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ENGF0004 detector
Creation date：2024-05-18 16:55:24
Belonging category：数学mathematics
detector

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Introduction


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
silicon.
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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
=
200

. . (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
mm.
The one-dimensional heat equation

=

2
2
. (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 (
W
m.K
), is its specific heat capacity (
J
kg.K
), and is the density of
the material (kg/m3).
The constant terms

can be combined into a single coefficient

= , called the diffusivity
constant.
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:
Thermoacoustic
cryocooler
Secondary Heat Shield,
ambient temperature: 20K
Stage 2:
Joule -Thompson
loop in cryocooler
Helium gas
MIRI
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
applied
• 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
language.

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
progresses.

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
exchanger.
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
practice.
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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