Variable-state methods of measuring the thermal properties of solids

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Variable-state methods of measuring the thermal properties of solids Harmathy, T. Z.

DIVISION OF BUILDING RESEARCH

VARIABLE-STATE METHODS OF MEASURING THE THERMAL PROPERTIES OF SOLIDS

by

T. Z. Harmathy

ANAL YZED

Internal Report No. 249 of the

Division of Building Research

Ottawa March 1962

PREFACE

The Fire Research Section of the Division has available the large furnaces required for the fire-endurance testing of wall and floor components

of bUildings. It is clear, however, that it is not

practical to repeat such costly large scale tests on every variation in wall and floor construction. There is an obvious need for further information on the pertinent properties determining the response of materials and constructions to fire exposure which can be used to extend the results from full

scale tests to the prediction of a wider range of

constructions. One of the physical properties

which it is desired to measure is thermal conductivity. A study, which is now reported, has been made of

various variable-state methods which may be of value

in the range of temperatures of interest in fire

endurance work.

The author, a mechanical engineer and a research officer in the Fire Research Section, is immediately in charge of fire resistance studies.

Ottawa

by

T. Z. Harmathy SUMMARY

Solids that are liable to undergo physico-chemical changes are not well suited to the steady-state methods of

thermal conductivity measurements. Two variable-state

methods have been developed which both offer the advantage of producing neglibible thermal disturbance in the solid during

the measurements. The first is a curve-fitting method with

which all thennal properties of the test specimen can be

determined from a test of less than lO-min duration. The

second is a single-point method, especially suited for use at elevated temperatures and for study of the effect of moisture content or sluggish reactions on the thermal diffusivity.

It is generally accepted that the steady-state methods of measuring the thermal conductivity of solids are

still unsurpassed in accuracy. These methods, however, have

several serious weaknesses that often make their application rather difficult or even impossible.

The four major weaknesses of the steady-state methods are as follows:

(a) A definite shape is required. In the case of very

hard materials, (e.g. certain rocks) the prepara-tion of the test specimen may be an extremely

time-consuming and costly operation. Very often even

finding a piece of material large enough for test

specimen is almost impossible.

(b) The time of reasonably closely approaching the

steady-state conditions is seldom less than

4

hours,

but may be longer than a day, Thus if the thermal

conductivity in a larger temperature interval is to be investigated, the measurements may take several

weeks. (It may be noted that a steady-state method

described by Sutton (1) can eliminate this difficulty; there are some doubts, however, about the accuracy of the method, especially when applied to materials of low thermal conductivity.)

2

-(c) The prolonged maintenance of a significant tempera-ture gradient across the test specimen may result in various phenomena that completely falsify the test results; e.g., in the case of porous materials in the 0 to 100°0 range migration of moisture will occur if the specimen is not completely dry. Some other materials undergo certain physico-chemical changes as a result of heating. These changes are faster on the hot side of the specimen, sluggish or completely non-existent on the cold side. If the heating is long enough, minerologically,

crystallographically, and even chemically different layers of the material may develop perpendicular to the direction of heat flow.

(d) If, within the range of the applied temperatures, the thermal conductivity is a markedly non-linear function of the temperature, correlating the

"average" thermal conductivity with the <ve r'age temperature is not permissible. If, on the other hand, the temperature gradient across the specimen is reduced to achieve Jpproximate linearity, the accuracy of the measurement will inevitably suffer. Some of these problems, but generally not all of them, can be eliminated by variable-state methods. The essence of all these methods is that at certain definite boundary conditions the variation of temperature at one or more points of the specimen is recorded. From these records, the thermal properties of the material (thermal diffusivity, thermal conductivity and specific heat) can be calculated.

The most frequently applied boundary conditions

are as follows: (i) stepwise variation of surface temperature, (ii) constant heat flux through the boundary, (iii) cooling according to Newton's law, (iv) instantaneous heat (or cold) pulse.

From the point of view of evaluating the tempera-ture records, these methods may be grouped as follows:

(i) 1t0urve-fitting method", i.e. to find the thermal constants of the material by comparing the complete temperature history at a point of the specimen with the curve obtained by an

analytical solution of the problem. (ii) ｬｾ･ｴｨｯ､ of the asymptote" which makes use of the fact that under certain

boundary conditions the temperature asymptotically approaches a simple curve. By plotting the asymptote on some convenient graph paper a straight line will result, whose slope and

intercept on the axes will yield equations for the calcula-tion of the thermal constants. (iii) "Single -point method!' It is sometimes possible to determine the thermal diffusivity simply by measuring the time coordinate of some characteristic point of the temperature versus time curve (e.g. maximum,

The curve-fitting methods generally offer the advantage of easy specimen preparation as well as short test periods, seldom more than 10 min and since the range of variation of temperature can be reduced to a few degrees, these methods are particularly well suited for studying such problems as the effect of moisture content or sluggish

physico-chemical changes on the thermal properties. The principal weakness of these methods is the sensitivity of the test results to the initial and boundary conditions. This fact is particularly noticeable when elevated tempera-ture measurements are being carried out. Even though the test proper does not take more than a few minutes, achieving a uniform temperature distribution within the specimen at the particular temperature level may require several hours.

Since the methods of the asymptote do not use the initial portion of the recorded temperature versus time curve, they are less sensitive to the initial temperature ､ｩｳｴｲｩｾ

bution than the curve-fitting methods, and thus are more suitable for elevated temperature measurements. As a rule, they require more elaborate and thus more expensive specimens, and longer time for the test proper.

As far as the sensitivity to initial conditions is concerned, the single-point methods are similar to the curve-fitting methods except when they are associated with a pulse method. An inherent weakness of these methods is that only the thermal diffusivity can be evaluated with moderate

accuracy from single-point measurements.

The rising popularity of the variable-state methods is manifested by the steadily increasing number of papers on this subject. The theoretical basis for almost all con-ceivable variable-state methods can be borrowed from the

fundamental work of Oarslaw and Jaeger (2). It is beyond the scope of this work to give even a brief review of all methods that have so far been published. Only those similar in some respect to the methods used by this author will be mentioned.

The primary purpose of the present investigations was to develop a simple method to determine the thermal

properties of bUilding materials in the 0 to 1000°0 range. Since most of the building materials are liable to undergo certain physico-chemical changes (desorption of moisture, dehydration, transformation, dissociation, etc.) in certain temperature intervals, and such changes are generally accom-panied by a variation of the thermal properties, a minimum thermal disturbance caused by the measurements seemed to be the most desirable feature of the method to be developed.

The asymptotical method described by Haupin (3) was the subject of the preliminary investigations. According to this method constant heat flux is applied to the specimen

4

-through a butt-welded thermocouple wire heated with alternating current. The thermocouple is inserted between two pieces of the specimen joined together along a plane. A filter network is used to block the alternating current in the thermocouple circuit, and the direct-current output is recorded on a

potentiometric recorder. After a short period, generally less than 60 sec, the temperature of the thermocouple wire becomes proportional to the logarithm of the time. From the factor of proportionality the thermal conductivity can be calculated.

The analytical solution of this heat flow problem

is available in various publications (2,

4, 5,

6). Unfortunately, the experimental difficulties are numerous and, althOUgh Haupin claimed an uengineering accuracy" for his method at any tempera-ture, this author found it impossible to obtain even reproducible temperature records. In most cases imperfect contact between the wire and the specimen was primarily responsible for the failure. However, even if this difficulty had been eliminated by moulding castable specimens ar8und the thermocouple wire, the slope of the temperature versus log time plot proved so small that the thermal conductivity could not be evaluated with any degree of accuracy. Vfuen the heat input was increased to increase the slope, the temperature gradient around the wire became excessive, and besides the development of the usual difficulties (onset of moisture migration or physico-chemical changes), the conditions on which the analytical solution had been based, became invalid.

2. A NEW CURVE-FITTING METHOD

2. (a) Theory

To eliminate any contact resistance between the heating element and the specimen, and at the same time to reduce the temperature gradient near the heater, it has been decided to use a metallic foil instead of wire to supply the heat. Unfortunately, with this arrangement the temperature history of the heating element (or rather the surfaces of the specimen in contact with the heating element) is not suited for the calculation of the thermal conductivity, so that the temperature history of a point (represented by a thermocouple

junction) at certain distance from the heating foil has to be recorded. The preparation of the specimen thus becomes more difficult; on the other hand, because of the absence of alternating current in the thermocouple circuit, the experi-mental difficulties are considerably reduced. Another

advantage of this method is that, from the temperature records, not only the thennal conductivity but also the specific heat can be determined.

Clarke and Kingston were probably the first to use metallic foil as a heating element

(7);

their method is,

however, an ｡ｳｾｾｰｴｯｴｩ｣｡ｬ one that requires elaborate specimens and a much longer testing period.

Since only an early phase of the heat conduction process is considered in the present ｭ･ｴｨｯｾｴｨ･ specimen can be regarded as an infinite solid with a constant heat supply in the plane x

=

O. The problem is represented by the

following equations:

*

dT

₌

d

2T for -oo<x <0 _(la)

ｾ

ｋｾ

and 0< x

<

0 0

-t-

=

-k

d

T _{when x = 0 , t> 0} (Lb )

oX

T

=

0 for

_00<

x

<00,

t

=

0

(Ie)

The solution for this case is known from (2) and is given by

T = ｾ

1f"iCt

_{V '..} ierfc x

v 2{Kt

or in dimensionless form,

and in the plane of heat supply

T=

.9.

J

Kt at x = 0 .

k

-rr

(2)

(2a)

(2b)

If T(t) is the temperature measured at some point at time t after the beginning of the test, and T(4t) is the temperature at the same point at a time 4t, it can be shown that

6

-g

2 ierfc ' x T(4t)

=

4" I<.t (3) T(t)

ｾｖｾ

ierfc

Similarly, for the T(2t)/T(t) ratio the following equation can be derived:

T{2tj

=

ｾｩ･ｲｦ｣

ｾ ｾ

T(t)

,I

2

,;-erfc

ｾ

"

ｾ

Equations (3) and (4) are plotted in Fig. 3. These

plots can be used to determine the Fourier numbers corresponding to some distance x and various times t, and hence to calculate the thermal diffusivity of the test specimen, after selecting a number of points on the experimentally determined

tempera-ture versus time curve. Knowing the Fourier number, the kT/qx

versus

ｾｴＯｸＲ

curve of Fig. 3 (which is a plot of equation (2a»

will yield the value of the thermal conductivity.

Alternatively, the graphically evaluated tangent to the temperature versus time curve can be used to calculate the

thermal conductivity. To facilitate such calculations the

ｾ ｾｾ

=

ｾｊｾＺ

ierfc

ｾｾｾＺ

+ ;\-

ｾ

erfc

ｾｨｾ

(5)

equation is also plotted in Fig. 3. It is seen that the tangent

has a maximum at Kt/X2 =

ｾＮ

At this point, therefore, the

temperature versus time curve must have an inflection. If the

point of inflection is easily recognizable, both the thermal diffusivity and thermal conductivity can be expressed from any two of the following three equations:

K.t _ I _(6a) 2 - '2 x kT at the point

-

_qx = 0.0834 _{of inflection} (6b) kx dT _{= 0.2421} (6c) qK

ax

This seldom applies, however, because as Fig. 3 shows, the slope of the tangent varies very little within

the 0.3

<

Ktjx2

<

1.0 interval. The more accurate way of

determining the thermal properties of the specimen by a single-point method is as follows:

At a distance x from the metallic foil, the tempera-ture continues to rise for a short while, even after the heat

supply is switched off. The time at which the maximum

tempera-ture is reached can generally be accurately determined from

the records. Also, it is easy to derive an analytical expression

for this point, starting from the equation

kT

ｾｴ

{o

,&2

R

-_qx = -_x2 J.erfc"2 -_Kt - 1- -t ierfc

This gives the variation of the temperature at point x after

the heat input has been switched off at t =j( (2). After

differentiating with respect to t and making the right-hand side of the resulting equation equal to zero, the following equation is obtained for the maximum:

-t:

Jet -

-k

t (8)

x2 - (1-

1:')

in

I_

t 1 I,

-t

from which, knOWing x,l(, and t (the time coordinate of the

maximum temperature),

k

can be calculated. Then, knOWing the

value of the maximum temperature, k can also be computed by

means of equation

(7).

2. (b) Experimental

The block diagram of the experimental set-up for

room temperature measurements is shovm in Fig. 1. Vfuen

carrying out measurements at elevated temperatures the specimen, together with the installed metal foil and thermocouple, is

included in a furnace in which the temperature is carefully

balanced at the desired level. The cold junction of the

thermocouple may also be moved from the Dewar flask into the furnace, where it shall be in contact with the surface of the

specimen, as shovm in Fig.

5.

In room temperature measurements, originally steel foils (shim stock), 0.001 in. thick and 10 em wide, were used

as heating element. Recently they have been replaced by

Constantan foils, 0.001 in. thick and Ｓｾ in. wide, which have

the advantage of exhibiting practically constant resistivity over a very wide temperature range, and are capable of being

8

-used in elevated temperature measurements as well. Palladium foils, 0.001 in. thick and 3 in. wide, are also available for high temperature measuremGnts. The electric resistivity of these materials is given in Table I.

TABLE I

ELECTRIC RESISTIVITY OF HEATING FOILS USED IN EXPERIMENTS

r ohm cm x 10 6 Material

Temperature

_°c

20 100 200 300 400 500 600 700 800

Steel shim stock 12.5 19.0

Constantana 48.2 48.2 48.2 48.3 48.5 49.2 50.1

Palladiumb 12.5 16.0 19.9 23.1 25.9 28.1c 30.0c 31.7c 33.0c a) Values reported in Ref. (8).

b) Values reported in Ref. (9). c) Extrapolated values.

Figure 2 shows the various specimens suitable for such tests. In the case of materials that harden by hydration

(concrete, plaster), the thermocouple wires are permanently embedded in the upper (or lower) test piece during the prepara-tion of the specimen, as shovm in Fig. 2a. Bricks, tiles,

ceramic or wood products can be tested as illustrated in Figs. 2b, 2c, and 2d. The surfaces in contact with the heating foil may be natural surfaces of the material. In Fig. 2b the upper piece has been cut into two slices by an abrasive disk. If the material is not too hard, it is easier to place the thermo-couple in a small bore parallel to the foil.

Preparation of the test specimen of hard rocks is slightly more laborious. ｾｦｬ･ surfaces in contact with the heating foil must be properly ground. A cleavage plane of the upper (or lower) piece may be used for the insertion of the thermocouple, as illustrated in Fig. 2e. If the material cannot be split easily, any breakage surface that is not

essentially perpendicular to the plane of the foil (as sho\Yn in Fig. 2c) can be utilized. To avoid significant heat leakage along the thermocouple wires it is very important that the

wires adjoining the junction be located in a plane parallel to the foil for a distance of at least 1 in.

Figure 2f shows a metallic specimen. The foil must be electrically insulated from the specimen by means of two pieces of paper sheet or thin layers of mica.

To minimize the disturbance of heat flow in the material by the presence of thermocouple wires, light gauge wires (say gauge No. 35) are recommended. Up to about 600 °0, the Chromel - Alumel thennocouple is probably the best choice; above this temperature platinum - platinum 10 per cent rhodium thermocouples are generally used. Since the maximum tempera-ture rise in the specimen is less than 20°C in most tests, it is not essential to use calibrated thermocouples.

In the cases shown in Figs. 2a, 2b, 2c, and 2e bare thermocouple wires can be installed. The junction is formed by butt welding. Several methods of performing butt welding on small wires are known (Refs. (3) and (10).

It has been found that the accuracy of the measure-ments is always satisfactory if the following two conditions are fulfilled: (i) the inflection occurs from 2 to 3.5 min from the beginning of heat input, (ii) the temperature rise at the point of inflection is from 1 to 2°0. From these observations, conditions for the optimum location of the thermocouple junction and the optimum heat input can be derived. By means of equations (6a) and (6b) one obtains

the optimum range of x: and

the optimum range of q:

15. 5

<

//c

<

20. 5 12

<

ｾ

c:

24

(9a)

(9b)

Because of the short duration of the test (generally less than 10 min), the boundary conditions will have no

significant effect on the results as long as the thickness of both the lower and upper test pieces is larger than twice the distance between the heating foil and the thermocouple junction. It is recommended that the length and width of the specimen be at least the same as its over-all thickness. In the case of metallic materials this condition is not essential; any bar at least 2 in. in diameter can be used as a test specimen and will probably yield sufficient accuracy.

The evaluation of a test record obtained at room temperature (23°C) for insulating fire brick, group 23, is given below as an example. The experimentally determined temperature versus time curve for x

=

1 cm is shown in

3Fig.

4.

The density of the material was found to be 0.811 g/cm. The heat was supplied through a steel shim stock 0.001 in. thick and 10 cm wide. As Fig. 4 indicates, the specimen was heated

(10)

ｾ 10

-for a

6·min

period. During this time the current decreased

slightly; the weighted average was 26.0 amps.

The heat supply can be calculated by the formula

I

2

r

q

=

ab 2

The average temperature of the steel foil is estimated at

31°C. At this temperature r

=

13.4 x 10-6 ohm cm, thus

=

262 x 13.4 x 10-6

=

0.03566 watt/cm2

q 0.00254 x 10 2

Further calculations are tabulated in Table II. TABLE II

CALCULATION OF ｔｈｅｭｶｾｌ DIFFUSIVITY

AND ｔｈｅｾＺｕｌ CONDUCTIVITY OF INSULATING FIRE BRICK

(on the basis of the curve in Fig. 4)

t _T(2t)/T(t) Kt K kT k 2" _cm2/sec - watt/cmoC sec _x qx 60 1.14/0.29

₌

3.93 0.198a 0.00330 0.0150a 0.00184 120 2.96/1.14 = 2.60 0.400a 0.00333 0.0590a 0.00185 180 4.63/2.05

=

2.26 0.590a 0.00328 0.1050a 0.00183 average 0.00330 0.00184

a) determined by means of Fig. 3.

The temperature rise of the heating foil at the end of the heat input period (6 min), according to equation (2b) is

T =

ＰＮＰＳＵＶｾ

10.00330 x 360

=

11.90C

0.00184

V

ｾ

and the average temperature of the foil is 23 + 2/3 x ＱＱＮＹｾ

31°C (because of the parabolic variation), exactly as

estimated. Thus the average temperature of the l-cm thick

2

= 0.00325 cm /sec

i (23 +227.63 + 31)

ｾ

ｾ 28.2°C •

This is, therefore, the temperature to which the above values of the thermal diffusivity and thermal conductivity shall be related.

It may be of interest to point out that if the time coordinate of the maximum of the experimental curve (431.5 sec) is used for the calculation of the thermal diffusivity, by

means of equation (8) the following value is obtained: 360

1

431.5

Ie =

2

x

431.5

x 350 A 1

1-

431.5

z-n

350

1-

_431•5

which is in good agreement with the value arrived at by the curve-fitting method.

The specific heat of the material is obtained as

0.00184 _ /

c

=

0.00330 x

0.811 -

0.688 watt sec gOC

converting to British units, the result of the calculations may be summarized as follows:

K = 0.0128 ft2/hr

k = 1.276 Btu/hr ft 2 of/in.

c = 0.1644 Btu/lb of

all at 83°F. In the Engineering Materials Handbook (11) the thermal conductivity of this material is given as 1.28 Btu/hr ft 2 of/in. at GOF.

Sometimes a definite increasing or decreasing tendency can be experienced in the values of the thermal diffusivity

calculated from successive portions of the experimental tempera-ture versus time curve. It has been found that an uneven

initial temperature distribution is most frequently responsible for such difficulties. As mentioned in the introduction, the sensitivity to the initial conditions is an inherent weakness of every curve-fitting method. The time spent on such incon-sistent tests is not completely wasted, however, because the

12

-thermal diffusivity can still be evaluated with fair accuracy from the position of the maximum.

Both experimental results and numerical analyses have proved that the time of the occurrence of maximum is

relatively insensitive to the initial temperature distribution. Because of this important feature of the maximum of temperature history curve, a new single-point method (a pulse method as regards the mode application of heat) has been devised and will now be described.

3.

A NEW SINGLE-POINT PULSE ｾｬｬｩｔｈｏｄ

3. (a) Theory

As the period of heat supply, ｾ is gradually reduced the value ｯｾ the expression on the right-hand side of equation

(8) decreases and tends ｴｯｾＮ ThUS, if a momentary heat pulse is supplied through the metal foil,

(K

t) _

...L

ｾ - 2

X h=O

• (ll)

The subscript h = 0 indicates that there is no heat flow through the x = 0 plane, i.e. the heat transfer coefficient at x

=

0 is zero. This applies when the test specimen con-sists of two pieces which surround the heating foil (at x

=

0) from both sides, as shown in Fig. 2. It is also possible to eliminate one of the pieces (the lower piece) and to use specimens similar to those illustrated in Fig. 6. If heat is supplied to the upper surface (plane x = 0) of such specimens (not necessarily by means of metallic foil) only part of the heat will penetrate into the specimen after the removal of the heat source; part of it will be transferred to the ambient atmosphere by convection and radiation. In such cases, therefore, the value of the heat transfer

coefficient at the x

=

0 plane has also a certain effect on the time in Which, after the application of heat, the maximum temperature will occur at some point x>

o.

The fact that the heat transfer coefficient does affect the time coordinate of the maximum, can easily be proven by assuming the extreme case that ｨｾｯｯＬ i.e. that immediately after termination of heating the temperature at x = 0 drops back to the value of the ambient atmosphere. The following equations now apply:

T

=

_To when

o<t<1

at x

=

0 T

=

0 when t

>7'

T

=

0 for t

=

0 _•

(12b)

(12c)

(13)

The solution of this problem has also been given by Carslaw and Jaeger (2), and is as follows:

i

_o = eric

ｾｊ ｾｾ

-

eric

ｾｊ ｬｃＨｴｾＧｴＧＩ

(14)

1

ｾ

1-,;-With the usual procedure it can be shovm that at a fixed value of x the temperature has a maximum when

r

1

t

=

0" (l-f)

l«

and that, if 17tｾ 0 (momentary heat supply)

By comparing equations

(8)

and

(14),

or equations

(11)

and (lS) one can see that as the heat transfer coefficient at the x

=

0 plane increases from 0 tooa, the time coordinate of the maximum temperature at point x decreases by a factor of

3.

As long as 1Jt

<

0.1 (i.e. if only short pulses are considered) the value of the expression containing "t/t in equations (8) and (14) approximately equals unity. It seems probable, therefore, that for practical values of h, the value of the Fourier number pertaining to the maximum is somewhere between 1/6 and 1/2. The calculations that have been performed to find a relationship between the Fourier number (at the temperature maximum) and the heat transfer coefficient are outlined below.

If at t

=

0 the temperature of the atmosphere in

contact with a semi-infinite solid along the x

=

0 plane is suddenly increased to To' and at t

=

'7:'

it is returned to the initial value (taken here as zero), the following equations will describe the process of heat transfer within and at the

14

-surface of the solid:

-k 3T

_oX

= h (To-T) k

ｾ

= hT for x'> 0 when O<t<"l at X

=

0 when (16a) (16b) T == 0 for t

=

0 (16c)

The solution of this problem is as follows (2) :

T _

ＡｲＮｊｾ

-eric

v;;t:;;

To - erfc

-k

K(t-'i1 -e e 4K(t-'i1 +e (11)

It may be seen that, for fixed values of x (and also ｾＬ k and

h),

this equation is of the form

ｾ

- ¢ (

t ) -

¢

(t-,;1

To

-If

1:'

is very small (only short pulses are considered),

,

i.e. only the first two terms of Taylor's series need be

taken into account. Thus

or, in full

T _ d [

T?-

CIt erfc

0"

After differentiation, this becomes

2[

(g)2

ｾ

-

ｾ

-k

!- + h Kt 2

T

!s-

=

e

41(1;

{Ie _

hie

e

Kt

k

IJKt:

ertciJ-Vx

+ h VlCt

ｾ

ｾ ｨｾ 1l't K \: Kt k ) .

(18)

After differentiating again with respect to t and making the right-hand side of the resulting equation equal to zero the following equation is obtained:

2

X2 _ XY +

y2 -

V.-,r

'[3 e(X+Y) erfc(X+Y) -

ｾ

= 0

(19)

where

&

_ I X ｘＭｾ

-

_let Y

=

ｾｙｋｴ

·

(19a)

(19b)

Equation

(19)

defines the X versus Y relationship corresponding

to the maximum of the temperature versus time curve.

The solution of equation

(19)

is presented graphically

in Fig.

7.

For convenience the

(Fourier number) and

16

-dimensionless groups have been chosen instead of X and Y, respectively, as the two variables when preparing the plot. It may be seen that as h-7>O,

ｋｴＯｸＲＮＮＭ［ＮｾＬ

and as

ｨｾｯ｣Ｌ

ｉｃｴＯｸＲｾＱＯＶＬ

in agreement with equations (11) and (15).

Although the plot in Fig.

7

has been arrived at on the assumption that the heat pulse was produced by an

instantaneous rise in the temperature of ambient atmosphere, it is obvious that the result is independent of the mode of generating the heat pulse. In room temperature measurements, the most common way of starting a pulse is by bringing a

piece of hot metal into contact with the surface x = O. Figure

7

offers a very simple way of calculating the thermal diffusivity from an experimentally determined value of the time coordinate of maximum temperature, provided that the heat transfer coefficient at the surface is known. Unfortunately, it is not always possible to estimate the value of h with sufficient accuracy. The main difficulty lies in the fact that h is not truly a constant but is a function of the surface temperature.

(20)

•

It may be useful to review briefly the way of

estimating the value of h. The coefficient of heat transfer by natural convection between a surface facing upward and the ambient air can be calculated by the following empirical formula (12, 13):

I

h c= O.000216I Ts\4

The coefficient of radiant heat transfer from a emissivitye to non-reflecting surroundings can from the Stefan-Boltzmann law, and is given by

Q

4 _

Q

4

h_r = ＨＬｾ s a L- Q _ Q s a surface of be derived (21) Which, if Q ｾ Q , becomes s a h_r = 4

«s»

_a3 (21a)

The coefficient of heat transfer by a combined convection-radiation mechanism is the sum of the coefficients for the two individual mechanisms, i.e.

Although the accuracy yielded by the above equations is very often quite satisfactory, the proper way of determining h is by experiment. For this purpose a material of known

thermal diffusivity shall be ｵｳ･､Ｎｾｾ Knowing t (from the

experiment), x, K, c, and f', h can be found by means of Fig. 7. This value can then be used for a larger group of materials with similar properties, prOVided that the thermocouple is located at the same distance from the surface and the "pulse intensity" (the product To"'Z") is roughly the same.

Because of the variation of h during the measurement, the "effective" value of h is as a rule, dependent on the

distance between the heated surface and the thermocouple junction. For example, for insulating fire brick at room temperature and at a pulse intensity of about 28000C

sec the following effective values have been measured:

for x

=

1 cm for x

=

2 cm

h_{e f f}

=

0.00077 watt

Icm

2

°c

h ef f

=

0.00057 watt

Icm

2

°c

The first value indicates an aVerage temperature difference of about 1.0

°c

between the surface and the ambient air. The second one is approximately equal to that which can be calcu-lated with equation (21a) alone, i.e. on the assumption that the surface temperature is equal to the temperature of the surroundings and he ｾ O.

If the intensity of the pulse is not too high, these assumptions are often permissible, even at room temperature. They are always permissible at elevated temperatures, for the following reasons: (i) the heat transfer by convection is negligible in comparison with that by radiation, and (ii) the coefficient of heat transfer by radiation is not significantly affected by slight changes in the surface temperature, so that it can be regarded as constant during the test.

2 As Fig. 7 shows, at high values of

htlfcx

the value of

Kt/x

varies very slowly with that of ｨｴＯｾ｣ｸ［ therefore at elevated temperatures often a rough estimate of h may on;Ly be needed. In the case of metallic materials, however, the high temperature range is the more critical one. At room tempera-ture the value of the

htlfcX

group is so small that taking

Kt/x 2

as equal to

ｾ

(at the temperature maximum) is generally permissible.

*

e.g. a material, the thermal diffusivity of which has been previously determined by the curve-fitting method.

18

-There is no need to estimate the value of h, if not the thermal diffusivity proper, but the effect of some slowly developing physico-chemical changes (e.g. desorption of moisture, dehydration, transformation, etc.) on the thermal diffusivity is of primary interest. This method is especially well suited for studying such transient states of the solids.

Two different pulse methods have been described recently (14, 15). Both are applicable only to metals, and none of them is comparable, in simplicity, with the present method.

3.

(b) Experimental

The block diagram of the experimental set-up for high temperature measurements is shovm in Fig.

5.

At high temperatures it is more convenient to apply a "cold" pulse that reaches the surface of the specimen in the form of a momentary jet of cool air issued from a fan-shaped horn. In room temperature measure-ments the use of a hot iron (the tool for pressing clothes) is recommended for starting a momentary heat pulse.

Various specimens suitable for the single-point method are ｳｨｯｾｭ in Fig. 6. Figure 6a shows the usual form

of specimens from castable materials (concretes, plasters, etc.). Bricks, tiles, ceramics and wood products can be tested in

shapes illustrated in Figs. 6b, 6c, and 6d. The common form of rock specimens and metallic specimens is shown in Figs. 6e and 6f, respectively.

The surface to be exposed to the heat or cold pulse (usually the upper surface) shall be reasonably flat, but not necessarily smooth.

The rules for selecting and preparing the

thermo-couples are essentially the same as those given in Section 2 (b). Since the actual value of the electromotive force is immaterial, in the case of metallic specimens the specimen itself can be used as one of the thermocouple materials.

The optimum range of x here, too, is roughly equal to that given by inequality (9a) in the list of equations. When testing insulating materials, the use of the upper half

of the range is recommended.

Since the exact temperature of the material or medium used as a heat or cold source is generally not kno?m, it is difficult to make recommendations concerning the intensity of the pulse. It has been found that the required minimum

intensity may vary between 200°C sec (for insulating materials) and 30000C

The calculation of the thermal diffusivity from the maximum of an experimental temperature versus time curve, shown in Fig. 8, is given below as an example of the applica-tion of the method. The test was carried out at 470°C on a specimen of insulating fire brick, group 23, the room tempera-ture properties of which have been calculated in Section 2 (b).

A cold pulse of about ｾＭｳ･｣ duration was applied.

Assuming that ｣ｾｏＮ 9 the approximate value of h can be calculated by means of equation (21a):

h

=

4 x 5.71 x 10-12 x 0.9 x (470 + 273)3 = 0.0084 watt /cm2

0c.

Also, as an approximation, one can use the values of

p

and c at room temperature (see Section 2 (b)). Thus, with x = 1 cm, and (from Fig. 7) t

=

62.5 sec

h t

pc x = 0.811 x0.0084 x 62.5

0.688

x 1

=

0.94

From Fig. 7 it may be seen that in this regime the variation of Kt/x2 is rather slow, so that even an approximate value of the ht/pcx group will yield sufficient accuracy. From the Figure kt/x2

=

0.229, therefore

ｾ = 0.229 x 1 = 0.00366 cm2/sec

62.5

A series of experiments and numerical analyses has been performed to investigate the effect of the frequency of pulses on the time coordinate of peak. It has been found that as long as the interval between two pulses is not less than 10 to 15 times the time coordinate of maximum, and the intensity of signal is high enough, the repetition of pulses will not significantly affect the test result. If the frequency of pulses is increased, some reduction in the time coordinate of the peaks will be experienced.

CONCLUSIONS

Two variable-state methods of measuring the thermal properties of solids have been described:

The first is a curve-fitting method. With this method all thermal properties of the test specimen can be determined from a simple test of less than 10-min duration. The experimental procedure is, however, not particularly suitable for high-temperature applications.

20

-The second is a single-point pulse method. This

is well suited for studying the thermal diffusivity of moist materials or the variation of the thermal diffusivity during

sluggish physico-chemical changes. It is also applicable to

high-temperature measurements, but its accuracy is. poorer than that of the curve-fitting method.

ACKNOWLEDGMENT

The author takes this opportunity to thank Mr. J.

Berndt for his collaboration in the experimental work. NOTATION

·a

₌

thiclmess of heating foil, cm

b

₌

width of heating foil, cm

c

₌

specific heat, watt sec/gOC

h = coefficient of heat transfer, watt/cm2

oc

I

₌

electric current, amp

k

₌

thermal conductivity, watt/ cm

°c

q

₌

rate of heat supply, watt/ cm2

r

=

electric resistivity, ohm cm

t

=

time, sec

T = temperature above the initial level, or above that of

the ambient atmosphere,

_°c

x

₌

distance, cm

X = dimensionless group, defined by equation (19a)

y

₌

_{dimensionless group, defined by equation (19b)}

00 ｾＲ erfcx =

ｾ

f

_{e- df} X 00 ierfcx =

f

･ｲｦ｣ｾ

_d

_S

)(

GREEK LETTERS

c

=

g

₌

K

=

P

=

(f

=

'7:

=

¢

₌

emissivity of surface, dimensionless

absolute temperature, oK

thermal diffusivity, cm2/sec density, g/cm3

Stefan-Boltzmann constant, 5.71 x 10-12

ｷ｡ｴｴＯ｣ｭＲＰｾ

period of heat supply, sec function SUBSCRIPrS a = c = 0 = r = s = of the surroundings by natural convection

of the surface or of the ambient atmosphere during the period of heat supply

by radiation of the surface REFERENCES

1. Sutton, W. H. Apparatus for measuring thermal conductivity

of ceramic and metallic materials to 12000C. J. Am.

Ceram. Soc., 43, (1960), p. 81-86.

2. Carslaw, H. S. and J. C. Jaeger. Conduction of heat in

solids. Second Edition, Oxford at the Clarendon

Press, 1959, p. 63, 74, 15, 345.

3. Haupin, W. E. Hot wire method for rapid determination of

thermal conductivity. Ceram. Bull., 39, (1960),

p. 139-141.

--4. Jaeger, J. C. Ｑｾ･ use of complete temperature - time curves

for determination of thermal conductivity with

particular reference to rocks. Austr. J. Physics,

12, (1959), p. 203-211.

5. Van der Held and Van ｄｲｾｭ･ｮＮ A method of measuring the

thermal conductivity of liquids. Physica, 15, (1945),

22

-6. Blackwell, J. II. A transient-flow method for determination of thennal constants of insulating materials in bulk. Bart 1. Theory. J. Appl. Physics, 25, (1954), p.

137-144.

--Ｗｾ Clarke, L. N. and R. S. T. Kingston. Equipment for the simultaneous determination of thermal conductivity and diffusivity of insulating materials using a variable-state method. Austr. J. Appl. Sci., 1,

(1950), p. 172-187.

-8. International Critical Tables, Vol. VI. First ｅ､ｩｴｩｯｮｾ

McGraw-Hill Co., Inc., (1929), p. 170.

9. Handbook of thermophysical properties of solid materials, Vol. 1 - Elements. Pergamon Press, New York, (1961), p. 1-487.

10. Radcliffe, S. V. and J. S. White. ｾｾｯ simple methods for spot welding wires. J. Sci. Instr., 38, (1961), p. 363-364.

11. Mantell, C. L. Engineering materials handbook. First

Edition. McGraw-Hill Book Co. Inc., (1958), p. 22-72. 12. Perry, J. H. Chemical engineers' handbook. Third Edition.

McGraw-Hill Book Co., Inc.,(1950), p. 414.

13. McAdams, W. H. Heat transmission. Second Edition. Me Graw-Hill Book Co., Inc., (1942), p. 240.

14. Woisand, E. L. Pulse method for the measurement of thermal diffusivity of metals. J. Appl. Physics, 32, (1961), p. 40-45.

15. Barker, W. J., R. J. Jenkins, C. P. Butler and G. L. Abbott. Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity. J. Appl. Physics, 32, (1961), p. 1619-1684.

[Z1

o 0 -o 0 0 0 Cu 1_ _ _ _ Cu ---, 7"", I I i5 H_2O Hg E_« 4 u « > 10

II

FIGURE I

BLOCK DIAGRAM OF EXPERIMENTAL SET-UP FOR ROOM TEMPERATURE MEASUREMENTS WITH CURVE - FITTING METHOD

I VARIABLE TRANSFORMER, 2 TRANSFORMER, 3 AMMETER, 4 METAL YOKE. 5 METAL FOIL, 6 SPECIMEN, 7 DEWAR FLASK, 8 D.C. AMPLIFIER, 9 POTENTIOMETRIC STRIP CHART RECORDER, Chl= CHROMEL WIRE, Aml= ALUMEL WIRE, Cu= COPPER WIRE

d)

th

b)

th th j ＭＭＭＭｾＬＭＭ｟

..

u

e)

f)

FIGURE 2

VARIOUS SPECIMENS SUITABLE FOR THE CURVE - FITTING METHOD

L= LOWER PIECE, U= UPPER PIECE, f= METALLIC FOIL, th= THERMOCOUPLE WIRES, j= HOT JUNCTION OF THERMOCOUPLE

10

ＧｾＭ

.. - .

I

l

J-ＭＭｾ •..-- ｾｾｾ

ｾｌＭ

- - ｾ.ｾｾＭ

ｾｴ

i -_.. --- --- ｾＮｾ _. 1 - ｾ ｾ ｾ Ｍ Ｍ ｾ Ｍ ｟ Ｎ - - -

"---I

I - I

r.---'"

T (4t) '\. "- _T(t) I

"'

ｾ <, I'- ,,-

1----<,

0...-

v ...

... !'_r--"",

-

-

. /V

-

. /

I

_'"

I T(2t) _ｾ

'"

1 T(t) / / ' ... kT 1/ qx

/

I III

L

1III I .."...- _{kx dT}

/

...r---... »>:_{'- qK} crt III

-e

... ｾｲＭＮＮ II

....

I ... 1/

_""

J ... I J

/

I I

/

I I I I I I

I

V

10

FIGURE 3

CURVES TO BE USED FOR CALCULATIONS ACCORDING

TO THE CURVE - FITTING METHOD

120 180 240 300 360 420 480 540 600 t sec 60 1--- _.- ---ｾ ｾ Ｎ ｟ .._- ._..._-" --- . - -_._----.e - - - ｾ Ｎ ｟ Ｎ ｟ Ｍ Ｍ Ｍ Ｍ --'"

-

10... c

-V

T

ｾ

-ｾ

/

ｾ::> --LL ｾ ｌｌｾ X V ｾ 0 «

/

Q. Q ｾ Q. ::> w

-7

en ::I: z U 0 ｾ !:: ｾ

/

w 3= u ::I: en w ...J LL

V

Z

V

ｾ

ｾｶ

- - _..ｾ

/

/

/

5 2 6 4

o

o

u o 3

I-FIGURE 4

EXPERIMENTAL CURVE OBTAINED FOR THE CURVE - FITTING METHOD.

INSULATING FIRE BRICK AT ROOM TEMPERATURE. X= I CM.

ｾｾ

ffi

[2J

uu 0 0 00 _{0 0} / 5 6 2 FIGURE 5

BLOCK DIAGRAM OF EXPERIMENTAL SET -UP FOR HIGH TEMPERATURE MEASUREMENTS WITH SINGLE - POINT METHOD

I SPECIMEN, 2 FURNACE, 3 FAN - SHAPED HORN, 4 VALVE, 5 D.C. AMPLIFIER, 6 POTENTIOMETRIC STRIP RECORDER, Chi; CHROMEL WIRE, Ami; ALUMEL WIRE

d)

th

a)

th

b)

th th

c)

f)

FIGURE 6

VARIOUS SPECIMENS SUITABLE FOR THE SINGLE - POINT METHOD

0·2 0·3 0·4 0'5 0·6 0·7 0·80·91·0 1·2 IA 1·6 ',8 2 2·5 3 4 5 6 8 10 20 5000 h t pc X 0,'

. +---±=--

---+---t----t__ ｴＭＭｲＭＭｴＭＭＭＭｾＭＫＭＭｾ

---｟ｾ . i . __ - - I ---f---t--'--t--.｟ｾＭＭＭＭＭＭＭＮＭ｟ＭＭ ｦＭＭＭＭｾ I i

--l+=i '

._.

1---+---'''',- - ' i ' , ' " ' I - - - - t - -

-+-r-1--

i ｾＫＭｲ｟ＭＭＭＭ｣ＭＭＭＮＭＭＭＭ

"

It-

1 , 1 : _ _ f - - " "

!

I I : , " ' " i --f-I_--+1 -1 1 " ' - - . 1 ' , i i I ...

-f-.-+-

'

I _ _+----+-_-+--'----,..--+- - - j I - -

L

i'-...; - I ｾＭｾＧＭＭＭｾＭＭＱ ｟ｾ I i ...1'-.. , I i •

I

i

I--r---.

I .

' . _

1 : I r - - t - _ _ . I

---+-

I , --t- ＮｾＢＢＢＢＭＮｉＮＬ ］ＭＭＭ｟ＭｉｦＭＭＭ｟ＭＮＴＫＭＭＭｾｾＭｲＭＬ --;---1 1 - - - '

---to

ｾ

.

--

-...:

i i i : I I - - , - - _ _ 1 - - - I j! I , i ! i i i 1--- " 1 - - : i I , I --+-+1---'----+-_ _---1 1 i i i I ! I---l---+---+---+-t-+----t--t--· - --- --

t--

I f - - - t - - - , - - - f----f----_t__ - i i ,

I

- - I I I , o o 0·3 0·1 0-2 OA ｾｉｎＩＨ FIGURE 7

.:

r-.

I

ｾ

r-.

::::> ｾ _{- - I - - .}

I

X

<,

<t ｾ

t-,

<,

r-.

r-;

...

r-;

ｾ

I

I

. .

u

u o I-3 2

o

o

60 120 t sec 180 240 300 1'>1\ＧＲｾＷｾﾷ｡

FIGURE 8

EXPERIMENTAL CURVE OBTAINED FOR THE SINGLE - POINT METHOD.

INSULATING FIRE BRICK AT 470°C. X= I CM.