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102 DOI: 10.7508/tpnms.2017.02.004

Numerical simulation of nanofluid flow over diamond-shaped elements in tandem

in laminar and turbulent flow

Hamed Safikhani1,2, *, Ebrahim Hajian3, Rafat Mohammadi1

1Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-88349, Iran 2Institue of Nanosciences & Nanotechnolgy, Arak University, ARAK, I. R. IRAN

3Department of Mechanical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran

Received 28 March 2016; revised 24 November 2016; accepted 29 November 2016; available online 10 June 2017

ABSTRACT: In this paper, the Al2O3-water nanofluid flow in laminar and turbulent flows inside tubes fitted with diamond-shaped turbulators is numerically modeled. The nanofluid flow is modeled by employing a two-phase mixture method and applying the constant heat flux boundary condition at tube walls. In the results, the effects of different parameters such as the geometry of turbulators, volume fraction and diameter of nanoparticles, etc. on the flow field in the tubes have been investigated. The obtained results indicate that, with the reduction of tail length ratio (TR) and increase of vertex angle of turbulators (θ), the heat transfer coefficient as well as the wall shear stress increase. Similarly, with the reduction of TR and increase of θ, the amount of secondary flows, vortices and the turbulent kinetic energy increase. Moreover, the increase in the volume fraction of nanoparticles and the reduction ofnanoparticles diameter lead to the increase of the heat transfer coefficient and wall shear stress.

KEYWORDS: Nanofluid; Diamond-shaped turbulators; CFD; Turbulent flow; Mixture model

INTRODUCTION

Increasing the amount of heat transfer in different engineering applications such as the cooling of electronic components, cooling and boiling of liquids, design and operation of heat exchangers, etc. is highly important. The methods of heat transfer improvement can be generally divided into two groups of active and passive techniques. The passive heat transfer enhancement method, due to its simplicity of use and lower cost, has greatly attracted the attention of industrialists and researchers in recent years [1-6].

A very important and useful way of passively increasing the amount of heat transfer is the use of turbulators in flow passage to induce and promote the flow turbulence and to increase the local Reynolds number. In this regard, numerous investigations have been conducted in recent years both numerically and experimentally. Eiamsa-ard and Promvonge [1] experimentally studied the flow of air in circular tubes fitted with diamond-shaped turbulators. They explored the effects of the tail length ratio and also the vertex angle of the diamond-shaped turbulators on the heat transfer coefficient and the friction factor of the fluid. Ayhan et al. [2] experimentally and numerically investigated the increase in the amount of heat transfer in circular tubes containing truncated hollow cone turbulators. Durmus [3] evaluated the improvement of heat transfer in 4 different cut out conical turbulators.

*Corresponding Author Email: h-safikhani@araku.ac.ir

Tel.: 09120686211; Note. This manuscript was submitted on March 28, 2016; approved on November 24, 2016; published online June 10, 2017.

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m d h M G c F d b p d G f f e p v n p e Nom a Acce Cp Speci C Cons dp Diam g Gravi h Local k Therm kB Boltz L Leng Lh Head Lt Tain Pr Prand q'' Heat

Re

Reyn T

Temp TR

Tail l

V Veloc shaped turbu simultaneously mechanisms, diamond-shape have been num

MATHEMA

Geometry

The geomet circular tubes Figure 1 show different geom behavior of n paper. The re different TR an

Fig.1. Detail Governing eq

In the prese flow is perfor fluid two ph equilibrium ov phase has its volume there nanoparticles.

Instead of phase separate energy equatio

enclature

leration (m s-2) ific heat (J kg-1 tant in Eq. (14) meter of nanopar

itational acceler l heat transfer c mal conductivit zmann constant th of tubes (m) d length of turbu

length of turbu dtl number (=αm

flux (W m-2) nolds number (=

perature (K) length ratio (=lt/ city (m s-1)

ulators. In t y applying tw namely, the ed turbulators merically invest

ATICAL MO

tries evaluated s fitted with ws the schemati

metrical param nanofluid flow esults will be nd 3 different θ

led definition of di quations

ent study, num rmed using mi hase approach ver spatial leng own velocity is a certain

utilizing the ely, it solves t ons for the mi

K-1)

rticles (m) ration (m s-2) coefficient (W/m

ty (W m-1 K-1) (=1.3807

10-ulator (m) ulator (m) m/νm)

=VDh/νm)

/lh)

this paper, wo heat transf use of nano in laminar and tigated.

ODELING

in this paper diamond-sha ic of these tube

eters such as w will be inv obtained and

θ values.

iamond-shaped tur

merical simulat ixture model w h. This meth gth scales. In t field, and in n fraction of

governing eq the continuity, xture, and the

H. S

m2K)

-23 J K-1)

the effects fer enhanceme ofluids and a

d turbulent flo

include differe aped turbulato

es. The effects TR and θ on t estigated in t

displayed for

rbulators in tube

tion of nanoflu which is a sing hod investiga

this method ea a given cont base fluid a

quations of ea momentum a volume fracti

Safikhani et al.

103 G β V δ D Φ N θ λf M

μ D ν K BF B dr D f F i I k I m M p N w W α T of ent lso ows ent ors. of the this r 3 uid gle ates ach trol and ach and ion equatio state co Continu   .(

Momen .( .( m n k    

Energy .( n k

m pC

  

Volume

.(

Where m

V

In Eq Greek Symbol Volumetric exp Distance betwe Nanoparticles v Semi vertex ang Mean free path Dynamic visco Kinematic visc Subscripts Base fluid Drift Fluid Inlet conditions Indices Mixture Nanoparticle ph Wall Thermal diffusi

on for nanopar onditions and tu

uity: 0 ) Vm m

ntum: , , 1 )

m m m

k k dr k dr

V V p

V V    

y: 1 ) k pk k k

pm

C V T

vt  

e fraction:

)

V

m P P



Vm is the mass

m n 1 k k k k

V

quation 2, Vdr,k

ls

pansion coeffici een particles (m volume fraction gle of turbulato h of water molec sity (N s m-2) osity (m2 s-1)

s

hase

ivity (=k/ρcp)

rticles. The eq urbulent mean , .( ) ) m t k m p g        

) .( m

T   k  T

V

.(

P

P dr,P

s average veloc

kis the drift vel

ient (K-1) m)

n ors

cular (m)

quations for t flow are as fol

)

P

city:

locity for nanop

(3)

104 m

k k ,

dr

V

V

V

(6)

The slip velocity (relative velocity) is calculated as the velocity of nanoparticles relative to the velocity of base fluid:

f p

pf

V

V

V

(7)

The relation between drift velocity and relative velocity is as follows:

fk n

1

k m

k k pf

p ,

dr V V

V

 

(8)

The relative velocity and drag function are calculated using Manninen et al. [13] and Schiller and Naumann [14] relations respectively, as follows:

a ) (

f 18

d V

P m P drag f

2 P P

pf

 (9)

 

0.687

1 0.15 Re Re 1000

0.0183 Re Re 1000

P P

drag

P P

for f

for

  

  

 (10)

The acceleration (a) in Eq. (9) is

m m

.

)

V

V

(

g

a

(11)

The shear stress relation is given by:

m m

V

  (12)

k k n

1 k

k k k

t

v v v

(13)

Turbulence modeling

In this paper, as suggested by Namburu et al. [15], the

k-ɛ model, proposed by Launder and Spalding [16], is investigated. This model introduces two new equations, one for the turbulent kinetic energy (k) and the other for turbulent dissipation rate (ɛ). The two equations are given by:

m m

k tm m m

m

V

k

)

.

(

)

k

G

.(

(14)

1 2

.( ) . ( )

( )

tm

m m m

m m

V

k

C G C

 

   

  

 

     

 

(15)

Where

2

m tm

k

C

  (16)

)

)

V

(

V

(

G

T

m m

tm

m

(17)

With C1=1.44, C2=1.92, Cμ=0.09, σK=1, σε=1.

Nanofluid mixture properties

In this paper the properties of Al2O3-water nanofluid are

calculated using the following expressions:

Density [17] :

f p

m



(1

)

   (18)

Specific heat capacity [18]:

f p p

p m

p ) ( C ) (1 )( C )

C

(

 

(19)

Dynamic viscosity [19] :

C 72

d

V 2

p B p f

m   (20)

Where VB and δ are the Brownian motion of nanoparticles

and the distance between nanoparticles respectively and are calculated basedon:

p p

B

p B

d

T

k

18

d

1

V



(21)

p

3

d

6

(22)

C in Equation 14 is defined as:

f

4 p 3 2

p

1d C ) (C d C )

C ( C

  

(23)

Where C1, C2, C3 and C4 are given as:

000000393 .

0 C , 00000009 .

0 C

000002771 .

0 C , 000001133 .

0 C

4 3

2 1

  

  

 

(24)

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W ( J T B t i l w t c N v t a v r g d b t e 0.9955 1. 1 64.7 Pr Re m f f f k

k  

Where Ref and

2 B f 3 T k Re     f f f Pr    

Where λf is

(λf = 0.17 nm),

J/K) and η can

.10 247.8, B T C A B C   Thermal expan

m (1

1           Boundary con

To numeric tubes fitted w is solved as a line is modele with constant h

At the in temperature an specified and a condition is ap

Numerical m

The numeri volume metho the convectiv algorithm is e velocity and pr results are in generated grid different direct been attempted the highest acc examined tube

0.7460 0.3

.2321

7 ( f )

p

f

d d

d Prf can be exp

f

T

  

f

the MFP (mea , kB is Boltzma

n be defined by

, 2.414 10

140

A 

 nsion coefficien f p p f ) 1        nditions

cally model th ith diamond-sh an axisymmetri

d as the axis a heat flux bound lets of tubes nd the volume at the outlets, pplied.

methods

ical simulation d. A second or ve and diffus employed to so

ressure fields. dependent of ds, several gri

tions has been d to consider f curacy and the es contain betw

3690( p)0.7476

f k k

pressed as:

an free path) of ann constant (kB

the following

5 0 ,  

nt [21]: f p 1 1 1              he nanofluid haped turbulat ic problem. Th and the tube w

dary condition s, the input e fraction of n

the constant pr

n is performed rder upwind m ive terms an olve the coupl To make sure the size and ids with diffe tested for each for each tube th e lowest comp ween 2 and 3 m

H. S

(2

(2

(2

f water molecu

B= 1.3807 × 10

equation:

(2

f

  

(2

flow in circu tors, the proble

he tube’s cent walls are model

.

flow veloci nanoparticles a ressure bounda

d using the fin method is used

nd the SIMPL ling between t

that the obtain d the number erent sizes alo

h tube; and it h he best grid, w

utation cost. T million elemen

Safikhani et al.

105 25) 26) 27) ular 0-23 28) 29) ular em tral led ity, are ary nite for LE the ned of ong has with The nts. Figure contain Valida

To a necessa data. F coeffici in a pla Kim et and Bia these fi availab

Fig.

numeri

Fig.

nume

2 shows a sa ning diamond-s

Fig. 2.

ations

attain the con ary to compare Figures 3 and

ient (h) in lam ain tube of pre al. [22] and E anco et al. [24 figures, the pre

le data.

.3.Comparison of l ical simulation wit tube, Re=1460,

4. Comparison of erical simulation w plain tube, Re=2

ample of grid shaped turbulat

A sample of grid

nfidence about e the results w d 4 compare minar and turb esent study wi Ebrahimnia et ] (turbulent fl esent simulatio

local heat transfer th the available dat

2089

f local heat transfer with the available d 20000 ، 500

d generation f tors.

generation

t the simulati with the availab

the local hea ulent flows re ith the availab t al. [23] (lam low). As is evi ons agree wel

r coefficient of the ta for laminar flow ،dp=20 nm, Φ=3

r coefficient  of the data for turbulent f 0 , dp=38 nm

for a tube

ions, it is ble related at transfer espectively ble data of minar flow) ident from l with the

present w in a plain

%

(5)

R a t e p l i t i p E o e a d a i f f l t t RESULTS

In this pape and turbulent f turbulators is effects of d parameters on laminar and investigations, turbulent flow in all of the parameter y+ is

Effect of turb

In this pape of turbulators evaluated. Sinc amount of hea drop and thus also importan investigated. S fluid flow th fluctuations, t locally in a fig table format. F the local heat t

Fig. 5. Effects θ=30°, 40

er, the Al2O3-w

flows inside tu numerically different geom

n the nanoflui turbulent flow the Reynolds s are 1000 and investigation s controlled an

bulator geome

er, the effects (TR and θ) on ce in the flow at transfer, the the amount o nt, the latter Since by placin he shear stres these shear s gure so their a Figure 5 shows

transfer coeffic

s of tail length rati 00 , dp=40 n

turbulen

water nanofluid bes fitted with modeled. In t metrical and id flow will ws in tubes. s numbers for d 10000, respec ns related to d it has approp

etry

of two geome nanofluid flow within tubes, i e amount of na of shear stress

r parameter ng the turbulato

s of walls u tresses canno average values s the effect of cient.

io on local heat tra nm, Φ=1.5%, (a) la nt Re=10000

d flow in lamin h diamond-shap this section, t non-geometri be explored In all of t the laminar a ctively. Moreov

turbulent flow priate values.

etrical paramet w field have be

in addition to t anofluid pressu

at tube walls has also be ors in the way undergoes seve

t be consider is presented in TR variations

ansfer coefficient, aminar Re=1000 (b

106 nar ped the cal for the and ver ws, ers een the ure s is een y of ere red n a on b) Also tube wa Effect Effect o Variabl TR θ(deg) Φ(%)

dp(nm)

Acco increase turbulen transfer diamon that the transfer walls, w with the in lami respecti the stre Figure figure, the vor turbulen conside conclud heat tr seconda Fig. 6 Simi coeffici by incr flows, Accord

, the impact o alls has been pr

t of different par of

le Value La 1 0 1.5 0 2 0 15 0 22.5 0 30 0 0 0 1.5 0 3 0 ) 20 0

40 0

ording to Fig es with the red nt flows. The r coefficient d nd-shaped turb

e reduction of r coefficient, a

which constitu e reduction of inar and turbul ively. The con eam lines in tu 6 for differen with the redu rtices existing nt kinetic en ering the resu ded that the re ransfer coeffic ary flows, vort

6. Effects of tail le

ilarly, the effe ient has been reasing the val the heat tra ding to Table

of this paramet resented in Tab

Table 1 rameters on the a

τ(pa) aminar Turbule 0.047 1.23 0.041 1.15 0.033 0.98 0.031 0.87 0.037 1.03 0.041 1.15 0.040 1.13 0.041 1.15 0.043 1.19 0.41 1.14 0.041 1.15

gure 5, the h duction of TR saw tooth fe diagram are d bulators in the TR, in additio also increases utes a disadva TR from 2 to 1 lent flows incr ntours of turbu urbulent flow nt TR values. uction of TR, t

g in flow an nergy increas ults of these eduction of TR cient to incre tices and mixin

ength ratio on turbu stream lines fect of θ on

depicted in Fi lue of θ in bot ansfer coeffici 1, the increase

er on the shea ble 1.

average wall she Changes i ent Laminar 0.00 -12.76 8 -29.78 7 0.00 19.35 32.25 0.00 2.50 9 7.50 4 0.00 0.00

heat transfer c in both the la eatures in the due to the ex tube. Table 1 on to increasin the shear str antage. As is 1, the wall she rease by 29% ulent kinetic e

have been illu As is indicat the secondary nd also the a se. By simu two figures, TR ultimately c

ease by incre ng of flow.

ulent kinetic energ

the local hea igure 7. As is th laminar and ient can be e of θ also lea

ar stress of

ear stress. in τ(%)

Turbulent 0.00 -6.50 -20.32 0.00 18.39 32.18 0.00 1.76 5.30 0.00 0.87 coefficient aminar and local heat istence of 1 indicates ng the heat ress at the observed, ar stresses and 20%, nergy and ustrated in ted in this flows and amount of ultaneously

it can be causes the easing the

gy (k) and

(6)

i f l

l d

t b h e

increase of she from 15o to 30

laminar and tu

Fig. 7. Effec TR=1.5, 4

The contour lines in a tur different θ valu

According secondary flow the amount of be concluded heat transfer existing second

Fig. 8. Effect

ear stress at th 0º, the amount urbulent flows i

cts of vertex angle 4000 , dp=4

(b) turbule rs of turbulent rbulent flow a

ues.

to this figure ws and the vor turbulent kinet that the incre coefficient to dary flows and

ts of vertex angle o strea

he walls. With of wall shear increases by ab

on local heat tran 0 nm, Φ=1.5%, (a ent Re=10000 t kinetic energy are illustrated

, with the inc rtices existing tic energy incr ase of θ ultim o increase by d vortices in flo

on turbulent kineti am lines

H. S

the increase o stress in both t bout 32%.

sfer coefficient, a) laminar Re=100

y and the strea in Figure 8

crease of θ, t in flow and a ease. Thus, it c mately causes t y increasing t ow.

ic energy (k) and

Safikhani et al.

107 f θ

the

0

am for

the lso can the the

Effect

In th nanopa diamon illustrat 0-3%) well as volume coeffici column volume stress in and 5.3

Fig. 9

coeffic Figur changes coeffici observe 100 to rather c Also change stress in 1.75% i

of nanopartic

his section, t article diamete nd-shaped turb

tes the effect o on the local h turbulent flow e fraction from

ient increases n in Table 1 in e fraction of na n laminar and 3%, respectivel

9. Effects of nanop cient, TR=1.5, θ=3

Re=10 re 10 shows th s from 20-40 ient in lamina ed, with the re

20 nm, the lo considerably.

, the fourth co of nanopartic ncreases very in turbulent flo

cle parameter

the effects of er on fluid flo bulators are of volume frac heat transfer c ws. As is observ

m 0% to 3%, rather signifi ndicates that w anoparticles, th turbulent flow ly.

particles volume fra 30°, 4000 000 (b) turbulent R he effect of nan 0 nm) on th ar as well as eduction of nan ocal heat trans

lumn in Table cle diameter, th slightly (2.43 ows).

s

f volume fra ow in tubes f investigated. ction (as it cha coefficient in l ved, with the i the local hea ficantly. Also, with this incre

he amount of w ws increases by

action on local hea ،dp=40 nm, (a

Re=10000 noparticle diam

he local hea turbulent flow noparticle diam

fer coefficient

1 indicates tha he amount of w

% in laminar

ction and fitted with Figure 9 anges from

laminar as ncrease of at transfer

the third ease in the wall shear y about 7.5

at transfer a) laminar

meter (as it at transfer

ws. As is meter from

t increases

(7)

V t

v d a f

Fig. 10. Effe coefficient, TR

Variations of turbulators w

vary on a tu designing of tu at the wall of flows have bee

Fig. 11. Wall str

ects of nanoparticl R=1.5, θ=30°,

Re=1000 (b) tu f different par wall Knowing

urbulator wall urbulators. Th f turbulator in en shown in Fi

ress distribution ac

les diameter on loc

4000 , Φ=

urbulent Re=10000 rameters on d

how the diff can be usef e amounts of l n laminar as w

gure 11.

cross one diamond

cal heat transfer =1.5%, (a) laminar

0

diamond-shap

ferent paramet ful in therma local shear stre well as turbule

d-shaped turbulator

108 r

ped

ers ally ess ent

rs

Acco bulges which c pressur kinetic turbulat observe similar parame valleys

Fig. 12

The and tur 13. Acc of press

Fig. 1

CONC

In th laminar diamon modelin two-pha flux bo

ording to this f (peaks) of th cause this type re loss. The de energy (k) a tor wall have ed, the change and the maxim eters respective

(dents).

2. Turbulent kinet distributions acr changes of pre rbulent flow re cording to this sure at the peak

3.Pressure distrib

CLUSION

his paper, the f r and turbul nd-shaped turbu

ng of nanoflu ase mixture m oundary cond

figure, the amo e turbulators e of turbulator etails of the f and turbulent e been shown es of both para mum and min ely appear at

tic energy (k) and t ross one diamond-essure over tu egimes have b

figure, nanofl ks.

bution across one d

flow field of A ent flows in ulators was nu uid was perfor method and app

dition at tube

ounts of wall sh display severe s to have a co fluctuations of dissipation ra n in Figure 1 ameters are qu nimum amount the peaks (bu

turbulent dissipatio -shaped turbulator urbulator wall i

been depicted luid has the lea

diamond-shaped tu

Al2O3-water na

nside tubes c umerically mod rmed by empl

plying the con walls. The

hear at the e changes,

nsiderable f turbulent ate (ɛ) on 12. As is ualitatively

ts of these ulges) and

on rate (ɛ) rs.

in laminar in Figure ast amount

urbulators

anofluid in containing deled. The loying the nstant heat

(8)

H. Safikhani et al.

109 different parameters such as the geometry of turbulators (TR and θ), volume fraction and diameter of nanoparticles, etc. on the flow field of nanofluid in the tubes were investigated. It was observed that, with the reduction of TR and increase of θ, the amounts of heat transfer coefficient as well as the wall shear stress increase. Similarly, with the reduction of TR and increase of θ, the number of secondary flows, vortices and turbulent kinetic energy in the flow increase. Also, the increase in the volume fraction of nanoparticles and the reduction of the nanoparticles diameter resulted in the increase of the heat transfer coefficient and wall shear stress. Moreover, the changes of shear stress, k and ε on turbulators were investigated and it was observed that the mentioned parameters have their highest values at the hills and their lowest values at the valleys of turbulators wall.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the Arak university fund for financial supporting of the project.

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[21] Khanafer K , Vafai K , Lightstone M. Buoyancy - driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. International journal of heat and mass transfer. 2003 Sep 30;46(19):3639-53.

[22] Kim D, Kwon Y, Cho Y, Li C, Cheong S, Hwang Y, Lee J, Hong D, Moon S. Convective heat transfer characteristics of nanofluids under laminar and turbulent flow conditions. Current Applied Physics. 2009 Mar 31;9(2):e119-23.

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110 International journal of heat and mass transfer. 2011 Sep 30;54(19):4376-88.

Figure

Figure 1 showws the schematiic of these tube
Fig. 2.A sample of grid
Fig. 5. Effects�� s of tail length rati00
Fig. 99.action on local hea�Reat transfer θ=3particles volume fracient, TR=1.5,  Effects of nanopcoefficRe=1030°, ��� � 4000 �000 (b) turbulent R��� ،dp=40 nm, (a=10000 a) laminar
+2

References

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