NATIONAL BUREAU OF STANDARDS REPORT

8301

THE NATIONAL BUREAU OF STANDARDS

HIGH TEMPERATURE ABSOLUTE CUT-BAR APPARATUS

FOR DETERMINATION OF THE THERMAL CONDUCTIVITY

OF SMALL SOLIDS

Complementary Report

March 1964

D. R. Flynn and H. E. Robinson

to the

Bureau of Ships

Department of the Navy

Washington, D. C.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

by
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1006-20-10461 4 March 1964 8301

THE NATIONAL BUREAU OF STANDARDS

HIGH TEMPERATURE ABSOLUTE CUT- BAR APPARATUS

FOR DETERMINATION OF THE THERMAL CONDUCTIVITY

OF SMALL SOLIDS

by

D. R. Flynn and H. E. Robinson

Heat Transfer Section

Building Research Division

Complementary Report to the

Bureau of Ships

Department of the Navy

Washington, D. C.

INDEX N0„ S-R007 12 01, Task No. 800

(BuShips 1700S-716-62). Code 2781

IMPORTANT NOTICE NATIONAL BUREAU 0T STAN

for use within the Government. Be and review. For this reason, the p whole or in part, is not authorize Bureau of Standards, Washington

the Report has been specifically pre

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

accounting documents intended

ijected to additional evaluation

ting of this Report, either in

ffice of the Director, National

; Government agency for which

;s for its own use.

11. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

THE NATIONAL BUREAU OF STANDARDS

HIGH TEMPERATURE ABSOLUTE CUT-BAR APPARATUS

FOR DETERMINATION OF THE THERMAL CONDUCTIVITY

OF SMALL SOLIDS

TABLE OF CONTENTS

PAGE

FORWARD

ABSTRACT 1

INTRODUCTION ...... 1

DESCRIPTION OF ABSOLUTE CUT- BAR METHOD 3

ANALYSIS OF METHOD ..... 7

RESULTS OBTAINED USING THE HIGH

TEMPERATURE ABSOLUTE CUT- BAR APPARATUS 10

ESTIMATE OF PRECISION AND ACCURACY 29

APPENDIX I 35

REFERENCES 38

LIST OF TABLES

Table 1. Thermal Conductivity, Electrical Resistivity,

and Lorens Function for a 60% Platinum--

40% Rhodium Alloy 13

Table 2. Chemical Compositions of Four

Nickel -Chromium Alloys, Weight Percent 21
TABLE OF CONTENTS (Continued)

PAGE

List of Tables (continued)

Table 3 • Thermal Conductivity of

Four Nickel -Chromium Alloys . . . 23

LIST OF FIGURES

Figure 1« Schematic of Working Parts of High

Temperature Absolute Cut -Bar Apparatus ..... 2

Figure 2. Simplified Section of the High

Temperature Portion of the

Apparatus with Its Surrounding Case ....... 4

Figure 3» Overall View of the Apparatus 4

Figure 4* Thermal Conductivity of a 60% Platinum- -40% Rhodium Alloy 12

Figure 5. Thermoelectric Power of Platinum-

407* Rhodium and Platinum- -10% Rhodium

Against Pure Platinum .............. 15

Figure 6 e Microstructures (11500) of Inconel 702

in' the Solution-Annealed State (A)

and in the Age -Hardened State (B) ........ 18

Figure 7® Departures of the Thermal Conductivity

Data for Solution-Annealed Inconel 702

from a Smooth Curve . 20

Figure 8„ Departures of the Thermal

Conductivity Data for AgeHardened Inconel 702 from a Smooth Curve .... 20

Figure 9® Thermoelectric Power of

Inconel 702 versus Platinums ........... 22
TABLE OF CONTENTS (Continued)

PAGE

List of Figures (continued )

Figure 10. Departures of the Thermal

Resistivity Data for

Pyroceram 9606 from a Straight Line 26

Figure 11. Departures of the Thermal

Conductivity Data for

Pyrex 7740 from a Smooth Curve 26
The National Bureau of Standards

High Temperature Absolute Cut-Bar Apparatus

for Determination of the Thermal Conductivity

of Small Solids

FORWARD

This report is an unmodified version of a paper

given by the authors at the 1963 Thermal Conductivity Conference sponsored by Oak Ridge National Laboratory and held

at Gatlinburg, Tennessee, on 16-18 October 1963. It is

being made available as an NBS Report in the belief that the

contents may be of general interest.

In this report, thermal conductivity data are presented which have not been formally published. All such

data are subject to review, editorially and otherwise, prior

to formal publication, and their preliminary status should

be recognized. The contents of this report may not be referenced.
THE NATIONAL BUREAU OF STANDARDS

HIGH TEMPERATURE ABSOLUTE CUT-BAR APPARATUS

D, R. Flynn and H. E. Robinson*

Heat Transfer Section

National Bureau of Standards

Washington, B. C.

ABSTRACT

A description is given of an apparatus used for determining

the thermal conductivities of small cylindrical specimens (2.54 cm

in diameter and 1.27 cm long) of materials of conductivity in the

range 0.01 to 0.1 W/cm deg at temperatures from 100° to 1,200® C.

The method utilizes longitudinal heat flow in a small cylindrical

sample held between two bars of platinum- rhodium alloy. Measurements have also been made on metal specimens in the form of longer

cylinders. Experimental results are presented for 607. platinum-

40% rhodium alloy, a nickel-chromium alloy (Inconel 702), a microcrystalline glass (Pyroceram 9606), and a borosilicate glass (Pyrex

7740). A discussion of the test method is given, with attention to

possible sources of error.

INTRODUCTION

We almost wrote an Introduction opening with a statement of

the object of the work described here and going on to give a justification for undertaking it. In addressing this august body, however,

such an introduction would be superfluous.

Of the "methods" papers presented at the previous two con- ferences, roughly half indicated or implied a definite limitation on specimen size, arising in the main from the inability to procure large

samples of adequate homogereity. In addition to measuring the thermal

conductivity using smaller and smaller specimens, most investigators

currently are striving for smaller and smaller uncertainties at higher

and higher temperatures.

Physicist, and Chief, respectively
FIGURE 1. SCHEMATIC OF WORKING PARTS OF HIGH

TEMPERATURE ABSOLUTE CUT-BAR APPARATUS

A Gold bar support

B Heaters

C Boundaries of region of

powder insulation in which

heat flow is analysed

D Thermocouples

E Cold bar

F Specimen

G Current leads to heaters

H Hot bar

I Insulation

J Hof bar support column

K Potential leads to

specimen heater
-3

-

During the initial discussion of the program leading to the

apparatus that is being described here, the questions of sample size,

temperature range, and limits of uncertainty were discussed at some

length. The principal area of discussion involved selection of specimen size, the sponsor wanting us to measure aspirin-tablet-sized

specimens while we were inclined toward brick- si zed specimens in

order to more easily meet the desired accuracy requirements. In com- promise, it was decided to develop a method and apparatus for steadystate thermal conductivity measurements at temperatures to 800° C and

above, and suitable for solids in the form of small specimens (1/2-in.

by 1-in. diameter disks). The overall objective was to provide sam- ples for use by other laboratories as thermal conductivity reference

specimens in connection with their measurements on solid semiconductors.

The apparatus being described is a third-generation model,

differing from the second-generation model chiefly in that it is built

of more refractory materials suitable for temperatures to 1200° C or higher. Because of the shortness of the specimen, an apparatus of

the cut-bar type, such as is frequently used in comparative measurements, was selected. To use the contacting bars accurately as heat

flowmeters, however, entailed numerous complications, not the least

of which was lack of a suitable reference standard of appropriate con- ductivity. There are other problems in the use of contacting bars as meters, some of which have been presented previously at these conferences (1, 2).* For these reasons it was decided that the cut-bar form

would be used but the heat flux measurement would be absolute.

DESCRIPTION OF ABSOLUTE CUT-BAR METHOD

The working portion of the thermal conductivity apparatus is

shown in Figure 1. The 2. 54-cm-diarneter specimen is interposed between a hot bar and a cold bar, fabricated from 60% platinum- -40% rhodium alloy. These bars are supported by an alumina tube from below

and an alumina rod from above. Surrounding the. inner assembly and

concentric with it is an alumina guard cylinder, which is in turn sur- rounded by a stainless steel case, as shown in Figure 2. The bar

assembly is supported on a post fixed to the base of a drill press, in

line with the spindle shaft, through which a dead-weight thrust is imposed. The guard and case are supported by the work table of the drill

References appear under the heading REFERENCES
-4-

THRUST

FIGURE 2. SIMPLIFIED SECTION

OF THE HIGH TEMPERATURE PORTION OF THE

APPARATUS WITH ITS

SURROUNDING CASE

FIGURE 3* OVERALL VIEW OF THE

APPARATUS

A Base,

B Drill press support column

C Support column for specimen

and bar assembly

D Table

E Counterweight

F Guard Case

G Spindle shaft

1 Spindle head

I Pulleys

J Dead weights
-5-

press, and thus can be lowered to afford access to the bar assembly.

Figure 3 is a "patent drawing" view of the overall setup. The space

inside the lower alumina support tube, that between the central assem- bly and the guard cylinder, and that between the guard cylinder and

the outer case are filled with finely divided alumina powder as thermal insulation.

The desired temperature distributions within the apparatus

are attained with the aid of a number of heaters, each of which was fabricated from 807. platinum--207> rhodium alloy wire.

a. The cold bar is brought to a desired temperature by means

of a small heater fabricated from 0.25-mm diameter wire

wound in a helical groove around the alumina bar immediately

above the cold bar.

b. The hot bar is raised to the desired temperature above the

cold bar by means of a small disk-shaped heater (which shall

be referred to as the specimen heater) between the hot bar

and the lower alumina support tube. This heater was fabricated from 0.25-mm diameter wire and has a resistance of approximately 2 ohms (at room temperature).

c. In order to minimize heat losses down the support tube from

the specimen heater, a small heater, fabricated of 0.25-mm

wire, was wound on the support tube at a position about

A cm below the hot bar.

d. A helical groove on the outer surface of the guard cylinder

was wound with two separate heaters of 0.5 ram wire. Between the heaters, the guard cylinder wall thickness is re- duced to permit a longitudinal temperature distribution

along the guard cylinder corresponding to that along the

inner assembly. A tap divides each of these heaters into

two sections which can be controlled separately.

Temperatures in the system are determined by means of platinum-107. rhodium :platinum thermocouples, which were fabricated from

0.38-mm (unless otherw-se stated) reference grade thermocouple

wire that had been calibrated by the NBS Temperature Physics Section.

Thermocouple positions are shown in Figure 1. There are two thermocouple stations in the hot bar and two in the cold bar. In the hot

bar support tube there are two thermocouple stations and one differential thermocouple. Eight thermocouples are installed to define the

temperature distribution along the guard cylinder. The number and

location of specimen thermocouples are determined by the nature and

geometry of the specimen.
Instrumentation

Power for the two guard heaters and the cold bar heater is

supplied by variable voltage transformers, which in turn are fed by a voltage regulating transformer* The cold bar heater and the guard

heater sections nearest the plane of the specimen are individually

regulated by thermocouple-actuated controllers. The guard heater sec- tions remote from the plane of the specimen are manually adjusted.

The specimen heater is powered by a 28-volt, 4-ampere regulated d-c power supply with a small bank of power resistors in series

for adjusting to the desired heater current.

The heater in the hot bar support tube is closely regulated,

in order to prevent heat gains or losses from the specimen heater

along the support tube. Using a small d-c potentiometer, a bias is

placed on the signal from a multiple junction differential thermocouple located in the hot bar support tube. The resultant signal is

amplified by a breaker- type d-c amplifier and fed into a d-c recorder.

The error signal from the control slidewire in the recorder is fed

into a current-adjusting-type proportional controller incorporating

automatic reset control and rate control. The output of this unit

regulates a magnetic amplifier which feeds power to the heater. The

operation of a control system of this type has been described in detail by West and Ginnings 0).

All heaters are equipped with separate isolating transfor- mers to minimize current leakage effects.

The noble metal leads of the thermocouples are, brought to an isothermal zone box at room temperature, A thermocouple with one junction in the zone box and one in an ice bath is placed in series

with a double-pole selector switch, so that each measuring thermocouple is automatically referenced against the ice bath (4), The zone box is also wired to enable determination of the emf developed between

similar leads of different thermocouples in the hot bar, in the specimen, and in the cold bar. For these determinations, a separate double

pole selector switch with no ice bath thermocouple in series is used.

Thermocouple emfs are read on a calibrated precision potentiometer, usually to 0.1 (iv for thermocouples in the specimen and the

hot and cold bar, and to 1,0 nv for thermocouples in the guard cylinder, Power input to the specimen heater is determined by measuring

the d-c current through the heater and the voltage drop across potential taps. These measurements are made using calibrated shunt and

volt boxes and measuring their output voltages by means of the precision potentiometer.
-7-

ANALYSIS OF METHOD

Determination of Heat Flow

Ideally , all of the heat generated in the specimen heater

should flow through the specimen with no augmentation from other sources. In order to minimize heat flow up or down the hot bar support

tube the temperatures at two longitudinal positions about 2.7 cm apart

are automatically matched by means of the control system described

above. The potential taps used to determine the voltage drop in the

specimen heater are located midway between these thermocouple stations

as shown in Figure 1. Of the total power output of this heater, only

about 0.5 percent is generated in the leads in the region between the

potential taps and the uppermost junction of the differential thermocouple; with the two jxmctions matched in temperature, it is felt that

substantially all of the power generated in the leads at positions

above the potential taps flows upward toward the specimen. For tests

in which the reading of the differential thermocouple is not zero, a correction is made for the slight heat flow through the support column.

Analysis of heat exchanges between the bar assembly and the

surrounding powder insulation is essential. Some insight into this

problem can be achieved by consideration of a rather crude model. The

hot bar, specimen, and cold bar can be thought of as three thermal re- sistances in series. Shunting each of these resistances is a resistance consisting of a portion of the surrounding powder insulation.

For the case in which the thermal conductivity of the specimen is different from that of the bars, there will be abrupt changes in the longitudinal temperature gradients in the bar assembly at the specimen

faces. Since the thermal conductivity of the insulation does not

change abruptly, the insulation must exchange heat with the bars and

specimen to conform to the bar assembly temperatures. In particular,

if the specimen has a lower conductivity than the bars, it is neces- sary for the insulation to steal heat from the bars and specimen on the

hotter side and return this heat on the colder side.

It is also necessary to originally establish a matching

temperature gradient in the insulation along the bar assembly. In the case where the bars are quite short, this matching gradient may not

become established, so that heat exchanges occur along the entire

length of the system. Even in the case where the specimen and the

bars have the same conductivity, there may be heat interchanges along

the entire system. In order to study the parasitic heat flows mentioned, it is necessary to perform a rather extensive analysis.
- 8 -

Although there may be both radial and longitudinal heat

flows in the powder insulation, it is the radial temperature gradient

at the surface of the bars and specimen that is of particular interest® In the ideal case, the radial gradient at this surface would be

zero® This is, however, not the case in general, so that an analysis

of the heat flow across this surface must be made® The heat flow across this surface in a longitudinal element of area is

where a is the radius of the bars and specimen,

k is the thermal conductivity of the powder insulation,

T is the temperature in the powder insulation,

r is the radial coordinate,

z is the longitudinal coordinate®

The net power flowing across the surface r = a between

z = 0 and z = i is

In order to evaluate this integral, it is necessary to have

a knowledge of the thermal conductivity of the powder insulation and

of (^T/3r)

r_a as a function of z® An exact treatment of the system

requires solution of the four body composite system involving the two

bars, the specimen, and the powder insulation® This has been done by

B* A® Peavy of the NBS Heat Transfer Section, The resultant solution

is very complicated, however, and was not used for the investigation

described in this report®

For this apparatus, (Sx/Br) rssa was determined by analysis of

the hollow cylinder of powder insulation with its boundaries at known

temperatures® The measured surface temperatures along the bars and

the guard cylinders were used to establish functions representing the

temperatures on the curved surfaces of the region® The ends of the

region, indicated by dashed lines in Figure 1, were assumed to be

closed by logarithmic radial temperature distributions, which provide

temperature continuity® Axial symmetry was assumed® The analytical

solution of the temperature distribution in the region established by

these boundary conditions was utilized to evaluate (&T/dr) raa in the

integral in (2)„ The thermal conductivity of the powdered alumina

insulation was measured in another apparatus (5).
-9-

Since the analytical solution of this problem is in the form

of an infinite Fourier series, a digital computer is used for numerical

evaluation of the data. The details of the mathematical development

are not amenable to brief treatment and, hence, are not included in

this presentation. A paper (1) giving a fairly detailed analysis of

heat flow in the insulation of a simplified cut-bar apparatus was given at an earlier conference.

The total heat flow through the specimen is calculated using

the expression

Q a P -b Ctf + p , (3)

is the total heat (per unit time) flowing in the specimen

is the measured electrical heat input

is the heat flow up or down the support column due to a unit temperature difference

is the temperature difference in the support column as indicated by the differential thermocouple

is the mean value of the total net heat interchange with

the surrounding insulation for a thermocouple span

in the specimen (see Equation (2)).

The measured power input, P, is known very accurately. The

contribution from heat flow in the support column is minimized by

keeping & small (usually less than 0.1 deg C). The heaters on the

guard cylinder are adjusted to minimize p.

where Q

*Dc

p

Temperature Gradients

Temperatures in the specimens are measured using noble metal

thermocouples, as described above. The particular method of thermocouple installation varies somewhat with the specimen being tested,

and will be discussed in the appropriate sections describing experimental results.

Since temperature differences in the specimen are rather

small, it is essential that the conversion of thermocouple emfs to

temperature not introduce any additional uncertainties. The equation

E “ 16 - 82614 (-T5oo-) " 10-5°487 (-ifoo-)

2

+ 7 - 91246 (-TSoo-J

- 2- 055° 5 (-I500-) (4>

- 2.85575(1.0 - exp[-^2

-j )
“ 10-

where T is temperature (®C) and E is emf (millivolt®), was found to

fit the platlnum--10%rhodium:plat inum thermocouple calibration data

quite closely over the temperature rang© 0® t© 1450® C. This equation

is used for conversion of all thermocouple voltages t© temperature.

Simultaneous Solution of Two Tests

For all tests, thermal conductivity values are computed by

simultaneous solution of two tests: 1) an "isothermal" test with no power input to the specimen heater, and 2) a "gradient" test with sufficient power input to the specimen heater to maintain the desired

longitudinal temperature gradient. This procedure tends to minimize

the effect of certain systematic errors in temperature or heat flow

measurements. A brief analysis of the advantages of this simultaneous

solution is given in Appendix I. All thermal conductivity values were

computed using Equation (A9). The use of this simultaneous solution

for computing results significantly improves the precision of the test

results and, it is believed, the accuracy as well.

RESULTS OBTAINED USING THE HIGH TEMPERATURE

ABSOLUTE CUT- BAR APPARATUS

In this section, thermal conductivity data are presented

which have not been formally published. All such data are subject to

review, editorially and otherwise, prior to formal publication.

Pi at inum- Rhodium Alloy

Measurements were made of the thermal conductivity of the

platinum-rhodium alloy from which the hot and cold bars in the high

temperature absolute cut-bar apparatus were to be fabricated.

Description of Spec imen . The specimen used for these determinations wa® in the form of a right circular cylinder 2. 5^9 cm in

diameter and 7,5 cm long with recesses at either end. The solid portion of the cylinder was 6.49 cm long. The geometry of this specimen

may be envisaged by reference to Figure 1. The specimen was later cut

into two pieces which were ground and optically polished to form the

hot and cold bars®
- 11 -

The specimen was machined by the manufacturer into the form

described. An NBS chemical analysis indicated that the material of

the specimen was 60.0 percent platinum and 40.0 percent rhodium by

weight*. The specimen was annealed at 1000° C prior to the tests

described below.

Test Procedure , The temperature distribution along the

platinum- rhodium alloy specimen was determined by means of four O. 38 - mm butt-welded platinum- 10% rhodium:platinum thermocouples pressed

into 0 . 3 -inm grooves in the convex surface of the bar. The distance

between the two extreme thermocouples was 3 ->94 cm.

Thermal conductivity measurements were made in increasing

order of temperature from 200® to 1200® C at 200-degree intervals, and

then in decreasing order of temperature at 800® and 400® C. Each con- ductivity determination involved two tests: ( 1 ) an "isothermal" test,

in which there was no power input to the specimen heater; ( 2) a "gradient" test with sufficient power input to the specimen heater to maintain a longitudinal temperature gradient in the specimen of about 2

deg/cm. In all of the tests, the guards were adjusted so that there was very little net heat exchange between the specimen bar assembly

and the surrounding insulation.

Thermal Conductivity. The thermal conductivity of the platinum-rhodium alloy as determined by simultaneous solution of the various

pairs of tests is shown in Figure 4. The points plotted are weighted

averages of the values obtained for the three thermocouple spans. Due

to uncertainty in the effective distances between thermocouples, there was some scatter in the individual values obtained for the spans. The

average of the three values, weighted according to the length of each

span, retains only about one-third of this uncertainty.

The circles shown in Figure 4 represent measurements made in

increasing order of temperature-,** and the inverted triangles those

made in decreasing order of temperature. The solid line shown is the

quadratic equation of least-mean-squares fit to all of the data points

*NBS spectrochemical analysis indicated the following impurity

elements: Fe 0.087„; Cu, Ir, Pd, Si, Zr 0. 001--0. 01%; B, Ca less

than 0 . 001 %.

**Prior to the tests shown in Figure 4, the specimen was annealed in

place at 1000 ® C; hence the designations "second heating" and "second cooling."
THERMAL

CONDUCTIVITY

,

W/cm

- 12-

shown. This equation is

k - 0.465 + 0.424 (-ilor) - 0-151 (-T5oo ’)

’ (5)

where k is thermal conductivity expressed in W/cm deg and T is temperature in °C. The dotted lines shown bound the region plus and

minus 2,2 percent of the conductivity, equivalent to twice the estimated standard deviation divided by the mean conductivity.

FIGURE 4. THERMAL CONDUCTIVITY OF A 60% FUTINUM—

40% RHODIUM ALLOY
-13-

Electrical Resistivity and Lorenz Function . When the 60%

platinum- "40% rhodium alloy bar was purchased, a 0. 5 -mm wire was drawn

by the manufacturer from the same material. The electrical resistivity

of a length of this wire was measured in air over the temperature

range —195° to 1500® C. Owing mainly to the small cross-sectional

area, the values are considered' to be uncertain by ±2 percent. The

electrical resistivity and thermal conductivity values and the corres- ponding Lorenz functions are given in Table 1. The apparent Lorenz

function appears to be substantially constant near the theoretical

value over the temperature range covered.

TABLE 1. THERMAL CONDUCTIVITY, ELECTRICAL

RESISTIVITY, AND LORENZ FUNCTION FOR

A 60% PLATINUM- -40% RHODIUM ALLOY

t P k L kp/T 100(L/L

( e c p$-cm W/cm deg

10

8

V

2/deg2 1

”200 11.9(b) -- --- —

0 16.8 — — --- 200 21.6 0.544 2.48 +1.5

400 26.4 0.611 2.40 -1.8

600 31.4 0,665 2.59 -2.1

800 36.5 0.708 2.41 -1.5

1000 41.5 0.738 2.41 -1.4

1200 46.5 0.756 2.39 -2.2

1400 51.6 — — —

1500 54.0 ... — ---

(a) a 2.443 x 10" 8

V

2/deg2

, where k * (8.6170

± 0.0012) x 10

5 eV/deg is Boltzmann's constant, and e is the

electronic charge.

(b) Extrapolated value,
-14-

Thermoelectric Powera A thermocouple fabricated from a length of the wire just mentioned and a length of O.jS-mm reference

grade platinum wire was calibrated by the NBS Temperature Physics Sec

tion over the temperature range 0° to 1100® C. The equation

E * 15 • 25682 (-T300-) - 2- 76979 (-I560-)

+ 6 - 09255 (-mo-)

3 - 79519 (;t5oo-)

4

(6 >

- 1.80869

(

1.0 -

-] )

where E is emf (millivolts) and T is temperature (°C) , was found to

fit the calibration data quite closely.

Differentiation of Equation (6) gives

dE

dT 13.257 5,540

(

1000 )

* 18.277

\

1000 /

8.112 exp

4, 5T 1

1000 J

( 7 >

as the thermoelectric power ((iV/deg) of 60% platinum- -40% rhodium:

platinum.

Similarly, differentiation of (4) gives

dE

dT 16.826 - 20.610 (^g_) * 23,737

8.212 ^ iqqq~J ” 11.415 ®x?|~

1000 J

( 3 )

as the thermoelectric power (uV/deg) of 907» platinum-- 10% rhodium:

platinum.
-15-

FIGURE 5. THERMOELECTRIC POWER OF PLATINUM- -407. RHODIUM AND

PLATINUM- -107. RHODIUM AGAINST PURE PLATINUM

The solid lines in Figure 5 show the thermoelectric power of

607. platinum- -407. rhodium and 90% platinum- -107. rhodium against platinum, as computed from (7) and (8), respectively. From the "Law of

Intermediate Metals" (6) it follows that the thermoelectric power of

60% platinum--40% rhodium versus 90% platinum—10% rhodium can be

obtained by subtracting Equation (8) from (/)

the thermoelectric

power of this combination is indicated by the dotted line in Figure 5.

For each of the tests presented in Figure 4, for both the

upper and lower thermocouple spans of the specimen, emfs were measured

between the respective platinum wires of the two thermocouples and

between the respective platinum—10% rhodium wires. Thus, the 60%

platinum- -4Q7. rhodium bar served as the central portion of four differential thermocouples. From the emf outputs of these differential
-isthermocouples and from £h@ temperature differences, as indicated by

the four platinum*- 10% rhedium: platinum thermocouples, the thermoelectric power of the specimen against 90% platinuas--10% rhodium and against

platinum was computed. The effect® of systematic errors were minimized

by simultaneous solution of the "gradient" and "isothermal" tests. In

figure 5, the plotted points give the average values obtained for thermoelectric power as a function of temperature. The circles and triangles correspond to the similar symbols in Figure 4 at the same temperatures, It can be seen that the measured values agree quite well

with the curves obtained from equations (7) and (8),

Nickel - Chromium Alloy

Measurements were made of the thermal conductivity of a nickel-chromium alloy (Inconel 702) which previously had been measured

in two models of the NBS metals apparatus (7), in the NBS modified

prototype absolute cut-bar apparatus 3 * and in the NBS steam calorimeter apparatus (8), Determinations made on this specimen material

served a dual purpose: (1) since the thermal conductivity of this

alloy had previously been measured over a large temperature range,

measurements on this material enabled a scrutiny of the accuracy of

the high temperature absolute cut-bar apparatus; and (2) they provided

additional measurements to elevated temperatures on a metal which was being considered as a possible thermal conductivity reference material.

Description of Specimen . The specimen used for these determinations was in the fora of a cylindrical bar nearly identical in

size and shape to the platinum- rhodium alloy specimen described above.

The specimen was machined at MBS from the same solutionannealed hot-rolled plate as were the specimens previously measured in

several other NBS apparatus. An NBS chemical analysis indicated that

the alloy was composed of Ni 79,5%, Cr 17,07., A1 2»57«, and several

other constituents in quantities less than 1 percent (by weight). The

detailed analysis is given in Table 2, along with analyses of several

similar alloys which will be discussed later.

lest Procedure . Temperatures along the specimen were determined by means of four thermocouple® installed in a similar manner to

those in the pXafcintsn-rhodium alloy.

*An earlier model of the present apparatus
-17-

Thermal conductivity measurements were made at 200° C, in

increasing order of temperature at 100-degree intervals from 400° to

1200® C, and then in decreasing order of temperature at 900°, 600°,

and 300® C. Following preliminary analysis of these data, an additional series of tests was conducted in which measurements were made

in increasing order of temperature at 200® C and 3OO 0 C, at 50-degree

intervals from 400® to 700® C, at 100-degree intervals from 800° to

1200® C, and then in decreasing order of temperature at 900®, 600®,

and 5OO 0 C. Each conductivity determination involved two tests:

( 1 ) an "isothermal" test, and ( 2) a "gradient" test with sufficient

power input to the specimen heater to maintain a longitudinal temperature gradient in the specimen of about 4 or 5 deg/ cm.

Thermal Conductivity, The Inconel 702 alloy from which the

specimen was fabricated was solution-annealed by the manufacturer

prior to purchase by NBS. The reported solution treatment for this

alloy is to hold the material at 1975® F (1080® C) for one hour, followed by rapid air cooling. If this alloy is held at temperatures in

the range from about 650® to 900® C, age hardening will occur due to

precipitation of gamma-prime particles from the super- saturated solid

solution. The size of the precipitate is highly dependent on the

aging temperature. After 900® C aging, the gamma-prime particles are quite coarse, having a maximum diameter of several thousand X; if the

solution-annealed alloy is aged only at lower temperatures, the precipitate particles are more abundant and finer in size and cannot be

resolved by the light microscope.

In the course of the thermal conductivity determinations

being discussed, the specimen was cycled twice between room temperature and 1200° C, Since the microstructure of this alloy is dependent

on thermal history, the thermal conductivity of the specimen might

also be expected to change somewhat, due to heat treatment. At the

onset of the measurements, the specimen was in the solution-annealed

state, with the gamma-prime phase in supersaturated solution. After

the specimen was heated above about 650® C, the gamma-prime phase

presumably began to precipitate. As the temperature was further increased, this precipitate became much coarser. Above about 1000® C

the precipitate again went into solution. After completion of testing

to 1200° C, the specimen was cooled to 900° C, at which time a coarse precipitation occurred. Thus, for the first heating cycle, the specimen was in the solution-annealed state up to about 650® C; from 650®

to 900° C, the gamma-prime precipitate was fine at the lower temperatures and became coarser as the temperature was increased; from 1000 ® to 1200° C, the gamma-prime phase was again in solution. For all of

the other tests (first cooling, second heating, second cooling), there
-18-

FIGURE 6. MICROSTRUCTURES (XI 500) OF INCONEL 702

IN THE SOLUTION-ANNEALED STATE (A)

AND IN THE AGE -HARDENED STATE (B)
-19-

was a coarse precipitate at 900® C and below. Since the exact temperatures corresponding to precipitation or solution are not known, the

temperatures mentioned should be considered approximate. Figure 6

shows the microstructures of this alloy in the solution-annealed, as received, condition and in the age-hardened condition after completion

of the described tests,

A cubic equation of least-mean- squares fit was used to represent the thermal conductivity values for the nickel-chromium alloy

as determined by simultaneous solution of the various pairs of tests

taken during the first heating cycle. This equation is

0,1198 + °- 07587 (-T500-) + °- m5 (-T5orj < 9 >

- 0,08265 >

where kg is thermal conductivity (W/cm deg) of the solution-annealed

alloy and T is temperature (®C). To enable closer scrutiny of the

deviations of individual determinations from the curve, these are shown

in Figure 7 as percent departures from the smooth curve plotted against

temperature. The dotted lines bound the region plus and minus two

standard deviations (estimated) from the solid curve. The maximum

departure of any data point from the solid curve is 0,6 percent.

The thermal conductivities obtained when the specimen was in an age-hardened condition can be represented by the equation

ka 0.1179 + 0.1149 + 0,1173 ( 10 )

- 0.05008 9

in the same units as above. This equation is that of least-meansquares fit to the data points obtained during the second heating and

second cooling. The deviations of individual determinations are shown

in Figure 8 as percent departures from smooth curve (10) plotted against

temperature. The dotted lines again bound the region plus or minus

two standard deviations (estimated, using data from second heating and

cooling tests only).
-20

.0 3.0 -

o 20

10

ClX

u

Cl -10 O

S

- 2 ° L

>

-3.0 -

i

1 r

o 8

c o

0

9 o

o

o

O 0

o

o

O FIRST HEATING

200 400 600 800 TEMPERATURE ,°C

1000 1200

FIGURE 7. DEPARTURES OF THE THERMAL CONDUCTIVITY DATA FOR

SOLUTION-ANNEALED INCONEL 702 FROM A SMOOTH CURVE

30

cn

i—o 2.0-

10 -

Ld1

CELU

CLX

UJ

• 0

®

© •

© —*

• • ® V 9

t

U- -l.o- 2

o

£

>

UJQ

-2 0 -

- 3 O'— V FIRST COOLING ® SECOND HEATING SECOND COOLING J i I 200 400 600 300

TEMPERATURE , °C

1000 1200

FIGURE B. DEPARTURES OF THE THERMAL CONDUCTIVITY DATA FOR

AGE-HARDENED INCONEL 702 FROM A SMOOTH CURVE
- 21 -

The apparent difference between the thermal conductivity of

this alloy in the solution- annealed state and that in the age-hardened

state is discussed in another paper being presented to this conference

by the same authors, in which some indication is given of an effect on thermal conductivity arising from a cause apparently different from

gamma-prime precipitation.

Since all of the results presented in Table 1 are from this

laboratory, results obtained on similar alloys by an independent laboratory may also be of interest. Powell and Tye of the National Physical Laboratory recently reported a series of measurements on a group

of nickel-chromium alloys somewhat similar in composition to the alloy

being investigated at NBS. The chemical composition of the NBS alloy

is presented in Table 2 along with the composition of three of the

NPL alloys.

TABLE 2. CHEMICAL COMPOSITIONS OF FOUR NICKELCHROMIUM ALLOYS, WEIGHT PERCENT

Constituent

Inconel

702

Nimonic

75

Nimonic

80(a)

Nimonic

90

Nickel 79.3 77.9(b) 73.7(b) 58.9(b)

Chromium 17.0 20.53 21.0 19.5

Aluminum 2.5 --- 1.2 1.4

Titanium 0.59 0.23 2.5 2.45

Iron 0.36 0.12 0.5 0.41

Silicon 0.19 0.79 0.5 0.65

Copper 0.14 0.06 — 0.14

Cobalt 0.08 — — 16.5

Manganese 0.04 0.27 0.6 0.03

Carbon 0.066 0.126 0.04 0.06

(a) Nominal composition

(b) By difference
SEEBECK

EMF

,

>iV/<Jeg

- 22-

200

- LAUBITZ

O FIRST HEATING

V FIRST COOLING • SECOND HEATING

SECOND COOLING

400 600 800

TEMPERATURE , °C

1000

FIGURE 9. THERMOELECTRIC POWER OF

INCONEL 702 VERSUS PLATINUM

T

9

_j

1200
-25-

Table 5 presents tabulated thermal conductivity values for

the NBS alloy as determined by this investigation and for the NPL

alloys as reported (9). Inspection of Table 3 reveals that the thermal conductivity values for the four alloys have a range, at any given

temperature, of about 13 percent at the lower temperatures and that

the values converge at higher temperatures so that the range at

800w C is only 3 percent.

TABLE 3* THERMAL CONDUCTIVITY OF

FOUR NICKEL-CHROMIUM ALLOYS

Temp. , °C

Inconel 702

Solution- Ageannealed hardenedNPL

Nimonic 75

NPL

Nimonic 80

NPL

Nimonic 90

200 0,142 0.145 0.157 0.138 0.146

300 ,157 .162 .175 .155 .165

400 .175 .179 .191 .168 .184

500 .194 .198 .210 .184 .200

600 .215 .218 .226 .210 .218

700 .237 .239 .243 .235 .237

800 .259 .259 .260 .255 .253

900 .281 .280 ---- .276 —

1000 .303 .300 ---- ---- 1100 .323 .323 ---- —-

1200 .341 .338 - “ - “ - * “ - - - - -

Thermoelectric Power . For each of the tests conducted, emfs were measured between similar legs of the thermocouples in the specimen. Using these data, the thermoelectric power of the specimen

against 90% platinum- -10% rhodium and against platinum was computed.

In Figure 9„ the plotted points give the average values for the thermoelectric power of the Inconel 702 specimen against platinum. The

symbols correspond to those in Figures (7) and (8). For comparison, we have taken the liberty of also showing the "equilibrium" values for

this quantity as presented by M. J. Laubits to this conference.
-24-

Microcrystalllne Glass

Measurements were made of the thermal conductivity of a microcrystalline glass (Pyroceram 9606; product of Coming Glass Works,

Corning, New York) e This material was selected with the hope that it

would come close to meeting the following criteria:

1® It should have a thermal conductivity in the range

0*01 to 0,05 W/ cm deg, comparable to thermal conductivities of thermoelectric materials.

2. It should be homogeneous and isotropic so that a variety of sizes and shapes of specimens can be made

up from one large stock of the material with confidence that all will have the same thermal values.

3. It should be stable up to 1200° C or higher.

4. It should be opaque to thermal radiation up to at least

1200° C.

5. Its cost should not be excessive.

Microcrystalline glass is first formed as a homogeneous glass

(incorporating a nucleating agent) which is transparent, so that any

defects can be readily detected visually. By suitable heat treatment,

the glass is later converted (by Coming) to a polycrystalline solid

almost opaque as a result of the large number of very small crystals.

The properties of such materials have been described in the literature

( 10 , 11 ).

Pyroceram 9606 appears to meet the above criteria fairly well,

the biggest uncertainty being its stability in the range 8QQ M to 1200° C.

There appears to be a small slow change in dimension at the high temperatures, but there is a reasonable possibility that this may not significantly affect the thermal conductivity.

The specimen was fabricated from one of a lot of 2- inch bars

supplied by Coming Glass Works to D. C. Ginninga for measurements of

thermal diffusivity, the results of which are being presented to this

conference by K» W. Flieger. A specimen was also prepared for measure- ment of thermal conductivity in our metals apparatus (7); the results

of these tests are given in another paper being presented to this con- ference by the present authors.
-25-

Description of Specimen . The specimen used for these determinations was in the form of a cylinder 2,540 cm in diameter and 1.269 cm in length* The density of the specimen material was determined at

NBS to be 2*599g g/cm3

at 22.71° C. The ends of the specimen were optically polished so as to be flat to within 1/10 light fringe (approx.

3 x 10"6 cm).

Test Procedure . The specimen was placed between the platinumrhodium hot and cold bars, as shown in Figure I. (The hot and cold

bars had been optically polished to be flat to within 1 light fringe.)

Temperatures along the Pyroceram specimen were determined by means of

three 0.20 -mm butt-welded platinum- 10% rhodium .’platinum thermocouples

pressed into 0.15 mm grooves in the convex surface of the specimen.

These grooves were equally spaced, one being at the mid-plane of the

specimen and the others 0.50 cm above and below the mid-plane.

Thermal conductivity measurements were made in increasing

order of temperature from 200° to 5Q0 W C at 100-degree intervals. The

specimen was then heated to 600° C, but no data were taken (due to a break in the water mains which supplied the cooling water); the specimen was then cooled to room temperature. On second heating, measure- ments were made at 300 ° C and at 100-degree intervals from 600° to

1000* C. The specimen was held at 1000° C for about 275 hours, sev- eral sets of data being taken during this time. Measurements were then

made in decreasing order of temperature at 900°, 600°, and 300° C.

Each conductivity determination involved an "isothermal" test and a "gradient" test.

Results of Thermal Conductivity Tests, The smoothed values

of thermal conductivity and thermal resistivity of the Pyroceram specimen are shown in our other paper. The values plotted tbire were calculated from the equation

w = 1/k 26.7 + ( 11 )

where w is thermal resistivity (cm deg/W) , k is thermal conductivity

(W/cm deg), and T is temperature (®C). Equation (11) is the linear

equation of least-mean-squares fit to the thermal resistivity values

obtained from the first and second heating cycles. The percent deviations of the experimental resistivity values from the smooth curve are shown in Figure 10. The points connected by a line (or in close juxtaposition) represent the individual values obtained in each pair of
DEVIATION

OF

EXPERIMENTAL

POINTS

, %

DEVIATION

OF

EXPERIMENTAL

POINTS,

-26-

FIGURE 10. DEPARTURES OF THE THERMAL RESISTIVITY DATA

FOR PYROCERAM 9606 FROM A STRAIGHT LINE

FIGURE 11. DEPARTURES OF THE THERMAL CONDUCTIVITY

DATA FOR PYREX 7740 FROM A SMOOTH CURVE
-27-

tests for the two thermocouple spans in the specimen. The horizontal

displacement of two connected points is indicative of the temperature

gradient in the specimen; for most of the tests this gradient was 30

to 45 deg/cm.* The dotted lines in Figure 10 bound the region plus or minus two standard percent deviations (estimated; first and second

heating only).

The data points obtained on second cooling are in substantial

agreement with the data taken before the specimen was held at 1000° C,

thus indicating that no appreciable change in the. thermal conductivity

of Pyroceram 9606 was perceived as a result of this heat treatment.

Borosilice t e Glass

Measurements were made of the thermal conductivity of a sam- ple of borosilicate glass (Pyrex Code 7740). This material is being

investigated as a possible thermal conductivity standard.

Description of Specimen . The specimen used for these determinations was in the form of a right circular cylinder 2.540 cm in diameter and 1.517 cm in length. The specimen was cut from a piece of oneinch commercial plate indicated by the supplier to be from the same original sheet as were specimens prepared for testing in the NBS guarded

hot plate (see earlier paper presented to this conference by Robinson

and Flynn). The ends of the specimen were optically polished so as to

be flat within 1/10 light fringe and then were given a "double-opaque"

coating of gold by evaporation technique. The flat surfaces ensured

good contact with the hot and cold bars; the gold coating minimized

direct radiative transmission through the specimen. After grinding to

size but prior to optical polishing, the specimen was carefully annealed

by the NBS Glass Section; a 60° prism cut from adjacent material was annealed to the same schedule.** The density of the prism was determined as 2.225g g/cm3 at 24.3° C prior to annealing and as 2.224^ g/cm3

at 25.0° C after annealing. The refractive index of this prism was measured on a precision spectrometer at controlled room temperatures

*Exceptions to this are as follows:

First hea t ing :

Second heating :

200° C, 20 deg C/cm (both pairs);

300 0 C, 20 deg C/cm (one pair);

600° C, 60 deg C/cm (one pair);

600® C, 10 deg C/cm (one pair);

1000° C, 15 deg C/cm (two pairs);

1000® C, 70 deg C/cm (one pair).

**The specimen and prism were heated to 570® C in one hour, held at 570‘

for two hours, coolad at 1-1/2 deg/hr to 435° C (90 hours), cooled at

10 deg/hr to 355® C (8 hours) and then furnace-cooled.
- 28 -

near 25 ®C and found to be 1.47257 (±0.00003) for the D lines of sodium

(5893 A) prior to annealing; subsequent to annealing the index was

found to be 1.47211 (±0.00003).

Test Procedure . The specimen was placed between the platinum- -rhodium hot and cold bars, as shown in Figures 1 and 2. Temperatures along the Pyrex specimen were determined by means of three 0.20-

mxn butt-welded platinum--107o rhodium: platinum thermocouples located in

0.22-mm grooves in the convex surface of the specimen. These grooves were equally spaced, one being at the midplane of the specimen and the

others 0.50 cm above and below the mid-plane.

Thermal conductivity measurements were made in increasing

order of temperature from 100° to 400° C at 50-degree intervals (Pyrex

glass should not be taken above about 450° C unless it can be very,

very slowly cooled, since its properties may undergo change). Measurements were then made in decreasing order of temperature at 3^0° , 200°,

and 100° C. Each conductivity determination involved an "isothermal"

test and a "gradient" test.

Thermal Conductivity . The thermal conductivities obtained

can be represented by the equation

where now k is expressed in mW/cm deg and T in °C. The percent departures of the experimental thermal conductivity values are shown in

Figure 11. The points connected by a line represent the individual

values obtained in each determination for the two thermocouple spans

in the specimen; the mid-point of the line represents the mean result

of the determination. The dotted lines in Figure 11 bound the region

plus or minus two standard ceviations (estimated). The points obtained

on cooling agree with those taken on heating.

A large number of data points were taken for this specimen in

order to study the effects of varying the guarding conditions in the

apparatus and the temperature gradient in the specimen. The consistency of the data, as indicated in Figure 11, is excellent, especially

when it is considered that for some of these tests the corrections due
-29-

to a net heat gain or loss to the insulation were purposely made as high as ten or fifteen percent of the total heat flow. This specimen

has not been removed from the apparatus--additional tests are planned.

In particular, the effects of using different insulations around the

specimen will be studied.

ESTIMATE OF PRECISION AND ACCURACY

We would like to introduce this section with a quotation from

a recent paper (12) by Churchill Eisenhart of the National Bureau of

Standards

"By the precision of a measurement process

we mean the degree of mutual agreement characteristic of independent measurements of a single

quantity yielded by repeated applications of the

process under specified conditions; and by its

accuracy the degree of agreement of such measure- ments with the true value of the magnitude of the

quantity concerned ..... accuracy has to do

with closeness to the truth ; precision , only with

closeness together."

We have not yet completed an analysis of all the uncertainties

involved in the apparatus being described. The precision of the measurements for a given specimen and installation is indicated by the deviations of the data points from smoothed curves as shown in the results

sections above. This component of the precision can probably be improved by further refinement of the corrections and calculation procedure. This is only one aspect of precision, however. Precision also

involves the question of how well the results would be reproduced if a new specimen, identical in thermal conductivity, were installed and

measurements made. We have not yet done this and hence cannot comment

meaningfully as to this aspect of precision, although we feel a reasonable confidence in this regard®

An estimate of accuracy entails evaluation of all systematic errors contained in the measurements. Again quoting Eisenhart (12):

"The overall systematic error of a measurement

process ordinarily consists of elemental systematic

errors due to both assignable and unassignable causes. Those of unknown (not thought of, not yet identified, or as yet undiscovered) origin are always to

be feared; allowances can be made only for those of

recognized origin."
-30-

With this in mind, the authors list all of the possible sources of experimental errors which they were able to imagine. In the

informal atmosphere of this conference , we invite your comments regarding additional sources of uncertainty and regarding the possible magnitudes of those which are listed®

The "true" value of the thermal conductivity, assuming linear

heat flow, is given by

k „ _ (Q r.A'lk.

0 A (AT - AT') (13) (A7)

where Q and Q' are the total "true" heat flows through the cross-sectional area A in two tests at substantially the same mean temperature,

and AT and AT' are the corresponding "true" temperature differences

between two parallel planes, normal to the direction of heat flow, a distance L apart (see Appendix I). By independent consideration of the

uncertainty in each quantity appearing in Equation (13), an estimate

can be made of the resultant uncertainty in thermal conductivity, as- suming this equation to be valid. Consideration must also be given to

the possibility that heat flow in the specimen may not conform to that

assumed in Equation (13) 5 i. e. , heat flow in the specimen may not be

linear.

Cross - Sectional Area . All thermal conductivity specimens are machined or ground to the form of a right circular cylinder of uniform

diameter. The diameter of the specimens is measured with sufficient

accuracy that the cross-sectional area at room temperature is uncertain

by less than 0.1 percent. No correction for thermal expansion has been

applied to the results given in the present paper; the accuracy of such

a correction is dependent on the accuracy of the coefficient of linear

thermal expansion used and cannot properly be charged against the

method of measuring thermal conductivity.

Thermocouple Separation. At discussed previously, all specimen thermocouples are pressed into thin transverse grooves in the con- vex surface of the specimen. The center- to-center distance between the

outermost grooves is determined to within 0*2 percent in the case of

metal specimens where this is about 4 cm, and to within 0.5 percent in

the case of the short specimens where the separation is about 1 cm. The butt-welded thermocouples are pressed firmly into the grooves and

it is felt that the effective separation between two thermocouples very

nearly coincides with their center-to-center separation. The maximum

possible uncertainty in effective thermocouple separation is one groove
“31-

width, i* e. , 0.8 percent for a 4 cm span with a 0.03 cm groove and

2.0 percent for a 1 cm span with a 0.02 cm groove; this is considered

to be an extremely unlikely occurrence, since it implies that the effective thermocouple positions coincide with the outermost edges of the

grooves. A more probable uncertainty level would be intermediate between these extremes. We have arbitrarily doubled the center-to-center

uncertainties to allow for possible variations of the effective locations of the thermocouples within the grooves; the estimated probable

uncertainties in thermocouple spacing are, then, 0.4 and 1.0 percent

for the long and short specimens.

Temperature Difference ® There are several sources of uncer- tainty involved in determining the overall uncertainty in the temperature difference between two positions in the specimen. We invite your

attention to the paper (14) given at last year’s conference by T. M»

D&uphinee for a comprehensive review of the factors involved. As shown

in Appendix I and pointed out by Bauphinee, the simultaneous solution

of two tests largely eliminates the effects of variations between individual thermocouples. All that is required i® that the slope of the

temperature versus em£ curve be essentially the same for the critical

thermocouples. That this is @0 is indicated by the fact that during an

’’isothermal test" at 1000®G all specimen thermocouples normally agreed

within much better than 0.1%,

It is difficult to estimate the uncertainties involved in

converting emf readings into temperatures. Representative thermocouples were given a secondary calibration by the NBS Temperature Physics Section, which with innate conservatism estimates that uncertainties in

the calibration are not more than 0.5 degree in the range 0° to 1100 6 C.

However, a plot of departures of the calibration points from the NBS

standard tables is smooth within 0*2 degree. The departures of the

calibration points from Equation (4) , which was used for conversion of

all emfs into temperature, are also less than 0.2 degree. We estimate

that the uncertainty in calibration is such that conversion of an emf

difference to a temperature difference introduces an uncertainty of

less than 0.4 percent® Since all thermocouples are pressed firmly into

the specimens and are carried around the specimen for some distance in an isothermal region, and since all thermocouples are subjected to es- sentially identical conditions, the errors due to lead conduction should

be negligible and are certainly less than 0.1 percent. For a 10-degree

temperature interval, the emf difference was read with less than 0.3

percent uncertainty (for a larger temperature interval this uncertainty

is less). Combining these uncertainties according to standard propagation of error formulae ( 13 ), the estimated uncertainty in determining a 10-degree temperature difference, is 0,5 percent.
-32-

Heat Flow Through the Specimen. The volt box, shunt box, and

potentiometer used in measuring the electrical power input were each

calibrated with an uncertainty of 0.01 percent. A correction is made

for current flow through the volt box. Hence the total power input to

the heater and leads above the potential taps is measured with an un- certainty of less than 0.05 percent.

About 0.5 percent of the total power is generated in the

leads above the potential taps and below the uppermost junction of the

differential thermocouple in the hot bar support column. When the two

junctions are matched in temperature, substantially all of the heat

generated in the leads above the potential taps flows upward toward the

specimen and that generated in the leads below the potential taps flows

downward. The uncertainty in electrical power input due to heat generation in the current leads is less than 0.3 percent.

As mentioned in the description of the apparatus, there are

two absolute thermocouple stations in the hot bar support column and

also a differential thermocouple. The former are used in analyzing

heat flows to and from the surrounding insulation; the latter is automatically controlled at a null reading in order to prevent heat flow

up or down the hot bar support column. It was originally felt that the

differential thermocouple would be less subject to errors arising from

contamination and hence this couple has been used to evaluate the

slight heat, flows along the support column. The simultaneous solution

procedure used, however, should substantially eliminate the effects of

individual variations between the two absolute (i. e. , reference temperature at the ice point) thermocouples* In general, the agreement

between readings of the temperature difference along the support column

using the differential thermocouple and using the two absolute thermocouples is not as good as would be desired. It was thought that the

uppermost junctions were sufficiently removed from the specimen heater

to be in a substantially isothermal region. In view of the indicated

disagreement, however, it appears that there may be angular or radial

temperature variations in the support column at the plane of the upper

junctions. There is a possibility that the powder insulation in the

support column has settled, leaving a cavity in which there is radiation

and possibly convection. We intend to investigate this difficulty, both

experimentally and analytically. In the meantime it is necessary to

assign an additional 1 percent uncertainty due to this unresolved difference.

The problem of heat flow in the insulation surrounding the

hot bar- - specimen- -co id bar assembly was discussed briefly above. In

order to evaluate this flow, Equation (2) is used, with the boundary

conditions previously discussed. Temperature distributions along the

outer surface of the bar assembly and along the inner surface of the

coaxial guard cylinder are derived by fitting polynomials through the
-33 -

measured temperatures along these surfaces. It is believed that these

functions serve quite well in defining longitudinal temperature distribution between measured points. The use of logarithmic functions to

define the radial temperature distribution across the ends of the hollow cylinder of insulation is believed to be a superior procedure to

that of attempting to accurately measure radial temperature variation

in a loose-fill insulating material. A mathematical analysis is planned to investigate how sensitive the mathematica! model is to the particular end boundary condition assumed.

Temperatures of the guard cylinder are actually measured at

locations within the wall of the guard and hence may not correspond

exactly to the interior surface temperature. Such radial differences,

which result in a slight error in the average temperature of the interior surface temperature of the guard, are essentially the same for two

tests at about the same mean temperature and hence cancel out under

simultaneous solution. Similarly, angular variations in temperatures

around the guard cylinder cancel out if they are the same for both

tests. Such is not the case for uncertainties in the longitudinal

thermocouple positions, since the temperature distribution along the

guard cylinder is different in the two testa. As Figure 2 shows, the

bottom of both the hot bar support column and of the guard cylinder are held in a fixed position and the tops are free to move due to thermal

expansion. Since the hot bar support column, the rod above the cold

bar, and the guard cylinder are all constructed of the same material,

they expand together such that relative longitudinal positions are quite well maintained. An investigation of the sensitivity of the apparatus to alignment of components will be made.

In general it is not possible, or at least it is not practical, to manage the temperature distribution along the guard cylinder in

such a manner that there is no augmentation or depletion of longitudinal heat flow in the bar assembly. It is necessary, therefore, to have

a rather good knowledge of the thermal conductivity of the powder insulation which fills the region between the bar assembly and the guard

cylinder. The conductivity of this insulation has been measured separately (5). In addition, it is possible to conduct several tests in

which the correction for heat flow through the insulation is varied over a small range about zero. From these data the thermal conductivity

of the insulation can be derived using consistency criteria. This has

been done in the case of Pyrex glass and significantly improved the

precision, and hopefully the accuracy, of the results. In fact, we feel

that thermal conductivity values inferred in this manner for the insulation are as good as some values found in the literature resulting

from direct measurement.

As can be appreciated, the analysis of heat flow in the insulation is rather complex. Pending further analysis, we will set at 1.5

percent the overall uncertainty introduced into the measurement by
-34-

possible departures between the experimental system and the mathematical

model, although w® feel it is considerably less than this.

Combining all of these uncertainties, the total heat flow

through the specimen is believed to be uncertain by less than 2.0 percent. It is believed that this uncertainty can be considerably reduced

by further analysis.

Departure From Linear Steady- State Heat Flow. In substantially all acceptable tests, the average temperature variation with time in

the specimen Goes not exceed 0.002 deg/min over the duration of a test,

a slow enough variation that heat absorption is less than 0.3 percent

of the total power input for the powers used*

Non-linearity of heat flow in the sample would be caused by

excessive heat exchanges with the powder insulation in the region of

the specimen, or by non-uniformity of contact between the specimen and

the hot and cold bars in the case of the short specimens, and by non- uniformity of heat flow near the specimen heater in the case of the long

specimens. As far as we know, these effects are negligible but will all

be investigated prior to formal publication.

Overall Accuracy. Combining all the known sources of possible

uncertainty, according to standard propagation of errors formulae,

leads to an overall uncertainty in the thermal conductivity values of

less than 2.5 percent for both short and long specimens. This estimate

is substantiated somewhat by agreement of results measured in this apparatus with results attained in our laboratory by other methods.
-35-

APPENDIX I

For one-dimensional steady-state heat flow, the total heat

flow, Q, through the specimen is

Q kAdT

dz

(Al)

where k is thermal conductivity,

A is the cross-sectional area of the specimen,

T is the temperature,

2 is the longitudinal coordinate*

For moderate temperature ranges, the thermal conductivity of the specimen is assumed to vary linearly with temperature; then (Al) becomes

Q - - k

0

ACi + a(T - T

0 )}~' , (A2)

where is the thermal conductivity at an arbitrary reference temperature, Tq, and CZ is its corresponding temperature coefficient.

Integration of (A2) yields, assuming Q to be constant,

k A

Q . -JL_ AT[1 + 05 (T - T

Q)J

where

AT Tj. - Tg

To (T

x + Tg)/2

, (A3)

, (A4)

T^ and Tg being the temperatures at two positions a distance L apart.

Consider two tests at approximately the same mean temperature;

let the heat flow in one test be given by (A3) and that in the other

test by a similar expression. The difference between these heat flows

will be

k A

Q - Q® a --“[AT - AT' 4* a{AT(f “ T

Q

) - AT® (T' “ T

Q

)}] , (A5)

where quantities of one test are distinguished from those of the other

by use of primes. The expression in braces in (A5) can be set equal to zero by proper choice of the reference temperature, T^,
If

MT - al' T

8 0 - “ AT “ (A6)

the thermal conductivity at the indicated reference temperature, Tq,

is given by

k .cq-q’jL

0

83 A (AT AT')

(A7)

Equation (A7) gives thermal conductivity in terns of the

"true" values of heat flows, temperature differences, and dimensions.

Mow let

Q . q “ s Q' aa q' — a *1“ *1“ S

1 *l' - V - «1 (A8)

Ts' ‘S' V “

‘a’ - *2 9

where the capital letters designate "true" values of the parameters;

the corresponding lower case letters are the measured values for these

parameters; and s, e, , and are the associated systematic errors.

It is assumed that, for two tests at about the same temperature, the

systematic errors associated with a particular parameter are the same;

experience indicates that this is a good approximation.

From (A7) and (A8) the thermal conductivity Is given by

k ^ fa T ,<QLK

0 A (At ” At ')

corresponding to the mean temperature

T

o

88 tQ * €

0

where Ati - At'r

At - At 1

6

0 "

At -

(A9)

(A10)

€

i

(t

l -

At'

(All)
"3

?“

At t

l t 2

(t

x + t 2)/2

(A12)

Equation (A9) gives thermal conductivity in terms of the

measured values of the parameters, all systematic errors of the type

being discussed having been eliminated. The effective mean temperature

will be in error by however, for most materials the effect of a slight error in mean temperature is negligible.
-38 -

REFERENCES

(1) Flynn, D® R. , "Thermal Guarding of Cut-Bar Apparatus," unpublished

proceedings of the Conference on Thermal Conductivity Methods held

at Battelle Memorial Institute, Columbus, Ohio (October 1961).

(2) Robinson, H. E. , "Thermal Conductivity Reference Standards," unpublished proceedings of the Second Conference on Thermal Conductivity

held at Division of Applied Physics, N. R. C., Ottawa (October 1962).

0) West, E® D. , and Ginnings, D. C», "Automatic Temperature Regulation

and Recording in Precision Adiabatic Calorimetry," Rev. of Sci.

Xnstru, , 28 (1957) pp 1070-1074.

(4) Rceser, W. F. , Thermoelectric Thermometry, J. Appl. Phys. , _11 (1940)

pp 404-405; also in Temperature . Its Measurement and Control in

Science and Industry . Reinhold Publishing Corporation, New York

(1941) p 202.

(5) Flynn, D. R, , "A Radial Flow Apparatus for Determining the Thermal

Conductivity of Loose-fill Insulations to High Temperatures," J.

Research NBS, 67C (Eng. and Instru.) No. 2 (April-June 1963) pp

129-137.

(6) Roeser, W. F. , op. cit., 389-390; 181-182.

(7) Watson, T. W. , and Robinson, H. E. , "Thermal Conductivity of Some

Commercial Iron-Nickel Alloys," ASME Jour, of Heat Transfer

(November 1961) pp 403-408.

(8) Flynn, D. R. , and Didion, D. A., "Steam Calorimeter Apparatus for

Refractories," unpublished proceedings of the Conference of Thermal

Conductivity Methods held at Battelle Memorial Institute, Columbus,

Ohio (October 1961).

(9) Powell, R. W. , and Tye, R. P. , "Thermal and Electrical Conductivities of Nickel-Chromium (Nimonic) Alloys," The Engineer (April 29,

1960) pp 729-732.

(10) Stookey, S. D. , "Glass-Ceramics," Chem, Eng. News, (June 19,

' 1961) p 116.

(11) Stookey, S. D. , "Glass-Ceramics," Mech. Eng., 82 (October 1960)

P 85.

(12) Eisenhart, Churchill, "Realistic Evaluation of the Precision and

Accuracy of Instrument Calibration Systems," J. Research NBS, 67C

(Eng. and Instru.) No. 2 (April-June 1963) pp 161-187.

(13) Davies, 0, L. , Statistical Methods in Research and Production . Hafner Publishing Company, New York (1957), 3r^ ed. , pp 40-41.

(14) Bauphinee, T. M. , "Thermocouple Measuring Techniques," unpublished

proceedings of the Second Conference on Thermal Conductivity held

at Division of Applied Physics, N. R. C* , Ottawa (October 1962).

1i))