HEAT TRANSFER ENHANCEMENT IN ROTATING DISK BOUNDARY-LAYER

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1 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp HEAT TRANSFER ENHANCEMENT IN ROTATING DISK BOUNDARY-LAYER by Ahmer MEHMOOD and Muhammad USMAN * Department o Mathematics and Statistics, FBAS, International Islamic University, Islamabad, Pakistan Original scientiic paper Introduction A generally admitted act about the nanoluids is the expedition o heat transer process in comparison to pure luids. The calculation o enhanced rate o heat transer depends strongly upon the nanoluid modeling. Following the experimental evidence most o the researchers assume the nanoluid to be a homogeneous mixture. However, this is a severe condition that results in under-prediction o heat transer rates. Due to the ongoing convection phenomena the nanoparticle concentration is actually non-homogeneous within the boundary-layer because o the presence o concentration gradients. The objective o this study is to calculate the heat transer enhancement in 3-D boundary-layer when the working luid is a nanoluid. The rotating disk geometry, which perhaps serves as the bench mark or the 3-D boundary-layers, have been chosen or the purpose here. The non-homogeneous nanoluid modeling has been utilized and a percent increase in Nusselt number has been calculated. Detailed analyses o low and heat transer phenomena or nanoluids have been conducted under the inluence o several physical parameters. Key words: nanoluid, rotating disk, heat transer enhancement, non-homogeneous model Heat and mass transer phenomena occur in diverse engineering applications like heat and mass exchangers, computer storage devices, luid low, and convective heat transer in rotor-stator systems (the rotor or a rotating disk and the stator or a stationary disk) are o great importance in turbomachinery and power engineering. Such situations require eicient heat transer mechanism so that the working machinery can continue without any collapse. Thereore the engineers and scientist are always interested in searching or situations in which higher rates o heat transer can be achieved. Dierent techniques are adopted, depending upon the suitability to the situation, in order to enhance the heat transer process. Among many choices, one o the modern technique is the selection o nanoluid as a cooling agent. The notion o nanoluids comes rom the act that metals have higher thermal conductivity than liquids. Due to the advent o nanotechnology it became possible to create metallic particles o nanoscale size that can be mixed in a luid (called base luid) to orm a nanoluid with enhanced thermal conductivity. Addition o small metallic particles has a big impact on thermophysical properties o the luid. The low and heat transer with microsize particles was irst studied by Maxwell * Corresponding author, usman75.iiui@gmail.com

2 468 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp [1] in Though he revealed some enhancement in heat transer but the particle dimensions caused sedimentation, abrasion and clogging []. In order to keep the mixture homogeneous, the size o the particles needed to be minimized. This can now easily be achieved with the help o modern nanotechnology, which can produce particles o the size between nm. Due to extremely small-sized particles, comparable to the size o the molecules o the base luid, they are easily accommodated by the luid and the aorementioned issues caused by the metallic particles get resolved. Choi [3] was the irst who introduced the word nanoluid or such luids. In most o the available literature on nanoluids, the researchers modeled the nanoluid as a homogeneous mixture by incorporating the role o thermophoressis and Brownian motion [4, 5]. Several authors investigated heat and mass transer phenomena through Buongiorno [4] and Tiwari and Das [5] models or variety o low geometries, a ew o them are mentioned here [6-9]. Based on this modeling Magia et al. [8] and Avramenko et al. [10, 11] pointed out that, due to homogeneous models, the heat transer coeicients o nanoluid are underpredicted. The reason is the incorrect incorporating o local concentration o the nanoparticles within the boundary-layer. The drawback with the Buongiorno [4] and Tiwari and Das [5] models is the consideration o uniorm distribution o nanoparticles in the base luid. This is actually done on the basis o experimental evidence. However, the homogeneous distribution o nanoparticles is not sustained within the boundary-layer due to the relative motion o nanoparticles and luid. Frank et al. [1] and Ding and Wen [13] theoretically proved the non-homogeneous distribution o nanoparticles in the nanoluid as a consequence o velocity and temperature gradients. The existence o concentration gradients is also conirmed through experiments [9]. Such a consideration also results in heat and mass transer rate augmentation. This issue has recently been addressed by Avramenko et al. [10] where they successully incorporated the eects o convective transport o nanoparticles on momentum, thermal and mass transer phenomena within the boundary-layer. For detailed inormation it is ruitul to ollow [14-16]. Rotating low phenomena occur in diverse engineering applications, namely, turbines o rocket, gas turbines and centriugal pumps, etc. These important applications in the leading areas o science and engineering motivate one to investigate heat and mass transer phenomena in rotating boundary-layers. Rotating disk systems can be used to model the low characteristics, together with heat and mass transer rates, that arise in practical turbomachinery or instance the low o waxy crude oils or oodstus in centriugal pumps, air crat engines, car brake systems, computer disk drivers and electrochemistry [17]. There are dierent reasons or choosing the rotating disk system as a prototype or practical rotating lows. In order to search a preerable system to study the transport phenomena o luid low in 3-D boundary-layers, the rotating disk is conirmed to be the best choice. It allows the complete 3-D sel-similar exact solution whereby reducing the coupled system o Navier-Stokes equations to a simple non-linear system o ODE. Despite the mentioning o above closely related engineering applications and the potential o mathematical simpliication the rotating disk low serves as the global paradigm or many 3-D lows, such as the low on swept wings. The generation o spiral vortices at the edge o swept wings is similar in nature to the class o low vortices developed due to the rotation o the disk in an ininite ambient luid. The presence o the point o inlection in the laminar velocity curve o the rotating disk low indicates the vulnerability o it to the cross-low instability. This ultimately makes the rotating disk model as an alternative laboratory towards the study and understanding o many boundary-layer eatures associated particularly to the 3-D lows. Based on these observations and great resemblances among the rotating disk boundary-layer and the 3-D boundary-layer Lingwood [18] preerred to choose the rotating disk model towards the study o absolute instability associated to the 3-D low. Her investigation concludes that

3 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp neither the Coriolis orce nor the streamline curvature aect the absolute instability in the disk low which conirms the application o the disk results to the swept wing case where the Coriolis eects are absent at all. Continuing with the Lingwood s choice o disk model Pier [19] proposed a new method in order to delay the onset o transition. This was accomplished by introducing a continuously supplied periodic orce to the unstable region in order to improve the ongoing sel-sustained non-linear dynamics. The laminar-turbulent transition in the boundary-layer lows can also be delayed, in some cases, because o the advantageous eects o the compliant walls. In order to investigate the supportive role o compliant wall in general, and to understand the drag reduction on the swimming dolphin s body due to the lexible nature o its skin in particular, Carpenter and Thomas [0] also preerred to choose the rotating disk model as the convenient low geometry. Following the same analogy between the disk low and 3-D boundary-layers, as utilized by the above researchers, Davies and Carpenter [1] also considered it as a proto type model o or the investigation o linear global behavior o the absolute instability in 3-D boundary-layers. Continuing in this way Imayama [] contributed suicient eorts towards the urther understanding o the laminar-turbulent transition in the rotating disk low. Further relevant, and o course interesting, studies can also be ound in the bibliographic items o []. The general reason behind the preerence to the rotating disk model in the above mentioned studies is the most general nature o disk low that resembles in large to the 3-D lows. The glimpses rom the bulk o available literature seem to be suicient or the reader in order to make him realize the wider scope and physical richness o the von Karman swirling low. Keeping this act in mind the rotating disk low has also been chosen here or the purpose. The rotating disk problem was irst investigated by von Karman [3] in 191. He transormed the well-known Navier-Stokes equations into coupled ordinary dierential equations using similarity transormations. Ater the work o von Karman, many o the researchers took interest in the study o rotating disk boundary-layers [4-7]. Bachok et al. [8] studied the eect o nanoluid on heat transer in rotating low near a porous disk. Turkyilmazoglu [9] considered the same low or a non-porous disk taking ive dierent types o nanoparticles. Enhanced heat transer rates were noticed in both the studies [8, 9] where the uniorm homogenous model o nanoluid was considered. Further results on rate o heat and mass transer enhancement can be ound in recent studies concerning rotating disk systems [30-3]. In all these studies [8-3] the nanoparticle concentration gradient as well as dependence o luid properties on nanoparticle distribution have not been taken into account. This provides a room or studying heat and mass transer enhancement in a rotating disk system using the idea o non-homogeneous distribution o nanoparticles in the boundary-layer. The objective o this article is to consider orced convection heat and mass transer phenomena in a rotating disk boundary-layer. The luid properties are considered as variable, depending upon the nanoparticle concentration. Enhanced rate o heat transer has been calculated and the eects o nanoparticle concentration on mass-low rate and moment coeicients are also highlighted. Results are interpreted through several graphs and tables. Mathematical ormulation Nanoluid modeling Consider a mixture o a pure luid and metallic particles o nano size in it. The concentration o nanoparticles in the base luid is ixed as φ (in percent). The presence o nanoparticles in the base luid alters the material properties o the mixture such as viscosity, density and thermal conductivity. Because o the diverse nature o nanoluids with regrad to the nature, size, shape and concentration o the nanoparticle within the base luid, it is quite hard to propose a

4 470 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp single model or every such property which would be applicable everywhere in general. So ar, the theory and experiment have not been succeeded in doing so. Consequently, there exist several empirical and theoretical models or the description o eective viscosity, density and thermal heat capacity in the nanoluid. These models have their own merits and demerits with in the limited domain o applicability. For example, a list o several models, based on theory and experiment, or the eective viscosity o nanoluids is given in a nice book by Minkowytz et al. [33]. Most o the models have particularly been developed or nanoparticles o spherical shape. The spherical shaped nanoparticle with no intra interaction are usually preerred where the nanoparticles are required to behave, to some extent, as the luid particles. Einstein [34] introduced the simplest model or eective viscosity o the nanoluid by using the hydrodynamic equations where the nanoparticle volume raction was limited to % only in the orm: µ = 1+.5φ µ (1) The Einstein s model (1) was urther improved by Brinkman [35], Batchelor [36], and Lundgren [37] in order to extend it to high-moderate concentration or to incorporate the Brownian motion eects. Lundgren [37] simply assumed the Einstein s model as the irst two terms o the Taylor series expansion o the orm: µ µ = = ( 1+.5φ + 6.5φ + ) µ 1.5φ and proposed the consideration o the complete series () instead o the irst two terms. The model is however still limited to the dilute suspension o nanoparticles. The Batchelor s model [36] improves the Einstein s one by including a one more term o the Taylor series () whose coeicient is taken as 6. instead o 6.5 as in eq. (). This model is actually based on the reciprocal theorem o Stokes low and includes the eects o Brownian motion. The Brinkman model [35] also considers the Taylor series o the orm: µ µ = (1.5 φ) whose irst two terms also cover the Einstein model. This model is applicable to the moderate and high concentration o the nanoparticle. In addition to these, there are several other nanoluid models or eective viscosity which are though limited to the spherical shaped nanoparticles but also include the radius o the spheres with some other restrictions. However literature is not limited to the spherical shaped nanoparticle but also includes some other shape such as cylinder, cone, brick, etc. Despite the presence o all such theoretical models the literature is also quite rich in the empirical models o the eective viscosity based wholly on the experimental data. Although, the empirical models are quite exact and have the capacity o predicting the very true results, but the problem with such models is that they are very limited in scope and apply only to those particular situations or which the corresponding experimental data have been collected. While in the theoretical analysis, as conducted in the current study, one is more inclined towards the general analysis o heat and mass transport phenomena in order to make a general qualitative analysis o the whole process. On this bases the theoretical models are usually being preerred having wider scope o applicability with some compromise on the qualitative measurements. Another aspect which is somehow more preerred by the theoretical analysits is the mathematical simpliication o the chosen models. In the current analysis we choose to assume the spherical shape o nanoparticle and the moderate range or nanoparticle concentration. Ow- /5 () (3)

5 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp ing to these properties and the mathematical ease the Brinkman model is observed to be the appropriate or current analysis. The eective density o the nanoluid based on the physical rule o mixture is given: ρ = (1 φ) ρ + φρ p (4) which also shows an excellent agreement with the experiment. Maxwell [38] was the irst to derive the eective thermal conductivity o the solid-liquid mixture and proposed that: k = k k + k + φ k k kp + k φ kp k p p which is applicable to the moderate concentration levels. The eq. (5) was extended to high concentration nanoluid by Bruggeman [39] in the orm: k p ( 3φ 1) + 31 ( φ) 1+ k k k ( 3φ 1) kp kp = = φ + k k which readily reduces to the Maxwell s one or the case o moderate concentration. The thermal heat capacity o the nanoluid based on the analytical model reads: ( 1 ) p ρc= φ ρc + φ ρc (6) It has been shown in Minkowytz et al. [33] that the expansion eq. (6) inds an excellent agreement with the experimental data. The Brownian motion diusion coeicient and thermophoresis parameters are given: D B kt B =, DT = βνφ 3πµ d where the thermal expansion parameter β is deined: p 1 p (5) (7) β = φ β + φβ (8) An alternative deinition o β is also available in literature with a certain modiied orm o the β given: ( 1 ) ρβ = φ ρβ + φ ρβ (9) In comparison to the experimental data both deinitions o β do not ind a good agreement; a poor approximation due to eqs. (8) and (9) can, however, not be denied. However, it has also been observed that the selection o eqs. (8) or (9) does not harm the results o skin riction and Nusselt number by any large. Staying on equal ooting regarding the experimental data, eqs. (8) or (9) are equally valid with eq. (8) having an edge o mathematical simplicity. Using the modiied material parameters or nanoluid the conservation laws governing the convective transport phenomena are given: ρv = 0 (10) p

6 47 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp v T ρ ( v ) v p + = + µ v+ v δ v t 3 h ( v ) h ( k T) T T ρ + = + ρpcp DB φ T + DT t T φ T + ( v ) φ = DB φ + DT (13) t T where v = ( vr, vϕ, vz) is the velocity vector, h = ct the enthalpy, v the velocity tensor gradient, and v T the conjugate tensor gradient o velocity. Governing equations We consider a uniorm lat disk o radius b resting at z = 0 in 3-D space. The disk is surrounded by the homogeneous nanoluid. The radius o the disk is assumed to be large enough so that it is larger than the boundary-layer thickness; edge eects have been ignored due to which the low resembles to that considered by von Karman or an ininite disk. z The disk rotates about the z-axis with an T δ angular speed ω. The disk temperature v φ v r is ixed as T w where the ambient temperature is assumed to be T such that T Tw < T. A schematic o the low geometry is shown in ig. 1. ωr b 0 φ T w The symmetry o the low geometry requires consideration o cylindrical r co-ordinates or urther analysis. In this ω way the conservation laws, eqs. (10)- (13), readily read as in cylindrical co-ordinates: Figure 1. Schematic o the ree rotating disk ( ρrv ) ( ρv ) 1 r z + = 0 r r z ( rv ) v v r vr p r vr ρ v ϕ r v µ + z = + + µ r r z r r r r z z ( rvϕ ) vϕ vv r ϕ vϕ µ vϕ ρ vr + + vz = + µ r r z r r r z z ( rv ) vz vz p µ z vz ρ vr + vz = + + µ r z z r r r z z (11) (1) (14) (15) (16) (17) h h T φ T DT T ρ vr + vz = k + ρpcp DB + r z z z z z T z (18)

7 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp φ φ φ DT T vr + vz = DB + r z z z T z and the appropriate boundary conditions are: φ D T z = 0: v = 0, v = rω, v = 0, T = T ( h= h ), D = z T z T r ϕ z w w B z= 0 z= 0 z : vr 0, vϕ 0, T T ( h h ), φ φ = = = = = = (19) (0) Sel-similar solution Due to its inherited nature, the von Karman swirling low o a nanoluid on a rotating disk also admits separable solution o the orm: vr = r ( z), vϕ = rg( z), vz = hv( z), p = p( z), h = he( z), φ = φ( z) (1) The use o dimensional analysis under the implementation o Buckingham Pi theorem results in the new dimensionless variables: ω η = z, = ωf( η), g = ωg( η), hv = ν ωhv( η), ν,, p = ρν ωp η h = h H η φ= Φ η E E, E The von Karman similarity transormations or velocity, pressure, enthalpy, and nanoparticles volume raction are usually obtained as a combination o eqs. (1) and (). Since the conventional notation or the dimensionless axial velocity in von Karman swirling low is H which is also used as a conventional notation or dimensionless enthalpy unction. In order to ollow the use o conventional notation and to avoid any conusion, the subscripts v and E have been introduced. In this way hv and H v represent the velocity unctions and he and H E correspond to enthalpy unctions in eqs. (1) and () and hence orth. The present ormulation requires sel-similar low rates in order to establish the similarity solution. In doing so the variable nature o the density unction is accommodated quite easily. Following Avramenko et al. [10] the sel-similar low rates are deined:,, r ϕ z v () ρv = ρ rωf η ρv = ρ rωg η ρv = ρ ν ωh η (3) where the ambient nanoulid density is used as the reerence density. The unctional dependence o thermophysical properties o nanoluid are described: ρ= ρφ, µ = µφ, D = D ( T), D = D ( φ), c= c( φ), k = k( φ) (4) B B T T Utilizing eqs. (1)-(4) the conservation eqs. (14)-(19) immediately transorm to the sel-similar orm: H + F = 0 (5) v R R R R M F F Φ + F Φ FΦ FΦ + R R R R + R ( M Φ ρhv ) F FΦ ρ( F G ) = 0 (6) R

8 474 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp R R R R M G G Φ + G Φ GΦ GΦ + R R R R R ( M Φ ρhv ) G GΦ + ρfg = 0 R (7) R R R R M Hv HvΦ Hv Φ + HvΦ HvΦ + R R R R R P + ( M Φ ρhv) Hv HvΦ = 0 R R (8) RC 1 R RC D ρprhv + Φ K + + K + + R Le R RC Le R RC 1 R RC KH E + H E' KΦ Φ K Φ D R RC Le + R RC + + H E RC R RC R RC + K + RC R RC R RC D H E + = 0 (9) Le H R RC HE DT ρ Φ 1 + D HE + Φ + DB ScHv + R RC DB H E R R RC DB D T RC R R RC HE H E Φ + HE + + D + D 0 + = R RC DB D B RC R R RC H E H E where ( Φ) E ρ 5 p ρcp R( Φ) = ( 1 Φ) + Φ, RC ( Φ) = ( 1 Φ) + Φ, M ( Φ) = ( 1 Φ), ρ ρc K kp + k + Φ kp k µ c µ =, Pr =, Sc HE ( η), k k Φ k k k = + ρ D p p ( η ) ( ρc) ( ρc) * D Sc H T ρ D =, Le HE ( η ), ρ D = = (31) Pr ρ B B Here the primes in the unctions F, G, Hv, HE, and Φ mean derivative with respect to η and primes in the unction R, RC, M, K, and D T mean derivative with respect to Φ and primes on DB mean derivative with respect to H E. From eq. (31) it is clear that the Prandtl number (as modeled) is independent o the inluence o the nanoparticles. The boundary conditions (0) also transorm to the sel-similar orm: p (30)

9 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp ( φ ) he, w R w H E RC η = 0 : F = 0, G =, Hv = 0, P = 0, HE =, = Φ D ρ h H RC E, η = : F = 0, G = 0, H = 1, Φ = φ where the subscripts w and with he denote the values o enthalpy at wall and ambient, respectively. In order to accommodate the ratio he,w / he, appearing in eq. (3), we replace enthalpy by temperature in energy eq. (18). In doing so, we also assume that speciic heat capacity o the nanoluid is uniorm c which is a valid assumption [10] and especially holds or large values o the Shmidt number, which is characteristic o low o nanoparticle. Further D B and D T are also taken as constant, and constant temperature equal to T in the denominator o last term in eqs. (18) and (19). Under these assumptions eqs. (18) and (19) are must to be simpliied to the ollowing orm: where E E (3) Φ D Kθ + θ + K Φ ρprhv + θ = 0 Le Le (33) ρ Φ ScHvΦ + Dθ = 0 R (34) T T µ c Sc ρ c T T D T T k Pr ρ c T D w w =, Pr =, Le =, D = θ η w T p p B Accordingly, the boundary data (3) also simpliies to: R( φ ) η = = = = = θ θ Φ ρ w 0 : F 0, G, Hv 0, P 0, = 0, D + = 0 η = : F = 0, G = 0, θ = 1, Φ = φ The analysis o current study strongly depends upon the role o some dimensionless physical parameters such as mass-low rate, moment coeicient, displacement thickness, tangent o the low swirl angle and the Nusselt number. The deinitions o these parameters speciic to the current study are given as the Nusselt number at the disk surace is deind by Nu = αr/ k, where α = k( T/ z) z = 0. The moment coeicient on both sides o disk is given by 5 CM = M/[(1/) ρnω b ], where is the radius o the disk (edge eects are ignored) and M is b the moment deined as M = π r τ d. 0 zϕ r The tangent o the low swirl angle at the disk surace is calculated as [18] α w = [( vr/ z)/( vϕ / z)] z= 0, the mass-low rate across the boundary-layer over the ree rotating disk is calculated as m d = πb ρ v d, z 0 r z and the displacement thickness is deined: = Numerical solution * 1 δ = v dz ωr ϕ z= 0 Numerical solution o the reduced boundary-layer eqs. (5)-(8), (33), and (34) corresponding to boundary data (35) is obtained through MATLAB built in sotware. Convergent (35)

10 476 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp solution is obtained ater several runs under the deault tolerance level o The accuracy o the present numerical procedure is checked by solving the sel-similar equations o pure luid over a ree rotating disk. A comparison o the present solution with that, already present in the literature [17] is given in tab. 1. The table shows an accurate match to our decimal places with the results obtained by Shevchuk [17] or dierent values o the Prandtl number. This authenticates the present solution procedure and allows or the solution o present equations using the same procedure. Table 1. Comparison between present results and Shevchuk [17] Pr θ'(0) F'(0) G'(0) Shevchuk [17] Present Shevchuk [17] Present Shevchuk [17] Present Result and discussion A detailed parametric analysis is made to understand the impact o nanoluid on heat transer phenomena in rotating disk boundary-layer. Von Karman similarity variables are also shown to be applicable to the present model and a sel-similar solution or varied values o the involved parameters, namely, Sc, φ, D, and Pr is obtained. In most o the results φ is varied rom 0.0 to 0.4 or Pr = 6 and strong convective heat transer is observed. Values o Schmidt number Sc range rom in order to assume the base luid as liquid. The diusion parameter D is taken as 0.05 or most o the study. The presence o nanoparticles in the luid also eects the momentum transport across the boundary-layer. The eect o Schmidt number on velocity components and pressure is shown in igs. and 4. Clearly, the inluence o Sc is very weak on all the velocity components and slightly stronger on pressure. Pressure decreases within the boundary-layer by increasing the values o Sc. Similarly, the temperature proiles are not aected by changing the values o Sc parameter as shown in ig. 3. However strong dependence o the concentration proile on Sc can be seen in ig. 3. Clearly, concentration boundary-layer thickness decreases by increasing Sc number which relects that the large values o Sc depreciate the mass transport phenomena across the boundary-layer and limit it to a very thin near-wall region. In comparison with momentum boundary-layer thickness, the concentration boundary-layer makes about 10% which means that over the most part o the momentum boundary-layer the nanoparticle concentration is uni Sc = 10 Sc = 100 Sc = 1000 Φ θ' Sc = 10 Sc = 100 Sc = Figure. Radial, transverse, and axial components o velocity or dierent Sc at φ = η η Figure 3. Temperature and concentration proile or dierent Sc at φ = 0.01 η

11 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp orm and is varying rapidly in the region close to the rotating disk. This is the reason that large values o Sc correspond to weak Brownian diusion o nanoparticles. In such a situation nanoluid behaves more or less as a homogenous mixure. Thereore simulation o transport processes with a homogeneous mixture is justiied or situations where Sc accepts large values. Figures 4 and 5 depict the eects o nanoparticle concentration φ on velocity components and pressure. It is clearly seen that all the velocity components are increased by increasing φ. P Sc = 1000, 100, 10 φ = 0, 0.1, 0. θ φ φ φ = 0.0 = 0. = 0.4 η Figure 4. Eects o Sc (at φ = 0.01) φ at Sc = 10 on pressure proile and C m /C m0 Nu/Nu0 η Figure 5. Radial, transverse and axial component o velocity and temperature proiles at Sc = 10 ṁ d /ṁ d0 Tanθ/Tanθ0 η Sc = 10 Sc = 100 Sc = 1000 Furthermore, the boundary-layer thickness also increases by increasing the concentration level o nanoparticles. This mean that the presence o nanoparticles in the luid enhances momentum transport across the boundary-layer. Figure 5 shows interesting impact o nanoparticles on the rotating disk boundary-layer. The downward axial velocity is enhanced by increasing the values o φ. This, in turn, relects the enhanced mass lux, which also avors heat transer enhancement. This can immediately be conirmed rom tab. that the nanoparticle concentration enhances the relative mass-low rate, and consequently the Nusselt number is also increased. Numerical values o the moment coeicient and the tangent o the low swirl angle have also been reported in tab. or varied values o nanoparticle concentration parameter φ and the Schmidt number. The results are very inormative and quite impressive, highlighting the signiicance o non-homogeneous concentration proile o nanoparticles across the boundary-layer. The relative Nusselt number increases by increasing φ but decreases by increasing the Schmidt number or the ixed values o φ. Furthermore, or small values o Schmidt number the rate o increase o Nu/Nu 0 with respect to φ is higher than that at large values o Schmidt number. This is because o the reason that at small Schmidt number values the non-homogeneous characteristic is somehow stronger and plays its role in heat transer phenomena whereas at large values o Schmidt number, the concentration is treated as almost homogeneous in most part o the momentum boundary-layer. This act highlights the importance o the present non-homogeneous model which correctly incorporates the contribution o nanoparticles towards momentum and thermal transport caused by their Brownian motion. In tab., it is also shown that moment o the disk is highly inluenced by nanoparticle concentration, meaning that the use o nanoluid in a rotating system increases its power as a consequence o enhanced moment. The eect o φ on the temperature proile is also shown in ig. 5. Analogous to the momentum boundary-layer the thermal boundary-layer also grows or increased values o φ. Figures 6 and 7 depict the curves o the parameters Nu/Nu 0, C m /C m0, α w /α w0, and m d/ m d0 against the nanoparticles concentration. Almost a linear relationship exists between φ and Nu/Nu 0 but non-linear or C m /C m0. The non-linearity becomes quite visible when the nanoparticle concentration becomes more than 0%, that is or φ > 0.. The tangent o the low swirl angle α w /α w0 does not depend upon the nanoparticle concentration, however, its value has been increased or nanoluid in comparison to base luid.

12 478 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp Table. Relative values o important physical parameters when Pr = 6, D = 0.05, ρp/ ρ = ϕ Sc = 10(Nu/Nu 0 ) Le (Nu/Nu 0 ) Le C m /C m α w /α w ṁ d /ṁ d δ * /δ * Sc = 50(Nu/Nu 0 ) Le (Nu/Nu 0 ) Le C m /C m α w /α w ṁ d /ṁ d δ * /δ * Sc = 100(Nu/Nu 0 ) Le (Nu/Nu 0 ) Le C m /C m α w /α w ṁ d /ṁ d δ * /δ * Sc = 1000(Nu/ Nu 0 ) Le (Nu/Nu 0 ) Le C m /C m α w /α w ṁ d /ṁ d δ * /δ * C m /C m0 Nu/Nu0 ṁ d /ṁ d0 Tanθ/Tanθ0 Sc = 10 Sc = 100 Sc = 1000 Figure 6. Relative moment coeicient and Nusselt number plotted against φ Figure 7. Relative mass-low rate and tangent o the low plotted against φ The relative mass-low rate m d/ m d0 or nanoluid also owns a non-linear relation with φ but the non-linearity weakens as φ grows. The quantative results o Nu/Nu 0, C m /C m0, α w /α w0, m d/ m d0, * * and δ / δ 0 are presented in tabs. -5 or speciic values o Sc, D, Pr and φ. Form tab., it is noticed that the relative Nusselt number, the tangent o the low swirl angle, moment coeicient, mass-low rate, and displacement thickness are increased on increasing the nanoparticle concentration ϕ. The values o all these parameters decrease by increasing the value o Schmidt num-

13 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp ber. The dependence o physical quantities o our interest such as Nusselt number, moment coeicient, tangent o angle o swirl, mass-low rate, and the displacement thickness on the base luid is an important aspect o this study. Present results help us to identiy a particular base luid that can be used in practical applications or optimal results. This analysis can be done rom tabs. 3 and 4 where we have listed results or various values o the Prandtl number. Dependence o the Nusselt number upon Prandtl number is demonstrated through tab. 3 by giving percent increase in the rate o heat transer. Clearly, at Sc = 10 the rate o heat transeris enhanced or luids o large Prandtl numbers at φ = 0.01 which is almost 18.51% when Prandtl number is increased rom 0.1 to 13 (130 times). But or the nanoluid with 0% nanoparticles the enhancement is not that signiicant, which is almost 3.18%. Almost the same situation persists with some depreciation or Sc = 100. On the other hand the water with 0% nanoparticles gives almost a 68% increase in the rate o heat transer in comparison to the pure luid. This act highlights the role o nanoparticles concentration towards heat transer enhancement. Table 3 concludes that the Nusselt number is increased by considering the base luid o large Prandtl number, but the major increase in the values o Nusselt number occurs at large concentration levels o nanoparticles. Relative values o the important physical parameters o current interest are listed in tab. 4 or dierent Prandtl numbers. From tab. 4 it is observed that, although the Nusselt number increases by increasing the Prandtl number, this increase is not signiicant. Similarly weak eects o Prandtl number are noted upon other parameters. It also reveals a weak dependence o above mentioned quantities on the Prandtl number, which means that a signiicant enhancement in the rate o heat transer (as listed in tab. 3) is simply due to the presence o nanoparticles only and the nature o the base luid does not matter considerably. However, at very low concentration levels, the luids o large Prandtl number serve as a good coolant. In a similar ashion the role o Table 3. Percent increase in the values o Nusselt number relative to the pure luid calculated at dierent Pr when D = 0.05, ρp/ ρ = Pr Sc =10 Sc =100 ϕ = 0.01 ϕ = 0.1 ϕ = 0. ϕ = 0.01 ϕ = 0.1 ϕ = % % % % % % % % % % % % % % %.8069% % % % % % % % % % % % % % % Table 4. Relative values o important physical parameters or dierent values o Pr when D = 0.05, ρ / ρ = , φ = 0.1 Sc = 10 Sc = 80 p Pr Nu/Nu C m /C m α w /α w ṁ d /ṁ d δ * /δ * Nu/Nu C m /C m α w /α w ṁ d /ṁ d δ * /δ *

14 480 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp diusion parameter D has been demonstrated in tab. 5. It is calculated that an increase o almost 8%, 5%, 14%, 8%, and 7% occurs in the values o Nusselt number, moment coeicient, tangent o the low swirl angle, displacement thickness, and mass-low rate, respectively, when the values o D were increased by 0 times. It reveals an appreciable role o the diusion parameter D towards low and heat transer enhancement. The discussion o tabs. 3-5 concludes that maximum gain in the rate o heat transer is obtained by increasing the nanoparticle concentration and urther enhancement is obtained by incorporating the role o nanoparticles diusion within the boundary-layer and the nature o the base luid does not aect largely. Table 5. Eects o D or Pr = 6, ρ / ρ = , φ = 0.1 Sc = 10 Sc = 80 p D Nu/Nu C m /C m α w /α w ṁ d /ṁ d δ * /δ * Nu/Nu C m /C m α w /α w ṁ d /ṁ d δ * /δ * Table 6 displays very important results regarding the non-homogeneous modeling o nanoluids which, o course, is the core issue highlighted in this communication. A comparison between the homogeneous (Tiwari and Das [5]) and non-homogeneous (Avramenko et al. [11]) models is made towards heat transer enhancement. Computed values relect that non-homogeneous model predicts increased rates o heat transer, as compared to the homogeneous model at all concentration levels or all Prandtl numbers but the dierence becomes wider when the nanoparticle concentration is 10% or higher. In particular, water with 0% nanoparticles produces almost 30% more gain in heat transer rate or non-homogeneous model as compared to homogeneous model. This act highlights the convection eects upon particles concentration within the boundary-layer which are usually ignored by the homogeneous models. Table 6. Percent increase in heat transer rate computed or homogeneous and non-homogeneous models Homogeneous Non-homogeneous (Sc = 10) Pr ϕ = 0.01 ϕ = 0.1 ϕ = 0. ϕ = 0.01 ϕ = 0.1 ϕ = Conclusions 0.00% 1.13 % 1.60 % 1.66% 1.68% 0.94% 10.91% % 16.53% 16.80% 4.93%.03% % 33.0% 33.77%.85% 3.77% 5.58% 6.19% 6.50% 30.% 31.60% 33.68% 34.34% 34.67% 66.57% 66.34% 67.81% 68.39% 68.69% Sel-similar solution or low and heat transer characteristics in a boundary-layer over a disk rotating in a nanoluid has been computed. The impact o nanoparticles is observed and calculated or various physical parameters o interest. The study reveals that the contribution o nanoparticles in the working luid over a ree rotating disk is manyold. In addition to the pre-

15 THERMAL SCIENCE: Year 018, Vol., No. 6A, pp sumed act that the nanoluid enhances the rate o heat transer, the rotating disk boundary-layer has beneited rom the nanoluid in variety o ways. The low and heat transer phenomena o nanoluid basically depend upon the Schmidt number and the nanoparticle concentration φ. Our analysis reveals that the role o Schmidt number in low and heat transer phenomena is not that signiicant but it strongly eects the concentration proile within the momentum boundary-layer. For large values o the Schmidt number the concentration layer becomes 10% o the momentum boundary-layer. However the variation in the values o φ contributes in dierent ways. The velocity proiles are enhanced due to the increased values o φ and stay unaltered by changing the Schmidt number. The pumping capacity o the disk is increased as a consequence o increased rate o mass-low or nanoluid and the mass-low rate urther increases by increasing the nanoparticle concentration φ. The presence o more nanoparticles in the nanoluid increases the moment coeicient o the disk. This act indicates the advantage or the rotating system towards increased power when the working luid is a nanoluid. The pressure alls or large the Schmidt number values whereas it grows in magnitude with the increase in concentration level o nanoparticles. The displacement thickness also increases with increased concentration. Furthermore, it is also observed that the variation o Prandtl number or some ixed φ also gives augmentation towards heat transer but it is not that signiicant as obtained by varying φ or any ixed Prandtl number. Overall 68% increase in the value o Nusselt number is observed when the base luid is water with a 0% nanoparticle concentration. The comparison between homogeneous and non-homogeneous models reveals that the non-homogeneous model predicts urther enhancement in heat transer processes as compared to the homogeneous model. In this way the non-homogeneous model is observed to be more useul or studying the convection phenomena with nanoluids. A considerable increase (o almost 30%) in rate o heat transer is computed through non-homogeneous modeling in comparison to homogeneous modeling when a mixture o water with 0% nanoparticles is used. This act highlights the role o non-homogeneous concentration o nanoparticles within the boundary-layer. Reerences [1] Maxwell, J. C., Treatise on Electricity and Magnetism, Oxord University Press, London, 1904 [] Keblinski, P., et al., Mechanisms o Heat Flow in Suspensions o Nano-Sized Particles (Nanoluids), International Journal o Heat and Mass Transer, 45 (00), 4, pp [3] Choi, S. U. S., Enhancing Thermal Conductivity o Fluids with Nanoparticle, in: Developments and Applications o Non-Newtonian Flows, (Eds. D. A. Siginer, H. P. Wang), ASME, New York, USA, C, 31 (1995), 66, pp [4] Buongiorno, J., Convective Transport in Nanoluids, Journal o Heat Transer, 18 (006), 3, pp [5] Tiwari, R. K., Das, M. K., Heat Transer Augmentation in a Two-Sided Lid-Driven Dierentially Heated Square Cavity Utilizing Nanoluids, International Journal o Heat and Mass Transer, 50 (007), 9-10, pp [6] Buschmann, M. H., Nanoluid in Thermosyphons and Heat Pipes: Overview o Recent Experiments and Modelling Approaches, International Journal o Heat and Mass Transer, 5 (009), Oct., pp [7] Rea, U., et al., Laminar Convective Heat Transer and Viscous Pressure Loss o Alumina-Water and Zirconia-Water Nanoluids, International Journal o Heat and Mass Transer, 5 (009), 7-8, pp [8] Magia, S. E. B., et al., Heat Transer Behaviors o Nanoluids in Uniormly Heated Tube, Supperlattices Microstructure, 35 (004), 3-6, pp [9] Merhi, D., et al., Particle Migration in Concentrated Suspensions Flowing between Rotating Plates: Investigations o Diusion Flux Coeicients, Journal o Rheology, 49 (005), 6, pp [10] Avramenko, A. A., et al., Sel-Similar Analysis o Fluid Flow and Heat-Mass Transer o Nanoluids in Boundary Layer, Physics o Fluids, 3 (011), 8, pp. 1-8 [11] Avramenko, A. A., et al., Symmetry Analysis and Sel-Similar Forms o Fluid Flow and Heat-Mass Transer in Turbulent Boundary Layer Flow o a Nanoluid, Physics o Fluids, 4 (01), 9, pp. 1-0 [1] Frank, M., et al., Particle Migration in Pressure Driven Flow o a Brownian Suspension, Journal o Fluid Mechanics, 493 (003), June, pp

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