The effects of these parameters were studied on the coefficient of convective heat transfer as well as the performance evaluation criterion.

- Owing to the severe impact of viscous dissipation on the temperature of the wall and fluid, in contrast with the macro-scale problem, as the Reynolds number increases, the convective heat transfer coefficient reduces. Therefore, the viscous dissipation effect that is induced by shear stresses cannot be ignored in microchannels. Furthermore, PEC reduces as the Reynolds number increases, which is the result of higher pressure losses in high values of Reynolds numbers.
- Augmentation in the volume fraction of a nanoparticle can increase the heat transfer coefficient, which is in expense of a lower PEC, although the variation of the PEC is not great. The PEC reduces from 2.05 for pure water to 1.77 for a volume fraction of 6%.
- Increasing the diameter of the nanoparticle can decrease heat transfer coefficient. Furthermore, it was observed that variations of the diameter of the nanoparticle have no effect on PEC for four different studied nanoparticle diameters and PEC equal to 1.98.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Br | Brinkman number |

c_{p} | Specific heat at constant pressure, J/kg K |

D | Diameter, m |

h | Heat transfer coefficient, W/m^{2} K |

k | Thermal conductivity, W/m K |

k_{b} | Boltzmann constant, J/K |

L | Length, m |

l_{b} | Mean free path, m |

$\dot{\mathrm{m}}$ | Mass flow rate, kg/s |

Pr | Prandtl number |

q | Heat flux, W/m^{2} |

Re | Reynolds number |

r | Spatial coordinates in radial direction, m |

T | Temperature, K |

u_{m} | Mean entrance velocity of nanofluid, m/s |

v | Velocity, m/s |

$\dot{\mathrm{V}}$ | Volumetric flow rate, m^{3}/s |

x | Axial Direction, m |

PEC | Performance Evaluation Criterion |

Greek symbols | |

Φ | Particle volume fraction,% |

μ | Viscosity, Pa.s |

ρ | Density, kg/m^{3} |

$\Pi $ | Pumping power, W |

ΔP | Pressure drop, Pa |

Subscripts | |

b | Bulk |

bf | Base fluid |

eff | Effective |

f | Fluid |

i | Inner |

o | Outer |

p | Particle |

s | Solid |

w | Wall |

## References

- Ramezanizadeh, M.; Alhuyi Nazari, M.; Ahmadi, M.H.; Lorenzini, G.; Pop, I. A review on the applications of intelligence methods in predicting thermal conductivity of nanofluids. J. Therm. Anal. Calorim.
**2019**, 138, 827–843. [Google Scholar] [CrossRef] - Ramezanizadeh, M.; Ahmadi, M.H.; Nazari, M.A.; Sadeghzadeh, M.; Chen, L. A review on the utilized machine learning approaches for modeling the dynamic viscosity of nanofluids. Renew. Sustain. Energy Rev.
**2019**, 114, 109345. [Google Scholar] [CrossRef] - Ramezanizadeh, M.; Alhuyi Nazari, M.; Hossein Ahmadi, M.; Chen, L. A review on the approaches applied for cooling fuel cells. Int. J. Heat Mass Transf.
**2019**, 139, 517–525. [Google Scholar] [CrossRef] - Ahmadi, M.H.; Ramezanizadeh, M.; Nazari, M.A.; Lorenzini, G.; Kumar, R.; Jilte, R. Applications of nanofluids in geothermal: A review. Math. Model. Eng. Probl.
**2018**, 5, 281–285. [Google Scholar] [CrossRef] - Abdollahzadeh Jamalabadi, M.; Ghasemi, M.; Alamian, R.; Wongwises, S.; Afrand, M.; Shadloo, M. Modeling of Subcooled Flow Boiling with Nanoparticles under the Influence of a Magnetic Field. Symmetry
**2019**, 11, 1275. [Google Scholar] [CrossRef] [Green Version] - Abdollahzadeh Jamalabadi, M.Y.; Alamian, R.; Yan, W.-M.; Li, L.K.B.; Leveneur, S.; Safdari Shadloo, M. Effects of Nanoparticle Enhanced Lubricant Films in Thermal Design of Plain Journal Bearings at High Reynolds Numbers. Symmetry
**2019**, 11, 1353. [Google Scholar] [CrossRef] [Green Version] - Karimipour, A.; D’Orazio, A.; Shadloo, M.S. The effects of different nano particles of Al
_{2}O_{3}and Ag on the MHD nano fluid flow and heat transfer in a microchannel including slip velocity and temperature jump. Phys. E Low-Dimens. Syst. Nanostruct.**2017**, 86, 146–153. [Google Scholar] [CrossRef] - Safaei, M.R.; Safdari Shadloo, M.; Goodarzi, M.S.; Hadjadj, A.; Goshayeshi, H.R.; Afrand, M.; Kazi, S.N. A survey on experimental and numerical studies of convection heat transfer of nanofluids inside closed conduits. Adv. Mech. Eng.
**2016**, 8, 168781401667356. [Google Scholar] [CrossRef] - Tso, C.P.; Hor, C.H.; Chen, G.M.; Kok, C.K. Heat induction by viscous dissipation subjected to symmetric and asymmetric boundary conditions on a small oscillating flow in a microchannel. Symmetry
**2018**, 10, 499. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; He, Y.-L.; Tang, G.-H.; Tao, W.-Q. Experimental and numerical studies of liquid flow and heat transfer in microtubes. Int. J. Heat Mass Transf.
**2007**, 50, 3447–3460. [Google Scholar] [CrossRef] - Lee, S.; Choi, S.U.-S.; Li, S.; Eastman, J.A. Measuring Thermal Conductivity of Fluids Containing Oxide Nanoparticles. J. Heat Transf.
**1999**, 121, 280. [Google Scholar] [CrossRef] - Eastman, J.A.; Choi, S.U.S.; Li, S.; Yu, W.; Thompson, L.J. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl. Phys. Lett.
**2001**, 78, 718–720. [Google Scholar] [CrossRef] - Halelfadl, S.; Adham, A.M.; Mohd-Ghazali, N.; Maré, T.; Estellé, P.; Ahmad, R. Optimization of thermal performances and pressure drop of rectangular microchannel heat sink using aqueous carbon nanotubes based nanofluid. Appl. Therm. Eng.
**2014**, 62, 492–499. [Google Scholar] [CrossRef] - Ahmadi, M.H.; Mirlohi, A.; Alhuyi Nazari, M.; Ghasempour, R. A review of thermal conductivity of various nanofluids. J. Mol. Liq.
**2018**, 265, 181–188. [Google Scholar] [CrossRef] - Jalali, E.; Akbari, O.A.; Sarafraz, M.M.; Abbas, T.; Safaei, M.R. Heat transfer of oil/MWCNT nanofluid jet injection inside a rectangular microchannel. Symmetry
**2019**, 11, 757. [Google Scholar] [CrossRef] [Green Version] - Safaei, M.R.; Togun, H.; Vafai, K.; Kazi, S.N.; Badarudin, A. Investigation of Heat Transfer Enhancement in a Forward-Facing Contracting Channel Using FMWCNT Nanofluids. Numer. Heat Transf. Part A Appl.
**2014**, 66, 1321–1340. [Google Scholar] [CrossRef] - Behnampour, A.; Akbari, O.A.; Safaei, M.R.; Ghavami, M.; Marzban, A.; Sheikh Shabani, G.A.; Zarringhalam, M.; Mashayekhi, R. Analysis of heat transfer and nanofluid fluid flow in microchannels with trapezoidal, rectangular and triangular shaped ribs. Phys. E Low-Dimens. Syst. Nanostruct.
**2017**, 91, 15–31. [Google Scholar] [CrossRef] - Tian, C.; Maleki, A.; Motie, S.; Yavarinasab, A.; Afrand, M. Generation expansion planning by considering wind resource in a competitive environment. J. Therm. Anal. Calorim.
**2019**. [Google Scholar] [CrossRef] - Zhang, W.; Maleki, A.; Rosen, M.A. A heuristic-based approach for optimizing a small independent solar and wind hybrid power scheme incorporating load forecasting. J. Clean. Prod.
**2019**, 241, 117920. [Google Scholar] [CrossRef] - Motie, S.; Keynia, F.; Ranjbar, M.R.; Maleki, A. Generation expansion planning by considering energy-efficiency programs in a competitive environment. Int. J. Electr. Power Energy Syst.
**2016**, 80, 109–118. [Google Scholar] [CrossRef] - Li, J.; Mohammadi, A.; Maleki, A. Techno-economic analysis of new integrated system of humid air turbine, organic Rankine cycle, and parabolic trough collector. J. Therm. Anal. Calorim.
**2019**. [Google Scholar] [CrossRef] - Maleki, A.; Rosen, M.A. Design of a cost-effective on-grid hybrid wind–hydrogen based CHP system using a modified heuristic approach. Int. J. Hydrogen Energy
**2017**, 42, 15973–15989. [Google Scholar] [CrossRef] - Celata, G.; Cumo, M.; Marconi, V.; McPhail, S.J.; Zummo, G. Microtube liquid single-phase heat transfer in laminar flow. Int. J. Heat Mass Transf.
**2006**, 49, 3538–3546. [Google Scholar] [CrossRef] - Bergman, T.L.; Incropera, F.P. Fundamentals of Heat and Mass Transfer; Wiley: Hoboken, NJ, USA, 2011; ISBN 9780470501979. [Google Scholar]
- Mital, M. Analytical analysis of heat transfer and pumping power of laminar nanofluid developing flow in microchannels. Appl. Therm. Eng.
**2013**, 50, 429–436. [Google Scholar] [CrossRef] - Faghri, M.; Sparrow, E.M. Simultaneous Wall and Fluid Axial Conduction in Laminar Pipe-Flow Heat Transfer. J. Heat Transf.
**1980**, 102, 58–63. [Google Scholar] [CrossRef] - Nonino, C.; Savino, S.; Del Giudice, S.; Mansutti, L. Conjugate forced convection and heat conduction in circular microchannels. Int. J. Heat Fluid Flow
**2009**, 30, 823–830. [Google Scholar] [CrossRef] - El Bécaye Maïga, S.; Tam Nguyen, C.; Galanis, N.; Roy, G.; Maré, T.; Coqueux, M. Heat transfer enhancement in turbulent tube flow using Al
_{2}O_{3}nanoparticle suspension. Int. J. Numer. Methods Heat Fluid Flow**2006**, 16, 275–292. [Google Scholar] [CrossRef] - Akbari, M.; Behzadmehr, A. Developing mixed convection of a nanofluid in a horizontal tube with uniform heat flux. Int. J. Numer. Methods Heat Fluid Flow
**2007**, 17, 566–586. [Google Scholar] [CrossRef] - Ahmadi, M.H.; Alhuyi Nazari, M.; Ghasempour, R.; Madah, H.; Shafii, M.B.; Ahmadi, M.A. Thermal conductivity ratio prediction of Al2O3/water nanofluid by applying connectionist methods. Colloids Surf. A Physicochem. Eng. Asp.
**2018**, 541, 154–164. [Google Scholar] [CrossRef] - Ahmadi, M.H.; Ahmadi, M.A.; Nazari, M.A.; Mahian, O.; Ghasempour, R. A proposed model to predict thermal conductivity ratio of Al
_{2}O_{3}/EG nanofluid by applying least squares support vector machine (LSSVM) and genetic algorithm as a connectionist approach. J. Therm. Anal. Calorim.**2018**. [Google Scholar] [CrossRef] - Ramezanizadeh, M.; Ahmadi, M.A.; Ahmadi, M.H.; Alhuyi Nazari, M. Rigorous smart model for predicting dynamic viscosity of Al
_{2}O_{3}/water nanofluid. J. Therm. Anal. Calorim.**2019**, 137, 307–316. [Google Scholar] [CrossRef] - Ramezanizadeh, M.; Alhuyi Nazari, M. Modeling thermal conductivity of Ag/water nanofluid by applying a mathematical correlation and artificial neural network. Int. J. Low-Carbon Technol.
**2019**, 14, 468–474. [Google Scholar] [CrossRef] - Hemmat Esfe, M.; Naderi, A.; Akbari, M.; Afrand, M.; Karimipour, A. Evaluation of thermal conductivity of COOH-functionalized MWCNTs/water via temperature and solid volume fraction by using experimental data and ANN methods. J. Therm. Anal. Calorim.
**2015**, 121, 1273–1278. [Google Scholar] [CrossRef] - Komeilibirjandi, A.; Raffiee, A.H.; Maleki, A.; Alhuyi Nazari, M.; Safdari Shadloo, M. Thermal conductivity prediction of nanofluids containing CuO nanoparticles by using correlation and artificial neural network. J. Therm. Anal. Calorim.
**2019**. [Google Scholar] [CrossRef] - Bagherzadeh, S.A.; Sulgani, M.T.; Nikkhah, V.; Bahrami, M.; Karimipour, A.; Jiang, Y. Minimize pressure drop and maximize heat transfer coefficient by the new proposed multi-objective optimization/statistical model composed of “ANN + Genetic Algorithm” based on empirical data of CuO/paraffin nanofluid in a pipe. Phys. A Stat. Mech. Its Appl.
**2019**, 527, 121056. [Google Scholar] [CrossRef] - Hemmat Esfe, M.; Wongwises, S.; Naderi, A.; Asadi, A.; Safaei, M.R.; Rostamian, H.; Dahari, M.; Karimipour, A. Thermal conductivity of Cu/TiO
_{2}–water/EG hybrid nanofluid: Experimental data and modeling using artificial neural network and correlation. Int. Commun. Heat Mass Transf.**2015**, 66, 100–104. [Google Scholar] [CrossRef] - Koo, J.; Kleinstreuer, C. A new thermal conductivity model for nanofluids. J. Nanopart. Res.
**2004**, 6, 577–588. [Google Scholar] [CrossRef] - Chon, C.H.; Kihm, K.D.; Lee, S.P.; Choi, S.U.S. Empirical correlation finding the role of temperature and particle size for nanofluid (Al
_{2}O_{3}) thermal conductivity enhancement. Appl. Phys. Lett.**2005**, 87, 153107. [Google Scholar] [CrossRef] - Ben Mansour, R.; Galanis, N.; Nguyen, C.T. Effect of uncertainties in physical properties on forced convection heat transfer with nanofluids. Appl. Therm. Eng.
**2007**, 27, 240–249. [Google Scholar] [CrossRef] - Li, J.; Peterson, G.; Cheng, P. Three-dimensional analysis of heat transfer in a micro-heat sink with single phase flow. Int. J. Heat Mass Transf.
**2004**, 47, 4215–4231. [Google Scholar] [CrossRef] - Lelea, D. Effects of temperature dependent thermal conductivity on Nu number behavior in micro-tubes. Int. Commun. Heat Mass Transf.
**2010**, 37, 245–249. [Google Scholar] [CrossRef] - Tso, C.P.; Mahulikar, S.P. Experimental verification of the role of Brinkman number in microchannels using local parameters. Int. J. Heat Mass Transf.
**2000**, 43, 1837–1849. [Google Scholar] [CrossRef] - Tso, C.P.; Mahulikar, S.P. The role of the Brinkman number in analysing flow transitions in microchannels. Int. J. Heat Mass Transf.
**1999**, 42, 1813–1833. [Google Scholar] [CrossRef] - Tso, C.P.; Mahulikar, S.P. The use of the Brinkman number for single phase forced convective heat transfer in microchannels. Int. J. Heat Mass Transf.
**1998**, 41, 1759–1769. [Google Scholar] [CrossRef] - Morini, G.L. Scaling Effects for Liquid Flows in Microchannels. Heat Transf. Eng.
**2006**, 27, 64–73. [Google Scholar] [CrossRef] - Safaei, M.R.; Mahian, O.; Garoosi, F.; Hooman, K.; Karimipour, A.; Kazi, S.N.; Gharehkhani, S. Investigation of micro- and nanosized particle erosion in a 90° pipe bend using a two-phase discrete phase model. Sci. World J.
**2014**, 2014, 740578. [Google Scholar] [CrossRef] [Green Version] - Lin, C. Application of the Symmetric Model to the Design Optimization of Fan Outlet Grills. Symmetry
**2019**, 11, 959. [Google Scholar] [CrossRef] [Green Version] - Chen, C.-W.; Lu, Y.-F. Computational Fluid Dynamics Study of Water Entry Impact Forces of an Airborne-Launched, Axisymmetric, Disk-Type Autonomous Underwater Hovering Vehicle. Symmetry
**2019**, 11, 1100. [Google Scholar] [CrossRef] [Green Version] - Qin, Y. Pavement surface maximum temperature increases linearly with solar absorption and reciprocal thermal inertial. Int. J. Heat Mass Transf.
**2016**, 97, 391–399. [Google Scholar] [CrossRef] - Qin, Y.; Tan, K.; Meng, D.; Li, F. Theory and procedure for measuring the solar reflectance of urban prototypes. Energy Build.
**2016**, 126, 44–50. [Google Scholar] [CrossRef] - Versteeg, H.; Malalasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method; Prentice Hall: Upper Saddle River, NJ, USA, 2007. [Google Scholar]
- Patankar, S. Numerical Heat Transfer and Fluid Flow; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Perry, R.H.; Green, D.W. Perry’s Chemical Engineers’ Handbook; McGraw-Hill: New York, NY, USA, 2008; ISBN 9780071422949. [Google Scholar]
- 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] - Roy, G.; Gherasim, I.; Nadeau, F.; Poitras, G.; Nguyen, C.T. Heat transfer performance and hydrodynamic behavior of turbulent nanofluid radial flows. Int. J. Therm. Sci.
**2012**, 58, 120–129. [Google Scholar] [CrossRef] - Lelea, D.; Nisulescu, C. The micro-tube heat transfer and fluid flow of water based Al
_{2}O_{3}nanofluid with viscous dissipation. Int. Commun. Heat Mass Transf.**2011**, 38, 704–710. [Google Scholar] [CrossRef] - Jung, J.Y.; Oh, H.S.; Kwak, H.Y. Forced convective heat transfer of nanofluids in microchannels. Int. J. Heat Mass Transf.
**2009**, 52, 466–472. [Google Scholar] [CrossRef]

**Figure 2.**Heat transfer coefficient of the numerical results with the work of [57] for different Brinkman numbers.

**Figure 3.**Coefficient of heat transfer for nanoparticle volume fraction of 2% with an average diameter of 10 nm.

**Figure 4.**Performance evaluation criterion (PEC) for nanoparticle volume fraction of 2% with 10 nm average diameter.

**Figure 5.**Heat transfer coefficient for the particles with an average diameter of 10 nm and Re of 1000.

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**MDPI and ACS Style**

Irandoost Shahrestani, M.; Maleki, A.; Safdari Shadloo, M.; Tlili, I.
Numerical Investigation of Forced Convective Heat Transfer and Performance Evaluation Criterion of Al_{2}O_{3}/Water Nanofluid Flow inside an Axisymmetric Microchannel. *Symmetry* **2020**, *12*, 120.
https://doi.org/10.3390/sym12010120

**AMA Style**

Irandoost Shahrestani M, Maleki A, Safdari Shadloo M, Tlili I.
Numerical Investigation of Forced Convective Heat Transfer and Performance Evaluation Criterion of Al_{2}O_{3}/Water Nanofluid Flow inside an Axisymmetric Microchannel. *Symmetry*. 2020; 12(1):120.
https://doi.org/10.3390/sym12010120

**Chicago/Turabian Style**

Irandoost Shahrestani, Misagh, Akbar Maleki, Mostafa Safdari Shadloo, and Iskander Tlili.
2020. "Numerical Investigation of Forced Convective Heat Transfer and Performance Evaluation Criterion of Al_{2}O_{3}/Water Nanofluid Flow inside an Axisymmetric Microchannel" *Symmetry* 12, no. 1: 120.
https://doi.org/10.3390/sym12010120