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: naimi@fstbm.ac.ma, naimima@yahoo.fr 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: 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: 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
© Scientific Technical Centre «TATA», 2012
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
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