Elsevier

Journal of Molecular Liquids

Volume 188, December 2013, Pages 155-161
Journal of Molecular Liquids

Heat transfer and flow analysis for SA-TiO2 non-Newtonian nanofluid passing through the porous media between two coaxial cylinders

https://doi.org/10.1016/j.molliq.2013.10.009Get rights and content

Highlights

  • Non-Newtonian nanofluid flow in porous medium between coaxial cylinders is modeled.

  • Sodium alginate (SA) and titanium dioxide (TiO2) nanoparticles are considered.

  • Nanofluid viscosity is considered as a function of temperature by Reynolds model.

  • Least Square Method (LSM), Collocation Method (CM) and numerical method are used.

  • Increasing the thermophoresis parameter makes an increase in nanofluid temperature.

Abstract

In this paper, heat transfer and flow analysis for a non-Newtonian nanofluid flow in the porous medium between two coaxial cylinders are investigated analytically and numerically. Sodium alginate (SA) is considered as the base non-Newtonian fluid and titanium dioxide (TiO2) nanoparticles are added to it. The viscosity of nanofluid is considered as a function of temperature by Reynolds model. Least Square Method (LSM), Collocation Method (CM) and fourth-order Runge–Kutta numerical method (NUM) are used to solve the present problem. The influences of the some physical parameters such as Brownian motion and thermophoresis parameters on non-dimensional velocity and temperature profiles are considered. The results show that increasing the thermophoresis parameter (Nt) caused an increase in temperature values in whole domain but it makes an increase in nanoparticle concentration near the inner cylinder wall.

Introduction

Non-Newtonian nanofluids are widely encountered in many industrial and technology applications, for example, melts of polymers, biological solutions, paints, asphalts and glues etc. Nanofluids appear to have the potential to significantly increase heat transfer rates in a variety of areas such as industrial cooling applications, nuclear reactors, transportation industry, micro-electromechanical systems, heat exchangers, chemical catalytic reactors, fiber and granular insulation, packed beds, petroleum reservoirs and nuclear waste repositories and biomedical applications [1].

In the review of its importance, the flow of Newtonian and non-Newtonian fluids through two infinite parallel vertical plates has been investigated by numerous authors. Ellahi et al. [1] used Reynolds and Vogel's model for viscosity of non-Newtonian nanofluid in a porous medium, also Ellahi and co-workers [2] studied the effect of magneto-hydrodynamic (MHD) on non-Newtonian nanofluid flow between two coaxial cylinders. Pawar et al. [3] carried out an experimental study on isothermal steady state and non-isothermal unsteady state conditions in helical coils for Newtonian and non-Newtonian fluids. They considered water, glycerol–water mixture as Newtonian fluids and dilute aqueous polymer solutions of sodium carboxymethyl cellulose (SCMC), sodium alginate (SA) as non-Newtonian fluids in their study. Investigation on non-Newtonian fluid flow in microchannels and flow characteristics of deionized water and the PAM solution over a wide range of Reynolds numbers has been done by Tang et al. [4].Yoshino et al. [5] presented a new numerical method for incompressible non-Newtonian fluid flows based on the lattice Boltzmann method (LBM). Their simulations indicate that the method can be useful for practical non-Newtonian fluid flows, such as shear-thickening (dilatant) and shear-thinning (pseudoplastic) fluid flows. Xu and Liao [6] studied the unsteady magnetohydrodynamic (MHD) viscous flows of non-Newtonian fluids caused by an impulsively stretching plate by means of homotopy analysis method to investigate the effect of integral power-law index of these non-Newtonian fluids on the velocity. In another experimental work Hojjat et al. [7] prepared three different types of nanofluids by dispersing Al2O3, TiO2 and CuO nanoparticles in a 0.5 wt% of carboxymethyl cellulose (CMC) aqueous solution. They measured thermal conductivity of the base fluid and nanofluids with various nanoparticle loadings at different temperatures. Their results show that the thermal conductivity of nanofluids is higher than the one of the base fluid and the increase in the thermal conductivity varies exponentially with the nanoparticle concentration [7].

There are some simple and accurate approximation techniques for solving differential equations called the Weighted Residuals Methods (WRMs). Collocation (CM), Galerkin (GM) and Least Square (LSM) are examples of the WRMs. Stern and Rasmussen [8] used collocation method for solving a third order linear differential equation. Vaferi et al. [9] studied the feasibility of applying of Orthogonal Collocation method to solve diffusivity equation in the radial transient flow system. Hendi and Albugami [10] used the Collocation and Galerkin methods for solving Fredholm–Volterra integral equation. Recently least square method is introduced by Aziz and Bouaziz [11] and is applied for prediction of the performance of a longitudinal fin [12]. They found that least squares method is simple compared with other analytical methods. Shaoqin and Huoyuan [13] developed and analyzed least-squares approximations for the incompressible magnetohydrodynamic (MHD) equations. Recently Hatami et al. [14] and Sheikholeslami et al. [15] applied LSM and CM on fin performance and nanofluid in porous channel respectively. Ellahi [16] used homotopy analysis method (HAM) analytical solution of MHD non-Newtonian nanofluid in a pipe. In another study, Rashad et al. [17] investigated the natural convection of non-Newtonian nanofluid around a vertical permeable cone.

The main aim of this paper is to investigate the problem of heat and fluid flow analysis for non-Newtonian nanofluid in porous media between two coaxial cylinders by least square, collocation and numerical methods. SA-TiO2 is considered as the non-Newtonian nanofluid which its viscosity is considered as a function of temperature according to Reynolds model. Introducing the governing equation and their application and solving them by two high efficient analytical techniques for the first time show the novelty and advantage of present study. Also the effects of some parameters such as Brownian motion and thermophoresis parameters on velocity and temperature profiles are investigated.

Section snippets

Description of the problem

Consider a steady, incompressible and non-Newtonian nanofluid in a porous media between two coaxial cylinders as shown in Fig. 1-a. Clearly a viscous fluid is governed by continuity and Navier–Stokes equations and when the fluid is considered to be incompressible, the conservation of momentum, total mass, thermal energy, and nanoparticles, are as follows [1],ρfVt+V.V=divTμφk1+λrtV+ϕρp+1ϕρf1βTθθwgρcfθt+V.θ=k2θ+ρcpDbϕ.θ+DTθwθ.θϕt+V.ϕ=Db2ϕ+DTθw2θ

Considering the boundary

Application of the problem in chemical science

As mentioned before, dilute aqueous polymer solutions of sodium carboxymethyl cellulose (SCMC) and sodium alginate (SA) as non-Newtonian fluids are used in experimental studies in chemical science in [3]. Also in another experiment research, Al2O3, TiO2 and CuO nanoparticles are added to these non-Newtonian fluids for creating nanofluids [7]. In this paper, SA with TiO2 nanoparticles are considered for non-Newtonian nanofluid which their properties are presented via Table 1. In this study

Applied methods

Least square (LSM) and Collocation methods (CM) which are two samples of the approximation techniques called weighted residual methods (WRMs) were firstly introduced by Ozisik [19] for solving differential equations in heat transfer problems. Many advantages of LSM and CM compared to other analytical and numerical methods make them more valuable and motivate researchers to use them for solving heat transfer problems. Some of these advantages are listed below [20]:

  • a)

    WRMs solve the equations

Results and discussion

In present study, the base fluid is considered sodium alginate (SA) as a non-Newtonian fluid and TiO2 nanoparticles are added to it and governing equations for nanofluid flow in porous medium between two coaxial cylinders are solved by LSM, CM and NUM. For solving Eqs. (13), (14), (15) by WRMs, because trial functions must satisfy the boundary conditions in Eq. (16), so each statement in v(r), θ(r) and ϕ(r) should contain (2  r) or (4  r2) to satisfy boundary condition in r = 2. In the other hand,

Conclusion

In this paper, two analytical approaches called Least Square Method (LSM) and Collocation Method (CM) along a numerical method have been successfully applied to find the most accurate solution of the heat transfer of SA-TiO2 non-Newtonian nanofluid flow in the porous area between two coaxial cylinders. As a main outcome from the present study, it is observed that the results of LSM are more accurate than CM and they are in excellent agreement with numerical ones, so LSM can be used for finding

References (26)

  • R. Ellahi et al.

    Math. Comput. Model.

    (2012)
  • S.S. Pawar et al.

    Exp. Thermal Fluid Sci.

    (2013)
  • G.H. Tang et al.

    J. Non-Newtonian Fluid Mech.

    (2012)
  • M. Yoshino et al.

    J. Non-Newtonian Fluid Mech.

    (2007)
  • H. Xu et al.

    J. Non-Newtonian Fluid Mech.

    (2005)
  • M. Hojjat et al.

    Int. J. Heat Mass Transfer

    (2011)
  • R.H. Stern et al.

    Comput. Biol. Med.

    (1996)
  • F.A. Hendi et al.

    J. King Saud Univ. (Sci.)

    (2010)
  • A. Aziz et al.

    Energy Convers. Manag.

    (2011)
  • G. Shaoqin et al.

    Acta Math. Sci.

    (2008)
  • M. Hatami et al.

    Energy Convers. Manag.

    (2013)
  • M. Sheikholeslami et al.

    Powder Technol.

    (2013)
  • R. Ellahi

    Appl. Math. Model.

    (2013)
  • Cited by (142)

    View all citing articles on Scopus
    View full text