Micro-component flow characterization

Transcription

1 Micro-component flow characterization Bruce A. Finlayon, Pawel W. rapala, Matt Gebhardt, Michael. Harrion, Bryan Johnon, Marlina Lukman, Suwimol Kunaridtipol, Treor Plaited, Zachary Tyree, Jeremy VanBuren, Albert Witara epartment of Chemical Engineering, Box Unierity of Wahington Seattle, WA Introduction Microfluidic deice inole fluid flow and tranport of energy and pecie. When deigning uch deice it i ueful for engineer to hae correlation that allow prediction before the deice i built, o that the deign can be optimized. Unfortunately, mot correlation in the literature are for high-peed, or turbulent, flow. Thu, it i ueful to deelop correlation for lower, laminar flow. While mot three-dimenional flow ituation can be modeled in laminar flow uing computational fluid dynamic (CF), the calculation are extenie and time-conuming. Correlation are epecially ueful when the CF capability i not preent. ue to the multidimenional character of microfluidic, it i important to undertand both the power and the limitation of concept baed on imple one- or two-dimenional cae. In thi chapter we outline correlation and pertinent theoretical concept for low peed flow and diffuion: preure drop, entry length, and mixing due to conection and diffuion. Mot of the work i conducted uing CF with the program Comol Multiphyic (formerly FEMLAB). Thi program i eay enough to ue that undergraduate can perform ueful work with only a bit of guidance. Undergraduate in the chemical engineering program at the Unierity of Wahington firt oled the problem decribed here oer the pat fie year. The cae hae been recomputed uing more element to improe the accuracy uing tate-of-the-art computer and program. Preure rop Friction Factor for low flow. Firt conider flow of an incompreible fluid in a traight pipe. The preure drop i a function of the denity and icoity of the fluid, the aerage elocity of the fluid, and the diameter and length of the pipe. The relationhip i uually expreed by mean of a friction factor defined a follow. f 4 L p ρ < > () The Reynold number i Re ρ < > η ()

2 The uual cure of friction factor i gien in Figure a in a log-log plot. For any gien Reynold number, one can read the alue of friction factor, and then ue Eq. () to determine the preure drop. In the turbulent region, for Re > 00, one correlation i the Blaiu formula. f = / Re 05. (3) In the laminar region, forre < 00, the correlation i f =6/Re (4) When the definition are inerted into Eq. (4) for laminar flow, one obtain for the preure drop L η < > p = 3 (5) Thu, in laminar flow in a traight channel, the denity of the fluid doe not affect the preure drop. To ue the correlation for friction factor, or Figure a, one mut know the denity. Thi problem i aoided if one plot the product of friction factor and Reynold number eru Reynold number, ince from their definition p fre = Lη < > (6) Thi i hown in Figure b. Notice that for Re < the only thing needed i the numerical contant, here 6. For fully deeloped pipe flow, the quantity fre doe not change a the Reynold number increae to 00, but it doe in more complicated cae. Thi coer the range that mot microfluidic deice operate in. To predict the preure drop, all that i needed i the ingle number for the geometry and flow ituation in quetion. The way thee number are determined i bet illutrated by looking at the mechanical energy balance for a flowing ytem. Mechanical energy balance for turbulent flow. Conider the control olume hown in Figure, with inlet at point and outlet at point. In the following, ariable mean the ariable at point (outlet) minu the ariable at point (inlet). The mechanical energy balance i [] 3 < > ( < > + + dp g h = W m E ρ (7) For no height change g h= 0; for no pump in the control olume Wm = 0; for an incompreible fluid with contant denity dp = ( p p) = p ρ ρ ρ

3 (a) Standard preentation of friction factor eru Reynold number (b) Low-Reynold number preentation of f Re Figure. Friction loe (preure decreae) for fully-deeloped pipe flow 3

4 Figure. Geometry and notation for mechanical energy balance Thu, under all thee aumption we hae 3 < > ( < > + p = E ρ (8) By continuity the ma flow rate i the ame, w = w. The definition of the icou loe i T E = τ : dv = η[ + ( ) ]: dv 0 and E V V = E w Rewrite Eq. (8) in the form 3 < > ( < > + = = p p E p ρ ρ (9) imenionle Mechanical Energy Balance We make all ariable non-dimenional by diiding them by a dimenional tandard (which ha to be identified in each cae) and denoting the tandard with a ubcript. For the elocity tandard, ue an aerage elocity omewhere in the ytem. =, < > We make the preure non-dimenional by diiding it by twice the kinetic energy p p" ρ The unit of icou diipation are 4

5 E 3 x E x [ ] x E η η = η = η and = [ = ] = x w ρx ρx Thu, define E x E ρ = η The non-dimenional form of the mechanical energy balance i then ( iiding by gie 3 < > x E p p ( " ") < > + η ρ = ( 3 3 < > ( " ") ( ( " ") < > + η = < > x E p p < > + E = p or p ρ Re (0) where Re = ρx / η. Thi i the approach one ue for high-peed flow, including all turbulent cae. For fully deeloped flow through a tube with a contant diameter,, the kinetic energy doe not change from one end to the other. We already hae a formula for the preure drop in term of the friction factor, o that p p" p" = f L ρ < > = Thu the icou diipation term in Eq. (0) i alo gien by the friction factor for flow in a tube. E = Re f L Mechanical energy balance for laminar flow. When the flow i low and laminar, we ue a different non-dimenionalization for the preure. iide the preure by the icoity time the elocity diided by a characteritic length. p η px p ; with p, p = p x η 5

6 Then the mechanical energy balance i ( Multiply by ρx 3 < > x E ( p p ) < > + η = η ρ ρx ρx ( η / η to get another non-dimenional form of the mechanical energy balance. 3 3 < > E ( p p ) R ( E ( p p ) < > + = or e < > < > + = () For fully deeloped flow through a tube with a contant diameter,, the kinetic energy doe not change from one end to the other. We already hae a formula for the preure drop in term of the friction factor, o that p p p = = η < > 3 L Thu the icou diipation term in Eq. () for laminar flow in a traight channel i: L E = 3 Eq. (0) and Eq. () expre the ame phyic. Eq. (0) i more ueful at high elocity in turbulent flow when f i a lowly arying function of Reynold number, and Eq. () i more ueful at low elocity in laminar flow. The non-dimenional preure in thee two equation are different, but they are related p = Re p" Preure drop for flow diturbance. Eq. () i the form of the mechanical energy balance that i ueful in microfluidic. Note that a Reynold number multiplie the term repreenting the change of kinetic energy, which i mall in microfluidic. Conider the cae when the Reynold number i extremely mall. Then any flow ituation i goerned by Stoke equation η = p In non-dimenional form thi equation i η x p px = p or = p x η 6

7 Table I. Frictional lo coefficient for turbulent flow* K = pexce / ρ < > 90º ell, tandard º ell, long radiu º bend,cloe return.5 Tee, branchng flow.0 * From Table 6-4, p. 6-8, Perry Chemical Engineer Handbook [] But with the definition of p = η / x = p Thu, the non-dimenional flow doe not depend upon the Reynold number. Thi mean that both ( 3 < > ) and E < > are contant, for the particular geometry being tudied, ince they can be calculated knowing the elocity field ditribution. For turbulent flow, a reaonable approximation i that the elocity i contant with repect to radial poition in the inlet and outlet duct, and then < 3 > / < >=< >. Thi i not true for laminar flow, although both term can be calculated for fully deeloped flow at the inlet and outlet. Furthermore, the kinetic energy term i negligible at mall Reynold number. To correlate the preure drop, the main tak i to correlate the icou diipation term, E. A conenient way to correlate data for exce preure drop when there are contraction, expanion, bend and turn, etc., when the flow i turbulent i to write them a p = exce K ρ < > Then, to calculate the preure drop for any gien fluid and flow rate, one only need to know the alue of K, a few of which are tabulated in Table I. A hown aboe, for low flow the preure drop i linear in the elocity rather than quadratic, and a different correlation i preferred. Write p exce = K L η < > Value of K L for different geometrie are gien in Table II a determined by finite element calculation. 7

8 Figure 3. Contraction flow geometry Table II. Coefficient K L for Re negligbily mall Column Picture K L Sharp bend, 9. Smooth bend, 90 degree, hort radiu, (centerline radiu = gap ize) Smooth bend, 90 degree, long radiu, (centerline radiu =.5 x gap ize) 9 (0.5) 9 (0.36) Bend, 45 degree, harp change, 5. (0.) Bend, 45 degree, long radiu,, (centerline radiu =.5 x gap ize) 4.3 (0.) Square corner, 3 4. (-.) Round pipe, (-0.) = aerage elocity, x = thickne or diameter The alue in ( ) i the alue if the entire length through the centerline i ued to calculate the fully deeloped preure drop. The exce preure drop beyond thi i negligible. Preure drop for contraction and expanion. Conider a large circular tube emptying into a maller circular tube, a hown in Figure 3. The preure drop in thi deice i due to icou diipation in the fully deeloped region of the large and mall channel, plu the extra icou diipation due to the contraction, plu the kinetic energy change. When the Reynold number i mall, the kinetic energy change i negligible. For the flow illutrated in Figure 3, we write the exce preure drop for the contraction a the total preure drop minu the preure drop for fully deeloped flow in the large and mall channel. The preure drop for fully deeloped laminar flow i known analytically, of coure [Eq. (5)]. p = p p p exce total l arg e channel mall channel 8

9 Figure 4. Laminar flow exce preure drop for a 3: contraction in a circular channel. K-L i the total preure minu the fully deeloped preure drop in the two region, expreed a K = p /η < >. L exce Table III. Coefficient K L for contraction and expanion for Re Column negligbily mall Picture K L : pipe/planar 7.3/3. 3: pipe/planar 8.6/4. 4: pipe/planar 9.0/ degree tapered, planar, 3: degree tapered, planar, 3: 0.8 3: quare (quarter of the geometry) 8. = aerage elocity, x = thickne or diameter, both in the mall ection 9

10 The exce preure drop i correlated uing pexce = KLη < > / and K L can be determined from finite element calculation. Note that one mut pecify which aerage elocity and ditance are ued in thi correlation, and the uual choice are the aerage elocity and the diameter or total thickne between parallel plate, all at the narrow end. The alue of K L depend upon the contraction ratio, too. Value are gien in Table III, a determined from finite element calculation. Table III indicate that the alue of K L i approximately 8-5 depending upon the geometry. When the Reynold number i not anihingly mall, K L depend on Reynold number, too, and the dependence i eaily determined uing CF. For example, for a 3: contraction in a pipe the reult are hown in Figure 4. Note that K L i approximately contant for Re <. The departure from a contant i due partially to extra icou diipation at higher Reynold number, but motly due to kinetic energy change. For the axiymmetric pipe or channel cae, in a contracting flow, the kinetic energy change i deried a follow. Let β = / be the area ratio ( < for contraction), and ue = [ ( r/ R) ]for the inlet and outlet. Then i 0i i R By continuity R 3 () r rdr / () r rdr = = < > 0 0 =, or = β The kinetic energy term i then < 0 > < 0 > =< 0 > β [ ] ( ) The total balance, Eq. (), i then p p = Re < > ( β ) + E, β = /, β < () If the elocity tandard i the aerage elocity in the narrow tube,< >=. The icou diipation i E = K. In dimenional term Eq. () i L or ( p p) ( ) < > = ρ < > β + η η η p p = ρ< > ( β ) + < > K L K L 0

11 When doing calculation or an experiment, it i the total preure drop that i calculated or meaured. When the analytic expreion for preure drop in both tube are ubtracted, what remain i the exce preure drop due to the contraction. The change of kinetic energy can be calculated o that the remaining term, icou diipation, i readily aailable. A een in Figure 4, the icou term predominate at low Reynold number (i.e. in microfluidic) wherea the kinetic energy term predominate at high Reynold number (>00). The equialent expreion for other geometrie are: Flat plate, contraction, x = H = thickne between plate, =< > o that < >= : 7 p p = Re < > ( β ) + E, β = H / H, β < (3) 35 or 7 η < > p p = ρ< > β + KL 35 ( ) H Flat plate with ymmetry condition uptream, contraction: 7 p p = Re < > ( β ) + E, β = H / H, β < (4) 35 7 η < > p p = ρ< > ( β ) + KL 35 H Expanion work in a imilar fahion, and the icou diipation at negligible Reynold number i the ame in a contraction and an expanion. For expanion, the aerage elocity in the mall tube i < > and we take =< > o that < >=. Circular tube, expanion: or p p = Re < > ( ) E +, β = > (5) β p p = ρ < > ( ) + K β Flat plate, expanion: L η < > 7 H p p = Re < > ( ) E +, β = > (6) 35 β H Flat plate with ymmetry condition uptream, expanion: 7 H p p = Re < > ( ) E +, β = > (7) 35 β H Howeer, in an expanion the kinetic energy change i oppoite in ign to that in a contraction, cauing the preure to increae due to the kinetic energy and decreae due to the icou diipation.

12 Figure 5. Manifold with quare hole (left) or flat plate (right) Manifold. When a flowing tream i approaching a manifold, with many hole, there are other effect that mut be conidered. A typical ituation i hown in Figure 5. Now calculation can be done for one cell of the manifold imply by uing a ymmetry condition uptream of the manifold, a illutrated. The cae conidered here i for a planar manifold, but 3 manifold could be analyzed in the ame way proided there wa ymmetry. Otherwie the entire deice mut be modeled. Thi ituation ha been analyzed by Kay [3] for turbulent flow, but he gie cure for laminar flow, too. Here we how that thoe cure are oer-implified for laminar flow. Kay [3] preent reult for a lo coefficient, K c, defined a p K c ρ < > = + σ (8) where the lo coefficient depend upon appleσ, the ratio of free area in the channel to the free area uptream (ee Figure 3, 6 in [3]). Howeer, there i only one cure for laminar flow, although there are multiple cure for different Reynold number in turbulent flow. The term σ repreent the kinetic energy change auming a flat elocity profile, which i not true for laminar flow between flat plate. Since the flow will be coniderably different at low Reynold number (when the preure change are linear in elocity) from the flow at higher, but till laminar, Reynold number (when the preure change are quadratic in elocity), multiple cure are expected for different laminar Reynold number, too. Finite element calculation are performed for ariou contraction ratio and the K c i calculated uing Eq. (8) along with the computed exce preure drop. Figure 6 how that there are different cure for different Reynold number, a expected, but that the cure for Re = 00 are cloe to thoe gien by Kay. Howeer, microfluidic deice eldom operate in thi regime. The ame calculated reult are plotted in Figure 7 in term of K L defined uing Eq. (4) and the computed exce preure drop. Now the alue approach an aymptote at mall Reynold number, which i more ueful for microfluidic flow.

13 Figure 6. K c defined by Eq. (8) for ariou planar contraction for a manifold Figure 7. K L defined by Eq. (4) for ariou planar contraction for a manifold; the cure for Re = 0 i inditinguihable from that for Re =. 3

14 Entry length When the diameter or ize of a flow channel change, the elocity profile mut change to adapt to the new ize. It take a certain length in the new ize for the flow to become fully deeloped (if that i poible). In a microfluidic deice, it i of interet to know how long that region i, o that the effect of a deeloping elocity field can be ignored if it i unimportant. The deeloping elocity field affect the preure drop and it alo complicate the diffuion of pecie in the deice. In thi ection the approach length and entry length are explained for laminar flow. Contraction flow. Conider flow into a circular channel or pipe with the elocity a contant at the entrance. A the fluid enter the channel, the elocity oon deelop a parabolic profile. The elocity at the centerline increae from <> (at the entrance) to <> (far downtream) while the elocity at the wall decreae uddenly from <> to zero. The entry length i defined a the length of channel it take before the centerline elocity i 99% of it ultimate alue, <>. Sometime the entry length i defined uing a 98% or 95% criterion. Atkinon et al. [, 4] oled thi problem uing the finite element method and correlated the reult with the equation L e = Re, % criterion Thi equation i baed on the Stoke flow olution (Re = 0, giing the 0.59) plu a boundary layer olution (giing the Re). It in t actually a cure fit of the reult for mall Reynold number. The calculation for Re = 0 in [4] and the experimental data in [5] gie the following correlation. L e = Re Re, Re 6.34, r =.0, % criterion For a 5% criterion, the correlation of Atkinon data i L e = Re Re, Re 6.34, r = , 5% criterion It i impoible to realize a contant elocity at the inlet to the circular channel unle the wall i paper thin and the flow can go pat the outer wall, too. A more realitic cae i where a larger channel decreae to a maller channel a hown in Figure 3. Then a fully deeloped elocity profile can be ued far uptream in the larger channel. The flow rearrangement occur before the contraction, a hown in Figure 8. Thi i important in laminar flow becaue orticity i generated at the corner and diffue uptream. A the elocity increae, thi region get puhed againt the inlet and become maller. Thu, for turbulent flow the effect i much horter. The approach length i preented in Figure 9 a a function of Reynold number. Inide the pipe the entry length for a 4: contraction in a circular channel or pipe i deried from finite element calculation including the uptream egment. L e = Re Re, Re, 0 r =.0, 5% criterion 4

15 Figure 8. Velocity rearrangement before contraction, 4: contraction, cylindrical geometry for different Reynold number baed on the downtream ection Figure 9. Approach length, 5% criterion, 4: contraction, cylindrical geometry 5

16 Figure 0. Entry length, 5% criterion, 4: contraction, cylindrical geometry Figure. iffuion in fully deeloped planar flow The effect of Reynold number i clearly diplayed in Figure 0. When the uptream region i included the entry length i much horter (L e / = 0.64 rather than 0.43) ince the elocity ha already rearranged omewhat uptream. iffuion Many microfluidic deice inole diffuion of concentration a well a fluid flow. The elocity croing a plane normal to the flow i eldom uniform, o the diffuion mut be examined in the midt of non-uniform elocity profile. The implet illutration of thi effect i with Taylor diperion. For implicity, conider flow between two flat plate a hown in Figure. The flow i laminar and the elocity profile i quadratic. Now uppoe the entering fluid ha a econd chemical with the concentration hown,.0 in the top half and 0 in the bottom half. A thi fluid moe downtream (to the right), the elocity i highet in the center, o the econd chemical moe fat there. Howeer, it can alo diffue ideway, and the chemical that doe firt diffue ideway then 6

17 Figure a. Fully deeloped elocity profile in a x 4 rectangular channel Figure b. Fully deeloped elocity profile in a x 8 rectangular channel, < >=, (0,0)=.68 Figure c. Fully deeloped elocity profile in a x 8 rectangular channel with lip on the ide wall, < >=, (0,0)=.500 moe lower down the flow channel. At a point not on the centerline, the concentration i determined by thi lower elocity in the flow direction and diffuion ideway becaue the concentration on the centerline i alway higher than the concentration in adjacent fluid path. Thi i called Taylor diperion [6]. Taylor [7] and Ari [8] howed how to model the aerage concentration a a function of length uing a impler equation (diperion in the flow direction only) and an effectie diperion coefficient gien by K Pe K = + <> <> = + = + 9 or 9 (9) 9 Next conider fully deeloped flow in a narrow channel, which i typical of a microfluidic deice. The normal iew i hown in Figure with flow going into the paper. The elocity arie inx and y according to the equation η p + = x y L 7

18 Figure 3. Three-dimenional rectangular channel Figure 4. Concentration profile in channel; Pe = 00, width = 8, height =, length = 30, < >=. Fully deeloped elocity profile determined with 8,60 element, 56,897 degree of freedom. Concentration determined with 46,46 element, 68,845 degree of freedom Typical olution are hown in Figure (a,b). If one ue a model ignoring the edge effect, one i eentially uing the elocity profile hown in Figure (c). When diffuion i added into the problem, thee two flow ituation (b and c) are different. Take the three-dimenional flow ituation hown in Figure 3, with the elocity profile being the ame for all x. Let the material on one ide hae a dilute concentration of ome pecie (taken = ), and on the other ide there i no concentration of that pecie (c = 0). What happen at different plane downtream? Typical concentration profile are hown in Figure 4 and 5, and a quantitatie decription i gien below. Characterization of mixing. In order to ealuate mixing it i neceary to hae a quantitatie meaure. The ariance from the mean i appropriate, but in a flowing ytem ome fluid element hae a higher elocity than other. The concept in chemical engineering that i ued to account for thi i the mixing cup concentration. Thi i the concentration of fluid if the flow emptied into a cup that wa well tirred. In mathematical term, the mixing cup concentration i c mixing cup = A cxyzxydxdy (,, ) (, ) A xydxdy (, ) (0) 8

19 Note that the aerage elocity time the mixing cup concentration gie the total mole/ma flux through an area at poition z. Then it i natural to define the ariance of the concentration from the mixing cup concentration. c ar iance = A [(,,) cxyz c ] (, xydxdy ) A mixing cup xydxdy (, ) () Aerage concentration along an optical path. Optical meaurement are frequently made through the thin layer of a microfluidic deice. The aerage concentration i then determined by the integral along the path, uch a L L optical = 0 0 c c( x. y, z) dy/ dy () When the elocity i ariable in the y-direction ome region of the fluid are moing fater than other. Thu, the optical meaurement, or integration along the optical path, may not be comparable to the mixing cup concentration. Indeed, it i entirely poible that eeral different flow and concentration ditribution can gie the ame aerage along an optical path. In that cae, computational fluid dynamic can be ued to interpret the optical meaurement. Peclet number. The conectie diffuion equation i c + c = c t If thi equation i made non-dimenional uing it i c t c =, =, = x, t =, c x Pe c c Pe c c c c t + = t + = or Pe (3) where x Pe (4) The Peclet number i a ratio of the time for diffuion, x /, to the time for conection, x/. Typically, Pe i ery large, and thi preent numerical problem. The econd form of Eq. () how that the coefficient of the term in the differential equation with highet deriatie goe to 9

20 zero; thi make the problem ingular. For large but finite Pe the problem i till difficult to ole. For a imple one-dimenional problem it can be hown [9] that the olution will ocillate from node to node unrealitically unle Pe x or x x, x Thi mean that a Pe increae, the meh ize mut decreae. Since the meh ize decreae, it take more element or grid point to ole the problem, and the problem may become too big. One way to aoid thi i to introduce ome numerical diffuion, which eentially lower the Peclet number. If thi extra diffuion i introduced in the flow direction only, the olution may till be acceptable. Variou technique include uptream weighting (finite difference [0]) and Petro- Galerkin (finite element []). Baically if a numerical olution how unphyical ocillation, either the meh mut be refined, or ome extra diffuion mut be added. Since it i the relatie conection and diffuion that matter, the Peclet number hould alway be calculated een if the problem i oled in dimenional unit. The alue of Pe will alert the chemit, chemical engineer, or bioengineer whether thi difficulty would arie or not. Typically i an aerage elocity, x i a diameter or height, and the exact choice mut be identified for each cae. Sometime an approximation i ued to neglect axial diffuion ince it i o mall compared with axial conection. If the flow i fully deeloped in a channel then one ole (for teady problem) c c c wxy (, ) = + (5) z x y Thi i a much impler problem ince the can be aborbed into the length, z, c c c wxy (, ) ( z) = x + y In non-dimenional term the equation i c c c w ( x, y ) ( z / Pe) = x + (6) y Then all reult hould depend upon z / Pe not z and Pe indiidually. The elocity i not a contant; it depend upon x and y. Thu, the problem i different from the following formulation < > = + = c c c c c + c z w t = ( z Pe x y x y, or where (7) / ) t' < w > Pe where < w >= w ( x, y ) dx dy / dx dy. A A 0

21 (a) Concentration at exit with fully deeloped elocity profile; ariance = 0.93 (b) Concentration at exit with elocity profile allowing lip on the ide wall, for the ame aerage elocity a (a); ariance = 0.95 (c) Concentration at exit with uniform elocity, for the ame aerage elocity a (a); ariance = 0.95 (d) Comparion of mixing cup and optical pathlength concentration Figure 5. Butterfly effect; concentration at exit at z = 30

22 Figure 6. T-enor; the knife edge can be added a hown iffuion in a rectangular channel. The quantitatie meaure are calculated for fully deeloped flow in rectangular channel. The concentration profile i hown in Figure 4. Figure 5a how contour of the concentration on the exit plane. The material near the upper and lower urface ha a lower elocity (ee Figure b); thu it ha a longer reidence time in the channel and ha more time to diffue ideway. In fact, the concentration profile reemble a butterfly, and thi effect wa called the butterfly effect by Kamholtz et al. []. If the elocity i changed to that hown in Figure c, but with the ame aerage elocity, the concentration profile are altered lightly but the butterfly effect till exit, Figure 5b. The peak elocitie are different in Figure b and c, though, een when the aerage elocitie are the ame, o that the butterfly effect would not be calculated correctly with a elocity allowing lip on the ide wall (Figure c). When Eq. (7) i ued with a uniform elocity, the butterfly effect diappear (Figure 5c) ince eery element of fluid ha the ame elocity, and hence the ame reidence time. The difference between the mixing cup concentration and the optical path concentration i illutrated in Figure 5d. The difference are not large (maximum of 8% difference for thi cae). iffuion in a T-enor [3, 4]. Conider next a two-dimenional cae with flow coming in top and bottom on the left (ee Figure 6). The fluid i mainly water, but a mall concentration of a econd pecie i in the upper flow. Along the top take c =, and along the bottom take c = 0. How will they mix? The ariance i hown a a function of length/peclet number in Figure 7. Point are computed when the problem i oled with different Peclet number and the cure are cloe to each other, jutifying Eq. (6). If thi problem i oled with a large Peclet number, ocillation appear ([5] p. 7) and the meh mut be refined or ome tabilization (like Petro- Galerkin or uptream weighting) mut be applied. The tabilization, of coure, moothe the olution and add unphyical diffuion. It i up to the analyt to decide if that effect can be tolerated. The effect of placing a knife-edge in the deice i ery mall; it i negligible becaue the amount of diffuion acro the line where the knife-edge would be i a ery mall portion of the total ma tranfer, and the flow field i affected only lightly. Serpentine mixer. A erpentine mixer (Figure 8) can be ued to mix two chemical in a horter length than happen in a traight channel. A typical olution for a Reynold number of.0 and Peclet number of 000 i hown in Figure 9. Comparing thee data with thoe for a T-enor

23 Figure 7. Variance for mixing in T-enor Figure 8. Serpentine mixer; width =, height = /8 3

24 Figure 9. Concentration on urface of the erpentine mixer; Re =, Pe = 000, 8,380 element, 45,94 degree of freedom for flow, 46,7 degree of freedom for concentration; ariance at exit i.6e-4 Table IV. Variance at different point in erpentine mixer Re Pe ariance Equialent length e e e e e e- 600 The equialent length i the length of traight channel of the ame dimenion that gie the ame ariance; the length are determined from Figure 7. All calculation hae a dimenionle preure drop of p = ph/ η < > =33, which mean that the dimenional preure drop increae linearly with Reynold number. Sc = 000. The cae with Pe = 8000 i oled with 88,986 element and 36,99 degree of freedom for the concentration. 4

25 Figure 0. Reactor with two inlet for two reagent and a catalyt (a) (b) Figure. Outlet condition, (a) elocity; (b) concentration of product (Figure 7) how that the total length/width neceary ha been reduced from 50 to 9.6 when the ariance i.6e-4. Another apect of the erpentine mixer i to define how much mixing occur for a gien preure drop. The flow olution gie the preure drop and the ariance gie the mixing. Table IV how reult at different elocitie for the erpentine mixer. Reactor ytem. The importance of diffuion for mixing (or the lack of it) i illutrated by an example of a reactor ytem deeloped at ow Chemical Company. The geometry of a mall part of the deice i hown in Figure 0. One chemical come in at the left, and another chemical plu a catalyt come in at the top inlet. The concentration ditribution are found by oling the conection-diffuion-reaction equation for the two chemical (A and B) plu the catalyt (C). The ytem i aumed to be dilute o that the total concentration doe not change appreciably. The reaction i taken a A+ B, in preence of C Needle to ay, the reaction only occur where both chemical and the catalyt are all in the ame place. Since little mixing occur in laminar flow, and diffuion i low, the product concentration 5

26 coming out i not uniform in pace (Figure b)) een though the elocity i fully deeloped at the exit (Figure a). In thi complicated three-dimenional cae only CF can gie the detail of mixing. Concluion Engineering correlation hae been preented for preure drop and entry length in common geometrical flow channel with laminar flow. The correlation differ from their counterpart with turbulent flow (aailable in handbook, depending upon denity and elocity) ince in low flow the preure drop i proportional to the elocity and the fluid icoity. The entry length in laminar flow i alo longer than i the cae for turbulent flow, which mean that ome ection of flow in microfluidic i in the entry region. Mixing i more difficult to achiee in laminar flow than in turbulent flow. The mixing cup ariance i introduced a a quantitatie meaure of mixing when there are flow and concentration ariation. iffuion i the mechanim in the T-enor, and the amount of mixing depend upon the Peclet number. In the erpentine mixer, howeer, mixing i due primarily to the intertwining of flow tream, with diffuion acro a hort ditance. A reaction ytem illutrated the importance of characterizing the flow and diffuion uing computational fluid dynamic. Reference [] R. B.Bird, W. E. Stewart, E. N. Lightfoot, Tranport Phenomena, nd ed., Wiley, 00. [] Perry Chemical Engineer Handbook, Perry, R. H., Green,. W. (ed.), 7th ed., McGraw-Hill, 997. [3] W. M. Kay, Tran. ASME 950, [4] B. Atkinon, M. P. Brocklebank, C. C. H. Card, J. M. Smith, AIChE Journal 969, 5, [5] B. Atkinon, Z. Kemblowki, J. M. Smith, AIChE Journal 967, 3, 7. [6] W. M. een, Analyi of Tranport Phenomena, Oxford, 998. [7] G. Taylor, Proc. Roy. Soc. Lond. A 953, 9, [8] R. Ari, Proc. Roy. Soc. Lond. A 956, 35, [9] B. A. Finlayon, Numerical Method for Problem with Moing Front, Raenna Park, 99. [0] R. Peyret, T.. Taylor, Computational Method for Fluid Flow, Springer Verlag, 983. [] T. Hughe, A. Brook, 9-35 in Finite Element Method for Conection ominated Flow, Ed. T. Hughe, ASME 979. [] A. E. Kamholz, B. H. Weigl, B. A. Finlayon, P. Yager, Anal. Chem. 999, 7, [3] B. H. Weigl, P. Yager, P., Scienc 999, 83, [4] A. Hatch, A. E. Kamholz, K. R. Hawkin,M. S. Munon, E. A. Schilling, B. H. Weigl, P. Yager, Nat. Biotechnol. 00, 9, [5] B. A. Finlayon, Introduction to Chemical Engineering Computing, Wiley,

27 Table of Nomenclature c [kmol m -3 ] [m] diameter of tube [m - ] diffuiity E [Pa m 3 - ] icou diipation E [Pa m 3 kg - ] icou diipation per ma flow rate, = E /w f [ ] friction factor, defined by Eq. () g h H [m - ] acceleration of graity [m] height aboe a datum [m] ditance between two flat plate K [ ] friction lo coefficient, = p ρ < > exce / K L [ ] friction lo coefficient for laminar flow, = pexcex /η K c [ ] friction lo coefficient for a manifold, defined by Eq. (8) L [m] length of channel L e [m] entry length for elocity to come within a percentage of the fullydeeloped alue p [Pa] preure Pe [ ] Pecle number, = x x / r [m] radial poition R [m] radiu of tube or correlation coefficient Re [ ] Reynold number, = ρx / η x Sc [ ] Schmidt number, = η/ ρ t [] time u [m - ] elocity in the x-direction [m - ] elocity, and elocity in the y-direction V [m 3 ] olume [m - ] ector elocity w [kg - ] ma flow rate or [m - ] elocity in z-direction W m [Pa m 3 kg - ] work of pump per ma flow rate x [m] coordinate y [m] coordinate z [m] coordinate Greek letter β [ ] ratio of diameter or height, defined by Eq. (-7) η ρ [Pa ] icoity [kg m -3 ] enity 7

28 σ τ [ ] ratio of free area in manifold to area uptream [Pa] hear tre Special ymbol non - dimenional preure, uing p = η / x " non - dimenional preure, uing p < > aerage oer an area gradient operator Laplacian operator = ρ Subcript mixing cup concentration defined by Eq. (0) optical concentration defined by Eq. () ariance ariance defined by Eq. () tandard quantity inlet alue outlet alue 0 centerline alue 8