Journal of Thermal Analysis and Calorimetry

, Volume 123, Issue 2, pp 1591–1599 | Cite as

A numerical study of heat transfer characteristics of CuO–water nanofluid by Euler–Lagrange approach



The two-phase Euler–Lagrange method is used here to study heat transfer characteristics of the CuO–water nanofluid in a straight tube under laminar flow regime. The comparison between two-phase and single-phase approaches shows that the Euler–Lagrange method presents more accurate results. The convective heat transfer coefficient increases with increment of particle concentration. Moreover, the amount of heat transfer enhancement increases along the tube length. The thermophoretic and Brownian forces affect the convective heat transfer and their effects become more prominent at greater distances from the tube inlet. In addition, the effects of these two forces on the convective heat transfer intensify at higher concentrations. The results obtained from the two-phase simulation reveal a higher particle concentration at the central regions as compared to regions near the wall. This can be one of the reasons for higher convective heat transfer coefficient obtained from the two-phase method in comparison with the single-phase approach. At low volume fraction, the nanofluid bulk temperature along the tube length obtained from single-phase and two-phase approaches does not differ considerably. However, at high volume fraction, the single-phase method is unable to predict the bulk temperature accurately. This can be attributed to the more intense interaction between the fluid and nanoparticles at higher concentrations.


Heat transfer Nanofluid Brownian Thermophoresis Particle distribution 

List of symbols


Brownian diffusion coefficient (m2 s−1)


Specific heat (J kg−1 K−1)


Cunningham correction factor







\( D \)

Tube diameter (m)


Particle diameter (m)

\( d_{\text {ij}} \)

Deformation tensor (s−1)

\( {\mathbf{F}} \)

Total force applied to particle (N kg−1)

\( {\mathbf{F}}_{\rm B} \)

Brownian force (N kg−1)

\( {\mathbf{F}}_{\rm D} \)

Drag force (N kg−1)

\( {\mathbf{F}}_{\rm L} \)

Lift force (N kg−1)

\( {\mathbf{F}}_{\rm T} \)

Thermophoretic force (N kg−1)

\( {\mathbf{F}}_{\rm V} \)

Virtual mass force (N kg−1)


Convective heat transfer coefficient (W m−2 K−1)


Thermal conductivity (W m−1 K−1)


Boltzmann constant (m2 kg s−2 K−1)


Knudsen number




Mass (kg)


Nusselt number


Pressure (Pa)


Prandtl number


Reynolds number


Spectral intensity basis


Spectral intensity

\( S_{{\rm p,e}} \)

Energy source term (W m−3)

\( {\mathbf{S}}_{\rm p,m} \)

Momentum source term (Pa m−1)


Time (s)


Temperature (K)


Axial direction

\( {\mathbf{v}} \)

Velocity (m s−1)

\( {\mathbf{v}}_{\rm T} \)

Thermophoretic velocity (m s−1)

Greek letters


Ratio of the nanolayer thickness to the original particle radius


Kronecker delta function


Zero-mean, unit-variance-independent Gaussian random number


Mean free path of the fluid (m)


Dynamic viscosity (kg m−1 s−1)


Kinematic viscosity (m2 s−1)


Density (kg m−3)


Particle volume fraction











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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.Mechanical Engineering Department, School of EnergyKermanshah University of TechnologyKermanshahIran

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