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The Diodicity Mechanism of

Tesla-Type No-Moving-Parts Valves

Ronald Louis Bardell

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

University of Washington

2000

Program Authorized to Offer Degree: Mechanical Engineering


Graduate 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 Böhringer

Date:

In presenting this dissertation in partial fulfillment of the requirements for the Doctorial

degree at the University of Washington, I agree that the Library shall make its copies

freely available for inspection. I further agree that extensive copying of the dissertation

is 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 to

Bell and Howell Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-

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Signature

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Abstract

The Diodicity Mechanism of

Tesla-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 diodicity

in 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-energy

conservation in regional control-volumes. The numerical method is carefully validated

by comparison with known analytical solutions and with experimental data from physical

realizations of two distinct designs of Tesla-type NMP valves. It predicts their diodicity

within 4% of measured values. It reveals their low-Reynolds-number diodicity mechanism

as the viscous dissipation surrounding laminar jets that have flow-direction-dependent lo-

cations and orientations. This diodicity mechanism is dominated by viscous forces, unlike

the high-Reynolds-number mechanism of macro-scale valves that is solely due to inertial

forces. Understanding of the diodicity mechanism is encapsulated in design guidelines

for laying out valve geometry and is demonstrated by developing an enhanced-diodicity

valve design solely by following these guidelines. The numerical method predicts with

95% confidence that the diodicity of this new design is a 27-47% improvement over the

original design. Clearly, knowledge of the low-Reynolds-number diodicity mechanism in

Tesla-type NMP valves leads directly to an improved design.

TABLE OF CONTENTS

List of Figures v

List of Tables xii

Chapter 1: Introduction 1

1.1 Nature and Scope of the Problem . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background for NMP Microvalves . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Diodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Laminar vs. Turbulent Flow . . . . . . . . . . . . . . . . . . . . . 3

1.2.3 Modeling the Dynamics of Microfluidic Systems . . . . . . . . . . 5

1.2.4 Steady-State Response . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.5 Transient Response . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Background for Tesla-Type NMP Valves . . . . . . . . . . . . . . . . . . . 11

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

Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 2: The Governing Equations 19

2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Kinetic Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Momentum Perspective . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2 Kinetic-Energy Perspective . . . . . . . . . . . . . . . . . . . . . . 28

i

Chapter 3: The Numerical Method 29

3.1 Methods to Quantify the Diodicity Mechanism . . . . . . . . . . . . . . . 30

3.2 Numerical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Physical Grid Layout and Grid Independence . . . . . . . . . . . . . . . . 31

3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Characteristic Parameters for Nondimensionalization . . . . . . . . . . . . 33

3.6 Calculation of the Terms in the Conservation Equations . . . . . . . . . . . 34

3.7 Verification via Analytical Solution for a 2-D Slot Flow . . . . . . . . . . . 35

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

4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Volume Flow-Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.2 Diodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.3 Prediction Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.4 Evidence of Laminar Flow . . . . . . . . . . . . . . . . . . . . . . 47

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

Chapter 5: Validation of the Numerical Method in Transient Flow 51

5.1 Harmonic Response of a 2-D Slot . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Harmonic Response of an NMP Valve . . . . . . . . . . . . . . . . . . . . 53

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter 6: Diodicity Mechanism of T45A Valve 55

6.1 Simulation Methods and Conditions . . . . . . . . . . . . . . . . . . . . . 55

6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.2.1 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.2.2 Pressure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2.3 Energy-Dissipation Field . . . . . . . . . . . . . . . . . . . . . . . 60

6.2.4 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . 64

6.2.5 Kinetic-Energy Conservation . . . . . . . . . . . . . . . . . . . . . 69

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

ii

Chapter 7: Diodicity Mechanism of T45C Valve 77


7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.2.1 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.2.2 Pressure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.2.3 Energy Dissipation Field . . . . . . . . . . . . . . . . . . . . . . . 82



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

Chapter 8: Valve Design Guidelines 99

8.1 Preliminary Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.2 How To Lay Out a Tesla-Type Valve . . . . . . . . . . . . . . . . . . . . . 101

Chapter 9: Enhanced Diodicity Mechanism of T45A-2 Valve 106


9.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.2.1 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

9.2.2 Pressure Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

9.2.3 Energy Dissipation Field . . . . . . . . . . . . . . . . . . . . . . . 111



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

Chapter 10: Conclusions 124

10.1 Develop a Numerical Method to Reveal the Diodicity Mechanism . . . . . 124

10.1.1 Verify Mathematically Correctness . . . . . . . . . . . . . . . . . . 125

10.1.2 Verify Steady-Flow Response Predictions . . . . . . . . . . . . . . 125

10.1.3 Verify Diodicity Prediction Accuracy . . . . . . . . . . . . . . . . 126

10.1.4 Verify Transient-Flow Response Predictions . . . . . . . . . . . . . 127

10.1.5 Reveal the Diodicity Mechanism in Low Reynolds Number Flow

in Tesla-Type NMP Valves . . . . . . . . . . . . . . . . . . . . . 127

10.2 The Low-Reynolds-Number Diodicity Mechanism is Dominated by Vis-

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

iii

10.3 Demonstrate Knowledge of the Diodicity Mechanism . . . . . . . . . . . . 129

10.3.1 Develop Valve Design Guidelines . . . . . . . . . . . . . . . . . . 129

10.3.2 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 137

B.1 Step Response of a 2-D Slot . . . . . . . . . . . . . . . . . . . . . . . . . 137

B.2 Step Response of a 2-D Slot . . . . . . . . . . . . . . . . . . . . . . . . . 139

B.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 149

F.1 T45A Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

F.2 T45C Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

F.3 T45A-2 Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

iv

LIST OF FIGURES

1.1 Flow separation and recirculation at Re 0 01 based on cavity depth (Taneda

1979). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 the

wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. The

white object is a human hair. . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Control volume for momentum analysis of F. Paul of reverse flow through

the 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 to

next time step, controlled by USRCVG.F. Note that all residuals have ceased

changing before a new time step begins: first the Mass residual, then the W

velocity residual, and finally the U and V velocity residuals. . . . . . . . . 31

4.1 T45A valve pressure drop vs. Reynolds number based on the hydraulic

diameters. Experimental and numerical data are shown as symbols. The

curves are the the fitted power-law relation (Eq. 4.1). The legends refer to

each valve name and its etch depth; the numerical simulations are marked

“sim”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 T45C valve pressure drop vs. Reynolds number based on the hydraulic

diameters. Experimental and numerical data are shown as symbols. The

curves are the the fitted power-law relation (Eq. 4.1). The legends refer to

each valve name and its etch depth; the numerical simulations are marked

“sim”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

v

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 legends

refer to each valve name and its etch depth; the numerical simulations are

marked “sim”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Diodicity following Eq. 1.1 versus Reynolds number of the valves in the

T45A Test Group. The symbols are numerical and experimental data.

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 and etch

depth; numerical simulations are marked “sim”. . . . . . . . . . . . . . . 44


T45C Test Group. The symbols are numerical and experimental data.

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 and etch

depth; numerical simulations are marked “sim”. . . . . . . . . . . . . . . 45


Deep T45C Test Group. The symbols are numerical and experimental

data. 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 and

etch depth; numerical simulations are marked “sim”. . . . . . . . . . . . . 45

4.7 Diodicity prediction error of the numerical method for all 11 tests of the

three test groups: the T45A Test Group, the T45C Test Group, and the

Deep 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-law

relation (Eq. 4.1). Legends refer to test name and etch depth; numerical

simulations are marked “sim”. . . . . . . . . . . . . . . . . . . . . . . . . 48

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

ing flow,λ 8 6 , from the numerical method (symbols) and the exact

solution (lines). The centerline of the slot is at zero slot height and the slot

wall is at slot height = 1. The legends note the phase of each profile with

respect to the applied pressure difference, a cosine function. . . . . . . . . 53

vi

5.2 Terms in the conservation of kinetic-energy equation (Eq. 2.12) for the

entire T45A valve over two cycles in a harmonic response simulation at

2.818 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 µl/min corresponding to

Re=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 µl/min corresponding to



6.4 Pressure field [atm] on the centerplane of a single-element, Tesla-type T45A

valve with a volume flow rate of 3710 µl/min corresponding to Re=528

based on the hydraulic diameter of the main channel. . . . . . . . . . . . . 61

6.5 Base 10 logarithm of the energy dissipation rate in forward flow on the

centerplane of a single-element, Tesla-type T45A valve with a volume flow

rate of 3710 µl/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 flow



6.7 Force vector terms in the integral form of the momentum conservation

equation for the T45A valve. Net pressure force and net momentum flux

into a control volume are positive. Viscous force is applied on the fluid

by the wall. X-vectors to the right and Y-vectors upward are positive and

consistent with the valve layout in Fig. 6.1 including the numbering of the

control volumes (blocks). . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

vii

6.8 Vector magnitudes of the terms in the momentum conservation equation

for the T45A valve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.9 Magnitude of the pressure work rate and the energy flux rate terms in

the kinetic-energy conservation equation from the T45A valve simulations

with a volume flow rate of 3710 µl/min corresponding to Re=528 based on

the hydraulic diameter of the main channel. . . . . . . . . . . . . . . . . . 70

6.10 Magnitude of the energy dissipation rate and the viscous work rate terms in

the kinetic-energy conservation equation from the T45A valve simulations



6.11 Magnitude of the terms in the kinetic-energy conservation equation from

the T45A valve simulations with a volume flow rate of 3710 µl/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


type T45C valve with a volume flow rate of 3640 µl/min corresponding to




type T45C valve with a volume flow rate of 3640 µl/min corresponding to



7.4 Pressure field [atm] on the centerplane of a single-element, Tesla-type T45C

valve with a volume flow rate of 3640 µl/min corresponding to Re=519



centerplane of a single-element, Tesla-type T45C valve with a volume flow



viii

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 rate

of 3640 µl/min corresponding to Re=519 based on the hydraulic diameter

of the main channel. One dimensionless unit equals 14 mW. . . . . . . . . 85


equation. Net pressure force and net momentum flux into a control volume

are positive. Viscous force is applied on the fluid by the wall. X-vectors to

the right and Y-vectors upward are positive and consistent with the valve

layout in Fig. 7.1 including the numbering of the control volumes (blocks). 87


for the T45C valve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.9 Magnitude of the pressure work rate and the energy flux rate terms in the

kinetic-energy conservation equation from the T45C valve simulations with

a volume flow rate of 3640 µl/min corresponding to Re=519 based on the

hydraulic diameter of the main channel. . . . . . . . . . . . . . . . . . . . 93

7.10 Magnitude of the energy dissipation rate and the viscous work rate terms in

the kinetic-energy conservation equation from the T45C valve simulations




the T45C valve simulations with a volume flow rate of 3640 µl/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 design

rules 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, and

side channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9.1 Division of the T45A-2 valve into regional control volumes. . . . . . . . . 107


type T45A-2 valve with a volume flow rate of 2987 µl/min corresponding

to Re=500 based on the hydraulic diameter of the main channel. One di-

mensionless unit equals 10 m/s. . . . . . . . . . . . . . . . . . . . . . . . . 108

ix


type T45A-2 valve with a volume flow rate of 2987 µl/min corresponding

to 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 2987µl/min corresponding to Re=500



centerplane of a single-element, Tesla-type T45A-2 valve with a volume

flow rate of 2987 µl/min corresponding to Re=500 based on the hydraulic

diameter 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 the

centerplane of a single-element, Tesla-type T45A-2 valve with a volume

flow rate of 2987 µl/min corresponding to Re=500 based on the hydraulic

diameter of the main channel. One dimensionless unit equals 14 mW. . . . 114


equation. Net pressure force and net momentum flux into a control volume

are positive. Viscous force is applied on the fluid by the wall. X-vectors to

the right and Y-vectors upward are positive and consistent with the T45A-2

valve layout in Fig. 9.1 including the numbering of the control volumes

(blocks). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116


for the T45A-2 valve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9.9 Magnitude of the pressure work rate and the energy flux rate terms in the

kinetic-energy conservation equation from the T45A-2 valve simulations



9.10 Magnitude of the energy dissipation rate and the viscous work rate terms

in the kinetic-energy conservation equation from the T45A-2 valve simu-

lations with a volume flow rate of 2987 µl/min corresponding to Re=500


x


the T45A-2 valve simulations with a volume flow rate of 2987 µl/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 and

T45C valves from both experiment and numerical simulation following

Eq.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 in

linear models where a single value is required. Note the similarity to the

characteristic curve of nonlinear friction for an object moving at low Re in

a fluid medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.1 The predicted volume flow-rate response to an applied pressure difference

in a 2-D slot from the numerical method (symbols) shows good agreement

with the exponential response from the series solution, Eq. C.8. The time

constant τ Re π2 is also shown. . . . . . . . . . . . . . . . . . . . . . . 140

B.2 Inertance versus Reynolds number from the step-response simulations via

Eq. B.5. Inertance shows some dependence on flow rate, but not on flow

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

D.1 Characteristic pressure drop versus volume flow rate curves for reverse and

forward flow in an NMP valve. Diodicity can be derived from either the

pressure-drop ratio or the flow-rate ratio. . . . . . . . . . . . . . . . . . . . 146

xi

LIST OF TABLES

2.1 Order of magnitude of the integrand in each term in the steady form of the

momentum conservation equation Eq. 2.5 for water in a straight duct and

an NMP valve at various Rep. . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Order of magnitude of each term in the kinetic-energy equation, Eq. 2.11

for various Rep and λ. For water in an NMP valve with DH 100µm,

1 f 10kHz corresponds to 8 λ 25 . . . . . . . . . . . . . . . . . 27

4.1 Etch depths and deviations from the mean depth of the valves tested in the

T45A group. The tested devices are representative samples of the set since

the deviations are much less than the measurement accuracy of 6.5%. . . . 38


T45C group. The tested devices are representative samples of the set since

the deviations are much less than the measurement accuracy of 6.5%. . . . 38


Deep T45C group. The tested devices are representative samples of the set

since the deviations are much less than the measurement accuracy of 6.5%. 38

4.4 Standard deviation σ 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 flow

directions were fitted separately. The overall mean standard deviation is

3.6%. A correlation coefficient of r2 1 0 is a perfect fit. . . . . . . . . . 43

4.5 Parameters β and n of the power-law fit of ∆P versus volume flow-rate

following Eq. 4.1. The measurements in the forward and reverse flow

directions were fitted separately. The units of β are Pasec m3. . . . . . . . 43

4.6 Mean prediction errors and standard deviations of the volume flow-rates

and 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 rewarding

one. In particular I wish to thank Prof. Fred K. Forster for introducing me to microfluidics

and steadfastly supporting me throughout my PhD studies. Without his encouragement I

would never have taken up the challenge. I also wish to thank Prof. James J. Riley for his

scientific advice and eagerness to help, Prof. Martin A. Afromowitz for his helpful insights,

Prof. George Kosály for guiding me through the Masters program, Prof. Karl F. Böhringer

for being on my Reading Committee, and Prof. Michael J. Pilat for his perspective and

advice. I’d 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 am

tempted to quote the last remarks of Huck Finn.

xiii

1

Chapter 1

INTRODUCTION

1.1 Nature and Scope of the Problem

MEMS (Micro-Electro-Mechanical Systems) devices often contain microfluidic systems

that 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, or

optical sensors to analyze chemical compounds outside the traditional medical laboratory

setting so that minitiaturization for portability is of prime concern. Another application

currently generating great research interest is the temperature control of three-dimensional

semiconductor 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 pumping

liquids 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-world

fluids that contain particles of several microns, or more, in diameter. These particles can

render the valve seats of active and passive microvalves dysfunctional; if the particles are

hard, they damage the valve seats; if they are soft, they can adhere to the seat and prevent

complete valve closure; if the particles are delicate, they can be damaged during passage

through the valve.

A more robust way to handle particle-laden flows is to use NMP valves that allow

free passage of particles and rely on fluidic instead of mechanical mechanisms to inhibit

reverse flow. There are many additional reasons to utilize NMP valves, such as ease of

manufacture, 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, where

the mechanism to inhibit reverse flow is based on fully-turbulent high-Reynolds-number

flow. This technology is of questionable value at the micro-scale, since microvalves are

2

low-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 fluidic

mechanisms that inhibit reverse flow in laminar low-Reynolds-number flow, resulting in a

dependence on the “build & test” method.

The goal of this research was to develop an understanding of the fluidic mechanism

of the Tesla-type NMP valve in low-Reynolds number flow, an understanding that can be

utilized 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 are

introduced in this section. The first is a parameter to characterize valve performance, the

valve diodicity, which is defined in Sec.1.2.1. The second subject is assessment of the flow

in 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 range

of operation. This is particularly true of the valves in this study. Thus, Sec. 1.2.2 describes

how 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 all

conditions, NMP valves provide a net forward flow in oscillatory flow conditions, and only

mildly inhibit reverse flow in steady flow conditions. Thus, they are always operated in

oscillatory 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 of

fluid flow in simple channels. These discussions form the basis for the analysis of NMP

microvalve dynamics in later chapters.

1.2.1 Diodicity

The figure of merit that characterizes the ability to pass flow in the forward direction while

inhibiting flow in the reverse direction is the diodicity of the valve. Since NMP valves

have more resistance to flow in the reverse direction than in the forward direction, they

produce a unidirectional net flow in the downstream direction even in the presence of a

backpressure. 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 (slosh

flow) and a direct current (net flow). The diodicity, Di, is defined as

3

Di ∆Preverse

∆Pforward 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 NMP

valves is relatively low, 1 Di 2. The pressure loss could be further broken down by the

dependence, or non-dependence, on flow direction as in

Di

∆Pindependent ∆Pdependent reverse

∆Pindependent ∆Pdependent forward Q

(1.2)

which shows that direction-independent pressure drop dilutes the diodicity and should be

avoided. By definition, Di 1, or the specification of forward and reverse directions would

be 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 and

viscous forces. The inertial losses are proportional to the square of the velocity and are due

to acceleration of the flow, for example, altering a plug-flow profile at the valve inlet to a

fully-developed, or even strongly-distorted, velocity profile at the outlet. In addition, local

accelerations distort the velocity profile in regions of separated flow that typically occur

where there are rapid changes in the channel crossectional area. Even flows with Reynolds

numbers 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 turbulent

flow. They become significant in laminar, separated flow where the velocity gradients are

large, for example, where jet flow occurs near the valve wall. This dissertation will show

that 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 flow

regime requires its own method of analysis, as the manner in which pressure loss varies

with volume flow rate depends on the flow regime. In laminar flow the friction factor of

a channel is proportional to the volume flow rate; in fully-turbulent flow it is relatively

constant despite changes in flow rate. The transition from laminar to turbulent flow in

a 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 still

holds true. In an early review of microfluidic devices, Gravesen [11] noted that not only

4

Figure 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 than

the entrance length for fully-developed laminar or turbulent flow. If the ratio of length and

width 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 to

inertial losses than viscous losses to be turbulent. Very few of the devices he reviewed were

marked by these rules as turbulent. Olsson [20] developed a formula to model pressure drop

as a function of the flow rate in his NMP valves, three of which are micro-scale and three

others are an order of magnitude larger in size. His relation for pressure drop contains two

terms, one representing laminar flow and a second representing turbulent, and he appears

to 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 by

either experimental or computational methods. Using experimental data, a laminar flow is

identified by a linear proportionality between the log of the pressure drop across the valve

and the volume flow rate, ie. a straight line on a log-log plot of pressure loss versus flow

rate. 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. Transition

to turbulence can also be identified using numerical methods to simulate a flow, because

as Hinze [13] states, turbulence is defined as irregular flow with random variation of flow

properties (eg. velocity, pressure, etc.) in both time and space coordinates simultaneously.

A numerical simulation based on solving the Navier-Stokes equations will not converge

to 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-

5

lution, the modeled flow is laminar. This also applies to the case of harmonic boundary

conditions; 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 or

turbulent, and analyzed accordingly.

1.2.3 Modeling the Dynamics of Microfluidic Systems

System dynamics is discussed here, because unlike classic macro-scale check valves, which

attempt to prevent reverse flow under all conditions, NMP valves operate only in oscillatory

flow conditions. Thus, modeling of the dynamics of flow in NMP valves is essential to

predict their performance; diodicity alone is insufficient to characterize an NMP valve.

There are additional reasons to use system dynamics modeling when microvalves are

implemented as components of a micropump with a piezoelectrically-activated membrane.

The pump diaphragm must be stiff enough to resist up to pressurize the pump chamber

to one atmosphere. Yet, it must be deformed sufficiently by the potential-driven strain

of the lead-zirconium-titanate (PZT) piezoelectric actuator to sweep out the neccessary

volume to supply the slosh flow through the valves. On the other hand, the electrical current

requirements should be kept low to minimize the size of the power supply and enhance

portability. As a result, the diaphragm displacement per volt of excitation of the PZT must

be maximized. This is achieved by designing the pump to operate at a minimally-damped

resonance.

According to Gravesen [11], the most common technique used to model the dynamics

of micropumps with microvalves of all types (active, passive, and NMP) is the lumped-

parameter method using the electrical-hydraulic analogy. For example, Voigt [29] used

this method to model a flap-valve micropump. Zengerle [32] also used this method to

address 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, and

capacitive elements were suggested for decoupling the micropump from the measurement

system. In another example, a complete linear model of a micropump with NMP valves was

introduced 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 in

Fig. 1.2. Lumped-parameter resistance and inertance elements were used to represent the

NMP valves: Rvi and Ivi for the inlet valve and Rvo and Ivo for the outlet valve. Another ex-

ample is Olsson’s [20] numerical design study based on a lumped-mass micropump model

6

Figure 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 of

the valve resistance and inertance elements is considered. Clearly the lumped-parameter

method is a useful technique, thus in addition to revealing the diodicity mechanism and

calculating diodicity, there are discussions of resistance and inertance in Tesla-type NMP

valves 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 simple

geometries, such as a rectangular channel, a pipe, and a two-dimensional slot. The fluid

resistance appropriate for a lumped-parameter element is developed forming the basis for

later discussions of NMP valve dynamics.

The resistance to fluid flow characterizes the steady-state response of the volume flow

rate to an applied pressure gradient between channel inlet and outlet. Since NMP valves

are typically formed in silicon by DRIE, they are planar structures with channel walls that

are vertical and flow crossections that are rectangular. As a result, the aspect ratio of the

channel, AR height width 1, becomes important in determining the resistance. Clearly

a channel with very large aspect ratio will have higher resistance to flow than a channel of

the same cross-sectional flow area with a unity aspect ratio. This variation in fluid resistance

7

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

DH

4Areawettedperimeter

2

1height 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 4τw

12 ρU2

∆P

12ρU2

DH

L (1.4)

is the ratio of the energy dissipated in shear and the kinetic energy, and is related to pressure

drop ∆P analogous to the Darcy-Weisbach equation for head loss [31]. Using the method of

O.C. Jones, Jr. [14], the laminar flow friction factor is easily approximated for rectangular

channels by

f 64φRe

whereRe ρUDHµ andφ

2

3 1124 AR

2 AR (1.5)

in which the factor φ accounts for the variation in aspect ratio and is within 2% of the

infinite series solution. Combining the definition of resistance as the ratio of pressure drop

and volume flow rate R ∆P Q ∆P AreaU with Eqs. 1.4 and 1.5 leads to

R 128µL

4AreaφD2H

(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 φ is set to unity, and

it also holds for a two-dimensional slot if the usual values for a slot are taken: DH as twice

the slot height, Area as the slot height times unit width, and AR 0 since height

width,

resulting in

R 12µL

height3 (1.7)

1.2.5 Transient Response

This section introduces the characterization of the transient response of flow in simple

geometries, 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 analytical

solutions for the velocity profiles and impedance of a harmonically-oscillating slot flow are

presented. These concepts form a basis for later discussions of NMP valve dynamics.

8

The fluid inertance is a measure of the transient response of the fluid in the channel

to a time-varying applied pressure gradient. The fluid inertance in the channel can be

approximated 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 Newton’s second law F ma) for the inertance of an

inviscid liquid in a simple straight channel is given by Ogata [18] in which the pressure

force F ∆PA applied to the fluid cross-sectional area A is balanced by the acceleration

a dQ Adt of the mass of fluid in the channel m ρLA as in

∆P ρLdQAdt

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

given by

I ∆P

dQ dt ρL 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 pressure

difference across the channel is an exponential function of time given by

Qt Qmax

1 exp t τ

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

where the resistance R is the ratio of

∆P 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 unsteady

flow in a slot driven by an oscillating pressure gradient. This discussion forms the basis for

the development of the exact solutions for the volume flow-rate and fluid impedance of an

oscillating slot flow in Chap. 5.

9

Figure 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 λ, sometimes referred to as the kinetic-Reynolds number, is described by

λ hν Ω 1 (1.9)

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

Fig. 1.3.

The analytical solution assumes the streamwise velocity, u, is a function of time and

the 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-direction

momentum equation becomes linear in u and is

∂u∂t

K cosΩt ν∂2u∂y2 (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 1ρ

dpdx max

∆PρL

(1.11)

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

y h, the initial conditions become uy h

t 0 at the wall and ∂u

∂y

y 0

t 0 at the

10

centerline.

If the variables are normalized by T Ωt, Y y h, and U u K Ω , (which scales

the velocity by the amplitude and frequency of the pressure oscillation), the dimensionless

form of the governing equation becomes

∂U∂t

cosT νΩh2

∂2U∂Y 2 (1.12)

which contains in the third term the reciprocal of the square of the kinetic-Reynolds number

λ2. The general solution for the velocity is

U real

iexpiT 1

cosh iλY cosh iλ (1.13)

Harmonic response of a general channel

If both the applied pressure difference and the resulting flow rate are sinusoidal, the fluid

impedance can be obtained from the ratio of their amplitudes. This method is appropriate

for 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 the

complex plane by

∆P Pm Re cosωt θP j sin

ωt θP Pm Re e jθPe jωt (1.14)

and

Q Qm Re cosωt θQ j sin

ωt θQ Qm Re e jθQe jωt

(1.15)

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

plane and is

Zjω R jωI

Pm e jθP

Qm e jθQ

Pm

Qm cosθP θQ j sin

θP θQ (1.16)

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

part and written as

R Pm

Qmcos

θP θQ (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.

and

I Pm

ωQmsin

θP θQ (1.18)

1.3 Background for Tesla-Type NMP Valves

There are a variety of NMP valve designs. Forster [9] presents techniques for design and

testing 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 and

the reverse flow sees converging walls. Gerlach [10] and Olsson [19] focus their research

solely on diffuser valves. Other designs are also feasible, such as a micro-scale version of

the 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 by

Nicola Tesla [28] and patented in 1920. It has a bifurcated channel that reenters the main

flow 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 400µm deep, 114µm wide, and at least 15-18 widths

long. 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 a

piezoelectric lead-zirconium-titanate (PZT) wafer, generally 50-95% of the diameter of

the diaphragm, that is bonded to its outer surface. Applying an alternating voltage to the

PZT results in a moment loading of the diaphragm such that it bows in and out of the

chamber creating a membrane pump. The inlet and outlet valves are often connected by

13-18 gauge stainless steel needles with B-D fittings to tri-fluoro-ethylene (TFE) or silicone

rubber 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 low

Reynolds 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 “valvular

conduit” subjected the reverse flow to rapid reversals of direction resulting in “friction

and 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 a

high-pressure combustion engine resulting in very high-speed turbulent flow. He found the

efficacy 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 stated

that 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 pressure

loss coefficient and minimizing the forward, and is a function of the valve geometry. He

concluded that the major pressure loss in the reverse flow direction is due to “confined

jet interacting flows”. He performed a control volume analysis of the junction where the

channels rejoin in reverse flow shown in Fig. 1.5. The pressure loss in the main channel

was proportional to the sum of the momentum flux out of the control volume and parallel

to the main channel as in

AP1 P3 m3V3 m1V1 m2V2 cosθ (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, θ is the intersection angle of

the 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 and

downstream locations in the main channel. He then assumed that the upstream locations

shared 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 experimental

data for prototype valves with θ 45 and overpredicted pressure loss by approximately

30% at θ 90 .

He did not model what he considered secondary losses: flow separation and turbulence

downstream of the intersection of the jets. His test fluid was air, and his valve was 3

orders of magnitude larger than NMP microvalves and within the turbulent flow regime, ie.

1700 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 channel

where the reversing flow impinged on the trunk channel flow. The vena contracta was

disabled in forward flow. His test fluid was water. He stated that the “best overall head

loss performance” occurred when the forward flow negotiated 45 bends, instead of the

10-20 bends suggested by Tesla. Eddies and other energy sinks were avoided “to utilize

the available energy to contract the flow and obtain a head loss”. He further teaches that

it was “extremely critical” to the performance of the valve to follow the exact shapes and

positioning of both the guide vane that separates the channels, and the cusp downstream of

the 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 each

special jet contraction or vena contracta acting to maintain high velocity and momentum,

which then effected a sizeable contraction at the next downstream jet impingement. In

addition, each vena contracta directed part of the impinged flow into the next downstream

reversing channel.

Reed notes that a considerable vacuum is formed in the separation region downstream

of the jet impingement, and that relieving that vacuum with a vent caused an additional

shrinkage of the vena contracta there and improved the pressure ratio by more than 20% in

one instance.

Overall, the discussion in the literature of the diodicity mechanism of Tesla-type valves

tends 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 developed

is 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 design

optimal NMP microvalves suffer from the lack of understanding of diodicity mechanisms

in 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 momentum

and 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-number

macro-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 establish

effective guidelines for the development of enhanced-diodicity valve designs. Attainment

of the following objectives is sufficient to substantiate this hypothesis:

1. develop a numerical method employing momentum and kinetic-energy conservation

in 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 inertial

forces,

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

by:

(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 NMP

valves, we have developed a numerical method that combines analysis of field variables

from valve flow simulations with analysis of momentum and kinetic-energy conservation

in 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 accuracy

of the numerical method, it is applied to the case of two-dimensional flow in a slot and

compared to the known analytical solution. To verify the predictive ability of the numerical

method, it is applied to two distinct designs of Tesla-type NMP valves and the predicted

volume-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 measurements

from 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 flow

in a slot.

The numerical method is consistently able to discern the fluidic mechanism responsible

for 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. This

diodicity mechanism is dominated by viscous forces, unlike the high-Reynolds-number

mechanism of macro-scale valves, which is solely due to inertial forces. Understanding

of 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 valve

designs. The value of these guidelines is demonstrated by using them as the sole method

to 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 often

used to model valves in system-dynamics analyses, valve resistance and inertance values

are derived. Valve resistance is direction-dependent and inertance is direction-independent.

For clarity, this dissertation is mainly composed of self-contained chapters, each with

its own methods, results, and discussion sections. The contents of the chapters are as

follows:

Chap. 1 presents an introduction to the problem, provides the background on NMP

valves and Tesla-type valves in particular, and lists the research objectives.

Chap. 2 presents the methods used to develop appropriate forms of the momentum

and kinetic-energy equations for regional-control-volume-based analyses, then

performs dimensional analysis. Results of approximation theory are presented

for each equation. Discussion of the evidence supporting objective #2 ends

the 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 the

numerical methods used to calculate each term of the equations developed in

Chap. 2. We verify the mathematical correctness of these methods to satisfy

research 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 of

steady 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 #1c

by demonstrating the accuracy of the diodicity predictions of the numerical

method.

Chap. 5 presents the methods used to verify the numerical method as time-accurate by

comparison of predictions and experimental measurements of transient flow in

the T45A and T45C, and displays the results. The discussion satisfies research

objective #1d by showing the numerical method correctly models transient

flow.

Chap. 6 presents the results of the numerical simulations of the Tesla-type T45A valve

using 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 discussion

includes comparisons with the T45A and also satisfies reseach objectives #1e

and #2.

Chap. 8 contains a preliminary discussion that extends key points on the diodicity

mechanism discussed in previous chapters, then satisfies reseach objective #3a

by presenting methods to obtain improved Tesla-type NMP valve designs in

the form of design guidelines. The guidelines are used to lay out a new valve

design, 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-2

valve design and shows the enhancement of the diodicity mechanism. The

discussion 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 in

first-order system models and shows valve resistance is direction-dependent.

Appendix B discusses lumped-parameter modeling of fluid inertance in NMP valves in

first-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 pressure

drops.

Appendix E contains the measured values of Reynolds number and valve diodicity from

tests 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-energy

conservation equations are developed. Then dimensional analysis and approximation the-

ory are applied to both equations to estimate the significance of each of their terms and

provide evidence supporting objective #2, ie. that viscous forces become dominant at low

Reynolds numbers.

2.1 Assumptions

Though these NMP valves are micro-scale with channels widths on the order of 100 µm,

this scale is still much greater than that required to hold a statistically significant number of

molecules, assuming the fluid is a liquid. (This would also hold true for a gas near standard

temperature and pressure.) Thus, the continuum hypothesis holds and molecular-averaged

properties 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 along

the 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 conditions

with water as the working fluid. Due to their low compression ratios, these pumps must

avoid 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 maximum

pressure 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 primarily

temperature-dependent, these assumptions define the fluid as having constant density and

viscosity, 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 Newton’s second law on a material region, a control volume moving

with the flow. This law states that the time rate of change of linear momentum in a material

region moving with the fluid (no mass flux across its boundaries) is equal to the forces

applied on its surfaces, which is

ddt

cv t ρudV

cs t τ ndA (2.1)

Then the Reynolds transport theorem,

ddt

cv t αdV

cv

∂α∂t

dV cs

αur ndA

which allows a material control volume moving at ur u to be fixed in space by accounting

for the flux of α across its boundaries, was applied to the left hand side of Eq. 2.1 with

α ρu, resulting in

cv

∂∂t

ρudV cs

ρuu n dA

cs

τ ndA

a vector force balance relation for a fixed control volume. The right hand side makes use

of the fact that the surface force concept is instantaneous and thus the surfaces forces are

identical on a moving material volume and a fixed control volume at the moment they are

coincident in time and space. The stress vector τ was separated into the thermodynamic

pressure and the viscous stress tensor τ , ∂

∂tρudV

ρuu n dA

PndA

τ ndA (2.2)

which for an incompressible fluid is τ µ∇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 flux

and shear force in the same direction.

PndA

ρu

u n dA

µ∇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] was

applied to each term in the momentum conservation equation, Eq. 2.3. The momentum

equation was normalized in a manner that assured the dependent variables were of order

unity, (O)=1, at the maximum value of their range. The magnitude of the resulting Πgroups and the magnitude of each term in the momentum equation were determined from

parameter values for typical operating conditions.

The dependent variables were first nondimensionalized by their characteristic values αin

x y

z x

y

z αx

A A α2x

V V α3x

u v

w u

v

w αu

t t αt P ∆P αp

τ ∇ u ταx µαu

(2.4)

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

LHS RHS#1 RHS#2 RHS#3 ραuα3

x

αt

ρα2

uα2x

αpα2

x

µαuαx

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

LHS RHS#1 RHS#2 RHS#3 ραuαx

αtαp

ρα2

u

αp

1

µαu

αpαx

For the case of a velocity response to a step input in pressure, the characteristic values

(in S.I. units) for the α coefficients were chosen to be

αp ∆P

αx DH

αu

∆P ρ

αt αx αu

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

22

αx was based on the hydraulic diameter of the main valve channel. The resulting Π groups

were

Π0

ραuαx

αtαp

1

Π1

ρα2u

αp

1

Π2 1

Π3

µαu

αpαx

ν

DH

∆P ρ

1

Rep

corresponding to the terms in the momentum equation, Eq. 2.2. All are unity except Π3

of the viscous force term, which is the reciprocal of a Reynolds number based on the char-

acteristic velocity,

∆P ρ. Inserting these Π groups, the normalized momentum equation

became ∂∂t

udV

uu n dA

PndA 1

Rep

∇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 f1 exp

t

as steady conditions

are approached the transient term becomes small, and the remaining terms can be gathered

into 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 in

the momentum equation, Eq. 2.5. Two cases were studied: fully-developed straight duct

flow and NMP valve flow in which the velocity gradients are an order-of-magnitude larger

due to the presence of separated flow with laminar jets. Applying the characteristic values

chosen above, the relative importance of each term shown in Table 2.1 depends strongly

on the Reynolds number. In both cases the momentum flux and pressure force terms are

dominant if the values of u and P 1, but at lower Reynolds numbers the viscous force

term 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 the

velocity 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 force

flux force duct valve1000, 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 ∂

∂tρudV

ρuu n dA

∇PdV

∇ τ dV 0

Further development of the desired integral form of the kinetic energy equation followed

Panton [22] starting with

∂∂t

ρu2

2dV

ρu2

2

u n dA

u ∇PdV

u ∇ τ dV (2.6)

where τ is the viscous stress tensor, ie. the stress tensor minus the thermodynamic pressure

terms on the diagonal. A more insightful form of the equation was obtained by applying

two identities

∇ Pu u ∇P P∇ u

∇ τ u u ∇ τ τ i j∂ui

∂x 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 heat

energy component. Utilizing these identities to replace the integrands on the R.H.S. of Eq.

2.6 resulted in

∂∂t

ρu2

2dV

ρ

u2

2

u n dA τ u ndA

P

u n dA

τ i j∂ui

∂x jdV

P∇ u dV (2.8)

24

after applying the divergence theorem to the pressure work rate and the viscous work rate

terms to obtain surface integrals. In steady flow the L.H.S. of Eq. 2.8 would be zero. In

incompressible flow the last term on the R.H.S., the compression work rate, is always zero

as it contains the divergence of the velocities. With these assumptions the kinetic energy

equation simplified to

P

u n dA

ρ

u2

2

u n dA

τ i j

∂ui

∂x jdV τ u ndA (2.9)

Going term by term from left to right, the L.H.S. is the pressure work rate and it is balanced

by 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 flux

rate would be zero if, for example, the cross-sectional flow area and the velocity profile of

the inlet were identical to those of the outlet. The viscous work rate concerns the viscous

forces normal to the wall, which are rarely significant. The energy dissipation rate term is

the volume integral of φ, the dissipation function. For incompressible flow, it is described

by White [30] as

φ τ ∂ui

∂x j

µ 2 ∂u∂y 2 2

∂v∂y 2 2

∂w∂z 2

µ ∂v∂x ∂u

∂y 2 ∂w∂y ∂v

∂z 2 ∂u∂z ∂w

∂x 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

Π groups and the magnitude of each term in the kinetic energy equation were determined

from typical values during operation of the valve.

The dependent variables of the kinetic-energy equation, Eq. 2.8, were normalized in

the same manner as the momentum equation, using Eq. 2.4. The approximation analysis

is more straightforward when performed on the volume integral form of the kinetic energy

equation, which is

∂∂t

ρu2

2dV

∇ ρ

u2

2udV

∇ τ u dV

∇ PudV

τ i j∂ui

∂x jdV (2.11)

25

in which the divergence theorem has been applied to the surface integral terms, and the

compression work rate has been neglected assuming incompressible flow. The resulting

dimensional coefficients for each term, respectively, in Eq. 2.11 are

LHS RHS#1 RHS#3 RHS#24

ρα2

uα3x

αt

ρα3

uα2x

αpαuα2

x

µα2

uαx

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

asLHS RHS#1 RHS#3 RHS#2

4

ραuαx

αtαp

ρα2

u

αp

1

µαu

αpαx

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 α coefficients were chosen to be

αp ∆P

αx DH

αt 1 Ω 1 2π f

αu

∆P ρ

in which αp was related to the maximum pressure differential applied across the valve, αx

was based on the hydraulic diameter of the main valve channel, and αt was based on the

time 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, resulting

in a velocity oscillation at twice the frequency of the pressure boundary conditions with a

magnitude of one-half the velocity amplitude, as can be seen from

ddt

u2

2

12

ddt

u0 sint 2

u0

4ddt

1 cos2t

uo

2sin2t

26

The resulting Π groups were

Π0 ραuαx

αtαp

λ2 Rep

Π1 ρα2

uαp

1

Π3 1

Π2 4 µαuαpαx

1 Rep

corresponding to the terms in the kinetic-energy equation, Eq. 2.11. The transient term

group Π0 is a ratio of the square of an unsteadiness parameter, (sometimes referred to as the

kinetic-Reynolds number or the Wommersley parameter), λ DH

ν Ω and a Reynolds

number Rep, which is based on the characteristic velocity αu

∆P ρ and the hydraulic

diameter of the main valve channel. The groups for the viscous work rate and the energy

dissipation rate, Π1 and Π2 are both the reciprocal of the Reynolds number Rep. Dropping

the superscript asterisks, the resulting form of the kinetic-energy equation is

λ2

Rep

∂∂t

u2

2dV

∇ u2

2udV 1

Rep

∇ τ u dV

∇ PudV 1

Rep

τ i j

∂ui

∂x jdV (2.12)

2.3.2 Results

Approximation theory was applied to determine the relative importance of each term in the

kinetic-energy equation, Eq. 2.12. Applying the characteristic values chosen above, the

dimensionless 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 separated

flow with laminar jets. The relative importance of each term in the kinetic-energy equation

depends on both Reynolds numbers (ie. Rep and λ) as shown by Table 2.2. The dominant

term 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 is

more important in the higher range of Rep, while the dissipation rate is more important in

the lower range.

27

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

Transient term Energy Viscous Pressure DissipationRep 1 kHz 10 kHz flux rate work rate work rate rate103 10 2 10 1 101 10 1 101 100

102 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 forces

between the surfaces, flow boundaries and fluid. In each control volume it is possible to

determine 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 the

three coordinate directions. The pressure forces on the valve calculated from steady-flow

simulations of reverse and forward flow provide an estimate of diodicity if the geometry of

the valve is appropriate. The inlet and outlet boundaries must have equal areas and surface

normals that are parallel, since the pressure forces are vectors and diodicity is a scalar. In

this case the diodicity can be estimated by

Di

∆Preverse

∆Pf orward Q PndA

reverse

PndAf orward Q

(2.13)

Steady-flow momentum conservation, Eq. 2.3, shows clearly two ways to enhance

diodicity by increasing the pressure force in the reverse-flow direction over that in forward

flow: increase the reverse-direction shear force (proportional to velocity) or ensure that

more momentum flux is leaving than entering the control volume (proportional to velocity

squared). The opposite tactics could be employed to increase diodicity by decreasing the

pressure force in the forward-flow direction.

In Table 2.1 the momentum flux and pressure force terms are dominant if the values

of u and P are near unity, but at lower Reynolds numbers the viscous force term becomes

larger than the momentum flux term. This suggests that prior researchers (see p. 12) were

correct in their neglect of the viscous force term when analyzing their macro-scale valve

28

flows with Re 1700. But NMP valves are micro-scale devices, and at typical operating

conditions the maximum slosh flow is in the range of 50 Re 500. Thus as the scale of

the 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 of

valve diodicity, a scalar quantity. The kinetic energy, also a scalar quantity, presents no such

difficulty. There is no involvement of surfaces or coordinate directions, only the transfer

of 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 calculated

from steady-flow simulations of reverse and forward flow. If the inlet and outlet surfaces

are 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 rate

provides a good estimate of the diodicity, the ratio of the corresponding pressure drops, as

shown in

Di

∆Preverse

∆Pf orward Q

Pu 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 energy

dissipation 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 forces

in valves of this scale. Secondly, this approximation analysis suggested that the transient

term 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 oscillating

at frequencies up to 10 kHz, including the relative importance of the dissipation and the

energy flux rate.

29

Chapter 3

THE NUMERICAL METHOD

The scale of microvalves presents challenges to understanding their physical behaviour

through experimentation. Flow visualization using particles, several microns in diameter

or less, will significantly affect the flow. Particles, one or two orders smaller, are difficult

to 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 which

whole-field data for a fluid flow oscillating at 1 5kHz must be taken, typically 10 50µs.

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 sensing

elements, (ie. vanes or cantilevered beams), are not yet available. Pressure transducers are

on the same scale as a microfluidic system itself, and pressure measurement by the height

of a fluid column profoundly alters the load seen by the system. Fortunately, the same

micro-scale that makes direct physical measurement difficult, makes numerical simulation

by computational fluid dynamics (CFD) more accurate, because unlike the macro-scale,

realistic flows are not turbulent, but laminar, and the exact governing equations can be

solved.

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 flux

and viscous force can be determined in any arbitrary control volume in the flowfield. This

has the potential of leading to complete understanding of the fluid mechanic behavior, even

in the case of transient or harmonic boundary conditions.

This chapter discusses the numerical method employed to predict the valve diodicity

and reveal the diodicity mechanism, including: algorithms, grid independence, boundary

conditions, and characteristic parameters for nondimensionalization. To satisfy research

objective #1a, it verifies the mathematical correctness of the numerical method via a 2-D

slot 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: a

field-variable approach and a regional control-volume approach, both based on numerical

simulations of forward and reverse flow in the valve. The field variable approach focuses on

the pressure, velocity, and energy dissipation fields and how they vary between the forward

and reverse flow directions. Study of the pressure fields reveals locations of significant

pressure gradients, which are coincident with the highest pressure losses. The velocity

fields 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 velocity

gradients are largest.

The regional control-volume approach determined the net value in each control volume

of each term in the momentum and energy conservation equations. In the momentum

perspective, the integral form of the momentum conservation equation was utilized in each

control volume to study the proportion of pressure force expended on creating momentum

flux in comparison to that applied to overcoming viscous force. A second perspective was

gained through the kinetic energy equation by comparing the rate at which pressure work

is 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 of

the flow features (eg. laminar jets, pressure gradients, energy-dissipation regions) seen in

the 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 possible

to uncover the nature of the diodicity mechanism, perceive it in terms of flow features, and

determine if it was due to changes in momentum or to viscous force, due to redistribution

of 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 and

the three velocity components: uv

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

31

0 100 200 300 400 500

10−4

10−2

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 average residual of the last 10 iterations. Once the change in each of the average

residuals m u v w is less than 1% of their magnitude, the main program is signalled to

proceed to the next time step, as shown in Fig. 3.1 from a typical simulation. The govern-

ing equations were the incompressible form of momentum conservation, which inherently

includes mass conservation, and a pressure correction following the “SIMPLEC” solution

algorithm. The discretization scheme CCCT was chosen, a quadratic upwind differencing

that uses two upwind points and one downwind. It is third-order accurate for the advection

terms and second order for the other terms including the diffusion terms. It is a modification

of the QUICK scheme in that it is bounded to prevent non-physical overshoots.

3.3 Physical Grid Layout and Grid Independence

The three-dimensional grids for the valve geometry were body-fitted grids in cartesian

coordinates with 22 by 15-20 cross-stream and 286-292 streamwise grid points, resulting

in totals of 94380-128480 finite volumes. The grid density was refined near walls and

at channel junctions, where large velocity gradients were expected. The dimensions of

the valve channels were extracted directly from the CIF format files containing the layout

drawings for the photolithography masks of micropumps with single valves at the inlet and

outlet. The sets of points defining the geometry of the T45A, T45C and T45A-2 valves are

included in Appendix F. The as-etched geometry varies from the mask design where cusps

32

and protruding corners exist, which etch at a slightly faster rate than the rest of the pattern.

The entrance to the inlet or outlet channel from the pump chamber or inlet/outlet port is not

sharp-edged, and was measured from a magnified image to have rounded corners of 15µm

radius. The nominal etch depth of the valve channels was 120µm, so the grid depth was

specified as 60µm assuming the flow was symmetrical about the valve centerplane. This

assumption allowed a 50% reduction in memory usage and reduced the simulation run time

by a factor of 3-4 to less than 24 hours of wall clock time.

The assessment of grid independence of the simulations was based on the accuracy of

the calculation of kinetic-energy conservation in Eq. 2.11. Since ideally the terms should

sum to zero, the error εKE was defined as the ratio of the sum of the terms and the root

of the sum of the squares of the terms. When the reduction in εKE was less than 5%

for a refined grid with at least 20% additional finite volumes, the solution was considered

grid independent. Note that the effect of this 20% grid refinement on the predicted value

of diodicity was less than 0.5% which is an order-of-magnitude less than the difference

between the predicted diodicity and that measured experimentally (see Chap 4).

The grid for the slot flow simulation used to verify the mathematical correctness of the

numerical method was laid out as a three-dimensional grid with a very high aspect ratio, 1

unit high by 100 units wide by 10 units long, to ensure that identical FORTRAN routines

could be used for the slot flow and the valve flow simulations. The grid was in cartesian

coordinates with 22 by 18 cross-stream and 100 streamwise grid points, resulting in a total

of 39600 finite volumes. A grid independence study was not done since the numerical

solution was compared directly to an analytical solution.

3.4 Boundary Conditions

Pressure boundaries were used at both inlet and outlet of the valve. They were located

beyond the inlet and the outlet in large, goblet-shaped plenums designed so that the velocity

at the pressure boundary was small, since specifying pressure boundary conditions also

applies zero spatial velocity gradients at that surface. The goblet-shaped plenums are an

approximation of a half-section the cylindrically-shaped inlet/outlet ports of the physical

devices.

The pressure boundaries for the slot flow were applied directly to the ends of the chan-

nels since zero velocity gradients were assumed in the streamwise direction.

33

Step-Response Simulations

In these simulations a differential pressure was applied, via the pressure boundaries at the

inlet and outlet, to an initially-zeroed velocity field. The downstream boundary was set to

zero pressure, thus the pressure field in the simulations was in terms of gauge pressure.

The pressures applied at the upstream boundary in the reverse-flow simulations were 0.1,

0.5, and 0.9 atm. Sufficiently lower pressures were applied at the upstream boundary in the

forward-flow simulations so that the volume-flow-rate responses were within less than 1%

of the reverse-flow value and the diodicity could be calculated via Eq. 1.1.

Harmonic-Response Simulations

These simulations also start with an initially-zeroed velocity field and a downstream bound-

ary set at zero pressure. The upstream pressure boundary was set each time step to the

average value over that time step following

Pupstream

Pmax

∆t

t

t ∆tsinΩt dt

Pmax

Ω∆t

cosΩ

t ∆t cosΩt

in the user-supplied subroutine USRBCS.F, where ∆t is the fixed width of the time steps,

t is the current time during the time step, and Ω is the radian frequency of oscillation.

To minimize the initial transient response and reach the long-term harmonic behavior

more quickly, Pmax is defined as a linear function of time during the first period, so that

Pmax0 t ∆t Pmax t Ω 2π

3.5 Characteristic Parameters for Nondimensionalization

The governing equations were nondimensionalized as described in Secs. 2.2 and 2.3 both

to generalize the solution to different geometries and to minimize roundoff error that would

have resulted from the using SI units for the spatial dimensions, which are on the order of

(O) = 10 4. The characteristic values chosen were:

αp = 101325 Pa = 1 atmosphere

αx 2

1 h 1 w 116 9 10 6m

the hydraulic diameter of the main valve channel

αu

αp ρ 10 066 m/s, a characteristic velocity

34

αt αx αu

11 616 10 6 s, a characteristic time.

Since the CFD software package (CFX 4.2) solves a dimensional equation set, values are

required for the following fluid properties in the CFX command file to implement the nondi-

mensionalization scheme:

density 1

viscosity 1Re

ναuαx

8 497 10 4, where ν 1 0 10 6m2 s.

3.6 Calculation of the Terms in the Conservation Equations

To calculate the terms of the momentum and kinetic-energy conservation equations, the

CFD simulations were modified by adding FORTRAN subroutines to accomplish two main

tasks: to export information, and to perform auxiliary computations after each time step.

To calculate the terms of the momentum equation, subroutines were written to export

grid geometry, fluid velocity, mass flux, and fluid pressure at the surfaces of the control

volumes. Matlab routines [12] were written to calculate the pressure force and momentum

flux through each surface from the relevant data on that surface: surface location, surface

area, pressures, and velocities. From the x and y components of the surface area and the

velocities, the surface normal and the flow rate normal to the surface were calculated. The

pressure force on the surface in each spatial direction and the momentum flux through

the surface in each direction were calculated by integrating the respective quantities over

the surface following Eq: 2.5. These were combined to calculate the pressure forces and

momentum fluxes on each of the six faces of each of the control volumes that make up the

valve.

A different programming strategy was employed to calculate the magnitudes of the

terms in the kinetic-energy conservation equation. FORTRAN subroutines were written to

perform auxiliary computations within the simulation after each time step and the magni-

tudes of the terms were then exported to disk files. Matlab routines were used to gather

the data from the forward-flow and reverse-flow simulations and make comparison plots.

The form of the kinetic-energy conservation equation used to calculate the energy flux rate,

work rates and dissipation rate was the volume integral form of Eq: 2.12 with two ex-

ceptions. Using the fact that in incompressible flow ∇ u 0

the energy flux rate was

calculated as

u ∇u2 2dV and the pressure work rate as

u ∇PdV . These formulations

reduced the use of velocity gradients, which are difficult to resolve accurately.

35

3.7 Verification via Analytical Solution for a 2-D Slot Flow

We have added numerical methods to the incompressible Navier-Stokes solver in CFX

4.2 to calculate kinetic-energy conservation by employing “user subroutines” written in

FORTRAN. These added routines must be verified as mathematically correct. A step-

response simulation of slot flow was performed to validate these routines. The differential

pressure applied was 0.01 atm and the resulting volume flow rate was equilibrated within

72 times steps at Re 115, based on the slot height.

The pressure work rate, energy flux rate, and the dissipation rate are the significant

terms in the kinetic-energy equation as discussed in Sec. 2.3. In incompressible flow in a

slot the energy flux rate is zero since the velocity profile is unchanged throughout a control

volume which contains only fully-developed flow. Thus the pressure work rate and the

dissipation rate must be equal and opposite.

The pressure work rate was calculated from

P

u n dA ∆PLz

udy ∆PLz

Ly

0

∆P2µLx

y y2 dy

where the dimensionless lengths are Lx 10

Ly

1Lz

100the differential pressure is

∆P 0 01and the dimensionless viscosity is µ 8 497 10 4 using the nondimension-

alization scheme of Sec. 3.5. The analytical solution for the pressure work rate is 0.0981,

the numerical solution was 0.09808, which is within 0.02%.

Since the velocity u uy is only a function of one coordinate direction, the dissipation

rate (see Eq. 2.10) was calculated using

τ i j

∂ui

∂x jdV LxLz

Ly

0µ

∂u

y

∂y 2

dy

As expected, the dissipation rate was equal to the pressure work rate. The numerical solu-

tion was 0.09807, which is within 0.03%.

The energy flux, as explained above, should be zero. The energy flux calculated in the

numerical simulation was 4 067 10 6, which is four orders-of-magnitude smaller than

the pressure work rate and the dissipation rate.

Thus the slot-flow simulation verifies that the added numerical methods used to cal-

culate the terms of the kinetic-energy equation are mathematically correct and research

objective #1a is satisfied.

36

Chapter 4

VALIDATION OF THE NUMERICAL METHOD IN STEADY

FLOW

Chapter 3 has described and discussed the implementation of the numerical method

used in this research to predict the flow-rate response to an applied pressure difference,

predict the valve diodicity, and reveal the diodicity mechanism. The mathematical cor-

rectness of this method (research objective #1a) was addressed in that chapter; it is the

task of this chapter to verify the numerical method has been correctly implemented in this

research and thus accurately predicts the steady-flow response and the diodicity of Tesla-

type NMP valves as required by research objectives #1b and #1c. Attainment of objectives

#1d and #1e, proof that the numerical method is time-accurate and can reveal the diodicity

mechanism, is left to following chapters. The current task is achieved by comparing vol-

ume flow-rate and diodicity predictions to 242 independent experimental observations from

physical-realized valves of three distinct groups of etch depths and two distinct Tesla-type

designs, the T45A and the T45C. By basing the comparison on data from multiple valve

geometries, we determine the accuracy of the numerical method’s diodicity predictions for

similar Tesla-type valves in a manner that is independent of geometry. In other words, we

are free to modify valve geometry (to the same extent as the variation between T45A and

T45C) and we still know the accuracy of the numerical method’s diodicity predictions.

4.1 Methods

4.1.1 Experimental Methods

To obtain physical devices for experimental data, T45A and T45C Tesla-type valves were

produced in silicon by deep reactive-ion etching DRIE at the Stanford Nanofabrication Fa-

cility and covered by a layer of Pyrex attached by anodic bonding. The etch depths of the

valves vary from wafer to wafer (approx. 25%) as well as by location on the wafer (approx.

15%) resulting in a range of etch depths of 90 h 150µm and a mean of h 120µm.

A technique for measuring the etch depth was developed using an MTI Photonic displace-

ment sensor (MTI, Latham, NY) and a Newport Corporation 3-axis stage with 855C pro-

37

grammable controller connected by GPIB interface to a 486PC running an existing FOR-

TRAN motion-control program. For each valve a series of n measurements (7 n 10) at

each of three locations (two up on the chip surface and one down in the valve inlet/outlet

port) were processed with a geometric-analysis Matlab routine to determine the valve-port

etch depth to within 5 micrometers. The valve-port depth was measured since the valve

channel is narrower than the spot size of the photonic sensor and cannot be measured di-

rectly with this technique. The valve port diameter is large enough to contain the sensor

spot, yet only 6.5 times larger than the channel width. To determine if the etch depths of

the valve channels are equal to the valve-port etch depths, four devices were milled and

polished and investigated with a microscope. The valve-channel etch depths were within

5% of the measured valve-port etch depth. The total etch-depth measurement error etotal

for the average-depth channel is a product of two independent errors: the valve-port etch

measurement error eport and the variation between valve-port etch and valve-channel etch

eport channel , and was determined by

etotal

e2

port e2port channel

5

120 2 052

065 or 6 5%

following Barlow [4].

An effort was made to select devices for testing that were close to the mean etch depth

of 120 µm. The resulting range of measured etch depths of the chosen devices was 109-116

µm, resulting in valve channels with slightly differing aspect ratios. To better understand

the impact of etch depth, three additional devices with depths of 144-152 µm were also

tested. To account for the variation in etch depth and aspect ratio when comparing the

characteristics of different valves, the pressure drop and resistance data for the Tesla-type

T45A and T45C valves are plotted versus Reynolds number instead of volume flow-rate.

The valve tests were divided into three groups considering etch depth and valve design:

the T45A group, the T45C group, and the Deep T45C group. The etch depths and devia-

tions from the mean depth of the valves in each test group are listed in Tables 4.1, 4.2, and

4.3. In each set the tested devices are representative samples since the deviations from the

mean etch are much less than the measurement accuracy of 6.5%.

The pressure drop ∆P across the valve was measured by driving de-ionized water at

known volume-flow rates through the valve using a syringe pump (Model 200, KD Sci-

entific, Boston, MA) and measuring the gauge pressure upstream of the valve using a

miniature pressure transducer (Model EPI-411-3.5B-/RTV, Entran, Fairfield, NJ) that was

38

Table 4.1: Etch depths and deviations from the mean depth of the valves tested in the T45Agroup. The tested devices are representative samples of the set since the deviations aremuch less than the measurement accuracy of 6.5%.

T45A Test Group etch µm deviation [%]

T45A Bi2 112 1.36T45A Bo2 109 -1.36T45A Ti2 110 -0.45T45A To2 111 0.45

Mean 110.5 0

Table 4.2: Etch depths and deviations from the mean depth of the valves tested in the T45Cgroup. The tested devices are representative samples of the set since the deviations aremuch less than the measurement accuracy of 6.5%.

T45C Test Group etch µm deviation [%]

T45C Li2 116 0.87T45C Lo2 114 -0.87T45C Li2t2 116 0.87T45C Lo2t2 114 -0.87

Mean 115 0

Table 4.3: Etch depths and deviations from the mean depth of the valves tested in the DeepT45C group. The tested devices are representative samples of the set since the deviationsare much less than the measurement accuracy of 6.5%.

Deep T45C Test Group etch µm deviation [%]

T45C Si 144 -2.92T45C So 149 0.45T45C Ri2 152 2.47

Mean 148.3 0

39

calibrated to a mercury manometer. Downstream of the valve the outlet tubing was open

to the atmosphere. Effects such as the elevation change between inlet and outlet, as well as

the pressure drop from flow resistance in the tubing, were calculated as less than 1% of the

total measured pressure drop. Data were recorded and fitted to a power-law function for

flow in each direction, forward (subscript F) and reverse (subscript R), by

∆PF βFQn

F and ∆PR βRQn

R (4.1)

using a linear least-squares method applied to the natural logarithm of the equation, ie.

ln∆P lnβ n lnQ

4.1.2 Numerical Methods

Steady-state solutions of forward and reverse flow in the T45A and T45C valves were

obtained by the numerical method described in Chap. 3. These steady-state solutions are

the final solution of step-response simulations that were allowed to proceed until steady

flow was achieved.

For study of the diodicity mechanism, the numerical simulations of the T45A and T45C

valves were performed at the nominal etch depth of 120µm. However, the flow rate in rect-

angular channels is dependent on channel width and height, and although the channel width

is 114µm for all devices, the etch depths of the tested valves vary from the 120µm etch used

for the diodicity mechanism study. In the T45A case the difference between the etch depth

of the numerical simulations and the mean etch depth of the T45A Test Group of experi-

mental devices is 120 110 5 9 5µm, or 8.5%. In the T45C Test Group the difference

is 120 115 5µm or 4.3%, which is within the 6.5% etch measurement accuracy. If the

flow in NMP valves behaved like fully-developed flow in straight rectangular channels, we

would expect the flow rate, in terms of Reynolds number, to be proportional to the hydraulic

diameter cubed for a given pressure gradient as in

Re ∝ φD3H (4.2)

which was derived from Eqs. 1.5 and 1.6 in which φ is a function of the channel height to

width aspect ratio and DH follows Eq. 1.3. But in Tesla-type valves the pressure drop at

the entrance and the bends in the flow path are dominant and the exponent in Eq. 4.2 is less

than 3. Instead of using a modification of Eq. 4.2 to adapt the results of the numerical sim-

ulations at the nominal etch depth to the etch depth of the physical devices, three additional

40

sets of numerical simulations were performed using the mean etch depth of each of the

test groups: h 110 5µm for the T45A Test Group, h 115µm for the T45C Test Group,

and h 148 3µm for the Deep T45C Test Group, so that experimental measurements and

numerical simulation results could be compared directly.

For each combination of valve design and etch depth, the numerical method employed

nine step-response simulations to characterize, via the power-law function of Eq. 4.1, the

flow-rate response to a range of applied pressure-differences limited by cavitation to ∆P 1

atm. The diodicity relation, Eq. 1.1, requires forward and reverse-flow pressure differences

that produce the same volume flow rate. To achieve pairs of forward and reverse-flow sim-

ulations with matching volume flow-rates, the following technique was employed. First,

reverse-flow simulations were performed, each using one of three applied pressure differ-

ences, ∆Preverse 0 1

0 5

and 0 9 atm. The mass flux m through the valve was recorded

for each of these three cases. Then, three corresponding forward-flow simulations were

performed by setting the normal velocity u at the inlet boundary to provide the correct

mass flux (ie. u m ρ) with Poutlet 0 at the outlet boundary. The average pressure de-

veloped at the inlet boundary Pinlet was recorded for each of these three cases. Finally,

three corresponding forward-flow simulations were performed using the applied pressure

differences ∆Pforward Pinlet Poutlet. The mass flux through the valve of each of these

forward-flow simulations was checked to ensure that it matched within less than 1% the

mass flux of the corresponding reverse-flow simulation. The applied pressure differences

from the first three∆Preverse and the last three simulations

∆Pforward were used directly

in the diodicity relation, Eq. 1.1, as each matched forward and reverse-flow pair produced

equal volume flow-rates.

4.2 Results

The four valve experiments in the T45A Test Group and the seven valve experiments in

the T45C Test Group and Deep T45C Test Group were conducted to validate the numerical

method. Each valve experiment contains 22 independent measurements of pressure drop

versus volume flow-rate for an overall total of 242 independent observations.

4.2.1 Volume Flow-Rate

The predicted valve-flow response to applied pressure from the numerical simulations at the

nominal etch depth of 120µm and at the mean etch depths of the T45A, T45C, and Deep

T45C Test Groups are compared with the corresponding experimental measurements in

41

−1000 −500 0 500 1000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Reynolds number

Pre

ssur

e dr

op, a

tm

Bi2 112umBo2 109umTi2 110umTo2 111umsimA 120umsimB 110um

Figure 4.1: T45A valve pressure drop vs. Reynolds number based on the hydraulic diame-ters. Experimental and numerical data are shown as symbols. The curves are the the fittedpower-law relation (Eq. 4.1). The legends refer to each valve name and its etch depth; thenumerical simulations are marked “sim”.

Figs. 4.1, 4.2, and 4.3 in terms of pressure drop versus Reynolds number. The experimental

and numerical data are plotted as symbols; the curves are the fitted power-law relation (Eq.

4.1).

A statistical analysis was done for the power-law function Eq. 4.1 as a model equa-

tion for the true relationship between pressure-drop ∆P and volume flow-rate. A linear

least-squares method was were used to fit the logarithm of the function to the data from

each experiment. The measurements in the forward and reverse flow directions were fitted

separately. The standard deviation σ and correlation coefficient r2 between the measured

∆P and the calculated ∆P from the power-law function are shown in Table 4.4. The overall

mean standard deviation is 3.6%. The correlation coefficients are r2 0 99 in all cases,

where r2 1 00 is a perfect fit. Figures 4.1, 4.2, and 4.3 allow visual comparison of the

data and the power-law fits. The power-law function is clearly a good representation of

the physical relationship between pressure-drop and flow-rate, and there is no justification

to seek a more complicated model equation. Thus the power-law function was also used

to represent the numerical method’s predictions of flow-rate response to applied-pressure.

The parameters β and n of the power-law fit are shown in Table 4.5.

42

−1000 −500 0 500 1000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Reynolds number

Pre

ssur

e dr

op, a

tm

Li2 116umLo2 114umLi2t2 116umLo2t2 114umsimA 120umsimB 115um

Figure 4.2: T45C valve pressure drop vs. Reynolds number based on the hydraulic diame-ters. Experimental and numerical data are shown as symbols. The curves are the the fittedpower-law relation (Eq. 4.1). The legends refer to each valve name and its etch depth; thenumerical simulations are marked “sim”.

−1000 −500 0 500 1000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Reynolds number

Pre

ssur

e dr

op, a

tm

Si 144umSo 149umRi2 152umsim 148um

Figure 4.3: Deep T45C valve pressure drop vs. Reynolds number based on the hydraulicdiameters. Experimental and numerical data are shown as symbols. The curves are the thefitted power-law relation (Eq. 4.1). The legends refer to each valve name and its etch depth;the numerical simulations are marked “sim”.

43

Table 4.4: Standard deviation σ and correlation coefficient r2 of the experimentally-measured ∆P with respect to the power-law fit of ∆P versus volume flow-rate followingEq. 4.1. The measurements in the forward and reverse flow directions were fitted sepa-rately. The overall mean standard deviation is 3.6%. A correlation coefficient of r2 1 0is a perfect fit.

Test σforward [%] r2forward σreverse [%] r2

reverse

T45A Bi2 4.098 0.9986 1.986 0.9999T45A Bo2 3.478 0.9989 3.494 0.9998T45A Ti2 2.994 0.9997 2.560 0.9995T45A To2 5.087 0.9978 4.161 0.9997T45C Li2 2.702 0.9995 2.227 0.9997T45C Lo2 3.224 0.9994 2.854 0.9991T45C Li2t2 4.439 0.9988 2.156 0.9996T45C Lo2t2 3.980 0.9996 2.651 0.9997T45C Si 4.728 0.9993 6.439 0.9968T45C So 6.621 0.9996 1.931 0.9998T45C Ri2 1.212 0.9998 1.768 0.9996

Table 4.5: Parameters β and n of the power-law fit of ∆P versus volume flow-rate fol-lowing Eq. 4.1. The measurements in the forward and reverse flow directions were fittedseparately. The units of β are Pasec m3.

Test βforward nforward βreverse nreverse

T45A Bi2 4.753e+15 1.533 3.746e+16 1.646T45A Bo2 5.568e+15 1.540 2.471e+16 1.618T45A Ti2 8.29e+15 1.563 3.952e+16 1.647T45A To2 2.145e+15 1.482 4.852e+16 1.658T45C Li2 6.442e+15 1.556 3.524e+16 1.643T45C Lo2 3.806e+15 1.520 4.263e+16 1.649T45C Li2t2 2.957e+15 1.509 3.674e+16 1.645T45C Lo2t2 6.028e+15 1.551 4.154e+16 1.650T45C Si 4.345e+15 1.568 6.195e+16 1.716T45C So 1.259e+16 1.633 6.642e+16 1.717T45C Ri2 6.988e+15 1.590 3.923e+16 1.681

44

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Reynolds number

Dio

dici

ty


Figure 4.4: Diodicity following Eq. 1.1 versus Reynolds number of the valves in the T45ATest Group. The symbols are numerical and experimental data. The curves are the ratio ofthe fitted power-law relations (Eq. 4.1) for the reverse and forward flow directions. Legendsrefer to test name and etch depth; numerical simulations are marked “sim”.

4.2.2 Diodicity

The predictions and experimentally-derived values of diodicity following Eq. 1.1 for the

120 µm nominal etch depth and the T45A, T45C, and Deep T45C Test Groups are shown

in Figs. 4.4, 4.5, and 4.6. The increase in scatter over that seen in the pressure versus

flow-rate test data is due to the difficulty of taking the ratio of two large numbers that are

nearly equal in value, ie. the ratio is very sensitive to measurement accuracy.

4.2.3 Prediction Accuracy

The numerical predictions and experimental data in Figs. 4.1 through 4.6 show good agree-

ment. To obtain a quantitative measure of the agreement, we utilized the power-law fits, our

representation of the true relationship between pressure drop and flow rate, as the basis for

comparison of experimental measurements and numerical predictions. One benefit of this

method is that it segregates the experimental data points from the numerical data points and

avoids conflating measurement errors with prediction errors, otherwise an experiment with

less scatter in the data would make the numerical prediction accuracy appear to improve.

45

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Reynolds number

Dio

dici

ty


Figure 4.5: Diodicity following Eq. 1.1 versus Reynolds number of the valves in the T45CTest Group. The symbols are numerical and experimental data. The curves are the ratio ofthe fitted power-law relations (Eq. 4.1) for the reverse and forward flow directions. Legendsrefer to test name and etch depth; numerical simulations are marked “sim”.

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Reynolds number

Dio

dici

ty


Figure 4.6: Diodicity following Eq. 1.1 versus Reynolds number of the valves in the DeepT45C Test Group. The symbols are numerical and experimental data. The curves are theratio of the fitted power-law relations (Eq. 4.1) for the reverse and forward flow directions.Legends refer to test name and etch depth; numerical simulations are marked “sim”.

46

Table 4.6: Mean prediction errors and standard deviations of the volume flow-rates and thediodicity for each Test Group, shown in percent.

Forward flow-rate Reverse flow-rate DiodicityTest group eRe σeRe eRe σeRe eDi σeDi

T45A -5.22 1.48 -7.94 1.06 5.43 1.95T45C -5.26 1.55 -6.17 1.55 2.21 1.19

Deep T45C -9.70 3.98 -11.45 3.68 4.78 2.93

The mean prediction errors (in percent) of the volume flow-rates and the diodicity for each

Test Group are shown in Table 4.6. Each flow-rate prediction error eRe is defined in terms

of Reynolds numbers by

eRe 100

Repredicted Remeasured

Remeasured ∆P(4.3)

in which the Re values both correspond to the same pressure-drop ∆P. Similarly, each

diodicity prediction error eDi is defined by

eDi 100

Dipredicted Dimeasured

Dimeasured Re(4.4)

in which the diodicity values both correspond to the same Reynolds number. Table 4.6

also includes the mean of the standard deviation of the error σe for each test in the test

group. The flow-rate response is underpredicted by the numerical method by 7.62% on

average over all three test groups: T45A, T45C, and Deep T45C, and the diodicities are

overpredicted by 4.14%.

Since determination of statistical significance of predicted diodicity improvement is

needed in later chapters, the confidence level estimates of diodicity prediction error were

calculated. There are N 11 valve tests contained in the three test groups: T45A, T45C,

and Deep T45C. Since N 25, the Student’s t distribution [4] was used to determine the

prediction error range that has a 95% probability of including the actual error of a diodic-

ity prediction. Instead of finding the mean error as in Table 4.6, the diodicity prediction

error eDi was plotted as a function of Reynolds number in Fig. 4.7. The diodicity mecha-

nism study in the following chapters utilizes numerical simulations in which the flow-rate

responses were within 5.6% of Re 500, so the 95% confidence band on diodicity predic-

tion error at that flow rate has special significance. At Re 500, the diodicity prediction

47

200 300 400 500 600 700 800−20

−15

−10

−5

0

5

10

15

20

Reynolds number

95%

Con

fiden

ce B

and,

per

cent

Upper limitMeanLower limit

Figure 4.7: Diodicity prediction error of the numerical method for all 11 tests of the threetest groups: the T45A Test Group, the T45C Test Group, and the Deep T45C Test Group.

error with 95% confidence is eDi 4 42 5 13% or 0 708 eDi 9 55%.

4.2.4 Evidence of Laminar Flow

It is important to show that the flow in the valve is laminar despite the presence of separated

flow and recirculation regions, since the governing equations employed in the numerical

method are the exact equations for modeling laminar flow. Identification of laminar flow

was discussed in Sec. 1.2.2 in which two methods were suggested for identifying the flow

in the valve as laminar over the Reynolds number range of the data. First, the data from a

laminar flow exhibit a linear proportionality between the log of the pressure drop and the

log of the volume flow rate. Figure 4.8 shows that the linear proportionality does hold for

data from both experiments and numerical simulations. Second, a numerical simulation

based on solving the Navier-Stokes equations will not converge to a steady solution if the

flow is randomly varying, ie. turbulent. All the valve simulations in this dissertation con-

verged to steady solutions without applying time-averaging of the flow properties, spectral

methods, or any other technique used to model a turbulent flow.

48

102

103

10−2

10−1

100

101

Reynolds number

For

war

d−flo

w p

ress

ure

drop

, atm


(a) Forward flow in T45A valves

102

103

10−2

10−1

100

101

Reynolds number

Rev

erse

−flo

w p

ress

ure

drop

, atm


(b) Reverse flow in T45A valves

102

103

10−2

10−1

100

101

Reynolds number

For

war

d−flo

w p

ress

ure

drop

, atm


(c) Forward flow in T45C valves

102

103

10−2

10−1

100

101

Reynolds number

Rev

erse

−flo

w p

ress

ure

drop

, atm


(d) Reverse flow in T45C valves

102

103

10−2

10−1

100

101

Reynolds number

For

war

d−flo

w p

ress

ure

drop

, atm


(e) Forward flow in Deep T45C valves

102

103

10−2

10−1

100

101

Reynolds number

Rev

erse

−flo

w p

ress

ure

drop

, atm


(f) Reverse flow in Deep T45C valves

Figure 4.8: Pressure drop versus Reynolds number based on hydraulic diameter. Symbolsare experimental and numerical data; curves are the fitted power-law relation (Eq. 4.1).Legends refer to test name and etch depth; numerical simulations are marked “sim”.

49

4.3 Discussion

The primary task of this chapter was to demonstrate that the numerical method has been

correctly implemented in this research and thus accurately predicts the flow-rate response to

applied pressure and the diodicity as required by research objectives #1b and #1c. This val-

idation was achieved by comparing volume flow-rate and diodicity predictions to 242 inde-

pendent experimental measurements from physical-realized valves of three distinct groups

of etch depths and two distinct Tesla-type designs, the T45A and the T45C. The tests of the

physical-realized valves were divided into three groups considering etch depth and valve

design: the T45A group, the T45C group, and the Deep T45C group. In each group the

tested devices are representative samples of the set since the deviations from the mean etch

are much less than the measurement accuracy of 6.5%.

For study of the diodicity mechanism in the following chapters, the numerical simula-

tions of the T45A and T45C valves were performed at the nominal etch depth of 120µm.

To ensure that experimental measurements and numerical simulation results could be di-

rectly compared, three additional sets of numerical simulations were performed using the

mean etch depth of each of the valve test groups: h 110 5µm for the T45A Test Group,

h 115µm for the T45C Test Group, and h 148 3µm for the Deep T45C Test Group.

It is also important to note that exactly the same numerical method was employed in all

cases. The only variation between simulations was the substitution of the physical grid

of the T45A or T45C scaled in the thickness dimension to match the appropriate mean

etch-depth.

Analysis of the experimental data showed that the power-law function of Eq. 4.1 pro-

vides a good estimate of the true functional relationship between measured pressure-drop

and volume flow-rate and there is no justification to use a more complicated function. Thus

the power-law function was used to represent the numerical method’s predictions of flow-

rate response to applied-pressure. Knowledge of a good representative function allowed

flow-rate predictions to be produced from fewer numerical simulations.

Analysis of the numerical simulation results showed that the flow-rate response is un-

derpredicted by the numerical method by 7.62% on average over all three test groups:

T45A, T45C, and Deep T45C. This is good agreement considering that the etch depth

measurement accuracy is 6.5% and that, in order to be a predictive tool, the numerical

model is gridded-up from the “ideal” mask layout drawing (see Ch. 3) instead of the as-

etched physical devices. Research objective #1b has been accomplished by verifying the

numerical method accurately predicts the steady-flow response to an applied pressure.

50

The diodicity is insensitive to any Reynolds-number underprediction that is a similar

proportion of both forward-direction and reverse-direction flow. The diodicity is overpre-

dicted by 4.14% on average over all three test groups: T45A, T45C, and Deep T45C. The

diodicity prediction error with 95% confidence is eDi 4 42 5 13% or 0 708 eDi

9 55% in the numerical simultations utilized in the diodicity mechanism study in the fol-

lowing chapters. By making comparisons between predicted and experimental values from

multiple valve geometries we demonstrated that we are free to modify valve geometry (to

the same extent as the variation between the T45A and T45C) and still obtain accurate flow

rate and diodicity predictions from the numerical method. Thus the numerical method is

validated as a firm basis on which to accept or reject valve designs for the enhancement

of diodicity. Research objective #1c has been accomplished by verifying the numerical

method accurately predicts valve diodicity. Proof that it can reveal the diodicity mecha-

nism remains the task of following chapters.

Finally, both the experimental measurements and numerical simulation results showed

that the flow in the valve is laminar despite the presence of separated flow and recirculation

regions, thus the governing equations employed in the numerical method are exact.

51

Chapter 5

VALIDATION OF THE NUMERICAL METHOD IN TRANSIENT

FLOW

This chapter verifies that the numerical method is time-accurate in harmonic-response

simulations. The approximation analysis shown in Table 2.2 and discussed in Sec. 2.4

suggests that the transient term of the kinetic-energy conservation equation is negligible for

oscillating water flows with frequencies as high as 10 kHz. This allows investigation of the

diodicity mechanism using inexpensive steady-state solutions to calculate the terms in the

conservation equation, (the steady form of Eq. 2.12), instead of computationally-intensive

harmonic-response solutions. To use the numerical method to verify that the transient

term is negligible, we first need to demonstrate that the numerical method produces time-

accurate harmonic-response simulations.

Since there are no analytical solutions for transient flow in an NMP valve nor exper-

imental methods available to directly measure it, the accuracy of the transient response

predictions of the numerical method was verified by modeling the harmonic response of

2-D slot flow for which an analytical solution does exist. Then the numerical method was

applied to model harmonic flow in a T45A valve and show that the kinetic-energy transient

term is negligible.

5.1 Harmonic Response of a 2-D Slot

The exact solution for the velocity response to an oscillating pressure gradient in a 2-D slot

was described in Sec. 1.2.5. The dimensionless velocity profiles are given by Eq. 1.13.

Proceeding with the nondimensional variables of Sec. 1.2.5, the volume flow-rate QT in

the slot was obtained by integrating the velocity profile over the slot height as in

QT

1

1UdY real

i expiT 2

1

01

cosh iλY cosh iλ dY

52

After integration, the volume flow rate is

QT 2real

iexpiT

1

tanh iλ iλ in which the only component of Q

T that has a dependence on time is the exponential.

The maximum value of the applied oscillating pressure gradient K cosT occurs at T

02π

4π

. The corresponding amplitude of the volume flow rate is

Q 2real

i

1

tanh iλ iλ (5.1)

The fluid impedance per unit length is a complex number that is the ratio of the applied

pressure gradient and the flow rate amplitude given by

Z K

i

1 tanh

iλ

iλ

(5.2)

which represents the particular case of oscillating flow in a slot, compared to Eq. 1.16

which represents the general case.

A harmonic response simulation was obtained for a 2-D slot following the numerical

methods described in Chap. 3, taking the slot height as 2h 90µm, the slot length as

L 1mm , and the viscosity of water µ 1 719 10 3 Ns m2 for 273 K. The amplitude of

the oscillating pressure was Pa 1 atm applied at a frequency of Ω 10 kHz. Convergence

was reached in 25 periods of 12 time steps each with 100 inner iterations per time step. The

predicted velocity profiles were compared to the analytical solution given by Eq. 1.13.

Following the nondimensionalization scheme of Sec. 1.2.5 as above, Fig. 5.1 shows

the excellent agreement between the nondimensional velocity profiles from the numerical

method’s harmonic-response simulation (symbols) and the exact analytical solution of Eq.

1.13 (lines) at 60 phase intervals of one cycle. The kinetic-Reynolds number is λ 8 6 and

the motion of the fluid near the wall is clearly out-of-phase with the flow near the centerline.

The predicted dimensionless volume-flow-rate amplitude was 1.854, which is 100.47% of

the exact solution obtained from Eq. 5.1. The phase lag θ of the flow rate response with

respect to the applied pressure difference across the slot was calculated by normalizing the

flow rate and pressure difference by their amplitudes (they are both sinusoidal), and finding

a θ that in a least squares sense minimized the error eθ Pnorm

Ωt Qnorm

Ωt θ over

53

0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Distance from centerline, Y

Vel

ocity

, U

060 sim060 exact120 sim120 exact180 sim180 exact240 sim240 exact300 sim300 exact360 sim360 exact

Figure 5.1: Comparison of the nondimensional velocity profiles in a slot with oscillatingflow,

λ 8 6 , from the numerical method (symbols) and the exact solution (lines). The

centerline of the slot is at zero slot height and the slot wall is at slot height = 1. The legendsnote the phase of each profile with respect to the applied pressure difference, a cosinefunction.

0 Ωt 2π The phase lag computed in this way was θ 84 96 , which is within 0.09%

of the exact solution determined from θ arctanimagZ realZ using Eq. 5.2. Clearly the

numerical solution agrees very well with the exact solution.

5.2 Harmonic Response of an NMP Valve

A harmonic response simulation of the T45A valve with a pressure amplitude of P 0 5

atm applied at 2.818 kHz for 6 cycles was performed using the numerical method described

in Chap. 3, including the nondimensionalization parameters of Sec. 3.5. The terms in the

conservation of kinetic-energy equation (Eq. 2.12) were calculated, including: the energy

dissipation rate (PHI), viscous work rate (VWR), energy flux rate (EFR), pressure work

rate (PWR), and the transient kinetic-energy (TKE). Figure 5.2 displays them and shows

that the transient kinetic-energy term is insignificant compared to the pressure work rate

and the dissipation rate.

54

4 4.5 5 5.5 6−0.04

−0.02

0

0.02

0.04

0.06

0.08

Cycle

Wor

k ra

te

PHIVWREFRPWRTKE

Figure 5.2: Terms in the conservation of kinetic-energy equation (Eq. 2.12) for the entireT45A valve over two cycles in a harmonic response simulation at 2.818 kHz, including: theenergy dissipation rate (PHI), viscous work rate (VWR), energy flux rate (EFR), pressurework rate (PWR), and the transient kinetic-energy (TKE). The dissipation and pressurework are dominant.

5.3 Discussion

The modeling of a 2-D slot has shown that the predictions of the harmonic simulation

agree with the analytical solution. This accomplishes research objective #1d to show that

the numerical method accurately predicts transient flow-rate response.

The approximation analysis for NMP valve flow shown in Table 2.2 and discussed

in Sec. 2.4 suggested that the transient term was negligible even for oscillating water

flows with frequencies as high as 10 kHz. This was corroborated by a harmonic response

simulation run at 2818 Hz that shows that the transient kinetic-energy term is insignificant

compared to the pressure work rate and the dissipation rate. The insignificance of transient

effects allows the use of relatively inexpensive steady-state simulations to calculate the

terms in the momentum and kinetic-energy conservation equations, (the steady forms of

Eqs. 2.5 and 2.12, respectively), instead of computationally-intensive harmonic-response

simulations. Thus steady-state simulations were used to study the diodicity mechanism in

the following chapters.

55

Chapter 6

DIODICITY MECHANISM OF T45A VALVE

6.1 Simulation Methods and Conditions

The diodicity mechanism of the T45A valve was analyzed by studying the results of the nu-

merical methods explained in Chap. 3 from two perspectives: first, analysis of the velocity,

pressure, and dissipation-rate fields, and second, analysis of the terms of the momentum

and kinetic-energy conservation equations applied in 9 regional control volumes as shown

in Fig. 6.1.

Steady-state solutions of forward and reverse flow in the T45A valve were obtained and

analyzed using the methods described in Chap. 3. These steady-state solutions were ob-

tained from the final solution of step-response simulations that were continued until steady

flow was achieved. In all cases the simulations converged to steady solutions without ap-

plying Reynolds averaging to the Navier-Stokes equations. As discussed in Sec. 1.2.2,

this identifies the flow in the valve as laminar over the Reynolds number range simulated,

Re 953, despite the presence of separated flow and recirculation regions. To assess grid

independence, the grid was refined from 96360 to 115632 finite volumes, an increase of

20%, resulting in a reduction in the kinetic-energy conservation error εKE 0 06 of less

than 5%. The accuracy of the flow-rate response predictions of the numerical method was

determined in Ch. 4.

To represent typical operating conditions a differential pressure of 0.5 atm was applied

to the pressure boundaries, producing an equilibrium volume flow rate in the reverse di-

rection of 3710µl min corresponding to ReD 528 based on the hydraulic diameter of

the main channel. To achieve the same flow rate in the forward-flow direction required a

differential pressure of only 0.394 atm. This corresponds to a diodicity according to Eq.

1.1 of 1.27. According to the analysis of diodicity prediction accuracy in Ch.4, there is a

95% probability that the true diodicity is within 1 15 Di 1 28.

56

Figure 6.1: Division of the T45A valve into regional control volumes.

6.2 Results

6.2.1 Velocity Field

From the step-response simulations, Fig. 6.2 shows the forward and Fig. 6.3 the reverse-

flow velocity fields on the symmetric centerplane of the valve. The velocity fields in the

forward and reverse flow cases are radically different. Flow separation and laminar jets

occur in three locations: where the channels separate, the channels recombine, and at the

valve exit.

In the forward-flow case the flow accelerates rapidly as it enters the main channel from

the goblet-shaped inlet plenum. As the velocity profile is developing it reaches the T-

junction. The flow begins to veer slightly into the side channel, and a small portion im-

pinging on the guide vane becomes a minor jet flowing into the side channel. However,

85% of the main channel flow is unperturbed and continues downstream becoming fully-

developed. The side channel jet spreads as it proceeds around the bend in the side channel,

and approaches the Y-junction as a low-velocity stream filling the entire width of the side

channel, moving at less than 20% of the bulk velocity of the main channel. However, it

has sufficient momentum to cause the main channel flow to begin to turn before it reaches

the far wall of the Y-junction. This may have a small effect in improving the diodicity as

it tends to increase the radius of curvature of the main channel flow as it travels around the

45 bend into the outlet channel. Downstream of the Y-junction, the flow separates from

the inner wall and forms a narrow, hi-speed jet next to the outer wall. The high velocity

gradient between the jet and the wall produces additional dissipation that lowers the diod-

57

BACK COLOUR LINE WIDTHMARKER SIZE TEXT SIZECOARSENESS

Z-SC-ROT CLK Z-SC-ROT ANTX-SC-ROT CLK X-SC-ROT ANTY-SC-ROT CLK Y-SC-ROT ANTLEFT MOVE RIGHT MOVEUP MOVE DOWN MOVENEAR MOVE FAR MOVEEXPAND SHRINKRESET VIEW

MENU WINDOW AXES OFF

0.0000E+001.4167E-012.8333E-014.2500E-015.6667E-017.0833E-018.5000E-01

Figure 6.2: Forward-flow velocity field on the centerplane of a single-element, Tesla-typeT45A valve with a volume flow rate of 3710 µl/min corresponding to Re=528 based on thehydraulic diameter of the main channel. One dimensionless unit equals 10 m/s.

58

icity of the valve. A significant portion of the channel is filled with a quiescent zone and

is essentially underutilized by the forward flow. It may improve diodicity to narrow this

section of the outlet channel so it cannot be utilized by the reverse flow. The flow leaves the

outlet channel and enters the goblet-shaped outlet plenum as a high-speed jet. Any of its

momentum flux that is dissipated increases the pressure drop in the forward-flow direction

and lowers diodicity.

In the reverse-flow case the flow accelerates rapidly as it enters the channel from the

goblet-shaped plenum, just as it does in the forward flow case. On reaching the Y-junction

the flow stream begins to veer slightly toward the main channel. As it impinges on the

cusp of the guide vane, 36% of the flow is deflected down the main channel where it travels

next to the guide vane wall, but the vast majority of the main channel is filled with a

large, slowly-moving recirculation zone. The side channel flow separates from the outer

wall and forms a laminar jet along the guide vane wall. The large velocity gradient there

increases dissipation, which improves diodicity. At the bend in the side channel, the jet

separates from the guide vane and attaches to the outer side-channel wall as it undergoes

radial acceleration. The flow emerges from the side channel as a laminar jet heading for

the opposite wall of the main channel. But the momentum of the smaller flow traveling

down the main channel turns the jet before it reaches the opposite wall. There is a large,

vigorous recirculation zone downstream of the T-junction, however the jet spreads to fill

the entire channel as it leaves the channel and enters the goblet-shaped plenum. Any of

the momentum flux at the channel exit that is eventually dissipated serves to increase the

diodicity.

6.2.2 Pressure Field

In contrast to the velocity fields there are similarities between the forward and reverse-flow

pressure fields of the T45A valve shown in Fig. 6.4, partly due to the symmetrical inlet

and outlet geometry. Significant pressure loss is apparent where the flow enters the channel

regardless of the flow direction. In fact this is a major source of pressure loss in either flow

direction. There is only one other signficant pressure loss in each case: in forward flow it

is at the 45 bend in the channel, in reverse flow it is just downstream of the T-junction.

Between these locations and their corresponding flow exits there is little additional pressure

drop; the momentum flux maintains the fluid velocity until it reaches the goblet-shaped

plenum and begins to diminish. Closer study of the pressure fields shows that pressure

drop at the 45 bend in forward flow is approximately 0.1 atm, and downstream of the

59




0.0000E+001.4167E-012.8333E-014.2500E-015.6667E-017.0833E-018.5000E-01

Figure 6.3: Reverse-flow velocity field on the centerplane of a single-element, Tesla-typeT45A valve with a volume flow rate of 3710 µl/min corresponding to Re=528 based on thehydraulic diameter of the main channel. One dimensionless unit equals 10 m/s.

60

T-junction in reverse flow is approximately 0.2 atm. This is the main source of diodicity.

Thus, as suggested by Eq. 1.2, if the direction-independent losses at the channel entrances

could be eliminated, the diodicity would be increased from 1 27 to 2 0. Though they

cannot be completely eliminated, it would be helpful to adjust the mouth of the channel

entrances so that the forward-flow entrance loss is less than that of the reverse flow.

6.2.3 Energy-Dissipation Field

The base 10 logarithm of the energy dissipation rate is shown in Figs. 6.5 and 6.6. The

red and yellow contours are the most significant and occur where the velocity gradients are

largest. In the forward-flow case, dissipation occurs along the walls of the main channel

and on both sides of the laminar jet downstream of the 45 bend. In the outlet goblet

plenum, the dissipation of the momentum flux leaving the exit is clearly visible. But the

most significant dissipation occurs at the convex surfaces of the channel mouth where the

flow accelerates as it enters the channel, at the upstream cusp of the guide vane, at the

inner corner of the 45 bend, and along the wall downstream of the 45 bend all the way to

channel exit. These locations correspond well to the locations of large pressure drop in Fig.

6.4, but it is not conclusive that dissipation is the most important source of that pressure

loss.

The locations of dissipation in the reverse flow case are quite different than in forward

flow. The only similar region is the dissipation along the walls of the channel from where

the flow enters the channel to the 45 bend. As the flow bifurcates at the Y-junction, the

dissipation is spread more evenly in the side channel than elsewhere, but very significant

dissipation occurs along the surfaces of the guide vane, both in the side channel and the

main channel, and especially at its leading cusp. There is additional major dissipation in

the side channel as the separated jet follows the curvature of the outer wall. This dissipa-

tion continues alongside the jet proceeding from the side channel into the main channel,

collocated with the large velocity gradient between the jet and the recirculation zone in the

main channel downstream of the T-junction. There is also significant dissipation where the

jet exiting from the side channel impinges on the opposite wall of the main channel. As in

the forward-flow case, part of the momentum flux leaving the channel is dissipated in the

goblet-shaped plenum.

61




<-8.0000E-021.0566E-021.1245E-012.1434E-013.1623E-014.1811E-015.2000E-01

(a) Forward flow




<-8.0000E-021.2308E-021.1615E-012.2000E-013.2385E-014.2769E-01

> 5.2000E-01

(b) Reverse flow

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

62




-8.0000E+00-6.6733E+00-5.3467E+00-4.0200E+00-2.6933E+00-1.3667E+00-4 .0000E-02

Figure 6.5: Base 10 logarithm of the energy dissipation rate in forward flow on the center-plane of a single-element, Tesla-type T45A valve with a volume flow rate of 3710 µl/mincorresponding to Re=528 based on the hydraulic diameter of the main channel. One di-mensionless unit equals 14 mW.

63




-8.0000E+00-6.6733E+00-5.3467E+00-4.0200E+00-2.6933E+00-1.3667E+00-4 .0000E-02

Figure 6.6: Base 10 logarithm of the energy dissipation rate in reverse flow on the center-plane of a single-element, Tesla-type T45A valve with a volume flow rate of 3710 µl/mincorresponding to Re=528 based on the hydraulic diameter of the main channel. One di-mensionless unit equals 14 mW.

64

6.2.4 Momentum Conservation

Diodic Effects of the Force Vector Interactions

The momentum equation is a vector relation, a summation of forces, so each of its terms

has both magnitude and direction sense. The X and Y components of these forces are

shown in Fig. 6.7 for both the forward and the reverse flow case. The figure caption

explains positive-negative direction sense and the X-Y coordinate system is shown in Fig.

6.1. Essentially, the figures show the viscous and pressure forces applied on the fluid by

the walls and inlet/outlet boundaries, and the fluid’s momentum flux response. Both the

momentum flux and viscous force are important in counterbalancing the pressure force.

The goblet-shaped inlet plenum (block 1) is oriented for flow in the X direction, so

as Fig. 6.7 shows, the Y-direction force components are negligible. In forward flow the

pressure force is to the right (positive) and it is expended by accelerating the fluid as it

approaches the inlet channel and by a small amount of viscous force. The momentum flux

is to the right (positive), but more is leaving the control volume than entering (defined as

negative), so its overall sign is negative. In reverse flow the pressure is already near ambient

when it enters the block 1 control volume, so the pressure force is negligible. The viscous

force applied by the walls is also very small, so the momentum flux is unchanged as the

flow passes through block 1.

The inlet channel (block 2) is also oriented in the X direction but due to its proximity

to the side channel (block 8) it has Y-direction force components during reverse flow. The

positive-Y pressure force is applied by the wall on the jet of fluid leaving the side chan-

nel, altering its direction so that it flows out of block 2 and into block 1. The momentum

of this jet is directed into the inlet channel (positive) but is flowing in the negative-Y di-

rection (negative) resulting in a net negative Y-component of the momentum flux. In the

X-direction in both forward and reverse flow, the upstream flow boundary applies the pres-

sure force needed to drive the fluid through the inlet channel offset by the viscous force

and momentum flux. In both flow directions the viscous force is larger than the momen-

tum flux. Unfortunately, the X-direction pressure force is larger in forward flow than in

reverse, which has an adverse impact on valve diodicity. It is not clear from the momentum

perspective what impact the Y-direction pressure force has on diodicity.

The T-junction (block 3) does not have significant Y-direction forces, a surprising result

since this control volume connects the side channel (block 8) to the inlet channel and the

main channel (block 4). However, inspection of Fig. 6.3 shows that the jet from the side

channel during reverse flow does not significantly interact with the wall in block 3, but

65

1 2 3 4 5 6 7 8 9

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Block number

For

war

d−F

low

X−

dire

ctio

n F

orce

s

Momentum FluxViscous ForcePressure Force

(a) X-direction, forward flow

1 2 3 4 5 6 7 8 9

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Block numberR

ever

se−

Flo

w X

−di

rect

ion

For

ces


(b) X-direction, reverse flow

1 2 3 4 5 6 7 8 9

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Block number

For

war

d−F

low

Y−

dire

ctio

n F

orce

s


(c) Y-direction, forward flow

1 2 3 4 5 6 7 8 9

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Block number

Rev

erse

−F

low

Y−

dire

ctio

n F

orce

s


(d) Y-direction, reverse flow

Figure 6.7: Force vector terms in the integral form of the momentum conservation equationfor the T45A valve. Net pressure force and net momentum flux into a control volume arepositive. Viscous force is applied on the fluid by the wall. X-vectors to the right and Y-vectors upward are positive and consistent with the valve layout in Fig. 6.1 including thenumbering of the control volumes (blocks).

66

passes through to block 2. However the X-direction forces are important in creating diod-

icity. In forward flow there is a small amount of pressure recovery as the momentum flux

drives the flow through the area expansion of block 3; this negative pressure force increases

diodicity as it lowers the net valve pressure force for forward flow. The reverse flow has a

much greater positive impact on diodicity; the momentum flux leaving block 3 (negative)

and flowing in the negative X direction is the jet from the side channel, and results in a

positive net momentum flux for block 3 as seen in Fig. 6.7b. A large pressure force is

needed to offset this momentum flux and this is a major source of diodicity.

The main channel (block4) has almost identical Y-direction force components in for-

ward and reverse flow. In both cases the upper wall, which forms part of the guide vane

or island of the valve, applies a pressure force in the negative-Y direction to the impinging

flow, which has just passed an opening to the side channel. This impingement can be seen

in both Figs. 6.2 and 6.3. It is not clear from the momentum perspective if this affects

diodicity. In the X direction the force components are very different between forward and

reverse flow. In forward flow most of the volume flow is carried by the main channel,

and as a result significant pressure force is needed to counteract viscous force due to wall

friction. This has a negative impact on diodicity as most of the reverse flow occurs in the

side channel and almost no pressure force is required to push the remaining reverse flow

through the main channel. The viscous force due to wall friction is supplied by reduction

in momentum flux.

The Y-connection (block 5) has negligible X-direction force components. In forward

flow the net Y-direction flux of momentum is out of the control volume (negative) and in

the negative Y direction resulting in a positive value, which is counteracted by the pressure

force applied by the outer wall. In reverse flow some of the flow is directed down the main

channel (block 4) by the guide vane, thus there is a net flux of momentum into the control

volume in the positive Y direction, so the momentum flux again has a positive Y sense.

The outlet channel (block 6) is similar in form and function to the inlet channel (block2),

except that it is oriented at 45 cw with respect to the positive X direction. Because of dif-

ferent orientations it is difficult to compare the forces in blocks 2 and 6 using Fig. 6.7, but

a comparison is made below using the vector magnitudes shown in Fig. 6.8 to assess the

impact on diodicity. Essentially, the pressure force applied at the upstream flow bound-

ary is roughly equally opposed by the viscous force applied by the walls and by the net

momentum flux leaving the control volume.

The goblet-shaped outlet plenum (block 7) is analogous to the inlet plenum (block 1)

except for its orientation of 45 cw with respect to the positive X direction. The forward flow

67

forces are small; the reverse-flow net pressure force accelerates the fluid as it approaches

the next downstream control volume (block 6). All the viscous forces are negligible.

The side channel (block 8) has negligible forces in forward flow because almost the

entire volume flow passes through the main channel instead. In reverse flow, Fig. 6.7d

shows a very large pressure force applied by the outer wall to alter the direction of the

impinging flow in the side channel. Accelerated by the outer wall (see Figs. 6.3 and 6.4), a

correspondingly large net momentum flux is propelled from the side channel into block 3.

Because these are Y-direction forces it is difficult to assess their impact on diodicity. The

last control volume (block 9) has significant forces only in the reverse-flow direction.

Forward-Flow vs. Reverse-Flow Force Vector Magnitudes

Another way to study the diodicity mechanism, especially for control volumes that are not

oriented along the X direction, was to study the vector magnitudes of the force components

in the control volumes as shown in Fig. 6.8, which directly compares the forward-flow and

reverse-flow components.

Especially pertinent is the comparison of the inlet and outlet channels (blocks 2 and 6)

and the inlet and outlet goblet-shaped plenums (blocks 1 and 7), which could not be directly

compared in the previous section because of their different orientations. Comparing the

goblet-shaped plenums, the viscous forces applied by the walls are small. The significant

forces are the pressure forces and momentum fluxes; those of the forward flow jet entering

block 1 and the reverse flow jet entering block 7 are an order of magnitude larger than all

other forces. Though it appears in Fig. 6.8a that slightly more pressure force is applied in

the plenums in forward flow than in reverse, the combined diodic effect of blocks 1 and 7

is actually slightly above unity, Di1 7 1 01

The inlet and outlet channels (blocks 2 and 6) function similarly as entrance and exit

channels due to their geometric similarity. One difference is the slightly greater net mo-

mentum flux into block 2 from the side channel jet during reverse flow compared to the

net momentum flux into block 6 from the main channel jet during forward flow; this can

be seen in Fig.6.8c. Another difference is the smaller viscous force applied by the walls of

block 2 during reverse flow, probably due to its isolation by the recirculation zone that can

be seen in Fig. 6.3. The net diodic effect is slightly above unity, though this is not obvious

on the pressure force magnitude plot of Fig.6.8a.

The main channel (block 4) and the Y-junction (block 5) have larger pressure force

magnitudes in forward flow than in reverse flow, and thus have a diodic effect less than

68

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

Block number

Pre

ssur

e F

orce

Mag

nitu

de

reverse flowforward flow

(a) Pressure force magnitude

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

Block number

Vis

cous

For

ce M

agni

tude


(b) Viscous force magnitude

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

Block number

Mom

entu

m F

lux

Mag

nitu

de


(c) Momentum flux magnitude

Figure 6.8: Vector magnitudes of the terms in the momentum conservation equation for theT45A valve.

69

unity. In block 4 this is because almost all the forward flow proceeds through the main

channel instead of bypassing it as it does in reverse flow. As a result the viscous force

applied by the wall is much almost twice as large in forward flow as it is in reverse. In

block 5 the larger pressure force in forward flow is required offset the larger momentum

flux and viscous force, which are due to the forward flow changing direction in block 5,

whereas in reverse flow it proceeds straight through toward the side channel.

The major contributors to diodicity are the T-junction (block 3) and the side channel

(blocks 8 and 9). The T-junction is where the side channel jet emerges and must be redi-

rected and forced out through block 2, the result is a large net increase in momentum flux

and additional viscous force applied by the walls that both must be overcome by the pres-

sure force. This pressure force needed to redirect the jet from the side channel is not needed

in forward flow, since the forward-flow jet in block 6 is already pointed downstream. The

large increase in pressure across block 3 can be seen in Fig. 6.4b. The side channel requires

much larger pressure force in reverse flow than in forward. In part this is due to there being

very little forward flow through the side channel, so of course there is much more viscous

force applied by the walls in reverse flow and the possibility of greater change in the mo-

mentum flux in the side channel. But the great increase in momentum flux is due to the

fluid impinging on the outer wall as it curves back toward the main channel. The jet that

emerges from the side channel has much higher velocity than when it entered as can be

seen in Fig. 6.3.

6.2.5 Kinetic-Energy Conservation

The calculation of the kinetic energy terms for the entire valve showed that two-thirds of

the pressure work is dissipated in the valve. The ratio of the dissipation rates (reverse flow

/ forward flow) is 1.13 showing that viscous effects within the valve are a significant source

of diodicity. The ratio of energy flux rates is 1.91 showing that the reverse-flow jet contains

more energy, and momentum, than the forward-flow jet. However, the energy flux rates are

only one-third the magnitude of the dissipation rates, so the overall valve diodicity is 1.27.

The diodicity prediction from the ratio of the pressure work rates as in Eq. 2.14 is 1.25,

which is within 2% of 1.27.

There are two alternate ways to look at the data. Figs. 6.9 and 6.10 show the three most

important terms of the kinetic energy equation in steady incompressible flow: the pressure

work rate, the energy flux rate, and the dissipation rate, which are shown for each of the

control volumes for both forward and reverse flow. Blocks 3, 6, 7, 8, and 9 were identified

70

1 2 3 4 5 6 7 8 9

−0.05

0

0.05

Block number

Pre

ssur

e W

ork

Rat

e


(a) Pressure work rate

1 2 3 4 5 6 7 8 9

−0.05

0

0.05

Block number

Ene

rgy

Flu

x R

ate


(b) Energy flux rate

Figure 6.9: Magnitude of the pressure work rate and the energy flux rate terms in thekinetic-energy conservation equation from the T45A valve simulations with a volume flowrate of 3710 µl/min corresponding to Re=528 based on the hydraulic diameter of the mainchannel.

71

as sources of diodicity, since the pressure work applied in the reverse flow direction is

larger than that applied in the forward direction. On the other hand, blocks 1, 2, 4, and

5 cause a significant reduction in diodicity. The energy flux rate has opposite sign (+ is a

net increase, - is a net decrease in the control volume) depending on flow direction in all

blocks except block 2. The dissipation rate also exhibits direction dependence, especially

in blocks 2, 6, and 8. The viscous work rate is inconsequential in all control volumes.

An alternate view of the kinetic energy in the valve is the contribution of each term of

the kinetic energy equation to the forward-flow direction and the reverse-flow direction as

shown in Fig. 6.11. Note that the pressure work rate typically must balance all the other

energy rate terms.

Blocks 1-2 and 6-7 fullfil similar roles as valve entrance and valve exit regions in for-

ward and reverse flow. Figure 6.11 shows that the pressure work rate is smaller in blocks 6

and 7 in reverse flow than in blocks 1 and 2 in forward flow. The ratio is 0.9, a reduction

in diodicity. They are geometric similar flow paths, both serving to accelerate the flow as

it enters the channel and develops its flow profile, but the forward-flow entrance, block 2 is

14% longer than block 6. This suggests block 2 should be shortened to improve diodicity.

However, blocks 2 and 6 also have a role as valve exits and in this role each contains a sep-

arated jet as can be seen in Figs. 6.2 and 6.3 with accompanying high rates of dissipation.

In fact the dissipation rate in the combined blocks 2 and 6 is 10% higher in forward than in

reverse flow; a negative impact on diodicity. Yet when the flow in the combined blocks 1,

2, 6, and 7 is considered, the ratio of pressure work rates in reverse flow relative to forward

flow is 1.08, an increase of diodicity. Study of Fig. 6.11 reveals that in forward flow the

energy flux rate into the outlet channel (block 6) is due to its laminar jet, whose momentum

flux is oriented approximately parallel to the downstream direction, but in reverse flow,

pressure work must be expended in block 2 to create energy flux. Figure 6.3 shows the

reason why; the laminar jet from the side channel is nearly perpendicular to the main chan-

nel, so the jet’s momentum flux does little to force the flow downstream. Thus diodicity

is created by the asymmetry; the forward-flow outlet-channel jet is aimed downstream but

the reverse-flow side-channel jet is not. If both blocks 2 and 6 were shortened, their large

dissipation rates would be diminished, and the positive diodic effect of the asymmetry of

energy flux rates and momentum flux directions, would not be so diluted.

The flow in the main channel (blocks 4 and 5) is radically different in the forward and

reverse directions. In the forward direction the main channel contains 85% of the valve

flow, and the pressure work is converted to energy dissipation and energy flux, 58% to

dissipation and 42% to flux. In the reverse direction only 36% of the valve flow is carried

72

1 2 3 4 5 6 7 8 9

−0.05

0

0.05

Block number

Dis

sipa

tion

Rat

e


(a) Energy dissipation rate

1 2 3 4 5 6 7 8 9

−0.05

0

0.05

Block number

Vis

cous

Wor

k R

ate


(b) Viscous work rate

Figure 6.10: Magnitude of the energy dissipation rate and the viscous work rate terms inthe kinetic-energy conservation equation from the T45A valve simulations with a volumeflow rate of 3710 µl/min corresponding to Re=528 based on the hydraulic diameter of themain channel.

73

1 2 3 4 5 6 7 8 9

−0.05

0

0.05

Block number

For

war

d−F

low

Pow

er

Energy Flux RateDissipation RateViscous Work RatePressure Work Rate

(a) Forward flow

1 2 3 4 5 6 7 8 9

−0.05

0

0.05

Block number

Rev

erse

−F

low

Pow

er


(b) Reverse flow

Figure 6.11: Magnitude of the terms in the kinetic-energy conservation equation from theT45A valve simulations with a volume flow rate of 3710 µl/min corresponding to Re=528based on the hydraulic diameter of the main channel.

74

by the main channel, and it is the energy flux that is converted to energy dissipation and

the pressure work needed to sustain the flow rate. The main-channel flow has a negative

impact on diodicity, but, as consequence of adopting a fixed geometry for the valve, it must

exist during reverse flow.

The most significant diodicity mechanism is located in the side channel and the T-

junction (blocks 3, 8, and 9) due to the radical asymmetry between the forward and reverse

flow there. In forward flow the only significant energy interchanges occur in the T-junction

(block 3) where 1/3 of the energy flux in the T-junction is dissipated and the remainder

is converted to pressure work to sustain the volume flow rate as the forward flow passes

through the area expansion of the T-junction and on into the main channel. In reverse

flow 1/3 of the pressure work rate of the entire valve is utilized to drive 64% of the valve

flow through the side channel and the T-junction. Virtually all the pressure work in the

side channel is dissipated in the large velocity gradient between the side-channel jet and

the outer wall of the side channel. The side-channel jet can be seen in Fig. 6.3, and the

corresponding dissipation in Fig. 6.6. Most of the pressure work (85%) in the T-junction is

converted to additional energy flux of the side-channel jet as it is turned to flow out through

block 2. The remainder of the pressure work is dissipated. The overall result is a strong

diodic effect as energy flux is transformed to pressure work in forward flow, but in reverse

flow 1/3 of the entire valve’s pressure work is dissipated or transformed to energy flux in

the side channel and T-junction alone. Considering the entire valve, 2/3 of the pressure

work is dissipated and 1/3 transformed to energy flux.

6.3 Discussion

From the preceeding qualitative analysis it is clear that the diodicity mechanism is located

in the side channel and T-junction (blocks 3, 8, 9), and that the rest of the valve adds to the

overall fluid resistance but has diodic effects near unity. As was suggested in Eq. 1.2, these

valve channels dilute the overall valve diodicity. For example, the goblet-shaped plenums

(blocks 1 and 7) and the inlet and outlet channels (blocks 2 and 6) exhibit a small amount

of diodicity, Di 1 1. Both the viscous forces and the dissipation rate are 10% higher in

forward flow than in reverse, but this adverse impact on diodicity is offset by the energy flux

rate and by the momentum flux because the laminar jet created in forward flow is pointed

downstream and the reverse flow jet is not. If these channels were not needed to direct the

fluid into and out of the valve, the overall valve diodicity would be closer to 2.0, instead of

1.27.

75

The underlying physical basis for the diodicity of the T45A valve is its ability to direct

the majority of the volume flow through alternate paths depending on flow direction. In

forward flow 85% of the fluid proceeds through the main channel (blocks 3 4 5); in

reverse flow 64% proceeds through the side channel (blocks 5 9 8 3). These two

alternate flow paths were compared quantitatively to determine the nature of the diodicity

mechanism. The mechanism is partly the viscous forces applied by the wall on the fluid;

the vector magnitude of the viscous force in reverse flow is 56% larger than in forward.

However, the energy dissipation rate is 97% larger in reverse flow than in forward. So the

diodicity mechanism is more due to dissipation in the fluid itself than to wall friction. The

energy flux rate is also 97% larger in reverse flow than in forward in these four blocks, but

it is only 12% of the rate of energy dissipation and its impact is small.

We satisfy research objective #1e, the description of the diodicity mechanism, by as-

sembling the results of the field variable analysis, the momentum perspective and the

kinetic-energy perspective. The velocity field showed that there is a forward-flow jet in

the outlet channel and a reverse-flow jet in the inlet channel. The pressure field showed

that a large pressure drop occurs at the T-junction in reverse-flow, but not in forward flow.

The energy-dissipation-rate field showed substantial dissipation in the side channel and in-

creased dissipation in the inlet channel during reverse flow. The momentum perspective

showed that substantial pressure force was needed to alter the direction of the reverse flow

jet, but not the forward flow jet, which is already pointed downstream. The kinetic-energy

perspective showed that most of the transformation of pressure work to energy dissipation

occurs in the shear gradient surrounding the laminar jets. Thus, the diodicity mechanism is

due to the asymmetry between a weaker forward-flow laminar jet pointed downstream and

a stronger reverse-flow laminar jet that is not pointed downstream and has a higher rate of

energy dissipation in the shear layer surrounding it.

Paul [23] performed a control-volume analysis of the region surrounding the T-junction

of a macro-scale Tesla-type valve with turbulent flow, see Sec. 1.3.1, in which he assumed

viscous forces were negligible and removed the viscous force term from the momentum

equation. He obtained an analytical solution for the pressure drop due solely to the change

in momentum flux, and agreement with his experimental data was good, showing the diod-

icity mechanism was due only to inertial forces. Let’s compare his macro-scale case with

flow in this micro-scale Tesla-type valve. Study of the pressure field from the T45A numer-

ical simulation shown in Fig. 6.4 reveals that the maximum pressure gradient occurs near

the T-junction where the channels recombine in reverse flow, the same location that Paul

assumed. In control volumes 2, 3, and 4, which are comparable to Paul’s control volume,

76

43% of the pressure force is required to overcome the viscous force in the x-direction, leav-

ing 57% to create momentum flux. Thus the viscous and inertial forces are of comparable

magnitude at Re 528 as anticipated by approximation theory and shown in Table 2.1.

In Paul’s macro-valve at Re 10000 there exists a turbulent jet instead of a laminar jet,

and the momentum flux would be four orders-of-magnitude larger than the viscous force,

but as the Reynolds number decreases below 1000, the viscous force becomes ever more

dominant over the momentum flux. Ignoring viscous forces in the low-Reynolds-number

flow of microvalves at this scale, ie. channel widths of 100 µm, would not be valid. In

fact, if the entire valve is considered, fully 2/3 of the pressure work is dissipated and only

1/3 is transformed to energy flux. Clearly, viscous effects dominate inertial effects even at

Re 500 in Tesla-type valves of this scale, satisfying research objective #2.

77

Chapter 7

DIODICITY MECHANISM OF T45C VALVE


The diodicity mechanism of the T45C valve was revealed by studying the steady-state

solutions of the numerical methods explained in Chap. 3 from two perspectives: first,

analysis of the velocity, pressure, and dissipation-rate fields, and second, analysis of the

terms of the momentum and kinetic-energy conservation equations applied in 9 regional

control volumes as shown in Fig. 7.1. It is important to note that the numerical methods

employed to analyze the T45C were exactly identical to those used in the T45A analysis.

The only variation is the substitution of the physical grid of the T45C for that of the T45A.

The flow behavior and characteristics are compared throughout this chapter with those of

the T45A valve.

Steady-state solutions of both forward flow and reverse flow in the T45C valve were

calculated using the numerical methods described in Chap. 3. These steady-state solutions

were obtained from the final solution of step-response simulations that were continued until

steady flow was achieved. In all cases the simulations converged to steady solutions without

applying Reynolds averaging to the Navier-Stokes equations. As discussed in Sec. 1.2.2,

this identifies the flow in the valve as laminar over the Reynolds number range simulated,

Re 748, despite the presence of separated flow and recirculation regions. To assess grid

independence, the grid was refined from 94380 to 127008 finite volumes, an increase of

34.6%, resulting in a reduction in the kinetic-energy conservation error εKE 0 05 of less

than 5%. The accuracy of the flow-rate response predictions of the numerical method was

determined in Ch. 4.



rection of 3640µl min corresponding to ReD 519 based on the hydraulic diameter of the

main channel. To achieve the same flow rate in the forward-flow direction required a differ-

ential pressure of only 0.377 atm. This corresponds to a diodicity Di 1 33 according to

Eq. 1.1. According to the analysis of diodicity prediction accuracy in Ch.4, there is a 95%

78

Figure 7.1: Division of the T45C valve into regional control volumes.

probability that the true diodicity is within 1 20 Di 1 34. The increase in predicted

diodicity over that of the T45A valveDi 1 27 is not statistically significant.

7.2 Results


From the step-response valve simulations, Fig. 7.2 shows the forward and Fig. 7.3 the

reverse-flow velocity fields on the symmetric centerplane of the valve. The velocity fields

in the forward and reverse flow cases are very similar to those in the T45A valve. Flow sep-

aration and laminar jets occur in three locations: where the channels separate, the channels

recombine, and at the valve exit.

In the forward-flow case the flow accelerates rapidly as it enters the main channel from

the goblet-shaped inlet plenum. As the velocity profile is developing it reaches the T-

junction. The flow begins to veer slightly into the side channel, and a small portion im-

pinging on the guide vane becomes a minor jet flowing into the side channel. However,

88% of the main channel flow (compared to 85% in the T45A) is unperturbed and con-

tinues downstream becoming fully-developed. The side channel jet spreads as it proceeds

around the bend in the side channel, and approaches the Y-junction as a low-velocity stream

filling the entire width of the side channel, moving at less than 11% of the bulk velocity

of the main channel (compared to 20% for the T45A). However, it has sufficient momen-

tum to cause the main channel flow to begin to turn before it reaches the far wall of the

Y-junction. This may have a small effect in improving the diodicity as it tends to increase

79




0.0000E+001.4167E-012.8333E-014.2500E-015.6667E-017.0833E-018.5000E-01

Figure 7.2: Forward-flow velocity field on the centerplane of a single-element, Tesla-typeT45C valve with a volume flow rate of 3640 µl/min corresponding to Re=519 based on thehydraulic diameter of the main channel. One dimensionless unit equals 10 m/s.

80

the radius of curvature of the main channel flow as it travels around the 45 bend. Down-

stream of the Y-junction, the main channel flow separates from the inner wall and forms a

narrow, hi-speed jet next to the outer wall. The high velocity gradient between the jet and

the wall produces additional dissipation that lowers the diodicity of the valve. However,

this channel downstream of the Y-junction is only 69% of the length of the corresponding

channel in the T45A and should contain relatively less dissipation. A significant portion of

the channel is filled with a quiescent zone and is essentially underutilized by the forward

flow. The flow leaves the channel and enters the goblet-shaped outlet plenum as a laminar

jet as in the T45A. Whatever momentum flux is dissipated increases the pressure drop in

the forward-flow direction and lowers diodicity.

In the reverse-flow case the flow accelerates rapidly as it enters the channel from the

goblet-shaped plenum, just as it does in the forward flow case. On reaching the Y-junction

the flow stream veers toward the main channel. As it impinges on the cusp of the guide

vane, 43% of the flow is deflected down the main channel (compared to 36% in the T45A)

where it travels next to the guide vane wall, but most of the main channel is filled with a

large, slowly-moving recirculation zone. The side channel flow separates from the outer

wall and forms a laminar jet attached to the guide vane wall. The large velocity gradient

there increases dissipation, which improves diodicity. At the bend in the side channel, the

jet separates from the guide vane, but unlike the jet in the T45A it does not attach to the

outer side-channel wall, but rather proceeds down the center of the side channel until it

reaches the T-junction. There it brushes past the protruding cusp and emerges from the side

channel as a laminar jet heading for the opposite wall of the inlet channel. The momentum

of the smaller flow traveling down the main channel turns the jet, but unlike the jet in the

T45A which never reaches the opposite wall, the jet in the T45C attaches to the opposite

wall of the inlet channel and remains attached through the rest of the valve. There is a large,

vigorous recirculation zone downstream of the T-junction, and the jet never spreads to fill

the entire channel as the jet in the T45A does.


The pressure fields of the T45C valve shown in Fig. 7.4 are very similar to those of the

T45A. A major pressure loss is occurs where the flow enters the valve channel regardless of

the flow direction. There is only one other signficant pressure loss in each flow direction:

in forward flow it is at the 45 bend in the channel, in reverse flow it is just downstream

of the T-junction. Between these locations and their corresponding flow exits there is little

81




0.0000E+001.4434E-012.8868E-014.3302E-015.7736E-017.2170E-01

> 8.5000E-01

Figure 7.3: Reverse-flow velocity field on the centerplane of a single-element, Tesla-typeT45C valve with a volume flow rate of 3640 µl/min corresponding to Re=519 based on thehydraulic diameter of the main channel. One dimensionless unit equals 10 m/s.

82

additional pressure drop; the momentum flux maintains the fluid velocity until it begins

to dissipate in the goblet-shaped plenum and beyond. Closer study of the pressure fields

shows that pressure drop at the 45 bend in forward flow is approximately 0.1 atm, and

downstream of the T-junction in reverse flow is approximately 0.24 atm, (compared to 0.20

in the T45A). This is the main source of diodicity. Thus, as suggested by Eq. 1.2, if the

direction-independent losses at the channel entrances could be eliminated, the diodicity

would be increased from 1 33 to 2 0. The outlet channel of the T45C is 30% shorter

than the T45A, and the inlet channel is 7% longer. The geometry of the mouths of the

channel entrances are identical.

7.2.3 Energy Dissipation Field



largest. Energy dissipation in the forward-flow case is very similar to the T45A; dissipation

occurs along the walls of the main channel and on both sides of the laminar jet downstream

of the 45 bend. In the outlet goblet plenum, the dissipation of the momentum flux leaving

the exit is clearly visible. But the most significant dissipation occurs at the convex surfaces

of the channel mouth where the flow accelerates as it enters the channel, at the upstream

cusp of the guide vane, at the inner corner of the 45 bend, and along the wall downstream

of the 45 bend all the way to channel exit. These locations correspond well to the locations

of large pressure drop in Fig. 7.4, but it is not conclusive from this field variable analysis

that dissipation is the most important source of that pressure loss.

The locations of dissipation in the reverse flow case are somewhat different than in

the T45A valve. As the flow bifurcates at the Y-junction, the dissipation is spread more

evenly in the side channel than elsewhere, but very significant dissipation occurs along the

surfaces of the guide vane, both in the side channel and the main channel, and especially

at its leading cusp. Unlike in the T45A, there is no large dissipation along the outer wall

of the side channel since the separated jet does not attach to it. There is very significant

dissipation at the protruding cusp at the T-junction, and this dissipation continues alongside

the jet as it proceeds from the side channel into the main channel, collocated with the large

velocity gradient between the jet and the recirculation zone in the main channel downstream

of the T-junction. There is also significant dissipation where the jet exiting from the side

channel impinges on the opposite wall of the inlet channel. But unlike in the T45A, the jet

remains attached to the opposite wall and this high rate of dissipation persists all along the

83




-8 .0000E-022.0000E-021.2000E-012.2000E-013.2000E-014.2000E-015.2000E-01

(a) Forward flow




<-8.0000E-021.0566E-021.1245E-012.1434E-013.1623E-014.1811E-015.2000E-01

(b) Reverse flow

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

84




-8.0000E+00-6.6733E+00-5.3467E+00-4.0200E+00-2.6933E+00-1.3667E+00-4 .0000E-02

Figure 7.5: Base 10 logarithm of the energy dissipation rate in forward flow on the center-plane of a single-element, Tesla-type T45C valve with a volume flow rate of 3640 µl/mincorresponding to Re=519 based on the hydraulic diameter of the main channel. One di-mensionless unit equals 14 mW.

85




<-8.0000E+00-6.7985E+00-5.4468E+00-4.0951E+00-2.7434E+00-1.3917E+00-4 .0000E-02

Figure 7.6: Base 10 logarithm of the energy dissipation rate in reverse flow on the center-plane of a single-element, Tesla-type T45C valve with a volume flow rate of 3640 µl/mincorresponding to Re=519 based on the hydraulic diameter of the main channel. One di-mensionless unit equals 14 mW.

86

wall until the plenum is reached. As in the forward-flow case, part of the momentum flux

leaving the channel is dissipated in the goblet-shaped plenum.



The momentum equation is a vector relation, a summation of forces, so each of its terms has

both magnitude and direction sense. The X and Y components of these forces are shown in

Fig. 7.7 for both the forward and the reverse flow case. The figure caption explains positive-

negative direction sense and the X-Y coordinate system is shown in Fig. 7.1. Essentially,

the figures show that the viscous and pressure forces applied on the fluid and the fluid’s

momentum flux response are similar to those of the T45A. The only differences are in the

outlet channel (block 6) in forward flow and the region of the T-junction (blocks 2, 3, 4,

and 8) in reverse flow.

The goblet-shaped inlet plenum (block 1) is oriented for flow in the X direction, so

as Fig. 7.7 shows, the Y-direction force components are negligible. In forward flow the

pressure force is to the right (positive) and it is expended by accelerating the fluid as it

approaches the inlet channel and by a small amount of viscous force. The momentum flux

is to the right (positive), but more is leaving the control volume than entering (defined as

negative), so its overall sign is negative. In reverse flow the pressure is already near ambient

when it enters the block 1 control volume, so the pressure force is negligible. The viscous

force applied by the walls is also very small, so the momentum flux is unchanged as the

flow passes through block 1.

The inlet channel (block 2) is also oriented in the X direction but due to its proximity

to the side channel (block 8) it has Y-direction force components during reverse flow. The

positive-Y pressure force is applied by the lower wall on the jet of fluid from the side chan-

nel, altering its direction so that it flows out of block 2 and into block 1. The momentum

of this jet is directed into the inlet channel (positive) but is flowing in the negative-Y di-

rection (negative) resulting in a net negative Y-component of the momentum flux. In the

X-direction in both forward and reverse flow, the upstream flow boundary applies the pres-

sure force needed to drive the fluid through the inlet channel offset by the viscous force and

momentum flux. In both flow directions the viscous force is larger than the momentum flux,

as in the T45A. However, the reverse-flow viscous force is 24% larger in the T45C than in

the T45A, because the laminar jet attaches to the lower wall instead of flowing nearer the

center of the channel as it does in the T45A. Unfortunately, the X-direction pressure force

87

1 2 3 4 5 6 7 8 9

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Block number

For

war

d−F

low

X−

dire

ctio

n F

orce

s



1 2 3 4 5 6 7 8 9

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Block numberR

ever

se−

Flo

w X

−di

rect

ion

For

ces



1 2 3 4 5 6 7 8 9

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Block number

For

war

d−F

low

Y−

dire

ctio

n F

orce

s



1 2 3 4 5 6 7 8 9

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Block number

Rev

erse

−F

low

Y−

dire

ctio

n F

orce

s



Figure 7.7: Force vector terms in the integral form of the momentum conservation equation.Net pressure force and net momentum flux into a control volume are positive. Viscous forceis applied on the fluid by the wall. X-vectors to the right and Y-vectors upward are positiveand consistent with the valve layout in Fig. 7.1 including the numbering of the controlvolumes (blocks).

88

is larger in forward flow than in reverse, which has an adverse impact on valve diodicity.

It is not clear from the momentum perspective what impact the Y-direction pressure force

has on diodicity.

The T-junction (block 3) does not have significant Y-direction forces, a surprising result

since this control volume connects the side channel (block 8) to the inlet channel and the

main channel (block 4). However, inspection of Fig. 7.3 shows that the jet from the side

channel during reverse flow does not significantly interact with the wall in block 3, but

passes through to block 2. However the X-direction forces are important in creating diod-

icity. In forward flow there is a small amount of pressure recovery as the momentum flux

drives the flow through the area expansion of block 3; this negative pressure force increases

diodicity as it lowers the net valve pressure force for forward flow. The reverse flow has a

much greater positive impact on diodicity; the momentum flux leaving block 3 (negative)

and flowing in the negative X direction is the jet from the side channel, and results in a

positive net momentum flux for block 3 as seen in Fig. 7.7b. The pressure force needed to

offset this momentum flux is 22% larger in the T45C than the T45A. This is a major source

of the increased diodicity of the T45C.

The main channel (block4) has almost identical Y-direction force components in for-

ward and reverse flow. In both cases the upper wall, which forms part of the guide vane

or island of the valve, applies a pressure force in the negative-Y direction to the impinging

flow, which has just passed an opening to the side channel. This impingement can be seen

in both Figs. 7.2 and 7.3. It is not clear from the momentum perspective if this affects

diodicity. In the X direction the force components are very different between forward and

reverse flow. In forward flow 88% of the volume flow is carried by the main channel, and

as a result significant pressure force is needed to counteract viscous force due to wall fric-

tion. This has a negative impact on diodicity as 57% of the reverse flow occurs in the side

channel and little pressure force is required to push the remaining reverse flow through the

main channel. The reverse-flow is 19% larger in the T45C than in the T45A, as a result

the viscous force applied by the wall in the T45C is correspondingly larger. Otherwise the

forces are essentially identical to those of the T45A.

The Y-connection (block 5) forces are essentially identical in the T45C to the T45A

forces. There are negligible X-direction force components. In forward flow the net Y-

direction flux of momentum is out of the control volume (negative) and in the negative Y

direction resulting in a positive value, which is counteracted by the pressure force applied

by the outer wall. In reverse flow some of the flow is directed down the main channel (block

4) by the guide vane, thus there is a net flux of momentum into the control volume in the

89

positive Y direction, so the momentum flux again has a positive Y sense.

The outlet channel (block 6) is similar in form and function to the inlet channel (block2),

except that it is oriented at 45 cw with respect to the positive X direction. Because of

different orientations it is difficult to compare the forces in blocks 2 and 6 using Fig. 7.7,

but a comparison is made below using the vector magnitudes shown in Fig. 7.8 to assess

the impact on diodicity. In the T45A, the pressure force applied at the upstream flow

boundary was roughly equally opposed by the viscous force applied by the walls and by

the net momentum flux leaving the control volume, but in the T45C the viscous force is

reduced because the outlet channel is 31% shorter, and thus the pressure force required in

both flow directions is 22% less in the T45C than in the T45A. This has a positive influence

on diodicity according to Eq. 1.2 by reducing direction-independent losses.

The goblet-shaped outlet plenum (block 7) is analogous to the inlet plenum (block 1)

except for its orientation of 45 cw with respect to the positive X direction. The forward flow

forces are small; the reverse-flow net pressure force accelerates the fluid as it approaches the

next downstream control volume (block 6). All the viscous forces are small. The forward-

flow momentum flux, though still small, is larger in the T45C than in the T45A, because the

outlet channel (block 6) is shorter and the velocity of the forward-flow jet entering block 7

is higher.

The side channel (block 8) has negligible forces in forward flow since it carries only

12% of the volume flow rate. In reverse flow, Fig. 7.7d shows a very large pressure force

applied by the outer wall to alter the direction of the flow in the side channel. Unlike in

the T45A, the flow is not attached to the outer wall (see Figs. 7.3 and 7.4), and in addition

there is 12% less flow in the side channel in the T45C than in the T45A. As a result, the

pressure force and momentum flux are 18% smaller in the T45C than in the T45A. Because

these are Y-direction forces it is difficult to assess their impact on diodicity. The last control

volume (block 9) has significant forces only in the reverse-flow direction.



oriented along the X direction, was to study the vector magnitudes of the force components



The inlet and outlet goblet-shaped plenums (blocks 1 and 7) perform similarly in the

T45C as in the T45A valve. The viscous forces applied by the walls are small, and the

90

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

Block number

Pre

ssur

e F

orce

Mag

nitu

de



1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

Block number

Vis

cous

For

ce M

agni

tude



1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

Block number

Mom

entu

m F

lux

Mag

nitu

de



Figure 7.8: Vector magnitudes of the terms in the momentum conservation equation for theT45C valve.

91

pressure forces and momentum fluxes are the significant forces with those of the forward

flow jet entering block 1 and the reverse flow jet entering block 7 an order of magnitude

larger than all other forces. The combined diodic effect of blocks 1 and 7 is again slightly

above unity, Di1 7 1 02

The inlet and outlet channels (blocks 2 and 6) function similarly as in the T45A. One

difference is a 20% reduction in the viscous forces in the shortened block 6 (31%) in the

T45C. Another difference is an increased viscous force applied by the walls of block 2

in the T45C during reverse flow, due to the reverse-flow jet that has attached to the wall

instead of remaining near the center of the channel as in the T45A; compare Figs. 7.3 and

6.3. The net diodic effect is slightly above unity, though this is not clear from the pressure

force magnitude plot of Fig. 7.8a.

The main channel (block 4) and the Y-junction (block 5) of the T45C perform identi-

cally as in the T45A. They have larger pressure force magnitudes in forward flow than in

reverse flow, and thus have a diodic effect less than unity. In block 4 this is because almost

all the forward flow proceeds through the main channel instead of bypassing it as it does in

reverse flow. As a result the viscous force applied by the wall is much almost twice as large

in forward flow as it is in reverse. In block 5 the larger pressure force in forward flow is

required offset the larger momentum flux and viscous force, which are due to the forward

flow changing direction in block 5, whereas in reverse flow it proceeds straight through

toward the side channel.

The major contributors to diodicity are the T-junction (block 3) and the side channel

(blocks 8 and 9). The T-junction is where the side channel emerges and must be redirected

and forced out through block 2, the result is a large net increase in momentum flux and

additional viscous force applied by the walls that both must be overcome by the pressure

force. The large increase in pressure across block 3 can be seen in Fig. 7.4b. The forward-

flow forces in the side channel and T-junction of the T45C are nearly identical to those in

the T45A. In reverse flow the T45C requires less pressure force in the side channel than

in the T45A since the flow rate is 11% less and the momentum flux is correspondingly

reduced. This reduces overall diodicity. However the net momentum flux in the T-junction

of the T45C is 22% higher than in the T45A, and the pressure force is 27% higher as seen

in Fig. 7.8. The net result is an increase in the diodicity of the T45C over the T45A.

92


The kinetic energy terms for the entire valve were compared to those of the T45A. Over

70% of the pressure work is dissipated in the valve compared to 66% in the T45A. The ratio

of the dissipation rates (reverse flow / forward flow) is 1.35, instead of 1.13 in the T45A,

due to a 12% reduction in forward-flow dissipation because of the shortening of the outlet

channel (block 6), and a 6% increase in reverse-flow dissipation because of the attachment

of the reverse-flow jet to the wall in the inlet channel (block 2). The ratio of energy flux

rates is 1.33, instead of 1.91 in the T45A, because the shortened outlet channel allows a

higher-velocity forward-flow jet to pass into the outlet plenum (block 7). The energy flux

rates are only one-third the magnitude of the dissipation rates, as in the T45A. The overall

valve diodicity of the T45C is 1.33. The diodicity prediction from the ratio of the pressure

work rates as in Eq. 2.14 is 1.34, which is within 1% of 1.33.

There are two alternate ways to look at the data. Figs. 7.9 and 7.10 show the two most


work rate and the dissipation rate, which are shown for each of the control volumes for

both forward and reverse flow. As in the T45A, blocks 3, 6, 7, 8, and 9 were identified as

sources of diodicity, since the pressure work applied in the reverse flow direction is larger

than that applied in the forward direction. On the other hand, blocks 1, 2, 4, and 5 cause

a significant reduction in diodicity. The only significant difference in the pressure work

rates for the T45C and the T45A is the reduced direction dependence of block 2, due to

an increase in the reverse-flow pressure work rate. The energy flux rate has opposite sign

(+ is a net increase, - is a net decrease in the control volume) depending on flow direction

in all blocks. The dissipation rate also exhibits significant direction dependence, except in

blocks 3, 4, and 5. There are three differences between the dissipation rates of the T45C and

T45A: the reverse-flow dissipation rate in block 2 is increased (due to the wall attachment

of the reverse-flow jet), the forward-flow dissipation rate in block 6 is decreased (due to

the shortening of block 6), and the reverse-flow dissipation rate in block 8 is decreased

(due to the reduced flow rate in the side channel). Once again, the viscous work rate is

inconsequential in all control volumes.



shown in Fig. 7.11. Note that the pressure work rate typically must balance all the other

energy rate terms.

Blocks 1-2 and 6-7 fullfil similar roles as valve entrance and valve exit regions in for-

93

1 2 3 4 5 6 7 8 9

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Block number

Pre

ssur

e W

ork

Rat

e



1 2 3 4 5 6 7 8 9

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Block number

Ene

rgy

Flu

x R

ate



Figure 7.9: Magnitude of the pressure work rate and the energy flux rate terms in thekinetic-energy conservation equation from the T45C valve simulations with a volume flowrate of 3640 µl/min corresponding to Re=519 based on the hydraulic diameter of the mainchannel.

94

1 2 3 4 5 6 7 8 9

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Block number

Dis

sipa

tion

Rat

e



1 2 3 4 5 6 7 8 9

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Block number

Vis

cous

Wor

k R

ate



Figure 7.10: Magnitude of the energy dissipation rate and the viscous work rate terms inthe kinetic-energy conservation equation from the T45C valve simulations with a volumeflow rate of 3640 µl/min corresponding to Re=519 based on the hydraulic diameter of themain channel.

95

1 2 3 4 5 6 7 8 9

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Block number

For

war

d−F

low

Pow

er


(a) Forward flow

1 2 3 4 5 6 7 8 9

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Block number

Rev

erse

−F

low

Pow

er


(b) Reverse flow

Figure 7.11: Magnitude of the terms in the kinetic-energy conservation equation from theT45C valve simulations with a volume flow rate of 3640 µl/min corresponding to Re=519based on the hydraulic diameter of the main channel.

96

ward and reverse flow. Figure 7.11 shows that the pressure work rate is smaller in blocks

6 and 7 in reverse flow than in blocks 1 and 2 in forward flow. The ratio is 0.85 compared

to 0.9 in the T45A, an even larger reduction in diodicity due to the shortening of block

6. They are geometric similar flow paths, both serving to accelerate the flow as it enters

the channel and develops its flow profile, but the forward-flow entrance, block 2 is 76%

longer than block 6 (compared to 14% longer in the T45A). This suggests block 2 should

be shortened as well to improve diodicity. However, blocks 2 and 6 also have a role as

valve exits and in this role each contains a separated jet as can be seen in Figs. 7.2 and 7.3

with accompanying high rates of dissipation. The dissipation rate in the combined blocks

2 and 6 is higher in reverse than in forward flow; a diodic effect of 1.28 compared to 0.89

in the T45A. Even when the flow in the combined blocks 1, 2, 6, and 7 is considered, the

ratio of pressure work rates in reverse flow relative to forward flow is 1.13 (compared to

1.09 in the T45A), an increase of diodicity.

In the T45A, there was an asymmetry between the forward-flow energy flux rate in

block 6 and the reverse-flow energy flux rate of block 2, see of Fig. 6.11, that had a

positive diodic effect. Since block 6 was shortened in the T45C, this asymmetry does

not exist (see Fig. 7.11), instead, additional momentum flux is expended in the high rate

of dissipation due to the wall attachment of the reverse-flow jet. In fact, block 2 has a

40% higher dissipation rate in the T45C than in the T45A, even though block 2 is only

7% longer in the T45C. This suggests that if block 2 was shortened as block 6 has been

(compared to the T45A), the forward-flow dissipation rate in block 2 would decrease more

than the reverse-flow dissipation rate, and some positive diodic effect of the asymmetry

of the dissipation rates may be retained. In any case, reduction of direction-independent

pressure losses should increase diodicity, following Eq. 1.2.

The flow in the main channel (blocks 4 and 5) of the T45C is very similar to that in the

T45A. In the forward direction the main channel contains 88% of the valve flow (3% more

than in the T45A), and the pressure work is converted to energy dissipation and energy

flux. In the reverse direction 43% of the valve flow is carried by the main channel (7%

more than in the T45A), and it is the energy flux that is converted to energy dissipation and

the pressure work needed to sustain the flow rate. The main-channel flow has a negative

impact on diodicity, but, as consequence of adopting a fixed geometry for the valve, it must

exist during reverse flow.

As in the T45A, the most significant diodicity mechanism is located in the side channel

and the T-junction (blocks 3, 8, and 9) due to the radical asymmetry between the forward

and reverse flow there. In forward flow the only significant energy interchanges occur

97

in the T-junction (block 3) where 1/3 of the energy flux is dissipated and the remainder

is converted to pressure work to sustain the volume flow rate as the forward flow passes

through the area expansion of the T-junction and on into the main channel. In reverse flow

35% of the pressure work rate of the entire valve (3% more than in the T45A) is utilized to

drive 57% of the valve flow through the side channel and the T-junction (7% less than in the

T45A). The increased pressure work rate in spite of a lower flow rate is due to the longer

path length of the side channel in the T45C. Unlike the T45A, in which virtually all the

pressure work in the side channel is dissipated by the large velocity gradient between the

side-channel jet and the outer wall, the T45C dissipates 76%, the remainder is transformed

into the energy flux of the side-channel jet. The side-channel jet is shown in Fig. 7.3,

and the corresponding dissipation in Fig. 7.6. Most of the pressure work (88%) in the T-

junction is converted to additional energy flux of the side-channel jet as it is turned to flow

out through block 2. The remainder of the pressure work is dissipated. The overall result is

a strong diodic effect as energy flux is transformed to pressure work in forward flow, but in

reverse flow 35% of the entire valve’s pressure work is dissipated or transformed to energy

flux.

7.3 Discussion

The identical numerical methods applied to the T45A were applied to the T45C. The only

change in the numerical computations was the substitution of the physical grid of the T45C

in place of the physical grid of the T45A. The T45C was shown to be functionally similar

to the T45A. The diodicity mechanism is located in the side channel and T-junction (blocks

3, 8, 9), and that the rest of the valve adds to the overall fluid resistance but has diodic

effects near unity. As was suggested in Eq. 1.2, these valve channels dilute the overall

valve diodicity. For example, the goblet-shaped plenums (blocks 1 and 7) and the inlet and

outlet channels (blocks 2 and 6) produce a diodic effect of 1.13 (compared to 1.09 in the

T45A). But if these channels were not needed to direct the fluid into and out of the valve,

the overall valve diodicity would be closer to 2.2, instead of 1.33.

As in the T45A, the underlying physical basis for the diodicity of the T45C valve is

its ability to direct the majority of the volume flow through alternate paths depending on

flow direction. In forward flow 88% of the fluid proceeds through the main channel (blocks

3 4 5); in reverse flow 57% proceeds through the side channel (blocks 5 9

8 3). The diodicity mechanism is partly due to the viscous forces applied by the wall

on the fluid; the vector magnitude of the viscous force in reverse flow is 60% larger than

98

in forward (compared to 56% in the T45A). However, the energy dissipation rate is 87%

larger in reverse flow than in forward, significantly less than the 97% of the T45A. So the

diodicity mechanism is again more due to dissipation in the fluid itself than to wall friction.

The reduced reverse-flow energy dissipation in the T45C is balanced by increased energy

flux; this additional energy in the side-channel jet enables it to cross the main channel and

attach to the far wall. Thus the T45A contains a reverse-flow jet that is attached to the

wall in the side channel with accompanying dissipation in the side channel; in the T45C

the reverse-flow jet doesn’t attach to any wall until it reaches the main channel, so the

accompanying dissipation occurs in the main channel.

As for the T45A, research objective #1e has been achieved. The diodicity mechanism

of the T45C valve is a reverse-flow laminar jet that is not pointed in the downstream di-

rection and has a high rate of energy dissipation in the shear layer surrounding it. This jet

requires significant pressure force to turn it downstream, and more pressure work to offset

the energy it dissipates than to offset the energy dissipated by the forward-flow jet. Due to

the combined effects of the shortening of the outlet channel (block 6) and the attachment

of the reverse-flow jet in the inlet channel (block 2), the T45C has an 11% lower dissipa-

tion rate in forward flow than the T45A, and a 6% increase in dissipation rate in reverse

flow, resulting in an overall increase in diodicity to 1.33. But, according to the analysis of

diodicity prediction accuracy in Ch.4, there is a 95% probability that the true diodicity is

within 1 20 Di 1 34. Thus the increase in predicted diodicity over that of the T45A

valveDi 1 27 is not statistically significant.

It is easy to answer research objective #2 for the T45C valve, that the diodicity mech-

anism is mainly due to viscous effects, since conveniently all four measures: the diodicity

and the ratios of the reverse-flow and forward flow pressure work, energy flux, and dis-

sipation are all within 2% of 1.33. Over 2/3 of the pressure work is expended in energy

dissipation, thus 2/3 of the diodicity is due to dissipation. Clearly viscous effects are dom-

inant.

99

Chapter 8

VALVE DESIGN GUIDELINES

This chapter is directed toward research objectives #3a: to demonstrate knowledge of

the diodicity mechanism of Tesla-type NMP microvalves by developing guidelines for the

creation of enhanced-diodicity valve designs, and #3b: to use these guidelines as the sole

method to modify a valve design to significantly increase its diodicity. We accomplish

this by modifying the lower-performing T45A valve according to the design guidelines to

produce a T45A-2 valve that has higher performance than the T45C.

Valve design is a difficult task, since most of the geometric design variables of a Tesla-

type NMP valve, variables a through h in Fig. 8.1, are interdependent; they impact the

distribution of flow between the main channel and the side loop, the foundation of the diod-

icity mechanism. Variation of any of these design variables will impact the optimal settings

of other variables. A parametric study of the 14-dimensional design space following Fig.

8.1 would require simulation of 14 14 196 valve designs even if only two values were

considered for each design variable. Even a fractional-factorial analysis would require the

simulation of at least 28 designs. To further add to the complexity of the design task, the

geometric layout determines not only the diodicity of the valve, but also its resistance and

inertance, which must be matched to the dynamics of the microfluidic system in which it

operates. A well-matched valve may provide higher performance than a poorly-matched

higher-diodicity valve. Thus, there is no single optimal valve design for all microfluidic

systems. Clearly, a more efficient method of valve design is needed than mapping diodicity

as each valve design variable is varied.

The solution is to use the knowledge gained from the kinetic-energy, momentum, and

field variable analyses of the diodicity mechanisms of the T45A and T45C valves to develop

valve design guidelines. These guide the design of a high-performing Tesla-type NMP

valve for the particular microfluidic system. Once the valve geometry has been laid out,

the impedance of the prototype valve design can be predicted by numerical simulations

and applied in lumped-parameter dynamics models of the complete microfluidic system to

predict performance [3][2][17].

100

8.1 Preliminary Discussion

To guide the layout of a valve, we reiterate and extend some of the key points from the

discussion sections of the previous chapters, in particular, Secs. 2.4, 6.3, and 7.3 in which

the diodicity mechanism of Tesla-type NMP valves was revealed.

In Tesla-type valves with their multi-directional flow, the vector nature of the momen-

tum perspective inhibits its use in estimating the diodicity, a scalar. However, the kinetic-

energy, itself a scalar, has no such difficulty. In Eq. 2.14 it was shown that the diodicity

could be estimated from the ratio of the pressure work rates in reverse and forward flow.

This ratio has predicted the diodicity of the both the T45A and T45C within 2%. And since

other terms of the kinetic-energy equation are insignificant, the pressure work rate is equal

to the sum of the energy flux rate out of the control volume and the energy dissipation rate

in the control volume. Valuable insight can be gained by estimating the diodicity from

Di

u2

2

u n dA 1

Rep

τ i j

∂ui∂x j

dV reverse

u2

2

u n dA 1

Rep

τ i j

∂ui∂x j

dV f orward

Q

(8.1)

in dimensionless variables. For high diodicity, the basic goal is to maximize the dissipation

rate and energy flux rate in reverse flow and minimize them in forward flow. This equation

also shows that the diodicity is a function of u3i and ∂ui ∂x j 2

, so that for a given flow

cross-sectional area, energy losses are exhibited in order of increasing losses by plug flow,

fully-developed parabolic profiles, and highly-asymmetric velocity profiles. The relative

importance of u3i and ∂ui ∂x j 2

is controlled by the Reynolds number; when Rep is high,

the energy flux rate dominates; when Rep is small, the dissipation rate dominates, as shown

in Table 2.2. Thus, diodicity is created by generating separated laminar jets with their

asymmetric velocity profiles and high velocity gradients (especially when the jet is attached

to the wall). To have high diodicity the reverse flow jet must dissipate more energy than

any, preferably nonexistent, forward-flow jet.

Though it is not as useful for estimating diodicity, additional insight into enhancing

diodicity is provided by the momentum perspective. Equation 2.13 relates diodicity to the

ratio of the pressure force in reverse and forward flow. Implicit is that this is a vector

equation. In other words, not only the velocity magnitude of the laminar jet is important,

but also its direction. Thus, a reverse-flow jet should be oriented counter to the downstream

direction, and if it is not possible to avoid creating a jet in forward flow, it should at least

be oriented downstream.

101

The diodicity relation Eq. 1.2 rewritten here

Di

∆Pdirection independent ∆Preverse direction dependent

∆Pdirection independent ∆Pf orward direction dependent

shows that direction-independent losses diminish diodicity. The most obvious implemen-

tation of this concept is the minimization of the lengths of the inlet and outlet channels.

Comparison of the T45A and T45C showed that shortening the outlet channel of the T45C

was partly responsible for its higher diodicity. Minimization of all fluid path lengths is

recommended when energy flux from the valve is certain to be dissipated.

It is significant that Eq. 8.1 assumes all energy flux will eventually be dissipated. In

other words, as the Reynolds number increases and less of the dissipation occurs within the

geometric boundaries of the valve itself, it becomes increasingly important to ensure that

channels upstream and downstream of the valve are dissipative not diffusive, (ie. change

cross-sectional flow area suddenly), transferring energy flux back into available pressure

work in the fluid.

An alternative strategy would choose the diodicity to be the ratio of the dissipation

within the valve, and attempt to recover the kinetic energy of the fluid by providing dif-

fusers at both ends of the valves, (ie. transform the energy flux back to available pressure

work). This would be a particularly appropriate strategy when it is desirable to reduce the

valve resistance and increase the valve inertance. It is an advantageous characteristic of

the T45C that its diodicity would increase 6% from 1.33 to 1.35 in this scenario, while the

ratio of reverse and forward dissipation rates in the T45A is such that its diodicity would

decrease 48% from 1.27 to 1.13. In the T45C it is the interaction of the reverse-flow jet

with the long inlet channel and the forward-flow jet with the short outlet channel that pro-

vide this direction-dependent internal energy dissipation. Up to 20% of the pressure work

could be recovered with a corresponding decrease in valve resistance if effective diffusing

channels were added to both ends of the T45C. The addition of diffusing channels would

also increase overall channel length increasing valve inertance and lowering the corner fre-

quency of the valve. This would be an appropriate strategy when energy flux from the valve

would not be dissipated in the surrounding structure.

8.2 How To Lay Out a Tesla-Type Valve

In this section we present design guidelines and demonstrate how to layout a Tesla-type

valve using a generic sketch of a Tesla-type valve shown in Fig. 8.1. As an example

102

b

c

hm

n

ag

Y

dido

Tinlet

side

main

outlet

f

e

l

k

ji

Figure 8.1: Sketch of generic Tesla-type NMP valve with dimensioning per design rulesfor high diodicity.

we will follow these guidelines to modify the geometry of the T45A valve to enhance its

diodicity mechanism resulting in a higher-performing valve we will call the T45A-2. This

new valve is analyzed in the following chapter by the numerical method to show that the

diodicity mechanism has indeed been enhanced. The sketch itself is an engineering tool;

there are other dimensioning schemes that could also be used, but the dimensioning used in

Fig. 8.1 is sufficient to allow us to manipulate the valve geometry to utilize our knowledge

of the diodicity mechanism.

Referring to Fig. 8.1, design of a Tesla-type valve begins with selection of angle a

between the main and side channels. The T45A and T45C both use angle a = 45 by

which they are named. Other angles have been used, from the 22 of the original design of

Tesla [28] up to 90 Angle a contains a design conflict; decreasing a weakens the forward-

flow jet in the outlet channel and diminishes forward-flow losses, increasing a increases

the proportion of the reverse flow directed into the side channel and increases reverse-flow

losses. Comparison of the T45A and T45C showed that a decrease in the proportion of

reverse flow passing through the side channel from 64% to 57% did not adversely affect

the diodicity. For the T45A-2 we will maintain angle a at 90

103

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

Widths b and c and the etch depth of the valve are chosen to achieve a desired Reynolds

number range and to allow particles of a desired size to pass (if no smaller constrictions

are included in the valve design). These widths were historically chosen as 114 µm and

the nominal etch depth as 120 µm. Calculation of the resistance of an equivalent-length

rectangular duct and knowledge of the maximum pressure difference to be applied across

the valve gives an estimate of the operational Reynolds number range, (Re < 1000 for the

T45A and T45C). The Reynolds number, which is based on the hydraulic diameter of the

main channel at the T-junction, determines whether the inertial or viscous portion of the

diodicity mechanism is dominant per Eq. 8.1. For the T45A-2 we will maintain width c as

114 µm and the nominal etch depth as 120 µm, but we will increase width b to 135 µm to

widen the main channel at the Y-junction and weaken the forward-flow jet.

Dimensions di and do are the offset of the inner side-channel radius e and the outer

side-channel radius f from the main channel. Together, these dimensions control the length

of the side channel, the angle between the side channel and main channel at the T-junction,

and the narrowing of the side channel as it approaches the T-junction. They also serve to

orient the jet emanating from the side channel counter to the downstream direction. Figure

8.2 shows an overlay of the T45A and T45C. In the T45A, di and do are approximately

zero so the side channel is a nearly constant width and is perpendicular to the main channel

at the T-junction. In the T45C, di and do are equal but nonzero such that the side channel

rejoins the main channel at the T-junction at a 45 angle. The increased path length of the

104

T45C side channel is clear, yet the reverse-flow jet from the side channel still has sufficient

momentum to reach the opposite wall of the main channel as seen in Fig. 7.3. Careful study

of the figure reveals a low velocity region where the inner wall of the side channel joins the

main channel. This suggests that di could be reduced in the T45C so that these walls join

perpendicularly, reducing the dimension g and diminishing the small jet that flows into the

side channel during forward flow seen in Fig. 7.2. Alternately, do could be increased to

narrow dimension g and orient the reverse-flow jet more counter to the downstream flow.

Note that 4 parameters are sufficient to define the side channel, so one of these dimensions

is redundant. For the T45A-2 we will maintain di at zero and e at 167 µm, but will increase

do from zero to 90 µm and decrease f from 284 to 225 µm to narrow dimension g and create

a side channel with converging walls in the reverse-flow direction. This will increase the

momentum of the reverse-flow jet so that it can reach the opposite wall of the main channel

as in the T45C, and also to reduce flow into the side channel during forward flow.

The angle h and the width b define the main channel. Angle h is zero in both the

T45A and the T45C. The rationale for including it here is an understanding of the diodicity

mechanism. In forward flow it has been shown that a laminar jet (and its accompanying

dissipation) are created downstream of the Y-junction. Diffusing the forward flow before it

reaches the Y-junction would diminish the energy flux rate into the outlet channel, slow the

forward-flow jet, and lessen forward-flow energy loss. Conversely, accelerating the reverse

flow in the main channel would increase the energy flux rate into the inlet channel. Both

effects enhance diodicity. We will implement this strategy in the T45A-2 by increasing b

to 135 µm (as previously mentioned) and setting h to 6 .

Length i and radii j and k define the inlet channel, and both radii are set as 15 µm in

the T45A and T45C. In the T45A, the inlet channel is a source of direction-independent

energy loss and should be minimized (i is 532 µm). The T45C exhibits a high dissipation

rate where the reverse flow jet is attached to the inlet channel wall. Shortening its inlet

channel (i is 566 µm) would result in less energy flux transferred to dissipation within the

valve itself, though all energy flux developed in the valve by pressure work will eventually

dissipate if the channels upstream and downstream of the valve are long enough and do

not provide a diffusive effect. We can increase the diodicity due to energy dissipation

in the valve by giving different values to radii j and k. Specifically, radius k should be

large enough to allow a gradual acceleration of the forward flow in the inlet channel to its

maximum value at the T-junction. This will minimize the forward-direction dissipation rate

by avoiding high velocity-gradients near the walls. We don’t need to worry about diffusing

the reverse-flow jet because we have already arranged via the side-channel geometry to

105

give the reverse-flow jet enough momentum to reach and attach to the wall on the opposite

side of the inlet channel. To increase the reverse-flow energy dissipation, we let the reverse-

flow jet dissipate itself along the wall of length i and set the radius j small enough to avoid

a diffusing effect. A reasonable value for j would be in the range of 50% to 100% of

dimension b, and i would be the difference between j and k. For the T45A-2 we will set j as

100 µm (75% of b) and k as 400 µm, and thus i will be 300 µm, which is a 44% reduction

in length compared to the T45A.

Radius l, angle m, and length n define the outlet channel. In the T45A and T45C the

radius l is 15 µm and angle m is 45 . The length n is 463 µm in the T45A and 321 µm

in the T45C. These dimensions address the forward-flow jet seen in Figs. 6.2 and 7.2.

The rounding of the corner via radius l minimizes the dissipation from the high velocity

gradient between the jet and the wall seen in Figs. 6.5 and 7.5. The radius l should be

large enough that the minimum outlet channel width is from 60% to 75% of width c, since

any additional flow cross-sectional area is not utilized by the forward flow and should be

made unavailable to the reverse flow. Angle m and length n define a diffusing section of the

outlet channel. The dissipation of the forward jet in the outlet channel would be more than

offset by the transfer of energy flux to pressure work, lessening the overall pressure loss in

the forward flow direction. In the reverse flow direction the increased energy flux into the

Y-junction due to the converging of the walls accomplished by angle m, combined with the

attachment of the flow to the outer wall of the side channel, would maximize the portion of

the flow proceeding into the side channel for any particular choice of angle a. The mouth

of the outlet channel may have sharp corners or protruding corners to further inhibit the

reverse flow. To follow these strategies in the T45A-2 we increase radius l to 85 µm and

angle m to 52 We shorten length n to 125 µm for the same reasons we shortened length i,

to reduce direction-independent dissipation. We choose sharp corners for the mouth of the

outlet channel.

We have accomplished research objective #3a by presenting design guidelines for the

geometrical layout of optimal Tesla-type NMP valves based on knowledge of the low-

Reynolds number diodicity mechanism. As an example we applied these guidelines to

modify the lower-performing T45A valve to produce a T45A-2 valve that has higher per-

formance than the T45C. We have also pointed out an additional utility of the kinetic-energy

perspective, which through Eq. 8.1 not only accurately estimates diodicity, but also shows

the contributions due to the energy flux rate and the dissipation rate. In the following

chapter we complete accomplishment of research objective #3b by applying the numerical

method to reveal the enhancement of the diodicity mechanism in the T45A-2.

106

Chapter 9

ENHANCED DIODICITY MECHANISM OF T45A-2 VALVE


The geometrical layout of the T45A-2 is a modification of the lower-performing T45A

valve. It was developed from the T45A by directly following the design guidelines pre-

sented in the previous chapter, which are based on knowledge of the low-Reynolds-number

diodicity mechanism revealed by the numerical method of Chap. 3. No other numerical

valve simulations of any kind were done varying the geometry of the T45A to determine

this new design. This design is also not an optimum version selected from a number of at-

tempts. The geometry of the T45A-2 is from a single application of the design guidelines to

improve the T45A. The results show that the design guidelines are effective and sufficient

to create a valve with an enhanced diodicity mechanism.

It is also important to note that the numerical methods employed to analyze the T45A-2

were exactly identical to those used in the T45A and T45C analyses. The only variation is

the substitution of the physical grid of the T45A-2 for that of the T45A or T45C. As was

done for these other valves, momentum and kinetic-energy conservation were applied in 9

regional control volumes as shown in Fig. 9.1. The flow behavior and characteristics are

compared throughout this chapter with those of the T45A valve.

Steady-state solutions of both forward flow and reverse flow in the T45A-2 valve were

calculated using the numerical methods described in Chap. 3. These steady-state solutions

were obtained from the solution of the final time step of step-response simulations that

were continued until steady flow was achieved. In all cases the simulations converged to

steady solutions without applying Reynolds averaging to the Navier-Stokes equations. As

discussed in Sec. 1.2.2, this identifies the flow in the valve as laminar over the Reynolds

number range simulated, Re 683, despite the presence of separated flow and recirculation

regions.



rection of 2987µl min corresponding to ReD 500 based on the hydraulic diameter of

107




2

8

3

9

4 5

6

7

1

x

y

Figure 9.1: Division of the T45A-2 valve into regional control volumes.

the main channel at the T-junction. To achieve the same flow rate in the forward-flow di-

rection required a differential pressure of only 0.287 atm. This corresponds to a diodicity

Di 1 74 according to Eq. 1.1. According to the analysis of diodicity prediction accu-

racy in Ch.4, there is a 95% probability that the true diodicity is within 1 57 Di 1 75

since the numerical method overpredicts the diodicity by 4 42 5 13%. To determine the

increase of diodicity over that of the T45A valve with 95% confidence, we take the ratio of

the diodicities and use the law of combination of errors [4] to obtain

DiT45A 2

DiT45A 1 74

1 27

1 e2

T45A 2 e2T45A

100

1 37 0 10

in which the 95% confidence band of the prediction errors are eT45A 2 eT45A 5 13%.

Thus the diodicity increase is a statistically significant 27-47%.

9.2 Results


From the step-response valve simulations, Fig. 9.2 shows the forward and Fig. 9.3 the

reverse-flow velocity fields on the symmetric centerplane of the valve. The velocity fields

in the forward and reverse flow cases are similar to those in the T45A valve. Flow sepa-

ration and laminar jets occur in three locations: where the channels separate, the channels

recombine, and where the flow leaves the valve.

108




0.0000E+001.4408E-012.8816E-014.3224E-015.7632E-017.2040E-018.6448E-01

Figure 9.2: Forward-flow velocity field on the centerplane of a single-element, Tesla-typeT45A-2 valve with a volume flow rate of 2987 µl/min corresponding to Re=500 based onthe hydraulic diameter of the main channel. One dimensionless unit equals 10 m/s.

109

In the forward-flow case the flow accelerates more slowly than in the T45A as it en-

ters the main channel from the goblet-shaped inlet plenum due to the increased radii of

the channel mouth. Since we have shortened the inlet channel, and the walls are still con-

verging, the velocity profile is still undeveloped as it reaches the T-junction. Because we

have narrowed the side channel at the T-junction, very little flow veers into the side chan-

nel; 94% of the main channel flow (compared to 85% in the T45A) is unperturbed and

continues downstream through the main channel. The diffusive strategy implemented in

the diverging walls of the main channel succeeds in decelerating the flow to minimize the

momentum of the forward-flow jet. Downstream of the Y-junction, the main channel flow

still separates from the inner wall and forms a forward-flow jet attached to the outer wall,

but the jet has 30% less momentum than the forward-flow jet in the T45A. Since we have

narrowed the outlet channel, a much smaller portion of the channel compared to the T45A

is filled with a quiescent zone that is essentially underutilized by the forward flow. The

flow leaves the channel and enters the goblet-shaped outlet plenum as a weaker jet than in

the T45A.

In the reverse-flow case the flow accelerates suddenly as it enters the channel from the

goblet-shaped plenum, just as it does in the T45A. On reaching the Y-junction the flow

stream veers toward the main channel. As it impinges on the cusp of the guide vane, 52%

of the flow is deflected down the main channel (compared to 36% in the T45A) where it

travels next to the guide vane wall, but most of the main channel is filled with a large,

slowly-moving recirculation zone. We have succeeded in creating a main channel that

is wholly available to the forward flow, but underutilized by the reverse flow. The side

channel flow separates from the outer wall and forms a laminar jet attached to the guide

vane wall. We have retained the large velocity gradient there that increases dissipation and

improves diodicity. As the side-channel flow approaches the T-junction it is accelerated by

the converging walls and emerges from the side channel as the reverse-flow jet heading for

the opposite wall of the inlet channel.This has increased the jet’s momentum sufficiently

to cross the main channel in spite of the momentum of the flow traveling down the main

channel and attach to the opposite wall of the inlet channel, unlike the reverse-flow jet in

the T45A that never reaches the opposite wall. There is a large separated zone downstream

of the T-junction, but the reverse-flow jet never spreads to fill the entire channel as the jet

in the T45A does, it remains attached to the lower wall until it reaches the plenum. This

high-velocity gradient between the wall and the jet is exactly the dissipative flow structure

the valve design guidelines anticipated.

110




0.0000E+001.4408E-012.8816E-014.3224E-015.7632E-017.2040E-018.6448E-01

Figure 9.3: Reverse-flow velocity field on the centerplane of a single-element, Tesla-typeT45A-2 valve with a volume flow rate of 2987 µl/min corresponding to Re=500 based onthe hydraulic diameter of the main channel. One dimensionless unit equals 10 m/s.

111


The pressure fields of the T45A-2 valve shown in Fig. 9.4 are similar to those of the T45A

in their major features. Approximately two-thirds of the pressure loss in forward flow

occurs in the inlet channel before the T-junction is reached. The pressure loss at the 45

bend in the channel has been reduced from that in the T45A by deceleration of the main

channel flow. In reverse flow there is an abrupt pressure loss as the flow enters the sharp-

cornered valve mouth as in the T45A, and the pressure loss has also been increased at the

T-junction. Between these locations there is little additional pressure drop. Closer study of

the pressure fields of the T45A-2 shows that about two-thirds of the pressure drop in each

flow direction occurs at the T-junction.

9.2.3 Energy Dissipation Field



largest. Energy dissipation in the forward-flow case is less than in the T45A, especially

along the diverging walls of the main channel and downstream of the 45 bend where we

have succeeded in reducing the momentum of the forward-flow jet. The dissipation rate is

reduced at the convex surfaces of the channel mouth where the flow accelerates as it enters

the inlet channel because we have provided such a large radius on the upper wall, knowing

that the reverse-flow jet would remain attached to the lower wall. The locations of high

dissipation rates correspond well to the locations of large pressure drop in Fig. 9.4.

The locations of dissipation in the T45A-2 in the reverse flow case are quite different

than in the T45A valve. High dissipation rates still occur along the wall as the flow enters

the valve where we chose sharp corners for the channel mouth and along the surfaces of the

guide vane both in the side channel and the main channel and especially at its leading cusp.

Unlike the T45A, there is no significant dissipation along the outer wall of the side channel

since the separated jet does not attach to it. As in the T45A there is a high dissipation rate

at the protruding cusp at the T-junction, and this dissipation continues alongside the jet as

it proceeds from the side channel into the main channel, collocated with the large velocity

gradient between the jet and the recirculation zone in the main channel downstream of the

T-junction. The most significant dissipation occurs where the reverse-flow jet exiting from

the side channel impinges on the opposite wall of the inlet channel. But unlike in the T45A,

the jet attaches to the opposite wall, as planned by the valve design guidelines, and this high

rate of dissipation persists all along that wall until the jet detaches as the plenum is reached.

112




-8 .7166E-021.1643E-021.1045E-012.0926E-013.0807E-014.0688E-015.0569E-01

(a) Forward flow




-8 .7166E-021.1644E-021.1045E-012.0926E-013.0807E-014.0688E-015.0569E-01

(b) Reverse flow

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

113




<-6.8789E+00-5.8255E+00-4.6404E+00-3.4553E+00-2.2702E+00-1.0851E+00

1.0001E-01

Figure 9.5: Base 10 logarithm of the energy dissipation rate in forward flow on the cen-terplane of a single-element, Tesla-type T45A-2 valve with a volume flow rate of 2987µl/min corresponding to Re=500 based on the hydraulic diameter of the main channel. Onedimensionless unit equals 14 mW.

114




-6.8789E+00-5.7158E+00-4.5526E+00-3.3895E+00-2.2263E+00-1.0631E+00

1.0001E-01

Figure 9.6: Base 10 logarithm of the energy dissipation rate in reverse flow on the cen-terplane of a single-element, Tesla-type T45A-2 valve with a volume flow rate of 2987µl/min corresponding to Re=500 based on the hydraulic diameter of the main channel. Onedimensionless unit equals 14 mW.

115

As in the forward-flow case, part of the momentum flux leaving the channel is dissipated

in the goblet-shaped plenum.



The momentum equation is a vector relation, a summation of forces, so each of its terms has

both magnitude and direction sense. The X and Y components of these forces are shown

in Fig. 9.7 for both the forward and the reverse flow case. The figure caption explains

positive-negative direction sense and the X-Y coordinate system is shown in Fig. 9.1.

Essentially, the figures show that the viscous and pressure forces applied on the fluid and

the fluid’s momentum flux response are significantly altered from those of the T45A. All

the forward-flow X-direction forces are reduced by as much as a factor of two, except for

those of the inlet channel (block2). The forward-flow Y-direction forces have been reduced

as well, including those in the outlet channel (block 6), the location of the forward-flow

jet, in which the pressure force has been reduced by 40%. The reverse-flow forces have

been reduced as well as a result of the reduction of direction-independent losses by the

shortening of the inlet and outlet channels. The only exceptions are the X-direction forces

in the T-junction (block 3), which are almost unchanged.



oriented along the X direction, is to study the vector magnitudes of the force components



This perspective shows how much direction-independent flow restrictions have been

decreased by shortening the inlet and outlet channels, while retaining the flow restrictions

due to the diodicity mechanism. The pressure force magnitudes have been decreased by

25-50% except for the forward-flow in the inlet channel (block 2). Even though the reverse-

flow pressure-force magnitude in the side channel (block 8) has been halved, this is coun-

terbalanced by the fact that the reverse-flow pressure-force magnitudes exceed those of

the forward-flow in all other control volumes except the inlet channel (block 2). The vis-

cous force magnitudes are all diminished approximately by half as well. Since the viscous

forces are comparatively small to the other forces, the momentum fluxes are quite similar

116

1 2 3 4 5 6 7 8 9−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Block number

For

war

d−F

low

X−

dire

ctio

n F

orce

s



1 2 3 4 5 6 7 8 9−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Block numberR

ever

se−

Flo

w X

−di

rect

ion

For

ces



1 2 3 4 5 6 7 8 9−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Block number

For

war

d−F

low

Y−

dire

ctio

n F

orce

s



1 2 3 4 5 6 7 8 9−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Block number

Rev

erse

−F

low

Y−

dire

ctio

n F

orce

s



Figure 9.7: Force vector terms in the integral form of the momentum conservation equation.Net pressure force and net momentum flux into a control volume are positive. Viscous forceis applied on the fluid by the wall. X-vectors to the right and Y-vectors upward are positiveand consistent with the T45A-2 valve layout in Fig. 9.1 including the numbering of thecontrol volumes (blocks).

117

1 2 3 4 5 6 7 8 90

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Block number

Pre

ssur

e F

orce

Mag

nitu

de



1 2 3 4 5 6 7 8 90

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Block number

Vis

cous

For

ce M

agni

tude



1 2 3 4 5 6 7 8 90

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Block number

Mom

entu

m F

lux

Mag

nitu

de



Figure 9.8: Vector magnitudes of the terms in the momentum conservation equation for theT45A-2 valve.

118

in magnitude and location to the pressure forces.


To assess the diodicity enhancement of the T45A-2, the sum of each kinetic energy term

over the entire valve was compared with that of the T45A. The energy dissipation rate in the

T45A-2 is reduced to 49% of the T45A rate in forward flow while 64% is retained in reverse

flow. The energy flux rate in the T45A-2 is reduced to 41% of the T45A rate in forward

flow while 78% is retained in reverse flow. The pressure work rate applied to the fluid in the

T45A-2 is reduced to 52% of the T45A rate in forward flow while 72% is retained in reverse

flow, and predicts a valve diodicity of 1.74 following Eq. 2.14. The diodicity determined

by Eq. 1.1 is also 1.74 compared to the 1.27 of the T45A. The diodicity enhancement has

greatly reduced the pressure work rate required for forward flow.

There are two alternate ways to look at the data. Figs. 9.9 and 9.10 show the three most


work rate, the energy flux rate, and the dissipation rate, which are shown for each of the

control volumes for both forward and reverse flow.

The enhancement of the diodicity mechanism by reduction of unnecessary flow re-

strictions is seen in the pressure-work rate. The forward-flow pressure-work rate has been

nearly eliminated in all regional control volumes except the inlet channel (block 2) in which

the flow is accelerated to enter the valve, and the outlet channel (block 6), the location of

the forward-flow jet, which has been reduced by 35% compared to the T45A. The reverse-

flow pressure-work rate has also been reduced, but not in the inlet channel or the T-junction

(block 3) where the reverse-flow jet is located. Some of these reductions can be attributed

to reductions in the energy-flux rate where the flow in either direction enters the valve. The

T45A-2 has a 42% reduction in the rate of energy flux creation in the inlet plenum and

channel (blocks 1 & 2) in forward flow and the outlet channel and plenum (blocks 6 & 7)

in reverse flow. The dissipation rate also shows the reduction of flow restrictions except for

those that are the diodicity mechanism. The dissipation rate has been greatly reduced in

all regional control-volumes except the inlet channel (block 2) where the reverse-flow jet

is attached to the wall, which has been reduced by only 24%. In comparison, the reduction

in the dissipation rate associated with the forward-flow jet in the outlet channel (block 6) is

72%. Once again, the viscous work rate is inconsequential in all control volumes.



119

1 2 3 4 5 6 7 8 9−0.05

0

0.05

Block number

Pre

ssur

e W

ork

Rat

e



1 2 3 4 5 6 7 8 9−0.05

0

0.05

Block number

Ene

rgy

Flu

x R

ate



Figure 9.9: Magnitude of the pressure work rate and the energy flux rate terms in thekinetic-energy conservation equation from the T45A-2 valve simulations with a volumeflow rate of 2987 µl/min corresponding to Re=500 based on the hydraulic diameter of themain channel.

120

1 2 3 4 5 6 7 8 9−0.05

0

0.05

Block number

Dis

sipa

tion

Rat

e



1 2 3 4 5 6 7 8 9−0.05

0

0.05

Block number

Vis

cous

Wor

k R

ate



Figure 9.10: Magnitude of the energy dissipation rate and the viscous work rate terms inthe kinetic-energy conservation equation from the T45A-2 valve simulations with a volumeflow rate of 2987 µl/min corresponding to Re=500 based on the hydraulic diameter of themain channel.

121

shown in Fig. 9.11. The forward-flow kinetic-energy terms are strikingly reduced from

those of the T45A. Only the terms associated with the acceleration of the fluid in the inlet

channel (block 2) and the forward-flow jet in the outlet channel and plenum (blocks 6

& 7) are of significance. Even the longest channel section, the main channel (block 4),

transforms very little pressure-work to the other forms. Its flow is driven by energy flux

created in the inlet channel (block 2). The reverse-flow kinetic-energy terms have been

somewhat reduced in comparison to the T45A, except for those associated with the reverse-

flow jet in the inlet region and T-junction (blocks 1, 2, & 3).

9.3 Discussion

The geometrical layout of the T45A-2 valve is a modification of the lower-performing

T45A obtained by directly following the valve design guidelines based on knowledge of the

low-Reynolds-number diodicity mechanism revealed by the numerical methods of Chap.

3. No other numerical valve simulations of any kind were done varying the geometry of

the T45A to determine this new design; this design is not an optimum version selected

from a number of attempts. The T45A-2 is the result of a single effort to apply the design

guidelines to improve the T45A. It is also important to note that the numerical methods

employed to analyze the T45A-2 were exactly identical to those used in the T45A and

T45C analyses. The only variation is the substitution of the physical grid of the T45A-2

for that of the T45A or T45C. The numerical results show that the design guidelines are

effective and sufficient to create a valve with an enhanced diodicity mechanism.

The T45A-2 was shown to be functionally similar to the T45A. The diodicity mecha-

nism is located in the side channel and T-junction (blocks 3, 8, 9). The underlying physical

basis for the diodicity of the T45A-2 valve is its ability to direct the majority of the vol-

ume flow through alternate paths depending on flow direction. In forward flow 94% of the

fluid proceeds through the main channel (blocks 3 4 5); in reverse flow 48% proceeds

through the side channel (blocks 5 9 8 3).

Analysis of the field variables of the T45A-2 simulations shows that following the de-

sign guidelines has enhanced the diodicity mechanism. The redesign of the inlet channel

has resulted in more gradual acceleration of the forward flow as it enters the valve, reducing

the pressure loss and high dissipation rate along the wall. The diverging walls of the main

channel have reduced the forward-flow velocity as well as the energy dissipation along the

walls. This has slowed and widened the forward-flow jet in the outlet channel reducing its

momentum. In the reverse-flow direction the pressure losses as the flow enters the valve

122

1 2 3 4 5 6 7 8 9−0.05

0

0.05

Block number

For

war

d−F

low

Pow

er


(a) Forward flow

1 2 3 4 5 6 7 8 9−0.05

0

0.05

Block number

Rev

erse

−F

low

Pow

er


(b) Reverse flow

Figure 9.11: Magnitude of the terms in the kinetic-energy conservation equation from theT45A-2 valve simulations with a volume flow rate of 2987 µl/min corresponding to Re=500based on the hydraulic diameter of the main channel.

123

have been increased by sharpening the corners at the mouth of the valve and narrowing the

outlet channel. The converging walls accelerate the reverse flow so that it impinges on the

guide vane with a high dissipation rate and creates a large recirculation in the main chan-

nel. The converging walls of the side channel accelerate and increase the momentum of the

reverse-flow jet sufficiently to cross the main channel and attach to the opposite wall of the

inlet channel. The diverging walls of the inlet channel provide a surface for dissipation of

the reverse-flow jet, yet offer a low pressure loss entry to the forward-flow.

The momentum perspective showed that the design guidelines correctly address the

minimization of direction-independent flow restrictions while retaining the flow restrictions

of the diodicity mechanism. The kinetic-energy perspective reinforces this conclusion by

showing that the pressure-work rate, energy-flux rate, and dissipation rate have all been

halved in forward flow in the T45A-2 compared to the T45A, while the reverse-flow rates

have only been reduced by one quarter. The direction-independent flow restrictions have

been minimized while those due to the diodicity mechanism have been maintained.

Research objective #3b has been completed by demonstrating the effectiveness of the

valve design guidelines of the previous chapter by using them as the sole method to obtain a

valve design in which the diodicity mechanism, and thus the diodicity, has been enhanced.

In the T45A-2 the direction-independent flow restrictions have been reduced, the forward-

flow jet has been weakened by decelerating the flow before it reaches the Y-junction, and

the reverse-flow jet has been accelerated sufficiently to cross the main channel and induce

a high rate of energy dissipation where it is attached to the opposite wall. By applying the

valve design guidelines to modify the geometry of the T45A, the T45A-2 design has been

created with a diodicity of 1.74 compared to 1.27 for the T45A, a statistically significant

diodicity increase of 27-47% with 95% confidence.

124

Chapter 10

CONCLUSIONS

Previous researchers of Tesla-type NMP valves have focused on inertial forces as the

sole source of the diodicity mechanism because their devices were macro-scale valves con-

taining high Reynolds-number flows. This level of understanding is of questionable value

at the micro-scale, since the physical dimensions of NMP microvalves lead to laminar

flows with Reynolds numbers well below 1000 at typical operating conditions. Thus, ef-

forts to design optimal NMP microvalves suffer from a lack of understanding of the fluidic

mechanisms that inhibit reverse flow in laminar low-Reynolds-number flow, resulting in a

dependence on the build & test method.

A numerical method using field variable analysis and momentum and kinetic-energy

conservation in regional control-volumes was developed from the governing equations to

reveal the low-Reynolds-number diodicity mechanism. The numerical method was ap-

plied to two distinct designs of Tesla-type NMP valves and was consistently able to discern

the fluidic mechanism responsible for the variation in pressure drop between forward and

reverse flow in each valve design. It revealed their low-Reynolds-number diodicity mecha-

nism as the viscous dissipation surrounding laminar jets that have flow-direction-dependent

locations and orientations. This diodicity mechanism is dominated by viscous forces, un-

like the high-Reynolds-number mechanism of macro-scale valves, which is solely due to

inertial forces. Understanding of the low-Reynolds-number diodicity mechanism was em-

ployed in the development of effective design guidelines for the geometrical layout of op-

timal Tesla-type NMP valves. The value of these guidelines was demonstrated by using

them as the sole method to obtain a valve design in which the diodicity was increased by a

statistically significant amount.

10.1 Develop a Numerical Method to Reveal the Diodicity Mechanism

Analysis of the velocity, pressure, and energy dissipation fields from the numerical simula-

tions provide qualitative information about the role that physical flow phenomena (laminar

jets, recirculation regions, regions of high pressure-gradient, and regions with high energy-

125

dissipation) play in creating valve diodicity. But a source of quantitative information is

neccessary to assess the significance of each of these phenomena to the diodicity mecha-

nism. To this end, each valve was divided into nine regional control volumes and integral

forms of the momentum conservation and kinetic-energy conservation equations were de-

veloped and applied in these control volumes. The magnitudes of the components of the

conservation equations in each control volume provide the quantitation that reveals which

phenomena are important and how their effects combine to create the diodicity mechanism.

The momentum perspective provides information about the interplay of forces between

the surfaces, flow boundaries and fluid. In each regional control volume it is possible

to determine the pressure force and shear force that each surface applies to the fluid, the

pressure force applied by each flow boundary, and the resulting momentum flux in each of

the three coordinate directions. Since Tesla-type NMP valves inherently have multiple flow

directions and the force components of the momentum perspective are vectors, it is often

difficult to relate them directly to valve diodicity, a scalar quantity. The kinetic energy,

also a scalar quantity, presents no such difficulty. The kinetic energy perspective provides

information on the transfer of energy across flow boundaries, work done on the fluid, and

dissipation within the fluid. There is a direct correspondence between diodicity and the

ratio of forward and reverse pressure-work rates when the inlet and outlet surfaces are

located such that the pressure on each surface has an approximately constant value. The

energy transfers and interplay of forces in each regional control volume are instrumental in

determining how the physical flow phenomena create the diodicity mechanism.

10.1.1 Verify Mathematically Correctness

The numerical method includes not only the incompressible Navier-Stokes solver imple-

mented in CFX 4.2, a mature commercial software package, but also “user subroutines”

written in FORTRAN to calculate kinetic-energy conservation. These were validated by

using the numerical method to model steady-flow in a slot and compute the magnitude of

the terms of the kinetic-energy conservation equation. These magnitudes agree very well

with those obtained from the analytical solution for steady-flow in a slot.

10.1.2 Verify Steady-Flow Response Predictions

The validation of the steady-flow modeling by the numerical method was achieved by com-

paring volume flow-rate and diodicity predictions to 242 independent experimental mea-

surements from physical-realized valves of three distinct groups of etch depths and two

126

distinct Tesla-type designs, the T45A and the T45C. The tests of the physical-realized

valves were divided into three groups considering etch depth and valve design: the T45A

group, the T45C group, and the Deep T45C group. Numerical simulations were performed

using the mean etch depth of each of the valve test groups: h 110 5µm for the T45A Test

Group, h 115µm for the T45C Test Group, and h 148 3µm for the Deep T45C Test

Group. It is important to note that exactly the same numerical method was employed in

all cases; the only variation between simulations was the substitution of the physical grid

of the T45A or T45C scaled in the thickness dimension to match the appropriate mean

etch-depth.

Analysis of the experimental data showed that the power-law function of Eq. 4.1 pro-

vides a good estimate of the true functional relationship between measured pressure-drop

and volume flow-rate and there is no justification to use a more complicated function. Thus

the power-law function was used to represent the numerical method’s predictions of flow-

rate response to applied pressure.

Statistical analysis of the numerical simulation results showed that the flow-rate re-

sponse is underpredicted by the numerical method by 7.62% on average over all three test

groups: T45A, T45C, and Deep T45C. This is good agreement considering that the etch

depth measurement accuracy is 6.5% and that, in order to be a predictive tool, the numeri-

cal model is gridded-up from the layout drawing of the photolithographic mask used in the

etching process instead of the as-etched shape of the physical devices.

10.1.3 Verify Diodicity Prediction Accuracy

The diodicity is insensitive to any Reynolds-number underprediction that is of similar pro-

portions in both forward and reverse-direction flow. The diodicity is overpredicted by

4.14% on average over all three test groups: T45A, T45C, and Deep T45C. For the particu-

lar flow-rate of Re 500 that is characteristic of the numerical simultations for the diodicity

mechanism study, the diodicity prediction error with 95% confidence is eDi 4 42 5 13%

or 0 708 eDi 9 55%.

By making comparisons between predicted and experimental values from the multiple

valve geometries of the T45A, T45C and Deep T45C Test Groups we demonstrated that we

are free to modify valve geometry (to the same extent as the variation between the T45A

and T45C) and still obtain accurate flow rate and diodicity predictions from the numerical

method. Thus the numerical method was validated as a firm basis on which to accept or

reject valve designs for the enhancement of diodicity.

127

10.1.4 Verify Transient-Flow Response Predictions

Since there are no analytical solutions for transient flow in an NMP valve nor experimental

methods available to directly measure it, the transient response predictions of the numer-

ical method were verified by modeling the harmonic response of 2-D slot flow for which

an analytical solution does exist. Since excellent agreement was shown, the numerical

method was applied to model harmonic flow in a T45A valve to show that the kinetic-

energy transient term is negligible. The insignificance of the transient term allowed the

investigation of the diodicity mechanism using steady-state numerical solutions instead of

the more computationally-intensive harmonic-response solutions.

10.1.5 Reveal the Diodicity Mechanism in Low Reynolds Number Flow in Tesla-Type

NMP Valves

The underlying physical basis for the diodicity of the T45A valve is its ability to direct

most of the forward flow (85%) through the main channel and most of the reverse flow

(64%) through the side channel. In each flow direction a laminar jet is created, surrounded

by a highly-dissipative shear layer. Most of the transformation of pressure work to energy

dissipation occurs in the shear gradient surrounding the laminar jets, and the shear layer

surrounding the reverse-flow jet in the side channel and the inlet channel is more dissipa-

tive than the shear layer surrounding the forward-flow jet in the outlet channel. Energy

dissipation in the fluid itself contributes more to diodicity than does wall friction. Energy

flux is also a contributor to diodicity, it is even more direction dependent than energy dis-

sipation, but it is an order of magnitude smaller. The vector nature of jet momentum also

contributes to diodicity. Substantial pressure force is needed to alter the direction of the

reverse flow jet to turn downstream, but not the forward flow jet, which is already pointed

downstream.

The T45C is functionally similar to the T45A. However, the T45A contains a reverse-

flow jet that attaches to the wall in the side channel and then proceeds down the center of

the inlet channel with a dissipative shear layer surrounding it; in the T45C the reverse-flow

jet doesn’t attach to the wall in the side channel and thus has enough additional energy to

cross the main channel and attach to the opposite inlet-channel wall. The dissipation rate

between the jet and the wall is greater than that experienced by the T45A jet at the center

of the channel. A second difference is the decreased length of the T45C outlet channel,

which decreases the energy dissipation between the forward flow jet and the outlet channel

wall where it is attached. As a result, the diodicity due solely to the ratio of reverse and

128

forward-flow dissipation rates is much higher in the T45C (1.35 instead of 1.13). But the

combination of effects is a statistically insignificant increase in diodicity from 1.27 in the

T45A to 1.33 in the T45C.

While macro-scale high-Reynolds-number valves also have direction-dependent jets,

the direction-dependent variation of energy dissipation rate, which is of such great impor-

tance in low Reynolds-number flow, would be negligible. Instead the directional variation

of the macrovalve jets’ inertial forces would be paramount.

10.2 The Low-Reynolds-Number Diodicity Mechanism is Dominated by Viscous Forces

In a classic analysis of a macro-scale Tesla-type valve with turbulent flow of Re 1700,

Paul [23] performed a control-volume analysis assuming viscous forces were negligible.

His analytical solution for pressure drop due solely to the change in momentum flux agreed

with experimental data, showing the diodicity mechanism was due only to inertial forces.

But in micro-scale Tesla-type valves the typical operating conditions generate a maxi-

mum flow-rate of Re 500. To assess the effect of Reynolds number on the balance be-

tween inertial and viscous forces, normalization and approximation theory were applied to

the momentum and kinetic-energy conservation equations. The results suggest that viscous

force in microvalve flows becomes larger than momentum flux for Re 100 and negligible

for Re 1000, and that energy dissipation is dominant over energy flux for Re 100 and

significant up to Re 1000. Thus as the scale of the valve decreases, the viscous forces

become ever more dominant.

Steady-flow numerical simulations of the T45A and T45C valves corroborate the results

of approximation theory. Even at Re 500, fully 2/3 of the pressure work is dissipated

and only 1/3 is transformed to energy flux. Ignoring viscous effects in the low-Reynolds-

number flow in valves would lead to invalid results. This is particularly clear for the T45C

valve, since conveniently all four measures: the diodicity and the ratios of reverse and

forward-flow pressure work, energy flux, and energy dissipation are all within 2% of 1.33.

Over 2/3 of the pressure work is expended in energy dissipation, thus 2/3 of the diodicity

is due to dissipation. Clearly viscous effects are dominant.

A further effect of the dominance of viscous forces over inertial forces is the direction-

dependence of valve resistance (see App. A) and the direction-independence of valve iner-

tance (see App. B).

129

10.3 Demonstrate Knowledge of the Diodicity Mechanism

Since the resistance and inertance of the valve must be matched to the dynamics of the

microfluidic system in which it operates, there is no single optimal valve design. However,

the knowledge gained from analysis of the diodicity mechanisms of the T45A and T45C

valves was applied to develop guidelines the design of high-performance Tesla-type NMP

valves. As an example we have applied these guidelines to modify the lower-performing

T45A valve to produce a T45A-2 valve that has higher performance than the T45C.

10.3.1 Develop Valve Design Guidelines

The diodicity can be cast in terms of the ratio of pressure work rate in the reverse and

forward direction, or cast in terms of the ratios of energy dissipation rate and energy flux

rate. In the latter case the diodicity is seen as a function of u3i and ∂ui ∂x j 2, so that for a

given flow cross-sectional area, energy losses are exhibited (in order of increasing losses)

by plug flow, fully-developed parabolic profiles, and highly-asymmetric velocity profiles.

The relative importance of u3i and ∂ui ∂x j 2

is controlled by the Reynolds number; when

Re 1000the energy flux rate dominates; when Re 100, the dissipation rate dominates.

Thus, the diodicity is strongly affected by the asymmetric velocity profiles of separated

laminar jets and their accompanying high velocity gradients (especially when the jet is

attached to the wall). And not only the velocity magnitude of the laminar jet is important,

but also its direction. Thus, a reverse-flow jet should be oriented counter to the downstream

direction, and if it is not possible to avoid creating a jet in forward flow, it should be oriented

downstream.

As the Reynolds number increases and less of the dissipation occurs within the geomet-

ric boundaries of the valve itself, it becomes increasingly important to ensure that channels

upstream and downstream of the valve are dissipative not diffusive, (ie. do not transfer

energy flux back into available pressure work in the fluid). If this is not possible, an al-

ternative strategy is to choose to create diodicity from the direction-dependent dissipation

rates within the valve only and attempt to recover kinetic energy in both flow directions by

providing diffusers at the ends of the valve. This is a particularly appropriate strategy when

it is desirable to minimize valve resistance.

Valve design guidelines were presented based on knowledge of the low-Reynolds-

number diodicity mechanism revealed by the numerical methods of Chap. 3. Their intent

is to enhance the diodicity mechanism by increasing the energy flux and dissipation rates

in the reverse-flow jet and decreasing them in the forward-flow jet, and also by decreasing

130

flow restrictions that are direction-independent.

10.3.2 Demonstrate the Effectiveness of the Guidelines

As an illustration of our understanding of the low-Reynolds-number diodicity mechanism

in Tesla-type NMP valves, a T45A-2 valve was created by modifying the geometry of the

T45A following the design guidelines. No additional technique was employed. The same

numerical methods employed to analyze the T45A and T45C were used to analyze the

T45A-2. The only variation was the substitution of the physical grid of the T45A-2 for

that of the T45A or T45C. Analysis of the T45A-2 simulations showed that the direction-

independent flow restrictions were minimized while retaining the flow restrictions of the

diodicity mechanism. The forward-flow jet was weakened by decelerating the flow before

it reaches the Y-junction; the reverse-flow jet was accelerated sufficiently to cross the main

channel and induce a high rate of energy dissipation where it attaches to the opposite wall.

The new inlet channel geometry provides a wall surface for dissipation of the reverse-flow

jet, yet offers little flow restriction in the forward-flow direction. In the reverse-flow di-

rection the flow restriction where the flow enters the valve was maintained by sharpening

the corners at the mouth of the valve and narrowing the channel. Overall, the pressure-

work rate, energy-flux rate, and dissipation rate have all been halved in forward flow in

the T45A-2 compared to the T45A, while the reverse-flow rates have only been reduced

by one quarter. By applying the valve design guidelines the diodicity mechanism has been

enhanced and the previous diodicity of 1.27 has been increased to 1.74, a statistically sig-

nificant diodicity increase of 27-47% using a 95% confidence level. These results show that

the valve design guidelines are effective and sufficient to create a valve with an enhanced

diodicity mechanism. The low-Reynolds-number diodicity mechanism in Tesla-type NMP

valves has been revealed; there is no longer any need to rely on the “build & test” method.

10.4 Future Work

Since the numerical method developed in this research to reveal the low-Reynolds-number

diodicity mechanism of Tesla-type valves is a general method that applies to all NMP

microvalves, it is recommended that this method be applied to reveal the diodicity mech-

anisms and develop guidelines for the optimal design of other types of NMP valves, such

as: diffuser valves, asymmetric conduits, and vortex diodes.

131

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134

Appendix A

VALVE RESISTANCE MODELING

The main purpose of this appendix is to show that the resistance to fluid flow through

a valve in response to an applied pressure difference is dependent on the flow direction in

the valve. This is anticipated by the dimensional analyses in Secs. 2.2 and 2.3 that show

that the diodicity mechanism of a microvalve is mainly due to viscous forces. In addition,

appropriate valve resistance values are suggested for linear and nonlinear system models.

Resistance as used in lumped-parameter system models was introduced in Sec. 1.2.3.

Fluid resistance is the change in pressure drop with respect to a change in volume flow rate,

so the local-slope valve resistance was calculated from the power-law relation Eq. 4.1 as

Rlocal slope

dPdQ

nβQn 1 (A.1)

in which the constants n and β have different values for the forward and reverse flow di-

rections, ie. nF nR and βF βR. It is refered to as local-slope resistance because the

fluid flow in the valve exhibits significant non-linear behavior, thus the slope dP dQ varies

dramatically over the typical operating range of flow rates.

Figure A.1 compares the local-slope resistances calculated by applying Eq. A.1 to

the numerical and experimental pressure versus flow-rate data of the T45A and the T45C

valves. The local-slope resistance of the T45A is overpredicted by 5% in the forward-flow

direction and by 7% in the reverse; the T45C by 5% in the forward and 9% in the reverse.

In each case the reverse-flow resistance is larger than the forward-flow resistance at any

measured Reynolds number, thus the dependence of resistance on flow direction, as well as

flow rate, is clear.

In linear system models a single constant resistance value is required and care should

be taken in its choice. If a fluid flow is oscillatory and the oscillation is a small perturbation

about a mean flow rate, then the local-slope R at the mean flow rate would be representative

of R across the flow-rate perturbation. However, in typical valve operation, the flow oscil-

lation is much larger than the mean flow rate, (ie. the slosh flow >> net flow), and using

the value of R at the slosh flow amplitude would not be representative of valve resistance

over the entire range of flow rate. In this situation Ogata [18] recommends using an aver-

135

−1000 −500 0 500 1000−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

12

Reynolds number

Loca

l−sl

ope

resi

stan

ce −

Pa*

s/m

3


(a) T45A Valves

−1000 −500 0 500 1000−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

12

Reynolds number

Loca

l−sl

ope

resi

stan

ce −

Pa*

s/m

3


(b) T45C Valves

Figure A.1: Local-slope resistance to fluid flow vs. Reynolds number in T45A and T45Cvalves from both experiment and numerical simulation following Eq.A.1, which is basedon the fitted power-law relation, Eq.4.1.

age resistance that is the slope of a line connecting the pressure ∆P at the slosh flow rate

amplitude Qa to the origin, which using the power-law fit introduced above is

Raverage

∆PQa

βQn

a

Qa

βQn 1a (A.2)

A more representative value of R was obtained by considering the harmonic nature of valve

flow Qt Qa sinΩt in typical operating conditions and computing the time-averaged

resistance R (simplified by a coordinate change from Ωt to θ) as

R 1π

π

0nβ

Qa sinθ n 1 dθ Rlocal slope

π

0

1π

sinn 1 θdθ (A.3)

which considered 1/2 of a period (eg. the forward-flow portion of a cycle). The integral

is a function of the power-law exponent n; a look-up table for 1 n 2 was numerically

integrated via Maple V software [8]. Ogata’s relation can be derived from a time-average

of Qt as a triangle wave. As can be seen in Fig. A.2, the time-averaged (sine) resistance

is higher than the average resistance, but significantly lower than the local-slope resistance.

For a linear system model containing an NMP valve with harmonic flow, the appropri-

ate value for R is the time-averaged resistance at the expected flow rate amplitude. For

136

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Res

ista

nce

Flow rate

Local slopeTime−ave (sine)Average

Figure 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 in linear modelswhere a single value is required. Note the similarity to the characteristic curve of nonlinearfriction for an object moving at low Re in a fluid medium.

nonlinear system models in which resistance can be a function of the slosh flow rate, the

fitted parameters β and n in Eq. A.1 are useful since the governing equation for a series

combination of resistor and inductor can be stated as

∆Pt βQn I

dQdt

(A.4)

We have developed methods to obtain appropriate values to characterize valve resis-

tance in both linear and nonlinear system dynamics models, and shown that the resistance

in a Tesla-type NMP valve is direction dependent.

137

Appendix B

VALVE INERTANCE MODELING

This appendix shows that the inertance that characterizes a Tesla-type NMP valve in a

lumped-parameter system dynamics model is independent of flow direction. Since there

are no analytical solutions for transient flow in an NMP valve and no experimental meth-

ods available to directly measure valve inertance, the accuracy of the transient response

predictions of the numerical method were verified by employing them to model the step re-

sponse of flow in a slot since those numerical results could be compared with an analytical

solution. Then the numerical method was applied to model step response in the T45A and

T45C valves and show that the inertance is independent of flow direction.

B.1 Step Response of a 2-D Slot

Analytical Solution

Relations to characterize the step-response of a channel in lumped-parameter dynamics

modeling were introduced in Secs. 1.2.4 and 1.2.5. Following Eq. 1.7 the resistance of a

slot of height h and length L is

R 12µL h3 (B.1)

and following Eq. 1.8 the inertance of inviscid flow in a slot with a cross-sectional area per

unit width of A h is

Iinviscid ρL h (B.2)

To rewrite these equations in nondimensional form we use the characteristic parameters

αp αu

αp ρ

and αt

αx αu of Sec. 3.5 except for the characteristic length, which

was taken as αx h 116 9µm. We find

L L αx R R

αp

αxαuand Iinviscid

I inviscidαpαt

αxαu

138

remembering that R and Iinviscid are on a per-unit-width basis. Substituting these into Eqs.

B.1 and B.2 gives us the nondimensional forms as

R 12L Re

and I inviscid L (B.3)

in which Re αuαx ν

But the inviscid-flow inertance is only approximate; when viscosity is considered the

crossectional shape of the channel becomes important. For starting flow in a pipe car-

rying viscous fluid White [30] presents an analytical solution utilizing Bessel functions.

Following his basic method a series solution for starting flow of a viscous fluid in a two-

dimensional slot was developed and is described in App. C. This series solution gives the

nondimensional inertance I in a slot as

I π2

8L 1 23L (B.4)

that is 23% larger than the inviscid flow inertance of Eq. B.3. Since the numerical method

considers viscosity, Eq. B.4 supplies the inertance value for comparison.

Numerical Solution

The step response of a 2-D slot was simulated using the numerical method described in

Chap. 3, the same method used to determine the step response of the NMP valves. Thus,

the slot simulation was performed as a 3-D rectangular channel with a 100 to 1 cross-

stream aspect ratio to ensure the 2-D character of the flow and a streamwise length of 10.

The same characteristic parameters used above for Eq. B.3 were used to nondimensionalize

the variables. A step pressure difference ∆P was applied at time t 0, and the simulation

was allowed to procede until steady-state was reached.

The governing equation for a first-order model that represents a fluid channel as a re-

sistor and an inductor in series is

∆P RQt IQ

t

At time t 0, the volume flow-rate is Q0 0. Solving for the inertance gives

I ∆P

Q0 (B.5)

139

The derivative of the volume flow-rate at t 0 was obtained from the numerically-derived

flow-rate vector Qt and time vector t via a O2 accurate three-point forward-differencing

method [1]

Q0

3Q1 4Q2 Q3

t3 t1

in which the subscripts refer to index position in the vector.

The inertance prediction of Eq. B.5 was used for comparison with the analytical solu-

tion, Eq. B.4.

B.2 Step Response of a 2-D Slot

The predicted volume flow-rate response to an applied pressure difference in a 2-D slot is

compared in Fig. B.1 with the series solution of Eq. C.8. At equilibrium the nondimen-

sional flow rate was Qtequil 0 098075 in response to an applied pressure difference of

∆P 0 01atm. This determined the proportionality constants β and n of the power-law

relation Eq. 4.1 as β ∆P Qtequil 0 10196 and n 1, since pressure drop is linearly

proportional to volume flow rate for laminar flow in a slot. The resulting numerical value

for inertance following Eq. B.5 is I 1 217L, which is 99% of the analytical solution of

Eq. B.4.

B.3 Step Response of an NMP Valve

Step-response simulations of the T45A and T45C valves were performed using the numer-

ical methods of Chap. 3. Because of the nonlinear response of the volume flow rate Qt ,

the inertance was determined from the numerically-calculated values by direct application

of Kirchoff’s voltage law for a resistor and inductor in series following Eq. B.5. The con-

stants β and n that describe the nonlinear relation between pressure drop across the valve

and the volume flow rate were obtained for each valve and each flow direction from power-

law fits of the numerical data from the step-response simulations following Eq. 4.1. Figure

B.2 shows that the inertance as a function of the Reynolds number of the T45A and T45C

valves has similar nonlinear behavior and magnitude. The inertance decreases slightly with

increasing Reynolds number but exhibits no significant dependence on flow direction.

140

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Time / Equilibrium time)

Flo

w r

ate

/ Equ

ilibr

ium

flow

rat

e

PredictedTime constantExponential

Figure B.1: The predicted volume flow-rate response to an applied pressure difference in a2-D slot from the numerical method (symbols) shows good agreement with the exponentialresponse from the series solution, Eq. C.8. The time constant τ Re π2 is also shown.

0 200 400 600 800 10000

5

10

15

20

25

Reynolds number

Iner

tanc

e

Reverse flowForward flow

(a) T45A valve

0 200 400 600 800 10000

5

10

15

20

25

Reynolds number

Iner

tanc

e

Reverse flowForward flow

(b) T45C valve

Figure B.2: Inertance versus Reynolds number from the step-response simulations via Eq.B.5. Inertance shows some dependence on flow rate, but not on flow direction.

141

Appendix C

SERIES SOLUTION FOR STARTING FLOW IN A SLOT

For starting flow in a pipe carrying viscous fluid White [30] presents an analytical so-

lution utilizing Bessel functions. Following his basic method a series solution for starting

flow of a viscous fluid in a two-dimensional slot was developed. This method plus a method

using image flows is also found in Sherman [25].

The starting point for the series solution for the starting flow of a viscous fluid in a

slot is the conservation of momentum equation assuming negligible cross-stream velocities

v w 0 and thus through continuity ∂u ∂x 0 resulting in

∂u∂t

dPdx 1

Re∂2u∂y2

(C.1)

a nondimensional equation in which the variables were nondimensionalized by the same

parameters as Eq. B.3 but we have dropped the asterisk notation. This partial differen-

tial equation models the development of the velocity profile over time at one streamwise

location in the slot.

The steady-flow solution for u was obtained by setting the transient term to zero and

integrating twice. Taking 0 y 1 as the dimensionless slot height, the steady velocity is

usteady

dPdx

Re2

y y2 (C.2)

leading to the maximum velocity of

umax

dPdx

Re8

and a steady volume-flow rate of

Qsteady

dPdx

Re2

1

0 y y2 dy

dPdx

Re12

(C.3)

To make Eq. C.1 homogeneous and utilize a transient heat conduction solution, the velocity

142

was separated into unsteady and steady parts using

u u 4umaxy y2 (C.4)

Substituting this relation for u into Eq. C.1 results in

∂u∂t

1

Re∂2u∂y2

(C.5)

which is analogous to unsteady heat conduction in a bar. Using separation of variables,

Boyce and DiPrima [6] give a series solution for initial value and boundary conditions of

uy

0 f

y

for 0 y 1

u0

t 0

u

1

t 0

for t 0

Thus the initial value is

fy 4umax y y2

which negates the steady component of u at t 0and the solution is

uy

t

∞

∑n 1

bn exp n2π2t Re sinnπy

where the coefficients are

bn 2

1

0f

y sinnπydy

8umax

π3

nπsinnπ 2cosnπ 2

n3

which was integrated with the aid of Maple V software [8]. Note that bn is not a function

of y.

We derive the unsteady portion of the volume-flow rate from the integral of

u over the

slot height as in

Q

1

0u

y

t dy

∞

∑n 1

bn exp n2π2t Re nπcosnπ 1

When n is even,cosnπ 1 0 and the summation is zero. When n is odd,

cosnπ

1 2 and the coefficients are bn 32umax

nπ 3 Thus the unsteady portion of the

143

volume flow rate is also written as

Q

∞

∑n 1 3 5

64umaxnπ 2 exp n2π2t Re

Because of the rapid growth of the denominator in the series expression and the rapid

decrease in the exponential with increasing n, only the first term of the series is significant.

Thus to good accuracy we have

Q

dPdx 8Re

π4 exp π2t Re (C.6)

Since the unsteady component of u is contained in u, the time derivative of the volume-flow

rate can be obtained from the time rate of change of

Q and is

Q ∂Q∂t

dPdx 8

π2 exp π2t Re (C.7)

the steady and unsteady components of the volume flow rate, Eqs. C.3 and C.6, were

combined to produce the volume flow rate as a function of time as

Q dPdx

Re

1

12

8π4 exp π2t Re (C.8)

Acknowledging that π4 8 12 18 12, (within 1.5%), allows simplification to

Q dPdx

Re12

1 exp π2t Re

(C.9)

The governing equation for a first-order model that represents a fluid channel as a re-

sistor and an inductor in series is

∆P RQt IQ

t

At time t 0, the volume flow-rate is Q0 0. Solving for inertance gives

I ∆P

Q0

144

and substituting Eq. C.7 evaluated at t 0 gives

I ∆P

dPdx

8π2

1 23L

since the constant pressure-gradient in fully-developed slot flow is dP dx ∆P L. This is

a 23% larger inertance than given by Eq. B.3 in which viscosity was neglected.

145

Appendix D

DIODICITY FROM A RATIO OF FLOW RATES

Diodicity can be calculated from a ratio of flow rates as well as from a ratio of pressure

drops. Figure D.1 shows the characteristic pressure drop versus volume flow rate curves

for reverse and forward flow in an NMP valve. Combining the power-law relation of Eq.

4.1 with Eq. 1.1 and referring to the points a, b, and c of the figure results in

Di

∆PR

∆PF Qb

∆Pc

∆Pa

βFQnF

c

βFQnFa

Qc

Qb nF

QF

QR nF

∆Pb

(D.1)

where subscript F is forward and R is reverse, showing that diodicity is determined from

either a ratio of pressure drops at the same flow rate or a ratio of flow rates at the same

pressure drop.

146

a

b

c

Volume flow rate

Pres

sure

dro

p Rev

erse

flow

Forw

ard

flow

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

147

Appendix E

VALVE DIODICITY MEASUREMENTS

T45A Test Group

The measured values of Reynolds number and valve diodicity of the T45A Test Group

corresponding to the data shown in Fig. 4.4.

T45A Bi2 T45A Bo2 T45A Ti2 T45A To2

Re Di Re Di Re Di Re Di

167.2 1.00 169.4 1.14 168.7 1.07 167.9 1.03

250.8 1.08 254.1 1.12 253.0 1.15 251.9 1.00

334.3 1.21 338.8 1.15 337.3 1.13 335.8 1.06

417.9 1.19 423.6 1.16 421.7 1.16 419.8 1.19

501.5 1.19 508.3 1.25 506.0 1.18 503.7 1.23

585.1 1.23 593.0 1.25 590.3 1.16 587.7 1.24

668.7 1.26 677.7 1.31 674.7 1.20 671.7 1.28

752.3 1.28 762.4 1.29 759.0 1.24 755.6 1.30

835.9 1.24 847.1 1.26 843.3 1.21 839.6 1.28

919.5 1.25 931.8 1.23 927.7 1.19 923.5 1.28

1003.0 1.22 1016.5 1.25 1012.0 1.24 1007.5 1.25

T45C Test Group

The measured values of Reynolds number and valve diodicity of the T45C Test Group


148

T45C Li2t2 T45C Li2 T45C Lo2t2 T45C Lo2

Re Di Re Di Re Di Re Di

165.8 1.04 165.8 1.16 168.7 1.13 165.3 1.07

248.6 1.19 248.6 1.22 253.1 1.25 248.0 1.21

331.5 1.27 331.5 1.22 337.4 1.30 330.7 1.28

414.4 1.25 414.4 1.29 421.8 1.27 413.3 1.28

497.3 1.29 497.3 1.30 506.2 1.34 496.0 1.37

580.2 1.31 580.2 1.32 590.5 1.33 578.7 1.34

663.0 1.36 663.0 1.37 674.9 1.34 661.3 1.26

745.9 1.39 745.9 1.38 759.2 1.38 744.0 1.43

828.8 1.34 828.8 1.34 843.6 1.35 826.7 1.39

911.7 1.34 911.7 1.31 928.0 1.37 909.3 1.38

994.5 1.37 994.5 1.36 1012.3 1.41 992.0 1.39

Deep T45C Test Group

The measured values of Reynolds number and valve diodicity of the Deep T45C Test Group


T45C Ri2 T45C Si T45C

Re Di Re Di Re Di

213.1 1.19 113.1 1.06 147.9 1.07

284.1 1.16 150.8 1.00 221.9 1.30

355.1 1.25 226.2 1.06 295.8 1.31

426.1 1.23 301.5 1.19 369.8 1.43

497.1 1.23 376.9 1.15 443.7 1.27

568.1 1.27 452.3 1.13 517.7 1.28

639.2 1.31 527.7 1.15 591.6 1.34

710.2 1.31 603.1 1.29 665.6 1.32

781.2 1.31 678.5 1.28 739.5 1.32

852.2 1.32 753.8 1.32 813.5 1.36

- - 829.2 1.33 887.4 1.37

- - 904.6 1.45 - -

149

Appendix F

VALVE LAYOUT POINTS

The xy points that define the geometry of the microvalves were directly extracted from

the CIF-format files from which the vendor created the photolithographic masks used in the

deep reactive-ion etching process that created the valves in silicon.

F.1 T45A Valve

All dimensions in millimeters.

inlet channel length = 0.531

outlet channel length = 0.462

71 points as (x,y,z)

-0.17 -0.001 0

-0.17 0.009 0

-0.168 0.026 0

-0.166 0.037 0

-0.161 0.054 0

-0.155 0.07 0

-0.146 0.087 0

-0.139 0.098 0

-0.126 0.114 0

-0.113 0.127 0

-0.102 0.136 0

-0.089 0.145 0

-0.072 0.154 0

-0.055 0.161 0

-0.041 0.165 0

-0.025 0.168 0

-0.019 0.169 0

-0.007 0.17 0

150

0.008 0.17 0

0.026 0.168 0

0.044 0.164 0

0.068 0.156 0

0.083 0.148 0

0.096 0.14 0

0.109 0.13 0

0.121 0.119 0

0.206 0.204 0

0.189 0.22 0

0.173 0.233 0

0.16 0.242 0

0.141 0.252 0

0.127 0.258 0

0.112 0.264 0

0.094 0.27 0

0.067 0.277 0

0.045 0.281 0

0.024 0.283 0

0.009 0.284 0

-0.009 0.284 0

-0.024 0.283 0

-0.042 0.281 0

-0.058 0.278 0

-0.075 0.274 0

-0.096 0.268 0

-0.13 0.255 0

-0.153 0.243 0

-0.179 0.227 0

-0.199 0.211 0

-0.211 0.199 0

-0.223 0.186 0

-0.236 0.169 0

-0.249 0.149 0

-0.26 0.128 0

151

-0.269 0.106 0

-0.274 0.088 0

-0.277 0.07 0

-0.28 0.049 0

-0.283 0.026 0

-0.284 -0.001 0

0.114 0.126 0

0.734 -0.494 0

0.904 -0.494 0

0.204 0.206 0

-0.94 -0.114 0

-0.284 -0.114 0

-0.284 0.0 0

-0.94 0.0 0

-0.631 -0.114 0

0.354 -0.114 0

0.354 0.0 0

-0.631 0.0 0

F.2 T45C Valve


inlet channel length = 0.566

outlet channel length = 0.321


-0.1 -0.141 0

-0.116 -0.128 0

-0.129 -0.115 0

-0.138 -0.104 0

-0.147 -0.091 0

-0.156 -0.074 0

-0.163 -0.057 0

-0.167 -0.043 0

-0.17 -0.027 0

-0.171 -0.021 0

152

-0.172 -0.009 0

-0.172 0.006 0

-0.17 0.024 0

-0.166 0.042 0

-0.158 0.066 0

-0.15 0.081 0

-0.142 0.094 0

-0.132 0.107 0

-0.121 0.119 0

-0.206 0.204 0

-0.222 0.187 0

-0.235 0.171 0

-0.244 0.158 0

-0.254 0.139 0

-0.26 0.125 0

-0.266 0.11 0

-0.272 0.092 0

-0.279 0.065 0

-0.283 0.043 0

-0.285 0.022 0

-0.286 0.007 0

-0.286 -0.011 0

-0.285 -0.026 0

-0.283 -0.044 0

-0.28 -0.06 0

-0.276 -0.077 0

-0.27 -0.098 0

-0.257 -0.132 0

-0.245 -0.155 0

-0.159 -0.149 0

-0.089 -0.144 0

-0.17 -0.001 0

-0.17 0.009 0

-0.168 0.026 0

-0.166 0.037 0

153

-0.161 0.054 0

-0.155 0.07 0

-0.146 0.087 0

-0.139 0.098 0

-0.126 0.114 0

-0.113 0.127 0

-0.102 0.136 0

-0.089 0.145 0

-0.072 0.154 0

-0.055 0.161 0

-0.041 0.165 0

-0.025 0.168 0

-0.019 0.169 0

-0.007 0.17 0

0.008 0.17 0

0.026 0.168 0

0.044 0.164 0

0.068 0.156 0

0.083 0.148 0

0.096 0.14 0

0.109 0.13 0

0.121 0.119 0

0.206 0.204 0

0.189 0.22 0

0.173 0.233 0

0.16 0.242 0

0.141 0.252 0

0.127 0.258 0

0.112 0.264 0

0.094 0.27 0

0.067 0.277 0

0.045 0.281 0

0.024 0.283 0

0.009 0.284 0

-0.009 0.284 0

154

-0.024 0.283 0

-0.042 0.281 0

-0.058 0.278 0

-0.075 0.274 0

-0.096 0.268 0

-0.13 0.255 0

-0.153 0.243 0

-0.179 0.227 0

-0.199 0.211 0

-0.211 0.199 0

-0.223 0.186 0

-0.236 0.169 0

-0.249 0.149 0

-0.26 0.128 0

-0.269 0.106 0

-0.274 0.088 0

-0.277 0.07 0

-0.28 0.049 0

-0.283 0.026 0

-0.284 -0.001 0

0.114 0.126 0

0.734 -0.494 0

0.904 -0.494 0

0.204 0.206 0

-0.994 -0.256 0

-0.338 -0.256 0

-0.338 -0.142 0

-0.994 -0.142 0

-0.484 -0.256 0

0.501 -0.256 0

0.501 -0.142 0

-0.484 -0.142 0

155

F.3 T45A-2 Valve



0.000000 0.000000 0.000000

0.118087 0.118087 0.000000

0.236174 0.000000 0.000000

0.316784 0.080610 0.000000

-0.167000 0.000000 -0.000000

0.148295 0.249099 0.000000

-0.010804 0.090000 -0.000000

-0.217020 0.000000 -0.000000

0.371174 -0.135000 0.000000

0.451784 -0.054390 0.000000

-0.167000 -0.078746 -0.000000

-0.217020 -0.078746 -0.000000

-0.467000 -0.078746 -0.000000

-0.217020 0.400000 -0.000000

-0.467000 -0.178746 -0.000000

-0.567000 -0.178746 -0.000000

-0.567000 0.206315 -0.000000

0.371174 -0.220000 0.000000

0.438155 -0.167669 0.000000

0.514539 -0.265436 0.000000

0.588684 -0.191290 0.000000

-0.888803 0.013785 -0.000000

-0.888803 -0.361215 -0.000000

-0.888803 0.388785 -0.000000

0.814172 -0.490924 0.000000

0.549007 -0.756089 0.000000

1.079337 -0.225759 0.000000

156

VITA

Born in Minneapolis in 1950, Ronald Louis Bardell began study of the classical piano

and string bass at the age of eight. In 1965 he switched to percussion and played rock&roll

and rhythm&blues at school dances, bars, and nightclubs. He left the music business at age

28 to attend the University of Minnesota and in 1983 received a Batchelor in Mechanical

Engineering degree. He worked in Minneapolis for the Onan Corporation, a division of

Cummins Engines, as a Design Engineer until 1986 when he moved to Sunnyvale, Califor-

nia, to work for Westinghouse Marine Division as a fluids mechanist in the Performance

Analysis group. At age 41 he moved to Seattle to attend the University of Washington, and

in 1994 received a Master of Science degree in Mechanical Engineering. He then became

involved in microfluidics research under Professor Fred K Forster, and was an author on

several proceedings papers and co-inventor on a patent. Since 1999 he has worked as a mi-

crofluidics consultant for biotechnology applications and has several patents pending. He

received the PhD in Mechanical Engineering from the University of Washington in 2000.

the diodicity mechanism of tesla-type no-moving · pdf file · 2004-08-26the...

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