Paper ID - TSME...The analytical study of heat transfer and pressure drop was carried out by Khan et al.[4, 6] with the help of boundary layer analysis for pin fin heat sinks experiencing

The 3rd TSME International Conference on Mechanical Engineering

October 2012, Chiang Rai

1

Paper ID TSF1009

Predicting the thermal performance characteristics of elliptical pin fin heat

sinks under combined natural and forced convection

P. A. Deshmukh

1,*, R. M. Warkhedkar

2

1Assistant Professor, JSPM’s Rajarshi Shahu College of Engineering, Pune (MS), India, 411 033.

2 Associate Professor, Government Engineering College, Karad (MS), India, 415 110.

*Corresponding Author: [email protected], +9120-22934344, +9120-22934084 Abstract

A comprehensive analytical and numerical study is carried out for predicting the thermal

performance of elliptical pin fin heat sink. An analytical model is formulated having capability of

predicting influence of various geometrical, thermal and flow parameters on the thermal resistance of the

heat sink. An experimental technique is developed for measuring the thermal performance of the heat sink

and the overall heat transfer coefficient for the fin bundle. Numerical simulations are carried out for wide

range of geometrical, thermal and flow parameters for pure natural convection and for combined natural

and force convection. The predictive capability of the analytical model is verified by comparison with

simulation data.

Symbol Illustration

a Major axis of elliptical pin fin, mm

b Minor axis of elliptical pin fin, mm

W Width of base plate, mm

L Length of base plate, mm

tb Thickness of base plate, mm

ST Transverse pitch, mm

SL Longitudinal pitch, mm

H Height of pin fin, mm

U∞ Approach velocity, m/s

Tb Surface temperature of pin fin, oC

Tfi Inlet temperature, oC

Tfo Outlet temperature, oC

q Heat flux, W/m2

Re Reynolds Number

Pr Prandtl Number

Gr Grashoff Number

Nu Nusselt Number

α Void Fraction

ϒ Aspect Ratio

1. Introduction

The continuing increase of power densities in

microelectronics and the simultaneous drive to

reduce the size and weight of electronic products

have led to the increased importance of thermal

management issues in this industry. The

temperature at the junction of an electronics

package (chip temperature) has become the

limiting factor determining the lifetime of the

package. The process industries and electronic

industries are taking great amount of efforts over

the years to reduce the size of the devices.

Advanced thermal architectures are required to

meet the future requirements of cooling.

The most common method for cooling

packages is the use of aluminum pin-fin heat

sinks. These heat sinks provide a large surface

area for the dissipation of heat and effectively

reduce the thermal resistance of the package.

They often take less space and contribute less to

the weight and cost of the product. For these

reasons, they are widely used in applications

where heat loads are substantial and/or space is

limited. The overall performance of a pin-fin heat

sink depends on a number of parameters

including the dimensions of the base plate and

pin-fins, thermal joint resistance, location and

concentration of heat sources. These parameters

make the optimal design of a heat sink very

difficult. Traditionally, the performance of heat

sinks is measured experimentally or numerically

and the results are made available in the form of

design graphs in heat sink catalogues. Analytical

and empirical models for the fluid friction and

heat transfer coefficients are used to determine

optimal heat sink design [1]. Several studies are

observed dealing with parametric analysis of heat

sinks with square or circular pin profile. Tahat et

al. [2] has carried out experimental investigation

for heat transfer and pressure drop across

shrouded circular pin fin heat sink arrays

experiencing forced convection. The heat transfer

performance of low aspect ratio pin fin was

investigated experimentally by Lyall et al.[3].

The analytical study of heat transfer and pressure

drop was carried out by Khan et al.[4, 6] with the

help of boundary layer analysis for pin fin heat

sinks experiencing forced convection. In this

study, they had compared circular, plate and



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Paper ID TSF1009

elliptical profiles for enhancement of heat

transfer. Forced convective experimental

investigations for inline and staggered

arrangement of square pin fin heat sink are

reported by Tzer-Mink Jeng et al.[5]. All the

above literature cited is dealing with either

experimental or analytical studies on heat sink

experiencing forced convection.

Very few studies are observed dealing with

mixed convection cooling. Ghosh et al. [7]

analyzed the mixed convection with a classical

approach along the semi-infinite vertical flat plate

by using boundary layer analysis. The effect of

fin density on heat transfer and fluid flow at low

Reynolds number as studied experimentally by

Selvarasu [8]. A comprehensive review of natural

and mixed convection models for electronic

applications was taken by Teertstra [9]. Mixed

convection air cooling of electronic components

is studied numerically by Hamouche [10]. Mixed

convection air cooling in electronic applications

with impinging flow are experimentally

investigated by Bhopte [11] and Kobus [12,13].

Both of these investigations are carried out for

circular pin fin heat sink resulting into a

parametric analysis.

According to the above literature review, no

attempt has been made so far to tackle the

problem of combined natural and forced

convection heat transfer from elliptical pin fin

heat sink either theoretically or experimentally.

The purpose of this work is to formulate a simple

theoretical model capable of predicting the

thermal performance characteristics of a pin fin

heat sink with elliptical profile under combined

natural and forced convection conditions, in terms

of various design parameters. Further, to develop

an experiential measurement technique that can

be used to indirectly measure the effective

thermal resistance of the heat sink and the

average convective heat transfer coefficient for

the fin array. The value of the simple model,

coupled with the heat transfer correlations for the

fin array, will be its ability to provide design

insight; including the existence of optimum fin

spacing.

2. Formulation of a theoretical model

A theoretical model for predicting the thermal

performance of a pin-fin array heat sink is

formulated by considering the heat sink to be

made up of a number of individual pin-fins

operating in parallel.

2.1 Assumptions:

This study assumes the following design

considerations:

1. Each pin is of uniform cross section and

height, H, with elliptical cross section.

2. The fin tips are adiabatic.

3. There is no airflow bypass, i.e. the heat

sink is fully ducted.

4. The airflow is normal to the pin-axis.

5. The approach velocity is uniform for each

row in a heat sink.

6. Flow is steady, laminar.

7. Radiation heat transfer is negligible.

8. There is no slip at the base plate and the

fin surface.

2.2 Mass Constraints

The shape of the elliptical pin fin is selected in

such a way that the mass of the elliptical and

circular fin is same. Assuming the material and

volume same for both the fins:

(a) (b)

Fig. 1 (a) Circular pin, (b) Elliptical pin

√ (1)

Equation (1) represents the equivalent diameter of

elliptical pin fin.

The aspect ratio (ϒ) and axis ratio (ϵ) for the

elliptical pin fin can be defined as:

(2)

Elliptical pins provide more general geometrical

configuration than circular pins. In the limiting

cases, they represent a horizontal plate fin

when , and a circular pin fin when .

Thus a systematic analytical investigation of

elliptical geometries can provide not only heat

transfer characteristics from elliptical fins but

also from circular fins and finite plate fins.



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Paper ID TSF1009

2.3 Theoretical Model

The well-known one-dimensional differential

equation governing the temperature distribution

for such a fin, covered in most introductory heat

transfer textbooks [1-2], is given by:

(

) ( ) (3)

(4)

where

√

(5)

The boundary conditions are,

and

(6)

Fig. 2 Single elliptical pin fin in an infinite flow

Solving the resulting differential equation,

subjected to appropriate boundary conditions,

yields the axial temperature distribution in the

representative fin as represented in “Eq. (7)”.

(7)

Using this temperature distribution, along

with Fourier model for conduction, the rate of

heat transfer from a single elliptical pin fin, ,

can be modeled as,

( ) (8)

Also in terms of the convective heat

transfer coefficient, the rate of heat transfer from

the part of heat sink base not occupied by fins,

can be expressed as,

(9)

where

Therefore, the total rate of heat transfer from

the heat sink, , which contains n fins, can be

expressed as,

(10)

Using “Eqs. (8)-(10)”, the effective thermal

resistance of the heat sink, , can be modeled

as ,

(11)

( ) ( ) (12)

(13)

Further “Eq. (13)” can be modified in least

number of parameters by considerable algebraic

manipulations as,

√ (14)

(√ ) (15)

where

is the efficiency of each pin

fin assuming negligible tip heat loss.

(16)

(17)

where

, is an important parameter

coefficient for a fin bundle and can be called as

the fin bundle void fraction. The void fraction, ,

of a fin bundle is that fraction of a cross sectional

area of the fin bundle that is occupied by air.

(a) Inline (b) Staggered

Fig. 3 Schematics of elliptical pin fin array.

The fin bundle void fraction, , in terms of

longitudinal pitch, SL, and transverse pitch, ST,

can be defined as,

.

“Eq. (17)” represents a theoretical model for

predicting the effective thermal resistance of heat

sink, , in terms of area of fin heat sink



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Paper ID TSF1009

base, , fin bundle void fraction, , number of

pin fins, n, efficiency of pin fin, , perimeter of

pin fin, P, height of pin fin, H, and the

convective heat transfer coefficient, h, between

the fins and flowing air. It is assumed in this

model that the convective heat transfer coefficient,

h, is the same for each fin and also for the base.

All the above physical and thermal parameters

are readily available, except one. The exception

being the convective heat transfer coefficient, h.

The convective heat transfer coefficient is the

result of a combination of a number of complex

physical mechanisms involving fin geometry, fin

spacing, free stream air velocity and direction,

buoyancy forces, fluid properties and the bundle

effect. The complexity of the physical mechanism

governing this particular parameter is such that

they can only partially be modeled. Therefore in

order to determine the required convective heat

transfer coefficient, h, there will be the need for

indirect measurement.

2.4 Indirect measurement of convective heat

transfer coefficient

In order to experimentally measure the

thermal performance of the finned heat sink, it is

essential that the rate of heat transfer between the

heat sink and the flowing air be accurately

measured. Also it should serve for indirect

measurement of convective heat transfer

coefficient, h, between the fins and flowing air.

Fig. 4 Schematic representation of experimental

set up with assisting flow

“Fig. 4” is representing the schematic of

experimental set up. It is divided in three different

sections namely:

1. Wind tunnel,

2. Test section,

3. Measurement and Control panel section

The main body of the rectangular cross-

section wind tunnel duct is manufactured from

wooden sheet and is 2 m high with a constant

internal width of 180 mm. However, the depth of

the duct, and hence the duct’s cross-sectional area,

can be varied by means of adjustable shroud.

Approximately half-way along the height of

the wind tunnel duct is the test section. A

transparent polycarbonate enclosure is used to

enable the pin fin array. Air straightener with

proper meshing is chosen to straighten the air.

The air with controlled velocity and properly

straightened is then passed on test section. Air

velocities are measured with Lutron make AM-

4204 hot wire anemometer. The test section

consists of aluminum elliptical pin fin heat sink.

Pin fins are mounted on the square base plate of

164 mm x 164 mm with 12 mm thickness. Each

elliptical pin fin has 12 mm major axis and 8 mm

minor axis.

The base of the heat sink is heated by a patch

heater with 400 W electrical-resistor strips as the

main heater. The assembly was firmly bolted

together to the bottom surface of the square base.

The lower horizontal surface and the sides of

main heater block are insulated thermally with 50

mm mineral wool blanket. A horizontal guard

heater, rated at 50 W, is positioned parallel to the

main heater, below the mineral wool blanket,

with yet another 20 mm thick layer of mineral

wool placed below it. The whole system of heat

sink base, main and guard heaters, with

associated thermal insulation, is located in a well

fitting open topped asbestos sheet box lined with

wooden sheets. Patch heater is sandwiched

between the base plate and supporting aluminum

plate. The thickness and bottom side of heater

arrangement is completely insulated by using

asbestos sheet insulation to avoid the heat loss.

The power supplied to the main heater could be

adjusted by altering the Variac setting and can be

measured by an in-line voltmeter and ammeter.

The heat input to the guard heater is to be

adjusted until the steady state temperature

difference, across the layer of insulant,

sandwiched between two heaters, is zero. Then,

in all test conditions employed, more than 98% of

the heat generated in the main heater, dissipated

to the air of the surrounding environment,

through the pin fin heat sink. The similar kind of

arrangement of heater assembly was used by

Tahat [2].



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Paper ID TSF1009

The steady state temperature at the base of the

fin array is to be measured by an appropriately

distributed set of five J-type (Iron-Constantan)

thermo-junctions, embedded within the base.

Each thermo junction is screwed in position, so

as to ensure a good thermal contact. The average

value obtained from this thermo-junction is

regarded as the mean overall base temperature.

The inlet and outlet air stream temperatures

across the test section in the wind tunnel are to be

measured by eight thermo-junctions: four are

located immediacy prior to the entrance to the pin

fin assembly and another four just downstream of

the array. All the thermocouples are connected to

the digital temperature indicator through a banana

socket. At a half hourly interval, observations are

to be recorded. When consecutive values are

identical, it is assumed that steady state

conditions are attained.

It is highly desirable to be able to indirectly

obtain an accurate measurement of the rate of

heat transfer from the heat sink to the flowing air,

by simply measuring the electrical power input to

the patch heater assuming a negligible heat loss

off the back side and the edges of the patch heater

assembly.

In order to verify the reliability of this

measurement technique, the independent

measurement of both the electrical power input to

the patch heater, P, and the actual rate of heat

transfer, Q, to the air by the heat sink are done.

The actual rate of heat transfer, Q, to the air by

the heat sink is done by doing the energy balance

on the air as it flows past the heat sink as,

(18)

The mass flow rate, , based on mean flow

velocity, ∞, of air in wind tunnel is defined as:

∞ (19)

Where

(20)

For

,

expression for specific heat of air at atmospheric

pressure [2] is,

[ (

)]

(21)

Due to some heat loss in polycarbonate

enclosure and in heater assembly, the Q value

will be slightly less than P value. The importance

of verifying this is that it provides confidence that

the power input, P, to the main heater is, for all

practical purposes, equal to the rate of heat

transfer, Qs, from the finned heat sink to the

flowing air; thus confirming the reliability the

proposed measurement technique.

After considerable algebraic manipulation,

“Eq. (17)” can be expressed as:

(22)

By using “Eqs. (18) and (22)”, the equation

for indirect measurement of the heat transfer

coefficient, h, can be expressed as:

(23)

The model in “Eq. (23)” provides a means for

indirectly measuring the convective heat transfer

coefficient, h, as a function of air velocity, .

It is recognized that because the fin

efficiency, , also involves the convective heat

transfer coefficient, h, an iteration scheme must

be used to solve for h for each experimental data

point, and therefore for each different air

velocity,

3. Design of experiment

“Eq. (17)” represents a theoretical model for

predicting the effective thermal resistance of heat

sink, , in terms of area of fin heat sink

base, , fin bundle void fraction, , number of

pin fins, n, efficiency of pin fin, , perimeter of

pin fin, P, height of pin fin, H, and the

convective heat transfer coefficient, h, between

the fins and flowing air. It is assumed in this

model that the convective heat transfer coefficient,

h, is the same for each fin and also for the base.

For doing the experimental investigation to

evaluate the performance of elliptical pin fin heat

sink in terms of thermal resistance, the

parameters like longitudinal pitch, SL, transverse

pitch, ST, fin bundle void fraction, α and aspect

ratio, ϒ, defined as H/D need to be varied in some

specific interval along with the approach velocity,

U∞.

A change in diameter has relatively little

influence on the effective thermal resistance for

the air velocities in the mixed convection region.

For very less velocities where natural convection

starts to dominate the heat transfer mechanism, a

25% change in diameter has 6% change in

thermal performance [12, 13]. The fin height has

a significant effect on air side performance of

heat sink [14-16] with a limitation on aspect ratio.

Too short fin can’t be modeled with adiabatic tip

which may lead to poor performance and



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Paper ID TSF1009

overheating of sink surface. Too long fins will

have to compromise upon the fin efficiency since

fin efficiency is the strong function of fin height.

Therefore, the variation in aspect ratio will be

done by varying the height of the pin fin with fin

efficiency close to 90%.

The parameters like longitudinal pitch, SL,

transverse pitch, ST and fin bundle void fraction, α,

has a strong effect on the pin density. Too wide

spacing in longitudinal and transverse directions

will lead to less dense structure resulting in less

number of pins on base plate which may have

poor effect on air side performance. The selection

and variation in approach velocity is the most

critical parameter in view of the current study of

mixed convection. A careful selection of velocity

and its variation should result in providing room

for both natural convection and forced convection.

The mixed convection parameter, Gr/Re2, should

be in the range of 0.1< Gr/Re2 <10 so that neither

natural convection nor the forced convection will

dominate the flow field.

For selection and variation of the parameters

like longitudinal pitch, SL, transverse pitch, ST, fin

bundle void fraction, α, aspect ratio, ϒ, and

approach velocity, U∞, the Taguchi [3] method of

an orthogonal array is used with five levels of

parameters as represented in Table. 1.

Table. 1 Parameters selected for analysis.

Levels

Aspect

Ratio ϒ

Approach

Velocity U∞ (m/s)

Void Fraction

α

Heat Flux

q

(W/m2)

1 5.1 0.1 0.534 2000

2 6.12 0.2 0.702 2500

3 7.14 0.3 0.793 3000

4 8.16 0.4 0.848 3500

5 9.18 0.5 0.884 4000

The corresponding values of longitudinal pitch, SL,

and transverse pitch, ST, for the resulting fin

bundle void fraction, α, are represented in Table.

2.

Table. 2 Longitudinal Pitch

SL (mm) Transverse Pitch

ST (mm) Void Fraction

α

18 9 0.534

22.5 11.25 0.702

27 13.5 0.793

31.5 15.75 0.848

36 18 0.884

As per Taguchi method of orthogonal array,

for four independent parameters with five levels,

L25 orthogonal array method was used to arrange

the input data in 25 combinations.

4. Results and discussion

The data was analyzed by using statistical

analysis software JMP 3.14. This software

analyzes the data by using analysis of variance

(ANNOVA). The outcome is presented in “Figs.

5 – 9”. “Fig. 9” is presenting the summary of the

influence of the selected independent parameter

on the response, thermal resistance.

(a) Inline

(b) Staggered

Fig. 5 Influence of approach velocity, U∞, on

thermal resistance



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Paper ID TSF1009

(a) Inline

(b) Staggered

Fig. 6 Influence of fin bundle void fraction, α, on

thermal resistance.

(a) Inline

(b) staggered

Fig. 7 Influence of aspect ratio, ϒ, on thermal

resistance.

(a) Inline

(b) Staggered

Fig. 8 Influence of input heat flux, q, on thermal

resistance.



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Paper ID TSF1009

The Pareto chart in “Fig. 9” show the

significant contribution of approach velocity on

the air side performance of heat sink. In mixed

convection, with assisting flow, the approach

velocity has a significant role to play. Also it is

observed that thermal resistance as a response has

a weak function with input heat flux (see “Fig.

7”) and aspect ratio (see “Fig. 8”).

(a) Inline

(b) Staggered

Fig. 9 Pareto charts showing %

significance of independent variables.

The regression analysis has provided the

platform for deciding the selection and omission

of parameters for further experimentation by

studying the Pareto charts. The Pareto charts

shown that input heat flux is having a very weak

function with the air side performance of heat

sink, i.e. the thermal resistance. Hence, the

further experimentation is performed at a constant

heat input. Also, it is revealed from the Pareto

charts that, the aspect ratio, both in inline and in

staggered arrangement is having a little influence.

But from the point of view of natural convection

assisting the forced convection, the fin height and

its variation can’t be ignored.

5. Conclusion

The main objective of numerical simulation

was to provide a physical insight of the mixed

convection mechanism and to study the influence

of number of independent parameters on the air

side performance of elliptical pin fin heat sink

both in inline and staggered arrangement.

The regression analysis has provided the

platform for deciding the selection and omission

of parameters for further experimentation by

studying the Pareto charts.

The Pareto charts shown that input heat flux is

having a very weak function with the airside

performance of heat sink, i.e. the thermal

resistance. Hence, the further experimentation can

be performed at a constant heat input.

Also, it is reveled from the Pareto charts that,

the aspect ratio, both in inline and in staggered

arrangement is having a little influence. But from

the point of view of natural convection assisting

the forced convection, the fin height and its

variation can’t be ignored.

6. References

6.1 Article in Journals

[1] Deshmukh, P. A., Warkhedkar, R. M (2011).

Thermal Performance of Pin Fin Heat Sinks-A

Review of Literature, International Review of

Mechanical Engineering, Volume 5. N. 4 May

2011, pp.-726-732

[2] Tahat, M., Kodah, Z.H., Jarrah, B. A., Probert,

S. D., Heat transfers from pin-fin arrays

experiencing forced convection, Applied Energy,

67 ,2000, pp. 419-442.

[3] Lyall, M. E., Thrift, A. A., Thole, K. A.,

Kohli, A., Heat Transfer From Low Aspect Ratio

Pin Fins, ASME Journal of Heat Transfer, 2011,

Vol. 133 011001-1-10.

[4] Khan, W. A., Culham, J. R., Yovanovic,M.

M., Modeling of Cylindrical Pin-Fin Heat Sinks

for Electronic Packaging, IEEE Transactions on

Components and Packaging Technologies, Vol.

31, No. 3, September 2008.

[5] Tzer-Ming Jeng, Sheng-Chung Tzeng.

Pressure drop and heat transfer of square pin-fin

arrays in in-line and staggered arrangements,

International Journal of Heat and Mass Transfer,

50, 2007, pp. 2364–2375.

[6] Khan, W. A., Culham, J. R., Yovanovic,M. M.

The Role of Fin Geometry in Heat Sink

Performance, ASME Journal of Heat Transfer,

2006, Vol. 128 , pp 324-330.

[7] Ghosh, M. S., Yao, L.S. Mixed convection

along a semi-infinite vertical flat plate with

uniform surface heat flux, ASME Journal of Heat

Transfer, Vol. 131, 2009, pp. 022502-1 to

022502-8.

[8] Selvarasu, N. K. C., Tafti, D. K., Blackwell, N.

E. Effect of Pin Density on Heat-Mass Transfer



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Paper ID TSF1009

and Fluid Flow at Low Reynolds Numbers in

Minichannels, ASME Journal of Heat Transfer,

2010, Vol. 126 061702-1-8.

[9] Teertstra, P., J.R., Culham, Yovanovich,

M.M. Comprehensive review of natural and

mixed convection heat transfer models for circuit

board arrays, Journal of Electronic

Manufacturing, 1997,Vol.7, No.2, pp 79-92.

[10] Hamouche , Bessaїh, R. Mixed Convection

Air Cooling of Electronic Components Mounted

In a Horizontal Channel, International Journal of

Theoretical and Applied Mechanics, V. 3 N.1,

2008, pp.53–64.

[11] Bhopte, S., Musa, S. A., Dereje, A., Gamal

Refai-Ahmed. Mixed Convection of Impinging

Air Cooling Over Heat Sink in Telecom System

Application, ASME Journal of Heat Transfer,

2004, Vol. 126 , pp 519 -523.

[12] Kobus, C.J., Oshio, T. Predicting the thermal

performance characteristics of staggered vertical

pin fin array heat sinks under combined mode

radiation and mixed convection with impinging

flow, International Journal of Heat and Mass

Transfer, V.48,2005, pp. 2684–2696.

[13] Kobus, C.J., Oshio, T. Development of a

theoretical model for predicting the thermal

performance characteristics of vertical pin fin

array heat sinks under combined forced and

natural convection with impinging flow,

International Journal of Heat and Mass Transfer,

2005, Vol.48, pp 1053-1063.

[14] Kai-Shing Yang, Wei-Hsin Chu, Ing-Yong

Chen, Chi-Chuan Wang, A comparative study of

the airside performance of heat sinks having pin

fin configurations, International Journal of Heat

and Mass Transfer, 2007, Vol.50, pp 4661–4667

[15] Sahray, D., Ziskind, G., Letan, R. Scale-Up

and Generalization of Horizontal-Base Pin-Fin

Heat Sinks in Natural Convection and Radiation,

ASME Journal of Heat Transfer, 2010, Vol. 132

pp. 112502-1-10.

[16] Dharma Rao, V., Naidu, S. V., Govinda Rao,

B., Sharma, K. V. Heat Transfer from a

Horizontal Fin Array by Natural Convection and

Radiation- A Conjugate Analysis, International

Journal of Heat and Mass Transfer, 49, 2006, pp.

3379-3391.

6.2 Books

[1] Cengel, Y. A.(2010). Heat and Mass Transfer-

A Practical Approach, Tata McGraw-Hill, New

Delhi, Tenth Reprint.

[2] Incropera FP, DeWitt DP.(1985).

Fundamentals of Heat and Mass Transfer, 2nd

edition, New York, John Wiley.

[3] Ginichi Taguchi,(1987). System of

Experimental Design, Quality Resources, Volume

2, pp. 1173.

Documents

Paper ID - TSME...The analytical study of heat transfer and pressure drop was carried out by Khan et al.[4, 6] with the help of boundary layer analysis for pin fin heat sinks experiencing