The study of heat transfer and laminar ﬂow of kerosene/ multiwalled carbon nanotubes (MWCNTs) nanoﬂuidin the microchannel heat sink with slip boundary condition
Abedin Arabpour
1
•
Arash Karimipour
2
•
Davood Toghraie
1
Received: 25 May 2017/Accepted: 12 August 2017/Published online: 4 September 2017
Akade´miai Kiado´, Budapest, Hungary 2017
Abstract
In this investigation, the laminar heat transfer of kerosene nanoﬂuid/multiwalled carbon nanotubes in themicrochannel heat sink is studied. The consideredmicrochannel is two layers in which the length of bottomlayer is truncated and is equal to the half of the length of bottom layer. The length of microchannel bottom layer is
L
=
3 mm, and the length of top layer is
L
1
=
1.5 mm.The microchannel is made of silicon, and each layer of microchannel has the thickness of
t
=
12.5
l
m. Along theexternal bottom wall, the sinusoidal oscillating heat ﬂux isapplied. The top external and lateral walls are insulated,and they do not have heat transfer with the environment.The results of this research revealed that in differentReynolds numbers, applying oscillating heat ﬂux signiﬁcantly inﬂuences the proﬁle ﬁgure of Nusselt number andthis impressionability is obvious in Reynolds numbers of 10 and 100. Also, by increasing the slip velocity coefﬁcienton the solid surfaces, the amount of minimum temperaturereduces signiﬁcantly which behavior remarkably entails theheat transfer enhancement.
Keywords
Heat transfer
Kerosene/multiwalled carbonnanotubes
Microchannel heat sink
Oscillating heat ﬂux
Slip velocity coefﬁcient
List of symbols
A
Area (m
2
)
B
=
b
/
H
Dimensionless slip velocity
C
f
Skin friction factor
C
p
Heat capacity (J kg

1
K

1
)
H
Microchannel height (m)
K
Thermal conductivity coefﬁcient(W m

1
K

1
)
L
Downlayer microchannel length(m)
L
1
Toplayer microchannel length(m)
Nu
Nusselt number
P
Fluid pressure (Pa)
Pe
=
(
u
s
d
s
/
a
f
) Peclet number
Pr
=
t
f
/
a
f
Prandtl number
q
00
(
X
) Oscillating heat ﬂux (W m

2
)
q
0
00
Constant heat ﬂux (W m

2
)
R
Thermal resistance (K W

1
)
Re
=
q
f
u
c
d
/
l
f
Reynolds number
T
Temperature (K)(
U
,
V
)
=
(
u
/
U
0
,
v
/
U
0
) Dimensionless velocitycomponents in
x
,
y
directions(
X
,
Y
)
=
(
x
/
d
,
y
/
d
) Cartesian dimensionlesscoordinates
u
,
v
Velocity components in
x
,
y
directions (m s

1
)
u
c
(m/s) Inlet velocity in
x
directions(m s

1
)
u
s
(m/s) Brownian motion velocity(m s

1
)
Greek symbols
b
Slip velocity coefﬁcient (m)
u
Nanoparticles volume fraction
&
Arash Karimipourarashkarimipour@gmail.com
1
Department of Mechanical Engineering, KhomeinishahrBranch, Islamic Azad University, Khomeinishahr, Iran
2
Department of Mechanical Engineering, Najafabad Branch,Islamic Azad University, Najafabad, Iran
1 3
J Therm Anal Calorim (2018) 131:1553–1566https://doi.org/10.1007/s109730176649x
k
l
=
L
1
/
L
Dimensionless length ration
l
Dynamic viscosity (Pa s

1
)
h
=
(
T

T
C
)/
D
T
Dimensionless temperature
q
Density (kg m

3
)
s
Shear stress (N m

2
)
t
Kinematics viscosity (m
2
s

1
)
Super and subscripts
Ave Averagec ColdEff Effectivef Base ﬂuid (pure water)H HotIn InletMax MaximumMin Minimumnf NanoﬂuidOut OutletS Solid nanoparticles
Introduction
The cooling of miniature equipment in the micro electromechanical and nanoelectromechanical industries hasincreased the need of understanding the ﬂuid ﬂow and heattransfer in the micro and nanogeometrics. The behavior of ﬂuid ﬂow and heat transfer in the miniature scales and byusing nanoﬂuid, due to the improvement in heat transfermechanisms in nano and microdimensions, comparing tothe custom scales, is far different. Numerous numerical andempirical studies have been done for investigating the ﬂowand heat transfer of custom ﬂuids and nanoﬂuid in themicrochannels whose main purpose is increasing the heattransfer [1–5]. The investigation of heat transfer enhance
ment in different industrial and experimental ﬂuids byusing novel methods has been expanded as the study ﬁeldsamong the adherents of this issue [6]. The microchannelheat sink as an applicable miniature equipment has highimportance in heat transfer of electronic industries. Thisequipment has been suggested by Tukerman and Pease [7]for cooling the electronic chips. In recent decades, thisequipment has been investigated and optimized byresearchers in different structures and arrangements forenhancing the cooling of electronic chips [8–10].
Kulkami et al. [11] numerically studied the multipurposedoptimization of doublelayer microchannel heat sink withthe crossﬁgured inlet section. Their results evidenced thatthe microchannel with narrower design has lower thermalresistance and higher pumping power and the pumpingpower by increasing the heat ﬂux reduces signiﬁcantly.Husain and Kim [12, 13] optimized the indented
microchannel heat sink and indicated that the thermalresistance of microchannel heat sink by optimizationreduces considerably. Xie et al. [14] studied the efﬁciencyof doublelayer microchannel heat sink with the wavy wallin the states of parallel and contrary ﬂows. They investigated the effects of wavy wall limitation and the ratio of mass ﬂow on the thermal resistance and pressure dropparameters. Seyf and Nikaaein [15] by using Al
2
O
3
, zincand Cu nanoparticles in the ethylene glycol/water ﬂuidnumerically studied the effects of nanoparticles dimensionsand Brownian motion of nanoparticles on the thermalperformance of a rectangular microchannel heat sink. Theirresults showed that the amount of nanoﬂuid conductivitywithout considering the Brownian motion reduces almostto 6.5%. Wu et al. [16] numerically studied the thermalresistance, pumping power and thermal distribution on thewall surface of doublelayer microchannel heat sink (DLMCHS). In their research, different parameters of microchannel dimensions and different ﬂow conditionshave been studied. The results of his study showed that theimprovement in total efﬁciency of doublelayermicrochannel heat sink depends on the pumping power.Chen and Chung [17] used the water/Cu nanoﬂuid. In theirinvestigation, the absorbed energy by the nanoﬂuid wasmore than the absorbed energy by water, and it has beenobserved that by enhancing volume fraction of nanoparticles, the hightemperature differences accomplish betweenthe inlet and outlet sections of microchannel heat sink in alow ﬂow rate. Jang and Choi [18] by using nanoﬂuidnumerically studied the cooling performance of amicrochannel heat sink. They reported that the nanoﬂuidcauses the reduction in thermal resistance and dimensionless temperature difference in microchannel heated walland cooling ﬂuid. Sui et al. [19] numerically investigatedthe ﬂuid ﬂow in the wavy microchannels. Their numericalresults indicated that with the uniform cross section, thethermal performance of wavy microchannel is higher thanthe rectangular ﬂat one. Ho et al. [20] studied the forcedconvection cooling performance of a Copper microchannelheat sink with water/Al
2
O
3
nanoﬂuid as the cooling ﬂuid.Their results showed that the heat sink cooled by nanoﬂuid,comparing to the heat sink cooled by water, has moreaverage heat transfer coefﬁcient. Till now, numerousresearches about the heat transfer in the microchannels andnanoﬂuid have been presented, and sometimes, the slipvelocity conditions, the effects of magnetic ﬁeld and theforced heat transfer under the inﬂuence of constant temperature or constant heat ﬂux have been investigated disparately [21–35]. Nikkhah et al. [36] numerically studied
the water nanoﬂuid/functional multiwalled carbon nanotubes in a twodimensional microchannel with slip andnoslip boundary conditions. They concluded that theaugment of solid nanoparticles weight fraction and slip
1554 A. Arabpour et al.
1 3
velocity coefﬁcient cause the increase in Nusselt number,and in higher Reynolds numbers, this enhancement is moreconsiderable. In their research, the computational ﬂuiddynamics and laminar heat transfer of kerosenenanoﬂuid/multiwalled carbon nanotubes in the doublelayer microchannel heat sink are simulated in the twodimensional domain. By considering the effect of slipboundary condition on the outcome results of numericalsimulation, in this study, the slip velocity boundary condition on the solid walls is used. The results of this researchare presented for different volume fractions of nanoparticles, slip velocity coefﬁcients and different ranges of Reynolds numbers. The main purpose of this study isinvestigating the behavior of temperature domain andhydrodynamic of laminar ﬂow of nanoﬂuid in the twodimensional doublelayer microchannel.
Problem statement
In the present study, the laminar ﬂow of kerosenenanoﬂuid/multiwalled carbon nano tubes in volume fractions of 0, 4 and 8% of nanoparticles is investigated. Figure 1 indicates the studied geometrics of this paper. In thisresearch, the material of microchannel is silicon. In Fig. 1,the bottom layer of microchannel is
L
=
3 mm and theheight is
H
=
50
l
m. The top layer of microchannel withthe length of
L
2
is equal to
L
2
=
1.5 mm, and by placing onthe bottom layer at the interface area, the heat transferswith it and in this region, the amount of heat generation isconstant and is equal to 100 kw/m
3
. In each layer of microchannel, the silicon material with the thickness of
t
=
12.5
l
m has surrounded the layers. The external areasof top layer with the length of
L
2
are insulated, and thebottom area of microchannel, on the external wall with thelength of
L
, is under the inﬂuence of sinusoidal ﬂux withthe equation of
q
00
X
ð Þ¼
2
q
00
0
þ
q
00
0
sin
p
X
4
in which theamount is calculated from the equation of (
q
0
00
). With thedeﬁnition of dimensionless slip velocity coefﬁcient as(
B
=
b
/
H
), the ratio of slip velocity coefﬁcient to theheight of microchannel, in this research, the numericalsimulation is done for the dimensionless slip velocitycoefﬁcients (
B
=
b
/
H
) of 0.001, 0.01 and 0.1 and Reynoldsnumbers of 1, 10 and 100. The inlet ﬂuid at the top andbottom layers enters with the temperature of 301 K asshown in Fig. 1. All of the internal walls which are incontact with ﬂuid have the slip velocity boundary condition. The used nanoﬂuid properties of this simulation andthe material of microchannel wall are described, respectively, in Table 1.In this simulation, the ﬂuid ﬂow and heat transfer areconsidered as laminar and fully developed. The nanoﬂuidproperties are considered as constant and independent fromthe temperature. The solid–liquid suspension in less densities is modeled as singlephased, and on the channelwalls, the oscillating heat ﬂux is applied. The slip boundarycondition is used on the microchannel. The numericalsimulation domain is two dimensional.
Governing equations
The dimensionless governing equations on the simulationdomain are deﬁned as follows [39, 40]:
Continuity equation
:
o
U
o
X
þ
o
V
o
Y
¼
0
ð
1
Þ
Momentum equation
:
U
o
U
o
X
þ
V
o
U
o
Y
¼
o
P
o
X
þ
l
nf
q
nf
m
f
1
Re
o
2
U
o
X
2
þ
o
2
U
o
Y
2
ð
2
Þ
U
o
V
o
X
þ
V
o
V
o
Y
¼
o
P
o
Y
þ
l
nf
q
nf
m
f
1
Re
o
2
V
o
X
2
þ
o
2
V
o
Y
2
ð
3
Þ
Energy equation
:
U
o
h
o
X
þ
V
o
h
o
Y
¼
l
nf
a
f
1
Re Pr
o
2
h
o
X
2
þ
o
2
h
o
Y
2
ð
4
Þ
For nondimensioning Eqs. (1)–(4), following parame
ters are used [36]:
Insoulation Solid Heat generation
L
1
L
1
LC B D H t A
U
in
,
T
in
U
in
,
T
in
q
′′
(
X
) = 2
q
0
′′ +
q
0
′′
sin
(
π
X
4
)
MWCNT//kerosene nanofluid
Xy
Fig. 1
The studied schematics of this research
Table 1
The thermophysical properties of base ﬂuid and nanoparticle of multiwalled carbon nanotubes and silicon [37, 38]
u
/%
q
/ kg m

3
C
p
/ J kg

1
K

1
k
/ W m

1
K

1
l
/Pa s
Pr
0 783 2090 0.145 0.001457 214 815 1989 0.265 0.001613 12.18 845 1895 0.390 0.001795 8.72Silicon 2329 702 124 – –The study of heat transfer and laminar ﬂow of kerosene/multiwalled carbon nanotubes (MWCNTs)
…
1555
1 3
X
¼
x H Y
¼
y H V
¼
vu
c
h
¼
T
T
c
D
T U
¼
t
u
c
B
¼
b
H
D
T
¼
q
00
0
H k
f
Pr
¼
t
f
a
f
P
¼
P
q
nf
u
2c
ð
5
Þ
Another parameter for investigating the microchannelperformance is the friction coefﬁcient which is calculatedfrom the following equation [41]:
C
f
¼
2
s
w
q
u
2in
ð
6
Þ
The average Nusselt number can be obtained as follows[42, 43]:
Nu
x
¼
h
H k
f
!
Nu
ave
¼
1
L
Z
L
0
Nu
x
X
ð Þ
d
X
ð
7
Þ
The amounts of thermal resistance [44, 45] of bottom
wall of microchannel and pressure drop are calculated fromthe following equation:
R
¼
T
max
T
min
q
00
0
A
¼
T
max
T
in
q
00
0
A
!
A
¼
W
L
!
R
W
¼
T
max
T
min
q
00
0
L
ð
8
Þ
D
P
¼
P
in
P
out
ð
9
Þ
In Eq. (9),
T
max
,
T
min
,
A
and
q
0
00
are, respectively, themaximum temperature of bottom wall, the minimum temperature (the temperature of inlet ﬂuid), cross section andthe applied heat ﬂux to the AB wall.
The governing boundary conditionson the problemsolving
The hydrodynamic and thermal boundary conditions usedin this problem are as follows:
U
¼
1
;
V
¼
0 and
h
¼
0 for
X
¼
0 and0
:
25
Y
1
:
25 and
X
¼
60
;
1
:
75
Y
2
:
75
V
¼
0 and
o
h
o
X
¼
o
U
o
X
for
X
¼
60 and0
:
25
Y
1
:
25 and
X
¼
30
;
1
:
75
Y
2
:
75
V
¼
0
;
U
¼
0 and
o
h
o
Y
¼
2
q
00
0
þ
q
00
0
sin
p
X
4
for
Y
¼
0 and 0
X
60
V
¼
0
;
U
s
¼
B
o
h
o
Y
and
k
nf
o
h
o
Y
¼
k
s
o
h
o
Y
for
Y
¼
0
:
25 and 0
X
60
V
¼
0
;
U
¼
0 and
o
h
o
Y
¼
0 for
Y
¼
1
:
5 and0
X
30 and
Y
¼
3 and 0
X
60
V
¼
0
;
U
s
¼
B
o
U
o
Y
and
k
nf
o
h
o
Y
¼
k
s
o
h
o
Y
for
Y
¼
1
:
25 and 0
X
60
V
¼
0
;
U
s
¼
B
o
U
o
Y
and
k
nf
o
h
o
Y
¼
k
s
o
h
o
Y
for
Y
¼
1
:
75 and 30
X
60
V
¼
0
;
U
s
¼
B
o
U
o
Y
and
k
nf
o
h
o
Y
¼
k
s
o
h
o
Y
for
Y
¼
2
:
75 and 30
X
60
ð
10
Þ
The mesh study and numerical solving procedure
In order to ensure the results independency of thisresearch, the rectangular organized grids have changedfrom the number of 30,000 to 100,000. The studiedparameters in the validation of present investigation areincluding Nusselt number along the AB wall and theamount of pressure drop. The changes in these twoparameters are investigated in Reynolds numbers of 10and 100 and volume fraction of 8% of nanoparticles inthe slip velocity coefﬁcient of 0.01. According to Table 2,by choosing grid number of 100,000, comparing to othergrid numbers, more accurate results can be obtained.However, the grid number of 63,000, compared to thegrid number of 100,000, has acceptable error and lessdemanded time for solving the numerical domain; therefore, in this numerical simulation, the grid number of 63,000 has been used. In this study, in order to enhancethe solving accuracy, to couple velocity and pressure,SIMPLEC algorithm [46, 47] has been used, and the
maximum loss for results convergence of this simulationhas been chosen 10

6
[48–50].
Table 2
The changes in studied grid numbers in the present study
Re
Parameters Grid point30,000 50,000 63,000 100,000
Re
=
100
Nu
ave
9.786 10.2103 10.4661 10.623Error 7.9% 3.89% 1.48% Base grid
D
P
/Pa 101,231 97,635 94,908.5 94,851Error% 6.72% 2.94% 0.06% Base grid
Re
=
10
Nu
ave
3.68 4.011 4.0248 4.101Error 10.27% 2.2% 1.86% Base grid
D
P
/Pa 9845 9271 9161.3 9100.5Error 8.2% 1.88% 0.67% Base grid1556 A. Arabpour et al.
1 3
Results and discussion
Validation
The results of the present study have been validated withthe numerical study of Nikkhah et al. [36] in Reynoldsnumber of 100 for the dimensionless temperature parameter at central section of ﬂow. Nikkhah et al. [36] numerically investigated the laminar ﬂow and heat transfer of water nanoﬂuid/functional carbon nanotubes in a rectangular microchannel with the ratio of length to the height of channel equal to 32. Their investigation has been done inReynolds numbers of 1–100 for volume fractions of 0–0.25% of nanoparticles. According to Fig. 2 and propercoincidence of the results of the present research with thestudy of Nikkhah et al. [36], it can be said that the solvingprocedure and the applied boundary conditions areaccurate.
X
051015202530
θ
H / 2
0.000.020.040.060.080.100.12
My study Nikkhah et al. [36]
ϕ
= 0.12
Re
= 100
Fig. 2
The validation with numerical study of Nikkhah et al. [36]
0.275091 0.825272 1.37545 1.92563 2.47582 3.026 4.126363.57618
B
= 0.1
B
= 0.01
B
= 0.001
0 0.001 0.002 0.003
x
/m
0 0.001 0.002 0.003
x
/m
0 0.001 0.002 0.003
x
/m
Fig. 3
The changes in dimensionless temperature in Reynoldsnumber of 1 and different dimensionless slip coefﬁcients in volumefraction of 0%
0.275091 0.825272 1.37545 1.92563 2.47582 3.026 4.126363.57618
B
= 0.1
B
= 0.01
B
= 0.001
0 0.001 0.002 0.003
x
/m
0 0.001 0.002 0.003
x
/m
0 0.001 0.002 0.003
x
/m
Fig. 5
The changes in dimensionless temperature in Reynoldsnumber of 1 and different dimensionless slip coefﬁcients in volumefraction of 8%
0.275091 0.825272 1.37545 1.92563 2.47582 3.026 4.126363.57618
B
= 0.1
B
= 0.01
B
= 0.001
0 0.001 0.002 0.003
x
/m
0 0.001 0.002 0.003
x
/m
0 0.001 0.002 0.003
x
/m
Fig. 4
The changes in dimensionless temperature in Reynoldsnumber of 1 and different dimensionless slip coefﬁcients in volumefraction of 4%The study of heat transfer and laminar ﬂow of kerosene/multiwalled carbon nanotubes (MWCNTs)
…
1557
1 3