NATIONAL BUREAU OF STANDARDS REPORT

7367

THERMAL CONDUCTIVITY OF SEMICONDUCTIVE SOLIDS;

METHOD FOR STEADY-STATE MEASUREMENTS ON

SMALL DISK REFERENCE SAMPLES

Technical Progress Report

for Period

July 1 to September 30, 1961

by

D. R. Flynn

Report to the

Bureau of Ships

Department of the Navy

Washington, D. C.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials, devices, and structures; advisory services to government agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; and the development of standard practices, codes, and specifications. The work includes basic

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July 1957 to June 1960 (Includes Titles of Papers Published in Outside Journals 1950 to 1959) (S2. 25); avail- able from the Superintendent of Documents, Government Printing Office, Washington 25, D. C.
NATIONAL BUREAU OF STANDARDS REPORT

THERMAL CONDUCTIVITY OF SEMICONDUCTIVE SOLIDS;

METHOD FOR STEADYtSTATE MEASUREMENTS ON

SMALL DISK REFERENCE SAMPLES

Technical Progress Report

for period

July 1 to September 30, 1961

D. R. Flynn

Heat Transfer Section

Building Research Division

Report to the

Bureau of Ships

Department of the Navy

Washington, D. C.

INDEX NO. S-R007 12 01. Task No. 800

(BuShips 1700S-696-61). Code 342B

NBS PROJECT

1006-20-10461 November 14, 1961

NBS REPORT

7367

IMPORTANT NOTICE

NATIONAL BUREAU OF ST/

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to reproduce additional copie

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Director of the National Institute of

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on October 9, 2015.

ogress accounting documents

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Such permission is not needed,

irepared if that agency wishes

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

Thermal Conductivity of Semi conductive Solids;

Method for Steady-State Measurements on

Small Disk Reference Samples

by

1.D R. Flynn

ABSTRACT

Thermal conductivity measurements were conducted

over the temperature range 200° to 1200°C on the bar of

60% platinum-40% rhodium alloy from which the hot and cool

contacting bars of the high-temperature apparatus are to be

fashioned. The tentative results, not yet corrected for

individual thermocouple variations, plot smoothly from

0. 57 w/cm-C at 200°C to 0.79 w/cm-C at 1200°C. A paper

"Thermal Guarding of Cut-Bar Apparatus," which was presented

at the Invitational Conference on Thermal Conductivity

Methods held at Battelle Memorial Ins titute , .Columbus , Ohio,

on October 26-28, 1961, is reproduced following the main

body of this report.

2.

STATEMENT OF PURPOSE

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 disks), with the

objective of providing samples for use by other laboratories

as thermal conductivity reference specimens in connection

with their measurements on solid semiconductors.

3.

WORK PERFORMED IN REPORTING PERIOD

(July 1 to September 30, 1961)

During this quarter additional measurements of the

thermal conductivity of the 60% platinum-hO% rhodium alloy

were completed, using the high temperature model of the

absolute cut-bar apparatus. (For a description of this

equipment, see NBS Report 7323.)

In the course of these measurements, it was found that

to maintain the desired temperature distributions along the

alumina guard cylinder, two additional heaters of platinum20% rhodium wire were required near the ends of the cylinder.

These heaters, shown in the schematic diagram of the apparatus

(Fig. 1), are controlled by variable voltage transformers to

minimize end heat losses from the central portion of the guard

cylinder

.
2

The thermal conductivity of the 60% platinum-hO% rhodium

alloy was remeasured over the temperature range 200° to

1200°C. The alloy was supported between the alumina tube

below and the alumina rod above. The thermal insulation used

throughout the apparatus was alumina powder. The power

generated in the specimen heater was determined by current

and potential drop measurements, using a calibrated volt box

and shunt box in conjunction with a precision potentiometer.

The temperature distribution along the platinum -rhodium

bar was determined by means of four butt -welded platinum:

platinum-10% rhodium thermocouples, fabricated from calibrated reference grade wire, pressed into four 0.3 mm grooves

in the convex surface of the bar. Each of the four specimen

thermocouples could be read against an ice junction, so that

temperature gradients could be determined by difference.

When the 60% platinum-40% rhodium alloy bar was purchased, a 0.5 mm wire was drawn by the manufacturer from the

same material. A thermocouple, fabricated from a length of

this wire and a length of 0.38 mm reference grade platinum

wire, was calibrated by the NBS Pyrometry Laboratory over the

temperature range 0°-1100°C. Using this calibration curve,

the thermoelectric power of the platinum-4-0% rhodium alloy

against platinum and against platinum-10% rhodium was derived.

The insulated room-temperature zone box was wired to

enable determination of the emf developed between similar

leads of different thermocouples. The emf was measured

between the platinum wires of the lower two junctions, the

platinum wires of the upper two junctions, the platinum-10%

rhodium wires of the lower two junctions, and the platinum10% rhodium wires of the upper two junctions. Thus, the 60%

platinum-hO% rhodium bar served as the central portion of

four differential thermocouples.'

Thermal conductivity measurements were made in increasing

order of temperature from 200° to 1200°C at 200 deg C intervals, and then in decreasing order of temperature at 800° and

400°C. Each conductivity determination involved two tests.

First, an "isothermal” test, in which there was no power

input to the specimen heater, was made to determine variations between specimen thermocouples. This test was followed

by a "gradient" test with' sufficient power input to the

specimen heater to maintain a longitudinal temperature gradient in the specimen of about 2 deg C/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.

The results of the gradient tests are plotted in Figure

2. The conductivities shown are computed from ,f re.w

,f dcits.

only. That is, the temperature readings from the gradient
3

tests have not been corrected for the thermocouple variations

determined by the isothermal tests. These corrections have

been postponed pending installation of improved digital computer facilities at NBS.

The solid symbols shown in Figure 2 represent measurements made with temperatures increased, and open symbols those

made with temperatures decreased. Thermal conductivity values

indicated represent the average of two conduct ivities i one

computed using the temperature difference between the lower

two specimen thermocouples and the other that between the

upper two specimen thermocouples. The circles (solid and

open) were obtained using temperature gradients as determined

by differences between the platinum : platinum-10^ rhodium

thermocouples read against an ice junction. The triangles

(solid and open) were obtained using temperature gradients as

determined by the platinum: plat inum-4-0^ rhodium : platinum differential thermocouples in which the specimen was the central

leg.

The solid line shown in Figure 2 is the least-mean-squares

quadratic through all of the points shown. The dashed lines

above and below the solid line bound the region plus and minus

5 percent of the solid line. The data shown are to be considered tentative, since corrections have not been made for

variations between thermocouples. When these corrections are

made, the data should be significantly smoother.

b. FUTURE ACTIVITIES

Measurements will begin in a few weeks on a specimen of

the nickel-chrome alloy that was measured in the modified

prototype absolute cut-bar apparatus. This material has also

been measured in two models of the NBS metals apparatus and

in the NBS steam calorimeter apparatus. Its conductivity is

believed to be well known (±2 percent) over the temperature

range -150° to 1000°C and hence this material should be quite

suitable for confirming the accuracy of the high temperature

absolute cut-bar apparatus.

A 1/2-inch by 1-inch diameter disk of Pyroceram 9606 is

currently being optically polished. This specimen will be

ultrasonically cleaned and the flat surfaces coated with

vapor-deposited platinum. Thermocouples will be affixed in

small grooves cut in the convex surface of the specimen to

enable determination of the temperature gradient in the specimen. A large specimen of the same material has been prepared

for measurement in the NBS metals apparatus. These independent measurements in the absolute cut-bar apparatus and the

metals apparatus will provide a cross-check on a material of

conductivity similar to that of many thermoelectric materials

of interest and will assist in determining the suitability of

this material for thermal conductivity reference purposes.
1+

5. USE OF A REFERENCE MATERIAL

The ultimate objective of this research is the provision,

for use by other laboratories, of suitable thermal conductivity reference specimens. Of equal importance, it is felt,

is the proper use of such standards.

The problems encountered in the design and operation of

the absolute cut-bar apparatus have pointed out some of the

difficulties arising in measurements of this type on small

samples. The two most serious difficulties are due to the

thermal resistances at the contacting surfaces of the specimen

and the extraneous exchange of heat with the surrounding insulation. In order to avoid the problem of attempting to measure

contact resistance, it is recommended that thermocouples be

placed in the specimen not too near its contact surfaces. In

order to evaluate heat exchanges with the insulation, a mathematical analysis was made. The general form of this analysis

was presented in Appendix A of NBS Report 7135 (April 28,

1961 ).

A detailed analysis of heat flow in the insulation of a

simplified cut-bar apparatus was presented by this author at

the Invitational Conference on Thermal Conductivity Methods

held at Battelle Memorial Institute, Columbus, Ohio, on

October 26-28, 1961. In view of the similarity of this analysis to that used for the NBS absolute cut-bar apparatus, this

paper "Thermal Guarding of Cut -Bar Apparatus" is reproduced

at the end of this report.

Many laboratories are currently utilizing various forms

of a comparative cut-bar apparatus for their thermal conductivity measurements. A typical example would be the apparatus

of Francl and Kingeryd). As has been pointed out, shunting

heat flows through the insulation are a serious problem in an

apparatus of this type. While these heat flows are evaluated

analytically in this laboratory, it is realized that many

laboratories will not find it convenient to resort to a digital

computer for evaluation of their particular apparatus.

The effects of these unwanted heat flows can be avoided

by means of a suitable calibration if suitable thermal conductivity reference specimens can be made available. The best

use of these reference specimens is summed up in another paper

presented at the above-mentioned conference. In his paper

"Current NBS Steady-State Thermal Conductivity Methods," H. E.

Robinson, Chief of the NBS Heat Transfer Section, concluded

by saying:

1. J. Am. Ceram. Soc. 2>Z [2] 80-6L ( 19 5V ) -
5

"The idea of a thermal conductivity reference standard

is manifest. Nevertheless , such standards involve other

problems than merely those of selecting suitable materials

and determining their conductivity as accurately as possible,

"For example, the best use of a reference material is

to use it as a substitute specimen, as is done for instance

in precise determinations of electrical resistance. When

the reference is used in this way in a particular apparatus,

it is presumably subject to the same unwanted heat flows,

resulting from what might be termed the accessory conditions

of the apparatus, as would be a test of specimen of equal

conductivity. If the reference is thus used to calibrate a

metering device of some sort, as it would be in a comparative

type of testing apparatus, the calibration of the meter would

inherently contain the effects of these perturbations. This

consideration is of special importance In the case of heat

flow measuring equipment, because there is no perfect thermal

insulation or guarding to eliminate .unwanted heat flows.

Such use of the reference as a substitute specimen carries

with it, however, the strict requirement that the accessory

conditions of the apparatus be duplicable in ordinary tests,

as in the calibrating tests. Thus, in this sense, the best

use of a reference standard involves the design of the apparatus to be calibrated with it, in regard to the precision

with which the accessory conditions can be controlled and

duplicated.

"As an illustration, it would not be the best use of a

reference material of known conductivity to employ it as the

heat flow meters in a series arrangement such as meterspecimen-meter. The better use would be to employ for the

metering bodies some suitable material, and obtain a calibration for them, by using the reference in place of the spec!

men. If reference substitute specimens covering a range of

conductivities were used, and accessory conditions were adequately reproduced, it is probable that test measurements on

unknown specimens of conductivities within the range covered

by the references could be made with results not seriously

inferior in absolute value to those of the reference specimens o.o..."
6

THRUST

Figure 1. High temperature absolute cut -bar apparatus for

determination of the thermal conductivity of small

solids ^
7

o

o

o

o

OJ

o

o

o

o

0

o

o BjJ

00 jr3

s

<

££

LlJ Q_

c 5

o LJ

CD 1-

o

o

o

o

CVJ

Figure

2.

Tentative

data

for

the

thermal

conductivity

of

a

60%

platinum-^O^

rhodium

alloy.

The

symbols

are

explained

in

the

text.
THERMAL GUARDING OF CUT-BAR APPARATUS

D. R. Flynn*

National Bureau of Standards

Washington, D. C.

ABSTRACT

A mathematical analysis of heat transfer in the insulation sur- rounding "cut -bar" thermal conductivity apparatus is presented. A general

solution is derived for arbitrary temperature distributions along the inner

and outer boundaries of the hollow cylinder of insulation.

For small heat exchanges between the insulation and metering bars

and specimen, the temperature distribution at the inner boundary is

approximated by three straight lines. Where temperature distribution along

the guard matches that along the meter bars and specimen, the heat gains or losses to the insulation are proportional to:

1') The thermal conductivity of the insulation,

2 ) The difference between the thermal resistivities of the

specimen and meter bar,

3) A geometrical factor.

The choice of a meter bar becomes critical for low specimen con- ductivities. Several conclusions are reached concerning the design and

operation of a cut -bar apparatus.

1. INTRODUCTION

The thermal conductivity of solid materials is of considerable

theoretical and practical interest. Unfortunately, published results for

this property are often widely divergent. While some variation among

literature data may be attributed to differences in sample composition and

mechanical properties, it is evident from the magnitude of these discrepancies that errors exist in the various methods of measurement. These

errors can frequently be attributed to unaccounted for heat losses or gains

in a measuring apparatus.

One common method of determining thermal conductivity Involves

the comparison of an unknown material with one of known conductivity. In

the comparative method, as usually employed, a sample of the unknown

material is placed in series with one or more samples of a known material

and a heat flow is established through this tandem assembly. A determination of the temperature gradient in th,e known material serves to define the

heat flow through the unknown sample.

A special case of the comparative method, sometimes designated as the "cut-bar" method, ‘'has been used extensively and is the subject of this

paper. An unknown .specimen is placed between two identical bars of a "known" material, which, in 4 actuality , serve as heat flow meters. (This

composite assembly, consisting of three bars in series will hereafter be

* Physicist, Heat Transfer Section
9

referred to as "the assembly.") The contacting surfaces of the meter bars

and of the specimen are made flat and parallel, and a thrust is exerted on

the assembly to assist good thermal contact. A temperature difference is

maintained between opposite ends of the assembly, causing a heat flow along

the bars and specimen. Thermocouples are employed to determine temperatures in the meter bars and specimen.

If data acquired by the cut -bar method are to be valid, care must

be taken to minimize and account for heat gains or losses from the curved

surfaces of the assembly. In this paper, analytical techniques are presented which assist in reducing and determining the magnitude of unwanted

heat exchanges in the cut-bar method.

In order to minimize - these .heat exchanges, it is customary to

provide a guard cylinder coaxial with the assembly, the space between the

assembly and the guard cylinder being filled with thermal insulation.

Usually, an attempt is made to maintain the same temperature distribution

along the guard cylinder as along the assembly, so that guard temperatures

match those of the assembly at corresponding longitudinal positions. This

paper shows that this guarding technique does not preclude unwanted heat

exchanges, and provides a means of computing and correcting for these

exchanges

.

2. MATHEMATICAL ANALYSIS

Figure 1 is a schematic of the assembly and surrounding insulation. For simplicity in the mathematical development, the meter bars are assumed to be identical in length and thermal conductivity.

Km? Ks , and Ki are the thermal conductivities of the meter bars,

the specimen,- and the insulation, respectively. The radius, r, of the

assembly is A; the inner radius of the guard is B; and the length of each

reference bar is M; that of the specimen is L; and the over-all length is

W=2M+L.

2.1 General Case

Consider the region consisting of the powder insulation between

the assembly and the guard cylinder. The boundary conditions for this

region can be expressed as:

1. r=A 0*z£W v=G(z)

2. r=B O^z^W v=H(z)

3 . Alr^B z=0 v=H(0)+{G(0)-H(0)}^|^

4 . AlrlB ?=W v=H(W)+{G(W)-H(W)}|^^

where temperature is denoted by the symbol, v. G(z) and H(z) are the temperature distributions along the .boundaries r=A and r=B, respectively. It

has been assumed that the radial temperature distribution at the ends of

the insulation, z=0 and z=W, can be represented logarithmically.

It is helpful to reduce all previously defined quantities to

dimensionless forms:
p=z/W ; m=M/W ,

’ <t=L/W

p=r/W ; a=A/W , b=B/W

0=(v-G(W) )/T, where T=G(0)

hs =Ks/Km , Pi=Ki/Kd

10

( 2 )

In these reduced variables, the boundary conditions become

1 .) p=a 0<p<l e=g(p)

2 .) p=b 0<p<l 9=h(p)

3.) a^p^b p =0

4.) a<p<b H=1

where all dimensionless temperatures, 9, have been referenced to g(l)=0 , g(0)=l.

The heat flow in the hollow cylinder, a<p<b, must satisfy

Laplace’s equation in cylindrical coordinates, which in dimensionless

parameters is, assuming angular symmetry

’ a

2 0

, l a 0 a

2 9 _ n

8 p

2 p bp 3p 2 oo

Assume as a

(i solution v 1 06

e=.[i+[h(°)-i}^s_e^] d-U +h ( 1 - +^fn (p)sin mrp )f^ (5)

in which fn ( P )

cnl o ( nmp

) +EnK0 ( nmp

)

• ( 6 ) The boundary conditions (3) and (h) are satisfied by Equation ( 5) • Io

Ko are the zero order modified Bessel functions of the first and second

kind, respectively .

(

2 )

and

If fn(a) and fn(b)' are the coefficients of the Fourier sine

series at p=a and p=b, respectively, then

fn(a)Kp (nmb)-fn(b)Ko(nrra)

Fo(n7ra;n7rb)

and

where

fn(b)Io(n7ra)-fn (a)Io(nTrb) n Fo(n7ra;n7rb.) F0 (x;y)=I 0 .(x)Ko(y)-Io(y)Ko(x)

,

(7)

( 8 ) and Equation ( 6 ) becomes

and'

r , n

\ _ fnCa)Fo(n7rp;nTrb)-fn (b)Fo(mrp;nTra) n P Fo(nva;mrb)

Boundary conditions (1) and (2) are satisfied if

fn(a)=2 f {g (p )1+p] sin rrrp dp

l

J ° fn(h)=2^

[h(l)-h( 0 )+[h( 0 )-h(l)}p]sinn7rp dp

(.9)

( 10 ) (ID

* References appear under the heading REFERENCES .
11

Equation

Since the heat flux crossing the surface p=a is of interest,

(5) is differentiated with respect to p, and evaluated at p=a:

90\ _ 9pp=

‘ [h(0)-l] (l-p)+h(l)p

a-tn h/

a

oo + ^n7rfn(a ) s in nrrp (12)

where fn(p)

fn (a)Fj (n7rp ;nTrb)-fn (b)F 1 (mrp ;n77;a)

Fq (nira ;n7rb) (13)

and Fi (x;y)=Ii (x)Kq (y)+Io (y)Ki (x) (il0

The radial flow of. heat at the surface, r=A, through a cylindrical element of' length dz is

dP=27rA Ki(|^) dz 3r r=a

(15)

In the dimensionless parameters, this becomes

dp=2iraOi’(~) dp

°P p=a

where dp=dP/(WTKm)

.

( 16 )

The total dimensionless power flowing across the surface p=a

between p=0 and p=p 1 , where Pi is any arbitrary value (0<Pi<l), is

p(Pi )=2TraOi

^ae

o 3p ) dp

P~a

(17)

, , , _ r 2 f h (o)-i]iu+{i+h(i)-h(o)}w2 pCm )=2TTa0l [ Ltnb/a

00 + £fn(a) (1-cos nrrp x )J n=l

(18)

Equation (18) is valid for any temperature distribution, g(p), at

p=a and any temperature distribution, h(p), at p=b. It will now be of

interest to select a particular temperature distribution along the curved

surface ' of the assembly, p=a.

2.2 Idealized Temperature Distribution on Inner Boundary

Returning to the apparatus of Figure 1, fn(a) will be evaluated

for a particular choice of g(p) (see Equation (10)). If the region A<r<B

contains a good inshlation, the longitudinal temperature distribution at

r=A will approximate that which would exist in the absence of heat gains or losses. If Sm denotes the constant longitudinal temperature gradient in

the meter bars and Ss that in the specimen (in the absence of any heat

gains or losses), the temperature distribution, G(z), at r=A is

O^zAM v=T-Smz

M^z^M+L v=T-SmM-S s (z-M) (19)

M+L^z^W v=T-SmM-S s L-S m (z-M-L)=Sm (W-z)

The dimensionless temperature

Q^p^m

m^p^m+t *

• m+i^p^l

distribution, g(p), at P=a is

9=l-tmP

0=l-^mm-ts (H-m)

0=tm (l-p)

( 20 )
where

12

o, =

' = i§W

YU) rjn T S £

Substituting g(p) from Equation (20) into Equation (10) and performing the indicated integration,

fn(a)

n

2

7r 2

nrr nul

cos—sin—

2 2

( 21 )

It will now be helpful to eliminate \]rm and \]rs from. Equation (21).

If 0 represents the total power that would be flowing longitudinally

through the assembly if there were no heat gain or loss, then

Q=7rA 2 KmSm=7rA2 Ks S s ,

. (22)

where ttA2 is the cross-sectional area of the assembly'. In dimensionless

parameters, this becomes

q = _J1_

WTKr = 7ra 2 tm=Tr,a 2 O s^ s

Since tym=bs1''s j 2m.+l=l, and 2\]qnm.-h]/s l=l, then

ds -- -i i 1

1m - ds + (l-bs )l

and \j/ s =

o s +(l-o s )t

(23)

(24)

which can be substituted into Equations (21) and (’23).

It is of interest. to investigate the net fraction of power lost

or gained between p=0 and

T(hi

)

q

(25)

2.3 Matched Guarding . Since it is customary in thermal conductivity measurements to

match the guard temperature distribution to that along the assembly, the

first case to be studied will be that of a guard exactly matched in this

respect. If the guard temperature distribution, H(z), is identically equal

to that along the bars and specimen, G(z), the following equalities hold:

h(n)=g(|i) , fn (b)=fn(a)

and Equation (5) becomes

oo 0=l-p- fn(p)sin mr\i

n=l-

(26

)

where now

fn ( o ) =fn ( a ) Fo ^ nTr P * nT,?1

?

) ~ Fo ( n7rP ; n7ra ) ^nv F 0 (nma jrarb) (27)

From Equation (25)

,

the net fraction of power lost or gained

between p=0 and q=Pi becomes

Y(di )=b 3 (1 - ~)D(p! )=Ki(j^ - j^)D(q1 ) ( 28 )
13

CO where D(p x ) )Un(l-cos 2mr\x 1 ) n=l

(29)

and < = 3^ (30)

since Fi (x;x)=l/x. Also, since cos rnr/2=0 for odd values of n, n has been

replaced by 2n wherever it appears.

2.4 Unmatched Guarding

Any type of guarding desired can be studied by evaluating fn(b)

in Equation (11) for an arbitrary guard temperature distribution, H(z).

However, in this paper attention will be confined to cases of matched

guarding.

3. NUMERICAL EXAMPLES FOR MATCHED GUARDING

The quantity of most interest in the above development is y(M-i)

(Equations (25) and (28)), which is the net fraction of power lost to or gained from the insulation between p=0 and p.=Pi

.

Since D(pi) in Equation (28) is a function of dimensionless

geometric parameters only, it represents a geometrical factor which depends

only on the shape of the particular apparatus being considered. . For a given geometry, the fractional parasitic heat flow is proportional to the

thermal conductivity of the surrounding insulation and to the difference of

the reciprocal thermal conductivities . (resistivities ) of the meter bars and

specimen.

A sphematic diagram of a particular cut-bar apparatus is shown in

the upper left-hand corner of Figure 2. The meter bars and specimen are equal in length; this length is equal to twice their diameter. The radius

of the guard has been chosed as 1.5 times that of the bars. For numerical

evaluation, the specimen and insulation have been chosen to have, respectively, thermal conductivities 1/5 and 1/100 that of the meter bars.

The graph in the lower left-hand corner of Figure 2 illustrates

the dimensionless longitudinal temperature distribution that. would exist

along the meter bars arid specimen in the absence of heat exchanges with the

insulation. The vertical dashed lines indicate the, positions of the interfaces between the meter bars and the specimen. Since, in this case, an

exactly matched guard is being considered, the temperature distribution

shown is also that which exists along the guard.

The graph in the upper right-hand 'corner' of Figure 2 illustrates

the dimensionless radial temperature gradient which exists in the insulation at the convex surface of the meter bars and specimen. The quantity

shown, which is plotted against dimensionless' length, is (30/3p)pr=a from

Equation (12), where appropriate substitutions have been made corresponding

to the case of matched guarding. In this particular case, the radial gradient is substantially zero , except in the regions near the interfaces. In

this graph, the peaks at the interfaces have arbitrarily been cut off at

-1.0 and +1.0.

The effect of such a radial gradient distribution is shown in the

graph in the lower right-hand corner of Figure 2.* The fractional power

change Y(p-i), from Equation (28), is plotted versus dimensionless length.

The quantity plotted is actually 100Y (pi ) , which expresses the percentage

change of power flowing longitudinally through the meter bars and specimen.
lb

In this graph, the power flowing in the first meter bar is seen

to remain constant until the interface is approached. The power drops

rapidly near the interface, then levels off in the specimen to a constant

value approximately 2.2 percent less than that which was flowing in the

first meter bar. Near the second interface, the longitudinal heat flow

increases so that its rate is the same in the second meter bar as it was in

the first.

3 • 1 Influence of Thermal Conductivity

It was seen in Equation (28) that, for matched guarding

Y(Hi)=Ki(-^ -^y)D(p) . (28)

where D(p) is a geometrical factor. To illustrate the errors that would

arise if a range of materials were measured using the same meter bar and

insulation, .the fractional power change is plotted in Figure 3 versus

dimensionless length for a variety of specimen thermal conductivities.

The geometry of the apparatus and the ratio of the thermal conductivity of

the insulation to that of the meter bars are the same as in Figure 2. The

quantity 100Y(pi) is shown for specimens having a thermal conductivity

ranging from one-tenth to ten times that of the meters.

The effect of varying specimen thermal conductivity for different

meter bars is shown in greater detail in Figure 4. Here the coefficient

conductivity-term of Equation (28)

Ki(_L

Km

.JL. Ks

)

is plotted against Ks , the thermal conductivity of the specimen, for four

values of, Km, the thermal conductivity of the meter bars. The thermal con- ductivity of the insulation is held fixed at 0.001 w/cm-C. The quantity

plotted goes through zero at Ks-Km, approaches -oo as Ks becomes quite

small, and asymptotically approaches Ki/Km. as Ks becomes large.

'As will be illustrated later, a typical geometrical factor, D(p),

would probably not exceed a value of 2.5* If the vertical scale in Figure

4 were multiplied by 2.5, each scale division, 0.02, would become 0 . 05 , or 5 percent. Hopefully, the geometrical .factor might be made as small as

0.5, or 1.0, so that each vertical scale division would become 1 or 2 percent, respectively.

If both the specimen and the meter bars have at least one hundred

times the thermal conductivity of the insulation,, the choice of a particular meter bar is not critical. If the specimen to be measured has a thermal conductivity on the order of ten times that of the insulation, or less,

the choice of a suitable meter bar becomes quite important.

It can be seen from Figure 4 that the conductivity of the meter

oars should be near the lower end of the range to be covered, if it is

desired to measure a wide range of specimen thermal conductivities,

utilizing a single pair of meter bars. ’ If

v -

2 ( ) max ( Ks ) min . m (Ks )max+ (Ks )min ^ where (Ks ) max and (Ks )min ar $ the maximum and minimum spe-cimen thermal con- ductivities to be measured, then the maximum error to be expected within

the range is

±

Kj 1

2 (Ks )max (Ks )min

)D( *l) (32)
15

For relatively large (Ks )max 5 Equation (31) reduces to

Km-- 2 (Ks) min (33)

and the maximum error over the range (Equation (32)) becomes

±

1

2

Kj

(Ks )m.in

D(p) (34)

It is of interest to investigate numerically the significances of

Equations (31) and (32). Table 1 gives several ranges of thermal conductivity, the preferred meter bar for each range, and the maximum error to be

expected for a given geometrical factor (1.0) and insulation thermal con- ductivity (0.001 w/cm-C).

TABLE 1. MAXIMUM ERROR FOR VARIOUS RANGES OF SPECIMEN

THERMAL CONDUCTIVITY

Ks Minimum Maximum Optimal Km

Maximum

Error

in Ks Range

w/cm-C w/cm-C w/ cm-C ± °/o"~ 0.005 0.01 0.006

7

' 5.0

.01 .015 .0120 1.7

.01 .02 .0133 2.5

.01 .05 .0167 • 4.0

.01 .10 .0182 4.5

.01 1.00 .0198 5.0

.05 .10 .0667 .50

. .05 .50 . .0909 .90

' .10 ' .50 .167 .40

.10 1.00 .182 .45

.10 10.00 .198 • 50

1.00 10.00 1.82 ’ .045

3 • 2 Geometrical Factor

Returning to Equation (28), it is of interest to investigate the

behavior of the geometrical factor D(p). The numerical evaluation of D(p)

from Equation (29) has been made with the use of a digital computer, and

required summing the first hundred terms of the series in order to obtain

sufficient accuracy.

Figure 5 shows the geometrical factor, D(p), plotted versus p,

the dimensionless length. The geometry chosen is such that the meter bars

and specimen are of equal length. The radius of the guard is taken as 1.5

times that of the bars'. The geometrical factor is shown for five values of

L/2A, the ratio of specimen length to specimen diameter. Due to the

boundary' conditions imposed, D(p) is zero at p=0 and p=l, regardless of the

particular geometry. Sinc.e these geometrical factors are symmetrical about

the center line of the specimen, they are not shown for p greater than 0.5.

For the case of L/2A=0*. 2, a stack of disks each only 1/5 as thick

as it is across, the geometrical factor is changing over the entire length

of the meter bars and specimen; there is no substantial region whe.re the

longitudinal, heat flow is constant. Similarly, there is no substantial

region within the meter bars and specimen free of radial temperature

gradients.
16

For the case of L/2A=0.5, the geometrical factor at the center of

the specimen is significantly larger than that for L/2A=0.2. However, as L/2A increases further, the geometrical factor, near the center of the

specimen approaches a limiting value of about 0.54.

As the ratio of specimen length to specimen diameter, L/2A,

increases, the fraction of the specimen length over which D(p) is substantially constant also becomes larger. Similarly, the fraction of the meter

bars over which D(p) is substantially zero becom.es larger. To illustrate

this effect, a somewhat arbitrary criterion has been chosen.

Consider the dashed curve superimposed on Figure 5 . The portion

of each D(p) curve above the point at which the dashed line intersects a D(p) curve is within 5 percent of the peak value of that curve. This

dashed_curve and its mirror image (not illustrated here) bound the fraction, L/L, of specimen length over which D(|i) is within 5 percent of D(l/2).

In the insert in Figure 5, the fraction of specimen length, L/L,

bounded by the dashed curve and its mirror image, is plotted against the

ratio, L/2A, of specimen length to specimen diameter. This graph indicates

that only 4l percent of the specimen length satisfies the arbitrary flatness criterion f or • L/2A=0. 2 , while for L/2A=5*0, 93 percent of the specimen

length satisfies the criterion.

Figure 6 is similar to Figure -5) except that the radius of the

guard is taken as 3.0 times that of the bars. The vertical scale is five

times as large as that in Figure 5* The curves shown do not reach their

limiting value until L/2A=5-0 and even then, only 72 percent of the specimen satisfies the flatness criterion.

Figure 7 shows the 'geometrical factor plotted against dimensionless length for a specimen length to specimen diameter ratio fixed at 2.0.

Curves are shown for guard to specimen radius ratios, B/A, ranging from

1.125 to 3-5. The geometrical factor at the center of_the specimen

increases rapidly with B/A. For small values of B/A, L/L is quite large,

but decreases rapidly with increasing B/A, as shown on the inset.

As was noted in Figures 5 and 6, D(p) reaches a limiting value

for a fixed' B/A as L/2A becomes quite large. This limiting value,

D(l/2) max , is shown plotted against B/A in Figure 8. Regardless of the

apparatus geometry, D(p) will never exceed D(l/2)u]ax for a given guard to

bar radius ratio.

Considerable insight into the nature of D(l/2) ffiax can be gained

by consideration of- a simplified model. It has been shown that, for sufficiently long meter bars and specimen, the power flowing longitudinally in

the meter bars is constant, until the interface with the specimen is

approached, and that the power flowing in the central portion of the specimen is also constant, but at a different value. The portion of the insulation near the bar assembly will exchange heat only with the assembly;

similarly ? the remainder of the insulation will exchange heat only with the

guard. The geometrical factor, D(l/2) max , can be regarded as the ratio of

the cross-sectional area of the insulation which exchanges heat only with . the assembly, to the cross-sectional area of the bar assembly.

4. DISCUSSION

The error due to neglecting the longitudinally shunting effect of

the insulation has been analyzed for a cut-bar apparatus having an exactly

matched guard. For such a guard, longitudinal heat flow in the insulation

can be minimized by making the radius of the guard very nearly that of the

bar. In practice, an exactly matched guard is not feasible, so that
17

radial heat exchange between the assembly and guard due to a mismatch also

must be considered.

Assume that the largest probable mismatch between corresponding

longitudinal positions of the bars and guard is e , a dimensionless temperature difference. The worst case would be that of the entire guard being

displaced by an amount, e, so that h(p)=g(p)+e . In this case, Equation

(11) is the same as Equation (10). Therefore, the radial heat flow between

the assembly and the guard is independent of the shunting heat flow and,

from Equation (18), the heat flow to the guard due to a uniform mismatch is

seen to be given by the usual logarithmic expression. The error due to

radial flow can be computed for various guard diameters and, by comparison

with the error due to shunting heat flow, a guard can be selected so that

the sum of these errors is approximately minimized.

An analysis similar to that presented in this paper is being used

at MBS to correct for heat exchanges in the absolute cut-bar apparatus.

The analysis has been very helpful in aiding our understanding of the problems involved, and is in fact regarded as essential in the effort to reduce

uncertainties in the results obtained with the apparatus.

It is hoped that the analysis and discussion presented .in this

paper will be of assistance to those who are utilizing a cut-bar apparatus.

It is realized that many laboratories will not find it convenient to resort

to a digital computer for evaluation of their particular geometrical

factors. Mr. Robinson pointed out the advantages of utilizing a substitution method for calibrating a thermal conductivity apparatus. By using

several references as substitute specimens, the effect of shunting heat

flows in the insulation can be included in the calibration, and the geometrical factor, if desired,' can be experimentally evaluated.

REFERENCES

(1) Carslaw and Jaeger, Conduction of Heat in Solids , Second Edition,

Oxford University Press (1959) PP 21H--229

(2) Watson, Theory of Bessel Functions , Cambridge University Press (19V+)

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II. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

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its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

front cover.

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