115

CHAPTER-5

CONVECTIVE HEAT

TRANSFER WITH GRAPHENE

NANOFLUIDS

116

Table of Contents

CHAPTER-5 CONVECTIVE HEAT TRANSFER WITH GRAPHENE

NANOFLUIDS 115 - 154

5.1 Introduction 117

5.2 Experimental studies on convective heat transfer 118

5.3 Thermo physical properties of nanofluids 123

5.3.1 Specific heat 123

5.3.2 Density 123

5.3.3 Base fluid viscosity 124

5.4 Heat transfer coefficient augmentation 128

5.5 Effect of thermal conductivity on fully developed heat transfer 136

5.6 Study on modelling of nanofluid flow 140

5.7 Governing equations 140

5.7.1 Steady wall temperature boundary condition 143

5.7.2 Constant wall heat flux boundary condition 144

5.8 Heat transfer coefficient improvement 149

5.9 Further study 151

5.9.1 Local Nusselt number 151

117

CHAPTER - 5

CONVECTIVE HEAT TRANSFER WITH GRAPHENE NANOFLUID

5.1 Introduction

Nanofluids are capable heat conveying fluids owing to their high thermal

conductivity. Thermal conductivity of nanofluids is already discussed item wise in

Chapter 4. In a bid to employ nanofluids in practical applications, their convective

heat transfer characteristics are required to be understood. For that reason, an

investigation is carried out on the convective heat transfer performance of nanofluids.

In this section, the deliberation is aimed on forced convection of nanofluids inside

circular tubes. For the study, laminar flow of the nanofluid inside a straight circular

tube under both constant wall temperature and constant wall heat flux boundary

conditions is considered. In addition to this, the effect of thermal conductivity of

nanofluids on fully developed flow heat transfer is studied.

There are several models in the literature related to the convective heat

transfer with nanofluids. In this literature survey, the conversation is focused on

Forced convection of Graphene nanofluids in circular tubes. The examination of

nanofluid convective heat transfer is investigated by comparing the combined results

with the experimental data in the literature.

118

1.2 Experimental Studies on convective heat transfer

Experimental Set up

Fig: 5.1: experimental set up to measure heat transfer

Fig: 5.2: Schematic view of experimental set up

Fig; 5.3: Insulated pipe with Nichrome heater inside

119

Fig: 5.4: Asbestos Insulated pipe

Fig: 5.5: Thermo generator

Fig: 5.6: DAQ system with 40 segments (KEITHLEY)

120

Fig: 5.7: DAQ connectors segments

NOMENCLATURE SPECIFICATIONS

Segments 40

Rating 50,60,400 Hz

Voltage 28 VA MAX

Temperature 200ºC max

Simulation soft ware ARGUS

DAQ-Manufacturer KEITHLEY

5.2.1 DESCRIPTION:

The experimental set up to measure the convective heat transfer

characteristics of the Graphene nanofluids is shown in the figure 5.1 It is planned to

conduct the experiment in two conditions at constant wall heat flux and constant wall

temperature, by experience and the literature survey it is impossible to maintain the

constant wall temperature and constant wall heat flux for the same length of the pipe

hence the pipe length and dimensions are changed for two different situations for

steady wall temperature boundary conditions the tube dia is 5mm and the length is

1m and for steady wall heat flux boundary condition the tube dia is 4.57mm and

length is 2m to achieve the same hydro dynamic boundary layer and thermal

boundary layer for the two situations under this arrangement, Apart from this the

remaining apparatus used in the experimental set up is a data acquisition system in

121

built with 40 segments(KEITHLEY) to collect the data, the copper-constantan

thermo couples (-50ºc to 400ºc) are arranged at the entrance and exit of the pipe and

also in between at interval of length, The thermo couples are arranged to measure the

temperature of the fluid flow, in addition to this a syringe pump is used to pump the

fluid into the tubes, The data given by the DAQ system is connected to a computer

run by ARGUS soft ware, To check the parallel arrangement of the pipe bulls eye test

(Spirit level) is conducted, “Shah correlation” is used to measure the constant wall

heat flux boundary condition heat transfer enhancement ratio, Where as “Sieder-Tate

correlation” is employed for constant wall temperature heat enhancement ratio,

Generally the heat transfer enhancement is caused by two methods i.e., Thermal

dispersion and Thermal diffusion, In the first technique, it is supposed that the

occurrence of nanoparticles in the stream influence the heat transfer only through the

distorted thermo physical properties. Then the governing equations of fluid flow and

heat convey for a conventional pure fluid can be used for the study of nanofluids by

substituting the relevant thermo physical properties. This emphasizes that the

established correlations of convective heat transfer for pure fluids can be utilized for

nanofluids. In the second technique, the nanofluid is considered as a single segment

fluid but the extra heat transfer improvement attained with nanofluids is measured by

modelling the diffusion phenomenon. It was identified that thermal dispersion

happens in nanofluid flow due to the irregular motion of nanoparticles. By allowing

the fact that this irregular motion generates small perturbations in temperature and

velocity, the effective thermal conductivity in the governing energy equation takes the

form.

,eff nf dk k k (5.1)

122

Where knf

is the thermal conductivity of the nanofluid. kd

is scattered thermal

conductivity and was proposed to be measured by using the following expression.

( ) ,d p nf x p ok c c u d r (5.2)

In this expression, cp

is specific heat, ρ is density, Ø is particle volume fraction, ux

is

axial velocity, r0is tube radius and d

p is nanoparticle diameter. C is an empirical

constant that should be calculated by matching experimental data. Subscript nf

represents the nanofluid. In one more study, based on this thermal diffusion model,

following correlation was proposed by Li and Xuan [73] for the prediction of Nusselt

number.

31 2 0.4

1 2(1 )Re Pr ,mm m

nf d nf nfNu c c pe (5.3)

The correlation was developed for forced convection of nanofluid flow inside circular

tubes. Pedis particle Peclet number, which is defined as

,m p

d

nf

u dPe

(5.4)

αnf

is nanofluid thermal diffusivity and um

is average flow velocity. Renf

and Prnf

are

the conventional Reynolds and Prandtl numbers; however the thermo physical

properties of the nanofluid have to be used in the connected calculations. c1, c

2, m

1, m

2

and m3are empirical constants that should be determined by using experimental data.

For the appropriate examination of nanofluid heat transfer, precise evaluation

of thermo physical properties of the nanofluid is significant. In order to emphasis the

significance of this problem, Combined equations are continuity equation for

nanofluid, continuity equation for nanoparticles, energy equation for nanofluid and

momentum equation for nanofluid. The variation in the fully developed velocity

123

profile in a circular tube due to the variation of viscosity in radial direction enhances

convective heat transfer of nanofluids.

Throughout the analysis, fully developed condition was studied for both

laminar and turbulent flow cases and nanofluids were taken as single phase fluids.

Several expressions used in the literature for the purpose of dynamic viscosity,

specific heat and thermal conductivity were accessed and difference between them

was illustrated.

5.3. Thermo physical Properties of Nanofluids

In the investigation of convective heat transfer of nanofluids, precise

determination of the thermo physical properties is a major issue. Measurements of

specific heat and density of nanofluids are comparatively straight forward, However

when it comes to thermal conductivity and viscosity, there is considerable Dis

agreement in both theoretical models and experimental results presented in the

literature.

5.3.1. Specific Heat

There are two expressions for evaluating the specific heat of nanofluids [10, 72]:

, , ,(1 ) ,p nf p p p fc c c (5.5)

( c ) ( ) (1 )( ) ,p nf p p p fc c (5.6)

It is said that Eq. (5.6) is theoretically more dependable since specific heat is a mass

specific quantity whose impression depends on the density of the ingredients of a

mixture.

5.3.2. Density

Density of nanofluids can be calculated by using the following expression [10].

(1 ) ,nf p f (5.7)

124

Here, Ø is particle volume fraction and subscripts nf, p and f correspond to nanofluid,

particle and base fluid, respectively. Pak and Cho [10] experimentally proved that Eq.

(5.5) is a precise expression for concluding the density of nanofluids.

5.3.3. Base fluid viscosity

Nanofluid viscosity is a significant constraint for practical applications since

it precisely influences the pressure drop in forced convection. Therefore, for extensive

use of nanofluids in practice, the amount of viscosity increase for nanofluids in

comparison to pure fluids must be carefully observed. In order to measure the

viscosity of base fluid and pure fluid Brookfield viscometer is used widely; calibrated

red wood viscometer has been used for measuring the viscosity of base fluid as well

as Graphene nanofluids,

Fig.5.8: Red wood Viscometer to measure viscosity

125

Table: 5.1 Temperature Vs Viscosity Pure Fluid and Base Fluid

S.No Temperature

(ºc )

Viscosity of

Water (CP)

Viscosity of Base

Fluid(CP)

1 25 0.985 1.43

2 33 0.843 1.32

3 38 0.732 1.26

4 42 0.652 1.2

5 52 0.534 1.001

6 56 0.523 0.96

Fig; 5.9: Graph of Temperature Vs Viscosity

To calculate the Graphene (Cn)/Water + EG nanofluid viscosity one should

use equation of Nguyen et al. [77]

2(1 2.5 150 ) ,nf f (5.10)

MODEL CALCULATION FOR VISCOSITY OF GRAPHENE NANOFLUID:

μnf = (1+2.5 x 0.04+150 x (0.04)2) x1.43=1.9162

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

20 25 30 35 40 45 50 55 60

visc

osi

ty(c

P)

Temperature (ºC)

viscosity vs Temperature

water

70% water+30%EG

126

Table: 5.2 Temperature Vs Viscosity of Graphene nanofluid

S.No Temperature

(ºc )

Viscosity of Base

Fluid(CP)

Viscosity of Graphene

nanofluid(Cp)

1 25 1.43 1.9162

2 33 1.32 1.7688

3 38 1.26 1.6884

4 42 1.2 1.608

5 52 1.001 1.3413

6 56 0.96 1.2864

Fig; 5.10: Graph of Temperature Vs Viscosity

5.3.3.1. Experimental Studies regarding viscosity

Compared to the argument of thermal conductivity, there is a considerable

Inconsistency in the experimental results concerning the viscosity of nanofluids.

However, the common inclination is that the rise in the viscosity by the addition of

nanoparticles to the conventional base fluid is important. The viscosity of Graphene

0

0.5

1

1.5

2

2.5

20 30 40 50 60

Vis

cosi

ty(c

p)

Temperature(ºc)

Base fluid

Graphene nano fluid

127

(Cn) /Water + ethylene glycol nanofluid is determined at room temperature. For the

fragment volume fraction of 3.5%, 40% increment in viscosity was noticed.

It is recognized that nanofluid viscosity depends on many points such as;

particle size, particle volume fraction, extent of clustering and temperature. Raising

particle volume fraction improves viscosity and this was supported by many studies

[63, 77, 78, 79].

Einstein [1906] suggested an expression for calculating the dynamic viscosity

of mitigate suspensions that hold spherical particles. In this case, the relations among

the particles are deserted. The linked expression is as follows.

(1 2.5 ) ,nf f (5.8)

Brinkman [80] proposed the following equation.

2.5

1,

(1 )nf f

(5.9)

In particular studies, the exchanges between particles were taken into consideration.

These chances widened the applicability choice of the models in terms of particle

volume fraction. Nguyen et al. [77] suggested viscosity equation for Al2O

3/water +

EG nanofluids are:

2(1 2.5 150 ) ,nf f (5.10)

The preceding relationship is suitable for the nanofluids with a particle size of 20-30

nm. Tentative studies show that nanoparticle size is a significant constraint that

influences the viscosity of nanofluids. Conversely at current, it is complicated to

attain a dependable set of tentative data for nanofluids that covers a wide range of

particle size and particle volume fraction. Therefore, for the time being, Eq. (5.10) can

be used as estimation for Graphene (Cn)/Water+ Ethyl glycol nanofluids with

128

different volumetric ratios. When it comes to temperature dependence of viscosity,

Nguyen et al. [77] concluded that for particle volume fractions below 4%, viscosity

enrichment ratio (viscosity of nanofluid divided by the viscosity of base fluid) does

not considerably change with temperature.

Precise study of convective heat transfer of nanofluids is significant for

reasonable usage of nanofluids in thermal devices. The methods propose that the

study can be made by considering the nanofluid flow as single phase flow because the

nanoparticles are very small and they mix easily [84]. The heat transfer

accomplishment of nanofluids can be elaborated by using traditional correlations

developed for the calculation of Nusselt number for the flow of pure base fluids. In

the calculations, one should replace the thermo physical properties of the nanofluid to

the associated expressions. This line of attack is a very sensible way of measuring

convective heat transfer of nanofluids [80, 81]. The afore said methods of using

conventional correlations was proposed by Shah [80] and Sieder-Tate [81] to

calculate constant wall temperature and constant wall heat flux temperature is

deliberated.

5.4 Heat Transfer Coefficient Augmentation

The speculative study of heat transfer obtained with nanofluids is presented in

two sections, according to the type of the boundary condition, namely; constant wall

heat flux boundary condition and constant wall temperature boundary condition.

5.4.1 Constant Wall Heat Flux Boundary Condition

The more frequently used empirical correlations for the calculation of

convective heat transfer in laminar flow method inside circular tubes is the “Shah

129

correlation [80]. The linked expression for the calculation of local Nusselt number is

as follows.

1/3 5

* *1.302 1 5 10xNu x forx

1/3 5 3

* *1.302 0.5 5 10 1.5 10 ,x for x (5.11)

410.506 3

* *4.364 0.263 1.5 10exx e forx

X* is defined as follows

*

/ / 1,

RePr x

x d x dx

Pe GZ (5.12)

Pr, Re and Pe are Prandtl number, Reynolds number and Peclet number, respectively.

d is tube diameter and x is axial position. At this spot, it must be prominent that for

the same tube diameter and same flow velocity, Pef and Pe

nf are unlike.

.

.

,nf f f nf p nf

f nf nf f p f

Pe k c

Pe k c

(5.13)

Improvement in density and specific heat increases Penf

where as thermal conductivity

enhances results in a diminishing in Penf

. Therefore, when the pure working fluid in a

system is replaced with a nanofluid, the flow velocity should be adapted in order to

activate the system at the same Peclet number. In this section of the examination, heat

transfer development attained with nanofluids are calculated by comparing the linked

results with the conventional pure fluid case for the same flow velocity and tube

diameter, so that the effect of the change in Peclet number is also examined In order

to study the heat transfer augmentation obtained with nanofluids for the same flow

velocity, axial position and tube diameter, by using Eq. (5.11) and obtain the local

Nusselt number enhancement ratio.

130

1/3

, *,

1/3

, *,

1.302 1

1.302 1

x nf nf

x f f

Nu x

Nu x

for

5

* 5 10 ,x

1/3

*,

1/3

*,

1.302 0.5

1.302 0.5

nf

f

x

x

for

5 3

*5 10 1.5 10 ,x (5.14)

*,

*,

410.506

*,

410.506

*,

4.364 0.263

4.364 0.263

nf

f

x

nf

x

f

x e

x e

for

3

* 1.5 10x ,

From the above equations it is evident that the value of the term x*,nf

increases with an increase in the density and specific heat of the nanofluid, which

subsequently increases local Nusselt number enhancement ratio. On the other hand,

enhancement in the thermal conductivity of the nanofluid reduces values of the terms

with x*,nf

and the local Nusselt number augmentation ratio reduces as a corollary. By

integrating the numerator and denominator of Eq. (5.14) along the tube and attain the

average Nusselt number augmentation ratio for a specialized case. The values of

thermo physical properties on average Nusselt number augmentation ratio are

qualitatively the same as the local Nusselt number case. For a pure fluid, average heat

transfer coefficient can be defined as follows.

,f f

f

Nu kh

d (5.15)

For a nanofluid, average heat transfer coefficient becomes the following expression.

,nf nf

nf

Nu kh

d (5.16)

By using Eqs (5.15, 5.16), one can gain the average heat transfer coefficient

enrichment ratio as follows.

131

,nf nf nf

f f f

h k Nu

h k Nu (5.17)

Eq. (5.14) can be used to execute the related integrations to arrive at average

Nusselt number augmentation ratio through Eq. (5.17). Although the enhancement in

thermal conductivity of the nanofluid reduces Nusselt number augmentation ratio, due

to the multiplicative result of thermal conductivity in the characterization of heat

transfer coefficient, it should be accepted that the growth in the thermal conductivity

of the nanofluid increases heat transfer coefficient augmentation ratio.

The present work is to study the convective heat transfer of Graphene

(Cn)/water + EG nanofluids for different particle volume fractions between 0.5% and

2%. Reynolds number was varied between 800 and 2200. At the steady wall heat flux

boundary condition, the heat transfer development of nanofluids improves. The

average Nusselt number for the forced convection of nanofluids inside circular tubes

is estimated using the thermal diffusion model as given below.

31 2 0.4

1 2(1 )Re Pr ,mm m

nf d nf nfNu c c Pe (5.18)

Pedis particle Peclet number which is defined as

Pe ,m p

d

nf

u d

(5.19)

um

is mean flow velocity. Renf

and Prnf

are the Reynolds and Prandtl numbers of the

nanofluids, but the thermo physical properties of the nanofluid must be used. For

steady wall heat flux boundary condition, Li and Xuan [73] suggested the empirical

constants c1, c

2, m

1, m

2 and m

3depend on their tentative study and combined

expression is the following.

132

0.754 0.218 0.333 0.40.4328(1 11.285 )Re Pr ,nf d nf nfNu Pe (5.20)

It can be believed that Eq. (5.20) is applicable in the series of the experimental data

[73]; 800 < Renf

< 2300 and 0.5% < φ < 2%. It might also be distinguished that tube

diameter is 1 cm and tube length is 1 m in the experiments.

MODEL CALCULATIONS:

Table: 5.3: Heat transfer coefficient enhancement ratio values with Re for

different volume fractions of Graphene (Cn)/ water + EG

S.No %Volume

added Ref hnf/hf

1

0.50

800

1.09

1.00 1.16

1.50 1.28

2.00 1.37

2

0.50

1200

1.1

1.00 1.179

1.50 1.3

2.00 1.39

3

0.50

1300

1.12

1.00 1.18

1.50 1.32

2.00 1.4

4

0.50

1500

1.13

1.00 1.186

1.50 1.34

2.00 1.43

5

0.50

1800

1.16

1.00 1.198

1.50 1.35

2.00 1.46

6

0.50

2000

1.165

1.00 1.22

1.50 1.355

2.00 1.47

7

0.50

2200

1.17

1.00 1.25

1.50 1.36

2.00 1.475

133

Figure: 5.11 Variation of average heat transfer coefficient enhancement

ratio with Reynolds number for different particle volume fractions

of the Graphene (Cn) 20nm/water+EG nanofluid.

5.4.2 Constant Wall Temperature Boundary Condition

The method followed in the earlier for constant wall heat flux boundary

condition is recurring in this segment for steady wall temperature boundary condition.

To conclude the average Nusselt number, one can use the classical Sieder-Tate

correlation [81].

0.14

0.8 0.3330.027 Re Pr

0.7 Pr 16700;Re 10,000 :

60

bD

w

D

Nu

L

D

(5.21)

By integrating the above equation

1/3

1.86 ,b

w

dNu Pe

L

(5.22)

Here L is tube length. μw

is dynamic viscosity at the wall temperature where as μbis

dynamic viscosity at the bulk mean temperature which is defined as:

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

800 1000 1200 1400 1600 1800 2000 2200

hn

f/h

f

Ref

0.50%

1.00%

1.50%

2.00%

134

,2

i ob

T TT

(5.23)

Ti

is inlet temperature and To

is outlet temperature. Neglecting the dissimilarity of

viscosity augmentation ratio of the nanofluid (μnf

/ μf) with temperature and using

Sieder-Tate correlation (Eq. 5.22) for a pure base fluid and nanofluid, following

equation can be obtained.

1/3 1/3

,

,

,nf nf f nf p nf

f f nf f p f

Nu Pe k c

Nu Pe k c

(5.24)

Using Eq. (5.17), heat transfer coefficient augmentation ratio becomes the following.

1/3 2/3

,

,

,nf nf p nf nf

f f p f f

h c k

h c k

(5.25)

By exploring this equation, it would be seen that the improvements in the thermo

physical properties of the nanofluid; specific heat, density and thermal conductivity,

develops the heat transfer coefficient. It should be eminent that the effect of thermal

conductivity development is more marked when compared to density and specific

heat. Their analysis part consists of a straight circular tube with an inner diameter of 5

mm and length of 1 m. The nanoparticles used in the nanofluid have a diameter of 20

nm. Peclet number was limited between 2500 and 6500 and the heat transfer

measurements were considered for different nanofluids with particle volumes

fractions changing between 1.0% and 2.5%.The related under forecast shows that

there should be added augmentation mechanisms associated to the convective heat

transfer of nanofluids which extended to improve the heat transport.

135

Table: 5.4: Heat transfer coefficient enhancement ratio with Pe number for

different particle volume fractions

S.No %Volume added Pef hnf/hf

1

1.0

2500

1.12

1.5 1.15

2.0 1.2

2.5 1.25

2

1.0

3200

1.13

1.5 1.16

2.0 1.22

2.5 1.26

3

1.0

3600

1.14

1.5 1.175

2.0 1.23

2.5 1.26

4

1.0

3800

1.15

1.5 1.178

2.0 1.235

2.5 1.27

5

1.0

4000

1.16

1.5 1.18

2.0 1.238

2.5 1.275

6

1.0

4100

1.186

1.5 1.186

2.0 1.239

2.5 1.283

7

1.0

4700

1.17

1.5 1.195

2.0 1.25

2.5 1.29

8

1.0

5000

1.18

1.5 1.22

2.0 1.27

2.5 1.3

9

1.0

5100

1.185

1.5 1.23

2.0 1.3

2.5 1.32

10

1.0

5950

1.19

1.5 1.24

2.0 1.32

2.5 1.35

11

1.0

6300

1.2

1.5 1.25

2.0 1.33

2.5 1.38

136

Figure 5.12: Variation of average heat transfer coefficient enhancement

ratio with Peclet number for different particle volume fractions

Of the Graphene (Cn)/water+EG nanofluid

5.5 Effect of Thermal Conductivity on Fully Developed Heat Transfer

In this supplementary segment, effect of thermal conductivity of nanofluids on

fully developed heat transfer coefficient values is studied. Parallel to the examination

in the previous sections, the nanofluid is treated as a pure fluid with improved thermo

physical properties. While this approach is shown to underrate the tentative results in

the preceding sections, yet it will be used to get a good understanding about the

consequence of thermal conductivity on heat transfer due to its plainness. As a

consequence of the Graetz solution for parabolic velocity profile under steady wall

temperature and constant wall heat flux boundary conditions. Nusselt number can be

obtained as follows [82]:

2

2

0

20

,

2

n

n

n

x nx

nnf

n n

A eh d

NuAk

e

For constant wall temperature (5.26)

1

1.1

1.2

1.3

1.4

1.5

2500 3500 4500 5500 6500

hn

f/h

f

Pef

1.00%

1.50%

2.00%

2.50%

137

2 1

41

11 1,

48 2

n

xx

nnf n n

h d eNu

k A

For constant wall heat flux (5.27)

Where ξ= x/ (ro Pe).

These terms are justifiable for the thermal admission region of a circular pipe with

hydro dynamically fully developed laminar nanofluid flow under the assumption of

treating nanofluids as pure fluids. Below the fully developed conditions, Nusselt

number becomes:

2 2(2.7043644)3.657,

2 2

ofdNu

for constant wall temperature, (5.28)

4.364,fdNu for constant wall heat flux. (5.29)

In organize to emphasize the importance of the exact strength of the thermal

conductivity of nanofluids, heat transfer coefficient of the laminar flow of Graphene

(Cn)/water+ ethyl glycol nanofluid inside a circular tube is studied by using the

exceeding mentioned asymptotic values of Nusselt number. Nanoparticles are

supposed to be spherical with a diameter of 20 nm. Different temperatures are

considered in the study at room temperature. Flow is both hydro dynamically and

thermally fully developed. Tube diameter is selected as 1 cm. For the determination of

the thermal conductivity of the nanofluids at room temperature,

Hamilton and Crosser [18] model (Eqs. 2, 3) is utilize and fully developed

heat transfer coefficients are resolute. A sample computation of this study is available

in Appendix. For the 4% vol Graphene (Cn)/water+EG nanofluid.

In Table 5.1, results of 1 and 4% Vol. Graphene (Cn)/water+ ethyl glycol

nanofluids are compared with pure water. As seen from the table, due to the

description of the Nusselt number (Nu=hd/k), the improvement in thermal

138

conductivity by the use of nanofluids openly results in the increment in heat transfer

coefficient. Thermal conductivity of the nanofluids is resolute by using the model of

Jang and Choi [65].

Table.5.5.Thermal conductivity and heat transfer coefficient values for pure

water and Graphene (Cn)/Water + ethyl glycol nanofluid at room temperature

Pure Water+EG 1 %vol

Graphene(Cn)/Water+

ethyl glycol

4 %vol

Graphene(Cn)/Water+

ethyl glycol

K[W/mK]

(Enhancement) 0.6060

(-)

1.2623

(2.7%)

1.4683

(11.1%)

hfd for constant

wall temp[w/m2k]

(Enhancement)

221.6

(-)

327.6

(2.7%)

367.2

(11.1%)

hfd for constant

wall Heat flux

[w/m2k]

(Enhancement)

264.5

(-)

356.6

(2.7%)

392.8

(11.15%)

The percentage values indicated are according to the expression 100(Knf – Kf)/Kf

Investigational data presented in the theoretical study part of Chapter 4

concludes that the convective heat transfer improvement of nanofluids overcomes the

augmentation expected due to the enhancement in the thermal conductivity. There are

various techniques newly projected to elucidate this bonus augmentation in

convective heat transfer; such as, thermal dispersion [72] and particle migration [74].

At this contemporary, there is disagreement about the comparative implication of

these mechanisms. Hence, further investigations are needed for the amplification of

this condition.

5.5.1 Stable Wall Temperature Boundary Condition

For steady wall temperature boundary condition, the planned view of the

arrangement is shown in Fig.5.13 In the study tube diameter is 5 mm and tube length

is 1 m. In the theoretical examination, depending on Peclet number, the field is

sometimes chosen to be longer than 1 m for getting thermally fully developed

139

situation at the exit, but only the 1-m part is measured in the purpose of heat transfer

parameters”.

Fig 5.13: Schematic view of the problem considered in the numerical study.

Boundary condition is stable wall temperature. Shaded region is the solution

domain.

5.5.2 Steady wall heat flux boundary condition

For stable wall heat flux boundary condition, the proposed view of the

configuration is shown in Figure.5.14 In order to get a proper assessment in

experimental data, tube size is chosen to be 4.57 mm and tube length is 2 m. In the

numerical study, depending on Peclet number, the domain is sometimes chosen to be

longer than 2 m for achieving thermally fully developed condition at the exit, but only

the 2-m part is measured in the calculation of heat transfer parameters.

Fig 5.14: Schematic view of the problem considered in the numerical study.

Boundary condition is stable wall heat flux. Shaded region is the solution

domain.

140

5.6 Study on modelling of Nanofluid Flow

5.6.1. Single phase approach

In the literature, there are mostly two techniques for the construction of

nanofluid flow. In the first method, the nanofluid is assumed as a single phase fluid

due to the reason that the particles are very small and they mix easily [75]. In this

method, the influence of nanoparticles can be considered into account by using the

thermo physical properties of the nanofluid in the governing equations. In the second

method, the problem is studied as a two-phase flow and the relations between

nanoparticles and the liquid matrix are constructed [76].

In the current study, the nanofluid is taken as a single phase fluid. Such an

attempt is an additional realistic way of studying heat transfer of nanofluids.

Conversely, the legality of the single phase guess needs authentication. It should be

noted that exclusively considering the thermo physical properties of the nanofluid to

the governing equations is not much diverse than using the conventional correlations

of convective heat transfer with thermo physical properties of the nanofluid. In the

theoretical examination part the single phase method needs some changes in order to

consider the additional enhancement. For this reason, the thermal diffusion model

proposed by Xuan and Rotzel [72] is used.

5.7. Governing Equations

For the study of heat transfer in the current problem, the governing equations

are the continuity, momentum and energy equations. For cylindrical coordinates,

incompressible and steady continuity equation is as follows [82].

1

0,xr ru uu u

r r r x

(5.30)

ux, u

r and u

θ are axial, radial and tangential parameters of flow velocity, respectively.

141

In the problem considered, the flow is hydro dynamically fully developed. Therefore,

the velocity of the flow does not vary in x direction and derivatives of the velocity

components in x-direction are zero. Moreover, while the flow is axisymmetric, all

terms with is also zero. Then the continuity equation becomes the following.

0,r ru u

r r

(5.31)

Noting that where r0is the tube radius, it can be obtained

0,ru (5.32)

r-momentum, θ-momentum and x-momentum equations in the lack of body forces for

cylindrical coordinates are as follows, respectively [82].

2

r r rr x

u uu u uu u

r r r x

2 2

2 2 2 2

1 1 1 2( ) ,r r

nf r

nf

uu upv ru

r r r r r r x

(5.33)

rr x

u u u u u uu u

r r r x

2 2

2 2 2 2

1 1 1 2,r

nf

nf

u uupv ru

r r r r r r x

(5.34)

x x xr x

u u u uu u

r r x

2 2

2 2 2

1 1 1,x x x

nf

nf

u u upv r

r x r r r r x

(5.35)

νnf

is kinematic viscosity of the nanofluid and p is pressure. It should be remembered

that pressure does not change with θ due to axisymmetric. Applying the

simplifications concerning the x- and θ-derivatives to Eq. (5.34):

1

0 ( ) ,rur r r

(5.36)

Noting that, it can be obtained urθ=0

142

0,u (5.37)

Eqs (5.33, 5.35) can also be re written by applying the calculations about the x- and θ-

derivatives and substituting Eqs. (5.32, 5.37):

1

0 ,nf

p

r

(5.38)

1 1

0 ,xnf

nf

upv r

r x r r r

(5.39)

0xu at 0,r r (5.40)

0xu

r

at 0r (5.41)

2

2

0

2 1 ,x m

ru u

r

(5.42)

Where u

m is mean flow velocity

After the resolution of the velocity distribution in the field, heat transfer in the system

can be discussed by allowing the energy equation which is as follows [82].

. ,p effnf

DTc k T q

Dt (5.43)

Where

,r x

uDu u

Dt t r r x

(5.44)

2

1 1. ,eff eff eff eff

T T Tk T rk k k

r r r r x x

(5.45)

22 2 2

1 1 12

2

x xrnf r

u u u uuu

r r x x r

221 1 1

,2 2

x r ru uu u

rr x r r r

(5.46)

Is the volumetric heat generation rate and Φ is the dissipation function. It should be

noted that the thermal conductivity term in the energy equation is changed by the

effective thermal conductivity (keff

) according to the thermal diffusion model.

143

Applying the calculations about the x- and θ-derivatives to Eq. (5.43) and substituting

Eqs. (5.32, 5.37, 5.42, 5.44-5.46)

22

2

0

12 1 ,x

p m eff eff nfnf

uT r T T Tc u k r k

t r x r r r x x r

(5.47)

The term with the time derivative is conserved in the energy equation since the

numerical solution technique used reaches the steady-state solution by marching in

time. Hence the problem is considered as transient, for the proper study of the

problem, non dimensionalization should be applied to Eq. (5.47). Non

dimensionalizations for steady wall temperature boundary condition and constant wall

heat flux boundary condition are a little different. Hence the connected discussion is

presented in two different sections.

5.7.1. Steady Wall Temperature Boundary Condition

For steady wall temperature boundary condition, following non dimensional

parameters are defined:

,W

i W

T T

T T

(5.48)

*

0

,x

xr

(5.49)

*

0

,r

rr

(5.50)

,*

2

0

,nf bt

tr

(5.51)

,*

,

,eff T

nf b

kk

k (5.52)

,

,mnf

nf b

u dPe

(5.53)

144

2

,

,

,nf b m

nf

nf b i W

uBr

k T T

(5.54)

Ti and T

ware inlet and wall temperatures, respectively and d is tube diameter. By using

these non dimensional parameters, Eq. (5.47) becomes:

*2 * * * *2

* * * * * * *

11 16 ,nf nfPe r k r k r Br

t x r r r x x

(5.55)

Brnf

is the nanofluid Brinkman number, which is a gauge of viscous effects in the

flow. For the current flow conditions, Brnf

is on the order of 10-7

, therefore viscous

dissipation is negligible. As a consequence, the final form of the energy equation is”:

*2 * * *

* * * * * * *

11 ,nfPe r k r k

t x r r r x x

(5.56)

*

0r

at

* 0,r (5.57)

0 at * 1,r (5.58)

1 at * 0,X (5.59)

5.7.2. Constant Wall Heat Flux Boundary Condition

For steady wall heat flux boundary condition, the expressions for the non

dimensional parameters, x*, r*, t*, k* and Penf

are similar to the steady wall

temperature condition, which are distinct from Eqs (5.49-5.53) correspondingly. But

the expressions for θ and Br differ as given below

"

0

,nf i

W

k T T

q r

(5.60)

2

"

0

,nf m

W

uBr

q r

(5.61)

It is optimistic when heat is transferred to the working fluid. Diligence of non

dimensionalization to the energy equation results in precisely the same differential

145

equation as in the condition of constant wall temperature (Eqs. 5.55, 5.56). The

boundary conditions are as follows:

*

0r

at

* 0,r (5.62)

* *

1

r k

at

* 1,r (5.63)

0 at * 0,x (5.64)

For both stable wall temperature and stable wall heat flux boundary

conditions, all of the thermo physical properties are measured at the bulk mean

temperature of the flow, which is shown by the subscript b in the linked expressions,

excluding the thermal conductivity. Nondimensional thermal conductivity, k*

, is clear

as the effectual thermal conductivity at the local temperature divided by the nanofluid

thermal conductivity at the bulk mean temperature. Bulk mean temperature is:

,2

i ob

T TT

(5.65)

It should be noted that k*

is a function of temperature and local axial velocity due to

Eqs. (5.52).

5.7.2.1. Stable Wall Heat Flux Boundary Condition

Comparable to the earlier case, the results of stable wall heat flux boundary

condition are compared with the predictions of the connected Graetz solution [83] for

parabolic velocity profile, which According to Graetz solution, local Nusselt number

is as follows [84]:

2 1

41

11 1,

48 2

n

xx

n n n

h d eNu

k A

(5.66)

o

x

r

Pe (5.67)

ξ is given by Eq. (5.67) and An, β

n values are got from Siegel et al. [84]. Fig. 5.15

provides the linked assessment in terms of the discrepancy of local Nusselt number in

146

axial direction for the flow of pure water for Pe = 2500, 4500 and 6500. When the

figure is observed, it is concluded that there is perfect conformity between the

numerical results and the predictions of the Graetz solution.

Table: 5.6: Experimental calculations of Graetz solution

S.No Pe Nux X*=X/ro

1 2500

7 15

6.6 25

6.1 75

5.8 100

5.4 125

5 150

4.7 175

4.5 225

4.5 400

4.5 600

4.5 800

4.5 1000

2 4500

6.8 25

5.95 100

5.4 175

5 225

4.7 400

4.7 600

4.5 800

4.5 1000

3 6500

6.9 50

6.7 75

5.95 100

5.9 125

5.6 150

5.5 175

5.2 225

4.7 400

4.7 600

4.5 800

4.5 1000

147

Table: 5.7: Numerical values of Graetz solution

S.No Pe Nux X*=X/ro

1 2500

7 15

6.6 25

6.2 50

5.95 100

5.8 100

5.4 125

5 150

4.7 175

4.5 225

4.5 400

4.6 600

4.5 800

2 4500

6.8 25

5.95 100

5.4 175

5 225

4.7 400

4.7 600

4.5 800

4.5 1000

3 6500

6.9 50

6.7 75

5.95 100

5.9 125

5.6 150

5.5 175

5.2 225

4.7 400

4.7 600

4.5 800

4.5 1000

148

Fig 5.15: dissimilarity of local Nusselt number in axial direction according to the

numerical results and Graetz solution for 20nm (Cn) of 1%, 1.5%, 2%, 2.5% Vol

Stable wall heat flux condition.

The discussion is presented in two main heading, stable wall temperature

boundary condition and stable wall heat flux boundary condition, correspondingly.

5.7.2.2 Stable Wall Temperature Boundary Condition

For the Stable wall temperature boundary condition, the results are first

discussed in terms of the standard heat transfer coefficient augmentation ratio (heat

4

4.5

5

5.5

6

6.5

7

0 200 400 600 800 1000

Nu

x

x*=x/ro

Pe=2500,Graetz

Pe=4500,Graetz

Pe=6500,Graetz

4

4.5

5

5.5

6

6.5

7

0 200 400 600 800 1000

Nu

x

X*=X/r0

Pe=2500,numerical

Pe=4500,numerical

Pe=6500,numerical

149

transfer coefficient of nanofluid divided by the heat transfer coefficient of

corresponding conventional base fluid). Then the change of local Nusselt number in

axial direction is studied for separate particle volume fractions. At last, influence of

particle size, heating and cooling is presented in terms of heat transfer coefficient

augmentation ratio.

5.8 Heat Transfer Coefficient improvement

The experimental data for heat transfer of nanofluid flow under stable wall

temperature boundary conditions. The results of the current study considered the heat

transfer characteristics of Graphene (Cn)/water+EG nanofluid in laminar flow. The

flow is hydro dynamically developed and thermally developing. Nanofluid flows

inside a circular tube with a diameter of 5 mm and length of 1 m.

The discrepancy of standard heat transfer coefficient augmentation ratio with

Peclet number for dissimilar particle volume fractions. Augmentation ratios are

considered by comparing the nanofluid with the pure fluid at the same Peclet number

in direct to concentrate on the special result of the improved thermal conductivity and

thermal diffusion. In direct to emphasis the significance of the claim of thermal

diffusion model,

150

Table: 5.8: Heat transfer enhancement ratio with Pe for different particle

volume ratios of Graphene (Cn) 20nm /water + EG

S.No %Volume

added Pef hnf/hf

1

1.0

2500

1.11

1.5 1.2

2.0 1.29

2.5 1.29

2

1.0

3000

1.12

1.5 1.22

2.0 1.282

2.5 1.3

3

1.0

3500

1.125

1.5 1.24

2.0 1.289

2.5 1.31

4

1.0

4000

1.13

1.5 1.265

2.0 1.33

2.5 1.33

5

1.0

4500

1.136

1.5 1.275

2.0 1.34

2.5 1.34

6

1.0

5000

1.145

1.5 1.278

2.0 1.334

2.5 1.35

7

1.0

5500

1.156

1.5 1.279

2.0 1.336

2.5 1.36

8

1.0

6000

1.186

1.5 1.286

2.0 1.33

2.5 1.38

9

1.0

6500

1.2

1.5 1.29

2.0 1.35

2.5 1.4

151

Fig 5.16: Variation of average heat transfer coefficient enhancement ratio with

Peclet number for different particle volume fractions of the Graphene (Cn)

20nm/water+EG nanofluid.

5.9 Further study

5.9.1 Local Nusselt Number

In this chapter, the same flow arrangement is analyzed theoretically in the

earlier sections is examined in terms of the axial variation of local Nusselt number. To

know the fully developed Nusselt number as well; the flow stream within a longer

tube is taken (5 m). Figure 5.17 shows the related results for the flow of pure water

and Graphene (Cn)/water+EG nanofluid at a Peclet number of 6500. In the figure, it is

observed that the local Nusselt number is higher for nanofluids throughout the tube.

This is mostly due to the thermal diffusion in the flow. Thermal scattering results in a

higher effective thermal conductivity at the middle of the tube which straightens the

radial temperature profile. Flattening of temperature profile raises the temperature

gradient at the tube wall and as a result, Nusselt number becomes superior when

compared to the flow of pure water. This is owing to the truth that the influence of

thermal distribution becomes more prominent with growing particle volume fraction.

1

1.1

1.2

1.3

1.4

1.5

1.6

2500 3000 3500 4000 4500 5000 5500 6000 6500

hn

f/h

f

Pe

Blue 1.0%

Red 1.5%

Green 2.0%

Violet 2.5%

152

It must be noted that the fully developed nanofluid Nusselt number values are

also superior to pure water case. Linked values for different particle volume fractions

of the Graphene (Cn)/water+EG nanofluid are offered in Table 5.10. It is observed

that rising particle volume fraction raises the fully developed Nusselt number. The

results existing in the table are for Pe = 6500 and since thermal diffusion is dependent

relative on flow velocity (Eq. 5.2), fully developed Nusselt number rises also with

Peclet number for the case of nanofluids. In Table 5.10 fully developed heat transfer

coefficient values are also presented. It should be noted that heat transfer coefficient

augmentation ratios are larger than Nusselt number augmentation ratios since the

previous shows the collective effect of Nusselt number augmentation and thermal

conductivity enrichment with nanofluids.

153

Table: 5.9: Calculation of local Nusselt number with dimensionless axial position

for pure water and Graphene (Cn) 20nm/water+EG nanofluid. Penf

=

Pef= 6500.

S.No Pure Water

Nux

Graphene

nanofluid

Volume

fraction (1.0%)

Graphene

nanofluid

Volume

fraction (2.0%) X

*=x/ro

Nux Nux

1 6 6 6 100

2 5.25 5.25 5.4 200

3 3.3 4.094 5 300

4 3.2 3.67 4.5 400

5 3.2 3.7 4.2 500

6 3.091 3.72 4.092 600

7 3.091 3.72 4.092 700

8 3.091 3.73 4.093 800

9 3.091 3.73 4.093 900

10 3.092 3.74 4.094 1000

11 3.092 3.74 4.094 1100

12 3.092 3.75 4.095 1200

13 3.093 3.75 4.095 1300

14 3.094 3.76 4.095 1400

15 3.095 3.76 4.096 1500

16 3.096 3.77 4.096 1600

17 3.097 3.77 4.097 1700

18 3.098 3.78 4.098 1800

19 3.099 3.78 4.099 1900

20 3.2 3.8 4.1 2000

154

Fig 5.17: Dissimilarity of local Nusselt number with dimensionless axial position

for pure water and Graphene (Cn)/water+EG nanofluid. Penf

= Pef=

6500.

Table 5.10 : Fully developed Nusselt number and heat transfer co efficient values

obtained from the numerical solution for pure water and Graphene(Cn) / Water

+ EG nanofluid with different particle volume fractions pef = penf = 6500

Fluid Nufd

Nu

Enhancement

Ratio(Nufd,nf

/Nufd,f)

hfd [W/m2K]

h Enhancement

ratio(hfd,nf/ hfd,f )

Water+EG

3.76 - 490 -

Nanofluid

1.0 %vol

1.5 %vol

2.0 %vol

2.5Vol %

3.98

4.25

4.36

4.72

1.069

1.078

1.082

1.098

589

641

662

693

1.362

1.571

1.728

1.931

3

3.5

4

4.5

5

5.5

6

0 500 1000 1500 2000

Nu

x

X*= X/ro

Blue 2.0 %

Pure Water

Green 1.0%