Modeling of gas flows through microchannel

Modeling of gas flows through microchannel configurations S. K. Stefanov, N. K. Kulakarni, and K. S. Shterev Citation: AIP Conference Proceedings 1561, 59 (2013); doi: 10.1063/1.4827214 View online: http://dx.doi.org/10.1063/1.4827214 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1561?ver=pdfcov Published by the AIP Publishing

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Modeling of Gas Flows through Microchannel Configurations S. K. Stefanov, N. K. Kulakarni and K. S. Shterev Institute of Mechanics, Bulgarian Academy of Sciences Acad. G. Bonchev Str., bl. 4, 1113 Sofia, Bulgaria Abstract. The present work is related to the study of the pressure driven isothermal gas flows through different microchannel configurations such as short straight micro-channels, 900 bend channels and T-shaped junctions. The direct simulation Monte Carlo (DSMC) method is used to calculate the gas flow through channel configurations in order to distinguish the flow regimes taking place in long and short microchannels. All the simulations were carried out in three dimensions but some of obtained results concern two dimensional configurations. In this case, the third dimension is confined between two specularly reflecting walls so that the flow in the third direction to be homogeneous. Implicit boundary conditions were applied to maintain a given pressure at inlet and outlet (I/O) boundaries. For small Knudsen number gas flow, the DSMC results are compared to the continuum numerical solution of the Navier-Stokes-Fourier equations obtained by using SIMPLE-TS finite volume method. Keywords: Kinetic theory, rarefied gas, microfluidics, DSMC PACS: 47.61.Fg, 47.45.-n

INTRODUCTION Gas flows are an inherent part of various technological processes and applications that are implemented in microelectro-mechanical systems (MEMS). These microsystems touch almost every industrial field (e.g. fluidic microactuators for active control of aerodynamic flows, vacuum generators for extracting biological samples, mass flow and temperature micro-sensors, pressure gauges, micro heat-exchangers for the cooling of electronic components or for chemical applications etc.). Single microchannels and more complex microchannel configurations are basic elements in such microsystem. In most applications, the flow regime in the microchannels is going beyond the continuum fluid dynamic description and kinetic theory models and methods must be applied for analysis of the gas flows [1-3]. Therefore, the gas flows through the micro-channels have been extensively studied over the past several years. There are different microchannel geometries that were considered to study the gas micro flows by means of simulations, and analytical methods. Some of these geometries are: short and long straight channels, circular pipes, channel bends at various angles (e.g., bend at 90 degrees), zigzag channels, and channels with various cross sectional aspect ratios and shapes such as rectangular, trapezoidal, triangular, hexagonal, etc. [ 4-9] . The concerned results were flow fields, flow rates, heat flux, and shear stress. The studies were conducted under assumed and applied conditions for isothermal/non-isothermal pressure driven flows, thermal driven flows and also shear driven flows. A special attention in these investigations had been paid to the analysis of the slip flow regimes realized in the microchannels [10, 11]. For example, S. Colin has presented models and reviews [12-14] on the validation of a second-order slip flow model in rectangular microchannels, on the rarefaction and compressibility effects on steady and transient gas flows in microchannels, and on the gas microflows in the slip flow regime with a critical review on convective heat transfer. Sharipov and Seleznev [10] have presented a review on the data on the internal rarefied gas flows, for the length ranging from zero to infinity, and for a wide range of Knudsen number. The paper [15] contains a study on the mass flow rate through a long rectangular channel, using the calculations based on the model kinetic equation for the whole range of the Knudsen number and in the wide range of the height-to-width ratio. The paper [16] is related to the study of the gas flow through a zigzag channel using the linearized Kinetic equation. The study was conducted over a wide range of rarefaction and for several cross sectional aspect ratios. Raju and Roy [17] performed simulations using a finite element method to simulate the gas flow in a microchannel with two 90 degree bends. A. Agarwal et al. [4] have used the lattice Boltzmann method to study flow Application of Mathematics in Technical and Natural Sciences AIP Conf. Proc. 1561, 59-67 (2013); doi: 10.1063/1.4827214 © 2013 AIP Publishing LLC 978-0-7354-1189-0/$30.00

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of a gas through a microchannel with a single 90 degrees bend. The computations were performed in twodimensions, for isothermal gas flow, considering some typical Reynolds and Knudsen numbers. G. L. Morini et al. [18] have theoretically studied the gas flow through rectangular, trapezoidal and doubletrapezoidal cross sections. An analysis of the friction factor for incompressible rarefied gas flow through microchannels was investigated with the aim to distinguish the conditions for the dominant rarefaction effects on the pressure drop. They have shown that, it is possible to predict the Knmin , the minimum value of the Knudsen number for which the rarefaction effects can be observed experimentally by taking into account the uncertainties related to evaluation of the friction factor. E.Arkillic et al. [19] have studied gaseous flow through long microchannels using experiments and also using the analytical expressions. A comparison of mass flow obtained from measurements and the analytical study was presented. P. Perrier et al. [20] and T. Ewart et al. [21] have performed experiments and investigated the gas flows in the slip regime, using the measurements, also performed comparative studies between measurement techniques, and between the results obtained from some approximated analytical solutions of the gas dynamics continuum equations and the results from their measurements. They have reported the tangential momentum accommodation coefficient (TMAC) in the 0.003–0.3 Knudsen number range. L.Szalmas et al. [22] have performed a comparative study between computational and experimental results for binary pressure-driven rarefied gas flows through long microchannels. The computations were based on the McCormack kinetic model and the experimental work is based on the Constant Volume Method, and the results are in the slip and transition regime. The molar flow rates of the noble gas (He–Ar) mixture flowing through a rectangular microchannel are estimated for a wide range of pressure drops, and several mixture concentrations of the noble gases. K. Ritos et al. [23] have studied the pressure and temperature driven flows through trapezoidal and triangular cross sections A combination of thermal creep and Poiseuille flow were computed the formulation based on the Shakhov model subjected to Maxwell boundary conditions, and solution was obtained using a finite- difference scheme in Physical space and discrete velocity method in the molecular velocity space. The flow rates in the whole range of Knudsen number, and pressure distribution along the channel were presented. The aim of the present paper is to study pressure driven isothermal gas flows through different three-dimensional microchannel configurations such as short straight micro-channels, 900 bend channels and T-shaped junctions, all with a rectangular cross-section of the microchanels. The direct simulation Monte Carlo (DSMC) code is used to calculate the isothermal flows but actually, close to the outlet a small variation of the temperature of the outgoing gas has been detected what was important in order to distinguish the flow regimes taking place in long and short microchannels. All the simulations were carried out in three dimensions but some of obtained results concern two dimensional configurations. In this case, the third dimension is confined between two specularly reflecting walls so that the flow in the third direction to be homogeneous. Implicit boundary conditions were applied to maintain a given pressure at inlet and outlet (I/O) boundaries. For small Knudsen number gas flow, the DSMC results are compared to the numerical solution of the continual Navier-Stokes-Fourier equations obtained by using SIMPLE-TS finite volume method [24].

PROBLEM FORMULATION AND COMPUTATIONAL CONSIDERATIONS We consider a general formulation of the problem of three-dimensional gas flow through a cross-junction configuration of microchannels with rectangular cross-section as shown in Figure 1. The configuration consist of four microchannel elements (1,2,3,4) with corresponding length Li and cross-section with height H and width W , attached to a central element (0), whose sides might be open or closed by solid walls depending on the configuration that we would like to realize. For example, if the sides to 2nd and 4th elements are closed the realized configuration is a straight channel consisting of elements 1, 0, and 3; by closing sides to 2nd and 3rd elements – vertical bend (elements 1, 0, and 4); by closing the side to the 2nd element, the configuration is a T-shaped junction (elements 1, 0, 3, and 4), etc. The walls of all elements are considered as diffuse reflective at constant temperature Tw . The end faces of elements 1, 2, 3, and 4 determine flow inlets and outlets depending on the boundary conditions imposed on them. In the present paper we restrict ourselves by consideration of pressure-driven flows and the inlets/outlets are kept at constant pressures P1 , P2 , P3 , and P4 , respectively. For small pressure differences the flow through the channel configuration is low speed isothermal one. The model gas consists of hard sphere molecules with mass m and diameter d and with an average initial number density n .

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Z

X

W

Y

H

1 4

0

L 2

3

FIGURE 1. The general 3D-microchannel geometry that is implemented in the DSMC code for simulation of gas flows through different microchannel configurations

The density n corresponds to one of outlet pressures P2 , P3 , and P4 . The basic non-dimensional flow parameter is the Knudsen number Kn defined as the ratio of mean free path O to the characteristic length equal to the channel height H . For the hard sphere model the mean free path is defined as

O

1 2S d 2 n

.

(1)

S 2Kn . However, to compare our The Knudsen number is related to the rarefaction parameter defined as G results to available in the literature data [14, 31], we have calculated the mean of the respective values at the inlet and the outlet of the micro-channel as G (G in  G out ) 2 where G in and G out are related to the corresponding inlet and outlet pressure. The Direct Simulation Monte Carlo (DSMC) method [2] is used for simulation of the gas flow in the microchannel configurations because it allows one to model a complete flow field at the molecular level. The method applies a large number of model particles (simulators) that requires large computer resources. Initially the DSMC was considered as a numerical technique that uses a finite set of model particles (simullators) denoted by their positions and velocities {xi , vi } , i=1,…,N, that move and collide in a computational domain to perform a stochastic simulation of the real molecular gas dynamics. The basic concept of the method is built on a discretization in time and space of the real gas dynamics process and splitting the motion into two successive stages of free molecular motion and binary intermolecular collisions within the grid cells each time step. The second stage of modeling the binary collisions in cells is more complicated. In general, a detailed mathematical description of the motion of a rarefied gas system can be given by an evolutionary kinetic equation in the following non-closed form with respect to the velocity distribution function f  t , x, v : w f  t , x, v =  D ª¬ f  t , x, v º¼  Q ª¬ f (2)  t , x, v, x* , v* º¼ , wt

(2)

where f  t , x, v = f (1) t , x, v and f (2)  t , x, v, x* , v* are one-particle and two-particle distribution functions of the particle velocities v and v* at time t and spacial coordinate x , D denotes a linear differential operator describing the free particle motion and Q is a non-linear integral operator describing the particle binary interactions. W We denote by operators SQW ,h and S D the numerical algorithms approximating the action of the collision and

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convective terms in (2), respectively. Operator SQW ,h acts locally on the particle subset N (l ) in each cell dx(l ) , while W operator S D acts on the total particle set within the entire computational domain D(x) .

If the operator evaluating the solution of (2) at tk  1 from the state at tk is denoted by SQW ,h D then the splitting method is expressed with the approximation SQW ,hD | SDW SQW , h | SQW , h SDW .

(3)

The splitting approximation (3) ilustrated nothing else but the traditional two-stage splitting scheme used in the DSMC method. Thus the algorithmic interpretation of the basic DSMC elements is given by the fllowing generalized flowchart: Start

Set initial state. Set particle indices

t m t W Move all particles. Compute interaction with boundaries Streaming operator

SQW

Reset particle cell Compute collisions in cells Collision operator

SQW ,h

Sample flow properties

Yes

t  Tend No End

FIGURE 2. General flowchart of the DSMC algorithm

Two collision schemes NTC (No Time Counter) (for more details see [2]) and SBT (Simplified Bernoulli trials) (see [25,26]) are applied to simulate the binary collisions in the grid cells. The first is used in calculations on coarser computational grid with large number of simulators per cell and the second in calculations on finer grid with small number of simulators per cell. The goal of the paper is to simulate a flow in straight channel, 900 bend, T-junction, and cross-junction. All simulations were carried out in three dimensions but they could be transformed easily to concern two dimensional configurations. In this case, the third dimension must be confined between two specularly reflecting walls so that the flow in the third direction is homogeneous. Implicit boundary conditions [27-29] are applied to maintain a given pressure at inlet and outlet (I/O) boundaries and maximally avoid the effects arising at the microchannel ends. We denote by subscript “in” all parameters related the upstream conditions at the microchannel end with the largest pressure (the end of element 1 with pressure P1 in our case). The inlet temperature Tin is maintained to be the same throughout the course of simulation. The outlet parameters for elements 2, 3, and 4 are denoted by subscript “out”. The temperature at the outlet is not imposed, rather calculated from the other flow properties, such as pressure and density. According to these conditions, inlet

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pressure Pin and temperature Tin and outlet pressure Pout are considered as known macroscopic variables. Unknown are the velocity profiles V characteristic equation du a

^u, v, w`

at both inlet and outlet and the temperature at the outlet. By using the 1-D

 d U U , where a

dp d U is the sound speed, the following implicit boundary

conditions are obtained for cell j adjacent to the outlet:

Uout , j

mnout , j

Uj 

Tout , j

Uout , j R

Pout  p j a 2j

Pout

(4)

p j  Pout

uout , j

uj 

vout , j

v j , wout , j

U jaj wj ,

where u must be the streamwise component of velocity, local sound speed is equal to a j

J RT j and the

variables with subscript j represent the macroscopic quantities averaged over a given computational time in the boundary cell j . The inlet boundary conditions for density and velocity components are computed as follows: Pin RTin

Uin , j

mnin , j

uin , j

uj 

vin , j

v j , win , j

Pin  p j

(5)

U jaj wj ,

where subscript j represents cells, adjacent to the inlet boundary. The computed macroscopic variables (4) and (5) are used for calculation of the number of injected through the boundary face A j of cell j particles and generation of their velocity. The formula for the number flux through cell-surface A j takes the form: Nj

where n j

nout , j , s j

E j uout , j , E j

n j Aj 2 SEj

1

^exp  s  2 j

`

S s j ª¬1  erf(s j ) º¼

2RTout , j for the outlet boundary and n j

(6) nin, j , s j

E j uin, j , E j

1

2RTin

for the inlet boundary. The implicit boundary conditions (4)-(6) allow considering gas flows through relatively short microchannels by using DSMC calculations and relate the results to those of the limit case of long channel when the macroscopic gradients are small.

NUMERICAL RESULTS The main goal of the present study is to simulate the gas flow through basic three-dimensional microchannel configurations as described in the previous section and try to find the limit of validity of the linear theory for long channels with regard to the channel length. We have focused mainly on two geometries: straight channel consisting of elements 1, 0 and 3 (see Figure 1) and 900 bent micro channel assembled by elements 1, 0 and 4. The calculations have been performed for different length L , where L L1  L0  L3 for straight channel and L L1  L0  L4 for bend, respectively. The cross-section aspect ratio W / H and the rarefaction parameter G are the other two parameters that were varied in a certain range. The results presented in the figures are scaled as follows: distance to height H , velocity to most probable velocity 2 RTin , temperature to Tin Tw , and pressure to Pout . For a straight

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channel with length L L1  L0  L3 9 and square cross-section W / H 1 the velocity and pressure profiles along the center line for a gas flow in slip regime at Knudsen number Kn 0.072 are shown in Figure 3. The DSMC results are compared to the numerical solution of the Navier-Stokes-Fourier equations, subject to first order slip boundary conditions. The continuum solution is obtained by using 3D FVM code SIMPLE-TS developed by Shterev and Stefanov [24]. As one can see, for small Knudsen numbers the agreement between continuum and DSMC solutions is very good. It is worth noting that the pressure distribution along the channel is nonlinear, and for pressure ratio P1 / P3 3.0 the non-linear character of the pressure drop is well-seen in Figure 3(b). If the length of the channel is mapped on the interval (0,1) for given pressure ratio the pressure distribution will be asymptotically same for an arbitrary length of a long enough channel. This fact illustrates the self-similarity of the flow behavior with respect to the channel length and we have used it to make prediction for the asymptotic flow behavior at large lengths with DSMC simulations of the flow through relatively short channels. Typical 3D fields of velocity and pressure in a straight channel and a 90-degree bend with square cross-section, both with length L 7 , are shown in Figure 4. One of the most important parameter of the gas flow through a micro-channel is the mass flow rate Qm through a cross section of the channel. The non-dimensional reduced flow rate G [15] for channels with rectangular crosssection is defined as 1/ 2

G

L § 2kBTin · 2 H W ( Pin  Pout ) ¨© m ¸¹

Qm .

(7)

Note that G does not depend on the channel length. The formula (7) is valid for long channels when the pressure gradient is small. The first our task is to calculate the reduce flow rate for both straight channel and bend for different lengths and find the length when the calculated by DSMC reduced flow rate approaches the theoretical value G obtained from the solution of the linearized BGK equation [15]. The results are shown in Figure 5. Naturally, for small lengths L the reduced flow rate, calculated directly by DSMC method, deviates from the theoretical value for long channel given in paper [15] for the straight channel with square cross-section at G 10.0 .

(a)

(b)

FIGURE 3. Comparison of profiles of velocity (a) and pressure (b) along the centerline of a straight channel with length L L1  L0  L3 9 (here, the length is transformed into interval (0,1)) and square cross-section W / H 1 at

Kn 0.072 and Pin / Pout

P1 / P3

3.0 , obtained by using the 3D continuum SIMPLE-TS code (solid line) and 3D DSMC

code (stars)

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vv

p

0 .3 2 0 .3 0 .2 8 0 .2 6 0 .2 4 0 .2 2 0 .2 0 .1 8 0 .1 6 0 .1 4 0 .1 2 0 .1 0 .0 8 0 .0 6 0 .0 4

2 .9 2 .8 2 .7 2 .6 2 .5 2 .4 2 .3 2 .2 2 .1 2 1 .9 1 .8 1 .7 1 .6 1 .5 1 .4 1 .3 1 .2 1 .1

Z

Z

Y

X

Y

X

Z

Z

Y

X

X

vv 0 .3 4 0 .3 2 0 .3 0 .2 8 0 .2 6 0 .2 4 0 .2 2 0 .2 0 .1 8 0 .1 6 0 .1 4 0 .1 2 0 .1 0 .0 8 0 .0 6 0 .0 4 0 .0 2

Y p 2 .9 2 .8 2 .7 2 .6 2 .5 2 .4 2 .3 2 .2 2 .1 2 1 .9 1 .8 1 .7 1 .6 1 .5 1 .4 1 .3 1 .2 1 .1

FIGURE 4. The velocity (left) and pressure (right) fields of gas flow in a straight channel (top) and bend (bottom) with length L 7 for Pin / Pout P1 / P3 P1 / P4 3.0 and G 10.0

1.34 1.32 1.3 1.28 1.26

G

1.24 straight bend Sharipov

1.22 1.2 1.18 1.16 1.14 1.12 1.1

2

4

6

8

L/H

10

12

14

FIGURE 5. The reduce flow rate G versus normalized length for a straight channel and bend with square cross-section for pressure ratio Pin / Pout 3.0 and rarefaction parameter G (G in  G out ) 2 10.0

In all calculated cases, from L 3 to L 13 , the DSMC data gradually approach from below the theoretical value of the reduced flow rate. Further increase of the channel length showed that the contribution to the flow rate was small and the obtained value at length L 13.0 could be considered as close enough to the limit of the reduced flow rate for long channel. It is interesting to note that for the given parameters the reduced flow rate through a bend is larger than the corresponding rate through a straight channel for all calculated lengths. This observation confirms the results obtained in papers [6,16,30] for 2D bend configurations. The final rates calculated by DSMC remain below the theoretical limit. This fact can be explained by the relatively large pressure ratio used in the calculations and the nonlinear character of the pressure distribution. Actually, the rarefaction parameter calculated as G (G in  G out ) 2

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is correct precisely for a linear pressure profile only. In Figure 3(b) one can conclude easily that the effective rarefaction parameter should take into account the non-linear character of the pressure distribution. The calculated by the DSMC reduced flow rate is closer to the theoretical value for smaller pressure ratio but this must be matter of more detailed investigation in the future. Here we present the limit cases In Table 1, the reduced flow rates calculated for straight channel and bend by the DSMC for different rarefaction parameters G and different cross-section aspect ratio W / H are compared with the theoretical estimations, given in papers [15, 31] for straight channel. From Table 1 a clear trend is seen that the reduced flow rate G is higher for larger aspect ratio W / H and larger rarefaction parameter G . An interesting problem that needs a further analysis concerns the relation between the reduced flow rates at different values of the rarefaction parameter G . For example, at G 20.0 the reduced flow rate G for the bend is smaller than that for the straight channel. However, at G 10.0 and G 5.0 it is slightly larger than the corresponding for the straight channel. TABLE 1. Reduced flow rate G for 3D straight channel and bend for different G , W / H and L / H Rarefaction parameter G 20 20 10 10 5 5 5 5 5 5

Type of channel (H,W) Aspect ratio L/H 15 15 13 13 21 21 21 21 21 21

straight (1,1) 900 bend(1,1) straight(1,1) 900 bend (1,1) straight (1,1) 900 bend (1,1) straight (1,2) 900 bend(1,2) straight (1,5) 900 bend(1,5)

DSMC simulated G 1.931 1.837 1.282 1,301 0.981 0.993 1.409 1.421 1.744 1.752

G from Sharipov [15] 2.0 -1.314 -0.9846 -1.413 -1.753 --

G from Loyalka et al. [ 31] ----0.9885 -----Z

Z

Y

X

(a)

X

Y

vv

p

0 .1 9 0 .1 8 0 .1 7 0 .1 6 0 .1 5 0 .1 4 0 .1 3 0 .1 2 0 .1 1 0 .1 0 .0 9 0 .0 8 0 .0 7 0 .0 6 0 .0 5 0 .0 4 0 .0 3

2 .9 2 .8 2 .7 2 .6 2 .5 2 .4 2 .3 2 .2 2 .1 2 1 .9 1 .8 1 .7 1 .6 1 .5 1 .4 1 .3 1 .2 1 .1

(b)

FIGURE 6. The velocity (a) and pressure (b) fields of gas flow in a T-shaped junction with length L L1  L0  L3 L1  L0  L4 7 for Pin / Pout P1 / P3 P1 / P4 3.0 and G 10.0

The straight channel and the bend are configurations with single inlet and outlet. More complicated is the gas flow through the T-shaped junction shown in figure 6. For the considered pressure ratio Pin / Pout 3.0 ( Pin P1 and Pout P3 P4 ) the T-shaped junction has one inlet and two outlets. Unlike the straight channel, where the bulk velocity increases monotonically along the longitudinal axis, in the T-shaped configuration the bulk velocity (figure 6a) increases along the axis of element 1, slows down in the bifurcation area (element 0) and increases again in the elements 3 and 4.

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CONCLUSIONS We have studied pressure driven isothermal gas flows through different three-dimensional microchannel configurations such as short straight micro-channels, 900 bends and T-shaped junctions, all with a rectangular crosssection. Implicit inlet/outlet boundary conditions have been applied at the channel ends in order to avoid the ends effects and study the transition from short to long channel configuration when the results of the linear kinetic theory can be used. It was found that a channel length of 13 to 15 times of the channel height gave reasonable results that deviated from the theoretical values with 2 to 3%. The presented results are illustrative and have a preliminary character. In a further work are going to analyze in detail the influence of the geometry of short channel configurations on the gas flow characteristics.

ACKNOWLEDGMENTS This work is supported by the NSF of Bulgaria under Grant No DID02/20 – 2009 and the European Commission 7FP HP-SEE - 2010, grant No 261499.

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