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Comment on the paper “Solar radiation assisted mixed convection MHD flow of nanofluids over an inclined transparent plate embedded in a porous medium, by Meisam Habibi Matin and Reza Hosseini, Journal of Mechanical Science and Technology, 28 (2014) 3885-3893”

Asterios Pantokratoras

School of Engineering, DemocritusUniversity of Thrace,

67100 Xanthi – Greece

e-mail:

The above paper investigates analytically and numerically the magneto-hydrodynamic (MHD) mixed convection flow of nanofluidover a nonlinear stretching inclined transparent plate embedded in a porous medium under the solar radiation. The two-dimensional governingequations are obtained considering the dominant effect of boundary layer and also in presence of the effects of viscous dissipationand variable magnetic field. These equations are transformed by the similarity transformation to two coupled nonlinear transformed equationsand then solved using a numerical implicit method called Keller-Box. The effect of various parameters such as nanofluid volumefraction, magnetic parameter, porosity, effective extinction coefficient of porous medium, solar radiation flux, plate inclination angle,diameter of porous medium solid particles and dimensionless Eckert, Richardson and Prandtl numbers have been studied on the dimensionlesstemperature and velocity profiles. Also the results are presented based on Nusselt number and Skin friction coefficient.

The partial differential equations of this problem, using the same symbols byMatin and Hosseini (2014) are

(1)

(2)

(3)

where u and v are the velocity components in the x and y-directions,μnf is the dynamic viscosityofthenanofluid, ρnf is the nanofluid density, K(ε) is the porous medium permeability, C(ε) is the porous medium inertia coefficient, σ is the electrical conductivity, B(x) is the strength of the magnetic field, βnfis the thermal expansioncoefficientof the nanofluid, g is the gravity acceleration, T is the nanofluid temperature, β is the plate inclination angle, kef is the thermalconductivityofthenanofluid, (ρCp)nf is theheatcapacitanceofthenanofluid and qrad is the radiation flux.

The above energy equation (3) is not correct for the following reason. The viscous dissipation term corresponds to a fluid without porous medium. In the flow inside a porous medium an extra term is needed ( Pantokratoras, 2009a, Barletta, 2015).

The boundary conditions are:

, , at y=0 (4)

, , as (5)

The radiation flux is given by the following equation

(6)

The transformed momentum equation is (equation 17 in Matin and Hosseini, 2014)

(7)

In equation (7) is the volume fraction, f is the stream function, ε is the porous medium porosity, dp is the nanoparticle diameter, Mn is the magnetic parameter and is the Richardson number. In equation (7) all quantities and the ratios , are non-dimensional except of dp and which are dimensional. It is reminded here that in a non-dimensional equation all parts should be non-dimensional. Otherwise the equation is invalid. The units of dp are meters (see nomenclature). Let us calculate the units of the Richardson number. It is given in equation (20) of Matin and Hosseini(2014) and is defined as

(8)

The units of gravity acceleration are

The units of the thermal expansion coefficient are Kelvin-1 . Therefore the units of theRichardson number are

This means that the Richardson number is dimensional and not dimensionless as the authors claim.

The thermophysical properties of water are given in table 3 of Matin and Hosseini (2014). However, the density, specific heat and thermal conductivity included in this table correspond to a temperature of 35 oC. At this temperature the water Prandtl number is approximately 4.80 (Jaluria and Torrance, 1986, page 351) whereas the results presented in the paper correspond to Pr=1. The Pr=1 does not correspond to water. In addition the value of thermal expansion coefficient given in table 3 of Matin and Hosseini (2014) is wrong. The correct value is βx10-5.

Two different symbols have been used in the paper for the radiation heat flux. The symbol qrad is included in the energy equation (3) and the symbol q'' in equation (6). However, both two quantitiesdisappeared from the rest of the paper. No subsequent equation or parameter exists including the radiation quantity.

Many times in the paper is mentioned that the problem is similar. However, on page 3888 it is mentioned that the presented results correspond to x=0.1. It is well known in Fluid Mechanics that in a similar problem the results do not depend on the longitudinal distance x, otherwise the problem is non-similar. And what means x=0.1?. 0.1 cm, 0.1 m or 0.1 km? There is no non-dimensional parameter x in the paper.

There is another problem in the paper. It is known in boundary layer theory that velocity and temperature profiles approach the ambient fluid conditions asymptotically and do not intersect the line which represents the boundary conditions. This requirement is expressed by the boundary conditions in equation (5) of the present paper which demands that velocity and temperature should reach the ambient values at infinity. However, most of velocity and temperature profiles presented in the work of Matin and Hosseini (2014) do not approach the ambient velocity and temperature asymptotically but intersect the horizontal axis at . This means that these profiles are truncated due to a small calculation domain used and are wrong.A much more calculation domain (greater than 5) is needed. This problem has been analyzed in detail by Pantokratoras (2009b).

Taking into account all the above the credibility of the results presented by Matin and Hosseini(2014) is doubtful.

References

  1. Meisam Habibi Matin and Reza Hosseini. Solar radiation assisted mixed convection MHD flow of nanofluids over an inclined transparent plate embedded in a porous medium, Journal of Mechanical Science and Technology, 28 (2014) 3885-3893.
  1. A. Pantokratoras (2009a). Comments on “Perturbation analysis of radiative effect on free convection flows in porous medium in the presence of pressure work and viscous dissipation, by A.M. Rashad, Communications in Nonlinear Science and Numerical Simulation, 2007”, Communications in NonlinearScience and Numerical Simulation,Vol. 14, pp. 345-346.
  1. A. Pantokratoras (2009b). A common error made in investigation of boundary layer flows, Applied Mathematical Modelling,Vol. 44, pp. 1187-1198.
  1. Y. Jaluria, K. E. Torrance (1986). Computational Heat Transfer, Hemisphere Publishing Corporation, Washington.
  1. A. Barletta (2015). On the thermal instability induced by viscous dissipation, International Journal of Thermal Science, Vol. 88, pp. 238-247.