111

Chapter 4

CONVECTIVE HEAT TRANSFER CHARACTERISTICS

OF NANOFLUIDS

Convective heat transfer analysis of nanofluid flowing inside a

straight tube of circular cross-section under laminar and turbulent

conditions is taken up. The flow and the thermal field are assumed

symmetrical with respect to the vertical plane passing through the

main axis. Further there exists no formulated theory known to

date that could reasonably predict the flow & heat transfer

behaviors of a nanofluid by considering it as multi-component

model. Therefore, it has been suggested that the particles may be

easily fluidized and consequently, can be considered as

conventional single-phase fluid, which posses effective physical

properties being function of the properties of both constituents and

their respective concentrations (Pak & Cho [44]; Xuan & Li [110]).

As a result, a direct extension from a conventional fluid to

nanofluid appears feasible, and one may then expect that the

classical theory developed for a conventional single-phase fluid can

be applied to nanofluid as well. Thus, all the equations of

conservation (mass, momentum and energy) as well known for

single-phase fluid can be directly applied to nanofluids.

0

y

v

x

u (4.1)

112

)(1

2

2

vuyy

u

dx

dp

y

vv

x

uu

(4.2)

)(2

2

Tvyx

T

y

Tv

x

Tu

(4.3)

4.1 Laminar Flow

Wen & Ding [113] reported experimental results of the convective

heat transfer of Al2O3/ water nanofluid flowing through a copper

tube in laminar regime. They compared their experimental results

with Shah correlation for laminar flow and found the theoretical

results are far less than the experimental values. This may be due

to considering the base fluid properties instead of nanofluid

properties. Heris et.al. [174] conducted experiments on laminar

flow of CuO/ water and Al2O3/ water nanofluids through copper

tube and proposed Seider-Tate equation to compare experimental

results and found to be far less than experimental results. This

could be due to wrong estimation of thermal conductivity and

viscosity of nanofluids. Thus, we propose a modified Seider- Tate

equation in the following form for Laminar flow as follows

333.0)Pr(ReL

DbNu nfnfnf (4.4)

The constant „b‟ is evaluated from the regression analysis of the

data obtained from [44,113,174] and calculated as 1.98.Thus the

correlation for calculation of Nusselt number for Laminar flow is

113

developed with an average deviation of 6% and standard deviation

of 7.4 % for all nanofluids.

333.0)Pr(Re98.1L

DNu nfnfnf (4.5)

The correlation gives good agreement with the experimental results

as shown in the Fig.4.1(a)-(b). The values of Renf and Prnf are to be

calculated using the properties of nanofluids as developed &

presented in sections 2 & 3. The variation of Nusselt number with

respect to Reynolds number, volume fraction, L/D ratio are

presented in graphical form as shown in Fig. 4.2 (a)–(d).

114

Figure 4.1(a) Laminar flow heat transfer comparison

115

v

Figure 4.1(b) Laminar flow heat transfer comparison for different L/D values

116

Figure 4.2 (a) Nusselt number vs Reynolds Number in Laminar flow

117

Figure 4.2(b) Effect of Volume fraction in Laminar flow

118

Figure 4.2(c) Variation of Nusselt number with L /D in Laminar flow

119

Figure 4.2(d) Heat transfer in laminar flow in different Nanofluids

120

4.2 Turbulent Flow

4.2.1 Model 1

S B Maiga et.al. [120] presented the numerical study of fully

developed turbulent flow of Al2O3 + water nanofluid in circular

tube at uniform heat flux of 50 W/cm2. The classical K- model

was used for turbulence modeling and a new correlation for

Nusselt number was derived as

35.071.0

PrRe085.0 ffnfNu (4.6)

for 54 10*5Re10;9.13Pr6.6

Pak and Cho[44] and Li and Xuan [110] has developed

correlations of a form similar to that of well known Dittus-Boelter

formula to characterize nanofluids heat transfer. Pak & Cho[44]

and Xuan & Li [110] proposed correlation treating it as a single

phase fluid for the calculation of nanofluid Nusselt number as

given in Eqs.(4.7) & (4.8). In these correlations the Reynolds

Number and Prandtl numbers are calculated by considering the

base fluid properties which will give estimated under results

compared with experimental values.

5.08.0 )(Pr)(Re021.0 ffnfNu (4.7)

4.09238.06886.0 PrRePrRe6286.71059.0

001.0

ff

p

ffnfD

dNu

4.8)

121

Hence the above considered single phase fluid approach is adopted

with a modification that instead of base fluid properties the

thermophysical properties of the Nanofluid itself are considered

while evaluating the non dimensional numbers.

Thus the Nusselt number is defined and evaluated as

nf

nfnf

k

DhNu 4.9)

The heat transfer coefficient of turbulent flow through circular tube

can be developed more appropriately in the following form

4.08.0 )(Pr)(Re nfnfnf aNu (4.10)

Where Renf and Prnf as defined as follows:

nf

nf

nf

Du

Re (4.11)

nf

nfnf

nfk

C Pr (4.12)

The Nusselt number data of the nanofluids obtained from the

experimental works of Pak et al.[44] and Xuan and Li[110] is

subjected to non-linear regression analysis and the constant „a‟ is

obtained as

0.0256 for Al2O3 + H2O

0.027 for Cu + H2O

122

Thus the correlations for calculation of Nusselt number are

developed as follows with an average deviation of 5% and standard

deviation of 6.4%

For circular tube

4.08.0 )(Pr)(Re0256.0 nfnfnfNu for Al2O3+ H2O (4.13)

4.08.0 )(Pr)(Re027.0 nfnfnfNu for Cu + H2O (4.14)

The above equation takes care of diameter of the nanoparticle,

concentration and temperature effects. The correlations give good

agreement with the experimental results as shown in the Fig.4.3

(a&b) & 4.4.(a&b) The values of Renf and Prnf are to be calculated

using the properties of nanofluids as developed & presented in

eqs.(2.1),(2.2),(2.8) & (3.11).

123

Figure 4.3(a) Turbulent heat transfer data comparison for =1.34

124

Figure 4.3(b) Turbulent heat transfer data comparison for =2.78

125

Figure 4.4(a) Turbulent heat transfer data comparison =1%

126

Figure 4.4(b) Turbulent heat transfer data comparison for =1.5%

127

4.2.2. Model 2

The earlier developed correlation is valid only when >0.01 and

purely base on experimental data. To overcome this limitation

another model is proposed in this section which will not only be

helpful to evaluate heat transfer rate but also for friction factor

under different conditions. The model was developed by

considering the experimental data of Xuan and Li [110] with

nanoparticles of copper and data of Pak & Cho [44] for Aluminum

in water medium as coolant. The data of Xuan and Li [110] and

Pak &Cho [44] is recast into the dimensionless parameters, Nu and

Re with the properties of the base fluid water being employed in

the computations.

A single correlation for both Cu + H2O & Al2O3 + H2O nanofluids is

developed and comparison with the experimental data are shown

in Fig. 4.5 with the predictions from the correlation eq (4.15) with

average deviation of 2.9% and standard deviation of 3.5%.

323.03/1

f

867.0

fnf )1(PrRe013.0Nu (4.15)

128

Figure 4.5 Validation of correlation with Experimental data

129

However, the variation of physical parameters of the base fluid

with as a variable is as follows.

For Al2O3 dispersion in the base fluid water, the property relations

are as follows:

Density ( Pak & Cho [44]):

fpnf )1( (4.16)

Dynamic Viscosity ( Pak & Cho [44]):

)9.53311.391( 2

fnf (4.17)

Thermal conductivity ( Pak & Cho [44]):

)772.4815.11(kk 2

fnf (4.18)

Specific heat ( Pak & Cho [44]):

nf

ffpp

nf

C)1(CC

(4.19)

Similarly for Cu dispersion in the base fluid water, the property

relations Density and Specific heat is calculated by using the above

eqs (4.16) & (4.19) and

Dynamic Viscosity (Chen et al. [45]):

)72.468645.3995.0( 2

fnf (4.20)

Thermal conductivity ( Chen et al. [45]):

)4.399065.01(kk 2

fnf (4.21)

130

4.3 Evaluation of eddy momentum diffusivity with

nanoparticle dispersion in the base fluid

As observed by Buongiorno [175] treating the dispersion of

nanoparticles as a continuum medium following Newtonian fluid

properties, the shear stress distribution for turbulent flow

conditions can be assumed by Boussinesq approximation for

turbulent flows.

In addition, the experimental observation is that the presence of

nanoparticles does not profoundly alter the momentum

characteristics. Though dispersion of nanoparticles in the medium

might significantly alter the viscous properties, it is an observed

fact that for viscous or non-viscous turbulent flows, the Darcy‟s

friction factor dependence on the Reynolds number is uniquely

defined by the relationship f = 0.0791/Re0.25. In other words, in the

estimation of friction losses, the low volumetric percentage

concentration of the nanoparticles can be totally ignored. Hence, it

can be treated totally as Newtonian base fluid ignoring the

presence of nanoparticles in the medium. The basis for further

development of the model is that the universal nature of the

velocity profile cannot be influenced by the volumetric percentage

concentration of the nanoparticles in the base medium. However,

the inclusion of nanoparticles would affect the temperature profiles

adjacent to the wall.

131

Thus in view of (Buongiorno [175])

y

u1

f

mf

(4.22)

Thus, the above equation can be converted into differential form

with assumption that the shear distribution is linear over the tube

cross section.

R

y1

w

(4.23)

Further it can be written in dimensionless form as follows:

f

m1

R

y1

y

u

(4.24)

The boundary condition is that at y+=0, u+=0

The eddy momentum parameter f

m

can be defined as per the

existing experiences for turbulent forced momentum exchange

mechanisms and we can employ the following Van Driest

expression [176].

2f

m )A/yexp(1By

(4.25)

Where B is considered as function of transitional Reynolds number

and A+=26. Its valve is determined as a dependent function of

transitional Reynolds number.

132

4.3.1 Numerical Procedure to evaluate B in Equation (4.25)

For the present case of internal flows in a tube, the value of B can

be estimated theoretically following the numerical procedure as

outlined in steps:

1. Obtain u+=f(y+) by solving equation (4.25) for an assumed

value of R+ and B.

2. Calculate Reynolds number from the formula

dy

R

y1u

4

ReR

0

3. Calculate

2

calRe

R8f

4. Calculate 25.0Blasius

Re

0791.0f

5. Compare fBlasius with fCal and check whether

01.0100f

ff

Blasius

Blasiuscal

6. If the criterion of accuracy is satisfied, the assumed value of

B is accepted as correct approximation.

Thus the numerical procedure the value of B is obtained close to

0.5 for 8000< Re< 80,000. The theoretical prediction of friction

coefficient is valid with Blasius relationship as shown in Fig.4.6.

Hence the momentum eddy diffusivity for nanoparticle inclusion

133

can be obtained by substituting the values of B=0.5 and A+=26 in

eq (4.25) with reasonable accuracy as

2f

m )A/yexp(1y5.0

(4.26)

4.4 Evaluation of Eddy thermal diffusivity

For turbulent convective heat transfer without nanoparticle

inclusion, the following expression is taken for heat transfer

calculations ( Incropera et al. [167]):

1PrforPr ff

f

m

f

H

(4.27)

In the present model it is thought that the presence of

nanoparticles will greatly influence the thermal eddy diffusivity and

such a modification can be considered by modification of the term

as follows

)(Pr

wheref

f

m

f

H (4.28)

Thus, hereafter the analysis is devoted to evaluate the functional

relationship between ξ and . The reason for such a postulation is

based on the conclusion by Buongiorno [175] that the thermal

penetration thickness is greatly influenced by the particles leading

to enhancement of heat transfer in the presence of nanoparticles.

134

For thermally developed flow conditions, the temperature profile

over the cross section can be obtained from the differential

equation

01

y

T

y f

H

(4.29)

As per the thermal model, the equation can be written as follows in

dimensionless form

0(Pr)1

y

T

y f

m

(4.30)

cw

w

TT

TTTwhere

The boundary conditions are as follows:

At y+=0, T+=0

At y+=R+, T+=1

4.4.1 Numerical procedure to evaluate ξ

This program is to be conjointly worked out with the earlier one in

which the velocity profiles are already obtained as u+ = F [y+, R+]

where R+=F [Re]

1. For the known values of R+ and Re, evaluate the temperature

profiles i.e. T+ from equation (4.30) with an assumed value of

ξ for given value of volumetric concentration .

135

2. For given value of R+ and from the computer run satisfying

the boundary conditions equation (4.30) , obtain 0

yy

T

3. Calculate the theoretical value of Nusselt number from the

equation :

bw

cw

yTTT

TT

y

TRNu

02 (4.31)

where

R

R

cw

bw

dyR

yu

dyR

yuT

TT

TT

0

0

1

1

4. Compare the theoretical value of NuT from the equation

(4.31) with the value calculated from correlation for

nanoparticle in the medium i.e. equation (4.15).

5. If the result of Nusselt number in step 3 is same as the one

in step 4 , then the assumed value of ξ, the exponent to

Prandtl number is considered as valid magnitude.

6. If the disagreement in the NuT and Nu correlation is not

within the limits of prescribed accuracy, then a proper

iteration technique is employed to achieve convergence.

136

Figure 4.6 Variation of friction coefficient with Reynolds Number

137

4.5 Velocity & Temperature profiles

The universality in velocity profiles for the ranges of 0.3% < < 3%

is uniquely depicted in Fig. 4.7 and the profiles are influenced by

the Reynolds number irrespective of the fact whether the base fluid

is homogenized with the nanoparticles or not. This observation can

be considered valid at least for the ranges of the concentrations of

nanoparticles investigated.

Typical plots depicting the temperature profiles near to the wall i.e.

12.0

R

y are shown plotted in Fig. 4.8. Evidently, the wall

temperature gradients are found to vary steeply with concentration

of the nanoparticles.

The profound influence of the concentration is more clearly

depicted for different values of versus Re (see Fig. (4.9).The

results of the plot can be comprehensively correlated with the help

of the equation as follows

0018.0338.0

0Re)1(102.0

ydy

dT (4.32)

The wall temperature gradients are shown as functions of Reynolds

number for different values of in Fig. 4.9. The equation obtained

by regression analysis for 72 points from this analysis is found to

possess the following limits of accuracy of a standard deviation of

0.73% and an average deviation of 0.59%.

138

Figure 4.7 Variation of velocity profile with Reynolds number

139

Figure 4.8 Effect of on temperature Profiles for different Re

140

Figure 4.9 Variation of Temperature gradients at the wall with Re

141

Figure 4.10 Variation of exponent to Prandtl number with volume fraction

142

4.6 Thermal Eddy Diffusivity

In Fig. 4.10, the variation of ξ, the exponent to Prandtl number of

the base fluid is shown plotted as a function of Re. The plot reveals

that ξ assumes asymptotic values for high Reynolds number. The

data of ξ from the theory can be represented by the relationship.

018.034.0 Re)1(883.0 (4.33)

However, from the nature of independency of ξ with Re (for high

magnitudes), ξ can be found to be purely a function of and

independent of Re. The asymptotic expression for the evaluation of

ξ in the eddy thermal diffusivity is given by the relationship as

follows:

2)1(036.0)1(372.0663.0 (4.34)

The magnitude of f

H

is found to be substantially affected with the

concentration in the base fluid (see Figure 4.11). The reason for

such increase in the magnitude of eddy thermal diffusivity is due

to enhanced mixing effects resulting from the Brownian motion of

the nanoparticles in the medium. The inclusion of nanoparticles

leads to profound change in thermal conductivities of the

dispersion relative to the base fluid.

143

Figure 4.11 Variation of eddy thermal diffusivity y+/ R+ with

Volume fraction as a parameter

144

4.7 Conclusion

This chapter is mainly focused on convective heat transfer of

nanofluids flowing inside a circular cross section tube.

Correlations to predict Nusselt number for nanofluids both in

laminar and turbulent have been developed separately. In laminar

flow to calculate Nusselt number for nanofluids a modified Seider-

Tate correlation as been developed by considering nanofluids

properties as

333.0)Pr(Re98.1L

DNu nfnfnf

The Nusselt number thus calculated by considering the above

correlation for nanofluids in Laminar flow found to be an average

deviation of 6% and standard deviation of 7.4 % with the

experimental data.

In Turbulent flow two models are developed to calculate Nusselt

number for Al2O3 + water and Cu + water Nanofluids. In first

model, correlation is developed by considering single phase fluid

approach. The correlation thus obtained is

4.08.0 )(Pr)(Re0256.0 nfnfnfNu for Al2O3+ H2O

4.08.0 )(Pr)(Re027.0 nfnfnfNu for Cu + H2O

with an average deviation of 5% and standard deviation of 6.4%

with the experimental data. However, this correlation is valid when

>0.01 as thermal conductivity used valid for >0.01.

145

In second model, a combined correlation for Al2O3 + water and Cu

+ water Nanofluids is developed by considering base fluid

properties. The correlations thus obtained is

323.03/1

f

867.0

fnf )1(PrRe013.0Nu

with average deviation of 2.9% and standard deviation of 3.5% with

experimental data. The correlations can be valid for 0 0.03.

Further, velocity and temperature profiles for nanofluids are

developed by considering turbulent forced momentum exchange

Van Driest expression below as been used.

2f

m )A/yexp(1By

Where B is considered as function of transitional Reynolds number

and A+=26 is calculated by numerical method and found as

0.5.Therefore the momentum and thermal eddy diffusivities are

The Momentum eddy diffusivity

2f

m )A/yexp(1y5.0

, where A+=26.

The Thermal eddy diffusivity

1PrforPr ff

f

m

f

H

018.034.0 Re)1(883.0