научная статья по теме ЕСТЕСТВЕННАЯ КОНВЕКЦИЯ ПЕРЕДАЧИ ТЕПЛА ДЛЯ НАНОЖИДКОСТЕЙ В НЕГЛУБОКОЙ ПРЯМОУГОЛЬНОЙ КАМЕРЕ С ВЕРТИКАЛЬНЫМИ СТЕНКАМИ ПОД ДЕЙСТВИЕМ ОДНОРОДНЫХ ПОТОКОВ ТЕПЛА Комплексное изучение отдельных стран и регионов

Текст научной статьи на тему «ЕСТЕСТВЕННАЯ КОНВЕКЦИЯ ПЕРЕДАЧИ ТЕПЛА ДЛЯ НАНОЖИДКОСТЕЙ В НЕГЛУБОКОЙ ПРЯМОУГОЛЬНОЙ КАМЕРЕ С ВЕРТИКАЛЬНЫМИ СТЕНКАМИ ПОД ДЕЙСТВИЕМ ОДНОРОДНЫХ ПОТОКОВ ТЕПЛА»

Статья поступила в редакцию 16.01.2012. Ред. рег. № 1184 The article has entered in publishing office 16.01.12. Ed. reg. No. 1184

УДК 532.5

ЕСТЕСТВЕННАЯ КОНВЕКЦИЯ ПЕРЕДАЧИ ТЕПЛА ДЛЯ НАНОЖИДКОСТЕЙ В НЕГЛУБОКОЙ ПРЯМОУГОЛЬНОЙ КАМЕРЕ С ВЕРТИКАЛЬНЫМИ СТЕНКАМИ ПОД ДЕЙСТВИЕМ ОДНОРОДНЫХ ПОТОКОВ ТЕПЛА

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) 5 23 48 51 12/22/82; Fax: (212) 5 23 48 52 01; E-mail: naimi@fstbm.ac.ma, naimima@yahoo.fr 2Университет султана Мулэя Слимана, полидисциплинарный факультет, междисциплинарная научно-исследовательская лаборатория (LIRST)

В.Р. 592, Бени-Меллал, Марокко

3Университет Кади Айада, Научный факультет Семлалия, кафедра физики, лаборатория механики жидкости и энергетики (LMFE)

В.Р. 2390, Маракеш, Марокко

Заключение совета рецензентов: 20.01.12 Заключение совета экспертов: 25.01.12 Принято к публикации: 31.01.12

Проведено комбинированное аналитическое и численное исследование двумерной устойчивой плавучей конвекции наножидкостей, удерживаемой в неглубокой прямоугольной камере, подвергавшейся воздействию тепловых потоков вдоль как коротких, так и длинных сторон при изолированных горизонтальных сторонах. Расчеты, ограниченные жидкостями на водной основе с числом Прандтля Pr = 7, проводились для основных параметров, варьируемых в диапазоне 1 < A < 8, для соотношения сторон камеры 102 < Ra < 107, для числа Релея и 0 < Ф < 0,2 для относительного объема твердых наночастиц. Проанализировано влияние этих параметров на поток и передачу тепла. Для A = 8 приближенное аналитическое решение выполнено на основе предположения параллельных потоков и подтверждено численно с помощью решения полных определяющих уравнений.

Ключевые слова: наножидкости, естественная конвекция, передача тепла, прямоугольная камера.

NATURAL CONVECTION HEAT TRANSFER FOR NANOFLUIDS IN A SHALLOW RECTANGULAR ENCLOSURE WITH VERTICAL SIDES SUBMITTED TO UNIFORM

HEAT FLUXES

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 Modeling (LAMET)

B.P. 523, Beni-Mellal, Morocco Tel.: (212) 5 23 48 51 12/22/82; Fax: (212) 5 23 48 52 01 E-mail: naimi@fstbm.ac.ma, naimima@yahoo.fr 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: 20.01.12 Expertise: 25.01.12 Accepted: 31.01.12

A combined analytical and numerical study has been conducted for two-dimensional steady state buoyancy driven convection of nanofluids confined in a shallow rectangular cavity subject to uniform heat fluxes along both the short vertical sides, while the long horizontal ones are insulated. The computations, which have been limited to water-based fluids whose Prandtl number value is, Pr = 7 have been carried out for governing parameters varying in the following range: 1 < A < 8, for the cavity aspect ratio, 102 < Ra < 107, for the Rayleigh number, and 0 < ® < 0.2, for the solid volume fraction of nanoparticles. The effects of these parameters on the flow and heat transfer are analyzed. For A = 8, an approximate analytical solution is developed on the basis of the parallel flow assumption and validated numerically by solving the full governing equations.

Keywords: nanofluids, natural convection, heat transfer, rectangular enclosures.

International Scientific Journal for Alternative Energy and Ecology № 01 (105) 2012

© Scientific Technical Centre «TATA», 2012

Л.

Mohamed Naïmi

Vita

National doctorate (INPL, Nancy, France, 1989)

State doctorate (Cadi Ayyad University, Marrakech, Morocco, 2001)

Professor of Mechanical Engineering (Sultan Moulay Slimane University, Faculty of Sciences and Technologies, Physics Department, Beni-Mellal, Morocco)

Director of Flows and Transfers Modelling Laboratory (LAMET)

Research Topics

- Natural convection in non-Newtonian fluids;

- Thermosolutal convection in non-Newtonian fluids.

Some Recent Publications

1. M. Lamsaadi, M. Naimi, M. Hasnaoui, A. Bahlaoui and A. Raji,Multiple steady state solutions for natural convection in a tilted rectangular slot containing non-Newtonian power law fluids and subject to a transverse thermal gradient, Numerical Heat Transfer, Part A, vol. 51, N° 3 & 4, pp.393-414, 2007.

2. M. Lamsaadi, M. Naimi, A. Bahlaoui A. Raji, M. Hasnaoui and M. Mamou, Parallel Flow Convection in a Shallow Horizontal Cavity Filled with non-Newtonian Power-law Fluids and Subject to Horizontal and Vertical Uniform Heat Fluxes, Numerical Heat Transfer, Part A, Vol. 53, N° 2, pp. 178-203, 2008.

3. T. Makayssi, M. Naimi, M. Lamsaadi, M. Hasnaoui, A. Raji and A. Bahlaoui, Effect of solutal buoyancy forces on thermal convection in confined non-Newtonian power-law fluids, International Scientific Journal for Alternative Energy and Ecology, Vol. 62, N° 6, pp. 77-86, 2008.

4. Makayssi, M. Lamsaadi, M. Naimi, M. Hasnaoui, A. Raji, and A. Bahlaoui, Natural double-diffusive convection in a shallow horizontal rectangular cavity uniformly heated and salted from the side and filled with non-Newtonian power-law fluids: the cooperating case, Energy Conversion and Management, Vol. 49, pp. 2016-2025, 2008.

5. M. Lamsaadi, M. Naimi, H. El Harfi, A. Raji and M. Hasnaoui, Double diffusive multiple solutions in a square cavity subject to cross gradients of temperature and concentration, International Scientific Journal for Alternative Energy and Ecology, Vol. 102, № 10, pp. 52-63, 2011

Nomenclature

Greek symbols

A

C g

H f

к

L

Nu

Nu

Pr

Ra t

T

T

AT* O, v)

(x У)

aspect ratio of the cavity, Eq. (20)

dimensionless temperature gradient in the x-direction

gravitational acceleration (m/s2)

height of the enclosure (m)

thermal conductivity of base fluid

dimensionless parameter, [= knf / kf ]

length of the rectangular enclosure (m)

local Nusselt number, Eqs. (21), (22) and (35)

average Nusselt number, Eqs. (23) and (35)

Prandtl number, Eq. (20)

constant heat flux per unit area (W/m2)

Rayleigh number, Eq. (20)

dimensionless time, [= (t af /H'2)]

dimensionless temperature, [= (T - T c)/AT ]

reference temperature at the geometric centre

of the enclosure (K)

characteristic temperature [= q'H' / kf] (K) dimensionless axial and transverse velocities, [= (U, v' )/(a/H')]

dimensionless axial and transverse co-ordinates

[= (X, y )/H ]

af thermal diffusivity (m2/s)

a dimensionless parameter, [= anf /af ]

Pf thermal expansion coefficient (1/K)

P dimensionless parameter, [= (pP' )n/(pP')f]

Vf kinematic viscosity of base fluid

V dimensionless parameter, [=vnf /Vf ]

ty dynamic viscosity of base fluid (Pa-s)

pf density of base fluid (kg/m3)

® nanoparticle volume fraction

y dimensionless stream function, [=y' /af ]

Q, dimensionless parameter, [=P/ (Va)

Superscript

' dimensional variable

Subscripts

c value relative to the centre of the enclosure

(x, y) = (A/2.1/2)

f base fluid

nf nanofluid

np nanoparticle

* characteristic variable

1. Introduction

The emergence of techniques allowing miniaturization of devices, like efforts to save energy, are certainly the main factors that have made the study of heat transfer an area of research more and more active during the last decade. It is well known that convection heat transfer can be enhanced passively by changing the flow configuration or by augmenting the exchange surfaces between the wall and the fluid (use of microchannel for example). But, such approaches have already shown their limits and new ways of optimization should be explored. Thus, the concept of nanofluids has been proposed as a route for surpassing the performance of heat transfer liquids currently available. They consist of nanometer-sized particles, called nanoparticles, suspended in based fluids [1]. Nanoparticles are typically made of metals, oxides, carbides, or carbon nanotubes, while base fluids include water, ethylene glycol and light oils [2]. Nanofluids, a term originally due to Choi [3], have novel properties that make them potentially useful in many applications in heat transfer including transportation, microelectronics, food, medical and manufacturing of many types [4]. In fact, they exhibit enhanced thermal conductivity and convective heat transfer coefficient compared to base fluids [5].

On the other hand, although nanofluids are solidliquid mixtures, the approach conventionally used in most studies handles the nanofluid as single-phase fluids [6]. In fact, since the suspended nanoparticles have usually small size and concentration, the hypothesis of a solideliquid mixture statistically homogeneous and isotropic can reasonably be advanced. This means that, under the further assumptions that the nanoparticles and base fluid are in local thermal equilibrium, and no slip motion occurs between the solid and liquid phases, to all intents and purposes the nanofluid can be treated as a pure fluid. Therfore, all the related classical theories, where physical properties of nanofluids are taken as functions of properties of both constituents and their concentrations, have been applied [5].

As an important mode of heat transport, occurring in several industrial processes, natural convection in nanofluids has become the target of lots of scientists and industrialists in the last years. A review-article recently compiled by Corcione [6] shows that the number of works carried out on this subject has been markedly increased, and most of them are related to rectangular cavities differentially heated. However, according to this author, the results of these studies lead to contra

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