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Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus
ICHMT-02843; No of Pages 14 International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
3Q1
Abbasali Abouei Mehrizi a,⁎, Kurosh Sedighi b, Mousa Farhadi b, Mohsen Sheikholeslami b a b
Young Researchers Club, Karaj Branch, Islamic Azad University, Karaj, Iran Mechanical Engineering, Faculty of Mechanical Engineering, Babol University of Technology, Babol, Islamic Republic of Iran
a r t i c l e
i n f o
a b s t r a c t
A numerical study for steady-state, laminar and natural convection in a horizontal annulus between a heated triangular inner cylinder and cold elliptical outer cylinder was investigated using lattice Boltzmann method. Both inner and outer surfaces are maintained at the constant temperature and air is the working fluid. Study is carried out for Rayleigh numbers ranging from 1.0 × 103 to 5.0 × 105. The effects of different aspect ratios and elliptical cylinder orientation were studied at different Rayleigh numbers. The local and average Nusselt numbers and percent of increment heat transfer rate were presented. The average Nusselt number was correlated. The results show that by decreasing the value of aspect ratio and/or increasing the Rayleigh number, the Nusselt number increases. Also the heat transfer rate increases when the ellipse positioned vertically. © 2013 Published by Elsevier Ltd.
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Available online xxxx
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Keywords: Lattice Boltzmann method (LBM) Free convection Triangular cylinder Elliptical cylinder Curve boundary Annuli
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7 8 9 11 10 12 13 14 15 16 17 18 19
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1. Introduction
35
Natural convection is one of the simplest mechanisms of heat transfer and has been used in many applications for purpose of cooling or heating in industry. The annular shape enclosure is one of the applicable geometry in engineering and industry; the geometry of the horizontal circular annulus is commonly found in solar collector-receiver, underground electric transmission cables and vapor condenser for water distillation, heat exchangers and food processing equipment. In the past, many studies were performed about natural convection in a horizontal annulus between two circular cylinders. Kuehn and Goldstein [1,2] reviewed the most reliable studies about the free convection in the annulus. Also, they conducted an experimental and theoretical analyzes of natural convection in concentric and eccentric horizontal cylindrical annuli. Vertical and horizontal eccentricity of the inner cylinder was examined by Glakpeet al. [3] and Guj and Stella [4] at constant heat flux and isothermal boundary conditions, respectively. Fewer studies were investigated natural convection in the non-circular cylinder, such as elliptical annuli and triangular cylinder as an inner or outer cylinder. Theoretical and experimental studies of steady-state natural convection in a symmetric annulus space between two concentric, confocal elliptic tubes were performed by Lee and Lee [5]. In the investigation, the natural convection formulation was developed in elliptical
44 45 46 47 48 49 50 51 52 53 54 55 56 57
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2
Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus☆
1
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (A.A. Mehrizi).
coordinate system and solved by finite difference method (FDM), and the results were compared with experimental data. Elshamy et al. [6] performed numerical simulations of free convection in a horizontal confocal elliptical annulus. Finite volume method (FVM) was used to simulate the problem, and the results were reported for local and average Nusselt numbers. Numerical investigations of buoyancy driven flows in horizontal concentric and eccentric elliptical geometry were carried out. The governing equation was discretized by body-fitted curvilinear coordinate transformation method and solved by finite volume method. Zhu et al. [7] utilized the differential quadrature (DQ) method to simulate steady natural convection in a symmetrical elliptic annuli shape. Mahfouz and Badr [8] investigated transient and steady natural convection heat transfer in an elliptical annulus. They used Fourier spectral method to solve mass, momentum and energy equations. Velocity and thermal fields and average Nusselt number were presented and discussed. Natural convection in triangular enclosures was investigated by many authors [9,10]. Triangle inner or outer cylinder was examined by Xu et al. [11,12]. They used finite volume approach to solve the governing equations for laminar natural convective heat transfer from a horizontal triangular cylinder to concentric cylindrical enclosure. The work examined different radius ratios and inclination angles for the inner triangular cylinder, and results were presented in term of streamlines, temperature contours and Nusselt number. They also used a horizontal cylinder inside a concentric triangular enclosure and examined the effect of different cross section of the inner cylinder. The lattice Boltzmann method is a numerical method that has been recently developed and used in simulation of complex phenomena in fluid mechanics [13] such as conjugate heat transfer[14,15], nonofluid
0735-1933/$ – see front matter © 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
T1:2 T1:3 T1:4 T1:5 T1:6 T1:7 T1:8 T1:9 T1:10 T1:11 T1:12 T1:13 T1:14 T1:15 T1:16 T1:17 T1:18 T1:19
A Ar B BR C F F G gy L Nu P Pr R
T1:39 T1:40 T1:41 T1:42 T1:43 T1:44 T1:45 T1:46 T1:47 T1:48 T1:49 T1:50 T1:51 T1:52 T1:53
Subscripts f first fluid nodes ff second fluid nodes b boundary nodes w wall nodes avg average c cold eff effective h hot in inner cylinder k lattice model direction L local out outer cylinder s sound
T1:54 T1:55 T1:56 T1:57 T1:58
Superscript eq equilibrium distribution function neq non-equilibrium distribution function ~ post-collision state of the distribution function.
w
f
b
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2. Lattice Boltzmann method
120
In contrast to the classical macroscopic Navier–Stokes (NS) approach, the lattice Boltzmann method (LBM) uses a mesoscopic model to simulate fluid flows. The LBM is a discretization of Boltzmann's transport equation (BTE). To develop the BTE, Boltzmann assumed a fluid made of particles that collide according to the laws of classical mechanics. In LBM, the domain is discretized in uniform Cartesian cells which each one holds a fixed number of distribution functions, which represent the number of fluid particles moving in discrete directions. For present work, the D2Q9 model is used, which consists of nine distribution functions. The distribution functions are calculated by solving the lattice Boltzmann equation (LBE), which is a special discretization of the kinetic Boltzmann equation. The LBM is based on simultaneously solving two distribution functions, one for velocity field and the other for the temperature field [29]. After introducing BGK (Bhatnagar– Gross-Krook) approximation, the Boltzmann equation without external forces can be formulated as below [30]:
121 122
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Greek symbols α thermal diffusivity coefficient [m2/s] β thermal expansion coefficient [1/°C] Δt lattice time step pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε dimensionless eccentricity of ellipse, ε ¼ 1−Ar 2 [−] jx −x j δ ¼ x −x fraction of an intersected link in the fluid region [−] j j ρ density [kg/m2] τt relaxation time for temperature equation [−] τv relaxation time for velocity equation [−] υ kinematic viscosity [m2/s] χ weighting factor [−] ω weighting factor [−] c θ ¼ TT−T Dimensionless temperature [−] −T c h
C
T1:25 T1:26 T1:27 T1:28 T1:29 T1:30 T1:31 T1:32 T1:33 T1:34 T1:35 T1:36 T1:37 T1:38
D
location vector
E
T1:22 T1:23 T1:24
velocity [m/s] ! ! ! x ¼ x i þy j
88 89
T
T1:20 T1:21
length of semi-major axis of ellipse [m] axis ratio of ellipse, Ar = b/a [−] length of semi-minor axis of ellipse [m] Aspect ratio, BR = b/R [−] discrete lattice velocity [m2/s] distribution function for density [−] total body force [−] distribution function for temperature [−] acceleration due to the gravity in (−y) direction [ms−2] characteristic length [m] Nusselt number [−] Perimeter of triangle [m] Prandtl number, Pr = υ/α [−] radius of the circumscribed circle of the equilateral triangle, [m] Ra Rayleigh number, Ra = gβΔT(a − R)3/αυ [−] T temperature [K] T Time [s] ! ! ! u ¼ u i þv j
[16,17] and porous media [18]. In comparison with the conventional CFD methods, the advantages of LBM include simple calculation procedure, simple and efficient implementation for parallel computation, easy and robust handling of complex geometries, etc. Various numerical simulations have been performed using different thermal LB models or Boltzmann-based schemes to investigate the natural convection problems [19–22]. Numerical investigation of natural convection heat transfer in a horizontal concentric annulus using lattice Boltzmann simulation was performed by Shi et al. [23]. They proposed a Finite difference-based LBGK model for heat transfer based on the discrete velocity model in curvilinear coordinates. Fattahi et al. [24] simulated natural convection heat transfer in eccentric annulus using lattice Boltzmann model (LBM) based on double-population approach. They successfully used an extrapolation method based on Guo et al. [25] and Mei et al. [26] velocity boundary condition approach. Also they investigated mixed convection heat transfer in an eccentric annulus based on multi-distribution function, double-population approach [27]. Osman et al. [28] investigated natural convection from a concentrically and eccentrically inner heated cylinder placed inside a cold outer cylinder. To solve the problem, a double-population thermal lattice Boltzmann was used. The D2Q4 and D2Q9 BGK models were selected to determine the temperature and velocity fields, respectively. The aim of the present study was a laminar natural convection heat transfer between the concentric inner heated triangular cylinder and outer elliptical enclosure using the lattice Boltzmann method (Fig. 1(a)). The study has been carried out for Rayleigh number in the range of 103 b Ra b 5 × 105, where the Prandtl number fixed at 0.71, and the aspect ratio was changed from 1.2 to 3. Correlations were suggested for average Nusselt number at different Rayleigh numbers and shape configurations.
F
Nomenclature
O
T1:1
R O
2
∂f ! ! 1 eq f −f þ c ∇f ¼ τv ∂t
92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
123 124 125 126 127 128 129 130 131 132 133 134 135 136
ð1Þ
In this equation, f is the distribution function. In the lattice Boltzmann method, Eq. (1) is discretized and assumed valid along specific directions and linkages. Therefore, the discrete Boltzmann equation can be written as ! ! f k eq x ; t −f k x ; t ! ! ! f k x þ c k Δt; t þ Δt −f k x ; t ¼ Δt τv
90 91
ð2Þ
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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Fig. 1. (a) Sketch of computational domain. (b) Characterized lattice nods and curved wall boundary.
152 153 154 155 156 157
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150 151
! where Δt denotes lattice time step, c k is the discrete lattice velocity in ! direction k, Fk is the external force in direction of lattice velocity c k , and τv denotes the lattice relaxation time. Hence, by applying a Chapman–Enskog analysis to Eq. (3), the equations of Navier–Stokes can be restored and the relation between τv and kinetic viscosity υ is defined as τv = (1/c2s )υ + 1/2. feq k is the equilibrium distribution function. The local equilibrium distribution function determines the type of problem that needs to be solved. Eq. (3) is usually solved in two steps:
E
149
fk
169
τv
! þ Δt: F k
ð4Þ
eq
3 ! ! 2 ! ! !! c u k c u 1 1 u:u7 6 ¼ ωk ρ41 þ k 2 þ − 5 2 2 c2s cs c4s
8 k ¼ 0 < ð0; 0Þ ; ! c k ¼ ð cos½ðk−1Þπ=2; sin½ðk−1Þπ=2Þ; pffiffiffi : ð cos½ð2k−9Þπ=4; sin½ð2k−9Þπ=4Þ 2
174 175 176
ð9Þ 178 177
! ! g k eq x ; t −g k x ; t ! ! ! g k x þ c k Δt; t þ Δt −g k x ; t ¼ Δt τt
ð10Þ 182 181
For thermal simulation, the equilibrium distribution functions could 183 be written as 184 " eq
ð5Þ
ð6Þ
k ¼ 1−4 k ¼ 5−8
173 172
The lattice Boltzmann equation for thermal energy distribution 179 could be written as 180
g k ¼ ωk T 1 þ
Eqs. (4) and (5) are called the collision and streaming steps, respectively, where fk and ef k denote the pre- and post-collision state of the distribution function. The collision step models various fluid particle interactions, which are calculated as follows: 2
166 167 168
! ! x ; t −f k x ; t
U
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! ! ! f k x þ c k Δt; t þ Δt ¼ ef k x ; t
161 160 162 163
N C O
eq fk ! ef ! k x ; t −f k x ; t ¼ Δt
159 158
τv
R
148 147
! þ Δt: F k ð3Þ
T
! ! ! f k x þ c k Δt; t þ Δt −f k x ; t ¼ Δt
! eq ! fk x ; t −f k x ; t
where ωk is a weighting factor. For this work, the D2Q9 model has been used. The values of w0 = 4/9 w1 − 4 = 1/9 and w5 − 9 = 1/36 are assigned, cs is the sound speed and ck discrete lattice velocity and is selected as
D
The general form of lattice Boltzmann equation with external force can be written as
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146
R
145
# ! ! ck u c2s
ð11Þ 185 186 187
where T is the fluid temperature and can be evaluated as T¼
X
eq g k k
ð12Þ 189 188
Furthermore, the temperature relaxation time is evaluated as func- 190 tion of the thermal diffusivity coefficients: 191 2 τt ¼ 1=cs α þ 1=2
ð13Þ 193 192
! where ρ and u are the macroscopic fluid density and velocity and are calculated as follows: Flow density : ρ ¼
X
fk
ð7Þ
In order to incorporate buoyancy force in the model, the Boussinesq approximation was applied, and radiation heat transfer is negligible; therefore, the force term in the Eqs. (3) and (4) need to be calculated as below in vertical direction (y):
194 195 196 197
k
171 170
! X ! Momentum : ρ u ¼ f k ck k
ð8Þ
! ! F y ¼ 3ωk g y βðT h −T c Þ
ð14Þ 198 199
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
Stream line
Isotherms
a)
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R O
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F
Ra =103
T
E
D
Ra =104
outer enclosure
C
inner cylinder
N
C
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R
R
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b)
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Fig. 2. Validation of natural convection in triangular cylinder to its concentric cylindrical enclosure. (a) Stream line and isotherms (FVM ) Ref [11], LBM (—). (b) Local Nusselt number along the triangular and circular cylinders and comparison with Ref [11], whenRo/Ri = 2.
200 201
! where g y is the acceleration of gravity acting in the y-direction of the lattice links, and β is the thermal expansion coefficient.
202
3. Curved boundary treatment
203 204
Generally, there are two important approaches was recommended to apply the curve boundary condition. The first approach was suggested by Fillipova and Haanel [31], which it is called FH boundary condition. This method is based on improvements of the bounce-back rules. Mei et al. [26] improved the stability of the FH scheme; the new scheme was called
205 206 207
MLS boundary condition. The second approach was developed by Guo and Zheng [25]. This method is based on the extrapolation scheme which was presented by Chen et al. [32]. In the present study, a second-order accuracy method was used to define curve boundary condition (MLS boundary condition). Fig. 1(b) shows a part of arbitrary curved wall geometry. In this Figure the white and gray sides show the fluid and solid regions, respectively. The clear white nodes in the solid region indicate the boundary ! ! nodes x b , the hatched nodes show the first fluid nodes x f .The solid gray nods to show the second fluid nods and the solid block nodes on
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
5
O
F
a)
Present study.
El-shanny et al. Ref [6]
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b)
Outer cylinder
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Inner cylinder
R
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! the boundary x w indicate the intersections of the wall with various lattice links. The fraction of an intersected link in the fluid region, δ, is determined by ! ! x f − x w δ ¼ ! ! x f − x b
222 221
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218 219
R
E
Fig. 3. Validation of natural convection in an elliptical annulus. (a) Stream line (left half) and isotherms (right half) at Ra = 104. (b) Local Nusselt number along the inner and outer ellipses and comparison with Ref [6], when Mr = 2.25, Ari = 0.436.
ð15Þ
The unknown distribution function ef k is calculated by linear interpo- 228 lation that was suggested by Fillipova and Hänel [31]. 229 3! ! ! ! ef ! x b ; t þ Δt ¼ ð1−χ Þef k x f ; t þ Δt þ χ f k x b ; t þ Δt þ 2ωk ρ 2 e k : u w k ck
ð16Þ 231 230
t3:1 t3:2
Table 1 Grid independency test in the case of BR = 1.5 and Ra = 104.
U
225 226
At the collision step, the fluid side distribution function on the fluid nod ef k is determined, but the solid side distribution function at the opposite direction ef k is unknown. On the other hand, for the streaming step, we need to know ef k at the boundary node xb.
223 224
227
3.1. Velocity in curved boundary condition
232
where ! ! f k x b ; t þ Δt ¼ ωk ρ x f ; t þ Δt " # 3 ! ! 9 ! ! 2 3 ! ! 1 þ !2 e k : u bf þ !4 e k : u f − !2 u f : u f ck 2ck 2ck ð17Þ
t3:3 t3:4 t3:5 t3:6 t3:7 t3:8 t3:9
Grid size 140 × 140 160 × 160 180 × 180 200 × 200 220 × 220
Horizontal ellipse
Vertical ellipse
Nuavg 3.652 3.665 3.672 3.673 3.673
Nuavg 3.514 3.527 3.534 3.536 3.537
! ! ! In Eq. (17), u f ≡ u x f ; t þ Δt , the fluid velocity near the wall u bf is the imaginary velocity for interpolations a unit vector in direction ! of k.χ is the weighting factor depends on u bf and at the MLS scheme it is calculated as, 3 ! 3 ! ! u þ u u bf ¼ 1− 2δ f 2δ w
and
χ¼
ð2δ−1Þ ðτ þ 1=2Þ
when δ≥ 1=2
ð18Þ
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
Ra =103
Ra =104
Ra =103
Ra =104
Ra =105
Ra =5×105
BR =1.2
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BR =1.5
P
BR =2
E
D
BR =3
T
Ra =105
Ra =5×105
242 241
χ¼
ð2δ−1Þ ðτ−2Þ
E
! ! ! ! u bf ¼ u ff ¼ u f x f þ ek Δt; t þ Δt and
when δb1=2
R
240 239
C
Fig. 4. Streamlines for different Rayleigh number and aspect ratios when Pr = 0.7.
ð19Þ
! where u ff is the velocity of second node on the fluid side and ek ≡ −ek .
244
3.2. Temperature in curved boundary condition
249
O
C
247 248
Treating the curve boundary conditions for the temperature field, the scheme suggested by Yan and Zu [33] was used. This scheme is based on extrapolation method with second order accuracy. Distribution function for temperature is divided into two parts, equilibrium and nonequilibrium.
N
245 246
R
243
251 250 252 253
U
eq neq e g k ðxb ; t Þ ¼ g k ðxb ; t Þ þ g k ðxb ; t Þ
ð20Þ
streaming equation By substituting Eq. (20) into temperature ! ! ! g k x þ c k Δt; t þ Δt ¼ e g k x ; t þ Δt :
1 neq eq g ðxb ; t Þ g k ðxb ; t Þ ¼ g k ðxb ; t Þ þ 1− τt k
258
To determine the value of g k ðxb ; t Þ, the value of equilibrium and nonequilibrium parts of distribution function should be defined. The equilibrium distribution function g eq ðxb ; t Þ can be expressed as k
3! ! eq g k ðxb ; t Þ ¼ ωk T b 1 þ 2 e k : u b c
h i ! ! ! u b ¼ u w þ ðδ−1Þ u f =δ; if δ≥0:75; h i h i ! ! ! ! ! u b ¼ u w þ ðδ−1Þ u f þ ð1−δÞ 2 u w þ ðδ−1Þ u ff =ð1 þ δÞ;
if
δb0:75;
ð23Þ T b ¼ ½T w þ ðδ−1ÞT f =δ;
263 262
if δ ≥0:75;
T b ¼ ½T w þ ðδ−1ÞT f þ ð1−δÞ½2T w þ ðδ−1ÞT ff =ð1 þ δÞ;
if δb0:75; ð24Þ
! ! where Tf, Tff, u f and u ff are temperature and velocity in the node xf and xff,respectively. The non-equilibrium distribution function at boundary nodes g neq k ðxb ; t Þ, could be approximated by x f ; t ; if δ≥0:75; neq neq neq g k ðxb ; t Þ ¼ δg k x f ; t þ ð1−δÞg k ðxff ; t Þ if δb0:75; neq
ð22Þ
265 264 266 267 268 269
neq
g k ðxb ; t Þ ¼ g k
ð25Þ
ð21Þ
255 254 256 257
ub ≡ u(xb,t) and Tb ≡ T(xb,t) are the velocity and temperature on bound- 259 260 ary nodes, respectively, and can be approximated by: 261
271 270
In this estimation, the second-order approximation of Chapman– 272 Enskog and Taylor series expansions are used, which was presented 273 by Yan and Zu [33]. 274 4. Problem definition
275
A natural convection heat transfer between a heated triangular 276 inner cylinder and elliptical outer cylinder was investigated in which 277
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
7
BR =1.2
R O
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BR =1.5
P
BR=2
E
D
BR =3
283 284 285 286 287 288 289
E
R
281 282
R
279 280
situations of an elliptical cylinder were assumed to be at vertical and horizontal orientation. Computational domain and boundary condition were shown in Fig. 1(a). Annulus shapes enclosure between a concentrically inner triangular and outer elliptical cylinder is considered. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The eccentricity for elliptical cylinder is given by ε ¼ 1−Ar2 , where Ar is the axis ratio of ellipse Ar = b/a. The inner cylinder is maintained at constant high temperature (Th), and the temperature of the outer cylinder is set at a constant low temperatures (Tc). Air was considered as working fluid with Pr = 0.7. BR is the aspect ratio, defined as the ratio of the minor axis of the ellipse to the radius of the circumscribed circle of the inner equilateral triangle, BR = b/R. The Rayleigh number (Ra) for the current problem is defined as follows:
N C O
278
C
T
Fig. 5. Isotherms for different Rayleigh number and aspect ratios when Pr = 0.7.
gβðT h −T c Þða−RÞ3 Ra ¼ αν
293 294
The Rayleigh number is defined to base on the distance between major axis of ellipse and radius R. Local Nusselt number is defined on the characteristic length L as ∂θ NuL ¼ L ! jwall ∂n
296 295 297 298 299 300 301 302
U
290 291 292
ð26Þ
ð27Þ
! where n is a normal vector on the wall, and θ is dimensionless temperature. The radius of the circumscribed circle of the inner triangular cylinder R and the perimeter of the outer elliptical cylinder Pe were selected as characteristic length L. The mean Nusselt number for inner and outer cylinder can be evaluated as 1 Nuin ¼ P
Z
P 0
NuL dp
ð28 aÞ
Nuout ¼
1 Pe
Z
Pe 0
NuL dpe
ð28 bÞ
304 303
306 305
where P is the perimeter of the triangular cylinder. And the average 307 Nusselt number for the two surfaces of the annulus is: 308 Nuavg ¼ Nuout þ Nuin =2
ð29Þ 310 309
5. Results and discussion
311
The effect of different aspect ratios on free convection heat transfers into the air-filled elliptical-triangular annuli space (Pr = 0.7) was investigated for four different Rayleigh numbers. The study was performed for horizontal and vertical orientations of the elliptical enclosure. Results were presented in the form of streamlines; isotherms, local and average Nusselt numbers. Natural convective heat transfers from a horizontal triangular cylinder to its concentric cylindrical enclosure by Xu et al. [11] and the natural convection between confocal horizontal elliptical cylinders by Elshamy et al. [6] were selected for validation of the present study. Validations were presented in the form of isotherms and streamlines for two different Reynolds numbers (Fig. 2(a)). Furthermore, the local Nusselt number was compared with Ref [11] for different Rayleigh numbers ranging from 1000 to 100000 (Fig. 2(b)). For the case of confocal horizontal elliptical annulus, the eccentricity of the inner and outer wall were taken 0.9 and 0.4, respectively, and the Rayleigh number is equal to10000 (Fig. 3(a)). The local Nusselt numbers of the inner and outer cylindrical ellipse for two Rayleigh numbers based upon the description in original paper were plotted at the Fig. 3(b). The LBM result indicates an acceptable agreement with the results that presented in Refs [6,11].
312
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331
338 339
Max
nþ1 n u − u ju j n
−6
≤10
;
Max
nþ1 n v − v jv j n
−6
≤10
;
θnþ1 −θn −6 Max ≤10 θn
ð30Þ 340 341 342
349
5.1. Horizontal orientation of the elliptical cylinder
350
For the constant aspect ratio, changing the Rayleigh number from 103 to 5 × 105 the size of vortices core along the side of the triangle is increased (Fig. 4). Moreover, at Ra = 105 and Ra = 5 × 105, the
E T C E R R O C
351 352
N
345 346
U
343 344
D
347 348
Where u and v are the velocity components in x and y direction, respectively. In all cases, results show that due to the buoyancy force produced by temperature gradient, the fluid moves up from the side of the triangle to the outer cylinder, and the flow becomes cool and denser by the outer cylinder so moves down along the ellipse walls. For this reason, two symmetrical recirculation cells are formed at the left and right of the triangle.
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streamlines are sharper and have different patterns with respect to the other Rayleigh numbers. With increment of the aspect ratio, the more deviation is observed on the streamlines near the side vertices of the triangle. Also, two new recirculation zones appear near the bottom section of the triangle sides at Ra = 105 and 5 × 105 for BR = 1.2. Fig. 5 shows the isotherms at different Rayleigh numbers and aspect ratios. It can be observed that the isotherms are symmetry like as streamlines. At Ra = 103, the isotherms change smoothly and conduction heat transfer became to dominate in all aspect ratios. With increasing the aspect ratio, the space between the sides of triangle and elliptical enclosure is restricted so the vortices cores extend and become stronger. Therefore, the isotherms have greater deflection. When the Rayleigh number increases to 105 and 5 × 105, the natural convection due to the gravity force is dominant mechanism for heat transfer, and the plume shape appears above the climax vertex (the upper vertex of the triangle), which causes to arise of thermal boundary layer at this point. With the increase of the aspect ratio, the size and shape of plume shape become smaller and sharper, respectively. The distribution of the local Nusselt number for inner and outer cylinders is shown in Fig. 6. The local Nusselt number decreases exponentially at the apex at P = 0 due to the separation of thermal boundary layer. Fig. 6(a) shows two local maxima at the side vertices of the triangle. At lower Rayleigh numbers due to conduction domination, the Nusselt number rises along the sides of the triangle with lower slope in comparisons with high Rayleigh number. However, it shows the linear mode at high Rayleigh number. At the base of triangle, conduction is dominant heat transfer. Hence, the slop of local Nusselt
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In this simulation, the uniform quadratic grid cell is used to simulation the problem. The grid independency test was examined for different sizes of grids. Table 1 shows the average Nusselt number for Ra = 104, BR = 1.5 and for tow horizontal and vertical orientation of the outer elliptical cylinder. As sown in Table 1, it was found that the calculated values of Nuavg are approximately equal after 180 × 180 grid size, so the 180 × 180 grid size was selected. The criterion of the convergency is selected as
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Fig. 6. Variation of the local Nusselt number along the wall of (a) inner triangular cylinder and (b) outer elliptical cylinder.
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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Fig. 6 (continued).
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Figs. 8 and 9 show the variation of streamlines and isotherms pattern for different Rayleigh number and aspect ratios when the outer cylinder orientation is changed to the vertical direction. As a general statement, the trend of variation of streamlines and isotherms are almost similar for both vertical and horizontal orientations of ellipse, with slight difference. At BR = 1.2, two new recirculation zones appear under the base of the triangle. These new recirculation zones disappear at high Rayleigh number and lower value of aspect ratios. The narrow gap between
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the vertices of triangle and ellipse walls causes for the phenomenon. The isotherm lines are denser near the wall on the top section of the ellipse and the boundary layer grows with moving along the wall to the bottom, but when BR = 1.2, the boundary layer is slightly denser and compresses near the side vertices. This effect is clear on the local Nusselt number distribution in Fig. 10(b). At height Rayleigh number and BR = 1.2, apart from the peak point at θ = 0, two new peak points appear on the local Nusselt number curves at the elliptical wall. These two new peak points are
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number distribution approximately becomes zero. By increasing the Rayleigh number and reducing the aspect ratio, the value of local Nusselt number increases. Fig. 6(b) shows variation of the local Nusselt number along the wall of outer elliptical cylinder. The isotherms on the upper section of the ellipse are become denser. Therefore, the local Nusselt number has greater value in this section, but it should be mentioned that the plume makes slightly jump on Nusselt number near this region. At Ra = 104, BR = 1.2 and Ra = 103 for all aspect ratio, the local Nusselt number in the bottom of ellipse at 150 ≤ θ ≤ 180 is greater than other Rayleigh numbers. The reason of this trend is the domination of conduction heat transfer; with augmentation of Rayleigh number, this value of local Nusselt number is limited to the zero. Fig. 7 shows the correlation of average Nusselt number versus the Rayleigh number. With an increase of Rayleigh number (augmentation of natural convection effect) and decrease of the aspect ratio the average Nusselt number increases.
Fig. 7. Adapted curve for the average Nusselt number as a function of the Rayleigh number.
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Fig. 8. streamlines for different Rayleigh number and aspect ratios when Pr = 0.7.
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formed near the side vertices at 90 ≤ θ ≤ 120 but at different Rayleigh numbers. However, the location of the peak point is slightly changed. At the lower Rayleigh number and higher aspect ratio, two new peak points merges into one peak point at the same location.
The distribution of the local Nusselt number for inner triangle also is shown in Fig. 10(a). Finally, the average Nusselt number can be correlated by using the least square method as shown in Fig. 11.
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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Fig. 9. streamlines for different Rayleigh number and aspect ratios when Pr = 0.7.
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The correlation functions were presented in Table 2 for both ellipse directions. Comparing the average Nusselt number for horizontal and vertical cylinders, it can be seen that the Nusselt number is greater when the ellipse is oriented horizontally. Table 3 shows variation of the average Nusselt number ratio between Ra = 5 × 105 and Ra = 103 for different aspect ratios
and different ellipse orientations. Furthermore, the percent of the average Nusselt increment ratio between two ellipse orientations is calculated. From the table, it is clear that the Nusselt number ratio is grater when the ellipse positioned vertically at all aspect ratios. The maximum increment of Nusselt number ratios occurs at AR = 3. The maximum increment of the Nusselt number ratio
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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Laminar natural convection in a concentric annulus was investigated by the lattice Boltzmann method. The curve boundary condition was implemented by MSL velocity boundary condition and a developed approach [31] based on extrapolation method was used to implement temperature boundary condition. The simulation data were validated by previously published works. The results indicate that lattice Boltzmann method in powerful method for simulation of natural convection heat transfer in complex geometry. The study showed that by increasing the Rayleigh number, the local and average Nusselt numbers increase for all aspect ratios. Also, at a constant Rayleigh number and with reduction the value of the aspect ratio, the average Nusselt number increases. Orientation of the ellipse has a small effect on streamlines and temperature contours but has a sensible effect on the number of recalculating zones. At vertical ellipse orientation, two new recirculation zones appear at BR = 1.2 for all Rayleigh numbers and at Ra = 103 for all aspect ratios except BR = 3. Due to the new vortices and narrower gap between the vertices side of triangle and ellipse wall, the number of peak points in the local Nusselt number for ellipse increases to three points. For the all aspect ratios and Ra numbers investigated, the average Nusselt number is greater for vertical orientation in comparison with horizontal orientation of the ellipse.
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[1] T.H. Kuehn, R.J. Goldstein, An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders, J. Fluid Mech. 74 (1976) 695–719. [2] T.H. Kuehn, R.J. Goldstein, An experimental study of natural convection heat transfer in concentric and eccentric horizontal cylindrical annuli, J. Heat Transf. 100 (4) (1978) 635–641. [3] E.K. Glakpe, C.B. Watkins Jr., J.N. Cannon, Constant heat flux solutions for natural convection between concentric and eccentric horizontal cylinders, Numer. Heat Transf. Part A-Appl. 10 (3) (1986) 279–295. [4] G. Guj, F. Stella, Natural convection in horizontal eccentric annuli: numerical study, Numer. Heat Transf. Part A -Appl. 27 (1) (1995) 89–105. [5] J.H. Lee, T.S. Lee, Natural convection in the annuli between horizontal confocal elliptic cylinders, Int. J. Heat Mass Transf. 24 (10) (1981) 1739–1742. [6] M.M. Elshamy, M.N. Ozisik, J.P. Coulter, Correlation for laminar natural convection between confocal horizontal elliptical cylinders, Numer. Heat Transf. 18 (1) (1990) 95–112. [7] Y.D. Zhu, C. Shu, J. Qiu, J. Tani, Numerical simulation of natural convection between two elliptical cylinders using DQ method, Int. J. Heat Mass Transf. 47 (4) (2004) 797–808. [8] F.M. Mahfouz, H.M. Badr, Heat convection between two confocal elliptic tubes placed at different orientations, Adv. Appl. Math. Mech. 1 (5) (2009) 639–663. [9] T. Basak, S. Roy, C. Thirumalesha, Finite element analysis of natural convection in a triangular enclosure: effects of various thermal boundary conditions, Chem. Eng. Sci. 62 (9) (2007) 2623–2640. [10] A. Koca, H.F. Oztop, Y. Varol, The effects of Prandtl number on natural convection in triangular enclosures with localized heating from below, Int. Commun. Heat Mass Transf. 34 (4) (2007) 511–519. [11] X. Xu, G. Sun, Z. Yu, Y. Hu, L. Fan, K. Cen, Numerical investigation of laminar natural convective heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure, Int. J. Heat Mass Transf. 52 (13–14) (2009) 3176–3186. [12] X. Xu, Z. Yu, Y. Hu, L. Fan, K. Cen, A numerical study of laminar natural convective heat transfer around a horizontal cylinder inside a concentric air-filled triangular enclosure, Int. J. Heat Mass Transf. 53 (1–3) (2010) 345–355. [13] A.A. Mohamad, Lattice Boltzmann method: fundamentals and engineering applications with computer codes, Springer-Verlag, London, 2011. [14] J. Wang, M. Wang, Z. Li, A lattice Boltzmann algorithm for fluid–solid conjugate heat transfer, Int. J. Therm. Sci. 46 (3) (2007) 228–234.
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between vertical and horizontal orientations of the ellipse is about 57.46 % at AR = 1.2.
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Fig. 10. Variation of the local Nusselt number along the wall of (a) inner triangular cylinder and (b) outer elliptical cylinder.
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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Fig. 10 (continued).
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Fig. 11. Adapted curve for the average Nusselt number as a function of the Rayleigh number.
Table 2 Correlations of the average Nusselt number based on curve fitting. Horizontal ellipse Maximum deviation(%) Vertical ellipse Maximum deviation(%)
Average Nusselt function
AR = 1.2 2.25 Ra0.102
AR = 1.5 0.12 Ra0.165
AR = 2 0.452 Ra0.193
AR = 3 0.303 Ra0.2
3.46
0.779
5.94
9.47
0.175
0.468 Ra 2.98
0.19
0.575 Ra 4.77
0.202
0.389 Ra 7.6
Nusselt augmentation
0.251 Ra0.212 15.1
AR = 1.2
AR = 1.5
AR = 2
AR = 3
t2:3
λH ¼
Nuavg jRa¼500000 Nuavg jRa¼10000
1.866
2.808
3.355
3.486
t2:4
Vertical ellipse
λV ¼
2.939
3.323
3.586
3.859
t2:5
Increasement (%)
λV −λH λV
Nuavg jRa¼500000 Nuavg jRa¼10000
57.465
18.342
6.909
10.701
t2:6
Horizontal ellipse
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Table 3 t2:1 Average Nusselt number ratio between Ra = 5 × 105 and Ra = 103 for different aspect ratios. t2:2
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[15] A. Tarokh, A.A. Mohamad, L. Jiang, Simulation of conjugate heat transfer using the lattice Boltzmann method, Numer. Heat Transf. Part A: Appl. 63 (3) (2013) 159–178. [16] A.A. Mehrizi, M. Farhadi, H.H. Afroozi, K. Sedighi, A.A. Darz, Mixed convection heat transfer in a ventilated cavity with hot obstacle: effect of nanofluid and outlet port location, Int. Commun. Heat Mass Transf. 39 (7) (2012) 1000–1008. [17] A.A. Mehrizi, M. Farhadi, S. Shayamehr, Natural convection flow of Cu–Water nanofluid in horizontal cylindrical annuli with inner triangular cylinder using lattice Boltzmann method, Int. Commun. Heat Mass Transf. 44 (2013) 147–156. [18] A.A. Mehrizi, M. Farhadi, K. Sedighi, M.A. Delavar, Effect of fin position and porosity on heat transfer improvement in a plate porous media heat exchanger, J. Taiwan Inst. Chem. Eng. 44 (3) (2013) 420–431. [19] P.H. Kao, R.J. Yang, Simulating oscillatory flows in Rayleigh–Benard convection using the lattice Boltzmann method, Int. J. Heat Mass Transf. 50 (17–18) (2007) 3315–3328. [20] A.A. Mohamad, R. Bennacer, M. El-Ganaoui, Double dispersion, natural convection in an open end cavity simulation via lattice Boltzmann method, Int. J. Therm. Sci. 49 (10) (2010) 1944–1953. [21] A.A. Mehrizi, M. Farhadi, H.H. Afrouzi, S. Shayamehr, H. Lotfizadeh, Lattice Boltzmann simulation of natural convection flow around a horizontal cylinder located beneath an insulation plate, J. Theor. Appl. Mech. 51 (3) (2013) 729–739. [22] A.A. Mohamad, A. Kuzmin, A critical evaluation of force term in lattice Boltzmann method, natural convection problem, Int. J. Heat Mass Transf. 53 (5–6) (2010) 990–996. [23] Y. Shi, T.S. Zhao, Z.L. Guo, Finite difference-based lattice Boltzmann simulation of natural convection heat transfer in a horizontal concentric annulus, Comput. Fluids 35 (1) (2006) 1–15.
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Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
3Q1
Abbasali Abouei Mehrizi a,⁎, Kurosh Sedighi b, Mousa Farhadi b, Mohsen Sheikholeslami b a b
Young Researchers Club, Karaj Branch, Islamic Azad University, Karaj, Iran Mechanical Engineering, Faculty of Mechanical Engineering, Babol University of Technology, Babol, Islamic Republic of Iran
a r t i c l e
i n f o
a b s t r a c t
A numerical study for steady-state, laminar and natural convection in a horizontal annulus between a heated triangular inner cylinder and cold elliptical outer cylinder was investigated using lattice Boltzmann method. Both inner and outer surfaces are maintained at the constant temperature and air is the working fluid. Study is carried out for Rayleigh numbers ranging from 1.0 × 103 to 5.0 × 105. The effects of different aspect ratios and elliptical cylinder orientation were studied at different Rayleigh numbers. The local and average Nusselt numbers and percent of increment heat transfer rate were presented. The average Nusselt number was correlated. The results show that by decreasing the value of aspect ratio and/or increasing the Rayleigh number, the Nusselt number increases. Also the heat transfer rate increases when the ellipse positioned vertically. © 2013 Published by Elsevier Ltd.
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Keywords: Lattice Boltzmann method (LBM) Free convection Triangular cylinder Elliptical cylinder Curve boundary Annuli
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1. Introduction
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Natural convection is one of the simplest mechanisms of heat transfer and has been used in many applications for purpose of cooling or heating in industry. The annular shape enclosure is one of the applicable geometry in engineering and industry; the geometry of the horizontal circular annulus is commonly found in solar collector-receiver, underground electric transmission cables and vapor condenser for water distillation, heat exchangers and food processing equipment. In the past, many studies were performed about natural convection in a horizontal annulus between two circular cylinders. Kuehn and Goldstein [1,2] reviewed the most reliable studies about the free convection in the annulus. Also, they conducted an experimental and theoretical analyzes of natural convection in concentric and eccentric horizontal cylindrical annuli. Vertical and horizontal eccentricity of the inner cylinder was examined by Glakpeet al. [3] and Guj and Stella [4] at constant heat flux and isothermal boundary conditions, respectively. Fewer studies were investigated natural convection in the non-circular cylinder, such as elliptical annuli and triangular cylinder as an inner or outer cylinder. Theoretical and experimental studies of steady-state natural convection in a symmetric annulus space between two concentric, confocal elliptic tubes were performed by Lee and Lee [5]. In the investigation, the natural convection formulation was developed in elliptical
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Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus☆
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☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (A.A. Mehrizi).
coordinate system and solved by finite difference method (FDM), and the results were compared with experimental data. Elshamy et al. [6] performed numerical simulations of free convection in a horizontal confocal elliptical annulus. Finite volume method (FVM) was used to simulate the problem, and the results were reported for local and average Nusselt numbers. Numerical investigations of buoyancy driven flows in horizontal concentric and eccentric elliptical geometry were carried out. The governing equation was discretized by body-fitted curvilinear coordinate transformation method and solved by finite volume method. Zhu et al. [7] utilized the differential quadrature (DQ) method to simulate steady natural convection in a symmetrical elliptic annuli shape. Mahfouz and Badr [8] investigated transient and steady natural convection heat transfer in an elliptical annulus. They used Fourier spectral method to solve mass, momentum and energy equations. Velocity and thermal fields and average Nusselt number were presented and discussed. Natural convection in triangular enclosures was investigated by many authors [9,10]. Triangle inner or outer cylinder was examined by Xu et al. [11,12]. They used finite volume approach to solve the governing equations for laminar natural convective heat transfer from a horizontal triangular cylinder to concentric cylindrical enclosure. The work examined different radius ratios and inclination angles for the inner triangular cylinder, and results were presented in term of streamlines, temperature contours and Nusselt number. They also used a horizontal cylinder inside a concentric triangular enclosure and examined the effect of different cross section of the inner cylinder. The lattice Boltzmann method is a numerical method that has been recently developed and used in simulation of complex phenomena in fluid mechanics [13] such as conjugate heat transfer[14,15], nonofluid
0735-1933/$ – see front matter © 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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T1:2 T1:3 T1:4 T1:5 T1:6 T1:7 T1:8 T1:9 T1:10 T1:11 T1:12 T1:13 T1:14 T1:15 T1:16 T1:17 T1:18 T1:19
A Ar B BR C F F G gy L Nu P Pr R
T1:39 T1:40 T1:41 T1:42 T1:43 T1:44 T1:45 T1:46 T1:47 T1:48 T1:49 T1:50 T1:51 T1:52 T1:53
Subscripts f first fluid nodes ff second fluid nodes b boundary nodes w wall nodes avg average c cold eff effective h hot in inner cylinder k lattice model direction L local out outer cylinder s sound
T1:54 T1:55 T1:56 T1:57 T1:58
Superscript eq equilibrium distribution function neq non-equilibrium distribution function ~ post-collision state of the distribution function.
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In contrast to the classical macroscopic Navier–Stokes (NS) approach, the lattice Boltzmann method (LBM) uses a mesoscopic model to simulate fluid flows. The LBM is a discretization of Boltzmann's transport equation (BTE). To develop the BTE, Boltzmann assumed a fluid made of particles that collide according to the laws of classical mechanics. In LBM, the domain is discretized in uniform Cartesian cells which each one holds a fixed number of distribution functions, which represent the number of fluid particles moving in discrete directions. For present work, the D2Q9 model is used, which consists of nine distribution functions. The distribution functions are calculated by solving the lattice Boltzmann equation (LBE), which is a special discretization of the kinetic Boltzmann equation. The LBM is based on simultaneously solving two distribution functions, one for velocity field and the other for the temperature field [29]. After introducing BGK (Bhatnagar– Gross-Krook) approximation, the Boltzmann equation without external forces can be formulated as below [30]:
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Greek symbols α thermal diffusivity coefficient [m2/s] β thermal expansion coefficient [1/°C] Δt lattice time step pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε dimensionless eccentricity of ellipse, ε ¼ 1−Ar 2 [−] jx −x j δ ¼ x −x fraction of an intersected link in the fluid region [−] j j ρ density [kg/m2] τt relaxation time for temperature equation [−] τv relaxation time for velocity equation [−] υ kinematic viscosity [m2/s] χ weighting factor [−] ω weighting factor [−] c θ ¼ TT−T Dimensionless temperature [−] −T c h
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location vector
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velocity [m/s] ! ! ! x ¼ x i þy j
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length of semi-major axis of ellipse [m] axis ratio of ellipse, Ar = b/a [−] length of semi-minor axis of ellipse [m] Aspect ratio, BR = b/R [−] discrete lattice velocity [m2/s] distribution function for density [−] total body force [−] distribution function for temperature [−] acceleration due to the gravity in (−y) direction [ms−2] characteristic length [m] Nusselt number [−] Perimeter of triangle [m] Prandtl number, Pr = υ/α [−] radius of the circumscribed circle of the equilateral triangle, [m] Ra Rayleigh number, Ra = gβΔT(a − R)3/αυ [−] T temperature [K] T Time [s] ! ! ! u ¼ u i þv j
[16,17] and porous media [18]. In comparison with the conventional CFD methods, the advantages of LBM include simple calculation procedure, simple and efficient implementation for parallel computation, easy and robust handling of complex geometries, etc. Various numerical simulations have been performed using different thermal LB models or Boltzmann-based schemes to investigate the natural convection problems [19–22]. Numerical investigation of natural convection heat transfer in a horizontal concentric annulus using lattice Boltzmann simulation was performed by Shi et al. [23]. They proposed a Finite difference-based LBGK model for heat transfer based on the discrete velocity model in curvilinear coordinates. Fattahi et al. [24] simulated natural convection heat transfer in eccentric annulus using lattice Boltzmann model (LBM) based on double-population approach. They successfully used an extrapolation method based on Guo et al. [25] and Mei et al. [26] velocity boundary condition approach. Also they investigated mixed convection heat transfer in an eccentric annulus based on multi-distribution function, double-population approach [27]. Osman et al. [28] investigated natural convection from a concentrically and eccentrically inner heated cylinder placed inside a cold outer cylinder. To solve the problem, a double-population thermal lattice Boltzmann was used. The D2Q4 and D2Q9 BGK models were selected to determine the temperature and velocity fields, respectively. The aim of the present study was a laminar natural convection heat transfer between the concentric inner heated triangular cylinder and outer elliptical enclosure using the lattice Boltzmann method (Fig. 1(a)). The study has been carried out for Rayleigh number in the range of 103 b Ra b 5 × 105, where the Prandtl number fixed at 0.71, and the aspect ratio was changed from 1.2 to 3. Correlations were suggested for average Nusselt number at different Rayleigh numbers and shape configurations.
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123 124 125 126 127 128 129 130 131 132 133 134 135 136
ð1Þ
In this equation, f is the distribution function. In the lattice Boltzmann method, Eq. (1) is discretized and assumed valid along specific directions and linkages. Therefore, the discrete Boltzmann equation can be written as ! ! f k eq x ; t −f k x ; t ! ! ! f k x þ c k Δt; t þ Δt −f k x ; t ¼ Δt τv
90 91
ð2Þ
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
137 138 139 140 141 142
144 143
3
R O
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F
A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
P
Fig. 1. (a) Sketch of computational domain. (b) Characterized lattice nods and curved wall boundary.
152 153 154 155 156 157
C
150 151
! where Δt denotes lattice time step, c k is the discrete lattice velocity in ! direction k, Fk is the external force in direction of lattice velocity c k , and τv denotes the lattice relaxation time. Hence, by applying a Chapman–Enskog analysis to Eq. (3), the equations of Navier–Stokes can be restored and the relation between τv and kinetic viscosity υ is defined as τv = (1/c2s )υ + 1/2. feq k is the equilibrium distribution function. The local equilibrium distribution function determines the type of problem that needs to be solved. Eq. (3) is usually solved in two steps:
E
149
fk
169
τv
! þ Δt: F k
ð4Þ
eq
3 ! ! 2 ! ! !! c u k c u 1 1 u:u7 6 ¼ ωk ρ41 þ k 2 þ − 5 2 2 c2s cs c4s
8 k ¼ 0 < ð0; 0Þ ; ! c k ¼ ð cos½ðk−1Þπ=2; sin½ðk−1Þπ=2Þ; pffiffiffi : ð cos½ð2k−9Þπ=4; sin½ð2k−9Þπ=4Þ 2
174 175 176
ð9Þ 178 177
! ! g k eq x ; t −g k x ; t ! ! ! g k x þ c k Δt; t þ Δt −g k x ; t ¼ Δt τt
ð10Þ 182 181
For thermal simulation, the equilibrium distribution functions could 183 be written as 184 " eq
ð5Þ
ð6Þ
k ¼ 1−4 k ¼ 5−8
173 172
The lattice Boltzmann equation for thermal energy distribution 179 could be written as 180
g k ¼ ωk T 1 þ
Eqs. (4) and (5) are called the collision and streaming steps, respectively, where fk and ef k denote the pre- and post-collision state of the distribution function. The collision step models various fluid particle interactions, which are calculated as follows: 2
166 167 168
! ! x ; t −f k x ; t
U
164 165
! ! ! f k x þ c k Δt; t þ Δt ¼ ef k x ; t
161 160 162 163
N C O
eq fk ! ef ! k x ; t −f k x ; t ¼ Δt
159 158
τv
R
148 147
! þ Δt: F k ð3Þ
T
! ! ! f k x þ c k Δt; t þ Δt −f k x ; t ¼ Δt
! eq ! fk x ; t −f k x ; t
where ωk is a weighting factor. For this work, the D2Q9 model has been used. The values of w0 = 4/9 w1 − 4 = 1/9 and w5 − 9 = 1/36 are assigned, cs is the sound speed and ck discrete lattice velocity and is selected as
D
The general form of lattice Boltzmann equation with external force can be written as
E
146
R
145
# ! ! ck u c2s
ð11Þ 185 186 187
where T is the fluid temperature and can be evaluated as T¼
X
eq g k k
ð12Þ 189 188
Furthermore, the temperature relaxation time is evaluated as func- 190 tion of the thermal diffusivity coefficients: 191 2 τt ¼ 1=cs α þ 1=2
ð13Þ 193 192
! where ρ and u are the macroscopic fluid density and velocity and are calculated as follows: Flow density : ρ ¼
X
fk
ð7Þ
In order to incorporate buoyancy force in the model, the Boussinesq approximation was applied, and radiation heat transfer is negligible; therefore, the force term in the Eqs. (3) and (4) need to be calculated as below in vertical direction (y):
194 195 196 197
k
171 170
! X ! Momentum : ρ u ¼ f k ck k
ð8Þ
! ! F y ¼ 3ωk g y βðT h −T c Þ
ð14Þ 198 199
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
Stream line
Isotherms
a)
P
R O
O
F
Ra =103
T
E
D
Ra =104
outer enclosure
C
inner cylinder
N
C
O
R
R
E
b)
U
Fig. 2. Validation of natural convection in triangular cylinder to its concentric cylindrical enclosure. (a) Stream line and isotherms (FVM ) Ref [11], LBM (—). (b) Local Nusselt number along the triangular and circular cylinders and comparison with Ref [11], whenRo/Ri = 2.
200 201
! where g y is the acceleration of gravity acting in the y-direction of the lattice links, and β is the thermal expansion coefficient.
202
3. Curved boundary treatment
203 204
Generally, there are two important approaches was recommended to apply the curve boundary condition. The first approach was suggested by Fillipova and Haanel [31], which it is called FH boundary condition. This method is based on improvements of the bounce-back rules. Mei et al. [26] improved the stability of the FH scheme; the new scheme was called
205 206 207
MLS boundary condition. The second approach was developed by Guo and Zheng [25]. This method is based on the extrapolation scheme which was presented by Chen et al. [32]. In the present study, a second-order accuracy method was used to define curve boundary condition (MLS boundary condition). Fig. 1(b) shows a part of arbitrary curved wall geometry. In this Figure the white and gray sides show the fluid and solid regions, respectively. The clear white nodes in the solid region indicate the boundary ! ! nodes x b , the hatched nodes show the first fluid nodes x f .The solid gray nods to show the second fluid nods and the solid block nodes on
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
5
O
F
a)
Present study.
El-shanny et al. Ref [6]
T
E
D
P
R O
b)
Outer cylinder
C
Inner cylinder
R
220
! the boundary x w indicate the intersections of the wall with various lattice links. The fraction of an intersected link in the fluid region, δ, is determined by ! ! x f − x w δ ¼ ! ! x f − x b
222 221
N C O
218 219
R
E
Fig. 3. Validation of natural convection in an elliptical annulus. (a) Stream line (left half) and isotherms (right half) at Ra = 104. (b) Local Nusselt number along the inner and outer ellipses and comparison with Ref [6], when Mr = 2.25, Ari = 0.436.
ð15Þ
The unknown distribution function ef k is calculated by linear interpo- 228 lation that was suggested by Fillipova and Hänel [31]. 229 3! ! ! ! ef ! x b ; t þ Δt ¼ ð1−χ Þef k x f ; t þ Δt þ χ f k x b ; t þ Δt þ 2ωk ρ 2 e k : u w k ck
ð16Þ 231 230
t3:1 t3:2
Table 1 Grid independency test in the case of BR = 1.5 and Ra = 104.
U
225 226
At the collision step, the fluid side distribution function on the fluid nod ef k is determined, but the solid side distribution function at the opposite direction ef k is unknown. On the other hand, for the streaming step, we need to know ef k at the boundary node xb.
223 224
227
3.1. Velocity in curved boundary condition
232
where ! ! f k x b ; t þ Δt ¼ ωk ρ x f ; t þ Δt " # 3 ! ! 9 ! ! 2 3 ! ! 1 þ !2 e k : u bf þ !4 e k : u f − !2 u f : u f ck 2ck 2ck ð17Þ
t3:3 t3:4 t3:5 t3:6 t3:7 t3:8 t3:9
Grid size 140 × 140 160 × 160 180 × 180 200 × 200 220 × 220
Horizontal ellipse
Vertical ellipse
Nuavg 3.652 3.665 3.672 3.673 3.673
Nuavg 3.514 3.527 3.534 3.536 3.537
! ! ! In Eq. (17), u f ≡ u x f ; t þ Δt , the fluid velocity near the wall u bf is the imaginary velocity for interpolations a unit vector in direction ! of k.χ is the weighting factor depends on u bf and at the MLS scheme it is calculated as, 3 ! 3 ! ! u þ u u bf ¼ 1− 2δ f 2δ w
and
χ¼
ð2δ−1Þ ðτ þ 1=2Þ
when δ≥ 1=2
ð18Þ
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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Ra =103
Ra =104
Ra =103
Ra =104
Ra =105
Ra =5×105
BR =1.2
R O
O
F
BR =1.5
P
BR =2
E
D
BR =3
T
Ra =105
Ra =5×105
242 241
χ¼
ð2δ−1Þ ðτ−2Þ
E
! ! ! ! u bf ¼ u ff ¼ u f x f þ ek Δt; t þ Δt and
when δb1=2
R
240 239
C
Fig. 4. Streamlines for different Rayleigh number and aspect ratios when Pr = 0.7.
ð19Þ
! where u ff is the velocity of second node on the fluid side and ek ≡ −ek .
244
3.2. Temperature in curved boundary condition
249
O
C
247 248
Treating the curve boundary conditions for the temperature field, the scheme suggested by Yan and Zu [33] was used. This scheme is based on extrapolation method with second order accuracy. Distribution function for temperature is divided into two parts, equilibrium and nonequilibrium.
N
245 246
R
243
251 250 252 253
U
eq neq e g k ðxb ; t Þ ¼ g k ðxb ; t Þ þ g k ðxb ; t Þ
ð20Þ
streaming equation By substituting Eq. (20) into temperature ! ! ! g k x þ c k Δt; t þ Δt ¼ e g k x ; t þ Δt :
1 neq eq g ðxb ; t Þ g k ðxb ; t Þ ¼ g k ðxb ; t Þ þ 1− τt k
258
To determine the value of g k ðxb ; t Þ, the value of equilibrium and nonequilibrium parts of distribution function should be defined. The equilibrium distribution function g eq ðxb ; t Þ can be expressed as k
3! ! eq g k ðxb ; t Þ ¼ ωk T b 1 þ 2 e k : u b c
h i ! ! ! u b ¼ u w þ ðδ−1Þ u f =δ; if δ≥0:75; h i h i ! ! ! ! ! u b ¼ u w þ ðδ−1Þ u f þ ð1−δÞ 2 u w þ ðδ−1Þ u ff =ð1 þ δÞ;
if
δb0:75;
ð23Þ T b ¼ ½T w þ ðδ−1ÞT f =δ;
263 262
if δ ≥0:75;
T b ¼ ½T w þ ðδ−1ÞT f þ ð1−δÞ½2T w þ ðδ−1ÞT ff =ð1 þ δÞ;
if δb0:75; ð24Þ
! ! where Tf, Tff, u f and u ff are temperature and velocity in the node xf and xff,respectively. The non-equilibrium distribution function at boundary nodes g neq k ðxb ; t Þ, could be approximated by x f ; t ; if δ≥0:75; neq neq neq g k ðxb ; t Þ ¼ δg k x f ; t þ ð1−δÞg k ðxff ; t Þ if δb0:75; neq
ð22Þ
265 264 266 267 268 269
neq
g k ðxb ; t Þ ¼ g k
ð25Þ
ð21Þ
255 254 256 257
ub ≡ u(xb,t) and Tb ≡ T(xb,t) are the velocity and temperature on bound- 259 260 ary nodes, respectively, and can be approximated by: 261
271 270
In this estimation, the second-order approximation of Chapman– 272 Enskog and Taylor series expansions are used, which was presented 273 by Yan and Zu [33]. 274 4. Problem definition
275
A natural convection heat transfer between a heated triangular 276 inner cylinder and elliptical outer cylinder was investigated in which 277
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
7
BR =1.2
R O
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F
BR =1.5
P
BR=2
E
D
BR =3
283 284 285 286 287 288 289
E
R
281 282
R
279 280
situations of an elliptical cylinder were assumed to be at vertical and horizontal orientation. Computational domain and boundary condition were shown in Fig. 1(a). Annulus shapes enclosure between a concentrically inner triangular and outer elliptical cylinder is considered. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The eccentricity for elliptical cylinder is given by ε ¼ 1−Ar2 , where Ar is the axis ratio of ellipse Ar = b/a. The inner cylinder is maintained at constant high temperature (Th), and the temperature of the outer cylinder is set at a constant low temperatures (Tc). Air was considered as working fluid with Pr = 0.7. BR is the aspect ratio, defined as the ratio of the minor axis of the ellipse to the radius of the circumscribed circle of the inner equilateral triangle, BR = b/R. The Rayleigh number (Ra) for the current problem is defined as follows:
N C O
278
C
T
Fig. 5. Isotherms for different Rayleigh number and aspect ratios when Pr = 0.7.
gβðT h −T c Þða−RÞ3 Ra ¼ αν
293 294
The Rayleigh number is defined to base on the distance between major axis of ellipse and radius R. Local Nusselt number is defined on the characteristic length L as ∂θ NuL ¼ L ! jwall ∂n
296 295 297 298 299 300 301 302
U
290 291 292
ð26Þ
ð27Þ
! where n is a normal vector on the wall, and θ is dimensionless temperature. The radius of the circumscribed circle of the inner triangular cylinder R and the perimeter of the outer elliptical cylinder Pe were selected as characteristic length L. The mean Nusselt number for inner and outer cylinder can be evaluated as 1 Nuin ¼ P
Z
P 0
NuL dp
ð28 aÞ
Nuout ¼
1 Pe
Z
Pe 0
NuL dpe
ð28 bÞ
304 303
306 305
where P is the perimeter of the triangular cylinder. And the average 307 Nusselt number for the two surfaces of the annulus is: 308 Nuavg ¼ Nuout þ Nuin =2
ð29Þ 310 309
5. Results and discussion
311
The effect of different aspect ratios on free convection heat transfers into the air-filled elliptical-triangular annuli space (Pr = 0.7) was investigated for four different Rayleigh numbers. The study was performed for horizontal and vertical orientations of the elliptical enclosure. Results were presented in the form of streamlines; isotherms, local and average Nusselt numbers. Natural convective heat transfers from a horizontal triangular cylinder to its concentric cylindrical enclosure by Xu et al. [11] and the natural convection between confocal horizontal elliptical cylinders by Elshamy et al. [6] were selected for validation of the present study. Validations were presented in the form of isotherms and streamlines for two different Reynolds numbers (Fig. 2(a)). Furthermore, the local Nusselt number was compared with Ref [11] for different Rayleigh numbers ranging from 1000 to 100000 (Fig. 2(b)). For the case of confocal horizontal elliptical annulus, the eccentricity of the inner and outer wall were taken 0.9 and 0.4, respectively, and the Rayleigh number is equal to10000 (Fig. 3(a)). The local Nusselt numbers of the inner and outer cylindrical ellipse for two Rayleigh numbers based upon the description in original paper were plotted at the Fig. 3(b). The LBM result indicates an acceptable agreement with the results that presented in Refs [6,11].
312
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331
338 339
Max
nþ1 n u − u ju j n
−6
≤10
;
Max
nþ1 n v − v jv j n
−6
≤10
;
θnþ1 −θn −6 Max ≤10 θn
ð30Þ 340 341 342
349
5.1. Horizontal orientation of the elliptical cylinder
350
For the constant aspect ratio, changing the Rayleigh number from 103 to 5 × 105 the size of vortices core along the side of the triangle is increased (Fig. 4). Moreover, at Ra = 105 and Ra = 5 × 105, the
E T C E R R O C
351 352
N
345 346
U
343 344
D
347 348
Where u and v are the velocity components in x and y direction, respectively. In all cases, results show that due to the buoyancy force produced by temperature gradient, the fluid moves up from the side of the triangle to the outer cylinder, and the flow becomes cool and denser by the outer cylinder so moves down along the ellipse walls. For this reason, two symmetrical recirculation cells are formed at the left and right of the triangle.
F
336 337
streamlines are sharper and have different patterns with respect to the other Rayleigh numbers. With increment of the aspect ratio, the more deviation is observed on the streamlines near the side vertices of the triangle. Also, two new recirculation zones appear near the bottom section of the triangle sides at Ra = 105 and 5 × 105 for BR = 1.2. Fig. 5 shows the isotherms at different Rayleigh numbers and aspect ratios. It can be observed that the isotherms are symmetry like as streamlines. At Ra = 103, the isotherms change smoothly and conduction heat transfer became to dominate in all aspect ratios. With increasing the aspect ratio, the space between the sides of triangle and elliptical enclosure is restricted so the vortices cores extend and become stronger. Therefore, the isotherms have greater deflection. When the Rayleigh number increases to 105 and 5 × 105, the natural convection due to the gravity force is dominant mechanism for heat transfer, and the plume shape appears above the climax vertex (the upper vertex of the triangle), which causes to arise of thermal boundary layer at this point. With the increase of the aspect ratio, the size and shape of plume shape become smaller and sharper, respectively. The distribution of the local Nusselt number for inner and outer cylinders is shown in Fig. 6. The local Nusselt number decreases exponentially at the apex at P = 0 due to the separation of thermal boundary layer. Fig. 6(a) shows two local maxima at the side vertices of the triangle. At lower Rayleigh numbers due to conduction domination, the Nusselt number rises along the sides of the triangle with lower slope in comparisons with high Rayleigh number. However, it shows the linear mode at high Rayleigh number. At the base of triangle, conduction is dominant heat transfer. Hence, the slop of local Nusselt
O
334 335
In this simulation, the uniform quadratic grid cell is used to simulation the problem. The grid independency test was examined for different sizes of grids. Table 1 shows the average Nusselt number for Ra = 104, BR = 1.5 and for tow horizontal and vertical orientation of the outer elliptical cylinder. As sown in Table 1, it was found that the calculated values of Nuavg are approximately equal after 180 × 180 grid size, so the 180 × 180 grid size was selected. The criterion of the convergency is selected as
R O
332 333
A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
P
8
Fig. 6. Variation of the local Nusselt number along the wall of (a) inner triangular cylinder and (b) outer elliptical cylinder.
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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9
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A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
380 381
E
Fig. 6 (continued).
397
5.2. Vertical orientation of the elliptical cylinder
398
Figs. 8 and 9 show the variation of streamlines and isotherms pattern for different Rayleigh number and aspect ratios when the outer cylinder orientation is changed to the vertical direction. As a general statement, the trend of variation of streamlines and isotherms are almost similar for both vertical and horizontal orientations of ellipse, with slight difference. At BR = 1.2, two new recirculation zones appear under the base of the triangle. These new recirculation zones disappear at high Rayleigh number and lower value of aspect ratios. The narrow gap between
388 389 390 391 392 393 394
399 400 401 402 403 404 405
R
386 387
N C O
384 385
the vertices of triangle and ellipse walls causes for the phenomenon. The isotherm lines are denser near the wall on the top section of the ellipse and the boundary layer grows with moving along the wall to the bottom, but when BR = 1.2, the boundary layer is slightly denser and compresses near the side vertices. This effect is clear on the local Nusselt number distribution in Fig. 10(b). At height Rayleigh number and BR = 1.2, apart from the peak point at θ = 0, two new peak points appear on the local Nusselt number curves at the elliptical wall. These two new peak points are
U
382 383
R
395 396
number distribution approximately becomes zero. By increasing the Rayleigh number and reducing the aspect ratio, the value of local Nusselt number increases. Fig. 6(b) shows variation of the local Nusselt number along the wall of outer elliptical cylinder. The isotherms on the upper section of the ellipse are become denser. Therefore, the local Nusselt number has greater value in this section, but it should be mentioned that the plume makes slightly jump on Nusselt number near this region. At Ra = 104, BR = 1.2 and Ra = 103 for all aspect ratio, the local Nusselt number in the bottom of ellipse at 150 ≤ θ ≤ 180 is greater than other Rayleigh numbers. The reason of this trend is the domination of conduction heat transfer; with augmentation of Rayleigh number, this value of local Nusselt number is limited to the zero. Fig. 7 shows the correlation of average Nusselt number versus the Rayleigh number. With an increase of Rayleigh number (augmentation of natural convection effect) and decrease of the aspect ratio the average Nusselt number increases.
Fig. 7. Adapted curve for the average Nusselt number as a function of the Rayleigh number.
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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Ra =103
Ra =5×105
Ra =105
Ra =104
R O
O
F
BR =1.2
C
T
E
D
P
BR =1.5
U
BR =3
N
C
O
R
R
E
BR =2
Fig. 8. streamlines for different Rayleigh number and aspect ratios when Pr = 0.7.
415 416 417 418
formed near the side vertices at 90 ≤ θ ≤ 120 but at different Rayleigh numbers. However, the location of the peak point is slightly changed. At the lower Rayleigh number and higher aspect ratio, two new peak points merges into one peak point at the same location.
The distribution of the local Nusselt number for inner triangle also is shown in Fig. 10(a). Finally, the average Nusselt number can be correlated by using the least square method as shown in Fig. 11.
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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A.A. Mehrizi et al. / International Communications in Heat and Mass Transfer xxx (2013) xxx–xxx
Ra =103
11
Ra =5×105
Ra =105
Ra =104
R O
O
F
BR =1.2
C
T
E
D
P
BR =1.5
U
BR =3
N C O
R
R
E
BR =2
Fig. 9. streamlines for different Rayleigh number and aspect ratios when Pr = 0.7.
423 424 425 426 427 428
The correlation functions were presented in Table 2 for both ellipse directions. Comparing the average Nusselt number for horizontal and vertical cylinders, it can be seen that the Nusselt number is greater when the ellipse is oriented horizontally. Table 3 shows variation of the average Nusselt number ratio between Ra = 5 × 105 and Ra = 103 for different aspect ratios
and different ellipse orientations. Furthermore, the percent of the average Nusselt increment ratio between two ellipse orientations is calculated. From the table, it is clear that the Nusselt number ratio is grater when the ellipse positioned vertically at all aspect ratios. The maximum increment of Nusselt number ratios occurs at AR = 3. The maximum increment of the Nusselt number ratio
Please cite this article as: A.A. Mehrizi, et al., Lattice Boltzmann simulation of natural convection heat transfer in an elliptical-triangular annulus, International Communications in Heat and Mass Transfer (2013), http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.08.009
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Laminar natural convection in a concentric annulus was investigated by the lattice Boltzmann method. The curve boundary condition was implemented by MSL velocity boundary condition and a developed approach [31] based on extrapolation method was used to implement temperature boundary condition. The simulation data were validated by previously published works. The results indicate that lattice Boltzmann method in powerful method for simulation of natural convection heat transfer in complex geometry. The study showed that by increasing the Rayleigh number, the local and average Nusselt numbers increase for all aspect ratios. Also, at a constant Rayleigh number and with reduction the value of the aspect ratio, the average Nusselt number increases. Orientation of the ellipse has a small effect on streamlines and temperature contours but has a sensible effect on the number of recalculating zones. At vertical ellipse orientation, two new recirculation zones appear at BR = 1.2 for all Rayleigh numbers and at Ra = 103 for all aspect ratios except BR = 3. Due to the new vortices and narrower gap between the vertices side of triangle and ellipse wall, the number of peak points in the local Nusselt number for ellipse increases to three points. For the all aspect ratios and Ra numbers investigated, the average Nusselt number is greater for vertical orientation in comparison with horizontal orientation of the ellipse.
457 458 459 460
T
455 456
C
453 454
E
451 452
R
449 450
R
447 448
O
445 446
C
443 444
N
441 442
U
439 440
462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497
F
438
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between vertical and horizontal orientations of the ellipse is about 57.46 % at AR = 1.2.
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Fig. 10. Variation of the local Nusselt number along the wall of (a) inner triangular cylinder and (b) outer elliptical cylinder.
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Fig. 10 (continued).
R N C O t1:1 t1:2 t1:3 t1:4 t1:5 t1:6 t1:7
Fig. 11. Adapted curve for the average Nusselt number as a function of the Rayleigh number.
Table 2 Correlations of the average Nusselt number based on curve fitting. Horizontal ellipse Maximum deviation(%) Vertical ellipse Maximum deviation(%)
Average Nusselt function
AR = 1.2 2.25 Ra0.102
AR = 1.5 0.12 Ra0.165
AR = 2 0.452 Ra0.193
AR = 3 0.303 Ra0.2
3.46
0.779
5.94
9.47
0.175
0.468 Ra 2.98
0.19
0.575 Ra 4.77
0.202
0.389 Ra 7.6
Nusselt augmentation
0.251 Ra0.212 15.1
AR = 1.2
AR = 1.5
AR = 2
AR = 3
t2:3
λH ¼
Nuavg jRa¼500000 Nuavg jRa¼10000
1.866
2.808
3.355
3.486
t2:4
Vertical ellipse
λV ¼
2.939
3.323
3.586
3.859
t2:5
Increasement (%)
λV −λH λV
Nuavg jRa¼500000 Nuavg jRa¼10000
57.465
18.342
6.909
10.701
t2:6
Horizontal ellipse
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Q2
Table 3 t2:1 Average Nusselt number ratio between Ra = 5 × 105 and Ra = 103 for different aspect ratios. t2:2
100
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