TM
The Hot Disk Thermal Constants Analyzer is an emerging technology that uses the transient plane source technique to measure the in-plane and through-plane thermal conductivity of an anisotropic material in the same test [1-5]. The sensors used in this test method consisted of a
10µm thick nickel foil embedded between two 25.4 µm thick layers of Kapton polyimide film.
The nickel foil was wound in a double spiral pattern and had a radius, R of either 3.189 mm or
6.403 mm. For the more conductive samples the sensor with the larger radius was used. The thermal conductivities were measured at 23 o C. Since the test specimens are anisotropic, this test method is useful for this project.
A picture of the Hot Disk Thermal Constants Analyzer apparatus is shown in Figure 1. Figure 2 and Figure 3 show the top view and the side view of the sample holder and sensor, respectively.
A picture of the covered sample holder can be seen in Figure 4. Figure 5 shows how the sensor is positioned between two samples of composite material.
In this experiment, the samples tested were composite disks with a diameter, D of 6.4 cm and a thickness, x of 3.2 mm. To help ensure that the assumption of an infinite sample domain was met and that heat was not penetrating completely through the sample in the axial direction, two of these composite disks were stacked together above the sensor and two more stacked below it, giving us a double thickness of sample. This stacking of disks allowed the generation of more reproducible data. For each formulation, typically 5 different sets of 4 disks (a total of 20 disks) were tested.
The sensor then had a constant electrical current (variable by sample from 0.03W – 1.25W) over a short period of time (variable by sample from 2.5s – 40s) passed through it. The generated heat dissipated within the double spiral was conducted through the Kapton insulating layer and into the surrounding sample, causing a rise in the temperature of the sensor and the sample.
From a theoretical standpoint, the double spiral pattern can be approximated by a series of concentric, equally spaced ring sources. The characteristic heat conduction equation, assuming radial symmetry in the sample, is then given as [1-5]:
  p

T
 t
 k in
1 r



 r r

T
 r


 k thru
 2
T
 z
2
  rings
Q r

( r
 r ' )

( z ) (1) where
ρ 3 ), C p
is the heat capacity of the sample (J/(kg · K)), T is the temperature of the sample (K), t is the time of the measurement (s), k in
and k thru
are the inplane and through-plane thermal conductivities of the sample (W/m · K),
δ
is the Dirac delta function, r ' is the radius of one of the ring sources, and Q r
is the power supplied to that ring per unit length of the ring (W/m). The total power for each ring is proportional to the circumference of the ring 2
 r ' , such that the total power supplied for all of the rings is Q (W). This total power Q is an input parameter to the Hot Disk Thermal Constants Analyzer. The first term in
Equation 1 represents accumulation of thermal energy, the second term radial (referred to as inplane in our experiments) heat conduction, the third term axial (referred to as through-plane in our experiments) heat conduction, and the final term is a heat source.
The sample can be approximated as an infinite domain if the experimental time is much less than the characteristic thermal diffusion time. For an anisotropic material in a cylindrical geometry,
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the experimental time must meet the following two criteria: t
 x
2
/
  thru

. In these formulas,
  k /(

C p
) t


D / 2
   in
and
, which is the thermal diffusivity of the composite material.
The average transient temperature increase of the sensor is simultaneously measured by recording the change in electrical resistance of the nickel sensor [1-5] according to:

T

1



R
R n no

1

 (2) where Δ T is the change in temperature at time t (K), β is the temperature coefficient of resistance
(TCR) of the material (1/K), R n
R no
is the electrical resistance of the nickel at time 0 (Ω). The temperature rise in Equation 2 is correlated with the in-plane and through-plane thermal conductivities through the solution of
Equation 1 as:

T

 3 / 2
R
P k in k thru
F (

) (3) where P is the power dissipation in the probe and F (

) is a dimensionless time dependent function of
   in t / R
2 given by an integral of a double series over the number of rings m :
F (

)

[ m ( m

1 )]

2

0
  
2


 l m  

1 k m

1 k exp


 l
2  k
4 m
2

2
2

 I
0 lk
2 m
2

2


 d

(4)
A more detailed derivation of Equations 3 and 4 is given by He [6]. Equations 1 through 4 are used to determine the in-plane and through-plane thermal conductivity of the composite being tested.
Additional information about the Hot Disk apparatus can be found at http://www.hotdisk.se/intro.html.
References
1. M. Gustavsson, E. Karawacki, and S.E. Gustafsson, Rev. Sci. Instru.
, 65(12) , 3856 (1994).
2. T. Log and S.E. Gustafsson, Fire and Materials , 19(1) , 43 (1995).
3. V. Bohac, M.K. Gustafsson, L. Kubicar, and S.E. Gustafsson, Rev. Sci. Instru.
, 71(6) , 2452
(2000).
4. Hot Disk Thermal Constants Analyzer Instruction Manual, Mathis Instruments, Ltd.,
Fredericton, New Brunswick, Canada (2001).
5. Transient Plane Source-Gustafsson Hot Disk Technique, standards for Contact Transient
Measurements of Thermal Properties. National Physical Laboratory, United Kingdom, accessed February 2006 at http://www.npl.co.uk/thermal/ctm.
6. Y. He, Thermochimica Acta, 436 , 122 (2005).
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Figure 1: Hot Disk Thermal Constants Analyzer Apparatus
Figure 2: Hot Disk Sample Holder and Sensor (Top View)
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Figure 3: Hot Disk Sample Holder and Sensor (Side View)
Figure 4:Hot Disk Sample Holder with Cover
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Figure 5: Schematic of Samples and Sensor for the Hot Disk
The insert at the lower left shows the double spiral heating element .
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