Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS 
Series:Mechanical Engineering Vol. 14, N

o
 2, 2016, pp. 169 - 177 

Original scientific paper 

BROWNIAN HEAT TRANSFER ENHANCEMENT 

IN THE TURBULENT REGIME 

UDC 536.2:532.5 

Suresh Chandrasekhar, Vaarin Majumdar Sharma 

Department of Mechanical-Mechatronics Engineering, LNM Institute of Information 

Technology, India 

Abstract. The paper presents convection heat transfer of a turbulent flow Al2O3/water 

nanofluid in a circular duct. The duct is a under constant and uniform heat flux. The 

paper computationally investigates the system’s thermal behavior in a wide range of 

Reynolds number and also volume concentration up to 6%. To obtain the nanofluid 

thermophysical properties, the Hamilton-Crosser model along with the Brownian 

motion effect are utilized. Then the thermal performance of the system with the 

nanofluid is compared to the conventional systems which use water as the working 

fluid. The results indicate that the use of nanofluid of 6% improves the heat transfer 

rate up to 36.8% with respect to pure water. Therefore, using the Al2O3/water 

nanofluid instead of water can be a great choice when better heat transfer is needed. 

Key Words: Nanofluid, Forced Convection, Heat Transfer Enhancement, Turbulence Flow 

1. INTRODUCTION 

In the past years, many different techniques were utilized to improve the heat transfer 

rate in order to get higher thermal efficiencies. Thus adding solid particles to a base fluid 

was suggested by Maxwell [1, 2]. However, large particles cause many serious problems. 

Choi [3] suggested the use of nano-sized particles.  
After that, the researcher tried to understand the effects of those nanoparticles on heat 

transfer efficiency of the system. In recent years, some numerical studies have been done 
on the forced or free convection of laminar flows of the nanofluids [4-7]. The results 
reveal that using nanofluid can enhance heat transfer performance of nanofluid compared 
to pure water. Also, some studies have focused on the turbulent flow of nanofluids. Heat 
transfer investigation of nanofluids flow in the turbulent regimes was numerically 

                                                           
Received April 5, 2016 / Accepted June 15, 2016 

Corresponding author: Suresh Chandrasekhar 

Department of Mechanical-Mechatronics Engineering, The LNM Institute of Information Technology, India  

E-mail:sur.chandrasekhar@gmail.com 



170 S. CHANDRASEKHAR, V. M. SHARMA 

performed. Manca et al. [8] numerically investigated the turbulent forced convection   
with nanofluids in 2D channel. They have used a single-phase approach and observed 
that the heat transfer enhancement increases with the particle volume concentration. 
Besides, Roy et al. [9] compared the performance of different nanofluids inside a typical 
radial flow cooling device. They reported that the heat transfer increases with volume 
concentration and Reynolds number. Bianco et al. [10] also numerically studied water-
Al2O3 nanofluids turbulent convection heat transfer inside a circular tube. They reported 
that the heat transfer enhancement increases with the particle volume concentration and 
Reynolds number; their results were in good agreement with those of other studies. 
Moreover, the heat transfer enhancement of TiO2 and water in a circular tube were 
investigated by Demir et al. [11]. A single-phase model having two-dimensional equations 
was employed for this simulation and they reported a noticeable jump in the heat transfer rate.  

In this study, a turbulent flow of Al2O3/water nanofluid passes through a tube which is 
under constant heat flux. To capture the effects of the temperature on the thermophysical 
properties of the nanofluid, the properties are considered temperature dependent. Also, 
the effect of the Brownian motion of nanoparticles is taken into account. To investigate 
the thermal performance of the system by using this new working fluid, the Nusselt 
number and the convective heat transfer coefficient are presented. 

2. MATHEMATICAL MODELING 

The geometry consists of a tube with diameter (D) of 0.01m and length of L=1m. The 

Al2O3/water nanofluid passes through this tube and a fixed heat flux is subjected to the tube 

wall. Volume concentration of nanoparticles () equals 0% (base fluid), 1%, 2%, 4% and 6%. 

2.1. Fluid properties 

In this study, the thermophysical properties of nanofluid are calculated based on 
temperature dependent models in order to capture the effects of temperature on the 
properties.  

The effective properties of the Al2O3/water nanofluid are defined as follows: 

 
(1 )

nf p p p bf
      

 (1) 

where p is the volume concentration of nanoparticles, while p, bf, nf are the density of 
particles, the density of the base fluid that is water and the density of nanofluid, respectively.  

Eq. (1) was originally introduced in [12] for determining the density and then widely 
employed in literature [4, 6, 10, 13].  

For defining the heat capacity, the following equation is used as in many other studies 
such as [4, 6, 5, 8, 10]: 

 

(1 )
p p p p bf bf

nf

nf

C C
C

   



 


 

(2)

 

In order to get the nanofluid thermal conductivity, many referential studies have used 
the Hamilton and Crosser model [14]: 

 

( 1) ( 1) ( )

( 1) ( )

nf p bf p bf p

bf p bf p bf p

k k n k n k k

k k n k k k





    


   
 

(3)

 



 Brownian Heat Transfer Enhancement in the Turbulent Regime 171 

In this model, n is an empirical shape factor which accounts for the effect of shape of 

particles; it can vary from 0.5 to 6.0. For spherical nanoparticles n equals 3. So this case 

of the Hamilton and Crosser model (n=3) is:  

 

2 2 ( )

2 ( )

nf p bf p bf p

bf p bf p bf p

k k k k k

k k k k k





  


  
 

(4) 

However, this equation only takes into account the nanoparticles shape and volume 

concentration; it cannot take into account the Brownian motion of nanoparticles inside 

the fluid. However, in this study, we consider the effect of the Brownian motion of 

nanoparticles. For this purpose, we need to modify the Hamilton-Crosser model by using 

the formals proposed by Koo and Kleinstreuer [15]:  

 

4
5 10 B

Brownian bf bf

p p

T
k C

d





 

 

(5)

 
where: 

 
0.0841

0.0017(100 ) , 1%  


 
 

(6)

 

Besides, the viscosity of nanofluid is calculated based on the Einstein model: 

 
(1 2.5 )

nf bf
   

 

(7)

 

2.2. Governing equations 

To capture the turbulence effect, standard - model is adapted. This - model has 

two additional equations for turbulent kinetic energy  and the rate of dissipation  in the 
following form: 

 

( ) ( )t
k

k

V P


    


 
      

   

(8)

 

 

2

1 2
( ) ( )t

k
V C P C

 



  
    

  

 
      

   

(9)

 

where the eddy viscosity is calculated by: 

 

2

t
C




 




 

(10)

 

 1 2
1.44, 1.92, 0.09, 1, 1.3

k
C C C

   
     

 
(11)

 

Besides, the governing equations for solving the problem are continuity, momentum 

and energy equations as follows: 

 
.( ) 0V 

 
(12)

 

 
' '

.( ) .( )VV P V v v       
 

(13)
 

 
' '

.( ) .( )
p p

C TV k T C t v     
 

(14)
 



172 S. CHANDRASEKHAR, V. M. SHARMA 

2.3. Numerical method 

In order to discretize the governing equations, a control volume approach is employed 

by using ANSYS Fluent. Also, a collocated grid system is used so that the grid near the 

wall is smaller in order to capture turbulence effects near the wall. Several unstructured 

grids have been created to assure consistency and accuracy of the numerical results. A 

grid independency test is done by comparison of the results obtained from employing 

different grids in terms of the local Nusselt number and the relevant errors showed that 

the 700150 non-uniform grid is good enough to ensure the independency of numerical 

results from the grid system. This grid consists of 150 nodes in radial (r) and 700 nodes 

in longitudinal direction (x).  

The boundary conditions consist of an uniform inlet temperature, Tin=293K also a 

uniform entrance velocity uin which can be calculated from Reynolds number based on 

the following equation, and the non-slip condition is set for the velocities at the walls. 

  

Re
in

u
D






 
(15)

 

Further, the constant intensity turbulence of 1% is imposed. Also, a uniform heat flux 

is applied at the pipe wall. Moreover, both turbulent kinetic energy and dissipation of 

turbulent kinetic energy are equal to zero. At the channel exit, the fully developed 

assumption is employed which means that all the axial derivatives are zero. 

3. RESULTS AND DISCUSSION 

In order to validate the numerical method, the current numerical Nusselt number of the 

pure water is compared to some models represented in the following equations presented in 

[13] versus Reynolds numbers in Fig. 1. This figure shows that there is a good agreement 

between the numerical results obtained from the current study and other models and studies.  

 

Fig.1 Comparison of the present numerical results with other works 



 Brownian Heat Transfer Enhancement in the Turbulent Regime 173 

Gnielinski: 

3

1/ 2

2 / 3

(Re 10 ) Pr
2

2
1 12.7 (Pr 1)

2

f

f
Nu

f

 
 

  
  

  
  

 

 (16) 

Dittus-Boelter: 
0.8 0.4

0.024 Re Pr       HeatingNu   
(17)

 

Petukhov: 
2 / 3

Re
8

1.07 12.7(Pr 1)
8

f
Pr

Nu
f

 
 
 



 

 (18) 

Fig. 2 shows the local heat transfer coefficient of water and nanofluids 1, 2, 4 and 6% 

for constant Reynolds number of 20000. This figure shows that by replacing the 

Al2O3/water nanofluid by pure water, the heat transfer coefficient increases. Also, by 

increasing the volume concentration of the dispersed nanoparticles in the base fluid, the 

heat transfer convective coefficient will increase as well. This enhancement in the heat 

transfer coefficient is because of enhancing the properties of the nanofluid due to adding 

some solid nano particles in the base fluid.  

 

Fig. 2 Local convective heat transfer coefficient of all working fluids 

As mentioned before, some studies have already explored the heat transfer performance 

of Al2O3/water nanofluid. Therefore, in Fig. 3, comparison was made between the present 

study and the previous studies for the working fluid of Al2O3/water nanofluid 1.0%. This 

comparison shows that the Maiga model defined by Eq. (19) overestimates the present 

numerical study results. 

 
0.71 0.35

0.085Re PrNu 
 

(19)
 

However, our results have a better agreement with the model developed by Pak and Cho 

in the following equation: 

 
0.8 0.5

0.021Re PrNu 
 

(20)
 

The reason for this difference is because Maiga et al. [16] have used a correlation on 

the experimental data by Wang et al. [17] to define the thermophysical properties of 



174 S. CHANDRASEKHAR, V. M. SHARMA 

Al2O3-water nanofluid. But those properties are noticeably different from those that Pak 

and Cho [12] have used in their work; the reason for this is in different nanoparticles 

sizes in these two studies. Besides, our results agree well with those of Bianco et al [10] 

and Takabi and Shokouhmand [13] since both the studies use a numerical simulation 

which is just like the method we use in this work.  

 

Fig. 3 Comparing Nu of the present study with other works 

Fig. 4 shows the Nusselt number of pure water and nanofluids 1 and 2% as a function 

of Reynolds number. This figure shows that when we use nanofluid instead of pure water 

as the working fluid, we can get a higher Nusselt number. Also, when we use higher 

volume concentrations of nanoparticles in the nanofluid, we will obtain a higher Nusselt 

number. Thus we can get higher thermal performance. Therefore, from this figure, it can 

be concluded that we can use nanofluid in a wide range of industrial applications, 

especially when we need higher thermal performance. For instance, nanofluid is a good 

choice to use in underwater pipelines [18], or in biomechanics [19], or where we need 

higher thermal rejection capacity [20,21] since we can improve thermophysical properties 

such as higher capacity by adding nano-sized solid particles. To be more specific, once 

nanoparticles are added to the base fluid, the mixture has a higher thermal capacity so 

that it can absorb shocks and external vibrations. In other words, the nanoparticles in a 

flow hit the wall and absorb some energy from the wall. So they can decrease extra 

energy resulting from vibration or thermal shocks on the wall. Besides, nanoparticles will 

 

Fig. 4 Nusselt number of pure water, nanofluid 1% and 2% as a function of Re 



 Brownian Heat Transfer Enhancement in the Turbulent Regime 175 

cause better thermal properties of the mixture. So, as can be seen in this figure, the slope 

for nanofluid 2% is steeper than for nanofluid 1% and water. Therefore, it means the 

nanofluid 2% has a better thermal performance than water or nanofluid 1%.  

In order to understand the effect of using nanofluid instead of water quantitatively, we 

use the average convective heat transfer coefficient ratio as the following equation which 

has already been used by [13] in a similar work.  

 

nf

r

bf

h
h

h


 (21) 

where
bf

h and 
nf

h are average convective heat transfer coefficients of the base fluid which is 

pure water and the nanofluid which is Al2O3/water nanofluid. Table 1 shows the average 

convective heat transfer coefficient ratio of Al2O3/water nanofluid with different volume 

concentrations. As can be seen, all the values are greater than 1. It means that using 

nanofluid will increase the convective heat transfer coefficient that is good for us. Also, the 

concentration for Al2O3/water nanofluid 1% is 1.091, while this value for nanofluid of 6% 

is 1.368. Therefore, it shows that by adding five more percent of the volume concentration, 

25.4% enhancement in the average convective heat transfer coefficient is achieved.  

 

Table 1 The average convective heat transfer coefficient ratio of nanofluid 

Nanofluid 1.0%   2.0%   
4% 

 
6.0%   

r
h  1.091 1.166 1.255 1.368 

 

Fig. 5 shows wall temperature and bulk temperature of water, nanofluid 1 and 2%. As 

can be seen, when we use nanofluid, the wall temperature decreases which is favorable. 

This phenomenon can be used in the laser eye surgery where the laser irradiation is so 

high that it could damage the cornea tissues. But the use of nanofluid could decrease the 

temperature on the cornea surface [22]. 

 

Fig. 5 Wall temperature and bulk temperature profile of water and nanofluid and 1.0% 



176 S. CHANDRASEKHAR, V. M. SHARMA 

This behavior is also reported by Takabi and Shokouhmand [13] and Bianco et al [10]. 

The decrease in the nanofluid wall temperature is caused by the fact that the nanoparticles 

in the flow hit the wall and absorb the thermal energy of the wall, as discussed earlier. 

Therefore, the wall temperature of nanofluid compared to water decreases. Also, the 

nanoparticles can enhance the fluid’s thermophysical properties. So nanofluid has a higher 

thermal capacity to absorb thermal energy. Furthermore, we can see that after the entrance 

region of the pipe, the wall temperature and the bulk temperature are getting parallel. It 

means the flow is fully developed in terms of heat transfer in that region which is in 

agreement with [23]. 

4. CONCLUSIONS 

In many industrial applications, we need higher thermal performance for different 

equipments. Thus we can use a better working fluid such as nanofluids. Nanofluids are 

mixtures of some nanoparticles into a base fluid. In this study, the effect of nanofluid with 

different volume concentration in a circular tube under constant heat flux in a wide range of 

turbulent flow with the Reynolds number between 10000 to 100000 is investigated. That is 

why a computational approach is used based on finite volume method. The thermophysical 

properties of the nanofluid are obtained by using the Hamilton-Crosser model along with 

the Brownian motion effect. Also for the turbulence effects, standard  model is used. 

The results reveal that the numerical current results are in a better agreement with other 

numerical studies by Bianco et al [10] and Takabi and Shokouhmand [13] than the model 

developed by Maiga and Pak-Cho. Also, using the nanofluid 6% can enhance the convective 

heat transfer coefficient up to 36.8% compared to the base fluid, while the nanofluid of 1% 

has an improvement of 16.6%. Therefore, using the higher volume concentrations for 

nanofluid can enhance the thermal performance more [24]. Besides, using the nanofluid can 

decrease the wall temperature which is a positive effect in the analysis.  

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