Simultaneous estimation of thermophysical properties by periodic hot-wire heating method

Simultaneous Estimation of Thermophysical Properties by Periodic Hot-Wire Heating Method I

Jun Fukai* Hiroshi Nogami Takatoshi Miura Shigemori Ohtani Department of Biochemistry and Engineering, Tohoku University, Sendai, Japan

• A new unsteady heat flux method is proposed to measure thermal conductivity and diffusivity simultaneously, the principle based on an analytical solution for an infinite hollow cylindrical system with a periodic heat source in the center. Thermal conductivity and diffusivity are determined from the amplitude and phase lag of the temperature response within the cylinder. The method also makes it easy to measure the temperature dependency of thermal properties during a continuous heating process. The measurement errors caused by the finite specimen size and the displacement of the thermocouple location are estimated numerically to confirm the accuracy of the measurement method. Effective thermal conductivity and diffusivity for the packed beds of aluminum oxide particles and potassium perchlorate particles are measured. The thermophysical properties measured by this method agree well with those measured by conventional methods such as the hot-wire, periodic heating, and continuous heating methods.

Keyworfls: instrumentation, measurement techniques,

thermophysical properties, thermal conductivity, thermal diffusivity

INTRODUCTION The unsteady heat flux method based on an analytical solution has generally been developed to measure such thermophysical properties as thermal conductivity, thermal ditfusivity, and specific heat. This method generally requires a steady state initially, so it takes a long time to measure the temperature dependency of the thermophysical properties over a wide temperature range. On the other hand, if the method can be applied to the continuous heating process in which the hole of the specimen is heated continuously, the measurement becomes easy and rapid, but it is necessary to pick out the transient temperature response from the measured temperature in which the transient temperature response is overlaid on the ramped temperature component produced by heating at a constant rate during the continuous heating process. Matsuda et al [1] measured the thermal conductivity under a crystal transformation and thermal decomposition reaction by using the differential thermal analysis (DTA) technique, which is one of the measuring methods used for the continuous heating process. The experimental procedure of the DTA technique is a little complicated, because of a couple of samples have to be prepared for the measurement and then heated at the same conditions. For the measurement of specific heat by ac calorimetry during the continuous heating process, the periodic component (ac signal) is detected only

*Present address: Department of Chemical Engineering, Kyushu University, Fukuoka, Japan 812.

by a lock-in amplifier [2] that has an analog high-pass filter. A digital high-pass filter has also been used for the conventional periodic heating method in which the sinusoidal heat flux is taken from the outer surface for thermal ditfusivity during the continuous heating process [3]. Thus the periodic temperature response can be more easily and more accurately picked out from the measured temperature than the other transient responses--the stepwise and pulse-wise responses. It is also known that the periodic heating method is very useful for measuring the thermophysical properties of materials accompanied with a mass transfer such as a wet porous material [4]. However, a useful periodic heating method has not yet been proposed to measure thermal conductivity. Most of the simultaneous determination methods of the thermophysical properties are also based on stepwise and pulse-wise heating [5-8], not on periodic heating. For the unsteady heat flux method, thermal diffusivity is determined only from the temperature response in the specimen. It is also necessary to measure the absolute heat flux value for thermal conductivity. This procedure makes it difficult to measure thermal conductivity by an unsteady heat flux method. The standard reference material is frequently used instead of measuring the absolute heat flux. However, an appropriate choice of standard material is very difficult [8]. A line heat source and a radiative heat source are very useful in determining the heat flux. In this study, a new periodic heating method is proposed to measure thermal conductivity and ditfusivity simultaneously, based on an analytical solution for an infinite hollow cylindrical system where a sinusoidal heat flux is taken from a line

Address correspondence to Prof. T. Miura, Department of Biochemistry and Engineering, Tohoku University, Katahira 2, Sendal 980, Japan.

Experimental Thermal and Fluid Science 1991; 4:198- 204 © 1991 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010

198

0894-1777/91/$3.50

Estimation of Tbermophysical Properties

199

and kei(~R b) = 0, Eqs. (7) and (8) in Table 2 can be simplified as shown by Eqs. (10) and (11), respectively. [ker 2 ( ~ R ) + kei 2 ( ~ R ) ] °5

(10)

{[ker '2 (~) + kei '2 (~)]21r/Fo} °'5

I

gp = tan_,[ kei'(~ )ker(~R) - ker'(~ )kei (~R) ]

I

ker' ( ~ )ker ( ~ R) + ke-~-(~ ) - - ~ e i - ( ~

- 7r

I I V

0 u

I o

O"

l

I

p4~

Figure

1. Principle of measurement method.

heat source at the center. The thermophysical properties of packed beds are measured during the continuous heating process at a temperature range of 350-650 K. Moreover, the experimental results are compared with those given by conventional measurement methods. MEASUREMENT PRINCIPLE The following assumptions are adopted in the theoretical analysis of the measurement principle: 1. The specimen is in a cylindrical cell that has an infinite length and diameter. 2. The temperature dependency of thermophysical properties is neglected. 3. The elapsed time from the beginning of the sinusoidal heating process is very long. 4. The periodic component can be picked out accurately from the measured temperature in which the ramped component is included in the continuous heating process. Figure 1 shows the measurement principle when a sinusoidal heat flux is given to a cylindrical sample from the inner surface (r = a) where tp is the heating period and qo is the amplitude. The radius of the cylinder b is large enough to be regarded as infinite. The heat conduction equation is given in Table 1 with the initial and boundary conditions based on the assumptions of 1 and 2. When the elapsed time from the beginning of the sinusoidal heating process is very long, the solutions of Eqs. (1)-(4) are given as shown in Table 2, where bei, ber, kei, and ker are Kelvin functions, Fo = rtp/a 2 (Fourier number), R = r/a, r= rt/a 2, Rb= b/a, and ~ = 27r/Fo. When the radius of the cylinder is large enough to satisfy the approximations of ker(~Rb) = 0

(11) The phase lag(0) of the temperature response is a function of Fo, which is, moreover, a function of thermal diffusivity (K). The amplitude (7) is a function of Fo and thermal conductivity (h). The thermal conductivity and diffusivity are hence obtained from the phase lag, and the amplitude of the temperature response from an arbitrary position within the specimen. The relations between 0 and Fo and between and Fo are shown in Figs. 2 and 3, respectively. When the temperature dependency of thermophysical properties is measured during the continuous heating process, the temperature response within the specimen is composed of a ramped component and a periodic one. The principle described above is evidently valid if the periodic component can be accurately picked out from the response in the specimen. E S T I M A T I O N OF M E A S U R E M E N T E R R O R The validity of the assumptions used for the constant thermophysical properties during the measurement period must be checked. Furthermore it is necessary to study the effect of thermocouple displacement as derived from the specimen shrinkage on the measurement accuracy by numerical experiment. The error due to the temperature dependency of the thermophysical properties has already been discussed from a mathematical point of view for the conventional periodic heating method in which the sinusoidal heat flux is taken from the outer surface of the cylindrical specimen in our previous work [9], so that the estimated error is less than 1% even if the thermophysical properties are increased by twice as much in a temperature rise of 100 K. In measuring a good electric conductor, proper insulation, such as a ceramic sheath, is required between the thermocoupie and the specimen. The existence of such insulation, however, interferes with the measurement of the original temperature signal. The effect of the insulation, in which the thermocouple is inserted, on the measurement of thermal

Table

--

Ot

1. Heat Conduction Equation = ~

[ ar z

+

--

(1)

r -~r

Initial conditions 0 = 0 Boundary conditions 00 2wt -X-- = qocos--

Or

tp

0 = 0

at t = 0

(2)

at r = a

(3)

at

(4)

r = b

200 Jun Fukai et al Table 2. Solution of Eqs. (1)-(4)

(5) (6)

0 = r/cos(27rr/Fo - (#) = (qoa/k)

= {[A ber(~R) + B bei(~R) + C k e r ( ~ R ) + D kei(~R)] 2 + [ - A bei (~R) + B ber (~R) - C kei (~R) + D ker (~R)]2} °5 - A bei (~R) + B ber (~R) - C kei (~R) + D ker (~R) (# = tanA ber (~R) + B b e i ( ~ R ) + Cker (~R) + D k e i ( ~ R ) where

ber'(~)

bei'(~)

ker'(~)

-bei'(~)

ber'(~j)

-kei'(~)

ber (~R,,) -bei(~m,o)

bei(~R,,) ber(~Ro)

ker (~R~) -kei(~R.)

(7)

(8)

kei'(~) ]rzl- I ker'(~) l [ ! ]

=

o]

kei (~R,,) ker(~R.)

(9)

and/j = (2a'/Fo) °'5

diffusivity has been studied, and the existence of the sheath has little effect on the measured thermophysical properties, even if the thermal conductivity and the diffusivity of the sheath are a hundred times as large as those of the specimen and the radius ratio of the sheath to the specimen reaches 0.08 [10]. The measurement error due to a finite length of specimen can be neglected when the ratio of the specimen length to the diameter is more than 3 [11]. It is then important to study the effect of the following two factors on the measurement of thermal conductivity and diffusivity: (1) a finite specimen diameter and (2) the displacement of the thermocouple.

eters by the following procedure. At first, a nondimensional temperature amplitude ( 9 and phase lag (d)) are calculated from Eqs. (7)-(9) for any R b and Fo. Next, ~& _ is computed by Eq. (10) from n and Foexp as calculated b y ' ~ t . (11). The relative error of the given properties (X and K) to the obtained values (kexp and Xexp) is defined by Eqs. (12) and (13), respectively.

Error due to Finite Radial Size of Specimen

The relation between the relative errors and the specimen size ( R b ) is shown in Fig. 4 as a function of Fo. The temperature is estimated at R = 50. As the specimen radius R b increases, the measurement error decreases sharply toward a minimum point, then gradually converges to zero. In order for the measurement error to converge to zero at the larger Fourier number, a larger radius of specimen is required. In our measurement range of Fo ( = 500-2000), the ratio of the specimen radius to the distance between the thermocouple and the central axis should be more than 3.

A measurement error is expected to result if the specimen radius is not large enough to satisfy the approximations of ker(~Rb) = 0 and kei(~Rb) = 0. In the estimation of the measurement error, the thermophysical properties computed on the basis of the measurement principle from the temperature distribution, obtained by calculating the finite difference of a heat conduction equation under the finite specimen radius are compared with those obtained under an infinite radius. This estimation is carried out using nondimensional param-

~X .

.

x

.

.

(12)

E, = - - ~ e x P - K = F%xp - F o

K

(13)

Fo

100 4rr 10 -I

313 R=20[-] 30 40 5060 80 100

7 10-2

"o ¢8

.~,2n

.O-

:!l

I~" 10-3

10-" |

10-5 I

101

I

10 2

10 3

10 4

Fo [ - ]

Figure 2. Variations of phase lag (~b) with Fourier number (Fo).

102

103

104

Fo [ - ]

Figure 3. Variations of nondimensional amplitude ( ~ with Fourier number.

Estimation of Thermophysical Properties 201

0.10

i

!

I

~

Ii.

0.05

I

i

R=50

Fo:2000

Hole

000

Heater Copper Wire k

C~

I

/

I ,

Thermo-

/

~ - - couple

W

~0.05 -0.10 0

' 50

' ' 100 150 Rb [ - ]

' 200

250

Figure 4. Effect of specimen diameter (Rb) on relative error

(~).

Error due to the Displacement of the Thermocouple The errors that are produced when the thermocouple deviates from its fixed position by the distance A R are estimated by the following procedures:

I. The nondimensional temperature amplitude ~ and the phase lag ~ are calculated at the displaced position of the thermocouplc, R(1 + AR), for any Fourier number (Fo). 2. ~c~ and Foe,,p are computed by Eqs. (I0) and (II), respectively. 3. The relativeerrors are given by Eqs. (12) and (13). The relation between the relative errors and A R is shown in Fig. 5 at R = 50 and Fo = 100. The Fourier number is found to have little effect on the calculated errors in the wide range of 100 < Fo < 2000, that is, the errors for thermal conductivity are less than 1%. Measurement of thermal conductivity is insensitive to the displacement of the thermocouple. The error for thermal diffusivity is, however, estimated to reach about 20% when the thermocouple is displaced by 10% from its fixed position. Therefore, displacement of the thermocouple should be prevented in the measurement of thermal diffusivity.

X - 0 2.__

-o'.1

0.2 _

l ......

-6.2

Figure 5. Effect of displacement on thermocouple location (A R) on relative error.

,omen

Figure 6. Details of test sections.

EXPERIMENTAL APPARATUS AND METHOD Figure 6 shows details of the test section. The chromel wire (0 0.015 mm) as a line heat source is placed at the center of a stainless-steel shell (inside diameter 36 ram × length 98 mm), and an alumel-chromel thermocouple (,k 0.015 nun) is placed at 5 mm from the center. Both are inserted into ceramic insulating sheaths (0.2 mm I.D., 0.5 mm O.D.). Copper wires are attached to the ends of the line heat source to measure the voltage loaded to the heater. To prevent displacement of the line source and the thermocouple, specimen particles are packed into the cell through the hole in the lid after fixing the insulating sheaths. Figure 7 shows the schematic diagram of the experimental apparatus. The heater at the center of the specimen is connected to a d c power controller and a digital galvanometer. The thermocouple and the copper wires are attached to the heater and are connected to a digital voltmeter through a scanner. The temperature of the enclosure is elevated at a constant heating rate within the electric furnace while the voltage of the line heat source at the center of the specimen is sinusoidally varied with time. The current and voltage given to the heater and the power of the electric furnace are all controlled automatically by a personal computer. The sampling of the temperature data in the specimen is carried out by using the same personal computer. The measured temperature data are recorded on a floppy disk. The voltage loaded to the heater is varied in proportion to [1 + cos(21ct/ti,)] °'5 in order to obtain the sinusoidal heat

202 Jun Fukai et al

Control' er7j"

[ GD~nal~mete 0 Thermocouple Speclmen Stlinless steel shell Electrlc furnlce Insulator Heater

I Floppy ~ LD=sk

Personal

I

L Co~0ute~

Figure 7. Schematic diagram of experimental apparatus.

Po*er Controller

generation at the line source. In the measurement, the inner radius of the insulating tube is taken as the radius of the heater, that is, a = 0.1 mm. After the measurement, the thermal conductivity and diffusivity are both calculated by using the personal computer. At first, the periodic component is picked out from the measured temperature by the Finite Impulse Response highpath filter. The amplitudes and phases for both the temperature response and the heat generation of the heater are determined on the basis of the Fourier transform method. The usefulness of the digital high-pass filter and the Fourier transform method has been pointed out in previous studies [3]. EXPERIMENTAL RESULTS AND DISCUSSION Figure 8 shows the variations of periodic temperature components, which are picked out from the measured temperature and the heat generation of the line heater. The open circles in the figure denote the measured values, and the solid lines show the results calculated by means of the Fourier transform method. The sinusoidal heat generation is confirmed from the measurement results. The temperatures also vary sinusoidally with time.

3.51

,

,

,

o

0.6

o

0

-3.5

0

. 10

.

. 20 t [s]

.

30

40

0.6

Figure 8. Variations of temperature and heat generation with time.

The measurements are carried out with a packed bed of aluminum oxide particles (mean particle diameter 0.03 mm and bulk density 1000 kg/m 3) at about 305 K or at a quasi-steady state. The measured results for various heating periods are shown in Table 3, where the subscript av denotes the average value of several measurements, ox and oK are the modified standard deviations defined in the table in order to compare both o's of k and K as shown in the nondimensional amount; n represents experimental times. The measurements are also carded out by using the conventional hot-wire method [12] and periodic heating method [3], so that the results of kexp = 1.70 W / ( m • K) at 300 K and rexp = 2.02 m2/s at 351 K are obtained. In Table 3, the heating period has little effect on the measured results because the average of the experimental results agree well with each other. The experimental results in Table 3 also agree well with the values measured by conventional methods. The modified standard deviations are only a small percent; those of thermal ditfusivity are particularly small. The modified standard deviations of thermal conductivity are two to three times those of thermal ditfusivity. The reason is that the measurement of the thermal conductivity includes not only the measurement error of the phase lag ~ but also one of the heat flux q0, because the thermal conductivity is determined from Fo, q0, and whereas the thermal diffusivity is obtained only from Fo. The modified standard deviations have a tendency to decrease with an increase in the heating period. As the heating period or Fo increases, the amplitude of the temperature response increase as shown in Fig. 3, and the measurement error of the temperature is reduced. If a larger heat flux is given to the specimen, the modified standard deviation is expected to decrease because a temperature having more figures is measured in the higher temperature region. The variations of the effective thermal conductivity and ditfusivity for a packed bed that is continuously heated at the rate of 0.05 K/s and a heating period of 32 s with the temperature are shown in Figs. 9 and 10, respectively. The results measured by using the conventional hot-wire method and the conventional periodic heating method are also shown in the figures by using solid lines. The results measured by this method agree well with those of the two methods mentioned above. These results are obtained by using the inner radius of the ceramic sheath (0.1 mm) as the inner radius of

Estimation of Thermophysical Properties 203

Table 3. Effect of Heating Period on Standard Deviations of Experimental Results at Quasi-steady State X

tp

r

(s)

X~v (W/(m • K))

ox

Kav (m2/s)

a,

32 40 60

1.67 x 10 - I 1.69 X 10 -~ 1.68 X 10 -1

3.2 X 10 -2 2.1 X 10 -2 1.0 X 10 -2

2.01 X 10 -7 2.13 × 10 -7 2.11 X 10 -7

1.3 X 10 -2 4.5 X 10 -2 4.0 X 10 -2

y = X or x.

try = [(1/n)Y,(YcxpYav)/yav]°'5;

i

the hollow cylinder (a). On the other hand, if the outer radius of the thermocouple or the outer radius of the ceramic sheath is adopted as the radius a, the shift of the derived result is less than 1%. The scatter of the thermal conductivity is also a little greater than that of the thermal diffusivity. The measurements are also carded out during the heating period from 20 s to 60 s, and then each measured thermophysical property is approximated by the least-squares method as a linear function of temperature, Table 4 shows the modified standard deviations between the measured thermophysical properties and the regressed values for each heating period. The modified standard deviations shown in Table 4 are 5 - 1 0 times as large as those in Table 3. The measurement error during the continuous heating process may be caused mainly by the electrical noise of the furnace and error of the digital filter. Figures 11 and 12 show the measured results of a packed bed of potassium perchlorate particles (bulk density 1300 k g / m 3, mean particle diameter 0.08 mm, and heating rate 0.05 K/s). The heating period is 32 s. The effective thermal diffusivity is also measured by a continuous heating method [9] in which the thermal conductivity of the specimen is determined from the temperature distribution in the specimen as shown by solid lines. The measured results also agree well with the conventional method. The effects of radiative and convective heat transfers that are induced by the central hot wire on the measurement are thus considered to be negligible within the present experimental conditions. As a result, the new method is quite precise and accurate even when it is applied to the continuous heating process. It can also be made to rapidly measure the temperature dependency of thermal conductivity and diffusivity over a wide

i

periodic heating method this method

E % 3 x

2

0

300

I

I

i

400

500

600

Figure 10. Variations of effective thermal diffusivity for a packed bed of aluminum oxide particles with temperature.

Table 4. Effect of Heating Period on Standard Deviations of Experimental Results of a Continuous Heating Process tp(S)

~rx

20 32

8.9 6.3 8.2 8.1

40

60

0.5

-

hot wire this

o

0.4

6.0 3.8 4.7 6.3

x 10 -2 x 10 -2 x 10 -2

i

x 10 -2

× 10 -2 x 10 -2 x 10 -2

i

hot wire method this method

o -

oK

x 10 -2

- 0.5

700

0 [K]

0.4

method

method

EO.3

,,(

_EO.3 ~0.2 o

o

o

o

,~0.2

°

o

°

•

oO

o

co o

0.1

0.1

I 0

300

400

i

i

500 0 [K]

600

700

Figure 9. Variations of effective thermal conductivity for a packed bed of aluminum oxide particles with temperature.

0 300

400

5()0

600

8 [K]

Figure 11. Variations of effective thermal conductivity for a packed bed of potassium perchlorate particles with temperature.

204 Jun Fukai et al i

- 4

t t

i

continuous heating method this method

time, s heating period, s

Greek Symbols AR

E 3 X

=~ 2

0

300

I

I

400

500

600

0 [K]

Figure 12. Variations of effective thermal ditfusivity for a packed bed of potassium perchlorate particles with temperature.

displacement of measurement position, dimensionless temperature amplitude, K temperature amplitude ( = hT1/qoa), dimensionless ex, e, relative errors defined by Eqs. (11) and (12), respectively, dimensionless 0 temperature, K r thermal diffusivity, m2/s X thermal conductivity, W / ( m • K) parameter [ = (2 ~r/Fo)°5], dimensionless a modified standard deviation (see Tables 3 and 4), dimensionless z time ( = r t / a 2 ) , dimensionless phase lag, radians

Subscript temperature range. Furthermore, wide application of the simultaneous thermophysical property determination method is expected, because of the ease with which the measurement apparatus can be manufactured and the procedure can be realized. On the other hand, the optimum conditions and the applicable range of this method are both determined from the size of the specimen, the distance between the two thermocouples, and the thermophysical properties by using the dimensionless parameters, for example Fo or ~/, that are used in the measurement error estimation. CONCLUSIONS A new method is proposed to measure thermal conductivity and thermal diffusivity simultaneously. This method has the following features: 1. The displacement of the thermocouple has little effect on the thermal conductivity. 2. The method for the continuous heating process is highly precise and accurate for the measured value. 3. With a continuous heating process, the method makes it easy to measure the temperature dependency of thermal conductivity and diffusivity over a wide temperature range in spite of the assumption of constant thermophysical properties. NOMENCLATURE

c/ radius of line heat source, m b bei, ber Fo kei, ker

Q qo q R Rb r

outside radius of specimen, m Kelvin functions, dimensionless Fourier number ( = r tp / a 2), dimensionless Kelvin functions, dimensionless heat generation, W / m 3 heat flux, W / m s amplitude of heat flux, W / m 2 radial distance ( = r / a ) , dimensionless outer radius of specimen (= b / a ) , dimensionless radial distance, m

av eft exp cal

average value effective value experimental value calculated value REFERENCES

1. Matsuda, H., Hasatani, M. and Sugiyama, S., Thermal Conductivities of Solids under Crystal Transformation and Thermal Decomposition Reaction, Kagaku Kogaku Ronbunshu, 1, 589-593, 1975. 2. Hatta, I., and Ikushima, A., Specific Heat of NaNO3 near its Transition Point, J. Phys. Chem. Solids, 34, 57-66, 1973. 3. Miura, T., Fukai, J., and Ohtani, S., Measurement of Thermal Diffusivity at Continuous Heating Rate, Kagaku Kogaku Ronbunshu, 9, 129-134, 1983. 4. Tanazawa, Y., A Method for the Measurement of Thermal Constants of Wet Material, J. JSME, 35, 390-399, 1932. 5. Katayama, K., Ohuchi, K. and Kotake, S., Transient Methods of Simultaneous Measurement of Thermal Properties, Trans. JSME B, 34, 2012-2018, 1968. 6. Kobayashi, K., and Kobayashi, T., A Measuring Method of Thermal Ditfusivity and Specific Heat of Solid by Square Wave Pulse, Trans. JSME, B, 46, 1318-1326, 1980. 7. Iida, Y., Shigeta, H., and Akimoto, H., Measurement of Thermophysical Properties of Solid by Arbitrary Heating, Trans. JSME, 48, 142-148, 1982. 8. Katayama, K., and Okada, M., Transient Comparison Methods of Simultaneous Measurement for Thermal Properties, Trans. JSME, B, 35, 832-839, 1970. 9. Miura, T., Fukai, J., and Ohtani, S., Effects of Measuring Method on the Effective Thermal Diffusivity of Packed Coal Column during Carbonization, Tetsu-to-Hagane, 70, 336-342, 1984. 10. Miura, T., Fukai, J., Tanno, S., and Ohtani, S., Effect of Thermocouple Sheaths on Temperature Response in Thermal Property Measurement, Kagaku Kogaku Ronbunshu, 9, 642-649, 1983. 11. Miura, T., Sugiyama, K., Fukai, J., Takashima, T., Iida, Y., and Ohtani, S., Measurement of Effective Thermal Diffusivity of Packed Coals during Carbonization, Tetsu-to-Hagane, 69, 538-544, t983. 12. Miura, T., Tajima, H., Miura, K., and Ohtani, S., Measurement of Effective Thermal Conductivity of Packed Coal during Carbonization, Kagaku Kogaku Ronbunshu, 8, 121-126, 1982. Received April 21, 1990; revised June 23, 1990