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Thermal Diffusivity of Plastic
James Fotheringham and Sagarika Jain
7743195 and 9587184
School of Physics and Astronomy
The University of Manchester
Second year laboratory report
November 2015

Abstract
A series of experiments were performed to calculate the thermal diffusivity of epoxy
resin
...
The result obtained for diffusivity was (8
...
42) ×
10−8 m2 ⁄s
...
The amplitude and phase lag from this
curve were then used to calculate diffusivity
...
12 ±
0
...


1
...
For example, carbon fibre
reinforced epoxy resin is used in the automotive and aerospace industries due to its
strength and low weight
...


A diglycidyl ether (DGEBA) is the most popular
[1]However,

it does not function well at high

temperatures, a situation common in aerospace application
...
[1] Therefore, measuring thermal properties of epoxy resin is
extremely important
...
It is necessary for the value of
thermal diffusivity to be accurate in order to ensure correct modelling of temperatures
in components, which determines the temperature ranges that are not suitable for the
usage of a particular epoxy resin
...
Theory
Diffusivity can be defined as the ratio between the rate of change of temperature and the
Laplacian of the temperature
...

In cylindrical coordinates, this can be expressed as
∂θ
∂t

∂2 θ

1 ∂2 θ

1 ∂θ

∂2 θ

= D( ∂r2 + r ∂r + r2 ∂ϕ2 + ∂z2 )
...
Hence, the formula now
becomes
∂θ
∂t

∂2 θ

1 ∂θ

= D ( ∂r2 + r ∂r )
...
1 Sudden Change in Temperature
The axial temperature is given by,
2

2

θ(0, t) = θ0 + (θ1 − θ0 ){1 − ∑∞ an e−λn Dt⁄a } ,[3]
n=1

(4)

where 𝜆 𝑛 are the positive roots of the Bessel function
...
Removing these terms
and rearranging gives
θ−θ1
∆θ

2

∝ e−λ1 Dt⁄a

2

,

(5)

where 𝜆1 ≃ 2
...


2
...

T

The axial temperature is represented as follows,
𝜃(0, 𝑡) = 𝑏0 + ∑∞ 𝑏 𝑛 𝑐𝑜𝑠(𝜔 𝑛 𝑡 − 𝜙 𝑛 )
𝑛=1

[3]

(8)

where,
an
bn

= |M0 (a√ωn /D)|

(9)

and,
ϕn = argM0 (a√ωn ⁄D)
...
Therefore, the change in axial
temperature is,
b1 =

2(θ2 −θ1 )
π|M0 (a√ωn /D)|

(11)

and
ϕ1 = argM0 (a√2π⁄TD)

(12)

where T is the time period
...

b3
b1

=

|M0 (a√ω1 /D)|
3|M0 (a√3ω1 /D)|

(14)

The peak-to-peak amplitude B is twice b1
...
Experimental Approach
Two beakers of water were set up
...
The
diffusivity was measured using a cylindrical piece of epoxy resin of radius (9
...
1)
mm
...
The chart recorder was
calibrated at a temperature of 273
...


Figure 1: Experimental setup for both experiments
...
1 Experiment A: A Sudden Change in Temperature
The cylinder was immersed in the ice water and was allowed to attain thermal
equilibrium at the ice point
...
The time of initial immersion was recorded on the graph paper
...
Graphs of ln

|θ−θ1 |
∆θ

versus

−a2
λ2 t
1

were plotted separately for an increase and

decrease in temperature ranging from 0 to 100℃
...


3
...
Several time periods were tried in order to obtain a sinusoidal curve
...


4
...
1 Experiment A
Two sets of values were obtained for the axial temperatures of the cylinder as it was
immersed in hot and cold water
...

Taking the logarithm of both sides of equation (5), we get
ln|

θ−θ1
|=
∆θ

lnK −

λ2 Dt
a2

(16)

where K is a constant of proportionality
...

The graphs for increasing and decreasing temperature as discussed in section 3 are
shown in Figure 2 and 3
...
(Reduced
Chi-Squared: 3
...
(Reduced
λ2 t
1

Chi- Squared: 1
...
9 × 10−8 m2 ⁄s and 1 × 10−7 m2 ⁄s,
represent the values of thermal diffusivity
...
7 ×
10−8 m2 ⁄s
...
2 Experiment B
This experiment was carried out three times, twice with a time period of 120 seconds
and once with a time period of 100 seconds
...

Using equation (15), from B, values of 𝑀 𝑜 were obtained and averaged
...
40, 7
...
84
...
Values for x at the calculated values of Mo were obtained using shapepreserving interpolation on Matlab
...
76, 5
...
89
...


(17)

Three values of diffusivity were calculated for different time periods and values of x,
which averaged to 2
...

The time lag was measured using the graphs from the chart recorder
...
The phase lag was then given by
∅=

2π∆t
T

(18)

Where ∆t was measured to be 105 seconds and T was 120 seconds
...
A graph was plotted between the given
values of Arg|𝑀 𝑜 |and x
...
35
...
10 × 10−8 m2 ⁄s
...
5℃ since it was half the
smallest easily measurable value
...
71℃
...
These uncertainties are depicted by the error
bars in Figures 2 and 3
...
These errors, when averaged, gave a
final uncertainty of ± 0
...

Experiment B
Error on B was given by adding in quadrature the individual uncertainties in measuring
the peaks on the graph
...
5 ℃
...
2, which was calculated using
Equation (14) and ranged from 1
...
46
...
The error on Mo was obtained by error propagation on equation
(15)
...


Error on x

Error on Mo
m

Figure 5: Calculating error on x
...
The difference between this value of x and its
average value gives the error on x
...
The calculated errors for the three values of Diffusivity were averaged to
get a final error of ±2
...

Error on the phase lag was obtained by error propagation on equation (18), 2
...
The
error on x was found using a plot ofArg|Mo | and x, similar to how it was calculated in the
previous section (Figure 5)
...
84 × 10−8 m2 ⁄s
...
70 × 10−8 m2 ⁄s

± 0
...
12 × 10−7 m2 ⁄s

±0
...
10 × 10−8 m2 ⁄s

±1
...
Discussion
The results obtained from both parts of the experiment, although very close, were not
within three standard deviations of each other
...
This implies
an underestimation of errors in part B of the experiment
...
Also, there were human
errors in the fact that the transfer of the cylinder from one beaker to another was not
instantaneous, leading to inaccuracies
...


6
...
All the results were
similar and close to the actual values of thermal diffusivity of epoxy resin, which are
between 1 and 2 x10-7m2s-1, depending on the type of epoxy resin considered

[4]
...


7
...
Multi ‐ component Epoxy Resin Formulation for High
Temperature Applications
...
The University of Manchester; 2014
...
Properties of Matter; 2000
...
Conduction of Heat in Solids
...
Oxford
University Press
...

[4] Chiguma, J et al
...
OJCM
...
Pg 51 – 62
...
scirp
...
aspx?PaperID=34468>
[Accessed on: 19th November 2015]



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