Analysis of Transient Heat Conduction and its Applications

Part 1: The Fundamental Analysis and Applications to Thermal Conductivity and

Thermal Diffusivity Measurements

By Morihiro Yoneda* and Sueo Kawabata, Members, TMSJ

Department of Polymer Chemistry, Kyoto University, Kyoto, 606

*Present Address: Department of Clothing Science, Nara Women's University, Nara, 630

Based on Journal of the Textile Machinery Society of Japan, Transactions, Vol. 34, No. 9, T183-T193 (1981-9)

Abstract

A theoretical analysis of a transient heat conduction is presented in a series of three papers to ex- plain the theoretical basis to link the thermal behaviour of fabrics with warm/cool feeling. In this paper, the fundamental analysis of the transient heat conduction in a sheet-like solid along its thick-

ness direction is analyzed in the case when a plate of good heat conducting properties having finite heat is placed on the top of a sheet-like solid of poor heat conducting properties such as fabrics.

The solution is obtained in those two cases, one is the case that the temperature at the bottom surface of the specimen is kept constant and another is that the bottom surface is thermally insulated.

From these analyses, the transient heat conduction under such conditions has been fully explained. A method for measuring thermal conductivity and thermal diffusivity from the transient phenome- non is presented as an application of this analysis. The feature of this method is that the measurement

can be carried out in a few seconds, and therefore, with less effect of the change of test condition, e.g., change of moisture content of specimen, on the measured values.

1. Introduction

There have been many researches by many workers on

the heat conduction properties of fiber assembly and they obtained their results according to their measurement

methods and philosophies. ~112 ] In this paper, a study on the heat conduction from a heat source having a finite heat

content to a fabric specimen in contact with heat source is analyzed. An experimental work by HolliesW3i has long

hcen the only work on heat conduction from heat source of

finite heat content to fabric until a study on the fabric warm/cool feeling was carried out recently by Kawabata

and Akagi' using this method with marked improvements in many technical points. The latter authors correlated the

maximum value of heat flux, gmax, to the fabric warm/cool feeling when a heat source having finite heat contacts fabric.

In view of above, the authors present in this study a theoreti-

cal basis for this method. Especially, theoretical basis for the correlation between the gmax and the fabric warm/cool

feeling is discussed. As an application of this theory, a quick

measuring method of thermal conductivity and thermal diffusivity of fabrics is presented. Using this method, the

heat conduction properties of sheet-like solid with poor

heat conduction such as polymer sheets and heat insulation

materials can also be measured.

This study will be presented in three papers. In Part 1, theoretical analysis on the transient heat con-

duction in this measuring system is made. The results are

confirmed by experiment and a high speed measuring

method of thermal conductivity and thermal diffusivity is

presented as an application of this analysis. In Part 2, the relationship between gmax, a measure of

fabric warm/cool feeling, and the transient heat conduction

phenomena in man skin when another body is touched is theoretically analyzed on the basis of the theory in Part 1.

In Part 3, the transient heat conduction phenomena of

two-layered body is analyzed both theoretically and experi-

mentally to shed light on the effects of specimen surface which influences the warm/cool feeling.

In this research, fabric is regarded as a simple solid with homogeneous material properties without going into the

structure of fabrics. The structural effects of the specimen

on the transient heat conduction will be treated in future

papers.

Vol. 29 No 4 (1983) 73

2. Basic Principle Measuring Method

The measuring method presented here is based on the

principle discussed in a previous paper~14~ through improve-

ments in several points and the measuring system is now

well established with easy handling and high accuracy.

Here, the basic principle of the method is presented without

reffering to the details of equipments. The measuring method

is further discussed in the experimental section of this paper.

The principle is shown in Fig. 1. A specimen first placed

on a plate of which temperature is kept constant. Before measurement, the temperature of the heat source plate made

of good heat conductor, actually copper, is raised up to

about 5-10 degree higher than that of the specimen and, in the measurement, the plate is rested on the upper surface of the specimen. The temperature fall of the heat source

plate due to the heat flow from the heat source plate is measured and recorded along passing time. The heat flux is obtained by differentiating electrically the signal of the

temperature fall curve. Experiments were carried out under conditions that the temperature of the bottom surface of

specimen is kept constant, called "the constant temperature condition", and that the bottom surface of the specimen

is thermally insulated, called "the insulated condition." This method enables the measurement of heat conduction

properties of fabrics such as thermal conductivity and ther-mal diffusivity. The heat flux, q(t), was found to exhibit a maximum peak value, gmax, which can also be measured,

and details of gmax will be presented in Part 2 of this series.

3. Theory

We consider a one-dimensional heat conduction in the

finite region (0<x<d) (Fig. 2). The temperature distribu-tion within the specimen, u(x, t), satisfies the following equation of heat conduction,

au 82 u _k .......................................... (1) at axe

General solution of eq. (1) and solutions under some special conditions are known.C16,173 Here, we denote the following.

Temperature and heat flux; u(x, t) : Temperature of the specimen at time t (sec) and posi-

tion x (cm);

y(t) : temperature of heat source plate at time t (sec) (deg); Yo : initial uniform temperature of the specimen (deg); q(t) : heat flux at the interface between heat source plate and

the specimen at time t (sec). (cal/cm2•sec). Material constants of the specimen;

K: Thermal conductivity (cal/cm•deg•sec) k : Thermal diffusivity (cm2/sec) C: Specific heat (at constant pressure) (cal/g•deg)

p : Density (g/cm3) d: Thickness (cm)

Here, k = K/ PC ................................. (2) Material constants of heat source plate (copper plate

with uniform thickness); Mo : Mass (g) Co : Specific heat (cal/g•deg)

Po : Density (g/cm3) do : Thickness (em) So : Contact area of interface between the specimen and the

heat source (cm)

ao : ao=MoCo/So=poCodo ..................... (3)

where ao denotes the heat content of the heat source plate

per unit area of contact surface (cal/cm2•deg). The purpose of this analysis is to obtain the heat flux

flowing out from the heat source plate to the specimen and relate it to the thermal properties of the specimen assuming that the heat source plate made of good heat conductor having finite heat content is rested on the upper surface of the specimen, which is usually poor heat condutor.

The heat flux, q(t), which is mainly used in this analysis, is given as follows.

a (t) =-ao • dy/dt ................................. (4)

The boundary and initial conditions are given as eqs.

(5)-(8). The unknown function, y(t), expressing the tempera-ture change of heat source plate (initial value, yo) appears in eqs. (6) and (7). Equation (1) is solved under the conditions (5), (6) and (8) and then y(t) is determined using eq. (7). u(x, t) and q(t) are derived from the y(t) thus ob-tained. u(0, t)=0 .......................................... (5)

u(d,t)=y(t) .......................................(6)

K • au/3x1 x=a=-ao • dy/dt .................. (7) u(x, 0)_0 .............................................(g)

Fig. 2 Mathematical model and its coordinate system

74 Journal of The Textile Machinery Society of Japan

Equation (5) shows that temperature is kept constant at x = 0 ("the constant temperature condition (A)"), and eq. (6) shows the continuity of temperature at the contact surface between the heat source and the specimen, eq. (7) shows the continuity of heat flux at the contact surface and eq. (8) shows that the temperature of the specimen is kept constant and uniform before the contact.

The partial differential equation (1) is solved by Laplace transform method and the inverse transform is carried out by Bromwich integral. Under the conditions (5)-(8), which we call the constant temperature condition (A), the solution is obtained as follows :

sin Sn . e-cant (A) u (x, t) =2yo 1

~nCOS~n+ 1+- sln~n

a

............... (9)

~ -lan t ° 1 ................. (10) y(t)=2y e

n=1 a9 +1+-

a

a (t) =-ao ' dy/dt ........................ (11)

where In : a series of positive roots of

cot = aj3 (n = 1, 2, 3...),......... (12)

a : a=poCodo/pCd ..................... (13)

I : l =k/d2 .............................. (14) Here, a denotes heat content ratio of heat source and speci-men per unit sectional area and l denotes the heat diffusion rate of specimen normalized with thickness of specimen , d.

In the next place, the solution under the condition that the specimen is thermally insulated at x = 0 ("the insulated condition (B)") is derived. In this case, eq. (5) is replaced by

(15) while (6), (7) and (8) remain unchanged.

au/axl x=o=0 .............................. (15)

Thus, we obtain;

(B) u(x, t) = y° 1 +2y° l+- a

COS yn • e_ -2 t \a I ........ (16)

-ynsinyn+(i+ 1 cosyn

a

z

y(t)=---+2y° e-~Ynt ... (17) 1+1 n-laa yn+l+1 a a

a(t) =-ao . dy/dt ........................ (18)

where yn : a series of positive roots of

tan y=-ay (n=1,2,3......) ..... (19)

For the purpose of comparison, the solution for the case

of semi-infinite solid is shown below. In this case, x = 0 is taken at the contact surface between heat source plate

and specimen with the x increasing in the downward direc-

tion. Eq. (5) in the case of (A) is replaced by eq. (20) in this case.

lim u (x, t) =0 .............................. (20) x=~

The solution is as follows :

(C) u (x, t) =yoeA2tkerf c (AT+x/2 kt)

.............. (21)

y (t) =yoe~'terf c (~~) ........................... (22)

q ,(t) =aoyo1 A/ art - A2eA2terf c (Af) ...... (23) where, A=pc%/T/ao= ptk/ao ............... (24)

erfc (x) is complementary error function which is defined as :

2 erfc(x)=- fe_ t'dt

4. Discussion of the Solution

4.1 The Solution under the Constant Temperature Throughout the present study, heat flux, q(t), flowing out

from heat source to the specimen is an important quantity in the analysis. Here, the relationship between q(t) and thermal properties of specimen under the constant tempera-ture condition (A) is discussed briefly.

Equation (11) can be rewritten as follows :

q(t) =aoy° ~Wne-t/rn .............................. (25) n=t

, where' Wn 2 19 ~ = 1 ........................ (26) a9+ 1+-

a

Tn=1/l/9„ ................................. (27)

Figure 3 shows the q(t) given by the eq. (25) for various thicknesses of the specimen under constant material proper-ties, K, k and pC. The vertical axis is scaled by common logarithm of q(t). In this section, material constants used in the calculation are those of plasticized PVC sheet.

K =1.88 X 10-4 ( cal/cm . deg . sec )

k =5.15X104 -( cm2/sec )

p C=0.366 ( cal/deg . cm3)

Each q(t) curve in Fig. 3 consists of two parts, that is, an initial curved part followed by a linear part. The initial curved part coincides with one unique curve regardless of changing d values and this unique curve is the solution (C) for semi-infinite solid (d = oo) under the same material con-stants and the initial curved part increases with increasing thickness, d. For small d, the linear part which may be ap-

proximated by a single exponential which constitute most of the q(t) curve. Figure 4 shows the transient temperature distribution, u(x, t), within the specimen. The u(x, t) curves for t = 0.2,1 sec clearly show that, during the initial curved

Vol. 29 No. 4 (1983) 75

Fig. 3 q(t) for various thicknesses in the case ao - 0.1, yo - 10 under the condition that temperature at x = 0 is kept con-stant. (material : p-PVC K = 1.88>< l0-~, k = 5.15 x 10-x,

pC = 0.336)

Fig. 4 Temperature distribution in specimen under the condition that temperature at x = 0 is kept constant. (material:

p-PVC d = 0.15 cm)

Table 1 Thickness dependence of fin, Pn, Wn, 2

an: cot /3 = afi, Pn = 2/(afn2+ 1 + 1/a), Wn = (21 /3n2)/(apn2+1 +1/a), n - 1// fn2

Material: p-PVC

d = 0.02, a = 13.66, l = 1.286

76 Journal of The Textile Machinery Society of J apan

part of q(t), the effect of heat flow from the heat source plate has not reached the bottom of the specimen and make the heat conduction in the initial period identical to that in the case of semi-infinite solid; that is, the initial curved part of

q(t) depends only on the thermal properties of the material and not on thickness. The temperature distribution within specimen at single exponential part of log q(t) (t = 10, 50, 100) are almost linear and this shows that steady heat flow is established in this time region.

In the next place, the effects of thickness, d, on the para-meters of eq. (25) are discussed. Table 1 shows the results of calculation. From this table, it is shown that a and l depend on thickness, and these parameters influence Wn and rn. Particularly, the first time constant, rl, and the degree of convergence of rn are sensitive to changes in d and consequently, they affect the behaviour of q(t). Thus, the q(t) curve has the characteristics of both semi-infinite solid solution and the single exponential. There are two limiting behaviours of q(t), one for very large d and another for very small d. For very large d (Fig. 3, d = oo), the limit-ing curve of q(t) for d = oo as given in eq. (10) coincides with the semi-infinite solid solution as given by eq. (22).

And this fact can be easily proved analytically. When d is very large, l and A have the relationship;

~ _,oCK_A a ao .....................................(28)

As far as a and l change their values under the condition of eq. (28) the initial curve of log q(t) coincide with a unique semi-infinite solid solution in spite of the variation of thick-ness, d. When d is very small, eq. (25) reduces to eq. (29).

q (t) y e-tit, ................................ (29) d

This relationship shows that the time delay caused by the

propagation of heat wave front within the specimen becomes very small when thickness of specimen is small.

4.2 The Solution under the Insulated Condition In Fig. 5, the temperature distribution within the specimen

under the insulated condition as given in eq. (16) is shown. After the initial transient, the temperature profile, u(x, t), approaches a uniform equilibrium temperature, y~. y~ is expressible as;

y~ a __ ..................................... (30) Yo a+1

Figure 6 compares the calculated q(t) under the three conditions, (A), (B) and (C). It can be seen that the initial curved part of log q(t) coincides with semi-infinite solid solution under all three conditions and the curved part extends with increasing thickness, d. This is because the initial part of q(t) is essentially identical with regardless of the boundary condition at the bottom of the specimen. This result is important as shown later.

Fig. 5 Temperature distribution in specimen under the condition that surface at x = 0 is insulated thermally. (material:

p-PVC d = 0.05 cm)

Fig. 6 q(t) for three conditions: Constant temperature at x = 0 (broken lines) Insulated at x = 0 (dots and dashes lines) Semi-infinite solid (solid line)

(material: p-PVC K = 1.88 x 10'4, k = 5.15 x 10'4, pC = 0.366)

Vol. 29 No. 4 (1983) 77

5. Application to the Measurement of Heat Conduction

Properties of Fabrics

Heat conduction properties such as K, k and pC can be

measured using the theoretical predictions discussed above.

The procedure of measurement are described in this section.

5.1 Measurement of zi under the Constant Temperature

Condition

Figure 7 shows the typical experimental curve of q(t) under the constant temperature condition. gmax is the peak value of heat flux, q(t), and appears shortly after contacting the heat source with specimen. (The physical meaning of

gmaz will be discussed in Part 2 of this series.) As time elapses, the transition from the curved part to the linear part is observed as predicted theoretically. The first term of the series in eq. (25) which is the linear part of q(t) are expressed as;

4(t) =aoyoWie-~`(31)

where r1=1/l,S,z ........................................... (32)

Taking common logarithm both sides of eq. (31), we obtain,

logic q (t) = logio aoyoW~ - t log,a a ................ (33)

This relationship shows that the first time constant, r, can be derived from the slope of the curve obtained experimental-ly, log q(t) vs. t under the constant temperature condition. That is;

r,=log,oe/f -0.4343/f ........... (34)

where f = d logy (t) /d t

ri depends on a and / because a has a correlation to 6i which is the first positive root of cot j9 = a j3.

5.2 a and the Solution under the Insulated Condition Parameter a can be determined using the solution (B) for

the insulated condition. Figure 8 illustrates the procedure to obtain a. As mentioned in section 4.2, eqs. (16) and (17) show that the temperature, u(x, t) and y(t), approach a constant value, T,,, as t becomes large. Broken line in Fig. 8 shows this process. The T,. can be expressible as;

T~/To=a/(a+.1) ........................... (35)

where T0: Initial temperature of heat source T~ : Temperature of heat source at t = Co.

From eq. (35), a can be determined as;

a= 1~/ (To-T) .................... (36)

Experimentally, however, complete insulation cannot be realized and for theat reason y(t) fails to settle to T~. The best practical insulation material available was hard poly-stylene foam which conduct a small amount of heat resulting in a slow and steady temperature fall shown by the solid line in Fig. 8. Fortunately, there is a convenient way to derive the equilibrium temperature, T~, for the case of complete insulation from the experimental y(t) curve. When the linear part of y(t) is extended to intersect the t = 0 line and the intersect gives the T. value.

5.3 Procedure for Obtaining Thermal Properties of Material From ri and a which are obtained from the experiments

under the constant temperature condition and the insulated condition respectively, the heat conduction properties of specimen, K, k and pC can be derived in the following manner. (1) Using the equation cot ~i = the relationship be-tween a and ai can be calculated numerically as shown in Table 2. From the experimental value of a obtained under the insulated condition, corresponding ~i (or 1/6i2) can be read off from the table. Using the value of 1 / Qi2 thus ob-tained and the experimental value of v obtained under the constant temperature condition, / can be determined by use of eq. (32)', that is,

Fig. 7 A typical curve in

temperature at x =

experiment under

0 is kept constant.

the condition that

Fig. 8 A typical curve in experiment under the

surface at x = 0 is insulated thermally.condition that

78 Journal of The Textile Machinery S ociety of Japan

l=1/r1

(2) Thermal diffusivity, k,

k=ld2

suffi

............................... (32Y

can be obtained using eq. (14)'.

.................................. (14),

(3) Specific heat per unit volume, pC, can be obtained using eq. (13)'.

(4) Using eq. (2)', thermal conductivity, K, can be obtained.

When d becomes large, measurement of ri becomes difficult and cient accuracy cannot be obtained for

reasons discussed in section 4.1. Therefore, there is an upper limit of thickness, d, of specimen. The upper limit of d is

given in terms of a by the approximate criterion a > 1. When d becomes too small, measurement of temperature

fall of heat source plate under the insulated condition be-

comes difficult. Therefore, there is a lower limit of thick-

ness of specimen to which this method is applicable and the

criterion is a> 10-20.

6. A Simplified Method to Obtain Thermal Conductivity

When parameter a is large, that is, when the heat content of the specimen is relatively small compared to that of heat source plate, thermal conductivity K can be obtained by a simplified method. In this method, experiments need be carried out only under the constant temperature condition to determine ri from which K value can be derived.

The first positive root of eq. (12) satisfies eq. (36).

cot,81=a91 .......................................... (37)

When a is large, j3i is very small. In this case,

tan fl>9. ............................................. (38)

Therefore,

cot,9 ="1/tan(3,-1//d, .......................... (39)

Substituting (39) into (37), we obtain;

a-1//912 ................................................ (40)

Within the range where the relationship (38) holds, eq. (32) can be rewritten as,

1 d 2 2 pCd 2 cod r, = 1~,2 = k~,2 =-K a---

From this equation, K is given as,

K=ao • d/r, ....................................... (41)

Equation (41) shows that the thermal conductivity, K, of the specimen can be obtained from the experimental values of ri and d.

The relative error( %) of l /j32 brought about by the simpli-fication depends on a and can be estimated using Table 2.

If 5 % relative error is allowed a> 6.7 must hold. If 10 % relative error is allowed a> 3.4 must hold.

When the heat source constant, ao, is 0.1, the upper limit of thickness for the simplified method is approximated as follows;

dc=0.1/pC• ac ....................................... (42)

where, a0 is the minimum a value given above correspond-ing to percentage of error allowed.

Using this relationship, dicision can be made as to whether the simplified method can be used. Estimated d~ values for fabric and polymer sheet are as follows;

In the case of typical fabrics having the parameter value

pC = 0.2; d0 = 0.75 mm if 5 % deviation in K from the correct value is allowed. d~ _ 1.5 mm if 10% deviation is allowed.

In the case of polymer sheet of pC 0.35; d~ _ 0.43 mm if 5 % deviation is allowed. d0 0.84 mm if 10 % deviation is allowed.

Table 2 Relation between a and 1 /j312 R.E. ; Difference between K values obtained by the simplified (K') and the strict (K) methods R.E. (%) _ (K-K')/K x 100

Vol. 29 No. 4 (1983) 79

[An example of calculation for obtaining thermal parameters] An example of calculation by the simplified method is

shown in this section. In an experiment using plasticized PVC specimen ofd = 0.0311 cm, v = 17.1 sec is obtained. Thermal conductivity is calculated as follows;

K=ao•d_O.1X0.0311 '1,814X10 r, 17.1 (cal/cm • deg • sec)

For comparison, calculation by the strict method given in section 5.3 is as follows. By carrying out the experiment under the insulated condition, a = 8.78 is obtained.

Procedure (1) : For the a = 8.78, l /j32 = 9.116 is ob-tained using Table 2.

l=1/r, •,8,2=9.116/17.1=0.533 (1/sec)

Procedure (2) :

k=1d2=0.533 x0.03112

=5.155 X 10-' (cm2/sec)

Procedure (3) :

ao _ 0.1 =0 .366 ( cal/deg•cm pC_ )

3

a • d 8.78 X 0.0311

Procedure (4) :

K=pC• k=0.366 X (5.155 X 10-4)

-1.887 X 10-4 (cal/cm • deg • sec)

7. Experiment

To confirm the agreement between thermal conductivities

measured by the transient method and by the steady state method, the following experiment was carried out. Block-

diagram of the apparatus for the transient heat conduction

measurement is shown in Fig. 9. This apparatus was devel-oped by Kawabata in the previous work~14,15~ and was

named "Thermo Labo". The dimension of the specimen used here is 3 cm x 3 cm

square. Thickness of the specimen ranges from 0.1 mm to

2 mm. The specimens are fabrics and polymer sheets. Before the measurement, the specimen is rested on the

"Water Box" whose temperature is kept constant by circulat-

ing water of room temperature to satisfy the boundary

condition (5).

The heat source plate is made of copper plate of 3 cm x 3 cm square and its thickness is about 1 mm so as to make ao equal to 0.1. The back side of the heat source plate which does not touch the specimen is thermally insulated by hard

polystylene foam. Temperature sensor is attached to this side. The heat source plate is heated up to a certain tem-

perature, which is higher than the room temperature, by putting it on the "BT Box" which consists of copper block whose temperature can be controlled by a heater control system. In taking the measurement, heat source plate ("T Box")

is taken out of "BT Box" and rested on the upper side of the specimen. The temperature fall of the "T Box" is detected by a platinum wire sensor having very quick response and the detected signal is transmitted to "Amp" part. In the "Amp" part , the signal is converted to q(t) by differentiat-ing the signal and then the q(t) signal is converted to log

q(t) using "Log Amp The peak value of the initial heat flux, gmaz, is stored by an electronic circuit for the prediction of warm/cool feeling, which will be discussed in the follow-ing paper. With this apparatus, measurement by the steady state method is also possible by putting the sample between the surfaces of the "Water Box" and the "BT Box" to measure thermal conductivity. Thus, we have two methods, the transient method and the steady state method for measuring thermal conductivity using the same apparatus.

The experiment were carried out under the conditions that water temperature was about room temperature (22-24°C) and the temperature difference between the heat source

plate and the "Water Box" was 10°C. The pressure applied to the specimen by "T Box" was 12.5 g/cm2. It took about 20 sec per one run of transient method.

On the other hand, in experiments under the insulated condition, hard polystylene foam board was used in place of the "Water Box" to place the specimen. In this case, ideal thermal insulation could not be realized as mentioned above. The temperature fall due to the leak of heat from the specimen to polystylene foamm was eliminated electrically by an electronic correction circuit. It takes about 5-6 minutes per one run under this condition.

Fig. 9 Block diagram of a main part of "Thermo Labo".

80 Journal of The Textile Machinery Society of Japan

8. Results and Discussion

8.1 Comparison between Transient Method and Steady

State Method

The specimens used in experiments were vulcanized Iso-

prene Rubber, plasticized PVC, PET and three kinds of

fabrics.

Table 3 summarizes material constants measured by this

equipment. The values of thermal conductivities measured

by the transient and the steady state methods have a good

agreement.

Figure 10 compares the experimentally measured q(t)

with theoretical q(t) given by eq. (25) using the K, k and pC determined by the transient method. The two are in good

agreement. The IR 2, 3 and 4 in Fig. 10 denote Isoprene Rubber

specimen which are molded to have various thicknesses.

The length of the initial curved part of log q(t) increases with increasing thickness. This trend is in agreement with

the theoretical prediction in section 4.1.

8.2 Results of the Simplified Method Table 4 summarizes the experimental values of thermal

Table 3 K values obtained by the transient heat conduction method (ao = 0.1, yo = 10)

conduction method compared by the steady heat

Table 4 Comparison of the values of thermal conductivity obtained by using the simplified

strict methods

and the

Vol. 29 No. 4 (1983) 81

conductivity, K, of fabrics and polymer sheets measured by the simplified method. Comparison of thermal conductivities measured by the strict and the simplified methods are also shown in Table 4. Relative error (%) or K obtained by the strict method to that of the simplified method is in agreement with what Table 2 predicts. This shows that K values ob-tained by the simplified method are reliable when a> a~ is satisfied.

8.3 The Measurement of Thermal Conductivity of Foam Material by the Simplified Method

Table 5 summarizes thermal conductivity of foam meas-ured by the simplified method and the steady state method. In the case of foam material, it is difficult to carry out the experiment under the insulated condition to obtain para-meter a. Therefore, the strict method cannot be used to measure the thermal conductivity of foam material. How-ever, because of the very small pC value of foam, parameter a is expected to be large enough to apply the simplified method when the thickness is 1-2 mm. In Table 5, estimated values of a of the foam assuming that pC 0.03 are shown. The agreement between K values obtained from the simpli-

fied method and those from the steady state method is good. This fact supports the good approximation ability of the simplified method.

Fig. 10 Experimental results of the transient heat conduction

(broken lines) The solid lines are the theoretically fitted curves using

measured thermal parameters.

Table 5 Thermal conductivity of foam materials obtained by the simplified method

Table 6 Comparison between the strict method and the simplified method

82 Journal of The Textile Machinery Society of Japan

8.4 Comparison between the Strict Method and the Simpli- fied Method

Table 6 summarizes the comparison between the strict and the simplified methods. With the simplified method, the K values can be obtained just by the experiment under the constant temperature condition. Meanwhile, in the strict method, experiments under the insulated condition must be carried out in addition to those under the constant tem-

perature condition but it adds only several minutes to one run.

Table 7 shows an example of reproducibility of the meas-ured K values due to this method. Relative error to the mean values is less than about 5 % and it may be concluded that the reproducibility is good.

When thermal diffusivity, k, and specific heat per unit volume, pC, have to be measured, the strict method is re-commended. On the other hand, when only thermal con-ductivity, K, is needed, the simplified method is recommend-ed. Advantages of the present method are as follows :

(1) The measurement can be carried out in a few seconds, therefore, with less effect of the change of test condition. For example, in the measurement of fabrics, which contain some amount of water in ordinary state, the heat conduc-tion properties of fabric can be measured in the moistured condition with less change in water content;

(2) The principle of the measurement method is very simple and the apparatus is also simple. In addition, the boundary condition is well defined. These combine to make

the physical meaning of the measured value very clear and

the reliability of the measured values high.

9. Conclusion

One-dimensional transient heat conduction from a heat

source having a finite heat content was analyzed mathe-

matically and the results were applied to the measurement

of heat conduction properties of sheet-like specimen.

Properties measured are thermal conductivity, K, thermal

diffusivity, k, and specific heat per unit volume, pC. The

values of K obtained by this transient method and that by the

steady state method were in good agreement. It is concluded

that the present transient method is applicable with high

reliability to the measurement of heat conduction properties

of fabrics and polymer sheets.

Table 7 Example of reproducibility of measurement

Material IR (d = 0.0742 cm)

References

[1] Naka, S., Motomura, H, and Kamata, Y.; J. Text. Mach. Soc. Japan, 26, T34 (1973).

[2] Naka, S. and Kamata, Y.; J. Text. Mach. Soc. Japan, 26, T43 (1973).

[3] Naka, S., Kamata, Y. and Yoshino, J.; Sen-i Gakkaishi, 30, T9 (1974).

[4] Naka, S. and Kamata, Y.; Sen-i Gakkaishi, 30, T43 (1974).

[5] Naka, S. and Kamata, Y.; J. Text. Mach. Soc. Japan, 30, T30 (1977).

[6] Kamata, Y.; Sen-i Gakkaishi, 31, T317 (1975). [7] Horikawa, A. and Matsuda, Y.; J. Text. Mach. Soc.

Japan, 27, T84 (1974).

[8] Nogai, T., Nozawa, Y. and Narumi, Y.; Sen-i Gakkai- shi, 33, T398 (1977).

[9] Nogai, T., Nozawa, Y. and Narumi, Y. ; Sen-i Gakkai- shi, 34, T301 (1978).

[10] Nogai, T.; Sen-i Gakkaishi, 36, T324 (1980). [I 1 ] Nogai, T. ; Sen-i Gakkaishi, 36, T389 (1980). [12] Nogai, T. and Ihara, M.; Sen-i Gakkaishi, 36, T427

(1980). [13] Hollies, N. R. S., Bogaty, H., Hintermaier, J. C. and

Harris, M.; Text. Res. J., 23, 763 (1953).

[14] Kawabata, S. and Akagi, Y.; J. Text. Mac. Soc. Japan, 30, T13 (1977)

[15] Kawabata, S.; Proceeding of the 33rd Annual Con- ference of Text. Mach. Soc. Japan (1980).

[16] Churchill, R. V.; "Operational Mathematics" 3rd ed. McGraw-Hill Kogakusha (1972).

[17] Carslaw, H. S. and Jaeger, J. C.; "Conduction of Heat in Solids" 2nd ed. Oxford University Press (1959).

Vol. 29 No. 4 (1983) 83