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Heat transfer enhancement in sodium alginate based magnetic and non-magnetic nanoparticles mixture hybrid nanofluid
Journal Pre-proof Heat transfer enhancement in sodium alginate based magnetic and non-magnetic nanoparticles mixture hybrid nanofluid Abid Hussanan, Muhammad Qasim, Zhi-Min Chen
PII: DOI: Reference:
S0378-4371(19)32193-4 https://doi.org/10.1016/j.physa.2019.123957 PHYSA 123957
To appear in:
Physica A
Received date : 21 April 2019 Revised date : 28 September 2019 Please cite this article as: A. Hussanan, M. Qasim and Z.-M. Chen, Heat transfer enhancement in sodium alginate based magnetic and non-magnetic nanoparticles mixture hybrid nanofluid, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2019.123957. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
*Highlights (for review)
Journal Pre-proof Highlights:
Flow of non-Newtonian sodium alginate-based Cu-Fe3O4 hybrid nanofluid is analyzed.
Constitutive Equations of Casson fluid model are used examined the viscoplastic
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characteristics of sodium alginate Effect of viscous dissipation is also accounted.
Governing equations are simplified using boundary layer approximations.
Exact solution of self-similar equations are computed in terms of hypergeometric
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functions.
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The effects of various physical parameters are discussed through graphs.
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Abid Hussanan1,2, Muhammad Qasim3 and Zhi-Min Chen1* School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China
Department of Mathematics, COMSATS University Islamabad (CUI) Park Road, Tarlai Kalan, Islamabad 455000, Pakistan *
Corresponding Author: [email protected]
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3
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Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen, 518060, China
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2
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1
Journal Pre-proof *Manuscript Click here to view linked References
HEAT TRANSFER ENHANCEMENT IN SODIUM ALGINATE BASED MAGNETIC AND NON-MAGNETIC NANOPARTICLES MIXTURE HYBRID NANOFLUID
School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China
2
Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and
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1
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Abid Hussanan1,2, Muhammad Qasim2, Zhi-Min Chen1*
Guangdong province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen, 518060, China 2
Department of Mathematics, COMSATS University Islamabad (CUI) Park Road,
*
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Tarlai Kalan, Islamabad 455000, Pakistan
Corresponding Author: [email protected]
Abstract: Sodium alginate (SA) based hybrid nanofluids are novel new generation of fluids for heat transfer. The thermo-physical properties of these fluids are very classic in comparison to common fluids. This study aims to examine the heat transfer enhancement in viscoplastic non-
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Newtonian based Cu-Fe3O4 hybrid nanofluid, flowing over a stretching/shrinking sheet. SA is being used as a non-Newtonian, viscoplastic base fluid with the addition of Cu and Fe3O4 as non-
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magnetic and magnetic nanoparticles. In the formulation of the mathematical model, Casson fluid model is exploited to examine the viscoplastic characteristics of SA. The effective thermal conductivity of Cu-Fe3O4 hybrid nanofluid calculated from the Maxwell model (for nanofluid). The exact solution of the nonlinear flow equation is obtained, and the solution of the heat transfer equation is expressed in hypergeometric function through Maple. The effects of
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consequential parameters such as magnetic parameter, Prandtl number, Casson parameter, Eckert number, and nanoparticles volume fraction on velocity and temperature field are examined. The result of this study suggests that SA based fluid should be used to obtain high rates of heat transfer.
Keywords: Hybrid nanofluid; sodium alginate; thermal conductivity; viscous dissipation.
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1.
INTRODUCTION
Most of the materials that we encounter in our daily lives are neither perfectly elastic
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solids nor simply Newtonian fluids and attempts often fail to describe these materials as being either fluid or solid. Examples of these materials include foams, syrups, nail polish, yogurt, cosmetics and toothpaste. These materials require critical shear stress to overcome before flow
p ro
can occur in the system. If stress magnitude less than the critical value of shear stress applied to the fluid, it exhibits like a solid. Thus, shear stress must exceed a critical value known as the yield stress for the fluid to flow. The idea of yield stress was originally introduced by Bingham in the 1920s. Papanastasiou [1] applied modified constitutive relation for the Bingham model in
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both yielded and rigid regions of the flow field. The linear stability analysis of Bingham fluid (when the yield stress approaches zero) was performed by Métivier et al. [2]. Park and Liu [3] investigated experimentally as well as analytically oscillatory pipe flows of Bingham yield stress fluid. Some related published data on Bingham fluid and yield stress can be found in Shepherd et al. [4]; Zengeni [5]; Dutta et al. [6]. One of the popular generalizations of the Bingham fluid model in shear flow is Casson fluid model [7]. Several Casson fluid models have been proposed, Dash et al. [8]; Qasim and Noreen [9]; Reddy et al. [10]. Mukhopadhyay [11] analyzed Casson
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fluid flow over a nonlinearly stretching sheet with yield stress. Khan et al. [12] examined the yield stress in Casson fluid flow over static and moving flat plates. Recently, the effect of yield
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stress on Rayleigh-Bénard convection in Casson fluid was investigated by Aghighi et al. [13].
A nanofluid is the suspension of nanoparticles into the conventional base fluid. The concept of nanofluid was first introduced by Choi [14]. In recent developments, Rashidi et al. [15] studied the heat transfer MHD nanofluid flow due to a rotating porous disk with entropy
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generation effect. Turkyilmazoglu [16] examined the unsteady convection heat transfer nanofluids flow over a vertical flat plate with both isothermal and isoflux conditions. Sheikholeslami et al. [17] studied steady forced convection Fe3O4/water nanofluid flow in a semi-annulus enclosure with the non-uniform magnetic field. Rashad et al. [18] numerically studied the effect of internal heat generation on Cu/water nanofluid free convection flow inside a rectangular cavity under a magnetic field. Turkyilmazoglu [19] studied two-dimensional laminar free nanofluid jet and for a circular axisymmetric free nanofluid jet with five different 2
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nanoparticles. Sheikholeslami et al. [20] studied the magnetic field effect on the convective Fe3O4/water nanofluid flow with elliptical heat fluxes. Sheikholeslami et al. [21] studied the CuO/water nanofluid PCM solidification through an enclosure with V-shaped fins. Their work
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was then updated by Sheikholeslami and Mahian [22] who studied the Lorentz force effects on the solidification of inorganic nanoparticles enhanced PCM using CuO/water nanofluid inside a porous annulus. Sheikholeslami [23] examined entropy generation in natural convection
p ro
Fe3O4/water ferrofluid inside a porous cavity. Sheikholeslami et al. [24] analyzed the entropy generation for turbulent heat transfer CuO/water nanofluid in a pipe. Further studies on heat transfer enhancement using nanofluid are to be found in Turkyilmazoglu [25]; Hussanan et al. [26]; Sheikholeslami [27]; Zyła et al. [28]; Sheikholeslami [29]; Turkyilmazoglu [30]; Nadeem
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et al. [31] and Sheikholeslami et al. [32].
Nowadays, hybrid nanofluids are the reasonable new generation of nanofluids, which can be developed either in mixture or composite form in the base fluids. A single material does not exhibit all the characteristics required for a particular purpose. For example, metal nanoparticles have high thermal conductivity as compared to metal oxide nanoparticles but less stability [33]. In that case, hybrid nanofluids may offer good heat transfer enhancements as compared to
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nanofluids. In the last few years, these nanofluids have been used widely in various heat transfer applications such as nuclear system cooling, cooling and heating in buildings, thermal storage,
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cooling of transformer oil and tubular heat exchanger [34]. Recently many experimental researches have been done on hybrid nanofluid. Esfe et al. [35] conducted an experimental study on thermal conductivity and viscosity of Ag-MgO nanoparticles dispersed into water with equal nanoparticle volume fraction (50% Ag and 50% MgO by volume). Their results showed that thermal conductivity and viscosity increased with increasing volume concentration. Afrand et al.
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[36] used Fe3O4-Ag/EG hybrid nanofluid at different shear rates with temperatures from 25 0C to 50 0C. Their results showed Newtonian behavior at solid volume fractions less than 0.3% and non-Newtonian at 0.6% to 1.2%. Moldoveanu et al. [37] studied the rheological behavior of Al2O3, TiO2 nanofluids and their hybrid, dispersed into water and found that nanofluids as well hybrid nanofluid acted like non-Newtonian at room temperature. Some other types of nanoparticles used in hybrid nanofluids followed by their articles are: Al2O3-MWCNTs/water,
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Nine et al. [38]; MgO-MWCNT/ethylene glycol, Soltani and Akbari [39]; Cuo-MWCNT/engine oil, Aghaei et al. [40]; Cu-Al2O3/water, Afridi et al. [41].
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Thermal properties of hybrid nanofluids depend on size, dispersibility of nanoparticles, preparation method, purity and compatibility of nanoparticles and the base fluids. The selection of base fluids is very important in the hybrid nanofluids heat transfer enhancement. Water,
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engine oil, ethylene glycol and ethylene/water mixtures are commonly used base fluids in the studies of, Hussanan et al. [42]; Zyła et al. [43]; Li et al. [44]; Sheikholeslami et al. [45]; Zyła et al. [46]; Hussanan et al. [47]; Swalmeh et al. [48] and Sheikholeslami et al. [49]. Additionally, Hayat and Nadeem [50] used Ag-CuO/water and Iqbal et al. [51] used Cu-CuO/water hybrid
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nanofluids for heat transfer enhancement. Hussain et al. [52] studied MHD mixed convection flow in an open cavity with a horizontal channel using Al2O3-Cu/water hybrid nanofluid. Numerical analysis of hybrid water based nanofluid in a semicircular cavity was studied by Chamkha et al. [53] with a focus on improving the heat transfer enhancement. Sheikhzadeh et al. [54] examined the heat transfer flow of Ag-MgO/water micropolar hybrid nanofluid in a permeable channel. Mollamahdi et al. [55] studied Al2O3-Cu/water micropolar hybrid in an expanding or contracting walls porous channel under a magnetic field. Amala and Mahanthesh
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[56] examined the effect of Hall current on Cu-Al2O3/water hybrid nanofluid in the existence of nonlinear convection. In all of the above-mentioned studies, the base fluid was assumed to be
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Newtonian.
Nowadays, sodium alginate, a non-Newtonian fluid, has gained much attraction due to its industrial and biomedicine applications such as textile, paper production, tissue formation, food industry and pharmaceuticals. Hatami and Ganji [57] investigated the SA-TiO2 nanofluid flow
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through a porous media between two coaxial cylinders, and it was found that the nanoparticle concentration increased near the inner cylinder wall. Hatami and Ganji [58] studied sodium alginate based Cu and Ag non-Newtonian nanofluid flow between two vertical flat plates, the effects of natural convection was also analyzed. Tlili et al. [59] examined two different types of nanofluids by dispersing Al2O3 and Cu nanoparticles in sodium alginate (SA) over a stretching cylinder. The impact of viscous dissipation on the unsteady boundary layer flow of viscoplastic
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Casson ferrofluid over a semi-infinite vertical plate with leading-edge accretion/ablation was investigated by Hussanan et al. [60].
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The aim of the present work is to explore the heat transfer and flow structure in nonNewtonian hybrid nanofluid with viscoplastic behavior flowing over a stretching/shrinking sheet. The base fluid is sodium alginate and nanoparticles, namely, Cu and Fe3O4 as non-magnetic and
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magnetic nanoparticles are added. Instead of the commonly used Newtonian fluid models, a viscoplastic non-Newtonian fluid model is employed for sodium alginate based Cu-Fe3O4 hybrid nanofluid which makes this study unique and the results are more realistic and practically useful. The thermophysical properties of Cu and Fe3O4 nanoparticles and SA based hybrid nanofluid
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have been summarized in Table 1.
Table 1: Thermophysical properties of Cu, Fe3O4 and SA.
kg m3
C p J kg K
k W mK
8933
385
401
5.96 107
5180
670
9.7
2.5 104
0.6376
2.6 10 4
0.613
0.05
Cu
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Fe3O4 SA H2 O 997.1 2.
4175
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989
4179
.m
1
PROBLEM FORMULATION
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Two-dimensional boundary layer steady flow of hybrid nanofluid (Cu-Fe3O4/SA) past a stretching/shrinking sheet is considered. The x and y are Cartesian coordinates taken along the sheet and normal to it, respectively. The stretched or shrunk velocity is assumed as uw ( x) ax , where a is a constant, as shown in Figs. 1(a) and 1(b). It is also assumed that the temperature of the sheet is Tw x T Ax2 [61]. Initially, Cu nanoparticles of volume fraction Cu range from 0.01 to 0.04 are dispersed into the sodium alginate (SA) to make Cu/AS nanofluid. Thus, to 5
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develop the hybrid nanofluid (Cu-Fe3O4/SA), magnetite Fe3O4 (iron oxide) nanoparticles with
different volume fraction Fe3O4
are suspended into Cu/AS nanofluid. The rheological behavior
of Cu-Fe3O4/SA hybrid nanofluid followed by [53, 56] can be described in the form of
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continuity, momentum, and energy equations V = 0
V
(1) (2)
E V E q . t
(3)
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hnf V V p div ij hnf b + J B, t
where represents viscous dissipation term and all other symbols and quantities are given in
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nomenclature. The effective density nf of convectional nanofluids is defined as
nf 1 f np ,
(4)
the above relation for effective density was firstly given by Pak and Cho [62] followed in the studies [41, 51]. The mixture density formula for Cu-Fe3O4/SA hybrid nanofluid [63] hnf is
hnf 1 f np np ,
(5)
np
and np (for hybrid nanofluid np 1, 2 ) correspond to base fluid (SA) and
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where f
nanoparticles (Cu, Fe3O4), is the combination of dispersed Cu nanoparticles volume fraction
and Fe3O4 nanoparticles volume fraction Fe3O4
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Cu
into sodium alginate based hybrid
nanofluid expressed as
Cu Fe O . 3
(6)
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In case of Cu-Fe3O4/SA hybrid nanofluid, equation (5) becomes
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hnf 1 Cu Fe O f Cu Cu Fe O Fe O . 3
4
3
4
3
4
(7)
The velocity vector for present flow problem is u u x, y V v v x , y . w 0
The constitutive relationship for viscoplastic non-Newtonian fluid, given by Casson [7], is 6
(8)
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y 2 B eij , c 2 ij , y eij , c 2 B 2 c
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(9)
Non-magnetic Cu nanoparticles
Size: 1-100 nm
y
Thermal boundary layer
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Magnetite Fe3O4 nanoparticles
Momentum boundary layer B0
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B0
x
u
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Figure 1(a): Stretching Case 0
y
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B0
B0
x u
Figure 1(b): Shrinking Case 0
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where eij eij and eij is the i, j th component of deformation rate is 1 vi v j , 2 x j xi
(10)
with 1 i, j 2 . In view of Ohm’s law
J = hnf E1 + V B
of
eij
(11)
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where E1 is the electric field, B is the combination of imposed magnetic field B0 and induced magnetic field b0 and hnf is the electrical conductivity of Cu-Fe3O4/SA hybrid nanofluid [64] can be written as
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hnf f
Cu Cu Fe3O4 Fe3O4 3 Cu Fe3O4 f 1 Cu Cu Fe3O4 Fe3O4 2 Cu Cu Fe3O4 Fe3O4 Cu Fe3O4 f f
,
(12)
here is given in equation (6), Cu and Fe3O4 are electrical conductivities of Cu and Fe3O4 nanoparticles, respectively. Equation (11), with the assumptions E1 = 0 and b0 = 0 , modifies to
J = hnf V B0
(13)
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As magnetic field of uniform strength is taken along y -axis, therefore, B0 0, B0 ,0 . Finally,
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J B hnf B02 V
(14)
In the absence of pressure gradient, body forces and using equations (9) and (14), the continuity
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and motion equations for steady incompressible flow are
u
u v 0, x y
u u v hnf x y
1 2u hnf 2 B u, 1 2 hnf 0 y
(15)
(16)
where B 2 c y is the non-Newtonian viscoplastic parameter, which depends on yield stress and flow only occurs in the system when critical shear stress is greater than the yield stress. The total energy per unit volume consists of internal energy and kinetic energy is
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V2 E hnf e . 2
(17)
The specific internal energy e defined by [65] is
p
hnf
.
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e C p ,hnf T Here
1 C p f Cu C p Cu Fe O C p Fe O
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C p ,hnf
3
4
3
4
hnf
.
(18)
(19)
is the heat capacity of Cu-Fe3O4/SA hybrid nanofluid obtained from heat capacity of nanofluid at constant pressure [66]. In terms of temperature field, the heat flux is
here
Kf
(20)
Cu KCu Fe O K Fe O 2 K f 2 Cu K Cu Fe O K Fe O 2 K f , Cu KCu Fe O K Fe O 2 K f Cu K Cu Fe O K Fe O K f 3
K hnf
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q K hnf T ,
4
3
3
4
4
3
3
4
3
4
(21)
4
3
4
3
4
is the thermal conductivity of Cu-Fe3O4/SA hybrid nanofluid which is obtained from Maxwell
equation (3) becomes
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model and is also given in [50]. The pressure is approximately uniform and T T x, y ,
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T T T u v t x x
1 2T K . hnf 2 y C p hnf
(22)
In the presence of viscous dissipation, above equation in case of steady flow modifies to
with
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K hnf T T u v x y C p
hnf
hnf
hnf 2T 2 y C p
1
Cu
f Fe3O4
2
hnf
2.5
1 u 1 , y
(23)
.
(24)
The boundary conditions are u uw ( x), v 0 at y 0; u 0 as y ,
9
(25)
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T Tw x T Ax2 at y 0; T T as y .
(26)
To simplify the equations (16) and (23), we introduce the similarity transformations as
f
, u axF , v a f F ,
T T . Tw T
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a
y
(27)
Using above similarity variables, equations (16) and (23) can be written as
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hnf f 1 f hnf 2 F 0, 1 F F F F M f hnf hnf f C p f 1 1 K hnf C p f F 2 F hnf 1 EcF 2 0, Pr K f C p f C p hnf hnf
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with boundary conditions are
(28)
(29)
F 0, F , 1 at 0,
(30)
F 0, 0 as ,
(31)
Here
C p f f f B02 uw2 a2 M , Pr , Ec . a f Kf A C p Tw T C p f f 2
number Nu
hnf
1 u 1 y
x T , Nu K hnf K f Tw T y y 0
1
K hnf 1 1/2 0 , 1 F 0 , Re x Nu Kf
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1 f uw2
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The quantity of engineering interest include the skin friction coefficients C fx and Nusselt
C fx
, y 0
(32)
in dimensionless form are
1
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Re1/2 x C fx
Cu
Fe2O3
2.5
(33)
where Re x uw x / f is the local Reynolds number.
3.
MOMENTUM EQUATION SOLUTION
According to Cortell [67] and Turkyilmazoglu [68], the assumed exact solution of equation (28) 10
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F
1 e ,
(34)
where Fe3O4
Cu
2.5
1
hnf M hnf f f
yields
F ( )
1
Cu Fe3O 4
2.5
1
hnf f
hnf 2.5 hnf 1Cu Fe3O4 M f 1 f 1 e . (35) M hnf f
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Finally, velocity expression is F ( ) e
F ( )
Cu
Fe3O4
1
1Cu Fe3O4
hnf hnf M f 1 f
2.5
hnf hnf M f f
,
(36)
1Cu Fe3O4 e
hnf hnf M f 1 f
2.5
.
(37)
ENERGY EQUATION SOLUTION
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4.
1
2.5
p ro
,
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1
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To find the solution of energy equation, we substitute F , F F given in equations (35) to (37) in equation (29), we obtain
1 Pr
k hnf kf
C p hnf e 2 e C p f
1 2 1 Ec e 0. f
(38)
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hnf
Taking t e , the above equation becomes
C p hnf 1 hnf 1 k hnf 1 t t 2 t t t t Pr k f C p 2 f f
1 2 1 Ec t 0,
(39)
and boundary conditions are
0 0, 1 1, 11
(40)
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where 1 2 and
3
4
2.5
hnf , 2 M 1 Cu Fe3O4 1 f
Using Maple toolbox, obtained solution of equation (39) is
2.5
hnf . 1 f
of
1 1 Cu Fe O
p ro
3 3 4 3 4 4 43 t t hypergeometric , , t e C1 3 3 3 3 4 4 43 43 t hypergeometric 3 , , t t e C2 3 3
where
3
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1 2 4t 5 3 24 , 2 4
(41)
C p hnf khnf 1 1 , 4 , 5 hnf 1 Ec 2 2 k f Pr f C p f
3 4 3 4 1 C1 5 3 2 4 hypergeometric 3 , ,0 2 4 3 3
1
1
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3 4 4 43 C2 hypergeometric 3 , , e 3 3 1
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3 4 4 43 1 5 3 4 hypergeometric 3 , , e 2 42 3 3
3 4 3 4 4 1 5 3 2 4 hypergeometric 3 , , 2 4 3 3 3 1
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4 4 hypergeometric 3 , 3 , . 3 3
Using boundary conditions (40), we obtained
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3 3 4 3 4 4 43 t 1 5 3 4 t hypergeometric , , t e 2 42 3 3 3
1
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3 4 3 4 1 5 3 4 2 4t hypergeometric 3 , , 0 2 42 3 3
1
4 4 43 43 t 3 4 4 43 hypergeometric 3 , 3 , t t e hypergeometric 3 , , e 3 3 3 3
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3 4 4 43 43 t hypergeometric 3 , , t t e 3 3
1
4 4 43 5 3 2 4 hypergeometric 3 , 3 , e 2 4 3 3
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3 4 4 43 43 t 3 4 4 hypergeometric 3 , , t t e hypergeometric 3 , , 3 3 3 3
3 4 3 4 4 1 5 3 2 4 hypergeometric 3 , , . 2 2 4 3 3 3
1
(42)
The expression for temperature t obtained in equation (42) is function of t . The temperature
GRAPHICAL RESULTS AND DISCUSSION
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5.
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solution in term of can be obtain by substituting t e in equation (42).
A Maple (software) toolbox has been used to explore the physical behaviors arising due to the existence of Casson parameter , magnetic parameter M , Eckert number Ec , Prandtl number Pr , Cu volume fraction Cu and Fe3O4 volume fraction Fe3O4 with the flow field. Figs.
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2(a) and 2(b) illustrate the variation of velocity at different values of Casson parameter along stretching sheet for Cu-Fe3O4 hybrid nanofluid when M 0 and M 0 , respectively. It is noted in Fig. 2(a), as increases the velocity decreases and the thickness of the boundary layer gets larger. The same behavior appears in Fig. 2(b) which concerns velocity variation, when M 0 . Further, the fluid velocity declines sharply in the presence of a magnetic field M 0 . The variations of velocity with Casson parameter for both M 0 and M 0 in shrinking case are
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illustrated in Figs. 3(a) and 3(b). As the strength of the magnetic field increases, the velocity of Cu-Fe3O4 hybrid nanofluid declines sharply for M 0 . Physically, the presence of a transverse magnetic field (normal to the flow direction) has a tendency to create the drag, (Lorentz force)
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which tends to resist the flow. The velocity variation of Fe3O4/SA nanofluid Cu 0 for both
M 0 and M 0 is shown in Figs. 4(a) and 4(b) for different values of Casson parameter , when 0 . It is seen in Fig. 4(a) that velocity is reduced with an increase in Casson parameter
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, for M 0 . Further, it is shown in Fig. 4(b) that as M increases M 0 , the impact of becomes more significant. We also noticed in all above figures that velocity of Cu-Fe3O4 hybrid nanofluid approaches to free stream condition at shorter distances, when M 0 as compared to
M 0 . As the value of Casson parameter increases, an increase in temperature is observed at
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M 0 (Fig. 5(a)). Moreover, there is also an increase in temperature, when M 0 , see Fig. 5(b). In addition, there is a large overshoot in temperature at M 0 through the variations of . Fig. 6(a) and 6(b) illustrate the variation of the temperature against the stretching parameter
0
for various values of Eckert number Ec , when M 0 and M 0 , respectively. These
figures reveal that in both cases, the temperature of Cu-Fe3O4 hybrid nanofluid rises with the enlargement of Eckert number Ec . Physically, the dissipative frictional forces between layers
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enhance with the enlargement of Ec and consequently, the temperature of Cu-Fe3O4 hybrid
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nanofluid rises.
Figs. 7(a-d) present the effect of the magnetic parameter M , Eckert number Ec and Casson parameter on Nusselt number Nu for Cu-Fe3O4 hybrid nanofluid. Nusselt number Nu reduces with augment in Casson parameter , deducing that sodium alginate based fluid should
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be used to obtain the highest rate heat transfer. Figs. 8(a) and 8(b) show the effect of Eckert number Ec and Cu volume fraction on Nusselt number Nu for Cu/SA nanofluid in both cases
M 0 and M 0 , respectively. The augments in Lorentz forces lead to stronger conduction mechanisms. Nusselt number Nu reduces with the increase of the magnetic field M 0 . In Figs. 9(a) and 9(b), the effects of the same parameters on Nusselt number Nu in case Fe3O4/SA nanofluid are depicted. The volume fraction of nanofluid induces the thermal boundary layer thickness to augment. Hence, Nusselt number Nu reduces with augment in Fe3O4 volume 14
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fraction. Further, it is found that Fe3O4 nanoparticles have low thermal conductivity
9.7 W
mK as compare to Cu 401 W m K , but give the highest rate of heat transfer, when
of
M 0. For showing the comparison of non-Newtonian based Cu-Fe3O4/SA and Newtonian
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based Cu-Fe3O4/water hybrid nanofluids, skin friction coefficient C fx and Nusselt number Nu are calculated in Tables 2 and 3, respectively. From Table 2, it is found that the skin friction coefficient C fx of Cu-Fe3O4/SA hybrid nanofluid decreases with the increase of Cu volume fraction when M 0 or M 0 . The opposite trend is noted for Fe3O4 volume fraction. Further,
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the results indicate that Nusselt number Nu tends to decrease as Cu volume fraction of CuFe3O4/SA hybrid nanofluid increases and the same behavior is observed for Cu-Fe3O4/SA hybrid for Fe3O4 volume fraction. Comparing the magnetic nanoparticles based Fe3O4/SA nanofluid with non-magnetic nanoparticles Cu/SA nanofluid, it is observed that Fe3O4/SA nanofluid has better heat transfer capabilities in comparison of Cu/SA nanofluid. Table 3 shows that Nusselt number Nu for Cu-Fe3O4/water hybrid nanofluid decreases with an increase of Cu and Fe3O4 volume fractions and the same behavior is observed, when M 0 . Further, it is similar to the
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trends of Nusselt number Nu noted in Table 2 for Cu-Fe3O4/SA hybrid nanofluid. Further, the utilization of smaller nanoparticles volume fraction achieves better thermal performance. It
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means that Cu and Fe3O4 nanoparticles volume fraction is a key factor for heat transfer enhancement in Cu-Fe3O4/water or Cu-Fe3O4/SA based hybrid nanofluids. Besides, the skin friction coefficient C fx of Cu-Fe3O4/water hybrid nanofluid also decreases with the rise of Cu volume fraction and Fe3O4 volume fraction, when M 0 or M 0 . The accuracy and validation
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of analytical results are tested by comparing present results for skin friction coefficient C fx and Nusselt number Nu with those obtained previously by Khan et al. [69] in the case of Newtonian Fe3O4/water nanofluid as shown in Table 4. From this tabular result, it is noticed that there is a very close agreement between present results and those of the existing literature.
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Table 2: Skin friction coefficients C fx and Nusselt number Nu in case of non-Newtonian CuFe3O4/SA hybrid nanofluid, when 1 (stretching sheet) and Pr 6.45 (sodium alginate).
0.03
0.1
0.5
0 0.01 0.02 0.03 0.03
0.1
0.1
0.004612649171 0.004573550124 0.004546083004 0.004528851155 0.004395986631 0.004436253520 0.004480547207 0.004528851155 0.007907778187 0.007739209239 0.007604724007 0.007499068263 0.007304716397 0.007361011092 0.007425839663 0.007499068263 0.004528851155 0.017712557610 0.036835949560 0.007499068263 0.029329221500 0.060994563690 -
3.887798336 3.799543407 3.740564882 3.683601483 3.854452822 3.796043929 3.739118140 3.683601483 3.878508194 3.790224954 3.731208963 3.674199691 3.845422269 3.786892544 3.786892544 3.674199691 3.683601483 3.667219625 3.651223497 3.674199691 3.646842850 3.619666228 3.683601483 3.683593824 3.683578505 3.674199691 3.674179150 3.674138068
0 0.01 0.02 0.03 0.03
2
0 0.01 0.02 0.03 0.03
0
0.03
0 0.01 0.02 0.03 0.03
2
0.03
0.03
0
0.03
2
0.03
0.1
0.03
0.1 0.2 0.3 0.1 0.2 0.3 0.1
0.03
0.1
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Nu
4
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2
C fx
3
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0
Ec
Fe O
Pr e-
0
Cu
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M
0.5 1.0 2.0 0.5 1.0 2.0
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Table 3: Skin friction coefficients C fx and Nusselt number Nu in case of Newtonian CuFe3O4/water hybrid nanofluid, when 1 (stretching sheet) and Pr 6.2 (water).
0
Nu
0.03
0.5
0 0.01 0.02 0.03 0.03
0.9790270381 0.9584905564 0.9405645173 0.9248772436 0.9333404589 0.9298948614 0.9270851387 0.9248772436 1.6790399570 1.6228056390 1.5744629990 1.5326910700 1.5517904190 1.5439683070 1.537632539 1.532691070 -
2.984670350 2.905523263 2.851447643 2.801541024 2.900673587 2.866278956 2.833284100 2.801541024 2.020893628 1.991379882 1.976334178 1.963203296 1.963905535 1.963687919 1.963484446 1.963203296 2.801541024 2.251321819 1.150883411 1.963203296 0.7647930430 0.2854289310
0 0.01 0.02 0.03 0.03
4
0
0.03
0 0.01 0.02 0.03 0.03
2
0.03
0.03
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2
0 0.01 0.02 0.03 0.03
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C fx
3
0.5 1.0 2.0 0.5 1.0 1.2
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2
Ec
Fe O
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0
Cu
Pr e-
M
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Table 4: Comparison of our numerical findings for skin friction coefficients C fx and Nusselt number Nu in case of Newtonian Fe3O4/water nanofluid with those of the existing literature,
Khan et al. [69]
Fe O
4
Ec
0 0 0 1 1 1
0.01 0.10 0.20 0.01 0.10 0.20
0.1 0.1 0.1 0.1 0.1 0.1
Present results
C fx
Nu
1.03366 1.35914 1.79238 1.44703 1.77443 2.22700
3.62729 3.91965 4.26287 3.42446 3.71411 4.59009
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Pr e-
3
C fx
Nu
1.033657 1.359136 1.792374 1.447032 1.774433 2.227001
3.627292 3.919651 4.262875 3.424463 3.714116 4.590091
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M
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when 1 , Pr 6.2 and .
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(a)
(b)
Figure 2: Velocity variation of Cu-Fe3O4/Sa hybrid nanofluid with .
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(b)
(b)
Figure 3: Velocity variation of Cu-Fe3O4/Sa hybrid nanofluid with .
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(a)
(b)
Figure 4: Velocity variation of Fe3O4/Sa nanofluid with .
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(a)
(b)
Figure 5: Temperature variation of Cu-Fe3O4/Sa hybrid nanofluid with .
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(a)
(b)
Figure 6: Temperature variation of Cu-Fe3O4/Sa hybrid nanofluid with Ec .
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(b) 0.3
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(a) 0.2
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(c) 0.4
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(d) 0.5
Figure 7: Effect of Eckert number Ec , magnetic parameter M and Casson parameter on Nusselt number Nu , when Pr 6.45 , 2 , Cu 0.01 and Fe3O4 0.01 .
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(a) M 0 (Cu)
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(b) M 2 (Cu)
Figure 8: Effect of Eckert number Ec , magnetic parameter M and Cu volume fraction Cu on
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Nusselt number Nu , when Pr 6.2 , 1 , 0.5 and Fe3O4 0.01 .
(a) M 0 (Fe3O4)
(b) M 2 (Fe3O4)
Figure 9: Effect of Eckert number Ec , magnetic parameter M and Fe3O4 volume fraction Fe3O4 on Nusselt number Nu , when Pr 6.45 , 1 , 0.5 and Cu 0.01 .
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CONCLUSIONS
Heat transfer enhancement characteristics of non-Newtonian SA based Cu-Fe3O4 hybrid nanofluid flow over a stretching/shrinking sheet is studied. Both magnetic (Fe3O4) and non-magnetic (Cu) nanoparticles are dispersed into the sodium alginate (SA) to make CuFe3O4 hybrid nanofluid. Outcomes of this study reveal that Fe3O4 nanoparticles have low
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thermal conductivity as compare to Cu, but give the highest rate of heat transfer in the presence of a magnetic field. The effect of viscous dissipation on the temperature field is
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more pronounced under the magnetic field. The SA based Cu- Fe3O4 hybrid nanofluid is found to have a higher rate of heat transfer, than water based hybrid nanofluid. Further, the utilization of smaller nanoparticles volume fraction achieves better thermal performance.
Nomenclature Positive constant
B0
Magnetic field intensity
b
Body force N
C p, f
SA heat capacity J kg 1 K 1
C p ,Cu
Cu heat capacity J kg 1 K 1
Pr e-
A
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C p ,Fe3O4 Fe3O4 heat capacity J kg 1 K 1
C p ,hnf Hybrid nanofluid heat capacity J kg 1 K 1 Eckert number
F
Stream function
F
Dimensionless velocity
Kf
SA thermal conductivity W m 1 K 1
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Ec
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K Fe3O4 Fe3O4 thermal conductivity W m 1 K 1 K Cu
Cu thermal conductivity W m 1 K 1
K hnf
Hybrid nanofluid thermal conductivity W m 1 K 1
M
Magnetic parameter
p
Pressure Pa
Pr
Prandtl number 26
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Tw
Sheet temperature K
T
Ambient temperature K
Stretching/shrinking parameter
Hybrid nanofluid volume fraction
Cu
Cu volume fraction
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T
Fe O Fe3O4 volume fraction 3
4
Viscous dissipation
B
Plastic dynamic viscosity kg m 1 s 1
f
SA dynamic viscosity kg m 1 s 1
hnf
Hybrid nanofluid dynamic viscosity kg m 1 s 1
f
SA density kg m 3
Cu
Cu density kg m 3 3
4
Pr e-
Fe O Fe3O4 density kg m 3
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Hybrid nanofluid density kg m 3
y
Yield stress kg m 1 s 2
f
1 SA electric conductivity .m
Cu
1 Cu electric conductivity .m
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hnf
1 Fe O Fe3O4 electric conductivity .m 3
4
1 Hybrid nanofluid electric conductivity .m
Viscoplastic parameter
Similarity variable
Dimensionless temperature
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hnf
Subscripts Cu
Copper
Fe3O4 Iron oxide 27
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Hybrid nanofluid
w
Wall
Infinity
Acknowledgement This research is supported by China Postdoctoral Science Foundation (2018M643156) and
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the National Natural Science Foundation of China (11571240).
[1]
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PII: DOI: Reference:
S0378-4371(19)32193-4 https://doi.org/10.1016/j.physa.2019.123957 PHYSA 123957
To appear in:
Physica A
Received date : 21 April 2019 Revised date : 28 September 2019 Please cite this article as: A. Hussanan, M. Qasim and Z.-M. Chen, Heat transfer enhancement in sodium alginate based magnetic and non-magnetic nanoparticles mixture hybrid nanofluid, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2019.123957. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
*Highlights (for review)
Journal Pre-proof Highlights:
Flow of non-Newtonian sodium alginate-based Cu-Fe3O4 hybrid nanofluid is analyzed.
Constitutive Equations of Casson fluid model are used examined the viscoplastic
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characteristics of sodium alginate Effect of viscous dissipation is also accounted.
Governing equations are simplified using boundary layer approximations.
Exact solution of self-similar equations are computed in terms of hypergeometric
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functions.
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The effects of various physical parameters are discussed through graphs.
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Abid Hussanan1,2, Muhammad Qasim3 and Zhi-Min Chen1* School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China
Department of Mathematics, COMSATS University Islamabad (CUI) Park Road, Tarlai Kalan, Islamabad 455000, Pakistan *
Corresponding Author: [email protected]
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3
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Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen, 518060, China
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Journal Pre-proof *Manuscript Click here to view linked References
HEAT TRANSFER ENHANCEMENT IN SODIUM ALGINATE BASED MAGNETIC AND NON-MAGNETIC NANOPARTICLES MIXTURE HYBRID NANOFLUID
School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China
2
Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and
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1
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Abid Hussanan1,2, Muhammad Qasim2, Zhi-Min Chen1*
Guangdong province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen, 518060, China 2
Department of Mathematics, COMSATS University Islamabad (CUI) Park Road,
*
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Tarlai Kalan, Islamabad 455000, Pakistan
Corresponding Author: [email protected]
Abstract: Sodium alginate (SA) based hybrid nanofluids are novel new generation of fluids for heat transfer. The thermo-physical properties of these fluids are very classic in comparison to common fluids. This study aims to examine the heat transfer enhancement in viscoplastic non-
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Newtonian based Cu-Fe3O4 hybrid nanofluid, flowing over a stretching/shrinking sheet. SA is being used as a non-Newtonian, viscoplastic base fluid with the addition of Cu and Fe3O4 as non-
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magnetic and magnetic nanoparticles. In the formulation of the mathematical model, Casson fluid model is exploited to examine the viscoplastic characteristics of SA. The effective thermal conductivity of Cu-Fe3O4 hybrid nanofluid calculated from the Maxwell model (for nanofluid). The exact solution of the nonlinear flow equation is obtained, and the solution of the heat transfer equation is expressed in hypergeometric function through Maple. The effects of
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consequential parameters such as magnetic parameter, Prandtl number, Casson parameter, Eckert number, and nanoparticles volume fraction on velocity and temperature field are examined. The result of this study suggests that SA based fluid should be used to obtain high rates of heat transfer.
Keywords: Hybrid nanofluid; sodium alginate; thermal conductivity; viscous dissipation.
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1.
INTRODUCTION
Most of the materials that we encounter in our daily lives are neither perfectly elastic
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solids nor simply Newtonian fluids and attempts often fail to describe these materials as being either fluid or solid. Examples of these materials include foams, syrups, nail polish, yogurt, cosmetics and toothpaste. These materials require critical shear stress to overcome before flow
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can occur in the system. If stress magnitude less than the critical value of shear stress applied to the fluid, it exhibits like a solid. Thus, shear stress must exceed a critical value known as the yield stress for the fluid to flow. The idea of yield stress was originally introduced by Bingham in the 1920s. Papanastasiou [1] applied modified constitutive relation for the Bingham model in
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both yielded and rigid regions of the flow field. The linear stability analysis of Bingham fluid (when the yield stress approaches zero) was performed by Métivier et al. [2]. Park and Liu [3] investigated experimentally as well as analytically oscillatory pipe flows of Bingham yield stress fluid. Some related published data on Bingham fluid and yield stress can be found in Shepherd et al. [4]; Zengeni [5]; Dutta et al. [6]. One of the popular generalizations of the Bingham fluid model in shear flow is Casson fluid model [7]. Several Casson fluid models have been proposed, Dash et al. [8]; Qasim and Noreen [9]; Reddy et al. [10]. Mukhopadhyay [11] analyzed Casson
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fluid flow over a nonlinearly stretching sheet with yield stress. Khan et al. [12] examined the yield stress in Casson fluid flow over static and moving flat plates. Recently, the effect of yield
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stress on Rayleigh-Bénard convection in Casson fluid was investigated by Aghighi et al. [13].
A nanofluid is the suspension of nanoparticles into the conventional base fluid. The concept of nanofluid was first introduced by Choi [14]. In recent developments, Rashidi et al. [15] studied the heat transfer MHD nanofluid flow due to a rotating porous disk with entropy
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generation effect. Turkyilmazoglu [16] examined the unsteady convection heat transfer nanofluids flow over a vertical flat plate with both isothermal and isoflux conditions. Sheikholeslami et al. [17] studied steady forced convection Fe3O4/water nanofluid flow in a semi-annulus enclosure with the non-uniform magnetic field. Rashad et al. [18] numerically studied the effect of internal heat generation on Cu/water nanofluid free convection flow inside a rectangular cavity under a magnetic field. Turkyilmazoglu [19] studied two-dimensional laminar free nanofluid jet and for a circular axisymmetric free nanofluid jet with five different 2
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nanoparticles. Sheikholeslami et al. [20] studied the magnetic field effect on the convective Fe3O4/water nanofluid flow with elliptical heat fluxes. Sheikholeslami et al. [21] studied the CuO/water nanofluid PCM solidification through an enclosure with V-shaped fins. Their work
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was then updated by Sheikholeslami and Mahian [22] who studied the Lorentz force effects on the solidification of inorganic nanoparticles enhanced PCM using CuO/water nanofluid inside a porous annulus. Sheikholeslami [23] examined entropy generation in natural convection
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Fe3O4/water ferrofluid inside a porous cavity. Sheikholeslami et al. [24] analyzed the entropy generation for turbulent heat transfer CuO/water nanofluid in a pipe. Further studies on heat transfer enhancement using nanofluid are to be found in Turkyilmazoglu [25]; Hussanan et al. [26]; Sheikholeslami [27]; Zyła et al. [28]; Sheikholeslami [29]; Turkyilmazoglu [30]; Nadeem
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et al. [31] and Sheikholeslami et al. [32].
Nowadays, hybrid nanofluids are the reasonable new generation of nanofluids, which can be developed either in mixture or composite form in the base fluids. A single material does not exhibit all the characteristics required for a particular purpose. For example, metal nanoparticles have high thermal conductivity as compared to metal oxide nanoparticles but less stability [33]. In that case, hybrid nanofluids may offer good heat transfer enhancements as compared to
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nanofluids. In the last few years, these nanofluids have been used widely in various heat transfer applications such as nuclear system cooling, cooling and heating in buildings, thermal storage,
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cooling of transformer oil and tubular heat exchanger [34]. Recently many experimental researches have been done on hybrid nanofluid. Esfe et al. [35] conducted an experimental study on thermal conductivity and viscosity of Ag-MgO nanoparticles dispersed into water with equal nanoparticle volume fraction (50% Ag and 50% MgO by volume). Their results showed that thermal conductivity and viscosity increased with increasing volume concentration. Afrand et al.
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[36] used Fe3O4-Ag/EG hybrid nanofluid at different shear rates with temperatures from 25 0C to 50 0C. Their results showed Newtonian behavior at solid volume fractions less than 0.3% and non-Newtonian at 0.6% to 1.2%. Moldoveanu et al. [37] studied the rheological behavior of Al2O3, TiO2 nanofluids and their hybrid, dispersed into water and found that nanofluids as well hybrid nanofluid acted like non-Newtonian at room temperature. Some other types of nanoparticles used in hybrid nanofluids followed by their articles are: Al2O3-MWCNTs/water,
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Nine et al. [38]; MgO-MWCNT/ethylene glycol, Soltani and Akbari [39]; Cuo-MWCNT/engine oil, Aghaei et al. [40]; Cu-Al2O3/water, Afridi et al. [41].
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Thermal properties of hybrid nanofluids depend on size, dispersibility of nanoparticles, preparation method, purity and compatibility of nanoparticles and the base fluids. The selection of base fluids is very important in the hybrid nanofluids heat transfer enhancement. Water,
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engine oil, ethylene glycol and ethylene/water mixtures are commonly used base fluids in the studies of, Hussanan et al. [42]; Zyła et al. [43]; Li et al. [44]; Sheikholeslami et al. [45]; Zyła et al. [46]; Hussanan et al. [47]; Swalmeh et al. [48] and Sheikholeslami et al. [49]. Additionally, Hayat and Nadeem [50] used Ag-CuO/water and Iqbal et al. [51] used Cu-CuO/water hybrid
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nanofluids for heat transfer enhancement. Hussain et al. [52] studied MHD mixed convection flow in an open cavity with a horizontal channel using Al2O3-Cu/water hybrid nanofluid. Numerical analysis of hybrid water based nanofluid in a semicircular cavity was studied by Chamkha et al. [53] with a focus on improving the heat transfer enhancement. Sheikhzadeh et al. [54] examined the heat transfer flow of Ag-MgO/water micropolar hybrid nanofluid in a permeable channel. Mollamahdi et al. [55] studied Al2O3-Cu/water micropolar hybrid in an expanding or contracting walls porous channel under a magnetic field. Amala and Mahanthesh
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[56] examined the effect of Hall current on Cu-Al2O3/water hybrid nanofluid in the existence of nonlinear convection. In all of the above-mentioned studies, the base fluid was assumed to be
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Newtonian.
Nowadays, sodium alginate, a non-Newtonian fluid, has gained much attraction due to its industrial and biomedicine applications such as textile, paper production, tissue formation, food industry and pharmaceuticals. Hatami and Ganji [57] investigated the SA-TiO2 nanofluid flow
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through a porous media between two coaxial cylinders, and it was found that the nanoparticle concentration increased near the inner cylinder wall. Hatami and Ganji [58] studied sodium alginate based Cu and Ag non-Newtonian nanofluid flow between two vertical flat plates, the effects of natural convection was also analyzed. Tlili et al. [59] examined two different types of nanofluids by dispersing Al2O3 and Cu nanoparticles in sodium alginate (SA) over a stretching cylinder. The impact of viscous dissipation on the unsteady boundary layer flow of viscoplastic
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Casson ferrofluid over a semi-infinite vertical plate with leading-edge accretion/ablation was investigated by Hussanan et al. [60].
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The aim of the present work is to explore the heat transfer and flow structure in nonNewtonian hybrid nanofluid with viscoplastic behavior flowing over a stretching/shrinking sheet. The base fluid is sodium alginate and nanoparticles, namely, Cu and Fe3O4 as non-magnetic and
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magnetic nanoparticles are added. Instead of the commonly used Newtonian fluid models, a viscoplastic non-Newtonian fluid model is employed for sodium alginate based Cu-Fe3O4 hybrid nanofluid which makes this study unique and the results are more realistic and practically useful. The thermophysical properties of Cu and Fe3O4 nanoparticles and SA based hybrid nanofluid
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have been summarized in Table 1.
Table 1: Thermophysical properties of Cu, Fe3O4 and SA.
kg m3
C p J kg K
k W mK
8933
385
401
5.96 107
5180
670
9.7
2.5 104
0.6376
2.6 10 4
0.613
0.05
Cu
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Fe3O4 SA H2 O 997.1 2.
4175
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989
4179
.m
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PROBLEM FORMULATION
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Two-dimensional boundary layer steady flow of hybrid nanofluid (Cu-Fe3O4/SA) past a stretching/shrinking sheet is considered. The x and y are Cartesian coordinates taken along the sheet and normal to it, respectively. The stretched or shrunk velocity is assumed as uw ( x) ax , where a is a constant, as shown in Figs. 1(a) and 1(b). It is also assumed that the temperature of the sheet is Tw x T Ax2 [61]. Initially, Cu nanoparticles of volume fraction Cu range from 0.01 to 0.04 are dispersed into the sodium alginate (SA) to make Cu/AS nanofluid. Thus, to 5
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develop the hybrid nanofluid (Cu-Fe3O4/SA), magnetite Fe3O4 (iron oxide) nanoparticles with
different volume fraction Fe3O4
are suspended into Cu/AS nanofluid. The rheological behavior
of Cu-Fe3O4/SA hybrid nanofluid followed by [53, 56] can be described in the form of
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continuity, momentum, and energy equations V = 0
V
(1) (2)
E V E q . t
(3)
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hnf V V p div ij hnf b + J B, t
where represents viscous dissipation term and all other symbols and quantities are given in
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nomenclature. The effective density nf of convectional nanofluids is defined as
nf 1 f np ,
(4)
the above relation for effective density was firstly given by Pak and Cho [62] followed in the studies [41, 51]. The mixture density formula for Cu-Fe3O4/SA hybrid nanofluid [63] hnf is
hnf 1 f np np ,
(5)
np
and np (for hybrid nanofluid np 1, 2 ) correspond to base fluid (SA) and
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where f
nanoparticles (Cu, Fe3O4), is the combination of dispersed Cu nanoparticles volume fraction
and Fe3O4 nanoparticles volume fraction Fe3O4
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Cu
into sodium alginate based hybrid
nanofluid expressed as
Cu Fe O . 3
(6)
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In case of Cu-Fe3O4/SA hybrid nanofluid, equation (5) becomes
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hnf 1 Cu Fe O f Cu Cu Fe O Fe O . 3
4
3
4
3
4
(7)
The velocity vector for present flow problem is u u x, y V v v x , y . w 0
The constitutive relationship for viscoplastic non-Newtonian fluid, given by Casson [7], is 6
(8)
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y 2 B eij , c 2 ij , y eij , c 2 B 2 c
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(9)
Non-magnetic Cu nanoparticles
Size: 1-100 nm
y
Thermal boundary layer
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Magnetite Fe3O4 nanoparticles
Momentum boundary layer B0
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B0
x
u
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Figure 1(a): Stretching Case 0
y
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B0
B0
x u
Figure 1(b): Shrinking Case 0
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where eij eij and eij is the i, j th component of deformation rate is 1 vi v j , 2 x j xi
(10)
with 1 i, j 2 . In view of Ohm’s law
J = hnf E1 + V B
of
eij
(11)
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where E1 is the electric field, B is the combination of imposed magnetic field B0 and induced magnetic field b0 and hnf is the electrical conductivity of Cu-Fe3O4/SA hybrid nanofluid [64] can be written as
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hnf f
Cu Cu Fe3O4 Fe3O4 3 Cu Fe3O4 f 1 Cu Cu Fe3O4 Fe3O4 2 Cu Cu Fe3O4 Fe3O4 Cu Fe3O4 f f
,
(12)
here is given in equation (6), Cu and Fe3O4 are electrical conductivities of Cu and Fe3O4 nanoparticles, respectively. Equation (11), with the assumptions E1 = 0 and b0 = 0 , modifies to
J = hnf V B0
(13)
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As magnetic field of uniform strength is taken along y -axis, therefore, B0 0, B0 ,0 . Finally,
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J B hnf B02 V
(14)
In the absence of pressure gradient, body forces and using equations (9) and (14), the continuity
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and motion equations for steady incompressible flow are
u
u v 0, x y
u u v hnf x y
1 2u hnf 2 B u, 1 2 hnf 0 y
(15)
(16)
where B 2 c y is the non-Newtonian viscoplastic parameter, which depends on yield stress and flow only occurs in the system when critical shear stress is greater than the yield stress. The total energy per unit volume consists of internal energy and kinetic energy is
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V2 E hnf e . 2
(17)
The specific internal energy e defined by [65] is
p
hnf
.
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e C p ,hnf T Here
1 C p f Cu C p Cu Fe O C p Fe O
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C p ,hnf
3
4
3
4
hnf
.
(18)
(19)
is the heat capacity of Cu-Fe3O4/SA hybrid nanofluid obtained from heat capacity of nanofluid at constant pressure [66]. In terms of temperature field, the heat flux is
here
Kf
(20)
Cu KCu Fe O K Fe O 2 K f 2 Cu K Cu Fe O K Fe O 2 K f , Cu KCu Fe O K Fe O 2 K f Cu K Cu Fe O K Fe O K f 3
K hnf
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q K hnf T ,
4
3
3
4
4
3
3
4
3
4
(21)
4
3
4
3
4
is the thermal conductivity of Cu-Fe3O4/SA hybrid nanofluid which is obtained from Maxwell
equation (3) becomes
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model and is also given in [50]. The pressure is approximately uniform and T T x, y ,
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T T T u v t x x
1 2T K . hnf 2 y C p hnf
(22)
In the presence of viscous dissipation, above equation in case of steady flow modifies to
with
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K hnf T T u v x y C p
hnf
hnf
hnf 2T 2 y C p
1
Cu
f Fe3O4
2
hnf
2.5
1 u 1 , y
(23)
.
(24)
The boundary conditions are u uw ( x), v 0 at y 0; u 0 as y ,
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(25)
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T Tw x T Ax2 at y 0; T T as y .
(26)
To simplify the equations (16) and (23), we introduce the similarity transformations as
f
, u axF , v a f F ,
T T . Tw T
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a
y
(27)
Using above similarity variables, equations (16) and (23) can be written as
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hnf f 1 f hnf 2 F 0, 1 F F F F M f hnf hnf f C p f 1 1 K hnf C p f F 2 F hnf 1 EcF 2 0, Pr K f C p f C p hnf hnf
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with boundary conditions are
(28)
(29)
F 0, F , 1 at 0,
(30)
F 0, 0 as ,
(31)
Here
C p f f f B02 uw2 a2 M , Pr , Ec . a f Kf A C p Tw T C p f f 2
number Nu
hnf
1 u 1 y
x T , Nu K hnf K f Tw T y y 0
1
K hnf 1 1/2 0 , 1 F 0 , Re x Nu Kf
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1 f uw2
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The quantity of engineering interest include the skin friction coefficients C fx and Nusselt
C fx
, y 0
(32)
in dimensionless form are
1
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Re1/2 x C fx
Cu
Fe2O3
2.5
(33)
where Re x uw x / f is the local Reynolds number.
3.
MOMENTUM EQUATION SOLUTION
According to Cortell [67] and Turkyilmazoglu [68], the assumed exact solution of equation (28) 10
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F
1 e ,
(34)
where Fe3O4
Cu
2.5
1
hnf M hnf f f
yields
F ( )
1
Cu Fe3O 4
2.5
1
hnf f
hnf 2.5 hnf 1Cu Fe3O4 M f 1 f 1 e . (35) M hnf f
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Finally, velocity expression is F ( ) e
F ( )
Cu
Fe3O4
1
1Cu Fe3O4
hnf hnf M f 1 f
2.5
hnf hnf M f f
,
(36)
1Cu Fe3O4 e
hnf hnf M f 1 f
2.5
.
(37)
ENERGY EQUATION SOLUTION
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4.
1
2.5
p ro
,
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1
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To find the solution of energy equation, we substitute F , F F given in equations (35) to (37) in equation (29), we obtain
1 Pr
k hnf kf
C p hnf e 2 e C p f
1 2 1 Ec e 0. f
(38)
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hnf
Taking t e , the above equation becomes
C p hnf 1 hnf 1 k hnf 1 t t 2 t t t t Pr k f C p 2 f f
1 2 1 Ec t 0,
(39)
and boundary conditions are
0 0, 1 1, 11
(40)
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where 1 2 and
3
4
2.5
hnf , 2 M 1 Cu Fe3O4 1 f
Using Maple toolbox, obtained solution of equation (39) is
2.5
hnf . 1 f
of
1 1 Cu Fe O
p ro
3 3 4 3 4 4 43 t t hypergeometric , , t e C1 3 3 3 3 4 4 43 43 t hypergeometric 3 , , t t e C2 3 3
where
3
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1 2 4t 5 3 24 , 2 4
(41)
C p hnf khnf 1 1 , 4 , 5 hnf 1 Ec 2 2 k f Pr f C p f
3 4 3 4 1 C1 5 3 2 4 hypergeometric 3 , ,0 2 4 3 3
1
1
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3 4 4 43 C2 hypergeometric 3 , , e 3 3 1
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3 4 4 43 1 5 3 4 hypergeometric 3 , , e 2 42 3 3
3 4 3 4 4 1 5 3 2 4 hypergeometric 3 , , 2 4 3 3 3 1
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4 4 hypergeometric 3 , 3 , . 3 3
Using boundary conditions (40), we obtained
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3 3 4 3 4 4 43 t 1 5 3 4 t hypergeometric , , t e 2 42 3 3 3
1
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3 4 3 4 1 5 3 4 2 4t hypergeometric 3 , , 0 2 42 3 3
1
4 4 43 43 t 3 4 4 43 hypergeometric 3 , 3 , t t e hypergeometric 3 , , e 3 3 3 3
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3 4 4 43 43 t hypergeometric 3 , , t t e 3 3
1
4 4 43 5 3 2 4 hypergeometric 3 , 3 , e 2 4 3 3
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3 4 4 43 43 t 3 4 4 hypergeometric 3 , , t t e hypergeometric 3 , , 3 3 3 3
3 4 3 4 4 1 5 3 2 4 hypergeometric 3 , , . 2 2 4 3 3 3
1
(42)
The expression for temperature t obtained in equation (42) is function of t . The temperature
GRAPHICAL RESULTS AND DISCUSSION
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5.
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solution in term of can be obtain by substituting t e in equation (42).
A Maple (software) toolbox has been used to explore the physical behaviors arising due to the existence of Casson parameter , magnetic parameter M , Eckert number Ec , Prandtl number Pr , Cu volume fraction Cu and Fe3O4 volume fraction Fe3O4 with the flow field. Figs.
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2(a) and 2(b) illustrate the variation of velocity at different values of Casson parameter along stretching sheet for Cu-Fe3O4 hybrid nanofluid when M 0 and M 0 , respectively. It is noted in Fig. 2(a), as increases the velocity decreases and the thickness of the boundary layer gets larger. The same behavior appears in Fig. 2(b) which concerns velocity variation, when M 0 . Further, the fluid velocity declines sharply in the presence of a magnetic field M 0 . The variations of velocity with Casson parameter for both M 0 and M 0 in shrinking case are
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illustrated in Figs. 3(a) and 3(b). As the strength of the magnetic field increases, the velocity of Cu-Fe3O4 hybrid nanofluid declines sharply for M 0 . Physically, the presence of a transverse magnetic field (normal to the flow direction) has a tendency to create the drag, (Lorentz force)
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which tends to resist the flow. The velocity variation of Fe3O4/SA nanofluid Cu 0 for both
M 0 and M 0 is shown in Figs. 4(a) and 4(b) for different values of Casson parameter , when 0 . It is seen in Fig. 4(a) that velocity is reduced with an increase in Casson parameter
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, for M 0 . Further, it is shown in Fig. 4(b) that as M increases M 0 , the impact of becomes more significant. We also noticed in all above figures that velocity of Cu-Fe3O4 hybrid nanofluid approaches to free stream condition at shorter distances, when M 0 as compared to
M 0 . As the value of Casson parameter increases, an increase in temperature is observed at
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M 0 (Fig. 5(a)). Moreover, there is also an increase in temperature, when M 0 , see Fig. 5(b). In addition, there is a large overshoot in temperature at M 0 through the variations of . Fig. 6(a) and 6(b) illustrate the variation of the temperature against the stretching parameter
0
for various values of Eckert number Ec , when M 0 and M 0 , respectively. These
figures reveal that in both cases, the temperature of Cu-Fe3O4 hybrid nanofluid rises with the enlargement of Eckert number Ec . Physically, the dissipative frictional forces between layers
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enhance with the enlargement of Ec and consequently, the temperature of Cu-Fe3O4 hybrid
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nanofluid rises.
Figs. 7(a-d) present the effect of the magnetic parameter M , Eckert number Ec and Casson parameter on Nusselt number Nu for Cu-Fe3O4 hybrid nanofluid. Nusselt number Nu reduces with augment in Casson parameter , deducing that sodium alginate based fluid should
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be used to obtain the highest rate heat transfer. Figs. 8(a) and 8(b) show the effect of Eckert number Ec and Cu volume fraction on Nusselt number Nu for Cu/SA nanofluid in both cases
M 0 and M 0 , respectively. The augments in Lorentz forces lead to stronger conduction mechanisms. Nusselt number Nu reduces with the increase of the magnetic field M 0 . In Figs. 9(a) and 9(b), the effects of the same parameters on Nusselt number Nu in case Fe3O4/SA nanofluid are depicted. The volume fraction of nanofluid induces the thermal boundary layer thickness to augment. Hence, Nusselt number Nu reduces with augment in Fe3O4 volume 14
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fraction. Further, it is found that Fe3O4 nanoparticles have low thermal conductivity
9.7 W
mK as compare to Cu 401 W m K , but give the highest rate of heat transfer, when
of
M 0. For showing the comparison of non-Newtonian based Cu-Fe3O4/SA and Newtonian
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based Cu-Fe3O4/water hybrid nanofluids, skin friction coefficient C fx and Nusselt number Nu are calculated in Tables 2 and 3, respectively. From Table 2, it is found that the skin friction coefficient C fx of Cu-Fe3O4/SA hybrid nanofluid decreases with the increase of Cu volume fraction when M 0 or M 0 . The opposite trend is noted for Fe3O4 volume fraction. Further,
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the results indicate that Nusselt number Nu tends to decrease as Cu volume fraction of CuFe3O4/SA hybrid nanofluid increases and the same behavior is observed for Cu-Fe3O4/SA hybrid for Fe3O4 volume fraction. Comparing the magnetic nanoparticles based Fe3O4/SA nanofluid with non-magnetic nanoparticles Cu/SA nanofluid, it is observed that Fe3O4/SA nanofluid has better heat transfer capabilities in comparison of Cu/SA nanofluid. Table 3 shows that Nusselt number Nu for Cu-Fe3O4/water hybrid nanofluid decreases with an increase of Cu and Fe3O4 volume fractions and the same behavior is observed, when M 0 . Further, it is similar to the
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trends of Nusselt number Nu noted in Table 2 for Cu-Fe3O4/SA hybrid nanofluid. Further, the utilization of smaller nanoparticles volume fraction achieves better thermal performance. It
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means that Cu and Fe3O4 nanoparticles volume fraction is a key factor for heat transfer enhancement in Cu-Fe3O4/water or Cu-Fe3O4/SA based hybrid nanofluids. Besides, the skin friction coefficient C fx of Cu-Fe3O4/water hybrid nanofluid also decreases with the rise of Cu volume fraction and Fe3O4 volume fraction, when M 0 or M 0 . The accuracy and validation
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of analytical results are tested by comparing present results for skin friction coefficient C fx and Nusselt number Nu with those obtained previously by Khan et al. [69] in the case of Newtonian Fe3O4/water nanofluid as shown in Table 4. From this tabular result, it is noticed that there is a very close agreement between present results and those of the existing literature.
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Table 2: Skin friction coefficients C fx and Nusselt number Nu in case of non-Newtonian CuFe3O4/SA hybrid nanofluid, when 1 (stretching sheet) and Pr 6.45 (sodium alginate).
0.03
0.1
0.5
0 0.01 0.02 0.03 0.03
0.1
0.1
0.004612649171 0.004573550124 0.004546083004 0.004528851155 0.004395986631 0.004436253520 0.004480547207 0.004528851155 0.007907778187 0.007739209239 0.007604724007 0.007499068263 0.007304716397 0.007361011092 0.007425839663 0.007499068263 0.004528851155 0.017712557610 0.036835949560 0.007499068263 0.029329221500 0.060994563690 -
3.887798336 3.799543407 3.740564882 3.683601483 3.854452822 3.796043929 3.739118140 3.683601483 3.878508194 3.790224954 3.731208963 3.674199691 3.845422269 3.786892544 3.786892544 3.674199691 3.683601483 3.667219625 3.651223497 3.674199691 3.646842850 3.619666228 3.683601483 3.683593824 3.683578505 3.674199691 3.674179150 3.674138068
0 0.01 0.02 0.03 0.03
2
0 0.01 0.02 0.03 0.03
0
0.03
0 0.01 0.02 0.03 0.03
2
0.03
0.03
0
0.03
2
0.03
0.1
0.03
0.1 0.2 0.3 0.1 0.2 0.3 0.1
0.03
0.1
of
Nu
4
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2
C fx
3
p ro
0
Ec
Fe O
Pr e-
0
Cu
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M
0.5 1.0 2.0 0.5 1.0 2.0
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Table 3: Skin friction coefficients C fx and Nusselt number Nu in case of Newtonian CuFe3O4/water hybrid nanofluid, when 1 (stretching sheet) and Pr 6.2 (water).
0
Nu
0.03
0.5
0 0.01 0.02 0.03 0.03
0.9790270381 0.9584905564 0.9405645173 0.9248772436 0.9333404589 0.9298948614 0.9270851387 0.9248772436 1.6790399570 1.6228056390 1.5744629990 1.5326910700 1.5517904190 1.5439683070 1.537632539 1.532691070 -
2.984670350 2.905523263 2.851447643 2.801541024 2.900673587 2.866278956 2.833284100 2.801541024 2.020893628 1.991379882 1.976334178 1.963203296 1.963905535 1.963687919 1.963484446 1.963203296 2.801541024 2.251321819 1.150883411 1.963203296 0.7647930430 0.2854289310
0 0.01 0.02 0.03 0.03
4
0
0.03
0 0.01 0.02 0.03 0.03
2
0.03
0.03
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0 0.01 0.02 0.03 0.03
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C fx
3
0.5 1.0 2.0 0.5 1.0 1.2
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Ec
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0
Cu
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M
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Table 4: Comparison of our numerical findings for skin friction coefficients C fx and Nusselt number Nu in case of Newtonian Fe3O4/water nanofluid with those of the existing literature,
Khan et al. [69]
Fe O
4
Ec
0 0 0 1 1 1
0.01 0.10 0.20 0.01 0.10 0.20
0.1 0.1 0.1 0.1 0.1 0.1
Present results
C fx
Nu
1.03366 1.35914 1.79238 1.44703 1.77443 2.22700
3.62729 3.91965 4.26287 3.42446 3.71411 4.59009
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Pr e-
3
C fx
Nu
1.033657 1.359136 1.792374 1.447032 1.774433 2.227001
3.627292 3.919651 4.262875 3.424463 3.714116 4.590091
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M
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when 1 , Pr 6.2 and .
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(b)
Figure 2: Velocity variation of Cu-Fe3O4/Sa hybrid nanofluid with .
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(b)
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Figure 3: Velocity variation of Cu-Fe3O4/Sa hybrid nanofluid with .
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Figure 4: Velocity variation of Fe3O4/Sa nanofluid with .
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(b)
Figure 5: Temperature variation of Cu-Fe3O4/Sa hybrid nanofluid with .
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(b)
Figure 6: Temperature variation of Cu-Fe3O4/Sa hybrid nanofluid with Ec .
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(b) 0.3
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(a) 0.2
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(c) 0.4
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(d) 0.5
Figure 7: Effect of Eckert number Ec , magnetic parameter M and Casson parameter on Nusselt number Nu , when Pr 6.45 , 2 , Cu 0.01 and Fe3O4 0.01 .
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(a) M 0 (Cu)
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(b) M 2 (Cu)
Figure 8: Effect of Eckert number Ec , magnetic parameter M and Cu volume fraction Cu on
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Nusselt number Nu , when Pr 6.2 , 1 , 0.5 and Fe3O4 0.01 .
(a) M 0 (Fe3O4)
(b) M 2 (Fe3O4)
Figure 9: Effect of Eckert number Ec , magnetic parameter M and Fe3O4 volume fraction Fe3O4 on Nusselt number Nu , when Pr 6.45 , 1 , 0.5 and Cu 0.01 .
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CONCLUSIONS
Heat transfer enhancement characteristics of non-Newtonian SA based Cu-Fe3O4 hybrid nanofluid flow over a stretching/shrinking sheet is studied. Both magnetic (Fe3O4) and non-magnetic (Cu) nanoparticles are dispersed into the sodium alginate (SA) to make CuFe3O4 hybrid nanofluid. Outcomes of this study reveal that Fe3O4 nanoparticles have low
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thermal conductivity as compare to Cu, but give the highest rate of heat transfer in the presence of a magnetic field. The effect of viscous dissipation on the temperature field is
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more pronounced under the magnetic field. The SA based Cu- Fe3O4 hybrid nanofluid is found to have a higher rate of heat transfer, than water based hybrid nanofluid. Further, the utilization of smaller nanoparticles volume fraction achieves better thermal performance.
Nomenclature Positive constant
B0
Magnetic field intensity
b
Body force N
C p, f
SA heat capacity J kg 1 K 1
C p ,Cu
Cu heat capacity J kg 1 K 1
Pr e-
A
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C p ,Fe3O4 Fe3O4 heat capacity J kg 1 K 1
C p ,hnf Hybrid nanofluid heat capacity J kg 1 K 1 Eckert number
F
Stream function
F
Dimensionless velocity
Kf
SA thermal conductivity W m 1 K 1
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Ec
Jo
K Fe3O4 Fe3O4 thermal conductivity W m 1 K 1 K Cu
Cu thermal conductivity W m 1 K 1
K hnf
Hybrid nanofluid thermal conductivity W m 1 K 1
M
Magnetic parameter
p
Pressure Pa
Pr
Prandtl number 26
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Tw
Sheet temperature K
T
Ambient temperature K
Stretching/shrinking parameter
Hybrid nanofluid volume fraction
Cu
Cu volume fraction
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T
Fe O Fe3O4 volume fraction 3
4
Viscous dissipation
B
Plastic dynamic viscosity kg m 1 s 1
f
SA dynamic viscosity kg m 1 s 1
hnf
Hybrid nanofluid dynamic viscosity kg m 1 s 1
f
SA density kg m 3
Cu
Cu density kg m 3 3
4
Pr e-
Fe O Fe3O4 density kg m 3
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Hybrid nanofluid density kg m 3
y
Yield stress kg m 1 s 2
f
1 SA electric conductivity .m
Cu
1 Cu electric conductivity .m
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hnf
1 Fe O Fe3O4 electric conductivity .m 3
4
1 Hybrid nanofluid electric conductivity .m
Viscoplastic parameter
Similarity variable
Dimensionless temperature
Jo
hnf
Subscripts Cu
Copper
Fe3O4 Iron oxide 27
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Hybrid nanofluid
w
Wall
Infinity
Acknowledgement This research is supported by China Postdoctoral Science Foundation (2018M643156) and
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the National Natural Science Foundation of China (11571240).
[1]
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