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Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance
Advanced Powder Technology xxx (2017) xxx–xxx
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Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt
Original Research Paper
Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance S.J. Hosseini a, A. Malvandi b,⇑, S.A. Moshizi c, M. Zamani d a
Department of Mechanical Engineering, School of Engineering, University of Tehran, Tehran, Iran Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran Young Researchers and Elite Club, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran d Young Researchers and Elite Club, Gonabad Branch, Islamic Azad University, Gonabad, Iran b c
a r t i c l e
i n f o
Article history: Received 29 August 2016 Received in revised form 24 October 2016 Accepted 1 November 2016 Available online xxxx Keywords: Thermal performance Anomalous heat transfer rate Nanoparticle migration Thermophoresis Brownian motion
a b s t r a c t This is a theoretical investigation on fully developed mixed convective flow of nanofluids inside microtubes subjected to a constant wall temperature (CWT). The modified Buongiorno model is used for the nanofluids which fully accounts for the distribution of nanoparticles concentration on thermophysical properties. The effect of nanoparticles migration originating from the nano-scale diffusivities including thermophoretic diffusion (temperature-gradient driven force) and Brownian diffusion (concentrationgradient driven force) on the thermophysical characteristics of nanofluids has been considered. A Navier’s slip condition is considered at the wall to model the non-equilibrium region at the fluid-solid interface in micro-scale channels. A scale analysis is performed to estimate the relative significance of the pertaining parameters that should be included in the governing equations. The effects of pertinent parameters including the ratio of Brownian motion to thermophoresis (NBT), slip parameter (k), mixed convective parameter (Nr), and bulk mean nanoparticle volume fraction (/B) on the flow and thermal fields are investigated. The figure of merit (FoM) is used to measure the thermal performance of equipment and finding the optimum thermal condition. It is shown that increasing the buoyancy force would enhance the heat transfer rate, especially for the larger nanoparticles. Also, larger nanoparticles enhance the thermal performance based on a required heat transfer rate with the lowest penalty in the pressure drop. Ó 2016 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
1. Introduction Nanofluids (colloidal suspensions of nanoparticles in base fluid) possess novel properties including the greater specific surface area, more stable colloidal suspension, lower pumping power for a specific heat transfer rate, reduced clogging compared to regular cooling colloids, and the ability to adjust the thermophysical properties of suspensions by changing the nanoparticle materials and physical conditions [1], volume fraction of particles, particles size, and their shape. These novel characteristics make nanofluids suitable for several industrial applications such as pharmaceutical processes (drug delivery), surfactant and coating, cooling in heat exchangers, fuel cells, hybrid-powered engines, solar PV, and microelectromechanical systems (MEMs). ⇑ Corresponding author. E-mail address: [email protected] (A. Malvandi).
Several theoretical models have been introduced so far to calculate the behavior of nanofluids on convective heat transfer. The proposed models, however, depend on certain inputs from experimental observations. Each model acquiring the best conformity to the experimental observations is construed as an accurate model by those researchers. In the literature, the heat transfer coefficients were determined by modeling the nanofluid as either single or two-phase flow. The most important findings from the experiments are: (a) an abnormal increase in the thermal conductivity of nanofluids with respect to the regular fluid [2]; (b) an abnormal increase in the viscosity of nanofluids relative to the regular fluid [3]; and (c) an abnormal single-phase heat transfer coefficient of nanofluids with respect to the regular fluid [4]. In 2006, Buongiorno [5] proved that the single-phase model as well as the dispersion models cannot accurately follow the experimental observations. Accordingly, he proposed a two-component (solid and fluid) four-equation (continuity, momentum, energy, and
http://dx.doi.org/10.1016/j.apt.2016.11.019 0921-8831/Ó 2016 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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Nomenclature cp dp DB DT FoM g h k kBO N Nr Nu NBT P qw Ro T u x; r
specific heat capacity (m2/s2 K) nanoparticle diameter (m) Brownian diffusion coefficient thermophoresis diffusion coefficient figure of merit gravity (m/s2) heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) Boltzmann constant (¼ 1:3806488 1023 m2 kg=s2 K) slip velocity factor mixed convective parameter Nusselt number ratio of the Brownian to thermophoretic diffusivities pressure (Pa) surface heat flux (W/m2) outer radius (m) temperature (K) axial velocity (m/s) coordinate system
nanoparticle flux) heterogeneous equilibrium model to illuminate the experimental findings. In the Buongiorno model, nanoparticle fluxes are considered in accordance with the two important slip mechanisms: Brownian diffusion (or Brownian motion) and thermophoresis (or thermophoretic diffusion). Next, after taking Buongiorno’s model into consideration in different geometries, several investigations are performed on the convective heat transfer in nanofluids; for instance, Sheremet et al. [6], Yadav and Linho [7], Sheikholeslami et al. [8,9], Sandeep and Malvandi [10], Garoosi et al. [11], and Nield and Kuznetsov [12]. Dinarvand et al. [13] probed the double-diffusive mixed convective boundary layer flow of a nanofluid near stagnation point region over a vertical surface with taking account of the Buongiorno’s model. Theoretical investigation on the effect of nanofluids has been systematically reported and well documented, which can be found in the open literatures [14–19]. Moreover, many researches such as Sheremet and Pop [20–22], Sheremet et al. [23] and Bondareva et al. [24] have been carried out on convective heat transfer in nanofluids under effects of the Brownian diffusion and thermophoresis. Comprehensive review papers on the application of nanofluids are conducted by Sheikholeslami and Ganji [25], Bahiraei [26], and Salman et al. [27]. Yang et al. [28] was modified the Buongiorno model to examine the impact of nanoparticle distribution on the thermal conductivity and viscosity of nanofluids. Their proposed modified model did not ignore the dependency of thermophysical properties of nanofluids to the nanoparticles volume fraction. Their results indicated that the non-uniformity of the thermophysical properties is the reason for the anomalous heat transfer enhancement. Malvandi and Ganji [29], then, used the modified model to examine the mutual impacts of buoyancy and nanoparticle migration on the mixed convection of nanofluids in vertical annuli. Subsequently, Malvandi and Ganji [30] investigated the laminar flow and convective heat transfer of alumina/water nanofluid inside a circular microchannel in the presence of a uniform magnetic field. Their results indicated that the nanoparticles migrate from the heated walls to the core region of the microchannel and form a nonuniform nanoparticles distribution. It was further observed that for smaller nanoparticles, the nanoparticle volume fraction is more uniform and abnormal variations in the heat transfer rate vanish. Hedayati and Domairry [31] investigated the effects of the nanoparticle migration on titania/water nanofluids in horizontal
Greek symbols / nanoparticle volume fraction c ratio of wall and fluid temperature difference to absolute temperature g transverse direction l dynamic viscosity (kg/m s) q density (kg/m3) k slip parameter Subscripts B bulk mean bf base fluid p nanoparticle w condition at the wall Superscript ⁄ dimensionless variable
and vertical channels. Recently, Malvandi and Ganji [32] took advantage of the modified Buongiorno model so as to evaluate the effects of temperature-dependent thermophysical properties on nanoparticle migration inside microchannels at the mixed convective heat transfer of nanofluids. The popularity of modeling the nanoparticle migration can be gauged from the numerous published literatures such as [33–38]. The current study is motivated by the need to examine the detailed behavior of nanoparticles movements for nanofluid flow inside circular microchannels and how they can tune the thermal performance and heat transfer rate. Recently, Malvandi and Ganji [39] demonstrated that because of thermophoresis, asymmetrically heated walls are able to tune and control the fluid flow and heat transfer characteristics of nanofluids. Moreover, the rate of heat transfer enhancement can be controlled by adjusting the heat flux at the boundaries. These observations show that thermal boundary condition is a salient factor on heat transfer characteristics of nanofluids. Consequently, in the current investigation, a constant wall temperature is prescribed at the wall, which is an important development for the modified Buongiorno model [28,29]. The fully developed governing equations of the modified Buongiorno model for the constant wall temperature are obtained for the first time and the results for the pressure drop and the heat transfer enhancement are presented versus varying pertinent parameters. Due to the low dimensional structures in microtubes, a linear slip condition is considered at the surfaces, which adequately represents the non-equilibrium region at the fluid/solid interface. To study the thermal performance, the figure of merit (FoM) is calculated to signify it. The impact of constant wall temperature at the wall on nanoparticle migration, the heat transfer rate as well as the thermal performance is of our particular interests.
2. Problem formulation The geometry of the problem under consideration with the adapted coordinate system is demonstrated in Fig. 1. To give a more physical outlook of the problem, the distributions of velocity and nanoparticle volume fraction are schematically illustrated. The cylindrical coordinates x and r were aligned parallel and perpendicular to the wall respectively. The slip velocity is considered at
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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1 @ @/ DT @T DB þ ¼0 r @r @r T @r
ð3Þ
where u, T, and p represent the axial velocity, local temperature, and pressure respectively. In addition, q, l, k, and cP are respectively the density, dynamic viscosity, thermal conductivity, and specific heat capacity of nanofluid obtained via [3,5]
q ¼ /qp þ ð1 /Þqbf ;
ð4Þ
l ¼ lbf ð1 þ al / þ bl /2 Þ;
ð5Þ
k ¼ kbf 1 þ ak / þ bk /2 ;
ð6Þ
cp ¼
/qp cpp þ ð1 /Þqbf cpbf ; /qp þ ð1 /Þqbf
ð7Þ
where the thermophysical properties of nanoparticles and base fluid (water) are provided in Table 1. Note that Eqs. (5) and (6) corresponds to the room-temperature viscosity and nanofluid thermal conductivity measured by Pak and Cho [4] respectively. Moreover, in Eq. (3), the Brownian motion coefficient DB and thermophoresis coefficient DT can be defined as [5]
DB ¼
kBO T 3plbf dp
ð8Þ
and Fig. 1. The geometry of physical model and coordinate system.
DT ¼ 0:26
the wall which allows the formation of axial velocity component at the wall. Also, the following assumptions have been made:
Incompressible flow, No chemical reactions, Negligible external forces, Dilute mixture, Negligible viscous dissipation, Negligible radiation, Local thermal equilibrium between the nanoparticles and base fluid.
Consequently, the incompressible conservation equations of the momentum, thermal energy, and nanoparticle volume fraction can be written in the following manner [5,30,39]: 1 d du dp rlð/Þ þ ½ð1 /B Þqbf bðT T B Þ ðqp qbf Þð/ /B Þg ¼ r dr dr dx
ð1Þ
@T 1 @ @T @/ DT @T @T rkð/Þ þ qp cPp DB þ qð/ÞucP ð/Þ ¼ @x r @r @r @r T @r @r
ð2Þ
Table 1 Physical properties of nanoparticles and base fluid (water) [33]. Physical properties
Fluid phase (water)
Al2O3
q (kg/m3)
998.2 4182 0.597 9.93 104
3880 773 36
cP (J/kg K) k (W/m K) l (kg/m s)
lbf kbf / 2kbf þ kp qbf
ð9Þ
respectively. kBO is the Boltzmann constant and dp is the nanoparticle diameter, which varies from 1 to 100 nm. To solve the governing equations, appropriate boundary conditions can be expressed as @T @r
r ¼ 0 ! du ¼ 0; dr
¼ 0;
r ¼ Ro ! u ¼ N du ; dr
@/ @r
¼ 0;
T ¼ Tw;
DB @/ þ DTT @r
@T @r
ð10Þ
¼ 0:
where N introduces the slip velocity factor. To describe the boundary conditions of Eq. (10) physically, it should be said that in the center of microtube there are zero gradients of temperature, velocity and nanoparticle concentration owing to symmetric annular boundaries. In the circumference of the microtube, the nanofluid flow is faced with the fluid-solid interface, thereby leading to a slip velocity related to the velocity gradient on the wall. In addition, the temperature of the nanofluid flow is constantly considered (Tw) and the nanoparticles flux is zero at the wall. 2.1. Scale analysis In this study, we take advantage of the scale analysis of the governing equations (1)–(3) which can be simplified effectively. Table 2 provides the device and material properties for the present physical conditions, typically occurring in the flow of nanofluids inside microtubes. Regarding Table 2, the order of each term in the governing equations can be estimated. Considering Eq. (1), the scale analysis can be written as follows
Dp Lref
l DU L2ref |ffl{zffl}
;
ð1 /Þqbf gbDT ; |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
ðqp qbf Þg D/ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
98
2:8104
ð11Þ
3:52103
Table 2 Device and material properties of alumina/water nanofluid. Reference variables Values
uref (m/s) 0.01
/ref 0.02
Lref ¼ Ro (m) 7:5 10
5
lref (kg/(m s))
g (m/s2)
DT (K)
0.00198
10
50
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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From Eq. (11), it can be deduced that the buoyancy effect due to the temperature gradient (second RHS term) can be neglected with respect to the buoyancy effect due to nanoparticles concentration gradient (third RHS term) and also the shear stress term (first RHS term). Thus, this term can be removed from Eq. (1). Accordingly, the scale analysis for Eq. (2) can be expressed asEq. (12) indicates that the LHS term is in the same order of magnitude with the first term of RHS and these are about 105 times more than the second RHS term. In fact, heat transfer associated with nanoparticle diffusion (third RHS term) can be neglected in comparison with the other terms. Hence, the governing equations (Eqs. (1)–(3)) can be written as:
axial conduction effect, the following relation can be determined for the axial temperature gradient [5]
@T T w T dT B ¼ ; @x T w T B dx
ð20Þ
where TB is the bulk temperature defined as
R Ro
R
T B R0 Ro 0
1 qð/Þcp ð/ÞuTr dr qð/Þcp ð/ÞuTð1 gÞdg : ¼ R0 1 qð/Þcp ð/Þur dr qð/Þcp ð/Þuð1 gÞdg 0
ð21Þ
Substituting Eq. (20) into Eq. (14) yields the energy equation as
qð/Þcp ð/Þu
T w T dT B 1 @ @T rkð/Þ ¼ T w T B dx r @r @r
ð22Þ
LHS
8 RHS DT > > > RHS1 ¼ kbf ð1 þ 7:47/B Þ 2 > > L > ref > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > 2:0861010 < U ref DT P 1 0 ð/B qp cpp þ ð1 /B Þqbf cpbf Þ > D x > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} B D D/ > > D T DT C C DT B B > > 2:0285108 þ C > RHS2 ¼ qp cpp B > T ref Lref A Lref > |fflffl{zfflffl}@ Lref > > |fflffl{zfflffl} |fflfflffl{zfflfflffl} |{z} : 6 310
1 d du dp r lð/Þ ðqp qbf Þð/ /B Þg ¼ 0 r dr dr dx
qð/ÞucP ð/Þ
1:97107
ð13Þ
@T 1 @ @T rkð/Þ ¼0 @x r @r @r
ðdp=dxÞR2o =lbf
:
ð16Þ
u00 þ Au ðgÞu0 þ Bu ðgÞ ¼ 0;
ð17Þ d dg
lbf ð1 Nrð/ /B ÞÞ: lð/Þ
1 A
Z A
C dA ¼
Z
1
p
R2o
0
Ro
C2pr dr ¼ 2
Z 0
1
Cð1 gÞdg
ð24Þ
B Finding dT from Eq. (23), substituting it into Eq. (22) and introducdx ing the following non-dimensional variables
T ¼
T TB ; Tw TB
HTC ¼
2Ro h ; kbf
ð25Þ
reduce Eq. (22) to
ð26Þ
⁄
where T is the dimensionless temperature and HTC stands for the heat transfer coefficient and
d/ d 1 ðLnðkð/ÞÞÞ ; dg d/ 1g kbf HTC qð/Þucp ð/Þ : BT ðgÞ ¼ C T ðgÞ ¼ hqð/Þucp ð/Þi kð/Þ
AT ðgÞ ¼
ð18Þ
where
gðqp qbf Þ : Nr ¼ ðdp=dxÞ
hCi
in which
d/ d 1 ðLnðlð/ÞÞÞ Au ðgÞ ¼ ; dg d/ 1g Bu ðgÞ ¼
ð23Þ
T 00 þ AT ðgÞT 0 þ BT ðgÞT þ C T ðgÞ ¼ 0
Thus, Eq. (1) reduces to
where 0 (prime symbol) represents
" # dT B 2 @T @T 2q ¼ w; rkð/Þ ¼ 2 rkð/Þ @r r¼Ro @r r¼0 dx Ro Ro
where
ð15Þ
The governing equations can be reduced to non-dimensional form via the following non-dimensional variables
u ¼
Eq. (22) can be reduced by integrating in the following manner
ð14Þ
2.2. Dimensionless momentum equation
u
6:67105
hqð/Þucp ð/Þi
1 @ @/ DT @T DB þ ¼0 r @r @r T @r
R r g¼ o ; Ro
5:54107
ð12Þ
ð27Þ
2.4. Dimensionless nanoparticle continuity
ð19Þ
According to Eq. (25), Eq. (15) can be written as
/0 ¼ 2.3. Dimensionless energy equation
ðT w T B Þ DT 0 T ½T ðT w T B Þ þ T B DB
ð28Þ
Introducing the following non-dimensional parameters Before the energy equation is solved, the thermally fully developed condition should be employed. For the nanofluids in a circular tube along with a prescribed wall temperature and ignoring the
c¼
Tw TB ; TB
NBT ¼
DBB /B ; DT B c
ð29Þ
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
yields Eq. (28) as
/0 þ A/ ðgÞT 0 ¼ 0
ð30Þ
A/ ðgÞ ¼
/ NBT ð1 þ cT Þ
2
:
g ¼ 0 : u ¼ k dudg ; T ¼ 1; / ¼ /w ;
ð33Þ
3. Numerical method and accuracy
To solve the non-dimensional governing equations, the apt boundary conditions should be derived. With regard to Eqs. (16) and (25), the boundary conditions in Eq. (10) can be written in the non-dimensional form as
g ¼ 1 : dudg ¼ 0;
hu /i : hu i
ð31Þ
2.5. Dimensionless boundary condition
(
rather than /w, the following relation is used as an additional constraint [28,33,40]
/B
where
5
@T @g
¼ 0;
ð32Þ
where k ¼ RNo . Eq. (32) represents the slip velocity and constant temperature at the wall. Since the value of /B is usually prescribed
The governing equations including Eqs. (17), (26), and (30) are the governing equations representing a system of nonlinear Ordinary Differential Equations (ODEs) which are going to be solved with the boundary conditions of Eq. (32). It is worth noting that the values of BT (g) and CT(g) in Eq. (26) are not specified in advance, because hqð/Þcp ð/Þui depends on the nanoparticle volume fraction profile and it is determined when the governing equations are solved. Furthermore, there is a similar condition for /w in Eq. (32). In order to convert the boundary-value problem to initial-value one, we used the method which is quite similar to the shooting method through which the boundary-value problem can turn to the initial-value problem. Therefore, a reciprocal procedure is required to solve the governing equations for the unknown
Fig. 2. Algorithm of the numerical method.
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
/B. This process is iterated until the prescribed value of /B is achieved, and the relative error between the assumed value of hqð/Þcp ð/Þui and the calculated one after solving the governing equations is lower than 106. To corroborate the accuracy of the numerical code, the velocity and temperature profiles for the laminar fully developed flow in circular channel with Nr = /B = k = 0 are compared to the reported results of Kays et al. [41] in Fig. 3. As is evident, the results are in the best agreement. Moreover, to verify the grid independence of the numerical results, they are presented in three different integration steps (dg) of 102, 5 103 and 103 in Table 3; obviously,
Fig. 3. Comparison of velocity (a) and temperature profile (b) for regular fluid with the results of Kays et al. [41].
Table 3 Grid independence test for different values of dg when /B ¼ 0:02; k ¼ 0:1; Nr ¼ 20 and N BT ¼ 1. dg
h=hbf
N p =N pbf
10
1.21493
1.86574
5 103
1.21495
1.86577
103
1.21496
1.86579
2
values of hqð/Þcp ð/Þui and /w. The algorithm of numerical approach is demonstrated in Fig. 2. Firstly, the values of parameters including /B, k, Nr and NBT are input to the code. Then, initial guesses for the unknown values of hqð/Þcp ð/Þui and /w have been considered. After the governing equations with the boundary conditions have been solved via the Runge-Kutta-Fehlberg method, the calculated values of hqð/Þcp ð/Þui and /B are compared with the assumed value of hqð/Þcp ð/Þui and the prescribed value of
Fig. 4. The effects of N BT on nanoparticle distribution (/=/B ), velocity u uB , and temperature (T ) profiles when /B ¼ 0:02; k ¼ 0:1 and Nr ¼ 20.
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
Fig. 5. The effects of /B on nanoparticle distribution ð/=/B Þ, velocity u uB , and temperature (T ) profiles when k ¼ 0:1; Nr ¼ 20 and N BT ¼ 1.
altering the grid size has insignificant effect on the results. It should be stated that all the following numerical results have been acquired using the integration step dg = 103.
7
Fig. 6. The effects of k on nanoparticle distribution (/=/B ), velocity u uB , and temperature (T ) profiles when /B ¼ 0:02; Nr ¼ 20 and N BT ¼ 1.
range of NBT from 0.25 to 10 is considered. The selected values of NBT lead to different modes of nanoparticle migration which will B be discussed later. In addition, the value of c ffi T wTT is assumed B
4. Results and discussion
to be 0.1 because of its trivial effect on the solution [28,29].
There are some background information which should be explained in advance. As was maintained in previous studies [30], for the nanoparticles with diameter around and less than 100 nm, NBT is in the order of 1; so, in the present study, a wide
4.1. Graphical results Figs. 4–7 provide graphical representation of dimensionless nanoparticle volume fraction, velocity, and temperature profiles
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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values of NBT, as illustrated in Fig. 4. For the higher values of NBT, the Brownian diffusion dominates the thermophoresis and eliminates the nanoparticle concentration gradient through the microtube. Also, it can be seen that the velocity and temperature profiles are completely affected by the nanoparticle volume fraction distribution. Indeed, as opposite to the regular fluids, the hydrodynamic and thermal characteristics of nanofluids are directly affected by the thermophoresis or nanoparticle migration. It is worth mentioning that NBT is inversely proportional to the nanoparticle diameter (dp); hence, the nanoparticle diameter is a key factor affecting the flow and thermal fields. Furthermore, it can be seen in the figure that the peak of velocity profile reduces for the larger nanoparticles (lower values of NBT) in which the nanoparticle migration is strong because a rise in the nanoparticle volume fraction in the central region of microtube intensifies the viscosity of the nanofluid, thereby decreasing the velocity. The same behavior can be observed for the thermal conductivity and the temperature. Fig. 5 reveals that an increase in /B declines the peak of nanoparticle concentration and velocity profiles; nonetheless, increasing /B grows the dip of the temperature. This is due to the fact that rising /B boosts the nanoparticle volume fraction, and therefore it increases the impact of the Brownian diffusion. This dynamic shifts the nanoparticles toward the central region of the microtubes, which has a higher kinetic energy (a higher velocity). In addition, due to a lower kinetic energy of nanoparticle near the wall, Brownian diffusion effect is negligible and there is no significant change there. The uniform distribution of nanoparticles at the higher values of /B has a direct impact on the viscosity and thermal conductivity of nanofluids; consequently, the velocity and temperature gradients reduce at the higher values of /B. Fig. 6 illustrates that growing the slip parameter (k) increases the slip velocity at the wall, followed by a reduction in the velocity peak in the central region of microtube. In fact, the velocity of the nanofluid accelerates near the walls and decelerates at the center of microtube. In addition, the migration of nanoparticles enhances for the higher values of k, thereby decreasing the dip of temperature profile. Fig. 7 depicts that a rise in Nr reduces the velocity peak in the central region of tube, however, it slightly accelerates the velocity at the walls. Also, it can be seen that increasing Nr leads to a gradual increase in the nanoparticle volume fraction at the core region of microtube. In fact, Nr enhances the non-uniformity of nanoparticle volume fraction. 4.2. Tabular data There are two main physical quantities of interest in heat exchange equipments, namely the heat transfer rate and the pressure drop. For nanofluids, it is beneficial to examine the heat transfer enhancement and pressure drop increment, demonstrating the merit of nanoparticle inclusion. For the laminar and fully developed flow inside a circular microchannel, the heat transfer coefficient (HTC) at the wall can be written as Fig. 7. The effects of Nr on nanoparticle distribution (/=/B ), velocity u uB , and temperature (T ) profiles when /B ¼ 0:02; k ¼ 0:1 and N BT ¼ 1.
for different values of NBT, /B, k, and Nr, respectively. In the figures, g = 0 denotes the surface condition while g = 1 represents the center of the microtube. Obviously, nanoparticle volume fraction nonuniformly distributed through the microtubes, especially for the lower values of NBT; a lower concentration of nanoparticles places at the wall (g = 0) which gradually increases to its peak at the center of microtube. Consequently, it can be deduced that nanoparticles migrate from the heated wall toward the center of microtubes. This dynamic can be weakened for the very greater
HTC ¼
hDh q00w 2Ro kw 0 ¼ ¼ 2T kbf ðT w T B Þkbf kbf
ð34Þ
where Dh = 2Ro. Therefore, the heat transfer enhancement can be given by
h HTC ¼ hbf HTC bf
ð35Þ
Also, the dimensionless pressure gradient can be provided by
dp Np ¼ dx
,
lbf
!
uB ð2Ro Þ
2
¼
4qB qu ðlbf =lw Þ
ð36Þ
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx Table 4 Heat transfer enhancement, pressure drop increment, and figure of merit for different values of Nr when /B ¼ 0:02; k ¼ 0:1. NBT
0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10
Nr = 10
Nr = 20
Nr = 40
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
1.143 1.156 1.174 1.186 1.192 1.195 1.196 1.196 1.195 1.182 1.173 1.168 1.165 1.162 1.161 1.159 1.158 1.157
1.075 1.104 1.185 1.290 1.405 1.523 1.638 1.750 1.854 2.578 2.949 3.167 3.310 3.410 3.485 3.542 3.588 3.625
1.115 1.118 1.110 1.089 1.064 1.039 1.014 0.992 0.973 0.862 0.818 0.795 0.782 0.772 0.766 0.761 0.757 0.753
1.190 1.198 1.211 1.218 1.221 1.221 1.220 1.217 1.215 1.193 1.181 1.174 1.169 1.166 1.164 1.162 1.161 1.160
1.115 1.138 1.212 1.311 1.423 1.539 1.653 1.762 1.866 2.583 2.951 3.169 3.311 3.411 3.485 3.543 3.588 3.625
1.147 1.148 1.136 1.113 1.085 1.058 1.032 1.008 0.987 0.869 0.823 0.799 0.784 0.775 0.768 0.762 0.758 0.755
1.438 1.355 1.319 1.304 1.293 1.284 1.275 1.267 1.260 1.216 1.196 1.185 1.178 1.174 1.170 1.168 1.166 1.267
1.646 1.415 1.373 1.428 1.515 1.615 1.718 1.819 1.916 2.603 2.962 3.175 3.315 3.414 3.488 3.545 3.590 3.611
1.218 1.207 1.187 1.158 1.126 1.094 1.065 1.038 1.015 0.884 0.833 0.806 0.790 0.779 0.772 0.766 0.761 0.826
Table 5 Heat transfer enhancement, pressure drop increment, and figure of merit for different values of nanoparticle volume fraction when Nr = 20, k ¼ 0:1. NBT
0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10
/B ¼ 0:01
/B ¼ 0:02
/B ¼ 0:04
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
1.099 1.104 1.111 1.114 1.114 1.114 1.113 1.111 1.110 1.097 1.091 1.087 1.085 1.083 1.082 1.081 1.081 1.080
1.047 1.059 1.095 1.142 1.192 1.242 1.290 1.335 1.376 1.639 1.764 1.835 1.880 1.912 1.935 1.953 1.967 1.978
1.083 1.083 1.077 1.065 1.051 1.036 1.022 1.009 0.998 0.930 0.903 0.888 0.879 0.873 0.868 0.865 0.862 0.860
1.190 1.198 1.211 1.218 1.221 1.221 1.220 1.217 1.215 1.193 1.181 1.174 1.169 1.166 1.164 1.162 1.161 1.160
1.115 1.138 1.212 1.311 1.423 1.539 1.653 1.762 1.866 2.583 2.951 3.169 3.311 3.411 3.485 3.543 3.588 3.625
1.147 1.148 1.136 1.113 1.085 1.058 1.032 1.008 0.987 0.869 0.823 0.799 0.784 0.775 0.768 0.762 0.758 0.755
1.366 1.377 1.400 1.415 1.423 1.426 1.425 1.423 1.419 1.381 1.358 1.345 1.337 1.331 1.326 1.323 1.321 1.318
1.318 1.350 1.502 1.733 2.015 2.324 2.645 2.969 3.287 5.770 7.206 8.096 8.695 9.125 9.447 9.698 9.898 10.062
1.246 1.246 1.222 1.178 1.127 1.076 1.030 0.990 0.954 0.770 0.703 0.670 0.650 0.637 0.627 0.620 0.615 0.611
Table 6 Heat transfer enhancement, pressure drop increment, and figure of merit for different values of slip parameter when Nr = 20, /B ¼ 0:02. NBT
0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10
k¼0
k ¼ 0:1
k ¼ 0:2
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
1.184 1.193 1.203 1.208 1.209 1.208 1.206 1.204 1.202 1.183 1.173 1.168 1.165 1.162 1.160 1.159 1.158 1.157
1.140 1.174 1.269 1.384 1.507 1.629 1.747 1.858 1.962 2.658 3.009 3.215 3.349 3.444 3.514 3.568 3.611 3.646
1.134 1.131 1.112 1.084 1.055 1.027 1.002 0.980 0.960 0.854 0.813 0.791 0.778 0.770 0.763 0.759 0.755 0.752
1.190 1.198 1.211 1.218 1.221 1.221 1.220 1.217 1.215 1.193 1.181 1.174 1.169 1.166 1.164 1.162 1.161 1.160
1.115 1.138 1.212 1.311 1.423 1.539 1.653 1.762 1.866 2.583 2.951 3.169 3.311 3.411 3.485 3.543 3.588 3.625
1.147 1.148 1.136 1.113 1.085 1.058 1.032 1.008 0.987 0.869 0.823 0.799 0.784 0.775 0.768 0.762 0.758 0.755
1.182 1.189 1.202 1.210 1.214 1.216 1.215 1.214 1.212 1.192 1.181 1.174 1.169 1.166 1.164 1.162 1.161 1.160
1.100 1.115 1.178 1.269 1.374 1.486 1.597 1.705 1.809 2.536 2.915 3.139 3.287 3.390 3.467 3.527 3.574 3.612
1.145 1.146 1.138 1.118 1.092 1.065 1.040 1.016 0.995 0.874 0.827 0.802 0.787 0.776 0.769 0.764 0.759 0.756
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
where qB is the bulk density. And the pressure drop increment can be written as
Ndp ¼
Np Npbf
ð37Þ
Moreover, to find the thermal performance, the figure of merit (FoM) is employed by heat transfer engineers. For the nanofluids, FoM can be calculate via
FoM ¼
h=hbf ðNp =Npbf Þ1=3
ð38Þ
Tables 4–6 provide information about the effects of Nr, /B, and k on the heat transfer enhancement, pressure drop increment and thermal performance (FoM) for different nanoparticle migration regimes (NBT). At first glance, it is obvious that increasing Nr increases both the heat transfer enhancement and pressure drop increment, especially at the lower values of NBT. Also, the thermal performance (FoM) is increased gradually. This means that the effects of heat transfer enhancement overwhelm the pressure drop increment, as Nr increases. Similarly, increasing /B leads to an increase in both the pressure drop increment and heat transfer enhancement. However, for the thermal performance, the effects of NBT should be considered. As is elucidated from Table 5, FoM increases with /B in the lower values of NBT while it decreases at the higher values of NBT. In other words, increasing /B is advantageous only for larger nanoparticles (lower values of NBT). According to Table 6, it can be observed that an increase in k reduces the pressure drop increment, whereas it leads to a rise in both the heat transfer rate of nanofluids and base fluid. Obviously, the heat transfer enhancement is increased when k increases from 0 to 0.1. It should be noted that further increase in k for lower values of NBT leads to a decrease in the heat transfer enhancement. This means that in this range, the heat transfer enhancement in the base fluid is much larger than the nanofluids. In addition, it is interesting to note that FoM increases for 0 < k < 0.1 and further increase in k has insignificant effects on the thermal performance.
5. Summary and conclusions The present study is a theoretical investigation of the figure of merit for the laminar fully developed mixed convection of nanofluids through microtubes at a constant wall temperature (CWT). The modified Buongiorno’s model is used for the nanofluid to assess the nanoparticle migration originating from the thermophoresis (nanoparticle slip velocity due to temperature gradient) and Brownian motion (nanoparticle slip velocity due to concentration gradient). The thermally fully developed governing equations are developed for a constant wall temperature, reduced to ordinary differential equations by appropriate dimensionless variables, and solved with a reciprocal numerical algorithm. It is shown that increasing nanoparticle concentration enhances the thermal performance for the larger nanoparticles while it reduces the thermal performance for the smaller ones. In addition, increasing the Buoyance force and nanoparticle diameter would enhance the thermal performance. Also, it is obtained that the thermal performance grows asymptotically with the slip parameter. References [1] A. Malvandi, A. Ghasemi, R. Nikbakhti, A. Ghasemi, F. Hedayati, Modeling and parallel computation of the non-linear interaction of rigid bodies with incompressible multi-phase flow, Comput. Math. Appl. 72 (2016) 1055–1065. [2] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Appl. Phys. Lett. 78 (2001) 718–720.
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Original Research Paper
Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance S.J. Hosseini a, A. Malvandi b,⇑, S.A. Moshizi c, M. Zamani d a
Department of Mechanical Engineering, School of Engineering, University of Tehran, Tehran, Iran Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran Young Researchers and Elite Club, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran d Young Researchers and Elite Club, Gonabad Branch, Islamic Azad University, Gonabad, Iran b c
a r t i c l e
i n f o
Article history: Received 29 August 2016 Received in revised form 24 October 2016 Accepted 1 November 2016 Available online xxxx Keywords: Thermal performance Anomalous heat transfer rate Nanoparticle migration Thermophoresis Brownian motion
a b s t r a c t This is a theoretical investigation on fully developed mixed convective flow of nanofluids inside microtubes subjected to a constant wall temperature (CWT). The modified Buongiorno model is used for the nanofluids which fully accounts for the distribution of nanoparticles concentration on thermophysical properties. The effect of nanoparticles migration originating from the nano-scale diffusivities including thermophoretic diffusion (temperature-gradient driven force) and Brownian diffusion (concentrationgradient driven force) on the thermophysical characteristics of nanofluids has been considered. A Navier’s slip condition is considered at the wall to model the non-equilibrium region at the fluid-solid interface in micro-scale channels. A scale analysis is performed to estimate the relative significance of the pertaining parameters that should be included in the governing equations. The effects of pertinent parameters including the ratio of Brownian motion to thermophoresis (NBT), slip parameter (k), mixed convective parameter (Nr), and bulk mean nanoparticle volume fraction (/B) on the flow and thermal fields are investigated. The figure of merit (FoM) is used to measure the thermal performance of equipment and finding the optimum thermal condition. It is shown that increasing the buoyancy force would enhance the heat transfer rate, especially for the larger nanoparticles. Also, larger nanoparticles enhance the thermal performance based on a required heat transfer rate with the lowest penalty in the pressure drop. Ó 2016 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
1. Introduction Nanofluids (colloidal suspensions of nanoparticles in base fluid) possess novel properties including the greater specific surface area, more stable colloidal suspension, lower pumping power for a specific heat transfer rate, reduced clogging compared to regular cooling colloids, and the ability to adjust the thermophysical properties of suspensions by changing the nanoparticle materials and physical conditions [1], volume fraction of particles, particles size, and their shape. These novel characteristics make nanofluids suitable for several industrial applications such as pharmaceutical processes (drug delivery), surfactant and coating, cooling in heat exchangers, fuel cells, hybrid-powered engines, solar PV, and microelectromechanical systems (MEMs). ⇑ Corresponding author. E-mail address: [email protected] (A. Malvandi).
Several theoretical models have been introduced so far to calculate the behavior of nanofluids on convective heat transfer. The proposed models, however, depend on certain inputs from experimental observations. Each model acquiring the best conformity to the experimental observations is construed as an accurate model by those researchers. In the literature, the heat transfer coefficients were determined by modeling the nanofluid as either single or two-phase flow. The most important findings from the experiments are: (a) an abnormal increase in the thermal conductivity of nanofluids with respect to the regular fluid [2]; (b) an abnormal increase in the viscosity of nanofluids relative to the regular fluid [3]; and (c) an abnormal single-phase heat transfer coefficient of nanofluids with respect to the regular fluid [4]. In 2006, Buongiorno [5] proved that the single-phase model as well as the dispersion models cannot accurately follow the experimental observations. Accordingly, he proposed a two-component (solid and fluid) four-equation (continuity, momentum, energy, and
http://dx.doi.org/10.1016/j.apt.2016.11.019 0921-8831/Ó 2016 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
Nomenclature cp dp DB DT FoM g h k kBO N Nr Nu NBT P qw Ro T u x; r
specific heat capacity (m2/s2 K) nanoparticle diameter (m) Brownian diffusion coefficient thermophoresis diffusion coefficient figure of merit gravity (m/s2) heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) Boltzmann constant (¼ 1:3806488 1023 m2 kg=s2 K) slip velocity factor mixed convective parameter Nusselt number ratio of the Brownian to thermophoretic diffusivities pressure (Pa) surface heat flux (W/m2) outer radius (m) temperature (K) axial velocity (m/s) coordinate system
nanoparticle flux) heterogeneous equilibrium model to illuminate the experimental findings. In the Buongiorno model, nanoparticle fluxes are considered in accordance with the two important slip mechanisms: Brownian diffusion (or Brownian motion) and thermophoresis (or thermophoretic diffusion). Next, after taking Buongiorno’s model into consideration in different geometries, several investigations are performed on the convective heat transfer in nanofluids; for instance, Sheremet et al. [6], Yadav and Linho [7], Sheikholeslami et al. [8,9], Sandeep and Malvandi [10], Garoosi et al. [11], and Nield and Kuznetsov [12]. Dinarvand et al. [13] probed the double-diffusive mixed convective boundary layer flow of a nanofluid near stagnation point region over a vertical surface with taking account of the Buongiorno’s model. Theoretical investigation on the effect of nanofluids has been systematically reported and well documented, which can be found in the open literatures [14–19]. Moreover, many researches such as Sheremet and Pop [20–22], Sheremet et al. [23] and Bondareva et al. [24] have been carried out on convective heat transfer in nanofluids under effects of the Brownian diffusion and thermophoresis. Comprehensive review papers on the application of nanofluids are conducted by Sheikholeslami and Ganji [25], Bahiraei [26], and Salman et al. [27]. Yang et al. [28] was modified the Buongiorno model to examine the impact of nanoparticle distribution on the thermal conductivity and viscosity of nanofluids. Their proposed modified model did not ignore the dependency of thermophysical properties of nanofluids to the nanoparticles volume fraction. Their results indicated that the non-uniformity of the thermophysical properties is the reason for the anomalous heat transfer enhancement. Malvandi and Ganji [29], then, used the modified model to examine the mutual impacts of buoyancy and nanoparticle migration on the mixed convection of nanofluids in vertical annuli. Subsequently, Malvandi and Ganji [30] investigated the laminar flow and convective heat transfer of alumina/water nanofluid inside a circular microchannel in the presence of a uniform magnetic field. Their results indicated that the nanoparticles migrate from the heated walls to the core region of the microchannel and form a nonuniform nanoparticles distribution. It was further observed that for smaller nanoparticles, the nanoparticle volume fraction is more uniform and abnormal variations in the heat transfer rate vanish. Hedayati and Domairry [31] investigated the effects of the nanoparticle migration on titania/water nanofluids in horizontal
Greek symbols / nanoparticle volume fraction c ratio of wall and fluid temperature difference to absolute temperature g transverse direction l dynamic viscosity (kg/m s) q density (kg/m3) k slip parameter Subscripts B bulk mean bf base fluid p nanoparticle w condition at the wall Superscript ⁄ dimensionless variable
and vertical channels. Recently, Malvandi and Ganji [32] took advantage of the modified Buongiorno model so as to evaluate the effects of temperature-dependent thermophysical properties on nanoparticle migration inside microchannels at the mixed convective heat transfer of nanofluids. The popularity of modeling the nanoparticle migration can be gauged from the numerous published literatures such as [33–38]. The current study is motivated by the need to examine the detailed behavior of nanoparticles movements for nanofluid flow inside circular microchannels and how they can tune the thermal performance and heat transfer rate. Recently, Malvandi and Ganji [39] demonstrated that because of thermophoresis, asymmetrically heated walls are able to tune and control the fluid flow and heat transfer characteristics of nanofluids. Moreover, the rate of heat transfer enhancement can be controlled by adjusting the heat flux at the boundaries. These observations show that thermal boundary condition is a salient factor on heat transfer characteristics of nanofluids. Consequently, in the current investigation, a constant wall temperature is prescribed at the wall, which is an important development for the modified Buongiorno model [28,29]. The fully developed governing equations of the modified Buongiorno model for the constant wall temperature are obtained for the first time and the results for the pressure drop and the heat transfer enhancement are presented versus varying pertinent parameters. Due to the low dimensional structures in microtubes, a linear slip condition is considered at the surfaces, which adequately represents the non-equilibrium region at the fluid/solid interface. To study the thermal performance, the figure of merit (FoM) is calculated to signify it. The impact of constant wall temperature at the wall on nanoparticle migration, the heat transfer rate as well as the thermal performance is of our particular interests.
2. Problem formulation The geometry of the problem under consideration with the adapted coordinate system is demonstrated in Fig. 1. To give a more physical outlook of the problem, the distributions of velocity and nanoparticle volume fraction are schematically illustrated. The cylindrical coordinates x and r were aligned parallel and perpendicular to the wall respectively. The slip velocity is considered at
Please cite this article in press as: S.J. Hosseini et al., Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology (2017), http://dx.doi.org/10.1016/j.apt.2016.11.019
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1 @ @/ DT @T DB þ ¼0 r @r @r T @r
ð3Þ
where u, T, and p represent the axial velocity, local temperature, and pressure respectively. In addition, q, l, k, and cP are respectively the density, dynamic viscosity, thermal conductivity, and specific heat capacity of nanofluid obtained via [3,5]
q ¼ /qp þ ð1 /Þqbf ;
ð4Þ
l ¼ lbf ð1 þ al / þ bl /2 Þ;
ð5Þ
k ¼ kbf 1 þ ak / þ bk /2 ;
ð6Þ
cp ¼
/qp cpp þ ð1 /Þqbf cpbf ; /qp þ ð1 /Þqbf
ð7Þ
where the thermophysical properties of nanoparticles and base fluid (water) are provided in Table 1. Note that Eqs. (5) and (6) corresponds to the room-temperature viscosity and nanofluid thermal conductivity measured by Pak and Cho [4] respectively. Moreover, in Eq. (3), the Brownian motion coefficient DB and thermophoresis coefficient DT can be defined as [5]
DB ¼
kBO T 3plbf dp
ð8Þ
and Fig. 1. The geometry of physical model and coordinate system.
DT ¼ 0:26
the wall which allows the formation of axial velocity component at the wall. Also, the following assumptions have been made:
Incompressible flow, No chemical reactions, Negligible external forces, Dilute mixture, Negligible viscous dissipation, Negligible radiation, Local thermal equilibrium between the nanoparticles and base fluid.
Consequently, the incompressible conservation equations of the momentum, thermal energy, and nanoparticle volume fraction can be written in the following manner [5,30,39]: 1 d du dp rlð/Þ þ ½ð1 /B Þqbf bðT T B Þ ðqp qbf Þð/ /B Þg ¼ r dr dr dx
ð1Þ
@T 1 @ @T @/ DT @T @T rkð/Þ þ qp cPp DB þ qð/ÞucP ð/Þ ¼ @x r @r @r @r T @r @r
ð2Þ
Table 1 Physical properties of nanoparticles and base fluid (water) [33]. Physical properties
Fluid phase (water)
Al2O3
q (kg/m3)
998.2 4182 0.597 9.93 104
3880 773 36
cP (J/kg K) k (W/m K) l (kg/m s)
lbf kbf / 2kbf þ kp qbf
ð9Þ
respectively. kBO is the Boltzmann constant and dp is the nanoparticle diameter, which varies from 1 to 100 nm. To solve the governing equations, appropriate boundary conditions can be expressed as @T @r
r ¼ 0 ! du ¼ 0; dr
¼ 0;
r ¼ Ro ! u ¼ N du ; dr
@/ @r
¼ 0;
T ¼ Tw;
DB @/ þ DTT @r
@T @r
ð10Þ
¼ 0:
where N introduces the slip velocity factor. To describe the boundary conditions of Eq. (10) physically, it should be said that in the center of microtube there are zero gradients of temperature, velocity and nanoparticle concentration owing to symmetric annular boundaries. In the circumference of the microtube, the nanofluid flow is faced with the fluid-solid interface, thereby leading to a slip velocity related to the velocity gradient on the wall. In addition, the temperature of the nanofluid flow is constantly considered (Tw) and the nanoparticles flux is zero at the wall. 2.1. Scale analysis In this study, we take advantage of the scale analysis of the governing equations (1)–(3) which can be simplified effectively. Table 2 provides the device and material properties for the present physical conditions, typically occurring in the flow of nanofluids inside microtubes. Regarding Table 2, the order of each term in the governing equations can be estimated. Considering Eq. (1), the scale analysis can be written as follows
Dp Lref
l DU L2ref |ffl{zffl}
;
ð1 /Þqbf gbDT ; |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
ðqp qbf Þg D/ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
98
2:8104
ð11Þ
3:52103
Table 2 Device and material properties of alumina/water nanofluid. Reference variables Values
uref (m/s) 0.01
/ref 0.02
Lref ¼ Ro (m) 7:5 10
5
lref (kg/(m s))
g (m/s2)
DT (K)
0.00198
10
50
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
From Eq. (11), it can be deduced that the buoyancy effect due to the temperature gradient (second RHS term) can be neglected with respect to the buoyancy effect due to nanoparticles concentration gradient (third RHS term) and also the shear stress term (first RHS term). Thus, this term can be removed from Eq. (1). Accordingly, the scale analysis for Eq. (2) can be expressed asEq. (12) indicates that the LHS term is in the same order of magnitude with the first term of RHS and these are about 105 times more than the second RHS term. In fact, heat transfer associated with nanoparticle diffusion (third RHS term) can be neglected in comparison with the other terms. Hence, the governing equations (Eqs. (1)–(3)) can be written as:
axial conduction effect, the following relation can be determined for the axial temperature gradient [5]
@T T w T dT B ¼ ; @x T w T B dx
ð20Þ
where TB is the bulk temperature defined as
R Ro
R
T B R0 Ro 0
1 qð/Þcp ð/ÞuTr dr qð/Þcp ð/ÞuTð1 gÞdg : ¼ R0 1 qð/Þcp ð/Þur dr qð/Þcp ð/Þuð1 gÞdg 0
ð21Þ
Substituting Eq. (20) into Eq. (14) yields the energy equation as
qð/Þcp ð/Þu
T w T dT B 1 @ @T rkð/Þ ¼ T w T B dx r @r @r
ð22Þ
LHS
8 RHS DT > > > RHS1 ¼ kbf ð1 þ 7:47/B Þ 2 > > L > ref > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > 2:0861010 < U ref DT P 1 0 ð/B qp cpp þ ð1 /B Þqbf cpbf Þ > D x > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} B D D/ > > D T DT C C DT B B > > 2:0285108 þ C > RHS2 ¼ qp cpp B > T ref Lref A Lref > |fflffl{zfflffl}@ Lref > > |fflffl{zfflffl} |fflfflffl{zfflfflffl} |{z} : 6 310
1 d du dp r lð/Þ ðqp qbf Þð/ /B Þg ¼ 0 r dr dr dx
qð/ÞucP ð/Þ
1:97107
ð13Þ
@T 1 @ @T rkð/Þ ¼0 @x r @r @r
ðdp=dxÞR2o =lbf
:
ð16Þ
u00 þ Au ðgÞu0 þ Bu ðgÞ ¼ 0;
ð17Þ d dg
lbf ð1 Nrð/ /B ÞÞ: lð/Þ
1 A
Z A
C dA ¼
Z
1
p
R2o
0
Ro
C2pr dr ¼ 2
Z 0
1
Cð1 gÞdg
ð24Þ
B Finding dT from Eq. (23), substituting it into Eq. (22) and introducdx ing the following non-dimensional variables
T ¼
T TB ; Tw TB
HTC ¼
2Ro h ; kbf
ð25Þ
reduce Eq. (22) to
ð26Þ
⁄
where T is the dimensionless temperature and HTC stands for the heat transfer coefficient and
d/ d 1 ðLnðkð/ÞÞÞ ; dg d/ 1g kbf HTC qð/Þucp ð/Þ : BT ðgÞ ¼ C T ðgÞ ¼ hqð/Þucp ð/Þi kð/Þ
AT ðgÞ ¼
ð18Þ
where
gðqp qbf Þ : Nr ¼ ðdp=dxÞ
hCi
in which
d/ d 1 ðLnðlð/ÞÞÞ Au ðgÞ ¼ ; dg d/ 1g Bu ðgÞ ¼
ð23Þ
T 00 þ AT ðgÞT 0 þ BT ðgÞT þ C T ðgÞ ¼ 0
Thus, Eq. (1) reduces to
where 0 (prime symbol) represents
" # dT B 2 @T @T 2q ¼ w; rkð/Þ ¼ 2 rkð/Þ @r r¼Ro @r r¼0 dx Ro Ro
where
ð15Þ
The governing equations can be reduced to non-dimensional form via the following non-dimensional variables
u ¼
Eq. (22) can be reduced by integrating in the following manner
ð14Þ
2.2. Dimensionless momentum equation
u
6:67105
hqð/Þucp ð/Þi
1 @ @/ DT @T DB þ ¼0 r @r @r T @r
R r g¼ o ; Ro
5:54107
ð12Þ
ð27Þ
2.4. Dimensionless nanoparticle continuity
ð19Þ
According to Eq. (25), Eq. (15) can be written as
/0 ¼ 2.3. Dimensionless energy equation
ðT w T B Þ DT 0 T ½T ðT w T B Þ þ T B DB
ð28Þ
Introducing the following non-dimensional parameters Before the energy equation is solved, the thermally fully developed condition should be employed. For the nanofluids in a circular tube along with a prescribed wall temperature and ignoring the
c¼
Tw TB ; TB
NBT ¼
DBB /B ; DT B c
ð29Þ
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
yields Eq. (28) as
/0 þ A/ ðgÞT 0 ¼ 0
ð30Þ
A/ ðgÞ ¼
/ NBT ð1 þ cT Þ
2
:
g ¼ 0 : u ¼ k dudg ; T ¼ 1; / ¼ /w ;
ð33Þ
3. Numerical method and accuracy
To solve the non-dimensional governing equations, the apt boundary conditions should be derived. With regard to Eqs. (16) and (25), the boundary conditions in Eq. (10) can be written in the non-dimensional form as
g ¼ 1 : dudg ¼ 0;
hu /i : hu i
ð31Þ
2.5. Dimensionless boundary condition
(
rather than /w, the following relation is used as an additional constraint [28,33,40]
/B
where
5
@T @g
¼ 0;
ð32Þ
where k ¼ RNo . Eq. (32) represents the slip velocity and constant temperature at the wall. Since the value of /B is usually prescribed
The governing equations including Eqs. (17), (26), and (30) are the governing equations representing a system of nonlinear Ordinary Differential Equations (ODEs) which are going to be solved with the boundary conditions of Eq. (32). It is worth noting that the values of BT (g) and CT(g) in Eq. (26) are not specified in advance, because hqð/Þcp ð/Þui depends on the nanoparticle volume fraction profile and it is determined when the governing equations are solved. Furthermore, there is a similar condition for /w in Eq. (32). In order to convert the boundary-value problem to initial-value one, we used the method which is quite similar to the shooting method through which the boundary-value problem can turn to the initial-value problem. Therefore, a reciprocal procedure is required to solve the governing equations for the unknown
Fig. 2. Algorithm of the numerical method.
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
/B. This process is iterated until the prescribed value of /B is achieved, and the relative error between the assumed value of hqð/Þcp ð/Þui and the calculated one after solving the governing equations is lower than 106. To corroborate the accuracy of the numerical code, the velocity and temperature profiles for the laminar fully developed flow in circular channel with Nr = /B = k = 0 are compared to the reported results of Kays et al. [41] in Fig. 3. As is evident, the results are in the best agreement. Moreover, to verify the grid independence of the numerical results, they are presented in three different integration steps (dg) of 102, 5 103 and 103 in Table 3; obviously,
Fig. 3. Comparison of velocity (a) and temperature profile (b) for regular fluid with the results of Kays et al. [41].
Table 3 Grid independence test for different values of dg when /B ¼ 0:02; k ¼ 0:1; Nr ¼ 20 and N BT ¼ 1. dg
h=hbf
N p =N pbf
10
1.21493
1.86574
5 103
1.21495
1.86577
103
1.21496
1.86579
2
values of hqð/Þcp ð/Þui and /w. The algorithm of numerical approach is demonstrated in Fig. 2. Firstly, the values of parameters including /B, k, Nr and NBT are input to the code. Then, initial guesses for the unknown values of hqð/Þcp ð/Þui and /w have been considered. After the governing equations with the boundary conditions have been solved via the Runge-Kutta-Fehlberg method, the calculated values of hqð/Þcp ð/Þui and /B are compared with the assumed value of hqð/Þcp ð/Þui and the prescribed value of
Fig. 4. The effects of N BT on nanoparticle distribution (/=/B ), velocity u uB , and temperature (T ) profiles when /B ¼ 0:02; k ¼ 0:1 and Nr ¼ 20.
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
Fig. 5. The effects of /B on nanoparticle distribution ð/=/B Þ, velocity u uB , and temperature (T ) profiles when k ¼ 0:1; Nr ¼ 20 and N BT ¼ 1.
altering the grid size has insignificant effect on the results. It should be stated that all the following numerical results have been acquired using the integration step dg = 103.
7
Fig. 6. The effects of k on nanoparticle distribution (/=/B ), velocity u uB , and temperature (T ) profiles when /B ¼ 0:02; Nr ¼ 20 and N BT ¼ 1.
range of NBT from 0.25 to 10 is considered. The selected values of NBT lead to different modes of nanoparticle migration which will B be discussed later. In addition, the value of c ffi T wTT is assumed B
4. Results and discussion
to be 0.1 because of its trivial effect on the solution [28,29].
There are some background information which should be explained in advance. As was maintained in previous studies [30], for the nanoparticles with diameter around and less than 100 nm, NBT is in the order of 1; so, in the present study, a wide
4.1. Graphical results Figs. 4–7 provide graphical representation of dimensionless nanoparticle volume fraction, velocity, and temperature profiles
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values of NBT, as illustrated in Fig. 4. For the higher values of NBT, the Brownian diffusion dominates the thermophoresis and eliminates the nanoparticle concentration gradient through the microtube. Also, it can be seen that the velocity and temperature profiles are completely affected by the nanoparticle volume fraction distribution. Indeed, as opposite to the regular fluids, the hydrodynamic and thermal characteristics of nanofluids are directly affected by the thermophoresis or nanoparticle migration. It is worth mentioning that NBT is inversely proportional to the nanoparticle diameter (dp); hence, the nanoparticle diameter is a key factor affecting the flow and thermal fields. Furthermore, it can be seen in the figure that the peak of velocity profile reduces for the larger nanoparticles (lower values of NBT) in which the nanoparticle migration is strong because a rise in the nanoparticle volume fraction in the central region of microtube intensifies the viscosity of the nanofluid, thereby decreasing the velocity. The same behavior can be observed for the thermal conductivity and the temperature. Fig. 5 reveals that an increase in /B declines the peak of nanoparticle concentration and velocity profiles; nonetheless, increasing /B grows the dip of the temperature. This is due to the fact that rising /B boosts the nanoparticle volume fraction, and therefore it increases the impact of the Brownian diffusion. This dynamic shifts the nanoparticles toward the central region of the microtubes, which has a higher kinetic energy (a higher velocity). In addition, due to a lower kinetic energy of nanoparticle near the wall, Brownian diffusion effect is negligible and there is no significant change there. The uniform distribution of nanoparticles at the higher values of /B has a direct impact on the viscosity and thermal conductivity of nanofluids; consequently, the velocity and temperature gradients reduce at the higher values of /B. Fig. 6 illustrates that growing the slip parameter (k) increases the slip velocity at the wall, followed by a reduction in the velocity peak in the central region of microtube. In fact, the velocity of the nanofluid accelerates near the walls and decelerates at the center of microtube. In addition, the migration of nanoparticles enhances for the higher values of k, thereby decreasing the dip of temperature profile. Fig. 7 depicts that a rise in Nr reduces the velocity peak in the central region of tube, however, it slightly accelerates the velocity at the walls. Also, it can be seen that increasing Nr leads to a gradual increase in the nanoparticle volume fraction at the core region of microtube. In fact, Nr enhances the non-uniformity of nanoparticle volume fraction. 4.2. Tabular data There are two main physical quantities of interest in heat exchange equipments, namely the heat transfer rate and the pressure drop. For nanofluids, it is beneficial to examine the heat transfer enhancement and pressure drop increment, demonstrating the merit of nanoparticle inclusion. For the laminar and fully developed flow inside a circular microchannel, the heat transfer coefficient (HTC) at the wall can be written as Fig. 7. The effects of Nr on nanoparticle distribution (/=/B ), velocity u uB , and temperature (T ) profiles when /B ¼ 0:02; k ¼ 0:1 and N BT ¼ 1.
for different values of NBT, /B, k, and Nr, respectively. In the figures, g = 0 denotes the surface condition while g = 1 represents the center of the microtube. Obviously, nanoparticle volume fraction nonuniformly distributed through the microtubes, especially for the lower values of NBT; a lower concentration of nanoparticles places at the wall (g = 0) which gradually increases to its peak at the center of microtube. Consequently, it can be deduced that nanoparticles migrate from the heated wall toward the center of microtubes. This dynamic can be weakened for the very greater
HTC ¼
hDh q00w 2Ro kw 0 ¼ ¼ 2T kbf ðT w T B Þkbf kbf
ð34Þ
where Dh = 2Ro. Therefore, the heat transfer enhancement can be given by
h HTC ¼ hbf HTC bf
ð35Þ
Also, the dimensionless pressure gradient can be provided by
dp Np ¼ dx
,
lbf
!
uB ð2Ro Þ
2
¼
4qB qu ðlbf =lw Þ
ð36Þ
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx Table 4 Heat transfer enhancement, pressure drop increment, and figure of merit for different values of Nr when /B ¼ 0:02; k ¼ 0:1. NBT
0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10
Nr = 10
Nr = 20
Nr = 40
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
1.143 1.156 1.174 1.186 1.192 1.195 1.196 1.196 1.195 1.182 1.173 1.168 1.165 1.162 1.161 1.159 1.158 1.157
1.075 1.104 1.185 1.290 1.405 1.523 1.638 1.750 1.854 2.578 2.949 3.167 3.310 3.410 3.485 3.542 3.588 3.625
1.115 1.118 1.110 1.089 1.064 1.039 1.014 0.992 0.973 0.862 0.818 0.795 0.782 0.772 0.766 0.761 0.757 0.753
1.190 1.198 1.211 1.218 1.221 1.221 1.220 1.217 1.215 1.193 1.181 1.174 1.169 1.166 1.164 1.162 1.161 1.160
1.115 1.138 1.212 1.311 1.423 1.539 1.653 1.762 1.866 2.583 2.951 3.169 3.311 3.411 3.485 3.543 3.588 3.625
1.147 1.148 1.136 1.113 1.085 1.058 1.032 1.008 0.987 0.869 0.823 0.799 0.784 0.775 0.768 0.762 0.758 0.755
1.438 1.355 1.319 1.304 1.293 1.284 1.275 1.267 1.260 1.216 1.196 1.185 1.178 1.174 1.170 1.168 1.166 1.267
1.646 1.415 1.373 1.428 1.515 1.615 1.718 1.819 1.916 2.603 2.962 3.175 3.315 3.414 3.488 3.545 3.590 3.611
1.218 1.207 1.187 1.158 1.126 1.094 1.065 1.038 1.015 0.884 0.833 0.806 0.790 0.779 0.772 0.766 0.761 0.826
Table 5 Heat transfer enhancement, pressure drop increment, and figure of merit for different values of nanoparticle volume fraction when Nr = 20, k ¼ 0:1. NBT
0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10
/B ¼ 0:01
/B ¼ 0:02
/B ¼ 0:04
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
1.099 1.104 1.111 1.114 1.114 1.114 1.113 1.111 1.110 1.097 1.091 1.087 1.085 1.083 1.082 1.081 1.081 1.080
1.047 1.059 1.095 1.142 1.192 1.242 1.290 1.335 1.376 1.639 1.764 1.835 1.880 1.912 1.935 1.953 1.967 1.978
1.083 1.083 1.077 1.065 1.051 1.036 1.022 1.009 0.998 0.930 0.903 0.888 0.879 0.873 0.868 0.865 0.862 0.860
1.190 1.198 1.211 1.218 1.221 1.221 1.220 1.217 1.215 1.193 1.181 1.174 1.169 1.166 1.164 1.162 1.161 1.160
1.115 1.138 1.212 1.311 1.423 1.539 1.653 1.762 1.866 2.583 2.951 3.169 3.311 3.411 3.485 3.543 3.588 3.625
1.147 1.148 1.136 1.113 1.085 1.058 1.032 1.008 0.987 0.869 0.823 0.799 0.784 0.775 0.768 0.762 0.758 0.755
1.366 1.377 1.400 1.415 1.423 1.426 1.425 1.423 1.419 1.381 1.358 1.345 1.337 1.331 1.326 1.323 1.321 1.318
1.318 1.350 1.502 1.733 2.015 2.324 2.645 2.969 3.287 5.770 7.206 8.096 8.695 9.125 9.447 9.698 9.898 10.062
1.246 1.246 1.222 1.178 1.127 1.076 1.030 0.990 0.954 0.770 0.703 0.670 0.650 0.637 0.627 0.620 0.615 0.611
Table 6 Heat transfer enhancement, pressure drop increment, and figure of merit for different values of slip parameter when Nr = 20, /B ¼ 0:02. NBT
0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10
k¼0
k ¼ 0:1
k ¼ 0:2
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
h/hbf
Ndp
FoM
1.184 1.193 1.203 1.208 1.209 1.208 1.206 1.204 1.202 1.183 1.173 1.168 1.165 1.162 1.160 1.159 1.158 1.157
1.140 1.174 1.269 1.384 1.507 1.629 1.747 1.858 1.962 2.658 3.009 3.215 3.349 3.444 3.514 3.568 3.611 3.646
1.134 1.131 1.112 1.084 1.055 1.027 1.002 0.980 0.960 0.854 0.813 0.791 0.778 0.770 0.763 0.759 0.755 0.752
1.190 1.198 1.211 1.218 1.221 1.221 1.220 1.217 1.215 1.193 1.181 1.174 1.169 1.166 1.164 1.162 1.161 1.160
1.115 1.138 1.212 1.311 1.423 1.539 1.653 1.762 1.866 2.583 2.951 3.169 3.311 3.411 3.485 3.543 3.588 3.625
1.147 1.148 1.136 1.113 1.085 1.058 1.032 1.008 0.987 0.869 0.823 0.799 0.784 0.775 0.768 0.762 0.758 0.755
1.182 1.189 1.202 1.210 1.214 1.216 1.215 1.214 1.212 1.192 1.181 1.174 1.169 1.166 1.164 1.162 1.161 1.160
1.100 1.115 1.178 1.269 1.374 1.486 1.597 1.705 1.809 2.536 2.915 3.139 3.287 3.390 3.467 3.527 3.574 3.612
1.145 1.146 1.138 1.118 1.092 1.065 1.040 1.016 0.995 0.874 0.827 0.802 0.787 0.776 0.769 0.764 0.759 0.756
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S.J. Hosseini et al. / Advanced Powder Technology xxx (2017) xxx–xxx
where qB is the bulk density. And the pressure drop increment can be written as
Ndp ¼
Np Npbf
ð37Þ
Moreover, to find the thermal performance, the figure of merit (FoM) is employed by heat transfer engineers. For the nanofluids, FoM can be calculate via
FoM ¼
h=hbf ðNp =Npbf Þ1=3
ð38Þ
Tables 4–6 provide information about the effects of Nr, /B, and k on the heat transfer enhancement, pressure drop increment and thermal performance (FoM) for different nanoparticle migration regimes (NBT). At first glance, it is obvious that increasing Nr increases both the heat transfer enhancement and pressure drop increment, especially at the lower values of NBT. Also, the thermal performance (FoM) is increased gradually. This means that the effects of heat transfer enhancement overwhelm the pressure drop increment, as Nr increases. Similarly, increasing /B leads to an increase in both the pressure drop increment and heat transfer enhancement. However, for the thermal performance, the effects of NBT should be considered. As is elucidated from Table 5, FoM increases with /B in the lower values of NBT while it decreases at the higher values of NBT. In other words, increasing /B is advantageous only for larger nanoparticles (lower values of NBT). According to Table 6, it can be observed that an increase in k reduces the pressure drop increment, whereas it leads to a rise in both the heat transfer rate of nanofluids and base fluid. Obviously, the heat transfer enhancement is increased when k increases from 0 to 0.1. It should be noted that further increase in k for lower values of NBT leads to a decrease in the heat transfer enhancement. This means that in this range, the heat transfer enhancement in the base fluid is much larger than the nanofluids. In addition, it is interesting to note that FoM increases for 0 < k < 0.1 and further increase in k has insignificant effects on the thermal performance.
5. Summary and conclusions The present study is a theoretical investigation of the figure of merit for the laminar fully developed mixed convection of nanofluids through microtubes at a constant wall temperature (CWT). The modified Buongiorno’s model is used for the nanofluid to assess the nanoparticle migration originating from the thermophoresis (nanoparticle slip velocity due to temperature gradient) and Brownian motion (nanoparticle slip velocity due to concentration gradient). The thermally fully developed governing equations are developed for a constant wall temperature, reduced to ordinary differential equations by appropriate dimensionless variables, and solved with a reciprocal numerical algorithm. It is shown that increasing nanoparticle concentration enhances the thermal performance for the larger nanoparticles while it reduces the thermal performance for the smaller ones. In addition, increasing the Buoyance force and nanoparticle diameter would enhance the thermal performance. Also, it is obtained that the thermal performance grows asymptotically with the slip parameter. References [1] A. Malvandi, A. Ghasemi, R. Nikbakhti, A. Ghasemi, F. Hedayati, Modeling and parallel computation of the non-linear interaction of rigid bodies with incompressible multi-phase flow, Comput. Math. Appl. 72 (2016) 1055–1065. [2] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Appl. Phys. Lett. 78 (2001) 718–720.
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