Stéphane Colin
email: stephane.colin@insa-toulouse.fr

GAS MICROFLOWS IN THE SLIP FLOW REGIME: A CRITICAL REVIEW ON CONVECTIVE HEAT TRANSFER

Keywords: Slip flow, convection, heat transfer, microchannel, microtube, rarefied flow, Nusselt number, viscous dissipation, constant heat flux, constant wall temperature, Brinkman number, Peclet number

Accurate modeling of gas microvection is crucial for a lot of MEMS applications (micro-heat exchangers, pressure gauges, fluidic microactuators for active control of aerodynamic flows, mass flow and temperature micro-sensors, micropumps and microsystems for mixing or separation for local gas analysis, mass spectrometers, vacuum and dosing valves…). Gas flows in microsystems are often in the slip flow regime, characterized by a moderate rarefaction with a Knudsen number of the order of 10 2 -10 1 . In this regime, velocity slip and temperature jump at the walls play a major role in heat transfer. This paper presents a state of the art review on convective heat transfer in microchannels, focusing on rarefaction effects in the slip flow regime. Analytical and numerical models are compared for various microchannel geometries and heat transfer conditions (constant heat flux or constant wall temperature). The validity of simplifying assumptions is detailed and the role played by the kind of velocity slip and temperature jump boundary conditions is shown. The influence of specific effects, such as viscous dissipation, axial conduction and variable fluid properties is also discussed.

INTRODUCTION

Due to the trend of miniaturization of electromechanical systems, there is an increasing need for micro heat exchangers. The applications concern for example the cooling of electronic components, MEMS or MOEMS. More generally, heat transfer is involved in most of microfluidic systems, and especially in those that use gases. In addition to compressibility, rarefaction of the flow at small scale has a significant impact on heat transfer.

Shrinking down the dimensions of fluidic microsystems submitted to internal gas flows leads to an increase of the Knudsen number

h Kn D   (1)
defined as the ratio of the mean free path  of the molecules over a characteristic length, for example the hydraulic diameter h D of a microchannel. The Knudsen number encountered in classic microsystems is frequently between 10  and 10  , which is the typical range of the well-known slip flow regime [START_REF] Colin | Rarefaction and compressibility effects on steady and transient gas flows in microchannels[END_REF]. In this moderate rarefied regime, velocity slip and temperature jump at the wall strongly influences heat transfer. In the past ten years, a number of theoretical and numerical studies have been aimed at modeling gas convective heat transfer in microchannels, taking into account rarefaction effects.

Reviews on this topic are dating, do not focus on gases and do not take into account the last research contributions. Sobhan and Garimella [START_REF] Sobhan | A comparative analysis of studies on heat transfer and fluid flow in microchannels[END_REF] presented in 2001 a compilation and analysis of the results from investigations on fluid flow and heat transfer in microand mini-channels, with special emphasis to experimental studies. The case of both liquids and gases was treated, but without analysis of rarefaction effects on gas microconvection. In 2001, Rostami et al. [START_REF] Rostami | Flow and heat transfer for gas flowing in microchannels: a review[END_REF] published a review devoted to gaseous flows in microchannels, but the analysis was essentially focused on hydrodynamics and not on heat transfer. More recently, in 2004, Morini [START_REF] Morini | Single-phase convective heat transfer in microchannels: a review of experimental results[END_REF] reviewed experimental studies on single-phase convective heat transfer in microchannels. Most of the papers listed in this review concerned heat transfer with liquids; only four studies were including data on Nusselt numbers for gases in microchannels. Moreover, the dimensions were such that rarefaction effects were negligible.

The objective of this paper is to present a detailed review of investigations on slip flow heat transfer in microchannels. The analysis is focused on the Nusselt number Nu defined with the hydraulic diameter h D as a reference length, and its dependence vis-à-vis rarefaction (Knudsen number Kn), viscous dissipation (Brinkman number Br or Br  ) and axial conduction (Peclet number Pe) effects.

NOMENCLATURE

SCOPE OF THE REVIEW

In this review, we consider pressure driven flows of gases with heat transfer in straight microchannels with constant cross-section (see Fig. 1). Heat transfer in circular microchannels is analyzed in section 4, the case of parallel-plate microchannels is discussed in section 5, rectangular and trapezoidal or triangular sections are considered in sections 6 and 7 respectively, whereas other sections are treated in section 8.

The flow is laminar and the regime is the slip flow regime. Investigations on thermally fully developed and thermally developing flows (TFDF and TDF), with constant heat flux (CHF) or constant wall temperature (CWT) are reviewed. The different slip flow and temperature jump boundary conditions are presented in section 3.

SLIP FLOW AND TEMPERATURE JUMP BOUNDARY CONDITIONS

From a theoretical point of view, the slip flow regime is particularly interesting because it generally leads to analytical or semianalytical models. In this regime, continuum Navier-Stokes equations are still valid, provided they are associated with velocity-slip and temperature-jump boundary conditions. Different forms of these boundary conditions can be found in the literature; the most frequently used and relevant for this review are listed below.

Velocity slip boundary conditions

First-order slip boundary conditions have been first written by Maxwell, and can be found on different simplified forms in the literature. The simplest one does not take into account thermal creep and assumes a totally diffuse reflection at the wall:

w u uu n      . ( 2
)
Although the tangential momentum accommodation coefficient u  is generally found to be close to unity [START_REF] Pitakarnnop | A novel experimental setup for gas microflows[END_REF], lower values can be considered using the boundary condition

2 u w u u uu n         (3)
and thermal creep effects due to an axial temperature gradient are taken into account in:

2 3 4 u w u u R T uu n p t            . (4)
In case of wall curvature, the previous equation should be extended as

2 3 4 u w u u v R T uu n t p t                     . ( 5
)
since the first term of the RHS derives from the tangential shear stress at the wall which also depends on the derivative vt  of the normal velocity [START_REF] Lockerby | Velocity boundary condition at solid walls in rarefied gas calculations[END_REF][START_REF] Barber | Isothermal slip flow over curved surfaces[END_REF]. This additional derivative should also be taken into account in microchannels with significant roughness inducing two components of the velocity in the vicinity of the wall. In some papers [START_REF] Khadem | Numerical simulation of roughness effects on flow and heat transfer in microchannels at slip flow regime[END_REF][START_REF] Hossainpour | Investigation of Fluid Flow and Heat Transfer Characteristics of Gases in Microchannels with Consideration of Different Roughness Shapes at Slip Flow Regime[END_REF][START_REF] Shojaeian | Three-dimensional numerical simulation of the slip flow through triangular microchannels[END_REF], however, the term dv dt is written du dt , leading to

2 3 4 u w u u u R T uu n t p t                     . ( 6
)
This form is not supported by any physical analysis and should be not be used, although [START_REF] Shojaeian | Three-dimensional numerical simulation of the slip flow through triangular microchannels[END_REF] refers -erroneouslyto the book of Hirschfelder et al. [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF] as a reference for Eq. ( 6). The second term of the RHS is often neglected in theoretical analysis, as it is of

  2 Kn  , while the first term is of   Kn 
. The accuracy of the first term could be improved, adding a correcting coefficient -of the order of 1.1-calculated from the kinetic theory. Including the actual value of u  , first-order slip boundary condition can be written in a more general form as:

1 wu u uu n      (7)
with 1 u  generally slightly higher than unity. Second-order boundary conditions have been proposed in the literature, in an attempt to increase the Knudsen range of applicability of the slip flow regime. In the specific case of a plane flow, they can be written on the general form [START_REF] Kandlikar | Heat transfer and fluid flow in minichannels and microchannels[END_REF]:

2 2 12 2 w u u uu uu n n             (8)
with different values proposed in the literature for 2 u  . Recent papers [START_REF] Cercignani | Variational derivation of second-order slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules[END_REF][START_REF] Lorenzani | Higher order slip according to the linearized Boltzmann equation with general boundary conditions[END_REF] provide accurate values of

    1 w n u au a u      ( 9
)
where a is the fraction of the surface covered by adsorbed atoms.

In the same way, temperature jump boundary conditions are proposed in different first-order or second-order forms:

2 1 w T TT Pr n       (10) 2 2 1 T w T T TT Pr n         (11) 1 wT T TT n      ( 12
) 2 2 12 2 w T T TT TT nn            (13)
    1 w n T aT a T      (14)
Equations ( 10), ( 11), ( 12), ( 13) and ( 14) are generally used in parallel with eqs. ( 2), ( 3), ( 7), ( 8) and ( 9), respectively.

HEAT TRANSFER IN CIRCULAR MICROCHANNELS

Heat transfer in circular microtubes has been extensively studied, both analytically and numerically, for gas flows in the slip flow regime (see Table 1). We consider in this section a microchannel with constant circular cross section, the radius of which is 0 r (see Fig. 1b).

Fully developed flow and uniform wall heat flux

The problem of hydrodynamically and thermally fully developed flow (HFDF and TFDF, respectively) in a microtube with uniform and constant heat flux (CHF) at the wall has been analytically solved in the slip flow regime by Sparrow and Lin [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF]. They used classic first-order boundary conditions (3)- [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF] and obtained an exact expression for the Nusselt number, which can be written as:

    1 2 C, ,CHF 2 * 11 64 * 96 * 2 48 1 4 * r r r r Nu              . ( 15
)
In this equation,

0 2 * u r u r       (16)
is the dimensionless coefficient of slip and

0 2 21 * 1 T r T Pr r        (17)
is the dimensionless temperature jump distance.

In

) 19
The same result has been demonstrated by Ameel et al. [START_REF] Ameel | Laminar forced convection in a circular tube with constant heat flux and slip flow[END_REF] in the case of full accommodation, with the boundary conditions ( 2) and [START_REF] Shojaeian | Three-dimensional numerical simulation of the slip flow through triangular microchannels[END_REF], which correspond to

  1 21 T Pr       and 1 1 u   .
It is also possible to include thermal creep effects, replacing the boundary condition (3) with the boundary condition (4). Sparrow and Lin [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF] showed that this leads to a correction of Eq. [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF].

Equation [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF] points out that the velocity slip tends to increase the Nusselt number, while the temperature jump decreases it. In most practical cases, the later effect is predominant, and the Nusselt number is reduced, as rarefaction increases.

Li et al. [START_REF] Li | Wall-adjacent layer' analysis for developed-flow laminar heat transfer of gases in microchannels[END_REF] studied the same problem of fully developed flow with constant heat flux at the wall. They kept classic boundary conditions, i.e. no velocity slip and no temperature jump, but they assumed a wall-adjacent layer in which the viscosity and thermal conductivity differ from those in the bulk flow. The thickness of this layer is about 3 to 5 times the mean free path of the molecules and the modification of viscosity and thermal conductivity in this layer is calculated from a simplified kinetic theory assuming hard sphere molecules and full accommodation ( 1 uT   ) at the wall. This analysis leads to:

      2 C,
   . ( 20
)
In this equation, the Nusselt number does not depend on the Prandtl number, neither on the ratio  of the specific heats. Although Eq.

(20) shows the same tendency as Eq. ( 19), it underestimates the decrease of the Nusselt number when rarefaction increases (see Fig.

Effects of viscous heating

Effects of viscous heating have been taken into account by Tunc and Bayazitoglu [START_REF] Tunc | Heat transfer in microtubes with viscous dissipation[END_REF] in 2001 and later by Aydın and Avcı [START_REF] Aydın | Heat and fluid flow characteristics of gases in micropipes[END_REF] and Jeong and Jeong [START_REF] Jeong | Extended Graetz problem including axial conduction and viscous dissipation in microtube[END_REF], both in 2006, then by Hooman [START_REF] Hooman | Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry[END_REF]

where u is the mean velocity and w  the uniform heat flux at the wall transmitted to the fluid. In [START_REF] Tunc | Heat transfer in microtubes with viscous dissipation[END_REF][START_REF] Aydın | Heat and fluid flow characteristics of gases in micropipes[END_REF][START_REF] Hooman | Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry[END_REF], boundary conditions

(2)- [START_REF] Shojaeian | Three-dimensional numerical simulation of the slip flow through triangular microchannels[END_REF] were considered, i.e. full accommodation was assumed, whereas in [START_REF] Jeong | Extended Graetz problem including axial conduction and viscous dissipation in microtube[END_REF], more general boundary conditions (3)- [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF] were considered. Tunc and Bayazitoglu [START_REF] Tunc | Heat transfer in microtubes with viscous dissipation[END_REF]

C Nu C C C C C Br C C           (24) with   2 1 8 2 uu C Kn     (25)
and

      3 2 2 1 TT C Kn Pr        . ( 26
)
For a heated surface ( 0 Br   ), an increase of viscous heating leads to a decrease of the Nusselt number, whereas for a cooled surface ( 0 Br   ), the opposite is observed. The data provided by Tunc and Bayazitoglu are in rough agreement with Eqs. [START_REF] Hooman | Effects of viscous heating, fluid property variation, velocity slip, and temperature jump on convection through parallel plate and circular microchannels[END_REF] and [START_REF] Barron | Evaluation of the eigenvalues for the Graetz problem in slip-flow[END_REF], although the effect of viscous dissipation they predicted is more pronounced for low Knudsen numbers, at least for 0.01 Br   and 0.01 Br   (see Fig. 2). Hooman [START_REF] Hooman | Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry[END_REF] also proposed an analytical solution of ( , ) Nu Kn Pr, Br  , but there is in all likelihood a typo in the equations reported in his paper, which does not allow a precise comparison with the previous studies. The plotted results for 0.01 Br   , however, seems to be in close agreement with those calculated by Eqs. ( 22) and [START_REF] Barron | Evaluation of the eigenvalues for the Graetz problem in slip-flow[END_REF].

The same problem of fully developed slip flow heat transfer with CHF and viscous heating has been recently extended by Çetin and Bayer [START_REF] Çetin | Evaluation of Nusselt number for a flow in a microtube using second-order slip model[END_REF], considering second-order boundary conditions ( 8)-( 13). The analytical solution should be tested with more appropriate values of the velocity slip and temperature jump coefficients  

1 2 1 2
, , ,

u u T T     .
Comments on the values recommended for these coefficients can be found in section 6.

Effects of radial fluid properties variation

Hooman and Ejlali [START_REF] Hooman | Effects of viscous heating, fluid property variation, velocity slip, and temperature jump on convection through parallel plate and circular microchannels[END_REF] recently extended their model, taking into account, in addition to viscous dissipation, the radial variation of the viscosity µ and the thermal conductivity k with the temperature. Their study is based on a perturbation technique, assuming

  1 w      , (27)
 

1 wk kk  , (28)
were  and k  are small parameters and  is the dimensionless temperature difference between the local temperature and the temperature in the fluid at the wall. The model is illustrated with data for 0.

2 k     , 0.7 Pr  , 1 .4 

Fully developed flow and uniform wall temperature

Sparrow and Lin [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF] studied the case of fully developed flow with uniform and constant wall temperature (CWT). The eingenvalue problem was numerically solved, and the derived Nusselt number is provided in Table 2 as a function of coefficients * Barron et al. [START_REF] Barron | Evaluation of the eigenvalues for the Graetz problem in slip-flow[END_REF][START_REF] Barron | The Graetz problem extended to slip-flow[END_REF] and Mikhailov and Cotta [START_REF] Mikhailov | Eigenvalues for the Graetz problem in slip flow[END_REF] studied the same problem, but they did not take into account temperature jump at the wall. As a consequence, they found that the Nusselt number increases with the Knudsen number, due to the slip of velocity (see Fig. 3). As the temperature jump, however, plays the main role in practical cases, a decrease of Nusselt number is expected when rarefaction increases, as predicted by Sparrow and Lin.

Effects of viscous heating

The effect of viscous heating has been taken into account by Tunc and Bayatoziglu [START_REF] Tunc | Heat transfer in microtubes with viscous dissipation[END_REF], Aydın and Avcı [START_REF] Aydın | Heat and fluid flow characteristics of gases in micropipes[END_REF], Jeong and Jeong [START_REF] Jeong | Extended Graetz problem including axial conduction and viscous dissipation in microtube[END_REF] and Hooman [START_REF] Hooman | Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry[END_REF]. The intensity of viscous dissipation is quantified by the Brinkman number

  2 0 w u Br k T T    , ( 29
)
where 0 T is the temperature of the fluid at the tube entrance and k its thermal conductivity. In the case of CHF, the fully developed Nusselt number depends on the value of the modified Brinkman number. Conversely, in the CWT case, when viscous dissipation is taken into account, whatever the Brinkman number, the fully developed Nusselt number is only function of the Knudsen number [START_REF] Jeong | Extended Graetz problem including axial conduction and viscous dissipation in microtube[END_REF][START_REF] Hooman | Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry[END_REF][START_REF] Çetin | Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation[END_REF]. According to the value of the Brinkmann number, however, the distance t from the entrance at which Nu reaches the fully developed value changes [START_REF] Çetin | Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation[END_REF]. Moreover, when 0 Br  , the fully developed Nusselt number can be explicitly obtained, while for 0 Br  , the implicit solution requires a numerical calculation. Jeong and Jeong [20] obtained the analytical solution

  1 C, ,CWT, 0 1 1 3 48 1 4 48 Br C Nu C C C    . (30)
Hooman [START_REF] Hooman | Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry[END_REF] found the same result in the case of full accommodation, for 1 uT   . For 0 Kn  , the solution tends to 48 5 9.6  .

Values obtained for   0;0.04;0.08 Kn  are confirmed by Çetin et al. [START_REF] Çetin | Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation[END_REF] (see Fig. 3). On the other hand, Tunc and Bayatoziglu [START_REF] Tunc | Heat transfer in microtubes with viscous dissipation[END_REF] found a limit value of 6.42, lower than the expected one.   , and for accommodation coefficients 1 uT   . The solution without viscous dissipation is given by Sparrow and Lin [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF], taking into account both velocity slip and temperature jump at the wall. The solution obtained by Barron et al. [START_REF] Barron | The Graetz problem extended to slip-flow[END_REF] does not take into account the temperature jump at the wall, and consequently, erroneously predicts an increase of the Nusselt number with rarefaction, instead of a decrease, at least in the case of full accommodation at the wall. This solution is plotted, however, in Fig. 3, as it allows quantifying the conjugate and opposite effects of velocity and temperature jump. When viscous dissipation is taken into account, the data obtained by Jeong and Jeong [START_REF] Jeong | Extended Graetz problem including axial conduction and viscous dissipation in microtube[END_REF], Hooman [START_REF] Hooman | Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry[END_REF] and Çetin et al. [START_REF] Çetin | Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation[END_REF] in the fully developed region are in close agreement. Although based on the same equations and the same definition [START_REF] Çetin | Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation -An extended Graetz problem[END_REF] of the Brinkman number, the results calculated by the integral transform technique by Tunc and Bayatoziglu [START_REF] Tunc | Heat transfer in microtubes with viscous dissipation[END_REF] are most likely erroneous and are not reported in Fig. 3. Aydın and Avcı [START_REF] Aydın | Heat and fluid flow characteristics of gases in micropipes[END_REF] analysed the effect of the viscous dissipation for the same problem using another definition of the Brinkman number: the temperature difference   0 w TT  in Eq. ( 29) was replaced with   wc TT  , where c T is the temperature on the centerline of the tube. For this reason, they found that the Nusselt number depends both on the Knudsen and on the Brinkman numbers, as in the case of CHF (see section 4.1).

A comparison of the values of fully developed Nusselt number

Effects of radial fluid properties variation

As in the CHF case, Hooman et al. [START_REF] Hooman | Scaling effects for flow in micro-channels: Variable property, viscous heating, velocity slip, and temperature jump[END_REF] took into account the radial variation of viscosity and thermal conductivity with the temperature, according to Eqs. ( 27) and [START_REF] Myong | The effect of gaseous slip on microscale heat transfer: An extended Graetz problem[END_REF]. They obtained an explicit expression of the Nusselt number. For 0.

2 k   , 0.7 Pr  , 1.4
  and 1

uT   , they observed that the effect of properties variation on the Nusselt number is negligible.

Developing flow: the extended Graetz problem

The problem of developing flow has been studied by several authors. It is an extension to the slip regime of the Graetz problem, which considers a hydrodynamically fully developed flow and a thermally developing flow (TDF), the gas experiencing heat transfer at the entrance of the microtube due to a constant heat flux or a constant temperature imposed at the wall. In the Graetz problem, the flow is assumed steady and incompressible, with constant fluid properties, no swirl component of velocity and negligible dissipation effects.

Moreover, an implicit assumption is that of a high Peclet number Pe, which means that the axial conduction is negligible in comparison with the axial convection. Several authors have extended the analysis of Graetz, who initially assumed 0 Kn  , 0 Br  and Pe , by taking into account rarefaction ( 0 Kn  ) and in some cases including viscous dissipation ( 0 Br  ) and axial conduction (finite Pe) effects.

Constant heat flux

Ameel et al. [START_REF] Ameel | Laminar forced convection in a circular tube with constant heat flux and slip flow[END_REF] calculated the velocity profile and the Nusselt number in the entrance region of a microtube for a thermally developing flow with a uniform temperature 0 T at the entrance ( 0 z  ) and a constant heat flux at the wall for 0 z  . They obtained an extension of Eq. [START_REF] Aydın | Heat and fluid flow characteristics of gases in micropipes[END_REF] for total accommodation at the wall. The thermal entrance length e z , defined as the heated length required for the Nusselt number to approach within 5 % the fully developed value, was found to increase with the Knudsen number.

  .

Xiao et al. [START_REF] Xiao | Microtube gas flows with second-order slip flow and temperature jump boundary conditions[END_REF] extended the analysis to the case of second-order slip flow and temperature jump boundary conditions, for hydrodynamically fully developed and thermally developing flow with CHF at the wall. They focused, however, on the specific boundary conditions early proposed by Deissler (Eqs. ( 47) and ( 48)), the coefficients of which are unfortunately not quantitatively correct (see a rapid discussion about these coefficients in section 6).

Effects of viscous dissipation and of axial conduction have been taken into account by Jeong and Jeong [START_REF] Jeong | Extended Graetz problem including axial conduction and viscous dissipation in microtube[END_REF] and by Çetin et al. [START_REF] Çetin | Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation -An extended Graetz problem[END_REF].

Coupled effects of Kn, Pe and Br  on Nu are discussed in [START_REF] Çetin | Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation -An extended Graetz problem[END_REF]. Data found in the fully developed region are in close agreement with those from [START_REF] Aydın | Heat and fluid flow characteristics of gases in micropipes[END_REF][START_REF] Jeong | Extended Graetz problem including axial conduction and viscous dissipation in microtube[END_REF]. The thermal entrance length increases as Pe decreases, due to the increased effect of axial conduction. Increasing the Brinkman number leads to a decrease of the Nusselt number, and the influence of viscous dissipation is less pronounced as rarefaction increases, which confirms what is observed in the fully developed region.

Constant wall temperature

A first analysis of the Graetz problem with constant temperature at the wall can be found in [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF], although the goal of the study from Sparrow and Lin was to determine the fully-developed Nusselt number value, obtained far from the entrance region. Data on Nusselt numbers in the entrance region were provided for the specific case 0.7 Pr  ,

  and 1

uT   . The same problem was more recently treated by Barron et al. [START_REF] Barron | Evaluation of the eigenvalues for the Graetz problem in slip-flow[END_REF][START_REF] Barron | The Graetz problem extended to slip-flow[END_REF], who experienced numerical instabilities issues fixed later by Mikhailov and Cotta [START_REF] Mikhailov | Eigenvalues for the Graetz problem in slip flow[END_REF]. In these papers, however, the authors did not take into account the temperature jump at the wall although they cited the boundary condition [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF]. As a consequence, they neglected the main contribution of rarefaction effects on the heat transfer and found an increase in the Nusselt number as the Knudsen number was increased. This is illustrated in Fig. 3, which shows the fully-developed Nusselt number corresponding to TFDF calculated in [START_REF] Barron | Evaluation of the eigenvalues for the Graetz problem in slip-flow[END_REF][START_REF] Barron | The Graetz problem extended to slip-flow[END_REF]. This issue was later fixed by Larrodé et al. [START_REF] Larrodé | Slip-flow heat transfer in circular tubes[END_REF], who used the first-order boundary conditions [START_REF] Barber | Isothermal slip flow over curved surfaces[END_REF] and [START_REF] Kandlikar | Heat transfer and fluid flow in minichannels and microchannels[END_REF]. Their results were presented in function of   and heat transfer decreases as rarefaction increases.

The effects of viscous dissipation were treated by Tunc and Bayazitoglu [START_REF] Tunc | Heat transfer in microtubes with viscous dissipation[END_REF]. Although the fully developed Nusselt number does not depend on the value of the Brinkman number Br, in the thermally developing region, an increase of Br leads to an increase of Nu. The values provided in [START_REF] Tunc | Heat transfer in microtubes with viscous dissipation[END_REF] should be considered with precaution, however, as the values calculated in the fully developed region differ from those calculated in [START_REF] Hooman | Entropy generation for microscale forced convection: Effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry[END_REF].

A more detailed analysis can be found in [START_REF] Çetin | Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation[END_REF], in which axial conduction is also considered. Taking into account axial conduction complicates the problem due to the presence of the non self-adjoint eingenvalue problem, which makes the linearly independent eingenfunctions non-orthogonal. Çetin et al. [START_REF] Çetin | Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation[END_REF] solved the energy equation numerically by using a finite difference scheme. They observed that for 0 Br  the local Nusselt number Nu increases with decreasing Pe , as it is the case for Nu  . The thermal entrance length also increases with decreasing Pe . For 0 Br  , corresponding to a cooling of the fluid, Nu experiences a jump due to viscous dissipation, the location of which depends on both Pe and Br . The fully developed value Nu  , however, is only function of Kn .

For negative values of the Brinkman number, corresponding to a heating of the fluid, there exists a singular point where Nu goes to the infinity, because the bulk mean temperature of the fluid becomes equal to the wall temperature. Beyond this location, the fluid heats the wall.

The Graetz problem with CWT has also recently been studied by Myong et al. [START_REF] Myong | The effect of gaseous slip on microscale heat transfer: An extended Graetz problem[END_REF], who employed Langmuir boundary conditions ( 9) and ( 14), and compared the results with those obtained from the classic Maxwell boundary conditions ( 3) and [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF]. They found very similar trends, but a slightly more pronounced reduction of the Nusselt number due to rarefaction was observed. As an example, for a Knudsen number 0.02 Kn  and a monatomic gas, the reduction for the fully-developed Nusselt number was 5.7% with Maxwell boundary conditions and 8.3 % with the Langmuir model. In addition, the authors investigated the influence of the axial heat conduction. For 10 Pe  , a significant increase of the Nusselt number is observed in the entrance region, and this effect decreases with increasing rarefaction.

Aziz and Niedbalski [START_REF] Aziz | Thermally developing microtube gas flow with axial conduction and viscous dissipation[END_REF] recently analyzed convective heat transfer with CWT conditions in a hydrodynamically fully developed and thermally developing flow in a microtube. They took into account viscous dissipation and axial conduction effects and compared Deissler second order [START_REF] Sun | Unsteady two-dimensional nitrogen flow in long microchannels with uniform wall heat flux[END_REF] with Maxwell first order (3) slip boundary conditions, but only considering first order (10) temperature jump conditions.

In a recent paper, Satapathy [START_REF] Satapathy | Slip flow heat transfer in an infinite microtube with axial conduction[END_REF] analyzed heat transfer for a thermally developing flow in a microtube with constant wall temperature, assuming uniform fluid velocity (slug flow) and taking into account axial conduction. The temperature boundary conditions were those of Eq. [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF]. There was no discussion, however, on how a slug flow of gas can be observed in a microtube, which limits the practical interest of this study.

Experimental data

There are very few experimental papers dealing with heat transfer with gases in circular microtubes, especially due to the difficulty to control thermal boundary conditions and to access local temperature measurements. Choi et al. [START_REF] Choi | Fluid flow and heat transfer in microtubes[END_REF] published experimental heat transfer data for a flow of nitrogen in microtubes with diameter of 9.7, 53 and 81.2 µm. The gas was heated at a temperature in the order of 60°C before entering the microtube and cooling was due to the lower temperature -approximately 20°C-of the air around the microtube. Thermocouples were mounted at 6 locations along the microtube to determine the bulk temperature distribution. In the laminar regime, a correlation was proposed for the fully developed Nusselt number depending both on the Prandtl and the Reynolds numbers. As the cooling conditions were not precisely controlled, it is difficult to draw sound conclusions on heat transfer from these experiments. Moreover, the inlet pressure was high (between 5.7 and 10 MPa) leading to negligible rarefaction effects with Knudsen numbers lower than 10  .

In a more recent study, Demsis et al. [START_REF] Demsis | Experimental determination of heat transfer coefficient in the slip regime and its anomalously low value[END_REF] provided first data on heat transfer with nitrogen in the slip flow regime. They investigated a range of the Knudsen number from 6.2 10   , however, were measured in the slip flow regime. As commented in [START_REF] Morini | A critical review of the measurement techniques for the analysis of gas microflows through microchannels[END_REF], one possible explanation of this very low value could be due to an erroneous estimation of the outlet bulk temperature linked to a default of insulation between the end of the heated section and the outlet side port.

HEAT TRANSFER IN PARALLEL PLATE MICROCHANNELS

A number of theoretical works have been devoted to heat transfer between parallel plates in the slip flow regime (see Table 3). In this section, the Knudsen number Kn, as well as the modified Brinkman number Br  , are still based on the hydraulic diameter, which is 4 h Dh  for parallel plates 2h away from each other (see Fig. 1c). Consequently, Eqs. ( 18) and ( 21) still hold, provided

Fully developed flow and uniform wall heat flux

Inman [START_REF] Inman | Laminar slip flow heat transfer in a parallel plate channel or a round tube with uniform wall heating[END_REF] calculated the analytical solution of heat transfer for a HFDF and TFDF between parallel plates with one insulated wall (1IW) and one heated wall with uniform heat flux. The boundary conditions were first-order velocity slip (3) and temperature jump [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF] conditions. He obtained the temperature profile and the Nusselt number:

   
            (34)
is the dimensionless temperature jump distance.

For a symmetrically heated microchannel, the solution is

    1 2 PP, ,CHF 2 * 17 84 * 105 * 4 140 1 3 * h h h h Nu              , ( 35
)
with the no-slip limit solution 140 17 8.24  . Zhu et al. [START_REF] Zhu | Analysis of heat transfer between two unsymmetrically heated parallel plates with microspacing in the slip flow regime[END_REF] extended the solution to the case of two unsymmetrically heated walls (UHW) by combining the solutions of two sub-problems similar to the above problem.

Li et al. [START_REF] Li | Wall-adjacent layer' analysis for developed-flow laminar heat transfer of gases in microchannels[END_REF] studied the same problem of fully developed flow with constant heat flux at the wall, assuming a wall-adjacent layer in which the viscosity and thermal conductivity differ from those in the bulk flow. They found for 1 uT   :

      2 PP, ,
   . ( 37
)
for a symmetrically heated microchannel. Although Eqs. ( 36) and [START_REF] Morini | A critical review of the measurement techniques for the analysis of gas microflows through microchannels[END_REF] show the same tendency as Eqs. ( 32) and [START_REF] Demsis | Experimental determination of heat transfer coefficient in the slip regime and its anomalously low value[END_REF], respectively, they underestimate the decrease of the Nusselt number when rarefaction increases (see Fig. 4).

Effects of viscous heating

Effects of viscous heating have been treated by Jeong and Jeong [START_REF] Jeong | Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel[END_REF], who obtained the following analytical expression of the fully developed Nusselt number:

      2 4 PP, ,
C Nu C C C C C Br C C           (38) with   4 1 12 2 vv C Kn     (39)
The same result has been found by Sadeghi and Saidi [START_REF] Sadeghi | Viscous Dissipation and Rarefaction Effects on Laminar Forced Convection in Microchannels[END_REF], who extended the analysis to the case of unsymmetrically heated walls. As for Aydın and Avcı [START_REF] Aydın | Analysis of laminar heat transfer in micro-Poiseuille flow[END_REF], they obtained in the case 1

             (40)
where 5 [START_REF] Colin | Rarefaction and compressibility effects on steady and transient gas flows in microchannels[END_REF] 12

C Kn  . ( 41
)
Eq. ( 40) gives exactly the same result as Eq. ( 38), but it is restricted to the specific case 1 32), ( 35), ( 36), ( 37) and ( 38) are plotted in Fig. 4, for a gas with 0.7 Pr  and 1.4

  , and for accommodation coefficients 1

uT   . Van Rij et al. [START_REF] Van Rij | The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer[END_REF] extended the solution to second-order boundary conditions.

Hooman and Ejlali [START_REF] Hooman | Effects of viscous heating, fluid property variation, velocity slip, and temperature jump on convection through parallel plate and circular microchannels[END_REF] took into account, in addition, the radial variation of the viscosity µ and the thermal conductivity k with the temperature, through Eqs. ( 27) and [START_REF] Myong | The effect of gaseous slip on microscale heat transfer: An extended Graetz problem[END_REF]. They drew similar conclusions as in the case of circular microtubes.

In the previously cited studies, the Nusselt number was deduced assuming that the heat flux at the wall

w T k n      ( 42
)
is fully transmitted by conduction in the fluid. Maslen [START_REF] Maslen | On heat transfer in slip flow[END_REF] early suggested, however, that in case of slip flow, the heat transfer at the wall should be modified to include the shear work done by the slipping fluid and should read

w Tu k µ u nn      . ( 43
)
Following Maslen, Sparrow and Lin [START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF] later suggested to extend their models for slip flow heat transfer in microtubes (Eq. ( 15) and Table 2) taking into account this new expression of the wall heat flux, but without quantifying its consequence on the Nusselt number.

Discussions about the role played by the shear work in slip flow were also conducted by Lockerby and Reese [START_REF] Lockerby | On the modelling of isothermal gas flows at the microscale[END_REF] and

Hadjiconstantinou [START_REF] Hadjiconstantinou | Dissipation in small scale gaseous flows[END_REF]. A clear demonstration of the origin of the additional second term in Eq. ( 43) can be found in [START_REF] Hong | Some considerations on thermal boundary condition of slip flow[END_REF]. This condition has not been, however, often implemented in slip flow heat transfer analysis. Miyamoto et al. [START_REF] Miyamoto | Choked flow and heat transfer of low density gas in a narrow parallelplate channel with uniformly heating walls[END_REF] analyzed the effect of viscous heating in CHF heat transfer between parallel plates in the slip flow regime using Eq. ( 43) and obtained an expression which is different from Eq. ( 38) calculated by Jeong and Jeong [START_REF] Jeong | Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel[END_REF]:

            1 22 1 1 1 2
                             (44)
Eq. ( 44) gives exactly the same results as Eq. [START_REF] Inman | Laminar slip flow heat transfer in a parallel plate channel or a round tube with uniform wall heating[END_REF] in the case of no rarefaction ( 0 Kn  ) and in the case of no viscous dissipation ( 0 Br   ), which was expected as shear work in the Knudsen layer scales with the Brinkman number [START_REF] Hadjiconstantinou | Dissipation in small scale gaseous flows[END_REF]. On the other hand, when rarefaction and viscous heating are both taken into account, the influence of rarefaction is stronger when shear work is taken into account (Fig. 4).

Fully developed flow and uniform wall temperature

The case of fully developed flow with uniform wall temperature was analyzed by Hadjiconstantinou and Simek [START_REF] Hadjiconstantinou | Constant-wall-temperature Nusselt number in micro and nano-channels[END_REF], who took into account axial heat conduction. The boundary conditions were initially Eqs. ( 7)-( 12), but the accommodation coefficients and correcting coefficients were absorbed in Kn, finally leading to the use of Eqs. ( 2)- [START_REF] Shojaeian | Three-dimensional numerical simulation of the slip flow through triangular microchannels[END_REF]. They compared the results from their analytical model with molecular simulations by the DSMC method, and found a good agreement between continuum and molecular results.

They concluded that slip flow models, neglecting viscous dissipation, expansion cooling and thermal creep were able to capture heat transfer. The effect of axial heat conduction was to increase the Nusselt number.

Effects of viscous heating

In the CWT case, when viscous dissipation is taken into account, whatever the Brinkman number, the fully developed Nusselt number is only function of the Knudsen number [START_REF] Jeong | Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel[END_REF]:

  4 PP, ,CWT, 0 4 4 3 140 1 7 140 Br C Nu C C C    , ( 45
)
with a limit at 17.5 when 0 Kn  . Hooman et al. [23] extended this solution, taking into account variable properties according to equations ( 27) and (28).

Extended Graetz problem

The extended Graetz problem was numerically studied by Kavehpour et al. [START_REF] Kavehpour | Effects of compressibility and rarefaction on gaseous flows in microchannels[END_REF] for both CWT and CHF conditions, using a control volume finite difference. They took into account compressibility effects and considered a thermally and hydrodynamically developing flow. They found that the Nusselt number was reduced taking into account rarefaction. Yu and Ameel [START_REF] Yu | Slip-flow Peclet number thermal entry problem within a flat microchannel subject to constant wall temperature[END_REF]   ; 0.7 Pr 

) is a representative value, however, for many engineering applications. In this case, heat transfer is significantly reduced when the temperature jump is taken into account. Mikhailov and Cotta [START_REF] Mikhailov | Mixed symbolic-numerical computation of convective heat transfer with slip flow in microchannels[END_REF] analytically solved the same problem.

Jeong and Jeong [START_REF] Jeong | Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel[END_REF] took in addition into account viscous dissipation and streamwise conduction, for both the CHF and CWT cases.

Chen [START_REF] Chen | Slip-flow heat transfer in a microchannel with viscous dissipation[END_REF] numerically solved the same problem in the CWT case, but found different results. For example, in the fully developed region, the limit value of the Nusselt number calculated by Jeong and Jeong (Eq. ( 45)) is 17.5 and Chen found 16.03. The origin of this deviation is still to be clarified.

Unsteady convection with CHF due to a sudden heat flux change or due to a sudden inlet pressure change were numerically investigated by Sun and Jaluria [START_REF] Sun | Unsteady two-dimensional nitrogen flow in long microchannels with uniform wall heat flux[END_REF], taking into account viscous dissipation and thermal creep effects.

Roughness effects

Roughness effects in the CWT case have been taken into account by Khadem et al. [START_REF] Khadem | Numerical simulation of roughness effects on flow and heat transfer in microchannels at slip flow regime[END_REF], with slip boundary conditions that should be adapted to curved surfaces, but unfortunately as mentioned in section 3, with an inappropriate velocity derivative term leading to Eq. ( 6) instead of the correct Eq. ( 5) . The authors also considered thermal creep in their numerical simulation. The wall roughness was modeled with periodically distributed triangular elements and random shaped micro peaks. It was found that roughness resulted in a decrease of Nusselt number and had more significant effect on higher Knudsen number flows with higher relative roughness. In another paper [START_REF] Hossainpour | Investigation of Fluid Flow and Heat Transfer Characteristics of Gases in Microchannels with Consideration of Different Roughness Shapes at Slip Flow Regime[END_REF], Hossainpour and Khadem investigated the role played by the roughness shape (rectangular, triangular, trapezoidal or make of random peaks) and shown that the Nusselt number is more sensitive to the roughness density than to the roughness shape.

As well as in [START_REF] Khadem | Numerical simulation of roughness effects on flow and heat transfer in microchannels at slip flow regime[END_REF], however, Eq. ( 6) was used instead of Eq. ( 5) and the consequence on the results should be further investigated. Ji et al. [START_REF] Ji | Numerical simulation of wall roughness on gaseous flow and heat transfer in a microchannel[END_REF] considered the same problem with second-order boundary conditions and drew qualitative similar conclusions. Croce and d'Agaro [START_REF] Croce | Compressibility and rarefaction effect on heat transfer in rough microchannels[END_REF] analyzed the competition between rarefaction and compressibility effects for high pressure drop flows in rough microchannels. They took into account viscous dissipation and covered a wide range of the Mach number. The slip boundary condition was similar to Eq. ( 5) but without the thermal creep term:

2 u w u uv uu nt              . ( 46
)
It was found that compressibility was the most dominant effect at high Mach number, yielding even inversion of heat flux, while roughness had strong effects when rarefaction was significant.

Experimental data

There are very few experimental data on convective slip flow heat transfer between parallel plates. The group of Miyamoto [START_REF] Shi | Choked flow of low density gas in a narrow parallel-plate channel with adiabatic walls[END_REF] measured the surface temperature distribution of a slip chocked flow of low-density air through a millimeter-size parallel plate channel with adiabatic walls. The same authors [START_REF] Miyamoto | Choked flow and heat transfer of low density gas in a narrow parallelplate channel with uniformly heating walls[END_REF] later published data on the same problem with uniformly heating walls. In both studies, the parallel-plate channel was 1 mm in depth, 50.5 mm in width and 120 mm in length. The cross-section had an aspect ratio of 0.02, and in that case, 3D effects are not totally negligible but remain limited, typically of the order of 2% [START_REF] Colin | High-order boundary conditions for gaseous flows in rectangular microchannels[END_REF]; for this reason, the experimental data were compared to a parallel plate model which takes into account viscous dissipation and shear work at the wall (Eq. ( 43)).

In the configuration with uniform heat flux at the wall [START_REF] Miyamoto | Choked flow and heat transfer of low density gas in a narrow parallelplate channel with uniformly heating walls[END_REF], the relative deviation between the measured and calculated wall temperature distribution was very small, of the order of 1 or 2% for the reported data, with an increased discrepancy for increasing Knudsen number. This very good agreement between model and experiment was later confirmed by Hong and Asako [START_REF] Hong | Heat transfer characteristics of gaseous flows in micro-channel with negative heat flux[END_REF] with a numerical analysis based on the same hypothesis. In the experiments the Knudsen number, however, was limited to values lower than 5×10 4 at the inlet and 5×10 3 at the outlet for the reported data and in this range slip flow and temperature jump effects are not very significant.

In the configuration with adiabatic walls [START_REF] Shi | Choked flow of low density gas in a narrow parallel-plate channel with adiabatic walls[END_REF], a few data were reported for higher Knudsen numbers, up to 6×10 3 at the inlet and 6×10 2 at the outlet, well in the slip flow regime. There is very good agreement with the simulated wall temperature distribution, taking into account shear work at the wall. On the contrary, when this shear work is not taken into account, the wall temperature can be underestimated with a deviation of a few percents. This analysis confirms that Eq. ( 43) should be used instead of Eq. ( 42) and that most of the analytical and numerical models proposed in the literature for convective heat transfer in gas microflows should be corrected to take into account the shear work at the wall.

HEAT TRANSFER IN RECTANGULAR MICROCHANNELS

Microchannels with rectangular cross-sections are frequently encountered in fluidic microsystems such as micro heat exchangers.

There are easily manufactured, for example, by deep reactive ion etching in silicon wafers. Hydrodynamics of gases in such microchannels in the slip flow regime were analytically modeled, both with first-order [START_REF] Ebert | Slip flow in rectangular and annular ducts[END_REF] and second-order [START_REF] Colin | High-order boundary conditions for gaseous flows in rectangular microchannels[END_REF] boundary conditions. These models have been validated by experiments for simple gases or binary mixtures of gases [START_REF] Pitakarnnop | A novel experimental setup for gas microflows[END_REF][START_REF] Colin | Validation of a second-order slip flow model in rectangular microchannels[END_REF][START_REF] Szalmás | Comparative study between computational and experimental results for binary rarefied gas flows through long microchannels[END_REF]. The problem of heat transfer in rectangular microchannels is rather complicated because it requires 2D and 3D analysis and involves complex thermal boundary conditions, the most frequent of which are classified as [START_REF] Spiga | Nusselt numbers in laminar flow for H2 boundary conditions[END_REF][START_REF] Morini | Analytical determination of the temperature distribution and Nusselt numbers in rectangular ducts with constant axial heat flux[END_REF]: i) Constant temperature at the wall (CWT);

ii) Uniform wall temperature along perimeter at a specified cross-section, with a linear evolution in the streamwise direction (H1);

iii) Constant wall heat flux (H2), with eight versions involving different combinations of heated and adiabatic walls: four heated sides (4), three heated sides and one adiabatic short side (3L), three heated sides and one adiabatic long side (3S), two heated long sides and two adiabatic short sides (2L), two heated short sides and two adiabatic long sides (2S), one short and one long heated sides (2C), one heated long side (1L) and one heated short side (1S).

In addition, the aspect ratio hb of the cross section (see Fig. 1d) is another parameter to take into account. Table 4 gives a summary of recent investigations of heat transfer in rectangular microchannels in the slip flow regime.

Van Rij et al. [START_REF] Van Rij | An evaluation of secondary effects on microchannel frictional and convective heat transfer characteristics[END_REF] have analyzed the role played by various types of boundary conditions, of first order ( 4)-( 11), of classic second order equations such as ( 8)-( 13) and more elaborate second order equations such as the equations proposed by Deissler [START_REF] Deissler | An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases[END_REF], which includes 3D effects that should be considered in rectangular sections:

2 2 2 2 2 2 2 2 9 2 16 u w u u u u u uu n n t s                          (47)
                                 (48)
It should be noted, however, that this form of second-order slip boundary conditions is not the best one. It takes into account 3D

effects but the values of first and second order coefficients should be corrected, taking into account numerical data obtained from the Boltzmann equation. For example, for 2D flows, it has been demonstrated [START_REF] Cercignani | Variational derivation of second-order slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules[END_REF][START_REF] Lorenzani | Higher order slip according to the linearized Boltzmann equation with general boundary conditions[END_REF] that the values of slip coefficients 1 1.11 u   and 2 0.24 u   should be considered for the second-order boundary condition [START_REF] Khadem | Numerical simulation of roughness effects on flow and heat transfer in microchannels at slip flow regime[END_REF] in case of diffuse reflexion for a simple gas, with a mean free path 2 RT p     . As the values of these coefficients directly depend on the definition of the mean path of the molecules, which can takes different forms, it is also possible to define the viscous and thermal slip coefficients using a unique definition of an equivalent mean free path; a discussion and accurate values of these coefficients can be found in [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF]. Corrections of Deissler's coefficients used in Eq. ( 47) have also been proposed in [START_REF] Pitakarnnop | A novel experimental setup for gas microflows[END_REF], showing a better agreement with both kinetic models and experimental data in isothermal slip flows through rectangular microchannels.

HEAT TRANSFER IN TRAPEZOIDAL OR TRIANGULAR MICROCHANNELS

Trapezoidal microchannels are also easily etched in silicon by chemical (for example with KOH) etching. The cross-section is defined by its depth h , its smallest width 2b , and its side angle  (see Fig. 1e). The aspect ratio of the section is

  2 hb  
. Slip flow heat transfer in such channels has been investigated by Niazmand et al. [START_REF] Niazmand | Developing slip-flow and heat transfer in trapezoidal microchannels[END_REF] who considered a developing flow (HDF and TDF) with constant wall temperature. A control-volume based numerical method was employed, assuming first-order boundary conditions ( 2)- [START_REF] Shojaeian | Three-dimensional numerical simulation of the slip flow through triangular microchannels[END_REF] ).

Kuddusi and Çetegen [START_REF] Kuddusi | Thermal and hydrodynamic analysis of gaseous flow in trapezoidal silicon microchannels[END_REF] considered the case of a microchannel etched by KOH in a silicon wafer, for which the crystalline structure of the substrate leads to an angle 54.74

  . The H2-4 constant heat flux problem for thermally developing flow was numerically solved in a square computational domain obtained by transformation of the trapezoidal geometry. Numerical data on the Nusselt number were provided for different values of the aspect ratio. More recently, Kuddusi [START_REF] Kuddusi | First and second law analysis of fully developed gaseous slip flow in trapezoidal silicon microchannels considering viscous dissipation effect[END_REF] analyzed the effect of viscous dissipation in fully developed flows in the same microchannel with H2-3S constant heat flux conditions. He observed that viscous dissipation effects could not be neglected when Triangular microchannel is a specific case of trapezoidal microchannel, for which   . Slip flow heat transfer in isosceles triangular microchannels was investigated by Zhu et al. [START_REF] Zhu | Analysis of the heat transfer in unsymmetrically heated triangular microchannels in slip flow regime[END_REF] who considered unsymmetrical heat conditions: one side insulated with two sides at constant temperature, or two sides insulated and one side at constant temperature. They finally proposed correlations for these two layouts in the case of an equilateral triangular section. Thermal creep effects have been taken into account in the numerical analysis of Shojaeian and Dibaji [START_REF] Shojaeian | Three-dimensional numerical simulation of the slip flow through triangular microchannels[END_REF], who investigated fully developed slip flows in triangular microchannels with various aspect ratios. Their slip flow boundary condition was given by Eq. ( 6) instead of Eq. ( 5) but as it was applied to a fully developed flow in a straight channel, the erroneous derivative ut  , as well as the exact one vt  , should be zero and should not modify the result.

HEAT TRANSFER IN MICROCHANNELS WITH OTHER SECTIONS

Microchannels with annular cross-sections

Slip-flow heat transfer with viscous dissipation in a micro-annulus, with uniform heat flux at one wall and adiabatic conditions at the other wall, was analytically studied by Avcı and Aydın [START_REF] Avci | Laminar forced convection slip-flow in a micro-annulus between two concentric cylinders[END_REF]. The case of uniform heat flux with different values of the flux on the two walls has been analyzed by Sadeghi and Saidi [START_REF] Sadeghi | Viscous Dissipation and Rarefaction Effects on Laminar Forced Convection in Microchannels[END_REF], who took into account viscous dissipation.

The same with constant wall temperature was numerically analyzed by Char and Tai [START_REF] Char | Effects of viscous dissipation on slip-flow heat transfer in a micro annulus[END_REF], taking into account viscous dissipation.

Duan and Muzychka [START_REF] Duan | Slip Flow Heat Transfer in Annular Microchannels With Constant Heat Flux[END_REF] published an explicit analytical solution for the fully developed Nusselt number in annular microchannels,

considering either CHF at one wall and adiabatic conditions at the other wall, or CHF on both walls.

Microchannels with other kinds of cross-sections

A semi-analytical model of fully developed heat transfer with CHF in microchannels with semi-circular cross-section is proposed in [START_REF] Languri | Slip flow forced convection in a microchannel with semi-circular cross-section[END_REF].

Developments on slip flow heat transfer with CWT in microchannels with rhombus section can be found in [START_REF] Shams | Numerical simulation of slip flow through rhombus microchannels[END_REF], whereas considerations on the modeling of slip flow heat transfer in microchannels of arbitrary cross sections are presented in [START_REF] Zhu | Heat transfer for laminar slip flow in a microchannel of arbitrary cross section with complex thermal boundary conditions[END_REF] and [START_REF] Hooman | A superposition approach to study slip-flow forced convection in straight microchannels of uniform but arbitrary cross-section[END_REF].

Developing slip flow CWT heat transfer in wavy or constricted microchannels is analyzed in [START_REF] Shokouhmand | Slip-flow and heat transfer of gaseous flows in the entrance of a wavy microchannel[END_REF] and [START_REF] Shokouhmand | Effects of Knudsen number and geometry on gaseous flow and heat transfer in a constricted microchannel[END_REF], respectively. The boundary conditions ( 4)-( 11) take into account the thermal creep and significant viscous dissipation effects are observed.

CONCLUSIONS AND RECOMMENDATIONS

The literature on slip flow heat transfer is now well stocked, with a series of available analytical solutions or numerical analysis for various heat conditions and microchannel geometries. The limits of the analytical solutions, based on simplifying assumptions, are not, however, always well documented. The influence of variations in gas properties has been analyzed in details [START_REF] Gulhane | Variations in gas properties in laminar micro-convection with entrance effect[END_REF], but there are few studies in the slip flow regime [START_REF] Qazi Zade | Heat transfer characteristics of developing gaseous slip-flow in rectangular microchannels with variable physical properties[END_REF]. When viscous dissipation has to be taken into account, the effect of shear work at the wall is generally not considered, and most of the analytical solutions should be extended to include its non negligible effect. In addition, the most accurate slip flow and temperature jump conditions are often not used, particularly when 3D flow are analyzed and/or when higher-order boundary conditions are considered.

On the other hand, although the theory of gas hydrodynamics in the slip flow regime is now supported by smart experiments [START_REF] Pitakarnnop | A novel experimental setup for gas microflows[END_REF][START_REF] Colin | Validation of a second-order slip flow model in rectangular microchannels[END_REF][START_REF] Ewart | Mass flow rate measurements in gas micro flows[END_REF], there is a crucial lack of experimental data concerning heat transfer in this regime. Providing accurate experimental data on slip flow heat transfer is a challenge for the next years, which would allow a real discussion on the validity of velocity slip and temperature jump boundary conditions, as well as on the limits of applicability of slip flow theory, in terms of degree of rarefaction.

CAPTION LIST FOR FIGURES

Refs. Year

Kind of Work

Heat Conditions

Assumptions BC [START_REF] Inman | Laminar slip flow heat transfer in a parallel plate channel or a round tube with uniform wall heating[END_REF] 1964

A E CHF-1IW HFDF, TFDF (3)
-( 11) [START_REF] Zhu | Analysis of heat transfer between two unsymmetrically heated parallel plates with microspacing in the slip flow regime[END_REF] 2002

A E CHF-UHW HFDF, TFDF (3)

Refs. Year

Kind of Work

Heat Conditions

Assumptions BC [START_REF] Yu | Slip-flow heat transfer in rectangular microchannels[END_REF] 2001

A I CWT HFDF, TDF (3)

v

  velocity component normal to the wall, m s 1Greek symbols  aspect ratio of trapezoidal section, dimensionless    heat flux, W m   angle of trapezoidal section, rad  ratio of specific heats, dimensionless 

  the Nusselt number tends to the classic value 3.66.

  studies cited above is shown in Fig.3, in the case of a gas with a Prandtl number 0

  pressures instead of low hydraulic diameters. The tube was a stainless steel tube with an inner diameter of 25 mm. It was heated by an outer flow of hot water. It was observed that the mean Nusselt number, based on a log mean temperature difference, decreased as Kn increased, which was expected from physical considerations. Very low values down to

4

 4

  considered a thermally developing flow with constant wall temperature, taking into account axial conduction. According to the value of the ratio 11 Tu     , they found an increase or a decrease of heat transfer along the microchannel when rarefaction increases. A transition value of 0.2 separates the region of heat transfer enhancement from the region of heat transfer reduction. The value 1

TABLE 4 :

 4 SUMMARY OF INVESTIGATIONS ON SLIP FLOW HEAT TRANSFER IN RECTANGULAR MICROCHANNELS.

  a fraction of the surface covered by adsorbed atoms, Langmuir's model b half-width of rectangular section or minimal half-width of trapezoidal section

	Br Brinkman number, dimensionless
	Br  modified Brinkman number, dimensionless
	c	p	specific heat capacity at constant pressure, J kg 1 K 1
	h D hydraulic diameter, m
	h		half-depth of parallel plate channel or of rectangular section; depth of trapezoidal section
	k		thermal conductivity, W m 1 K 1
	Kn Knudsen number, dimensionless
	n		normal coordinate, m
	Nu Nusselt number, dimensionless
	p		pressure, Pa
	Pe Peclet number, dimensionless
	Re Reynolds number, dimensionless
	Pr Prandtl number, dimensionless
	0 r	radius of circular section, m
	R		specific gas constant, J kg 1 K 1
	s		spanwise tangential coordinate, m
	t		streamwise tangential coordinate, m
	T		temperature, K
	u		streamwise velocity, m s 1

  1

																u   and	T 	1	  21      : Pr
													6
	Nu		PP, ,CHF,	Br 	0		1	1	 1 84	Br		66	Br	12	Br	12	Kn
							2 5	5 C	35	2 5 C		35	3 5 C	35	4 5 C	  1	Pr

  1

	u   and	T 	1	  21      . For this Pr
	reason, Eq. (38) should be preferred, as it allows using more realistic boundary conditions, i.e. more realistic values of 1 u  and 1 T  ,
	simply modifying the expression of coefficients 3 C and 4 C .
	Nusselt numbers calculated from Eqs. (

  Kn is increased in the entrance region. The following correlation was proposed for the fully developed Nusselt number:

		,	1 Pr  and	 	1.4	. Different channel aspect ratios and side angles were investigated. Heat transfer is significantly reduced as
	 Nu	TR , ,CWT 			0.26 22 4.8exp 3.9       0.21 2.87                    	12 G G	(49)
	with
	1 G	1 1.75  	0.64 Kn	  1 0.72 tanh 2   		(50)
	and
	2 G	 1 0.075 1  			  exp 0.45 	Re Pr		.	(51)
	This correlation is in agreement within 10% with the numerical simulations for all considered aspect ratios (	 	 0.25;0.5;1;2 	) and
	side angles (	 		n	with	n 	 2;3;4;6 

TABLE 1 :

 1 1: Different cross sections of microchannels considered in this review.FIGURE 2: Nusselt number for a fully developed flow in a microtube with uniform wall heat flux as a function of Knudsen number. Summary of investigations on slip flow heat transfer in microtubes.

	Pr 	0.7	,	 	1.4	,	uT  	1
	FIGURE 3: Nusselt number for a fully developed flow in a microtube with uniform wall temperature as a function of Knudsen
	number.	Pr 	0.7	,	 	1.4	,	uT  	1
	FIGURE 4: Fully developed Nusselt number for a parallel plate microchannel flow with uniform heat flux as a function of Knudsen
	number.	Pr 	0.7	,	 	1.4	,	uT  	1
	CAPTION LIST FOR TABLES

TABLE 2 :

 2 Nusselt number for fully developed flow in a microtube with uniform wall temperature, in function of slip flow and temperature jump coefficients. Data derived from Sparrow and Lin[START_REF] Sparrow | Laminar heat transfer in tubes under slip-flow conditions[END_REF].

TABLE 3 :

 3 Summary of investigations on slip flow heat transfer in parallel plate microchannels.

TABLE 4 :

 4 Summary of investigations on slip flow heat transfer in rectangular microchannels.

TABLE 5 :

 5 Summary of investigations on slip flow heat transfer in trapezoidal or triangular microchannels.

TABLE 1 : SUMMARY OF INVESTIGATIONS ON SLIP FLOW HEAT TRANSFER IN MICROTUBES.

 1

	Refs.	Kind	Heat	Assumptions	BC
	Year	of	Conditions	Particular Effects
		Work		taken into account
	[15]	A E	CHF	HFDF, TFDF	(3)-(11)
	1962			+ thermal creep	(4)-(11)
		A I	CWT	HFDF, TFDF	(3)-(11)
	[16]	A E	CHF	HFDF, TFDF	(2)-(10)
	1997	A I		HFDF, TDF
	[17]	A E	CHF	HFDF, TFDF,	no u-slip,
	2000			variable µ and k near	no T-jump
				the wall
	[18]	A I	CHF	HFDF, TDF, viscous	(2)-(10)
	2001		CWT	heating
	[19]	A E	CHF	HFDF, TFDF, viscous	(2)-(10)
	2006		CWT	heating
	[20]	A E	CHF	HFDF, TDF, viscous	(3)-(11)
	2006		CWT	heating, axial
				conduction
	[21]	A E	CHF	HFDF, TFDF, viscous	(2)-(10)
	2007		CWT	heating
	[22]	A I	CHF	HFDF, TFDF, viscous
	2010			heating,  
	[23]	A E	CWT	HFDF, TFDF, viscous	(3)-(11)
	2009			heating,  	kT
	[24-26]	A I	CWT	HFDF, TDF	(2)
	1996-97					no T-jump
	[27]	A I	CWT	HFDF, TDF	(7)-(12)
	2000
	[28]	A I	CWT	HFDF, TDF	(9)-(14)
	2006			axial conduction
	[29]	A I	CHF	HFDF, TDF, viscous	(2)-(11)
	2009			heating, axial conduction
	[30]	N	CWT	HFDF, TDF, viscous	(2)-(11)
	2008			heating, axial
				conduction
	[31]	A I	CWT	HFDF, TDF, axial	slug flow
	2010			conduction		(10)
	[32]	A E	CHF	HFDF, TFDF, viscous	(8)-(13)
	2011			heating
	[33]	A I	CHF	HFDF, TDF	(47)-(48)
	2009
	[34]	A I	CWT	HFDF, TDF, viscous	(3)-(10)
	2011			heating, axial	and
				conduction		(47)-(10)
	[35]	E	Outer	Stainless steel tubes
	2009		heating flow

T  ,   kT (3)-(11) T  ,  

TABLE 3 : SUMMARY OF INVESTIGATIONS ON SLIP FLOW HEAT TRANSFER IN PARALLEL PLATE MICROCHANNELS.

 3

ACKNOWLEDGMENTS

The support of the European Community under grant PITN-GA-2008-215504, 'Gas flows in Micro Electro Mechanical Systems' is gratefully acknowledged.

TABLE 5: SUMMARY OF INVESTIGATIONS ON SLIP FLOW HEAT TRANSFER IN TRAPEZOIDAL OR TRIANGULAR MICROCHANNELS.

Refs. Year

Kind of Work

Heat Conditions

Assumptions BC [START_REF] Niazmand | Developing slip-flow and heat transfer in trapezoidal microchannels[END_REF] 2008

- [START_REF] Hirschfelder | Molecular theory of gases and liquids[END_REF]