Pin Fin Seminar Report (2)

8/10/2019 Pin Fin Seminar Report (2)

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CHAPTER 1

1

INTRODUCTION

1.1 INTRODUCTION

Electronic devices need highly effective cooling technology for ensuring their

excellent performance and reliability at all operating conditions. This is very

significant in recent electronic innovations. Because nowadays size of electronic

appliances is becoming far small, and consequently volume decreases to a large

extent. But their heat generation remains the same for that particular application.This results in the fact that effective heat generation per unit volume of the

electronic device increases drastically. Hence to transfer the heat generated in

small volume is the most challenging tas in front of designers.

!f heat transfer mechanism is not effective, it can result in excessive

heating of electronic devices. This may lead to serious problems which will mae

the device non"functional. Excessive heat generation will result in abnormal

operating temperature which can damage the entire electronic circuits. #uch high

temperatures will cause the burn out of components. $lso high temperature will

decrease the reliability of the device. Especially in case of measuring devices,

controlling devices etc. accuracy of operation is getting reduced which will cause

serious faulty outcomes. $ %& degree rise in the temperature can reduce the

reliability of an electronic device to half of its original value.

'ins or extended surfaces are one of the technologies used in transferring

heat generated in electronic devices. $s the heat generation per effective volume

increases ordinary fins may be inappropriate to transfer the excess heat.

!ncreasing the length of the fin beyond a critical value will decrease the heat

transfer rate. Besides this, large fins are difficult to be accommodated in small

electronic device. (e cannot allocate more volume or size only for the purpose of

cooling. $n effective electronic cooling systems must be capable of transferring

more heat using limited available size and must be light weight, low cost and

compatible with the design of the device in which it is to be used. #o the tas is to


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increase heat transfer rate for a specified fin volume or to decrease fin volume for

a given heat transfer rate.

1.1 POROUS PIN FINS

)orous pin fins can be a best substitute for ordinary fins used in electronics. Theyare found to be more effective in transferring heat than the ordinary fins of same

dimensions. This can avoid design problems along with increasing net heat

transfer rate.

)orous pin fins are extended surfaces of finite length and have circular cross

section. Their entire volume is consisting of numerous pores or void channels.

*enerally pores are observed along the length. This allows surrounding fluid to

enter in to the fin body and passes through it. #uch penetration of fluid through fin

volume can bring about net increase in heat transfer rate between fin and

surrounding fluid. This can be the result of two facts, porosity increases the net

surface area available for convective heat transfer. $nother fact is, fluid flow gets

intensified by the presence of pores. This results in higher value of convective

heat transfer coefficient.

1.2 PROBLEM DEFINITION

!n this study, an analytical methodology followed by $domian

decomposition method is applied to solve the nonlinear class of governing energy

equations of a porous pin fin attached to a vertical isothermal wall. +arcy model

is used to analyse the porous pin fin. *overning energy equations are formulated

using the +arcy model and

The present approximate analytical technique is a very useful and practical

method for solving any class of nonlinear governing equations without adopting

linearization or perturbation technique. !t provides an analytical solution in the

form of power series where the temperature on the fin surface can be expressedexplicitly as a function of position along the length of the fin. Thus, the

temperature distribution and its performances are easily being determined for a

wide range of design variables of porous fins.


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CHAPTER 2

LITERATURE REVIEW

undu and Bhan-a %/ developed an analytical model for determination of the

performance and optimum dimensions of porous fin of rectangular shape. 0ecently,

undu et al. 1/ wored on the performance and optimum design analysis of porous fin of

various profiles operating in convection environment.

2u and 3hen 4/ performed a study on optimization of circular fin with variable

thermal parameter. #aedodin and 5lan 6/ investigated the temperature distribution over

fin surface and compared the results with conventional fins. 'or the analysis they have

selected a pin fin sub-ect to heat transfer in natural convection condition.

$+7 8$domian polynomial method9 :/ is employed to solve differential

equations which gives accurate results than ordinary Taylor series expansion.


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CHAPTER 3

MODELLING OF THE PROBLEM

3.1 MODEL

'igure given below shows a straight porous pin fin having uniform cross"section,

length ; and diameter +. 'in is attached to a vertical isothermal wall. Heat flow is

directed from the wall to the fin by means of conduction and from fin to atmosphere

through natural convection. #urrounding fluid 8air9 can penetrate into fin interior since

the fin is provided with numerous pores. The porous fin increases the effective surface

area of the fin through which the fin convects heat to the woring fluid 8air9.

'ig 4.%


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3.2 ASSUMPTIONS

'or maing the analysis convenient, following assumptions were madesing +arcy model, we can get flow velocity, flow rate etc. of penetrating


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7

3.4 FORMATION OF GOVERNING EQUATION

$ small element of length is considered and energy balance equations are applied

to it.

Total convective heat transfer from the porous fin is taing place in two ways. 'irst

way is natural convection between solid surface and ambience. This is common to all

fins irrespective of their type. #econd way is peculiar to porous fins which is the main

factor that increases heat transfer rate in porous fins. This is due the interaction of

fluid particles penetrating through porous medium with solid fin material. +arcy?s

law should be applied to calculate heat transfer due to this method. Total heat transfer

from fin surface is the sum of heat transfer due to both methods. By applying an

energy balance to the differential segment of the porous fin with considering only

convection, mathematically it yields

() ( + ) = ( ) + (1 )( ) (1)( (( (( (( (( (( (( (( (( (( (( (

Here in 0H#, first term comes from the +arcy model and second term is applicable

for any fin with insulated tip and finite length, but multiplied by a factor 1 . This

factor is used because some area is not available for heat transfer because of porosity.

'rom +arcy?s formulation we get,

7ass flow rate of fluid passing through pores,

=

(2)

The fluid velocity can be estimated from +arcy model. !t yields,


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8

= {( ( ))} (3)((((((((((((((((((((((

@ow substituting these values to eq. 8%9, we can rewrite the ;H# of it, by applying


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9

'ourier?s law of conduction. This is purely based on the assumption that heat transfer

rate through the solid due to conduction is balanced by the total convective heat

transfer rate. That is, all heat conducted is dissipated to atmosphere by convection.

@ow,

() (+) =

(4)

Here $ is the cross sectional area of pin fin

=4

2

(5)

@ow dividing by throughout, and differentiating with respect to , we get,

2

2

4 (

) 2

4 ( 1 ) (

)

= 0

(6)

Eq. 8A9 is the one dimensional energy equation of porous pin fin.


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1

CHAPTER 4

NON-DIMENSIONALISATION (SCALING)

4.1 INTRODUCTION

#ome quantities are better measured relative to some other appropriate unit called

quantities intrinsic to the system. This can recover the characteristics properties if the

system. $lso it is very useful where systems are described by differential equations.

>sing non"dimensionalisation, measurement in one system can be compared with

common property measurement in other system which has same intrinsic property as the

first system. This technique, can suggest the parameters which should be used to analyse

the system. But anyway a starting equation is needed.4.2 STEPS IN NON DIMENSIONALISATION

!dentify the independent and dependent variables in the starting equation.

0eplace them with scaled 8non" dimensionalised9 quantity.

+ivide throughout by coefficient of highest order polynomial or derivative

term.

3hoose -udiciously to minimize the number of coefficients.

0ewrite in terms of the scaled quantities.

4.3 SCALING THE ENERGY EQUATION OF POROUS PIN FIN

By defining following scaled quantities,

(; ; ; ) = [;

;

;

]

(7)

(; ; ) = [

( )3

;

;

2]

(8)

(12) = [2

;

Eq. 8A9 can be written as

(1 ) 2 ] ; =


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1

= + (1 (9))

2

2 = 1+ 2

(10)

2


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1

(ith boundary conditions

= 0, = 0

(11)

$nd

= 1, = 1

(12)

@ow it is well understood that non dimensionalisation can yield governing

differential equations which are simple and easy to solve. This also reduces the number

of terms involved in differential equation.

$s we want study the performance of porous pin fin, we need to calculate the

following quantities

$ctual heat transfer rate per unit area of pin fin

= Ideal heat transer rate !er "n#t area $ !#n #n

=%nnned heat transer rate !er "n#t area $ !#n #n 'in Efficiency

'in Effectiveness

!t is very effective to calculate aforesaid quantities in terms of dimensionless

quantities defined earlier. This will reduce the efforts and can give solution rapidly.

=

( )/= (

)

=1

(13)

={+ (1 )}

(14)

= 0.5 (15)


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1

'in Efficiency, =

(16)

'in Effectiveness, =

(17)


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1'

CHAPTER

ADOMIAN DECOMPOSITION METHOD (ADM)

.1 ADM

$+7 is a modern methodology employed to solve governing equations of many

current systems. They are preferred over Taylor?s series expansion nowadays. $+7 was

developed by mathematician *eorge $domian during %CD&"%CC&.

The $+7 which can accurately compute the series solution, is of great interest to

applied sciences. The method can provide the resulting solution as a quicly converging

series with components that are elegantly computed.

$dvantages of this method over other methods are ! can be applied to any

differential or integral equations without considering whether they are

linear or nonlinear

homogeneous or inhomogeneous

with constant coefficients or with variable coefficients

Besides this, this method is highly capable of minimizing the size of computation

wor while still maintaining high accuracy of the numerical solution.

.2 GENERAL FORM OF ADM

;et there be a differential equation say,

2 +

2

2

= 1 (18)2

et there e a l#near se*$nd $rder d#erent#al $!erat$r , s"*h that =@ow Eq. 8%49 can be written as

2

() + 2

= 1

(19)

1

$ssume that inverse operator 1


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1

exists and

= ( )

1 2 1 2

$ =

( ) =

2

(1 ) (20)


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1-

.3 SOLUTION USING ADM

Eq. 8%&9 can be written using operator as

= 12

+ 2

(21)

@ow apply inverse operator 8two fold integral operator9 on both sides of Eq. 8%A9 which

yields,

= (0) + (

0)

+ 11(2) +

21

()

(22)

(here (0) is the dimensionless tip temperature of the fin, noted as0.@ow,

1 1

= = 0 +

1=0

[] +

2=0

[]

(23)=0

. 1 /nd #s the /d$#an !$ln$#al *$rres!$nd#n t$ n$n&l#near ter 2

$lso

= 1

1(1) +

21

(1)

(24)

#o after collecting $domian polynomials, they can be written as,

22

(

0

; 1

; 2

; 3

; ! ! !)

=(

0

; 21

0

; 22

0

+ 1

; 22

1

; ! ! !)

(25)

>sing Eqs. 8C9" 8%%9, non"dimensional temperature expression () can be found out as,

2 4

() = 0 + (1 0 + 20) 24 + (21 0 + 3120 + 2 0 ) '4 + (2-)(((((((((((

@ow from Eq. 81A9 we can get the non"dimensional temperature distribution of porous

pin fin in terms of non"dimensional wall temperature 0. The above relation is very

useful in order to calculate various performance parameters of porous pin fin.

2 2 2


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17

CHAPTER !

RESULTS AND DISCUSSION

!n this section, our prime interest is to plot different characteristic curves of

porous pin fin used in electronic cooling. These are plots with performance parameters on

2 axis and thermo physical and thermo geometric properties of porous pin fin on axis.

5b-ective behind these plots are to analyse the improvements made by using

porous medium and to investigate actual causes behind those phenomena. $lso this plots

can be effective in predicting the optimum values of design parameters for best

performance of fins.

'ig A.%

'ig A.% shows variation of dimensionless temperature with dimensionless length

5 6#th ar#$"s $ther !araeters l#e a, :", , and 'rom graph it is observed that an increase inRa improves the effective convective

heat transfer coefficient between the fin and the woring fluid which enhances the heat

transfer rate by convection. $nd thus dimensionless temperature declines as predicted in

'ig. A.% 8a9.

#ame trend was observed with porosity parameter . $ctually, a high porosity

decreases the effective thermal conductivity of the porous fin due to the removal of solid


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19

'ig A.4

'in performances as a function of porosity parameter and +arcy number is shown

in 'ig. A.4. The main viewpoint behind using porous fins is to increase the effective

surface area through which heat is convected to the surrounding fluid. (hen the value of

approaches to a unit value fin performance parameters become zero as effective thermal

conductivity is reduced to a very less in magnitude.

$s the permeability of the porous fin increases, i.e., increasing Da number, the

woring fluid ability to penetrate through the fin pores and to convect heat increases but

side by side it increases the ideal heat transfer rate also as defined in Eq. 8%A9 and 8%D9.

Thus a reduction in fin efficiency is noticed. 5n the other hand there is no impact of this

parameter in calculating heat transfer rate in un"finned condition and thus fin

effectiveness is remarably increased.


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20

'ig A.6

'ig. A.6 8a9 shows the effect of Da and Nu on temperature gradient at fin base as a

function of. $s mentioned earlier that a high Da number indicates mainly high

permeability of the porous fin which means more woring fluid can pass through it and

thus creates a higher temperature gradient at the fin base. )orous fins having small Da

number behave as solid fins due to their small permeability. The effect of Nu number also

shows the same trend because it increases the heat transfer coefficient over the fin

surface. 5n the other hand, the dimensionless actual heat transfer rate through the porouspin fin surface as a function of, Ra and

is depicted in 'ig. A.6 8b9. !t is clear from this

figure that actual heat transfer rate enhances with the increase of these parameters. 'or a

particular fluid, with increasing the parameter, thermal conductivity of the fin material

is also increased that reduces the conductive resistance in the fin surface and thus heat

transfer rate is enhanced.


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21

CHAPTER "

CONCLUSIONS

$n effort has been made to determine the temperature distribution, fin performance and

heat transfer rate over a straight porous pin fin that may help in optimum design analysis.

The fin dissipates heat to the environment through natural convection. 'or the

aforementioned conditions, an approximate analytical technique, namely, $domian

decomposition method 8$+79 has been proposed for the solution of governing fin

equation. This method provides solution in the form of infinite power series and it has

high accuracy and fast convergence. Thus, fin performance parameters and heat transfer

rate can easily be obtained from the explicit form of the temperature distribution. Thefollowing concluding remars can be drawn from the present study

Pin Fin Seminar Report (2)

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