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Figure 1. An image in the mid-infrared light spectrum of the 'Pillars of Creation", which astronomers call an incubator for
new stars, captured with the MIRI optical module of the James Webb Space Telescope.

The James Webb Space Telescope is a remarkable feat of engineering, only possible due to
innovations in almost all areas related to its development. The $10 billion space observatory was built
to capture images of the first galaxies and stars in the universe, and extend our knowledge of the birth
of stars, galaxies and even the universe to unprecedented levels (Figure 1).
One of the instrument making these observations is the Mid-infrared Instrument (MIRI), which is capable
of detecting light at wavelengths up to 28.5 microns. Video 1 demonstrates the path the light takes
inside the instrument, from entering to reaching the optical detector which is made from Arsenic-doped
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Video 1. Video of the light path inside the MIRI instrument before it reaches the detector. Click on the video to be directed to
YouTube, where you can watch it in full (1 minute watch). Source: ESA

Because all colder objects (room temperature and below) glow with infrared light due to their heat, MIRI
is especially sensitive to thermal noise, or in other words, disruption due to the heat of its detector and
surrounding parts. For this reason, it needs to be kept exceptionally cold at temperatures below 7 K by
means of a cooling system. This is delivered through a cryocooler, which is itself remarkably innovative,
relying on thermoacoustics and the cooling of gases upon adiabatic expansion (the Joule-Thompson
effect) to obtain this level of cooling.
In this coursework, you will explore different heat transfer processes inside MIRI, developing different
mathematical models to study them and discuss the implications of your findings.

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Model 1: Heat conduction inside the MIRI detector [90 marks]

The MIRI detector is designed to detect infrared light up to 28.5 m in wavelength. The
relationship between its maximum operating temperature ( measured in degrees Kelvin,
K) and the maximum detectable wavelength (measured in m) is governed by the equation

. . (1)
To guarantee this temperature is maintained, the detector is cooled by the cooler system
described in Figure 2. The MIRI detector is made of arsenic-doped silicon, and its depth is 35
The one-dimensional heat equation. (2)
can be used to model the heat conduction through the depth of the MIRI detector, where
(, ) is the temperature in K and which is dependent on both space and time, is the thermal
conductivity of the material (
), is its specific heat capacity (
), and is the density of
the material (kg/m3).
The constant terms

can be combined into a single coefficient

= , called the diffusivity
Your analysis in the first model, guided through the three questions, will be aimed at predicting
at what time, following the start of maximum cooling, MIRI reaches operating temperature.
Figure 2. Cooler system of the MIRI instrument. Two shields provide passive cooling to ambient temperatures of 40K and
20K respectively. The two-stage cryocooler ensures the detector is kept at operating temperatures, by means of
refrigeration with Helium gas cooled to 18K during the first stage and to 6K during the second stage.
Stage 1:
Secondary Heat Shield,
ambient temperature: 20K
Stage 2:
Joule -Thompson
loop in cryocooler
Helium gas
detector 17K 6K
Primary Heat Shield,
ambient temperature: 40K
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Question 1 [10 marks]
As a starting point in analysing the temperature variation in the MIRI detector (a diagram for
which is provided in Figure 3), a number of simplifying initial and boundary conditions can be
Initially, the detector’s temperature is determined by the passive cooling provided by
the secondary shield,
the side which interfaces with the cooling liquid is kept at 6K,
through the detecting side of the detector no heat transfer occurs.

a) [5 marks] Research the literature to determine the coefficients describing the thermal
properties of silicon: and , its density , and the resulting diffusivity constant .
Remember to include units and your sources.
b) [5 marks] Write the initial and boundary conditions described above in mathematical

Figure 3. Diagram of the cross-section of the MIRI detector.

Question 2 [55 marks]
If boundary conditions are equal to zero when using the separation of variables method to
solve PDEs analytically, the process of determining coefficient values in the general solution
is simplified. In order to make use of this, it is often convenient to perform a variable
transformation in which the dependent variable is represented by two parts: a steady-state
part , and a transient part ,
= + .
The conditions described in Question 1 would lead to a steady-state solution of constant 6K
for the temperature ( = 6K) throughout the detector.
a) [10 marks] Derive the one-dimensional heat equation and initial and boundary
conditions in terms of the transient temperature , using the two-part representation
of the temperature as a starting point.
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b) [10 marks] Apply the separation of variables method to split the solution for in a time
and space-dependent component, and obtain the two resulting ODEs.
c) [20 marks] Solve analytically these ODEs and obtain a solution for the transient
temperature and from there for the temperature .
d) [15 marks] Implement the obtained solution for in MATLAB and use graphs to report
the time when the entire detector has reached operating temperature levels (below the
as defined by Eq. (1)). Discuss briefly the behaviour of the solution as time

Question 3 [25 marks]
The assumptions for the boundary conditions so far have been simplified to allow the analytical
study of the system’s behaviour. If we employ a numerical solution scheme, we may be able
to find solutions with more varied initial and boundary conditions such as accounting for
internal heat generation inside the detector, for example, as it is hit by photons.
As a first step when implementing a numerical solution scheme, it is important to validate its
accuracy by comparing its solution to a solution of a simplified problem which can be solved
analytically. This is your task in this question.
a) [25 marks] Set up a numerical solution scheme for the heat equation and solve Eq.(1)
for , given the initial and boundary conditions prescribed in Question 1. Validate the
accuracy of this solution for your chosen size of the space-step and time-step, by
comparing it to the analytical solution you obtained in Question 2.

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Model 2: Heat exchanger model [90 marks]
Now that you have explored the temperature variation in the detector, it is useful to study the
system which delivers this cooling itself. The cryocooler is made up of three consecutive heat
exchangers, bringing the temperature of the helium gas running through it from 300K to 17K.
A final Joule-Thompson loop can be activated to provide maximum cooling to of the Helium to
6K. Since this is a very complex system in its entirety, you will focus your analysis on a small
part of the whole: the heat transfer and temperature change of the one half of the first heat
To simplify the analysis of this process, it is possible to represent it through an equivalent
electric circuit, as shown in Figure 4. This approach is common when modelling heat
processes and is similar to the analogy between second order spring-mass-damper
mechanical systems and RLC (resistor-inductor-capacitor) electrical circuits. This method of
finding simpler, well-studied equivalent models is common and very useful in engineering

Table 1. Analogy between the thermal and electrical quantities.
Thermal Quantity Electrical Quantity
Parameter Unit Parameter Unit
Temperature, K Voltage, V
Heat flux, ? W Current, A
Thermal resistance, / K.m/W Resistance, Ω
Heat capacity, J/kg.K Capacitance, F
Time, s Time, s

When such an equivalence is used, an analogy between the variables describing the thermal
and the electrical circuits can be drawn: temperature difference is equivalent to potential
difference (in other words voltage), heat capacity is equivalent to capacitance, thermal
resistance (the inverse of thermal conductivity, 1/) is equivalent to electric resistance. More
details are provided in Table 1.

Figure 4. The equivalent electric circuit to the thermal system, modelling heat transfer through one side of a heat
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The equivalent electrical circuit in Figure 4 can be shown to be modelled by the first-order
ordinary differential equation (Eq. (3)).

+ = , . (3)
where is the resistance, is the capacitance, is the input voltage and is the output

Figure 5. Diagram of the James Webb telescope orbit around Lagrange point 2 (L2) and its position relative to the Earth and
As the James Webb space telescope orbits around Lagrange point 2 (L2, a stable orbital
position in the orbits of three bodies – the Sun, Earth and James Webb telescope, depicted in
Figure 5), small changes in the solar and internal equipment conditions occur, causing a time-
dependent variation in the initial temperature profile in the heat exchanger.

Figure 6. Plot of the square wave describing the time-dependent variation of the input voltage, , away from the intended
stable voltage , as a square wave with a period T and a maximum amplitude + . This can be simplified to a square
wave with amplitude above 0.
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Figure 6 shows the representative equivalent input voltage which describes this time-
dependent variation away from the intended stable voltage , as a square wave with a period
and a maximum amplitude + . This can be simplified to a square wave from 0 to ,
which you should use in your analysis henceforth.
Note, this is a variation above the intended stable voltage (corresponding to a temperature
of 300K) of the initial stage of the cryocooler. It will be passed on to the subsequent stages in
the cryocooler system in a similar manner until it reached the MIRI detector itself.

Question 1 [25 marks]
Find the Fourier series of the square wave function which describes . Use the simplified
version of the square wave. Show in your solution the first four non-zero terms. Support your
solution with an appropriately labelled graph, produced in MATLAB.

Question 2 [35 marks]
Given () is described by the square wave in Question 1, use Laplace transforms to solve
equation Eq.(3), where the initial condition for () are (0) = 0. Support your solution with
an appropriately labelled graph, produced in MATLAB.
Note that because this is a linear system, the principle of superposition applies. The response
of the system, the output voltage , to a sequence of inputs () is
= 1 + 2 + 3 + 4 + ? + ?,
where 1 is the system’s response to the first term of (), 2 is the response to the second
term, and similarly for each following term.
Use = 100s and = 100s.

Question 3 [30 marks]
Modify your solution accordingly (most efficiently done if you implement your solution for
Question 2) and explore the relationship between and , if
a) has a period of = 100s, but is changed to 10s and 1s.
b) remains 100s, but the period of , is changed to 10s, 1,000s and 10,000s.
Produce appropriately labelled graphs for the new solutions and comment on the form of
the resulting waveforms for 0. Discuss the relationship between the values of the parameters
and , and the changes in the form of the system response, , in comparison to its input,
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Summary and Reflection [20 marks]
Now that you have performed these two pieces of analysis, you have some understanding of
the heat transfer inside the MIRI cryocooler and the operational conditions of the MIRI
a) [20 marks] Discuss how your findings about the time-varying conditions on one side of
the cryocooler due to the periodically changing environmental conditions may impact
the operation of MIRI, which requires it is maintained at a very stable temperature
(variation smaller than 0.02K over 1000s). How will you design the cryocooler and its
properties (electrical equivalent properties) to ensure MIRI is operational if = 1K?
Limit your discussion to 100-200 words. To support your answer, you may include or refer to
a previous figure, equation, or solution result. Please limit the answer to this question to one
page (2 at the very maximum).

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