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5Статья 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 contradictory conclusions, thus leaving still unanswered the question if the use of nanoparticle suspensions for natural convection applications is actually advantageous with respect to pure liquids. This shows that the problem of natural convection in a differentially heated vertical enclosure filled with a nanoparticle suspension is an issue still far from being completely solved, which can be mainly imputed to ambiguous evaluations of the nanofluid effective thermophysical properties.
To the best of our knowledge, the problem of natural convection heat transfer of nanofluids in an enclosure subject to Neumann boundary conditions for temperature was not sufficiently analyzed. The only study which was conducted, in this respect, is that of Alloui et al. [7], who reported analytical and numerical results related to natural convection in a shallow rectangular cavity, filled with (Ag, Cu, CuO, А120з, TiO3) - water nanofluids, where constant heat fluxes are applied to its long horizontal walls, while the two short vertical ones are maintained adiabatic. In their study, the parameters governing the problem are the Rayleigh number, Ra, the type of nanoparticles and the volume fraction of the solid nanoparticles, Ф. An analytical solution for finite amplitude convection was derived on the basis of the parallel flow approximation valid in the limit of shallow enclosure. It was, particularly, found that an increase of Ф reduces, in general, the flow intensity, but influences differently, depending on the range of values of Ra, the heat transfer rate, Nu . In fact, Nu appears to be lower or greater than that related to the base fluid, depending on whether that Ra is less or more than a relatively small value, which depends upon both Ф and the type of nanoparticle. These authors observed, also, that nanoparticles with a higher thermal conductivity, such as Cu and Ag, give rise to a greater enhancement in the rate of heat transfer, and that the less efficient nanoparticle seems to be TiO2 (Table 1).
Таблица 1
Термофизические свойства воды и наночастиц [7]
Table 1
Thermophysical properties of water and nanoparticles [7]
P (kg/тз) Cp (Jkg"1K) к (Wm"'K"') ß105 (K-1)
H2O 997.1 4179 0.613 21
Ag 10500 235 429 1.89
Cu 8933 385 401 1.167
CuO 6320 531.8 76.5 1.8
AI2O3 3970 765 40 0.85
TiO3 4250 686.2 8.9538 0.9
Then, in order to bring more information about the effect of the boundary conditions kind on natural convection heat transfer within nanofluids, the present paper focuses attention on such a problem within a two-dimensional horizontal rectangular enclosure filled with (Ag, Cu, CuO, M2O3, TiOs) - water nanofluids, like [7], and submitted to uniform heat fluxes at the short vertical sides, while the long horizontal ones are maintained adiabatic. A numerical solution of the full governing equations is obtained for wide ranges of the controlling parameters. An analytical solution, based on the parallel flow approximation, is also proposed. The results, in
International Scientific Journal for Alternative Energy and Ecology № 01 (105) 2012
© Scientific Technical Centre «TATA», 2012
terms of stream function and temperature profiles and heat transfer rates, are presented for different values of the Rayleigh number, Ra, and the solid volume fraction of nanoparticles, A unique useful correlation of heat transfer results is also proposed, in wide ranges of Ra and for all the kinds of nanofluids considered here.
dv_ 'dv_ =___
dt' dx' dy' p , dy'
dp'
V
n/
n/
av av
-+-
dx' dy'
+-L(pß)„/g(-TQ_); (3)
pnf
2. Mathematical formulation
The studied configuration is sketched in Fig. 1. It is a rectangular enclosure of height H' and length LL, filled with (Ag, Cu, CuO, Al2O3, TiO3) - water nanofluids. 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 and impermeable. 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 assumed 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.
Рис. 1. Эскиз полости и системы координат Fig. 1. Sketch of the cavity and co-ordinates system
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:
— + duT_) = а/VT'. (4)
dt' dx' dy' nf
To close the problem, the following appropriate boundary conditions are applied:
r\Tr f
u' = v' = 0 and-+ = 0 for x' = 0 and x' = L' ;(5)
dx' k
n/
dT'
u' = v' = 0 and-= 0 for y' = 0 and y' = H'. (6)
dy'
To model the effective physical properties of the nanofluid, appearing in the above equations, we use the following formulas:
Pnf =(l f +^„p for the effective density, as shown in [2];
Ц /
Ц/ =
(l -Ф)2
(V)
(S)
for the effective dynamic viscosity, which is due to Brinkman [8];
(pß)f = (l-ф)И) f+ФИ)„
(9)
for the thermal expansion coefficient [7];
(pCp)f =(1 -®)(pCp)f +®(pCp)np (10)
for the heat capacity [2];
hf _K + 2kf-2^(kf-K)
kf knp + 2kf + 2Ф(( - k„p )
(ll)
for the effective thermal conductivity, due to Maxwell-Garnett [9] which is a restriction of the Hamilton-Crosser model [10] to spherical nanoparticles; and
k.
а nf =
n/
(pcp)
(l2)
du' + dv_=Q. dx' + dy' ~ '
du' ,du' ,du'
-+ u -+ v-
dt' dx' dy'
_l_ dp_
Pnf dx' Pnf
dx'2 dy'
(l)
(2)
for the thermal diffusivity [7].
On the other hand, using the characteristic scales
H', H'2/af
аf /H'.
\'H' / kf and а f
corresponding to length, time, velocity, temperature and stream function, respectively, the dimensionless governing equations and the corresponding boundary conditions are
du dv 0. dx dy
du du du 1 dp — + u — + v— = + v Pr
dt dx dy p dx
d u + o u
dx2 3"2
dy2
(13)
(14)
dv dv dv 1 dp — + u — + v— = -=^7-+v Pr dt dx dy p dy
d2v d2v
äx2+ э7
dr dr dr _
--+ u--+ v— = а
dt dx dy
d2r d2r
dx2 +dy2
+ = PrRar ; P
(15)
(16)
rkT 1
u = v = — + = = 0 for x = 0 and A; (17) dx к
u = v = — = 0 for y = 0 and y = 1, (18)
dy
к = knf/ к f,
а = а nf / а f,
V = Vnf / vf ,
dw dw u = —¡- and v =--
dy dx
(19)
A = Ü ,Pr =V H' а
f and Ra = **'H'
а f V fkf
(20)
Nu (y ) =
A
1
knf АГ '/ L'
кДГ кАГ / A
(21)
difference between two vertical sections, far from the end sides, as suggested by Lamsaadi et al. [11]. Thus, by analogy with Eq. (21), and considering two infinitesimally close sections, the local Nusselt can be defined by
Nu (y ) =1 lim =1 lim -—1—- =
V k 8T k 0 ( / Sx)
1
1
к (dr / dx)
(22)
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, as follows:
Nu = jNu(y)dy .
(23)
where
P = (pP)„/ /(pP)f and p = pn/ / pf are parameters
depending on according to models given above. In addition, to analysis the flow structure, the stream function related to the velocity components via the following relationships:
with y = 0 at all boundaries is used.
The above equations let appears some dimensionless parameters that govern the problem, namely, the solid volume fraction the aspect ratio of the enclosure, A, the Prandtl, Pr, and Rayleigh, Ra, numbers. For the last three, the expressions are
As an additional check of the results accuracy, energy balance systematically has verified for the system at each numerical code running. Thus, the overall heat transfer, through each vertical plane, has been evaluated and compared with the quantities of heat furnished to the system at x = 0. For the results reported here, the energy balance has been satisfied within 2% as a maximum difference.
3. Numerics
Eqs. (13)-(16) associated with Eqs. (17)-(19) have been solved by using a finite volume method and SIMPLE algorithm in a staggered uniform grid system (Patankar [12]). A second order back-wards finite difference scheme has been employed to discretize the temporal terms appearing in Eqs. (14)-(16). A line-byline 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
XI fj - j < 10-5 £ I fk/1, where fj stands for the
Naturally, the type of nanoparticle considered is to be specified which introduce another parameter.
The local heat transfer, through the nanofluid-filled cavity, can be expressed in terms of the local Nusselt number defined as
where AT = T (0, y)- T (A, y) is the side to side
dimensionless local temperature difference. This definition is, however, notoriously inaccurate owing to the uncertainty of the temperature values evaluated at the two vertical walls (edge effects). Instead, the Nusselt number is calculated on the basis of a temperature
value of u, v, p or T at the kth iteration level and grid location (t, j) 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 with the parallel flow ones within reasonable accuracy. 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, St, has been varied in the range 10-7 < St < 10-4, depending on the values of the governing parameters. More precisely, small values of St have been used for high values of Ra.
International Scientific Journal for Alternative Energy and Ecology № 01 (105) 2012
© Scientific Technical Centre «TATA», 2012
Таблица 2
Исследования шага координатных сеток для Cu наножидкостей на водной основе, Ra = 105, A = 8 и различных значений Ф
Table 2
Grids size tests conducted for water based Cu nanofluids, Ra = 105, A = 8 and various values of Ф
Grids Ф w Nu
(140x20) 0.0 S.5Q6 30.604
0.1 S.263 17.600
0.2 7.793 10.129
(120x40) 0.0 S.373 30.994
0.1 S.243 17.633
0.2 7.SQ1 10.119
(140x40) 0.0 S.37S 30.990
0.1 S.25Q 17.614
0.2 7.7S7 10.139
(160x40) 0.0 S.379 30.991
0.1 S.25Q 17.614
0.2 7.7S3 10.131
l40x 60) 0.0 S.367 30.993
0.1 S.253 17.611
0.2 7.7S1 10.130
4. Approximate parallel flow analytical solution
As can be seen from Fig. 2, which has been obtained for Си-water nanofluid, A = 8 and various values of Ra and Ф, the flow (stream lines) and temperature (isotherms) fields exhibit a parallel aspect and a linear stratification in the most part of the cavity, respectively. This allows the following simplifications:
u(x, y) = u(y), v(x, y) = 0, and T(x, y) = C(x - A/2) + 9(y),
(24)
where С is unknown constant temperature gradient in the x-direction. Therefore, the non-dimensional governing equations become
^ = Ä Ra dT = !_ RaC ; dy pv dx
PV
d2e(y) , ч
а—yp- = Cu (y)
dy
(25)
(2б)
i
J u (y )dy = Q
(2S)
as boundary and return flow conditions, respectively.
Using such an approach, the solution of Eqs. (25) and (26), satisfying Eqs. (27) and (28) and taking into account of the centro-symmetry of the dynamical and thermal fields in the core region, is
(y) = Ц ORaC (2y3 - 3y2 + y) ;
e(y ) = — ORaC w 12
5 4 3 i
2| У__У_ + У___1
1Q 4 б 12Q
(29)
(3Q
The expression of the stream function, ^(y), can be deduced from Eq. (29) by integration of Eq. (19) with the corresponding boundary conditions, which gives:
v(y )=12 °RaC ( £ - y3 + .
(31)
Therefore, the stream function at the center of the enclosure, which is a measure of the flow intensity, is
¥c =v(y = 1/2) =
аORaC 3S4
(32)
where ^ = p/avp .
On the other hand, C has been evaluated from thermal boundary condition imposed on the end walls. Because of the turning flow at the end regions of the fluid layer, the boundary conditions in the x-direction, Eq. (17), could not be satisfied by the parallel flow approximation. Instead, the expression of C has been determined by matching the core solution, Eq. (24), to the integral solution for the end regions, which consists of the integration of Eq. (16), together with the boundary conditions (17) and (18), by considering the arbitrary control volume of Fig. 1, which yields:
C+1 =1 ju(y)e(y)dy . (33)
k a •
The substitution of Eq. (33) to Eqs. (29) and (30) gives the following transcendental equation:
с+1с 3 = q, k 362SSQ
(34)
with
which has been solved by the Newton-Raphson method, for each given value of Ra and leading to the value of C.
Finally, taking into account of Eqs. (22) and (23) and substituting to them Eq. (34), the Nusselt number is constant and can be expressed as
u = — = Q for y = Q and 1;
dy
(27)
— 1 1 (ORaC )2 Nu = Nu = -=—= 1 +-i-'k C 362SSQ
(35)
Рис. 2. Линии обтекания (слева) и изотермы (справа) для Cu наножидкостей на водной основе A = 8 и различных значений Ф (a - Ф = 0; b - Ф = 0,1 и c - Ф = 0,2) и Ra (1 - Ra = 104, 2 - Ra = 105 и 3 - Ra = 106) Fig. 2. Streamlines (left) and isotherms (right) for water based Cu nanofluids, A = 8 and various values of Ф (a - Ф = 0; b - Ф = 0,1 и c - Ф = 0,2) and Ra (1 - Ra = 104, 2 - Ra = 105 and 3 - Ra = 106)
5. Results and discussion
The fact of imposing uniform heat flux, as boundary conditions, leads to flow characteristics independent on, A , when this parameter is large enough. The approximate analytical solution, developed in the preceding section, on the basis of the parallel flow assumption, is thus valid asymptotically in the limit of a shallow cavity (A >> 1). Therefore, numerical tests have been performed to determine the smallest value of A leading to results
reasonably close to those of the large aspect ratio approximation. Moreover, since the study has been limited to water based solutions, it has been assumed that Pr = 7. Anyway, as it was demonstrated by Lamsaadi et al. [11] and many others, the present solution is rather independent of Pr provided that Pr > 1. This finding is confirmed by the approximate analytical solution, developed in the previous section which, in its range of validity, is independent on Pr. Therefore, the natural convection flow developed inside the enclosure is
International Scientific Journal for Alternative Energy and Ecology № 01 (105) 2012
© Scientific Technical Centre «TATA», 2012
governed only by three dimensionless parameters, namely Ra, ® and the type of nanoparticles. The effects of these parameters on the flow and thermal fields and the resulting heat transfer will be now discussed.
Let us mention that the numerical simulations have been performed with the pertinent thermophysical properties given in Table 1.
5.1. Determination of the value of A satisfying the large aspect ratio approximation
The goal of this subsection is to determine the smallest value of A leading to convection heat transfer
results that agree reasonably with the analytical ones. In this respect, the evolution of (a) and Nu (b), versus A, are displayed in Fig. 3, for water based Cu nanofluids and various values of ® and Ra = 106.
It is easy to see that, for all explored values of the asymptotic limits of and Nu, which coincide with the analytical ones, are largely reached for A = 8; value already obtained by Lamsaadi et al. [11] for pure fluid (O = 0) and assumed to be independent on the type of nanoparticles considered.
20
16
12-
к . . " . " " " . . """" т Î . ~t . . ....!
Numerical solution
-Ф = о
Я' • Ф = 0.1
. ■ тФ = 0,2
Analytical solution
--Ф = 0
Ф = 0.1
-1--- Ф = 0.2 т-1-г---1---
12
Nu
180
120-
60
Numerical solution Analytical solution
. Ф = о ф = о
• Ф = 0 1 -Ф = 0.1
- Ф = 0.2 Ф = 0.2
■ - '
. Ш
■
т
b
A 12
Fig. З
Рис. 3. Эволюция функции течения (a) и числа Нуссельта (b) в центральной части полости с соотношением сторон для Cu наножидкостей на водной основе, Ra = 106 и различных значений Ф Evolution of the stream function (a) and the Nusselt number (b) in the central part of the cavity with the aspect ratio for water based Cu nanofluids, Ra = 106 and various values of Ф
a
5.2. Validation of the approximate parallel flow
analytical solution To validate the approximate analytical solution, the numerical results (full circles), obtained by solving the full governing equations, are compared to those obtained analytically (solid lines) for water based Cu nanofluids,
A = 8, Ra = 105 and various values of O. As can be seen, from Fig. 4, displaying y(y: a) and T(y: b), calculated at x = A/2, the agreement between the two kinds of results is excellent.
10 M 6
2-
Numerical solution .0 = 0
• Ф = 0.1
• Ф = 0.2 JP Analytical solution <
Ф = 0 Ф = 0.1 Ф = 0.2
0.2
0.4
0.6
0.8
-0.09
b
Fig. 4. the stream function (a) and temperature (b) profiles at mid-length of the cavity, along the vertical coordinate for water based
Cu nanofluids, A = 8, Ra = 105 and various values of Ф Рис. 4. Профили потоковой функции (a) и температуры (b) на половине длины полости вдоль вертикальной координаты для
Cu наножидкостей на водной основе, A = 8, Ra = 105 и различных значений Ф
a
In addition, the quantitiesy(x: a) and T(x: b), obtained at y = 1/2, with both approaches, and plotted in Fig. 5, confirm the perfect agreement and the validity of the parallel flow hypothesis in the range 2 < x < 6, i.e. far from the vertical sides. Last, computed and calculated
values of \ (flow intensity: a) and Nu (mean heat
transfer rate: b) presented in Figs. 6-8, show a generalization of the accordance between the analytical and numerical results for a wide range of Ra, various values of ® and different types of nanoparticles.
10
1* ■ t
Numerical solution
■ Ф = 0
* ■ Ф = 0.1 »
* Ф = 0.2 r
Analytical solution
Ф = 0
Ф = 0.1
Ф = 0.2 -1-
0.4
0.2
-0.2
-0.4
¡W
Numerical solution
• Ф = 0
• ф =0.1 - Ф = 0.2
Analytical solution Ф = 0 Ф = 0.1 Ф = 0.2
""•M:;:,
Рис. 5. Fig. 5.
Профили потоковой функции (а) и температуры (b) на половине длины полости вдоль горизонтальной координаты
для Cu наножидкостей на водной основе The stream function (а) and temperature (b) profiles at mid-height of the cavity, along the horizontal coordinate for water based Cu nanofluids, A = 8, Ra = 105 and various values of Ф
Numerical solution
—' * Ag
» Tio,
• Cu
Ф = Ox^K? = AI,Oj * CuO
Analytical solution Ag -Tio,
Ф = 0,2 Cu ■ - CuO
10*
10=
105
Ra 107
700 Nu 100
10
Numerical solution
• Ag
• Tio, - Cu
■ a'A
• CuO
Analytical solution Ag
Iю*
Cu
"
CuO
10г
b
Ra
107
Рис. 6. Эволюция функции течения (а) и числа Нуссельта (b) в центральной части полости с числом Рэлея для различных
типов наночастиц, различных значений Ф и A = 8 Fig. 6. Evolution of the stream function (a) and the Nusselt number (b), in the central part of the cavity, with the Rayleigh number for
different types of nano-particules, various values of Ф and A = 8
a
25
M
15
5
Numerical solution Analytical solution
' Ag Ag
= 14 Tio,
' ?" Cu
CuO
A'A
CuO
Ra = 10s
Ra = 10s
*......о .
Ra = 10'
0.05
0.1 a
0,15
Ф
0.2
Numerical solution Analytical solution
• Ag Ag
c ï°.< Tio
* Cu Cu
" Al A AIA
CuO
b
Рис. 7. Эволюция функции течения (а) и числа Нуссельта (b) в центральной части полости с Ф для различных типов наночастиц, различных значений Ra и A = 8 Fig. 7. Evolution of the stream function (a) and the Nusselt number (b) in the central part of the cavity with Ф for different types of
nano-particules, various values of Ra and A = 8
International Scientific Journal for Alternative Energy and Ecology № 01 (105) 2012
© Scientific Technical Centre «TATA», 2012
10°..........................
10* 105 108 Ra 10'
Рис. 8. Эволюция числа Нуссельта в центральной части полости с числом Рэлея для Cu наножидкостей на водной основе A = 8 и различных значений Ф Fig. 8. Evolution of the Nusselt number, in the central part of the cavity, with the Rayleigh number for water based Cu nanofluids, A = 8 and various values of Ф
5.3. Articulated effects of the nanoparticles volume fraction and Rayleigh number
Typical numerical results in terms of streamlines (left) and isotherms (right) are presented in Fig. 2 for A = 8 and various values of ® and Ra. Their inspection informs about the articulated effects of ® and Ra, on the flow and temperature fields within the cavity. The unicellular nature of the flow is always preserved for ® and Ra varying in their respective ranges. By and large, except the end regions, the flow and temperature fields remain parallel to the long sides and linearly stratified in the horizontal direction, respectively. These observations are behind the simplifications made above (cf. section 4), to transform the governing partial differential equations to ordinary ones. Qualitatively, the isotherms seem to be more sensitive to the variations of ® or Ra than the streamlines. Their inclination with respect to the vertical direction, which is that of the conductive regime, gets more and more important by decreasing ® or increasing Ra. These qualitative observations are consistent with the evolutions of the flow intensity, c| (a), and heat
transfer rate, Nu (b), plotted in Figs. 6, respectively, for different types of nanoparticles and various values of Indeed, two trends of evolution can be seen for each quantity:
- For 102 < Ra < 104, c| and Nu increase,
respectively, relatively rapidly and slowly with Ra. In fact, in this range of Ra, although the convection flow is initiated, the heat transfer is mainly dominated by conduction, which explains the slow augmentation of Nu with Ra. For this range of Ra, both c| and Nu
undergo a decrease with This can be explained by the fact that, in such a situation, the viscous effects operate in complicity with the aspect ratio of the cavity, which is large enough, to dominate the thermal diffusion, which results in delaying the flow and reducing the heat transfer.
- For Ra > 104, and Nu increase monotonically because of the obvious contribution of the buoyancy effects in promoting the convection. But, in this case,
| is seen to be quite unconcerned about the variations of ® (all the curves coincide) than Nu which remains a decreasing function of such a parameter (all the curves are parallel). In fact, according to Eq. (15), the buoyancy term becomes preponderant compared to the momentum diffusion one, which inhibits the effect of ® on the flow by mixing phenomenon, whereas in Eq. (16) the importance of the diffusion term (right member), which is pondered by a, diminishes with like a, leading to a reduction of heat transfer. In such a situation, all goes as if the temperature and velocity fields are decoupled. These findings can be confirmed by Fig. 5, giving y(x: a) and T(x: b), obtained at y = 1/2, where a very slight diminution of the flow intensity and a strong augmentation of the absolute value of the horizontal temperature gradient with ® can be observed. Note that, as the flow and heat transfer take place essentially in the horizontal direction, it is not obvious to find precious information, in this respect, from Fig. 4 displaying y(y: a) and T(y: b), calculated at x = A/2.
Another analysis of the conjugated effects of @ and Ra can be done from Figs. 7, a-b representing the evolutions of c| (a) and Nu (b) with for various
values of Ra and different types of nanoparticles. It is easy to observe the almost linear tendency that exhibits c| , when varying with whose weak slope takes a little importance, depending on the type of nanoparticles, as Ra increases. In contrast, Nu appears, in general, as a monotonic function of ® whose decrease with this parameter gets more and more important with an increasing Ra. A plausible explanation of this behavior can be given as follows:
- at low Ra, the presence of nanoparticles, whatever their type, encourages more the conduction regime and reinforces the viscosity, reducing, thus, the convection heat transfer with
- at high Ra, the viscosity forces are dominated by the buoyancy ones, which offers the possibility to the thermal diffusion, whose importance increases with to reduce the heat transfer without been able to affect significantly the flow.
Finally, the nanoparticles kind seems to affect slightly, in progressive manner, c| and Nu with
increasing ® and Ra.
To render the obtained results more useful, the heat transfer results are correlated in terms of Nu versus adequate combinations, including ® and Ra > 104, in the following mathematical form:
Nu = (0.019 - 0.103® + 0.193®2)Ra0646 , (36)
which remains valid for any type of nanoparticle, since this is without effect on Nu .
The results obtained with Eq. (36), for water based Cu nanofluids (for example) and various values of Ф, are presented in Fig. 8 with dashed lines. They are seen to be in good agreement with both the parallel flow solution and the numerical results with a maximum difference not exceeding 3%.
In view of these results, it is clear that the Maxwell-Garnett model chosen, for the dependence of the effective conductivity on the volume fraction of nanoparticles, leads to deterioration of heat transfer instead of its enhancement, against all odds. This paradoxical behavior may be imputed to the fact that such a model does not take into account some physical phenomena that occur in the presence of nanoparticles within the base fluid when flowing under buoyancy-driven effects. The degradation observed may also depend on the particle density, concentration, viscosity as well as the aspect ratio of the cavity. Such a behavior was observed experimentally by Putra et al. [13], while studying natural convection of nanofluids inside a horizontal cylinder heated and cooled from the two ends respectively, and numerically by Abouali and Falahatpisheh [14] in their study of the same phenomenon in a vertical annuli with the inner and outer walls differentially heated .
5. Conclusion
In this paper a numerical and analytical study on natural convection in a two-dimensional horizontal shallow enclosure, filled with nanofluids, has been performed in the case where both short vertical sides are submitted to uniform heat fluxes while the long horizontal ones are insulated.
The full partial differential equations, governing the problem, have been solved numerically using a finite volume method. The numerical results presented have been limited to water-based solutions, i.e Pr = 7. The computations have been conducted with five different types of nanoparticles (Ag, Cu, CuO, Al2O3, TiO3) and with the governing parameters, A, Ra and Ф, varying, respectively, in the ranges 1 < A < 8, 102 < Ra < 107, 0 < Ф A < 0.2. The main findings of such an investigation are summarized as follows:
- The fluid flow and heat transfer characteristics have been found insensitive to any increase of the aspect ratio when this parameter is large enough A > 8.
- The approximate analytical solution, developed on the basis of the parallel flow hypothesis in the core region of the cavity, agrees perfectly with the numerical solution, obtained by solving the full governing equations.
- Increasing the Rayleigh number is associated with increasing the strength of buoyant flow and therefore enhancing the heat transfer rate.
- Against all odds, the addition of nanoparticles into the pure water leads to a deterioration of natural convection heat transfer.
Based on this later point, which is paradoxal and difficult to argue, further investigations in this subject are necessary in order to be able to use nanofluids for natural convective heat transfer enhancement.
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International Scientific Journal for Alternative Energy and Ecology № 01 (105) 2012
© Scientific Technical Centre «TATA», 2012
