COMBINED LID AND BUOYANCY DRIVEN EFFECTS IN A SHALLOW RECTANGULAR CAVITY CONFINING NANOFLUIDS AND UNIFORMLY HEATED AND COOLED FROM ITS VERTICAL SIDES Текст научной статьи по специальности «Физика»

ТЕРМОДИНАМИЧЕСКИМ АНАЛИЗ В АЛЬТЕРНАТИВНОЙ ЭНЕРГЕТИКЕ

THERMODYNAMIC ANALYSIS IN RENEWABLE ENERGY

Статья поступила в редакцию 31.07.12. Ред. рег. № 1379 The article has entered in publishing office 31.07.12. Ed. reg. No. 1379

УДК 669.791

КОМБИНИРОВАННОЕ ВЛИЯНИЕ КРЫШКИ И ПЛАВУЧЕСТИ В МЕЛКОЙ ПРЯМОУГОЛЬНОЙ ПОЛОСТИ, УДЕРЖИВАЮЩЕЙ НАНОЖИДКОСТИ И РАВНОМЕРНО НАГРЕТОЙ И ОХЛАЖДЕННОЙ

ПО ВЕРТИКАЛЬНЫМ СТОРОНАМ

Х. Эль Харфи1, М. Наими1, М. Ламсаади2, А. Раджи1, М. Хаснауи3

'Университет султана Мулэя Слимана, научно-технический факультет, кафедра физики, лаборатория моделирования потоков и перемещений (LAMET), В.Р. 592, Бени-Меллал, Марокко Tel.: (212) (0) 5 23 48 51 12/22/82; Fax: (212) (0) 5 23 48 52 01; E-mail: [email protected], [email protected] 2Университет султана Мулэя Слимана, полидисциплинарный факультет, междисциплинарная научно-исследовательская лаборатория (LIRST), В.Р. 592, Бени-Меллал, Марокко

^Университет Кади Айада, Научный факультет Семлалия, кафедра физики, лаборатория механики жидкости и энергетики (LMFE),

В.Р. 2390, Маракеш, Марокко

Заключение совета рецензентов: 15.08.12 Заключение совета экспертов: 20.08.12 Принято к публикации: 25.08.12

В работе описано численное исследование смешанной конвекции в мелкой закрываемой крышкой прямоугольной полости с наножидкостями на водной основе, равномерно нагретой теплыми потоками вдоль вертикальных стенок, с помощью полного решения основного уравнения через метод конечного объема и алгоритм СИМПЛЕРА, а также аналитически с помощью допущения параллельных потоков. Найдено хорошее совпадение результатов двух подходов в ограничении исследуемых величин основных параметров (числа Рейнольдса и Ричардсона) и относительного объема твердой фазы наночастиц. Исследуются и описываются эффекты этих параметров на поток и температуру полей, а также на перенос тепла.

Ключевые слова: наножидкость, смешанная конвекция, перенос тепла, закрытый объем, параллельный поток, метод конечного объема.

COMBINED LID AND BUOYANCY DRIVEN EFFECTS IN A SHALLOW RECTANGULAR CAVITY CONFINING NANOFLUIDS AND UNIFORMLY HEATED AND COOLED FROM ITS VERTICAL SIDES

H. Elharfi1, M. Naimi1, M. Lamsaadi2, A. Raji1, M. Hasnaoui3

'Sultan Moulay Slimane University, Faculty of Sciences and Technologies, Physics Department, Laboratory of Flows and Transfers Modelling

(LAMET), B.P. 523, Beni-Mellal, Morocco Tel.: (212) (0)5 23 48 51 12/22/82; Fax: (212) (0)5 23 48 52 01 E-mail addresses: [email protected]; [email protected] 2Sultan Moulay Slimane University, Polydisciplinary Faculty, Interdisciplinary Laboratory of Research in Sciences and Technologies (LIRST),

B.P. 592, Beni-Mellal, Morocco

3Cadi Ayyad University, Faculty of Sciences Semlalia, Physics Department, Laboratory of Fluid Mechanics and Energetics (LMFE),

B.P. 2390, Marrakech, Morocco

Referred: 15.08.12 Expertise: 20.08.12 Accepted: 25.08.12

Mixed convection, in a shallow lid-driven rectangular cavity filled with water-based nonofluids and subjected to uniform heat flux along the vertical side walls, has been studied numerically, by solving the full governing equations via the finite volume method and the SIMPLER algorithm, and analytically, by using the parallel flow assumption. A good agreement has been found between the results of the two approaches in the limit of the explored values of the governing parameters, which are the Reynolds, the Richardson numbers and the solid volume fraction of nanoparticles. The effects of these parameters, on the flow and temperature fields, and the heat transfer, have been examined and discussed.

Keywords: nanofluid, mixed convection, heat transfer, lid-driven enclosure, parallel flow, finite volume method.

International Scientific Journal for Alternative Energy and Ecology № 10 (114) 2012

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Nomenclature

A - aspect ratio of the cavity, Eq. (22) C - dimensionless temperature gradient in the x-direction

g - gravitational acceleration (m/s2) Gr - Grashof number H - height of the enclosure (m) h - heat exchange coefficient (W/m2K) k - thermal conductivity of fluid (W/mK) k - dimensionless parameter, [= knfjkf J

L - length of the rectangular enclosure (m) Nu - local Nusselt number, Eqs. (25), (26) and (41) Nu - average Nusselt number, Eqs. (27) and (41) Pr - Prandtl number, Eq. (23) and (24) q - constant heat flux per unit area (W/m2) Re - Reynolds number, Eq. (22) and (23) Ri - Richardson number, Eq. (22) and (23) t - dimensionless time, [= t'U'0 ¡H']

T - dimensionless temperature, [= (T' - Tc')/AT * ] Tc - reference temperature at the geometric centre of the enclosure (K)

AT * - characteristic temperature [= q'H'/kf ] (K) (u, v) - dimensionless axial and transverse velocities

[=(uV)/U' J

U ' lid-velocity (m/s) (x, y) - dimensionless axial and transverse co-ordinates [=(x',y' )/H' J

Greek symbols

a - thermal diffusivity (m2/s)

a - dimensionless parameter, [= anflaf J

P - thermal expansion coefficient (1/K)

P - dimensionless parameter, [= (pP' ) nfl (pP ') f J

v - kinematic viscosity (m2/s)

v - dimensionless parameter, [=vnf/ v f J

^ - dynamic viscosity (Pa-s) p - density of base fluid (kg/m3) O - nanoparticle volume fraction ^ - dimensionless stream function, [= ^'/a f J

Q - dimensionless parameter, [= p(pva)J

Superscript

' - dimensional variable

Subscripts

c - value relative to the centre of the enclosure or critical value f - base fluid m - minimum value nf - nanofluid np - nanoparticle * - characteristic variable

1. Introduction

During the last decade, nanofluids has attracted lots

of researchers encouraged by their critical importance and promising role, as new advanced heat transfer fluids, to take up challenges. Therefore, numerous studies, on convection heat transfer, have been conducted, and most of them have dealt with forced convection, indicating that nanoparticle suspensions have unquestionably a great potential for heat transfer enhancement, as reported in a recent paper by Corcione [1]. Among them mixed convection in lid-driven cavities, which has not received much consideration in view of the related number of works, although it finds applications in many industrial processes. The interaction between the lid driven flow due to and buoyancy driven flow is quite complex, which necessitates a comprehensive analysis to understand the physics of the resulting flow and heat transfer process. In this respect, different configurations and combinations of thermal and dynamical boundary

conditions have been considered and analyzed by some investigators such as Tiwari and Das [2], who studied heat transfer enhancement in a nanofluid-filled square cavity, with the vertical sides moving and differentially heated, while the horizontal ones are insulated and motionless. Three situations, depending on the direction of the moving walls, were examined, and a model taking into account the solid volume fraction of nanoparticles was developed to analyze the nanofluids behavior. With only one uniformly moving wall, from left to right, first, it is to bring up the research of Abu-Nada and Chamkha [3] dealing with mixed convection flow in an inclined square enclosure filled with a nanofluid. The left and right walls are kept insulated while the bottom and the moving top ones are maintained at constant cold and hot temperatures, respectively. Mahmoodi [4] investigated mixed convection fluid flow and heat transfer in rectangular enclosures filled with a nanofluid. The left and right walls as well as the top one are maintained at a constant cold temperature. The moving bottom is kept at

a constant hot temperature. In the case of a nanofluid-filled square cavity with cold sides, a partially heated (with constant heat flux heater) and insulated bottom, and a moving cold top, Mansour et al. [5] examined the effects of Reynolds number, type of nanofluids, size and location of the heater and the volume fraction of the nanoparticles in their study related to mixed convection. Muthtamilselvan et al. [6] studied heat transfer enhancement of nanofluids in rectangular enclosures, where the moving top is at higher constant temperature than the bottom whereas the left and right boundaries are insulated. Nemati et al. [7] investigated heat transfer performance of a moving top square cavity, filled with nanofluids and subject to different side wall temperatures. As for Talebi et al. [8], they conducted an investigation on mixed convection flows in a square lid-driven cavity, having left and right sides heated and cooled, respectively, and moving top and bottom both adiabatic, utilizing nanofluids. Finally, like Tiwari and Das [2], Sheikhzadeh et al. [9] were interested in laminar mixed convection of a nano-fluid in two sided lid-driven enclosures. The moving left and right walls are maintained at constant cold and hot temperatures, respectively, while the horizontal ones are insulated.

All these investigations, where the thermal boundary conditions are of imposed temperature type, are of numerical nature using various single-phase models to describe effective conductivity and viscosity of the considered nanofluids, which are principally Al2O3 or Cu-water. Therefore, in order to know more about the effect of the boundary conditions kind on flow and heat transfer within confined nanofluids, the present paper is concerned with mixed convection within a two-dimensional shallow rectangular enclosure, filled with Cu-water nanofluids, whose short vertical sides are submitted to uniform heat fluxes while the long horizontal ones are maintained adiabatic with the top moving in the direction of the imposed heat flux. Two ways are explored to examine flow and heat transfer in such a system: a numerical solution based on a finite volume method and an analytical one, using a parallel flow approximation.

2. Mathematical formulation

The studied configuration is sketched in Fig. 1. It is a shallow rectangular enclosure of height H' and length L', filled with copper (Cu)-water nanofluid, whose thermophysical properties of Cu and water are given in Table 1, according to Abu Nada et al. [10]. The long horizontal walls are adiabatic, while the vertical short ones are submitted to a uniform density of heat flux, q'. All these boundaries are rigid, impermeable and motionless apart from the top one which moves in its own plane from left to right at uniform velocity. The main assumptions made here are those commonly used, i.e.:

- The base fluid and the nanoparticles are in thermal equilibrium and they flow at the same velocity (i.e. no slip occurs between them or the nanoparticles are

uniformly dispersed within the base fluid so that the resulting nanofluid can be considered a single-phase fluid);

- The nanoparticles are spherical;

- The nanofluid is Newtonian and incompressible;

- The thermophysical properties of the considered nanofluids are constant except for the density in the buoyancy term, which obeys the Boussinesq approximation;

- The flow is two-dimensional, laminar and steady;

- The radiation heat transfer between the sides of the cavity is negligible when compared with the other mode of heat transfer.

д T' д /

= 0

u:

у > v

и

L'_

T1

H' ±

q

x, u

д T' д у'

= 0

Рис. 1. Диаграмма оболочки и системы координат Fig. 1. Sketch of the enclosure and co-ordinates system

Таблица 1

Термофизические характеристики воды и наночастиц

Table 1

Thermophysical properties of water and nanoparticles

P Cp (J/(kg-K)) к ß-105

H2O 997.1 4179 0.613 21

Cu 8933 385 401 1.67

Therefore, the equations describing the conservation of mass (1), momentum (2)-(3) and energy (4), written in terms of velocity components (u', v'), pressure (p') and temperature (T), are:

du' + ы _0.

dx' + dy' ~ '

(1)

du' du' du' 1 dP' Kf I d2u' dV Ï

—+u — + v —_---+—I —r + —г ;

dt' dx' dy' Pf dx' Pf Idx'2 Эу'2 J'

(2)

dV_ ,dV_ ,dV_ 1 dP' dt' u dx' V Эу' " Pf dy'

International Scientific Journal for Alternative Energy and Ecology № 10 (114) 2012

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0

дГ д^'Т') д^'Т') 2 ,

-+ \ . + \ . = а^V2T'. (4)

дt, дx,

дy/

To close the problem, the following appropriate boundary conditions are applied:

7\Tf nr

u' = v' = О and-+ — = О for x' = О and x' = L'

дx, k

nf

дT'

(5)

u = v' = 0 and-= 0 for y' = 0; (6)

u' - U'a = v' = 0 and-= 0 for y' = H'. (7)

To model the effective physical properties of the nanofluid, appearing in the above equations, the following formulas are used:

Pnf = (1 -O) p f + OPnp (8)

for the effective density, as shown in [11];

M f

Ц nf =

(1-Ф)2

(9)

for the effective dynamic viscosity, which is due to Brinkman [11];

(pp)nf = (1 -O)(pp) f +O(pp)np (10)

for the thermal expansion coefficient [12];

(pCp)nf = (1 - O)(pCp) f + O(pCp)np (11)

for the heat capacity [11];

knL = knp + 2kf - 2Ф(^ - knp ) kf knp + 2kf +ф(kf - knp )

(12)

k_,

аnf =

(pCp)

(13)

nf

дu дv — + — = О ; дx дy

(14)

дu дu дu 1 дP v f д^ д2u . ,, „ч — + u — + v— = -=—+ —I—- + —- ! ; (15) дt дx дy p дx Re i дx дy

дv дv дv 1 дP V fд2v д2v^ — + u — + v— = -=— + —I — + —- ^ RiT ; дt дx дy p дy Re iдx дy J p

(16)

дт дт дт а f д2т д2t

— + u — + v— = —I —--— ! ; (17)

дt дx дy Peiдx2 дy J

дТ 1

u = v = — + = = О for x = О and A; (1S)

дx k

дТ

u = v = — = О for y = О; дТ

u -1 = v = — = О for y = 1,

(19)

(2О)

а

; = Vnf/ V f

where k = kr^fJkf , a = a rffl

P = (pP)rf/(pP)f and P = P„f/Pf are parameters depending on 0, according to models given above. In addition, to analysis the flow structure, the stream function, related to the velocity components via

дш дш

u = — and v = —— дy дx

(with y = О on all boundaries)

(21)

is used.

The above equations let appears some dimensionless parameters that govern the problem, namely, the solid volume fraction Ф, the aspect ratio of the enclosure, A, the Peclet, Pe, Reynolds, Re, and Richardson, Ri, numbers. For the last four, the expressions are

л L' „ U'H' „ U'H' , A = —, Pe ^ —0—, Re = —— and

H

а

gßfqH

Ri =

ff 2

for the effective thermal conductivity, due to Maxwell-Garnett, which is a restriction of the Hamilton-Crosser model to spherical nanoparticles [11];

kfU О2

Note that

Pe = PrReandRi =

Gr Ra

Re2 PeRe

(22)

(23)

where

for the thermal diffusivity [13].

On the other hand, using the characteristic scales H', pfU02, H'/U'0 , U' and q'H'jkf , corresponding to

length, pressure, time, velocity and temperature, respectively, the dimensionless governing equations and the corresponding boundary conditions are

Gr =

gß fqH

vfkf

Pr = and Ra = Pr Gr (24)

а

are the Grashof, Prandtl and Rayleigh numbers, respectively.

The local heat transfer, through the nanofluid-filled cavity, can be expressed in terms of the local Nusselt number defined as

Nu( y) =

hL q L L AT* A

1

AT' kf H' AT' AT AT j A

(25)

where h is the heat exchange coefficient, AT * = q'H' jkf a characteristic temperature and AT = T (0,y)- T (A, y)

the side to side dimensionless local temperature difference. This definition is based on the thermal conductivity of the base fluid, kf, which seems logical

since, according to Corcione [1], Nu that would describe the thermal performance of the enclosure, with immediacy, should vary in the same manner as h and vice versa. However, Eq. (25) is notoriously inaccurate owing to the uncertainty of the temperature values evaluated at the two vertical walls (edge effects). Instead, Nu is calculated on the basis of a temperature difference between two vertical sections, far from the end sides, as suggested by Lamsaadi et al. [14]. Thus, by analogy with Eq. (25), and considering two infinitesimally close sections, Nu can be expressed by

Sx 1 Nu(у) - lim— - lim---—-

sx^ 0 ST (ST/Sx )

(T/ дx)

(26)

- A¡ 2

where 8x is the distance between two symmetrical sections with respect to the central one. The corresponding average Nusselt number is calculated, at different locations, from

Nu - J0 Nu(у)—у .

Таблица 2

Шаг координатной сетки тестов, проведенных с A = 8, Re = 1, Ri = 103 и различными значениями Ф

Table 2

Grids size tests conducted with A = 8, Re = 1, Ri = 103 and various values of Ф

(27)

Grids Ф Ve Nu

0.0 0.521 6.7S1

160x20 0.1 0.503 5.217

0.2 0.465 4.157

0.0 0.520 6.701

120x40 0.1 0.505 5.219

0.2 0.471 4.203

0.0 0.462 4.196

160x40 0.1 0.519 6.701

0.2 0.501 5.209

0.0 0.519 6.701

200x40 0.1 0.505 5.211

0.2 0.471 4.202

0.0 0.519 6.6SS

160x60 0.1 0.505 5.209

0.2 0.471 4.196

3. Numerics

Eqs. (14)-(17) associated with Eqs. (18)-(20) have been solved by using a finite volume method and SIMPLER algorithm in a staggered uniform grid system [15]. A second order back-wards finite difference scheme has been employed to discretize the temporal terms appearing in Eqs. (15)-(17). A line-by-line tridiagonal matrix algorithm with relaxation has been used in conjunction with iterations to solve the nonlinear discretized equations. The convergence has been

considered as reached when f kj+ - f j <

•. i

< 10-5 ^ |1, where fkd stands for the value of u, v, p

•. i

or T at the kth iteration level and grid location (•, i) in the plane (x, y). The mesh size has been chosen so that a best compromise between running time and accuracy of the results may be found. The procedure has been based on grid refinement until the numerical results agree, within reasonable accuracy, with the analytical ones, obtained from the parallel flow approach developed in the next section. Hence, as shown in Table 2, a uniform grid of 140x40 has been selected for A = 8 (value used for the numerical computations) and has been estimated sufficient to model accurately the flow and temperature fields within the cavity. The time step size, 8t, has been varied in the range 10-7 <8t < 10-4, depending on the values of the governing parameters.

4. Approximate parallel flow analytical solution

As can be seen from Figs. 2-4, displaying streamlines (left) and isotherms (right), the flow and temperature fields exhibit a parallel aspect and a linear stratification, respectively, in the most part of the cavity, for A = 8 and various values of Re, Ri and Accordingly, the following simplifications

u( x,y) = u( y), v (x,y) = 0, x,y) = y) and

T(x,y) = C (x - A/2) + 0(y), (28)

where C is unknown constant temperature gradient in the x-direction, are possible, which leads to the ordinary non-dimensional governing equations:

73 7\T

—u = aQ ReRi — = aQ Re RiC (29)

dy dx

with

—0

u -1 = — = 0 fory = 0 and 1; (31)

—y

1

J u (y)dy = 0; (32)

0 1

J0(y)dy = 0 (33)

0

as boundary, return flow and mean temperature conditions, respectively.

3Q International Scientific Journal for Alternative Energy and Ecology № 10 (114) 2012

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Using such an approach, the solution of Eqs. (29) and (30), satisfying Eqs. (31), (32) and (33), is

u(y) = 12QReRiC(2y3 - 3y2 + y) + (3y2 - 2y) ; (34)

1

12

e(y ) = — QRaC 2I y— ^ + y---— | +

1

1О

12О

+PC fy_ - +_L

а I 4 3 30

(35)

On the other hand, according to Bejan [16], the energy balance in x-direction is

J-d-dy + ^ JuTdy = J-f^1 dy . (38)

0 dx a 0 0 Vdx )x = 0 or A

In particular, in the parallel flow region and with the application of Eq. (18), Eq. (38) becomes:

^ Pe г 1

C +I ue dy = -=. а J k

(39)

The expression of the stream function, ^(y), can be deduced by integration of Eq. (21), taking into account of the corresponding boundary conditions and Eq. (34), which gives:

Ш(y) = 12ПReRiCI y4 - y3 + y- | + (y3 - y2 ), (36)

where П = —— . Therefore, the flow intensity is pаv

Шс = |ши

(37)

which, when substituted to Eqs. (34) and (35), gives the following transcendental equation:

1

Pe2

C -

QPeRa n2Ra2 -C +-

= +I 1 + -7 , -

k i 105а2 I 3360а 362SS0

C3 = О,

(4О)

whose solution, via Newton-Raphson method, for given Pe, Ra and leads to C.

Finally, taking into account of Eqs. (26) and (27), the Nusselt number is constant and can be expressed as

It corresponds to the maximum value of k(y)| in the central vertical section of the enclosure ( x = A/ 2 ).

Nu = Nu = - —.

C

(41)

Y __y J J

't J J J J , J

с (3)

Рис. 2. Направления течений (слева) и изотермы (справа) для A = 8, Re = 1 и различных значений Ф: (а) Ф = 0.0, (b) Ф = 0.1 и (с) Ф = 0.2 и Ri: (1) Ri = 10, (2) Ri = 102 и (3) Ri = 103 Fig. 2. Streamlines (left) and isotherms (right) for A = 8, Re = 1 and various values of Ф: (а) Ф = 0.0, (b) Ф = 0.1 и (с) Ф = 0.2 и Ri: (1) Ri = 10, (2) Ri = 102 and (3) Ri = 103

a

b

0,0'

-0,1 • Y

-0,2'

-0,3. -0,4-0,5-

-0,6'

Analytical solution

® 5 а в в «-8'

Numerical solution • Ф = 0.0 ° Ф= 0.1 ■ Ф= 0.2

0,0

0,2

0,4 y 0,6

0,8

1,0

0,0'

-0,1 ■

Y

-0,2.

-0,3. -0,4. -0,5. -0,6.

x. ------------ • • • •

Analytical solution Numerical solution • Ф=0 о Ф=01 • Ф=0.2

У

0,0

-0,1

Y

-0,2

0,0 Рис.

0,4 y

0,30,2-T . 0,10,0-0,1-0,2-0,3-(

0,3'

0,2 T 0,1

0,0

-0,1

-0,2

Analytic solution

Ri=10

.-o-o-o-o-

Numerical solution

• Ф=0 о Ф=01 ■ Ф=0.2

0,0

0,2

0,4 y 0,6

0,8

1,0

-0,3

. Analytic solution

' Ri=l0

Numerical solution

• Ф=0

о Ф=01

■ Ф=0.2

0,0

0,2

0,4 y 0,6

0,8

1,0

0,3

0,2' T 0,1

0,0-0,1

-0,2. -0,3

' Analytical solution

' Ri=l0

Numerical solution

• Ф=0

о Ф=01

■ Ф=0.2

0,0

0,2

0,4 y 0,6

0,8

1,0

3. Профили функции потока (слева) и температуры (справа) в полости средней длины вдоль вертикальной координаты для A = 8, Re = 1 и различных значений Ф и Ri Fig. 3. Stream function (left) and temperature (right) profiles at mid-length of the cavity, along the vertical coordinate, for A = 8, Re = 1 and various values of Ф and Ri

10

w

Analytical solution

Numerical solution • Ф = 0.0 о Ф = 0.1 ■ Ф = 0 2

Nu

103

10=

10'

10[

ю-!

Analytical solution

Numerical solution • Ф = 0.0

О ф = 0.1

- Ф = 0.2

10°

103

b

Ri

10s

Рис. 4. Эволюция интенсивности потока (а) и скорость передачи потока (b) с Ri, для A = 8, Re = 1 и различных значений Ф Fig. 4. Evolution of the flow intensity (a) and heat transfer rate (b) with Ri, for A = 8, Re = 1 and various values of Ф

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a

5. Results and discussion

The results presented here are obtained for A = 8, Re = 1, 10-3 < Ri < 106, 0 <$< 0.2 and Pr = 7 (water based mixtures), which means that the only governing parameters are Ri and O, defined before. Their effects are discussed below.

Typical streamlines (left) and isotherms (right) are displayed in Fig. 2. First of all, it is to observe that the flow is unicellular and clockwise, as a result of the cooperating aspect of buoyancy and shear effects which act from left to right. Also, as mentioned in section 4, except in the end sides, the flow is parallel to the horizontal boundaries and the temperature is linearly stratified in the horizontal direction. On the other hand, the shear effect is so important that the flow symmetry, which is synonymous of a dominant buoyancy effect, necessitates an important increase of Ri. In such a situation, the role of ® remains almost inexistent. The corresponding isotherms are more affected but undergo opposite changes with Ri and

To check the validity of the approximate analytical solution, the numerical results (full circles) are compared to those obtained analytically (solid lines), as displayed in Fig. 3, giving stream function (left) and temperature (right) profiles along y-axis at the mid-length of the cavity, A / 2, y) and T (A /2, y), respectively. As can be seen, the two types of results agree well, confirming thus the existence of an analytical solution and validating mutually each other. An additional confirmation of such an agreement can be found in Fig. 4 displaying analytical and numerical values of the stream function, and the mean Nusselt number, Nu , at the vertical central section of the cavity, for a wide range of Ri, and various values of

On the other hand, although the results of Fig. 4 are related to the core region, where the parallel flow concept is valid, they inform amply about the flow and thermal fields than those related to Fig. 2. Thus, the presence of a single relative minimum in the stream function profile in all cases indicates that the flow is unicellular clockwise, driven by both lid and buoyancy cooperating effects. Moreover, the temperature profile presents, in general, two portions, with negative and positive signs, whose amplitude depends on the magnitude of the above mentioned aiding effects. In fact, the resulting clockwise flow makes warm the top, by transporting the heat from the left hot side, and cold the bottom, after passing near the right cold one, which explains why the sign of the lower portion is negative and that of the upper one is of positive. These results show, also, the opposing effects of Ri and expressed by an increase of A /2, y) and |T (A /2, y)| with Ri

and a decrease of these quantities with

The evolution of the flow intensity, (top), and heat transfer rate, Nu (bottom), which are reported, against

Ri, in Figs. 8-10, for each Re and various reveal in general two distinct convection regimes, namely:

- A weak convection regime, where and Nu are nearly constant up to Ri ~ 10. This is related essentially to the fact that the viscosity effect is still dominant for an enclosure of large aspect ratio (shallow enclosure) that inhibits the inertia effects and favors, at the same time, momentum and heat diffusions. This regime is also characterized by a quasi-independence of on because of its low circulation. In fact, an increase of ® leads to an increase of the effective viscosity, which makes more negligible the inertia effects and slows down the flow. In contrast, increasing ® increases Nu , due to the fact that the thermal conductivity of nanoparticles is higher than that of the base fluid.

- A second regime, dominated by convection, that manifests its self from a value of Ri > 10 and depending, generally, on both Such a regime is characterized by an increase of and Nu with Ri, which starts slightly and reaches an asymptotic linear trend from a certain value of Ri, because buoyancy becomes, gradually, the mainly driving force for the fluid motion with an increasing Ri. Moreover, the effect of ® is such that an increase of this parameter leads to a decrease of and Nu. For the reason is that the addition of nanoparticles, in the base fluid, augments the viscosity whose slowing down role of the motion is well known. This appears only for a range of Ri, beyond which the inertia of the nanoparticles becomes comparable with that of the fluid, due to large values of Ri, minimising significantly the effect of which explains the meeting

of curves in such a situation. For Nu, the decrease with ® is the consequence of the conflict between effective conductivity and viscosity. In fact, the former tends to enhance heat transfer while the later tends to reduce it by slowing down the fluid motion, particularly near the thermally active walls, whose role in terms of heat transport is primordial.

6. Conclusion

Mixed convection in a two-dimensional horizontal shallow enclosure, filled with Cu-water nanofluid, has been studied, by both numerical and analytical ways, in the case where both short vertical sides are submitted to uniform heat fluxes while the long horizontal ones are assumed to be insulated, with the top one uniformly moving in the direction of the imposed heat flux.

The simulations have been carried out for A = 8, Pr = 7, Re = 1, 1 < Ri < 5-104 and 0 <$< 0.2, as values attributed to governing parameters.

It has been found that in the limit of the selected values of the governing parameters, analytical results, agree perfectly with the numerical ones.

The evolution of the flow intensity and the heat transfer rate with Ri reveals two behaviour types. Hence, in the range of low values of Ri, and Nu are

insensitive to any increase of Ri, but addition of nanoparticles increase Nu without affecting Beyond this range, these quantities undergo an increase with Ri, but additional nanoparticles lead in general to decrease of and Nu, i.e. to deterioration of mixed convection heat transfer, which is not expected.

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International Scientific Journal for Alternative Energy and Ecology № 10 (114) 2012

© Scientific Technical Centre «TATA», 2012