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ENGF0004 Mathematical Modelling and Analysis II

2022/2023

Coursework 1

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 it here)

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 work here and how to avoid plagiarism here. 

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· Do not publish ENGF0004 assessment material: We expect students not to share ENGF0004 assessment materials on external online forums, including tutoring or "homework" help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

	Learning Objective
Model 1	Show an understanding of boundary conditions from description in text and/or in diagram.
	Show an understanding of a physical problem and an ability to interpret it mathematically.
	Apply a technique: Separation of variables method to reduce PDEs to ODEs.
	Apply a technique: General solution of ODEs through the auxiliary equation.
	Apply a technique: Apply initial and boundary conditions appropriately to find a particular solution to the ODEs and PDE.
	Model computationally: Implement computationally solutions for system behaviour.
	Communicate technical information: Present clearly and concisely theoretical or computational methods used to study a system. Use clear and labelled graphs to assist in this communication by displaying solutions or other key points.
	Show an understanding of the implications of mathematical results about the behaviour of the system.
Model 2	Apply a technique: Fourier series formula to represent a discontinuous function as an infinite sum of sinusoid terms.
	Apply a technique: Laplace transforms and inverse Laplace transforms to solve ODEs.
	Show an understanding of the linear nature of the Laplace transform and apply the principle of linear superposition.
	Model computationally: Implement an efficient computational model of a system.
	Model computationally: Explore the behaviour of a system computationally, by observing the effect of different parameter values.
	Communicate technical information: Present clearly and concisely theoretical or computational methods used to study a system. Use clear and labelled graphs to assist in this communication by displaying solutions or other key points.
	Show an understanding of the implications of mathematical results about the behaviour of the system.

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.

 

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]

 

 

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

 

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  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 .

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.

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:  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  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.

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.

  

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.

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,
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, ) 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 exchanger.

The equivalent electrical circuit in Figure 4 can be shown to be modelled by the first-order ordinary differential equation (Eq. (3)).

 

where  is the resistance,  is the capacitance,  is the input voltage and  is the output voltage.

 

Figure 5. Diagram of the James Webb telescope orbit around Lagrange point 2 (L2) and its position relative to the Earth and Sun.

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.

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 ) 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 . 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 

where  is the system’s response to the first term of ,  is the response to the second term, and similarly for each following term.

Use  and .

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 , but  is changed to  and .

b)  remains , but the period of ,  is changed to ,  and .

Produce appropriately labelled graphs for the new solutions and comment on the form of the resulting waveforms for . 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, .

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 detector.

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 ?

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).