Accepted Manuscript Title: Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet Author: Kai-Long Hsiao PII: DOI: Reference:
S1359-4311(16)00033-8 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.12.138 ATE 7556
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
10-10-2015 31-12-2015
Please cite this article as: Kai-Long Hsiao, Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.12.138. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
Stagnation Electrical MHD Nanofluid Mixed Convection with Slip Boundary on a Stretching Sheet Kai-Long Hsiao Department of the Digital Recreation and Game Design, Taiwan Shoufu University, 168, Nansh Li, Madou District, Tainan City, Taiwan, Republic of China. Email:
[email protected],
The tel/fax numbers: 886911864791/88662896139
Highlights: ► Stagnation EMHD nano energy conversion for extrusion stretching sheet processing. ► Conjugate heat conduction, convection and mass transfer problem have been solved. ► Flow and heat mass transfer of nanofluids important parameters have been examined. ► Excellent calculation results have been achieved and compared with others study. Abstract In this study, the stagnation nano energy conversion problems have been completed for conjugate mixed convection heat and mass transfer with Electrical Magneto Hydrodynamic (EMHD) and heat source/sink effects nanofluid flow field over a slip boundary stretching sheet surface. The physical phenomena are varied which are depended on different kinds of factors. All of the important nano energy conversion parameters dominance
of
the
S0
,M, E, Gt, Gc, λ, Pr, Sc, Ncc, S and δ which have been represented the
magnetic
energy
effect,
electric
effect,
mixed
convection
effect,
heat
generation/absorption energy effect, heat transfer effect, mass diffusion effect, heat conduction-convection effect and slip boundary effects, respectively. The similarity transformation and a modified Finite-Difference method are used to analyze present nano energy conversion system thermal energy conversion problem. The non-linear ordinary equations of the corresponding flow field momentum, temperature, concentration equations and plate sheet heat conduction equation are derived by employing the similarity transformation technology. The dimensionless non-linear ordinary equations have been composed of momentum, temperature, concentration and plate sheet heat conduction equations which have been solved numerically by an improved finite difference technique. Keywords: EMHD, Stagnation, Nanofluid, Mixed convection, Heat source/sink
1
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1.
Introduction Stagnation nano energy conversion problem about nanofluids passing a slip stretching sheet have been
become of increasing importance in the last few years. Qualitative analyses of studies have significant bearing on several industrial applications, such as polymer sheet extrusion from a dye, drawing of plastic films, etc. It needs cooling the stretching sheet when the manufacturing processing at very high temperature. There are some ways cooling the stretching surface, the present study just is chosen that nano fluid as necessary for more effective cooling of the stretching surface. Maybe it is not easily available, but for specific requirement it is still important for using. And also, the fluids have been processed many types of effects (i.e. magnetic energy, electrical energy, buoyancy energy and heat generation/absorption energy) into the problem. At first, there are some applied energy applications for related heat and mass transfer aspect. Hsiao [1-2] studied about heat and mass mixed convection for MHD viscoelastic fluid past a stretching sheet with Ohmic dissipation. Hsiao [3] investigated conjugate heat transfer for mixed convection and Maxwell fluid on a stagnation Point problem. Hsiao [4] provided for energy conversion conjugate conduction-convection and radiation over non-linearly extrusion stretching sheet with physical multimedia effects. Hayat et al. [5] studied about inclined magnetic field effect in stratified stagnation point flow over an inclined cylinder problem, it is considered for an inclined cylinder thermal energy system application to industry application. The study of non-Newtonian fluid flows past stretching sheet was provided by Hartnett [6]. Aman and Ishak [7] studied the problem of mixed convection boundary layer flow adjacent to a stretching vertical sheet in an incompressible electrically conducting fluid. These are related studies to the present investigation about nanofluid flow. There were some studies about similar boundary layer problem, the non-liner stretching sheet heat and fluid energy conversion systems investigation. Vajravelu and Soewono [8] had solved the fourth order non-linear systems arising in combined free and forced convection flow over a stretching sheet. Afify [9] studied heat and mass transfer of a viscous, incompressible and electrically conducting fluid over a stretching surface. Cortell [10] studied effects of radiation on the thermal boundary layer over a non-linearly stretching sheet. Kechil and Hashim [11] studied series solution of flow over non-linearly stretching sheet with chemical reaction and magnetic field. Cortell [12] studied viscous flow and heat transfer over a nonlinearly stretching sheet. Vajravelu [13] investigated viscous flow over a nonlinearly stretching sheet. Liu and Liu [14] studied Stokes’ Second Problem with arbitrary initial phase flow. Recently, the slip boundary layer problems for flow passing the sheet were studied by some researchers. Vajravelu et al. [15] studied second order slip flow and heat transfer over a stretching sheet with non-linear Navier boundary condition problem. Ibrahim and Shankar [16] studied MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with 2
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velocity, thermal and slip boundary conditions. Alin et al. [17] investigated flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip. Das [18] studied slip flow and convective heat transfer of nanofluids over a permeable stretching surface. Turkyilmazoglu [19] investigated heat and mass transfer of MHD second order slip flow. Turkyilmazoglu [20] studied dual and triple solutions for MHD slip flow of non-Newtonian fluid over a shrinking surface. Raftari and Vajravelu [21] solved the problem for homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall. Hayat et al. [22] studied about slip flow and heat transfer of a second grade fluid past a stretching sheet through a porous space. Khan and Pop [23] investigated boundary-layer flow of a nano fluid past a stretching sheet problem. All of above studies [15 – 23] are not considered for the conjugate heat transfer problem, but towards the slip boundary layers fluid flow for many kinds of features. There are some related boundary layer studies for the exponentially shrinking sheet. Bhattacharyya and Vajravelu [24] investigated stagnation-point flow and heat transfer over an exponentially shrinking sheet. Sajid and Hayat [25] studied influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet. Pop et al. [26] studied boundary layer flow and heat transfer over an exponentially shrinking vertical sheet. For nanofluid flow, Khan and Pop [27] studied boundary-layer flow of a nanofluid past a stretching sheet. For applied energy fields, some of studies about latent-heat storage and photo catalytic effectiveness problems were provided by Chai et al. [28]. Jiang et al. [29] designed magnetic microcapsules based on n-eicosane core and Fe3O4/SiO2 hybrid shell for dual-functional phase change materials. Jiang et al. [30] studied about thermal properties of paraffin microcapsules modified with nano-Al2O3. Yu et al. [31] investigated calcium carbonate shell for enhancement of thermal conductivity problems. There were many energy conversion problems about fins becoming importance, Shi and Dong [32] investigated entropy generation and optimization of laminar convective heat transfer in a micro channel with staggered arrays of pin fin structure problem. Elsafi and Gandhidasan [33] studied about comparative study of double-pass flat and compound parabolic concentrated photovoltaic–thermal systems with and without fins and obtained a good result for different conditions. Syed et al. [34] numerically studied of an innovative design of a finned double-pipe heat exchanger with variable fin-tip thickness and find them a good effect. Hazarika et al. [35] analytically studied the solution to predict performance and optimum design parameters of a constructed T-shaped fin with simultaneous heat and mass transfer. Turkyilmazoglu [36] studied about stretching/shrinking longitudinal fins of rectangular profile and heat transfer and related parameter effects. From Refs. [32] to [36] were investigated the energy conversion problems about the fins, but present study was an extension work and application to an extrusion slip stretching sheet energy conversion problem. For the heat source/sink feature 3
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studies, Mahantesh et al. [37] investigated heat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperature. Mahantesh et al. [38] investigated studied for heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink problems. For the nanofluids heat transfer problems, Noghrehabadi et al. [39] studied Analyze of fluid flow and heat transfer of nanofluids over a stretching sheet near the extrusion slit. Hsiao [40] investigated a nofluid flow with multimedia physical features for conjugate mixed convection and radiation. For the thermal energy conversion problems, Cortell [41] studied magneto-hydrodynamic flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption, a very complex heat energy conversion problem in a non-Newtonian flow field. Haro et al. [42] investigated about heat transfer and entropy generation in the parallel plate flow of a power-law fluid with asymmetric convective cooling related energy conversion problem. Hachicha et al. [43] considered about wind speed effect on the flow field and heat transfer around a parabolic trough solar collector couple energy conversion problem. Vajravelu et al. [44] investigated about flow and heat transfer of nanofluids at a stagnation point flow over a stretching/shrinking surface in a porous medium with thermal radiation related energy conversion heat transfer problem. Hayat et al. [45] studied boundary layer flow of Carreau fluid over a convectively heated stretching sheet energy conversion related problems, it is also a novel non-Newtonian fluid flow couple with thermal boundary layer heat transfer problems. Liu and Andersson [46] investigated heat transfer over a bidirectional stretching sheet with variable thermal conditions; it is mainly focus on the thermal energy conversion related problems at the couple boundary layer fluid flow system.
Recently, there are some of studies about the nano energy heat transfer problems.
Hayat et al. [47] investigated Soret and Dufour effects in peristaltic transport of physiological fluids with chemical reaction problems. In the present investigation, a novel stagnation nanofluid nano energy conversion study for conjugate conduction-convection and heat source/sink problem have been performed. The term “nanofluid” refers to a liquid containing a suspension of submicron solid particles (nanoparticles). There are different model to handle such kind of problems. The nanoparticles have been used in nanofuids are typically made of nano metals (Al, Cu), carbides (SiC), nitrides (AlN, SiN), etc. The suspended particles are able to increase the thermal conductivity and heat transfer performance. In this study, using Brownian motion model to nanofluid and its properties used, From Eqs. (10) and (14) to solve the problem. All of the properties have been divided into fluid parts for example
kf ( c p ) f
and nanoparticle parts for example
( c p ) p
. A conjugate mixed convection heat and
mass transfer problem with the magnetic energy effect, electrical conductivity effect and heat source/sink 4
Page 4 of 26
energy effects passing a slip stretching surface have been processed. From the results, finding some better heat transfer effects which were made by some important parameters, such as
S0
, M, Gc, Gt, Pr, Sc, Nt, Nc, λ and
Ncc. 2.
Theory and Analysis
2.1 Flow field analysis In this study, consider an incompressible nanofluid past a slip boundary stretching sheet. The generalized continuity equation, momentum equation, fluid energy equation, fluid concentration equation and sheet conduction equation in three dimensional are formed on the basis of Ohm’s law and Maxwell’s equation with study of Electrical Magneto Fluid Dynamics (EMHD). The steady EMHD free convection and mass transfer flow of an electrically conducting viscous incompressible nanofluid through an infinite vertical sheet plate has been considered. The flow is assumed to be in the X-direction which is taken along the thin plate in the upward direction and Y-axis is normal to it. For whole flow field the assumptions which were made by (i) 2D incompressible boundary layer stagnation steady flow (ii) Brownian motion for nanofluid and mass diffusion (iii) mixed convection of thermal expansion by using Boussinesq approximation model (iv) heat source/ sink fluid flow (v) electrical and magnetic control flow (vi) one dimensional thin stretching sheet heat conduction model. The flow direction is toward the sheet, and then it is a stagnation flow phenomenon. From above assumptions the well-known Boussinesq approximation has been used to represent the buoyancy term. The heat source/sink, electrical conductivity effect and the magnetic force effect are also included. It has been also applied to the nano energy conversion thermal system. The fluid motion within the film is due to a slip boundary layer for stretching sheet. This study is divided two parts, the fluid and solid. All of fluid part properties have been indicated by the subs-nomenclature such as ( c p ) f
and the solid part properties have been indicated by the subs-nomenclature such as
( c p ) p
. Present
study is used the Brownian motion and thermophoresis effects model, so that it is one kind of nanoparticle mixed with fluid thermal system. The geometry of the problem is shown in Figure 1. The steady two-dimensional boundary layer equations for this fluid, in the usual notation, are u x
u
u x
v y
v
0,
u y
u
(1)
u x
u 2
f
y
2
B0 f
2
( U u ) g x t (T T ) g x c (C C )
E 0B0 f
(2)
These equations are governing for the electrical magneto fluid dynamics flow field, it is also a novel work for this related fields. Where (x, y) denotes the Cartesian coordinates along the sheet and normal to it, u and v are 5
Page 5 of 26
the velocity components of the fluid in the x and y directions, U is the free stream fluid velocity, respectively. The physical significance of the term
u
u x
is the fluid flow towards the vertical plat sheet by the free stream
velocity normally. It is a very larger and 100 percent fluid flow rate to the plat sheet, so that the heat transfer and fluid flow effect are good than any other oblique flow field. The term may change as
u
u x
S in
, when
θ=0 the term will be vanish, and it will be a flat plate flow , and θ=90 degree, then it is a stagnation flow, the term is
B0
u
u x
. The electrically conducting fluid past a flat heated sheet with a variable magnetic parameter
which has been applied normal to the sheet and σ is the electrical conductivity (assumed constant),
the electric field factor,
gx
is the magnitude of the gravity,
t
is the coefficient of thermal expansion,
c
E0
is
is the
coefficient of mass diffusion, T is the fluid temperature and C is the concentration, ∞ is the free stream notation, f
f
is the fluid density and
is the kinematic viscosity of the fluid. The boundary conditions to the problem
are u u w (x ) cx L
u 0
,
u y
y
as
at
ｖ= v w = a x ,
,
y =0,
(3)
,
Where L is slip parameter related to the surface stretching reference velocity, w is the wall notation, a and c are the proportional constants. Defining new similarity variables as y
a f
,
u a x f '( ),
ｖ= - a f f ( )
(4)
Defining the non-dimensional temperature and concentration
T T TW T C C CW C
and
and
T T A x ( )
,
C C B x()
(5)
,
Substituting into Eqs. (1) and (2) give f ''' S 0 ff '' f ' M (1 f ') G t G c M E 0 2
(6) B0
2
Where
U
S0 b [ 2
c L
a
] is
w
the stagnation parameter,
3
f
M
fa
is the dimensionless magnetic
''
f (0 )
6
Page 6 of 26
E0
E
parameter,
is the dimensionless electric parameter,
B0U
G
t
uw
Ag x t (
c L f ''( 0 )
thermal free convection parameter,
G
c
uw
Bg xc (
c L f ''( 0 )
) a
) a
is the dimensionless
3
f
is the dimensionless mass free convection parameter,
3
f
where a prime denotes differentiation with respect to the independent similarity variable η. The boundary conditions (3) becomes f S , f ' 1 f '' at 0
f 0
as
Where
S L
'
(7)
1
'
f (0 )
L
is the dimensionless slip parameter,
f
a f
is the dimensionless shear stress at the
stretched surface is defined as u w f y w
(8)
and we obtain form (4) and (8) a
W f
Where
3
(9)
x f ''( 0 )
f
is the dynamic viscosity of the fluid.
2.2 Heat convection analyses By using usual boundary layer approximations, the equation of the energy for temperature T in the presence of heat source/sink and viscous dissipation, is given by: u
T x
v
T y
kf
T
( c p ) f y
Where ρ is the density,
fluid,
( c p ) p
2
2
cp
Q0 ( c p ) f
( T T )
( c p ) p ( c p ) f
C T y y
DT T
(
T y
2
heat generation/absorption coefficient, ( c p ) p ( c p ) f
DB
m
=
k ( c p ) f
(10)
) ]
is the specific heat of a fluid at constant pressure,
is heat capacity of the nanoparticle,
diffusion coefficient,τ=
[D B
( c p ) f
is the heat capacity of the
is the thermal diffusivity, Q 0 is the dimensional
is the Brownian diffusion coefficient,
DT
is the thermophoresis
is the shear stress. Similarity solutions of Eq. (10) can be found by choosing
appropriate boundary conditions. For slip boundary conditions are
7
Page 7 of 26
T Tw k 1
T T
Where
T
Pr
(11)
is slip parameter related to the surface stretching reference temperature, from the non-dimensional
temperature
1
y=0;
y
as k1
as
y
T T
and using Eqs. (4) and (11) into Eq. (10), we get.
TW T
(12)
'' f ' N b ' ' N t ' 0 2
0 1 1 ' 0 ,
Where
1 k 1
Pr
a f
f
(13)
0,
is the Prandtl number,
f
Q0 a ( c p ) f
is the dimensionless heat source or sink parameter,
B[
is dimensionless temperature slip parameter,
N
b
( c p ) p ( c p ) f
[c L
A[
parameter and
N
t
( c p ) p ( c p ) f
]D B u w ( 0 ) a
is the Brownian motion
3 ''
f (0 )]
]D T ( 0 )
is a dimensionless thermophoresis parameter. The physical significance a
T (c L
3
''
f (0 ))
of Brownian motion coefficient which is about the diffusion equation yields an approximation of the time evolution of the probability density function associated to the position of the particle going under a Brownian movement under the physical definition. At present study the physical significance of the term are composed by many factors, so that it is expressed the solid particles motion at the fluid phenomena. From the Brownian motion
coefficient
it
is
made
by
many
compose
factor
values
B , ( c p ) p , ( c p ) f , D B , u w , ( 0 ), , c , a , f ( 0 ) , and comprise of by the governing equations of this study, ''
so that it can be effected to the heat mass transfer and fluid flow. On the other hand, how was the Brownian motion coefficient evaluated? It is depended on the Brownian motion coefficient its composed factors to B[
evaluate. Such as for example:
N
b
( c p ) p ( c p ) f
[c L
]D B u w ( 0 )
is the Brownian motion coefficient, then all of the a
3
''
f (0 )]
compose factor values B , ( c p ) p , ( c p ) f , D B , u w , ( 0 ), , c , a , f ( 0 ) may produce the Brownian motion ''
coefficient value. Some items are unknown such as ( 0 ),
''
f (0 )
, they are still needed to through the numerical
calculation processing then can be obtained the unknown terms ( 0 ), 8
''
f (0 )
. The thermophysical properties of Page 8 of 26
water and nanoparticle materials are given in Table 1. 2.3 Mass diffusion analysis By using usual boundary layer approximations, the equation of the energy for temperature T in the presence of mass diffusion, is given by: u
C x
v
C
C 2
DB
y
y
2
DT T 2
T y
(14)
2
Where C is the concentration,
DB
is the Brownian diffusion coefficient,
DT
is the thermophoresis diffusion
coefficient. The boundary conditions to the problem are C Cw k2
C C
Where
C y
at y =0, y ,
as k2
is slip parameter related to the surface stretching reference concentration. From the non-dimensional
temperature
'' S c f '
(15)
C C
Nt
CW C
and substituting into Eq. (14) give
'' 0
(16)
Nb
0 1 2 '0 ,
Where
Sc f / D
(17)
0,
is the Schmidt number and
a
2 k2
f
is the dimensionless diffusion slip parameter.
2.4 Heat conduction analysis: The proper is designed for cooling a thin stretching sheet, ample evidence is based on finite difference solutions showing that a one-dimensional model is adequate for calculation and the results are quite well. The stretching sheet temperature at any x location serves as the wall temperature for the adjacent fluid and has denoted as T f ( x ) . The physical formulation of stretching sheet nano energy conversion heat conduction equation can be expressed as 2
d Tf dx
2
h ( T Te ) k t f f
(18)
In which, k f is the thermal conductivity of the stretching sheet, h is the average heat convection coefficient, t is the thickness of the stretching sheet, T f is the sheet temperature and T e is the fluid temperature. For the solutions of equation (18) recasts in a dimensionless form by the substitutions 9
Page 9 of 26
X x / L,
f ( Tf Te ) / ( T0 Te )
(19)
Where T 0 is the base temperature of the stretching sheet, L is the length of the stretching sheet, so that 2
d f dX
h N cc f
2
(20)
with boundary conditions f 1
df
(X=0),
0
(21)
dX
Where N c c is the conduction-convection number and h is the modified convective coefficient, defined as N cc
k ( T0 Te )
h h
;
k f tL
L
(22)
k
The quantity h can be written as '
h ( 0 )(
G
t
Gc
)
1/ 4
(23)
2
3.
Numerical method
In the present thermal energy application problem, an energy conversion system for heat conduction, convection system has been investigated. Since equations (6), (7), (12), (13), (16), (17), (20) and (21) are highly nonlinear system, and it is difficult to find the closed form or exact solution. In present study was used an approximate solution method to solve the problem. An improved Finite-Difference method has been developed to solve the set of similarity equations (6), (7), (12), (13), (16), (17), (20) and (21). These non-linear ordinary differential equations have been discretized by an accurate central difference method [48]. Hsiao et al. [49-51] and Vajravelu [52] had used the similar analytical and numerical solutions to solve the related problems. Seeing the numerical method was presented by Chapra and Canale [53]. 4.
Results and Discussions The model for nanofluids passing a slip boundary stretching sheet nano energy conversion thermal system
has been presented in this study. Present nano energy conversion effects of dimensionless parameters including stagnation parameter ( S 0 ), magnetic parameter (M), electric parameter (E), Prandtl number (Pr), heat source/sink parameter (λ), Brownian diffusion coefficient (Nb), thermophoesis diffusion coefficient (Nt), free convection number (Gt and Gc), Schmidt number (Sc), conduction-convection parameter (Ncc) and slip parameters ( S, , 1 , 2 ) are mainly interested of the study. Flow and temperature fields of the stretching sheet flow have been analyzed by utilizing the boundary layer concept to obtain a set of coupled fluid momentum
10
Page 10 of 26
equation, energy equation. On the other hand, for the sheet plate heat conduction part was used a heat conduction equation to solve the problem. A similarity transformation has been used to convert the non-linear, coupled partial differential equations to a set of non-linear, coupled ordinary differential equations. An accurate modified finite difference method has been used to obtain solutions of those equations. Multimedia refers to content that uses a combination of different content forms. The sentence "content that uses a combination of different content forms" exactly meaning is the showing paper content by different type forms, such as texts, figures, tables, etc. This contrasts with media that use only rudimentary computer displays such as text-only or traditional forms of printed or hand-produced material. The present study is using the multimedia method which are included a combination of text, tables, images, formulas typing, etc. So that the study is using a computer numerical calculation processing couple with the different software to solve the nano energy conversion heat transfer problems, the method consider multimedia related items stage by stage to complete the research work. Table 2 is a comparison with Khan and Pop [23] and has a good agreement result for present method. Table 3 shows that the different values of skin friction −f″(0) and Nusselt number −θ′(0) for different values of physical parameters λ, M, Pr, Gt, Gc, when
S0
=0, E=0, δ =
1 2
= S = Sc = Nt= Nb = 0.1.
Figure 2 to Figure 8 and Figure 10 to Figure 11 depict flow field dimensionless temperature profiles
vs. .
The Figure 2, Figure 3, Figure 6, Figure 7, Figure 8 and Figure 11 are revealed that the increasing of buoyancy parameters Gt, Gc, Prandtl number Pr, Brownian diffusion coefficient Nb and thermophoesis diffusion coefficient Nt and stagnation parameter S0 result in the decreasing of temperature distribution at a particular point of the flow region. This is because there would be a decreasing of the thermal boundary layer thickness with the increasing of values of Prandtl number Gt, Gc, Nb, Nt,
S0
or Pr. Figure 2, Figure 3 Figure 6, Figure 7,
Figure 8 and Figure 11 are revealed that the increasing of the parameters Gt, Gc, Pr, Nb, Nt or
S0
results in the
decreasing of temperature distribution at a particular point of the flow region. This is because there would be a decreasing of the thermal boundary layer thickness with the increasing of value of Gt, Gc, Pr, Nb, Nt or The results have been shown that a larger value of Gt, Gc, Pr, Nb, Nt or
S0
S0
.
will be produced a larger heat
transfer effect. On the other hand, the Figure 4, Figure 5 and Figure 10 are revealed that the increasing of magnetic parameter M, heat source/sink parameter λ and electric parameter E result in the decreasing of temperature distribution at a particular point of the flow region. This is because there would be a decreasing of the thermal boundary layer thickness with the increasing of values of Prandtl number heat, E or M. The Figure 4, Figure 5 and Figure 10 are revealed that the increasing of the parameters λ, E or M result in the decreasing of 11
Page 11 of 26
temperature distribution at a particular point of the flow region. This is because there would be a decreasing of the thermal boundary layer thickness with the increasing of value ofλ, E or M. The results have been shown that a larger value of λ, E or M will be produced a lower heat transfer effect. Figure 9 and Figure 12 depict dimensionless mass diffusion profiles vs.
for varies parameters. The
effects of Sc or Nb/Nt on mass transfer process may be analysis from Figure 9 and Figure 12 for the case of prescribed temperature phenomena. Figure 9 and Figure 12 are shown that the increasing of values of parameter Sc or Nb/Nt results in the increasing of dimensionless stretching sheet mass diffusion distribution as a result of increasing of the mass transfer boundary layer thickness with the increasing values of Sc or Nb/Nt. The larger parameters Sc or Nb/Nt will be produced a larger effect to the mass transfer processing. Figure 13 depicts that dimensionless temperature profiles f vs. X for varies parameters as S0=0.1, Pr=10, M=0.1, E=0.1, Nt=0.1, Nb=0.1, λ=0.1, Gc=0.1, Gt=0.1, Sc=0.1, S=0.1, δ=0.1, 1
0 .1, 2 0 .1
and
Ncc = 5,
10, 15, 20, 25. The effects of conduction-convection parameter Ncc on heat transfer process may be analysis from Figure 13 for the case of prescribed temperature phenomena. Figure 13 is shown that the increasing of values of parameter Ncc results in the decreasing of dimensionless stretching sheet temperature distribution as a result of decreasing of the thermal boundary layer thickness with the increasing values of Ncc. The larger parameter Ncc number will be produced a good effect to the heat conduction processing. 5. Conclusions A modified finite-difference method has been successfully applied to perform a comparative study of nano energy conversion problem about conjugate conduction-convection and heat source/sink over slip extrusion stretching sheet. This study is based on a novel nano energy conversion problem about extrusion stretching sheet heat transfer. On the other hand, for expressing the whole system’s function, all of the physical meaning about different kinds of status, the paper using the multimedia methods to investigate the related problems, and obtain the precision numerical calculation results at the same time. So that present study is not only a novel study work but also a correct result for a nano energy conversion system problem. There are some important conclusions as: (1) It has been seemed that the increasing of magnetic parameter M or electric parameter E resulting in the increasing of temperature distribution at a particular point of the flow region. (2) The effect of free convection parameters Gt, Gc and Prandtl number Pr on heat transfer process may show that the increasing of value of free convection parameters Gt, Gc or Pr results in the decreasing of temperature distribution as a result of decreasing of the thermal boundary layer thickness with the increasing values of Gt, Gc or Pr and then obtain higher heat transfer effects. 12
Page 12 of 26
(3) For heat generation/absorption heat nano energy conversion important parameter λ, it will be produced a higher effect at a lower value of λ. (4) The effect of nanofluid parameters Nt and Nb on heat transfer process may show that the increasing of values of Nt or Nb and obtain lower heat transfer effects, so that the parameters had better not in a larger values for a good nano energy conversion purpose. (5) For heat conduction energy conversion part, the conduction-convection parameter Ncc will be produced a higher heat conduction effect clearly. (6) For convection nano energy conversion, the free convection nano energy conversion effect is better than the forced convection clearly. (7) For stagnation effect is an importance to this study, when stagnation parameter is larger its heat transfer effect is larger too.
ACKNOWLEDGEMENTS The author would like to acknowledge the reviewers valuable comments to this work and to acknowledge Ministry of Science and Technology, R.O.C for the financial support through Grant
MOST
104-2221-E-434-001-.
13
Page 13 of 26
References: [1] Kai-Long Hsiao, Corrigendum to ‘‘Heat and mass mixed convection for MHD viscoelastic fluid
past a
stretching sheet with Ohmic dissipation’’ [Commun Nonlinear Sci Numer Simulat 15 (2010) 1803–1812], Communications in Nonlinear Science and Numerical Simulation, doi:10.1016/j.cnsns.2015.04.004, Volume 28, Issues 1–3, November 2015, Page 232 [2] Kai-Long Hsiao, Heat and Mass Mixed Convection for MHD Viscoelastic Fluid Past a Stretching Sheet with Ohmic Dissipation, July 2010, Communications in Nonlinear Science and Numerical Simulation, 15 (2010) 1803–1812 [3] Kai-Long Hsiao, Conjugate Heat transfer for Mixed Convection and Maxwell Fluid on a Stagnation Point, Arabian Journal for Science and Engineering, DOI 10.1007/s13369-014-1065-z, Volume 39, Issue 6, June 2014, Pages 4325-4332 [4] Kai-Long Hsiao, Energy conversion conjugate conduction-convection and radiation over non-linearly extrusion stretching sheet with physical multimedia effects, Energy, Volume 59, 15 September 2013, Pages 494-502 [5] T Hayat, M Farooq, A Alsaedi Inclined Magnetic Field Effect in Stratified Stagnation Point Flow Over an Inclined Cylinder, Zeitschrift für Naturforschung A, 2015 [6] Hartnett J.P., Viscoelastic fluids: a new challenge in heat transfer, Transactions of ASME, 296-303, (1992). [7]. Fazlina Aman and Anuar Ishak, Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet with prescribed surface heat flux, Heat Mass Tran., Volume 46, Number 6, Pages 615-620 (2009) [8] Vajravelu K. and E. Soewono, Fourth order non-Linear systems arising in combined free and forced convection flow of a second order fluid, Int. J. Non-Linear Mech. Vol. 29, No. 6, 861-869 (1994). [9] Ahmed A. Afify, MHD free convective flow and mass transfer over a stretching sheet with chemical reaction, Heat Mass Tran., Volume 40, Numbers 6-7, Pages 495-500 (2003) [10] Anuar Ishak, MHD Boundary Layer Flow Due to an Exponentially stretching sheet with radiation effect, Sains Malaysiana, 40(4)2011, 391-395 [11] Awang Kechil S. and Hashim I., Series solution of flow over nonlinearly stretching sheet with chemical reaction and magnetic field, Phy. Letters A, Volume 372, Issue 13, 24 March 2008, Pages 2258-2263 [12] Cortel R. l, Viscous flow and heat transfer over a nonlinearly stretching sheet, Appl. Math Comput., Volume 184, Issue 2, 15 January 2007, Pages 864-873 [13] Vajravelu K., Viscous flow over a nonlinearly stretching sheet, Appl. Math. Comput, Volume 124, Issue 3, 15 December 2001, Pages 281-288 14
Page 14 of 26
[14] Liu Chi-Min and Liu I-Chung, A Note on The Transient Solution of Stokes’ Second Problem with Arbitrary Initial Phase, J. Mech., Vol.22, pp.349-354 (2006) [15] Mahantesh M. Nandeppanavar, K. Vajravelu, M. Subhas Abel, M.N. Siddalingappa, Second order slip flow and heat transfer over a stretching sheet with non-linear Navier boundary condition, International Journal of Thermal Sciences, Volume 58, August 2012, Pages 143-150 [16] Wubshet Ibrahim, Bandari Shankar, MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions, Computers & Fluids, Volume 75, 20 April 2013, Pages 1-10 [17] Alin V. Roşca, Ioan Pop, Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip, International Journal of Heat and Mass Transfer, Volume 60, May 2013, Pages 355-364 [18] Kalidas Das, Slip flow and convective heat transfer of nanofluids over a permeable stretching surface, Computers & Fluids, Volume 64, 15 July 2012, Pages 34-42 [19] M. Turkyilmazoglu, Heat and mass transfer of MHD second order slip flow, Computers & Fluids, Volume 71, 30 January 2013, Pages 426-434 [20] M. Turkyilmazoglu, Dual and triple solutions for MHD slip flow of non-Newtonian fluid over a shrinking surface, Computers & Fluids, Volume 70, 30 November 2012, Pages 53-58 [21] Behrouz Raftari, Kuppalapalle Vajravelu, Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall, Communications in Nonlinear Science and Numerical Simulation, Volume 17, Issue 11, November 2012, Pages 4149-4162 [22] T. Hayat, T. Javed, Z. Abbas, Slip flow and heat transfer of a second grade fluid past a stretching sheet through a porous space, International Journal of Heat and Mass Transfer, Volume 51, Issues 17–18, August 2008, Pages 4528-4534 [23] W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer 53 (2010) 2477–2483. [24] Krishnendu Bhattacharyya, Kuppalapalle Vajravelu, Stagnation-point flow and heat transfer over an exponentially shrinking sheet, Communications in Nonlinear Science and Numerical Simulation, Volume 17, Issue 7, July 2012, Pages 2728-2734 [25] M. Sajid, T. Hayat, Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet, International Communications in Heat and Mass Transfer, Volume 35, Issue 3, March 2008, Pages 347-356 [26] Azizah Mohd Rohni, Syakila Ahmad, Ahmad Izani Md. Ismail, Ioan Pop, Boundary layer flow and heat 15
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transfer over an exponentially shrinking vertical sheet with suction, International Journal of Thermal Sciences, Volume 64, February 2013, Pages 264-272 [27] W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer 53 (2010) 2477–2483. [28] Luxiao Chai, Xiaodong Wang, Dezhen Wu, Development of bifunctional microencapsulated phase change materials with crystalline titanium dioxide shell for latent-heat storage and photocatalytic effectiveness Applied Energy, Volume 138, 15 January 2015, Pages 661-674 [29] Fuyun Jiang, Xiaodong Wang, Dezhen Wu, Design and synthesis of magnetic microcapsules based on n-eicosane core and Fe3O4/SiO2 hybrid shell for dual-functional phase change materials, Applied Energy, Volume 134, 1 December 2014, Pages 456-468 [30] Xiang Jiang, Ruilian Luo, Feifei Peng, Yutang Fang, Tomohiro Akiyama, Shuangfeng Wang, Synthesis, characterization and thermal properties of paraffin microcapsules modified with nano-Al2O3 Applied Energy, Volume 137, 1 January 2015, Pages 731-737 [31] Shiyu Yu, Xiaodong Wang, Dezhen Wu, Microencapsulation of n-octadecane phase change material with calcium carbonate shell for enhancement of thermal conductivity and serving durability: Synthesis, microstructure, and performance evaluation Applied Energy, Volume 114, February 2014, Pages 632-643 [32] Zhongyuan Shi, Tao Dong, Entropy generation and optimization of laminar convective heat transfer and fluid flow in a microchannel with staggered arrays of pin fin structure with tip clearance, Energy Conversion and Management, Volume 94, April 2015, Pages 493-504 [33] Amin M. Elsafi, P. Gandhidasan, Comparative study of double-pass flat and compound parabolic concentrated photovoltaic–thermal systems with and without fins, Energy Conversion and Management, Volume 98, 1 July 2015, Pages 59-68 [34] K.S. Syed, Muhammad Ishaq, Zafar Iqbal, Ahmad Hassan, Numerical study of an innovative design of a finned double-pipe heat exchanger with variable fin-tip thickness, Energy Conversion and Management, Volume 98, 1 July 2015, Pages 69-80 [35] Saheera Azmi Hazarika, Dipankar Bhanja, Sujit Nath, Balaram Kundu, Analytical solution to predict performance and optimum design parameters of a constructal T-shaped fin with simultaneous heat and mass transfer, Energy, In Press, Corrected Proof, Available online 26 March 2015 [36] M. Turkyilmazoglu, Stretching/shrinking longitudinal fins of rectangular profile and heat transfer, Energy Conversion and Management, Volume 91, February 2015, Pages 199-203 [37] Mahantesh M. Nandeppanavar, K. Vajravelu, M. Subhas Abel, Chiu-On Ng, Heat transfer over a 16
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nonlinearly stretching sheet with non-uniform heat source and variable wall temperature, International Journal of Heat and Mass Transfer, Volume 54, Issues 23–24, November 2011, Pages 4960-4965 [38] Mahantesh M. Nandeppanavar , K. Vajravelu , M. Subhas Abel, Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink Commun Nonlinear Sci Numer Simulat 16 (2011) 3578-3590 [39] Aminreza Noghrehabadi, Ehsan Izadpanahi, Mohammad Ghalambaz, Analyze of fluid flow and heat transfer of nanofluids over a stretching sheet near the extrusion slit, Computers & Fluids, Volume 100, 1 September 2014, Pages 227-236 [40] Kai-Long Hsiao, Nanofluid flow with multimedia physical features for conjugate mixed convection and radiation, Computers & Fluids, Volume 104, 20 November 2014, Pages 1-8 [41] Rafael Cortell, MHD (magneto-hydrodynamic) flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption, Energy, Volume 74, 1 September 2014, Pages 896-905 [42] M. López de Haro, S. Cuevas, A. Beltrán, Heat transfer and entropy generation in the parallel plate flow of a power-law fluid with asymmetric convective cooling, Energy, Volume 66, 1 March 2014, Pages 750-756 [43] A.A. Hachicha, I. Rodríguez, A. Oliva, Wind speed effect on the flow field and heat transfer around a parabolic trough solar collector, Applied Energy, Volume 130, 1 October 2014, Pages 200-211 [44] Dulal Pal, Gopinath Mandal, K. Vajravelu, Flow and heat transfer of nanofluids at a stagnation point flow over a stretching/shrinking surface in a porous medium with thermal radiation, Applied Mathematics and Computation, Volume 238, 1 July 2014, Pages 208-224 [45] T. Hayat, Sadia Asad, M. Mustafa, A. Alsaedi, Boundary layer flow of Carreau fluid over a convectively heated stretching sheet, Applied Mathematics and Computation, Volume 246, 1 November 2014, Pages 12-22 [46] I-Chung Liu, Helge I. Andersson, Heat transfer over a bidirectional stretching sheet with variable thermal conditions, International Journal of Heat and Mass Transfer, Volume 51, Issues 15–16, 15 July 2008, Pages 4018-4024 [47] T. Hayat, Humaira Yasmin, Maryem Al-Yami, Soret and Dufour effects in peristaltic transport of physiological fluids with chemical reaction: A mathematical analysis, Computers & Fluids, Volume 89, 20 January 2014, Pages 242-253 [48] Brewster M.Q., Thermal Radiative Transfer Properties, Wiley, New York,1972. [49] Hsiao Kai-Long, MHD mixed convection for viscoelastic fluid past a porous wedge, Int. J. Non-Linear 17
Page 17 of 26
Mech., January 2010, 46 (2011) 1–8 [50] Hsiao Kai-Long, Numerical Conjugate Heat Transfer for Free Convection along a Vertical Plate Fin, J. Therm. Sci., July 2010, Vol.19, No.4 (2010), P337-345 [51] Hsiao Kai-Long, Viscoelastic Fluid over a Stretching Sheet with Electromagnetic Effects and Non-Uniform Heat Source/Sink, Math. Probl. Eng., vol. 2010, Article ID 740943, 14 pages, January (2010) [52] Vajravelu K., Convection Heat Transfer at a Stretching Sheet with Suction and Blowing, J. Math. Analy. Appl., 188, 1002-1011(1994). [53] Chapra and Canale, Numerical Methods for Engineers, McGRAW-HILL, 2ed, 1990
18
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Fig 1.
The physical model for stagnation nano energy conversion conjugate mixed convection flow with EMHD and heat generation/absorption effects of incompressible nanofluid over an extrusion stretching sheet
Fig 2 vs. for varies parameters as Pr=10, λ=0.1, Nb=0.1, Nt=0.1, Gc=0.1, M=0.1, Ncc=0.1, S=0.1, S0
=0.1, E=0.1,δ=0.1, 1
0 .1, 2 0 .1
and
19
Gt= 5, 20, 40, 60, 80
Page 19 of 26
Fig. 3 vs. for varies parameters as Pr=10, λ=0.1, Nb=0.1, Nt=0.1, M=0.1, Gt=0.1, Ncc=0.1, S=0.1, S0
=0.1, E=0.1,δ=0.1, 1
0 .1, 2 0 .1
and
Gc= 5, 20, 40, 50, 60
Fig. 4 vs. for varies parameters as Pr=10, λ=0.1, Nb=0.1, Nt=0.1, Gc=0.1, Gt=0.1, Ncc=0.1, S=0.1, S0
=0.1, E=0.1,δ=0.1, 1
0 .1, 2 0 .1
and 20
M= 0.5, 1, 5, 10, 15 Page 20 of 26
Fig. 5 vs. for varies parameters as Pr=10, M=0.1, Nb=0.1, Nt=0.1, Gc=0.1, Gt=0.1, Ncc=0.1, S=0.1, S0
=0.1, E=0.1,δ=0.1, 1
0 .1, 2 0 .1
Fig. 6 vs. for varies parameters as S0
=0.1, E=0.1, S=0.1, δ=0.1, 1
and
λ= 0.05, 0.1, 0.2, 0.3, 0.4
Nb=0.1, M=0.1, Nt=0.1, λ=0.1, Gc=0.1, Gt=0.1, Ncc=0.1,
0 .1, 2 0 .1
21
and
Pr = 10, 15, 20, 25, 30 Page 21 of 26
Fig. 7 vs. for varies parameters as Pr=10, M=0.1, Nt=0.1, λ=0.1, Gc=0.1, Gt=0.1, Ncc=0.1, S=0.1, S0
=0.1, E=0.1,δ=0.1, 1
0 .1, 2 0 .1
and
Nb = 1, 1.5, 2, 2.5, 3
Fig. 8 vs. for varies parameters as Pr=10, M=0.1, Nb=0.1, λ=0.1, Gc=0.1, Gt=0.1, Ncc=0.1, S=0.1, S0
=0.1, E=0.1,δ=0.1, 1
0 .1, 2 0 .1
and
Nt = 1, 1.5, 2, 2.5, 3
22
Page 22 of 26
Fig. 9 vs. for varies parameters as as Pr=10, M=0.1, Sc=0.1, λ=0.1, Gc=0.1, Gt=0.1, Ncc=0.1, S=0.1, S0
=0.1, E=0.1,δ=0.1, 1
0 .1, 2 0 .1
and
Nb/Nt = 0.08, 0.10, 0.15, 0.20, 0.25
Fig.10 vs. for varies parameters as Pr=10, λ=0.1, Nb=0.1, Nt=0.1, Gc=0.1, Gt=0.1, Ncc=0.1, M=0.1, S0
=0.1, S=0.1, δ=0.1, 1
0 .1, 2 0 .1
and
E= 1, 2, 10, 20, 30
23
Page 23 of 26
Fig. 11 vs. for varies parameters as Pr=10, λ=0.1, Nb=0.1, Nt=0.1, Gc=0.1, Gt=0.1, Ncc=0.1, E=0.1, M=0.1, S=0.1, δ=0.1, 1 0 .1, 2 0 .1 and S 0 = 1, 1.5,2, 2.5, 3.0
Fig. 12 vs. for varies parameters as Pr=10, M=0.1, Nt=0.1, Nb=0.1, λ=0.1, Gc=0.1, Gt=0.1, Ncc=0.1, S0
=0.1, E=0.1, S=0.1, δ=0.1, 1
0 .1, 2 0 .1
and
24
Sc = 5, 6, 7, 8, 9
Page 24 of 26
Fig. 13 f vs. X for varies parameters as Pr=10, M=0.1, Nt=0.1, Nb=0.1, λ=0.1, Gc=0.1, Gt=0.1, Sc=0.1, S0
=0.1, E=0.1, S=0.1, δ=0.1, 1
0 .1, 2 0 .1
and
25
Ncc = 5, 10, 15, 20, 25
Page 25 of 26
Table 1. Thermophysical properties of water and some nanoparticle materials Properties
Fluid(water)
CuO
Cu
K(W/mK)
0.613
76.5
400
ρ
997.2
6320
8940
Cp
4179
532
385
Table 2. Comparison the values of parameters when Nt -1.5 0 1 1 0 0 1 1 0
0.1 0.2 0.3 0.4 0.5 0.1 0.1 0.1 0.1
(0 )
and
'
(0 ) '
Khan and Pop [23] 0.9524 2.1294 0.6932 2.2740 0.5201 2.5286 0.4026 2.7952 0.3211 3.0351 0.5056 2.3819 0.2522 2.4100 0.1194 2.3997 0.0543 2.3836
0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.4 0.5
'
(0 )
for different values of physical parameters and fixed
=Gt=Gc=λ=M=E=δ=
S0
(0 )
Nb
'
1
2
(0 )
=S=0 and Sc=Pr=10 with Khan and Pop [23]
(0 )
'
'
Present Present 0.9524 0.6932 0.5201 0.4026 0.3211 0.5056 0.2521 0.1194 0.0542
2.1294 2.2740 2.5287 2.7952 3.0352 2.3819 2.4100 2.3997 2.3836
Table 3. The different values of skin friction −f″(0), Nusselt number −θ′(0), and mass diffusion number ( 0 ) for different values of physical parameters λ, M, Pr, Gt, Gc, when δ = S0
1 2
= S = Sc = Nt= Nb = 0.1,
=0 and E=0 m -1.5 0 1 1 0 0 1 1
λ 0 0 0 0 0 0 1 1
M
Pr
0 0 0 0 0 1 0 1
10 20 30 50 60 70 80 90
Gt, Gc 1 2 3 4 5 6 7 8
f″(0)
(0 ) '
-0.8873 -3.8317 0.1116 -5.6054 0.7039 -6.9830 1.2374 -9.0908 1.7808 -10.0421 1.7523 -10.8141 2.9933 -7.8518 2.9081 -8.2833
26
(0)
0.1166 0.1277 0.1415 0.1581 0.1782 0.1787 0.2319 0.2281
Page 26 of 26
