Copyright: © 2019
|Pages: 63

DOI: 10.4018/978-1-5225-7595-5.ch005

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TopA steady, constant property, two-dimensional flow of an incompressible nanofluid through a homogenous porous medium with permeability of *K*, over a stretching surface with linear velocity distribution, i.e., is assumed (Figure 1) (Sheikholeslami, Ellahi, Ashorynejad, Domairry and Hayat, 2014).

The fluid is a water based nanofluid containing different types of nanoparticles: Cu, Al_{2}O_{3}, Ag and TiO_{2}. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. The thermo physical properties of the nanofluid are given in Table 1.

The transport properties of the medium can be considered independent from the temperature when the temperature difference between wall and ambient is not significant (Starov and Zhdanov, 2001). The origin is kept fixed while the wall is stretching and the *y*-axis is perpendicular to the surface. Using the above-mentioned assumptions, the continuity equation is:

The effective density, the effective dynamic viscosity , the heat capacitance and the thermal conductivity of the nanofluid are given as:

Here, is the solid volume fraction.

The hydrodynamic boundary conditions are:

The following thermal boundary conditions are considered:

The power-law temperature and heat flux distribution, described in Equations (9) and (10), resent a wider range of thermal boundary conditions including isoflux and isothermal cases. For example, by setting *n* equal to zero, Equations (9) and (10) yield isothermal and isoflux, respectively.

Second law of thermodynamics analysis of porous media is found to be more complicated compared to the clear fluid counterpart due to increased number of variables in governing equations (Nield and Kuznetsov, 2005). In the non-Darcian regime, there are three alternative models for the fluid friction term which are the clear-fluid compatible model, the Darcy model, and the Nield model or the power of drag model. Following the entropy generation function introduced by Nield and Kuznetsov (2005) the volumetric entropy generation rate, , reads:

Using boundary layer approximations, Equation (11) reduces to:

Using the stream function, , the continuity equation is satisfied:

The hydrodynamic boundary layer thickness scales with . This can be found through a scale analysis between the first and the second terms on the right hand side of Equation (2), i.e., the viscous and the Darcy terms. Therefore, instead of the other similarity parameters reported in the literature, the following dimensionless similarity parameter is defined

The *u*-velocity is assumed to be correlated to , a dimensionless similarity function as:

Substituting from and into Equations (2) and (4), one will find the following differential equation for the -momentum equation:

is the injection parameter. Positive/negative values of show suction/injection into/from the porous surface, respectively.

The wall shear stress is the driving force that drags fluid flow along the stretching wall. The wall shear stress term can then be found, in terms of the similarity function, as:

Introducing a similarity function, , as:

Which are subjected to the following boundary conditions:

Power-law temperatureFor power-law temperature and heat flux boundary conditions, respectively. Employing the definition of convective heat transfer coefficient, the local Nusselt numbers, become:

Finally, the local volumetric entropy generation rate for the above cases, respectively, reads:

The last term () is the contribution of fluid friction irreversibility to the total entropy generation. Not only the wall and fluid layer shear stress but also the momentum exchange at the solid boundaries (pore level) contributes to . In terms of the primitive variables, and become

One can also define the Bejan number, , as

The Bejan number shows the ratio of entropy generation due to heat transfer irreversibility to the total entropy generation so that a *Be* value more/less than 0.5 shows that the contribution of to the total entropy generation is higher/less than that of . The limiting value of shows that the only active entropy generation mechanism is *HTI* while represents no contribution to the total entropy production.

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