To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-Nanofluid with parameters control method

Accepted Manuscript To Promote Radiation Electrical MHD Activation Energy Thermal Extrusion Manufacturing System Efficiency by Using Carreau-Nanofluid with Parameters Control Method

Kai-Long Hsiao PII:

S0360-5442(17)30740-5

DOI:

10.1016/j.energy.2017.05.004

Reference:

EGY 10806

To appear in:

Energy

Received Date:

20 February 2017

Revised Date:

23 April 2017

Accepted Date:

01 May 2017

Please cite this article as: Kai-Long Hsiao, To Promote Radiation Electrical MHD Activation Energy Thermal Extrusion Manufacturing System Efficiency by Using Carreau-Nanofluid with Parameters Control Method, Energy (2017), doi: 10.1016/j.energy.2017.05.004

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ACCEPTED MANUSCRIPT Dear Editor: Highlights: ►A novel radiation MHD Activation energy Carreau and nanofluid effects of thermal energy system have been investigated. ► Energy conversion is application for thermal extrusion energy system. ►A stagnation point MHD with Activation Energy mixed convection Carreau nanofluid to increase the efficiency. ► To promote energy and economic efficiency with thermal energy enhancement by various values of pertinent parameters. 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]

ACCEPTED MANUSCRIPT To Promote Radiation Electrical MHD Activation Energy Thermal Extrusion Manufacturing System Efficiency by Using Carreau-Nanofluid with Parameters Control Method Kai-Long Hsiao Taiwan Shoufu University, 168, Nansh District, Madou Jen, Tainan, Taiwan Email Address: [email protected], Tel./Fax numbers: 886-911864791/886-6-2896139

Abstract In this study, a thermal energy extrusion system was made by an improved parameters effect controlling method to promote the manufacturing economic efficiency. The present investigation problem is composed of activation energy electrical MHD Ohmic dissipation and mixed convection of a viscoelastic non-Newtonian Carreau-Nanofluid on a stagnation-point energy conversion problem. The governing equations for thermal energy extrusion system are solved by analysis and implicit finite difference method. The thermal system is composed of flow velocity field, temperature field, mass diffusion field and heat conduction-convection field. The related important parameters have been produced as function of the fluid material parameter (λ), activation energy chemical reaction parameter (  A ), Prandtl number (Pr) and mixed convection buoyancy parameters (Gc, Gt), etc. The results are shown that it will be provided greater thermal system effects with larger or lower values of those parameters, and have been divided six degree sequences to show their importance at this system. At last, it can be obtained a higher efficiency thermal energy extrusion system and can be promoted the system’s economic efficiency. Keywords: Parameters control method; Conjugate heat transfer; Activation energy; Stagnationpoint; Carreau-Nanofluid; Economic efficiency

1

ACCEPTED MANUSCRIPT 1. Introduction The flow of viscoelastic non-Newtonian Carreau-Nanofluid flow is importance to industrial applications for increasing energy efficiency. For example, it has been used in the extrusion of a polymer sheet from a die or in the drawing of plastic films. Present work is promoted a thermal system energy efficiency to obtain manufacturing economic efficiency. At first, CarreauNanofluid with Parameters Control Method are mainly used to control present study work, and have something difference with other works. In this work, we can find that some importance parameters such as velocity parameter (λ), source/sink parameter (E), Weissenberg parameter (We), magnetic parameter (M), mixed convection parameter (Gr, Gc), Prandtl number (Pr), Eckert number (Ec), heat source/sink parameter (AL), radiation parameter (R0), electric parameter (E1), mass diffusion Schmidt number (Sc), Brownian motion parameter (Nb), thermophoresis parameter (Nt), activation energy chemical reaction rate parameter (  A ), activation energy parameter ( E A ) and conduction-convection number (Ncc) which can be made the thermal energy effects lower or higher, then it can be controlled those parameters one by one at the manufacturing thermal system. Secondly, the present study work is towards about the manufacturing extrusion thermal system, the parameters control methods can help to obtain a higher system energy efficiency directly, so that arrive a saving energy purpose and it can be reduced the money payment. At last, it can be obtained a good thermal energy system efficiency which can be become a good economic manufacturing system. The viscoelastic non-Newtonian fluid flow has been developed for industry production application and can be produced high effects. There are many studies in this ways. Recently, Hsiao [1] investigated about combined activation energy electrical MHD heat transfer thermal energy extrusion system using Maxwell fluid with radiative and viscous dissipation effects for applied thermal engineering application. Yousefi-Lafouraki et al. [2] studied about entropy generation analysis of a confined slot impinging jet in a converging channel for a shear thinning nanofluid 2

ACCEPTED MANUSCRIPT problem. Hsiao [3] investigated stagnation-point activation energy electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet for thermal system application. Cortell [4] provided magneto-hydrodynamic flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption at energy aspect. Abbas [5] studied about numerical solution of binary chemical reaction on stagnation-point flow of Casson fluid over a stretching/shrinking sheet with thermal radiation for energy application. Vajravelu et al. [6] investigated for homotopy analysis of the magneto hydrodynamic flow and heat transfer of a second grade fluid in a porous channel at energy field application. Hsiao [7] studied about energy conversion conjugate conduction–convection and radiation over non-linearly extrusion stretching sheet with physical multimedia effects for energy efficiency application. The energy efficiency is one kid of application methods to save energy and to produce more well products for industry manufacturing. Some of studies are provided by different ways. Apergis et al. [8] analyzed energy efficiency of selected OECD countries for slacks based model with undesirable outputs problems. Parker and Liddle [9] presented about energy efficiency in the manufacturing sector of the OECD: Analysis of price elasticity aspect. Chen et al. [10] investigated for energy efficiency potentials: Contrasting thermodynamic, technical and economic limits for organic Rankine cycles within UK industry problems. Ivelisse et al. [11] examined economic analysis of alternatives for optimizing energy use in manufacturing company problem. Mukherjee [12] addressed about energy using efficiency in U.S. manufacturing by a nonparametric analysis method. E et al. [13] studied about effects of inlet pressure on wall temperature and exergy efficiency of the micro-cylindrical combustor with a step problem. Zhang et al. [14] investigated for a real options model for renewable energy investment with application to solar photovoltaic power generation in China related aspects. May et al. [15] considered energy management in manufacturing toward eco-factories of the future by a focus group study ways. Nataf and Bradley [16] investigated an economic comparison of battery energy storage to 3

ACCEPTED MANUSCRIPT conventional energy efficiency technologies in Colorado manufacturing facilities. Cassettari et al. [17] developed about energy resources intelligent management using on line real-time simulation: A decision support tool for sustainable manufacturing problem. Pintaldi et al. [18] investigated for energetic evaluation of thermal energy storage options for high efficiency solar cooling systems application. Above studies are investigated many related manufacturing efficiency events but not consider for present energy conversion Carreau fluid thermal energy extrusion system manufacturing problems. Recently, the most efficiency to increase such kinds of manufacturing field is applied viscoelastic non-Newtonian Carreau fluid flow to promote the performance. Raju and Sandeep [19] discussed about unsteady three-dimensional flow of Casson–Carreau fluids past a stretching surface. Sulochana et al. [20], G. P. considered transpiration effect on stagnation-point flow of a Carreau-Nanofluid in the thermophoresis and Brownian motion event. Babu and Sandeep [21] presented MHD non-Newtonian fluid flow over a slendering stretching sheet in cross-diffusion effects. Akbar et al. [22] investigated MHD stagnation-point flow of Carreau fluid toward a permeable shrinking sheet problem. Mohamed [23] employed the dual solutions in hydromagnetic stagnation point flow and heat transfer towards a stretching/shrinking sheet with non-uniform heat source/sink and variable surface heat flux thermal system. Riaz et al. [24] addressed about peristaltic transport of a Carreau fluid in a compliant rectangular duct problem. Mahanthesh et al. [25] investigated for unsteady three-dimensional MHD flow of a Nano Eyring-Powell fluid past a convectively heated stretching sheet with thermal radiation, viscous dissipation and Joule heating problem. Ferdows et al. [26, 27] studied about MHD Mixed convective boundary layer flow of a nanofluid through a porous medium due to an exponentially stretching sheet, and numerical study of transient magnetohydrodynamic radiative free convection nanofluid flow from stretching permeable surface problems. Beg et al. [28] investigated for explicit numerical study of unsteady hydromagnetic mixed convective nanofluid flow from an exponential stretching sheet in porous 4

ACCEPTED MANUSCRIPT media problem. The nanofluid model has been applied the most effect Brownian motion theory to present nanofluid thermal conversion management system. Brownian motion theory is not only can be applied to fluid flow or heat transfer but also can be used to the economic field. The application to thermal energy problems are increasing importance for many studies to investigate. The other kinds of ways are by using Magneto-Hydrodynamic with radiation or nanofluid related effects application to this aspect. It is suited to many industry thermal system applications. Ahmad et al. [29] were using numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet. The latest studies about Carreau nanofluid by Khan et al. [30-32]. Hayat et al. [33] investigated about magneto hydrodynamic three-dimensional flow of nanofluid over a convectively heated nonlinear stretching surface. Abbas et al. [34-35] investigated about hydromagnetic slip flow of nanofluid over a curved stretching surface with heat generation and thermal radiation and numerical solution of binary chemical reaction on stagnation point flow of Casson fluid over a stretching/shrinking sheet with thermal radiation problems. Imtiaz et al. investigated about homogeneous heterogeneous reactions in MHD flow due to an unsteady curved stretching surface. Makinde et al. [36] investigated MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat. The related heat conduction efficiency about fins was investigated by He et al. [37-39]. Up to now, no investigation has been made which can provide the numerical solution for the steady two-dimensional stagnation point flow of non-Newtonian Carreau-Nanofluid towards a stretching surface with energy conversion Carreau fluid thermal energy extrusion system parameter control problem. From above, it has been provided the ideas for application to make a novel extrusion manufacturing processing thermal system and to produce a high effect device to promote the manufacturing economic efficiency. The present study is to investigate for thermal energy extrusion system energy conversion problem with electric hydro magnetic heat and mass mixed 5

ACCEPTED MANUSCRIPT convection in an incompressible second grade Carreau-Nanofluid with radiation and viscous dissipation effects. On the other hand, present study novelty is provided a novel Activation Energy Carreau and nanofluid effects on the thermal extrusion energy system. The Activation Energy Carreau and nanofluid effects are controlled by some importance parameters, such as material parameter (λ), activation energy chemical reaction parameter (  A ), Prandtl number (Pr) and mixed convection buoyancy parameters (Gc, Gt), etc. It can be used those related parameters to control the thermal extrusion energy system efficiency higher or lower. It is a novel method application to such manufacturing system. 2. Methodology 2.1 Theory and Analysis The present geometry physical model for steady mixed convection Carreau-Nanofluid on stagnation-point thermal forming stretching sheet energy conversion thermal system is shown in Fig. 1. The thermal extrusion sheet can be used for many industry products, such as plastic sheet, food sheet, metal sheet, etc. Present design the device is finding a high efficiency thermal energy conversion manufacturing process. Because of the extrusion sheet is very hot thin plate, and need to cooling by the working fluids, so that it can be selected a good fluid flow to do this job. On the other hand, it can be selected a vertical stagnation flow to arrive a good effect. Many useful factors are considered to improve this manufacturing production. It is a combined fluid field, stretching sheet conduction-convection thermal energy system. The related physical theory has been introduced as following. 2.1.1 Fluid Field Analysis For steady-state incompressible boundary layer fluid flow, the boundary layer equations for viscoelastic non-Newtonian Carreau flow is referred to previous section referred papers, nanofluid and some other physical effects are also referred to previous section related studies. Besides, the present study has been added some different effect items, such as magnetic, electric, stagnation6

ACCEPTED MANUSCRIPT point point, etc. It is made an energy thermal conversion management system for application to industry field. The Carreau fluid was introduced by some studies. Wahiduzzaman et al. [40] investigated about MHD convective stagnation-point flow of nanofluid over a shrinking surface with thermal radiation, heat generation and chemical reaction problem. Akbar et al. [41] developed MHD stagnation-point flow of Carreau fluid toward a permeable shrinking sheet: dual solutions for thermal system application. Akbar [42] and Nadeem investigated for combined effects of heat and chemical reactions on the peristaltic flow of Carreau fluid model in a diverging tube problem. The Carreau fluid can be used the model ij  0 [1 

(n  1) . 2 . (  ) ]  ij 2

Where ij is the extra stress tensor, 0 is the zero shear rate viscosity,  is the time constant, n is .

power law index and  ij is defined as .

 ij 

1  i 2

.

1 II 2

.

   j ij ji

Were II is the second invariant strain tensor. For a first order chemical reaction activation energy have provided by Arrhenius law which is usually used with the following form [43]: [

 Ea

K A  BA (T  T ) w e k(T T )

]

Where K A is the rate constant of chemical reaction (1/sec) and BA is the pre-exponential factor ( K  w / sec ),

simply pre-factor (constant), is based on the fact that increasing the temperature

frequently causes a marked increase in the rate of reactions. E a is the activation energy (eV) and k= 8.61 × 10−5 eV/K is the Boltzmann constant which is the physical constant relating energy at the individual particle level with temperature observed at the collective or bulk level. such as geothermal, the governing equations are

u v  0 x y

(1) 7

In areas

ACCEPTED MANUSCRIPT

u

u u dU 2u 3(n  1) 2 u 2  2 u v U  f 2  f ( ) x y dx y 2 x y 2

g*  T  T   g**  C  C  

u

(2)

B02 E 0 B0 (U  u)  f f

kf 2T q (c p ) p T T q C T DT T 2 v   (T  T )  r  [D B  ( ) ] 2 x y (c p )f y (c p )f y (c p )f y y T y

(3)

E

a [ ] C C  2C DT   2T  k(T T ) w u v  DB 2   B (T  T ) e (C  C ) A    x y y T  y 2 

(4)

Where u, v are the velocity components in the x and y directions, U is the free stream velocity, T is the temperature, n is power law index, g is the magnitude of the gravity,  f is the fluid kinematic viscosity, q is the heat generation parameter, q r is the radiation heat parameter, λ is the material parameter, E is the viscoelastic parameter, * is the coefficient of thermal expansion, ** is the coefficient of mass diffusion expansion, T is the temperature of the ambient fluid, f is the fluid density, c p is the specific heat at constant pressure,  is the electrical conductivity, B0 is the magnetic field factor, k is the conductivity, C is the concentration, D B is the Brownian diffusion coefficient, DT is the thermophoresis diffusion coefficient. The well-known Boussinesq approximation is used to represent the buoyancy mixed term. By using Rosseland approximation the radiation heat flux has given by

qr = 

4* T 4 3k * y

(5)

The equation items * is the Stephan-Boltzmann constant, k * is mean absorption coefficient. The item T 4 can be expanded by a Taylor series. It can be expanded T 4 about T and neglecting higher order terms, it can be obtained 8

ACCEPTED MANUSCRIPT T 4  4T3 T  3T4

(6)

From Equations (5), (6) and Equation (3) becomes u

k f  2T 16*T3  2 T (c p ) p T T q C T DT T 2 v   (T  T )   [D B  ( ) ]  2 * 2 x y (c p )f y (c p )f 3k y (c p )f y y T y

(7)

The corresponding boundary conditions to the problem are x u  u w  cx , v=0, T  Tw  T  A( ) L

u  u e (x)  bx,

T  T

at

y  0,

(8)

at y  

From above Equation (7) the items Tw and T are constant wall temperature and ambient fluid temperature, A is the proportional constant, x is the position along the x-axial and L is the characteristic length, respectively. A similarity solution for fluid velocity can be obtained if introduce a set of transformations, such as u

 , y

v

 x

(9)

  (c /  f )1/2 y ,   (c f )1/2 xf ()    

T  T , T  T  A1x   , Tw  T

   

C  C , C  C  B1x   , C w  C

(10)

From the transformations it can be obtained u and v as

u  cxf ' (), v  (c f )1/2 f ()

(11)

Equation (9)-(11) has satisfied the continuity equation (1), Substituting Equations (9)-(11) into Equations (2, 4) and (7), it can be obtained 3(n  1)We 2 ''' '' 2 f  (f  ff )  1  f (f )  [G r   G c   M(1  f ' )  ME1] 2 '''

'2

''

''  Pr R 0 [f'   2f  AL    E c (f '' ) 2  N b '  ' N t  '2 ]  0

 '' Scf '

E ( A) Nt  '' Sc  A w e    0 Nb

(14) 9

(12) (13)

ACCEPTED MANUSCRIPT The items for equations (12), (13) and (14), it can be obtained that We  Weissenberg parameter,

b3 x 2  2 is the 

G r  g* (Tw  T ) / cu w and G c  g** (C w  C ) / cu w are the mixed

convection parameter, M  B0 2 / c is the magnetic parameter, E1  E 0 / B0 u w is the electric parameter, Pr=  /  is the Prandtl number, E c  u w 2 / c p (Tw  T ) is the Eckert number, L is the wall thickness of the stretching sheet. R=

and AL 

B1[ Nb 

(c p ) p (c p )f

q c p A1 ]D B u w (0)

c3 '' [c  L f (0)] 

16*T3 3R and R 0  is the radiation parameter * 3k k   3R  4 

is the heat source/sink parameter, Sc 

is the Brownian motion parameter,

A1[ Nt 

 is the Schmidt number, DB (c p ) p (c p )f

]DT (0)

c3 '' T (c  L f (0)) 

is a dimensionless

thermophoresis parameter.  A  BA (T  T ) w / c is the dimensionless activation energy chemical reaction rate parameter, E A 

Ea is the dimensionless activation energy parameter. The k(TW  T )

corresponding boundary conditions become f(0)=E,

f ' (0)   ,

f ' ( )  1 ,

()  0 ,

(0)  1 , (0)  1

(15)

()  0

Where E   v w / (cv)1/2 and λ=c/b 2.1.2 Conjugate Heat Transfer Analysis The average Nusselt number Nu L is defined by Nu L 

qw L hL  kf Tw -T k f

(16)

This expression can be written as Nu L 

qw L hL   ' (0)(G r1/4  G c1/4 ) kf Tw -T k f

(17)

For the solutions of stretching sheet heat conduction equation, can be expressed as 10

ACCEPTED MANUSCRIPT

d 2 Tf dx

2

 h    (Tf  Te )  ks t 

(18)

Equation (18) recasts in a dimensional form, x is the position along x-axis of the stretching sheet, h is local surface heat flux transfer coefficient, k s is the thermal conductivity of the stretching

sheet, Tf is the dimensional stretching sheet temperature and Te is the edge temperature. By the substitutions X  x / L,

(19)

f  (Tf  Te ) / (T0  Te )

Where X dimensionless length of the stretching sheet, L is the total length of the stretching sheet,

f is dimensionless stretching sheet temperature and T0 is the base dimensional temperature of the stretching sheet. We have its dimensionless stretching sheet heat conduction equation d 2 f dX

2



 h N cc f

(20)

With boundary conditions f  1

(X=0),

df 0 dX

(X=1)

(21)

Where N cc is the heat conduction-convection number and is defined as

N cc  (k f / Lk s t)

(22)



The quantity h is a dimensionless form of the convective heat transfer coefficient and can be written as   hL  h   kf 

(23)

2.1.3 Numerical Technique The numerical technique is developed to 11

ACCEPTED MANUSCRIPT solve present thermal energy extrusion energy conversion management problem, for a conjugate heat transfer system which is composed by radiation electrical magneto hydrodynamic mixed convection for viscoelastic non-Newtonian Carreau-Nanofluid towards stagnation-point flow field thermal forming extrusion stretching sheet with radiation, viscous dissipation and Ohmic heating effects. The set of non-linear similarity equations (12), (13), (14) and (20) and their boundary conditions (15) and (21) are solved by an improved numerical finite-difference method. Vajravelu [44] and Hsiao [45] are also used the similar methods to solve the related problems. The present improved numerical finite difference method was similar to Keller’s box method and was used by Chapra and Canale [46]. The systems of linearized non-linear difference equations are iteratively solved. The step size of Δh= 0.01 and the convergence criterion of 106 is applied to take into account the boundary layer effect. Convergence and stability are importance for numerical calculation Khan et al. [47] these two characteristics of a numerical scheme in the solution of nonlinear ordinary differential equations can be analyzed simultaneously as illustrated next. Convergence means that the finite-difference solution approaches the true solution to the differential equation as the increments ∆ η go to zero. Stability means that the error caused by a small perturbation in the numerical solution remains bound. Present study is followed above ways, trying a suitable ∆ η with many separation points. On the other hand, we have used suitable parameter values to obtain reasonable numerical results. 3. Results and Discussion the In this study, it is provided multimedia physical feature to investigate a thermal energy extrusion energy conversion management system which are combined conjugate heat transfer and stagnation-point flow field. The hot extrusion sheet is cooled by viscoelastic non-Newtonian Carreau-Nanofluid with various effective parameters. The related dimensionless parameters including velocity parameter (λ), source/sink parameter (E), Weissenberg parameter (We), magnetic parameter (M), mixed convection parameter (Gr, Gc), Prandtl number (Pr), Eckert 12

ACCEPTED MANUSCRIPT number (Ec), heat source/sink parameter (AL), radiation parameter (R0), electric parameter (E1), mass diffusion Schmidt number (Sc), Brownian motion parameter (Nb), thermophoresis parameter (Nt), activation energy chemical reaction rate parameter (  A ), activation energy parameter ( E A ) and conduction-convection number (Ncc) are mainly investigated by this study. At first, we have used similarity transformation method to convert the set of nonlinear, coupled partial differential equations to a set of nonlinear, coupled ordinary differential equations. An improved finite difference method has been used to obtain solutions of those equations. The related thermophysical properties of water and some nanoparticle materials are shown in Table 1 and Table 3 is to show that the related values of f '' (0) , - ' (0) , - ' (0) at fixed parameters R0=0.01, Gc=Sc=Ncc=0.5 and Pr=10, and with varied parameters E, Gr, M, AL, Ec and E1. Table 2 is a comparison results of f '' (0) for different values of physical parameters at fixed parameters M=Gc =Gt =We =E1=0 and n=1 with Sulochana et al. [20] and is obtained a good agreement. 3.1 Conjugate heat transfer effects Figs. 2, 5, 6, 11 and 12 to depict stretching sheet dimensionless temperature profiles f vs. X. From Figs. 2, 5, 6,11 and 12 are revealed that the increasing of buoyancy parameter Gr, Prandtl parameter Pr, conduction-convection number Ncc, buoyancy parameter Gc, or velocity parameter λ result in the decreasing of temperature distribution at a particular point of the flow region. This is because there would be decreased of the thermal boundary layer thickness with the increasing of values of buoyancy parameters buoyancy parameter Gr, Prandtl parameter Pr, conductionconvection number Ncc, buoyancy parameter Gc, or velocity parameter λ. Figs. 2, 5, 6, 11 and 12 are revealed that the increasing of the parameters Gr, Gc, Ncc, Pr or λ result in the decreasing of temperature distribution at a particular point of the flow region. This is because there would be decreased of the thermal boundary layer thickness with the increasing of values of Gr, Gc, Ncc, Pr or λ. The results have been shown that a larger value of Gr, Gc, Ncc, Pr or λ will be produced a larger combined heat transfer effect. On the 13

ACCEPTED MANUSCRIPT other hand, the Figs. 3, 4, 7, 9, 15 and 16 are revealed that the increasing of magnetic parameter (M), Eckert number (Ec), heat source/sink parameter (AL), radiation parameter (R0), thermophoresis parameter (Nt) or Brownian motion parameter (Nb) result in the decreasing of temperature distribution at a particular point of the flow region. This is because there would be decreased of the thermal boundary layer thickness with the increasing of values of Eckert number Ec, heat source/sink parameter AL and radiation parameter R0. 3, 4, 7, 9, 15 and 16 are revealed that the increasing of the parameters M, Ec, AL, R0, Nt or Nb result in the decreasing of temperature distribution at a particular point of the flow region. This is because there would be decreased of the thermal boundary layer thickness with the increasing of value of M, Ec, AL, R0, Nt or Nb. The results have been shown that a larger value of M, Ec, AL, R0, Nt or Nb will be produced a lower conjugate heat and mass diffusion heat effect. The third status is another phenomenon that are shown as Figs. 8 and 10, the related parameters source/sink parameter (E) or electric parameter (E1) which results are not in a fixed larger or lower but changed with non-linear phenomena. 3.2 Mass diffusion effects For mass diffusion problem is depicted by Figs. 15 and 16 at those figures are shown that the larger mass diffusion effects are made by the larger parameter Sc or  A , the Sc or  A value larger then produce the larger mass diffusion effect, all of the physical phenomena are very clearly by smooth curves. On the other hand, the related parameters Nt or Nb have a different result, the larger values of Nt or Nb will be reduced the thermal effect. 3.3 The conduction-convection thermal effects Parameter conduction-convection number Ncc is an importance parameter for present thermal system. From fig. 15 can be observed that one of the statuses for thermal system effect is good for higher system efficiency. When the Ncc values are larger, then the curves are lower as the sheet temperature lower, so that it will be increased the heat transfer efficiency for this status. 14

ACCEPTED MANUSCRIPT 3.4 The parameters effects to promote economic efficiency In this study, it is one kind of numerical simulation about the extrusion processing for manufacturing application. On the physical aspect, the geometry outlook was shown at Figure 1. From Figure 1 can see physical model for energy conversion combined radiation activation energy electrical MHD mixed convection of stagnation-point with non-Newtonian Carreau-Nanofluid flow on a thermal forming stretching sheet. The hot sheet is very thin and long and cooling with high effect fluid flow, so that suit to the boundary layer theory. Present study is combined conduction-convection simulation the problem. For convection part, including the momentum and energy equations, for heat conduction-convection part, including the total length stretching sheet along the x-axis direction. The extrusion sheet is importance application to many industry products, extrusion commonly is extruded materials including metals, polymers, ceramics, concrete, play dough, and foodstuffs, etc. After the extrusion processing, it can be obtained a very hot thin sheet product and needs to an efficiency cooling by some ways. In this study, we have selected a good flow field toward the sheet by stagnation-point ways. On the other hand, we have considered an efficiency non-Newtonian flow field by using many parameters to improve the extrusion production efficiency. From present study results are shown that some parameters need to lower and some parameters need to higher, then it can be obtained more well effects. We can use parameters to control the manufacturing processing to improve the whole thermal system’s efficiency. On the other hand, as the thermal system’s efficiency become good, then we can obtain a more well economic efficiency. The whole thermal energy extrusion stretching sheet energy conversion efficiency influence factors are included the electric energy conversion efficiency, extrusion device mechanical efficiency, manufacturing processing efficiency, etc. For obtain a good system’s efficiency, it needs to provide many aspects in a good effect status. Present study is towards the related parameters control ways to improve the system efficiency. 3.5 The parameters effects importance sequence analysis 15

ACCEPTED MANUSCRIPT There are some deeply investigation about the thermal system’s efficiency related to parameters discussion were made by Zuo et al. [48] for emitter efficiency of the micro-cylindrical combustor with a step, and Feng et al [49] for hydropower system operation optimization by discrete differential dynamic programming based on orthogonal experiment design. Present study has the similar status about the importance parameters investigation. From Fig. 2 the parameter Gr=0.1, 0.5, 1.0, 1.5, 2.0 is varied from 0.1 to 2.0, so that this parameter needs a small gape parameter Δ Gr=0.5 to produce a clear influence to the system. By the similar ways, we find that for all of other importance parameters on the other Figs. We can obtain a Table 4 to show it. From the table 4 we can find that there are some importance parameters effects which were produced by numerical figs. results. From the numerical calculation we can find the effect gapes by different importance parameters, and selected their impact sequences by the small effect gape ways. At last, we can obtain six grades impact sequences for our study as the table 4. The first sequence parameter is R0, the second sequence parameter is λ, the third sequence parameters are Gr, Gc, E1, Nt and Nb, the fourth sequence parameter is Ec, the fifth sequence parameters are M and Pr, the sixth sequence parameters are Ncc, Sc and  A . 4. Conclusion The present thermal system is couple with heat and mass transfer effects, it can be controlled the importance parameters to produce a high efficiency extrusion products by manufacturing process. Dimensionless conjugate heat and mass diffusion parameters are related to λ, E, We, M, Gr, Gc, Pr, Ec, AL, R0, E1, Sc, Nb, Nt,  A , E A and Ncc, those parameters are the important factors in this study, and we can find their importance sequence by a parameters effect gape ways. At last, there are eight important results for this study obtained as: (1)

Present study is used the parameters control method for a conjugate heat mass transfer mixed convection of viscoelastic non-Newtonian Carreau-Nanofluid on a stagnation-point flow field towards extrusion thin sheet thermal system. 16

ACCEPTED MANUSCRIPT (2)

The present implicit finite difference technique with stability and convergence ways has been employed as a solution technique to complete the simulation of the present steady finitedifference model.

(3) For heat transfer energy conversion management part, the phenomena were made by different related parameters control method, and the related parameters are Gr, Gc, Pr, Ec, AL, R0 and E1. (4) For mass diffusion energy conversion management part, the larger values parameters Sc or  A will produce the higher mass diffusion effects at last. (5) For heat conduction-convection energy conversion management part, the conductionconvection parameter Ncc will be produced a higher heat conduction effect clearly. (6) For convection energy conversion management phenomena, the Viscoelastic non-Newtonian Carreau-Nanofluid mixed convection energy conversion management effect is better than the forced convection clearly. (7) In this study, we have made a comparison with other similar numerical results and obtained a good agreement with others study. (8) The importance parameters have been investigated deeply, and find some news about their characteristic and importance for present thermal energy system and divided six degree sequences to show their importance at this system.

ACKNOWLEDGEMENTS The author Kai-Long Hsiao like to acknowledge Ministry of Science and Technology, R.O.C for the financial support through Grant MOST 104-2221-E-434-001-, and like to thank the valuable comments which provided by the reviewers.

17

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ACCEPTED MANUSCRIPT stretching surface with heat generation and thermal radiation, Journal of Molecular Liquids, 215( 2016) 756-762 [35] Z. Abbas, M. Sheikh, S.S. Motsa, Numerical solution of binary chemical reaction on stagnation point flow of Casson fluid over a stretching/shrinking sheet with thermal radiation, Energy, 95(15) (2016) 12-20 [36] O.D. Makinde, F. Mabood, W.A. Khan, M.S. Tshehla, MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat, Journal of Molecular Liquids, 219( 2016) 624-630 [37] WW Wang, LB Wang, YL He, Parameter effect of a phase change thermal energy storage unit with one shell and one finned tube on its energy efficiency ratio and heat storage rate, Applied Thermal Engineering 93(2016)50-60 [38] H Wang, Y Liu, P Yang, R Wu, Y He, Parametric study and optimization of H-type finned tube heat exchangers using Taguchi method, Applied Thermal Engineering 103(2016)128138 [39] P. Yuan, GB Jiang, YL He, WQ Tao, Performance simulation of a two-phase flow distributor for plate-fin heat exchanger, Applied Thermal Engineering 99(2016)1236-1245 [40] M. Wahiduzzaman, Md. Shakhaoath Khan, Ifsana Karim, MHD convective stagnation flow of nanofluid over a shrinking surface with thermal radiation, heat generation and chemical reaction, Proc. Eng. 105 (2015) 398–405. [41] N.S. Akbar, S. Nadeem, R.U.I. Haq, S. Ye, MHD stagnation point flow of Carreau fluid toward a permeable shrinking sheet: dual solutions, Ain Shams Eng. J. 5 (2014) 1233–1239. [42] Akbar NS, Nadeem S., Combined effects of heat and chemical reactions on the peristaltic flow of Carreau fluid model in a diverging tube. Int J Numer Methods Fluid 2011;67:1818– 32. [43] M. Tencer, J. S. Moss, and T. Zapach, “Arrhenius average temperature: the effective 22

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Table and Figure Captions Table 1. Thermophysical properties of water and some nanoparticle materials Table 2. Comparison results of f '' (0) for different values of physical parameters at fixed parameters M=Gc =Gt =We =E1=0 and n=1 Table 3. Numerical results of - f '' (0) , − ' (0) and - ' (0) for different values of physical parameters at fixed parameters R0=0.01, λ=Gc=Sc=Ncc=0.5 and Pr=10 Table 4 Importance parameters effects impact sequences made by effect gape Fig 1.

A sketch of physical model for energy conversion combined radiation activation energy electrical MHD mixed convection of stagnation Carreau fluid flow on a thermal forming stretching sheet.

Fig. 2 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature E1=0.1,

f vs. X as Pr=10, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1, Sc=0.1,

AL=0.1,

R0=0.01,

λ=0.1,

Nt=0.1,

Nb=0.1,

n=0.1,

We=0.1,

w=1,

E A  1,  A  0.1 and Gr = 0.1, 0.5, 1.0, 1.5, 2.0 Fig. 3 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.01, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1, Sc=0.1,

E1=50, AL=0.1, R0=0.01,λ=0.1, Nt=0.1, Nb=0.1, n=1.1, We=0.1, w=1, E A  1,  A  0.1 and M = 1, 5, 10, 15, 20 Fig. 4 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Gr=1, M=0.1, Pr=100, E=0.1, Ncc = 10, Gc=0.1, Sc=0.1,

E1=0.1, AL=0.1, R0=0.01,λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Ec = 4, 5, 6, 7, 8 Fig. 5 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless 24

ACCEPTED MANUSCRIPT temperature

f vs. X as Gr=1, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1, Sc=0.1,

E1=0.1, AL=0.1, R0=3, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Pr = 1, 5, 10, 20, 30 Fig. 6 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.1, M=0.1, Ec=0.1, E=0.1, Gc=0.1, Sc=0.1, E1=0.1,

AL=0.1, R0=0.01,λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 , Ncc=100, 150, 200, 250, 300 Fig. 7 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.1, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1,

Sc=0.1, E1=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and AL= 10, 20, 30, 40, 50 Fig. 8 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.1, M=0.1, Ec=0.1, Gc=0.1, Sc=0.1, E1=0.1,

AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and (E=1, Ncc=5), (E=5, Ncc=10), (E=10, Ncc=15), (E=50, Ncc=20), (E=100, Ncc=25), Fig. 9 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.01, M=0.1, E=0.1, Ncc = 100, Gc=0.1, Sc=0.1,

E1=0.1, AL=0.1, Ec=1, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and R0 = 0.1, 0.2, 0.25, 0.30, 0.35 Fig. 10 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=1000, Gr=0, M=1000, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1,

Sc=0.1, AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=1.1, We=2.1, w=1, E A  1,  A  0.1 and E1 = 0.1, 0.5, 1.0, 1.5, 2.0 Fig. 11 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.1, M=0.1, R0=0.01, Ec=0.1, E=0.1, Ncc = 10,

Sc=0.1, E1=0.1, AL=0.1,λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Gc = 0.1, 0.5, 1.0, 1.5, 2.0 Fig. 12 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=1000, Gr=16, M=0.1, Ec=0.1, E=0.1, Ncc = 100, Gc=0.1, 25

ACCEPTED MANUSCRIPT Sc=0.1, E1=0.1, AL=0.1, R0=0.01, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 andλ=0.1, 0.2, 0.3, 0.6, 1.0 Fig. 13 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature Sc=0.1,

f vs. X as Pr=100, Gr=10, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1,

E1=0.1,

AL=0.1,

R0=0.01,

λ=0.1,

Nb=0.1,

n=0.1,

We=0.1,

w=1,

E A  1,  A  0.1 and Nt=1, 0.5, 1.0, 1.5, 2.0 Fig. 14 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=100, Gr=10, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1,

Sc=0.1, E1=0.1, AL=0.1, R0=0.01,λ=0.1, Nt=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Nb=1, 0.5, 1.0, 1.5, 2.0 Fig. 15 Dimensionless stretching sheet mass diffusion profiles ɸ vs. η as Pr=10, Gr=0.1, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1, E1=0.1, AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Sc = 40, 60, 100, 150, 200 Fig. 16 Dimensionless stretching sheet mass diffusion profiles ɸ vs. X as Pr=10, Gr=0.01, M=0.1, Ec=0.1, E=0.1, Gc=0.1, Sc=0.1, E1=0.1, AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1 and  A = 1, 50, 100, 150, 200

26

ACCEPTED MANUSCRIPT

Tables and Figures Table 1. Thermophysical properties of water and some nanoparticle materials Properties

Fluid (water)

CuO

Cu

K (W/mK)

0.613

76.5

400

 (kg / m3 )

997.2

6320

8940

C p (J / kgK)

4179

532

385

Table 2. Comparison results of f '' (0) for different values of physical parameters at fixed parameters M=Gc =Gt =We =E1=0 and n=1 λ

f '' (0)

f '' (0)

Sulochana et al.[20]

0.1 0.2 0.3

1.232612 1.146610 1.051101

Present

1.2321 1.1529 1.0672

Table 3. Numerical results of - f '' (0) , − ' (0) and - ' (0) for different values of physical parameters at fixed parameters R0=0.01, λ=Gc=Sc=n=We=Nt=Nb=0.1 and Pr=10 E 0.1 0.2 0.3 0.4 0.5 0.6

Gr 0.1 0.2 0.3 0.4 0.5 0.6

M 0.1 0.2 0.3 0.4 0.5 0.6

AL Ec 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6

E1 0.1 0.2 0.3 0.4 0.5 0.6

f '' (0) 1.4715 1.5957 1.7314 1.8789 2.0386 2.2111

-  (0) 1.0047 1.0066 1.0075 1.0069 1.0047 1.0006

-  (0) 1.0076 1.0109 1.0154 1.0212 1.0287 1.0382

'

'

Table 4 Importance parameters effects impact sequences made by effect gape Importance parameters effects

Effect Gape

Impact sequences

Fig. 2; Gr=0.1,0.5,1.0,1.5,2

ΔGr〜0.5

3

Fig. 3;M = 1, 5, 10, 15, 20

ΔM〜5

5

Fig. 4;Ec = 4, 5, 6, 7, 8

ΔEc〜1

4

Fig. 5;Pr = 1, 5, 10, 20, 30

ΔPr〜5

5

Fig.6;Ncc=100,150,200,250,300

ΔNcc〜50

6

Fig.9; R0=0.1,0.2,0.25, 0.3,0.35

ΔR0〜0.05

1

Fig. 10; E1= 0.1,0.5, 1.0,1.5,2.0

ΔE1〜0.5

3

Fig. 11; Gc = 0.1,0.5,1.0,1.5,2.0

ΔGc〜0.5

3

27

ACCEPTED MANUSCRIPT

Fig 1.

Fig. 12;λ=0.1, 0.2, 0.3, 0.6,1.0

Δλ〜0.1

2

Fig. 13; Nt=0.1, 0.5, 1.0, 1.5, 2.0

ΔNt〜0.5

3

Fig. 14; Nb=0.1, 0.5, 1.0, 1.5, 2.0

ΔNb〜0.5

3

Fig. 15; Sc =40,60,100,150,200

ΔSc〜50

6

Fig. 16;  A = 1, 50,100,150,200

Δ  A 〜50

6

A sketch of physical model for thermal energy conversion combined with radiation activation energy electrical MHD mixed convection of stagnation-point Carreau fluid flow on a thermal forming stretching sheet.

Fig. 2 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, 28

ACCEPTED MANUSCRIPT M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1, Sc=0.1, E1=0.1, AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Gr = 0.1, 0.5, 1.0, 1.5, 2.0

Fig. 3 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.01, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1, Sc=0.1,

E1=50, AL=0.1, R0=0.01,λ=0.1, Nt=0.1, Nb=0.1, n=1.1, We=0.1, w=1, E A  1,  A  0.1 and M = 1, 5, 10, 15, 20

Fig. 4 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Gr=1, M=0.1, Pr=100, E=0.1, Ncc = 10, Gc=0.1, Sc=0.1,

E1=0.1, AL=0.1, R0=0.01,=0.1,Nt=0.1, 29

ACCEPTED MANUSCRIPT Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Ec = 4, 5, 6, 7, 8

Fig. 5 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Gr=1, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1, Sc=0.1,

E1=0.1, AL=0.1, R0=3, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Pr = 1, 5, 10, 20, 30

Fig. 6 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.1, M=0.1, Ec=0.1, E=0.1, Gc=0.1, Sc=0.1, E1=0.1,

AL=0.1, R0=0.01,λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 , 30

ACCEPTED MANUSCRIPT Ncc=100, 150, 200, 250, 300

Fig. 7 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.1, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1,

Sc=0.1, E1=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and AL= 10, 20, 30, 40, 50

Fig. 8 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.1, M=0.1, Ec=0.1, Gc=0.1, Sc=0.1, E1=0.1,

AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and (E=1, Ncc=5), (E=5, Ncc=10), (E=10, Ncc=15), (E=50, Ncc=20), (E=100, Ncc=25), 31

ACCEPTED MANUSCRIPT

Fig. 9 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.01, M=0.1, E=0.1, Ncc = 100, Gc=0.1, Sc=0.1,

E1=0.1, AL=0.1, Ec=1, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and R0 = 0.1, 0.2, 0.25, 0.30, 0.35

Fig. 10 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=1000, Gr=0, M=1000, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1,

Sc=0.1, AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=1.1, We=2.1, w=1, E A  1,  A  0.1 and E1 = 0.1, 0.5, 1.0, 1.5, 2.0

32

ACCEPTED MANUSCRIPT

Fig. 11 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=10, Gr=0.1, M=0.1, R0=0.01, Ec=0.1, E=0.1, Ncc = 10,

Sc=0.1, E1=0.1, AL=0.1,λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Gc = 0.1, 0.5, 1.0, 1.5, 2.0

Fig. 12 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=1000, Gr=16, M=0.1, Ec=0.1, E=0.1, Ncc = 100, Gc=0.1,

Sc=0.1, E1=0.1, AL=0.1, R0=0.01, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 andλ=0.1, 0.2, 0.3, 0.6, 1.0

33

ACCEPTED MANUSCRIPT

Fig. 13 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature Sc=0.1,

f vs. X as Pr=100, Gr=10, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1,

E1=0.1,

AL=0.1,

R0=0.01,

λ=0.1,

Nb=0.1,

n=0.1,

We=0.1,

w=1,

E A  1,  A  0.1 and Nt=0.1, 0.5, 1.0, 1.5, 2.0

Fig. 14 The energy conversion Carreau fluid thermal energy extrusion sheet dimensionless temperature

f vs. X as Pr=100, Gr=10, M=0.1, Ec=0.1, E=0.1, Ncc = 10, Gc=0.1,

Sc=0.1, E1=0.1, AL=0.1, R0=0.01,λ=0.1, Nt=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Nb=0.1, 0.5, 1.0, 1.5, 2.0

34

ACCEPTED MANUSCRIPT

Fig. 15 Dimensionless stretching sheet mass diffusion profiles ɸ vs. η as Pr=10, Gr=0.1, M=0.1, Ec=0.1, E=0.1, Gc=0.1, E1=0.1, AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1,  A  0.1 and Sc = 40, 60, 100, 150, 200

Fig. 16 Dimensionless stretching sheet mass diffusion profiles ɸ vs. X as Pr=10, Gr=0.01, M=0.1, Ec=0.1, E=0.1, Gc=0.1, Sc=0.1, E1=0.1, AL=0.1, R0=0.01, λ=0.1, Nt=0.1, Nb=0.1, n=0.1, We=0.1, w=1, E A  1 and  A = 1, 50, 100, 150, 200

35