Abstract
Flow field, heat transfer and entropy generation of forced convection of CuOwater nanofluid is investigated in a parallel plate microchannel in the presence of magnetic field. Two vertical micromixers are attached on the hot walls of the microchannel. To consider the effect of the Brownian motion of the nanoparticles, the KKL model is utilized to estimate thermal conductivity of the nanofluid. The governing equations, which are accompanied with the slip velocity and temperature jump boundary conditions, are solved by the finite volume method (FVM) and SIMPLER algorithm. The study is conducted for the Reynolds numbers in the range of 10 < Re < 100, Hartmann numbers in the range of 0 < Ha < 40, Knudsen numbers ranging of 0 < Kn < 0.1 and volume fraction of nanoparticles ranging of 0 < φ < 0.04. The results show that when the Hartmann or Reynolds numbers, or the volume fraction of nanoparticles increase, the average Nusselt number and the total entropy generation rate increase. Furthermore, when Knudsen number increases, the total entropy generation rate decreases.
This is a preview of subscription content, access via your institution.
Abbreviations
 a _{j} :

Coefficient values of \(g^{\prime}\) function
 B _{0} :

Magnetic field intensity (N/m A)
 c _{i} :

Coefficients of fully developed velocity profile
 c _{p} :

Specific heat capacity (J/kg K)
 d _{np} :

Nanoparticles diameter (nm)
 D _{h} :

Hydraulic diameter (m)
 h :

Local heat transfer coefficient (W/m^{2} K)
 Ha:

Hartmann number
 k :

Thermal conductivity (W/m K)
 Kn:

Knudsen number
 L :

Unit length (m)
 n :

Unit normal vector
 Nu:

Nusselt number
 p :

Pressure (N/m^{2})
 P :

Dimensionless pressure
 Po:

Poiseuille number (≡f Re)
 Pr:

Prandtl number
 Re:

Reynolds number
 R _{bf} :

Thermal interfacial resistance (Km^{2}/W)
 \(\dot{s}^{\prime\prime\prime}\) :

Volumetric rate of entropy generation (W/m^{3} K)
 \(\dot{S}^{\prime\prime\prime}\) :

Dimensionless volumetric rate of entropy generation
 \(\dot{S}\) :

Dimensionless total rate of entropy generation
 T :

Temperature (K)
 (u,v) :

Velocity components (m/s)
 (U,V) :

Dimensionless velocity components
 (x,y) :

Coordinates (m)
 (X,Y) :

Dimensionless coordinates
 α :

Thermal diffusivity (m^{2}/s)
 γ :

Heat capacity ratio (≡ c _{p}/c _{v})
 θ :

Dimensionless temperature
 κ _{b} :

Boltzmann constant (=1.38064852 × 10^{−23} kg m^{2}/s^{2} K)
 λ :

Mean free path (m)
 μ :

Dynamic viscosity (kg/m s)
 ν :

Kinematic viscosity (m^{2}/s)
 ρ :

Density (kg/m^{3})
 σ :

Electrical conductivity (1/Ω m = A^{2} s^{3}/kg m^{3})
 σ _{T} :

Thermal accommodation coefficient
 σ _{v} :

Momentum accommodation coefficient
 χ :

Irreversibility distribution ratio
 \(\phi\) :

Nanoparticles volume fraction
 0 :

Reference state value
 av:

Average
 bf:

Base fluid
 Brownian:

Brownian motion component of thermal conductivity
 eff:

Effective
 fd:

Fully developed region
 h :

Hot walls
 in:

Inlet flow
 nf:

Nanofluid
 np:

Nanoparticles
 static:

Static component of thermal conductivity
 w :

Walls
References
 1.
Yu S, Ameel TA (2001) Slipflow heat transfer in rectangular microchannels. Int J Heat Mass Transf 44:4225–4234
 2.
Renksizbulut M, Niazmand H, Tercan G (2006) Slipflow and heat transfer in rectangular microchannels with constant wall temperature. Int J Therm Sci 45:870–881
 3.
Zhu X, Liao Q (2006) Heat transfer for laminar slip flow in a microchannel of arbitrary cross section with complex thermal boundary conditions. Appl Therm Eng 26:1246–1256
 4.
Barkhordari M, Etemad SG (2007) Numerical study of slip flow heat transfer of nonNewtonian fluids in circular microchannels. Int J Heat Fluid Flow 28:1027–1033
 5.
Nayinian SMM, Shams M, Afshar H, Ahmadi G (2008) Two phase analysis of heat transfer and dispersion of nano particles in a microchannel. In: ASME 2008 Heat Transfer Summer Conference collocated with the Fluids Engineering, Energy Sustainability, and 3rd Energy Nanotechnology Conferences, pp 457–463
 6.
Niazmand H, Renksizbulut M, Saeedi E (2008) Developing slipflow and heat transfer in trapezoidal microchannels. Int J Heat Mass Transf 51:6126–6135
 7.
Hooman K, Hooman F, Famouri M (2009) Scaling effects for flow in microchannels: variable property, viscous heating, velocity slip, and temperature jump. Int Commun Heat Mass Transfer 36:192–196
 8.
Shojaeian M, Dibaji SAR (2010) Threedimensional numerical simulation of the slip flow through triangular microchannels. Int Commun Heat Mass Transfer 37:324–329
 9.
Buonomo B, Manca O (2010) Natural convection slip flow in a vertical microchannel heated at uniform heat flux. Int J Therm Sci 49:1333–1344
 10.
Languri EM, Hooman K (2011) Slip flow forced convection in a microchannel with semicircular crosssection. Int Commun Heat Mass Transfer 38:139–143
 11.
Adesanya SO (2015) Free convective flow of heat generating fluid through a porous vertical channel with velocity slip and temperature jump,” Ain Shams Engineering Journal
 12.
Tuckerman DB, Pease R (1981) Highperformance heat sinking for VLSI. Electron Device Lett IEEE 2:126–129
 13.
Hettiarachchi HM, Golubovic M, Worek WM, Minkowycz W (2008) Threedimensional laminar slipflow and heat transfer in a rectangular microchannel with constant wall temperature. Int J Heat Mass Transf 51:5088–5096
 14.
Shams M, Shojaeian M, Aghanajafi C, Dibaji S (2009) Numerical simulation of slip flow through rhombus microchannels. Int Commun Heat Mass Transfer 36:1075–1081
 15.
Akbarinia A, Abdolzadeh M, Laur R (2011) Critical investigation of heat transfer enhancement using nanofluids in microchannels with slip and nonslip flow regimes. Appl Therm Eng 31:556–565
 16.
Ebrahimi A, Roohi E, Kheradmand S (2015) Numerical study of liquid flow and heat transfer in rectangular microchannel with longitudinal vortex generators. Appl Therm Eng 78:576–583
 17.
Karimipour A (2015) New correlation for Nusselt number of nanofluid with Ag/Al2O3/Cu nanoparticles in a microchannel considering slip velocity and temperature jump by using lattice Boltzmann method. Int J Therm Sci 91:146–156
 18.
Abbassi H (2007) Entropy generation analysis in a uniformly heated microchannel heat sink. Energy 32:1932–1947
 19.
Guo J, Xu M, Tao Y, Huai X (2012) The effect of temperaturedependent viscosity on entropy generation in curved square microchannel. Chem Eng Process 52:85–91
 20.
Ibáñez G, López A, Pantoja J, Moreira J, Reyes JA (2013) Optimum slip flow based on the minimization of entropy generation in parallel plate microchannels. Energy 50:143–149
 21.
Ibáñez G, López A, Pantoja J, Moreira J (2014) Combined effects of uniform heat flux boundary conditions and hydrodynamic slip on entropy generation in a microchannel. Int J Heat Mass Transf 73:201–206
 22.
Anand V (2014) Slip law effects on heat transfer and entropy generation of pressure driven flow of a power law fluid in a microchannel under uniform heat flux boundary condition. Energy 76:716–732
 23.
Ting TW, Hung YM, Guo N (2014) Entropy generation of nanofluid flow with streamwise conduction in microchannels. Energy 64:979–990
 24.
Hooman K (2007) Entropy generation for microscale forced convection: effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry. Int Commun Heat Mass Transfer 34:945–957
 25.
Sohel M, Saidur R, Hassan N, Elias M, Khaleduzzaman S, Mahbubul I (2013) Analysis of entropy generation using nanofluid flow through the circular microchannel and minichannel heat sink. Int Commun Heat Mass Transfer 46:85–91
 26.
Pourmahmoud N, Soltanipour H, Mirzaee I (2014) The effects of longitudinal ribs on entropy generation for laminar forced convection in a microchannel. Thermal Science, pp 110–110
 27.
Hajialigol N, Fattahi A, Ahmadi MH, Qomi ME, Kakoli E (2015) MHD mixed convection and entropy generation in a 3D microchannel using Al2O3–water nanofluid. J Taiwan Inst Chem Eng 46:30–42
 28.
Aminossadati SM, Raisi A, Ghasemi B (2011) Effects of magnetic field on nanofluid forced convection in a partially heated microchannel. Int J Non Linear Mech 46:1373–1382
 29.
Baheri Islami S, Dastvareh B, Gharraei R (2014) An investigation on the hydrodynamic and heat transfer of nanofluid flow, with nonNewtonian base fluid, in micromixers, International Journal of Heat and Mass Transfer 78:917–929
 30.
Aghaei A, Khorasanizadeh H, Sheikhzadeh G, Abbaszadeh M (2016) Numerical study of magnetic field on mixed convection and entropy generation of nanofluid in a trapezoidal enclosure. J Magn Magn Mater 403:133–145
 31.
Arefmanesh A, Aghaei A, Ehteram H (2016) Mixed convection heat transfer in a CuO–water filled trapezoidal enclosure, effects of various constant and variable properties of the nanofluid. Appl Math Model 40:815–831
 32.
Heidary H, Hosseini R, Pirmohammadi M, Kermani MJ (2015) Numerical study of magnetic field effect on nanofluid forced convection in a channel. J Magn Magn Mater 374:11–17
 33.
Mahian O, Kianifar A, Kleinstreuer C, A.N. Moh’d A, Pop I, Sahin AZ, Wongwises S (2013) A review of entropy generation in nanofluid flow, International Journal of Heat and Mass Transfer 65:514–532
 34.
Oueslati F, BenBeya B, Lili T (2013) Doublediffusive natural convection and entropy generation in an enclosure of aspect ratio 4 with partial vertical heating and salting sources. Alex Eng J 52:605–625
 35.
Li J (2009) Computational Analysis of Nanofluid Flow in Microchannels with Applications to Microheat Sinks and BioMEMS, Ph.D. dissertation, MAE Department, NCSU, Raleigh, NC
 36.
Koo J, Kleinstreuer C (2005) Laminar nanofluid flow in microheatsinks. Int J Heat Mass Transf 48:2652–2661
 37.
Brinkman HC (1952) The viscosity of concentrated suspensions and solution. J Chem Phys 20:571–581
Acknowledgments
The authors wish to thank the Energy Research Institute of the University of Kashan for their support regarding this research (Grant No. 55806).
Author information
Affiliations
Corresponding author
Additional information
Technical Editor: Francis HR Franca.
Rights and permissions
About this article
Cite this article
Abbaszadeh, M., Ababaei, A., Abbasian Arani, A.A. et al. MHD forced convection and entropy generation of CuOwater nanofluid in a microchannel considering slip velocity and temperature jump. J Braz. Soc. Mech. Sci. Eng. 39, 775–790 (2017). https://doi.org/10.1007/s4043001605787
Received:
Accepted:
Published:
Issue Date:
Keywords
 CuOwater nanofluid
 Entropy generation
 Parallel plate microchannel
 Slip velocity and temperature jump
 Magnetic field
 KKL Brownian motion model