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Free convection in a triangle cavity filled with a porous medium saturated with nanofluids with flush mounted heater on the wall
International Journal of Thermal Sciences 50 (2011) 2141e2153
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Free convection in a triangle cavity filled with a porous medium saturated with nanofluids with flush mounted heater on the wall Qiang Sun a, *, Ioan Pop b, * a b
Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore University of Cluj, Faculty of Mathematics, R-400082 Cluj-Napoca, CP 253, Romania
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 January 2011 Received in revised form 31 May 2011 Accepted 4 June 2011 Available online 7 July 2011
Steady-state free convection heat transfer behavior of nanofluids is investigated numerically inside a right-angle triangular enclosure filled with a porous medium. The flush mounted heater with finite size is placed on the left vertical wall. The temperature of the inclined wall is lower than the heater, and the rest of walls are adiabatic. The governing equations are obtained based on the Darcy’s law and the nanofluid model proposed by Tiwari and Das [1]. The transformed dimensionless governing equations were solved by finite difference method and solution for algebraic equations was obtained through Successive Under Relaxation method. Investigations with three types of nanofluids were made for different values of Rayleigh number Ra of a porous medium with the range of 10 Ra 1000, size of heater Ht as 0.1 Ht 0.9, position of heater Yp when 0.25 Yp 0.75, enclosure aspect ratio AR as 0.5 AR 1.5 and solid volume fraction parameter f of nanofluids with the range of 0.0 f 0.2. It is found that the maximum value of average Nusselt number is obtained by decreasing the enclosure aspect ratio and lowering the heater position with the highest value of Rayleigh number and the largest size of heater. It is further observed that the heat transfer in the cavity is improved with the increasing of solid volume fraction parameter of nanofluids at low Rayleigh number, but opposite effects appear when the Rayleigh number is high. Ó 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Porous media Nanofluid Free convection Triangular enclosure
1. Introduction Heat and fluid flow in cavities filled with porous media are wellknown natural phenomenon and have attracted interest of many researchers due to its many practical situations. Among these insulation materials, geophysics applications, building heating and cooling operations, underground heat pump systems, solar engineering and material science can be listed. These are reviewed in several books: Pop and Ingham [2], Bejan et al. [3], Ingham and Pop [4], Nield and Bejan [5], Vafai [6,7], Vadasz [8] and in the papers by Varol et al. [9], Varol et al. [10] and Basak et al. [11,12]. A technique for improving heat transfer is using solid particles in the base fluids, which has been used recently. The term nanofluid, first introduced by Choi [13], refers to fluids in which nanoscale particles are suspended in the base fluid. He suggested that introducing nanoparticles with higher thermal conductivity into the base fluid results in a higher thermal performance for the resultant nanofluid. It is expected that the presence of the
* Corresponding authors. E-mail addresses: [email protected] (Q. Sun), [email protected] (I. Pop). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.06.005
nanoparticles in the nanofluid increases its thermal conductivity and therefore, substantially enhances the heat transfer characteristics of the nanofluid [13]. Use of metallic nanoparticles with high thermal conductivity will increase the effective thermal conductivity of these types of fluid remarkably. However, the increase in the thermal conductivity depends on the shape, size and thermal properties of the solid particles [14]. It must, however, be noted that heat transfer enhancement by means of nanofluids is still a controversial issue. Contradictory studies have also been reported in the literature, which argue that the dispersion of nanoparticles in the base fluid may results in a considerable decrease in the heat transfer [15,16]. It has been demonstrated that the augmentation or mitigation of the heat transfer found in the numerical studies depends on the existing models used to predict the properties of the nanofluids [17,18]. Buongiorno [19] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He has shown that in the absence of turbulent effects, it is the Brownian diffusion and the thermophoresis that are important and he has written down conservation equations based on these two effects. There are several numerical and experimental studies on the forced and natural convection using nanofluids related with differentially
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Nomenclature AR Cp g h H Ht k K NuY Nu Ra T u, v U, V W x, y
enclosure aspect ration W/H specific heat at constant pressure (J kg1 K1) gravitational acceleration (m s2) size of heater (m) height of enclosure (m) dimensionless size of heater thermal conductivity (W m1 K1) permeability of porous medium (m2) local Nusselt number average Nusselt number Rayleigh number for porous medium dimensional fluid temperature (K) dimensional components of velocity (m s1) dimensionless components of velocity width of enclosure (m) dimensional coordinates
heated enclosures and we mention here those by Khanafer et al. [20], Maïga et al. [21], Tannehill et al. [22], Oztop and Abu-Nada [23], Muthtamilselvan et al. [24], Ghasemi and Aminossadati [25,26], Popa et al. [27], Mahmoudi et al. [28,29], etc. The book by Das et al. [30] and the review papers by Daungthongsuk and Wongwises [31], Ding et al. [32], Wang and Mujumdar [33,34], and Kakaç and Pramuanjaroenkij [35] present excellent collection of up to now published papers on nanofluids. It is obvious from the foregoing review that most of the studies are performed considering the water-based nanofluids in cavities. Very little research is performed considering a porous medium filled with nanofluids. Recently, Nield and Kuznetsov [36] have studied the Cheng and Minkowycz’s problem [37] for natural convective boundary layer flow over a vertical flat plate embedded in a porous medium filled with nanofluid taking into account the combined effects of heat and mass transfer in the presence of Brownian motion and thermophoresis as proposed by Buongiorno [19]. In another paper, Kuznetsov and Nield [38] have provided numerical solution to the problem of natural convective heat transfer in the boundary layer flow of a nanofluid past a vertical flat plate embedded in a viscous (Newtonian) fluid using the same
X, Y yp Yp
dimensionless coordinates dimensional location of heater center (m) dimensionless location of heater center
Greek symbols a fluid thermal diffusivity (m2 s1) b thermal expansion coefficient (K1) f solid volume fraction of nanofluid Q dimensionless fluid temperature J dimensionless flow stream function r density (kg m3) m dynamic viscosity (kg m1 s1) Subscripts nf nanofluid f fluid s solid
Buongiorno’s model [19]. Also, Khan and Pop [39], and Bachok et al. [40] have studied the steady boundary layer flow of a nanofluid past a stretching surface using Buongiorno’s nanofluid model [19], while Ahmad and Pop [41] have considered the steady mixed convection boundary layer flow over a vertical flat plate embedded in a porous medium saturated with a nanofluid using the nanofluid model proposed by Tiwari and Das [1]. However, Buongiorno [19] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He considered in turn seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity settling. In the present study, the problem of steady free convection heat transfer in a triangular enclosure filled with a nanofluid, where the enclosure, with a heater on its vertical wall and filled with a water Cu nanofluid considered by Ghasemi and Aminossadati [25] has been extended to a triangular cavity filled with a porous medium and saturated by nanofluids using the nanofluid model proposed by Tiwari and Das [1]. Three different types of nanoparticles are considered, namely Cu, Al2O3 and TiO2. The present study has been motivated by the need to determine the detailed flow and temperature characteristics as well al the local and average Nusselt numbers. To the best knowledge of the authors, no study which considers this problem has yet been reported in the literature. As such, the focus of this paper is to examine the effects of pertinent parameters such as Rayleigh number for a porous medium, solid volume fraction parameter of nanofluids, heater size and position, and enclosure aspect ratio.
2. Physical model and basic governing equations The physical domain for the free convection in a triangle cavity is sketched in Fig. 1 with dimensions and boundary conditions
Table 1 Thermalephysical properties of fluid and nanoparticles [23].
Fig. 1. Sketch of the physical model.
Physical properties
Fluid phase (water)
Cu
Al2O3
TiO2
Cp (J kg1 K1) r (kg m3) k (W m1 K1) a 107 (m2 s1) b 105 (K1)
4179 997.1 0.613 1.47 21
385 8933 400 1163.1 1.67
765 3970 40 131.7 0.85
686.2 4250 8.9538 30.7 0.9
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
10.5
Table 2 Comparison of average Nusselt number Nu for f ¼ 0.0 (pure fluidewater) when AR ¼ 1.0 and Ra ¼ 1 103.
NuY|Y=0.5
10
9.5
9
8.5
8 31
2143
51
71
91
111
131
151
171
number of grid points Fig. 2. Dependence of the local Nusselt number NuY at Yp ¼ 0.5 on the number of grid points for Cuewater nanofluid when Ra ¼ 1000, f ¼ 0.1, AR ¼ 1.0, Ht ¼ 0.8 and Yp ¼ 0.5.
according to different positions of the heater. In this model, heater size can be changed, which is denoted by h. The position of the heater is expressed by yp, which is measured from the middle point of the heater to the bottom wall of the cavity. For the enclosure
Literature
Nu
Bejan [43] Goyeau et al. [44] Baytas and Pop [45] Varol et al. [9,10] Present study
15.800 13.470 14.060 13.564 13.575
(cavity), length of the bottom wall and height of the vertical wall are shown by W and H, respectively. Inclined wall of the cavity has constant cold temperature Tcold, while the heater has constant hot temperature Thot, and the remained walls are adiabatic. The fluid within the cavity is a water-based nanofluid. Three different types of nanoparticles are considered, namely Cu, Al2O3 and TiO2, which thermalephysical properties are listed in Table 1. In this study, the nanofluid flow is set to be incompressible and laminar. It is presumed that the base fluid (i.e. water) and the nanoparticles are in thermal equilibrium and no slip occurs between them. Meanwhile, the Boussinesq approximation is employed and homogeneity and local thermal equilibrium in the porous medium is assumed. It is also assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. As a result, in keeping with the Darcy’s law and adopting the nanofluid model proposed by Tiwari and Das [1], the basic continuity, momentum, and energy equations can be written as
Fig. 3. (a, c) Local NuY profile for Cuewater nanofluid; (b, d) Dependence of average Nusselt numbers on Rayleigh number Ra for different nanoparticles when f ¼ 0.1, AR ¼ 1.0 and Yp ¼ 0.5.
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Fig. 4. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when f ¼ 0.1, Ht ¼ 0.8, AR ¼ 1.0 and Yp ¼ 0.5.
vu vv þ ¼ 0; vx vy
mnf K
mnf K
vp u ¼ ; vx v ¼
i vp h þ frs bs þ ð1 fÞrf bf gðT Tcold Þ; vy
(1)
(2)
(3)
an advective term and a Forchheimer quadratic drag term do not appear in the Darcy’s Eqs. (2) and (3). The viscosity of the nanofluid mnf can be approximated as viscosity of a base fluid if containing dilute suspension of fine spherical particles, which is given by Brinkman [42] as
mnf ¼
mf ð1 fÞ2:5
:
(5)
and
Meanwhile, the thermal diffusivity of the nanofluid is defined by Oztop and Abu-Nada [23] as
! vT vT v2 T v2 T : u þv þ ¼ anf vx vy vx2 vy2
anf ¼ nf ; rCp nf
k
(4)
In Eqs. (1)e(4), x and y are Cartesian coordinates measured along the horizontal and vertical walls of the cavity respectively, u and v are the velocity components along the x- and y- axes respectively, T is the fluid temperature, p is the fluid pressure, g is the gravity acceleration, K is the permeability of the porous medium, f is the solid volume fraction of the nanofluid, bf and bs are the coefficients of thermal expansion of the fluid and of the solid fractions respectively, rf and rs are the densities of the fluid and of the solid fractions respectively, mf is the viscosity of the fluid fraction, mnf is the viscosity of the nanofluid, and anf is the thermal diffusivity of the nanofluid. The flow is assumed to be slow so that
(6)
where ðrCp Þnf is the heat capacity of the nanofluid, which is given by Khanafer et al. [20] as
rCp
nf
¼ ð1 fÞ rCp f þf rCp s ;
(7)
and knf stands for the effective thermal conductivity of the nanofluid that can be obtained according to the Maxwell-Garnetts model
ks þ 2kf 2f kf ks knf : ¼ kf ks þ 2kf þ f kf ks
(8)
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
a
3.5
b
2145
3.8
3.6
3.2
3.4
Nu
Nu
2.9 3.2
2.6 3 2.3
2
2.8
0
0.05
0.1
φ
0.15
2.6
0.2
0
0.05
Ra = 50
c
0.1
φ
0.15
0.2
0.15
0.2
Ra = 100
d
7.9
12 11.5
7.5
11 10.5
Nu
Nu
7.1
6.7
10 9.5
6.3 9 5.9
5.5
8.5
0
0.05
0.1
φ
0.15
0.2
8
0
0.05
Ra = 500
0.1
φ
Ra = 1000
Fig. 5. Dependence of average Nusselt numbers on solid volume fraction parameter f for different nanoparticles when Ht ¼ 0.4, AR ¼ 1.0 and Yp ¼ 0.5.
In the above two equations, kf and ks are the thermal conductivities of the fluid and the solid fractions respectively, ðrCp Þf stands the heat capacity of the fluid fraction, and ðrCp Þs is the heat capacity of the solid fraction respectively. The following dimensionless variables
x y H H T Tcold X ¼ ; Y ¼ ; U ¼ u; V ¼ v; Q ¼ ; af af Thot Tcold H H
(9)
are introduced, and the dimensionless stream function J is defined in the usual way as
U ¼ vJ=vY;
V ¼ vJ=vX:
ð1 fÞ2:5
! i vQ h v2 J v2 J ¼ Ra ð1 fÞ þ f rs =rf bs =bf þ ; vX vX 2 vY 2 (11)
! anf v2 Q v2 Q vJ vQ vJ vQ ; þ ¼ af vX 2 vY 2 vY vX vX vY
8 Alongthe vertical wall : on heater; Q ¼ 1; J ¼ 0: > > < Along the verticalwall : on the unheated part;vQ=vX ¼ 0; J ¼ 0: On the adiabaticðbottomÞwall;vQ=vY ¼ 0; J ¼ 0: > > : On the inclinedwall; Q ¼ 0; J ¼ 0: In the meantime, the length and position of the heater(13) are respectively nondimensionalized as
(10)
When Eqs. (9) and (10) are substituted into Eqs. (2)e(4), and the pressure terms are eliminated from Eqs. (2) and (3), it is found that
1
of the fluid and the impermeability of the cavity walls, non-slip condition is composed along the whole boundary of the calculation domain in this model. Consequently, the corresponding dimensionless boundary conditions are converted as
Ht ¼
yp h and Yp ¼ : H H
(14)
The definitions for local Nusselt number NuY on the left vertical wall and average Nusselt number Nu are
k vQ NuY ¼ nf ; kf vX X¼0
(15)
and
(12)
where Ra is the Rayleigh number for a porous medium, which is defined as Ra ¼ gK rf bf ðThot Tcold ÞH=ðmf af Þ. Due to the viscosity
Z1 Nu ¼
NuY dY; 0
respectively.
(16)
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a
6.5
b
6.3
6 6 5.5
Nu
Nu
5.7 5
4.5
5.1
4
3.5
5.4
4.8
0
0.05
0.1
φ
0.15
0.2
0
0.05
Ra = 50
c
φ
0.15
0.2
0.15
0.2
Ra = 100
d
9.9
14
13.5
9.6
13
Nu
9.3
Nu
0.1
9
8.7
12.5
12
8.4
11.5
8.1 0
0.05
0.1
φ
0.15
0.2
Ra = 500
11
0
0.05
0.1
φ
Ra = 1000
Fig. 6. Dependence of average Nusselt numbers on solid volume fraction parameter f for different nanoparticles when Ht ¼ 0.8, AR ¼ 1.0 and Yp ¼ 0.5.
3. Numerical procedure Finite difference method [22] was adopted to solve numerically the governing Eqs. (11) and (12) together with the boundary conditions in Eq. (13). The diffusion terms within Eqs. (11) and (12) were replaced by the second order central differencing schemes, while the second order upwind differencing scheme was chosen to approximate the convective term in order to make the numerical procedure stable. The solution for the corresponding linear algebraic equations was obtained through the Successive Under Relaxation (SUR) method. The temperature function Q and the stream function J were calculated through iteration when the initial guess was made. The iteration process is terminated when the following criterion is satisfied
PM
i¼1
nþ1 ci;j cni;j 106 ; PN nþ1 j ¼ 1 ci;j
PN
PM
i¼1
j¼1
points were examined from 31 31 to 171 171. As can been seen in Fig. 2, the calculation results become mesh independent when the number of grid points are higher than 91 91. Meanwhile, in order to verify the rationality and accuracy of the results, the above numerical simulation procedure was tested with the classical natural convection heat transfer problem in a differentially heated square porous enclosure (AR ¼ 1.0) in context with the pure fluid (i.e. water), which is equivalent with f ¼ 0.0 in the presented model. The obtained results for the average Nusselt number, as defined in Eq. (16), were compared with those given by different authors, as listed in Table 2. It can be seen that the results obtained here are in good agreement with the results that have been reported. 4. Results and analysis
n 1;
(17)
in which c reflects either Q or J, M and N are the number of grid points in the X- and Y- directions respectively, and n is the iteration step. In this study, the grid points with uniform spaced mesh were generated along both X- and Y- directions. The number of the grid
Numerical simulation results on flow field, temperature distribution, and Nusselt number are presented in this section. The effects of different values of Rayleigh number Ra of a porous medium, solid volume fraction parameter f of nanofluid, types of the nanoparticles, heater size Ht, heater position Yp and enclosure aspect ratio AR are analyzed. Enclosure aspect ratio AR changes from 0.5 to 1.5. Dimensionless size of the heater Ht varies from
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
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Fig. 7. Dependence of average Nusselt numbers on aspect ratio AR for different nanoparticles when f ¼ 0.1 and Yp ¼ 0.5.
0.1 to 0.9, and dimensionless heater position Yp varies from 0.25 to 0.75. We assume, as in Oztop and Abu-Nada [23], that the range for solid volume fraction parameter f of nanoparticles is taken from 0.0 to 0.2. We assume also that Rayleigh number Ra for porous medium varies from 10 to 1000. Rayleigh number Ra is a very important parameter that has effects on heat transfer within a porous medium. The results in Fig. 3a and c present the distribution of local Nusselt number NuY along the heater for different values of Ra. From Fig. 3b and d, one can see that average Nusselt number Nu is improved when the Rayleigh number is increased. This conclusion is supported by Varol et al. [9] for pure fluid. Meanwhile, Fig. 3b and d also show that the average Nusselt number of Cuewater nanofluid is the highest among the three kinds of nanofluids, which is probably due to that the thermal conductivity of nanoparticle Cu is higher than those of the other two nanoparticles (Al2O3 and TiO2). Fig. 4 presents the streamlines and isotherms for Cuewater nanofluid under different values of the Rayleigh number. Fig. 4a, b and c demonstrate that a single circulation flow cell is formed in the clockwise direction for all values of Ra that have been tested. When the Rayleigh number increases, the flow convection is strengthened, the width of flow cell is increased as egg-shaped cell is extended to triangle-shaped cell, and the boundary layers become more significant. Solid volume fraction parameter f is a key factor to study how nanoparticles affect the heat transfer of nanofluids. Figs. 5 and 6 present the average Nusselt number for different values of the Rayleigh number and the solid volume fraction parameter. Once again, the average Nusselt number is elevated when the Rayleigh
number increases and nanoparticle Cu is used. However, the effects of solid volume fraction parameter f on the heat transfer of nanofluids are complicated. When Rayleigh number Ra is low, the average Nusselt number increases as solid volume fraction parameter f increases (Figs. 5a and 6a, b). However, elevating f has adverse effects on the heat transfer of nanofluids when Ra is high, as shown in Figs. 5c, d and 6d. One possible explanation for the above phenomena is as following. The density of nanofluid is defined by Oztop and Abu-Nadaas [23] as
rnf ¼ frs þ ð1 fÞrf :
(18)
Based on the definitions in Eqs. (5) and (18), and considering the physical values listed in Table 1, the density and viscosity of nanofluids are elevated relative to the pure fluid (water) as f increases, which means that the inertial and viscous resistances for nanofluids are higher than those for the pure fluid (water). When the Rayleigh number is low, the flow convection is insignificant. The heat transfer in the cavity is dominated by conduction. Along the increase of f, the thermal conductivity of nanofluids is increased relative to that of pure fluid (water) due to the high thermal conductivity of nanoparticles. Consequently, the heat transfer of nanofluids is improved compared with pure fluid (water), as depicted in Figs. 5a and 6a, b. As the Rayleigh number increases, the flow convection becomes stronger, and the heat transfer in the cavity is improved, as displayed in Fig. 3b and d. However, as f increases, the elevated inertial and viscous resistances of
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Fig. 8. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1 and Yp ¼ 0.5.
nanofluids would compromise the flow convection, which is able to lead to the adverse effects on the heat transfer in the nanofluids compared with the pure fluid (water). Nevertheless, when f is upon some level, the positive effects of increased thermal conductivity of nanofluids can appear again to overtake the adverse effects of the elevated inertial and viscous resistances, which results in the value of average Nusselt number Nu to rise after falling. The curves in Fig. 5b and solid line in Fig. 6c reflect the above analysis. When Rayleigh number Ra is high (Ra ¼ 1000), convection dominates the fluid movement. Under such a circumstance, heat transfer, for all three types of nanofluids, is decreased along the increasing of the solid volume fraction parameter due to the elevated inertial and viscous resistances of the nanofluids, as shown in Figs. 5c, d and 6d. It is possible that when solid volume fraction parameter f is above 0.2, sedimentation sets in and the fluid loses the regular (Darcian) character as it was concluded by Muthtamilselvan et al. [24] for a copperewater nanofluids in a lid-driven enclosure. Enclosure aspect ratio AR is another feature that has influence on the heat transfer within the cavity. Normally, the value of
average Nusselt number Nu is reduced along the increase of AR, as shown in Fig. 7a, b and d. This is understandable because when AR increases, the distance between the heater and cold wall of the cavity is enlarged, which can lead to the decrease in temperature gradients. However, the effects of AR on Nu are also affected by other physical parameters. For example, when Ra ¼ 1000, the heat transfer within the cavity with small size of heater (Ht ¼ 0.4) falls after rising along the increasing of AR, as shown in Fig. 7c. Nevertheless, when the heater size is enlarged as Ht ¼ 0.8, the heat transfer inside the enclosure again decreases consistently as AR increases (Fig. 7d). The comparisons for streamlines and isotherms of Cuewater nanofluid with different values of AR are presented in Figs. 8 and 9. The heat transfer can also be affected by the position of heater. When Ra ¼ 1000, convection leads the flow movement within the cavity. Considering the shape of the enclosure and the boundary conditions, when the position of heater is high, for example Yp ¼ 0.75, the flow and heat transfer cannot be fully developed inside the entire enclosure, as shown in Fig. 12. Under this circumstance, heat transfer within the cavity is decreased along the increase of Yp, as displayed in Figs. 10b and 11b. Nevertheless, when Ra is low as
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
Fig. 9. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1 and Yp ¼ 0.5.
a
b
Fig. 10. Dependence of average Nusselt numbers on center position of heater Yp for different nanoparticles when f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.
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Fig. 11. Local Nusselt number profile of Cuewater nanofluid for different center position of heater Yp when f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.
Ra ¼ 1000, the flow convection is insignificant, and the heat transfer inside the cavity is mainly performed by conduction, as presented in Fig. 13. In such a case, considering the shape of the enclosure, when the position of heater is high, the distance between the heater and
cold wall is shortened, which leads to the improvement of the heat transfer inside the cavity, as shown in Figs. 10a and 11a. It is easily expected that enlarging the size of heater Ht is able to improve the heat transfer inside the enclosure. Fig. 14 reflects that
Fig. 12. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
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Fig. 13. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 100, f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.
expectation for three types of nanofluids. It is also can be seen that the difference of the average Nusselt number becomes significant for larger size of the heater since flow convection becomes stronger. These results are consistent with the findings reported by Oztop
and Abu-Nada [23] for the convection in a rectangular cavity of homogeneous nanofluids. In the meantime, the streamlines and isotherms of Cuewater nonafluid for different values of Ht are plotted in Fig. 15.
Fig. 14. Dependence of average Nusselt numbers on size of heater Ht for different nanoparticles when f ¼ 0.1, AR ¼ 1.0, and Yp ¼ 0.5.
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Fig. 15. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1, AR ¼ 1.0, and Yp ¼ 0.5.
5. Conclusions A numerical study has been performed in this paper to investigate the free convection heat transfer problem in a partly heated triangle cavity filled with a porous medium saturated with nanofluids. The governing equations were solved by finite difference method. It is found that the maximum value of the average Nusselt number can be achieved for the highest Rayleigh number, the largest heater size. Meanwhile, under that circumstance, lowering the heater position and decreasing the aspect ratio of the enclosure are beneficial to the heat transfer in the cavity. Among the three types of nanofluids, the highest value of the average Nusselt number is obtained when using Cu nanoparticles. The effects of the solid volume fraction parameter on the heat transfer of nanofluids within a porous medium are complex. When the Rayleigh number is low, increasing the value of the solid volume fraction parameter of nanofluids can improve the value of the average Nusselt number, while if Rayleigh number is high, elevating the solid volume fraction parameter of nanofluids reduces the value of the average Nusselt number. An optimization investigation might be required in future to search for the best value of the solid volume fraction parameter of nanofluids to achieve the most efficient way of improving heat transfer within a porous medium, which is nevertheless beyond the scope of this study. It is worth mentioning to this end that the study of nanofluids is still at its early stage and it
seems that it is very difficult to have a precise idea on the way the use of nanoparticles acts in convective flow in porous media and complementary work is necessary to understand the heat transfer characteristics of nanofluids and identify new applications for these fluids. References [1] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transf. 50 (2007) 2002e2018. [2] I. Pop, D. Ingham, Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media. Pergamon, Oxford, 2001. [3] A. Bejan, I. Dincer, S. Lorente, A.F. Miguel, A.H. Reis, Porous and Complex Flow Structures in Modern Technologies. Springer, New York, 2004. [4] D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media III, Elsevier, Oxford, 2005. [5] D. Nield, A. Bejan, Convection in Porous Media, third ed. Springer, New York, 2006. [6] K. Vafai, Handbook of Porous Media. Taylor & Francis, New York, 2005. [7] K. Vafai, Porous Media: Applications in Biological Systems and Biotechnology. CRC Press, New York, 2010. [8] P. Vadasz, Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York, 2008. [9] Y. Varol, H.F. Oztop, A. Varol, Free convection in porous media filled rightangle triangular enclosures, Int. Comm. Heat Mass Tran. 33 (2006) 1190e1197. [10] Y. Varol, H.F. Oztop, I. Pop, Natural convection in porous media-filled triangular enclosure with a conducting thin fin on the hot vertical wall, Proc. I. M. E. Part C, J. Mech. Eng. Sci. 222 (2008) 1735e1743.
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153 [11] T. Basak, S. Roy, D. Ramakrishna, I. Pop, Visualization of heat transport due to natural convection for hot materials confined within two entrapped porous triangular cavities via heatline concept, Int. J. Heat Mass Transf. 53 (2010) 2100e2112. [12] T. Basak, S. Roy, D. Ramakrishna, I. Pop, Visualization of heat transport during natural convection within porous triangular cavities via heatline approach, Numer. Heat Transfer, Part A: Appl. 57 (2010) 431e452. [13] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: Proc. 1995 ASME Int. Mech. Engng. Congress and Exposition (1995), pp. 99e105 66, ASME, FED 231/MD, San Franciscos, USA. [14] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol- based nanofluids containing copper nanoparticles, Appl. Phys. Lett. 78 (2001) 718e720. [15] N. Putra, W. Roetzel, S.K. Das, Natural convection of nano-fluids, Heat Mass Transf. 39 (2003) 775e784. [16] D. Wen, Y. Ding, Formulation of nanofluids for natural convective heat transfer applications, Int. J. Heat Fluid Flow 26 (2005) 855e864. [17] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transf. 51 (2008) 4506e4516. [18] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of Al2O3ewater nanofluid on heat transfer enhancement in natural convection, Int. J. Heat Mass Transf. 30 (2009) 679e690. [19] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240e250. [20] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 3639e3653. [21] S.E.B. Maïga, C.T. Nguyen, N. Galanis, G. Roy, Heat transfer behaviours of nanofluids in a uniformly heated tube, Superlattice. Microst. 35 (2004) 543e557. [22] J.C. Tannehill, D.J. Anderson, R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, second ed. Taylor & Francis, United States of America, 1997. [23] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326e1336. [24] M. Muthtamilselvan, P. Kandaswamy, J. Lee, Heat transfer enhancement of copperewater nanofluids in a lid-driven enclosure, Commun. Nonlinear. Sci. Numer. Simul. 15 (2010) 1501e1510. [25] B. Ghasemi, S.M. Aminossadati, Brownian motion of nanoparticles in a triangular enclosure with natural convection, Int. J. Therm. Sci. 49 (2010) 931e940. [26] B. Ghasemi, S.M. Aminossadati, Mixed convection in a lid-driven triangular enclosure filled with nanofluids, Int. Comm. Heat Mass Tran. 37 (2010) 1142e1148.
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[27] C.V. Popa, S. Fohanno, C.T. Nguyen, G. Polidori, On heat transfer in external natural convection flows using two nanofluids, Int. J. Therm. Sci. 49 (2010) 901e908. [28] A.H. Mahmoudi, M. Shahi, F. Talebi, Effect of inlet and outlet location on the mixed convective cooling inside the ventilated cavity subjected to an external nanofluid, Int. Comm. Heat Mass Tran. 37 (2010) 1158e1173. [29] A.H. Mahmoudi, M. Shahi, A.H. Raouf, A. Ghasemian, Numerical study of natural convection cooling of horizontal heat source mounted in a square cavity filled with nanofluid, Int. Comm. Heat Mass Tran. 37 (2010) 1135e1141. [30] S.K. Das, S.U.S. Choi, W. Yu, T. Pradet, Nanofluids: Science and Technology. Wiley, New York, 2007. [31] W. Daungthongsuk, S. Wongwises, A critical review of convective heat transfer nanofluids, Renew. Sustain. Energ. Rev. 11 (2007) 797e817. [32] Y. Ding, H. Chen, L. Wang, C.-Y. Yang, W. He, W.P. Yang, L. Lee, R. Zhang, R. Huo, Heat transfer intensification using nanofluids, KONA 25 (2007) 23e38. [33] X.-Q. Wang, A.S. Mujumdar, A review on nanofluids - part i: theoretical and numerical investigations, Braz. J. Chem. Eng. 25 (2008) 613e630. [34] X.-Q. Wang, A.S. Mujumdar, A review on nanofluids - part ii: experiments and applications, Braz. J. Chem. Eng. 25 (2008) 631e648. [35] S. Kakaç, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transf. 52 (2009) 3187e3196. [36] D.A. Nield, A.V. Kuznetsov, The ChengeMinkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transf. 52 (2009) 5792e5795. [37] P. Cheng, W.J. Minkowycz, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res. 82 (1977) 2040e2044. [38] A. Kuznetsov, D. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2010) 243e247. [39] W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transf. 53 (2010) 2477e2483. [40] N. Bachok, A. Ishak, I. Pop, Boundary-layer flow of nanofluids over a moving surface in a flowing fluid, Int. J. Therm. Sci. 49 (2010) 1663e1668. [41] S. Ahmad, I. Pop, Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, Int. Comm. Heat Mass Tran. 37 (2010) 987e991. [42] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571. [43] A. Bejan, On the boundary layer regime in a vertical enclosure filled with a porous medium, Lett. Heat Mass Tran. 6 (1979) 93e102. [44] B. Goyeau, J.P. Songbe, D. Gobin, Numerical study of double-diffusive natural convection in a porous cavity using the darcy-brinkman formulation, Int. J. Heat Mass Transf. 39 (1996) 1363e1378. [45] A.C. Baytas, I. Pop, Free convection in a square porous cavity using a thermal nonequilibrium model, Int. J. Therm. Sci. 41 (2002) 861e870.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Free convection in a triangle cavity filled with a porous medium saturated with nanofluids with flush mounted heater on the wall Qiang Sun a, *, Ioan Pop b, * a b
Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore University of Cluj, Faculty of Mathematics, R-400082 Cluj-Napoca, CP 253, Romania
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 January 2011 Received in revised form 31 May 2011 Accepted 4 June 2011 Available online 7 July 2011
Steady-state free convection heat transfer behavior of nanofluids is investigated numerically inside a right-angle triangular enclosure filled with a porous medium. The flush mounted heater with finite size is placed on the left vertical wall. The temperature of the inclined wall is lower than the heater, and the rest of walls are adiabatic. The governing equations are obtained based on the Darcy’s law and the nanofluid model proposed by Tiwari and Das [1]. The transformed dimensionless governing equations were solved by finite difference method and solution for algebraic equations was obtained through Successive Under Relaxation method. Investigations with three types of nanofluids were made for different values of Rayleigh number Ra of a porous medium with the range of 10 Ra 1000, size of heater Ht as 0.1 Ht 0.9, position of heater Yp when 0.25 Yp 0.75, enclosure aspect ratio AR as 0.5 AR 1.5 and solid volume fraction parameter f of nanofluids with the range of 0.0 f 0.2. It is found that the maximum value of average Nusselt number is obtained by decreasing the enclosure aspect ratio and lowering the heater position with the highest value of Rayleigh number and the largest size of heater. It is further observed that the heat transfer in the cavity is improved with the increasing of solid volume fraction parameter of nanofluids at low Rayleigh number, but opposite effects appear when the Rayleigh number is high. Ó 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Porous media Nanofluid Free convection Triangular enclosure
1. Introduction Heat and fluid flow in cavities filled with porous media are wellknown natural phenomenon and have attracted interest of many researchers due to its many practical situations. Among these insulation materials, geophysics applications, building heating and cooling operations, underground heat pump systems, solar engineering and material science can be listed. These are reviewed in several books: Pop and Ingham [2], Bejan et al. [3], Ingham and Pop [4], Nield and Bejan [5], Vafai [6,7], Vadasz [8] and in the papers by Varol et al. [9], Varol et al. [10] and Basak et al. [11,12]. A technique for improving heat transfer is using solid particles in the base fluids, which has been used recently. The term nanofluid, first introduced by Choi [13], refers to fluids in which nanoscale particles are suspended in the base fluid. He suggested that introducing nanoparticles with higher thermal conductivity into the base fluid results in a higher thermal performance for the resultant nanofluid. It is expected that the presence of the
* Corresponding authors. E-mail addresses: [email protected] (Q. Sun), [email protected] (I. Pop). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.06.005
nanoparticles in the nanofluid increases its thermal conductivity and therefore, substantially enhances the heat transfer characteristics of the nanofluid [13]. Use of metallic nanoparticles with high thermal conductivity will increase the effective thermal conductivity of these types of fluid remarkably. However, the increase in the thermal conductivity depends on the shape, size and thermal properties of the solid particles [14]. It must, however, be noted that heat transfer enhancement by means of nanofluids is still a controversial issue. Contradictory studies have also been reported in the literature, which argue that the dispersion of nanoparticles in the base fluid may results in a considerable decrease in the heat transfer [15,16]. It has been demonstrated that the augmentation or mitigation of the heat transfer found in the numerical studies depends on the existing models used to predict the properties of the nanofluids [17,18]. Buongiorno [19] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He has shown that in the absence of turbulent effects, it is the Brownian diffusion and the thermophoresis that are important and he has written down conservation equations based on these two effects. There are several numerical and experimental studies on the forced and natural convection using nanofluids related with differentially
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Nomenclature AR Cp g h H Ht k K NuY Nu Ra T u, v U, V W x, y
enclosure aspect ration W/H specific heat at constant pressure (J kg1 K1) gravitational acceleration (m s2) size of heater (m) height of enclosure (m) dimensionless size of heater thermal conductivity (W m1 K1) permeability of porous medium (m2) local Nusselt number average Nusselt number Rayleigh number for porous medium dimensional fluid temperature (K) dimensional components of velocity (m s1) dimensionless components of velocity width of enclosure (m) dimensional coordinates
heated enclosures and we mention here those by Khanafer et al. [20], Maïga et al. [21], Tannehill et al. [22], Oztop and Abu-Nada [23], Muthtamilselvan et al. [24], Ghasemi and Aminossadati [25,26], Popa et al. [27], Mahmoudi et al. [28,29], etc. The book by Das et al. [30] and the review papers by Daungthongsuk and Wongwises [31], Ding et al. [32], Wang and Mujumdar [33,34], and Kakaç and Pramuanjaroenkij [35] present excellent collection of up to now published papers on nanofluids. It is obvious from the foregoing review that most of the studies are performed considering the water-based nanofluids in cavities. Very little research is performed considering a porous medium filled with nanofluids. Recently, Nield and Kuznetsov [36] have studied the Cheng and Minkowycz’s problem [37] for natural convective boundary layer flow over a vertical flat plate embedded in a porous medium filled with nanofluid taking into account the combined effects of heat and mass transfer in the presence of Brownian motion and thermophoresis as proposed by Buongiorno [19]. In another paper, Kuznetsov and Nield [38] have provided numerical solution to the problem of natural convective heat transfer in the boundary layer flow of a nanofluid past a vertical flat plate embedded in a viscous (Newtonian) fluid using the same
X, Y yp Yp
dimensionless coordinates dimensional location of heater center (m) dimensionless location of heater center
Greek symbols a fluid thermal diffusivity (m2 s1) b thermal expansion coefficient (K1) f solid volume fraction of nanofluid Q dimensionless fluid temperature J dimensionless flow stream function r density (kg m3) m dynamic viscosity (kg m1 s1) Subscripts nf nanofluid f fluid s solid
Buongiorno’s model [19]. Also, Khan and Pop [39], and Bachok et al. [40] have studied the steady boundary layer flow of a nanofluid past a stretching surface using Buongiorno’s nanofluid model [19], while Ahmad and Pop [41] have considered the steady mixed convection boundary layer flow over a vertical flat plate embedded in a porous medium saturated with a nanofluid using the nanofluid model proposed by Tiwari and Das [1]. However, Buongiorno [19] noted that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (that he calls the slip velocity). He considered in turn seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity settling. In the present study, the problem of steady free convection heat transfer in a triangular enclosure filled with a nanofluid, where the enclosure, with a heater on its vertical wall and filled with a water Cu nanofluid considered by Ghasemi and Aminossadati [25] has been extended to a triangular cavity filled with a porous medium and saturated by nanofluids using the nanofluid model proposed by Tiwari and Das [1]. Three different types of nanoparticles are considered, namely Cu, Al2O3 and TiO2. The present study has been motivated by the need to determine the detailed flow and temperature characteristics as well al the local and average Nusselt numbers. To the best knowledge of the authors, no study which considers this problem has yet been reported in the literature. As such, the focus of this paper is to examine the effects of pertinent parameters such as Rayleigh number for a porous medium, solid volume fraction parameter of nanofluids, heater size and position, and enclosure aspect ratio.
2. Physical model and basic governing equations The physical domain for the free convection in a triangle cavity is sketched in Fig. 1 with dimensions and boundary conditions
Table 1 Thermalephysical properties of fluid and nanoparticles [23].
Fig. 1. Sketch of the physical model.
Physical properties
Fluid phase (water)
Cu
Al2O3
TiO2
Cp (J kg1 K1) r (kg m3) k (W m1 K1) a 107 (m2 s1) b 105 (K1)
4179 997.1 0.613 1.47 21
385 8933 400 1163.1 1.67
765 3970 40 131.7 0.85
686.2 4250 8.9538 30.7 0.9
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
10.5
Table 2 Comparison of average Nusselt number Nu for f ¼ 0.0 (pure fluidewater) when AR ¼ 1.0 and Ra ¼ 1 103.
NuY|Y=0.5
10
9.5
9
8.5
8 31
2143
51
71
91
111
131
151
171
number of grid points Fig. 2. Dependence of the local Nusselt number NuY at Yp ¼ 0.5 on the number of grid points for Cuewater nanofluid when Ra ¼ 1000, f ¼ 0.1, AR ¼ 1.0, Ht ¼ 0.8 and Yp ¼ 0.5.
according to different positions of the heater. In this model, heater size can be changed, which is denoted by h. The position of the heater is expressed by yp, which is measured from the middle point of the heater to the bottom wall of the cavity. For the enclosure
Literature
Nu
Bejan [43] Goyeau et al. [44] Baytas and Pop [45] Varol et al. [9,10] Present study
15.800 13.470 14.060 13.564 13.575
(cavity), length of the bottom wall and height of the vertical wall are shown by W and H, respectively. Inclined wall of the cavity has constant cold temperature Tcold, while the heater has constant hot temperature Thot, and the remained walls are adiabatic. The fluid within the cavity is a water-based nanofluid. Three different types of nanoparticles are considered, namely Cu, Al2O3 and TiO2, which thermalephysical properties are listed in Table 1. In this study, the nanofluid flow is set to be incompressible and laminar. It is presumed that the base fluid (i.e. water) and the nanoparticles are in thermal equilibrium and no slip occurs between them. Meanwhile, the Boussinesq approximation is employed and homogeneity and local thermal equilibrium in the porous medium is assumed. It is also assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. As a result, in keeping with the Darcy’s law and adopting the nanofluid model proposed by Tiwari and Das [1], the basic continuity, momentum, and energy equations can be written as
Fig. 3. (a, c) Local NuY profile for Cuewater nanofluid; (b, d) Dependence of average Nusselt numbers on Rayleigh number Ra for different nanoparticles when f ¼ 0.1, AR ¼ 1.0 and Yp ¼ 0.5.
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Fig. 4. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when f ¼ 0.1, Ht ¼ 0.8, AR ¼ 1.0 and Yp ¼ 0.5.
vu vv þ ¼ 0; vx vy
mnf K
mnf K
vp u ¼ ; vx v ¼
i vp h þ frs bs þ ð1 fÞrf bf gðT Tcold Þ; vy
(1)
(2)
(3)
an advective term and a Forchheimer quadratic drag term do not appear in the Darcy’s Eqs. (2) and (3). The viscosity of the nanofluid mnf can be approximated as viscosity of a base fluid if containing dilute suspension of fine spherical particles, which is given by Brinkman [42] as
mnf ¼
mf ð1 fÞ2:5
:
(5)
and
Meanwhile, the thermal diffusivity of the nanofluid is defined by Oztop and Abu-Nada [23] as
! vT vT v2 T v2 T : u þv þ ¼ anf vx vy vx2 vy2
anf ¼ nf ; rCp nf
k
(4)
In Eqs. (1)e(4), x and y are Cartesian coordinates measured along the horizontal and vertical walls of the cavity respectively, u and v are the velocity components along the x- and y- axes respectively, T is the fluid temperature, p is the fluid pressure, g is the gravity acceleration, K is the permeability of the porous medium, f is the solid volume fraction of the nanofluid, bf and bs are the coefficients of thermal expansion of the fluid and of the solid fractions respectively, rf and rs are the densities of the fluid and of the solid fractions respectively, mf is the viscosity of the fluid fraction, mnf is the viscosity of the nanofluid, and anf is the thermal diffusivity of the nanofluid. The flow is assumed to be slow so that
(6)
where ðrCp Þnf is the heat capacity of the nanofluid, which is given by Khanafer et al. [20] as
rCp
nf
¼ ð1 fÞ rCp f þf rCp s ;
(7)
and knf stands for the effective thermal conductivity of the nanofluid that can be obtained according to the Maxwell-Garnetts model
ks þ 2kf 2f kf ks knf : ¼ kf ks þ 2kf þ f kf ks
(8)
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
a
3.5
b
2145
3.8
3.6
3.2
3.4
Nu
Nu
2.9 3.2
2.6 3 2.3
2
2.8
0
0.05
0.1
φ
0.15
2.6
0.2
0
0.05
Ra = 50
c
0.1
φ
0.15
0.2
0.15
0.2
Ra = 100
d
7.9
12 11.5
7.5
11 10.5
Nu
Nu
7.1
6.7
10 9.5
6.3 9 5.9
5.5
8.5
0
0.05
0.1
φ
0.15
0.2
8
0
0.05
Ra = 500
0.1
φ
Ra = 1000
Fig. 5. Dependence of average Nusselt numbers on solid volume fraction parameter f for different nanoparticles when Ht ¼ 0.4, AR ¼ 1.0 and Yp ¼ 0.5.
In the above two equations, kf and ks are the thermal conductivities of the fluid and the solid fractions respectively, ðrCp Þf stands the heat capacity of the fluid fraction, and ðrCp Þs is the heat capacity of the solid fraction respectively. The following dimensionless variables
x y H H T Tcold X ¼ ; Y ¼ ; U ¼ u; V ¼ v; Q ¼ ; af af Thot Tcold H H
(9)
are introduced, and the dimensionless stream function J is defined in the usual way as
U ¼ vJ=vY;
V ¼ vJ=vX:
ð1 fÞ2:5
! i vQ h v2 J v2 J ¼ Ra ð1 fÞ þ f rs =rf bs =bf þ ; vX vX 2 vY 2 (11)
! anf v2 Q v2 Q vJ vQ vJ vQ ; þ ¼ af vX 2 vY 2 vY vX vX vY
8 Alongthe vertical wall : on heater; Q ¼ 1; J ¼ 0: > > < Along the verticalwall : on the unheated part;vQ=vX ¼ 0; J ¼ 0: On the adiabaticðbottomÞwall;vQ=vY ¼ 0; J ¼ 0: > > : On the inclinedwall; Q ¼ 0; J ¼ 0: In the meantime, the length and position of the heater(13) are respectively nondimensionalized as
(10)
When Eqs. (9) and (10) are substituted into Eqs. (2)e(4), and the pressure terms are eliminated from Eqs. (2) and (3), it is found that
1
of the fluid and the impermeability of the cavity walls, non-slip condition is composed along the whole boundary of the calculation domain in this model. Consequently, the corresponding dimensionless boundary conditions are converted as
Ht ¼
yp h and Yp ¼ : H H
(14)
The definitions for local Nusselt number NuY on the left vertical wall and average Nusselt number Nu are
k vQ NuY ¼ nf ; kf vX X¼0
(15)
and
(12)
where Ra is the Rayleigh number for a porous medium, which is defined as Ra ¼ gK rf bf ðThot Tcold ÞH=ðmf af Þ. Due to the viscosity
Z1 Nu ¼
NuY dY; 0
respectively.
(16)
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Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
a
6.5
b
6.3
6 6 5.5
Nu
Nu
5.7 5
4.5
5.1
4
3.5
5.4
4.8
0
0.05
0.1
φ
0.15
0.2
0
0.05
Ra = 50
c
φ
0.15
0.2
0.15
0.2
Ra = 100
d
9.9
14
13.5
9.6
13
Nu
9.3
Nu
0.1
9
8.7
12.5
12
8.4
11.5
8.1 0
0.05
0.1
φ
0.15
0.2
Ra = 500
11
0
0.05
0.1
φ
Ra = 1000
Fig. 6. Dependence of average Nusselt numbers on solid volume fraction parameter f for different nanoparticles when Ht ¼ 0.8, AR ¼ 1.0 and Yp ¼ 0.5.
3. Numerical procedure Finite difference method [22] was adopted to solve numerically the governing Eqs. (11) and (12) together with the boundary conditions in Eq. (13). The diffusion terms within Eqs. (11) and (12) were replaced by the second order central differencing schemes, while the second order upwind differencing scheme was chosen to approximate the convective term in order to make the numerical procedure stable. The solution for the corresponding linear algebraic equations was obtained through the Successive Under Relaxation (SUR) method. The temperature function Q and the stream function J were calculated through iteration when the initial guess was made. The iteration process is terminated when the following criterion is satisfied
PM
i¼1
nþ1 ci;j cni;j 106 ; PN nþ1 j ¼ 1 ci;j
PN
PM
i¼1
j¼1
points were examined from 31 31 to 171 171. As can been seen in Fig. 2, the calculation results become mesh independent when the number of grid points are higher than 91 91. Meanwhile, in order to verify the rationality and accuracy of the results, the above numerical simulation procedure was tested with the classical natural convection heat transfer problem in a differentially heated square porous enclosure (AR ¼ 1.0) in context with the pure fluid (i.e. water), which is equivalent with f ¼ 0.0 in the presented model. The obtained results for the average Nusselt number, as defined in Eq. (16), were compared with those given by different authors, as listed in Table 2. It can be seen that the results obtained here are in good agreement with the results that have been reported. 4. Results and analysis
n 1;
(17)
in which c reflects either Q or J, M and N are the number of grid points in the X- and Y- directions respectively, and n is the iteration step. In this study, the grid points with uniform spaced mesh were generated along both X- and Y- directions. The number of the grid
Numerical simulation results on flow field, temperature distribution, and Nusselt number are presented in this section. The effects of different values of Rayleigh number Ra of a porous medium, solid volume fraction parameter f of nanofluid, types of the nanoparticles, heater size Ht, heater position Yp and enclosure aspect ratio AR are analyzed. Enclosure aspect ratio AR changes from 0.5 to 1.5. Dimensionless size of the heater Ht varies from
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
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Fig. 7. Dependence of average Nusselt numbers on aspect ratio AR for different nanoparticles when f ¼ 0.1 and Yp ¼ 0.5.
0.1 to 0.9, and dimensionless heater position Yp varies from 0.25 to 0.75. We assume, as in Oztop and Abu-Nada [23], that the range for solid volume fraction parameter f of nanoparticles is taken from 0.0 to 0.2. We assume also that Rayleigh number Ra for porous medium varies from 10 to 1000. Rayleigh number Ra is a very important parameter that has effects on heat transfer within a porous medium. The results in Fig. 3a and c present the distribution of local Nusselt number NuY along the heater for different values of Ra. From Fig. 3b and d, one can see that average Nusselt number Nu is improved when the Rayleigh number is increased. This conclusion is supported by Varol et al. [9] for pure fluid. Meanwhile, Fig. 3b and d also show that the average Nusselt number of Cuewater nanofluid is the highest among the three kinds of nanofluids, which is probably due to that the thermal conductivity of nanoparticle Cu is higher than those of the other two nanoparticles (Al2O3 and TiO2). Fig. 4 presents the streamlines and isotherms for Cuewater nanofluid under different values of the Rayleigh number. Fig. 4a, b and c demonstrate that a single circulation flow cell is formed in the clockwise direction for all values of Ra that have been tested. When the Rayleigh number increases, the flow convection is strengthened, the width of flow cell is increased as egg-shaped cell is extended to triangle-shaped cell, and the boundary layers become more significant. Solid volume fraction parameter f is a key factor to study how nanoparticles affect the heat transfer of nanofluids. Figs. 5 and 6 present the average Nusselt number for different values of the Rayleigh number and the solid volume fraction parameter. Once again, the average Nusselt number is elevated when the Rayleigh
number increases and nanoparticle Cu is used. However, the effects of solid volume fraction parameter f on the heat transfer of nanofluids are complicated. When Rayleigh number Ra is low, the average Nusselt number increases as solid volume fraction parameter f increases (Figs. 5a and 6a, b). However, elevating f has adverse effects on the heat transfer of nanofluids when Ra is high, as shown in Figs. 5c, d and 6d. One possible explanation for the above phenomena is as following. The density of nanofluid is defined by Oztop and Abu-Nadaas [23] as
rnf ¼ frs þ ð1 fÞrf :
(18)
Based on the definitions in Eqs. (5) and (18), and considering the physical values listed in Table 1, the density and viscosity of nanofluids are elevated relative to the pure fluid (water) as f increases, which means that the inertial and viscous resistances for nanofluids are higher than those for the pure fluid (water). When the Rayleigh number is low, the flow convection is insignificant. The heat transfer in the cavity is dominated by conduction. Along the increase of f, the thermal conductivity of nanofluids is increased relative to that of pure fluid (water) due to the high thermal conductivity of nanoparticles. Consequently, the heat transfer of nanofluids is improved compared with pure fluid (water), as depicted in Figs. 5a and 6a, b. As the Rayleigh number increases, the flow convection becomes stronger, and the heat transfer in the cavity is improved, as displayed in Fig. 3b and d. However, as f increases, the elevated inertial and viscous resistances of
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Fig. 8. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1 and Yp ¼ 0.5.
nanofluids would compromise the flow convection, which is able to lead to the adverse effects on the heat transfer in the nanofluids compared with the pure fluid (water). Nevertheless, when f is upon some level, the positive effects of increased thermal conductivity of nanofluids can appear again to overtake the adverse effects of the elevated inertial and viscous resistances, which results in the value of average Nusselt number Nu to rise after falling. The curves in Fig. 5b and solid line in Fig. 6c reflect the above analysis. When Rayleigh number Ra is high (Ra ¼ 1000), convection dominates the fluid movement. Under such a circumstance, heat transfer, for all three types of nanofluids, is decreased along the increasing of the solid volume fraction parameter due to the elevated inertial and viscous resistances of the nanofluids, as shown in Figs. 5c, d and 6d. It is possible that when solid volume fraction parameter f is above 0.2, sedimentation sets in and the fluid loses the regular (Darcian) character as it was concluded by Muthtamilselvan et al. [24] for a copperewater nanofluids in a lid-driven enclosure. Enclosure aspect ratio AR is another feature that has influence on the heat transfer within the cavity. Normally, the value of
average Nusselt number Nu is reduced along the increase of AR, as shown in Fig. 7a, b and d. This is understandable because when AR increases, the distance between the heater and cold wall of the cavity is enlarged, which can lead to the decrease in temperature gradients. However, the effects of AR on Nu are also affected by other physical parameters. For example, when Ra ¼ 1000, the heat transfer within the cavity with small size of heater (Ht ¼ 0.4) falls after rising along the increasing of AR, as shown in Fig. 7c. Nevertheless, when the heater size is enlarged as Ht ¼ 0.8, the heat transfer inside the enclosure again decreases consistently as AR increases (Fig. 7d). The comparisons for streamlines and isotherms of Cuewater nanofluid with different values of AR are presented in Figs. 8 and 9. The heat transfer can also be affected by the position of heater. When Ra ¼ 1000, convection leads the flow movement within the cavity. Considering the shape of the enclosure and the boundary conditions, when the position of heater is high, for example Yp ¼ 0.75, the flow and heat transfer cannot be fully developed inside the entire enclosure, as shown in Fig. 12. Under this circumstance, heat transfer within the cavity is decreased along the increase of Yp, as displayed in Figs. 10b and 11b. Nevertheless, when Ra is low as
Q. Sun, I. Pop / International Journal of Thermal Sciences 50 (2011) 2141e2153
Fig. 9. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1 and Yp ¼ 0.5.
a
b
Fig. 10. Dependence of average Nusselt numbers on center position of heater Yp for different nanoparticles when f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.
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Fig. 11. Local Nusselt number profile of Cuewater nanofluid for different center position of heater Yp when f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.
Ra ¼ 1000, the flow convection is insignificant, and the heat transfer inside the cavity is mainly performed by conduction, as presented in Fig. 13. In such a case, considering the shape of the enclosure, when the position of heater is high, the distance between the heater and
cold wall is shortened, which leads to the improvement of the heat transfer inside the cavity, as shown in Figs. 10a and 11a. It is easily expected that enlarging the size of heater Ht is able to improve the heat transfer inside the enclosure. Fig. 14 reflects that
Fig. 12. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.
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Fig. 13. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 100, f ¼ 0.1, Ht ¼ 0.4, and AR ¼ 1.0.
expectation for three types of nanofluids. It is also can be seen that the difference of the average Nusselt number becomes significant for larger size of the heater since flow convection becomes stronger. These results are consistent with the findings reported by Oztop
and Abu-Nada [23] for the convection in a rectangular cavity of homogeneous nanofluids. In the meantime, the streamlines and isotherms of Cuewater nonafluid for different values of Ht are plotted in Fig. 15.
Fig. 14. Dependence of average Nusselt numbers on size of heater Ht for different nanoparticles when f ¼ 0.1, AR ¼ 1.0, and Yp ¼ 0.5.
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Fig. 15. (a, b, c) Streamlines and (e, d, f) isotherms for Cuewater nanofluids when Ra ¼ 1000, f ¼ 0.1, AR ¼ 1.0, and Yp ¼ 0.5.
5. Conclusions A numerical study has been performed in this paper to investigate the free convection heat transfer problem in a partly heated triangle cavity filled with a porous medium saturated with nanofluids. The governing equations were solved by finite difference method. It is found that the maximum value of the average Nusselt number can be achieved for the highest Rayleigh number, the largest heater size. Meanwhile, under that circumstance, lowering the heater position and decreasing the aspect ratio of the enclosure are beneficial to the heat transfer in the cavity. Among the three types of nanofluids, the highest value of the average Nusselt number is obtained when using Cu nanoparticles. The effects of the solid volume fraction parameter on the heat transfer of nanofluids within a porous medium are complex. When the Rayleigh number is low, increasing the value of the solid volume fraction parameter of nanofluids can improve the value of the average Nusselt number, while if Rayleigh number is high, elevating the solid volume fraction parameter of nanofluids reduces the value of the average Nusselt number. An optimization investigation might be required in future to search for the best value of the solid volume fraction parameter of nanofluids to achieve the most efficient way of improving heat transfer within a porous medium, which is nevertheless beyond the scope of this study. It is worth mentioning to this end that the study of nanofluids is still at its early stage and it
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