Home

SearchCollections

Journals

About

Contact us

My IOPscience

A New Method for Simultaneous Determination of Anisotropic Thermal Conductivities Basedon Two-Dimensional Analysis

This content has been downloaded from IOPscience. Please scroll down to see the full text.
2001 Jpn. J. Appl. Phys. 40 388(http://iopscience.iop.org/1347-4065/40/1R/388)
View the table of contents for this issue, or go to the journal homepage for more
Download details:
IP Address: 160.29.75.151
This content was downloaded on 06/06/2017 at 08:46
Please note that terms and conditions apply.

You may also be interested in:
Thermoelectric Properties of Li-Doped CuONaoki Yoshida, Tomoyuki Naito and Hiroyuki Fujishiro
Temperature Dependence of Sound Velocity in High-Strength Fiber-Reinforced Plastics
Ryuji Nomura, Keiichi Yoneyama, Futoshi Ogasawara et al.
Simulation of the van der Pauw Measurement for Electrically Anisotropic Semiconductors
Finite-Element Method
Yuichi Sato and Susumu Sato
An Oxide Single Crystal with High Thermoelectric Performance in Air
Ryoji Funahashi, Ichiro Matsubara, Hiroshi Ikuta et al.
Low thermal conductivity porous Si at cryogenic temperatures for cooling applications
K Valalaki and A G Nassiopoulou

Using theJpn. J. Appl. Phys. Vol. 40 (2001) pp. 388–392
Part 1, No. 1, January 2001c
2001
The Japan Society of Applied Physics

A New Method for Simultaneous Determination of Anisotropic Thermal Conductivities
Based on Two-Dimensional Analysis
Hiroyuki F UJISHIRO, Tatsuya O KAMOTO, Manabu I KEBE and Koichi H IROSE
Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan(Received May 31, 2000; revised manuscript received August 28, 2000; accepted for publication October 12, 2000)

A new experimental setup and analysis method for simultaneous determination of the anisotropic thermal conductivities (κx , κ y )is proposed for materials with uniaxial anisotropy or a layered structure. By applying heat power diagonally through the sample
from a heater to a cold finger, temperature differences (Tx , Ty ) along the two directions were monitored. The monitored
temperature differences were compared with Tx and Ty , which were calculated by solving a two-dimensional heat diffusion
equation. Calculations were repeated until Tx and Ty agreed with Tx and Ty , to determine κx and κ y . This technique
was applied to a carbon-fiber-reinforced plastic (CFRP) from 40 K to 200 K and a satisfactory agreement was achieved with theκx and κ y values and those determined from independent measurements of κx and κ y .
KEYWORDS: anisotropic materials, anisotropic thermal conductivity, low temperature, heat diffusion equation, identical experimental setup

1. Introduction
In the field of low-temperature physics and cryogenic engineering, many structurally anisotropic materials [e.g. single crystals of high-Tc superconductors and fiber-reinforced
plastics (FRPs)] are used as functional and constructive materials. It is very important to investigate their anisotropic
natures in terms of their mechanical, electrical and thermal transport properties. The thermal conductivity κ(T ) isa valuable parameter for practical use such as in the design of cryostats and power leads for superconducting magnets.1) As shown in Fig. 1(a), κ(T ) is usually measured by a
one-dimensional (1D) steady-state heat flow method using along rectangular-shaped sample, which is based on the relation κ = (Q/T )(L/S),2) where Q is the applied heat power,
S the cross section of a sample, and T and L the temperature
difference and distance between thermometers, respectively.
For anisotropic materials, anisotropic κ values (e.g. κx , κ y ,κz ) are measured in separate experimental setups using long
rectangular-shaped samples cut along each direction. In layered compounds, for example, only a plate-like single crystal
can be obtained and it is often difficult to prepare long samples to realize a homogeneous 1D heat flow for perpendicular
conductivity (κ⊥ ) measurement. Figure 1(b) shows a conventional experimental setup for κ⊥ measurement perpendicular
to the sample face of a plate-like material.3) A sapphire platewith a higher thermal conductivity is attached to the sample
face and a heater is adhered to the sapphire plate. Because of
the short distance L of the sample, the thermocouples cannot
be attached directly to the sample and temperatures have to
be measured on both the sapphire plate and the cold finger.
In this case, which is referred to as a thermally two-terminal
method, the contact thermal resistivity Wc between the sample and the sapphire (the cold finger) significantly influencesthe T measurement. Furthermore, the wide contact area between the sapphire plate (or the cold finger) and the sample
may prevent homogeneous 1D heat flow because of the inhomogeneity of Wc in the contact area. Therefore, it is desirable
to measure κ⊥ (T ) without the influence of Wc .
In this paper, we propose a new method to enable simultaneous determination of the anisotropic thermal conductivities(κx , κ y ) in samples having two-dimensional (2D) anisotropy

using an identical experimental setup which is free from theinfluence of Wc . This method has been applied to a carbonfiber-reinforced plastic (CFRP) from 40 K to 200 K and wasverified to obtain κx and κ y with high precision.
2. Experimental Procedure
Figure 2 shows a schematic view of the experimental setup
around the sample. A rectangular sample with uniaxial
anisotropy is situated parallel to the x- and y-axes. The sample is thermally connected to a heater and a cold finger andFig. 1. (a) Conventional experimental setup for κ(T ) measurement by asteady-state heat flow method using a long sample. (b) Conventional experimental setup for the perpendicular conductivity (κ⊥ ) measurement. A
sapphire plate is attached to the sample face and a heater is adhered to thesapphire plate.

388

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 1

Fig. 2. Schematic diagram of the new experimental setup around the sample in this study.

heat power Q is applied diagonally through the sample. In thissituation, temperatures at three measuring points, P1 , P2 andP3 , increase due to heat flow, and the temperature rise (T1 ,T2 , T3 ) at each point is determined by thermal conductivities (κx , κ y ) along the x- and y-directions and the applied heat
power Q. The temperature differences, Tx = T1 − T2and Ty = T1 − T3 , also change depending on the magnitudes of κx , κ y and Q. The measured temperature differences (Tx and Ty ) after the application of heat power arecompared with calculated ones (Tx and Ty ) obtained bysolving a two-dimensional heat diffusion equation including
different pairs of anisotropic thermal conductivities (κx and

H. F UJISHIRO et al.

389

κ y ). This procedure is continued until Tx = Tx andTy = Ty are achieved at each measuring temperature andthe thermal conductivities along the two directions, κx (T ) andκ y (T ), can be determined.
Table I shows sizes and lengths in the experimental setup
shown in Fig. 2. The measured sample was a CFRP witha fiber volume fraction of Vf = 0.6. The high-κ carbon
fiber was aligned along the y-direction. The sample size
was 5.7 × 5.5 × 30 mm3 in the x-, z- and y-directions.
A Gifford-McMahon (GM) cycle helium refrigerator wasused as a cryostat and a copper cold finger was attachedto the cold head of the GM refrigerator.4) The sample
was adhered to both a metal film resistance heater (1 k;
2.0 × 5.5 × 2.5 mm3 ) and the copper cold finger using Ag
paste. AuFe(0.07 at.%)-chromel thermocouples with a 76 µm
φ were used to measure temperatures at the three positions(P1 , P2 and P3 ). The detection limit of the temperature difference T was ≈0.005 K. The measurement was performed
from 40 to 200 K. The sample was enclosed by a radiation shield of Ni-plated copper which was thermally anchoredto the cold head. The sample chamber was evacuated to<1 × 10−5 Torr by an oil diffusion pump to prevent heat convection and conduction by remanent gas. After heat power Q
was applied to the CFRP sample and a steady heat-flow state
was realized, the temperature differences Tx and Ty were
measured.
3. Numerical Solution of 2D Heat Diffusion Equation
In the case that heat power Q is applied diagonally to auniaxially anisotropic material using the experimental setup
shown in Fig. 2, the time dependence of the temperature variation at a certain position T (x, y, t) can be given by solving
the following 2D heat diffusion equation,5)

∂T
∂2T
∂2T
κx ∂ 2 Tκy ∂ 2 T= αx 2 + α y 2 =
+
,
∂t
∂x
∂y
dC ∂ x 2
dC ∂ y 2(1)where αx and α y (cm2 /s) are the thermal diffusivities along the x- and y-directions, respectively, d (g/cm3 ) is the mass density
and C (J/cm3 K) is the specific heat per volume of the sample. The sample lengths L X and L Y and time t are divided into
equal intervals x, y and t, respectively, so that the x-y-t space is covered by a grid of rectangles. Equation (1) can be
represented by the following finite-difference solution using an explicit method,
κx T (i + 1, j, t) − 2T (i, j, t) + T (i − 1, j, t)
T (i, j, t + t) − T (i, j, t)
=
t
dC
x 2
κ y T (i, j + 1, t) − 2T (i, j, t) + T (i, j − 1, t)
+
,
(2)
dC
y 2
where T (i, j, t) is a temperature at a grid point (i, j) at time t with i and j being integers, and T (i, j, t + t) is a temperature
at the point (i, j) after a time interval t. It is to be noted that the spacial coordinates of the grid point (i, j) are x = i × x
and y = j × y, respectively. We can rearrange eq. (2) as follows:κx tT (i, j, t + t) = T (i, j, t) +{T (i + 1, j, t) − 2T (i, j, t) + T (i − 1, j, t)}dC x 2κ y t+{T (i, j + 1, t) − 2T (i, j, t) + T (i, j − 1, t)}.
(3)dC y 2Thus, the unknown temperature T (i, j, t + t) at the point(i, j) after the time interval t can be represented in termsof the five known temperatures, T (i + 1, j, t), T (i, j, t),
T (i − 1, j, t), T (i, j + 1, t) and T (i, j − 1, t). In order to illuminate the principle and procedure of the proposed method,
we present typical results of numerical solutions of eq. (3).

The temperature rise at a given point i, Ti , was normalized by the maximum temperature difference in the sample,
TH − TC , where TH is the temperature at the sample surface
area where the heater is attached and TC is the temperature at
the sample surface area where the cold finger is attached. This
normalized temperature rise was then denoted Ti . Accord-

390

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 1

H. F UJISHIRO et al.

Table I. Meanings of symbols and lengths in experimental setup shown in Fig. 2.
SymbolLength (mm)Remarks

LY
LX

30.0
5.7

T
LLY
LCY
LHY
YY

5.5
12.8
6.4
2.5
20.1

length of the sample along the y-direction
distance between P1 and P2(length of the sample along the x-direction)
length of the sample along the z-direction
distance between P1 and P3
contact length with cold finger (y-direction)
contact length with heater (y-direction)
distance between P1 and the upper surface of cold finger

Fig. 3. Calculated isothermal lines (every T  = 0.1) in the sample for (a)A(= κ y /κx ) = 1, (b) A = 3, (c) A = 5 and (d) A = 10. Sample size andthe lengths in the experimental setup are set as shown in Table I. Temperature difference T  is reduced by the maximum temperature differenceTmax (=TH − TC ) between the temperatures of the sample surfaces at theheater (TH ) and at the cold finger (TC ) (see text).

ingly, the normalized temperature difference between eachpoint of the sample was denoted by Tx = T1 − T2
and Ty = T1 − T3 . A constant heat flow condition wasapplied to the surface area where the heater is attached anda constant temperature condition was applied to the surface
area where the cold finger is attached. An adiabatic condition (∂ T /∂ x = 0 or ∂ T /∂ y = 0) was applied to the othersurface areas. The temperature differences Tx and Ty
in the steady heat flow state were calculated confirming thesteady values of the calculated temperatures after a sufficient
time lapse. The contact thermal resistivities Wc between thesample and the heater and between the sample and the cold
finger do not affect Tx and Ty in the steady state. In thepresent calculation, x, y and t were set to be 0.025 mm,
0.04 mm and 0.005 s, respectively.
Figure 3 depicts some examples of calculated isothermallines. The sample sizes and lengths are summarized inTable I. The given dimensions correspond to the experimental
setup in the following section. The isotherm lines are drawn
at intervals of T  = 0.1. The thermal conductivity ratio
A = κ y /κx was changed from A = 1 in Fig. 3(a) to A = 10
in Fig. 3(d). These figures suggest that the measuring position P1 should be located as close to the heater as possible
because the temperature difference between P1 and P2 is thenvery sensitive to the thermal conductivity ratio A.
Figure 4 shows the calculated reduced temperature differ-Fig. 4. Calculated normalized temperature differences Tx (between P1
and P2 ) and Ty (between P1 and P3 ) as a function of the anisotropic
thermal conductivity ratio A. Tx and Ty are normalized by Tmax .

ences Tx and Ty as a function of anisotropy ratio A. It canbe confirmed that the temperature difference Tx increases
with increasing the anisotropy ratio A. On the other hand,Ty decreases with increasing A. In actual experiments, thetemperature differences Tx and Ty measured at each measuring temperature are determined by the heat power Q andthe thermal conductivities κx and κ y along the x- and y-axes,
respectively. We uniquely determine the κx and κ y values bycomparing the measured Tx and Ty with Tx and Ty
(not reduced) which were numerically calculated on the basis of eq. (3). A conceptional view of the present method ispresented in the following figure.
Figures 5(a) and 5(b) present the calculated curves of Txand Ty as a function of the thermal conductivity κ y forseveral fixed κx values under a constant applied heat power
Q. We discuss the estimation method only in the κ y > κx
(A > 1) region. The temperature difference Tx increases
with increasing κ y for a fixed κx value but decreases withincreasing κx for a fixed κ y value. Conversely, Ty values
decrease with increasing κ y for a fixed κx value and also decrease with increasing κx for a fixed κ y value. In Fig. 5(a),a set of points at which a calculated Tx value is equal toa measured Tx value forms a curve parallel to the κx -κ yplane. A similar curve can be also obtained for Ty = Ty in
Fig. 5(b). Figure 5(c) schematically shows the projections ofboth sets of points to the κx -κ y plane. The cross-point of thetwo projection curves can thus be uniquely obtained and theanisotropic thermal conductivities (κx , κ y ) can be determined.

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 1

H. F UJISHIRO et al.

391

Fig. 6. Temperature dependence of measured Tx /Q and Ty /Q values
for the CFRP sample under the experimental setup shown in Fig. 2.

Fig. 5. Conceptional view for determination of the anisotropic thermal
conductivities κx , κ y . The calculated temperature differences Tx andTy are given for several pairs of the thermal conductivities κx , κ y undera constant applied heat power Q. In (a) and (b), the projections of a setof points to the κx -κ y plane are drawn schematically where Tx = Txand Ty = Ty are achieved. In (c), a cross-point of the two projection
curves can be uniquely obtained and the anisotropic thermal conductivities(κx , κ y ) can be decided.

4. Experimental Results and Summary
Figure 6 shows the temperature dependence of the measured values of Tx /Q and Ty /Q for the CFRP sample under the experimental setup shown in Fig. 2. The Tx /Q andTy /Q (K/mW) values are the temperature differences along
the x- and y-directions normalized by the applied heat power
Q. The value Ty /Q is larger than Tx /Q over the entire
temperature region and increases with decreasing temperature. The Tx /Q values slightly increase as temperature decreases to 50 K and then sharply decrease with further decrease of temperature. In the present experiment, no Tx /Q
values could be obtained at T < 40 K because the Tx values
became very small, which are beneath the detection limit ofour system. These results indicate that both κx and κ y values
decrease and that anisotropic ratio A = κ y /κx also decreases

Fig. 7. Thermal conductivities κx (T ), κ y (T ) of the CFRP sample as afunction of T determined by the present technique. Dashed lines show thethermal conductivities κx0 (T ), κ y0 (T ) which were measured separately
by a 1D steady-state heat flow method using slender rectangular-shaped
CFRP samples for each direction.

with decreasing temperature, as inferred from Fig. 4.
Figure 7 shows the temperature dependence of the thermal
conductivities κx (T ), κ y (T ) of the CFRP sample determinedby the technique used in this study. In this figure, dashed lines
show the thermal conductivities κx0 (T ), κ y0 (T ) which were
measured by the 1D steady-state heat flow method using two
slender rectangular-shaped CFRP samples cut from the samebulk sample for each direction. The κx (T ) and κ y (T ) values obtained by the present technique are in good agreementwith the values measured by the 1D heat flow method. Thus
the anisotropic thermal conductivities κx (T ) and κ y (T ) can be
determined simultaneously and uniquely using a single experimental setup. It is again to be noted that in the present technique, the contact thermal resistivity Wc between the sample
and the heater (or the cold finger) does not influence the determination of κx (T ) and κ y (T ) because the temperatures at
P1 , P2 and P3 are directly measured on the sample surface andthus the temperatures at the heater and the cold finger do notappear in the analyses of the heat flow.
In summary, we proposed a new technique for simultaneous determination of the anisotropic thermal conductivities

392

Jpn. J. Appl. Phys. Vol. 40 (2001) Pt. 1, No. 1(κx , κ y ) for materials with uniaxial anisotropy using a single
experimental setup. This technique can be applied to materials with a wide range of anisotropic thermal conductivity
ratios A = κ y /κx and the various ratios of sample length
L y /L x . The validity and usefulness of this method wereverified using a carbon-fiber-reinforced plastic (CFRP) in thetemperature range from 40 K to 200 K.
Acknowledgment
The authors wish to thank T. Kashima and A. Yamanaka
of TOYOBO Co., Ltd., in the preparation of the CFRP sample. This work is partially supported by a Grant-in-Aid for

H. F UJISHIRO et al.

Scientific Research from the Ministry of Education, Science,Sports and Culture (No. 09555018).

1) H. Fujishiro, M. Ikebe, K. Noto, M. Matsukawa, T. Sasaoka, K. Nomura,
J. Sato and S. Kuma: IEEE Trans. Mag. 30 (1994) 1645.
2) R. Berman: Thermal Conduction in Solids (Clarendon Press, Oxford,
1975).
3) S. J. Hagen, Z. Z. Wang and N. P. Ong: Phys. Rev. B 40 (1989) 9389.
4) M. Ikebe, H. Fujishiro, T. Naito and K. Noto: J. Phys. Soc. Jpn. 63 (1994)
3107.
5) J. Crank: The Mathematics of Diffusion (Clarendon Press, Oxford,
1975).