Copyright: © 2019
|Pages: 113

DOI: 10.4018/978-1-5225-7595-5.ch002

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TopThe considered physical geometry with related parameters and coordinates are shown in Figure 1(a). A rectangular body with height *t *and width is placed in the center of the enclosure, is supposed to be isothermal at higher temperature than two vertical isothermal walls while the top and bottom walls are insulated. Also, it is also assumed that the uniform magnetic field () of constant magnitude is applied, where and are unit vectors in the Cartesian coordinate system. The orientation of the magnetic field form an angle with horizontal axis such that . The electric current *J *and the electromagnetic force *F* are defined by and , respectively. In this section, is equal to zero (Sheikholeslami & Ganji, 2015).

(a) Geometry of the problem; (b) Discrete velocity set of two-dimensional nine-velocity (D_{2}Q_{9}) model

One of the novel computational fluid dynamics (CFD) methods which is solved Boltzmann equation to simulate the flow instead of solving the Navier–Stokes equations is called Lattice Boltzmann methods (LBM) (or Thermal Lattice Boltzmann methods (TLBM)). LBM has several advantages such as simple calculation procedure and efficient implementation for parallel computation, over other conventional CFD methods, because of its particulate nature and local dynamics. The thermal LB model utilizes two distribution functions, *f* and *g*, for the flow and temperature fields, respectively. It uses modeling of movement of fluid particles to capture macroscopic fluid quantities such as velocity, pressure and temperature. In this approach, the fluid domain discretized to uniform Cartesian cells. Each cell holds a fixed number of distribution functions, which represent the number of fluid particles moving in these discrete directions. The D_{2}Q_{9} model was used and values of for (for the static particle), for and for are assigned in this model (Figure 1(b)). The density and distribution functions i.e. the *f* and *g*, are calculated by solving the lattice Boltzmann equation, which is a special discretization of the kinetic Boltzmann equation. After introducing BGK approximation, the general form of lattice Boltzmann equation with external force is as follow:

For the flow field:

For the temperature field:

In order to incorporate buoyancy forces and magnetic forces in the model, the force term in the Equation (2) need to calculate as below (Sheikholeslami & Ganji, 2015):

For natural convection, the Boussinesq approximation is applied and radiation heat transfer is negligible. To ensure that the code works in near incompressible regime, the characteristic velocity of the flow for natural regime must be small compared with the fluid speed of sound. In the present study, the characteristic velocity selected as 0.1 of sound speed.

Finally, macroscopic variables calculate with the following formula:

According to Bejan (1982), one can find the volumetric entropy generation rate as

In terms of the primitive variables, *HTI* and *FFI* become

One can also define the Bejan number,, as

Note that a value more/less than 0.5 shows that the contribution of *HTI* to the total entropy generation is higher/lower than that of *FFI*. The limiting value of shows that the only active entropy generation mechanism is *HTI *while represents no HTI contribution.

The dimensionless form of entropy generation rate, Ns, is defined as

The dimensionless viscous dissipation function, addressed in Equation (11), takes the following form

Here, Ge is the Gebhart number which is defined as (Sheikholeslami, Ashorynejad, & Rana, 2016).

Average is denoted by, where the angle brackets show an average taken over the area, as:

Selecting the fluid, trapped between the heated plate and the cavity, as the thermodynamic system, one observes that the amount of heat entered through the heated plate is equal to the one transferred to the surroundings via the isothermal walls. Moreover, one notes that the total volumetric entropy generation rate is obtainable as

Applying perturbation techniques for small values of , say , one has

The dimensionless entropy generation number can be obtained as

In order to simulate the nanofluid by the lattice Boltzmann method, because of the interparticle potentials and other forces on the nanoparticles, the nanofluid behaves differently from the pure liquid from the mesoscopic point of view and is of higher efficiency in energy transport as well as better stabilization than the common solid-liquid mixture. For modeling the nanofluid because of changing in the fluid thermal conductivity, density, heat capacitance and thermal expansion, some of the governed equations should change. The effective density , the effective heat capacity , thermal expansion and electrical conductivity of the nanofluid are defined as:

and are obtained according to Koo–Kleinstreuer–Li (KKL) model (Sheikholeslami & Kandelousi, 2014):

In order to compare total heat transfer rate, Nusselt number is used. The average Nusselt numbers are defined as follows:

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