# Boundary Layer Flow and Heat Transfer of FMWCNT/Water Nanofluids over a Flat Plate

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations for Nanofluids Heat Transfer

_{p}is the heat capacity, β is the coefficient of thermal expansion, and k is the effective heat conductivity of nanofluid.

## 3. Nanofluid Properties

^{−5}T)

^{−5}T

^{2})

## 4. Boundary Conditions

## 5. Numerical Procedure

## 6. Validation

## 7. Grid Independence

## 8. Results and Discussion

^{1/2}. That is, the boundary layer thickness decreases with the increase of free stream velocity and increases with the increase of effective kinematic viscosity. These trends are observed from Figure 3 and Figure 4 for the nanofluids as well as water.

^{1/2}. It is also seen that as the free stream velocity decreases or solid weight percentage increase, the thermal boundary layer thickness increase. However, the variation of the thermal boundary layer thickness with changes in the free stream velocities is more significant when compared with those for the nanotube weight percentages for concentrations of less or equal 0.2 percent.

## 9. Conclusions

- (1)
- The Nusselt number of a nanofluid is augmented by increasing the weight percentage of nanotubes and the free stream velocity.
- (2)
- For the range of concentration less than 0.2% FMWCNT, the effect of the nanotube weight percentage on wall shear stress is insignificant.
- (3)
- Under similar conditions, the boundary layer thickness increases as free stream velocity decreases or particle volume fraction increases.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

T | temperature (K) |

k | thermal conductivity (W·m^{−2}·K^{−1}) |

t | time (s) |

P | pressure (Pa) |

C_{p} | specific heat capacity (J·kg^{−1}·K^{−1}) |

u, v | Velocity components (m·s^{−1}) |

Greek Symbols | |

δ | Boundary layer thickness (m) |

ρ | Density (kg·m^{−3}) |

μ | Dynamic viscosity (Pa·s) |

υ | Kinematic viscosity (m^{2}·s^{−1}) |

ϕ | Weight percentage of nanotubes |

Subscripts | |

Bf | Base fluid |

nf | Nanofluid |

∞ | Infinity |

## References

- Hassan, M.; Sadri, R.; Ahmadi, G.; Dahari, M.B.; Kazi, S.N.; Safaei, M.R.; Sadeghinezhad, E. Numerical Study of Entropy Generation in a Flowing Nanofluid Used in Micro-and Minichannels. Entropy
**2013**, 15, 144–155. [Google Scholar] [CrossRef] - Asadian, A.M.; Abouali, O.; Yaghoubi, M.; Ahmadi, G. The Effect of Temperature Dependent Electrical Conductivity on the MHD Natural Convection of Al
_{2}O_{3}–Water Nanofluid in a Rectangular Enclosure. In Proceedings of the ASME 2014 12th International Conference on Nanochannels, Microchannels and Minichannels, Chicago, IL, USA, 3–7 August 2014. - Herwig, H.; Hausner, O. Critical view on “new results in micro-fluid mechanics”: An example. Int. J. Heat Mass Transf.
**2003**, 46, 935–937. [Google Scholar] [CrossRef] - Goodarzi, M.; Safaei, M.R.; Vafai, K.; Ahmadi, G.; Dahari, M.; Kazi, S.N.; Jomhari, N. Investigation of nanofluid mixed convection in a shallow cavity using a two-phase mixture model. Int. J. Therm. Sci.
**2014**, 75, 204–220. [Google Scholar] [CrossRef] - Yarmand, H.; Ahmadi, G.; Gharehkhani, S.; Kazi, S.N.; Safaei, M.R.; Alehashem, M.S.; Mahat, A.B. Entropy generation during turbulent flow of zirconia-water and other nanofluids in a square cross section tube with a constant heat flux. Entropy
**2014**, 16, 6116–6132. [Google Scholar] [CrossRef] - Goshayeshi, H.R.; Goodarzi, M.; Safaei, M.R.; Dahari, M. Experimental study on the effect of inclination angle on heat transfer enhancement of a ferrofluid in a closed loop oscillating heat pipe under magnetic field. Exp. Therm. Fluid Sci.
**2016**, 74, 265–270. [Google Scholar] [CrossRef] - Safaei, M.R.; Ahmadi, G.; Goodarzi, M.S.; Safdari Shadloo, M.; Goshayeshi, H.R.; Dahari, M. Heat Transfer and Pressure Drop in Fully Developed Turbulent Flows of Graphene Nanoplatelets–Silver/Water Nanofluids. Fluids
**2016**, 1, 20. [Google Scholar] [CrossRef] - Malvandi, A.; Ganji, D.D.; Hedayati, F.; Yousefi Rad, E. An analytical study on entropy generation of nanofluids over a flat plate. Alex. Eng. J.
**2013**, 52, 595–604. [Google Scholar] [CrossRef] - Ahmad, S.; Pop, I. Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Int. Commun. Heat Mass Transf.
**2010**, 37, 987–991. [Google Scholar] [CrossRef] - Hamad, M.A.A.; Pop, I.; Md Ismail, A.I. Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate. Nonlinear Anal. Real World Appl.
**2011**, 12, 1338–1346. [Google Scholar] [CrossRef] - Malvandi, A.; Hedayati, F.; Ganji, D.D. Slip effects on unsteady stagnation point flow of a nanofluid over a stretching sheet. Powder Technol.
**2014**, 253, 377–384. [Google Scholar] [CrossRef] - Safaei, M.R.; Gooarzi, M.; Akbari, O.A.; Shadloo, M.S.; Dahari, M. Performance Evaluation of Nanofluids in an Inclined Ribbed Microchannel for Electronic Cooling Applications. Available online: http://www.intechopen.com/books/electronics-cooling/performance-evaluation-of-nanofluids-in-an-inclined-ribbed-microchannel-for-electronic-cooling-appli (accessed on 20 September 2016).
- Bejan, A. Convection Heat Transfer; John Wiley & Sons: Hoboken, NJ, USA, 2013; Volume 4. [Google Scholar]
- Aboulhasan Alavi, S.M.; Safaei, M.R.; Mahian, O.; Goodarzi, M.; Yarmand, H.; Dahari, M.; Wongwises, S. A Hybrid Finite-Element/Finite-Difference Scheme for Solving the 3-D Energy Equation in Transient Nonisothermal Fluid Flow over a Staggered Tube Bank. Numer. Heat Transf. Part B Fundam.
**2015**, 68, 169–183. [Google Scholar] [CrossRef] - Goodarzi, M.; Safaei, M.R.; Karimipour, A.; Hooman, K.; Dahari, M.; Kazi, S.N.; Sadeghinezhad, E. Comparison of the finite volume and lattice boltzmann methods for solving natural convection heat transfer problems inside cavities and enclosures. Abstr. Appl. Anal.
**2014**, 2014, 762184. [Google Scholar] [CrossRef] - Nikkhah, Z.; Karimipour, A.; Safaei, M.R.; Forghani-Tehrani, P.; Goodarzi, M.; Dahari, M.; Wongwises, S. Forced convective heat transfer of water/functionalized multi-walled carbon nanotube nanofluids in a microchannel with oscillating heat flux and slip boundary condition. Int. Commun. Heat Mass Transf.
**2015**, 68, 69–77. [Google Scholar] [CrossRef] - Amrollahi, A.; Rashidi, A.M.; Lotfi, R.; Emami Meibodi, M.; Kashefi, K. Convection heat transfer of functionalized MWNT in aqueous fluids in laminar and turbulent flow at the entrance region. Int. Commun. Heat Mass Transf.
**2010**, 37, 717–723. [Google Scholar] [CrossRef] - Safaei, M.R.; Togun, H.; Vafai, K.; Kazi, S.; Badarudin, A. Investigation of heat transfer enhancement in a forward-facing contracting channel using FMWCNT nanofluids. Numer. Heat Transf. Part A
**2014**, 66, 1321–1340. [Google Scholar] [CrossRef] - Wasistho, B.; Geurts, B.; Kuerten, J. Simulation techniques for spatially evolving instabilities in compressible flow over a flat plate. Comput. Fluids
**1997**, 26, 713–739. [Google Scholar] [CrossRef] - Patankar, S.V. Numerical heat transfer and fluid flow, Series in Computational Methods in Mechanics and Thermal Sciences; Hemisphere Publishing Corporation: Washington, DC, USA, 1980. [Google Scholar]
- Safaei, M.R.; Goodarzi, M.; Mohammadi, M. Numerical modeling of turbulence mixed convection heat transfer in air filled enclosures by finite volume method. Int. J. Multiphys.
**2011**, 5, 307–324. [Google Scholar] [CrossRef] - Rahmanian, B.; Safaei, M.R.; Kazi, S.N.; Ahmadi, G.; Oztop, H.F.; Vafai, K. Investigation of pollutant reduction by simulation of turbulent non-premixed pulverized coal combustion. Appl. Therm. Eng.
**2014**, 73, 1222–1235. [Google Scholar] [CrossRef] - Togun, H.; Ahmadi, G.; Abdulrazzaq, T.; Shkarah, A.J.; Kazi, S.; Badarudin, A.; Safaei, M. Thermal performance of nanofluid in ducts with double forward-facing steps. J. Taiwan Inst. Chem. Eng.
**2015**, 47, 28–42. [Google Scholar] [CrossRef] - Karimipour, A.; Afrand, M.; Akbari, M.; Safaei, M.R. Simulation of Fluid Flow and Heat Transfer in the Inclined Enclosure. Int. J. Mech. Aerosp. Eng.
**2012**, 6, 86–91. [Google Scholar] - Schlichting, H. Boundary-Layer Theory; McGraw-Hill: New York, NY, USA, 1968. [Google Scholar]
- White, F.M.; Corfield, I. Viscous Fluid Flow; McGraw-Hill: New York, NY, USA, 2006; Volume 3. [Google Scholar]
- Ahmadi, G. On mechanics of saturated granular materials. Int. J. Non-Linear Mech.
**1980**, 15, 251–262. [Google Scholar] [CrossRef] - Ahmadi, G. On the mechanics of incompressible multiphase suspensions. Adv. Water Resour.
**1987**, 10, 32–43. [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] [PubMed]

**Figure 2.**Temperature profiles versus distance from the plate at x = 0.5 m for different free stream velocities and nanotube weight percentages. (

**a**) Full profile; (

**b**) Enlarged segment near the plate.

**Figure 3.**Velocity profiles versus distance from the plate at x = 0.5 m. (

**a**) Full profile; (

**b**) Enlarged segment near the plate.

**Figure 4.**Contours of velocity magnitude for U = 0.0017 m/s and different nanofluids. (

**a**) Distilled water; (

**b**) φ = 0.1 wt. % MWCNT; (

**c**) φ = 0.2 wt. % MWCNT.

**Figure 7.**Temperature Contours of for different nanofluids and inlet velocities. (

**a**) Distilled water, U = 0.00017 m/s; (

**b**) Distilled water, U = 0.001 m/s; (

**c**) Distilled water, U = 0.0017 m/s; (

**d**) φ = 0.1 wt. % MWCNT, U = 0.0017 m/s; (

**e**) φ = 0.2 wt. % MWCNT, U = 0.0017 m/s.

Wt. % FMWCNT/Water | Density (kg·m^{−3}) | Viscosity (kg·m^{−1}·s^{−1}) | Thermal Conductivity (W·m^{−1}·K^{−1}) | ||||
---|---|---|---|---|---|---|---|

20 °C | 27 °C | 33 °C | 20 °C | 27 °C | 33 °C | ||

0 | 995.8 (26.6 °C) | 0.000980 | 0.000860 | 0.000765 | 0.59 | 0.61 | 0.62 |

0.1 | 1003 (23.4 °C) | 0.000998 | 0.000881 | 0.000781 | 0.62 | 0.64 | 0.66 |

0.2 | 1006 (23.1 °C) | 0.00103 | 0.000901 | 0.000790 | 0.67 | 0.69 | 0.71 |

U (m/s) | Exact Solution | Present Study | |||
---|---|---|---|---|---|

C_{f} | τ (Pa) | Nu | τ (Pa) | Nu | |

0.00017 | 0.051 | 7.37 × 10^{−7} | 8.326 | 7.79 × 10^{−7} | 8.798 |

0.001 | 0.021 | 1.05 × 10^{−5} | 20.193 | 1.12 × 10^{−5} | 21.538 |

0.0017 | 0.016 | 2.30 × 10^{−5} | 26.328 | 2.45 × 10^{−5} | 27.998 |

Local Coefficient of Friction | Number of Grids |
---|---|

0.02342 | 50 × 25 |

0.02286 | 75 × 40 |

0.02252 | 100 × 50 |

0.02249 | 110 × 55 |

0.02247 | 125 × 65 |

0.02245 | 150 × 75 |

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**MDPI and ACS Style**

Safaei, M.R.; Ahmadi, G.; Goodarzi, M.S.; Kamyar, A.; Kazi, S.N.
Boundary Layer Flow and Heat Transfer of FMWCNT/Water Nanofluids over a Flat Plate. *Fluids* **2016**, *1*, 31.
https://doi.org/10.3390/fluids1040031

**AMA Style**

Safaei MR, Ahmadi G, Goodarzi MS, Kamyar A, Kazi SN.
Boundary Layer Flow and Heat Transfer of FMWCNT/Water Nanofluids over a Flat Plate. *Fluids*. 2016; 1(4):31.
https://doi.org/10.3390/fluids1040031

**Chicago/Turabian Style**

Safaei, Mohammad Reza, Goodarz Ahmadi, Mohammad Shahab Goodarzi, Amin Kamyar, and S. N. Kazi.
2016. "Boundary Layer Flow and Heat Transfer of FMWCNT/Water Nanofluids over a Flat Plate" *Fluids* 1, no. 4: 31.
https://doi.org/10.3390/fluids1040031