Effect of angle of applied magnetic field on natural

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Sep 18, 2017 - A simplified double distribution function (DDF)-thermal lattice Boltzmann ... B. Uniform magnetic field, N/(m A) c. Lattice streaming speed, m/s. Cp .... heated and cooled walls were kept vertical (left and right); while for another case, these ... known to author) has explored combined effect of angle/orientation.
chemical engineering research and design 1 2 7 ( 2 0 1 7 ) 22–34

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Effect of angle of applied magnetic field on natural convection in an open ended cavity with partially active walls Krunal M. Gangawane ∗ Department of Chemical Engineering, College of Engineering Studies, University of Petroleum and Energy Studies, Dehradun, 248007 Uttarakhand, India

a r t i c l e

i n f o

a b s t r a c t

Article history:

In this work, 2D, steady, natural convection in a partially active/heated open ended square

Received 7 July 2017

cavity subjugated to the magnetic field for incompressible, Newtonian fluid is studied

Received in revised form 5

and presented. A simplified double distribution function (DDF)-thermal lattice Boltzmann

September 2017

method (TLBM) based on single relaxation time (SRT) is utilized for solving field controlling

Accepted 7 September 2017

equations. One of the vertical walls of open cavity is exposed to the heat source (i.e., heater)

Available online 18 September 2017

partially; while another vertical wall is open to ambient. In particular, the influence of various geometric as well as parametric conditions, such as heater size (LH = 0.25, 0.5, 0.75), angle

Keywords:

of magnetic field ( M = 0◦ , 45◦ and 90◦ ), Hartmann number (0 ≤ Ha ≤ 100) and Rayleigh number

Lattice Boltzmann method

(103 ≤ Ra ≤ 105 ) on local and global convection features have been investigated. The depen-

MHD

dence of average Nusselt number with magneto-convective parameter ( = Ha2 /Ra) has also

Open ended cavity

been illustrated. It is observed that cavity with the applied magnetic field at  M = 45◦ offers

Rayleigh number

highest heat transfer restriction than other considered cases. Effect of heater size remains

Nusselt number

effective only at higher Ra (≥105 ).

Partial heating

1.

© 2017 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Introduction

transfer as well as to stabilize the fluid circulation (Sivakumar et al., 2015). Recently, the application of MHD-nanofluid to study melting heat

MHD flow has wide occurrence in nature, for instance, solar flares,

transfer has also been studied (Sheikholeslami and Rokni, 2017).

solar corona, sunspots, solar wind, etc., as well as in various industrial activities (e.g., advanced propulsion, tokamak fusion reactors, metal-

Established CFD methods (finite difference method (FDM), finite element method (FEM) and finite volume method (FVM)) are grounded on the discretization of macroscopic continuum equations (i.e.,

lurgy, etc.) (Zhang et al., 2017; Hayat et al., 2017). Fundamentally, MHD couples Maxwell’s electromagnetic equations (for electrically conducting fluids) with traditional fluid dynamics (Navier–Stokes) equations, thereby generating modified body force term called ‘magnetic force’. The

mass, momentum, and energy equations); whereas lattice Boltzmann method (LBM), which belongs to class of mesoscopic approach, is based on mesoscopic kinetic equation (Boltzmann equation). The key

magnetic force causes a resisting impact on fluid flow (Sarris et al., 2006). Magnetic field along with the natural convection heat transfer in the systems modifies the magnetic force to interact with buoyancy force. It shapes the convective fluid flow pattern in the system, which

aspect of LBM method is to build simple kinetic models. Such mod-

may be useful for chemical engineering process, such as, crystallization ing fluid circulation by convection), blankets of liquid metal utilized

models are simply the discrete distribution functions or population f(x,e,t) that constitutes the probability of finding a particle moving with lattice speed ‘e’ at position ‘x’ and at time ‘t’. Kinetic nature

for fusion reactors (Sarris et al., 2006; Sivasankaran et al., 2016). The magnetic field, therefore, can be used to suppress the convection heat

of LBM has numerous benefits over conventional CFD methods, few of them are, simplicity in the algorithm, implementation of bound-

in semiconductors production (maintain crystal growth and minimiz-



els compromise the essential physics of microscopic or mesoscopic processes, such that, microscopic averaged properties are the suitable representation of the desired macroscopic equations. These kinetic

Fax: +91 135 2776090. E-mail addresses: [email protected], [email protected] https://doi.org/10.1016/j.cherd.2017.09.006 0263-8762/© 2017 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

chemical engineering research and design 1 2 7 ( 2 0 1 7 ) 22–34

Nomenclature AR B c Cp e f feq F g geq Gr H h Ha K L LH Ma Nu Nu P Pr Ra Re Ri t T Tc Tref Ux, Uy, V X,Y T Greek ϕ ␮ ␣ ␤ ␴ ␪m ␯ ␳  ␺ 

Aspect ratio, dimensionless Uniform magnetic field, N/(m A) Lattice streaming speed, m/s Heat capacity, J/(kg K) Lattice link direction, dimensionless Flow particle distribution function, dimensionless Flow particle distribution function at equilibrium, dimensionless External force term, N Thermal particle distribution function, dimensionless Thermal particle distribution function at equilibrium, dimensionless Grashof number, dimensionless Height of cavity, m Heat transfer coefficient, W/(m2 K) Hartmann number, dimensionless Thermal conductivity, W/(m K) Length of cavity, m Length of partial heater, m Mach number, dimensionless Local nusselt number, dimensionless Average nusselt number, dimensionless Pressure, N/m2 Prandtl number, dimensionless Rayleigh number, dimensionless Reynolds number, dimensionless Richardson number, dimensionless Lattice time step, s Temperature, dimensionless Ambient temperature, dimensionless Reference temperature, K Velocity components, m2 /s Characteristic velocity, m/s Co-ordinates, m Temperature difference, K

Vorticity, m2 /s Dynamic viscosity, N s/m2 Thermal diffusivity, m2 /s Coefficient of thermal expansion, 1/K Electrical conductivity, S/m Angle of magnetic field Kinematic viscosity, m2 /s Density, kg/m3 Average density determined at Tref , K Stream function, m2 /s Angular rotational speed, RPM

Subscripts H Heated Lattice link direction k f Flow field Thermal field t Cartesian coordinates x,y Superscript Equilibrium condition eq

23

ary conditions at leisure, ease of parallel computing and handling of complex geometries, pressure field calculation is extremely simple (solving the equation of state). Furthermore, the convection operator is linear, and microscopic distribution functions can be transferred to macroscopic physical quantities by simple arithmetic calculations (Gangawane et al., 2015a). Although, a large number of studies have revealed some interesting behavior in systems under the influence of magnetic field, much less is known about the impact of the orientation of magnetic field. On the other hand, buoyancy driven flow and heat transfer in the cavity (of different types) encompasses a rich scientific phenomenon owing to its broad spectrum of applications in scientific/engineering practices, such as electronic equipment, processing of polymer and materials, solar energy collectors, etc. (Gangawane, 2015, 2017; Gangawane and Manikandan, 2017). Among the immense literature available on MHD buoyancy driven convection in the cavity, Bilgen and Oztop (2005) studied the natural convection in an inclined partially open cavity for the range of Rayleigh number (Ra) and partial opening sizes. Koca (2008) delineated conjugate heat transfer in an open square cavity with the partial opening. They reiterated that the position of ventilation has a remarkable impact on heat transfer rate. Shahi et al. (2010) numerically elucidated the mixed convection in the square cavity with partially heated bottom wall for nanofluids for a broad range of parametric conditions (Reynolds number, Richardson number, solid volume fraction, etc.). Rahman et al. (2013) studied the influence of magnetic field in a channel with the complete as well as partially heated wall. Consequently, Jing et al. (2013) presented three-dimensional study of natural convection of liquid metal under magnetic field. They studied the influence of three different magnetic field directions on heat transfer characteristics. Souza et al. (2014) experimentally explored the impact of the presence of partial heater in the cavity. They reiterated that the presence of plate makes fluid circulation to become two-dimensional in the plane, therefore, making it suitable for 2D numerical methods. Rahman et al. (2015) used heatline-massline analysis to understand the unsteady heat and mass transfer in the partially-heated open cavity. Recently, few studies also have analyzed the thermal characteristics due to buoyancy driven convection in the partially heated open cavity (Gangawane et al., 2015c,b, 2016). Moreover, the impact of Prandtl number on heat transfer rate can be found in Gangawane et al. (2015c), while the influence of heating location and heater size on free convection can be found in Gangawane et al. (2015b), Gangawane et al. (2016). Arbin et al. (2016) carried out the numerical simulation of doublediffusive convection in an open cavity with the partially active wall by using heatline analysis. They noted significant variation of the length of a partial heater on the rate of heat as well as mass transfer. Sheremet et al. (2016) interpreted MHD free convection in a porous tall open cavity with a wavy wall filled with nanofluids. They investigated the influence of corner heater on the rate of heat transfer. Arici et al. (2017) delineated the effect of partially heated walls on melting of nanoparticle-enhanced paraffin wax in a rectangular cavity by considering two cases for numerical experimentation. In one case, partially heated and cooled walls were kept vertical (left and right); while for another case, these were horizontal (top and bottom). Lately, Karatas and Derbentli (2017) presented an experimental study of natural convection in rectangular cavities having one active vertical wall. On the other hand, extensive research has been documented on natural convective flow and heat transfer in cavity subjugated to magnetic force. For an instance, Kefayati (2013) utilized thermal lattice Boltzmann method (TLBM) to study convection in the enclosure under magnetic field for nanofluids and a wide range of pertaining parameters. With the increase in the magnetic field inclination, the rate of heat transfer increases. Yu et al. (2013) numerically examined the MHD in the cavity for various directions of the magnetic field for a wide range of Ra, Ha at Pr = 0.025. A significant effect of inclination angle on convective heat flow rate was observed especially when AR = / 1. Sheikholeslami et al. (2014) used TLBM solver for exploring MHD natural convection in the cavity for nanofluids. The rate of heat transfer increases with Ra and nanoparticle fraction; while it deteriorates with Ha. Hossain and Abdul Alim (2014) studied similar physical process within the trapezoidal shaped cavity with bottom wall subjected to

24

chemical engineering research and design 1 2 7 ( 2 0 1 7 ) 22–34

non-uniform heating. Kefayati (2015) conducted a numerical study to explore the MHD natural convection in partially-differentially heated cavity for non-Newtonian fluids (based on power law). Author reported that there is heat transfer augmentation with power law index. Similar study was performed by Mejri and Mahmoudi (2015) for sinusoidal boundary condition. Bondareva et al. (2016) analyzed heatline visualization of MHD free convection in an open-tilted cavity for nanofluids. Hayat et al. (2016) included thermal radiation as well as inclined magnetic field effect for nanofluids. On the other hand, Kefayati (2016) studied combined effect of non-Newtonian power law index and magnetic field on overall hydrodynamics and thermal characteristics in cavity by using finite difference-lattice Boltzmann method (FD-LBM). The major finding of the study was the opposite effect of Hartmann number on total entropy generation. Selimefendigil and Oztop (2016) delineated the influence of rotating cylinder on natural convection heat transfer under magnetic field in a differentially heated cavity with partial heater. For shear thinning fluids, the rate of heat transfer increases for clockwise rotating cylinder than motionless cylinder. Similarly, the influence of heat radiation is investigated on MHD convection due to buoyancy is explored by Sheikholeslami et al. (2016). The working fluid for their study was Al2 O3 nanofluid. Increase in average Nu with radiation parameter was observed. Lately, Zhang and Che (2016) used multiple relaxation time (MRT) lattice Boltzmann method for analyzing MHD natural convective characteristics in an inclined cavity having four square heat sources for range of governing parameters. They reported nonlinear variation of average Nu with Ra, Ha and inclination angle. Recently, Feng et al. (2016) studied melting of solid gallium in bottom heated rectangular cavity under the influence of magnetic field by using TLBM. Some of the recent advancements in the MHD free convection for nanofluids are presented by Sheikholeslami and Seyednezhad (2017), Sheikholeslami (2017a), Sheikholeslami and Shehzad (2017), Sheikholeslami (2017b), Sheikholeslami and Sadoughi (2017). Therefore, analysis of available literature points availability of extensive data on natural convection in cavities (of varying type), but portion of literature documenting thermal characteristics in cavity with partial heating subjugated to magnetic field is quite modest. Most of the MHD-convection studies have covered differentially heated closed cavity, very few studies have considered an open ended cavity. Acknowledging the fact that natural convection in open ended has cavity has pragmatic relevance, such as oven, furnaces, refrigerators, cooling system of electronic devices, etc. it is important to find the optimum heater location such systems. Moreover, it is known that magnetic field can be used to suppress the convective flow and heat transfer, which can be used in several industrial applications. From the review of available literature, it is observed that no study (as much known to author) has explored combined effect of angle/orientation of magnetic field direction and size of heater in partially heated open ended cavity on natural convection characteristics. This has inspired author to conduct present research to find out impact of heater size as well as magnetic field strengths (at different angles) on overall heat transfer and fluid structure in open cavity. Therefore, in this work, the dependency of flow governing parameters, such as heater

Fig. 1 – Schematics of problem under consideration (Open ended cavity subjected to partial heating under magnetic field). of cavity and at the center/middle of vertical wall. Right wall (X = 1) is open to ambient, but to make the problem computationally feasible, well known, slip condition is applied on right wall. Horizontal walls of cavity (bottom and top) as well as the region other than heater of left/no-slip wall are thermally insulated. The Boussinesq approximation is used to express dependency of density with temperature. In order to justify this approximation, the temperature difference is maintained small (Here, T = 1). Further, thermo-physical properties of working fluid (Pr = 0.71) are assumed to be constant. The effects of radiation, heat dissipation (due to friction) and compression work are neglected. It should be noted that although air can not necessarily be called as electrically conducting fluid, it can be used for numerical simulation purpose (Chatterjee and Gupta, 2017). Moreover, as per Sarris et al. (2006), Pr = 0.7 can be used for nuclear mixed oxide (molten) fuels [(U0.8 Pu0.2 )O2 ].

3.

Mathematical formulation

Continuity, momentum (−x and −y) and energy equations for MHD natural convection for earlier mentioned assumptions are given below in non-dimensional form. • Continuity equation

size (LH = 0.25,0.5,0.75), angle of magnetic field ( m = 0◦ , 45◦ , 90◦ ), Hart-

∂Uy ∂Ux + =0 ∂X ∂Y

mann number (0 ≤ Ha ≤ 100) and Rayleigh number (103 ≤ Ra ≤ 105 ) for Pr = 0.71 on convection characteristics have been explored. The location

• Momentum equation-x component

of heater is maintained at the center of no-slip vertical wall. The prob-

(1)



lem statement and governing equations are presented in subsequent section.

Ux

2.

• Momentum equation-y component

Problem statement

The schematic of problem under consideration with Cartesian coordinate (origin at left-bottom corner of cavity) system is depicted in Fig. 1. It is consists of square (L/H = 1) open ended  cavity exposed to uniform magnetic field of B = Bx + By at angle of m =

  Bx By

(Yu et al., 2013) from horizontal axis. A

heater LH = 0.25/0.5/0.75 is kept along left, no-slip wall (X = 0)

∂P ∂Ux ∂Ux + Uy =− + Pr ∂X ∂Y ∂X

∂Uy ∂Uy ∂P Ux + Pr + Uy =− ∂Y ∂X ∂Y +RaPrT − Ha2 PrUy

∂2 Ux ∂2 Ux + 2 ∂X ∂Y 2



∂2 Uy ∂X 2

+



∂2 Uy

(2)



∂Y 2 (3)

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chemical engineering research and design 1 2 7 ( 2 0 1 7 ) 22–34

• Energy equation

Ux

∂T ∂T + Uy = ∂X ∂Y



4. ∂2 T ∂2 T + 2 ∂X ∂Y 2

 (4)

Mathematical description of boundary treatments (for flow and thermal fields) for open ended cavity in non-dimensional form for each wall can be written as, • Left wall (X = 0),

Thermal lattice Boltzmann method

Thermal lattice Boltzmann method (TLBM) originated from lattice gas automata (LGA) and based upon a simplified double distribution function (DDF) is used for solving flow and thermal fields. Thermal lattice Boltzmann equations (LBE) along with equilibrium distribution functions for flow and thermal fields and for unit time step (Gangawane et al., 2015c,b, 2016) are given below: eq

fk (x + ek , t + 1) − fk (x, t) = −

fk (x, t) − fk (x, t)

f

+F

(12)

Ux = Uy = 0;

⎧ ∂T ⎪ = 0 for 0 ≥ Y ≥ h1 , ⎪ ⎪ ⎪ ⎨ ∂x

eq

gk (x + ek , t + 1) − gk (x, t) = − (6)

T = TH for h1 ≥ Y ≥ h2 , ⎪ ⎪ ⎪ ⎪ ⎩ ∂T = 0 for h ≥ Y ≥ 1, 2

∂x

(13)

t

where, f = 3.0␯ + 0.5 ; t = 3.0˛ + 0.5 (Gangawane et al., 2015c,b, 2016) are the relaxation parameters for flow and thermal fields, respectively.

 where, h1 and h2 are the start and end point of heater, respectively.

gk (x, t) − gk (x, t)

eq fk

2

e ·u (e · u) u·u 1+3 k2 +9 k 4 +3 2 2c c 2c

= wk



eq

gk = Twk 1 + 3

• Right wall (X = 1),

eq

ek · u c2

 (14)



eq

⎪ ⎩ ∂T = 0 if Ux ≥ 0

where, k, fk , gk , ek , f , t , x (ek t), c(x/t = 1) and Fk are lattice link direction, flow and thermal field equilibrium distribution function, discrete velocity, relaxation time for flow field and thermal fields, space step, streaming speed and force term, respectively (Gangawane et al., 2015b, 2016; Kefayati, 2013). For D2Q9 lattice model, wk is weight function for direction k, is given as,

• Bottom, top walls (i.e., horizontal walls),

w0 =

∂Uy ∂Ux = = 0; ∂X ∂X

⎧ ⎪ ⎨ T = 0 if Ux < 0

(7)

∂X

At(0 ≤ Y ≤ 1), Ux = Uy = 0;

∂T =0 ∂y

(8)

The dimensionless parameters, i.e., Rayleigh number (Ra), Prandtl number (Pr) and Hartmann number (Ha), appearing in governing equations (Eqs. (1)–(4)) are expressed as follows (Kefayati, 2013; Yu et al., 2013): Ra =

gˇ TH3





; Pr =

Cp

= ; Ha = BH k ˛



The solution of Eqs. ((1)–(8)) yields the velocity, pressure and temperature fields, which are further processed to estimate the important flow and heat transfer quantities (stream function, Nusselt number, etc.). These parameters can be estimated as per following definitions (Gangawane et al., 2015c,b, 2016; Kefayati, 2013; Yu et al., 2013). • Stream function:

4 1 1 ; w1−4 = ; w5−8 = 9 9 36

Similarly, the lattice link based velocity vectors for D2Q9 lattice model are represented as, ekx = [0.0, 1.0, 0.0, −1.0, 0.0, 1.0, −1.0, −1.0, 1.0]

• Nusselt number: Nu =

Effect of magnetic as well as buoyancy force can be incorporated by adding force (F) term in the collision equation of LBE. It takes into account the magnetic field intensity (B) and Boussinesq approximation term (Yu et al., 2013; Sheikholeslami, 2017b; Sheikholeslami and Sadoughi, 2017). Fk = Fkx + Fky



∂x

H

• Characteristic velocity: V =

Heatedwall





Fkx = 3wk  Z(uy sin m cos m − ux sin2 m )





Fky = 3wk  gy ˇ(T − T) + Z(ux sin m cos m − uy cos2 m ) where, Z =

 ∂T 

(16)

eky = [0.0, 0.0, 1.0, 0.0, −1.0, 1.0, 1.0, −1.0, −1.0]



= Ux dy (9)

(15)

 Ha 2 H

(17)



dY (10)

gˇ THorV =

H



Ra Pr

(11)

Governing equations with pertinent boundary conditions are solved by using in-house developed thermal lattice Boltzmann method solver, the description of which is presented in subsequent section.

The macroscopic quantities such as density, velocity, and temperature are determined as follows: = u= T=



f ; k k

1 



k

g k k

fk ek ;

(18)

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4.1.

LBM boundary conditions

Well known bounce back of non-equilibrium distribution functions (Gangawane et al., 2015c,b, 2016) is widely acknowledged in lattice Boltzmann studies. The different boundary treatments employed in this work are expressed as given below (Gangawane et al., 2015b, 2016). • Velocity conditions eq

eq

No-slip : f1,5,8 − f1,5,8 = f3,7,6 − f3,7,6

(19)

Slip : f3,6,7,nx = f3,6,7,nx−1 • Temperature conditions eq

eq

Isothermal : g1,5,8 − g1,5,8 = g3,7,6 − g3,7,6 Adiabatic : gk,n = gk,n−1



Slip :

g3,6,7,nx = −g1,8,5,nx ifUx < 0

(20)

g3,6,7,nx = g3,6,7,nx-1 ifUx > 0

Lattice Boltzmann algorithm consists of two basic steps, viz., collision (RHS of Eq. (12)) and streaming (LHS of Eq. (12)). More details of lattice Boltzmann algorithm can be found in Authors’ earlier experience (Gangawane et al., 2015b, 2016). The structure of LBM include steps such as initialization, collision, streaming, kinetic boundary treatments, estimation of macroscopic variables, followed by check of convergence. In this work, relative error criteria based upon the fundamental variables (Ux , Uy and T) is utilized for assuring convergence. It is expressed as follows:

   ς n+1 − ς n   ij ij   n+1  ≤ 10−8  ςij 

(21)

4.2. Choice of optimum parameters and code validation The influence of different uniform lattice sizes along the characteristic length on average Nusselt number value have described herein. In particular, the grid effect on Nu has been ascertained by using five uniform lattice/grid sizes (N1 :612 , N2 :812 , N3 :1012 , N4 :1212 , N5 :1412 ) for LH = 0.75;  m = 45◦ , Ra = 103 , 105 and Ha = 100. The effect of different grid sizes at Ha = 0 and otherwise similar conditions can be found in Authors previous experience (Gangawane et al., 2016). Fig. 2(a) and (b) depicts the variation of average and local Nusselt number, respectively, for various lattice sizes. The careful analysis of paltry variation can be observed after lattice size of N3 :1012 , and as per Authors’ earlier experience (Gangawane et al., 2015b, 2016) also, it is safe to carry out numerical computations with lattice size of N3 :1012 to get the accurate results for considered range of pertaining parameters. Moreover, the exactitude of the same problem under the influence of uniform magnetic field (B) is verified herein. Extra simulations are carried for the problem of partially-differentially heated square cavity (Yu et al., 2013; Kandaswamy et al., 2008; Rudraiah et al., 1995) for Gr = 2 × 104 ; 2 × 105 , Ha = 50,100 and  m = 90 for Pr = 0.71. Table 1 shows the comparison of the average Nu obtained from present LBM numerical code (uniform lattice size of 101 × 101). The

Fig. 2 – Grid independence test for different LH and given Ra, Ha and ␪m based on (a) average Nusselt number and (b) local Nusselt number.

bracketed values represent the deviation of present numerical values with literature results (Yu et al., 2013; Kandaswamy et al., 2008; Rudraiah et al., 1995). It can be seen that except for the one case (Kandaswamy et al., 2008) at higher Grashof number and Ha = 100 (where the deviation is 15.1%) else show remarkable likeness with the present results. Minimum and maximum deviations are observed to be 0.19% and 3.53% (one higher deviation of 15.1% with (Kandaswamy et al., 2008)), respectively. Additionally, the compassion with (Rudraiah et al., 1995; Mohamad et al., 2009) is shown in Fig. 3 for natural convection with magnetic field (a) and without magnetic field (b), respectively. Fig. 3(a) shows the comparison of profiles of vertical velocity component (Uy ) along vertical geometric half of cavity for Ha = 10, Gr = 2 × 104 , and Pr = 0.733 (Rudraiah et al., 1995) with magnetic field acting parallel to gravitational acceleration. While Fig. 3(b) depict the comparison of isotherm profile of Mohamad et al. (2009) for Ra = 104 and Pr = 0.71 for open ended cavity. The accuracy of present LBM code can surely be approved by analyzing Fig. 3(a,b). Slight variances detected in the results are, rather common in modeling and simulation studies. Many factors (numerical scheme, grid size, criteria for convergence, etc.) are responsible for minor discrepancies observed in Table 1 and Fig. 3. Therefore, considering all these inadvertent constituents affecting the numerical simulation results as well as the comparison shown in Table 1 and Fig. 3 assures the confidence in the reliability and accuracy of the present in-house TLBM solver, which can be used to obtain new results based on problem considered.

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chemical engineering research and design 1 2 7 ( 2 0 1 7 ) 22–34

Table 1 – Validation of present TLBM code for MHD natural convection in partially-differentially heated cavity with the few previous works at Pr = 0.71. Gr = 2 × 104

Source

Present (101 × 101) Yu et al. (2013) (129 × 129) Kandaswamy et al. (2008) (51 × 51) Rudraiah et al. (1995) (41 × 41)

Gr = 2 × 105

Ha = 50

Ha = 100

Ha = 50

Ha = 100

1.083 1.076 (0.64%) 1.099 (2.13%) 1.086 (1.18%)

1.009 1.006 (0.29%) 1.025 (1.58%) 1.011 (0.19%)

2.895 2.994 (3.41%) 2.978 (2.86%) 2.844 (1.76%)

1.413 1.463 (3.53%) 1.632 (15.5%) 1.432 (1.34%)

Fig. 3 – (a) Comparison of vertical velocity profile (Uy ) plotted at X = 0.5 with the results of Rudraiah et al. (1995); (b) comparison of isotherm profiles with that of Mohamad et al. (2009) at Ra = 104 and Pr = 0.71.

5.

Results and discussions

This research illustrates Magnetohydrodynamics (MHD) natural convection in a partially heated open ended square cavity by using in-house developed TLBM solver (in C++ programming language). The influence of three heater sizes (size, LH = 0.25, 0.5 and 0.75) on local as well as global convection physics is elucidated for range of relative parameters, such

as, Rayleigh number (Ra = 103 , 104 , 105 ), Hartmann number (Ha = 0, 25, 50, 75 and 100) and angle of magnetic field orientation ( m = 0◦ ,45◦ and 90◦ ) for Pr = 0.71. The results of impact of heater sizes as well as locations on convection characteristics in open ended cavity without magnetic field (Ha = 0) has already been covered in Authors’ previous experience (Gangawane et al., 2016). Therefore, the same has not been repeated here.

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chemical engineering research and design 1 2 7 ( 2 0 1 7 ) 22–34

Fig. 4 – Isotherms and streamlines contours for west wall completely heated (a,b) and partially heated (c,d) for given Ra, Pr = 0.71, LH = 0.5 and Ha = 0.

5.1.

Detailed convection characteristics

The influence on partial heating (LH = 0.5) on flow and temperature distribution inside open cavity is shown in Fig. 4. Isotherms and streamlines for given Ra with complete heating are depicted in Fig. 4(a,b), which shows excellent likeness with profiles given by Mohamad et al. (2009). On the other hand, the consequence of flow pertinent parameters, such as

Ha, Ra for different heater sizes (LH ) on streamline profiles are depicted in Figs. 5–7 for angle of magnetic field direction,  m = 0◦ ,45◦ , and 90◦ , respectively. As can be seen from Fig. 5 (for  m = 0), Ha and Ra remarkably shapes the flow circulation in cavity. For instance, at LH = 0.25 (heater size, 25% of height of cavity) attributes for low buoyancy force. At Ha = 50, which represents moderate strength of electromagnetic force, therefore, flow structure seems to slightly identical to cavity

chemical engineering research and design 1 2 7 ( 2 0 1 7 ) 22–34

29

Fig. 5 – Variation of streamlines in partially heated open cavity at  m = 0.

Fig. 6 – Variation of streamlines in partially heated open cavity at  m = 45. without magnetic field (Ha = 0) (Gangawane et al., 2015b, 2016). At Ha = 50 and Ra = 103 , buoyancy force is weak (as dominant is conduction over convection) resulting flow bifurcation. The flow is observed to be partitioned into two parts, one major zone (close to heated wall) and another (minor zone) near open end of cavity. If heat intensity is increased (increase in Ra to 105 ), the flow circulation takes place in clockwise pattern (fluid entering from open end, reaching heated part of cavity and then fluid exit from open end). This flow circulation is found to be limited toward the walls of cavity, thereby creating a vortex in the center of cavity. It can also be revealed that the heater size significantly shapes the flow structure in cav-

ity than other pertinent parameters. For higher heater sizes (LH = 0.5 and 0.75), buoyancy forces are found to be substantial, therefore, enabling fluid to be circulated throughout the cavity. Nearly no change in observed in streamline structures for Ha = 100. The flow splitting is found to be identical irrespective of heater size. When magnetic field direction is changed to  m = 45◦ (Fig. 6), significant changes in flow patterns are noted. In particular, the flow structure seems to be confined towards the heated part and minor vortex is formed near open end. This splitting of fluid zones creates stagnant fluid motion in the center of cavity. This is indication that magnetizing force is higher for  m = 45◦ . The shape of vortex shells is observed

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Fig. 7 – Variation of streamlines in partially heated open cavity at  m = 90. to be elongated as compared with the case of  m = 45◦ . For  m = 90◦ (Fig. 7), flow circulation inside the cavity remains more or less similar with  m = 45◦ . Comparatively fewer bifurcations can be seen for vertical magnetic field, which is indication of higher convective flow as that of  m = 45◦ . Also, very limited effect of heater size is observed on flow pattern implying weak buoyancy driven flow. Similar characteristics are revealed from the analysis of temperature profiles inside cavity (Figs. 8–10). At  m = 0◦ , isotherm profiles are shown in Fig. 8. The increase in heater size is expected to increase the strength of buoyancy force, therefore isotherms occupy most of the cavity region. At LH = 0.25, as low buoyancy force is exerted, heat transfer remains conduction dominant. At Ra = 105 and Ha = 50, the boundary layer region can be clearly visible. It stresses the fact that temperature gradient at this condition is higher, resulting in higher heat transfer rate. The augmentation in Ha causes rise in magnetizing force. It results in diminution of isotherm crowding. The dense cluster of isotherms along heated wall vanishes, implying decline in heat transfer rate. If  m is changed to 45◦ , remarkable change in isotherm patterns is observed. As compared to horizontal or vertical magnetic field, applying magnetic field at  m = 45◦ offers higher suppression of buoyancy driven flow, which is evident from more evenly spread isotherm throughout cavity. As per Fig. 8, at Ha = 0, Ra = 103 , only LH = 0.25 shown conduction dominant heat transfer; while for  m = 45◦ (Fig. 9), for all LH , heat transfer is found to be dominated by conduction. Thermal profile at  m = 90◦ (Fig. 10) is found to more or less similar to  m = 0◦ . The change in angle of magnetic field from horizontal to vertical slimly affects the flow and thermal patterns in cavity. Hence, magnetic field applied at  m = 45◦ can effectively fulfill the criterion of restriction of convective flow. Also, as expected, the impact of heater size diminishes with increase in Ha. Therefore, thermal patterns in cavity are more predominantly altered by Ha, followed by  m , Ra and lastly LH . After systematic evaluation of physical insights of system (by streamlines, isotherms), the influence of pertinent parame-

ter on heat transfer characteristics has shown in subsequent section

5.2.

Heat Transfer characteristics

Nusselt number is acknowledged as one of the significant heat transfer parameter intended to express the rate of heat transfer by convection. In this part, the dependence of average Nusselt number (Nu) with the problem pertaining parameters (Ha, Ra,  m , LH ) is presented and discussed. Fig. 11 depicts dependence of Nuwith LH for different Ha and Ra as function of  m . Insignificant changes are observed on Nu pattern for different Ha at Ra = 103 . At this Ra, for most of the cases, heat transfer is governed by conduction. Nu values remain nearly unaffected by variation of heater size. Heat transfer rate is found to be maximum at horizontal magnetic field orientation, followed by vertical and lastly at  m = 45◦ . The reason remains same as per our earlier discussion. Also, increase in magnetic field strength reduces the significance of heater size on Nusselt number. Heat transfer rate shows augmentation with heater size only at Ra = 105 (higher buoyancy force, resulting in higher fluid circulation between hot and cold walls). Therefore, in order to supress fluid flow due to convection, magnetic field can be applied at  m = 45◦ . An attempt has been made to utilize parameter called magneto-convective parameter (␭). The importance of the parameter dwells in the fact that it can compile essence of Hartmann number and Rayleigh number into a single plot. The importance of magneto-convective parameter has already been presented earlier (Yu et al., 2013). The mathematical definition of the parameter is given below:

=

Ha2 Ra

(22)

The significance of the parameter lies in the fact that it gives comparative strength of magnetic field to buoyancy driven forces. When  → 0 it indicates weak magnetic field; while it draws close to infinity ( → ∞), strong magnetic field

chemical engineering research and design 1 2 7 ( 2 0 1 7 ) 22–34

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Fig. 8 – Variation of isotherms in partially heated open cavity at  m = 0.

Fig. 9 – Variation of isotherms in partially heated open cavity at  m = 45. is pointed. Fig. 12 delineates the effect of  m on average Nusselt number with  for a given Ra. The values of  for given Ra and Ha are given in Table 2. The Nusselt number pattern is seen to be oscillating with . The higher and lower peaks of Nu are obtained due to rise of buoyancy strength (Table 2). The rate of heat transfer is higher when 0.5 ≤ ≤1. Maximum Nusselt number values are estimated for  m = 0 and LH = 0.75. Impact of heater size nearly diminishes at around  = 0.1. When magnetic field is applied at  m > 0, higher peaks are detected near

 = 1 (magnetic field strength matches buoyancy force). The patterns of Nu with  remains equivalent for  m = 45 and 90, but relatively higher heat transfer rate is observed at  m = 90 at  ≈ 10. Also, as interpreted from Fig. 12 and Table 2, the rate of heat transfer in considered geometry enhances with  only for  m . The increase in angle of magnetic field the complexity involved in the heat transfer increases. But looking at the overall heat transfer characteristics, magnetic field applied at

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Fig. 10 – Variation of isotherms in partially heated open cavity at  m = 90.

Fig. 11 – Variation of average Nusselt number for different considered parameters.

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Fig. 12 – Variation of average Nusselt number (Nu) with magneto-convection parameter ( =

References

Table 2 – Values of magneto-convective parameter ( = Ha2 /Ra) for different Ha and Ra. Ha

25 50 75 100

 Ra = 103

Ra = 104

Ra = 105

0.625 2.5 5.625 10

0.0625 0.25 0.5625 0.1

0.00625 0.025 0.05625 0.01

 m = 45 (irrespective of LH ) can be effective from heat transfer deterioration point of view.

6.

Ha2 Ra ).

Concluding remarks

Thermal lattice Boltzmann method (TLBM) is used for exploring MHD-natural convection in partially heated open ended cavity. Influence of heater sizes (LH = 0.25, 0.5 and 0.75) is investigated on flow physics due to natural convection. Numerical computations were performed for extensive range of parameters, such as Hartmann number (0 ≤ Ha ≤ 100), magnetic field orientation (0◦ ≤ ␪m ≤ 90◦ ), Rayleigh number (103 ≤ Ra ≤ 105 ,) and Prandtl number (Pr = 0.71). The accuracy of numerical solver is assured by deliberate choosing of grid size (performing thorough grid independence test) and validation with literature. The angle of magnetic field ( m ) observed to play significant role in controlling the rate of heat transfer in cavity. Higher heat transfer rates can be achieved by using magnetic field application at  m = 0◦ or 90◦ . Further, the magnetizing forces observed to have higher effect at  m = 45◦ . Average Nusselt numbers are plotted as a function of magneto-convective parameter (). This parameter can take into account the impact of both Ha and Ra into a single graph. Limited influence of LH on heat transfer rate is seen at lower Ra (≤104 ). The impact of LH on heat transfer characteristics becomes significant only at Ra = 105 . Therefore, it can be concluded that magnetic field applied at  m = 45◦ can be effective from restriction of heat transfer rate due to convection.

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