From Deep Sea to Laboratory 3
Illustration representative of the book: the Challenger expedition (route, vol. 1), physical measurements (samples, vol. 2) and the compressibility of liquids (globes, vol.3)
From Deep Sea to Laboratory 3
From Tait's Work on the Compressibility of Seawater to Equations-of-State for Liquids
Frédéric Aitken Jean-Numa Foulc
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com
Cover image © John Steven Dews (b.1949), H.M.S. Challenger in Royal Sound, Kerguelen Island, in the Southern Ocean (oil on canvas).
© ISTE Ltd 2019 The rights of Frédéric Aitken and Jean-Numa Foulc to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2019943766
British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-376-9
Contents
Foreword ...... vii
Preface ...... xi
Notations ...... xv
Chapter 1. The Compressibility of Liquids and Tait’s Equation-of-State ...... 1 1.1. Introduction ...... 2 1.2. Concepts of compressibility ...... 3 1.3. The first instruments to measure the compressibility of liquids ...... 5 1.4. The piezometers used onboard the Challenger ...... 21 1.5. Sources of pressure measurement errors ...... 24 1.5.1. Apparent compressibility of water and mercury ...... 24 1.5.2. Apparent compressibility of liquid and piezometer ...... 27 1.6. Compressibility of fresh and salt water ...... 32 1.6.1. Results on fresh water compressibility ...... 34 1.6.2. Results on seawater compressibility ...... 38 1.6.3. Results on the compressibility of saline solutions ...... 40 1.6.4. Equilibrium of a water column ...... 42
Chapter 2. Interpretations of the Parameters of Tait’s Equation ...... 45 2.1. Introduction ...... 46 2.2. Comparison and analogy with the Boyle–Mariotte equation-of-state ...... 46 2.3. Comparison and analogy with the Hirn equation-of-state ...... 54 vi From Deep Sea to Laboratory 3
2.4. Comparison and analogy with the van der Waals equation-of-state ...... 84 2.4.1. The molecular motion model ...... 88 2.4.2. Establishing the van der Waals equation ...... 94 2.4.3. The different expressions and interpretations of covolume ...... 111
Chapter 3. Tait–Tammann–Gibson Equations-of-State ...... 147 3.1. Introduction ...... 148 3.2. Examples of compressibility equations-of-state ...... 150 3.3. Evolution of the parameters of the mixed modulus ...... 155 3.3.1. Application in the case of fresh water ...... 160 3.3.2. Application in the case of standard seawater ...... 168 3.3.3. Application in the case of helium-4 ...... 179 3.3.4. Application in the case of helium-3 ...... 192 3.3.5. Density anomalies ...... 199 3.3.6. Compressibility anomalies ...... 201 3.4. Discussion and conclusion ...... 207
Chapter 4. The Modified Tait Equation ...... 245 4.1. Introduction ...... 246 4.2. Development of a complete equation-of-state ...... 249 4.3. Study of the adiabatic elastic modulus ...... 255 4.3.1. Application in the case of fresh water ...... 255 4.3.2. Application in the case of helium-3 ...... 264 4.3.3. Application in the case of helium-4 ...... 271
Conclusion ...... 279
Appendices ...... 283
Appendix A. Compressibility of a Straight Tube ...... 285
Appendix B. Virial Theorem ...... 291
References ...... 335
Index ...... 343
Summary of Volume 1 ...... 347
Summary of Volume 2 ...... 351
Foreword
It is a beautiful adventure that Frédéric Aitken and Jean-Numa Foulc have undertaken, using physical data from the Challenger expedition, the first major oceanographic expedition, sponsored by the British Admiralty in the 1870s. Indeed, this data, temperature and pressure readings at various depths and at multiple points of the world, was relatively little used at the time despite the visionary intuition of one of the initiators of the expedition, Professor Carpenter, that this data would allow for the reconstruction of ocean circulation. The authors attribute this relative lack of interest to the fact that most scientists on the expedition were naturalists, and that from the point of view of biology, the total benefits were already huge, with, for example, the discovery of life at a great depth.
Exploiting data is not the least interesting of the physicist’s tasks. To deal with the problem, we simplify the situation and try not to delete anything essential. The terms of the equations are evaluated, keeping only the most important, and then two situations may arise. Let us say that the discrepancy with the data is clear: we are generally convinced that it has been oversimplified, but where? We are tempted in bad faith to defend our idea, even if it means becoming the Devil’s advocate and destroying what we have built. We go back to the overlooked terms one by one, and, with some luck, this may lead to a new effect. We make do with what we know; the battle is tough, and this is its appeal.
Let us say that the similarity is acceptable. This is when a good physicist is suspicious: is it not a coincidence that two important effects are not offset by any chance? It would be necessary to make a prediction, and to repeat the experiment in different conditions, but it is not always possible. Another boat was not sent out with 200 people around the world for three years! The rigor with which experiments have been conducted, and the confidence that can be placed in the measures, are essential. The experimenters have had to multiply the situations blindly, without knowing viii From Deep Sea to Laboratory 3 which ones would be used as a test, with the sole aim of doing their best every time, by describing their protocol for future use.
The development of the measurement protocol is part of the experiment’s design, as was instrument construction. At that time, a physicist worth his salt would never have used an instrument that he did not know how to build. How can one measure a temperature in a place that one cannot reach oneself (2,000 m below the surface of the sea, for example)? We can record the maximum and minimum temperatures reached during the descent (I found, with much emotion, the description of the maximum and minimum thermometer used by my grandfather in his garden). But what to do for intermediate temperatures? How to make sure that the line does not break in bad weather under the boat’s blows? How to decide the real depth despite currents, and the fact that the line continues to run under its own weight once the sensor is at the bottom? The design phase of the experiment can be exciting: I knew a physicist who was ready to sabotage a barely built experience (under the pretext, of course, of improving it) to be able to move more quickly to the design of the following experiment.
Despite all the attention given to the design, sometimes an error is suspected in the measurements. This is the case here. Having reached unexpected depths (they discovered the Mariana Trench), the Challenger scientists wondered if their measurements had not been distorted by contraction of the glass envelopes. After their return, they assigned Peter Tait, a physicist from Edinburgh, the task of assessing these errors. One thing leading to another, he raised questions about the compressibility of seawater, and other liquids, and so about their equation-of-state, connecting pressure, temperature and density (and even salinity). The result of his studies left a lasting mark on the physics of liquids. Estimating errors, a task hated and despised by the typical physics student, yielded new knowledge.
From the same period as the van der Waals equation, Tait’s efforts were part of the first trials to represent the equation-of-state of dense, liquid and solid bodies by continuous functions. The goal was twofold: metrological, to interpolate between experimental results, and to provide experimenters and engineers with the most accurate characterization of the thermodynamic and physical properties of the fluids they use. But also more fundamental, in the wish to have a better understanding of the underlying physical mechanisms: formation of molecular aggregates, local crystalline order, shape of interaction potentials, etc. These two interests, pragmatism and rigor, are often in conflict, as is clear from the authors’ account, who apply the ideas from that time to fluids that were not of concern then, such as the fluid phases of the two stable isotopes of helium.
Foreword ix
Many aspects of this scientific adventure are thus universal, and it is touching to see how the value codes of the scientific approach have been transmitted over decades, or almost centuries. But our step back in time gives us an advantage: the ability to judge the ideas from that period in light of the extraordinary sum of knowledge that has been accumulated since. However, a direct comparison would be unfair and clumsy. It is much more interesting to put us in the mindset of the players of that era, to share their doubts, their hesitations and even their mistakes. This is an aspect that is too often absent from our education. For the sake of efficiency, we do not mention brilliant ideas that have led to a stalemate. Yet these ideas may contribute elsewhere. There may be some hesitation in mentioning great names such as Clausius, Joule and van der Waals, who fill us not only with humility in the face of the mastery that allowed them to find the right path, but also with confidence when faced with our own doubts. The variety of players and points of view that have marked this period show how much science is a collective adventure.
It is all of this that I found in this book by Frédéric Aitken and Jean-Numa Foulc, and even more: the human adventure that was this trip of three years around the world, the incidents, drama and joys, what it revealed about the personality of each participant, their lives which, for some, are also described, the moving relay that is transmitted when a change of assignment, or worse, death, interrupts a task. There is also the welcome reserved for the expedition, sometimes idyllic (ah! the difficulty of leaving Tahiti), sometimes colder, the importance of the band and personal talent of the participants, not to mention the providence that the Challenger represented for the Robinsons, abandoned on an island by a boat that was unable to come back for them. After reading the story based on the logbook, how can we not mention Jules Verne’s novels? It is the same period, that of a thirst for knowledge about our environment, accessible to all of us, acquired by real yet so human adventurers, so close to us. The credit goes to the authors for having dedicated so much time, energy and enthusiasm to this humanist and complete book, with the spirit of this laboratory where I had the pleasure to come for discussions during my years at Grenoble.
Bernard CASTAING Member of the French Academy of Sciences
Preface
In May 1876, the oceanographic expedition of the H.M.S. Challenger reached England after having sailed the seas of the world for more than three years. The main objectives of this expedition were to study animal life in depth, examine the ocean floor to improve knowledge of undersea reliefs, and observe the physical properties of the deep sea in order to establish the link between ocean temperatures and currents. The naturalist William Carpenter, one of the investigators of the Challenger expedition, suggested this previous point. However, although work on animal life was widely promoted after the expedition, the same was not true of the physical observations accumulated throughout the expedition because the theoretical knowledge of ocean dynamics was almost non-existent back then.
Another person played a decisive role after the return of the Challenger. It was the physicist Peter Tait, who was asked by the scientific leader of the expedition to solve a tricky question about evaluating the temperature measurement error caused by the high pressure to which the thermometers were subjected. On this occasion, Peter Tait used a new high-pressure cell that allowed him to accurately determine the correction to be made to the temperatures collected by the Challenger. Later, he embarked on more fundamental research on the compressibility of liquids and solids that led him, nine years later, to formulate his famous equation-of-state. Analysis of the properties of the compressibility of liquids is the second challenge of this book.
From Deep Sea to Laboratory has three volumes. The first volume relates the H.M.S. Challenger expedition and addresses the issue of deep-sea measurement. The second and third volumes offer a more scientific presentation that develops the two points raised earlier: the correlation between the distribution of temperature and ocean currents (Volume 2) and the properties of compressibility of seawater and, more generally, that of liquids (Volume 3). xii From Deep Sea to Laboratory 3
Presentation of Volume 3
Chapter 1 begins with a history of liquid compressibility measurement techniques and provides some details on the piezometers used during the Challenger’s expedition. This naturally leads us to present Tait’s work, starting from 1879, on the measurement of the compressibility of fresh water, seawater, mercury and glass, and we discuss his famous equation-of-state parameterized by two quantities.
Chapter 2 examines the physical evolutions and interpretations of the two parameters of Tait’s equation by using comparison and analogy techniques to discuss the best-known equations-of-state of the time, especially van der Waals’, to get a picture of the “structure” of compressed liquid media.
Chapter 3 proposes an in-depth study of the Tait–Tammann–Gibson equation (related to the isothermal mixed elastic modulus) and leads us to propose new equations-of-state that describe in particular the liquid phase of fresh water, seawater and helium-3 and 4. We show that these new relationships have a precision comparable to that of current reference equations. Different “anomalies” of these environments are then highlighted and discussed. Finally, we emphasize the difficulties encountered with various other approaches, other than Tait, Tammann and Gibson’s, in reproducing the compressibility properties of liquids in a simple way.
Chapter 4 focuses on the equation-of-state called the “modified Tait equation”, which is Tait’s ideas on the isothermal secant elastic modulus applied to the adiabatic tangent elastic modulus. It is an equation that is particularly well-suited for describing shock wave phenomena because it is a complete equation-of-state. After an in-depth theoretical study of the thermodynamic functions that can be deduced from the equation of the adiabatic tangent module, new equations-of-state are proposed to describe in particular the liquid and supercritical states of fresh water and helium-3 and 4. We also show here that these new relationships have a precision similar to that of reference equations. “Anomalies” on the adiabatic compressibility of these media are then identified and discussed.
Overview of Volumes 1 and 2
Volume 1 presents the context, organization and conduct of the expedition of the H.M.S. Challenger. The detailed account of the cruise is embellished with numerous illustrations (maps, photographs, etc.) that are rarely presented together. The key role of the officers and scientists involved in this cruise is highlighted, and a brief biography of each of them is presented. In the first volume, we also discuss the Preface xiii problem of deep-sea sounding, which at the time was a delicate and not always well-controlled operation. A theoretical approach to the immersion velocity of a lead is given and compared to the experiment. We end with a presentation of some results of bathymetric surveys and physical observations made by the Challenger’s scientists. Bathymetric surveys are used to represent typical and known seabed reliefs (e.g. the Mariana Trench, South-Atlantic ridges, etc.), and physical observations appear in the form of temperatures, salinities and densities depending on the depth.
In Volume 2, we examine the measurement and distribution of temperature within the ocean and its relationship with the ocean circulation. We begin by describing the evolution of temperature measurement techniques in the 19th Century, by recalling the impact of pressure (at great depths) on measurements. After pointing out that the ocean is composed of different strata, we develop a simplified model of the thermocline in interaction with other ocean layers. This proposed model is limited to thermal aspects (water temperature variation between the equator and the poles) and mechanical aspects (effect of the Earth’s rotation and wind action on surface layers) to establish a link between the cartography of major ocean currents and the distribution of ocean temperatures. The Challenger’s observations and physical data collected in the Atlantic, Pacific and Indian Oceans are analyzed for the first time and compared with more recent works. We end with a general presentation of the mechanisms leading to the global mixing of ocean waters, called the thermohaline circulation.
The book describes a “journey over and through water” with a cross-examination of human history, the history of science and technology, terrestrial and undersea geography, ocean dynamics and thermics, and the sciences dealing with the physical properties of liquids. Curious readers, attracted by travel, science and history, will discover the background and conduct of a great scientific expedition in Volume 1. Students, engineers, researchers and teachers of physics, fluid mechanics and oceanography will also find subjects to deepen their knowledge in Volumes 2 and 3.
We would like to warmly thank Bernard Castaing, a former professor at the Joseph Fourier University of Grenoble (France) and at the École Normale Supérieure of Lyon, France, for carefully reading the manuscript and for his pertinent remarks. We express our gratitude to Ferdinand Volino and André Denat, Senior Researchers at the CNRS, and Jacques Bossy, CNRS researcher, who kindly shared their observations and advice during the preparation of the manuscript and read the final manuscript. We warmly thank Armelle Michetti, head of the library of physics laboratories of the CNRS campus in Grenoble, for her contribution to the search for often old and restricted documents that enabled us to illustrate and support the historical and scientific parts of the book. xiv From Deep Sea to Laboratory 3
We also thank the people who gave us special support: Michel Aitken, Philippe Vincent, Yonghua Huang, Glenn M. Stein and J. Steven Dews.
Finally, we would like to thank the organizations and their staff who have graciously allowed us to use some of their iconographic holdings, and in particular the Natural History Museum in London, the National Portrait Gallery in London, the United Kingdom Hydrographic Office in London, the University of Vienna (Austria), the scientific museum of the Lycée Louis-le-Grand in Paris and Orange/DGCI Company.
Bibliographical references on specific points appear in footnotes and those of a more general nature are collated in the references section at the end of each volume. The footnote reference numbers always correspond to footnotes of that chapter.
Frédéric AITKEN Jean-Numa FOULC June 2019
Notations
a van der Waals parameter
A parameter of the Tait equation b covolume (or atomic volume) bHirn Hirn covolume bVdW van der Waals covolume
Β Tait parameter or the modified Tait equation
B2 second viral coefficient c sound celerity
C i Ginell parameter representing the total number of j-mers per unit volume
Cp heat capacity at constant pressure
CV isochoric heat capacity
D available volume e specific internal energy
E Young’s modulus xvi From Deep Sea to Laboratory 3
Ec kinetic energy
Ep potential energy
F internal energy
j-mer aggregate of j molecules (or atoms) of liquid ~ J parameter of the Tammann equation
k B Boltzmann constant
KS adiabatic compressibility
KT isothermal compressibility
LV latent heat of vaporization
m mass of a molecule
n number of particles (molecules or atoms) or parameter of the modified Tait equation
N Avogadro number
P absolute pressure Pp, amount of movement
Pc critical pressure
Q amount of heat exchanged by a system
R perfect gas constant
s salinity
S (a) surface area
(b) entropy Notations xvii t temperature in Celsius degree (°C)
T temperature in Kelvin (K)
Tc critical temperature
U internal energy
V volume at a given P and T
= V0 volume along the isobar P0 ; generally, P0 1 atm v molecular speed v 2 mean square velocity of molecules
V specific volume (V = 1/ρ)
t V i inner virial
W works exchanged by a system
Z n Ginell parameter representing the degree of association
------
α average coefficient of thermal expansion isovolume
β T isobaric thermal expansion coefficient
β average coefficient of isobaric thermal expansion
χ ()T Carnot function
Δ space available in a cluster of molecules
φ Ginell function
Γ total pressure in the Hirn model xviii From Deep Sea to Laboratory 3
Γ Grüneisen coefficient
η (a) stacking density of molecules
(b) efficiency
η c Carnot cycle efficiency
κ modulus of elasticity in volume
κ T tangent modulus
κ T secant modulus
κ~ T mixed modulus
λ e thermal conductivity of water
μ dynamic viscosity
Π parameter of the Tait equation ~ Π parameter of the Tammann equation
Πint internal pressure ρ density
ρ e density of seawater
σ (a) diameter of the spherical molecules
(b) radius of exclusion of molecules
ω volume of a molecule
Ω solid angle
ϒ free volume
1
The Compressibility of Liquids and Tait’s Equation-of-State
Peter Tait’s high pressure cell (source: Scientific report of the H.M.S. Challenger, narrative II, 1882)
From Deep Sea to Laboratory 3: From Tait's Work on the Compressibility of Seawater to Equations-of-State for Liquids, First Edition. Frédéric Aitken and Jean-Numa Foulc. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc. 2 From Deep Sea to Laboratory 3
1.1. Introduction
The temperature-measuring devices used by H.M.S. Challenger scientists have been presented in Volume 2. As pointed out, the high pressures in the seabed caused disruptions and led to errors in temperature measurements. After the return of the expedition, at the request of C.W. Thomson (scientific leader of the Challenger expedition), Peter Tait examined the conditions of use of the devices and carried out a detailed study to evaluate the temperature measurement errors due to the contraction of the solid and liquid components of the thermometers. It is on the basis of this particular study that we develop and present more general notions concerning the measurement of high pressures, the compressibility of liquids and the equations- of-state, with a particular focus on Tait’s equation-of-state in Volume 3. But first, let us look at a practical application using pressure measurement.
Challenger sailors found that ocean depth measurement using a sampling line was not always highly reliable, especially at great depths (see Volume 1, Section 4.4.3). The idea of using a pressure measurement to determine depth therefore appeared to be a relevant alternative. Indeed, a quasi-proportionality relationship exists between these two quantities:
P*()z z = [1.1] ρ × e ()P g
* zP )( P* = 0)0( where denotes the pressure gradient of the water, at depth z (with ), += * P is the absolute pressure ( a , PPPP a being the atmospheric pressure at the ρ () sea surface), g is the gravity field and e P is the density of the seawater.
ρ Knowing that the compressibility of seawater is relatively low (at 200 atm, e ρρ≅= decreases by less than 1%), we can consider that ee()P Const and that the relationship [1.1] becomes:
Pz*() zKPz≅=×*() [1.2] ρ × e g
The depth is therefore almost proportional to the pressure, with K ≅10 m atm-1 .
To measure pressure, an instrument capable of recording the contraction of a fluid or solid material subjected to pressure must be used. This, of course,
The Compressibility of Liquids and Tait’s Equation-of-State 3 presupposes that the response of the stressed material (dependence of its density or relative contraction as a function of the applied pressure) is known. A good knowledge of the compressibility of the materials used to manufacture pressure- (and temperature-) measuring instruments is therefore essential. In Volume 2, section 1.3, we have seen how the contraction of materials (glass, ebonite) affects temperature measurement errors. Peter Guthrie Tait’s work has thus made it possible to better understand the impact of pressure on the reading of Miller-Casella thermometers. In doing so, Tait also realized, on the one hand, that his predecessors’ results on low-pressure fresh water compressibility were very scattered and, on the other hand, that the study of salt solution compressibility was practically non-existent. The study of fresh water compressibility was therefore absolutely necessary because this liquid was widely used in piezometers and the study of the compressibility of salt solutions was also essential to be able to deduce with precision the depth of the sea from a pressure measurement.
Tait’s work after the Challenger’s return allowed him to further develop more general theoretical and experimental studies in the field of fresh water and saline compressibility, the effect of pressure on the maximum density of fresh water, and equations-of-state for liquids. In this chapter, we begin by describing Tait’s approach that led him to write his famous equation-of-state1. This relationship contains two parameters that change with temperature. In Chapter 2, we continue by comparing Tait’s equation with the equations-of-state of the same period by discussing parameter interpretations. This analysis is extended in Chapter 3 by applications to a few specific fluids that will provide us with the various parameter evolutions. Chapter 4 deals with the adiabatic compressibility module, which allows for modeling supercritical states up to very high pressures.
1.2. Concepts of compressibility
Compressibility is a general property of a material that causes anything to reduce its volume under the effect of pressure. This property is characterized by coefficients that can be different depending on the material concerned (gas, liquid or solid). In the case of a liquid (usually a state of matter that cannot withstand static shear stress without flow), the only modulus that can be defined is its modulus of elasticity in volume κ, also called the tangent modulus in volume.
1 If a phase of a system is transformed so that its intensive (density, temperature, etc.) and extensive (mass, volume, etc.) state parameters vary continuously, only some of them can be chosen arbitrarily. The others are functions of these arbitrary parameters. These relationships constitute the equations-of-state of the considered phase. 4 From Deep Sea to Laboratory 3
A specific volume V of liquid that is subjected to a hydrostatic pressure variation −=Δ PPP 0 (P is the applied pressure and P0 the reference pressure) undergoes a volume decrease equal to ΔV; its deformation in volume is: -ΔV/V. The modulus of elasticity in volume is then, by definition:
ΔP dP κ ≡−=−lim VV [1.3] Δ→P 0 ΔVdV
The reference pressure P0 is often taken as 1 atmosphere. In practice, since the pressures applied to measure the compressibility of liquids are much higher than 1 atm, the pressure variation is considered to be equal to the pressure applied (i.e. P =Δ P ).
The value of the module κ depends on the speed at which pressure variations occur. If the pressure is applied slowly, the liquid will remain at a constant κ temperature and, under these conditions, we will have an isothermal module T. If the pressure variations are so rapid that there can be practically no heat exchange κ between the liquid and its environment, then we will have an adiabatic module S.
The isothermal KT and adiabatic KS compressibilities are defined respectively as the inverse of the isothermal and adiabatic moduli of elasticity:
1 ∂V 1 ∂V K ≡− and K ≡− [1.4] T ∂ S ∂ VPT VPS
Thermodynamic relationships show that adiabatic compressibility is related to the celerity of sound c by the relationship:
= ρ 2 KcS 1 [1.5] and that the isothermal compressibility can be deduced from the adiabatic compressibility and other thermoelastic coefficients using the following equality:
β 2 =+T VT KKTS [1.6] CP
β where T is the coefficient of isobaric thermal expansion and CP is the heating capacity at constant pressure of the fluid. The Compressibility of Liquids and Tait’s Equation-of-State 5
The effect of liquid compressibility occurs in many processes such as the development of products under high pressure, the generation of shock waves, etc. In the electrical industry, the study of the behavior of insulating liquids subjected to very high voltages shows the appearance of streamers (channels of ionized vapor) leading to liquid breakdown (cover page illustration, Chapter 4). The propagation rate of these streamers is partly related to the compressibility properties of the medium (gas and liquid phases)2.
The first experimental studies of liquid compressibility encountered two major difficulties: the implementation of cells resistant to high pressures and the measurement of pressures within these same cells. As was done in Volume 2 on temperature measurement, we present below a brief history of the evolution of liquid compressibility measurement techniques3.
1.3. The first instruments to measure the compressibility of liquids
Until the 17th Century, it was accepted by physicists that gases and solids were compressible. For liquids, the answer was less clear-cut, but the most widespread opinion was that liquids were incompressible4. From then on, physicists were keen on carrying out experiments to determine with certainty whether or not liquids were compressible. – In his book Novum Organum, which was published in 1620 and which proposes a new approach to science, Francis Bacon5 addressed various subjects dealing with the laws of nature and, in particular, that of the contraction and
2 This type of study is carried out in particular at the Grenoble Electrical Engineering Laboratory (UMR 5269/INP Grenoble-UGA-CNRS), in the Dielectric and Electrostatic Materials team to which the authors of the book are attached. 3 Other contributions from researchers, not presented here, are referenced in the bibliography. 4 The reference to the particular structure of water, which at the time seemed different from that of gases and solids, argued in favor of its incompressibility, as suggested by Huygens C. (1666–1695), Œuvres complètes, vol. 19, Mécanique théorique et physique, Martinus Nijhoff, 1937 and The Hague, Perrault, Du ressort et de la dureté des corps. Essais de physique, vol. 1, pp. 51–57, Imprimeur Jean-Baptiste Coignard, Paris, 1680. 5 English philosopher, scientist and politician, born on January 22, 1561, in Strand near London, and died on April 9, 1626, in Highgate (London). Francis Bacon was a precocious child with a great vivacity of mind. At the age of 12, he began his studies at Cambridge University and at 16, he wrote a book to mark his opposition to Aristotle’s philosophy based on the supremacy of theory over experience. F. Bacon held eminent political responsibilities in England before becoming Grand Chancellor. He is also considered the father of “modern empiricism”, a philosophical theory based on the acquisition of knowledge from experience and the discovery of the laws of nature. Biographical note and account of the experience of crushed balls in Lorquet A., Novum Organum, new translation, Librairie de L. Hachette et Cie, Paris, 1857. 6 From Deep Sea to Laboratory 3 expansion of the bodies. On that occasion, he described an experiment, in which he participated, to assess the compressibility of water. He had a hollow lead globe prepared, with a thick wall and a volume of about 1 liter, which he filled with water before blocking the filling opening with molten lead. The globe was then heavily struck with a large hammer and placed under a press to reduce its volume and compress the water it contained. The water eventually escaped through the wall of the globe in the form of a fine dew. The calculation of the inner volume of the globe at the end of the experiment made it possible to estimate the water’s tensile strength, but not its compressibility because the pressure applied was not known. The conclusion drawn from this experiment was that water was extremely weak and difficult to compress. – Approximately 1660, members of the Accademia del Cimento6 embarked on the study of water compression. Three experiments were undertaken. The first two, only qualitative (expansion and compression of the water by thermal effect; experiment of balls filled with water and struck by a hammer, see representative illustration of the book), brought no new results. The third experiment, although it did not produce a convincing result either, nevertheless had the merit of presenting complete equipment capable of measuring compressibility. This device (see Figure 1.1a) consists of an AB glass container about 70 cm high and 8 cm in diameter, with a side outlet (CH nozzle) and an open EF crystal tube about 2 m high and 1.5 cm in diameter, welded to the top of the A container and whose lower part does not touch the bottom of the container.
In the experiment, water is poured into the container to CD height, then mercury is added to the tube until the water pushed by the mercury completely fills the container and the nozzle, and finally the hole H is closed with a flame. After determining the altitude A0 at which the mercury level was positioned, additional mercury is introduced into the tube up to the maximum altitude AM (point E). Under mercury pressure, the water contracts and rises to a value A E (dotted line in Figure 1.1b).
Finally, the observers failed to detect any contraction of the water and concluded that its compressibility could not be made sensitive by the experiment.
6 The Accademia del Cimento (Florentine Academy), Europe’s first scientific society, was founded in 1657, by Leopold and Ferdinand II de Médicis. This academy, which remained active for about 10 years, undertook a vast program of experimentation of physical and astronomical phenomena (thermometry, areometry, hydrostatics, etc.). It is in this context that work on compressibility and fluid density was carried out and published in Saggi di naturali esperienze fatte dall’ Academia del Cimento (Firenze): “Esperienze intorno alla compressione dell acqua”, pp. 197–205, Florence, 1667. The Compressibility of Liquids and Tait’s Equation-of-State 7
(a) (b)
hM
hE
Water Mercury F ΔhE
Figure 1.1. Compressibility measuring instrument of the Accademia del Cimento. (a) Original instrument; (b) schematic view of the instrument defining the quantities to be measured for the calculation of compressibility (source: from “Esperienze intorno alla compressione dell acqua”, 1667). For a color version of this figure, see www.iste.co.uk/aitken/deepsea3.zip
Let us analyze this experiment more closely, taking into account current knowledge. By denoting PM as the pressure exerted in the tube by mercury, PE the pressure exerted on the water in the container, h M the height of the mercury Δ column in the tube and hE the difference in water level in the container caused by PE , we get: = ρ ρ 1) PgMMM h; M , mercury density; g, acceleration of gravity, = = hAAMM0- ; PEMP ; 8 From Deep Sea to Laboratory 3
Δ= = 2) hAAhKPEE0EE– T ; K T , isothermal compressibility coefficient of water.
= = ρ =×33= 2 By taking: h.M 185 m; h.E 059 m; M 13. 5 10 kg/m ; g 10 m/s and =×−4-1 K.T 045 10 atm , we obtain:
≅ Δ ≅ P.M 25 atm; hE 70 μm
It can therefore be seen that, in this experiment, the contraction of water remains low. Nevertheless, for experimenters at the time, trained to detect very small variations in mercury levels, it can be assumed that an amplitude deviation of 70 µm could have been detected.
NOTE.– The device described above is most probably the first complete instrument for measuring the compressibility of a liquid, the principle of which was adopted a century and a half later by Œrsted.
Other experiments, using a metal container filled with water, clogged, and then compressed, were carried out during the 18th Century, but the only one that seems to prove that water is compressible is that of Robert Boyle who, having pierced the container with a needle after its compression, found that a jet of water was coming out with force7. – It was not until 1761 and John Canton’s8 experiments that the compressibility of liquids was proven [CAN 62, CAN 64]. In his first approaches, J. Canton appropriately used a thermometer to evaluate the contraction of water following a pressure increase.
7 Boyle R., Nova Experimenta Physico-Mechanica, Experiment XX, pp. 122–126, Pofttema ed. 1669. 8 John Canton was an English physicist born on July 13, 1718, in Stroud, and died on March 22, 1772, in London. At the age of 9, his father withdrew him from school believing that he knew enough to become an apprentice weaver. But not giving up his irresistible desire to learn, J. Canton educated himself and was favorably noticed by his entourage, in particular by Henry Miles, pastor, scientist and future member of the Royal Society. When Canton reached the age of 18, Miles finally convinced Canton’s father to let his son go to London so that he could study. Shortly afterwards, Canton became a school assistant teacher and Miles introduced J. Canton to the London scientific community. In 1744, J. Canton married Penelope Colebrooke, from a wealthy family of bankers who opened the doors of high society to him. During his career, J. Canton undertook scientific work in electrostatics, magnetism and chemistry, and studied the properties of liquids (water compressibility). In 1749, he was elected a member of the Royal Society of London, which twice awarded him the Copley Medal for his work on magnet making and water compressibility. Biographical note in Herbert K.B.H. John Canton FRS (1718–72), Physics Education, vol. 33, No. 2, pp. 126–131, 1998. The Compressibility of Liquids and Tait’s Equation-of-State 9
Portrait 1.1. John Canton (1718–1772). English physicist (© National Portrait Gallery, London)
The instrument (see Figure 1.2) consisted of a spherical tank at its base and a long vertical capillary tube of inner section S, graduated from bottom to top and open at its end. The principle of measuring the relative contraction of water Δ VV )/( at a given temperature T and pressure variation ΔP is as follows:
i) the instrument at temperature T is partially filled with water (total volume V ) under atmospheric pressure. The height of the water in the capillary indicates a value h0; ii) the tank is heated until the capillary is completely filled (the heated water has expanded) and then the end of the tube is welded; iii) the heating is switched off and we wait for the instrument to return to its initial temperature T. The height of the water in the capillary then indicates a value −=Δ h1 . Since the water is now subjected to zero pressure, we obtain: V 01 )( Shh for P =Δ 1 atm .
With this instrument, J. Canton measured the compressibility of different liquids (rainwater, seawater, mercury, liquors, etc.) at different temperatures and pressures limited to 3 atm. He measured a contraction / VV =Δ 1/10870 for water at t = 10°C and P =Δ 2 atm . These observations lead to an isothermal compressibility coefficient of about 0.46 × 10-4 atm-1, a value very close to that obtained by Kell and Whalley in 1965 (0.48 × 10-4 atm-1) [KEL 65]. 10 From Deep Sea to Laboratory 3
– About 60 years then passed between Canton’s initiating work and the appearance of the first operational compressibility measuring instruments proposed independently by Perkins and Œrsted.
Closed P = 1 atm tube P = 0
(a) (b) (c) h1
h0
T0 T1 > T0 T0
Figure 1.2. Compressibility measuring instrument by J. Canton. (a) The tank contains water at temperature T0 and subjected to atmospheric pressure; (b) water heated to temperature T1 fills the capillary, which is then welded at its end; (c) water brought to temperature T0 is subjected to zero pressure
In 1819, Jacob Perkins9 designed a new instrument to study liquids at very high pressure (P > 100 atm) [PER 20, PER 26]. Its experimental cell consists of a vertically positioned gun barrel (see Figure 1.3a) whose upper part is hermetically sealed with a screw cap. In the center of the cap, a valve transmits the pressure supplied by an external hydraulic pump and informs the operator when the preset
9 Jacob Perkins was an American mechanical inventor born on July 9, 1766, in Newburyport (USA), and died on July 30, 1849, in London. He started his professional life as an apprentice goldsmith and filed numerous patents during his career. In 1819, he emigrated to England and then ran a banknote and postage stamp production company. He brought real innovations in steam engines and refrigeration systems and invented a device to measure water compression. Biographical note in Encyclopedia Britannica, vol. 21, p. 173, 1911.