COMPRESSIBILITY EFFECT ON SLIP FLOW IN NON-CIRCULAR

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Nanoscale and Microscale Thermophysical Engineering, 11: 259–272, 2007 Copyright Ó Taylor & Francis Group, LLC ISSN: 1556-7265 print / 1556-7273 online DOI: 10.1080/15567260701715321

COMPRESSIBILITY EFFECT ON SLIP FLOW IN NON-CIRCULAR MICROCHANNELS Zhipeng Duan and Y.S. Muzychka Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada

Microscale fluid dynamics has received intensive interest due to the emergence of microelectro-mechanical systems (MEMS) technology. When the mean free path of the gas is comparable to the channel’s characteristic dimension, the continuum assumption is no longer valid and a velocity slip may occur at the duct walls. Non-circular cross sections are common channel shapes that can be produced by microfabrication. The non-circular microchannels have extensive practical applications in MEMS. Compressibility effect on slip flow in noncircular microchannels has been examined and simple models are proposed to predict the pressure distribution and mass flow rate for slip flow in most non-circular microchannels. The effects of momentum changes are also considered in the models. The accuracy of the developed models has been examined with some experimental measurements and numerical analysis. The model predictions and experimental data are in good agreement. The developed models may be used to predict mass flow rate and pressure distribution of slip flow in noncircular microchannels. KEY WORDS: slip flow, microchannels, pressure distribution, mass flow rate

INTRODUCTION Fluid flow in microchannels has emerged as an important research area. This has been motivated by their various applications such as medical and biomedical use, computer chips, and chemical separations. The advent of micro-electro-mechanical systems (MEMS) has opened up a new research area where non-continuum behavior is important. MEMS are one of the major advances of industrial technologies in the past decades. MEMS refers to devices that have a characteristic length of less than 1 mm but greater than 1 m, which combine electrical and mechanical components and are fabricated using integrated circuit fabrication technologies. Micron-size mechanical and biochemical devices are becoming more prevalent both in commercial applications and in scientific research. Microchannels are the fundamental part of microfluidic systems. Understanding the flow characteristics of microchannel flows is very important in determining pressure distribution, mass flow rate, heat transfer, and transport properties of the flow. Microchannels can be defined as channels whose characteristic dimensions are from The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Address correspondence to Zhipeng Duan, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Newfoundland, A1B 3X5, Canada. E-mail: [email protected] 259


260

Z. DUAN AND Y.S. MUZYCHKA

NOMENCLATURE A a a b b c Dh f Kn Kn* L Ma m? m* P PO p R Re r ri

2

flow area, m major semi-axis of ellipse or rectangle, m base width of a trapezoidal duct, m minor semi-axis of ellipse or rectangle, m height of a trapezoidal duct, m short side of a trapezoidal duct, m hydraulic diameter, 4A/P Fanning friction factor,

= 12 u2

Knudsen number, /Dh modified Knudsen number, Kn(2-)/ channel length, m Mach number, u/Vs mass flow rate, kg/s normalized mass flow rate perimeter, m Poiseuille number, Dn = u pressure, N/m2 specific gas constant, J/kgK. Reynolds number, uDh/v dimensionless radius ratio ri/ro inner radius of a concentric duct, m

ro T u u Vs x,y z

outer radius of a concentric duct, m temperature, K velocity, m/s average velocity,pm/s ffiffiffiffiffiffiffiffiffi speed of sound, RT Cartesian coordinates, m coordinate in flow direction, m

Greek Symbols constants g ratio of specific heats n eigenvalues " aspect ratio molecular mean free path, m dynamic viscosity, Ns/m2 v kinematic viscosity, m2/s tangential momentum accommodation coefficient average wall shear stress, N/m2 t F half angle, rad momentum flux correction factor Subscripts c continuum i inlet o outlet

one micron to one millimeter. Above one millimeter the flow exhibits behavior that is the same as continuum flows. The non-circular cross sections such as rectangular, elliptic, and trapezoidal are common channel shapes that may be produced by microfabrication. These cross sections have wide practical applications in MEMS [1–3]. The Knudsen number (Kn) relates the molecular mean free path of gas to a characteristic dimension of the duct. Knudsen number is very small for continuum flows. However, for microscale gas flows where the gas mean free path becomes comparable with the characteristic dimension of the duct, the Knudsen number may be greater than 103. Microchannels with characteristic lengths on the order of 100 m would produce flows inside the slip regime for gas with a typical mean free path of approximately 100 nm at standard conditions. The slip flow regime to be studied here is classified as 103 , Kn , 101. LITERATURE REVIEW Rarefaction effects must be considered in gases in which the molecular mean free path is comparable to the channel’s characteristic dimension. The continuum assumption is no longer valid and the gas exhibits non-continuum effects such as velocity slip


SLIP FLOW IN NON-CIRCULAR MICROCHANNELS

261

and temperature jump at the channel walls. Traditional examples of non-continuum gas flows in channels include low-density applications such as high-altitude aircraft or vacuum technology. The recent development of microscale fluid systems has motivated great interest in this field of study. There is strong evidence to support the use of Navier-Stokes and energy equations to model the slip flow problem, while the boundary conditions are modified by including velocity slip and temperature jump at the channel walls. The small length scales commonly encountered in microfluidic devices suggest that rarefaction effects are important. For example, experiments conducted by Pfalher et al. [4, 5], Harley et al. [6], Choi et al. [7], Arkilic et al. [8, 9], Wu et al. [10], and Araki et al. [11] on the transport of gases in microchannels confirm that continuum analyses are unable to predict flow properties in microsized devices. Arkilic et al. [8, 9] investigated helium flow through microchannels. The results showed that the pressure drop over the channel length was less than the continuum flow results. The friction coefficient was only about 40% of the theoretical values. The significant reduction in the friction coefficient may be due to the slip flow regime, as according to the flow regime classification by Schaaf and Chambre [12], the flows studied by Arkilic et al. [8, 9] are nearly within the slip flow regime. When using the Navier-Stokes equations with slip flow boundary condition, the model was able to predict the flow accurately. Araki et al. [11] investigated frictional characteristics of nitrogen and helium flows through three different trapezoidal microchannels whose hydraulic diameter is from 3 to 10 m. The measured friction factor was smaller than that predicted by the conventional theory. They concluded that this deviation was caused by the rarefaction effects. Liu et al. [13] have also proved that the solution to the Navier-Stokes equation combined with slip flow boundary conditions show good agreement with the experimental data in microchannel flow. THEORETICAL ANALYSIS Duan and Muzychka [14] examined slip flow in non-circular microchannels and presented the following expressions for pressure distribution and mass flow rate in non-circular microchannels: p 2 Kno þ ¼ po

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 pi 2 pi 2 2 pi z Kno þ Kno ð1Þ 2 1 þ 2 1 L po po po

where ¼ 11:97 10:59" þ 8:49"2 2:11"3

ð2Þ

and denotes tangential momentum accommodation coefficient, which is usually between 0.87 and 1 [15]. The most usual conditions correspond to 1; therefore, may be assumed have a value of unity. The same procedure is valid even if 6¼ 1, defining a modified Knudsen number as Kn* ¼ Kn(2)/.


262


The mass flow rate in the rectangular microchannel was given by m_ ¼ uA ¼

4ab3 dp dz

" # sinh "n 2" sin2 n n 4 n 4 ð n þ sin n cos n Þ " cosh "n þ 2 n¼1 n 1þ" Kn n sinh "

1 X

ð3Þ

in which the eigenvalues, n, can be obtained from n tan n ¼ 2

1

ð4Þ

4 1þ" Kn

Taking derivative of the Eq. (1) for pressure and evaluating at outlet z ¼ L, we obtain h

m_ ¼

2ab3 po 2 pi 2 LRT po 2

1 þ 2 2 Kno

pi po

1

i

1 þ 2 Kno " # 1 2 X sinh "n 2" sin n n 4 n 4 ð n þ sin n cos n Þ " cosh "n þ 2 n¼1 n 1þ" Kno n sinh "

ð5Þ

Examination of the single term and two terms of the series solutions reveals that the greatest error occurs when Kn ¼ 0.001 and " ¼ 1 in slip regime, which gives a value 2.6 and 0.3% below the exact value, respectively. Therefore, using the first term of the series or two terms at most is accurate enough to obtain the mass flow rate. It can be demonstrated that the limit of Eq. (5) for " ! 0 corresponds to parallel plates channel m_ ¼

2ab3 po 2 pi 2 2 pi Kn 1 þ 24 1 o 3LRT po 2 po

ð6Þ

Similarly, the mass flow rate in the annular microchannel was given by " 2 # rð1 r2 Þ 1 þ 4 2 Kn

r2o r2i r2o dp 2 2 2 1 þ r þ 8ðr r þ 1Þ Kn þ m_ ¼ ð7Þ 8 dz r ln r 2ð1 r2 Þ 2 Kn Taking derivative of the Eq. (1) for pressure and evaluating at outlet z ¼ L, we get h

m_ ¼ "

ðr2o r2i Þr2o po 2 pi 2 16LRT po 2

1 þ 2 2 Kno

1 þ 2 Kno

pi po

1

i

2 # rð1 r2 Þ 1 þ 4 2 2 2 2 Kno Kno þ 1 þ r þ 8ðr r þ 1Þ r ln r 2ð1 r2 Þ 2 Kno

ð8Þ



263

where ¼ 8 when r ¼ 0 and 12 for r . 0.1. The limit of Eq. (8) for r ! 0 reduces to circular tubes m_ ¼

r4o po 2 pi 2 2 pi Kn 1 þ 16 1 o 16LRT po 2 po

ð9Þ

Furthermore, Duan and Muzychka [14] presented the simple expression for mass flow rate that is suitable for most non-circular microchannels 0 B m_ ¼ uA ¼ B @

1

5=2 p2o AP 12RTz

C p2i p2 2 pi p C Kn þ 2 o A p o po p2o p2o

ð10Þ

pffiffi

"ð1þ"Þ 1192" 5 tanhð2"Þ

The definition of aspect ratio proposed by Muzychka and Yovanovich [16] is summarized in Table 1 for a number of geometries. The aspect ratio for regular polygons is unity. The aspect ratio for all singly connected ducts is taken as the ratio of the maximum width to maximum length such that 0 , " , 1. For the trapezoid, annular sector, and the doubly connected duct, simple expressions have been derived to relate the characteristic dimensions of the duct to a width to length ratio. In the above analysis for the pressure distribution and mass flow rate in microchannel flows, momentum changes are neglected as the pressure force is mostly utilized to overcome the friction force against the walls, and very little is spent in accelerating the flow. The effect of the momentum changes will become important when Mach number is increased. The effects of momentum changes on pressure distribution and mass flow rate will be discussed as follows. A similar analysis on pressure distribution was done by Ebert and Sparrow [17] for flow in annular tubes and Sreekanth [18] for flow in circular tubes. As it is exceedingly difficult to solve the Navier-Stokes equations to determine the actual velocity distribution of the compressible gas flow in non-circular microchannels, the flow is assumed to be locally fully developed and isothermal. Because the pressure drop is due to viscous effects and not to any free expansion of the gas, the isothermal assumption should be reasonable. The locally fully developed flow assumption means that the velocity field at any cross section is the same as that of a fully developed flow at the local density and the wall shear stress also takes on locally fully developed values.

Table 1 Definitions of aspect ratio [16] Geometry Regular polygons Singly connected Trapezoid Annular sector Circular annulus

Aspect ratio "¼1 " ¼ ba 2b " ¼ aþc 1r " ¼ ð1þrÞ 1r " ¼ ð1þrÞ


264


The momentum balance on a control volume with axial length dz is given by: Z Adp w Pdz ¼ d u2 dA ð11Þ A

The momentum flux may be written as Z d

u2 dA

¼

A

Z Re2 2 A 1 u 2 dA d Dh 2 A u A

ð12Þ

The wall shear force can be expressed as follows: 1 2 PoRe2 u ¼ w ¼ f 2 Dh 2

ð13Þ

Substituting Eqs. (12) and (13) into (11) and utilizing the perfect gas equation of state p ¼ RT, with some algebraic manipulation, we get p p p Po z d þ Re2 d d ¼0 þ 4Poc Re po po po p=po Poc Dh

ð14Þ

where 1 ¼ A

Z 2 u dA A u

ð15Þ

The Knudsen number may be calculated based on the following formula Kn ¼ ¼ Dh Dh

rffiffiffiffiffiffiffiffiffiffi

2RT

ð16Þ

therefore, ¼

2 RT 2 ¼ Kno 2 po 2 D h 2

ð17Þ

The Poiseuille number reduction depends on the geometry of the cross section. It is convenient that the Poiseuille number results are expressible by the relation Po 1 ¼ Poc 1 þ 2 Kn

ð18Þ

It is clear that Po/Poc and depend on the Knudsen number, or on the pressure. The derivation of is pretty complicated and the expressions for are also very complex and therefore are omitted here. Figures 1 and 2 show the values of as a function of Knudsen number for circular tubes and rectangular microchannels, respectively. It is seen that the variation of with Knudsen number is very gentle. It



265

Figure 1. Momentum flux correction factor for circular tubes.

Figure 2. Momentum flux correction factor for rectangular ducts.

is reasonable that the approximation ¼ constant as is a weak function of Knudsen number. Figures 1 and 2 may be used to choose an estimated value of for most noncircular microchannels for practical application calculation. By noting that pKn is constant for isothermal flow, a reference state that is taken at the outlet can be introduced Kn ¼

Kno po p

ð19Þ


266


Equation (14) may be rearranged to "

p 2 2 Kno þ Re2 Kno 2 po

2 z ¼ 4Poc Re Kno 2 d

Dh

2 Kno 1 þ 2 p=po ðp=po Þ

!# d

p po

ð20Þ

An expression of the pressure difference between some upstream position z and the outlet position z ¼ L can be found by integration Eq. (20) (known outlet conditions) " # p 2 2 p Kno 1 þ 2 1 po po 4 2 po p 16 Lz þ Re2 Kno 2 Kno 1 ln ¼ Poc ReKno 2

po

Dh p

ð21Þ

Similarly, taking a reference state at the inlet, the pressure difference between some downstream position z and the inlet location z ¼ 0 is (known inlet conditions) " # p 2 2 p Kni 1 þ 2 1 pi pi 4 2 2 pi p 16 z 2 þ Re Kni Kni 1 ln ¼ Poc Re Kni 2

pi

Dh p

ð22Þ

Equations (21) and (22) may be applied to any duct. It is seen that variations of do not bring obvious differences in pressure distribution in practical microchannel flows where Re Kn Ma is small. Sreekanth [18] also showed that the approximation ¼ constant had little effect on pressure distribution covered in his experiments. Taking as a constant is completely reasonable for practical microchannel flows application. Integrating Eq. (20) and employing Eq. (10), we obtain 5=2

m_ ¼

p2o AP

n

p2i p2o

2

pp2 þ 2 2 Kno o

pi po

h

io po po pi ppo þ 4 Re2 Kno 2 2 Kno pi p ln p

12RTz pffiffi

"ð1þ"Þ 1192" 5 tanhð2"Þ

ð23Þ Letting z ¼ L gives: 5=2

m_ ¼

p2o AP

n

p2i pi 1 þ 2 2 Kno po 1 p2o

þ 4 Re2 Kno 2

12RTL pffiffi

"ð1þ"Þ 1192" 5 tanhð2"Þ

h io po pi 2 Kn 1 ln o pi po

ð24Þ



267

It is seen that, in microchannel flows, compressibility affects the pressure distribution and mass flow rate primarily through the viscous shear rather than through momentum flux. However, when Re Kn Ma increases, the effect of the momentum flux becomes gradually significant. Especially for low pressure ratio, the effect of the momentum flux is comparatively pronounced. The continuum flow mass flow rate is given by: 0 B m_ c ¼ uA ¼ @

5=2 p2o AP 12RTL

pffiffi

"ð1þ"Þ 1192" 5 tanhð2"Þ

1

2 C pi A 21 po

ð25Þ

The effect of slip may be illustrated clearly by dividing the slip flow mass flow Eq. (24) by the continuum flow mass flow Eq. (25). m_ ¼1þ m_ c

2 2 Kno

pi po

h i po pi 1 þ 4 Re2 Kno 2 2 Kn 1 ln o pi po p2i p2o

ð26Þ

1

It is seen that the rarefaction increases the mass flow for low Mach number microchannel flows and that the effect of rarefaction becomes more significant when the pressure ratio decreases. This could be interpreted as a decrease of the gas viscosity. For comparatively high Mach number microchannel flows, the rarefaction may decrease the mass flow. Combining Eq. (23) and Eq. (24), we obtain the implicit expression for pressure distribution: p2i p2 2 pi p 4 2 2 po po pi 2 Kno Kno þ2 þ Re Kno ln

p2o p2o po po pi p po

2 ð27Þ z pi 2 pi 4 2 2 po pi 2 Kno Kno 1þ2 1 þ Re Kno 1 ln ¼ L p2o

po pi po In the limit of Re Kn ! 0, Eq. (27) reduces to the explicit form Eq. (1). The deviations of the nonlinear pressure distribution from the linear distribution are given by p pi pi z 2 Kno 1 ¼ po po L po sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pi 2 pi 2 2 pi z pi pi z Kno þ Kno þ þ 2 1þ2 1 1 L po L po po po po

ð28Þ

Taking derivative of Eq. (28) and letting it equal zero, we obtain the location of maximum deviation from linearity as 3 ppi þ 4 2 z Kno þ 1

¼ o p L 4 i þ 2 2 Kno þ 1 po

ð29Þ


268


It is seen that the location of maximum deviation from linearity is between 0.5 and 0.75. The location approaches 0.5 for low-pressure ratio and approaches 0.75 for high-pressure ratio. RESULTS AND DISCUSSION The mass flow rate models Eq. (5) and Eq. (24) have been examined using Arkilic et al.’s [9] experimental data. Figure 3 presents the normalized mass flow rate m ¼

_ mRTL 2ab3 po 2

as a function of the pressure ratio. It is found that the predictions agree with Arkilic et al.’s [9] experimental data within 9.8%. It is seen that there is a significant mass flow rate increase due to rarefaction effects from this figure. The experimental data and model predictions are in good agreement. Figure 4 demonstrates the pressure distribution comparison between the proposed model Eq. (27) and Pong et al.’s [19] experimental data. It is found that the model predictions agree with Pong et al.’s experimental data within 2.2%. Figure 5 presents the pressure distribution comparison between the proposed model Eq. (27) and Lui et al.’s [13] experimental results. The model predictions are in agreement with Lui et al.’s experimental results within 1.6%. Jang and Wereley [20] experimentally investigated pressure distributions of slip flow in uniform rectangular microchannels. Figure 6 shows the pressure distribution comparison between the proposed model Eq. (27) and Jang and Wereley’s [20]

Figure 3. Normalized mass flow rate comparison for Arkilic et al.’s [9] experimental data.



269

Figure 4. Pressure distribution comparison for Pong et al.’s [19] experimental data.

Figure 5. Pressure distribution comparison for Liu et al.’s [13] experimental data.

experimental data. The model predictions are in agreement with Jang and Wereley’s experimental data within 0.4%. The corresponding experimental conditions for Arkilic et al., Pong et al., Lui et al., and Jang and Wereley are given in Table 2. From an inspection of the above figures, it is seen that the pressure distribution exhibits a nonlinear behavior due to the compressibility effect. Pressure drop required


270


Figure 6. Pressure distribution comparison for Jang and Wereley’s [20] experimental data.

Table 2 Experimental conditions for microchannel flows Parameter Length (m) Width (m) Height (m) Gas Kno

Arkilic et al. [9]

Pong et al. [19]

Liu et al. [13]

Jang and Wereley [20]

7500 52.25 1.33 Helium 0.08460

3000 40 1.2 Nitrogen 0.02833

4500 40 1.2 Helium 0.05665

5020 104.63 38.57 Air 0.001232

is less than that in a conventional channel. The deviations of the pressure distribution from the linear distribution decrease with an increase in Knudsen number. The nonlinearity increases as the pressure ratio increases. The effects of compressibility and rarefaction are opposite, as Karniadakis et al. [2] demonstrated. When the pressure ratio is very small, the pressure distribution is nearly linear, which is close to an incompressible flow. Karniadakis et al. [2] simulated nitrogen flow in a microchannel. The microchannel is 1.25 m high and 40 m wide. Figure 7 shows the deviation from linear pressure distribution comparison between the proposed model Eq. (28) and Karniadakis et al.’s [2] simulation results. Equation (28) agrees with Karniadakis et al.’s [2] simulation results very well. It is clear that the pressure distribution may be predicted from Eq. (21), Eq. (22), or Eq. (27) according to different known conditions. The pressure distribution models can provide quite accurate results. The mass flow rate may be predicted from Eq. (24) for most non-circular microchannels, provided that an appropriate definition of the aspect ratio is chosen. The maximum deviation of exact values is less than 10%.



271

Figure 7. Deviation from linear pressure distribution comparison for Karniadakis et al.’s [2] numerical data.

CONCLUSION This article investigated compressibility effects on slip flow in non-circular microchannels. Simple models were developed for predicting the pressure distribution and mass flow rate in non-circular microchannels for slip flow. The effects of momentum changes were considered in the models. The effect of the momentum flux becomes gradually significant with an increase in Mach number. The developed pressure distribution and mass flow rate models can provide very accurate results. The accuracy of the developed models has been examined with some experimental measurements and numerical analysis. As for slip flow, no solutions exist for most geometries, and the developed models may be used to predict mass flow rate and pressure distribution of slip flow in non-circular microchannels. REFERENCES 1. M. Ged-el-Hak, MEMS Handbook, CRC Press, Boca¨ Raton, FL, 2001. 2. G.E. Karniadakis, A. Beskok, and N. Aluru, Microflows and Nanoflows, Springer, New York, 2005. 3. N.T. Nguyen, and S.T. Wereley, Fundamentals and Applications of Microfluidics, Artech House, Norwood, MA, 2003. 4. J. Pfahler, J. Harley, H. Bau, and J.N. Zemel, Gas and Liquid Flow in Small Channels, Micromechanical Sensors, Actuators, and Systems ASME, vol. 32, pp. 49–58, 1991. 5. J. Pfahler, J. Harley, H. Bau, and J.N. Zemel, Gas and Liquid Transport in Small Channels, Microstructures, Sensors and Actuators, ASME, vol. 19, pp. 149–157, 1990. 6. J. Harley, Y. Huang, H. Bau, and J.N. Zemel, Gas Flows in Micro-Channels, Journal of Fluid Mechanics, vol. 284, pp. 257–274, 1995. 7. S.B. Choi, R.F. Barron, and R.O. Warrington, Fluid Flow and Heat Transfer in Microtubes, Micromechanical Sensors, Actuators, and Systems ASME, vol. 32, pp. 123–134, 1991. 8. E.B. Arkilic, K.S. Breuer, and M.A. Schmidt, Gaseous Flow in Microchannels, ASME Application of Microfabrication to Fluid Mechanics, vol. FED-197, pp. 57–66, 1994.


272


9. E.B. Arkilic, K.S. Breuer, and M.A. Schmidt, Gaseous Slip Flow in Long Microchannels, IEEE Journal of Microelectromechanical Systems, vol. 6, pp. 167–178, 1997. 10. S. Wu, J. Mai, Y. Zohar, Y.C. Tai, and C.M. Ho, A. Suspended Microchannel with Integrated Temperature Sensors for High Pressure Flow Studies, Proceedings of IEEE Workshop on Micro Electro Mechanical Systems, Heidelberg, Germany, pp. 87–92, 1998. 11. T. Araki, M.S. Kim, I. Hiroshi, and K. Suzuki, An Experimental Investigation of Gaseous Flow Characteristics in Microchannels, Proceedings of International Conference on Heat Transfer and Transport Phenomena in Microscale, Begell House, New York, pp. 155–161, 2000. 12. S.A. Schaaf, and P.L. Chambre, Flow of Rarefied Gases, chap. 1, Princeton University Press, Princeton, NJ, 1958. 13. J. Liu, Y.C. Tai, and C.M. Ho, MEMS for Pressure Distribution Studies of Gaseous Flows in Microchannels, IEEE International Conference on Micro Electro Mechanical Systems, Amsterdam, Netherlands, pp. 209–215, 1995. 14. Z.P. Duan, and Y.S. Muzychka, Slip Flow in Non-Circular Microchannels, Microfluidics and Nanofluidics, vol. 3, pp. 473–484, 2007. 15. W.M. Rohsenow, and H.Y. Choi, Heat, Mass, and Momentum Transfer, chap. 11, PrenticeHall Inc., Englewood Cliffs, NJ, 1961. 16. Y.S. Muzychka, and M.M. Yovanovich, Laminar Flow Friction and Heat Transfer in NonCircular Ducts and Channels: Part I-Hydrodynamic Problem, Compact Heat Exchangers, A Festschrift on the 60th Birthday of Ramesh K. Shah, eds. G.P. Celata, B. Thonon, A. Bontemps, and S. Kandlikar, Edizioni ETS, Pisa, Italy, Grenoble, France, pp. 123–130, 2002. 17. W.A. Ebert, and E.M. Sparrow, Slip Flow in Rectangular and Annular Ducts, Journal of Basic Engineering, vol. 87, pp. 1018–1024, 1965. 18. A. Sceekanth, Slip Flow through Long Circular Tubes, Proceedings of the Sixth International Symposium on Rarefied Gas Dynamics, eds. L. Trilling and H.Y. Wachman, Academic Press, New York, pp. 667–680, 1969. 19. K. Pong, C. Ho, J. Liu, and Y. Tai, Nonlinear Pressure Distribution in Uniform Microchannels, ASME Application of Microfabrication to Fluid Mechanics, vol. FED-197, pp. 51–56, 1994. 20. J. Jang, and S.T. Wereley, Pressure Distributions of Gaseous Slip Flow in Straight and Uniform Rectangular Microchannels, Microfluidics and Nanofluidics, vol. 1, pp. 41–51, 2004.

COMPRESSIBILITY EFFECT ON SLIP FLOW IN NON-CIRCULAR

COMPRESSIBILITY EFFECT ON SLIP FLOW IN NON-CIRCULAR

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