Natural Convection Heat Transfer of Al2O3 Nanofluid Through Packed Beds

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1 Natural Convection Heat Transer o AlO3 Nanoluid Through Packed Beds 1 G. Srinivas Rao 1 Deartment o Mechanical Engineering, Kakatiya Institute o Technology and Science, Warangal, Andhra Pradesh, India. K.V. Sharma Deartment o Mechanical Engineering, JNTU College o Engineering Kukatally, Hyderaad , Indi. Astract:- Natural convection heat transer through acked eds has een resented in this aer. The heat transer exerimental data as resented and comared ith data rom general correlations roosed in the literature. The unsteady and steady state natural convection heat transer o a nanoluid in vertical acked column is investigated. The Darcy model is used to ormulate the rolem. The nonlinear governing equations are solved using an unconditional stale inite dierence scheme. The heat transer enhancement and luid temerature at the ed outlet or AlO3 nanoluid is comared ith that o ater or dierent orosities o the ed. It is oserved that heat transer rate increases ith a decrease in ed articles in the range o 6mm to 14.6mm diameter or the nanoluid concentration range o 0% to 5% and increases ith concentration and lo rate. Keyords: Darcy model, heat transer, natural convection, acked ed, nanoluid 1. INTRODUCTION: The analysis o natural convection heat transer in luid saturated orous media lays an imortant role in many alications. These include geothermal engineering, thermal insulation systems, acked ed chemical reactors, orous heat exchangers, oil searation rom sand y steam, underground disosal o nuclear aste materials, ood storage, and electronic device cooling to name a e alications. Due to the characteristic qualities o high surace area, luid mixing qualities, high thermal conductivity and alicaility in a ide range o industrial alications, natural convection through orous media has gained considerale attention o various researchers in the ast e decades. Nield and Bejan [199] and Kavinay [1995] have resented the main rincile o heat transer through orous medium. With the orosities in the range o 4-6 most o the researchers studied ree convection through orous medium, acked eds and granular materials. Prasad and Kulacki [1984], Prasad [1986] carried out studies on natural convection in a vertical orous annulus or isothermal heating y alying a constant heat lux at the inner all or a much ider range o Rayleigh numers, asect ratios and radius ratios than those considered y revious authors. The natural convection, in a vertical annulus ithout orous media, has een extensively investigated in the literature or uniorm or discrete heating. Natural convection in a orous square cavity ith an isolux and isothermal discrete heater laced at the alls has een numerically studied y Saeid and Po [005] using the Darcy model. They ound that maximum heat transer can e achieved hen the heater is laced near the ottom o the cavity. Saeid [006] numerically studied the natural convective lo induced y to isothermal heat sources on a vertical late channel illed ith a orous layer. Natural convection heat transer in a square orous enclosure due to non-uniormly heated alls has een investigated in the literature y Basak et al. [007] and Sathiyamoorthy et al. [007]. Using Bejan s heat lines method, Kaluri et al. [009] analyzed the otimal heating in a square cavity illed ith a luid saturated orous medium or three dierent thermal conditions. Natural or ree convection in acked eds as carried out y Beckman [1931] and later extended y Kraussold [1934]. The exerimental study o natural convection in acked eds as undertaken y Elder [1967], ho roosed a correlation or the estimation o mean Nusselt numer, develoed as a unction o Rayleigh numer. Kuehn and Goldstein [1978] have undertaken exerimental studies using Mach-Zehnder intererometer and conducted numerical simulations using the inite elements technique. Charrier-Mojtai et al. [1979], Farouk and Guceri [1981] solved or natural convection under turulent lo using a to-equation model to redict the Nusselt numer leading to imroved accuracy in redicting the thermal erormance o storage systems. Garon and Goldstein [1973], Caltagirone [1976], Poulikakos [1984], Bejan [1987], Oosthuizen and Naylor [1989] roosed correlations or the estimation o natural convection heat transer in a orous medium. The comined eect o heat and mass transer ith orous medium have een reorted y Nield [1968], Khan and Zei [1981]. Calmidi and Mahajan [1999], Zhao et al. [004, 005] oserved natural convection using steel alloy oam ith air as the interstitial luid. Other researchers like Phanikumar et al. [00], Krishnan et al. [004], Zhao et al. [004, 005], and Edimilson et al. [006] have reorted IJERTV5IS09083.ijert.org 91

2 studies on natural convection using metal oams or attaining imroved thermal storage erormance. The conventional heat transer luids, such as oil, ater and ethylene glycol mixture are oor heat transer luids. The thermal conductivity o luids eteen the heat transer medium and the heat transer surace, lay an imortant role in determining the heat transer coeicient. An innovative technique or imroving heat transer, y using ultra ine solid articles in the luids, has een develoed and used extensively during the last years. Certain investigations or the estimation o heat transer coeicient o Al O 3 and CuO nanoluids in a circular tue under laminar and turulent lo conditions have een reorted y several researchers. Comared ith conventional luids, nanoluids have een reorted to have higher heat transer rates. Sundar et al. [007] undertook the exerimental determination o heat transer coeicients y measuring the viscosity and thermal conductivity, using Al O 3 nanoluid, or lo in a lain tue and ith tisted tae insert. They concluded that nanoluids enhance heat transer coeicients hich can reduce the size o thermal storage system. Rao et.al. [011], [01] have estimated ressure dro and the heat transer coeicient in acked eds ith the same nanoluid and concluded that nanoluids erormed etter than conventional luids such as air and ater. The ojective o the ork is to determine heat transer coeicient and ressure dro under natural convection lo o luid in a acked ed illed ith glass eads and eles rom unsteady to steady state o temerature. The governing equations o continuity, momentum and energy in non dimensional orm are solved sujected to relevant oundary conditions. The exeriments values at various lo rates, inlet temeratures and nanoluid concentrations are comared ith the numerical results. The roerties o nanoluids are adoted rom the generalized equations o Sharma et.al [01]. The results o unsteady temeratures are comared ith the numerical solutions o Burmeister [1993], Bejan [1995], Hornung [1997] and those o steady state ith that o Oosthuizen [1989].. MATHEMATICAL MODEL Mathematical models are emloyed y Burmeister [1993], Bejan [1995], Hornung [1997] or the determination o transient temeratures in a luid saturated acked ed storage system under natural convection at lo lo rate o the luid. The assumtions made y these investigators include: i. Isotroic orous medium ii. Thermal equilirium eteen the hases iii. iv. Negligile internal heat generation Negligile temerature gradient in radial direction The governing equations are 1 1 rv u 0 (1 t r r z- comonent u u u u u v u ( t r- comonent v v v v v v u (3 t Energy Equation T T T 1 T T C v u k (4 t r here the non-dimensional terms are: * r z T T, and K,,, u v L t, R R R, Z, R L T i Kg L ( T T *, * i u v, Ra o U 0 U 0 and reresents the ermeaility o ed, density, thermal diusivity, coeicient o thermal exansion, and asolute viscosity o the luid resectively. T In the resent exerimental setu, as the diameter is small comared to length o the ed, the variation o temerature in the radial direction is ignored. The Eqs. (1, (, (3 and (4 are transormed into non dimensional terms as (5 IJERTV5IS09083.ijert.org 9

3 u Z 0 (6 R u L Z R Ra L 0 R u Z L Equations (6 and (7 are constrained y the olloing oundary conditions along axial direction: At Z 0, 0, u 1 The initial condition is given y At Z 1, u 0, 0 The equations (6 - (8 are solved numerically ith exlicit inite dierence aroximate analysis sujected to oundary conditions (9 and (10 or the determination o non-dimensional temerature roiles. The exerimental values are shon in comarison ith the theoretical results and lotted through Figs. (3 - (6. 3. EXPERIMENTAL SETUP The exerimental setu consists o 4cm diameter and 50 cm length o a ed column. Figures 1 and sho the exerimental lo rocess and instrumentation diagram. An immersion heater heats the ater, hich is connected to eed ater storage tank o 50 liters caacity. The tank is setu at a height o m rom the ground level to maintain suicient head to conduct the exeriment under ree convection mode. The orking luid (ater/nanoluid is heated in the elevated tank and enters the test section due to gravity at a redetermined temerature. The interaction eteen the cold ed and the hot luid takes lace. As a result, the luid temerature decreases at the ed outlet. The orking luid is umed to the storage tank or recirculation in a closed circuit. The orking luid los through a helical coil immersed in the hot ater tank ith the aid o a um, having lo control and yass valves, entering the test section under gravitational orce. The lo rate o orking luid and the variation o axial temerature are measured and recorded using suitale instrumentation. It achieves the desired temerature eore C n n Z (9 0, 1, u 0 (10 it enters the test section. When the ed reaches a steady state, the temeratures along the ed length are otained rom ersonal comuter through a data logger or to dierent glass eads o sizes 6 mm and 14.6 mm diameter. Exeriments are undertaken ith to ed article sizes o 6mm and 14.6 mm ith glass eads using ater and nanoluid or lo rates o 15 and 30LPH ith Rayleigh numers in the range o Ra = 6.0E6 to.0e6 or oth 6mm and 14.6mm glass eads or ater and nanoluid at dierent concentrations. Exerimental investigations are undertaken or inlet temeratures o 40 to 55 0 C in stes o 5 0 C. The orking luid ater and nanoluid at 0, 1 and 5% volume concentrations are emloyed in the investigations ith loer lo rates at 15 and 30 LPH considered or etter understanding o lo henomena. The exeriments are conducted till the steady state is reached. The exerimental Nusselt numers are comared ith the values otained rom equations availale in the literature. (7 (8 4. THERMO PHYSICAL PROPERTIES OF NANOFLUID The density and seciic heat o nanoluid at dierent volume concentrations are estimated using the olloing relations valid or homogeneous mixture given y 1 (11 C C (1 IJERTV5IS09083.ijert.org 93

4 The exerimental data o viscosity availale in the literature or Cu, CuO, Al O 3, TiO, SiO, ZrO, ZnO nanoarticles disersed in ater is sujected to regression y Sharma et al. [01] or , 0 T 70, d. The equations are otained ith a deviation o less than 10% here is in ercent, and d in nanometer is given y T in 0 C n C Tn d (13 Sharma et al. [01] develoed an equation or the determination o thermal conductivity o Cu, CuO, Al O 3, TiO, SiO, ZrO, ZnO nanoarticles disersed in ater valid in the range , 0 T 70, k k n Tn 70 0 d 70. The equation resented y them is used to determine the thermal conductivity o nanoluid or the exerimental conditions given y d 150 (14 5. EXPERIMENTATION The data o temerature variation o the ed or dierent concentrations o the nanoluid and ater as noted at the 60 th minute and the values ere comared ith those otained using the Eqs. 6 and 7 shon through Figs. 3 to 5. The temerature variation o the ater and ed along its length or our inlet temeratures o 40, 45, 50, and 55 0 C is determined at dierent lo rates. Temerature roiles or inlet conditions o 40 and 55 0 C, or nanoluid concentration o 0 and 5 are resented. The variation o ed temerature along the length or 14.6 mm articles is resented in Fig.. The temerature o the ed at 15LPH lo rate is loer comared ith the value otained at a lo rate o 30LPH as shon in Fig. This is due to loer heat transer exchange eteen the luid and the ed. The eect o concentration on temerature roiles are shon in Fig.3. The temerature gradients ith nanoluid concentration o 5% at 30LPH lo rate are greater than ater or an identical inlet temerature o 55 0 C. It can e due to greater values o nanoluid thermal conductivity comared to ater at the same temerature. Under similar oerating conditions or oth articles as resented in Fig. 4, the gradients ith small size ed articles comared ith larger diameters are greater due to the greater surace area. The temerature gradients estimated ith the theoretical model are oserved to e in satisactory agreement ith the exerimental values. The steady state values o exerimental Nusselt numers are comared ith numerical results emloying Eqs. (15 to (17. Garon and Goldstein [1973] roosed an equation or the estimation o Nusselt numer or ater saturated in metal oam or ree convection given y Eq. (15. Nu. Ra (15 G Oosthuizen and Naylor [1989] roosed a generalized correlation or ree convection in rectangular enclosures ith orous medium given y Eq. (16 Nu O 5 508Ra / AR (16 Equation (17 is develoed y Cheng [1980] or the determination o ree convection heat transer in orous medium. It is given as Nu C Ra The exerimentation values are estimated ith the energy alance relation Q m C T T (18 h ex ( i o 5 (17 Q / A ( T T (19 s s here T s and T are the average surace and ulk temeratures. The values o exerimental Nusselt numers are in good agreement ith the values estimated ith Eqs. (15 to (17. IJERTV5IS09083.ijert.org 94

5 6 RESULTS AND DISCUSSION Exeriments are undertaken ith to ed article sizes o 6mm and 14.6mm ith glass eads using ater and nanoluid at lo rates o 15 and 30LPH. The temerature variation o the ed or dierent concentrations o the nanoluid and ater lo determined at the 60 th minute is shon through Figures 3 to 5. The temerature variation o the ater and ed along its length or to inlet temeratures o 40 0 C and 55 0 C is resented in Figure 3. Higher inlet temeratures sho greater nondimensional temeratures or the luid and the ed or 30LPH. The eect o concentration on the temerature roiles are shon in Figure 4. At 5% nanoluid concentration, or a lo rate o 30LPH, an inlet temerature o 55 0 C, the temerature gradients or nanoluids are greater than ater y a 5% to 7%; this is due to larger values o thermal conductivity o nanoluid comared to ater at the same temerature. Under similar oerating conditions or to article sizes resented in Figure 5, the gradients or smaller size ed articles are higher due to the greater surace area comared to articles o larger diameter. The temerature gradients estimated ith theoretical models are oserved to e in satisactory agreement ith the exerimental values. Fig 6 reveals the eect o lo rates on Nusselt numer comared ith Darcy-modiied Rayleigh numer, Ra or 14.6mm glass eads using ater as the orking luid. The exerimental values o Nusselt numer at an asect ratio AR =1. or dierent values o Ra are determined. The Nusselt numer is oserved to increase ith the lo rate o the orking luid. The variation o temerature and nanoluid concentration on the values o Nusselt numer is shon in Figure 7. The exeriments are undertaken at various temeratures in the range o 40 to 55 0 C at a constant lo rate o 15LPH ith 14.6mm glass eads as ed articles. The Nusselt numers otained are comared ith Eq.(16 o Oosthuizen [1989]. The exerimental values o ater otained ith 6mm and 14.6mm glass eads are lotted and comared ith the regression Equations o Chang [1980], Oosthuizen [1989] and Garon & Goldstein [1973] or steady state natural convection are shon in Figure 7. Exeriments are undertaken ith nanoluid at dierent mass lo rates and concentration ith glass eads o 14.6mm diameter are resented in Figure 9. The Nusselt numer increases ith lo rate and concentration o the nanoluid. The exerimental heat transer coeicients are estimated ith the Eqs.(18 and (19 and comared ith the values rom theory given y Eq. (16. It can e oserved rom Figures (6 to (9 that the exerimental Nusselt numers are in satisactory agreement ith the values otained ith Oosthuizen correlation (1989 Eq. (16. A regression equation alicale or Raleigh numer in the range o 13.3 x 10 6 to.1 x 10 6, ulk temerature eteen 40 0 C and 55 0 C or ed article diameter alicale eteen 6.0 and 14.56mm is given y Nu 01151Ra (1 (0 valid in the volume concentration range o 0< ϕ< 5% 7. CONCLUSIONS The exerimental results are in good agreement comared ith the numerical results or the ase liquid ater. The enhancement in Nusselt numer ith nanoluid is greater y 10 % ith 5% nanoluid concentration. The olloing conclusions are made. 1. The temerature gradient increases ith time and concentration o the nanoluid.. The temerature gradient is redicted oth exerimentally and theoretically. Mathematical models are emloyed y Burmeister [1993], Bejan [1995], Hornung [1997] or the determination o transient temeratures in a luid saturated acked ed storage system under natural convection associated ith lo lo rates. The exerimental NOMENCLATURE gradients are in good agreement ith theoretical models ith 10 to 0% o deviation. The temerature gradient increases ith time and concentration o the nanoluid. 3. The nanoluid heat transer coeicient is greater ith 6mm comared to 14.56mm ed articles, at the same lo rate. The heat transer coeicient or 5 % nanoluid ith 6mm articles is 1.7 times higher at loer temeratures and 1.85 times higher at 55 0 C ith Raleigh numer rom 13.9x10 6 to.1x10 6 rom loer to higher temeratures. 4. The thermal energy storage caacity o the acked ed system is enhanced ith increasing concentration o nanoluid y 10% to 15% y using nanoluid at 5% o volume concentration. C seciic heat, J / KgK d diameter K D a Darcy numer, D t h convective heat transer coeicient, W / m K IJERTV5IS09083.ijert.org 95

6 h ex Exerimental heat transer coeicient k thermal conductivity, W / mk K Permeaility L length o the tue, m Nu Nusselt numer ased on the luid roerties, Pr Prandtl Numer, C / k r Radius m mass lo rate * R Ra Ra Nondimensional radius Rayleigh numer ased on the luid roerties g D ( T 3 t h ( T Darcy modiied Rayleigh numer, g KD ( T t ( h c T T Temerature, K Z non dimensional axial length z L,v, u Velocity comonents in axial, radial directions U 0 c Reerence velocity / suericial velocity o luid Greek symols Thermal diusivity, Void raction k C Dimensionless luid temerature, Thermal exansion coeicient dynamic viscosity, kg/m s T T Ti T density, kg/m 3 non dimensional time volume concentration o nanoarticles, % Suscrits ase luid e eective nanoarticle th theoretical ater luid n Nanoluid i Inlet 0 Outlet REFERENCES [1] Basak T., Roy S.,Takhar H.S, Eects o non-uniormly heated all(s on a natural- convection lo in a square cavity iled ith orous medium, Journal Numerical Heat Transer Part A: A 51 ( [] Burmeister. L., Convective heat transer, second ed., 1993, John Wiley & Sons Inc. [3] Bejan, A, Convective Heat Transer Second ed., 1995, Wiley, Ne York. [4] Bejan, A., Convective heat transer in orous medium Handook o Single hase convective heat transer, Chater16, 1987, Wiley, Ne York. [5] Beckman,W., Die Warmeuertragung in Zylindrischen Gasschichtenei Naturlicher Konvektin" Forsch.Ge. Ing.Wes., [6] Charrier-Mojtai M.C. Mojtai, A. Caltagirone, J.P, Numerical solution o lo due to natural convection in horizontal cylindrical annulus, J. Heat Transer vol. 101, 1979, [7] Calmadi, V.V, Mahajan, R.L, The eective thermal conductivity o high orous irous metal orm, J. Heat transer, Vol.11, [8] Calatigirone,J.P,Farie,P. Natural convection in a orous medium at high Rayleigh numers, Part- 1.Darcy smodel.eur. J. Mechanical Engineering, Vol. 8, 1976, [9] Edimilson J. Braga, Marcelo J.S. de Lemos, Simulation o turulent natural convection in a orous cylindrical annulus using a macroscoic to-equation model, International Journal o Heat and Mass Transer vol.49, 006, [10] Elder,J.W, Steady ree convection in a orous medium heated rom elo,j.fluid Mech.7, 1967, [11] Farouk, B., Guceri, S.I., Laminar and turulent natural convection in the annulus eteen concentric cylinders, in Symosium Volume on Natural Convection, 0th National Heat Transer Conerence, Milaukee, Wisconsin, 1981, [1] Garon, A.M and Goldstein, R.J., Velocity and heat transer measurements in thermal convection, Phys Fluids, vol.16, 1973, [13] Hornung U., Homogenization in orous media, Sringer Verlag, Ne York [14] Khan.A.A.,Zei A., Doule diusive in staility in a vertical layer o a orous medium., J. Heat Transervol.103,1981, [15] Krishnan, S., Murthy, J.Y., Garimella, S.V., A to temerature model or the analysis o assive thermal control systems, J Heat Transer , [16] Kaluri R.S.,. Basak T,. Roy S, Bejan s heatlines and numerical visualization o heat lo and thermal mixing in various dierentially heated orous square cavities, Numerical Heat Transer Part A: Al. 55 (5 ( [17] Kuehn, T.H. Goldstein R.J, An exerimental study o natural convection heat transer in concentric and centric horizontal cylindrical annuli, J. Heat Transer vol , [18] Kaviany, M., Princiles o Heat Transer in Porous Media, second ed., Sringer-Verlag, Ne York, [19] Kraussold.H., WarmeagadeVonZylindrischenFlussigkeitsschi chtenei Naturlicher Konvektion, Forsch. Ge. Ing.Wes , [0] Nield, D.A., Onset o thermal convection in orous in a orous medium Journal o ater Resources Res.11, 1968, [1] Nield D.A, Bejan A, Convection in Porous Media, Sringer, Ne York, 199. IJERTV5IS09083.ijert.org 96

7 [] Oosthuizen, P.H., and David Naylor, An Introduction to convective heat transer analysis, TMH, chater10, [3] Poulokakos,D. and Bejan, A Penetrative convection in orous medium ounded y a horizontal all ith hot and cold sots, Int. J. Heat mass transer vol.7, 1984, [4] Prasad., V.; Kulacki., F.A., Natural convection in a vertical orous annulus, Int. J. Heat Mass Transer 7 ( [5] Prasad., V., Numerical study o natural convection in a vertical, orous annulus ith constant heat lux on the inner all, Int. J. Heat Mass Transer 9 ( [6] Phanikumar, M.S., Mahajan, R.L., Non-Darcy natural convection in high orosity metal oams, Int. J. Heat Mass Transer vol45, 00, [7] Sharma, K.V. Sarma, P.K., Azmi W.H., Rizalman Mamat., Noor M.M, Kadirgama K., [8] Estimation o turulent orced convection heat transer coeicients o ater ased nanoluids, ICMER-1 Vol.1 01, [8] Syam Sundar L., Sharma. K.V., and Ramanathan.S Exerimental investigation o Heat Transer Enhancements ith Al O 3Nanoluidand Tisted Tae Insert in a Circular Tue,International Journal o Nanotechnology and Alications Numer [9] Srinivasrao. G. Sharma. K.V., Naryana K.V.S., Chary. S.P Exerimental Study o Pressure Dro and Heat Transer Coeicients in a Packed Bed Column ith Glass Beads International Journal o Alied Engineering and Research volume6, Numer 39(011, [30] Srinivasa Rao, G., Sharma. K.V, Chary. S.P., Bakar. R.A., Rahman M. M., Kadirgama K. and Noor M.M. Exerimental Determination o Heat Transer Coeicient And Friction Factor O Al 0 3 Nanoluid In A Packed Bed Column Journal o Mechanical Engineering and Sciences (IJMES e-issn: ; Volume 1,. 1-15, Decemer 011 FKM, Universiti Malaysia Pahang. [31] Srinivasrao. G; Sharma. K.V.;. Chary S.P,;. Naik M.T.; Forced Convective Heat Transer Eects and Flo Resistance in a Packed Bed Column ith AlO3 Nano Fluid International Journal o Engineering, Science and Metallurgy Vol.1, No [3] Saeid N.H,. Po I, Natural convection rom discrete heater in a square cavity illed ith a orous medium, J. Porous Media 8 (1 ( [33] Saeid N.H., Natural convection rom to thermal sources in a vertical orous layer, ASME J.Heat Transer18 ( [34] Sathiyamoorthy M,. Basak T. Roy, S. Po,, I Steady natural convection lo in a square cavity illed ith a orous medium or linearly heated side all(s, Int. J. Heat Mass Transer 50 ( [35] Zhao C.Y., Kim T.J., Lu T.J., Hodson, H.P, Thermal transort in high orosity cellular metal orm, J.Thermo hysical Heat transer,1,.,004a, [36] Zhao C.Y.,. Lu T.J, Hodson H.P., Jackson J.D., The temerature deendence o eective thermal conductivity o oen-celled steel alloy oams, Material Science Eng., Vol. 367, 004, [37] Zhao C.Y., Lu T.J, Hodson, H.P, Natural convection in metal oams ith oen cells, Int. J. Heat Mass Transer , Figure. 1 Photograh o the exerimental setu Figure 3 Eect o mass lo rate and inlet temerature on temerature distriution or solid and luid. IJERTV5IS09083.ijert.org 97

8 1 m Part list: 1. Suly tank.. Test section 3. Collecting tank 4.Pum 5.Data Acquisition System Figure. Flo diagram or natural convection Figure4: Eect o nanoluid concentration on temerature distriution or luid and solid Fig. 5: Eect article size o article and concentration o nanoluid on nondimensional temerature distriution or solid and luid Figure. 6 Eect o lo rate and temerature on Nusselt numer or validation o exerimental setu ith ree convection IJERTV5IS09083.ijert.org 98

9 Fig.7 Eect o luid inlet temerature and concentration on Nusselt numer at loer lo rates Fig. 9 Eect o lo rate and concentration o nanoluid on Nusselt numers Figure 8: Eect o article size and luid inlet temerature on Nusselt numers IJERTV5IS09083.ijert.org 99