Natural convection from a discrete heater in enclosures filled with a micropolar fluid

International Journal of Engineering Science 43 (2005) 1409–1418

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Natural convection from a discrete heater in enclosuresfilled with a micropolar fluid

Orhan Aydin a,*, Ioan Pop b

a Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkeyb Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

Received 15 April 2005; accepted 4 June 2005

Abstract

In this study, we numerically investigate the steady laminar natural convective flow and heat transfer of micropolar flu-ids in enclosures with a centrally located discrete heater in one of its sidewalls by applying a finite difference method. Theother sidewall is kept at isothermal conditions, while horizontal walls are assumed to be insulated. Computations are car-ried out to investigate effects of the dimensionless heater length, the material parameter of the micropolar fluid, the Ray-leigh number and the Prandtl number both for weak and strong concentration cases. Local results are presented in theform of streamline and isotherm plots as well as the variation of the local Nusselt number through the discrete heater.It was shown that micropolar fluids presented lower heat transfer values than those of the Newtonian fluids. An increaseat the material parameter, K is shown to decrease the heat transfer. The results for K = 0, which corresponds to the New-tonian fluid case is compared with those available in the existing literature and, an excellent agreement is obtained.� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

Micropolar fluids have been receiving a great deal of research focus and interest due to their application in anumber of processes that occur in industry. Such applications include the extrusion of polymer fluids, solid-ification of liquid crystals, cooling of a metallic plate in a bath, animal bloods, exotic lubricants and colloidaland suspension solutions, for example, for which the classical Navier–Stokes theory is inadequate. The theoryof micropolar fluids is first proposed by Eringen [1,2].

In many engineering applications and naturally occurring processes, natural convection plays an importantrole as a dominating mechanism. Besides its importance in such processes, due to the coupling of fluid flowand energy transport, the phenomenon of natural convection remains an interesting field of investigation. Thisfact is reflected by numerous studies in the existing literature dedicated to this topic during the past few dec-ades. Some excellent comprehensive review articles on this subject are given by Ostrach [3], Yang [4], and

0020-7225/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijengsci.2005.06.005

* Corresponding author. Tel.: +90 462 377 3182/2974; fax: +90 462 325 5526/3205.E-mail address: [email protected] (O. Aydin).

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1410 O. Aydin, I. Pop / International Journal of Engineering Science 43 (2005) 1409–1418

Fusegi and Hyun [5]. Most of the studies on natural convection in enclosures have focused on either a hori-zontally or vertically imposed temperature difference. However, departures from this basic situation are oftenencountered. Localized heating or heating by a discrete heater in enclosures can be given as an example. Aydinand Yang [6] reviewed natural convection studies in enclosures with localized heating. In their study, theynumerically analyzed natural convection of air in a two-dimensional, rectangular enclosure with localizedheating below and symmetrical cooling from the sides. Al-Bahi et al. [7] studied the laminar natural convectionheat transfer in an air filled square cavity differentially heated with a single isoflux discrete heater on one sidewall. In a recent study, Saeid and Pop [8] numerically studied natural convection in a two-dimensional cavityfilled with a porous medium, when one of its vertical walls was differentially heated by isoflux or an isothermaldiscrete heater.

Most of the previous studies on natural convection in enclosures have been related to Newtonian fluids.Despite the importance of the micropolar fluids mentioned above, there are only a few research efforts on nat-ural convection of these fluids in enclosures. Hsu and Chen [9] numerically investigated the Rayleigh–Benardconvection of a micropolar fluid in an enclosure using the cubic spline collocation method. They performedparametric studies on the effects of microstructure of heat and fluid flow. They disclosed that the heat transferrate of micropolar fluids was smaller than that of the Newtonian fluid. Natural convection of micropolar fluidsin a completely partitioned enclosure heated from below was investigated by Hsu and Tsai [10]. In anotherstudy, Hsu et al. [11] studied natural convection of micropolar fluids in a tilting enclosure equipped with asingle or multiple uniform heat sources.

The aim of the present article is to numerically study natural convective heat transfer of micropolar fluids inan enclosure having an isoflux discrete heater on one of its sidewalls. The effects of main governing parameterson the transport phenomena are studied and also compared with those for Newtonian fluids.

2. Analysis

2.1. Mathematical formulation

Consider the unsteady natural convection flow in a square cavity of the height H filled with a micropolarfluid, as depicted in Fig. 1, where the coordinates �x and �y are chosen such that �x measures the distance alongthe bottom horizontal wall, while �y measures the distance along the left vertical wall, respectively. The size ofthe cavity in the z-direction is assumed to be infinitely long. The left-hand sidewall has a centrally located dis-crete heater with the length of l which is assumed to dissipate a constant heat flux, qw. The right-hand side isisothermally cooled at a constant temperature, Tc, while the left-hand side wall, except for the heater section,

Fig. 1. The geometry of the problem.

O. Aydin, I. Pop / International Journal of Engineering Science 43 (2005) 1409–1418 1411

and the horizontal (i.e. upper and lower) surfaces are considered to be adiabatic. The micropolar fluid isassumed to be incompressible with constant properties except the density in the buoyancy term (i.e. The Bous-sinesq approximation is invoked). Under these assumptions, the basic unsteady equations of motion andenergy are

o�uo�x

þ o�vo�y

¼ 0 ð1Þ

qo�uo�t

þ �uo�uo�x

þ �vo�uo�y

� �¼ � o�p

o�xþ ðlþ jÞ o2�u

o�x2þ o2�u

o�y2

� �þ j

oNo�y

ð2Þ

qo�vo�t

þ �uo�vo�x

þ �vo�uo�y

� �¼ � o�p

o�yþ ðlþ jÞ o

2�vo�x2

þ o2�v

o�y2

� �� j

oNo�x

þ qgbðT � T 0Þ ð3Þ

qjoNo�t

þ �uoNo�x

þ �voNo�y

� �¼ c

o2N

o2�xþ o2N

o2�y

� �� 2jN þ j

o�vo�x

� o�uo�y

� �ð4Þ

oTo�t

þ �uoTo�x

þ �voTo�y

¼ ao2To�x2

þ o2To�y2

� �ð5Þ

subject to initial and boundary conditions

�t < 0 : �u ¼ �v ¼ 0; N ¼ 0; T ¼ T 0 any 0 6 �x; �y 6 L�t P 0 :

�u ¼ �v ¼ 0; N ¼ �no�uo�x

;oTo�x

¼ 0 on �x ¼ 0; 0 6 �y <H � l2

andH þ l2

< �y 6 H

�u ¼ �v ¼ 0; N ¼ �no�uo�x

; qw ¼ cons. on �x ¼ 0;H � l2

6 �y 6H þ l2

ð6Þ

�u ¼ �v ¼ 0; N ¼ �no�uo�x

; T ¼ T c on �x ¼ L; 0 6 �y 6 L

�u ¼ �v ¼ 0; N ¼ �no�vo�y

;oTo�y

¼ 0 on �y ¼ 0 and �y ¼ L; 0 6 �x 6 L

where �u and �v are the velocity components along �x and �y axes, T is the fluid temperature, N is the componentof the microrotation vector normal to the �x–�y plane, �t is the time, g is the magnitude of the acceleration due togravity, q is the density, l is the dynamic viscosity, j is the vortex viscosity, c is the spin-gradient viscosity, j isthe microinertia density and n is a constant, 0 6 n 6 1. It should be mentioned that the case n = 0, calledstrong concentration of microelements [12], indicates N = 0 near the walls. It represents concentrated particleflows in which the microelements close to the wall surface are unable to rotate [13]. The case n = 1/2, on theother hand, indicates the vanishing of anti-symmetric part of the stress tensor and denotes weak concentration[13]. The case n = 1, as suggested by Peddieson [14] is used for the modeling of turbulent boundary layer flows.Further, we shall assume that c has the following form as proposed by Ahmadi [15] and used by Rees and Pop[16] for the problem of free convection boundary layer flow over a vertical flat plate embedded in a micropolarfluid

c ¼ lþ j2

� �j ¼ l 1þ K

2

� �j ð7Þ

where K is called the material parameter.We eliminate now the pressure terms from Eqs. (2) and (3), and introduce the following non-dimensional

variables

x ¼ �x=L; y ¼ �y=L; t ¼ ðt=L2Þ�t; h ¼ ðT � T cÞ=ðqwH=kÞN ¼ ðL2=tÞN ; x ¼ ðL2=tÞ�x

ð8Þ

where t is the kinematic viscosity and �x is the vorticity function. Substituting (8) into Eqs. (1)–(5), we get


o2wox2

þ o2woy2

¼ �x ð9Þ

oxot

þ uoxox

þ voxoy

¼ ð1þ KÞ o2xox2

þ o2xoy2

� �� K

o2Noy2

þ o2Nox2

� �þ Gr

ohox

ð10Þ

oNot

þ uoNox

þ voNoy

¼ 1þ K2

� �o2Nox2

þ o2Noy2

� �� 2KN þ K

ovox

� ouoy

� �ð11Þ

ohot

þ uohox

þ vohoy

¼ 1

Pro2hox2

þ o2hoy2

� �ð12Þ

where Pr is the Prandtl number and w is the non-dimensional stream function which is defined in the usualway as

u ¼ owoy

; v ¼ � owox

ð13Þ

The initial and boundary conditions (6) become

t < 0 : u ¼ v ¼ 0;N ¼ 0; h ¼ 0 any 0 6 x; y 6 1

t P 0 :

u ¼ v ¼ 0; N ¼ �nouox

;ohox

¼ 0 on x ¼ 0; 0 6 y <1� e2

and1þ e2

< y 6 1

u ¼ v ¼ 0; N ¼ �nouox

;ohox

¼ �1 on x ¼ 0;1� e2

6 y 61þ e2

ð14Þ

u ¼ v ¼ 0; N ¼ �nouox

; h ¼ �0:5 on x ¼ 1; 0 6 x 6 1

u ¼ v ¼ 0; N ¼ �novoy

;ohoy

¼ 0 on y ¼ 0 and y ¼ 1; 0 6 x 6 1

where e = l/H. One quantity of physical interest is the local, Nu given by

Nu ¼ 1

hjx¼0

ð15Þ

2.2. Method of solution

The systems of coupled differential equations under the boundary conditions described above are numer-ically solved using the finite difference method. The vorticity transport, microrotation and energy equationsare solved using the ADI (alternating direction implicit) method and the stream function equation is solvedby the successive over-relaxation (SOR) method. The over-relaxation parameter is chosen to be 1.8 for streamfunction solutions. In order to avoid divergence in the solution of the energy, microrotation and vorticityequations, an under-relaxation parameter of 0.5 is employed. The hybrid differencing is used with the convec-tive terms, while diffusive and buoyancy terms are discretized by employing the central differences. First-order-accurate forward differences are used with the time derivative. The following criterion is employed to check forthe steady state solution
P
i;jj/nþ1i;j � /n

i;jjPi;jj/

ni;jj

6 ERMAX ð16Þ

where / stands for w, x, N or h; n refers to time; and i and j refer to space coordinates. The value of ERMAXis chosen as 10�7. Convergence of iterations for the stream function solution is obtained at each time step. Thetime step used in the computations is varied between 10�5 and 4 · 10�3, depending on the value of Ra and thegrid size. All the computations are carried out on a PC. A non-uniform grid structure is employed, which isconstructed using finer grid spacings near the walls and the mixed boundary points at the edges of the discrete


heater. The grid structure is changed for each e because of the shifted positions of the mixed boundary points.For each case, a grid refinement study is conducted and finally the grid structure on which finer grid refine-ment does not have a significant effect on the results is chosen. As an example, for e = 1/5 at Ra = 106, itis observed that increasing the grid size from 41 · 41 to 61 · 61 changed wmax and Nu less than 0.1 percent.Thus, 41 · 41 grid size is approved to be sufficient to resolve the velocity and temperature fields for the relatedcase.

2.3. On the validity of the computer code developed

The validity of the computer code developed has been already verified for the problem of natural convec-tion in a square cavity having differentially heated vertical walls [17]. Here, for the local heating case, we com-pare our results with those by Chadwick et al. [18] who studied natural convection in a discretely heatedenclosure experimentally and theoretically for single and multiple heater configurations. We simulated theChadwick et al. [18]�s case of a single heater configuration. For an aspect ratio of 5, s/H = 0.5 and forGr = 5.16 · 105, the streamlines and isotherms obtained are shown in Fig. 2. The comparison of these profileswith those of Chadwick et al. [18] (Fig. 3 in that article) presents an excellent agreement.

3. Results and discussion

Computations are carried out for the following values of the governing parameter both for the weak con-centration case (n = 0) and for the strong concentration (n = 0.5) cases: e = 1/5, 2/5, 3/5, 4/5 and 1; Pr = 0.71;Ra = 103, 104, 105 and 106; K = 0, 0.1, 0.5 and 2. Note that the case of K = 0 represents the Newtonian fluid.

In the following, the combined effects of the dimensionless length of the discrete heater and the materialparameter on heat transfer for different Rayleigh numbers are studied. For each e, the effect of K on themomentum and energy transport determined by increasing Ra. Figs 3 and 4 illustrate the streamlines andthe isotherms for K = 0 and 2 and e = 1/5, 2/5, 3/5, 4/5 and 1 at Ra = 103 and 106, respectively. As seen,

Fig. 2. Validity check of the computer code developed against Ref. [18].

Fig. 3. Streamlines an isotherms for different e and K at Ra = 103.


for each case, the bulk fluid heated by the isoflux discrete heater ascend through it, then faces the adiabaticupper wall through which it moves horizontally toward to the cold wall, descends along this cold wall andfinally moves horizontally along the lower adiabatic wall, thereby completing the main recirculation flow celloccupied by the enclosure. At Ra = 103, the circulation inside the enclosure is so weak that the viscous forcesare dominant over the buoyancy force. Isotherms deviate slightly from the structure of the conduction regimefor which isotherms are parallel to the surfaces of the isoflux heater and cold sidewall by exhibiting a region ofthermally stable stratification. With increasing Rayleigh number, the intensity of the recirculation inside in theenclosure increases and the hydrodynamic and thermal boundary layers near the walls become thinner. For


the higher values of Ra, advection of the circulating bulk fluid becomes the dominant mechanisms of the heattransfer.

However, for a constant Rayleigh number, it is too difficult to see and explain the effect of the materialparameter of the micropolar fluid on heat and fluid flow from the figures above since related functions arevery concentrated near the heated and cooled boundaries. Based on the examination of these figures, onemay say that, at a constant Ra, the differences at streamlines and isotherms for K = 0 and K = 2 seem tobe negligible. This is misleading and therefore, heat transfer results are suggested to be analyzed. ForK = 0, the effect of varying Ra on the local Nusselt number along the discrete heater dissipating a constantheat flux is shown in Fig. 5. As seen, the local Nusselt number takes its maximum value at the leading edgeof the isoflux discrete heater. Then it decreases along the heater up to a minimum after which it increases

Fig. 4. Streamlines an isotherms for different e and K at Ra = 106.


again. This trend becomes much more discernible at higher values of Ra. Also, with an increase at Ra, thepoint of the minimum Nusselt number moves shifts upward.

In order to see the effect of the material parameter on the local Nusselt number, as an illustrative example,Nu is plotted for different K at e = 3/5 in Fig. 6. As seen an increase at K reduces the Nusselt number.

All the above computations obtained for the weak concentration case (n = 0) are repeated for the strongconcentration case (n = 0.5). And interestingly, no difference is observed between the results of the weak

2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nu

Y

Ra=103, 104, 105, 106

ε =1/5

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nu

Y

Ra=103, 104, 105, 106

ε =2/5

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nu

Y

Ra=103, 104, 105, 106ε =3/5

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nu

Y

Ra=103, 104, 105, 106

ε =4/5

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nu

Y

Ra=103, 104, 105, 106

ε =1

(a) (b)

(c) (d)

(e)

Fig. 5. Variation of the local Nusselt number for different Ra at (a) e = 1/5, (b) e = 2/5, (c) e = 3/5, (d) e = 4/5 and (e) e = 1.

1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.950

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nu

Y

K=2, 1, 0ε = 3/5

Ra=103

4 5 6 7 8 9 10 11 12 13 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nu

Y

K=2, 1, 0ε = 3/5

Ra=106

(b)

(a)

Fig. 6. Variation of the local Nusselt number for different K at e = 3/5 (a) Ra = 103 and (b) Ra = 106.


and strong concentration cases. This is attributed to the symmetrical boundary conditions for the microtationboth in x and y directions.

4. Conclusions

In this study, the natural convection heat transfer of micropolar fluids in a discretely heated square enclo-sure is computationally studied using the finite difference method. Simulations are performed to investigate theeffects of the length of the isoflux discrete heater, the Rayleigh number, Ra, Prantl number, Pr and the mate-rial parameter, K on the momentum and heat transfer. As expected, it is found that the mean Nusselt number


increases with increasing Rayleigh numbers. On the other hand, it is disclosed that an increase at the materialparameter reduces the heat transfer.

References

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Transfer, John Wiley, New York, 1987, Chapter 13.[5] T. Fusegi, J.M. Hyun, Laminar and transitional natural convection in an enclosure with complex and realistic conditions, Int. J. Heat

Fluid Flow 15 (1994) 258–268.[6] O. Aydin, W.-J. Yang, Natural convection in enclosures with localized heating from below and symmetrical cooling from sides, Int. J.

Numer. Meth. Heat Fluid Flow 10 (5) (2000) 51–529.[7] A.M. Al-Bahi, A.M. Radhwan, G.M. Zaki, Laminar natural convection from an isoflux discrete heater in a vertical cavity, Arabian J.

Sci. Eng. 27 (2C) (2002) 149–164.[8] N.H. Saeid, I. Pop, Natural convection from a discrete heater in a square cavity filled with a porous medium, J. Porous Media 8 (1)

(2005) 1–9.[9] T.-H. Hsu, C.-K. Chen, Natural convection of micropolar fluids in a rectangular enclosure, Int. J. Eng. Sci. 34 (4) (1996) 407–415.[10] T.-H. Hsu, S.-Y. Tsai, Natural convection of micropolar fluids in a two-dimensional enclosure with a conductive partition, Numer.

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233.[13] S.K. Jena, M.N. Mathur, Similarity solutions for laminar free convection flow of a thermomicropolar fluid past a nonisothermal flat

plate, Int. J. Eng. Sci. 19 (1981) 1431–1439.[14] J. Peddieson, An application of the micropolar fluid model to the calculation of turbulent shear flow, Int. J. Eng. Sci. 10 (1972) 23–32.[15] G. Ahmadi, Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite flat plate, Int. J. Eng. Sci. 14

(1976) 639–646.[16] D.A.S. Rees, I. Pop, Free convection boundary-layer flow of a micropolar fluid from a vertical flat plate, IMA J. Appl. Math. 61

(1998) 179–197.[17] O. Aydin, A. Unal, T. Ayhan, Natural convection in rectangular enclosures heated from one side and cooled from above, Int. J. Heat

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Natural convection from a discrete heater in enclosures filled with a micropolar fluid