IJITCE November 2012

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.2 NO.11 NOVEMBER 2012


UK: Managing Editor International Journal of Innovative Technology and Creative Engineering 1a park lane, Cranford London TW59WA UK E-Mail: [email protected] Phone: +44-773-043-0249

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IJITCE PUBLICATION

INTERNATIONAL JOURNAL OF INNOVATIVE

TECHNOLOGY & CREATIVE ENGINEERING

Vol.2 No.11

November 2012

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From Editor's Desk

Dear Researcher, Greetings! Research article in this issue discusses about Robust Scan Matching, Al6061/SiC Metal Matrix Composite by Taguchi’s Technique. Let us review research around the world this month; Software revamp guard grid against cyberattack. Gas pipelines, factories and power stations aren't necessarily the first targets that spring to mind. But such facilities are often controlled by software that lacks the appropriate defences. Now the cyber security company Kaspersky Labs, based in Moscow, Russia, is developing an operating system that it claims will block hacker or malware attacks on critical infrastructure. Research has shown that different colours of light affect more than just our conscious vision system. A set of receptors in our eyes responds to blue light by suppressing production of sleep-inducing melatonin, so the naturally blue-rich light of daytime keeps us alert, while reddish evening light lets us ease into sleep. Fluorescent bulbs contain a lot of blue light, so being exposed to them late in the day or at night can contribute to sleep problems. Many LEDs bulbs perpetuate the problem, because they generate white light using blue LEDs coated in compounds that emit longer wavelengths when illuminated. The Hue bulb instead contains red, green and blue LEDs. That's a more expensive way to generate white light, but the level of each colour can be adjusted, meaning it's possible to produce a broad gamut of colours, including white mixtures that contain very little blue light. NASA is developing similar lights for the International Space Station because astronauts have trouble sleeping more than 6 hours a night. The lights will switch from blue-rich to keep the astronauts alert during their working day to red-rich light when they are relaxing before bed. Tiny engine runs on single hydrogen molecules. Power packs don't get much smaller than this. The random movements of single hydrogen molecules have powered a tiny, vibrating springboard, mimicking molecular machines in nature. Heat from their surroundings causes all molecules to move randomly, but engineers tend to regard such movements as noise to be avoided like the plague. Jose Ignacio Pascual at the Free University of Berlin in Germany and colleagues took inspiration from the natural world, where random motion powers structures such as proteins that move cargo around inside cells. They placed a quartz springboard, weighing a fraction of a milligram, next to a slab of copper coated with hydrogen molecules. When a molecule changed its orientation, the force between the molecule and the board changed, setting the board vibrating. The team could keep the board vibrating by injecting electrons that encouraged the molecules to move, one at a time. It has been an absolute pleasure to present you articles that you wish to read. We look forward to many more new technology-related research articles from you and your friends. We are anxiously awaiting the rich and thorough research papers that have been prepared by our authors for the next issue. Thanks, Editorial Team IJITCE


Editorial M embers

Dr. Chee Kyun Ng Ph.D Department of Computer and Communication Systems, Faculty of Engineering, Universiti Putra Malaysia,UPM Serdang, 43400 Selangor,Malaysia. Dr. Simon SEE Ph.D Chief Technologist and Technical Director at Oracle Corporation, Associate Professor (Adjunct) at Nanyang Technological University Professor (Adjunct) at Shangai Jiaotong University, 27 West Coast Rise #08-12,Singapore 127470 Dr. sc.agr. Horst Juergen SCHWARTZ Ph.D, Humboldt-University of Berlin, Faculty of Agriculture and Horticulture, Asternplatz 2a, D-12203 Berlin, Germany Dr. Marco L. Bianchini Ph.D Italian National Research Council; IBAF-CNR, Via Salaria km 29.300, 00015 Monterotondo Scalo (RM), Italy Dr. Nijad Kabbara Ph.D Marine Research Centre / Remote Sensing Centre/ National Council for Scientific Research, P. O. Box: 189 Jounieh, Lebanon Dr. Aaron Solomon Ph.D Department of Computer Science, National Chi Nan University, No. 303, University Road, Puli Town, Nantou County 54561, Taiwan Dr. Arthanariee. A. M M.Sc.,M.Phil.,M.S.,Ph.D Director - Bharathidasan School of Computer Applications, Ellispettai, Erode, Tamil Nadu,India Dr. Takaharu KAMEOKA, Ph.D Professor, Laboratory of Food, Environmental & Cultural Informatics Division of Sustainable Resource Sciences, Graduate School of Bioresources, Mie University, 1577 Kurimamachiya-cho, Tsu, Mie, 514-8507, Japan Mr. M. Sivakumar M.C.A.,ITIL.,PRINCE2.,ISTQB.,OCP., ICP Project Manager - Software, Applied Materials, 1a park lane, cranford, UK Dr. Bulent Acma Ph.D Anadolu University, Department of Economics, Unit of Southeastern Anatolia Project(GAP), 26470 Eskisehir, TURKEY Dr. Selvanathan Arumugam Ph.D Research Scientist, Department of Chemistry, University of Georgia, GA-30602, USA.

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Contents

Dirichlet series and approximate method for the sol ution of axisymmetric flow over a stretching sheet by Vishwanath B. Awati ……......…………...…………..………………………… ……………………………………………[1]

Optimization of Tribological Parameters in Al6061/S iC Metal Matrix Composite by Taguchi’s Technique by Ashok Kr Mishra, Rakesh Sheokand, Gopal Pr Yadav, D r. B. C. Sharma, Dr. R. K. Srivastava ………………[10]


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Dirichlet series and approximate method for the solution of axisymmetric flow over a

stretching sheet Vishwanath B. Awati

Department of Mathematics,Rani Channamma University,Belagavi - 591156, India. e-mail: [email protected]

Abstract— We study the boundary layer flows induced by

the axisymmetric stretching of a sheets are studied using more suggestive schemes. The equation of motion of a axisymmetric flow over a stretching sheet and the sheet is stretched with a velocity is proportional to the distance from the vertical axis. The governing nonlinear differential equations are reduced to nonlinear linear ordinary differential equations (ODEs) by using similarity transformations. The resulting nonlinear ODEs are solved by using fast convergent Dirichlet series method and an approximate analytical method by Method of stretching of variables. These methods have advantages over pure numerical methods for obtaining the derived quantities accurately for various values of the parameters involved at a stretch and these are valid in much larger domain as compared with the classical numerical schemes.

Keywords: Axisymmetric flow; boundary layer equations; Stretching sheet; Dirichlet series; Powell’s method; Least square approximation

I. INTRODUCTION

In this discussion, we consider boundary layer flow over axisymmetric stretching of a sheet which are of significant interest in the recent years due to their applications. The third order nonlinear ordinary differential equation over an infinite interval with parameter M, Hartman number (magnetic field) is of special interest and in very few specific cases they have analytical solutions. The flow of a viscoelastic fluid over a stretching sheet was investigated by Rajagopal et al. [1], Sarpkaya [2] who probably the first to consider the MHD flow of non-Newtonian fluids. Andersson [3] and Mamaloukas et al. [4] have obtained similarity solution of the boundary layer equation governing the flow of a viscoelastic and a second grade fluid past a stretching sheet in the presence of an external magnetic field. The fluid occupies the space above the sheet and the motion

is caused by stretching sheet in opposite directions with the velocity is proportional to the distance from the fixed axis studied by Crane [5], and also he has given an elegant solution of the problem . The more interesting fact is that the problem still admits an exact analytical solution. The other effects are taken into account, such as suction at the sheet was discussed by (Gupta and Gupta [6]), viscoelasticity of the fluid by Ariel [7,8], partial slip at the boundary by Wang [9]. The problem of flow due to the radial stretching of the sheet (i. e the velocity of the sheet is proportional to the distance from a vertical rather than a horizontal axis) does not have an exact solution. For this reason, this problem has received much less attention in the literature. Wang [10], discussed the numerical solution of the flow due to the radial stretching of the sheet. The effects of visco-elasticity were studied by Ariel for an elastico-viscous fluid [11] and the second grade fluid by [12]. Areal [13], discussed the axisymmetric flow due to stretching of a sheet in hydromagnetics as the prototype problem for the non-iterative algorithm and also develops an algorithm for solving the problems of the flow induced by the moving boundaries in hydromagnetics. Hayat et al.[14] has used modified decomposition method and Pade’ approximants, for the solution of equation third order nonlinear ODE with infinite interval arsing in MHD. Shahzad et al [15], investigated the exact solution for axisymmetric flow and heat transfer over nonlinearly radially stretching sheet by HAM. Recently, Khan and Shahzad [16] have analysed the axisymmetric flow of sisko fluid over a radially stretching sheet using HAM.

The present investigation is to analyze the boundary layer flow induced by axisymmetric stretching of a sheet given by Mirgolbabaei et al [17]. The solution of the resulting third order nonlinear boundary value


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problem with infinite interval is obtained by Dirichlet series method and approximation method. We seek solution of the general equation of the type

20f A f f B f C f′′′ ′′ ′ ′+ + + = (1)

with the boundary conditions

( ) ( ) ( )1 10 , 0 , 0f f fα β′ ′= = ∞ = ) (2)

where A, B and C are constants and prime denotes derivative with respect to the independent variable η .

This equation admits a Dirichlet series solution; necessary conditions for the existence and uniqueness of these solutions may also be found in [18, 19]. For a

specific type of boundary condition i.e. ( ) 0f ′ ∞ = , the

Dirichlet series solution is particularly useful for obtaining the derived quantities. A general discussion of the convergence of the Dirichlet series may also be found in Riesz [20]. The accuracy as well as uniqueness of the solution can be confirmed using other powerful semi-numerical schemes. Sachdev et al. [21] have analyzed various problems from fluid dynamics of stretching sheet using this approach and found more accurate solution compared with earlier numerical findings. Recently, Awati et al [22, 23] and Kudenatii et al [24] have analysed the problems from MHD boundary layer flow with nonlinear stretching sheet using the above methods and found more accurate results compared with the classical numerical methods. Dirichlet series solution and MSV which we present here is more attractive than adapted variational iteration method (AVIM) discussed by Mirgolbabaei et al [17].

The present work is structured as follows. In section 2 the mathematical formulation of the proposed problem with relevant boundary conditions is given. Section 3 is devoted to semi-numerical method for the solution of the problem using Dirichlet series. In section 4 the solution of the proposed problem by an approximate analytical method using the method of stretching of variables (MSV). In section 5 detailed results obtained by the novel method explained here are compared with the corresponding numerical schemes. Section 6 Conclusions.

II. MATHEMATICAL FORMULATION OF THE PROBLEM

Consider the equation of motion of electrically conducting, viscous incompressible fluid caused by radial stretching sheet at z=0 in the presence of transverse magnetic field. The stretching velocity of the sheet is proportional to the distance from the origin of

the sheet. In the cylindrical polar coordinates ( , , )r zθ

and the flow takes place in the upper half plane 0z > . In view of the rotational symmetry of the flow all physical

quantities are independent of θ i.e 0θ∂ ∂ ≡ . The

equation of motion for steady, laminar, axisymmetric flow and continuity are of the form (Mirgolbabaei et al [17])

2 2

0

2 2 2

1,

u u pu w

r z r

Bu u u uu

r rr z r

ρ

σµρ

∂ ∂ ∂ + = − + ∂ ∂ ∂ ∂ ∂ ∂+ + − − ∂∂ ∂

(3)

2 2

2 2

1,

w w p

u w

r z r

w w w

r rr z

ρ

µ

∂ ∂ ∂ + = − ∂ ∂ ∂ ∂ ∂ ∂+ + + ∂∂ ∂

(4)

0 ,u u w

r r z

∂ ∂+ + =∂ ∂ (5)

where is fluid density, µ is coefficient of viscosity, σ is

the electrical conductivity of the fluid, p is the pressure

and ( ),0,u w are the velocity components along

( ), ,r zθ directions. The boundary conditions for the

above flow situations are

, 0 at 0

0 as

u c r w z

u z

= = = → → ∞

(6)

where c > 0 is the constant of proportionality relating to the stretching of the sheet. The boundary layer Eq. (3)-


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.(6) admit the similarity solution (Wang [9]),

( ) ( ), 2 , ( )u crf w c f c zη υ η η υ′= =− = (7)

where υ µ ρ= is the kinematic viscosity of the fluid

and prime denotes differentiation with respect to η , eventually reduce the Navier-Stokes equation to an ODE. From Eq. (3), we get

2 22

pc r f f f f M f

r

ρ∂ ′′′ ′′ ′ ′ = + − − ∂ (8)

where 0M B cσ ρ= is the magnetic parameter. On the

other hand Eq. (4), gives

4 2p

c f f c fµ µη

∂ ′ ′′= − −∂ (9)

which when integrated with respect to η yields,

( )22 2 ,p c f c f g rµ µ ′=− − + (10)

where g(r) is an arbitrary function of r. Substitute for p Eq. (10) into Eq. (8), we obtain

( ) 2

22

g rf f f f M f

c rρ′

′′′ ′′ ′ ′= + − − (11)

Since in Eq. (11), the left hand side is a function of r only, and the right hand side is a function of η only, in

order for it to be consistent, each side must be constant, say A1. Hence we have

( ) 2

1g r c r Aρ′ =

(12)

Its integration with respect to r, gives

( ) ( ) 2 2

0 11/2g r p c r Aρ= +

(13)

where 0p is a constant. Substitution of g(r) from Eq.(13)

into Eq.(10) leads to

( ) 2 2 2

0 11/2 2 2p p c r A c f c fρ µ µ ′= + − − (14)

Since the entire motion of the fluid is caused due to stretching of the sheet, the pressure far away from the sheet must be given by the Bernoulli’s equation, i.e., Matching of the pressure from Eq. (14), gives A1=0. Hence from equation (11), we get the following DE for f (Mirgolbabaei et al [17])

22 0f f f f M f′′′ ′′ ′ ′+ − − = (15)

Also the pressure p at any point in terms of the physical variables is

2

0

1

2

w

p p w

z

ρ µ ∂= − +∂ (16)

The boundary conditions of the problem are

( ) ( ) ( )0 0, 0 1, 0f f f′ ′= = ∞ = (17)

III. DIRICHLET SERIES APPROACH TO THE BOUNDARY VALUE PROBLEMS OVER AN

INFINITE INTERVAL

We seek a Dirichlet series solution of equation (1) satisfying the last boundary condition of Eq.(2)

automatically i.e ( ) 0=∞′f in the form of (Kravchenko

and Yablonskii [18, 19])

1

1

6

i i

i

i

f b a e

A

γηγγ∞

−

=

= + ∑ (18)

where γ and a are parameters. Substituting (18) into

(1), we get


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{ }

( ){ }

2 3 2

1

1

2 1

2

2 1

6 0

i iy

i

i

i

i i y

k i k

i k

i A i C i b a e

A k B k i k b b a e

A

η

η

γ γγ

γ

∞−

=

∞ −−

−= =

− + −

+ + − =

∑

∑∑ (19)

For i=1, we have 2

1

C

A

γγ += (20)

Substituting (20) into (19) the recurrence relation for obtaining coefficients is given by

( ){ } { }2 1

2

2

1

6( )

1

i

i k i k

k

b A k B k i k b b

A i i i C

γγ

−

−=

= + −− − ∑

(21)

For i = 2, 3, ..... . If the series (18) converges absolutely

when 0γ > for some 0

η , this series converges

absolutely and uniformly in the half plane

0Re Reη η≥ and represents an analytic

( )2 iπ γ periodic function ( )0f f η= such that

( ) 0=∞′f (Kravchenko & Yablonskii [19]). The series

(18) contains two free parameters namely a andγ .

These unknown parameters are determined from the

remaining boundary conditions (2) at 0η =

( )2

1

1

60

i

i

i

Cf b a

A A

γ γ αγ

∞

=

+= + =∑ (22)

and ( ) ( )2

1

1

60

i

i

i

f i b a

A

γ β∞

=

′ = − =∑ (23)

The solution of these transcendental equations (22) and (23) yield, constants a and γ . The solution of these

transcendental equations is equivalent to the unconstrained minimization of the functional

( )2 2

2 2

1 1

1 1

6 6

i i

i i

i i

C

b a i b a

A A A

γ γ γα βγ

∞ ∞

= =

+ + − + − −

∑ ∑ (24)

We use Powell’s method of conjugate directions (Press et al [25]) which is one of the most efficient techniques for solving unconstrained optimization problems. This

helps in finding the unknown constants a and γ uniquely for different values of the parameters A, B,

C,1

α and1

β . Alternatively, Newton’s method is also

used to determine the unknown parameters a andγ accurately. The shear stress at the surface of the problem is given by

( ) ( )2

1

60

i

i

i

f b a i

A

γ γ∞

=

′′ = ∑ (25)

The velocity profiles of the problem is given by

( )2

1

6 ( ) i i

i

i

f i b a e

A

γ ηγη∞

−

=

′ = −∑ (26)

IV. METHOD OF STRETCHING OF VARIABLES

Many nonlinear ODE arising in MHD problems are not amenable for obtaining analytical solutions. In such situations, attempts have been made to develop approximate methods for the solution of these problems. The numerical approach is always based on the idea of stretching of variables of the flow problems. Method of stretching of variables is used here for the solution of such problems. In this method, we have to choose

suitable derivative function H ′ such that the derivative boundary conditions are satisfied automatically and

integration of H ′ will satisfy the remaining boundary condition. Substitution of this resulting function into the given equation gives the residual of the form

( ),R ξ α which is called defect function. Using Least

squares method, the residual of the defect function can be minimized. For details see (Ariel, [26]). Using the

transformation Fffw

+= into Eq. (1), we get

( ) 20,

w

F A f F F B F C F′′′ ′′ ′ ′+ + + + = 27)

and the boundary conditions (2) become

( ) ( ) ( )0 0, 0 1, 0F F F′ ′= = ∞ = (28)

We introduce two variables ξ and Gin the form


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( ) ( )G Fξ α η= andξ αη= (29)

whereα >0, is an amplification factor. In view of Eq.(29), the system (27-28) are transformed to the form

( )2 20,

w

G A f G G BG CGα α′′′ ′′ ′ ′+ + + + = (30)

and the boundary conditions in Eq. (28) become

(0) 0, G (0) 1, G ( ) 0G ′ ′= = ∞ = (31)

We choose a trail velocity profile

exp( )G ξ′ = − (32)

which automatically satisfies the derivative conditions in

Eq.(31). Integrating Eq. (32) with respect toξ from 0 to

ξ using the first boundary conditions in (31) and then

substituting this into Eq. (30), we get the residual of the defect function

( )( )

2( , )

exp( ) exp( 2 )

w

R A f A C

A B

ξ α α α

ξ ξ

= − − +

− + + − (33)

By using the least squares method as discussed in Ariel [26], the equation (33) can be minimized for which

( )2

0

, 0R dξ α ξα

∞∂ =∂ ∫ . (34)

Substituting (33) into equation (34) and solving cubic equation inα for a positive root, we get

( )2 213 3 4 8 12 3

6w w

A f A B C A fα = ± − − +

and 2

w

A fα = .

(35)

Once the amplification factor is calculated, then using Eq.(27), original function f can be written as

( )11 exp( )

w

f f αηα

= + − − . (36)

with α defined in Eq. (35). Thus Eq. (36) gives the

solution of Eq. (1) for all A , B , C and w

f .

V. RESULT AND DISCUSSION

In the present paper, the axisymmetric flow over a stretching of a sheet is discussed by using semi-numerical method and approximate analytical method. The Eq. (15) and (17) are solved semi-numerically using one of the powerful techniques due to Dirichlet series method and the method of stretching of variables. We have given an exact analytical solution of the boundary value problem in more general form. In this semi-numerical method and, it is important to note that the edge boundary condition automatically satisfied an also we have given analytical solution by approximate method.

Case I. Consider the flow of a fluid with no magnetic

field Eq. (15) reduces to 22 0f f f f′′′ ′′ ′+ − = and the

boundary conditions are same as of Eq.(17). We check the validity of our solution by comparing it with the exact solution. The measure of the physical quantity viz. shear

stress at the sheet is ( )0f ′′− . The exact value

of ( )0f ′′− is 1.173721 and the value obtained by

Dirichlet series is 1.173721 and MSV is 1.15470.The error being very less as compared to AVIM.

Case II. In this case, consider the flow of an electrically conducting, viscous incompressible fluid over a radially stretching sheet in presence of a transverse magnetic filed. The governing equation is same as Eq. (15) and

M is the Hartmann number. As the value of M

increased, Hartmann layers start in at 0η = causing the

great difficulties in obtaining the numerical solution. We have presented much better solution than AVIM by using Dirichlet series and MSV for arbitrary values of M which are comparable with exact numerical solutions by Ariel [26] which are listed in Table 1. The above said methods are capable for providing the solutions in the presence of Hartmann layers near the stretching sheet.


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Case III. The massive transfer of the fluid across the boundary, the type of the boundary layer is manifested near the boundary. The semi-numerical and approximation method are able to handle the suction boundary layer in an efficient manner with Hartmann layer. The suction takes place across the sheet, the boundary conditions (17) changes to

( ) ( ) ( )0 , 0 1, 0w

f f f f′ ′= = ∞ = , where w

f is the suction

parameter given by ( ) ( )02

w

f w cρ µ= , where

0w is the suction velocity. The governing equation for f is

same as Eq. (15). The values of ( )0f ′′− are presented

for various a vales of w

f using Dirichlet series method

and MSV are given in Table 2. and these values are comparable with numerical solution given by Ackroyd [27].

Case IV. In this case, we consider the flow of an electrically conducting viscous incompressible fluid due to radial stretching of a sheet in the presence of the transverse magnetic field and also the suction at the sheet. The governing equation is same as Eq. (15) and

relevant boundary conditions ( ) ( ) ( )0 , 0 1, 0w

f f f f′ ′= = ∞ = .

Numerical computations are performed by using the above said methods for various values of the physical parameters involved in the equation viz., Hortmann

number M, and mass suction parameterw

f . The present

solutions are then validated by comparing it with the previously published work of Migolbabaei et al [17] as

shown in the Tables 3.

Table1. Comparison of Dirichlet series method, Method of stretching of variables (MSV) with exact solution and Adapted variational Iteration method (AVIM) for the flow in the presence of magnetic field.

M Dirichlet Series Method Exact

( )0f ′′−

MSV

( )0f ′′−

AVIM

( )0f ′′− a γ ( )0f ′′−

0 -0.56719 1.50299 1.17372 1.17372 1.15470 1.182125

0.01 -0.18857 1.50659 1.17606 1.17783 1.15902 1.192910

0.04 -0.18627 1.51758 1.17902 1.17901 1.17189 1.209962

0.25 -0.16798 1.57558 1.25906 1.27303 1.25831 1.298851

1.0 -0.12258 1.77491 1.53417 1.53571 1.52752 1.556834

4.0 -0.05861 2.46267 2.31048 2.31172 2.30940 2.322880

25.0 -0.01249 5.19738 5.13178 5.13181 5.13160 5.151863

100 -0.00328 10.09967 10.06632 10.06647 10.06645 10.087672

500 -0.00066 22.40537 22.39211 --- 22.39047

1000 -0.00033 31.65439 31.64398 --- 31.64385

Table 2.Comparison of Dirichlet series method, Method of stretching of variables (MSV) with exact solution and Adapted variational Iteration method (AVIM) for the flow in the presence of magnetic field.


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A Dirichlet Series Method Exact

( )0f ′′−

MSV

( )0f ′′−

AVIM

( )0f ′′− a γ ( )0f ′′−

0 -0.56719 1.50299 1.17372 1.17372 1.15470 1.182125

0.1 -0.16335 1.59707 1.27865 1.28242 1.25902 1.329105

0.2 -0.14159 1.69558 1.41216 1.40236 1.37189 1.488416

0.5 -0.09085 2.04339 1.79567 1.79867 1.75931 2.057265

Table 3. Comparison of Dirichlet series method, Method of stretching of variables(MSV) with exact solutionand Adapted variational Iteration method (AVIM) for MHD flow with sucion.

M A Dirichlet Series Method Exact

( )0f ′′−

MSV

( )0f ′′−

AVIM

( )0f ′′− a γ ( )0f ′′−

1

0 -0.12258 1.77491 1.53417 1.53571 1.52753 1.556834

0.1 -0.10843 1.87345 1.64267 1.64312 1.63079 1.683059

0.2 -0.09608 1.97844 1.75491 1.75664 1.74056 1.821750

0.5 -0.06724 2.32755 2.13048 2.13194 2.10728 2.328433

4

0 -0.05861 2.46267 2.31049 2.31172 2.30940 2.322880

0.1 -0.05386 2.56390 2.41458 2.41586 2.41156 2.444133

0.2 -0.04948 2.66927 2.52468 2.52422 2.51804 2.576313

0.5 -0.03855 3.00932 2.87325 2.87403 2.86291 3.051377

25

0 -0.01249 5.19738 5.13178 5.13181 5.13160 5.15186

0.1 -0.01202 5.29831 5.23308 5.23319 5.23258 5.289863

0.2 -0.01156 5.40116 5.33646 5.33653 5.33549 5.435615

0.5 -0.01029 5.72122 5.65798 5.65812 5.65590 5.92565


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VI. CONCLUSIONS

In this article, we describe the analysis of boundary value problem for third order nonlinear ODEs over an infinite interval arising in axisymmetric flow over a stretching sheet. The semi-numerical and an approximate analytical scheme described here offer some advantages over solutions by HAM, HPM, Adomain decomposition methods and AVIM etc. The convergence of the Dirichlet series methods is given. The results are presented in Tables.

ACKNOWLEDGEMENTS

The author is grateful to University Grant Commission, New Delhi for the financial support through Major Research Project (F. No. 41-781/2012 (SR)) for this work.

REFERENCES

[1] K. R. Rajagopal, T.Y. Na and A.S. Gupta, AS., Flow of viscoelastic fluid over a stretching sheet, Rheology Acta.,vol. 23,pp. 213-215, 1984.

[2] T. Sarpkaya, Flow of non-Newtonian fluids in a magnetic field AIChE. J.,vol 7,pp. 324-328, 1961.

[3] H.I. Andersson, MHD Flow of a Viscoelastic Fluid past a Stretching Surface. Acta Mechanica, , vol 95, pp. 227–230, 1992.

[4] Ch. Mamaloukas, S.Spartalis, Z. Manussaridus, Similarity approach to the problem of second grade fluid flows over a stretching sheet, ICNAAM, Wiley-VCH, ESCMSE, pp. 239-245, 2004.

[5] L.Crane, Flow past a stretching sheet. ZAMP, vol. 21, pp. 645-647, 1970.

[6] P. Gupta, A. Gupta , Heat and Mass Transfer on a Stretching Sheet with Suction or Blowing. Can. J. Chem Eng., vol. 55, pp. 744-746,1977.

[7] P.D. Ariel, A Hybrid Method for Computing the Flow of Viscoelastic Fluids. Int. Jl. Numerical Methods in Fluids, vol. 14, pp. 757–774, 1992.

[8] P.D.Ariel, A Numerical Algorithm for Computing the Stagnation Point Flow of a Second Grade Fluid with or

withoutSuction. Journal of Computational and Applied Mathematics, vol. 59,pp. 9–24, 1995.

[9] C. Wang , Flow due to a Stretching Boundary with Partial Slip-An Exact solution of the Navier-Stokes equations.Chemical Engineering Science, vol. 57, pp. 3745–3747 2002.

[10] C.Wang, The three dimensional flow due to a stretching flat surface. Physics of Fluids, vol. 27,pp. 1915–1917, 1984.

[11] P.D.Ariel, Computation of flow of viscoelastic fluids by parameter differentiation. International Journal for Numerical Methods in Fluids, vol. 15, pp.1295–1312, 1992.

[12] P.D.Arie. Axisymmetric flow of a second grade fluid past a stretching sheet. International Journal of Engineering Science, vol. 39,pp. 529–553, 2001.

[13] P.D. Ariel, Computation of the MHD flow due to moving boundary. Trinity Western University. Dept. of Mathematical Sci. Tech. Report, 2004. MCS-2004-01.

[14] T.Hayat, Q.Hussain and T. Javed , The modified decomposition method and Pade’ approximants for the MHD flow over nonlinear stretching sheet, Nonlinear Analysis: Real World Applications, vol. 10,pp. 966-973, 2009.

[15] A .Shahzad, R. Ali, M. Khan, On the exact solution for axisymmetric flow and heat transfer over nonlinear radially stretching sheet, Chin. Phys. Lett., Vol. 29(8), pp. 084705-1-4, 2012.

[16] M.Khan, A. Shahzad, On axisymmetric flow of sisko fluid over a radially stretching sheet, Int. Jl.Nonlin. Mech.,vol. 47, pp. 999-1007, 2012.

[17] H.Mirgolbabaei, D.D.Ganji, M.M. Etghani and A. Sobati, Adapted variational iteration method and axisymmetric flow over a stretching sheet, World Jl. Modelling and Simulation, vol.5(4), pp. 307-314, 2009.

[18] T.K.Kravchenko, A.I.Yablonskii, Solution of an infinite boundary value problem for third order equation, Differential’nye Uraneniya, vol. 1,pp. 327, 1965.

[19] T.K. Kravchenko, A.I.Yablonskii, A boundary value problem on a semi-infinite interval, Differential’nye Uraneniya, vol. 8(12), pp. 2180-2186, 1972.

[20] S.Riesz , Introduction to Dirichler series, Camb. Univ. Press, 1957.

[21] P.L.Sachdev, N. M.Bujurke and V. B. Awati, Boundary value problems for third order nonlinear ordinary differential equations, Stud. Appl. Math., vol. 115, pp. 303-318, 2005.


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[22] V.B.Awati , N.M.Bujurke and R.B.Kudenatti, An exponential series method for the solution of the free convection boundary layer flow in a saturated porous medium. AJCM, vol. 1 , pp. 104-110, 2011.

[23] V.B.Awati ,N.M. Bujurke and R.B. Kudenat, Dirichlet series method for the solution of MHD flow over a nonlinear stretching sheet. IJAMES, vol. 5(1), pp.07-12, 2011.

[24] R.B.Kudenatti, V.B.Awati and N.M.Bujurke, Exact analytical solutions of class of boundary layer equations for a stretching surface, Appl Math Comp, vol. 218, pp. 2952-2959, 2011.

[25] W.H.H.Press, B.P.Flannery, S.A.Teulosky and W.T.Vetterling, Numerical Recipes in C, Camb. Univ. Press, UK, 1987.

[26] P.D.Ariel, Stagnation point flow with suction ; an approximate solution. Journal of Applied Mechanics, vol. 61(4), pp. 976-978, 1994.

[27] J. A. D. Ackroyd, A series method for the solution of laminar boundary layers on moving surface, ZAMP, vol. 29, pp.729-741, 1978.


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Optimization of Tribological Parameters in Al6061/SiC Metal Matrix Composite by

Taguchi’s Technique Ashok Kr Mishra1, Rakesh Sheokand2, Gopal Pr Yadav3, Dr. B. C. Sharma4, Dr. R. K. Srivastava5

1. Astt. Prof. in BRCM College of Engineering & Technology, Bahal 2. Astt. Prof. in Jind Institute of Engineering & Technology, Jind

3. Prof. & Head Deptt. of Mechanical, BRCM CET, Bahal 4. Principal & Prof. in BRCM CET, Bahal

5. Prof. & Head Deptt. of Production, B. I. T. Sindri, Jharkhand Corresponding Author1: email: [email protected]

Abstract

Tribological behavior of aluminium alloy Al 6061 reinforced with silicon carbide (10%) fabricated by stir casting process was investigated. Dry sliding wear test was conducted to understand the tribological behavi or of samples. This study is carried out to optimize the tribological properties: wear rate & frictional for ce of Al / SiC metal matrix composite. The experiments were conducted as the Taguchi design of experiment. A L 9 orthogonal array was selected for analysis of the d ata. The wear parameters chosen for the experiment were: sli ding speed, applied load & sliding distance. Each parame ter was assigned three levels. Signal to Noise ratio an alysis has been carried out to determine optimal parametri c condition, which yields minimum wear rate & frictio nal force. Investigation to find the influence of appli ed load, sliding speed and sliding distance on wear rate, as well as the frictional force during wearing process was car ried out using ANOVA and regression equations for each respo nse were developed. Objective of the model was chosen a s “smaller the better” characteristics to analyze the dry sliding wear resistance. Result show that sliding d istance and load have the highest influence on minimum wear rate and frictional force respectively. Finally, confirm ation tests were carried out to verify the experimental r esults and Scanning Electron Microscopic (SEM) studies wer e done on the wear surfaces.

Keywords: Metal matrix composites; Silicon Carbide; Taguchi Technique; Signal to Noise

I. INTRODUCTION In the last two decades, Researcher has shifted from monolithic materials to composite materials to meet the global demand for light weight, high performance, environmental friendly, wear & corrosion resistant materials. Particulate reinforced metal matrixes Composites have combination of low density, improved stiffness and strength, high wear resistance and isotropic properties [5]. These properties of MMCs

enhance their uses in automotive and tribiological application in the field of automobile,MMCs are used for piston, Brake Drum and Cylinder block because of better corrosion and wear resistance [7,8].

The principle tribiological parameter effects the friction and wear performance of dis-continously reinforced aluminium composite are surface interaction, Mechanical characteristic (Extrinsic to materials), Material characteristic (Intrinsic to material) and tribo-contact condition. Most frequently encountered factors which are associated with four tribological parameter such as mechanical parameter, surface interaction, material parameter and tribo contact which yields wear and friction mechanism. The mechanical parameter depends upon the different factor such as loading condition, contact geometry and duration of interaction. Surface interaction parameter depends upon the various operations such as rolling, sliding, impact, erosion, fretting and reciprocating. Material parameter associated with types of chemical bonding, thermal properties, fracture toughness, hardness and yield strength. Tribo contact parameter concern with chemical reaction during dry or lubricated sliding. Dry sliding wear behavior of Al Metal Matrix Composite has been reported (and abrasive wear of Al Composites has extensively reviewed by Deuis et al [13]. Uyyuru et al [14] studied the effect of reinforcement volume fraction and size distribution of reinforced the tribological behavior of Al-Composites and Martin investigated the temperature effect on the wear behavior of particulate reinforced Al based composite[15] and influence of heat treatment on the wear behavior of Al Composites by M. A. Chowhury et al[12]. Effect of aging on behavior of MMCs was carried out by Guo et al [16] The result shows that SiC particles are effective agents in increasing dry sliding wear resistance of Al6061/SiC Composite.


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Much research carried out to understand wear behavior of composite material. Meager information is available regarding the optimization of tribological parameters: Sliding wear rate and Frictional force of the metal matrix composite. In this light, this study is carried out to optimize tribological parameters of Silicon Carbide (10%) reinforced Al Metal Matrix Composite using Taguchi’s Parameter design methodology.

1.1 Design of Experiment

Quality characteristic of a product under investigation in response to a factor introduced in the experimental design is the ‘signal’ of the desired effect. The effect of external factors (uncontrollable factor) on the outcome of quality characteristics under test is termed as ‘noise’. The S/N Ratio measures the sensitivity of the quality characteristic being investigated in a controlled manner to those of external influencing factors (noise factor) not under control. The S/N Ratio is transformed figure of merit, created from the loss function. S/N Ratio combines both the parameters (the mean level of the quality and the variation around this mean) in a signal metric. The aim in any experiment is always to determine the highest possible ratio for the result (wear rate and frictional force) a high value of S/N Ratio implies the signal is much higher then the random effect of noise factor.

II. EXPERIMENTAL DETAILS

2.1 Test Materials

In the present investigation, Al 6061 was chosen as the base matrix since its properties can be tailored through heat treatment process. The reinforcement of SiC was, average size of 150 to 160 microns, and there are sufficient literatures including the improvement in wear properties through the addition of SiC. Due to the property of high hardness and high thermal conductivity, SiC after accommodation in soft ductile aluminium base matrix, enhance the wear resisting behaviour of the Al / SiC metal matrix composite.

Table 1 Chemical composition of matrix alloy Al – 6061

C he m i c a l

c o m p o s i ti o n S i F e C u M n M g C r Z n T i A l

% 0 .4 -0 .8 0 .7 0 .1 5 - 0 .4 0 0 .1 5 0 .8 -1 .2 0 .0 4 - 0 .3 5 0 .2 5 0 .2 B a l a nc e

2.2 Composite Preparation In order to achieve high level of mechanical properties in the composite, a good interfacial bonding (wetting) between the dispersed phase and the liquid matrix has to be obtained. Stir-casting technique is one such simplest and cost effective method to fabricate metal matrix composites which has been adopted by many researchers. This method is most economical to fabricate composites with discontinuous fibres and particulates and was used in this work to obtain as the cast specimens. Care was taken to maintain an optimum casting parameter of pouring temperature (650°C) an d stirring time (15 min). The reinforcements were preheated prior to their addition in the aluminium alloy melt. Degassing agent (hexachloro ethane) was used to reduce gas porosities. The molten metal was then poured into a permanent cast iron mould of diameter 26mm and length 300mm. The die was released after 6 hours and the cast specimens were taken out. 2.3 Wear Behaviour The aim of the experimental plan is to find the important factors and combination of factors influencing the wear process to achieve the minimum wear rate and frictional force. The experiments were developed based on an orthogonal array, with the aim of relating the influence of sliding speed, applied load and sliding distance. These design parameters are distinct and intrinsic feature of the process that influence and determine the composite performance [17]. Taguchi recommends analysing the S/N ratio using conceptual approach that involves graphing the effects and visually identifying the significant factors. 2.4 Plan of experiment using orthogonal array Dry sliding wear test was performed with three parameters: applied load, sliding speed, and sliding distance and varying them for three levels. According to the rule that degree of freedom for an orthogonal array should be greater than or equal to sum of those wear parameters, a L9 Orthogonal array which has 9 rows and 3 columns was selected. Table 2 Process parameters and levels

L e v e l L o a d (N ) S l i d i n g S p e e d ( m /s ) S l i d i n g D i s t a n c e (m )

1 1 0 2 1 0 0 0

2 2 0 3 1 7 5 0

3 3 0 4 2 5 0 0


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2.5 Experimental Procedure

To evaluate the tribological performance of the composites under dry sliding condition, wear tests were carried out on a pin on disc type friction and wear monitoring test rig. Specimens of size 10 mm diameter and 25 mm length were cut from the cast samples, and then machined. The contact surface of the cast sample (pin) was made flat so that it should be in contact with the rotating disk. During the test, the pin was held pressed against a rotating EN31 carbon steel disc (hardness of 65HRC) by applying load that acts as counterweight and balances the pin. The track diameter was varied for each batch of experiments in the range of 50 mm to 100 mm and the parameters such as the load, sliding speed and sliding distance were varied in the range given in Table 2. A LVDT (load cell) on the lever arm helps determine the wear at any point of time by monitoring the movement of the arm. Once the surface in contact wears out, the load pushes the arm to remain in contact with the disc. This movement of the arm generates a signal which is used to determine the maximum wear and the coefficient of friction is monitored continuously as wear occurs and graphs between co-efficient of friction and time was monitored for the specimens i.e., 10 % SiC/ Al-6061 MMCs.

Table 3 Results of L9 Orthogonal array for Al – 6061 / 10% SiC MMC

S . N o . L (N ) S (m /s ) D (m ) F r i c t i o na l

F o rc e (F F ) N

W e a r r a t e (m m

3

/m )

S /N ra t i o (F F ) d b

S /N r a t i o (w e a r ra t e ) d b

1 1 0 2 1 0 0 0 3 .1 1 0 .0 0 4 8 1 - 9 .8 5 5 2 4 6 .3 5 7 1

2 1 0 3 1 7 5 0 2 .9 0 0 .0 0 3 6 - 9 .2 7 7 9 4 8 .8 7 3 9

3 1 0 4 2 5 0 0 2 .7 7 0 .0 0 1 7 8 - 8 .8 4 9 6 5 4 .9 9 1 6

4 2 0 2 1 7 5 0 7 0 .0 0 4 2 2 - 1 6 .9 0 2 4 7 .4 9 3 8

5 2 0 3 2 5 0 0 6 .8 6 0 .0 0 2 2 2 - 1 6 .7 2 6 5 5 3 .0 7 2 9

6 2 0 4 1 0 0 0 7 .4 4 0 .0 0 3 7 - 1 7 .4 3 1 5 4 8 .6 3 6 0

7 3 0 2 2 5 0 0 1 0 .8 0 .0 0 2 9 6 - 2 0 .6 6 8 5 5 0 .5 7 4 2

8 3 0 3 1 0 0 0 1 2 .3 0 .0 0 3 7 - 2 1 .7 9 8 1 4 8 .6 3 6 0

9 3 0 4 1 7 5 0 1 1 .7 0 .0 0 2 5 4 - 2 1 .3 6 3 7 5 1 .9 0 3 3

III. RESULT DISCUSSION

3.1 Signal to Noise Ratio

The experimental observations are transformed into Signal to Noise ratio. There are several S/N ratios available depending on the type of characteristics under study. The wear rate & frictional force are coming ‘smaller is better’ type of quality characteristic and the respective S/N ratio is calculated using formula given below.

S/N

Where n = no. of tests in a trial

For the present study n = 2

The aim of this experiment is to determine the highest possible Signal to Noise ratio for the parameters under study. A high value of S/N ratio implies that signal is much higher than the random effects of noise factors. The S/N ratio was computed using above formula for each response of the Table 3.

3.2 Results of Statistical Analysis of Experiments

The results for various combinations of parameters were obtained by conducting the experiment as per the orthogonal array. The measured results were analysed using the commercial software MINITAB 15 specifically used for design of experiment applications [23]. Table 3 shows the experimental results average of two repetitions for wear rate and frictional force. To measure the quality characteristics, the experimental values are transformed into signal to noise ratio. The influence of control parameters such as load, sliding speed, and sliding distance on wear rate and frictional force has been analysed using signal to noise response table. The ranking of process parameters using signal to noise ratios obtained for different parameter levels for wear rate & frictional force are given in Table 3 and Table 4 for 10% reinforced SiC MMCs. The control factors are statistically significant in the signal to noise ratio and it could be observed that the sliding distance is a dominant parameter on the wear rate and frictional force followed by applied load significantly. Figure 1 to Figure 4 shows for 10% influence of process parameters on wear rate and coefficient of friction graphically. The analysis of these experimental results using S/N ratios gives the optimum conditions resulting in minimum wear rate and coefficient of friction. The optimum condition for wear rate and frictional force as shown in Table 8.

3.3 Analysis of Variance Results for Wear Test

The experimental results were analysed with Analysis of Variance (ANOVA) which is used to investigate the influence of the considered wear parameters namely, applied load, sliding speed, and sliding distance that significantly affect the performance measures. By performing analysis of variance, it can be decided which independent factor dominates over the other and the percentage contribution of that particular independent variable. Table 6 and Table 7 shows 10% SiC MMCs of the ANOVA results for wear rate and coefficient of


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friction for three factors varied at three levels and interactions of those factors. This analysis is carried out for a significance level of •=0.05, i.e. for a confidence level of 95%. Sources with a P-value less than 0.05 were considered to have a statistically significant contribution to the performance measures. Table 4 Response Table for Signal to Noise Ratios Smaller is better (Wear Rate)

L e v e l L o a d S p e e d D i s t a n c e

1 5 0 . 0 7 4 8 . 1 4 4 7 . 8 8

2 4 9 . 7 3 5 0 . 1 9 4 9 . 4 2

3 5 0 . 3 7 5 1 . 8 4 5 2 . 8 8

D e l ta 0 .6 4 3 .7 0 5 .0 0

R a n k 3 2 1

Table 5 Response Table for Signal to Noise Ratios Smaller is better (friction force)

L e v e l L o a d S p e e d D i s t a n c e

1 - 9 .3 2 8 - 1 5 . 8 0 9 - 1 6 . 3 6 2

2 - 1 7 . 0 2 0 - 1 5 . 9 3 4 - 1 5 . 8 4 8

3 - 2 1 . 2 7 7 - 1 5 . 8 8 2 - 1 5 . 4 1 5

D e l ta 1 1 . 9 4 9 0 .1 2 6 0 .9 4 7

R a n k 1 3 2

D IS T A N C E

L O A D V E L O C IT Y

M a i n E ff e c t s P l o t f o r S N ra t i o s

D a ta M e a n s

1 0 0 0 1 7 5 0 2 5 0 0

- 2 0 .0

- 1 7 .5

- 1 5 .0

- 1 2 .5

- 1 0 .0

- 2 0 .0

- 1 7 .5

- 1 5 .0

- 1 2 .5

- 1 0 .0

1 0 2 0 2 3 0 3 4

Mean o

f S

N r

atio

s

S ig n a l - to - n o is e : S m a l l e r to b e tte r

Fig.1 Main effects plot for S/N ratios –Frictional force

D IS T A N C E


M a in E f f e c t s P l o t f o r M e a n s

D a ta M e a n s

1 0 0 0 1 7 5 0 2 5 0 0

4

6

8

1 0

1 2

4

6

8

1 0

1 2

1 0 2 0 2 3 0 3 4

Mean of M

eans

S ig n a l - to - n o i s e : S m a l l e r to b e tte r

Fig.2 Main effects plot for Means –Frictional force

D I S T A N C E


M a in E f f e c t s P l o t f o r M e a n s

D a ta M e a n s

1 0 0 0 1 7 5 0 2 5 0 0

0 .0 0 2 5

0 .0 0 3 0

0 .0 0 3 5

0 .0 0 4 0

1 0 2 0 2 3 0 3 4

Mean o

f M

ean

s

S ig n a l - to - n o is e : S m a l le r to b e tte r

0 .0 0 2 5

0 .0 0 3 0

0 .0 0 3 5

0 .0 0 4 0

Fig.3 Main effects plot for Means –Wear Rate

D IS T A N C E


M a i n E ff e c t s P l o t f o r S N ra t i o s

D a ta M e a n s

1 0 0 0 1 7 5 0 2 5 0 0

4 8 .0

4 9 .2

5 0 .4

5 1 .6

5 2 .8

4 8 .0

4 9 .2

5 0 .4

5 1 .6

5 2 .8

1 0 2 0 2 3 0 3 4

Mea

n o

f S

N r

atio

s

S ig n a l - to - n o is e : S m a l le r to b e tte r

Fig.4 Main effects plot for S/N ratios –Wear Rate

Table 6 Analysis of Variance for Means (Wear Rate)

S o u rc e D F S e q S S A d j S S A d j M S F P P r (% )

L o a d 2 0 . 6 0 9 5 0 .6 0 9 5 0 .3 0 4 7 0 .2 8 0 .7 8 3 0 .9 7 0 2 2 8

S p e e d 2 2 0 . 6 3 8 1 2 0 . 6 3 8 1 1 0 . 3 1 9 0 9 .3 7 0 .0 9 6 3 2 . 8 5 2 6

D i s t a n c e 2 3 9 . 3 6 9 7 3 9 . 3 6 9 7 1 9 .6 8 4 9 1 7 . 8 7 0 .0 5 3 6 2 . 6 7 0 3 5

E rro r 2 2 . 2 0 3 1 2 .2 0 3 1 1 .1 0 1 5 3 .5 0 6 9 8 7

T o ta l 8 6 2 . 8 2 0 3 1 0 0

Table 7 Analysis of Variance for Means (Frictional force)

S o u rc e D F S e q S S A d j S S A d j M S F P P r(% )

L o a d 2 2 2 0 .0 7 7 2 2 0 .0 7 7 1 1 0 .0 3 9 3 8 9 6 . 4 0 0 .0 0 0 9 9 . 3 5 5 3 2

S p e e d 2 0 .0 2 4 0 .0 2 4 0 .0 1 2 0 .4 2 0 .7 0 3 0 .0 1 0 8 3 5

D is ta n c e 2 1 .3 4 8 1 .3 4 8 0 . 6 7 4 2 3 . 8 6 0 .0 4 0 0 .6 0 8 5 6 4

R e s id u a l E r ro r 2 0 .0 5 6 0 .0 5 6 0 .0 2 8 0 .0 2 5 2 8 2

T o ta l 8 2 2 1 .5 0 5 1 0 0 It can be observed that for aluminium 10% SiC Metal Matrix Composites, from the Table 6,that the sliding distance has the highest influence (Pr =62.5%) on wear rate. Hence sliding distance is an important control factor to be taken into consideration during wear process followed by applied loads (P=0.99%) & sliding speed (Pr=32.85%) respectively. The pooled error is very low accounting for only 3.5%. In the same way from the Table 7 for frictional force, it can observe that the load has the highest contribution of about 99.35%,


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followed by sliding distance (0.6%) & sliding speed (0.01%) for Al-6061 with 10% SiC metal matrix composites means that negligible effect of rest both parameters. The interaction terms have little or no effect on wear rate & frictional force & the pooled errors accounts only 3.5% & 0.025%. From the analysis of variance & S/N ratio, it is inferred that the sliding distance and applied load have the highest contribution on wear rate & frictional force respectively.

IV. MULTIPLE LINEAR REGRESSION MODEL

A multiple linear regression model is developed using statistical software “MINITAB 15”. This model gives the relationship between an independent / predicted variable & a response variable by fitting a linear equation to observe data. Regression equation thus generated establishes correlation between the significant terms obtained from ANOVA analysis namely applied load, sliding speed & sliding distance.

The regression equation developed for Al-6061 / (10%) SiC MMCs wear rate (Wr) and frictional force (FF) are as follows

Wr = 0.00764 - 0.000016 L - 0.000662 S - 0.000001 D Eq (1) FF = - 1.02 + 0.433 L + 0.167S - 0.000538 D Eq (2) From Eq (1), it is observed that the applied load, sliding speed & sliding distance increases or decreases at any parametric value, it will be decrease the wear rate of the value of 0.00764mm3/m. But in case of frictional force Eq (2), applied load plays a major role as well as followed by sliding speed and sliding distance. Overall for the 10% reinforced SiC in Al-6061 MMCs regression equation gives the clear indication about frictional force is highly influenced by applied load. Following are the observation of optimum level process parameter for wear rate and frictional force.

Table 8 Optimum level Process Parameters for Wear Rate and Frictional Force

S r . N o . L o a d (N ) S l i d i n g

S p e e d (m /s )

S l id in g

D i s ta n c e (m )

W e a r R a t e

(m m3

/m )

F ri c t i o n a l

f o r c e (N )

S /N R a ti o

(d b )

1 3 0 3 1 0 0 0 0 .0 0 3 7 4 8 .6 3 6 0

2 1 0 2 1 7 5 0 3 .1 1 4 9 .9 7 8 8

From Eq (1), observed that the negative value of coefficient of speed reveals that increase in sliding speed decreases the wear rate of 10% reinforced SiC

MMCs. This can be attributed to the oxidation of aluminium alloy Al – 6061 which forms an oxide layer at higher interfacial temperature thus preventing the sliding, thereby decreases the wear rate & frictional force which is represented in Eq (2) and a similar behaviour has been observed [17].

From Eq (2), it is observed that the positive value of coefficient of applied load & sliding speed reveals that increase in applied load & sliding speed increases the wear rate and frictional force of 10% reinforced SiC metal matrix composites. This can be related to the reinforcement of weight percentage of silicon carbide in Al – 6061 MMCs from 10% resulted the hard & tough property of the material. Wear rate & Frictional force are largely governed by the interaction of two sliding surfaces.

To understand the wear mechanism of composites for 10% SiC, the worn surfaces were examined by Scanning Electron Microscope. During sliding, the entire surface of the pin has contact with the surface of the steel disc & machine marks can also be observed. Fig: 5 shows the microstructure of the worn surfaces of composites (for 10% SiC) at an applied load 30 N, sliding distance of 1000 m for sliding speed of 2 m/s, 3 m/s and 4 m/s respectively. Grooves were mainly formed by the reinforcing particles of SiC. As the sliding speed increases, the number of grooves also increases & the reinforcements are projecting out from the surface due to ploughing action counterface & pin and formation of wear debris was also observed in 10% SiC reinforced Al-6061 MMCs.

The negative value of distance is indicative that increase in sliding distance decreases the wear rate as well as frictional force for both MMCs, the presence of hard SiC particle which provides abrasion resistance, resulting in enhanced dry sliding wear performance.

V. CONFIRMATION TEST A confirmation experiment is the final step in the Design process. A dry sliding wear test was conducted using a specific combination of the parameters & levels to validate the statistical analysis.


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After the optimal level of testing parameters have been found, it is necessary that verification tests are carried out in order to evaluate the accuracy of the analysis & to validate the experimental results.

Table 11 Confirmation Experiment for Wear Rate and Coefficient of Friction

MMCs Exp. No.

Load(N) Sliding

Speed(m/s) Sliding

Distance(m)

Al-6061+10%

SiC

1 13 2.4 1200

2 19 2.8 1800

3 28 3.5 2200

Table 12 Result of Confirmation Experiment and their comparison with Regression

M M C s E x p . W e a r

R a te (m m3

/m )

R e g . M o d e l E q ( 1 ) ,

W e a r R a t e (m m3

/m ) % E r ro r

E x p . F r i c t i o n a l

F o rc e (N )

R e g . M o d e l E q ( 2 ) ,

F r i c ti o na l F o r c e (N ) % E rr o r

A l6 0 6 1 + 1 0 % S iC

0 .0 0 5 0 .0 0 4 6 4 7 .8 9 4 .1 4 1 5 4 .3 6 4 2 5 .1 1

0 .0 0 3 8 9 0 .0 0 3 6 8 5 .7 6 .1 4 1 3 6 .7 0 6 2 8 .1 2 5

0 .0 0 3 0 8 0 .0 0 2 7 7 1 1 .2 3 1 0 .1 5 9 3 1 0 .5 0 4 9 3 .2 9

The experimental value of wear rate is found to be varying from wear rate calculated in regression equation by error percentage between 5.7% to 11.23%, while for frictional force it is between 3.29% to 8.125% for 10% weight percentage of SiC reinforced with Al-6061 MMCs.

6. Conclusions

Following are the conclusions drawn from the study on dry sliding wear test using Taguchi’s technique.

• Sliding distance (62.67%) has the highest influence on wear rate followed by sliding speed(32.85%) and applied load (0.97%) and for frictional force, the contribution of applied load is 99.36%, sliding distance is 0.608% for Al – 6061/ 10% SiC metal matrix composites.

• From the above conclusion we predict that sliding distance & applied load have the highest influence on wear rate & frictional force in composites.

• Regression equation generated for the (10% SiC MMCs) present model was used to predict the wear rate & frictional force of Al – 6061/(10%) SiC MMCs for intermediate conditions with reasonable accuracy.

• Confirmation experiment was carried out & made a comparison between experimental values showing an error associated with dry sliding wear rate & frictional force in composites varying from 5.7% to 11.23%, and 3.29% to 8.125% respectively. Thus design of experiments by Taguchi method was successfully used to optimize the parameter for tribological behavior of composites.

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