Numerical investigation of combined convection flow in a cavity subjected to a nanofluid with an inside hot obstacle: effect of diameter of nanoparticles and cavity inclination angles

Hemmat Esfe, Mohammad; Saedodin, Seyfolah

doi:10.22075/jhmtr.2016.428

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Numerical investigation of combined convection flow in a cavity subjected to a nanofluid with an inside hot obstacle: effect of diameter of nanoparticles and cavity inclination angles

Articles in Press, Accepted Manuscript , Available Online from 20 June 2016

Document Type: Full Lenght Research Article

DOI: 10.22075/jhmtr.2016.428

Authors

Mohammad Hemmat Esfe

; Seyfolah Saedodin

Semnan University

Abstract

The present paper focuses on problem of mixed convection fluid flow and heat transfer of Al₂O₃-water nanofluid with temperature and nanoparticles concentration dependent thermal conductivity and effective viscosity inside Lid-driven cavity having a hot rectangular obstacle. The governing equations are discretized using the finite volume method while the SIMPLER algorithm is employed to couple velocity and pressure fields. Using the developed code, the effects of cavity inclination angle, diameter and solid volume fraction of the Al₂O₃ nanoparticles on the flow and thermal fields and heat transfer inside the cavity are studied. The obtained results show that the average Nusselt number for all range of solid volume fraction decreases with increase in diameter of nanoparticles. Also the results have clearly indicated that With increasing slope of the cavity to 90 degree, heat transfer continuously decreases at all studied Richardson numbers

Keywords

Nanofluid; Variable properties; Solid volume fraction; lid-driven cavity; Diameter of nanoparticles

Main Subjects

Nano Heat transfer

Full Text

Numerical investigation of combined convection flow in a cavity subjected to a nanofluid with an inside hot obstacle: effect of diameter of nanoparticles and cavity inclination angles

Mohammad Hemmat Esfe¹, Seyfolah Saedodin¹

¹Faculty of mechanical engineering, Semnan University, Semnan, Iran

Abstract:

Keywords: Nanofluid, Variable properties, Solid volume fraction, lid-driven cavity, Diameter of nanoparticles

*Greek symbols*		*Nomenclature*
thermal diffusivity, m² s	α	specific heat, J kg^-1 K^-1	c_p
thermal expansion coefficient, K^-1	β	Grashof number	Gr
dimensionless temperature	θ	gravitational acceleration, m s^-2	g
dynamic viscosity, Kg m^-1 s^-1	μ	heat transfer coefficient, W m^-2 K^-1	h
kinematic viscosity, m² s^-1	ν	enclosure length, m	L
density, kg m^-3	ρ	thermal conductivity, W m^-1 K^-1	k
volume fraction of the nanoparticles	φ	Nusselt number	Nu
		pressure, N m^-2	p
		dimensionless pressure	P
*Subscripts*		Prandtl number	Pr
cold	c	heat flux, W m^-2	q
effective	eff	Reynolds number	Re
fluid	f	Richardson number	Ri
hot	h	dimensional temperature, K	T
nanofluid	nf	dimensional velocities components in x and y direction, m s^-1	u, v
solid particles	s	dimensionless velocities components in X and Y direction	U, V
wall	w	lid velocity	U₀
		dimensional Cartesian coordinates, m	x, y
		dimensionless Cartesian coordinates	X, Y

1. introduction:

Nanofluids are created by suspending nanometer-sized particles (less than 100 nm) in a pure fluid such as water, ethylene glycol or propylene glycol. The first to coin the “nanofluids” for these fluids with superior thermal properties was Choi (1995). Existence of high thermal conductivity metallic nanoparticles (e.g., copper, aluminum, silver and Titanium) increases the thermal conductivity of such mixtures, thus enhancing their overall heat transfer capability (Xuan et al. 2003). In recent years, Nanofluids have attracted attention as a new innovation of heat transfer fluids in building heating, in various heat exchangers, in plants and in automotive cooling applications, because of their excellent thermal performance. Various benefits of the application of nanofluids include: improved heat transfer, heat transfer system size reduction, minimal clogging, micro channel cooling and miniaturization of systems (choi, 1995).

Numerous investigates have been conducted on the thermophysical properties of nanoﬂuids (effective dynamic viscosity, thermal conductivity and etc.) and the energy transport in nanofluids. Study of thermophysical properties of nanoﬂuids can be observed in several literatures such as Lee et al. (2000), Xie et al. (2002), Patel et al. (2005), and Chang et al. (2005), Also many theoretical, numerical and experimental studies on influence of existence of nanoparticle in convective heat transfer have been reported.

On the other hand, fluid flow and heat transfer in a cavity filled by pure fluid which is driven by buoyancy and shear have been studied extensively in literature. Mixed Convection (a kind of convection including both natural and forced convection) has significant role in many applications in industry and engineering has been specified. lake and reservoirs (Imberger and Hamblin, 1982), food processing, crystal growth (Moallemi and Jang, 1992), electronic cooling devices, drying technologies, solar ponds (Cha and Jaluria, 1984) solar collectors(Ideriah, 1980) and float glass production (Pilkington, 1969), are among its current applications.

Several investigates on mixed convection in single or double lid-driven enclosure flow and heat transfer including different cavity geometries and configurations, different base fluids and boundary condition have been reported. Particularly in recent years some interesting researches have been done such as Talebi et al. (2010), Abunada et al.(2010), Mahmoodi(2011), Sadodin et al.(2011), Abbasian Arani et al.(2012), Fereidoon et al.(2013) and Zarei et al.(2013).

Ghasemi and Aminossadati(2010), Arefmanesh and Mahmoodi(2011) and Rabbani bidgoli et al. (2012).Another work that was done by Nikfar and Mahmoodi (2012) also approved the above statement. They studied about natural convection in a square cavity filled with Al₂O₃–water nanoﬂuid. The horizontal walls of the cavity were insulated while left and right wavy side walls of cavity were maintained at high and low constant temperatures. They demonstrated that increase in the volume fraction of the nanoparticles, the average Nusselt number of the hot wall also increases.

Effect of existence a obstacles within the cavity is one of the interesting investigations for researchers. Recently free convection fluid flow and heat transfer investigated numerically by Mahmoodi and Mazrouei(2012). Their work was included of Cu-water nanofluid around the adiabatic square bodies at the center of a square cavity. They illustrated that for most Rayleigh numbers the Nusselt number increases with increase in the volume fraction of the nanoparticles. They also showed that at low Rayleigh numbers by increasing the size of the adiabatic square body, the rate of heat transfer decreases and opposite is true at high Rayleigh numbers. Mixed convection of Al₂O₃-water nanofluid in cavity with hot moving bottom wall and cold right, left, and top walls is investigated numerically by Mahmoodi (2012). Also in valuable study, the effect of nanofluid variable properties on mixed convection in a rectangular cavity has been analyzed by Mazrouei Sebdani et al. (2012).

Motivated by the investigations mentioned above, the purpose of the present work is to consider mixed convection flows of Al₂O₃–water nanofluid in an inclined square cavity with an inside heated obstacle and a moving upper lid that moves uniformly in the horizontal plane. Also the effects of the Richardson number, diameter and solid volume fraction of the Al₂O₃ nanoparticles on the flow and thermal fields and heat transfer inside the cavity are studied.

2. physical modeling:

Fig. 1 displays a two-dimensional inclined Lid-driven square cavity with an inside heated obstacle. The height and the width of the square cavity are denoted L. The cavity is filled with a suspension of Al₂O₃ nanoparticles in water. Left and horizontal walls are insulated whereas the right wall is kept at low temperature T_c. in order to induce the buoyancy effect, An obstacle with a relatively higher temperature, T_h, is located on the bottom wall of the cavity.

Fig. 1. Schematic diagram of current study

The nanofluid in the enclosure is Newtonian, incompressible and laminar. In addition, it is assumed that both the fluid phase and nanoparticles are in the thermal equilibrium state and they flow with the same velocity. The density variation in the body force term of the momentum equation is satisfied by Boussinesq’s approximation. The thermophysical properties of nanoparticles and the water as the base fluid at T = 25˚C are presented in Table1.

Table 1. Thermophysical properties of water and nanoparticles at T =25˚C.

Physical properties	Fluid phase (Water)	Solid (Al₂O₃)
Cp(J/kg k)	4179	765
(kg/m3)	997.1	3970
K (W m-1 K-1)	0.6	25
β×10^-5 (1/K)	21.	0.85
μ×10^-4(Kg/ms)	8.9	……..
d_p (nanometer)	---	47

The thermal conductivity and the viscosity of the nanofluid are taken into consideration as variable properties; both of them change with volume fraction and temperature of nanoparticles. Under the above assumptions, the system of governing equations is:

(1)

(2)

(3)

and

(4)

The dimensionless parameters may be presented as

(5)

Hence,

(6)

The dimensionless form of the above governing equations (1) to (4) become

(7)

(8)

(9)

and

(10)

2.1. Thermal diffusivity and effective density

Thermal diffusivity and effective density of the nanofluid are

	(11)
	(12)

2.2. Heat capacity and thermal expansion coefficient

Heat capacity and thermal expansion coefficient of the nanofluid are therefore

	(13)
	(14)

2.3. Viscosity

The effective viscosity of nanofluid was calculated by:

(15)

This well-validated model is presented by Jang et al. (2007) for a fluid containing a dilute suspension of small rigid spherical particles and it accounts for the slip mechanism in nanofluids. The empirical constant and are _0.25 and 280 for Al₂O₃, respectively.

It is worth mentioning that the viscosity of the base fluid (water) is considered to vary with temperature and the flowing equation is used to evaluate the viscosity of water,

where .

(16)

2.4. Dimensionless stagnant thermal conductivity:

The effective thermal conductivity of the nanoparticles in the liquid as stationary is calculated by the Hamilton and crosser(H-C model)(1962), which is:

(17)

2.5. Total dimensionless thermal conductivity of nanofluids:

(18)

This model was proposed by Xu et al. (2006) and it has been chosen in this study to describe the thermal conductivity of nanofluids. c is an empirical constant(e.g. c = 85 for the deionized water and c = 280 for ethylene glycol) but independent of the type of nanoparticles. Nu_p is the Nusselt number for liquid flowing around a spherical particle and equal to two for a single particle in this work. The fluid molecular diameter df =4.5*10^-10(m) for water in present study. The fractal dimension D_f is determined by:

where d_p,maxand d_p,minare the maximum and minimum diameters of nanoparticles, respectively. Ratio of minimum to maximum nanoparticles d_p,min/d_p,max is R.

3. Numerical approach:

Governing equations for continuity, momentum and energy equations associated with the boundary conditions in this investigation were calculated numerically based on the finite volume method and associated staggered grid system, using FORTRAN computer code. The SIMPLER algorithm is used to solve the coupled system of governing equations. The convection terms is approximated by a hybrid-scheme which is conducive to a stable solution. In addition, a second-order central differencing scheme is utilized for the diffusion terms. The algebraic system resulting from numerical discretization was calculated utilizing TDMA applied in a line going through all volumes in the computational domain. To verify grid independence, numerical procedure was carried out for nine different mesh sizes.

Fig. 2. Mesh valid at X=0.6L, φ=0.05, γ=90˚

As can be observed, 101 × 101 uniform grid size yields the required accuracy and was hence applied for all simulation exercises in this work as presented in the following section.

To ensure the accuracy and validity of the this new model, we analyze a square cavity filled with base fluid with Pr = 0.7 and different Ra numbers. Table 2 shows the comparison between the results obtained with the our new model and the values presented in the literature. The quantitative comparisons for the average Nusselt numbers indicate an excellent agreement between them.

Table 2.code validation shows the comparison between the results in present study and other research
Tiwari and Das (2007)	Present study

		(b) Ra = 10⁴
16.1439	16.052	u_max
0.822	0.817	Y
19.665	19.528	v_max
0.110	0.110	X
2.195	2.215	Nu_ave
		(c) Ra = 10⁵
34.30	36.812	u_max
0.856	0.856	Y
68.7646	68.791	v_max
0.05935	0.062	X
4.450	4.517	Nu_ave
		(d) Ra = 10⁶
65.5866	66.445	u_max
0.839	0.873	Y
219.7361	221.748	v_max
0.04237	0.0398	X
8.803	8.795	Nu_ave

4. Results and discussion:

In this paper, thermal characteristics and flow patterns inside an inclined cavity filled with nanofluid with hot barrier are investigated. Some parameters such as diameter of nanoparticles, cavity inclination angles, solid volume fraction and Richardson number are considered and their effects on streamlines, isotherms and total heat transfer are studied. Grashof number is assumed constant and equal to 10⁴ while

Figure 3 shows flow and temperature behavior in different cavity inclination angles at . Flow pattern for horizontal situation of the cavity shows presence of a clockwise primary cell in upper parts of the cavity while one small vortex is formed at right side of hot barrier. Presence of vortexes results from two buoyancy force due to temperature difference and shear force due to upper lid movement. In horizontal situation, buoyancy and shear forces assist each other. With increasing slope of this cavity, buoyancy acts in a direction opposite to shear forces and formation of a small vortex near upper lid proves this fact. It is expected that increasing cavity inclination angles to 90^oresults in relative neutralization of buoyancy forces and shear forces and consequently vortex forces and heat transfer inside the cavity decrease. Isotherm lines in horizontal position of the cavity also show density of lines near isotherm walls. With increasing slope of the cavity, density of isotherm lines and thereupon temperature gradient near walls decrease and this decrease is expected to reduce heat transfer inside the cavity.

Fig. 3 . streamlines and isotherms in different inclination angles in Ri=100, φ=0.05, d_p=20 nm

Figure 4 portrays streamlines and isotherms to demonstrate the effect of nanoparticles diameter for Ri=1, T=300 and . 4 different diameters (20, 40, 60 and 80 nano meter) are shown in this figure. As it is observed, changes in nanoparticles diameter produces no substantial changes in flow patterns and temperature. Flow pattern in this parameter range shows formation of two clockwise and anti-clockwise vortexes which upper clockwise vortex is stronger than lower and occupies more spaces of the cavity. The strength of vortexes slightly decreases with increasing nanoparticles diameter but their core and primary form is maintained. Temperature lines also demonstrate formation of a thermal boundary layer inside the cavity and near the barrier. Isotherm lines are slightly separated from each other with increasing diameter of nanoparticles and therefore, temperature gradient decreases. According to isotherm lines it is expected that increasing diameter of nanoparticles causes total heat transfer inside the cavity to decrease slightly.

		d_p=20 nanometer
		d_p=40 nanometer
		d_p=60 nanometer
		d_p=80 nanometer
Fig. 4. streamlines and isotherms in different diameter of nanoparticles at Ri=1, φ=0.05, T=300, γ=30

Variation of isotherms and streamlines versus volume fraction of nanoparticle inside the cavity are demonstrated in figure 5 at Ri=0.01, dp=40, T=300, . Streamlines show formation of a strong vortex in upper and middle parts of the cavity while two strong and weak vortexes can be observed at sides of hot rectangular barrier. No significant changes occur in flow pattern with increasing volume fraction of nanoparticles. Temperature lines in this range of parameters present intense density of isotherm lines near isothermal walls. Density of lines decreases very slightly with increasing volume fraction and temperature gradient reduces. Despite a reduction in temperature gradient with increasing volume fraction, no accurate prediction can be made for total heat transfer inside the cavity. Increasing solid concentration increases thermal conductivity of nanofluid while it slightly decreases temperature gradient.

Fig. 5. streamlines and isotherms in different solid volume fraction at Ri=0.01, dp=40, T=300,

Figure 6 illustrates heat transfer inside the cavity versus changes of slope and Richardson number for dp=20, T=300, . As it is obvious in this diagram, with increasing Richardson number and (buoyancy forces dominate shear force) Nusselt number and consequently heat transfer inside the cavity decrease. On the other hand, with increasing cavity inclination angles from horizontal position, buoyancy force and shear force counteract each other and as a result total heat transfer inside the cavity decreases. This matter was discussed earlier when figure 2 was studied.

Fig. 6. Variation of Nusselt number versus cavity inclination angle and Richardson number for dp=20, T=300, .

Figure 7 shows variation of Nusselt number versus Richardson number for different diameters of nanoparticles dispersed in water for T=300^o, and . At this condition, values of Nusselt number for all diameters of nanoparticles decrease with increasing Richardson number. With increasing nanoparticles diameter from 20 nm to 40 nm, a relatively significant change occur in heat transfer while with increasing diameter more than 40 nm, no significant changes occur in heat transfer and Nusselt number.

Fig. 7. variation of Nusselt number versus Richardson number for different diameters of nanoparticles dispersed in water for T=300^o, and .

Figure 8 illustrates variation of Nusselt number versus changes in Richardson number and volume fraction of nanoparticles at d_p=40, T=300^o and . As it is seen in this diagram, adding nanoparticles to base fluid causes a considerable increase in heat transfer inside the cavity and this increasing trend continues with increasing volume fraction. On the other side, in this range of parameters increasing Richardson number (i.e. buoyancy force dominates shear force) also results in a reduction in Nusselt number and consequently heat transfer inside the cavity.

Fig. 8. variation of Nusselt number versus Richardson number and volume fraction of nanoparticles at d_p=40, T=300^o and

Conclusion:

In this paper thermal behavior of nanoparticles inside an inclined cavity with a hot barrier was investigated. Effects of some important parameters including nanoparticle diameter, cavity inclination angles, volume fraction and Richardson number were studies and below results were obtained:

1- With increasing Richardson number and predominance of buoyancy force over shear force, Nusselt number and heat transfer decrease.

2- In horizontal situation that buoyancy and shear forces have the same direction, the maximum values are obtained for Nusselt number and heat transfer.

3- With increasing slope of the cavity to 90 degree, heat transfer continuously decreases at all studied Richardson numbers.

4- With decreasing diameter of dispersed nanoparticles in water, heat transfer increases.

5- Adding nanoparticles to base fluid in addition to maintaining flow pattern causes a significant increase in heat transfer inside the cavity.

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