Heat and Mass Transfer

, Volume 53, Issue 6, pp 2061–2072 | Cite as

Two-phase nanofluid condensation and heat transfer modeling using least square method (LSM) for industrial applications

  • M. Hatami
  • S. Mosayebidorcheh
  • D. Jing


In this paper, two-phase Nanofluid condensation and heat transfer analysis over a vertical plate under gravity and between two parallel plates under magnetic force are investigated respectively using Least Square Method (LSM) and numerical method. After presenting the governing equations and solving them by LSM, the accuracy of results is examined by fourth order Runge–Kutta numerical method. Modeling results show that the condensate film thickness after condensation is reduced and therefore, the rate of heat transfer is enhanced by the addition of nanoparticles to the regular fluid. Effect of different nanoparticles and constant numbers on the temperature/velocity/concentration profiles as well as Nusselt number and boundary layer thickness, are also investigated. For instance, it was found that TiO2 and Ag have maximum boundary layer thicknesses and Nusselt number, respectively. By considering the magnetic field effect, it is also found that nanoparticles concentration can be controlled by changing the Hartmann number which, in turn, leads to different condensation and heat transfer properties.


Heat Transfer Nusselt Number Little Square Method Nanoparticles Concentration Hartmann Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Ratio of thermophoretic effects to Brownian diffusion effects of nanofluid


Concentration of nanofluid


Specific heat

\(D_{T} ,\,D_{B}\)

Parameters in the Eqs. (810)

\(F\left( \varphi \right)\)

Function of nanoparticle volume fraction


Gravity acceleration


Thermal conductivity


Thermal conductivity of nanofluid


Thermal conductivity of pure fluid


Thermal conductivity of nanoparticle


Rotation parameter


Magnetic parameter


Brownian motion parameter


Thermophoresis parameter


Nusselt number




Prandtl number

Greek symbols


Condensate film thickness


Nanoparticle volume fraction

\(\eta ,\,\xi\)

Dimensionless distance


Viscosity of nanofluid


Dimensionless temperature




Density of nanofluid


Density of pure fluid


Density of nanoparticle



The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Nos. 51422604, 21276206) and the National 863 Program of China (No. 2013AA050402). This work was also supported by the China Fundamental Research Funds for the Central Universities.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.International Research Center for Renewable Energy, State Key Laboratory of Multiphase Flow in Power EngineeringXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Mechanical EngineeringEsfarayen University of TechnologyEsfarayenIran
  3. 3.Young Researchers and Elite Club, Najafabad BranchIslamic Azad UniversityNajafabadIran

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