Document Type : Full Lenght Research Article
Authors
^{1} school of engineering, Damghan university, Damghan, Iran
^{2} School of Engineering, Damghan University, Damghan, Iran
^{3} School of Engineering, Damghan University, Damghan,Iran
^{4} 2 Shanghai Automotive Wind Tunnel Center, Tongji University, No.4800, Cao’an Road, Shanghai, China, 201804
Abstract
Keywords
Main Subjects
Semnan University 
Journal of Heat and Mass Transfer Research Journal homepage: http://jhmtr.semnan.ac.ir 

Natural Convection Heat Transfer of AgMgO/Water Micropolar Hybrid Nanofluid Inside an FShaped Cavity Equipped by Hot Obstacle
Rasul Mohebbi[*]^{,a}, Abbas Abbaszadeh^{a}, Mahsa Varzandeh^{a}, Yuan Ma*^{,b,c}
^{a}School of Engineering, Damghan University, Damghan 3671641167, Iran
^{b}Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China
^{c} Shanghai Automotive Wind Tunnel Center, Tongji University, No.4800, Cao’an Road, Shanghai 201804. China
Paper INFO 

ABSTRACT 
Paper history: Received: 20200903 Revised: 20210703 Accepted: 20210708 
This paper presents a series of numerical simulations of nanofluid natural convection inside an Fshaped enclosure equipped by heat source. A hybrid nanofluid consisting of Ag and MgO nanoparticles and water as base fluid was used. Lattice Boltzmann method (LBM) was applied and the effects of Raleigh number (10^{3} ≤ Ra ≤ 10^{6}), solid volume fraction of nanoparticle (0 ≤ ϕ ≤ 0.02), and heat source location (0 ≤ S ≤ 0.9) on the flow field, distribution of temperature and heat transfer performance were analyzed according to streamlines, isotherms, and profiles of average Nusselt numbers. The results indicated that the average Nusselt number enhances by increasing the ϕ, although the addition of nanoparticles cannot change the flow pattern and the thermal field significantly. At low Ra, the effect of Ra on average Nu is weak. However, for high Ra, the heat transfer increases significantly by increasing the Ra. The position of heat source also affects the average Nu. The S = 0.6 is the best position of the hot obstacle for enhancing the heat transfer and S = 0.9 is the worst choice. This trend cannot be affected by Ra and ϕ. DOI: 10.22075/JHMTR.2021.21269.1303 



Keywords: Nanofluid; Hot obstacle; Lattice Boltzmann Method; Natural convection; Fshaped cavity.



© 2021 Published by Semnan University Press. All rights reserved. 
Heat transfer of natural convection has been widely used in many engineering applications due to its easiness, low cost, small size, and dependability of heat transfer. The natural convection in cavities has attracted more and more attention in the past few decades[1,2]. From another perspective, because of the lack of energy, improving energy efficiency is most important. In the heat transfer’s field, the applications of nanofluids enhance heat transfer and reduce the waste of energy[3,4]. Therefore, it is natural to think of using nanofluids to enhance the natural convection heat transfer in the enclosures. Alloui et al.[5] studied nanofluid natural convection in a rectangular enclosure. In their work, two horizontal walls were Neumann boundary conditions and two vertical walls were insulated. According to the results, the authors considered that nanofluid reduces the flow strength and the phenomenon is more evident at lower Rayleigh number. Natural convection heat transfer of nanofluid studied by Sheikhzadeh et al.[6] in a square cavity. They found that the average Nusselt number increases by augmention of Ra and ϕ. Besides, for the different Ra, the locations of thermal active parts for maximum average Nusselt numbers are different.
In addition to regular rectangular cavities, the nanofluids heat transfer of natural convection in irregularly shaped cavities has also been extensively studied. Sheremet et al.[7] investigated Cu/water nanofluid heat transfer in an inclined wavy cavity. In their cavity, the left bottom corner was heated and the top wavy wall was fixed at a low temperature. In addition, when the cavity inclination angle was changed, the positions of the heater and cooler were changed, which lead to the essential of fluid flow and heat transfer became different. Makulati et al.[8] investigated the Al_{2}O_{3}/water nanofluid natural convection in a Cshaped enclosure. Almeshaal et al.[9] performed a threedimensional analysis on the CNTAl_{2}O_{3}/water hybrid nanofluid inside a Tshaped cavity. They results showed that regardless of the cavity size, the heat transfer in the cavity with hybrid nanofluid is higher than that using pure fluid at high Ra.
The flow field, distribution of temperature and heat transfer performance inside the cavity can be affected by adopting obstacles. The location, size, and thermal boundary conditions of obstacles have significant effects on the flow and temperature fields. Kalidasan et al.[10] numerically studied CuTiO_{2}/water hybrid nanofluid in a Cshaped cavity with an isothermal obstacle. Izadi et al.[11] performed a numerical simulation of MWCNTFe_{3}O_{4}/water hybrid nanofluid natural convection inside a ┴ shaped enclosure equipped by a heat source. They found that the increase of heat source aspect ratio leads to better cooling performance. Hatami and Safari[12] studied nanofluid heat transfer inside a wavywall cavity with a heated obstacle. They found that when the heated obstacle locates at the center, the heat transfer of both sidewalls is enhanced. Ahmed et al.[13] studied Cu/water nanofluid natural convection inside a fined triangular cavity. They found that when the fin locates near the left wall, the heat transfer is better than the other positions.
Due to the diversity of nanoparticle types, the types of nanofluids are also diverse. In addition to some of the abovementioned nanofluids, there are still many different types of nanoparticles, such as MgO[14], ZnO[15,16], SiO_{2}[17], CuO[18,19], Fe_{3}O_{4}[20]. Besides, there are also many kinds of hybrid nanofluid, which include more than one kind of nanoparticles, such as CuZn[21], ZnOTiO_{2}[22], CuAg[23], AgMgO[24], Fe_{3}O_{4}CNT[25], Al_{2}O_{3}Cu[26,27]. In the present study, nanofluid natural convection heat transfer inside an Fshaped enclosure was simulated and analyzed. The Fshaped cavity consists of one vertical narrow space and two horizontal spaces. There is a hot block on the left wall, which is not only a heat source but also an obstacle, to affect the flow and temperature fields. Besides, AgMgO/water hybrid nanofluid is used as heat transfer medium inside the cavity to increase the heat transfer performance. It should be mentioned that the heat transfer by natural convection inside an Fshaped enclosure can be found in many practical applications, especially the cooling of electronic electronics, such as the computer or television with many capacitances.
The definition of the problem showed in Figure 1. An “F” shaped cavity with a hot square obstacle (T_{h}=1) by length “W” and height “H” (H = 2W) configured in this paper. The ratios of L/W=0.3 and K/W=0.7 also the dimensions of hot obstacle defined by length “a” and height “b”, so that the obstacle aspect ratio is AR=a/b=0.1. The sidewalls of the cavity (BC, DE, FG, HI) are cold (T_{c}=0) and other sides are adiabatic. The position of hot obstacle (S=S_{y}/H) changes between 0 to 0.9. Water is considered as base fluid, while an AgMgO Micropolar Hybrid is considered as nanoparticles in which their thermophysical properties listed in Table 1. Moreover, the viscous dissipation and thermal radiation are neglected.
In the present study, the effective dynamic viscosity, effective thermal conductivity, the density and the specific heat capacity of nanofluid are expressed as follows respectively[24]:

(1) 

(2) 

(3) 

(4) 
where subscripts "p", "f", and "nf" indicate the nanoparticle, base fluid, and nanofluid, respectively. In addition, the thermal diffusivity of nanofluid and Prandtl number are as[2931]:
, 
(5) 
, 
(6) 
Figure 1. definition of the problem
Table 1. Properties of the pure fluid and the nanoparticles[28]
Property 
Fluid phase (Water) 
Ag 
MgO 
C_{p} (J kg^{1} K^{1}) 
4179 
235 
955 
ρ (kg m^{3}) 
997.1 
10500 
3560 
K (W m^{1} K^{1}) 
0.613 
429 
45 
β×10^{5} (K^{1}) 
1.67 
1.89 
1.13 
µ×10^{4} (kg m^{1} s^{1}) 
8.55 


ν (m^{2} s^{1}) 
0.79 


The D_{2}Q_{9} model was used for simulation the flow field and heat transfer. The governing equations for flow and thermal field are as follow:

(7) 

(8) 
where F is external forces and Δt denotes lattice time, g_{i}^{eq} and f_{i}^{eq} denotes the equilibrium distribution function, τ_{c} and τ_{v} indicate the temperature and flow relaxation time, respectively^{],[}.

(9) 

(10) 
where is the speed of sound, c=1 and v is kinetic viscosity. In addition, can be found by[32]:

(11) 

(12) 
where the ρ is the local density. The weight function w_{i} has the value of , , and the discrete particle velocity vectors e_{i} in equations (7) and (8) are defined by:

(13) 
The force term in Eq. (7) in vertical direction (y) calculated by:

(14) 
The macroscopic variables can be found by:

(15) 
For the boundary conditions, the bounceback scheme is used for the solid walls. In addition, the walls of obstacles and BCDEFGHI walls are set to be unity (T_{h}=1) and zero (T_{c}=0), respectively. The other remaining walls (ABAJJIHGEFDC) imposed adiabatic.
The local Nusselt number is very important in heat transfer problems, so for the hot wall the local Nu numbers derived as:
For vertical walls:

(16a) 
For horizontal walls:

(16b) 
where and the averaged Nu number calculates by integrating the local Nu along obstacle.
For study the grid independence of homemade code by FORTRAN, three different meshes developed and the averaged Nu number calculate for S=0 and ϕ=0.01 at different Ra number. The results showed in Table 2. As can be found, the grid 200 × 200 is appropriate for the study. The accuracy of the present numerical model checked by study of Matori et. al and a very good addoption found between the results, see Figure 2.
Natural convection of AgMgO/water hybrid nanofluid inside an Fshaped cavity with a hot obstacle is studied using LBM. The effects of nanoparticle volume fraction (0 ≤ ϕ ≤ 0.02), Rayleigh number (10^{3} ≤ Ra ≤ 10^{6}) and obstacle position (0 ≤ S ≤ 0.9) on the flow pattern, temperature distribution and heat transfer performance are investigated. The results are displayed by the contour maps of streamlines and isotherms and profiles of average Nusselt number.
Figure 3 shows the effects of ϕ and Ra on the streamlines inside the cavity for S = 0. It is found that the fluid around the hot source is heated and moves upward. As a result, a clockwise vortex is established inside the cavity. At Ra = 10^{3}, the vortex is relatively small and locates on the bottom side of the enclosure. The fluid on the upper side of the enclosure is almost static and no obvious streamlines of the fluid can be found because at low Ra, the buoyancy force is relatively small, leading to the weak intensity of natural convection. As for Ra = 106, the vortex inside the enclosure becomes larger in size and occupies more space. Actually, the main heat transfer mechanism at low Ra is conduction heat transfer. However, at higher Ra, the natural convection caused by buoyancy force is enhanced. Thus, the occupied area by vortex becomes larger. Convection heat transfer becomes the prominent heat transfer mechanism. Besides, two secondary vortices are established inside the large vortex.
There is an interesting point that even though at the high Rayleigh number (Ra = 10^{6}), the fluid in the two horizontal parts is also almost static. This is decided by the position of heat source and thermal boundary conditions. In fact, the vertical walls (BC and FG) of two horizontal space are fixed at a low temperature, and when the heat source locates on the bottom (S = 0) of sidewall (AJ), vertical walls (BC and FG) and hot sources are not directly connected. Accordingly, no distinct vortex can be found inside the horizontal spaces. As for the effect of nanoparticle volume fraction on the streamlines, one can found that the flow pattern inside the enclosure does not change by increasing nanoparticle volume fraction.
Figure 2. Comparison of average Nusselt number for the present study and Matori et. al (2019) for different aspect ratios and position at Ra=106 and ϕ=0.003
Figure 4 displays the effects of ϕ and Ra on the temperature distribution inside the enclosure. At Ra = 10^{3}, when the primary heat transfer mechanism is conduction heat transfer, the isothermal lines can be found around the heat obstacle. They are smooth and equidistant. The fluid on the upper part of the enclosure is almost kept at T = 0, which is similar to streamlines. When the Rayleigh number increases to Ra = 10^{6}, the isothermal lines diffuse towards the upper part and become crooked. Besides, the fluid in the two horizontal spaces becomes warm and the fluid in the low horizontal space has a higher temperature than high horizontal space, which is due to the fact that the low horizontal space is closer to the heat source. There isnt any obviuse distinctive between the pure fluid and nanofluid.
ϕ = 0.00 


ϕ = 0.02 



Ra = 10^{3} 
Ra = 10^{6} 
Figure 3. Effects of ϕ and Ra on the streamlines at S=0.
Table 2. Effect of the mesh size on average Nusselt number, S=0 and ϕ=0.01
Ra 
Number of nodes 
Average Nusselt number 
Percentage of error 
1000 
100 100 200 200 300 300 
1.89238 1.86035 1.85971 
1.7217 0.0344 
1000000 
100 100 200 200 300 300 
2.41684 2.35791 2.35716 
2.4992 0.0318 
ϕ = 0.00 


ϕ = 0.02 



Ra = 10^{3} 
Ra = 10^{6} 
Figure 4. Effects of ϕ and Ra on the isotherms at S=0.
Figure 5 indicates the streamlines inside the Fshaped enclosure when the heat source moves up (S = 0.3) for different ϕ and Ra. In contrast to the case of S = 0, the flow pattern inside the enclosure changes significantly. At Ra = 10^{3}, two small vortices form adjacent to the hot obstacle. Similarly, the fluid far away from the hot obstacle is also almost static due to the conduction primary heat transfer mechanism. Compared with the flow pattern for Ra = 10^{3 }and S = 0, in which only one vortex can be found inside the enclosure, at Ra = 10^{3 }and S = 0.3, the two vortices form due to two different reasons. One can found that when obstacle moves up, the temperature gradient between the hot obstacle and the wall DE, causes the upper vortex. The other vortex is established by the temperature difference between the hot obstacle and wall HI.
When the Ra increases to 10^{6}, because of the strong natural convection, the vortex enlarges and occupies almost the whole enclosure. It should be mentioned that no secondary vortex can be found inside the large vortex and the single vortex core locates on the left top of the hot source. Because the hot obstacle is closer to the horizontal space, the fluid convection heat transfer there intensifies.
ϕ = 0.00 


ϕ = 0.02 



Ra = 10^{3} 
Ra = 10^{6} 
Figure 5. Effects of ϕ and Ra on the streamlines at S=0.3.
ϕ = 0.00 


ϕ = 0.02 



Ra = 10^{3} 
Ra = 10^{6} 
Figure 6. Effects of ϕ and Ra on the isotherms at S=0.3.
As for the temperature field, Figure 6 shows the isothermal lines for different ϕ and Ra at S = 0.3. At Ra = 10^{3}, the isothermal lines surround the hot obstacle and there is almost no difference of the isotherm gap above and below the heat source, which indicates that the temperature gradients above and below the heat source are similar. However, as Ra increasing, the convection intensity increases. It is obvious that the temperature gradient below the heat source is significantly higher than that above it. Moreover, the temperature gradient in the upper horizontal space is higher than that in the lower horizontal space. This phenomenon can be explained by the stronger vortex inside the upper horizontal space, which is shown in Figure 5.
Figure 7 presents the streamlines for different ϕ and Ra at S = 0.6. Due to the high value of S, the hot obstacle locates between two horizontal spaces. At Ra = 10^{3}, a vortex is established on the right of the obstacle. Due to the small gap between the heat source and wall DE, the vortex takes the shape of a dumbbell and two secondary vortices can be found inside the vortex. Moreover, due to the weak convection, the vortex is almost symmetrical. The flow pattern inside the Fshaped enclosure shows that fluid in the lower part of the cavity cannot affect the upper fluid flow pattern. That’s to say, at Ra = 10^{3}, the flow pattern in the Fshaped enclosure is similar to that inside the Cshaped enclosure. As the Rayleigh number becomes 10^{6}, the flow pattern changes significantly. One large vortex forms adjacent to the hot obstacle and occupies the upper region of the cavity. No secondary can be found inside the large vortex. Besides, one small vortex can be found in the lower horizontal space. Different from the clockwise large vortex, the smaller vortex is anticlockwise. This is due to the fluid viscosity and the larger vortex drags the smaller vortex.
Figure 8 shows the effect of ϕ and Ra on the isotherms at S=0.6. When the Rayleigh number is 10^{3}, the isothermal lines are almost symmetrical. The fluid in three regions, including two horizontal spaces and one lower region of the enclosure, are static, whose temperature is close to 0. For Ra = 10^{6}, the symmetry of isothermal lines disappear, and the isotherms below and on the right of the hot obstacle are more crowded. The temperature gradient above the obstacle is relatively small. The fluid temperature in the upper horizontal space is higher than that in the lower horizontal space.
ϕ = 0.00 


ϕ = 0.02 



Ra = 10^{3} 
Ra = 10^{6} 
Figure 7. Effects of ϕ and Ra on the streamlines at S=0.6.
ϕ = 0.00 


ϕ = 0.02 



Ra = 10^{3} 
Ra = 10^{6} 
Figure 8. Effects of ϕ and Ra on the isotherms at S=0.6.
ϕ = 0.00 


ϕ = 0.02 



Ra = 10^{3} 
Ra = 10^{6} 
Figure 9. Effects of ϕ and Ra on the streamlines at S=0.9.
Figure 9 and Figure 10 show the effect of ϕ and Ra on the streamlines and isothermal lines at S=0.9, respectively. Due to the high location of hot obstacle (S = 0.9), the positions of the vortices for different Rayleigh numbers (10^{3} and 10^{6}) are same. For Ra = 10^{6}, when the buoyancy force caused by temperature difference is large, the natural convection in other regions except the upper horizontal space is also weak. This is because, except wall BC, the other cold walls (DE, FG, and HI) locate lower than the heat source, which means the direction of the temperature gradient is same to the direction of gravity force. However, for the upper horizontal space, the obvious vortex can be found. As for the temperature distribution, one can found that the difference between isotherms for Ra = 10^{3} and Ra = 10^{6} at S = 0.9 is not as significant as those at S = 00.6. This is since the weak convection heat transfer when the hot obstacle locates on the top wall (S = 0.9).
Figure 11 presents the variations of average Nusselt numbers by Rayleigh number and S at different nanoparticle volume fractions. The Ra, ϕ, and S affect the average Nusselt number significantly. By increasing Ra, the average Nu increases, but the growth trend relies on the Rayleigh number. At low Rayleigh number (10^{3} ≤ Ra ≤ 10^{5}), the Rayleigh number influence average Nusselt number weakly. This is a consequence of the primary conduction heat transfer mechanism. When conduction is more important than convection on heat transfer, the effect of changing Ra on heat transfer by affecting convection can be neglected. However, as Ra = 10^{6}, the primary heat transfer mechanism is convection. As a result, the heat transfer increases significantly by increasing Rayleigh number. As for the nanoparticle volume fraction, one can found that the average Nusselt number increases by increasing ϕ, regardless of Ra and S.
When the position of heat source changes, the average Nusselt number changes significantly. The order of the average Nusselt number for different S is (S = 0.6) > (S = 0.3) > (S = 0) > (S = 0.9). This trend cannot be affected by Ra and ϕ. When S = 0.6, the heat source locates between two horizontal spaces with cold walls (BC and FG). Consequently, this arrangement is good for the heat transfer of heat source. However, at S = 0.9, the position of heat source is higher than the cold walls and the convection heat transfer is impeded. Therefore, its average Nusselt number is the least.
ϕ = 0.00 


ϕ = 0.02 



Ra = 10^{3} 
Ra = 10^{6} 
Figure 10. Effects of ϕ and Ra on the isotherms at S=0.9.


ϕ =0 
ϕ =0.01 


ϕ =0.02 
Figure 11. Effects of Ra and S on the average Nusselt number at different ϕ.
All figures and tables should be numbered with Arabic numerals and must be mentioned in the manuscript. They must be placed as close as possible to the first reference to them in the paper. In figures, number and caption should be typed below and in tables those should be typed above. Figures and tables must be aligned in the center of column and sized appropriately as width as one column. Although, large figures and tables that takes up more than 1 column width should be placed at the top or bottom of a page.
In the present research, the AgMgO/water hybrid nanofluid natural convection inside an Fshaped enclosure with a heat source was simulated by the LBM. The effects of Ra, ϕ and heat source location on the flow pattern, temperature distribution and heat transfer performance were investigated. To demonstrate the flow field and heat transfer characteristics, the streamlines, isotherms and profiles of average Nusselt numbers were introduced. The results showed that:
Conflict of Interest Statement
There is no conflict of interest.
Nomenclature
a,b 
length and height of obstacle, respectively 

AR 
the obstacle aspect ratio 

c 
lattice speed 

cp 
speci 

cs 
speed of sound, (m/s) 

ei 
streaming speed for particle 

f 
density distribution function 

feq 
equilibrium density distribution function 

g 
energy distribution function 

H, W 
Height and length of cavity, respectively 

I 
Exergy destruction rate [KJ/Kg] 

geq 
equilibrium energy distribution function 

k 
thermal conductivity, (W/mK) 

Nu 
Nusselt number 

Pr 
Prandtl number 

S 
position of hot obstacle 

T 
fluid temperature, K 

t 
Time, s 

u 
velocity vector, (m/s) 

x 
Cartesian coordinates, m 

Greek symbols 


ω_{i} 
weight function in direction i 

ϕ 
solid volume fraction of nanoparticles 

τ_{c} 
relaxation time for heat transfer 

α 
thermal diffusivity, (m2/s) 

ρ 
density, (kg /m3) 

τ_{v} 
relaxation time for flow 

μ 
dynamic viscosity, (kg/ms) 

Subscripts 

f 
fluid 

H 
Hot 

i 
move direction of singleparticle 

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[*]Corresponding Author: Rasul Mohebbi, School of Engineering, Damghan University, Damghan 3671641167, Iran.
Email: rasul_mohebbi@du.ac.ir
[*]Corresponding Author: Yuan Ma, Shanghai Automotive Wind Tunnel Center, Tongji University, No.4800, Cao’an Road, Shanghai 201804. China
Email: 1510812@tongji.edu.cn