Modeling Radial Thermal Diffusion
Ryan Phillips, Tim Gfroerer, and Peter Rossi
Physics Department, Davidson College, Davidson, NC
Abstract
Thermal conduction is governed by the thermal diffusivity
constant of the conducting material. In our experiment, we
use a thermal camera to gather temperature data from metal
plates, which are heated at their center by a cylindrical rod.
We fit our measured radial thermal profiles to Gaussians,
and use the results to obtain the diffusivity via the analytical
solution to the heat equation. To confirm the accuracy of our
approach, we model the system with a radial finite difference
simulation. Our results are further verified by comparison
with thermal diffusivity values reported in the literature. [2]
Theory
Experimental Design
Thermal
Camera
Reflection
Shield
Metal
Sheet
Rod Motion
Simulation
Rod
Axis
r
Heating
Rod
Method of Finite Differences
æ Ti+1, j - Ti, j ö
æ Ti, j - Ti-1, j ö
ri+1/2 ç
÷ - ri-1/2 ç
÷
Ti, j+1 - Ti, j
è Dr ø
è Dr ø
=a
Dt
ri Dr
http://www.matter.org.uk/matscicdrom/manual/df.html
J  T
Fick’s 1st Law:
Torch
A heated cylindrical rod briefly touches the center of a metal sheet.
On the other side of the sheet, a thermal camera monitors the
temperature of the metal as heat diffuses.
α = Measured Diffusivity
The heat flux across an interface depends on the temperature
gradient and the thermal diffusivity α of the medium.
Fick’s
2nd
T
Law:
  2T
t

T    T 


r

g
r
,t


t
r r  r 
Using the diffusivity, the temperature distribution at a given
time step can be expressed as a linear transformation of the
temperature distribution in the previous time step.
If the heat flux into a volume element differs from the heat flux
out, the temperature will change in accordance with Fick’s 2nd law.
Given the azimuthal symmetry of the experiment, the heat
equation reduces to the cylindrical coordinate form shown above,
assuming negligible dependence on the z-coordinate.
T ( r, t ) µ e
-2 r 2 /( R 2 +8a t )
b2 = R2 + 8at
The analytic solution to the heat equation contains a Gaussian term,
which describes the shape of the time-dependent radial profile. [1]
We obtain Gaussian fits and equate their parameters to the
exponential term in the analytic solution. The slope of the result vs.
time can be used to identify experimental diffusivity values.
Time Delay
0.3s
0.9s
1.4s
1.8s
2.3s
100
• We achieve a high degree of accuracy and precision in
our analysis without any adjustable parameters.
10-2
Cu:
a =1.23´10-4 ± 6 ´10-6 m
-6 m2
s
2
s
Al:
alit = 6.70 ´10-5 m
Cu:
alit =1.17´10
s
40
0.0012
6061 Al
110 Cu
0.0010
0.0009
0.0008
0.0007
50
Acknowledgments
Acknowledgment is made to the Donors of the American
Chemical Society Petroleum Research Fund for partial
support of this research.
Time Delay
0.3s
0.5s
0.9s
1.2s
1.6s
s
Obtaining Thermal Diffusivity Through b2 Parameter
0.0011
30
Comparison of 110 Cu Simulation and Experimental Profiles
2
-4 m2
20
Radial Distance (mm)
Temperature Difference (K)
a = 7.0 ´10 ± 5´10
-5
10
Literature Diffusivity Values
Al:
100
10-1
References
[1] In-plane thermal diffusivity evaluation by infrared
thermography, F. Cernuschi, A. Russo, L. Lorenzoni, and
A. Figari, Rev. Sci. Instrum. 72, 3988 (2001)
10-2
[2] Thermal Diffusivity Imaging, Tim Gfroerer, Ryan Phillips
(Davidson '16), and Peter Rossi (Davidson '15), American
Journal of Physics (in press).
0.0006
10-3
0.0005
0
0.0004
10
20
30
40
50
60
Radial Distance (mm)
0.0003
0.0002
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (s)
2.2
2.4
2.6
2.8
3.0
• The thermal camera offers 1mK resolution in about
105 pixels, allowing us to gather data with
unprecedented precision. Moreover, the frame rate of
10Hz minimizes the loss of accuracy due to cooling –
we collect all data within three seconds of rod contact.
• Our experimental design enables us to obtain
diffusivity values for Aluminum and Copper alloys
with less than 5% deviation from literature values.
Using our experimental diffusivities, we simulate the
radial temperature profiles with an efficient, onedimensional finite difference method.
10-1
0
Measured Diffusivity Values
Conclusion
Comparison of 6061 Al Simulation and Experimental Profiles
Temperature Difference (K)
Analytic Solution for
Instantaneous Heating:
This representative thermal image shows the temperature
distribution shortly after contact of the rod. The color scale on the
right identifies temperature in degrees Celsius.
Results
Obtaining Diffusivities
b2 (m2)
Present Time Step
Next Time Step
Measured radial profiles in aluminum and copper are compared with
simulated results (solid lines). The precision of the analysis is
highlighted by the logarithmic scale.