Copyright: © 2019
|Pages: 71

DOI: 10.4018/978-1-5225-7595-5.ch003

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TopThe physical model along with the important geometrical parameters and the mesh of the enclosure used in the present CVFEM program are shown in Figure 1 (Sheikholeslami, 2014). The inner circular wall is maintained at constant hot temperature , the two horizontal walls are thermally isolated and other walls are maintained at constant cold temperature (>). To reach the shape of inner circular and outer rectangular boundary which consists of the right and top walls, a supper elliptic function can be used as follows;

When and the geometry becomes a circle. As increases from 1 the geometry would approach a rectangle for and square for . In this study we assumed that .

For the expression of the magnetic field strength it can be considered that two magnetic sources at points and . The components of the magnetic field intensity () and the magnetic field strength () can be considered as (Sheikholeslami and Rashidi, 2015):

(a) Geometry and the boundary conditions; (b) the mesh of enclosure considered in this work.

The flow is two-dimensional, laminar and incompressible. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. Using the Boussinesq approximation, the governing equations of heat transfer and fluid flow for nanofluid can be obtained as follows:

The terms and in (6) and (7), respectively, represent the components of magnetic force, per unit volume, and depend on the existence of the magnetic gradient on the corresponding *x* and *y* directions. These two terms are well known from FHD which is the so-called the Kelvin force. The term s and appearing in (6) and (7), respectively, represent the Lorentz force per unit volume towards the *x* and *y* directions and arise due to the electrical conductivity of the fluid. These two terms are known in MHD. The principles of MHD and FHD are combined in the mathematical model presented in Sheikholeslami and Ganji (2014) and the above mentioned terms arise together in the governing equations (6) and (7). The term in *Equation* (8) represents the thermal power per unit volume due to the magneto caloric effect. Also the term in (8) represents the Joule heating. For the variation of the magnetization , with the magnetic field intensity and temperature , the following relation is used Loukopoulos and Tzirtzilakis (2004):

In the above equations, is the magnetic permeability of vacuum , is the magnetic field strength, is the magnetic induction and the bar above the quantities denotes that they are dimensional. The effective density () and heat capacitance of the nanofluid are defined as:

The dynamic viscosity of the nanofluids is given is

The effective thermal conductivity of the nanofluid can be approximated by the Maxwell–Garnetts (MG) model as (Sheikholeslami, 2014):

By introducing the following non-dimensional variables:

Thermo physical properties of water and nanoparticles

(Sheikholeslami and Ganji, 2014)

The stream function and vorticity are defined as:

The stream function satisfies the continuity *Equation* (18). The vorticity equation is obtained by eliminating the pressure between the two momentum equations, i.e. by taking y-derivative of *Equation* (20) and subtracting from it the x-derivative of *Equation* (19).

The boundary conditions as shown in Figure 1 are:

on the inner circular boundary on the outer wall boundary on all solid boundariesThe values of vorticity on the boundary of the enclosure can be obtained using the stream function formulation and the known velocity conditions during the iterative solution procedure.

The local Nusselt number of the nanofluid along the hot wall can be expressed as:

To estimate the enhancement of heat transfer between the case of and the pure fluid (base fluid) case, the heat transfer enhancement is defined as:

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