Microfluidics, Microscale Fluid Mechanics, MEMS, Microsystems, Micro-Chem-Lab. ; do h m n r t u Uz. (7 tangential momentum accommodation coefficient

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1 > @v&zLMo-dw+7L Proceedngs of ASME FEDSM OO ASME 2000 Fluds Engneerng Dvson Summer Meetng June Boston Massachusetts FEDSMOO- NVESTGATON OF GAS FLOW N 6$J.$ LONG AND NARROW % 6 (vyfj C. Channy Wong Trace L. Zoeller Douglas R. Adkns John N. e s A % 0. ~~m Sanda Natonal Laboratores Engneerng Scences Center Albuquerque New Mexco Tel: (505) Fax (505) E-mal: ccwona@?sanda.aov KEYWORDS:. Mcrofludcs Mcroscale Flud Mechancs MEMS Mcrosystems Mcro-Chem-Lab. ABSTRACT To mnmze the vscous flow losses n a ~crosystem for chemcal analyss we have nvestgated gas flow n long caplhuy tubes and mcrochannels to characterze the flow behavor. Both expermental results and theoretcal predctons ndcate that for gas flow n long and narrow channels as n capllary tubes or rectangular channels compressblty effect s very mportant. Ths leads to a hgher mass flow rate than predcted by the ncompressble flow model. Dfferent computatonal flud dynamcs (CFD) codes have been appled to smulate ths flow problem. Whle some exstng CFD codes have dfllcultes to model ths problem other codes such as MPSalsa predct a converged and reasonably accurate soluton. Ths dffculty may be caused by the numercal soluton technque n these computer codes beng optmzed for ncompressble flow problems rather than for compressble lowspeed flow problems. NOMENCLATURE L Kn Ma P Re T a length Knudsen number Mach number pressure Reynolds number temperature speed of sound ; do h m n r t u Uz z average thermal speed of gas molecules Boltzmann constant dameter or wdth heght molecular mass number densty of gas per unt volume radal dstance tme velocty axal velocty axal dstance Greek $ sothermal compressblty coeffcent mean free path of gas molecules P shear vscosty of gas P densty (7 tangental momentum accommodaton coeffcent Subscrpt nlet o outlet NTRODUCTON Recent development of Mcrosystems for chemcal and bologcal applcatons has opened up a new era of mcroscale flud mechancs research. At Sanda Natonal Laboratores we have developed an ntegrated mcrosystem to detect trace 2000 by ASME

2 1 1 DSCLAMER Ths report was prepared as an account of work sponsored by an agency of the Unted States Government. Nether the Unted States Government nor any agency thereof nor any of ther employees make any warranty express or mpled or assumes any legal lablty or responsblty for the accuracy completeness or usefulness of any nformaton apparatus product or process dsclosed or represents that ts use would not nfrnge prvately owned rghts. Reference heren to any specfc commercal product process or servce by trade name trademark manufacturer or otherwse does not necessarly consttute or mply ts endorsement recommendaton or favorng by the Unted States Government or any agency thereof. The vews and opnons of authors expressed heren do not necessarly state or reflect those of the Unted States Government or any agency thereof. ( 1 t e. ~w> $.>

3 : 8 DSCLAMER 1 Portons of ths document may be llegble n electronc mage products. mages are produced from the best avalable orgnal document.

4 1 chemcal agents for antterrorsm and nonprolferaton applcatons (pchemlabm). Ths pchemlabm s a hand-held unt contanng both lqud and gas phase detecton [Frye- Mason 1998]. One crtcal component n the ~ChernLabm gas phase detecton s the gas chromatography (GC) column for analyte separaton (Fg. 1). The current desgn of thk GC column s an open column that conssts of a long and narrow channel wth a statonary phase coated on the wall (Fg. 2) to provde physochemcal nteracton for the separaton of analyte [Grant 1995]. To ft ths GC column nto the confned space n the pchemlab M (1 cm2) our goal s to etch a rectangular hgh-aspect rato mcrochannel n a slcon wafer. The baselne desgn for ths mcrochannel s 40 pm wde 200 pm deep and 1 m long. Dfferent desgns (deeper wder and longer mcrochannels) have been studed to evaluate the transport and separaton performance [Hudson 1998]. n ths paper our focus s to understand the vscous flow losses by characterzng the gas flow behavor n these mcrochannels and assessng modelng and smulaton capabltes. Gas flow n a long and narrow channel - e.g. a capllary tube wth a dameter of 100 pm total length of 1 m and the length-to-dameter rato of at a subsonc speed s a unque flud dynamcs problem. Even though the flow velocty s much smaller than the speed of sound the compressble effect has to be consdered. Snce the channel s very small the vscous effect becomes domnant and the nerta effect becomes less mportant (but not neglgble as n lqud flow where the Stokes Flow can be appled). The pressure requred to pump gas across the channel ncreases as the cross secton decreases. When the gas s travelng from the nlet whch has a much hgher statc pressure towards the outlet whch has a much lower statc pressure t goes through an enormous change n pressure and densty. To satsfy the law of mass conservaton flow wll accelerate sgnfcantly near the outlet. When studyng ths phenomenon a compressble creep flow we need to consder the compressble effect by ncorporatng a large change n pressure and densty even though gas s travelng at a very low subsonc speed. DMENSONAL ANALYSS For a flud flow problem to determhe whether the flow can be treated as ncompressble conventonal wsdom s to compute the local Mach number. That s f the Mach number s less than 0.3 the flow can be treated as ncompressble [Anderson 1984]. However ths crteron of Mae 0.3 s only a necessary condton but t s not a sufllcent condton that the assumpton of an ncompressble flow s always vald [Panton 1996]. A dmensonal analyss can be performed to nvestgate what condtons are necessary for gas flow n mcrochannels to be treated as ncompressble. n many flud flow problems densty can be expressed as: p=p~+ap where p. s a reference densty; and Ap s the local departure from ths reference densty. f AppO<Cl the term p(dudt) n the momentum equaton can be wrtten as pjdudt). A smlar expresson can be made to those terms n the contnuty and scalar transport equatons that contan the densty varable. Ms s the procedure to treat the flow as ncompressble [Trtton 1989]. For an sothermal or adabatc gas flow problem the densty varaton wth respect to ts reference densty s equvalent to the pressure varaton as follows: AplpO- PAP (1) where ~ s the sothermal compressblty coeffcent expressed as: The crteron of Ma < 0.3 s derved by performng dmensonal analyss and comparng the nerta term wth the pressure gradent term -.e. AWL - pau2l. For gas flow n a long and narrow channel the vscous effect can be domnant t scales nversely to the square of the characterstc length. Thus as the sze of channel decreases t s necessary to consder the pressure term versus the vscous term. Performng a dmensonal analyss and consderng the pressure term versus vscous term t wll lead to the followng expresson: AWL - pazddo2 (3) where do and L are the dameter and length of the tube respectvely. Further analyss wll produce the followng expresson hence AP -.@ do )(Au do); (4) AplpO- AP(pa2) - (Udo)(A4a2LRe). (5) Therefore when the vscous effect becomes domnant as n gas flow n a long and narrow capllary tube t s necessary to consder a modfed crteron (Ldo)(Ma2A?e)e< 1 n order to determne whether the flud can be treated as ncompressble. For the gas flow n a mcrochannel beng consdered n the pchemlabm a l-m long capllary tube wth an nner dameter of 100 pm and a velocty of 10 ms the Reynolds number wll be 55.6 and the Mach number wll be less than 0.3. Wth Udo = Ltd(f the modfed crteron s: (Zdo)(Ma2~e) = Hence the flow n ths problem cannot be treated as ncompressble. From the knetc theory of gas the dynamc bulk vscosty of the gas can be expressed as: p= ncmlj3 (6) where n s the number densty of molecules per unt volume m s the molecular mass X s the mean free path and c s the average thermal speed of gas molecules gven as: c = {8kT(zm)}0 5. (7) 2000 by ASME..=

5 t By substtutng these two expressons nto the modfed crteron the crteron can be rewrtten as: Apl p. - (Zdo)(Ma.Kn). (8) Hence by checkng the Mach number the Knudsen number and the length-to-dameter rato we can determne f the assumpton of ncompressble flow s vald for the problem of nterest. For a l-m long tube wth 100 ~ dameter a flow velocty of 10 nds wll lead to: (L4do)(Ma.Kn) = 0.2. Hence the assumpton of an ncompressble flow wll not be vald for thk stuaton. The dmensonal analyss presented here has demonstrated that gas flow n a long and narrow channel has a unque characterstc that t s necessary to consder the compressblty effect when analyzng ths problem. n addton to consderng compressblty analyzng gas flow n mcrodomans may requre analysts to model nerta pressure and vscous effects altogether. All these effects can be equally mportant. One nterestng fndng regardng the nonlnearty of the axal pressure dstrbuton along the capllary tube s that the nonlnear behavor appears only when the compressblty effect becomes mportant. f the tube dameter s relatvely large - e.g. greater than mm - the flow can be treated as ncompressble and the pressure dstrbuton s lnear. However f the tube dameter s small enough that t s comparable wth the mean free path of gas the flow wll beat the slp flow regme. The slp velocty at the wall s as follows: u~r=ro) = (2-0)10* A * (6 u. lf3r).wll (12) Under thk condton the pressure dstrbuton along the tube wll be dfferent because of the addtonal terms wth respect to the slp at the wall. P(Z) = Cop. + [(C#O)2 + (1-Z?L).2C#P~ + (22 L).2c#.2 + (1-m). P? + (z). Po2]o 5 where CO=8 * (2-0)0 * Kno (13) AN ANALYTCAL COMPRESSBLE FLOW MODEL Mass Flow - Pressure Drop Relatonshp Assumng a fully developed lamnar flow n a long capllary tube f only consderng the global densty varaton but not the local densty varaton wthn the nfntely small secton one can ntegrate the governng momentum equaton to obtan the average flow rate as a functon of pressure drop. Hence the relatonshp between the mass flow rate and the pressure drop across the capllary tube can easly be obtaned. where and AmAt = nd:(256pl)*(pjpj *(Pt - P02) (9) P and P. are nlet and outlet pressure respectvely; p. s the outlet denst~ p s the shear vscosty of gas; do s the tube dameteq L s the total length of the capllary tube. Clearly ths expresson mples a nonlnear relatonshp between the mass flow rate and pressure drop across the capllary tube. Only f the pressure drop s small when compared wth the outlet pressure the expresson can be smplfed to show a lnear relatonshp as n the analytcal ncompressble flow model: AmlAt = ~do~(128@)*(po) *(P- Po). Pressure Dstrbuton along the Capllarv Tubes (lo) By relatng the local pressure gradent wth the local average velocty across the tube and ntegratng the pressure gradent from the tube entrance (z=o) to an axal locaton (z.=z) the pressure dstrbuton along the capllary tube can be found. P(z) = [(l-z)* P? + (m) * Po2]05. (11) EXPERMENTAL MEASUREMENT A few flow measurement data for mcrochannels exst [Arkllc 1993; Arkllc 1997; Shh 1996; Cho 1991; Baley 1995]. Most of the data are flow measurements for short (n mm or less) but shallow mcrochannels etched n slcon wafers; hence ther length-to-dameter rato s always less than 500. A wde range of channel sze wth dfferent shapes has been studed. One common fndng of these studes s that even though the flow s wtln the contnuum flow regme the compressble effect can be mportant and the streamwse pressure dstrbuton s nonlnear. Snce the channel sze for pchemlabm s moderately dfferent from the exstng data (relatvely bgger and much longer) t wll be worthwhle to conduct addtonal gas flow experments. n addton to acqurng more data there s a desre to valdate the above anrdyss of compressble effect on gas flow n a long and narrow channel. We have conducted experments usng capllary tubes and mcrochannels etched n a slcon wafer. For the flow n capllary tubes three seres of experments have been performed usng and 30- mcrometer dameter capllary tubes that are one meter long. Ntrogen argon and helum gases have been studed. The setup of the experment s shown n Fgure 3. The test procedure s as follows: the nlet gas pressure s read; then the gas s fed through the capllary tube and the flow rate s measured usng a bubble flow meter and a stopwatch. Bubble flow meters have been used as a standard n calbratng chromatography equpment. A soap bubble s njected nto a precson bore tubq gas from the test stream moves the bubble between two markers on the tube. Fve tube dameters have been appled to ensure that an accurate tmng can be made of the bubble translaton. The uncertanty n flow rate (whch prmarly s attrbuted to tmng) s less than 1 percent. The tests 2000 by ASME

6 : are repeated several tmes and an average value s calculated for each dfferent pressure drop across the mcrochannel. Pressure drops of 34 to 310 cpa (5 to 45 ps) have been nvestgated. The data then wll be compared to the analytcal models for each case. TEST RESULTS. Effect of Tube Dameter Gas flow n dfferent mcrochannels has been studed. Our objectve s to measure the mass flow rate versus pressure drop for dfferent geometry (fhsed slca capllary tubes wth varous dameters). Fgure 4 shows the results for gas flow n these capllary tubes wth nner dameters of and 200 pm respectvely and a length of 1 m. The behavor of the curves (n the sem-logarthmc plot) ndcates that the relatonshp between the mass flow rate and the pressure drop s nonlnear. Effect of Dfferent Gases Measurement has been made for three dfferent gases (ntrogen helum and argon) flowng through a long and narrow channel. Fgure 5 shows the ext volumetrc flow rate versus pressure drop. These volumetrc flow rate data reveal that ther behavor follows closely wth the vscosty of gas as n Equaton (3). The hgher the vscosty the smaller the flow rate wll be. The vscosty of the gases examned s as follows: 17.6 ppa-sec for ntrogen 19.7 PPa-sec for helum and 22 ppa-sec for argon respectvely. COMPARSON BETWEEN DATA AND MODEL Expermental data for gas flow n the l-m long capllary tubes wth varous nner dameters has been used to assess the predctons from the analytcal model. Only data for the tube wth nner dameter of 30 pm wll be shown here. Fgures 6 and 7 compare the measured and predcted mass flow rate of ntrogen and helum as a functon of pressure drop across the capllary tube. Ths comparson ndcates that compressble effect s an mportant factor for gas flow n mcrochannels and t needs to be consdered n any gas flow analyss. Assumng an ncompressble flud flow wll lead to an naccurate predcton; t wll sgnfcantly underpredct the flow rate. COMPARSON BETWEEN DATA AND COMPUT- ATONAL SMULATONS Snce the present desgn of the gas chromatography column for the pchemlabm s a hgh-aspect rato rectangular channel etched. n a slcon wafer n a spral confguraton analyzng gas flow n ths complex geometry (.e. 3-D curvature effect) requres an advanced computatonal flud dynamcs (CFD) code. However the analytcal model presented here can stll be used as a valuable tool to gude the desgn and optmzaton. To assess our smulaton capablty we have analyzed ths gas flow n capllary tubes problem wth dfferent advanced computatonal flud dynamcs (CFD) codes whch can solve the full Naver-Stokes equatons. Among all of the computatonal smulatons that we have performed those smulatons usng the MPSalsa code show promsng results (Table 1). The MPSaka predctons are consstent wth the compressble flow model (Fgure 8). MPSalsa predcts a slghtly hgher mass flow rate than the analytcal compressble flow model because the code models the flow entrance effect whereas the analytcal model assumes that the flow s flly developed. Other computer codes have dffculty to obtan a converged soluton even after many teratons (n term of hundred thousands of teratons or tmesteps). lk dffculty may be caused by the soluton technques n these CFD codes whch have been optmzed mostly for ncompressble flow problems. Wth some adjustments or applyng the compressble low-speed flow solver as n the case of FDAP analyss these codes may work as well. Hence any computer code users should be aware of te characterstcs of the flow problems that they are analyzng. STUDY OF FLOW N MCRO GC COLUMNS The baselne desgn of the mcro GC column s a hghaspect rato rectangular channel that s 40 pm wde 250 pm deep and 1 m long. Results of the gas flow n capllary tubes have demonstrated that the gas compressblty effect s mportant and should be ncluded n the model. Usng a smlar approach as n the capllary tube the mass flow rate for the rectangular channel can be derved as follows Nong 1998]: where AmAt = hd:(24@)*(pjpo) flp:-p:)*(l-arc) (14) ARC = (192Az?)* (djh) *.Xtanh(hM2do) s wth = $ and do and h are the wdth and depth of the rectangular channel. Fgure 9 shows the results for gas flow n these mcro GC columns and the comparson between the data and the analytcal model. The favorable comparson allows us to use the analytcal model to further desgn and optmze the mcro GC column for better transport and separaton. SUMMARY We have nvestgated gas flow n long and narrow channels to understand mcroscale transport n GC columns. Both expermental results and theoretcal predctons ndcate that for gas flow n long and narrow channels the compressblty effect s mportant. Thk leads to a nonlnear streamwse pressure dstrbuton along the capllary tube and sgnfcant flow acceleraton near the outlet. Dfferent computatonal flud dynamcs codes have been appled to smulate ths flow problem. Whle some computer codes have dffculty to produce a converged soluton those predctons by the MPSalsa code are reasonably accurate. Results of ths study wll buld up by ASME

7 .. -. our confdence n usng modelng and smulaton approaches to analyze speces transport and separaton n the GC columns. However more analyss and code valdaton especally studyng GC columns wth dfferent rectangular geometry and spral confguraton are recommended. ACKNOWLEDGMENTS Ths work s funded by the Sanda Natonal Laboratores Laboratory-Drected Research and Development (LDRD) program. Sanda s a multprogram laboratory operated by Sanda Corporaton a Lockheed Martn Company for the Unted States Department of Energy under contract number DE- AC04-94AL t s a team effort to conduct ths nvestgaton and to accomplsh ths progress. We would lke to acknowledge the followng ndvduals and ther contrbutons: Crag Gnn Amala Lopez Kambz Salar Seung Cho Jeff Payne & Todd Sterk of Sanda Natonal Laboratores; Perry Daley of CFD Research Corporaton and Scott mlay of Amtec Engneerng. REFERENCES Anderson J. D Fundamentals of Aerodynamcs McGraw-Hll Book Company New York N. Y. pp. 36. Arkllc E. B. and Breuer K. S Gaseous Flow n Small Channels AAA AZAA Shear Flow Conference Orlando FL. Arkllc E.B. Schmdt M. A. and Breuer K. S Gaseous Slp Flow n Long Mcrochannels Journal of Mcro- Electro-Mechancal Systems Vol. 6 No. 2 pp Baley D. K. Arneel T. A. Warrngton R. O. Savoe T Sngle Phase Forced Convecton Heat Transfer n Mcrogeometres - a Revew ASME Proceedngs of the 30th ntersocety Energy Converson Engneerng Conference Vol. 2 Orlando Florda pp CFDRC 1993 CFD-ACE Theory Manual CFD Research Corporaton Huntsvlle Alabama. Cho S. B. Barron R. F. and Warrngton R. O Flud Flow and Heat Transfer n Mcrotubes ASME DSC-VO1. 32 Mcromechancal Sensors Actuators and Systems pp FD 1993 FDAP 7.0 Users Manual Flud Dynamcs nternatonal nc. Evanston llnos. Frye-Mason G C. Kottenstette R. J. Heller E. J. Matzke C. M. Casalnuovo S. A. Lews P. R. Mangnell R. P. Schubert W. K. and Hletala V. M ntegrated Chemcal Analyss Systems for Gas Phase CW Agent Detecton Proceedngs of the 3t Zntematonal Symposum on Mcro Total Analyss System Banff Alberta Canada pp Grant D.W Capllary Gas Chromatography John Wley and Sons New York. Hudson M. L. Kottenstette R. J. Matzke C. M. Frye- Mason G C. Shollenberger K. A. Adkns D. R. and Wong C. C Desgn Testng and Smulaton of Mcroscale Gas Chromatography Columns ASME DSC-VO1. 66 McroElectroMechancal Systems (MEMS) pp Arntec 1995 ZZ?CAVerson 2.0 3D Mult-2%ne Naver- Stokes Flow Analyss wth Fnte-Rate Chemstry Amtec Engneerng nc. Bellevue Washngton. Panton R. L ncompressble Flow 2nd Edton John Wley &Sons nc. New York N.Y. pp O Rourke P.J. and Sahota M. S NO-UTOPA The Flow Solver for the CHAD Computer Code LA-* Los Alamos Natonal Laboratory New Mexco. Shadd J. S. Moffat H. K. Hutchnson S. A. Hennngan G L. Devne K. D. Salnger A.G 1996 MPSalsa a Fnte Element Computer Program for Reactng Flow Problems Part 1 - Theoretcal Development SAND Sanda Natonal Laboratores New Mexco. Shh J. C. Ho C. M. Lu J. and Ta Y. C Monatomc and Polyatomc Gas Flow Through Unform Mcrochannels ASME DSC-VO. 59 McroElectroMechancal Systems (MEMS) pp Trtton D. J Physcal Flud Dynamcs 2ndEdton Oxford Scence Publcatons New York N.Y. pp Wong C. C. and Adkns D. R Gaseous Flow n Long and Narrow Channels - Expermental Results Zntemal Memo to Dstrbuton Sanda Natonal Laboratores Albuquerque New Mexco. Table 1: Results of Computatonal Smulatons of Gas Flow n Long Capllary Tubes. Computer Codes MPSalsa [Shadd 1996] CFD-ACE [CFDRC 1993] [0 R~u%1999] NCA [Amtec 1995] Soluton Technque SUPG method wth GMRES lnear solver Converged after ~ 100k teratons Yes or Tme Steps \ 1 4 Predctons Good No 1 No 1 Yes LU-SGS scheme wth c@ relaxaton No NA by ASME

8 ( Fgure 1 Pcture of Crtcal Components n pchemlabw: proceedng clockwse a U.S. dme (18 mm dameter) preconcentrator for sample collecton SAW array sensors for detecton and GC column for separaton. # Fgure 2 Cross Secton SEM of a Gas Chromatography Column Fabrcated n Slcon by Bosch Reactve on-etchng Process. Outlet To Pressure + Atmospherc Transducer t ~ Oas supply Regulator Bypass Vent Valve Bubble Flow Meter # [ Fgure 3 Expermental Setup to Measure Gas Flow Rate versus Pressure Drop n Caplla~ Tubes. by ASME ~

9 t. CVhyTt50100 &2oOumd. 1 mhxarfhv t CapllaryTub= 100 rrcmnda.0.5t5 m long ErrorEtmates pressure4-15%. flowmda#0.8% PP - C7-0 ~ ~.- Q- a Q- ~z. - ~zm~ x--. -Q-- looun oum 1o j f ~%sure m Drop (kpa) Suo Fgure 4 Measured Mass Flow Rate versus Pressure Drop for Gas Flow n Capllary Tubes wth Dfferent Dameters. Fgure 5 Measured Ext Volumetrc Flow Rate versus Pressure Drop for Gas Flow n Capllary Tube wth Dfferent Gases. CepllaryTubrx 30 urnda. 1 m long; Ntrogenflow Ce@ryTube 30 umda 1 m onq HelumFlov T ~ncomprassble 0.00C05 ~compress ble -- A --expermental ~ E 0.C0C04 -- ~ ~ g > 0 2 o.coco CC07 0.0QO06 O.cocol Praaaure Drop (kpa) o S00 Pressure Drop (k%) Fgure 6 Comparng the Mass Flow Rate versus Pressure Drop for Ntrogen between the Measurement and Model Predctons (Compressble and ncompressble Analytcal Models). Fgure 7 Comparng the Mass Flow Rate versus Pressure Drop for Helum between the Measurement and Model Predctons (Compressble and ncompressble Analytcal Models). byasme s ?... ~

10 .. Capllary Tuba: 100 um dal m bn~ arfbw F ~ Ex#erfmental cbta A MPSalsa Pnxlbfon o.cm Anayfblcampmssbb flow mabl A Analytcal ncompmssblofbw mocbl ~ EO.c05 f g 4 ;0.004 ; & Fgure 8 Comparng the Mass Flow Rate of Ar versus Pressure Drop between the Measurement Analytcal Modes and Smulatons. mode~ L=O.3 m. modch dm u o data L=O.Sm chtad m 13 Fgure 9 Comparng the Ext Volumetrc Flow Rate n Spral Rectangular Columns versus Pressure Drop between the Measurement and Analytcal Model.