J. Cent. South Univ. (2016) 23: 983-990

DOI: 10.1007/s11771-016-3146-5

Kerosene-alumina nanofluid flow and heat transfer for cooling application

M. Mahmoodi¹, Sh. Kandelousi²

1. Department of Aerospace Engineering, Malek-Ashtar University of Technology, Karaj, Tehran, Iran;

2. Department of Mechanical Engineering, Babol University of Technology, Babol, Iran

 Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: Kerosene-alumina nanofluid flow and heat transfer in the presence of magnetic field are studied. The basic partial differential equations are reduced to ordinary differential equations which are solved semi analytically using differential transformation method. Velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number are determined analytically. The influence of pertinent parameters such as magnetic parameter, nanofluid volume fraction, viscosity parameter and Eckert number on the flow and heat transfer characteristics is discussed. Results indicate that skin friction coefficient decreases with increase of magnetic parameter, nanofluid volume fraction and viscosity parameter. Nusselt number increases with increase of magnetic parameter and nanofluid volume fraction while it decreases with increase of Eckert number and viscosity parameter.

Key words: magnetic field; nanofluid; heat transfer; differential transformation method

1 Introduction

Different types of cooling techniques are being used to protect the chamber and nozzle walls. In a typical cooling technique, like regenerative cooling, one of the propellants is passed through the coolant passage surrounding the wall of the nozzle. An innovative cooling system for semi-cryogenic engine needs to be explored by improving thermo-physical properties of kerosene, which can enhance the heat transfer capacity of kerosene. A recent way of improving the performance of these systems is to suspend metallic nanoparticles in the base fluid. SHEIKHOLESLAMI and ELLAHI [1] studied three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid. They found that thermal boundary layer thickness increases with increase of Lorentz forces. Force convective heat transfer of magnetic nanofluid in a lid driven semi annulus enclosure has been investigated by SHEIKHOLESLAMI et al [2]. Their results showed that Nusselt number has direct relationship with Reynolds number, nanoparticle volume fraction while it has reverse relationship with Hartmann number. Ferrofluid heat transfer treatment in the presence of variable magnetic field has been studied by SHEIKHOLESLAMI and RASHIDI [3]. They found that Nusselt number increases by considering magnetic field dependent viscosity. HATAMI et al [4] investigated the magnetohydrodynamic Jeffery-Hamel nanofluid flow in non-parallel walls. They found that skin friction coefficient is an increasing function of Reynolds number, opening angle and nanoparticle volume friction but decreasing function of Hartmann number. HATAMI et al [5] simulated the flow and heat transfer of nanofluid flow between two parallel plates. They showed that in order to reach the maximum Nusselt number, copper should be used as nanoparticle. Recently, several authors studied different passive methods to enhance the thermal performance [6-30].

Most phenomena in our world are essentially nonlinear and are described by nonlinear equations. Nonlinear differential equations usually arise from mathematical modeling of many physical systems. One of the semi-exact methods which doesn’t need small parameters is the differential transformation method. Therefore, same as the HAM and the HPM, the DTM can overcome the foregoing restrictions and limitations of perturbation methods. This method constructs an analytical solution in the form of a polynomial. It is different from the traditional higher-order Taylor series method. The Taylor series method is computationally expensive for large orders. The differential transform method is an alternative procedure for obtaining an analytic Taylor series solution of differential equations.  The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization and therefore, it is not affected by errors associated to discretization. The concept of DTM was first introduced by ZHOU [31], who solved linear and nonlinear problems in electrical circuits.  HATAMI et al [32] used multi-step differential transformation method in order to investigate motion of a spherical particle in plane couette fluid flow. Semi analytical methods have been applied in various problems [33-44].

In this work, kerosene-alumina nanofluid flow and heat transfer in channel in the presence of magnetic field are investigated. DTM is applied to find the approximate solutions of nonlinear differential equations. The effects of the magnetic parameter, viscosity parameter, Eckert number and nanofluid volume fraction on flow and heat transfer characteristics are investigated.

2 Problem statement

Consider steady flow of nanofluid between two horizontal parallel plates (see Fig. 1). The origin is located at the lower plate, and the plates are located at y=0 and y=h. The lower plate is being stretched by two equal opposite forces so that the position of the point (0,0,0) remains unchanged. A uniform magnetic flux with density of B₀ is acting along y-axis. Under these assumptions, the Navier–Stokes and energy equations are:

                                (1)

 (2)

            (3)

           (4)

where u, v and w denote the fluid velocity components along the x, y and z directions, respectively; p^* is the modified fluid pressure; T is temperature and the physical meanings of the other quantities are mentioned in the nomenclature. The corresponding boundary conditions of Eqs. (1)-(4) are:

            (5)

The effective density (ρ_nf), the effective dynamic viscosity (μ_nf), the effective heat capacity (ρC_p)_nf, the effective thermal conductivity (k_nf)  and the electrical conductivity (s_nf) of the nanofluid are defined as

                         (6)

Fig. 1 Physical model along with coordinate system

The thermo-physical properties of the nanofluid are given in Table 1.

Table 1 Thermo-physical properties of water and nanoparticles

The following non-dimensional variables are introduced:

   (7)

where the prime denotes differentiation with respect to η.

Therefore, the governing momentum and energy equations for this problem are given in dimensionless form by

          (8)

       (9)

The dimensionless quantities in these equations are:

              (10)

where R, M, Pr and Ec are viscosity parameter, magnetic parameter, Prandtl number and Eckert number, respectively.

The boundary conditions are:

                  (11)

The physical quantity of interest in this problem is the skin friction coefficient C_f along the stretching wall, which is defined as

                              (12)

The Nusselt number at the lower plate is defined as

                           (13)

3 Differential transform method (DTM)

3.1 Basic of DTM

Basic definitions and operations of differential transformation are introduced as follows. Differential transformation of the function f(η) is defined as

                       (14)

In Eq. (14), f(η) is the original function and F(k) is the transformed function which is called the T-function (it is also called the spectrum of the f(η) at η-η₀, in the k domain). The differential inverse transformation of F(k) is defined as

                      (15)

by combining Eqs. (14) and (15), f(η) can be obtained as

             (16)

Equation (16) implies that the concept of the differential transformation is derived from Taylor’s series expansion, but the method does not evaluate the derivatives symbolically. However, relative derivatives are calculated by an iterative procedure that is described by the transformed equations of the original functions. From the definitions of Eqs. (14) and (15), it is easily proven that the transformed functions comply with the basic mathematical operations shown in below. In real applications, the function f(η) in Eq. (16) is expressed by a finite series and can be written as

                     (17)

Equation (16) implies that   is negligibly small, where N is series size.

Theorems to be used in the transformation procedure, which can be evaluated from Eqs. (14) and (15), are given in Table 2.

3.2 Solution with differential transformation method

Now, differential transformation method has been applied into governing equations (Eqs. (8) and (9)). Taking the differential transforms of equations (8) and (9) with respect to χ and considering H=1 gives:

   (18)

           (19)

(20)

                           (21)

where F[k] and Θ[k] are the differential transforms of f(η), θ(η), respectively; and a₁, a₂, a₃ are constants which can be obtained through boundary condition. This problem can be solved as follows:

                          (22)

                           (23)

The above process is continuous. By substituting eqs. (22) and (23) into the main equation Eq. (17) based on DTM, it can be obtained that the closed form of the solutions is:

         (24)

                    (25)

by substituting the boundary condition from Eq. (11) into eqs. (24) and (25) at point η=1, it can be obtained the values of a₁, a₂, a₃. By substituting obtained a₁, a₂, a₃ into eqs. (24) and (25), it can be obtained the expression of F(η) and Θ(η).

Table 2 Some of basic operations of differential transformation method

4 Results and discussion

In this work, kerosene-alumina nanofluid flow and heat transfer in presence of magnetic field are investigated. The basic partial differential equations are reduced toordinary differential equations. Differential transformation method is used to solve the governing equations. Effects of magnetic parameter, alumina volume fraction, viscosity parameter and Eckert number on flow and heat transfer are investigated. In order to verify the accuracy of the present results, we have compared the results for the temperature profiles with those reported by SHEIKHOLESLAMI and GANJI [45] when f=0 (regular or Newtonian fluid). This comparison shows an excellent agreement (Fig. 2).

Fig. 2 Comparison of temperature profiles between present work and Ref. [45] when λ=0. 5, M=1, R=0.5 and Kr=0.5

Effects of magnetic number and viscosity parameter on velocity profiles and skin friction coefficient are shown in Figs. 3 and 4. As magnetic parameter increases, Lorentz forces increases. So, flow suppressed and in turn skin friction coefficient decreases with increase of magnetic field. Velocity boundary layer thickness near the lower plate increases with increase of viscosity parameter and in turn skin friction coefficient decreases with increase of this parameter. Also, it can be found that skin friction coefficient decreases with increase of nanofluid volume fraction.

Fig. 3 Effects of magnetic parameter and viscosity parameter on velocity profiles when f=0.04:

Effects of magnetic number, nanofluid volume fraction, viscosity parameter and Eckert number on temperature profile and Nusselt number are shown in Figs. 5 and 6, respectively. Effects of magnetic number and nanofluid volume fraction on temperature profile are similar together. It means that thermal boundary layer thickness near the hot wall decreases with increase of these parameters. So, Nusselt number increases with rise of magnetic number and nanofluid volume fraction. As viscosity parameter increases, temperature increases and in turn rate of heat transfer decreases with augment of viscosity parameter. Temperature increases with increase of vicious dissipation. So, Nusselt number decreases with increase of Eckert number.

Fig. 4 Effects of magnetic parameter, nanofluid volume fraction, and viscosity parameter on skin friction coefficient:

Fig. 5 Effects of magnetic parameter, nanofluid volume fraction, viscosity parameter and Eckert number on temperature profiles when Pr=21.976:

Fig. 6 Effects of magnetic parameter, nanofluid volume fraction, viscosity parameter and Eckert number on Nusselt number when Pr=21.976:

5 Conclusions

kerosene-alumina nanofluid flow and heat transfer in a channel in the presence of magnetic field are investigated. Differential transformation method is used to solve the governing equations. The effect of the magnetic parameter, nanofluid volume fraction, viscosity parameter and Eckert number on heat and fluid flow are investigated. The following results have been obtained. Differential transformation method has good agreement. Using kerosene-alumina nanofluid as fuel of liquid rocket engine can improve the cooling process of chamber and nozzle walls. Nusselt number increases with increase of alumina volume fraction in kerosene but opposite trend is observed for skin friction coefficient. Nusselt number increases with increase of magnetic parameter while it decreases with increase of Eckert number.

Nomenclature

A₁, A₂, A₃, A₄, A₅

dimensionless constants

C_f

skin friction coefficients

C_p

specific heat capacity at constant pressure

Ec

Eckert number

f(η)

similarity function

h

distance between plates

K

thermal conductivity

M

Magnetic parameter

Nu

Nusselt number

p

modified fluid pressure

Pr

Prandtl number

R

Viscosity parameter

u,n

velocity components along x, y, respectively

Greek symbols

α

thermal diffusivity

f

nanoparticle volume fraction

η

Dimensionless variable

μ

dynamic viscosity

υ

kinematic viscosity

θ

dimensionless temperature

s

Electrical conductivity

ρ

fluid density

τ_w

skin friction or shear stress along stretching surface

Subscripts

nf

nanofluid

f

base fluid

s

nano-solid-particles

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(Edited by YANG Hua)

Received date: 2015-03-25; Accepted date: 2015-07-31

Corresponding author: Sh. Kandelousi, Lecture; M. Mahmoodi, Assistant Professor; Tel: +98-2636102789; E-mail: mostafamahmoodi@mut.ac.ir (M. Mahmoodi), m.Kandelousi.sh@gmail.com (Sh. Kandelousi)