Trans. Phenom. Nano Micro cales, 3(1): 37-45, Winter - pring 15 DOI: 1.758/tpnms.15.1.4 ORIGINAL REEARCH PAPER. MHD Natural Convection and Entropy Generation o Variable Properties Nanoluid in a Triangular Enclosure A. Aghaei 1,*, G.A. heihzadeh 1, H.R. Ehteram 1, M. Hajiahmadi 1 1 Mechanical engineering Department, university o Kashan, Kashan, I.R. Iran ARTICLE INFO. Article history Received 16 May 14 Accepted 7 January 15 Keywords Bejan Number Brownian Motion Entropy Generation Hartman Number Nanoluid Triangular Enclosure Abstract Natural convection heat transer has many applications in dierent ields o industry; such as cooling industries, electronic transormer devices and ventilation equipment; due to simple process, economic advantage, low noise and renewed retrieval. Recently, heat transer o nanoluids have been considered because o higher thermal conductivity coeicient compared with those o ordinary luids. In this study; natural convection and entropy generation in a triangular enclosure illed by Al O 3 water nanoluid aected by magnetic ield considering Brownian motion is investigated numerically. Two inclined walls are maintained at constant cold temperature (T c ) while the bottom wall is ept at constant high temperature (T h ) with (T h >T c ). In order to investigate natural convection, a computer program (FORTRAN language) based on inite volume method and IMPLER algorithm has been used. Analyses is perormed or volume raction o nanoparticles,.,.4, Hartmann number, 5,1, Rayleigh numbers 1 3,1 4,1 5 and angle o inclined walls 45. In investigated angles and Rayleigh numbers; average Nusselt number is increased by enhancement o volume raction o nanoparticles in a ixed Hartmann number. It is also observed that total entropy generation variations by increasing volume raction o nanoparticles are similar to that o Nusselt number. By the results; eect o riction is always insigniicant on generated entropy. It is observed that natural convection o nanoluid is decreased by enhancement o Hartmann number and its behavior is close to thermal conduction. It is also concluded that average Nusselt number and total generated entropy are decreased. 1. Introduction Natural convection heat transer has many applications in dierent ields o industry such as cool *Corresponding author Email address: AlirezaAghaei1@gmail.com -ing industries, electronic transormer devices and ventilation equipment; due to simple process, economic advantage, low noise and renewed retrieval. Because o higher thermal conductivity, Heat transer o nanoluids has been considered recently compared with ordinary luids. 37
Aghaei et al./ TPNM 3 (15) 37-45 Nomenclature Gree ymbols B Magnetic ield α Thermal diusivity coeicient (ms-1) c P peciic heat (Jg-1K-1) β Thermal expansion coeicient(k-1) d Diameter o water molecule (nm) μ viscosity (gm-s-1) d p Diameter o nanoparticle (nm) υ Kinematic viscosity(ms-1) L Height o the enclosure (m) θ Dimensionless temperature Ha Hartmann number ρ density (gm-3) Thermal conductivity (Wm-1K-1) φ Volume raction o nanoparticles p pressure (pa) ubscripts P r Prandtl number Avg average R a Rayleigh number c cold s Entropy luid m Entropy generation o magnetic ield ( ) h hot T Temperature (K) nanoluid T H Temperature o the hot wall (K) p particle T c Temperature o the cold wall (K). Ghasemi and Aminossadati [1] investigated mixed convection o Al o 3 -Water nanoluid in a right angle triangular enclosure. Based on their results, heat transer increases with increasing the volume raction o nanoparticles. Ho et al. [] perormed a numerical study to investigate ree convection heat transer o Al o 3 - water in a square enclosure by inite volume method. They ound that dierent models o viscosity lead to predict various values o Nusselt number. Oztop and Abu-Nada [3] analyzed the luid low and heat transer due to natural convection in a heated cavity. Their results showed that using nanoluid in low aspect ratio cavities mae higher enhancement in heat transer. Mahmoodi and Hashemi [4] investigated natural convection heat transer and luid low o Cu- Water nanoluid in a c-shape enclosure. They observed that average Nusselt number increased with increasing Rayleigh number and volume raction o nanoparticles. Because the entropy generation is a measure o destruction o available wor o the system, determination o entropy generation is important to enhance system eiciency, Bejan [5]. Recently, entropy generation minimization has been considered a lot to reach optimum design o systems, Mahian et al. [6]. Dagtein et al. [7], investigated numerically the entropy generation in natural convection luid low in Γ-shape enclosure. Their results showed that entropy generation duo to heat transer is the main part o total entropy. Ilis et al. [8]carried out a numerical wor to investigate iluence o aspect ratio on entropy generation or natural convection luid low in rectangular enclosures. They observed that by increasing aspect ratio the entropy generation increases up to a maximum value and then decreases. Varol et al. [9] perormed a numerical study to analyze the natural convection and entropy generation. Based on their results with decreasing Rayleigh number, Bejan number that is deined as the ratio o generated entropy due to heat transer to the total entropy generation, decreases. One considerable point in study o convective heat transer is disagreement between the experimental results, Abbasian et al. [1] and [11], and numerical studies or nanoluids. Maybe one reason or this discoormity is to ignore some phenomenon lie Brownian motion. In Brownian motion, luid molecules are constantly hitting nanoparticles and disperse them inside the luid. In present study the eect o this phenomenon is considered. Through a comprehensive investigation it is observed that ew studies has been done on entropy generation o nanoluid with various properties in triangular enclosures. Novelty o this article is investigation magnetic ield eect on triangular enclosure, entropy generation, using the variable properties or viscosity and thermal conductivity coeicient estimation, and Brownian motion o nanoparticles eect. This simulation can be a model o cooling an electronic device (or example at computer boards). At present study iluence o luid low, heat transer and entropy generation is investigated in natural 38
Aghaei et al./ TPNM 3 (15) 37-45 convection o Al o 3 -Water nanoluid in a triangular enclosure with hot bottom wall and cold sidewalls or various Rayleigh numbers, Hartmann numbers and volume raction o nanoparticles and at θ s =45 o..governing equations and boundary conditions chematic view o the problem and its boundary conditions is shown in igure 1. y B Fig. 1. schematic view o the enclosure with boundary conditions The bottom wall is ept at hot temperature (T H ) and sidewalls are maintained at cold temperature (T c ). All the walls are assumed to be electrically insolated. tudy is done or the Rayleigh numbers o 1 3,1 4,1 5, Hartmann numbers o, 5,1, angle o inclined sidewalls with horizon 45 o and nanoparticles volume raction o -.4. The height o the enclosure, H is presumed to be constant. The value o θ s is θ s =45 o. Thermophysical properties o water as the base luid and Al o 3 nanoparticles are presented in table 1 Abu-Nada et al. [1]. Table 1 Thermophysical properties o base luid and the nanoparticles, Abu-Nada et al. [1]. water Al O 3 X y β (K -1 ).1 1-4.85 1-5 T c x L (Wm -1 K -1 ).613 5 c p (jg -1 K -1 ) 4179 765 ρ (gm -3 ) 997.1 397 The iluence o magnetic ield as Lorentz volumetric orce is taen into account in momentum equation in term o = where is magnitude o magnetic ield and is density o electrical low. The T h θ s T c H continuity equation, entropy generation and stream unction aected by magnetic ield are given by: u v x y u u 1 p 1 u u v [ ( ) x y x x x u ( )] B vcossinusin y y v v 1 p 1 v u v [ ( ) x y y x x v ( ) ( )] gt ( Tc ) y y B ucossin vcos T T 1 T T u v [ ( ) ( )] x y ( c ) x x y y gen p T T u T [ x y T x v u v ] s y y x m (1) () (3) (4) (5) udy (6) (x,y) By employing dimensionless variables and considering horizontal magnetic ield in the x- direction, dimensionless equations o continuity, momentum, energy conservation and entropy generation are obtained rom -6. x y ul vl pl X, Y, U, V, P L L T Tc Th Tc, T, T Th Tc, Th Tc 3 gβδtl Ra=, Ha B H να (7) 39
Aghaei et al./ TPNM 3 (15) 37-45 ubtitles o and n, represent base luid and nanoluid, respectively. thermal conductivity coeicient, Maxwell [15], are obtained by 15-4 relations, respectively: U X V Y (8) (1 ) (15) s U U P 1 U V X Y X X U U X Y Y V V P 1 U U V [ X Y Y X X Y U ] Ra Pr Ha Pr V cos Y (9) (1) ( cp ) (1 )( cp ) ( cp ) s (16) ( ) (1 )( ) ( ) (17) s ( c ) (18) p 1 U V [ X Y c X X ] Y Y p Dimensionless stream unction is deined as ollows: (, ) (11) XY UdY (1) Total entropy generation that includes both o entropy due to heat transer and entropy due to luid riction is obtained by this relation, Bejan [13]: gen X Y U X m V U V Y Y X T ( ) LT Dimensionless boundary conditions are given by: U V,, On the sidewalls o enclosure U, V, 1, On the hot side (13) (14) Nanoluid properties such as; density, heat capacity, thermal expansion coeicient, thermal diusivity coeicient, static viscosity, Brinman [14], and static (1 ).5 (19) tatic tatic ( s ) ( s) ( s ) ( s) () (1) e tatic Brownian () e tatic Brownian Where μ Brownion and Brownion are, Koo [16]: T Brownian 4 5 1 ( T, ) srs 4 Brownian 5 1 p, (, ) srs (3) T c T (4) Iluence o Brownian motion in terms o temperature is observed at viscosity (3) and thermal conductivity coeicient (4) (Viscosity and thermal conductivity coeicient change with temperature). ρ s and R s are density and radius o nanoparticles (3.5 1-9 ), respectively and is the Boltzmann constant (K=1.387 1-3 (J/)). For Al o 3 -Water nanoluid unction ζ that is estimated experimentally is deined as, Li [17]: =[a+bln(dp)+cln( )+dln( )ln(dp)+bln(dp) ] ln(t)+(g+h ln(dp)+iln( )+jln( )ln(dp)+ ln(dp) ) (5) 4
Aghaei et al./ TPNM 3 (15) 37-45 Constants o equation (19) or Al o 3 -Water is shown at table, Li [17]. Table Li [17], Constants o relation (19). constant a b c d e amount 5.8135 6.1156.6956.417.1769 constant g h i j Convection heat transer coeicient is: h q T T h c amount -98.1981-34.537-3.953 -.354 -.9991 (6) Nusselt number that it characteristic length measured by the height o the enclosure is: h H Nu (7) Heat lux per unit area is given by: q T T h H c Y Y (8) ubstituting relations (6) and (8) in (7), gives the Nusselt number: Nu Y (9) Y The average Nusselt number on the hot wall is: Nu Avg 1 1 Nu dx L (3) 3. Numerical implementation Governing equations by using the inite volume method and IMPLER algorithm and computer program (FORTRAN language) are solved numerically. In order to validate the results o the computer program, the numerical simulation o reerence, Oztop and Abu-Nada [3], was used. The results o this simulation is compared with the results o Oztop and Abu-Nada [3], and is given at table 3. Table 3 Comparisons o the present results or average Nusselt number With the results o Oztop and Abu- Nada [3]. Ra Present study amount 1 3. 1.8 1.4.5 1.119 1.1 1 5. 3.993 3.983.5 4.34 4.71 In order to determine a proper grid or the numerical simulation, a grid study is conducted or the Al o 3 -Water nanoluid, average Nusselt numbers rom the dierent uniorm grids are presented in table 4. It is observed rom the results that a 111 61uniorm grid is suiciently ine to ensure a grid independent solution. Hence, this grid is used to perorm all subsequent calculations. Table 4 Average Nusselt number at the walls o the enclosure or Al o 3 -Water nanoluid at θ s =45, Ra=1 5, φ=.4. Number o Nodes 71 41 91 51 111 61 131 71 151 81 171 91 Nu avg 1.6 1.34 1.48 1.5 1.51 1.51 The convergence criterion or pressure, temperature and velocity is obtained rom ollowing relation Where m and n are the number o grid points in x- and y- direction, respectively, ζ is any o the computed ield variables, is the iteration number. M N 1 i, j i, j i 1 j 1 6 M N 1 Error 1 i 1 j 1 i, j 4. Results and discussions (31) Figure shows the streamlines, isotherms and total 41
Aghaei et al./ TPNM 3 (15) 37-45 entropy generation at dierent Rayleigh and Hartmann numbers. For a constant Rayleigh number, with increasing Hartmann number, the eyes o the eddies tend toward the bottom vertices o the triangular enclosure. This is because o the Lorentz orce due to magnetic ield which prevents natuaral convection o nanoluid and its moving in vetical direction o the enclosure. For a constant Hartmann number, with increasing Rayleigh number, eddies tend toward the center o the enclosure and stretch in vertical direction and become more powerull. For Rayleigh number(=1 3 ) and all investigated Hartmann numbers, streamlines have slight curves which shows that natural convection o nanoluid is wea. At Ra =1 4 and Ha= isotharms have more curves, this indicates more powerul natural convection o nanoluid. At this Rayleigh and with increasing Hartmann number, isotherms is similar to its status at Ra =1 3 that shows thermal conduction domination compared with behavior o nanoluid. In Ra=1 5, isotherms have beore curve attain to Ha=5. That because the ilence o Lorentz orce at this Hartmann=5 is not high adequately to prevent natural convection o nanoluid and cause conduction dominate. For Ra =1 5 and Ha =1 the Lorentz orce is high enough to prevent natural convection o nanoluid and cause isotherms be similar to isotherms at Rayleigh(=1 3,1 4 ). Total entropy lines are more densely paced adjacent the bottom vertices o the enclosure or Rayleigh number (=1 3,1 4 ). In thease regions, isotherms are denser,too. That indicates more temperature gradient in these regions. At next sections it will be shown that main contribution o total entropy is generated entropy due to heat transer, consequently, total entropy lines are more densely paced at this regions. This behavior is observed or all investigated Rayleigh and Hartmann numbers, except or Ra=1 5 and Ha =. For Ra =1 5 and Ha=, because natural convection is dominant and causes entropy generation due to riction to have more contribution and also leads to more heat transer, total entropy lines are observed adjascent o the cold walls and large portion o bottom hot wall. Total entropy are denser at this regions that indicates more entropy generation.tables 5 and 6, show amounts o max as the criterion o the low power at dierent Hartmann numbers and volume ractions or Rayleigh numbers (1 3-1 5 ). At Ra =1 3 and or all investigated Hartmann numbers, max decreases with increasing the volume raction o nanoparticles. Because o low movement o nanoluid at Ra=1 3, with increasing the viscosity o nanoluid aected enhancement o volume raction o nanoparticles, nanoluid motion be prevented and consequently, max decreased. Also it is observed that or Ra=1 4 natural convection o nanoluid is insigniicant because o the low amount o Rayleigh number lie Ra=1 3, so with increasing the volume raction o nanoluid max has similar behavior. Table 5 Amounts o max at dierent Ha and volume ractions or Ra=1 3...4 Ha=.153.1117.15 Ha=5.151.14.19 Ha=1.49.45.4 Table 6 Amounts o max at dierent Hartmann numbers and volume ractions or Ra=1 4...4 Ha=.9191.568.417 Ha=5.167.1463.1341 Ha=1.498.459.45 Table 7 shows amounts o max at dierent Hartmann numbers and volume ractions or Ra=1 5. At Rayleigh =1 5 unlie the Rayleigh numbers (=1 3,1 4 ), max increases with increasing volume raction o nanoparticles or Hartmann=. By enhancement o volume raction o nanoparticles, thermal conductivity coeicient and viscosity o nanoluid increase. Increment o the thermal conductivity coeicient cause appropriate heat transer and consequently enhanced convection. Whereas, increase o the viscosity prevent rom suitable convection.at this Rayleigh number and or Ha=, the increasing eect o the thermal conductivity coeicient on natural convection dominates on the decreasing eect o thermal viscosity. For Ra=1 5 and Ha=5,1 iluence o increasing volume raction on max is similar to Ra=1 3,1 4, because the Lorentz orce due to magnetic ield weaens natural convection o nanoluid. For all investigated Rayleigh numbers, Hartmann numbers and volume ractions, maximum value o max is 16.3 and occurs or Ra=1 5, volume max is 16.3 and occurs or Ra=1 5, volume raction=.4. 4
Aghaei et al./ TPNM 3 (15) 37-45 Ha= Ha=5 Ha=1 Ra=1 5 θ Ra=1 4 θ Ra=1 3 θ Fig.. treamlines, isotherms and total entropy at dierent Rayleigh and Hartmann numbers Whereas minimum value o max is.15 and occur or Ra=1 3 and φ=.4. Table 7 Amounts o max at dierent Hartmann numbers and volume ractions or Ra=1 5...4 Ha= 15.598 16.8 16.38 Ha=5.7777.45.41 Ha=1.5898.53.47 The variations o average Nusselt number and total entropy generation with respect to the volume raction or dierent Rayleigh and Hartmann numbers are shown in igure 3 For all investigated Rayleigh and Hartmann numbers, average Nusselt number increases with increasing the volume raction o nanoparticles. By increasing volume raction o nanoparticles, thermal conductivity coeicient increases, so increment o the average Nusselt number is reasonable. Thereore, by enhancement o Hartmann 43
Aghaei et al./ TPNM 3 (15) 37-45 number, the heat transer regime is conductiondominated. ubsequently, with increasing the Hartmann number, iluence o Rayleigh number on motion o nanoluid becomes low. For Ha=5, the variations o Nusselt number in Ra= 1 3,1 4 are the same and this variation or Ra=1 5 is similar to two mentioned Rayleigh numbers. At Ha=1, the variation ranges o Nusselt number is the same or all Rayleigh numbers and lines coincide. By enhancement o Hartmann, maximum decrease o average Nusselt number is 5.3% that occurs or Ra=1 5 and φ=. Whereas, with increasing Hartmann number,minimum decrease o average Nusselt number is.6 % occurs or Ra=1 4 and volume raction o.4. As it can be observed rom Fig.3, or investigated Rayleigh and Hartmann numbers, by enhancement o volume raction o nanoparticles total entropy generation increases. Entropy generation is due to heat transer and luid riction. As it can be seen rom tables 8-1 the amounts o entropy generation due to riction are insigniicant and main contribution o entropy generation is due to heat transer. Consequently, the variation o total entropy should be similar to variation o average Nusselt number with increasing the volume raction. The maximum value o total entropy generation is 33.16 and occurs or Ra=1 5, Ha= and φ=. Whereas the minimum value o total entropy generation is 19.73 which occurs at Ra=1 3, Ha=1 and φ=. As it observed rom tables 8-11 with increasing the volume raction or all investigated Rayleigh and Hartmann numbers, the contribution o entropy generation due to riction become less, because heat transer increases by enhancement o volume raction and, subsequently, the contribution o entropy generation due to heat transer is more. However, as mentioned beore, behavior o max at Ra=1 5 and Ha= variations o entropy generation due to heat transer is inverse o other situations. Because at Ra=1 5 and Ha=, max increases with increasing the volume raction, consequently entropy generation due to riction become more, too. Table 8 Entropy generation due to riction or dierent volume ractions and Hartmann numbers at Ra=1 3...4 Ha= 4.87E-3 4.1E-3 3.69E-3 Ha=5.4E-4.4E-4 1.87E-4 Ha=1 4.1E-5 3.85E-5 3.54E-5 Table 9 Entropy generation due to riction or dierent volume ractions and Hartmann numbers at Ra=1 4...4 Ha=.9 1-1.66 1-1. 1 - Ha=5.4 1-4.17 1-4 1.96 1-4 Ha=1 4.9 1-5 3.91 1-5 3.58 1-5 Table 1 Entropy generation due to riction or dierent volume ractions and Hartmann numbers at Ra=1 5...4 5. Conclusion Ha= 8.76 1-3 9.4 1-3 9.56 1-3 Ha=5 5.13 1-4 4.18 1-4 3.37 1-4 Ha=1 5.9 1-5 4.58 1-5 4.5 1-5 At present study the mixed convection low, heat transer and entropy generation o Al o 3 -Water nanoluid with various properties aected by magnetic ield in triangular enclosure was investigated. ide walls o the enclosure are cold and bottom wall is hot. This study is carried out or Ra=1 3,1 4,1 5, Ha=, 5,1, angle o inclined sidewalls =45 and volume raction o nanoparticle=-.4. Numerical results show that: 1. with applying and increasing the magnetic ield (enhancement o Hartmann number) velocity o nanoluid and, subsequently, stream unction magnitude in enclosure decreases because o the Lorentz orce.. With increasing the Hartmann number, natural convection becomes wea and heat transer regime is conduction-dominated. 3. or all investigated Rayleigh and Hartmann numbers and volume ractions, maximum absolute value max is 16.3 that occurs at Ra=1 5 and φ=.4. Whereas the minimum absolute value o max is.15 occurs or Ra=1 3 and φ=.4. 4. For all investigated Rayleigh and Hartmann numbers, average Nusselt number increases with increasing the volume raction o nanoparticles. 5. Maximum decrease o average Nusselt number is 5.3% that occurs or Ra=1 5 and φ= with increasing the Hartmann number. Whereas, minimum decrease o average Nusselt number is.6% occur or Ra=1 4 and φ=.4 with increasing the Hartmann number. 6. For all investigated Rayleigh and Hartmann number, total entropy generation increases by enhancement o volume raction o nanoparticles. 7. Value o the maximum total entropy generation is 33.16 that occurs or Ra=1 5, Ha= and φ=. Whereas, 44
Aghaei et al./ TPNM 3 (15) 37-45 The amount o the minimum total entropy generation is 19.73 that occurs or Ra=1 3, Ha=1 and φ=. 8. In all investigated situations, total entropy generation due to riction is insigniicant and main contribution o entropy generation is due to heat transer. o, the variations o total entropy generation is similar to variation o average Nusselt number. Reerences [1] B. Ghasemi,.M. Aminossadati, Mixed convection in a lid-driven triangular enclosure illed with nanoluids, Int. Commun. in Heat Mass Transer 37 (1) 114 1148. [] CJ. Ho, MW. Chen, ZW. Li, Numerical simulation o natural convection o nanoluid in a square enclosure: eects due to uncertainties o viscosity and thermal conductivity, Int. J. Heat Mass Transer 51 (8) 456 4516. [3] H. Oztop, E. Abu-Nada, Numerical study o natural convection in partially heated rectangular enclosures illed with nanoluids, Int. J. Heat Fluid Flow, 9, (8),136 1336. [4] M. Mahmoodi,.. Hashemi, Numerical study o natural convection o a nanoluid in C-shaped enclosures, Int. J. Thermal ciences 55 (1) 76-89. [5] A. Bejan, econd law analysis in heat transer, Energy 5 (198) 71-73. [6] O. Mahian,. Mahmud, I. Pop, Analysis o irst and second laws o thermodynamics between two isothermal cylinders with relative rotation in the presence o MHD low, Int. J. Heat Mass Transer 55 (1) 488-4816. [7] E. Dagtein, H.F. Oztop, A. Bahloul, Entropy generation or natural convection in Γ-shaped enclosures, Int. Commun. Heat Mass Transer 34 (7) 5 51. [8] G.G. Ilis, M. Mobedi, B. unden, Eect o aspect ratio on entropy generation in a rectangular cavity with dierentially heated vertical walls, Int. Commun. Heat Mass Transer 35(8) 696 73. [9] Y. Varol, H.F. Oztop, A. Koca, Entropy generation due to conjugate natural convection in enclosures bounded by vertical solid walls with dierent thicnesses, Int. Commun. Heat Mass Transer 35 (8) 648 656. [1] A. A. Abbasian Arani, J. Amani, Experimental study on the eect o TiO water nanoluid on heat transer and pressure drop, Experiment. Thermal Fluid cience 4 (1) 17-115. [11] A. A. Abbasian Arani, J. Amani, Experimental investigation o diameter eect on heat transer perormance and pressure drop o TiO water nanoluid, Experiment. Thermal Fluid cience 44 (13) 5-533. [1] E. Abu-Nada, Z. Masoud, A. Hijazi, Natural convection heat transer enhancement in horizontal concentric annuli using nanoluids, Int. Commun. Heat Mass Transer 35(8) 657 665. [13] A. Bejan, Entropy Generation Minimization, CRC Press New Yor (1995). [14] H.C. Brinman, The viscosity o concentrated suspensions and solution, J. Chemical Physics (195) 571 581. [15] J.C. Maxwell-Garnett, Colours in metal glasses and in metallic ilms, Philos, Trans. Roy. oc. A 3 (194) 385-4. [16] J. Koo, C. Kleinstreuer, a new thermal conductivity model or nanoluids, J.Nanoparticle Research 6 (4) 577 588. [17] J. Li, C. Kleinstreuer, Thermal perormance o nanoluid low in microchannels, Int. J. Heat Fluid Flow 9 (8) 11 13 45