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Experimental and numerical study of natural convection in a square enclosure filled with nanofluid
International Journal of Heat and Mass Transfer 78 (2014) 380–392
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Experimental and numerical study of natural convection in a square enclosure filled with nanofluid Yanwei Hu a, Yurong He a,⇑, Cong Qi a, Baocheng Jiang a, H. Inaki Schlaberg b a b
Harbin Institute of Technology, Harbin 150001, China North China Electric Power University, Beijing 102206, China
a r t i c l e
i n f o
Article history: Received 1 January 2014 Received in revised form 21 June 2014 Accepted 1 July 2014
Keywords: Two phase Lattice Boltzmann model Interaction forces Nanofluid Natural convection
a b s t r a c t The coefficient of thermal conductivity and viscosity of Al2O3–water nanofluid is measured, and its heat transfer is experimentally investigated in a square enclosure. In addition, a 2D two-phase Lattice Boltzmann model considering interaction forces (gravity and buoyancy force, drag force, interaction potential force and Brownian force) between nanoparticles and base fluid is developed for natural convection of nanofluid, and is applied to simulate the flow and heat transfer of Al2O3–water nanofluid in the square enclosure by coupling the density distribution (D2Q9) and the temperature distribution with 4-speeds. In this paper, the effects of different nanoparticle volume fractions (u = 0.25%, u = 0.5%, u = 0.77%) and different Rayleigh numbers (Ra = 30,855,746 and Ra = 63,943,592 for u = 0.25%, Ra = 38,801,494 and Ra = 67,175,834 for u = 0.5% and Ra = 55,888,498 and Ra = 70,513,049 for u = 0.77%) on heat transfer in the transition region are experimentally and numerically discussed. The numerical results have a good agreement with the experimental results. It is found that the heat transfer of nanofluid is more sensitive to the thermal conductivity than viscosity at low nanoparticle fractions and it is more sensitive to the viscosity than the thermal conductivity at high nanoparticle fractions. In addition, the forces between water and nanoparticles are analyzed, and the nanoparticle volume fraction distribution is investigated. It is found that the temperature difference driving force makes the greatest contribution to the nanoparticle volume fraction distribution, and nanoparticle volume fraction distribution is opposite to that of the water phase density distribution. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection is applied in more and more fields, for example, in heat exchangers, cooling of electronics, crystal growth and so on. Due to the fact that nanofluid has a higher thermal conductivity compared to the base fluid such as pure water or oil, thus in order to enhance the heat transfer of natural convection, nanofluid is used as the medium instead of just the base fluid. Gradually, researchers began to experimentally and numerically investigate the natural convection of nanofluid. Researchers have performed extensive experiments on the natural convection of nanofluid in recent years. Ho et al. [1] experimentally studied the natural convection heat transfer of a nanofluid in vertical square enclosures of different sizes, and the effects of nanoparticle volume fractions and Rayleigh numbers are investigated. Xuan et al. [2] experimentally studied the flow and heat transfer of Cu–water nanofluid in a tube, and obtained ⇑ Corresponding author. Tel./fax: +86 451 86413233. E-mail address: [email protected] (Y. He). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.001 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
the conclusion that the nanofluid has a larger heat transfer coefficient than that of water and that the heat transfer feature of the nanofluid increases with nanoparticle volume fraction. Williams et al. [3] experimentally investigated the natural convection of alumina–water and zirconia–water nanofluids in horizontal tubes, and discussed the effects of velocity, temperature, heat flux and volume fraction. Ding et al. [4] experimentally studied the heat transfer of aqueous suspensions of multi-walled carbon nanotubes (CNT nanofluid) in a horizontal tube, and the effects of flow conditions, CNT concentration and the PH on the enhancement are discussed. Chang et al. [5] experimentally investigated the natural convection of alumina–water nanofluid in an enclosure at angles of inclination to the horizontal of 90°, 30°and 0°, and the effects of nanoparticle volume fractions, Rayleigh numbers and the angles are discussed. Usually, the natural convection of nanofluid with different volume fractions (u = 1–5%) at different Rayleigh numbers (Ra = 103–105) is investigated. However, there are few studies on natural convection of Al2O3–water nanofluid in a square enclosure with a small volume fraction at high Rayleigh numbers. In this paper, the natural convection of nanofluid with different mass
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Nomenclature a A Ba c cs cp ea far fareq 0 F ra F ra FH FD FA FS g G Gi ha b H k kB Lcc Ma mr ni Nu Pr r Ra t T ra T ra eq T T0 TH TC ur uc
radius of nanoparticle (m) Hamaker constant adjustable coefficient reference lattice velocity lattice sound velocity specific heat capacity (J/kg K) lattice velocity vector density distribution function local equilibrium density distribution function dimensionless external force in direction of lattice velocity dimensionless total interparticle interaction forces dimensionless gravity and buoyancy force dimensionless drag force dimensionless interaction potential force dimensionless buoyancy force due to temperature difference dimensionless gravitational acceleration dimensionless effective external force Gaussian random number convective heat transfer coefficient (W/(m2 K)) dimensionless characteristic length of the square cavity thermal conductivity coefficient (W m1 K1) Boltzmann constant center-to-center distance between particles (m) Mach number mass of a single nanoparticle (kg) number of the particles within the adjacent lattice i Nusselt number Prandtl number position vector Rayleigh number time (s) temperature distribution function local equilibrium temperature distribution function dimensionless temperature dimensionless average temperature (T0 = (TH + TC)/2) dimensionless hot temperature dimensionless cold temperature dimensionless macro-velocity dimensionless characteristic velocity of natural convection
fractions (wt% = 1%, wt% = 2% and wt% = 3%, which are equivalent to the volume fractions: u = 0.25%, u = 0.5%, u = 0.77%) in a square enclosure at different Rayleigh numbers (Ra = 30,855,746 and Ra = 63,943,592) is investigated. Computational fluid dynamics (CFD) is becoming more accessible to graduate engineers for research and development in industries [6]. With the development of nanotechnology, nanofluid has been widely used in the enhancement of heat transfer and different models are applied to simulate this kind of problems [7–11]. In order to investigate the mechanisms and the microscopic details of natural convection, several research groups began to use various numerical methods to simulate the natural convection characteristics of nanofluids [12–16]. Among these methods, the Lattice Boltzmann method is a new way to investigate natural convection of nanofluid. The method has many merits, for example, the algorithm is simple, the boundary conditions are easily dealt with, and the transform between macroscopical equations and microscopic equations is easily achieved. Hence, the Lattice Boltzmann method is widely applied in the study of natural convection [17–19].
VA V wa x, y
dimensionless interaction potential volume of a single lattice (m3) weight coefficient dimensionless coordinates
Greek symbols br thermal expansion coefficient (K1) qr density (kg/m3) m kinematic viscosity (m2 s1) g dynamic viscosity (Pa s) v thermal diffusion coefficient (m2 s1) c surface tension (N/m) u nanoparticle volume fraction dx lattice step dt time step t r components (r = 1, 2, water and nanoparticles) sf dimensionless collision-relaxation time for the flow field sT dimensionless collision-relaxation time for the temperature field DT dimensionless temperature difference (DT = TH TC) Dq0 dimensionless mass density difference between nanoparticles and base fluid Du dimensionless velocity difference between nanoparticles and base fluid Uab dimensionless energy exchange between nanoparticles and base fluid Error1 maximal relative error of velocities between two adjacent time layers Error2 maximal relative error of temperatures between two adjacent time layers Subscripts
a avg C nf H w p
lattice velocity direction average cold nanofluid hot base fluid nanoparticle
Barrios et al. [20] proposed a Lattice Boltzmann model and numerically investigated the natural convection in a square enclosure with a partially heated left wall. Peng et al. [21] proposed a simple Lattice Boltzmann model without gradient term, which is easily applied, on the assumption that there is no thermal diffusion. He et al. [22] developed a new Lattice Boltzmann model, which introduced an energy density distribution function to simulate the temperature field, and the simulation result has good agreement with the benchmark solution. Nemati et al. [23] simulated the natural convection of lid driven flow in a square filled with nanofluid, and the effects of nanoparticle fraction and Reynolds number on heat transfer were discussed. Kefayati et al. [24,25] simulated natural convection in an open cavity with magnetic field, a cavity with sinusoidal temperature distribution, and also turbulent natural convection. Sheikholeslami et al. [26,27] took the simulations of natural convection considering different kind of nanoparticles, the effects of magnetic field, and various figures of enclosures. Dixit et al. [28] simulated the natural convection in a square cavity at high Rayleigh number by a thermal Boltzmann method based on the BGK model, which used the
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double population approach to respectively simulate the hydrodynamic and thermal fields. Bararnia et al. [29] simulated natural convection in a partitioned square cavity and different parameters such as Rayleigh number, the length and the orientation of the partition were analyzed. Peng et al. [30] proposed a 3D incompressible thermal Lattice Boltzmann model with two different velocity models of D3Q15 and D3Q19, and investigated the natural convection of air in a cubical enclosure by the model. However, all above Lattice Boltzmann models are for a single phase. All the single phase models assumed that there are no slip velocities between nanoparticles and fluid molecules and assumed that the nanoparticle concentration is uniform [31]. It is accurate enough to simulate pure fluid and nanofluid at a low concentration for the macroscopical heat transfer characteristics. But when taking the interaction forces between the base fluid and the nanoparticles into account, the single phase model will be powerless. Therefore, researchers proposed two phase Lattice Boltzmann models to investigate the characteristics of two phase flow and heat transfer. Xuan et al. [32] proposed a two phase Lattice Boltzmann model to investigate sudden-start Couette flow and convection in parallel plate channels without researching the effect of forces on volume fraction distribution of nanoparticles. In addition, Zhou et al. [33] simulated the sudden-start Couette flow of Cu–water nanofluid in a microchannel and laminar flow of the Cu–water nanofluid in a channel within parallel plates with two phase Lattice Boltzmann models. However, there are few two phase Lattice Boltzmann models on natural convection of nanofluid in an enclosure. In this paper, a two phase Lattice Boltzmann model for natural convection of nanofluid in an enclosure is developed by considering the external and internal forces acting on nanoparticles and base fluid. It is applied to the simulation of natural convection heat transfer of Al2O3–water nanofluid in an experimental enclosure using the same conditions as the actual experiment (the same nanoparticle fractions and Rayleigh numbers).
The flow and temperature of nanofluid in the enclosure will reach a balance when the temperatures of the two walls do not change any more. The Nusselt number of the nanofluid can be calculated by the temperatures of the two walls and the heating power in the balanced state. The heating power is calculated as follows:
Q ¼ UI
ð1Þ
The temperature of the back of the left wall (copperplate) is calculated as follows:
Tb ¼
T1 þ T2 þ T3 þ T4 þ T5 5
ð2Þ
The temperature of the left wall in contact with the water is calculated as follows:
Qd Akw
TH ¼ Tb
ð3Þ
where d is the thickness of the copperplate, A is the area of the copperplate, and kw is the thermal conductivity of copperplate. The temperature of the outside right wall is calculated as follows:
To ¼
T 6 þ T 7 þ T 8 þ T 9 þ T 10 5
ð4Þ
Because the temperature of water in the cooling cavity is constant, only two copperplates should be considered. Then the temperature of the inside right wall contacting water is calculated as follows:
TL ¼ To
2Qd Akw
ð5Þ
The heat transfer coefficient is calculated as follows:
h¼
Q AðT H T L Þ
ð6Þ
The Nusselt number is calculated as follows: 2. Experimental study A two-step method was used to formulate the Al2O3–water nanofluid and the Al2O3 nanoparticles with an average diameter about 30 nm were procured from Tansail Company. The thermal conductivity coefficients of nanofluid were measured based on the transient hot-wire method (TC3000L Series, Xiatech Electronic Technology). The thermal conductivity coefficient meter has a measurement range from 0.001 to 5.0 W/(m K) and an accuracy of ±2%. The rheological properties of nanofluids are measured in present work by a super rheometer (Kinexus Pro, Malvern Instruments). Fig. 1(a) and (b) show the experimental setup for natural convection heat transfer and the structure of the square enclosure. There are five main parts which include heating system (silica gel heater and DC power), cooling system (constant temperature water bath), experimental system (enclosure), measuring system (data acquisition instrument, computer, thermocouple, and heat flow meter) and insulating system (adiabatic nano-board). The fluid in the enclosure is Al2O3–water nanofluid. The size of the enclosure is: 180 mm (length) 80 mm (width) 80 mm (height). The left wall is heated by a silica gel heater, and the right wall is cooled by a constant temperature water bath. There are ten T-type thermocouples located in the left and right walls to measure the wall temperatures. The temperature values of the two walls can be read by the data acquisition instrument (Agilent 34972A). The heating power of the silica gel heater can be obtained from the voltage U and electric current I of the DC power supply.
Nu ¼
hH k
ð7Þ
where H is the width of the enclosure and k is the conductive coefficient of the nanofluid. Uncertainty of experimental results was determined by measurement deviation of the parameters, including heating power, thermal conductivity, temperature, and the area of the heating wall. Taking the logarithm of Eq. (1)
lnQ ¼ ln U þ ln I
ð8Þ
Then the error propagation formula of heating power Q is
@ ln Q DQ @ ln Q DU DI ¼ D U þ @I DI ¼ U þ I Q @U
ð9Þ
Taking the same step, we got the error propagation formula of Tb is
@T b DT 1 þ @T b DT 2 þ @T b DT 3 þ @T b DT 4 þ @T b DT 5 DT b ¼ @T 1 @T 2 @T 3 @T 4 @T 5 1 ¼ ðDT 1 þ DT 2 þ DT 3 þ DT 4 þ DT 5 Þ ð10Þ 5 And the error propagation formula of Tn is
@ ln T n DT n @ ln T n Dd þ @ ln T n DA þ @ ln T n Dkw ¼ D Q þ @d @A @kw Tn @Q DQ Dd DA Dkw ¼ þ þ þ Q d A kw
ð11Þ
Y. Hu et al. / International Journal of Heat and Mass Transfer 78 (2014) 380–392
383
(a)
(b) Fig. 1. Experimental apparatus (a) experimental installation, (b) structure of the square enclosure.
The error propagation formula of area is
@ ln A DA @ ln A DL DW ¼ D L þ @W DW ¼ L þ W A @L
ð12Þ
The error propagation formula of TH is
@T H DT b þ @T H DT n ¼ DT b þ DT n DT H ¼ @T b @T n
ð13Þ 3. Numerical study
The error propagation formula of To is
@T o DT 6 þ @T o DT 7 þ @T o DT 8 þ @T o DT 9 þ @T o DT 10 DT o ¼ @T @T @T @T @T 6 7 8 9 10 1 ¼ ðDT 6 þ DT 7 þ DT 8 þ DT 9 þ DT 10 Þ ð14Þ 5
ð15Þ
The error propagation formula of h is
@ ln h Dh @ ln h DA þ @ ln h DðT H T L Þ ¼ D Q þ @A @ðT T Þ h @Q H L DQ DA DðT H T L Þ ¼ þ þ Q A ðT H T L Þ
Continuous equation
ð16Þ
ð19Þ
where p is the pressure, v is the viscosity, g is the local gravity, b is the thermal expansion coefficient, T0 is the reference temperature and q0 is the fluid density at the temperature of T0. Energy equation
@T þ r ðuTÞ ¼ r ðvrTÞ @t ð17Þ
ð18Þ
where u is the velocity vector. Momentum equation
@u rp þ r ðuuÞ ¼ þ mr2 u gbðT T 0 Þ @t q0
And the error propagation formula of Nu is
@lnNu DNu @lnNu DH þ @lnNuDknf ¼ Dh þ Nu @h @H @knf Dh DH Dknf ¼ þ þ h H knf
3.1. Governing equations
ru¼0
The error propagation formula of TL is
@T L DT o þ @T L DT n ¼ DT o þ 2DT n DT L ¼ @T @T o n
In present work, the measuring error of Q is ±0.65 W, the measuring error of Tb is ±0.001 °C, the measuring error of To is ±0.001 °C, the measuring errors of the length and the width are ±0.5 mm, and the relative error of thermal conductivity is ±2%. So the error of heat transfer coefficient of natural convection is 5.65% and the error of Nusselt number is 8.275%.
v is the thermal diffusion coefficient.
ð20Þ
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3.2. Lattice Boltzmann method
In the case of no internal forces and external forces, the macroscopic temperature, density and velocity are calculated as follows:
A nanofluid is a kind of two phase fluid which includes a liquid phase and a solid phase. The macroscopic density and velocity fields are still simulated using the density distribution function:
far ðr
þ ea dt ; t
0
F ra ¼ G
þ dt Þ far ðr; tÞ
1 ¼ r far ðr; tÞ fareq ðr; tÞ
sf 2srf 1 F ra dt ea 0 þ þ dt F ra Ba c2 2srf
ðea ur Þ req fa p
3
a ¼ 1; . . . ; 4
ð21Þ
ð23Þ
8 ð0; 0Þ a¼0 > < p p a ¼ 1; 2; 3; 4 ea ¼ c cos ða 1Þ 2 ; sin ða 1Þ 2 > : pffiffiffi a ¼ 5; 6; 7; 8 2c cos ð2a 1Þ p4 ; sin ð2a 1Þ p4
ð24Þ
" ¼ q wa
wa ¼
84 > <9
1 > 91 : 36
ea ur ðea ur Þ2 ur2 1þ þ 2 c2s 2c4s 2cs
# ð25Þ
a¼0 a ¼ 1; . . . ; 4 a ¼ 5; . . . ; 8
ð26Þ
2
where c2s ¼ c3 is the lattice sound velocity, wa is the weight coefficient. The macroscopic temperature field is simulated using the temperature distribution function:
1 T ra ðr þ ea dt ; t þ dt Þ T ra ðr; tÞ ¼ r T ra ðr; tÞ T ra eq ðr; tÞ s
ð27Þ
T
where sT is the dimensionless collision-relaxation time for the temperature field. The temperature equilibrium distribution function is chosen as follows:
" T ra eq ¼ wa T r 1 þ 3
8 1 X ur ¼ r far ea q
ð31Þ
a¼0
Since the nanofluid is a kind of two phase fluid, there are some forces between nanoparticles and the base fluid. Considering the internal and external forces, the macroscopic velocities for the nanoparticles and base fluid are changed to:
F p Dtea 2qr DtF W ¼ uw þ 2Lx Ly qw
upnew ¼ up þ
ð32Þ
uwnew
ð33Þ
where Fp is the total force acting on the nanoparticles, FW is the total force acting on base fluid, and LxLy is the total number of lattices. When the internal and external forces are considered, energy is exchanged between nanoparticles and base fluid, and the macroscopic temperature for nanoparticles and base fluid is given as:
T rnew ¼ T r þ dt sT
dT ¼ T r þ dt sT Uab dt
2
ea ur ðea ur Þ ur2 þ 4:5 1:5 2 c2 2c4 2c
#
ð34Þ
where Uab is the energy exchange between nanoparticles and base fluid, Uab ¼
hab ½T b ðx;tdt ÞT a ðx;tdt Þ , qa cpa aa
and hab is the convective heat trans-
fer coefficient of the nanofluid. The corresponding kinematic viscosity and thermal diffusion coefficients are respectively defined as follows:
1 1 dt 3 2
1 1 vr ¼ c2 srT dt 3 2
ð35Þ ð36Þ
For natural convection the important dimensionless parameters are the Prandtl number Pr and the Rayleigh number Ra defined by:
mr
The density equilibrium distribution function is chosen as follows: r
ð30Þ
mr ¼ c2 srf
For the two-dimensional nine-velocity LB model (D2Q9) [34] considered herein, the discrete velocity set for each component a is:
fareq
8 X far
a¼0
ð22Þ
12 a ¼ 5; . . . ; 8
ð29Þ
a¼0
qr ¼
where srf is the dimensionless collision-relaxation time for the flow field, ea is the lattice velocity vector, the subscript a represents the lattice velocity direction, r is the component (r = 1, 2, respectively represent the water and nanoparticle components), far ðr; tÞ is the distribution function of the nanofluid with velocity ea (along the direction a) at lattice position r and time t, fareq ðr; tÞ is the local equilibrium distribution function, dt is the time step, F ra is the total interparticle interaction force, Ba is one of the adjustable coefficients, c = dx/dt is the reference lattice velocity, dx is the lattice step, the order numbers a = 1, . . ., 4 and a = 5, . . ., 8, respectively represent the rectangular directions and the diagonal directions of a lattice, F ra 0 is the total interparticle interaction force, F ra is the external force term in the direction of lattice velocity without interparticle interaction, G = b(Tnf T0)g is the effective external force, where g is the gravity acceleration, b is the thermal expansion coefficient, Tnf is the temperature of the nanofluid, T0 is the mean value of the high and low temperatures of the walls. The adjustable coefficient Ba is given as:
Ba ¼
8 X T ra
Tr ¼
Pr ¼ r v Ra ¼
ð37Þ
gbr DTH3 Pr
mr2
ð38Þ
where DT is the temperature difference between the high temperature wall and the low temperature wall, H is the characteristic length of the square cavity. Another dimensionless parameter Mach number (Ma) is defined by:
Ma ¼
urc cs
ð39Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where urc ¼ gbr DTH is the characteristic velocity of natural convection. For natural convection the Boussinesq approximation is applied. To ensure that the code works in a near incompressible regime, the characteristic velocity must be small compared to the fluid speed of sound. In the present study, the characteristic velocity is selected as 0.1 times the sound of speed. The dimensionless collision-relaxation time sf and sT are respectively given as follows:
srf ¼ 0:5 þ ð28Þ
srT ¼ 0:5 þ
pffiffiffiffiffiffiffiffi MaH 3Pr pffiffiffiffiffiffi c2 dt Ra r 3m Prc2 dt
ð40Þ ð41Þ
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Y. Hu et al. / International Journal of Heat and Mass Transfer 78 (2014) 380–392
The two phase Lattice Boltzmann model is applied to simulate the natural convection heat transfer in a square cavity which is shown in Fig. 2. The square cavity is filled with Al2O3–water nanofluid. The thermo-physical properties of water and Al2O3 are given in Table 1. The height and the width of the enclosure are both H. The left wall is kept at a high constant temperature (TH), and the right cold wall is kept at a low constant temperature (TC). The boundary conditions of the other walls (top wall and bottom wall) are all adiabatic. The initial conditions of the four walls are given as follows:
x ¼ 0 u ¼ 0; T ¼ 1;
x ¼ 1 u ¼ 0; T ¼ 0
y ¼ 0 u ¼ 0; @T=@y ¼ 0; y ¼ 1 u ¼ 0; @T=@y ¼ 0
ð42Þ
In the simulation, a nonequilibrium extrapolation scheme is adopted to deal with the boundary, and the criteria for the flow field convergence and temperature field are given as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 2 h P r ux ði;j;t þ dt Þ urx ði;j;tÞ þ ury ði;j;t þ dt Þ ury ði;j;tÞ i;j rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < e1 Error 1 ¼ i P h r 2 2 r i;j ux ði;j;t þ dt Þ þ uy ði;j;t þ dt Þ ð43Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P r 2 r i;j ½T ði;j;t þ dt Þ T ði;j;tÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qP < e2 Error2 ¼ 2 r i;j T ði;j;t þ dt Þ
ð44Þ
where e is a small number, for example, for Ra = 1 103, e1 = 106, e2 = 106. The physical parameters used in the simulation are: time step: t = 1 105 s, lattice step: d = 1 105 m, radius of nanoparticle: a = 20 nm. However, in the simulation, all the units are lattice units, and the international units have to be converted to lattice units. The units with ‘‘0 ’’ represent lattice units, and the units without ‘‘0 ’’ represent international units, the relationships between international units and lattice units are as follows:
D ¼ D0 L;
L2 m ¼ m0 ; T
q ¼ q0
G
ð45Þ
3
L
T¼
3
m0 D 2 q D ; G¼ 0 q D0 m D0
ð46Þ
adiabatic surface
3
q (kg/m ) cp (J/kg k) v (m2/s) k (W m1 K1)
Fluid phase (H2O)
Nanoparticles (Al2O3)
997.1 4179 0.001004 0.613
3970 765 / 25
The lattice units – time, length, velocity, acceleration, mass, and force are respectively given as:
8 < t 0 ¼ t ; l0 ¼ l ; u0 ¼ u T ¼ u m0 D0 ; a0 ¼ a T 2 ¼ a m0 2 D0 3 T L L m D L m D : m0 ¼ m ¼ m q0 D0 3 ; F 0 ¼ F T 2 ¼ F q0 m0 2 G q D GL q m
g TH
Nu ¼
hH knf
ð48Þ
The heat transfer coefficient is computed from:
h¼
qw TH TL
ð49Þ
The thermal conductivity of the nanofluid is defined by:
qw @T=@x
knf ¼
ð50Þ
Substituting Eqs. (49) and (41) into Eq. (48), the local Nusselt number along the left wall can be written as:
@T H Nu ¼ @x T H T L
ð51Þ
The average Nusselt number is determined from:
Nuavg ¼
Z
1
NuðyÞdy
ð52Þ
3.3. Interaction forces between base fluid and nanoparticles As noted before a nanofluid is actually a kind of two phase fluid. There are interaction forces between liquid and nanoparticles, which affect the behavior of the nanofluid. The external forces include gravity and buoyancy force FH, and the interparticle interaction forces which here comprise drag force (Stokes force) FD, interaction potential FA, and Brownian force FB. We introduce them as follows. The gravity and buoyancy force is given as:
TC
4pa3 g Dq 0 3
ð53Þ
where a is the radius of a nanoparticle, Dq0 is the mass density difference between the suspended nanoparticle and the base fluid. The drag force (Stokes force) is given as:
F D ¼ 6plaDu
H
adiabatic surface
Y
ð47Þ
All the physical parameters in the simulation can be changed into lattice units according to Eq. (47). The Nusselt number can be expressed as:
FH ¼
H
D ; D0
Physical properties
0
where D is the nanoparticle diameter, m is the kinematic viscosity of the base fluid, and q is the density of the base fluid, L is the length dimension, T is the time dimension, and G is the mass dimension. The length dimension, time dimension and mass dimension can be obtained from Eq. (45):
L¼
Table 1 Thermo-physical properties of water and Al2O3 [13].
X Fig. 2. Schematic of the square simulation cavity.
ð54Þ
where l is the viscosity of the fluid, and Du is the velocity difference between the nanoparticle and the base fluid. The interaction potential is presented as [35]:
1 2a2 2a2 L2 4a2 VA ¼ A 2 þ 2 þ cc 2 2 6 Lcc 4a Lcc Lcc
! ð55Þ
where A is the Hamaker constant, and Lcc is the center-to-center distance between particles.
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Y. Hu et al. / International Journal of Heat and Mass Transfer 78 (2014) 380–392
The interaction potential force is shown as:
FA ¼
8 X
ni
1.030
@V A @ri
1.025
where ni is the number of the particles within the adjacent lattice i, ni = qrV/mr, mr is the mass of a single nanoparticle, and V is the volume of a single lattice. The Brownian force is calculated as [36]:
1.020
i¼1
rffiffiffiffiffi C F B ¼ Gi dt
knf/kf
ð56Þ
nðF H þ F D þ F A þ F B Þ Fp ¼ V
1.015 1.010
ð57Þ
where Gi is a Gaussian random number with zero mean and unit variance, which is obtained from a program written by us, and C = 2ckBT = 2 (6pga)kBT, c is surface tension, kB is the Boltzmann constant, T is absolute temperature, g is dynamic viscosity. The total per unit volume forces acting on nanoparticles of a single lattice is:
Maxwell model 20°C 25°C 30°C 35°C 40°C
1.005 1.000 0.000
0.002
0.004
0.006
0.008
0.010
vol Fig. 4. Thermal conductivity coefficient ratio between nanofluid and water.
ð58Þ 0.0020
where n is the number of particles in the given lattice, and V is the lattice volume. In a nanofluid, the forces acting on the base fluid are mainly drag force and Brownian force. Thus the forces acting on the base fluid in a given lattice is:
All the forces above are calculated in lattice units, which leads to the forces dimensionless.
)
ð59Þ
0.0016
μ( ⋅
F w ¼ nðF D þ F B Þ
wt%=0% wt%=1% wt%=2% wt%=3%
0.0018
0.0014 0.0012
4. Results and discussion
0.0010
4.1. Experimental result 0.0008 0
There are two contradictory factors for natural convection heat transfer of nanofluid. One advantageous factor is the high conductivity coefficient and a disadvantageous factor is the high viscosity. Here, Figs. 3 and 5 give the thermal conductivity coefficient and the viscosity. The thermal conductivities for all the samples were measured for five times, and the maximum repeatability error was less than 0.92%. Fig. 3 presents the thermal conductivity coefficient of the nanofluid measured by a thermal conductivity coefficient instrument (TC3000L). It can be seen that the thermal conductivity
10
20
30
40
50
γ (s
60
70
80
90
)
-1
(a) 0.0015 0.0014 0.0013
μ(Pa⋅s)
0.0012 0.66 wt%=0% wt%=1% wt%=2% wt%=3%
0.65
knf(W/m2⋅K)
0.64
0.0011 0.0010 0.0009
0.63
0.0008 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Mass fraction %
0.62
(b
0.61
Fig. 5. Viscosity of nanofluid at different shearing velocities (a) viscosity of nanofluid (b) error bars.
0.60 0.59 290
295
300
305
310
T(K) Fig. 3. Thermal conductivity coefficient of nanofluid.
315
coefficient of the nanofluid increases with temperature and nanoparticle mass fraction. The thermal conductive coefficient of nanofluid is higher than that of pure water. From Fig. 4, it can be seen that the thermal conductivity coefficient ratio between nanofluid and water is in good agreement with the Maxwell model.
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4.2. Numerical simulation result
25 24 23
Nuavg
22
water literature [1] wt%=0% (exp) wt%=1% (exp) wt%=2% (exp) wt%=3% (exp) wt%=1% (sim) wt%=2% (sim) wt%=3% (sim)
21 20 19 18 17 2E7
4E7
6E7
8E7
1E8 1.2E8
Ra Fig. 7. Average Nusselt number of nanofluid at different Rayleigh numbers.
1.1
critical line wt=1% (exp) wt=2% (exp) wt=3% (exp)
1.08 1.06 1.04
Nunf/Nuf
Fig. 5(a) and (b) show the viscosity of nanofluid at different shearing velocities measured by a Kinexus Pro. rheometer (made by Malvern Instruments Ltd) and the error bars. It can be seen that the viscosity of nanofluid is higher than that of water, and it increases with nanoparticle mass fraction. As a whole, changes in viscosity of nanofluid at different shearing velocities are not large, which explains that nanofluid can be approximatively seen as a Newtonian fluid in this work. In order to validate the experimental installation, Fig. 6 presents the average Nusselt number of water at different Rayleigh numbers. It can be seen from Fig. 6 that the experimental results with water are in good agreement with those in the literature [1]. The natural convection experiment was carried out for three times, and there was a maximum repeatability error about 4.6%. Fig. 7 shows the average Nusselt number of nanofluid at different Rayleigh numbers. Fig. 8 presents the average Nusselt number ratio between nanofluid and water. From Figs. 7 and 8, it can be seen that the heat transfer of nanofluid shows an enhancement compared with that of pure water at low nanoparticle mass fraction (wt% = 1%), it is almost the same at wt% = 2%, and there is a reduction of heat transfer at high nanoparticle mass fraction (wt% = 3%). For a low nanoparticle mass fraction, the high conductivity coefficient of nanoparticles plays a major role. As the mass fraction increases, the high viscosity of nanofluid begins to play the main role and weakens the heat transfer. From this, it can be concluded that the heat transfer of nanofluid is more sensitive to thermal conductivity than viscosity at a low nanoparticle fractions and it is more sensitive to the viscosity than the thermal conductivity at a high nanoparticle fraction.
1.02 1 0.98
As shown in Table 2, a grid independence test is performed in a square enclosure using successively sized grids, 128 128, 192 192, 256 256 and 320 320 at Ra = 1 105, Pr = 0.7. It can be seen from Table 2 that there is a bigger difference between the result with grid 128 128, 192 192 and the results in the literature [37], than with the results obtained using grids 256 256 and 320 320. In addition, the result with grid 256 256 and the result with grid 320 320 are very close. In order to accelerate the numerical simulation, a grid size of 256 256 is chosen as suitable and being able to guarantee a grid independent solution. In order to validate the Lattice Boltzmann model proposed in this work, the temperature distribution at the mid-sections of the enclosure at Ra = 1 105, Pr = 0.7 is compared with numerical
0.96 0.94 3E7
4E7
5E7
6E7
7E7
8E7
Ra Fig. 8. Average Nusselt number ration between nanofluid and water.
1.0
present simulation Khanafer et al [38] (numerical) Krane and Jessee [39] (experimental)
0.8
27 25
0.6
literature Ho[1] experimental pure water
T
26
0.4
24 23
Nuavg
22
0.2
21 20
0.0 0.0
19
0.4
0.6
0.8
1.0
X
18 17 2E7
0.2
Fig. 9. Temperature distribution at mid-sections of the enclosure (Ra = 105, Pr = 0.7).
4E7
6E7
8E7
1E8
Ra Fig. 6. Average Nusselt number of water at different Rayleigh numbers.
results by Khanafer et al. [38] and experimental results by Krane et al. [39] in Fig. 9. It can be seen that the results of this paper
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Table 2 Comparison of the mean Nusselt numbers with different grids (Ra = 105, Pr = 0.7). Physical properties
128 128
192 192
256 256
320 320
Literature [37]
Nuavg
4.5466
4.5251
4.5220
4.5218
4.5216
are in good agreement with the numerical [38] and experimental results [39]. The results of this paper are closer to the experimental results [39] than the numerical results [38]. Figs. 10 and 11 show the comparison of isotherms and streamlines for different mass fractions of nanoparticles and different Rayleigh numbers (Ra = 30,855,746 and Ra = 63,943,592 for wt% = 1%, Ra = 38,801,494 and Ra = 67,175,834 for wt% = 2%, and Ra = 55,888,498 and Ra = 70,513,049 for wt% = 3%). When the
Rayleigh number increases, the isotherm becomes more crooked and the centralization of the streamlines in the core of the cavity tends toward the hot wall, which indicates that convection heat transfer is stronger at a higher Rayleigh number. The forces involved are gravity–buoyancy force, drag force, interaction potential force, Brownian force between water and nanoparticles and temperature difference driving force. These forces are presented in Table 3. From Table 3, it can be seen that
Ra=30855746
Ra=63943592
(a) wt%=1%
Ra=38801494
Ra=67175834
(b) wt%=2%
Ra=55888498
Ra=70513049
(c) wt%=3% Fig. 10. Comparison of the isotherms at various Rayleigh numbers and mass fractions of nanoparticles.
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Ra=30855746
Ra=63943592
(a) wt%=1%
Ra=38801494
Ra=67175834
(b) wt%=2%
Ra=55888498
Ra=70513049
(c) wt%=3% Fig. 11. Comparison of the streamlines at various Rayleigh numbers and mass fractions of nanoparticles.
the temperature difference driving force (Fs) is the biggest one. Correspondingly, the effect of the temperature difference driving force on the nanoparticle volume fraction distribution is the greatest, and the effects of other forces on the nanoparticle volume fraction distribution can be ignored. Other forces also have effects on
the flow and heat transfer of nanofluid. Gravity–buoyancy force and driving force give rise to movement and velocity of the nanofluid in the enclosure. Drag force between nanoparticles and water molecules decreases the velocity of water, which is a disadvantage for heat transfer. Interaction potential force keeps
Table 3 Comparison of different forces of Al2O3–water nanofluid (Ra = 30855746, u = 0.01). Forces
FS
FA
FBx
FBy
FH
FDx
FDy
Minimum Maximum
5E06 5E06
3.2E19 2E20
5E13 5E13
2E14 2E13
9E19 1E19
5E16 5E15
1.4E15 1.6E15
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(a)
(b)
Fig. 12. Temperature difference driving force distribution, wt% = 1% (a) Ra = 30,855,746 (b) Ra = 63,943,592.
(a)
(b)
Fig. 13. Water phase density distribution of nanofluid, wt% = 1% (a) Ra = 30,855,746 (b) Ra = 63,943,592.
(a)
(b)
Fig. 14. Nanoparticle volume fraction distribution of nanofluid, wt% = 1% (a) Ra = 30,855,746 (b) Ra = 63,943,592.
the nanoparticles from gathering together. The irregular motion of nanoparticles due to Brownian force can disturb the laminar boundary layer, which can enhance the heat transfer of natural convection. The driving force distribution is given in Fig. 12. From Fig. 12, it can be seen that the driving force in the area near left and top walls is large and upward, and it is large and downward near right and bottom walls, which drives the nanofluid to revolve in the enclosure. In addition, it is found that equivalent driving force lines are more crooked at the high Rayleigh number than at the low Rayleigh number. Because the driving force is due to the temperature gradient, the bigger temperature gradient causes equivalent driving force lines to become more crooked. It can be seen from Fig. 11 that isotherms are more crooked at the higher Rayleigh
number, and changes in the isotherms correspond to changes of temperature gradient. The driving force and the temperature difference give rise to the large density changes of the water phase in the nanofluid. Fig. 13 shows the water phase density distribution of the nanofluid with wt% = 1% at Ra = 30,855,746 and Ra = 63,943,592. For Ra = 30,855,746, it can be seen that density of the water phase in the area near the top wall is small while in the area near the bottom wall it is large. The water near the left wall is heated and its density decreases. Then the low density water flows upward, causing a low density water region near the top wall. The water near the top wall is then cooled and the density becomes larger. Then the higher density water flows downward, causing a high density
Y. Hu et al. / International Journal of Heat and Mass Transfer 78 (2014) 380–392
water region near the bottom wall. For Ra = 63,943,592, it can be found that the density of water near the top at the high Rayleigh number (Ra = 63,943,592) is smaller than it is for the low Rayleigh number (Ra = 30,855,746). As the Rayleigh number increases, the velocity of the nanofluid becomes bigger and the convection heat transfer of the nanofluid is enhanced. Hence, more low density water rises to the top wall, and the density of water near the top becomes much lower. Since the temperature of water near the top is higher than that near the bottom wall, the Brownian movement of nanoparticles is more drastic, which leads to the density distribution of water phase not smooth. Fig. 14 presents the nanoparticle volume fraction distribution with wt% = 1% at Ra = 30,855,746 and Ra = 63,943,592. From Fig. 14, it can be seen that the nanoparticle volume fraction distribution near the top is bigger than it is near the bottom, and the nanoparticle volume fraction distribution near the top at the high Rayleigh number (Ra = 63,943,592) is bigger than it is at the low Rayleigh number (Ra = 30,855,746). The reason for this is that the temperature difference driving force has the greatest effect on the nanoparticle volume fraction distribution. As the Rayleigh number increases, the corresponding driving force becomes larger and drives more nanoparticles to the top. In addition, it can be seen from Fig. 13 that the water phase density is low near the top, and the nanoparticle fraction distribution is opposite to that of the water phase density distribution. Hence, the nanoparticle volume fraction is large near the top, and increases with a higher Rayleigh number. Average Nusselt number of the nanofluid at different Rayleigh numbers is presented in Fig. 7. It shows that at the lowest nanoparticle mass fraction (wt% = 1%), the heat transfer characteristic is enhanced. While with increasing the mass fraction, the effect of heat transfer enhancement becomes weaker. When the mass fraction of nanoparticles reaches 2%, the Nusselt number is almost same as water. When the mass fraction of nanofluid achieves 3%, the heat transfer is weakened. Comparing the results of simulation with that of experiment, it can be seen that the numerical simulation results are in good agreement with the experimental results. It proves that the two phase Lattice Boltzmann model developed in this paper is suitable to simulate natural convection of nanofluid.
5. Conclusion The flow and heat transfer of Al2O3–water nanofluid in a square enclosure are experimentally and numerically investigated. The following conclusions are obtained: A two phase Lattice Boltzmann model considering interaction forces for natural convection of nanofluid in an enclosure is developed by coupling the density distribution function and the temperature distribution function. Numerical results show a satisfactory agreement with experimental results. Compared to pure water, the heat transfer of nanofluid shows an enhancement at a low nanoparticle mass fraction (wt% = 1%), is almost the same at wt% = 2%, and shows a reduction at a high nanoparticle mass fraction (wt% = 3%), which leads to the conclusion that the heat transfer of nanofluid is more sensitive to thermal conductivity than to viscosity at a low nanoparticle fraction and is more sensitive to viscosity than thermal conductivity at a high nanoparticle fraction. Among the forces, the temperature difference driving force is the biggest one and has the largest effect on nanoparticle volume fraction distribution. The nanoparticle volume fraction is large near the top and is small near the bottom, and increases with the Rayleigh number near the top. The temperature difference driving force, Brownian force, Interaction potential force and gravity–buoyancy force have a positive
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effect on the enhancement of natural convection heat transfer, while the effect of the drag force is negative. Conflict of interest None declared. Acknowledgments This work is financially supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51121004) , the Science Creative Foundation for Distinguished Young Scholars in Harbin (Grant No. 2014RFYXJ004) and the Fundamental Research Funds for the Central Universities (Grant No. HIT.BRETIV.201315). References [1] C.J. Ho, W.K. Liu, Y.S. Chang, C.C. Lin, Natural convection heat transfer of alumina–water nanofluid in vertical square enclosures: an experimental study, Int. J. Therm. Sci. 49 (8) (2010) 1345–1353. [2] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, ASME J. Heat Transfer 125 (1) (2003) 151–155. [3] W. Williams, J. Buongiorno, L.W. Hu, Experimental investigation of turbulent convective heat transfer and pressure loss of alumina/water and zirconia/ water nanoparticle colloids (nanofluids) in horizontal tubes, ASME J. Heat Transfer 130 (4) (2008) 1–7. [4] Y. Ding, H. Alias, D. Wen, R. Williams, Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids), Int. J. Heat Mass Transfer 49 (1–2) (2006) 240–250. [5] B.H. Chang, A.F. Mills, E. Hernandez, Natural convection of microparticle suspensions in thin enclosures, Int. J. Heat Mass Transfer 51 (5–6) (2008) 1332–1341. [6] J. Tu, G.H Yeoh, C. Liu, Computational Fluid Dynamics: a Practical Approach, Butterworth, Heinemann, 2007, pp. 1–27. [7] A.H. Mahmoudi, K. Hooman, Effect of a discrete heat source location on entropy generation in mixed convective cooling of a nanofluid inside the ventilated cavity, Int. J. Exergy 13 (3) (2013) 299–319. [8] G. Imani, M. Maerefat, K. Hooman, Lattice Boltzmann simulation of conjugate heat transfer from multiple heated obstacles mounted in a walled parallel plate channel, Numer. Heat Transfer Part A: Appl. 62 (10) (2012) 798–821. [9] G.H.R. Kefayati, Natural convection of ferrofluid in a linearly heated cavity utilizing LBM, J. Mol. Liq. 191 (2014) 1–9. [10] M. Sheikholeslami, M. Gorji-Bandpy, Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field, Powder Technol. 256 (2014) 490–498. [11] M. Sheikholeslami, D.D. Ganji, M. Gorji-Bandpy, S. Soleimani, Magnetic field effect on nanofluid flow and heat transfer using KKL model, J. Taiwan Inst. Chem. Eng. 45 (3) (2014) 795–807. [12] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid filled enclosure, Eur. J. Mech. – B/Fluids 28 (5) (2009) 630–640. [13] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of Al2O3– water nanofluid on heat transfer enhancement in natural convection, Int. J. Heat Fluid Flow 30 (4) (2009) 679–690. [14] K.S. Hwang, J.H. Lee, S.P. Jang, Buoyancy-driven heat transfer of water-based Al2O3 nanofluids in a rectangular cavity, Int. J. Heat Mass Transfer 50 (19–20) (2007) 4003–4010. [15] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, P. Rana, S. Soleimani, Magnetohydrodynamic free convection of Al2O3–water nanofluid considering thermophoresis and Brownian motion effects, Comput. Fluids 94 (2014) 147–160. [16] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, Natural convection heat transfer in a cavity with sinusoidal wall filled with CuO– water nanofluid in presence of magnetic field, J. Taiwan Inst. Chem. Eng. 45 (1) (2014) 40–49. [17] G.R. Kefayati, Simulation of ferrofluid heat dissipation effect on natural convection at an inclined cavity filled with kerosene/cobalt utilizing the Lattice Boltzmann method, Numer. Heat Transfer Part A: Appl. 65 (6) (2014) 509–530. [18] M. Sheikholeslami, M. Gorji-Bandpy, S.M. Seyyedi, et al., Application of LBM in simulation of natural convection in a nanofluid filled square cavity with curve boundaries, Powder Technol. 247 (2013) 87–94. [19] G.H. Kefayati, S.F. Hosseinizadeh, M. Gorji, H. Sajjadi, Lattice Boltzmann simulation of natural convection in tall enclosures using water/SiO2 nanofluid, Int. Commun. Heat Mass Transfer 38 (6) (2011) 798–805. [20] G. Barrios, R. Rechtman, J. Rojas, R. Tovar, The Lattice Boltzmann equation for natural convection in a two-dimensional cavity with a partially heated wall, J. Fluid Mech. 522 (2005) 91–100. [21] Y. Peng, C. Shu, Y.T. Chew, Simplified thermal Lattice Boltzmann model for incompressible thermal flows, Phys. Rev. E 68 (2) (2003) 026701.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Experimental and numerical study of natural convection in a square enclosure filled with nanofluid Yanwei Hu a, Yurong He a,⇑, Cong Qi a, Baocheng Jiang a, H. Inaki Schlaberg b a b
Harbin Institute of Technology, Harbin 150001, China North China Electric Power University, Beijing 102206, China
a r t i c l e
i n f o
Article history: Received 1 January 2014 Received in revised form 21 June 2014 Accepted 1 July 2014
Keywords: Two phase Lattice Boltzmann model Interaction forces Nanofluid Natural convection
a b s t r a c t The coefficient of thermal conductivity and viscosity of Al2O3–water nanofluid is measured, and its heat transfer is experimentally investigated in a square enclosure. In addition, a 2D two-phase Lattice Boltzmann model considering interaction forces (gravity and buoyancy force, drag force, interaction potential force and Brownian force) between nanoparticles and base fluid is developed for natural convection of nanofluid, and is applied to simulate the flow and heat transfer of Al2O3–water nanofluid in the square enclosure by coupling the density distribution (D2Q9) and the temperature distribution with 4-speeds. In this paper, the effects of different nanoparticle volume fractions (u = 0.25%, u = 0.5%, u = 0.77%) and different Rayleigh numbers (Ra = 30,855,746 and Ra = 63,943,592 for u = 0.25%, Ra = 38,801,494 and Ra = 67,175,834 for u = 0.5% and Ra = 55,888,498 and Ra = 70,513,049 for u = 0.77%) on heat transfer in the transition region are experimentally and numerically discussed. The numerical results have a good agreement with the experimental results. It is found that the heat transfer of nanofluid is more sensitive to the thermal conductivity than viscosity at low nanoparticle fractions and it is more sensitive to the viscosity than the thermal conductivity at high nanoparticle fractions. In addition, the forces between water and nanoparticles are analyzed, and the nanoparticle volume fraction distribution is investigated. It is found that the temperature difference driving force makes the greatest contribution to the nanoparticle volume fraction distribution, and nanoparticle volume fraction distribution is opposite to that of the water phase density distribution. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection is applied in more and more fields, for example, in heat exchangers, cooling of electronics, crystal growth and so on. Due to the fact that nanofluid has a higher thermal conductivity compared to the base fluid such as pure water or oil, thus in order to enhance the heat transfer of natural convection, nanofluid is used as the medium instead of just the base fluid. Gradually, researchers began to experimentally and numerically investigate the natural convection of nanofluid. Researchers have performed extensive experiments on the natural convection of nanofluid in recent years. Ho et al. [1] experimentally studied the natural convection heat transfer of a nanofluid in vertical square enclosures of different sizes, and the effects of nanoparticle volume fractions and Rayleigh numbers are investigated. Xuan et al. [2] experimentally studied the flow and heat transfer of Cu–water nanofluid in a tube, and obtained ⇑ Corresponding author. Tel./fax: +86 451 86413233. E-mail address: [email protected] (Y. He). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.001 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
the conclusion that the nanofluid has a larger heat transfer coefficient than that of water and that the heat transfer feature of the nanofluid increases with nanoparticle volume fraction. Williams et al. [3] experimentally investigated the natural convection of alumina–water and zirconia–water nanofluids in horizontal tubes, and discussed the effects of velocity, temperature, heat flux and volume fraction. Ding et al. [4] experimentally studied the heat transfer of aqueous suspensions of multi-walled carbon nanotubes (CNT nanofluid) in a horizontal tube, and the effects of flow conditions, CNT concentration and the PH on the enhancement are discussed. Chang et al. [5] experimentally investigated the natural convection of alumina–water nanofluid in an enclosure at angles of inclination to the horizontal of 90°, 30°and 0°, and the effects of nanoparticle volume fractions, Rayleigh numbers and the angles are discussed. Usually, the natural convection of nanofluid with different volume fractions (u = 1–5%) at different Rayleigh numbers (Ra = 103–105) is investigated. However, there are few studies on natural convection of Al2O3–water nanofluid in a square enclosure with a small volume fraction at high Rayleigh numbers. In this paper, the natural convection of nanofluid with different mass
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381
Nomenclature a A Ba c cs cp ea far fareq 0 F ra F ra FH FD FA FS g G Gi ha b H k kB Lcc Ma mr ni Nu Pr r Ra t T ra T ra eq T T0 TH TC ur uc
radius of nanoparticle (m) Hamaker constant adjustable coefficient reference lattice velocity lattice sound velocity specific heat capacity (J/kg K) lattice velocity vector density distribution function local equilibrium density distribution function dimensionless external force in direction of lattice velocity dimensionless total interparticle interaction forces dimensionless gravity and buoyancy force dimensionless drag force dimensionless interaction potential force dimensionless buoyancy force due to temperature difference dimensionless gravitational acceleration dimensionless effective external force Gaussian random number convective heat transfer coefficient (W/(m2 K)) dimensionless characteristic length of the square cavity thermal conductivity coefficient (W m1 K1) Boltzmann constant center-to-center distance between particles (m) Mach number mass of a single nanoparticle (kg) number of the particles within the adjacent lattice i Nusselt number Prandtl number position vector Rayleigh number time (s) temperature distribution function local equilibrium temperature distribution function dimensionless temperature dimensionless average temperature (T0 = (TH + TC)/2) dimensionless hot temperature dimensionless cold temperature dimensionless macro-velocity dimensionless characteristic velocity of natural convection
fractions (wt% = 1%, wt% = 2% and wt% = 3%, which are equivalent to the volume fractions: u = 0.25%, u = 0.5%, u = 0.77%) in a square enclosure at different Rayleigh numbers (Ra = 30,855,746 and Ra = 63,943,592) is investigated. Computational fluid dynamics (CFD) is becoming more accessible to graduate engineers for research and development in industries [6]. With the development of nanotechnology, nanofluid has been widely used in the enhancement of heat transfer and different models are applied to simulate this kind of problems [7–11]. In order to investigate the mechanisms and the microscopic details of natural convection, several research groups began to use various numerical methods to simulate the natural convection characteristics of nanofluids [12–16]. Among these methods, the Lattice Boltzmann method is a new way to investigate natural convection of nanofluid. The method has many merits, for example, the algorithm is simple, the boundary conditions are easily dealt with, and the transform between macroscopical equations and microscopic equations is easily achieved. Hence, the Lattice Boltzmann method is widely applied in the study of natural convection [17–19].
VA V wa x, y
dimensionless interaction potential volume of a single lattice (m3) weight coefficient dimensionless coordinates
Greek symbols br thermal expansion coefficient (K1) qr density (kg/m3) m kinematic viscosity (m2 s1) g dynamic viscosity (Pa s) v thermal diffusion coefficient (m2 s1) c surface tension (N/m) u nanoparticle volume fraction dx lattice step dt time step t r components (r = 1, 2, water and nanoparticles) sf dimensionless collision-relaxation time for the flow field sT dimensionless collision-relaxation time for the temperature field DT dimensionless temperature difference (DT = TH TC) Dq0 dimensionless mass density difference between nanoparticles and base fluid Du dimensionless velocity difference between nanoparticles and base fluid Uab dimensionless energy exchange between nanoparticles and base fluid Error1 maximal relative error of velocities between two adjacent time layers Error2 maximal relative error of temperatures between two adjacent time layers Subscripts
a avg C nf H w p
lattice velocity direction average cold nanofluid hot base fluid nanoparticle
Barrios et al. [20] proposed a Lattice Boltzmann model and numerically investigated the natural convection in a square enclosure with a partially heated left wall. Peng et al. [21] proposed a simple Lattice Boltzmann model without gradient term, which is easily applied, on the assumption that there is no thermal diffusion. He et al. [22] developed a new Lattice Boltzmann model, which introduced an energy density distribution function to simulate the temperature field, and the simulation result has good agreement with the benchmark solution. Nemati et al. [23] simulated the natural convection of lid driven flow in a square filled with nanofluid, and the effects of nanoparticle fraction and Reynolds number on heat transfer were discussed. Kefayati et al. [24,25] simulated natural convection in an open cavity with magnetic field, a cavity with sinusoidal temperature distribution, and also turbulent natural convection. Sheikholeslami et al. [26,27] took the simulations of natural convection considering different kind of nanoparticles, the effects of magnetic field, and various figures of enclosures. Dixit et al. [28] simulated the natural convection in a square cavity at high Rayleigh number by a thermal Boltzmann method based on the BGK model, which used the
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double population approach to respectively simulate the hydrodynamic and thermal fields. Bararnia et al. [29] simulated natural convection in a partitioned square cavity and different parameters such as Rayleigh number, the length and the orientation of the partition were analyzed. Peng et al. [30] proposed a 3D incompressible thermal Lattice Boltzmann model with two different velocity models of D3Q15 and D3Q19, and investigated the natural convection of air in a cubical enclosure by the model. However, all above Lattice Boltzmann models are for a single phase. All the single phase models assumed that there are no slip velocities between nanoparticles and fluid molecules and assumed that the nanoparticle concentration is uniform [31]. It is accurate enough to simulate pure fluid and nanofluid at a low concentration for the macroscopical heat transfer characteristics. But when taking the interaction forces between the base fluid and the nanoparticles into account, the single phase model will be powerless. Therefore, researchers proposed two phase Lattice Boltzmann models to investigate the characteristics of two phase flow and heat transfer. Xuan et al. [32] proposed a two phase Lattice Boltzmann model to investigate sudden-start Couette flow and convection in parallel plate channels without researching the effect of forces on volume fraction distribution of nanoparticles. In addition, Zhou et al. [33] simulated the sudden-start Couette flow of Cu–water nanofluid in a microchannel and laminar flow of the Cu–water nanofluid in a channel within parallel plates with two phase Lattice Boltzmann models. However, there are few two phase Lattice Boltzmann models on natural convection of nanofluid in an enclosure. In this paper, a two phase Lattice Boltzmann model for natural convection of nanofluid in an enclosure is developed by considering the external and internal forces acting on nanoparticles and base fluid. It is applied to the simulation of natural convection heat transfer of Al2O3–water nanofluid in an experimental enclosure using the same conditions as the actual experiment (the same nanoparticle fractions and Rayleigh numbers).
The flow and temperature of nanofluid in the enclosure will reach a balance when the temperatures of the two walls do not change any more. The Nusselt number of the nanofluid can be calculated by the temperatures of the two walls and the heating power in the balanced state. The heating power is calculated as follows:
Q ¼ UI
ð1Þ
The temperature of the back of the left wall (copperplate) is calculated as follows:
Tb ¼
T1 þ T2 þ T3 þ T4 þ T5 5
ð2Þ
The temperature of the left wall in contact with the water is calculated as follows:
Qd Akw
TH ¼ Tb
ð3Þ
where d is the thickness of the copperplate, A is the area of the copperplate, and kw is the thermal conductivity of copperplate. The temperature of the outside right wall is calculated as follows:
To ¼
T 6 þ T 7 þ T 8 þ T 9 þ T 10 5
ð4Þ
Because the temperature of water in the cooling cavity is constant, only two copperplates should be considered. Then the temperature of the inside right wall contacting water is calculated as follows:
TL ¼ To
2Qd Akw
ð5Þ
The heat transfer coefficient is calculated as follows:
h¼
Q AðT H T L Þ
ð6Þ
The Nusselt number is calculated as follows: 2. Experimental study A two-step method was used to formulate the Al2O3–water nanofluid and the Al2O3 nanoparticles with an average diameter about 30 nm were procured from Tansail Company. The thermal conductivity coefficients of nanofluid were measured based on the transient hot-wire method (TC3000L Series, Xiatech Electronic Technology). The thermal conductivity coefficient meter has a measurement range from 0.001 to 5.0 W/(m K) and an accuracy of ±2%. The rheological properties of nanofluids are measured in present work by a super rheometer (Kinexus Pro, Malvern Instruments). Fig. 1(a) and (b) show the experimental setup for natural convection heat transfer and the structure of the square enclosure. There are five main parts which include heating system (silica gel heater and DC power), cooling system (constant temperature water bath), experimental system (enclosure), measuring system (data acquisition instrument, computer, thermocouple, and heat flow meter) and insulating system (adiabatic nano-board). The fluid in the enclosure is Al2O3–water nanofluid. The size of the enclosure is: 180 mm (length) 80 mm (width) 80 mm (height). The left wall is heated by a silica gel heater, and the right wall is cooled by a constant temperature water bath. There are ten T-type thermocouples located in the left and right walls to measure the wall temperatures. The temperature values of the two walls can be read by the data acquisition instrument (Agilent 34972A). The heating power of the silica gel heater can be obtained from the voltage U and electric current I of the DC power supply.
Nu ¼
hH k
ð7Þ
where H is the width of the enclosure and k is the conductive coefficient of the nanofluid. Uncertainty of experimental results was determined by measurement deviation of the parameters, including heating power, thermal conductivity, temperature, and the area of the heating wall. Taking the logarithm of Eq. (1)
lnQ ¼ ln U þ ln I
ð8Þ
Then the error propagation formula of heating power Q is
@ ln Q DQ @ ln Q DU DI ¼ D U þ @I DI ¼ U þ I Q @U
ð9Þ
Taking the same step, we got the error propagation formula of Tb is
@T b DT 1 þ @T b DT 2 þ @T b DT 3 þ @T b DT 4 þ @T b DT 5 DT b ¼ @T 1 @T 2 @T 3 @T 4 @T 5 1 ¼ ðDT 1 þ DT 2 þ DT 3 þ DT 4 þ DT 5 Þ ð10Þ 5 And the error propagation formula of Tn is
@ ln T n DT n @ ln T n Dd þ @ ln T n DA þ @ ln T n Dkw ¼ D Q þ @d @A @kw Tn @Q DQ Dd DA Dkw ¼ þ þ þ Q d A kw
ð11Þ
Y. Hu et al. / International Journal of Heat and Mass Transfer 78 (2014) 380–392
383
(a)
(b) Fig. 1. Experimental apparatus (a) experimental installation, (b) structure of the square enclosure.
The error propagation formula of area is
@ ln A DA @ ln A DL DW ¼ D L þ @W DW ¼ L þ W A @L
ð12Þ
The error propagation formula of TH is
@T H DT b þ @T H DT n ¼ DT b þ DT n DT H ¼ @T b @T n
ð13Þ 3. Numerical study
The error propagation formula of To is
@T o DT 6 þ @T o DT 7 þ @T o DT 8 þ @T o DT 9 þ @T o DT 10 DT o ¼ @T @T @T @T @T 6 7 8 9 10 1 ¼ ðDT 6 þ DT 7 þ DT 8 þ DT 9 þ DT 10 Þ ð14Þ 5
ð15Þ
The error propagation formula of h is
@ ln h Dh @ ln h DA þ @ ln h DðT H T L Þ ¼ D Q þ @A @ðT T Þ h @Q H L DQ DA DðT H T L Þ ¼ þ þ Q A ðT H T L Þ
Continuous equation
ð16Þ
ð19Þ
where p is the pressure, v is the viscosity, g is the local gravity, b is the thermal expansion coefficient, T0 is the reference temperature and q0 is the fluid density at the temperature of T0. Energy equation
@T þ r ðuTÞ ¼ r ðvrTÞ @t ð17Þ
ð18Þ
where u is the velocity vector. Momentum equation
@u rp þ r ðuuÞ ¼ þ mr2 u gbðT T 0 Þ @t q0
And the error propagation formula of Nu is
@lnNu DNu @lnNu DH þ @lnNuDknf ¼ Dh þ Nu @h @H @knf Dh DH Dknf ¼ þ þ h H knf
3.1. Governing equations
ru¼0
The error propagation formula of TL is
@T L DT o þ @T L DT n ¼ DT o þ 2DT n DT L ¼ @T @T o n
In present work, the measuring error of Q is ±0.65 W, the measuring error of Tb is ±0.001 °C, the measuring error of To is ±0.001 °C, the measuring errors of the length and the width are ±0.5 mm, and the relative error of thermal conductivity is ±2%. So the error of heat transfer coefficient of natural convection is 5.65% and the error of Nusselt number is 8.275%.
v is the thermal diffusion coefficient.
ð20Þ
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3.2. Lattice Boltzmann method
In the case of no internal forces and external forces, the macroscopic temperature, density and velocity are calculated as follows:
A nanofluid is a kind of two phase fluid which includes a liquid phase and a solid phase. The macroscopic density and velocity fields are still simulated using the density distribution function:
far ðr
þ ea dt ; t
0
F ra ¼ G
þ dt Þ far ðr; tÞ
1 ¼ r far ðr; tÞ fareq ðr; tÞ
sf 2srf 1 F ra dt ea 0 þ þ dt F ra Ba c2 2srf
ðea ur Þ req fa p
3
a ¼ 1; . . . ; 4
ð21Þ
ð23Þ
8 ð0; 0Þ a¼0 > < p p a ¼ 1; 2; 3; 4 ea ¼ c cos ða 1Þ 2 ; sin ða 1Þ 2 > : pffiffiffi a ¼ 5; 6; 7; 8 2c cos ð2a 1Þ p4 ; sin ð2a 1Þ p4
ð24Þ
" ¼ q wa
wa ¼
84 > <9
1 > 91 : 36
ea ur ðea ur Þ2 ur2 1þ þ 2 c2s 2c4s 2cs
# ð25Þ
a¼0 a ¼ 1; . . . ; 4 a ¼ 5; . . . ; 8
ð26Þ
2
where c2s ¼ c3 is the lattice sound velocity, wa is the weight coefficient. The macroscopic temperature field is simulated using the temperature distribution function:
1 T ra ðr þ ea dt ; t þ dt Þ T ra ðr; tÞ ¼ r T ra ðr; tÞ T ra eq ðr; tÞ s
ð27Þ
T
where sT is the dimensionless collision-relaxation time for the temperature field. The temperature equilibrium distribution function is chosen as follows:
" T ra eq ¼ wa T r 1 þ 3
8 1 X ur ¼ r far ea q
ð31Þ
a¼0
Since the nanofluid is a kind of two phase fluid, there are some forces between nanoparticles and the base fluid. Considering the internal and external forces, the macroscopic velocities for the nanoparticles and base fluid are changed to:
F p Dtea 2qr DtF W ¼ uw þ 2Lx Ly qw
upnew ¼ up þ
ð32Þ
uwnew
ð33Þ
where Fp is the total force acting on the nanoparticles, FW is the total force acting on base fluid, and LxLy is the total number of lattices. When the internal and external forces are considered, energy is exchanged between nanoparticles and base fluid, and the macroscopic temperature for nanoparticles and base fluid is given as:
T rnew ¼ T r þ dt sT
dT ¼ T r þ dt sT Uab dt
2
ea ur ðea ur Þ ur2 þ 4:5 1:5 2 c2 2c4 2c
#
ð34Þ
where Uab is the energy exchange between nanoparticles and base fluid, Uab ¼
hab ½T b ðx;tdt ÞT a ðx;tdt Þ , qa cpa aa
and hab is the convective heat trans-
fer coefficient of the nanofluid. The corresponding kinematic viscosity and thermal diffusion coefficients are respectively defined as follows:
1 1 dt 3 2
1 1 vr ¼ c2 srT dt 3 2
ð35Þ ð36Þ
For natural convection the important dimensionless parameters are the Prandtl number Pr and the Rayleigh number Ra defined by:
mr
The density equilibrium distribution function is chosen as follows: r
ð30Þ
mr ¼ c2 srf
For the two-dimensional nine-velocity LB model (D2Q9) [34] considered herein, the discrete velocity set for each component a is:
fareq
8 X far
a¼0
ð22Þ
12 a ¼ 5; . . . ; 8
ð29Þ
a¼0
qr ¼
where srf is the dimensionless collision-relaxation time for the flow field, ea is the lattice velocity vector, the subscript a represents the lattice velocity direction, r is the component (r = 1, 2, respectively represent the water and nanoparticle components), far ðr; tÞ is the distribution function of the nanofluid with velocity ea (along the direction a) at lattice position r and time t, fareq ðr; tÞ is the local equilibrium distribution function, dt is the time step, F ra is the total interparticle interaction force, Ba is one of the adjustable coefficients, c = dx/dt is the reference lattice velocity, dx is the lattice step, the order numbers a = 1, . . ., 4 and a = 5, . . ., 8, respectively represent the rectangular directions and the diagonal directions of a lattice, F ra 0 is the total interparticle interaction force, F ra is the external force term in the direction of lattice velocity without interparticle interaction, G = b(Tnf T0)g is the effective external force, where g is the gravity acceleration, b is the thermal expansion coefficient, Tnf is the temperature of the nanofluid, T0 is the mean value of the high and low temperatures of the walls. The adjustable coefficient Ba is given as:
Ba ¼
8 X T ra
Tr ¼
Pr ¼ r v Ra ¼
ð37Þ
gbr DTH3 Pr
mr2
ð38Þ
where DT is the temperature difference between the high temperature wall and the low temperature wall, H is the characteristic length of the square cavity. Another dimensionless parameter Mach number (Ma) is defined by:
Ma ¼
urc cs
ð39Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where urc ¼ gbr DTH is the characteristic velocity of natural convection. For natural convection the Boussinesq approximation is applied. To ensure that the code works in a near incompressible regime, the characteristic velocity must be small compared to the fluid speed of sound. In the present study, the characteristic velocity is selected as 0.1 times the sound of speed. The dimensionless collision-relaxation time sf and sT are respectively given as follows:
srf ¼ 0:5 þ ð28Þ
srT ¼ 0:5 þ
pffiffiffiffiffiffiffiffi MaH 3Pr pffiffiffiffiffiffi c2 dt Ra r 3m Prc2 dt
ð40Þ ð41Þ
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Y. Hu et al. / International Journal of Heat and Mass Transfer 78 (2014) 380–392
The two phase Lattice Boltzmann model is applied to simulate the natural convection heat transfer in a square cavity which is shown in Fig. 2. The square cavity is filled with Al2O3–water nanofluid. The thermo-physical properties of water and Al2O3 are given in Table 1. The height and the width of the enclosure are both H. The left wall is kept at a high constant temperature (TH), and the right cold wall is kept at a low constant temperature (TC). The boundary conditions of the other walls (top wall and bottom wall) are all adiabatic. The initial conditions of the four walls are given as follows:
x ¼ 0 u ¼ 0; T ¼ 1;
x ¼ 1 u ¼ 0; T ¼ 0
y ¼ 0 u ¼ 0; @T=@y ¼ 0; y ¼ 1 u ¼ 0; @T=@y ¼ 0
ð42Þ
In the simulation, a nonequilibrium extrapolation scheme is adopted to deal with the boundary, and the criteria for the flow field convergence and temperature field are given as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 2 h P r ux ði;j;t þ dt Þ urx ði;j;tÞ þ ury ði;j;t þ dt Þ ury ði;j;tÞ i;j rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < e1 Error 1 ¼ i P h r 2 2 r i;j ux ði;j;t þ dt Þ þ uy ði;j;t þ dt Þ ð43Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P r 2 r i;j ½T ði;j;t þ dt Þ T ði;j;tÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qP < e2 Error2 ¼ 2 r i;j T ði;j;t þ dt Þ
ð44Þ
where e is a small number, for example, for Ra = 1 103, e1 = 106, e2 = 106. The physical parameters used in the simulation are: time step: t = 1 105 s, lattice step: d = 1 105 m, radius of nanoparticle: a = 20 nm. However, in the simulation, all the units are lattice units, and the international units have to be converted to lattice units. The units with ‘‘0 ’’ represent lattice units, and the units without ‘‘0 ’’ represent international units, the relationships between international units and lattice units are as follows:
D ¼ D0 L;
L2 m ¼ m0 ; T
q ¼ q0
G
ð45Þ
3
L
T¼
3
m0 D 2 q D ; G¼ 0 q D0 m D0
ð46Þ
adiabatic surface
3
q (kg/m ) cp (J/kg k) v (m2/s) k (W m1 K1)
Fluid phase (H2O)
Nanoparticles (Al2O3)
997.1 4179 0.001004 0.613
3970 765 / 25
The lattice units – time, length, velocity, acceleration, mass, and force are respectively given as:
8 < t 0 ¼ t ; l0 ¼ l ; u0 ¼ u T ¼ u m0 D0 ; a0 ¼ a T 2 ¼ a m0 2 D0 3 T L L m D L m D : m0 ¼ m ¼ m q0 D0 3 ; F 0 ¼ F T 2 ¼ F q0 m0 2 G q D GL q m
g TH
Nu ¼
hH knf
ð48Þ
The heat transfer coefficient is computed from:
h¼
qw TH TL
ð49Þ
The thermal conductivity of the nanofluid is defined by:
qw @T=@x
knf ¼
ð50Þ
Substituting Eqs. (49) and (41) into Eq. (48), the local Nusselt number along the left wall can be written as:
@T H Nu ¼ @x T H T L
ð51Þ
The average Nusselt number is determined from:
Nuavg ¼
Z
1
NuðyÞdy
ð52Þ
3.3. Interaction forces between base fluid and nanoparticles As noted before a nanofluid is actually a kind of two phase fluid. There are interaction forces between liquid and nanoparticles, which affect the behavior of the nanofluid. The external forces include gravity and buoyancy force FH, and the interparticle interaction forces which here comprise drag force (Stokes force) FD, interaction potential FA, and Brownian force FB. We introduce them as follows. The gravity and buoyancy force is given as:
TC
4pa3 g Dq 0 3
ð53Þ
where a is the radius of a nanoparticle, Dq0 is the mass density difference between the suspended nanoparticle and the base fluid. The drag force (Stokes force) is given as:
F D ¼ 6plaDu
H
adiabatic surface
Y
ð47Þ
All the physical parameters in the simulation can be changed into lattice units according to Eq. (47). The Nusselt number can be expressed as:
FH ¼
H
D ; D0
Physical properties
0
where D is the nanoparticle diameter, m is the kinematic viscosity of the base fluid, and q is the density of the base fluid, L is the length dimension, T is the time dimension, and G is the mass dimension. The length dimension, time dimension and mass dimension can be obtained from Eq. (45):
L¼
Table 1 Thermo-physical properties of water and Al2O3 [13].
X Fig. 2. Schematic of the square simulation cavity.
ð54Þ
where l is the viscosity of the fluid, and Du is the velocity difference between the nanoparticle and the base fluid. The interaction potential is presented as [35]:
1 2a2 2a2 L2 4a2 VA ¼ A 2 þ 2 þ cc 2 2 6 Lcc 4a Lcc Lcc
! ð55Þ
where A is the Hamaker constant, and Lcc is the center-to-center distance between particles.
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The interaction potential force is shown as:
FA ¼
8 X
ni
1.030
@V A @ri
1.025
where ni is the number of the particles within the adjacent lattice i, ni = qrV/mr, mr is the mass of a single nanoparticle, and V is the volume of a single lattice. The Brownian force is calculated as [36]:
1.020
i¼1
rffiffiffiffiffi C F B ¼ Gi dt
knf/kf
ð56Þ
nðF H þ F D þ F A þ F B Þ Fp ¼ V
1.015 1.010
ð57Þ
where Gi is a Gaussian random number with zero mean and unit variance, which is obtained from a program written by us, and C = 2ckBT = 2 (6pga)kBT, c is surface tension, kB is the Boltzmann constant, T is absolute temperature, g is dynamic viscosity. The total per unit volume forces acting on nanoparticles of a single lattice is:
Maxwell model 20°C 25°C 30°C 35°C 40°C
1.005 1.000 0.000
0.002
0.004
0.006
0.008
0.010
vol Fig. 4. Thermal conductivity coefficient ratio between nanofluid and water.
ð58Þ 0.0020
where n is the number of particles in the given lattice, and V is the lattice volume. In a nanofluid, the forces acting on the base fluid are mainly drag force and Brownian force. Thus the forces acting on the base fluid in a given lattice is:
All the forces above are calculated in lattice units, which leads to the forces dimensionless.
)
ð59Þ
0.0016
μ( ⋅
F w ¼ nðF D þ F B Þ
wt%=0% wt%=1% wt%=2% wt%=3%
0.0018
0.0014 0.0012
4. Results and discussion
0.0010
4.1. Experimental result 0.0008 0
There are two contradictory factors for natural convection heat transfer of nanofluid. One advantageous factor is the high conductivity coefficient and a disadvantageous factor is the high viscosity. Here, Figs. 3 and 5 give the thermal conductivity coefficient and the viscosity. The thermal conductivities for all the samples were measured for five times, and the maximum repeatability error was less than 0.92%. Fig. 3 presents the thermal conductivity coefficient of the nanofluid measured by a thermal conductivity coefficient instrument (TC3000L). It can be seen that the thermal conductivity
10
20
30
40
50
γ (s
60
70
80
90
)
-1
(a) 0.0015 0.0014 0.0013
μ(Pa⋅s)
0.0012 0.66 wt%=0% wt%=1% wt%=2% wt%=3%
0.65
knf(W/m2⋅K)
0.64
0.0011 0.0010 0.0009
0.63
0.0008 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Mass fraction %
0.62
(b
0.61
Fig. 5. Viscosity of nanofluid at different shearing velocities (a) viscosity of nanofluid (b) error bars.
0.60 0.59 290
295
300
305
310
T(K) Fig. 3. Thermal conductivity coefficient of nanofluid.
315
coefficient of the nanofluid increases with temperature and nanoparticle mass fraction. The thermal conductive coefficient of nanofluid is higher than that of pure water. From Fig. 4, it can be seen that the thermal conductivity coefficient ratio between nanofluid and water is in good agreement with the Maxwell model.
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4.2. Numerical simulation result
25 24 23
Nuavg
22
water literature [1] wt%=0% (exp) wt%=1% (exp) wt%=2% (exp) wt%=3% (exp) wt%=1% (sim) wt%=2% (sim) wt%=3% (sim)
21 20 19 18 17 2E7
4E7
6E7
8E7
1E8 1.2E8
Ra Fig. 7. Average Nusselt number of nanofluid at different Rayleigh numbers.
1.1
critical line wt=1% (exp) wt=2% (exp) wt=3% (exp)
1.08 1.06 1.04
Nunf/Nuf
Fig. 5(a) and (b) show the viscosity of nanofluid at different shearing velocities measured by a Kinexus Pro. rheometer (made by Malvern Instruments Ltd) and the error bars. It can be seen that the viscosity of nanofluid is higher than that of water, and it increases with nanoparticle mass fraction. As a whole, changes in viscosity of nanofluid at different shearing velocities are not large, which explains that nanofluid can be approximatively seen as a Newtonian fluid in this work. In order to validate the experimental installation, Fig. 6 presents the average Nusselt number of water at different Rayleigh numbers. It can be seen from Fig. 6 that the experimental results with water are in good agreement with those in the literature [1]. The natural convection experiment was carried out for three times, and there was a maximum repeatability error about 4.6%. Fig. 7 shows the average Nusselt number of nanofluid at different Rayleigh numbers. Fig. 8 presents the average Nusselt number ratio between nanofluid and water. From Figs. 7 and 8, it can be seen that the heat transfer of nanofluid shows an enhancement compared with that of pure water at low nanoparticle mass fraction (wt% = 1%), it is almost the same at wt% = 2%, and there is a reduction of heat transfer at high nanoparticle mass fraction (wt% = 3%). For a low nanoparticle mass fraction, the high conductivity coefficient of nanoparticles plays a major role. As the mass fraction increases, the high viscosity of nanofluid begins to play the main role and weakens the heat transfer. From this, it can be concluded that the heat transfer of nanofluid is more sensitive to thermal conductivity than viscosity at a low nanoparticle fractions and it is more sensitive to the viscosity than the thermal conductivity at a high nanoparticle fraction.
1.02 1 0.98
As shown in Table 2, a grid independence test is performed in a square enclosure using successively sized grids, 128 128, 192 192, 256 256 and 320 320 at Ra = 1 105, Pr = 0.7. It can be seen from Table 2 that there is a bigger difference between the result with grid 128 128, 192 192 and the results in the literature [37], than with the results obtained using grids 256 256 and 320 320. In addition, the result with grid 256 256 and the result with grid 320 320 are very close. In order to accelerate the numerical simulation, a grid size of 256 256 is chosen as suitable and being able to guarantee a grid independent solution. In order to validate the Lattice Boltzmann model proposed in this work, the temperature distribution at the mid-sections of the enclosure at Ra = 1 105, Pr = 0.7 is compared with numerical
0.96 0.94 3E7
4E7
5E7
6E7
7E7
8E7
Ra Fig. 8. Average Nusselt number ration between nanofluid and water.
1.0
present simulation Khanafer et al [38] (numerical) Krane and Jessee [39] (experimental)
0.8
27 25
0.6
literature Ho[1] experimental pure water
T
26
0.4
24 23
Nuavg
22
0.2
21 20
0.0 0.0
19
0.4
0.6
0.8
1.0
X
18 17 2E7
0.2
Fig. 9. Temperature distribution at mid-sections of the enclosure (Ra = 105, Pr = 0.7).
4E7
6E7
8E7
1E8
Ra Fig. 6. Average Nusselt number of water at different Rayleigh numbers.
results by Khanafer et al. [38] and experimental results by Krane et al. [39] in Fig. 9. It can be seen that the results of this paper
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Table 2 Comparison of the mean Nusselt numbers with different grids (Ra = 105, Pr = 0.7). Physical properties
128 128
192 192
256 256
320 320
Literature [37]
Nuavg
4.5466
4.5251
4.5220
4.5218
4.5216
are in good agreement with the numerical [38] and experimental results [39]. The results of this paper are closer to the experimental results [39] than the numerical results [38]. Figs. 10 and 11 show the comparison of isotherms and streamlines for different mass fractions of nanoparticles and different Rayleigh numbers (Ra = 30,855,746 and Ra = 63,943,592 for wt% = 1%, Ra = 38,801,494 and Ra = 67,175,834 for wt% = 2%, and Ra = 55,888,498 and Ra = 70,513,049 for wt% = 3%). When the
Rayleigh number increases, the isotherm becomes more crooked and the centralization of the streamlines in the core of the cavity tends toward the hot wall, which indicates that convection heat transfer is stronger at a higher Rayleigh number. The forces involved are gravity–buoyancy force, drag force, interaction potential force, Brownian force between water and nanoparticles and temperature difference driving force. These forces are presented in Table 3. From Table 3, it can be seen that
Ra=30855746
Ra=63943592
(a) wt%=1%
Ra=38801494
Ra=67175834
(b) wt%=2%
Ra=55888498
Ra=70513049
(c) wt%=3% Fig. 10. Comparison of the isotherms at various Rayleigh numbers and mass fractions of nanoparticles.
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Ra=30855746
Ra=63943592
(a) wt%=1%
Ra=38801494
Ra=67175834
(b) wt%=2%
Ra=55888498
Ra=70513049
(c) wt%=3% Fig. 11. Comparison of the streamlines at various Rayleigh numbers and mass fractions of nanoparticles.
the temperature difference driving force (Fs) is the biggest one. Correspondingly, the effect of the temperature difference driving force on the nanoparticle volume fraction distribution is the greatest, and the effects of other forces on the nanoparticle volume fraction distribution can be ignored. Other forces also have effects on
the flow and heat transfer of nanofluid. Gravity–buoyancy force and driving force give rise to movement and velocity of the nanofluid in the enclosure. Drag force between nanoparticles and water molecules decreases the velocity of water, which is a disadvantage for heat transfer. Interaction potential force keeps
Table 3 Comparison of different forces of Al2O3–water nanofluid (Ra = 30855746, u = 0.01). Forces
FS
FA
FBx
FBy
FH
FDx
FDy
Minimum Maximum
5E06 5E06
3.2E19 2E20
5E13 5E13
2E14 2E13
9E19 1E19
5E16 5E15
1.4E15 1.6E15
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(a)
(b)
Fig. 12. Temperature difference driving force distribution, wt% = 1% (a) Ra = 30,855,746 (b) Ra = 63,943,592.
(a)
(b)
Fig. 13. Water phase density distribution of nanofluid, wt% = 1% (a) Ra = 30,855,746 (b) Ra = 63,943,592.
(a)
(b)
Fig. 14. Nanoparticle volume fraction distribution of nanofluid, wt% = 1% (a) Ra = 30,855,746 (b) Ra = 63,943,592.
the nanoparticles from gathering together. The irregular motion of nanoparticles due to Brownian force can disturb the laminar boundary layer, which can enhance the heat transfer of natural convection. The driving force distribution is given in Fig. 12. From Fig. 12, it can be seen that the driving force in the area near left and top walls is large and upward, and it is large and downward near right and bottom walls, which drives the nanofluid to revolve in the enclosure. In addition, it is found that equivalent driving force lines are more crooked at the high Rayleigh number than at the low Rayleigh number. Because the driving force is due to the temperature gradient, the bigger temperature gradient causes equivalent driving force lines to become more crooked. It can be seen from Fig. 11 that isotherms are more crooked at the higher Rayleigh
number, and changes in the isotherms correspond to changes of temperature gradient. The driving force and the temperature difference give rise to the large density changes of the water phase in the nanofluid. Fig. 13 shows the water phase density distribution of the nanofluid with wt% = 1% at Ra = 30,855,746 and Ra = 63,943,592. For Ra = 30,855,746, it can be seen that density of the water phase in the area near the top wall is small while in the area near the bottom wall it is large. The water near the left wall is heated and its density decreases. Then the low density water flows upward, causing a low density water region near the top wall. The water near the top wall is then cooled and the density becomes larger. Then the higher density water flows downward, causing a high density
Y. Hu et al. / International Journal of Heat and Mass Transfer 78 (2014) 380–392
water region near the bottom wall. For Ra = 63,943,592, it can be found that the density of water near the top at the high Rayleigh number (Ra = 63,943,592) is smaller than it is for the low Rayleigh number (Ra = 30,855,746). As the Rayleigh number increases, the velocity of the nanofluid becomes bigger and the convection heat transfer of the nanofluid is enhanced. Hence, more low density water rises to the top wall, and the density of water near the top becomes much lower. Since the temperature of water near the top is higher than that near the bottom wall, the Brownian movement of nanoparticles is more drastic, which leads to the density distribution of water phase not smooth. Fig. 14 presents the nanoparticle volume fraction distribution with wt% = 1% at Ra = 30,855,746 and Ra = 63,943,592. From Fig. 14, it can be seen that the nanoparticle volume fraction distribution near the top is bigger than it is near the bottom, and the nanoparticle volume fraction distribution near the top at the high Rayleigh number (Ra = 63,943,592) is bigger than it is at the low Rayleigh number (Ra = 30,855,746). The reason for this is that the temperature difference driving force has the greatest effect on the nanoparticle volume fraction distribution. As the Rayleigh number increases, the corresponding driving force becomes larger and drives more nanoparticles to the top. In addition, it can be seen from Fig. 13 that the water phase density is low near the top, and the nanoparticle fraction distribution is opposite to that of the water phase density distribution. Hence, the nanoparticle volume fraction is large near the top, and increases with a higher Rayleigh number. Average Nusselt number of the nanofluid at different Rayleigh numbers is presented in Fig. 7. It shows that at the lowest nanoparticle mass fraction (wt% = 1%), the heat transfer characteristic is enhanced. While with increasing the mass fraction, the effect of heat transfer enhancement becomes weaker. When the mass fraction of nanoparticles reaches 2%, the Nusselt number is almost same as water. When the mass fraction of nanofluid achieves 3%, the heat transfer is weakened. Comparing the results of simulation with that of experiment, it can be seen that the numerical simulation results are in good agreement with the experimental results. It proves that the two phase Lattice Boltzmann model developed in this paper is suitable to simulate natural convection of nanofluid.
5. Conclusion The flow and heat transfer of Al2O3–water nanofluid in a square enclosure are experimentally and numerically investigated. The following conclusions are obtained: A two phase Lattice Boltzmann model considering interaction forces for natural convection of nanofluid in an enclosure is developed by coupling the density distribution function and the temperature distribution function. Numerical results show a satisfactory agreement with experimental results. Compared to pure water, the heat transfer of nanofluid shows an enhancement at a low nanoparticle mass fraction (wt% = 1%), is almost the same at wt% = 2%, and shows a reduction at a high nanoparticle mass fraction (wt% = 3%), which leads to the conclusion that the heat transfer of nanofluid is more sensitive to thermal conductivity than to viscosity at a low nanoparticle fraction and is more sensitive to viscosity than thermal conductivity at a high nanoparticle fraction. Among the forces, the temperature difference driving force is the biggest one and has the largest effect on nanoparticle volume fraction distribution. The nanoparticle volume fraction is large near the top and is small near the bottom, and increases with the Rayleigh number near the top. The temperature difference driving force, Brownian force, Interaction potential force and gravity–buoyancy force have a positive
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effect on the enhancement of natural convection heat transfer, while the effect of the drag force is negative. Conflict of interest None declared. Acknowledgments This work is financially supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51121004) , the Science Creative Foundation for Distinguished Young Scholars in Harbin (Grant No. 2014RFYXJ004) and the Fundamental Research Funds for the Central Universities (Grant No. HIT.BRETIV.201315). References [1] C.J. Ho, W.K. Liu, Y.S. Chang, C.C. Lin, Natural convection heat transfer of alumina–water nanofluid in vertical square enclosures: an experimental study, Int. J. Therm. Sci. 49 (8) (2010) 1345–1353. [2] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, ASME J. Heat Transfer 125 (1) (2003) 151–155. [3] W. Williams, J. Buongiorno, L.W. Hu, Experimental investigation of turbulent convective heat transfer and pressure loss of alumina/water and zirconia/ water nanoparticle colloids (nanofluids) in horizontal tubes, ASME J. Heat Transfer 130 (4) (2008) 1–7. [4] Y. Ding, H. Alias, D. Wen, R. Williams, Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids), Int. J. Heat Mass Transfer 49 (1–2) (2006) 240–250. [5] B.H. Chang, A.F. Mills, E. Hernandez, Natural convection of microparticle suspensions in thin enclosures, Int. J. Heat Mass Transfer 51 (5–6) (2008) 1332–1341. [6] J. Tu, G.H Yeoh, C. Liu, Computational Fluid Dynamics: a Practical Approach, Butterworth, Heinemann, 2007, pp. 1–27. [7] A.H. Mahmoudi, K. Hooman, Effect of a discrete heat source location on entropy generation in mixed convective cooling of a nanofluid inside the ventilated cavity, Int. J. Exergy 13 (3) (2013) 299–319. [8] G. Imani, M. Maerefat, K. Hooman, Lattice Boltzmann simulation of conjugate heat transfer from multiple heated obstacles mounted in a walled parallel plate channel, Numer. Heat Transfer Part A: Appl. 62 (10) (2012) 798–821. [9] G.H.R. Kefayati, Natural convection of ferrofluid in a linearly heated cavity utilizing LBM, J. Mol. Liq. 191 (2014) 1–9. [10] M. Sheikholeslami, M. Gorji-Bandpy, Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field, Powder Technol. 256 (2014) 490–498. [11] M. Sheikholeslami, D.D. Ganji, M. Gorji-Bandpy, S. Soleimani, Magnetic field effect on nanofluid flow and heat transfer using KKL model, J. Taiwan Inst. Chem. Eng. 45 (3) (2014) 795–807. [12] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid filled enclosure, Eur. J. Mech. – B/Fluids 28 (5) (2009) 630–640. [13] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of Al2O3– water nanofluid on heat transfer enhancement in natural convection, Int. J. Heat Fluid Flow 30 (4) (2009) 679–690. [14] K.S. Hwang, J.H. Lee, S.P. Jang, Buoyancy-driven heat transfer of water-based Al2O3 nanofluids in a rectangular cavity, Int. J. Heat Mass Transfer 50 (19–20) (2007) 4003–4010. [15] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, P. Rana, S. Soleimani, Magnetohydrodynamic free convection of Al2O3–water nanofluid considering thermophoresis and Brownian motion effects, Comput. Fluids 94 (2014) 147–160. [16] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, Natural convection heat transfer in a cavity with sinusoidal wall filled with CuO– water nanofluid in presence of magnetic field, J. Taiwan Inst. Chem. Eng. 45 (1) (2014) 40–49. [17] G.R. Kefayati, Simulation of ferrofluid heat dissipation effect on natural convection at an inclined cavity filled with kerosene/cobalt utilizing the Lattice Boltzmann method, Numer. Heat Transfer Part A: Appl. 65 (6) (2014) 509–530. [18] M. Sheikholeslami, M. Gorji-Bandpy, S.M. Seyyedi, et al., Application of LBM in simulation of natural convection in a nanofluid filled square cavity with curve boundaries, Powder Technol. 247 (2013) 87–94. [19] G.H. Kefayati, S.F. Hosseinizadeh, M. Gorji, H. Sajjadi, Lattice Boltzmann simulation of natural convection in tall enclosures using water/SiO2 nanofluid, Int. Commun. Heat Mass Transfer 38 (6) (2011) 798–805. [20] G. Barrios, R. Rechtman, J. Rojas, R. Tovar, The Lattice Boltzmann equation for natural convection in a two-dimensional cavity with a partially heated wall, J. Fluid Mech. 522 (2005) 91–100. [21] Y. Peng, C. Shu, Y.T. Chew, Simplified thermal Lattice Boltzmann model for incompressible thermal flows, Phys. Rev. E 68 (2) (2003) 026701.
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