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Numerical modeling of three-dimensional compressible gas flow in microchannnels

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2006 J. Micromech. Microeng. 16 292

(http://iopscience.iop.org/0960-1317/16/2/014)

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING

J. Micromech. Microeng. 16 (2006) 292–302 doi:10.1088/0960-1317/16/2/014

Numerical modeling of three-dimensionalcompressible gas flow in microchannnelsV Jain and C X Lin

Applied Research Center, Florida International University, 10555 West Flagler Street,EC 2100, Miami, FL 33174, USA

E-mail: chengxian.lin@arc.fiu.edu

Received 10 August 2005, in final form 9 December 2005Published 9 January 2006Online at stacks.iop.org/JMM/16/292

AbstractNitrogen gas flow in long microchannels with square cross-sections wassimulated numerically with a three-dimensional continuum model with slipand no-slip boundary conditions. The governing equations of the modelwere solved by a control volume method. The numerical model wasvalidated with the available experimental and numerical results. Forincompressible flow, it was found that when Dh was less than 60 µm, a slipboundary condition must be applied. An analytical expression fornormalized friction coefficients, C∗

IC, i.e. the ratio of f Re (slip) to f Re(no-slip), was developed on the basis of incompressible flow behavior. Forcompressible flow, a parametric study was conducted for Dh = 1 µm,L/Dh = 200 and with varying pressure ratios (PR = 1.5–5.0). It was foundthat as the pressure ratio increased from 1.5 to 5.0, compressibility effectsincreased while the rarefaction effects started diminishing. Slip effects alsoplayed an important role in the friction characteristics of microchannelflows. An analytical expression for normalized friction coefficients, C∗

C,i.e. the ratio of f Re (compressible) to f Re (incompressible), was developedon the basis of flow behavior for compressible flow. A comparative study oftwo-dimensional and three-dimensional flows was also conducted, and itwas shown that the two-dimensional assumption for the compressible flowwas not valid since it predicted 15–45% higher flow velocities, and 7–12%lower friction factors than those predicted by the 3D models.

Nomenclature

A area (m2)Dh hydraulic diameter (m)f friction factorKn Knudsen numberH channel height (m)L channel length (m)n normal coordinates pointing inward from the

channelP pressure (Pa)Q flow rate (kg s−1)Re Reynolds numberT temperature (K)U, V, W u, v,w velocities in x, y, z directions

respectivelyum mean velocity (m s−1)

W channel width (m)x, y, z x, y, z coordinatesC∗

IC slip f Re/no-slip f Re

C∗C compressible f Re/incompressible f Re

PR Pin/Pout, ratio of inlet pressure to outlet pressureαv momentum accommodation coefficientαT thermal accommodation coefficientµ molecular viscosity (kg ms−1)ρ fluid density (kg m−3)

Subscripts

g gash hydraulicm meanw wall

0960-1317/06/020292+11$30.00 © 2006 IOP Publishing Ltd Printed in the UK 292

Numerical modeling of three-dimensional compressible gas flow in microchannnels

0 0.01 0.1 1 10 100

Collisionless Boltzmanequation

Discrete or molecular model

Euler Eqs. Continuum model Navier-Stokes

Boltzmann Equation

Conservation Equations do not form a closed set

Figure 1. The Knudsen number limits on the mathematical models (Bird [2]).

1. Introduction

All devices with characteristic dimensions between 1 µmand 1 mm are called micro-devices. Micro mechanicaldevices are rapidly emerging technologies, where newpotential applications are continuously being developed. Thecompactness and high surface-to-volume ratios of microscalefluid devices make them alternatives to conventional flowsystems. With the technological development in micro-flowdevices (MFD), the manufactures of micro-pumps, micro-valves, micro-cold plates, micro-heat exchangers, and othermicro-components and sensors used in flow measurements,electronic cooling, microelectromechanical system (MEMS),chemical analysis and biomedical diagnostics are becomingrealities [1].

These emerging fields have created strong interest inunderstanding the physical aspects of these micro systems andcomponents and the physics governing them. Microchannelfluid flow is currently modeled using either the continuumapproach or the non-continuum approach most commonlycalled molecular approach. The continuum model solvesgoverning equations for macroscopic fluid properties as afunction of spatial coordinates and is used widely for fluidflow applications. As the length scale of a physical systemdecreases, the validity of the standard continuum approachcould diminish.

The Knudsen number, a nondimensional parameter,determines the degree of appropriateness of the continuummodel. It is defined as the ratio of the fluid mean free path λ

to the macroscopic length scale of a physical system �,

Kn = λ

�(1)

where λ = 16µ/(5ρ√

2πRT ), using the Chapman–Enskogresult for the coefficient of dynamic viscosity µ in a hardsphere gas at temperature T. And, � = ρ/∂ρ/∂x, where ρ isthe fluid density. The Knudsen number can also be interpretedbased on the ratio of the Mach number Ma = u/γRT to theReynolds number Re = ρuD/µ as

Kn =√

πγ

2

Ma

Re=

√π

2RT

µ

ρD(2)

where γ is the specific heat ratio, u is the free-streamfluid velocity, R is the specific gas constant and D is thecharacteristic dimension of the system.

The local Knudsen number can be used as a measure ofthe degree of rarefaction of gases encountered in flows througha microchannel. Figure 1 describes different regimes of fluidflow depending on the Knudsen number. As the value ofKnudsen number increases, rarefaction effects become more

important. In this study, we deal with continuum gas flows inthe regimes of both slip and no-slip flows.

Experimental results related to liquid flow inmicrochannels are more easily available than the compressiblegas flow in microchannels [1]. Liquid flow in longmicrochannels becomes fully developed after relatively shortentrance length as compared to the compressible gas flow,which seems to be never fully developed [3, 4]. The liquidflow can be modeled as two-dimensional flow [5], while in thecase of gases, compressibility and rarefaction are two majorfactors governing the gas flow; therefore, two-dimensionalnumerical predictions may not be correct physically.

Morini [1] recently reviewed experimental results ofsingle-phase convective heat transfer in microchannels. Mostof them are related to incompressible liquid flows. Heconcluded that further systematic studies were required togenerate a sufficient body of knowledge of the transportmechanism responsible for the variation of the flowstructure and heat transfer in microchannels. Morini [12]numerically evaluated the trapezoidal and double trapezoidalmicrochannels and suggested that the Navier–Stokes equationshave to be considered valid for microchannels having ahydraulic diameter greater than 30 µm. He reported thatthe deviations in the available results were decreasing dueto better fabrication of microchannels and the increase inreliability/accuracy of the more recent experimental dataquoted in the literature.

For the investigations on compressible gas flows inmicrochannels, the published experimental data are veryfew due to high pressure and the associated difficultiesinvolved in the test set-up [13]. Choi et al [13] measuredthe friction factors of nitrogen gas for both laminar andturbulent flows in microtubes having diameters from 3 to81 µm. They reported that the measured f Re was 53 forlaminar flow, and was smaller than conventional correlationpredictions for turbulent flows. Sobhan and Garimella [3]conducted an exhaustive study of the available experimentaldata. They did comparative study of the experimental resultsand theoretical investigation available in the literature onflows and heat transfer in microchannels. They concludedthat there was no evidence that continuum assumptions wereviolated for the microchannels tested, most of which havehydraulic diameters of 50 µm or more. They reportedthat there was a little agreement between the experimentalresults reported by various scientists. They also reportedthat the results reported by various investigators could notbe distinguished by fluid or substrate type or by microchanneldimensions and shapes. They reported that the informationin the literature did not point out to unequivocal trends invariation or reasons between the flow and heat transfer in

293

V Jain and C X Lin

microchannels and that in channels of conventional sizes.They concluded that a reliable prediction of heat transfers andpressure drop in microchannels was not possible and additionalstudies were necessary. Arkillic and Schmidt [6] conductedanalytic and experimental investigation into gaseous flowwith slight rarefaction through microchannels; they deriveda theoretical formula to predict pressure, velocity and massflow rate of gaseous flow in microchannels. The analyticresults matched well the experimental results. The studydemonstrated that both compressibility and non-continuumeffects were present in microchannel. However this analyticalsolution was a two-dimensional solution, and it can only beused for microchannels with a very small aspect ratio (channelheight/channel width).

Pfalher et al [14–16] experimentally studied the gas flowsof helium and nitrogen in trapezoidal microchannels 100 µmin width and 0.5–50 µm in depth. They reported thatthe measured friction factor was consistently lower than thetheoretical predictions. Pong et al [17] tested the nitrogen andhelium gas flows in rectangular microchannels with a depth of1.2 µm, a width of 5–40 µm and a length of 4500 µm. Withthe observation of nonlinear pressure distribution along themicrochannel, they suggested that this could be due to therarefaction and the compressibility effects, and indicatedthe need of slip boundary conditions for such gas flows inmicrochannels. Harley et al [18] tested the microchannelswith hydraulic diameters ranging from 1.01 to 35.91 µm.They also observed a smaller friction factor with respect to thepredictions by conventional theory. Harley et al explained thisphenomenon by using an isothermal ‘locally fully developed’first order slip flow model.

Chen and Lee [19] simulated flow in microchannels usingNavier–Stokes equations and reported that for a very smallReynolds number, the fluid in a microchannel was indeedisothermal due to the extraordinarily large momentum andheat diffusion in the smallest dimension direction. Howeverthis study was a two-dimensional simulation. Kavehpourand Faghri [20] simulated gaseous flow in two-dimensionalmicrochannels. They found that the Nusselt number andfriction coefficient were substantially reduced for the slipflows compared with the continuum flows. Guo and Wu[21] solved the Navier–Stokes equations to simulate gas flowsin microtubes. They reported that the neglect of the flowcompressibility effect might be one of the reasons for thediscrepancy between experimental results of flow and heattransfer in microtubes among different investigators.

Hsieh et al [22] conducted an experimental and theoreticalstudy of low Reynolds number compressible gas flow in amicrochannel. The results were found in good agreementwith those predicted by analytical solutions. However theyused a 2D continuous flow model although they applied slipboundary conditions. They claimed that using slip boundaryconditions with the 2D model, one could well predict the flowcharacteristics for a three-dimensional physical system. Royet al [24] used finite element based microscopic formulationto efficiently predict the overall flow characteristics up to thetransition regime for reasonably high Knudsen number flowinside two-dimensional microchannels. Their results werecomparable to the numerical and experimental data publishedin the literature. They reported a nonlinear distribution for

pressure and velocity inside the microchannel and as thepressure ratio increased the nonlinearity was also grown.

Male et al [25] simulated gas flow in a squaremicrochannel that was heated from one side. Correlationsfor Nusselt and Sherwood numbers as a function of the Graetznumber were derived for laminar and plug flow conditions.The laminar flow case agrees well with the experimental dataobtained from a 500 µm square microchannel. Chen [5]simulated three-dimensional steady compressible flow in amicrochannel using a reduced form of compressible Navier–Stokes equations. He compared calculated mass flow rateand pressure distribution with a known analytical solutionfor a 2D microchannel flow and experimental data. Chenreported that slip effects played an important role in the frictioncharacteristics of microchannel flow. However, his work waslimited to compressible flow only under slip conditions.

The objectives of the present work were to use the full3D Navier–Stokes equations to understand and predict thecompressibility and rarefaction effects for three-dimensionalgas flow in square microchannels, to investigate the differencebetween slip and no-slip boundary condition effects oncompressible gas flow in square microchannels, and toevaluate the deviation of two-dimensional models from three-dimensional models for square microchannels, which hadnot yet been reported in the literature to the best of theauthors’ knowledge. The simulations were also done to studythe effects of varying hydraulic diameter, effect of varyingchannel length on flow parameters, such as f Re, for the three-dimensional incompressible flow in microchannels.

2. Physical problem description

Nitrogen gas flow in a 3D square microchannel, as shown infigure 2, was studied numerically. For the three-dimensionalsteady laminar incompressible flow in the microchannel,the channel outlet pressure was maintained at atmosphericpressure with a constant pressure ratio of 1.0001. The channellength over the hydraulic diameter ratio was kept as L/Dh =1000 and 200 with the channel hydraulic diameter varyingfrom Dh = 100 to 0.1 µm. As the pressure ratio was verysmall, the compressibility effect was negligible. For thecompressibility and rarefaction effects study in compressibleflows, the microchannel dimensions were L/Dh = 200 andDh = 1 µm. The microchannel outlet pressure was maintainedat atmospheric pressure, with a pressure ratio varying from1.5 to 5.0. For such high pressure ratios, the compressibilityeffects were expected to be crucial. The inlet was located atx = 0 in the coordinate system shown in figure 2. The completephysical model description and fluid properties used in thepresent numerical study are tabulated in table 1.

3. Governing equations

The simulated 3D gas flow and heat transfer is assumed to besteady and laminar. The continuity, momentum and energyequations for the compressible gas flow are expressed asfollows.

Continuity∂ρu

∂x+

∂ρv

∂y+

∂ρw

∂z= 0 (3)

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Numerical modeling of three-dimensional compressible gas flow in microchannnels

x

y

z

L

H W

Gas Flow

Figure 2. Microchannel geometry.

Table 1. Physical model dimensions and fluid properties.

Parameter Range or mean

Working fluid NitrogenLength, L/Dh 200–1000 µmWidth, W = H 0.01–200 µmPressure ratio, PR = Pin/Pout 1.0001, 1.5–5.0Outlet pressure, Pout 101.325 kPaInlet temperature, Ti 297 KWall temperature, Tw 297 KKnudsen number, Kn 0.001–0.065Absolute viscosity, µ 1.663 × 10−5 Ns m−2

Specific gas constant, R 296.7 J kg−1 K−1

Ratio of specific heat, γ 1.4Molecular mass 28.013 kg kmol−1

Momentum

u∂ρu

∂x+ v

∂ρu

∂y+ w

∂ρu

∂z+

∂P

∂y− µ

{∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2

+1

3

(∂2u

∂z2+

∂2v

∂x∂y+

∂2w

∂x∂z

)}= 0

u∂ρv

∂x+ v

∂ρv

∂y+ w

∂ρv

∂z+

∂P

∂y− µ

{∂2v

∂x2+

∂2v

∂y2+

∂2v

∂z2

+1

3

(∂2v

∂y2+

∂2u

∂x∂y+

∂2w

∂y∂z

)}= 0

u∂ρw

∂x+ v

∂ρw

∂y+ w

∂ρw

∂z+

∂P

∂z− µ

{∂2w

∂x2+

∂2w

∂y2+

∂2w

∂z2

+1

3

(∂2w

∂z2+

∂2u

∂x∂z+

∂2v

∂y∂z

)}= 0 (4)

Energy

∇ · (→V (ρE + p))− ∂

∂x

(k∂T

∂x

)− ∂

∂y

(k∂T

∂y

)− ∂

∂z

(k∂T

∂z

)

+2

3µ

(∂u

∂x+

∂v

∂y+

∂w

∂z

)2

− µ

[2

(∂u

∂x

)2

+ 2

(∂v

∂y

)2

+ 2

(∂w

∂z

)2]

− µ

[(∂u

∂y+

∂v

∂x

)2

+

(∂u

∂z+

∂w

∂x

)2

+

(∂v

∂z+

∂w

∂y

)2]

= 0 (5)

Slip boundary conditions [23]

Uw − Ug =(

2 − αv

αv

)Kn

∂U

∂n(6)

Tw − Tg = 2

(2 − αT

αT

)Kn

∂T

∂n(7)

The gas velocity and temperature under the slip boundarycondition were governed by equations as indicated above.All walls were assumed to be adiabatic under slip or no-slip conditions. For the case where flow was consideredincompressible, the density term was considered constant.Pressure boundary conditions were used at both inlet and outletboundaries, with outlet boundary at atmospheric pressure andinlet boundary at specified static pressures.

Numerical method

The governing Navier–Stokes equations were solved withthe finite volume method. The continuity and momentumequations were solved with a second order upwind implicitscheme, with the pressure–velocity coupling by SIMPLEC[28] algorithm. The energy equation was also solved bythe second order upwind implicit scheme. As the hydraulicdiameters of the simulated channels were mostly of the orderof 10−6 m, the scaled convergence criterion was set fourorders less than the hydraulic diameter for all parametercalculations. The 3D microchannel domain was divided intosmall hexahedral mesh volumes for numerical calculations.

Grid independence study, as shown in table 2 for a typicalcase, was performed to ensure that a sufficient number ofgrids had been generated near the walls. The grid of size 30 ×30 × 600 was chosen as an optimum grid, as with the furtherincrease of the grid size it resulted in less than 2% increase inthe average outlet velocity.

4. Results and discussions

The numerical technique was validated first with analytical,experimental and numerical results from the literature, thesimulations were carried out for different hydraulic diametersfor the incompressible gas flow to study the effects ofhydraulic diameter on gas properties, then compressibility andrarefaction effects were studied for the compressible gas flow,and finally the 3D effects were studied.

Numerical technique validation

To validate the numerical technique, the numerical simulationswere first performed on a conventional-size channel [6]. Thechannel height was fixed at 1 cm. The channel width wasvaried to obtain different aspect ratios. The channel lengthwas 1000 times the hydraulic diameter. The working fluidwas nitrogen. The channel outlet pressure was atmospheric

295

V Jain and C X Lin

Table 2. Grid-independent study.

Grid size Vout Difference (%) Grid size Vout Difference (%)

10 × 10 × 600 0.689 014 5.90 30 × 30 × 150 0.734 5011 4.520 × 20 × 600 0.727 9103 0.80 30 × 30 × 300 0.734 2024 2.330 × 30 × 600 0.733 5787 0.60 30 × 30 × 600 0.733 5787 0.140 × 40 × 600 0.735 5692 0.40 30 × 30 × 900 0.733 74 0.0150 × 50 × 600 0.736 4773 0.00 30 × 30 × 1200 0.733 75 0

pressure and the pressure ratio, PR, was 1.0001. The frictionfactor for the flow in conventional size channels was calculatedfor different aspect ratios and the calculated f Re werecompared with the analytical solution and results obtainedby Chen [5] as shown below.

An analytical solution for the fully developedincompressible laminar channel flow is given below [19]; thesymbols um, ρm are the average density and velocity over across section, respectively.

um = − W 2

12µ

dP

dx

[1− 192

π5

(W

H

) ∞∑n=1,3,...

1

n5tanh

(nπH

2W

)]

(8)

The analytical solutions for the mass flow rate Q, friction factorf and Reynolds number Re were calculated using the analyticalum.

Q = ρumA (9)

f =(

−dP

dx

)2Dh

ρu2m

(10)

Re = ρumDh

µ(11)

f Re = 96(1 + W

H

)2

1[1 − 192W

π5H

∑∞n=1,3,5,...

tanh( nπH2W

)

n5

] (12)

The gas flow was considered incompressible since the pressureratio used in the numerical simulations was very small. Thenumerical simulations done were in good agreement withboth the numerical simulations reported by Chen [5] and theanalytical results reported by Arkillic [6], within about 1%.

To further validate the numerical program, we thensimulated compressible microchannel flows and compared thesimulated pressure distribution with the available analyticalsolution [6] and the experimental data [6]. The dimensions ofthis microchannel were 1.2 × 40 × 3000 µm. To comply withthe test conditions, the microchannel outlet pressure was fixedat the atmospheric pressure. The pressure ratio was 2.708 withworking fluid being nitrogen. Due to high pressure, the flowwas considered compressible and the slip effects were alsoconsidered. As shown in figure 3 about the local pressure ratioalong the microchannel, the experimental [6] and numerical[5] results match well present numerical results, thus havevalidated our numerical technique. In the present simulationthe Reynolds number for the flow was 0.104.

Effect of geometrical properties

In the following study of incompressible gas flow in the squaremicrochannels, the geometrical properties such as hydraulic

Figure 3. The longitudinal local pressure (P/Pout) distribution fornitrogen at PR = 2.708.

diameter and channel length were varied and their effects onflow properties were reported. The hydraulic diameter is equalto the height or width of the square microchannel. The pressureratio PR is defined as Pin/Pout.

In the study by Mo et al [23], the square microchannel hadL/Dh = 1000, PR = 1.0001. The channel hydraulic diameterwas decreased from Dh = 100 to 0.1 µm. They reportednumerical results of f Re = 54.8, which differed from theanalytical solution [23] by 3.85%. They reported that the slightdifference could be resulted from the neglect of sidewall effectsin the 2D analytical solutions. Mo et al [23] also reported thatwhen the gas flow was in the slip regime, the friction factorwould reduce due to the slip flow effect compared to the no-slip wall condition. The numerically calculated f Re for slipflow conditions started to decrease at Dh = 60 µm with thedecrease of Dh, where the flow was still in the continuum flowregime.

Present numerical results were obtained for the squaremicrochannel with L/Dh = 200, PR = 1.0001. The channelhydraulic diameter in our case was also decreased from Dh =100 to 0.1 µm to investigate the effect of L/Dh.

In the present study, the normalized friction coefficient,C∗

IC =f Reslip/f Renoslip, was calculated for the incompressiblegas flow as studied by Mo et al [23]. C∗

IC was calculated basedon the constant f Re = 54.8 from the no-slip model [23].As shown in figure 4, our results matched well the previousresults [23]. We can conclude that for the incompressible gasflow reducing the L/Dh from 1000 to 200 had negligible effecton the normalized friction coefficient. So to save expensivecomputational time we could use L/Dh = 200 for the developedincompressible flow calculation in the microchannel. Fromfigure 4, we could also conclude that as the hydraulic diameterincreased the effect of slip boundary conditions reduced.

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Numerical modeling of three-dimensional compressible gas flow in microchannnels

Figure 4. C∗IC versus hydraulic diameter, C∗

IC = f Reslip/f Renoslip.

Table 3. Friction factor for conventional size channel.

f Ref Re f Re (numerical,

Re H/W (numerical [5]) (analytical [5]) present)

116 1 57.08 56.91 57.45212 0.5 62.44 62.19 62.89

Table 4. Normalized friction coefficients comparison.

C∗IC C∗

IC fromDh(µm) numerically equation (13) % Difference

1 0.622 0.625 0.5862 0.765 0.769 0.638

10 0.941 0.943 0.12420 0.969 0.971 0.06540 0.985 0.985 0.05350 0.987 0.988 0.04460 0.989 0.991 0.03380 0.992 0.993 0.026

100 0.994 0.995 0.022

An analytical expression for the normalized frictioncoefficient was developed for the incompressible flow basedon the results in figure 4.

C∗IC = Dh

Dh + c(13)

Within the present investigated parameter ranges, the valuesof constants c were ∼0.6 ± 0.05. In table 4, a comparison ismade between C∗

IC calculated numerically and C∗IC calculated

from our analytical expression from equation (13).

Compressibility and rarefaction effects

To study the compressibility effects for slip and no-slipboundary conditions on the three-dimensional microchannelflows, we simulated flow in square microchannels as indicatedin table 1. The inlet pressure varied from 1.5 to 5.0.

The pressure drop in the channel takes place in order toovercome the frictional forces along the walls. The effect ofslip and no-slip boundary conditions on the pressure variation

Figure 5. The longitudinal pressure distribution at various pressureratios.

(here P/Pout is the ratio of the local pressure to the outletpressure) along the centerline of the channel is shown infigure 5 for compressible flow. It could be observed thatwith the increase in the pressure ratio (PR = Pin/Pout), thedivergence in the pressure distribution between slip and no-slipwall conditions increases. The maximum per cent differencein the pressure distribution between the two wall conditionswas almost 6% at PR = 5.0. The linear pressure distribution ofthe incompressible no-slip flow is also depicted in the figure forcomparison purpose. As shown in figure 5, when the pressureratio was small, the pressure distribution was almost linear forall the flow conditions because the compressibility effect wasnegligible. But with the increase of the pressure ratio, thepressure distribution became more and more nonlinear due tothe compressibility effects. In this study, we found that whenthe pressure ratio PR is larger than 1.5, the compressibilityeffects need to be considered. We could predict that thenonlinearity of the pressure distribution and the correspondingcompressibility effect increased with the increasing pressureratio for the 3D flow conditions. As observed in figure 5, thepressure distribution for the no-slip flow was more nonlinearcompared to the slip flow. The reason could be that theslip flow reduced the wall friction and therefore diminishedthe nonlinearity of the pressure distribution compared to theno-slip flow.

As the pressure ratio PR increased, the mass flow rateincreased nonlinearly for compressible slip and no-slip gasflows, while it increased linearly for incompressible no-slipsimulations, as shown in figure 6. Since as the pressure ratioincreases, the compressibility effect becomes more dominantand thus affects the mass flow rate more substantially. Asobserved in figure 6, the mass flow rate is higher withthe slip boundary conditions than with the no-slip boundaryconditions. This was the result of the lower friction resistancein the microchannel due to slip flow.

Since for the compressible gas flow the density decreasesfrom inlet to outlet, the Knudsen number varies inversely withdensity. Therefore the Knudsen number of a microchannelflow increases along the longitudinal axis and reaches themaximum at the outlet, as shown in figure 7. Figure 7 showsthe local Knudsen number variation from the inlet to the outlet

297

V Jain and C X Lin

Figure 6. Mass flow rates at various pressure ratios.

Figure 7. Local Knudsen number variation along the channellength.

for two pressure ratios for the compressible flow. The behaviorwas similar for all other pressure ratios. The Knudsen numbervariation from the inlet to the outlet for the incompressible gasflow was very minimal and thus can be assumed to be constant.

Figure 8 shows the average Knudsen number behaviorwith increasing pressure ratio. The mass weighted averageKnudsen number was calculated for the microchannel at eachpressure ratio. As could be seen from the figure, with theincrease of the pressure ratio, the average Knudsen numberwas reducing for the compressible case. While for theincompressible case it remains almost constant with the changeof the pressure ratio. The difference between the averageKnudsen number for the slip compressible flow and that for theno-slip compressible flow was negligible. Within the presentinvestigated parameter ranges, it could be observed that as thepressure ratio increased from 1.5 to 5.0, the average Knudsennumber in the microchannel was almost in the no-slip regime,Kn = 0.012.

As the pressure ratio increased, the average mass weightedMach number increased. As shown in figure 9, thecompressible no-slip case had the least average Mach number

Figure 8. Average Knudsen number with varying pressure ratios.

Figure 9. Average Mach number with varying pressure ratios.

than the compressible slip case. At relatively lower values ofPR = Pin/Pout, the average Mach number for the compressibleslip flow was the highest, while at relatively higher valuesof Pin/Pout, the incompressible no-slip flow’s average Machnumber was the highest. The reason could be that as thepressure ratio increased, the compressibility effects becamemore dominant in the microchannel.

A normalized friction coefficient, C∗C = f Recomp/

f Reincomp, was calculated for varying pressure ratios as shownin figure 10. The f Re for compressible flow was normalizedusing f Re for the incompressible no-slip fully developedflow, which was a constant. The compressible no-slip casehad higher friction coefficients and it increased slightly withincreasing pressure ratios. The compressible slip flow hadfriction coefficients lower than those for the compressible no-slip flow. The results showed that the slip effect indeed reducedthe wall friction significantly. The results also proved that thef Re value of slip flow depends strongly on the pressure ratio(or Reynolds number), especially when Pin/Pout was low. Asmentioned earlier, the slip effects decreased with increasingpressure ratio, since at higher pressure ratio the gas densitywas increased and the mean free path of gas molecules was

298

Numerical modeling of three-dimensional compressible gas flow in microchannnels

Figure 10. C∗C versus varying pressure ratios, C∗

C = f Recomp/f Reincomp.

reduced. The Knudsen number and the slip effect diminishwith decreasing mean free path of the gas molecules.

(a) (b)

(d)(c)

in outin out

in out in out

Figure 11. Downstream streamwise velocities at centerline for various pressure ratios, at channel exit.

An analytical expression for the normalized frictioncoefficient was developed for the compressible slip flow basedon the results shown in figure 10.

C∗C = k1 ek2(

PinPout

)−0.5

(14)

where k1 and k2 were constants. For our cases, they werek1 ∼1.31 ± 0.05 and k2 ∼ −0.78 ± 0.02.

Figure 11 shows the streamwise velocity profiles forpressure ratios of 2.0, 3.0, 4.0 and 5.0 at the microchannelexit. Results under both slip and no-slip flow conditions forcompressible flows as well as that under no-slip flow conditionfor incompressible flow are compared. For the pressure ratioless than 3.0, the compressible slip flow conditions had themaximum centerline velocity as shown in figures 11(a) and(b). But as the pressure ratio increased from 3.0 to 5.0the compressible no-slip flow condition had the maximumcenterline velocity as shown in figures 11(c) and (d). Asdiscussed earlier this kind of behavior could be due to thediminishing slip effects with increasing pressure ratio. Thoughthe compressible flow in the microchannel was not fullydeveloped, traditional parabolic types of curves exist as shownin the figures.

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V Jain and C X Lin

in out

in out

in out

Figure 12. Centerline velocity distribution normalized with speedof sound at the inlet Cin for various pressure ratios.

Figure 13. Slip and no-slip U-velocity at different cross-sectionsalong the y-direction.

The velocity variations at the channel centerline alongthe flow direction are plotted in figure 12. The slip conditionsolutions consistently show a higher magnitude of the velocitythan the no-slip condition, due to lower shear stress at the walls.The nonlinear distribution of the velocity was dependent onthe pressure distribution and hence varies at different pressureratios. As the pressure ratios increased, the nonlinearity of thestreaming velocity distribution became prominent.

Figure 13 shows the flow development along themicrochannel for PR = 3.5. As could be observed, theincompressible flow was fully developed but the compressibleflow was not fully developed but ‘developing’ in the flowdirection for both slip and no-slip conditions. Figure 13also shows the increase in the streamwise velocity with thecorresponding rise in the wall velocity due to slip. As Knincreased due to lower density downstream, the slip effects inthe flow direction increased.

Three-dimensional (3D) effects

To investigate the 3D effects, compressible gas flows in 2D and3D microchannels were simulated to compare with each other.

in out

in

in

out

out

Figure 14. Centerline U-velocity distribution for compressible gasflow with slip flow boundary conditions.

Table 5. The 2D and 3D flow U-velocity comparison.

Pin/Pout 2D (U, m s−1) 3D (U, m s−1) % Difference

2.0 6.55 3.79 42.23.0 14.89 9.01 39.494.0 26.50 16.12 39.175.0 40.81 24.24 40.61

Table 6. The 2D and 3D flow V-velocity comparison.

Pin/Pout 2D (V, m s−1) 3D (V, m s−1) % Difference

2.0 1.61 × 10−4 8.84 × 10−5 45.13.0 2.94 × 10−4 1.89 × 10−4 35.74.0 3.83 × 10−4 2.53 × 10−4 33.95.0 4.35 × 10−4 2.92 × 10−4 32.9

The geometric configurations of the microchannel consideredfor this study were L/Dh = 200, Dh = 1 µm. The followingcomparative study was conducted under slip flow boundaryconditions.

Figure 14 compares the velocity solutions along thechannel centerline. As can be observed from the figure, underexactly the same conditions, the U-velocity component alwayshad a higher magnitude for 2D flow than for 3D flow. Thedifference between mean U-velocity components for 3D and2D flows was in the range of 15–20%. The difference betweenU-velocity components for 3D and 2D flows at x/L = 1.0 forvarious pressure ratios is provided in table 5, which shows thedifference up to 42%.

This magnitude of deviation in the velocity componentbetween the 2D and 3D models shows that for compressibleflows the assumption of 2D microchannel is not accurate.Under certain conditions, it may not able to produce acceptableresults as expected. A similar behavior was observed forV-velocity component in the y-direction along the channelcenterline, as shown in figure 15. The difference betweenmean V-velocity components for 3D and 2D flows was inthe range of 20–25%. The difference between V-velocitycomponents for 3D and 2D flows at y/H = 0.25 for variouspressure ratios is provided in table 6, which shows thedifference up to 45%.

300

Numerical modeling of three-dimensional compressible gas flow in microchannnels

in out

in out

Figure 15. V-velocity at centerline cross-section along they-direction for compressible gas flow with slip flow boundaryconditions.

Figure 16. W-velocity at centerline cross-section along thez-direction for compressible gas flow with slip flow boundaryconditions.

Figure 16 shows the W-velocity component along themicrochannel centerline for the 3D case at the exit of themicrochannel. The magnitude of W-velocity, which iscomparable to the V-velocity, increased with the increase ofpressure ratio. The results about the velocity in the z-direction,W-velocity, can only be obtained by a three-dimensionalmodel. This velocity as shown in the figure, which could besubstantial in magnitude under certain conditions, was actuallyneglected in any two-dimensional study.

Figure 17 compares C∗C for 2D and 3D cases. As could be

seen from this figure, the 2D microchannel always gave lowerC∗

C, consistently deviating from the 3D values. The differencein the 3D and 2D results about C∗

C was in the range of 7–12%.This further validated our point that one could not use two-dimensional models to predict accurately the compressibleflow in microchannels that are three-dimensional.

Figure 17. C∗C versus varying pressure ratios, C∗

C = f Recomp/f Reincomp.

5. Conclusions

In this paper, the full 3D Navier–Stokes equations were used toinvestigate the compressibility and rarefaction effects on gasflow in square microchannels under slip and no-slip boundaryconditions, as well as the deviation of two-dimensional modelsfrom three-dimensional models for square microchannels.

The continuum numerical technique employed in thisstudy was validated using conventional channels andmicrochannels with available analytical, experimental andnumerical results.

For the incompressible model, the effect of variation ofL/Dh on f Re or C∗

IC was found negligible. As the hydraulicdiameter was varied from Dh = 0.1 to 100 µm, the frictionfactor increased. An analytical correlation was developed topredict the normalized friction factor variation with change inpressure ratio for incompressible flows in microchannels.

Compressibility and rarefaction effects in the three-dimensional gas flow were studied with Reynolds numbervarying from 0.0001 to 1.2 and pressure ratio varying from1.5 to 5.0, while Knudsen number varying from 0.001 to 0.06.Some of the findings include the following.

• Due to compressibility effects, the pressure drop forthe compressible flow exhibits an unusual nonlinearbehavior as compared to that for incompressible flows.Rarefactions effects were important but as the pressureratio (or Reynolds number) increased, the compressibilityeffects became the dominant factor though the Machnumber was still very small.

• Knudsen number increased from inlet to outlet forcompressible flow, but decreased as the pressure ratioincreased.

• With increasing pressure ratio, the normalized frictionfactor increased for both slip and no-slip flows. Withinthe simulated range of pressure ratios, the f Re forthe compressible slip flow is less than that for theincompressible case.

• An analytical correlation was developed to predict thenormalized friction factor variation with the change ofpressure ratio for the compressible slip flow.

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V Jain and C X Lin

• Two-dimensional assumption was not valid for study ofcompressible flow in three-dimensional microchannelssince it gave much higher velocities (15–45% higher than3D), and much lower normalized friction factor (7–12%lower than 3D).

Acknowledgment

This research is supported by the Air Force Office of ScientificResearch (AFOSR) under grant award F49620-03-1-0314.

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