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# UNDERGRAD RESEARCH CHEM E 499

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This 13 page Class Notes was uploaded by Paula Koss on Wednesday September 9, 2015. The Class Notes belongs to CHEM E 499 at University of Washington taught by Staff in Fall. Since its upload, it has received 25 views. For similar materials see /class/192157/chem-e-499-university-of-washington in Chemical Engineering at University of Washington.

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Date Created: 09/09/15

Evaluation of Concentration Variance as a Function of Z Pe in a Micro uidic Device Chem B 499 Mentor Professor Emeritus Bruce Finlayson Jordan Flynn Friday June 6 2008 IA I A I III III I 39I39I39II e In VII 39Vilgllli vn wlg gilll I v rm m t ht wry 17543 3911 WIIIA E IIIquot ii a V V Introduction The purpose of this research is to characterize the mixing in a micro uidic device This will be achieved by looking at the variance a measure of mixing versus the length of the device divided by the Peclet number z Pe From looking at graphs of variance versus Z Pe we expect to see a relationship To test this relationship the length of the device will be varied from one half unit in length to two and a half units in length The Peclet numbers will be chosen at different lengths so that the ratio of Z Pe will be identical for ve points one at each of the different lengths Each of the ve data sets will then be graphed and a relationship should be present between these ratios of Z Pe and the variance Another graph will be created with Peclet numbers ranging from 10 to 1000 in different length devices to create a large range of values to see if a consistent trend is followed in a wider range of Z Pe values The 2D geometry will be compared to a 3D geometry to assess the differences in mixing that takes place in a 3D mixer as compared to the 2D version of the device In addition the variance data will be compared to the variance data for a T sensor to compare how efficient of a mixer my micro uidic device is compared to a T sensor The Problem in Detail To solve this problem the Incompressible NavierStokes equation and the Convection and Diffusion equation will be solved simultaneously in their non dimensional forms For the NavierStokes equation we start out with the dimensional form of the equation a pairipu Vu VpuV2u To nondimensionalize this equation we then define the following quantities u i p 961 VxSV s P x This equation can then be arranged to give the following nondimensional form Bu 8 Reu V 39 V39p V392 u t In the non dimensional form of the NavierStokes equation the dynamic viscosity 7 is set equal to one and the density J is set equal to one A normal in ow velocity of one is used at the inlet of the device The outlets have been set to a pressure of zero with no viscous stress All other boundaries of the device are walls with a no slip boundary condition For the Convection and Diffusion Equation we will solve the equation in a non dimensional form The Convection and Diffusion equation we start with is vVcDV2c a To make the equation dimensionless we define the following quantities v tv v V39xxVt 5 v x s s C The Convection and Diffusion equation then becomes l acvvVvaVv20y at39 Pe 1 Since the nond1mens1onal form of the equation Will be used D P and we can 6 simply put in for diffusivity to vary our Peclet number in the sub domain settings Also in the sub domain settings the X and y velocities are set to u and V respectively which are quantities solved for in the NavierStokes Equations The upper half of the inlet is set to a concentration of zero and the lower half is set to a concentration of one in the boundary conditions The outlets are set to convective fluX and all other walls are set to insulation symmetry A picture of the device follows with the dimensions of the device Suvface Velanlv eld ms lnlet Width Constant Figure I The dimensions on this device are one unit for the inlet width and 1 units for the width of each outlet The length in the above device was varied from 05 units to 25 units while all other dimensions were kept constant This gure shows the velocity eld present in a device with aPeclet number of 200 and a length of one unit To calculate the variance in the device the mixing cup concentration must be determined using the following equation Icxyzvxydxdy mexmgeup A Iv x ydxdy A In the mixing cup concentration equation c is concentration and V is the velocity This equation is integrated over A the area of the boundary Since half the inlet is at a concentration of zero and half at a concentration of one the cmixing cup will be equal to 05 From this the variance can be calculated using the following equation JTCI xyz mexmgeup2 v1 xydA 7 11 A ln 96506114 11 A Boundary integrations are performed across all outlet boundaries and the inlet boundary to give the desired quantities Results Figure two shows a plot of Z Pe versus the variance for the ve cases with the same Z Pe ratios in ve different length devices Variance as a function of Z39IPe 1 EIEIEEIEI Length 05 3 3 3 I I Length10 WEIEIEVEH A Length15 gt i A length20 Len th25 1 m n D m EM 1 Z39IPe Figure 2 This Figure shows a plot ofthe variance versus the length overPeclet number Notice that the relationship is consistent for all devices despite d erent dimensions of the device The results of the calculations presented in the graph above have been verified by hand for the case with a length of 25 and a Peclet number of 100 See Appendix Identical calculations were used to obtain the results for the other cases The typical mesh solved for has approximately 1600 elements and around 13000 degrees of freedom The effects of mesh re nement were determined by re ning the mesh of a particular case and examining its effects on the calculated variance Effects of a More Refined Mesh L1 Pe25 Case 1 Case 2 Elements 1612 6448 Degrees of Freedom 12946 46844 Solution Time s 4204 17062 Variance 955E02 989E02 Percent Error From Best Solution 396 055 Figure 3 This table compares the results of calculating the variance with a more re ned mesh The Percent error is determined by Error 100 The Bestpossible solution was obtained var1 var2 var2 by continuing to re ne the mesh and solve the problem until the computer ran out of memory to solve the problem This table shows that the error in using an unre ned mesh has a less than 5 error from the best solution obtainable The results for a larger range of Z Pe data have been tabulated and presented below in Figure 4 Variance as a Function of Z39IPe 100E00 Q I A a x x 5 100501 I XX 0 Length05 g Length1 g Length15 g x Length2 100502 x Length25 A X X 100E03 i i i 00001 0001 001 01 1 Z39IPe Figure 4 Thisfigure presents data on Pe in a wider range The Peclet numbers in this data were rangedfrom 10 to 1000 The data presented in gure four show that the variance follows a distinct curve regardless of the dimensions or Peclet number of the device The only quantity that matters is Z Pe which characterizes the mixing of the device To examine the effects of a 3D geometry versus a 2D geometry the calculations presented in the graph of gure ve were carried out Variance 2D vs 3D Variance 100E00 i 139 c 3D Variance DOEO1 1 I 2D Variance 100E02 I I I I 00001 0001 001 01 1 Z39lPe Figure 5 T his figure compares the variance of a 2D model against a 3D model The calculations were carried out at a range of Peclet numbers with a length of 05 units The 3D geometry has been extruded by one unit From the results of gure ve a conclusion can be drawn that there is not a signi cant difference between the 3D and 2D models The small change in variance of a 3D model can be attributed to the no slip conditions applied to four surfaces instead of two which slows down the ow and allows slightly more diffusion than the 2D geometry To compare the mixing obtained in my micro uidic device to that of a T sensor a plot of variance comparing both devices has been constructed Variance Variance Comparison between a Tsensor and Microfluidic Device 1000000 7 0100000 0010000 0001000 7 o 0 Figure 817 Data I Experimental Data 0000100 00 0000010 7 01 0001 001 0000001 7 Z39lPe F Figure 6 T his figure compares the variance of a T sensor to the research geometry The actual geometries are shown at the right of the variance graph In gure siX it is seen that the variance of a T sensor is lower and therefore the T sensor is a better mixer than the geometry chosen for research Comparisons to Literature Data For the results obtained above it is important that they compare to the paper Generating fixed concentration arrays in a micro uidic device by Holden ET All Comparisons can be drawn between the authors of this paper and my own simulations to verify the simulations that l have performed are correct I have verified the papers results by plotting the wall concentration and examining the effects of an increasing flow rate on concentration plots The wall concentration from the literature results and my own simulation are presented in figure seven 05 K00052 v 2D 3D 04 Annly cat 03 9 02 01 905 04 a Figure 7 The wall concentrations plotted across half of the inlet channel at the start of the first micro channel outlet The definition of 7 is given in the experimental data left which is compared to the literature data X 05 and Pe25 0 for the experimental data The diffusion of the dye in the literature data has diffused out from the wall farther than in the experimental data This result is consistent because of the values of As the Peclet number increases the value decreases allowing less diffusion to take place Since the experimental results plot a lower concentration across the half channel these results are consistent Another check between the literature and experimental data is the comparison of concentration plots The literature plots concentration as the flow rate of the device is increased from 50 to 500 nLmin To simulate this change in ow rate I have created plots of concentration across the channels with varying Peclet numbers The results show that the literature and the experimental data agree qualitatively As the Peclet number and flow rate increases the mixing that takes place in the device decreases This decrease in mixing is evident by the amount of outlets with concentrations around zero and one Near the edges of the device as the ow rate increases less time is available for diffusion to occur in the device creating a steeper variation of concentrations that is present in both literature and experimental data These results are presented in the Appendix Conclusions As can be seen from the graphs above as Z Pe increases the variance becomes smaller This trend is consistent for various lengths and could be extended to approximate the relationship for other devices of this type The effect on variance by using a 2D versus 3D model for this micro uidic device is very small and well within an order of magnitude Since the 2D and 3D results are comparable a 2D geometry may be used to approximate results that would be obtained in 3D modeling of similar devices Finally the micro uidic device chosen is a less successful mixer than a T sensor Appendix Results Tables Table of Results V dth5 Used in Sample Calculation z Dimensionless Peclet Number Dimensionless z39Pe Dimensionless Variance Dimensionless 05 000777 05 40 00125 002711 05 60 000833 004895 05 80 000625 006786 05 100 0005 008335 1 40 0025 000831 1 80 00125 003314 1 120 000833 005747 1 160 000625 007663 1 200 0005 009138 15 60 0025 002010 15 120 00125 004532 15 180 000833 007019 15 240 000625 008963 15 300 0005 010447 2 80 0025 002046 2 160 00125 004732 2 240 000833 007266 2 320 000625 009193 2 400 0005 010638 25 100 0025 001974 25 200 00125 004501 25 300 000833 006844 25 400 000625 008678 25 500 0005 010121 2D Data at Chosen Ratios Width 1 Figure 4 Z39Pe V 39 Pe arianoe Pressure Drop Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless 00250 484E02 2974 05 40 00125 893E02 2974 05 60 00083 114E01 2974 0 5 80 00063 131 E01 2974 0 5 100 00050 143E01 2974 1 40 00250 642E02 2981 1 80 00125 110E01 2981 1 120 00083 134E01 2981 1 160 00063 150E01 2981 1 200 00050 161E01 2981 1 5 60 00250 725E02 2986 1 5 120 00125 119E01 2986 1 5 180 00083 143E01 2986 1 5 240 00063 158E01 2986 15 300 00050 168E01 2986 2 80 00250 776E02 2991 2 160 00125 124E01 2991 2 240 00083 148E01 2991 2 320 00063 163E01 2991 2 400 00050 174E01 2991 2 5 100 00250 812E02 3007 2 5 200 00125 127E01 3007 2 5 300 00083 150E01 3007 25 400 00063 163E01 3007 25 500 00050 172E01 3007 2D Data at a Large Range Width1 Figure 4 Figure 6 Pe Z39Pe V oe Z arian Pressure Drop Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless 00199 2974 05 100 0005 01456 2974 05 500 0001 02000 2974 05 1000 00005 02112 2974 1 10 01 00075 2980 1 100 001 01234 2980 1 500 0002 01974 2980 1 1000 0001 02148 2980 15 10 015 00039 2986 15 100 0015 01067 2986 15 500 0003 01874 2986 1 000 100 1 000 T Sensor Data Figure 6 Z39Pe 00000 00050 00100 00150 00200 00250 00000 00071 00143 00214 Variance 01876 01359 01026 00780 00593 00453 01745 01144 00772 00523 00355 00242 01347 00543 00222 00091 00037 00016 00581 00054 00005 00001 00000 00000 02048 01689 01443 01248 01085 00946 00015 002 0004 0002 025 0025 0005 00025 02084 00027 00935 01868 02126 00026 00819 01787 02038 Literature Concentration Plots 50 ILmin 100 nLmjn 500 nLmin 250 nLmin L8 06 06 g g V 04 04 02 072 l torment m c m0 m31 Cnn entramn c mmm3 cancent on c mmm3 EX enmental Concentration Plots Sample Calculation of Variance Calculation of the Variance of Figure 1 geometry with Width05 L25 and Pe100 Start with the Equation for Variance Erincl xyz mexmgcup2 v1 xydAl 11 A 21V 96506114 11 A We will evaluate the top half of the integral by doing boundary integrations on each outlet with the cmixing cup equal to 5 2 Integration 1 01 xy Cmmngcup v1 xyab41 002421 A1 Integration 2 0002163 Integration 3 0001626 Integration 4 0001126 Integration 5 7258641e4 Integration 6 439248e4 Integration 72459387e4 Integration 8 132477e4 Integration 9 6957116e5 Integration 10 3768569e5 Integration 11 2435543e5 Integration 12 198145e5 Integration 13 1827043e5 u minim Next the total is computed by summing all of the 13 channels xyz cmmgmpT v x ydA 9049225e 3 ompute the bottom ofthe integral we use continuity to note that iIvl xydAl IvmdAm 45833 Variance is therefore 9 mg V1 Wm A Iv dA entry in the table above for variance with a width of 05 Pe100 and a length25 19748 2 This hand calculation agrees with the

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