Nanofluid flow and forced convection heat transfer over a stretching ...

Eur. Phys. J. Plus (2015) 130: 155 DOI 10.1140/epjp/i2015-15155-8

THE EUROPEAN PHYSICAL JOURNAL PLUS

Regular Article

Nanofluid flow and forced convection heat transfer over a stretching surface considering heat source M. Mohammadpour1 , P. Valipour2,a , M. Shambooli1 , M. Ayani3 , and M. Mirparizi4 1 2 3 4

Department Department Department Department

of of of of

Mechanical Engineering, Babol University of Technology, Babol, Iran Textile and Apparel, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran Mechanical Engineering, Khaje Nasir Toosi University of Technology, Tehran, Iran Mechanical Engineering, University of Yazd, Yazd, Iran

Received: 12 May 2015 / Revised: 4 July 2015 c Societ` Published online: 30 July 2015 –  a Italiana di Fisica / Springer-Verlag 2015 Abstract. In this paper, magnetic field effects on the forced convection flow of a nanofluid over a stretching surface in the presence of heat generation/absorption are studied. The equations of continuity, momentum and energy are transformed into ordinary differential equations and solved numerically using the fourthorder Runge-Kutta integration scheme featuring the shooting technique. Different types of nanoparticles as copper (Cu), silver (Ag), alumina (Al2 O3 ) and titania (TiO2 ) with water as their base fluid has been considered. The influence of significant parameters, such as magnetic parameter, volume fraction of the nanoparticles, heat generation/absorption parameter, velocity ratio parameter and temperature index parameter on the flow and heat transfer characteristics are discussed. The results show that the values of temperature profiles increase with increasing heat generation/absorption and volume fraction of the nanoparticles but they decrease with increasing velocity ratio parameter and temperature index parameter. Also, it can be found that selecting silver as nanoparticle leads to the highest heat transfer enhancement.

Nomenclature A1 , A2 , A3 a b Cf f k M Nu Pr qw Q0 Rex T T∞ (u, v) (x, y)

a

Constants parameters Stretching sheet parameter Free stream velocity parameter Skin friction coefficient Dimensionless stream function Thermal conductivity Magnetic parameter Nusselt number Prandtl number Surface heat flux Dimensional heat generation or absorption coefficient Local Reynolds number Fluid temperature Ambient temperature Velocity components in the (x, y) directions, respectively Cartesian coordinates along x, y axes, respectively

e-mail: [email protected]

Greek symbols α η θ ρ φ μ υ τw ψ λ

Thermal diffusivity Similarity parameter Similarity function for temperature Density nanoparticle volume fraction Dynamic viscosity Kinematic viscosity Wall shear stress Stream function Velocity ratio parameter

Subscripts w ∞ nf

Condition at the surface Condition at infinity Nanofluid

f s

Base fluid Nano-solid-particles

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Eur. Phys. J. Plus (2015) 130: 155

1 Introduction The study of heat transfer in the boundary layer over a stretching surface finds applications in the extrusion of plastic sheets, polymer, spinning of fibers, cooling of elastic sheets, etc. The quality of the final product depends on the rate of heat transfer and therefore the cooling procedure has to be controlled effectively. Elbashbeshy and Basid [1] studied flow and heat transfer in a porous medium over a stretching surface with internal heat generation and suction/blowing when the surface is kept at constant temperature. The study of magnetic field effects has important applications in physics and engineering, such as the cooling of reactors and many metallurgical processes involving the cooling of continuous tiles. Also, in several engineering processes, materials manufactured by extrusion processes and heat-treated materials traveling between a feed roll and a wind up roll on convey belts possess the characteristics of a moving continuous surface. In recent years, we find several applications in the polymer industry (where one deals with stretching of plastic sheets) where hydro-magnetic techniques are being used. In view of these applications, Chakrabarti and Gupta [2] studied the MHD flow of Newtonian fluids initially at rest, over a stretching sheet at different uniform temperatures. Borkakoti and Bharali [3] studied the two-dimensional channel flows with heat transfer analysis of a hydromagnetic fluid where the lower plate was a stretching sheet. The term “nanofluid” refers to a liquid containing a suspension of submicronic solid particles (nanoparticles). The term was coined by Choi [4]. Sheikholeslami et al. [5] studied the natural convection in a concentric annulus between a cold outer square and heated inner circular cylinders in the presence of a static radial magnetic field. They reported that the average Nusselt number is an increasing function of the nanoparticle volume fraction as well as the Rayleigh number, while it is a decreasing function of the Hartmann number. The effect of a static radial magnetic field on the natural convection heat transfer in a horizontal cylindrical annulus enclosure filled with a nanofluid is investigated numerically using the Lattice Boltzmann method (LBM) by Ashorynejad et al. [6]. They found that the average Nusselt number increases as nanoparticle volume fraction and Rayleigh number increase, while it decreases as the Hartmann number increases. Soleimani et al. [7] studied natural convection heat transfer in a semi-annulus enclosure filled with a nanofluid using the control volume-based finite element method. They found that the angle of turn has an important effect on the streamlines, isotherms and maximum or minimum values of the local Nusselt number. Sheikholeslami and Abelman [8] used the two-phase simulation of a nanofluid flow and heat transfer in an annulus in the presence of an axial magnetic field. Sheikholeslami and Rashidi [9] studied the effect of space-dependent magnetic field on the free convection of a Fe3 O4 -water nanofluid. They showed that the Nusselt number decreases with increasing the Lorentz forces. Sheikholeslami et al. [10] applied LBM to simulate the three-dimensional nanofluid flow and heat transfer in the presence of a magnetic field. They indicated that adding magnetic field leads to a decrease in the rate of heat transfer. Sheikholeslami Kandelousi [11] studied the effect of a spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. He found that enhancement in the heat transfer decreases with increasing Rayleigh number and magnetic number but it increases with an increase in the Hartmann number. Sheikholeslami et al. [12] investigated ferrofluid hydrothermal behavior in the presence of thermal radiation. Sheikholeslami Kandelousi [13] used KKL correlation for the simulation of nanofluid flow and heat transfer in a permeable channel. He found that heat transfer enhancement has a direct relationship with the Reynolds number when the power law index is equal to zero but an opposite trend is observed for other values of power law index. Hatami et al. [14] investigated the magnetohydrodynamic Jeffery-Hamel nanofluid flow in non-parallel walls. They found that the skin friction coefficient is an increasing function of Reynolds number, opening angle and nanoparticle volume friction but a decreasing function of the Hartmann number. Various methods were used for improving the rate of heat transfer [15–29]. In the present study, we study the magnetic field effects on forced convection flow of a nanofluid over a stretching surface in the presence of heat generation/absorption using the fourth-order Runge-Kutta integration scheme featuring a shooting technique. The effects of the parameters governing the problem are studied and discussed.

2 Problem formulation We consider the steady, two-dimensional flow of a nanofluid near the stagnation point on a stretching sheet saturated in the presence of transverse magnetic field and volumetric rate of heat generation/absorption as shown in fig. 1. The stretching velocity Uw (x) and the free stream velocity U∞ (x) are assumed to vary proportional to the distance x from the stagnation point, i.e. Uw (x) = ax and U∞ (x) = bx, where a and b are constants with a > 0 and b ≥ 0. It is also assumed that the surface of the sheet is subjected to a prescribed temperature Tw (x) = T∞ + cxn , where T∞ is the ambient fluid temperature and c and n are constants with c > 0 (heated surface). Further, a uniform magnetic field of strength B0 is assumed to be applied in the positive y-direction normal to the stretching sheet. The magnetic Reynolds number is assumed to be small, and thus the induced magnetic field is negligible. The fluid is a water-based nanofluid containing different types of nanoparticles: Cu, Al2 O3 , Ag and TiO2 . It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. The

Eur. Phys. J. Plus (2015) 130: 155

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Fig. 1. Schematics of the geometry. Table 1. Thermo-physical properties of water and nanoparticles [22]. Pure water Copper (Cu) Silver (Ag) Alumina (Al2 O3 ) Titania (TiO2 )

ρ (kg/m3 ) 997.1 8933 10500 3970 4250

Cp (j/kg k) 4179 385 235 765 686.2

k (W/m·k) 0.613 401 429 40 8.9538

β × 105 (K−1 ) 21 1.67 1.89 0.85 0.9

thermo-physical properties of the nanofluid are given in table 1. Under these assumptions, the governing equations are ∂u ∂υ + = 0, ∂x ∂y   ∂u ∂u dU∞ ∂2u +υ − U∞ = μnf 2 + σnf B02 (U∞ − u), ρnf u ∂x ∂y dx ∂y   ∂T ∂T ∂2T +υ = knf 2 + Q0 (T − T∞ ), (ρCp )nf u ∂x ∂y ∂y

(1) (2) (3)

subject to the boundary conditions y = 0 : u = Uw (x), y → ∞ : u → U∞ (x),

υ = 0, T = Tw (x) T → T∞ ,

(4)

where u and υ are the velocity components along the x and y axes, respectively, T is the fluid temperature and Q0 is the dimensional heat generation or absorption coefficient. The effects of nanoparticles on the viscosity of nanofluids are introduced by the so-called apparent viscosity, which is indicated by μapp . Therefore, the effective viscosity can be defined as follows [29]: ρs VB d2s . (5) 72Cδ Let us consider a nanoparticle that is suspended in the base fluid moving relative to the base fluid. The Brownian motion is an important parameter that creates a relative velocity between the nanoparticle and the base fluid in nanofluids [30]. The Brownian velocity and the corresponding Reynolds number are defined as [31]  1 18Kb T , Kb = 1.38 × 10−23 J/K VB = ds πρs ds  1 18Kb T ReB = . (6) υf πρs ds μnf = μf + μapp = μf +

It is worth noting that the Reynolds number is very small for the nano-size particle diameter.

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Eur. Phys. J. Plus (2015) 130: 155

Also, δ is the approximate distance between the centers of particles and it is obtained from [29]  π δ= 3 ds , 6φ 1 [(c1 ds + c2 ) + (c3 ds + c4 )] , C= μf c1 = −0.000 001 133; c2 = −0.000 002 771; c3 = 0.000 000 09; c4 = −0.000 000 393,

(7)

The effective thermal conductivity of the nanofluid is calculated via Patel et al. [32] model as follows: knf ks As As =1+ + cks P e kf kf Af kf Af us d s df φ As , Pe = = , Af ds 1 − φ αf

us =

2Kb T , πμf d2s

c = 25000.

(8)

The effective density ρnf , the heat capacitance (ρCp )nf and the electrical conductivity (σnf ) of the nanofluid are given as ρnf = ρf (1 − φ) + ρs φ, (ρCp )nf = (ρCp )f (1 − φ) + (ρCp )s φ, 3(σs /σf − 1)φ σnf . =1+ σf (σs /σf + 2) − (σs /σf − 1)φ

(9)

The continuity equation (1) is satisfied by introducing a stream function, ψ, such that u=

∂ψ ∂y

and

υ=−

∂ψ . ∂x

(10)

The momentum and energy equations can be transformed into the corresponding ordinary differential equations by the following transformation: η=

 a 1/2 υ

y,

f (η) =

ψ , (aυ)1/2 x

θ(η) =

T − T∞ . Tw − T∞

(11)

The transformed ordinary differential equations are f  + f f  − f 2 + λ2 +

M (λ − f  ) = 0, A1

(12)

S 1 A1 · A3 · (1 − φ)2.5  · θ + f θ − nf  θ + θ = 0, Pr A2 A2

(13)

subject to the boundary conditions (4), which become f (0) = 0,

f  (0) = 1,



θ(∞) → 0.

f (∞) → λ,

θ(0) = 1, (14)

Here prime denotes differentiation with respect to η, λ = b/a is the velocity ratio parameter, P r = (μf (ρCp )f )/(ρf kf ) is the Prandtl number and M = σf B02 /(ρf a) is the magnetic parameter and A1 , A2 and A3 are parameters having the following form: A1 = (1 − φ) +

ρs φ, ρf

A2 = (1 − φ) +

(ρCp )s φ, (ρCp )f

A3 =

knf ks + 2kf − 2φ(kf − ks ) . = kf ks + 2kf + 2φ(kf − ks )

(15)

3 Results and discussion The ordinary differential equations (12) and (13) with the boundary conditions (14) have been solved numerically for some values of the governing parameters, magnetic parameter, volume fraction of the nanoparticles, heat generation/absorption parameter, velocity ratio parameter, temperature index parameter and different kinds of nanoparticles using the fourth-order Runge-Kutta integration scheme featuring the shooting technique. The Runge-Kutta method

Eur. Phys. J. Plus (2015) 130: 155

Page 5 of 8 2.4 Cu-Water

2

'

f(η)

1.6

{ {

λ=0.1

1.2

λ=2

0.8

0.4

M= 0 M= 1 M= 5 M = 10 M= 0 M= 1 M= 5

φ = 0.1

M = 10

0 0

1

η

2

3

Fig. 2. Effect of the magnetic parameter on the velocity profile. 1 M= 0 M= 1 M= 5

Cu-Water

0.8

M = 10

φ=0.1

θ(η)

0.6

0.4

0.2

0 0

1

2

η (a)

3

4

(b)

Fig. 3. Effect of the magnetic parameter on the temperature distribution for different values of λ when S = 0.1, n = 1 and P r = 6.2.

is a method of numerically integrating ordinary differential equations using a trial step at the midpoint of an interval to cancel out lower-order error terms. Figure 2 displays the effect of the magnetic parameter on the velocity profile. The magnetic parameter represents the effect of the magnetic field on the flow. The presence of a transverse magnetic field sets in the Lorentz force, which results in a retarding force on the velocity field and, therefore, as the magnetic parameter increases, so does the retarding force and, hence, the thickness of the boundary layer decreases. When λ > 1, the flow has a boundary layer structure and, due to this reason, the acceleration of the external stream is increased and this leads to thinning of the momentum boundary layer, which implies an increase in the straining motion near the stagnation point region. On the other hand, when λ < 1, the flow has an inverted boundary layer structure, which results from the fact that the stretching velocity, ax, of the surface exceeds the velocity, bx, of the external stream when b/a < 1. For this case, too, the thickness of the boundary layer decreases with M , which implies increasing behavior of the magnitude of the velocity gradient at the surface. The increase in the value of λ implies that free stream velocity increases in comparison to stretching velocity, which results in the increase in pressure and straining motion near the stagnation point and hence thinning of boundary layer takes place. The effect of the magnetic parameter on the temperature distribution for different values of λ, when S = 0.1, n = 1 and P r = 6.2 is shown in fig. 3. Different characteristics are observed in fig. 3, when λ < 1, the temperature is found to increase as M increases, but for λ > 1 increase in M leads to decreasing values of temperature profiles. Also, it can

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Eur. Phys. J. Plus (2015) 130: 155 1

1 φ=0 φ = 0.05

Cu-Water

φ = 0.1

0.8

φ=0 φ = 0.05

Cu-Water

φ = 0.1

0.8

θ(η)

' f(η)

0.6

0.6

0.4 0.4 0.2 0.2 0 0

1

η (a)

2

3

0

1

2

η

3

4

(b)

Fig. 4. Effect of the nanoparticle volume fraction (φ) on (a) the velocity profile and (b) the temperature distribution when M = 1, S = 0.1, n = 1, λ = 0.1 and P r = 6.2.

Fig. 5. Effect of the heat generation/absorption parameter on the temperature distribution when λ = 0.1, n = 1, M = 1 and P r = 6.2.

be seen that the temperature decreases as the distance from the surface increases and, finally, vanishes at some large distance from the surface. Figure 4 shows the effect of the nanoparticle volume fraction (φ) on the velocity profile and temperature distribution, when M = 1, S = 0.1, n = 1, λ = 0.1 and P r = 6.2. When the volume fraction of the nanoparticles increases, velocity and temperature increase. Figure 5 shows the effect of the heat generation/absorption parameter (S) on the temperature distribution when λ = 0.1, n = 1, M = 1 and P r = 6.2. The values of the temperature profile increase due to the increase in S. Figure 6 shows the effect of the velocity ratio parameter (λ) on the temperature distribution when S = 0.1, n = 1, M = 1 and P r = 6.2. Figure 7 displays temperature distribution for different values of the temperature index parameter (n) when S = 0.1, M = 1, λ = 0.1 and P r = 6.2. Temperature profiles subside uneventfully to zero as η increases, and the magnitude of the temperature gradient at the surface increases with increasing λ or n, due to the decreasing behavior of the thermal boundary layer thickness with increasing these parameters. Figure 8 displays the behavior of the temperature profile using different nanofluids when φ = 0.1, M = 5, Re = 1 and P r = 6.2. The minimum of temperature is obtained with adding titania to the fluid, while choosing silver as the nanoparticle maximum value of temperature observed. This means that the nanofluids will be important in the cooling and heating processes.

Eur. Phys. J. Plus (2015) 130: 155

Page 7 of 8 1 λ = 0.1 λ = 0.5 λ = 1.5

Cu-Water

0.8

λ=2

φ = 0.1

θ(η)

0.6

0.4

0.2

0 0

1

η

2

3

Fig. 6. Effect of the velocity ratio parameter on the temperature distribution when S = 0.1, n = 1, M = 1 and P r = 6.2. 1

Cu-Water

0.8

φ = 0.1

n= n= n= n=

-1 0 1 2

θ(η)

0.6

0.4

0.2

0 0

1

η

2

3

Fig. 7. Temperature distribution for different values of the temperature index parameter (n) when S = 0.1, M = 1, λ = 0.1 and P r = 6.2.

Fig. 8. Temperature distribution for different types of nanofluids when M = 1, S = 0.1, n = 1, λ = 0.1, φ = 0.1 and P r = 6.2.

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Eur. Phys. J. Plus (2015) 130: 155

4 Conclusions In the present paper the two-dimensional laminar-forced convection flow of a nanofluid over a stretching surface in the presence of heat generation/absorption has been numerically studied. The effects of the magnetic parameter, volume fraction of the nanoparticles, heat generation/absorption parameter, velocity ratio parameter and temperature index parameter on the flow and heat transfer characteristics are determined for four kinds of nanofluids: copper (Cu), silver (Ag) alumina (Al2 O3 ) and titania (TiO2 ). The working fluid is water with the Prandtl number P r = 6.2. The results showed that the momentum boundary layer thickness decreases due to an increase in the magnetic parameter, but for the thermal boundary layer thickness different characteristics are observed. When λ > 1, the temperature increases with the magnetic parameter, whereas an opposite behavior is noted when λ < 1. The values of temperature profiles increase with increasing heat generation/absorption and volume fraction of the nanoparticles but they decrease with increasing velocity ratio parameter and temperature index parameter.

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