ENGF0004探测器硅

詹姆斯·韦伯太空望远镜是一项非凡的工程壮举，只有在几乎所有与其发展相关领域的创新。耗资100亿美元的太空天文台建成捕捉宇宙中第一批星系和恒星的图像，并扩展我们对诞生的了解恒星、星系甚至宇宙的数量达到了前所未有的水平（图1）。进行这些观测的仪器之一是中红外仪器（MIRI），它能够能够检测波长高达28.5微米的光。视频1演示了光的路径在仪器内部，从进入到到达由砷掺杂制成的光学探测器硅

Figure 1. An image in the mid-infrared light spectrum of the 'Pillars of Creation", which astronomers call an incubator fornew 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 toinnovations in almost all areas related to its development. The $10 billion space observatory was builtto capture images of the first galaxies and stars in the universe, and extend our knowledge of the birthof 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 capableof detecting light at wavelengths up to 28.5 microns. Video 1 demonstrates the path the light takesinside the instrument, from entering to reaching the optical detector which ismade from Arsenic-dopedsilicon.ENGF0004Video 1. Video of the light path inside the MIRI instrument before it reaches the detector. Click
Because all colder objects (room temperature and below) glow withinfrared light due to their heat, MIRIis especially sensitive to thermal noise, or in other words, disruption due to the heat of itsdetector andsurrounding parts. For this reason, it needs to be kept exceptionally cold at temperatures below 7 K bymeans of a cooling system. This is delivered through acryocooler,which is itself remarkably innovative,relying on thermoacoustics and the cooling of gases upon adiabatic expansion(the Joule-Thompsoneffect) to obtain this level of cooling.In this coursework, you will explore different heat transfer processes inside MIRI, developing differentmathematical models to study them and discuss the implications of your findings.
To guarantee this temperature is maintained, the detector is cooled by the cooler systemdescribed in Figure 2. The MIRI detector is made of arsenic-doped silicon, and its depth is 35mm.
Question 1 [10 marks]As a starting point in analysing the temperature variation in the MIRI detector (a diagram forwhich is provided in Figure 3), a number of simplifying initial and boundary conditions can beapplied Initially, the detector’s temperature is determined by the passive cooling provided bythe 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 thermalproperties 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 mathematicallanguage.
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 tosolve PDEs analytically, the process of determining coefficient values in the general solutionis simplified. In order to make use of this, it is often convenient to perform a variabletransformation in which the dependent variable is represented by two parts: a steady-statepart , and a transient part ,= + .
The conditions described in Question 1 would lead to a steady-state solution of constant 6Kfor the temperature ( = 6K) throughout the detector.a) [10 marks] Derive the one-dimensional heat equation and initial and boundaryconditions in terms of the transient temperature , using the two-part representationof the temperature as a starting
b) [10 marks] Apply the separation of variables method to split the solution for in a timeand space-dependent component, and obtain the two resulting ODEs.c) [20 marks] Solve analytically these ODEs and obtain a solution for the transienttemperature and from there for the temperature .d) [15 marks] Implement the obtained solution for in MATLAB and use graphs to reportthe time when the entire detector has reached operating temperature levels (below theas defined by Eq. (1)). Discuss briefly the behaviour of the solution astimeprogresses.
Question 3 [25 marks]
The assumptions for the boundary conditions so far have been simplified to allow the analyticalstudy of the system’s behaviour. If we employ a numerical solution scheme, we may be ableto find solutions with more varied initial and boundary conditions such as accounting forinternal 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 itsaccuracy by comparing its solution to a solution of a simplified problem which can be solvedanalytically. 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 theaccuracy of this solution for your chosen size of the space-step and time-step, bycomparing 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 thesystem which delivers this cooling itself. The cryocooler is made up of three consecutive heatexchangers, 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 to6K. Since this is a very complex system in its entirety, you will focus your analysis on a smallpart of the whole: the heat transfer andtemperature change of the one half of the first heatexchanger.
To simplify the analysis of this process, it is possible to represent it through an equivalentelectric circuit, as shown in Figure 4. This approach is common when modelling heatprocesses and is similar to the analogy between second order spring-mass-dampermechanical systems and RLC (resistor-inductor-capacitor) electrical circuits. This method offinding simpler, well-studied equivalent models is common and very useful in engineeringpractice.
Question 1 [25 marks]
Find the Fourier series of the square wave function which describes . Use the simplifiedversion of the square wave. Show in your solution the first four non-zero terms. Support yoursolution 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 solveequation Eq.(3), where the initial condition for () are (0) = 0. Support your solution withan appropriately labelled graph, produced in MATLAB.Note that because this is a linear system, the principle of superposition applies. The responseof 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 secondterm, 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 forQuestion 2) and explore the relationship between and , ifa) 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 ofthe resulting waveforms for 0. Discuss the relationship between the values of the parametersand , 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 ofthe heat transfer inside the MIRI cryocooler and the operational conditions of the MIRIdetector.
a) [20 marks] Discuss how your findings about the time-varying conditions on one side ofthe cryocooler due to the periodically changing environmental conditions may impactthe 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 itsproperties (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 toa previous figure, equation, or solution result. Please limit the answer to this question to onepage (2 at the very maximum).
WX：codehelp