Experimental determination of the molecular diffusion coefficient of gases by enhanced dispersion in oscillatory flows

103 Pages·1995·3.2 MB·English

by  Zachrich, Gregory Allen, 1955-

EXPERIMENTAL DETERMINATION OF
THE MOLECULAR DIFFUSION COEFFICIENT OF GASES
BY ENHANCED DISPERSION IN OSCILLATORY FLOWS
By
GREGORY ALLEN ZACHRICH
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1995
.
ACKNOWLEDGEMENTS
Dr. Ulrich H. Kurzweg, chair of my committee, lead me
through this research project. Although he performed many
required tasks, specifically I thank him for reviewing this
paper's various preliminary versions suggesting several
modifications to improve the final copy. Also, his
continued effort to keep me focused on the main topic of
this dissertation is much appreciated.
Dr. E. Rune Lindgren, whom I consider a friend,
prompted me to further my education. I thank him for
allowing me to discuss many of my thoughts in confidence
with him.
Dr. Marc Jaeger was extremely helpful in providing his
laboratory equipment and assisting in the collection of
data. Without his cooperation this work would not have been
possible
I thank the remainder of my committee, Drs. Shyy and
Lear, for serving in their respective capacities as member
and external member.
I truly appreciate the valued advice and guidance of
Shirley Robinson, who, among other things, reminded me of
the numerous deadlines that arose.
11
My mother, father, brothers and sister provided
unquantifiable moral support during this long endeavor, for
which I am most grateful.
iii
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS ii
LIST OF TABLES vi
LIST OF FIGURES vii
LIST OF SYMBOLS ix
ABSTRACT xii
CHAPTER 1 INTRODUCTION 1
CHAPTER 2 ESTIMATING THE DIFFUSION AND VISCOSITY
COEFFICIENT IN GASES 5
Estimating the Diffusion Coefficient 5
Estimating the Viscosity Coefficient 9
CHAPTER 3 ANALYTICAL SOLUTION FOR ENHANCED DISPERSION
IN OSCILLATING FLOW 14
Velocity Profile Solution 14
Concentration Solution 23
TDiedtaelrmiDniisnpglactehmeenEtnhaSnocleudtioDniffusion Coefficient... 2362
Enhanced Diffusion Coefficient in Final Form 38
Graphical Presentation 39
CHAPTER 4 EXPERIMENTAL PROCEDURE 41
Enhanced Diffusion Process 41
Experimental Procedure 44
Experimental Assumptions 48
Diffusion Coefficient is Independent of
Concentration 48
Incompressible Gases 49
Linear Concentration Gradient 50
Pressure Gradient is a Function of a Pure
Sine Wave in Time 50
Cross-Sectional Velocity Profile is
Fully Developed 53
Transverse Diffusion Effects Overwhelms
Axial Diffusion 53
Flow is Completely Laminar 54
TSutbeeadyisStCaotmepleCtoenldyitiSotnrsaighhatve been Achieved.. 5545
IV
All Measurements are Accurate 55
No Leaks To or From Tube System 56
Tidal Displacement Remains Small 57
Deriving the Non-dimensional Expressions Used in
the Experimental Measurements of the Two Setups.. 57
Determining Density and Viscosity Coefficients
Experimentally 63
CHAPTER 5 EXPERIMENTAL RESULTS 64
Setup One 64
Setup Two 69
Pressure-Displacement 73
CHAPTER 6 CONCLUSIONS 75
Concluding Remarks 75
Future Work 77
APPENDIX 79
TABLE OF REFERENCES 84
BIOGRAPHICAL SKETCH 87
v
LIST OF TABLES
Table page
2.1 Tabulated Values of Parameters for
Various Gases 8
2.2 Predicted and Reference Diffusion Coefficient 8
2.3 Predicted and Reference Viscosity 11
2.4 Diffusion Collision Integral 12
2.5 Viscosity Collision Integral 13
5.1 Raw Data Setup 1 67
5.2 Results Setup 1 68
5.3 Raw Data Setup 2 70
5.4 Results Setup 2 71
5.5 Pressure-Displacement Results 74
vi
TABLE OF FIGURES
Figure page
3.1 Plot of Log J0((-i)5 x) 19
3.2 Plot of Log J,((-i)5 x) 20
3.3 Plot of ber0, bei0, ber,, and bei, 21
3.4 Velocity Profiles at 0 Degree 22
3.5 Velocity Profiles at 30 Degree 22
3.6 Velocity Profiles at 60 Degree 22
3.7 Velocity Profiles at 90 Degree 22
3.8 Velocity Profiles at 120 Degree 22
3.9 Velocity Profiles at 150 Degree 22
3.10 Pressure-Displacement Relationship
(Smaller a) 35
3.11 Pressure-Displacement Relationship
(Larger a) 36
3.12 Phase Between Pressure Gradient and Velocity 37
3.13 Contour Plot of X = Deff /w Ax2 verses a and S 40
4.1 Schematic of Setup 1 46
4.2 Schematic of Setup 2 46
4.3 Photograph of Setup 2 47
4.4 PShiontusoogirdaaplh AoxfiaLlineDairspMloatcoermenUtssedoftoFPluriodduce
within Conduit 50
4.5 Photograph of Pressure Variation and
Tidal Volume 51
vii
4.6 Contour Plot of N(a,S) 61
5.1 Experimental Data Setup 1 69
5.2 Experimental Data Setup 2 71
5.3 Experimental Data Pressure-Displacement
Relationship 74
vm
KEY TO SYMBOLS
a Womersley Number, a = a*(oj / v)5 (non-dimensional)
7 Concentration Gradient (g/cm4)
Ax Tidal Displacement (cm)
e Intermolecular Distance (cm'8 = A)
r/ Radial Distance, rj = r/a (non-dimensional)
6 Angular Direction (radian)
X Dispersion Function (non-dimensional)
M Dynamic Viscosity (g/cm s)
/iAIR Dynamic Viscosity of Air (1 cP = 0.01 g/cm s)
v Kinematic Viscosity (cm2/s)
p Density (g/cm3)
Atomic Diffusion Volume Increments
aAB Intermolecular Distance (A)
a; Angular Frequency (1/s)
Collision Integral for Viscosity
^d,ab Collision Integral for Diffusion Coefficient
a Radius of Tube (cm)
A Cross-Sectional Area (cm2
)
c Acoustic Velocity (cm/s)
C Concentration (g/cm3 or %)
Dab Molecular Diffusion Coefficient (cm2/s)
IX
D Effective Enhanced Diffusion (cm2/s)
eff
Dm Molecular Diffusion Coefficient (cm2/s)
dS Circumferencial Distance dS = rj dd (non-dimensional)
f(r?) Non-dimensional Velocity Function
F(rj) Non-dimensional Concentration Function
g(r?) Non-dimensional Concentration Function
G(rj) Non-dimensional Concentration Function
h(rj) Non-dimensional Concentration Function
J Bessel Function of First Kind of Zero Order
0
Bessel Function of First Kind of First Order
kb Boltzman's Constant (1.38054 x 10"23 J/molecule °K)
K Magnitude of Pressure Grad. / density (Atm cm2/g)
L Length of Tube (cm)
M Molecular weight (g/gmole)
N Non-dimensional Function
P Pressure (Atmospheres)
Q Volumetric Flow Rate (cm3/s)
r Radial Direction (cm)
R Universal Gas Constant (8.3143 J/gmole °K)
S Schmidt Number, S = v / DM (non-dimensional)
t Time (s)
T Temperature (°K)
T Critical Temperature (°K)
c
u Velocity (cm/s)
U Mean Velocity (cm/s)
0
V Volume of Isolation Chamber (cm3
)
x