- Email: [email protected]
Figure of merit for optimization of nanofluid flow in circular microchannel by adapting nanoparticle migration
Accepted Manuscript Figure of merit for optimization of nanofluid flow in circular microchannel by adapting nanoparticle migration A. Malvandi, M. Zamani, S.J. Hosseini, S.A. Moshizi PII: DOI: Reference:
S1359-4311(17)31135-3 http://dx.doi.org/10.1016/j.applthermaleng.2017.02.081 ATE 9962
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
13 March 2016 16 February 2017 19 February 2017
Please cite this article as: A. Malvandi, M. Zamani, S.J. Hosseini, S.A. Moshizi, Figure of merit for optimization of nanofluid flow in circular microchannel by adapting nanoparticle migration, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng.2017.02.081
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Figure of merit for optimization of nanofluid flow in circular microchannel by adapting nanoparticle migration A. Malvandi a, *, M. Zamani b, S. J. Hosseini c, S. A. Moshizi d a
Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran b Young Researchers and Elite Club, Gonabad Branch, Islamic Azad University, Gonabad, Iran c Department of Mechanical Engineering, School of Engineering, University of Tehran, Tehran, Iran d Young Researchers and Elite Club, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran
Abstract In this paper, the laminar fully developed flow of alumina/water nanofluid inside circular microchannels subjected to a constant wall temperature (CWT) is theoretically investigated. The effect of nanoparticles migration originating from thermophoretic diffusion (temperaturegradient driven force) and Brownian diffusion (concentration-gradient 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 order to assume a hydrodynamically and thermally fully developed flow, the governing equations are reduced to a system of ordinary differential equation and solved using the appropriate reciprocal algorithm. The effects of pertinent parameters including the ratio of Brownian motion to thermophoresis (NBT), slip parameter (λ) and bulk mean nanoparticle volume fraction (ϕB) on the flow and thermal fields are investigated. In addition, the results are compared with the case of constant heat flux (CHF) at the wall. The figure of merit (FoM) is used to measure the thermal performance of equipment and finding the optimum thermal condition. It is shown that the anomalous heat transfer enhancement depends on the thermal boundary condition as well as the nanoparticles diameter. Furthermore, the optimum value of NBT for the case of constant wall temperature (approximately 1) is found to be greater than that of the constant wall heat flux (approximately 0.5). Thus, it can be concluded that the optimum diameter of nanoparticles for the case of constant wall temperature should be smaller than that of constant wall heat flux. Key Words: Figure of merit; nanofluid; nanoparticle migration; thermophoresis; Brownian
motion.
*
Corresponding author, Email: [email protected]
1
Nomenclature cp
specific heat capacity (m2/s2 K)
dp
nanoparticle diameter (m)
Dh
Hydraulic diameter, Dh=2Ro (m)
DB
=
DT
= 0.26
FoM
Figure of Merit
h
heat transfer coefficient (W/m2.K)
k
thermal conductivity (W/m.K)
kBO
Boltzmann constant ( 1.3806488
N
slip velocity factor
Nu
Nusselt number
N BT
ratio of the Brownian to thermophoretic diffusivities
P
pressure (Pa)
qw
surface heat flux (W/m2)
T
temperature (K)
u
axial velocity (m/s)
x, r
coordinate system
k BOT , Brownian diffusion coefficient (m2/s) 3bf d p k bf bf , thermophoresis diffusion coefficient (m2/s) 2k bf k p bf
Greek symbols
2
1023 m2 kg s2 K )
nanoparticle volume fraction
ratio of wall and fluid temperature difference to absolute temperature
transverse direction
dynamic viscosity (kg/m.s)
density (kg/m3)
slip parameter
internal angle inside cross-section of microtube
Subscripts B
bulk mean
bf
base fluid
p
Nanoparticle
w
condition at the wall
Superscripts *
dimensionless variable
3
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, 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).
1.1
Theoretical modeling of nanofluids
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 [1]; b) an abnormal increase in the viscosity of nanofluids relative to the regular fluid [2, 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 nanoparticle flux) heterogeneous equilibrium 4
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 et al. [7], Sheikholeslami and Ellahi [8], Garoosi et al. [9], and Nield and Kuznetsov [10]. Theoretical investigation on the effect of nanofluids has been systematically reported and well documented, which can be found in the open literature [11-15]. An excellent review paper on the application of nanofluids in microchannels is conducted by Salman et al. [16].
1.2
Nanoparticle migration effects
In 2013, Yang et al. [17, 18] modified the Buongiorno’s model to consider the impact of nanoparticle distribution on the thermal conductivity and viscosity of nanofluids. In fact, the proposed modified model does not ignore the dependency of thermophysical properties of nanofluids to the nanoparticles concentration. Their results indicated that the non-uniformity of the thermophysical properties is the reason for the anomalous heat transfer enhancement. Malvandi et al. [19], 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 [20] investigated the impacts of the nanoparticle migration as well as asymmetric heating at the walls on the forced convective heat transfer of magnetohydrodynamic alumina/water nanofluid in microchannels. Hedayati and Domairry [21] investigated the effects of the nanoparticle migration on titania/water nanofluids in horizontal and vertical channels. The popularity of modeling the nanoparticle migration can be gauged from the numerous published works of literature such as [22-27]. 5
1.3
Motivation and novelty
From the previous explanations, the utilization of the modified Buongiorno model has been limited to a special thermal boundary condition that provides the constant heat flux at the boundaries. Recently, Malvandi and Ganji [20] demonstrated that because of thermophoresis, asymmetrically heated walls are able to control the fluid flow and heat transfer characteristics of nanofluids. In addition, the amount of enhancement in the heat transfer rate can be controlled by adjusting the heat flux at the boundaries. These outcomes pointed out that thermal boundary conditions are a key factor in the fluid flow and heat transfer characteristics of nanofluids. Accordingly, in this paper, a prescribed wall temperature as a widespread thermal boundary condition is imposed on the wall, which is an important development for the modified Buongiorno model [28, 29]. The fully developed governing equations of the modified Buongiorno model are obtained and the results for the pressure drop and the heat transfer enhancement are presented versus varying pertinent parameters. Because of low dimensional structures in microchannels, a linear slip condition is considered at the surfaces, which appropriately represents the non-equilibrium region near the fluid/solid interface. To study the thermal performance, the figure of merit (FoM) is calculated to signify it. The impacts of the thermal boundary condition, nanoparticle migration, as well as the slip velocity on thermal performance are of our particular interests.
2. Problem description and governing equations The geometry of the problem with the coordinate system is shown in Figure 1. To give a more physical outlook of the problem, the distribution of velocity and nanoparticle volume fraction is schematically illustrated. The following assumptions are considered in the present paper:
6
1. The nanofluid is assumed incompressible, on the ground that the density of nanofluid can be approximated by that of the base fluid, i.e.,
bf
, since the volumetric fraction of
nanoparticles is only a few percent [5, 30, 31]. 2. There are some important assumptions for mathematical modeling including incompressible flow, a dilute mixture, negligible external force and radiative heat transfer, no chemical reactions and negligible viscous dissipation. 3. The modified Buongiorno’s model is considered so as to simulate the effects of Brownian diffusion and thermophoresis. 4. Adapted cylindrical coordinates x and r were aligned parallel and perpendicular to the wall respectively. 5. The slip velocity is considered in the wall which allows the formation of the axial velocity component at the wall. 6. Flow and thermal fields are assumed to be fully developed. 7. There is a thermal equilibrium between the nanoparticles and the base fluid. 8. All the fluid properties vary with the nanoparticle volume fraction. The momentum and energy equations in general form can be written as:
. V 0,
(1)
V . V V p . V , t
(2)
D cPT . cP VT . kT . cP pTDB p cP DT T cPT . p DB p T T , t T
(3)
D p . pV . p DB p T T , t T
(4)
7
where u, T, and p represent the axial velocity, local temperature, and pressure respectively. In addition, ρ, µ, k, and cP are respectively the density, dynamic viscosity, thermal conductivity, and specific heat capacity of nanofluid obtained via [2, 5]
p 1 bf ,
(5)
bf 1 a b 2 , a 39.11, b 533.9 for alumina
(6)
k kbf 1 ak bk 2 , ak 7.47, bk 0 for alumina
(7)
cp
p c p 1 bf c p p
p 1 bf
bf
,
(8)
where the thermophysical properties of the alumina nanoparticles and base fluid (water) are determined in the following manner: Base fluid:
c pbf 4182 J/(kg K), bf 998.2 kg / m3 , kbf 0.597 W/(m K), bf 9.93 104 kg/(m s)
(9)
Alumina nanoparticles:
c p p 773 J/(kg K), p 3880 kg / m3 , k p 36 W/(m K)
(10)
Moreover, in Eqs. (3) and (4), the Brownian motion coefficient DB and thermophoresis coefficient DT can be defined as [5] DB
k BOT 3bf d p
, DT 0.26
kbf
bf
2kbf k p bf
(11)
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
8
du T 0 , 0 , 0 , dr r r
r 0
r Ro u N
(12)
du DT T , T Tw , DB 0. dr r T r
where N introduces the slip velocity factor. Buongiorno [5] corroborated that the second and third terms on the right-hand side of Eq. (3) presenting the heat transfer related to the nanoparticle dispersion (i.e., nanoparticle diffusion flux) is always insignificant in comparison with convection and conduction terms. So, the term vanishes for the case of fully developed flows. Thus, the energy equation becomes identical to that of a pure fluid, except that all properties strongly depend on the spatial distribution of [5, 17]. In addition, in the studied case, the diffusion mass flux of the nanoparticles remains constant across the tube since the wall is impermeable. Thus, the Brownian diffusion flux and thermophoretic diffusion flux cancel out everywhere in the tube. As a consequence, the term on the left-hand side of Eq. (4) is equal zero [17]. Furthermore, since the mass flux inside channels is constant (so
uA cte . ),
it results in x 0 with considering hydrodynamic fully
developed condition [17, 23, 32, 33]. In addition, the flow field is assumed to be thermally and hydrodynamically fully developed ( ur 0 , ux x 0 and ux u ) and steady state ( t 0 ). Accordingly, the incompressible conservation equations of the momentum, thermal energy, and nanoparticle fraction can be written in the following manner [5, 34, 35]:
dp 1 d du r ( ) 0 dx r dr dr
( )uc p ( )
T 1 T rk ( ) x r r r
9
(13) 0
(14)
1 DT T DB 0 r r r T r
2. 1
(15)
Dimensionless momentum equation
The governing equations can be reduced to non-dimensional form via the following nondimensional variables
Ro r Ro
, u*
u dp dx Ro 2 W
(16)
Thus, Eq. (13) reduces to
u* Au u* Bu 0, where ‘ (prime symbol) represents
Au
(17)
d in which d
d d 1 Ln ( ) d d 1
Bu
,
W ( )
(18)
Moreover, u B* is the bulk velocity defined as
u * B
u *
(19)
where R
2. 2
1
1 1 o dA 2 rdr 2 1 d A A Ro 2 0 0
(20)
Dimensionless energy equation
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
10
ignoring the axial conduction effect, the following relation can be determined for the axial temperature gradient [5]
T Tw T dTB , x Tw TB dx
(21)
where TB is the bulk temperature defined as
TB
( )c p ( )uT ( )c p ( )u
( )c ( )uT 1 d . ( )c ( )u 1 d 1
p
0
(22)
1
p
0
Substituting Eq. (21) into Eq. (14) yields the energy equation as
( )c p ( )u
Tw T dTB 1 T rk ( ) Tw TB dx r r r
(23)
Eq. (23) can be reduced by integrating in the following manner
( )uc p ( )
Finding
dT B 2 T T 2 k ( )r k ( )r dx Ro r r Ro r
2qw . R o r 0
(24)
dTB from Eq. (24), substituting it into Eq. (23) and introducing the following nondx
dimensional variables
T*
T T B Tw T B
,
HTC
qw 2Ro k hD h w 2T * , k bf k bf Tw T B k bf
(25)
reduce Eq. (23) to
T * AT T * BT T * CT 0
(26)
where T* is the dimensionless temperature and HTC stands for the heat transfer coefficient and
AT
2. 3
d d 1 Ln k ( ) d d 1
,
( )uc p ( ) BT CT ( )uc p ( )
Dimensionless nanoparticle continuity 11
kbf HTC . k ( )
(27)
According to Eq. (25), Eq. (15) can be written as T T DT * * w B T T Tw TB TB DB Introducing the following non-dimensional parameters
Tw TB TB
N BT
,
DBB B , DTB
(28)
(29)
yields Eq. (28) as
A T * 0
(30)
where
A
2. 4
N BT 1 T
* 2
.
(31)
Dimensionless boundary condition
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. (12) can be written in the non-dimensional form as
du* * 0 : u , T * 1 , w , d * * 1 : du 0 , T 0, d where
(32)
N . Eq. (32) represents the slip velocity and constant temperature at the wall. Since Ro
the value of ϕB is usually prescribed rather than ϕw, the following relation is used as an additional constraint
B
u * u*
.
(33)
3. Numerical method and accuracy The equations (17), (26), and (30) are the governing equations representing a system of nonlinear Ordinary Differential Equations (ODEs) which should be solved with the boundary conditions of 12
Eq. (32). It is worth noting that the values of BT (η) and CT (η) in Eq. (26) are not specified in advance, because ( )c p ( )u depends on the nanoparticle volume fraction profile and it is determined when the governing equations are solved. Furthermore, there is a similar condition for ϕw in Eq. (32). Therefore, a reciprocal procedure is required to solve the governing equations for the unknown values of
( )c p ( )u
and ϕw. The algorithm of numerical approach is
demonstrated in Figure 2. Firstly, the values of parameters including ϕB, λ, and NBT are input to the code. Then, initial guesses for the unknown values of
( )c p ( )u and ϕw have been
considered. After the governing equations with the boundary conditions have been solved via the Runge-Kutta-Fehlberg method, the calculated values of ( )c p ( )u and ϕB are compared with the assumed value of ( )c p ( )u and the prescribed value of ϕB. This process is iterated until the prescribed value of ϕB is achieved, and the relative error between the assumed value of
( )c p ( )u and the calculated one after solving the governing equations is lower than 10 -6. To corroborate the accuracy of the numerical code, the velocity and temperature profiles for the laminar fully developed flow in the circular channel with ϕB=λ=0 are compared to the reported results of Kays et al. [36] in Figure 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 (dη) of 10-2, 5×10-3 and 10-3 in Table 1; obviously, altering the grid size has the insignificant effect on the results. It should be stated that all the following numerical results have been acquired using the integration step dη=10-3.
4. Results and discussion Here, there is some background information which should be explained in advance. As it was shown in previous studies [37], for the nanoparticles with diameter around and less than 100nm, 13
NBT is in the order of 1; so, in the present study, a wide range of NBT from 0.25 to 10 is considered. The selected values of NBT lead to different modes of nanoparticle migration which will be discussed later. In addition, the value of
Tw T B is assumed to be 0.1 because of its TB
trivial effect on the solution [17, 19]. In justification of the slight values of
Tw T B T w 1, TB TB
it can be seen that when the flow is thermally fully developed, the discrepancy between Tw and
T B decreases, after which it reaches a plateau [36]. So, has an insignificant slope related to x and possesses the low values.
4. 1 Flow and thermal fields Figures 4-6 show the impact of pertinent parameters including NBT, ϕB, and λ on dimensionless nanoparticle volume fraction, velocity, and temperature profiles. In the figures, η=0 denotes the condition at the wall while η=1 represents the center of the tube. The figures indicate that the smallest amount of nanoparticle concentration is at the wall (η=0) and gradually increases to the highest amount at the center of the microchannel, indicating that the nanoparticles move from the wall toward the central region of the tube. This dynamic definitely proves the impact of the thermophoresis, which pushes the nanoparticles from the hot region (η=0) toward the colder one (η=1) and makes a non-uniform distribution of nanoparticles. This phenomenon can be strengthened or weakened for different parameters which will be discussed hereinafter. But, it should keep in mind that the non-uniform distribution of nanoparticles leads to the non-uniform distribution of thermophysical properties, Eqs. (5)-(8); hence, the velocity and temperature profiles are significantly affected by the nanoparticle distribution. In fact, despite the regular
14
fluids, the hydrodynamic and thermal characteristics of nanofluids are directly affected by the thermophoresis or nanoparticle migration. The effect of NBT as a principal parameter is demonstrated in Figure 4. It is observed that increasing NBT weakens the nanoparticle migration such that the nanoparticle concentration becomes approximately uniform for NBT =10. This is due to the fact that an increase in NBT decreases the thermophoresis, the driving force of nanoparticles. However, as NBT decreases, the effect of thermophoresis elevates, which makes the nanoparticles to move toward the middle of the microchannel. In this case, there is a nanoparticle depleted region at the wall and a nanoparticle accumulated region at the central region. Notably, NBT is inversely proportional to the nanoparticle diameter (dp); therefore, the larger nanoparticle diameter leads to a higher migration of nanoparticles, while the smaller one results in a more uniform nanoparticle distribution. Moreover, based on the aforementioned explanations with regard to the dependence of TB to axial coordinate resulting in a reduction of as x increases, it can be concluded that the values of NBT would elevate. The results for the higher values of NBT indicate that the heat transfer coefficient becomes almost constant as NBT increases and the slope of pressure drop increment is almost constant. In other words, for the higher values of NBT, the non-homogeneous model of nanofluids leads to a homogeneous model (check Eq. (31) when NBT grows). This physic can also be seen in Figure 4a (for NBT=10 the nanoparticle volume fraction becomes constant and the nanofluids flow is completely homogeneous). Figure 4 also signifies that the peak of velocity profile reduces for the larger nanoparticles in which the nanoparticle migration is strong since increasing the nanoparticle concentration in the central region of microchannel grows the viscosity of the nanofluid, decreasing the velocity. The same behavior can be seen for the thermal conductivity and the temperature. Figure 5 illustrates that an increase in ϕB declines
15
the peak of nanoparticle concentration and velocity profiles; however, it increases the dip of the temperature. This is because a rise in ϕB increases the nanoparticle volume fraction, and hence it grows the Brownian diffusion effect. This dynamic pushes the nanoparticles into the central region of the tube, which has a higher kinetic energy (a higher velocity), in the direction of decreasing the concentration gradient. Therefore, the nanoparticle volume fraction becomes more uniform in the central region. Nevertheless, 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 ϕB has a direct impact on the viscosity and thermal conductivity of nanofluids; consequently, the velocity and temperature gradients reduce at the higher values of ϕB. Figure 6 illustrates that growing the slip parameter (λ) increases the slip velocity at the wall, followed by a reduction in the velocity peak in the central region of the microchannel. In fact, the velocity of the nanofluid accelerates near the walls while, due to a constant mass flux in the tube, the velocity decelerates at the core region. In addition, the migration of nanoparticles enhances for the higher values of λ, thereby decreasing the dip of the temperature profile.
4. 2 Heat transfer enhancement and pressure drop increment In heat exchange equipment, two main physical quantities of interest should be provided, namely heat transfer rate and pressure drop. In nanofluids, it is worthwhile to consider the heat transfer enhancement and pressure drop increment, which demonstrates 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 defined as
HTC
hDh qw 2 Ro k w 2T * kbf kbf Tw TB kbf 16
(34)
where Dh=2Ro. Hence, the heat transfer enhancement may be evaluated as
h HTC hbf HTCbf
(35)
Similarly, the dimensionless pressure gradient can be written as
Np
4B uB dp bf 2 dx 2Ro u* bf w
(36)
where ρB is the bulk density. And the pressure drop increment can be given by
N dp
Np N pbf
(37)
Figures 7 and 8 show the effect of pertinent parameters including ϕB and λ on the heat transfer enhancement and pressure drop increment via different nanoparticle migration regimes (NBT). The figures show that the heat transfer enhancement has a peak around NBT≈1 for different parameters, called hereinafter as critical NBT. The existence of critical NBT signifies different modes of heat transfer rate as well as the importance of nanoparticle migration on enhancing heat transfer rate. To explain, the nanoparticle migration has two opposing effects on the heat transfer rate of nanofluids: a) the thermal conductivity reduction at the wall due to nanoparticle depletion and b) momentum augmentation at the wall due to viscosity decrement there. In fact, the thermal conductivity and viscosity of nanofluids reduce as the nanoparticles migrate toward the central region. The thermal conductivity decrement has a negative impact on the heat removal ability of nanofluids while the viscosity reduction at the wall results in a momentum augmentation at the wall which has a positive impact on the heat transfer rate. When the value of NBT is higher than the critical one, the heat transfer enhancement slightly decreases and asymptotically reaches to a constant value. This is due to the fact that for the higher values of NBT, the nanoparticles uniformly distribute in the tube and more changes in the value of NBT would not have any 17
influence on the distribution of nanoparticles and the thermal conductivity. For the value of NBT lower than the critical NBT, the nanoparticle depletion at the wall leads to a significant reduction in the thermal conductivity at the wall, thereby reducing the heat transfer coefficient. In addition, the figures reveal that the pressure drop increment significantly increases for the higher values of NBT. Thus, the larger nanoparticles (the smaller values of NBT) lead to a lower increase in the pressure drop. With regard to Figure 7, it can be seen that a rise in ϕB increases the heat transfer enhancement as well as the pressure drop ratio, which is due to the thermal conductivity and viscosity increments. From Figure 8, it can be seen that increasing λ has a trivial effect on the heat transfer rate for the lower values of NBT, whereas it leads to a rise in the heat transfer rate for the higher values of NBT. Also, increasing λ decreases the pressure drop ratio at low NBT. As NBT increases, the changes in pressure drop ratio become insignificant.
4. 3 Thermal performance To characterize the performance of heat exchange equipment, the figure of merit (FoM) is used as the comparative measure of the heat transfer enhancement with respect to the penalty in pumping power. For nanofluids, FoM can be defined as
FoM
h hbf
1/3
N p N pbf
(38)
For FoM>1, the thermal performance of the system is increased while it is the opposite for FoM<1. Figures 9 and 10 show the computed figures of merit via NBT for different values of ϕB and λ respectively. The figures reveal the nanofluids have a merit for the lower values of NBT, where the thermophoresis dominates the Brownian diffusion and the nanoparticles migrate significantly. In addition, there is an optimum value for NBT around 0.4 in which the FoM reaches its peak. Since NBT is inversely proportional to the nanoparticle diameter (dp), it can be stated that
18
there is an optimum nanoparticle diameter in which the performance of the system is maximum. In addition, Figure 9 shows that increasing ϕB significantly enhances the FoM for the lower values of NBT, whereas, at the higher values of NBT it has an opposite effect. Thus, increasing the nanoparticle volume fraction is beneficial for larger nanoparticles when the thermophoresis effect dominates the Brownian motion one. Furthermore, Figure 10 indicates that the thermal performance (FoM) slightly increases as λ grows; however, this trend reduces for very large and very small values of NBT. As a result, slippage at the fluid-solid interface has a positive effect on the performance.
4. 4 Constant wall temperature versus constant wall heat flux The thermal boundary condition is one of the consequential issue affecting the behavior of nanofluids [20, 38]. Figure 11 illustrates a comparison between two common thermal conditions at the wall namely, constant heat flux (CHF ) [17] and constant wall temperature (CWT) of the present study on the anomalous heat transfer enhancement (HTCkbf/kB). Firstly, it can be observed that the constant wall heat flux leads to a greater heat transfer enhancement in nanofluids. In addition, the effect of nanoparticle migration (NBT) for the two cases (CHF and CWT) are the same. Obviously, there are optimum NBT for the heat transfer enhancement in both cases, in which the heat transfer enhancement reaches its peak. However, the optimum NBT for the case of CWT is greater (about 1) than that for the case of CHF (about 0.5). Thus, it can be concluded the optimum diameter of nanoparticles for the case of CWT should be smaller than the case of CHF.
5. Summary and Conclusions The present study is a numerical investigation of the figure of merit for the laminar fully developed flow and heat transfer of nanofluids through a circular microchannel subjected to a 19
constant wall temperature (CWT). The modified Buongiorno's model is used for the nanofluid to simulate 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. The major outcomes of the present study can be classified as follows: The nanofluids have a merit for the lower values of NBT, where the thermophoresis dominates the Brownian diffusion and the nanoparticles migrate significantly. In addition, there is an optimum value for NBT around 0.4 in which the FoM reaches its peak. Since NBT is inversely proportional to the nanoparticle diameter (dp), it can be stated that there is an optimum nanoparticle diameter in which the performance of the system is maximum. The heat transfer enhancement has a peak around the critical NBT≈1 for different parameters. When the value of NBT is higher than the critical one, the heat transfer enhancement slightly decreases and asymptotically reaches a constant value. For the values of NBT lower than the critical NBT, the nanoparticle depletion at the wall leads to a significant reduction in the thermal conductivity there, which reduces the heat transfer enhancement. Increasing ϕB leads to a rise in the nanoparticle concentration, thereby growing the Brownian diffusion effect. This dynamic makes the nanoparticles in the central region of the tube, which has a higher kinetic energy, move in the direction of reducing the
20
concentration gradient. Thus, the nanoparticle volume fraction becomes more uniform in the central region, as ϕB increases. Increasing the nanoparticle volume fraction is beneficial for larger nanoparticles when the thermophoresis effect dominates the Brownian motion one. Furthermore, slippage at the fluid-solid interface has a positive effect on the performance. The constant heat flux (CHF) at the wall leads to a greater heat transfer enhancement in nanofluids than that of the constant wall temperature (CWT). Furthermore, there is an optimum NBT for the heat transfer enhancement in both cases (CHF and CWT), in which the heat transfer enhancement reaches its peak. However, the optimum NBT for the case of CWT is greater (about 1) than that for the case of CHF (about 0.5). Thus, it can be concluded the optimum diameter of nanoparticles for the case of CWT should be smaller than that for the case of CHF.
Acknowledgement The authors wish to acknowledge the anonymous reviewers for their careful reading of our manuscript and their constructive comments which helped us to improve the manuscript.
References [1] 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, Applied Physics Letters, 78 (2001) 718-720. [2] H.C. Brinkman, The Viscosity of Concentrated Suspensions and Solutions, The Journal of Chemical Physics, 20 (1952) 571-571. [3] G.K. Batchelor, The effect of Brownian motion on the bulk stress in a suspension of spherical particles, Journal of Fluid Mechanics, 83 (1977) 97-117.
21
[4] B.C. Pak, Y.I. Cho, HYDRODYNAMIC AND HEAT TRANSFER STUDY OF DISPERSED FLUIDS WITH SUBMICRON METALLIC OXIDE PARTICLES, Experimental Heat Transfer, 11 (1998) 151-170. [5] J. Buongiorno, Convective Transport in Nanofluids, Journal of Heat Transfer, 128 (2006) 240-250. [6] M.A. Sheremet, T. Groşan, I. Pop, Steady-state free convection in right-angle porous trapezoidal cavity filled by a nanofluid: Buongiorno’s mathematical model, European Journal of Mechanics B/Fluids, 53 (2015) 241-250. [7] D. Yadav, R. Bhargava, G.S. Agrawal, Numerical solution of a thermal instability problem in a rotating nanofluid layer, International Journal of Heat and Mass Transfer, 63 (2013) 313-322. [8] M. Sheikholeslami, R. Ellahi, Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid, International Journal of Heat and Mass Transfer, 89 (2015) 799-808. [9] F. Garoosi, L. Jahanshaloo, M.M. Rashidi, A. Badakhsh, M.E. Ali, Numerical simulation of natural convection of the nanofluid in heat exchangers using a Buongiorno model, Applied Mathematics and Computation, 254 (2015) 183-203. [10] D.A. Nield, A.V. Kuznetsov, The onset of double-diffusive convection in a nanofluid layer, International Journal of Heat and Fluid Flow, 32 (2011) 771-776. [11] S. Soleimani, M. Sheikholeslami, D.D. Ganji, M. Gorji-Bandpay, Natural convection heat transfer in a nanofluid filled semi-annulus enclosure, International Communications in Heat and Mass Transfer, 39 (2012) 565-574. [12] A. Sabaghan, M. Edalatpour, M.C. Moghadam, E. Roohi, H. Niazmand, Nanofluid flow and heat transfer in a microchannel with longitudinal vortex generators: Two-phase numerical simulation, Applied Thermal Engineering, 100 (2016) 179-189. [13] N.S. Akbar, S.U. Rahman, R. Ellahi, S. Nadeem, Nano fluid flow in tapering stenosed arteries with permeable walls, International Journal of Thermal Sciences, 85 (2014) 54-61. [14] S.A. Moshizi, Forced convection heat and mass transfer of MHD nanofluid flow inside a porous microchannel with chemical reaction on the walls, Engineering Computations, 32 (2015) 2419-2442. [15] A. Malvandi, A. Ghasemi, D.D. Ganji, I. Pop, Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology, 27 (2016) 1941-1948. [16] B.H. Salman, H.A. Mohammed, K.M. Munisamy, A.S. Kherbeet, Characteristics of heat transfer and fluid flow in microtube and microchannel using conventional fluids and nanofluids: A review, Renewable and Sustainable Energy Reviews, 28 (2013) 848-880. [17] C. Yang, W. Li, Y. Sano, M. Mochizuki, A. Nakayama, On the Anomalous Convective Heat Transfer Enhancement in Nanofluids: A Theoretical Answer to the Nanofluids Controversy, Journal of Heat Transfer, 135 (2013) 054504-054504. [18] C. Yang, W. Li, A. Nakayama, Convective heat transfer of nanofluids in a concentric annulus, International Journal of Thermal Sciences, 71 (2013) 249-257.
22
[19] A. Malvandi, D. Ganji, Magnetic field effect on nanoparticles migration and heat transfer of water/alumina nanofluid in a channel, Journal of Magnetism and Magnetic Materials, 362 (2014) 172179. [20] A. Malvandi, D.D. Ganji, Effects of nanoparticle migration on hydromagnetic mixed convection of alumina/water nanofluid in vertical channels with asymmetric heating, Physica E: Low-dimensional Systems and Nanostructures, 66 (2015) 181-196. [21] F. Hedayati, G. Domairry, Effects of nanoparticle migration and asymmetric heating on mixed convection of TiO2–H2O nanofluid inside a vertical microchannel, Powder Technology, 272 (2015) 250259. [22] A. Malvandi, D.D. Ganji, I. Pop, Laminar filmwise condensation of nanofluids over a vertical plate considering nanoparticles migration, Applied Thermal Engineering, 100 (2016) 979-986. [23] C. Yang, Q. Wang, A. Nakayama, T. Qiu, Effect of temperature jump on forced convective transport of nanofluids in the continuum flow and slip flow regimes, Chemical Engineering Science, 137 (2015) 730-739. [24] A. Malvandi, S. Moshizi, D. Ganji, Two-component heterogeneous mixed convection of alumina/water nanofluid in microchannels with heat source/sink, Advanced Powder Technology, 27 (2016) 245-254. [25] M. Bahiraei, S.M. Hosseinalipour, Particle migration in nanofluids considering thermophoresis and its effect on convective heat transfer, Thermochimica Acta, 574 (2013) 47-54. [26] A. Malvandi, A. Ghasemi, D.D. Ganji, Thermal performance analysis of hydromagnetic Al2O3water nanofluid flows inside a concentric microannulus considering nanoparticle migration and asymmetric heating, International Journal of Thermal Sciences, 109 (2016) 10-22. [27] S.A. Moshizi, I. Pop, Conjugated Effect of Joule Heating and Magnetohydrodynamic on Laminar Convective Heat Transfer of Nanofluids Inside a Concentric Annulus in the Presence of Slip Condition, International Journal of Thermophysics, 37 (2016) 1-22. [28] S.A. Moshizi, M. Zamani, S.J. Hosseini, A. Malvandi, Mixed convection of magnetohydrodynamic nanofluids inside microtubes at constant wall temperature, Journal of Magnetism and Magnetic Materials, 430 (2017) 36-46. [29] S.J. Hosseini, A. Malvandi, S.A. Moshizi, M. Zamani, Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology. [30] D.Y. Tzou, Instability of Nanofluids in Natural Convection, Journal of Heat Transfer, 130 (2008) 072401-072401. [31] M.M. MacDevette, T.G. Myers, B. Wetton, Boundary layer analysis and heat transfer of a nanofluid, Microfluidics and Nanofluidics, 17 (2014) 401-412. [32] W. Zhang, W. Li, A. Nakayama, An analytical consideration of steady-state forced convection within a nanofluid-saturated metal foam, Journal of Fluid Mechanics, 769 (2015) 590-620. [33] D. Song, D. Jing, B. Luo, J. Geng, Y. Ren, Modeling of anisotropic flow and thermodynamic properties of magnetic nanofluids induced by external magnetic field with varied imposing directions, Journal of Applied Physics, 118 (2015) 045101. 23
[34] A. Malvandi, S.A. Moshizi, D.D. Ganji, Nanoparticle transport effect on magnetohydrodynamic mixed convection of electrically conductive nanofluids in micro-annuli with temperature-dependent thermophysical properties, Physica E: Low-dimensional Systems and Nanostructures, 88 (2017) 35-49. [35] S.A. Moshizi, A. Malvandi, Magnetic field effects on nanoparticle migration at mixed convection of MHD nanofluids flow in microchannels with temperature-dependent thermophysical properties, Journal of the Taiwan Institute of Chemical Engineers, 66 (2016) 269-282. [36] W. Kays, M. Crawford, B. Weigand, Convective Heat & Mass Transfer W/ Engineering Subscription Card, McGraw-Hill Companies,Incorporated, 2005. [37] A. Malvandi, D.D. Ganji, Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field, International Journal of Thermal Sciences, 84 (2014) 196-206. [38] A. Malvandi, D.D. Ganji, Effects of nanoparticle migration and asymmetric heating on magnetohydrodynamic forced convection of alumina/water nanofluid in microchannels, European Journal of Mechanics - B/Fluids, 52 (2015) 169-184.
24
*Tables
Table 1. Grid independence test for different values of d when B 0.02, 0.1 and N BT 1 d
h hbf
N p N pbf
102
1.17610
1.43151
5 103
1.17611
1.43154
103
1.17612
1.43156
Fig. 1. The geometry of physical model and coordinate system
25
Take Start
B , , N BT Guess the value of
w , ( )c p ( )u
guess
Solve
u* A u * B 0 u u * * * T AT T BT T CT 0 * A T 0 with B.C.
du * * 0 : u , T * 1 , W d * * 1 : du 0 , T 0 d Find
B cal . ( )c p ( )u
End
u * u*
1
cal .
2 ( )c p ( )u(1 )d 0
Yes if
(B )cal . B
( )c p ( )u
cal .
( )c p ( )u 26
No guess
Fig. 2. Algorithm of the numerical method
27
Fig. 3. Comparison of velocity (a) and temperature profile (b) for regular fluid with the results of Kays et al. [36]
28
29
Fig. 4. The effects of N BT on (a) nanoparticle distribution ( / B ), (b) velocity ( u* uB* ), and (c) temperature ( T * ) profiles when B 0.02 and 0.1 .
30
31
Fig. 5. The effects of B on (a) nanoparticle distribution ( / B ), (b) velocity ( u* uB* ), and (c) temperature ( T * ) profiles when 0.1 and N BT 1 .
32
33
Fig. 6. The effects of on (a) nanoparticle distribution ( / B ), (b) velocity ( u* uB* ), and (c) temperature ( T * ) profiles when B 0.02 and N BT 1 .
34
Fig. 7. The effects of B on (a) the heat transfer coefficient ratio 35
and (b) the pressure drop ratio when 0.1 .
36
Fig. 8. The effects of λ on (a) the heat transfer coefficient ratio and (b) the pressure drop ratio when B 0.02 .
Fig. 9. The effects of B on the figure of merit (FoM) when 0.1 .
37
Fig. 10. The effects of λ on the figure of merit (FoM) when B 0.02 .
38
Fig. 11. Anomalous heat transfer rate for both cases of constant heat flux (CHF) [17] at the wall and constant wall temperature (CWT) when 0 .
* Highlights
Thermally fully developed flow of nanofluid in circular microchannels at constant wall
temperature
Effect of nanoparticle migration on fluid flow and heat transfer characteristics
Thermal performance gets its peak around NBT=0.4
Anomalous heat transfer rate occurs for NBT=1
Optimum value of NBT for the case of CWT is greater than that of CHF
39
S1359-4311(17)31135-3 http://dx.doi.org/10.1016/j.applthermaleng.2017.02.081 ATE 9962
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
13 March 2016 16 February 2017 19 February 2017
Please cite this article as: A. Malvandi, M. Zamani, S.J. Hosseini, S.A. Moshizi, Figure of merit for optimization of nanofluid flow in circular microchannel by adapting nanoparticle migration, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng.2017.02.081
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Figure of merit for optimization of nanofluid flow in circular microchannel by adapting nanoparticle migration A. Malvandi a, *, M. Zamani b, S. J. Hosseini c, S. A. Moshizi d a
Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran b Young Researchers and Elite Club, Gonabad Branch, Islamic Azad University, Gonabad, Iran c Department of Mechanical Engineering, School of Engineering, University of Tehran, Tehran, Iran d Young Researchers and Elite Club, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran
Abstract In this paper, the laminar fully developed flow of alumina/water nanofluid inside circular microchannels subjected to a constant wall temperature (CWT) is theoretically investigated. The effect of nanoparticles migration originating from thermophoretic diffusion (temperaturegradient driven force) and Brownian diffusion (concentration-gradient 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 order to assume a hydrodynamically and thermally fully developed flow, the governing equations are reduced to a system of ordinary differential equation and solved using the appropriate reciprocal algorithm. The effects of pertinent parameters including the ratio of Brownian motion to thermophoresis (NBT), slip parameter (λ) and bulk mean nanoparticle volume fraction (ϕB) on the flow and thermal fields are investigated. In addition, the results are compared with the case of constant heat flux (CHF) at the wall. The figure of merit (FoM) is used to measure the thermal performance of equipment and finding the optimum thermal condition. It is shown that the anomalous heat transfer enhancement depends on the thermal boundary condition as well as the nanoparticles diameter. Furthermore, the optimum value of NBT for the case of constant wall temperature (approximately 1) is found to be greater than that of the constant wall heat flux (approximately 0.5). Thus, it can be concluded that the optimum diameter of nanoparticles for the case of constant wall temperature should be smaller than that of constant wall heat flux. Key Words: Figure of merit; nanofluid; nanoparticle migration; thermophoresis; Brownian
motion.
*
Corresponding author, Email: [email protected]
1
Nomenclature cp
specific heat capacity (m2/s2 K)
dp
nanoparticle diameter (m)
Dh
Hydraulic diameter, Dh=2Ro (m)
DB
=
DT
= 0.26
FoM
Figure of Merit
h
heat transfer coefficient (W/m2.K)
k
thermal conductivity (W/m.K)
kBO
Boltzmann constant ( 1.3806488
N
slip velocity factor
Nu
Nusselt number
N BT
ratio of the Brownian to thermophoretic diffusivities
P
pressure (Pa)
qw
surface heat flux (W/m2)
T
temperature (K)
u
axial velocity (m/s)
x, r
coordinate system
k BOT , Brownian diffusion coefficient (m2/s) 3bf d p k bf bf , thermophoresis diffusion coefficient (m2/s) 2k bf k p bf
Greek symbols
2
1023 m2 kg s2 K )
nanoparticle volume fraction
ratio of wall and fluid temperature difference to absolute temperature
transverse direction
dynamic viscosity (kg/m.s)
density (kg/m3)
slip parameter
internal angle inside cross-section of microtube
Subscripts B
bulk mean
bf
base fluid
p
Nanoparticle
w
condition at the wall
Superscripts *
dimensionless variable
3
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, 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).
1.1
Theoretical modeling of nanofluids
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 [1]; b) an abnormal increase in the viscosity of nanofluids relative to the regular fluid [2, 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 nanoparticle flux) heterogeneous equilibrium 4
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 et al. [7], Sheikholeslami and Ellahi [8], Garoosi et al. [9], and Nield and Kuznetsov [10]. Theoretical investigation on the effect of nanofluids has been systematically reported and well documented, which can be found in the open literature [11-15]. An excellent review paper on the application of nanofluids in microchannels is conducted by Salman et al. [16].
1.2
Nanoparticle migration effects
In 2013, Yang et al. [17, 18] modified the Buongiorno’s model to consider the impact of nanoparticle distribution on the thermal conductivity and viscosity of nanofluids. In fact, the proposed modified model does not ignore the dependency of thermophysical properties of nanofluids to the nanoparticles concentration. Their results indicated that the non-uniformity of the thermophysical properties is the reason for the anomalous heat transfer enhancement. Malvandi et al. [19], 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 [20] investigated the impacts of the nanoparticle migration as well as asymmetric heating at the walls on the forced convective heat transfer of magnetohydrodynamic alumina/water nanofluid in microchannels. Hedayati and Domairry [21] investigated the effects of the nanoparticle migration on titania/water nanofluids in horizontal and vertical channels. The popularity of modeling the nanoparticle migration can be gauged from the numerous published works of literature such as [22-27]. 5
1.3
Motivation and novelty
From the previous explanations, the utilization of the modified Buongiorno model has been limited to a special thermal boundary condition that provides the constant heat flux at the boundaries. Recently, Malvandi and Ganji [20] demonstrated that because of thermophoresis, asymmetrically heated walls are able to control the fluid flow and heat transfer characteristics of nanofluids. In addition, the amount of enhancement in the heat transfer rate can be controlled by adjusting the heat flux at the boundaries. These outcomes pointed out that thermal boundary conditions are a key factor in the fluid flow and heat transfer characteristics of nanofluids. Accordingly, in this paper, a prescribed wall temperature as a widespread thermal boundary condition is imposed on the wall, which is an important development for the modified Buongiorno model [28, 29]. The fully developed governing equations of the modified Buongiorno model are obtained and the results for the pressure drop and the heat transfer enhancement are presented versus varying pertinent parameters. Because of low dimensional structures in microchannels, a linear slip condition is considered at the surfaces, which appropriately represents the non-equilibrium region near the fluid/solid interface. To study the thermal performance, the figure of merit (FoM) is calculated to signify it. The impacts of the thermal boundary condition, nanoparticle migration, as well as the slip velocity on thermal performance are of our particular interests.
2. Problem description and governing equations The geometry of the problem with the coordinate system is shown in Figure 1. To give a more physical outlook of the problem, the distribution of velocity and nanoparticle volume fraction is schematically illustrated. The following assumptions are considered in the present paper:
6
1. The nanofluid is assumed incompressible, on the ground that the density of nanofluid can be approximated by that of the base fluid, i.e.,
bf
, since the volumetric fraction of
nanoparticles is only a few percent [5, 30, 31]. 2. There are some important assumptions for mathematical modeling including incompressible flow, a dilute mixture, negligible external force and radiative heat transfer, no chemical reactions and negligible viscous dissipation. 3. The modified Buongiorno’s model is considered so as to simulate the effects of Brownian diffusion and thermophoresis. 4. Adapted cylindrical coordinates x and r were aligned parallel and perpendicular to the wall respectively. 5. The slip velocity is considered in the wall which allows the formation of the axial velocity component at the wall. 6. Flow and thermal fields are assumed to be fully developed. 7. There is a thermal equilibrium between the nanoparticles and the base fluid. 8. All the fluid properties vary with the nanoparticle volume fraction. The momentum and energy equations in general form can be written as:
. V 0,
(1)
V . V V p . V , t
(2)
D cPT . cP VT . kT . cP pTDB p cP DT T cPT . p DB p T T , t T
(3)
D p . pV . p DB p T T , t T
(4)
7
where u, T, and p represent the axial velocity, local temperature, and pressure respectively. In addition, ρ, µ, k, and cP are respectively the density, dynamic viscosity, thermal conductivity, and specific heat capacity of nanofluid obtained via [2, 5]
p 1 bf ,
(5)
bf 1 a b 2 , a 39.11, b 533.9 for alumina
(6)
k kbf 1 ak bk 2 , ak 7.47, bk 0 for alumina
(7)
cp
p c p 1 bf c p p
p 1 bf
bf
,
(8)
where the thermophysical properties of the alumina nanoparticles and base fluid (water) are determined in the following manner: Base fluid:
c pbf 4182 J/(kg K), bf 998.2 kg / m3 , kbf 0.597 W/(m K), bf 9.93 104 kg/(m s)
(9)
Alumina nanoparticles:
c p p 773 J/(kg K), p 3880 kg / m3 , k p 36 W/(m K)
(10)
Moreover, in Eqs. (3) and (4), the Brownian motion coefficient DB and thermophoresis coefficient DT can be defined as [5] DB
k BOT 3bf d p
, DT 0.26
kbf
bf
2kbf k p bf
(11)
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
8
du T 0 , 0 , 0 , dr r r
r 0
r Ro u N
(12)
du DT T , T Tw , DB 0. dr r T r
where N introduces the slip velocity factor. Buongiorno [5] corroborated that the second and third terms on the right-hand side of Eq. (3) presenting the heat transfer related to the nanoparticle dispersion (i.e., nanoparticle diffusion flux) is always insignificant in comparison with convection and conduction terms. So, the term vanishes for the case of fully developed flows. Thus, the energy equation becomes identical to that of a pure fluid, except that all properties strongly depend on the spatial distribution of [5, 17]. In addition, in the studied case, the diffusion mass flux of the nanoparticles remains constant across the tube since the wall is impermeable. Thus, the Brownian diffusion flux and thermophoretic diffusion flux cancel out everywhere in the tube. As a consequence, the term on the left-hand side of Eq. (4) is equal zero [17]. Furthermore, since the mass flux inside channels is constant (so
uA cte . ),
it results in x 0 with considering hydrodynamic fully
developed condition [17, 23, 32, 33]. In addition, the flow field is assumed to be thermally and hydrodynamically fully developed ( ur 0 , ux x 0 and ux u ) and steady state ( t 0 ). Accordingly, the incompressible conservation equations of the momentum, thermal energy, and nanoparticle fraction can be written in the following manner [5, 34, 35]:
dp 1 d du r ( ) 0 dx r dr dr
( )uc p ( )
T 1 T rk ( ) x r r r
9
(13) 0
(14)
1 DT T DB 0 r r r T r
2. 1
(15)
Dimensionless momentum equation
The governing equations can be reduced to non-dimensional form via the following nondimensional variables
Ro r Ro
, u*
u dp dx Ro 2 W
(16)
Thus, Eq. (13) reduces to
u* Au u* Bu 0, where ‘ (prime symbol) represents
Au
(17)
d in which d
d d 1 Ln ( ) d d 1
Bu
,
W ( )
(18)
Moreover, u B* is the bulk velocity defined as
u * B
u *
(19)
where R
2. 2
1
1 1 o dA 2 rdr 2 1 d A A Ro 2 0 0
(20)
Dimensionless energy equation
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
10
ignoring the axial conduction effect, the following relation can be determined for the axial temperature gradient [5]
T Tw T dTB , x Tw TB dx
(21)
where TB is the bulk temperature defined as
TB
( )c p ( )uT ( )c p ( )u
( )c ( )uT 1 d . ( )c ( )u 1 d 1
p
0
(22)
1
p
0
Substituting Eq. (21) into Eq. (14) yields the energy equation as
( )c p ( )u
Tw T dTB 1 T rk ( ) Tw TB dx r r r
(23)
Eq. (23) can be reduced by integrating in the following manner
( )uc p ( )
Finding
dT B 2 T T 2 k ( )r k ( )r dx Ro r r Ro r
2qw . R o r 0
(24)
dTB from Eq. (24), substituting it into Eq. (23) and introducing the following nondx
dimensional variables
T*
T T B Tw T B
,
HTC
qw 2Ro k hD h w 2T * , k bf k bf Tw T B k bf
(25)
reduce Eq. (23) to
T * AT T * BT T * CT 0
(26)
where T* is the dimensionless temperature and HTC stands for the heat transfer coefficient and
AT
2. 3
d d 1 Ln k ( ) d d 1
,
( )uc p ( ) BT CT ( )uc p ( )
Dimensionless nanoparticle continuity 11
kbf HTC . k ( )
(27)
According to Eq. (25), Eq. (15) can be written as T T DT * * w B T T Tw TB TB DB Introducing the following non-dimensional parameters
Tw TB TB
N BT
,
DBB B , DTB
(28)
(29)
yields Eq. (28) as
A T * 0
(30)
where
A
2. 4
N BT 1 T
* 2
.
(31)
Dimensionless boundary condition
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. (12) can be written in the non-dimensional form as
du* * 0 : u , T * 1 , w , d * * 1 : du 0 , T 0, d where
(32)
N . Eq. (32) represents the slip velocity and constant temperature at the wall. Since Ro
the value of ϕB is usually prescribed rather than ϕw, the following relation is used as an additional constraint
B
u * u*
.
(33)
3. Numerical method and accuracy The equations (17), (26), and (30) are the governing equations representing a system of nonlinear Ordinary Differential Equations (ODEs) which should be solved with the boundary conditions of 12
Eq. (32). It is worth noting that the values of BT (η) and CT (η) in Eq. (26) are not specified in advance, because ( )c p ( )u depends on the nanoparticle volume fraction profile and it is determined when the governing equations are solved. Furthermore, there is a similar condition for ϕw in Eq. (32). Therefore, a reciprocal procedure is required to solve the governing equations for the unknown values of
( )c p ( )u
and ϕw. The algorithm of numerical approach is
demonstrated in Figure 2. Firstly, the values of parameters including ϕB, λ, and NBT are input to the code. Then, initial guesses for the unknown values of
( )c p ( )u and ϕw have been
considered. After the governing equations with the boundary conditions have been solved via the Runge-Kutta-Fehlberg method, the calculated values of ( )c p ( )u and ϕB are compared with the assumed value of ( )c p ( )u and the prescribed value of ϕB. This process is iterated until the prescribed value of ϕB is achieved, and the relative error between the assumed value of
( )c p ( )u and the calculated one after solving the governing equations is lower than 10 -6. To corroborate the accuracy of the numerical code, the velocity and temperature profiles for the laminar fully developed flow in the circular channel with ϕB=λ=0 are compared to the reported results of Kays et al. [36] in Figure 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 (dη) of 10-2, 5×10-3 and 10-3 in Table 1; obviously, altering the grid size has the insignificant effect on the results. It should be stated that all the following numerical results have been acquired using the integration step dη=10-3.
4. Results and discussion Here, there is some background information which should be explained in advance. As it was shown in previous studies [37], for the nanoparticles with diameter around and less than 100nm, 13
NBT is in the order of 1; so, in the present study, a wide range of NBT from 0.25 to 10 is considered. The selected values of NBT lead to different modes of nanoparticle migration which will be discussed later. In addition, the value of
Tw T B is assumed to be 0.1 because of its TB
trivial effect on the solution [17, 19]. In justification of the slight values of
Tw T B T w 1, TB TB
it can be seen that when the flow is thermally fully developed, the discrepancy between Tw and
T B decreases, after which it reaches a plateau [36]. So, has an insignificant slope related to x and possesses the low values.
4. 1 Flow and thermal fields Figures 4-6 show the impact of pertinent parameters including NBT, ϕB, and λ on dimensionless nanoparticle volume fraction, velocity, and temperature profiles. In the figures, η=0 denotes the condition at the wall while η=1 represents the center of the tube. The figures indicate that the smallest amount of nanoparticle concentration is at the wall (η=0) and gradually increases to the highest amount at the center of the microchannel, indicating that the nanoparticles move from the wall toward the central region of the tube. This dynamic definitely proves the impact of the thermophoresis, which pushes the nanoparticles from the hot region (η=0) toward the colder one (η=1) and makes a non-uniform distribution of nanoparticles. This phenomenon can be strengthened or weakened for different parameters which will be discussed hereinafter. But, it should keep in mind that the non-uniform distribution of nanoparticles leads to the non-uniform distribution of thermophysical properties, Eqs. (5)-(8); hence, the velocity and temperature profiles are significantly affected by the nanoparticle distribution. In fact, despite the regular
14
fluids, the hydrodynamic and thermal characteristics of nanofluids are directly affected by the thermophoresis or nanoparticle migration. The effect of NBT as a principal parameter is demonstrated in Figure 4. It is observed that increasing NBT weakens the nanoparticle migration such that the nanoparticle concentration becomes approximately uniform for NBT =10. This is due to the fact that an increase in NBT decreases the thermophoresis, the driving force of nanoparticles. However, as NBT decreases, the effect of thermophoresis elevates, which makes the nanoparticles to move toward the middle of the microchannel. In this case, there is a nanoparticle depleted region at the wall and a nanoparticle accumulated region at the central region. Notably, NBT is inversely proportional to the nanoparticle diameter (dp); therefore, the larger nanoparticle diameter leads to a higher migration of nanoparticles, while the smaller one results in a more uniform nanoparticle distribution. Moreover, based on the aforementioned explanations with regard to the dependence of TB to axial coordinate resulting in a reduction of as x increases, it can be concluded that the values of NBT would elevate. The results for the higher values of NBT indicate that the heat transfer coefficient becomes almost constant as NBT increases and the slope of pressure drop increment is almost constant. In other words, for the higher values of NBT, the non-homogeneous model of nanofluids leads to a homogeneous model (check Eq. (31) when NBT grows). This physic can also be seen in Figure 4a (for NBT=10 the nanoparticle volume fraction becomes constant and the nanofluids flow is completely homogeneous). Figure 4 also signifies that the peak of velocity profile reduces for the larger nanoparticles in which the nanoparticle migration is strong since increasing the nanoparticle concentration in the central region of microchannel grows the viscosity of the nanofluid, decreasing the velocity. The same behavior can be seen for the thermal conductivity and the temperature. Figure 5 illustrates that an increase in ϕB declines
15
the peak of nanoparticle concentration and velocity profiles; however, it increases the dip of the temperature. This is because a rise in ϕB increases the nanoparticle volume fraction, and hence it grows the Brownian diffusion effect. This dynamic pushes the nanoparticles into the central region of the tube, which has a higher kinetic energy (a higher velocity), in the direction of decreasing the concentration gradient. Therefore, the nanoparticle volume fraction becomes more uniform in the central region. Nevertheless, 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 ϕB has a direct impact on the viscosity and thermal conductivity of nanofluids; consequently, the velocity and temperature gradients reduce at the higher values of ϕB. Figure 6 illustrates that growing the slip parameter (λ) increases the slip velocity at the wall, followed by a reduction in the velocity peak in the central region of the microchannel. In fact, the velocity of the nanofluid accelerates near the walls while, due to a constant mass flux in the tube, the velocity decelerates at the core region. In addition, the migration of nanoparticles enhances for the higher values of λ, thereby decreasing the dip of the temperature profile.
4. 2 Heat transfer enhancement and pressure drop increment In heat exchange equipment, two main physical quantities of interest should be provided, namely heat transfer rate and pressure drop. In nanofluids, it is worthwhile to consider the heat transfer enhancement and pressure drop increment, which demonstrates 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 defined as
HTC
hDh qw 2 Ro k w 2T * kbf kbf Tw TB kbf 16
(34)
where Dh=2Ro. Hence, the heat transfer enhancement may be evaluated as
h HTC hbf HTCbf
(35)
Similarly, the dimensionless pressure gradient can be written as
Np
4B uB dp bf 2 dx 2Ro u* bf w
(36)
where ρB is the bulk density. And the pressure drop increment can be given by
N dp
Np N pbf
(37)
Figures 7 and 8 show the effect of pertinent parameters including ϕB and λ on the heat transfer enhancement and pressure drop increment via different nanoparticle migration regimes (NBT). The figures show that the heat transfer enhancement has a peak around NBT≈1 for different parameters, called hereinafter as critical NBT. The existence of critical NBT signifies different modes of heat transfer rate as well as the importance of nanoparticle migration on enhancing heat transfer rate. To explain, the nanoparticle migration has two opposing effects on the heat transfer rate of nanofluids: a) the thermal conductivity reduction at the wall due to nanoparticle depletion and b) momentum augmentation at the wall due to viscosity decrement there. In fact, the thermal conductivity and viscosity of nanofluids reduce as the nanoparticles migrate toward the central region. The thermal conductivity decrement has a negative impact on the heat removal ability of nanofluids while the viscosity reduction at the wall results in a momentum augmentation at the wall which has a positive impact on the heat transfer rate. When the value of NBT is higher than the critical one, the heat transfer enhancement slightly decreases and asymptotically reaches to a constant value. This is due to the fact that for the higher values of NBT, the nanoparticles uniformly distribute in the tube and more changes in the value of NBT would not have any 17
influence on the distribution of nanoparticles and the thermal conductivity. For the value of NBT lower than the critical NBT, the nanoparticle depletion at the wall leads to a significant reduction in the thermal conductivity at the wall, thereby reducing the heat transfer coefficient. In addition, the figures reveal that the pressure drop increment significantly increases for the higher values of NBT. Thus, the larger nanoparticles (the smaller values of NBT) lead to a lower increase in the pressure drop. With regard to Figure 7, it can be seen that a rise in ϕB increases the heat transfer enhancement as well as the pressure drop ratio, which is due to the thermal conductivity and viscosity increments. From Figure 8, it can be seen that increasing λ has a trivial effect on the heat transfer rate for the lower values of NBT, whereas it leads to a rise in the heat transfer rate for the higher values of NBT. Also, increasing λ decreases the pressure drop ratio at low NBT. As NBT increases, the changes in pressure drop ratio become insignificant.
4. 3 Thermal performance To characterize the performance of heat exchange equipment, the figure of merit (FoM) is used as the comparative measure of the heat transfer enhancement with respect to the penalty in pumping power. For nanofluids, FoM can be defined as
FoM
h hbf
1/3
N p N pbf
(38)
For FoM>1, the thermal performance of the system is increased while it is the opposite for FoM<1. Figures 9 and 10 show the computed figures of merit via NBT for different values of ϕB and λ respectively. The figures reveal the nanofluids have a merit for the lower values of NBT, where the thermophoresis dominates the Brownian diffusion and the nanoparticles migrate significantly. In addition, there is an optimum value for NBT around 0.4 in which the FoM reaches its peak. Since NBT is inversely proportional to the nanoparticle diameter (dp), it can be stated that
18
there is an optimum nanoparticle diameter in which the performance of the system is maximum. In addition, Figure 9 shows that increasing ϕB significantly enhances the FoM for the lower values of NBT, whereas, at the higher values of NBT it has an opposite effect. Thus, increasing the nanoparticle volume fraction is beneficial for larger nanoparticles when the thermophoresis effect dominates the Brownian motion one. Furthermore, Figure 10 indicates that the thermal performance (FoM) slightly increases as λ grows; however, this trend reduces for very large and very small values of NBT. As a result, slippage at the fluid-solid interface has a positive effect on the performance.
4. 4 Constant wall temperature versus constant wall heat flux The thermal boundary condition is one of the consequential issue affecting the behavior of nanofluids [20, 38]. Figure 11 illustrates a comparison between two common thermal conditions at the wall namely, constant heat flux (CHF ) [17] and constant wall temperature (CWT) of the present study on the anomalous heat transfer enhancement (HTCkbf/kB). Firstly, it can be observed that the constant wall heat flux leads to a greater heat transfer enhancement in nanofluids. In addition, the effect of nanoparticle migration (NBT) for the two cases (CHF and CWT) are the same. Obviously, there are optimum NBT for the heat transfer enhancement in both cases, in which the heat transfer enhancement reaches its peak. However, the optimum NBT for the case of CWT is greater (about 1) than that for the case of CHF (about 0.5). Thus, it can be concluded the optimum diameter of nanoparticles for the case of CWT should be smaller than the case of CHF.
5. Summary and Conclusions The present study is a numerical investigation of the figure of merit for the laminar fully developed flow and heat transfer of nanofluids through a circular microchannel subjected to a 19
constant wall temperature (CWT). The modified Buongiorno's model is used for the nanofluid to simulate 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. The major outcomes of the present study can be classified as follows: The nanofluids have a merit for the lower values of NBT, where the thermophoresis dominates the Brownian diffusion and the nanoparticles migrate significantly. In addition, there is an optimum value for NBT around 0.4 in which the FoM reaches its peak. Since NBT is inversely proportional to the nanoparticle diameter (dp), it can be stated that there is an optimum nanoparticle diameter in which the performance of the system is maximum. The heat transfer enhancement has a peak around the critical NBT≈1 for different parameters. When the value of NBT is higher than the critical one, the heat transfer enhancement slightly decreases and asymptotically reaches a constant value. For the values of NBT lower than the critical NBT, the nanoparticle depletion at the wall leads to a significant reduction in the thermal conductivity there, which reduces the heat transfer enhancement. Increasing ϕB leads to a rise in the nanoparticle concentration, thereby growing the Brownian diffusion effect. This dynamic makes the nanoparticles in the central region of the tube, which has a higher kinetic energy, move in the direction of reducing the
20
concentration gradient. Thus, the nanoparticle volume fraction becomes more uniform in the central region, as ϕB increases. Increasing the nanoparticle volume fraction is beneficial for larger nanoparticles when the thermophoresis effect dominates the Brownian motion one. Furthermore, slippage at the fluid-solid interface has a positive effect on the performance. The constant heat flux (CHF) at the wall leads to a greater heat transfer enhancement in nanofluids than that of the constant wall temperature (CWT). Furthermore, there is an optimum NBT for the heat transfer enhancement in both cases (CHF and CWT), in which the heat transfer enhancement reaches its peak. However, the optimum NBT for the case of CWT is greater (about 1) than that for the case of CHF (about 0.5). Thus, it can be concluded the optimum diameter of nanoparticles for the case of CWT should be smaller than that for the case of CHF.
Acknowledgement The authors wish to acknowledge the anonymous reviewers for their careful reading of our manuscript and their constructive comments which helped us to improve the manuscript.
References [1] 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, Applied Physics Letters, 78 (2001) 718-720. [2] H.C. Brinkman, The Viscosity of Concentrated Suspensions and Solutions, The Journal of Chemical Physics, 20 (1952) 571-571. [3] G.K. Batchelor, The effect of Brownian motion on the bulk stress in a suspension of spherical particles, Journal of Fluid Mechanics, 83 (1977) 97-117.
21
[4] B.C. Pak, Y.I. Cho, HYDRODYNAMIC AND HEAT TRANSFER STUDY OF DISPERSED FLUIDS WITH SUBMICRON METALLIC OXIDE PARTICLES, Experimental Heat Transfer, 11 (1998) 151-170. [5] J. Buongiorno, Convective Transport in Nanofluids, Journal of Heat Transfer, 128 (2006) 240-250. [6] M.A. Sheremet, T. Groşan, I. Pop, Steady-state free convection in right-angle porous trapezoidal cavity filled by a nanofluid: Buongiorno’s mathematical model, European Journal of Mechanics B/Fluids, 53 (2015) 241-250. [7] D. Yadav, R. Bhargava, G.S. Agrawal, Numerical solution of a thermal instability problem in a rotating nanofluid layer, International Journal of Heat and Mass Transfer, 63 (2013) 313-322. [8] M. Sheikholeslami, R. Ellahi, Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid, International Journal of Heat and Mass Transfer, 89 (2015) 799-808. [9] F. Garoosi, L. Jahanshaloo, M.M. Rashidi, A. Badakhsh, M.E. Ali, Numerical simulation of natural convection of the nanofluid in heat exchangers using a Buongiorno model, Applied Mathematics and Computation, 254 (2015) 183-203. [10] D.A. Nield, A.V. Kuznetsov, The onset of double-diffusive convection in a nanofluid layer, International Journal of Heat and Fluid Flow, 32 (2011) 771-776. [11] S. Soleimani, M. Sheikholeslami, D.D. Ganji, M. Gorji-Bandpay, Natural convection heat transfer in a nanofluid filled semi-annulus enclosure, International Communications in Heat and Mass Transfer, 39 (2012) 565-574. [12] A. Sabaghan, M. Edalatpour, M.C. Moghadam, E. Roohi, H. Niazmand, Nanofluid flow and heat transfer in a microchannel with longitudinal vortex generators: Two-phase numerical simulation, Applied Thermal Engineering, 100 (2016) 179-189. [13] N.S. Akbar, S.U. Rahman, R. Ellahi, S. Nadeem, Nano fluid flow in tapering stenosed arteries with permeable walls, International Journal of Thermal Sciences, 85 (2014) 54-61. [14] S.A. Moshizi, Forced convection heat and mass transfer of MHD nanofluid flow inside a porous microchannel with chemical reaction on the walls, Engineering Computations, 32 (2015) 2419-2442. [15] A. Malvandi, A. Ghasemi, D.D. Ganji, I. Pop, Effects of nanoparticles migration on heat transfer enhancement at film condensation of nanofluids over a vertical cylinder, Advanced Powder Technology, 27 (2016) 1941-1948. [16] B.H. Salman, H.A. Mohammed, K.M. Munisamy, A.S. Kherbeet, Characteristics of heat transfer and fluid flow in microtube and microchannel using conventional fluids and nanofluids: A review, Renewable and Sustainable Energy Reviews, 28 (2013) 848-880. [17] C. Yang, W. Li, Y. Sano, M. Mochizuki, A. Nakayama, On the Anomalous Convective Heat Transfer Enhancement in Nanofluids: A Theoretical Answer to the Nanofluids Controversy, Journal of Heat Transfer, 135 (2013) 054504-054504. [18] C. Yang, W. Li, A. Nakayama, Convective heat transfer of nanofluids in a concentric annulus, International Journal of Thermal Sciences, 71 (2013) 249-257.
22
[19] A. Malvandi, D. Ganji, Magnetic field effect on nanoparticles migration and heat transfer of water/alumina nanofluid in a channel, Journal of Magnetism and Magnetic Materials, 362 (2014) 172179. [20] A. Malvandi, D.D. Ganji, Effects of nanoparticle migration on hydromagnetic mixed convection of alumina/water nanofluid in vertical channels with asymmetric heating, Physica E: Low-dimensional Systems and Nanostructures, 66 (2015) 181-196. [21] F. Hedayati, G. Domairry, Effects of nanoparticle migration and asymmetric heating on mixed convection of TiO2–H2O nanofluid inside a vertical microchannel, Powder Technology, 272 (2015) 250259. [22] A. Malvandi, D.D. Ganji, I. Pop, Laminar filmwise condensation of nanofluids over a vertical plate considering nanoparticles migration, Applied Thermal Engineering, 100 (2016) 979-986. [23] C. Yang, Q. Wang, A. Nakayama, T. Qiu, Effect of temperature jump on forced convective transport of nanofluids in the continuum flow and slip flow regimes, Chemical Engineering Science, 137 (2015) 730-739. [24] A. Malvandi, S. Moshizi, D. Ganji, Two-component heterogeneous mixed convection of alumina/water nanofluid in microchannels with heat source/sink, Advanced Powder Technology, 27 (2016) 245-254. [25] M. Bahiraei, S.M. Hosseinalipour, Particle migration in nanofluids considering thermophoresis and its effect on convective heat transfer, Thermochimica Acta, 574 (2013) 47-54. [26] A. Malvandi, A. Ghasemi, D.D. Ganji, Thermal performance analysis of hydromagnetic Al2O3water nanofluid flows inside a concentric microannulus considering nanoparticle migration and asymmetric heating, International Journal of Thermal Sciences, 109 (2016) 10-22. [27] S.A. Moshizi, I. Pop, Conjugated Effect of Joule Heating and Magnetohydrodynamic on Laminar Convective Heat Transfer of Nanofluids Inside a Concentric Annulus in the Presence of Slip Condition, International Journal of Thermophysics, 37 (2016) 1-22. [28] S.A. Moshizi, M. Zamani, S.J. Hosseini, A. Malvandi, Mixed convection of magnetohydrodynamic nanofluids inside microtubes at constant wall temperature, Journal of Magnetism and Magnetic Materials, 430 (2017) 36-46. [29] S.J. Hosseini, A. Malvandi, S.A. Moshizi, M. Zamani, Fully developed mixed convection of nanofluids in microtubes at constant wall temperature: Anomalous heat transfer rate and thermal performance, Advanced Powder Technology. [30] D.Y. Tzou, Instability of Nanofluids in Natural Convection, Journal of Heat Transfer, 130 (2008) 072401-072401. [31] M.M. MacDevette, T.G. Myers, B. Wetton, Boundary layer analysis and heat transfer of a nanofluid, Microfluidics and Nanofluidics, 17 (2014) 401-412. [32] W. Zhang, W. Li, A. Nakayama, An analytical consideration of steady-state forced convection within a nanofluid-saturated metal foam, Journal of Fluid Mechanics, 769 (2015) 590-620. [33] D. Song, D. Jing, B. Luo, J. Geng, Y. Ren, Modeling of anisotropic flow and thermodynamic properties of magnetic nanofluids induced by external magnetic field with varied imposing directions, Journal of Applied Physics, 118 (2015) 045101. 23
[34] A. Malvandi, S.A. Moshizi, D.D. Ganji, Nanoparticle transport effect on magnetohydrodynamic mixed convection of electrically conductive nanofluids in micro-annuli with temperature-dependent thermophysical properties, Physica E: Low-dimensional Systems and Nanostructures, 88 (2017) 35-49. [35] S.A. Moshizi, A. Malvandi, Magnetic field effects on nanoparticle migration at mixed convection of MHD nanofluids flow in microchannels with temperature-dependent thermophysical properties, Journal of the Taiwan Institute of Chemical Engineers, 66 (2016) 269-282. [36] W. Kays, M. Crawford, B. Weigand, Convective Heat & Mass Transfer W/ Engineering Subscription Card, McGraw-Hill Companies,Incorporated, 2005. [37] A. Malvandi, D.D. Ganji, Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field, International Journal of Thermal Sciences, 84 (2014) 196-206. [38] A. Malvandi, D.D. Ganji, Effects of nanoparticle migration and asymmetric heating on magnetohydrodynamic forced convection of alumina/water nanofluid in microchannels, European Journal of Mechanics - B/Fluids, 52 (2015) 169-184.
24
*Tables
Table 1. Grid independence test for different values of d when B 0.02, 0.1 and N BT 1 d
h hbf
N p N pbf
102
1.17610
1.43151
5 103
1.17611
1.43154
103
1.17612
1.43156
Fig. 1. The geometry of physical model and coordinate system
25
Take Start
B , , N BT Guess the value of
w , ( )c p ( )u
guess
Solve
u* A u * B 0 u u * * * T AT T BT T CT 0 * A T 0 with B.C.
du * * 0 : u , T * 1 , W d * * 1 : du 0 , T 0 d Find
B cal . ( )c p ( )u
End
u * u*
1
cal .
2 ( )c p ( )u(1 )d 0
Yes if
(B )cal . B
( )c p ( )u
cal .
( )c p ( )u 26
No guess
Fig. 2. Algorithm of the numerical method
27
Fig. 3. Comparison of velocity (a) and temperature profile (b) for regular fluid with the results of Kays et al. [36]
28
29
Fig. 4. The effects of N BT on (a) nanoparticle distribution ( / B ), (b) velocity ( u* uB* ), and (c) temperature ( T * ) profiles when B 0.02 and 0.1 .
30
31
Fig. 5. The effects of B on (a) nanoparticle distribution ( / B ), (b) velocity ( u* uB* ), and (c) temperature ( T * ) profiles when 0.1 and N BT 1 .
32
33
Fig. 6. The effects of on (a) nanoparticle distribution ( / B ), (b) velocity ( u* uB* ), and (c) temperature ( T * ) profiles when B 0.02 and N BT 1 .
34
Fig. 7. The effects of B on (a) the heat transfer coefficient ratio 35
and (b) the pressure drop ratio when 0.1 .
36
Fig. 8. The effects of λ on (a) the heat transfer coefficient ratio and (b) the pressure drop ratio when B 0.02 .
Fig. 9. The effects of B on the figure of merit (FoM) when 0.1 .
37
Fig. 10. The effects of λ on the figure of merit (FoM) when B 0.02 .
38
Fig. 11. Anomalous heat transfer rate for both cases of constant heat flux (CHF) [17] at the wall and constant wall temperature (CWT) when 0 .
* Highlights
Thermally fully developed flow of nanofluid in circular microchannels at constant wall
temperature
Effect of nanoparticle migration on fluid flow and heat transfer characteristics
Thermal performance gets its peak around NBT=0.4
Anomalous heat transfer rate occurs for NBT=1
Optimum value of NBT for the case of CWT is greater than that of CHF
39