2nd year laboratory script
G2 Ångström’s Bar
Department of Physics and Astronomy
nd
2 Year Laboratory
G2 Ångström’s Bar
Scientific aims and objectives
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To determine an accurate value for the thermal diffusivity of copper,
and compare it with accepted values
To solve a dispersive one-dimensional wave equation for a real world
system
Learning Outcomes
•
•
•
To fit complex equations by manipulating data to the form y = mx + c
To be able to use χ2 to fit an errors weighted straight line
To understand how a Fourier series expansion simplifies a wave
propagation problem
Apparatus
•
•
•
•
•
•
Copper rod with attached
coloured thermocouples
Acrylic cover tube
Thermocouple selector box
Heater-cooler unit
P.C.
Water cooling (pre-attached)
September 2010
Safety
•
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Do not touch the bar during
operation – it gets hot
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2nd year laboratory script
G2 Ångström’s Bar
Task 1 – Pre-session questions
You are required to complete the tasks found at the back of this script before starting your
laboratory work. These questions will prepare you for the experiment by teaching you how to
manipulate relevant complex equations to yield a simple straight line fit, prove the solution to
the wave equation and understand Fourier series.
Task 2 – The Experiment
A copper rod with pre-attached thermocouples at different positions is provided. You MUST
turn the green tap on at least 10 minutes before taking measurements to ensure a dynamic
thermal equilibrium between the rod, heater and surroundings.
The objective is to determine the diffusivity of copper by comparing the wave properties
(namely amplitude and phase) at different points along the rod.
By measuring the amplitude as a function of distance one is able to determine the attenuation
constant α, and similarly the phase shift – distance relation directly provides the wave
number k. These parameters directly yield an experimental value of diffusivity D. In making
plots of amplitude and phase versus thermocouple distance you will have to assume that the
higher harmonics of the overall waveform have only a negligible contribution and you should
attempt to explain why this is the case.
The experimental thermal diffusivity D can then be compared with that derived from
established values (where K is thermal conductivity, ρ is density, and s is specific heat of
copper sCu = 385 J kg-1 K-1 ) Ensure that you quote the uncertainty of both experimental and
literature values of D and cite where your literature value originated from. Experimental
uncertainty of D can be determined by estimating the range over several wavelengths of raw
data as collected by the computer. However the raw data will need converting into a straight
line graph in order to determine α and k and the propagation of errors on doing so, and on
further propagation to yield D will need to be calculated.
Task 3 – Reporting
You should explain precisely the methods that you used to analyse your recorded data and
any assumptions that you make in this analysis. You should also explain how you arrived at
your experimental uncertainty. You should present both raw waveform data and manipulated
straight line graphs. You should make an assessment of the precision of your final value for D
and explain how you do this. You will be likely to need compound errors formula and you
should show how you apply them. Explaining how the high frequency wave components can
be neglected is also an important discussion point that you might like to explore.
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2nd year laboratory script
G2 Ångström’s Bar
Appendix of supporting information
1.
2.
One-dimensional temperature wave propagation
Solving the wave equation
Table of compound errors
Instructions for use of data logging equipment
Pre-lab questions
3.
4.
5.
1.
One-dimensional temperature wave propagation
The flow of heat
∂Q
at any point x along a bar is given by
∂t
∂Q
∂θ
= − AK
∂t
∂x
(A1)
with A the cross-sectional area, K the thermal conductivity, and θ the temperature. In
other words, heat flow is proportional to the temperature gradient.
The difference in heat flow between two points separated by ∆x is
 ∂Q

 ∂t
  ∂Q
 − 
x   ∂t

∂ 2θ
 = AK 2 ∆x
∂x
x + ∆x 
(A2)
The difference in heat flow has gone to both heating the bar and losses to the
surroundings. Assuming that the difference between the temperature of the bar,
θ(x,t), and the surrounding temperature, θs is small, one can apply Newton’s law of
cooling which states that the heat loss of the bar is proportional to φ = θ − θ c .
The factors controlling this proportionality are the surface area of the bar S, and the
emissivity β that characterises the heat loss process. Mathematically, the loss of heat
from the bar due to Newton’s law of cooling can be given as:
∂Q
= 2πr∆xβφ
∂t loss
(A3)
Finally, the change in heat flow between position x and position x+∆ x is the sum of
the heat used to produce a temperature rise in the bar, and the heat that is radiated
from the bar to its surroundings (as given by Newton’s law of cooling (A3)):
Change in heat flow = raise in bar temperature + loss to surroundings
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2nd year laboratory script
 ∂Q

 ∂t
AK
  ∂Q
 − 
x   ∂t
G2 Ångström’s Bar

∂θ
∂ 2θ
 = AK 2 ∆x = Aρs
∆x + 2πr∆xβφ
∂t
∂x
x + ∆x 
∂ 2θ
∂θ
= Aρs
+ 2πrβφ
2
∂t
∂x
(A4)
(A5)
K ∂ 2θ 2πrβ
∂θ
φ=
−
2
AρDs
∂t
ρs ∂x
(A6)
∂ (φ + θ s ) ∂φ
∂ 2θ ∂ 2 (φ + θ s ) ∂ 2φ
=
= 2 and similarly
=
A6 can be further simplified
2
2
∂t
∂t
∂x
∂x
∂x
K
2πrβ 2 β
, and substituting γ =
=
to
by introducing the definition of diffusivity D =
ρs
AρDs Kr
give
Since
∂ 2φ
1 ∂φ
− γφ =
2
D ∂t
∂x
(A7)
This is a dispersive wave equation you will encounter similar forms of in quantum
mechanics (one dimensional time-dependent Schrödinger’s equation) and
electromagnetism.
2.
Solving the wave equation
If the bar is heated sinusoidally at position x = 0, then a sinusoidal temperature wave
is expected to travel up the rod. However this type of sinusoidal temperature wave is
difficult to produce experimentally. In the real experiment the thermo junction either
heats or cools at a constant rate, and the resulting temperature wave in the bar close
to the thermo junction is triangular in form.
Fourier’s theorem states that any cyclic wave can be expressed as the sum of
individual sine and cosine waves.
φ ( x , t ) = ∑ φ n ( x, t )
(A8)
n
Each harmonic of the input wave takes the form
φn = Bn e −α x e − i ( nωt −k x )
n
(A9)
n
But physically all harmonics other than the fundamental (n=1) are rapidly damped out
leaving only one wave of interest. This waveform can be substituted into A7 and solved to
find the equation for diffusivity
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2nd year laboratory script
D=
G2 Ångström’s Bar
ω
v
=
2αk 2α
(A7)
So to determine D, you must find α and k.
From equation A9, the decay in amplitude with increasing x is given by the first
exponential term, namely
A = Be −αx
(A8)
Therefore an appropriate straight line plot of amplitude against distance yields a
suitable measurements of the amplitude at different values of x, combined with the
graphical treatment covered by question 1 of the pre-session work allows the
attenuation coefficient to be determined.
Heat transfer is not instantaneous along the bar. As one moves along the bar the peak
of each tenmperature cycle lags that of the previous thermocouple. This phase shift θ
is included within the second part of equation A6, namely kx and is given by θ ( x) = kx .
θ is calculated from the raw amplitude-time data as the difference in time at the peak
of the wave from the reference of the heating square wave as determined by the
heater square wave ( θ peak = 2πt peak T ). A graph of θ versus x directly yields k.
3.
Table of compound Errors
f
x
2
Proportional error
∆f 2∆x
=
f
x
∆f n∆x
=
f
x
xn
∆f = nx n −1∆x
x+ y
(∆f ) 2 = (∆x) 2 + (∆y ) 2
(∆f ) 2 ( ∆x) 2 + (∆y ) 2
=
f2
( x + y) 2
xy
(∆f ) 2 = y 2 (∆x) 2 + x 2 (∆y ) 2
(∆f ) 2  ∆x   ∆y 
=   +  
f2
 x   y 
x/ y
4.
Absolute error
∆f = 2 x∆x
(∆f ) 2 =
(∆x) 2 x 2
+ 4 (∆y ) 2
y2
y
2
2
2
2
(∆f ) 2  ∆x   ∆y 
=   +  
f2
 x   y 
Instructions for using the datalogging hardware and software
Since the temperature of the copper bar is continually varying as the bar is heated
and cooled it is important that the system is switched on and allowed to dynamically
equilibrate before measurements are taken. If such time is not taken to equilibrate
then the temperature of the thermocouples will drift outside the range of the
instrument during a single measurement cycle.
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2nd year laboratory script
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Both the heater power and the as-selected thermocouple are shown as arbitrary
values on the datalogging software screen. The record of heater waverform is useful
in order to standardise the initial phase of the different thermocouples. As soon as
the datalogging program is opened it will begin recording data.
The dynamic range of the temperature measurement should be maximised by
adjusting both the gain and offset of the thermocouples input. This optimisation
should be made for each thermocouple in turn. The data can be saved by pressing
“log” option on the toolbar, after which point the data is saved once an entire screen
width has been collected. While the data is collecting, click on the menu button
beneath the toolbar and click on “log file”. Here you will need to give a name to the file
that the data will be stored in – it is recommended that you call the file after the
colour of the thermocouple you are using, prefaced with your initials – this will make
identification of your data easier.
Each time the trace reaches the end of the screen, the data will automatically be
stored in that file as a .log file type – if you take more than one full screen trace, the
files will be numbered sequentially. Once you have finished collecting the data, click
on “log” on the toolbar and select “dump data”.This process should be repeated for all
the thermocouples. When the data collection is complete, the data files can be found
on the C-Drive, in a folder called tti_vips. To view these files in Excel, use the
following process:1) Open Excel
2) Go to “Open File” and select “All Files” from the “File Type” option. This will show
you the .log files.
3) Choose “de-limited” from the options in Step 1, “Space” from the options in
Step 2 and “do not import” from the options in Step 3.
The data will now be presented in the usual Excel Workbook format, which can be
saved as a .xls file.
5.
Pre lab questions
1. The amplitude of a dispersive wave is given by A = Be −αx . In an experiment that
measures the amplitude A at different positions x, which manipulation
correctly allows the attenuation constant α to be determined graphically?
A
− x =α
a.
B
b.
log10 A = log10 B − αx
c.
Ae x
α
B
d.
ln A = ln B − αx
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2nd year laboratory script
G2 Ångström’s Bar
∂ 2φ
1 ∂φ
can be any wave of the form
− γφ =
2
D ∂t
∂x
φ = Be −αx e − i (ωt −kx ) . By substituting this solution back into the originating wave
equation and equating real and imaginary terms, determine which of the
following relationships are correct.
2. The solution to the wave equation
a.
k2 =α 2 −γ
b.
αk =
c.
2α 2 = γ ± γ 2 + ω 2 D 2
ω
2D
3. A triangular wave is comprised of a sum of waves of different frequencies nω
where n is an integer. The Fourier expansion is illustrated by a java applet at
http://www.falstad.com/fourier/. Use the applet to estimate the number of
terms required to adequately describe a triangle wave. Is it:
a.
b.
c.
d.
September 2010
One Sine term and no cosine terms
Two cosine terms and two sine terms
Three sine terms and no cosine terms
Eight sine terms and no cosine terms
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