Thermodynamics of Irreversible Processes in Liquid Metals

104 Pages·1966·2.945 MB·German

by  Hans Knof (auth.)

Thermodynamics
of Irreversible Processes
in Liquid Metals
Thermodynamics
of Irreversible Processes
in Liquid Metals
HANS KNOF
University of Hamburg
SPRINGER FACHMEDIEN WIESBADEN GMBH
This work won a prize in the Jubilee Prize
Competition held by the Verlag Friedr. Vieweg & Sohn.
The English translation has been made by the author.
ISBN 978-3-663-06421-3 ISBN 978-3-663-07334-5 (eBook)
DOI 10.1007/978-3-663-07334-5
©
1966 by Springer Fachrnedien Wiesbaden
Originally published by Friedr. Vieweg & Sohn G m b H, Braunschweig 1966.
Published in Great Britain in 1966 by Heywood Books for I1iffe Books Ltd .•
Dorset House, Stamford Street, London, S.E.I
To GISELA
Preface
This book is intended to be a comprehensive survey of irreversible
phenomena in liquid metals. Experiments and special theoretical consid
erations can be brought into a general scheme by means of the phenome
nological theory of thermodynamics of irreversible processes. In this
way well-investigated phenomena such as electrolysis in liquid alloys,
and new fields of research such as isotopic separation in liquid metals
by action of an electrical direct current, can be given the same theo
retical foundation. The goal in this book is to deduce the theory of
a special phenomenon as well as to describe experiments connected
with the resulting effects. On the other hand no kinetic theory is given
since to date no general kinetic theory exists for liquids as it does exist
for gases. In general, kinetic models seem to agree with experiments
only if the results of the measurements are known beforehand.
In Chapter 1 the Onsager symmetry relations are deduced in general.
Chapter 2 gives the general phenomenological theory of liquid metals.
These two chapters serve as a foundation for the remaining chapters,
where the general theory is applied to sO.me cases of special physical
interest. The scalar phenomena of relaxation are given in Chapter 3
for the case of nuclear magnetic resonance. Vector phenomena are
discussed in Chapters 4 - 6. Chapter 7 deals with viscosity effects as
examples of tensor effects. The last chapter describes coupling effects
between vector and tensor forces which occur in liquid metals flowing
through capillaries and membranes.
The literature references are restricted to books and general articles.
In addition the primary works and the most recent articles are men
tioned separately.
Throughout this book extensive use is made of tensor notation and
simple tensor algebra. Since most of the recent books on thermodynamics
of irreversible processes have an appendix on this topic, there is no
need to repeat it in this book.
The first manuscript of this book was presented in German on the
occasion of the "Vieweg-JubiHiums-Preisausschreiben" in Spring 1962.
The English edition is completely revised and contains the results of
some very recent research.
The author is indebted to Prof. Richard P. Wendt, Baton Rouge,
Louisiana, for reading the English manuscript and making corrections
of style and grammar.
College Park, Maryland
Hanau Fall 1963 Hans Knof
7
Contents
1. Onsager Symmetry Relation 1
1.1. Irreversible Processes. . . . 1
1.2. Onsager Relation in Scalar Processes 4
1.3. Onsager Relation for Processes in a Magnetic Field 7
2. The Phenomenological Relations . . . . . . . . . 11
2.1. Assumption of Local and Mechanical Equilibrium 11
2.2. Conservation Laws and Balance Equations. 14
2.3. Phenomenological Equations 20
3. Nuclear Magnetic Resonance 24
3.1. Theory of Nuclear Magnetic Resonance 24
3.2. Knight Shift and Relaxation Times . . 25
4. Galvanomagnetic and Thermomagnetic Phenomena 28
4.1. Phenomenological Theory of Pure Metals 28
4.2. Definition of Effects . . . . . . . 32
4.3. Coefficients for Measurable Effects 35
4.4. Hall Effect in Liquid Metals . . • 40
s. Diffusion and Thermodiffusion in Alloys 42
5.1. Transformation Equations for the Diffusion Flux . 42
5.2. Isothermal Diffusion in Binary Alloys. . . . . 44
5.3. Diffusion in Multi-Component Systems 47
5.4. Thermo Diffusion and Diffusion Thermo Effects 52
5.5. Experimental Work on Self Diffusion . 55
5.6. Diffusion Experiments in Binary Alloys . . . . 58
6. Isothermal Electrical Phenomena in Metals and Alloys 63
6.1. Binary Alloys . . . . . . . . . . . . . . . . . . 63
6.2. Electrolysis in Mercury Amalgams . . . . . . . . 68
6.3. Electromagnetic Migration in Liquid Gold Amalgam 70
6.4. Isotopic Separation by Electrolysis and Electromagnetic Migration 72
7. Viscosity Phenomena in a Magnetic Field. 74
7.1. Phenomenology of Viscous Flow . . . . 74
7.2. Viscosity Effects in a Magnetic Field . . 76
7.3. Measurements of Viscous Flow in a Magnetic Field 80
9
8. Effects between Vector and Tensor Forces 82
8.1. Electrokinetic Effects. . . . . . . . . . 82
8.2. Electroosmosis in Mercury . . . . . . . 84
8.3. Pressure-Thermo and Pressure-Diffusion Effects. 87
Literature 90
Index . . 92
10
List of Symbols
Symbols printed in italics (a) are scalar quantities, in boldface type (A) are vectors,
and in boldface sans-serif characters (D) are dyadics and matrices. The number in
parentheses gives the section where the symbolized quantity is first defined.
a,Qi variables (1.1.) k Boltzmann constant (1.1)
aI, a2 phenomenological coefficients (3.1) L Lorentz number (4.2)
A,AK affinity (2.2) L external angular momentum (7.1)
A vector potential (4.1) Ljk phenomenological coefficients (1.1)
b variable (odd function) (1.3) Lpq phenomenological tensors (1.3)
b (electrolytic) mobility (1.1), (6.2) M magnetization (2.2)
B magnetic induction (1.3) nk molar fraction of component k
c velocity oflight (2.2) (5.1)
c concentration (5.5) Nk molar density of component k
Ck mass fraction of component k (2.2) (5.1)
C p, C pi molar heat, partial molar heat of p hydrostatic pressure (2.1)
component i (5.4) pI, p~ isothermal, adiabatic Nernst coef-
D diffusion coefficient (5.2) ficient (4.2)
DT thermo diffusion coefficient (5.4) pt Ettingshausen coefficient (4.2)
D diffusion matrix (5.3) P pressure tensor (2.2)
e total charge density (2.2) Q partition function (2.1)
ee, ei specific charge density of electrons, Q* reduced heat of transfer (4.3)
ions (4.1) Q* molar transfer heat (5.4)
ek charge density of componentk(2.2) Q) , Q~ isothermal, adiabatic Ettingshau
E electric field strength (2.2) sen-Nernst coefficient (4.2)
f.fk specific potential energy, of com- Q~, Q! isothermal, adiabatic Nernst
ponent k (2.2) coefficient (4.2)
Fk external force on component k R gas constant (5.2)
(2.1) R Ohmic resistivity (5.6)
g specific Gibbs function (5.3) R}, R~ isothermal, adiabatic electric re
H magnetic field strength (2.2) sistivity (4.2)
conduction current density (2.2) Rf, R! isothermal, adiabatic Hall coeffi-
I electrical current (5.6) cient (4.2)
j mass flux (5.5) s;se,si; specific entropy (2.1); of electrons,
J total angular momentum (7.1) SA, sB ions (4.1); of metal A, B (4.3)
Ji flux density of component i (2.1) S entropy (1.1)
st
Ik molar flux of component k (5.1) Righi-Leduc coefficient (4.2)
Jq heat flow (2.2) S intrinsic angular momentum (7.1)
Js entropy flow (2.2) time (1.1)
II
T temperature (1.1) 'I]i phenomenological coefficients
T electromagnetic tension tensor (i = 1,2, ... ,5) (7.2)
(2.2) 'YJv, 'YJ, 'YJr volume, shear, rotational viscosity
u specific internal energy (2.1) (7.1)
v, Vk specific volume, of component k {} local entropy production (2.2)
(5.1) e average moment of inertia (7.1)
v barycentric velocity (2.1) " heat conductivity (5.3)
Ai> Aa isothermal, adiabatic heat con-
1'0 mean volume velocity (5.1)
ductivity (4.1)
Vk velocity of component k (2.2) ft chemical potential (2.2)
I'm mean molar velocity (5.1)
fto vacuum permeability (2.2), (3.1)
W probability (1.1) fi electrochemical potential (6.1)
W energy density of electromagnetic fl Jacobian matrix for the chemical
field (2.2) potentials with respect to mass
Wy energy per unit volume of electro- densities (5.3)
magnetic field (2.2) n viscous pressure tensor (2.2)
xi, yt: conjugated variables (1.3) lIxx,lIicoordinates of viscous pressure
Xi thermodynamic forces (2.2) tensor (7.1)
Y j thermodynamic fluxes (2.2) fls, na symmetric, antisymmetric pressure
tensor (7.1)
a deviation from equilibrium
(even function) (1.3) (I, (Ii mass density, of component i (2.1)
ai thermodynamic coefficients (2.2) G, Gk intrinsic angular momentum (2.2)
Ii diffusion tensor of intrinsic angu-
f3 deviation from equilibrium (odd
lar momentum (2.2)
function) (1.3)
7: time (1.1)
I' gyro magnetic ratio (2.2)
7:10 '2 longitudinal, transverse relaxation
or! Kronecker symbol (1.2) time (3.1)
o(r-r') Dirac's a-function (1.3) cp scalar electrical potential (4.1)
C total energy density (2.2) X susceptibility (2.2)
cxx, Ci coordinates of velocity gradient OCr) general differential operator (1.3)
tensor (7.2) w mean angular velocity (7.1)
12
1. Onsager Symmetry Relation
1.1. Irreversible Processes
In nature, every process is irreversible. The restriction on reversible
processes, as done in classical thermodynamics, gives a limiting case
only. That is, a process is not reversible unless it proceeds infinitesimally
slowly and therefore is independent of time. In nature, however, all
processes proceed with finite speed and therefore are time dependent.
The second law of classical thermodynamics can be stated as follows:
The entropy increases if an irreversible process is occurring. One can
state this sentence in the opposite way as well: The irreversible process
occurs because it is connected with an increase of entropy. Here the
increase of entropy is looked upon as force which generates the thermo
dynamic process. With this picture it is possible to get a relation between
an increase of entropy and the rate of the thermodynamic process [1, 2].
Let a be a state variable of some system, and let us suppose the entropy
is a known function of this variable, i.e., S = Sea). Then the time de
rivative of the entropy is
S=d oS (1)
oa
In this equation oSjoa can be looked upon as force which causes the
change in a. At equilibrium oSjoa = O. For small deviations from
equilibrium we set
d=C oS (2)
oa
Here the proportionality factor C is independent of a. The entropy
has a maximum at the equilibrium value ao of a. Thus we get
(OS) = 0, (3)
oa ao
Near equilibrium (2) can also be written as
(4)
1