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Xerox University Microfilms
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1 V 2 M 5 4 3 !
LD3907 /
.G7 Storm, Martin Lee, I923-
1951 Ileat conduction in single
• S8lj- m etals.
v ,ll5p. diagrs.-
Thesis (Ph.D.) - N.Y.U.,
Graduate School, 1951.
Bibliography: p .109-112, :
C75305
Shell M51
Xerox University Microfilms, Ann Arbor, Michigan 48106
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.
LIBRARY OY
REIT YORK UNIVERSITY
UNIVERSITY HEIS-HT9
HEAT CONDUCTION IN SIMPLE METALS
MARTIN L# STORM
A dissertation in the department of physics sub
m itted in p a rtial fulfillm ent of the requirements
for the decree of Doctor of Philosophy a t New York
U niversity
ACKNOWLEDGMENT
The author wishes to take th is opportunity to thank
Dr. George E. Hudson, Dr. Hartmut Kallman, and Dr. F ritz Reiche
for th eir constant help and encouragement. This dissertation
was prepared under the guidance of Dr. Hudson.
The author also wishes to acknowledge his indebted
ness to the late Dr. J. K. L. MacDonald who firs t suggested
the problem to him and helped him overcome the in itia l mathe
m atical d iffic u ltie s.
1 0 's 1 i r
ii
TABLE OF CONTENTS
Section
1 Introduction............................................................................ 1
1.1 Purpose of Thesis ................................................ 1
1.2 The D ifferential Equation of Heat
Conduction in an Isotropic Solid . . . . . . . 1
1.3 Formulation of the Problem ............................ 3
2 History........................................................................................ 8
2.1 Previous Work on the Non-Linear Equation
of Heat Conduction ............................................... 8
2.1.1 Van Dusen ..................................................... 8
2.1.2 Sawada ................................................................. 10
2.1.3 Sawada ................................................................. 12
2.1.4 Awbery ................................................................. 13
2.1.3 Hopkins .............................................................. 16
2.1.6 Barrer ................................................................. 20
2.1.7 MacDonald......................................................... 21
2.1.8 E llion ................................................................. 23
2.1.9 Discussion of Previous Work on the
Non-Linear Equation ................................... 28
2.2 Relations Previously Proposed to Relate
the Thermal Parameters ..................................... 29
2.2.1 Bidwell ............................................................... 29
2.2.2 Hume-Rothery ....................................... 31
2.2.3 Bidwell ............................................................... 31
2.2.4 Powell ................................................................. 33
i i i
Section Page
3 A Discussion of the Solid State ........................... 35
3*1 In tro d u ctio n ........................................................... 35
3*2 The Electron Theory of M etals................ 35
3.2.1 Calculation of the Mean Free Path ... 38
3*3 The Equation of State for an Isotropic
S o lid ............................................................................... 4l
3.3*1 In tro d u ctio n .............................................. 4l
3.3*2 Quantum T heory......................................... 42
3. 3«2.1 Debye's Theory ........................................ 42
3.3*2.2 Monochromatic Theory .......................... 46
3.3.3 C lassical Theory .................................................... 47
4 Investigation of the Relations Between the
Thermal Parameters .................................................................... 50
4.1 Derivation of an Expression for the
Product K S................................................................... 50
4.1.1 Investigation of the Temperature
Dependence o fK S .................. 56
4.2 Investigation of the Temperature
Dependence of ^ ^ S lo g /^ .................... 62
4.3 Comparison of Theory with Available Data 65
4.3.1 In tro d u ctio n .......................................................... 65
4.3-2 Examination of Data for Some Simple
Metals ........................................................ 68
4.3-2.1 Discussion of Data and Comparison
with T heory...................................... 75
4.3.3 Examination of Data of Fused Quartz . 78
4.3.4 Examination of the Data for Iron and
.80# Carbon Steel • • *.................................. 79
iv
Section Page
5 Treatment of the One-Dimensional, Non-Linear
P artial D ifferential Equation.of Heat
Conduction.......................................................................... •. 83
5*1 Transformation of the E quation..................... 83
5*2 Case of a Sem i-Infinite Metal with a
Constant Heat F lu x ................................................. 87
5*3 Discussion of the Mathematical Trans
formation ...................................*............................... 93
5*4 A pplication of Solution for the
Temperature D istribution in a Semi-
In fin ite Metal with a Constant Heat Flux 98
5*4.1 Introduction of Dimensionless
Variables into the S o lu tio n .................. 98
5.4.2 Calculation of a Numerical Example .. 99
5.4.3 Comparison with the Results of the
Usual Linearized Theory ................................ 101
6 Summary of Results and Suggestions for Future
W ork................................................................................................... 105
6.1 Summary of R e su lts....................................... 105
6.2 Suggestions for Future W ork................................ 107
B ibliography............................................................................ 109
Appendix: The Laplace Transformation ......................... 114
v
SECTION 1
INTRODUCTION
1.1 Purpose of Thesis
I t is an aim of th is thesis to investigate th eo reti
cally the relations existing between the thermal parameters
of simple m etals, and to check these relations on the basis of
available experimental data. The m otivation of th is in v esti
gation is the discovery that the constancy of a certain combi
nation of the thermal parameters is a mathematical condition
for the linearization of the one-dimensional, non-linear,
p a rtial d ifferen tial equation of heat conduction. The relatio n
ship of the above mentioned combination of thermal parameters
to results derivable from the theory of solids w ill be in v esti
gated; and applications of the resulting linearized equation
to problems in heat conduction w ill be considered.
1.2 D ifferential Equation q£_ Heat, Pan&W&inP In an
IgflfcggPlff Solid
The fundamental hypothesis for the mathematical
theory of heat conduction in an isotropic solid is that the
rate at which heat crosses from the inside to the outside of
an Isotherm al surface per unit area per unit time at a point
is equal to
-k |2- 1.2.1
in
where K is the thermal conductivity of the substance, T the
tem perature, and ^ denotes differen tiatio n along the outward-
drawn normal to the surface. 1
Consider a solid through which heat is flowing but
in which no heat is being generated. The general d ifferen tial
equation for non-steady state heat conduction is obtained from
the fundamental hypothesis 1.2.1 and the conservation of heat
energy. Consider an element of volume with area dydz and
thickness dx:
The heat entering in time dt along the x axis is
-Kdydz^dt
dx
and that leaving is
-dydZ[Ki l + jt(icffl)te]dt
The difference between that entering and leaving
along a ll three axes is equated to that stored in
the element, or
todydztj^Kil) ♦ +-i(KJI.)]dt = dxdydz[fcpil]d t
so that the d ifferen tial equation for non-steady
state heat conduction expressed in rectangular
coordinates is
JL (k^) t ^-(K|3L) + = pc ££ 1. 2.2
dx dx dy dy dz dz r pH
In the above, T is the temperature of the solid at
time t and position (X ,y,z), K the thermal conductivity, f the
density, and cp the specific heat a t constant pressure. If we
le t S = fcp 1.2.3
then 1.2.2 can be w ritten as
V'[KVT] * s|S- 1.2.4
When the term "thermal parameters" Is used, I t Is meant to
refer specifically to the two quantities K and S. In the
e .g .s. system the units of K are cal./cm .sec.°C , and the units
of S are cal./cm^°C.
Equation 1.2.4 is non-linear since the thermal
parameters are functions of tem perature. In the usual mathe
m atical treatm ent of heat conduction i t is assumed that the
thermal parameters are constant and solutions of the resulting
linear equation
1.2.5
v K d t
have been thoroughly investigated. 1 However, in the case of
m etals, th is approximation holds for lim ited ranges of tempera
ture only, and discrepancies between the measured and calculated
tem peratures are usually attributed to the neglect of the
variation of the thermal param eters.
1.3 Formulation of the Problem
We shall see, in section 2.1, that previous in v esti
gators who allowed the thermal parameters to vary had more
success in handling the steady state heat conduction equation
than in solving the corresponding problem of non-steady heat
conduction. In the la tte r case the methods of investigation
can be roughly divided into the following categories:
(a) Investigations in which the only criterio n
governing the choice of functions used to
represent the thermal parameters was the fact
