Using Nanofluid in Lid Driven Shallow Enclosure at Particular Richardson Number: Investigation the Effect of Velocity Ratio
Arash Karimipour
1*, Seyed Sadegh Mirtalebi
1and Masoud Afrand
11Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iran; [email protected]
Abstract
In this study, using the finite volume method, a numerical investigation on mixed convection flow and heat transfer in a two-sided lid-driven shallow cavity which is subjected to Al2O3-water nanofluid was applied. The study was conducted in different velocity ratio and the mean particle diameter is 80 nm. Thermal conductivity and dynamic viscosity of abovemen- tioned nanofluid were calculated by utilizing H-C and Jang models, respectively. The impact of different ratio of velocity at the particular Richardson number of the nanofluid is discussed, with respect to hydrodynamic and thermal characteristics.
The results declared great enhancement in heat transfer by increasing the velocity ratio. Besides, it can be points out that heat transfer enhanced considerably by addition of Al2O3 nanoparticles compared with that of the water.
*Author for correspondence
1. Introduction
The thermal conductivity of nanofluid, which is a combi- nation of nano-sized particles suspended in a base fluid such as water, ethylene gelychol and oil, is higher than the base fluid. Due to this feature, the rate of heat transfer in industrial application develops considerably. Therefore, different sides of nanofluids, such as combined convec- tion heat transfer in enclosures filled with nanofluids, have been conducted by many researchers.
Oztop and Dagtekin1 and Alleborn et al.2 were ones who established a study on combined convection in binary sided lid-driven enclosures and stated that the important factor to combined convection was moving direction of the lid. Chamkha3 conducted an investigation on lid- driven enclosures with vertical moving lids which contain hydromagnetic combined convection with internal heat absorption or generation. And also some researches have been made that simulate the natural convection heat transfer using nanofluid in other different geometrical
Keywords: Al2O3-water, Heat Transfer, Isotherm Lines, Shallow Square Enclosure, Streamline
configuration by other researchers4–8. A numerical inves- tigation on buoyancy-driven convection in an enclosure with partly thermally active vertical walls was made by Kandaswamy et al.9 The study indicated that when the heating spot is at middle of the hot wall, heat transfer rate is gone up. a study on the steady-state, laminar and fully developed combined convection of double non-reacting gas mixture flowing upward in a vertical-plate chan- nel, analytically, was established by Boulama et al.10. The heat transfer feature of nanofluids combined convection within tubes or driven lid cavity have been subjected of a few other investigations11–14, as its empirical importance.
Many patterns developed since the deficiency of an advanced theory for calculating the thermal con- ductivity of nanofluids which mainly concentrated on several parameters, for instance: temperature or Brownian motion, interaction between based fluid and nanopar- ticles and geometry of nanoparticles. The model given by Hemanth et al.15 have been improved by Patel et al.16 Three contributions have been considered for heat flow
by them: advection owing to Brownian motion of the particles and conduction through liquid and through solid. The model is able to estimate the thermal conduc- tivity over a vast range of particle concentrations (1– 8)
%, particles size (10–100)nm, different base fluids and temperatures. Opposing and aiding mechanism in a buoyancy and shear-driven enclosure were compared by Aydin17. A numerical study on the laminar natural convection heat transfer in a differentially heated square enclosure filled with copper-water nanofluid was con- ducted by Santra et al.18 and two parameter power law model for an incompressible non-Newtonian fluid was considered by them. A numerical study in the realm of horizontally moving lid in enclosures was made by some researchers19–23 and all of the authors declared uniformly that the most crucial factor to flow field, temperature dis- tribution and heat transfer was the ratio of lid velocity to power of buoyancy force.
2. Physical Modeling and Governing Equations
Figure 1 illustrates an enclosure considered for this inves- tigation.The enclosure is filled with Al2O3-water nanofluid and having adiabatic vertical moving lids.
Right wall is kept in high temperature (Th) while the Top wall is maintained at cold temperature (Tc). Velocity of left and right moving lids are different and denoted by Uo and U0*V.R, respectively.
The fluid (Al2O3- water) in the square cavity is incom- pressible and Newtonian. Also, the density variation in
the body force term of the momentum equation is sat- isfied by Boussinesq’s approximation. Some of useful thermophysical properties of solid and liquid phases are mentioned in Table1.
Also, in this investigation, new variable properties correlation is utilized for nanoparticle with diameter size of 80 nanometers and R=0.007.
The calculation of thermal conductivity and dynamic viscosity of Al2O3- water nanofluid were assume as vari- able properties. Particle concentration and temperature of nanofluid influence on thermal conductivity and dynamic viscosity. The system of governing equations is presented below:
Continuity equation:
Figure 1. Schematic diagram of square cavity in this study.
Table 1. Thermophysical properties of fluid and disperse solid
Thermophysical
properties Base fluid (water)
Solid particle (Al2O3)
Cp(J/kg k) 4179 765
ρ(kg/m3) 997.1 3970
K (W m-1 K-1) 0.6 29
β×10-5 (1/K) 21 0.85
μ×10-4(Kg/ms) 8.9 ---
dp(nanometers) --- 80 And energy equation
And momentum equations in horizontal and vertical directions:
(3)
X xL Y y
L V v
u U u
u
T T T T T
T P p
h c c u
nf
= = = =
= − =
−
=
, , ,
, , .
0 0
02
∆
∆
q r
The dimensionless parameters is assumed as follow:
∂
∂ +
∂
∂ u = x
v
y 0, (1)
(2)
u Tx v T
y nf T
∂
∂ +
∂
∂
=α ∇2 . (4)
(5) u ux v u
y
p
x u g T Sin
nf
nf
nf nf
∂
∂ +
∂
∂
= −
∂
∂
+ ∇ +
1 2
r u rb
r
( ) ∆ . ( )γ
u vx v v y
p
y v g T Cos
nf nf
nf nf
∂
∂ +
∂
∂
= −
∂
∂
+ ∇ +
1 2
r u rb
r
( ) ∆ . ( )γ
2.1 Density and Thermal Diffusivity of Nanofluid
Thermal diffusivity and effective density of the nanofluid may be presented as:
2.2 Thermal Expansion and Specific Heat Capacity of Nanofluid
Heat capacity and thermal expansion coefficient of the nanofluid are therefore
2.3 Viscosity
Estimation of dynamic viscosity of nanofluid was done by using Jang et al.24 Model. This model is devoted to fluid containing suspended small rigid particles.
є and η are empirical constant and equal to -0.25 and 280 for alumina, respectively.
It should be noted that the dynamic viscosity of water is considered to change with temperature and the following equation is utilizing to compute the viscosity of base fluid:
2.4 Thermal conductivity of Al
2O
3- nanofluid
Stationary part of the thermal conductivity of the Al2O3– water is evaluated by the Hamilton and Crosser25, which is:
Thermal conductivity of nanofluids was obtained using the model proposed by Xu et al.26
c is equal to 85 for water and independent of the type of nanoparticles. The fractal dimension Df is calculated with follow correlation:
where, are the maximum and minimum diameters of nanoparticles are denoted by dp,max and dp,min in this cor- relation.
R is equal to 0.007 and indicate the ratio of minimum to maximum nanoparticles dp,min/dp,max
so,
Re ,
Pr.Re , , Pr .
=r = = =
m
b u
f u
f
f
f f
f f
u L0 Ri Ra Ra g TL
2
∆ 3
α α (6)
governing equations (1) to (4) in dimensionless form become continuity:
∂
∂ +
∂
∂ U = X
V
Y 0 (7)
Momentum in X and Y directions:
U UX V U Y
P
X nf U Ri
f
nf f
∂
∂ +
∂
∂
= −
∂
∂
+u ∇ +
u
b
b q
1 2
Re.
Pr. ∆ .sin( )γ (8) U VX V V
Y P
Y nf V Ri
f
nf f
∂
∂ +
∂
∂
= −
∂
∂
+u ∇ +
u
b
b q
1 2
Re.
Pr. ∆ .cos( )γ (9) And energy equation
U X V Y
nf f
∂
∂ + ∂
∂
= ∇
q q α q
α
2 (10)
αnf nf
p nf
k
= c (r )
rnf =jrs+(1−j r) f (12) (11)
(rcp nf) =j r( cp s) (+ 1−j r)( cp f) ( )rb nf =j rb( ) (s+ 1−j rb)( )f
(13) (14)
µ = µ ϕ η ϕ ε
−2ε
eff f(1 2 5+ . ) 1+ 23 1
(
+)
d
L
p (15)
µ × ×
× ×
H o rc rc
rc rc
T T
T T
2 1 2723 8 736 33 708 246 6 518 78
5 4
3 2
= − +
− +
( . .
. .
. ××T ×
T Log T
rc
rc
+
= −
1153 9 10
273 . ) 6
( ).
Where
(16)
k k
k k k k
k k k k
stationary f
s f f s
s f f s
=
+ − −
+ + −
2 2
2 j j
( )
( ) (17)
k k
k k
k k
k k k k
k k k k
c
nf f
stationary f
c f
s f f s
s f f s
= + =
+ − −
+ + −
+
2 2
2 j j
( )
( )
N
Nu d D D D
d d d d
p f f f
f
Df
2 1
1
2
1 2
(
−)
(
−)
−
−
Pr max
m
max min
iin
−
2−
1 1
Df
dp
(18)
D d
d
f
p p
= −
2 ln
ln ,min
,max
ϕ
d d D
D d d
d d D
D
p p f
f p p
p p f
f ,max
,min ,max
,min
.
.
=
−
=
− 1 −
1
1
3. Numerical Method
Based on the finite volume method and staggered grid system, using FORTRAN software code, the equations of continuity, momentum and energy with respect to bound- ary conditions were calculated numerically. Numerical procedure was conducted for 9 different mesh sizes so as to verify grid independence. Figure 2 reveals the average Nu of the hot wall obtained for 2 different situations for each grid size.
4. Results and Discussion
In this article, fluid flow diagrams and isotherm lines of a lid-driven shallow enclosure are studied and discussed.
The enclosure has studied in two cases: one subjected to nanofluid with solid volume fraction of 6% and another subjected to distilled water as a pure fluid. The enclosure walls have different and constant velocity. In addition, the width of enclosure is half of its length and the impact of velocity ratio of moving lids and existence of dispersion nanoparticles in fluid on hydrodynamic properties and thermal characteristics is investigated.
Variations in flow and thermal behavior of water and Al2O3-water with the changes of velocity ratio are shown in Figure 3. The comparison between the fluid behav- ior in various ratios of velocity in this figure and other figures (presented in this study) declare the impact of changes in velocity ratio of moving lids. Fig3 illustrates flow pattern and isotherm lines of Al2O3-water and water in various ratios of velocity in Re=100, A.R=2 and Ri=0.1.
Streamlines illustrates the formation of two vortices with various intensity in two sides of enclosure. In this range of studied parameters, the major contributor of formation of vortices is force due to the motion of lids on the fluid adjacent to the lid. Therefore the dimension and intensity of vortices are depending on the velocity of the horizontal walls. Because of alignment of shear and buoyancy forces
Figure 3. Streamlines and isotherms for base fluid(solid line) and nanofluid(dash line) at Ri=0.1 , Re=100 , A.R=2 a) V.R=0.2 b) V.R=0.7 c) V.R=1 d) V.R=2.
Figure 2. Mesh valid.
and overcoming of shear force of lids on buoyancy, only two vortices are made inside the enclosure.
Vortex produced by motion of left lid is much vigor- ous than other vortices in velocity ratios less than one.
On the other hand, with increasing of ratio of velocity, the vortex around right wall is corroborated.
Temperature lines illustrate convection heat transfer is in entire the enclosure. Thickness of lines (temperature gradient) produced in the proximity of the upper and lower walls increased with the rise of ratio of velocity.
In the given Richardson number, the exchanging ratio of natural convection to force convection is negligible.
Also, in this case, for sure the dominant heat transfer mechanism is force convection. Owing to the absence of intermediate vortex caused by buoyancy force in this case, the flow velocity in the middle compartment of the cavity is low. Therefore, it results in decreasing the overall heat transfer within the cavity in lower Richardson numbers.
Figure 4 shows the flow patterns and thermal behav- ior of base and nanofluid in various velocity ratios as well as A.R=2, Ri=1 and Re=100. In the section a of figure 4, formation of central vortex in cavity is due to increase of Ri relative to Figure 3.The formation of vortex, made flow pattern quite different. However, the strength intensity of the double vortex in right side of cavity is not consid- erable. So, with composition of 3 vortices with different strength and size, flow pattern has formed. In addition,
the intensity of isotherm lines close to walls with constant temperatures indicates that increasing of ratio of veloc- ity and strength of central vortex lead to increasing of temperature gradient near of these walls. it is expected to make a huge augment in Al2O3-water heat transfer with increasing of velocity ratio due to intensity and compres- sion of isotherm line near the walls. Exact and accurate analysis of heat transfer augmentation requires investigat- ing of Nusselt number plots.
The average Nusselt number figure against the veloc- ity ratio in different Richardson numbers of the nanofluid and base fluid in Re=100 and A.R=2 is shown in Figure 5.
There is a very sharp increase in Nusselt number and subsequently heat transfer ratio which is stem from an increase in Richardson number at a constant velocity ratio.
Moreover, the results indicated that while the Richardson number increasing from 0.1 to 10, Nusselt number shows a considerable increase up to 44%. Thus, it displays the substantial effect of Richardson number in this particu- lar geometry and boundary conditions. Also, as for the heat transfer ratio of the base fluid, it is observed that in the same range of Richardson number, the heat transfer increases up to 40.6%. Furthermore, the heat transfer ratio within the cavity increases by increasing the velocity ratio at a constant Richardson number’s value. This result is common for both of the base fluid and nanofluid. As far as the nanofluid at the solid volume fraction of 6%
Figure 4. Streamlines and isotherms for base fluid(solid line) and nanofluid with solid volume fraction of 0.06(dash line) at Ri=1 , Re=100 , A.R=2 a) V.R=0.2 b) V.R=0.7 c) V.R=1 d) V.R=2.
6. References
1. Oztop HF, Dagtekin I. Mixed convection in two-sided lid- driven differentially heated square cavity. Int J Heat Mass Transfer. 2004; 47:1761–69.
2. Alleborn N, Raszillier H, Durst F. Lid-driven cavity with heat and mass transport. Int J Heat Mass Transfer. 1999;
42:833–53.
3. Chamkha AJ. Hydromagnetic combined convection flow in a vertical lid driven cavity with internal heat generation or absorption. Numer Heat Transfer. 2002; 41 Part A:529–46.
4. Santra AK, Sen S, Chakraborty N. Study of heat transfer due to laminar flow of copper–water nanofluid through two is thermally heated parallel plates. International Journal of Thermal Science. 2009; 48:391–400.
5. Abu-Nada E, Oztop HF. Effects of inclination angle on natural convection in enclosures filled with Cu–water nanofluid. International Journal of Heat and Fluid Flow.
2009; 30(4):669–78.
6. Akbarnia A. Impacts of nanofluid flow on skin friction fac- tor and Nusselt number in curved tubes with constant mass flow., International Journal of Heat and Fluid Flow. 2008;
29:229–41.
7. Akbarnia A, Behzadmehr A. Numerical study of lami- nar mixed convection of a nanofluid in horizontal curved tubes. Applied Thermal Engineering. 2007; 27:1327–37.
8. Mirmasoumi S, Behzadmehr A. Effects of nanoparticles mean diameter on the particle migration and thermo- hydraulic behavior of laminar mixed convection of a nanofluid in an inclined tube. 2012; 48:1297–308.
and Richardson equal to 0.1 is concerned, the velocity ratio varying from 0.2 to 2 and also the Nusselt number increases up to 42.1%. On the other hand, Nusselt num- ber increases up to 22.6% at Richardson equal to 10. On the whole, it can be stated that increasing the Richardson number leads to restriction in the heat transfer ratio.
5. Conclusions
In this study, a numerical investigation on the flow pat- tern and thermal behavior of a lid-driven square shallow cavity was performed. Also, the impact of the velocity ratio and existence of nanoparticles in base fluid were investigated.
Based on the investigation, the conclusion can be drawn as follow:
1. While the velocity ratio increases in all cases, with the exception of the very low values of the Reynolds number, the vortex in the vicinity of the right wall enhances and the heat transfer ratio increases.
2. There is a very sharp increase in the Nusselt number and subsequently in the heat transfer ratio while the Richardson number increases at a constant velocity ratio.
3. Adding the nanoparticles to the base fluid, results in increasing the average Nusselt number. It also leads to increase the rate of heat transfer.
Figure 5. Average Nusselt diagram versus velocity ratio for different Ri number at Re=100, A.R=2 for a) nanofluid with solid volume fraction of 0.06 b) base fluid.
9. Kandaswamy P, Sivasankaran S, Nithyadevi N. Int J Heat Mass Trans. 2007; 50:942–48.
10. Boulama KG, Galanis N. J Orfi Int J Thermal Sci. 2006;
45:51–9.
11. Akbarnia A, Behzadmehr A. Appl Therm Eng. 2007;
27:1327–37.
12. Mirmasoumi S, Behzadmehr A. Int J Heat Fluid Flow. 2008;
29:557–66.
13. Muthtamilselvan M, Kandaswamy P, Lee J. Comm Nonlinear Sci Numer Simul. 2010; 15:1501–10.
14. Talebi F, Mahmoudi AH, Shahi M. Int Comm Heat Mass Transf. 2010; 37:79–90.
15. Hemanth KD, Patel HE, Kumar VRR, Sundararajan T, Pradeep T, Das SK. Model for heat conduction in nanoflu- ids. Phys Rev Lett. 2004; 93:144301–1–4.
16. Patel HE, Pradeep T, Sundararajan T, Dasgupta A, Dasgupta N, Das SK. A micro convection model for thermal con- ductivity of nanofluid. Pramana-Journal of Physics. 2005;
65:863–69.
17. Aydin O. Aiding and opposing mechanisms of mixed con- vection in a shearand buoyancy-driven cavity. Int Commun Heat Mass Transfer. 1999; 26:1019–28.
18. Santra AK, Sen S, Chakraborty N. Study of heat transfer in a differentially heated square cavity using copper–water nanofluid. International Journal of Thermal Science. 2008;
48:1113–22.
19. Sharif MAR. Laminar mixed convection in shallow inclined driven cavities with hot moving lid on top and cooled from bottom. Appl Therm Eng. 2007; 27:1036–42.
20. Morzynski M, Popiel CO. Laminar heat transfer in a two- dimensional cavity covered by a moving wall. Numer Heat Transfer. 1988; 13:265–73.
21. Freitas CJ, Street RL. Non-linear transport phenomena in a complex recirculation flow: a numerical investigation. Int J Numer Methods Fluids. 1988; 8:769–802.
22. Iwatsu R, Hyun JM, Kuwahara K. Convection in a differen- tially-heated square cavity with a torsionally-oscillating lid.
Int J Heat Mass Transfer. 1992; 35:1069–76.
23. Mohamad AA, Viskanta R. Stability of lid-driven shallow cavity heated from below. Int J Heat Mass Transfer. 1989;
32:2155–66.
24. Jang SP, Lee JH, Hwang KS, Choi SUS. Particle concentra- tion and tube size dependence of viscosities of Al2O3-water nanofluids flowing through micro- and minitubes. Appl Phys Lett. 2007; 91:24–31.
25. Hamilton RL, Crosser OK. Thermal conductivity of hetero- geneous two component systems. Indus Eng Chem Fund.
1962;1:187–91.
26. Xu J, Yu B, Zou M, Xu P. A new model for heat conduction of nanofluids based on fractal distributions of nanoparti- cles. J Phys D. 2006; 39:4486–90.