Proceedings of the International Conference on Mechanical Engineering and Mechatronics
Toronto, Ontario, Canada, August 8-10 2013
Paper No. XXX (The number assigned by the OpenConf System)
A New Sensor Design for the Measurement of Thermal
Conductivity
Ali Nouri-Borujerdi, Navid Sarikhani
Sharif University of Technology, Department of Mechanical Engineering
Azadi Ave., Tehran, Iran
anouri@sharif.edu; nsarikhani@mech.sharif.edu
Abstract - A new sensor has been fabricated from copper wire of diameter 33 micrometers, for the measurement of
both thermal conductivity and thermal diffusivity, rapidly and simultaneously. The method is a variant of Hot Wire
technique and uses a resistive element both as heater and temperature sensor. The main advantages of this new
design are its simplicity, non-destructive nature, sensitive and durable heater and small sample size requirements.
Some measurements have been carried out by the present sensor in the range of 0.15-14.6 W m-1K-1 for thermal
conductivity and 0.09-3.9 mm2s-1 for thermal diffusivity on samples of polypropylene, high density polyethylene
and AISI-304 stainless steel. The uncertainty of the measurements is estimated to be less than 10% for thermal
conductivity and 20% for thermal diffusivity.
Keywords: thermal conductivity, thermal diffusivity, sensor, disk heat source
1. Introduction
There are two main approaches for the measurement of thermal conductivity, steady-state approach
and transient approach. Guarded Hot Plate (ISO 8302) and Guarded Heat Flow Meter (ASTM E 1530) are
well known and standardized examples of the steady-state one. In steady-state methods, temperature at
two points in the test specimen and corresponding steady heat flux are measured. Then thermal
conductivity is determined by using the Fourier’s conduction law. The need for large samples and long
measurement times are the main drawbacks of the steady-state methods. In transient methods, on the
other hand, thermal conductivity is measured by analyzing the temperature response of the test specimen
to a transient heat source. Due to the transient nature, these methods are fast and require small samples.
Among the transient methods the Hot Wire family is the most successful one and is widely used (Mathis,
2000). The principle of this family of transient methods is the same. A direct electrical current passes
through a resistive element, which is placed between two pieces of material sample under investigation.
The resistive element serves as both heater and temperature sensor. By the measurement of instantaneous
mean temperature of the heater, along with an appropriate theoretical model it is possible to deduce both
thermal conductivity and thermal diffusivity from a single transient recording experiment. The variation
in this family is primarily due to the shape of sensor. The sensor may be a simple straight wire (ISO
8894), or a strip (Gustafsson, 1983), or a foil cut in a spiral shape (ISO 22007-2).
In this study we have designed and fabricated a new sensor from copper wire. The wire is bent in
such a way that it covers a flat circular area. Compared to other designs, fabrication of our sensor is
simple and inexpensive. The electrical resistance of the sensor can be controlled and adjusted during the
fabrication procedure and sensor with arbitrary size and resistance can be made. The result is a more
sensitive and durable sensor. Due to small sample size requirement and rapid measurement time, our
method is especially suitable for expensive materials with a low thermal conductivity, such as nanocomposites and nano-fluids.
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2. Design and Experimental Setup
Figure 1 shows a schematic of the sensor. The total resistance of the sensor at room temperature is
6.51  and has been made of copper wire of diameter 33 m and length 424.025 mm.
Fig. 1. Schematic of the sensor
Starting from a copper wire of diameter 120 m and purity of 99.9%, at the first step wire was etched
in a dilute ferric chloride solution (5 gr FeCl3/95 gr H2O) for about one hour. The instantaneous resistance
of the wire was monitored during the etching process using an ohmmeter with a resolution of 0.01 .
After attaining the desired resistance, wire was washed out and bent in a zigzag shape to cover a circular
area of diameter 16 mm. Two ends of the copper wire then was soldered on copper foils of thickness 20
m. Finally sensor was covered in both sides with 30 m polyamide thin films (Kapton tape), to ensure
its mechanical strength and planarity, as well as keep it electrically insulating. Other parts such as
handgrip and connector cables then have been attached.
The sensor serves as both heat source and temperature sensor. Instantaneous mean temperature
increase of the sensor is monitored by recording its instantaneous electrical resistance in an electrical
Wheatstone bridge (Figure 2).
Fig. 2. Wheatstone bridge for monitoring the instantaneous electrical resistance of the sensor.
The electrical resistance of the sensor varies with temperature as
R (t )  R H (1   T )
(1)
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where, RH and R(t) are the initial and instantaneous sensor resistance, T is the instantaneous mean
temperature increase of the sensor and 0.0039 K-1 is the temperature coefficient of resistance of copper.
Using basic circuit analysis, the relation between T and voltage imbalance in the bridge, U ,
may be obtained as follows
T 
U (R H  R S )2
 R H V R S  U (R H  R S )
(2)
where, V is the voltage of power supply and RS is a standard constant resistance, which is in series with
the sensor.
If one considers the ADC branch in Figure 2, using the perturbation theory with   R H  T as
small parameter, it can be shown that



 2R S  2
RS 
1
1
P (t )  P0 1 
1
1 
 
   ...
2 
RH 
 R S  R H  R H  (R S  R H ) 

(3)
where, P0 and P(t) are initial and instantaneous output power of the sensor.
To maintain the output power of the sensor as constant as possible, standard resistance RS, should be
chosen close to the sensor resistance RH. So, the second term in the right hand side of Eq. (3) would be
negligible and the variation of the sensor output power becomes very small of the order of 2. Thus the
sensor can be considered as a disk heat source with a constant output power of P0, where
 V

P0  R H 

 R H  RS 
2
(4)
To measure the thermal conductivity, the sensor is placed between the flat surfaces of two sample
pieces of the material under investigation. The analytical solution of the instantaneous mean temperature
increase of a disk heat source developed by Pinsker, 2006 and Nouri 2013 is as follows
T ( )  T i  T s  T i 
D ( ) 

P0
D ( )
 ak
(5a)
4

1

 1 
 1 


 (2  3 2 )I 0  2   (2   2 )I 1  2  
3
 3   exp  1  
 2 
 2  
 2
 2 
t
(5b)
(5c)
a
In the above equations, k and  are thermal conductivity and thermal diffusivity of the medium,
respectively. T is the instantaneous mean temperature increase of the sensor. T i is the temperature
increase over the polyamide insulating thin films covered the both sides of the sensor. T s is the
instantaneous temperature increase of the specimen surface. a and P0 are the radius and output power of
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the sensor,  is dimensionless time and D ( ) is dimensionless specific time function. Finally, I0 and I1
are the first kind modified Bessel functions of zeroth and first order.
Figure 3a shows a typical plot of the mean temperature increase of the sensor obtained from voltage
recording. In this case, 200 data points have been collected in 125 seconds. Data points from 15 to 150
were used in calculations as the experimental time window. The first 15 data points were excluded
because data points at the beginning of the experiment may be interrupted by polyamide films covered the
both sides of the sensor. Nevertheless, the heat conduction across this layer becomes steady after a short
time and its effect would be eliminated. Data points at the end of transient recording also may be
influenced by physical boundary of the sample. The heat penetration depth (Arpaci, 1966) can be
calculated from 5t and sample size should be large enough to fulfil the infinite medium condition,
during the time that the experiment is being performed.
Fig. 3. Typical mean temperature increase of the sensor as a function of (a) time and (b) specific time
function.
To determine the thermal diffusivity, first an approximate interval for its value, such as [ min ,  max ] ,
is selected. Thermal diffusivity of most materials is in the range of 0.01 to 100 mm2s-1 and thus this
interval can be used. Then the interval is divided to m  1 segments. For each
 i   min  i  max   min  / m , i  0,1,..., m , all pairs of data points [t j , T (t j )],
j  1, 2,..., n is


transformed to  D ( ij ), T ( ij )  by using the equations of (5b) and (5c). Now the correlation coefficient
(Ross, 2009) for each  i is calculated according to
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n
Ci 
 (D (
j 1
ij
)  D ( ij )) (T ( ij )  T ( ij ))
n
n
2 
2
  (D ( ij )  D ( ij ))     (T ( ij )  T ( ij )) 
 j 1
  j 1

,
 ij 
i t j
a
(6)
The correct value for thermal diffusivity is the value with C i  1 , which implies a linear relationship
between T ( ij ) and D ( ij ) , j  1, 2,..., n . After finding the thermal diffusivity,  i , employing the
least square method, a line is fitted to the data pairs of  D ( ij ), T ( ij )  , j  1, 2,...n . Thermal
conductivity then can be determined from the slop of the fitted line by using the Eq. (5a).
3. Results and Discussion
Two identical samples of each three different materials, polypropylene, high density polyethylene
and AISI-304 stainless steel were prepared. All samples were in disk shape and large enough to provide
an infinite medium at the experimental time window. Polypropylene (PP) was an isotactic homopolymer
supplied by Sabic (Saudi Basic Industries Corporation, Europe). Commercial-grade high density
polyethylene (HDPE) was also obtained from Imam Khomeini Petrochemical Complex, Mahshahr, Iran.
A digital voltmeter with the resolution of 100 V and sampling rate of two data points per second
was used for recording of the voltage imbalance in the Wheatstone bridge. The radius, resistance and
temperature coefficient of resistance of the sensor were a  8 mm, R H  6.51  and   0.0039 K-1,
respectively. A standard resistance with R S  6.04  and 0.1% tolerance was placed in series with the
sensor. In another branch of Wheatstone bridge a potentiometer with R P  1024  was used. The power
was supplied by three Ni-MH batteries with the total voltage of V  3.6  0.006 V. All the measurements
were carried out at room temperature, 22 ˚C.
For polypropylene and polyethylene, 200 data points in 120 seconds were recorded at each
experimental run and an interval of about 85 seconds (data points from 15 to 150) were used as
experimental time window. For stainless steel 30 data points were recorded and the data points from 5 to
25 were used.
The results are summarized in Table 1. In all cases the correlation coefficient is greater than 0.999.
As one can see, the standard deviation for thermal conductivity is less than 10% and for thermal
diffusivity is less than 20%.
Table 1. Thermal conductivity and thermal diffusivity of three different material samples at 22 ˚C.
Experimental
Run
1
2
3
4
5
Mean
StDev.
StDev. (%)
Polypropylene
k

(W m-1 K-1) (mm2 s-1)
0.16
0.10
0.16
0.09
0.14
0.08
0.15
0.09
0.14
0.07
0.15
0.09
0.01
0.01
6.7
13.3
High Density Polyethylene
k

(W m-1 K-1) (mm2 s-1)
0.44
0.19
0.49
0.25
0.49
0.26
0.48
0.21
0.45
0.20
0.47
0.22
0.02
0.03
5.0
14.0
AISI-304 Stainless Steel
k

(W m-1 K-1) (mm2 s-1)
15.23
4.33
15.43
4.71
14.59
3.92
13.07
3.41
12.25
3.05
14.1
3.9
1.39
0.67
9.9
17.3
Recommended values for thermal conductivity and thermal diffusivity of homopolymer
polypropylene (Mark, 2007) are 0.15 Wm-1 K-1 and 0.09 mm2 s-1, which are in excellent agreement with
our results. For high density polyethylene recommended values (Mark, 2007 and Web-1) are 0.46±0.04
Wm-1 K-1 for thermal conductivity and 0.23 mm2s-1 for thermal diffusivity. The tabulated values of
thermal conductivity and thermal diffusivity of AISI-304 stainless steel (Touloukian et al., 1970 & 1973
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and Web-1) also scatter around 14.6 Wm-1K-1 in the interval of ±1% and 3.7 mm2s-1 in the interval of
±7%, respectively.
The work is in progress to improve the accuracy of the measurements by using a more accurate
voltmeter with a higher sampling rate. Also, an additional precision voltage regulator for power supply is
planned to reduce the voltage perturbations below 100 V. However, an important advantage of our
sensor design is the possibility of fabrication of sensors with very high resistance per unit area. Present
sensor which is made of 33 m copper wire has 0.032 ohm.mm-2 resistance per unit area. New sensor of
15 m copper wire with 0.157 ohm.mm-2 resistance per unit area is under development. The concept of
our design can be easily extended to use other metallic wires, such as nickel and molybdenum wires, too.
6. Conclusions
A new sensor has been designed and fabricated for the measurement of thermal conductivity and
thermal diffusivity of materials. The idea of using the etched wire, allows for the fabrication of sensor to
be simple and low cost. As well as, sensor size and resistance can be adjusted precisely in a wide range of
values, during the fabrication process. The fabricated sensor is highly sensitive and durable. The
performance of this sensor was assessed by the measurements on polypropylene, high density
polyethylene and AISI-304 stainless steel. The results obtained coincide well with the reference values
found in literature. The accuracy of the measurements is estimated to be better than 10% for thermal
conductivity and 20% for thermal diffusivity.
Acknowledgements
This work has been financially supported by the Iran National Science Foundation (Project:
INSF-89004000), which is gratefully acknowledged.
References
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[2] ASTM E 1530, Standard Test Method for Evaluating the Resistance to Thermal Transmission of
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Transfer 7th edition” Wiley.
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[13] Touloukian Y.S., Powell R.W., Ho C.Y. and Klemens P.G. (1970). “Thermophysical Properties of
Matter: Thermal Conductivity, Metallic Elements and Alloys vol. 1” New York/Washington:
IFI/Plenum.
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[14] Touloukian Y.S., Powell R.W., Ho C.Y. and Klemens P.G. (1973). “Thermophysical Properties of
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Web sites:
Web-1: http://www.thermtest.com/ consulted 20 Jan. 2013.
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