Double-diffusive natural convection and entropy generation of Carreau fluid in a heated enclosure with an inner circular cold cylinder (Part I: Heat and mass transfer)

Highlights

• The increase in power-law index provokes heat and mass transfer to drop gradually.

• The increase in Rayleigh number enhances heat transfer.

• The increase in Eckert number affects heat and mass transfer.

• The rise of Carreau number causes heat and mass transfer to drop gradually.

Abstract

In this paper, double-diffusive natural convection, studying Soret and Dufour effects and viscous dissipation in a heated enclosure with an inner cold cylinder filled with non-Newtonian Carreau fluid has been simulated by Finite Difference Lattice Boltzmann Method (FDLBM). This study has been conducted for certain pertinent parameters of Rayleigh number (Ra = ${10}^4$ and ${10}^5$), Carreau number (Cu = 1, 10, and 20), Lewis number (Le = 2.5, 5 and 10), Dufour parameter ($D_f$ = 0, 1, and 5), Soret parameter ($S_r$ = 0, 1, and 5), Eckert number (Ec = 0, 1, and 10), the Buoyancy ratio (N = −1, 0.1, 1), the radius of the inner cylinder ($R_d$ = 0.1 L, 0.2 L, 0.3 L, and 0.4 L), the horizontal distance of the circular cylinder from the center of the enclosure ($\mathrm{\Omega }$ = −0.2 L, 0 and 0.2 L), the vertical distance of the circular cylinder from the center of the enclosure ($\delta$ = −0.2 L, 0 and 0.2 L). Results indicate that the increase in Rayleigh number enhances heat transfer for various studied parameters. The increase in power-law index provokes heat and mass transfer to drop gradually. The increase in the Lewis number declines the mass transfer considerably while causes heat transfer to drop marginally. The heat transfer increases with the rise of the Dufour parameter and the mass transfer enhances as the Soret parameter increases for different Rayleigh numbers. The augmentation of the buoyancy ratio number enhances heat and mass transfer. The increase in Eckert number affects heat and mass transfer; especially, at Ra = ${10}^5$. The rise of Carreau number causes heat and mass transfer to drop gradually. The movement of the center of the cylinder from the bottom to the top side of the enclosure vertically ($\delta$ = −0.2 L, 0 and 0.2 L) decreases heat and mass transfer significantly while the effect of power-law index drops. The increase in the radius of the cylinder enhances heat and mass transfer. The alteration of the center of the cylinder horizontally ($\mathrm{\Omega }$) to the left and right sides enhance heat and mass transfer although this augmentation is different in various power-law indexes.

Introduction

Analysis of natural convection in enclosures has been extensively conducted using different numerical techniques and experiments because of its wide applications and interest in engineering e.g. nuclear energy, double pane windows, heating and cooling of buildings, solar collectors, electronic cooling, and so on. The wide range of studies into this topic has led to the natural convection in a cavity to become a common benchmark among researchers in the field of CFD (Computational Fluid Dynamics). It consists of a two-dimensional cavity and the temperature of the heated section on the left is maintained at a higher temperature and the right wall is held at a lower temperature. The horizontal walls are considered to be adiabatic and the density variation is approximated by the standard Boussinesq model. The natural convection flow of a Newtonian fluid has been studied numerically by de Vahl Davis [1], Quere and de Roquefort [2], Quere [3]. Many studies have conducted the effect of the presence of isothermal bodies inside the enclosure on the natural convection phenomena and focused on the diverse body shapes, e.g. circular, square and triangular cylinders. Kim et al. [4] carried out numerical calculations for natural convection induced by a temperature difference between a cold outer square enclosure and a hot inner circular cylinder. They investigated the effect of the inner cylinder location on the heat transfer and fluid flow. Further, the location of the inner circular cylinder was changed vertically along the center-line of square enclosure. Mehrizi et al. [5] investigated a numerical study for steady-state, laminar natural convection in a horizontal annulus between a heated triangular inner cylinder and cold elliptical outer cylinder, using lattice Boltzmann method. Both inner and outer surfaces were maintained at the constant temperature and air was the working fluid. Park et al. [6] studied the natural convection induced by a temperature difference between a cold outer square enclosure and two hot inner circular cylinders. A two-dimensional solution for natural convection in an enclosure with inner cylinders was obtained using an accurate and efficient immersed boundary method. The immersed boundary method based on the finite volume method was used to handle inner cylinders located at different vertical centerline positions of the enclosure for different Rayleigh numbers. Mehrizi and Mohamad [7] utilized Lattice Boltzmann method to simulate steady-state, laminar, free convection in two-dimensional annuli between a heated triangular inner cylinder and elliptical outer cylinder. The study was performed for different inclination angles of inner triangular and outer elliptical cylinders. Mun et al. [8] conducted two-dimensional numerical simulations to investigate the natural convection heat transfer induced by the temperature difference between cold walls of the tilted square enclosure and a hot inner circular cylinder for different prandtl numbers. Seo et al. [9] conducted two-dimensional numerical simulations for the natural convection phenomena in a cold square enclosure with four hot inner circular cylinders. The immersed boundary method (IBM) was used to capture the virtual wall boundary of the four inner cylinders based on the finite volume method (FVM). Zhang et al. [10] investigated a numerical study for steady-state natural convection in a cold outer square enclosure containing a hot inner elliptic cylinder using the variational multiscale element free Galerkin method (VMEFG). In the cited studies, the fluids have been assumed to be Newtonian fluids while most materials demonstrates non-Newtonian behavior. Natural convection of non-Newtonian power-law fluids and Bingham fluids in an enclosure recently have been studied by some researchers [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. However, natural convection of Carreau fluids in an enclosure have not been considered thus far. Carreau fluid is a special sub-class of non-Newtonian fluids in which follows the Carreau model [22]. This model was introduced in 1972 and has been applied extensively up to date. Carreau models have been employed to simulate various chemicals, molten plastics, slurries, paints, blood, etc. Some limited isothermal and non-isotermal problems of Carreau fluids have been studied. Shamekhi and Sadeghy [23] analyzed Lid-driven cavity flow of a purely-viscous non-Newtonian fluid obeying Carreau-Yasuda rheological model numerically using the PIM meshfree method combined with the Characteristic-Based Split-A algorithm. Results were reported for the velocity and pressure profiles at Reynolds numbers as high as 1000 for a non-Newtonian fluid obeying Carreau-Yasuda rheological model. Bouteraa et al. [24] performed a linear and weakly nonlinear analysis of convection in a layer of shear-thinning fluids between two horizontal plates heated from below. The shear-thinning behavior of the fluid was described by the Carreau model. Shahsavari and McKinley [25] studied The flow of generalized Newtonian fluids with a rate-dependent viscosity through fibrous media with a focus on developing relationships for evaluating the effective fluid mobility. They conducted a numerical solution of the Cauchy momentum equation with the Carreau or power-law constitutive equations for pressure-driven flow in a fiber bed consisting of a periodic array of cylindrical fibers. Pantokratoras [26] considered the flow of a non-Newtonian, Carreau fluid, directed normally to a horizontal, stationary, circular cylinder. The problem was investigated numerically using the commercial code ANSYS FLUENT with a very large calculation domain in order that the flow could be considered unbounded.

Lattice Boltzmann method (LBM) has been demonstrated to be a very effective mesoscopic numerical method to model a broad variety of complex fluid flow phenomena [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42]. This is because the main equation of the LBM is hyperbolic and can be solved locally, explicitly, and efficiently on parallel computers. However, the specific relation between the relaxation time and the viscosity has caused LBM not to have the considerable success in non-Newtonian fluid especially on energy equations. In this connection, Fu et al. [43], [44] proposed a new equation for the equilibrium distribution function, modifying the LB model. Here, this equilibrium distribution function is altered in different directions and nodes while the relaxation time is fixed. Independency of the method to the relaxation time in contrast with common LBM provokes the method to solve different non-Newtonian fluid energy equations successfully as the method protects the positive points of LBM simultaneously. In addition, the validation of the method and its mesh independency demonstrates that is more capable than conventional LBM. Huilgol and Kefayati [45] derived the three dimensional equations of continuum mechanics for this method and demonstrated that the theoretical development can be applied to all fluids, whether they be Newtonian, or power law fluids, or viscoelastic and viscoplastic fluids. Following the study, Huilgol and Kefayati [46] developed this method for the cartesian, cylindrical and spherical coordinates. Kefayati [47] simulated double-diffusive natural convection with Soret and Dufour effects in a square cavity filled with non-Newtonian power-law fluid by FDLBM while entropy generations through fluid friction, heat transfer, and mass transfer were analysed. Kefayati [48], [49] analysed double diffusive natural convection and entropy generation of non-Newtonian power-law fluids in an inclined porous cavity in the presence of Soret and Dufour parameters by FDLBM. Kefayati and Huilgol [50] conducted a two-dimensional simulation of steady mixed convection in a square enclosure with differentially heated sidewalls when the enclosure is filled with a Bingham fluid, using FDLBM. The problem was solved by the Bingham model without any regularisations and also by applying the regularised Papanatasiou model. Kefayati [51] simulated double-diffusive natural convection, studying Soret and Dufour effects and viscous dissipation in a square cavity filled with Bingham fluid by FDLBM. In addition, entropy generations through fluid friction, heat transfer, and mass transfer were studied. The problem was solved by applying the regularised Papanastasiou model.

The main aim of this study is to simulate double diffusive natural convection of Carreau fluid in a heated enclosure with an inner cold cylinder. The innovation of this paper is studying heat and mass transfer in the presence of Soret and Dufour and the viscous dissipation effect on Carreau fluid for the first time. An innovative method based on LBM has been employed to study the problem numerically. Moreover, it is endeavored to express the effects of different parameters on heat and mass transfer. The obtained results are validated with previous numerical investigations and the effects of the main parameters (Rayleigh number, Lewis number, buoyancy ratio number, Eckert number, Carreau number, Soret parameter, and Dufour parameter) are researched.