# 辅导ENGF0004、MATLAB编程设计辅导

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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

silicon.

ENGF0004

P a g e 5 | 12

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.

ENGF0004

P a g e 6 | 12

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)

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

ENGF0004

P a g e 7 | 12

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.

ENGF0004

P a g e 8 | 12

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.

ENGF0004

P a g e 9 | 12

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, 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

exchanger.

ENGF0004

P a g e 10 | 12

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

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.

ENGF0004

P a g e 11 | 12

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,

.

ENGF0004

P a g e 12 | 12

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 = 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).

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.

ENGF0004

P a g e 5 | 12

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.

ENGF0004

P a g e 6 | 12

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)

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

ENGF0004

P a g e 7 | 12

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.

ENGF0004

P a g e 8 | 12

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.

ENGF0004

P a g e 9 | 12

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, 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

exchanger.

ENGF0004

P a g e 10 | 12

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

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.

ENGF0004

P a g e 11 | 12

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,

.

ENGF0004

P a g e 12 | 12

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 = 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).