Heat transfer from a nano-sphere with temperature and velocity discontinuities at the interface

International Journal of Heat and Mass Transfer 55 (2012) 6491–6498

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Heat transfer from a nano-sphere with temperature and velocity discontinuities at the interface Zhi-Gang Feng a, Efstathios E. Michaelides b,⇑ a b

Department of Mechanical Engineering UTSA, San Antonio, TX 78259, USA Department of Engineering TCU, Fort Worth, TX 76132, USA

a r t i c l e

i n f o

Article history: Received 26 January 2012 Received in revised form 4 May 2012 Accepted 18 June 2012 Available online 19 July 2012 Keywords: Nanoparticle Nanofluid Temperature slip Interface Nusselt number

a b s t r a c t A singular perturbation method has been used to derive a general equation for the rate of heat transfer from a sphere at low Knudsen number. The final expression includes both velocity slip and temperature slip at the interface and applies to a general Stokesian flow regime. The asymptotic analysis was carried up to the order Pe3ln(Pe). By choosing an expression for the drag multiplier, the derived expression for the Nusselt number may be applied to solid, fluid as well as porous spheres, which are special cases of the general solutions. Comparisons with known results for these special cases indicate the accuracy and wide range of applicability of the derived general expression. The inclusion of the temperature slip at the interface makes this equation applicable to particles, bubbles and drops of nanometer sizes, in the continuum or the slip-flow regime, that is for Knudsen number Kn < 0.1. Our results show that the velocity slip at the interface does not affect significantly the overall Nusselt number, Nu. However, the temperature slip affects the heat transfer significantly. If the temperature discontinuity becomes large, the sphere becomes almost adiabatic. This indicates that, if a temperature slip is possible at the interface of nanospheres, it must be taken into account by using the derived expression for Nu. Our results at the limit of Pe = 0 are compared very well with experimental results found in the literature. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The problem of the heat transfer from a sphere is as old as the subject of heat transfer itself and has its origin at the series of short papers that formed the doctoral thesis of Fourier [1] and were published later in his famous book on heat transfer. His motivation was to determine the age of the earth. Fourier’s work was considered as one of the most important scientific works of the nineteenth century as well as the intellectual stimulation for the methods adopted in many other scientific fields including the flow of electric currents [2]. In a twentieth-century treatise of the subject, Carslaw and Jaeger [3] essentially extended Fourier’s ideas on the transient conduction from a solid sphere and other simple geometrical shapes and presented several solutions on more modern applications of transient heat transfer at vanishing Peclet numbers. While the original work on the steady heat transfer from a sphere, preceded the studies on the hydrodynamic force from a solid sphere, both in viscous and potential flow [1], most of the recent studies on the heat and mass transfer from spheres are based on the analogous studies for the determination of the hydrodynamic ⇑ Corresponding author. Tel.: +1 817 257 6226; fax: +1 817 257 4582. E-mail addresses: [email protected] (Z.-G. Feng), [email protected] (E.E. Michaelides). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06.049

force. For example, for the calculation of the steady rate of heat transfer from a solid sphere at finite but small Peclet numbers, Acrivos and Taylor [4] used an asymptotic method similar to the one used by Proudman and Pearson [5] and conducted a study with the energy equation to derive an expression for the Nusselt number. Several other early empirical studies in the 1960’s and 1970’s derived correlations for the convective heat transfer coefficients at transient or steady-state processes, which are similar to the corresponding expressions for the steady-state drag coefficients. Taylor [6] followed a singular perturbation method and extended the solution by Acrivos and Taylor [4] to small spheres with a temperature slip at the interface under a Stokesian flow regime (Re  1). Two decades later, Brun [7] extended the solution of [4] to small but finite Peclet numbers and obtained the Nusselt number in cases of a sphere in various Stokes flows where Pe < 1. However, it has been observed that the solution by Taylor [6] at vanishing temperature slip does not agree with the solution by Brun [7] at finite velocity slip and vanishing temperature slip. Since the solution by Brun [7] has been verified, it was suggested that the original asymptotic study [6] and its final results may not be correct. Regarding the transient equation for the heat transfer from a sphere, Michaelides and Feng [8] used a method similar to the method used by Maxey and Riley [9] for the determination of the

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transient hydrodynamic force on a sphere. They obtained the first complete analytical solution for the unsteady energy equation at creeping flow conditions and showed that the general form of the transient energy equation from a sphere also contains a history term. This term is a consequence of the diffusion of the temperature gradients in fluid; it is similar in form to the analogous term in the Boussinesq/Basset expression and decays as t1/2 (t is a time variable) A subsequent study [10] followed the analytical method used [11] on the transient hydrodynamic force and showed that the energy equation becomes significantly different at small but finite values of the Peclet number and that the history term decays faster than t1/2 when Pe is small but finite. The subject of heat and mass transfer from very small spheres has attracted considerable attention in the recent past with the applications of nanofluids to the heat transfer in micro-channels. Nanofluids are composed of nanoparticles whose sizes are of the order of tens or hundreds of nanometers. Several experimental studies have shown that the effective thermal conductivity of nanofluids, keff, is significantly higher than the thermal conductivity of the base fluid, kb [12,13]. These observations make the nanofluids ideal cooling fluids for miniaturized electronic components and other applications with inherently limited heat transfer area. Key to the performance of the nanofluids is the heat transfer from the nanoparticles to the base fluid. Because the size of nanoparticles is of the order of nanometers, the pertinent Reynolds numbers, based on the relative velocity of the nanoparticles, are typically much less than one (Re  1). The pertinent Peclet numbers, do not always vanish, but they are invariably less than one (Pe < 1). In addition and because of the small size of the particles, the Knudsen number of the nanoparticles is not negligibly small. This signifies that the nanoparticles operate in a flow regime where velocity and temperature discontinuities (slip) at the fluid–solid interface are expected. Particle–fluid velocity and temperature discontinuities have been examined in the past because they are characteristics in the flow and energy exchange of particles in rarefied gases. Millikan [14] postulated the existence of a velocity slip that modifies the drag coefficient and Epstein [15] derived an expression for the velocity slip in terms of the reflections from the surface of a sphere. Springer and Tsai [16] devised a method for the experimental determination of the heat transfer between a sphere and a rarefied gas. They also delineated the four energy transfer modeling regimes of a particle in a rarefied gas, which are, in decreasing Knudsen numbers: (a) Free molecular energy transfer; (b) transitional; (c) temperature jump approximation; and (d) continuum approximation. Mikami et al. [17] use this classification and derived experimental data for the heat transfer coefficients of a small sphere (thermistor) in the temperature jump regime in the range 0.008 < Kn < 0.4. More recently, Barber and Emerson [18] examined non-equilibrium flows with applications to nanofluids and MEMS where interphase slip occurs. They also point out the dependence of the momentum and slip coefficients on the mean free path, the thermal accommodation coefficient and the stress at the solid surface. This study pertains to the heat transfer from small spheres at Re  1 and Pe < 1, which are the characteristics of nanofluids. The study is based on a singular perturbation method through which the inner and outer general solutions for the fluid temperature field around the sphere are obtained. A matching process at the region of overlap helps determine the coefficients of the general solution. From the constructed temperature field, an expression for the Nusselt number is obtained, which is valid for all values of temperature slip at the liquid-nanoparticle interface for a sphere immersed in a Stokes flow. The method used here employs a general factor for the hydrodynamic drag on the sphere, F. This makes the method and the derived results applicable to

cases other than solid spheres, such as bubbles, droplets, porous spheres, etc. The results of this study agree with the result derived [7] in the case of a finite velocity slip and zero temperature slip. This signifies that the final solution derived by this study is correct and an improvement to the one derived in [6]. The present results of the special case of conduction heat transfer from a sphere are compared very well with experimental data from Mikami et al. [17] and Takao [25]. 2. Problem description – governing equations Let us consider a small sphere of radius a immersed in a uniform flow of velocity U. The sphere is small enough for the condition Re  1 to be satisfied. The flow around the sphere is Stokes flow and the velocity field of the fluid may be obtained analytically. By using a as the reference length, U as the reference velocity, the dimensionless Stokesian velocity field in the spherical system of coordinates (r, h, u) that are chosen for this study is:



   3F 3F 1  1  3 3 cosh 2r r r

ð1Þ

  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3F 1 3F 1 1  ðcoshÞ2 ; þ  uh ¼  1  4r 2 4 r3

ð2Þ

ur ¼

1

and

where F is a parameter that represents the hydrodynamic drag on the sphere, which is made dimensionless by the Stoksian drag 6lpUa. This is a very general solution of flow velocity, with different drag expressions representing different types of flow regimes and was used in [7]. For example, the proper choice of F will make the final solution applicable to fluid spheres (bubbles and drops) as well as spheres of porous materials. For inviscid, potential flow, F = 0; for a solid sphere without slip, F = 1; for a fluid droplet, lo F ¼ 33lli þ2 where li and lo are the viscosity of internal fluid and i þ3lo external fluid, respectively. In the case of a solid sphere with surface r, where r is a dimensionless parameter slip at the interface, F ¼ 1þ2 1þ3r that characterizes the extent of slip. Basset [19] was the first to l , quantify this velocity slip with a dimensionless number, r ¼ ab

where b is the coefficient of the sliding friction of the sphere. A more complex expression for F in the case of a permeable sphere is given in [20,20], where the Stokes resistance of the sphere is 2

2ðb þkbÞ found to F ¼ 4kbþ2b be, with k and b being the properties re2 þ3kb1 þ1

lated to the permeable sphere [20]. It must be noted that an alternative way to model the slip, using statistical mechanics is via the specular reflection coefficient, the mean free path of the molecules, the tangential stress and the heat flux. However, several of these parameters are not known a priori. For this reason, a common practice is to use a single parameter, such as r, F, and determine it either experimentally or in terms of the transport properties [6,7,15,21]. The energy transport equation for the sphere in dimensionless form is described by the following governing equation:



r2 T ¼ Pe ur

 @T uh @T þ ; @r r @h

ð3Þ

where the Peclet number is defined as:, Pe ¼ qcUa ; q, c, and k are k fluid density, specific heat, and thermal conductivity, respectively. Because the particles considered here are nanoparticles (a < 106 m) and the relative velocity are very small, we consider solutions at small but finite Peclet numbers (Pe < 1). Of course this solution also covers the case Pe  1, which implies Pe ? 0. We will assume that the sphere has a constant temperature equal to 1 in dimensionless form, and that the undisturbed fluid temperature is equal to 0. The isothermal boundary condition assumption is valid when small spheres of relatively high

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conductivity are present and implies that the Biot number is very small (Bi  1). Close to the surface of the sphere, the fluid temperature is influenced by the temperature of the sphere. Without any temperature jump/slip, the temperature of fluid at r = a is equal to the temperature of the sphere. The slip or jump of the temperature at the interface modifies this condition at the boundary of the sphere and the surrounding fluid. Under the temperature discontinuity condition or slip, the boundary condition may be expressed in dimensionless form terms of a dimensionless parameter k as follows:

T 1¼k

@T at r ¼ 1: @r

ð4Þ

The dimensionless parameter k is related to other parameters as follows ([6]),

k¼

2ð2  aÞ-l 4ð2  aÞ-Kn ¼ ; ð- þ 1Þa Pr a ð- þ 1Þa Pr

ð5Þ

where which a is the accommodation coefficient, Pr the Prandtl number, - the ratio of specific heats, l the mean free path of the fluid, and Pr the Prandtl number. The Knudsen number, Kn, is defined as the ratio between molecular free path length to the diameter of a sphere. It must be noted that, from the definition of k, Eq. (4) is the same as one of the expressions suggested by Barber and Emerson [18] which was attributed to Smoluchowski. Also, that the temperature slip is not independent of the velocity slip. The two slip coefficients are related through the mean free path of the fluid, the heat flux and the shear stress [18]. Expressions for r and k, which are in terms of the transport coefficients of the fluid are parameterizations that have proven to be convenient as well as accurate with continua [19–21]. Far away from the sphere the fluid temperature field is undisturbed. Hence the far field boundary condition is:

T ¼ 0 as r ! 1:

ð6Þ

It is well known [4] that the solution to the energy equation for a sphere may not be obtained by a regular perturbation method. Instead, a singular perturbation method must be used to solve this problem. Similar singular perturbation methods have been successfully applied to determine the hydrodynamic force on a sphere at small but finite Reynolds numbers under various conditions [11,23], as well as heat transfer from a solid sphere at low but finite Peclet number[24]. In this study, we follow a singular perturbation method, we construct solutions that apply to an inner region and an outer region, and match these two solutions inside the common region where the solutions overlap and the derived expressions must match.

Thus, we obtain the following conditions for the coefficients:

tk ¼ k

at r ¼ 1 for k P 0:

ð9Þ

Similarly, the asymptotic solution in the outer region may be written as:

T ¼ F 0 ðPeÞT 0 þ F 1 ðPeÞT 1 þ F 2 ðPeÞT 2 þ    :

ð10Þ

The undisturbed fluid temperature far away from the sphere is:

T k ¼ 0 at r ! 1 for k P 0:

ð11Þ

The functions fi and Fi(i = 0,1,2, . . . ) are the so called gauge functions and will be determined later by matching of the inner solution and the outer region in their common region of overlap. For the asymptotic analysis to converge, the gauge functions must satisfy the following conditions:

fkþ1 ðPeÞ ! 0 and fk ðPeÞ

F kþ1 ðPeÞ ! 0 ðk P 0Þ as Pe ! 0: F k ðPeÞ

ð12Þ

3.2. Inner solution and coefficients Having obtained the form of the first order expansion, we now proceed to obtain the zeroth-order solution for the inner region temperature, t0. At first we note that, because the Peclet number is very small and the velocity field Stokesian the advection effects in the inner region are negligible in comparison to the diffusion effects for both momentum and heat transfer [4,6,11]. Hence, the governing equation in the inner region is simply the conduction equation:

r2 t0 ¼ 0:

ð13Þ

A general solution to this Laplace equation is [22]:

t0 ¼

1   X A0k rk1 þ B0k r k Pk ðlÞ:

ð14Þ

k¼0

Where Pk(l) is the Legendre Polynomial of kth-degree, and l = cosh. In addition, the boundary condition and at the interface (r = 1) requires: 1  1    X X 0 A0k þ B0k Pk ðlÞ  1 ¼ k ðk  1ÞA0k þ kBk Pk ðlÞ: k¼0

ð15Þ

k¼0

This yields the following equations:

A00 ¼

B0k þ 1 ð1 þ kÞ

ð16Þ

and

A0k ¼

3. Construction of the asymptotic solutions

@t k @r

B0k ð1  kkÞ ; 1 þ ðk þ 1Þk

k > 0:

ð17Þ

It follows that the inner solution, which satisfies the boundary conditions at the interface of the sphere, may be written as:

3.1. First-order expansion terms We start the singular perturbation method by assuming that the inner solution of the temperature field may be given by the following series:

t0 ¼

!   1 X B00 þ 1 1 1 þ kk k1 þ B00 þ r B0k þ r k P k ðlÞ: 1þk r 1 þ ðk þ 1Þk k¼1 ð18Þ

t ¼ f0 ðPeÞt 0 þ f1 ðPeÞt 1 þ f2 ðPeÞt 2 þ    :

ð7Þ

At the limit Pe = 0, the solution should reduce to the pure conduction solution. Therefore we must have f0(Pe) = 1. It is also noted that the boundary condition at the surface of the sphere, r = 1, yield the following equation:

t 0 þ f1 ðPeÞt 1 þ f2 ðPeÞt 2 þ     1 ¼ k

@ðt0 þ f1 ðPeÞt1 þ f2 ðPeÞt2 þ   Þ : @r ð8Þ

B0k ðk

The determination of the coefficients ¼ 0; 1; . . .Þ is accomplished by the matching conditions of the inner and the outer solutions in the area of overlap. 3.3. Outer solution and expansion coefficients In this region the characteristic radial distance is obtained by q the scaling: r ¼ Pe . For the outer solution, the governing equation may be written in dimensionless form as follows:

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r2 T 0 ¼ l

@T 0 1  l2 @T 0 þ : @q q @l

ð19Þ

A general solution to this equation may be obtained in the following form:

rffiffiffiffiX q p1 0 1 T 0 ¼ e2ql C K P ðlÞ; q k¼0 k nþ1=2 2 k

ð20Þ

q 2

rffiffiffiffi

¼

n p 12q X ðn þ mÞ! : e q m¼0 ðn  mÞ!m!qm

ð21Þ

!   1 X B00 þ 1 Pe 1 þ kk Pekþ1 qk1 þ Pek qk Pk ðlÞ þ B00 þ B0k 1þk 1 þ ðk þ 1Þk q k¼1 ( ) 1 q p 0 12qðl1Þ 12ql X 0 þ  $ F 0 ðPeÞ C e þe C k K nþ1=2 Pk ðlÞ þ  ð22Þ q 0 2 k¼1 One may expand the exponential functions as:

ð23Þ

and use the last two equations to obtain the following solutions for the gage functions:

F 0 ðPeÞ ¼ Pe; B00 ¼ 0 and C 00 ¼

ð24Þ 1 ð1 þ kÞp

ð25Þ

and

B0k ¼ 0 and C 0k ¼ 0 ðk P 1Þ:

ð26Þ

Thus, the complete solutions for the zeroth-order inner and outer regions are:

 t0 ¼







1 1 1 1 qðl1Þ e2 : : and T 0 ¼ 1þk r 1þk q

  1 3F 3k þ 1 3F þ  1 ; ð1 þ 2kÞA11 þ ð1  kÞB11 ¼  2ð1 þ kÞ 4 4ð1 þ kÞ 2 ½1 þ ðk þ 1ÞkA1k þ ð1  kkÞB1k ¼ 0; k P 2:

A matching condition is required to determine the coefficients. Since Pe < 1, the matching of the two solutions at the overlap region (r ? 1 and q ? 0) yields the conditions:

1 1 e2qðl1Þ ¼ 1 þ qðl  1Þ þ    2

k¼2

The application of the boundary conditions at the interface r = 1 yields the following conditions for the coefficients:

ð1 þ kÞA10 þ B0 A10 ¼ 0;

where K nþ1=2 ðq2Þ is the modified Bessel function defined as:

K nþ1=2

( )     A10 1 1 3F 1 3F 1 A11 1 1 þ 3 þ B1 r P1 ðlÞ  þ 1 t1 ¼ þ B0 þ 1 þ k 2 4r 4 2 r3 r r 1   X A1k r k1 þ B1k r k Pk ðlÞ: ð31Þ þ

ð27Þ

ð32Þ ð33Þ ð34Þ

The matching of the outer solution, under the condition Pe ? 0, yields the following expression:

ðt 0 þ Pe t1 Þ as r ! 1 ¼ Pe T 0

as q ! 0:

ð35Þ

q

By letting r ¼ Pe, the last equation yields:

     1 Pe A1 1 1 3F 1 3F 1  Pe þ Pe3 1 þ Pe2 0 þ PeB10 þ Pe 1þk q 1þk 2 4q 4 2 q3 q ) 1 1   X A q P 1 ðlÞ þ Pe Pekþ1 A1k qk1 þ Pek B1k qk Pk ðlÞ þPe2 12 þ B11 q Pe k¼2   h i 1 1 q q þ  $ Pe 1  þ P 1 ðlÞ þ  ð36Þ 1þk q 2 2



From which the following coefficients are determined:

B10 ¼ 

1 ; 2ð1 þ kÞ

B1k

¼ 0 ðk P 1Þ; 1 ; A10 ¼ 2ð1 þ kÞ2 3Fð3 þ 5kÞ  6ð1 þ kÞ A11 ¼ 8ð1 þ kÞð1 þ 2kÞ

ð37Þ ð38Þ ð39Þ ð40Þ

and

A1k ¼ 0 ðk P 2Þ:

ð41Þ

The above result in the following general expressions for the inner temperature field:

    1 1 1 1 3F 1 3F 1  þ  þ 1 1 þ k 2ð1 þ kÞr 2 2 4r 4 2 R3



3Fð3 þ 5kÞ  6ð1 þ kÞ 1 P 1 ð lÞ : þ 8ð1 þ 2kÞ r2

t1 ¼ 3.4. Second-order expansion terms The governing equation for the second order expansion terms may be derived to be as follows:

r2 ðt0 þ f1 ðPeÞt1 þ   Þ

  @ðt 0 þ f1 ðPeÞt 1 þ   Þ uh @ðt0 þ f1 ðPeÞt1 þ   Þ : þ ¼ Pe ur @r @h r

ð28Þ

By choosing the first gage function as: f1(Pe) = Pe, we may obtain the governing equation for t1 as:

r2 t 1 ¼ 

    1 1 3F 3F 1 P1 ðlÞ:   1  1 þ k r 2 2r2 2 r5

ðt1 Þp ¼

    1 1 3F 1 3F 1 P1 ðlÞ:  þ 1 1 þ k 2 4r 4 2 r3

The second-order expansion terms in the outer region, T1, can be obtained following a similar procedure outlined in [4]. The governing equation in this region is:

r2 T 1 ¼ l

    @T 1 1  l2 @T 1 3 l @T o 3 1  l2 @T 0 þ   2 @q q @l 2 q @q 4 q @l

ð30Þ

From the latter we may obtain the general solution for t1 in the following form:

ð43Þ

and the general solution of this equation is: rffiffiffiffiX q p1 1 F P k ð lÞ þ ½R0 ðqÞ þ R1 ðqÞP 1 ðlÞ þ R2 ðqÞP 2 ðlÞ: T1 ¼ C k K kþ1=2 1þk q k¼0 2

ð29Þ

A particular solution to this equation may be obtained as follows:

ð42Þ

ð44Þ The functions of, R0(q), R1(q), and R2(q) have been derived in [4]. As q ? 0, these terms reduce to the following:

  ln c ln q 1 ln c   þ ; 2 2q 2 4 3 3 3 ðln c  1Þ ðln c  1Þ   R1 ðqÞ ! 2q2 4q 16

R0 ðqÞ ! 

ð45Þ ð46Þ

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and

The corresponding particular solution will be of the following form:

3

3 1 : R2 ðqÞ ! 3 ð3  ln cÞ  ð3  ln cÞ þ 8q 24 q

ð47Þ

    1 F F 1 3F 1þk 1  ln r þ 1 þ A0 F 2 1þk 6 2 24 2 4 r     3F F 3F ð1 þ kÞA0 þ 1   1 2 24r 3 2 12r 4 )  2 3F 1 þ 1 ; 2 60r5



1 Pe 1 1 1 þ Pe2  Pe 1þk q 2ð1 þ kÞ 2ð1 þ kÞ2 q     1 1 3F 1 3F 1 þ Pe  Pe 1 þ Pe3 1þk 2 4q 4 2 q3

3Fð3 þ 5kÞ  6ð1 þ kÞ1 P1 ðlÞ þPe2 8ð1 þ kÞð1 þ 2kÞq2   h i 1 1 q q 1  þ P1 ðlÞ þ    $ Pe 1þk q 2 2   p 12 1 p12q 1 2 P ðlÞ e C0 þ C1 1 þ þ F 1 ðPeÞ q q q 1   1 q X p 1 6 12 C 1k K kþ1=2 P k ð lÞ þ e2q C 12 2 þ þ 2 P2 ðlÞ þ q q q 2 k¼3

F þ ½R0 ðqÞ þ R1 ðqÞP1 ðlÞ þ R2 ðqÞP2 ðlÞ þ    1þk

L0 ðrÞ ¼

L1 ðrÞ ¼ A0

1 þ Fð1 þ kÞ ln c

2pð1 þ kÞ2 3Fð1  ln cÞ C 11 ¼ 4pð1 þ kÞ

ð50Þ ð51Þ

and

C 12 ¼ 

Fð3  ln cÞ : 4pð1 þ kÞ

ð52Þ

3.5. Higher-order expansion terms The equation for the third expansion term of the inner solution t2 is calculated to be:         3F 3F 1 @t1 3F 1 3F 1 1 @t1 þ 1 þ r2 t 2 ¼ 1   1  :  P1 ðlÞ ð1  l2 Þ 3 @r 2r 2 r 4r 2 4 r3 r @l

ð53Þ

Upon the substituting the solution for t1 and also noting that l2 ¼ 2P2 ð3lÞþ1, the re-arrangement of the equations leads to the following governing equation for the higher order expansion term t2:,

r2 t 2 ¼ z0 ðrÞ þ z1 ðrÞP1 ðlÞ þ z2 ðrÞP 2 ðlÞ;

ð59Þ

     1 r 5F 9 1 5 3F  þ 2ð1 þ kÞA1 þ F 2 þ 1 1 þ k 12 24 8 6r 24 2      5 1 F 3F ln r 3F ð1 þ kÞA1 1 þ 1  ð1 þ kÞA1 F 2  8 r 8 2 r3 2 6r4  2 ) 5 3F 1 : ð60Þ 1 þ 168 2 r5

ð48Þ

ð49Þ

;

 

1 3F 3F 1   1 2 4r 2 4r 3

L2 ðrÞ ¼

and the coefficients are found to be:

C 10 ¼

ð58Þ

and

and the latter yields the gage function F1:

F 1 ðPeÞ ¼ Pe2

ð57Þ

where

The Euler constant is: lnc = 0.577215 . . . . The matching condition at the overlap region requires:



ðt 2 Þp ¼ L0 ðrÞ þ L1 ðrÞP1 ðlÞ þ L2 ðrÞP2 ðlÞ;

ð54Þ

where,

    1 1 F 1 3F 1þk 1 þ  2þ 1 A0 F 4 1 þ k 3r 2r 12 2 2 r )      2 3F F 3F ð1 þ kÞA0 3F 1 ;  1 þ 1  þ 1  2 4r5 2 2 3r 7 r6  

1 3F 3F 1 ð55Þ z1 ðrÞ ¼ A0 2  3  1  r 2r 2 r5

z0 ðrÞ ¼

Based on this particular solutions we may now compose the general solution for t2 as follows:

" # A20 A21 2 2 t2 ¼ L0 ðrÞ þ þ B0 þ L1 ðrÞ þ 2 þ B1 r P 1 ðlÞ r r " # 1   X A2 A2k r k1 þ B2k r k Pk ðlÞ: þ L2 ðrÞ þ 32 þ B22 r P 2 ðlÞ þ r k¼3

The coefficients in the general solution may be obtained from the application of the boundary condition at r = 1:

h i k L00 ð1Þ  A20 ¼ L0 ð1Þ þ A20 þ B20 ; h i k L01 ð1Þ  2A21 þ B21 ¼ L1 ð1Þ þ A21 þ B21 ; h i k L02 ð1Þ  3A22 þ B22 ¼ L2 ð1Þ þ A22 þ B22

ð62Þ ð63Þ ð64Þ

and

½1 þ ðk þ 1ÞkA2k þ ð1  kkÞB2k ¼ 0 for k > 2:

ð65Þ A2k

B2k

Additional equations for determining constants and may be obtained from the matching requirement. However, as argued in [4], the net contribution of O(Pe2lnPe) arising from L0 (r) cannot be matched in the outer solution. This implies that the inner solution must contain a term in its expansion of the order of Pe2lnPe. By substituting the overall form of the asymptotic solution,

t ¼ t 0 þ Pe t 1 þ Pe ln Pe t 2 þ Pe2 t 2 þ    ;

ð66Þ

into the heat transport equation, one may easily determine that the governing equation for this expansion is simply,

r2 t2 ¼ 0:

ð67Þ

Its solution of the last equation may be readily obtained to be:

and

     1 1 5F 9 1 5 3F  þ 2  2ð1 þ kÞA1 þ F 2 3 þ  1 z2 ðrÞ ¼ 1þk 3r 4r 8 r 6 2      5 1 5 3F 1 3F ð1 þ kÞA1 þ 1 þ ð1 þ kÞA1 F 4 þ F 1  2 r 8 2 r5 2 r6 )  2 5 3F 1 : ð56Þ 1 ;þ 12 2 r7

ð61Þ

t2 ¼

  F 1 1 : 2ð1 þ kÞ ð1 þ kÞr

ð68Þ

Accordingly, the new matching requirement becomes:

t0 þ Pe t 1 þ Pe ln Pe t2 þ Pe2 t2 $ Pe T 0 þ Pe2 T 1 :

ð69Þ

The matching requirement will result in additional equations to determine the constants A2k and B2k . However, the only constant coefficients that contribute to the total heat transfer rate, or

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equivalently the Nusselt number, are the terms associated with P0(l) in the expansions for the inner solution. Since we are only interested in the rate of heat transfer and not on the intrinsic details of the temperature field, we only list the equations derived from the matching requirements for A20 and B20 , which are:

B20

¼

1 þ Fð1 þ kÞlnc 4ð1 þ kÞ2

  1 ln c F F   þ 2 4 1 þ k 8ð1 þ kÞ

kL00 ð1Þ  L0 ð1Þ  B20 : 1þk

ð70Þ

ð71Þ

On the other side, the presence of the t2 expansion term results in the following contribution to the inner solution during the matching:

F

2

Pe ln Pe

3

2ð1 þ kÞ2

þ Pe ln Pe

F

1

2ð1 þ kÞ3 q

:

ð72Þ

The first term is already matched to a term in the expansion of t2. However, the second term stemming from the t 2 expansion cannot be matched in the inner solution. This requires an outer expansion in the form of Pe3lnPe, which must follow the expansion term of Pe2. Thus the expression for T in the outer field will become:

T ¼ Pe T 0 þ Pe2 T 1 þ Pe3 lnPe T 2 þ    :

ð73Þ

It must be noted that the form of the governing equation for T 2 is the same as the form for T0. Following the same method as before, we obtain:

F

T 2 ¼

1

4ð1 þ kÞ q 2

q

e2ðl1Þ :

ð74Þ

When this term is used in the expansion, it produces a term O(Pe3lnPe), which is matched in the inner expansion, t 3 , with

r2 t 3 ¼ 0:

ð75Þ

The solution of this term, t3, is obtained to be as follows:

t 3 ¼



F 4ð1 þ kÞ

2

1 1 ð1 þ kÞr

2 1 F þ Pe þ Pe2 lnPe 1 þ k 2ð1 þ kÞ2 4ð1 þ kÞ2 ( 1 156 þ 148F  152k þ 341kF þ 129F 2 þ 528kF 2 þ 960ð1 þ 2kÞ 2ð1 þ kÞ2

1 F þ ðln c  ln2 þ 2kln cÞF Pe2 þ Pe3 lnPe þ OðPe3 Þ; ð79Þ 2 16ð1 þ kÞ3

Nu ¼

and

A20 ¼

particulate flows and, especially, in the literature for nanofluids to define the Peclet number in terms of the diameter of the sphere, d = 2a. Under this new and more practical definition for the Peclet number, the Nusselt number is given by the following expression:

 ð76Þ

and this completes the singular asymptotic expansion process and the matching of the corresponding terms.

with Pe ¼ 2qkcUa. 4.1. Special cases Eq. (79) is the most general one for the heat or mass transfer of a sphere in a Stokesian flow regime. The equation may be applied to a solid sphere, a liquid droplet, or a permeable sphere, using the pertinent value of the drag multiplier, F. The velocity boundary condition applies to both the no-slip and slip cases. The temperature boundary condition also may be continuous or discontinuous with slip at the interface. In the next sections we discuss some special cases for the applicability of this expression and compare it to known solutions that have been derived in the past. 4.1.1. A solid sphere with slip velocity and slip temperature The Nusselt number for this case can be obtained by taking r in Eq. (79). It is also a case that has been studied in [6], F ¼ 1þ2 1þ3r whose solution has been disputed by Brunn [7]. 4.1.2. A sphere with no-slip temperature boundary condition By letting k = 0 in Eq. (79), we obtain

   1 1 1 13 1 37 Nu ¼ 2 þ Pe þ FPe2 ln Pe þ  þ ln c þ  ln 2 F 2 4 2 80 2 120  43 2 F 2 3 3 F Pe þ Pe ln Pe þ OðPe Þ: ð80Þ þ 320 16 This solution is identical to the one derived in [7] for no-slip temperature boundary condition. It must be noted that the author in [7] defines the Peclet number in terms of the radius of the sphere and in Eq. (79) Pe ¼ 2qkcUa.

4. Heat transfer rate from a sphere with slip For the calculation of the heat transfer, the average Nusselt number, Nu, around the sphere Nu can be calculated by the integral:

  @t Nu ¼  dl: 1 @r r¼1 Z

1

ð77Þ

From the overall solution for the temperature field around the sphere in the inner expansions, we obtain the following expression for Nu:

2 1 F þ Pe þ Pe2 ln Pe 1 þ k ð1 þ kÞ2 ð1 þ kÞ2 ( 1 0 L0 ð1Þ 1 þ Fð1 þ kÞlnc L ð1Þ  þ þ2  1þk 0 1þk 4ð1 þ kÞ2 )   1 ln c F F F Pe2 þ þ Pe3 ln Pe: ð78Þ   2 4 ð1 þ kÞ2 8ð1 þ kÞ2 2ð1 þ kÞ3

4.1.3. A solid sphere with no-slip velocity but finite temperature slip By letting F = 1, k – 0, we obtain:

Nu ¼

2 1 1 þ Pe þ Pe2 ln Pe 1 þ k 2ð1 þ kÞ2 4ð1 þ kÞ2

1 121 þ 869k 1 þ ðln c  ln 2 þ 2k ln cÞ Pe2 þ 2 960ð1 þ 2kÞ 2 2ð1 þ kÞ þ

1 16ð1 þ kÞ3

Pe3 ln Pe þ OðPe3 Þ: ð81Þ

Nu ¼

An explicit form of the functions L00 ð1Þ; L0 ð1Þ, and A11 can be determined in the previous sections. In all the above equations, the Peclet number, Pe, was defined in terms of the radius of the sphere, a. However, it is more common in applications of

4.1.4. A solid sphere with no-slip velocity and no-slip temperature conditions In this case, we set F = 1, and k = 0 in Eq. (79). Then the Nusselt number reduces to:



1 1 1 121 þ ðln c  ln 2Þ Nu ¼ 2 þ Pe þ Pe2 ln Pe þ 2 4 4 480 1 3 2 3 Pe ln þOðPe Þ: Pe þ 16 This is identical to the result obtained in [4].

ð82Þ

Z.-G. Feng, E.E. Michaelides / International Journal of Heat and Mass Transfer 55 (2012) 6491–6498

4.1.5. A sphere with infinite-slip velocity and slip temperature This case may be applicable to inviscid bubble flow and heat transfer. In this case, we set F = 0 and the following solution is obtained:

Nu ¼



2 1 1 39  38k þ Pe2 þ OðPe3 Þ Pe  1 þ k 2ð1 þ kÞ2 ð1 þ kÞ2 480ð1 þ 2kÞ ð83Þ

4.1.6. A sphere with infinite velocity slip and no-slip temperature condition This is the simplest case, by letting F = 0, k = 0, we arrive the following result for the Nusselt number:

1 13 2 Pe þ OðPe3 Þ: Nu ¼ 2 þ Pe  2 160

ð84Þ

6497

5. Comparison with experimental data While there are not any data in the literature at finite Peclet numbers with surface slip, there are some sets of data for the case of conduction in rarefied gases, that is at Pe  1 [17,25]. The theoretical solution of Eq. (79), predicts for the case of simple conduction from a rarefied gas to a sphere:

Nu ¼

2 ; 1þk

ð85Þ

aÞ-Kn where k ¼ 4ð2 . Therefore, the last expression may be written in ð-þ1ÞaPr terms of the Knudsen number as follows:

Nu ¼

2 aÞ- 1 1 þ 4ð2 a Kn ð-þ1ÞPr

ð86Þ

A comparison was made of the last expression with the solution that was derived by Mikami et al. [17]. Their solution is as follows in terms of the Knudsen number:

2 : 1 þ 7:5a1 Kn

4.2. Numerical results with intermediate values of temperature slip

Nu ¼

In order to find out the effect of the temperature-slip to the rate of heat transfer, we chose three different temperature-slip conditions, corresponding to, k = 0, 0.2, and 0.5. For each temperatureslip coefficient, we also selected three different Stokes resistance parameters, F = 0, 0.5, and 1.0. The results are plotted in Fig. 1. We see in this figure that the Stokes resistance coefficient has a very small effect on the Nusselt number at small Peclet number (Pe < 0.7). Even at Pe = 1.0, the difference of Nu between F = 0 and F = 1 is less than 10% for the slip coefficients considered here (k 6 0.5). However, the temperature slip parameter, k, has a significant effect on the Nusselt number and the rate of heat transfer as it may be seen in Fig. 1, where Nu is reduced from 2 to 1.3 when k is increased from 0 to 0.5 (for Pe = 0). It must also be noted that at the limit Pe = 0 (conduction), Nu = 0 when k = 1. We also considered the effect of temperature-slip coefficient at different Peclet numbers. Fig. 2 shows that the Nusselt number is more sensitive to the value of k when the latter is in the range 0.1 to 10. However, as k > 10, the Nusselt numbers become very small and both the Peclet number and the drag multiplier, F, have a small effect on Nu. The results show that for k > 100, the sphere may be considered as adiabatic, since Nu is very close to 0. Fig. 2 also shows that the effect of Stokes resistance factor to the Nusselt number increases as the Peclet number increases, a result observed in Fig. 1 as well.

It is observed that the correlation derived by Takao [25] is of the similar form as our Eq. (86). We performed a comparison for the case of hydrogen, which has Pr = 0.7, and specific heat ratio = 1.4. The accommodation coefficient for this gas varies from a = 0.2 to 0.4, according to Mikami et al. [17]. We choose a = 0.25, which yields k = 23.3 Kn, for the accommodation coefficient. The results of the comparison are depicted in Fig. 3, which demonstrates that there is a very good agreement between the experimental data and the analytical expression derived in this study. Takao [25] also studied the heat transfer from a brass sphere in air. In order to compare our theoretical expression to his results, we choose Pr = 0.7, specific heat ratio, - = 1.4, and accommodation coefficient a = 0.8. These values yield k = 5 Kn. It is observed in Fig. 3 that the analytical data, emanating from Eq. (86), fit perfectly well with this set of experimental results. It is also observed that the agreement with the data extends to values of Kn close to 1. Small deviation is seen for Kn > 1. It must be pointed out that the x axis in the original graph of [25] is the ratio of the radius of a sphere to the molecular free path length, which is the inverse of half of Kn defined in this paper. The excellent agreement of the analytical expression derived in this study at the limit of zero Peclet number with the experimental data reported in the literature, is an indication of the validity of the analysis and the slip temperature condition used for the derivation

Fig. 1. Nusselt number vs. Peclet number at different temperature slip conditions and dimensionless Stokes resistance.

Fig. 2. Nusselt number vs. temperature slip coefficient at different dimensionless Stokes resistance coefficient and Peclet number.

ð87Þ

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Acknowledgements This work was partly supported by a grant from the DOE, through the National Energy Technology Laboratory, (DENT0008064) Mr. Steven Seachman project manager; by a Grant from NSF (HRD-0932339); and by a supplemental Grant from the NSF (HRD-1137764) to the SiViRT Center, Dr. Richard Smith, project manager. References

Fig. 3. Comparison of the results of Eq. (79) at Pe = 0 with the experimental data and correlations of Mikami et al. [17] and Takao [25].

of the general expression (79) for the heat transfer in the presence of slip. Fig. 3 also shows that the analytical solution is valid for Kn < 1. This observation would impose a limit on the dimensionless temperature slip factor k, as follows:

k<

4ð2  aÞ: ð- þ 1ÞaPr

ð88Þ

The actual numerical value of k depends on the accommodation coefficient a, the specific heat ratio -, and the Prandtl number Pr. In the experiment by Mikami et al. [17] when H2 was used, the slip factor k = 23.3 at Kn = 1; on the other hand, the experiment by Takao [25] was performed with air under the conditions k = 5 and Kn = 1. 6. Conclusions Using a singular perturbation method a general equation for the rate of heat transfer from a sphere has been derived, which included both velocity slip and temperature slip at the interface under a Stokesian flow regime. The asymptotic analysis applies up to the order Pe3lnPe. By choosing an expression for the drag multiplier, this expression may be applied to solid, fluid and porous spheres, which are special cases of the general solutions. Comparisons with some known results for these special cases have shown the accuracy and wide range of applicability of the derived general expression. The inclusion of the temperature slip at the interface makes this equation applicable to particles, bubbles and drops of nanometer sizes. It has been observed that the velocity slip at the interface does not affect significantly the overall Nusselt number. However, the temperature slip is a more important parameter in the heat transfer and affects significantly the rate of heat transfer and Nu. This leads to the conclusion that, if a temperature slip is possible at the interface, it must be taken into account by using the appropriate closure equation that emanates from Eq. (79). The slip temperature boundary condition used for the present analysis is further validated by comparing conduction heat transfer results from a sphere in rarefied gases.

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