Thermal Diffusivity of Plastic

John Barber

10138600

School of Physics and Astronomy

The University of Manchester

Second year laboratory report

May 2023

This experiment was performed in collaboration with Jiaqi Hu

Abstract
The heat flow through Poly-Ether Ether Ketone (PEEK), a high temperature plastic, was
investigated to find the thermal diffusivity using two methods: (a) using a sudden change in
temperature and (b) using periodic changes. The diffusivity (D) for (a) was found to be
D = 0.188 ± 0.001m m 2 s −1 . The D for (b) found for a time period, τ = 300s , was
D = 0.206 ± 0.015m m 2 s −1. The value for D found online is 0.2m m 2 s −1 (Phillip, Pech-May et
al, 2019). The values found for (a) and (b) both agree with the value found online to 6% and 3%
percentage error respectively, however, the uncertainty for (a) was underestimated and the true
value does not lie within its range. Despite this, the experiment overall was a success.

[1] Phillip, A., Pech-May, N., et al. Direct Measurement of the In-Plane Thermal Diffusivity of
Semitransparent Thin Films by Lock-In Thermography: An Extension of the Slopes Method,
Analytical Chemistry, Volume 91, Issue 13, 8476-8483, 2019.
1. Introduction
Poly-Ether Ether Ketone (PEEK) is a thermoplastic polymer first developed in 1978. PEEK is a
high-temperature plastic used widely in science and industry due to its valuable properties. It is a
good electrical and thermal insulator, it is very durable and it is biocompatible, giving it
applications in many fields (Drake Plastics Ltd. Co., 2023). Much of the groundwork for
understanding thermal diffusivity and conductivity was laid by Joseph Fourier. In 1822, he
published his work in The Analytical Theory of Heat, in which he reasoned that “the heat flow
between two small deviations in distance was proportional to the small difference in their
temperatures and he made the first experimental determination of thermal conductivity in a solid”.
He proposed his differential equation for the conductive diffusion of heat, now known as Fourier’s
law or the conduction equation. The experimental procedure he used to investigate the conduction
of heat involved heating a small sphere of known specific heat, density and radius up to a constant
temperature, which was then allowed to cool in a temperature-controlled environment
(Narasimhan, 2010). Today, determinations of thermal diffusivity involve using lasers to heat thin
films of the material in question in a vacuum, using thermal imaging cameras to map the flow of
heat across the surface (Phillip, Pech-May et al, 2019).

2. Theory
Letting the temperature at time t and position r in a medium be T(r, t), considering the heat flow in
and out of a volume element leads to the conduction equation (Carslaw, Jaeger, 1959)
∂T K 2
= ∇ T (1)
∂t ρC
where K is the thermal conductivity, ρ is the density and C is the specific heat. Together they make
up the thermal diffusivity
K
D= (2)
ρC
In this experiment, the sample of PEEK used is cylindrical so cylindrical polar coordinates have
been used, along with the approximation that as the length is much greater than the radius, ϕ and z
dependence have been ignored. This gives the conduction equation in the form (Fourier, 2009)
∂T ∂ 2T 1 ∂T
= D[ 2 + ] (3)
∂t ∂r r ∂r
Using this approximation and treating D as independent of variation in temperature and position,
the equation can be solved for the cylinder and the diffusivity can be found. Two cases are
explored in this experiment.

2.1 Case A - A Sudden Change in Temperature

If the cylinder is considered to be at thermal equilibrium at temperature T (0) at time t = 0 when the
temperature at the surface, r = a, is suddenly changed to a temperature T (1). The axial temperature
behaves according to the solution of (3) which is a series of exponential terms (Carslaw, Jaeger,
1959)

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 ∞ −λ 2n Dt
(0) (1) (0)
∑
T (0,t) = T + (T − T )[1 − an e a2 ] (4)
n=1

where λn are the positive roots of the Bessel function J0(x) = 0, and an are constants of order unity.
If long enough time has passed then any terms for n ≥ 2 become so small that they can be ignored
and the graph of ln(T (1) − T ) plotted against time will become linear.

2.1 Case B - A Periodic Change in Temperature

This case considers a periodic change in temperature, assuming that any transients have died away.
Equation 3 is first solved for sinusoidal variation and then the arbitrary periodic variation is
constructed using Fourier techniques. The amplitude and phase lag are described by complex
algebra so solutions are of the form:
T (r, t) = f (r)e iωt (5)

Substituting for T into equation 3 then and making the change of variables z = r −iω /D gives:
d2 f 1 df
+ +f = 0 (6)
dz 2 z dz
This is Bessel’s equation of order zero. The solution of which, for a complex argument of
z = r −iω /D or z = x −i where

x =r ω /D (7)

is the Kelvin function, M0(x). From this the ratio of the surface temperature is

T (a, t) M0(r ω /D)

= (8)
T (0,t) M0(0)
and the ratio of amplitudes is

T (a, t) M0(a ω /D)

| |=| | = |M0(a ω /D)| (9)
T (0,t) M0(0)
and the relative phase is

T (a, t) M0(r ω /D)

arg = arg = arg(M0 a ω /D) (10)
T (0,t) M0(0)
If the temperature at the surface of the cylinder is considered to be a square wave of period τ,
alternating between temperatures T (1) and T (2), it can be expressed as a Fourier series. The
resulting analysis of this gives the equations
2(T (2) − T (1)) 2(T (2) − T (1))
b= = (11)
π|M0(a ω /D)| π|M0(a 2π /D τ)|

ϕ = arg|M0(a 2π /D τ)| (12)

where b is amplitude and ϕ is the phase lag. The final equation used in calculations is two times
equation 11 as the peak-to-peak amplitude was used to decrease the size of the uncertainty.

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3. Experimental approach

Fig. 1. Two thermocouples are connected to the PEEK cylinder, one embedded in the
centre and one on the circumference. These are then connected to the control unit
which outputs the measurements to the external PC.

The internal and external thermocouples are connected to the control unit, which displays the
temperatures on an LCD screen and outputs the measurements to a program running on an external
PC. Beakers of boiling water and ice water were prepared, the ice water beaker was kept in a
styrofoam bucket to insulate it from the surrounding temperature and the beaker of boiling water
was prepared using a hot plate. The sample time can be changed using the program and in this
experiment, it was set to 500 ms. When the cylinder is submerged in the ice, it must be stirred
constantly to ensure the surface temperature reaches the ice temperature. This is not necessary
when submerged in the boiling water as convection currents disperse the thermal energy evenly.
The diameter of the cylinder was measured using Vernier callipers over 3 points along the length.
An average of these 3 values was taken and divided by two to find the radius. The standard
deviation of the 3 measurements was calculated to be used as the uncertainty because the resolution
of the callipers was much smaller than the deviations in the measurements. The value obtained was
a = 11.9 ± 0.035m m.

3.1 Case A
For case A, the PEEK cylinder was allowed to reach thermal equilibrium in the beaker of ice water,
then it was submerged into the boiling water. It was allowed to reach thermal equilibrium again
and then the procedure was reversed. The diffusivity was then calculated from the readings in
Labview.

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3.2 Case B
For case B, the PEEK cylinder was periodically submerged into the ice and boiling water. The
theoretical analysis requires that the lowest harmonic dominates for the axial temperature to appear
sinusoidal. A range of periods were used to experimentally find a response in the axial temperature
that appeared reasonably sinusoidal.

4. Results and Discussion

The data for case A is presented in Fig. 2. It shows the natural log of the modulus of the difference
in the axial and external temperature plotted against time. The line of best fit, fitted using least
squares regression is shown in the top right, the gradient of which is m = − 0.0077 . As the
relationship between the two plotted variables is linear, the gradient of the line is equal to the
exponent given in equation 4 giving this relationship:
−λn2 D
m= (11)
a2
where a is the radius of the cylinder and λn is the first positive root of the Bessel function
J0(x) = 0. Rearranging for D gives the diffusivity which was calculated to be
D = 0.188 ± 0.001m m 2 s −1. The uncertainty on this value was calculated by plotting the data into
the program LSFR on Python and finding the uncertainty in the gradient. This was then combined
with the uncertainty on the radius to get the final value of 0.48%. This gives ± 0.0009 which was
rounded to 0.001 as an uncertainty can never be more precise than the measurement.

Fig. 2. This shows a graph of ln(|T(a xial) − T(external)|) against time for case A,
going from the ice water to the boiling water with the equation of the least
squares regression line of best fit labelled in the top right.

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The data for Case B is presented in Fig. 3. It shows the axial temperature plotted against time for a
period of 300 seconds. The first 450 seconds were not used in the calculation of the diffusivity as
the cylinder needed time for the temperature variation to become sinusoidal. The peak-to-peak
amplitude was then measured at 3 points along the sinusoid and the standard deviation was
calculated to be the uncertainty of this measurement. The value for this was B = 25.38 ± 0.9 ℃.
This amplitude was then plugged into equation 11 and rearranged to find |M0(a ω /D)| . Then,
using tabulated values for the Kelvin function, a linear relationship was plotted between
|M0(a ω /D)| and x so that values for x could be interpolated for any given value of
|M0(a ω /D)|. The value for |M0(a ω /D)| was calculated to be 3.792 which corresponded to
an x value of 4.034. Rearranging the substitution made in equation 7 to find D gave a value of
D = 0.206 ± 0.015m m 2 s −1. The uncertainty on this value was calculated from the combination
of the uncertainty on the axial temperature, the external temperature, the radius, the time period and
the standard deviation of the amplitude using standard error propagation techniques.

Fig. 3. This shows the graph of axial temperature plotted against time for a time
period of 300 seconds for case B.

Fig. 4. This shows the graph of natural log of | M0| plotted against x, used to
interpolate x values from calculated | M0| values.

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The value found for the diffusivity in case A is quite accurate and does agree with the true value of
0.2m m 2 s −1, with a percentage error of only 6%. However, the error on this value has been
underestimated as the true value does not lie within the uncertainty range and it is a tiny percentage
of only ≈0.5%. The value for case B was more accurate, with a percentage uncertainty of just 3%.
The uncertainty on this value has also been estimated better. The percentage uncertainty is 7% and
this put the true value within the uncertainty range. The result for case A could be improved by
estimating the uncertainty better. The temperature was never perfectly constant whilst taking
measurements and fluctuated by a fraction of a degree. A standard deviation for the fluctuations
could have been calculated and used instead of using the resolution of the temperature readings and
this would have given a more accurate estimate of the uncertainty. For case B, although the value
and uncertainty are quite accurate, the kelvin function could have been plotted in Python instead of
using the plot made for interpolation. This would have also allowed for the uncertainty to be
calculated easier as the derivative of the function could be calculated easily using Python. The
diffusivity in this experiment was calculated by the real part of the kelvin function, however, it also
could have been calculated by using the imaginary part or the argument of the function and the
phase lag using equation 12. This would have provided another way to calculate the diffusivity and
a way to check the results using different data.

6. Conclusion
In conclusion, this experiment was mostly a success. The value for the diffusivity of PEEK plastic
was found using two different methods to good degrees of accuracy, having a 6% and 3%
percentage difference for cases A and B respectively. Three ways this experiment could be
improved is through the methods mentioned in the discussion; the standard deviation in the
temperature fluctuations should have been used to calculate the uncertainty for case A and for case
B the kelvin function should have been plotted to improve the interpolated relationship between x
and | M0|. This would have improved the accuracy of the final result and improved the accuracy of
the estimate of the uncertainty. The argument of the kelvin function should have also been
considered as another way to check the results using different data as there was data that did not
end up being used.

References
[2] Carslaw, H. S., Jaeger, J. C., (1959) Conduction of Heat in Solids, Clarendon Press, Oxford,
Chapter 7.

[3] Fourier, J. B. J., (2009). “Of the Diffusion of Heat,” In The Analytical Theory of Heat,
Cambridge Library Collection - Mathematics, chapter, Cambridge, Cambridge University Press,
pp. 333–466.
[4] Phillip, A., Pech-May, N., et al., (2019). Direct Measurement of the In-Plane Thermal
Diffusivity of Semitransparent Thin Films by Lock-In Thermography: An Extension of the Slopes
Method, Analytical Chemistry, 91, 13, 8476-8483, 2019.

[5] https://drakeplastics.com/peek/, PEEK, Drake Plastics Ltd. Co., Date accessed (08/05/2023).

[6] Narasimhan, T. N., (2010). Thermal conductivity through the 19th century, Physics Today, 63, 8,
36-41, 2010.

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