The Diodicity Mechanism ofTesla-Type No-Moving-Parts Valves

Ronald Louis Bardell

A dissertation submitted in partial fulfillmentof the requirements for the degree of

Doctor of Philosophy

University of Washington

2000

Program Authorized to Offer Degree: Mechanical Engineering

University of WashingtonGraduate School

This is to certify that I have examined this copy of a doctoral dissertation by

Ronald Louis Bardell

and have found that it is complete and satisfactory in all respects,and that any and all revisions required by the final

examining committee have been made.

Chairperson of the Supervisory Committee::

Fred K Forster

Reading Committee:

Fred K Forster

James J Riley

Karl F Bhringer

Date:

In presenting this dissertation in partial fulfillment of the requirements for the Doctorialdegree at the University of Washington, I agree that the Library shall make its copiesfreely available for inspection. I further agree that extensive copying of the dissertationis allowable only for scholary purposes, consistent with fair use as prescribed in the U.S.Copyright Law. Requests for copying or reproduction of this dissertation may be referred toBell and Howell Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346, or to the author.

Signature

Date

University of Washington

Abstract

The Diodicity Mechanism ofTesla-Type No-Moving-Parts Valves

by Ronald Louis Bardell

Chairperson of the Supervisory Committee:

Professor Fred K ForsterMechanical Engineering

Microvalves are needed for micropumps that can move particulate-laden fluids in MEMS(Micro-Electro-Mechanical Systems) devices. No-moving-parts (NMP) valves are espe-cially qualified for this task, yet no knowledge of the mechanism that creates valve diodicityin the low-Reynolds number regime so characteristic of microfluidics has been available.As a result, the design of NMP valves has relied on the "build & test" method.

We have developed a numerical method that accurately predicts the diodicity and re-veals the diodicity mechanism of NMP microvalves by combining analysis of field vari-ables from numerical valve simulations with analysis of momentum and kinetic-energyconservation in regional control-volumes. The numerical method is carefully validatedby comparison with known analytical solutions and with experimental data from physicalrealizations of two distinct designs of Tesla-type NMP valves. It predicts their diodicitywithin 4% of measured values. It reveals their low-Reynolds-number diodicity mechanismas the viscous dissipation surrounding laminar jets that have flow-direction-dependent lo-cations and orientations. This diodicity mechanism is dominated by viscous forces, unlikethe high-Reynolds-number mechanism of macro-scale valves that is solely due to inertialforces. Understanding of the diodicity mechanism is encapsulated in design guidelinesfor laying out valve geometry and is demonstrated by developing an enhanced-diodicityvalve design solely by following these guidelines. The numerical method predicts with95% confidence that the diodicity of this new design is a 27-47% improvement over theoriginal design. Clearly, knowledge of the low-Reynolds-number diodicity mechanism inTesla-type NMP valves leads directly to an improved design.

TABLE OF CONTENTS

List of Figures v

List of Tables xii

Chapter 1: Introduction 11.1 Nature and Scope of the Problem . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background for NMP Microvalves . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Diodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Laminar vs. Turbulent Flow . . . . . . . . . . . . . . . . . . . . . 31.2.3 Modeling the Dynamics of Microfluidic Systems . . . . . . . . . . 51.2.4 Steady-State Response . . . . . . . . . . . . . . . . . . . . . . . . 61.2.5 Transient Response . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Background for Tesla-Type NMP Valves . . . . . . . . . . . . . . . . . . . 111.3.1 Prior Research on the Diodicity Mechanism in Tesla-Type NMP

Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 2: The Governing Equations 192.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Kinetic Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Momentum Perspective . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Kinetic-Energy Perspective . . . . . . . . . . . . . . . . . . . . . . 28

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Chapter 3: The Numerical Method 293.1 Methods to Quantify the Diodicity Mechanism . . . . . . . . . . . . . . . 303.2 Numerical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Physical Grid Layout and Grid Independence . . . . . . . . . . . . . . . . 313.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Characteristic Parameters for Nondimensionalization . . . . . . . . . . . . 333.6 Calculation of the Terms in the Conservation Equations . . . . . . . . . . . 343.7 Verification via Analytical Solution for a 2-D Slot Flow . . . . . . . . . . . 35

Chapter 4: Validation of the Numerical Method in Steady Flow 364.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . 364.1.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 Volume Flow-Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.2 Diodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.3 Prediction Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.4 Evidence of Laminar Flow . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 5: Validation of the Numerical Method in Transient Flow 515.1 Harmonic Response of a 2-D Slot . . . . . . . . . . . . . . . . . . . . . . 515.2 Harmonic Response of an NMP Valve . . . . . . . . . . . . . . . . . . . . 535.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter 6: Diodicity Mechanism of T45A Valve 556.1 Simulation Methods and Conditions . . . . . . . . . . . . . . . . . . . . . 556.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.2.1 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2.2 Pressure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2.3 Energy-Dissipation Field . . . . . . . . . . . . . . . . . . . . . . . 606.2.4 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . 646.2.5 Kinetic-Energy Conservation . . . . . . . . . . . . . . . . . . . . . 69

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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Chapter 7: Diodicity Mechanism of T45C Valve 777.1 Simulation Methods and Conditions . . . . . . . . . . . . . . . . . . . . . 777.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.2.1 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2.2 Pressure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.2.3 Energy Dissipation Field . . . . . . . . . . . . . . . . . . . . . . . 827.2.4 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . 867.2.5 Kinetic-Energy Conservation . . . . . . . . . . . . . . . . . . . . . 92

7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Chapter 8: Valve Design Guidelines 998.1 Preliminary Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.2 How To Lay Out a Tesla-Type Valve . . . . . . . . . . . . . . . . . . . . . 101

Chapter 9: Enhanced Diodicity Mechanism of T45A-2 Valve 1069.1 Simulation Methods and Conditions . . . . . . . . . . . . . . . . . . . . . 1069.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.2.1 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.2.2 Pressure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.2.3 Energy Dissipation Field . . . . . . . . . . . . . . . . . . . . . . . 1119.2.4 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . 1159.2.5 Kinetic-Energy Conservation . . . . . . . . . . . . . . . . . . . . . 118

9.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Chapter 10: Conclusions 12410.1 Develop a Numerical Method to Reveal the Diodicity Mechanism . . . . . 124

10.1.1 Verify Mathematically Correctness . . . . . . . . . . . . . . . . . . 12510.1.2 Verify Steady-Flow Response Predictions . . . . . . . . . . . . . . 12510.1.3 Verify Diodicity Prediction Accuracy . . . . . . . . . . . . . . . . 12610.1.4 Verify Transient-Flow Response Predictions . . . . . . . . . . . . . 12710.1.5 Reveal the Diodicity Mechanism in Low Reynolds Number Flow

in Tesla-Type NMP Valves . . . . . . . . . . . . . . . . . . . . . 12710.2 The Low-Reynolds-Number Diodicity Mechanism is Dominated by Vis-

cous Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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10.3 Demonstrate Knowledge of the Diodicity Mechanism . . . . . . . . . . . . 12910.3.1 Develop Valve Design Guidelines . . . . . . . . . . . . . . . . . . 12910.3.2 Demonstrate the Effectiveness of the Guidelines . . . . . . . . . . 130

10.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Bibliography 131

Appendix A: Valve Resistance Modeling 134

Appendix B: Valve Inertance Modeling 137B.1 Step Response of a 2-D Slot . . . . . . . . . . . . . . . . . . . . . . . . . 137B.2 Step Response of a 2-D Slot . . . . . . . . . . . . . . . . . . . . . . . . . 139B.3 Step Response of an NMP Valve . . . . . . . . . . . . . . . . . . . . . . . 139

Appendix C: Series Solution for Starting Flow in a Slot 141

Appendix D: Diodicity From a Ratio of Flow Rates 145

Appendix E: Valve Diodicity Measurements 147

Appendix F: Valve Layout Points 149F.1 T45A Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149F.2 T45C Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151F.3 T45A-2 Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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LIST OF FIGURES

1.1 Flow separation and recirculation at Re 0 01 based on cavity depth (Taneda1979). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Circuit diagram for linear model of complete micropump. . . . . . . . . . 6

1.3 Velocity vector field shows center flow out-of-phase with flow nearer thewall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Closeup photo of a single-element, Tesla-type (T45A) outlet valve connect-ing the pump chamber on the right and the outlet port on the upper left. Thewhite object is a human hair. . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Control volume for momentum analysis of F. Paul of reverse flow throughthe T-junction of a Tesla-type valve. Reverse flow is from right to left. . . . 13

3.1 Typical residuals plot showing termination of interations and procession tonext time step, controlled by USRCVG.F. Note that all residuals have ceasedchanging before a new time step begins: first the Mass residual, then the Wvelocity residual, and finally the U and V velocity residuals. . . . . . . . . 31

4.1 T45A valve pressure drop vs. Reynolds number based on the hydraulicdiameters. Experimental and numerical data are shown as symbols. Thecurves are the the fitted power-law relation (Eq. 4.1). The legends refer toeach valve name and its etch depth; the numerical simulations are markedsim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 T45C valve pressure drop vs. Reynolds number based on the hydraulicdiameters. Experimental and numerical data are shown as symbols. Thecurves are the the fitted power-law relation (Eq. 4.1). The legends refer toeach valve name and its etch depth; the numerical simulations are markedsim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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4.3 Deep T45C valve pressure drop vs. Reynolds number based on the hy-draulic diameters. Experimental and numerical data are shown as symbols.The curves are the the fitted power-law relation (Eq. 4.1). The legendsrefer to each valve name and its etch depth; the numerical simulations aremarked sim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Diodicity following Eq. 1.1 versus Reynolds number of the valves in theT45A Test Group. The symbols are numerical and experimental data.The curves are the ratio of the fitted power-law relations (Eq. 4.1) for thereverse and forward flow directions. Legends refer to test name and etchdepth; numerical simulations are marked sim. . . . . . . . . . . . . . . 44

4.5 Diodicity following Eq. 1.1 versus Reynolds number of the valves in theT45C Test Group. The symbols are numerical and experimental data.The curves are the ratio of the fitted power-law relations (Eq. 4.1) for thereverse and forward flow directions. Legends refer to test name and etchdepth; numerical simulations are marked sim. . . . . . . . . . . . . . . 45

4.6 Diodicity following Eq. 1.1 versus Reynolds number of the valves in theDeep T45C Test Group. The symbols are numerical and experimentaldata. The curves are the ratio of the fitted power-law relations (Eq. 4.1)for the reverse and forward flow directions. Legends refer to test name andetch depth; numerical simulations are marked sim. . . . . . . . . . . . . 45

4.7 Diodicity prediction error of the numerical method for all 11 tests of thethree test groups: the T45A Test Group, the T45C Test Group, and theDeep T45C Test Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.8 Pressure drop versus Reynolds number based on hydraulic diameter. Sym-bols are experimental and numerical data; curves are the fitted power-lawrelation (Eq. 4.1). Legends refer to test name and etch depth; numericalsimulations are marked sim. . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1 Comparison of the nondimensional velocity profiles in a slot with oscillat-ing flow,

8 6 , from the numerical method (symbols) and the exactsolution (lines). The centerline of the slot is at zero slot height and the slotwall is at slot height = 1. The legends note the phase of each profile withrespect to the applied pressure difference, a cosine function. . . . . . . . . 53

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5.2 Terms in the conservation of kinetic-energy equation (Eq. 2.12) for theentire T45A valve over two cycles in a harmonic response simulation at2.818 kHz, including: the energy dissipation rate (PHI), viscous work rate(VWR), energy flux rate (EFR), pressure work rate (PWR), and the tran-sient kinetic-energy (TKE). The dissipation and pressure work are dominant. 54

6.1 Division of the T45A valve into regional control volumes. . . . . . . . . . 56

6.2 Forward-flow velocity field on the centerplane of a single-element, Tesla-type T45A valve with a volume flow rate of 3710 l/min corresponding toRe=528 based on the hydraulic diameter of the main channel. One dimen-sionless unit equals 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3 Reverse-flow velocity field on the centerplane of a single-element, Tesla-type T45A valve with a volume flow rate of 3710 l/min corresponding toRe=528 based on the hydraulic diameter of the main channel. One dimen-sionless unit equals 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.4 Pressure field [atm] on the centerplane of a single-element, Tesla-type T45Avalve with a volume flow rate of 3710 l/min corresponding to Re=528based on the hydraulic diameter of the main channel. . . . . . . . . . . . . 61

6.5 Base 10 logarithm of the energy dissipation rate in forward flow on thecenterplane of a single-element, Tesla-type T45A valve with a volume flowrate of 3710 l/min corresponding to Re=528 based on the hydraulic diam-eter of the main channel. One dimensionless unit equals 14 mW. . . . . . . 62

6.6 Base 10 logarithm of the energy dissipation rate in reverse flow on the cen-terplane of a single-element, Tesla-type T45A valve with a volume flowrate of 3710 l/min corresponding to Re=528 based on the hydraulic diam-eter of the main channel. One dimensionless unit equals 14 mW. . . . . . . 63

6.7 Force vector terms in the integral form of the momentum conservationequation for the T45A valve. Net pressure force and net momentum fluxinto a control volume are positive. Viscous force is applied on the fluidby the wall. X-vectors to the right and Y-vectors upward are positive andconsistent with the valve layout in Fig. 6.1 including the numbering of thecontrol volumes (blocks). . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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6.8 Vector magnitudes of the terms in the momentum conservation equationfor the T45A valve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.9 Magnitude of the pressure work rate and the energy flux rate terms inthe kinetic-energy conservation equation from the T45A valve simulationswith a volume flow rate of 3710 l/min corresponding to Re=528 based onthe hydraulic diameter of the main channel. . . . . . . . . . . . . . . . . . 70

6.10 Magnitude of the energy dissipation rate and the viscous work rate terms inthe kinetic-energy conservation equation from the T45A valve simulationswith a volume flow rate of 3710 l/min corresponding to Re=528 based onthe hydraulic diameter of the main channel. . . . . . . . . . . . . . . . . . 72

6.11 Magnitude of the terms in the kinetic-energy conservation equation fromthe T45A valve simulations with a volume flow rate of 3710 l/min corre-sponding to Re=528 based on the hydraulic diameter of the main channel. . 73

7.1 Division of the T45C valve into regional control volumes. . . . . . . . . . 78

7.2 Forward-flow velocity field on the centerplane of a single-element, Tesla-type T45C valve with a volume flow rate of 3640 l/min corresponding toRe=519 based on the hydraulic diameter of the main channel. One dimen-sionless unit equals 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.3 Reverse-flow velocity field on the centerplane of a single-element, Tesla-type T45C valve with a volume flow rate of 3640 l/min corresponding toRe=519 based on the hydraulic diameter of the main channel. One dimen-sionless unit equals 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.4 Pressure field [atm] on the centerplane of a single-element, Tesla-type T45Cvalve with a volume flow rate of 3640 l/min corresponding to Re=519based on the hydraulic diameter of the main channel. . . . . . . . . . . . . 83

7.5 Base 10 logarithm of the energy dissipation rate in forward flow on thecenterplane of a single-element, Tesla-type T45C valve with a volume flowrate of 3640 l/min corresponding to Re=519 based on the hydraulic diam-eter of the main channel. One dimensionless unit equals 14 mW. . . . . . . 84

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7.6 Base 10 logarithm of the energy dissipation rate in reverse flow on the cen-terplane of a single-element, Tesla-type T45C valve with a volume flow rateof 3640 l/min corresponding to Re=519 based on the hydraulic diameterof the main channel. One dimensionless unit equals 14 mW. . . . . . . . . 85

7.7 Force vector terms in the integral form of the momentum conservationequation. Net pressure force and net momentum flux into a control volumeare positive. Viscous force is applied on the fluid by the wall. X-vectors tothe right and Y-vectors upward are positive and consistent with the valvelayout in Fig. 7.1 including the numbering of the control volumes (blocks). 87

7.8 Vector magnitudes of the terms in the momentum conservation equationfor the T45C valve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.9 Magnitude of the pressure work rate and the energy flux rate terms in thekinetic-energy conservation equation from the T45C valve simulations witha volume flow rate of 3640 l/min corresponding to Re=519 based on thehydraulic diameter of the main channel. . . . . . . . . . . . . . . . . . . . 93

7.10 Magnitude of the energy dissipation rate and the viscous work rate terms inthe kinetic-energy conservation equation from the T45C valve simulationswith a volume flow rate of 3640 l/min corresponding to Re=519 based onthe hydraulic diameter of the main channel. . . . . . . . . . . . . . . . . . 94

7.11 Magnitude of the terms in the kinetic-energy conservation equation fromthe T45C valve simulations with a volume flow rate of 3640 l/min corre-sponding to Re=519 based on the hydraulic diameter of the main channel. . 95

8.1 Sketch of generic Tesla-type NMP valve with dimensioning per designrules for high diodicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.2 Overlay of the T45A (solid red lines) and the T45C (dashed blue lines)showing the variation in path lengths: inlet channel, outlet channel, andside channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9.1 Division of the T45A-2 valve into regional control volumes. . . . . . . . . 1079.2 Forward-flow velocity field on the centerplane of a single-element, Tesla-

type T45A-2 valve with a volume flow rate of 2987 l/min correspondingto Re=500 based on the hydraulic diameter of the main channel. One di-mensionless unit equals 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . 108

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9.3 Reverse-flow velocity field on the centerplane of a single-element, Tesla-type T45A-2 valve with a volume flow rate of 2987 l/min correspondingto Re=500 based on the hydraulic diameter of the main channel. One di-mensionless unit equals 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . 110

9.4 Pressure field [atm] on the centerplane of a single-element, Tesla-type T45A-2 valve with a volume flow rate of 2987l/min corresponding to Re=500based on the hydraulic diameter of the main channel. . . . . . . . . . . . . 112

9.5 Base 10 logarithm of the energy dissipation rate in forward flow on thecenterplane of a single-element, Tesla-type T45A-2 valve with a volumeflow rate of 2987 l/min corresponding to Re=500 based on the hydraulicdiameter of the main channel. One dimensionless unit equals 14 mW. . . . 113

9.6 Base 10 logarithm of the energy dissipation rate in reverse flow on thecenterplane of a single-element, Tesla-type T45A-2 valve with a volumeflow rate of 2987 l/min corresponding to Re=500 based on the hydraulicdiameter of the main channel. One dimensionless unit equals 14 mW. . . . 114

9.7 Force vector terms in the integral form of the momentum conservationequation. Net pressure force and net momentum flux into a control volumeare positive. Viscous force is applied on the fluid by the wall. X-vectors tothe right and Y-vectors upward are positive and consistent with the T45A-2valve layout in Fig. 9.1 including the numbering of the control volumes(blocks). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

9.8 Vector magnitudes of the terms in the momentum conservation equationfor the T45A-2 valve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9.9 Magnitude of the pressure work rate and the energy flux rate terms in thekinetic-energy conservation equation from the T45A-2 valve simulationswith a volume flow rate of 2987 l/min corresponding to Re=500 based onthe hydraulic diameter of the main channel. . . . . . . . . . . . . . . . . . 119

9.10 Magnitude of the energy dissipation rate and the viscous work rate termsin the kinetic-energy conservation equation from the T45A-2 valve simu-lations with a volume flow rate of 2987 l/min corresponding to Re=500based on the hydraulic diameter of the main channel. . . . . . . . . . . . . 120

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9.11 Magnitude of the terms in the kinetic-energy conservation equation fromthe T45A-2 valve simulations with a volume flow rate of 2987 l/min cor-responding to Re=500 based on the hydraulic diameter of the main channel. 122

A.1 Local-slope resistance to fluid flow vs. Reynolds number in T45A andT45C valves from both experiment and numerical simulation followingEq.A.1, which is based on the fitted power-law relation, Eq.4.1. . . . . . . 135

A.2 Fluid resistance vs. volume flow rate in a typical NMP valve. The time-average and average R are approximations of the local-slope R for use inlinear models where a single value is required. Note the similarity to thecharacteristic curve of nonlinear friction for an object moving at low Re ina fluid medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.1 The predicted volume flow-rate response to an applied pressure differencein a 2-D slot from the numerical method (symbols) shows good agreementwith the exponential response from the series solution, Eq. C.8. The timeconstant Re pi2 is also shown. . . . . . . . . . . . . . . . . . . . . . . 140

B.2 Inertance versus Reynolds number from the step-response simulations viaEq. B.5. Inertance shows some dependence on flow rate, but not on flowdirection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

D.1 Characteristic pressure drop versus volume flow rate curves for reverse andforward flow in an NMP valve. Diodicity can be derived from either thepressure-drop ratio or the flow-rate ratio. . . . . . . . . . . . . . . . . . . . 146

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LIST OF TABLES

2.1 Order of magnitude of the integrand in each term in the steady form of themomentum conservation equation Eq. 2.5 for water in a straight duct andan NMP valve at various Rep. . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Order of magnitude of each term in the kinetic-energy equation, Eq. 2.11for various Rep and . For water in an NMP valve with DH 100m,1 f 10kHz corresponds to 8 25 . . . . . . . . . . . . . . . . . 27

4.1 Etch depths and deviations from the mean depth of the valves tested in theT45A group. The tested devices are representative samples of the set sincethe deviations are much less than the measurement accuracy of 6.5%. . . . 38

4.2 Etch depths and deviations from the mean depth of the valves tested in theT45C group. The tested devices are representative samples of the set sincethe deviations are much less than the measurement accuracy of 6.5%. . . . 38

4.3 Etch depths and deviations from the mean depth of the valves tested in theDeep T45C group. The tested devices are representative samples of the setsince the deviations are much less than the measurement accuracy of 6.5%. 38

4.4 Standard deviation and correlation coefficient r2 of the experimentally-measured P with respect to the power-law fit of P versus volume flow-rate following Eq. 4.1. The measurements in the forward and reverse flowdirections were fitted separately. The overall mean standard deviation is3.6%. A correlation coefficient of r2 1 0 is a perfect fit. . . . . . . . . . 43

4.5 Parameters and n of the power-law fit of P versus volume flow-ratefollowing Eq. 4.1. The measurements in the forward and reverse flowdirections were fitted separately. The units of are Pasec m3. . . . . . . . 43

4.6 Mean prediction errors and standard deviations of the volume flow-ratesand the diodicity for each Test Group, shown in percent. . . . . . . . . . . 46

xii

ACKNOWLEDGMENTS

There are many who have helped make my experience as a graduate student a rewardingone. In particular I wish to thank Prof. Fred K. Forster for introducing me to microfluidicsand steadfastly supporting me throughout my PhD studies. Without his encouragement Iwould never have taken up the challenge. I also wish to thank Prof. James J. Riley for hisscientific advice and eagerness to help, Prof. Martin A. Afromowitz for his helpful insights,Prof. George Kosly for guiding me through the Masters program, Prof. Karl F. Bhringerfor being on my Reading Committee, and Prof. Michael J. Pilat for his perspective andadvice. Id like to thank all my fellow graduate students who made learning science a joy,especially Nigel R. Sharma, Chris Morris, Brian Williams, Ling-Sheng Jang, James Pate,Robert Penney, Bill Constantine, Bent Wiencke, Tina Toburen, and Paul Galambos. I amtempted to quote the last remarks of Huck Finn.

xiii

1Chapter 1

INTRODUCTION

1.1 Nature and Scope of the Problem

MEMS (Micro-Electro-Mechanical Systems) devices often contain microfluidic systemsthat are designed to move very small quantities of fluid, such as microliters or even nano-liters, within the device. These devices are often testing units using chemical, electrical, oroptical sensors to analyze chemical compounds outside the traditional medical laboratorysetting so that minitiaturization for portability is of prime concern. Another applicationcurrently generating great research interest is the temperature control of three-dimensionalsemiconductor devices that would greatly benefit from internal cooling by a high heat ca-pacity liquid. A variety of micropumps for moving liquids using active, passive, or no-moving-parts (NMP) valves have been developed in recent years, as reported by Shoji [26]and Gravesen [11].

Many of these micro-scale devices have been developed to pump gases, but pumpingliquids entails additional difficulties. Most applications do not involve a closed-loop sys-tem in which the working fluid can be highly filtered, but require pumping of real-worldfluids that contain particles of several microns, or more, in diameter. These particles canrender the valve seats of active and passive microvalves dysfunctional; if the particles arehard, they damage the valve seats; if they are soft, they can adhere to the seat and preventcomplete valve closure; if the particles are delicate, they can be damaged during passagethrough the valve.

A more robust way to handle particle-laden flows is to use NMP valves that allowfree passage of particles and rely on fluidic instead of mechanical mechanisms to inhibitreverse flow. There are many additional reasons to utilize NMP valves, such as ease ofmanufacture, simplicity of operation, robustness due to lack of moving parts, low cost,etc. However, the technology of NMP valves was developed for the macro-scale, wherethe mechanism to inhibit reverse flow is based on fully-turbulent high-Reynolds-numberflow. This technology is of questionable value at the micro-scale, since microvalves are

2low-Reynolds number devices by nature of the small dimensions of their channels. Thus,efforts to design optimal NMP microvalves suffer from the lack of understanding of fluidicmechanisms that inhibit reverse flow in laminar low-Reynolds-number flow, resulting in adependence on the build & test method.

The goal of this research was to develop an understanding of the fluidic mechanismof the Tesla-type NMP valve in low-Reynolds number flow, an understanding that can beutilized to design improved valves and break the dependency on the build & test method.

1.2 Background for NMP Microvalves

For the reader to understand the operation of all NMP microvalves, several subjects areintroduced in this section. The first is a parameter to characterize valve performance, thevalve diodicity, which is defined in Sec.1.2.1. The second subject is assessment of the flowin the valve as laminar or turbulent, since each requires its own method of analysis. Mi-crovalves are typically small enough in scale to contain laminar flow throughout their rangeof operation. This is particularly true of the valves in this study. Thus, Sec. 1.2.2 describeshow to differentiate laminar from turbulent flow. The final subject is system dynamics.Unlike classic macro-scale check valves, which attempt to prevent reverse flow under allconditions, NMP valves provide a net forward flow in oscillatory flow conditions, and onlymildly inhibit reverse flow in steady flow conditions. Thus, they are always operated inoscillatory conditions, and understanding the dynamics of flow in NMP valves is essen-tial. Sections 1.2.3, 1.2.4, and 1.2.5 introduce characterization of the dynamic response offluid flow in simple channels. These discussions form the basis for the analysis of NMPmicrovalve dynamics in later chapters.

1.2.1 Diodicity

The figure of merit that characterizes the ability to pass flow in the forward direction whileinhibiting flow in the reverse direction is the diodicity of the valve. Since NMP valveshave more resistance to flow in the reverse direction than in the forward direction, theyproduce a unidirectional net flow in the downstream direction even in the presence of abackpressure. The remaining portion of the instantaneous flow is the oscillatory slosh flow.In an electrical analogy, the instantaneous current is a sum of an alternating current (sloshflow) and a direct current (net flow). The diodicity, Di, is defined as

3Di

PreversePforward Q

(1.1)

in which the ratio of pressure loss in the reverse-flow direction to loss in the forward di-rection is taken at identical volume flow rates. The typical diodicity for micro-scale NMPvalves is relatively low, 1 Di 2. The pressure loss could be further broken down by thedependence, or non-dependence, on flow direction as in

Di

Pindependent Pdependent reversePindependent Pdependent forward Q

(1.2)

which shows that direction-independent pressure drop dilutes the diodicity and should beavoided. By definition, Di 1, or the specification of forward and reverse directions wouldbe interchanged. Note that a unity diodicity device produces zero net flow.

The differential pressure loss that creates the diodicity of a valve is due to inertial andviscous forces. The inertial losses are proportional to the square of the velocity and are dueto acceleration of the flow, for example, altering a plug-flow profile at the valve inlet to afully-developed, or even strongly-distorted, velocity profile at the outlet. In addition, localaccelerations distort the velocity profile in regions of separated flow that typically occurwhere there are rapid changes in the channel crossectional area. Even flows with Reynoldsnumbers as low as 0.01 can exhibit separation, as shown in Fig. 1.1 from Taneda, et al,[27]. Viscous losses are proportional to velocity, and relatively unimportant in turbulentflow. They become significant in laminar, separated flow where the velocity gradients arelarge, for example, where jet flow occurs near the valve wall. This dissertation will showthat viscous forces are a major source of valve diodicity in low-Reynolds-number flows.

1.2.2 Laminar vs. Turbulent Flow

It is neccessary to differentiate flow in a microvalve as laminar or turbulent flow. Each flowregime requires its own method of analysis, as the manner in which pressure loss varieswith volume flow rate depends on the flow regime. In laminar flow the friction factor ofa channel is proportional to the volume flow rate; in fully-turbulent flow it is relativelyconstant despite changes in flow rate. The transition from laminar to turbulent flow ina pipe is generally considered to occur at approximately 2000 Re 2300. For non-circular channels of micro-scale dimensions there has been some concern whether this stillholds true. In an early review of microfluidic devices, Gravesen [11] noted that not only

4Figure 1.1: Flow separation and recirculation at Re 0 01 based on cavity depth (Taneda1979).

are the channel width and height of small scale, but typically the channel length is less thanthe entrance length for fully-developed laminar or turbulent flow. If the ratio of length andwidth was large enough L DH 70, he considered flows with Re 2300 to be turbulent.He modeled shorter devices as orifices and labeled flows with more pressure drop due toinertial losses than viscous losses to be turbulent. Very few of the devices he reviewed weremarked by these rules as turbulent. Olsson [20] developed a formula to model pressure dropas a function of the flow rate in his NMP valves, three of which are micro-scale and threeothers are an order of magnitude larger in size. His relation for pressure drop contains twoterms, one representing laminar flow and a second representing turbulent, and he appearsto apply both terms simultaneously to obtain least-squares fits of his experimental data.

There is no need for this uncertainty; a flow can be identified as laminar or turbulent byeither experimental or computational methods. Using experimental data, a laminar flow isidentified by a linear proportionality between the log of the pressure drop across the valveand the volume flow rate, ie. a straight line on a log-log plot of pressure loss versus flowrate. If the flow transitions to turbulence at higher flow rates, the same linear proportional-ity would no longer hold and the slope of the line would change at that flow rate. Transitionto turbulence can also be identified using numerical methods to simulate a flow, becauseas Hinze [13] states, turbulence is defined as irregular flow with random variation of flowproperties (eg. velocity, pressure, etc.) in both time and space coordinates simultaneously.A numerical simulation based on solving the Navier-Stokes equations will not convergeto a steady solution if the flow is randomly varying. Time-averaging of the flow proper-ties, spectral methods, or some other technique must be used to model a turbulent flow.If a Navier-Stokes-based simulation without Reynolds averaging converges to a steady so-

5lution, the modeled flow is laminar. This also applies to the case of harmonic boundaryconditions; if the simulation produces a steady, harmonic solution, the flow is laminar.By these means, the flow in a microvalve can be accurately differentiated as laminar orturbulent, and analyzed accordingly.

1.2.3 Modeling the Dynamics of Microfluidic Systems

System dynamics is discussed here, because unlike classic macro-scale check valves, whichattempt to prevent reverse flow under all conditions, NMP valves operate only in oscillatoryflow conditions. Thus, modeling of the dynamics of flow in NMP valves is essential topredict their performance; diodicity alone is insufficient to characterize an NMP valve.

There are additional reasons to use system dynamics modeling when microvalves areimplemented as components of a micropump with a piezoelectrically-activated membrane.The pump diaphragm must be stiff enough to resist up to pressurize the pump chamberto one atmosphere. Yet, it must be deformed sufficiently by the potential-driven strainof the lead-zirconium-titanate (PZT) piezoelectric actuator to sweep out the neccessaryvolume to supply the slosh flow through the valves. On the other hand, the electrical currentrequirements should be kept low to minimize the size of the power supply and enhanceportability. As a result, the diaphragm displacement per volt of excitation of the PZT mustbe maximized. This is achieved by designing the pump to operate at a minimally-dampedresonance.

According to Gravesen [11], the most common technique used to model the dynamicsof micropumps with microvalves of all types (active, passive, and NMP) is the lumped-parameter method using the electrical-hydraulic analogy. For example, Voigt [29] usedthis method to model a flap-valve micropump. Zengerle [32] also used this method toaddress the interaction between micropumps and their connected fluid system, and devel-oped a methodology for modeling each of the components as well as the entire system.The interaction was shown to be especially prominent for pumps with pulsatile flow, andcapacitive elements were suggested for decoupling the micropump from the measurementsystem. In another example, a complete linear model of a micropump with NMP valves wasintroduced by Bardell, et al. [3] and used to predict the resonance frequency, membrane-displacement amplitude and slosh-flow-rate amplitude. The circuit diagram is shown inFig. 1.2. Lumped-parameter resistance and inertance elements were used to represent theNMP valves: Rvi and Ivi for the inlet valve and Rvo and Ivo for the outlet valve. Another ex-ample is Olssons [20] numerical design study based on a lumped-mass micropump model

6Figure 1.2: Circuit diagram for linear model of complete micropump.

with six coefficients used to match performance data from experiments. In contrast, Morris[17] shows that fitting the model to data is not necessary if the frequency-dependence ofthe valve resistance and inertance elements is considered. Clearly the lumped-parametermethod is a useful technique, thus in addition to revealing the diodicity mechanism andcalculating diodicity, there are discussions of resistance and inertance in Tesla-type NMPvalves in Apps. A and B, respectively.

1.2.4 Steady-State Response

This section introduces the characterization of the steady-state response of flow in simplegeometries, such as a rectangular channel, a pipe, and a two-dimensional slot. The fluidresistance appropriate for a lumped-parameter element is developed forming the basis forlater discussions of NMP valve dynamics.

The resistance to fluid flow characterizes the steady-state response of the volume flowrate to an applied pressure gradient between channel inlet and outlet. Since NMP valvesare typically formed in silicon by DRIE, they are planar structures with channel walls thatare vertical and flow crossections that are rectangular. As a result, the aspect ratio of thechannel, AR height width 1, becomes important in determining the resistance. Clearlya channel with very large aspect ratio will have higher resistance to flow than a channel ofthe same cross-sectional flow area with a unity aspect ratio. This variation in fluid resistance

7is often modeled by using the hydraulic diameter, DH , which is

DH 4Area

wettedperimeter

21

height 1

width(1.3)

and a friction factor that is a function of aspect ratio as well as Reynolds number.The Darcy friction factor,

f 4w12 U2

P12U2

DHL

(1.4)

is the ratio of the energy dissipated in shear and the kinetic energy, and is related to pressuredrop P analogous to the Darcy-Weisbach equation for head loss [31]. Using the method ofO.C. Jones, Jr. [14], the laminar flow friction factor is easily approximated for rectangularchannels by

f 64Re

whereRe UDH and

23

1124 AR

2 AR (1.5)

in which the factor accounts for the variation in aspect ratio and is within 2% of theinfinite series solution. Combining the definition of resistance as the ratio of pressure dropand volume flow rate R P Q P AreaU with Eqs. 1.4 and 1.5 leads to

R 128L

4AreaD2H(1.6)

for resistance to fluid flow in a channel of rectangular crossection and length L. This rela-tion holds for a circular pipe if DH is taken as the pipe diameter and is set to unity, andit also holds for a two-dimensional slot if the usual values for a slot are taken: DH as twicethe slot height, Area as the slot height times unit width, and AR 0 since height width,resulting in

R 12L

height3 (1.7)

1.2.5 Transient Response

This section introduces the characterization of the transient response of flow in simplegeometries, such as a rectangular channel, a pipe, or a two-dimensional slot. The inviscid-flow inertance appropriate for a lumped-parameter element is developed, and the analyticalsolutions for the velocity profiles and impedance of a harmonically-oscillating slot flow arepresented. These concepts form a basis for later discussions of NMP valve dynamics.

8The fluid inertance is a measure of the transient response of the fluid in the channelto a time-varying applied pressure gradient. The fluid inertance in the channel can beapproximated by analogy with either of two standard electrical circuit analysis methods:transient response to a step input, or steady-state forced response to a sinusoidal excitation.

Step response of a channel

A one-dimensional model (based on Newtons second law F ma) for the inertance of aninviscid liquid in a simple straight channel is given by Ogata [18] in which the pressureforce F PA applied to the fluid cross-sectional area A is balanced by the accelerationa dQ Adt of the mass of fluid in the channel m LA as in

P L dQAdt

The inertance I is the ratio of the change in pressure and the resulting change in flow rategiven by

I P

dQ dt L A (1.8)

If a channel filled with inviscid liquid is modelled as a first-order system, (ie. a resis-tor and inductor in series), the response of the volume flow rate to a step input pressuredifference across the channel is an exponential function of time given by

Q t Qmax

1 exp t

At Q Qmax 0 632, the time constant is t I R

where the resistance R is the ratio ofP Qmax

These relations are valid for inviscid flow, but only approximate when viscosity is con-sidered and the crossectional shape of the channel becomes important. The analytical so-lution for impedance in a two dimensional slot is developed in App. C.

Harmonic response of a two-dimensional slot

An exact analytical solution for the velocity profiles is given by Panton [21] for unsteadyflow in a slot driven by an oscillating pressure gradient. This discussion forms the basis forthe development of the exact solutions for the volume flow-rate and fluid impedance of anoscillating slot flow in Chap. 5.

9Figure 1.3: Velocity vector field shows center flow out-of-phase with flow nearer the wall.

If the viscous diffusion length is much less than the channel height 2h, the dimension-less parameter , sometimes referred to as the kinetic-Reynolds number, is described by

h

1 (1.9)

and the center-channel flow becomes out-of-phase with the flow near the wall, as shown inFig. 1.3.

The analytical solution assumes the streamwise velocity, u, is a function of time andthe cross stream direction, y, but not the streamwise direction, x. The cross stream velocity,v, is assumed zero everywhere, because it is zero at the wall. As a result the x-directionmomentum equation becomes linear in u and is

ut

K cost

2uy2 (1.10)

where the amplitude of the oscillating pressure gradient is due to the peak differential pres-sure P across a slot of length L resulting in

K 1

dpdx

max

PL (1.11)

Assuming no slip along the wall and symmetry about the centerline at y 0, since h y h, the initial conditions become u

y h

t 0 at the wall and uy

y 0

t 0 at the

10

centerline.

If the variables are normalized by T t, Y y h, and U u

K , (which scalesthe velocity by the amplitude and frequency of the pressure oscillation), the dimensionlessform of the governing equation becomes

Ut

cosT

h22UY 2 (1.12)

which contains in the third term the reciprocal of the square of the kinetic-Reynolds number2. The general solution for the velocity is

U real

iexpiT 1 cosh iY cosh i

(1.13)

Harmonic response of a general channel

If both the applied pressure difference and the resulting flow rate are sinusoidal, the fluidimpedance can be obtained from the ratio of their amplitudes. This method is appropriatefor determining impedance from the results of a harmonic numerical simulation of oscil-lating flow.

The pressure difference and the flow rate can be represented as the real parts in thecomplex plane by

P Pm Re cos

t

P j sin

t

P Pm Re e jPe jt (1.14)

andQ Qm Re cos

t

Q

j sin t

Q Qm Re e jQe jt (1.15)

The fluid impedance is the ratio of the pressure difference and the flow rate in the complexplane and is

Z j R

jI Pm ejP

Qm e jQ

PmQm cos

P Q j sin

P Q (1.16)

in which the fluid resistance is the real part of Z j and the fluid inertance is the imaginary

part and written asR

PmQm cos

P Q (1.17)

11

Figure 1.4: Closeup photo of a single-element, Tesla-type (T45A) outlet valve connectingthe pump chamber on the right and the outlet port on the upper left. The white object is ahuman hair.

andI

PmQm sin

P Q (1.18)

1.3 Background for Tesla-Type NMP Valves

There are a variety of NMP valve designs. Forster [9] presents techniques for design andtesting of NMP valves including the Tesla-type (discussed below) and the diffuser valve,which is a simple flat diffuser oriented such that the forward flow sees diverging walls andthe reverse flow sees converging walls. Gerlach [10] and Olsson [19] focus their researchsolely on diffuser valves. Other designs are also feasible, such as a micro-scale version ofthe classic vortex diode [5].

The focus of this research is on Tesla-type NMP valves, which as explained by Forster[9] are expected to provide higher diodicity than diffuser-type valves. The simplest config-uration is shown in Fig. 1.4 which is roughly similar to that designed in the macro-scale byNicola Tesla [28] and patented in 1920. It has a bifurcated channel that reenters the mainflow channel perpendicularly when the flow is in the reverse direction. In the forward direc-tion, the majority of the flow is carried by the main channel with reduced pressure losses.The valve channels are typically 60 400m deep, 114m wide, and at least 15-18 widthslong. These smoothly curving shapes are etched in silicon by deep reactive ion etching

12

(DRIE) to attain independence from the crystal planes and achieve vertical sidewalls.NMP valves are used in micropumps, and form the inlet and the outlet to the central

pump chamber, which is generally 3-10 mm in diameter and sealed by a sheet of anodically-bonded Pyrex. The Pyrex serves as the pump diaphragm and is typically actuated by apiezoelectric lead-zirconium-titanate (PZT) wafer, generally 50-95% of the diameter ofthe diaphragm, that is bonded to its outer surface. Applying an alternating voltage to thePZT results in a moment loading of the diaphragm such that it bows in and out of thechamber creating a membrane pump. The inlet and outlet valves are often connected by13-18 gauge stainless steel needles with B-D fittings to tri-fluoro-ethylene (TFE) or siliconerubber tubing.

1.3.1 Prior Research on the Diodicity Mechanism in Tesla-Type NMP Valves

Previous researchers have experimented with macro-scale NMP valves, in which velocity-squared losses are more significant than viscous losses. Paul [23] reported diodicity val-ues up to 4.07 for 0.3 m diameter single-element, momentum-interaction valves and Reed[24] reported values of 12.5 for six-element valves. However, at the micro-scale the lowReynolds number flows result in lower diodicities, typically 1 Di 2.

In his 1920 patent, Tesla [28] claimed that the recesses in the walls of the valvularconduit subjected the reverse flow to rapid reversals of direction resulting in frictionand mass resistance and causing violent surges and eddies which interfere very materi-ally with the flow through the conduit. His test fluid was hot, compressible gas from ahigh-pressure combustion engine resulting in very high-speed turbulent flow. He found theefficacy of device was: first, the reverse flow resistance being larger than the forward; sec-ond, the number of valve elements; third, the character of the gas impulses. He also statedthat the ratio of reverse flow resistance to forward was as high as 200.

Paul [23] states that diodicity is achieved by maximizing the reverse flow total pressureloss coefficient and minimizing the forward, and is a function of the valve geometry. Heconcluded that the major pressure loss in the reverse flow direction is due to confinedjet interacting flows. He performed a control volume analysis of the junction where thechannels rejoin in reverse flow shown in Fig. 1.5. The pressure loss in the main channelwas proportional to the sum of the momentum flux out of the control volume and parallelto the main channel as in

A

P1 P3 m3V3 m1V1

m2V2 cos (1.19)

13

V3 V1

V2

Figure 1.5: Control volume for momentum analysis of F. Paul of reverse flow through theT-junction of a Tesla-type valve. Reverse flow is from right to left.

where A is the crossectional flow area of the main channel, is the intersection angle ofthe side channel with the main channel measured clockwise from horizontal, subscript

2

refers to a location in the side channel, and

1 and

3 are, respectively, upstream anddownstream locations in the main channel. He then assumed that the upstream locationsshared a stagnation zone, and thus had the same static and total pressures, which for incom-pressible flow resulted in V1 V2. His pressure loss predictions agreed with experimentaldata for prototype valves with 45 and overpredicted pressure loss by approximately30% at 90 .

He did not model what he considered secondary losses: flow separation and turbulencedownstream of the intersection of the jets. His test fluid was air, and his valve was 3orders of magnitude larger than NMP microvalves and within the turbulent flow regime, ie.1

700 Re 17

000 and 0 01 Ma 0 13.Reed [24] claimed that the reverse flow losses were due to a vena contracta formed in

the trunk channel at the junction of the trunk flow channel and the reversing flow channelwhere the reversing flow impinged on the trunk channel flow. The vena contracta wasdisabled in forward flow. His test fluid was water. He stated that the best overall headloss performance occurred when the forward flow negotiated 45 bends, instead of the10-20 bends suggested by Tesla. Eddies and other energy sinks were avoided to utilizethe available energy to contract the flow and obtain a head loss. He further teaches thatit was extremely critical to the performance of the valve to follow the exact shapes andpositioning of both the guide vane that separates the channels, and the cusp downstream ofthe junction of the trunk flow channel and the reversing flow channel.

Reed also stated that, unexpectedly, the head loss performance of the valve improved

14

nonlinearly as additional valve elements were added, resulting in pressure ratios of 4, 10,and 12 in 1, 3, and 6 element valves, respectively. He claimed this effect was caused by eachspecial jet contraction or vena contracta acting to maintain high velocity and momentum,which then effected a sizeable contraction at the next downstream jet impingement. Inaddition, each vena contracta directed part of the impinged flow into the next downstreamreversing channel.

Reed notes that a considerable vacuum is formed in the separation region downstreamof the jet impingement, and that relieving that vacuum with a vent caused an additionalshrinkage of the vena contracta there and improved the pressure ratio by more than 20% inone instance.

Overall, the discussion in the literature of the diodicity mechanism of Tesla-type valvestends to be qualitative and always focused on the role inertial losses play in creating diod-icity, and correctly so, as they are dominant in turbulent flow.

1.4 Research Objectives

Since NMP valves originated as high-Reynolds-number devices, previous research has fo-cused solely on inertial forces as the source of diodicity. The technology that was developedis of questionable value for micro-scale devices, since microvalves are low-Reynolds num-ber devices by virtue of the small dimensions of their channels. Thus, efforts to designoptimal NMP microvalves suffer from the lack of understanding of diodicity mechanismsin laminar low-Reynolds-number flow, resulting in a dependence on build & test meth-ods.

It is the hypothesis of this research that a numerical method employing momentumand kinetic-energy conservation in regional control-volumes accurately predicts the diod-icity and reveals the low-Reynolds-number diodicity mechanism of Tesla-type NMP mi-crovalves, that this diodicity mechanism is unlike the mechanism of high-Reynolds-numbermacro-valves in that viscous forces, not inertial forces, are dominant, and that this under-standing of the diodicity mechanism is necessary and sufficient knowledge to establisheffective guidelines for the development of enhanced-diodicity valve designs. Attainmentof the following objectives is sufficient to substantiate this hypothesis:

1. develop a numerical method employing momentum and kinetic-energy conservationin regional control-volumes, and verify that it:

(a) is mathematically correct,

15

(b) accurately predicts flow-rate response to applied pressure in steady-flow condi-tions,

(c) accurately predicts valve diodicity,(d) accurately predicts transient flow-rate response to step input and harmonically-

varying pressures,

(e) reveals the low-Reynolds-number diodicity mechanism of Tesla-type NMP mi-crovalves,

2. show that this diodicity mechanism is dominated by viscous forces, unlike the high-Reynolds-number mechanism of macro-scale valves, which is solely due to inertialforces,

3. demonstrate knowledge of the diodicity mechanism of Tesla-type NMP microvalvesby:

(a) developing guidelines for the creation of enhanced-diodicity valve designs,(b) using these guidelines as the sole method to modify a valve design to signifi-

cantly increase its diodicity.

1.5 Summary

To predict the diodicity and reveal the low-Reynolds-number diodicity mechanism of NMPvalves, we have developed a numerical method that combines analysis of field variablesfrom valve flow simulations with analysis of momentum and kinetic-energy conservationin regional control-volumes. The numerical method is developed from the governing equa-tions and the relative importance of each of their integral components is studied by dimen-sional analysis.

The numerical method is extensively validated. To verify the mathematical accuracyof the numerical method, it is applied to the case of two-dimensional flow in a slot andcompared to the known analytical solution. To verify the predictive ability of the numericalmethod, it is applied to two distinct designs of Tesla-type NMP valves and the predictedvolume-flow-rates resulting from applied pressure-gradients are compared to 242 experi-mental measurements from physically-realized valves. To verify the accuracy of the diodic-ity predictions of the numerical method, they are compared to 121 diodicity measurementsfrom the same physical devices. To verify that the numerical method is time accurate, its

16

predicted volume-flow-rate is compared to the exact analytical solution for oscillating flowin a slot.

The numerical method is consistently able to discern the fluidic mechanism responsiblefor the variation in pressure drop between forward and reverse flow in each valve design.It reveals their low-Reynolds-number diodicity mechanism as the viscous dissipation sur-rounding laminar jets that have flow-direction-dependent locations and orientations. Thisdiodicity mechanism is dominated by viscous forces, unlike the high-Reynolds-numbermechanism of macro-scale valves, which is solely due to inertial forces. Understandingof the low-Reynolds-number diodicity mechanism is employed in the development of ef-fective design guidelines for the geometrical layout of improved Tesla-type NMP valvedesigns. The value of these guidelines is demonstrated by using them as the sole methodto modify a valve design to enhance its diodicity mechanism. The numerical method pre-dicts the diodicity of the improved design is 27-47% higher than the original with 95%confidence.

Since NMP valves operate in oscillatory flow and lumped-parameter elements are oftenused to model valves in system-dynamics analyses, valve resistance and inertance valuesare derived. Valve resistance is direction-dependent and inertance is direction-independent.

For clarity, this dissertation is mainly composed of self-contained chapters, each withits own methods, results, and discussion sections. The contents of the chapters are asfollows:

Chap. 1 presents an introduction to the problem, provides the background on NMPvalves and Tesla-type valves in particular, and lists the research objectives.

Chap. 2 presents the methods used to develop appropriate forms of the momentumand kinetic-energy equations for regional-control-volume-based analyses, thenperforms dimensional analysis. Results of approximation theory are presentedfor each equation. Discussion of the evidence supporting objective #2 endsthe chapter.

Chap. 3 is a methods chapter that explains how the numerical simulations were per-formed, covering such topics as: algorithms, grid layout, boundary conditions,and the characteristic parameters for nondimensionalization. We present thenumerical methods used to calculate each term of the equations developed inChap. 2. We verify the mathematical correctness of these methods to satisfyresearch objective #1a.

17

Chap. 4 presents the methods used to verify the correct implementation of the numer-ical method by comparison of predictions and experimental measurements ofsteady flow in the T45A and T45C, and displays the results. The discus-sion satisfies research objective #1b by demonstrating the accuracy of flow-rate response predictions in steady flow and satisfies research objective #1cby demonstrating the accuracy of the diodicity predictions of the numericalmethod.

Chap. 5 presents the methods used to verify the numerical method as time-accurate bycomparison of predictions and experimental measurements of transient flow inthe T45A and T45C, and displays the results. The discussion satisfies researchobjective #1d by showing the numerical method correctly models transientflow.

Chap. 6 presents the results of the numerical simulations of the Tesla-type T45A valveusing the field variable approach and the regional control volume approach.The following discussion assembles the results to prove reseach objectives#1e and #2.

Chap. 7 presents the results of the T45C valve analogous to Chap. 6. The discussionincludes comparisons with the T45A and also satisfies reseach objectives #1eand #2.

Chap. 8 contains a preliminary discussion that extends key points on the diodicitymechanism discussed in previous chapters, then satisfies reseach objective #3aby presenting methods to obtain improved Tesla-type NMP valve designs inthe form of design guidelines. The guidelines are used to lay out a new valvedesign, the T45A-2, by modifying the T45A valve design to enhance its diod-icity mechanism.

Chap. 9 presents the results of application of the numerical method on the T45A-2valve design and shows the enhancement of the diodicity mechanism. Thediscussion includes comparisons with the T45A and completes the accom-plishment of reseach objective #3b.

Chap. 10 presents the overall conclusions of the research by focusing on the accom-plishment of each of the research objectives.

18

Appendix A discusses lumped-parameter modeling of fluid resistance in NMP valves infirst-order system models and shows valve resistance is direction-dependent.

Appendix B discusses lumped-parameter modeling of fluid inertance in NMP valves infirst-order system models and shows valve inertance is direction-independent.

Appendix C derives a series solution for starting flow in a slot.

Appendix D shows how to calculate diodicity from a ratio of flow rates instead of pressuredrops.

Appendix E contains the measured values of Reynolds number and valve diodicity fromtests of the physical devices.

Appendix F contains the geometrical information used to establish the physical dimen-sions of the T45A, T45C, and T45A-2 in the numerical models.

19

Chapter 2

THE GOVERNING EQUATIONS

In this chapter, useful integral forms of the momentum conservation and kinetic-energyconservation equations are developed. Then dimensional analysis and approximation the-ory are applied to both equations to estimate the significance of each of their terms andprovide evidence supporting objective #2, ie. that viscous forces become dominant at lowReynolds numbers.

2.1 Assumptions

Though these NMP valves are micro-scale with channels widths on the order of 100 m,this scale is still much greater than that required to hold a statistically significant number ofmolecules, assuming the fluid is a liquid. (This would also hold true for a gas near standardtemperature and pressure.) Thus, the continuum hypothesis holds and molecular-averagedproperties of the fluid and of the flow can be defined at any point in the valve. Additionally,since the scale of the valves is large enough, Knudsen number effects (slip velocity alongthe wall) are negligible.

Since NMP valves operate only in oscillatory flow conditions as explained in Sec. 1.2.3,this suggests additional assumptions about the fluid. These valves are typically used in mi-cropumps that operate at resonance as open systems in standard atmospheric conditionswith water as the working fluid. Due to their low compression ratios, these pumps mustavoid creating gas bubbles by cavitation, or most of the pump stroke is lost compress-ing bubbles instead of moving fluid through the valves. Also, since these pumps oper-ate at system resonance, it is reasonable to assume harmonic oscillation and a maximumpressure amplitude of 0.9 atm, which limits the variation of density to less than 5.6% as-suming isothermal conditions and water as the working fluid. Since viscosity is primarilytemperature-dependent, these assumptions define the fluid as having constant density andviscosity, and the flow as incompressible.

20

2.2 Momentum Conservation

The integral form of the momentum equation was obtained following Panton [22] and Leal[16] by application of Newtons second law on a material region, a control volume movingwith the flow. This law states that the time rate of change of linear momentum in a materialregion moving with the fluid (no mass flux across its boundaries) is equal to the forcesapplied on its surfaces, which is

ddt

cv t udV

cs t ndA (2.1)

Then the Reynolds transport theorem,

ddt

cv t dV

cv

t dV

csur ndA

which allows a material control volume moving at ur u to be fixed in space by accountingfor the flux of across its boundaries, was applied to the left hand side of Eq. 2.1 with u, resulting in

cv

t udV

csu

u n dA

cs ndA

a vector force balance relation for a fixed control volume. The right hand side makes useof the fact that the surface force concept is instantaneous and thus the surfaces forces areidentical on a moving material volume and a fixed control volume at the moment they arecoincident in time and space. The stress vector was separated into the thermodynamicpressure and the viscous stress tensor ,

t udV

u

u n dA

PndA

ndA (2.2)

which for an incompressible fluid is u In steady flow the momentum in the control volume is constant, so that the pressure

force normal to the control volume surfaces is equal to the corresponding momentum fluxand shear force in the same direction.

PndA

u

u n dA

u ndA (2.3)

Note that if we had used the divergence theorem to transform the surface integrals into a

21

volume integral, the integrand would be the Navier-Stokes equation for steady, incompress-ible flow.

2.2.1 Dimensional Analysis

To determine their relative magnitudes, approximation theory following Kline [15] wasapplied to each term in the momentum conservation equation, Eq. 2.3. The momentumequation was normalized in a manner that assured the dependent variables were of orderunity, (O)=1, at the maximum value of their range. The magnitude of the resulting groups and the magnitude of each term in the momentum equation were determined fromparameter values for typical operating conditions.

The dependent variables were first nondimensionalized by their characteristic values in

x

y

z x

y

z x

A A 2x

V V 3xu

v

w u

v

w u

t t t

P P p u x u

(2.4)

resulting in dimensional coefficients for each term, respectively, in Eq. 2.2 of

LHS RHS#1 RHS#2 RHS#3

u3xt

2u2x

p2x

ux

Subsequently normalizing by the coefficient of term #2 produced the generic groups of

LHS RHS#1 RHS#2 RHS#3

uxtp

2up

1

upx

For the case of a velocity response to a step input in pressure, the characteristic values(in S.I. units) for the coefficients were chosen to be

p P

x DH

u

P t

x u

in which p was related to the maximum pressure differential applied across the valve, and

22

x was based on the hydraulic diameter of the main valve channel. The resulting groupswere

0 uxtp

1

1 2up

1

2 1

3 u

px

DH

P

1Rep

corresponding to the terms in the momentum equation, Eq. 2.2. All are unity except 3of the viscous force term, which is the reciprocal of a Reynolds number based on the char-acteristic velocity,

P . Inserting these groups, the normalized momentum equationbecame

t udV

u

u n dA

PndA

1Rep

u ndA (2.5)

in which the asterisk superscripts have been dropped. Since the velocity exhibits an expo-nential response to the step input in pressure, u u f

1 exp

t

as steady conditionsare approached the transient term becomes small, and the remaining terms can be gatheredinto a single surface integral and their individual magnitudes assessed.

2.2.2 Results

Approximation theory was applied to determine the relative importance of each term inthe momentum equation, Eq. 2.5. Two cases were studied: fully-developed straight ductflow and NMP valve flow in which the velocity gradients are an order-of-magnitude largerdue to the presence of separated flow with laminar jets. Applying the characteristic valueschosen above, the relative importance of each term shown in Table 2.1 depends stronglyon the Reynolds number. In both cases the momentum flux and pressure force terms aredominant if the values of u and P 1, but at lower Reynolds numbers the viscous forceterm becomes larger than the momentum flux term.

2.3 Kinetic Energy Conservation

The conservation of kinetic energy equation was derived from the scalar product of thevelocity and the transient momentum conservation equation, Eq. 2.2, after applying the

23

Table 2.1: Order of magnitude of the integrand in each term in the steady form of themomentum conservation equation Eq. 2.5 for water in a straight duct and an NMP valve atvarious Rep.

Rep

u

P Momentum Pressure Viscous forceflux force duct valve

1000, 1, 1 100 100 10

3 10

2

100, 0.1, 0.1 10

2 10

1 10

3 10

2

10, 0.01, 0.01 10

4 10

2 10

3 10

2

divergence theorem to the right hand side terms,

u

t udV

u

u n dA

PdV

dV 0

Further development of the desired integral form of the kinetic energy equation followedPanton [22] starting with

t

u22

dV

u2

2

u n dA

u PdV

u dV (2.6)

where is the viscous stress tensor, ie. the stress tensor minus the thermodynamic pressureterms on the diagonal. A more insightful form of the equation was obtained by applyingtwo identities

Pu u P

P

u

u u

i juix j

(2.7)

that separate the respective work rates into kinetic energy and heat energy components.The first identity represents the pressure work rate, and the second, the viscous work rate.The first term on the R.H.S. is the kinetic energy component, and the second term, the heatenergy component. Utilizing these identities to replace the integrands on the R.H.S. of Eq.2.6 resulted in

t

u22

dV

u2

2

u n dA

u ndA

P

u n dA

i juix j

dV

P

u dV (2.8)

24

after applying the divergence theorem to the pressure work rate and the viscous work rateterms to obtain surface integrals. In steady flow the L.H.S. of Eq. 2.8 would be zero. Inincompressible flow the last term on the R.H.S., the compression work rate, is always zeroas it contains the divergence of the velocities. With these assumptions the kinetic energyequation simplified to

P

u n dA

u2

2

u n dA

i juix j

dV

u

ndA (2.9)

Going term by term from left to right, the L.H.S. is the pressure work rate and it is balancedby the energy flux rate, the energy dissipation rate, and the viscous work rate on the R.H.S.The dominant terms are the pressure work rate and the dissipation rate. The energy fluxrate would be zero if, for example, the cross-sectional flow area and the velocity profile ofthe inlet were identical to those of the outlet. The viscous work rate concerns the viscousforces normal to the wall, which are rarely significant. The energy dissipation rate term isthe volume integral of , the dissipation function. For incompressible flow, it is describedby White [30] as

uix j 2

uy

2

2

vy

2

2

wz

2

vx

uy

2

wy

vz

2

uz

wx

2

(2.10)

2.3.1 Dimensional Analysis

Approximation theory was also applied to the kinetic-energy equation to determine the rel-ative magnitude of each term. The equation was normalized for the case of a harmonic ve-locity response to harmonic pressure boundary conditions. The magnitude of the resulting groups and the magnitude of each term in the kinetic energy equation were determinedfrom typical values during operation of the valve.

The dependent variables of the kinetic-energy equation, Eq. 2.8, were normalized inthe same manner as the momentum equation, using Eq. 2.4. The approximation analysisis more straightforward when performed on the volume integral form of the kinetic energyequation, which is

t

u22

dV

u2

2udV

u dV

PudV

i juix j

dV (2.11)

25

in which the divergence theorem has been applied to the surface integral terms, and thecompression work rate has been neglected assuming incompressible flow. The resultingdimensional coefficients for each term, respectively, in Eq. 2.11 are

LHS RHS#1 RHS#3 RHS#2

4

2u3xt

3u2x

pu2x

2ux

Normalizing these by the coefficient of term #3 produced generic groups for each termas

LHS RHS#1 RHS#3 RHS#2

4

uxtp

2up

1

upx

which are identical to those obtained previously for the momentum equation.

For the case of harmonic velocity response to harmonic pressure boundary conditions,the characteristic values (in S.I. units) for the coefficients were chosen to be

p P

x DH

t 1 1 2pi fu

P

in which p was related to the maximum pressure differential applied across the valve, xwas based on the hydraulic diameter of the main valve channel, and t was based on thetime period of the pressure boundary oscillation.

To approximate the magnitude of the first term of Eq. 2.11, the basic assumption of har-monic velocity response, u uo sint, was utilized to evaluate the time-derivative, resultingin a velocity oscillation at twice the frequency of the pressure boundary conditions with amagnitude of one-half the velocity amplitude, as can be seen from

ddt

u2

2

12

ddt

u0 sint 2 u04

ddt

1 cos2t uo2

sin2t

26

The resulting groups were

0 uxtp 2 Rep

1 2u

p 1

3 12 4 upx

1 Rep

corresponding to the terms in the kinetic-energy equation, Eq. 2.11. The transient termgroup 0 is a ratio of the square of an unsteadiness parameter, (sometimes referred to as thekinetic-Reynolds number or the Wommersley parameter), DH

and a Reynoldsnumber Rep, which is based on the characteristic velocity u

P and the hydraulicdiameter of the main valve channel. The groups for the viscous work rate and the energydissipation rate, 1 and 2 are both the reciprocal of the Reynolds number Rep. Droppingthe superscript asterisks, the resulting form of the kinetic-energy equation is

2Rep

t

u2

2dV

u2

2udV

1Rep

u dV

PudV 1Rep

i juix j

dV (2.12)

2.3.2 Results

Approximation theory was applied to determine the relative importance of each term in thekinetic-energy equation, Eq. 2.12. Applying the characteristic values chosen above, thedimensionless u and P are unity. In NMP valve flow the velocity gradients are an order-of-magnitude larger than in fully-developed straight-duct flow due to the presence of separatedflow with laminar jets. The relative importance of each term in the kinetic-energy equationdepends on both Reynolds numbers (ie. Rep and ) as shown by Table 2.2. The dominantterm is the pressure work rate; the transient term and the viscous work rate are negligible;and the remaining terms exchange importance depending on Rep: the energy flux rate ismore important in the higher range of Rep, while the dissipation rate is more important inthe lower range.

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Table 2.2: Order of magnitude of each term in the kinetic-energy equation, Eq. 2.11 forvarious Rep and . For water in an NMP valve with DH 100m, 1 f 10kHz corre-sponds to 8 25

Transient term Energy Viscous Pressure DissipationRep 1 kHz 10 kHz flux rate work rate work rate rate103 10

2 10

1 101 10

1 101 100102 10

3 10

2 10

2 10

2 100 10

1

101 10

4 10

3 10

5 10

3 10

1 10

2

2.4 Discussion

2.4.1 Momentum Perspective

The momentum perspective offers a wealth of information about the interplay of forcesbetween the surfaces, flow boundaries and fluid. In each control volume it is possible todetermine the pressure force and shear force that each surface applies to the fluid, the pres-sure force applied by each flow boundary, and the resulting momentum flux in each of thethree coordinate directions. The pressure forces on the valve calculated from steady-flowsimulations of reverse and forward flow provide an estimate of diodicity if the geometry ofthe valve is appropriate. The inlet and outlet boundaries must have equal areas and surfacenormals that are parallel, since the pressure forces are vectors and diodicity is a scalar. Inthis case the diodicity can be estimated by

Di

PreversePf orward Q

PndA

reverse

PndA

f orward Q (2.13)

Steady-flow momentum conservation, Eq. 2.3, shows clearly two ways to enhancediodicity by increasing the pressure force in the reverse-flow direction over that in forwardflow: increase the reverse-direction shear force (proportional to velocity) or ensure thatmore momentum flux is leaving than entering the control volume (proportional to velocitysquared). The opposite tactics could be employed to increase diodicity by decreasing thepressure force in the forward-flow direction.

In Table 2.1 the momentum flux and pressure force terms are dominant if the valuesof u and P are near unity, but at lower Reynolds numbers the viscous force term becomeslarger than the momentum flux term. This suggests that prior researchers (see p. 12) werecorrect in their neglect of the viscous force term when analyzing their macro-scale valve

28

flows with Re

1

700. But NMP valves are micro-scale devices, and at typical operatingconditions the maximum slosh flow is in the range of 50 Re 500. Thus as the scale ofthe valve decreases the viscous forces become important.

2.4.2 Kinetic-Energy Perspective

Since the Tesla-type NMP valves by design have multiple flow directions, it becomes some-what complicated to utilize the vector-based momentum perspective to obtain a measure ofvalve diodicity, a scalar quantity. The kinetic energy, also a scalar quantity, presents no suchdifficulty. There is no involvement of surfaces or coordinate directions, only the transferof energy across flow boundaries, work done on the fluid, and dissipation within the fluid.And there is a correspondence between the diodicity and the pressure work rate calculatedfrom steady-flow simulations of reverse and forward flow. If the inlet and outlet surfacesare located such that the pressure on each surface has an approximately constant value,then the ratio of the reverse and the forward-flow pressure work rates at the same flow rateprovides a good estimate of the diodicity, the ratio of the corresponding pressure drops, asshown in

Di

PreversePf orward Q

P

u n dA reverse

P

u n dA f orward Q (2.14)

The approximation analysis of the kinetic-energy equation had two important out-comes. First, it suggested that although the viscous work rate was negligible, the energydissipation rate was not, and indeed dominates over the energy flux rate in lower Re p flows.This supports hypothesis #2 that the diodicity mechanism is dominated by viscous forcesin valves of this scale. Secondly, this approximation analysis suggested that the transientterm was negligible even for oscillating water flows with frequencies as high as 10 kHz.This is important in that it suggests that steady-state simulations are sufficient to under-stand the role of the terms in the kinetic-energy equation when modeling flows oscillatingat frequencies up to 10 kHz, including the relative importance of the dissipation and theenergy flux rate.

29

Chapter 3

THE NUMERICAL METHOD

The scale of microvalves presents challenges to understanding their physical behaviourthrough experimentation. Flow visualization using particles, several microns in diameteror less, will significantly affect the flow. Particles, one or two orders smaller, are difficultto see. Fluorescent dyes are useful if they do not affect important fluid properties, ie.viscosity. However, any flow visualization method is hampered by the time scale in whichwhole-field data for a fluid flow oscillating at 1 5kHz must be taken, typically 10 50s.Direct measurement of physical properties, such as instantaneous pressure and flow rate,are difficult because miniaturized flow meters that do not disrupt the flow with sensingelements, (ie. vanes or cantilevered beams), are not yet available. Pressure transducers areon the same scale as a microfluidic system itself, and pressure measurement by the heightof a fluid column profoundly alters the load seen by the system. Fortunately, the samemicro-scale that makes direct physical measurement difficult, makes numerical simulationby computational fluid dynamics (CFD) more accurate, because unlike the macro-scale,realistic flows are not turbulent, but laminar, and the exact governing equations can besolved.

Numerical simulations also have the advantage of offering complete velocity and pres-sure field information, so that energy flux and dissipation rate, as well as momentum fluxand viscous force can be determined in any arbitrary control volume in the flowfield. Thishas the potential of leading to complete understanding of the fluid mechanic behavior, evenin the case of transient or harmonic boundary conditions.

This chapter discusses the numerical method employed to predict the valve diodicityand reveal the diodicity mechanism, including: algorithms, grid independence, boundaryconditions, and characteristic parameters for nondimensionalization. To satisfy researchobjective #1a, it verifies the mathematical correctness of the numerical method via a 2-Dslot flow that has an analytical solution.

30

3.1 Methods to Quantify the Diodicity Mechanism

To identify the diodicity mechanism of an NMP valve, two methods were employed: afield-variable approach and a regional control-volume approach, both based on numericalsimulations of forward and reverse flow in the valve. The field variable approach focuses onthe pressure, velocity, and energy dissipation fields and how they vary between the forwardand reverse flow directions. Study of the pressure fields reveals locations of significantpressure gradients, which are coincident with the highest pressure losses. The velocityfields show flow separation and laminar jets occurring where channels separate or recom-bine and at the valve exit; the energy dissipation rate is most significant where the velocitygradients are largest.

The regional control-volume approach determined the net value in each control volumeof each term in the momentum and energy conservation equations. In the momentumperspective, the integral form of the momentum conservation equation was utilized in eachcontrol volume to study the proportion of pressure force expended on creating momentumflux in comparison to that applied to overcoming viscous force. A second perspective wasgained through the kinetic energy equation by comparing the rate at which pressure workis expended in energy dissipation to the energy flux rate out of the control volume.

The regional control-volume approach enabled the assessment of the significance ofthe flow features (eg. laminar jets, pressure gradients, energy-dissipation regions) seen inthe field-variable approach as momentum or energy loss mechanisms. From the correspon-dence between the field-variable and the regional control-volume analyses it was possibleto uncover the nature of the diodicity mechanism, perceive it in terms of flow features, anddetermine if it was due to changes in momentum or to viscous force, due to redistributionof energy or to dissipation of energy.

3.2 Numerical Algorithms

The simulations of fluid flow in NMP valves were produced with the finite-volume com-putational fluid dynamics package CFX 4.2 from AEA Technology Engineering Software,Inc., Pittsburgh, PA, [7]. These simulations were performed using the laminar, isother-mal, incompressible, transient flow model. The time-stepping algorithm was backward-difference with fixed time steps of length t. During each time step, the number of iter-ations 50 n 1000 performed depended on the change in the residuals of mass m andthe three velocity components: u

v

and w A user routine USRCVG.F was written that com-

31

0 100 200 300 400 500

104

102

100

iteration

resi

dual

UVWMass

Figure 3.1: Typical residuals plot showing termination of interations and procession to nexttime step, controlled by USRCVG.F. Note that all residuals have ceased changing before anew time step begins: first the Mass residual, then the W velocity residual, and finally theU and V velocity residuals.

putes the a