- Email: [email protected]
Natural convection analysis employing entropy generation and heatline visualization in a hollow L-shaped cavity filled with nanofluid using lattice Boltzmann method- experimental thermo-physical properties
Accepted Manuscript Natural convection analysis employing entropy generation and heatline visualization in a hollow L-shaped cavity filled with nanofluid using lattice Boltzmann methodexperimental thermo-physical properties Alireza Rahimi, Abbas Kasaeipoor, Emad Hasani Malekshah, Ali Amiri PII:
S1386-9477(17)31183-9
DOI:
10.1016/j.physe.2017.10.004
Reference:
PHYSE 12931
To appear in:
Physica E: Low-dimensional Systems and Nanostructures
Received Date: 5 August 2017 Revised Date:
16 September 2017
Accepted Date: 7 October 2017
Please cite this article as: A. Rahimi, A. Kasaeipoor, E.H. Malekshah, A. Amiri, Natural convection analysis employing entropy generation and heatline visualization in a hollow L-shaped cavity filled with nanofluid using lattice Boltzmann method- experimental thermo-physical properties, Physica E: Lowdimensional Systems and Nanostructures (2017), doi: 10.1016/j.physe.2017.10.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Natural convection analysis employing entropy generation and heatline visualization in a hollow L-shaped cavity filled with nanofluid using lattice Boltzmann method- Experimental
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thermo-physical properties
Alireza Rahimi 1, Abbas Kasaeipoor 2, Emad Hasani Malekshah3*, Ali Amiri4
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1) Engineering faculty, Kashan University, Kashan, Iran E-mail: [email protected]
2) Faculty of Engineering, Department of Mechanical Engineering, University of Isfahan, Hezar Jerib
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Avenue, Isfahan 81746-73441, Iran. E-mail: [email protected]
3) Corresponding author: Department of Mechanical Engineering, Imam Hossein University, Tehran, IR.Iran, Email: [email protected], [email protected] 4) Micro and Nano Mechanical Theory, Department of Mechanical Engineering, Tsinghua University,
Abstract
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100084 Beijing, China. Email: [email protected]
The natural convection heat transfer and fluid flow is analyzed using lattice Boltzmann numerical method. The entropy generation analysis and heatline visualization are used to study the convective
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flow field comprehensively. The hollow L-shaped cavity is considered and filled with SiO2TiO2/Water-EG (60:40) hybrid nanofluid. The thermal conductivity and dynamic viscosity of nanofluid are measured experimentally. To use the experimental data of thermal conductivity and dynamic
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viscosity, two sets of correlations based on temperature for six different solid volume fractions of 0.5, 1, 1.5, 2, 2.5 and 3 vol% are derived. The influences of different governing parameters such different aspect ratios, solid volume fractions of nanofluid and Rayleigh numbers on the fluid flow, temperature filed, average/local Nusselt number, total/local entropy generation and heatlines are presented.
Keywords: Lattice Boltzmann simulation; natural convection; entropy generation; heatline visualization; hollow L-shaped cavity
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1. Introduction The natural convection phenomenon has many engineering and industrial applications causing attracting the researchers to analyze the influence of different parameters on the fluid flow and heat transfer in different enclosures due to natural convection. Some of these applications can be
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mentioned as nuclear reactors, lead-acid batteries, furnaces, passive cooling, buildings ventilation, double-pane window, heat exchangers, MEMs, solar collectors, etc.[1-13].
Due to poor heat transfer performance of conventional operating fluids such as water, air and oil, the modern fluids are developed with strong heat transfer performance. To improve the heat transfer
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performance of fluids, the researchers added the metallic and non-metalic nanoparticles such as Cu [14], CuO [15], Al2O3 [16], TiO2 [17], MgO [18], Fe2O3 [19], Fe3O4 [20], CNTs (carbon nanotubes)
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[21], SWCNTs (single wall carbon nanotubes) [22] , DWNTs (double wall carbon nanotubes) [23, 24], MWCNTs (multi wall carbon nanotubes) [25, 26], etc. have been added to the base fluids such as water [27], oil [17] and ethylene glycol [28].
Many investigators studied the natural convection fluid flow and heat transfer in different-shaped enclosures filled with different kinds of nanofluids [26, 29, 30]. In an experimental and numerical investigation, Malekshah and Salari [31] analyzed the natural convection heat transfer in a three-
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dimensional cuboid enclosure filled with two immiscible fluids of water and air. They considered two main parameters of Rayleigh number and liquid height aspect ratio on the fluid flow and heat transfer. They obtained the temperature distribution and average Nusselt number experimentally and compared them with numerical simulations. The results show close agreement between experimental
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and numerical observations. Sheremet et al. [32] analyzed the magnetohydrodynamic natural convection within a wavy open tall cavity filled with porous media. The studied fluid and the main
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governing parameters were water nanofluid and effects of corner heater, respectively. The enclosure was heated from right bottom corner, cooled form left and the bottom wall was considered adiabatic wall. The single-phase nanofluid approach in the non-dimensional variables such as vorticity, streamfuction and temperature was used for different governing parameters such as Rayleigh number (10 < < 10 ), Hartmann number (0 < < 100), solid volume fraction of nanoparticles
(0 < < 0.05), inclination angle of magnetic field (0 <γ < π). Finally, it was concluded that the heat transfer rate increases and reduces with increasing of Rayleigh number and Hartmann number. Da Silva et al. [33] numerically investigated the natural convection within a trapezoidal enclosure with different physical and geometric parameters. They proposed a correlation for the average Nusselt 2
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number based on and numbers and the inclination angle of upper surface for each baffle height. Sheremet et al. [34] presented the result of their study about steady-state natural convection in porous trapezoidal cavity filled by a nanofluid by applying Buongiorno’s mathematical model. It was found that the trapezoidal geometry enhances the fluid flow and heat transfer considerably compared
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with square cavity. Varol [35] studied the natural convection within a porous trapezoidal cavity filled with hot materials to simulate a cell of solar collector. He presented the results for different value of governing parameters such as thermal conductivity ratio between the middle horizontal wall and fluid, aspect ratio of two entrapped trapezoidal cavity and Darcy-modified Rayleigh umber. Moreover, he
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obtained the heat transfer rate based on the local and mean Nusselt number.
To improve the accuracy of the numerical simulations, the lattice Boltzmann numerical method is
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being used by many researchers instead of the conventional numerical methods. In this context, Rahimi et al. [36] studied the natural convection and entropy generation in a square cavity filled with nanofluid. They analyzed the thermal conductivity and dynamic viscosity of nanofluid experimentally and developed two sets of correlations connecting these parameters to temperature and solid volume fraction. They used the experimental data in the numerical simulations using these correlations. They concluded that the average Nusselt number and entropy generation enhances with increasing of
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Rayleigh number. Mejri et al. [37] studied fluid flow and heat transfer due to the natural convection within an inclined triangular cavity filled with water using Lattice Boltzmann method. They examined the effects of inclination angle and Rayleigh number on the streamlines, isotherms as well as Nusselt number. It was found that the heat transfer function strengthen with enhancing of Rayleigh number.
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Furthermore, they concluded that the inclination has influences on heat transfer rate so that highest and lowest heat transfer rates occur at = 0° and = 135° respectively. Kefayati et al. [38] used Lattice Boltzmann method to simulate natural convection within the tall cavities filled with SiO2-water
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nanofluid. They examined the effects of different parameters such as Rayleigh number (10 < <
10 ), solid volume fraction (0 < < 0.04) and aspect ratio (0.5 and 2) of the enclosure. They obtained
flow structures and temperature filed and average Nusselt number. It was concluded that the average Nusselt number enhances with increasing of solid volume fraction in all range of Rayleigh number and aspect ratio. Furthermore, the nanoparticle play more important role to increase heat transfer function in higher value of aspect ratio of the enclosure. The main purpose of the present work is to use the lattice Boltzmann method to analyze the natura; convection in a hollow L-shaped cavity filled with nanofluid. The thermo-physical properties of SiO23
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TiO2/Water-EG (60:40) hybrid nanofluid are obtained using experimental data, and used in the numerical simulations using developing correlations. The influences of different parameters such as four different arrangements of internal active bodies, four different Rayleigh numbers and six solid volume fractions (φ=0.5%,1%,1.5%,2%,2.5% and 3%) on the flow structure, temperature field,
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heatlines and entropy generation are studied comprehensively.
Nomenclature
!"
#
#!" #%
&
&
@
fluid friction irreversibility heat transfer irreversibility
internal energy distribution functions
Grashof number '& = #(∆ * ⁄+ Gebhart number '& = #( ⁄. -
/
dimensionless entropy generation
9, : ; <
=
+
F
( G
location of partially heater and cooler
thermal diffusivity solid volume fraction dimensionless viscous dissipation
dimensionless temperature ' = ' − D -⁄' E − D --
kinematic viscosity fluid density stream function thermal expansion coefficient dynamic viscosity
Subscripts
total entropy generation
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0, 8
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/0123 average Nusselt number
B
acceleration of gravity
height/width of the cavity
A
equilibrium internal energy distribution functions
Prandtl number
length of partially heater and cooler
Greek symbols
equilibrium distribution
5 43!6
?
discrete lattice velocity in direction
width of internal heater
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>
speed of sound
length of internal heater
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Bejan number
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fluid temperature
velocity components
H8#
Cartesian coordinates
I
width of cavity
height of cavity
J
thermal conductivity specific heat capacity 4
cold hot average nanofluid base fluid solid particles
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2.1. Problem statement The considered geometry and the related boundary conditions and dimensions are depicted in Fig.1. The width and height of the cavity are denoted by and equal to unity. Also, all dimensions which
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are denoted by K are equal to K = ⁄2. Except top and bottom walls which are insulated, other side walls have constant and uniform cold temperature. The no-slip boundary condition is applied at the surface of all internal surfaces of cavity. Four different ratios of H = >⁄K are considered in the
present investigation as H = 0.1, 0.2, 0.3 I? 0.4. The considered nanofluid is SiO2- TiO2/Water-EG
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with experimental thermo-physical properties which are used as a single-phase fluid in the simulations. The flow of nanofluid within the enclosure is assumed to be laminar, Newtonian, and
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incompressible. The thermo-physical properties of the nanofluid are assumed constant except for a variation of the density which is determined based on Boussinesq approximation, dynamic viscosity
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and thermal conductivity which are based on experimental correlations.
Figure 1. The considered geometry for each case
2.2. Governing equations
The continuity, momentum and energy equations for the laminar and steady state natural convection in two dimensional forms can be written as follows [39-41]: LM LN
LP
O LQ = 0
(1)
5
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LM
Ov
LM
=
u
LP
Ov
LP
=
LM
Ov
LM
= α[\ ]
u
LN
LN
LN
LQ
LQ
LQ
L^ M
T
X−
LY
O μ[\ ]
T
X−
LY
O μ[\ ]
UVW
UVW
LN
LN
L^ h
L_^
O ^
L_
L^ M
L_^
O
L^ h LQ^
`
O
L^ M LQ^
`a
L^ M LQ^
(2)
` O 'ρβ-g'T − Tc-a
(3)
(4)
By introducing the following dimensionless parameters: 8 n̅ m= , = A6l F6l A6l
' − p +6l #(6l q s˝ , = , = , q = +6l A6l q r A6l u6l
2.3. The Lattice Boltzmann Method
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=
: 0 j = , k = , A6l
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i i = ,
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u
It is worth to mention that the lattice Boltzmann method has some pronounced advantages convincing the researchers to use this method, although it is computationally expensive respect to conventional
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methods such as FV and FD. The advantages of LBM can be mentioned as linear stability in parallel computing which makes this method more accurate due to locally collision calculation. Moreover, this method is able to handle complex geometry and simulate the fluid flow and heat transfer of incompressible flows accurately. Overall, The major advantages of LBM over other conventional CFD
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methods due to the fact that the solution for the particle distribution functions is explicit, easy for parallel computation and implementation of boundary conditions on complex boundaries is simple.
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Two distribution functions, f and g, are utilized by the thermal LB model for the flow and temperature fields, respectively[42]. The thermal Lattice Boltzmann Method applies modeling of the movement of the fluid particles in order to capture fluid quantities in macroscopic scale such as velocity, temperature and pressure. Moreover, the fluid region will be discretized to Cartesian cells uniformly. Also, the probability of finding each particle in a specific range of velocities and locations replaces tagging each particle as in the computationally-intensive molecular dynamics simulation approach. In Lattice Boltzmann Method, each uniform cell has a fixed and specific number of distribution functions presenting the number of the fluid particles movement in the discrete directions. In the present study, the D2Q9 model is employed which is shown in Fig.2. In this model the values of parameters are 6
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assigned as: vw = 4⁄9 for |pw | = 0 for the static particles, vTz{ = 1⁄9 for |pTz{ | = 1 and vz| = 1⁄36
for |pz|| = √2. It must be noted that the density and distributions functions (f and g) are calculated by discretizing the kinetic Boltzmann equation (solving the Lattice Boltzmann equation). The general
For the flow filed: ∆
!" '9, - − '9, - O ∆p
For the temperature filed: ∆
#!" '9, - − # '9, -
(6)
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# '9 O p ∆, O ∆- = # '9, - O
(5)
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'9 O p ∆, O ∆- = '9, - O
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form of Lattice Boltzmann equation with external force is as follows [43-45]:
Where p is the discrete lattice velocity in the direction of , q presents the lattice time step, and D
represent lattice relaxation times for the flow and temperature fields respectively, is the external
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force in the direction of the lattice velocity.
Fig.2. Discrete particle velocity vectors for D2Q9 model
The external force in the natural convection problems is given by Eq.(7)
= ^ . p
(7) 7
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Where in the total external body force.
respective relaxation times by Eq.(8) + = ' − 1⁄2-,
A = 'D − 1⁄2-
(8)
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Where is the lattice speed of sound and equal to = p ⁄√3.
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Moreover, the kinetic viscosity '+- and thermal diffusivity 'A - are defined as a function of their
It should be noted that the limitation > 0.5 must be satisfied for both relaxation timed in order to be
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ensure that the viscosity and thermal diffusivity are positive. The type of simulated problem is determined by the local equilibrium distribution function. Also, it models the equilibrium distribution functions calculated with Eqs. (5) and (6) for flow and temperature fields, they are the local equilibrium distribution functions that have an appropriately prescribed functional dependence on the local hydrodynamic properties, respectively: !"
= v F X1 O
#!" = v X1 O
D . D^
D . D^
O
T 'D . -^
a
D
−
(10)
T ^ D^
a
(9)
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Here F is lattice fluid density and v is waiting factor.
model:
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The force term in Eq. (5) is considered in the vertical y-direction to apply the buoyancy force in the = 3v F#% ( ' − -
(11)
Where #% is gravitational acceleration, and ( is thermal expansion coefficient. The Boussinesq approximation is applied for the natural convection simulation in the cavity. The characteristic velocity of fluid flow of natural convection regime defined as m6 = (#% ∆ must be small compared with the fluid speed of sound to be ensured that thermal Lattice Boltzmann
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Method code works well. For this purpose, the characteristic velocity is considered 0.1 of sonic
Flow density
:
F = ∑ ,
Momentum
:
F0 = ∑ ,
Temperature
:
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speed. The macroscopic variables are calculated with the following formulations:
(12)
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= ∑ # .
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2.4. Boundary conditions
2.4.1. Fluid flow
For all solid boundaries, bounce-back boundary conditions are applied which means that the incoming boundary populations are equal to outgoing populations after collision. It was proven by
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Chen at al. [46]that the bounce-back boundary condition renders more accurate numerical results as the LBM approach is applied. As an example, the following conditions are considered for the east boundary:
(13)
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,6 = T,6 , ,6 = ,6 , ,6 = ,6
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Here n denotes the note of the lattice.
2.4.2. Temperature
The bounce-back boundary condition for thermal boundary condition representing adiabatic condition is applied for the north and south walls. As an example, the following boundary condition is applied at the north wall: #,6 = #,6zT , #{,6 = #{,6zT , #,6 = #,6zT
(14)
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It should be noted that the thermal boundary condition at the rectangular body and it’s active walls are known. For this purpose, at the west wall the dimensionless temperature is = 1, and the unknowns
parameters are # , # , and # as D2Q9 is used:
# = E v'6- O v '8- − #
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# = E v'3- O v '1- − #T (15)
2.5. Entropy generation
5 = O 43!6
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The total entropy generation rate is defined as follows[47]:
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# = E v'7- O v'5- − #
(16)
Where HTI represents the heat transfer irreversibility in the direction of finite temperature gradient and FTI denotes the fluid friction irreversibility.
In terms of the basic variables, HTI and FFI are defined as follows: =
•
=
'∇¡ . ∇¡¢£
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•
¡^
(17)
¡
E¡¤
= E¡¤¥¦¦¤
(18)
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And the Bejan number is defined as follows:
/J =
^
§ 5 ] ` ©ª« ¨
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The dimensionless form of total entropy generation, /J, is defined as follows: (19)
it can be extended it as follows: /J =
¬ ^ ¬ ^ ` ¥] ` ¬® ¬¯ ° ^ '° ±² ¥r-^
]
O
³!£
´° ^ '° ±² ¥r-
(20)
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Where represents the Rayleigh number, and µ is the non-dimensional temperature difference which is defined as follows: ¡¶ z¡
(21)
¡
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µ=
The dimensionless dissipation function which is used in the Eq. (12) is as follows: ·2
·¸
·¸
· ·¸
O
·2
·¸
`
(22)
Where & denotes the Gebhart number defined as follows: 3¹E
(23)
Dº
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& =
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·
= 2] ` O 2] ` O ]
The average value of dimensionless entropy generation is determined by 〈/J〉. The angle brakets show an average taken at a specific area as follows: ¾
〈/J〉 = ½1
1
?H
(24)
It is clear that the entering heat energy will be equal to transferred heat energy from the cooled walls.
¿¿
5 〉=" ]T − 〈43!6 E
¡
T
¡§
`
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As such, the total entropy generation can be written as follows: (25)
E^
°^
T¥°
(26)
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5 〉 = 4/0 〈43!6
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It can be obtained by terms of Nusselt number as follows:
For small value of µ, such as µ ≪ 1, and by applying perturbation techniques, the above-mentioned equation can be written as follows: 5 〉 ≈ {¾ ° 〈43!6
^ 'Tz°-
E^
(27)
The dimensionless form of total entropy generation can be obtained as follows: 〈/J〉 =
{ ¾ T¥°
(28)
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2.6. Lattice Boltzmann model for nanofluid It must be noted that the nanofluid behavior is totally different, in both hydrodynamic and thermal properties, from pure fluids because of existence of inter-particle potentials and other common forces
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acting between nanoparticles and pure fluids. Using nanoparticles in the pure fluids cause higher rate of energy transport and efficiency. The governing equations are as same as Eqs. (1)-(6) as the pure fluid is the operating fluid in the problem. On the other hand, the governing equations must be modified when the nanofluid is being used because of different thermo-physical properties such as
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density, thermal conductivity, heat capacitance and thermal expansion. The considered nanofluid in the present work is SiO2- TiO2/Water-EG. The simulations are performed in single-phase model, and
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the solid particles and the base fluid are in thermal equilibrium with no-slip condition between them. The local and average Nusselt numbers at the surface of heater are given by the following: /0 = Â
Ã«Ä ·¡ ÃÄ
Å
·¸
T
Æ 9 = 0, 1 I? /02 = ½w /0 ?:
2.7. Heatline visualization
(25)
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The heat line can be used to visualize the path-line and intensity of heat flow which is similar to streamlines. The heatlines are applicable to visualize and identify the heat flow from heat sources to heat sinks in the cavities. The heat filed within a two-dimensional cavity for convective transport process was mathematically studied by Kimura and Bejan [48]. The
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heatlines are represented by heatfunctions (K) which can be applied to plot the heatlines in the ·¡
·¡
cavity and also obtained from the conductive heat fluxes (− ·¸ , − ·%) and convective heat
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fluxes (0 , + ). The heat heatfunction parameter (K) satisfies the energy conservation equation for fluid media and nanofluids: ·¡
·¡
·^¡
·^ ¡
0 ·¸ O + ·% = A ]·¸ ^ O ·% ^`
(35)
So,
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·Ç
·¡
(36a)
·¡
(36b)
= Fp= 0 ' − w - − u ·¸
·%
·Ç
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− ·¸ = F+ ' − w - − u ·%
To make dimensionless form of heatfunction È, the above-mentioned dimensionless variables are used: ·r
= k − ·Ë,
−
= m −
·É ·Ë
(37a)
·r ·Ê
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·Ê
, (37b)
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·É
Which can be written in a single equation as follows: ·^É ·Ë ^
O
·^É ·Ê ^
=
·
·Ê
'k - −
·
·Ë
'm -
(38)
It should be noted that the counter clockwise circulation is represented by positive sign of È,
3. Material and methods
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3.1. Nanofluid preparation
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and the clockwise circulation is represented by negative sign of È.
The nanofluid preparing process in briefly presented here. In this study a modern nanofluid is
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selected for analyzing its thermo-physical properties and using in the experimental setup to study the natural convection heat transfer. The SiO2 and TiO2 nanoparticles with high purity of 99%, obtained from US research nanomaterials, Inc, are dispersed in the base fluid. The base fluid is considered a mixture of pure water and ethylene glycol with volume ratio of 60:40. Different solid volume fractions are considered in this study such as 0.5%, 1%, 1.5%, 2%, 2.5%, and 3%. The obtained nanofluids at different concentration of nanoparticles and a TEM image of SiO2 and TiO2 nanoparticles are presented in Fig.3 and Fig.4, respectively. It should be noted that the color of obtained nanofluid is faded as the solid volume fraction reduces which is not clear in the picture. Furthermore, the properties of supplied nanoparticles and base fluids are presented in Table.1. 13
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Figure 3. The obtained nanofluids at different concentration of nanoparticles
TiO2 nanoparticles
SiO2 SiO2 nanoparticles
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Figure 4. TEM image of SiO2 and TiO2 nanoparticles
Material
Purity (%)
Color
Size (nm) Density (kg/m3)
TiO2
99
White
30-50
4230
SiO2
99.99
Colorless
22
2220
Colorless
-
-
Table 1. Properties of utilized nanoparticles and base fluids.
Water/Ethylene glycol 99.5
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3.2. Stability analysis For all solid volume fractions for this nanofluid, the appropriate amounts of SiO2 and TiO2 nanoparticles are added to the base fluid. Afterwards, the particles and water are mixed with a magnetic stirrer for 2.5 hours. After mixing of the nanoparticles and base fluid, the suspension is
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inserted to an ultrasonic processor (Hielscher Company, Germany) with the power of 400W and frequency of 24 kHz for 5 h. this process prevents the agglomeration between the nanoparticles and sedimentation. The stability of nanofluid is observed for at least one week with any sedimentation.
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3.3. Thermal conductivity and Dynamic viscosity measuring
In order to measure the thermal conductivity of nanofluid, the reliable and fast method of transient
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hot-wire (THW) technique is utilized. In this context, a KD2 pre thermal properties analyzer (Decagon devices, Inc., USA) is utilized shown in Fig 5. The thermal conductivity analyzer device is calibrated with distilled water before starting the measurements, and the maximum error is measured equal to ±5%. It should be noted that the each measurement is repeated three times to verify the obtained results. The Brookfield viscometer of Brookfield engineering laboratories of USA is utilized, shown in Fig.6, which has high accuracy of ±5%. The viscometer is calibrated by using distilled water before
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starting the measuring process. All of the measurements are performed at room temperature and
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repeated three times to be ensured that the results are reliable.
Fig 5. KD2 pre thermal properties analyzer (Decagon devices, Inc., USA).
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Fig 6. Brookfield viscometer of Brookfield engineering laboratories of USA.
4. Results and discussion
Entropy generation analysis and heatline visualization of natural convection heat transfer in a hollow
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L-shaped cavity is analyzed using lattice Boltzmann method. Effects of different governing parameters such as Rayleigh number, solid volume fraction and aspect ratio on the local/total entropy generation, heatlines and local/total entropy generation have been investigated comprehensively. The dynamic viscosity and thermal conductivity of SiO2-TiO2/Water-EG hybrid nanofluid are obtained
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experimentally, and two sets of correlations based on temperature and solid volume fraction are developed and used in numerical simulations. For validation analysis of the present code with
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previous work, two works are selected. The natural convection heat transfer analysis performed by Oztop and Abu-Nada [49] in a partially heated cavity filled with SiO2-TiO2/Water-EG nanofluid. The comparison of present results with the data of mentioned literature is shown in Fig.7(A). Also, isotherms of the present numerical work (colored section) and experimental study performed by Calcagni et al. (gray section) [50] are compared in Fig.7(B). The comparison, shown in Fig.7, shows close agreement between the results. The CPU-times based on iteration for different Rayleigh numbers and aspect ratios of internal refrigerant are presented in Table.1. It can be observed that the CPU-time will be considerably influenced by different parameters.
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B
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A
Fig 7.Comparison between the present results and the mean Nusselt number in ÌÍ = ÎÏÐ for different solid volume fractions done by Oztop and Abu-Nada[49] and isotherms of the present work (colored section) and experimental numerical study performed by Calcagni et al. (gray section) [50].
Table 2. CPU-time of calculations at different case studies.
= 10 10732 9747 8513 7452
Rayleigh number = 10 10871 9806 8692 7591
= 10 10964 9957 8716 7786
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Case A Case B Case C Case D
= 10 10538 9628 8424 7381
{
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Case
4.1. Dynamic viscosity analysis
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The influence of the temperature and solid volume fractions on the dynamic viscosity is analyzed. The dynamic viscosity is analyzed at six different solid volume fractions ' = 0.5, 1, 1.5, 2, 2.5 I? 3 8ÑÒ%-
and a temperature range of 30 to 80°C. The value of dynamic viscosity 'G- variation with respect to
temperature as a function of solid volume fraction is depicted in Fig.8. As it can be seen in Fig.8, the dynamic viscosity of nanofluid reduces with increasing of temperature. It is due to the fact that the intermolecular forces reduce with augmenting of temperature in the fluid. As such, different layers of nanofluid can move easier as temperature increasing with respect to lower temperature because of lower shear stress between two layers. The experimental thermo-physical properties of SiO2TiO2/Water-EG nanofluid are utilized in the numerical simulation. In this context, the correlations 17
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between dynamic viscosity of nanofluid and temperature at all six different solid volume fractions are
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presented in Table.2.
Fig 8. Dynamic viscosity at different temperature as a function of solid volume fractions.
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Table 2. Derived correlations for dynamic viscosity at different solid volume fractions.
Solid volume fraction = 0.5 8ÑÒ%
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= 2.5 8ÑÒ%
u = 4.9161 − 0.0787 O 0.0004
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= 2 8ÑÒ%
R-squared
u = 5.0438 − 0.0865 O 0.0005
= 1 8ÑÒ%
= 1.5 8ÑÒ%
Correlation
= 3 8ÑÒ%
u = 4.9801 − 0.0802 O 0.0004 u = 4.8547 − 0.0691 O 0.0003 u = 4.8509 − 0.0665 O 0.0003 u = 4.6687 − 0.0557 O 0.0002
0.9993 0.9902 0.9907 0.9893 0.987 0.9707
4.2. Thermal conductivity analysis The influence of the temperature and solid volume fractions on the thermal conductivity is analyzed. The
thermal
conductivity
is
analyzed
at
six
different
solid
volume
fractions
' = 0.5, 1, 1.5, 2, 2.5 I? 3 8ÑÒ%- and a temperature range of 30 to 80°C.The value of thermal 18
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conductivity 'u - with respect to temperature as a function of solid volume fraction is depicted in Fig.9. It can be observed that the thermal conductivity increases considerably at high solid volume fractions since the number of collision between the solid particles augments, as a result of greater number of solid particles in a specific volume of base fluid and Brownian motion. On the other hand, the
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temperature increment boosts the kinetic energy of solid particles which has resulted in significant motions of solid particles and according collisions with other particles including base fluid and solid particles. As such, the temperature increment has more significant influence on the enhancing of effective thermal conductivity at high concentration. The experimental thermo-physical properties of
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SiO2- TiO2/Water-EG nanofluid are utilized in the numerical simulation. As such, it is needed to identify the relationship between the thermal conductivity with temperature at different solid volume
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fractions. In this context, the correlations between effective thermal conductivity of nanofluid and
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temperature at all six different solid volume fractions are presented in Table.3.
Fig 9. Thermal conductivity at different temperature as a function of solid volume fractions.
Table 3. Derived correlations for thermal conductivity at different solid volume fractions.
Solid volume fraction = 0.5 8ÑÒ% = 1 8ÑÒ%
Correlation u = 0.3939 O 0.001 − 2 Ô 10z u = 0.4281 − 8 Ô 10z O 10z
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R-squared 0.9975 0.9985
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= 1.5 8ÑÒ%
u = 0.4346 O 8 Ô 10z O 10z
= 2 8ÑÒ%
0.9951
u = 0.4421 − 5 Ô 10z O 10z
= 2.5 8ÑÒ%
0.9939
u = 0.437 O 0.0004 − 10z
= 3 8ÑÒ%
0.9884
u = 0.4531 O 0.0002 − 10z
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0.9923
4.3. Flow structure
The fluid flow in the enclosure due to the natural convection is influenced by different parameters such as temperature difference, physical geometry of the enclosure, gravity acceleration, and thermo-
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physical properties of the operating fluid. In the present work, the influences of the temperature difference in terms of Rayleigh number and different arrangements internal heaters and refrigerant
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bodies on the flow structure are presented in Figs.10-13. It can be observed that the nanofluid stream ascends at the adjacent of internal hot walls. It is due to the fact that the temperature of nanofluid increases and its density decreases. As a result if this matter, the buoyancy force creating upstream force. On the contrary, the nanofluid stream descends at the adjacent of cold side walls.
ÌÍ = ÎÏÕ
ÌÍ = ÎÏÖ
ÌÍ = ÎÏ×
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Fig 10. Flow structure for different arrangements of rigid bodies at Case A and Ø = Ù ÚÛÜ%
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4.4. Influence of Rayleigh number on flow structure The influence of the Rayleigh number on the nanofluid flow structure for different aspect ratios is presented graphically in Figs.10-13. It should be noted that the difference in Rayleigh number caused by temperature difference which is main parameter which has considerable influence on flow
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structure. As Rayleigh number enhances, the strength of fluid flow enhances which can be concluded by compacted streamlines. Moreover, the flow structure becomes more complex and irregular with more secondary eddies in higher Rayleigh number. 4.5. Influence of aspect ratio on flow structure
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The influence of different aspect ratio on the nanofluid flow structure for different Rayleigh numbers is presented in Figs.10-13. It can be seen that the aspect ratio has no considerable influence on the
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flow structure. At higher aspect ratio, the fluid flow becomes stronger due to higher total heat energy importing from internal walls. It can be concluded by more compacted streamlines with respect to lower aspect ratio.
ÌÍ = ÎÏÕ
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Fig 11. Flow structure for different arrangements of rigid bodies at Case B and Ø = Ù ÚÛÜ%
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4.6. Temperature field The isothermal maps for different Rayleigh numbers and different aspect ratios are presented graphically in Figs.10-13. It can be observed that the aspect ratio of cavity has no pronounced effects
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on the structure of temperature field. It is due to the fact that the thermal and physical boundary conditions are similar at different aspect ratios. As the Rayleigh number increases, the isothermal lines patterns become irregular with higher temperature gradient. It is due to the fact that the fluid flow becomes stronger, in high Rayleigh number, which is able to transfer the heat energy.
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Fig 12. Flow structure for different arrangements of rigid bodies at Case C and Ø = Ù ÚÛÜ%
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ÌÍ = ÎÏÕ
ÌÍ = ÎÏÖ
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Fig 13. Flow structure for different arrangements of rigid bodies at Case D and Ø = Ù ÚÛÜ%
4.7. Local fluid friction irreversibility (FFI)
The second parameter constituting the total entropy generation is the fluid friction irreversibility. The
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friction irreversibility maps for different Rayleigh numbers '10 < < 10 - and different aspect ratios are depicted graphically in Fig.14. It is worth to mention that the solid volume fraction has no considerable influence on the fluid friction irreversibility maps; as a result, one specific solid volume fraction is selected which its fluid friction irreversibly maps are fairly similar to other cases. It can be seen that the fluid friction irreversibility maps are accumulated at the adjacent of active walls due to significant velocity magnitude at these regions. For instance, between left cold wall and internal hot wall, the fluid friction irreversibility is sensible due to strong circulation at the gap. As Rayleigh number enhances, the fluid friction irreversibility maps are accumulated near the active walls, and the value of fluid friction irreversibility becomes uniform in computational domain. 23
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Case A
Case B
Case C
Case D
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ÌÍ = ÎÏÐ
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ÌÍ = ÎÏ×
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Fig 14. Fluid friction irreversibility map for different aspect ratios and Rayleigh numbers of rigid bodies at Ø = Ù ÚÛÜ%
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Case A
Case B
Case C
Case D
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ÌÍ = ÎÏÐ
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ÌÍ = ÎÏ×
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Fig 15. Heat transfer irreversibility map for different aspect ratios and Rayleigh numbers of rigid bodies at Ø = Ù ÚÛÜ%
4.8. Local heat transfer irreversibility (HTI) One of two main effective parameters constituting the total entropy generation is heat transfer irreversibility. The heat transfer irreversibility caused by the temperature gradient in the computational domain. As such, it can be concluded that the heat transfer irreversibility will have higher magnitude 25
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at the regions with high temperature gradient. The local heat transfer irreversibility maps for different Rayleigh numbers and different aspect ratios for one specific solid volume fraction of = 1 8ÑÒ% is depicted in Fig.15. It should be noted that the nanoparticle concentration has no pronounced influence on the heat transfer irreversibility maps; as a result, the heat transfer irreversibility maps for
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one specific solid volume fraction are obtained. It can be observed that the heat transfer irreversibility is dominant at the adjacent of internal hot walls. Also, changes in the heat transfer irreversibility near the corners of both hot and cold walls are significant. Moreover, the influence of aspect ratio of cavity in the structure of heat transfer irreversibility maps is negligible. As Rayleigh number increases, the
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isotherms are compacted at the adjacent of internal active walls.
4.9. Local Nusselt variation
The Nusselt number is a dimensionless parameter which shows the share of each heat transfer mechanism of conduction and convection in the natural convection heat transfer. As Nusselt number increases, it can be concluded that the share of convection heat transfer augments with respect to
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conduction mechanism. The local Nusselt variation maps for different Rayleigh numbers '10 < <
10 - and different aspect ratios for specific solid volume fraction of = 1 8ÑÒ% are depicted graphically in Fig.16. It is worth to mention that the solid volume fraction has no considerable influence on the local Nusselt variation maps; as a result, one specific solid volume fraction is
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selected which its local Nusselt variation maps are fairly similar to other cases. It can be seen that the maximum and minimum values of Nusselt variations locate at the adjacent of active and adiabatic
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walls, respectively. It is due to the fact that the heat transfer occurs at the active walls, and the insulated walls are unable to transfer heat energy. As Rayleigh number enhances, the structure of local Nusselt variation becomes irregular as a result of strong fluid flow. Furthermore, the isotherms of Nusselt variation are compacted near the active walls.
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Case A
Case B
Case C
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Fig 16. Local Nusselt variation map for different aspect ratios and Rayleigh numbers of rigid bodies at Ø = Ù ÚÛÜ%
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Fig 17. Heatlines for different arrangements of rigid bodies at Ø = Ù ÚÛÜ% and ÎÏÐ < ÌÍ < ÎÏÕ .
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4.10. Heatline visualization The heatlines inside the cavity for two different Rayleigh numbers '10 < < 10{ - and different aspect ratios are depicted in Fig.17. In order to visualize the heat flow within the cavity, the heatline
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visualization approach is employed allowing to detect the heat sources and heat sinks and determine how the heat energy paths between them. It can be observed that the heat energy is transferred from heat sources to heat sinks with specific pattern at each case. It is due to the fact that the thermal boundary conditions determine the heatlines patterns. In lower Rayleigh number, the heat energy
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move from heat sources to heat sinks without any strong distortion. On the contrary, the heatlines are deviated in higher Rayleigh number, and the heat energy is entrapped to circulations due to strong
4.11. Average Nusselt number
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main eddies.
The values of average Nusselt number with respect to Rayleigh number '10 < < 10 - as a function of solid volume fractions ' = 0.5, 1, 1.5, 2, 2.5 I? 3 8ÑÒ%- for four aspect ratios are
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presented in Fig.18. As Rayleigh number enhances, the average Nusselt number increases. It is due to the fact that the strength of fluid flow augments in higher Rayleigh number. As a result of this matter, the heat transfer coefficient enhances causing increasing of convective heat transfer and average Nusselt number. In addition, the rate of enhancement of average Nusselt number augments
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at higher Rayleigh number since the influence of adding nanoparticles on the heat transfer performance becomes highlighted because of higher collision number of nanoparticles to other
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nanoparticles and base fluid. In addition, the average Nusselt number augments as solid volume fraction of nanofluid increases due to improved thermo-physical properties of nanofluid in higher nanoparticle concentration. Also, as the aspect ratio of the cavity enhances the average Nusselt number increases because of vaster active walls.
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Fig 18. Average Nusselt number with respect to Rayleigh number as a function of Rayleigh number for different aspect ratios at the surface of left hot wall.
4.12. Entropy generation
The total entropy generation causes by two parameters of heat transfer irreversibility and fluid friction irreversibility. The values of total entropy generation with respect to Rayleigh number '10 < <
10 - as a function of solid volume fraction ' = 0.5, 1, 1.5, 2, 2.5 I? 3 8ÑÒ%- for different aspect ratios
are presented in Fig.19. The value of average Nusselt number enhances as Rayleigh number
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increases. It is due to the fact that the value of both heat transfer irreversibility and fluid friction irreversibility enhances due to higher temperature gradient and velocity gradient, respectively. Also, the nanoparticle concentration causes changing of the value of total entropy generation. As the solid volume fraction of nanofluid increases, the total entropy generation reduces due to weaker velocity
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magnitude of nanofluid stream and reduction in the temperature gradient. Finally, the value of total
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entropy generation has direct relationship with the aspect ratio of the cavity.
Fig 19. Total entropy generation with respect to Rayleigh number as a function of solid volume fraction for different aspect ratios.
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Fig 20. Bejan number with respect to Rayleigh number as a function of solid volume fraction for different aspect ratios.
4.13. Bejan number
The share of each parameter of fluid friction irreversibility and heat transfer irreversibility on forming the total entropy generation can be identified by Bejan number. For this purpose, the values of Bejan number with respect to Rayleigh number as a function of different solid volume fractions of nanofluid for different aspect ratios of the cavity are presented in Fig.20. As Rayleigh number increases, the value of Bejan number reduces because of increasing value of fluid friction irreversibility. In addition, 32
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the Bejan number has direct relationship with the solid volume fraction. It is due to the fact that the fluid flow becomes weaker and heat transfer performance improves as the solid volume fraction augments.
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5. Conclusions The lattice Boltzmann method is used to study the natural convection phenomenon in a hollow Lshaped cavity filled with SiO2- TiO2/Water-EG (60:40) hybrid nanofluid. Different approaches are employed such as heatline visualization and entropy generation to analyze the influence of different
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parameters comprehensively. The cavities are filled with SiO2- TiO2/Water-EG (60:40) hybrid nanofluid which its thermal conductivity and dynamic viscosity are measured experimentally at six
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different solid volume fractions of 0.5%, 1%, 0.05%, 1.5%, 2%, 2.5% and 3% and temperature range of 30 to 80 (°C). Two sets of correlations for ther mal conductivity and dynamic viscosity based on temperature and solid volume fraction are developed and used in the numerical simulations. The influence of different parameters such as Rayleigh number '10 < < 10 -, solid volume fraction of
nanofluid '0.5%, 1%, 0.05%, 1.5%, 2%, 2.5% I? 3%- and four different aspect ratios of cavity on the fluid flow, heat transfer, total/local entropy generation, average/local Nusselt number and heatlines
•
The lattice Boltzmann numerical method renders more accurate results compared with conventional methods.
•
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are presented systematically. The results can be listed as:
Changing the dimensions of internal geometry has significant influence on fluid flow and heat
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transfer performance.
Using variable properties of operating fluid in the numerical simulations increases the
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accuracy of results. •
The average Nusselt number enhances with increasing of Rayleigh number.
•
The average Nusselt number enhances with increasing of solid volume fraction of nanofluid.
•
The total entropy generation increases with increasing of Rayleigh number.
•
The total entropy generation reduces with enhancing of solid volume fraction of nanofluid.
•
The local fluid friction and heat transfer irreversibility are dominant at the adjacent of internal active walls.
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Highlights
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Lattice Boltzmann Simulation of natural convection and entropy generation The cavity is filled with nanofluid Experimental measurements thermo-physical properties of SiO2-TiO2/Water-EG Influences of aspect ratio on the flow structure, heat transfer performance and entropy generation
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• • • •
S1386-9477(17)31183-9
DOI:
10.1016/j.physe.2017.10.004
Reference:
PHYSE 12931
To appear in:
Physica E: Low-dimensional Systems and Nanostructures
Received Date: 5 August 2017 Revised Date:
16 September 2017
Accepted Date: 7 October 2017
Please cite this article as: A. Rahimi, A. Kasaeipoor, E.H. Malekshah, A. Amiri, Natural convection analysis employing entropy generation and heatline visualization in a hollow L-shaped cavity filled with nanofluid using lattice Boltzmann method- experimental thermo-physical properties, Physica E: Lowdimensional Systems and Nanostructures (2017), doi: 10.1016/j.physe.2017.10.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Natural convection analysis employing entropy generation and heatline visualization in a hollow L-shaped cavity filled with nanofluid using lattice Boltzmann method- Experimental
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thermo-physical properties
Alireza Rahimi 1, Abbas Kasaeipoor 2, Emad Hasani Malekshah3*, Ali Amiri4
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1) Engineering faculty, Kashan University, Kashan, Iran E-mail: [email protected]
2) Faculty of Engineering, Department of Mechanical Engineering, University of Isfahan, Hezar Jerib
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Avenue, Isfahan 81746-73441, Iran. E-mail: [email protected]
3) Corresponding author: Department of Mechanical Engineering, Imam Hossein University, Tehran, IR.Iran, Email: [email protected], [email protected] 4) Micro and Nano Mechanical Theory, Department of Mechanical Engineering, Tsinghua University,
Abstract
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100084 Beijing, China. Email: [email protected]
The natural convection heat transfer and fluid flow is analyzed using lattice Boltzmann numerical method. The entropy generation analysis and heatline visualization are used to study the convective
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flow field comprehensively. The hollow L-shaped cavity is considered and filled with SiO2TiO2/Water-EG (60:40) hybrid nanofluid. The thermal conductivity and dynamic viscosity of nanofluid are measured experimentally. To use the experimental data of thermal conductivity and dynamic
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viscosity, two sets of correlations based on temperature for six different solid volume fractions of 0.5, 1, 1.5, 2, 2.5 and 3 vol% are derived. The influences of different governing parameters such different aspect ratios, solid volume fractions of nanofluid and Rayleigh numbers on the fluid flow, temperature filed, average/local Nusselt number, total/local entropy generation and heatlines are presented.
Keywords: Lattice Boltzmann simulation; natural convection; entropy generation; heatline visualization; hollow L-shaped cavity
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1. Introduction The natural convection phenomenon has many engineering and industrial applications causing attracting the researchers to analyze the influence of different parameters on the fluid flow and heat transfer in different enclosures due to natural convection. Some of these applications can be
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mentioned as nuclear reactors, lead-acid batteries, furnaces, passive cooling, buildings ventilation, double-pane window, heat exchangers, MEMs, solar collectors, etc.[1-13].
Due to poor heat transfer performance of conventional operating fluids such as water, air and oil, the modern fluids are developed with strong heat transfer performance. To improve the heat transfer
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performance of fluids, the researchers added the metallic and non-metalic nanoparticles such as Cu [14], CuO [15], Al2O3 [16], TiO2 [17], MgO [18], Fe2O3 [19], Fe3O4 [20], CNTs (carbon nanotubes)
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[21], SWCNTs (single wall carbon nanotubes) [22] , DWNTs (double wall carbon nanotubes) [23, 24], MWCNTs (multi wall carbon nanotubes) [25, 26], etc. have been added to the base fluids such as water [27], oil [17] and ethylene glycol [28].
Many investigators studied the natural convection fluid flow and heat transfer in different-shaped enclosures filled with different kinds of nanofluids [26, 29, 30]. In an experimental and numerical investigation, Malekshah and Salari [31] analyzed the natural convection heat transfer in a three-
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dimensional cuboid enclosure filled with two immiscible fluids of water and air. They considered two main parameters of Rayleigh number and liquid height aspect ratio on the fluid flow and heat transfer. They obtained the temperature distribution and average Nusselt number experimentally and compared them with numerical simulations. The results show close agreement between experimental
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and numerical observations. Sheremet et al. [32] analyzed the magnetohydrodynamic natural convection within a wavy open tall cavity filled with porous media. The studied fluid and the main
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governing parameters were water nanofluid and effects of corner heater, respectively. The enclosure was heated from right bottom corner, cooled form left and the bottom wall was considered adiabatic wall. The single-phase nanofluid approach in the non-dimensional variables such as vorticity, streamfuction and temperature was used for different governing parameters such as Rayleigh number (10 < < 10 ), Hartmann number (0 < < 100), solid volume fraction of nanoparticles
(0 < < 0.05), inclination angle of magnetic field (0 <γ < π). Finally, it was concluded that the heat transfer rate increases and reduces with increasing of Rayleigh number and Hartmann number. Da Silva et al. [33] numerically investigated the natural convection within a trapezoidal enclosure with different physical and geometric parameters. They proposed a correlation for the average Nusselt 2
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number based on and numbers and the inclination angle of upper surface for each baffle height. Sheremet et al. [34] presented the result of their study about steady-state natural convection in porous trapezoidal cavity filled by a nanofluid by applying Buongiorno’s mathematical model. It was found that the trapezoidal geometry enhances the fluid flow and heat transfer considerably compared
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with square cavity. Varol [35] studied the natural convection within a porous trapezoidal cavity filled with hot materials to simulate a cell of solar collector. He presented the results for different value of governing parameters such as thermal conductivity ratio between the middle horizontal wall and fluid, aspect ratio of two entrapped trapezoidal cavity and Darcy-modified Rayleigh umber. Moreover, he
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obtained the heat transfer rate based on the local and mean Nusselt number.
To improve the accuracy of the numerical simulations, the lattice Boltzmann numerical method is
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being used by many researchers instead of the conventional numerical methods. In this context, Rahimi et al. [36] studied the natural convection and entropy generation in a square cavity filled with nanofluid. They analyzed the thermal conductivity and dynamic viscosity of nanofluid experimentally and developed two sets of correlations connecting these parameters to temperature and solid volume fraction. They used the experimental data in the numerical simulations using these correlations. They concluded that the average Nusselt number and entropy generation enhances with increasing of
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Rayleigh number. Mejri et al. [37] studied fluid flow and heat transfer due to the natural convection within an inclined triangular cavity filled with water using Lattice Boltzmann method. They examined the effects of inclination angle and Rayleigh number on the streamlines, isotherms as well as Nusselt number. It was found that the heat transfer function strengthen with enhancing of Rayleigh number.
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Furthermore, they concluded that the inclination has influences on heat transfer rate so that highest and lowest heat transfer rates occur at = 0° and = 135° respectively. Kefayati et al. [38] used Lattice Boltzmann method to simulate natural convection within the tall cavities filled with SiO2-water
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nanofluid. They examined the effects of different parameters such as Rayleigh number (10 < <
10 ), solid volume fraction (0 < < 0.04) and aspect ratio (0.5 and 2) of the enclosure. They obtained
flow structures and temperature filed and average Nusselt number. It was concluded that the average Nusselt number enhances with increasing of solid volume fraction in all range of Rayleigh number and aspect ratio. Furthermore, the nanoparticle play more important role to increase heat transfer function in higher value of aspect ratio of the enclosure. The main purpose of the present work is to use the lattice Boltzmann method to analyze the natura; convection in a hollow L-shaped cavity filled with nanofluid. The thermo-physical properties of SiO23
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TiO2/Water-EG (60:40) hybrid nanofluid are obtained using experimental data, and used in the numerical simulations using developing correlations. The influences of different parameters such as four different arrangements of internal active bodies, four different Rayleigh numbers and six solid volume fractions (φ=0.5%,1%,1.5%,2%,2.5% and 3%) on the flow structure, temperature field,
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heatlines and entropy generation are studied comprehensively.
Nomenclature
!"
#
#!" #%
&
&
@
fluid friction irreversibility heat transfer irreversibility
internal energy distribution functions
Grashof number '& = #(∆ * ⁄+ Gebhart number '& = #( ⁄. -
/
dimensionless entropy generation
9, : ; <
=
+
F
( G
location of partially heater and cooler
thermal diffusivity solid volume fraction dimensionless viscous dissipation
dimensionless temperature ' = ' − D -⁄' E − D --
kinematic viscosity fluid density stream function thermal expansion coefficient dynamic viscosity
Subscripts
total entropy generation
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0, 8
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/0123 average Nusselt number
B
acceleration of gravity
height/width of the cavity
A
equilibrium internal energy distribution functions
Prandtl number
length of partially heater and cooler
Greek symbols
equilibrium distribution
5 43!6
?
discrete lattice velocity in direction
width of internal heater
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>
speed of sound
length of internal heater
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Bejan number
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fluid temperature
velocity components
H8#
Cartesian coordinates
I
width of cavity
height of cavity
J
thermal conductivity specific heat capacity 4
cold hot average nanofluid base fluid solid particles
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2.1. Problem statement The considered geometry and the related boundary conditions and dimensions are depicted in Fig.1. The width and height of the cavity are denoted by and equal to unity. Also, all dimensions which
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are denoted by K are equal to K = ⁄2. Except top and bottom walls which are insulated, other side walls have constant and uniform cold temperature. The no-slip boundary condition is applied at the surface of all internal surfaces of cavity. Four different ratios of H = >⁄K are considered in the
present investigation as H = 0.1, 0.2, 0.3 I? 0.4. The considered nanofluid is SiO2- TiO2/Water-EG
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with experimental thermo-physical properties which are used as a single-phase fluid in the simulations. The flow of nanofluid within the enclosure is assumed to be laminar, Newtonian, and
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incompressible. The thermo-physical properties of the nanofluid are assumed constant except for a variation of the density which is determined based on Boussinesq approximation, dynamic viscosity
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and thermal conductivity which are based on experimental correlations.
Figure 1. The considered geometry for each case
2.2. Governing equations
The continuity, momentum and energy equations for the laminar and steady state natural convection in two dimensional forms can be written as follows [39-41]: LM LN
LP
O LQ = 0
(1)
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LM
Ov
LM
=
u
LP
Ov
LP
=
LM
Ov
LM
= α[\ ]
u
LN
LN
LN
LQ
LQ
LQ
L^ M
T
X−
LY
O μ[\ ]
T
X−
LY
O μ[\ ]
UVW
UVW
LN
LN
L^ h
L_^
O ^
L_
L^ M
L_^
O
L^ h LQ^
`
O
L^ M LQ^
`a
L^ M LQ^
(2)
` O 'ρβ-g'T − Tc-a
(3)
(4)
By introducing the following dimensionless parameters: 8 n̅ m= , = A6l F6l A6l
' − p +6l #(6l q s˝ , = , = , q = +6l A6l q r A6l u6l
2.3. The Lattice Boltzmann Method
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=
: 0 j = , k = , A6l
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i i = ,
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u
It is worth to mention that the lattice Boltzmann method has some pronounced advantages convincing the researchers to use this method, although it is computationally expensive respect to conventional
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methods such as FV and FD. The advantages of LBM can be mentioned as linear stability in parallel computing which makes this method more accurate due to locally collision calculation. Moreover, this method is able to handle complex geometry and simulate the fluid flow and heat transfer of incompressible flows accurately. Overall, The major advantages of LBM over other conventional CFD
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methods due to the fact that the solution for the particle distribution functions is explicit, easy for parallel computation and implementation of boundary conditions on complex boundaries is simple.
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Two distribution functions, f and g, are utilized by the thermal LB model for the flow and temperature fields, respectively[42]. The thermal Lattice Boltzmann Method applies modeling of the movement of the fluid particles in order to capture fluid quantities in macroscopic scale such as velocity, temperature and pressure. Moreover, the fluid region will be discretized to Cartesian cells uniformly. Also, the probability of finding each particle in a specific range of velocities and locations replaces tagging each particle as in the computationally-intensive molecular dynamics simulation approach. In Lattice Boltzmann Method, each uniform cell has a fixed and specific number of distribution functions presenting the number of the fluid particles movement in the discrete directions. In the present study, the D2Q9 model is employed which is shown in Fig.2. In this model the values of parameters are 6
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assigned as: vw = 4⁄9 for |pw | = 0 for the static particles, vTz{ = 1⁄9 for |pTz{ | = 1 and vz| = 1⁄36
for |pz|| = √2. It must be noted that the density and distributions functions (f and g) are calculated by discretizing the kinetic Boltzmann equation (solving the Lattice Boltzmann equation). The general
For the flow filed: ∆
!" '9, - − '9, - O ∆p
For the temperature filed: ∆
#!" '9, - − # '9, -
(6)
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# '9 O p ∆, O ∆- = # '9, - O
(5)
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'9 O p ∆, O ∆- = '9, - O
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form of Lattice Boltzmann equation with external force is as follows [43-45]:
Where p is the discrete lattice velocity in the direction of , q presents the lattice time step, and D
represent lattice relaxation times for the flow and temperature fields respectively, is the external
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force in the direction of the lattice velocity.
Fig.2. Discrete particle velocity vectors for D2Q9 model
The external force in the natural convection problems is given by Eq.(7)
= ^ . p
(7) 7
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Where in the total external body force.
respective relaxation times by Eq.(8) + = ' − 1⁄2-,
A = 'D − 1⁄2-
(8)
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Where is the lattice speed of sound and equal to = p ⁄√3.
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Moreover, the kinetic viscosity '+- and thermal diffusivity 'A - are defined as a function of their
It should be noted that the limitation > 0.5 must be satisfied for both relaxation timed in order to be
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ensure that the viscosity and thermal diffusivity are positive. The type of simulated problem is determined by the local equilibrium distribution function. Also, it models the equilibrium distribution functions calculated with Eqs. (5) and (6) for flow and temperature fields, they are the local equilibrium distribution functions that have an appropriately prescribed functional dependence on the local hydrodynamic properties, respectively: !"
= v F X1 O
#!" = v X1 O
D . D^
D . D^
O
T 'D . -^
a
D
−
(10)
T ^ D^
a
(9)
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Here F is lattice fluid density and v is waiting factor.
model:
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The force term in Eq. (5) is considered in the vertical y-direction to apply the buoyancy force in the = 3v F#% ( ' − -
(11)
Where #% is gravitational acceleration, and ( is thermal expansion coefficient. The Boussinesq approximation is applied for the natural convection simulation in the cavity. The characteristic velocity of fluid flow of natural convection regime defined as m6 = (#% ∆ must be small compared with the fluid speed of sound to be ensured that thermal Lattice Boltzmann
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Method code works well. For this purpose, the characteristic velocity is considered 0.1 of sonic
Flow density
:
F = ∑ ,
Momentum
:
F0 = ∑ ,
Temperature
:
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speed. The macroscopic variables are calculated with the following formulations:
(12)
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= ∑ # .
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2.4. Boundary conditions
2.4.1. Fluid flow
For all solid boundaries, bounce-back boundary conditions are applied which means that the incoming boundary populations are equal to outgoing populations after collision. It was proven by
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Chen at al. [46]that the bounce-back boundary condition renders more accurate numerical results as the LBM approach is applied. As an example, the following conditions are considered for the east boundary:
(13)
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,6 = T,6 , ,6 = ,6 , ,6 = ,6
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Here n denotes the note of the lattice.
2.4.2. Temperature
The bounce-back boundary condition for thermal boundary condition representing adiabatic condition is applied for the north and south walls. As an example, the following boundary condition is applied at the north wall: #,6 = #,6zT , #{,6 = #{,6zT , #,6 = #,6zT
(14)
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It should be noted that the thermal boundary condition at the rectangular body and it’s active walls are known. For this purpose, at the west wall the dimensionless temperature is = 1, and the unknowns
parameters are # , # , and # as D2Q9 is used:
# = E v'6- O v '8- − #
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# = E v'3- O v '1- − #T (15)
2.5. Entropy generation
5 = O 43!6
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The total entropy generation rate is defined as follows[47]:
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# = E v'7- O v'5- − #
(16)
Where HTI represents the heat transfer irreversibility in the direction of finite temperature gradient and FTI denotes the fluid friction irreversibility.
In terms of the basic variables, HTI and FFI are defined as follows: =
•
=
'∇¡ . ∇¡¢£
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•
¡^
(17)
¡
E¡¤
= E¡¤¥¦¦¤
(18)
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And the Bejan number is defined as follows:
/J =
^
§ 5 ] ` ©ª« ¨
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The dimensionless form of total entropy generation, /J, is defined as follows: (19)
it can be extended it as follows: /J =
¬ ^ ¬ ^ ` ¥] ` ¬® ¬¯ ° ^ '° ±² ¥r-^
]
O
³!£
´° ^ '° ±² ¥r-
(20)
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Where represents the Rayleigh number, and µ is the non-dimensional temperature difference which is defined as follows: ¡¶ z¡
(21)
¡
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µ=
The dimensionless dissipation function which is used in the Eq. (12) is as follows: ·2
·¸
·¸
· ·¸
O
·2
·¸
`
(22)
Where & denotes the Gebhart number defined as follows: 3¹E
(23)
Dº
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& =
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·
= 2] ` O 2] ` O ]
The average value of dimensionless entropy generation is determined by 〈/J〉. The angle brakets show an average taken at a specific area as follows: ¾
〈/J〉 = ½1
1
?H
(24)
It is clear that the entering heat energy will be equal to transferred heat energy from the cooled walls.
¿¿
5 〉=" ]T − 〈43!6 E
¡
T
¡§
`
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As such, the total entropy generation can be written as follows: (25)
E^
°^
T¥°
(26)
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5 〉 = 4/0 〈43!6
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It can be obtained by terms of Nusselt number as follows:
For small value of µ, such as µ ≪ 1, and by applying perturbation techniques, the above-mentioned equation can be written as follows: 5 〉 ≈ {¾ ° 〈43!6
^ 'Tz°-
E^
(27)
The dimensionless form of total entropy generation can be obtained as follows: 〈/J〉 =
{ ¾ T¥°
(28)
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2.6. Lattice Boltzmann model for nanofluid It must be noted that the nanofluid behavior is totally different, in both hydrodynamic and thermal properties, from pure fluids because of existence of inter-particle potentials and other common forces
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acting between nanoparticles and pure fluids. Using nanoparticles in the pure fluids cause higher rate of energy transport and efficiency. The governing equations are as same as Eqs. (1)-(6) as the pure fluid is the operating fluid in the problem. On the other hand, the governing equations must be modified when the nanofluid is being used because of different thermo-physical properties such as
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density, thermal conductivity, heat capacitance and thermal expansion. The considered nanofluid in the present work is SiO2- TiO2/Water-EG. The simulations are performed in single-phase model, and
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the solid particles and the base fluid are in thermal equilibrium with no-slip condition between them. The local and average Nusselt numbers at the surface of heater are given by the following: /0 = Â
Ã«Ä ·¡ ÃÄ
Å
·¸
T
Æ 9 = 0, 1 I? /02 = ½w /0 ?:
2.7. Heatline visualization
(25)
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The heat line can be used to visualize the path-line and intensity of heat flow which is similar to streamlines. The heatlines are applicable to visualize and identify the heat flow from heat sources to heat sinks in the cavities. The heat filed within a two-dimensional cavity for convective transport process was mathematically studied by Kimura and Bejan [48]. The
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heatlines are represented by heatfunctions (K) which can be applied to plot the heatlines in the ·¡
·¡
cavity and also obtained from the conductive heat fluxes (− ·¸ , − ·%) and convective heat
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fluxes (0 , + ). The heat heatfunction parameter (K) satisfies the energy conservation equation for fluid media and nanofluids: ·¡
·¡
·^¡
·^ ¡
0 ·¸ O + ·% = A ]·¸ ^ O ·% ^`
(35)
So,
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·Ç
·¡
(36a)
·¡
(36b)
= Fp= 0 ' − w - − u ·¸
·%
·Ç
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− ·¸ = F+ ' − w - − u ·%
To make dimensionless form of heatfunction È, the above-mentioned dimensionless variables are used: ·r
= k − ·Ë,
−
= m −
·É ·Ë
(37a)
·r ·Ê
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·Ê
, (37b)
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·É
Which can be written in a single equation as follows: ·^É ·Ë ^
O
·^É ·Ê ^
=
·
·Ê
'k - −
·
·Ë
'm -
(38)
It should be noted that the counter clockwise circulation is represented by positive sign of È,
3. Material and methods
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3.1. Nanofluid preparation
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and the clockwise circulation is represented by negative sign of È.
The nanofluid preparing process in briefly presented here. In this study a modern nanofluid is
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selected for analyzing its thermo-physical properties and using in the experimental setup to study the natural convection heat transfer. The SiO2 and TiO2 nanoparticles with high purity of 99%, obtained from US research nanomaterials, Inc, are dispersed in the base fluid. The base fluid is considered a mixture of pure water and ethylene glycol with volume ratio of 60:40. Different solid volume fractions are considered in this study such as 0.5%, 1%, 1.5%, 2%, 2.5%, and 3%. The obtained nanofluids at different concentration of nanoparticles and a TEM image of SiO2 and TiO2 nanoparticles are presented in Fig.3 and Fig.4, respectively. It should be noted that the color of obtained nanofluid is faded as the solid volume fraction reduces which is not clear in the picture. Furthermore, the properties of supplied nanoparticles and base fluids are presented in Table.1. 13
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Figure 3. The obtained nanofluids at different concentration of nanoparticles
TiO2 nanoparticles
SiO2 SiO2 nanoparticles
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Figure 4. TEM image of SiO2 and TiO2 nanoparticles
Material
Purity (%)
Color
Size (nm) Density (kg/m3)
TiO2
99
White
30-50
4230
SiO2
99.99
Colorless
22
2220
Colorless
-
-
Table 1. Properties of utilized nanoparticles and base fluids.
Water/Ethylene glycol 99.5
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3.2. Stability analysis For all solid volume fractions for this nanofluid, the appropriate amounts of SiO2 and TiO2 nanoparticles are added to the base fluid. Afterwards, the particles and water are mixed with a magnetic stirrer for 2.5 hours. After mixing of the nanoparticles and base fluid, the suspension is
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inserted to an ultrasonic processor (Hielscher Company, Germany) with the power of 400W and frequency of 24 kHz for 5 h. this process prevents the agglomeration between the nanoparticles and sedimentation. The stability of nanofluid is observed for at least one week with any sedimentation.
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3.3. Thermal conductivity and Dynamic viscosity measuring
In order to measure the thermal conductivity of nanofluid, the reliable and fast method of transient
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hot-wire (THW) technique is utilized. In this context, a KD2 pre thermal properties analyzer (Decagon devices, Inc., USA) is utilized shown in Fig 5. The thermal conductivity analyzer device is calibrated with distilled water before starting the measurements, and the maximum error is measured equal to ±5%. It should be noted that the each measurement is repeated three times to verify the obtained results. The Brookfield viscometer of Brookfield engineering laboratories of USA is utilized, shown in Fig.6, which has high accuracy of ±5%. The viscometer is calibrated by using distilled water before
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starting the measuring process. All of the measurements are performed at room temperature and
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repeated three times to be ensured that the results are reliable.
Fig 5. KD2 pre thermal properties analyzer (Decagon devices, Inc., USA).
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Fig 6. Brookfield viscometer of Brookfield engineering laboratories of USA.
4. Results and discussion
Entropy generation analysis and heatline visualization of natural convection heat transfer in a hollow
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L-shaped cavity is analyzed using lattice Boltzmann method. Effects of different governing parameters such as Rayleigh number, solid volume fraction and aspect ratio on the local/total entropy generation, heatlines and local/total entropy generation have been investigated comprehensively. The dynamic viscosity and thermal conductivity of SiO2-TiO2/Water-EG hybrid nanofluid are obtained
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experimentally, and two sets of correlations based on temperature and solid volume fraction are developed and used in numerical simulations. For validation analysis of the present code with
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previous work, two works are selected. The natural convection heat transfer analysis performed by Oztop and Abu-Nada [49] in a partially heated cavity filled with SiO2-TiO2/Water-EG nanofluid. The comparison of present results with the data of mentioned literature is shown in Fig.7(A). Also, isotherms of the present numerical work (colored section) and experimental study performed by Calcagni et al. (gray section) [50] are compared in Fig.7(B). The comparison, shown in Fig.7, shows close agreement between the results. The CPU-times based on iteration for different Rayleigh numbers and aspect ratios of internal refrigerant are presented in Table.1. It can be observed that the CPU-time will be considerably influenced by different parameters.
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A
Fig 7.Comparison between the present results and the mean Nusselt number in ÌÍ = ÎÏÐ for different solid volume fractions done by Oztop and Abu-Nada[49] and isotherms of the present work (colored section) and experimental numerical study performed by Calcagni et al. (gray section) [50].
Table 2. CPU-time of calculations at different case studies.
= 10 10732 9747 8513 7452
Rayleigh number = 10 10871 9806 8692 7591
= 10 10964 9957 8716 7786
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Case A Case B Case C Case D
= 10 10538 9628 8424 7381
{
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Case
4.1. Dynamic viscosity analysis
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The influence of the temperature and solid volume fractions on the dynamic viscosity is analyzed. The dynamic viscosity is analyzed at six different solid volume fractions ' = 0.5, 1, 1.5, 2, 2.5 I? 3 8ÑÒ%-
and a temperature range of 30 to 80°C. The value of dynamic viscosity 'G- variation with respect to
temperature as a function of solid volume fraction is depicted in Fig.8. As it can be seen in Fig.8, the dynamic viscosity of nanofluid reduces with increasing of temperature. It is due to the fact that the intermolecular forces reduce with augmenting of temperature in the fluid. As such, different layers of nanofluid can move easier as temperature increasing with respect to lower temperature because of lower shear stress between two layers. The experimental thermo-physical properties of SiO2TiO2/Water-EG nanofluid are utilized in the numerical simulation. In this context, the correlations 17
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between dynamic viscosity of nanofluid and temperature at all six different solid volume fractions are
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presented in Table.2.
Fig 8. Dynamic viscosity at different temperature as a function of solid volume fractions.
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Table 2. Derived correlations for dynamic viscosity at different solid volume fractions.
Solid volume fraction = 0.5 8ÑÒ%
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= 2.5 8ÑÒ%
u = 4.9161 − 0.0787 O 0.0004
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= 2 8ÑÒ%
R-squared
u = 5.0438 − 0.0865 O 0.0005
= 1 8ÑÒ%
= 1.5 8ÑÒ%
Correlation
= 3 8ÑÒ%
u = 4.9801 − 0.0802 O 0.0004 u = 4.8547 − 0.0691 O 0.0003 u = 4.8509 − 0.0665 O 0.0003 u = 4.6687 − 0.0557 O 0.0002
0.9993 0.9902 0.9907 0.9893 0.987 0.9707
4.2. Thermal conductivity analysis The influence of the temperature and solid volume fractions on the thermal conductivity is analyzed. The
thermal
conductivity
is
analyzed
at
six
different
solid
volume
fractions
' = 0.5, 1, 1.5, 2, 2.5 I? 3 8ÑÒ%- and a temperature range of 30 to 80°C.The value of thermal 18
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conductivity 'u - with respect to temperature as a function of solid volume fraction is depicted in Fig.9. It can be observed that the thermal conductivity increases considerably at high solid volume fractions since the number of collision between the solid particles augments, as a result of greater number of solid particles in a specific volume of base fluid and Brownian motion. On the other hand, the
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temperature increment boosts the kinetic energy of solid particles which has resulted in significant motions of solid particles and according collisions with other particles including base fluid and solid particles. As such, the temperature increment has more significant influence on the enhancing of effective thermal conductivity at high concentration. The experimental thermo-physical properties of
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SiO2- TiO2/Water-EG nanofluid are utilized in the numerical simulation. As such, it is needed to identify the relationship between the thermal conductivity with temperature at different solid volume
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fractions. In this context, the correlations between effective thermal conductivity of nanofluid and
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temperature at all six different solid volume fractions are presented in Table.3.
Fig 9. Thermal conductivity at different temperature as a function of solid volume fractions.
Table 3. Derived correlations for thermal conductivity at different solid volume fractions.
Solid volume fraction = 0.5 8ÑÒ% = 1 8ÑÒ%
Correlation u = 0.3939 O 0.001 − 2 Ô 10z u = 0.4281 − 8 Ô 10z O 10z
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R-squared 0.9975 0.9985
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= 1.5 8ÑÒ%
u = 0.4346 O 8 Ô 10z O 10z
= 2 8ÑÒ%
0.9951
u = 0.4421 − 5 Ô 10z O 10z
= 2.5 8ÑÒ%
0.9939
u = 0.437 O 0.0004 − 10z
= 3 8ÑÒ%
0.9884
u = 0.4531 O 0.0002 − 10z
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0.9923
4.3. Flow structure
The fluid flow in the enclosure due to the natural convection is influenced by different parameters such as temperature difference, physical geometry of the enclosure, gravity acceleration, and thermo-
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physical properties of the operating fluid. In the present work, the influences of the temperature difference in terms of Rayleigh number and different arrangements internal heaters and refrigerant
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bodies on the flow structure are presented in Figs.10-13. It can be observed that the nanofluid stream ascends at the adjacent of internal hot walls. It is due to the fact that the temperature of nanofluid increases and its density decreases. As a result if this matter, the buoyancy force creating upstream force. On the contrary, the nanofluid stream descends at the adjacent of cold side walls.
ÌÍ = ÎÏÕ
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Fig 10. Flow structure for different arrangements of rigid bodies at Case A and Ø = Ù ÚÛÜ%
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4.4. Influence of Rayleigh number on flow structure The influence of the Rayleigh number on the nanofluid flow structure for different aspect ratios is presented graphically in Figs.10-13. It should be noted that the difference in Rayleigh number caused by temperature difference which is main parameter which has considerable influence on flow
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structure. As Rayleigh number enhances, the strength of fluid flow enhances which can be concluded by compacted streamlines. Moreover, the flow structure becomes more complex and irregular with more secondary eddies in higher Rayleigh number. 4.5. Influence of aspect ratio on flow structure
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The influence of different aspect ratio on the nanofluid flow structure for different Rayleigh numbers is presented in Figs.10-13. It can be seen that the aspect ratio has no considerable influence on the
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flow structure. At higher aspect ratio, the fluid flow becomes stronger due to higher total heat energy importing from internal walls. It can be concluded by more compacted streamlines with respect to lower aspect ratio.
ÌÍ = ÎÏÕ
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Fig 11. Flow structure for different arrangements of rigid bodies at Case B and Ø = Ù ÚÛÜ%
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4.6. Temperature field The isothermal maps for different Rayleigh numbers and different aspect ratios are presented graphically in Figs.10-13. It can be observed that the aspect ratio of cavity has no pronounced effects
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on the structure of temperature field. It is due to the fact that the thermal and physical boundary conditions are similar at different aspect ratios. As the Rayleigh number increases, the isothermal lines patterns become irregular with higher temperature gradient. It is due to the fact that the fluid flow becomes stronger, in high Rayleigh number, which is able to transfer the heat energy.
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Fig 12. Flow structure for different arrangements of rigid bodies at Case C and Ø = Ù ÚÛÜ%
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ÌÍ = ÎÏÕ
ÌÍ = ÎÏÖ
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Fig 13. Flow structure for different arrangements of rigid bodies at Case D and Ø = Ù ÚÛÜ%
4.7. Local fluid friction irreversibility (FFI)
The second parameter constituting the total entropy generation is the fluid friction irreversibility. The
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fluid friction irreversibility is caused by the velocity gradient at the computational domain. As such, the fluid friction irreversibility will have high value at the regions with high velocity gradient. The fluid
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friction irreversibility maps for different Rayleigh numbers '10 < < 10 - and different aspect ratios are depicted graphically in Fig.14. It is worth to mention that the solid volume fraction has no considerable influence on the fluid friction irreversibility maps; as a result, one specific solid volume fraction is selected which its fluid friction irreversibly maps are fairly similar to other cases. It can be seen that the fluid friction irreversibility maps are accumulated at the adjacent of active walls due to significant velocity magnitude at these regions. For instance, between left cold wall and internal hot wall, the fluid friction irreversibility is sensible due to strong circulation at the gap. As Rayleigh number enhances, the fluid friction irreversibility maps are accumulated near the active walls, and the value of fluid friction irreversibility becomes uniform in computational domain. 23
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Case A
Case B
Case C
Case D
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ÌÍ = ÎÏÐ
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Fig 14. Fluid friction irreversibility map for different aspect ratios and Rayleigh numbers of rigid bodies at Ø = Ù ÚÛÜ%
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Case A
Case B
Case C
Case D
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ÌÍ = ÎÏÐ
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Fig 15. Heat transfer irreversibility map for different aspect ratios and Rayleigh numbers of rigid bodies at Ø = Ù ÚÛÜ%
4.8. Local heat transfer irreversibility (HTI) One of two main effective parameters constituting the total entropy generation is heat transfer irreversibility. The heat transfer irreversibility caused by the temperature gradient in the computational domain. As such, it can be concluded that the heat transfer irreversibility will have higher magnitude 25
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at the regions with high temperature gradient. The local heat transfer irreversibility maps for different Rayleigh numbers and different aspect ratios for one specific solid volume fraction of = 1 8ÑÒ% is depicted in Fig.15. It should be noted that the nanoparticle concentration has no pronounced influence on the heat transfer irreversibility maps; as a result, the heat transfer irreversibility maps for
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one specific solid volume fraction are obtained. It can be observed that the heat transfer irreversibility is dominant at the adjacent of internal hot walls. Also, changes in the heat transfer irreversibility near the corners of both hot and cold walls are significant. Moreover, the influence of aspect ratio of cavity in the structure of heat transfer irreversibility maps is negligible. As Rayleigh number increases, the
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value of the heat transfer irreversibility becomes uniform at the computational domain, and some
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isotherms are compacted at the adjacent of internal active walls.
4.9. Local Nusselt variation
The Nusselt number is a dimensionless parameter which shows the share of each heat transfer mechanism of conduction and convection in the natural convection heat transfer. As Nusselt number increases, it can be concluded that the share of convection heat transfer augments with respect to
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conduction mechanism. The local Nusselt variation maps for different Rayleigh numbers '10 < <
10 - and different aspect ratios for specific solid volume fraction of = 1 8ÑÒ% are depicted graphically in Fig.16. It is worth to mention that the solid volume fraction has no considerable influence on the local Nusselt variation maps; as a result, one specific solid volume fraction is
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selected which its local Nusselt variation maps are fairly similar to other cases. It can be seen that the maximum and minimum values of Nusselt variations locate at the adjacent of active and adiabatic
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walls, respectively. It is due to the fact that the heat transfer occurs at the active walls, and the insulated walls are unable to transfer heat energy. As Rayleigh number enhances, the structure of local Nusselt variation becomes irregular as a result of strong fluid flow. Furthermore, the isotherms of Nusselt variation are compacted near the active walls.
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Case A
Case B
Case C
Case D
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ÌÍ = ÎÏÐ
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Fig 16. Local Nusselt variation map for different aspect ratios and Rayleigh numbers of rigid bodies at Ø = Ù ÚÛÜ%
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ÌÍ = ÎÏÐ
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Fig 17. Heatlines for different arrangements of rigid bodies at Ø = Ù ÚÛÜ% and ÎÏÐ < ÌÍ < ÎÏÕ .
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4.10. Heatline visualization The heatlines inside the cavity for two different Rayleigh numbers '10 < < 10{ - and different aspect ratios are depicted in Fig.17. In order to visualize the heat flow within the cavity, the heatline
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visualization approach is employed allowing to detect the heat sources and heat sinks and determine how the heat energy paths between them. It can be observed that the heat energy is transferred from heat sources to heat sinks with specific pattern at each case. It is due to the fact that the thermal boundary conditions determine the heatlines patterns. In lower Rayleigh number, the heat energy
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move from heat sources to heat sinks without any strong distortion. On the contrary, the heatlines are deviated in higher Rayleigh number, and the heat energy is entrapped to circulations due to strong
4.11. Average Nusselt number
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main eddies.
The values of average Nusselt number with respect to Rayleigh number '10 < < 10 - as a function of solid volume fractions ' = 0.5, 1, 1.5, 2, 2.5 I? 3 8ÑÒ%- for four aspect ratios are
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presented in Fig.18. As Rayleigh number enhances, the average Nusselt number increases. It is due to the fact that the strength of fluid flow augments in higher Rayleigh number. As a result of this matter, the heat transfer coefficient enhances causing increasing of convective heat transfer and average Nusselt number. In addition, the rate of enhancement of average Nusselt number augments
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at higher Rayleigh number since the influence of adding nanoparticles on the heat transfer performance becomes highlighted because of higher collision number of nanoparticles to other
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nanoparticles and base fluid. In addition, the average Nusselt number augments as solid volume fraction of nanofluid increases due to improved thermo-physical properties of nanofluid in higher nanoparticle concentration. Also, as the aspect ratio of the cavity enhances the average Nusselt number increases because of vaster active walls.
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Fig 18. Average Nusselt number with respect to Rayleigh number as a function of Rayleigh number for different aspect ratios at the surface of left hot wall.
4.12. Entropy generation
The total entropy generation causes by two parameters of heat transfer irreversibility and fluid friction irreversibility. The values of total entropy generation with respect to Rayleigh number '10 < <
10 - as a function of solid volume fraction ' = 0.5, 1, 1.5, 2, 2.5 I? 3 8ÑÒ%- for different aspect ratios
are presented in Fig.19. The value of average Nusselt number enhances as Rayleigh number
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increases. It is due to the fact that the value of both heat transfer irreversibility and fluid friction irreversibility enhances due to higher temperature gradient and velocity gradient, respectively. Also, the nanoparticle concentration causes changing of the value of total entropy generation. As the solid volume fraction of nanofluid increases, the total entropy generation reduces due to weaker velocity
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magnitude of nanofluid stream and reduction in the temperature gradient. Finally, the value of total
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entropy generation has direct relationship with the aspect ratio of the cavity.
Fig 19. Total entropy generation with respect to Rayleigh number as a function of solid volume fraction for different aspect ratios.
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Fig 20. Bejan number with respect to Rayleigh number as a function of solid volume fraction for different aspect ratios.
4.13. Bejan number
The share of each parameter of fluid friction irreversibility and heat transfer irreversibility on forming the total entropy generation can be identified by Bejan number. For this purpose, the values of Bejan number with respect to Rayleigh number as a function of different solid volume fractions of nanofluid for different aspect ratios of the cavity are presented in Fig.20. As Rayleigh number increases, the value of Bejan number reduces because of increasing value of fluid friction irreversibility. In addition, 32
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the Bejan number has direct relationship with the solid volume fraction. It is due to the fact that the fluid flow becomes weaker and heat transfer performance improves as the solid volume fraction augments.
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5. Conclusions The lattice Boltzmann method is used to study the natural convection phenomenon in a hollow Lshaped cavity filled with SiO2- TiO2/Water-EG (60:40) hybrid nanofluid. Different approaches are employed such as heatline visualization and entropy generation to analyze the influence of different
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parameters comprehensively. The cavities are filled with SiO2- TiO2/Water-EG (60:40) hybrid nanofluid which its thermal conductivity and dynamic viscosity are measured experimentally at six
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different solid volume fractions of 0.5%, 1%, 0.05%, 1.5%, 2%, 2.5% and 3% and temperature range of 30 to 80 (°C). Two sets of correlations for ther mal conductivity and dynamic viscosity based on temperature and solid volume fraction are developed and used in the numerical simulations. The influence of different parameters such as Rayleigh number '10 < < 10 -, solid volume fraction of
nanofluid '0.5%, 1%, 0.05%, 1.5%, 2%, 2.5% I? 3%- and four different aspect ratios of cavity on the fluid flow, heat transfer, total/local entropy generation, average/local Nusselt number and heatlines
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The lattice Boltzmann numerical method renders more accurate results compared with conventional methods.
•
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are presented systematically. The results can be listed as:
Changing the dimensions of internal geometry has significant influence on fluid flow and heat
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transfer performance.
Using variable properties of operating fluid in the numerical simulations increases the
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accuracy of results. •
The average Nusselt number enhances with increasing of Rayleigh number.
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The average Nusselt number enhances with increasing of solid volume fraction of nanofluid.
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The total entropy generation increases with increasing of Rayleigh number.
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The total entropy generation reduces with enhancing of solid volume fraction of nanofluid.
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The local fluid friction and heat transfer irreversibility are dominant at the adjacent of internal active walls.
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Highlights
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Lattice Boltzmann Simulation of natural convection and entropy generation The cavity is filled with nanofluid Experimental measurements thermo-physical properties of SiO2-TiO2/Water-EG Influences of aspect ratio on the flow structure, heat transfer performance and entropy generation
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• • • •