Simulation of nanoliquid thermogravitational convection within a porous chamber imposing magnetic and radiation impacts

Journal Pre-proof Simulation of nanoliquid thermogravitational convection within a porous chamber imposing magnetic and radiation impacts M. Sheikholeslami, Mikhail A. Sheremet, Ahmad Shafee, Iskander Tlili

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S0378-4371(19)32243-5 https://doi.org/10.1016/j.physa.2019.124058 PHYSA 124058

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Physica A

Received date : 20 April 2019 Revised date : 29 September 2019 Please cite this article as: M. Sheikholeslami, M.A. Sheremet, A. Shafee et al., Simulation of nanoliquid thermogravitational convection within a porous chamber imposing magnetic and radiation impacts, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.124058. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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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Simulation of nanoliquid thermogravitational convection within a porous chamber imposing magnetic and radiation impacts M. Sheikholeslami a,b, Mikhail A. Sheremet 1,c, Ahmad Shafee d, Iskander Tlili e,f a

Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran

Renewable Energy Systems and Nanofluid Applications in Heat Transfer Laboratory, Babol Noshirvani University of Technology, Babol, Iran c d

Laboratory on Convective Heat and Mass Transfer, Tomsk State University, Tomsk 634050, Russia

Public Authority of Applied Education & Training, College of Technological Studies, Applied Science Department,

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Shuwaikh, Kuwait e

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b

Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam f

Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Abstract

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Innovative approach for simulation of nanoliquid natural convection through a permeable wavy space was scrutinized. Analysis has been conducted within inclined porous rectangular cavity having cold side borders under the impacts of heated internal wavy horizontal cylinder and uniform tilted magnetic field. The considered boundary-value problem devised on the basis of the Darcy linear law, Boussinesq relation, Rosseland approach and Lorentz force has been resolved by the finite volume technique. The impacts of nanomaterials' shape, buoyancy force, radiation term and Hartmann number on nanoliquid hydrodynamic treatment have been investigated. Outcomes have

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proved that greater quantities of shape factor result in the greater convection. By enhancing magnetic forces, convection becomes weaker.

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Keywords: Darcy model; Radiation; Finite volume method; Shape factor; Free convection; Magnetic force; Nanofluid; Wavy cylinder. 1. Introduction

Convective energy transport in irregular enclosures has an essential value in different

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industrial fields including electronics cooling, chemical reactors, heat exchangers, solar collectors and others [1–5]. Intensification of thermal transmission within such regions can be achieved by addition of porous medium and nanoliquids [6–9]. Porous media allow increasing the heat transfer   1

Corresponding author:

Email: [email protected] (Mikhail A. Sheremet), [email protected] (Iskander Tlili), [email protected] (M. Sheikholeslami)

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surface and as a result to intensify the energy transport [5–7]. Nanoliquids as a combination of host fluid and nano-sized particles of metal or metal oxide allow augmenting effective thermal conductivity and heat transfer coefficient [8–13]. Comprehensive review on convective heat transfer from wavy surfaces was published by Shenoy et al. [1]. Alsabery et al. [14] have scrutinized numerically convective energy transport inside the region with internal solid cylinder. Employing the finite element technique, the authors have shown that a augment of cylinder thermal

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conductivity leads to the reduction of high nanoparticles volume fraction areas. Sheremet et al. [15] have computationally investigated free convection of nanoliquid within an open porous region

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under the Lorentz force influence. Obtained results have shown the energy transport strengthening with nanoparticles volume concentration for moderate and high Hartmann numbers. Al-Srayyih et al. [16] have analyzed numerically permeable chamber with involve of buoyancy forces with the linear temperature profile along the left side wall. They have revealed that low quantities of thermal conductivity ratio between porous medium and nanoliquid characterize greater thermal transmission

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augmentation than large quantities of the thermal conductivity ratio. Haq et al. [17] scrutinized flow style a wavy permeable region with magnetic field impact. It has been ascertained the heat intensification with a growth of the wavelength parameter. The problem of double diffusion and magnetic field impact on nanoliquid in a stretching region has been solved by Bilal et al. [18]. They have demonstrated that fraction rises but Sherwood number reduces with increasing Dufour number. Hashemi et al. [19] have worked out the problem of magnetic free convection of micropolar nanoliquid within porous Darcy region with inner elliptic heater employing the nonequilibrium conditions. They found that a rise of the medium porosity augments the circulation

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strength, while total Nusselt number is increased also. Sheikholeslami et al. [20] have analyzed MHD natural convective-radiative thermal transmission of water-based nano-suspension within a

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porous area. They have shown that the porous medium permeability growth reflects the rise of the Nu. Impact of magnetic dipole on behavior of ferrofluid was evaluated by Kausar et al. [21] and considered induced magnetic role on flow style of nanomaterial. Izadi et al. [22] have calculated the thermogravitational copper-water nano-suspension convection within porous irregular chamber using the Buongiorno’s nanofluid approach and LTNE model. They have ascertained that the mean

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Nu diminishes with the thermophoresis coefficient. Other practical outputs were found in [23–32]. Taking into account the brief review presented above, the convective nanoliquid circulation and energy transport in corrugated porous regions is a very important and useful topic. Unfortunately, nowadays there are no data about MHD thermogravitational convection of H2Oformed nano-suspension within porous rhombic region with internal heated wavy cylinder. The present study is overcome abovementioned gap using the single-phase nanoliquid model with experimentally-obtained nanoliquid physical correlations and the Darcy porous medium approach. 2   

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2. Problem explanation Free convection within porous cavity depicted in Fig. 1 with inclined magnetic force has been analyzed. Darcy law has been employed for permeable terms with local thermal equilibrium conditions for host liquid, nano-sized particles and porous solid matrix. The impact of magnetic field is described by the uniform Lorentz force. The borders of the external rhombic cavity were kept at low temperature Tc, but the constant heat flux q is kept at internal wavy surface. Needed

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boundary conditions are demonstrated in Fig. 1. Numerical modeling was performed using the

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CVFEM. 3. Mathematical formulation and computational technique

3.1. Control PDEs

Using the abovementioned description of the considered physical problem the following partial differential equations were employed for explanation of transport processes for energy and within

u v  0 x y

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the permeable zone under the Lorentz influence.

P 2  uK 1nf  B 02  sin   u   sin  v  cos     nf  0   x

P  K 1v nf   nf T T c   nf g y  nf B

2 0

 cos    sin  u   cos  v 

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2 1   T qr  T T   2T  1  u v   k C     C p nf y  x y  nf p nf  y 2 x 2 

(2)

(3)

(4)

4 e T 4 . 3 R y

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where T 4  4Tc3T  3Tc4 , qr  

(1)

Here the single-phase nanoliquid model is applied with the Darcy law for porous medium, Rosseland approximation for thermal radiation and Lorentz force for the uniform magnetic field. Physical properties of the nano-suspension have been presented in [33].

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Description of the effective dynamic viscosity and thermal conductivity has been performed employing the experimentally-based relations with the Brownian motion impact as follows

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

k Brownian f   static Prf kf



G  1   3 ln     4 ln d p  ln   Ln T

   ln d



  6  10 ln d p    9 ln d p  ln     8 ln     7 ln d p  5

2

p



2



  2 ln d p  Ln T





bT   pd p f

k Brownian  5  104 c p ,f G 

While for knf includes the nano-sized particles shape

k  1  m   S   k f  k nf  p , kf k p  S   k f 1  m 

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S  kf k p,

(5)

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eff 

(6)

Tables 1–3 reported different m, related coefficient and nano-suspension properties [33]. We supposed the Eq. (7) to find final form. T   x , y  / L ,  

T c , T  Lk f1q  T

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   /  nf ,  X ,Y

(7)

As a result, the final formulation is:

 2  A A  A 2 2 Ra    6 Ha   sin 2   3 2 2 2 A5 Y X X Y     A 4 A 5 X 2 A  2 2   2  Ha 6  sin     cos    2 Y X 2 A5  

(8)

 2  4  knf  1  X 2  3  k f 

(9)

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1   2       0  Rd  2   Y Y X Y X  

Utilized parameters in above formulas are:

A4 

g K    f L T

, Rd  4 eTc3 /   R k f

f f

  CP nf   nf  nf nf , A2  , A3  , A5  , f f   CP  f    f knf kf

, A6 

 (10)

 nf f

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A1 

f

, Ra 

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Ha  K

B02 f

In addition, we can mention briefly the borders conditions for the half part of the computational domain

  0.0 at outer surface;

  0.0 at all surfaces for half part of the cavity;

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 n  1 at inner surface

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For the understanding the thermal transmission intensity the Nusselt number has been defined as follows Nuloc

1  4k  1   nf   3  k f 

1  k   nf  Rd      k f   

(12)

s

1 Nuloc ds S 0

(13)

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Nuave 

3.2. Considered finite volume technique

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For solution to the considered boundary-value problem the finite volume technique with triangular elements has been applied. Detailed description of this technique can be found in [33]. Liner interpolation with considering triangular elements has been utilized to calculate scalars. 4. Code validation and Mesh analysis

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Good numerical simulation should have optimized mesh size which can support reaching outputs with lower grids. So, we tested several grids listed in Table 4 for special case to reach the best mesh. The written code has great accuracy and this fact can be seen in Fig. 2 because the deviation with previous benchmark [34] is very low. 5. Results and discussion

In considered manuscript, the roles of radiation, magnetic force and Nanomaterials shape on

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nanoliquid hydrothermal treatment have been simulated. The porous domain has been saturated with Al2O3–H2O. Roles of radiation term (Rd = 0–0.8), Nanomaterials shape, Lorentz strength form.

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(Ha = 0–20), fraction of nano-sized particles ( = 0.0–0.04) were illustrated in diagram (Figs. 3–6) Figure 3 pictures isolines of  and  for m = 5.7, Rd = 0.8, Ha = 0, Ra = 600and various quantities of the nanoparticles volume concentration. Taking into account heating of the central corrugated cylinder and cooling of the external borders, convective cells are formed within wavy

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troughs with up-going currents near the hot surface and top-down currents near the cooling walls. The formed eddies characterize the zones of convective thermal transmission, while in the bottom cavity part heat conduction is a dominating regime owing to upper warming and bottom refrigeration. More essential thermal plume is appeared in the tope zone of the chamber over the upper corrugated crest. An addition of nanoparticles reflects the weakening of convective circulation, while the flow nature does not change. More essential heating can be found in distribution of isotherms due to raised effective thermal conductivity (see Eq. (6)). Taking into 5   

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account low value of the Rayleigh number (Ra = 600) there are no recirculations within wavy troughs. The influence of Ha on isolines of  and  is presented in Fig. 4. At Ha = 0, flow characterizes a generation of strong convective eddy in the left part of the region, while isotherms do not reflect an origin of thermal plume. Such weak convective flow can be explained by low Ra (Ra = 100). A rise of the Hartmann number (Ha = 20) demonstrates essential attenuation of the

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convective circulation, while thermal regime is a heat conduction and temperature values are increased near the corrugated cylinder surface.

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An increment of Ra up to Ra = 600 results in the strengthen of the circulation within the nanoliquid gap, where for Ha = 0 the intensive convective cell displaces to the upper cavity part (see Fig. 5). In this case more stable thermal plume can be found in the top chamber zone. More significant heating can be found in the bottom wavy trough. In this case a raise of Ha characterizes an augment in the temperature within the tank, but circulation intensity reduces. Heating from the

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internal corrugated surface becomes more homogeneous with heat conduction dominating mode and major circulation occurs in the left cavity corner.

Figure 6 demonstrates the behavior of Nuave with considered governing parameters. Using this figure, it is possible to conclude that Nuave increases with thermal radiation parameter, Ra and nanoparticles shape coefficient, whilst it diminishes with Ha. An influence of the Hartman number becomes essential for high Ra, but for high quantities of Ha the effects of Ra and Rd are weak. Obtained results allow extracting the following relation for Nuave in the wide considered ranges of governing parameters:

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Nuave  1.51  0.037 m  0.51Rd  0.23Ra  0.18 Ha

6. Conclusions

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3.6  103 m Ha  0.12 Rd Ha  0.23RaHa  4.58 105 m 2

Thermogravitational convection and thermal radiation of alumina-water nanoliquid within the rhombic porous region of cooling borders with internal corrugated surface of constant positive heat flux under the uniform tilted magnetic field impact has been studied. The single-phase

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nanosuspension model has been employed with experimentally-obtained dependence for the physical parameters of the considered “smart” medium. The porous medium is assumed to satisfy the Darcy law and local thermal equilibrium conditions. Computational investigation has been performed for large variety of control characteristics using the developed finite volume technique. Obtained results have allowed concluding that a rise of the Rayleigh number intensify convective circulation in the upper part of the porous region with an origin of stable temperature plume over the top wavy crest of internal corrugated surface. Moderate and low quantities of Ra reflect a 6   

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development of circulations in the left and right rhombic cavity corners. The average dimensionless thermal transmission parameter is an enhancement function of Ra, thermal radiation parameter and nano-sized particles shape parameter, whilst it is a reducing function of Ha. Acknowledgements: Dr. Sheremet acknowledges the financial support from the Ministry of

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Fig. 1. Sketch of the considered region

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Gr  2 105

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Fig. 2. Variation of Nuave at Pr = 0.733

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Streamlines

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Isotherms

2 4

3 3

5

3 2 1

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5

0 .2

0.6 0.4

0.2

0.6

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Fig. 3. Impacts of  on nanoliquid behavior ( = 0.0 (- - -) and  = 0.04 (––))

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for Rd = 0.8, Ra = 600, Ha = 0, m = 5.7                         12   

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Streamlines

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1.4 1.3 1.2 1 0.7 0.5 0.4 0.18

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Ha = 0

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

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Isotherms

0.04 0.03 0.02 0.01

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Ha = 20

2.2 2 1.8 1.6 1.4 1.2 1 0.6 0.2

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Fig. 4. Influence of Lorentz on nanoliquid behavior for m = 5.7, Rd = 0.8, Ra = 100             13   

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Streamlines

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Isotherms

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Ha = 0

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

0.25 0.2 0.15 0.1 0.05

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Ha = 20

2.2 2 1.8 1.4 1 0.6 0.4 0.2

5 4.6 4.3 4 3 2 1

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Fig. 5. Influence of Lorentz on nanoliquid behavior when m = 5.7, Rd = 0.8, Ra = 600    

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Ha = 10, m = 4.35,  = 0.04

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Ra = 350, Rd = 0.4,  = 0.04

Rd = 0.4, Ha = 10,  = 0.04

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Ra = 350, Ha = 10,  = 0.04

Rd = 0.4, m = 4.35,  = 0.04

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Ra = 350, m = 4.35,  = 0.04

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Ha = 10, Rd = 0.4,  = 0.04

Ha = 10, m = 4.35,  = 0.04

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Ra = 350, Rd = 0.4,  = 0.04

Pr e-

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Ra = 350, Ha = 10,  = 0.04

Ra = 350, m = 4.35,  = 0.04

Rd = 0.4, m = 4.35,  = 0.04

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Fig. 6. Influences of scrutinized variables on Nuave             16   

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Table 1. The quantities for alumina-water nanoliquid

Quantities

1

52.813488759

2

6.115637295

3

0.6955745084

4

0.0417455552786

5

0.176919300241

6

–298.19819084

7

–34.532716906

8

–3.9225289283

9

–0.2354329626

10

–0.999063481

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Coefficients

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Table 2. Physical parameters of Al2O3 and host liquid

Water Aluminum oxide

Cp (J/kg/K)

 (kg/m3)

k (W/m/K)

105 (1/K)

 (1//m)

4179

997.1

0.613

21

0.05

3970

25

0.85

10-10

765

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Table 3. Quantities of m shape

Cylinder

4.8

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m

Brick

3.7

Platelet

5.7

Spherical

3

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Table 4. Vaues of Nuave for various grids at Rd = 0.8,  = 0.04, Ha = 0 and Ra = 600

Grid parameters

41121

51151

61181

71211

81241

2.7908

2.7976

2.8021

2.8068

2.8091

17   

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Highlights • Innovative approach for simulation of nanoliquid free convection within a permeable wavy space has been scrutinized. • The boundary-value problem formulated on the basis of the Darcy linear law,

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Boussinesq approximation, Rosseland approach and Lorentz force has been resolved.

• Large values of shape factor result in the stronger convection.

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• Heat conduction becomes more sensible for intensive magnetic field.

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Tomsk State University Laboratory on Convective Heat and Mass Transfer

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Tomsk, 29.09.2019

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CONFLICT OF INTEREST STATEMENT

This is to confirm that no any conflict of any kind interest exists in what concerns the

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work entitled “Simulation of nanoliquid thermogravitational convection within a porous chamber imposing magnetic and radiation impacts”, submitted for publication in PHYSICA A: STATISTICAL MECHANICS AND ITS

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APPLICATIONS.

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Very sincerely yours, Dr. Mikhail Sheremet Head of the Laboratory on Convective Heat and Mass Transfer, Tomsk State University, Russia