maximization and modeling matching of convective heat ...no total number of pin fin n' linear...

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:12 No:04 11

121104-9898-IJMME-IJENS © August 2012 IJENS I J E N S

Abstract-- In the present study, a numerical model has been developed to investigate thermal performance and optimization of relevant thermogeometric parameters involved in a convective cooling system. The heat removal system comprises of flow of fluid or nanofluid through a rectangular channel with its lower surface subjected to constant heat flux and the upper surface equipped with pin fins having variable distributions. The governing energy equation and momentum equation have been solved to find an optimum profile for variable fin density. Comparison of Nusselt numbers for non-uniform fin distribution with those for uniformly distributed fins reveals 2.7% improvement in heat transfer rate for the former case. This increase was associated with Peclet umber of Pe=1.5, No=0.9, Nf=0.001 and θα= –0.5 for a variable fin density profile with B=0.045 and Φo=0.6. Furthermore, two nanofluids one containing copper oxide nanoparticles dispersed in water and the other in a base fluid of 60:40 (by mass) ethylene glycol and water mixture (60:40 EG/W) have been used in present computations with particle volumetric concentration ranging from 0 to 6%. Since heat flux is provided at the lower surface of the channel, Nusselt numbers were calculated only at the lower surface. The analyses reveal about 65% increase in average Nusselt number at low Peclet number (Pe=1.0) and about 27% increase in that at high Peclet number (Pe=200). Moreover, Effort has also been made to explore the possibility of using some published dispersed model for nanofluids to fit a published experimental thermal conductivity model of nanofluids. It is shown that both models can be excellently matched based on equal local or average Nusselt numbers at specific dispersion model constant coefficients. These coefficients are mainly depending on nanoparticles material, particles volumetric concentration and Peclet number. Finally, a correlation for the dispersion model coefficients as functions of the relevant parameters has also been proposed.

N.I. Abdulhafiz is serving in Department of Mechanical Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

A.-R.A. Khaled is serving in Department of Mechanical Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Tel: +966 2 6402000 Ext. 68185; Fax: +966 2 6952182.

E-mail Address: [email protected] A.Y. Bokhary is serving in

Department of Mechanical Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Index Term-- Heat transfer , conversion , nanofluid ; Modeling ; numerical. NOMENCLATURE Ac cross-sectional area of a pin fin, πd2/4, m2 Ar channel aspect ratio, H/L As upper/lower surface area of the channel, W.L, m2 B Fin density profile parameter, bH b exponent in Eq.(8) cp specific heat at constant pressure, J/kg.K d pin fin diameter, m H channel height, m h convective heat transfer coefficient, W/m2.K ho heat transfer coefficient of the channel surface

exposed to the ambient, W/m2.K hd fin surface convective heat transfer coefficient,

W/m2.K hL upper surface convective heat transfer coefficient,

W/m2.K H channel height, m i x-direction grid point j y-direction grid point k thermal conductivity of nanofluid particles, W/m.K L channel length kbf thermal conductivity of base fluid or pure fluid,

W/m.K m maximum number of grids in x-direction n maximum number of grids in y-direction no total number of pin fin n' linear density of the fins along x-direction, aebx Nu average Nusselt number, hH/kbf NuL average lower surface Nusselt number, hL H/kbf No dimensionless heat transfer coefficient of the channel

surface exposed to the ambient, hoH/kbf Nf dimensionless heat transfer coefficient of fin,

bf

dd

kH

dkh

P Peremeter, πd, m Pe Peclet number, Re.Pr Pr Prandtl number, μcp/kbf q" heat flux at lower surface of the channel, W/m2

Maximization and Modeling Matching of Convective Heat Transfer with Nanofluids

and Non-uniform Fin Distribution N.I. Abdulhafiz, A.-R.A. Khaled and A.Y. Bokhary



Re Reynolds number, ρum H/μ, um H /ν T fluid temperature, K Ti fluid temperature at inlet, K Tm mean fluid temperature at any axial location, K Tα free stream temperature, K u axial velocity, m/s um mean axial velocity, m/s U dimensionless axial velocity, u/um = 6Y(1-Y) x axial coordinate along the direction of flow X dimensionless axial length, x/H y vertical coordinate normal to the direction of flow Y dimensionless vertical height, y/H W channel width GREEK SYMBOLS θ dimensionless fluid temperature, kbf (T-Ti)/Hq", θ∞ dimensionless free stream temperature kbf (T∞ -

Ti)/Hq" θm dimensionless mean fluid temperature kbf (Tm -

Ti)/Hq" θL,m maximum lower wall temperature of the channel μ dynamic viscosity, N.s/m2 ν kinematic viscosity, m2/s ρ fluid density, kg/m3 λ thermal conductivity ratio, k/kbf Φ variable area ratio, Ac(x)/As

Φo fixed area ratio, WL4

dn 2oπ

ζo ratio of Nusselt number with fin to that without fin ζ ratio of Nusselt number with variable fin density to

Nusselt number with uniform fin density ζnf ratio of Nusselt number with nanofluid to that with

pure fluid

I. INTRODUCTION

Efficient use of energy sources is one of the most effective ways to reduce the energy demand. Heat exchanger as a device of energy utilization is widely applied in power engineering, petroleum refineries, chemical industries, food industries, electronic industries and so on. The augmentation of heat transfer in thermal systems is of great importance in many engineering applications mainly with a view to reducing the size, weight, and cost of heat exchangers. Otherwise the purpose may be an upgrading of the capacity of an existing heat exchanger. And again one might prefer to lower the approach temperature difference or even, sometimes, to reduce the pumping power required for a given application.

It may be mentioned that low value of Peclet number is concerned with liquid metal cooling technology. Liquid metal cooling is of great significance where use of fans, water-cooling, or airflow is not possible or a matter of concern from the point of view of weight, size, or reliability. For example, avionics and military applications involve high-

power electronic devices packed in confined spaces which lead to high thermal densities.

Generally, heat transfer augmentation techniques are classified in three broad categories: (a) active method, (b) passive method, (c) compound method. The active techniques require external power, while passive techniques require roughed surface geometries or other special fluid additives without external power. Passive techniques are by far more important than active in most of the applications. Widely employed passive enhancement techniques are: 1)Treated surfaces 2)Rough surfaces 3)Extended surfaces 4)Displaced enhancement devices 5)Swirl flow devices 6)Coiled tubes 7)Surface tension devices 8)Additives for fluids.

Khaled [3-9], Khaled and Vafai [10-13], and Vafai and Khaled [14] have conducted extensive study on heat transfer enhancement using different techniques. The aforementioned studies [3-14] have concisely been presented by Siddique et.al.[15] who have recently reported a detailed review of literature on heat transfer augmentation. They [15] have identified 16 possible mechanisms of heat transfer enhancement. They have opined that at least one of these sixteen mechanisms is involved in heat transfer enhancement. Their emphasis was on paying more attention towards single phase heat transfer enhancement with microfins in order to minimize the discrepancies between the studies of the different authors. They have concluded that the systematic thermal-hydraulic modeling of flow inside porous media could be helpful in achieving an in-depth knowledge of the mechanism of heat transfer augmentations due to nanofluids. Finally, they have recommended that significant contributions from researchers are required towards further modeling of thermal-hydraulic behavior inside convective media supported by flexible/flexible-complex seals. Siddique et.al.[15] have reviewed about 170 articles in preparing their report which includes almost all the recent studies on heat transfer augmentations. Few other recent studies on heat transfer augmentation are reported in [16-19].

Compound augmentation method is one of the most promising heat transfer enhancement techniques, in which both active and passive methods are used in combination. This method usually leads to heat transfer enhancement which is usually larger than the individual techniques applied separately. Thermal performance of an electronic cooling system involving compound augmentation method can further be motivated by using nanofluids.

A nanofluid is a suspension of nanometer-sized particles in a conventional base fluid which significantly enhances the heat transfer characteristics of the original fluid. It is important to note that use of nanofluids incurs little or no penalty in pressure drop. Over a decade ago, researchers focused on measuring and modeling the effective thermal conductivity. Recently a number of important research works on convective heat transfer have been published in the open literatures on the augmentation of heat transfer using nanofluids.



A systematic review of literature on researches on nanofluid has been presented by Kakaç and Pramuanjaroenkij[20]. They have reviewed about 70 articles and the survey reveals that further theoretical modeling and experimental works on the effective thermal conductivity of nanofluids are needed .They also have suggested that further investigation is required for the treatment of nanofluids as a two-phase flow since slip velocity between the particle and base fluid plays important role on the heat transfer performance of nanofluids.

Siddique et al.[15] have recently reviewed the literatures on heat transfer with nanofluids. They[15] have discussed about the agreement and possible reasons of disagreement among the models proposed by many authors[21-42]. They[15] have identified Xuan and Roetzel[1] model as one of the elementary models of the effective thermal conductivity which suggests that effective thermal conductivity of the nanofluid is the sum of two terms. First term is the effective thermal conductivity of the nanofluid under stagnant conditions where the bulk velocity is equal to zero. The second term corresponds to the dispersion effect of the nanofluid. Recently, Vajjha and Das[2] have proposed a new model based on their experimental investigations. It may mentioned that Xuan and Roetzel[1] model is a purely theoretical model whereas Vajjha and Das[2] is based on experimental investigation.

As mentioned above, a large number of papers presented the study on the heat transfer enhancement in channels with various geometrical configurations. However, none of these papers reported on the heat transfer enhancement with compound augmentation method using nanofluids. In the present study, modeling of a special kind of thermal system involving compound augmentation method using pure fluid with uniform as well as variable fin density and also, using nanofluid with uniform fin density has been undertaken. The present work is also concerned with investigating how the dispersion model proposed by Xuan and Roetzel[1] is related to the model proposed by Vajjha and Das[2].

II. FORMULATION AND PROBLEMS STATEMENT

Consider a steady-state fully developed laminar flow of an incompressible Newtonian fluid through a channel, as shown in Fig. 1. For the formulation of the problem, thermally developing forced convection flow has been assumed with constant average properties except for the thermal conductivity. It is also assumed that the channel is large in both the stream-wise x-direction and the span-wise z-direction (perpendicular to the x-y plane), compared to the cross-flow y-direction which is the height of the channel, H. The length of the channel in the stream-wise x-direction is L. The flow is considered to be hydrodynamically symmetric across the mid-plane(x, y=H/2). In order to enhance heat transfer rate, fins are employed on the upper surface of the channel. Pin fin with

diameter d has been selected in the present formulation. The problem can then be considered as two-dimensional in the (x, y) plane with channel width W in the span-wise z-direction.

Fig. 1. Physical model and the coordinate system. The governing equations for 2-D, steady, laminar boundary layer flow, forced convection heat transfer with negligible viscous dissipation terms expressed in Cartesian coordinates are: x-direction momentum equation

2

2

yu

xP

yuv

xuu

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂ μρ (1)

Conservation of Energy

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

yTk

yxTucpρ (2)

The set of governing equations are to be solved for simulating the flow situation with the specified boundary conditions. The boundary conditions in the present problem are: 1) Inlet (x=0): u=u(y), T=Ti ( inlet temperature) (3) 3) Lower surface (0



⎟⎟⎠

⎞⎜⎜⎝

⎛

−Φ=

−=Φ

114)(

2

bL

bx

obx

bLo

ebebL

WLdn

x eeπ (10)

where, WL

dnoo 4

2π=Φ (11)

While nondimensionalizing the governing equations, the following dimensionless parameters are used. X=x/H, Y=y/H, U=u/um, θ= k (T–Ti)/Hq", θ∞ = kbf (T∞ –Ti)/Hq" (12) Dimensionless form of x-direction momentum equation Eq.(1) leads to the expression for velocity profile as, U = 6Y(1–Y) (13) Dimensionless form of the Energy equation, Eq.(2), is

( ⎟⎠⎞

∂∂

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

YYXPeU θλθ (14)

Thermal boundary condition at the lower surface of the channel, Eq.(4), can be nondimensionalized as

10,

−=∂∂

=YXYθλ (15)

While the dimensionless form of the thermal boundary condition at the upper surface of the channel, Eq.(5), is expressed as

[ ]αθθθλ −=∂∂

−=

=1

1Y

Y

DY

(16)

where, D = [(1–Φ(X)) No + 2 Φ Nf] (17) Nondimensional form of Eq.(10) is,

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

=

1e

eAB

)X(rA

B

BX

roΦΦ (18)

Nusselt number on the lower surface of the channel can be determined from the convection equation,

"m0yLq)TT(h =−

= which is nondondimesionalized as

( )mYLNu

θθλ −=

=0

1 (19)

Similarly, Nusselt number on the upper surface, is determined from the following convection

equation,Hy,x

mHyU yTk)TT(h

== ∂

∂−=−− (20)

Dimensionless form of Eq.(20) is

mY

YU

YNuθθ

θ

−⎭⎬⎫

⎩⎨⎧

∂∂

==

=

1

1 (21)

where, ∫=

=

=1Y

0Ym dYUθθ (22)

III. NUMERICAL METHODS

Having specified the governing equations and the boundary conditions, it is necessary to construct a finite difference formulation of the equations. Rectangular grids with grid size ΔX in horizontal (axial) direction and ΔY in vertical direction have been considered. Total number of grids in x-direction is m and that in y-direction is n.. Correspondingly, number of grid points in x and y directions are m+1 and n+1 respectively. Hence, ΔX=L/m and ΔY=H/n. The dimensionless equations for energy, momentum, lower surface thermal boundary condition and upper thermal boundary condition have been descritized using three-point central differencing in the transverse direction (Y-direction). Whereas, two-point differencing has been used in the axial direction (X-direction). The finite difference equation for momentum equation, Eq.(13) is: Ui,j = 6Yi,j (1–Yi,j ) (23) The finite difference equation for energy equation, Eq.(14) is:

jijijijijijijiji dcba ,,1,,,,1,, θθθθ =++ +− (24) Where,

21,,

, 2 Ya jijiji Δ

+= −

λλ (25)

XjiPeU

Yb jiji Δ

−Δ

−=),(2

2,

,

λ (26)

21,,

, 23

Yc jijiji Δ

−= −

λλ (27)

XjiPeU

d jiji Δ−= − ,1,

),( θ (28)

In the present analysis, the Thermal conductivity model of nanofluids presented by Vajjha and Das [41] has been used. The model is:

( )( )

( ) ( )( )

( )ψρ

κ

ρβψ

ψψ

λ

,Tfd

)j,i(T

Ck

kkkkkkkk

)j,i(

pp

bfpbfbf

pbfbfp

pbfbfp

×

×+

−++−−+

=

41051

2222

(29) where,



( ) ( )

( )32

32

109112331006693

1091731081272

−−

−−

×−×−+

⎟⎟⎠

⎞⎜⎜⎝

⎛×+×=

..

TT..,Tf

o

ψ

ψψ

(30) where κ is the Boltzmann constant ( KJ /10381.1 23−×=κ ), To is a reference temperature ( K273To = ), ( )bfpC is the specific heat of the base fluid,

bfρ is the base fluid density, pρ is the density of the

nanoparticle and pd is the nanoparticle diameter. The finite difference equation for thermal boundary condition at the lower surface of the channel is:

Yiiii Δ−=+− 2,1,1,1, θλθλ (31) Hence, the coefficients in the descritization equation for lower surface are:

0a j,i = (32)

1,, ijib λ−= (33)

1,, ijic λ= (34) Yd j,i Δ−= (35)

Similarly, finite difference equation for thermal boundary condition at the upper surface of the channel is:

αθθΔλ

θΔλ

DDYY n,in,i

1n,in,i =⎟⎟

⎠

⎞⎜⎜⎝

⎛++− − (36)

Hence, coefficients in the descritization equation for upper surface are:

Ya n,ij,i Δ

λ−= (37)

DY

b n,ij,i += Δλ

(38)

0c j,i = (39)

αθDd j,i = (40) Dispersed model as proposed by Xuan and Roetzel [1] has been used for the determination of thermal conductivity of nanofluid for comparison purpose. According to this model,

( )( ) dpbfbfp

pbfbfp

kk2k2kkk2k2k

λφφ

λ +−++

−−+= (41)

where,

)Y21(PeY6)c()c(

Cbfp

nfp*d −= ρ

ρλ (42)

where C* is an unknown constant which should be determined by matching experimental data. The experimental data of Vajjha and Das [2] has been used to generate the value of C* for different Peclet number and volumetric

concentration of nanoparticle. For CuO nanofluid, the average deviation is 5.74%. Descritization equation for Nusselt number on the lower surface is:

)(1

,1,1, imiiLNu θθλ −

= (43)

Nusselt number on the upper surface,

i,mn,i

1n,in,in,i

UY

Nuθθ

Δθθ

λ

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

=

−

(44)

where, ∑=

=

=nj

1jj,ij,ii,m YU Δθθ (45)

The tridiagonal matrix algorithm (TDMA), also known as the Thomas algorithm, was used to solve the set of descritization equations evolving from Eq.24. The present numerical code has been benchmarked with the fully developed Nusselt number[43] for flow through parallel plates with one surface at uniform heat flux and the other being insulated. The code was further tested by comparing the fully developed Nusselt number [43] for flow through parallel plates with one surface at constant wall temperature and the other being adiabatic. In both the cases, results obtained with the present code were in excellent agreement with the published results [43]

IV. RESULT AND DISCUSSIONS

In the numerical investigations, all computations were performed with uniform grids in x and y coordinates and an aspect ration of Ar=0.05. Grid independency test showed that a uniform grid size of ΔXxΔY = 0.02 x 0.0025 was good enough to handle the present problem.

A. UNIFORM FIN DENSITY Computations were performed to find out the

optimum values of heat transfer design parameters such as Pe, No, Nf, θα. These results were obtained by investigating the variation of ζo with Φo. Simultaneous variation of θL,m with Φo was also studied.

The effect of dimensionless base heat transfer coefficient, No, on the variation of Nusselt number ratio ζo with the fraction of fin area at the base Φo for Pe=1.5, Nf=0.01 and θα= –0.5 is presented in Fig.2. It is seen in Fig.2 that No plays an important role. For No



for Pe=1.5, Nf=0.01 and θα= –0.5 is shown in Fig.3. At low value of No, θL,m decreases with increasing Φo. However, the trend changes at higher values of No as clear from Fig.3.

Fig. 2. Effect of dimensionless base heat transfer coefficient, No, on the variation of Nusselt number ratio, ζo , with the fraction of fin area at the base Φo.

Fig. 3. Effect of dimensionless base heat transfer coefficient, No, on the

variation of dimensionless maximum lower wall temperature of the channel, θL,m with the fraction of fin area at the base Φo.

The effect of Pe on the variation of Nusselt number

ratio ζo with the fraction of fin area at the base Φo for No=0.4, Nf=0.01 and and θα= –0.5 is presented in Fig.4. It can be seen in Fig.4 that irrespective of values of Pe, ζo increases with increasing Φo. However, effect of Pe seems to be maximum at Pe=2. Hence, optimum results are obtained at Pe=2. Figure 5 presents the effect of Pe on the variation of dimensionless temperature θL,m with the fraction of fin area at the base Φo for Pe=1.5, Nf=0.01 and θα= –0.5. At low value of Pe, θL,m increases with increasing Φo. However, the trend changes at higher values of Pe and, as clear from Fig.5, at very high values of Pe, θL,m becomes insensitive to Φo.

The effect of dimensionless fin heat transfer coefficient, Nf, on the variation of Nusselt number ratio ζo with the fraction of fin area at the base Φo for No = 0.4, Pe =1.5 and and θα=–0.5 is presented in Fig.6. It may be noticed in Fig.6 that for Nf≤0.01, increasing Nf does not have much impact on the variation of ζo with Φo. Also, maximum change in ζo with respect to Φo occurs at about Nf=0.01. For Nf≥0.01, as Nf increases, ζo decreases with increasing Φo. At higher values of Nf, ζo even starts decreasing with increasing Φo. This indicates that use of fins does not contribute to heat

transfer enhancement at higher values of Nf with the given other design parameters.

Fig. 4. Effect of Peclet number, Pe, on the variation of Nusselt number ratio,

ζo, with the fraction of fin area at the base Φo.

Fig. 5. Effect of Peclet number, Pe , on the variation of dimensionless

maximum lower wall temperature of the channel, θL,m with the fraction of fin area at the base Φo.

Figure 7 shows the influence of dimensionless fin heat transfer coefficient, Nf, on the variation of dimensionless lower wall temperature of the channel, θL,m with the fraction of fin area at the base Φo for No=0.4, Pe=1.5 and and θα= –0. is presented in Fig.7. At low values of Nf(0.0001-0.1), θL,m increases with increasing Φo. For higher values of Nf (1-100), θL,m is found to decrease with increasing Φo.

The effect of dimensionless free stream temperature, θα, on the variation of Nusselt number ratio ζo with the fraction of fin area at the base Φo for No=0.4, Pe=1.5 and and Nf= 0.01 is shown in Fig.8. It is evidenced from Fig.8 that θα does not have much effect on the variation of ζo with Φo. A value of θα= −0.5 has been selected for use along with other design parameters. Fig.9 shows the effect of dimensionless free stream temperature, θα, on the variation of dimensionless lower wall temperature of the channel, θL,m with the fraction of fin area at the base Φo for No=0.4, Pe=1.5 and and θα= –0.5. It is clear from Fig.9 that θL,m increases with increasing Φo irrespective of value of θα. B. VARIABLE FIN DENSITY



In order to investigate the effect of variable fin density on heat transfer enhancement, different fin distribution profiles have been considered by varying the parameter B in Eq.(18). Investigation for uniform fin density in the preceding subsection provides the optimum values of thermogeometric parameters such as No=0.4, Pe=1.5, θα=−0.5, and Nf=0.01. Using these optimum values as guess values, the effect of area ratio Φo on the variation of Nusselt number ratio ζ with fin distribution profile parameter B is plotted in Fig.10. It is evident from Fig.10 that maximum enhancement (2.3%) in heat transfer corresponds to profile with B=0.045 with Φo=0.6. Different possible profiles with Φo=0.6 is shown in Fig.11. For B=0.045, which corresponds to maximum heat transfer augmentation, uniform fin density with Φo=0.6 has just been redistributed in an exponential fashion such that it varies from fin density with Φo=0.37 to fin density with Φo=0.91. It is important to note that simple redistribution of fins with proper set of relevant design parameters leads to enhancement in heat transfer.

Fig. 6. Effect of dimensionless fin heat transfer coefficient, Nf, on the

variation of Nusselt number ratio, ζo , with the fraction of fin area at the base Φo.


variation of dimensionless lower wall temperature of the channel, θL,m with the fraction of fin area at the base Φo.

Fig. 8. Effect of dimensionless free stream temperature, θα, on the variation of

Nusselt number ratio ζo with the fraction of fin area at the base Φo. The effect of dimensionless base heat transfer coefficient, No, on the variation of Nusselt number ratio ζ with Peclet number, Pe for Φo=0.6, B=0.045, Nf=0.01 and θα= –0.5 is presented in Fig.12. It is seen in Fig.12 that for a given value of No=0.9, with increasing Pe, ζ increases up to a certain value, attains a peak (ζ =1.025) at Pe=1.5 and then starts decreasing. It is also interesting to note that ζ versus Pe plots are highly influenced by the values of No. For No≤0.9, ζ increases with increasing No. However, for No≥0.9, ζ decreases with increasing No. Hence, for the case of variable fin density, optimum value of No can be considered as 0.9.

Fig. 9. Effect of dimensionless free stream temperature, θα, on the variation of dimensionless temperature θL,m with the fraction of fin area at the base Φo.

Fig. 10. Variation of Nusselt number ratio (ζ) with fin distribution parameter,

B, for different values of the fraction of fin area at the base Φo.



0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 4 8 12 16 20

Φ

X

Φo = 0.6, Ar = 0.05

B=0.005

B=0.01

B=0.015

B=0.02

B=0.025

B=0.03

B=0.035

B=0.04

B=0.045

Fig. 11. Variation of the fraction of fin area at the base Φ(X) along the axial

direction at different values of fin distribution parameter, B.

Effect of No on the variation of dimensionless temperature θL,m with Peclet number, Pe for Φo=0.6, B=0.045, Nf=0.01 and θα= –0.5 is shown in Fig.12. θL,m is found to decreases with increasing Pe irrespective of value of No.


variation of Nusselt number ratio, ζ, with Peclet number, Pe.


variation of dimensionless lower wall temperature of the channel, θL,m with Peclet number, Pe.

The effect of dimensionless fin heat transfer coefficient, Nf, on the variation of Nusselt number ratio ζ with Peclet number, Pe for Φo=0.6, B=0.045, No=0.9 and θα= –0.5 is plotted in Fig.14. It is evidenced from Fig.14 that for Nf≤0.1 (0.0001-.1) ζ increases with increasing Pe, attains a maximum near Pe=1.5 and then starts decreasing. However, for Nf>0.1 (0.5-12.5), with increasing Pe, ζ first decreases slightly, and

then increases. Based on the results presented in Fig.14, optimum value of Nf comes out to be about 0.001. Effect of Nf on the variation of dimensionless temperature θL,m with Peclet number, Pe for Φo=0.6, B=0.045, Nf=0.01 and θα= –0.5 is presented in Fig.15. θL is seen to be decreasing with increasing Pe irrespective of value of Nf.


variation of Nusselt number ratio, ζ, with Peclet number, Pe.

Fig. 15. Effect of dimensionless base heat transfer coefficient, Nf, on the

variation of dimensionless lower wall temperature of the channel, θL,m with Peclet number, Pe.

C. UNIFORM FIN DENSITY WITH NANOFLUID In order to investigate the effect of using nanofluids on heat transfer augmentations, two nanofluids, one containing copper oxide nanoparticles dispersed in water and the other in a base fluid of 60:40 (by mass) ethylene glycol and water mixture (60:40 EG/W). The correlations suggested by Vajjha and Das [41] have been used to compute the temperature dependent thermal conductivities of the base fluids. Also, for calculating thermal conductivities of nanofluids, correlation proposed by Vajjha and Das [41] has been used. Present computations were performed with particle volumetric concentration ranging from 0 to 6% due to the limitation that the correlation proposed by Vajjha and Das [41] is valid for 0≤ ψ≤ 0.06. Other input data are detailed in Table 1. Since heat flux is provided at the lower surface of the channel, Nusselt numbers were calculated only at the lower surface. Temperature dependent thermal conductivities of water and 60:40 EG/W were calculated by using the correlations developed by Vajjha and Das [41] as mentioned in Table 2.



Table I INPUT DATA USED IN THE PRESENT STUDY

Type of nanofluid

Dia

met

er o

f na

nopa

rticl

e (n

m)

Inle

t te

mpe

ratu

re

(oC

) H

eat f

lux

at

low

er

surf

ace

of

chan

nel

(W/m

2 )

Cha

nnel

he

ight

(m

m)

Nanofluid-1 (Copper oxide dispersed in water)

29 30 3500 2

Nanofluid-2 (Copper oxide dispersed in 60:40 EG/W)

29 30 2000 2

Table II TEMPERATURE DEPEND THERMAL CONDUCTIVITY OF BASE

FLUIDS (T in Kelvin) Nanofluid-1 kwater= –8E-06 T2 + 0.0064 T – 0.5831

Nanofluid-2 k60:40EG/W = –3.2E-06 T2 + 0.00251188 T

–0.105411

The effect of particle volumetric concentration,ψ, on the axial variation of local Nusselt number for Pe=1.5, Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5 is presented in figure 16. It is seen in Fig.16 that for a given value of ψ=0 (pure fluid), the local Nusselt number gradually decreases from a maximum near the entrance and tends to attain a constant value as the flow develops. It can be seen that with the increase in ψ, the local Nusselt number increases at given value of X. Significant increase in local Nusselt number is noticed with the use of nanofluid.

Fig. 16. Effect of particle volumetric concentration (ψ) in water on the axial variation of Nusselt number for Pe=1.5, Nf=0.001, No=0.4, Φo=0.6 and θα=

−0.5.

The influence of particle volumetric concentration (ψ) in water on the variation of average Nusselt number ratio (ζnf) with Peclet number for Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5 is shown in Fig.17. This figure shows the quantitative role of nanofluid in increasing heat transfer rate.

It is evident from Fig.17 that for a given value of ψ, the averege Nusselt number ration is maximum at low value of Peclet number. As Peclet number increases, the Nusselet number ratio decreases and at higher values of Peclet number, average Nusselt number ratio tends to attain a constant value. For ψ=0.01, at Pe=1.0, about 22% increase in heat transfer rate is observed due to use of nanofluid. For a given value of Peclet number, as the particle volumetric concentration is increased, significant heat transfer enhancement is noticed. For example, as clear from the Fig.17, for ψ=0.06, about 50% increase in average Nusselt number is found at Pe=1.0. At higher value of Peclet number (Pe=200) also, for ψ=0.06, about 23% increase in average Nusselet number is seen.

Fig. 17. Effect of particle volumetric concentration (ψ) in water on the variation of average Nusselt number ratio (ζnf) with Peclet number for

Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5. Figure 18 presents the influence of particle

volumetric concentration (ψ) in water on the variation of maximum lower channel surface temperature (TL,m) with Peclet number for Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5. Maximum lower channel surface temperature is the temperature at the exit of channel. It is important to maintain this temperature well below the boiling point of nanofluid. It is seen in Fig.18 that TL,m decreases with increasing Peclet number and becomes almost constant at higher values of Peclet number. No significant effect of particle volumetric concentration is noticed on TL,m.

The effect of particle volumetric concentration (ψ) in 60:40 EG/W on the variation of average Nusselt number ratio (ζnf) with Peclet number for Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5 is shown in Fig.19. This figure shows, from the qualitative point of view, nanofluid-1 and nanofluid-2 have the same characteristic. However, from quantitative point of view, for a given set of parameters, nanofluid-2 exhibits higher heat transfer augmentation than nanofluid-1. For ψ=0.01, at Pe=1.0, about 30% increase in heat transfer rate is observed. As evident from the figure 19, for ψ=0.06, about 65% increase in average Nusselt number is noticed at Pe=1.0 and 23% increase in average Nusselet number is seen at Pe=200.



Fig. 18. Effect of particle volumetric concentration (ψ) in water on the

variation of maximum lower channel surface temperature (TL,m) with Peclet number for Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5.

Fig. 19. Effect of particle volumetric concentration (ψ) in 60:40 EG/W on the

variation of average Nusselt number ratio (ζnf) with Peclet number for Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5.

Figure 20 shows the effect of particle volumetric concentration (ψ) in 60:40 EG/W on the variation of maximum lower channel surface temperature (TL,m) with Peclet number for Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5. It can be seen in Fig.20 that, variation of the maximum lower channel surface temperature, TL,m, with Peclet number for nano-fluid-2 is similar to the case of nanofluid-2 quantitatively as well as qualitatively.

Fig. 20. Effect of particle volumetric concentration (ψ) in water on the

variation of maximum lower channel surface temperature (TL,m) with Peclet number for Nf=0.001, No=0.4, Φo=0.6 and θα= −0.5.

D. MATCHING XUAN AND ROETZEL [1]

MODEL with VAJJHA AND DAS[2] All computation were performed with Nf=0.001, No=0.4, Φo=0.6, θα= −0.5, and with a heat flux at lower surface being equal to 3500 W/m2.

Effect of particle volumetric concentration (ψ) on the variation of the constant term of dispersed model (C*) with Peclet number (Pe) is presented in Fig.21. For a given value of ψ and Pe, the constant C* was obtained by matching the average Nusselt number (dispersed model) with the average Nusselt number (Das model). For all C* values, matching was within ± 1.0%. It is evident from Fig.21 that for a given value of ψ, C* decreases with increasing Pe and becomes fairly constant at higher values of Pe. It is also noticed that the effect of ψ is to increase the value of C*. Using the C*, ψ and Pe data, following correlation was developed,

⎟⎠⎞⎜

⎝⎛ ++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++

+= ψψ 1098765

43

21*

1

aaa aaaaa

aaPePePe

PeC (46)

The constants of the correlation (5) are:

4.0463 a9.61951 a0.0958397 a7.19964 a

0.372105 a1.76294a15.8318a25.3598a1.0756a16.5008a

54

32

====

======

10

987

6

1

The above correlation fits the data with a maximum of 12.7%.

Fig. 21. Effect of particle volumetric concentration (ψ) on the variation of the

constant term of dispersed model (C*) with Peclet number (Pe).

Validity of the dispersed model was further checked by comparing the local Nusselt number (Nux) and local dimensionless temperature (θ) for ψ=0.03 and Pe=10,50 and 100. Comparison of local Nusselt number (Nux) is presented in Fig.22 for and the comparison of local dimensionless temperature (θ) is shown in Fig.23. It is clear from these two figures that local Nusselt number and local dimensionless temperature obtained from dispersed model are in excellent agreement with relevant values of Das model.



Fig. 22. Comparison of dispersed model and Das model showing the variation of local Nusselt number (Nu) along the length of the channel (X) for ψ=0.03.

Fig. 23. Comparison of dispersed model and Das model showing the variation

of local dimensionless temperature (θ) along the transverse direction (Y) at X=16 for ψ=0.03.

V. CONCLUSIONS A model has been developed to study thermal performance and optimization of relevant thermogeometric parameters involved in a convective cooling system. The model is applicable for both pure fluids and nanofluids. In the present study, results have been obtained for pure fluid with uniform as well as variable fin density, and for nanofluids with uniform fin density only. The present work has also been intended to explore the possibility of matching dispersed model as proposed by Xuan and Roetzel [1] with the thermal conductivity model presented by Vajjha and Das[2]. Based on the results obtained in the present investigations, following conclusions may be made: 1. It has been observed that use of fins or increasing the

number of fins is not always favorable to heat transfer enhancement. Thermogeometric parameters like Peclet number, dimensionless heat transfer coefficients for fin (Nf) as well as bare surface (No), fraction of total surface area covered by the fin (Φ) were found to play an important role in achieving maximum value of Nusselt number.

2. Redistribution of the uniformly embedded fins in an exponential fashion led to increase in Nusselt number. Interestingly, this increase in Nusselt number comes

without any penalty in terms of pressure drop and cost. Rather, this increase will lead to more compactness of a heat exchanger.

3. A combination of Pe=1.5, No=0.9, Nf=0.001 and θα= –0.5 for a variable fin density profile with B=0.045 and Φo=0.6 was found to give the best thermal performance (ζ=1.027). In other words, Nusselt number increases by 2.7% simply by redistributing the fins in an exponential manner. It may be mentioned that low value of Peclet number is concerned with liquid metal cooling technology. Liquid metal cooling is of great significance where use of fans, water-cooling, or airflow is not possible or a matter of concern from the point of view of weight, size, or reliability. For example, avionics and military applications involve high-power electronic devices packed in confined spaces which lead to high thermal densities.

4. Use of nanofluids was found to enhance Nusselt number significantly in comparison to pure fluid.

5. Enhancement in Nusselt number was found to be higher at lower Peclet number.

6. Nanofluid containing copper oxide nanoparticles dispersed in a base fluid of 60:40 EG/W was found to exhibit better enhancement than the nanofluid containing copper oxide nanoparticles dispersed in water.

7. With increasing particle volumetric concentration, Nusselt number was also found to increase. The analyses reveal about 65% increase in average Nusselt number at low Peclet number (Pe=1.0) and about 27% increase in that at high Peclet number (Pe=200) with nanofluid containing copper oxide nanoparticles dispersed in a base fluid of 60:40 EG/W.

8. It has been observed that dispersed model as proposed by Xuan and Roetzel [1] excellently matches the thermal conductivity model presented by Vajjha and Das[2].

9. With proper selection of the constant term C* in dispersed model[1], average Nusselt number was found to match within ± 1% with the average Nusselt number obtained from Vajjha and Das[2]. Since, Vajjha and Das[2] model is based on experimentally obtained data, present analysis very well establishes the validity of dispersed model[1].

10. Validity of the dispersed model was further strengthened by comparing the local Nusselt number (Nux) and local dimensionless temperature (θ). Local Nusselt number (Nux) and local dimensionless temperature (θ), both were found to agree well with the Vajjha and Das[2] model.

11. A correlation has been developed to predict the value of C* as a function of ψ and Pe.



VI. ACKNOLEDGEMENT

The authors acknowledge the full support of this work by King Abdulaziz City for Science and Technology (KACST) under project no. 8-ENE192-3.

REFERENCES

[1] Y. Xuan and W. Roetzel, “Conceptions for heat transfer correlation of nanofluids,” International Journal of Heat and Mass Transfer, Vol.43 pp. 3701–3707, 2000.

[2] R. S. Vajjha and D. K. Das, “Experimental determination of thermal conductivity of three nanofluids and development of new correlations,” International Journal of Heat and Mass Transfer, Vol.52, pp. 4675–4682, 2009.

[3] A.-R. A. Khaled, “The role of expandable thermal systems in improving performance of thermal devices, ” International Journal of Thermal Sciences, Vol.46, pp. 413–418, 2007.

[4] A.-R. A. Khaled, “Heat transfer analysis through solar and rooted fins,” Journal of Heat Transfer – Transactions of the ASME, Vol.130, Article ID 074503, 2008.

[5] A.-R. A. Khaled, “Heat transfer enhancement in hairy fin systems,” Applied Thermal Engineering, Vol.27, pp. 250–257, 2007

[6] A.-R. A. Khaled, “Maximizing heat transfer through joint fin systems,” Journal of Heat Transfer - Transactions of the ASME, Vol.128, pp. 203–206, 2006.

[7] A.-R. A. Khaled, “Analysis of heat transfer through Biconvection fins, ” International Journal of Thermal Sciences, Vol.48, pp. 122–132, 2009.

[8] A.-R. A. Khaled, “Investigation of heat transfer enhancement using permeable fins,” Journal of Heat Transfer - Transactions of the ASME, Vol.132, Article ID 034503, 2010.

[9] A.-R. A. Khaled, “Analysis of heat transfer inside flexible thin film channels with nonuniform height distributions,” Journal of Heat Transfer - Transactions of the ASME, Vol.129, pp. 401–404, 2007.

[10] A.-R. A. Khaled and K. Vafai, “Flow and heat transfer inside thin films supported by soft seals in the presence of internal and external pressure pulsations,” International Journal of Heat and Mass Transfer, Vol.45, pp. 5107–5115, 2002.

[11] A.-R. A. Khaled and K. Vafai, “Cooling enhancements in thin films supported by flexible complex seals in the presence ultrafine suspensions,” Journal of Heat Transfer - Transactions of ASME, Vol.125, pp. 916–919, 2003.

[12] A.-R. A. Khaled and K. Vafai, “The role of porous media in modeling flow and heat transfer in biological tissues,” International Journal of Heat and Mass Transfer, Vol.46, pp. 4989–5003, 2003.

[13] A.-R. A. Khaled and K. Vafai, “Analysis of thermally expandable flexible fluidic thin-film channels,” Journal of Heat Transfer - Transactions of the ASME, Vol.129, pp. 813–818, 2007.

[14] K. Vafai and A.-R. A. Khaled, “Analysis of flexible microchannel heat sink systems,” International Journal of Heat and Mass Transfer, Vol.48, pp. 1739–1746, 2005.

[15] M. Siddique, A.-R. A. Khaled, N. I. Abdulhafiz, and A. Y. Boukhary, “Recent Advances in Heat Transfer Enhancements: A Review Report,” International Journal of Chemical Engineering, Vol.2010, Article ID 106461, 2010.

[16] G. Sachdeva, K.S. Kasana and R. Vasudevan, “Analysis on Heat Transfer Enhancement by Using Triangular Shaped Inserts as Secondary Fins in Cross Flow Plate Fin Heat Exchanger,” International Journal of Dynamics of Fluids, ISSN 0973-1784 6 (2010), no. 1 pp. 41–47.

[17] A. V. N. Kapatkar, B. Dr. A. S. Padalkar and C. S. Kasbe, “Experimental Investigation on Heat Transfer Enhancement in Laminar Flow in Circular Tube Equipped with Different Inserts,” Proc. of. International Conference on Advances in Mechanical Engineering, (2010).

[18] Y. S. Muzychka, E. J. Walsh, “Heat Transfer Enhancement Using Laminar Gas-Liquid Segmented Plug Flows,” Journal of Heat Transfer - Transactions of the ASME, Vol.133, Article no. 041902, 2011.

[19] P. Promvonge, S. Sripattanapipat, S. Tamna, S. Kwankaomeng, C. Thianpong “Numerical investigation of laminar heat transfer in a square channel with 45° inclined baffles,” International Communications in Heat and Mass Transfer, Vol.37, pp. 170–177, 2010.

[20] S. Kakaç and A. Pramuanjaroenkij, “Review of convective heat transfer enhancement with nanofluids,” International Journal of Heat and Mass Transfer, Vol.52, pp. 3187-3196, 2009.

[21] A. Akbarinia, M. Abdolzadeh and R. Laur, “Critical investigation of heat transfer enhancement using nanofluids in microchannels with slip and non-slip flow regimes,” Applied Thermal Engineering, Vol.31, pp. 556-565, 2010.

[22] E. Abu-Nada, “Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step,” International Journal of Heat and Fluid Flow, Vol.29, pp. 242–249, 2008.

[23] B.C. Pak and Y.I. Cho, “Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles,” Experimental Heat Transfer, Vol.11, pp. 151–170, 1998.

[24] Y.M. Xuan and Q. Li, “Investigation on convective heat transfer and flow features of nanofluids,” Journal of Heat Transfer - Transactions of the ASME, Vol.125, pp. 151–155, 2003.

[25] A. Akbarinia and A. Behzadmehr, “Numerical study of laminar

mixed convection of a nanofluid in horizontal curved tubes,” Applied Thermal Engineering, Vol.27, pp. 1327–1337, 2007.

[26] A. Akbarinia and R. Laur, “Investigation the diameter of solid particles affects on a laminar nanofluid flow in a curved tube using a two phase approach,” International Journal of Heat Fluid Flow, Vol.30, pp. 706–714, 2009.

[27] S.M.S. Murshed, K.C. Leong and C. Yang, “Thermophysical and electrokinetic properties of nanofluids – a critical review,” Applied Thermal Engineering, Vol.28, pp. 2109–2125, 2008.

[28] A. Akbarinia, “Impacts of nanofluid flow on skin friction factor and Nusselt number in curved tubes with constant mass flow,” International Journal of Heat and Fluid Flow, Vol.29, pp. 229–241, 2008.

[29] C.T. Nguyen, F. Desgranges, N. Galanis, G. Roy, T. Maré, S. Boucher and H. Angue Mintsa, “Viscosity data for Al2O3–water nanofluid—hysteresis: is heat transfer enhancement using nanofluids reliable? ,” International Journal of Thermal Sciences, Vol.47, pp. 103–111, 2008.

[30] J. Koo and C. Kleinstreuer, “Laminar nanofluid flow in microheat-sinks,” International Journal of Heat and Mass Transfer, Vol.48, pp. 2652–2661, 2005.

[31] S.P. Jang and S.U.S. Choi, “Cooling performance of a microchannel heat sink with nanofluids,” Applied Thermal Engineering, Vol.26, pp. 2457–2463, 2006.

[32] R. Chein and G. Huang, “Analysis of microchannel heat sink performance using nanofluids,” Applied Thermal Engineering, Vol.25, pp. 3104–3114, 2005.

[33] H. Abbassi and C. Aghanajafi, “Evaluation of heat transfer augmentation in a nanofluid-cooled microchannel heat sink,” Journal of Fusion Energy, Vol.25, pp. 187–196, 2006.

[34] T.H. Tsai and R. Chein, “Performance analysis of nanofluid-cooled microchannel heat sinks,” International Journal of Heat and Fluid Flow,Vol.28, pp. 1013–1026, 2007.

[35] J. Li and C. Kleinstreuer, “Thermal performance of nanofluid flow in microchannels,” International Journal of Heat and Fluid Flow, Vol.29, pp. 1221–1232, 2008.

[36] R. Chein and J. Chuang, “Experimental microchannel heat sink performance studies using nanofluids,” International Journal of Thermal Sciences, Vol.46, pp. 57–66, 2007.

[37] J.Y. Jung, H.S. Oh and H.Y. Kwak, “Forced convective heat transfer of nanofluids in microchannels,” International Journal of Heat and Mass Transfer, Vol.52, pp. 466–472, 2009.



[38] C.J. Ho, L.C. Wei and Z.W. Li, “An experimental investigation of forced convective cooling performance of a microchannel heat sink with Al2O3-water nanofluid, Applied Thermal Engineering, Vol.30, pp. 96–103, 2010.

[39] S.E.B. Maiga, C.T. Nguyen, N. Galanis and G. Roy, “Heat transfer behaviors of nanofluids in a uniformly heated tube,” Superlattice Microstructure, Vol.35, pp. 543–557, 2004.

[40] C.H. Chon, K.D. Kihm, S.P. Lee and S.U.S. Choi, “Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement,” Applied Physics Letters, Vol.87, pp. 1–3, 2005.

[41] D. C. Tretheway and C. D. Meinhart, “Apparent fluid slip at hydrophobic microchannel walls,” Physics of Fluids, Vol.14, pp. L9–L12, 2002.

[42] A.K. Santra, S. Sen and N. Chakraborty, “Study of heat transfer

due to laminar flow of copper-water nanofluid through two isothermally heated parallel plates,” International Journal of Thermal Sciences, Vol.48, pp. 391–400, 2009.

[43] W. M. Kays and M.E. Crawford, “Convection Heat and Mass

Transfer,” 3rd ed. McGraw-Hill, New York, 1993.

maximization and modeling matching of convective heat ...no total number of pin fin n' linear...

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