Effects of Nano-Fluids Types, Volume Fraction of Nano-Particles, and Aspect Ratios on Natural Convection Heat Transfer in Right-Angle Triangular Enclosure.

This study inve stigates natural convection heat transfer and f luid flow characteristic of water based nano-fluids in a right-angle triangular enclosure, where the left vertical wall is insulated, the right inclined wall is cooled, and the horizontal wall is heated by spatially varying temperature. Governing eq uations are solved using stream-vorticity formulation in curvilinear c oordinates. Streamlines, i sotherms, local a nd average Nusselt number, more over to NUR factor are used to present the corresponding flow an d thermal fields inside the triangular enclosure. Calculation were performed for three aspect ratio of enclosure geometry (AR=0.5, 1, 2), solid volume fractions of nano-particles ranging from PHI =0, to 4%, a nd Rayleigh number varying from 10 4 to 10 6 . Three types of nan o-particles are taken into c onsideration: Cu, Al 2 O 3 , and TiO 2 . The results show that, the average heat transfer rate increases significantly as particle volume fraction and Rayleigh number increase. Als o, the type of nano-fluid is a key factor for heat transfer enhancement where the high values are obtained when using Cu, TiO 2 , and Al 2 O 3 nano-particles res pectively. Finally, it is observed tha t the aspect rat io of t he enclosure is one of the most important on flow and heat transfer. Increasing the AR leads that to increase the flow strength and heat transfer rate.


Introduction
The primary limitation of conventional fluids such as water, ethylene glycol or propylene glycol is their low thermal conductivity.Therefore in recent years nano-fluids have attracted more attention for cooling in various industrial applications.Such fluids consist of suspended nano-particles which have a better suspension stability compared to millimeter or micrometer sized ones.Use of metallic -particles with high thermal conductivity will increase the effective thermal conductivity of these types of fluid remarkably.For instance just 0.3% volume fraction of copper nano-particles with 10 nm diameter led to an increase of up to 40% in the thermal conductivity of ethylene glycol [1], indeed when nanosized particles are added to liquid flow.
Nano-fluids, as a kind of new engineering material consisting of nanometer sized additives and base fluids.Have attracted great attention of investigation for its superior thermal properties and many potential applications.Many investigations on nano-fluids were reported and especially some interesting phenomena.Yanjiao et.al.,2009 [2] shows a review of recent development in research on synthesis and characterization of stationary nano-fluids and try to find some challenging issues that need to be solved for future research.Eiyad, 2009 [3] studied the heat transfer enhancement in horizontal annuli using variables properties of Al 2 O 3 -water nano-fluid.It was observed that for 4 10 ≥ Ra , the average Nusselt number was reduced by increasing the volume fraction of nanoparticles.Also, the results show that, the difference in Nusselt number between the Maxwell-Garnett and Chon et.al. models prediction is small, but there was a deviation in prediction at 3 10 = Ra and this deviation becomes more significant at high volume fraction of nano-particles.[4] showed a numerical analysis of the effect of inclination angle on natural convection heat transfer and fluid flow in a twodimensional enclosure filled with Cuwater nano-fluid.The performance of nano-fluids is tested inside an enclosure by taking into account the solid particle dispersions.The angle of inclination was used as a control parameter for flow and heat transfer, results showed that, percentage of heat transfer enhancement using nano-particles decreases for higher Rayleigh numbers.).Results show that, the Nusselt number was not sensitive to the volume fraction at low Rayleigh number but it sensitive to the aspect ratio.Also, they found that, at high Rayleigh number the average Nusselt number was more sensitive to the viscosity models than to the thermal conductivity models.

Eiyad et. al., 2009
Elif, 2009 [6] investigate natural convection heat transfer for water based nano-fluids in an inclined square enclosure where the left vertical side is heated with a constant heat flux, the right side is cooled, and the other sides are kept adiabatic.The governing equations are solved using polynomial differential quadrature method.Calculation were performed for inclination angles from o 0 to o 90 , solid volume fractions ranging from 0% to 20%, constant heat flux heaters of lengths 0.25, 0.5 and 1., Rayleigh number varying from 10 4 to 10  presented the periodic natural convection in an enclosure filled with nano-fluids.Whilst a heat source with oscillating heat flux is located on the left wall of the enclosure, the right wall is maintained at a relatively low temperature and the other walls are thermally insulated.Based upon numerical predictions, the effects of pertinent parameters such as Rayleigh number, solid volume fraction, heat source position, type of nano-particles an oscillation period are examined.A periodic behavior is found for the flow and temperature fields as results of the oscillating heat flux.The results of this study can be used in the design of an effective cooling system for electronic components to help ensure effective and safe operational conditions.In fact, convective heat transfer is affected by the thermophsical properties of the nano-fluid such as viscosity and thermal conductivity.Simultaneous effects of the buoyancy force, centrifugal force, and nano-particles concentration has been presented and discussed.The nanoparticle volume fraction dose not has a direct effect on the secondary flow, axial velocity and the skin friction coefficient.
Nano-fluids are a new kined of heat transfer fluids containing a small quantity of non-sized particles (usually less than 100 μm) that are uniformly and stably suspended in a liquid.Therefore, the scope of the current research is to present the effects of nano-particles types (Cu, Al 2 O 3 , TiO 2 ), volume fraction of nano-particles (PHI=0, 0.1, 0.2, 0.4), aspect ration of the right angle triangular enclosure (AR=0.5, 1, 2), Rayleigh number (Ra=10 3 to 10 6 ) on flow field, temperature distribution and natural convection heat transfer enhancements.
Based on above literature survey and to the Author's knowledge, no previous study takes these parameters on this geometry of the enclosure (right-angle triangular).

Geometrical Description
The proposed physical model for a two-dimension right-angle triangular enclosure (height H, width W) filled with nano-fluid which is Newtonian, incompressible, and laminar as shown in fig.
(1.B).The study applied for three types of nano-fluids (Cu-water, Al 2 O 3water, TiO 2 -water), three aspect ratio of the enclosure geometry (AR=0.5, 1, 2), different values of nano-particles volume fraction (PHI=0, 0.1, 0.2, 0.4), for wide range of Rayleigh number (Ra=10 3 -10 6 ).The vertical wall of the enclosure assumed to be insulated and the bottom wall considered to be heated wall and spatially varying with sinusoidal temperature T h as the expression below: [ ] The effective density of the nano-fluid is given as: The heat capacitance of the nano-fluid is expressed as [17, 18]: The effective thermal conductivity of the nano-fluid is approximated by Maxwell-Garnetts model: The use of this equation is restricted to spherical nano-particles where it dose not account for other shapes of nano-particles.This model is found to be appropriate for studying heat transfer enhancement using nano-fluids [16,17].The viscosity of the nano-fluid can be approximated as viscosity of a base fluid f µ containing dilute suspension of fine spherical particles and is given by [19]: The two components of velocities are given by the following relations, respectively: The following dimensionless groups are introduced: & … (10) By using the dimensionless parameter the equations are written as: The dimensionless boundary conditions are written as:

On the bottom (Heated) wall:
[ ] On the vertical wall:

Grid Generation
It is of great importance to implement the surrounding boundaries of arbitrary curvature in the governing equations and to become a part of solution.The proper choice of the used technique to transfer the physical domain into computational domain has a great influence on the solution.Elliptical Partial differential equations method is the most general, applicable and programmable method.
There are two types of generating system, Laplace equation type and Poisson equation type.
The second type was used in this study.
The transformation function is individually obtained by solving the following two elliptic Poisson equations: Where P and Q are two arbitrary function specified to adjust the local density of the grids.Meanwhile, the orthogonality of the generated grids system can be improved by carefully setting the boundary conditions.Figure (1.C) show symbol of curvilinear grid system applied in this study.Then, the set of non-dimensional transformed equation are:

Grid Testing and Code Validation
Figure (2) demonstrates the influence of number of grid points for a test case of fluid confined within the present configuration at Ra=10 5 , AR=1., and PHI=0.Figure shows the distribution of u, v, and θ for point located in the mid sections of the enclosure with grid number, it is clear that, the grid system of 41 41 * is fine enough to obtain accurate results.Therefore, adopted a grid system of 41 41 * .
In order to validate the present code, the streamlines and isotherms are calculated as shown in figure (3) for natural convection in a differentially heated enclosure filled with water (Pr=6.2) for case study of [7] as a present in figure (1.A).In this test the enclosure filled with Cu-water nano-fluid with (Ra = 10 5 , y p = 0.5, AR = 1, h = 0.1, PHI = 0.1).The present results are in good agreement with results of [7] as shown in figure (3), therefore the numerical procedure is reliable and efficient.

Numerical implementation
The governing equation in the curvilinear coordinates (equations 15, 16, and 17) as well as boundary conditions were discretized by finite difference method.In this study the finite difference equation were derived by using central difference approximation for the partial derivatives except the convective terms for which upwind difference formula was employed.Derivative at the boundary were approximated by three point forward or backward difference.The explicit method was chose for the solution of flow and energy fields, while relaxation method was chose for stream function calculation.
In order to evaluate how the presence of the nano-fluids affect the heat transfer rate along the hot wall according to the parameters Rayleigh number, nanoparticles volume fraction, and aspect ratio it is necessary to observe the variation of the local Nusselt number on the hot wall.In generalized coordinate the local and average Nusslet number defined as: The above integral was calculated using Simpson's rule 1/3 method.
To show the effect of the nanofluids on heat transfer rate, we introduce a variable called Nusselt number ratio (NUR) with its definition given as: If the value of NNR greater than 1 indicated that the heat transfer rate is enhanced on that fluid, whereas reduction of heat transfer is indicated when NUR is less than 1.

Results and Discussion
Results presented into two sections, the first section will focus on flow and temperature fields, which contents streamlines, isotherms for pure fluid and nano-fluids types.Another case of results represented the distribution of local Nusselt number on the heated wall, average nusselt number, and NUR factor.

Flow and Temperature Fields
Flow and temperature fields are simulated using streamlines and isotherms for selected parameters of nano-fluids types, nano-particles volume fraction, and geometry.Effects of Rayleigh number on streamlines and isotherms are presented in fig.(4)  three vortices are formed for nano-fluid case (plotted by dashed lines) and two vortices are for pure fluid case (plotted by solid lines).This is due to different thermal physical properties in theses cases and to non uniform heating from below, so the heated flow moves up from the horizontal heated wall of the enclosure and impinges to the cold inclined wall.The first vortex is formed between heated horizontal wall and the inclined cooled wall of the triangular enclosure, which rotates in clockwise direction.Its shape and length are almost same in this range of Rayleigh number.The second vortex located between the heated wall and insulated wall which rotates in counterclockwise direction.
In this range of Rayleigh number ( Aspect ratio AR of the triangular enclosure is important parameter for flow and temperature fields.This changed from 0.5 to 2 along the present study.Thus, fig. (7) shows the effect of AR on streamlines (on the left) and isotherms (on the right) for Cu-water nano-fluid at Ra=10 6 and PHI=0.2.Two vortices are observed for AR=0.5, the right vortex rotates in clockwise direction while left one in counterclockwise direction.The separated line between them take place after mid point of the length of bottom wall of the enclosure due to non uniform heating from below and different boundaries about heated horizontal wall.These vortices are have an egg-shaped recirculation moreover to be strong flow strength became here in nano-fluid case comparing with pure fluid case.
For AR=1. , there are two vortices weak one on the right side and big vortex in the other side of the enclosure in pure fluid case, but in nano-fluid case this description could be changed where a small vortex located in the left side and big strong vortex located in the other side with high intensity of streamlines and strong flow strength.In AR=2., three vortices distributed in the enclosure with different direction and strength, on in the right side, left side, and on the top region of the enclosure.But in nano-fluid case, these vortices could grouping and become one strong vertex in the middle region of the enclosure.This phenomena at streamlines which effects on isotherms which are uniform nearly at AR=0.NUR factor are calculated to obtain the case of heat transfer enhancement could be happen, this results presented as a percentage values in table (2).

Conclusion
In these analysis, the results of the study of natural convection in triangular enclosure filled with nano-fluids with non uniform heating from bellow are presented for main parameters of interested as Ra, AR, nano-particles type, and nano-partical volume fraction.In view of the obtained results, following findings may be summarized:   PDF created with pdfFactory Pro trial version www.pdffactory.com[Ψ ] [Ψ] AR=2. [Ψ] ) While the other wall kept at cooled temperature T c .Governing Equation and Formulation Figure (1.B) shows a schematic diagram of right-angle triangular enclosure.The fluid in the enclosure is a water-based nano-fluid containing different types of nano-particles such as (Cu, Al 2 O 3 , TiO 2 ).It is assumed that the base fluid (water) and nano-particles are in thermal equilibrium and no slip accurse between them.The thermo-physical properties of the nano-fluid are given as shown in Table (1) [7]: The thermo-physical properties of the nano-fluid are assumed to be constant except for the density variation which is approximation by the Boussinesq model.The governing equations for the laminar PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.28, No.16, 2010 Effects of Nanofluids types, Volume Fraction of Nanoparticles, and Aspect Ratios on Natural Convection Heat Transfer in Right-Angle Triangular Enclosure 5369 and steady state natural convection in terms of the stream function-vorticity formulation are: Stream function-vorticity pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.28, No.16, 2010 Effects of Nanofluids types, Volume Fraction of Nanoparticles, and Aspect Ratios on Natural Convection Heat Transfer in Right-Angle Triangular Enclosure 5370 And the boundary conditions become:PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.28, No.
between nanofluid case and pure fluid case are the strong values of streamlines and a third vortex located in the top corner of the enclosure in nano-fluid case.Then, for 6 10 = Ra the two vortex are exist in the enclosure for nano-fluid and pure fluid cases, but the location and direction are opposite moreover to the strong convection will become here.These effects in heat transfer could be match on isotherms (right colum) in figure (4) where the isotherms are regular and uniform distribution in the enclosure for low Rayleigh number.Where the quasiconduction heat transfer regime is formed clearly for low Rayleigh number, but if Rayleigh numbers increase the convection regime become strong and clear.Also, for high Rayleigh number the isotherms become non uniform and random due to increasing of energy exchange rates in the fluid.Generally, isotherms change and deform according to change in streamlines.If nano-particles changed as shown in fig.(5) for Al 2 O 3 and fig.(6) for TiO 2 the same behavior could be see for streamlines and isotherms with decreasing in values of absolute maximum streamlines according to the reduction in heat transfer due to the decreasing in the thermo-physical properties of Al 2 O 3 and TiO 2 nano-particles comparing with Cu nano-particle.

l
5 and become random and irregular in the enclosure If AR increase due to vortices and non uniform heating.This description could be see in fig.(8) and fig.(9) for PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.28, No.respectively, with decreasing in flow strength and special case in AR=1 for Al 2 O 3 in fig.(8) where, three vortices in the enclosure comparing with the other cases.Nusselt number and NUR factor The changes in l Nu for water based Cu, Al 2 O 3 , TiO 2 nano-particles at different Ra, AR, and PHI presented in figures (10), (11), and (12) respectively.Nu are parable and max.value of l Nu located after mid point of the axis.If 6 10 = Ra , the distribution will become fluctuation along the heated wall due to the circulation strength increases as a result of higher buoyancy force this results in an increase of the l Nu moreover to strong heat exchange between the particles and base fluid .Fig. (13) presents the l Nu distribution along the heated wall for different nano-particles and AR at Ra=10 5 , and PHI=0.1 .Results show that, high l Nu accurse at Cu, TiO 2 , Al 2 O 3 , nano-particles respectively.Results show the important nano-fluid technology in engineering application.Fig. (14) shows the variation of ave Nu with nano-particle volume fraction for different AR, Ra, and nano-particles type.Where, the ave Nu increase with increase PHI for all cases, due to increase the heat exchange between the solid particles and based fluid for this range of PHI.Other results can be seen in fig.(15) for ave Nu with nano-particle volume fraction, where as Ra increase the circulation strength increase as results of increasing in buoyancy force, the relation nearly is linear.

Figure ( 12 )Figure ( 14 )
Figure (12) Variation of local Nusselt number along the heated wall for different aspect ratio, and Rayleigh number for Al 2 O 3 -water nanofluid Left side: PHI=0.2 , Right side: PHI=0.4

Figure ( 13 )Figure ( 15 )
Figure (13) Variation of local Nusselt number along the heated wall for different aspect ratio, and Nanoparticles type for Ra=10 5 , PHI=0.1 PDF created with pdfFactory Pro trial version www.pdffactory.com

& Tech. Journal, Vol.28, No.16, 2010 Effects of Nanofluids types, Volume Fraction of Nanoparticles, and Aspect Ratios on Natural Convection Heat Transfer in Right-Angle Triangular Enclosure 5367
. Five types of nano-particles are taken into consideration Cu, Ag, CuO, Al 2 O 3 , and TiO2.The results show that, the average heat transfer rate increases PDF created with pdfFactory Pro trial version www.pdffactory.comEng.

Tech. Journal, Vol.28, No.16, 2010 Effects of Nanofluids types, Volume Fraction of Nanoparticles, and Aspect Ratios on Natural Convection Heat Transfer in Right-Angle Triangular Enclosure 5372
for Cu-water nano-fluid, AR=1., and PHI=0.2,visualization are given from Ra=104 to Ra=106.It can be seen from PDF created with pdfFactory Pro trial version www.pdffactory.com

& Tech. Journal, Vol.28, No.16, 2010 Effects of Nanofluids types, Volume Fraction of Nanoparticles, and Aspect Ratios on Natural Convection Heat Transfer in Right-Angle Triangular Enclosure 5374
g Gravitational acceleration 2 / s m PDF created with pdfFactory Pro trial version www.pdffactory.comEng.

Variation of local Nusselt number along the heated wall for different aspect ratio, and Rayleigh number for Cu-water nanofluid. Left side
: PHI=0.2 , Right side: PHI=0.4PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.

Variation of local Nusselt number along the heated wall for different aspect ratio, and Rayleigh number for TiO 2 -water nanofluid Left side
: PHI=0.2 , Right side: PHI=0.4PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.