Entropy generation of viscous dissipative nanofluid flow in microchannels

International Journal of Heat and Mass Transfer 55 (2012) 4169–4182

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Entropy generation of viscous dissipative nanofluid flow in microchannels Wei Han Mah, Yew Mun Hung ⇑, Ningqun Guo School of Engineering, Monash University, 46150 Bandar Sunway, Malaysia

a r t i c l e

i n f o

Article history: Received 28 February 2011 Received in revised form 28 February 2012 Accepted 28 February 2012 Available online 18 April 2012 Keywords: Entropy generation Nanofluid Viscous dissipation

a b s t r a c t An analytical study on the viscous dissipation effect on entropy generation in laminar fully developed forced convection of water–alumina nanofluid in circular microchannels is reported. In the first-law analysis, closed form solutions of the temperature distributions in the radial direction for the models with and without viscous dissipation term in the energy equation are obtained. The results show that the heat transfer coefficient decreases with nanoparticle volume fraction largely in the laminar regime of nanofluid flow in microchannel when the viscous dissipation effect is taken into account. In the second-law analysis, the two models are compared by analyzing their relative deviations in entropy generation for different Reynolds number and nanoparticle volume fraction. When the viscous dissipation is taken into account, the temperature distribution is prominently affected and consequently the entropy generation ascribable to the heat transfer irreversibility is significantly increased. The increase of entropy generation induced by the increase of nanoparticle volume fraction is attributed to the increase of both the thermal conductivity and viscosity of nanofluid which causes augmentation in the heat transfer and fluid friction irreversibilities, respectively. By incorporating the viscous dissipation effect, both thermal performance and exergetic effectiveness for forced convection of nanofluid in microchannels dwindle with nanoparticle volume fraction, contrary to the widespread conjecture that nanofluids possess advantage over pure fluid associated with higher overall effectiveness from the aspects of first-law and secondlaw of thermodynamics. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction To enhance the thermal performance of microchannel, the choice of working fluid has been a key issue and the use of suspended ultrafine solid particles in the base fluid to improve the heat transfer characteristics of conventional fluids is one of the innovative methods adopted. In 1995, Choi [1] first coined the term ‘‘nanofluid’’ for this type of fluid, which is shown to exhibit anomalously increased effective thermal conductivity with small volume fraction (in general less than 10%) of the ultrafine nanoparticles, which may be either metallic or nonmetallic. A number of review articles on heat transfer characteristics of nanofluids have been well-documented [2–8]. Besides the analysis based on the basic conservation laws, the second-law analysis is crucial in understanding the entropy generation attributed to the thermodynamic irreversibility. This is useful for studying the optimum operating conditions in designing a system with less entropy and exergy destruction, in accordance to the Gouy–Stodola theorem stating that the lost available work is directly proportional to the entropy generation [9,10]. Bejan [10] referred this method of engineering research as Entropy Generation Minimization (EGM) and discussed ⇑ Corresponding author. Tel.: +603 5514 6251; fax: +603 5514 6207. E-mail addresses: [email protected] (W.H. Mah), hung.yew.mun@ monash.edu (Y.M. Hung), [email protected] (N. Guo). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.03.058

its derivations and applications in a vast coverage of applied thermal engineering. Viscous dissipation which features as a source term in the fluid flow generates appreciable rise in the fluid temperature due to the conversion of mechanical energy to thermal energy associated with the viscous forces. This effect is of particular significance in the heat and fluid flow in microchannel whose length-to-diameter ratio is considerably large [11–14]. By incorporating the effect of viscous dissipation, the convection heat transfer of nanofluid flow in microchannels was analyzed and it was concluded that the Nusselt number is overestimated when the viscous dissipation is neglected, and subsequently, the thermal performance of the microchannel is overrated [15]. While most of the previous studies on nanofluid primarily focused on the heat transfer characteristics from first-law point of view, until very recently only a very small number of studies that dealt with entropy generation related problems in nanofluid flow are in existence. Most recently, the entropy generation of alumina nanofluids flow in laminar and turbulent regimes through an isoflux circular pipe was analytically investigated [16]. There exists another analytical study on the entropy generation of nanofluid in microchannel, minichannel and conventional channel in the laminar and turbulent regimes [17]. Both studies employed a simplified averaged model where the spatial information of the entropy generation is unavailable, and on top of that the viscous dissipation term was neglected in the

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Nomenclatures Ac Be Br0 Ck Cl cp D d k NS NS Pr qw r r0 R Re S_ 000 gen T T u  u

cross-sectional area (m2) Bejan number Brinkman number based on uniform heat flux condition thermal conductivity ratio, as defined in Eq. (2) viscosity ratio, as defined in Eq. (3) specific heat (J kg1 K) inner diameter of microchannel (m) diameter of nanoparticle (m) thermal conductivity (W m1 K1) dimensionless entropy generation average dimensionless entropy generation Prandtl number uniform wall heat flux (W m2) radial coordinate (m) internal radius of microchannel (m) dimensionless radial coordinate Reynolds number volumetric rate of entropy generation (W m3 K1) temperature (K) bulk mean temperature (K) fluid velocity (m s1) mean velocity of fluid (m s1)

energy equation (first-law analysis) for the fluid phase in these analyses. On the other hand, a numerical analysis for steady laminar nanofluid flow in trapezoidal microchannels was carried out with a commercial code ANSYS CFX 11 [18]. Although the viscous dissipation term was included in the energy equation, this study investigated the relations of the channel aspect ratio and Reynolds number range with the entropy generation rate rather than the effect of viscous dissipation. By performing the second-law analysis on the forced convective heat transfer in microchannels, it has been pointed out that the effects of viscous dissipation on entropy generation are significant albeit not dominant [19,20]. Judging from this, it is instructive to take the effect of viscous dissipation into account in the derivation of the entropy generation rate of nanofluid in microchannels. Employing the models with and without viscous dissipation term in the energy equation, an analytical model is developed from first principles for a microchannel filled with nanofluid as working fluid under fully developed condition. The analysis emphasizes details of the entropy generation rate variations with the governing parameters such as Nusselt number and nanoparticle volume fraction. The relative deviations between the two models are analysed for scrutinizing the changes entailed in the entropy generation due to the incorporation of the effects of viscous dissipation and nanoparticle suspensions. 2. Mathematical formulation 2.1. First-law analysis Three main transport properties involved in nanofluids which induce significantly different heat transfer performance from conventional fluids are thermal conductivity, heat capacity, and viscosity. The most common water-alumina nanofluids are employed as the model fluid. Thermal conductivity is a key parameter in enhancing the heat transfer performance. The suspended nanoparticles with higher thermal conductivity than the base fluids are responsible in enhancing the thermal conductivity of nanofluids. To estimate the effective thermal conductivity of water–Al2O3 nanofluids, the correlation proposed by Prasher et al. [21] for the

^ u x

dimensionless fluid velocity longitudinal coordinate (m)

Greek symbols D relative deviation, as defined in Eq. (40) (%) d relative deviation, as defined in Eq. (38) (%) j thermal conductivity ratio l dynamic viscosity (N s m2) h dimensionless temperature q fluid density (kg m3) / nanoparticle volume fraction w dimensionless heat flux x entropy generation ratio Subscripts eff effective FF of fluid friction f of base fluid HT of heat transfer nf of nanofluid p of nanoparticle w value at wall

Brownian motion-induced convection from multiple nanoparticles is employed:

keff ¼ C k kf ;

ð1Þ

where Ck is defined as

  jð1 þ 2aÞ þ 2 þ 2/½jð1  aÞ  1 1=3 C k ¼ 1 þ ARem / ; b Pr jð1 þ 2aÞ þ 2  /½jð1  aÞ  1

ð2Þ

with / the nanoparticle volume fraction of the nanofluid. The parameter j = kp/kf is the thermal conductivity ratio of the thermal conductivity of the particle kp to the thermal conductivity base fluid kf; a = 2Rbkf/d is the Biot number of the particle, with d the diameter of nanoparticle and Rb the interfacial resistance; and Reb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 18kb T=pqp d=mf is the Brownian–Reynolds number, with kb the Boltzmann constant and mf the fluid kinematic viscosity. For Al2O3 nanoparticles in water as base fluid, with Rb = 0.77  108 K m2 W1, it is found that m = 2.5 and A = 40,000 [19]. In the light of the effects of inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity settling [22], nonzero slip velocity between the nanoparticles and the base fluid was reported. It has been pointed out that the conventional Einstein model failed to capture these rheological features of nanofluids [22–25]. Following this, the effective viscosity of nanofluids is modelled using a modified Einstein model taking into account the slip velocity of nanoparticle with respect to the base fluid as [24]

leff ¼ C l lf ;

ð3Þ

where lf is the dynamic viscosity of the base fluid and the ratio Cl is defined as

"

#  2e d 2=3ðeþ1Þ ; C l ¼ ð1 þ 2:5/Þ 1 þ g / D

ð4Þ

with e = 1/4 and g = 280 the empirical constants for Al2O3 nanoparticles, and D the inner diameter of microchannel. In the absence of theoretical formulas to satisfactorily predict the thermal conductivity and the viscosity of nanofluids, for a sufficiency of addressing the effect induced by viscous dissipation and without loss of generality, the effective thermal conductivity and viscosity in the present

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study are assumed to follow the semi-empirical and empirical models in Eqs. (2) and (4), respectively. However, by modifying the parameters Ck in Eq. (2) and Cl in Eq. (4), the results obtained in the present study are easily transformed to suit other semi-empirical or empirical correlations in the existing literature. The specific heat of nanofluids is defined based on mass fraction as [26]

cp;nf ¼

qf cp;f ð1  /Þ þ qp cp;p / ; qf ð1  /Þ þ qp /

ð5Þ

and the effective density is volume-averaged, expressed as [17]

qnf ¼ qf ð1  /Þ þ qp /:

ð6Þ

In dealing with fluid flows in microchannels, the flowing fluids are treated as continuous media and the continuum modeling approach is applicable in most of the practical cases for channels larger than 1 lm [27]. Viscosity measurements revealed that the nanofluid manifests itself as Newtonian fluid [28,29]. Therefore, for steady and axisymmetric flow in a circular microchannel with internal radius r0, and assuming that the fluid phase and nanoparticles are in thermal equilibrium with zero relative velocity, the equation of motion in the axial direction can be written in the form

  1 @ @u 1 @p r ¼ : r @r @r leff @x

ð7Þ

Based on the fact that op/ox = constant, the velocity distribution yields the Hagen–Poiseuille expression as

"

 2 # r  1 u ¼ 2u ; r0

2 r 20

Z

ur dr:

ð9Þ

0

For constant fluid properties, the energy equation is expressed as

   2 @T k @ @T @u qnf cp;nf u ¼ eff ; r þ leff @x @r @r r @r

ð10Þ

where T is the temperature of the nanofluid. Taking into account the heat generation due to viscous dissipation, this effect is incorporated as the second term on the right-hand side in Eq. (10). The thermal boundary conditions are expressed as

keff

 @T  ¼ qw ; @r r¼r0

 @T  ¼ 0; @r r¼0

ð11Þ

where qw is the uniform heat flux applied at the wall. The bulk mean temperature T is defined as

T¼

2  r 20 u

Z

r0

uTr dr:

ð12Þ

0

For uniformly heated wall condition and thermally fully developed flow, the temperature gradient along the axial direction is a constant given as

@T dT ¼ ; @x dx

ð13Þ

which is independent of the radial direction. Since the temperature gradient is a constant, the axial conduction term keff o2T/ox2 in the energy equation would be equal to zero. Therefore, under a fully developed thermal condition with uniformly heated boundary wall, the axial conduction term is absent in Eq. (10) and its contribution to the net energy transfer is negligible [30]. Integrating Eq. (10) over the cross section of the microchannel gives

Z

r0

uTr dr ¼ keff r

0

r Z r0  2 @T  0 @u þ leff r dr:  @r 0 @r 0

dT 2  2 Þ; ¼ ðq r þ 4leff u  r 20 w 0 dx qnf cp;nf u

ð14Þ

ð15Þ

which conforms to the energy balance over the cross section of the microchannel. In the case of thermally fully developed condition with uniform wall heat flux, from Eq. (15), it is observed that the axial temperature gradient is reduced to a constant. By utilizing the energy balance relation in Eq. (15) and introducing the following dimensionless variables

R¼

r ; r0

^¼ u

u ;  u

h¼

keff ðT  T w Þ ; qw D

ð16Þ

with Tw denotes the wall temperature, Eq. (10) is nondimensionalized as

   2 ^ 1 d dh du ^  Br 0 R ¼ ð1 þ 8Br 0 Þu : R dR dR dR

ð17Þ

In Eq. (17), the term on the left-hand side corresponds to radial conduction while the first term and second term on the right-hand side are related to convection and viscous dissipation, respectively. The modified Brinkman number is defined as

ð8Þ

r0

@ @x

By utilizing Eqs. (8), (11), and (13), Eq. (14) can be simplified as

Br 0 ¼

 is the mean velocity of the base fluid over the cross section where u area of the microchannel defined as

¼ u

qnf cp;nf

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leff u2

ð18Þ

qw D

which is based on the uniform heat flux condition, a measure of the importance of the impact of viscous dissipation. From the requirements that T(r0) = Tw and the symmetric condition at the center, the dimensionless boundary conditions are, respectively, given by

hð1Þ ¼ 0;

dhð0Þ ¼ 0: dR

ð19Þ

Substituting the dimensionless velocity profile and its gradient into Eq. (17) and solving it yields the closed form dimensionless temperature profile as

  1 1 3 16 0 1þ Br : h1 ðRÞ ¼  ð1 þ 16Br 0 ÞR4 þ ð1 þ 8Br 0 ÞR2  8 2 8 3

ð20Þ

For the case when the viscous dissipation term is neglected in the energy equation, i.e. Br0 = 0, the dimensionless temperature profile becomes

1 1 3 h2 ðRÞ ¼  R4 þ R2  : 8 2 8

ð21Þ

Hereafter, for the purpose of comparison of results, the model with viscous dissipation term incorporated in the energy equation is denoted as Model 1 (with subscript 1) and that without viscous dissipation term in the energy equation as Model 2 (with subscript 2). We define the heat transfer coefficient between the wall and the nanofluid,

h¼

qw ; ðT w  TÞ

ð22Þ

and subsequently the average Nusselt number which characterizes the heat transfer rate between the wall and the nanofluid can be derived as

Nu ¼

hD 48 ¼ : keff 48Br 0 þ 11

ð23Þ

2.2. Second-law analysis Entropy is generated due to the presence of irreversibility, and entropy generation is adopted as a quantitative measure of the

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irreversibility associated with a process. The volumetric rate of entropy generation which arises due to the heat transfer and fluid friction losses, is expressed as [10]

keff l _ 000 _ 000 ðrTÞ2 þ eff U S_ 000 gen ¼ SHT þ SFF ¼ T T2

ð24Þ NS2

where U is the viscous dissipation function. The heat transfer irreversibility S_ 000 HT corresponding to the first term on the left-hand side of Eq. (24) is attributed to the heat transfer in the direction of finite temperature gradients and the second term on the left-hand side, fluid friction irreversibility S_ 000 FF , originates from the viscous effects of fluid friction. In the present study, the volumetric rate of entropy generation which arises due to the heat transfer and fluid friction losses, can be derived as

keff S_ 000 gen ¼ T2

"

@T @x

2 þ

dissipation term is absent in the energy equation (Model 2), by following the same procedures described previously, the dimensionless entropy generation, its two components and Bejan number are derived, respectively, as

 2 #   @T l @u 2 : þ eff @r @r T

# 2 " 2   2 ^ w dh2 4 w du þ Br 0 ¼ þ 2 1 þ wh2 1 þ wh2 dR dR ðRePrÞ n ¼ 16 1  8Br 0 R6 þ 4ð8Br 0 þ 1ÞR4 þ 4½2Br 0 ð8=w  3Þ þ 1R2 o. þ16=ðRePrÞ2 ðR4  4R2 þ 3  8=wÞ2 ; ð30Þ 

# 2 " 2 w dh2 4 þ 1 þ wh2 dR ðRePrÞ2 h i. ¼ 16 R6  4R4 þ 4R2 þ 16=ðRePrÞ2 ðR4  4R2  8=w þ 3Þ2 ;

NHT2 ¼

ð25Þ



ð31Þ

The dimensionless entropy generation (for Model 1) is derived as

NS1

 2 #  2 " 2 2 S_ 000 w dh1 1 þ 8Br 0 gen r 0 ¼ ¼ þ4 1 þ wh1 keff dR RePr   2 ^ w du þ Br 0 1 þ wh1 dR n ¼ 16 ð16Br 0 þ 1Þð8Br 0 þ 1ÞR6  4ð8Br 0 þ 1Þ2 R4 o  þ4 32Br 0 þ 2Br 0 ð8=w þ 5Þ þ 1 R2 þ 16ð8Br 0 þ 1Þ2 =ðRePrÞ2

h i2 ð16Br 0 þ 1ÞR4  4ð8Br 0 þ 1ÞR2 þ 16Br 0  8=w þ 3 ; ð26Þ

where w = qwD/(keffTw) is denoted as dimensionless heat flux, and Pr = cp,nfleff/keff is the Prandtl number. The dimensionless entropy generation due to conductive heat transfer in radial and axial directions is correspondingly expressed as

# 2 " 2 w dh1 4ð1 þ 8Br 0 Þ2 þ 1 þ wh1 dR Pe2 n ¼ 16 ð16Br 0 þ 1Þ2 R6  4ð16Br 0 þ 1Þð8Br 0 þ 1ÞR4 o þ4ð8Br 0 þ 1Þ2 R2 þ 16ð8Br 0 þ 1Þ2 =ðRePrÞ2

h i2 ð16Br 0 þ 1ÞR4  4ð8Br 0 þ 1ÞR2 þ 16Br 0  8=w þ 3 ; ð27Þ

NHT1 ¼



and the dimensionless form of entropy generation contributed by fluid friction is written as



 2 ^ w du NFF1 ¼ Br 0 1 þ wh1 dR .h i ¼ 128Br 0 R2 16Br 0 þ 1 R4  4ð8Br 0 þ 1ÞR2 þ 16Br 0  8=w þ 3 : ð28Þ The ratio of entropy generated due to heat transfer irreversibility to total entropy generation rate is denoted as Bejan number, which is given as 2

2

N HT1 ðdh1 =dRÞ þ ½2ð8Br0 þ 1Þ=RePr ¼ 2 N S1 ðdh1 =dRÞ þ ½2ð8Br 0 þ 1Þ=RePr2 þ Br0 ð1 þ wh1 Þðdu ^ =dRÞ2 =w h ¼ ð16Br 0 þ 1Þ2 R6  4ð16Br 0 þ 1Þð8Br0 þ 1ÞR4 þ 4ð8Br 0 þ 1Þ2 R2 i.n 2 2 2 þ16ð8Br0 þ 1Þ =ðRePrÞ ð16Br0 þ 1Þð8Br 0 þ 1ÞR6  4ð8Br0 þ 1Þ R4 o 2 2 þ4½32Br02 þ 2ð5 þ 8=wÞBr0 þ 1R2 þ 16ð8Br 0 þ 1Þ =ðRePrÞ :

Be1 ¼

ð29Þ Be = 1 is the limit at which the heat transfer irreversibility dominates while Be = 0 is the opposite limit where the irreversibility is solely attributed to fluid friction. For the case when the viscous

NFF2 ¼ Br 0



w 1 þ wh2



 ^ du dR

¼ 128Br 0 R2 =ðR4  4R2  8=w þ 3Þ;

ð32Þ

NHT2 ðdh2 =dRÞ2 þ ð2=RePrÞ2 ¼ 2 NS2 ^ =dRÞ2 =w ðdh2 =dRÞ þ ð2=RePrÞ2 þ Br 0 ð1 þ wh2 Þðdu .n ¼ ½R6  4R4 þ 4R2 þ 16=ðRePrÞ2  ð1  8Br 0 ÞR6 þ 4ð8Br 0 þ 1ÞR4 o ð33Þ þ4½2Br 0 ð8=w  3Þ þ 1R2 þ 16=ðRePrÞ2 :

Be2 ¼

It should be noted that in Model 2, the viscous dissipation term is neglected in the energy equation (first-law analysis) only but not in the expression of entropy generation (second-law analysis). The average dimensionless entropy generation and average Bejan number over the circular cross section of the microchannel are, respectively, given as

NS ¼ Be ¼

1 Ac 1 Ac

Z Z

NS dAc ¼ 2

Ac

Ac

Be dAc ¼ 2

Z Z

1

NS R dR;

ð34Þ

BeR dR:

ð35Þ

0 1

0

Numerical integration using the midpoint Riemann sum is performed in Eqs. (34) and (35). 3. Results and discussion All the results are reported for fully developed laminar flow of water-based nanofluids containing alumina nanoparticles and the input parameters required for the computation are listed in Table 1. Following this, the Reynolds number for the nanofluid, Re ¼ qnf uD=lnf , is less than 2300. For this range of Reynolds number, we identify the following scales for the changes in the variables: qw  105, T  102 , D  1045–104, qnf  103, leff  103, keff  100,   101  102 . Based on the order of magnitude analysis, the and u modified Brinkman number varies as Br 0  101  100 , in which case the effect of viscous dissipation is considered to be significant [15]. We have assumed that no slip boundary condition is used in Eq. (8) and the nanofluid inside the microchannel can be regarded as a continuum. To assess the validity of this assumption, the Knudsen number Kn which is defined as

Kn ¼

k D

ð36Þ

is calculated where k is the fluid molecule mean free path and D is the characteristic length of the microchannel. The water molecule effective size and mean free path in liquid water are both of the

W.H. Mah et al. / International Journal of Heat and Mass Transfer 55 (2012) 4169–4182

order of 3 Å (0.3 nm) [21] and in the similar manner, the mean free path of the nanoparticle is considered to be of the same order to its effective size which is 60 nm. For the dimension of microchannel of interest (70 lm), the maximum Knudsen number is relatively small (Kn < 9  104), and the continuum assumption is justified. To

Table 1 Input parameters used in the analysis. Parameters

Values

Bulk mean temperature, T Uniform heat flux, qw Channel inner diameter, D Particle diameter, d Base fluid density, qf Base fluid conductivity, kf Base fluid viscosity, lf Base fluid, cp,f Particle density, qp Particle conductivity, kp Particle, cp,p

300 K 1  105 W/m2 70 lm 60.4 nm 997 kg/m3 0.613 W/mK 8.55  104 Ns/m2 4179 kJ/kg K 3880 kg/m3 42.34 W/mK 729 kJ/kg K

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avoid plotting multiple graphs and to draw general conclusions from the analytical study, we resort to nondimensional variables henceforth. 3.1. First-law analysis As the viscous dissipation term is proportional to the square of the mean velocity, the Brinkman number and the Reynolds number are the key factors which determine the impact of viscous dissipation [13]. Fig. 1 depicts the relationship between the modified Brinkman number Br0 and the Reynolds number Re for different nanoparticle volume fraction /. It is observed that Br0 increases in a parabolic manner with respect with Re, indicating that the viscous dissipation effect is more pronounced at higher Reynolds number in the laminar flow regime for both clear fluid and nanofluids. The viscous dissipation effect becomes more prominent when the nanoparticle volume fraction is increased due to the increase in the effective viscosity of the nanofluid. In order to show the variations of the heat transfer coefficient h with the Reynolds number, the modified Nusselt number CkNu = hD/kf is defined such that h is proportional to CkNu. Fig. 2 depicts

Fig. 1. Variations of modified Brinkman number with Reynolds number in the fully developed region for water-based alumina nanofluids, nanoparticle volume fraction being a parameter.

Fig. 2. Variations of modified Nusselt number with Reynolds number in the fully developed region for water-based alumina nanofluids for models with and without viscous dissipation effect.

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the variations of CkNu with Re for the models with and without viscous dissipation effect. For Model 2 (without viscous dissipation), the modified Nusselt number (and the heat transfer coefficient) remains constant for all values of Reynolds number, agreeing with the classical results. Model 1 (with viscous dissipation) suggests substantial decrease in heat transfer coefficient when the Reynolds number increases. As depicted in Eq. (23), an increase in Br0 induces a decrease in the Nusselt number. Since Br0 increases in a

parabolic manner with respect with Re as shown in Fig. 1, increase in Reynolds number significantly decreases the Nusselt number and hence the heat transfer coefficient. This is because the amount of heat conducted from the wall to the fluid and then transported by convection is compensated by the internal heat generation through increasing viscous dissipation, as illustrated in Eq. (10). When the viscous dissipation is neglected, the heat transfer coefficient increases with nanoparticle volume fraction, indicating that

Fig. 3. Averaged and local radial values of entropy generation rate ratio of Model 1 and Model 2 as a function of volume fraction. Approximate values evaluated using simplified model of Ref. [17] are included.

Fig. 4. Total entropy generation contours for pure fluid and nanofluid flows at Re = 2000 of (a) Model 1, and (b) Model 2.

W.H. Mah et al. / International Journal of Heat and Mass Transfer 55 (2012) 4169–4182

Fig. 5. Entropy generation contours due to fluid friction irreversibility for pure fluid and nanofluid flows at Re = 2000 of (a) Model 1, and (b) Model 2.

Fig. 6. Entropy generation contours due to heat transfer irreversibility for pure fluid and nanofluid flows at Re = 2000 of (a) Model 1, and (b) Model 2.

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the thermal performance of the nanofluid is improved when / becomes larger. However, when the viscous dissipation is taken into account, this is only true when the Reynolds number is smaller than 300, exceeding which, the trend is reversed. Higher / induces higher effective viscosity of the nanofluid. Since the viscous dissipation term is proportional to the viscosity, it can be reasoned out that at higher Reynolds number, the heat transfer coefficient is diminished by the increase of nanoparticle volume fraction which augments the viscous dissipation effect. By comparing Model 1 and Model 2, at Re = 2300, a maximum decrease of 95% in the variation of heat transfer coefficient is observed for nanofluid with / ¼ 8% while for pure water there is a decline of 30%. 3.2. Second-law analysis For the second-law analysis, we turn our attention to the contributions of nanoparticle suspensions and viscous dissipation and on the entropy generation of nanofluid flow. Results for different volume fraction for both Model 1 and Model 2 are analysed. For comparison purpose, an entropy generation rate ratio is defined as follows:

S_ 000

x ¼ _ 000gen;i ; i ¼ 1; 2; S

ð37Þ

gen0;i

where S_ 000 gen0 is evaluated at / ¼ 0, which is the case of pure fluid. Fig. 3 depicts the effect of volume fraction on the entropy generation rate at different radial distances and its averaged counterpart for both Model 1 and Model 2. The approximated values obtained by considering the reduced entropy generation ratio equation for laminar flow in microchannel from Ref. [17] are also included for comparison. It can be observed that for both models the entropy generation ratio is above unity and increases with /. This shows that from the second-law point of view, the exergetic effectiveness of nanofluid in microchannel is depleted with the addition of nanoparticle. This is more obvious when the viscous dissipation effect is taken into account. While it was qualitatively correctly pointed out in Ref. [17] that the ratio of entropy generation rate of nanofluids to the entropy generation rate of pure fluid is always greater than unity for laminar flow in microchannel, the entropy generation ratio derived from the reduced model of Ref. [17] deviates quantitatively from those calculated in the present study. The deviation becomes more significant when the volume fraction of nanoparticle

Fig. 7. (a) Dimensionless entropy generation distributions, and (b) Bejan number distributions, in the radial direction of pure fluid and nanofluid flows for Model 1 and Model 2, with Reynolds number being a parameter.

W.H. Mah et al. / International Journal of Heat and Mass Transfer 55 (2012) 4169–4182

increases. We can observe that x of the reduced model only agrees well with that of Model 2 of the present study at R = 0.000017 when the viscous dissipation effect is not taken into account. This shows that the entropy generation ratio of the reduced model proposed in [16,17] is accurate only at the proximity to the center line but not the entire cross section of the channel, attributable to the fact that the reduced model might be led astray by the loss of spatial information of the entropy generation and the neglect of viscous dissipation effect in the energy equation. 3.2.1. Entropy generation contours To investigate the entropy generation rate in the microchannel, 2 the modified dimensionless entropy generation, C k N S ¼ S_ 000 gen r 0 =kf , is defined such that S_ 000 is proportional to C N . Similarly, k S gen 2 _ 000 2 C k N FF ¼ S_ 000 FF r 0 =kf and C k N HT ¼ SHT r 0 =kf are also defined in the same manner for its two components. To illustrate the variation of the entropy generation, the entropy contours of the modified dimensionless entropy generation C k N S for pure fluid and nanofluid flows of Model 1 and Model 2 are charted in Fig. 4. Due to the symmetric nature of circular cross-section, only contours of one quarter of the cross-section are depicted. At a fixed Reynolds number of 2000, we

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can observe that the total entropy generation increases tremendously from zero at the center of the channel to a maximum value at the wall due to the zero velocity and temperature gradients at the centerline, and the comparatively high values of these gradients in the vicinity of the wall. This phenomenon can be further explained by Figs. 5 and 6, which show the distributions of dimensionless entropy generation contours for Model 1 and Model 2 at Re ¼ 2000 due to fluid friction irreversibility, C k N FF , and heat transfer irreversibility, C k N HT , respectively. For both models, C k N FF displays zero value at the centerline and reaches a maximum value at the wall of the channel. The high value of C k N FF adjacent to the wall is contributed by the high near wall velocity gradient. The difference of C k N FF1 and C k N FF2 is not significant, showing the viscous dissipation effect induced on the fluid friction irreversibility is negligible. Comparing Figs. 5 and 6, it is observed that the magnitude of C k N HT is relatively small compared to C k N FF , indicating that the fluid friction irreversibility dominates over the heat transfer irreversibility in the contribution to the total entropy generation. The observation that the heat transfer irreversibility is smaller than the fluid friction irreversibility is common in both macro- and micro-channels. However, comparing Model 1 and

Fig. 8. Dimensionless entropy generation distributions of pure fluid and nanofluid flows due to irreversibility of (a) fluid friction, and (b) heat transfer, for Model 1 and Model 2, with Reynolds number being a parameter.

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Model 2, C k N HT1 is significantly higher than C k N HT2 . This is attributed to the fact that the heat transfer irreversibility is strongly dependent on the temperature field, as expressed in Eqs. (27) and (31). Neglecting viscous dissipation effect in the first-law analysis will indirectly induce significant deviation on the heat transfer irreversibility. 3.2.2. Entropy generation distributions To further investigate the variation of entropy generation in the radial direction, Fig. 7 plots the corresponding radial dimensionless entropy generation and Bejan number distributions for different Reynolds number of Model 1 and Model 2. As depicted in Fig. 7(a), the total entropy generation of both models increases with increasing Reynolds number. Comparing to that of pure fluid, the magnitude of entropy generation of nanofluid which increases with volume fraction of nanoparticle is much higher, indicating that the exergetic effectiveness of nanofluid in microchannel is depleted with the addition of nanoparticle. The increase of entropy generation induced by the increase of nanoparticle volume fraction is attributed to the increase of both the thermal conductivity and viscosity of nanofluid which causes augmentation in the heat transfer and fluid friction irreversibilities, respectively, as depicted

in Eq. (25). From Fig. 7(b), we can observe that the Bejan number is close to zero at the wall and peaks at the center of the channel. This trend of Bejan number indicates that the maximum entropy produced at the wall is mainly due to the fluid friction irreversibility contributed by the high near wall velocity gradient whereas towards the center of the channel with zero velocity gradient, the entropy generation due to fluid friction diminishes. It is known that _ 000 S_ 000 HT will only surpass SFF if Be > 0.5. Therefore, the contribution to the total entropy generation is mainly due to fluid friction irreversibility while the contribution of heat transfer irreversibility is minor. Comparing Model 1 and Model 2 for nanofluid, Be1 is significantly higher than Be2 at high Reynolds number. As shown in Fig. 8, the loss contributed by fluid friction and heat transfer irreversibility for both models increases with Reynolds number. As the effect of viscous dissipation on the heat transfer irreversibility is significant, the difference of heat transfer irreversibility between Model 1 and Model 2 is enlarged with increasing Reynolds number. This phenomenon is more prominent in nanofluid compared to that in pure fluid. Due to the increase of heat transfer irreversibility, the total entropy generation increases with Reynolds number and it is larger in magnitude when the viscous dissipation is taken into account in nanofluid.

Fig. 9. (a) Average dimensionless entropy generation distributions, and (b) average Bejan number distributions, as a function of Reynolds number for Model 1 and Model 2, with volume fraction of nanoparticle being a parameter.

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3.2.3. Average entropy generation Fig. 9(a) plots the average modified dimensionless entropy generation as a function of Reynolds number for different nanoparticle volume fraction. To investigate the contribution of heat transfer irreversibility and fluid friction irreversibility to the entropy generation, Fig. 9(b) depicts the corresponding average Bejan number for both models as a function of Reynolds number for different nanoparticle volume fraction. Over the range of laminar flow, we can observe from Fig. 9(a) that the entropy generation for both models increases with Reynolds number and decreases with nanoparticle volume fraction. For a given Reynolds number, nanofluid with largest nanoparticle volume fraction, / ¼ 8%, provides the largest entropy generation. Fig. 9(b) shows that the average Bejan number for both models approaches unity when Reynolds number is very close to zero and decreases to a minimum value, after which the average Bejan number for Model 1 increases with small gradient with Reynolds number whereas for Model 2, the average Bejan number slightly decreases as Reynolds number increases. Therefore, at a given Reynolds number, we can observe that the Bejan number decreases with increasing / at low Reynolds number while the trend is reversed at higher Reynolds number. One can deduce that most of the entropy generated in the microchannel (for both models) is attributed to the fluid friction irreversibility. At low Reynolds number the nanofluids reduce the contribution of the heat

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transfer irreversibility while at high Reynolds number (within the laminar regime), the heat transfer irreversibility is intensified with / .This phenomenon can be further explained by Fig. 10, which shows the distributions of dimensionless entropy generation by fluid friction contribution and dimensionless heat transfer contribution (for Model 1 and Model 2) for different Brinkman number. C k N FF1 and C k N FF2 are almost overlapping especially when the volume fraction of nanoparticle is small, showing that the viscous dissipation effect on the fluid friction irreversibility is negligible. For higher /, the viscous dissipation effect induced on the fluid friction irreversibility is also not obvious in this case. Comparing Fig. 10(a) and (b), it is observed that the magnitude of C k N HT is relatively small compared to C k N FF , indicating that the contribution of the heat transfer irreversibility to the total entropy generation is negligible in this case. As discussed earlier, similarly for the average values, C k N HT1 is significantly higher than C k N HT2 , as observed by comparing Model 1 and Model 2, due to the fact that the heat transfer irreversibility is strongly dependent on the temperature field. Therefore, the viscous dissipation affects the heat transfer irreversibility more significantly rather than the fluid friction irreversibility. 3.2.4. Effects of viscous dissipation and nanoparticle suspensions In order to scrutinize the effect of viscous dissipation on the entropy generation and its components, by using the value of Model 1

Fig. 10. Average dimensionless entropy generation distributions due to irreversibility of (a) fluid friction, and (b) heat transfer, as a function of Reynolds number for Model 1 and Model 2, with volume fraction of nanoparticle being a parameter.

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as the basis for comparison, the relative deviation of each parameter between Model 1 and Model 2 is quantified as

d¼

p1  p2  100%; p1

_ 000 _ 000 p ¼ S_ 000 gen ; SFF ; SHT ;

ð38Þ

which simply states the relative error incurred by neglecting the viscous dissipation effect in the energy equation. The relative deviation (in percentage) of the total entropy generation as a function of Reynolds number is observed in Fig. 11(a). The relative deviation at a fixed Reynolds number increases with the nanoparticle volume fraction. Subsequently, this observation allows us to deduce from the formula

p1  p2 d ; ¼ 1d p2

_ 000 _ 000 p ¼ S_ 000 gen ; SFF ; SHT ;

ð39Þ

that the absolute deviation, p1–p2, also increases with the Reynolds number, as does the total entropy generation and its components, as shown in Figs. 9 and 10. For pure fluid (/ ¼ 0), the total entropy generation of Model 1 is slightly larger (less than 2%) than that of Model 2. Although this deviation for pure fluid is small enough to

trigger an attention in the performance analysis, it might not be the same case for nanofluids, as deduced from the earlier first-law analysis of nanofluid in Fig. 2. When the volume fraction is increased to 8%, the deviation of total entropy generation is significantly enhanced with d exceeding 10% at Reynolds number more than 2000. Therefore, it can be asserted that the effect of viscous dissipation on the total entropy generation in nanofluid is pronounced when the nanoparticle volume fraction and Reynolds number are high, within the range of practical applications. By referring to Fig. 11(b), d of S_ 000 HT is significantly higher than that of _ 000 for nanofluids increases with nano. The relative deviation of S S_ 000 FF HT particle volume fraction and Reynolds number. It can be observed that for nanofluids, the d of S_ 000 HT is significant for the flow regime of Reynolds number larger than 300, where the viscous dissipation has to be considered in the analysis. Further increase in the Reynolds number and nanoparticle volume fraction eventually results in the scenario that d of S_ 000 HT approaching its asymptote of d = 100%. This suggests that the viscous dissipation effect is accrued by the increase of amount of nanoparticle which incurs more im-

Fig. 11. The relative deviations of (a) total entropy generation, and (b) its components, between Model 1 and Model 2, as a function of Reynolds number, with volume fraction of nanoparticle being a parameter.

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pact on the heat transfer irreversibility, attributed to the increase of thermal conductivity of the nanofluid. Although simultaneously the increase of amount of nanoparticle increases the viscosity of nanofluid, the viscous dissipation effect on the total entropy generation is mainly governed by the heat transfer irreversibility, despite of the fact that the total entropy generation is dominated by the fluid friction irreversibility. To examine further the contribution of the nanoparticle suspensions to the total entropy generation and its two components, we consider the relative deviation D, which is defined as follows:

D¼

pi  p0;i  100%; pi

_ 000 _ 000 p ¼ S_ 000 gen ; SFF ; SHT ;

i ¼ 1; 2;

ð40Þ

where p0 is evaluated at / ¼ 0, which is the case of pure fluid. Eq. (40) simply expresses the relative difference in the entropy generation between the nanofluid and the pure fluid. Fig. 12(a) and (b) show the relative deviations (in percentage) of the total entropy generation and its two components, as a function of Reynolds number, respectively. Analogous to the observation in Eq. (39), the absolute deviation, pi–p0,i, also increases with the Reynolds number, as does the total entropy generation and its components. As depicted

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in Fig. 12(a), for the case of a very small Reynolds number, _ 000 S_ 000 gen  Sgen0 is very small for both models, indicating that the effect of nanoparticle suspensions to entropy generation is negligible. As the Reynolds number is increased, the contribution of nanoparticle suspensions becomes more prominent. We observe that the relative deviation D is a function of the volume fraction of nanoparticle, and, as expected, is greater for higher volume fraction. The relative deviations approach their asymptotes when Reynolds number increases. However, this asymptotic value depends on the nanoparticle volume fraction, being greater for the higher volume fraction and vice versa. The existence of the asymptotes can be understood from the _ 000 fact that when the Reynolds number is increased slightly, S_ 000 gen  Sgen0 increases rapidly, with a higher rate for a higher volume fraction, and vice versa. We now turn our attention to the effect of nanoparticle suspensions on the fluid friction and heat transfer irreversibilities. From Fig. 12(b), we observe that for both models, the nanoparticle suspensions intensify S_ 000 FF of nanofluids, up to a significant level more than 80%. While D, interpreted here as the deviation incurred by the exclusion of the nanoparticle suspensions, decreases very slightly with the Reynolds number, it becomes greater as the nanoparticle volume fraction is increased. On the

Fig. 12. The relative deviations due to the contribution of the nanoparticle suspensions to the (a) total entropy generation, and (b) its components, for Model 1 and Model 2, as a function of Reynolds number, with volume fraction of nanoparticle being a parameter.

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other hand, for the case of heat transfer irreversibililty, we observe that the variation of D with Reynolds number is more drastic for Model 2 when the viscous dissipation term is excluded in the energy equation. For Model 2, D for S_ 000 HT is always negative and in this case it is understood that the magnitude of S_ 000 HT of nanofluid is always smaller than that of pure fluid. This shows that for Model 2, the effect of nanoparticle suspensions on the heat transfer irreversibility is more pronounced when the Reynolds number increases. When the viscous dissipation effect is taken into account (Model 1), at a given Reynolds number, we observe that the D for S_ 000 HT increases as the Reynolds number is increased from a negative value to zero at Reynolds number of about 300 and becomes positive henceforth. This shows that the magnitude of S_ 000 HT of nanofluid is smaller than that of pure fluid for Reynolds number less than 300 while the converse is true for Reynolds number more than 300. For higher nanoparticle volume fraction, an asymptote value of 100% is reached at higher Reynolds number. Following this, referring to Fig. 10 again, we can assert that the increase of entropy generation induced by the increase of nanoparticle volume fraction is attributed to the increase of both the thermal conductivity and viscosity of nanofluid which causes augmentation in the heat transfer and fluid friction irreversibilities, respectively. 4. Conclusions A mathematical model based on the first law and second law of thermodynamics of water-alumina nanofluids in circular microchannels during steady state operation has been developed, to investigate, primarily, the effect of the viscous dissipation on the entropy generation assessments and a better understanding of the physical problem is reached. This investigation provides interesting insights into the phenomena which take place in the comparison between the models with and without viscous dissipation effect on the assessments of entropy generation and overall exergetic effectiveness of convection heat transfer of nanofluids in microchannels. The forms of entropy generation associated in a microchannel under forced convection are ascribable to fluid friction irreversibility and heat transfer irreversibility. When the viscous dissipation is taken into account, the temperature distribution is prominently affected and consequently the entropy generation ascribable to the heat transfer irreversibility is intimately associated with the effect of viscous dissipation. On the other hand, the effect of viscous dissipation on the entropy generation due to fluid friction irreversibility is not significant. As the total entropy generation is dominated by its component from fluid friction irreversibility, therefore, the effect of viscous dissipation on the total entropy generation is also not significant. The increase of entropy generation induced by the increase of nanoparticle volume fraction is attributed to the increase of both the thermal conductivity and viscosity of nanofluid which causes augmentation in the heat transfer and fluid friction irreversibilities, respectively. On the other hand, the first-law analysis shows that the heat transfer coefficient decreases with nanoparticle volume fraction largely in the laminar regime of nanofluid flow in microchannel when the viscous dissipation effect is taken into account. It can be concluded that by incorporating the viscous dissipation effect, both thermal performance and exergetic effectiveness for forced convection of nanofluid in microchannels dwindle with nanoparticle volume fraction, contrary to the widespread conjecture that nanofluids possess advantage over pure fluid associated with higher overall effectiveness from the aspects of first-law and second-law of thermodynamics.

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