MIXED CONVECTION HEAT TRANSFER OF NANOFLUIDS IN A LID DRIVEN SQUARE CAVITY: A PARAMETRIC STUDY

Transcription

1 International Journal of Mechanical and Materials Engineering (IJMME), Vol. 8 (2013), No. 1, Pages: MIXED CONVECTION HEAT TRANSFER OF NANOFLUIDS IN A LID DRIVEN SQUARE CAVITY: A PARAMETRIC STUDY Z. Said 1,*, H.A. Mohammed 2, R. Saidur 1 1 Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia 2 Department of Thermofluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, UTM Skudai, Johor Bahru, Malaysia *Corresponding author s zaffar.ks@gmail.com Received 20 December 2012, Accepted 19 March 2013 ABSTRACT Laminar mixed convection heat transfer in a lid driven square cavity filled with nanofluids is investigated numerically. In this current study, the vertical walls are purported at steady but dissimilar temperatures, whereas the bottom and top horizontal walls are insulated. The top wall is set with uniform velocity and insulated. Al 2 O 3, CuO, SiO 2 and TiO 2 are the nanoparticles used with diameters 25nm, 40nm and 60nm. Results are presented in terms of streamlines, isotherms and Nusselt number for different parameters such as, the Reynolds number ( 700 Re 1600), the Rayleigh number 4 6 (10 Ra 10 ) and the volume fraction It was found that Al 2 O 3 -H 2 O has the highest Nusselt number, followed by TiO 2 -H 2 O, SiO 2 - H 2 O and lastly CuO-H 2 O. Nusselt number is higher for nanofluids compared to pure water. It was noted that with the increment in volume fraction, Nusselt number increases but with larger diameter of nanoparticles it reduces. Increase in the Reynolds number resulted in higher Nusselt number. Higher difference between the hot and cold walls also results in higher Nusselt numbers. Keywords: Mixed convection, Square cavity, Heat transfer, CFD, Nanofluids. NOMENCLATURE C p specific heat at constant pressure ( ) d diameter (nm) g gravitational acceleration ( ) Gr Grashof number k thermal conductivity ( Boltzmann s constant, kb = kr conductivity ratio N Avogadro s number, [ ] Pr Prandtl number ( Ra Rayleigh number ( ) Re Reynolds number ( ) Ri T Richardson number cold wall temperature (K) hot wall temperature (K) dimensional temperature U 0 top wall driven velocity u,υ velocity components in the x- and y-directions U, V dimensionless velocity components X, Y dimensionless coordinates Greek symbols thermal diffusivity volumetric coefficient of thermal expansion fluid kinematic viscosity T Tc Th Tc dimensionless time dimensionless temperature, Subscripts C cold eff effective f fluid p nanoparticle H hot nanofluid 1. INTRODUCTION It is significant from both theoretical and practical point of view that, mixed convection problem with lid-driven flows in enclosures are confronted in a number of engineering applications such as cooling of electronic devices, lubrication technologies, chemical processing equipment, drying technologies, etc. Suspending metallic nanoparticles in fluids is an innovative technique to enhance the thermal conductivities. This suspension would result in considerably larger thermal conductivity in comparison with conventional fluids. These nanoparticles significantly enhance the effective thermal conductivity and subsequently increase the heat transfer characteristics. Cheng (2011) investigated numerically the systematic characteristics of flow and heat transfer in a 2-D square cavity. A flow induced by the shear force results in due combined motion between the buoyancy force and upper lid due to bottom heating. Perpendicular walls were adiabatic while the bottom and top moving wall were isothermally purported, with the bottom wall having greater temperature than the top moving-wall. A positive effective of the solid concentration was noticed on the heat transfer enhancement for a given Reynolds number 48

2 and Rayleigh number by Talebi et al. (2010) for a liddriven square cavity. Higher volume fractions of nanoparticles reduce both fluid temperature and motion; however, the corresponding average Nusselt number increase was observed by Mansour et al. (2010). The two dimensional flow is characterized by the vortices that are initiated by the existence of buoyancy effect was concluded by Hussein and Hussain (2010). Mahmoudi et al. (2010) found that the average Nusselt number surged linearly with the increasing solid volume fraction of nanoparticles with fixed heat source geometry and a given Rayleigh number. Sivasankaran et al. (2010) observed that the average Nusselt number enhanced at first and then reduced when increasing the phase deviation from 0 to π. Bilgen and Oztop (2005) studied partially inclined open square cavities, which were formed by adiabatic walls and a partial opening. Abu- Nada and Chamkha (2010) showed that adding of Al 2 O 3 nanoparticles resulted in improved heat transfer by keeping modest and higher Ra in the enclosure. Shahi et al. (2010a) concluded that at a lower Re, higher effect of Richardson number was observed on the average bulk temperature. Tiwari and Das (2007) concluded that for the governing parameters Ri and χ the nanoparticles when immersed in a fluid were able to enhance the heat transfer capacity of base fluid. Koca (2008) showed that the heat transfer improved as the Ra number increased. Jahanshahi et al. (2010) concluded that the indecision associated with different formulas used for the effective thermal conductivity of the nanofluid had a great effect on the natural convection heat transfer characteristics in the enclosure. Magnetohydrodynamic (MHD) mixed convection in a lid driven cavity along with joule heating was studied numerically by Rahman et al. (2009). Rahman et al. (2010a) concluded that in the presence of a clean forced convection the obstacle had important effect on the flow field in the pure mixed convection region, and on the thermal field. The fluid flows and heat transfer induced by the combined effects of mechanically driven lid and buoyancy force within a rectangular cavity was investigated by Rahman et al. (2010b). Moraga et al. (2009) concluded that a higher heat transfer was found when the external flow with Ri = 1 had a lower Reynolds number (Re = 200). Oztop et al. (2008) found heat transfer to be a reducing function of Richardson number and thermal conductivity ratio and it also resulted in a reducing function for the wall thickness. Mamun et al. (2010a) concluded that the flow and thermal fields had strong dependence on diameter of the hollow cylinder in the cavity. Mixed convection heat transfers in a two-dimensional trapezoidal cavity with constant heat flux at the heated bottom wall while the isothermal moving top wall in the horizontal direction has been studied numerically (Mamun et al., 2010b). Santra et al. (2008) found a considerable decrement in the heat transfer was observed with the increase in solid volume fraction for any Ra number. Shahi et al. (2010b) observed that the copper water nanofluid was heated and for the preheating purposes during the daytime or night time it was stored in the storage tank. Moallemi and Jang (1992) computed that the average Nusselt number at the hot bottom wall resulted in heat transfer upsurged rapidly with increasing both Re and Gr for Ri = 0.01 but the same was not observed for 0.5 Ri 100. In literature, the mixed convection flows has been studied extensively in many studies due to the significant nature of the flow and heat transfer involved. The extensive application of nanofluids in industry has inspired the present study to examine mixed convection flows in a square lid-driven cavity filled with nanofluids. 2. NUMERICAL MODEL The geometry of the problem being investigated is depicted in Figure 1. The figure shows a twodimensional square cavity of length W and height H which its aspect ratio is taken to be equivalent to one unit. Both the vertical walls are kept at steady temperatures. The left vertical wall is confined at a greater temperature in order to produce the buoyancy effect. The two horizontal walls are insulated and the top wall slides from left to right with constant velocity. Both the fluid phase and nanoparticles are expected to be in a thermal equilibrium. The properties of nanoparticles and fluid are taken to be constant with the exception of density. 2.1 Boundary conditions and governing equations For this particular study, the dimensionless governing equations can be written as Conservation of mass (Oztop and Dagtekin, 2004): Momentum conservation in x-direction: Y-direction: Energy Conservation: (1) (2) ( ) (3) (4) With the Ra and Pr numbers presented as (Moallemi and Jang 1992): (5) 49

3 For the heat transfer development to be evaluated the Nusselt number (Nu) for the vertical hot wall is calculated as: ii) f (8) Effective Thermal Conductivity (9) As for the effective static thermal conductivity of the nanofluid,, it is calculated by (( ) ( )) [ ] (10) (( ) ( )) As for the effective Brownian thermal conductivity of the nanofluid, it is calculated by Figure 1 Schematic diagram of computational model. f (11) Where, is the thermal conductivity of nanofluid and is the thermal conductivity of base fluid, u v 0 on all the rigid walls. The temperature of the cold wall is set to be 293K. The working temperature of the fluid is K. forms: The boundary conditions are in the following u = v = 0 at x = 0, W 0 y H u = v = 0 at y = 0, 0 x W u = U 0,v = 0 at y = H, 0 x W T = T H at x = 0 0 y H T=T C at x = W 0 y H T y 0 at y = 0, H 0 x W 2.2 Nanofluids thermophysical properties The following are the equations used to calculate the properties for the nanofluids, all the values obtained are similar except for the useful viscosity and helpful thermal conductivity (Khanafer, Vafai et al. 2003). i) Effective Viscosity (6) As for the effective Static viscosity of the effective viscosity of the nanofluid,, it is calculated by As for the effective Brownian viscosity of the effective viscosity of the nanofluid,, it is calculated by (7) Where is the Boltzmann constant = , Modelling function, for > 1%., for 1% and 300. The effective mass density of the nanofluid, by (Brinkman 1952) is given (12) Where and are the mass densities of the base fluid and the solid particles, respectively. Then, the effective specific heat at constant pressure of the nanofluid,, is calculated by ( ) ( ( ) ( ) ) ( ) (13) As for the effective coefficient of thermal expansion of the nanofluid, it is calculated by ( ) ( ) (14) 3. NUMERICAL IMPLEMENTATION AND CODE VALIDDATION Various grid sizes are considered in this study, which are 11 11, 21 21, 31 31, 41 41, 51 51, 61 61, and The surface Nusselt numbers of the hot wall are plotted using these grids. Reynolds number of 1500 and Rayleigh number of is used. Figure 2 shows the plots of the grids. It s noticed from the figure which shows smooth lines after grid size 41 41, the rest of the lines after this grid are actually overlapping each other. Theoretically, this would mean that any one of the last five grids starting 50

4 from till can be used. However, enhanced precision in terms of results is for grid sizes with more nodes. But this would result in packed cells, which would require longer computational time. Thus, the finest grid size must strike a balance between the two aspects. Therefore, the grid size is used, which is the finest in terms of both aspects. Comparisons with previous work of Shahi et al. (2010) & Davis (1983) for local Nusselt number results is shown in Table 2. Their study involves the numerical investigation on mixed convection flow in a square lid- driven cavity having a hot and a cold wall. However, the working fluid is air, which is a convectional heat transfer fluid. A good balance must be achieved between the results obtained from the current code and those of the above mentioned authors. Excellent concurrence was observed among the results of the present work and the former work. An increasing pattern with almost comparable values is observed from the table. The obtained values are not inconsistent, which means that the results achieved are precise. Therefore, we can conclude that the present code is in very good agreement with those achieved by Shahi et al. (2010) & Davis (1983). 4. RESULTS AND DISSCUSSION Aluminium Oxide (Al 2 O 3 ), Copper Oxide (CuO) Silicon dioxide (SiO 2 ), Titanium dioxide (TiO 2 ) are the nanofluids considered for this specific study, with pure water as a base fluid. All parameters are kept stable in order to check the influence of different nanoparticles and its dissimilar effects. Results are presented for different nanoparticles concentrations (1%, 2%, 3% and 4%) and different Reynolds numbers in the range of 700 to The effect of different nanofluids Figure 4 represents the Nusselt number for different nanofluids. Higher Nusselt number is achieved for all the four nanofluids in contrast to pure water, indicating that the heat transfer in pure water is relatively lower. From the simulations results, Al 2 O 3 is observed to give better heat transfer rate, followed by SiO 2, TiO 2 and lastly CuO. This result is obtained due to the fact that Al 2 O 3 has the greatest heat capacity and the lowest thermal conductivity, therefore resulting in an enhanced heat capacity. Thus, different nanofluids give different heat transfer augmentation rate, with Al 2 O 3 -H 2 O nanofluids evolving as the greatest among the four nanofluids investigated. 4.2The effect of different nanoparticle volume fractions Nanofluids with larger volume fractions comprise of more nanoparticles within the base fluid. Pure base fluid (water) has no nanoparticles present in it. Figure 6 shows that the Nusselt number enhances with the enhancement in volume fraction. Concentration of nanoparticles in the fluid is varied from 0 % to 4%. In this context, higher volume fractions intensify the velocity and thus leading to a favourable convection that result in the increase of Nusselt number. Therefore, nanofluids with higher volume fraction results in greater heat transfer enhancement. 4.3 Effect of different diameter of nanoparticles Apart from the different types of nanofluids and its different volume concentration, the different diameter of nanoparticles, dp are used for this study. Range of diameter of particles which are chosen and investigated are 25nm, 40nm and 60nm. Figure 7 shows that when the diameter of the nanoparticles is decreased an increase in the Nusselt number is achieved. Similar to previous results, pure water shows the smallest Nusselt number in comparison to the nanofluids studied. The highest surface Nusselt number is obtained for the nanoparticles with a diameter of 25nm followed by 40nm and the smallest is 60nm. Due to the decrease in the size of nanoparticles, the ratio of surface area-to-volume of nanoparticles enhances and results in letting the nanoparticles to absorb and transfer heat more efficiently. Increase in consistency of the nanoparticles distribution is attained when the smallest particle size is used. 4.4 Effect of Re number Nusselt number at different Reynolds numbers for the first, second and third configurations are shown in Figure 8, respectively. It is observed from these three cases that the Nusselt number increases with the increase in Reynolds number. The Nusselt number for the three cases gets lower by enhancing the Reynolds number at the cold wall but the average Nusselt number of the hot wall enhances with the enhancement in Reynolds number. Therefore it can be stated that the more the increase in the velocity of the moving wall, the more the amount of heat can be evacuated from the cavity outside. 4.5 Effect of Rayleigh number In this simulation, the Rayleigh Number is varied for Al 2 O 3, where the concentration and the diameter of the nanofluid are kept constant at 4% and 25nm respectively. The Rayleigh Number values used for this simulation are 10 4, 10 5 and 10 6 as shown in Figure 9. Higher temperature dissimilarities between the hot wall and cold wall results in higher Ra number. High Rayleigh numbers are used to investigate the effect of the increased strength of buoyancy driven flow on Heat transfer. A faster vibration is observed by the nanoparticles in the cavity with the increase in Ra number. Therefore, the vibrations of these particles improve the acceleration of the nanofluid flow resulting in the enhancement of heat transfer around the surface area. Hence, the increased strength of buoyancy driven flow can assist and enhance the heat transfer rate. The result shows that by using Al 2 O 3 as the nanofluid with concentration of 4% and diameter of 25nm, Rayleigh number of 10 6 results in the highest Nusselt number. 51

5 Table 1 Thermo-physical Properties at 300K. Property Pure water Al 2 O 3 CuO SiO 2 TiO 2 3 kg / m N. m/ s 1 x k W m. K C p J kg. K K 207x x x x x10-6 Figure 2 Dissimilarity of Nusselt number versus position for different grid sizes. Table 2 Comparing present results with Shahi et al. (2010) and Davis (1983). Present Shahi et al. Error % Present Davis Error % Ra=10⁴ Ra=10⁵ Ra=10⁶

6 Figure 3 Comparison of isotherms between the results of Davis (1983) (left) with the results of present work (right) for air in square cavity showing contour maps of stream function: A) Ra =10 3, a) Ra =10 3, B) Ra =10 4 b) Ra =10 4 & contour maps of temperature : C) Ra=10 3,c) Ra=10 3, D) Ra=10 4, d) Ra= Figure 4 Evaluation of streamlines between the present work (right) with the results of Davis (1983) (left) for Air in Square Cavity. 53

7 Nusselt Number, Nu Nusselt Number, Nu Water CuO,φ=0.04 SiO2,φ=0.04 TiO2, φ=0.04 Al2O3, φ= Position, x(m) Figure 5 Variation against position of Nusselt number for various nanofluids Water, Ra=10^4 φ=0.01 φ=0.02 φ=0.03 φ= Position, x(m) Figure 6 Different volume fractions for nanofluids. 54

8 Nusselt Number, Nu Nusselt Number, Nu Water dp=60nm dp=40nm dp=25nm Position, x(m) Figure 7 Effects of different nanoparticle diameters Re=700 Re=1000 Re=1500 Re= Position, x(m) Figure 8 Effects of different Re numbers. 55

9 Nusselt Number, Nu Water dp=25nm, Ra=10^4 dp=25nm, Ra=10^5 dp=25nm, Ra=10^ Position, x(m) Figure 9 Effect of different Ra number. 5. CONCLUSION In this paper a numerical study on mixed convection heat transfer in a lid driven square cavity filled with nanofluids, has been conducted in the case where the vertical walls are maintained at steady but different temperatures, whereas the horizontal walls are insulated, with the top one uniformly moving in the same direction with the heat flux. The full partial differential equations, governing the problem, have been solved numerically. The nanofluids investigated in this research are: Al 2 O 3 -H 2 O, CuO-H 2 O, SiO 2 -H 2 O and TiO 2 -H 2 0. Considering the obtained results, the following findings may be summarized. 1. Nusselt number enhances with the enhancement in Rayleigh number. 2. Nanoparticle with gives the largest Nusselt number. 3. Nanoparticles with smallest diameters i.e. 25 nm give the highest Nusselt number. 4. Geometry plays a significant role on Nusselt number using nanofluids. 5. Higher Reynolds number results in greater heat transfer, regardless of the types of nanofluids. 6. Larger temperature differences between the hot and cold wall results in greater heat transfer. 7. Al 2 O 3 has the highest Nusselt number when compared with other nanofluids. ACKNOWLEDGEMENT The authors would like to acknowledge the financial support from the Ministry of higher education (MoHE), project no: FP A. REFERENCES Abu-Nada, E. and Chamkha, A.J Mixed convection flow in a lid-driven inclined square enclosure filled with a nanofluid, European Journal of Mechanics-B/Fluids 29 (6): Bilgen, E. and Oztop, H Natural convection heat transfer in partially open inclined square cavities, International Journal of Heat and Mass Transfer 48 (8): Cheng, T Characteristics of mixed convection heat transfer in a lid-driven square cavity with various Richardson and Prandtl numbers, International Journal of Thermal Sciences 50 (2): Davis, D.V Natural convection of air in a square cavity, a bench mark numerical solution, International Journal For Numerical Methods in Fluids 3: Hussein, A. and Hussain, S Mixed Convection through a Lid-Driven Air Filled Square Cavity with a Hot Wavy Wall, International Journal of Mechanical and Materials Engineering 5 (2):

10 Jahanshahi, S.H., Alipanah, M., Dehghani, A. and Vakilinejad, G Numerical simulation of free convection based on experimental measured conductivity in a square cavity using water/sio 2 nanofluid, International Communications in Heat and Mass Transfer 37 (6): Koca, A Numerical analysis of conjugate heat transfer in a partially open square cavity with a vertical heat source, International Communications in Heat and Mass Transfer 35 (10): Mahmoudi, A.H., Shahi, M., Raouf, A.H. and Ghasemian, A Numerical study of natural convection cooling of horizontal heat source mounted in a square cavity filled with nanofluid, International Communications in Heat and Mass Transfer 37 (8): Mamun, M., Rahman, M., Billah, M. and Saidur, R. 2010a. A numerical study on the effect of a heated hollow cylinder on mixed convection in a ventilated cavity, International Communications in Heat and Mass Transfer 37 (9): Mamun, M., Tanim, T., Rahman, M., Saidur, R. and Nagata, S. 2010b. Mixed Convection Analysis in Trapezoidal Cavity with a Moving Lid, International Journal of Mechanical and Materials Engineering 5 (1): Mansour, R.M., Abd-Elaziz, M. and Ahmed, S Numerical simulation of mixed convection flows in a square lid-driven cavity partially heated from below using nano-fluid, International Communications in Heat and Mass Transfer 37 (10): Moallemi, M. and Jang, K Prandtl number effects on laminar mixed convection heat transfer in a lid-driven cavity, International Journal of Heat and Mass Transfer 35 (8): Moraga, N.O., Riquelme, J.A. and Jauriat, L.A Unsteady conjugate water/air mixed convection in a square cavity, International Journal of Heat and Mass Transfer 52 (23-24): Oztop, H.F. and Dagtekin, I Mixed convection in two-sided lid-driven differentially heated sqeare cavity, International Journal of Heat and Mass Transfer 47 (8-9): Oztop, H.F., Sun, C. and Yu, B Conjugatemixed convection heat transfer in a lid-driven enclosure with thick bottom wall, International Communications in Heat and Mass Transfer 35 (6): Rahman, M., Alim, M. and Sarker, M. 2010a. Numerical study on the conjugate effect of joule heating and magnato-hydrodynamics mixed convection in an obstructed lid-driven square cavity, International Communications in Heat and Mass Transfer 37 (5): Rahman, M., Billah, M., Mamun, M., Saidur, R. and Hasanuzzaman, M. 2010b. Reynolds and Prandtl Numbers Effects on MHD Mixed Convection in a Lid-Driven Cavity Along with Joule Heating and a Centered Heat Conducting Circular Block, International Journal of Mechanical and Materials Engineering 5 (2): Rahman, M., Mamun, M., Saidur, R. and Nagata, S Effect of a heat conducting horizontal circular cylinder on MHD mixed convection in a lid-driven cavity along with joule heating, International Journal of Mechanical and Materials Engineering 4 (3): Santra, A.K., Sen, S. and Chakraborty, N Study of heat transfer augmentation in a differentially heated square cavity using copperwater nanofluid, International Journal of Thermal Sciences 47 (9): Shahi, M., Mahmoudi, A.H. and Talebi, F. 2010a. Numerical study of mixed convective cooling in a square cavity ventilated and partially heated from the below utilizing nanofluid, International Communications in Heat and Mass Transfer 37 (2): Shahi, M., Mahmoudi, A.H. and Talebi, F. 2010b. Numerical simulation of steady natural convection heat transfer in a 3-dimensional single-ended tube subjected to a nanofluid, International Communications in Heat and Mass Transfer 37 (10): Sivasankaran, S., Sivakumar, V. and Prakash, P Numerical study on mixed convection in a lid-driven cavity with non-uniform heating on both sidewalls, International Journal of Heat and Mass Transfer 53 (19-20): Talebi, F., Mahmoudi, A.H. and Shahi, M Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid, International Communications in Heat and Mass Transfer 37 (1): Tiwari, R.K. and Das, M.K Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, International Journal of Heat and Mass Transfer 50 (9-10):