Peak-time method for measuring thermal diffusivity of small solid specimens

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Peak-time method for measuring thermal diffusivity of small solid

specimens

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S e r

T H l

N2lr2

no.

457 c . 2

BLDG

NATIONAL

RESEARCH COUNCIL OF CANADA

A

~

J

A

L

~

CONSEIL NATIONAL DE RECHERCHES DU CANADA

PEAK-TIME METHOD FOR MEASURING

THERMAL DIFFUSNITY OF SMALL

SOLID SPECIMENS

BY

T.

Z. Harmathy

Reprinted From

Journal, American Institute of Chemical Engineers

Volume

17,

Number

1,

January 1971

Pages

1 98-20

1

RESEARCH

PAPER

NO.

457

OF THE

Dlvlslo~

OF BUILDING RESEARCH

O T T A W A

January 7977

MAR

1s

193 I

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METHODE DE TEMPS DE POINTE POUR MESURER LA

DIFFUSIVITE THERMIQUE DE PETITS SPECIMEN SOLIDES

SO MMAIRE

Une mCthode d ' e s s a i s i m p l e e s t dCveloppCe pour dCterminer l a

diffusivith thermique d e s matCriaux solides e t g r a n u l a i r e s e n

m e s u r a n t l e t e m p s auquel l a pointe s u r v i e n t s u r un r e c o r d t e m -

pCrature temps. E l l e p e r met l ' u s a g e de petits s p e c i m e n s

facilement prCparCs e t d e s spCcimens de for me irrCguli'ere

(i.

e. p i e r r e s ) . L e s e s s a i s peuvent Gtre effectuCs

'a

une tempCra-

t u r e normale a u s s i bien qu"a d e s t e m p C r a t u r e s ClevCes, e t peuvent

t t r e rCpCtCs

'a

c o u r t e s i n t e r v a l l e s .

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Peak-Time Method for Measuring

Thermal

Diff

usivi

ty

of Small

Solid

Specimens

N a t i o n a l Research C o u n c i l o f C a n a d a , O t t a w a , C a n a d a

A

simple t e s t method i s developed for determining the thermal d i f f u s i v i t y of s o l i d s and granular materials by measuring the t i m e a t w h i c h the peak on a temperature-time record occurs.

It

a l l o w s t h e use of small e a s i l y prepared specimens and specimens of irregular shape (for example, stones). The t e s t s can be performed a t room temperature a s w e l l as a t h i g h temperatures, and can be repeated i n short t i m e intervals.

The principal difficulties of measuring tlie thermal dif-

fusivity of solids are associated with specimen prepara-

tion. Very often a specimen has to be machined or ground

to a certain shape, and for solids of considerable hardness

this operation may be extremely time-consuming and costly.

The difficulties are further increased by the fact that it i s

often next to impossible to find a piece of the solid suf-

ficiently large to serve a s a specimen.

This latter problem i s especially acute with some con-

crete aggregates that are often unavailable in s i z e s larger

than about

1% in. As the thermal conductivity of aggre-

gates i s known to he the primary factor in the conductivity

of concrete (1, 2), inability to determine this property seri-

ously hinders the design of concrete mixtures in applica-

tions where thermal characteristics are of principal impor-

tance.

The method

to

be described for measuring thermal diffu-

sivity of solids i s notable for its simplicity with regard

to

both specimen preparation and the performance of the test.

It is especially suitable for nonmetallic materials, and can

be

carried out on very small specimens.

THEORY

It will be shown that a method developed earlier

( 3 )

for

the measurement of thermal diffusivity, using idealized

semi-infinite solids, can be extended to include the use of

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solids of more convenient geometry, such a s cube, sphere,

and even

to

solids of irregular shape.

If

heat i s supplied for a short period to the surface of a

solid body, initially in equilibrium with i t s surroundings,

the temperature a t any point inside the solid will exhibit a

maximum a t some time following cessation of the heat

supply. T h e thermal diffusivity of the solid can be deter-

mined by measuring the time a t which the maximum tem-

perature occurs. (For this reason i t seems appropriate to

refer to this method a s peak-time method.)

By generalization of the results in reference 3, one can

assume that, a t l e a s t approximately, the following relation

i s applicable

to

the peak

where the coefficient

A

depends on the geometry of the

solid body and on t h e location of the point of interest in-

side the body. As the

cp,

and

cp,

functions are such that

the two functions are actually correction terms applicable

when the ~ / t ,

4

0 and h t / k

-

m

conditions cannot be

approximated closely enough in the experiment. (Approxi-

mate expressions for

cp,

and

cp,

will be presented later.)

The coefficient

A

can be determined by examining the

solution of the following heat conduction problem:

on the condition that

T +

0. If T ( x / t , y / t , z / t , ~ t / t ' ) represents the solution ob-

tained when the above boundary conditions [Equations (6)

and (7)l are replaced by

T ( x , y , z ) = T o + l when t > O , x , y , z o n S

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then the solution to the problem stated by Equations (4) to

(7) i s obtained a s (4)

or, in the limit a s T

4

0, a s

Because at the maximum dT/dt

=

0 , the time coordinate of

the maximum temperature (in the limit a s T

4

0) can be

expressed from the equation

and the resulting expression i s of the form

It i s logical to select the most important point of the

solid, i t s center of gravity, a s the point a t which the tem-

perature variation i s to be observed. With this choice

A

becomes a unique function of the shape of the body.

Fig. 1. V a l u e s of t h e coefficient A (referred to c e n t e r of g r a v i t y

l a c a t i o n s ) for d i f f e r e n t solid bodies.

S o l i d Body S p h e r e I n f i n i t e C i r c u l a r C y l i n d e r

6Q

C u b e

E

D

I n f i n i t e S q u a r e P r i s m

k@iP

I n f i n i t e S l a b S e m i - i n f i n i t e S o l i d

In Figure

1 the values of

A

(referred-to center of

gravity locations) are listed for a number of body shapes

that may be considered in thermal diffusivity studies.

These values were calculated by using in Equation (11) the

expressions of T developed by Carslaw and Jaeger ( 4 ) for

these shapes. The interpretation of the characteristic di-

mension i s a l s o illustrated in the figure.

The coefficient

A

for a semi-infinite solid, t h e value of

which was derived earlier ( 3 , i s also included in Figure

1. By comparing the l a s t two items one may conclude that

A

=

'4

holds good for any point inside a slab; or in other

words that it i s not affected by the other surface s o long a s

the characteristic length i s taken a s the distance between

the point and the nearest surface.

As pointed out in connection with Equation ( I ) , q , i s a

correction term applicable whenever the T/tm

4

0 condi-

tion i s not fulfilled closely enough in the experiment.

An expression derived earlier (3) for

cp,

for a semi-

infinite solid c a n be approximated a s follows:

A 0.0918 0. 1170 0. 1261 0. 1399 0. 1666 0. 1667

which i s sufficiently accurate to about ~ / t ,

=

0.15. In

Appendix

A

i t will be proved that Equation (13) i s applica-

ble to solids of any geometry.

The explicit form of the other correction term, function

cp,,

for semi-infinite solids can be evaluated from the graph

in Figure 7 of reference 3, which was arrived a t theoreti-

cally. (In this plot the product of ~ t , / & ' and h t / k was

used a s the independent variable instead of the group

ht/k.) This graph indicates that for finite values of

h t / k ,

cp,

>

1. For h t / k

>

10 the following empirical equa-

tion seems to b e applicable:

The results of many computer calculations indicate thqt

this equation i s a l s o applicable to solids of other geome-

tries,* a t l e a s t when h t / k

>

10. In thermal diffusivity

'According to Equation (14)

cp,

m a s h t / k

-

0.

For points a t center of gravity locations i n simple bodies having three p l a n e s of symmetry, t h i s is t h e expected behavior of

cp,.

s i n c e t,-+ m a s h

-

0.

For points in a semi-infinite s o l i d , on the other hand,

cp2-

3 a s h e / k -

0.

T h i s may be s e e n from Figure 7 of reference 3.

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measurements, one should nevertheless endeavor t o make

the group h&/k a s high a s possible-at

l e a s t higher than

30. Methods of achieving this will be discussed.

EXPERIMENT

The cube i s undoubtedly the most convenient shape for use a s a specimen ( s e e Figure 1). Figure 2 a illustrates a cubic specimen prepared for thermal diffusivity test. To ensure high output and reduce heat conduction along the wires, very light gauge ( s a y B and S gauge 36) chromel-alumel thermocouples are recommended. The twin-hole porcelain tube of '/,-in. O.D. provides re- liable electrical insulation for the wires, both inside and outside the specimen, and a strong handle for manipulation. Extreme care should be exercised to ensure that the thermocouple junction i s at the center of the specimen and that i t i s in good contact with the specimen material.

Thirty seconds seems to be a convenient order of magnitude for t,. The characteristic specimen dimensions corresponding to this value have been calculated for several groups of materials and are listed in Table 1. Smaller specimens can also be used if the laboratory p o s s e s s e s facilities (for example, oscillograph) for recording rapid temperature changes. ( I t should be remembered that t, i s proportional to the square uf the characteristic dimension.)

(corresponding to t, = 30 sec.)

Group No. Material group

4 ,

cm.

1 Insulating materials, wood, s o i l s 0.5- 1.0

2 Most building materials 0.9- 2.0

3 Rocks 1.3- 2.8

4 Most ceramics 1.2- 5.2

5 Iron, lead, brass 5.3- 9.0

6 Aluminum, copper, silver 14.0-20.0

Because of the large specimen s i z e s required, the present method i s not particularly well suited for measuring the thermal diffusivity of most metals. Although fast-response recording de- vices permit the selection of considerably smaller specimens, with the reduction of

-L

the group h&/k also diminishes and t, may become increasingly dependent on the function

cp,

[see Equation (I)], the exact nature of which i s not yet known for lower values of hd/h.

To keep the group'h&/k at a conveniently high level, it i s recommended that the specimen be submerged in a liquid metal bath (primary bath) during the pretesting periot' (that is, for

t

<

0, when the specimen acquires uniform temperature at the desired level, To) and after the application of the temperature pulse (that is, for t 2 0 . It i s immaterial whether the temperature pulse i s a heat pulse or a cold pulse, that i s , whether Tp

>

To or Tp

<

To.

In fact, it i s very convenient t o start the pulse by submerging the specimen (for 0

<

t

<

T ) in a bath of the same liquid metal, referred to a s the secondary bath, at some lower temperature level, Tp<< To. In addition t o providing high heat transfer coefficients, liquid metals a l s o offer the advantage that they do not penetrate the pores of the specimen owing to their high contact angles.

In the DBR/NRC laboratory mercury baths are used for room temperature tests. (The experimenter should consult reference 5 concerning the toxic characteristics of liquid metals.) The secondary bath i s kept at the freezing point of mercury (-38.g°C.) by means of an alcohol jacket cooled by occasional addition of solid carbon dioxide.

With mercury the effective value of the heat transfer coefficient in the primary bath i s of the order of 0.1 cal./(sq. cm.)(sec.)(OC.). This becomes several times higher when the specimen i s gently shaken while submerged in the mercury. With h = 0.1 and with values of

-L

listed in Table 1, h&/k i s generally .greater than 30 for the first two groups of materials. At such high values of the Nusselt number,

cp

is sufficiently close to 1 and the effect on t, of shaking the speiimen in the primary bath i s not detectable. For materials in the l a s t three groups of Table 1 to achieve sufficiently high Nusselt numbers, mechanical agitation of the primary bath i s necessary, even if full size specimens are used.

For measurements above 23Z°C. liquid tin baths s e e m to be most convenient. Within the,20° to 23Z°C. temperature range a low-melting alloy may be selected. (Without proper safety mea- sures mercury must not be used at elevated temperatures.)

When using cubic specimens of s i z e s listed in Table 1 and mercury baths (primary a t To = 20°C.. secondary at Tp =

-38.g°C.), it is recommended that the thermocouple output be slightly amplified ( s a y up to x 10) in order t o have ~ / t ,

<

0.1 and still obtain on a 1-mv. recorder a temperature-time record the maximum of which can be clearly determined. [AS Equation (10) indicates, the thermocouple output i s approximately proportional to the product T ( Tp

-

TJ.]

The t e s t s can be repeated a s soon a s the temperature of the specimen in the primary bath (after the application of the pulse) ceases to show significant variation. This time i s approximately eight to ten times t,, s o that if t,= 30 sec. the t e s t can be repeated in approximately 4 to 5 min. Easy repeatability i s an- other advantage of the present method.

In Figure 3 the thermocouple output i s shown for a test performed on a limestone rock specimen of cubic shape ( & = 1.27 cm.). During this t e s t the primary bath (mercury, To = 2Z0c.) was agitated with an impeller stirrer. Submersion of the specimen in the secondary bath ( T p =-38.g°C.) lasted approximately 1 sec. From the output versus time record (Figure 3) t, = 18.4 sec. Hence ~ / t , = 0.054 and

cp,

= 1.027. Assuming temporarily that

cp,

= 1.0, the thermal diffusivity for t h i s rock i s obtained according to Equation ( l ) , with A = 0.1261 ( s e e Figure 1) a s

X = 0.1261

x

1.027 x 1.27,

18.4 = 0.0114 sq. cm./sec.

It may be noted that with a different measurement technique ( 3 ) performed on a 20-cm. by 10-cm. by 10-cm. piece of the same rock, the thermal diffusivity was found t o be 0.0115 sq. cm./sec.

The density of the rock i s 2.65 g./cu. cm. I t s specific heat i s estimated at 0.2 cal./(g.N0C.), so that its thermal conductivity i s k = 0.0114

x 2.65

x

0.2 = 0.0060 c a l . / ( ~ m . ) ( s e c . ) ( ~ C . ) . Owing to the stirring of the primary bath, the heat transfer coefficient i s

Page

200 AlChE Journal

January,

1971

0.16 I I I I I I 8 E

-

I

-

> ; 0.12

-

a &

-

a 0

-

L" 2 & 3 3

₂

0.08

-

0 2i =

-

,.,

r

-

0 b c 0.04

-

1 S p l c l a ~ n 2 T h e r m ~ c o u p l ~ wlrm,

-

3 T h l r m o c o ~ p I ~ J Y n c l l l n 4 Twin Holo Coramlc Tub, 3 P l n g - p o n s 8.11 S h e l l 6 P l u g

1 Plugs 0 10 20 30 40 50 60 T I M E . SEC.

Fig. 2. Specimens for thermal d i f f u s i v i t y tests; ( a ) cubic

specimen, ( b ) irregular shaped specimen, (c) specimen o f F i g . 3. Thermocouple output i n a t e s t performed on a

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probably of the order of 1.0 cal./(sq. cm.)(sec.)(OC.); thus hL/k =

1.0 x 1.27/0.0060 = 212, which i s sufficiently high to validate the assumption that

cp,

= 1.0.

The accuracy of this peak-time method of measuring the thermal diffusivity of solids depends on several factors: the accuracy of evaluating t, from the temperature-time.record; the accuracy of locating the thermocouple junction in the center of the specimen; whether a good contact h a s been achieved between the junction and the specimen material; the amount o f f oreign material (ceramic tube, thermocouple wire) along some short heat flow path; the value of the group, h U k , if it i s lower than about 30.

Experience shows that t, can be determined to an accuracy of about 3%. Errors associated with locating the thermocouple junction and achieving good thermocouple-to-specimen contact depend entirely on the care taken in the experiment, and are difficult to a s s e s s . ( T h e s e errors may be reduced hy radiographic inspection of the specimens.) Errors associated with the fourth and fifth factors depend, to some extent, on the s i z e of the specimen. To reduce them, the largest available piece of the material should be

selected for the specimen, preferably such that L

>

1.0 cm. For specimens of elongated shape (for example, cylindrical, prismatic, ellipsoidal), it i s advisable to introduce the ceramic tube and the thermocouple wires along some long heat flow path. An example i s shown in Figure 26.

It i s good practice to stir the primary bath well and to check the magnitude of the Nusselt number after the calculation of thermal diffusivity.

Experience gained in the DBR/NRC laboratory indicates that for materials listed in the first four groups of Table 2 an overall accuracy better than f 7 % can be achieved if great care i s exercised in preparing the specimen and in performing the test.

Whenever possible, the t e s t specimen should be machined or ground to one of the regular s h a p e s shown in Figure 1. If, how- ever, the material i s very hard or available only in small pieces, it i s possible to t e s t irregular shaped specimens (Figure 26).

Figure 1 reveals that the range of variation of coefficient A i s relatively narrow. It i s possible, therefore,. to estimate its value for shapes other than those listed if only an approximate value of the thermal diffusivity i s of interest.

For irregular shaped specimens half of the smallest diameter of the body should be taken a s the characteristic dimension

4 .

Consequently, the thermocouple junction h a s to be located half way between these two governing surfaces (Figure 26) and A should be a s s e s s e d on the b a s i s of their curvature and the near-

-

T

=

unit step response temperature, dimensionless

V

=

volume bounded by surface

S

=

space coordinates, cm.

Greek Leners

K

=thermal diffusivity, sq. cm./sec.

T

=

period of heat or cold pulse, sec.

cp

=

function

Subscripts

rn

=

at the maximum of

T

M

=

a t the maximum of

T

when

T

-+

0 o

= a t

t

= 0

p

=

of the medium starting the pulse

L I T E R A T U R E C I T E D

1. Boulder Canyon Project Final Reports, Pt. VII. Cement and Concrete Investigations. Bull. 1. Thermal Properties of Con- crete. U.S. Dept. Interior Bureau of Reclamation, Denver. Colo. Colo. ( 1940).

2. Harmathy, T.

:,.,

"Thermal Properties of Concrete a t Elevated Temperatures. Journal of ibfaterials, 5, 47 ( 1970).

3.

-

,

J. Appl. Phys., 35, 1190 (1964).

4. Carslaw, H. S., and J . C. Jaeger, "Conduction of Heat in Solids," 2nd edit., pp. 31, 60, 99, 173. 185, 198, 233, Oxford at the Clarendon P r e s s , London (1959).

5. Sax, N. I., "Dangerous Properties of Industrial Materials." 2nd edit., Reinhold, New York (1963).

6. DeVries, D. A., Annexe 1952-1, Bull. Inst. Intern. Froid, 115 (1952).

7. Gorring, R.

L.,

and S. W. Churchill, Chem. Eng. Progr., 57, 5 3 (1961).

8. Hamilton, R. L.. and 0.

K.

Crosser, Ind. Eng. Cl~ern. Funrfa- mentals, 1, 187 (1962).

9. Godbee, H. W., and W. T. Ziegler, J . Appl. Phys., 37, 56

( 1966). n e s s of the other surfaces, making use of the values listed in

Figure 1.

APPENDIX A: V E R I F I C A T I O N OF EQUATION (13)

To obtain an accurate value of the thermal diffusivity of irregu-

lar shaped specimens it i s necessary first to determine the value The defining equation for the function

rpl

is obtained by of A for the particular shape and thermocouple location from a dividing Equation ( 1 ) (with

q,

= 1, that i s , for h -+ _by separate experiment. A copy of the specimen must be made (us- Equation (12):

ing some simple molding technique) from a convenient reference

material (for example, beeswax) whose thermal diffusivity h a s (PI = t m / t ~ r ( l a )

previously heen determined to a high degree of accuracy. Into At t,, aT/at = 0; using Equation ( 9 ) one obtains this copy specimen the thermocouple junction i s installed in the

same way a s in the original specimen. After determining t, for _{A T ( t , ) - d T ( t , - ~ )} _{= O}

this copy specimen (for h t / k -+ 4, A can be expressed from at at

( 2 a ) ~ ~ u a t i o n (

U.

Figure 2c illustrates the application of the present method to The Taylor s e r i e s expansion of T ( t ) in the neighborhood of tM the measurement of thermal diffusivity of granular materials. A i s

ping-pong ball may be used to provide a spherical shell for the -

-

₁

_{a Z -}

granules. Naturally, the t e s t result i s applicable only to a cer- .T(t) = T ( ~ M ) i d T ( t n i ) ( t - tM)

+ -

7 T ( t M ) ( t - tM)2

tain degree of compaction of the granular material. It should also at 2 at

be remembered that, in general, there i s no simple relationship 1

a 3 -

between the thermal diffusivity of a two-phase system, such a s a

+ -

7 ~ ( t M ) ( t - t M ) 3 + . . . (3rd

granular material, and those of the constituent phases (6 to 9 ) . 6 at

where the third term on the right side of the equation i s zero on

ACKNOWLEDGMENT

account of Equation (11). By differentiating

The author thanks E. Oracheski for h i s collaboration in the ~at T ( ~at ) = ~ 2 at3 T ( ~ ~ )t - t M ) 2 + . . . + ~ ~ T ( ~( 4 a ) ~ ~ ) ( experimental work. The computer calculations to verify Equa-

tion ( 14) were performed by C. J . Shirtliffe. If this expression, with the retention of the above two t e r m only. This paper i s a contribution from the Division of Building Re- i s applied to t = t, and t = t,

-

T, and the two expressions are search, National Research Council of Canada, and i s published substituted into Equation ( 2 ~ ) . the following equation results: with the approval of the Director of t h e Division.

( t m - t M ) 2 - ( t , ~ ~ - t h f ) 2 = 0 (5a)

NOTATION

from which

A

=

coefficient, dimensionless

- -

t, _{- I + -}1

_-

T _(6a)

h

= h e a t transfer coefficient (in the primary bath), c a l l

t~ 2 t~

(sq. c m . ) ( s e ~ . ) ( ~ C . )

_{By virtue of Equation ( l a ) and of the fact that t, = tM for}

k

=thermal conductivity, c a l . / ( c m . ) ( ~ e c . ) ( ~ C . )

small values of T/trn, this equation provides proof for the general

&

=

characteristic dimension, cm.

applicability of Equation (13).

S

=

surface of solid

t =

time, sec.

T

=temperature, OC.

1969; paper accepted July Manuscript received May 1 , 1969; revision received July 11, 1969. 7,

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