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Free convection in a porous horizontal cylindrical annulus with a nanofluid using Buongiorno’s model
Computers & Fluids 118 (2015) 182–190
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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Free convection in a porous horizontal cylindrical annulus with a nanofluid using Buongiorno’s model Mikhail A. Sheremet a,b, Ioan Pop c,⇑ a
Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia c Department of Applied Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania b
a r t i c l e
i n f o
Article history: Received 10 December 2014 Received in revised form 1 April 2015 Accepted 8 June 2015 Available online 21 June 2015 Keywords: Free convection Horizontal annulus Porous medium Nanofluids Buongiorno model Numerical method
a b s t r a c t Natural convection flow in a porous concentric horizontal annulus saturated with a water based nanofluid is numerically investigated. The mathematical model used is of single-phase and is formulated in dimensionless stream function and temperature taking into account the Darcy–Boussinesq approximation and the nanofluid model proposed by Buongiorno. The transformed dimensionless partial differential equations have been solved using a second-order accurate finite-difference technique. The results indicate that inclusion of nanoparticles into pure water changes the flow structure at low values of the Rayleigh number. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Porous media appear everywhere. Examples range from civil, chemical and geological engineering until an essential part of our daily lives to live comfortably [1]. A few examples of porous media are: soils, aquifers, sands, clothing, filters and catalytic converters in cars, ground water flow, etc. Convective flow and heat transfer is, generally, prevalent in fields of physics and engineering, such as geothermal reservoirs, float glass production, flow and heat transfer in solar ponds, air-conditioning in rooms and cooling of electronic devices [2]. Natural or free convection in a porous cavity has received considerable attention in recent years because of its relation to the thermal performance of many engineering installations [3]. The phenomenon of natural convection in porous media is a fundamental transport mechanism encountered in a wide range of engineering, geophysics, and scientific applications, such as packed bed solar energy storage, fibrous and granular insulation systems, water reservoirs and post-accident cooling of nuclear reactors [4]. Despite the fact that buoyant convection in this system was first studied about 40 years ago, there has lately been renewed interest in such flow in porous cavities owing to its importance in environmental and energy management problems ⇑ Corresponding author. Tel.: +40 264 405300; fax: +40 264 591906. E-mail address: [email protected] (I. Pop). http://dx.doi.org/10.1016/j.compfluid.2015.06.022 0045-7930/Ó 2015 Elsevier Ltd. All rights reserved.
in current scientific and geo-political context [4]. Application and control concept of flow through porous media to oil and gas reservoir simulation, geothermal energy, or groundwater remediation are important research topics in porous media nowadays [5]. As traditional fluids used for heat transfer applications such as water, mineral oils and ethylene glycol have a rather low thermal conductivity, nanofluids with relatively higher thermal conductivities have attracted enormous interest from researchers due to their potential in enhancement of heat transfer with little or no penalty in pressure drop [6]. It seems that Choi [7] is the first who introduced the term nanofluid to describe the mixture of nanoparticles and base fluid such as water, mineral oils and ethylene glycol. The addition of nanoparticles into the base fluid is able to change the flow and heat transfer capability of the liquids and indirectly increase the low thermal conductivity of the base fluid which is identified as the main obstacle in heat transfer performance. This mixture has attracted the interest of numerous researchers because of its many significant applications such as, for example, in the medical applications, transportations, microelectronics, chemical engineering, aerospace and manufacturing [8]. The convective heat transfer characteristic of nanofluids depends on the thermo-physical properties of the base fluid and the ultra fine particles, the flow pattern and flow structure, the volume fraction of the suspended particles, the dimensions and the shape of these particles. The utility of a particular nanofluid for a
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heat transfer application can be established by suitably modeling the convective transport in the nanofluid [6]. Many authors such as Kakaç and Pramuanjaroenkij [9], Wong and Leon [10], Fan and Wang [11], Jaluria et al. [12], Mahian et al. [13], Hajmohammadi et al. [14], Soleimani et al. [15], Ashorynejad et al. [16] and Seyyedi et al. [17] have presented comprehensive literature review on nanofluids. For example, Hajmohammadi et al. [14] analyzed nanofluid flow and heat transfer over a permeable flat plate using convective boundary condition. It was shown that the increase in skin friction is a considerable drawback imposed by Cu/water and Ag/water nanofluids, especially in the case of injection. Soleimani et al. [15] investigated numerically natural convection inside the semi-annulus cavity filled with a nanofluid. They found that the effect of the nanoparticles is more pronounced at low Rayleigh number than at high Rayleigh number because of greater amount of enhancement and increasing Rayleigh number leads to a decrease in ratio of heat transfer enhancement. Ashorynejad et al. [16] studied numerically on the basis of the Lattice Boltzmann method the effect of magnetic field on natural convection in a horizontal cylindrical annulus filled with a nanofluid. It has been shown that the average Nusselt number is an increasing function of nanoparticle volume fraction and Rayleigh number, while it is a decreasing function of Hartmann number. Seyyedi et al. [17] using the finite volume method numerically analyzed the natural convective heat transfer in an annulus filled with a Cu/water nanofluid. It has been found that the angle of turn for the boundary condition of the inner cylinder essentially affects the average Nusselt number. However, these papers are based on the mathematical nanofluid models proposed by Khanafer [18], and Tiwari and Das [19] for the two-phase mixture containing micro-sized particles. On the other hand, one should also mention the mathematical nanofluid model proposed by Buongiorno [20] used in many papers pioneered by Kuznetsov and Nield [21] for the free convection boundary layer flow along a vertical flat plate embedded in a porous medium and, Nield and Kuznetsov [22] for the problem of thermal instability in a porous medium layer saturated by a nanofluid. In this model, the Brownian motion and thermophoresis enter to produce their effects directly into the equations expressing the conservation of energy and nanoparticles, so that the temperature and the particle density are coupled in a particular way, and that results in the thermal and concentration buoyancy effects being coupled in the same way. We also mention here the very recently published review paper by Sakai et al. [23], where a macroscopic set of the governing equations for describing heat transfer in nanofluid saturated porous media were rigorously derived using a volume averaging theory, for possible heat transfer applications of metal foams filled with nanofluids as high performance heat exchangers. Metal foams filled with nanofluids can be one of the most promising candidates for a high performance heat exchanger needed for highly concentrated heat generating devices [24]. Literatures indicate that convection heat transfer inside concentric and eccentric annuli has many applications in science and engineering, such as electrical motor and generator, completion of an oil source and heating and cooling of underground electric cables. Recently, investigation of the effect of eccentricity on heat transfer has become a subject of interest to most researchers working in the area of convection heat transfer problems. It seems that the first person who worked on eccentric annuli was Heyda [25], and he applied a fundamental solution known as Green’s function on solving the momentum equation for a laminar flow inside an eccentric annulus. A valuable list of references on this problem can be found in the paper by Matin and Pop [26]. It is, however, worth mentioning that El-Amin et al. [27] have founded a mathematical model of nanoparticles transport in two-phase flow in
porous media based on the formulation of fine particles transport in two-phase flow in porous media proposed by Liu and Civian [28]. The principal objective of the present paper is to analyze the steady natural convection in a porous horizontal cylindrical annulus with a nanofluid using the single-phase mathematical nanofluid model proposed by Buongiorno [20]. To our best of knowledge this problem has not been considered before, so that the reported results are new and original. It is worth noting that in the present paper we use the numerical method for solution to the boundary value problem for the partial differential equations. At the same time it is possible to transform these equations to ODE’s and to solve by the semi-analytical methods [29–33]. Such algorithms [30,34,35] can allow to study the stability of the convective flows. 2. Basic equations Consider the steady free convection flow and heat transfer in a porous horizontal annulus filled with a water based nanofluid. It is assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. This prevents nanoparticles from agglomeration and deposition on the porous matrix (see Kuznetsov and Nield [21]; Nield and Kuznetsov [22]). A schematic geometry of the problem under investigation is shown in Fig. 1, where r and c are the polar system of coordinates and r2 r1 is the size of the annulus. It is assumed that the internal surface r ¼ r1 is heated and maintained at the constant temperature T h , while the external surface r ¼ r 2 is cooled and has the constant temperature T c . If we assume that the angular coordinate is measured clockwise from the vertically down position and that the problem is symmetric about a vertical plane passing through the axis of the cylinder, then consideration will be confined to the range 0 < c < p (or p < c < 2p). The basic equations for the flow, heat transfer and nanoparticles can be written in the following form (see Buongiorno [20]; Kuznetsov and Nield [21]; Nield and Kuznetsov [22]),
rV ¼0
l
h
i
V þ C qp þ ð1 C Þqf 0 ð1 bðT T c ÞÞ g K @T r þ ðV rÞT ¼ am r2 T þ d½DB rC rT þ ðDT =T c ÞrT rT @t @C 1 qp þ ðV rÞC ¼ r jp @t e 0 ¼ rp
Fig. 1. Physical model and coordinate system.
ð1Þ ð2Þ ð3Þ ð4Þ
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where V is the Darcy velocity vector, T is the temperature, C is the nanoparticle volume fraction, t is the time, p is the pressure, g is the gravitational vector, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, jp ¼ qp ½DB rCþ ðDT =T c ÞrT is the nanoparticles mass flux, qf 0 is the reference density of the fluid, am ; l; qp are the effective thermal diffusivity of the porous medium, the dynamic viscosity and nanoparticle mass den sity, d is an additional parameter defined by d ¼ e qC p p = qC p f ; C p
The mathematical model can be formulated in terms of the dimensionless variables. By using r 1 as the length scale, T h T c as the temperature scale, am as the stream function scale, C 0 as the nanoparticle volume fraction, the following dimensionless variables have been introduced:
is the heat capacity, b is the coefficient of volumetric thermal expansion, r ¼ qC p m = qC p f ; qC p f is heat capacity of the base fluid, qC p p is effective heat capacity of the nanoparticle material, qC p m is effective heat capacity of the porous medium. Further on we can linearize the momentum equation as described in detail previously [2] and write Eq. (2) as
and substituting (15) into Eqs. (12)–(14), we obtain
l
0 ¼ rp V K h i þ C qp qf 0 þ qf 0 ð1 bðT T c Þð1 C 0 ÞÞ g
ð5Þ
Eqs. (1), (3)–(5) for the problem under consideration can be written in dimensional cylindrical coordinates r; c as Þ @ v @ ðru þ ¼0 ð6Þ @r @c o l @p n ð7Þ 0¼ u C qp qf 0 þ qf 0 ½1 bðT T c Þð1 C 0 Þ g cosðcÞ @r K o l 1 @p n 0¼ v þ C qp qf 0 þ qf 0 ½1 bðT T c Þð1 C 0 Þ g sinðcÞ ð8Þ r @ c K " # @T @T v @T 1@ @T 1 @2T r þ u þ ¼ am r þ 2 2 @t @r r @c r @r @r r @c ( " 2 2 #) @C @T 1 @C @T DT @T 1 @T þ d DB þ 2 þ 2 þ ð9Þ r @ c @r @r r @ c @ c @r Tc " # @C 1 @C v @C 1@ @C 1 @2C þ r þ u ¼ DB þ 2 2 r @r @r r @ c @t e @r r @ c " # DT 1 @ @T 1 @2T r þ þ 2 2 ð10Þ r @ c T c r @r @r
; v are the velocity components along r ; c directions, where u respectively. defined by Taking into account a stream function w
¼ u
1 @w ; r @ c
v
@w ¼ @r
ð11Þ
Eq. (6) is satisfied identically. Therefore the steady governing equations in variables stream function, temperature and nanoparticle volume fraction can be written as follows: qf 0 bð1 C 0 ÞK @T qp qf 0 @C cosðcÞ 1@ @w 1 @2 w r K þ 2 2¼ g r @r @r r @ c r @c @c l l qf 0 bð1 C 0 ÞK @T qp qf 0 @C þ K g sinðcÞ @r @r l l " # @T 1 @ w @T 1 @w 1@ @T 1 @2T r ¼ am þ 2 2 r @ c @r r @r @ c r @r @r r @ c
@C @T 1 @C @T þ d DB þ 2 @r @r r @ c @ c " 2 2 #) DT @T 1 @T þ þ 2 r @ c @r Tc " # @C 1 1 @ w @C 1 @ w 1@ @C 1 @2C r ¼ DB þ 2 2 r @r @r r @ c e r @ c @r r @r @ c " # DT 1 @ @T 1 @2T þ þ 2 2 r r @ c T c r @r @r
am ; h ¼ ðT T c Þ=ðT h T c Þ; w ¼ w= u ¼ C=C 0 ; R ¼ r2 =r1 r ¼ r =r1 ;
ð15Þ
1 @ @w 1 @2w @h @u þ 2 sinðcÞ ¼ Ra Nr r r @r @r r @ c2 @r @r @h @ u cosðcÞ Nr þ Ra @c r @c 2 1 @w @h 1 @w @h 1 @ @h 1 @ h þ 2 2 r ¼ r @ c @r r @r @ c r @r @r r @c @ u @h 1 @ u @h þ Nb þ 2 @r @r r @ c @ c " 2 # 2 @h 1 @h þ Nt þ 2 @r r @c " # 1 @w @ u 1 @w @ u 1 1 @ @u 1 @2u þ 2 r ¼ r @ c @r r @r @ c Le r @r r @ c2 @r " # Nt 1 @ @h 1 @2h þ 2 2 r þ Le Nb r @r @r r @c
ð16Þ
ð17Þ
ð18Þ
where Ra is the Rayleigh number for the porous media, Le is the Lewis number, Nr is the buoyancy-ratio parameter, Nb is the Brownian motion parameter and Nt is the thermophoresis parameter, which are defined as
Ra ¼
ð1 C 0 ÞgK qf 0 bDTr1
am l
;
Nr ¼
qp qf 0 C 0
qf 0 bDT ð1 C 0 Þ dDB C 0 dDT DT am Nb ¼ ; Nt ¼ ; Le ¼ am am T c eDB
; ð19Þ
The boundary conditions for the formulated problem (16)–(18) are as follows.
w ¼ 0;
h ¼ 1;
Nb @@ru þ Nt @h ¼ 0 on r ¼ 1 @r
w ¼ 0;
h ¼ 0;
Nb @@ru þ Nt @h ¼ 0 on r ¼ R @r
w ¼ 0;
@h @c
¼ 0;
@u @c
ð20Þ
¼ 0 on c ¼ 0 and c ¼ p
The local Nusselt Nu and Sherwood Sh numbers can be defined as
@h Nu ¼ lnðRÞ ; @r r¼1
Sh ¼ lnðRÞ
@u @r r¼1
ð21Þ
and the average Nusselt Nu and Sherwood Sh numbers are ð12Þ
Nu ¼
1
Z p
p
0
Nudc;
Sh ¼
1
Z p
p
0
Shdc
ð22Þ
It should be noted here that for an analysis of Sherwood numbers it is possible to study only Nusselt numbers because at inner Nt @h cylinder surface we have @@ru ¼ Nb taking into account boundary @r conditions for u (Eq. (19)) as described in detail previously [2]. ð13Þ
3. Numerical method
ð14Þ
The partial differential Eqs. (16)–(18) with corresponding boundary conditions (20) were solved by the second order finite difference method (see Aleshkova and Sheremet [36]; Sheremet and Trifonova [37]; Sheremet and Pop [38,39]; Sheremet et al.
M.A. Sheremet, I. Pop / Computers & Fluids 118 (2015) 182–190
[40]) using non-uniform mesh for the radial coordinate. The solution for the corresponding linear algebraic equations was obtained through the successive under relaxation method. Optimum value of the relaxation parameter was chosen on the basis of computing experiments [38,39]. The computation is terminated when the residuals for the stream function get bellow 106 . The numerical methodology was coded in C++, and to check its validity, a comparison with selective data from the published literature was carried out [2,38–40]. The performance of the porous horizontal cylinder part of the model was tested against the results of Charrier-Mojtabi [41] and Khanafer et al. [42] for steady-state free convection in a porous horizontal annulus. Fig. 2 shows a good agreement between the obtained streamlines and temperature contour plots and the results by Charrier-Mojtabi [41] for Ra = 200. It is well known that different flow structures and isotherms may appear for the same values of Ra and R depending on the initial conditions introduced in the computations (see Charrier-Mojtabi [41]). Streamlines and isotherms presented in Fig. 2 have been obtained using the following initial conditions: motionless fluid and mean temperature inside the annulus between isothermal walls. Fig. 3 shows a good agreement between the obtained streamlines and temperature contour plots and the results by Charrier-Mojtabi [41] and Khanafer et al. [42] for R = 2.0 and Ra = 200. In this case streamlines and isotherms have been obtained using the following initial conditions: flow and temperature structures for R = 2.0 and Ra = 100. Table 1 shows the values of the average Nusselt number for different Rayleigh numbers in comparison with other authors for the natural convection problem in the annulus between two horizontal concentric cylinders presented in Fig. 2 using the following initial
Fig. 2. Comparison of streamlines and isotherms for R = 2.0 and Ra = 200 with motionless fluid: numerical results of Charrier-Mojtabi [41] – a, present study – b.
185
conditions: motionless fluid and mean temperature inside the annulus between isothermal walls. For the purpose of obtaining grid independent solution, a grid sensitivity analysis is performed. The grid independent solution was performed by preparing the solution for steady-state free convection in a porous horizontal annulus filled with a water based nanofluid at Ra = 200, Le = 1, Nr = 0.1, Nb = 0.1, Nt = 0.1. Four cases of non-uniform grid along r-coordinate are tested: a grid of 100 100 points (Dr min ¼ 0:00087; Dr max ¼ 0:038 and Dc ¼ 0:031), a grid of 150 150 points (Drmin ¼ 0:00022; Drmax ¼ 0:0333 and Dc ¼ 0:021), a grid of 200 200 points (Drmin ¼ 0:00019; Drmax ¼ 0:024 and Dc ¼ 0:016), and a grid of 250 250 points (Drmin ¼ 0:00015; Drmax ¼ 0:0196 and Dc ¼ 0:013). Table 2 shows an effect of the mesh on the average Nusselt number of the hot wall. On the basis of the conducted verifications the non-uniform grid of 200 200 points has been selected for the following analysis. 4. Results and discussion Numerical investigation of the boundary value problem (16)–(20) has been carried out at the following values of key parameters: Rayleigh number (Ra = 50–500), Lewis number (Le = 1–50), the buoyancy-ratio parameter (Nr = 0.1–0.4), the Brownian motion parameter (Nb = 0.1–0.4), the thermophoresis parameter (Nt = 0.1–0.4). Particular efforts have been focused on the effects of these key parameters on the fluid flow, heat and mass transfer characteristics. Figs. 4 and 5a show streamlines, isotherms and isolines of nanoparticles volume fraction for a water based nanofluid at different values of the Rayleigh number for Le = 10.0, Nr = Nb = Nt = 0.1. It should be noted that these distributions have been obtained in case of motionless fluid and mean temperature as initial conditions. The classical crescent-shaped vortex is formed in an annulus between two horizontal concentric cylinders for Ra 6 200 [48]. As indicated by streamlines, the flow inside the left (right) half part of the annulus includes counterclockwise (clockwise) convective cell that is in agreement with the prescribed thermal boundary conditions. These cells are symmetrical with respect to the vertical axis of the cavity and reflect a formation of thermal plume along the upper part of the vertical line of symmetry. An increase in the Rayleigh number from 50 to 200 leads to an intensification of convective flow and as a result to an increase in the temperature of the upper part of the considered annulus. The former can be confirmed by the maximum absolute values of the stream function as following ¼ 5:56 < jwjRa¼100 ¼ 9:96 < jwjRa¼200 ¼ 16:28. jwjRa¼50 max max max Also an increase in Ra leads to a decrease in the thermal boundary
Fig. 3. Comparison of streamlines and isotherms for R = 2.0 and Ra = 200 with initial conditions at Ra = 100: numerical results of Charrier-Mojtabi [41] – a, Khanafer et al. [42] – b, present study – c.
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Table 1 Comparison of the average Nusselt number of the hot internal cylinder. Ra
Authors Caltagirone [43]
Rao et al. [44]
Bau [45]
Facas [46]
Facas and Farouk [47]
Present results
49 49
10 10
30 44
50 50
25 25
50 50
100 100
200 200
1.328 1.829
1.341 1.861
1.335 1.844
1.342 1.835
1.362 1.902
1.3455 1.8752
1.3447 1.8721
1.3435 1.8689
Grid size
50 100
Table 2 Variations of the average Nusselt number of the heat wall with the non-uniform grid. Non-uniform grids 100 100 150 150 200 200 250 250
Nu 2.8673 2.8722 2.8712 2.8709
D¼
jNuij Nu200200 j Nuij
100%
0.14 0.03 – 0.01
layer thickness close to the bottom part of the inner cylinder and upper part of the outer cylinder. That can be explained by more intensive motion of the nanofluid inside the annulus due to an increase in the buoyancy force magnitude. As for a formation of ascending plume over the cylinder, it is interesting to note that due to an effect of the buoyancy force the boundary layer is torn from the surface of the inner cylinder with motion from the bottom part to the upper part of this cylinder. An increase in the Rayleigh number leads to a displacement of this separation point close to middle section c ¼ p=2. The reason for such behavior has been mentioned above. At the same time distribution of nanoparticles inside the annulus is non-homogeneous. Since the heat conduction regime enhances the effect of the thermophoresis phenomenon [49], therefore the nanoparticles distribution is highly non-homogeneous for small values of Ra (Fig. 4a). An increase in the Rayleigh number leads to an expansion
Fig. 5. Streamlines w, isotherms h and isoconcentrations u at Ra = 500, Le = 10.0, Nr = Nb = Nt = 0.1, initial conditions: motionless fluid and mean temperature – a, flow and temperature structures at Ra = 200 – b.
Fig. 4. Streamlines w, isotherms h and isoconcentrations u at Le = 10.0, Nr = Nb = Nt = 0.1: Ra = 50 – a, Ra = 100 – b, Ra = 200 – c.
M.A. Sheremet, I. Pop / Computers & Fluids 118 (2015) 182–190
187
Fig. 6. Variation of the local Nusselt number – a and average Nusselt number – b at inner cylinder surface with Rayleigh number for Le = 10.0, Nr = Nb = Nt = 0.1 (in figure b with different initial conditions: motionless fluid and mean temperature for Ra = 500 – solid line, flow and temperature structures at Ra = 200 for Ra = 500 – dashed line).
of the homogeneous area. Simultaneously one can find a decrease in the non-homogeneous zones and their locations are close to the upper and bottom parts of the vertical line of symmetry. Taking into account the obtained distribution of nanoparticles it is possible to consider that nanoparticles move to the bottom part of the annulus. An increase in the Rayleigh number to Ra = 500 leads to both a formation of two counter-rotating vortices in each half part of the annulus (left and right) and an intensification of convective flow, taking into account the stream function values of the main vortex ¼ 26:12 (Fig. 5a). It is worth noting here that the flow jwjRa¼500 max configuration, for example in the left part of the annulus, consists of the main counterclockwise convective vortex with ascending flows along the heated inner cylinder and descending flows along the cooled outer cylinder. The secondary small clockwise convective cell is located in the upper part of the annulus where one can find the thermally unstable region with heated bottom wall and cooled top wall that was described previously for pure fluid [48]. Such flow configuration characterizes an essential displacement of the boundary layer separation point close to the middle cross-section of the inner cylinder. A formation of thermal plume occurs between the main and secondary convective cells. At the same time the distribution of nanoparticles inside the annulus becomes essentially non-homogeneous in comparison with small values of the Rayleigh number. It is well known (see Charrier-Mojtabi [41]) that for natural convection in a porous horizontal annulus different flow structures and isotherms may appear for the same values of Ra and R depending on the initial conditions introduced in the computations. The presented local parameters (Figs. 4 and 5a) have been obtained when at initial time the flow was motionless and the temperature was constant h ¼ 0:5 for all internal nodes. In case of the results for Ra = 200 as initial conditions the single cell vortex is formed in the annulus like for Ra 6 200 with essentially intensive convective flow ¼ 27:92 . Distributions of nanoparticles here are similar jwjRa¼500 max to data for Ra = 200 (more homogeneous in comparison with results presented in Fig. 5a). The results in Fig. 6a present the distribution of local Nusselt number along the internal cylinder surface for different values of Ra. A dashed line with three dotes (Ra = 500⁄) corresponds to single cell convective structure (Fig. 5b). A formation of unicellular flow at Ra 6 200 and Ra = 500⁄ reflects an increase in the local Nusselt
number with c that characterizes a decrease in the thermal boundary layer with c and simultaneously an increase in the temperature gradient. It should be noted that boundary layer separation for Ra = 500 occurs in the upper part of the annulus. This separation leads to non-monotonic behavior of the local Nusselt number. Minimum value of Nu corresponds to the presence of thermal plume due to the boundary-layer separation. Dependences of the average Nusselt number for the inner cylinder surface on the Rayleigh number are presented in Fig. 6b. An increase in the Rayleigh number leads to an increase in the average Nusselt and Sherwood numbers. Formation of unicellular convective structure characterizes a reduction of Nu. Figs. 4c and 7 illustrate streamlines, isotherms and isoconcentrations at different values of the Lewis number. An increase in Le leads on the one hand to more intensive convective heat transfer
Fig. 7. Streamlines w, isotherms h and isoconcentrations u at Ra = 200, Nr = Nb = Nt = 0.1: Le = 1.0 – a, Le = 50.0 – b
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inside the annulus and on the other hand to transfer from bicellular to unicellular flow structures. It should be noted that, with respect to Fig. 7, low values of the Lewis number characterize non-homogeneous distributions of the nanoparticles inside the annulus. Also an increase in Lewis number leads to an essential decrease in the thickness of concentration boundary layers at cylindrical surfaces. It physically means that flow with large Lewis number prevent spreading the nanoparticles in the base fluid. An effect of the Lewis number on the local Nusselt number along the inner cylindrical surface is presented in Fig. 8. Non-monotonic distribution of Nu occurs for Le = 1, that characterizes an appearance of two convective cells in each vertical part of the annulus. An increase in Le leads to a formation of an increasing profile of local Nusselt number along c. Taking into account boundary conditions for the nanoparticles volume fraction (19) distribu-
Fig. 8. Variation of the local Nusselt number along the internal cylinder surface with the Lewis number for Ra = 200, Nr = Nb = Nt = 0.1.
tions of Sh are similar to distributions of Nu. Effects of the parameters Nr, Nb and Nt on the local Nusselt number along the inner cylindrical surface are presented in Fig. 9. An increase in Nr leads to inessential changes in the
Fig. 9. Variation of the local Nusselt number along the internal cylinder surface at Ra = 200, Le = 10 with: buoyancy-ratio parameter, Nb = Nt = 0.1 – a, Brownian motion parameter, Nr = Nt = 0.1 – b, thermophoresis parameter, Nr = Nb = 0.1 – c.
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distributions of streamlines, isotherms and isoconcentrations. At the same time an increase in Nr leads to a decrease in Nu in the bottom part of the inner cylinder while the average Nusselt number is a weakly decreasing function of Nr. Changes in Nb do not lead to changes in the local Nusselt number (Fig. 9b). An increase in Nt leads to a decrease in the local Nusselt number (Fig. 9c). The fact that the changes in Nb do not lead to changes in the local Nusselt number is in agreement with the findings reported by Keblinski et al. [50] that Brownian motion of nanoparticles contributes much less than other factors since Brownian motion of nanoparticles is too slow to transport significant amount of heat through a nanofluid.
5. Conclusions Steady natural convective flow and heat transfer of a water based nanofluid in a porous annulus between two isothermal horizontal concentric cylinders has been numerically investigated using the nanofluid model proposed by Buongiorno. Distributions of streamlines, isotherms, isoconcentrations, local and average Nusselt numbers at a wide range of key parameters such as Ra = 50–500; Le = 1–50; Nr = 0.1–0.4; Nb = 0.1–0.4; Nt = 0.1–0.4 have been obtained. It has been found that an insertion of nanoparticles leads to a transfer from bicellular flow structure to unicellular flow structure for small values of the Rayleigh number. This result characterizes a displacement of the bifurcation point 2-D unicellular flow to 2-D bicellular flow for R = 2. It has been shown a formation of unicellular and bicellular flows for Ra = 500 at different initial conditions. Acknowledgement This work of M.A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K. References [1] Narasimhan A. Essentials of heat and fluid flow in porous media. New York: CRC Press; 2013. [2] Sheremet MA, Pop I. Natural convection in a square porous cavity with sinusoidal temperature distributions on both side walls filled with a nanofluid: Buongiorno’s mathematical model. Transp Porous Media 2014;105:411–29. [3] Nield DA, Bejan A. Convection in porous media. 4th ed. New York: Springer; 2013. [4] Bagchi A, Kulacki FA. Natural convection in superposed fluid-porous layers. New York: Springer; 2014. [5] Jansen JD. A systems description of flow through porous media. New York: Springer; 2013. [6] Basak T, Chamkha AJ. Heatline analysis on natural convection for nanofluids confined within square cavities with various thermal boundary conditions. Int J Heat Mass Transfer 2012;55:5526–43. [7] Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. Proceedings of the 1995 ASME international mechanical engineering congress and exposition, vol. 66. San Francisco (USA): ASME, FED 231/MD; 1995. p. 99–105. [8] Li Y, Zhou J, Tung S, Schneider E, Xi S. A review on development of nanofluid preparation and characterization. Powder Technol 2009;196:89–101. [9] Kakaç S, Pramuanjaroenkij A. Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transfer 2009;52:3187–96. [10] Wong KV, Leon OD. Applications of nanofluids: current and future. Adv Mech Eng 2010:11. Article ID 519659. [11] Fan J, Wang L. Review of heat conduction in nanofluids. ASME J Heat Transfer 2011;133:040801. [12] Jaluria Y, Manca O, Poulikakos D, Vafai K, Wang L. Heat transfer in nanofluids. Adv Mech Eng 2012:2. Article ID 972973. [13] Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S. A review of the applications of nanofluids in solar energy. Int J Heat Mass Transfer 2013;57:582–94. [14] Hajmohammadi MR, Maleki H, Lorenzini G, Nourazar SS. Effects of Cu and Ag nano-particles on flow and heat transfer from permeable surfaces. Adv Powder Technol 2015;26:193–9.
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[15] Soleimani S, Sheikholeslami M, Ganji DD, Gorji-Bandpay M. Natural convection heat transfer in a nanofluid filled semi-annulus enclosure. Int Commun Heat Mass Transfer 2012;39:565–74. [16] Ashorynejad HR, Mohamad AA, Sheikholeslami M. Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method. Int J Therm Sci 2013;64:240–50. [17] Seyyedi SM, Dayyan M, Soleimani S, Ghasemi E. Natural convection heat transfer under constant heat flux wall in a nanofluid filled annulus enclosure. Ain Shams Eng J 2015;6:267–80. [18] Khanafer K, Vafai K, Lightstone M. Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transfer 2003;46:3639–53. [19] Tiwari RK, Das MK. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transfer 2007;50:2002–18. [20] Buongiorno J. Convective transport in nanofluids. ASME J Heat Transfer 2006;128:240–50. [21] Kuznetsov AV, Nield DA. The Cheng–Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid: a revised model. Int J Heat Mass Transfer 2013;65:682–5. [22] Nield DA, Kuznetsov AV. Thermal instability in a porous medium layer saturated by a nanofluid: a revised model. Int J Heat Mass Transfer 2014;68:211–4. [23] Sakai F, Li W, Nakayama A. A rigorous derivation and its applications of volume averaged transport equations for heat transfer in nanofluid saturated metal foam. In: Proc 15th int heat transfer conference, IHTC-15. Kyoto, Japan; August 10–15, 2014. 15 p. [24] Dukhan N. Metal foams: fundamentals and applications. 1st ed. DEStech Publications, Inc.; 2012. [25] Heyda JF. Green function solution for the laminar incompressible flow between nonconcentric cylinders. J Franklin Inst 1959;267:25–34. [26] Matin MH, Pop I. Numerical study of mixed convection heat transfer of a nanofluid in an eccentric annulus. Num Heat Transfer, Part A 2014;65:84–105. [27] El-Amin MF, Salama A, Sun SS. Numerical and dimensional analysis of nanoparticles transport with two-phase flow in porous media. J Petroleum Sci Eng 2015;128:53–64. [28] Liu XH, Civian F. A multiphase mud fluid infiltration and filter cake formation model. In: SPE-25215, SPE int sympos oilfied chemistry. New Orleans (LA, USA); 1996. [29] Hajmohammadi MR, Nourazar SS, Manesh AH. Semi-analytical treatments of conjugate heat transfer. J Mech Eng Sci 2012;227:492–503. [30] Hajmohammadi MR, Nourazar SS. On the solution of characteristic value problems arising in linear stability analysis; semi analytical approach. Appl Math Comput 2014;239:126–32. [31] Khan ZH, Gul R, Khan WA. Effect of variable thermal conductivity on heat transfer from a hollow sphere with heat generation using homotopy perturbation method. In: ASME 2008 heat transfer summer conference. p. 301–9. [32] Gul R, Khan ZH, Khan WA. Heat transfer from solids with variable thermal conductivity and uniform internal heat generation using homotopy perturbation method. In: ASME 2008 heat transfer summer conference. p. 311–9. [33] Hajmohammadi MR, Nourazar SS. Conjugate forced convection heat transfer from a heated flat plate of finite thickness and temperature-dependent thermal conductivity. Heat Transfer Eng 2014;35:863–74. [34] Hajmohammadi MR, Nourazar SS. On the insertion of a thin gas layer in micro cylindrical Couette flows involving power-law liquids. Int J Heat Mass Transfer 2014;75:97–108. [35] Hajmohammadi MR, Nourazar SS, Campo A. Analytical solution for two-phase flow between two rotating cylinders filled with power law liquid and a micro layer of gas. J Mech Sci Technol 2014;28:1849–54. [36] Aleshkova IA, Sheremet MA. Unsteady conjugate natural convection in a square enclosure filled with a porous medium. Int J Heat Mass Transfer 2010;53:5308–20. [37] Sheremet MA, Trifonova TA. Unsteady conjugate natural convection in a vertical cylinder partially filled with a porous medium. Numer Heat Transfer, Part A 2013;64:994–1015. [38] Sheremet MA, Pop I. Thermo-bioconvection in a square porous cavity filled by oxytactic microorganisms. Transp Porous Media 2014;103:191–205. [39] Sheremet MA, Pop I. Conjugate natural convection in a square porous cavity filled by a nanofluid using Buongiorno’s mathematical model. Int J Heat Mass Transfer 2014;79:137–45. [40] Sheremet MA, Grosan T, Pop I. Free convection in shallow and slender porous cavities filled by a nanofluid using Buongiorno’s model. ASME J Heat Transfer 2014;136:082501. [41] Charrier-Mojtabi MC. Numerical simulation of two- and three-dimensional free convection flows in a horizontal porous annulus using a pressure and temperature formulation. Int J Heat Mass Transfer 1997;40:1521–33. [42] Khanafer K, Al-Amiri A, Pop I. Numerical analysis of natural convection heat transfer in a horizontal annulus partially filled with a fluid-saturated porous substrate. Int J Heat Mass Transfer 2008;51:1613–27. [43] Caltagirone JP. Thermo-convective instability in porous medium bounded by two concentric horizontal cylinders. J Fluid Mech 1976;76:337–62. [44] Rao YF, Fukuda K, Hasegawa S. Steady and transient analysis of natural convection in a horizontal porous annulus with Galerkin method. ASME J Heat Transfer 1987;109:919–27.
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[45] Bau HH. Thermal convection in a horizontal, eccentric annulus containing a saturated porous medium – an extended purturbation expansion. Int J Heat Mass Transfer 1984;27:2277–87. [46] Facas GN. Natural convection from a buried pipe with external baffles. Numer Heat Transfer, Part A 1995;27:595–609. [47] Facas GN, Farouk B. Transient and steady state natural convection in a porous medium between two concentric cylinders. ASME J Heat Transfer 1983;105: 660–3.
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Contents lists available at ScienceDirect
Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Free convection in a porous horizontal cylindrical annulus with a nanofluid using Buongiorno’s model Mikhail A. Sheremet a,b, Ioan Pop c,⇑ a
Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia c Department of Applied Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania b
a r t i c l e
i n f o
Article history: Received 10 December 2014 Received in revised form 1 April 2015 Accepted 8 June 2015 Available online 21 June 2015 Keywords: Free convection Horizontal annulus Porous medium Nanofluids Buongiorno model Numerical method
a b s t r a c t Natural convection flow in a porous concentric horizontal annulus saturated with a water based nanofluid is numerically investigated. The mathematical model used is of single-phase and is formulated in dimensionless stream function and temperature taking into account the Darcy–Boussinesq approximation and the nanofluid model proposed by Buongiorno. The transformed dimensionless partial differential equations have been solved using a second-order accurate finite-difference technique. The results indicate that inclusion of nanoparticles into pure water changes the flow structure at low values of the Rayleigh number. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Porous media appear everywhere. Examples range from civil, chemical and geological engineering until an essential part of our daily lives to live comfortably [1]. A few examples of porous media are: soils, aquifers, sands, clothing, filters and catalytic converters in cars, ground water flow, etc. Convective flow and heat transfer is, generally, prevalent in fields of physics and engineering, such as geothermal reservoirs, float glass production, flow and heat transfer in solar ponds, air-conditioning in rooms and cooling of electronic devices [2]. Natural or free convection in a porous cavity has received considerable attention in recent years because of its relation to the thermal performance of many engineering installations [3]. The phenomenon of natural convection in porous media is a fundamental transport mechanism encountered in a wide range of engineering, geophysics, and scientific applications, such as packed bed solar energy storage, fibrous and granular insulation systems, water reservoirs and post-accident cooling of nuclear reactors [4]. Despite the fact that buoyant convection in this system was first studied about 40 years ago, there has lately been renewed interest in such flow in porous cavities owing to its importance in environmental and energy management problems ⇑ Corresponding author. Tel.: +40 264 405300; fax: +40 264 591906. E-mail address: [email protected] (I. Pop). http://dx.doi.org/10.1016/j.compfluid.2015.06.022 0045-7930/Ó 2015 Elsevier Ltd. All rights reserved.
in current scientific and geo-political context [4]. Application and control concept of flow through porous media to oil and gas reservoir simulation, geothermal energy, or groundwater remediation are important research topics in porous media nowadays [5]. As traditional fluids used for heat transfer applications such as water, mineral oils and ethylene glycol have a rather low thermal conductivity, nanofluids with relatively higher thermal conductivities have attracted enormous interest from researchers due to their potential in enhancement of heat transfer with little or no penalty in pressure drop [6]. It seems that Choi [7] is the first who introduced the term nanofluid to describe the mixture of nanoparticles and base fluid such as water, mineral oils and ethylene glycol. The addition of nanoparticles into the base fluid is able to change the flow and heat transfer capability of the liquids and indirectly increase the low thermal conductivity of the base fluid which is identified as the main obstacle in heat transfer performance. This mixture has attracted the interest of numerous researchers because of its many significant applications such as, for example, in the medical applications, transportations, microelectronics, chemical engineering, aerospace and manufacturing [8]. The convective heat transfer characteristic of nanofluids depends on the thermo-physical properties of the base fluid and the ultra fine particles, the flow pattern and flow structure, the volume fraction of the suspended particles, the dimensions and the shape of these particles. The utility of a particular nanofluid for a
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heat transfer application can be established by suitably modeling the convective transport in the nanofluid [6]. Many authors such as Kakaç and Pramuanjaroenkij [9], Wong and Leon [10], Fan and Wang [11], Jaluria et al. [12], Mahian et al. [13], Hajmohammadi et al. [14], Soleimani et al. [15], Ashorynejad et al. [16] and Seyyedi et al. [17] have presented comprehensive literature review on nanofluids. For example, Hajmohammadi et al. [14] analyzed nanofluid flow and heat transfer over a permeable flat plate using convective boundary condition. It was shown that the increase in skin friction is a considerable drawback imposed by Cu/water and Ag/water nanofluids, especially in the case of injection. Soleimani et al. [15] investigated numerically natural convection inside the semi-annulus cavity filled with a nanofluid. They found that the effect of the nanoparticles is more pronounced at low Rayleigh number than at high Rayleigh number because of greater amount of enhancement and increasing Rayleigh number leads to a decrease in ratio of heat transfer enhancement. Ashorynejad et al. [16] studied numerically on the basis of the Lattice Boltzmann method the effect of magnetic field on natural convection in a horizontal cylindrical annulus filled with a nanofluid. It has been shown that the average Nusselt number is an increasing function of nanoparticle volume fraction and Rayleigh number, while it is a decreasing function of Hartmann number. Seyyedi et al. [17] using the finite volume method numerically analyzed the natural convective heat transfer in an annulus filled with a Cu/water nanofluid. It has been found that the angle of turn for the boundary condition of the inner cylinder essentially affects the average Nusselt number. However, these papers are based on the mathematical nanofluid models proposed by Khanafer [18], and Tiwari and Das [19] for the two-phase mixture containing micro-sized particles. On the other hand, one should also mention the mathematical nanofluid model proposed by Buongiorno [20] used in many papers pioneered by Kuznetsov and Nield [21] for the free convection boundary layer flow along a vertical flat plate embedded in a porous medium and, Nield and Kuznetsov [22] for the problem of thermal instability in a porous medium layer saturated by a nanofluid. In this model, the Brownian motion and thermophoresis enter to produce their effects directly into the equations expressing the conservation of energy and nanoparticles, so that the temperature and the particle density are coupled in a particular way, and that results in the thermal and concentration buoyancy effects being coupled in the same way. We also mention here the very recently published review paper by Sakai et al. [23], where a macroscopic set of the governing equations for describing heat transfer in nanofluid saturated porous media were rigorously derived using a volume averaging theory, for possible heat transfer applications of metal foams filled with nanofluids as high performance heat exchangers. Metal foams filled with nanofluids can be one of the most promising candidates for a high performance heat exchanger needed for highly concentrated heat generating devices [24]. Literatures indicate that convection heat transfer inside concentric and eccentric annuli has many applications in science and engineering, such as electrical motor and generator, completion of an oil source and heating and cooling of underground electric cables. Recently, investigation of the effect of eccentricity on heat transfer has become a subject of interest to most researchers working in the area of convection heat transfer problems. It seems that the first person who worked on eccentric annuli was Heyda [25], and he applied a fundamental solution known as Green’s function on solving the momentum equation for a laminar flow inside an eccentric annulus. A valuable list of references on this problem can be found in the paper by Matin and Pop [26]. It is, however, worth mentioning that El-Amin et al. [27] have founded a mathematical model of nanoparticles transport in two-phase flow in
porous media based on the formulation of fine particles transport in two-phase flow in porous media proposed by Liu and Civian [28]. The principal objective of the present paper is to analyze the steady natural convection in a porous horizontal cylindrical annulus with a nanofluid using the single-phase mathematical nanofluid model proposed by Buongiorno [20]. To our best of knowledge this problem has not been considered before, so that the reported results are new and original. It is worth noting that in the present paper we use the numerical method for solution to the boundary value problem for the partial differential equations. At the same time it is possible to transform these equations to ODE’s and to solve by the semi-analytical methods [29–33]. Such algorithms [30,34,35] can allow to study the stability of the convective flows. 2. Basic equations Consider the steady free convection flow and heat transfer in a porous horizontal annulus filled with a water based nanofluid. It is assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. This prevents nanoparticles from agglomeration and deposition on the porous matrix (see Kuznetsov and Nield [21]; Nield and Kuznetsov [22]). A schematic geometry of the problem under investigation is shown in Fig. 1, where r and c are the polar system of coordinates and r2 r1 is the size of the annulus. It is assumed that the internal surface r ¼ r1 is heated and maintained at the constant temperature T h , while the external surface r ¼ r 2 is cooled and has the constant temperature T c . If we assume that the angular coordinate is measured clockwise from the vertically down position and that the problem is symmetric about a vertical plane passing through the axis of the cylinder, then consideration will be confined to the range 0 < c < p (or p < c < 2p). The basic equations for the flow, heat transfer and nanoparticles can be written in the following form (see Buongiorno [20]; Kuznetsov and Nield [21]; Nield and Kuznetsov [22]),
rV ¼0
l
h
i
V þ C qp þ ð1 C Þqf 0 ð1 bðT T c ÞÞ g K @T r þ ðV rÞT ¼ am r2 T þ d½DB rC rT þ ðDT =T c ÞrT rT @t @C 1 qp þ ðV rÞC ¼ r jp @t e 0 ¼ rp
Fig. 1. Physical model and coordinate system.
ð1Þ ð2Þ ð3Þ ð4Þ
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where V is the Darcy velocity vector, T is the temperature, C is the nanoparticle volume fraction, t is the time, p is the pressure, g is the gravitational vector, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, jp ¼ qp ½DB rCþ ðDT =T c ÞrT is the nanoparticles mass flux, qf 0 is the reference density of the fluid, am ; l; qp are the effective thermal diffusivity of the porous medium, the dynamic viscosity and nanoparticle mass den sity, d is an additional parameter defined by d ¼ e qC p p = qC p f ; C p
The mathematical model can be formulated in terms of the dimensionless variables. By using r 1 as the length scale, T h T c as the temperature scale, am as the stream function scale, C 0 as the nanoparticle volume fraction, the following dimensionless variables have been introduced:
is the heat capacity, b is the coefficient of volumetric thermal expansion, r ¼ qC p m = qC p f ; qC p f is heat capacity of the base fluid, qC p p is effective heat capacity of the nanoparticle material, qC p m is effective heat capacity of the porous medium. Further on we can linearize the momentum equation as described in detail previously [2] and write Eq. (2) as
and substituting (15) into Eqs. (12)–(14), we obtain
l
0 ¼ rp V K h i þ C qp qf 0 þ qf 0 ð1 bðT T c Þð1 C 0 ÞÞ g
ð5Þ
Eqs. (1), (3)–(5) for the problem under consideration can be written in dimensional cylindrical coordinates r; c as Þ @ v @ ðru þ ¼0 ð6Þ @r @c o l @p n ð7Þ 0¼ u C qp qf 0 þ qf 0 ½1 bðT T c Þð1 C 0 Þ g cosðcÞ @r K o l 1 @p n 0¼ v þ C qp qf 0 þ qf 0 ½1 bðT T c Þð1 C 0 Þ g sinðcÞ ð8Þ r @ c K " # @T @T v @T 1@ @T 1 @2T r þ u þ ¼ am r þ 2 2 @t @r r @c r @r @r r @c ( " 2 2 #) @C @T 1 @C @T DT @T 1 @T þ d DB þ 2 þ 2 þ ð9Þ r @ c @r @r r @ c @ c @r Tc " # @C 1 @C v @C 1@ @C 1 @2C þ r þ u ¼ DB þ 2 2 r @r @r r @ c @t e @r r @ c " # DT 1 @ @T 1 @2T r þ þ 2 2 ð10Þ r @ c T c r @r @r
; v are the velocity components along r ; c directions, where u respectively. defined by Taking into account a stream function w
¼ u
1 @w ; r @ c
v
@w ¼ @r
ð11Þ
Eq. (6) is satisfied identically. Therefore the steady governing equations in variables stream function, temperature and nanoparticle volume fraction can be written as follows: qf 0 bð1 C 0 ÞK @T qp qf 0 @C cosðcÞ 1@ @w 1 @2 w r K þ 2 2¼ g r @r @r r @ c r @c @c l l qf 0 bð1 C 0 ÞK @T qp qf 0 @C þ K g sinðcÞ @r @r l l " # @T 1 @ w @T 1 @w 1@ @T 1 @2T r ¼ am þ 2 2 r @ c @r r @r @ c r @r @r r @ c
@C @T 1 @C @T þ d DB þ 2 @r @r r @ c @ c " 2 2 #) DT @T 1 @T þ þ 2 r @ c @r Tc " # @C 1 1 @ w @C 1 @ w 1@ @C 1 @2C r ¼ DB þ 2 2 r @r @r r @ c e r @ c @r r @r @ c " # DT 1 @ @T 1 @2T þ þ 2 2 r r @ c T c r @r @r
am ; h ¼ ðT T c Þ=ðT h T c Þ; w ¼ w= u ¼ C=C 0 ; R ¼ r2 =r1 r ¼ r =r1 ;
ð15Þ
1 @ @w 1 @2w @h @u þ 2 sinðcÞ ¼ Ra Nr r r @r @r r @ c2 @r @r @h @ u cosðcÞ Nr þ Ra @c r @c 2 1 @w @h 1 @w @h 1 @ @h 1 @ h þ 2 2 r ¼ r @ c @r r @r @ c r @r @r r @c @ u @h 1 @ u @h þ Nb þ 2 @r @r r @ c @ c " 2 # 2 @h 1 @h þ Nt þ 2 @r r @c " # 1 @w @ u 1 @w @ u 1 1 @ @u 1 @2u þ 2 r ¼ r @ c @r r @r @ c Le r @r r @ c2 @r " # Nt 1 @ @h 1 @2h þ 2 2 r þ Le Nb r @r @r r @c
ð16Þ
ð17Þ
ð18Þ
where Ra is the Rayleigh number for the porous media, Le is the Lewis number, Nr is the buoyancy-ratio parameter, Nb is the Brownian motion parameter and Nt is the thermophoresis parameter, which are defined as
Ra ¼
ð1 C 0 ÞgK qf 0 bDTr1
am l
;
Nr ¼
qp qf 0 C 0
qf 0 bDT ð1 C 0 Þ dDB C 0 dDT DT am Nb ¼ ; Nt ¼ ; Le ¼ am am T c eDB
; ð19Þ
The boundary conditions for the formulated problem (16)–(18) are as follows.
w ¼ 0;
h ¼ 1;
Nb @@ru þ Nt @h ¼ 0 on r ¼ 1 @r
w ¼ 0;
h ¼ 0;
Nb @@ru þ Nt @h ¼ 0 on r ¼ R @r
w ¼ 0;
@h @c
¼ 0;
@u @c
ð20Þ
¼ 0 on c ¼ 0 and c ¼ p
The local Nusselt Nu and Sherwood Sh numbers can be defined as
@h Nu ¼ lnðRÞ ; @r r¼1
Sh ¼ lnðRÞ
@u @r r¼1
ð21Þ
and the average Nusselt Nu and Sherwood Sh numbers are ð12Þ
Nu ¼
1
Z p
p
0
Nudc;
Sh ¼
1
Z p
p
0
Shdc
ð22Þ
It should be noted here that for an analysis of Sherwood numbers it is possible to study only Nusselt numbers because at inner Nt @h cylinder surface we have @@ru ¼ Nb taking into account boundary @r conditions for u (Eq. (19)) as described in detail previously [2]. ð13Þ
3. Numerical method
ð14Þ
The partial differential Eqs. (16)–(18) with corresponding boundary conditions (20) were solved by the second order finite difference method (see Aleshkova and Sheremet [36]; Sheremet and Trifonova [37]; Sheremet and Pop [38,39]; Sheremet et al.
M.A. Sheremet, I. Pop / Computers & Fluids 118 (2015) 182–190
[40]) using non-uniform mesh for the radial coordinate. The solution for the corresponding linear algebraic equations was obtained through the successive under relaxation method. Optimum value of the relaxation parameter was chosen on the basis of computing experiments [38,39]. The computation is terminated when the residuals for the stream function get bellow 106 . The numerical methodology was coded in C++, and to check its validity, a comparison with selective data from the published literature was carried out [2,38–40]. The performance of the porous horizontal cylinder part of the model was tested against the results of Charrier-Mojtabi [41] and Khanafer et al. [42] for steady-state free convection in a porous horizontal annulus. Fig. 2 shows a good agreement between the obtained streamlines and temperature contour plots and the results by Charrier-Mojtabi [41] for Ra = 200. It is well known that different flow structures and isotherms may appear for the same values of Ra and R depending on the initial conditions introduced in the computations (see Charrier-Mojtabi [41]). Streamlines and isotherms presented in Fig. 2 have been obtained using the following initial conditions: motionless fluid and mean temperature inside the annulus between isothermal walls. Fig. 3 shows a good agreement between the obtained streamlines and temperature contour plots and the results by Charrier-Mojtabi [41] and Khanafer et al. [42] for R = 2.0 and Ra = 200. In this case streamlines and isotherms have been obtained using the following initial conditions: flow and temperature structures for R = 2.0 and Ra = 100. Table 1 shows the values of the average Nusselt number for different Rayleigh numbers in comparison with other authors for the natural convection problem in the annulus between two horizontal concentric cylinders presented in Fig. 2 using the following initial
Fig. 2. Comparison of streamlines and isotherms for R = 2.0 and Ra = 200 with motionless fluid: numerical results of Charrier-Mojtabi [41] – a, present study – b.
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conditions: motionless fluid and mean temperature inside the annulus between isothermal walls. For the purpose of obtaining grid independent solution, a grid sensitivity analysis is performed. The grid independent solution was performed by preparing the solution for steady-state free convection in a porous horizontal annulus filled with a water based nanofluid at Ra = 200, Le = 1, Nr = 0.1, Nb = 0.1, Nt = 0.1. Four cases of non-uniform grid along r-coordinate are tested: a grid of 100 100 points (Dr min ¼ 0:00087; Dr max ¼ 0:038 and Dc ¼ 0:031), a grid of 150 150 points (Drmin ¼ 0:00022; Drmax ¼ 0:0333 and Dc ¼ 0:021), a grid of 200 200 points (Drmin ¼ 0:00019; Drmax ¼ 0:024 and Dc ¼ 0:016), and a grid of 250 250 points (Drmin ¼ 0:00015; Drmax ¼ 0:0196 and Dc ¼ 0:013). Table 2 shows an effect of the mesh on the average Nusselt number of the hot wall. On the basis of the conducted verifications the non-uniform grid of 200 200 points has been selected for the following analysis. 4. Results and discussion Numerical investigation of the boundary value problem (16)–(20) has been carried out at the following values of key parameters: Rayleigh number (Ra = 50–500), Lewis number (Le = 1–50), the buoyancy-ratio parameter (Nr = 0.1–0.4), the Brownian motion parameter (Nb = 0.1–0.4), the thermophoresis parameter (Nt = 0.1–0.4). Particular efforts have been focused on the effects of these key parameters on the fluid flow, heat and mass transfer characteristics. Figs. 4 and 5a show streamlines, isotherms and isolines of nanoparticles volume fraction for a water based nanofluid at different values of the Rayleigh number for Le = 10.0, Nr = Nb = Nt = 0.1. It should be noted that these distributions have been obtained in case of motionless fluid and mean temperature as initial conditions. The classical crescent-shaped vortex is formed in an annulus between two horizontal concentric cylinders for Ra 6 200 [48]. As indicated by streamlines, the flow inside the left (right) half part of the annulus includes counterclockwise (clockwise) convective cell that is in agreement with the prescribed thermal boundary conditions. These cells are symmetrical with respect to the vertical axis of the cavity and reflect a formation of thermal plume along the upper part of the vertical line of symmetry. An increase in the Rayleigh number from 50 to 200 leads to an intensification of convective flow and as a result to an increase in the temperature of the upper part of the considered annulus. The former can be confirmed by the maximum absolute values of the stream function as following ¼ 5:56 < jwjRa¼100 ¼ 9:96 < jwjRa¼200 ¼ 16:28. jwjRa¼50 max max max Also an increase in Ra leads to a decrease in the thermal boundary
Fig. 3. Comparison of streamlines and isotherms for R = 2.0 and Ra = 200 with initial conditions at Ra = 100: numerical results of Charrier-Mojtabi [41] – a, Khanafer et al. [42] – b, present study – c.
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Table 1 Comparison of the average Nusselt number of the hot internal cylinder. Ra
Authors Caltagirone [43]
Rao et al. [44]
Bau [45]
Facas [46]
Facas and Farouk [47]
Present results
49 49
10 10
30 44
50 50
25 25
50 50
100 100
200 200
1.328 1.829
1.341 1.861
1.335 1.844
1.342 1.835
1.362 1.902
1.3455 1.8752
1.3447 1.8721
1.3435 1.8689
Grid size
50 100
Table 2 Variations of the average Nusselt number of the heat wall with the non-uniform grid. Non-uniform grids 100 100 150 150 200 200 250 250
Nu 2.8673 2.8722 2.8712 2.8709
D¼
jNuij Nu200200 j Nuij
100%
0.14 0.03 – 0.01
layer thickness close to the bottom part of the inner cylinder and upper part of the outer cylinder. That can be explained by more intensive motion of the nanofluid inside the annulus due to an increase in the buoyancy force magnitude. As for a formation of ascending plume over the cylinder, it is interesting to note that due to an effect of the buoyancy force the boundary layer is torn from the surface of the inner cylinder with motion from the bottom part to the upper part of this cylinder. An increase in the Rayleigh number leads to a displacement of this separation point close to middle section c ¼ p=2. The reason for such behavior has been mentioned above. At the same time distribution of nanoparticles inside the annulus is non-homogeneous. Since the heat conduction regime enhances the effect of the thermophoresis phenomenon [49], therefore the nanoparticles distribution is highly non-homogeneous for small values of Ra (Fig. 4a). An increase in the Rayleigh number leads to an expansion
Fig. 5. Streamlines w, isotherms h and isoconcentrations u at Ra = 500, Le = 10.0, Nr = Nb = Nt = 0.1, initial conditions: motionless fluid and mean temperature – a, flow and temperature structures at Ra = 200 – b.
Fig. 4. Streamlines w, isotherms h and isoconcentrations u at Le = 10.0, Nr = Nb = Nt = 0.1: Ra = 50 – a, Ra = 100 – b, Ra = 200 – c.
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Fig. 6. Variation of the local Nusselt number – a and average Nusselt number – b at inner cylinder surface with Rayleigh number for Le = 10.0, Nr = Nb = Nt = 0.1 (in figure b with different initial conditions: motionless fluid and mean temperature for Ra = 500 – solid line, flow and temperature structures at Ra = 200 for Ra = 500 – dashed line).
of the homogeneous area. Simultaneously one can find a decrease in the non-homogeneous zones and their locations are close to the upper and bottom parts of the vertical line of symmetry. Taking into account the obtained distribution of nanoparticles it is possible to consider that nanoparticles move to the bottom part of the annulus. An increase in the Rayleigh number to Ra = 500 leads to both a formation of two counter-rotating vortices in each half part of the annulus (left and right) and an intensification of convective flow, taking into account the stream function values of the main vortex ¼ 26:12 (Fig. 5a). It is worth noting here that the flow jwjRa¼500 max configuration, for example in the left part of the annulus, consists of the main counterclockwise convective vortex with ascending flows along the heated inner cylinder and descending flows along the cooled outer cylinder. The secondary small clockwise convective cell is located in the upper part of the annulus where one can find the thermally unstable region with heated bottom wall and cooled top wall that was described previously for pure fluid [48]. Such flow configuration characterizes an essential displacement of the boundary layer separation point close to the middle cross-section of the inner cylinder. A formation of thermal plume occurs between the main and secondary convective cells. At the same time the distribution of nanoparticles inside the annulus becomes essentially non-homogeneous in comparison with small values of the Rayleigh number. It is well known (see Charrier-Mojtabi [41]) that for natural convection in a porous horizontal annulus different flow structures and isotherms may appear for the same values of Ra and R depending on the initial conditions introduced in the computations. The presented local parameters (Figs. 4 and 5a) have been obtained when at initial time the flow was motionless and the temperature was constant h ¼ 0:5 for all internal nodes. In case of the results for Ra = 200 as initial conditions the single cell vortex is formed in the annulus like for Ra 6 200 with essentially intensive convective flow ¼ 27:92 . Distributions of nanoparticles here are similar jwjRa¼500 max to data for Ra = 200 (more homogeneous in comparison with results presented in Fig. 5a). The results in Fig. 6a present the distribution of local Nusselt number along the internal cylinder surface for different values of Ra. A dashed line with three dotes (Ra = 500⁄) corresponds to single cell convective structure (Fig. 5b). A formation of unicellular flow at Ra 6 200 and Ra = 500⁄ reflects an increase in the local Nusselt
number with c that characterizes a decrease in the thermal boundary layer with c and simultaneously an increase in the temperature gradient. It should be noted that boundary layer separation for Ra = 500 occurs in the upper part of the annulus. This separation leads to non-monotonic behavior of the local Nusselt number. Minimum value of Nu corresponds to the presence of thermal plume due to the boundary-layer separation. Dependences of the average Nusselt number for the inner cylinder surface on the Rayleigh number are presented in Fig. 6b. An increase in the Rayleigh number leads to an increase in the average Nusselt and Sherwood numbers. Formation of unicellular convective structure characterizes a reduction of Nu. Figs. 4c and 7 illustrate streamlines, isotherms and isoconcentrations at different values of the Lewis number. An increase in Le leads on the one hand to more intensive convective heat transfer
Fig. 7. Streamlines w, isotherms h and isoconcentrations u at Ra = 200, Nr = Nb = Nt = 0.1: Le = 1.0 – a, Le = 50.0 – b
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inside the annulus and on the other hand to transfer from bicellular to unicellular flow structures. It should be noted that, with respect to Fig. 7, low values of the Lewis number characterize non-homogeneous distributions of the nanoparticles inside the annulus. Also an increase in Lewis number leads to an essential decrease in the thickness of concentration boundary layers at cylindrical surfaces. It physically means that flow with large Lewis number prevent spreading the nanoparticles in the base fluid. An effect of the Lewis number on the local Nusselt number along the inner cylindrical surface is presented in Fig. 8. Non-monotonic distribution of Nu occurs for Le = 1, that characterizes an appearance of two convective cells in each vertical part of the annulus. An increase in Le leads to a formation of an increasing profile of local Nusselt number along c. Taking into account boundary conditions for the nanoparticles volume fraction (19) distribu-
Fig. 8. Variation of the local Nusselt number along the internal cylinder surface with the Lewis number for Ra = 200, Nr = Nb = Nt = 0.1.
tions of Sh are similar to distributions of Nu. Effects of the parameters Nr, Nb and Nt on the local Nusselt number along the inner cylindrical surface are presented in Fig. 9. An increase in Nr leads to inessential changes in the
Fig. 9. Variation of the local Nusselt number along the internal cylinder surface at Ra = 200, Le = 10 with: buoyancy-ratio parameter, Nb = Nt = 0.1 – a, Brownian motion parameter, Nr = Nt = 0.1 – b, thermophoresis parameter, Nr = Nb = 0.1 – c.
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distributions of streamlines, isotherms and isoconcentrations. At the same time an increase in Nr leads to a decrease in Nu in the bottom part of the inner cylinder while the average Nusselt number is a weakly decreasing function of Nr. Changes in Nb do not lead to changes in the local Nusselt number (Fig. 9b). An increase in Nt leads to a decrease in the local Nusselt number (Fig. 9c). The fact that the changes in Nb do not lead to changes in the local Nusselt number is in agreement with the findings reported by Keblinski et al. [50] that Brownian motion of nanoparticles contributes much less than other factors since Brownian motion of nanoparticles is too slow to transport significant amount of heat through a nanofluid.
5. Conclusions Steady natural convective flow and heat transfer of a water based nanofluid in a porous annulus between two isothermal horizontal concentric cylinders has been numerically investigated using the nanofluid model proposed by Buongiorno. Distributions of streamlines, isotherms, isoconcentrations, local and average Nusselt numbers at a wide range of key parameters such as Ra = 50–500; Le = 1–50; Nr = 0.1–0.4; Nb = 0.1–0.4; Nt = 0.1–0.4 have been obtained. It has been found that an insertion of nanoparticles leads to a transfer from bicellular flow structure to unicellular flow structure for small values of the Rayleigh number. This result characterizes a displacement of the bifurcation point 2-D unicellular flow to 2-D bicellular flow for R = 2. It has been shown a formation of unicellular and bicellular flows for Ra = 500 at different initial conditions. Acknowledgement This work of M.A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K. References [1] Narasimhan A. Essentials of heat and fluid flow in porous media. New York: CRC Press; 2013. [2] Sheremet MA, Pop I. Natural convection in a square porous cavity with sinusoidal temperature distributions on both side walls filled with a nanofluid: Buongiorno’s mathematical model. Transp Porous Media 2014;105:411–29. [3] Nield DA, Bejan A. Convection in porous media. 4th ed. New York: Springer; 2013. [4] Bagchi A, Kulacki FA. Natural convection in superposed fluid-porous layers. New York: Springer; 2014. [5] Jansen JD. A systems description of flow through porous media. New York: Springer; 2013. [6] Basak T, Chamkha AJ. Heatline analysis on natural convection for nanofluids confined within square cavities with various thermal boundary conditions. Int J Heat Mass Transfer 2012;55:5526–43. [7] Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. Proceedings of the 1995 ASME international mechanical engineering congress and exposition, vol. 66. San Francisco (USA): ASME, FED 231/MD; 1995. p. 99–105. [8] Li Y, Zhou J, Tung S, Schneider E, Xi S. A review on development of nanofluid preparation and characterization. Powder Technol 2009;196:89–101. [9] Kakaç S, Pramuanjaroenkij A. Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transfer 2009;52:3187–96. [10] Wong KV, Leon OD. Applications of nanofluids: current and future. Adv Mech Eng 2010:11. Article ID 519659. [11] Fan J, Wang L. Review of heat conduction in nanofluids. ASME J Heat Transfer 2011;133:040801. [12] Jaluria Y, Manca O, Poulikakos D, Vafai K, Wang L. Heat transfer in nanofluids. Adv Mech Eng 2012:2. Article ID 972973. [13] Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S. A review of the applications of nanofluids in solar energy. Int J Heat Mass Transfer 2013;57:582–94. [14] Hajmohammadi MR, Maleki H, Lorenzini G, Nourazar SS. Effects of Cu and Ag nano-particles on flow and heat transfer from permeable surfaces. Adv Powder Technol 2015;26:193–9.
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