- C'lg P

NATIONAL BUREAU OF STANDARDS REPORT

7135

THERMAL CONDUCTIVITY OF SEMICONDUCTIVE SOLIDS;

METHOD FOR STEADY-STATE MEASUREMENTS ON

SMALL DISC REFERENCE SAMPLES

Interim Technical Report

Covering Period

February 23, 1959, to March 31, 196

1

by

D. R. Flynn and H. E. Robinson

Report to the

Bureau of Ships

Department of the Navy

Washington, D. C.

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NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1006-20-104-61 April 28, 1961 7135

THERMAL CONDUCTIVITY OF SEMICONDUCTIVE SOLIDS;

METHOD FOR STEADY-STATE MEASUREMENTS ON

SMALL DISC REFERENCE SAMPLES

Interim Technical Report

Covering Period

February 23 i 1959, to March 31', 1981

by

D* R. Flynn and H. E. Robinson

Heat Transfer Section

Building Research Division

Report to the

Bureau of Ships

Department of the Navy

INDEX NO. S-R007 12 01 « Task No. 800

(BuShips 1700S-64-5-59). Code 34-2B

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gress accounting documents

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epared If that agency wishes

IL S. DEPARTMENT OF COMMERCE

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.
TABLE OF CONTENTS

Page

ABSTRACT 1

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

2. METHOD OF APPROACH. ...... ..... 2

2.1 Parameters Involved. ........ 2

2.2 Cut-Bar Method ......... 3

2.3 The Absolute Cut-Bar Method *+

3. RESULTS OBTAINED USING THE ORIGINAL

PROTOTYPE APPARATUS. 5

3.1 Description of Pyrex Specimens ... 5

3.2 Test Results for Pyrex Glass 6

b. MODIFIED PROTOTYPE DESIGN ....... 10

5. RESULTS OBTAINED USING THE MODIFIED APPARATUS ... 10

6. ADDITIONAL RESULTS. . 13

6.1 Addition of Thermocouples 13

6.2 Measurements on Pyroceram. 14-

7. FUTURE ACTIVITIES ................. 15

APPENDIX A

GUARDING OF A THERMAL CONDUCTIVITY APPARATUS A-l

OF THE CUT -BAR TYPE. .................

APPENDIX B

CALCULATION PROCEDURE ................. B-l

LIST OF FIGURES

Figure 1. A Schematic Cut -Bar Apparatus Utilizing

Reference Bars to Determine Heat Flow

Through the Specimens. 17
LIST OF FIGURES (cont'd)

Pagj

Figure 2. The Original Prototype Absolute Cut-Bar

Apparatus Utilizing the Measured Power

Input to an Electrical Heater to Determine Heat Flow Through the Specimen 18

Figure 3. Four of the Pyrex Glass No. 7740 Specimens

Used in the Evaluation of Contact

Resistance 19

Figure 4. Total Thermal Resistance, Including Contact

Resistance, for Several Specimens of Pyrex

Glass No. 7740 20

Figure 5. Over-all Thermal Conductivity of Pyrex Glass

No. 7740, No Corrections Having Been Made

for the Contact Resistance of Three Types

of Interfaces. .... 21

Figure 6

.

Total Thermal Resistance, Including Contact

Resistance of Platinum Foil Interfaces, of

Two Specimens of Pyrex Glass No. 7740. . . . 22

Figure 7. Thermal Conductivity of Pyrex Glass No. 7740

as a Function of Temperature, Including

Corrections for Contact Resistance 23

Figure 8. Thermal Conductivity of Pyrex Glass No. 7740

(tentative data) Compared With Literature

Values. The Circles Indicate Additional

Results Derived from the Data of Figure 4. . 24

Figure 9 . The Modified Prototype Absolute Cut -Bar

Apparatus 25

Figure 10. The Nickel-Chrome Alloy Specimen Used to

Check the Modified Prototype Apparatus . . . 26

Figure 11. Thermal Conductivity of a Nickel-Chrome

Alloy as Determined Using Two Methods of

Thermocouple Installation in the Specimen. . 27

Figure 12. Departures of the Data for the Nickel-Chrome

Alloy from the Smoothed Data from the NBS

Metals Apparatus 28

Figure 13. Thermal Conductivity of Pyroceram Code 9606

Versus Temperature, Before and After

Correcting for Contact Resistance 29
THERMAL CONDUCTIVITY OF SEMI -CONDUCTIVE SOLIDS;

METHOD FOR STEADY-STATE MEASUREMENTS ON

SMALL DISC REFERENCE SAMPLES

D. R. Flynn and H. E. Robinson

ABSTRACT

An absolute cut-bar method of measuring the

thermal conductivity of solids is described,

suitable for small specimens of semiconduct ive

materials. Measurements have been made at temperatures from 100° to 800°C on Pyrex, Pyroceram,

and a nickel-chrome alloy, and representative

tentative results are given. The data for the

nickel-chrome alloy, over the common temperature

range 120° to 600°C, agree with the smoothed data

from measurements made on the same material by an

independent absolute method, with a standard

deviation of 0.7 percent. The data for Pyrex and

Pyroceram may be subject to errors because of the

uncertainties introduced by thermal contact resistance at the surfaces of the specimens.

It is shown that the use of a matched guard

in a cut-bar apparatus does not reduce extraneous

heat exchanges between the bars and specimen and

the surrounding powder insulation to the small

magnitude desirable for good accuracy of measurement. A mathematical analysis of the system indicates a superior guarding condition which greatly

reduces these extraneous heat flows, and which

was used in the measurements mentioned.

Using the knowledge acquired by these investigations, a final model of the apparatus is being

built for measurements on small specimens (1/2-in.

by 1-in. diameter discs) at temperatures in excess

of 1200°C.

1. INTRODUCTION

The thermal conductivity of thermoelectric materials is

of great importance in determining the performance of a thermoelectric device. If such a device is to have a high

efficiency, the thermal conductivity of the thermoelectric

elements must be as low as possible in the temperature range

in which they are to operate. Reliable thermal conductivity
- 2 -

data are required at temperatures ranging from room temperature or below to 2000°C, since it is advantageous to operate

a thermoelectric generator at as high a temperature as

possible

Reliable thermal conductivity data at elevated temperatures are not generally available, partly because of the

difficulties of making such measurements. Many semiconductor materials of interest are available only in small samples because of the difficulty of obtaining them with

adequate uniformity. Consequently, it is necessary that a method of measuring the thermal conductivity of such

materials be one suitable for small specimens and capable

of use at high temperatures.

It is the purpose of this investigation to develop a method and apparatus for steady-state 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 discs), with the objective of providing

samples for use by other laboratories as thermal conductivity reference specimens in connection with their

measurements on solid semiconductors.

This report discusses the apparatus which is being

developed for this purpose and gives a summary of the

findings of this investigation to date.

2. METHOD OF APPROACH

2.1 Parameters Involved

The steady-state measurement of thermal conductivity

requires a determination of the heat flux through the

specimen and of the temperature gradient within the specimen.

For thermal conductivity measurements of the so-called

absolute type, the heat flux is determined either by

supplying heat to the specimen at a known rate by means of

an electrical heater or by calorimetrically measuring the

heat flow leaving the specimen. For thermal conductivity

measurements of the comparative type, the temperature

gradient in a known standard is used to determine the heat

leaving or entering the specimen. Since available high

temperature insulating materials have a thermal conductivity

only one or two orders of magnitude less than that of semiconductor materials, it becomes increasingly difficult to

insulate against extraneous heat flows as temperature

increases. Thus, the problem is not only to measure the

rate of heat input or putput , but also to insure that the
- 3

-

heat flow measured coincides with that through the specimen,

or that adequate corrections for extraneous heat flows are

made

.

Because of their small size, thermocouples are

generally the most satisfactory elements for measuring

temperatures in a specimen or system with minimum disturbance to the heat flow pattern. However, in specimens

of small volume, even a thermocouple may introduce some

disturbance, and for this reason many investigators have

preferred to deduce the temperature at the contact surface

of a specimen by extrapolating temperatures observed in

the contacting body. If this is done, the difficult problem of the thermal resistance at the contacting interfaces

is encountered. In attempting highly accurate measurements,

the selection of the best method to use depends upon the

parameters of the system, and must be guided by expermental

findings

.

2.2 Cut-Bar Method

The method most frequently used for determining the

thermal conductivity of small solid specimens is a special

form of the comparative method, sometimes known as the

cut-bar method. The more important features of this design

are illustrated in Figure 1. The disc-shaped specimen is

interposed between two rods of known thermal conductivity,

along which heat is caused to flow by raising the temperature at the far end of one bar. The rod-like assembly is

surrounded by an axially concentric guard cylinder; the

space between them is filled with a powder-type thermal

insulation to restrict heat interchanges. Thermocouples

in the "known'8 bars indicate temperatures from which the

heat flow and temperature at each of the ends contacting

the specimen can be calculated.

A study of the characteristics of the method led to

the following principal conclusions.

(1) A major practical limitation of the general

method is that, because of the temperature gradient required

in the bar, the maximum mean temperature of the specimen is

likely to be one hundred or more degrees C lower than that

at the hotter end of the hotter bar. It is undesirable,

in view of the importance of high temperature in this program, to have the maximum specimen temperature so much

reduced below the maximum apparatus temperature obtainable

with the electrical heaters used.
(2) The proper guarding of a cut-bar type of

apparatus is discussed in Appendix A. It is shown that

the conventional method of guarding such an apparatus

(i.e., maintaining the same temperature distribution along

the guard cylinder as along the reference bars and specimen, so that guard temperatures match those of the

reference bars at corresponding longitudinal positions)

does not eliminate extraneous heat exchanges between the

reference bars and specimen and the surrounding powder

insulation. While it is theoretically possible to place

a temperature distribution on the guard which will eliminate parasitic heat flows, to do so would be very difficult. A mathematical analysis, such as that in Appendix

A, of a practical compromise guarding condition would

serve for evaluating the heat gains or losses into the

powder insulation reasonably well. However, these heat

exchanges affect the heat flows within the bars, which are

used as heat flow meters, and an accurate treatment of

their effect requires solution of the four-body composite

system involved. Mathematically, this problem is intractable because of the many continuity conditions which must

be satisfied. The importance of the problem increases as

the conductivity of the powder insulation increases with

temperature

.

For these major reasons, an alternative method was

devised, capable of measurements on disc specimens by an

absolute method.

2.3 The Absolute Cut-Bar Method

In order to meet the objections to the cut-bar method

given in the preceding section, it was decided to utilize

the general cut-bar arrangement, but to make an absolute

power measurement rather than to use the bars as references

for determining the heat flow. This permits placing the

heater much nearer the specimen so that the specimen mean

temperature is only slightly lower than the maximum

obtainable apparatus temperature. Furthermore, since the

bars are no longer used as heat flow meters, the elimination of extraneous heat flows from them is not so critical.

It was decided to build a prototype apparatus first,

with which to study the various parameters involved.

Figure 2 shows schematically the prototype apparatus as it

was first constructed. The disc specimen is interposed

between the hot bar and cold bar shown in the figure. The

power input to the hot bar heater is accurately measured

potentiometrically , and is adjusted to raise the temperature on the hot side of the specimen by the desired amount

over that on the cold side. The desired temperature on the
- 5

-

cold side of the specimen is established by regulating the

cold bar heater at some temperature in excess of the

coolant temperature.

The temperature of the hot and cold guards are controlled by their respective heaters. By varying the guard

temperatures, extraneous heat flows can be regulated.

(The effect of various guarding conditions is shown in

Appendix A.

)

Heat flow to or from the hot bar along the hot bar

support is prevented by regulating the heater at the base

of the support to maintain zero temperature difference

between differential thermocouple junctions in the support.

The prototype apparatus was constructed of stainless

steel. Calibrated 28-gage chromel-alumel thermocouples

were used for temperature measurement. The method of computing thermal conductivity from the data acquired is outlined in Appendix B.

3. RESULTS OBTAINED USING THE ORIGINAL

PROTOTYPE APPARATUS

3.1 Description of Pyrex Specimens

Pending availability of zirconia and alumina samples to

be furnished, exploratory development tests of the prototype

apparatus were conducted using specimens of Pyrex No. 77b0.

This material was chosen because of its homogeneity and low

thermal conductivity, and because some data on its conductivity are available in the literature.

While optically-f lat specimens were being prepared,

tests were conducted on several Pyrex specimens having

factory-polished surfaces. These tests were valuable in

refining the operation of the control assembly. They also

provided information regarding contact resistance and parasitic heat flows, but nothing that was not better determined

using the specimens described below. Hence, the results of

these tests are not included in this report.

To evaluate contact resistance and, at the same time,

study the effects of heat flow into the powder insulation,

five specimens were prepared of thicknesses in the range

0.25 cm to 2.25 cm. All specimens were cut from the same

piece of Pyrex and ground to a diameter of 2.5^0 ± 0.001 cm.

The contacting surfaces were optically polished to be flat

to within 1/10 fringe of sodium light (approx. 2.5xl0~ 6 cm)

and parallel to within 0.001 cm. Four of these specimens

are shown in Figure 3 .
- 6

-

3.2 Test Results for Pyrex Glass

Previous to the tests mentioned in Section 3.1, it was

thought that heat exchanges between the bars and the powder

insulation would be minimized if the guard temperatures

were matched to the bar temperatqres at points near the

specimen. The mathematical analysis given in Appendix A

showed, however, that matching these temperatures does not

yield a minimum heat exchange with the insulation, because

of longitudinal heat flow in the insulation. This analysis

indicated the possibility of a superior procedure, i.e., to

raise the hot guard to a temperature greater than that of

the hot bar and to control the cold guard at a temperature

less than that of the cold bar. The mathematical analysis

predicts that at properly selected guard temperatures, the

heat flow through the specimen will be equal to the measured

power input. All tests discussed hereafter utilize this

procedure. Because the exact guard temperatures desired

cannot always be obtained in practice, corrections are made

for the small amount of parasitic heat flow calculated

mathematically (see Appendices A and B).

Representative results obtained in the study of contact thermal resistance are shown in Figure b. The total

resistance per unit area, R-p , is plotted against specimen

thickness, L. The four curves shown represent specimen

mean temperatures of 200°, 300°, 400° , and 500°C, respectively. For the tests shown, diatomaceous earth was used

as insulation, the pressure exerted on the specimen was

10 kg/cm 2 (approx. 142 psi), and the specimen temperature

gradient was approximately 25°C/cm.

The total thermal resistance per unit area should plot

linearly versus the specimen thickness, provided the contact

resistance is substantially the same at a given temperature

and the conductivity of the specimens does not vary. Under

these conditions, the points for a given mean temperature

should fall on a straight line having an ordinate intercept

equal to the total contact resistance of the two interfaces.

The inverse slope of the line is equal to the thermal conductivity of the specimen at the mean temperature concerned.

The lines shown are least mean square lines fitted for

the four specimens 0.75 cm and greater in length. The data

obtained at 200° and 300°C conform closely to straight lines

for all five specimens, while at 400° and 500°C there is

more departure from linearity. The resistance of the 0.25-

cm specimen is substantially greater than that indicated by

the corresponding line, which may be due to one or both of

two possible causes: a) the error involved in determining
-7-

a small temperature difference as the difference between two

high measured temperatures, b) a decrease in radiative heat

transfer near the ends of a specimen, which becomes proportionately more significant as specimens become shorter, and

as the specimen mean temperature increases.

Because of the uncertainty of the resistances obtained

for the 0.25 cm specimen, the lines were drawn ignoring

them. However, at 200° and 300°C, the points for this specimen conform closely to the respective lines, although the

departure becomes greater at 400° and 500°C.

The ordinate intercepts of the lines, which should indicate the total contact resistance, are in fair agreement for

the 200°, 300°, and 400°C lines. The intercept for the 500°C

line is markedly greater, which would not be expected, since

the contact resistance theoretically should diminish slightly

with increasing temperature. It is thought that this large

intercept implies inexactness in the resistances obtained at

500°C, which with our present information is ascribed to the

effect of significant radiative heat transfer in the specimens at 500 ° C.

The inverse slopes of the four lines give the apparent

conductivity (including radiation) of the Pyrex material,

which was determined from these measurements as

Temperature

App. Cond., mw/cm-°C

For the reason given above,

500°C is questionable.

200 ° 300 ° 400° 500 0

13.3 l4.1 16.4- 20 .7

the apparent conductivity at

Many materials such as alumina become non-opaque to

thermal radiation at elevated temperatures and therefore it

was felt that more investigation of the radiation component

of heat transfer at the lower temperatures possible with

Pyrex would be helpful in analyzing the relative conductive

and radiative components of heat transfer in other materials

at elevated temperatures. For an infinite body, the radiation conductivity has been derived independently by several

investigators and is given in the literature as

n , _ 16 n

2 dT 3 Kp -

3 a where n = the index of refraction of the medium

h = S tefan-Boltzmann constant

T = absolute temperature

a = absorption coefficient.
- 8

-

For short specimens such as those being investigated in

this project, the problem becomes much more complex. In the

specimen, near the hot and cold bars, the radiative heat

transfer depends on the emittances of the contacting surfaces and throughout the specimen is a function of both the

temperature and the temperature gradient in the specimen.

In order to investigate the effect of contact surface

emittance on the apparent conductivity, a series of tests

was conducted on a specimen of 1.2 57 -cm thickness. The

tests were made at several mean temperatures from 200° to

550°C with the specimen in contact with (a) the oxidized

stainless steel faces of the hot and cold bars, (b) 1-mil

shiny annealed platinum foil, and (c) 1-mil platinized

(blackened) annealed platinum foil.

In this series of tests, the insulation used was diatomaceous earth, the pressure applied was 10 kg/cm 2

, and the

temperature gradient across the specimen was approximately

25°C/cm. The results are shown in Figure 5» The "over-all

thermal conductivity" of the specimen (which includes the

effect of contact resistance) is plotted against temperature

for each of the three curves shown. Judging from the relative positions of the three curves at the lower temperatures,

the contact resistances were least for the platinized foil

and greatest for the oxidized stainless steel. Although two

additional interfaces were added when foil was used, better

thermal contact appears to have been made because of the

plasticity of the foil. The curve for the platinized foil

lies above both of the other curves. The explanation of

this is that the foil makes excellent contact, and radiation

into the specimen is increased because of the higher surface

emittance of the platinized foil.

Before giving further consideration to the results in

Figure 5? it is necessary to have more information concerning contact resistance when annealed platinum foil is

used as an interface medium. Accordingly, additional tests

were made on a 0 . 252 -cm specimen at 200°, 300°, 5-00°, and

500 ° C with a shiny foil interface, the results of which are

shown in Figure 6 along with previous results obtained with

the 1.257-cm specimen. Comparing Figure 6 and Figure 4, it

is seen that the contact resistances were made smaller when

shiny annealed platinum foil was introduced as an interface

between the specimen and the oxidized stainless steel bars.

Again, a relative increase of the ordinate intercept at

500°C was found, as was first seen in Figure h.

Resuming consideration of the data of Figure 5? Figure

7 presents apparent conductivities obtained by making corrections for the contact resistances, using the values from
-9-

Figure b for the oxidized stainless steel interfaces, and

those from Figure 6 for the bright platinum foil interfaces.

Because contact resistances were not determined separately

for the platinized foil, the conductivity in Figure 7 for

that condition was adjusted to agree with that for the other

curves at the lower temperatures. As higher temperatures

are approached, the three curves diverge from a straight

line and from each other. Curve A rises most rapidly,

indicating a relatively large radiation component due to the

high surface emittance of the oxidized stainless steel.

Curve B is the flattest of the three curves, indicating a lower radiation component due to the low surface emittance

of the shiny foil. Curve C, for platinized foil interfaces, lies between the other two curves, indicating that

its emittance was intermediate (which probably was so, since

only a very thin layer of platinum black was present on

these foils). These three curves represent apparent thermal

conductivities, including both conductive and radiative components as governed by the emittances of the interfaces. It

must be emphasized that curves A, B, and C are valid only

for specimens of 1 . 257 -cm thickness under the conditions of

the tests.

If the conductive component is assumed to be linear, as

it appears to be at lower temperatures, curve D would represent the conductive thermal conductivity. Values of thermal

conductivity for Pyrex glass taken from published literature

agree with the curves shown in Figure 7 to within ± 10 percent over the temperature range shown. Some literature

values—'/ are indicated in Figure 8 , along with curve D of

Figure 7* A linear extrapolation of this curve agrees to

within '± 3 percent of values near room temperature obtained

in this laboratory using the NBS guarded hot plate (ASTM

C-177)? as indicated in Figure 8 .

As a further comparison, the data from the table on

page 7 are plotted in Figure 8 as open circles. These

points, which represent the apparent conductivity (including

radiation) of the glass, were obtained by taking the inverse

slopes of the lines in Figure 6 . It is seen that the agreement with the curve from Figure 7 is good at lower

1. F. Birch and H. Clark, The thermal conductivity of rocks

and its dependence upon temperature and composition, Am.

J. Sci. 238 , 529 (19^0).

R. W. B. Stephens, The temperature variation of the

thermal conductivity of Pyrex glass, Phil. Mag. l4, 89 7

(1932).

C. F. Lucks, H. W. Deem, and W. D. Wood, Thermal properties of six glasses and two graphites, Am. Ceram. Soc.

Bui. 33, 313 (I960).
- 10 -

temperatures . The apparent conductivity at 500°C shows a substantial component of radiation conductivity.

Later developments and modifications to be discussed

below indicate that the above data may have been subject

to a systematic error, which probably increased with temperature. Thus, there may be inaccuracy in the absolute

magnitudes obtained, but the qualitative discussion given

above is considered valid.

4. MODIFIED PROTOTYPE DESIGN

As was mentioned above, when the prototype absolute

cut-bar apparatus was first designed and built, it was

believed that optimum guarding could be achieved by matching

the temperature profile along the guard to that along the

bars and the specimen. Later developments have shown, both

theoretically and experimentally, that this procedure does

not yield optimum guarding. As discussed in Section 3-2 and

Appendix A, a more satisfactory procedure is to raise the

hot guard to a temperature greater than that of the hot bar

and to control the cold guard at a temperature less than

that of the cold bar.

The design of the prototype apparatus rendered the

attainment of these better guard temperatures difficult,

especially at lower temperatures, because of the presence

of metallic "bridges" between the hot and cold bars and

their respective guards (see Figure 2).

To cope with this difficulty, the prototype was modified (see Figure 9)* Metallic "bridges" were eliminated.

Thin-walled stainless steel tubing was used to support the

hot guard from below and the cold guard from above, and as

a support for the hot bar. A ceramic spacer ring was used

to separate the hot and cold guards. To enable free access

to the specimen, the assembly, consisting of the guards,

spacer ring, surrounding powder insulation, and case, was

mounted so that it coule be readily lowered away from the

hot and cold bars.

5. RESULTS OBTAINED USING THE MODIFIED APPARATUS

In order to check the modified prototype apparatus in

respect to heat flow and temperature measurement, a series

of comparison tests was conducted. Two materials were

selected: (a) a metal whose conductivity had been determined by an independent absolute longitudinal heat flow

method; (b) the 1.257-cm sample of Pyrex No. 77*+0, which had

been measured in the apparatus before it was modified.
- 11 -

A specimen of nickel-chrome alloy (see Figure 10) was

fabricated from the same slab of material used to fabricate

a specimen measured in the NBS metals apparatus. The specimen was installed in the absolute cut-bar apparatus without

an interface material. A stainless steel spacing ring was

installed between the hot and cold guards and diatomaceous

earth was used as insulation.

The thermocouples shown on the centerline of the specimen were installed in ceramic tubing inserted in drill holes

to the center of the specimen. The thermocouples shown at the

edge of the specimen were peened into small holes in the surface of the specimen.

Conductivity determinations were made at eight temperatures ranging from about 115° to 800°C. The thermal conductivity was calculated separately for each of the four thermocouple spans indicated in Figure 10. The results of these

calculations are plotted in Figure 11. The circles refer to

the peened thermocouples. The solid line represents a quadratic equation of least squares fit through all of the points

obtained. It will be noted that the data for the peened

thermocouples are much smoother than those for the inserted

thermocouples. The dashed lines above and below the solid

line bound the region plus and minus 5 percent of the solid

line

.

In Figure 12, the data of Figure 11 have been replotted

as departures from the smoothed data obtained using the NBS

metals apparatus. The superiority of the peened thermocouples

is quite evident in this plot. It is seen that the agreement

of the absolute cut-bar apparatus with the NBS metals apparatus is excellent.

These data for the nickel-chrome alloy indicate that the

absolute cut-bar apparatus is capable of accurate thermal

conductivity measurements if the temperature gradient in the

specimen can be precisely determined. The absolute power

measurement appears to be satisfactory and the parasitic

heat flows, both through the insulation and through the hot

bar support, are adequately evaluated.

It should be noted, however, that this nickel-chrome

specimen is not similar to the specimens for which this

apparatus is designed. In the measurements previously

described for Pyrex glass prior to modification of the apparatus, thermocouples were not placed in the specimens. The

temperature gradient in these specimens was evaluated by

means of thermocouples located in the hot and cold bars.

In order to determine this gradient, it was necessary to

evaluate thermal contact resistance by means of an elaborate
- 12 -

series of measurements on specimens of several thicknesses.

Although the data acquired for Pyrex appear to have internal consistency, there remains the possibility of a systematic error in determining the temperature gradient. This

will be further discussed below.

The thermal conductivity of thermoelectric materials

must be as low as possible if the thermoelectric device is

to have a high efficiency. As was the case for Pyrex, the

heat flow through a specimen of low conductivity will be much

less than that through the nickel-chrome specimen. Consequently, the evaluation of parasitic heat exchanges assumes

far greater importance for these low conductivity specimens.

The purpose of the above discussion is to point out

that the measurement of thermal conductivity is much more

difficult for thermal semiconductors than for metals. The

excellent agreement of the data for the nickel-chrome alloy

with that from the NBS metals apparatus should not be taken

as proof of this apparatus and method for materials having

a conductivity much closer to that of the surrounding insulation. It is our belief that the method will prove to be

more than adequate for such materials, but it must be emphasized that the agreement of the nickel-chrome data with

another absolute method and the internal consistency of the

Pyrex results are not sufficient proof of the method. Further tests are necessary to establish the accuracy of the

method for low conductivity materials.

In order to check the modified prototype apparatus on

materials of low conductivity, the 1 . 257 -cm specimen of

Pyrex glass No. 77^0, which had been measured in the apparatus before it was modified, was installed. Diatomaceous

earth was used as insulation, the pressure on the specimen

was 10 kg/cm 2

, and the temperature gradient was about

25°C/cm. No interfacial material was used. Conductivity

determinations were made at temperatures ranging from 100°

to 500°C. An attempt was made to duplicate as closely as

possible the test conditions attained for this specimen

prior to modification of the prototype apparatus.

The results obtained on this specimen of Pyrex did not

agree with those obtained previously and discussed under

Section 3*2. More scatter was observed in the data than

before. An extrapolation of the newly acquired data to room

temperature agreed substantially with the previous room temperature extrapolation and with the values obtained using

the NBS guarded not plate (see Figure 8). However, the

slope of the thermal conductivity curve was more nearly that

of the NBS guarded hot plate, or that of Lucks, Deem, and

Wood. This resulted in a thermal conductivity value that
-13-

was 10 or 15 percent lower at 300°C than the values previously obtained (tentative data of Figure 8).

Numerous checks and tests were taken in an effort to

isolate this difficulty. No instrumentation difficulties

were found. The data for the nickel-chrome alloy indicated

that the power measurement was satisfactory. It was therefore concluded that the temperature gradient in the specimen was in question.

Since the thermal conductivity of the stainless steel

hot and cold bars was known closely, it was possible to compute the temperature gradients in the bars by using this

conductivity and the measured power input in the formula

41 = Q

Az ss KssA

where AT/Az is the temperature gradient in the hot or cold

bar

Q is the measured power input corrected for

extraneous heat exchanges

Kss is the thermal conductivity of the stainless

steel bars

A is the cross-sectional area of the bars.

The results of this calculation were not in satisfactory agreement with the bar temperature gradients as determined by the inserted thermocouples in the bars. This would

seem to indicate that the temperatures in the bars were not

known accurately, and hence the temperature gradient through

the Pyrex specimen was in error. A similar analysis of the

earlier data discussed under Section 3*2 indicated that there

may have been errors in determining the specimen temperature

gradient prior to modification of the prototype apparatus.

Owing to the difficulties of temperature gradient

measurement, it was not possible to determine which, if any,

of the data acquired for Pyrex glass are correct. As was

stated at the end of Section 3.2, the qualitative conclusions drawn in that section are considered to be correct.

However, the thermal conductivities shown in Figure 7 and

Figure o may be in quantitative error.

6. ADDITIONAL RESULTS

6.1 Addition of Thermocouples

A further change in the apparatus was made in view of

the results obtained above. As was noted, the data acquired

from the peened thermocouples (Fig. 11 and 12) are much
-14-

smoother than those for the inserted thermocouples. Consequently, it was decided to install peened thermocouples in

the hot and cold bars of the prototype apparatus in an

effort to better determine temperatures in these bars.

The thermocouples shown on the centerline of the bars

in Figure 9 are the original thermocouples which were

installed in two-hole ceramic tubing inserted to the center

of the bars . The thermocouples shown at the surface of the

hot and cold bars are the new chromel-alumel thermocouples

which were peened into small holes in the surface of the

bars

.

6.2 Measurements on Pyroceram

Owing to the difficulties associated with thermal

radiation through the Pyrex glass specimen, it was decided

to undertake thermal conductivity determinations on small

specimens of an opaque material. Two specimens of Pyroceram

Code 9606 were prepared having thicknesses of approximately

1.26 cm and 0.77 cm, respectively. These specimens were not

optically polished, but had a fine surface grind on the contacting faces.

Determinations were made on the two specimens at mean

temperatures of approximately 200°, 300°, 400°, 500°, and

600°C. For these tests, diatomaceous earth was used as

insulation, the pressure on the specimen was 10 kg/cm 2

, and

the temperature gradient was 20° to 25°C/cm. Platinized

foil was used as an interfacial material to improve contact.

The results of these tests are shown in Figure 13 . The

lower two curves show apparent thermal conductivity plotted

against temperature for the two Pyroceram specimens. The

limits shown above and below each datum point indicate the

variation of the conductivity if various combinations of the

thermocouples in the hot and cold bars are used to determine

the temperature drop across the specimen; the data points

are the arithmetic averages of the three values obtained in

each test.

The upper curve in Figure 13 displays the thermal conductivity of Pyroceram Code 9606 after contact resistance

adjustments are made to the lower curves. In making these

contact resistance adjustments, it was assumed that the contact resistances were the same for the two specimens. The

platinized foils were not in very good condition, especially

for the tests on the 0.77-cm specimen, so that the assumption may not be strictly correct. For a material of low

conductivity, such as Pyrex, the contact resistance adjustment is small, so that an error would not seriously affect

the resulting conductivity value. However, for a material
-15-

of higher conductivity, like Pyroceram, an error in contact

resistance evaluation would more seriously affect the conductivity values. The conductivity shown for Pyroceram is

therefore considered tentative.

A series of measurements was conducted on a 0.77-cm

specimen of a commercial alumina. The data acquired were

smooth from 200° to 600°C<> Dense alumina has a high thermal conductivity in this temperature range. Consequently,

contact resistance adjustments are very large. Comparing

the results of these measurements with literature values

for similar material, it was observed that the resistance

of the two contacts exceeded that of the specimen. Accurate

determination of such relatively large contact resistances

would require accurate thermal resistance measurements on

specimens of several thicknesses. It is questionable

whether or not contact resistance would reproduce sufficiently well from one specimen to the next to enable precise

evaluation. Due to the uncertainty of contact resistance

adjustment, the data for the 0.77-cm alumina specimen are

not presented in this report.

An investigation of the temperature gradients in the

hot and cold bars was made for both the inserted and the

peened thermocouples. The peened thermocouples were noted

to be superior, but were not entirely satisfactory. At

least part of this difficulty is attributed to the use of

chromel-alumel thermocouple wire.

7. FUTURE ACTIVITIES

It is felt that sufficient information has been acquired

in the period covered by this report to permit design and

construction of the high temperature model of the absolute

cut-bar apparatus. Design work on this apparatus has been

substantially completed and construction has begun.

The design of the high temperature model will parallel

that of the modified prototype apparatus. The hot and cold

bars will be fabricated from 60% platinum-^0% rhodium alloy.

The remaining high temperature structural portions of the

apparatus are to be constructed of a high purity (+99%)

alumina. All heaters will be 80% platinum-20% rhodium

alloy. Thermocouples are to be reference grade platinum/

platinum-10% rhodium.

Jn order to check the apparatus when it is completed,

it has been decided to conduct measurements on two materials

of known thermal conductivity. The nickel-chrome alloy

measured in the modified prototype apparatus will be the

high conductivity material. For the low conductivity
16 -

specimen, a material having a thermal conductivity similar

to that of many thermoelectric materials would be preferable.

The measurements made on Pyroceram seem to indicate that it

would be a satisfactory material. A specimen of Pyroceram

Code 9606 is currently being fabricated for measurement in

the NBS metals apparatus. Specimens from the same bar will

be measured in the high temperature absolute cut-bar apparatus. Dr. D. C. Ginnings of the Heat Measurements Section,

NBS, intends to measure the thermal diffusivity of this

material sometime soon, thus providing additional valuable

cross-checks

.

It has been pointed out in this report that the two

quantities which must be determined in a steady-state

measurement of thermal conductivity are the heat flux

through the specimen and the temperature gradient in the

specimen. For the absolute cut-bar method, the results

shown and the analysis of heat flow given in Appendix A

indicate that the heat flux through the specimen can be

determined with sufficient accuracy. It is felt, however,

that the temperature gradient in the specimen has not been

adequately determined for the measurements given in this

report. It was originally felt that measurements could be

made expeditiously, utilizing thermocouples exterior to the

specimen and making corrections for contact resistance.

This has not proved to be the case. The contact resistance

is too large, and possibly too uncertain, to be handled in

this manner, even with the use of optically flat surfaces.

In the apparatus presently being constructed, a different method of measuring specimen temperature gradients

will be attempted. Thermocouples will be affixed in small

grooves cut in the convex surface of the specimen, and these

thermocouples will be utilized to determine the temperature

gradient. In order to assure uniform heat flow, contact

resistance will be reduced by spraying the ends of the specimen with platinum. It is hoped that these modifications will

enable more accurate determination of temperature gradients,

and thus remove the remaining largest source of inaccuracy

involved in these measurements.
- 17 -

thermocouple

position

Fig. 1. A schematic cut-bar apparatus utilizing reference

bars to determine heat flow through the specimens.
-18-

Fig. 2. The original prototype absolute cut-bar apparatus

utilizing the measured power input to an electrical

heater to determine heat flow through the specimen.
of

contact

resistance.
- 20 -

Fig. b. Total thermal resistance, including contact resistance, for several specimens of Pyrex glass No. 77^+0.
- 21-

O

ooU

Ih*

00 N- <X> Do-ujo/muO Fig..

5°

Over-all

thermal

conductivity

of

Pyrex

glass

No.

77^0,

no

corrections

having

been

made

for

the

contact

resistance

of

three

types

of

interfaces.
- 22 -

L CcmD

Fig. 6. Total thermal resistance, including contact resistance of platinum foil interfaces, of two specimens

of Pyrex glass No. 77^0.
23"

c\j ^ QQ cd uo ^ ro [5,,-UiO/Mlu] >j Fig.

7-

Thermal

conductivity

of

Pyrex

glass

Wo.

77^0,

as

a

function

of

temperature,

including

corrections

for

contact

resistance.
-2k

[5 o- UIO/ MUJ] » x

1ig

.

8.

Thermal

conductivity

of

Pyrex

glass

No.

77^+0

(tentative

data)

compared

with

literature

values.

The

circles

indicate

additional

results

derived

from

thi

data

of

Figure

k.
-2 5

'

THRUST

Fig . 9- The modified prototype absolute cut-bar apparatus.
-26-

1.40

NOTES:

1. X DENOTES THERMOCOUPLE

POSITION.

2. LENGTHS GIVEN IN CM.

Fig. 10. The nickel-chrome alloy specimen used to check the modified prototype apparatus.
WATTS

CM—

°C

-27-

T - E°CJ 11. Thermal conductivity of a nickel-chrome alloy as

determined using two methods of thermocouple

installation in the specimen.

Fig.
-28-

[%11-Snj.VHVddV S1Vi3W S8N WOdd SNOI1VIA3Q Fig.

12.

Departures

of

the

data

for

the

fiiekel-chrome

alloy

from

the

smoothed

data

from

the

NBS

metals

apparatus.
-29-

C

Do — IAIO MIAJ -M

Fig.

13*

Thermal

conductivity

of

Pyroceram

Code

9606

versus

temperature,

before

and

after

correcting

for

contact

resistance.

A-l

APPENDIX A

GUARDING OF A THERMAL CONDUCTIVITY APPARATUS

OF THE CUT -BAR TYPE

The general physical arrangement of a comparative "cutbar n apparatus for determination of thermal conductivity was

shown in Figure 1. The modified prototype absolute n cut-bar"

apparatus, shown in Figure 9 ? has a similar physical arrangement. It is the purpose of this discussion to consider heat

exchanges between the bars and specimen and the surrounding

powder insulation in an apparatus of this type.

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 resistances

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 a corresponding change in the longitudinal temperature

gradient in the adjacent insulation. Since the thermal conductivity of the insulation does not change materially, the

insulation must exchange heat with the bars and specimen.

In particular, if the specimen has a lower conductivity than

the bars, it is necessary for the insulation to steal heat

from the bars and specimen on the hotter side and return this

heat on the cooler side.

It is also necessary to originally establish a longitudinal gradient in the insulation near the hotter bar. In

the design shown in Figure 1, this gradient will be established near the lower end of the bar so that there will be

no further exchange of heat until the specimen is approached.

However, in the case where the bars are quite short the

longitudinal 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

.

Although there may be both radial and longitudinal

heat flows in the powder insulation, it is the radial

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
A-2

this surface must be made. The heat flow across this surface in a longitudinal element of area will be

where a is the radius of the bars and specimen

k is the thermal conductivity of the powder insulation

9 is the temperature in the system

r is the radial coordinate

z is the longitudinal coordinate.

The net power flowing across the surface r=a between

z=Zi and z-z 2 is

In order to evaluate this integral it is necessary to

have a knowledge of the thermal conductivity of the powder

insulation and the temperature distribution within the insulation. An exact treatment of the system requires solution

of the four body composite system involving the two bars,

the specimen, and the powder insulation. Mathematically,

this problem is intractable because of the many continuity

conditions which must be satisfied.

For this apparatus the region consisting of the powder

insulation was analyzed as a boundary-value problem. The

measured surface temperatures along the bars and the guard

cylinders were used to establish simple linear functions

representing the temperatures on the curved surfaces of the

region. The ends of the region were assumed to be closed

by logarithmic radial temperature distributions , which provide temperature continuity. The analytical solution of the

temperature distribution in the region established by these

boundary conditions was utilized to evaluate (90/5r) r =a in

the integral in Eq. (A2). The thermal conductivity of the

powder insulation was measured in another apparatus.

Since the analytical solution of this problem is in

the form of an infinite Fourier series, it was necessary to

employ the use of a digital computer for numerical evaluation of the data. The details of this mathematical development are not amenable to brief treatment and hence have been

excluded from this report. It is planned to publish the

mathematical analysis of a cut-bar thermal conductivity

apparatus soon.

(Al)

( A2

)
A-3

Prior to the mathematical analysis outlined above, it

was thought that heat exchanges between the bars and the

powder insulation would be minimized if the same temperature distribution was maintained along the guard cylinder

as along the reference bars and specimen, so that guard

temperatures matched those of the bars at corresponding

longitudinal positions. It was soon found that such a procedure does not eliminate extraneous heat exchanges between

the bars and specimen and the surrounding powder insulation.

For a Pyrex glass specimen 1/2 inch thick at 500°C, using

diatomaceous earth as insulation in the prototype apparatus,

heat flow through the specimen was about 13 percent less

than the measured power input for a test in which guard temperatures approximately matched those in the bars. It is

seen that the matched-guard condition does not reduce

extraneous heat flows to the small magnitude desirable for

good accuracy of measurement.

For a material such as glass, or many semiconductor

materials having a thermal conductivity much less than that

of the bars, there will be a radial heat flow from the

hotter bar near the specimen to establish the larger longitudinal gradient necessary in the adjacent insulation. This

radial flow can be greatly reduced by raising the temperature

of the guard immediately opposite. Similarly, the heat flow

from the insulation into the colder bar can be lessened by

lowering the temperature of the guard opposite the colder

bar. What this does, in effect, is to permit the guard

cylinder to set up most of the longitudinal gradient in the

powder insulation adjacent to the bars and specimen. The

exact temperature distribution that should be placed on the

guard cylinder has not been determined. Theoretically, it

should be possible to guard the system so that the radial

gradient at the surface of the bars and specimen is exactly

zero. Such a guarding condition would be almost impossible

to achieve experimentally at elevated temperatures.

What was done in practice was to establish a compromise

temperature distribution along the guards to yield a small

radial heat flow from the bars and specimen. By astute

adjustment of the guards, it is possible to make the heat

flow through the specimen closely equal to the measured

power input at the heater. Although there are small heat

gains and losses along the assembly, the net total of these

interchanges can be made nearly zero, for the length of the

assembly up to the specimen, by means of suitable guard

temperatures

.

For example, for the 1/2-inch Pyrex glass specimen at

500°C, using diatomaceous earth as insulation in the prototype apparatus, heat flow through the specimen was within
A-4-

1/2 of 1 percent of the measured power input using the compromise guarding condition described above. Furthermore,

the mathematical analysis corrects for even this small error.

All of the data presented in this report were computed

using the above mentioned analysis to evaluate parasitic

heat flows. It is believed that, by the use of such an

analysis, the precision and accuracy of the heat flow determinations were greatly improved.
B-l

APPENDIX B

CALCULATION PROCEDURE

The numerical calculation of the data was effected by

means of a digital computer. The over-all thermal conductivity of the specimen, including the contacts, was calculated using the following equation.

k

.. _Q_L_

AAT

(Bl)

where k =

T

= L = A

= AT =

Q =

thermal conductivity (watts cm

-1 deg -1 C) at temperature T

mean temperature of specimen (°C) _

specimen thickness (cm) at temperature T

specimen cross-sectional area (cm 2

) at temperature T

temperature difference (deg C) across specimen and

contacts

average heat-flow (watts) through the specimen.

This equation is valid for steady-state longitudinal heat

flow over a small temperature interval. The determination of

the various factors in the conductivity equation will now be

discussed individually.

1. Specimen thickness (L).

The specimen thicknesses were carefully measured at

room temperature, using a metric micrometer which could be

read to 0.0005 cm. For the nickel-chrome alloy, the distance between thermocouple junctions was used. Thickness as

used in Eq. (1) was determined by the following:

L = L 25 [l + cc (f - 25)] (B2

)

where L2 5 is the measured thickness (cm) at room temperature

(25°C)

a is the coefficient of linear thermal expansion

(deg -1 C) of the specimen material

2. Specimen cross-sectional area (A).

The diameter of each specimen was carefully

measured at room temperature and the area computed. Crosssectional area as used in Eq. (1) was determined by the

following

A = A 25 [ 1 + a(T -25)]

2 (B3)
B-2

where A2

5

is the area (cm 2

) at room temperature.

3. Temperature difference [AT].

The temperature difference across the specimen is

determined by measuring two temperatures in the hot bar and

extrapolating linearly to determine the temperature of the

face of the hot bar, and similarly measuring two temperatures in the cold bar and extrapolating linearly to determine the temperature of the face of the cold bar. The difference between these temperatures represents the temperature drop across the specimen and the two interfaces. The

temperature difference as used in Eq. (Bl) was determined

by the following:

AT = TH - Tc (b4)

where Th is the inferred temperature (°C) of the hot bar face

Tc is the inferred temperature (°C) of the cold bar face.

Th and Tc were determined by linear extrapolation of

temperatures measured in the hot and cold bars, respectively.

For the measurements on Pyroceram and alumina, various combinations of the bar thermocouples were used.

4. Heat flow (Q)

.

The power input to the hot-bar heater was measured

by means of a precision volt box and shunt box in conjunction with a d-c precision potentiometer. Corrections were

made for the small current flow through the volt box.

The heat produced could flow, if permitted, in

several directions:

a) Up the hot bar, through the specimen and into

the cold bar

b) Up or down the hot bar support column

c) Across the convex surfaces of the hot bar and

the specimen.

These three classifications can best be treated

separately:

a) Ideally, the heat should flow entirely through

the specimen. The heat flows b) and c) should therefore

be minimized.
B-3

b) The heat flow up or down the support column was

minimized by keeping the chromel-alumel differential thermocouple located in the column as nearly as possible at a zero

reading. For tests in which the reading of the differential

thermocouple was not zero, a correction was made for the

slight heat flow through the support column. The area of

metal and its conductivity, and the area of porcelain tubing

and its conductivity, gave a value for the thermal conductance between the differential junctions . The reading of

the differential thermocouple was used to compute the heat

flow up or down the support column. This correction was

confirmed experimentally.

c) The heat flow across the convex surfaces of the

hot bar and the specimen was analyzed as presented in Appendix A. This analysis was utilized to determine the net heat

flow across the inner surface of the hollow cylinder of powder insulation, taking place up to various longitudinal positions of interest on the assembly.

The net heat gain or loss across the convex surface

was evaluated at both ends of the specimen and at the center

of the specimen. In short, Eq. (A2) was used to compute the

net heat interchange with the powder insulation over the

region of the bar and specimen assembly up to selected

longitudinal positions z 2 . These were chosen to correspond

to the positions of the two ends of the specimen, and its

center, and the weighted average of these three heat interchanges yields the average net gain or loss of heat, p, between the heater and the specimen.

The heat flow as used in Eq. (1) was

following

Q — P + CA t + p

determined by the

(B5)

where P

k

a

P

At

C

is the measured power input (watts)

is the conductivity of the powder insulation

(watts-cm“‘ -deg”1 C)

is the radius of the specimen (cm)

is the weighted average of the net heat interchange

from the assembly for the specimen

is the temperature difference in the support column

as indicated by the differential thermocouple (deg C)

is the heat flow up or down the support column due

to a unit temperature difference (watts-deg” 1 C)

.

USCOMM-NBS-DC
'
U. Si DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Aatin, Director THE NATIONAL BUREAU OF STANDARDS

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