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DECLARATION

Except where acknowledgements are made in the text, all the material contained in this thesis is the work of the candidate.

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Abstract

A diaphragm cell has been designed and used for self-diffusion 5 -2

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ACKNOWLEDGEMENTS

The author is greatly indebted to his supervisor, Dr. L.A. Woolf, for his stimulation and encouragement during the course of this

research. In addition thanks are also due to Dr. A.F. Collings,

Dr. R.O. Watts, Mr. R.A. Fisher and Dr. R. Mills for helpful discussions. The technical assistance of the workshop staff of the Research School of Physical Sciences, under the supervision of Mr. C. Steele and in particular that of Mr. D. Hall is gratefully acknowledged. I would also like to thank The Australian National University for a postgraduate scholarship.

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TABLE OF CONTENTS

Declaration Abstract

Acknowledgements List of Tables List of Figures Chapters:

1 Introduction 1

2 Techniques Used to Study Liquid Diffusion 4 Under Pressure

2.1 Introduction 5

2.2 Self-diffusion Techniques at Atmospheric Pressure 5 2.3 Self-diffusion Techniques at Pressures Greater 6

than Atmospheric

2.3.1 The Nuclear Magnetic Resonance Spin Echo Technique 6

2.3.2 The Diaphragm Cell Technique 7

2.3.3 The Capillary Cell Technique 8

2.3.4 Diffusion Column Technique 10

2.3.5 Fritted Disc Techniques 11

2.4 Comparison of Pressure Diffusion Data from 13 Different Techniques

2.5 Summary 14

3 The Diaphragm Cell 16

3.1 Introduction 17

3.2 Diaphragm Cell 17

3.3 Theory of Self-diffusion Measurements with a 19 Diaphragm Cell

3.4 The Design Requirements of the Pressure Cell 23

3.5 The First Pressure Cell 24

3.6 The Second Pressure Cell 24

3.6.1 Construction 25

3.6.2 The Diaphragm 30

3.6.3 The Bottom Compartment 32

3.6.4 The Top Compartment 33

3.6.5 The Stirrers 33

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TABLE OF CONTENTS (Continued)

Chapters:

4 Experimental Techniques 36

4.1 Introduction 37

4.2 Pressure Techniques 37

4.3 Temperature Technique 42

4.4 Rotating Magnets 44

4.5 Safety Precautions 45

4.6 Purification of Chemicals 45

4.7 Determination of Volumes 47

4.7.1 Volume of Diaphragm 47

4.7.2 Volume of Top Compartment 43

4.7.3 Volume of Bottom Compartment 50

4.8 Procedures Required for a Typical Experiment 59

4.8.1 Cleaning the Cell 53

4.8.2 Adjustment of Top Compartment Volume 53

4.8.3 Test of Stirrers 51

4.8.4 Filling the Cell 52

4.8.5 Calibration Equipment 52

4.8.6 Pressure Experiment 53

4.8.7 Analysis of Samples Extracted from Cell 54

5 Results 59

5.1 Introduction 60

5.2 Calibration Experiments 60

5.3 Diffusion Experiments 66

5.4 Sources of Error 75

5.4.1 Errors Inherent in the Diaphragm Cell Technique 75

5.4.2 Errors Due to Temperature Effects 76

5.4.3 Errors Due to Pressure Effects 33

5.4.4 Summary of Errors 34

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TABLE OF CONTENTS (Continued)

Chapters:

6 Discussion 87

6.1 Introduction 88

6.2 Comparison of Reported Data with Literature Values 88

6.2.1 Atmospheric Pressure Studies 88

6.2.2 Pressure Studies 91

6.3 Hydrodynamic Theory 96

6.4 Activation and Volume Theories 103

6.5 Significant Structure Theories 112

6.6 More Rigorous Theories 115

6.6.1 Longuet-Higgins and Pople 115

6.6.2 Corresponding States Principle 116

7 Conclusions and Recommendations 119

7.1 Conclusions 120

7.2 Recommendations 120

Appendices:

A Procedures Required to Calculate the Diffusion 122 Coefficient of a Pressure Experiment

A.l Introduction 123

A.2 Calculation of the Volumes of the Compartments 125 A.3 Calculation of the Cell Constant at the Experimental 127

Pressure

A.4 Calculation of the Term („C^ - „C.) / („C^ - 0C_) 127 Z 1 Z D J 1 J D

A.4.1 Assumptions Required 128

A.4.2 Calculation of the Volume of Bulk Flow 130 A.4.3 Calculation of the Mass of Radioactive Liquid 131 A.5 The Time Length of a Diffusion Experiment 136 A.6 Choice of the Time Length of a Pressure Experiment 136 B Experimental Data Listed from a Computer Output 142 C A Diaphragm Cell for Diffusion Measurements in Liquids 146

Under Pressure

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List of Tables

Table:

3.1 Characteristics of Diaphragms 31

4.1 Components of Pressure System 39

4.2 Density of Purified Liquids 46

4.3 Volumes of Compartments of Cells 49

5.1 Calibration Results 62

5.2 Self-Diffusion Ceofficients of Benzene 67 5.3 Self-Diffusion Coefficients of Carbon Tetrachloride 69 5.4 Self-Diffusion Coefficients of Cyclohexane 71

5.5 Compressibility Data 82

6.1 Self-Diffusion at Atmospheric Pressure 89

6.2 Viscosity Data 99

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List of Figures

Figure:

3.1 The Stokes Type Diaphragm Cell 3.2 Assembled Pressure Diaphragm Cell 3.3 Exploded View of Cell

3.4 Cross Section of Cell

3.5 Engineering Drawing of Cell 3.6 Stirrers for Cell

4.1 Arrangement of Equipment 4.2 Pressure Bomb

4.3 Platinum Resistance Thermometer for Pressure Cell 5.1 Variation of Cell Constant with Experimental Time 5.2 Variation of Cell Constant with Age

5.3 Benzene: D vs P

5.4 Carbon Tetrachloride; D vs P 5.5 Cyclohexane; d v s P

5.6 Increase in Temperature Inside the Cell with Application of Pressure

6.1 Log D vs 1/T at Atmospheric Pressure 6.2 Benzene: log D vs P

6.3 Carbon Tetrachloride:log D vs P 6.4 Cyclohexane: log D vs P

6.5 D /D vs n /n 1 p p i

6.6 Test of Stokes Einstein Equation

, 1/2

6.7 Carbon Tetrachloride: log (D/T ) vs 1/T 1/2

6.8 Benzene: log (D/T ) vs 1/T 1/2

6.9 Cyclohexane: log (D/T ) vs 1/T

6.10 Constant Volume Activation Energy vs Density 1/2

6.11 Carbon Tetrachloride: D vs T at constant density 6.12 Carbon Tetrachloride: log D vs T at constant density 6.13 Carbon Tetrachloride: Test of Significant Structure

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F i g u r e :

List of Figures (Continued)

A.l Four Stages of a Pressure Experiment 124

A . 2 Application of Pressure to Liquid in

the Cell

126

A . 3 Movement of Flow 135

A . 4 The Effect of Pressure on the Bulk Flow

for Cell B with Benzene at 55°C

Correction 137

A . 5 Relationship Between the Ratio'C /C and

Percentage Change in D Due to Bulk Flow

the 138

A . 6 Relationship Between the Relative Error

V CB

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Chapter One

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2. Of the three states of matter the liquid state is by far the

least understood. Liquids have neither the ordered structure typical of the solid state nor the absence of multiple intermolecular

interactions which typifies dilute gases.Processes of diffusion in the liquid state are due to spontaneous molecular movement caused by the thermal energies of the molecules and are of fundamental

importance in their theoretical significance and practical application. From the viewpoint of theory an understanding of these transport processes should lead to a satisfactory theory of liquids. "For a complete evaluation of present theories of transport in liquids it is essential that experimental transport coefficients be available for simple liquids over a range of temperature, and pressure. Certainly there is a large body of data for the shear- viscosity of liquids as a function of temperature and pressure for this purpose. Although the situation is not completely satisfactory with thermal conductivity measurements it is in the field of diffusion that most work needs to be done. There are few reliable studies of the temperature dependence of the self­ diffusion coefficient, D, of liquids and the paucity of data is worst of all in the case of the pressure dependence of D." (Collings, Hall, McCool and Woolf,1971). Repeated references have been made in the literature to this lack of data (Najarajan and Bockris 1966, Cleaver 1967, Tham and Gubbins 1970).

Studies of the pressure dependence of diffusion are also of considerable practical importance. There are high pressure processes in nature where diffusion is thought to play an important, if not dominating role. These include migration of mineral fluids in rocks

(Collins 1961) and formation of marine sediments (Horne et. al 1969). In the industrial field such processes include distillation and

diffusion-controlled chemical reactions.

The lack of data referred to above is undoubtedly due to the

experimental difficulties inherent in studying diffusion under pressure. Probably the most accurate method of measuring self-diffusion coefficients is the diaphragm cell (Mills 1961).

This thesis describes how a diaphragm cell has been developed and used for high pressure studies. Although a preliminary design of a diaphragm cell for these studies had been made earlier in these

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3

as we were able to show) that no bulk flow would occur in the cell at any pressures and consequently the equations for the conventional diaphragm cell as used at atmospheric pressure were to be employed to calculate the diffusion coefficient. In our cell, however, bulk

flow at pressures greater than atmospheric was a deliberate feature of the design. Consequently it was necessary to develop equations to accurately calculate the bulk flow and also to modify the equations used to

calculate the diffusion coefficient.

The design of the present cell and the necessity to establish the validity of the techniques imposed two restrictions on the choice of

liquids which could be studied. These limitations were (a) that the liquids could be handled at room temperature and (b) that their compressibilities were known. Because of these experimental

restrictions the liquids chosen were: benzene, carbon tetrachloride and cyclohexane.

In Chapter Two the methods that have been used previously to study the effect of pressure on the self-diffusion will be discussed. It will be shown that the method most likely to give accurate data at high pressures is the diaphragm cell. Chapter Three describes how the cell design was modified for high pressure work. In Chapter Four the auxiliary equipment employed for the experimental techniques used will be described. The results obtained from the measurements are tabulated and the errors in the technique discussed in Chapter Five. The data are then examined against current theories in Chapter Six, and the conclusions of the thesis and recommendations for future development

of the pressure diaphragm cell are given in Chapter Seven, There are three appendices. Appendix A discusses the effect of the non-diffusional flow upon the self-diffusion coefficient and describes how this flow is

calculated and minimised. Appendix B lists the computer output for the experimental data and finally to conclude the thesis a preprint of a publication dealing with the design of the cell is given in Appendix C

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4

.

Chapter Two

Plan of 2.1

2.2

2.3

2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.4

2.5

Techniques Used to Study Liquid Diffusion Under Pressure

Chapter

Introduction

Self-diffusion techniques at atmospheric pressure Self-diffusion techniques at pressures greater than atmospheric

The nuclear magnetic resonance spin echo technique The diaphragm cell technique

The capillary cell technique Diffusion column technique Fritted disc techniques

Comparison of pressure diffusion data from different techniques

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5

. 2.1. Introduction

In this chapter a brief outline of the techniques that have been used to study the self-diffusion of liquids at atmospheric pressure will be given. It will be shown how these techniques have been employed to study the effect of pressure on the self-diffusion coefficient. Finally the methods will be compared by reference to data reported for a number of liquids.

2.2 Self-Diffusion Techniques at Atmospheric Pressure

In the last twenty years the experimental techniques for measuring diffusion coefficients have been discussed by a number of authors.

These have included books by dost (1952) , Ottar (1958), Robinson and Stokes (1959) and Tyrrell (1961) and reviews by Johnson and Babb (1956) and Gosting (1956). The techniques for measuring the self-diffusion coefficient (i.e. diffusion of a species in its own environment) can be divided into two classes:

(a) Techniques that measure an actual self-diffusion coefficient Methods that measure the actual diffusion coefficient have been developed only recently. By far the most extensively used technique is that of nuclear magnetic reasonance spin echo (NMR). Other methods that have been used include neutron - and light-scattering techniques.

(b) Techniques that measure a pseudo self-diffusion coefficient These techniques are modifications of long established methods that have been used to determine the diffusion coefficients of solutions. In these techniques the experiment is arranged so that the diffusion takes place at an interface or a series of interfaces between solutions of different concentrations. The diffusion coefficient is determined by measuring the change in concentration with time either at the

interface or in the solutions. However, self-diffusion is the diffusion of a single species in its own environment (i.e. no concentration

changes are present). So as to obtain a concentration gradient some of the molecules are labelled and consequently can be physically distinguished from the bulk liquid of which they are part. It is assumed that this

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6

.

The most popular tracer methods for measuring self-diffusion coefficients are the diaphragm cell and the capillary cell. Less widely used techniques include the diffusion column and fritted disc techniques.

2.3 Self-Diffusion Techniques- at Pressures Higher than Atmospheric

In this section we will discuss the methods which have been used to determine self-diffusion of liquids'under pressure. The non-tracer technique (i.e. NMR spin-echo) will be discussed first

followed by the tracer methods.

2.3.1 The'Nuclear Magnetic Resonance^ Spin' Echo Technique

This method was originated by Hahn (1950) and developed by Carr and Purcell (1954). In this technique a magnetic field gradient is placed across a region in which the sample under investigation is located. The effective magnetic field seen by the nucleus can be calculated. Now during the time of observation the average molecule will diffuse a certain distance and hence experience a spread in the effective field. Due to this effect there is a decrease in the apparent spin-spin relaxation time. From this decrease and the gradient field the diffusion coefficient can be calculated.

The main advantages of this technique are that (i) isotopic labelling is not required, (ii) the measurements of the sample can be monitored in situ, and (iii) the*'time required for the complete

experiment is relatively small (-30 mins). The main disadvantages are that (i) the techniques can be used to study only species whose nuclear spin is greater than zero and (ii) that the errors inherent in the techniques (±5-10%) are relatively large- compared with the capillary or diaphragm cell techniques.

This method was first employed for pressure studies by Benedek and Purcell (1954). In their technique the sample surrounded by the

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7

. they reported their results by tabling the diffusion coefficient at 25°C and atmospheric pressures,and an activation volume Av4- (at 28°C) where

Avt

= s - RT (3 ln D/3P)

T

R - gas constant

T temperature

D = diffusion coefficient

P = pressure

In a series of later papers (1959 b, 1961, 1963) these authors obtained the pressure dependence of D in a similar manner for water, nitromethane, acetone, benzene, cyclohexane, isopentane, neopentane and tetramethyl silane. They reported that the activation volume

(Av1~) for these liquids was independent of pressure except for nitromethane, isopentane and neopentane. In these cases the activation volumes decreased with increasing pressure. McCall, Anderson and Huggins (1961) measured the pressure dependence of the. self "-diffusion of various linear dimethyl-siloxanes at 29C C and showed that the activation volumes of these

liquids were independent of pressure.

Wade and Waugh (1965) employed this technique to measure the self-diffusion of liquid ethane for pressure between vapour pressure

(-40 atms) and 2200 atms. for a series of temperatures between 155°K and 298°K. They showed that a plot of ln D vs. P did not give a straight line for any of these temperatures.

2.3.2 The Diaphragm-Cell Technique

This technique is probably the most accurate one for studying tracer diffusion.Mills (1961) reported that errors in this technique are of the order of ±0.2% to 0.5%. In this method diffusion is confined to a porous diaphragm on either side of which'are solutions of different concentrations. It appears that only one attempt has been made to use a diaphragm

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8

. 2.3.3 The Capillary- Cell Technique

Ottar (1958) reported that Wroblewski (1881) was the originator of this method.' The technique was developed^for modern usage by Anderson and Saddington (1949). The'cell consists essentially of a small capillary closed at one end and filled with a labelled sample of liquid. It is immersed in a bath of the non-active

liquid so that the top of the capillary is just above the surface of the liquid. After temperature equilibrium is reached the capillary is lowered gently beneath the surface of the liquid, and diffusion takes place from the capillary into the bath. At the end of the experiment the capillary is removed from the bath for analysis. The average concentrations

in the capillary at the beginning and- the end of the experiment are

determined and the diffusion coefficient calculated. In these calculations it is assumed that the concentration of the labelled liquid around the top of the capillary is zero throughout:the experiment.

Errors can arise in this techniques due to end-effects at the top of the capillary. If the bath;is not"stirred a build up in concentration can occur at the end of the capillary during the experiment. In this case the measured diffusion coefficient will be less than the true diffusion coefficient. To overcome this problem a number of workers (see

for example Tyrrell 1961) have stirred the bath. Mills (1955) pointed out extensive errors can arise from the type of stirring employed. He showed that the data obtained from a capillary experiment only coincided with the data of a diaphragm cell experiment when a streamline flow of liquid occured across the open end of the capillary at a rate of 1-3 mm/sec. A flow rate less than this gave a low diffusion coefficient»whilst a greater

flow rate gave a higher diffusion coefficient because liquid is scooped out of the capillary. By using the proposed flow.rate Mills improved the precision of successive measurements from ±2% to ±0.8%.

A second source of error in this technique-is due to the accuracy with which the concentration in the capillary can be determined.

Difficulty is encountered in the complete removal of the small volumes (-10^1) of the liquid in the capillary. Mills and Goldbole (1958) avoided this problem by monitoring the concentration change in the capillary

throughout an experiment. The capillary was surrounded by a scintillation crystal which could be viewed by a photomultiplier. By using a y emitter

22 +

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9.

The capillary cell technique was first used for pressure studies by Watts, Alder and Hildebrand (1955). They studied the self-diffusion

36

of carbon tetrachloride (using Cl) at three temperatures at atmospheric pressure and'also at 200 atms. Their apparatus consisted of three

capillaries located in an unstirred bath in a pressure bomb. After reaching equilibrium the capillaries were lowered beneath the liquid surface, the pressure bomb'sealed and:pressure applied, Diffusion was allowed to take place for about three days before the pressure was released and the'concentration of radioactive carbon tetrachloride determined.

As well as the errors due to the end effects of an unstirred bath and the analytical procedures, this technique will also have errors due to pressure effects. Application of pressure causes a contraction in the volume of the active liquid in the capillary and hence reduces the

column length of*active liquid. Th« measured:diffusion coefficient

would consequently be less than the true diffusion coefficient. This error would be offset to a limited extent b y the*'rise in temperature (and hence diffusion rate) due to the heat'of compression of the liquid. The

release of pressure will give rise to similar errors. These workers estimated their errors to be of the order of ±2%.

Hiraoka et. al (1958 , 1959) used'a: similar apparatus to study the diffusion of-benzene and methanol for a*series of temperatures and pressures. In calculating the diffusion coefficient these authors made an allowance for the initial reduction in the length of active liquid in the capillary due to the pressure. The correction is to replace the length Ü used in the equation to calculate the diffusion coefficient by the expression

Z (1-3P) where P is the pressure of the experiment and 3 is the

compressibility of the liquid. However this correction may be inadequate. The boundary conditions of the diffusion equation for the capillary

cell state that the concentration of liquid at the end of the capillary (i.e. at the interface between the active and bulk liquid) must be zero. At the start of the experiment before pressure is applied this will be so. On the application of pressure this'interface will move down the capillary. This contraction in volume will cause an additional end effect due to the restricted diffusion in the capillary before diffusion takes place

into the bulk liquid. Thus the boundary conditions of the equation for which the diffusion coefficient is calculated will not be observed.

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1 0.

the true diffusion coefficient. 'Increasing the pressure of the

experiment will give an increase in the magnitude of'the effect and hence an increase in the difference between'the measured'and the actual

diffusion coefficient. Hiraoka et. al estimated their overall error to be between ±1% to ±4%.

Naghizadeh'and Rice (1962) used a capillary technique to study the self-diffusion of liquid argon, krypton, xenon and methane up to 140 atms'. and at temperatures from the boiling point to the critical

point. Their apparatus differed in two major features from that described above: (i) Because of the low temperatures involved, a valve was

built into the apparatus so that diffusion could be initiated only after the bath reached pressure and temperature equilibrium, and (ii) at the beginning of the experiment the bath contained the active liquid and the capillary the inactive.

2.3.4 Diffusion Column Techniques

This technique was developed by Clack (1908). The apparatus consisted of a column containing a solution of the liquid under study. By observing the changes in concentration at different levels throughout the column the diffusion coefficient could be calculated-. Walker (1950) analyzed the concentration by a radioactive technique; the active and the inactive liquids were separated in the column by a mesh~sieve and the radioactivity monitored by a Geiger-Muller tube.

Careri, Paoletti and Salvetti (1954) used a similar technique

to measure the self-diffusion of liquid indium. They melted the radioactive and inactive indium on top of each other in a capillary. Diffusion was allowed to take place for a suitable time. At the end of the experiment the cell was cooled so that the indium solidified. The activity along the indium cylinder was measured and the diffusion coefficient calculated.

This technique has been used to measure the self-diffusion of liquid metals under pressure by a number of workers. Nachtrieb and Petit

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11. to take place. At the conclusion of the experiment the discs were

rotated so that the capillaries were separated. Samples were removed from the discs and analysed for the radioactive liquid. Nachtrieb and Petit estimated their errors to be between ± 2 % to ±4% for the pressure experiments. They reported data for a number of liquid metals including mercury and gallium.

R.C. Robinson (1965) employed a similar technique to study the diffusion of liquid carbon dioxide and:' propane. This apparatus consisted of - four rotatable discs which'contained the fluid in a set of aligned holes. After termination of a run the contents of the four holes were individually swept out into sample bombs. The samples were expanded into an ionization chamber for counting of the

radioactive tracer at about atmospheric pressure. Diffusion experiments were performed with carbon-14 labelled’carbon dioxide and carbon-14 labelled propane for three temperatures and a series of pressures up to 168 atmospheres.

The difficulty with this technique is to be sure that mass transfer occurs in the column only by diffusion. Any thermal

gradients that may be present could cause convection and give rise to higher diffusion coefficients. A number of workers have avoided this problem by using a packing in the column. These techniques will be discussed in the next section.

2.3.5 Fritted Disc Techniques

This technique was originated by Schultz (1914) and developed by Aten and Dreve (1948) and Wall et. al (1952). A thick, sintered disc containing one solution is immersed in a bath containing another solution. After a short period the disc is removed and its contents analysed. The difference between the concentrations at the beginning and the end of the experiment are used to determine the diffusion coefficient. However it is necessary to know the diffusion

characteristics of the disc. These are determined by a calibration experiment with a species of known diffusion coefficient.

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12. liquid diffused into the disc. Mysels and Stiger (1953) used two

fritted discs containing solutions of"different concentrations. The discs were placed in contact with each other and diffusion allowed to take place. At the end of this experiment the contents of each

disc were analysed and the diffusion coefficient calculated.

This technique was used for high pressure studies by Drickamer and co-workers for a series of liquids and liquid mixtures. Timmerhaus and Drickamer (1952) reported the diffusion coefficient of the

14

system GO^ - CC^ in the pressure range 100-1000 atms. and at temperatures of 0,° 25°, and 50°C. In their apparatus two fritted discs were located in a bomb in such a way that if the top one was rotated 180° the upper and lower surfaces of the respective discs would be in contact with each other. The top disc contained active liquid and the lower disc inactive liquid. After pressure and temperature

equilibrium had been reached the top disc was rotated 180° and diffusion allowed to take place from the top to the bottom disc. The increase in radioactivity of the contents of the bottom disc was monitored by a scintillation technique. To calculate the diffusion coefficient it was necessary to determine the diffusion characteristics of the discs by a calibration technique.

Koeller and Drickamer (1953) adapted this technique to measure the 35

diffusion of the system CS2~CS S at pressures up to 10,000 atms. and temperatures of 0 , 20° and 40°C. The modification of the technique introduced a further complication. This was that the discs containing the active and inactive liquids were placed in contact before the pressure was applied. After the application of pressure the active liquid was assumed to flow into the bottom disc by a plug type flow. It was then

necessary to estimate the distance from the active/inactive interface to the bottom of the lower disc. This calculation could introduce further errors. Cuddeback, Koeller and Drickamer (1953) used this apparatus to study the diffusion of the system TH0 - H00 up to 10,000 atms and at temperatures of 0°, 25° and 50°C.

Cova and Drickamer (1953) with a similar technique studied the

self-diffusion of liquid sulphur under pressure using a series of fritted discs. Radioactive liquid was added to the top disc and at the end of the experiment the discs were separated and their activity determined. Calibration of the discs was obta ned from an experiment carried out at

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2.4 Comparison of Pressure Diffusion Data from Different Techniques

13.

One of the difficulties in estimating the errors inherent in the techniques described above is that there is very little data available which could be used for comparison. There are, in fact, only a few liquids whose diffusion coefficient has been measured

at high pressures by more than one method. These include water, carbon dioxide and benzene.

Three different groups of workers have reported high pressure diffusion data for water. Benedek and Purcell (1954) and McCall, Douglass and Anderson (1959b) employed NMR spin echo techniques whilst Cuddeback, Koeller and Drickamer (1953) used a fritted disc. Benedek and Purcell (1954) reported a normalised diffusion coefficient (D /D )

o 2 8 —2 ^

at 28.8 C and seven pressures up to 9200 kg/cm (- 9 x 10 Nm ) with errors of the order of ±5%; McCall, Douglass and Anderson (1959b) reported an activation volume at 28°C and pressures up to 700 atms

(- 7 x 10^ Nm “); finally Cuddeback, Koeller and Drickamer (1953) reported the diffusion coefficient of tritiated water in inactive water at 0°, 25°, and 50°C at pressures up to 10,000 atms ( - l x 10^

_2

Nm ). Although the three sets of data were reported at different temperatures (25°, 28°, and 28.8°C) it is assumed that the temperature effect on the pressure dependence will be small. The three sets of

5 5 - 2

data overlap from 1 x 10 to 700 x 10 Nm . Up to this pressure the data of McCall et. al reproduces earlier measurements of Benedek and Purcell. At higher pressures there is considerable disagreement between the data of Cuddeback et. al and Benedek and Purcell. Thus at 1000 x 10^

-2 5 -2

Nm the difference is of the order of 20% whilst at 9000 x 10 Nm it is over 100%. In addition to this disagreement there is also considerable difference in the reported change in diffusion coefficient with pressure. Cuddeback et. al reported an increase in the diffusion coefficient from

5 - 2 5 - 2

atmospheric pressure ( 1 x 10 Nm ) up to 1000 x 10 Nm , whilst the other two sets of data showed no change in this pressure range.

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14. The effect of pressure on the self-diffusion of liquid carbon

dioxide has been studied with a fritted disc technique (Timmerhaus and Drickamer, 1952) and a segmentated column technique (R.C. Robinson 1965). These two sets of data generally are not in agreement. Thus at 273°K and 1.38 x 10^ Nm ^ Timmerhaus and Drickamer report the

-9 2 -1

diffusion coefficient D to have a value 0.4 x 10 m sec . whilst -9 2 -1 an interpolated value from Robinson's data would be 1.0 x 10 m sec Similarly at 298°K and 1.14 x 10^ NM ^ Timmerhaus and Drickamer report

-9 2 -1

a value of 0.9 x 10 m sec whilst Robinson gives a value of 1.8 x -9 2 -1

10 m sec

Data for benzene has been obtained with the spin echo technique (McCall, Douglass and Anderson 1959b) and the capillary cell method (Hiraoka et. al 1958), McCall et al report an activation volume at 28°C and for pressures up to 700 atms whilst Hiraoka et. al report the diffusion coefficient at 25°C and at three pressures from 1 to 470 atms. Within experimental error (± 5 - 10%) the results for the effect of pressure are in close agreement. This agreement is discussed and illustrated in Chapter Six (see also figure 6.2).

2.5 Summary

From the proceeding discussion it can be seen that experimental work on the self-diffusion of liquids is limited. The NMR spin echo technique offers the best prospects of obtaining data quickly at various pressures. However until the errors associated with this

technique have been significantly reduced it would appear that better data could be obtained with a tracer method. At atmospheric pressure the

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15.

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Chapter Three

The Diaphragm Cell Plan of Chapter

3.1 Introduction 3.2 Diaphragm cell

3.3 Theory of self diffusion measurements with a diaphragm cell

3.4 The design requirements of pressure cell 3.5 The first pressure cell

3.6 The second pressure cell 3.6.1 Construction

3.6.2 The diaphragm

3.6.3 The bottom compartment 3.6.4 The top compartment 3.6.5 The stirrers

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3.1 Introduction

The general principles that have determined the development of the diaphragm cell method for measuring diffusion coefficients will be discussed in this chapter. It will be shown how both the theoretical and practical requirements have been satisfied in the design of the high pressure diaphragm cell. Finally, the individual components of the cell and the manner in which they are assembled will be described.

3.2 The Diaphragm Cell

Important reviews of the diaphragm cell method include those of Gordon (1945), Stokes (1950), Janz and Mayer (1966) and Mills and Woolf (1968). The use of this technique to measure diffusion coefficients was first proposed by Northrup and Anson in 1928. In this method a diaphragm separates two solutions of different

concentrations. The fundamental feature of this technique is that mass transfer occurs in the diaphragm only by isothermal diffusion.

It is the function of the diaphragm to exclude any other possible contributions such as those due to thermal or mechanical transport. The concentration changes on both sides of the diaphragm are

determined over a period of time and used to calculate the diffusion coefficient.

Since 1928 developments in both the practical and theoretical aspects of the method have improved the accuracy of the technique. The conventional form of a diaphragm cell, described by Stokes in 1950, is shown in figure 3.1. This cell consists of a glass cylinder divided into two equal compartments by a horizontal sintered glass diaphragm. The diameter of the pores of this diaphragm should lie between 2 to 10 microns so as to obtain reproducible results

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18.

AUUUUWW

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3.3 Theory of Self Diffusion Measurements with a Diaphragm C e l l .

In this section a method of calculating the diffusion

coefficient from the experimental data will be developed. The

self diffusion experiment can be considered to be an

investigation of a binary liquid system, whose components are

isotopic forms of the liquid. The diffusion coefficient is

assumed to be independent of the isotopic form of the liquid

and hence remains constant throughout the experiment. A further

assumption is that the mixing of the isotopes during the experiment does not give rise to any significant changes in volume or density. For all simple isotopic species, except those of low mass number, these two assumptions are reasonable.

A simple analysis of the diaphragm cell experiment can be developed for any two component system in which these assumptions

are valid. There is^however, one major assumption that must be

made: There is a linear concentration gradient of both components in the diaphragm throughout the experiment.

The rate of change of concentration in a compartment is determined by the total flow into and out of the compartment

through the diaphragm. Thus in the case where diffusion takes

place from the top compartment to the bottom compartment.

dC,

t = - J(t) A (3 .1)

dt V,

T

dC

B = J(t) A (3 .2)

dt

where

C = concentration in top compartment

C = concentration in bottom compartment

D

t = time

J(t) = diffusional flow, which is a function

of the time t, and is in units of mass per unit area.

(30)

Combining equations (3.1) and (3.2)

d(W

-J(t)A

VT VB

(3.3)

From Fick’s first law the diffusional flow can be related to the concentration gradient by the diffusion coefficient D.

J(t) - D dc

dx

Applying (3.4) to diffusion in the diaphragm J(t) ~ - DI

(3.4)

(3.5)

where l = the vertical length of thh diaphragm. Substitution of (3.5) in (3.3) gives

d (C - C )

T B D ( CT - Cß ) A

Rearranging (3.6) d In (CT - Cfi)

-

*

i|

T B

J

D A

T

1

+

1

n

a

vm

v

L T B J

dt - T B

Intergrating between the start (t = 0) and end (t experiment.

where

c

-

c

o T o B I D A t 1 +

, c

c

a

V

I T - 1 B J w x

0CT - concentration

the start of

0 CB = concentration

at the start

1CT = concentration

finish of the

1CB = concentration

finish of the

U

(3.6)

(3.7)

t) of the

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21.

Rearranging

A

1CT

A

1CB 1 +

V

(3.9)

The quantities inside the brackets in the denominator of equation 3.9 are all characteristic of the geometry of a particular diaphragm cell. Consequently this term is known as the cell

constant,

3 +

VT

(3.10)

Throughout the above elementary theory it has been assumed that the concentration gradient in the diaphragm remains linear throughout the experiment. Barnes (1934) showed that this cannot be the case, and for various initial conditions in the diaphragm he derived exact

mathematical expressions to obtain D. Using the limitations that the volumes of the compartments are equal and that the ratio of the

diaphragm volume to the total volume of the compartments is less than 0.02 his expressions can be simplified as follows.

Gradient-filled Diaphragm

In the case of an initial linear concentration gradient in the diaphragm (experimentally known as the gradient-filled diaphragm technique), the assumption of a linear gradient produces negligible error when equation 3.9 is used to calculate D.

Solvent-filled Diaphragm

He considered a second case in which one compartment and the diaphragm initially contain zero concentration of one component, and the other compartment contains a solution of the components,

(32)

2 2.

In this case equation (3.9) must be replaced by

In o n H

1

o V

[i

- a ~

Li c t - icb

J

l_

6

J

D

t A

£

1 + V_

where 2 V

V V T + B V

D = volume of diaphragm.

(3.11)

Solution-filled Diaphragm

Mills, Woolf and Watts (1968) using Barnes' analysis obtained an expression for a third case in which the diaphragm and one

compartment are filled initially with a solution of both components and there is zero concentration of one of the components in the other compartment. (This is known as a solution-filled diaphragm technique). They further reported that an experimental investigation of the same system using the three initial conditions, gradient-, solvent-, and solution-filled, in the diaphragm, resulted in the same value of D.

Of the three techniques considered above, the most convenient for self-diffusion measurements is that of the solvent-filled diaphragm. In this technique the diaphragm cell is filled with pure liquid. At the start of an experiment a quantity of a labelled species of the liquid is injected into one compartment

of the cell. After a suitable interval the top and bottom compartments are emptied and the concentration of the labelled species determined.

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23.

Toor (1960) pointed out that in practice the pores are actually meandering paths which give rise to local diffusion in other planes.

However he proved mathematically that in both cases the result is the same .

Thus, in equation (3.10) th e ’A' and terms refer to "effective" areas and lengths and so usually cannot be measured directly. They are determined using a calibration technique. This involves a diaphragm cell diffusion experiment using a system whose diffusion coefficient is known.

3.4 Design Requirements of Pressure Diaphragm Cell

From the discussion in section (3.3) the requirements for a diaphragm cell to measure the self-diffusion of liquid are:

(a) The diaphragm must have a pore diameter between 2-10 microns.

(b) The volumes of the top and bottom compartments should be closely matched.

(c) X must be less than 0.02.

(d) It is necessary to prevent the formation of surface layers on the diaphragm and ensure the contents of each compartment are homogenous.

In addition to these requirements a pressure cell must also have the ability to withstand the applied pressure and transfer it. to the liquid inside the cell. Although it is possible to combine the diaphragm cell and a pressure bomb as a single entity it is more convenient to separate them. Thus the diaphragm cell will not then have to withstand a high differential pressure.

To transfer pressure from outside to inside the cell a number of techniques have been developed. These include bellows, pistons and mercury (Bridgman 1949). Although pistons have been used in an apparatus of similar design (Eckert and Grieger, 1970)

(34)

2 4.

For a high pressure diaphragm cell experiment a desirable feature would be the ability to monitor the concentration changes in the compartments throughout the experiment. This would

eliminate errors that may be caused by the application and release of pressure. To develop these techniques would require considerable investment in time and equipment. Even at atmospheric pressures previous attempts to develop suitable equipment to monitor

3 radiation have not been successful (Tilley, 1967). Consequently it was decided in the first instance to construct a cell whose concentrations would be analysed after the pressure was released.

3.5 The First Pressure Diaphragm Cell.

A cell was constructed according to the principles outlined in section (3.4). In this design both top and bottom compartments consisted of metal bellows. A series of diffusion experiments were performed at constant pressure. These showed that the results obtained were dependent upon the loading conditions at the start of the experiment. Results differing by an order of 40% were obtained if the experiment was started by injecting the radioactive liquid

into the bottom as opposed to the top compartment. Further experiments showed that the initial application of pressure definitely caused

non-diffusional flow from one compartment to the other, indicating that the bellows in each compartment reacted differently to the pressure.

Because of these results this cell had to be modified as described in section 3.6.

3.6 The Second Pressure Diaphragm Cell

The design of the first pressure cell was significantly modified and a second cell constructed. In the modification one of the

(35)

25

.

The experiment can be set up in such a way that the size of this correction is minimised. It involves starting the experiment

by injecting the labelled species into the top, or non-compressible compartment.

It can be seen from the requirements outlined in section (3.4) that it is necessary to have the volumes of the top and bottom compartments closely matched. As the volume of the bellows

compartment is dependent on the pressure, the other compartment is designed so that its volume can be set at a pre-determined value before pressure is applied. The feature which allows this is discussed

in section (3.6.4).

3.6.1 Construction of the Pressure Diaphragm Cell

Photographs of the high pressure cell are shown in figures 3.2 and 3.3. The cell is approximately 6M long and 1.5" diameter. A cross- sectional diagram is shown in figure 3.4 and an engineering drawing in figure 3.5. For convenience in construction and use the cell was made in five main parts. These are the diaphragm, the top and bottom

compartments and the top and bottom stirrers. When assembled, these parts are held together by six screws. On each metal to metal junction a groove was machined to hold a PTFE square-sectioned sealing ring. The dimensions are given in figure 3.5 and in the following discussion of the individual components only the dimensions of importance are quoted. In the series of experiments reported here three cells were used.

(36)

26

.

(37)

2 7

.

TOP

PLUG

M O N E L THREADED BODY

VOLUME ADJUSTMENT RING

M A I N BODY OF TOP

TOP STIRRER

COMPARTMENT

DIA PH R A GM

BOTTOM PLUG

V

(38)

28.

Top plug

Threaded top section

Dual paddle stirrer

Volume adjustment ring

Fixed top section

Stirrer support

Sinter

Sinter support ring

Bellows

Bottom plug

Bellows guide

(39)

29

F

i

g

u

r

e

3

.

5

Eng

i

ne

er

i

ng

D

r

a

w

i

n

g

o

f

P

r

e

s

s

u

r

e

C

el

(40)

3.6.2 The Diaphragm

The diaphragm in a Stokes-type cell is usually sealed along the edges by glass walls. In the high pressure cell these walls are made of stainless steel in the form of a location ring

(referred to as ring M ) . The diaphragm is sealed in this ring. The diaphragm to be used in this work must be both robust and able to be machined and the edges sealed in stainless steel. Platinum and stainless steel have previously been found to be suitable diaphragms for the diaphragm cell (Ivielsen 1952).

Throughout this work three diaphragms, one of platinum and two of

stainless steel were used. Their characteristics, including pore sizes are given in cable 3.1.

Difficulty was encountered sealing the diaphragm into the metal ring. Initially the sealing was accomplished by embedding the edges of the diaphragm in PTFE and holding the sintered disc by means of a threaded ring as shown in Appendix C. This technique was only successful in a few cases and was used for the platinum sinter of cell A.

The seal was tested by containing the assembled ring in a horizontal position between two "perspex tubesP The top tube

(or compartment) was partially filled with water so that the top surface of the diaphragm was covered with a layer about 1" deep. The pressure of air in the bottom compartment was slowly increased until bubbles of air appeared on the top surface of the diaphragm. The air pressure was held constant so that a slow stream of

(41)

T a b l e 3 . 1

C h a r a c t e r i s t i c s of D i a p h r a g m s

M a t e r i a l

P o r e S i z e (m )

T h i c k n e s s ( i n c h e s )

S u p p l i e r

P l a t i n u m

1 x 1 0 " °

3 / 3 2

Mott M e t a l l u r g i c a l C o r p . , H a r t f o r d C o n n . , U . S . A.

S t a i n l e s s S t e e l

5 x 1 0 ~ b

1 / 1 6

P a l l ( U . K . ) L t d . L o n d o n ,

(42)

32

.

A second method of sealing was found to be more successful and was used for cells B and C. The sintered material was

machined into a disc 0.030" larger than required to fit into the sealing ring M. The disc was then clamped between two identical stainless steel rings whose diameters were slightly smaller than the sintered disc. The overhanging edge of the disc was then welded to the rings using a tungsten inert gas technique (T.I.G). During the welding operation copper heat sinks (copper rods in contact with the surface of the sintered disc) were placed in such a position that heat was quickly removed from the surface of the diaphragm. These heat sinks ensured that melting only took place at the edge of the sinter. The excess weld was removed to leave a smooth surface. In this way the diaphragm was made an integral part of a single stainless steel ring. The outside diameter of the disc was then about 0.0015" larger than the

inside diameter of the metal ring M. The disc was then contracted by liquid nitrogen, placed in position in the ring M and allowed to expand to form a seal. When assembled the diaphragm and the locating ring were tested for leaks by the method described above.

3.6.3 Bottom Compartment

The bottom compartment consists of four separate stainless steel components: a ring, a bellows, a valve body and a plug as shown in figure 3.3 . The ring, the bellows and the valve body were T.I.G. welded in a manner similar to that used for the diaphragm.

(43)

33

.

The entrance to the bottom compartment is through a threaded hole in the valve body. This is sealed by a stainless steel plug. To prevent buckling of the bellows during the application of pressure, movement of the valve body is guided by a stainless steel tube. This guide tube is slotted to allow easy circulation of the hydraulic fluid.

A grub screw, located in a threaded hole in the- bellows guide, is used to lock the bellows in place. This prevents the bellows from twisting when the sealing plug is screwed into, or out of, the entrance hole in the valve body. The bellows are also locked with the grub screw during the procedures used to determined the volume of the bottom

compartment and during a calibration experiment. This ensured that the volume of the bottom compartment for a calibration experiment was fixed precisely.

3.6.4 Top Compartment

The volume of the top compartment must remain constant throughout the application of pressure. However the compartment was designed so that its volume could be pre-set to a required value before an experiment. This was achieved by incorporating into the inside wall of the compartment

a stainless steel ring. This ring can be replaced by others of different thicknesses. The ring is located between a stainless steel body and a

threaded monel valve body. The metal to metal surfaces of the cylinder, ring and valve body are sealed by PTFE rings. Four holes were machined in the top of the valve body so that it could be screwed down onto the PTFE rings by means of a special tool. The valve body was made from monel rather than stainless steel to eliminate the possibility of the threads on the valve body and stainless steel cylinder binding together. The inside surfaces of the valve body, rings and cylinder were highly polished.

3.6.5 The Stirrers

(44)

34.

stirrers in the pressure diaphragm cell are significantly different from the Stokes stirrers. They must be (i) able to withstand high pressures, and (ii) function over a range of densities.

As shown in figure 3.6 each stirrer consists of two blades of magnetic stainless steel on either end of a shaft. This shaft is located in a stirrer support which fits into the location ring M. In operation a long flat blade set about 0.010" from the diaphragm sweeps the diaphragm face while a shorter but much wider blade ensures stirring of the contents of the compartment away from the diaphragm.

Two design of stirrers were used in this study.. In the first design the central shaft rotated freely upon PTFE bushes. After repeated experiments and cleaning procedures including washing in hot solvent (see Chapter Four), the PTFE bushes

tended to become distorted and stop the blade from rotating.

To overcome this difficulty the bushes and shaft were removed from each stirrer and replaced by a lightly polished stainless steel shaft which had a spiral groove cut on the surface so that liquid flows between the shaft and the stirrer support thus acting as a lubricant.

3.6.6 Assembly

During assembly of the cell it is vital that the individual components be aligned correctly. The components are placed together in the correct order, and the screws are located. The cell is then placed in a clamp and mechanical pressure applied so that the PTFE sealing rings are uniformly compressed. Before releasing the pressure the six screws are tightened using an Allen key.

(45)

3 5.

(46)

Chapter Four

Experimental Techniques Plan of Chapter

4.1 Introduction

4.2 Pressure techniques 4.3 Temperature techniques 4.4 Rotating magnets

4.5 Safety precautions

4.6 Purification of chemicals 4.7 Determination of volumes 4.7.1 Volume of diaphragm

4.7.2 Volume of top compartment 4.7.3 Volume of bottom compartment

4.8 Procedures required for a typical experiment 4.8.1 Cleaning the cell

4.8.2 Adjustment of top compartment volume 4.8.3 Test of stirrers

4.8.4 Filling the cell 4.8.5 Calibration equipment

00

Pressure experiment

(47)

37

.

4.1 Introduction

The procedures that are employed in a diaphragm cell experiment are discussed in this chapter. Initially we will describe the equipment and the techniques employed to ensure that the diaphragm cell remains at constant pressure and temperature throughout an experiment. The

magnets, used to rotate the stirrers inside the cell, will then be described. Because of the toxic nature of the liquids used in these experiments it is necessary to take special precautions when handling them. An outline of these precautions and of the method used to purify the liquids will be given. This will be followed by a description of the methods used to determine the volumes of the compartments and the diaphragm.

The remainder of the chapter will be a discussion of the procedures necessary for a typical experiment. These procedures can be outlined as follows:

(i) Initially the cell was cleaned, assembled (see Chapter Three) and the volume of the top compartment adjusted to the required volume;

(ii) The stirrers were then tested to see if they functioned correctly and the cell filled;

(iii) An experiment, either a calibration or a pressure experiment, was then started;

(iv) At the end of an experiment the compartments were emptied and the solutions analysed.

4.2 Pressure Techniques

(48)

3 8.

B X 2 X A

/ * / / / / / • / i t / 1 / / / / / /

4

K ey

1 . 4 0 ,0 0 0 p s i H y d r a u lic Pum p,

2 . T w o w ay v a lv e s : A is o la tin g v a lv e , B r e le a s e v a lv e .

3

. P r e s s u r e gauge (0 - 25 , ooo psi o r 0 -

3

,0 0 0 p s i ) , 4 . W a te r b a th ,

5 . O il bath ,

6 . B a r m agnets and fr a m e ,

7 . S t i r r e r m o t o r ,

8 . 5 0 ,0 0 0 p s i p re s s u r e v e s s e l,

9 . B ath s t i r r e r m o to r.

(49)

T a b l e 4 .1

C o m p o n e n t s of P r e s s u r e S y s t e m

P r e s s u r e C o m p o n e n t

S u p p l i e r P r e s s u r e R a t i n g

( p . s . i )

D i m e n s i o n s

B o m b ( B e - C u) J . L y s a g h t ( A u s t ) L t d . S y d n e y . A u s t .

0 - 5 0 , 0 0 0 8 " l o n g

2 3 / 8 " o u t s i d e d i a . 1

\

" i n s i d e d i a .

P u m p E n e r p a c

B u l t e r , Wis . , U. S . A.

0 - 4 0 , 0 0 0

T u b i n g A m i n c o

S i l v e r S p r i n g , Ma. , U. S . A.

0 - 6 5 , 0 0 0 3 / 3 2 " i n s i d e d i a .

V a l v e s A m i n c o

S i l v e r S p r i n g , Ma. , U. S . A .

0 - 6 0 , 0 0 0

G a u g e s B u d e n b e r g C o . L t d .

M a n c h e s t e r , U . K .

0 - 2 5 , 0 0 0 5 " d i a .

N a t i o n a l I n s t r u m e n t C o . 0 - 3 , 0 0 0 M e l b o u r n e . A u s t .

(50)

40. The maximum pressure used in the diffusion experiments was 25,000 p.s.i.

-2

(17.2 MN m ). As a safety measure the pressure ratings of all components were considerably higher than this. These components will now be discussed individually.

A Blackhawk hydraulic hand pump with a maximum pressure

rating of 40,000 p.s.i. was used. For the majority of the experiments a commercial grade hydraulic fluid, Blackhawk PLS, was used as the pressure transmitting medium. However in certain cases, discussed later, this fluid was replaced by methyl cyclohexane.

Two pressure gauges, both of the Bourdon type, were used

throughout the experiment. One gauge covered a range 0-3000 p.s.i. whilst the other covered the range 0-25,000 p.s.i. They were obtained

from N.I.C. Pty. (Australia) and Budenberg Co. Ltd. (Great Britain) respectively and were calibrated to ± 1%.

The pressure tubing, connections and taps made by Aminco were capable of withstanding pressures up to 60,000 p.s.i. The tubing was attached to other components using a standard high pressure technique. A collar was located on a left hand thread near the pressure tube’s end, which in turn was coned to 60°. A hexagonal nut, which fitted over the collar and the tube, screwed tightly

into the body of the tap (or relevant component).

A diagram of the pressure bomb is shown in figure 4.2. The cylindrical bomb was attached to a horizontal stainless steel base plate by four screws. These could be adjusted so that the bomb was vertical. The pressure tubing passed through a hole in the plate and was connected to the bomb using the technique described above. The base plate was in turn suspended in the water bath by three rods.

The bomb, which was made from beryllium copper, had a wall thickness of 7/16". It was pressure tested at the National Physics Laboratory, Sydney, to 50,000 p.s.i. The outside surface of the

(51)

4 1.

-Screw

Closure Nut

Sealing Plug

Sealing O Ring

Support Rod

Pressure Bomb

Pressure Feed-through

Support Screw

Support Plate

References

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