Guidelines:

Failure to follow this guidance might result in a penalty of up to 10% on your marks.

I.      Submit a single PDF document with questions in ascending order. This can be produced for example in Word, LaTeX or MATLAB Live Script. Explain in detail your reasoning for every mathematical step taken. Include units for final answers where possible.

II. Do not write down your name, or student number, or any information that might help identify you in any part of the coursework. Do not write your name or student number in the title of your coursework  document  file. Do not copy  and  paste  the  coursework  questions  into  your submission – simply rewrite information where necessary for the sake of your argument.

III.       Insert relevant graphs or figures, and describe any figures or tables in your document. All figures must be labelled, with their axes showing relevant parameters and units.

IV.      You will need MATLAB coding to solve some questions. Include all code as pasted text (for the purposes of plagiarism checks) in an Appendix at the end of your document. Remember to comment on your code, explaining your steps .

This coursework counts towards 20% of your final ENGF0004 grades and comprises two questions, referred to as models. Each model is worth a total of 100 marks, both making up 50% of your CW 1.

On Academic Integrity (Read more about ithere)

Academic integrity means being transparent about our work.

• Research: You are encouraged to research books and the internet. You can also include and paraphrase any solution steps accessible in the literature and online content if you reference them.

• Acknowledge others: We are happy when you acknowledge someone else's work. You are encouraged to point out if you found inspiration or part of your answers in a book, article or teaching resource. Read more about how to reference someone else's workhereand how to avoid plagiarismhere.

• Academic misconduct:

• Do not share and do not copy: We expect students not to share and not to copy assessment solutions or MATLAB code from their peers, even if partially.

• Do not publish ENGF0004 assessment material: We expect students not to share ENGF0004 assessment materials on external online forums, including tutoring or "homework" help websites.

Students found in misconduct can receive a 0 mark in that assessment component and have a record of misconduct in their UCL student register. In some extreme cases, academic misconduct will result in the termination of your student status at UCL.

Learning Objectives

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 ofthe 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.

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

[https://www.esa.int/ESA_Multimedia/Videos/2021/09/Webb_MIRI_imaging-mode_animation/(lang)/en]

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.

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

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 17K during the first stage and to 6K during the second stage.

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

Tmax = 200 Eq. (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

=

can be used to model the heat conduction through the depth of the MIRI detector, where T(x, t) is the temperature in K and which is dependent on both space and time, k is the thermal

conductivity of the material (), c is its specific heat capacity (), and p is the density of

the material (kg/m3).

The constant terms can be combined into a single coefficient = a, 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.

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 gas 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: k and c, its density p, and the resulting diffusivity constant a . 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 T is represented by two parts: a steady-state part v, and a transient part U,

T = v + U.

The conditions described in Question 1 would lead to a steady-state solution v of constant 6K for the temperature (v = 6K) throughout the detector.

a)  [10  marks]  Derive  the  one-dimensional  heat  equation  and  initial  and  boundary conditions in terms of the transient temperature U, using the two-part representation of the temperature T as a starting point.

b)  [10 marks] Apply the separation of variables method to split the solution for U 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 U and from there for the temperature T.

d)  [15 marks] Implement the obtained solution for T in MATLAB and use graphs to report the time when the entire detector has reached operating temperature levels (below the Tmax  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 T, 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.

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.