NATIONAL BUREAU OF STANDARDS REPORT

9387

MEASUREMENTS OF THE THERMAL CONDUCTIVITY AND

ELECTRICAL RESISTIVITY OF PLATINUM FROM 373 TO 1373° K

by

D. R. Flynn

Environmental Engineering Section

Building Research Division

Institute for Applied Technology

National Bureau of Standards

and

M. E. O' Hagan

School of Engineering and Applied Science

The George Washington University

DO NOT REFERENCE THIS DOCUMENT

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

421.03-40-4212231 9387

August 22, 1966

MEASUREMENTS OF THE THERMAL CONDUCTIVITY AND

ELECTRICAL RESISTIVITY OF PLATINUM FROM 373 TO 1373 °K

by

D. R. Flynn

Environmental Engineering Section

Building Research Division

Institute for Applied Technology

National Bureau of Standards

and

M. E. O' Hagan

School of Engineering and Applied Science

The George Washington University

DO NOT REFERENCE THIS DOCUMENT

IMPORTANT NOTICE NATIONAL BUREAU OF ST l for use within the Government,

and review. For this reason, the whole or in part, is not author! Bureau of Standards, Washingto

the Report has been specifically

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

s accounting documents intended

subjected to additional evaluation

listing of this Report, either in Office of the Director, National

the Government agency for which

pies for its own use.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU Of STANDARDS
ABSTRACT

Measurements have been made of the thermal conductivity and the electrical resistivity of commercial grade platinum (99.98% pure) in the temperature

range 373 to 1373 °K. The measurements have been made with a view to providing

accurate data on the thermal conductivity of platinum to serve as a basis for

establishing platinum as a thermal conductivity standard reference material.

Two methods of measuring the thermal conductivity have been employed,

one an electrical method and the other a non-electrical method. In the electrical method, a direct current passed through a necked-down portion of the specimen and the thermal conductivity was determined in terms of the temperature

and electrical potential distribution in the necked-down region. The second

method was of the absolute guarded longitudinal heat flow type. The experiment

was designed to permit measurements by both methods in the same apparatus and

on the same specimen thereby providing as direct a comparison as possible

between the methods.

The data given by the two methods agree within experimental error and

show the thermal conductivity of platinum to be a smoothly increasing function

of temperature in the measured range. The data are considered to be accurate

to better than 1% and indicate that platinum could be adopted as a thermal

conductivity standard reference material.

ii
TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS v LIST OF TABLES . . . ....... V £ LIST OF ILLUSTRATIONS

vii

INTRODUCTION .................. 1

Chapter

I. PREVIOUS INVESTIGATIONS 4

II. DESCRIPTION OF THE APPARATUS ....... 24

Mechanical Configuration 27

Specimen 27

Furnace 34

Environmental System 36

Thermal Configuration 37

Specimen 37

Guard 42

Instrumentation 45

Temperature Control 45

Temperature Measurement 46

Power Measurement 46

Resistance Measurement 46

III. THEORY OF THE INDIRECT METHOD 50

IV. CALCULATION PROCEDURES 63

Indirect Method 63

Direct Method 73

V. SPECIMEN CHARACTERIZATION 85

Source 85

Fabrication and Thermal History 85

Hardness 88

Spectrographic Analysis . 88

Metallography 89

Density 89

Residual Resistance 91

Ice-Point Resistivity 92

EMF vs Pt 27 93

- iii -
VI. TEST PROCEDURES 94

Direct Method 95

Resistivity 97

Indirect Method 98

VII. RESULTS 101

Electrical Resistivity 101

Thermal Conductivity ....... ....... 106

VIII. DISCUSSION OF RESULTS 116

Accuracy ....... ..... 116

Electrical Resistivity . 116

Thermal Conductivity ....... 117

Comparison with Other Investigations . . 130

Electrical Resistivity ..... ..... 130

Thermal Conductivity 131

Appendix

A. Machining the Neck and Wiring the Specimen . 138

B. Potential Perturbations . ..... 141

C. The Thomson Effect . 148

D. Alternative Calculation Procedure for the Indirect Method . . 150

E. A Simplified Expression for the Maximum Temperature Rise in the

Neck in the Presence of Heat Losses 152

F. Derivation of the Equation for Heat Flow in the Current Leads

of the Specimen Heater 154

G. Errors in Temperature Measurements Due to Heat Conduction along

Thermocouple Wires 157

H. Departures from One -Dimensional Heat Flow in the Direct Method 160

I. Tempering of the Heater Current Leads 162

References 164

- iv -
ACKNOWLEDGMENTS

This research was the subject of a dissertation submitted by Malcolm

O'Hagan to the Faculty of The School of Engineering and Applied Science, The

George Washington University, in partial satisfaction of the requirements for

the degree of Doctor of Science.

This work was supported in part by Engelhard Industries, Inc., Newark,

New Jersey, who supplied the platinum and financial assistance in the form of

a Grant to the National Bureau of Standards. We thank Henry G. Albert and

A. V. Lincoln, both of Engelhard Industries, Inc., for their support and advice.

Malcolm E. O'Hagan was supported in part by Grant AFOSR-1025-66 from

the Air Force Office of Scientific Research and in part by the Institute for

Industrial Research and Standards, Dublin, Ireland.

We thank Henry E„ Robinson, Chief, Environmental Engineering Section,

NBS, and Adjunct Professor of Engineering, The George Washington University

for his advice and support.

The intricate and difficult machining required for this project was

done by Raymond Chidester, Section 145.01 NBS. Much good advice in matters

of fabrication came from John Hettenhouser , Section 145.01 NBS.

- v -
LIST OF TABLES

Table Page

1. Sample characterization of platinum samples used by

various investigators 17

2. Thermal conductivity of platinum in the temperature range

273-373 °K as reported by several investigators ... 18

3. Hardness values as measured on the platinum bar .... 88

4. Spectrographic analysis as determined on platinum wire

No. 175105 89

5. Resistance values for three samples of platinum at icepoint and helium-point temperatures and the computed re- sistance ratios at zero current 92

6. EMF vs Pt 27 as determined on 20-mil platinum wire. . . 93

7. Experimental values for the thermal conductivity of

platinum as determined by the direct method 107

8. Typical set of data from measurements of the thermal con- ductivity by the direct method 108

9. Experimental values for the thermal conductivity of

platinum as determined by the indirect method .... 113

10. Comparison of smoothed resistivity values for platinum

with those of other investigators 130

11. Thermal conductivity of platinum 132

12. Values for the relative magnitude of the Thomson Term . 149

13. Values of U(T) o0 m) for arguments from 0.0 to 0.5 . . . . 151

vi -
LIST OF ILLUSTRATIONS

Figure Page

1. Conductor Carrying an Electric Current 5

2. Necked-down Sample 7

3. Thermal Conductivity of Platinum as Reported by

Several Investigators .... 19

4. Low Temperature Thermal Conductivity of Platinum

as Reported by Several Investigators 20

5. Lorenz Number for Platinum as Reported by Several

Investigators 21

6. The Apparatus 28

7. Upper End of the Platinum Specimen 29

8. Close-up View of the Specimen Heater 29

9. Details of the Suspension System and the Upper Assembly. . , 30

10. Device for Constraining the Upper Part of the Specimen ... 32

11. Details of the Apparatus . 32

12. Heater and Thermocouple Locations 38

13. The Guard 43

14. Guard Heater 43

15. General View of the Apparatus 48

16. Close-up View of the Apparatus 49

17. An Electric Conductor of Arbitrary Geometry 56

18. The Neck, Surrounded by Powder Insulation 65

19. Temperature Distributions in the Necked-down Region .... 68

20. Circuit Diagram for The Specimen Heater 75

21. Arrangement of Potential Leads for The Specimen Heater ... 76

22. Percentage Deviations from Equation (7-2) of Electrical

Resistivity Data Points Obtained in The Necked-down Region

of The Platinum Specimen 103

- vii -
23 . Percentage Deviations from Equation (7-2) of Electrical

Resistivity Data Points Obtained in The Lower Part of

The Platinum Specimen 104

24. Effective Thermal Conductivity in Argon, Helium and Air of

The Powder Insulation Surrounding The Specimen 109

25. Percentage Deviations from Equation (7-4) of Thermal

Conductivity Data Points Obtained by The Direct Method . . Ill

26. Percentage Deviations from Equation (7-6) of Thermal

Conductivity Data Points Obtained by The Indirect Method . 114

27. Thermal Conductivity of Platinum: Comparison between

Existing High Temperature Data and Those of The Present

Investigation 134

28. Thermal Conductivity of Platinum: Comparison between

Existing Low Temperature Data and Those of The Present

Investigation . ..... 135

29 . Lorenz Number for Platinum: Comparison between

Existing Data and Those of The Present Investigation . . . 136

30. The Specimen Being Machined 139

31. The Machining Set-up for Machining the Neck in The Specimen 139

32. Semi-infinite Solid 141

33. Local Departures from The Average Value of The Potential

Gradient Caused by The Presence of Strip Sources at

z =0. (x/b =0.0) 143

34. Local Departures from The Average Value of The Potential

Gradient Caused by The Presence of Strip Sources at

z = 0. (x/b =1.0) 144

35. Semi-infinite Cylinder 145

36. Local Departures from The Average Value of The Potential

Gradient Caused by A Circular Source of Radius, a, at The

End, z = 0, of a Semi-infinite Cylinder of Radius, b.

(r/b =1.0) 147
INTRODUCTION

Standards and standard reference materials are the basis of a consistent

and accurate measuring system. The need for standard reference materials in

thermal conductivity measurements is two-fold. In the first place, such

materials are required for comparative measurements in which the thermal

conductivity of the material under test is determined in terms of that of

the standard reference material. Secondly, such materials are required in

evaluating the accuracy of apparatus designed for thermal conductivity

measurements. The degree to which the measured value of the thermal conductivity of the standard reference material agrees with the accepted value

is a check on the accuracy of the apparatus in which the measurements were

made

.

The basic requirements for any standard reference material are that

it be stable, reproducible and appropriate for the measurements at hand,

and that the property in question be uniform throughout the material . In

the case of standard reference materials for thermal conductivity other

desirable requirements are that the standard be usable over a wide range of

temperature, that it be chemically inert so as not to be affected by or

affect other materials in the system and that the thermal conductivity of

the reference material be close in value to that of the materials which are

to be measured in terms of it. For a more complete discussion of thermal

conductivity reference standards the reader is referred to reference [l ] .—

The advantages of using pure platinum as a thermal conductivity reference

material have been pointed out by Powell and Tye [2] and by Slack [3]

.

\J Numbers in square brackets refer to references listed at the end of the

paper.

1
Platinum is a f ace-centered-cubic metal available in high purity (easily

evaluated by electrical resistivity measurements) in pieces of substantial

size. It has a fairly high melting point (2042°K on the 1948 IPTS) , is

optically opaque, and is relatively stable chemically in air and other

atmospheres, wi th the exception of hydrogen, even at high temperatures [4, 5].

It is readily machinable and is not prohibitively expensive. Its thermal

conductivity, although relatively high for use as a standard with non-metals,

is about the geometric mean for metals and alloys (circa 0.7 W/ cm deg); it

has no known transition points. Powell and Tye [2] have pointed out the

particular advantages of using a reference material that is also one of the

most frequently used thermocouple elements.

An obvious objection one might raise concerning the use of platinum as

a thermal conductivity reference material is that the thermal and electrical

conductivities of ultra-pure elements at low temperatures are markedly

affected by the first traces of impurity. However, small amounts of impurity

would not be expected, on present theoretical grounds, to seriously affect

thermal or electrical conductivity at temperatures in the vicinity of the

Debye temperature or above. The stability of the electrical conductivity

of platinum is evidenced by the fact that the International Practical

Temperature Scale is defined by a platinum resistance thermometer in the

temperature range -f90.18 to 903.65°K [6, 7], and that studies are currently

under way at NBS [8] and other laboratories to investigate the possibility

of extending to the gold point (1336. 15°K) the range over which a platinum

resistance thermometer is used to define the temperature scale. The work

at NBS, although not complete, appears to indicate that a properly constructed

platinum resistance thermometer can be used at temperatures up to the gold

2
point with very good stability. The electrical resistance of very fine

platinum wires (0.00127-cm diameter) has been observed [ 9 ] to be very stable

in several different atmospheres at temperatures up to 1300°K. Changes in

room- temperature resistance of fine wires which had been heated to temperatures

in excess of 1300°K have been observed [ 9 ], however.

Stability studies on the platinum-10% rhodium: pi atinum thermocouple,

which is currently used to define the International Practical Temperature

Scale from 903.65 to 1336. 15°K, have indicated that the th ermoelectric power

of 0.05-cm diameter platinum wire is quite stable to 1500°K in air, inert

atmospheres, or vacuum [lO, 1 1 ]

.

Platinum appears to be, in every way save one, an ideal material to use

as a thermal conductivity reference standard. The exception is that the

spread among the literature values for the thermal conductivity of platinum

above room temperature is considerable, to say the least.

Referring to Figure 3 we see that at 1200°K the spread in the data is

25%. The need for further independent and more accurate determinations of

the thermal conductivity of platinum above room temperature is obvious and

it is this need that has led to the present investigation. The objective of

this investigation then is to provide accurate data on the thermal conductivity

of platinum above room temperature to permit the use of platinum as a thermal

conductivity standard reference material. Data of this nature are also of

interest to theoretical physicists in checking theoretical predictions.

3
CHAPTER 1

PREVIOUS INVESTIGATIONS

The thermal conductivity of platinum has been the subject of a number

of investigations with the first reported values dating back to the nineteenth

century. While a great variety of methods have been employed by the different

investigators it is possible to classify them into two general groups. In

the first group an electric current flows in the specimen generating heat

therein. In the second group there is no current flowing in the specimen, and

the heat is supplied from an external heat source as opposed to being generated

within the specimen. For the purposes of this paper methods of the first

group will be referred to as electrical methods or indirect methods since

they require a knowledge of the electrical resistivity of the specimen,

whereas methods of the second group will be referred to as non-electrical

methods or direct methods.

The methods in the first group are based on the differential equations

describing the steady-state distribution of temperature and electrical potential in a conductor carrying an electric current. In the absence of thermoelectric effects these equations can be written [see Chapter 3] as:

where k is the thermal conductivity of the material, p its electrical

resistivity, T the temperature and 0 the electrical potential. If heat losses

from the surface of the conductor are rendered negligible (1-1) and (1-2)

reduce to a simple equation:

P d-1)

V • I V0 = o,

p

( 1 - 2 )

T

V

2 = 8j

m

kpdT

T

o

- 4

(1-3)
where V is the voltage drop measured between two isothermal surfaces in the

conductor at temperature T , and T is the maximum temperature in the

o m

r conductor

.

Figure 1. Conductor carrying an electric current.

Equation (1-3) is developed in detail in Chapter 3. It is independent

of geometry provided heat losses from the surface, (C) , are negligible. From

(1-3) we see then that, in the case of negligible heat losses, the thermal

conductivity is determined by measuring the voltage drop, V, in the specimen,

the end temoeratur 1 es , T , the maximum temperature, T , and the electrical

o

r m resistivity, p.

The works of Jaeger and Diesselhorst [l 2] , Meissner [l 3 ] , Holm and

Stormer [l4] , Hopkins [ 1 5 ] , Hopkins and Griffith [l 6] , and Cutler, et al [ 1 7

]

are all based on equation (1-3). Differences between these investigations

arise mainly with respect to the following:

1. Generation and measurement of T . o

2. Measurement of T . m

3. Method of rendering heat losses negligible.

4. Analysis of the data, e.g. treatment of kp as a function of temperature, etc.

5
Jaeger and Diesselhorst LI 2 ] measured on a platinum rod 1.5 cm in

diameter using currents up to 350 amps. The specimen was covered with thermal

insulation to minimize heat losses from its surface. The ends of the rod

were in thermal contact with two constant temperature water baths thereby

maintaining the ends at constant temperature T .

Meissner [ 1 3 ] measured on a wire 2 mm in diameter and 6.3 cm long. The

measurements were made in vacuum in the temperature range from 20.7 to 373°K

where heat losses by radiation are negligible. The maximum temperature rise,

(T - T

q

) , was measured by using the specimen as its own resistance thermometer. To a first approximation it can be shown [l3, 18] that

0 =|

-R - R a R

o o (1-4)

where 9 = T -T , R is the measured resistance at current I, R is the

m m o o resistance extrapolated to I = 0 , or in other words the resistance of the

specimen at T = T , and oi is the temperature coefficient of resistance at r o o

r

T = T . More exact expressions relating 9 to the measured resistances are

o m developed by Meissner [l3], Holm [l 8] and in Chapter 3 of this paper.

Holm and Stormer [l4] were the first to apply the method to high

temperature thermal conductivity measurements, measuring in the temperature

range from 292 to 1293°K. While Meissner could employ a long specimen due

to small radiation losses at the lower temperatures, a long specimen can no

longer be used at the higher temperatures where radiation becomes significant.

This consideration led Holm and Stormer to adopt a short necked-down specimen

- 6 -

as shewn in Figure 2.
V

Figure 2 , Necked-down Sample

Their sample was 1.0 cm diameter by 2.5 cm long with a neck 0.7 nrn diameter

by 5 mm long machined at the center of it. The specimen was located in a

furnace where the whole specimen could be brought to the temperature T . The

o temperatures T

q were measured with Pt/Pt 10% Rh thermocouples and the maximum

temperature rise T^ - T^ was measured using the specimen as its own resistance

thermometer similar to Meissner. The data analysis differs from that of

Meissner, and is developed in full by Holm [ 1 9 1 . The measurements were made

in a nitrogen atmosphere using currents up to 60 amps. A correction was

applied for heat losses from the neck by convection and radiation.

In their experiments at high temperatures (1473 - 2580°K) Hopkins [l 5

]

and Hopkins and Griffith [l6] used a simulated necked-down sample. A platinum

wire 5 cm long and 0.193 mm diameter was clamped symmetrically between

platinum clamps over nearly half its length at each end so as to leave 0.5 mm

exposed at the middle thereby simulating a necked-down sample as used by

Holm and Stormer. The clamps consisted of pairs of platinum semicylinders.

(0.635 cm diameter and 2.5 cm long) fitted accurately into semicyl indrical

channels in water-cooled copper blocks which maintained the ends of the

specimen at the temperature of the cooling water, (T

q

) . The maximum temperature, T , was measured by optical pyrometry. Measurements were made in vacuum,

in air, in nitrogen and in hydrogen, using currents up to 87 amps. It is

7
worth noting the extreme temperature gradients and current densities that

existed in the neck, the former reaching values of 9200°C/cm and the latter

5 2 being as high as 3 x 10 amps/cm . In reducing their data Hopkins and

Griffith differentiate (1-3) and write:

V If = 4kp

(T ) (1 " 5) m v nu

where kp is evaluated at T = T . The thermal conductivity is thus determined m

by measuring V and p as functions of T , T being held constant.

m o Cutler and his associates [l 7 , 20, 21, 22, 23, 24] simulated a neckeddown sample by welding a section of 5-mil platinum wire between two blocks

of the same material. The sample was suspended in a furnace which was used

to bring the specimen to temperature, T . The maximum temperature T^ was

measured using the specimen as its own resistance thermometer. The change in

resistance per se was not measured, but, rather, the corresponding voltage

change. They write:

and

» TS

dR

dT ’

6v = -4- d «n R> 1 2kpT d(4n T)

( 1 - 6 )

(1-7)

where d(^n R)/d(>£n T) is nearly unity.

Equation (1-6) follows from (1-3) and (1-4) using the approximation

V

2 = 8kP9

m

. With the eat ire specimen at uniform temperature, T^ (measured

byaPt/Pt 107o Rh thermocouple) the current is switched on. There is an

instantaneous voltage drop V across the specimen before it has time to heat

up. Superimposed on this is a transient voltage due to the change in
resistance as the neck heats up. The change in voltage 6v corresponding to

the change in resistance 6R is observed on an oscilloscope as the peak of

the transient. Measurements were made in vacuum from room temperature to

1400°K.

In the case of one-dimensional conduction where heat losses are present

(1-1) and (1-2) can be written in the form [25, Sec. 4.1C)]:

2 2 7T + P A - 2"af(T-Ta ) - 0,

dx TTa (1-8)

where a is the radius of the conductor, f(T-Ta

) is the heat loss per unit

area, I the electric current and T a the ambient temperature. The first term

in (1-8) represents the net amount of heat conducted from an element of the

specimen, the second term represents the Joule heat generated in the element

and the third term represents the heat loss from the surface of the element.

The works of Kannuluik and Carman [26, 27], Krishnan and Jain [28], and

Bode [29] are based essentially on equation (1-8). In all cases the measurements were conducted in vacuum so that heat losses from the surfaces of the

conductor were by radiation and an appropriate form for f(T-Ta ) was used.

In their experiments Kannuluik and Carman [26, 27] used a platinum

wire that was nominally 1.5 mm diameter and 11.6 cm long. The wire was

mounted in a closed coaxial cylindrical tube so that the ends of the wire

and the walls of the tube were at the same temperature, T = T . The tube

was evacuated. Measurements were made at five fixed-point temperatures in

the range 90 to 579°K; the thermal conductivity apparatus being immersed in

fixed-point baths at each of the five temperatures. Kannuluik and Law [30]

developed a solution to (1-8) in terms of measured resistances rather than

9
temperatures, and that solution was applied in deriving the thermal conductivity of platinum.

Krishnan and Jain [28, 31 ] measured on a platinum wire 0.5 mm diameter.

They give no details of their apparatus. They give two approximate solutions

to (1-8), one in terms of the temperature distribution near the center of a

short filament, and the other in terms of the temperature distribution in

regions slightly removed from the center of a long filament. Measurements

were made in the temperature range 1300 to 1800°K using alternating current

and measuring the temperature distributions along the wires with an optical

pyrometer

Bode [29] derived a solution to (1-8) giving the thermal conductivity

in terms of the rate of change with current of the resistance and the maximum

temperature in the conductor. The heat losses were evaluated by adjusting

the current through the specimen until its temperature was uniform throughout

and there was no temperature gradient in it. Under these conditions the

first term in (1-8) is zero and the third term in (1-8), the heat loss term,

is readily evaluated. The measurements were made on a wire 1 mm in diameter

and 5-10 cm long, using direct currents ranging from 8 to 13 amps. Tne wire

was mounted concentrically in an evacuated, water-cooled glass cell and the

temperature distribution along it measured with an optical pyrometer.

The second group of methods mentioned at the beginning of the chapter

can be considered in terms of the partial differential equation describing

the one-dimensional flow of heat in a rod or bar in the absence of an

electric current [25, Chapter 4 ]:

10
d-9) TTa2 k " 2TTaf(T-T ) = TTa2 wc ~ , 9x2 3 5t

where w is the density of the material, c its specific heat and dT/dt the

time variation of temperature. The other parameters are as defined above.

Equation (1-9) describes a transient state with the term on the right hand

side representing the rate of accumulation of heat. The solution to (1-9)

yields the thermal diffusivity, h, = k/wc, whence the thermal conductivity is

determined from a knowledge of the density, w, and the specific heat, cWheeler [32], Martin and Sidles [33], Zolotukhin [34], Schulze [35] mea sured

the thermal diffusivity of platinum and used their results to compute thermal

conductivity.

In his measurements Wheeler used a method proposed by Cowan [36] . The

specimen, in the form of a thin solid plate, is mounted in a vacuum chamber

and heated uniformly to incandescence by bombarding one face with a beam of

electrons from an electron gun. The electron beam is amplitude modulated to

vary sinusoidally with time, and the phase difference between the resulting

temperature fluctuations on the two faces of the plate yields a solution for

the thermal diffusivity. Heat loss is by radiation and the fourth power

radiation law is assumed. The platinum plate was 1 mm thick with face area

of 0.3 - 1.3 cm . The phase difference was measured in terms of the phase

difference between the signals from two photocells trained on the opposite

surfaces of the plate. Temperature measurements were made with an optical

pyrometer

.

Martin and Sidles [33] used a modified Angstrom method in measuring the

thermal diffusivity of platinum. A sinusoidally varying temperature was

11
applied to one end of a long rod (10-12 inches long and 3/16 inches in

diameter). The temperature oscillations produced at the end of the rod were

propagated along it. From temperature measurements at two locations along

the rod the velocity of propagation of the temperature wave was determined

as well as the amplitude decrement of the wave between the two measuring

stations. The amplitude of the temperature oscillation in the specimen did

not exceed 5°C so that Newton cooling could be assumed. Under these conditions

Sidles [37] has shown that the solution to (1-9) yields a simple expression

for the thermal diffusivity:

K = LV

( 1 - 10 ) 2 An q

’

where L is the separation of the measuring stations, V is the velocity of

propagation of the temperature wave and qis the amplitude decrement of the

wave between the measuring stations. The period of the oscillations was

60 seconds, and temperatures were measured with Pt/Pt 10% Rh thermocouples.

Measurements were carried out from room temperature to 1200°K on two samples

of platinum, one of commercial purity and one of high purity. The measurements were made in vacuum except at the lower temperatures where a helium

atmosphere was used.

Zolotukhin [34] apparently used a method of the Angstrom type [38]

but no details of his work are available to the author.

In measuring the thermal diffusivity of platinum at room temperature

Schulze [35] allowed the specimen to assume a constant temperature. One end

of the rod was then cooled suddenly and the temperature observed at regular

time intervals a short distance from that end. The solution to (1-9) under

12
these conditions is

T = 1 erf

x ijvx

( 1 - 11 )

where erf denotes the error function. A discussion of this method and the

solution is given in reference [25, Sec. 2.10].

For steady-state conditions the term on the right-hand side of (1-9) is

zero and we can write:

na

2 k ——

J- - 2TTaf(T-T

a ) = 0. (1-12)

dx

Barratt [39] measured the thermal conductivity of platinum on a wire

1 mm in diameter and 35 cm long at 290 and 373°K. Heat was supplied at a

constant rate to one end of the wire. The heat was conducted down the wire

and lost from the surface of the wire to the surrounding medium. The temperature of the hot end of the wire did not exceed that of the surrounding medium

by more than about 10°C so that Newton cooling could be assumed, i.e. f(T-T a )=

(T-T a)H where H is a constant. Under these conditions the solution to (1-12)

gives

k q

2

2n 2

a3H(TH

-Ta ) 2 coth 2rL (1-13)

where Q is the amount of heat flowing into the wire at the hot end, T is the

H

temperature of the wire at the hot end, L is the length of the wire and

/

2H

ka

The temperatures were measured with platinum resistance thermometers.

The surface conductance H was determined in a second experiment. All the

heat generated in the center span of the wire when an electric current passed

through it was assumed to be lost to the surrounding medium. The mean

13
temperature rise of the span above that of the surrounding medium was

determined using the wire as its own resistance thermometer. Then

H = q'/Q* where Q* is the heat generated and 0* is the mean temperature rise.

Barratt and Winter [40] used essentially, the same method as Barratt had

used earlier with the exception that a shorter wire was employed and both its

ends were held at fixed temperatures. For the solution see [25, Sec. 4.5].

If heat losses from the specimen are rendered negligible (by use of a

guard), (1-9) further simplifies to the basic equation defining thermal

conductivity

“na2k £ =Q > (1 " 14)

where Q is the heat flowing in the specimen and dT/dx is the temperature

gradient therein. Thermal conductivity is measured most directly in terms

of (1-14). In longitudinal heat flow methods a measured amount of heat flows

in a bar of diameter large enough to permit accurate determination of its

cross-sectional area. The temperature distribution along the bar is measured

and used in computing the temperature gradient. A cylindrical tube is located

coax ial with, the specimen and bears a temperature distribution matching

that on the specimen thereby minimizing lateral heat losses from the specimen.

The space between the specimen and the guard cylinder can either be filled

with thermal insulation, as is generally done at medium and high temperatures,

or be evacuated, as is generally done at low temperatures. Any heat losses

are then by conduction or radiation and can be satisfactorily analyzed. The

guarded longitudinal heat flow method is potentially the most accurate means

of measuring thermal conductivity, but, despite its apparent simplicity, it

14
is also one of the most difficult methods experimentally. Bode [41], Powell,

Tye and Woodman [42] and Powell and Tye [ 2 ] measured the thermal conductivity

of platinum in this way. Gruneisen and Goens [ 43 ], Mendelssohn and

Rosenberg [44] and White and Woods [45] conducted their measurements at low

temperatures in vacuum where use of a guard is not necessary.

Grtmeisen and Goens [ 43 ] measured the thermal conductivity at 21.2 and

83.2°K on the same platinum as had been used earlier by Meissner [ 13 ]. A

heater was attached to one end of the specimen and the temperatures were

measured with thermocouples.

Mendelssohn and Rosenberg [44] measured on a rod about 5 cm long and

1 - 2 mm in diameter over the temperature range 2.4 to 31.7°K. A small

electric heater was wound around one end of the rod and the other end was

connected to the expansion chamber of a Simon helium liquifier. Helium gas

thermometers were attached to the specimen by copper contacts 3 cm apart.

The pressure difference in the thermometers corresponding to their temperature

difference was read on a differential oil manometer.

White and Woods [45] used the same method as Mendelssohn and Rosenberg.

Their specimen was 1.5 mm in diameter and 5 - 7 cm long.

Bode [4l] measured the thermal conductivity of platinum on a bar 5 cm

in diameter and 7 cm long in the temperature range 273 t0 3?3°K. Details of

the apparatus are given by Bode and Fritz [46]. The heat input was measured

in terms of the electric power dissipated in a heater located at the hot

end of the specimen. The temperature distributions in the specimen and

guard were measured with copper-constantan thermocouples.

15
Powell, Tye and Woodman [42] and Powell and Tye [ 2 ] have made an

extensive series of measurements on the thermal conductivity of platinum.

They carried out their measurements on two samples using several variations

of a comparative guarded longitudinal heat flow method. The first specimen

was 0.635 cm in diameter and 6.1 cm long, and the second specimen 1.269 cm

in diameter and 10.16 cm long. The thermal conductivities of both specimens

were measured from 320 to 520°K using a comparative method in which the

specimen was attached to a rod of Armco iron of known thermal conductivity

at its upper end and to a water-flow calorimeter at its lower end. The heat

in-flow to the specimen was measured in terms of the temperature gradient

established in the iron, and the heat out-flow by means of the water-flow

calorimeter. The thermal conductivity of the platinum was thus determined

in terms of that of the Armco iron, with heat loss corrections applied from

the two heat-flow measurements.

For measurements above 520°K the calorimeter was not used, and the

assembly was inverted so that the platinum was uppermost and at the higher

temperature. Two series of measurements were made. In one the Armco

iron rod had a diameter of 0.371 cm, in the other an Armco iron rod of

1.273 cm diameter was used. The measurements were made in vacuum with

powder insulation between the specimen assembly and the guard.

A further series of measurements were made on the second specimen from

273 to 800°K. The assembly was the same as described above for the high

temperature measurements but with a water-flow calorimeter attached to the

lower end of the Armco iron rod to give a second measure of heat flow.

- 16
Other reported values of the thermal conductivity of platinum include

those of Carter [47], Johansson and Linde [48] and Mikryukov [49]. No details

of their works are available to the author. It is possible that the values

reported by Carter have been referenced from Jaeger and Diesselhorst [l 2]

The available information pertaining to the characterization of the

platinum specimens used by various investigators is summarized in Table 1

below so that differences in their specimens may be allowed for in comparing

their results.

Table 1. Sample Characterization of Platinum Samples Used

By Various Investigators.

Investigator Ref c 1/ Source— Purity 2/

p 2 7 3— „ 3/ Po— Density

Pt7o pQ-cm pQ-cm gm/ cm^

Jaeger and Diesselhorst 12 10.12 21.39

Schul ze 35 FH 10.0

Barratt 39 JM "pure" 9.77

Meissner 13 FH "high" 9.81 0.0162

Grtineisen and Goens 43 FH "high" 9.81 0.0162

Holm and Stormer 14 FH 99.95 9.90

Kannuluik and Carman 26 9.87

27 9.97

Mendelssohn and Rosenberg 44 JM 99.999 ~9 .

-0.01

Krishnan and Jain 28 JM " Spectro"

White and Woods 45 Baker 99.99 9.79 0.0125

Hopkins and Griffith 16 JM 99.999

Mikryukov 49 99.99

Bode 29 Degussa 99.9

Cutler, et al 17 "• 9 4/ Powell and Tye 2 JM 99.99 9— 9,85 0.013 21.5

Bode

^

j

Martin and Sidles—

41 99.98it/ 21 .32

33 Bishop 99.999

TTK , 6/ Wheel er*- 32 * JM

99.9

99.95 21 .5

1 / FH - Firma Heraeus., Germany; JM - Johnson, Matthey and Co., England;

Baker - Baker Platinum Co., USA; Degussa, Germany; Bishop - J. Bishop

and Co .

,

USA.

2/ Ice-point resistivity

- 17 -
3/ Residual resistivity.

4/ Spectrographic analysis given in reference.

5/ Martin and Sidles computed thermal conductivity from diffusivity using

assumed density of 21.37 gm/ cm" and specific heat data of Jaeger and

Rosenbohm [5 0]

6/ Wheeler computed thermal conductivity from diffusivity using measured

density of 21.5 gm/ cm

3 and specific heat data of Kubaschewski and

Evans [51 ]

The thermal conductivity data on platinum are summarized in Figures 3

and 4 and in Table 2. Reported values of the Lorenz Number are plotted

in Figure 5

Table 2. Thermal Conductivity of Platinum in the

Temperature Range 273-373°K as Reported

by Several Investigators.

Temperature °K

Investigator Ref

273 2921/ 373

Jaeger and Diesselhorst 12 0.696 0.725

Schul ze 35

Barratt 39 0.691 0.690 0.711

Meissner 13 0.699 0.701 0.706

Johansson and Linde 48 0.690

Holm and Stormer 14 0.699 0.715-

Kannuluik and Carman 26 0.702 0.720

Mikryukov 49 .720

Cutler, et al 17 .609

Powell and Tye 2 .733 .730

Bode 41 .702 .705 .710

Martin and Sidles 33 .701

\J Nominal temperature - data points range from 290 to 294°K.

2/ Value interpolated from equation given in reference.

18
»Csp ujo/m *A J.IAIXOO ONOO 1VWH3HX

19

FIG.

3,

THERMAL

CONDUCTIVITY

OF

PLATINUM

AS

REPORTED

BY

SEVERAL

INVESTIGATORS.
20.0

»6ap ujo/m ‘All AllOnQNOD TVWH3H1

- 20

FIG.

4,

LOW

TEMPERATURE

THERMAL

CONDUCTIVITY

OF

PLATINUM

AS

REPORTED

BY

SEVERAL

INVESTIGATORS.
>-

CO

o

&y

I—K

O

ft. ftj £C

Cft <t

H

O

K

tsJ CD

m

N

Ul

K

O

16

d

tl

10 ro cm CM CSJ CM CM

,M 6®p/2 A

4 a38N0N ZN3B01

21

SEVERAL

INVESTIGATORS
The agreement between the reported values for the thermal conductivity

of platinum in the temperature range 273 - 373°K is quite good. Representative of these values are the results of Bode [4l] which are quoted as being

accurate to 0.5%. All the investigators, with two exceptions, Cutler, et al

and Powell and Tye, agree with Bode to within 2%. Cutler, et al report four

values at room temperature with scatter of l47o indicating the inaccuracy of

these results. The results of Powell and Tye, representing as they do a

large number of determinations by well established methods [52, 53], are

about 4% higher than those of Bode. This may be due to the higher purity

and higher density of the samples used by Powell and Tye.

Some of the high temperature thermal conductivity data on platinum are

plotted in Figure 3. The data of Meissner, Kannuluik and Carman, and

Mikryukov all fall within 2% of Holm and Stormer 1 s curve. The data of

Bode [29] which he refers to as preliminary, and those of Cutler, et al, show

considerable scatter but tend to agree also with the results of Holm and

Stormer. As shown in Figure 3, values of thermal conductivity computed from

the Lorenz number reported by Hopkins and Griffith, using the resistivity

values of Vines [79], tend to confirm those of Holm and Stormer at the higher

temperatures. The results of Holm and Stormer are thus representative of

those of a number of people all of whom measured the thermal conductivity by

an electrical method. However, these results as a group differ significantly

from the results of Powell and Tye, Martin and Sidles, and Wheeler, all of

whom used a non-electrical method. The results of Krishnan and Jain fall in

a category by themselves showing the thermal conductivity of platinum to be

a decreasing function of temperature.

22
The low temperature thermal conductivity results are in good agreement

when one allows for differences in sample purity as the thermal conductivity

is strongly dependent on purity at low temperatures. The data of Gruneisen

and Goens [43] are not shown in Figure 2 but they are essentially in agreement

with the results of Meissner.

23
CHAPTER It

DESCRIPTION OF THE APPARATUS

It has been pointed out in Chapter 1 that all the data above room

temperature, with the exception of that of Krishnan and Jain, derived from

investigations using electrical methods are in fairly good agreement.

However, the values given by the electrical methods are considerably higher

than those given by non-electrical methods. Although there is a great deal

of variation in published data, there is some evidence to indicate that

electric methods tend to give higher values of thermal conductivity than do

non-electric methods for some of the other transition metals also, notably

tungsten [38] . This immediately raises the question as to whether or not

there are some inherent differences in the values of thermal conductivity

yielded by the two types of methods. Two possible reasons for such

differences come to mind.

1. The analyses of electrical methods may be subject to errors of

omission or logic.

2. Thermal conductivity may be significantly dependent on electric

current density, at least for some materials.

In view of the differences just noted it was decided to conduct the

present investigation using both an electrical and a non-electrical method.

As we have seen in Chapter 1, a necked-down sample is an appropriate form

for high temperature measurements by an electrical method. Moreover, this

is the sample configuration utilized by Holm and St'ormer, Hopkins and Griffith

and Cutler, et al , with whom we wish to compare data. Consequently, it was

decided to use such a sample configuration for the electrical method.

24
As regards a non-electrical method it was decided to use the guarded

longitudinal heat flow method. This method, while cumbersome, is the most

direct and most accurate method of measuring thermal conductivity for high

conductivity materials.

To give a direct and accurate comparison between the two methods it

was considered desirable to combine both sets of measurements in one apparatus

and on the same specimen thereby eliminating a number of uncertainties which

would arise in comparing data derived from measurements in different apparatus

and on different specimens.

The above considerations led to the specimen configuration shown in

Figure 12. The specimen (A) is raised to the desired temperature level by

means of the heaters and . In the non-electrical method, or the direct

method as it will be referred to in the remainder of the paper, the heater,

, located slightly above the center of the bar is energized so as to

cause heat to flow down the bar. Heat from this heater is prevented from

flowing up the bar by adjusting the top heater so that there is no temperature

difference across the necked-down region of the specimen. Lateral heat losses

from the bar are minimized by matching the temperature distribution along the

guard to that along the specimen. Thermal conductivity is determined from

the measured temperature distribution along the lower portion of the bar,

the power input to the central heater and the geometry.

In the electrical method or the indirect method as it will be referred

to, an electric current is passed through the specimen. The electrical

potential drop across the necked-down region of the specimen is measured

25
as a function of current, while the maximum temperature rise in the neck

is determined from the change in electrical resistance (due to a given

change in current) and the temperature coefficient of resistance of the

material. The thermal conductivity is determined from the voltage drop

across the neck, the maximum temperature rise in it, and the electrical

resistivity of the material. The electrical resistivity is determined

from potential drops in the lower part of the specimen at the lowest

current setting.

The apparatus is described in detail in this chapter and the methods

are discussed at length in Chapters 3 and 4.

26 -
Mechanical Configuration

The mechanical configuration of the apparatus is illustrated diagrammatically in Figures 6 and 9 and described in detail below.

Specimen : The specimen (A) is a bar 0.7864 inch (~ 2 cm) diameter by

7.25 inches long with a 0.045-inch diameter by 0.130-inch long neck machined

in it 1.622 inches from the upper end (Figure 7). A special technique,

described in Appendix A, had to be developed for machining the neck due to

its structural weakness.

Molybdenum extensions (B) of the same diameter as the specimen are

screwed to the specimen (A) at both ends. These extensions consist of bars

approximately 3 1/2 inches long bored from one end to a depth of 3 inches,

leaving a 0.040-inch thick wall. The open ends are brazed to copper blocks

(C) that act as heat sinks. The molybdenum extensions are filled with high

purity coral alumina.

The lower end of the specimen assembly (B-A-B) is bolted to, but

electrically insulated from, a brass flange which is welded to a water-cooled

brass column (B) . This column is firmly bolted to plate (E) which serves as

a base for the apparatus.

One of the major problems with the necked-down specimen is that of

protecting the neck from mechanical strain due to tension, compression,

torsion or bending. Any clamp supporting the neck would have to be

electrically insulated from the specimen and differential thermal expansion

between the clamp and specimen would introduce strain in the neck. As an

alternative to a clamp it was decided to counterbalance the load on the

- 27 -
28
Figure 7 • Upper end of the platinum specimen showing

the neck, the outer set of thermocouples and the speci- men heater. The upper heater and part of the upper

molybdenum extension also appear.

! Figure 8. Close up view of the specimen heater. The

j two current leads can also be seen along with the two

potential leads welded to the right-hand current lead,

j The two wires coming from above the specimen are thermocouple wires.

.1

- 29 -
J

s

Figure 9. Details of the Suspension System and the Upper

Assembly. (Components are Identified in the Text).

- 30 -
neck due to the weight above it so that there would be no net force on the

neck. The weight of platinum above the center of the neck was computed

from dimensional measurements and from the measured density of the specimen,

and the components extending from the upper end of the specimen were weighed

before assembly.

The counterweight (W) is suspended from a string which passes over two

pulleys and attaches to an aluminum hanger (H) . The pulley wheels are mounted

on ultra-high-precision, low-friction, bearings of 1/8-inch diameter bore and have

a starting force of less than 1 g each. The hanger pulls on two pins (J)

projecting from a split aluminum collar (M) clamped to a molybdenum well (F)

which is brazed to the upper copper block (C) . The molybdenum well passes

through a 3/4- inch linear bearing (L) which serves to maintain the upper part

of the specimen in precise alignment with the lower part, and presents

negligible resistance to the free vertical motion of the specimen resulting

from thermal expansion. The bearing is mounted on the upper plate (E) but

electrically insulated from it. The two 15 3/4-inch diameter by 1-inch thick

aluminum plates (E,E) are connected by three 1-inch diameter stainless steel

tie bars (K) to form a rigid framework for maintaining proper specimen

alignment. The overall length from the inside of one plate to the inside

of the other is 30 1/2 inches. One of the pins (J) projecting from the

aluminum collar is constrained in a device (Z) consisting essentially of

two fixed vertical pillars and two horizontal plates which can be moved up

and down (Figures 9 and 10). The vertical pillars prevent lateral motion of

the pin and thereby prevent the upper part of the specimen from rotating

31
Figure 10 . Device for constraining the upper part

of the specimen. The end of the pin can be seen between the two horizontal plates and the two

vertical pillars.

Figure 11. Close up view showing the linear bearing

housing (L), the upper end of the molybdenum well (F),

the clamp (M), the hanger (H), the molybdenum electrode

(0) and the copper support (S).

32
relative to the lower part. The horizontal plates can be located so as to

prevent up or down motion of the pin and consequently of the upper part of

the specimen. The horizontal plates are maintained in the constraining

position to protect the specimen except when tests are in progress and the

specimen must be free to contract and expand thermally.

Current is introduced to the specimen through a 3/8 -inch diameter copper

rod (N) at the lower end and through a molybdenum electrode (0) at the upper

end (Figures 9 and 11). The electrode is a 1/4-inch outside diameter by 3/16-

inch inside diameter tube closed at one end and brazed to a copper support (S)

at the other. The copper support is fastened to the upper plate (E) but

electrically insulated from it. The molybdenum well (F) contains a liquid

metal into which the electrode dips thereby affording a flexible current

connection. The buoyant force of the liquid metal on the electrode contributes

to the load on the neck and must be compensated for in the counterweight.

The electrode is fixed but the molybdenum well moves upwards with the specimen

due to thermal expansion during test runs. 'This changes the buoyant force and

consequently puts a load on the neck. The maximum change in buoyant force is

only 3 grams however and so the neck is not strained significantly. The

liquid metal used was a gall ium- indium eutectic alloy chosen primarily for

its low vapor pressure and its comparatively low freezing temperature of

15.7°C. The requirement for low vapor pressure was dictated by a need to

evacuate the system. Preliminary tests were run to evaluate the uncertainty

in buoyant force due to surface tension and sticking of the gall ium- indium

to the molybdenum surfaces. A very definite hysteresis effect was observed

33
as the electrode was moved relative to the well and then returned to its

initial position. The largest uncertainty in buoyant force was determined

to be about 4 grams. The choice of molybdenum as the electrode and well

material stems from its compatabil ity with gallium which reacts with most

other metals, and from the fact that molybdenum is wetted by gallium.

The current feed-in system serves also as a heat sink for the upper

part of the specimen assembly. Water flows down a 1/8-inch diameter thin-wall

stainless steel tube (T) within and concentric with the molybdenum tube and

flows back out between the two so that the electrode acts as a heat exchanger

transferring the heat from the liquid metal to the cooling water. The inner

tube is also brazed to the copper support as shown in Figure 9. The water

is pumped from a tank through a closed system and its temperature is always

well above the freezing point of the gallium-indium eutectic alloy.

Furnace : The furnace skeleton (Figure 6) consists of two brass plates

(P) , 1/2-inch thick by 15 3/4 inches in diameter, joined by three 1-inch

diameter stainless steel tie rods (R) . The outer shell of the furnace (X)

is a brass tube 81/2 inches outside diameter by 1/8-inch wall thickness.

It slips over a ring fastened to the lower brass plate but is not connected

to the ring. Copper coils for water cooling the furnace are soft soldered

to both plates and to the outer shell.

The inner core (G) of the furnace, or the guard as it is called, is a

molybdenum tube 15 inches long by 2 1/4 inches inside diameter. Molybdenum

was selected as the most suitable metal from a standpoint of high- temperature

thermal and mechanical properties, compatabil ity with other materials in the

34 -
system, and cost. Since the inner core acts as a thermal guard to prevent

heat losses from the specimen, it was considered more desirable to make it

from metal rather than from ceramic so that the temperature distribution

along it could be more easily controlled and more accurately measured. In

the gradient section at each end and in the central gradient section, the

wall thickness of the guard is 1/16 inch. For a span of 3 1/4 inches opposite

the neck region on the specimen the wall thickness increases to 3/8 inch to

assist in making that region isothermal as desired in the experiment. There

is a second span of 1/2 inch where the wall thickness is 3/8 inch to allow

grooves to be machined in the tube to accommodate a heater winding. The

space between the specimen and the molybdenum guard is filled with high

purity aluminum oxide powder of very low thermal conductivity (0.43 mW/cm

3

deg G in air at 22°C)and low density (0.16 gm/ cm ). The guard sits on a brass

plate which is fastened to the lower furnace plate, and can be removed for

inspection or repair without disturbing the remainder of the furnace.

Attached to the upper end of the guard is a brass ring (a) having the same

inside and outside diameters as the guard itself. The ring has a copper

cooling coil soldered to it to maintain the end of the guard at room

temperature. It also has a terminal strip screwed to it to which the heater

2/ leads are connected. Two-mil— indium foil is used at all interfaces to

enhance thermal contact.

The outer furnace core (V) is an aluminum oxide tube 5 inches inside

diameter by 5 3/4 inches outside diameter by 10 inches long supported on

three 3/8-inch diameter aluminum oxide rods (U)

— 1 mil = 0.001 inch.

35
Attached to the upper plate of the furnace is a split nut. This nut

engages a lead screw mounted between the plates (E,E) and by turning the

screw the furnace can be moved up or down. Ready access to the specimen is

thereby afforded. The three tie rods (K) act as guide rods for the furnace.

Six linear bearings (Y) attached to the furnace plates ensure alignment and

permit the furnace to move up and down freely. The correct vertical location

of the guard relative to the specimen is determined when a probe attached

to the upper end of the guard makes electrical contact with a plate attached

to the molybdenum well at the upper end of the specimen assembly. When the

furnace has been positioned the indicating probe attached to the guard is

removed.

Envi ronmental System : The entire apparatus is mounted on the base plate

of a 4-inch vacuum station, the base plate being 26 inches in diameter by 1

inch thick. A 4-inch high by 24-inch diameter aluminum feed-through ring

7ith twelve ports accommodates power, thermocouple, water, gas and vacuum-gage

feed-throughs. The thermocouple feed-throughs were made by passing the wires

with flexible fiber-glass sleeving on them through a brass elbow section and

3/ filling it with hot low vapor pressure vacuum wax— which formed a vacuum

seal upon solidification. A 24-inch diameter by 26-inch high stainless steel

bell jar sits on the feed-through ring and encloses the complete apparatus.

The bell jar has two 4- inch diameter viewing ports and is counterweighted

to enable it to be raised and lowered.

The system can be evacuated and measurements made in vacuum. However,

it The wax used was Hard "W" Apiezon wax.

36
preliminary tests run in vacuum on the type of heater used for heating the

specimen, as described below, showed that the mean temperature of the heater

winding was approximately 1000*C above that of the test bar in which the

heater was located when the test bar was at room temperature and the heater

was delivering only 7 watts. Similar tests in nitrogen atmosphere showed the

mean temperature of the heater winding to be approximately 600°C above that of

the test bar with the latter at 300°C when the heater was delivering 110 watts.

In view of these tests, it was obvious that the specimen could not be run in

vacuum. As molybdenum is easily oxidized, an inert atmosphere had to be used.

Argon was chosen on account of its low thermal conductivity.

Thermal Configuration

Specimen ; The heater and thermocouple locations on the specimen and

guard are shown in Figure 12.

End heaters}, and Q^,are located in the molybdenum extensions (B) , 3/8

inch from the platinum-molybdenum interfaces. Each heater consists of six

helical coils of 8-mil diameter Pt 307o Rh wire, the coils having an outside

diameter of 31 mils and a pitch of 15 mils. The individual coils are

slipped into single-bore, high purity, alumina tubes of 32 mils bore and

47 mils outside diameter. The tubes are placed in 48-mil diameter holes

eloxed in the molybdenum extensions and located so as to uniformly introduce

the heat over the cross-sectional area. The tubes are a snug fit in the

holes giving reasonably good thermal contact. The coils are torch welded

to 15-mil diameter platinum jumpers which connect them in series. The

jumpers are insulated in alumina tubes. Current leads of 20-mil diameter

37
f 6

6 |_

Q<

Q,

9 X

•••••« 8 x

7 x

6 x

5x

4x

Q, ••••••

3x

B

X—- THERMOCOUPLE

<— CONTROL COUPLE

® — HEATER WINDING

Figure . 12, Heater and Thermocouple Locations.

38
platinum wire are welded to the heaters at each end and are insulated by

24-mil bore alumina tubing. Eight-mil Pt 30% Rh potential taps are also

welded to the heaters so that the heaters may be used as resistance

thermometers to measure their own mean temperature. Twenty-mil Platinel —

control couples with butt-welded junctions are pressed into slits in the

bars adjacent to the heaters.

The central heater, Q a , is contained in six holes drilled through the

platinum bar similar to the end heaters, Q1

, and Q3 , (See Figure 8). The

inner four holes are 40 mils in diameter and accommodate helical elements

contained in thin-wall ceramic tubes. The elements are made from 5-mil

diameter Pt 10% Rh wire, the outside diameter of the helix being 18 mils

and the pitch 10 mils. The outer two holes are 63 mils in diameter and

accommodate "swaged elements" having Pt 10% Rh sheaths insulated from

Pt 10% Rh heater wires by compacted MgO powder insulation. The swaged

elements are a snug fit in the holes so that there is good thermal contact

between the sheath and the bar, and consequently good thermal coupling

_4/ Platinel is a new all noble metal thermocouple developed by Englehard

Industries, Inc. It has high thermal emf, approximately that of

Chrome 1-Alumel, and can be used up to 1300® C. The negative leg of the

thermocouple is 65% Au, 35% Pd alloy (Platinel 5355) and the positive

leg is 55% Pd, 31% Pt and 14% Au (Platinel 7674). For further information see references [54, 55, 56 and 57].

39
between the heater and the bar. The six elements are connected in series

as in heaters and Q^. Platinum heater leads, 20 mils in diameter, are

welded to the ends of the swaged elements. The good thermal contact in

the swaged elements ensures that the temperature at the ends of the heater

closely approximates that of the specimen. Moreover, the current leads

extend radially from the heater in an isothermal plane. The combined result

is to minimize heat losses out the leads. Two 8-mil Pt 10% Rh potential

leads are welded to each of the current leads, one at the junction of the

heater and the current lead, and the other about 1/2 inch back along the

current lead. The two Pt 10% Rh potential leads together with the intervening

section of platinum current lead serve as a differential thermocouple to

determine the temperature gradient in the current lead. By taking potential

readings with the current flowing in the forward and reverse directions the

IR drop in the leads can be eliminated and the temperature gradients therein

determined. This data is used in computing heat flows along the leads. The

potential drop across the inner taps is used in computing the power generated

in the heater. The distances from the heaters to the nearest thermocouples

and potential taps are such that perturbations in heat flow and electric

current flow generated by the presence of the heaters have decayed to an

insignificant level at the position of the thermocouples or potential taps

as discussed in Appendix B.

Located in the gradient zone of the specimen are five 8-rail Pt/Pt 107o Rh

thermocouples spaced 0.787 inch (one specimen diameter) apart, with the

first one 0.787 inch from the cold end of the specimen.

40
The thermocouple wires are annealed in air at 1450° C for 1/2 hour and

then butt-welded ~ together. They are pressed into 7-mil wide by 9-mil deep

slits in the specimen thereby replacing the metal removed in machining the

slits. By virtue of the fact that the specimen is fairly pure platinum with

essentially the same absolute thermoelectric power as the Pt leg of the

thermocouple the junction of the thermocouple is effectively at the point

where the Pt 10% Rh leg first makes contact with the specimen, and the

temperature measured is the temperature at that point. Since the Pt leg of

the thermocouple leaves the specimen at another point there is only the

Pt 10% Rh leg to conduct heat away from the junction. Temperature measurement is consequently more accurate. The Pt 10% Rh wire emerging from its

groove extends a short way around the specimen in the same isothermal plane--

insulated from the specimen in broken ceramic tubing--so as to minimize the

amount of heat conducted away from the junction.

Similar thermocouples are located in the molybdenum extensions, three

in each, to measure the temperature distribution along them. This information

is essential to the mathematical analysis of the system.

Additional thermocouples are located on either side of the neck, one

57 To butt-weld the thermocouples, the two wires are held in a jig with a small gap between their ends. The ends of the wires are heated with an oxygen-gas torch and expand toward each other. If the original spacing

is correct and the heat is applied carefully the wires will just be at

the right temperature for them to fuse when thermal expansion brings

them together . Junctions made in this way were so even that it was

hard to find them even under a microscope.

41
set 0.875 inch from the center of the neck, the other set 0.090 inch from

the center of the neck. These are 15-mil Pt/Pt 10% Rh thermocouples contained in 14-mil wide by 16-mil deep slits. In addition to measuring

temperature, the couples are cross-wired at the selector switches so that

the Pt legs can be used to measure voltage drops across the neck when an

electric current is flowing. With no current flowing the Pt 107, Rh legs,

in conjunction with the platinum neck, can be used as a differential

thermocouple to control the differential temperature across the neck in

the direct mode of measurement

Guard: The guard (Figure 13) has three heaters Q. , Qc , and (k at

positions corresponding to those on the specimen assembly. All three are

swaged heaters with Pt 10% Rh sheaths and heating elements, and MgO insulation. They are pressed into grooves machined in the guard (Figure 14) giving

good thermal contact. The end heaters (Q/ and Q^) have a 62-mil O.D. sheath

and a 20-mil diameter heating wire and they are used to bring the guard to

the desired temperature. The central heater (Q,.) has a 40-mil O.D. sheath

with a 15-mil heating wire and it is used to produce a temperature gradient

in the guard matching that in the specimen. Forty-mil platinum heater leads,

insulated in ceramic tubing go to the end heaters while 30-mil leads go to

the central heater. The guard is electrically grounded but the heaters are

floating.. Platinel control thermocouples are peened into the guard adjacent

to each of the heaters.

Twelve thermocouples are located on the guard, three in the gradient

zone, three in the isothermal zone and three in each of the end zones. All

the thermocouples are 15-mil Pt/Pt 107, Rh with the junctions pressed into

42
Figure i r, , The guard The thermocouple wires emerge

from slits and go around the guard in ceramic tubing

Figure 1 . Guard heater. The swaged heater can be seen pressed into grooves machined in the guard. The

heater leads are welded to the ends of the heater and are insulated in ceramic tubing.

43
slits machined in the guard. The thermocouple wires are taken one turn

around the guard in broken ceramic tubing in an isothermal plane and

cemented to the guard with high purity alumina cement. This helps to

temper the thermocouple leads and reduce the amount of heat conducted away

from the junction by the leads. Within the furnace all the thermocouples

are insulated in single-bore ceramic tubes. For the remainder of their

lengths the wires are insulated in flexible fiber-glass sleeving.

All the thermocouples, both from the guard and the specimen assembly,

go to a junction box mounted on the inside of the feedthrough ring. There

they are torch welded to identical wires which are taken through the vacuum

feedthroughs to an ice bath.

All the Platinel control couples go to terminal strips on the upper

plate of the furnace. There they are spot-welded to Chromel -Alumel wires

coming from the controllers.

The alumina core (V) has a 40-mil molybdenum heater winding (Qy) located

in 1/16-inch wide by 3/32-inch deep by 3/8-inch pitch grooves terminated

1/2 inch from each end of the core. The winding is cemented over with alumina

cement. The function of this heater is to bring the furnace as a whole to

temperature and to reduce the heat losses from the molybdenum guard and the

power load on its heaters. The heater has four current taps, one at each end

of the winding and one 1- inch from each end, to enable the power density at

the ends of the winding to vary from that over the central span, and thereby

compensate for end heat losses. A Platinel control couple is mounted on the

core

44 -
Instrumentation

T empe ra ture Con t rol : In the direct method of measuring the thermal

conductivity the Pt 10% Rh legs of the outer pair of thermocouples in the

neck region are used in conjunction with the necked-down portion of the

specimen as a differential thermocouple to control the power to heater Q

and maintain zero temperature differential across the neck. The signal

from this thermocouple is amplified by a breaker-type d-c amplifier and fed

into a current-adjusting- type proportional controller incorporating automatic

reset control and rate control. The output of the proportional controller

regulates the power to the heater by means of a transistorized power amplifier

fed by a regulated d-c power supply. In the indirect mode an electric current

flows through the specimen and the above system of control cannot be employed.

In this case the Platinel control couple adjacent to heater is put in

series opposition with a signal from an adjustable constant voltage source

and the resultant signal fed to the proportional controller which regulates

the power to . The external signal is adjusted to give zero temperature

differential across the neck.

The specimen heater (C^) is fed constant voltage from a regulated 10 A d-c

power supply. Regulation is better than 0.01%.

Power to heater Q and to the three guard heaters (Q^, and Q

&

) is

supplied by variable-voltage transformers, which in turn are fed by voltageregulating isolation transformers. Power to each heater is pulsed high-low

by individual thermocouple-actuated controllers.

45
Power to the heater winding (Q^) on the alumina core is supplied by a

variable-voltage transformer fed by a voltage-regulating isolation transformer.

The current is manually ratioed among the three heater sections. The total

power to this heater is pulsed high-low by a single thermocouple-actuated

controller

All heaters are supplied by separate isolating transformers or power

supplies to minimize current leakage effects.

Temperature Measurement : The noble metal leads of the thermocouples

are brought to an ice bath, where they are individually joined to copper

leads. The copper leads go in shielded cables to a bank of double-pole

selector switches of the type used in precision potentiometers. The selector

switches are housed in a thermally insulated 3/8-inch thick aluminum box.

The copper leads are thermally grounded to the switch box to minimize heat

transfer directly to the switches. The emfs of the specimen thermocouples

are read on a calibrated 6-dial high-precision potentiometer to 0.01 |ulV, using

a photocell galvanometer amplifier and a secondary galvanometer as a null

detector. The emfs of all other themocouples are read on a second precision

potentiometer to 0.1 jj.V using an electronic null detector.

P ower Measurement : Power input to the specimen heater (Q^) is determined

by measuring the d-c current through the heater and the voltage drop across

the potential taps. These measurements are made by means of the 6-dial

potentiometer

Resis tance Measurement : All electrical resistance measurements are made

by measuring the current (from a 0-100 A regulated d-c power supply) flowing

through the specimen, utilizing a calibrated 0.001 ohm shunt, and measuring

46
the appropriate voltage drop in the specimen. The selector switches mentioned

above are cross-wired so that the voltage drops between corresponding legs of

the specimen thermocouples may be read on either potentiometer.

47
Figure 15 • General view of the apparatus. The apparatus

proper stands on a round table with the vacuum system suspended

below the table. The furnace is shown in the down position

just prior to taking the clamp off the specimen. The bottom

of the bell jar appears at the top of the picture. The ice

bath can be seen in the lower center of the picture. The

thermocouple feed-throughs are just above it. The 6-dial

potentiometer and a small section of the instrumentation are housed in the relay rack to the left of the picture. The

second potentiometer appears in the lower left-hand corner.

48
Figure 16', Close up view of the apparatus. The specimen is in

place with the clamp still on. After the clamp is removed, the

furnace is cranked up to the operating position. The furnace

has already been filled with insulating powder. The counterweight hangs in position at the top of the picture. The

pulley stand, the copper support, the hanger and the constraining device can all be seen.

49
CHAPTER III

THEORY OF THE INDIRECT METHOD

In the Indirect method the specimen supports an electrical potential

gradient, as well as a temperature gradient, and the corresponding thermoelectric effects must be analyzed. This is done most conveniently by the

methods of irreversible thermodynamics, applying the Onsager Reciprocal

Relations [58, 59, 60, 61, 62, 63 , 64, 65, 66]. The application of the

relations to steady-state processes involves an approximation; but as

Callen [58] points out, it is an excellent approximation and is completely

Justifiable in this case.

We define a set of current densities J^:

= s Ltj Xj , (5-1)

—

where Xj are the "conjugate forces," such that

R(S) = 2 , (3-2)

where R(S) is the rate of production of entropy in the system. Then the

Onsager Reciprocal Theorem [67, 68] states that

L

ij = Lji (3-3)

in the absence of a magnetic field. The rate of entropy production R(S)

is uniquely defined by the system under consideration, but since R(S) can

be split into a sum of products in many ways, one is left with a choice of

current densities and conjugate forces [59],

^ We define an electric current density J, an energy current density W,

and an entropy current density S, so that the divergence of each of these

current densities is the rate of change per unit volume of the corresponding

50
thermodynamic variable. Since thermal expansion is negligible in our system

we can write [58, equation 12]:

TS (3-4)

where T is the absolute temperature, p. is the electrochemical potential,

e is the electronic charge and J, the electric current density, is equal to

the electron current density times the electronic charge. We are considering the electron current to be the only mass current. Mass transport and the

associated entropy flows due to electromigration (preferential migration of

ions in a solid when a direct current is passed through the solid) and due

to the Soret effect (mass transport in a solid due to a temperature gradient)

are not considered. The reader is referred to [69, 70, 7l] for discussions

of these two effects.

We are concerned with steady-state conditions for which:

v-w = o (3-5)

V-J = o (3-6)

and V-S - R(s) . (3-7)

From (3-4) and (3-7) we can write

r(s) - j{- -±- 7 (4-) - -4- 7 (rr)] + ”- 7 (-r) • (3-8)

Using the identity

7 (~r) - - ie 7T • (3-9)

we can write the current densities in terms of the "conjugate forces:"

J

—W

( 3

- 10 )

(3-11)

51
The electrochemical potential M- is given by

H = Q + e0 , (3-12)

where 0 is the electrostatic potential, and Q is the chemical potential,

which in the case of a metal is simply the Fermi energy [72]. Since the

Fermi energy is a function of temperature only, we can write

v (-r) -r

71 + 70 • o-u)

We now proceed to evaluate the coefficients Ln , Li S and l^ a

.

The electrical conductivity is defined under isothermal conditions

a =

0) as

r

- j 1

" -J

“ L ^0 7

(l)

( 3

- 14)

since for VT = 0 , V^-gj! - V0. From (3-10) it follows that

Lll » aT . (3-15)

The absolute thermoelectric power is defined (see [ 73]) as

„abs

i> =

_ VT J %0

from ( 3

- 10 ) it follows that

Lis + TS

abs

]

The thermal conductivity is defined as

k r

-q 1 r -wiL VT J J=0 L VT J J=0

( 3

- 16 )

( 3

- 17 )

( 3

- 18 )

where Q is the heat current density, since when J a 0, W = Q.

52 -
From (3-10) and (3"H) we get

Las * kT2 + aT + TS

abs

J

(3-19)

Substituting for , L^ a and Lsa in (3-10) and (3-11), we can write

“4 — J and W in terms of the defined thermoelectric parameters:

j „ - CT [v (-S-) + s ebsv T

]

i

( 3

- 20 ) w - - a (-£- + Ts

abs

) v (-t-) - [asabs(-i- + TS

abs

) + k] VT

- + TS

abs

]

J - kVT . (3-21)

Taking the divergence of W and using (3-5) and 0- 6)

V«W J»V + TS

abs )- V«kVT = 0 ( 3

- 22)

Therefore

{' (-4-) WSabs + S

abs

VT

] - V«(kVT) » 0 (3

- 23 )

,abs In a homogeneous and isotropic medium, the parameter S is a function of

temperature only, and we can write

J-[v (“-) + (t - d

|T ) VT + S absVT] - V.(kVT) = 0 .(3-24)

The coeffieient T

dS

abs

OT

is defined [58] as the Thomson coefficient T.

From (3-20) and (3-24)

+ TVT] - V-(kVT) 0 (3-25)

At this point, it is convenient to introduce a pseudo-potential i|r

such that . - crv* ( 3

- 26 )

- 53 -
and Vr|f mV (~^“) + S absVT (3-27)

Writing 0-25) in terms of i|r we get

7. (kVT) + a7i|r .Vi|r + crrVy.VT - 0 . 0-28)

Equation 0-28) can be considered as the generalized equation relating the

temperature distribution and the electric potential distribution in an

isotropic homogeneous medium carrying an electric current. We now proceed

to solve that equation.

The second term in (3-28) can be written as V« (iJfCjVijf) since

V * (TjrcrVij/ ) = \|rV* (aViJr) + aV\|r*Vi|f

= since V»(aV^) s — V* J a 0 . (3-29)

The third term in 0-28) can also be written in the form of a divergence,

since

v{

a

(

C

TdT

) v+l

- a7* ,vJ^TdT +

(

C

TdT

>

( V - (o-ViJr

)

rT = O'Vijr .7

1 ^TdT = (JTV^.VT J (3-30)

where T* is any arbitrary fixed temperature.

Using these identities 0-28) can be written as

V^kVT + +ct Q^Tdl)

7^J o 0 (3-31)

Integrating

kVT + <j\|rVijr + a Q* TdT^ Vi)f s V\ +

—

C > (3-32)

where \ is any potential satisfying the Laplacian V^X « 0 and c is a vector

54
constant. Since V»(aVf) »« 0, we can write

V\ - t 0oV* (3-33)

where ^ 0

is a constant which must satisfy the boundary conditions. Substituting for V* in ( 3

- 32), we get

kVT + a(t - t c

) Vf 4- (aVf)J TdT « c (3-34)

If c 0 , it can be seen from (3»34) that VT and Vijr must be parallel

at all points. This implies that if there is no electric current flow

across a boundary, there can be no heat flow across that same boundary.

Moreover, equipotential surfaces are also isothermal. More explicitly, the

condition that c » 0 requires that all the heat generated in an electrically

insulated conductor must flow out the ends of the conductor, there being no

heat losses from the sides. This condition can never actually be met in

practice since there is no perfect or even near-perfect thermal insulator.

However, by appropriate choice of geometry, as employed in the present

experiments, heat losses can be made very small so that we can take "c =* 0

as a valid statement of the boundary condition. Equation (3 _ 3^) then

reduces to

kpVT + (f " t 0)vt + WfT

TdT « 0

J <pc (3-35)

where p is the electrical resistivity, p « 1/a.

If there is a point in the medium where the temperature has a maximum

value Tm , the gradient is zero at that point. Further, if we let

T* » Tm , (3-36)

the first and third terms in (3“35) are zero at T = Tm

. Since there is a

- 55
current flowing aV\]f £ 0 and (3 - 35) is satisfied only if i(f — t|f 0 = 0 at

T = Tm

. In other words

* = t 0

at T ° Tm > (3-37)

so that if

is the value of at the point of maximum temperature.

Integrating (3 “3 5) along a line from one isothermal surface )

to another isothermal surface S 2 (T2,i|r 2) an<3 substituting Tm

for T*

r*Sp -» [»S 2 -» (*S 2 _

TcpVT*dr + I (ilr - * )V*.dr + Vijr-dr TdT = 0 , ( 3

- 38) J Si JSj, 0 J

Si J Tm

or, since VF*dr = dF

(3-39)

An arbitrary isotropic and homogeneous conductor is represented in

Figure 17.

Figure 17. An electrical conductor with arbitrary geometry

56
There is no electric current flow across the boundary C and the geometry

is selected to make heat losses from C negligible so that the condition for

—

c =* 0 discussed above is satisfied. The isothermal surfaces Sx and S 2 are

taken to be at the same temperature, T0 , and the surface of maximum temperature is represented by Sm. In the absence of the Thomson effect, the

surface of maximum temperature would bisect the reduced resistance between

Si and S 2 , the reduced resistance--a geometrical quantity-being the resistance of the medium for electrical conductivity of unity. However, the

Thomson effect moves the surface of maximum temperature away from the symmetrical location, this effect being referred to as "the Thomson shift"

[74]. It is analyzed in detail by Davidson [74] and discussed by Holm [l8]

and Llewellyn Jones [75], It is convenient to define >|r Q * 0 and consider

the integral 0-39) from Sm

where T “ ^m and i|r ** 0 to any other surface S

where \|r = \)r , T T. Rearranging we have

0-40)

It is convenient to write 0-40) in the following form

ijr 2 — 2ei}r — F

2 = 0 0- 41 )

where 0- 42)

and 0-43)

Solving the quadratic equation (3-41) , we get

i|r - e ± (e 2 + F

2

) 1 / 2 e ± F(1 + (e/F) 2

) 1 / 2 0- 44)

57
The Thomson effect is relatively small as shown in Appendix C, so that the

second-order term (e/F) 2 in (3-51) can be neglected. Then

V = e ± F , 0-45)

the plus and minus signs referring to the high and low potential sides of

Sm respectively.

Referring to 0"43) we see that although T is a function of if,

fTm

TdT is independent of the sign of i|r since Tm

is always greater than T.

Consequently e is an even function of , having the same value at corresponding points on the high and low potential sides of Sm . At Si we have

x - e(T0

,Tm) - F(T0,Tm) (3-46)

and at S 2

+2 - e(T0 ,Tm) + F(T0

,Tm) . (3-47)

Therefore

t 2 - * x - 2F(T0 ,Tm) . (J-48)

From (5-27) and (3

- 13) we have

V\|r = V0 + -|^- VT + S abSVT . 0-49)

Integrating from Si to S 2 >

2 - = 02 - + ~T tC 2 - C x

3 + J%

absdT . (3-50)

sb s In an isotropic and homogeneous medium Q and S are functions of temperature

only. When the terminal temperatures are the same, as in the case we are

considering; i. e., when Ti = T2 = Tos we can write from (3-48) and 0“5O)

= 02-01 = V = 2F(T0

,Tm) , (3-51)

where V is the voltage drop between the surfaces Si and S 2 . If the potential probes are not at the same temperature, a correction must be made as

58
indicated in (3-50). Writing F explicitly we finally get from (3-51) and

(3-42)

~T~ “ J^dT “

J^'”

k*,d9 - 0-52)

where 9 » T — T^, © jn TM -^m — To*

Equation 0-52) shows that the maximum temperature rise 9 m in an

electrically-heated conductor with negligible lateral heat losses and with

both its ends held at the same temperature is, to first order, a function

only of the voltage drop V across it, and of the thermal and electrical

conductivities of the material, and is independent of the geometry of the

conductor provided the geometry is such as to make lateral heat losses

negligible. If the maximum temperature rise can be measured as a function

of the applied potential, kp is readily determined.

Hopkins and Griffith [15,16] maintained the ends of their specimen at

room temperature and measured the maximum temperature I’m directly by sighting

on the specimen with an optical pyrometer.

The maximum temperature rise can also be measured indirectly using the

specimen as its own resistance thermometer. This method was employed by

Holm and Stormer [l4] and Cutler et al. [l7] and requires that the temperature coefficient of resistance of the material be known over the temperature

range of interest and that it be large enough to yield sufficient sensitivity. Platinum satisfies both requirements very well and the resistance

method of measuring the maximum temperature rise was adopted in the present

experiments. We now proceed to relate the maximum temperature rise in a

conductor to the measured resistance of the conductor. This problem is

discussed at length by Holm [l8, paragraph 18],

59
Consider two geometrically identical conductors, each carrying the same

current, and differing only in that one conductor has temperature-dependent

properties p =» p(©) and k = k(9), while the other conductor has temperature

independent or constant properties P = P Q

and k = kD where 9 = T — T0

is the

temperature measured from any arbitrary reference temperature T0

. Let if and

9 be the potential and the temperature respectively in the conductor having

temperature-dependent properties, and let f and 0 Q refer to the corresponding variables in the conductor having constant properties. Then if d^r and

d| 0

are the potential differences between corresponding equipotential surfaces

in the two conductors, we can write from (3-26)

d

+o -

_Pq

df p

( 3

- 53 )

Neglecting terms arising from the Thomson effect, we have from (3-41):

and from (3 "3 5) :

t = ±F (3-54)

Combining the last three equations,

d

+o

kp d©

± —

(3-55)

(3-56)

Integrating,

J!

m kp od0

(3-57)

Since both conductors are assumed to carry the same current,

Ro

(3-58)

60
and we have

Ro 1 P®m kP 0dG

R

=

F J

0 F

(3-59)

Assuming that k and p are both linear functions of temperature--a valid

assumption over small temperature intervals--we can write:

P " Po^ 1 + ao®) 0-60)

k = R0 ( 1 + 3 ©9) (3-61)

and kp 13 koPo0 + T] o0) , (3-62)

where ot Q

is the temperature coefficient of resistance evaluated at T » TQ ,

g o

is the temperature coefficient of thermal conductivity evaluated at

T » Tq and TJ0 » aQ + 3©* The term in ao$o02 can be neglected. Substituting

the above value for kp in (3 - 42) and integrating we get

F(e,em) - + [ekoPo {sm (i + -|

2-

e

m) - s(i +

e)}]

1/2

. (5-63)

If the integration is carried out from Si to S 2 where T » T

05 or 0 =0,

we have, from 0-63) and (3-51),

F = ^ 2k0p o0 m ^1 + g ®m)J 0-64)

and V

2 - 8koP 0

8 m (l + -g2- 8m)

. (3-65)

Substituting F from (3~64) into (3-59) and performing the indicated integration we have for 8 0,

Ro

R

ftp uo

Tlo 'Ho arctan G 0- 66 )

61
where

2tlo9m (l + G -jr- 9m) 0-67) 2 -

Equation (3

- 66 ) gives 0 m implicitly in terms of known and measured

parameters, and its value can be determined by iteration. This value can

then be plugged into 0-65) to solve for Icq.

62 -
CHAPTER IV

CALCULATION PROCEDURES

Indirect Method ;

We have shown in Chapter IH that , in the indirect method, if the resistance of the neck and the voltage drop across it are measured with a current

passing through the specimen, the thermal conductivity can be determined

from these measured parameters using equations (3-65), (3-66) and (3-67).

Greater precision can be achieved, however, if measurements are made at a

number of different current levels. In the present investigation measurements were made at four or five current levels, and the way in which the

data have been analyzed is presented in this section.

For each current level we can write

V * 8 kQ p Q 0 m ( 1 + 0 m) (3-65)

It was impractical in our experimenta' set-up to hold TQ at exactly the same

value for different current settings. If we let Tr be an arbitrary constant

reference temperature then we can write:

po pr L

1 + ar ( T0

- T r)J , (4-1)

o

1! rH + 3r (To - Tr) _

> (4-2)

and

r I

k0 Po “ krPr L

1 + T] r (To “ Tr)J (4-3)

where k,. and pr are evaluated at T = T

r

, and T r

=> Ctf- + 3 r

• The higher order

term in a r 0 r is neglected. Substituting this expression for k 0 p 0 in (3-65)

we get ~ r 'i V

2 = 8 kr pr

e m (1 + e m) (i + 7) r (Tc

- T

r) j .

- 63 - (4-4)
We do not measure 0 m directly but, rather, use the neck as a resistance

thermometer, the relation between the electrical resistance of the neck, R,

and the temperature rise,0 m , being given by (3-66) and (3 -67). In the limit

of small 0Co 9m , (3-66) reduces to — ° ~9m (4-5) % R

o

3

where R

q

is the resistance the neck would have if it were uniformly at temperature T

q

. We define an adjusted resistance, R

1 , for the general case

where a o ®m is not sufficiently small for (4-5) to be valid and where T

q may differ from :

R’ - Rr (i + 6 m ) . (4-6)

In the limiting case where and 0 m < <1, (4-6) reduces to (4-5)

and R

1 *» R.

We also define an adjusted voltage, V

1 , given by

V

t 3

Substituting (4-6) and (4-7) i

V

2

[l +

^ ©m] [l + fyr (To ~ Tr )]

i f //i \

v

' s - 12

^ < 4- 8 >

Differentiating (4-8) and rearranging, we obtain

v , «r «r i V*

r ——— ______

12 pr

d R'

The thermal conductivity is thus determined from

R

1 curve. From (4-6) and (4-7) we see that

(4-9)

the slope of the V' 2 vs.

d V

,a o Lim d V

3

d R* CHo^m^O d R

To -* Tr

(4-10)

64
In the analysis given in Chapter III and also that given just above, we

explicitly assumed that there was no loss or gain of either electric current

or heat across the surface C of the conductor. In the present investigation

a necked-down sample is employed, the neck and the specimen as a whole being

surrounded by an insulating powder. Since this insulating powder has an

electrical conductivity many orders of magnitude lower than that of the speci

men material, the assumption of no flow of electric current across the boundaries is completely valid. The powder surrounding the neck prevents heat

loss by convection, and radiation through the powder is negligible. However,

the powder conducts heat away from the neck and it is necessary to analyze

this heat loss and develop an appropriate correction for it.

Since most of the temperature rise in the necked-down region is in the

neck itself, we will consider only heat losses from the neck and will neglect other heat exchanges. We idealize the neck and its environment as

shown in the sketch below and as described in the following discussion.

rr/////////

insulation'

W////Z,

1

i

) 2a NECK 2b

t

/

W777T/,

INSULATION

////////Z , r z

Figure 18. The neck surrounded by powder insulation.
The region -&< z < l

,

a<r<b contains thermal insulation of thermal

conductivity The surfaces r ** b and z » ±i are maintained at zero

temperature. The neck region, r < a, has thermal conductivity ko, and is

heated by electric current, I. We assume the neck to be isothermal across

any given cross section, and that there is continuity of temperature and

heat flux at the interface between the neck and the insulation.

The solution to this problem is given by Carslaw and Jaeger [25, p. 221,

eq. (25)], and, in our nomenclature, is

m

u* = ^CnF0[Rn ;Bn]cos Zn » (4-11)

n-1,3,5,

where the notation n = 1,3,5, indicates that the summation is to be taken

only over odd values of n, and where

c -

4(-l) 2 P 0 I 2 - c^qX2

] F0 [An ;Bn] - 2n2a 2

Anft 0F l [An ;Bn

FoCxjy] = I 0 (x)K0 (y) - K0 (x)I0 (y) , (4-13)

F x [x;y] = I

1 (x)K0 (y) + Kx (x)I 0 (y) , (4-14)

a = nrra w = nrrb _ nTTr nnz ~ 2JL ’ 2C ’ Kn ~ 2JL

* ^ 2SL*

p = Po (l + <* 0 u*)is the electrical resistivity of the neck material,

Im = modified Bessel function of first kind and order m,

Km = modified Bessel function of second kind and order m,

u* = excess temperature above that at the boundaries r = b and z ±Jb

66
The temperature in the neck itself is given by

u* ^

CnF0[An

;Bn ]cos 2n n=l,3,5,

(4-15)

Since the Thomson effect is small, we can assume a symmetrical temperature distribution with the maximum temperature, Ujn*, occurring at the center

of the neck, z = 0. Evaluation of (4-15) at z = 0 yields, after some

rearrangement

,

I6l z

p 0

lz

TT 5

a4

kn,

(-D

: (4-16)

y

n=l ,5,5, ns T 1 - ^^qPo12 _ 1

L nWk0 nna k0 Fo[ An ;Bn]

-I

as the maximum temperature in the presence of heat losses.

For the parameter values appropriate to our experiment, the term in

#oI2 and the term in ft 0/k0

in the denominator of the right-hand side of

(4-16) are small compared to unity and, as shown in Appendix E, (4-16)

reduces to the form

ftp 1281 (rra/ 2£) 1

kQ

* rr 4

a Ko(rra/21 ) J

(4-17)

where u^ is the maximum temperature rise the neck would have if there were

no heat losses, i.e., if = 0.

The temperature distribution along the center-line of the specimen (not

to scale) is shown schematically in Figure 19.

67
The quantities 0^ and 0 m

refer to the maximum temperature rises between

potential taps in the presence and absence, respectively, of heat losses

-B- from the neck. The quantities and um refer to the temperature rises

only in the neck itself. The solid curve indicates the actual temperature

distribution; the dotted curve indicates the temperature distribution that

would exist in the absence of heat losses.

- 68
For the purpose of evaluating heat losses from the neck we can neglect

the temperature-dependence of kp and write an expression, analogous to

(3-65), for the temperature rise in the neck in the absence of heat losses

V

2 v n SkoPo (4-18)

where Vn

is the voltage drop in the neck itself. For o£ m < < 1, Vn is

given by:

21 'n TTa' Ro

where V is the total voltage drop between the potential taps,

(4-18) and (4-19)

%

V*

2ft 2a

4 V

(4-19)

Thus from

(4-20)

From (4-17) we can write

in - e* ^

1281 Ki(tra/ae)

ko TT

4, a Ko(TTa/2e)

(4-21)

which, after substitution from (4-20) and (3-65), becomes

where C is a constant given by

and the geometrical factor, £2, is given by

n

64 / 24 N

3 K1 (na/2£) P Q

2

TTsas ’ \ TTa / Ko(TTa/24) I^2

with the term in T|o© m from (3-65) neglected.

(4-22)

(4-23)

(4-24)

69
We define an adjusted resistance, R* (analogous to R

1 ), as

R* - Rr

(l + -|- « ra m*) (4-25)

In the absence of heat losses 9^** 9 m

and R* * R' . From (4-6), (4-22) and

(4-25) we get, in the case of small Qf r

0 m

R' - R*(l + ~~ G? rC

m*) (4-26)

For small ar

0 m voltage drops are proportional to resistances and we can

write

V' = V*f(l1 + -|- arc8*

) (4-27)

where V' is the value V* would have in the absence of heat losses. The

thermal conductivity in the absence of heat losses is given by (4-9) and is

proportional to dV 1 ~/dR' . When heat losses are present the measured resistances and voltage drops are too low and must be corrected to correspond to

the values they would have in the absence of heat losses. We can write for

small ^x^m

dV' dV

dR' dR*

dR*

dR'

dR* d

dR* dR* [v*2

(l + ( 4- 28 )

Using the definitions of R' and R* and carrying through

reduces to

dV 1 2

dR'

1 +

1 + C 1 +

dV*g

dR*

the algebra (4-28)

(4-29)

If G < < 1 and also ®r®tn ^ ^ 1 5 the terms involving products of these

quantities can be neglected and (4-29) becomes simply

70
(4-30)

dV 1

'

dR'

1 dV*‘

1 + C dR*

Thus in the presence of small heat losses from the neck, the apparent

value of kr obtained by assuming dV' /dR 1 dV*2/dR* must be multiplied by

the factor 1/(1 + C) to get the true value

ar^r dV*2

1 + C 12p r dR* (4-31)

where C is given by (4-23) > if C goes to zero (4-31) reduces to (4-9)

.

The calculation procedure used is as follows: let us assume that, at

any given nominal temperature, measurements are made at n current levels.

Let (T0)j, (To^ 2» * * * (T0) nbe tbe en<f temperatures at each of the n

current settings, and let (Rc) l

, (R^g, • • • ( Ro)n be tbe resistances the

neck would have if it were uniformly at these temperatures. Further, let

(R)j^ (R)g> • • • (R) n be the actual measured resistances of the neck at the

different current settings. Then, the reference temperature Tr

is taken as

the average value of the end temperatures

n ±1*0h (4-32)

i-1

An approximate value for the resistance Rr

is computed from the resistance

(R) corresponding to the lowest current level for which Joule heating is

minimal, and the neck is nearly isothermal at temperature (T0 ) i so that

(R)

x « (R0 ) 1

:

Hr » OOjl + «i(Tr - (To),)] . ( 4_33)

where is evaluated at temperature (To)^. Values of cn were computed from

the data obtained in the resistivity measurements [see Chapter VIl].

For each experimentally determined resistance (R)^ we compute a

71
corresponding approximate value of 9 ^ using (4-5) and a value for (R0) given

by (R

0)j. - RrLl + ar ((I0)i

- T.)] , (4-?4)

where o' is evaluated at T = Tv . Using this value of 9* as the first trial

value, Newton-Raphson iteration is used to compute the value of 0* which

J>/

satisfies (3-66). Values for R* and V*3 are then computed from (4-6) and (4-7)

respectively. A least-squares straight line through these values of R* and

V*3 gives as the ordinate intercept and dV*s /dR* as the slope. This value

for Rt

is a more accurate value than the initial value used, as it is based

on an extrapolation to I = 0. Improved values for (R0)^ are now computed from

the new R^ using (4-35) . The calculation is then repeated using these values

of (R ) . . This in turn leads to improved values for R and dV*3 /dK* . The

o' i r iteration converges in a few passes and the final values for R^ and dV^s/dR*

are used in (4-31) to compute the apparent thermal conductivity.

The above calculation procedure was effected using a high-speed digital

computer. It would not be practical for hand calculations since iterative

solution of (3-66) for 9 * would be too tedious. An alternative calculation

m

procedure, suitable for use with a desk calculator, is given in Appendix D.

Using the apparent value of , as computed above, and equation (4-23)

and (4-24) the correction factor for heat losses from the neck is computed

and plugged into (4-32) to give the true value for the thermal conductivity.

_6/ Values for 0 , computed from the thermal conductivity data obtained in

the NBS Metals Apparatus on the same specimen T76], were used as first

trial values, and these values were also iterated on.

72 -
Direct Method

In the direct method of measuring the thermal conductivity each datum

point is computed by simultaneous solution of three tests:

1. an "isothertial" test with no power input to the specimen heater

and with the temperature distribution on the guard adjusted to closely

match that on the specimen.

2. a "matched"gradient test with sufficient power input to the

specimen heater to maintain the desired longitudinal temperature

gradient in the specimen and with a matched temperature distribution

on the guard.

3. an "unmatched "gradient test with the power input to the specimen

heater and the temperature at the center of the measuring span the

same as in the "matched "gradient test, and with the temperature distribution on the guard parallel to that on the specimen but 10° below it.

For each of these tests, assuming one-dimensional steady-state heat

flow in the specimen (an assumption justified in Appendix H) , the total

heat flow, Q, through the specimen is

Q

= -k A -H- (4-36)

dz

where k is the thermal conductivity, A is the cross-sectional area of the

specimen, T is temperature and z is the longitudinal coordinate. For

moderate temperature ranges, the thermal conductivity of the specimen can

be assumed to vary linearly with temperature; then (4-36) becomes

Q

= -k

rA {1 + 0 r (T-Tr )J -g- (4-37)

where k^. is the thermal conductivity of the specimen at a reference temperature, Tr , and

p r

is its corresponding temperature coefficient. The difference between the heat flows in two tests is given by

73
(4-38)

where quantities of one test are distinquished from those of the other by

use of primes. If we define the reference temperature as

Tr = T (dT/dz) - T' (dT/dz) ' (dT/dz) - (dT/dz)

(4-39)

the second term on the right-hand side of (4-38) vanishes, and the thermal

conductivity at the reference temperature, Tr , is given by

k,- ~ A

ro

- (Q-Q 1 ) 0 dT/dz “ dT/dz')

' (4-40)

Inspection of (4-40) reveals that systematic errors in the measured heat

flow, and in the measured temperature gradient, which are not dependent on

these quantities, are eliminated by simultaneous solution of two tests having

different heat flows. The most obvious errors of this type are thermocouple

errors. In a simultaneous solution each thermocouple in effect measures

a temperature difference so that errors in calibration of the thermocouples

cancel out to first order. Further possible sources of error will become

evident below. Determination of k involves measurement of the cross - r sectional area of the specimen, the total heat flow in the specimen and the

longitudinal temperature gradient in the specimen for each of two tests.

These quantities are evaluated at the position of the middle thermocouple

in the measuring span.

Cross-sectional area

The effective diameter of the specimen at 25°C was computed from the

measured diameter and surface roughness as described in the section on

density measurements in Chapter v • The diameter at temperature T was

computed from that at 25°C using the equation

74
(4-41) D

T D

25 (°‘ 99978 + 8. 876x10 -e T + 1.511xlO-9 T

2

)

This equation was fitted to smoothed thermal expansion data for platinum

[77j. The cross-sectional area was then computed from the diameter*

Heat Flow

The total heat-flow through the specimen at the position of the center

thermocouple was calculated using the expression

,Q = P-q a -qb ~qn

-q

i -q

c (4-42)

where P is the measured electrical power input to the specimen heater;

qa and q^ are the heat losses along the two leads carrying current to

the specimen heater;

qn

is the heat loss across the necked-down portion of the specimen;

q^ is the heat loss into the insulation surrounding the specimen; and

q c is the heat loss down the thermocouple wires and ceramic tubes next

to the specimen.

Each of the quantities in (4-42) is considered separately below.

Power input to Specimen Heater (P) : The electrical circuit for the

specimen heater is illustrated diagrammatical ly in Figure 20 below.

Figure 20. Circuit diagram for the specimen heater.

- 75 -
The power input to the heater is measured in terms of the current flowing

through it and the potential drop across it, and is given by

P = nEv (1 + R /Ry) (E

s /R

g

- n yRv) (4-43)

where is the voltage drop across the output of the volt box as measured

with a potentiometer, n is the resistance ratio of the volt box, E

g is the

voltage drop across the standard resistor, Rq , in the shunt box, as measured

with a potentiometer, R is the total resistance of the potential leads

and R^ is the total resistance of the volt box.

Heat Flow in Current Leads ( q^g^) : The circuit diagram in Figure 20

shows only one set of potential leads coming from the heater. In fact there

were two sets as shown in Figure 21, but only the inner set, i.e. the

potential taps closer to the heater, was used in measuring the power input

to the heater. The two sets of potential leads were required in measuring

heat conduction along the current leads.

HEATER

AT\ Z j 2

Figure 21. Arrangement of potential leads for the specimen heater.

The current leads were platinum and the potential leads Pt 10 Rh. Consequently, each current lead could be used along with its two potential

leads as a differential thermocouple to measure the temperature drop, AT,

between the potential taps. With current flowing to the heater the voltage

drop measured between 1-2 or 3-4 is the algebraic sum of the XR drop

between adjacent potential taps (where I is the current and R the resistance

76
of the current lead between the potential taps) and the Seebeck emf due

to the temperature drop, AT, between the potential taps. Assuming no

heat losses from the current leads, it can be shown (Appendix F) that

the heat conducted along the leads is given by

q a , b

I

3 R

2

( 4 - 44 )

where p is the electrical resistivity of the current lead and k its

thermal conductivity. By taking measurements with the current flowing forward and reverse, R and Al can be determined for each lead, and consequently qa and qfe

.

The derivation of (4-44) is based on the assumption that there is no

heat flow across the surface of a current lead. In fact, however, heat lost

from the current leads into the surrounding powder insulation is not

necessarily negligible, but a significant portion of it flows back to the

specimen and so in effect is not lost. The rest of the heat lost into

the powder either flows to the guard or flows longitudinally in the powder

insulation. It is very difficult to analytically derive a correction for

the effective heat loss from the current leads into the insulation and no such

correction has been applied to the data presented in this paper. The

heat conducted along the leads, as determined from (4-44), was less than

0.05% of the heat flowing in the specimen and the error due to neglecting

heat loss from the surfaces of the current leads into the insulation is

discussed in Chapters VII and VIII.

Heat Flow across Neck (q n) : Referring to Figure 18, it is seen that

heat can be conducted across the necked-down region of the specimen by the

neck itself and by the powder insulation surrounding it. The conductance

of the powder is ftn(b2 - a

2

)/2 £ where ft is its thermal conductivity and the

77
other parameters are defined in Figure 18. The conductance of the specimen

between the two thermocouples (marked by x in Figure 18) is kF where k

is the thermal conductivity of the specimen and F is the geometric factor.

The factor, F, can be determined from the corresponding equation for electrical conductance, I = o-FV where I is the current, (j the electrical conductivity and V is the voltage drop between the thermocouples. This data

is available from measurements by the indirect method.

The heat flow across the neck is then given by

^n

= kF + rrCb2 - a?)

'

11

(4-45)

where it is assumed that the temperature differential across the insulation

is the same as that measured between the thermocouples.

Heat Loss into the Insulation (q^) : In order to determine q^, the

heat exchange between the specimen and the surrounding insulation, it was

necessary to perform an extensive mathematical analysis. If the temperature

distribution along the guard exactly matched that along the specimen there

would be no radial heat exchange between the specimen and the guard. However, there would still be an exchange of heat between the specimen and the

surrounding insulation in order to provide the longitudinal heat flow in

the insulation adjacent to the specimen.

The heat flow from an elemental length of the surface of the specimen is

dq t = 2na *(-^-) dz

, (4-46)

where a is the radius of the specimen, ft the thermal conductivity of the

insulation, 0 the temperature in the insulation relative to an arbitrary

fixed temperature, r the radial coordinate and z the longitudinal coordinate.

78
The net heat flowing across the surface r = a between z, and z is 1 2

^i^ 2

! > Z

2

) = 2TTa dz

= a (4-47)

where ^ is, in general, temperature dependent. Let us define a new potential, f, that satisfies the relation

fe r V ? = ft(0) V9 , where ' = fj(o)

Integrating (4-48) 9

1

r> 9 — | ft(e) de

r

(4-48)

(4-49)

where we have selected the integration constant so that £ = 0 when 9=0.

Writing (4-47) in terms of § we get

IjL

(

z

i

> z

2 ) = ftrD ( z

x > z p) (4-50)

where

(4-51)

In order to evaluate D(z, , z

?

) one must know the temperature dependence

of the thermal conductivity of the powder insulation as well as the radial

temperature gradient at the surface of the specimen. The thermal conductivity of the insulation was assumed to be a linear function of temperature

ft = ft r

(l + Yr 9) , (4-52)

where and y^ are evaluated at 9 = 0, Y being the temperature coefficient

of the thermal conductivity of the insulation. Then, from (4-49)

£ = 0 + ~!-£- 9

s (4-53)

2

79
The radial temperature gradient at the surface of the specimen was determined by Fourier analysis of the hollow cylinder of powder insulation

between the specimen and the guard with known temperature distributions

at its boundaries. At the surface of the specimen, r = a, the potential

distribution was represented by

CO (£) = g G + (g - gQ

) — + Y An sln (4-54)

r = a ° (JD ° cjo L (i) n=l

and at the inner surface of the guard, r=b, the potential distribution

was represented by

(O r_h = h

o + (h ' hn

) “

+ Y

Bn sin > (4 " 55) r-b ° (jo ° od /Li n uo n=l

where ^(r,z) has the values ^(a,0) = g

Q

, ^(a,oo) = g^, f(b,0) = h

Q

,

§(b,oo) = h , at the ends of the region, z=0 and z=uo, which correspond to

the outermost thermocouples on the guard. At these ends the radial temperature distribution in the powder insulation was assumed to be logarithmic :

<?>„„ = go + (h

o

- *o)

In r/a

^n b/a

(5) ~ - s» ^ - V -ttff '(JO (JO (JO

(4-56)

(4-57)

The Fourier coefficients A and B in (4-54) and (4-55) were deter

n n mined from the measured temperature distributions along the specimen

assembly and the guard cylinder. In the gradient region of the specimen,

and in the upper and lower specimen extensions, the observed temperature

and thermocouple locations (corrected for thermal expansion) were used to

derive quadratic equations giving Q vs. z, where Q was measured relative to

the temperature at the center of the gradient region of the specimen. In

80
the neck region, which is nearly isothermal at all times, a linear equation

was used. The three quadratics in Q transform to quartics in £ and the

linear equation in Q transforms to a quadratic in | with application of

(4-53). In the intervening heater regions, smoothing cubics TlOOl were used

which provided continuity of temperature and longitudinal temperature

gradients . Thus the ^-distribution along the specimen was described by

seven smoothly- joined polynomials: quartic (lower extension), smoothing

cubic (lower' extension heater), quartic (gradient region of specimen),

smoothing cubic (specimen heater), quadratic (neck region of specimen),

smoothing cubic (upper extension heater), quartic (upper extension).

Equating this set of polynomials to the right-hand side of (4-54) and

making use of orthogonality enabled calculation of the A

1 s . A similar

n set of polynomials was used to represent the ^-distribution along the

guard, from which the B^'s were calculated. It would serve no useful

purpose to explicitly display the expressions for A

n

or in this paper:

they are quite lengthy and rather complex.

With the boundary conditions (4-54), (4-55), (4-56), and (4-57) the

potential, at any point in the powder insulation is given by

? * [s0 + < h

o

s0

) In r/a

In b/

a

f

1 + (h

cu In r/a ~l z /nb/a cm

V

1 An

Fo( nTTr/ a°» nTT b/u)) " Bn

Fo( nrrr/yo > nTTa/uu) nnz

> — — ——— sin n= 1 F

0

(nirra/a); nrrb/oo) w

,(4-58)

where

F

0

(x,y) = I

0

(x> K

o

(y) - K

q

(x) IQ

(y)

I = modified Bessel function of first kind and order m,

m

K = modified Bessel function of second kind and order m.

81
Substitution of (4-58) into (4-51) finally yields D (z^z ):

D(zi>Zs ) = 2rraj |(h

Q

- g

0

)(z

s - z

x ) + (h^- h

Q

- g+ gQ

)

z

32 - z 2

2(o

® A^F^nna/cu; nnb/yj) - (u)/nTra)Bn f mz nTTZ ' ^ _ CQS —

0) 03 '

: •I » nE 1 F

q

(nn a /o) ; nrr b /to)

ain b/a

1 M-59)

where F

i (x,y) = ^

(x) K

Q

(y) + (x) XQ(y) . (4-60)

Since the heat flow in the specimen is to be evaluated at the position

of the center thermocouple in the gradient region all the heat loss to the

insulation from the location of the heater down to the position of the

center thermocouple must be considered. In addition, heat losses in the

region between the specimen heater and the neck must be considered since

these have to be provided by the specimen heater. The neck in effect, can

be considered as the upper end of the specimen for purposes of this analysis,

since any heat exchanges between the powder and the specimen above the neck

do not affect that part of the specimen below the neck as long as zero temperature differential is maintained across the neck. Therefore, in evaluating q^, the limits of integration for D(z

i ,z

j

) are the position of the

center thermocouple (z

x ) and the position of the center of the neck (z^)

Heat loss along thermocouple wires and insulator (q£

) : The heat loss

qc

, along the thermocouples and ceramic insulators next to the specimen was

computed from the expression n

q

c (4-61)

i=l

where is the longitudinal thermal conductance of the i

tJl wire and its

insulator, n is the total number of wires crossing the plane where the

thermal conductivity is evaluated, and dT/dz is the temperature gradient at

that plane. Each was computed from the thermal conductivities and dimen

sions of the wire and insulator.

82
Tempera ture Gradient

The temperature gradient in the specimen was computed from the

measured temperatures at the five thermocouple positions in the gradient

region. The separations between thermocouple grooves at room temperature

were accurately measured before the thermocouples were installed; the

separation at elevated temperature was computed using (4-41) . Since

temperature gradients in the specimen were rather small (less than 5°c/cm)

it was essential that the conversion of thermocouple emfs to temperature

not introduce any additional uncertainties. The equation

where T is temperature (°C) and E is emf (mV), was found to fit to within

l|~iV the calibration data for the Pt/Pt lQ%Rh thermocouple wire from which

the specimen thermocouples were fabricated. This equation was used for converting the thermocouple voltages to temperatures.

The temperature gradient was computed by evaluating the slope, at the

center thermocouple location, of the quadratic equation of least-squares

fit to the five temperatures and thermocouple positions.

Appendix G gives a brief analysis of temperature measurement errors

due to conduction of heat along the thermocouple wire.

In analysing the data the "isothermal" test is combined with the

"matched gradient" test and with the "unmatched gradient" test to give two

equations of the form (4-40) . Inspection of equations (4-40) , (4-42) and

E = 15.83952

/ T \4

1.92753 (-1 ± j v 1000 7

- 2.50480 (1.0 - expf-4. 18312 (T/1000) 1) , (4-62)

83
(4-50) shows that kj. is a linear function of ^ . The effective thermal

conductivity of the insulation surrounding the specimen depends on the

density of the powder and the pressure and type of gas present, and is best

determined under experimental conditions. This is done by simultaneous

solution of the two equations of the form (4-40) which yields values both

for the thermal conductivity of the specimen, kp, and the thermal conductivity of the insulation

Electrical Resistivity :

The electrical resistivity of the gradient region of the specimen was

determined by measuring the voltage drops between platinum legs of the

thermocouples in that region with a 10A d-c current passing through the

specimen. The specimen was held isothermal to better than 1°C during this

measurement. In order to minimize thermoelectric effects, voltage drops

were measured, with the current flowing forward and reversed. Resistivity

values were computed from these voltages and from the measured current

and geometry. The effective mean temperature was computed by Simpson's

rule integration of the observed temperatures.

84
CHAPTER V

SPECIMEN CHARACTERIZATION

In order for a truly valid comparison to be made between the results of

different investigators who measure on a particular kind of material it is

necessary that their specimens be characterized as extensively as possible

so that differences in specimens may be accounted for. For this reason a

number of pertinent measurements have been made in an attempt to characterize

the specimen used in the present investigation. These measurements and the

results thereof are described below. Additional appropriate information

is also provided.

Source : The platinum was provided by Engelhard Industries, Inc., and

was classified as being of commercial purity or according to Engelhard

classification as being E-2 grade. Spectrographic analyses are given below.

Three samples were provided, all three coming from the same casting as

described below. The first sample was a bar 0.805 inch in diameter and

12.200 inches long from which the thermal conductivity specimen was later

machined. The second sample was a rod 0.150 inch in diameter and 9.84 inches

long to be used for resistivity measurements. The third sample consisted

of 20-mil diameter wire to be used in resistivity measurements and in

measurements of thermoelectric power versus Pt 27 [78],

Fabrication and Thermal History of Specimen : Commercial purity platinum

sponge (approx. 200 ounces) was induction-melted in air in a zirconium

silicate crucible. The metal was slightly over the melting point when it

was poured into a clean graphite mold (casting 11/4 inches in diameter

and 15 inches long).

85
The cast bar was swaged cold to a diameter of 1 inch . The bar was then

machined to 0.90-inch diametero Next, a piece 12 3/4 inches long was cut

from the middle of the bar, acid-cleaned in hot aqua regia and annealed at

700°C for 1 hour. This piece was then machined to a solid cylindrical bar,

0.805 inch in diameter and 12.200 inches long with a 0.531-inch diameter

by 2. 156- inch deep flat-bottomed hole drilled coaxially in one end.

The bottom end of the original bar was then worked down to a diameter

of 0.162 inchs and a piece 12 inches long was removed to supply the 0.150-inch

specimen. This piece was cleaned in boiling aqua regia and approximately

57o (by weight) of the surface was removed. It was then drawn to 0.150 inch

in diameter by 9.84 inches long through clean carbide dies.

A portion of the remaining material was cold swaged to 80-mil diameter,

drawn through clean carbide dies to 50-mil diameter, and drawn through

diamond dies to 25-mil diameter. This wire was acid cleaned in hot aqua

regia to 22-mil diameter. It was then drawn through clean diamond dies to

a diameter of 20 mils.

The three samples were cleaned at finished size in the following manner:

U1 trasonicsl ly cleaned, then pickled for 10 minutes in 50% hot nitric acid.

Washed and then pickled for 10 minutes in 50% hot hydrochloric acid, washed

in distilled or demineralized water. The fabrication and cleaning described

above were carried out at the Engelhard Plant.

The as-received 0.805-inch bar was annealed in air for 51/2 hours

at 770°C in a horizontal tubular furnace and furnace cooled at a rate of

approximately 120°C per hour. Shortly thereafter the bar was accidentally

dropped causing it to deform slightly at one end. After correcting the

damage the bar was reannealed for 1 1/2 hours at 680°C and furnace cooled

at a rate of approximately 90°C per hour. Its thermal conductivity was

86 -
then measured In the NBS Metals Apparatus over the temperature range -160°

to +810°C. Following these measurements the hollow end of the bar was cut

off and the remainder of the bar machined and ground to a solid bar 10

inches long by 0.786 inch diameter. The electrical resistance of this bar

was measured at ice and liquid helium temperatures (see below) . The thermal conductivity specimen (7.25 inches long by 0.786 inch diameter) was

then cut from one end of this bar. A length of 2.5 inches was cut from

the other end for low temperature thermal conductivity measurements , (see

Discussion of Results). The remaining disc, approximately 0.4 inch long,

was reserved for metal lographic and spectrographic analyses. Two 1/4-20

holes were chased 1/4 inch deep in the specimen, one at each end, and the

thermocouple slits, heater holes and neck were machined in the specimen.

Steel cutting tools were used in machining the neck, while a mixture

of trichlorethylene and oil served as a cutting fluid. To cleanse the neck

of oil and any contamination contracted from the cutting tools, the following cleaning procedure was followed. Degreasing was effected by immersion

for half an hour in trichlorethylene vapor. The neck was then pickled for

10 minutes in hot 50% nitric acid. Following this the neck was washed in

distilled water, pickled for 10 minutes in 50% hot hydrochloric acid, and

again washed in distilled water. There was equal likelihood of the rest of

the specimen having slight surface contamination as a result of machining

but its effect on the bulk properties of the specimen would not be nearly

as grave as the corresponding effect in the neck region. It was considered

sufficient to clean the surface of the specimen with toluene and carbon

tetrachloride.

87
Hardness: Hardness measurements were made on the platinum bar in the

as-received condition as well as after each of the two anneals. Measurements

were made with a Vickers Pyramidal Diamond Tester using a 10 Kg load and a

2/3-inch objective. The results are tabulated below.

Table 3. Hardness Values as Measured on the Platinum Bar

3

2 4

Location* 12Vickers Hardness Number

3 4 5

As received 69.6 68 .

65.8 53.3 65.2

After first anneal 35.0 35.2 36,0 38.2 36.3

After second anneal 46.7 40.3 36.5 38.0 36.5

* The locations indicated above do not represent specific

locations but merely general locations where the hardness

measurements were made.

The higher values obtained at locations 1 and 2 after the second anneal,

occurring as they do at the end that was deformed slightly when the bar

fell, tend to indicate that not all of the hardening resulting from the

fall was annealed out. However, no measurements were made on that end

of the bar so that there was no need to worry about it.

Spectrographic Analysis : A quantita-t ive spectrographic analysis on the

20-mil wire was made by Engelhard Industries, and the results of their

analysis are tabulated below. A general qualitative spectrographic analysis

was performed on the platinum disc at the National Bureau of Standards. The

qualitative analysis, being less sensitive, detected only Ag (10-100 ppm),

Mg (< 10 ppm), Pd (10-100 ppm), and Fe (< 10 ppm). The higher concentration

of iron indicated for the wire may result from the extra fabrication.

88
Table 4. Spectrographic Analysis as Determined on Platinum

Wire No. 175105

Quantitative Analysis No. 3312 By Engelhard Industries

Impurity ppm Impurity ppm

Rh 3 Ca 0.4

Pd 73 A1 <1

Ir <5 Ni <1

Ru ND Cr ND

Os ND Mn <1

Au <2 Sb ND

Ag 3 B ND

Pb ND Co ND

Sn ND As ND

Zn ND Bi ND

Fe 39 Cd ND

Cu 0.4 Mo ND

Si 3 Te ND

Mg <1 Na ND

Pt* 99.987+ 7o. < less than ND - Not Detected * This value, obtained by difference, represents only the

metallic percent, no gas analysis having been done on the sample.

Metallography : Photomicrographs were made of the platinum disc cut from

the bar sample. They showed grain size to be of the order of 0.01 inch..

No evidence appeared of inclusions of foreign matter or of any irregularities

in microstructure.

Density : The density was determined on the solid bar, when it was 10

inches long, by mass and dimensional measurements.

The diameter was measured at three angular orientations, approximately

120° apart, at each of six locations along the bar giving a total of 18

measurements. The value thus determined for the diameter of the bar at 21°C

was

0.78643 ± 0.00005 inches (3 ct limits).

Five measurements were made on the length of the bar by comparison with a

set of gage blocks using an electronic gage block comparator. The value

89
determined for the length of the bar at 21 J C was

10.01300 ± 0.00003 inches ( 3 cr limits).

The surface roughness was measured on a Talysurf machine against an absolute

datum with a 30 mg load and a stylus of radius less than 100 micro-inches.

The value recorded was

29 micro-inches AA at 0.03-inch cutoff.

Assuming that the surface can be characterized by a sinusoidal profile , which

is a fairly reasonable assumption, the amplitude of the sine wave is calculated as follows.

The arithmetic average deviation, automatically measured on the Talysurf

machine, is defined as

AA - J; | y | dx (5-1)

L Jo

where y is the deviation of a point in the surface from a reference datum

located such that the net area enclosed by it and the surface profile is

zero, x is the position along the surface and L is the total distance

traversed. In the case of a sine wave

2 fE 9 AA a ~ p h sin x dx 2 — h (5-2)

TT Jq tt

h = *2 AA (5-3)

Assuming that peak to peak values were measured in measuring the diameter

of the bar the correction to be applied in order to arrive at the effective

diameter is

D c Dmeasured ^ ^2

= 0.78643 - 0.00009

D b 0,78634 inch = 1.9973 cm

90
The bar was weighed on a balance (B-l) in the Mass Section of the

National Bureau of Standards and the true mass computed to be

1703.975 ± 0.010 gnu

The density was computed from the above measurements to be

21,384 ± 0.002 gm/cm3 at 21°C.

The density just quoted is an average value for the whole bar and there is

no guarantee that the density is uniform to that degree throughout the bar.

R273.15°K/R4.2°K; The ratio of the resistance at the ice-point

temperature to that at the boiling point of helium at atmospheric pressure

is a measure of the extent and condition of impurities in a material and

of the crystallographic state of the material. This ratio was determined

on each of the three samples provided by Engelhard. The 0.786-inch diameter

bar was 10 inches long and had been annealed as described in the section on

Fabrication and Thermal History above. The other two samples were also

annealed prior to these measurements. The 0.150-inch diameter rod was

annealed in air for one hour at 850°C and allowed to slowly furnace-cool.

The 20-mil diameter wire was annealed in air by passing a dc current

through it using its change in resistance as a measure of its mean temperature.

The ice-point resistances were measured both before and after the helium

point measurements. The current was supplied from a regulated dc power

supply and was measured using a shunt box and precision potentiometer. The

voltage drops in the specimens were measured on a high precision 6-dial

potentiometer. In each case the resistance was measured at three or four

different current levels and the value corresponding to zero current

obtained by plotting resistance as a function of current squared and

extrapolating to zero current. The measured values of the resistances are

91
presented in Table 5, each value representing the average of four measurements, two with the current flowing forward and two with the current reversed.

The values for the resistance ratios are also given, these values being good

to within better than 1% for the two smaller samples and probably good to

within 1% for the large bar.

Table 5. Resistance Values for Three Samples of Platinum at

Ice-Point and Helium-Point Temperatures, and the

Computed Resistance Ratios at Zero Current.

Temperature 273 . 15°K Temperature 4.2°K

Nominal Resistance Resistance Nominal Resis- ResisCurrent Before After Current tance tance

Sampl e Amps Amps uP Ratio

o* 58.96 0* 0.150 393

10 58.96 58.97 30 0.151

0.786 in. 20 58.99 58.99 60 0.150

diam. bar 30 59.01 59.01 80 0.150

40 59.04 59.06 95 0.150

0* 1545. 0* 3.612 428

1 1545. 1545. 10 3.618

0.150 in. 2 1546. 1545. 20 3.615

diam. rod 3 1546. 1545. 30 3.612

40 3.612

0* 90280. 0* 233.5 387

0.2 90320. 0.8 233.5

0.020 in. 0.5 90430. 90450. 1.5 233.5

diam. wire 0.8 90560. 90550. 3.0 233.6

1.0 90650. 90620.

* The values tabulated at zero current are extrapolated values <>

Ice-Point Resistivity : A set of knife edges of known separation was

fastened to the 0.786-inch bar during the first set of ice-point resistance

measurements. The knife edges acted as potential taps. Since the separation

of the knife edges and the cross sectional area of the bar were known it was

92
possible to compute the ice-point resistivity. The value obtained, corrected

to 0°C dimensions, was

po = 9.847 ±0*010 |jP cm.

E. M, F. vs Pt 27 : The following table gives the values of the electromotive force of the 20 -mil platinum wire versus the platinum standard Pt-27

corresponding to the temperature of the measuring junction when the reference

junctions are at 0°C as determined at the National Bureau of Standards.

Table 6'. E. M. F. vs Pt 27 as determined on 20-mil Platinum Wire*

National Bureau of Standards Test No. 3.1d/8404A.

Degrees C

Clnt. 1948)

Absolute

Microvolts

Degrees C

(Int. 1948)

Absolute

Microvolts

0 0 600 +6

100 +1 700 +8

200 +2 800 +10

300 +3 900 +13

400 +4 1000 +14

500 +5 1100 +15

The platinum wire was electrically annealed in air for one hour at about

1450°C before testing.

4fi

93
CHAPTER VI

TEST PROCEDURES

Preliminaries

The furnace was heated to 150°C and the system evacuated to 3 x 10 4 torr

Initial pump-down was through a needle valve to give a sufficiently slow

rate so as not to disturb the very light power insulation. When the pressure

had fallen below 100 microns the diffusion pump was turned on, pumping

initially through the needle valve, and , when the pressure was below 10" 2 torr,

through the 4 inch port. After pumping for 24 hours the pumps were turned

off and the system back-filled with high purity ( 99 . 99%) argon. The argon

was bled in slowly through a needle valve to avoid disturbing the powder.

The pressure was allowed to build up to almost one atmosphere before the

valves were shut off and the cycle of evacuating and backfilling repeated.

The final argon pressure after the second backfilling was about threequarters of an atmosphere.

The cooling-water flows to the system were adjusted to the desired level

as indicated on flow-meters. The water was pressure-regulated to maintain

constant flow rate. A control thermocouple on the specimen was wired to

shut all the heaters off if its temperature exceeded a pre-determined level.

This was a safety precaution in the event of an interruption in the coolingwacer flow.

The test procedures are described in detail below. To facilitate the

discussion let us refer to Figure 12. The region of the specimen below the

specimen heater, Q2 , will be referred to as the lower part of the specimen,

and the region above Q2 as the upper part of the specimen.

94
Direct Method

irMatched" Gradient Test ; The furnace temperature was raised

by means of heater Q7 . The power to the specimen heater, Q8 , was adjusted

to give a temperature gradient of 5°C/cm in the lower part of the specimen.

This heater was fed from a constant voltage dc power supply with regulation

of ±0.005% or better. The output voltage ranged from 10 to 15 V with a current of about 1 A. Power to Q3 was automatically controlled, using a

proportional 'controller, to maintain minimum temperature drop across the neck.

The PtlO%Rh legs of thermocouples 9 and 12 were used in conjunction with

the necked-down portion of the specimen as a differential thermocouple

activating the proportional controller. The temperature drop across the

neck never exceeded 0.1°C. The specimen was maintained at the required

mean temperature by thermostatting the power high-low to the lower

heater QiThe temperature distribution along the guard was forced to match that

along the specimen by controlling the power to heaters Q4 , Qs and Q6 .

Temperatures at corresponding locations on the specimen and the guard

generally agreed to within 1°C.

After allowing time for the system to come to equilibrium readings were

taken and recorded. The readings were made in the following sequence:

(a) The emfs of the guard thermocouples were read to 0.1p,V.

(b) The emfs of thermocouples 1 through 3 and 9 through 15 on the specimen

assembly were to read to 0.1p,V (thermocouples 1, 2, 14 and 15 are not

shown in Figure 12 but refer to the thermocouples on the molybdenum

extensions)

.

(c) The emfs of thermocouples 4 through 8 on the specimen were read to

O.Ol^V.

95
(d) The differential temperature T u - T l0 across the neck was read to

O.OlpV using the PtlO%Rh legs of thermocouples 11 and 10 and the

necked-down region of the specimen as a differential thermocouple.

(e) The current to the specimen heater, Q2 , and the voltage drop across

the inner taps were read to five significant figures.

The above sequence of readings was taken three times to check the equilibrium, each set of readings taking about three-quarters of an hour. The temperatures never drifted more than a few hundredths of a degree from one set

of readings to the next and the three sets of data were averaged. When

the drift between the first and second sets of readings was less than 0.01°C

the third set of readings was not taken. On completion of the last set of

readings the following additional readings were made t

(f) The voltage drop across the outer taps of the specimen heater was

measured to O.lmV.

(g) The potential drops in the current leads, Ei 2 and E34 (see Figure 21),

were measured to lpV.

The current through the specimen heater was then reversed and readings (e) ,

(f) and (g) repeated.

"Unmatched" Gradient Test : Upon completion of the "matched" gradient

test the guard heaters Q4 , Q5 and Q6 were adjusted to lower the temperatures

on the guard by 10°C while maintaining the temperature distribution parallel

to that on the specimen. With the guard at the lower temperature heat

losses from the specimen to the surrounding insulation were significantly

increased and the heat flow in the specimen reduced. As a result, the

temperature gradient in the specimen and its mean temperature were decreased.

96
The power to heater Qi was adjusted to restore the specimen to the initial

mean temperature. The system was then allowed to equilibrate and

the same series of readings were taken as for the "matched" gradient test.

"Isothermal" Test ; For the "isothermal" test heater Q2 was shut off

and heater Qi adjusted until the specimen was approximately isothermal.

No adjustments had to be made to Q3 as the controller automatically

adjusted to maintain Tn - T10 ** 0. The guard heaters were likewise adjusted

until the temperature distribution on the guard once again matched that on

the specimen. When the system was in equilibrium the readings listed above

for the gradient tests, with the exception of items (e) and (f) , were taken.

At 1100°C an additional "matched" gradient test was run after the

three regular tests were completed. The extra data obtained in this test

served as a check on drifts in thermocouple calibrations during the testing

period

.

Resistivity

After completion of the thermal conductivity measurements by the direct

method, with the specimen still isothermal (to within 1°C), a 10A dc

current was passed through the specimen. The current was supplied from a

dc power supply with current regulation of better than 0.017o . The

resistance of the lower part of the specimen was determined by simultaneous

measurement of the current and of the voltage drops between the platinum legs

of the thermocouples. Readings were taken in the following sequence;

(a) The emfs of thermocouples 4 through 8 were read to O.OlpV.

(b) The voltage drops between the platinum legs of the thermocouple

- 97
combinations 4-8, 4-6, and 6-8 were read to O.OlpV while the current

was read simultaneously to 0.1mA.

(c) All of readings (a) and (b) were taken in the reverse sequence.

(d) The current was reversed and readings (a), (b) and (c) repeated.

With current flowing through the specimen differential temperature control

at the neck could not be employed and Q3 was controlled absolutely, as

described in Chapter II, to maintain Tn as close to T10 as possible.

Indirect Method

Measurements by the indirect method were made immediately following

the resistivity measurements. Power to heaters Qx and Q3 was controlled to

maintain T10 and Inconstant with | Tn- T10| less than 0„2°C. Temperatures

along the guard in the region opposite the neck were maintained at the

same value as T10 . The current through the neck and the voltage drop across

it were measured with a dc current passing through the specimen. The

current was supplied by a 100A dc power supply with current regulation

of 0.05% or better. With 10A passing through the specimen the following

readings were taken

(a) The emfs of the guard couples in the neck region were read to O.lpV.

(b) The emfs of thermocouples 9, 12, 10 and 11 on the specimen were read

in that sequence to O.OlpV.

(c) The voltage drops between the platinum legs of thermocouples 10

and 11, and 9 and 12, were measured to five significant figures with

the current measured simultaneously to five significant figures.

(d) All of readings (b) and (c) were taken in reverse sequence.

(e) Steps (b), (c) and (d) were repeated and finally

- 98 -
(f) Readings (a) were repeated.

In this way each quantity, with the exception of the guard temperatures, was

measured four times with the averages all corresponding to the same mean time.

The temperatures never drifted more than a few hundredths of a degree during

a test (approximately three-quarters of an hour), and although the current

showed drifts of 0.05% the measured resistances remained constant to 0.0l7o

with current and voltage being read simultaneously.

To compensate for thermoelectric effects and IR drops in the thermocouples

the current was reversed, the above sequence of measurements repeated and

data from the two tests averaged. During tests at 300°C a quick check was

made on the magnitude of IR drops in thermocouples 9 through 12. The emfs

of these thermocouples were read with a current of 50A flowing forward. The

current was reversed and the emfs read immediately again. The emfs changed

by amounts ranging from 2.5 to 13 mV.

With current flowing through the specimen the Peltier effect gives rise

to heat generation at one of the platinum-molybdenum interfaces and heat

absorbtion at the other depending on the direction of the current flow. For

example, at 1000°K with 100A flowing through the specimen the Peltier heat

generated or absorbed at a platinum-molybdenum interface is approximately

4 watts. When the current is reversed the processes are reversed, and, at

the interface where heat was generated, it is now absorbed and vice versa.

Consequently when the current was reversed heaters Q and Q3 had to be adjusted

to compensate for the Peltier effect, and time had to be allowed for the system

to reach a new equilibrium.

The measurements made at 10 amps were repeated at 58, 82, and 100 amps

giving four sets of data at approximately even increments of is

.

99
All of the measurements described above were made at a number of temperatures. Tests were first run in air at 100°C S then in argon at 100, 3 00 , 500,

700, 600, 400, and 200 °C in that order. One of the guard heaters would short

out at about 750°C and consequently the upper limit on the first run was 700°C.

After completing that run in argon the system was evacuated and backfilled with

helium (99.99% pure) to a pressure of about three-quarters of an atmosphere.

The helium, having a much higher thermal conductivity than argon, changed the

effective thermal conductivity of the insulation surrounding the specimen.

Tests were run in helium at 200 and 400°C to experimentally evaluate heat

losses from the specimen to the insulation. The system was then opened up

and the trouble with the guard heater rectified. The furnace was filled with

fresh powder insulation, the powder being packed lightly around the neck to

ensure that the necked-down region was uniformly filled with insulation. The

system was refilled with argon as described at the beginning of this section

and tests were run at J>00 , 500, 700, 900, and 1100*0. Heater Q3 burned out

during the indirect test at 1100®C and that test was incomplete. The thermocouples started drifting at 1100*0 and losing their calibration due to contamination. Consequently it was decided to terminate the tests at that

point.

100 -
CHAPTER VII

RESULTS

Electrical Resistivity

The ice-point resistivity was determined on the platinum bar before

the neck was machined in it (see Chapter V) . The value obtained was

= 9.849 ± 0.010 (j,Q cm (7-1)

corresponding to room temperature (25 °C) dimensions.

In the indirect method of measuring thermal conductivity the resistance

of the necked-down region of the specimen at each temperature was determined

at a number of current levels. The resistance, ? at temperature, T, was

evaluated by extrapolation to zero current as described in Chapter IV. In

the first series of tests, or the first run as it will be referred to, measurements were made at 40 and 100 °C in air, at 100, 300, 500, 700, 600, 400, and

200 °C in argon and at 200 and 400 °C in helium in that order. The apparatus

was then opened up for repair of the guard heater as described in Chapter VI.

In the second series of tests, or the second run as it will be referred to,

measurements were made at 300, 700, 900, and 1100 °C in argon.

The values for the resistance of the neck obtained during the second

run were about 1 ^% higher than the values obtained at corresponding temperatures in the first run. This increase in resistance was observed both for

measurements at the inner potential taps (10-11) and for measurements at the

outer potential taps (9-12) indicating that the increase in values were due

to a change in the geometric factor of the neck. What happened is not known

but, apparently, the neck suffered some slight geometric change when repairs

were being made to the guard heater. In determining the electrical resistivity,

data from the first run and that from the second run had to be treated separately

in view of the change in the geometric factor.

101
A least-squares quadratic equation fits the data from the first run

with a standard deviation of 0.03%. The equation, extrapolated to 0 °C and

normalized to the ice-point resistivity, (7-1), gives

P = 9.849(1 + 3.953x 10-3T - 0. 5744x 10“8 T

2

) (7-2)

as the electrical resistivity of the neck from 0 to 700 °C where T is in °C.

Departures of the data points from this equation are plotted in Figure 22

where the overall scatter is seen to be ±0.05%. All but two of the points

fall within 0.025% of the curve and there is no significant difference between

the values obtained on heating and cooling.

A second quadratic equation was fitted to the data points at 300, 700,

and 900 °C obtained in the second run. The equation, normalized to the value

given by (7-2) at 700 °C is

P = 9.797(1 + 4.001 x 10“2 T - 0.6043 x 10“S T2) (7-3)

While the geometric factor of the neck changed between the first and second

runs there is no reason to believe the resistivity would have changed. Any

strains introduced when the geometric factor changed would have annealed out

at 700 °C„ Thus normalizing the data from the second run to that from the

first run at 700 °C is felt to be justified. The departures of the normalized

data points at 300, 700, 900, and 1100 °C from the values given by equation (7-2)

are indicated by triangles in Figure 22. The 1100 °C data point was not used

in deriving (7-3) . The thermocouples started to drift at 1100 °C due to contamination so that greater uncertainty had to be assigned to data obtained at

that temperature. The 1100 °C data point agrees with the extrapolated value

given by equation (7-3) to within 0.02%. The deviations from the quadratic

equation (7-2) at the higher temperatures as shown in Fig, 22 are in line

with what one would expect. Recent measurements at NBS f80] in conjunction

with work toward extending platinum resistance thermometry to the gold point

showed that the measured resistance of a certain high purity platinum

102
+

0.20

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103

obtained

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of

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specimen.
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104

the

lower

part

of

the

platinum

specimen

from

equation

(7-2).
resistance thermometer at the gold point (1063 °C) was about 0.07% below

the value obtained by extrapolating the Cailendar equation [ 7 ] for that

thermometer to the gold point. Laubitz £81] in measuring the electrical

resistivity of high purity (99.999% pure) platinum found that between 800

and 1200 °C his experimental results fell consistently below the values

extrapolated from an equation similar to (7-2) by an average amount of 0.10%.

Values for the electrical resistivity, P, and its temperature coefficient (y “

1 dp

dT

used in computing the thermal conductivity by the indirect

method were calculated from (7-2) for the first run and from (7-3) for the

second run.

Electrical resistivity data were also obtained from measured resistances

and dimensions in the lower part of the specimen using the defining equation

p = RA/L, where R is the resistance of a length L of specimen of cross-sectional area A. Measurements of the cross-sectional area are described in

Chapter V. The length, L, was the spacing between the potential taps. The

Pt legs of the thermocouples were used as potential taps. The thermocouple

wires were pressed into slits machined in the specimen. The slits were 8milF "Wide

and the spacing between slits was measured optically prior to installing the

thermocouples. Resistances were determined by simultaneous measurements of

the current and the voltage drops between locations 4-8, 4-6, and 6-8. (see

Figure 12) . The average value of the three resistances was used giving the

first one twice as much weight as each of the other two. The mean temperature

was computed from the five temp era ttmres T^ through Tg by Simpson's Rule integration. Values obtained with the current flowing forward and reverse were

averaged.

A least-squares parabola fits the resistivity data obtained in the

lower part of the specimen with a standard deviation of 0.14%, and the value

given by extrapolation of the equation to 0 °C is 9.88 as compared with 9.85

105
for the measured ice-point resistivity. The values obtained on cooling are

consistently lower than those obtained on heating in the first run, and the

overall scatter is much greater than that obtained from measurements in the

necked-down region. The resistance of the neck was 15 times larger than thac

of the lower part of the apecimen so that its value could be more accurately

determined. Moreover the resistance values determined for the neck represent

an extrapolation to zero current whereas the values determined in the lower

part of the specimen are based solely on the data for 10A. In view of the

above factors the resistivity values determined from the measured resistances

in the neck and from the ice point resistivity are considered to be more

accurate. A discussion of the accuracy of the resistivity data will ensue in

the next Chapter. Departures from the values given by (7-2) of the data

obtained in the lower part of the specimen are plotted in Figure 23.

Thermal Conductivity

Direct Method : The experimental values obtained for the thermal

conductivity of platinum by the direct method are shown in Table 7. Thermal

conductivity values are given both corrected for and uncorrected for thermal

expansion (the values discussed below refer to values uncorrected for thermal

expansion; corrections can easily be made with the use of equation (4-41)).

Values are also given for the effective thermal conductivity of the insulation, these values havir.g been determined by simultaneous solution of the

'’matched" and "unmatched" gradient tests as described in Chapter IV. A

typical set of data is shown in Table 8. In all but the tests at 100°C the

temperature gradient was nominally 5°C/cm. In the tests at 100°C the

gradient was nominally 2°C/cm. The temperature drop across the necked-down

region was always less than 0.1°C and q while temperature dependent was

always less than 7mW. The heat flow in the current leads, q and q were

a b

106 -
TABLE 7 : Experimental values for the thermal conductivity of platinum

as determined by the direct method: values both corrected for (1^) and uncorrected for (k) thermal expansion are given along with values for the thermal

conductivity (^) of the powder insulation and the temperature gradient (grad.)

in the specimen.

T grad._l/ h K k , Conditions

°C °C/cm mW/cm °C mW/cm^C mW/ cm 'Ll

99.63 1.85 0.715 0.715 Argon, first run 99.82 1.91 0.715 0.716 Argon, first run

201.72 4.84 0.504 0.721 0.722 Argon, first run

301.01 5.08 0.729 0.730 Argon, first run 400.19 4.96 0.838 0.740 0.743 Argon, first run

501.47 4.94 0.753 0.756 Argon, first run

601.32 5.27 1.256 0.769 0.773 Argon, first run

701.19 4.91 0.784 0.790 Argon, first run

300.03 5.04 0.667 0.729 0.731 Argon, second run

701.32 4.84 1.553 0.786 0.791 Argon, second run

900.04 5.10 2.070 0.822 0.829 Argon, second run 1100.80 5.47 0.857 0.867 Argon, second run

199.64 5.01 1.655 0.721 0.722 Helium, first run 400.61 5.09 2.156 0.740 0.742 Helium, first run

i/ grad . - (-£-) - (-£-)

107
TABLE8: Typical set of date from measurements of the thermal conductivity by the direct method. The data are from the second test run in argon

at 700 °C.

Test

"Matched"

gradient "Isothermal"

"Unmatched"

gradient'-

Temperature distribution t4 682.2 700.1 683.5

along the specimen - deg C T

5 691.8 700.0 692.3

T

6 701.3 699.7 701.1

T

7 710.9 699.6 710.2

T

8

720.4 699.4 719.3

Temperature gradient at

the center of the specimen dT 4.75 -0.09 4.42

- deg G/ cm dz

Power generated in the

specimen heater - W

P 12.137 0.0 12.133

Heat flow across the

necked-down region - W

qu -0.005 -0.006 -0.005

Heat flow along the cur- ^a 0.007 0.001 0.009

rent leads - W 3b 0.0002 0.001 0.008

‘Heat flow to the insulation - W

3i + 0.013 + 0.038 + 0.739

Heat flow along thermocouple

wires and insulators - W 3C - 0.045 -0.001 0.042

108
o

-r-l CD T3C

COE

D

<L> si

c

o

bO

co

C'i

0)U

3

C>0

l-(

109

powder

insulation

surrounding

the

specimen.
always of the same order of magnitude being essentially independent of

temperature. The heat flow to the insulation, q^, being directly proportional to the thermal conductivity of the insulation, had approximately

the same temperature dependence as it. The values obtained for the thermal

conductivity of the powder insulation in air, in argon and in helium are

plotted in Figure 24. The heat flow along the thermocouple wires and the

ceramic tubing insulating them should have decreased with increasing

temperature as a result of the decrease in the conductance of the ceramic

tubing

A least squares cubic equation was found to fit the thermal conductivity

data from the direct method with a standard deviation of 0.08%. This was

significantly less than the standard deviation from a parabola. The equation

is

k = 0.713 + 0 . 130x10*“* T + 0 . 175xl0~6

T® - 0 . 505xl0"10 T

3 (7-4)

where T is temperature in °C. The corresponding equation in °K is

k= 0.723 - 0.936x10“^ + 0. 216xlO"s

'I3 - 0 . 504x1 0"10 T

3 (7-5)

Departures of the data points from (7-4) are plotted in Figure 25. With the

exception of the point at 1100°C all the data points, including the two

obtained in helium fall within ±0.10% of the curve. There are no significant

differences between the values obtained on heating and cooling in the first

run and none between the values from the first run and those from the second

run. At 1100°C the thermocouples started drifting giving rise to significant

uncertainties in temperature measurement. After completing the "matched"

gradient, "unmatched" gradient and "isothermal" tests at 1100°C a second

"matched" gradient test was run. Two values for the thermal conductivity

were obtained by simultaneous solution of each of the two "matched" gradient

tests with the "isothermal" test. The first value (0.833) was below the

value predicted by (7-4) and the second value (0.896) was above it.

110
T" i r~~i 1

1 1 r~

CD

0

* *

1

•

• • - • -

*

m # -

# •

# • —

• •

__l ! _J 1 l l l l

o

o

00

o

o

CO

yj

oc3

<t

QL

Ui

CL

s

LlI hO

o

o

o

CM

O to o 40 O £> o 40 O

CM

d d d o

d

O

d d d

CM

d

+ + + + i 1

i

1 %

1 N0IXVIA3G

in

Figure

25

.

Percentage

deviations

from

equation

(

7

- 4)

of

thermal

conductivity

data
Assuming that the temperature drifts were linear with time, interpolation

to the time of the "isothermal" test gives a value of 0.867 for the thermal

conductivity of the specimen at 1100°C. This value is 0.46% below the value

given by extrapolation of (7-4) to 1100°C. The 1100°C point, due to the

larger uncertainty associated with its value, was not used in deriving (7-4).

Indirect Method ? The experimental values obtained for the thermal

conductivity of platinum by the indirect method are given in Table 9 along

with some other pertinent data. The values for p and a were computed from

equations (7-2) in the first run and from (7-3) in the second run. The values

given for the maximum temperature rise in the neck are the actual values.

The values that would exist in the absence of heat losses are given by (4-22).

The values for S show that thermal conductivity values computed on the assumption that the resistance of the neck is related to the maximum temperature

3 ... in it by R 4- R

q

(1 + "3Gy o0m) would be too high by amounts ranging from 0.36%

at 100°C to 2.85% at 900°C. The heat loss correction is seen to vary from

0.24% at 100°C to 1.15% at 900°C, and the Lorenz number varies from 2.63

to 2.78.

A least-squares cubic equation was found to fit the thermal conductivity

data obtained in argon from the indirect method with a standard deviation of

0.31 percent, the equation being

k= 0.717 - 0.205xl0"B T + 0.193x10“®^ - 0.832xl0

_1° T

3 (7-6)

where T is temperature in °C. The 1100°C data point was not used in

deriving (7-6) as the indirect test at that temperature was incomplete. The

value shown in Table 9 for ft at 1100°C is based only on measurements at 10A

forward and reverse and 58A forward, and consequently the uncertainty

associated with this value is greater. Deviations of the data points from

112
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114 -

Figure

26.

Percentage

deviations

from

equation

(7-6)

of

thermal

conductivity

data
equation (7-6) are plotted in Figure 26 where the scatter in the data is seen to be within ±0.4 percent. The scatter is random and there are no significant differences between values obtained on heating and cooling or between the first run and the second run. The value obtained in helium at

200°C agrees within 0.01% of the value given by (7-6) but the helium point

at 400°C shows a deviation of +0.94 percent. The measurements in helium were

made during the first run and there is a possibility that the powder insulation

did not uniformly and completely fill the necked-down region. If the powder

did not completely fill the space the boundary conditions assumed in deriving

the heat loss corrections (see Chapter IV, p. 65) would not have been met.

Any errors in the heat loss correction would be amplified in helium particularly at the higher temperature due to the high thermal conductivity of this

gas. In the second run the powder was carefully packed around the neck so

that the necked-down region was completely and uniformly filled with insulation .

115
CHAPTER VIII

DISCUSSION OF RESULTS

(8-1)

Accuracy

Electrical Resistivity

The electrical resistivity values were computed from

\

p = p ° (Mo

where p^ is the independently-measured ice-point resistivity, Rr

is the

resistance of the necked-down region at zero current, and (Rr

) is the

value of R

r extrapolated to the ice point. The fractional total uncertainty

in p is given by

Ap

P

A p

°j

P o +

|ARr A(Rr ) 0

(

P-r ) o (8-2)

where j Ap

o

/p 0 | is the fractional uncertainty in the ice-point resistivity.

The quantities ARr and A(Rr

) 0 appear inside the same absolute value sign since

they are correlated quantities rather than being independent.

The uncertainty in p Q was not more than 0.1%. The uncertainties in the

measured values of R did not exceed 0 . 057o . However, an additional uncertainty must be assigned to the values of R^ due to the uncertainty in the

temperature to which these values correspond. Quite conservatively, temperatures were known to within 0.5°C, corresponding to 0.14% of R^ at 100°C and

0.04% at 900°C. The uncertainty in (Rr

) Q> which is based on an extrapolation,

is estimated to be less than 0.25%. Values of AR^. corresponding to the lower

temperatures are correlated with A(R

r

) o so that these quantities tend to

cancel at lower temperatures but not necessarily at higher temperatures. The

overall uncertainty in electrical resistivity is estimated asj_AP_ < q. 10% at

0°C and Ap

< 0.44% at 900°C

116
Thermal Conductivity

Direct Method :

relation

k =

r

The thermal conductivity values were computed from the

--(Q ~ Q')

A[ (dT/dz) - (dT/dz)']

(4-40)

as discussed in Chapter IV. The fractional total uncertainty in is given

by

£]&| . I A(Q - O')

I (Q - Q')

M

A

(dT/dz) - (dT/dz)'

}

(dT/dz) - (dT/dz)']

(8-3)

where j &A/A | is the fractional uncertainty in area, etc. We now consider

each of the terms in (8-3) individually.

Heat Flow t From (4-42) we write

a(q-q')

= a (p-p * ) -a

(

qa -ga

1 ) -a

(

q b

-q

b

1 ) -a

(

qn

-qn

1 ) -a

(

qj-qj 1 ) -a

(

q c -q

c

'

)

(Q-Q') (P-P')-(q

a-q

a

, )-(q

b-q

b

')-(qn

-q

n

, )-(qi-qi"')- (qc-qc

7

)

For purposes of error analysis, we can neglect the correction terms in the

denominator of (8-4); furthermore, for our tests P'=0 ("isothermal" test)

and we can write

A(0-0M |ap! MVOI + + + +

(Q-Q’) “I P| + — 5

(8-5)

The following error estimates are based on the value of P necessary to

maintain a temperature gradient of 5°C/cm in the specimen.

The potentiometer, voltbox, and shunt box were each calibrated to 0.01

percent or better. The emf of the standard cell was known to 0.01 percent

or better. The correction terms in (4-43) for the voltage drop in the potential leads and for the current through the voltbox were small (a few tenths

117
of a percent) and uncertainties in these corrections could not have introduced more than 0.01 percent additional error in P. Thus the percentage

uncertainity in the measured power input was | AP/P | < 0.05%.

The corrections (q -q ') and (q -q ’) for heat flow along the current

a a b b

leads were each about 0.05 percent or less of P. The derivation of (4-44),

the equation used to compute q and q was based on the assumption that

3 D there was no heat flow from the leads into the powder insulation. If the

leads were at the same temperature as the specimen at the point where they

enter the specimen, the assumption of no heat loss from the leads would

be a good assumption. Preliminary tests with the same style of heater as

the specimen heater had shown that the mean temperature of the heater

winding would be significantly higher than that of the adjacent specimen

material. In order to minimize the corresponding temperature rise in the

current leads, swaged elements of relatively low electrical resistance were

installed (see Chapter II, p. 39) for the purpose of tempering, or thermally grounding, the leads.

In Appendix I it is shown that the tempering attained by these elements

was sufficiently good to limit heat losses out the current leads to 0.05%

of P. The corrections q a and q^ probably compensate for much of this,

but to cover the case where they don't an uncertainty of 0.05% due to heat

flow out each current lead is adopted. This uncertainty, it should be noted,

would always be a heat loss and not a gain and hence must be added to the

root-mean-square of the other uncertainties which have approximately equal

probability of being either positive or negative.

The correction (q -q ' ) for heat flow across the necked-down region of n n

the specimen was always less than 0.05% of P. Errors in the temperature

drop across the neck due to possible inhomogenieties in the thermocouple

118
leads or to stray thermal emfs would, for the most part, cancel since they

were common to q and q

' . At 300°C a series of tests were run in which the

n n temperature drop across the neck was held in turn at about -5°C, 0°C, and

+5 C, a range 20 times larger than that which occurred during normal

measurements. The corresponding values for (qn

-q ') were about +1%, 07,

and -17, respectively, of P. The three thermal conductivity values obtained

using (4-45) to effect the correction for heat flow across the neck, fell

within a range of less than ±0.027,. For normal tests, in which the temperature difference across the necked-down region was maintained quite small,

it is felt that } A(qn-q

n

')/P | < 0.027,.

The correction (q c -q

1 ) for shunting of heat by the thermocouples and

ceramic tubing was rather large, falling from about 0.87, at 100°C to 0.37o

at 1100°C, this falloff being due to the rapidly decreasing thermal conductivity of the ceramic tubing. Although the thermal conductivity of the

thermocouple wires were known fairly accurately (57,) the thermal conductivity

of the ceramic tubing was known only approximately (157,) . Furthermore, the

cross-sectional areas of the wires and tubing were not accurately known (57,).

Thus the total conductance of the wires and the ceramic tubing was only

known to about 257,. This implies j A(q r -q ')/P j < 0.207 at 100°C and

^ c <0.087 at 1100°C.

The correction (q^-q,') for heat exchange with the powder insulation

was potentially a large source of error and considerable effort was expended

to, first, keep this correction small and, second, evaluate it accurately.

Evaluation of this correction as seen from (4-50) requires a knowledge of

the integral, D(z, , z

2

) , and of the thermal conductivity, %, , of the

insulation.

119
Numerous factors could adversely effect the determination of D(z1} z

a ).

The use of logarithmic functions to define the radial temperature distribution

across the ends of the hollow cylinder of insulation is an approximation. However, an argument similar to that developed in Appendix B shows that the potential distribution near the specimen is not affected by the boundary conditions

at the remote ends of the extensions. The details of the actual temperature

distributions in the heater regions where smoothing cubics are used could conceivably influence D(z! ,za ) (see Chapter IV, pp. 79-82). Any such effects should

be small, involving as they do only second and higher order derivatives. Furthermore, except for the regions of Qg and such effects should be about the

same in the gradient and in the isothermal tests. In the mathematical analysis

it is assumed that the temperatures on the inner surface of the guard are the

same as the temperatures measured on the outer surface. The molybdenum guard

has high thermal conductivity so that any radial temperature gradients in the

guard would be small and the associated errors would tend to cancel on simultaneous solution. Angular variations in the temperature distribution on the

guard could arise if the specimen and guard were not concentric or if the insulation between the specimen and the guard was not packed uniformly. Great

care was taken to avoid both of these conditions. Any angular variations would

be approximately the same for two tests at the same mean temperature and so

the associated errors would in large part cancel under simultaneous solution.

Such would not be the case for errors arising from uncertainties in the

longitudinal positions of the thermocouples since the temperature distribution

along the guard cylinder in the gradient test differs from that in the

isothermal test. The necessary steps were taken to ensure that the longitudinal position of the guard was accurately known relative to that of the

specimen, and the location of thermocouple slits both on the guard and

120 -
on the specimen assembly were measured accurately prior to installation of

the thermocouples.

As discussed in Chapter IV, the effective thermal conductivity of the

powder insulation was determined in place by making measurements with the

guard unmatched. The values thus obtained are shown in Fig. 24 . The

precision of the data is quite good, especially considering that they were

obtained essentially as a by-product. The data labeled "air" were obtained

[82] on the NBS guarded hot plate apparatus on a sample of alumina powder

from the same lot and packed to about the same density.

If there was a significant heat exchange between the specimen and the

insulation that was not being adequately corrected for, one would expect

a systematic difference between the values obtained for the thermal

conductivity of the specimen in helium and those obtained in argon, due to

the large difference between the thermal conductivity of the powder in the

different atmospheres. In fact, however, the values measured in helium fell

within the scatter band of those measured in argon indicating that any

uncorrected heat exchanges were certainly less than 0.2%, the width of the

scatter band.

For all sets of tests | q^-q^, '

j < 0.1% of P. It is felt that D(zx , z

2 )

and were each known to better than 10%, and hence | A(q^-q^') | / P < 0 . 02%.

However, in view of all the factors which conceivably could influence this

correction, an uncertainty of | ACq^-q^') | / P < 0.10% is assigned.

The ratio of heat absorbed (or released) to that conducted in the

specimen is given approximately by

wcAL(dT/dt) = L (dT/dt) (8-6)

kA(dT/dz) K (dT/dz)

121
where w is density, c is specific heat, A is area, L is total length of

specimen below necked-down region, dT/dt is time rate of temperature change,

k is thermal conductivity, dT/dz is temperature gradient, and x=k/wC is

thermal diffusivity. Temperatures in the system did not drift at a rate

greater than 0.03°C/hr (i.e., 10 °C/sec); the length, L, was about 14cm;

the temperature gradient was 5°C/cm; and the thermal diffusivity of platinum

in the temperature range 0-1100°C is always greater than 0.2cm3 /sec r 8

3

~1 -

Hence the ratio of heat absorbed (or released) to that conducted was less

than ±0.02%.

The above uncertainties in heat flow sum to an estimated maximum

uncertainity in ( Q- Q

' ) of 0.49 percent at 100°C and 0.37 percent at 1100°C,

with an estimated probable uncertainty (i.e., root-mean- square) of 0.33

percent at 100°C and 0.24 percent at 1100°C.

Area : The cross sectional area of the specimen at room temperature is

believed to have been known to within ±0.02% (see Chapter V, pp. 89-90)

Temperature Gradient : The separations between the slits in which the

thermocouples were located were measured with an uncertainty much less than

the half-width of the slits so that the effective position of a thermocouple

in a slit was the controll ing fector in the uncertainty of thermocouple

separations. In determining the temperature gradient at the center of the

specimen a least-squares parabola was fitted to the five temperatures.

However, the temperature distribution was almost linear, and for the purpose

ci error analysis, it can be assumed that it was. For five thermocouples

evenly spaced at points a distance f apart, each with an uncertainty j [

in position, the maximum error in the slope of a straight line through these

points due to the uncertainties in their positions is — L _ 3

122
The thermocouples were spaced 2cm apart and | td | < 0.01 cm so that the

maximum possible uncertainty in the slope is 0.33 percent. A value of 0.01cm

is considered to be an extremely conservative estimate of | hi \ since it

implies that the effective position of a thermocouple coincides with the

edge of the groove in which it is located. A more probable uncertainty in

the gradient due to uncertainties in effective thermocouple positions is

0.2 percent.

As mentioned in Chapter IV, the simultaneous solution of a gradient and

an isothermal test largely eliminates the effects of variations between the

calibrations of individual thermocouples. All that is required is that the

slope of the temperature versus emf curve be the same for the different

thermocouples. Evidence indicating the fulfillment of this requirement is

provided by the fact that during an "isothermal" test the thermocouple

readings all agreed quite closely. For example, at 900°C the temperatures

indicated by the five thermocouples in the lower part of the specimen were

within 0.1°C of each other or within 0.01% of the temperature. These

differences of 0.1°C or less should have been compensated for to better than

0.01°C by the simultaneous solution of the two tests. It is estimated that

the conversion of thermocouple emfs to temperatures introduced uncertainties

of less than 0.2% of the AO^C temperature drop in the specimen. The errors

in reading the emf differences did not introduce an uncertainty of more than

0.05% (for a 40°C drop). As discussed in Appendix G, errors due to heat

conduction along thermocouple leads should have been negligible and were

certainly less than 0.05%.

The estimated maximum uncertainty in the temperature gradient is 0.59%

and the estimated probable uncertainty is 0.29%.

123 -
In addition to the above uncertainties, all of which are associated

with (4-40) , there is the uncertainty in the temperatures to which the

thermal conductivity values correspond. For a 0.5°C uncertainty in temperature, the associated uncertainty in thermal conductivity is less than 0.001%

at 100°C and less than 0.02% at 900°C.

Summarizing all of the above uncertainty, the estimated maximum

uncertainty in the thermal conductivity values from the direct method is

| A^ r /^

r | < 1.10% at 100°C and < 1.00% at 900 c C. The estimated probable

uncertainties (i.e., r.m.s. value) are 0.44% at 100°C and 0.38% at 900°C.

Indirect Method : In the indirect method, the thermal conductivity

values were computed from

1

Q' r Rr dV*2

kr 1+C 12p dR* (4-32)

as discussed in Chapter IV. The fractional total uncertainty in k is

given by

Ak

r AC

+

Ao, r + A(Rr /Pr ) +

A (dV*2

/ dR*)

k

r 1+C "r Rr/pr (dV*2 /dR*)

(8-7)

The uncertainty in the sensitivity, <y

_^ , of the specimen as a resistance

thermometer is essentially the uncertainty in the sensitivity of the thermocouples used plus a small uncertainty in the resistance measurements and the

thermocouple emf measurements. That this is so is seen by considering that

we measure R

r as a function, say R = f(E), of the thermocouple emf, E. Then

a.

1 dR

r

1 df (E) dE dE

R^ dT~

=

f (E) ~dE dT

= dT

(8-8)

124 -
and

(8-9)

Ao'r A (dE/dT) +

A (g(E)

)

Oi r dE/dT 8(E)

The uncertainty in dE/dT is estimated to be less than 0.20% while the

uncertainty in the resistance and voltage measurements is less than 0.05%,

so that | Ao, r

/oiT \ < 0.25%.

The quantity R / pr

is uncertain only by the scatter in the individual

R

r points around a smoothed curve, since pr was derived by smoothing the R

values. Therefore j A (R

r /pr ) / (R

r /pr ) i < 0.05%.

The voltage drop across the neck and the resistance of the neck were

each measured with an uncertainty of about 0.02%. The change in the

resistance of the neck for the different current levels was small compared

to the resistance itself, so that any error in measuring resistance would

be greatly magnified in computing the rate of change of resistance as a

function of the voltage drop. For all of the data taken, the departures of

the R* values from the least-squares straight line fitted to the R* and V*2

values were less than 0.01% and for a majority of the tests they were less

than 0.005%. However, since (R-R ) « R, the small scatter in R is highly

magnified in calculating dV^/dR*. What departures did exist tended to be

systematic rather than random and tended to indicate that the plot of R* vs

V*2 was very slightly concave downward rather than linear, as assumed. Such

an effect could possibly be due to neglecting higher order terms in the

temperature dependence of the thermal conductivity and the electrical

resistivity. As a result of these departures there was an uncertainty in

dV*2 /dR* that is estimated to be | A (dV*2 /dR*) / (dV*c /dR*) | < 1.50%.

In evaluating the correction, C, for heat loss from the neck it was

assumed that the boundaries z = ±1 and r = b (see Figure 18) were isothermal

whereas in fact they were not. Moreover, while the correction accounts for

125
the cooling of the neck by heat losses through the powder insulation, it does

not account for the fact that some of the heat lost from the neck goes back

to the specimen through the surfaces z ±& thereby heating the specimen at

| z j > H. The replacement of (4-16) by (4-17) does not introduce an error of

more than 10% in C. The thermal conductivity of the powder insulation

surrounding the neck was assumed to be the same as that determined from the

tests by the direct method. However, the packing of the powder insulation

around the necked-down region, especially during the first run, may have led

to a different density, and therefore a somewhat different thermal conductivity, than that in the region between the specimen and the guard. All in

all, the uncertainty in C may have been as large as 50% of C. For the tests

in argon this corresponds to jAC/(l+C)| < 0.12% at 100®C and < 0.55% at 900°C.

For the tests in helium |AC/(1+C)| < 0.53% at 200°C and < 0.66% at 900 dC

The above-mentioned uncertainties, associated with (4-32), lead to an

estimated maximum uncertainty in k^ due to systematic errors of |Ak

r /kr | <

1.92% at 1Q0 6 C and < 2.35% at 900°C, in argon, and estimated probable

uncertainties of < 1.78% at 100°C and < 2.11% at 900°C.

In the derivation of the mathematical expressions used to compute

thermal conductivity in the indirect method, there were a number of explicit

or implicit assumptions made which could lead to erroneous results if these

assumptions were not valid.

It was shown in Chapter III (pp. 57-58) and in Appendix C that the

Thomson effect cancels out to first order provided the temperatures at the

two potential taps (used to measure V) are approximately the same. The

requirement that the potential taps be at the same temperature is also

necessary for the Fermi energy and Seebeck emf terms in (3-50) to drop out.

126
In all of our measurements, the temperatures at the inner potential taps

(10-11) agreed within 0.2°C or better, as compared to values of 0 m (for 100A

current) ranging from 35°C at 100° C to 100°C at the highest temperatures.

The integral term involving the Seebeck emf in (3-50) reduces to

3. b s (Ta - T^S , provided (Ts - Tx ) is much less than the absolute temperature.

sb s For (Ta - Tx ) < 0.2°C, (T2 - ^ ) S is less than 2|j,V at 100°C and less than

5 |_lV at 1100°C as compared with an IR drop in the necked-down region of

5 x lO^V at 1'00°C and 2 x lO^V at 1100°C for a current of 10A, the worst

case. Thus, neglecting the Seebeck term in (3-50) involved an error of less

than 0.05%.

We can make an order of magnitude estimate of the term in (3-50)

involving the Fermi energy by considering a metal with free electrons obeying

Fermi-Dirac statistics, for which [84, p. 16]

C = Co - r^k2 ^

12Co

( 8

- 10 )

where is the value of the Fermi energy (chemical potential) at absolute

zero and k is the Boltzmann constant. Thus

TT^k2

12eC c Of T|>

tt3 k

2 T

6 eC^o

(Ti Ta> (8-11)

for small values of (Tx - T2

) . For platinum £

/k is probably greater than

10 , 000°K [84, p. 277] so that i(£ 3

- < 2p,V at 100°C and < 4jj,V at

1100°C. Thus this term is comparable with the Seebeck terms and involves an

error of less than 0.05%.

It is explicitly assumed in Chapter III that we are dealing with a

homogeneous and isotropic medium; this should be a valid assumption for pure

127
platinum.

As stated on p, 51, we are considering the electron current to be the

only mass current. In principle it would be possible for platinum, or any

impurities which might be present, to migrate under the influence of the

electric potential gradient (electromigration) or under the influence of the

temperature gradient (Soret effect). If such mass motion existed there would

be an associated entropy flow and equation (3-4) and all following equations

in Chapter III would have to be modified to include a term involving the

mass current density and its associated entropy transport. A rigorous

analysis including mass migration would be quite complex and was felt to be

beyond the scope of the present investigation. We can, however, give a

qualitative discussion of the probable effect of such mass motion if it were

to occur.

In electromigration, the energy flow is proportional to the electric

field [71]. Similarly, the energy flow due to the Thomson effect is essentially proportional to the electric field. Thus if one were to introduce an

electromigration term into (3-4) and rigorously go through the analysis, the

equivalent expression to (3-31) would include an electromigration term of a

form analogous to (i.e., proportional to the electric field) the term in

(3-31) involving the Thomson coefficient. Provided this term were small

compared with the first two terms in (3-35) it would, to first order, cancel

in a similar manner to the way the Thomson term cancelled, provided the

temperature at the two potential taps were the same and the medium was

homogeneous and isotropic. Physically what happens is that the energy transport due to electromigration adds to the energy transport by conduction on

one side of the surface of maximum temperature (Sm

in Figure 17) and

subtracts on the other side, with no net effect on the maximum temperature

128
rise in the conductor or on the voltage drop between the potential taps, at

least to first order. To reemphasize, the temperatures at the taps must be

the same for this cancellation to occur and also the medium must be

homogeneous and isotropic.

In the Soret effect, the energy transport is proportional to the

temperature gradient. Thus the ratio of energy transport due to the Soret

effect to that due to heat conduction by other mechanisms is independent of

the temperature gradient. The Soret effect is a diffusion process, similar

to heat conduction. For a homogeneous medium, therefore, the Soret effect

simply behaves as an additional mechanism for "heat conduction" and thus is

a legitimate augmentation (or depletion, depending on the sign of the heat

of transport) of the thermal conductivity and should be included in the total

thermal conductivity value.

For the conditions of our experiments, there would be no errors involved

due to electromigration or Soret effect provided the medium remained

homogeneous and isotropic. Both of these effects can change the distribution

of impurities and vacancies in a solid. In our experiments the current was

in one direction through the specimen about one-half of the time and in the

other direction about one-half of the time. Thus, it is doubtful if there

was significant inhomogenity introduced due to electromigration effects, even

if they were occurring in our temperature range. The Soret effect, if large

enough, could cause a redistribution of impurities in the specimen.

For platinum of the purity we used, the thermal conductivity at high

temperatures would not be expected to be significantly affected by changes

in the impurity and vacancy distribution. All of the data at low temperatures, where impurity concentration could affect the thermal conductivity,

were taken before the specimen was heated to temperatures where significant

129
mass migration was likely to occur. In view of the above discussion it is

felt that neither electromigration nor Soret effect had any adverse effect

on the results.

Comparison with Other Investigations

Electrical Resistivity

Smoothed values for the electrical resistivity of platinum based on

equations (7-2) and (7-3) are compared with the values of Roeser [85] and

Laubitz [81 3 in Table 10.

Table 10. Comparison of smoothed resistivity values for platinum

with those of Roeser [85] and Laubitz [81],

Investigation Roeser Laubitz Present

T, °C pR PL

P p -p

R P-PL

0 9.83 9.82 9.85 0.02 0.03

100 13.68 13.67 13.69 0.01 0.02

200 17.43 17.41 17.41 -0.02 0.00

300 21.06 21.04 21.02 -0.04 -0.02

400 24.57 24.54 24.52 -0.05 -0.02

500 27.96 27.94 27.90 -0.06 -0.04

600 31.24 31.22 31.17 -0.07 -0.05

700 34.40 34.38 34.33 -0.07 -0.05

800 37.44 37.43 37.38 -0.06 -0.05

900 40.38 40.36 40.28 -0.10 -0.08

1000 43.21 43.18 43.07 -0.14 -0.11

1100 45.93 45.98 45.78 -0.15 -0.20

130 -
Roeser probably used resistance thermometry grade (99.999) platinum

and Laubitz used platinum that was 99.9991 pure. The platinum used in the

present investigation was only 99.98% pure and its resistivity would be

expected to be higher than the values reported by Roeser and Laubitz. The

resistance ratio R

273 2°K

// ^4°K Laubitz 's platinum was 1890 so that the

residual resistivity was 9.82/1890 = 0.005|j,Qcm. The resistance ratio of the

platinum used in the present investigation was 390 so that the residual

resistivity was 9.85/390 = 0.025p,Qcm. Thus on the basis of Matthiessen ' s

rule, according to which the ideal and residual resistivities are additive,

one would expect the values obtained in the present investigation to be

higher than those of Laubitz by the constant amount of 0.02p,Qcm. As can be

seen in Table 10 the converse is true with the present investigation giving

values lower than those of Laubitz. Laubitz reports an estimated error of

±0.3% and the estimated uncertainties in the present work range from 0.1% at

0°C to 0.44% at 900°C so that differences between the two sets of data fall

well within the estimated limits of experimental error.

Thermal Conductivity

Values for the thermal conductivity of platinum based on equations (7-4)

and (7-6) are compared with the smoothed values of Watson and Flynn [ 76 ]

and Halpern and Flynn [86,A] in Table 11. The measurements by Watson and

Flynn were made on the bar from which the specimen for the present investigation was machined. The measurements by Halpern and Flynn were made on a

sample cut from the same bar. Watson and Flynn made their measurements in

air in the NBS Metals Apparatus £87] which is a longitudinal heat flow

apparatus. Halpern and Flynn conducted their measurements in vacuum in the

NBS Low Temperature Apparatus [88] which is also a longitudinal apparatus.

131
While these two methods are in principal the same as the direct method of the

present investigation the experimental variations and differences in analysis

make each of the methods unique. Thus the values in Table 11 for the thermal

conductivity of platinum represent values obtained by four different methods.

Table 11. Thermal conductivity of platinum: values are given for the

thermal conductivity as determined by the direct method and

the indirect method; the values of Watson and Flynn

determined on the bar from which the specimen was machined,

and those of Halpern and Flynn, determined on a sample cut

from the same bar, are also shown.

T Watson & Flynn

°C

Halpern & Flynn Present Investigation

Direct Method Indirect Method

-150 0.712 0.729

-100 0.711 0.719

- 50 0.710 0.714

0 0.711 0.712 0.713* 0.717*

50 0.712 0.714 0.714* 0.717*

100 0.714 0.716 0.718

200 0.720 0.722 0.723

300 0.729 0.731 0.731

400 0.742 0.743 0.741

500 0.758 0.756 0.754

600 0.777 0.772 0.767

700 0.799 0.790 0.781

800 0.825 0.809 0.796

900 0.829 0.811

1000 0.850 0.825

1100 0.871* 0.838*

* Extrapolated values.

132
The values from the present investigation extrapolated to 50°C agree

with each other and with the results of the other two investigations to

within 0.7%. The values given by the indirect method are slightly higher

(0.3%) than those given by the direct method at 100°C and are lower by 2.2%

at 900°C. As shown above the estimated probable uncertainty in the direct

method ranges from 0.44% at 100°C to 0.38% at 900°C while the estimated

probable uncertainty in the indirect method ranges from 1.42% at 100°C to

1.85% at 900°'c. The results of the two methods are thus seen to agree within

the combined estimated uncertainties of the two methods.

The results of the indirect method agree with those obtained in the NBS

Metals Apparatus to within 0.3% from 100 to 500°C. Above 500°C the results

of Watson and Flynn shown a greater increase of thermal conductivity with

temperature, the value at 800°C being higher by 1.7% than that given by the

direct method. This value is still well within the combined estimated

uncertainties of the two methods.

Values for the thermal conductivity of platinum given by the direct

method are compared with those of other investigators in Figures 27 and 28.

At the higher temperatures the results of the present investigation show the

same slope as those of Martin, Sidles and Danielson [83] and Holm and

Stormer [ 14]. The value at 1100°K is 3% higher than that of Martin, Sidles

and Danielson. Values reported by Laubitz [l C l]f or the thermal conductivity

of platinum are not shown in Figure 27 but they agree within experimental

error with those of present investigation.

The values given by the direct method at lower temperatures are confirmed by the results of Watson and Flynn and those of Halpern and Flynn

[ 86 , A]. They are also in good agreement with the results of Bode [41 ],

133
T T

' \

\\

\

O

ID

o

o

ro

o O ILlS q:3

<

or. Id

CL

ill I—

1 i l _L

LO o in o in o m

CD CD 00 00 N CD

d d d d d d d

>1 6ap Uio/M AllAilOnQNOD 1VIAIU3H1

134

Figure

27-

Thermal

conductivity

of

platinum:

comparison

between

existing
0.77

»6spujo/M AllAllOnaNOO "IVI«iy3Hl

LU

a:

z>5

qc

Ld

CL

Ld

I—

135

Figure

28.

Thermal

conductivity

of

Platinum:

comparison

between

existing

low

temperature

data

and

those

of

the

present

investigation.
d

0 0 co X

CVJ 0*

0) X *» d

(0 UJ

Q

si d0 cr

XId X

x X 3

x

X

CO H u

a) x

X

t>0 X O <

CO E CD 1-1 X O oc

ai x O CO X 0 H a) yj

XI x X a> X

> 0.

GO X d d

co X

d

cd

CO H

d X CO X yj

d cd N 1—

0) d d X xt l-H 0) r- X e (U 0) co X H & a) a. cu 1

0X0

P-, 5

0K

X

(X 0

o

rd

00

cvi CD

cvi

y\ 69 p/ a ‘daaiAsnN

w 2

cvi ZN3d01

CVS

cvi

O

o

CO

o

o

LO

o

o

CD

o

o

fO

O

cvi

136

Figure

29.

Lorenz

number

for

platinum:

comparison

between

existing

data

and

those

of

the

present

investigation.
Kannuluik and Carman [ 27], Meissner [ 13J and Moore [89] although the results

of Moore would be expected to be higher due to the higher purity of his

plntfnum. At the lower temperatures one must consider the purity of the

samples since impurities have a significant effect on thermal conductivity

values. At the higher temperatures no significant effects are to be expected

due to small amounts of impurities. The purity of the specimen used in the

present investigation and in those of Watson and Flynn and Halpern and

Flynn [8 6, A] was 99.98%. Bode's specimen was of the same purity. The

measurements by Moore were made on the same specimen as used by Tye [90],

the platinum being 99.999% pure. The purity of the second sample measured

by Flynn and Halpern [86, B] was almost 99.999%, , and the difference in values

from their first sample are considered to be due in large part to the

differences in purities.

Values obtained for the Lorenz number are compared with those of other

investigators in Figure 29. The Lorenz number was found to increase with

temperature contrary to theoretical predictions. However, this increase with

temperature is considered to be real in view of the high precision with

which the Lorenz number was measured. Consequently it would seem that the

theory requires modification to account for this effect in a transition

metal such as platinum.

137
APPENDIX A

Machining the Neck and Wiring the Specimen

Due to the structural weakness of a small neck in a large bar a special

technique, described below, had to be developed for machining the neck in

the specimen.

The platinum bar was held in a precision-bored jig that consisted

essentially of two clamps--one clamping the bar on either side of the neck--

rigidly coupled together by four ribs which bridged the neck region (see

Figure 30). With the jig mounted between centers and one of the ribs removed

a passing cut was made with a 0.125-inch diameter end-mill (Figure 3l) . The

jig was then rotated through an angle of six degrees and another passing

cut made. This process was repeated sixty times as the jig was rotated

through 360° replacing and removing ribs as needed but always having at

least three ribs in place. Successively deeper series of cuts were made

until the desired neck size was reached. The cross-sectional configuration

of the neck was that of a sixty-sided polygon. Several prototype specimens

were fabricated out of lead to test this technique on a weak material. One

of the prototype specimens had a neck as small as 20 mils in diameter.

The jig described above extended the whole length of the specimen and

had to be removed in order to install the heater and thermocouples. It

was necessary to design a second fixture to support the neck while the

specimen was being wired and installed. A split brass collar was clamped

to the bar on the lower side of the neck. A split aluminum collar was

precision bored so that it clamped on the upper part of the bar and

on the brass collar simultaneously. Dowel pins projecting from the brass

138 -
Figure 80. The specimen being machined. One of the

four ribs connecting the two halves of the jig has been

removed and the end mill is being positioned for a final passing cut.

Figure . The machining set-up for machining the

neck in the specimen. The specimen is held in the

machining jig with the latter mounted between centers.

Three of the ribs connecting the two halves of the jig can be seen. The milling head is centered over the

neck which has just been machined. - 139 -
collar guided the aluminum collar into place fixing the relative vertical

position of the two. This clamp was attached to the specimen before

removing it from the machining jig. When the specimen was mounted in the

apparatus and ready for testing, the clamp was then removed carefully.

The two thermocouples adjacent to the neck were installed while the

specimen was still in the machining jig but before the support clamp was

attached. The support clamp had slots machined in it to accomodate the

thermocouple wires. With the support clamp in place the specimen was

removed from the machining jig and mounted on vee blocks. The molybdenum

extensions were screwed into the ends of the specimen. The remainder of

the thermocouples and the heater were installed with the specimen assembly

resting on the vee blocks. The thermocouples were pressed into the slits

using a small screw driver with head-width approximately that of the slit.

All the heater welds were made with an oxygen-gas torch using a very fine

tip.

140
APPENDIX B

Potential Perturbat ions

The procedures used for calculating the electrical resistivity and the

thermal conductivity (direct method) assume that the equipotential surfaces

along the specimen are parallel and also that the isothermal surfaces along

the specimen $re parallel. The discrete nature of the specimen heater

introduces perturbations in the heat and electric current flow which cause

the isothermal and equipotential surfaces to be distorted in the vicinity

of the heater. Also, at the cold end of the specimen nonuniform contact with

the molybdenum extension may give rise to distortions. It is the purpose

of this Appendix to show that these distortions decay to negligible proportions in the region where temperatures are measured.

We can approximate the discrete heater problem by considering a semi=>

infinite solid with heat or electric current flowing through uniformly

spaced strips on the surface (Figure 32).

The boundary conditions are:

= -- TM , 0<x< a , z = 0 (B-l)

3 Z <T

, a<x<b , z = 0 (B-2)

3z

141
— =0 , x-0 and x-b, 0£=z<co (B-3)

dx

where 0 is electric potential or temperature, J is flux of current or heat,

a is electrical or thermal conductivity, 2a Ls the width of the region through

which flux passes, and 2b is the center-to-center distance between the regions

passing flux.

The potential or temperature distribution satisfies y

2

0 = 0 and the

solution is easily shown to be

0 = 2b8

tt a Y-4 sin

nna

cos nnx -nuz/b ^

b

6

-J. (B-4)

n=l

The gradient in the z -direction at any point is given by

38 -

a J

fl + JL V J_ sin JTM- cos J22L e'

lmz/b l (B-5)

TTa L n

n= 1

The average value of the gradient in the z-direction is given by

b

(

Bz/ b

£9,

o

v 3z

dx = - (B-6)

The fractional difference between the local value of the gradient and its

average value is given by

J£L . /jlA

B

z X Sz/

"i§)

= +

2b

rra

UJ Y

n = l

sin rrrra cos

nrrx -mrz/b

(B-8)

Values of the percentage difference^ lOOf, for different values of a/b, are

plotted in Figure 33 for x/b = 0.0 and in Figure 34 for x/b = 1.0. It is

142
z/b

Figure 33. Local departures from the average value

of the potential gradient caused by the

presence of strip sources at z *= 0. (x/b = 0.0)

- 143 -
Figure 34. Local departures from the average value of the

potential gradient caused by the presence of

strip sources at z = 0. (x/b = 1.0)

144
seen that, regardless of the magnitude of the original disturbance, the error

in the gradient has decayed to the order of 0.1 percent at positions only

2b away from the heater. In the present experiment the nearest thermocouple

was about 4b away from the heater.

At the cold end of the specimen a molybdenum stud joins the specimen

to the lower extension and this stud might cause preferential flow of heat

or electrical current through the central region. We approximate this problem by considering a semi-infinite cylinder, of radius b, which is perfectly

insulated except for a circle of radius a at z = 0 (Figure 35).

rs b

Figure 35. Semi-infinite cylinder.

The boundary conditions are

59 = _ 5z

J

CT

, 0<r<a , N

II O (B-9)

ii CDS N

0 , a<r< b , N

II O (B-10)

J&- = G , r = b 0 ^ Z< co (B-ll)

Sr

The solution is easily shown to be

e s

2bs

y

J^o^a/h) J0 (en

r/b)

a n=l

^

J

o

S K>

-Q>n

z/b-|

J 9 (B-12)

where JQ

and are Bessel functions of the first kind of zero and first

th

order, respectively, and cvn

is the n root of (o^) = 0. Differentiating,

145
-2a

Bz a b2

1 +

As before 9 J a

2 (B-14)

and

(B-15)

Values of the percentage difference, 100 f, for different values of a/b, are

plotted in Figure 36 for r/b - 1 (temperatures were measured at r = b) . It

is seen that, regardless of the magnitude of the original disturbance, the

error in the gradient has decayed to the order of 0.1 percent at positions

only one diameter (i.e. 2b) away from the disturbance. In the present investigation the disturbance was probably small to begin with (certainly nowhere

nearly as drastic as assumed in the statement of the problem above) and the

nearest thermocouple was one specimen diameter away from end of the specimen.

146
ERROR

ISM

z/c

Figure 36. Local departures from the average value of the

potential gradient caused by a circular source

of radius, a, at the end, z = 0, of a semiinfinite cylinder of radius, b. (r/b = 1.0)

147
APPENDIX C

The Thomson Effect

We wish here to evaluate the magnitude of the Thomson effect represented

by the second term on the right-hand side of (3“40) :

0-40)

The Thomson coefficient T is to a good approximation a linear function of

the absolute temperature T [ 91 ]

T - YT , (C-l)

where 7 is a constant. Consequently

(C-2)

Since T is a function of tjr we must express (C-2) in terms of i]f before the

Thomson term in 0-40) can be integrated.

Since the Thomson effect is in general a second-order effect, we can to

a first approximation write

^dijr = J

T

\p dT

J 0 , ® T

(C-3)

Moreover in the case of a pure metal such as platinum we can write

kp = LT , (C-4)

where L is the Lorenz number. This is a good approximation if instead of

using the theoretical value of the Lorenz number we use the value of L at

T + Tjn

temperature 0 as determined experimentally, and assume that L is constant over the temperature interval (Tm — T) , that temperature interval being

small in comparison with the absolute temperature T. Substituting (C-4) into

148
(C-3) and integrating we get

t 2 = L(Tffi2 - T2

) Using (C-2) and (C-5) in 0-40) and integrating

The relative magnitude of the Thomson term is thus given by Yt/3L- Some

representative values are shewn below. The values shown for correspond to

the maximum current (lOOA) passed through the specimen, the variations in

i{? reflecting the variation in resistance of the specimen with temperature.

(C-5)

(C-6)

Table 12. Values for the relative magnitude of the Thomson term

in equation (3-40)

.

T Y

L \|r ^ V/2 Yt/3L

°K (V/°C2 )xl08 (V/°C) 2 xlQ8 V

300 0OO i

—

1

1 2.62 .02 .003

600 -1.08 2.69 .04 .004

900 t-l oCO 2.74 .06 .008

1200 1—* oOO 2.79 .08 .oio1500 1 1

1 oOO 2.82 .10 .013

For the purpose of evaluating the Thomson effect it was assumed that ijj =* V/2

where V is the measured voltage drop across the neck. The relative magnitude

of the Thomson term is seen to be less than one percent over the entire

temperature range. Moreover, in a conductor with the ends held at the same

temperature the Thomson effect in one half of the conductor cancels that in

the other half to first order so that the net Thomson effect is of the order

of only 0.02 percent.

- 149
APPENDIX D

Alternative Calculation Procedure for the Indirect Method

Let us rewrite ( 3

- 66 ) in the form

»oR S' ^o^mU

where

and

D “ D<1)°8m) "

"a)o0 m t

1 " “g" arctan G]

G

2 - 2T|o0m(l + Vm)

Our definition of R* is

R

1 = Rr(l + 2

(D-l)

(D-2)

(3-67)

(D-3)

Eliminating 0 m between (D- 1) and (D-3) yields, after some manipulation,

(a r/^o)Rr(R - Ro) ~ UR(R - Rr) R' c R 1 +

UR"

] (D-4)

For Tq = Tr , aQ = ar , Ro Rr , and

(l-Rr/R)(U - Rr/R)

R' [’ R 1 U ] (D 5)

In the present investigation U was approximately constant with a value of

0.935, and Rr /R ranged from O .936 to 1.000. The maximum temperature rise

was 10(?C and jl0 — Tr | was generally less than 0.1°C. Thus for © m < 100 the

second term in brackets in (D-4) is quite small so that in order to calculate

a value for U it is sufficient to use the value for 9 m

given by 9 m =

(R - Ro)

2/3 Q^qRo

. Likewise 0 m need be known only approximately for calculation

- 150 -
of V' 3 from (4-7) provided 9 m

is not too large. Table 13 gives values of

U(Tlo9 m) for arguments from 0.0 to 0.5.

Table 13« Values of U(T) o

0 m) for arguments from 0.0 to 0.5.

.=3 s 0.00 0.01 0.02 0.03 0 .04 0 .05 0.06 0 .07 0 .08 0 .09

0.00 1 .000 0.995 0.986 0.979 0 . 973; 0 . 966 0.960 0 . 953 0 . 947 0 . 941

0.10 0 .935 0.929 0.923 0.917 0 . 911 0 . 905 0.900 0 . 894 0 . 889 0 . 00 00V* 0.20 0 .878 0.873 0.867 0.862 0, 857 0 . 852 0.847 0 . 842 0 . 837 0 . OO VJJ ro

O.JO 0

00 CVi 00

« 0.823 0.818 0.814 0 . 809 0 . 805 0.800 0 . 796 0 . 792 0 . 787

0.40 0 .785 0.779 0.775 0.771 0 . 767 0 . 763 0.759 0 . 755 0 . 751 0 . 747

- 151
APPENDIX E

A Simplified Expression for the Maximum Temperature

Rise in the Neck in the Presence of Heat Losses

In Chapter IV we derived the expression

Jm

TTB a

4 k„ I

n=l,3,5

n3

D

n-l

(-D 2

(4-16)

^nl3 bo

n

3

rr4 a

4 k^ nrra k,-.

Fi [An;B^}

j

F 0 ^An;Bn]

as the maximum temperature in the presence of heat losses. For the parameter

values appropriate to the present experiment, the term in cj,qI

z and the term

in bo/k0 in (4-16) are small compared to unity. Using the relation

(1 - x)"

1 S3 1 + x for small x, and the identities [ 92 ,p 64]

1 1 1

1 — + t^3 y3

TT32 (E-l)

1 - 36 +^6—75 + e, = J?IL_ , (E-2)

1536

we can write

= 0

2r? a

4

k.

1+ 50o^£po^ )

128X V

n-1

Y, r An ;Bn ] 6 2^a4‘k

Q

TT4 a k0 n-1, 3, 5 n

4 F 0 [AnJBn]

(E"3)

In the geometry used in the present experiment b > > a, and (E-3)

reduces to

* P 0

I

3

“m

=

2TT3

a4

k.

1 +

5 Qq ( f PqI

5 ^

1281 *0 \

6 ^ 2na a

4 k/ n

4 a k Q 4=1, 3,

(-i) 2 K^nna/ae)

K0 (rma/ 2JL)

(E-4)

152
Owing to the very rapid convergence of the series in (E-4) , we need retain

only the leading term of the series. We can then write

um

1281

rr4 a

K, (na/2l) ~| K0 (na/2l)J

(4-17)

where is the maximum temperature rise in the neck in the absence of heat

losses, i.e. for = 0;

^PqJ3

fi + sogf

2n3 a

4 k0 6 ^

The second order term in aQ

lP is neglected in

fPo? \]

2TT*a

4 k/J

(4-17).

(E-6)

153
APPENDIX F

Derivation of the Equation for Heat Flow

in the Current Leads of the Specimen Heater

In this appendix equation (4-44) is derived.

Several papers f 93, 94, 95, 96, 97, 98, 99 ] in the recent literature

discuss the problem of heat flow along current-carrying leads. With the

assumption of constant thermal conductivity and electrical resistivity,

(4-44) can be derived from various expressions given in these papers. However, an extension of the mathematical analysis already given in Chapter 3

for the indirect method leads quite naturally to a general expression for

heat flow along a current-carrying conductor.

Consider a conductor through which an electric current is passing, and

which has its surface perfectly insulated both thermally and electrically.

The geometry may be arbitrary. The heat flux, Q, at any point in the

conductor is

Q

= -k V T (F-l)

where k is the thermal conductivity of the conductor and T the temperature.

Neglecting thermoelectric effects, we can write from (3-35) and (3-49)

0 V 0

= -kp v T , (F-2)

and 0

2 = 2 kpdT (F-3)

where 0is the electrical potential, p the electrical resistivity of the

conductor, and T

‘ a fixed reference temperature.

From (F-l) and (F-2)

_> 1 Q

= 0V0 = -0J , (F-4)

- 154 -
— where J is the electric current density.

Integrating (F-4) over an equipotential surface, S, we get

?

= “ 0 ? » (F-5)

where q is the total heat flow through S and T is the total electric

current through S. Substituting for 0 from (F-3), (F-5) becomes

- WpH i . (F-6)

Consider two equipotential, and hence isothermal, surfaces at temperatures and T

2 , with total heat flows qx and q^ » Since the surface of the

conductor is perfectly insulated we can write

PR- q, - , (F-7)

where R is the total electrical resistance between the two surfaces and

?ii » ^nd I are the magnitudes of , qo , and I, respectively.

Utilizing (F-6), (F-7) becomes

p p

-L ^ 2 r m* 1 pi

I R - I 2j k p dTj ~ I

* rp 2j k P dTl . m —

>

(F-8)

T T

s 1

Rearranging (F-8) and squaring both sides

2 f T

k p dT - af

1 k p dT + 2 Ir[

2J

A k p dlj~ + fit? , (F-9)

from which

r r-T f 1 j» 2 k D dT

_ t r T IR i.

2 k p dT + iS. . M 2

'I Substitution of (F-10) into (F-6) yields

(F-10)

%

-

dT +

I

2 R

2

(F-ll)

155
From (F-7) and (F-ll),

% (F-12)

Equations (F-ll) and (F-12) are identical to equations (8) and (9) in

the paper by Neighbor P 94 J. For constant k and p they reduce to

?! ” -^-(V Ti> + (F-13)

and

q

= Je_<T - T ) 25- , (P-14)

2 R

a 2

which is identical with (4-44)

156
APPENDIX G

Errors in Temperature Measurements Due To

Heat Conduction Along Thermocouple Wires

The purpose of this Appendix is to estimate the error in temperature

measurement due to heat conduction along the wires of the thermocouple

making the measurement. The specimen can be considered as a semi- infinite

solid of thermal conductivity k, perfectly insulated over the plane z = 0,

except for a circular area of radius a (representing a thermocouple wire)

through which heat flows at a constant rats, Q, per unit area, per unit

time

.

Assuming circular symmetry, the Laplacian of the temperature, Q, can

be written as

cP8 l a?

+ 7 a?

» 0

(G-l)

A solution to (G-l) for z ^ © is

e - e J©(\r) (G-2)

for any value of X, where J© is the Bessel function of the first kind and

order zero.

9 - [

05 e

“XZ

J@(Xr)/(\)dX (G-3)

J o

is also a solution to (G-l) for any X. The function /(X) must be chosen to

satisfy the boundary conditions:

k & « +q

dz

- 0

, e * 0, 0 <r <a

, * « 0, r > a (G-4)

157
Using the relation [ 94, sec. 13.42]

J0

(Xr)J! (Xa)d\ - 0

“ o

®* l/2a

“ 1/a

, r > a

, r = a

r < a

(G-5)

where is the Bessel function of the first kind and order one, we find that

/(X) “ - -P” Jj, (Xa) satisfies the boundary conditions and we can write

0 ~ J

® J0 (Xr) Jx (Xa)i&

k O K (G-i)

The average temperature, < 0 > over o<r<aatz=ois

< 9 > » - JLQ_ f J1 S (Xa)iA » . (G-7)

V j « 1 3rrk

At regions remote from o<r<a, z=o, the temperature v = o. The total

heat flow q

«* rfa^ Q, so that, from (G-7)

< 0 > » :?A- * (G-8) 3Ak

The above development is from ©arslaw and Jaeger [25, pp 214-217].

Let us assume that the thermocouple wire supports a temperature gradient j

G so that the heat flow in it is given fey

q » ~rras kw

G , (G-9)

where is the thermal conductivity of the wire. Eliminating q between

(G-8) and (@-9) , we get

< 9 > t T- G > (G- 10>

158
where < @ > can be considered as the effective change in temperature of the

junction due to heat conduction along a thermocouple wire. If G is negative

th® junction is cooled an amount < 0 >.

In the present investigation only the Pt 10% Rh wires affected the

temperatures at the measuring junctions on the specimen, as discussed in

Chapter II. In the gradient region these were 8-mil wires. Assuming that

they had the same thermal conductivity as the platinum specimen, a conservative estimate, and that they supported the same temperature gradient as

the specimen in th® gradient tests, i.e. 5*C/cm, the error introduced in

th© measured temperature du® to heat conduction along the thermocouple wire

was 0.085®C or 0.2% of the temperature drop between thermocouples 4 and 8

(see fig. 12). Since th® thermocouple wire was wrapped part-way around the

specimen in the isothermal plan® of its junction the temperature gradient in

the wire was probably much less than 5°C/cm . Furthermore since all the

thermocouples were installed in a similar manner they would all give rt»e

to about the same error both in sign and magnitude, and, hence in taking

temperature differences, the errors would in large part cancel. Consequently

it is felt that a conservative estimate of the maximum error in the measured

temperature gradient due to heat conduction along the Pt 10% Rh thermocouple

wires is 0.05%.

159
APPENDIX H

Departures From One-Dimensional Heat Flow in the Direct Method

The purpose of this Appendix is to show that radial temperature gradients

in the specimen were negligibly small for the guarding conditions used in

the direct method. To accomplish this we consider a circular bar of radius

b, length i and thermal conductivity k, with a constant heat flux q through

the cylindrical surface of the bar. The boundary conditions are:

, z - 0 , 0< r< b ;

, z s l , 0 < r< b ;

, 0 <z <i , r « b

The use of isothermal boundary conditions at each end of the bar does not

affect the radial temperature distribution at positions near the center of

the bar since, as shown in Appendix B, the effect of the end boundaries decays

very rapidly and is negligibly small at positions one diameter away.

The solution to the above problem Is

9 = 0

9 “ T

- ^.q y,

1 IQ

(nrrr/je) ^ nrrz

i T^k Ll n

2

X

i (nrrb/i) l

(H-l)

n“V3,5,

where I

c and 3^ are the modified Bessel functions of the first kind, and

of zero and first order, respectively; the notation n = 1, 3, 5, indicates

that the summation is over odd values of n only. The longitudinal gradient is

30 T 4q $2, 1 I (nrrr/4 ) nrrz ——~ - — - —— \ ——* —I .. —' cos . (H-2)

3z i rrk ^ n I (xmb/jg) l n=1,3,5,

The average value of ^q/^z at any longitudinal position is

- 160 -

(H-3)
Substituting (H-2 ) into (H- 3) and integrating yields

8/q ^

_1_

n^b k n» cos

HTTZ

l y (H-4)

which sums to

T

T BIT (i “ 2z) (H-5)

The fractional difference between (39 /§z) and <59 /^z> is r ® d

f =

Sqi & JL_

n2b k L n2

1 9 3 , 5

1 »

nrrb Io(imb/i) -

24 ^

(nrrb/x) cos

nrrz

T

l

(4 - 2z)

bk

>. (H-6)

Since, for our experiments, q^/bk< <

r£/l i we can write

f = JL. F (b/i; z/4) , kT(H-7)

where

F(b/j? ; z/i) =

84

r I

1 rrr 2 b n2 n=l, 3 , 5 .

1 - nub IQ

(nTTb/4 ) -j

H I

x (nrrb/4) ^

nrrz cos (H-8)

Evaluation of F and f for the conditions of our tests indicated that

f < 0.02% for the "unmatched" gradient tests and was considerably smaller

still for the "matched" gradient and the "isothermal" tests.

161
APPENDH. i

Tempering of the Heater Current Leads

Consider a current-carrying wire of radius* a* surrounded by insulation

of radius, b, all imbedded in a semi-infinite medium of high thermal conductivity which is maintained at zero temperature. The differential equation describing the temperature distribution in the wire is

JLIL - h

3 © - - M

4z

2 (I-D

where © is the excess temperature of the wire above its surroundings.

2

M

2ft &

2 k ftnb/a

»3

TTS a4 k

( 1-2 )

(1-3)

ft is the thermal conductivity of the ins u 1 at ion , k is the thermal

conductivity of the wire,

p is the electrical resistivity of the wire, and

I is the current passing through the wire. We assume the boundary condition

0 - V , z s 0.

The solution to (I-l) which satisfies (1-4) for z > 0 is

H U>

The temperature gradient in the wire i© given by

JL . Vv - -!t) ,

, s / az (j, while the heat flow along the wire is given bjr

P , d© 2 (xt M \ -uz

q

® »

Tra

2 k —— s3 tt* 4(V - I e ^

j_ ' 'd / • az u,

(1-4)

(1-5)

( 1-6 )

162

(1-7)
The above analysis is based on an infinitely long tempering lead while

we used tempering leads of finite length. An approximate analysis of the

finite case indicated that the analysis for the infinite case is a good

approximation for the conditions present in our measurements. For the

swaged elements (described in Chapter II) which were used to temper the leads,

a sa 0. 025 cm and b «« 0.050 cm. so that £nb/a 0.7. We estimate that

& ^ 0.004 W/cm deg and k ^ 0.8 W/cm deg at all temperatures. Thus

|is "> 20 cm"3 at all temperatures. Also at all temperatures pi2 < 6 x 10“5 Wcm,

so that M < 20 deg/cn^ at all temperatures . The length of swaged element used

to temper each lead was 1.3 cm. Plugging all of these numbers into (1-5), (1-6),

and (1-7) we obtain for the colder end of the swaged element

0 < 1 + 0.0033(V - 1) °C

dfl

I —• | < 0. 015(V - 1) *C/cm

dz

q < 0.000011(V - 1) W

The temperature rise in the heater winding was always less than 200 °C; therefore

q < 0.0025 W

and the combined heat flow out both current leads was less than 0.05% of the

power input to the heater. This estimate is in concordance with the small

values found for qa and q^.

163
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171
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172