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Eccentricity effects of heat source inside a porous annulus on the natural convection heat transfer and entropy generation of Cu-water nanofluid
International Communications in Heat and Mass Transfer 109 (2019) 104367
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Eccentricity effects of heat source inside a porous annulus on the natural convection heat transfer and entropy generation of Cu-water nanofluid
T
Payam Gholamalipoura, Majid Siavashia, , Mohammad Hossein Doranehgardb ⁎
a b
Applied Multi-Phase Fluid Dynamics Laboratory, School of Mechanical Engineering, Iran University of Science & Technology, Tehran, Iran Department of Civil and Environmental Engineering, School of Mining and Petroleum Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada
ARTICLE INFO
ABSTRACT
Keywords: Eccentricity Annulus Porous media Natural convection Nanofluid Entropy generation
This study presents a numerical investigation on free convection heat transfer and entropy generation of Cuwater nanofluid inside an annulus, fully filled with a porous foam, in presence of a cylindrical heat source. For accurate nanofluid flow simulation, the two-phase mixture model is applied, and the viscosity and thermal conductivity of the mixture are computed based on the empirical Corcione's models. The effects of the vertical and horizontal heat source eccentricity (−0.4 ≤ ζ ≤0.4) and nanoparticles volume fraction (0 ≤ ϕ ≤ 0.04) for different values of Darcy (10−4 ≤ Da ≤ 10−1) and Rayleigh (103 ≤ Ra ≤ 106) numbers on the heat transfer and entropy generation are discussed. Results indicate that for various figures of eccentricity, heat transfer can be improved or deteriorated, depending on the value of Da, Ra and direction of the inner cylinder movement. Consequently, for each Rayleigh number, optimal values of Da and ζ exist to meet the maximum average Nu number. In addition, an appropriate eccentricity – around ζ =0.1 and 0.2 – exists to minimize the total entropy generation. Moreover, the downward eccentric annulus expresses the best performance in which the highest heat transfer and lowest entropy generation occur.
1. Introduction Free convection, in a thermodynamic system, is a mechanism of heat transport in which the fluid motion is not generated by an external force (exerted by tools such as pumps, fans or suction devices), but happens by density differences in the fluid occurring as a result of temperature gradients [1,2]. Due to the low costs of maintenance and minimum noises, this type of heat transfer has a wide range of applications [3–5]. In this regard, many scientists focused on natural convection to investigate its various aspects and many papers have been published in this field. Furthermore, for responding to the increasing demands for energy and limited energy resources, researchers have obliged to use several approaches in order to enhance heat transfer. Among the lunched methods, as coming in the following, using nanoparticles and porous foams have received a lot of attention due to their simplicity of usage and acceptable efficiency [6]. Research works show that adding metallic nanoparticles to the pure base fluid can significantly increase the thermal conductivity of the produced mixture (known as nanofluid) [7]. For instance, Eastman et al. [8] observed that thermal conductivity values of Al2O3/water and CuO/water nanofluids with 5% nanoparticles volume fraction are respectively 29% and 60% higher than that of the base fluid. At this point, ⁎
Corresponding author. E-mail address: [email protected] (M. Siavashi).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104367
0735-1933/ © 2019 Elsevier Ltd. All rights reserved.
to simulate nanofluid flow, there are two general models: 1- Singlephase model in which it is assumed that a homogeneous single-phase mixture is composed of nanoparticles and the base fluid with effective properties; 2- Two-phase models in which different velocities can be defined for nanoparticles and the carrying base fluid. Two-phase models give more precise results but requires higher computational costs. These models can be generally categorized into Eulerian-Eulerian and Eulerian-Lagrangian models. Goktepe et al. [9] showed that for nanofluid flows with low nanoparticles concentration, the Eulerian twophase mixture model provides acceptable results without need to high computational resources. This model has been recently implemented for various nanofluid flow studies and is also used in the present study [10–14]. Mahian et al. [15,16] wrote two review papers and discussed comprehensively on the theory and applications of nanofluid flows. In addition to using nanoparticles, implementing porous foams, due to improving the fluid-surface interface and increasing the random movements of fluid through porous regions, results in heat transfer enhancement [17–23]. Rong et al. [24] investigated the effects of several parameters, such as porosity and Darcy number on the flow and heat transfer enhancement inside a partially porous pipe. They concluded that adjusting porous media thickness, while the Darcy number is set to be 10−3, can provide the highest improvement in the heat
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Nomenclature Cp dp D Da fdrag g Κ k kb Nu Nu⁎ P Pr q” Ra Sgen S⁎ T T0 u, v x, y
Subscripts
specific heat (J kg−1 K−1) diameter of the nanoparticle (m) Diameter of the outer cylinder Darcy number friction factor gravity (ms−2) permeability (m2) thermal conduction (W m−1 K−1) Boltzmann constant (J K−1) Nusselt number Nu to concentric Nu ratio pressure (Pa) Prandtl number heat flux (W m−2) Rayleigh number entropy generation rate (W m−3 K−1) relative entropy generation temperature (K) bulk temperature (K) velocity components, (m s−1) Cartesian coordinator (m)
avg c dr f eff h m p s tot
average cold drift base fluid effective hot mixture (nanofluid) nanoparticle solid medium total
Greek letters α β ε ζ θ η μ ρ ϕ
transfer. Torabi et al. [25] observed the same result for a partially porous channel. In last decade, many researchers have investigated natural convection flow and heat transfer enhancement inside different geometries by concurrently using nanoparticles and porous media. Bourantas et al. [26] numerically analyzed nanofluid free convection inside a porous cavity using a meshless technique. A numerical investigation on natural convection of Cu-water nanofluid in an inclined porous square cavity is performed by Yaghoubi-Emami et al. [27]. They used two-phase mixture model and showed that use of nanofluid in the presence of porous media can improve the heat transfer at low Ra numbers, however, the opposite fact was observed at high Ra numbers. Sun and Pop [28] have studied natural convection of different nanofluids inside a porous triangular cavity in the presence of a flush mounted heater with finite size. They observed that for a low Rayleigh number, increasing volume fraction of nanofluid mixture has improved the heat transfer in the cavity. On the other hand, for high Rayleigh numbers, this trend is vice versa. Toosi and Siavashi [3] in a numerical investigation analyzed the heat transfer of natural convective nanofluid in a square cavity with
thermal diffusivity (m2/s) thermal expansion coeff. (K1) porosity Dimensionless eccentricity dimensionless temperature eccentricity of inner cylinder dynamic viscosity (kg m‐1 s‐1) Density (kg m‐3) nanoparticles volume fraction
porous layers on its walls. Their outcomes show that for different conditions of Ra and Da, the heat transfer can be maximized through optimal selection of the porous layer thickness and the nanofluid concentration. Siavashi et al. [29] have numerically investigated the impact of simultaneous use of nanoparticles and porous fins on free convection flow in a square chamber. They showed that using porous fins can boost heat transfer, from one side, and decrease irreversibility, from the other side. Some researchers have studied the impact of the geometrical parameters on heat transfer. Mohebbi et al. [4] investigated nanofluids natural convection in porous cavity. Qi et al. [30] analyzed the effect of different aspect ratios of the enclosure on the CueGa nanofluid free convection heat transfer. They observed that the average Nu has opposite relation with the enclosure aspect ratio. Selimefendigil and Oztop [31] investigated nanofluid free convective flow through numerical simulation and studied the problem from the viewpoint of the first and second laws of thermodynamics. In their study, they consider obstacles with different shapes, internal heat generation, and also existence of a magnetic field. Ma et al. [32] studied the effect of cavity's
Fig. 1. Computational domain and boundary conditions of three different configurations 2
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Table 1 Water and Cu nanoparticles thermo-physical properties (T=310 K) [39]. ρ (kg/m3) Water Cu
k (W/m.K)
993.4 8933
β (K‐1)
Cp (J/kgK)
0.635 401
‐5
4178 385
α (m2/s)
μ (kg/m.s) ‐4
36×10 1.67×10‐5
6.95×10 ‐
1.44×10 ‐
dp (nm) ‐7
‐ 20
Table 2 Present results comparison with the outcomes of Nithiarasu et al. [45] (Pr=1.0, ε = 0.4 and 0.9) Da
‐2
10 10‐2 10‐2
Ra
6
10 104 5×105
Nuavg (ε = 0.4)
Difference%
Nuavg (ε = 0.9)
Difference%
Nithiarasu et al. [48]
Present work
2.14
Nithiarasu et al. [48]
Present work
1.40 2.98 4.99
1.43 2.91 4.84
2.34 3.00 2.14
1.64 3.91 6.70
1.62 3.85 6.44
1.21 1.53 3.88
thermal characteristics and entropy generation are numerically studied. 2. Problem description Fig. 1 depicts the geometry of the studied problem, a circular eccentric annulus with outer and inner diameters of D and D/2, respectively. Dimensionless eccentricity is defined as ζ = D in which η is the 2
eccentricity distance. The inner cylinder is assumed to have horizontal or vertical eccentricity and its value is in the range of ‐0.4≤ ζ ≤ 0.4. The outer wall (cold wall) has the constant temperature of Tc = 305 K. While, the inner wall is hot and has a constant temperature (Th = 315 K). The space between the inner and the outer cylinders is fully filled by a porous foam with porosity of ε=0.9. Some reports exist that nanofluids with high amount of nanoparticles volume fraction could exhibit non-Newtonian behavior, even with a Newtonian base fluid. However, in several numerical and experimental studies in which the nanofluid volume fraction is less than 4%, it is mentioned that the nanofluid with a Newtonian base fluid behaves as a Newtonian fluid [35–38]. In the present study, a water based Newtonian nanofluid containing Cu nanoparticles with volume fractions of 0% to 4% has saturated the porous region between two annuli. Table 1 [39] depicts the thermo-physical properties of water and nanoparticles.
Fig. 2. The average Nu values of the present work are compared with results of Jahanshahi et al. [38].
aspect ratio on free convection of nanofluid within a U-shaped chamber equipped with a heating obstacle. They observed that as the aspect ratio increased, the positive impact of nanoparticles on the heat transfer was enhanced. Magami et al. [33] focused on effects of width ratio of a cavity on heat transfer in a square porous-annulus. They observed that Nu has a direct relationship with the height of the cavity. Kefayati and Tang [34] studied the effect of horizontal and vertical distance of the inner cylinder from the center of a heated enclosure on natural convection and entropy generation of Carreau fluid. Their results showed that horizontal or vertical alteration of the center of the cylinder has a remarkable impact on heat and mass transfer. Based on the above literature review, studying the effects of horizontal and vertical eccentricities of fluid flow inside annular cavities seems to be important. However, to the authors' knowledge, the impact of horizontal and vertical eccentricity on free convective heat transfer and entropy generation of nanofluid flow in a cylindrical annular chamber containing a porous foam, has not been studied and is worthy of investigation. Accordingly, this study aims to investigate this problem in the presence of a heat source by using the two-phase mixture model, for nanofluid, and also considering Darcy-BrinkmanForchheimer equation for flow inside the porous foam. Hence, the effects of nanoparticles volume fraction, horizontal and vertical movements of the inner cylinder, Rayleigh and Darcy number on hydro-
3. Mathematical modeling Regarding mathematical modeling, governing equations for fluid flow of a Newtonian nanofluid in porous media will be discussed. 3.1. Governing equations For nanofluid flow through porous media, the non-dimensional form of the mass conservation equation is given by [27,40]:
um v + m =0 x y
(1)
The momentum equations are presented by: um =
um u + vm m x y 2 p +
x
µ u Pr m m + µf x
f m
um | Vm| + 2 (1
x )
f m
udr , f
udr , f x
y
µ u Pr m m µf y
2 Pr
udr , f
p
+ vdr , f
y
f µm um Da m µf
+ 2
m
udr , p
udr , p x
1.75 150 + vdr , p
1/2
Da udr , p y
(2) 3
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Fig. 3. Comparisons of streamlines and isotherm contours of the present work and those of experimental data (modified from Jahanshahi et al. [38]) for Ra= 106 and ϕ =0.04. Table 3 Grid independence study for the case with ζ=0.4, ϕ = 0.04, Ra = 105 and Da = 10−2. Mesh size
45×320
50×376
54×450
58×550
65×660
70×800
75×960
Nuavg
14.257
14.252
14.250
14.242
14.240
14.238
14.237
um
mixture. Dimensionless variables in the aforementioned equations are defined as follows [27]:
vm v + vm m x y 2 p +
=
m
x
vm | Vm| + 2 (1 udr , p
vdr , p x
µ v µ v Pr m m + Pr m m µf x y µf y
f
y
)
f m
+ vdr , p
vdr , f
udr , f
vdr , p y
x
+ vdr , f
vdr , f
f µm vm Da m µf
2 Pr
+ 2
y
1.75 150
1/2
Da
x =
p m
( )m f + 2 RaPr ( )f m
x
( ) + vm
y
( )=
keff x
kf
x
+
Pr =
kf
y
( vm ) + y
( udr , f ) x
+
( vdr , f ) y
=0
T Th
Tc Tc
(6)
, Ra =
g
f
Tc ) D3
(Th f
f
, Da =
K D2
(7)
More details about the Eulerian two-phase mixture model can be found in [27,40]. Corcione [41] used a wide variety of experimental data, available in the literatures, to propose correlations for estimation of nanofluid’s effective dynamic viscosity and thermal conductivity. The data used for thermal conductivity consists of various nanoparticles in the range between 10 nm and 150 nm, with water and ethylene glycol (EG) as the
(4)
And the volume fraction equation is:
( um ) + x
f f
keff y
=
Also, Prandtl (Pr), Rayleigh (Ra) and Darcy (Da) numbers are defined by the following relations:
(3)
The energy equation is as follows:
um
pD 2 uD x y vD ,y = ,u= ,v = ,p = , 2 D D f f f f
(5)
where, subscripts p and m respectively represent the particle and 4
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Concerning entropy generation, the entropy generation rate is defined by the following relation [42,43]: (13)
stot = sT + sF sT′′′
where, is known as the thermal entropy generation rate and is written as:
sT =
km T02
T x
2
T y
+
2
(14)
sF′′′
is called the frictional entropy generation rate and can be evaluated as follows [44]:
sF =
µm T0
2
um x
2
+2
m
y
2
+
m
x
+
um y
2
(15)
The dimensionless entropy generation rate (Stot′′′) is defined by:
Stot =
S=
base fluid. In order to propose the correlation for the effective dynamic viscosity the data of nanoparticles used in nanofluids is in the range of 25 nm to 200 nm. This correlation is the best fit in the mentioned range of used data, however, it could be even valid for a wider range and other nanofluids with spherical nanoparticles. In this study Cu nanoparticles with 20 nm diameter are used, and Corcione’s correlation for estimation of the effective thermal conductivity is valid. Also, the size of our nanoparticles is very close to the range of data used by Corcione for the effective dynamic viscosity and could provide results with enough accuracy. Accordingly, Corcion’s correlations are used in this study and his relation for the effective dynamic viscosity is presented as follows:
dp
34.87
1.03 p
T Tfr
kp kf
4.2. Grid independence check To check the grid independency of numerical algorithm, an analysis for grid sensitivity is done to provide answers with enough precisely and an appropriate computational cost. As is illustrated in Table 3, seven structured computational grids, including 45×320, 50×376, 54×450, 58×550, 65×660, 70×800, and 75×960 grid cells (number of cells is counted in θ- and r-directions, respectively) are used to conduct the grid independence study. It’s worthy to mention that the case with ζ=0.4, ϕ = 0.04, Ra = 105, and Da = 10−2 is investigated for grid independence. Finally, according to the reported average Nu values in Table 3, the grid with 65×660 cells could result in grid independent outcomes (as is shown in Fig. 4).
0.03 0.66 p
(9)
where ReB, Pr, Tfr, and kp are the nanoparticle Reynolds number, the Prandtl number of the base liquid, the freezing point of the base liquid, and the nanoparticle’s thermal conductivity, respectively. The nanoparticle Reynolds number is given by:
ReB =
p uB dp
µf
(10)
ρf and μf, are the mass density and dynamic viscosity of the base fluid, respectively, and uB is the mean Brownian velocity which is defined by:
uB =
2kB T µf dp2
5. Results and discussion As previously mentioned, the convective heat transfer and entropy generation of nanofluid flow in a cylindrical annulus, filled with porous media, have been numerically investigated considering the effects of four parameters: nanoparticles volume fraction (0 ≤ ϕ ≤ 0.04), horizontal and vertical eccentricity of the inner cylinder (‐0. 4 ≤ ζ ≤ 0. 4), Rayleigh number (103 ≤ Ra ≤ 106), and Darcy number (10‐4 ≤ Da ≤ 10‐1). Results of this section are presented in three subsections: first, the impact of using nanoparticles on heat transfer
(11)
In the above relation, kB is Boltzmann constant of the nanoparticle. The average Nusselt number is calculated by the following relation:
Nuavg , wall =
qwall D kf (Th
Tc )
(17)
To validate the implemented numerical solver, two problems are considered. First, natural convection of a pure fluid flowing in a porous square chamber is simulated. The results of this simulation with those of a similar research work done by Nithiarasu et al. [45] are compared in Table 2. As can be seen, a good agreement is observed in the average Nu number. In the next step, free convection of nanofluid (SiO2-water) flow in a square cavity is simulated with the two-phase mixture model. The results are compared with those of Jahanshahi et al. [38] (see Fig. 2). Also, the stream functions and isotherms of the results have been compared in Fig. 3. According to this figure, the results are in a satisfactory agreement.
(8)
10
Stot dV
4.1. Validation
where dp, df, and ϕp are the nanoparticle diameter, the equivalent diameter of the base fluid molecule, and volume fraction of the nanoparticles, respectively. The correlation for the effective thermal conductivity is proposed by:
keff = kf 1 + 4.4ReB0.4 Pr 0.66
V
4. Numerical procedure and validation
0.3
df
(16)
The total entropy generation in the non-dimensional form is calculated by:
Fig. 4. The computational structured-grid with 65×660 cells to solve the problem
µm = µ f / 1
T02 H 2 stot kf (Th Tc ) 2
(12) 5
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Fig. 5. Variation of the average Nusselt number with different Ra, ζ, and ϕ values for Da=10‐1.
vertical directions, at Da=10‐1. Figs. 6 and 7 respectively show the impact of nanoparticles addition (0≤ ϕ ≤0.04) to the base fluid on the patterns of isotherms and stream functions of case B (horizontal movement of the inner cylinder), at various ζ and Rayleigh numbers. According to Fig. 5, it can be seen that for the case with horizontal eccentricity and for all Ra, Da and ζ values, by increasing ϕ, the average Nusselt number is increased and its maximum value is observed at ϕ =0.04. From Fig. 6, it can be observed that by increasing Ra for the case with horizontal eccentricity, the isotherms deviate from their circular curved shape which represents the increasing heat convection and decreasing heat conduction. This fact is also endorsed by Fig. 7 as by
improvement is investigated, then, the effect of eccentricity on the heat transfer characteristics of the nanofluid is analyzed for different cases, and finally, based on the second law of thermodynamics the results are discussed. 5.1. Nanofluid effects on heat transfer enhancement In this part, the impact of nanoparticle addition into the base fluid on heat transfer is discussed. At this point and for briefness, only horizontal eccentricity has been studied. Fig. 5 presents the average Nu values in terms of ϕ, ζ, Ra and eccentricity values in horizontal and 6
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Fig. 6. Contour of isotherms for Da=10‐1 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
increasing Ra for the horizontal eccentricity, ψnfmax has been increased. According to Fig. 6, increasing ζ from 0 to 0.4, in addition to the increment in the convection heat transfer in the left side of the inner cylinder (regarding to the isotherms deviated from their spiral shape), leads to the improvement of heat conduction in the right side of the inner cylinder as the isotherms are more condensed and their density is much higher between the two walls. This outcome is confirmed by Fig. 7 as by increasing ζ, isotherms in the left side of the inner cylinder
become closer to each other. As depicted in Figs. 5 and 6, the effect of raising Rayleigh number on heat transfer and isotherms variations – due to the increment in the nanofluid volume fraction – is different for various amounts of ζ. Increasing ϕ from 0 to 0.04 at Ra=105, the maximum variation between isotherms of the base fluid and the nanofluid (the maximum improvement caused by adding nanoparticles to the base fluid) for the cases with horizontal eccentricity values of ζ =0, 0.1 and 0.2 can respectively 7
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Fig. 7. Contour of stream functions at Da=10‐1 (solid lines for ϕ=0.04 and dashed lines for ϕ= 0.0)
lead to 12%, 12.2%, and 11.7% enhancement in the heat transfer. In addition, at Ra=104 the maximum effect of nanoparticle addition on enhancing the heat transfer are 13% and 12.3%, respectively, that corresponds to the cases with ζ=0.3 and 0.4. Fig. 7 confirms these
outcomes as for the aforementioned cases (in which the differences between isotherms are maximum), due to the positive effects of nanoparticles, stream functions are compressed near the walls which this fact leads to the velocity increment and improvement of the heat 8
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Fig. 8. Variations of the relative Nusselt number, Nu⁎avg, with different Da, Ra, and ζ for ϕ =0.04.
convection process. In all of the above-mentioned cases, according to the isotherms exhibited in Fig. 6, the remarkable positive effect of increasing the nanofluid volume fraction on the heat transfer is mainly due to the fact that the convection is not extremely powerful in the way that viscous effects become sensible. In addition, conduction effects are relatively considerable so that the increment in the heat conductivity of the base fluid, due to the nanofluid volume fraction increment, plays an important role in the total heat transfer rate. However, by increasing Ra to 106 (at a constant amount of ζ), as the convection increases, the nanofluid concentration enhancement leads to decline in the positive effects of nanofluid on heat transfer, as a result of the increasing viscosity. Fig. 6 shows that for Ra=103 and 104, by increasing ζ, the dashed lines are more deviated and, hence, the maximum impact of nanoparticles on isotherms can be observed at the highest amount of eccentricity (ζ =0.4). On the other hand, at Ra=105 and 106 this phenomenon is vice versa and the impact of increment of nanoparticles concentration on heat transfer is slightly decreased by increasing ζ, due to the existence of relatively powerful natural convection and high viscosity dissipations. Fig. 7 endorses this result as for Ra=103 and 104, by increasing ζ, isotherms are more compressed to the walls, and hence the heat convection is boosted.
mainly includes horizontal and vertical movements of the cylinder at different Darcy and Rayleigh numbers for ϕ =0.04, in which the maximum heat transfer enhancement is obtained. In this regard, Fig. 8 depicts variations of the Nu⁎avg (the ratio of Nuavg for the eccentric annulus to that of the concentric annulus) as a function of Ra and ζ, for different Da numbers at ϕ =0.04. In addition, Figs. 9 and 10 represent non-dimensional isotherm and stream function contours for various Ra and ζ numbers at Da=10‐4, respectively. The same contours are also exhibited in Figs. 11 and 12 for Da=10‐1. As can be seen in Fig. 8, results of the leftward- and rightwardhorizontal eccentricity are the same, as the consequence of the symmetry. Figs. 9 and 10 also depict the similarity of the streamlines and isotherms of these two states. As exhibited, horizontal eccentricity enhances the heat transfer rate. In contrast, for the vertical eccentricity, the upward and downward movement of the hot cylinder will lead to different outcomes. Investigation of Fig. 8 reveals that for majority of the cases, the eccentricity would enhance the heat transfer rate. However, the verticalupward eccentricity can act inversely and slightly reduce the Nusslet number. For the cases with a weak flow field (low Da or Ra numbers), this negative impact cannot be observed. However, by enhancing the flow strength (higher Da and Ra values), the negative impact of upward eccentricity on the heat transfer rate will be appeared, and the worst condition is observed when ζ =0.2. Results of Figs. 10 and 12 also confirm the mentioned point, since for the cases with upward eccentricity the maximum strength of the stream function is reduced with respect to the concentric case. In contrast, downward eccentricity
5.2. Eccentricity effects on heat transfer In this section, the eccentricity of the inner cylinder and its effects on heat transfer, isotherms and streamlines are studied. This study 9
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Fig. 9. Contour of isotherms at Da=10‐4 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
strengthens the flow field and improves the heat transfer rate. Eccentricity of the inner cylinder in presence of the metal foam can affect the heat transfer with two mechanisms. In the first mechanism, it affects the conduction through the porous medium. Increasing ζ will make a short connection between the hot and cold surfaces and provides a high conductive region for heat transfer. Hence, for the cases with dominant conduction heat transfer (low Da and Ra numbers), increasing ζ improves the heat transfer rate. In the second mechanism, the increased eccentricity can influence the convection field of natural
convection. As depicted in Figs. 10 and 12, only downward eccentricity will strengthen the convection field, and in the other cases the eccentricity weakens the strength of the convection field. Accordingly, for convection dominated flows (large Da and Ra), increasing ζ slightly affects the heat transfer rate, while, the highest enhancement due to increasing the eccentricity is observed in the conduction dominated cases (low Da or Ra values). The balance of conduction and convection and their roles in the overall heat transfer rate will result in the value of Nu⁎avg. Results of Fig. 8, depicts between 10% to 80% enhancement in 10
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Fig. 10. Contour of stream functions for Da=10‐4 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
the heat transfer due to eccentricity. In addition, the highest enhancement corresponds to the downward movement of the inner cylinder. For Da=10‐4, the Ra effect on Nu⁎avg is minimum and by increasing Da,
the role of Ra in Nu⁎avg becomes more evident. As the consequence, the range of variation of Nu⁎avg for various Ra numbers is confined for Da=10‐4, while, changing Ra at Da=10‐1 substantially affects Nu⁎avg. 11
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Fig. 11. Contour of isotherms at Da=10‐1 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
5.3. Eccentricity effects on entropy generation
total entropy generation is considerably lower than the role of thermal irreversibility, hence, the total entropy generation is mostly influenced by the temperature gradients. As was observed in Fig. 9, high eccentricity values led to high temperature gradient zones. For instance, at |ζ| = 0.4, the temperature gradient is much greater than that of the case with |ζ| = 0.2. Therefore, as is illustrated in Fig. 14, a high entropy generation rate region is formed around the hot surface and the entropy generation rate of the cases with |ζ| = 0.4 is higher than that of the cases with |ζ| = 0.2. Increasing Ra number will increase the frictional
Fig. 13 shows S* values (the ratio of the total entropy generation of the eccentric annulus to that of the concentric annulus) as a function of Ra and ζ, for various Da numbers at ϕ =0.04. Through this figure, the effect of eccentricity on entropy generation can be investigated. Figs. 14 and 15 illustrate the total entropy generation rate in the annulus, for various Ra numbers at Da=10‐4 and 10‐1, respectively. In low Ra numbers, the role of friction entropy generation in the 12
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Fig. 12. Contour of stream functions for Da=10‐1 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
irreversibility and results in the higher entropy generation rate (see Fig. 14). Increasing the eccentricity highly affects the temperature gradients, hence, for the cases with dominant thermal irreversibility it
would lead to the higher entropy generation. Accordingly, it is expected to see higher S* values at lower Ra numbers, in which the thermal irreversibility is dominant (see Fig. 13). 13
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Fig. 13. Variations of the relative entropy generation, S⁎, with different Da, Ra and ζ for ϕ =0.04
• The two-phase mixture model provides acceptable results relative to
Increasing Da number will improve the natural convection field and makes higher the frictional entropy generation rate. As the consequence, the high entropy generation rate regions for the cases with Da=10‐1 (see Fig. 15) are appeared in the stagnation points and boundary layers, where the velocity gradients are high. An important point that can be deduced Fig. 13 is that an optimal eccentricity value exists to minimize the total entropy generation. This optimal value is around |ζ| = 0.1 and 0.2, depending on the value of Da and Ra numbers. Increasing the absolute value of eccentricity can increase the flow and temperature gradients in one side and decrease them on the other side of the inner cylinder. Hence, it is expected to have an optimal point to meet the minimum entropy generation.
• • •
6. Conclusion
•
Effects of the vertical and horizontal eccentricity of the heat source and nanoparticles volume fraction for various Rayleigh (Ra) and Darcy (Da) numbers on the heat transfer and entropy generation of natural convection nanofluid flow inside a fully porous annulus are studied. Two-phase mixture model and Darcy-Brinkman-Forchheimer relations are implemented to simulate nanofluid flow and heat transfer through porous media. Numerical results were validated via comparison with those of experimental works and results are illustrated in terms of Nusselt number, isotherms, stream functions, and entropy generation contours. According to the obtained results, the following remarks can be listed:
• •
14
the experimental data [38] for different Rayleigh numbers (105 − 107) and volume fraction (0 % − 4%). In all Ra and Da numbers for both horizontal and vertical eccentricity, by increasing the volume fraction of nanoparticles, ϕ, the average Nusselt number is increased and its maximum value is achieved at ϕ =0.04. Downward eccentricity will strengthen the convection field, while the horizontal and upward eccentricity will weaken it. For weak convection fields (low Da or Ra numbers), increasing the eccentricity will increase Nu number, while, for strong convection fields (high Da and Ra numbers), increasing the eccentricity will decrease Nu. The only exception is the downward eccentricity, in which the heat transfer rate is always a direct relation with the eccentricity. Use of eccentric annuli is more beneficial at low Ra numbers, where higher heat transfer enhancement could be obtained. The highest improvement due to eccentricity (80%) is observed in the case with Da=10‐1 and Ra=104 inside a downward eccentric annulus with ζ= − 0.4. An optimal eccentricity (around |ζ| = 0.1 and 0.2) exists to minimize the entropy generation. For low convection fields (low Da or Ra numbers), increasing the eccentricity highly affects the entropy generation. However, for strong convection fields (high Ra and Da numbers), the eccentricity effect on entropy generation is less.
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Fig. 14. Contours of the total entropy generation rate for Da=10‐4 and ϕ=0.04
• Use of the downward eccentric annulus with the optimal eccen-
According to this study, it was shown that the source eccentricity in annuli could substantially affect the heat transfer and flow characteristics. Hence, it is suggested to focus more on its effects when designing annular heat exchangers. Also, the effect of nanofluid sedimentation in
tricity and filled with porous media is always recommended to minimize the entropy generation and improve the heat transfer rate.
15
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Fig. 15. Contours of the total entropy generation rate for Da=10‐1 and ϕ =0.04
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naturally driven flows [46], and also the other affecting forces such as thermophoresis and Brownian effects can be considered in eccentric annuli [47].
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Loukopoulos, et al., Heat transfer and natural convection of nanofluids in porous media, Eur. J. Mech. B/Fluids 43 (2014) 45–56. [27] R. Yaghoubi Emami, M. Siavashi, G. Shahriari Moghaddam, The effect of inclination angle and hot wall configuration on Cu-water nanofluid natural convection inside a porous square cavity, Advanced Powder Technology 29 (2018) 519–536. [28] Q. Sun, I. Pop, 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) 2141–2153. [29] M. Siavashi, R. Yousofvand, S. Rezanejad, Nanofluid and porous fins effect on natural convection and entropy generation of flow inside a cavity, Advanced Powder Technology 29 (2018) 142–156. [30] C. Qi, L. Liang, Z. 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Tang, Three-dimensional Lattice Boltzmann simulation on thermosolutal convection and entropy generation of Carreau-Yasuda fluids, International Journal of Heat and Mass Transfer 131 (2019) 346–364. [35] X. Wang, X. Xu, S.U. S. Choi, Thermal conductivity of nanoparticle-fluid mixture, Journal of thermophysics and heat transfer 13 (4) (1999) 474–480. [36] R. Prasher, D. Song, J. Wang, et al., Measurements of nanofluid viscosity and its implications for thermal applications, Applied physics letters 89 (13) (2006) 133103. [37] İ.O. Sert, N. Sezer-Uzol, Numerical analysis of convective heat transfer of nanofluids in circular ducts with two-phase mixture model approach, Heat and Mass Transfer 52 (9) (2016) 1841–1850. [38] M. Jahanshahi, S.F. Hosseinizadeh, M. Alipanah, et al., Numerical simulation of free convection based on experimental measured conductivity in a square cavity using Water/SiO2 nanofluid, Int. Commun. Heat Mass Transf. 37 (2010) 687–694. [39] F. Garoosi, F. Hoseininejad, Numerical study of natural and mixed convection heat transfer between differentially heated cylinders in an adiabatic enclosure filled with nanofluid, Journal of Molecular Liquids 215 (2016) 1–17. [40] M. Siavashi, H.R. Talesh Bahrami, E. Aminian, Optimization of heat transfer enhancement and pumping power of a heat exchanger tube using nanofluid with gradient and multi-layered porous foams, Applied Thermal Engineering 138 (2018) 465–474. [41] M. Corcione, Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, Energy Conversion and Management 52 (2011) 789–793. [42] G. Kefayati, Heat transfer and entropy generation of natural convection on nonNewtonian nanofluids in a porous cavity, Powder technology 299 (2016) 127–149. [43] M. Salari, E.H. Malekshah, M.H. Esfe, Three dimensional simulation of natural convection and entropy generation in an air and MWCNT/water nanofluid filled cuboid as two immiscible fluids with emphasis on the nanofluid height ratio's effects, Journal of Molecular Liquids 227 (2017) 223–233. [44] D.B. Ingham, I. Pop, Transport phenomena in porous media III, Elsevier, 2005. [45] P. Nithiarasu, K. Seetharamu, T. Sundararajan, Natural convective heat transfer in a fluid saturated variable porosity medium, International Journal of Heat and Mass Transfer 40 (1997) 3955–3967. [46] C.J. Ho, D.S. Chen, W.M. Yan, et al., Buoyancy-driven flow of nanofluids in a cavity considering the Ludwig–Soret effect and sedimentation: Numerical study and experimental validation, Int. Commun. Heat Mass Transf. 57 (2014) 22–26. [47] C.J. Ho, D.S. Chen, W.M. 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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] T.L. Bergman, F.P. Incropera, D.P. DeWitt, et al., Fundamentals of heat and mass transfer, John Wiley & Sons, 2011. [2] H.F. Öztop, P. Estellé, W.-M. Yan, et al., A brief review of natural convection in enclosures under localized heating with and without nanofluids, Int. Commun. Heat Mass Transf. 60 (2015) 37–44. [3] M.H. Toosi, M. Siavashi, Two-phase mixture numerical simulation of natural convection of nanofluid flow in a cavity partially filled with porous media to enhance heat transfer, Journal of Molecular Liquids 238 (2017) 553–569. [4] R. Mohebbi, S.A.M. Mehryan, M. Izadi, et al., Natural convection of hybrid nanofluids inside a partitioned porous cavity for application in solar power plants, J. Ther. Anal. Calorim. (2019) 1–15. [5] J. Belabid, S. Belhouideg, S. Allali, et al., Numerical simulation for impact of copper/water nanofluid on thermo-convective instabilities in a horizontal porous annulus, J. Ther. Anal. Calorim. (2019) 1–11. [6] M. Siavashi, H.R.T. Bahrami, H. Saffari, Numerical investigation of flow characteristics, heat transfer and entropy generation of nanofluid flow inside an annular pipe partially or completely filled with porous media using two-phase mixture model, Energy 93 (2015) 2451–2466. [7] K.C. Lin, A. Violi, Natural convection heat transfer of nanofluids in a vertical cavity: Effects of non-uniform particle diameter and temperature on thermal conductivity, International Journal of Heat and Fluid Flow 31 (2010) 236–245. [8] J. Eastman, U. Choi, S. Li, et al., Enhanced thermal conductivity through the development of nanofluids, MRS Online Proc. Libr. Arch. 457 (1996). [9] S. Göktepe, K. Atalık, H. Ertürk, Comparison of single and two-phase models for nanofluid convection at the entrance of a uniformly heated tube, International Journal of Thermal Sciences 80 (2014) 83–92. [10] M. Siavashi, A. Rostami, Two-phase simulation of non-Newtonian nanofluid natural convection in a circular annulus partially or completely filled with porous media, International Journal of Mechanical Sciences 133 (2017) 689–703. [11] M. Vahabzadeh Bozorg, M. Siavashi, Two-phase mixed convection heat transfer and entropy generation analysis of a non-Newtonian nanofluid inside a cavity with internal rotating heater and cooler, International Journal of Mechanical Sciences 151 (2019) 842–857. [12] M. Siavashi, K. Karimi, Q. Xiong, et al., Numerical analysis of mixed convection of two-phase non-Newtonian nanofluid flow inside a partially porous square enclosure with a rotating cylinder, Journal of Thermal Analysis and Calorimetry 137 (1) (2019) 267–287. [13] P. Hanafizadeh, M. Ashjaee, M. Goharkhah, et al., The comparative study of single and two-phase models for magnetite nanofluid forced convection in a tube, Int. Commun. Heat Mass Transf. 65 (2015) 58–70. [14] H.A. Mohammed, I.A.M. Ali Abuobeida, H.B. Vuthaluru, et al., Two-phase forced convection of nanofluids flow in circular tubes using convergent and divergent conical rings inserts, Int. Commun. Heat Mass Transf. 101 (2019) 10–20. [15] O. Mahian, L. Kolsi, M. Amani, et al., Recent advances in modeling and simulation of nanofluid flows-Part I: Fundamentals and theory, Physics Reports 790 (2019) 1–48. [16] O. Mahian, L. Kolsi, M. Amani, et al., Recent advances in modeling and simulation of nanofluid flows—Part II: Applications, Physics Reports 791 (2019) 1–59. [17] D.S. Bondarenko, M.A. Sheremet, H.F. Oztop, et al., Mixed convection heat transfer of a nanofluid in a lid-driven enclosure with two adherent porous blocks, Journal of Thermal Analysis and Calorimetry 135 (2019) 1095–1105. [18] M. Sheikholeslami, H.F. Öztop, N. Abu-Hamdeh, et al., Nanoparticle transportation of CuO-H2O nanofluid in a porous semi annulus due to Lorentz forces, Int. J. Num. Methods Heat Fluid Flow 29 (2019) 294–308. [19] F. Selimefendigil, F. Bayrak, H.F. Oztop, Experimental analysis and dynamic modeling of a photovoltaic module with porous fins, Renewable Energy 125 (2018) 193–205. [20] S. Rashidi, M.H. Kashefi, K.C. Kim, et al., Potentials of porous materials for energy management in heat exchangers – A comprehensive review, Applied Energy 243 (2019) 206–232. [21] S. Rashidi, J.A. Esfahani, N. Karimi, Porous materials in building energy technologies—A review of the applications, modelling and experiments, Renewable
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Eccentricity effects of heat source inside a porous annulus on the natural convection heat transfer and entropy generation of Cu-water nanofluid
T
Payam Gholamalipoura, Majid Siavashia, , Mohammad Hossein Doranehgardb ⁎
a b
Applied Multi-Phase Fluid Dynamics Laboratory, School of Mechanical Engineering, Iran University of Science & Technology, Tehran, Iran Department of Civil and Environmental Engineering, School of Mining and Petroleum Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada
ARTICLE INFO
ABSTRACT
Keywords: Eccentricity Annulus Porous media Natural convection Nanofluid Entropy generation
This study presents a numerical investigation on free convection heat transfer and entropy generation of Cuwater nanofluid inside an annulus, fully filled with a porous foam, in presence of a cylindrical heat source. For accurate nanofluid flow simulation, the two-phase mixture model is applied, and the viscosity and thermal conductivity of the mixture are computed based on the empirical Corcione's models. The effects of the vertical and horizontal heat source eccentricity (−0.4 ≤ ζ ≤0.4) and nanoparticles volume fraction (0 ≤ ϕ ≤ 0.04) for different values of Darcy (10−4 ≤ Da ≤ 10−1) and Rayleigh (103 ≤ Ra ≤ 106) numbers on the heat transfer and entropy generation are discussed. Results indicate that for various figures of eccentricity, heat transfer can be improved or deteriorated, depending on the value of Da, Ra and direction of the inner cylinder movement. Consequently, for each Rayleigh number, optimal values of Da and ζ exist to meet the maximum average Nu number. In addition, an appropriate eccentricity – around ζ =0.1 and 0.2 – exists to minimize the total entropy generation. Moreover, the downward eccentric annulus expresses the best performance in which the highest heat transfer and lowest entropy generation occur.
1. Introduction Free convection, in a thermodynamic system, is a mechanism of heat transport in which the fluid motion is not generated by an external force (exerted by tools such as pumps, fans or suction devices), but happens by density differences in the fluid occurring as a result of temperature gradients [1,2]. Due to the low costs of maintenance and minimum noises, this type of heat transfer has a wide range of applications [3–5]. In this regard, many scientists focused on natural convection to investigate its various aspects and many papers have been published in this field. Furthermore, for responding to the increasing demands for energy and limited energy resources, researchers have obliged to use several approaches in order to enhance heat transfer. Among the lunched methods, as coming in the following, using nanoparticles and porous foams have received a lot of attention due to their simplicity of usage and acceptable efficiency [6]. Research works show that adding metallic nanoparticles to the pure base fluid can significantly increase the thermal conductivity of the produced mixture (known as nanofluid) [7]. For instance, Eastman et al. [8] observed that thermal conductivity values of Al2O3/water and CuO/water nanofluids with 5% nanoparticles volume fraction are respectively 29% and 60% higher than that of the base fluid. At this point, ⁎
Corresponding author. E-mail address: [email protected] (M. Siavashi).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104367
0735-1933/ © 2019 Elsevier Ltd. All rights reserved.
to simulate nanofluid flow, there are two general models: 1- Singlephase model in which it is assumed that a homogeneous single-phase mixture is composed of nanoparticles and the base fluid with effective properties; 2- Two-phase models in which different velocities can be defined for nanoparticles and the carrying base fluid. Two-phase models give more precise results but requires higher computational costs. These models can be generally categorized into Eulerian-Eulerian and Eulerian-Lagrangian models. Goktepe et al. [9] showed that for nanofluid flows with low nanoparticles concentration, the Eulerian twophase mixture model provides acceptable results without need to high computational resources. This model has been recently implemented for various nanofluid flow studies and is also used in the present study [10–14]. Mahian et al. [15,16] wrote two review papers and discussed comprehensively on the theory and applications of nanofluid flows. In addition to using nanoparticles, implementing porous foams, due to improving the fluid-surface interface and increasing the random movements of fluid through porous regions, results in heat transfer enhancement [17–23]. Rong et al. [24] investigated the effects of several parameters, such as porosity and Darcy number on the flow and heat transfer enhancement inside a partially porous pipe. They concluded that adjusting porous media thickness, while the Darcy number is set to be 10−3, can provide the highest improvement in the heat
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Nomenclature Cp dp D Da fdrag g Κ k kb Nu Nu⁎ P Pr q” Ra Sgen S⁎ T T0 u, v x, y
Subscripts
specific heat (J kg−1 K−1) diameter of the nanoparticle (m) Diameter of the outer cylinder Darcy number friction factor gravity (ms−2) permeability (m2) thermal conduction (W m−1 K−1) Boltzmann constant (J K−1) Nusselt number Nu to concentric Nu ratio pressure (Pa) Prandtl number heat flux (W m−2) Rayleigh number entropy generation rate (W m−3 K−1) relative entropy generation temperature (K) bulk temperature (K) velocity components, (m s−1) Cartesian coordinator (m)
avg c dr f eff h m p s tot
average cold drift base fluid effective hot mixture (nanofluid) nanoparticle solid medium total
Greek letters α β ε ζ θ η μ ρ ϕ
transfer. Torabi et al. [25] observed the same result for a partially porous channel. In last decade, many researchers have investigated natural convection flow and heat transfer enhancement inside different geometries by concurrently using nanoparticles and porous media. Bourantas et al. [26] numerically analyzed nanofluid free convection inside a porous cavity using a meshless technique. A numerical investigation on natural convection of Cu-water nanofluid in an inclined porous square cavity is performed by Yaghoubi-Emami et al. [27]. They used two-phase mixture model and showed that use of nanofluid in the presence of porous media can improve the heat transfer at low Ra numbers, however, the opposite fact was observed at high Ra numbers. Sun and Pop [28] have studied natural convection of different nanofluids inside a porous triangular cavity in the presence of a flush mounted heater with finite size. They observed that for a low Rayleigh number, increasing volume fraction of nanofluid mixture has improved the heat transfer in the cavity. On the other hand, for high Rayleigh numbers, this trend is vice versa. Toosi and Siavashi [3] in a numerical investigation analyzed the heat transfer of natural convective nanofluid in a square cavity with
thermal diffusivity (m2/s) thermal expansion coeff. (K1) porosity Dimensionless eccentricity dimensionless temperature eccentricity of inner cylinder dynamic viscosity (kg m‐1 s‐1) Density (kg m‐3) nanoparticles volume fraction
porous layers on its walls. Their outcomes show that for different conditions of Ra and Da, the heat transfer can be maximized through optimal selection of the porous layer thickness and the nanofluid concentration. Siavashi et al. [29] have numerically investigated the impact of simultaneous use of nanoparticles and porous fins on free convection flow in a square chamber. They showed that using porous fins can boost heat transfer, from one side, and decrease irreversibility, from the other side. Some researchers have studied the impact of the geometrical parameters on heat transfer. Mohebbi et al. [4] investigated nanofluids natural convection in porous cavity. Qi et al. [30] analyzed the effect of different aspect ratios of the enclosure on the CueGa nanofluid free convection heat transfer. They observed that the average Nu has opposite relation with the enclosure aspect ratio. Selimefendigil and Oztop [31] investigated nanofluid free convective flow through numerical simulation and studied the problem from the viewpoint of the first and second laws of thermodynamics. In their study, they consider obstacles with different shapes, internal heat generation, and also existence of a magnetic field. Ma et al. [32] studied the effect of cavity's
Fig. 1. Computational domain and boundary conditions of three different configurations 2
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Table 1 Water and Cu nanoparticles thermo-physical properties (T=310 K) [39]. ρ (kg/m3) Water Cu
k (W/m.K)
993.4 8933
β (K‐1)
Cp (J/kgK)
0.635 401
‐5
4178 385
α (m2/s)
μ (kg/m.s) ‐4
36×10 1.67×10‐5
6.95×10 ‐
1.44×10 ‐
dp (nm) ‐7
‐ 20
Table 2 Present results comparison with the outcomes of Nithiarasu et al. [45] (Pr=1.0, ε = 0.4 and 0.9) Da
‐2
10 10‐2 10‐2
Ra
6
10 104 5×105
Nuavg (ε = 0.4)
Difference%
Nuavg (ε = 0.9)
Difference%
Nithiarasu et al. [48]
Present work
2.14
Nithiarasu et al. [48]
Present work
1.40 2.98 4.99
1.43 2.91 4.84
2.34 3.00 2.14
1.64 3.91 6.70
1.62 3.85 6.44
1.21 1.53 3.88
thermal characteristics and entropy generation are numerically studied. 2. Problem description Fig. 1 depicts the geometry of the studied problem, a circular eccentric annulus with outer and inner diameters of D and D/2, respectively. Dimensionless eccentricity is defined as ζ = D in which η is the 2
eccentricity distance. The inner cylinder is assumed to have horizontal or vertical eccentricity and its value is in the range of ‐0.4≤ ζ ≤ 0.4. The outer wall (cold wall) has the constant temperature of Tc = 305 K. While, the inner wall is hot and has a constant temperature (Th = 315 K). The space between the inner and the outer cylinders is fully filled by a porous foam with porosity of ε=0.9. Some reports exist that nanofluids with high amount of nanoparticles volume fraction could exhibit non-Newtonian behavior, even with a Newtonian base fluid. However, in several numerical and experimental studies in which the nanofluid volume fraction is less than 4%, it is mentioned that the nanofluid with a Newtonian base fluid behaves as a Newtonian fluid [35–38]. In the present study, a water based Newtonian nanofluid containing Cu nanoparticles with volume fractions of 0% to 4% has saturated the porous region between two annuli. Table 1 [39] depicts the thermo-physical properties of water and nanoparticles.
Fig. 2. The average Nu values of the present work are compared with results of Jahanshahi et al. [38].
aspect ratio on free convection of nanofluid within a U-shaped chamber equipped with a heating obstacle. They observed that as the aspect ratio increased, the positive impact of nanoparticles on the heat transfer was enhanced. Magami et al. [33] focused on effects of width ratio of a cavity on heat transfer in a square porous-annulus. They observed that Nu has a direct relationship with the height of the cavity. Kefayati and Tang [34] studied the effect of horizontal and vertical distance of the inner cylinder from the center of a heated enclosure on natural convection and entropy generation of Carreau fluid. Their results showed that horizontal or vertical alteration of the center of the cylinder has a remarkable impact on heat and mass transfer. Based on the above literature review, studying the effects of horizontal and vertical eccentricities of fluid flow inside annular cavities seems to be important. However, to the authors' knowledge, the impact of horizontal and vertical eccentricity on free convective heat transfer and entropy generation of nanofluid flow in a cylindrical annular chamber containing a porous foam, has not been studied and is worthy of investigation. Accordingly, this study aims to investigate this problem in the presence of a heat source by using the two-phase mixture model, for nanofluid, and also considering Darcy-BrinkmanForchheimer equation for flow inside the porous foam. Hence, the effects of nanoparticles volume fraction, horizontal and vertical movements of the inner cylinder, Rayleigh and Darcy number on hydro-
3. Mathematical modeling Regarding mathematical modeling, governing equations for fluid flow of a Newtonian nanofluid in porous media will be discussed. 3.1. Governing equations For nanofluid flow through porous media, the non-dimensional form of the mass conservation equation is given by [27,40]:
um v + m =0 x y
(1)
The momentum equations are presented by: um =
um u + vm m x y 2 p +
x
µ u Pr m m + µf x
f m
um | Vm| + 2 (1
x )
f m
udr , f
udr , f x
y
µ u Pr m m µf y
2 Pr
udr , f
p
+ vdr , f
y
f µm um Da m µf
+ 2
m
udr , p
udr , p x
1.75 150 + vdr , p
1/2
Da udr , p y
(2) 3
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Fig. 3. Comparisons of streamlines and isotherm contours of the present work and those of experimental data (modified from Jahanshahi et al. [38]) for Ra= 106 and ϕ =0.04. Table 3 Grid independence study for the case with ζ=0.4, ϕ = 0.04, Ra = 105 and Da = 10−2. Mesh size
45×320
50×376
54×450
58×550
65×660
70×800
75×960
Nuavg
14.257
14.252
14.250
14.242
14.240
14.238
14.237
um
mixture. Dimensionless variables in the aforementioned equations are defined as follows [27]:
vm v + vm m x y 2 p +
=
m
x
vm | Vm| + 2 (1 udr , p
vdr , p x
µ v µ v Pr m m + Pr m m µf x y µf y
f
y
)
f m
+ vdr , p
vdr , f
udr , f
vdr , p y
x
+ vdr , f
vdr , f
f µm vm Da m µf
2 Pr
+ 2
y
1.75 150
1/2
Da
x =
p m
( )m f + 2 RaPr ( )f m
x
( ) + vm
y
( )=
keff x
kf
x
+
Pr =
kf
y
( vm ) + y
( udr , f ) x
+
( vdr , f ) y
=0
T Th
Tc Tc
(6)
, Ra =
g
f
Tc ) D3
(Th f
f
, Da =
K D2
(7)
More details about the Eulerian two-phase mixture model can be found in [27,40]. Corcione [41] used a wide variety of experimental data, available in the literatures, to propose correlations for estimation of nanofluid’s effective dynamic viscosity and thermal conductivity. The data used for thermal conductivity consists of various nanoparticles in the range between 10 nm and 150 nm, with water and ethylene glycol (EG) as the
(4)
And the volume fraction equation is:
( um ) + x
f f
keff y
=
Also, Prandtl (Pr), Rayleigh (Ra) and Darcy (Da) numbers are defined by the following relations:
(3)
The energy equation is as follows:
um
pD 2 uD x y vD ,y = ,u= ,v = ,p = , 2 D D f f f f
(5)
where, subscripts p and m respectively represent the particle and 4
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Concerning entropy generation, the entropy generation rate is defined by the following relation [42,43]: (13)
stot = sT + sF sT′′′
where, is known as the thermal entropy generation rate and is written as:
sT =
km T02
T x
2
T y
+
2
(14)
sF′′′
is called the frictional entropy generation rate and can be evaluated as follows [44]:
sF =
µm T0
2
um x
2
+2
m
y
2
+
m
x
+
um y
2
(15)
The dimensionless entropy generation rate (Stot′′′) is defined by:
Stot =
S=
base fluid. In order to propose the correlation for the effective dynamic viscosity the data of nanoparticles used in nanofluids is in the range of 25 nm to 200 nm. This correlation is the best fit in the mentioned range of used data, however, it could be even valid for a wider range and other nanofluids with spherical nanoparticles. In this study Cu nanoparticles with 20 nm diameter are used, and Corcione’s correlation for estimation of the effective thermal conductivity is valid. Also, the size of our nanoparticles is very close to the range of data used by Corcione for the effective dynamic viscosity and could provide results with enough accuracy. Accordingly, Corcion’s correlations are used in this study and his relation for the effective dynamic viscosity is presented as follows:
dp
34.87
1.03 p
T Tfr
kp kf
4.2. Grid independence check To check the grid independency of numerical algorithm, an analysis for grid sensitivity is done to provide answers with enough precisely and an appropriate computational cost. As is illustrated in Table 3, seven structured computational grids, including 45×320, 50×376, 54×450, 58×550, 65×660, 70×800, and 75×960 grid cells (number of cells is counted in θ- and r-directions, respectively) are used to conduct the grid independence study. It’s worthy to mention that the case with ζ=0.4, ϕ = 0.04, Ra = 105, and Da = 10−2 is investigated for grid independence. Finally, according to the reported average Nu values in Table 3, the grid with 65×660 cells could result in grid independent outcomes (as is shown in Fig. 4).
0.03 0.66 p
(9)
where ReB, Pr, Tfr, and kp are the nanoparticle Reynolds number, the Prandtl number of the base liquid, the freezing point of the base liquid, and the nanoparticle’s thermal conductivity, respectively. The nanoparticle Reynolds number is given by:
ReB =
p uB dp
µf
(10)
ρf and μf, are the mass density and dynamic viscosity of the base fluid, respectively, and uB is the mean Brownian velocity which is defined by:
uB =
2kB T µf dp2
5. Results and discussion As previously mentioned, the convective heat transfer and entropy generation of nanofluid flow in a cylindrical annulus, filled with porous media, have been numerically investigated considering the effects of four parameters: nanoparticles volume fraction (0 ≤ ϕ ≤ 0.04), horizontal and vertical eccentricity of the inner cylinder (‐0. 4 ≤ ζ ≤ 0. 4), Rayleigh number (103 ≤ Ra ≤ 106), and Darcy number (10‐4 ≤ Da ≤ 10‐1). Results of this section are presented in three subsections: first, the impact of using nanoparticles on heat transfer
(11)
In the above relation, kB is Boltzmann constant of the nanoparticle. The average Nusselt number is calculated by the following relation:
Nuavg , wall =
qwall D kf (Th
Tc )
(17)
To validate the implemented numerical solver, two problems are considered. First, natural convection of a pure fluid flowing in a porous square chamber is simulated. The results of this simulation with those of a similar research work done by Nithiarasu et al. [45] are compared in Table 2. As can be seen, a good agreement is observed in the average Nu number. In the next step, free convection of nanofluid (SiO2-water) flow in a square cavity is simulated with the two-phase mixture model. The results are compared with those of Jahanshahi et al. [38] (see Fig. 2). Also, the stream functions and isotherms of the results have been compared in Fig. 3. According to this figure, the results are in a satisfactory agreement.
(8)
10
Stot dV
4.1. Validation
where dp, df, and ϕp are the nanoparticle diameter, the equivalent diameter of the base fluid molecule, and volume fraction of the nanoparticles, respectively. The correlation for the effective thermal conductivity is proposed by:
keff = kf 1 + 4.4ReB0.4 Pr 0.66
V
4. Numerical procedure and validation
0.3
df
(16)
The total entropy generation in the non-dimensional form is calculated by:
Fig. 4. The computational structured-grid with 65×660 cells to solve the problem
µm = µ f / 1
T02 H 2 stot kf (Th Tc ) 2
(12) 5
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Fig. 5. Variation of the average Nusselt number with different Ra, ζ, and ϕ values for Da=10‐1.
vertical directions, at Da=10‐1. Figs. 6 and 7 respectively show the impact of nanoparticles addition (0≤ ϕ ≤0.04) to the base fluid on the patterns of isotherms and stream functions of case B (horizontal movement of the inner cylinder), at various ζ and Rayleigh numbers. According to Fig. 5, it can be seen that for the case with horizontal eccentricity and for all Ra, Da and ζ values, by increasing ϕ, the average Nusselt number is increased and its maximum value is observed at ϕ =0.04. From Fig. 6, it can be observed that by increasing Ra for the case with horizontal eccentricity, the isotherms deviate from their circular curved shape which represents the increasing heat convection and decreasing heat conduction. This fact is also endorsed by Fig. 7 as by
improvement is investigated, then, the effect of eccentricity on the heat transfer characteristics of the nanofluid is analyzed for different cases, and finally, based on the second law of thermodynamics the results are discussed. 5.1. Nanofluid effects on heat transfer enhancement In this part, the impact of nanoparticle addition into the base fluid on heat transfer is discussed. At this point and for briefness, only horizontal eccentricity has been studied. Fig. 5 presents the average Nu values in terms of ϕ, ζ, Ra and eccentricity values in horizontal and 6
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Fig. 6. Contour of isotherms for Da=10‐1 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
increasing Ra for the horizontal eccentricity, ψnfmax has been increased. According to Fig. 6, increasing ζ from 0 to 0.4, in addition to the increment in the convection heat transfer in the left side of the inner cylinder (regarding to the isotherms deviated from their spiral shape), leads to the improvement of heat conduction in the right side of the inner cylinder as the isotherms are more condensed and their density is much higher between the two walls. This outcome is confirmed by Fig. 7 as by increasing ζ, isotherms in the left side of the inner cylinder
become closer to each other. As depicted in Figs. 5 and 6, the effect of raising Rayleigh number on heat transfer and isotherms variations – due to the increment in the nanofluid volume fraction – is different for various amounts of ζ. Increasing ϕ from 0 to 0.04 at Ra=105, the maximum variation between isotherms of the base fluid and the nanofluid (the maximum improvement caused by adding nanoparticles to the base fluid) for the cases with horizontal eccentricity values of ζ =0, 0.1 and 0.2 can respectively 7
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Fig. 7. Contour of stream functions at Da=10‐1 (solid lines for ϕ=0.04 and dashed lines for ϕ= 0.0)
lead to 12%, 12.2%, and 11.7% enhancement in the heat transfer. In addition, at Ra=104 the maximum effect of nanoparticle addition on enhancing the heat transfer are 13% and 12.3%, respectively, that corresponds to the cases with ζ=0.3 and 0.4. Fig. 7 confirms these
outcomes as for the aforementioned cases (in which the differences between isotherms are maximum), due to the positive effects of nanoparticles, stream functions are compressed near the walls which this fact leads to the velocity increment and improvement of the heat 8
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Fig. 8. Variations of the relative Nusselt number, Nu⁎avg, with different Da, Ra, and ζ for ϕ =0.04.
convection process. In all of the above-mentioned cases, according to the isotherms exhibited in Fig. 6, the remarkable positive effect of increasing the nanofluid volume fraction on the heat transfer is mainly due to the fact that the convection is not extremely powerful in the way that viscous effects become sensible. In addition, conduction effects are relatively considerable so that the increment in the heat conductivity of the base fluid, due to the nanofluid volume fraction increment, plays an important role in the total heat transfer rate. However, by increasing Ra to 106 (at a constant amount of ζ), as the convection increases, the nanofluid concentration enhancement leads to decline in the positive effects of nanofluid on heat transfer, as a result of the increasing viscosity. Fig. 6 shows that for Ra=103 and 104, by increasing ζ, the dashed lines are more deviated and, hence, the maximum impact of nanoparticles on isotherms can be observed at the highest amount of eccentricity (ζ =0.4). On the other hand, at Ra=105 and 106 this phenomenon is vice versa and the impact of increment of nanoparticles concentration on heat transfer is slightly decreased by increasing ζ, due to the existence of relatively powerful natural convection and high viscosity dissipations. Fig. 7 endorses this result as for Ra=103 and 104, by increasing ζ, isotherms are more compressed to the walls, and hence the heat convection is boosted.
mainly includes horizontal and vertical movements of the cylinder at different Darcy and Rayleigh numbers for ϕ =0.04, in which the maximum heat transfer enhancement is obtained. In this regard, Fig. 8 depicts variations of the Nu⁎avg (the ratio of Nuavg for the eccentric annulus to that of the concentric annulus) as a function of Ra and ζ, for different Da numbers at ϕ =0.04. In addition, Figs. 9 and 10 represent non-dimensional isotherm and stream function contours for various Ra and ζ numbers at Da=10‐4, respectively. The same contours are also exhibited in Figs. 11 and 12 for Da=10‐1. As can be seen in Fig. 8, results of the leftward- and rightwardhorizontal eccentricity are the same, as the consequence of the symmetry. Figs. 9 and 10 also depict the similarity of the streamlines and isotherms of these two states. As exhibited, horizontal eccentricity enhances the heat transfer rate. In contrast, for the vertical eccentricity, the upward and downward movement of the hot cylinder will lead to different outcomes. Investigation of Fig. 8 reveals that for majority of the cases, the eccentricity would enhance the heat transfer rate. However, the verticalupward eccentricity can act inversely and slightly reduce the Nusslet number. For the cases with a weak flow field (low Da or Ra numbers), this negative impact cannot be observed. However, by enhancing the flow strength (higher Da and Ra values), the negative impact of upward eccentricity on the heat transfer rate will be appeared, and the worst condition is observed when ζ =0.2. Results of Figs. 10 and 12 also confirm the mentioned point, since for the cases with upward eccentricity the maximum strength of the stream function is reduced with respect to the concentric case. In contrast, downward eccentricity
5.2. Eccentricity effects on heat transfer In this section, the eccentricity of the inner cylinder and its effects on heat transfer, isotherms and streamlines are studied. This study 9
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Fig. 9. Contour of isotherms at Da=10‐4 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
strengthens the flow field and improves the heat transfer rate. Eccentricity of the inner cylinder in presence of the metal foam can affect the heat transfer with two mechanisms. In the first mechanism, it affects the conduction through the porous medium. Increasing ζ will make a short connection between the hot and cold surfaces and provides a high conductive region for heat transfer. Hence, for the cases with dominant conduction heat transfer (low Da and Ra numbers), increasing ζ improves the heat transfer rate. In the second mechanism, the increased eccentricity can influence the convection field of natural
convection. As depicted in Figs. 10 and 12, only downward eccentricity will strengthen the convection field, and in the other cases the eccentricity weakens the strength of the convection field. Accordingly, for convection dominated flows (large Da and Ra), increasing ζ slightly affects the heat transfer rate, while, the highest enhancement due to increasing the eccentricity is observed in the conduction dominated cases (low Da or Ra values). The balance of conduction and convection and their roles in the overall heat transfer rate will result in the value of Nu⁎avg. Results of Fig. 8, depicts between 10% to 80% enhancement in 10
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Fig. 10. Contour of stream functions for Da=10‐4 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
the heat transfer due to eccentricity. In addition, the highest enhancement corresponds to the downward movement of the inner cylinder. For Da=10‐4, the Ra effect on Nu⁎avg is minimum and by increasing Da,
the role of Ra in Nu⁎avg becomes more evident. As the consequence, the range of variation of Nu⁎avg for various Ra numbers is confined for Da=10‐4, while, changing Ra at Da=10‐1 substantially affects Nu⁎avg. 11
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Fig. 11. Contour of isotherms at Da=10‐1 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
5.3. Eccentricity effects on entropy generation
total entropy generation is considerably lower than the role of thermal irreversibility, hence, the total entropy generation is mostly influenced by the temperature gradients. As was observed in Fig. 9, high eccentricity values led to high temperature gradient zones. For instance, at |ζ| = 0.4, the temperature gradient is much greater than that of the case with |ζ| = 0.2. Therefore, as is illustrated in Fig. 14, a high entropy generation rate region is formed around the hot surface and the entropy generation rate of the cases with |ζ| = 0.4 is higher than that of the cases with |ζ| = 0.2. Increasing Ra number will increase the frictional
Fig. 13 shows S* values (the ratio of the total entropy generation of the eccentric annulus to that of the concentric annulus) as a function of Ra and ζ, for various Da numbers at ϕ =0.04. Through this figure, the effect of eccentricity on entropy generation can be investigated. Figs. 14 and 15 illustrate the total entropy generation rate in the annulus, for various Ra numbers at Da=10‐4 and 10‐1, respectively. In low Ra numbers, the role of friction entropy generation in the 12
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Fig. 12. Contour of stream functions for Da=10‐1 (solid lines for ϕ =0.04 and dashed lines for ϕ = 0.0)
irreversibility and results in the higher entropy generation rate (see Fig. 14). Increasing the eccentricity highly affects the temperature gradients, hence, for the cases with dominant thermal irreversibility it
would lead to the higher entropy generation. Accordingly, it is expected to see higher S* values at lower Ra numbers, in which the thermal irreversibility is dominant (see Fig. 13). 13
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Fig. 13. Variations of the relative entropy generation, S⁎, with different Da, Ra and ζ for ϕ =0.04
• The two-phase mixture model provides acceptable results relative to
Increasing Da number will improve the natural convection field and makes higher the frictional entropy generation rate. As the consequence, the high entropy generation rate regions for the cases with Da=10‐1 (see Fig. 15) are appeared in the stagnation points and boundary layers, where the velocity gradients are high. An important point that can be deduced Fig. 13 is that an optimal eccentricity value exists to minimize the total entropy generation. This optimal value is around |ζ| = 0.1 and 0.2, depending on the value of Da and Ra numbers. Increasing the absolute value of eccentricity can increase the flow and temperature gradients in one side and decrease them on the other side of the inner cylinder. Hence, it is expected to have an optimal point to meet the minimum entropy generation.
• • •
6. Conclusion
•
Effects of the vertical and horizontal eccentricity of the heat source and nanoparticles volume fraction for various Rayleigh (Ra) and Darcy (Da) numbers on the heat transfer and entropy generation of natural convection nanofluid flow inside a fully porous annulus are studied. Two-phase mixture model and Darcy-Brinkman-Forchheimer relations are implemented to simulate nanofluid flow and heat transfer through porous media. Numerical results were validated via comparison with those of experimental works and results are illustrated in terms of Nusselt number, isotherms, stream functions, and entropy generation contours. According to the obtained results, the following remarks can be listed:
• •
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the experimental data [38] for different Rayleigh numbers (105 − 107) and volume fraction (0 % − 4%). In all Ra and Da numbers for both horizontal and vertical eccentricity, by increasing the volume fraction of nanoparticles, ϕ, the average Nusselt number is increased and its maximum value is achieved at ϕ =0.04. Downward eccentricity will strengthen the convection field, while the horizontal and upward eccentricity will weaken it. For weak convection fields (low Da or Ra numbers), increasing the eccentricity will increase Nu number, while, for strong convection fields (high Da and Ra numbers), increasing the eccentricity will decrease Nu. The only exception is the downward eccentricity, in which the heat transfer rate is always a direct relation with the eccentricity. Use of eccentric annuli is more beneficial at low Ra numbers, where higher heat transfer enhancement could be obtained. The highest improvement due to eccentricity (80%) is observed in the case with Da=10‐1 and Ra=104 inside a downward eccentric annulus with ζ= − 0.4. An optimal eccentricity (around |ζ| = 0.1 and 0.2) exists to minimize the entropy generation. For low convection fields (low Da or Ra numbers), increasing the eccentricity highly affects the entropy generation. However, for strong convection fields (high Ra and Da numbers), the eccentricity effect on entropy generation is less.
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Fig. 14. Contours of the total entropy generation rate for Da=10‐4 and ϕ=0.04
• Use of the downward eccentric annulus with the optimal eccen-
According to this study, it was shown that the source eccentricity in annuli could substantially affect the heat transfer and flow characteristics. Hence, it is suggested to focus more on its effects when designing annular heat exchangers. Also, the effect of nanofluid sedimentation in
tricity and filled with porous media is always recommended to minimize the entropy generation and improve the heat transfer rate.
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Fig. 15. Contours of the total entropy generation rate for Da=10‐1 and ϕ =0.04
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naturally driven flows [46], and also the other affecting forces such as thermophoresis and Brownian effects can be considered in eccentric annuli [47].
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