THE EVAPORATION STABILITY A DROPLET IN A …...The Evaporation and Stability of a Droplet in a Finite System Alan James Hastings McGaughey A ... F2.Measured and Calculated Data Used

THE EVAPORATION AND STABILITY OF A

DROPLET IN A FINITE SYSTEM

Alan James Hastings McGaughey

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate De partment of Mechanical and Industrial Engineering University of Toronto

8 Copyright Alan James Hastings McGaughey 2000

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MSTRACT The Evaporation and Stability of a Droplet in a Finite System

Alan James Hastings McGaughey

A thesis submitted for the degree of Master of Applied Science. Graduate Department of Mechanical and Industrial Engineering, University of Toronto.

2000

The results of recent experiments indicate that a temperature discontinuity exists at a

liquid-vapour interface undergoing steady evaporation. In order to investigate this phe-

nomenon in an unsteady situation, an experimental apparatus that can study the evapora-

tion of a single one-component liquid droplet in a finite system has been built. The

measured temperatures predict the existence of a temperature discontinuity at the liquid-

7 vapour interface in agreement with the previous experiments. The D- -1aw of droplet evap-

oration gives a reasonable estimate of the evaporation rate when the temperature disconti-

nuity is considered. Using the measured temperatures and droplet size, the pressure in the

vapour during the evaporation process can be predicted using Statistical Rate Theory. For

each of 80 sets of measurements, the largest difference between the prediction and the

measured value is 9 Pa (less than 05%). Adsorption on the surface of the confining vol-

ume is found to play a significant role in the evaporation process. The stability of a droplet

in a finite system has been investigated theoretically with consideration of the solid-vapour

interface. The results indicate that a stable droplet can exist if the pressure at which filrn-

wise condensation is predicted to occur is greater than the saturation pressure. The stable

equilibrium size has not been observed with the expenmental apparatus, and this is attrib-

uted to the nature of the solid surface. However, a quasi-steady state has been observed,

which could serve as a well defined initial condition for an evaporation experiment.

My academic supervisor, Dr. Charles A. Ward, provided the motivation for this work.

and help with both the theoretical and experimental aspects. In my time in the Themody-

namics and Kinetics Laboratory, we have had many stimulating discussions. and I have

leamed much frorn him about thermodynamics and research. Barbara Ward helped with

the organizational aspects of many of my endeavon. Dennis Stanga was of invaluable

assistance with the design of the experimental apparatus and procedures. The staff of the

Mechanical and Industrial Engineering machine shop built a number of the components of

the experimental apparatus. Nazir Kherani at Ontario Power Generation provided the data

acquisition hardware, and helped in the development of the software.

The friends I have made in the lab, the department and the university have provided me

with help, support, encouragement and distraction. In particular. 1 would like to thank

Payarn Rahimi, Yarnini Ramarnoorthy, Mahtab Roshanmehr. Dennis Stanga and Frank van

den Bosch. My parents, Anne and David McGaughey, have without fail encounged my

academic endeavors. Their support and interest in my work has always been appreciated.

This work was supported by the Natural Sciences and Engineering Research Council

of Canada and the Canadian Space Agency.

..* .......................................................................................................... Acknowledgements ui

TabIe of Contents ...e........e................................................................................................ iv

List of Tables .................................................................................................................... Lx ................................................................................................................... List of Figures si

Nomenciature ................................................................................................................. mi

Chapter 1:Introduction ..e............................................................................................... *el

1 . 1 .Motivation ......................................................................................................... 1

7 I . 2.Droplet Evaporation Background .................... .. ............................................ .- 1.3 . Droplet Stability Background ..................................... .. .................................... -5

1.3 . 1 .The Equilibriurn Conditions at a Liquid-Vapour Interface ............... 5

1.3.2.Theoreticai Predictions ..................................................................... 6

1.3.3 . Experirnental investigations ............................................................. -7

1 AScope of The Thesis ........................................................................................... 7

Chapter 2:The Evaporation of a Dropiet in a Finite System .........e....e...................... ..9

2.1 . Experîrnental Apparatus ..................................................................................... 9 22Adsorption Experiments ................................................................................. 13

22.1 .Motivation ....................................................................................... 13

2.2.î.The BET Isotherm ........................................................................... 14

2.2.3.Method ............................................................................................ 15

2.2.4.Results and Andysis ....................................................................... 16

2.3 . Evaporation Experiments ................................................................................. 21

2.3.1 . Method .......................................................................................... 21

............................................................................................. 2.3 2.Results 23

...................................................................... . 2.3 1.1 Size History 23

............................................. 2.3 22.Temperatures and Pressure 23

23.3 Anaiysis Using Statistical Rate Theory .......................................... 36

2.3.3.1 Jntroduction ...................................................................... 36

23.3 2.Calculation of the Mass Flux ........................................... 36

2.3.3.3.Calculation of the Temperature Field .............................. 37

2.3.3 A.Comparison of the Measured Pressure to That Predicted by

............................................................................... SRT 4 3 ?

2.3.3 5 . Companson of the Results to the D- -Law ...................... 46

2.3.3.6.Comparison to S teady Evaporation Ex penments ............ 53

2.4.Summary of the Evaporation Work ................................................................. 54

Chapter 3:The Stabiüty of a Droplet in a Finite Sptem: Thcoq .m......................... 3 6

...................................................................................... 3.1 .Description of System 56

32.Detemination of the Equilibrium Conditions ................................................. 57 33.Modeling the Solid-Vapour Interface .............................................................. 63

3.4.Formation of the Helmholtz Potential .............................................................. 64

35.Investigation of the Behaviour of the System as A Function of the Reference

............................................................................................................ Pressure 67

........... 3.5.1.The Reference Fressure is Equal to the Saturation Pressure 67

35.2.The Reference Pressure is Greater than the Saturation Pressure .... 73

3 5.3 . Changing One of the Independent Variables When the Reference

Pressure is Greater Than the Saturation Pressure ............................ 77

3.6.The Effect of Neglecting the Solid-Vapour Interface ...................................... 81

3.7.The Effect of a Second Component ................................................................. 88

3.8A System With Multiple Droplets .................................... .... ....................... 1

4.1 Arst Experiment ............................................................................................... 97

. ............................................................................................ 4.1.1 Method 97

.......................................................................................... 4.2.Second Expeciment 98

............................................................................................ . 42.1 Method 98

42.2.Results ............................................................................................. 99

........................................................................................... 4.3 . Third Experiment 1 03

...................................................................................... 4.3.1 . Method 1 0 4

........................................................................................... 4.3.2.Results 104

4.3.3 .Discussion ..................................................................................... 104

4.4.Discussion of the Stability Experiments ........................................................ 105

45 . Using the Quasi-Steady State as The Initial Condition for an Evaporation Ex-

perirnent ......................................................................................................... 1 O?

..................................................... 45.1 Design of an Ideal Experirnent 108

......................................................................................... 4.5 2.Method -1 09

. 43.3 Results ........................................................................................... 1 10

......................*...... ........*..................*..............*.*........ 45.4.Discussion .., 1 13

4.6.Summary of the Stability Experiments .......................................................... 1 13

A2.The Test Section ............................................................................................. 1 19

A3 .The Support Frame ........................................................................................ 1 26

A4.The Valve Assembly ..................................................................................... 1 28

A5.The Vacuum Pump Assembly ........................................................................ 1 28

A6.The Degassing Flask Assembly ..................................................................... 130

A7.The Ultra-High Vacuum System .............................................................. 132

A8 .T he Syringe Pump Assembly ......................................................................... 133

A9.The Water Bath Assembly ............................................................................. 1 35

A lO.The Temperature and Pressure Measurement Systems ................................ 137

............................................. .................................... A 10.1 .Hardware ,... 137

................. A102Software ................,,,.................................................. 138

A l 1 .The Optical Measurement System ............................................................... 138

Appendix B:Cai.ibration of Measurement Equipment aeeaa.*a.m*aaaa.aammmmaa*amaaa*aaaa*m.m* a. aem*140

B 1 . Cdibration of the Pressure Transducer .............................................. ..... ... 1 4

B2.Caiibration of the TherrnocoupIes ............................................................... -143

B3 . Optical Measurements .................................................................................... 145

.................................................................... B3.1 .Cali bration Procedure 145

8 3 2Droplet Measurements .................................................................. 145

Appendix C:Systern Tests .....................................................................~..*.................. .1JS ....................................................................................... C 1 .Bath Performance 1 4 8

C2.Checking the Tightness of the System ........................................................... 148

Appendix D:Experimental Methods ................................................................~.......... 151

................................................................................. D 1 h m p i n g on the Sy stam ljl

MAlignment of the Needle and the Droplet Thermocouple .............................. 151

...................................................................................... D3 . Water Preparation 1 5 3

W.Water Degassing ............................................................................................ 153 . d

D5.Syringe Filling ............................................................................................... 133

D6.Preparation of the Test Section .................................................................. 158

D7.Forrnation and Placement of the Droplet ...................................................... 158

........................................................................................ Appendix E:Expansion tests 161

Appendix F:Data From Evaporation Experiments ................... ......*.... ............. 166

. FI Size Data ......................................................................................................... 166

..... F2.Measured and Calculated Data Used in Statistical Rate Theory Analysis 169

............................................................................................. F2.1 .Test I 169

............................................................................................. F22.Test 2 173

. ..............................................****.*.................................*....... F2.3 Test 3 175

....... Appendix G:Statisticai Rate Theory Expression for the Rate of Evaporation 177

........................................... Appendir H:Classical Thermodynamics Background m**179

..................................................................................................... H I .De finitions 179

..................................................................... H 1.1 .Simple Huid System 179

H 1.2.Types of Systems ....................................................................... 179

H 1 3 . Properties ...................................................................................... 180

Hl AEquilibrium State .......................................................................... 180

Hl 5.Internal Energy .............................................................................. 180

................................................................................................... H2.Postulates 1 8 0

H2.1 .The Errst Postulate ............................................................... 1 8 0

H2.2 .T he Second Postulate .................................................................... 181

vii

H2.3 .T hird Postulate ............................................................................ 182

................. H3Developrnent of Important Relations For a Simple Ruid System 183

....... H3.1 Definition of Temperature. Pressure and Chernical Potential 183

........................................ H3 2.First Order Homogeneous Properties 1 8 4

H3.3 .The Euler Relation ........................................................................ 185

H3.4.The Gibbs-Duhem Relation .......................................................... 186

H3 5.Description Of A Surface Phase ................................................... 157

H4.The Use of Thermodynamic Potentials ..................................................... 1 8 8

H4.1 .Introduction ................................................................................... 188

H4.2.The Helmholtz Potential ............................................................ 189

H4.3.Description of a Work Interaction ................................................ 190

H4.4.Specific Example: The Helmholtz Potential ................................. 192

Appendix 1:The Unstable Droplet ........................ e o o o o o ~ o o o o o o o o o e o o e e e e e o o e ~ ~ o o ~ 8 e o ~ e e ~ o ~ o o e o o ~ o o ~ ~ o o ..194 ..................................................................................... 1 1 .Description of System 194

i2.Development of the D Potential ...................................................................... 195

13.Determination of the Equilibrium Conditions ................................................. 196

14.Formation of the Potential ............................................................................... 199

................................................................................................ Appendix J:Nucleation 201

J L . Homogeneous Nucleation ............................................................................. 1

J2.Heterogeneous Nucleation ............................................................................. 202

References .................................e.................................................................................... 205

... V l l l

Table 2.1 . The extent of the test section. approximated geometrically ............................ 13

Table 2.2 . Expenmental adsorption isotherm data ........................................................... 18

Table 2.3 . Set up parameters for the evaporation expenments ........................................ 22

Table 2.4 . Comparison of the three evaporation tests ...................................................... 24

Table 2.5 . The pressure breakdown data from a droplet diameter of 1.50 mm to when the

droplet had completely evaporated ................................................................. 32

Table 2.6 . A sarnple of the parameters used to predict the temperature discontinuity at the

liquid-vapour interface for test 2 .The bath temperature is 26.87'C ............... 43

Table 2.7 . A sample of the comparison between the measured and predicted pressures for

test 2. ............................................................................................................... 44

........................ . Table 2-8 Cornparison of the predictions of the -law to the experiments 48

Table 2-9 . The difference between the pressure required to bring the predicted values of

into agreement with the experimental data and the saturation pressure corre-

................................................................. sponding to the liquid temperature 52

Table 2- 10 . A comparison of the range of the temperature, mass fluxes and radii encoun-

................................................................ tered in the different investigations 54

Table 3-1 . The change in the average slope of the Helmholtz potential as a function of the

.................................................................................................. droplet radius 70

. ...........*................. Table 3-2 Companson of the results of the three systems considered 86

Table 3-3 . The effect of a second component on the equilibrium radius and the partial pres-

....................................... sure of component 1 in the stable equilibrium state 91

Table 4- 1 . System parameten at the beginning at end of the quasi-steady time period . The . . elapsed ttme 1s 1926 seconds ...................................................................... 1 1 1

Table C-1 . Leak rate into the test section for different configurations ........................... 149

Table F- 1 . Size data for test 1 ............................................. ........................................... 166

Table F-2 . Size data for test 2 ...................................................................................... 167

Table F-3 . Size data for test 3 ........................................................................................ 168

Table F 4 . Measured data from test 1 . The bath temperature is 26.78 'C ...................... 169

. Table F-5 CaicuIated data fiom test 1 ........................................................................... 171

Table F.6 . Measured data from test 2.The bath temperature is 26.87'C. ....................... 173

............................................................................ Table F.7 . Calculated data from test 2 174

Table F.8 . Measured data from test 3 . The bath temperature is 26.80'C. ...................... 175

Table F.9 . Calculated data from test 3 .................................... ,, . 176

Table G-l . Parameters needed to evaluate the Statistical Rate Theory expression for the

evaporative flux ............................................................................................ 178

Figure 2-1A schematic diagram of the front view of the experimental apparatus ........... 10

Figure 2-2.The configuration of the droplet and vapour thermocouples. ......................... 1 1

Figure 2-3 A typical image of a droplet. its diameter is measured as shown. and in this case has a value of 152 mm. ...........,,.....,~....................................................... 12

SV Figure 24.The BET isotherm. The solid surface is taken as Pyrex with C = 47.86 and - 9 - 1.13 x 1017 rnJmœ. ...................................................................... 15

Figure 2-5 A sample adsorption pressure-time history . This curve corresponds to the point on the isotherm with a pressure ratio of 0522 ............................................... 17

Figure 2-6.The experirnentally determined adsorption isotherm. The BET fit has panme- SV 7 ters C = 0.9 1 and n, = 1.1 x 1020 mdm- ................................................. . 18

Figure 2-7.The size history for the evaporation tests. The top graph is a plot of the entire evaporation process. The solid markers correspond to when the droplet shape is symrnetric. The hollow markers indicate when this is no longer valid. The bottom graph plots the square of the diameter up to the point where the droplet loses symmetry. Also plotted are best fit lines for each of the data sets. ............ 25

Figure 2-8.The measured temperature and pressure histories for test 1. .......................... 26

Figure 2-9.The measured temperature and pressure histones for test 2. ................ .. . .. .... . 27

Figure 2- 1O.The measured temperature and pressure histories for test 3. ........................ 28

Figure 2- 1 1 .The error in the temperature rneasurements for test 1. The error bars are plotted for every fortieth data point. .............................................................. ........ 29

Figure 2-12.The measured difference between the readings of the droplet and vapour thermocouples during each of the three tests. The data is plotted up to the point w here the droplet evaporated . . . . ..... ............. ... .... .. ... .... .. . . . . . . . . . . . . . . . . . 30

Figure 2- 13 .The pressure breakdown for test 2 ........................................................... 32

Figure 2-14.The measured pressure and the saturation pressure corresponding to the measured liquid temperature for test 2 ............................................................... 34

Figure 2-1S.The mode1 used to calculate the temperature profile in the vapour. ............. 37

Figure 2-16.The predicted temperature discontinuity for test 2. ...................................... 42

Figure 2-17.The pressure predicted by Statistical Rate Theory ploaed against the measured pressure for test 1. .................................................................................... 45

Figure 2-18.The pressure predicted by Statistical Rate Theory ploned against the measured pressure for test 2. ............................................................................ . ......... 45

Figure 2-19.The pressure predicted by Statistical Rate Theory plotted against the measured pressure for test 3. .................................... : ............................................... 46

Figure 2-20.Comphson of rhz prêdictzd and rnéasurêd values of K .............................. 49

Figure 2-21 A plot of the predicted value of K using SRT and the measured value for test 2 .......................................................................................................... . . 52

Figure 2-22.The temperature discontinuity for the current work and three other data sets plotted against the rneasured pressure in the vapour. A = Ref. 1. B = Ref. 6. C = Ref.48 ........................................................................................ 53

Figure 3-1 A one component droplet in its own vapour. The walls are diathermal. rigid and non-permeable. The surrounding reservoir is at a constant tempenture. .. ..... 56

Figure 3-2.The critical radius as a function of the droplet radius when Pr = P z . R, s 0, and there is therefore no equilibrium size for R > 0. ..................................... 68

Figure 3-3.The Helmholtz function as a function of the droplet radius when Pr = P x . Note the sudden change in the slope of the curve around R = 0.0008 m. ........... .. 7 1

Figure 34.The mass of the adsorbed phase and the pressure in the vapour as a function of the droplet radius when Pr = P, . ................................................................ 71

Figure 3-%The difference between the liquid and vapour chemical potentials as a function of the droplet radius when Pr = P , .............................................................. 73

Figure 3-6.The critical radius as a function of the droplet radius when Pr = P, + 0.4 Pa. . . There are two equilibnum states. .................................................................. 75

Figure 3-7.The difference in the chemical potentials of the liquid and vapour phases when Pr = P, + 0.4 Pa. .......................................................................................... 75

Figure 3-8.The Helmholtz potentiai as a function of the droplet radius when Pr = P, + 0.4 Pa. The first equilibrium states is unstable, while the second is stable. ......... 76

Figure 3-9.The effect of raising the system temperature by 05 "C. The stable equilibrium size decreases ..................................................................................... 78

xii

Figure 3- [O.The effect of raising the system temperature by 1 5'C. There is no longer a stable equilibrium size, and the system will corne to equilibrium as a superheated vapour in equilibrium with the adsorbed phase. .................... .. ................... 79

Figure 3-1 1 .The effect of changing the system temperature on the number of equilibrium states. As the temperature increases, the system shifts from having two equilib-

............................................................................. rium States to having none. 80

Figure 3-12.The difference between the reference pressure and the saturation pressure required for a stablz equilibrium siz2 to sxist as a function of the mas of the 5)s-

tem ............................................................................................................ 8 1

Figure 3-13.The critical radius plotted as a function of the droplet radius when the adsorbed phase is not considered. There are two equilibrium States .............................. 82

Figure 3-14.The Helmholtz potential for the finite sized system without consideration of the adsorbed phase plotted as a function of the droplet radius. R,! is at a maximum, and corresponds to an unstable equilibnum state. RC2 is at a minimum. and corresponds to a stable equilibriurn state. ............................................. 83

Figure 3- 1,

Figure 3- 1

SeThe critical radius of the droplet as a function of the pressure in the vapour in Gibbs' initial analysis of droplet stability. Note that R, > O only for v P > P,(T) .................................................................................................... 85

6.The thenodynamic potential for Gibbs' system plotted as a function of the dropiet radius for a pressure of 4245 Pa. The only equilibrium state occurs at and is a maximum, and is therefore unstable. ................................................. 85

Figure 3-17.The Helmholtz potential for a finite systern containing two droplets. .......... 94

Figure 4- l .The square of the droplet diameter as a function of time for the second stability expenment. Also plotted are best fit lines over the two distinct regions. ..... 100

Figure 4-2.The measured temperature and pressure histones for the second stability experiment. Note the similarity in shape between the iiquid temperature and vapour pressure curves, ............................................................................................. IO2

Figure 4-3 .Schematic of an ideal experiment for observing the evaporation of a droplet from a well defined initiai condition .......................................................... 109

Figure 4-4.The measured size of the droplet. The sparsity of readings between 7000 and 1 2 0 seconds is due to vibrations in the droplet caused by an unknown source

....................................... which made it difficult to make the measurement. 1 1 1

Figure 4-5.The temperature and pressure histones for the evaporation experiment run using a well defined initial condition. Both the temperature and pressure respond quickly after the expansion ........................................................................ 1 12

xiii

Figure A- 1 . Block diagram of the experimental apparatus ............................................. 120

Figure A.2.Schematic of the test section ......................................................................... 120

Figure A3.The confiiguration of the droplet and vapour thermocouples ........................ 122

Figure A4.Configurations of the droplet on the thermocouple ...................................... 124

Figure A-5 . Schematic of the support frame .................................................................. 127

Figure A.6.Schematic of the valve assembly .................................................................. 129

Figure A-7 . Schematic of the vacuum pump assembly .................................................... 130

Figure A.8.Schematic of the degassing flask assembly ................................................... 131

Figure A.9.Schematic of the syrinp valve assembly ...................................................... 135

Figure B- 1 .Pressure transducer caiibration curve ........................................................... 142

Figure B-2A sarnple thermocouple calibration curve . This is the thermocouple that is used to monitor the temperature in the bath ........................................................... 14-4

Figure B.3.A typical image used to measure the droplet size ......................................... 146

Figure B 4 A n examination of the sphericity of the droplet ............................................ 147

Figure D-1 .The configuration of the needle and the droplet thermocouple before placing a droplet on the thennocouple ....................................................................... 153

Figure D-2.Droplet transfer procedure . A:The thermocouple is threading the needle and a suitably sized droplet has been formed . B:The needle has been withdrawn to the point just before the droplet disengages . C:The droplet has successfully been placed on the thenocouple ........................................................................... 160

Figure E- 1 .The pressure and temperature time histories around the expansion for the nitrogen expansion test ...................................................................................... 163

Figure E-2.The pressure and temperature time histories for the water vapour expansion test . The pressure graph is for the entire time of the test . The ternperature graph is only around the expansion .......................................................................... 164

Figure H- I .An isolated system composed of a reservoir and a composite system . There are no restrictions on the boundary between the reservoir and the composite system . The reservoir boundaries are adiabatic. rigid and non.permeable ................ 190

xiv

Figure 1- 1 A single one-component liquid droplet in its own vapour. The pressure in the vapour and the temperature are maintained by the surrounding reservoir. The walls are non-permeable. .......................................................................... 194

Figure J-1 A sessile one-component droplet in its own vapour. The pressure in the vapour and the temperature are maintained by the surrounding reservoir. The walls are

............................................................................................... non-permeable 202

Figure J-2.The barrier for nucleation for heterogeneous nucleation normalized by the bar- 3 rier for homogcneous nacleation as a function of the contact angle .............. -04

surface area

fitting parameter in BET isothen

a constant

çpecific heat a? constant pressure

D potential (derived in Appendix I), droplet diameter

an effective droplet diameter

droplet diameter at the beginning of an evapontion experiment

molec ular diameter

Helmholtz potential

Gibbs potential

gravitational constant

specific enthalpy

mass flux

7 parameter in the D- -law

Boltzmann constant. thermal conductivi ty

Mach number

molecular mass

number of molecules

number of molecules in a finite system

average Nusselt number for natural convection around a sphere of diameter D

number of droplets, number of molecules per unit area

monolayer coverage per unit area in BET isotherm. a fitting parameter

pressure

leak rate

pressure at equilibrium in an isolated system used to formulate the SRT expression for the rate of evaporation

xvi

final pressure measured in an adsorption experiment

a pressure used in the evaporation experiments

a pressure used in the evaporation experiments

maximum pressure read in an adsorption experiment

measured pressure used in pressure breakdown analysis

reference pressure used in Bi3 isoihrrm

Peclet number

Prandtl num ber

vi brational partition function

droplet radius

critical radius

the critical radius in the first equilibrium state

the critical radius in the second equilibrium state

the stable equilibrium radius in a one droplet system

the stable equilibrium radius in an 11 droplet system

regression parameter

= r / r y , a dirnensionless le@

Rayleigh number for a sphere of diameter D

entropy

specific entropy

distance from the origin of a spherical coordinate system

tempe rature

a reference temperature at which properties are evaluatrd

the temperature measured by the vapour thennocouple

dimensionless

ambient vapour temperature

temperature

xvii

time

interna1 energy

volume

volume of a finite system

specific volume

the ratio of the mass of the condensing cornponent to the total mass is a two component system

thermal diffusivity

ratio of ideal gas specific heats. surface tension

pressure change due to the evaporation of the dropiet

pressure change due to adsorption

pressure change due to leakage

temperature difference between the droplet surface and the ambient vapour

mass diffusion coefficient

- P T )) , used in SRT expression 1 for evaporative flux

contact angle, vi brational temperature

chernical potential, dynamic viscosity

kinematic viscosity

density

B refers to the bath

L refers to the Iiquid phase

LV refers to the liquid-vapour interface

R refers to the reservoir

SL refers to the solid-liquid interface

xviii

refen to the solid-vapour interface

refers to the vapour phase

refers to the position of the wall of the test section with respect to the centre of the droplet

refers to the properties of a composite system

refen tu the position d the liquid-vûpour interface with respect to the centre of the droplet

indicates that properties are to be evaluated at the liquid- vapour interface refers to the main chamber

refers to a reference condition

refus to the test section

refers to the position of the vapour thennocouple with respect to the centre of the droplet refers to the equal divisions of a composite system

refers to saturation conditions

refers to the condensing component in a two component system refers to the non-condensing cornponent in a two component system

xix

1.1 MOTIVATION

The conditions existing at a liquid-vapour interface undeqoing phase change have

become controversial. Specifically, there are different claims as to the nature of the tem-

perature profile near and across the interface. In the past, such a system has been theoreti-

cally analyzed using either the continuum approach or Classical Kinetic Theory (CKT).

an^' presents a detailed review of this work. There is no known experimental work to

suppon the claims of either of these approaches.

The results of recent experiments have questioned the validity of both of these

approaches. For a steadily evaporating system with water as the working Ruid. the temper-

ature in the vapour at the interface was always found to be higher than that in the liquid.

Differences as large as 8 5 'C were measured14. This difference is also referred to as n

temperature discontinuity. For the same conditions, one approach developed using CKT"

predicts a discontinuity of 0.007 'C in the opposite direction. In the experimental work.

temperatures were measured to within a few mean free paths of the interface. denying a

potential argument that the temperature discontinuity predicted by CKT would only be

apparent within the Knudsen layer. Similar experiments have been done on a steadily con-

densing system using water as the working fluid6. There. the temperature discontinuity

measured was found to be in the same direction to that predicted by CKT (higher in the

vapour than in the liquid) and of a greater magnitude. Temperature discontinuities as large

as 0.6 "C were measured.

An alternate theoretical approach called Statistical Rate ~ h e o r ~ ' (SRT) can be

used to analyze the problem. Statistical Rate Theory is based on the transition probability

concept as defined in quantum mechanics and uses the Boltzmann definition of entropy to

introduce a thermodynamic description of the systern. In the past, SUT has been applied to

rate processes such as gas-solid a d s ~ r p t i o n ~ - ~ ~ . penneation in ionic channels in biological

rnernbranes13 and crystal growth from solution'? It can be used to predict that the mass

flux across a Iiquid-vapour interface will be a function of the temperature at the interface

in each phase, the pressure in the vapour at the interface and the radius of curvature of the

interface3. It can be s h o w that the calculated value of the mass flux is most sensitive to

the pressure in the vapourJ. With this in mind. the expression for the mass flux can be used

to predict the pressure in the vapour using the four other known parameters. and this value

cornpared with the experirnental measurement. In the experiments of Refs. 1 4 and 6. al1

five of these parameten were independently measured. and good agreement was found

between the predicted and measured pressures.

A logical next step in an investigation into the conditions at a liquid-vapour inter-

face undergoing phase change is to consider an unsteady systern. A problem which rneets

this critena is droplet evaporation. It is a fundamental physical problem with many appli-

cations such as atmospheric physics, combustion and thermal sprays.

1.2 DROPLET EVAPORATION BACKGROUND The ~ ' - l awl ' - '~ was the fint widely accepted theory developed to mode1 droplet

evaporation and combustion. It predicts that the surface area of the droplet will decrease at

a constant rate described by

where D is the droplet diameter, r is tirne and K is a constant. It models a stationary. sin-

gle droplet in a gaseous phase which may or may not contain its own vapour in the initial

state. The mode1 assumes that:

1. Spherical symmetry exists, and convection effects can be neglected.

2. The pressure in the ambient gashapour phase is constant.

3. Diffusion is the rate controlling process.

4. Transport properties are constant.

5. The Cas phase is quasi-steady.

6. The droplet temperature is uniform and constant.

7. The temperature profile across the liquid-vapour interface is continuous.

8. The partial pressure of the vapour at the liquid-vapour interface is equal to the satura-

tion pressure corresponding to the liquid temperature.

9. Gravitational effects are ignored.

From the conservation equations, K is predicted to bel9

where the superscripts G and L refer to the gas and liquid phases respectively. p is den-

sity, d is the mass diffusion coefficient of the vapourizing species in the gas phase. (., is

specific heat at constant pressure, AT is the difference in temperature between the droplet

surface and the ambient gas and hfg is the specific enthalpy of vapourization. ~ a w l9 indi-

cates that the estimate of K from Eq.(l-2) is 'crude' and that due to the difficulty in speci-

Qing some of the transport properties of the gadvapour phase, the measured value of K is

G G often used as a fitting parameter to specify p 6 in expenmental analyses. While the pre-

dicted value of K may not be useful in al1 cases, the trend predicted by Eq.(l- 1) has con-

sistently been seen in experimental data. Reviews by ~ a w ' ~ and sirignanoM give a

detailed account of the basis of this mode1 , its limitations. and attempts made to improve

it.

To try and develop better models. investigaton have used different methods to

solve the conservation equations which do not make al1 the assumptions inherent in the

7 D- - l a d 2 l'? Consideration has been given to factors suc h as variable thermal propçr-

ties, non-ideal phase behaviour. radiation effects. and separate rnodeling of the continuum

and Knudsen regions. Very few investigations have considered the possibility of a temper-

ature discontinuity at the liquid-vapour interface during droplet evaporation. young5 has

modeled droplet evaporation and condensation with a temperature discontinuity predicted

by CKT. As discussed in Section 1.1. in evaporation. the temperature in the liquid at the

interface is predicted to be greater than that in the vapour. No cornparison is made to

expenrnental results. Elpenn and ~rasovitov'' develop a droplet evaporation model takinp

into account a temperature discontinuity as proposed by Yalamov et al3. The temperature

discontinuity is shown to affect the late stages of the evaporation process. increasing the

predicted total evaporation time of srnaIl droplet ( D - p i ) by 10%. The time scale for the

evaporation is on the order of milliseconds. There is no known expenrnental work which

has aied to rneasure or predict a temperature discontinuity in droplet evaporation.

Much of the expenmental work in the literanire deals with droplet combustion.

The expenmental parameten of interest are the initial droplet size and composition. the

total evaporation time, and the arnbient conditions. Droplets are typically Iess than 1 mm

in diameter, and due to the high ambient temperatures used, the evaporation times are on

the order of seconds or minutes. Little consideration is

tion process itself. Because of this, the new theoretical

given to the details of the evapora-

tools developed are generall y only

compared to droplet size histones, as linle data is available with respect other aspects of

3 the system behaviour. As the trend of the D--law is generally present, it is difficult to

comment on the impact of the new models.

The initial condition in a droplet evaporation experiment is difficult to define. Ide-

ally, an experiment would be run where a droplet is initially in stable equilibrium with ifs

surroundings. The system could then be perturbed in a controlled manner to start the evnp-

oration process. The question then aises as to whether a droplet can be in a stable equilib-

rium state.

1.3.1 THE EQUILIBRIUM CONDITIONS AT A LIQUID-VAPOUR INTERFACE

The equilibriurn conditions at an isothermal. curved. one component liquid-vapour

interface can be predicted by the Kelvin relation. It is generaily expressed as26

V where P is the pressure in the vapour. P , is the saturation pressure at the system

LV L temperature T, y is the surface tension of the liquid-vapour interface. v , is the specific

volume of the liquid at the saturation conditions, R is the radius of curvature of the inter-

face and k is the Boltzmann constant. When the liquid is on the concave side of the inter-

face, such as for a droplet, R is greater than zero. The derivation of the Kelvin relation

assumes that mechanicai and thermal equilibrium exist at the interface, and that there is no

net mass transfer between phases. For a Rat interface ( R -, = ) at equilibrium, thermody-

narnics predicts that the pressures in both phases will be the same and equal to the satura-

tion pressure corresponding to the system temperature. This is seen in the Kelvin Relation.

For the case when R is greater than zero, the Kelvin Relation predicts that the pressure in

the vapour at equilibnum will be always higher than the saturation pressure.

1.3.2 THEORETICAL PREDICTIONS

The stability of a single one-cornponent droplet was fi rst investigated by ~ ibbs" .

He showed that in an unbounded expanse of its own vapour at a pressure higher than the

saturation pressure corresponding to the system temperature, such a droplet will have one

equilibriurn size, and that this state is unstable. The radius in this state is often referred to

as the critical radius. and has applications to nucleation theory. The Kelvin Relation c m

predict its value. The equilibrium state is unstable because mass transfer to or from the

droplet has no effect on the pressure in the vapour.

A second equilibrium state, which is stable. is predicted to exist in a one compo-

nent systern of finite size and r n a s ~ ~ ' ~ ~ . ~ h e equilibrium size is stable because mass trans-

fer to or from the droplet has an effect on the system pressure. The pressure in the vapour

when the droplet is in the stable equilibrium state is always above the saturation pressure.

consistent with the Kelvin relation. In that analysis, the effect of the solid vapour interface

was ignored. Recently, some confusion seerns to have developed in the literature. Claims

have been made that "the Kelvin equilibrium for a single component drop is always iorrrn-

bit?"''.

1.3.3 EXPERIMENTAL INVESTIGATIONS

There have been many attempts to experimentally venfy the Kelvin Relation. but

to this point, none have gotten agreement within several percen$O. One of the difficulties

is that the effect is only significant for very small values of R . As well. sorne investiga-

tions considered systems which are not at equilibnum, while the Kelvin relation applies

only to an equilibriurn state. In terms of droplet stability, there have been no known exper-

imental studies to test the predictions for a finite system. The results of Lamer and

~ n i e n ~ l have been interpreted3' as an example of a stable droplet size when the droplet

contains a non-volatile component and the pressure in the sunounding vapour is constant.

This interpretation is open for discussion.

1.4 SCOPE OF THE THESIS

The purpose of the present work is to begin a theoretical and experimental investi-

gation of droplet stability and evaporation in a finite system.

In Chapter 2. the experimental apparatus used in the investigation is presented. The

results of a series of ihree evaporation experirnents of a single, stationary one-component

droplet in a finite system are then presented and analyzed. Consideration is given to the

size of the droplet as it evaporates, the temperatures in the liquid and vapour phases and

7

the pressure in the vapour. The droplet size history is compared with the D- -law. The tem-

perature profile in the system is detenined, and the predichons of the conditions at the

liquid-vapour interface are compared with those predicted by CKT and those measured in

steady evaporation systems. The pressure in the vapour is predicted with SRT using the

measured temperatures and droplet size, and compared to the measured pressure. The

effect of the adsorbed phase is analyzed.

In Chapter 3, the stability of a single one-component droplet in a finite system is

investigated theoretically with the inclusion of the adsorbed phase. Special consideration

is given to the pressure at which filmwise condensation is predicted to occur on the soiid

surface. The predictions of the theory are compared to those of simpler models. The effect

of a second. non-volatile component on the droplet stability is considered. A system with

multiple droplets is analyzed. and the prediction compared with that of another investiga-

tion.

In Chapter 4. the results of a series of experiments run to try and observe the pre-

dicted stable equilibrium site are presented. The observed behaviour is analyzed by con-

sidering the adsorption on the solid surface of the system and the results of the evaporation

experiments.

2.1 EXPERIMENTAL APPARATUS

The objective of the experimental investigation is to hang a liquid droplet on a

thermocouple bead and measure its site and temperature. and the pressure and tempera-

ture of the sunounding vapour as the droplet evaporates. Water is chosen as the working

fluid. Droplets with diameters up to 1.80 mm can be investigated. A schematic of the

experimental apparatus that was designed and built is shown in Figure 2- 1. A detailed

description of the apparatus is provided in Appendix A

The central component of the apparatus is the test section. It is comprised of two

sections: the main charnber (mc in the figure) and the auxiliary c hamber (ac in the figure).

which are separated by a valve. Through a connection to a turbo-rnolecular pump. the

pressure in the test section can be reduced to lo4 Pa. Water vapour at a desired pressure

less than ambient saturation conditions can be introduced to the test section from the

depssing flask without the possibility of contamination by air. Once an experiment is

setup. the test section can be immersed in a water bath that can maintain a constant tem-

perature. The temperature in the water bath is measured with a thermocouple.

The thermocouple on which the droplet hangs is located in the main chamber. The

thermocouple bead is approximately spherical with diameter 0.35 mm. A second thermo-

couple is located 4.30 mm away. These thermocouples rvill be referred to as the droplet

thermocouple and the vapour thermocouple. They are mounted on a linear-rotary motion

feedthrough. A picture of the configuration of the two thermocouples when there is a drop-

Figure 2-1. A schematic diagram of the front view of the experimental apparatus.

let present in the system is shown in Figure 2-2. The droplet is introduced to the system by

a needle connected to a syrinp that is mounted on a syringe pump that provides fine con-

vol of the flow rate. The pump is mounted on a three degree of freedom position mani pu-

lator. The syringe can be filled with liquid water from the degassing flask. The needle

enters and exits the system through a septum which ensures that there is no air leakage.

When the needle is inside the test section, it can be isolated from the syringe by a valve

and pumped down to vacuum conditions with the test section. while the syringe remains

full of water. When a droplet is to be formed on the needle. a valve between the needle and

the syringe is opened. The droplet is transferred from the needle to the droplet themiocou-

ple by adjusting the position of the needle and the thermocouple using the feedthrough and

the position manipulator.

Figure 2-2. The configuration of the droplet and vapour thennocouples.

I ?

Dunng an experirnent, measurements of the temperatures read by the droplet ther-

mocouple, the vapour thermocouple and the thermocouple in the bath are made using a

data acquisition system. The calibration procedure used for the thermocouples is presented

in Appendix B. The absolute error associated with the measurements is iû.10 'C and the

relative error between measurements is &.O5 OC. The temperatures encountered in the

experiments are between 19 'C and 30 O C . The pressure is measured with a pressure trans-

ducer that is also connected to the data acquisition system. The calibntion of the trans-

ducer is discussed in Appendix B. The error associated with the pressure readings is t20

Pa. The pressures encountered in the experiments are between 2000 Pa and 3500 Pa. The

data acquisition system can take measurements at one second intervals for long periods of

time, and at intervals as small as 30 ms in short bursts. The droplet size is measured using

a solid state video camera that is connected to image processing software. A discussion of

the optical calibration and the droplet size measurernent technique are presented in

Appendix B. The error associated with the droplet size measurements is M.04 mm. A typ-

ical image of a droplet used to make a measurement is shown in Figure 2-3. The droplet

diameter is taken as the largest horizontal dimension that can be measured.

Figure 2-3. A typical image of a droplet. Its diametei this case has a value of 152 mm.

is measured as shown, and in

A series of tests were done to quantify the performance of the apparatus with

respect to the temperature control of the bath and the amount and nature of leakage into

the test section in different configurations. The results are presented in Appendix C. It was

found that the bath could maintain a temperature in a &.03'C range. The leakage into the

system is generally insignificant, and will be discussed more specifically in the context of

the different experiments that were run.

Due to the complex nature of the many valves and fittings in the test section. it is

dificult to accurately determine its extent. The volume and surface area of the test section

were approxirnated geornetricaliy to allow certain calculations to be made. The resulis are

presented in Table 2- 1.

-- 1 Auxiliary Chamber 1 3.2 1 - 3.2 1

Table 2-1 .The extent of the test section. approxirnated geometrically.

The methods used in the expenmental investigations outlined in this chapter and in

Chapter 4 were dependent on the type of experiment that was being run. The method for

each type of experiment will be presented at the appropriate time. For al1 experiments.

water is prepared by filtering, distilling, deionizing and degassing. A detailed description

of the procedures developed is presented in Appendix D.

Section

Main Charnber

2.2.1 MO~VATION

In a finite system where a droplet is evaporating, the pressure in the vapour will be

afTected not oniy by the evaporation process, but also by the adsorption of vapour ont0 the

Volume (m3 x IO")

4.1

Surface Area (m2 x 1 O-')

4.4

solid surface of the system. The amount of adsorption will be dependent on the nature of

the solid surface and on the pressure and temperature in the vapour. It will also depend on

the amount of mass already in the adsorbed phase. It order to undentand the expenmental

results, it will be necessary to have a model of the adsorption that takes place. The inside

surface of the test section is primarily stainless steel. with some glass, rubber and vacuum

epoxy. As well as being non-unifonn, the surface is also not smooth. It is l i kely that the

stainless steel surface has oxidized and has pores of many different sizes on it which make

it difficult to predict how it will behave with respect to adsorption. An isotherm for the test

section was detemined experimentally. Before descri bing the adsorption experirnents. a

standard adsorption model will be presented that will be used as a cornparison.

2.2.2 THE BET ISOTHERM

An approach that has been developed to analyze a solid-vapour interface at equilib-

riurn is the BET i ~ o t h e n n ~ ~ . It predicts the number of rnoiecules. NS" . that will adsorb on

a smooth homogeneous sotid surface at equilibrium as a function of the pressure in the

vapour, pV :

SV where n, is the monolayer adsorption per unit area, C is a constant and Pr is a reference

SV pressure. The parameters n, and C must be determined empirically, and Pr must be

specified. The mode1 assumes that multilayer adsorption occun, and that there is no inter-

action between stacks of adsorbed molecules. When the pressure in the vapour is equal to

the reference pressure, the number of adsorbed layen becomes infinite and filmwise con-

SV densation will occur. A plot of Eq.(2-1) is shown in Figure 2-4. The ordinate is ti . which

is equal to and the abscissa is pV/ P, . The solid surface is taken as Pyrex. with

SV the ernpirical parameten n, and C taken from Ref. 34. Note that the amount of adsorp-

v tion increases dramatically as P appmaches P r .

Figure 2-4. The BET isotherm. The solid surface is taken as Pyrex with C = 47.86 and SV

n, = 1.13 x 1 0 ~ ~ r n j r n ~ .

2.2.3 METHOD

The test section was prepared for an adsorption expenment by immersing it in the

water bath and purnping it down to a pressure of IO-' Pa for at least 12 hours. Following a

previous adsorption experiment, it takes at least 12 houn of purnping to get the pressure to

this level. Using the vapour supply, a pre-determined pressure of water vapour was then

established in the test section. During this tirne, measurements of the pressure were taken

every 0.15 seconds to ensure that the highest pressure reached was recorded. The test sec-

tion was then isolated.

The system was then left to corne to steady state. During this time. the pressure. the

bath temperature and one temperature inside the test section were recorded every ten min-

utes. It was found for the higher initial pressures that a steady state was not reached after

up to three days of observations. For this reason, a criterion was developed for when to end

a test. Steady state is assumed to exist when the total change in pressure for each of a

series of eight 100 minute intervals (a total tirne of 13.3 hours) was less than one percent

of the total pressure change up to that point. Thus, if the experirnent were left to run for

another 13.3 houn, the change in pressure would be less than eight percent of the total

change, and more likely less than that. as the rate of adsorption decreases with time. This

critena was used on the three points with the highest value of the pressure ratio.

2.2.4 RESULTS AND ANALYSIS

An example of a pressure versus time curve is shown in Figure 2-5. For this case.

the procedure described in the previous section was used to end the test. Frorn the results

of the leak tests described in Appendix C, any significant leakage into the systern will be

water vapour. Assuming that the highest pressure, P,,, . read during the filling of the test

section is a reflection of the initiai mass in the system, and that water vapour behaves as an

ideal gas, then the total number of molecules in the system at the end of a test is

where V , is the volume of the test section obtained from Table 2-1, P is the leak rate

given in the 3rd column of Table C-1, t is the total time of the experiment, and T is the

Tirne (s)

Figure 2-5. A sample adsorption pressure-tirne history. This curve corresponds to the point on the isotherm with a pressure ratio of 0.522.

average of al1 the temperature readings taken. As mentioned. adsorption begins immedi-

ately when vapour enters the test section. It takes between 2 and 13 seconds to get the

pressure in the test section up to the desired level. The rate of adsorption is highest at this

time, and the initial pressure reading will therefore be an underestimation of the initial

mass in the system. The effect increases with increasing initial pressure as it takes longer

to fil1 the test section. The ratio of the calculated leaked mass to the total mass is always

less than 0.09. As discussed in Appendix C, the Ieak rate may be less than that used in the

calculations because of the adsorption. This will cause an overestimation in the value of

NT. If the final pressure in the system is P / V , then the number of molecules in the

adsorbed phase wiI1 be

where NT is given in Eq.(2-2). The criteria used to end the latter tests will cause the calcu-

lated value to be an underestimation of the adsorbed mass. To account for the errors dis-

SV cussed, a relative error of 15% is assigned to the values of N .

The adsorption isothem is shown in Figure 2-6 and the data used to constmct it is

presented in Table 2-2. Note that. for convenience, the reference pressure is chosen as the

saturation pressure for the temperature of each data point.

Table 2-2.Experimental adsorption isotherm data.

Figure 2-6. The expenmentally determined adsorption isotherm. The BEï fit has

Point

1

SV parameters C = O S 1 and N , = 1.1 x 1020 m,jm2

P / V / P , O. 173

~ , ( m , x ioZ0) 1.3

r ~ ~ ~ ( r n , x 10")

0.2 1

N ~ ~ / N ~

O. 16 T ( O C )

26.92

The data points fa11 within a temperature range of 0.06'C, which corresponds to a

range of 13 Pa in the saturation pressure. This is 0.4% of the value of the saturation pres-

sure at 26.9'C. The maximum initiai pressure that can be supplied by the degassing Ras k is

about 3400 Pa, which based on these experiments would give a final pressure ratio on the

order of 0.75. The conditions that are of interest for the evaporation and stability experi-

rnents include pressures close to saturation conditions. With the current set up. getting

points on the isotherm in this area is difficult. It might be possible to introduce more

vapour to the system as the adsorption takes place. but then there would be even more

uncertainty in the calculation of the total mass in the system.

M e n a BET fit is applied to the data points, the discrepancies between the data

points and the curve are between 0.0% and 18.8%. The fit is shown in Figure 2-6. The

BET parameten are C = 0.91 and N:~' = 1.1 x 1do m,. The parameter C is q u a 1 to the

exponential of the difference between the differential heat of adsorption of the first

adsorbed layer compared to that of the other layers divided by k ~ " . This differencr is

always positive. and therefore C should be greater than one. In fact. the difference is such

that the magnitude of C should always be large when compared to oneJ3. This is not seen.

With the surface area of the test section approximated as 0.076 rnt (from Table 2-1). the

SV monolayer coverage per unit area n , is 15 x 102' rn2m2. This is four orders of magni-

tude greater than the monolayer coverage found for Pyrex. From this. it can be inferred

that the area calculated geometrically is not a good indication of the surface area rhat

adsorption takes place on. This is not unexpected, as the stainless steel surface is not

smooth. However. this factor is not enough to explain the results. Even though the BET

equation gives a reasonable fit to the data in the range considered, due to the values of the

equation parameters determined, caution must be used in assuming that the adsorption

mechanism in the test section is the same as that used to formulate the theory. Also. exper-

imentally, it has been found that the BET isotherm is only valid for pressure ratios up to

0.3, beyond which it predicts an excessively large nurnber of adsorbed rnole~ules~~. As ri

final point to consider, there is a significant amount of adsorption on the walls at relatively

low pressure ratios. Using the results of Chapter 3, it can be shown that for a system mod-

eled using the isotherm in Figure 2-4, that at a pressure ratio of 0.99994, the amount of

mass in the adsorbed phase is 17% of the total mass in the system. For the experimental

apparatus. at a pressure ratio of 0.639, the adsorbed phase represents 28% of the total

mass .

Based on the above discussion, it is cIear that the BET isotherm is not the correct

theoretical model for the adsorption in the test section. This is not unexpected due to the

nature of the surface, which is neither srnooth nor homogeneous. There are no known the-

oretical models which accurately predict experirnental results for pressure ratios near one.

or for surfaces which are not smooth3'. In order to qualitatively understand how the solid

surface behaves, it is necessary to consider a different model of the adsorption process.

One such mode1 is capillary condensation3839, which is the condensation of liquid in the

pores of a solid surface. It will be further discussed in the context of the stability experi-

ments in Chapter 4.

2.3 EVAPORA~ON EXPERIMENTS A series of evaporation experiments were run to assess the validity of the rneasure-

ments and to gain a better undentanding of how the system behaves.

2.3.1 METHOD

The nzadk was piacrd in the tesr secuon, and die system was pumped to 10 '' Pa

for at least 12 hours. Water vapour was introduced to the test section. and a droplet was

placed on the droplet thennocouple. The pressure in the test section was raised to slow

down the evaporation. The test section was imrnersed. and the pressure reduced to a pre-

detennined Ievel by pumping it down with the vacuum pump. When this pressure was

reached, the main chamber was isolated and the evaporation process was observed.

Based on the results of Appendix E. the effect of the pump down. which can be

interpreted as an expansion. will be gone in two minutes after the valve is shut. Follow ing

this, temperature and pressure measurements were begun at ten seconds intervals.

Depending on the expected rate of evaporation. the droplet size was measured at different

time intervals. The results of three such experiments are presented in this section. The set

up parameters for each of the experiments are presented in Table 2-3. PH is the pressure

that the test section was raised to prior to being immersed. PL is the pressure that the

main chamber was pumped to after the immersion was cornplete. D,,,, is the diameter of

the droplet at the time after which the expansion effects are no longer important. In order

to compare the three tests, the size history will be considered starting at a diameter of 150

mm. As the initial size of each droplet was not the sarne, some time passed to reach the

desired state. The pressure at the start of the observation penod was different for the three

tests.

As seen in Table 2-3, the set up of each of the experiments was slightly different.

For test 1. the fint droplet that was formed on the needle was of a suitable size and was

transferred to the thermocouple immediately. For the tests 2 and 3. more time was

required. For these tests. some droplets were formed that were too big to be supponed by

the needle, and fell to the bottom of the main chamber. Sorne time was needed for them to

evaporate. The time required was decreased by pumping down the test section to 2000 Pa.

letting the liquid evaporate, and repeating until it was al1 gone. Also. for both of these

tests, some time was required once a suitable droplet formed on the needle to let it evapo-

rate to a size at which it could be safely transferred to the thermocouple.

The time to place the droplet is important with respect to the leakage of air into the

system. Based on the results presented in Appendix C, the leak rate when the test section is

not immersed and the needle is inside is 6.83 Pafhr. For tests 1,2 and 3, this corresponds to

a total Ieak in Pa of 1-84, 11 .L8 and 10.05 respectively. For tests 2 and 3, the amal amount

of air will be less due to the pumping that was done to hasten the evaporation of the fallen

droplets. Even if this is not considered, the mass fraction of air based on PH that would be

present in the test section after the droplet was placed is at most 0.3%. Once the needle is

removed, the test section is immersed. This takes about ten minutes. In this time. based on

the results of the leak test, there will be leakage of 0.47 Pa into the test section, giving a

Table 2-3.Set up parameters for the evaporation experiments.

test

1

2

3

time needed to place

droplet(s)

970

5860

5300

P a t D = 150 nim

(Pa) 3200

2578

2252

3298

3305

P ~ ( P a )

3060

2593 i

3329 1 2091

*,lm

(mm) time to D = 1.50 mm (s)

156 240

1.73

1.69

4500

1600

maximum possible partial pressure of air of 1 1.65 Pa. For the lowest value of P L . thiç cor-

responds to a mass fraction of 058%. The pressure effect is iess than the accuracy of the

pressure reading , and will be ignored.

2.32.1 SEE HISTORY

The size histories for the three tests starting at a diameter of 1 50 mm are shown

together in Figure 2-7. For tests 1 and 2, the droplet size was rneasured every three min-

utes. For test 3, the droplet size was measured every two minutes. In the top graph. al1 the

rneasured data is shown. All the data is tabulated in Appendix F. The solid markers corre-

spond to when the droplet shape was symmetric. The hollow markers start when this was

no longer valid. As described in Appendix B, the transition point varies between experi-

rnents, but is generally around 0.90 mm.The last data point for each set corresponds to the

diameter of the thermocouple bead. As would be expected. the rate of evapontion

increases as the pressure in the vapour is decreased. In the bottom graph. the square of the

diameter venus time is plotted for each test. Only the data points which correspond to a

syrnmetric droplet are considered. Also shown on the graph are best fit lines for each test.

Some of the important parameten for these curves are presented in Table 2 4 . The param-

eter K is the slope of the best fit line through the data on the bonom graph. and is the

7 3 parameter used in the Do-law. RD is the regression parameter provided by Microsoft

Excel. In al1 three tests, the measured data agrees well with a linear fit. The resul ts wi ll be

compared to the predictions of the D' -law in Section 2.3 3 5

Table 24Comparison of the three evaporation tests.

3.3 23 TEMPERATURES AND PRESSURE

test 1

The measured temperature and pressure histories for each of the experiments are

shown in Figure 2-8, Figure 2-9 and Figure 2-10. The data is plotted for the entire rvapo-

ration process and 2000 seconds aftenvards. The shapes of the curves are similar for al1

three expenments. Note the different scales for each of the three tests. For al1 three tem-

peratures. the curves have been smoothed using a moving average trendline to eliminate

noise in the signal caused by the ice reference. The plotted measurement at a given tirne is

the average of a set number of the preceding measurements. The number of measurernents

chosen to take the average ranged from 5 to 15. and was selected based on the highest

number that maintained the shape of the curve.

When there is still liquid present on the droplet thermocouple, its temperature is

always less than the temperature in the vapour, which is always less than the bath temper-

ature. The differences between the three temperatures increases with the evaporation rate.

As the evaporation takes place, both the liquid and vapour temperatures increase. The rate

of this increase rises with the evaporation rate.

total evaporation

time (s) 13080

tirne for evaporation frorn 1 50 mm to

1 .O0 mm (s)

8670

K (IO-'

rnm2/s)

L A94

R'

0.9980

Time (s)

-

O 2000 4000 6000 8000 Loo00 12000 I W O

Time (s)

Fiere 2-7. The size history for the evaporation tests. The top graph is a plot of the enhre evaporation process. The solid markers correspond to when the droplet shape is symmetric. The hollow markers indicate when this is no longer valid. The bottom graph plots the square of the diameter up to the point where the droplet Ioses symmetry. Aiso plotted are best fit lines for earh of the data sets.

bath

droplet 1

25.0 - -

O 5000 1OOOO 15000

Tirne (s)

3 1% -

O 5000 Loo00 15000

Time (s)

Figure 2-8. The measured temperature and pressure histories for test 1 .

27.0- bath

26.5 - -

evaporated 23 .O - - - - -. . .

Time (s)

evaporated

2860 --

O 2000 4000 6000 8000

Tirne (s)

Figure 2-9. The measured temperature and pressure histories for test 2.

droplet evaporated

2100 --

O 1000 2000 3000 4000 5000 6000

Time (s)

Figure 2- 10. The measured temperature and pressure histones for test 3.

To show the magnitude of the error in the temperature measurements compared to

the measured temperature differences. the temperature data for test 1 is replotted along

with error bars at every fortieth data point in Figure 2-1 1. This is the test where the tem-

perature differences are the smallest. While the droplet is evaporating, there is a rneasur-

able difference between the three temperatures. When the droplet has evaporated. there is

no measurable difference between the three readings.

Figure 2- 1 1. The error in the temperature measurements for test 1. The error bars are plotted for every fortieth data point.

The temperature profile in the vapour is such that heat is being transferred towards

the droplet. in the opposite direction of the mass fiux due to the evaporation. The tempera-

ture in the vapour increases because the droplet temperature is rising, but also because the

liquid surface is receding. The distance between the vapour thermocouple and the droplet

surface increases with time, and it will be in a different place in the temperature profile. It

is not until the droplet has cornpletely evaporated that the temperature in the test section

rises to the bath temperature. This point is clearly seen on the temperature graphs as where

the droplet thermocouple temperature rises rapidly. The vapour itself heats up quickly. but

some time is required to heat the thermocouple beads. After this tirne, there is no measur-

able difference between the three temperatures. To give a cornpanson between the temper-

ature measurements in the three tests, a plot of the temperature difference between the

droplet and the vapour themocouples is shown in Figure 2- 12 up to the point where the

droplet evaporated. Before discussing the liquid temperature, it is necessary to present

some discussion of the observed pressure curves.

test 1 l I

Tirne (s)

rigure 2-12. The measured difference between the readings of the droplet and vapour thermocouples dunng each of the three tests. The data is plotted up to the point where the droplet evaporated.

The pressure history for each of the three experiments follows a similar trend. As

the droplet evaporates, the pressure in the system increases. The amount of increasr is dif-

ferent for the three experiments, even though the initiai droplet size was the same. To

quantify this efFect, the measured pressure history can be broken down into the factors that

contribute to it. Tbere are three separate processes to consider: the evaporation of the drop

let, adsorptioddesorption from the solid surface. and leakage into the system. The time

history of the measured pressure, Pm,, , starting from a time r, can be expressed in r e m s

of these factors:

# 7 4 .

'rnrar = P o + 'Pcvop + 'PudJ + "lest \ - - t

where Po is the measured pressure at time r , , AP,,, is the pressure change due to the

evaporation of the droplet, AP,,, is the pressure change due to adsorption. and API,,, is

the pressure change due to leakage. The pressure change due to rvapontion can be

expressed as

where Do is the diameter at time r, , D is the diameter at time r and V,, is the volume of

the main chamber. The temperature T is evaluated as the bath temperature. The pressure

change due to leakage can be expressed as

APleok = Pt (2 -6

where P is the leak rate calculated in Appendix C, which for this case is 9.62 Pa/hr. The

only unknown in Eq.(2-4) is the adsorption term, which can be isolated to give

The values of Pm,, - P o . APeWp, -APud5 and APleak for the entire evaporation process

starting at a diarneter of 150 mm for each of the three experirnents are presented in

Table 2-5. There is a small difference in the AP,,, term as the initial size measurement

was not aiways exactiy 150 mm. A plot of the four t ems for test 2 as functions of time is

shown in Figure 2-13. The plot extends beyond the tirne when the droplet had completely

evaporated to show how the systern behaviour changes. Note that -AP,,, is the amount of

adsorption that has taken place. The size of the droplet is only available at certain times.

and each of the t ems are evaluated at these times. While there will be some error gener-

ated by the size measurements below 0.90 mm. the observed trends are still valid.

Table 2-5.The pressure breakdown data from a droplet diameter of 150 mm to w h m the droplet had completely evaporated.

O 2000 4000 6000 8000

Time (s)

Figure 2- 13. The pressure breakdown for test 2

In dl three experiments, the leakage is not significant. The effect of the evapora-

tion is constant, as the droplet starts and ends in the same state for each test. although the

test P ,,,, - Po (Pal A P , , (Pa) -APuds (Pa) bPleak (Pa)

time scales are different. It is in the adsorption where big differences are seen. The differ-

ences in the systern parameten for each of the three tests have been presented. The impor-

tant parameter to consider when analyzing this data is the absolute pressure in the system.

The amount of adsorption during the evaporation of the droplet increases as the pressure in

the main chamber increases. This can be explained by considering the adsorption isotherm

for the test section as shown in Figure 2-6. It was found that the amount of mass that can

adsorb on the surface increases significantly as the pressure in the system increases.

To understand how the system behaves. consider what happens during a srnail time

penod G r . The droplet evaporates, and a small amount of mass enters the vapour. causing

the pressure to go up. Some of this mass wil1 adsorb on the solid surface of the test section.

During an experirnent, the adsorbed phase will not be in equilibrium with the vapour. as

not enough time has passed for this state to be reached. More of the mass will adsorb if the

pressure in the systern is higher. The adsorption will result in a decrease in the measured

pressure. The total effect on the measured pressure for these three tests is an increase. and

the amount of the increase depends on the pressure in the vapour. If the pressure in the sys-

tem was high enough. it would be possible for the pressure to actually decrease as the

droplet evaporated. To further complicate the issue. the amount of adsorption that takes

place is not only a function of the pressure, but also of how much adsorption has already

taken place. The behaviour will be very different if the adsorbed phase is empty compared

to if it is near equilibrium with the surrounding vapour.

When the evaporation is complete. the measured pressure starts ro decrease. as

adsorption is the only significant factor left that affects the system pressure. The curve

would then follow a similar trend to that shown in Figure 2-4, where the pressure in the

system for an adsorption experirnent was plotted as a function of time.

The last measurement that must be considered is the temperature of the liquid. One

7

of the assumptions in the D--1aw is that the pressure in the vapour at the droplet surface is

equal to the saturation pressure corresponding to the droplet surface temperature. Further-

more, the temperature in the droplet is assumed to have no spatial variation. To this end.

the measured pressure and the saturation pressure corresponding to the measured tempera-

ture in the liquid for test 2 are plotted together as functions of tirne in Figure 2-14 The

plot covers the time from a droplet diameter of 1 J O mm to just before al1 the liquid evapo-

rated.

2850 -- - - - - -

O IO00 2000 3000 4000 500 6000

Time (s)

Figure 2- 14. The measured pressure and the saturation pressure corresponding ro the measured liquid temperature for test 2

There is a very strong agreement between the ~ v o curves. The pressure in the

vapour phase follows the saturation pressure corresponding to the liquid temperature very

closely. Any differences are covered by the experimental enors. It is therefore possible to

predict the pressure in the system while the droplet is evaporating with only the measure-

ment of the liquid temperature. Sirnilar agreement is found for tests 1 and 3. The agree-

ment between the two curves gives strong support for there being no significant air content

in the system. It is important to note that we are not saying that the pressure in the vapour

is equal to the saturation pressure corresponding to the liquid temperature. only that exper-

imentally, it would be impossible to distinguish between the two. This wil1 be important

when the data is analyzed using Statistical Rate Theory.

It has been established that the adsorbed phase to a great extent regulates the pres-

sure in the vapour. It now appean that the pressure in the vapour controls the droplet tern-

perature. For test I . where the measured vapour pressure changes very little during the

evaporation due to the laqe amount of adsorption that can take place. the temperature in

the liquid phase also changes very little. For test 3. where there is a laqe change in the

measured vapour pressure as less adsorption takes place. there is a large change in the liq-

uid temperature.

The measured temperature in the liquid can also be interpreted from a heat transfer

perspective. At any time, the tempemture gradient in the vapour will cause a certain

amount of energy to be transferred to the droplet. When the droplet temperature is seen to

rise, as in test 3, some of this eneqy is used to heat the droplet. while the rest is used for

phase change. When the droplet temperature is steady, as seen in test 1. al1 the heat trans-

ferred from the vapour is used to bring about the phase change.

23.3.1 INTRODUC~ION

The experimental results can be analyzed using Statistical Rate Theory (SR?').

which was introduced in Chapter 1. Statistical Rate Theory can be used to predict that the

mass flux, j across a liquid-vapour interface will be given by

L v where T i is the temperature in the liquid at the interface, Ti is the temperature in the

v vapour at the interface. Pi is the pressure in the vapour at the interface and R is the radius

of curvature of the interface, which in this case is the radius of the droplet. which cvill be

denoted as rD . The full expression for j is given in Appendix F. In order to perform the

necessary analysis, experimental values of these five parameters are required.

L Having assurned that the liquid temperature at any instant in time is uniform. T ,

has been directly measured. We assume that the pressure in the system is also uniform and

v as there is no significant air content, Pi has also been directly measured. The droplet

diameter, D has been measured, and thus rD is also known.

2.3.3.2 CALCU~ATION OF THE MASS FLUX

To calculate the mass flux, the time history of the droplet size is used. The evapora-

tive mass flux can be expressed as

where is the liquid density. At a given point, it is a function of the dope of the droplet

size time history. The droplet diameter as a function of tirne is fairly linear over small time

intervals between diameters of 150 mm and 1.00 mm as seen in Figure 2-7. The flux at a

measurement point is calculated by finding the slope of the best fit line through that point

and the two points on either side of it, for a total of five points. That is,

L

where SLP is a function which retums the slope of a least squares fit to the data. Thus. the

mass flux can be calculated starting at the second measurernent after a diameter of 1.50

mm up to two measurements before a diameter of 1 .O0 mm. A sample of the calculated

mass fluxes for test 2 are shown in Table 2-6. Al1 of the measured and calculated parame-

ters used in the SRT analysis are presented in Appendix F.

2.3.3.3 CALCULATION OF THE TEMPERATURE FIELD

To calculate T;. an approxirnate temperature profile is detennined in the vapour

phase using the measured temperatures. The systern of interest in shown in Figure 2- 15.

C

main chamber waI1

vapour thermocouple

droplet T ~ L O 0

PD Figure 2-15. The mode1 used to calculate the temperature profile in the vapour.

v In Figure 2- 15, T , is the temperature measured by the vapour thermocouple and r , is the

distance between the centre of the droplet and the centre of the vapour thermocouple. The

B temperature in the vapour at the wall is assumed to be equal to the bath temperature. T

and the distance from the centre of the droplet to this point is rB .

The steady, one dimensional, constant properties energy equation in spherical

coordinates without viscous damping is

where c p is the specific heat at constant pressure. v, is the radial velocity and iC. is the

thermal conductivity. Defining the dimensionless length R and temperature 7 as

and

the energy equation can be recast as

where Pe is the Peclet number and is given by

To assess the relative magnitudes of the left and right sides of Eq.(2- 14), an order of m a g

nitude andysis will be performed. We are interested in the importance of the convection

term. Convection effects will be greatest at the surface of the droplet, and for this reason.

the region between the droplet and the vapour thermocouple is considered. Assuming that

the temperature profile between the two points is linear, the left side of Eq.(2- 14) becomes

The right side becomes

Thus, to compare the magnitudes of the two sides. we must consider the ratio

Both the numerator and denominator of Eq.(2-18) are functions of the position R . As

mentioned, convection effects will be most signifiant at the droplet surface. where the

radial velocity will be the highest. To evaluate the Peclet number. the properties of water

vapour are evaluated at their saturation values at 26.85 O C . and the mass Aux is chosen as

the highest encountered in the evaporation tests, which is 1.4 x 10-' kg/m2s. The radius of

the droplet in this case was 5.60 x IO-' m. When is evaluated at R D , Eq.(2- 18) evaluates

to 0.0006. It is therefore justifiable to neglect the convection term in the formulation of the

temperature profile. This analysis is equivalent to expanding the solution of Eq.(Z-14) in

powen of the Peclet number, which is small compared to one, and only considering the

first order terrns.

The possibility of naturd convection must also be considered. The average Nusselt

nurnber for natural convection around a sphere is given bya

where Pr is the Prandtl number and RaD i s the Rayleigh number. given for an ideal gas

by

where g is the gravitational constant. AT is the difference in temperature between the sur-

v face of the sphere and the ambient temperature T , . a is the thermal diffusivity and v is

the kinematic viscosity. For the three tests, the largest value of AT is 7 K. With the vapour

properties evaluated at saturation conditions at 26.85 OC. the Nusselt number is found to be

2.13. A value of 2 corresponds to pure conduction. For this extrerne case. natural convec-

tion will account for about 6% of the observed heat transfer. It will be neglected.

To find the temperature profile in the vapour, one must then solve

with the boundary conditions

and

The solution is

As the droplet evaporates, the distance to the vapour thermocouple increases. Also. the

droplet tends to move up the thermocouple wires during this time. To account for these

effects, r , is determined separately for every calculation, based on the known distance

between the two thermocouple beads and the relative location of the centre of the droplet

and the droplet thermocouple bead. The value of r , generally varies between 3.89 mm

and 4.45 mm. r , is taken as the shortest distance between the droplet thermocouple and

the side of the main charnber. This has been rneasured as 0.027 m. Due to the spherical

nature of the system under consideration. the results are not sensitive to small changes in

B rB that would be brought about by the movernent of the droplet. T is taken as the aver-

C' age bath temperature for the total measurement period. To determine Ti . Eq.(2-22) must

be evaluated at the radius of the droplet. A sample of the results of these calculations for

test 2 are shown in Table 2-6 along with the other SRT parameters. All of the data for al1

V B t three tests is presented in Appendix F. Note that r D . T,, , T and T , are measured. while

v Ti is calculated. The calculated temperature measurements were taken at ten second

intemals, and do not aiways conespond to the time at which the droplet size was mea-

sured. In these cases, the temperatures used are those which are closest to the size mea-

surement tirne. The error will be at moçt five seconds. This is srna11 compared to the total

time of any of the evaporation processes. To show the temperature discontinuity graphi-

cally, the temperature history for test 2 is replotted along with the predicted temperature in

the vapour at the interface in Figure 2-16. The error ban are an indication of the uncer-

tainty brought about by the temperature and length measurements.

The temperature in the vapour at the interface is found to always be higher than

that in the liquid. This is consistent with previous experimental results of steady

evaporationL4". In that work, the difference was referred to as a temperature discontinuity.

or temperature jump. There is some scatter in the data, which c m be panly amibuted to

noise in the temperature measurement signal as discussed in Appendix B. As s h o w in

Appendix F, similar results are found for tests I and 3. In test 1, where the mass flux was

lower, the temperature difference is smaller. In test 3, where the rnass flux was higher. the

temperature is larger. Using the experirnental data, the largest discontinuity predicted by

one approach using Classical Kinetic ~ h e o r y ~ is -6 x iod 'C, which is rnuch less than the

srnailest predicted value, and in the opposite direction.

predicted temperature in the vapour at the interface T

-r

/droplet

23 .O 1 t 1 1

O 2000 4000 6000 8000 Tirne (s)

Figure 246. The piedicted temperature discontinuity for test 2.

Table 2-6. A sample of the parameten used to predict the temperature discontinuity at the liquid-vapour interface for test 2. The bath temperature is 26.87'C

- -

r , (mm) T L ( O C ) T: ( O C ) T L (OC)

2.3.3.4 COMPAREON OF THE MEASURED PRESSURE TO THAT PREDIC~ED BY S RT

It has been found that the SRT expression for j is most sensitive to the pressure in

the vapouf. For this reason, the theory was tested by using the experimental values of j .

v L v Ti .Ti and rD to predict Pi . and then comparing this to the experimental rneasurement.

The same procedure is used for the current investigation. Plots of the predicted pressure

from SKï against the measured pressure for al1 three tests are shown in Figure 2- 1%

Figure 2- 18 and Figure 2-19. Note the different pressure scales and ranges.The soiid line

in dl three graphs is at 45'. and would indicate perfect agreement. A sample of the data

used to make the graph for test 2 is presented in Table 2-7. Al1 of the data for ail three tests

is presented in Appendix F.There is a strong agreement between the measurements and the

predictions for al1 three tests. The largest difference between the predictions and the mea-

surements is 9 Pa, which is within the rneasurement error of the pressure. The predicted

pressure is always very close to the saturation pressure corresponding to the liquid temper-

ature, but is not equal to it. The difference between the two is generally less than 0.05 Pa.

This difference could not be measured experirnentally.

Table 2-7A sample of the comparison between the measured and predicted pressures for test 2.

Measured Pressure IDifferencel (Pa)

Measured Pressure (Pa)

Figure 2-19. The pressure predicted by Statistical Rate Theory plotted against the measured pressure for test 3.

2.3.3 5 COMPARISON OF THE RESULTS TO THE D~ -LAW '1

In Section 1.2, the D- -law was presented. It predicts that the droplet surface area

will regress at a constant rate denoted by K . which. for a droplet evaporating in its o w

vapour. is predicted to be

As discussed, the prediction is considered to be approximate, and the experimentally mea-

v V sured value of K is often used as a fitting parameter to specify p 8 . To compare Eq.(2-

23) to the experirnentai results, the fiuid properties must be specified.

As the system has only one component, the rnass diffusion coefficient 8' is a self

diffusion coefficient. From Kinetic Theory, it is predicted to be4'

where rn is the molecular mass and d is the molecular diameter in the hard sphere approx-

imation. The molecular diameter can be estimated by using the dynamic viscosity of the

where CIv- is the dynarnic viscosity. Combining Eq.(2-24) and Eq.(2-25) gives

Substituting Eq.(2-26) into Eq.(2-23) gives

For the subsequent calculation e vapour dynarni c viscosity and specific heat are rvalu-

ated at the saturation conditions corresponding to a reference temperature T,,/ recorn-

mended by Abramzon and Sirignano as"

The liquid density and the latent heat of vapourization are evaluated at the saturation con-

ditions corresponding to the rneasured droplet temperature. The temperature difference in

Eq.(2-27) is specified in the D'-law as the difference between the ambient temperature

and the temperature at the surface of the droplet. The temperature of the water bath. 9 . is

taken as the ambient temperature. It has been found that there is a temperature discontinu-

L C ' ity at the droplet surface, and a question anses as to what temperature to use: T, or T , .

The analysis will be carried out for both cases.

For each of the three tests, K will be predicted at each of the points where the

droplet size was measured between diarneten of 150 mm and 1 .O0 mm. The results. along

with the experimental measurements of K are presented in Table 2-8 and Figure 2-20. As

the properties and the temperature difference in Eq.(2-27) change during the evapontion

process, the predicted value of K also changes. The predicted values in Table 2-8 and the

points plotted in Figure 2-20 correspond to the average value of K for each test. For the

B L case of the temperature difference taken as T - Ti , the bars in Figure 2-20 indicate the

range of the predictions. For test 1, where there was linle change in the ternperatures. there

is little variation in K . For test 3. where the temperatures changed significantly. there is n

larger range in the predicted values of K.

Table 2-8. Cornparison of the predictions of the D'-law to the experirnents.

L When the temperature discontinuity is taken into account by using Ti in the AT

expression, there is good agreement between the predicted and measured values of K. The

disagreement between the average predicted value and the measured value is at rnost 122.

When the temperature discontinuity is not considered, the theory underpredicts the evapo-

ration rate by 30% to 70%.

3 While the D6-law has been found to give a reasonable prediction of the rate of

evaporation, in other experimental studies this has not necessarily been found. To under-

O 1 2 3 4 5 4 2 Km, (IO mm 1s)

Figure 2-20. Cornparison of the predicted and measured values of K

stand why it works well for the current experiments but not others. some of the assump-

2 tions in the D -law rnust be considered. In sorne experiments. the difference in

temperature between the droplet and the ambient gas mixture can be on the order of hun-

dreds of degrees Celsius. This will cause there to be a large variation in the transport prop-

7 erties of the gas phase, while the Do-law assumes that they have a constant value. In the

current investigation. the range of temperatures and pressures encountered is srnall enough

that there are no significant effects of property variation at any point in time. When there

are large temperature gradients, there can be a significant arnount of droplet heating during

7 the evaporation process. The Dm-law assumes that there is no droplet heating. The arnount

of energy required to bnng about the heating can be on the same order as the energy

required to evaporate the droplet. The ratio of the energy associated with the droplet heat-

ing and the droplet evaporation between radii of r , and r , is given by

where h is the liquid specific enthalpy. In the current expenments, this ratio has a value of

at most 0.05 for values of r , and r , of 150 mm and 1 .O0 mm respectively. The droplet

heating is therefore not significant.

In the current experiments, the magnitude of the temperature discontinuity is of the

same order as the difference between the ambient temperatures and those at the surface.

and therefore has a significant effect on the prediction of K . In an experiment where the

difference between the arnbient and the surface was very large, the discontinuity rnight not

play as important a role.

A prediction of K can also be made using Statistical Rate Theory. The time rate of

change of the square of the droplet size as a function of time can be expressed as

Wth the definition of mass flux frorn Eq.(2-9). Eq.(2-30) becomes

The left side of Eq.(2-3 1) has been rneasured, and is a constant for each of the evaporation

experiments. It is equal to -K. The nght side can be evaluated using SRT to predict the

value j, and the rneasured droplet radius at each of the measurement points. It is a predic-

tion of -K. As discussed, the SRT expression for the mass flux is very sensitive to the

pressure in the vapour. if the nght side of Eq.(2-3 1) is evaluated using the measured pres-

sure, the result show a large scatter and no consistency. It has been obsewed that the mea-

sured pressure in the vapour agrees well with the saturation pressure corresponding to the

liquid temperature. With this in mind, the mass flux in Eq.(2-3 1) was evaluated by assum-

ing that

v pi = P,(T~)

The results of the ulations for test 2 are pw sented i i i Figure 2-2 1. Noie [hat the wlid

data points are the predicted value of K . Also ploaed is a straight line corresponding to the

measured value of K . There is scatter in the results. but the predicted values of K are of

3 the sarne order as those of the D- -1aw. A sarnple of the difference between the pressure

needed to collapse each of the data points ont0 the experimental value and the saturation

pressure corresponding to the liquid temperature, which was used to make the predictions.

is listed in Table 2-9. The difference is extremely small and could not be measured experi-

mentally. Similar results are found for tests 1 and 3. This is a further indication of the sen-

sitivity of the SRT expression for the mass flux on the pressure in the vapour. The fact thai

the difference required to collapse the data decreases as the droplet evaporates can be

attributed to the changing conditions dunng the evaporation process

a predicted - measured

O -- -

O 1000 2000 3000

Time (s)

Figure 2-21. A plot of the predicted value of K using SRT and the measured value for test 2.

Table 2-9.The difference between the pressure required to bring the predicted values of K into agreement with the experimental data and the saturation pressure corresponding to the liquid temperature.

r , (mm)

Difference between the saturation pressure and that needed to collapse the data ont0

the measured value of K (Pa)

2.3.3.6 COMPARISON TO STEADY

To try and correlate the

results, the temperature jump was

EVAPORATION EXPERIMENTS

current data with three sets of existing experimental

plotted as a function of the pressure for al1 the data sets.

As the results are most sensitive to the pressure, it is the parameter most likeiy to correlate

the data. The results are shown in Figure 2-22. Data sets A. B and C were obtained in the

study of the steady evaporation of water in Refs. 1.6 and 46 respectively. For these exper-

iments, the temperature discontinuity was rneasured. In the current work. the temperature

discontinuity has been predicted. The range of the three other parameters for each set is

presented in Table 2- 10. The range of al1 the parameters is significantly different between

the steady experiments and the current investigation.While the current work does not over-

lap with the existing results. the data appears to follow a consistent trend.

+ A u t e s t 1

* B test z + C A test 3

Pressure (Pa)

Figure 2-22. The temperature discontinuity for the current work and three other data sets plotted against the measured pressure in the vapour. A = Ref. 1. B = Ref. 6. C = Ref. 48.

Table 2-10A comparison of the range of the tempenture, mass fluxes and radii encountered in the different investigations.

1 current 1 26.0 -> 26.8 ( 19.8 -> 25.3 1 0.03 -> 0.11 1 0.50 -> 0.75 1

2.4 SUMMARY OF THE EVAPORATION WORK

The evaporation of a single water droplet in a finite system of its own vapour has

been investigated experimentally. For three different evaporation rates. the size and tem-

perature of the droplet. and the temperature and pressure in the surrounding vapour were

measured.

The results show that the temperature of the droplet is controlled by the pressure in

the vapour, which to a great extent is controlled by the adsorption process on the solid sur-

face of the system. The adsorption was independently investigated. It was found that the

solid surface. which is primarily stainless steel, can adsorb far more mass than a smooth

surface such as glass, and for this reason plays a dominant role in the evaporation process.

This is the effect of the finite system.

A temperature discontinuity has been predicted to exist at the liquid-vapour inter-

face. The predicted discontinuity correlates well with the measurements of steady evapo-

ration experirnents, and disagrees with the predictions of Classical Kinetic Theory. The

results of the evaporation experiments have been anaiyzed using Statistical Rate Theory.

and good agreement is found between the measured pressure and that predicted by the the-

ory. The sensiüvity of the SRT expression to the pressure in the vapour has been further

investigated.

The square of the droplet size history agrees well with a linear fit, as is predicted

3 by the Dm-law. Including the temperature discontinuity has a significant effect on the pre-

3 diction of the rate by the Dm-law, and considerably better agreement is found with the

experimental results when it is considered. In order to accurately predict the rate of evapo-

ration using Statistical Rate Theory, the pressure in the system would need to be measured

to within less than 1 mPa, which is not possible with any known measurernent rquipment.

The thermodynamics approach used in the development of the theory is not con-

ventional. A brief review of the postulates and the important basic relations is presented in

Appendix H.

3.1 DESCRIPTION OF SY STEM

The system of interest is an isolated system composed of a composite system sur-

rounded by a reservoir as shown in Figure 3- 1.

I rese rvo i r

Figure 3- 1. A one component droplet in its own vapour. The walls are diathermal. risid and non-permeable. The surrounding reservoir is at a constant temperature.

The reservoir has a constant temperature of rR. The walls of the composite systern are

rigid, non-permeable and diathermal. Therefore, when the isolated systern is at equilib-

riurn, the composite system will have the same temperature as the reservoir which will be

denoted by T. The composite system consists of a single one-component spherical droplet

of radius R in its own vapour. The total number of molecules in the composite system is

NT. The volume of the composite system is V , and the solid surface has an ara of A ~ .

Field effects are ignored, and thermophysical properties are assumed to be only a function

of temperature.

Based on the description of the composite system, it can be inferred h m the ther-

SV modynamic postulates that its independent variables are T, V , , A and N T . These are

the independent variables of the Helmholtz potential, F, which is discussed in Appendix

H. For a bulk phase with r components.

where P is the pressure and pi is the chemical potential of the i th species. For a surface

phase,

where y is the surface tension.

The Helmholtz. potential is the appropriate thermodynamic potential to use for the

analysis of this problem. A minimum of F corresponds to a stable equilibrium state and a

maximum corresponds to an unstable equilibrium state.

3.2 DETERMINATION OF THE EQUILLBRIUM CONDITIONS

For the system under consideration.

where the superscripts V , L , L V and SV correspond to the vapour phase. the liquid

phase, the liquid-vapour interface and the solid-vaporir interface respectively. By consider-

ing Eq.(3-1) and Eq.(3-2). Eq.(3-3) can be written as

V V V V L t L L LV LV LV LV SV SV S\/ Sr; F = - P V + p N - P V + p N + y A + p N + y A +;i iV ( 3 - 4

The necessary conditions for equilibriurn States are defined where the ton! differcntiat of

F p e s to zero:

d~ = - P " ~ V ~ - v V d p V + &i~'+ ~~d~~ - pLdvL - v L d p L + + (3-5)

N'dp' + y L V d A ' ~ + + + NLL.'dp'V + Y S V d A S v + ,45b''I.ISC

For a system at constant temperature, the Gibbs-Duhem relation. Eq.(H-22). can be writ-

ten for a bulk phase and a surface phase as

-VdP+Ndp = O

and

Ady + Ndv = O

respectively. W~th these two equations, Eq.(3-5) becomes

d~ = - pVdvV + &d'- pLdvL + y L d ~ L + y L V d ~ L V + p L Y d ~ L V + y S V d ~ S Y (3-8)

The system constraints are

= NT

and

w here C is a constant. These relations give

dvV = -dV L

d ~ ' = -dNL-dNLv-dNSV

and

d ~ ' ~ = O

The droplet is spherical, and thus.

and

7 = 4nR-

and therefore ,

and

2 d v L d~~~ = 8rcRdR = - R

(3-15)

Substituting Eq.(3-12). Eq.(3-13), Eq.(3-14), Eq.(3-17) and Eq.(3- 18) into Eq.(3-8) gives

For virtuai displacements about the equilibrium state. dF goes to zero. and the equilib-

rium conditions must then be

v L LV SV P = C L = C I = CL

and

The first condition states that there is no net rnass transfer between the three phases. The

second condition is the Laplace equation, which describes the mechanical equilibrium

between two fluid phases separated by a curved interface. For the droplet. the liquid pres-

sure will aiways be higher the vapour pressure as the radius must be greater than zero.

Thermal equilibrium has been assumed with the use of the Helmholtz potential.

Solving these two equations sirnultaneously will give an expression for the equilib-

rium radius. This radius is often called the critical radius. and is denoted by R, .

If the liquid phase is approximated as a slightly compressible liquid. and the

vapour phase is approximated as an ideal gas then their chernical potential can be

expressed as.'*

and

V v v p (T, P ) = p (T, P , ( T ) ) + kTln

respectively, where the subscript x refers to saturation conditions. rv is specific volume

and k is the Boltzmann constant. The chemical potential of the liquid and vapour phases

are equai at saturation conditions4. That is,

L P (T9 U T ) ) = c('(T, P , ( T ) ) (3-24)

Wïtirh Eq.(3-ZZj, Eq.(3-231, and Eq.(3-24), the equivalence of the liquid and vapotir phase

chernicd potentials in Eq.(3 -20) c m be sirnpli fied to

L Al1 fluid properties are assurned to be evaluated at T unless noted. Substituting in for P

from Eq.(3-21), an expression for the cntical radius is found

C' Generally, the first term in the denominator is much Iarger than the second. If ( P x - P )

is ignored, Eq.(3-26) can be recast as

which we recognize as the Kelvin relation, Eq.(l-3).

It is desirable to have the expression for the critical radius in tems of the indepen-

dent variables. As the vapour has been approxirnated as an ideal Cas,

By considering Eq.(3-9) and Eq.(3- IO), and &y neglecting the mass of the Iiquid-vapour

interface, Eq.(3-28) can be written as

The number of molecules in the liquid phase is

With the Iiquid volume as given in Eq.(3- H), this can be expressed as

Therefore. with Eq.(3-29) and Eq.(3-3 1), the vapour pressure may be w ritten as

In Eq.(3-26) and Eq.(3-32) we have two equations in four unknowns: R, . R . N ~ " and

v P . At equilibrium, we require that R, be equal to R , which reduces the problem to two

equation in three unknowns. It is imponant to distinguish between R and R, . The critical

radius, R, is a thermodynamic property of the system, and is only a function of its inde-

pendent variables. It defines an equilibnum condition, and exists whether or not there is a

droplet actually present in the system. The radius of an existing droplet, R . can take on

any value as long as it satisfies the system constraints. In order to close the series of equa-

tions, a model of the solid-vapour interface must be considered.

3.3 MODELING THE SOLID-VAPOUR INTERFACE The solid-vapour interface will be modelled using the BE3 isotherm. which was

presented in Section 2.22. While the use of this rnodel near saturation conditions is ques-

tionable. there are no other models which provide a significantly better agreement with

experimental observations. and are suitable for the analysis that will be performed. Any

model which predicts that filmwise condensation will occur at a pressure Pr will exhibit

similar behaviour. Recall that the B€ï model predicts the amount of adsorption that will

take place on a smooth homogeneous solid surface at isothermal conditions as a function

of the pressure in the vapour at equilibrium:

As it is an equilibrium model. the chemical potential of the solid-vapour interface and the

vapour phase will be equal:

I*Sv = p Y (3-34)

which is the sarne as part of Eq.(3-20). one of the equilibrium conditions for the system

being investigated. Thus, results predicted by the BET approach can be used for equili b-

rium calculations. Eq.(3-32) and Eq.(3-33) are two equations with the three unknowns

v tVSV, P" and R . By specifying R , they can be solved to %ive P and hlSV as functions of

R .The critical radius c m now be expressed as

SV Rc = Rc(T, Vo , NT, A R )

With the requirement that R, be equal to R in an equilibrium state, the equations have

been closed. Ic is not possible to solve the equations algebraically. A graphical solution

must be used. Before investigating this, the Helmholtz potential will be formed.

3.4 FORMATION OF THE HELMHOLTZ POTENTIAL

The Helmholtz potential can be expressed as wrinen in Eq(3-4) . As i t stands. there

is no way of ploning F vs. R . However, this problem can be resotved by introducing a ref-

erence condition F,. Let F, be defined as the state where there is no droplet present in

the system and the vapour is in equilibrium with the adsorbed phase. Thus.

w here

SV v and No and No are the number of molecules in the solid-vapour interface and the

SV vapour phase respectively in the reference state. iV, can be found by solving Eq.(3-32)

and Eq.(3-33) with R equal to zero. is the solid-vapour surface tension in the refer-

ence state.

As the two phases are in equilibrium in this state,

SV c ~ ~ ( T ~ PU) = p ( T , yOV)

and by conservation of mass,

N:+ = N T

and therefore,

SV v SV SV F , = - (N, - N, ) k ~ + p (T, P , V ) N ~ + y, A

Subûsicting F , from F gives

As the vapour has been approximated as an ideal gas,

v t' v p," p (T, Po 1 = p t ' ( ~ , P + k T h ~ ( ~ )

and by applying Eq.(3-9) and Eq.(3- 10), E q . ( 3 4 ) can be sirnplified to

where

= p L V ( ~ )

and

SV p = C I S v ( ~ , y S v ) .

At equilibrium, Eq.(3-20) and Eq.(3-21) can be applied, and by considering Eq.(3- 15) and

Eq.(3- 1@, Eq.(3-42) reduces to

' t v 3 LV 8 x R y SV F - F , = 41tR-1 - (3-43)

3 4

v v SV where R, . Pu . P and N, have been defined previously.

To determine an expression for the last term in Eq.(3-43) in terms of the system's

independent variables, a rnethod developed by Ward et. al? is used. For a solid-vapour

interface under isothermal conditions the Gibbs Duhem relation may be written as

As the chemical potential in the solid-vapour interface is equal to that of the vapour. and

the vapour has been assumed to behave ideally, Eq.(3-44) can be written as

With N'' from Eq.(3-33), this expression can be integrated to give

and the Helmholtz potential can be expressed as

SV F - F , = F - F , ( T , V , , N T , A , R ) .

3.5 INVESTIGATION OF THE BEHAVIOUR OF THE SY STEM AS

A FUNCTION OF THE REFERENCE PRESSURE

3.5.1 THE REFERENCE PRESSURE IS EQUAL TO THE SATURATION PRESSURE

Recall that the reference pressure Pr is the pressure at which filmwise condensa-

tion will occur on the solid surface. The value of the reference pressure Pr used in the

BET isotherm is usually chosen to be the saturation pressure for the isotherm temperature.

at satura- This cornes from the basic thermodynamic idea of a two phase system existin,

tion conditions.

With this assumption, piots of Eq.(3-26) and Eq.(3-43) c m be made. In Figure 3-2.

SV a plot of R, vs. R is shown. The values of T , NT . V , and A are chosen as 27 OC. 4.2

x 10" rn,, 0.0041 m3 and 0.038 m' respectively. At this temperature. the saturation pres-

sure is 3567 Pa. These are values typical of the experimental investigation. Thermody-

namic properties for these calculations and al1 that follow are taken from Ref. 43. The

critical radius is always less than zero, and it will never intersect with the line

There are therefore no equilibrium states. Other values of the independent variables were

considered, and the result was always the same.

Droplet Radius (m)

Figure 3-2. The critical radius as a function of the droplet radius whrn Pr = P, . R, s O . and there is therefore no equilibrium size for R > 0 .

To understand why no equilibnum states exists. consider the equili brium condi-

tions that were denved. The equivalence of the liquid and vapour chernical potentials and

the Laplace equation were combined to give the Kelvin relation. It predicts that for an iso-

thermal, curved liquid-vapour interface with the liquid on the concave side. that the pres-

sure in the vapour at equilibrium rnust be higher than the saturation pressure. To account

for the equivalence of the chernical potentials of the solid-vapour interface and the vapour

phase, the BFT isotherm was used with Pr equal to P,(T) , which says that the pressure

in the vapour cannot be greater than the saturation pressure, or filmwise condensation will

occur. The two equilibrium conditions are rnutually exclusive, and it is therefore impossi-

ble for there to be any equilibrium states, stable or unstable. Note that the behaviour of the

system will be the similar if the reference pressure is Iess than the saturation pressure.

To understand how such a system might behave, consider the plot of F - F , versus

R that is shown in Figure 3-3. The difference between the Helmholtz potential and the ref-

erence condition is a mcnotonically increasing function of R , and always greater than or

equal to zero. The only equilibriurn state is when there is no droplet present. This is the

reference state F , .

At this point, it is useful to discuss the justification of the use of thermodynarnic

potentials away from equilibrium states. There is an implicit assumption in the theory

developed that the expression for F - F, is valid away from the equilibnum state. cven

though the equili brium conditions were used to form it. This is justified by the follow ing

argument: a value of R is picked and assumed to be an equili brium state. The potential is

evaluated at this value of R .The system is being allowed to corne to equilibrium whiie the

size of the droplet is constrained. This procedure is repeated over a suitable range of R

values. When this is complete, maximums and minimums of R can be identified as equi-

librium states, and the use of F - F, at these points is justified. It is also justifiable to

apply F - F , for virtual displacements about the equilibriurn states. The system rnust

have some way to move between equilibrium states. Physically. the droplet can gow or

shrink. In terms of the F - F, curve, there can be no extrema or inflection points between

the equilibnum states that have been established. The function must be either monotoni-

cally increasing or decreasing between these states. It is therefore reasonable to assume

that the system will follow a 'path' similar to that generated by the F - F , curve. For this

reason, thermodynamic potentials are often considered to be the dnving force behind the

movement of a system towards an equilibrium state. The slope of the potential curve can

be taken as an indication of how fast the system will move towards equili brium. Suppose a

droplet has a radius of0.09 m. When introduced to the system, it will irnmediately smrt

to evaporate, moving to the left on the curve. If F - F , is approximated as being linear

over small intervals for values of R between O m and 0.0009 m, a qualitative idea of the

rate of evaporation can be established. The slope for five such segments is listed in

Table 3-1. The driving force slows down tremendously as the droplet evaporates. It is

therefore reasonable to assume that the rate at which the droplet evapontes will also

decrease significantly. At some point, the rate may decrease to such a small value that the

droplet would appear to be in a steady state. where its size was not changing with time.

Very large time scales would be needed to observe any change. However. it is important to

note that this is not an equilibrium state.

Table 3-1 .The change in the average slope of the Helmholtz potential as a fiinction of the droplet radius.

1' To gain a better understanding of what is happening. consider the graphs of P

and N'' vs. R shown in Figure 3 4 . As with the F - F, curve. both of these plots have

two distinct regions, divided at approximately the same point. Starting again with a droplet

of radius 0.0009 m, as it evaporates, the pressure in the vapour increases until it reaches a

value just below the saturation pressure. After this, the pressure increases very slowly as

the droplet continues to evaporate. The pressure in the final state is 3566.77 Pa. The num-

ber of molecules in the adsorbed phase remains small compared to the total mass of the

system (about 0.01%) during the penod when the pressure is noticeably rising. Only when

O 0.0002 0.0004 0.0006 0.0008 0.00 10 Droplet Radius (m)

Figure 3-3. The Helmholtz function as a function of the droplet radius when Pr = Pz . Note the sudden change in the slope of the curve around R = 0.0008 m.

SV the pressure gets close to the saturation pressure does N become significant. This is

consistent with the BFT adsorption model. In this region. the adsorbed phase represents

about 10% of the mass in the system. The droplet is still evaporating. but the mass of the

vapour is staying almost constant. The mass is being transferred from the droplet to the

solid-vapour interface through the vapour.

Recalling the justification of the use of thermodynamic potentials away from equi-

librium States, in this case the solid-vapour interface and the vapour phase will be in equi-

librium at every value of R chosen. but the equilibrium state itself is changing. As the

Kelvin relation cannot be satisfied, there will always be some difference in the chemical

potential of the liquid and vapour phases. However, as the vapour pressure approaches the

saturation pressure, this difference will decrease, and the rate of mass flow will also

decrease. This is a more 'physical' explanation of why the Helmholtz potential suddenly


Droplet Radius (m)

Figure 3-4. The mass of the adsorbed phase and the pressure in the vapour as a function of the droplet radius when Pr = P ,

L v pts very Rat. To further illustrate this poinr, a plot of p - y vs. R is presented in

Figure 3-5. Note that

v v where P is calculated as described previously. Note that while - is always greater

than zero, indicating that the droplet will always evaporate, it is not a monotonically

increasing function. The evaporation rate will actually speed up once the droplet gets

below a certain size. With respect to Eq.(3-47), this is the point where the R term domi-

v nates over the P tenn.

1 1 L

0,0002 0.0004 0.0000 0.0008 0,0010 Droplet Radius (m)

Figure 3-5. The difference between the liquid and vapour chernical potentials as n function of the droplet radius when Pr = P, .

3.5.2 THE REFERENCE PRESSURE IS GREATER THAN THE SATURAIION PRESSURE

The choice of the saturation pressure as the reference pressure is arbitrary. Recall-

ing that for a liquid vapour interface to be in equilibrium, the vapour pressure must be

greater than the saturation value. It is a logicai step to check if the theory described to

mode1 the adsorbed phase is affected by raising the reference pressure above the saturation

pressure.

As discussed in Appendix J, a supersanirated vapour is in a metastable equilibrium

state. The event that will cause it to move away from this state is the nucleation of a liquid

droplet of a critical size. The critical size of the droplet increases as the supersaturation

v decreases, and at values of P /Pa very close to one, the probability of such an event is

essentially zero. The reference pressure chosen could be greater than the saturation pres-

sure, and condensation would not necessarily take place.

For the sarne independent variables used in the previous section. the curves of R, . ,LL' - I rV and F - F, as functions of R are replotted for

P, = P , + 0.4 Pa

in Figure 3-6, Figure 3-7 and Figure 3-8.To highlight certain aspects of the F - F, curve.

it is ploned twice. as the scales involved with the important feanires are different. There is

a significant difference compared to the results of the previous section. The critical radius

is now positive in a certain region, and one finds two solution points with the line

R, = R

There are thus two equilibrium radii. This is not unexpected. as both the Kelvin relation

and the BET isotherm can be satisfied as the pressure in the system can be above the satu-

ration value without the possibility of filmwise condensation. The equilibrium radii are

2.16 x 10" m and 5.87 x IO-' m. From Figure 3-8, the first equili brium state is unstable.

while the second is stable. The pressure in the vapour in the unstable equilibrium state is

3567.17 Pa and in the stable equilibrium size is 3567.01 Pa. Both of these pressures are

slightly above the saturation pressure, consistent with the Kelvin relation. and below the

v pressure at which filrnwise condensation will occur. The curves for P and NS" as func-

tions of R are almost identical to those found in figure 3-4.

O 0.0002 0.0004 0.0006 0.0008 0.00 1 O Droplet Radius (m)

Figure 3-6. The critical radius as a function of the droplet radius when Pr = P, + 0.4 Pa.There are two equilibrium States.

-2E-25 1 a . O 0.0002 0.0004 0.0006 0.0008 0.0010

Droplet Radius (m)

Figure 3-7. The difference in the chernical potentials of the Iiquid and vapour phases when Pr = P, + OAPa.

LE-IO

O 0.00001 0.00002 0.00003 0.00004 ~.otlûo5

Dmpiet Radius (m)

Droplet Radius (m)

Figure 3-8. The Helmholtz potential as a function of the droplet radius when Pr = P, + 0.4 Pa.The fint equilibnum States is unstable. while the second is stable.

3 - 5 3 CHANGING ONE OF THE INDEPENDENT VARIABLES WHEN THE

REFERENCE ~ S S U R E IS GREATER THAN THE SATURATION PRESSURE

For al1 of the cases investigated when Pr was greater than P, . the differencr

between the saturation pressure and the pressure at the stable equilibrium size was always

much less than 1 Pa. Experimentally, the equilibriurn vapour pressure could never be dis-

tinguished from the saturation pressure. W~th this in mind, it is possible to predict how the

equilibrium state will change if one of the independent variables is changed.

Suppose that the temperature of the system discussed in Section 35.2 is nised to

275 OC while V,, A'" and NT are held constant. The adsorption isotherm used is

assumed to be valid over a small range of temperatures without significant error. The dif-

ference between the reference pressure and the saturation pressure is constant at 0.4 Pa. As

the saturation pressure for a pure substance is a monotonically increasing function of rem-

perature, the saturation pressure used in the calculations will increase. For equilibrium to

be restored, the pressure in the vapour will need to rise to a value slightly above the new

saturation pressure. The only source of mass is the droplet, and it will evaporate until the

new equilibrium pressure is reached. The stable equilibrium size should decrease. and this

is what is seen. The mass in the adsorbed phase does not change considerabiy. A plot

showing F - F, vs. R for temperatures of 27 and 275 O C is shown in Figure 3-9. Also

indicated is the path that the droplet would follow as it moved between equilibrium States.

If the temperature was instantaneously changed, the thermodynarnic state of the system

would go from being at a minimum on the 27 O C curve to a point to the right of a minimum

on the 275 'C curve. The system wants to minimize F - F, , and this is accomplished by

the droplet getting smaller. If the system temperature was lowered, a sirnilar argument

could be made to show that the stable equilibrium size would increase.

-4E-6 1 a


Figure 3-9. The effect of raising the system temperature by 0 5 OC. The stable equili bium size decreases

Alternately, if T and NT were held constant while V , was changed. n sirnilar

approach c m be used to predict the direction of change of the stable equilibrium size. If

the volume was increased, the pressure in the vapour would decrease. For equilibrium to

be restored. the pressure would need to rise back above P , . The only way this can happen

is for the droplet to evaporate, and the critical radius will decrease.

Now consider the case of changing the temperature of the example system to 28.5

O C . The droplet will respond by evaporating, trying to raise the pressure in the vapour.

However. in this case. the droplet is not big enough to bring about the necessary pressure

change. It will completely evaporate, and the resulting equilibrium state will be a homoge-

neous superheated vapour in equilibrium with the adsorbed phase. A plot of the two

F - F, functions is shown in Figure 3-8. Note that for the 285 'C curve, there is a global

minimum when the radius is equal to zero, and at this point the curve has a slope of zero.

It is a stable equilibriurn state. This reinforces the idea that a stable droplet cannot exist

unless the vapour pressure is above the saturation pressure. A final state such as this could

also be obtained if the volume of the system was made sufficientiy large while keeping the

other independent variables constant.

4E-6 I I 1 1 I 1

O 0.0002 0.0004 0.0006 0,0008 0.00 10 Dropiet Radius (m)

Figure 3- 10. The effect of raising the system temperature by 15'C. There is no longer a stable equilibrium size. and the system will corne to equilibrium as a superhented vapour in equilibrium with the adsorbed phase.

A more theoretical understanding of this behaviour can be gained by considering

how the expression for the cntical radius, Eq.(3-26), changes when the temperature is

changed. A conceptual plot of the critical radius as a function of the droplet radius for

three different temperatures is shown in Figure 3-1 1. As the temperature increases. the

system goes from having two equilibrium states to having none. There is a transition point

where there is one equilibriurn state. This corresponds to an inflection point on the F - F(,

curve, and is therefore an unstabie equilibrium state.

At this point, there is no physical significance to the reference pressure. It is cho-

sen as the pressure at which condensation would start to occur, which is unknown. For any

no equilibrium states >

one eqtrilibn'um state

/ two eauilibrium states 'd increasing temperature

Droplet Radius

Figure 3-1 1. The effect of changing the system teniperature on the numbrr of equilibrium states. As the temperature increases. the system shifts from hnving two equilibrium states to having none.

given set of independent variables. there is a minimum value of the reference pressure that

will brhg about the stable equilibrium size. For a temperature of 27 'C and a volume of

0.00041 m3, a plot of the difference between the reference pressure and the saturation

pressure required for a stable equilibrium state as a function of the mass in the system is

shown in Figure 3-12. As the mas in the system decreases, the required difference

increases asymptotically. There is a minimum mass that the system must have to allow for

a stable equilibrium state to exist. This is andogous to the discussion of Figure 3- 1 1 .

where the effect of changing the system temperature was shown to affect the number of

equilibrium states. As the mass of the system increases beyond this minimum value. the

pressure difference required decreases quickly.

Figure 3- 12. The difference between the reference pressure and the saturation pressiire required for a stable equiiibriurn size to exist as a function of the mass of the system.

3.6 THE EFFECT OF NEGLECTING THE SOLID-VAPOUR INTERFACE

If no adsorption is assumed to occur, Eq.(3-26) and Eq.(3-43) sirnplify to

and

where

and

In this case, the reference state is defined as where al1 the mass is in the vapour. This is the

result obtained by Rao and ~erne" and ~ o ~ e l s b e r ~ e ; ~ . It can be investigated in the same

rnanner as done in the previous section. The same values of the independent variables are

used. Plots of R, and F - F, as functions of R are shown in Figure 3- 13 and Figure 3- 14.

1 E-9 1 E-8 1 E-7 L E-6 I E-5 1 E 4 1 E-3 1 €2 Droplet Radius (m)

Figure 3-13. The critical radius plotted as a function of the droplet radius when the adsorbed phase is not considered. There are two equilibriurn States.

Droplet Radius (m)

O 0.0002 0.0004 0.0006 0.0008 0.0010

Droplet Radius (m)

Figure 344. The Helmholtz potential for the finite sized system without consideration of the adsorbed phase plotted as a function of the droplet radius. R, , is at a maximum. and corresponds to an unstable equilibrium state. RC2 is ar a

minimum, and corresponds to a stable equilibrium state.

The behaviour is similar to the case where Pr is greater than P , considered previously.

The R, curve is shown in more detail than that in Figure 3-6 to show the existence of the

two equilibrium States.

In the limit of

and

Eq(3-48) through Eq.(3-5 1) reduce to

' LV F - F , = 4nR-y - 8 n ~ 3 y LV

3 R c

and

This is the systern that was investigated by ~ i b b s ~ ~ . and corresponds to a case where the

vapour pressure is held constant. Note that the expression for the critical radius is explicit

in tems of the system parameters. This problem is formulated and discussed in Appendix

1. The correct thennodynarnic potential to consider in this case is actuaily the D potential.

which is given by

D = U - T S + P ~ V (3-55)

v Plots of R, as a function of P and D - Do as a function of R at P" q u a i to

4245 Pa for this system are shown in Figure 3-15 and Figure 3-16.

Figure 3-15. The critical radius of the droplet as a function of the pressure in the vapoiir

in Gibbs' initial analysis of droplet stability. Note that R, > O only for P" > P,(T)

Radius (m)

Figure 3-16. The thermodynarnic potential for Gibbs' system plotted as a function of the droplet radius for a pressure of 4245 Pa. The only equilibrium state occun at R = R, and is a maximum. and is therefore unstable.

A companson of the three system considered is given in Table 3-2.

Table 3-Xompanson of the results of the three systems considered.

current work (solid- vapour interface

Ref. 41 (vapour is at a constant

pressure)

in a finite system,

NIA

Refs. 44-45 (finite svstem with non-

To develop and understanding of the behaviour of the three systems. first consider

the simplest case. which is the system investigated by Gibbs. From Figure 3- 15. as Rc.

C' must be greater than or equal to zero. an equilibrium state only exists w hen P is greater

than P, (T) . If this is true. the vapour will be supersaturated. The extremum in Figure 3-

16 is a maximum. and therefore corresponds to an unstable equilibnum state.

To get a physical undentanding of the unstable nature of the equilibriurn. recall

that Eq.(3-52) can be recast as the Kelvin relation.

where R, has been replaced by the physical radius R . Now suppose that R is equal to R,

and a molecule of vapour condenses on the surface. The radius will increase, and by the

Kelvin relation, the vapour pressure required for equilibrium will decrease. The systern

v responds to this by trying to decrease P .The only way this can happen is for more mole-

v cules to condense on the surface. However, in tiiis system, P is constant. In order to rees-

tablish equilibrium, the droplet tries to get bigger, but this has no effect. As well. as its size

increases. the equilibrium pressure required keeps decreasing. It will continue to grow

until the system has become a homogeneous liquid phase. A similar argument can be made

for the case of one rnolecule evaporating from the surface. The final state in this case is a

homogeneous vapour phase.

For the finite sized system with non-adsorbing walls, RCI occurs at a maximum.

and is therefore unstable. This state is analogous to the unstable size discussed in the pre-

vious paragraph. This can be seen by considering Eq.(3-48) for

and

In this case, the expressions for the cntical radius and the thennodynamic potential

becornes equivalent to those of Gibbs for the same temperature and a pressure of PO. For

v this case, Po is equal to 4245 Pa, the value used as the reservoir pressure in the calcula-

tions for the first system. The values of the radius and the potential at the unstable equilib-

rium are the same in both cases. The finite size of the system has no effect and the vapour

phase acts like a pressure reservoir. If al1 the liquid were to completely evaporate. there

would not be a significant change in the vapour pressure.

The second equilibrium state, RC2 occurs at a minimum, which means that it is sta-

ble. The vapour is slightly supenanirated, which is consistent with the preliminary ideas

of a possible equilibrium state. The stable nature of this equilibrium state can be argued by

considering displacements of the Helmholtz potentiai about RCz and knowing that F is a

minimum in the stable equilibriurn state. Physically. the equilibriurn state is stable because

mass transfer to or from the droplet will have an effect on the system pressure. If a droplet

is in the stable equilibrium state and some liquid evapontes, the pressure in the vapour

will increase. To retum to equilibrium, the pressure must decrease, and this is accom-

plished by having some of the vapour condense on the droplet. bnnging it back to size

R C 2 . A similar argument can be made for the case of some condensation occurring on the

liquid surface.

While the behaviour exhibited in both finite systems is similar. the conditions at

both the unstable and stable equilibrium States are different. The unstable radius for the

system where there is no adsorption occurs at a radius four ordea of magnitude laqer than

that when the adsorbed phase is not considered. Accordingly. the value of the Helmholtz

potential is also much greater. This occurs because the pressure in the system is being

restncted by the adsorbed phase. The unstable equilibrium must occur at a pressure just

above the saturation value. It can still be considered as a bamer to nucleation. which in

this case will be impossible to overcome. The stable size when the adsorbed phase is con-

sidered is smaller, and this is because of the mass in the solid-vapour interface. The value

of the potential is accordingly smaller.

3.7 THE EFFECT OF A SECOND COMPONENT

In an experiment. there will always be some impurities in the system. For the case

of using water as the working Auid, the gas phase will consist of water vapour and possi-

bly air, and the liquid water phase may have air dissolved in it. The effect of a second com-

ponent which cannot condense can be anaiyzed in a sirnilar method to that shown in

Section 3.2 and Section 3.4. The amount of nitrogen adsorption on the walls of the experi-

mental apparatus has been found to be at least two orders of magnitude less than the

arnount of water vapour adsorption. It is therefore justifiable to ignore the mass of the sec-

ond component that will adsorb. For this reason, and the fact that the behaviour of the sys-

tem with no adsorption is the same when there is adsorption and Pr is greater than P , .

the theory developed in this section i s done for the simpler case. The following expres-

sions are found for the critical radius and the Helmholtz potential:

and

w here

and

and the subscript 1 refers to the working Ruid, the subscnpt 2 refers to the second compo-

nent, and KH is the Henry constant for the system. The ratio of IV,, to N I + iV2, wiil

be denoted by x . The gases are assumed to behave as an ideal gas mixture and the dis-

solved gas is treated as a weak solution in the droplet.

The values of R,, - for values of x between 0.2 and 1. for the values of T . \gr,, and

NT, used in the previous calcuIations are listed in Table 3-3. The ratio of the amount of

dissolved gas in the droplet to the amount of liquid present is less than 5 x 1 0 . ~ for al1

cases. The assumption of a weak solution is therefore valid. Also listed for each value of .r

is the difference between the partial pressure of component I and its saturation pressure at

equilibrium. The difference in the value of the stable equilibrium size w hen the second

cornponent is present compared to the case of having only the pure substance could not be

measured experimentally. The resolution of the droplet measurement in the experirnental

investigation is on the order of 0.01 mm. Any air present in the system will be due to leak-

age. In the experîmental investigation, the maximum arnount of air present is determined

to corresponds to a minimum r value of 0.995, and it is therefore reasonable to ignore the

effect of a second component.

Table 3-3. The effect of a second component on the equilibrium radius and the partial pressure of component 1 in the stable equilibriurn state

3.8 A SYSTEM WITH MULTIPLE DROPLETS The theory developed can be extended to a system of n droplets. vogelsbergcr*

states that Eq.(3-49) can be applied to a systern of n droplets by wtiting it as

4

where

This predicts that at equilibriurn, al1 the droplets will have the sarne size, and that the total

volume of liquid at equilibrium will be independent of 11 . Thus. the stable equilibrium

droplet size for a system with n droplets, R,, can be predicted if the stable equilibnum

size for single droplet, R, is known:

1/3 R r n = ri 4 . 1 (3-44)

From a thermodynamic standpoint, this is not correct. To keep consistent with the

work of Vogelsberger, the current theory is applied neglecting the adsorption. The physical

radius of each droplet can take on different values, and must be considered independently

in the formulation of R, and F - F , . Wlth this in mind. one finds

n LV 8xyLV " F - F , = 4ny c RL-2 R'-

3 R -

where

Thus. we have for the Helmholtz potential, that

F-F, = F - F , ( T , V , , N T , R I , ..., Rn)

and not

F - F , = F - F , ( T , V , , N , , R , > t )

as predicted by Vogelsberger.

For the case of two droplets, a three dimensional plot of F - F, vs. R , and R , is

shown in Figure 3- 17. The independent variables are the same as those used previously.

The surface has four extrema. There is a local maximum at A. a saddle point at B

and minima at C and D. We are interested in the minima, which correspond to stable equi-

libriurn States. The stable equilibrium state at both C and D corresponds to only one drop-

let being present in the system. The radius in this state is the same critical radius found

previously. There is no stable state where both droplets c m exist simultaneously.

The behaviour of the system can be predicted by considering the shape of the

F - F, surface. The Helmholtz potentiai is a minimum in a stable equilibrium state. and

will take the 'easiest route' to get there. The system will always move in the direction of

the ,-dient of the surface so as to decrease F - F, . If, initially, R , is greater than R , . the

stable equilibrium state will correspond to point D, where R I is equal to R,. , and R, is

0.00 io- 0

Figure 3- 17. The Helmholtz potential for a finite systern containing two droplets.

equal to O. Or, if initially R, is greater than R , , the stable equilibrium state will corre-

spond to point C, where R I is equal to zero and R, is equal to R , , . The surface is syrnmetric about the line where R , is equal to R, . Perpendicular to

the line, the surface always decreases. If the two droplets start with the same size. thermo-

dynamics is unable to predict the final configuration of the system, other than that it will

contain only one droplet of the equilibrium size. Which droplet takes on tliis size is not

known. Fiuctuations on the molecular level will determine which droplet is left. Once one

droplet gets bigger than the other, by the net transfer of one molecule from the vapour. the

rest of the path is defined.

The most important thing to take from this analysis is that the critical radius is a

thermodynamic property of the system, and only a function of the independent variables

T. V, and NT. It is not dependent on the initial state of the system. This result will be

used in Section 4.4 when the experimental results of La Mer and Ciruen3 are discussed.

3.9 SUMMARY OF THE THEORETICAL DEVELOPMENT

In this chapter, the stability of a single one-cornponent droplet was discussed for

three different situations.

For a droplet placed in a volume of its own vapour maintained at a constant pres-

sure. there is one possible equilibrium state that is unstable. This state is unstable because

mass transfer to or from the droplet has no effect on the vapour pressure. It exists only

when the pressure in the vapour is higher than the saturation pressure corresponding to the

system temperature.

For a droplet placed in a finite sized system of its own vapour. when there is no

adsorption on the solid surface. a second equilibrium state, which is stable, can exist. The

pressure in the vapour in this state is higher than the saturation pressure corresponding to

the system temperature. The difference between the two is found to be so small that it

could not be measured experimentaily. The stable equilibrium state exists because mass

transfer to or from the droplet has an effect on the pressure in the vapour.

To consider the adsorbed phase, the solid vapour interface was modeled with the

BFT theory of adsorption. With the standard assumption of the reference pressure used in

the mode1 taken as the saturation pressure, it is found that are no equilibrium States. This is

because it is impossible to satim the BET mode1 and the Kelvin relation at the same time

for this case. When the reference pressure is taken as being slightly above the saturation

pressure, the stable equilibrium size was once again found to exist. From thermodynarnics.

it can be shown that a supersaturated vapour is in a metastable equilibrium state. and that

for very low supersaturations the possibility of condensation is extremely small. It is

therefore justifiable to consider values of the reference pressure greater than the saturation

pressure. Any adsorption mode1 which predicts that filmwise condensation wili occur at a

specified reference pressure will exhibit the same kind of behaviour.

The presence of a small amount of a non-condensing substance was found to

reduce the stable equilibrium size. but the change was smaller than the optical measure-

ment equipment used in the experimental investigation could resolve. A system with mul-

tiple droplets was considered. and it was found that only one droplet can be present in the

stable equilibrium state.

A senes of experiments were run using the apparatus discussed in Section 2.1 to

investiga~e whether or not the prcdicted stable droplet s i x cxists. For al1 r2m the b a ~ h

temperature was set at 26.85 IO. 10°C. To prepare for an experiment, the needle was placed

in the test section, and the system was pumped to IO-' Pa for at least 12 hours. Based on

the theoretical calculations and the adsorption and evaporation experiments. the key issue

was to get enough mass in the system so the stable size exists for the system temperature

and volume.

4.1.1 METHOD

A droplet was placed on the thermocouple, and the test section was immersed. The

water in the degassing flask was heated, and used to raise the pressure in the test section to

3472 Pa. The main chamber was isolateci and the system was observed. Measurements of

the temperatures and pressure were made at 10 second intervals and the size of the droplet

was measured every five minutes. The initial size of the droplet was 1.36 mm.

4.1.2 RESULTS

The droplet completely evaporated. The total evaporation time was 24500 S. and

over the range where the droplet shape was symmetric, the K value was 0.640 x IO-'

mm2/s. With the slow evaporation rate there were penods of 35 and 25 minutes in the early

stage of the evaporation process where there was no measurable change in the droplet size.

The droplet temperature started at 2658 'C, and decreased during the evaporation to 26.10

OC. This is the opposite trend to what was observed in the evaporation expenments of Sec-

tion 2.3. During the evaporation, the pressure in the vapour decreased from 3472 Pa to

3364 Pa. This is also the opposite trend to what was observed in the evaporation experi-

ments. The pressure continued to decrease after the droplet had evaporated. The observed

decrease in both the liquid temperature and the pressure is consistent with the discussion

in Section 2.32.2, where it was proposed that the pressure in the vapour controls the drop-

let temperature. In this case, due to the high pressure in the system, adsorption dominated

over the evaporation process, and the overall effect was a decrease in pressure. The mea-

sured pressure in the vapour is in good agreement with the saturation pressure correspond-

ing to the liquid temperature during the evaporation process.

4.1.3 DISCUSSION

The stable droplet size was not observed. The initial pressure was 0.98 of the satu-

ration value, but there was not enough mass to bring the adsorbed phase into equili brium

with the vapour.

4.2.1 METHOD

A droplet was placed on the syringe. Its initial diameter was 1.47 mm. In order to

get more mass into the system, a large droplet was fomed on the syringe, and was placed

on the bottom window of the main chamber by vibrating the needle manually h m outside

the test section. It was hoped that the extra mass would bnng the system into a configura-

tion where the stable equilibnurn size would exist. As the bottom window is in direct con-

tact with the bath, any liquid placed on it should evaporate faster than the droplet. The

approxirnate rnass of the bulk Iiquid phase on the bottom window was 3 5 I 0.5 x 10-' kg.

If it were to completely vapounze, it would increase the systern pressure by 11830 Pa.

Note that the effect brought about by a droplet with diameter 150 mm is about 600 Pa.

The test section was immersed. and the pressure was raised to 3474 Pa. Measurements of

the temperatures and pressure were made at ten second intervals.

4.2.2 RESULTS

The droplet completely evaporated. but not before the liquid on the bottom had

fint disappeared. The behaviour in this test was different than in the first experirnent. In

Figure 4- 1. the square of the droplet diameter is ploned as a function of tirne. Only those

points where the droplet shape was symmetric are included. The data falls into two distinct

regions. Through each of these regions a best fit line was plotted. In the first region. the K

value is 0.276 x lo4 mm%. The evaporation was slow enough that there were periods of

up to 60 minutes where there was no measurable change in the size of the droplet. In this

region the diameter was measured every five minutes. In the second region. the K value is

0.616 x lo4 mm%. In this region, the diameter was measured every 30 minutes. While a

measurement of the amount of liquid on the bottom window of the test section as a func-

tion of time cannot be made, this liquid phase was observed throughout the evaporation. It

was around the time when it had cornpletely evaporated that the droplet began to evaporate

faster. Note that the dope of the curve for the second region is very close to that for the

first stability experiment. The total evaporation time was about 12 hours.

Time (s)

Figure 4- 1. The square of the droplet diameter as a function of time for the second stability experiment. Also plotted are best fit iines over the two distinct regions.

In Figure 4 2 . the measured temperature and pressure histories are shown. As with

the droplet diameter. the data falls into two distinct regions. separated by the point at

which the bulk liquid phase disappeared. In the fîrst region. the temperatures and pressure

al1 remain essentially constant. During this period, the amount of evaporation of the two

liquid phases rnust have been equal to the amount of adsorption onto the solid surface. In

the second region, the pressure and liquid temperature decrease in a manner very simiiar

to the first stability experiment. The observation of the relation between the liquid ternper-

atwe and the pressure in the vapour is even clearer than in the previous experiments.

While, in the previous cases, the two quantities were either rnonotonically increasing or

decreasing, in this case there is correlation during the steady region when there are srnaIl

fluctuations in the measurements. These fluctuations are likely brought about by the sec-

ond liquid phase. This phenornena is indicated on the graph. The results indicate that both

liquid phases are at the same temperature.

4.2.3 D~SCUSSION

While the stable droplet size was not observed, there were extended periods in the

observations nhere there was nu rneasurable cliange in the dropiet size. the rhree rempera-

tures or the pressure. While equilibrium has clearly not been established. these periods

could be considered as a quasi-steady state. and could be used as an initial condition for an

evaporation experiment with a suitable apparatus.

Based on the initial and final pressures, and the initial sizes of the droplet and the

liquid on the bonom window, the mass of Ruid in the adsorbed phase was almost three

times that in the vapour at the end of the experiment. No condensation was visible at any

time during the experiment. In order for this to be physically possible, capillary condensa-

tion rnust be occumng. This is consistent with the discussion of the adsorption experi-

ments of Section 2.2.

The theory developed in Chapter 3 can be used to predict that an increase in the

mass in a finite system will result in an increase of the stable equilibrium radius. Leakage

of water vapour into the system could therefore have the effect of slowing down the evap-

oration rate by increasing the equilibrium radius, thus bringing about the observed quasi-

steady state. To assess the effect of the leakage, the increase in the critical radius during

one of the quasi-steady states should be compared to the srnallest rneasurable change in

the droplet radius. For this experiment, the longest quasi-steady state lasted for 60 min-

utes. In this time, based on the results of the leak tests presented in Appendix C. the num-

ber of molecules in the system will increase by 95 x 10" mC. From the theory developed

26.8 -. bath

bottom droplet

Time (s)

Figure 4-2. The measured temperature and pressure histories for the second stability experiment. Note the similarity in shape between the liquid ternperature and vapour pressure curves,

in Chapter 3, with the independent variables taken as the values used in those calculations,

and the solid surface taken as Pyrex, this will result in a change in the stable equilibrium

radius of 0.007 mm. For that case. the ratio of the amount of leakage to the system mass is

0.002. In the real experiment, there is more mass in the system due to the presence of the

bu1 k liquid phase. In that case. the ratio of the amount of leakage in one hour to the sy stem

mass is 0.0006. This will cause the increase in the equilibrium radius to be less. Also. the

effect will be less as more mass cm be adsorbed on the solid surface of the test section

than on Pyrex. There is no quantitative mode1 by which the solid surface can be modeled.

but the increase in the stable equilibrium radius will most likely be less than 0.001 mm.

The smallest change that can be observed in the droplet radius is 0.01 1 mm. which is an

order of magnitude greater than the leakage effect. The change in the predicted stable

equilibriurn size due to leakage can therefore be neglected as a possible cause of the

observed quasi-steady state. The very slow evaporation rate is primarily due to the second

liquid phase. which causes the pressure in the vapour to be maintained near the saturation

conditions corresponding to the bath temperature, bringing about a small temperature dif-

ference between the droplet and the surrounding vapour,

4.3 THIRD EXPERIMENT

The stable equilibrium size was not observed in either of the tirst two experiments.

One possible expianation for this was that there was not enough rnass in the system. To

this end. a third experiment was run.

4.3.1 METHOD

A droplet was placed on the syringe. The initial droplet diameter was 1.64 mm. As

in the second experirnent, in order to get more mass into the system, liquid was piaced on

the bottom window of the main chamber. The approximate mass of the liquid on the bot-

tom window in this case was 1.35 * 0.25 x 1 0 ~ kg, and if it were to completely vapourize.

it would increase the systern pressure by 45640 Pa. The test section was immersed and

measurements of the temperatures and pressure were made at ten second intervals. The

droplet size was initially measured every five minutes. When the evaporation process was

observed to be very slow, the measurement interval was increased to 30 minutes.

4.3.2 RESULTS

Once again, the droplet completed evaporated. The total evaporation time for this

experirnent was 67 hours. The observed trends were very similar to those for the second

experiment, with the exception of the longer tirne scale. There were time periods of up to

two and a half hours when there was no measurable change in the droplet size. the temper-

anires or the pressure. For much of the initiai evaporation period, while the bottom liquid

was still present. the difference between the three measured temperatures was less than

O.1O0C. The ratio between the measured pressure and the saturation pressure correspond-

ing to the bath temperature during this time was always near 0.99. These observations are

consistent with the 'quasi-steady state' described in Section 4.2.3

4.3.3 DISCUSSION

The original hypothesis as to why the stable dropiet size was not being observed

was that there was not enough mass in the system to bring the adsorbed phase into equilib-

rium with the vapour. This may still be tme, but other factors must also be considered. A

very large amount of adsorption has been observed, which cannot be explained unless

there is capillary condensation in the system. Based on the initial amount of Buid in the

test section, the mass of the adsorbed phase at the end of this experiment is more than ten

times that of the vapour. No condensate was visible in the main chamber at any point. A

large part of the inside surface can be observed through the three viewing windows. Other

than the pressure transducer, ail of the inside surface is in contact with the bath water.

whose temperature does not fluctuate by more than 0.0S0C. If there was condensation in

the pressure transducer. this would have affected the pressure readings. and this does not

appear to be the case.

4.4 DISCUSSION OF THE STABILITY EXPERIMENTS In order to explain the observations, there rnust be capillary condensation on the

solid surface. Capillary condensation383g is the condensation of liquid in the pores of a

solid surface. The BET mode1 assumes that multilayer adsorption takes place in columns

of molecules. and that there is no interaction between the columns. While this rnay be a

valid argument for a smooth surface at low pressure ratios, for a surface which is not

smooth it would be very difficult to justify. To undentand the mechanism by which capil-

lary condensation takes place, suppose that a surface has a pore in which there are a few

columns of adsorbate As the colurnns grow, they may come in contact. If enough columns

come into contact with each other, they can combine to form a liquid surface, and a liquid

phase. In this way, a large rnass of liquid c m accumulate on the solid surface without film-

wise condensation taking place.

It is interesting to note that it would not be possible to have a droplet in equilibrium

with a system in which there was capillary condensation. The Kelvin relation cannot be

satisfied for both cases.The pores require a pressure less than the saturation pressure to be

in equilibnum. When the pressure rises to the saturation pressure for the system. filrnwise

condensation will occur. The droplet, for which the cu~a tu re of the liquid-vapour inter-

face is in the opposite direction. requires that the pressure in the vapour at equilibrium be

greater than the saturation value. If the reference pressure is less than the saturation value.

the stable size is not predicted to exist. If the reference pressure is higher than saturation.

the fluid in the pores cannot be in equilibrium. If it is indeed capillary condensation that is

taking place in the system. then it may be impossible to observe a stable droplet with the

current apparatus. One would need an apparatus with a smooth surface. where a traditional

adsorption model might be more appropriate.

The very slow evaporation rate observed in the second and third experiments is

somewhat consistent with the results of Section 35.1. where the stability of a droplet was

investigated when the reference pressure is equal to the saturation pressure. There it was

shown that the evaporation rate would be very slow. The experimental system is different

from the model in that there are two liquid phases present. and the adsorption mechanisrn

is different. However. in both cases, the slow evaporation rate is brought about by pres-

sures very close to the saturation value corresponding to the ambient conditions. Based on

the expenmentd results of both the evaporation and stability experiments. a pressure near

the saturation value will result in a liquid temperature near the ambient temperature. and a

slow evaporation rate.

Wtth the results of the theory developed and the stability expenments, some corn-

ment can be made on the experiments of Larner and l ni en^'. The addition of a srnall

amount of a non-volatile component to a droplet in an expanse of its own vapour kept at a

constant pressure can lead to the existence of a second equilibrium state which is stable".

The stable equilibnum size exists because the vapour pressure of the solvent required for

equilibrium is decreased by the presence of the solute to below the saturation value. In

their experiments, Lamer and Gruen observed a constant droplet size in a monodisperse

aerosol of dioctyl phthalate and benzene for penods of 30 minutes or more using light

scattering techniques.The droplets, whose radius was on the order of 1 p. were in a ben-

zene vapour phase which was in contact with a bulk liquid phase which contained the two

substances. While the authon make no comment with respect to stability issues. Defay et

interpret their observation as a confirmation of the stable equilibrium size. This is

questionable for two reasons. Fïrst. as shown in the second and third stability experiments.

the presence of a bulk liquid phase can have the effect of slowing down the rate of evapo-

ration of droplets to the point where there is no measurable change in size for up to two

and a half hours. Lamer and Gruen may not have observed their system for long enough to

see changes. Second, the theory developed does not take into account the interaction of the

droplets. As shown in Section 3.8, the effect of droplet interaction in a finite system is that

there can be only one droplet present in the stable equilibrium case. While the system

being modeled assumes a constant pressure in the aerosol, there will be local effects

between droplets of different sizes.

4.5 USING THE QUASI-STEADY STATE AS THE INITIAL CONDITION FOR AN EVAPORATION EXPERIMENT

As discussed in the Chapter 1. part of the motivation for determining if a stable

droplet size exists was to use it as the initial condition for an evaporation experiment.

While the stable size has not been observed, the quasi-steady behaviour observed could be

also be used as an initial condition.

4.5.1 DESIGN OF AN IDEAL EXPERIMENT

The experimental apparatus has been designed so as to allow the volume of the

system to be suddenly changed. ivhile keping the mass and tempzratuw sonaani. This is

accomplished by isolating the auxiliary chamber from the main chamber during an exper-

iment, and pumping it down to [O-' Pa. By then opening the valve between the two cham-

bers, there will be a sudden expansion and decrease in pressure with no change in mass.

while the temperature is controlled by the water bath. An investigation of the transient

period around this event has been done, and the results are presented in Appendix E. As

has been shown. the evaporation process is dominated by the pressure in the vapour. and

this change in the independent variables could be used to begin an evaporation process

from the quasi-steady state.

The presence of a second liquid phase, which will be referred to as the bu1 k liquid

phase, had been shown to retard the evaporation process to a point where there has been no

measurable change in the size of the droplet, the three measured temperatures or the pres-

sure in the vapour for penods of up to two and half houn. This can be interpreted as a

quasi-steady equilibrium, and could be used as the initial condition for an evaporation

expenment. Idedly, the bulk liquid phase would be larger in extent than the droplet. so

that one would not have to worry about it cornpletely evaporating when the steady state

was being established. However, if the procedure outlined in the previous paragraph was

followed, when the valve was opened, there wouid be a sudden drop in pressure. but part

of the second liquid phase would quickly evaporate, and bring the pressure back to near

saturation conditions. This type of behaviour has been observed on a nurnber of occasions.

Wth these ideas in mind, an ideal experiment can be conceptualized. A schematic

of the apparatus if shown in Figure 4-3. Initially, valve 1 is closed. and chamber A is under

vacuum. There is a droplet in charnber B, and valve 2 is open. The bulk liquid phase in

chamber C will cause the droplet to evaporate very slowly, and exhibit what has been

termed as a quasi-steady state. When this has been achieved, valve 2 is closed. and imme-

diately after. valve 1 is opened, bringing about an expansion of the vapour. and initiating

the evaporation process.

O droplet 1 bulkliquid 1

Figure 4-3. Schematic of an ideal experiment for observing the evaporation of a droplet from a well defined initial condition.

The current apparatus does not allow for such an experiment to be run. but has the

possibility to be modified accordingly. Using the current setup, a evaporation experiment

such as this was attempted using a modified version of the proposed method.

4.5.2 METHOD

The presence of a second liquid phase has been shown to slow down the evapon-

tion of the droplet, and produce a quasi-steady state. To use this successfully as the initia1

condition for a droplet evaporation experiment, there can be no other liquid present in the

system at the time when the evaporation is begun. To satisfy these conditions with the cur-

rent apparatus, an expenment was nin where a droplet was placed on the thermocouple.

and another droplet was placed on the bottom of the main chamber. M i l e the second liq-

uid phase is present, the rate of evaporation has been found to be slow, and when it disap-

pears, the results of the stability expenments show that the rate of evaporation will

increase. Therefore. if the expansion of the vapour were to take place at the same time as

the bulk liquid phase completely evaporated. it might be possible to take advantage of the

quasi-steady state .

4.5.3 RESULTS

The droplet size during the experirnent is shown in Figure 4-4. The initial droplet

diameter was 1.64 mm. All measurement are included. Readings were taken at rhree

minute intervalsJhe rime at which the bulk liquid phase evaporated and the expansion

took place is noted. In the 30 minutes leading up to the expansion. the droplet diameter

decreased by 1 pixel, or 0.0 18 mm. Following the expansion, the droplet evaporated corn-

pletely with a K value of 0597 x IO-' rnmZls over the range where the shape was symmet-

rict

The temperature and pressure histories are shown in Figure 4-5. Measurements

were taken at 10 second intervals. During the time when it was hoped to observe the quasi-

steady state, there are clezrly changes in the droplet temperature and the temperature and

pressure in the vapour. A summary of the measurements during this period are presented

in Table 4-1. The changes in the panimeters are srnall, but still measurable for T~ and P I ' .

The reason for the changes is that the adsorption was the dominant process during this

- Valve to awriliary

Tirne (s)

Figure 44. The measured size of the droplet. The sparsity of readings between 7000 and 12000 seconds is due to vibrations in the droplet caused by an unknown source which made it difficult to make the rneasurement.

time period, which caused the pressure in the system to decrease. and the rest of the

parameters were affected accordingly.

Table 4-1 .System parameters at the beginning at end of the quasi-steady time period. The elapsed time is 1926 seconds.

M e r the expansion, the pressure in the vapour quickly retumed to close to its ini-

tial value. The same is hue for the temperatures. In fact, if the region around the expansion

is ignored, both the measured pressure and liquid temperature follow a similar trend

before and after. Again, this behaviour is explained by consideration of the adsorbed

phase. There was sufficient mass in the adsorbed phase to bnng the pressure back up to

start

end

26.79

26.66

26.95 26.92

35 10

3480

1 64

1.63

Valve to auxiliary unber opened

1 bath

Valve to auxiliary 3600 - c arnberopened !

2900 O lm 20000 30000

Time (s)

Figure 4-5. The temperature and pressure histones for the evaporation experiment run using a well defined initial condition. Both the temperature and pressure respond quickiy after the expansion

near its initial value after the expansion, even though the surface area of the system

increased after the expansion. The extra mass in the adsorbed phase came primarily from

the bulk liquid phase.

4.5.4 DISCUSSION

To do this kind of experiment properly, the adsorbed phase needs to be in a near

equilibrium with the vapour. This could be accomplished by initially introducing a bigger

droplet on the bottom of the chamber. However, as has been shown. the mass of Ruid in the

adsorbed phase can be many times that in the vapour. When there is a signifiant amount

of adsorption on the system walls when the expansion takes place, the pressure in the

vapour quickly cornes back up to the value it had before, and the evaporation continues at

a slow rate. The goal of this type of experiment is for the evaporation rate be much greater

after the expansion. so that the assumption of the quasi-steady state can be justified.

4.6 SUMMARY OF THE STABILITY EXPERIMENTS An experimental investigation was undertaken to try and observe the stable equi-

librium size. A stable droplet was not observed. This is attributed to the nature of the solid

surface of the apparatus. As shown in Chapter 2, the stainless steel surface is capable of

adsorbing a large amount of water vapour. In these experiments, it was found that the rnass

in the adsorbed phase could be more than ten times that in the vapour, and still not be in

equilibrium at pressures near saturation conditions. It was not possible to get enough mass

into the system to bring the independent variables to values where the stable size rnight

exist. To explain the amount of adsorption, capillary condensation must be occumng on

the walls. If this is m e , then it was argued that it would not be possible to observe a stable

droplet in the current experimentai apparatus.

In the stability expenments, the evaporation rate of the droplet was slowed down to

such an extent that periods of up to two and a half hours were observed w here there w as no

measurable change in the droplet diameter and temperature, and the pressure and tempera-

ture in the vapcur. While this is not the stable equilibrium size that was desired for an ini-

tial condition in an evaporation experiment, it can be considered as a quasi-steady state.

and used for the same purpose. An experiment such as this was attempted. but was unsuc-

cessful due to the limitations of the current apparatus and the effect of the adsorbed phase.

CHAPTER 5: SUMMARY AND CONCLUSIONS The evaporation and stability of a single, stationary one-component droplet in a

finite system have been investigated theoretically and experimentally. The effect of the

finite size of the systern is rnanifest through the adsorption of vapour ont0 the solid sur-

face. In both the theory and the experiments, the sdsorbed phase is found to play a signifi-

cant role in the behaviour of the systern.

The pressure in the vapour during the evaporation process is a function of both the

evaporation of the droplet and adsorption on the solid surface. At higher pressures. more

mass is found to adsorb, reducing the rneasured change in pressure for a given rxperiment.

The measured pressure in the vapour is found to correlate very strongly with the saturation

pressure corresponding to the measured temperature in the droplet. In this way. the pres-

sure in the vapour controls the temperature of the liquid.

Using the measured temperatures. the temperature in the vapour at the interface

has been predicted to be higher than that in the liquid. Temperature discontinuities

between 0.4"C and 1.4'C have been predicted. This result is consistent with those of

steady evaporation expenments. To try and correlate the results, the temperature disconti-

nuity was plotted as a function of the measured pressure in the vapour as shown in

Figure 2-22. There is no overlap between the range of pressures for the data sets, but a

generai trend is evident.

Statistical Rate Theory can be used to predict that the rate of evaporation at a liq-

uid-vapour interface will be a function of the temperatures in the liquid and vapour at the

interface. the pressure in the vapour at the interface and the radius of curvature of the inter-

face. Previous work has shown that the mass flux is most sensitive to changes in the

vapour pressure. With this in mind, the experimental values of the four other parameten

were used to predict the pressure in the vapour, and the result was compared to the mea-

sured pressure. As shown in Figure 2-17, Figure 2-18 and Figure 2- 19, over 80 sets of

measurements from three different experiments, the largest difference between the two

values was found to be 9 Pa (less than 05% of the measured pressure). which is well

within the uncertainty of the pressure measurement. The difference between the predicted

pressure in the vapour and the saturation pressure corresponding to the liquid temperature

was found to be on the order of 0.1 Pa. These two pressures could never be distinguished

experimentally. Based on this observation, the SRT expression for the mass flux may actu-

ally be most sensitive to the difference between these two pressures and not just the pres-

sure in the vapour itself.

In the evaporation experiments, the square of the droplet diameter as a function of

7 tirne displayed a strong linear trend. as is predicted by the D--1aw. The dope of these

7 curves predicted by the D- -law (Eq.(2-27)) was found to agree well with the predictions

when the temperature discontinuity was taken into account. While the predictions of SRT

have been shown to be consistent with the measurements. in order to use it to predict the

evaporation rate, the pressure in the vapour would need to be measured to within less than

1 mPa (see Table 2-9), which would not be possible with any known measurement equip-

ment.

To extend the evaporation work further, a logical step would be to solve the conser-

vation equations for droplet evaporation by including the possibility of a temperature dis-

continuity. The problern would need to be fomulated usinp the SRT expression for the

mass flux as a boundary condition at the liquid-vapour interface.

The stability of a droplet in a finite system has k e n modeled for a more physicd

system than previous investigations. The nature of the adsorbed phase is the dominant fac-

tor in detemiining how the system behaves. This aspect had not been previously consid-

ered. By speciQing the pressure at which filmwise condensation is predicted to occur. one

can significantly alter the behaviour of the system. For pressures up to and equal to the sat-

uration pressure, there is no possibility for an equilibrium state, as the Kelvin relation can-

not be satisfied. If this pressure is chosen as being above the saturation pressure. by less

than even 1 Pa, a stable equilibrium is predicted to exist. Examining this theory experi-

rnentally proved to be a difficult task. and the stable size was not observed. Dur to the

nature of the solid surface, it was not possible to Cet the pressure in the system to higher

than 99% of the saturation pressure. In these conditions. the amount of adsorbed mass was

found to be an order of magnitude greater than the mass in the vapour. without equilibrium

having been established.

Based on the experimental results, it is not possible to say whether or not the stable

equilibrium size can exist. Due to the difficulty in understanding the behaviour of a Auid

and how it interacts with a solid surface near its saturation pressure. any experimental

attempt would need io be carefully pianned. To fully undentand the behaviour of the solid

surface, and to be able to use it to predict how the system will behave, a better defined sur-

face is required. The existing surface is stainless steel, and due to its porous structure. can

adsorb a large amount of vapour through capillary condensation. As well as knowing how

the solid-vapour interface behaves at equilibrium, it would also be necessary to predict its

transient response. A smooth surface such as glass might be more easily modeled. A

smdler system may also be advantageous as it would reduce the arnount of adsorption that

wouid take place. A necessary first step would be to investigate the ceference pressure

closely, and determine whether or not it can be greater than the saturation pressure. Based

on the results of the experimental investigation, it is not surprising that there has been so

Little documented work in this area.

From the stability experiments came the observation of what was termed a 'quasi-

steady' state. It was found that in the presence of a bulk liquid phase, the rate of evrpora-

tion of the droplet became so slow that there was no measurable change in its size for peri-

ods of up to two and a half hours. Dunng this time, there was also no significant change in

the measured temperatures or pressure. The slow rate of evaporation cornes about because

the bulk liquid phase is able to rnaintain the system pressure near the saturation pressure

coi~esponding to the ambient conditions. resulting in srnall temperature gradients in the

system. Such a state could be used as a well defined initial condition for an evaporation

experirnent.

A 1 INTRODUCTION

A block diagram of the experirnental apparatus is shown in Figure A-l . Arrows

indicate the possible directions of mass flow. The test section is the central component of

the apparatus. It is surrounded by the water bath, and has connections to the valve assem-

bly, the ultra-high vacuum (UHV) system and the syringe pump assembly. The valve

assembly connects to the degassing Rask assembly and to the vacuum pump assembly.

There are further connections between the subsections as noted. The letters A through G

are used to specify some of the connection points. and will be seen in more detail in subse-

quent sections. Also indicated are the temperature measurernent system. the pressure mea-

surement system and the optical measurement system. The subsections are mounted on or

located near the support frame. Al1 fittings and valves used are made of stainless steel

unless noted.

A2 THE TEST SECTION

A schematic of the test section is shown in Figure A-2. The test section is predom-

inantly made of Conflat finings and flanges. These components are made of 304L stainless

steel and are designed for use in U W systems. While the experiments do not require a

W environment, the modularity of the fittings and flanges available makes them very

useful. The Contlat components are fastened together with six 6.35 mm bolts and a seal is

established by cornpressing a copper gasket between adjoining flanges. The al1 metal seal

is able to hold pressures of less than 1 0 - ~ Pa.

temperature and pressure

I

- - - + - - assemblv + - T g - - -

- I

test section t t vacuum pump '4 assembly 1

Figure A- 1. BIock diagram of the experimentd apparatus.

Fi y r e A-2. Schematic of the test section.

The test section has two main parts: the main chamber and the auxiliary chamber.

The auxiliary chamber is a cross which has a 69.85 mm Conflat Range at the end of each

a m . The inside diameter of the tubing is 38.10 mm. Unless otherwise specified. al1 Con-

Bat cornponents have these characteristics. The bottom flange is blanked off. The top

Range connects to a UHV valve. The valve seals with a copper gasket. The other side of

the U W valve is connecred to the ultra-high vacuum system. The left Range connecrs to a

19.05 mm thick spacer flange whose other side is connected to a blank Aange. Part of the

spacer Range has been rnachined to a flat surface and tapped. A similar design concept is

used in an number of other places in the test section. There is a groove cut in the spacer

Aange above the tap where an o-ring can be placed to assure a good seal. The rubber wi l l

not provide as tight a seal as the copper gaskets, but is sufficient for the purposes of the

experiments. A Varian Type 053 1 thermocouple pressure gauge is attached to the auxiliary

chamber at this point. Its measurement is displayed on an NRC 80 1 dia1 gauge. The blank

fiange outside the spacer Range has been tapped for a 3.18 mm bal1 valve. This valve con-

nects to the valve assembly and will be refened to as valve 1. This is connection point A. I t

is through this connection that vapour can enter the system from the water supply. and

where the system can be purnped down with the vacuum pump. The right flange is con-

nected to a flange which converts from 69.85 mm Conflat to 6.35 mm Swage-Lok. A 6.35

mm bal1 valve is connected to this flange by way of a Swage-Lok fitting. On the other side

of the valve is a simiiar Conflat to Swage-Lok conversion flange, which is mounted on the

left flange of the main chamber. The valve allows the main chamber to be isolated from the

auxiliary charnber.

The main charnber is a double cross. There are six Banges to which components

can be attached. As indicated, the left flange is used to link the main chamber to the auxil-

iary chamber. The front and back Ranges have viewing windows mounted on them. It is

through these windows that the droplet is obsewed. The bottom flange also has a viewing

window mounted on it. In some cases, a liquid droplet may fall or be placed on the Win-

dow during the setup of an experirnent. The window is used to monitor the amount of liq-

uid present, and also to check for condensation during an experiment. Two 25.40 mm thick

spacer flanges are mounted on the top Range, which is sealed with a linear-rotary motion

feedthrough. On the vacuum side, the feedthrough has a 6.35 mm diameter rod whose ver-

tical position and angle of rotation can be independently controlled. There is unlimited

rotational ability and 50.80 mm of linear motion. Two thennocouples are mounted on the

feedthrough as shown in the picture in Figure A-3. They will be referred to as the droplet

. The configuration of the droplet and vapour thermocouples.

thermocouple and the vapour thermocouple. The sizes of the two spacer flanges were cho-

sen so that the range of motion of the thermocouples mounted on the feedthrough would

correspond to their being visible though the viewing windows. The thermocouples are

type K and were made using a TL1 Select Amp Welder. The thermocouple wires have a

123

diameter of 0.08 mm. The beads on the droplet and vapour thermocouples are approxi-

mately spherical with diameters of 0.35 mm and 0.40 mm respectively. The distance

between the beads is 4.30 mm.

The droplet thermocouple was chosen for the way in which a liquid droplet hangs

on it. Approxirnately ten thermocouples were made initially, and each tested to determine

which one would be used. The welding procedure does not offer high repeatability. The

criteria used to select the droplet thermocouple were:

The bead needed to be big enough to support a droplet of diameter 1.80 mm. but small

enough that it would not occupy a significant volume until the droplet diameter was less

than 0.9 mm, at which point the shape of the droplet begins to deviate significantly

from being spherical and symmetric. Beads of diameter 0.30 mm to 0.45 mm were

found to satisfy these conditions.

The thermocouple had to have a high success rate for the transfer of a droplet from the

needle. While this is a function of the bead size. it is also dependent of how the bead is

oriented on the thermocouple wires.

The position of the droplet on the thermocouple as it evaporated had to meet certain

requirements. The thermocouple bead had to be fully immersed in the liquid at al1 tirnes

to ensure that the temperature read corresponded to the liquid temperature. In Figure A-

4, the droplet-thermocouple bead configurations that are seen in an experimental run

are shown. Generally, when a droplet is first placed on the bead, it adopts configuration

A, where it is hanging only on the bead. As it evaporates and gets smaller, gravitational

forces decrease, and surface tension forces become important. The droplet starts to

slowly move up the thermocouple wires. forcing the bead further into the liquid as

shown in configuration B. The shape of the droplet should be affected as little as possi-

ble by this movement. Eventually, the droplet will reach a point where the bead is at its

bottom as shown in configuration C. Evaporation will continue with a spherical shape

until the diameter reaches around 0.90 mm. The exact point where the sphericity is lost

varies between experiments, and is judged by visually inspecting the images taken. This

will be further discussed in Appendix B.

A B

Figure A-4. Configurations of the droplet on the thermocouple.

The themocouple chosen met the above cnteria. Occasionaiiy. the initial configuration of

the droplet on the bead will not meet the third criteria, and part of the bead will not be

immersed. This happens if the droplet is initially srnaIl.

The four thermocouple wires are stripped of their insulation only around the beads.

It is not desirable to have any contact between the wires and another electncal conductor

other than at the bead. The insulated wires are bonded to the bottom of the feedthrough

with Torr Seal vacuum epoxy. The wires run up the feedthrough, and are bonded to it again

near the top. The wires end shortiy past this point, where each is individually welded to a

type K thermocouple wire of diameter 0.25 mm. The weld is sealed with vacuum epoxy so

that no bare metal is exposed. Each of these wires is then removed from the system

through a 3.18 mm Swagelok union mounted on the upper spacer Range. The union is

sealed to the main charnber using the tapped holeto-ring senip descnbed before. The wires

are sealed with two layers of vacuum epoxy. First, with the insulation still attached to the

wires, the exposed end of the union was filled with vacuum epoxy and the four wires were

separated. The fint 5 mm of each wire coming out of the fitting was then stripped, and a

seal was made with epoxy around al1 the exposed bare wire, so that only insulated wire

was visible. This was done as it was found that air could enter the system through the gap

between the bare wire and the insulation. Some slack is left inside the main chamber to

allow for the rotation and translation of the feedthrough. The larger wires are used because

it was found to be very difficult to seal the smaller wires as described. The thermocouple

wires are connected to the temperature measurement system.

A 6.35 mm bal1 valve is mounted on the upper spacer Range using the tapped hole1

O-ring setup. An Omega PX8 1 1-OOSAV pressure transducer is mounted on the valve. The

valve aliows the transducer to be isolated from the system. and for the system to be kept

under vacuum if the transducer is removed. The transducer is connected to the pressure

measurement system.

A 6.35 mm bal1 valve is mounted on a blank flange on the right side of the main

chamber using the tapped holelo-ring setup. The purpose of this valve is to allow for pas-

sage of the needle through which Iiquid enten the systern while providing a seal between

the main chamber and its surroundings. This is accomplished by a two step seal comprised

of the valve itself and a septum. The septum is made of three layers of rubber and has a

total thickness of 3.9 mm and a diameter of 8.0 mm when uncompressed. The septum

diameter is slightly larger than the internai diarneter of the male connection pon of the

valve. It can be fit into this port, and then compressed and fixed in place by screwing on

the corresponding female fitting. The thread is sealed with nylon tape. Two washers are

placed on the inside of the female fitting to increase the surface area cornpressing the sep-

tum. This compression assures that a good seal exists when the needle is inserted through

the septum and when it is not present. It has been found that the same septum can be used

for a number of expenments without any effect on the quality of the seal. It is desirable to

pierce the same hole when using the same septum repeatedly.

In order to ensure clean surfaces within the test section. where possible al1 of the

Conflat fittings and Ranges were put through a cleaning procedure4g. The component is

soaked in acetone for 24 houn. It is then soaked in an Alconox detergent mixture for 24

hours. Following this. it is rinsed with high purity water, and then soaked in chrornic acid

for 24 hours. The component is again rinsed with high purity water and left to dry. When

dry, the cornponents are assembled and put under vacuum as soon as possible to prevent

contamination frorn the atmosphere. When the cleaning procedure was not possible. corn-

ponents from the test section or those which might be exposed to the working Ruid were

either rinsed with or soaked in acetone andior the detergent.

A3 THE SUPPORT FRAME

A schematic of the support frame and the vanous subsections of the appantus is

shown in Figure A-5. The support frame is an open steel table. Each Ieg has been extended

with a length of rectangular steel tubing which has a levelling screw at its base. An alumi-

num channel section is mounted across the top of the left section of the table. Three verti-

cal aiuminum support bars are attached to the bottom of the channel. The bottom of each

. Figure A-5. Schematic of the support frame.

of these pieces has been machined to fit over the top half a 69.85 mm Conflat flange. The

flange is held in place on the bar with a ring clamp. The three support bars are positioned

on the channel to intersect the test section at the left and right sides of the auxiliary cham-

ber and at the left side of the main chamber. The main chamber is supponed on its right

side by a length of stainless steel tubing that is attached to the channel.

A4 THE VALVE ASSEMBLY

A schematic of the valve assembly is shown in Figure A-6. It acts as a link

between the test section (connection point A), the vacuum pump assembly (connection

point B). the degassing Rask assembly (connection point C). the syringe pump assembly

(connection points D and E), and a nitrogen supply. The functionality of the valve assem-

bly will be discussed when each of those sections is described. It is made up of sections of

3.18 mm tubing, Swage-Lok tees and Swage-Lok bal1 valves. As indicated in the figure.

the valves will be referred to by the Roman nurnerals 11.111 and IV.

A5 THE VACUUM PUMP ASSEMBLY A schematic of the vacuum pump assembly is shown in Figure A-7. 1t is made up

of brass fittings. 6.35 mm brass valves and a dial pressure puge. The valves will be

referred to by the Roman numerals V through VI11 as noted. At connection point B. the

tubing from the valve assembly connects to a valve V. This valve can be opened and closed

to isolate the valve assembly from the vacuum pump assembly. Valve VI connects to the

vapour portion of the degassing Rask at connection point G. This connection is used when

the water is degassed. A connection to the vapour section of the degassing flask is not

to vdve V in the vacuum pump assembly

B

to valve X in the degassing flask assembly

C

II

A to valve I in icsi ssction

to valve XII1 in the syringe pump assembly

the

I to the nitrogen

Figure A-6. Schematic of the valve assembly.

made through the valve assembly, as it would involve a longer length of a smaller diameter

tubing. Valve VII connects to the inlet of the cold trap. The cold t n p is made of glass. and

sits in a styrofoam container that can be filled with dry ice. The outlet of the cold trap con-

nects to the vacuum pump. Valve VI11 is used as a vent to atmosphere. It is desirable to

degas as fast as possible. The dia1 pressure gauge is used as a rough indication of the vac-

uum being pulled by the vacuum pump. If the pump has been off and then restarted. it may

not be desirable to expose the rest of the system to it until it is pulling a sufficient vacuum.

to the valve assembly

B

to valve IX in the c v degassing flask

to the cold ûap and the vacuum pump

assembly 1

vent to atmosphere Figure A-7. Schematic of the vacuum pump

A6 THE DEGASSING FLASK ASSEMBLY

A schematic of the degassing flask assembly is shown in Figure A-8. Water is

stored in the degassing flask. It is made of glass, has an open top. and has a 6.35 mm diarn-

eter glas tube of lene@ 25 mm extending horizontally from near its base. There is a small

tefion covered magnet in the degassing flask. It is used during the degassing procedure to

pmmote the nucleation of air bubbles. The degassing Rask sits on top of a Tek Stir Hot-

plate H2395- 1 hotplate and magnetic stiner. Anached to the glass tube is an UItra-torr fit-

ting whose other end is a section of 635 mm diameter stainless steel tubing. This end of

the fining has been tapped so that a 159 mm Swage-Lok union can be screwed into it. It is

O dia1 pressure gauge

to valve VI in the

to valve XII in the syringe pump assembly

F rnagnet

r

stirrer-heater

- - - - - -- - - - . --

Figure A-8. Schematic of the degassing flask assembly.

sealed with nylon tape. A length of 1.59 mm stainless steel tubing is attached to the other

side of the union. It is through this tube that liquid water is transferred to the syringe pump

assembly. This is connection point H. The top of the degassing flask is sealed with a

tapered rubber cork that has a piece of 6.35 mm tubing going through it. The tubing con-

nects to a 6.35 mm Swage-Lok cross. A dia1 pressure gauge is mounted on the top of the

cross. Each of the horizontal branches of the cross connects to an 3.18 mm Swage-Lok

bal1 valve by way of a 6.35 mm to 3.18 mm union. These valves will be referred to as

valves IX and X. Valve IX connects io the vacuum purnp assembly. This is connection

point G. The valve is opened when the water is degassed. Valve X connects to the valve

assembly. This is connection point C. The valve is opened when water vapour is trans-

ferred to the test section. There is no circumstance under w hich valves IX and X would be

open at the same tirne. Unless an experiment is being run or the water is being degassed.

both valves should be closed to prevent possible air leakage into the degassing flask.

A7 THE ULTRA-HIGH VACUUM SYSTEM

The UHV systern is connected to the test section by the UHV valve. It is mounted

on a table next to the support frame. The important cornponents are the turbo-molecular

pump and the Pirani and Penning pressure gauges. Al1 the components are UHV fittings

and Ranges. On the UHV system alone, the turbo molecular purnp is capably of pulling a

vacuum of IO-' Pa. When brought into contact with the test section. it can pull a vacuum of

IO-' Pa. The difference is due to the O-ring and epoxy seals present in the test section.

Keeping the test section at this pressure ensures that the solid surfaces stay free from con-

tamination. Whenrver possible, the test section is purnped on by the turbo-molecular

pump. The turbo-molecular pump cannot pump down from atmospheric conditions. For

this reason, an initial evacuation by a vacuum purnp is required. For the ultra-high vacuum

systern alone, this is provided by a vacuum pump downstrearn of the turbo-molecular

pump. This vacuum pump also keeps the pressure at the outlet of the turbo pump around I

Pa. The initial evacuation of the test section is done by the vacuum pump in the vacuum

pump assembly. There is a therinocouple pressure gauge installed at the outlet of the turbo

pump. It is the same mode1 as the one mounted on the auxiliary chamber of the test sec-

tion. The reading provided by this gauge is used to determine when the system can be

turned over to the turbo pump. A pressure of 10 Pa is used as a crossover point. Due to its

long start up and shut down times. and a desire to keep it clean, the turbo-molecular pump

is kept on at al1 times.

The pressure in the ultra-high vacuum system. and in the test section when the two

are in contact, is measured by either a Penning or a Pirani gauge, both of which are con-

nected to a digital readout-controller. Each of the gauges is suited to a specific range of

vacuum conditions. and the controller switches between them as necessary. As there is no

way of cali brating the readings, they are only taken as being accurate to w ithin an order of

magnitude.

A8 THE SYRINGE PUMP ASSEMBLY

The sytinge pump assembly consists of an aluminum table. a three degree of free-

dom motion rnanipulator, a syringe pump, a syringe and an arrangement of supports.

valves, fittings and tubing. The motion manipulator is clamped to the table. which sits

inside the support frame. The syringe pump is attached to an aluminum frame that is

mounted on the motion manipulator. The syringe is held in place by a clamp on the pump.

The syringe has been significantly altered from its initial state for the purposes of

the expenment. Because the location where it is desired to form the liquid droplet is far

from the syringe body, and a system by which to fil1 the syringe with liquid water at satu-

ration pressures is required, the needle does not attach directiy to the synnge as its design

intends. The metal needle fitting has been removed, and the tefion on which it rnounts has

been dnlled through by hand so that a piece of 159 mm tubing can be inserted through it

into the shaft of the syringe. The portion of this teflon piece inside the syringe has been

tapered to where the tubing ends to encourage the expulsion of vapour and air pockets dur-

ing the syringe filling procedure. The tubing is sealed to the syringe with vacuum epoxy.

At times. the contents of the syringe and its accompanying valve assembly will be under

vacuum. It was found that the seal between the piston and the syringe wall was not suffi-

cient to prevent air from leaking in from the bottom. The bottom portion of the syringc has

been modified to allow for a connection to the vacuum pump, through the valve assembly.

This is connection point D. This allows the volume between the piston and the bonom of

the synnge to be kept under vacuum.

The tube coming out the top of the synnge connects to a valve assernbly. which is

shown in detail in Figure A-9. It consists of 1 J 9 mm sections of tubing. bal1 valves. and

vanous Swag-Lok fittings. The valves will be referred to by the Roman numerals XI. XII

O corn- and XIII. Valve XI connects to the needle. Valve XII connects to the 159 mm tubin,

ing from the liquid portion of the water supply. This is connection point H. Valve XII I c m

connect to the vacuum pump assembly through the valve assernbly (connection point E).

or can be used as a vent to atmosphere for flushing of the system. The valve assembly is

kept in position by an aluminurn angle mounted on the aluminum frame. The angle has

two holes in it where valves XI and XII can be fixed in place.

The needle is made of three telescoped sections of stainless steel tubing. The t in t

has an outside diameter of 159 mm and exposed length of 10 mm, and connects to valve

XII. The second section has outside diameter 0.90 mm. exposed length 120 mm and is

sealed to the first section with vacuum epoxy. The third section is the needle cut from a

metal needle fitting similar to what originally came with the syringe. The outside diameter

is 0.48 mm and the exposed leneth is 20 mm. It is sealed to the second section by vacuum

to the liquid pomon of degassing flask

F I

needle

E to valve assembly

draidvent to atmosphere

I to valve assernbly

Figure A-9. Schematic of the syringe valve assernbly.

epoxy. The needle has a 05 mm diameter hole on its side, 3 mm frorn its tip. This is w here

the droplet is formed. The hole is oriented upward. Only the second and third sections of

the needle enter the test section when an experiment is being setup.

A9 THE WATER BATH ASSEMBLY

The water bath is made of 12.7 mm thick clear colourless acrylic. The bonom and

four side walls are held together with stainless steel screws and sealed with silicone. Each

of these five walls is insulated with styrofoam.The insulation is not directly bonded to the

walls. The side and back pieces are held in place by the support frame and with duct tape.

The front piece is easily rernoved so that the test section can be observed. There is a srna11

mirror placed on the bottom of the bath below the bottom Range of the main chamber to

allow for observation of the inside of the test section through the viewing window

mounted on that fiange. A Lauda MS immersion heating circulator sits on the back risht

corner of the bath. The bath-insulation assembly sits on a piece of 4.76 mm steel that is the

same size as the bonom piece of insulation.

The test section is fixed in place. Therefore, to immene it. the bath must be moved

up and down. There are three tables of different heights that the assembly can sit on. The

tables are on wheels to allow for their easy placement and removal. The assembly is raised

and lowered between these heights with four MotoMaster 8 ton hydraulic jacks. The bot-

tom piece of steel is used as the jacks would simply break through the insulation if they

were brought into direct contact with it.The lowest table pu& the top of the bath below the

test section and in a position where it can be removed from the support frame. The middle

table is used when an experiment is being set up. The top of the right side wall of the bath

and its corresponding piece of insulation each have a notch cut it them to allow the needle

to enter the main chamber when the bath is in this position. This height was chosen so that

the height difference between the middle and top tables was less than the maximum range

of motion of the jacks. The top table is used when an experiment is being mn. The height

of the top table was chosen for two reasons. First, to rninimize the surface area of projec-

tions out from the bath. Second, to put the test section in a position where coveting it with

water will correspond to a situation where the circulator heating coils are fully immersed

and where the water level in the tank is as low as possible. It is not desirable to have to

and circulate any more water than necessary. The level of coverage was chosen as

being slightly above the top of the feedthrough flange. As well as trying to minimize the

arnount of water needed, it is not desirable to immerse some of the components such as the

pressure transducer. The front and back pieces of insulation have holes in them for the col-

limator and the CCD camera which are described in Section 1. L J O . It takes about ten min-

utes to raise the bath from the middle table to the top table.

As there are a number of things that must corne out of the bath (the connection to

the ultra-high vacuum system. the alurninum ban that attach to the support frame. the top

of the feedthrough and the wires for the thermocouples and pressure transducer). i t is not

possible to seal it cornpletely. The lid is comprised of a few pieces of acrylic and styro-

foam that cover most of the open area. Large gaps are filled with loose insulation. Distilled

water is used in the bath to prevent the growth of algae. There is a 952 mm diameter hole

on each of the side walls into which a tube fitting is rnounted. They are used for draining

the bath and for the possible attachment of an extemal circulator.

A 10 THE TEMPERATURE AND PRESSURE MEASUREMENT SY STEMS

A10.1 HARDWARE

The temperature and pressure signals are output from the test section as voltages.

The pressure signal goes to an Omega DWl-S display unit, which also excites the trans-

ducer. The display has an analog output which is connected to an Hewlett Packard 39702A

data acquisition system. The two thermocouples from the test section, and three more from

the bath and the surroundings are referenced against an Omega ice point, and the wiring

p e s directiy to the 39702A. The 39072A is connected to an Apple IIci cornputer by an RS

232 connection, and the data is collected and manipulated by a Labview virtual instru-

ment.

A10.2 SOFI-WARE

A number of Labview virtual instruments have been created which convert the

measured voltage signais to pressure and temperature readings based on know ri caii brütion

curves. The calibration of the thennocouples and the pressure transducer is described in

Appendix B. This data is continuously displayed in graphical and numerical formats and is

written to a file at a specified time interval.

There are two types of virtual instrument used to take measurements. The first is

used to record continuous measurements at time intervals of one second or greater for an

unlimited amount of tirne. The second type, which run as a sub-vi inside the first type. are

able to record measurements every 30 ms, and are limited time-wise by the buffer size of

the 39072A. This is referred to as a ' burst' .

A 1 1 THE OPTICAL MEASUREMENT SYSTEM

A significant portion of the experirnental procedure and analysis relies on being

able to see the thennocouples inside the test section. As discussed, they are very srnall. and

magnification is required. The steps taken in a pamcular expenment to transfer the liquid

from the needle to the droplet thermocouple are govemed by being able to see the config-

uration of those components. In ternis of the analysis, the size of the droplet is measured

ophcdly.

Light is provided by either a fluorescent Iight or a small light bulb and a bielles

Griot 25 mm collimator. The Auorescent Iight is used in the set up procedure as it provides

a wider field of illumination. The srnall bulb and collirnator are used when measurements

are taken, as they provide a more unifom source of light. The collirnator is mounted on a

three dimensional motion manipulator that is rigidly rnounted on the support frame. This

allows its position to be adjusted depending on the location of the droplet. or what is being

observed. The thermocouples are viewed with either a Spindler Hoyer telescope or a Cohu

solid state camera with a Canon FD 10014.0 Macro Lens and a Kenko No. 4 close up lens.

The telescope is used for some of the set up procedures. It is mounted on a three dimen-

sional motion manipulator that is rigidly mounted on the support frame. The motion

manipulator can resolve to 0.01 mm. As it has a much narrower focus range than the Cam-

era, the telescope is useful for positioning the needle with respect to the thermocouples.

The camera is also mounted on a three dimensional motion manipulator that is rigidly

mounted on the support frarne. The telescope and the camera cannot be in place on the

support frame at the same time. wth the lens used, the carnera has a wider field of view

and focus range than the telescope, which is advantageous in many situations. The camera

output is fed into an Appie Quadra computer and viewed with NIH [mage5'. The software

cm provide live footage of the setup, and capture images. This setup is used when the

droplet is placed on the themocouple and to take measurements of the droplet size. The

calibration and measurement procedure using the carnera and Image is discussed in

Appendix B.

B 1 CALIBRATION OF THE PRESSURE TRANSDUCER The pressure transducer has a rated error of O. 1% of its full scale. or d5 Pa. The

documentation indicates that it is free from hysteresis. The display can be calibrated to

show a reading of zero when the vansducer is put under a high vacuum. However. there is

a drift in the reading at al1 times which will cause an offset in the recorded measurement.

Part of this can be amibuted to the electronics. and another part to temperature effects. To

quanti@ the effect, the test section was put under a vacuum of [O-' Pa and immersed in the

water bath. The transducer itself is not immened. but is covered by the lid. Its tempernture

is monitored by a thermocouple. Readings were taken every minute for a period of 18

hours. The pressure reading stayed in a 2 5 Pa range while the temperature stayed within a

0.4 'C range. As the pressures that will be recorded on are on the order of hundreds and

thousands of Pa. this error can be neglected provided the transducer remains in a sirnilar

temperature range.

The experimental method developed causes the transducer to be at a different tem-

perature at the beginning and end of an expenment. Initially, the test section is not

immersed, and the transducer temperature is regulated by arnbient conditions. When the

experiment is ninning. the transducer is under the lid of the bath and at a temperature near

that of the bath. At the end of an expenment, the offset is recorded, and this is used to cor-

rect the readings. The temperature of the pressure transducer is recorded during the exper-

iment. It is generally found to stay within the same range as the test mentioned above. For

tests that do not use the bath, the offset of the transducer is recorded at 104 Pa before and

after the test to check the isotherrnal assumption. When it not valid, a correction is made to

the measured pressures by assuming that the drift was a linear function in time. This cor-

rection is generally less than 10 Pa.

The pressure transducer was calibrated against a rnercury manometer using nitro-

gen. The manometer is made of glass and has an inside diameter of 8 mm. Before being

filled with mercury, it was put through the cleaning procedure described in Appendix A.

The mercury used has a maximum level of impurity of non-volatile substances of 5 pprn.

The manorneter was attached to the test section at the same point as the pressure trans-

ducer using additional fittings. To check its validity, the difference in height between the

two columns of mercury was mensured when the system was under at a pressure of [O-'

Pa. Observations were made using a F ï I cathetometer with a reading precision of 0.01

mm. The reading accuracy is taken as &.O5 mm to account for rneasurernent error and dif-

ferences in the contact angle at each of the levels. As two measurernents are made. the

pressure reading provided therefore has an associated error of 113.3 Pa. The difference

between the two levels was measured to be 0.04 mm, As this is on the same order as the

accuracy of the readings, it is neplected.

Nitrogen was then introduced to the test section in increments of approximateiy

500 Pa up to 4000 Pa. At each step, the system was lefi to settle for ten minutes. and then

four readings were taken with the cathetometer. An average value of the readout of the

pressure transducer display was recorded. Once the readings at 4000 Pa were complete.

the pressure in the test section was reduced in increments of 1ûûû Pa back to zero. Read-

ings were done at al1 stages. This was done to check for hysteresis effects. The system was

brought back to a pressure of 104 Pa. The manometer read a difference of 0.02 mm, which

is less than the measurement error, and is therefore neglected. At al1 points. the manometer

readinp were within at most an 1 I Pa range, which is within the assigned measurement

error. The reading of the pressure transducer changed by at most 2 Pa at each measurement

point. A plot of the pressure measured by the pressure transducer versus the pressure mea-

sured with the mercury manometer is shown in Figure B-1. The measurement erron are

within the size of the data points. Also included in the plot is a 45 degree line which woiild

indicate perfect agreement. There is no appreciable hysteresis but the curve does not

Hg manometer reading (Pa)

Figure B- 1. Pressure transducer calibration curve.

match the 45' line. The difference between the two lines increases as the pressure rises,

and is not covered by the expenmentai errors at the top of the range. A linear least squares

fit was applied to the data using Microsoft Excel, and gave an R' value of 09999. The

standard deviation of the difference between the best fit line and the points used to make it

is 11 2 Pa. This calibration is used in al1 subsequent pressure measurements. Note that an

initial correction must always be made to account for the offset of the transducer before

applying the calibration. With the error associated with the manometer readings. and the

spread of the data about the calibration curve for the transducer. the error associated with

the pressure measurement is taken to be I 20 Pa, which is less than the rated error. For

rneasurements in the range of 2000 Pa to 3600 Pa. which is typical of the experirnental

investigation, this is a relative error on the order of one percent.

B2 CALIBRATION OF THE THERMOCOUPLES

Three type K thermocouples were calibrated against a mercury thermometer with

O.laC divisions. The reading error is I0.05 OC. It has been checked against a calibrated

Okazaki Pt 100 AS7885 W D temperature probe. The error associated with the calibration

is I0.05 'C. The thermocouples were placed in a small intemal circulating bath. The tem-

perature maintained by the bath was raised from 20 'C to 30 'C in increments of 0.5 O C . At

each temperature. the bath was left to corne to equili brium for ten minutes. Voltage read-

ings were then taken using the temperature measurement system every ten seconds for ten

minutes. The temperatures read by the data acquisition system show a periodic drift with

amplitude 0.035 'C and period 45 S. This length of time allows for the drift to cancel out.

Note that the ice point has a rated stability of k 0.04'C. The mercury thermometer was

monitored during this penod. The temperature was increased to the next level only if there

was no change in the temperature read by the thermometer during the measurement inter-

val. To construct a calibration curve. the voltage measurements for each ten minute period

were averaged and plotted with the thermometer temperature. The data is in swng agree-

ment with a linear fit.A sarnple of one of the calibration curves is shown in Figure B-2. Al1

of the standard deviations of the voltage readings are l e s than 106 V (- 0.025'C). Both

this and the error associated with the temperature readings from the thennometer are cov-

ered by the size of the data points.

5 b

Figure

19 - ----- -- -- - -

0.0007 0.0008 0.0009 0.0010 0.001 1 0.0012 0.0013 Voltage (V)

B-2. A sample therrnocouple calibration curve. This is the thermocouple that is used to monitor the temperature in the bath.

Due to the procedure needed to put together the test section. it was not possible to

calibrate the two thermocouples inside it in the manner described in the previous pan-

graph. Instead, they were cali brated against the three other themiocouples. The test section

was immened in the bath, and filled with nitrogen at 3000 Pa. The bath thermocouples

were spread out around the main chamber of the test section. The heating circulator was

then used to establish a constant temperature in the bath. When the temperature was con-

stant, readings were taken of a11 five thermocoupIes every ten seconds for ten minutes.

This was done in 1 'C increments between 20 O C and 30 O C . The readings of the three bath

thermocouples were averaged over t irne and each other, and this used against the voltage

readings of the hvo test section thermocouples. Calibration curves were made as before.

and a linear fit was again found to be in strong agreement with the data. To account for the

errors associated with the calibraiion procedure, the error associated with the temperature

readings is taken as IO.OS0C. This implies that measured temperature differences of

0.05'C or greater are significant. Al1 of the measurements have a a. 1'C absolute error

associated with them.

B3.1 CALIBRATION PROCEDURE

The accuracy of the measurements read from the micromanipuiator that the rele-

scope is mounted on was investigated by checking it against the scale on a pair of vernier

calipers. Over a distance of 18 mm, the two measurements agreed to within 0.13 mm. or

0.72%. On a measurement of 1.60 mm, this would correspond to an error of 0.0 15 mm.

The images taken with the carnera have dimensions of pixels. With the collirnator

in place, interfaces can be resolved to within one pixel. In order to convert these to a dis-

tance, a caiibration is required. In order to rninirnize the error associated with the calibra-

tion, as large an object as possible should be used. The shaft of the feedthrough is chosen.

Its diameter at a given angle of rotation and distance from its base was measured with the

telescope. For every experiment. an image of the feedthrough in this position is taken and

used as the reference for al1 measurements.

B3.2 DROPLET MEASUREMENTS

The droptet size is determined from images taken with the solid state camera. An

example of what the image looks like is shown in Figure 8-3. As mentioned in

Section B3.1, the interface can be located to within one pixel. With one pixel generaliy

king 0.015 mm, this corresponds to an error of M.03 mm on the measurement of the

droplet size. Combined with the calibration error, the error in the droplet size measure-

ment is taken as I0.04 mrn.The droplet diameter is measured as the longest horizontal dis-

tance visible. This is shown in the figure as DH.

DH Figure 8-3. A typical image used to measure the droplet size.

As the droplet is in a gravitational field, it will not have a spherical shape. In

Figure B-4, the vertical dimension. D V . of an evaporating droplet is ploaed against its

horizontal dimension. D, . Also plotted is a 45' line. which would correspond to a sphere.

Dv is harder to specify because of the thermocouple bead and wires. When the droplet is

in a position sirnilar to that in Figure 8-3, D, is taken as a point between the two rhermo-

couple wires and the bottom of the drop. As the droplet evaporates and moves up the

wires, it becomes easier to specify the top location.

There are two distinct regions in the data, separated at around where DH is equal

to 1 mm. When LIH is greater than I mm. gravity causes Dv to be greater than L I H . The

effect decreases as the droplet gets srnaller. When DH is less than 1 mm, the gravitational

effects have decreased, but the thermocouple wires and bead start to distort the shape. As

well as becoming oblong, the droplet loses syrnmetry, and the volume of the thermocouple

bead becomes significant with respect to the liquid volume. Generally, below 0.9 mm.

measurements on the droplet size are no longer valid. Note that the smallest measurement

made corresponds to the bead size.

Horizontal Measurement (mm)

Figure B-4. An examination of the sphericity of the droplet.

Assuming that vertical cross-sections through the droplet centre are identicai. the

droplet volume can be approximated as

and an effective droplet diameter, corresponding to a spherical droplet with the same vol-

ume, is then

For al1 cases considered in the experimental work, the difference between DH and Defi

was at most 3%. It is therefore justifiable to ignore the gravity effect for the calculations

that were done.

C 1 BATH PERFORMANCE

The theory developed assumes that the system is isothermal. Experimentally. this

is achieved using the water bath and the heating circulator. To assess how tightiy the

heater-circulator can control the system temperature, the bath was left at a set point of 24.1

OC for 15 houn. The three bath thennocouples were positioned around the test section.

Measurements were taken every ten seconds on each of the therrnocouples. and averaged

over ten minute intervals. On each interval. the averaged temperatures agreed to within at

most 0.03 OC. This is an indication of the temperature uniformity within the bath and sup-

ports the calibration procedure. Over the 15 hours, each averaged temperature remained

within at most a 0.04 'C interval. This is an indication of the stability of the temperature

control provided by the bath.

C2 CHECNNG THE TIGHTNESS OF THE SYSTEM

One of the assumptions in the theory developed is that the system has a constant

mass. While no expenmental apparatus can be hermetically sealed, every effort was made

to reduce the possibility of leaks. To test this, a series of expenments were performed to

establish the ieak rate into the system in different configurations.

The test section was pumped down to a pressure of IO-' Pa. Part of the test section

was then isolated. The tests were run starting from a vacuum to minirnize adsorption

effects. Pressure rneasurements were taken every ten seconds for at least four hours. At the

end of the allotted time, the system was pumped back to IO-' Pa. The pressure data was

plotted, and agreed well with a linear fit. The dope for each of the configurations consid-

ered is listed in Table C-L. The temperature of the bath was rnaintained at 26.85 O.I0C

for each of these tests.

Table C- 1. Leak rate into the test section for different configurations.

1 Entire test section. out of bath. needle outside 1 2.84 1 5.1 , 1017

System Configuration

1 Entire test section, out of bath, needie inside 1 6.83 1 12 1018

LeakRate (Paihr)

I Entire test section, in bath

Main chamber. in bath

When the apparatus is irnmersed,the mass leak rate into the main chamber and

into the entire test section are indistinguishable when the measurement mors are consid-

ered. The major source of leakage is therefore in the main chamber. The only parts of the

main chamber which are not submerged are the top of the feedthrough and the pressure

transducer. The feedthrough is a UHV component, and it is unlikely that there is any sig-

nificant leakage through it. To determined if the pressure transducer is the source of the

ieak, it was isolated, and monitored for four hours. The result of the test is shown in the

last row of Table C-1. The calculation of the volume of the transducer and its f i t t i n g was

rough, and a maximum possible value was calculated to give an upper bound on the leak

rate. Any leakage through the pressure transducer is insignificant compared to what is

observed in the main charnber. Based on knowledge of the system. it is most likely that the

ieak is where the thennocouple wires exit the main chamber. Any leakage into the system

while it is irnmersed must therefore be water vapour.

LeakRate (mdhr)

Pressure transducer, in bath

5 55

9.62

9.9 x 10"

9.5 l0l7

< 3 < 3 x ld5

The leakap into the system is higher when immersed than when it is exposed to

the ambient air with the needle removed. When it is not immersed, any leakage w il1 be air.

As determined in the previous paragraph, any leakage into the system while it is immersed

will be water vapour. Differences in the two rates could be due to the higher mobility of

the lighter water molecule andfûr because of the greater pressure differential when the test

section is irnmersed.

Starting from a vacuum will give the maximum leak rate into the system. as the

pressure difference between the inside and outside is greatest. If. as will be discussed in

the next section. there is capillary condensation when water vapour is inside the test sec-

tion. it is possible that it may slow down the leak rate. This is difficult to quantify as the

adsorption process takes a losg time at high pressures.There is no way to separate the two

effects. Leakage into the system was dealt with in different ways for different aspects of

the experimental investigation.

D 1 PUMPING ON THE SYSTEM The experiments are mn at pressures in the range of 3000 to 4000 Pa. Ideally. there

wilf be no substances other than water present. To meet this requirernent, it is necessary to

be able to remove al1 the volatile mass in the system before introducing the water. This is

accomplished using the vacuum pump assembly and the ultra-high vacuum system. This

section will descnbe how the test section can be taken from atmospheric pressure to an

ultra-high vacuum.

The first stage of the pumping is done with the vacuum pump in the vacuum pump

assembly.Valve VI1 is opened, and the pump is turned on. The pump is brought in contact

with the test section by opening valves 1, II and V. The pressure in the test section is rnoni-

tored with the thennocouple pressure gauge. While its accuracy is not high. it offers more

resolution than the pressure transducer at pressures less than 25 Pa. The vacuum pump is

left to pump on the test section until its pressure is less than 10 Pa. At this point. the pump-

ing can be tumed over to the turbo pump by closing valve 1 and opening the UHV valve. It

is left in this state until needed. The rest of the cornponents of the system are brought into

contact with the vacuum pump by opening valves [II. IV and VI.

D2 ALIGNMENT OF THE NEEDLE AND THE DROPLET THERMOCOUPLE

One of the steps in the experimental procedure is the transfer of liquid from the

needle to the dropiet thermocouple. To prepare for this procedure, the needle and the drop-

let thermocouple m u t be positioned precisely. The position of the droplet thermocouple is

manipulated with the linear-rotary motion feedthrough. The position of the needle is

adjusted with the three degree of freedom motion manipulator that the syringe pump is

mounted on.

The first step is to get the needle inside the test section. The tip of the needle is

positioned at the centre of the septum located in the valve on the right side of the main

chamber. It is then slowly moved fonvard so that it pierces the septum, and advanced until

the smallest section is inside the valve. The valve is then opened. If the systern is under

vacuum, this will cause the pressure to rise as the needle has been at atmospheric pressure.

However, the volume of the needle is sufficiently small that there will be no detrimental

effect on the turbo-molecular pump if it is pumping on the test section at this time. The

needle is then moved further into the main chamber until it is visible in the front viewing

window. As the needle is long and slender, it has a tendency to buckle as it is pushed

through the septum. In order to counteract this, the needle is held in a horizontal position

using a clean glove as it entes

The needle is now in the middle of the main chamber. The desired arrangement of

the needle and the droplet themocouple is shown in Figure D-1. We want the needle to

thread the thermocouple past the opening on its side. The two components are observed

using the telescope because of its tight focus range. The droplet thermocouple is put in

position by rotating the feedthrough to 135' and adjusting its vertical position to align with

the needle. The horizontal location of the needle is adjusted slightly using the motion

manipulator until it cornes in line with the thermocouple. The needle is then advanced to

the desired position. As the needle is not completely straight at this tirne, moving it for-

ward may cause some up or down movernent. This must be compensated for by adjusting

the vertical position of the thermocouple as the needle threads it.

droplet thermocouple

vacuum epoxy /

\

needle

Figure D- 1. The configuration of the needle and the droplet thermocouple before placing a droplet on the thermocouple.

D3 WATER PREPARATION

The theory developed is based on a pure system. In order to ensure this. high purity

water must be used. Tap water is filtered through a Bamstead High Purity Demineralizer

Canridge. It is then distilled. The water is then passed through a Barnstead NANOpure

Water Purification System. The final purîty is at least 18 Megaohrns-cm. This water is then

placed in the degassing flask.

D4 WATER DEGASSING The water preparation process consists of filtering, distilling and de-ionizing.

When complete, there will still be dissolved air in the water as it has been in contact with

the atmosphere throughout the process. In order to get rid of the air, the water must be

degassed. Both degassed liquid and pure vapour are required for an experiment. Dunng

the procedure, the test section is pumped on by the turbo-molecular pump.

The 0.062 mm tubing connecting the liquid portion of the degassing flask to the

syringe pump assembly cannot be isolated from the degassing flask. Due to its smali size.

it is possible that air could get caught in it during the degassing procedure which is unde-

sirable for the needle filling procedure. It is therefore beneficial to flush it through with

liquid before degassing. The flushing is accomplished by opening valves XII and XIlI and

letting water flow through the tubing. The flow is dnven by the static head difference

between the degassing flask and the synnp pump assembly. Liquid is left to flow for 10

minutes. Valve XII is closed.

The degassing is accomplished by putting the top portion of the degassing Rask

under vacuum. The degassing is started by opening valve IX. bringing the degassing Hask

into contact with the vacuum pump. The stirrer is tumed on. causing the teflon coated

rnagnet to spin. The teflon surface promotes heterogeneous nucleation. and the spinning

causes the bubbles to break free of the surface. Generally. no air bubbles will be visible

after about 30 minutes. The degassing procedure is continued for at least three hours.

Valve IX is then closed. isolating the degassing flask.

[t is not desirable to have water vapour flow though the vacuum pump. It may con-

dense out and contaminate the oil. The purpose of the cold trap is to prevent this from hap-

penning. Before degassing, the styrofoam container that the cold trap sits in is filled with

dry ice. Most of the water vapour that passes through will condense and then freeze on the

cold surface. The cold trap can easily be removed from the vacuum pump assembly when

the degassing procedure is cornplete so that the ice can be removed. It is important not to

degas for too long as the cold trap may become plugged with ice, blocking the suction pro-

vided by the vacuum pump.

The temperature of the liquid in the degassing flask will be less than the arnbient

temperature during the degassing procedure. Energy is required to drive the evaporation. If

the liquid is suficiently cold, if is possible to have vapour nucleation in the horizontal tube

at the bottom of the fiask. This area will be at a higher temperature than the Rask as it is

away from the evaporation surface and in closer proxirnity to the surroundings. The pres-

sure here will be iess than the saturation pressure for ambient conditions. A vapour bubble

will appear to grow and shnnk from the Ultra-torr fitting at a frequency of about 2 Hz.

Occasionally the bubble will get big enough so that it can escape from the tube and rise to

the surface. It is important to let the Rask corne to equilibnum with the surroundings

before proceeding. The presence of vapour in the tubing will result in an unsuccessful f i l l -

ing of the syringe.

D5 SYRINGE FILLING

The first step in the syringe filling procedure is to remove the air and water which

may be in the syringe valve assembly. This is done by bringing the syringe valve assembly

into contact with the vacuum pump by way of the valve assembly between valves IV and

XIII. Valve IV should be closed while the connection is made. The syringe valve assembly

is isolated from the surroundings by closing valve XI. Valve X was closed during the

degassing procedure. Once the connection to the vacuum pump has been made. valves IV

and XIII are opened, and the system is evacuated. There is no way to accurately monitor

the pressure in the system during this tirne, so the pumping is left to take place for 30 min-

utes.

It is not desirable to let liquid into the valve assembly through valve XII at this

point. The filling procedure works better if the syringe valve assembly is full of water

vapour. This is accomplished by bringing this assembly into contact with the vapour sec-

tion of the degassing flask. The tubing between these two sections is isolated from the rest

of the apparatus by closing valve II. By opening valve X, water vapour will Row from the

degassing fiask into the syringe valve assembly. The piston is moved to a position a few

millimeters below its top dead centre position. The synnge valve assembly is then isolated

by closing valve XIII. Valve IV is closed, isolating the tubing that goes from the valve

assembly to the syringe valve assembly. The tubing is disconnected at valve XIII. Valve X

is closed, isolating the vapour section of the degassing Bask.

The syringe valve assembly can now be filled with liquid. If the degassing proce-

dure has been done properiy, there should be liquid water immediately above valve XII.

When valve XII is opened, the water there will be at a higher pressure than the vapour in

the syringe valve assembly because of the stîtic head provided by the degassing flask. It

will Bow into the syringe valve assembly, causing the vapour to compress andlor con-

dense. Some liquid will find its way to the syringe. and fa11 in. The piston should be high

enough so that the first few droplets that enter wet its surface, and allow any vapour or air

present to rise to the top. It is not desirable to have any gas pockets trapped by the liquid.

Only a certain arnount of liquid will enter at this point, and there will be a liquid-vapour/

air interface present in the syringe. By slowly withdrawing the piston by hand in a number

of small intervals, more liquid will corne into the synnge drop by drop. The piston is wi th-

drawn as far as it can go. It is important to do this slowly so that the liquid droplets can fall

to the interface, and push air and vapour to the top. If the preparation and the initial fil1

have been done properly, a maximum of a few millimeters of vapoudair will be visible at

the top of the syringe.

The next step is to flush out the air. First, the synnge valve assembly is isolated

from the liquid section of the degassing flask by closing valve XII. The system is then

pressurized by opening valve XIII, which has one side open to atmosphere as the connec-

tion to the valve assembly has been removed. The pressurization will cause any air bubbles

present to compress. The contents of the syringe valve assembly are then forced through

valve XII1 by ninning the syrinp pump in the fast forward mode. A few liquid droplets

should corne out. The piston should be stopped a few millimeten below its top dead centre

position. Valve XIII is closed, isolating the syringe valve assembly.

Following the same procedure outlined above, the needle is then refilled with liq-

uid. If when valve XII is opened, bubbles are seen to flow into the bottom of the degassing

ff ask. the filling procedure should be stopped as the tubing is no longer continuously filled

with liquid. When the fi lhg is complete, the amount of air visible at the top of the syringe

should be less than before. The liquid is then flushed through, and the syringe FiIled again.

This process should be repeated until it seems as though the amount of air in the system is

constant. It is impossible to get al1 the air out simply by flushing. In order to get rid of the

rest of the air, the syringe is left filled with valve XIII open so that the its contents are pres-

surized. The high pressure will increase the solubility of air in water. The systern is left

ovemight. The next day, the liquid is flushed out a few times using the procedure described

above. At this point, there should be very little, if any, air left. If there is still air present.

the pressurization and fiushing should be continued. Once the needle has been filled. it is

possible to keep it in this state indefinitely. By flushing it through on a daily basis and

leaving it open to the atmosphere, it is ready for use at any time. Before running an exper-

iment. it is Bushed through two times, and valve XIII is not opened after the last fill . so

that the liquid pressure inside is on the order of the saturation pressure.

D6 PREPARATION OF THE TEST SECTION

Before mnning an experiment, degassed water must be available and the syringe

must be 611ed. and left iso!ated. The needle is positioned as described in Section D2. The

bath starts on the middle table, and the immersion circulator-heater is set at 27 O C . The test

section and needle are pumped to IO-' Pa for a penod of at least eight hours. The CCD

camera is put in place so that the droplet thermocouple and the needle can be viewed in

reai tirne. The valve assembly is isolated from the vacuum pump assembiy by closing

valve II. Water vapour is introduced to the test section by opening valves 1 and X. Adsorp-

tion begins immediately. and the pressure will be less than saturation.

D7 FORMATION AND PLACEMENT OF THE DROPLET

The formation of a suitably sized droplet on the needle is the most difficult aspect

of the experimental procedure. If the droplet is too big. the thenocouple will not be able

to support its weight. If it is too small, the thermocouple bead will distort its shape. The

ideal initial droplet has a radius of about 0.85 mm.

The liquid in the syringe is brought into contact with the test section by opening

valve XI. The syringe pump is run at a rate of 10 pUrnin until liquid cornes out of the nee-

dle. A large droplet forrns, and if the pump is immediately stopped. it will not fall off the

needle. It will often be too big to transfer to the thermocouple, but by Ieaving it sitting in

the test section, it will slowly evaporate until it reaches a suitable size. To speed up the

evaporation, the test section can be pumped to a lower pressure. In doing this. one must be

careful not to pump too low, as it will cause more liquid to come out of the needle and

increase the droplet size.

Once the droplet has reached a suitable size, the next task is to tnnsfer it to the

droplet thermocouple. Recall that at this stage, the two components are in the configura-

tion shown in Figure D- 1. The droplet will have rotated to the bottom of the needle.

Figure D-2 shows a sequence of pictures of this procedure. First, the thennocouple's verti-

cal position is adjusted so that the bead is just below the bonom of the needle. The needle

is slowly withdrawn until the thermocouple wires hit the droplet. The needle is then with-

drawn very slowly, and the droplet is held back by the thermocouple. Once the thennocou-

ple has cleared the needle, the droplet will have two solid contact points: on the

thermocouple bead and at the tip of the needle. In order to transfer the droplet completely

to the thermocouple, the thermocouple is raised slightly. and the needle withdrawn as

slowly as possible. At some point. the droplet will disengage from the needle and either

faIl to the bottom of the main chamber or sit on the thermocouple bead. If the droplet was

initially too big. it will fall, but this is not the only reason why this happens. It cm also

happen if the procedure is done too quickly, or if the components are not aligned properly.

The procedure is extremely delicate.

The needle is then fully withdrawn from the system. It is important that there be no

liquid on its surface that might faIl or nib off on something when the needle is removed. If

any liquid is visible on the needle, it is left to evaporate before removing the needle. The

valve on the right side of the test section is closed before the needle is fully removed to

ensure a good seal. In some experiments, a quantity of liquid may be placed on the bottom

of the main charnber. This is accomplished by advancing the syringe pump further while

the needle is still above the bottom window. This step will be further discussed when the

particular experiments are presented.

Figure D-2. Droplet transfer procedure. A:The thermocouple is threading the needle and a suitably sized droplet has been formed. B:The needle has been withdrawn to the point just before the droplet disengages. C:The droplet has successfully been placed on the thermocouple.

APPENDIX E: EXPANSION TESTS

Al1 of the tests were done with the bath temperature set at 26.85 I O. lO0C. The test

section was purnped to IO-' Pa. It was then filled with nitrogen at 3235 Pa. The main

chamber was isolated, and the auxiliary chamber pumped back to [O-' Pa and isolated.

The nitrogen adsorption takes place in about ten minutes. After fifteen minutes. the prer-

sure in the main chamber stabilized at 3220 Pa. After another five minutes, the valve

between the two chambers was opened, and the system left to come to equilibrium. During

this time, measurements of the pressure and the two system themocouples were made

every second. The expenment was then repeated, with a burst run on the three measure-

ments. It was necessary to run the tests separately due to the nature of the burst program.

With three measurements being taken, the time interval between the data points for one

channel is 0.15 S. The initial steady state pressure for this test was 33 15 Pa. The data for

the burst is shown in Figure E-1. The pressure in the system stabilizes very quickly. From

the burst data, the time for the pressure to come to a new steady value is about 0.75 sec-

onds. which is on the same order as the time needed to open the valve. The response of the

themocouples is slower. The droplet thermocouple temperature decreased by 2.6'C. com-

pared to 2.0°C for the vapour thermocouple. This is because it is smaller, and therefore has

a lower thermal mass. The time taken for the two thermocouples to come back to the ambi-

ent temperature is about ten seconds. For both tests, the pressure before and after the

expansion remained within a L Pa interval for at least five minutes. The initial and final

pressure readings were obtained by averaging the readings over the steady interval. From

the two tests that were done, the ratios of the final pressure to the initial pressure are

05899 and 05898. if the nitrogen is assumed to behave as an ideal gas, then this ratio rep-

resents the ratio of the volume of the main chamber to the volume of the entire test section.

Based on the geometric approximations of the volumes in Table 2-1, this ratio is predicted

to be 0.56. From the observed pressure ratios, the volume calculations appear to be reason-

able.

The same experiment was then repeated, but this time using water vapour. Adsorp-

tion will now play an important role. The pressure was raised to 3400 Pa. the main cham-

ber was isolated and left until it reached a pressure of 3321 Pa. Dunng this time. the

auxiliary charnber was pumped back to loJ Pa and isolated. The valve between the two

chamben kvas then opened. The same experiment was repeated. with a bunt taken around

the expansion. The results of the second test are shown in Figure E-2. The top graph shows

the total time history of the pressure. The bottom graph shows what happened to the tem-

peratures around the expansion. The behaviour is significantly different than what was

seen with nitrogen. During the lead up to the expansion, the pressure in the system is

decreasing because of the adsorption. When the valve is opened, the pressure drops. and

water desorbs frorn the walls of the main chamber and starts to adsorb on the wall of the

auxiliary charnber. The net effect is an increase in pressure in the system with time. The

temperature drop and response time of the thermocouples is greater than observed with

nitrogen. This is likely due to sorne capillary condensation on the beads which evaporates

when the pressure is suddenly reduced.

From these expenments, the effect of the adsorbed phase has been shown to be sig-

nificant on the behaviour of the system. Also, after an expansion has taken place. the sys-

tem takes tirne to respond. Some of the observed response tirne is due to the finite thermal

mas of the thennocouple beads. It is difficult to anaiyze how the presence of the droplet

affects the response time. as the evaporation process is ongoing, and cannot be separated

1500 - - -- - - A - -- --

O .O 05 1 .O 15 2 .O 2.5 3 .O

Time (s)

Time (s)

Figure E- 1. The pressure and temperature tirne histones around the expansion for the nitrogen expansion test.

2200 - --- "-- -

O Io00 2000 3000 4000

Time (s)

- Droplet TC

V a p o u r TC

Time (s)

Figure E-2. The pressure and temperature time histones for the water vapour expansion test. The pressure graph is for the entire time of the test. The temperature graph is only around the expansion.

lrom the expansion. To eiiminate the effects of an expansion, a minimum time period of

two minutes is chosen, only after which measurements are considered to be significant.

This is an order of magnitude larger than the observed response time frarne, and should be

sufficient .

F1 SIZE DATA Table F- 1 .Size data for test 1

Time (s) Diameter (mm) II Tirne (s) 1 Diarneter (mm)

Table F-2. Size data for test 2

Time (s)

O 180

360

Diameter (mm)

1-19

1.16

1. 15

Diarneter (mm)

1.50

1 .48

1.47

Time (s) 2700

2880 3060

Table F-3. Size data for test 3

Time (s)

O 120

240

Diameter (mm) 1.51

1.47

1.44

Time (s) 1800

1920

2040

Diameter (mm) 1 .O9

1 .O5

1 .O4

F2 MEASURED AND CALCULATED DATA USED IN

F2.1 TEST 1

Table F4. Measured data from test 1. The bath temperature is 26.78 OC.

170

Table F-4. Measured data from test 1 . The bath temperature is 26.78 O C .

pV (Pa)

Table F-5. Calculated data from test 1.

IpV - p;,,I (Pa) ry (OC)

2.68 E-OS

2.68 E-05 3.12E-05 4.07 E-OS 3.26E-05

4.07 E-O5 3.26E-05 2.85 E-05 2.85 E-OS 2 -44E-05

Tabie F-5. Calcuiated data from test 1 .

2.85E-05 25 .O2

1.63 E-OS 25.80

2.04E-05 26 .O0

3.26E-O5 25.95

3 26E-05 25.8 1

2.85E-05 25.95

3.67 E-OS 25.98

4.48 E-05 25.70

4.48 E-05 25 .70

F2.2 TEST 2

Table F-6. Measured data from test 2.The bath temperature is 26.87'C.

r f (OC) pV (Pa) r , (ml


1

2

3 4

5

j (kglm2s) 5.3 1 E-05

6.12E-05

6.12E-05

6.12E-05

6.12E-05

TV (OC) 24.30

2455

24.39

24.46

24.63

TV - T: (OC)

0.87

1 .O5

0.85

O .86 1 .O0

P!&, (Pa)

2885

2897

2905

29 16

2920

F2.3 TEST 3

Table F-8. Measured data from test 3. The bath temperature is 26.80aC.

pV (Pa)

176


APPENDIX G: STATISTICAL RATE THEORY EXPRESSION FOR THE RATE OF

EVAPORA~ON Statistical Rate Theory can be used to predict that the rate of evaporation at a liq-

uid-qmur interface at an instant in timz in an isolatéd system is given bj '

w here

and P: must satim

The terms are the vibrational temperatures for the fluid, which for water are 3650 K.

1590 K, and 3760 K. A description of the parameten is given in Table G- 1

Table G-1. Parameters needed to evaluate the Statistical Rate Theory expression for the evaporative flux.

Parameter

j

1 pressure in the vapour at the interface (Pa)

Description (units)

mass Rux (kglm2s) I

--

k ' Boltzrriann coiisiant (Jiiii,K j

m molecular mass (mJkg)

P,(T')

ps

temperature in the vapour at the interface (K)

saturation pressure corresponding to the liquid temperature at the interface (Pa)

pressure in the isolated system at equilibrium (Pa) L

R

~f

- --

specific volume of liquidat saturation conditions corresponding to the liquid temperature at the

radius of curvature at the interface (m)

temperature in the liquid at the interface (K)

y L V

interface (rn4m3)

surface tension, evaluated at the liquid temperature at the interface (Nlrn)

The material discussed in this section is based on the course MIE 1 10 1 : Thermo-

dynamics 2, offered at the University of Toronto by C. A. ~ a r d ?

H l DEFINITIONS

Classical thermodynamics is based on a series of established postulates. Before

stating the postulates, a nurnber of definitions will be presented.

H l . 1 SIMPLE FLUID SYSTEM

A simple fluid system has the following attributes:

It is hornogeneous

It is isotropic

8 It may contain an arbitrary number of chemical species.

8 No chemical reactions take place in the system.

Surface and field effects are neglected.

H 1 -2 TYPES OF SY STEMS

A system can be classified by how it interacts with its surroundings. This interac-

tion is a function of the nature of the boundary between the system and its surroundings. A

closed system is one in which there is no mass transfer to or from the systern. The bound-

aries are said to be non-permeable. An isolated system is one in which there is no mass or

heat transfer to or f'rom the system. The boundaries are said to be non-permeable and adia-

batic. The system itself may be a simple fluid system, or a collection of simple fluid sys-

tems, which is cailed a composite systern.

H 1.3 PROPERTES

An extensive property of a simple fluid system is dependent on the arnount of mat-

ter present. An intensive property of a simple fluid system is independent of the arnount of

matter present. A thermodynarnic property is only a function of the state of the system.

and not of how the system reached that state. Themodynamic properties are also referred

to as path independent properties.

H 1.4 EQUILIBRIUM STATE

A simple fluid system is said to be in equilibrium if there are no spatial gradients in

any of its independent properties and if no spontaneous macroscopic changes take place in

a time period t .The magnitude of r determines the specific nature of the equilibrium state

with respect to its thermodynamic stability.

For r -. O . the system is in an unstable equilibrium state.

For t finite. the system is in a metastabie equilibrium state.

For r -. m , the system is in a stable equilibrium state.

H 1.5 INTERNAL ENERGY

The intemal energy of a simple fluid system is denoted by U . It represents the

energy the system possesses due to the motion and interactions of its constituent particles

with respect to a chosen reference.

H2.1 THE FIRST POSTULATE

For a simple fluid system:

1. Its intemal energy, U , and entropy, S , are extensive thermodynamic properties.

2. If it is in an equilibrium state, then it has independent variables S. V and

NL, ..., N,, where V is the system volume, N i is the number of moles of the ith species

present in the system and r is the number of different species present.

The second statement implies that the macroscopic state of a simple Ruid system is

characterizcd completely by the noted variables. The first statement says that U is an

extensive thermodynamic property of the system. which was defined in Section H 1.3.

Therefore, it follows that in an equilibrium state, U can be expressed in ternis of the inde-

pendent variables as

U = U(S, V, N I , ..., N,) (H-1)

Eq.(H-1) is called a fundamental relation. It contains al1 the information needed ro corn-

pletely describe the macroscopic equilibrium States of the simple Buid system.

H2.2 THE SECOND POSTULATE

1. The intemal energy is differentiable with respect to each if its independent vari-

ables.

Implicit in this statement is that the interna1 energy is a continuous function of its

inde pendent variables.

2. Eq.(H-1) can be inverted to obtain

S = S(U, V , Ni, ..., N,) (H-2)

When Eq.(H-1) is written as Eq.(H-2), it is still called a fundamental relation. The

first statement of the Second Postulate also applies to Eq.(H-2).

3. For a composite system,

and

where the subscript c refers to the composite system and the subscript i refers to the ith

simple Ruid system in the composite system.

This statement implies that the intemal energy and entropy of a simple fluid sys-

tem are first order homogeneous. This will be proved in Section H3.2.

4. Of ail the possible configurations that a composite system rnay take on. the one

that exists when equilibtium has been established after an internai constraint has been

removed is that which corresponds to a global maximum of the entropy.

It is this final statement of the Second Postulate that allows many different thermo-

dynamic systems to be analyzed and understood. The entropy function can be maximized

in order to determine equilibrium conditions. This statement is often referred to as the

entropy postulate.

H2.3 THIRD POSTULATE

1. The entropy of a simple fluid system is a monotonically increasing function of

its intemal energy. That is,

where f i represents N, , . . ., N, .

2. If as = O then the enrropy in that state is zero.

H3.1 DEFINITION OF TEMPERATURE. PRESSURE AND CHEMICAL POTENTIAL

From Eq.(H-I), the total differentiai of the internai energy of a simple ff uid system

is

where N i is ail the N terms except for N i .

Define the intensive properties of temperature. 7'. pressure, P , and the chernical

potential of the ith mass species. pi, as

and

The appropnateness of these definitions is seen when the equilibrium conditions of a com-

posite system are found. For now, consider T as a potential for heat transfer, P as a poten-

tiai for mechanical work and pi as a potential for mass transfer of the i th species. Eq.(H-

6) can now be written as

H3.2 FIRST ORDER HOMOGENEOUS PROPERTIES

The third statement of the Second Postulate implies that U and S for a simple

fluid system are first order hornogeneous functions. This is equivalent to saying that a fun-

damental relation is first order homogeneous. That is.

AU(ST V , NIT ..a, N,) = L i ( h . S h V h N , ,... AiV,) (H-1 1 )

where A is a constant greater than or equal to zero. To prove Eq.(H-1 1). consider a simple

fluid system at equilibnum with known S , V and N i . The system boundaries are rigid.

non-permeable and adiabatic. The interna1 energy for this system can be expressed by

Eq.(H-1). Now imagine that the system is divided into h sections of equal volume Vie

where the subscript h. refers to the properties of each of the sections. By the definition of

equilibrium, the entropy and number of moles of each substance will be the same in each

section. Denote these quantities as Sh and N i A respectively. Each of these sections is in

itself a simple fluid system, and rogether they form a composite system. From Eq.(H-3)

and Eq .(H-4),

(H- 12)

and

185

(H- 13)

and from the above statements,

and

(H- 15)

Thus, from Eq.(H-13), Eq.(H- 14) and Eq.(H-lS), Eq.(H- 1 ) can be written as

U = U ( S , V , NI, ..., N , ) = U ( L S A A V k h N , , , . . . AN,,) (H-16)

For each of the A simple Ruid systems in the composite system. Eq.(H-1) can be written

as

U A = U,(S,, v,, N , , , * * * , N,,)

With Eq.(H-17), Eq.(H-11) becomes

U = LU, = hu(Sh ,VA ,Nik9*** Nd) (H- 18)

The left sides of Eq.(H- 16) and Eq.(H- 18) are the same quantity. and the interna1 energy is

therefore first order homogeneous. A similar procedure could be followed to show that the

entropy is also first order homogeneous.

H3.3 THE EULER RELATION

Differentiating both sides of Eq.(H- 1 1) with respect to h. gives

(H- 19)

Conside ring Eq.(H-7), Eq .(H-8) and Eq.(H-9) gives

which is known as the Euler relation.

H3.4 THE GIBBS-DUHEM RELATION

The total differential of the Euler reIation is

d u = TdS + SdT - PdV - VdP + x p,dNi + 2 Nidp . I

Substituting Eq.(H- 10) for dU gives

SdT - VdP + 2 N i d p i = O

which is known as the Gibbs-Duhem relation. For a one cornponent system. Eq.(H-22)

becomes

SdT- V d P i N d p = O (H-23)

For a one component system, the intensive properties of specific entropy and specific vol-

ume are defined as

and

v v a -

N (H-25)

Dividing Eq.(H-23) through by N and by considering Eq.(H-24) an Eq.(H-25), an inten-

sive form of the Gibbs-Duhem relation is found

d p = - s d T + v d P (H-26)

This is often a more convenient form to use. Note that Eq.(H-26) implies that

P = N T , P) ( H-27)

It is important to remember that this is only true for a single cornponent system

H3.5 DESCRIPTION OF A SURFACE PHASE

Surface phases (also called surfaces or interfaces) such as that which exists

between a liquid droplet and the vapour andlor gas that surrounds it, are not simple fluid

systems. They will be analyzed using the Gibbs model. The surface is assumed to have

zero thickness. but has a finite mass. The gradient for intensive properties is infinite at the

surface. The behaviour of the surface is assumed to be independent of curvriture. The pos-

nilates presented previously can be modified to deal with surfaces by considering the area.

A . of the surface phase as opposed to the volume of a simple fluid system. Eq.(H- 1) and

Eq.(H-2) then become

LI = U ( S , A&, ..., N , )

and

S = S(U, A, N,, ..., N,)

The total differentiai of the internai energy is then

The surface tension. y , is defined as

allowing Eq.(H-30) to be written as

From Eq.(H-20), the Euler relation for a surface phase can be inferred to be

and from Eq.(H-22), the Gibbs-Duhem relation for a surface phase can be inferred to be

SdT + Ady + 2 N i d p i = O

H4.1 INTRODUC~~ON

For some composite systems, it is more convenient to use a thermodynamic poten-

tial to determine the equilibrium conditions than the entropy postulate. This is because not

al1 systems have II, V, and N I , ..., N r as their independent variables. Also. thermody-

namic potentids allow the amount of work a system could perfom in a certain situation to

be determined. In the fint part of this section, the thermodynamic potential relevant to pan

of this work will be formed. Following that, the behaviour of a composite system and its

surroundings undergoing a work interaction will be analyzed, and the usefulness of ther-

modynamic potentials will be shown.

H4.2 THE HELMHOLTZ POTENTIAL

The independent variables of a simple fluid system can be changed from U. C'.

and N I , ..., iV, to T , V , and N',, ..., N, by lemng

where F is called the Helmholtz potential. W~th Eq.(H-7). F can be expressed as

F = U - T S (H-36)

By taking the total differential of F and considering Eq.(H-IO), one can prove that

F = F(T, V , N ,, . . ., N , ) . Note that by considenng the Euler relation. the Helmholtz

potential for a simple fluid system can be written as

and for a surface phase as

It will be shown Iater that for a systern with independent variables T . V . and IV , , . . .. N, .

extrema of F correspond to equilibrium States.

H4.3 DESCNPTION OF A WORK INTERACTION

To understand the usefulness of the Helmholtz potential , first consider a very gen-

eral isolated system that consists of two parts: a composite system and a reservoir that sur-

rounds it as show n in Figure H-1

Figure H- 1. An isolated system composed of a reservoir and a composite system. There are no restrictions on the boundary between the reservoir and the composite system. The reservoir boundaries are adiabatic, ngid and non-permeable.

A reservoir is defined as a systern where changes in its extensive properties do not change

its intensive properties. Roperties pertaining to the reservoir are denoted by a superscript

R . Properties pertaining to the composite system have no supencript. The reservoir has a

R R constant temperature and pressure of T and P respectively. There are no restrictions on

the nature of the boundary between the composite systern and the reservoir. The reservoir

boundary is adiabatic and non-permeable, as follows from the fact that it is part of an iso-

lated system, and is rigid. Suppose that the isolated system is at equili brium, and that a

certain amount of work W is done on it, causing it to go to a new equilibrium state. We are

interested in describing the new equilibrium state. By conservation of energy,

where A represent the change from the initial to the final state. AU^ can be expressed as

the difference in the Euler relation for the reservoir between the initial and final States to

give

Based on the system constraints, the total volume and mass are constant. Thus.

A V + A V ~ = O

and

and therefore ,

From the first statement of the Third Postulate, we can write

A S + A & O

and therefore

This is the most general fonn of the work interaction, as no restrictions have been placed

on the boundaries of the composite system.

H4.4 SPECIFIC EXAMPLE: THE HELMHOLTZ POTENTIAL

Now suppose that the composite system walls are diathermal, rigid and non-per-

meable. It can exchange energy with the reservoir, but has a fixed volume and mass.

Therefore,

A V = O

and

ANi = 0,i = 1 .... r

and at equilibrîurn

T = T~

With Eq.(H-46), Eq.(H-47) and Eq .(H-48). Eq .(H45)becomes

WaA(U-TS) (H-49)

which we recognize as the Helmholtz potential. This is not unexpected, as the independent

variables of the composite system are temperature. volume and mass. Therefore.

W Z A F IH-50)

Thus, the effect of the work interaction on the isolated system is somehow related to the

change of the Helmholtz potentiai of the composite system. More specifically, the value of

the work puts an upper limit on the arnount that the Helmholtz potential can change.

Now consider an isolated system compnsed of a composite system and a reservoir

that is subject to Eq.(HSO). Furthemore, suppose that there is an additional constraint

imposed on the composite system that keeps it in a constrained equilibrium. For example.

the composite system may have two volumes that are separated by a ngid piston that is

pinned in place. This equilibrium state is not necessarily the equilibriurn state that would

exist if the piston were unpinned. Suppose for now that the constrained equilibrium state

and the unconstrained equilibrium state are not the same. If the pin is removed for a penod

of time S r . the piston will move towards the unconstrained equilibrium state. No work

interaction takes place outside the isolated system, and therefore

W = A U + A U ~ = O

From Eq.(HJO). it follows that

A F s O (H-52)

Letting AF = Ff- F i , where Ff and Fi are the Helmholtz potential of the composite

system in the initial and final States,

F / s Fi (H-43)

The Helmholtz potential decreased as the composite system tried to move to the uncon-

strained equilibnum state. If the procedure of removing the pin for a time 61 is repeated.

the Helmholtz potential will once again decrease. If this pmcess is repeated until the pis-

ton reaches the unconstrained equilibrium state, the Helmholtz potential will reach a mini-

mum constant value. That is, at some point,

Ff = Fi (H-54)

Note that this equilibrium state will be stable. A similar statement to the entropy postdate

can now be made: "Of al1 the possible configurations that a composite system may take

on, the one which exists when equilibriurn has been established after an intemal constraint

has been removed is that which corresponds to a global minimum of the Helmholtz poten-

tial ."

I l DESCRIP~ION OF SYSTEM

The system of interest is an isolated system composed of a composite system sur-

rounded by a reservoir as shown in Figure 1- 1. The reservoir has a constant temperature

R and pressure of T~ and P respectively. The walls of the composite system are non-per-

meable and diathermal. Therefore, when the isolated systern is in equilibrium. the corn-

posite system will have the same temperature as the reservoir. One of the composite

system walls is a piston that is free to move. Because of this, the pressure on both sides of

the piston will be the sarne when the isolated system is in an equilibrium state. and equal

R to P . The composite system consists of a single one-component spherical droplet of

radius R in its own vapour and the surface phase that separates them. The total number of

molecules in the system is NT. The solid-vapour interface is neglected. As the pressure

and temperature of the vapour are constant, the rnass of fluid in the adsorbed phase will be

constant. Field effects are ignored, and thermophysical properties are assumed to be only a

function of temperature. This is the systern discussed by ~ibbs".

reservo i r

Figure 1- 1. A single one-component liquid droplet in its own vapour. The pressure in the vapour and the temperature are maintained by the surrounding reservoir. The walls are non-permeable .

Based on the description of the composite system, it can be inferred that its inter-

na1 energy and volume are not two of its independent variables. In fact, it can also be

R inferred that the independent variables are Iikely to be T~ , P and N T . For this reason. it

will be convenient to use a therrnodynamic potential to determine the equilibriurn condi-

tion as opposed to the entropy postdate.

12 DEVELOPMENT OF THE D POTENTIAL

Eq.(H-45) descnbes a general work interaction involving an isolated system con-

sisting of a composite system and its surroundings. For the system described in Section I l .

A N i = O (1-1 1

Also. based on the nature of the composite system walls. at equilibnum. let

R T = T

and

pV = P R (1-3)

where P' is the pressure in the vapour phase of the composite system. With the above

constraints, Eq.(H-45) becomes

W ~ A U - T A S - P ' A V

v As T and P are constants for equilibrium States,

W ~ A ( U - T S + P ~ V ) (r-5)

It is important to note that the term inside the brackets is not the Gibbs potentiai. which is

defined as

G = u - T S + P V

With the Euler relation, Eq.(H-2O), the Gibbs potential can be wrinen as

The pressure term in the Gibbs potential implies a constant pressure throughout the system

being described. The pressure term in Eq.(I-5) only deals with the pressure in the vapour

phase.As will be shown later, the curvature of the liquid-vapour interface forces the pres-

sure in the liquid to be greater than that in the vapour for this system. Use of the Gibbs

potential to analyze the problern would be incorrect. Define the thermodynamic potential

and therefore, from Eq .(I-5).

W z A D (1-9)

Using a similar argument to that developed when discussing the Helmholtz potential in

Appendix A, it can be shown that a minimum of the D potential corresponds to a stable

equilibrium state.

13 DETERMINATION OF THE EQUILIBRIUM CONDITIONS

For the system under consideration,

D = D " + D ~ + D ~ ' (I- 10)

As stated, D will be a minimum when the composite system is in an equilibrium state.

Therefore, in such a state, it is required that

d D = O

and

wth Eq.(H-36). Eq.(H-37), Eq.(H-38). Eq.(I-6), Eq.(I-7). and Eq.(I-8). and noting that

vLV = O from the mode1 of the interface k i n g used, the D potential c m be txpressed as

v v L L L L L V L V L V L V V L D = p N - P V + p N + y A + p N + P V

Taking the total differential of Eq.(I-13) and expanding gives

d~ = + ~ ' d ~ ' - f L d v L - v L d p L + + ~ ~ d ~ ~ + y L V d ~ L " + (1- 14)

LV LV A dy + p L v d ~ L V + ~ ~ ~ d ~ ~ ~ + p Y d v L + v L d

At constant temperature, the Gibbs-Duhem relation, Eq.(H-23). for the liquid phase

reduces to

- v L d p L + tvLdyL = O (1- 15)

Likewise, for a liquid-vapour interface at constant temperature. the appropriate fom of the

Gi bbs-Duhem relation. Eq .(H-34). reduces to

LV LV A dy + f f L V d P L V = O (1- 16)

For the vapor phase, which is at constant temperature and pressure, Eq.(H-23) reduces to

~ ' d $ ' = O (1- 17)

With Eq.(I-15), Eq.(I-16), and Eq.(I-17), and noting that dpV = O, Eq.(I-14) can be

expressed as

d~ = p V d ~ V - pLdvL + p L d ~ L + y L V d ~ L V + p L V d ~ L V + p V d v L (1-1s)

The droplet is sphencal and can be modeled by Eq.(3-15) through Eq.(3-18). The system

is closed to mass transfer, and thus

N ' + N ~ + N ~ ' = NT

and as NT 1s constant,

L dNV = -dN -dN LV

Eq.(I- 18) can now be expressed as

For equilibrium, we require dD = O . The only way that this is guaranteed by Eq.(I-2 I I is

if al1 of the arguments of the differential terms go to zero. That is.

and

These are the equilibrium conditions. Thermal equilibrium has been assumed with the use

of the D potential. Solving these two equations simultaneously will give an expression for

the equilibrium radius for a given set of system parameters. If the liquid phase is approxi-

rnated as a slightly compressible liquid, and the vapour phase is approximated as an ideal

pas, with Eq.(3-23), Eq.(3-24) and Eq.(3-25), Eq.(I-22) can be written as

L Substituting in for P from Eq.(I-23), an explicit expression for the critical radius in terms

of the reservoir temperature and pressure is found:

Note that the expression is independent of the mass of the system.

14 FORMATION OF THE D POTENTIAL

The O potential can be expressed as written in Eq.(I-13). As it stands. there is no

way of plotting D vs. R, . However, this problem can be resolved by introducing a refer-

ence condition Do. Let Do be defined as the state where there is no droplet present in the

system for a given set of independent variables. Thus. from Eq.(I- 13) and Eq.(I- 19).

v v v L LV D o = p N T = p ( N + N + N ) (1-26)

v As T and P are constant. v V = p V ( ~ , P') will also be constant, independent of the

existence or size of a droplet. Subtracting Do from D gives

L V L L 1' L L LV LV D - L I , = (p -p )N +(p - C < V ) ~ L V + ( ~ V - ~ )V + y A (1-37)

At equilibrium, Eq.(I-22) and Eq.(I-23) hold, and by considering these conditions as well

as the spherical nature of the droplet, Eq.(I-27) gives

w here R, is given by Eq.(I-25). As Do is a constant, extrema of Eq.(I-28) w il1 occur at the

same values of R that they would if only D were considered.

It is of interest to consider the nature of the homogeneous liquid and vapour states

in the system discussed in Appendix H. The homogeneous vapour state ( R = O ) occurs at

a minimum of the D - Do curve. However, it is not the global minimum for the potential.

which occun when the system is a homogeneous liquid and

The Second Postulate states that the stable equilibrium state for a system occurs at the glo-

bal minimum of its governing thermodynamic potential. Thus. for this system. the stable

equilibrium state is the hornogeneous liquid. The question then anses of how to interpret

the homogenous vapour state. Recall from Appendix H that a metastable equilibrium was

defined as a state in which the system only stays for a finite period of time. On a molecular

level, a vapour can be analyzed as a collection of free moving particles that intenct. At

times. these particles may corne together and form a small cluster of liquid. Based on the

previous discussion, if the radius of the cluster is less than R , . it will evaporate and the

system will return to being a hornogeneous vapour. However, if the radius of the cluster is

greater than R, , it will start to grow, and in the limit the system will go to the homogenous

liquid state. This event is called nucleation, and foms the basis for a large field of

research. The homogeneous vapour c m therefore be classified as a metastable equilibriurn

state. Given enough time, a cluster of the critical size will nucleate. The energy associated

with a cluster of radius R, is

This is often referred to as the barrier to nucleation. From and, nucleation cm only take

v place when P > Pm . The size of the critical radius decreases as the pressure increases.

and from Eq.(J-Z), the barrier to nucleation will also decrease. Nucleation is more likely to

v occur as the pressure increases beyond the saturation value. Ln tact, as P approaches P,

from the right side, the barrier to nucleation increases dramatically.

52 HETEROGENEOUS NUCLEATION

Oeneous As the solid-vapour interface is being considered. the possibility of hetero,

nucleation must be considered. We are concemed with the unstable equilibriurn state.

which can lead to nucleation, which has been shown to be equivalent for both the D

potential and the Helmholtz potential under certain conditions. For simplicity. the stability

analysis will be done using the D potential. The system in question is shown in Figure J- 1.

I reservoir

Figure J-1. A sessile one-component droplet in its own vapour. The pressure in the vapour and the temperature are maintained by the surrounding reservoir. The walls are non-permeable.

The contact angle at the three phase line is denoted by 0 .The solid-liquid interface

is denoted by the supencnpt SL. For such a system. the equilibrium conditions can be

shown to be

and

SV SL LV y - y = y COS^ (J-5)

Eq.(J-5) is known as the Young equation. It describes the equilibrium conditions at a three

phase line. The critical radius is once again found to be

In the sarne manner as Appendix H. the D potential can be shown to be

w here

and

3û4

(J- IO)

from the geometry.

We are interested in how the surface affects the barrier to nucleation. Notin; that

0 =L8O0 corresponds to a free droplet. a plot of the ratio of the bamer to nucleation for a

given contact angle to the barrier for the free droplet (homopeneous nucleation) as a func-

tion of the contact angle can be made. This is shown in Figure J-2 for the case of T = 27

'C and P" = 4245 Pa.

Figure 5-2. The barrier for nucleation for heterogeneous nucleation normalized by the bamer for homogeneous nucleation as a function of the contact angle.

Clearly, heterogeneous nucleation has a lower banier than homogeneous nucle-

ation. For a contact angle of Oa, there is no bamer, and filmwise condensation will occur.

For the theory developed, this is not desirable. For any experirnental work. a material

should be chosen with a contact angle greater than zero to eliminate this possibility. As

discussed in Appendix B, the unstable radius for a very slightly supenaturated vapour is

large, and nucleation is very improbable. However, different issues can arise in an expen-

mental investigation, and precautions should be taken to avoid any potential problems.

1 . G. Fang. Rate of Liquid Evaporation: Statistical Rate Theos Approach, Ph .D. thesis. Department of Mechanical and Industrial Engineering, University of Toronto (1998).

2. G. Fang and CA. Ward, 'Temperature measured close to the interface of an evaporating liquid". Physical Review E 79,417 ( 1999).

3. C.A. Ward and G. Fang. "Expression for predicting liquid evapontion ff ux: Statistical Rate T heory Approach". Physical Review E 79.429 ( 1999).

4. G. Fang and CA. Ward, "Examination of the statistical rate theory expression for l iquid evapontion". Physical Review E 79,441 ( 1999).

5. J. B. Young, 'The condensation and evaporation of liquid droplets in pure vapor at arbitrary Knudsen nurnber," Int. J. Heat Mass Transfer 34. 1649 ( 199 1 ).

6. C. A. Ward and D. Stanga, submitted for publication.

7. C. A. Ward. 'The rate of $as absorption ar a liquid interface". J. Chem. Phys. 67( 1 ). 129 (1977).

8. J.A.W. Elliott and CA. Ward, "Statistical Rate Theory and the Material Propenies Con- trolling Adsorption Kinetics on Well Defined Surfaces". Eqriilibria and Dwnmics Gas Adsorption on Heterogeneoris Solid Srirfaces, W. Rudzinski. W.A. Steele. G. Zgrablich Eds., Studies in Surface Science and Catalysis. 104.285-333 ( 1995).

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14. M. Dejmek and C. A. M d . "Study of Interface Concentration During Crystai Growth of Dissolution", J. Chem. Phys. 108,8698 (1998).

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29. W. Vogelsberger. "On the use of chernical potentials in the condensation of a pure vapor", Chem. Phys. Letters 63(3) 301 (1979).

30. H. Reiss and G. J. M, Koper, "The Kelvin Relation: Stability, Fluctuation and Factors Involved in Measurement", J, Phys. Chem. 99.7837 (1995).

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33. S. Brunauer, P. H. Emmett and E. Teller. "Adsorption of gases in multimolecular lay - ers", J. Am. Chem. Soc. 60,309 (1938).

34. C. A. Ward, P. Rahirni and A. Keshavarz. "Contact angle hysteresis generated by tlie space shuttle". to appear in the proceedings of Spacebound 2000.

35. V. Ponec, 2. Knor and S. Cerny, Adsorption on Solids. Buttenvorths. London. 1974. p.36 1.

36. ibid., p. 362.

37. ibid., p. 364.

38. ibid. pp. 372-378.

39. R. Defay et al.. op. cit.. pp. 222-227.

40. A. Bejan. Heat Transfer. John W11ey and Sons. Lnc. New York (1993). p. 36 1.

41. R. D. Present. Khetic Theoc of Gares, McGraw-Hill. New York ( 1958). p.55.

42. ibid., p 44.

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45. M. J. Moron and H. N, Shapiro, Fundamentals of Engineering Thernzodynonzics. John Wtley and Sons, Inc., New York (2000), p. 804.1. H. Shames, Mechanics of Flriids. McGraw-HM, Inc., New York (1992), p. B4.

46. W. Vogelsberger. 'Themodynamics of finite systems: a possibility for interpretarion of nucleation and condensation experiments", J. Coli. Int. Sci. 88(1), 17 ( 1982)

47..R. Defay et al.. op. cit.. p. 262-266

48. D. Stanga, personal communication. The data cornes from a special experiment run Marc h 28,2000.

49. A. Keshavarz, An Investigation of rhe Stable Eqiiilibriiini State of a Blrbblr in n Finiir Volume of o Water-Nitrogen Sokition, MA.Sc thesis, Department of Mechanical and Industrial Engineering, University of Toronto ( 1998).

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