Investigation of flow and heat transfer characteristics of rarefied gaseous slip flow in nonplanar micro-Couette configuration

International Journal of Thermal Sciences 54 (2012) 262e275

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Investigation of flow and heat transfer characteristics of rarefied gaseous slip flow in nonplanar micro-Couette configuration Mehdi Shamshiri, Mahmud Ashrafizaadeh*, Ebrahim Shirani Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 April 2011 Received in revised form 28 November 2011 Accepted 29 November 2011 Available online 29 December 2011

In the present study the mechanism of transport phenomena in a low-speed rarefied gas flow confined between a shaft and its concentric housing, namely, the cylindrical Couette problem, is investigated analytically in the slip-flow regime. The incompressible NaviereStokeseFourier (NSF) equations in the cylindrical polar coordinate reference frame are employed while the simultaneous effects of viscous dissipation and rarefaction phenomenon are taken into account. Two different thermal boundary conditions are considered: the Uniform Heat Flux (UHF) and the Constant Wall Temperature (CWT). Solutions for the velocity and temperature distributions, the Nusselt number and the wall heat flux are obtained for different values of the Knudsen and Brinkman numbers and curvature parameter. Ó 2011 Elsevier Masson SAS. All rights reserved.

Keywords: Microscale Rarefied Couette flow Analytical solution Velocity distribution Temperature distribution

1. Introduction Gas flows in microscale devices have received particular attention over the past two decades with the rapid development in the branch of the so-called micro-electro-mechanical-systems (MEMS). This is largely driven by the need for new technical tools to accurately predict microscale transport processes and to design devices with enhanced performance. Hence, a special understanding of the physics associated with flow and heat transfer in miniaturized devices ewhich is significantly different from that of their macroscale counterpartse seems to be absolutely necessary. Many experimental studies confirm the already mentioned deviation from the continuum behavior at microscales (e.g. [1e6]). Araki et al. [4], for instance, performed an empirical investigation to study frictional characteristics of nitrogen and helium flows in microchannels. They reported that the frictional resistance of gas flow in microchannels is smaller than that in conventional channels. More recently, some experimental studies were conducted by Demsis et al. [5,6] to investigate heat transfer features of rarefied gaseous flows in a circular tube under constant wall temperature boundary condition. Their promising studies fall into the first

* Corresponding author. Tel.: þ98 9131104623; fax: þ98 311 3913919. E-mail addresses: [email protected] (M. Shamshiri), [email protected] (M. Ashrafizaadeh). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.11.023

empirical heat transfer measurements in low rarefied conditions and reveal that the measured values of the Nusselt number are twofive orders of magnitude smaller than the corresponding values in the continuum regime. As a microscale point of view, different flow regimes may be encountered as the mean free path of the molecules, l e the flight distance of the molecules before colliding into each other e becomes comparable with the characteristic length of the system, Lc. In other words, the main attributes of the microscale flows can be characterized through a nondimensional parameter called Knudsen number which is defined as the ratio of the mean free path of the molecules to the characteristic length of the system isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi related to the mac(Kn ¼ l/Lc). It should be noted that l, itself,p roscopic quantities of the flow via l ¼ (m/P) pkB T=2m where m is the viscosity, P is the pressure, m is the molecular mass, kB is the Boltzmann’s constant, and T is the absolute temperature. The case of Kn < 0.01 is referred to as the continuum regime, in which the conventional hydrodynamic equations, i.e., the NaviereStokes equations with no velocity slip-boundary conditions and the Fourier heat conduction equation with no temperature jump boundary condition, are the appropriate governing equations. The values of 0.01  Kn  0.1 are related to the slip-flow regime in which the already mentioned boundary conditions seem to fail and velocity slip and temperature jump will appear on the solid boundaries and the NaviereStokeseFourier (NSF) equations should be solved subject to the slip/jump boundary conditions. As the

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

Nomenclature A Br cp De DE h hu,hQ k kB Kn Lc LD m Nu P Pr qw r/R r* T U uq CWT NSF

cross-section area [m2] Brinkman number [e] specific heat at constant pressure [kJ/kg K] equivalent diameter [m] dimensionless equivalent diameter [e] heat transfer coefficient [W/m2 K] Eq. (14) thermal conductivity [W/m K] Boltzmann’s constant [kJ/kg] Knudsen number [e] characteristic length [m] dimensionless characteristic length [e] molecular mass [kg] Nusselt number [e] pressure [Pa] Prandtl number [e] wall heat flux [W/m2] dimensional/dimensionless radial coordinate [m/e] aspect ratio of the configuration [e] temperature [K] dimensionless velocity [e] tangential velocity [m/s] constant wall temperature NaviereStokeseFourier

Knudsen number approaches higher values, after a transition period, the free molecular flow regime is experienced [7,8]. Numerous research works have been carried out to study flow and heat transfer characteristics of rarefied gas flows in micro geometries. The readers may refer to the available excellent review researches in the literature (e.g. Sharipov and Seleznev [9], Ho and Tai [10], Palm [11], Sobhan and Garimella [12], Gad-el-Hak [13], Obot [14], Rostami et al. [15], Guo and Li [16,17], Morini [18]). However, the current study is analytical approach oriented and tends to find simplified models for micro-thermo fluidic structures based on the NSF equations, which can find extended utilization in engineering applications and other practical areas. Hence, in the following, we review some of the previous analytical investigations describing transport phenomena in some typical gaseous micro geometries such as straight micro channels/Couettes, micro tubes and micro annuli. Analytical hydrodynamic aspects of internal slip flow are presented in Refs. [19e23]. Ameel et al. [24] analytically treated the problem of laminar gas flow in microtubes. Imposing uniform heat flux and considering velocity slip and temperature jump boundary conditions at the wall, they showed that the fully developed Nusselt number decreased with an increase in rarefaction degree. Zhang and Bogy [25] analytically solved the NSF equations with discontinuous boundary conditions to obtain flow and heat transfer characteristics of a thin slider/disk micro air bearing. They stated that the heat transfer between the slider and air bearing depends on both heat conduction and viscous dissipation effects. Moreover, the increase/decrease of the heat conduction/viscous dissipation effects with the decrease of the flying height (or disk rotation speed) is reported. Yu and Ameel [26] analytically investigated the effects of rarefaction and fluidewall interaction on the heat transfer for laminar forced convection flow in rectangular microchannels. Assuming hydrodynamically fully developed flow, they applied a modified generalized integral transform technique to solve the energy equation. After all, they determined the critical point at which the heat transfer reduces after experiencing a

UHF

263

uniform heat flux

Greek symbols percentage of the diffusely scattered particles specific heat ratio [e] mean free path of the molecules [m] dynamic viscosity [kg/m s] momentum/energy accommodation coefficient [e] angular velocity [rad/s] viscous slip/temperature jump coefficient [e] Eq. (35) Eq. (8) overall temperature rise factor [e] dimensionless temperature [e]

ad g l m su/sT u xu/xT j z D Q

Subscripts and superscripts i,o inner and outer cylinders, respectively m mean max maximum s fluid properties at the wall w wall q tangential direction * modified dimensionless temperature (except for r*) ^ dimensionless heat flux walls with CWT/UHF boundary condition .000 /.

period of enhancement. Tunc and Bayazitoglu [27] analytically treated steady laminar hydrodynamically developed and thermally developing slip flow in microtubes with different boundary conditions (uniform temperature and uniform heat flux at the walls) using the integral technique. Effects such as viscous heating and temperature jump condition are also considered within their study. Tunc and Bayazitoglu [28] investigated both hydrodynamically and thermally fully developed slip flow in rectangular microducts under constant wall heat flux boundary condition, using integral transform method. Aydin and Avci [29,30] analyzed laminar forced convective heat transfer of a Newtonian fluid in a micropipe and microchannel considering hydrodynamically and thermally fully developed flow case with constant heat flux and constant wall temperature boundary conditions. Taking the viscous effect, the velocity slip and the temperature jump at the wall into account, the associated temperature distribution and Nusselt number for each type of boundary condition are analytically determined as functions of the Brinkman and Knudsen numbers. Hooman [31] presented the closed-form solutions for fully developed temperature distribution and entropy generation due to forced convection in MEMS in the slip-flow regime. Two different cross-sections, namely, microducts and micropipes are analyzed. Different types of thermal boundary conditions are imposed, i.e., isothermal and isoflux walls. Avci and Aydin [32] studied forced convection heat transfer in a microannulus considering a hydrodynamically and thermally fully developed flow with viscous dissipation. Within the study, issues such as viscous heating, rarefaction and aspect ratio of annular geometry and their effects on the Nusselt number are discussed (for the special case of one wall being adiabatic and the other having constant heat flux). Applying a Fourier series approach, closed-form solutions for fully developed velocity and temperature distributions in a poroussaturated microduct of rectangular cross-section in the slip-flow regime are presented by Hooman [33,34]. Expressions for the friction factor, slip coefficient and Nusselt number in terms of some key parameters are also extracted. Hooman [35] presented a

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superposition approach to investigate forced convection in microducts of arbitrary cross-section, subject to different boundary conditions, in the slip-flow. It is shown that applying an average slip velocity and temperature jump definition, one can still use the no-slip/no-jump results with some minor modifications. Hooman et al. [36] reported closed-form solutions, based on perturbation techniques, for fully developed, both hydrodynamically and thermally, slip-flow forced convection in both parallel plate and circular microchannels subject to isothermal wall boundary condition. Scaling effects, including variable property, viscous dissipation, velocity slip, and temperature jump are studied in detail. Sadeghi and Saidi [37] investigated hydrodynamically and thermally fully developed laminar forced convection heat transfer of a rarefied gas flow in two microgeometries, namely, microannulus and parallel plate microchannel. They concluded that for both geometries, as Brinkman number increases, the Nusselt number decreases. However, the effect of viscous heating on the Nusselt number at greater values of Knudsen number becomes insignificant. Forced convection heat transfer in a porous annular microduct for hydrodynamically and thermally fully developed flow of a dilute gas is studied analytically by Hosseini Hashemi et al. [38]. Brinkman extended Darcy equation is used and the porous medium shape parameter, velocity slip and temperature jump at the walls are taken into account. Two distinct thermal boundary conditions are analyzed: constant heat flux at the outer cylinder and insulated inner cylinder and vice versa. Explicit expressions for velocity distribution, temperature distribution and the Nusselt number have been obtained. Investigating cylindrical Couette flow of a rarefied gas has been of interest over the years due to its wide application in the design and fabrication process of rotation-base micro technologies such as micro-motors, -pumps, -turbines and micro bearings. Previous analytical and numerical studies [39e41] have clarified that under certain conditions of rarefaction, an anomalous behavior known as ‘inverted velocity profile’ would exist while the inner cylinder is rotating, which is the velocity profile between the cylinders reverses direction so that the gas moves faster near the stationary outer cylinder. The phenomenon was first predicted by Einzel, Panzer and Liu (EPL) [39] who developed a generalized slipboundary condition for flows over curved surfaces but not necessarily for a rarefied gas. Recasting EPL’s formulation, Tibbs et al. [40] presented a new version which could be applied to rarefied gases. They also stated that the velocity-inversion process only occurs for small values of tangential momentum accommodation coefficient (TMAC). Aoki et al. [41] conducted an asymptotic analytical solution at low Knudsen numbers and a direct numerical solution of the Boltzmann equation at higher Knudsen numbers and concluded that the occurrence of a velocity-inversion depended on the ratio of TMAC to Kn. Lockerby et al. [42] and Barber et al. [43] utilized a generalized version of Maxwell’s slip equation [44] to investigate low Knudsen number isothermal flow over walls with substantial curvature. The generalized slip equation is written in terms of the tangential shear stress to overcome the limitations of the conventional slip-boundary treatment. Yuhong et al. [45] scrutinized the velocity-inversion phenomenon in rarefied cylindrical Couette gas flow using a new boundary treatment derived from the Maxwell’s slip model. They also investigated the influence of different values of TMAC at the inner and outer walls and reported that all velocity profiles pass through a common point. Myong et al. [46] theoretically investigated the velocity slip on the solid surfaces of microscale cylindrical Couette flow using a simple slip model based on the gasesurface molecular interaction known as the Langmuir adsorption model and concluded that the velocity-inversion phenomenon in such a rotating microstructure can be described by the Langmuir slip model based on the concept

of gaseous adsorption onto solids. More recently, Agrawal and Prabhu [47] performed an analysis to extend the applicability of the NaviereStokes equations to high Knudsen number regimes. They used the available experimental data in the literature on two concentric rotating cylinders with a low pressure gas in the gap to deduce the slip coefficient in the slip and transition flow regimes. The authors concluded that the NaviereStokes equations can be used for analysis of the transition flow regime by appropriately tuning the values of the slip coefficient. Also, several studies have been concerned with investigating heat transfer characteristics of the nonplanar rarefied Couette flow problem. Sharipov and Kremer [48], for instance, numerically investigated the momentum and heat transfer through a rarefied gas confined between two concentric cylinders, rotating with different angular velocities. The solution, which is based on the kinetic equation, considers a wide range of the Knudsen number. However, in this study the special case of cylinders with constant temperature has been examined and the effect of external heating is not included. Moreover, the Nusselt number, which is a crucial parameter for analyzing heat transfer process, is not discussed. Rarefied gas flow between two coaxial cylinders caused by evaporation and condensation on the system boundaries is studied by Sharipov et al. [49]. The hydrodynamical equation has been treated analytically, while the kinetic equation has been solved by the discrete velocity method modified for the discontinuous distribution function. The distributions of the density, velocity and temperature are then presented. More recently, Titarev and Shakhov [50] proposed an implicit second-order numerical method based on the kinetic theory for analyzing the spiral Couette flow of a rarefied gas between coaxial cylinders. The cylinders are assumed to have constant temperature. The authors have reported that the numerical algorithm proposed is much easier to implement in a software code and is simpler to generalize to more complicated boundary conditions. Analytical solutions not only provide helpful tools for benchmarking numerical computations, but are very useful for parametric studies. In fact, in the situations where numerical results cannot be experimentally validated, or no experiments are possible due to the limitations usually encountered, analytical approaches happen to be the best choice for verification, especially when a large number of parameters are involved. The main objective of the present study is to analytically investigate the transport phenomena associated with rarefied gaseous flows within a shaft-housing micro-configuration. Hence, in the following we aim at presenting closed-form solutions of the governing momentum and energy equations, under the influence of external heating condition, and we are also concerned with understanding the corresponding effects of the associated influential parameters on the transport phenomena within the above mentioned configuration. 2. Mathematical formulation Consider a shaft of radius ri and angular velocity ui concentrically positioned inside its hosting of radius ro and angular velocity uo (Fig. 1). The gap is filled with a rarefied gas while the system boundaries are under the influence of either Constant Wall Temperature (CWT) or Uniform Heat Flux (UHF) conditions. For the problem under consideration, the incompressible Naviere StokeseFourier equations including viscous heating effect can be written as

duq ¼ 0 dq

(1)

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

265

Eq. (4), known as the Maxwell relation, is the earliest slip model, in which the velocity slip is proportional to the mean free path and the constitutive relation is linear. This linear model, however, is prone to failure in special circumstances such as flows with high rarefaction degrees or systems with considerable curvature or roughness effects [51]. Hence, recently, efforts have been devoted to presenting more accurate slip models. As a result, nonlinear slip models have been introduced for which the constitutive relations are modified so that the stress is expressed in a more realistic way and also the mean free path is modified by effective expressions [51]. In addition, since Eq. (5), referred to as the Smoluchowski relation, suffers from some fundamental inaccuracy, improved models satisfying a detailed balance are proposed [52,53]. Nevertheless, due to their simplicity and reasonable accuracy, these conventional slip/jump relations have been widely utilized as boundary conditions for the analysis of rarefied flow in the literature [20e27,29e38,43e47]. The earliest forms of the viscous slip and temperature jump coefficients proposed by Maxwell and Smoluchowski, respectively, are as follows [7]

xu ¼

2  su

3

su

3

Fig. 1. Schematic diagram of the configuration and imposed boundary conditions.

xT ¼

2  sT

3

sT

3

d2 u q d uq  ¼ 0 þ 2 dr r dr

(2)

    kd dT duq uq 2 r þm ¼ 0  r dr dr dr r

(3)

where uq is the tangential velocity component, T denotes the temperature, k is the thermal conductivity, m is the dynamic viscosity and r represents the radial direction. Beyond the continuum limit, the conventional no slip/jump boundary conditions seem to fail and the presence of slight velocity slip and temperature jump is inevitable. More precisely, due to the presence of rarefaction phenomenon in microscales the fluid particles adjacent to the solid surface no longer attain the velocity and temperature of the surface. The gas at the solid boundary slips along the surface and there is a temperature discontinuity, or jump, between the solid surface and the adjacent gas. Hence, in the present study slip/jump boundary conditions for the velocity and temperature fields are applied to the shaft and housing surfaces which read [24,42e45]

uq ðr3 Þ  uw ¼ xu Lc Kn 3

Tðr3 Þ  Tw

3

  duq ðr3 Þ uq ðr3 Þ  dr r

  dTðr3 Þ ¼ xT Lc Kn dr

(6)

where su and sT are the tangential momentum and energy accommodation coefficients, respectively, g is the ratio of the specific heats, and Pr shows the Prandtl number defined as Pr ¼ mcp/k (here cp is the specific heat at constant pressure). The accommodation coefficients depend on various factors that affect surface interaction, such as the gas and surface compositions and temperatures, local pressure, the velocity and the mean direction of the local flow, chemical state and roughness [7,24,32]. Moreover, the values of su and sT are not necessarily equal and they are usually tabulated for some common gases and surfaces [7]. According to Refs. [20,21,27,37,54], for most engineering applications the values of the accommodation coefficients are close to unity. It is necessary to remark that more recent investigations have focused on the presentation of more accurate correlations between the slip/jump and the accommodation coefficients. As a result, other dependencies of xu and xT on the momentum and energy accommodation coefficients have been reported. In particular, it is found that xT should be considered as a function of both su and sT (see Ref. [55] for more details). Alternatively, based on the diffuse specular scattering kernel, the viscous slip and temperature jump coefficients for polyatomic gases are presented as [55]

(4)

xu ¼

(5)

xT ¼

in which the positive and negative signs refer to the shaft and the housing, respectively. The first and second terms in the left hand side of the above equations denote the velocity and temperature of the fluid at the walls and those of the moving walls, respectively. Moreover, xu and xT are referred to as viscous slip and temperature jump coefficients, respectively, the subscript 3 (¼i,o) represents the inner and outer walls, Lc¼(ro  ri) is the characteristic length of the configuration and Kn ¼ (l/Lc) is the Knudsen number.

2g 1 g þ 1 Pr

2  ad

ad

3

1 þ 0:1366ad



3

3

 2  ad

ad

3

 þ 0:17

3

2g 1 g þ 1 Pr

(7)

where ad is the percentage of the incident particles which are scattered diffusely. In the present study, we have opted for the forms presented in Eq. (7) as the definitions of the boundary condition coefficients. In order to avoid dimensionality, the following nondimensional quantities are introduced,

R ¼

r ; ro

r* ¼

ri ; ro

U ¼

uq ; umax

Q ¼

T  Tðr3 Þ

z

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where umax ¼ maxjuw j while uw ¼ r3 u3 ; 3 ¼ i; o and the parameter z e which depends on the imposed thermal boundary condition (Cases A, B and C in the following) e and 3 used for defining the dimensionless temperature read 3

Case A : 3 ¼ o;

z ¼

Case C : 3 ¼ o;

q_ wi ro k

1 þ 2huo

a1 ¼ 

s a1

a3 ¼ 

1 þ 2huo

s¼

q_ wo ro k

z ¼

Case B : 3 ¼ i;

3



;

a2 ¼ r* a1 þ 1

;

a4 ¼ r* a3

 1 2hui * þ  r ð1 þ 2h Þ uo 2 r* r*

(17)

3.2. Thermal field solution

z ¼ Twi  Two

(8)

The governing Eqs. (1)e(3) are then modified into the following dimensionless form

Substituting the flow field solution of Eq. (15) into the energy equation of (11) and then integrating twice, leads to the following general form for the temperature distribution

dU ¼ 0 dq

QðRÞ ¼ 

(9)

  d2 U d U ¼ 0 þ dR2 dR R

(10)

    1 d dQ dU U 2 R þ Br ¼ 0  R dR dR R dR

(11)

where Br is the Brinkman number given as

Br ¼

mu2max zk

Consequently, the nondimensional slip/jump boundary conditions can be expressed as follows

UðR3 Þ  Uw ¼ hu 3

QðR3 Þ  Qw

3

3

  dUðR3 Þ UðR3 Þ  dR R3

(13)

hQ ¼ xT LD Kn

(14)

3

3 3

C3 ¼ 

¼ i ¼ o

3.1. Flow field solution Here, we should state that although the velocity distribution for the current problem has been investigated previously in Ref. [45], a nondimensional solution is derived here, to be incorporated into Eq. (11). The velocity profile can be determined by solving the equation of motion, i.e. Eq. (10), subject to the slip-boundary condition given by Eq. (12) which reads

C2 R

(15)

where the coefficients C1 and C2 are given by

C1 ¼ a1 Uwi þ a2 Uwo C2 ¼ a3 Uwi þ a4 Uwo with ai(i ¼ 1e4) and s specified as

(19)

2BrC22  r* ; r *2

C4 ¼ BrC22

(20)

For the Case B, the nondimensional thermal boundary conditions for the energy equation can be expressed as

3. Solution procedure

UðRÞ ¼ C1 R þ

dQ ¼ 1 dR R ¼ 1/Q ¼ 0

Under such conditions, the unknown constant coefficients of the temperature distribution can be specified this way

where hu and hQ are defined as   r* ; and ultimately, LD¼(Ro  Ri) where R3 ¼  1;

where C3 and C4 are the unknown coefficients which will be determined according to the imposed thermal boundary conditions. As stated before, using different forms of thermal boundary condition, three thermal cases are considered here which are shown in Fig. 1: Case A: uniform heat flux at the shaft and constant temperature at the housing, i.e., ðq_ wi ¼ const & Two ¼ constÞ; Case B: uniform heat flux at the housing and constant temperature at the shaft, i.e., ðTwi ¼ const & q_ wo ¼ constÞ; Case C: constant temperature at both walls, i.e. ðTwi ¼ const & Two ¼ constÞ. Hereafter, we adopt that qw is a positive quantity when its direction is to the fluid (from the hot wall), otherwise it is a negative one (to the cold wall). It is also assumed that qw has a nonzero value. For the Case A, the thermal boundary conditions in dimensionless form are as follows

R ¼ r* /

3

3

(18)

(12)

  dQðR3 Þ ¼ hQ dR

hu ¼ xu LD Kn;

BrC22 þ C3 lnðRÞ þ C4 R2

(16)

R ¼ r * /Q ¼ 0 dQ ¼ 1 R ¼ 1/ dR

(21)

Regarding the above conditions, the unknown coefficients C3 and C4 take the form

C3 ¼ 1  2BrC22 ;

C4 ¼

  BrC22  C ln r* 3 r *2

(22)

And finally, for the Case C, the dimensionless thermal boundary conditions read

   dQ dQ  R ¼ r * /Q ¼ 1 þ hQi  h  Qo dR R¼Ri dR R¼Ro R ¼ 1/Q ¼ 0

(23)

The corresponding expressions for the unknown coefficients of the temperature profile are then

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

        2hQi hQi 1 *   C ln r C3 ¼ 1 þ BrC22 þ  2h þ h 4 Q Q o o r *3 r *2 r* C4 ¼ BrC22

(24)

It is possible to rewrite the already extracted relations for the temperature profile e which are in terms of Ts (the temperature of the fluid at the wall) e into equations in terms of the wall temperature Tw using the temperature jump boundary condition of (13). To clarify more, we have

Q* ðRÞ ¼ QðRÞ  Qw ¼ 3

T  Tðr3 Þ

z



Tw  Tðr3 Þ 3

z

¼

Subsequently, the general forms of the Nusselt number Nu for different types of thermal boundary condition, namely, Uniform Heat Flux Boundary Condition (UHF B.C.) and Constant Wall Temperature Boundary Condition (CWT B.C.) are obtained through the combination of Eqs. (28) and (29).

q_ w~ De DE  ¼ UHF B:C: : Nu~3 ¼  * * Q w~  Qm k Tw~  Tm 3

 dT  H  dr 

3

(25) where Qw ¼ hQε ðdQ=dRÞjR¼R3 Consequently, the transformed relations for the above mentioned thermal boundary conditions take the form

BrC22 ~ þ C3 lnðRÞ þ C 4 R2

Q* ðRÞ ¼ 

(26)

where

  2 ~ ¼ C þh Case A : C 4 4 Qo 2BrC2 þ C3 2BrC22 C3 ~ ¼ C þh þ * Case B : C 4 4 Qi r *3 r

(27)

3.3. Nusselt number and heat flux Despite the fact that no convective term is present in the energy equation of (11) and the mechanism of heat transfer for this configuration is pure conduction, providing an ordinary expression for the Nusselt number may lead to a better understanding of the transport phenomena in such a concentrically rotating micro structure and also a beneficial comparative criterion for more complicated geometries. The Nusselt number based on the equivalent diameter of the configuration De ¼ 2(ro  ri) can be expressed as

Nu~3 ¼

(28)

k

where h is the heat transfer coefficient. Balancing the Fourier’s conduction law and Newton’s law of cooling leads to the following relation

qw~ 3

3

*

i

r¼r~3 ;~3 ¼jo

  ¼ h~3 Tw~  Tm 3

Q*w~  Q*m

(32)

3

*

 ~3 ¼ i/Nui ¼ ½Eq: ð32Þ With Q*w ¼ 0 i    CaseB:  dQ*  * * ~3 ¼ o/Nuo ¼ ½Eq: ð31Þ With Qw ¼ Q ðRo ÞhQo  o  dR    ~3 ¼ i/Nu ¼ ½Eq: ð32Þ With Q* ¼ 1 i  wi Case C :   ~3 ¼ o/Nuo ¼ ½Eq: ð32Þ With Q* ¼ 0 wo

R¼Ro

(33)

Note here that Q*m is given by

ZRo

Q*m ¼

Tm  Tw

z

3

¼

* UðRÞQ ðRÞdR

Ri

(34)

ZRo UðRÞdR

  r* where R3 ¼  1

¼ i ¼ o As the final investigation in this section, the expressions for the heat fluxes at the walls with constant temperature are derived which in dimensionless form read

(29)

  dQ*   ~3 ¼ i   q_ w~  dR  * R¼r   ¼ ¼  * j  dQ  ~3 ¼ o   dR  3

(35)

R¼1

where j ¼ ðzk=ro Þ.

Zro

Z

3

3

000

b q w~

where Tm is the mean or bulk temperature given by

uq ðrÞTðrÞdr

uTdA ¼ udA

$DE

where Qw and Qm are the dimensionless wall and bulk temperature, respectively and DE ¼ 2(Ro  Ri) is the dimensionless equivalent diameter of the clearance. The resulting form of the Nusselt number in terms of the nondimensional quantities for each case reads

3

Tm ¼ Z

¼

i

R¼R~3 ;~3 ¼jo

Ri

h~3 De

 dT  ¼ Hk  dr 

$De

  Tw~  Tm

CWT B:C: : Nu~3 ¼

   

*

dQ H dR

*  ~3 ¼ i/Nui ¼ ½Eq: ð31Þ With Q* ¼ Q* ðRi ÞhQ dQ  w i i Case A :  dR R¼Ri ~3 ¼ o/Nuo ¼ ½Eq: ð32Þ With Q*wo ¼ 0

!

  2 ~ ¼ C þh Case C : C 4 4 Qo 2BrC2 þ C3

i

r¼r~3 ;~3 ¼jo

3

(31)

3

3

T  Tw

z

267

ri

(30)

Zro uq ðrÞdr ri

4. Results and discussion To this end the main features of the current analytical approach have been discussed and the relevant expressions for the associated

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M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

properties are presented. In the following, we investigate the interactive effects of the Brinkman number Br, Knudsen number Kn, rotation mode and curvature on fluidic and thermal major variables such as velocity, temperature, Nusselt number and surface heat fluxes. The gas considered here is air with Pr ¼ 0.7 and g ¼ 1.4. It is worth mentioning that although for nonplanar flows of this type there are many other efficacious parameters (e.g., ad, gas properties, inner versus outer cylinder rotating, etc.) to be considered, to keep the number of the presented figures to a reasonable quantity, the effect of such parameters are skipped. Hence, for the rest of the analysis we set the value of ad to unity by assuming fully diffuse reflection for both cylinders and for convenience, the aspect ratio of the annuli r* is considered to have the value of 0.5, except for the calculation of the Nusselt number for which the value falls into the range of 0.2  r*  0.8. In addition, here nonnegative values of the Brinkman number are undertaken, unless while calculating heat fluxes. The two primary hydrodynamic cases we examine here are (1) the shaft rotating and the housing at rest (uo ¼ 0) and (2) the housing rotating while the shaft is at rest (ui ¼ 0). Moreover, as stated earlier, two different thermal boundary conditions are considered in the present study, i.e., Uniform Heat Flux (UHF) and Constant Wall Temperature (CWT) cases. Throughout the combination of these boundary conditions, three thermal case studies are constructed, namely, Cases A, B and C which have already been discussed and are treated separately in the following. It should be noted that while examining the effect of the Knudsen number on the major variables, the Brinkman number is set to unity and when investigating the influence of the Brinkman number, the Knudsen number is assumed to be at the accepted upper limit of the slip-flow regime, namely, Kn ¼ 0.1, unless otherwise stated. Before discussing the results for the problem under consideration, it is informative to state that there are several choices for the characteristic length of the system in the definition of Kn, namely, the gap size (ro  ri), the radius of the shaft ri and the radius of the housing ro. Accordingly, the criterion of the degree of rarefaction and also the validity of the continuum approach, i.e., the Knudsen number is not uniquely defined. However, following the work of Ref. [45], we base the definition of this dimensionless parameter on the gap size of the annuli and adjust this factor so that desired values of Kn are obtained. In other words, since the gas properties are assumed to be constant, the Knudsen number varies according to a change in the clearance size rather than the value of the mean free path of the molecules l.

4.1. Knudsen number variation Fig. 2 shows the nondimensional velocity profile for different rotation modes and illustrates the effect of rarefaction on the velocity slip at system boundaries. Note that Kn ¼ 0 holds for the macroscale case, while Kn > 0 represents the microscale counterpart. As can be seen, for both rotation modes, an increase in Kn results in increasing velocity slip at the boundaries and a general flattening of the velocity profiles. Another noticeable point is that the velocity profiles for all values of the Knudsen number intersect at a common location apart from which the effect of Kn on velocity distribution changes. Fig. 3 illustrates how the nondimensional temperature and Nusselt profiles are affected by the Knudsen number at a fixed Brinkman number (Br ¼ 1) when the shaft is rotating. For the Case A, increasing Kn intensifies the temperature jump at both boundaries and tends to flatten the temperature curves. As a result, the absolute temperature gradient at the housing decreases for the whole range of r* (not shown here), while at the shaft surface remains constant. Similar to the velocity profiles, there is an intersection for the temperature profiles of this case that can be referred to as a critical point which reverses the effect of Kn on temperature distribution. Next to be discussed is the Nusselt number which, from the engineering point of view, is an important factor that determines the heat transfer rate. At the shaft, as can be seen from the figure, the effect of Kn on the Nusselt number depends on the value of the aspect ratio of the annuli r*. More precisely, for small quantities of r*, Nui increases with increasing Kn, while the opposite is true for the large values of the parameter. To explain this, as the medium becomes more rarefied, increasing values of velocity slip bring about more reduction in the shear stress at the shaft, in the whole range of the aspect ratio of the annuli. This, along with increasing amounts of the temperature jump at this surface, results in a reduction in the temperatures of both the shaft and the bulk flow. The shear stress at the shaft, however, is not a monotonous function of the aspect ratio of the system. More precisely, this quantity exhibits a minimum as r* approaches higher values. Despite the increasing trend of the shear stress at the shaft at higher values of the aspect ratio of the annuli, the shaft and the bulk temperatures do not undergo an increasing procedure and still decrease. The reason must be sought in the fact that as r* increases, heat conduction dominates heat transfer process rather than rarefaction. This corresponds to more heat transfer from the bulk flow to the outer wall and consequently, higher heat fluxes at the housing surface for greater values of r*. These observations

1.0

1.0 Kn = 0.0 Kn = 0.025 Kn = 0.05 Kn = 0.075 Kn = 0.1

Kn = 0.0

0.8

0.8

0.6

0.6

U

U

Kn = 0.1

0.4

0.4

0.2

0.2

Kn = 0.1 Kn = 0.0 Kn = 0.025 Kn = 0.05 Kn = 0.075 Kn = 0.1

Kn = 0.0

0.0

0.0 0.5

0.6

0.7

0.8

R

0.9

1.0

0.5

0.6

0.7

0.8

R

Fig. 2. Effect of Kn on the velocity distribution. Left: inner wall rotating; Right: outer wall rotating.

0.9

1.0

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

269

Fig. 3. Effect of Kn on the temperature and Nusselt profiles for the case of inner wall rotating. Left: Case (A); Middle: Case (B); Right: Case (C).

*

*

lead to a reduction in Qwi  Qm with increasing Kn for smaller values of the aspect ratio, while an increasing trend is observed for larger values of the quantity (not shown here). Considering the above points and regarding the reduction of the dimensionless equivalent diameter, DE, with an increase in r*, the illustrated dependency of Nui pertained to the Case A on the Knudsen number and the aspect ratio can be easily justified through Eq. (31). The last figure presented for the Case A in Fig. 3 examines the effect of rarefaction and curvature on the Nusselt number at the outer cylinder Nuo. The plot shows that Nuo decreases/increases monotonously with increasing Kn/r*. In addition, the rate of departure from macroscale Nusselt number (Nuo at Kn ¼ 0) increases with an increase in the value of r*. To clarify more, increasing Kn for any specific value of r*, on one hand, leads to a decrease in bulk temperature and consequently in absolute temperature difference between the housing surface and the bulk flow and on the other hand, to a decrease in the absolute temperature gradient at this boundary. Thus, although the absolute values of both numerator and denominator in Eq. (32) decrease, the net effect is a drop in Nuo. The effect of the Knudsen number on the heat transfer characteristics of the Case B in the rotation mode (1) is shown in the second column of Fig. 3. For this case, an increase in rarefaction

degree is accompanied by an increase in the temperature through the annuli and also in temperature jump at the boundaries. Unlike the corresponding velocity profiles in Fig. 2, the temperature curves do not intersect in a common location. Turning to the Nusselt number at the shaft, one can deduce from the plot that increasing Kn and r* monotonously decrease Nui, progressing toward uniformly distributed profiles. To explain this behavior, we shall refer to the definition of the Nusselt number in Eq. (32) again. The effect of increasing rarefaction is, on one hand, to increase the bulk temperature and consequently the absolute value of Q*wi  Q*m , and on the other hand, to decrease the absolute temperature gradient at the shaft. Therefore, both numerator and denominator in Eq. (32) are in accordance with the general behavior of Nui. In addition, although the magnitude of temperature difference between the inner wall and the bulk flow decreases monotonously with increasing r* in a specific value of Kn, but that is not the case for the absolute temperature gradient at the wall. This quantity reduces at lower values of r* while tends to increase at higher values of the quantity. These, along with the occurrence of a reduction in the equivalent diameter of the system with increasing r* are the reasons why the departure rate of Nui from its macroscale counterpart (Nui at Kn ¼ 0) changes significantly as the aspect ratio of the annuli increases. Despite the inner Nusselt number, the outer one increases gradually as Kn approaches its upper limit. The reason

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Table 1 Variation of wall heat flux at the outer wall with Kn, Br and rotation mode for the Case A. b q wo

Br Kn Mode 1.0 0.5 0.0 0.5 1.0

0.0

0.025

0.05

0.075

0.1

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

2.17 0.83 0.50 1.83 3.17

0.17 0.17 0.50 0.83 1.17

1.76 0.63 0.50 1.63 2.76

0.07 0.22 0.50 0.78 1.07

1.45 0.47 0.50 1.47 2.45

0.01 0.26 0.50 0.74 0.99

1.19 0.35 0.50 1.35 2.19

0.08 0.29 0.50 0.71 0.92

0.98 0.24 0.50 1.24 1.98

0.13 0.31 0.50 0.68 0.87

must be sought in the reduction in jQ*wo  Q*m j with increasing Kn through the entire range of r*. The last column in Fig. 3 shows how the corresponding heat transfer variables of the Case C are affected by the Knudsen number. As can be deduced from the first plot, due to the presence of an intersection point, the influence of Kn on the temperature profile depends on the radial distance from center of the annuli. If the distance is less than that of the intersection, temperature decreases with increasing Kn, otherwise, the temperature profiles experience an increasing procedure. Similar to the previous cases, temperature jump at both cylinders increases as the area becomes more rarefied. Moreover, temperature gradients at the shaft and the housing exhibit different behaviors as Kn increases. For the special case of r* ¼ 0.5 shown here, temperature gradient at the shaft approaches more negative values with increasing Kn, while the opposite is true for the quantity at the housing. Next plot in this column shows how the Nusselt number at the inner wall varies versus the aspect ratio of the annuli when the Knudsen number is increased. The first noteworthy feature is that Nui is negative for a wide range of r* and Kn. Here is the reason. Since the Brinkman number is considered to be unity and due to the definition of Br for this case, the shaft has a higher temperature than the housing, resulting in positive temperature differences between the shaft and the bulk flow, i.e. Q*wi  Q*m , for the given values of r* and Kn. An analogous behavior is exhibited by the temperature gradient at the wall except for higher values of r* and Kn at which the quantity is negative. Consequently, with regard to the definition of the Nusselt number for this case (Eq. (32)), the already mentioned behavior of Nui can be easily interpreted. Another point of interest is that the Knudsen number appears to have little effect on Nui as r* approaches its upper limit. To put it differently, at lower values of r*, increasing Kn tends to flatten Nui profiles and departure from macroscale Nusselt number is significant, while at higher values of this quantity effect of Kn on Nui is negligible and all profiles reach the value associated with Kn ¼ 0. For this case, the bulk temperature decreases monotonously with increasing Kn for all values of r* and as a result, the absolute value of Q*wo  Q*m tends to decrease. On the other side, magnitude of the slope of the temperature profiles at the housing surface monotonously increases/decreases with increasing r*/Kn, respectively. Accordingly, although the absolute value of both

numerator and denominator in Eq. (32) decreases, the conclusive effect is a decrease in Nuo due to increased Kn. As the final investigation in this section, the effect of the Knudsen number on the heat fluxes at the boundaries for the above thermal cases and for the special case of r* ¼ 0.5 is summarized in Tables 1e4. These tables also contain the corresponding effects of the other parameters including Br and rotation mode, which will be discussed in the following sections. The results show that for the case of Br ¼ 0, the increased Knudsen number does not affect the heat fluxes associated with the Cases A and B, while tends to decrease the absolute value of b q w at both boundaries for the Case C. As for other values of the Brinkman number, depending on the value of this parameter, different trends for the variation of j b qwj with increasing Kn may be observed. 4.2. Brinkman number variation Shown in Fig. 4 is the effect of the Brinkman number, Br, on heat transfer characteristics of the flow when the shaft is rotating. The left column in the figure is associated with the Case A in which a uniform heat flux is imposed to the inner wall, while the other wall has a constant temperature. The first plot in this column demonstrates how temperature distribution for the special case of r* ¼ 0.5 is affected by different values of the Brinkman number in the range of 0  Br  1. As can be deduced from the figure, the Brinkman number e which is a measure for magnitude of the viscous dissipation term e severely influences the temperature profiles, * enforcing Q to admit higher values with increasing Br through the whole channel gap. Pondering the problem from a different angle, viscous dissipation, as an energy source, increases the temperature due to its contribution to the internal heating of the fluid. Another remarkable point is that the bulk temperature and the temperature of the shaft Q*wi significantly increase with increasing Br; especially at lower values of r*. Subsequently, the net effect is an increase in * * jQwi  Qm j which causes Nui to undergo a reducing procedure. Furthermore, unlike the inner Nusselt number, the outer one exhibits a gradual growth in magnitude while increasing the Brinkman number. The reason is that according to Eq. (32), at a fixed value of r*, the Nusselt number at the outer wall is under the influence of two different factors, namely, the temperature gradient

Table 2 Variation of wall heat flux at the inner wall with Kn, Br and rotation mode for the Case B. b q wi

Br Kn Mode 1.0 0.5 0.0 0.5 1.0

0.0

0.025

0.05

0.075

0.1

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

3.33 0.67 2.0 4.67 7.33

0.67 1.33 2.0 2.67 3.33

2.53 0.26 2.0 4.26 6.53

0.87 1.43 2.0 2.57 3.13

1.89 0.05 2.0 3.95 5.89

1.03 1.51 2.0 2.49 2.97

1.38 0.31 2.0 3.69 5.38

1.15 1.58 2.0 2.42 2.85

0.97 0.52 2.0 3.48 4.97

1.26 1.63 2.0 2.37 2.74

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

271

Table 3 Variation of wall heat flux at the inner wall with Kn, Br and rotation mode for the Case C. b q wi

Br Kn Mode 1.0 0.5 0.0 0.5 1.0

0.0

0.025

0.05

0.075

0.1

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

6.15 4.52 2.88 1.25 0.38

3.70 3.29 2.88 2.48 2.07

5.26 3.93 2.61 1.28 0.04

3.27 2.94 2.61 2.28 1.95

4.58 3.48 2.38 1.29 0.19

2.93 2.66 2.38 2.11 1.83

4.03 3.10 2.19 1.27 0.35

2.65 2.42 2.19 1.96 1.73

3.60 2.81 2.03 1.24 0.46

2.42 2.22 2.03 1.83 1.64

Table 4 Variation of wall heat flux at the outer wall with Kn, Br and rotation mode for the Case C. b q wo

Br Kn Mode 1.0 0.5 0.0 0.5 1.0

0.0

0.025

0.05

0.075

0.1

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

0.41 0.92 1.44 1.96 2.48

1.18 1.31 1.44 1.57 1.70

0.37 0.84 1.30 1.77 2.24

1.07 1.19 1.30 1.42 1.54

0.34 0.77 1.19 1.62 2.04

0.98 1.08 1.19 1.30 1.40

0.33 0.71 1.10 1.48 1.86

0.90 0.10 1.10 1.19 1.29

0.32 0.66 1.01 1.36 1.71

0.84 0.93 1.01 1.10 1.189

Fig. 4. Effect of Br on the temperature and Nusselt profiles for the case of inner wall rotating. Left: Case (A); Middle: Case (B); Right: Case (C).

272

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275 *

*

at the housing surface and Qwo  Qm . Even though both the former and the latter are found to increase in magnitude with increasing Br in the whole range of r*, the effect of the numerator in the above mentioned equation dominates and ultimately, the result is an increase in Nuo. The second column in Fig. 4 depicts the considerable effect of the Brinkman number on heat transfer characteristics associated with the Case B in the rotation mode (1). Again, as expected, increasing values of Br tend to positively shift the temperature curves. Moreover, the quantity of variables such as jQ*w~  Q*m j~3 ¼i;o , Q*m and the absolute temperature gradient at the shaft surface monotonously increases with increasing Br for all values of r*. In addition, while the two first quantities decrease monotonously with increasing r* in a specific Br, the third one exhibits a minimum as r* approaches higher values, except for Br ¼ 0. Needless to say, an increase in r* also leads to a decrease in the value of the equivalent diameter. On the basis of the above, the behavior of the Nusselt number at the shaft and the housing can be easily interpreted with regard to Eqs. (31) and (32), respectively. The last column in Fig. 4 depicts the effect of the Brinkman number on variables such as Q*, Nui and Nuo for the Case C while the shaft is exposed to a clockwise rotation and the housing is at rest. For this case, as is shown in the first plot, the main result of increasing Br is a considerable change in the curvature of the 3

temperature profiles and also in the amount of Q*. At lower values of the Brinkman number, the temperature curves are in a slight concave shape, gradually turning to convex counterparts as Br increases. Another consequential outcome of positively growing values of Br is a monotonous increase in the bulk temperature for the whole range of r*. Note again that according to the definition of Br for this case, positive values of the Brinkman number correspond to higher temperatures at the shaft compared to the housing. Regarding this and due to the fact that for the parameters considered here, the bulk temperature is lower/higher than that of the inner/outer cylinder, a decrease/an increase in the value of * * * * jQwi  Qm j=jQwo  Qm j seems to be inevitable with increasing Br * for all values of r . Meanwhile, the absolute value of temperature gradient at the inner/outer wall decreases/increases monotonously with increasing the Brinkman number in the whole range of the aspect ratio of the annuli. Although both the numerator and denominator in Eq. (30) decrease/increase in magnitude, the conclusive effect is a drop/rise in Nui/Nuo. Since the effect of the Brinkman number on the heat fluxes at the boundaries has been previously described through explaining the behavior of the Nusselt number, we avoid discussing the issue further and just for more illustration, we present detailed data about b q w for r* ¼ 0.5 via Tables 1e4. Note here that positive values of Br correspond to wall heating, while the negative values

Fig. 5. Effect of Kn on the temperature and Nusselt profiles for the case of outer wall rotating. Left: Case (A); Middle: Case (B); Right: Case (C).

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

represent the cooling mode and Br ¼ 0 gives an account of the case without the effect of viscous dissipation. 4.3. Rotation mode variation The effect of rotation mode on the velocity distribution is shown through Fig. 2. As can be observed from the plots, the velocity profile associated with the rotation mode (1) has a clear concave shape while the other rotation mode exhibits a slightly convex shape. Furthermore, it can be concluded that rotation mode dramatically changes the influence of the Knudsen number on the velocity profiles. To elucidate more, in the case of inner cylinder rotating, the velocity decreases with increasing Kn throughout much of the clearance and increases in a small region confined between the intersection point and the housing, while the opposite takes place for the other rotation mode. A comparison between the results shown in Fig. 3 with their counterparts in Fig. 5 depicts that rotation mode can have a significant effect on heat transfer characteristics of the flow. In other words, it not only changes the magnitude of the major variables and the corresponding profile shapes, but also can change the way other influential parameters, e.g., Kn, affect the temperature and Nusselt profiles. In the interest of the brevity, our discussion is * restricted to just two plots, namely, the temperature profile, Q , for

273

the Case A and the Nusselt number at the housing, Nuo, for the Case B. Regarding the values of Q* in the first column of Figs. 3 and 5 which are related to the Case A at the special value of r* ¼ 0.5, it is clear that higher temperature values and correspondingly bulk temperatures are exhibited by the rotation mode (1). Moreover, the range at which Q* decreases with increasing Kn, is more extensive in the rotation mode (1) than the other mode, which implies that rotation mode affects the influence of the Knudsen number on the temperature profiles of this case through changing the position of the intersection point. Considering the outer Nusselt number Nuo for the Case B, one may realize that how dramatic the effect of rotation mode on the heat transfer characteristics can be. Higher Nuo for the case of outer wall rotating compared with the other case in the whole range of r* and at any specific value of Kn signifies that the absolute temperature difference between the housing surface and the bulk flow, which is the main deriving mechanism for the heat transfer from the wall to the fluid, in the rotation mode (1) is more than that in the rotation mode (2). It can be physically interpreted as entering more heat from the housing surface to the flow domain for the first rotation mode in comparison with that of the second one. Another interesting point is that, unlike the rotation mode (1) for which Nuo increases with increasing Kn, the other rotation mode demonstrates a decreasing trend for Nuo while rarefaction increases. This

Fig. 6. Effect of Br on the temperature and Nusselt profiles for the case of outer wall rotating. Left: Case (A); Middle: Case (B); Right: Case (C).

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

2.0

λ = 0.067 × 10 -6 m μ = 0.0000185 Pa.s k = 0.0257 W/m.K Pr = 0.7 γ = 1.4 σui = σuo= 1.0 σTi = σTo= 1.0

3.0

1.5

1.0

0.5

0.0

0.0

λ = 0.067 × 10 m μ = 0.0000185 Pa.s k = 0.0257 W/m.K Pr = 0.7 γ = 1.4 σui = σuo= 1.0 σTi = σTo= 1.0 -6

Kn = 0.01 Kn = 0.05

2.5

o

2.5

o

Overall Temperature Rise ( C )

3.0

Overall Temperature Rise ( C )

274

2.0

Kn = 0.01 Kn = 0.05

1.5

1.0

0.5

0.0

1.0

2.0 3.0 Heat Flux (Kw/m2)

4.0

5.0

0.0

1.0

2.0 3.0 Heat Fux (Kw/m2)

4.0

5.0

Fig. 7. Overall temperature rise versus the applied heat flux at different rarefaction degrees. Left: Case (A); Right: Case (B).

is because despite for the case of inner wall rotating, jQ*wo  Q*m j increases with increasing Kn for the rotation mode (2) in the whole range of r* (not shown here). The effect of rotation mode on the behavior of thermal variables at different values of the Brinkman number can be inferred through comparing Figs. 4 and 6. It is obvious that rotation mode considerably distorts the way Br affects the temperature and Nusselt profiles through changing the velocity profile. To state the matter differently, since the temperature profile is tightly under the influence of the velocity distribution, any change in the latter may lead to severe alternations in the former. Depending on the corresponding rotation mode, any of the velocity profiles in Fig. 2 may be incorporated into Eq. (11) to form the temperature distribution. Furthermore, the effect of the Brinkman number on the thermal variables may differ significantly from one rotation mode to another one. One may refer to Nui for the Cases A and C and Nuo for the Case C, as some examples. Tables 1e4 involve the influence of rotation mode on b q w for the special value of r* ¼ 0.5. As can be seen, the only valid common statement for all above cases is that at Br ¼ 0 and for any value of Kn, rotation mode does not play any role in the calculation of heat fluxes passing through the walls. However, the effect of rotation mode on general behavior of b q w depends on the value of other parameters such as Kn and Br, and differs from one thermal case to another one.

adjacent to the heated boundary and the wall with constant temperature. Fig. 7 examines the variation of the overall temperature rise factor with the applied wall heat flux and the Knudsen number for Cases A and B. The values of q_ w chosen here and shown in the figure correspond to the Brinkman number in the range of 0.01  Br  1.0. As the figure depicts, for both cases, D is almost a linear function of the imposed heat flux, increasing with increased values of this quantity (decreased values of the Brinkman number). In addition, this quantity is found to reduce as the medium becomes more rarefied. However, at lower values of the applied heat flux (greater Brinkman numbers) D becomes independent of the Knudsen number. One can readily infer from the plots that, for both cases and with the values opted for the parameters, the maximum overall temperature rise factor attained throughout the analysis is D y 2.31  C. This temperature rise is indeed remarkable compared to the operating temperature of the system. However, it signifies that the assumption of constant properties made at the first stage is a reasonable approximation. Note here that for very high values of the overall temperature rise factor the thermophysical properties of the flow are not constant anymore and for accurate flow measurements, their dependence on the temperature must be taken into consideration [56]. Another mentionable point is that the resemblance of the values and trends mentioned above for the Cases A and B is completely incidental and may vanish with a change in the quantity of the system parameters.

4.4. Application 5. Conclusions To provide a sense of practical operating conditions, a dimensional study for the Cases A and B is carried out in this section. As mentioned before, the fluid considered here is air at 20  C with properties presented in Fig. 7. While the shaft is kept stationary, the housing is considered to set the fluid in motion with an angular velocity of uo ¼ 5  105 rpm. The radius of the housing is fixed at ro ¼ 1 mm, while that of the shaft varies with the Knudsen number and can be calculated via the definition of this parameter. With the air mean free path at atmospheric pressure and standard temperature being l ¼ 0.067  106 m, the shaft radii for Kn ¼ 0.01 and Kn ¼ 0.05 are estimated as ri y 0.993 mm (corresponding to a configuration with Lc ¼ 7 mm and r* ¼ 0.993) and ri y 0.998 mm (pertained to a system with Lc ¼ 2 mm and r* ¼ 0.998), respectively. Of interest are the effects of the influential parameters on the overall temperature rise factor, D, which is defined as the temperature difference between the fluid

In this study, we have obtained analytical solutions for flow and heat transfer fields of a rarefied gaseous flow confined between two concentrically rotating cylinders throughout the entire slip-flow regime. The incompressible NaviereStokeseFourier (NSF) equations including viscous dissipation effects in the cylindrical polar coordinate reference frame are analytically solved subject to the socalled slip/jump boundary conditions to obtain the associated velocity and temperature distributions. Throughout a combination of the UHF and CWT boundary conditions, three thermal case studies, namely, Cases A, B and C are constructed and then treated separately. The effects of some parameters such as the Knudsen number, the Brinkman number, rotation mode and the aspect ratio of the annuli on the velocity and temperature distributions, as well as on the Nusselt number and wall heat fluxes have been studied. Here, the corresponding influences of some parameters (e.g., ad and

M. Shamshiri et al. / International Journal of Thermal Sciences 54 (2012) 262e275

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