Scholarly article on topic 'Numerical investigation of pure mixed convection in a ferrofluid-filled lid-driven cavity for different heater configurations'

Numerical investigation of pure mixed convection in a ferrofluid-filled lid-driven cavity for different heater configurations Academic research paper on "Materials engineering"

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{"Mixed convection" / Ferrofluid / "Lid-driven cavity" / "MHD convection" / "Bottom heater"}

Abstract of research paper on Materials engineering, author of scientific article — Khan Md. Rabbi, Sourav Saha, Satyajit Mojumder, M.M. Rahman, R. Saidur, et al.

Abstract Mixed convection has been a center point of attraction to the heat transfer engineers for many years. Here, pure mixed convection analysis in cavity is carried out for two different geometric heater configurations under externally applied magnetic field. Ferrofluid (Fe3O4–water) is considered as working fluid and modeled as single phase fluid. The heaters at the bottom wall are kept at constant high temperature while vertical side walls are adiabatic. The top wall is moving at a constant velocity in both geometric configurations and is kept at constant low temperature. Galerkin weighted residuals method of finite element analysis is implemented to solve the governing equations. The analysis has been carried out for a wide range of Richardson number (Ri =0.1–10), Reynolds number (Re =100–500), Hartmann number (Ha =0–100) and solid volume fraction (φ =0–0.15) of ferrofluid. The overall heat transfer performance for both the configurations is quantitatively investigated by average Nusselt number at the heated boundary wall. It is observed that higher Ri enhances the heat transfer rate, although higher Ha decreases heat transfer rate. Moreover, at higher Ri and lower Ha, semi-circular notched cavity shows significantly better (more than 30%) heat transfer rate.

Academic research paper on topic "Numerical investigation of pure mixed convection in a ferrofluid-filled lid-driven cavity for different heater configurations"

Alexandria Engineering Journal (2016) 55, 127-139

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Alexandria University Alexandria Engineering Journal

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ORIGINAL ARTICLE

Numerical investigation of pure mixed convection c^Ma* in a ferrofluid-filled lid-driven cavity for different heater configurations

Khan Md. Rabbia, Sourav Sahaa, Satyajit Mojumdera, M.M. Rahman b*, R. Saidurc, Talaat A. Ibrahim d'e

a Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh b Universiti Brunei Darussalam, Mathematical and Computing Sciences Group, Faculty of Science, Brunei c Centre of Research Excellence in Renewable Energy, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

d Vice Presidency for Projects, King Saud University, P.O. 70908, 11577 Riyadh, Saudi Arabia e Mechanical Power Dept., Faculty of Engineering-Mattaria, Helwan University, Cairo 11718, Egypt

Received 3 August 2015; revised 8 December 2015; accepted 20 December 2015 Available online 12 January 2016

KEYWORDS

Mixed convection; Ferrofluid; Lid-driven cavity; MHD convection; Bottom heater

Abstract Mixed convection has been a center point of attraction to the heat transfer engineers for many years. Here, pure mixed convection analysis in cavity is carried out for two different geometric heater configurations under externally applied magnetic field. Ferrofluid (Fe3O4-water) is considered as working fluid and modeled as single phase fluid. The heaters at the bottom wall are kept at constant high temperature while vertical side walls are adiabatic. The top wall is moving at a constant velocity in both geometric configurations and is kept at constant low temperature. Galerkin weighted residuals method of finite element analysis is implemented to solve the governing equations. The analysis has been carried out for a wide range of Richardson number (Ri = 0.1-10), Reynolds number (Re = 100-500), Hartmann number (Ha = 0-100) and solid volume fraction (u = 0-0.15) of ferrofluid. The overall heat transfer performance for both the configurations is quantitatively investigated by average Nusselt number at the heated boundary wall. It is observed that higher Ri enhances the heat transfer rate, although higher Ha decreases heat transfer rate. Moreover, at higher Ri and lower Ha, semi-circular notched cavity shows significantly better (more than 30%) heat transfer rate.

© 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

* Corresponding author. Tel.: +673 2463001x1355; fax: +673 2461502.

E-mail address: mustafizur.rahman@ubd.edu.bn (M.M. Rahman). Peer review under responsibility of Faculty of Engineering, Alexandria University.

1. Introduction

Magneto-hydrodynamic (MHD) convection is the heat transfer mechanism in fluid under external magnetic field. This topic has turned out to be a focal point of enormous amount of

http://dx.doi.org/10.1016/j.aej.2015.12.021

1110-0168 © 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

a notch spacing length (m)

g acceleration due to gravity (m s~2) Greek symbols

k thermal conductivity of fluid (W m_1 K_1) a thermal diffusivity (m2 s_1)

L length of the square cavity (m) ß volume expansion coefficient (K_1)

Nu average Nusselt number on heater surface u solid volume fraction of ferrofluid

p pressure (Pa) i dynamic viscosity (Ns m~2)

P dimensionless pressure V kinematic viscosity (m2 s_1)

Pr Prandtl number P density (kg m~3)

Ha Hartmann number r electrical conductivity ((X m)_1)

Re Reynolds number w stream function

Gr Grashof number 0 dimensionless temperature

Ri Richardson number

s heater wall perimeter Subscripts

T temperature (K) h hot

u velocity at x-direction (m s_1) c cold

U dimensionless velocity at x-direction f base fluid (water)

v velocity at y-direction (m s_1) ff ferrofluid

V dimensionless velocity at y-direction p ferrous particle

X dimensionless distance along x-coordinate

Y dimensionless distance along y-coordinate

researches from the last three decades because of its wide range of application in petroleum industries, plasma physics, geophysics, MHD pump, MHD flow meter, cooling of nuclear reactors and various thermal systems [1-3].

Previously, several investigations on natural, forced and mixed convection have been performed under external magnetic field. It was reported by many researchers that natural convection can be affected by external magnetic field [4-7]. Rudraiah et al. [8] investigated the effect of magnetic field on natural convection in a rectangular enclosure numerically and presented a correlation formula of Nusselt number. They found that the effect of the magnetic field decreased the rate of heat transfer at any Grashof number. Similar studies have been performed for different geometries such as square enclosure [9,10], semi-annulus enclosure [11,12], and trapezoidal enclosure [13]. It is also noticeable that many researchers investigated MHD natural convection using nanoparticles improvised in a base fluid [14-20]. Mahmoudia et al. [21] investigated MHD natural convection and entropy generation in a trapezoidal enclosure using Cu-water nanofluid. They found that enhancement of Nusselt number occurred with increasing Hartmann number at Ra = 104 and 105, but at even higher Rayleigh number, opposite occurred. Moreover, fer-rofluid also has become very popular while achieving higher heat transfer rate [22,23]. Recently, few researchers investigated natural convection in different geometric configurations using ferrofluids [24-26]. Gavili et al. [27] investigated thermo-magnetic convection process in two-dimensional enclosure filled with ferrofluid. It was concluded that average Nusselt number increased when magnetic field and temperature gradient had same direction. Lajvardi et al. [28] carried out experimental study on natural convection using ferrofluid under external magnetic field. It was reported that significant enhancement of heat transfer was present at lower strength of magnetic field. Bondareva et al. [29] investigated the effect

of uniform magnetic field on laminar flow regimes of natural convection in an enclosure. It was observed that Rayleigh number, Hartmann number and orientation of magnetic field had impact on velocity distribution, temperature field and average Nusselt number.

Many researchers tried to figure out how external magnetic field affected heat transfer rate while analyzing mixed convection [30-32]. Oztop et al. [33] studied MHD mixed convection in an open channel with a fully or partially heated cavity. It was evident that higher value of Ha did not lead to any change in flow field for any length of heater. It was also reported that at any values of Richardson number, higher amount of nanoparticles increased heat transfer rate. Some related investigations for different cavities and boundary conditions can be found in [34-36]. Many researchers reported that nanofluids and ferrofluids had significant influence on mixed convection [37,38]. Ahmed et al. [39] investigated MHD mixed convection of a lid-driven cavity filled with nanofluid. It was investigated that the existence of an inclined magnetic field decreased the fluid movement and hence decreased the heat transfer rate. Sheremet et al. [40] studied mixed convection in a lid-driven square cavity with nanofluid inside. It was evident that Prandtl, Reynolds number, Grashof number, Lewis number, buoyancy-ratio, the Brownian motion, and the thermophoresis had effect on flow and heat transfer. Aminfar et al. [41] investigated numerically the hydro-thermal characteristics of a fer-rofluid in a vertical rectangular duct which was exposed to a non-uniform transverse magnetic field.

The present study is concerned with the comparison of heat transfer performance between semi-circular and triangular notches in lid-driven cavities. The purpose of this investigation is to study heater geometry effect on pure mixed convection heat transfer. Ferrofluid has been considered as the working fluid. The flow and thermal field have been studied and discussed by streamline and isothermal contours respectively.

Figure 1 Schematic diagram of (a) triangular and (b) semi-circular notched cavity.

2D plots are included to show the effect of different pertinent parameters such as Hartmann number (Ha), Reynolds number (Re), Grashof number (Gr), and Richardson number (Ri) for different heater geometries. Finally the consequence of varying solid volume fraction (u) of ferrofluid has been illustrated and discussed.

2. Problem formulation

2.1. Physical model

The geometries of two models are shown in Fig. 1 with appropriate dimensions and boundary conditions. The models considered here are two dimensional square enclosures with sides of length L and non-dimensional notch spacing of a/L = 0.5. Both enclosures, one with triangular notch and the other with semi-circular notch, are filled with Fe3O4-water ferrofluid. The horizontal top walls are moving with a constant velocity of Uo in both the cases. Both the notches (triangular and semi-circular) are kept at high temperature of Th whereas the opposite top walls are kept at low temperature of Tc. The left and right side walls and also the bottom walls except notch surfaces are kept insulated in both enclosures. Moreover, a uniform magnetic field is applied horizontally normal to the vertical wall. Gravity is assumed to act along the negative y-direction. The radiation, pressure work and viscous dissipation are assumed to be negligible. Fe3O4 nanoparticles are assumed to have a uniform shape and size. It is also assumed that the nanoparticles and water are in thermal equilibrium, and the ferrofluid is Newtonian and incompressible. For mathematical modeling, the fluid is assumed to be of single phase continuum. The flow is considered to be steady, two dimensional and laminar. The displacement currents, induced magnetic field, viscous dissipation and radiation heat transfer are also neglected.

2.2. Mathematical modeling 2.2.1. Governing equation

The governing equations of the problem specification are conservation of mass, momentum and energy equations. Boussi-

nesq approximation is assumed to hold true for buoyancy effect. Under these assumptions, the governing equations for the steady, two-dimensional, laminar and incompressible mixed convection flow are expressed as follows:

dx dy '

du dv dx dy

1 dp / d2u d21 Pff dx ^ dx2 ^ dy2

dv dv 1 dp id v d v\ „ .„ „, rtt _2

u +v dydy+f @x2+w)+ fg(T - Tc)-fBvv

de de _ id2в d2&\

dx dy 4\dx2 dy2 )

The effective thermo-physical properties of ferrofluids are defined by using the following formula:

The effective density of ferrofluid is obtained by,

Pff = (1 - U)Pf + UPp-

Under the thermal equilibrium conditions, the specific heat of ferrofluid is given as,

{pCp)ff — (1 - u){pCp)f + u{pCp)p. (6)

Electrical conductivity of ferrofluid [43] is given as follows

°//—(1 - U)rf + Urp- (7)

Thermal expansion coefficient of ferrofluid is given by,

(ßp)ff =(1 - v)(ßp)bf + u(Pb)p- (8)

The dynamic viscosity of ferrofluid is expressed by Brink-man model [44],

Iff = 1f(1 - u)-

The thermal conductivity of the ferrofluid is obtained by Maxwell-Garnett model [45] and stated as

kff — kf

Г(кр 4 -2kf)- 2u(kf - kp)'

l_ (kp 4 2kf) 4 u(kf - kp)

Eqs. (1)-(4) can be converted to the dimensionless form by using the following dimensionless parameters:

Table 1 Present boundary condition in the nondimensional form.

Boundary

Flow field

Top wall Left vertical wall Right vertical wall Bottom wall

0 6 X 6 0.25; 0.75 6 X 6 1 Triangular notched wall Semi-circular notched wall

U =1, V = 0 U = V =0 U = V =0 U = V =0

U= V= U= V=

Thermal field

в — 0

M-0 gx — 0

gx — 0

ge _ 0 @y — 0

в—1 в1

Table 2 Thermo-physical properties of Fe3O4 (nanoparticles) and water (base fluid) [42].

Property Fe3O4 Water (base

(nanoparticles) fluid)

Heat capacitance 670 4179

(JKg-'K-1)

Density (Kg m~3) 5200 997.1

Thermal conductivity 6 0.613

(Wm_1 K~')

Thermal expansion coefficient 1.18 x 10-5 2.1 x 10-4

Dynamic viscosity (Ns m 2) - 0.001003

x y u v

X — -, Y — У, U — -, V — -,

L L U0 U0

2 ' PnpU2

Th — Tc

Gr —

gßf(Th - Tc)L3

Ha — B0H

Pr — V, a—

Ri —

Therefore using the above parameters leads to dimension-less forms of the governing equations as below:

@U 8_V_0

gx+@y—0; @u au gp 1 Vff (g2u g2u\

U--h V— =---1---—I--1--

gx @y gx Re vf \gx2 gy2)'

T,gV+VfiV- gP+ 1 Vff (g2V gV\ ßff Hê V

ge ge 1 a.ff (g2 в g2e\

U--h V-=--—\--1--

gx^ gy RePr a.f dY2)'

Local Nusselt number can be expressed as,

_ kff ge Nu — -Т—^Т ■

дв dY

The characteristics of heat transfer are obtained by average Nusselt number and can be expressed as

Nu — - Nuds. s Jq

Here, s signifies the circumference. For triangular notch, L

s — —¡= 2

and for semi-circular notch, L

s — p 2

The stream function can be given by,

вф gy '

gx ■

2.2.2. Boundary conditions

Boundary conditions illustrated in Fig. 1 can be written in dimensionless forms and are presented in Table 1. Both flow and thermal boundary conditions have been shown in tabular form at top wall, vertical walls, bottom wall and notch (heater) wall.

2.3. Properties of ferrofluid

As described in the introduction section, the present investigation has considered ferrofluid (Fe3O4-water) as the working fluid. The thermo-physical properties of nanoparticle (Fe3O4) and base fluid (water) have been summarized in Table 2. These properties have been considered constant except for density variation according to Boussinesq approximation.

3. Numerical procedure

The entire domain is fragmented into non-overlapping triangular mesh elements of different sizes. To solve the system along with specified boundary conditions, Galerkin weighted residual method of finite element analysis has been employed. Non-dimensional continuity, momentum and energy equations are converted into set of algebraic equations. P2-P1 Lagrange finite elements are used to discretize pressure and velocity components and Lagrange-quadratic finite elements are chosen for temperature. Iterative procedure is used to find the solution until the convergence occurs. The criterion of convergence is set to 10~6, so that |zp+1 - zp| 6 10~6, where z is the general dependent variable and p is the number of iteration. When the difference exceeds the criterion, iteration terminates.

3.1. Grid independency test

A grid independency test has been carried out to find numerical accuracy while solving the system. The test is done for Gr = 104, Ri = 0.1, Ha = 25 and u = 0.15. Fig. 2 illustrates the effect of mesh elements on mid-plane velocity profile (U) for both triangular and semi-circular notched cavities. From Fig. 2(a) it is evident that the velocity profile almost coincides for mesh elements of 2178 and 6112. Again Fig. 2(b) illustrates that mid-plane velocity profile for 2072 mesh elements almost matches with the same for mesh elements 5992. Therefore, mesh elements of 6112 and 5992 have been selected to carry out the numerical simulation for triangular and semi-circular notched cavities respectively.

0.75 -

0.75 -

0.50 -

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 U

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 U

Figure 2 Mid-plane velocity profile for different element numbers for (a) triangular and (b) semi-circular notched cavity at Gr = 10 Ri = 0.1, Ha = 25 and u = 0.15.

Streamline

Isotherm

o 13 O

<u m <u

Figure 3 Comparison of streamline and isotherms for Ra = 105, Re = 100 and Ha = 50 between numerical investigation by Oztop et al. [46] and present study.

3.2. Code validation

The present code is validated with the previous investigation by Oztop et al. [46] for lid-driven cavity with corner heater under external magnetic field. Validation is established through streamlines and isotherm contours. Comparison of

the flow and thermal field is illustrated in Fig. 3 for Ra = 105, Re = 100, Ha = 50. From the figure it is significant that the present code conforms to previous study and a good comparison is registered. According to this, the present numerical code and solution procedure are perfectly reliable. Hence, numerical simulation can be carried out for the given study.

Ha =10 Ha = 25 Ha = 50

u = 0 (—) and 0.15 (- - -).

4. Result and discussion

Primary objective of this numerical investigation is to compare convective heat transfer scenario between a triangular and a semi-circular notched cavity filled with Fe3O4-water ferrofluid in the presence of external magnetic field. Results from numerical simulation are presented in terms of streamlines, isotherms and average Nusselt number distributions at the heated boundary for various values of Richardson number (Ri), Hartmann number (Ha), and nanoparticle volume fraction (u). Ri measures relative importance of the natural and forced convection. By varying its numerical value (0.1 6 Ri 6 10), mode of convection heat transfer is altered from forced to mixed to natural. Ha represents strength of external magnetic field as body force. To observe magnetic response of flow, Ha is varied from 0 to 100.

Particular emphasis is put on observing pure mixed convection regime of heat transfer (Ri = 1). To compare effectiveness of ferrofluid solid volume fraction of u = 0 (pure water) and of u = 0.15 (ferrofluid) is selected for comparison of heat transfer scenario.

4.1. Effect of Hartmann number

Effect of Hartmann number on flow and thermal fields, for both ferrofluid and pure water, is depicted in terms of stream function and isothermal contours in Figs. 4 and 5 for triangular and semi-circular notched cavities respectively. Pure mixed convective heat transport is selected as representative case. For Ha = 10, a clockwise vortex is observed in the cavity and a secondary vortex is observed near the left bottom end with opposite sense of rotation. The pattern is similar for both types of cavity and independent of fluid type. Increment of Ha to 25 generates two more secondary vortices around notch surface due to the intense effect of magnetic field. However, the intensity of these cells is negligible when compared with the main circulation cells. Physically, high strength of magnetic field tries to counter-balance the impact of forced convection. Thus at high Ha, natural convection becomes marginally more dominant compared to forced flow. Secondary vortices are the results of secondary convection (natural mostly) inside the cavity. When Ha = 50, the streamlines are stretched more, horizontally, with lower strength and the secondary circulations cover the notch surface. Streamlines also cluster around the

Ha =10 Ha = 25 Ha = 50

Figure 5 Effect of Hartmann number (Ha) on isothermal lines for (a-c) triangular and (d-f) semi-circular notched cavity at Ri = 1, U = 0 (—) and 0.15 (- - -).

top surface. It is noticed that with the increase of Hartmann number, value of stream function decreases. As increasing Hartmann number induces stronger magnetic field, which in turn produces Lorentz force in flow field, it decreases the strength of flow (i.e. weak convection current) in the cavity. The secondary circulation for ferrofluid (u = 0.15) stretches more as it is more susceptible to magnetic field than pure water (u = 0) in case of Ha = 25 and 50.

Stream function values in semi-circular notched cavity are higher than the triangular one for high Hartmann numbers. This denotes superiority of convection heat transfer in semicircular notched cavity than that in triangular one.

Impact of Hartmann number on thermal field is represented by isotherm contours (see Fig. 5) for pure water (solid line) and ferrofluid (dotted line) in triangular and semi-circular notched cavities respectively.

For Ha =10, approximately parallel isotherms are densely distributed near the right half of heated notch due to the formation of thermal boundary layer (i.e. weak convection). Irregular, distorted and less dense isotherms evolve with the approach from right to left half due to the increasing convec-tive heat transfer. Higher Hartmann numbers, Ha = 25 and 50, show more parallel isotherms throughout the cavity. These

horizontal isotherm contours indicate weaker convection and domination of conduction heat transfer. It is observed that with the increase of Ha, the isotherm contours become more horizontal for ferrofluid compared to pure water as conduction is dominant in ferrofluid than pure water.

As the Hartmann number increases, the effect of magnetic field enhances. Thus, the more the Ha, the more the Lorentz force dominates on the buoyancy force. Therefore, the strength of flow and thermal field varies with the strength of magnetic field.

4.2. Effect of Richardson number

Figs. 6 and 7 show evolution of streamline and isotherms with Ri in triangular and semi-circular notched cavities respectively at Ha = 25. To vary Ri, Grashof number is kept fixed while changing Reynolds number through the lid velocity uo. Ri value varies from 0.1 to 10 describing the transition between forced and natural convection analysis.

For Ri = 0.1 one primary clockwise vortex and two secondary vortices on either side of the notch appear for triangular and semi-circular notched cavities. Forced convection,

Ri = 0.1 Ri =1.0 Ri =10

(—) and 0.15 (- - -).

which begets from inertia force due to driven velocity uo, is dominant in this case. Further increase of Richardson number (i.e. Ri = 1 and 10) gives augmented vortices. Both CW vortices merge together and cover larger area of the cavity. With the increase of Ri value, vortices absorb more energy from heat source and become enlarged. Streamlines spread throughout the cavity with higher stream function value due to higher buoyancy force at high Ri. This indicates stronger convection at higher Ri.

It is interesting to observe that for ferrofluid both the CW and CCW vortices spread more compared to pure water. Indeed faster heat transfer occurs for ferrofluid. It causes enhancement of the buoyancy force and hence the flow intensity. It is also noticed from analysis that the streamlines of main cavity spread much in semi-circular notched cavity compared to triangular one at higher strength. This indicates higher convection effect in semi-circular than in triangular notched cavity.

Fig. 7 shows that at Ri = 0.1, isotherms are almost uniformly distributed near the center of cavity which points to dominance of conduction heat transfer. It is noticed that isotherms are bunched near the heat source and stretched toward

right and left adiabatic walls. This pattern emerges due to higher temperature gradient near the source and decreasing temperature gradient away from source. For Ri = 1, the solid isotherm contours (u = 0) become distorted. However, the dotted isotherm contours (u = 0.15) are still almost uniform. Further increase of Ri to 10 presents completely distorted isotherm patterns because of better convection. Isotherms seem to follow the fluid flow path as we see that those are more densely packed on left of heated notches.

Comparison of isotherms for ferrofluid and pure water reveals that the temperature of ferrofluid is lower than that of pure water at the vicinity of heat source and in right half of cavity, but in left half of cavity lower temperature is obtained for pure fluid. This is due to the presence of nanopar-ticles of ferromagnetic materials enhances interaction among recirculation zones.

Typically, the natural convection is negligible when Ri < 0.1, forced convection is negligible when Ri >10, and neither is negligible when 0.1 < Ri < 10. Therefore, 0.1 < Ri < 10 is used to describe the effect of both natural convection and forced convection on the flow and thermal field.

Ri = 0.1

Ri = 1.0

Ri = 10

Figure 7 Effect of Richardson number (Ri) on isothermal lines for (a-c) triangular and (d-f) semi-circular notched cavity at Ha = 25, U = 0 (—) and 0.15 (- - -).

4.3. Impact of Ri on mid-plane velocity

Fig. 8 depicts effect of Ri on mid-plane x-velocity at different Ha (10, 25 and 50) for triangular ((a), (c), and (e)) and semicircular ((b), (d), and (f)) notched cavities at Gr = 104 and U = 0.15. As we see, increment of Ha, lessens the x-velocity in lower half of the cavity while raising it on upper half. Same pattern is observed for both kinds of notched cavities. Flow behavior is entirely responsible for such velocity distribution. Moreover, low Ri gives rise to overall high velocity of flow inside the cavity.

4.4. Combined effect of Ha, Re and Ri on average Nusselt number for ferrofluid (u = 0.15)

Average Nusselt number (Nu), as a function Ha and Re for fer-rofluid (u = 0.15) at Gr = 103 and Gr = 104, is analyzed in Figs. 9 and 10 for triangular ((a), (c)) and semi-circular ((b), (d)) notched cavities respectively.

We see from Fig. 9(a)-(d) Nu decreases with the increment of Ha. It conforms to the previous observation where it was

found that Ha had an adverse effect on the flow strength inside the cavity. Nu decreases rapidly up to Ha = 20. At higher value of Hartmann number (Ha P 40), value of Nu becomes almost constant. At higher Reynolds number (Re « 500), initial average Nusselt number is high (Nu « 10) which decreases sharply up to Ha = 20. But at low Reynolds number (Re « 100), initial average Nusselt number is low (Nu « 5) which further decreases with Ha. Interestingly pattern of variation is quite similar for both types of cavities and both values of Gr. Since taking only two values of Gr cannot be deemed as conclusive, we can only comment on insensitivity of geometry of notch on flow physics but not for increment of heat transfer rate. This remark is backed by streamline plots previously shown.

To observe impact of Ri further analysis is done by varying its value at different Ha at Gr = 104 and u = 0.15 (see Fig. 10 (a) and (b)). It is observed that increasing Ri actually aids heat transfer. Therefore, in case of convection in notched cavity, lid velocity should be such that entire heat transfer process remains more or less natural convective. Interestingly, forced convection, which generally is supposed to augment heat transfer rate, proves to be futile for the present case.

0.75 -

О >-II

°с 0.50

0.75 -

0.50 -

Figure 8 Effect of Richardson number (Ri) on mid-plane x-velocity for (a, c, e) triangular and (b, d, f) semi-circular notched cavity at Gr = 104 and u = 0.15 .

4.5. Effect of solid volume fraction (u) on average Nusselt number

Fig. 11 depicts the variation of average Nusselt number with the increment of solid volume fraction of ferrofluid. It is significant that Nu increases with u no matter what strength the external magnetic field possess. In case of triangular notched

cavity, Fig. 11(a) represents that better heat transfer rate (more than 14%) is obtained (Nu « 15) at lowest strength of magnetic field (Ha = 10). Moreover, Fig. 11(b) illustrates the same for semi-circular shaped heater geometry. It is observed that more than 30% of heat transfer rate can be achieved at Ha = 10 with ferrofluid inside (u = 0.15). This phenomenon is evident mainly because of the augmentation of thermal

Figure 9 Variation of average Nusselt number (Nu) with Hartmann number (Ha) and Reynolds number (Re) for (a, c) triangular and (b, d) semi-circular notched cavity at u = 0.15.

Figure 10 Variation of average Nusselt number (Nu) with Richardson number (Ri) for (a) triangular and (b) semi-circular notched cavity at Gr = 104 and u = 0.15.

-a- Ha = 10

-A- Ha = 25

- Ha = 50

0.00 0.05 0.10

0.05 0.1

Figure 11 Variation of average Nusselt number (Nu) with solid volume fraction (u) for (a) triangular and (b) semi-circular notched cavity at Gr = 104 and Ri = 1.

conductivity and energy transfer with the increment of solid concentration. Increment of flow intensity is also obtained with higher solid volume fraction. As the solid volume fraction is increased, micro-convection becomes more prominent which contributes to overall heat transfer rate.

Thus it can be concluded that implementation of ferrous particles on water in an enclosure of any heater geometry suddenly enhances the heat transfer while analyzing pure mixed convection (Ri = 1).

5. Conclusions

In present analysis, both geometric and physical behaviors of mixed convection in a square notched cavity under magnetic field are highlighted in brief manner. This study reveals some key aspects of such convective flow. It is revealed that Ha has negative impact on convection. This conclusion is absolute and is not dependent on notch geometry or convection regime. Ferrofluid is superior in terms of heat transfer augmentation to pure water in all cases of analyzed scenario. Interestingly, in notch induced convection, natural convection proved to be more conducive mode of heat transfer compared to forced convection. Impact of Ha is most severe among other factors. So by tuning Ha as key parameter, heat transfer rate can be controlled. Finally, it is concluded that semi-circular notched cavity seems to produce higher value of Nusselt number compared to triangular notched cavity. This is expected since, semi-circular heater has more area for heat transfer for identical geometry. In fact, heat transfer increases up to 14% and 30% due to the implementation of ferrofluid for triangular and semi-circular notched cavities respectively. Present analysis can be taken ahead further by implementing experimental measures to observe the actual scenario.

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