Ames Laboratory ISC Technical Reports Ames Laboratory

12-1951

Thermal conductivity of metals at hightemperaturesPaul H. SidlesIowa State College

G. C. DanielsonIowa State College

Follow this and additional works at: http://lib.dr.iastate.edu/ameslab_iscreports

Part of the Ceramic Materials Commons, Engineering Physics Commons, and the MetallurgyCommons

This Report is brought to you for free and open access by the Ames Laboratory at Iowa State University Digital Repository. It has been accepted forinclusion in Ames Laboratory ISC Technical Reports by an authorized administrator of Iowa State University Digital Repository. For moreinformation, please contact digirep@iastate.edu.

Recommended CitationSidles, Paul H. and Danielson, G. C., "Thermal conductivity of metals at high temperatures" (1951). Ames Laboratory ISC TechnicalReports. 36.http://lib.dr.iastate.edu/ameslab_iscreports/36

Thermal conductivity of metals at high temperatures

AbstractA new method of measuring thermal diffusivity and hence thermal conductivity of metals is suggested. Likepreviously reported dynamic methods, this method uses a heat source, whose temperature varies sinusoidallylocated at one end of an effectively infinite rod. Unlike these methods only one period of the heat wave isrequired to eliminate the unknown coefficient determining the heat lost by radiation since both velocity andamplitude decrement of the heat wave are measured. The new method is faster in taking data and simpler incomputation. The thermoelectric potentials from two thermojunctions are amplified and plotted on a Brown"Electronic" recorder in order to obtain a permanent record of all necessary data for computing the thermaldiffusivity. Results for copper over the temperature range 0-560°C and for thorium over the temperature range0-430°C are given.

KeywordsAmes Laboratory

DisciplinesCeramic Materials | Engineering | Engineering Physics | Materials Science and Engineering | Metallurgy |Physics

This report is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/ameslab_iscreports/36

ISC-198

THERMAL CONDUCTIVITY OF METALS AT HIGH TEMPERATURES

By Paul H. Sidles G. C. Danielson

December 1951

Ames Laboratory

II T uh n; <a I In fa' m a,; on S uv; <e, Oak R; d ge, Ten ne n u

METALLURGY AND CERAMICS

Reproduced direct from copy as submitted to this office.

PRINTED IN USA PRICE 15 CENTS

Available from the Office of Technical Services

Department of Commerce Washington 25, D. C.

Work performed under Contract No. W-7405-eng-82.

AEC, Oak Ridge, Tenn.-W22419

3 ISC-198

TABLE OF CONTENTS

Page

I. ABSTRACT -·-<"" 4 . II. INTRODUCTION • •· 4

III. THEORY • . • . • .. . • . • • • 5

A. Basic Differential Equation • 5 B. Theory of Velocity l1ethod • • 7 c. Theory of Amplitude Decrement Method 8 D. Theory of New Method. 8

IV. APPARATUS 9

A. Samples . 9 B. Sinusoidal Heat Source 11 c. Bucking Potential and Thermocouple Switching 14 D. Amplifier and Recorder . 14 E. Furnace . 14

V. PROCEDURE 14

A. Calibration . ·". • 14 B. !1easurement . 17 c. Calculation • 19

VI. RESULTS . . . ..... .. ... • -. 19

A. Copper. 19 B. Thorium • • • · .. • • • 21

VII. DISCUSSION . • .. • • 23

VIII. REFERENCES • 24

THERMAL CONDUCTIVITY ·oF METfU.S AT HIGH TEMPERATURES*

by

Paul-H. Sidles ann G. C. Danielson

I ABSTRACT

A new method of measuring thermal diffusivit.y and. hence thermal conductivit.y of metals is suggested. Like previously reported dynamic methods, this method uses a heat source, whose temperature varies sinusoidallyi located at one end of an effectively infinite rod. Unl:i.ke t4ese methods only one period of the heat wave is required to eliminate the unknown coefficie~t determining the heat lost by radiation since both velocity and amplitude decrement of the heat wave are measured. The new method is faster in taking data and simpler in computation. The thermoelectric potentials from two thermojunctions are amplified and plotted on a Brown "Electronic" recorder in order to obtain a permanent record of all necessary data for computing the thermal diffusivity . Results for copper over the temperature range 0-560°C and for thorium over the temperature range 0-4300C are given.

II. INTRODUCTION

. Many methods have been suggested for measuring the thermal con­ductivittes of metals at high temperatures . These methods may be classified into two basic categories~ static and dynamic. The earliest measurements· at high temperatures were made by Forbes (l) who used a static method. The Forbes method~ though modified considerably by

·Baillie (2), Lees (3), Bidwell (4) andcthersi consists essentially of heating _one. end of a bar, cooling the other end, and measuring ~he steady state temperature distribution along the bar. In order to calculate the thermal conductivity the heat lost from the surface of the bar must be either measured experimentally or reduced to insignificance. Reduction of the surface heat loss can be accomplished by thermal lagging or by enclosing the S8.lilple in a vacuum. However, the problem of experimentally eliminating or measuring the surface heat loss limits the temperature · range over which the Forbes method is useful. At low temperatures radiation losses are n~gligible and the metal specimen can be enclosed in a vacuum .to prevent conduction and convection losses.

"*This report is based on a masterqs thesis by Paul H. Sidles, submitted December, 1951.

5 ISC-198

Therefore, no appreciable heat loss lvill occur and the Forbes method can be used ui th little difficulty. .itt higher temperatures, especially at tenperatures above a fe1v- hundred de c;rees centigrade, radiation losses become larEe and additional precautions are necessary. Guard rings maintained at the averat~e temperature of the sample are often used to reduce this radiation loss.~, ~ Ilouever, f:Uard rings are difficult to maintain at a constant temoerature at high temperatures and are not conpletely effective at any temperature since the bar possesses a temperature r;radient not _ possessed by the guard rinG.

Another sta.tic method ~-lhich has been employed over a la.rt;e range of tc~:rperatures is that proposed by Kohlrausch (5) and used 1-rl th various modifications by Jac~::er and Diesselhorst (6)", Angell (7), 1'1eissner (8) and others. Basically this method consists of medsurinc either the r adial or the longitudinal temperature- distribution in a cylindrical sa-<ple ~r:1ich is heated by an electric current aDci cooled by surface radiation or electrode conduction. This method also requires that elaborate precautions be taken to eliminate heat loss corrections.

A dynar.j_c ~net!10d L'irst used by Kine (9) <md later :nodified by Starr (10 ) is t~e method 1rith 1·!hich this investi ;:_:ation is concerned. This nethod uill be discussed in detail. In this case the surface ;-;.eat loss need not be either 1:1easured or reduced in r.J.agni tude in order to c.:J.lcLll:ite the t:1cr:n<!l conductivity.

E:;-.'"}Jerime ntally all dyn<ulic j!J.et:-IOds are s i ;:ri.lcir. •>- heat source :!!lose t em;)cra ture va ries s inusoidally i !'lpresse5 a sinusoidal heat 1·JaVe at one end of a lonr~ rod. Dy samplinb this hca.t ::ave a s it r:1oves dmm the rod the necessar"J inforndtion for deterr:lining tl1e thernal conductivity of t!1e rod may be obtained.

III. THEOHY

A. Basic Differential Equation

The differentic.l equation for hea t ;lo:; in an infinite rod is

k a2• - = ~

vrhcre g is the ternperature, t is the time, and x is the distance

(1)

neasured along the rod. By definition, the thermal diffusivity lc is

k = K/cd (2)

1-1hcre K is the thermal conductivity , c is the specific heat and d is

6 ISC-198

the density of the material. and the temperature difference small enough for the heat loss the differential equation .._rill

If the rod is .radiating to its surroundings bet\veen the rod and its surroundings is to be a linear function of temperature, include the heat loss term pe

(3)

where ~io~he coefficient of surface heat loss. is given by the equation

This coefficient

jJ : Ep/Acd (4)

where E is the emissivity of the surface of the rod, p is the perimeter, A is the cross-sectional area and c is the specific heat of the rod.

The solution of this differential equation under the boundary conditions

x = o, e = 6]. ... e2 cos w t

x = oo~ g = 0

is

lvhere

o= F ~ ~Vw 1') 0( - 2: - + (A} 1'

-

fi - ~2~ ( V? 2 + (A)2 - }I)

(.5)

( 6)

(7)

(8)

(9)

An examination of either ex or f3 shows that lvith the exception of all of the quantities necessary to calculate k can be measured experi-mentally uithout difficulty. The quantity fl, however, becomes in-creasingly difficult to measure accurately or to requce to a negligible

7 ISC-193

value as the temperature is increased. An alternative to measuring J.1 is to measure other quanti ties necessary to obtain t1w expression inp and k and thus eliminate f'. Three Hays of acconplishine this will n011 be discussed • .

B. Theory of Velocity Hethod

In the r.tethod :1roposed by King (9) a sinusoidal heat source is placed at one enc;l of a long rod and the velocity v of the heat Have is measured in passinz bet1Jeen tlvo fixed points on the rod for tuo different periods T of the sinusoidal bo'.lnda:rJ condition. This deter-mines the quantities fJf, and p2 corresponding to ev1 and ev2• By elimina.tine p bet\-;een t .1ese e)-.-pressions u. value for the thermal diffusivity k is obtained.

If ~ is the 1-lavelenc;t h , eqm~tion ~ G ) sho:,;s tl1at

px = (x/A ) 21THhencc v = )/T = 21T/T f or j3 = 21T • vT

Using cquatio!r'(lO) to repL:iCe f3 1 . and ft 2 by v1 and v2, after eliminatinc; fl ,

k =

(10)

(11)

<

The therr:ul cond-,tetiTlty is then i.'ound by :n-.Jltiplyinc k by the product of the s :1ecific !1eat -::J.nd the density of t:1e rod.

K = ked (12)

'· .:.:··

.t.

8 ISC-198

C. Theory of Amplitude Decrement l1ethod

Starr (10) measured the amplitude decrement q of the heat wave between tvm points on the rod for two different periods T1 and T of the sinusoidal boundar,y condition. The two resulting expresgions for 0( and W >vere used to eliminate the radiation constant }/• By definition,

e -ocXJ. q = e4i<X2 whence ln q = o<(x2-x1 ) = o<L or o<=

lnq L

Using equation ~13) to replace~ and 0<2 by q1 and q2, after eliminating }I,

k

where

b

(13)

(14)

(15)

The thermal Conductivity is given by equation (12) as in King's method.

D. Theory of NeH Method

Both of the above methods, in addition to reqUlrlng lengthy calculations, also require that measurements be made over a considerable period of time. After the necessary data for period T1 has been taken, the period of the sinusoidal boundar,y condition must be changed and the rod allovmd to reach an equilibrium condition for the net-T period T2 before the r emaining data can be taken. During this comparatively long period of time required to reach a second equilibrium and take a second

ISC-19S

)

set of data, errors ~nay enter altering the rd.diation loss. from the sinusoidal heater to

into the e:h.-peri ment ·due to surface reactions AlsoJ the efficiency of heat transfer

the sample may ch<lnge.

To reduce this possibility of errors arising from the variation of experir.1ental conditions 11:i. th tb1e and to allmv measurements to be :;-ude :nore r a:?idly, the follo;1·lng method 1ms used. Instead of eit,her the velocity or the a~1~litude decrement being neasured for two periods of the heat Have, both the velocity and the amplitude decrement Here measured for a single period.

The anplitude decrement q, £:Jeasured between tHo points on the rod separated by u. distance L, determine the quantity o< by equation (13). The velocity v and the period T determines the quantity f3 by equation (10). EliminatinG t:-1e unkno:m radiation constant f1 by equating the product of «and f_from equations (8) and (9) to the product of a<and /}from equations (10) a .. :1d (13)

or re?l:lcing tAJ by ( 21T /T),

k =

(A)

2k

Lv

. (;;r )(¥)

• 2 ln q

i'ii th this vill. ue of k the thermal conductivity is determined from equation (12) as before. The expression for k u:;;ins this ne11

(16)

(17)

method is :~1Uch :::;i:npler in i'orJJ V1an for either of the :)receding methods and !;.:J.s the added advantdce that it is independent of the per:Lod of t::1e sinusoidal bounda.rJ condition. The nu;uber oi' qu-1nti ties ~i'1 ich must be :'ncasured for a determination of k is reduced, from the four quanti tics required by ei thcr King 1 s 1:1ethod or Starr 1 s :ncthod, to the t~ro quantities q and v. 111is nmr method, Hhich uses only one period of the heat >:rave , is therefore not only superior in reliability but also sinpler in ta.kine clute.~ and si::1pler in computation.

IV. APPARATUS

A. Sanples

j~ block diagram of the apparatus for !Tleasuring thermal conductivities usinr, thi5 ncu nethoci is shmm in Figure 1. The s<:Ll;::>les that 1..rere meawrcd <!ere app roximately 1/El" in diameter and at least 50 em in length.

SINUSOIDAL BUCKING POTENTIAL : ) HEAT SOURCE ) AND

THERMOCOUPLE

SWITCHING

VACUUM PUMPS AND D.C. AMPLIFIER

GAUGES 0 /<= (<:: ;>

FURNACE ./ K RECORDER · -POWER SUPPLY v

----- -

APPARATUS FOR MEASURING THERMAL CONDUCTIVITY ·

Fig. 1

'

1--' 0

H

~ I

1--' \() co

ll ISC-198

Ttm American \·Jire gauce 28 chromel-alunel thermocouples ;Jere peened into holes drilled in the sar)1ple •·Iith a No. 70 drill. 'l'he thermo-couples were separated a distance 1 Hhose magnitude 1vas chosen according to the thermal conductivity of the sa11ple. i<'or metals <lith high thermal conductivity the l1eat 1-.rave travels -:nth a higher velocity than for those ',Jith loH thermal conductivity. Therefore the thermocouple separation 1 •-ras selected so that the time of travel and hence the velocity v could be most conveniently measured. The maxhmm separation ivas 16. 82 en in the -case of copper and the rrininum >-Ias 5. 32 em in the case of thorium.

A distcnce o.!.' several centimeters separated the sinusoidal heater at one end of the rod and the nearest thermocouple. Thus any irregularity in the cross-sectional distribution of the heat ~vuvc in the vicinity of the hcuter •muld be s :aoothed out by the time it reached the firnt thermocouple. The re;aainder of the s-i·.:ple 1vas coiled into a helix as shoun in Figure l so that it could be acCOl'lJJodatcd in a small furnace. The sample ''at enclosed in a conkincr uhich Has evucuatcd to a pressure of 5 x lc- l'lt''l Hg.

B. Sinusoidal Ilcat Source

Before the 1}--.cory >lhich has been developed can be ap:Jlied to the measurement of thcrm.:.:.l cond"J.ctivi ty a r1ec.ms must be devised to cause the te :·1~erature at one end of a lon,s rod to vary 1-1ith time according to the cx~)ression 8 = ~ .,. G2 cos wt. Ideall;;r a sa;nple heater ~~hose terr~per.:1turc varies u.s a cosine function should be used. Hm-1ever, it is not possible to generate this function using electric heaters since such heaters can supply heat but cannot extract hee1t i'rom the sa:nple. The sinusoidal heat source used in this invcstigution is shmm in i"ieure

· 2. The dis~;lacement of a ~oint on a disk 11hich is rotatinc; .nth constant aneuhr velocity is a sinusoidal function. Using the apparatus detailed in Fu~;ure 2 this displacement uas used to rotdte a Hodel ll6U Pm-rerstat V;.triable transforr;ter so t:"1a t its output volt<J.e;e varied a:s (1-cos wt). 'l'his potential \Jas a_LJ~;lied to a resistance heater, ulso detailed in .i.•'~01re 2, so that the ~oat develo:)cd in the heater v:J.ried as (1-cos wt)~. ~ii th tl1is heat function applied at one e:1d of the rod the 1:rc.;.ve l'orm of the res:ilting temperature at the nearest thermocouple -h,.c.iS exu:rined. 'I'his uu.ve i'or;7t 1-i'<.i.S, to a r,ood Cl.pproximation, sinusoidal in form and t.."le required boundary condition -v;as therefore satisfied.

IIOV AC.

Powers tat Model

116 u

SV filament J transformer

Po•eratat Shaft 3.0cm. --~66cm 1 w

~ I...

DETAIL OF POWERSTAT DRIVE

Powerstat Model

116 u

--®-~ Sample Heater

Gear Box Shaft

Leatha

Gear Box c 78248 .. ...... ( 100• I ( Synchro , .... Speed ..... ...

Reduction Transmitter

Variable Speed Driving Unit

~-------3.0cm--------~ Q()()41 w wire

1.7mm ~2~~:...:..-----~ .Pyrex glaa

DETAIL OF SAMPLE HEATER

SINUSOIDAL HEAT SOURCE

Fig. 2

~

H (/) 0 I

~ (X)

13 ISC-198

As a further check on the justification of this procedure, an examination 1..ras made of the effect of different periods and different heat inputs on the measured values of thermal conductivity. This examination shoued that the effect of vai'"'fing these quanti ties over a •·ride r ange had very little effect on the measured value of thermal conductivity. Hmvever, it 1:1as found that, when the period of the heat function Has creater than 3 minutes, laree pmver inputs caused an increase in.the value of k. For this reason most of the measurements of k 1..rere made using periods shorter than 2 rninutes and power inputs less than one-half that required to cause an increase in k. A typical operating condition would be a period of 110 seconds and a pmver input of 0.30 1-1atts. These conditions -vwuld result in a sinusoidal temperature variation at the thermocouple nearest the heater of 0.5°C.

In the early stages of this investigation heaters 1·1ere either uound tightly on a smooth sample or wound in threads cut on the outside of the S:.l ''lple . These heaters ;mr e insulated from the sa'11ple by the oxides 1hi.ch were fonne:l m .the hmters -whm tmy •·a-e reata:J:. in air • . Trese oxfu:ioo .a~d to l::eccine unsiahle wten .. tl:E :rU:l ms heate:l 1..'1 va.c:w.m to hi£:11 terperatures md. 03USEr.L the h::ater' .to S:xrl to the S:lznPI.B o ~ For this rea.smthis tyrem lmter~ct:andcn~ . Imters,-m.ich ~ s:p:Uated.f'ran 'tm L: ~·c:1ple by a sm:tll distance, uer·e also tried. , These heaters ;:>erform.ed satisfactorily when making measurements near room temperature in an inert. atmosphere. Hmvever, under vacuu!Tl conditions vmere radia tive heat tra'1sfe r predominates, this tJ~e of heater was unso.tisfactor.r due to ineffi cient heat transfer to the sample. Since the solid angle subtended by that :portion of the sa:nple directly beneath the heater <vas less than 27T, more than !1ali of the heat supplied to the heater Hi:J.S lost to the sample cont<:tiner, to the remainder of the sample, and to the t hermocouple leads.

To overcome these undesirable conditions the type of heater shovJn in detail in Figure 2 1-ras developed. These heaters were 1.vound 1vi th 4 mil tunesten wire using a No. 70 drill as a mandrel. Jlfter winding, one end of the wire 1-Jas pulled through the coil and insula ted from it by a fine Pyrex glass capillary. This assembly vw.s placed inside a Pyrex tube 3 mm in inside diameter. This tube was then heated to the softening point and :pulled dm·m around the heater in order to have a thin layer of Pyrex glass on the outside. This layer of glass served to support the windings and to insulate them from each other and from the Sillnple. The completed heater vJas dropped into a hole drilled axially with a No. 50 drill into the end of the sample. This t)~e of heater has performed satisfactorily in vacuum up to a temperature of 650°C, the maximum tem­perature attained in this investigation. No material was placed between the heater and the sample to improve the thermal contact. The solid angle subtended by the sample with this type of heater is nearly 4~ so that nearly all of the energy supplied to the heater is transferred to the sample.

ISC-198

C. Bucking Potential and Thermocouple Switching

· The outputs of the thermocouples >vere fed into the circuit shown schematically in Figure 3. Here the thermocouple outputs were opposed by de potentials 1.ffiich corresponded to the ambient temperature at vlhich the measurements were beine made. Only the sinusoidal component orthe thermocouple output was left to be amplified and recorded. This circuit Has contained in a constant temperature box to minimize thermal strays and to provide a constant cold junction temperature. Matched copper and low thermal solder were used in the connection from this unit to the de amplifier. By using a 10 K Helipot in each bucking potential circuit a convenient means was provided for independently positioning the outputs of the thermocouples on the recorder chart.

D. Amplifier and Recorder

A Liston-Folb Hodel 10 de amplifier and a high speed 0-12. mv Brown Electronik strip chart recorder Here used to make a record of the data necessary for calculatine the thermal conductivity. The Liston-Folb am?lifier had a maximum gain of 80 db . and a frequency response sufficient to follow a one cycle per second signal. The Bro1m recorder had a full scale pen travel time of 2 seconds ~•hich uas more than ample for it to follm.r the sienals used in this investigation.

Built-in calibration circuits in the Liston-Folb amplifier lvere used to calibrate the ' amplifier-recorder system. Since the output voltaee of this a:nplifier is c;reater than 12 mv a voltage divider viaS required bet-v1een the a::tplifier and the recorder.

E. Furnace

The furnace for heating the sample >·las a conventional resistance furnace. The temperature of the furnace 1vas controlled by the apparatus shmm schematically in Figure 4. With this apparatus the temperature of the sample could be held sufficiently constant during the time interval required to take the necessary data.

V. PROCEDURE

A. Calibration

The temperature of the furnace containing the sample was set at the temperature at Hhich the measurement 1vas to be made and the control system v1as set to hold the furnace at this temperature. When an equili­brium condition had been reached the following calibration procedure was

Thermocoupi. Cold Junctions 2'" I+ 1- 2+

I ' I

Some carcuit as ~5V. shown for Thermo- 10 10 -couple I.

~ ~

~ 10 ~ ~a; ~ 10

Q 4 l ---------1_ 1.5V""'" ~ + +

- ~

!!:?

8 8

+ -·-r----r4 --• t

Special 1.~. lo

0 ~ ::::E C'

~ ..,

Q ::::E

4 I D.C. Amplifier To Galvanometer

Fig. 3

~ ~4 56 -18 -3 : g~

~2 I io-u: • ~ ~ ~ 10 us 2 !II: <i 10 !(') C\1 t:

~ .... 56 7

~4 8 3 9~

-2 10-I lie

!II: CD -~ ,.... 0 8 Q CD C\i 'ii (1) n C\1 !(') C\1 :J: !:

~ 0

BUCKING POTENTIAL AND THERMOCOUPLE SWITCHING.

!II: !II: C\1 Q ~ !!:?

§ CD ;:::

!II: CD' -::

~~

I-' \J1

H

~ I I-' \0 ():)

10 v ~.c. I A

~

Power stat Model 1126

Powerstat Model 1126

' ~ M-H

M904E Motor

8 z I 0000 A

000

II II Brown Electronic

Proportional

Controller

FURNACE POWER SUPPLY

Fig. 4

c 0

c Chrome I c

Alumel c

Thermo~

~ c

0

c

Furnace

f-J 0\

H (I) 0 I

(0 ())

J!..7 ISC-198

carried out. The gain selector of the amplifie:r- 1.J'as set at the desired value. The output of one of the thermocouples was switched to the amplifier input and the position control for that thermocouple used to position the recorder pen at the lm.J'er end of the recorder chart. A knmm calibrating potential vras then inserted in the thermocouple circuit using the built-in calibrating equipment of the Liston-Folb ampli.fier. This caused the recorder pen to be deflected up-scale. The calibrating potential was then removed and the recorder allowed to return to its original position. This procedure uas then followed for other gain settings of the amplifier and for the other thermocouple circuit. When all necessary amplifier gain-settings were calibrated the chart was removed from the recorder and the pen displacements· were measured as accurately as possible. Dividing each calibration voltage by the pen displacement caused by that voltage gave the sensitivity of the amplifier­recorder system for that particular gain setting of the amplifier.

B. :tvieasurement

After the system had been calibrated the sinusoidal boundary condition was applied and the sample allo1-1ed to come to equilibrium. When equilibrium had been reached and the outputs of both thermocouples had been positioned on the recorder chart the output of the thermocouple nearest the heater \•:<..t.s recorded for several cycles. Immediately after either a maximum or a minimum vias recorded the amplifier gain v1as changed and the recorder sHitched to the other thermocouple in order to record several cycles of its output. A tJ~ical record is shown in Figure 5. The a:-1plitudes of every cycle :.v-ere measured and the average amplitude for each thermocouple output was calculated. The calibration data previously obtained allowed conversion of these distance amplitudes to voltage amplitudes. The ratio of the amplitude for the thermocouple nearest the heater to the amplitude for the more distant thermocouple is the amplitude decrement q. This is one of the two quantities which must he measured in order to determine the thermal diffusivity k.

The other quantity r:.1hich must be measured is the velocity v which · Has found in the follo>·ling manner. The one-quarter, half and three­

quarter points uere geometrically determined on the chart for the nearest cycle on each side of the point at which the thermocouples \vere switched. The six points thus determined for the output of the thermocouple nearest the heater >vere then advanced along the chart a distance corresponding to one cycle. The distance m along the chart separating each of these points from the . correspondine point on the output of the farthest thermo­.couple is a measure of the velocity of the heat wave in passing between the t-wo thermocouples. This distance, together ivith the chart speed and the thermocouple separation, determines tne velocity.

18 ISC-198

· · ··r .. : --. "P:liJft: r::t~-:-· - : .. -.:ttl.{ ' . 7 . ' I ' ' I I .

. . . i::· ~l ·' j -+*lii' . 1 : . 'l i ; j! ')

. . . : ' ' : . ' ' .. I I !. I I . ' I ! . ~ , . ,,, : ... . . . : : .-! _:.~-~-'- . . . : : ! ! I ! . . . ! .· . ' . I . . ! : . •

,, . :\I . - t : ,' f ~ - ~. ..~ . . 6 . . .

I . .! ~ ,...~ I ~ . ~- ·--....:_:__ . . .. . - . . . . •

- ~ oi :1 . . .. . ~~ . . . r .o·

: ~ I;~ l _! .l . __ ... -s ··""~~-~ ~ ~ -· •• ~ . . I . I .. ____ , .. -. •

. l . . - i,.--- . ' : ; i i : : ; •

•• · · · 1 · -~ : ~-1:1:'--;~~a · :·: ;o·: :r J: : ·l~m~· ··· .. .. I~, ;u: • ~ I 1: . . :-..~ ' . ' . ' : ! I 11 I • • .. . . . I· I ! .. I · :.d~-~ ~ lTr ill • : :. - - I! ~ .. : . . ' . ' : : lTD: ' !! i : • . ,. . . I ~- I . i . ~n-. ! II : ~,. I • • I : . --3-~ ~ ' ~ . : I ; ! ! 1-1 • ! ' ,; ; ' ' I I • -"! ... : _ -:-r ...,.--::-_ . ·-;; .. . . .. .. .. .. ·

1._ .. ~:~· ~-: ... 1-1rf •

1 · · I ~ - ' · . rT1 ' : ' i I I • •• ~ ·~ _ I :r ... ,n a , ;•Jo :iii .. :: li ·~.

: .. · 1 ·X ~ - 9 , -: : · r : t; -r !i 1 :

~ i ··- j .. 2 . . : . : I I I .J.tl • <. : .. i . ' .. > . "-'~~ . -·--·: - ·---T~- t- --~. :-1-tl :! ' ( •

::! .. ... . .l . ~~=~:±~ -U~~ iTi I_ -· : I ' ! -~~~ . : • ! ~ . ! : I I ,I .• ~-! ' •lo ;!' •• :.: . a~:~:f: : j•o i i i i••i .·~~

~~ -~ :. ~ i , :1 ·r-· .1.: .. -tl ... II .. - -- ' I : :I: I i j : I ; 1

--~ '1 !!I: i !'l · I

- ~ -~~~ ;··· · . rtt ritr +t l ·· • ··. ~ll ~ .2 1 ,,

. ~ ........ ··· : ~:.~:. : ;·: ; · t .. tt·: +r .. .. ~#'. . . I--1--H--H-H-+++-H-H+I .. •

. t~i:.: ···:·!Ji' ··t, l i ,•.Jj i ~ ~~ . ~~ ~ - ... Ul __ . .. . ,.. . : .~~- . ... ~~L., . u: f' t~ .. -~ ·: +~~ --+.q ..... r~ .. L. : :· ....... ...... ..... .... - ... ... ... .... . .. .. •

. r~ .. , t+ -~ ...... , .. I l !-+· .. l·Ff . .. 1·· ++-1H++-' H' -H-... 1-ft:,.-+.F +: H: : : . ~ + :. ; +-4- kf .. J. L.,.. _ ~rr ...... ~+R.. . --+-+-++++-++-+ ___ ;. .... . .. ... . :::: .: ~ , ... . :.· •

• • ... ~. - -~~ .. --;, .... ·.i ~--·.mt· rr .. ~ ...... +± #t.t'~ I j ~~ ... ~ .. : . . ... ..... :. ~- . ::: ..... _;; -~ j !: : ... : : - ..... .1 ~ ~.-~ . ; i i i - · r-+~...l ....... -· .. I "t . ... .. I ·I ' . . .. . . .. . . . . . . .... ... . .... .... ... ; ·''· ..... . : 0

: ; ... :!! : - ::: ·~ , : ; . ,j····"' r::~"""' - ~~ . .. ~- ·- ~---· : .. ... 1"!~: .: ; .1:~~~ 1 r-1-...:.....!... ----- : · : ~d IU 1, . -N- ~ ~

-~-~:· :- . : ,.; o~ .. , ."~ - ... ~ - - · . ·\ · l : . . . . '• . ~ . . l

i .. -.. __; >' ..

·' I;

t

I'

I'

:'\ ...... .. . - (•

• • • • •

ISC-198

C. Calculation

Since the velocity v = Ls/m l·lhere 1 . is the thermocouple separation, s is the chart sp~ed, and m is the linear separation of corresponding points' 1-.Te can lvri te

k ·= 1v .,--:::--2 ln q

= 2 m ln q

. 12 = :~ 2 ll ~. Lin 1n q:=J

).

(18)

The thermal diffusivi ty k Has thEm multipiied by the specific heat and the density of the sam.ple, using the best available data, to obtain the thermal conductivity K.

VIo RESULTS

A. Copper

The results of usinr, this new 1nethod for measuring the thermal conductivity of copper are shmm in tabular for:n in Table 1 and in graphical fonn in Figure 6. 'l'he points at 36°C andl36°C on the thermal diffusivity curve a1~ the result of three independent measurements •mile the remaining points are the result.of five independent determinations. The thermal conductivity of copper was calculated by multiplying these values for the · thermal diffusivity by the product of the specific heat and the density of the sample. The International Critical 'l'etb.les (11) give cp = 2h.33 + 6.63 x lo-3 t joules as the best value deduced from all available information for thempl~~ecific heat of copper over the temperature range from 0 to 500°C. ~ihile no accuracy 1vas given for this value it ,.;as stated that specific heat accuracies for metals are rarely better than one per -cent and ~~certainties of several per cent are not unusual. ii. value of d. 92 em/ cm3 at 20°C is given by the International Critical Tables as the density of copper. These values for specific heat and density 1·tere those used :in. calculating the thermal conductivity K from the themal diffusivity k. The results of this calculation, after small corrections have been made for the change of density d and thermo­couple separation L 1-vi th teP'lperature, <lrc given in the right hand colunm of Table 1 and in the lovmr portion of :Figure 6. The correction for changes of density and thermocouple separation 1vith temperature Here 6 made using a value f or the coefficient of thermal e.xpansion of ~OT;x: 10-per degree centigrade as given by the Ibndbook of Chemistry and Physics(l2) for the ter:~erature range 0-625°C.

A literature sm·vey sho~ved that ver;- fevr measurements of the thermal conductivity of copper as a function of temperature have been reported. For purposes of comparison the values for the therm:~.l conductivity of ·· copper as taken from Smithells' Hetals Reference Book (13) are also shown

-r L - THERMAL DIFFUSIVITY OF COPPER (a)

!lJ -~ > i1S

~ I ~ 4.0

I -~ THERMAL CONDUCTIVITY OF COPPER (b)

3.90~ ~ I f------- - ~ I 1\)

0

-~ .....

~ __ , 0 CALCULATED FROM K•llcd

() SMITHEU.S -METALS REFERENCE BOOK(I949)

e WLKINS a BUNN -COPPER AND

I I COPPER BASE ALLOYS (1943)

~

0 100 200 300 400 ~ 600

TEMPERATURE, (•c) H

F5 Fig. 6 I

1-' \0 ():)

Temperature (OC)

36 136 246 323 414 476 561

21

' Table 1

Thermal Diffusivity and Thermal Conductivity of Copper

Thermal Diffusivity (k) Thermal Conductivity (cm2/sec°C) ( vJatts/ cm°C)

+ + 1.157 0.014 3.988 . - 0.049 1.094 + 0.013 3.864 + o.o45 -1.039 :t 0.020 3. 770 !' 0.069 1.014 "' 0.011 3.747 + 0.038 - -0.9745 + 0.0078 3.677 + 0.027 - ... 0.9363 .f. 0.0063 3.582

,. 0.022

0.9024 + 0.0025 3.517 + 0.009 -

ISC-198

(K)

in the lmver portion of Figure 6. These values 1r1ere taken from a table of thennal conductivities of metals ·tvhich had been compiled from data taken from International Critical Tables and subsequent papers. The single point shmm at 20°C 1v-as also taken from the Hetals Reference Book and lias credited to 1-lilkins' and Bunn1 s book Coppe·r and Corper Base Alloys (14). Since small variations in the amount of i mpurity in a metal can appreciably change its physical properties the result s obt ained for copper are in good agreement with those of other workers.

B. Thorium

The results of measuring the thermal conductivity of high purity thorium are shmm in Figure 7. The room temperature point on this curve is the result of a series of 22 independent measurements. Considerably fewer independent determinations were made for the other points on this curve. The large root-mean-square deviation for these points is at least ~artially due to the fact that less refined techniques and apparatus were used for these measurements than Here used for copper. The smaller thermal conductivity of thorium is probably not responsible for these deviations. ·

The points on the thermal conductivity curve shmm in the lower portion of Figure 7 v<ere obtained from those on the thermal diffusivity curve using .the relation K = ked. The specific heat of thorium as determined by- C. F. Miller (15) of the Arne~ Laboratory of the Atomic

THERMAL DIFFUSIVITY AND CONDUCTIVITY OF THORIUM

.3 6~ >- 0.342 ±o.o34 •c - .3 4_ :~

0.303±0.032 0.3 6±0.036

., ::t .32 - • -~ CJ - 0.~00! 0.001 0 G ~ .30

N

o e .28 e u ~

' ... G .26 .&:. 1-~

.24 -I

.50 ~

>- .48 I--:~ (.) .46

0.428±0.047

- . 0.416±0.044 0.420±.0.049

u -::t 0

~ e A 5 u .4 -o;;) - .42 - 0~12 ±0.001 --o ~ ~ .40

(

... "-0 .&:. 1- .38- ,

l ~

.36 I I I I 0 100 200 300 400

Temperature oc

Fig. 7

I 500

1\) 1\)

H ({.) 0 I f-' \0 CD

2'3 ' . ISC-198

Energy Cormnission is 0.1188 joules/gm~C-:'in the te:-:1perature r9-n~e 0°C to 200°C. The density of thorium was measured as 11.558 grn/c~. Since specific heat and density data were not available for elevated temper­atures, these quanti ties '1-mre assumed to be constant iri calct¥-a ting the thermal conductivity. No thermal conductivity measurements of thorium by other obs~rvers were available for comparison with these results.

VI:4 DISCUSSION

The 22 independent determinations for thorium at room temperature were done under different conditions of the sinusoidal boundary conditions. Several different periods and heat inputs Here ·used. By using the infor­mation obtained 1vith the same heat input but for t,..m different periods of the boundarJ condition, calculations 1vere made for the thermal diffusivity of thorium using both the velocity and the amplitude decrement methods. The results of these calcu~tions using the velocity method gave a value for k of 0.293 : 0 .025 em /sec°C for 16 independent calculations. Using the a~nlitude decrement method a value of 0.281 : 0.025 cm2/secoC from 16 ind~pendent calculi ti ons Has obtained. These values for k agree with the value obtained using the ne-vr method Hi thin their root-mean-square deviations. A comparison of the root-mean-square deViations for the velocity and amplitude decrer:1ent methods >vi th that of the method used in this investigation indicates that the new method is inherently capable of r,reater accuracy under the same experimental conditions. Further investigations of this tT!)e should be made to confirm this point.

Inadequate irwestigation Has made of the effect of changing the wave­form of the peti.Q.dic boundar;:,r conditions on the measured value of thennal conductivity. . If . the -.nearest thermocouple vlas separated from the sample heater by only a few centimeters it :<as found that the 'ilaveform of the heat Have ap:glied at the end of the rod did not have to be sinusoidal for the temperature dt the near2st thermocouple to be sinusoidal in form. Further investic;ation of the effect on k of changing the vraveform of the input to the -sqmple heater is suggested.

24 ISC-198

ifi~ R.Elt'ERENCES

1. J. D. l~orbes, Trans. Hoy. Soc. Edinb. 23 133 (1865).

2. T. C. Baillie, Trans. Roy. Soc. Edinb. 39 361 (1397-98).

3. C. II. Lees, Phil. Trans. Roy.Soc., LondonA208 381 (1908 ).

4. C. C. Did11ell, Phys. Rev. 28 584 (1926).

5. F. Kohlrausch, Arill. d. Phys. ! 132 (1900).

6. Jaeger and Diesselhorst, a.bhand. d. Reichsanstalt 3 269 (1900). OriE;inal not availab~e for exa~rri.nation; ci ~e~ i~ J:- K. Rol;erts~ Heat and Thermodyn~~cs (Black~e and Son L~~tea, London, 1940J.

?. H. F. Angell, Phys. Rev. 33 421 (1911).

8. vi. IJeissner, Ber. deut. ~hys . Ges. 12 262 (1914).

9. R. w. King, Phys. Rev. §_ 437 (1915).

10. c. Starr, Rev. Sci. Inst. 8 61 (193?).

11. E. W. -~'lashburn ed., International Critical Tables, 1st. ed. (}1cGravr-Hill Book Co., Ne1-1 York, 1927).

, .. 12. C. D. Hode;man, ed. Handbook of Chemistry and ?nysics~-3~st ed.

(Chemical Rubber Publis:-1ine Co., Cleveland, 1949). . : -~ . ~ 'L. t$ ' ~

13. C. J. Smithells, Hetals Reference Book (Interscience Po:13fi:shers Inc., Nevr York, 1949). ' '•ri.i ' : '4 <i ~ '

14. R. A. Wilkins and E. S. Bunn, Copper and Copper Base 1il.llbys' (HcGraw­Hill Book Co., Inc., Nevi York, 1943).

• 15. C. F. Biller, private connunication.

Ei'ID OF DOCUMENT