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ENGF0004 Mathematical Modelling and Analysis II
2023/2024
Coursework 2
Standard Formula Handbook: Assessment section on MMA II Moodle site
Release Date: 3 March 2024
Submission Deadline: 4 April 2024, 2 pm UK time
Estimated Coursework Return: 4 working weeks after deadline
Topics Covered: Topics 5 – 8
Expected Time on Task: 15 hours
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 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 is worth a total of
100 marks.
LONDON’S GLOBAL UNIVERSITY
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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.
Understanding Academic integrity: complete this course here to ensure you
understand how to uphold academic integrity.
Generative AI: This is a Category 2 assignment, meaning that AI tools can be used in
an assistive role for this assignment. Please note none of the tasks require the use of
AI tools and that AI tools should not be used instead of textbooks and other reputable
resources. Find out how best to engage with AI tools here.
Do not pass materials generated by AI tools as your own work. Refer to the
library resources here on how to reference AI resources.
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.
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Coversheet
Please complete this coversheet to declare you have followed good academic practice. Please tick the
boxes that apply.
1. I have carefully read and understood Section 9 of the academic manual: Student Academic
Misconduct Procedure https://www.ucl.ac.uk/academic-manual/chapters/chapter-6-student_x005f casework-framework/section-9-student-academic-misconduct-procedure.
Yes No
2. I have made sure to correctly reference external resources incl. any teaching material.
Yes No
3. Have you used any AI tools to support your assignment
Yes No
4. If so, which one(s)
Answer: ________________________
5. I have made sure to correctly reference the use of any AI tools in the entirety of my report.
Yes No
Declaration
I declare that all the information provided is true and understand that failure to comply with section
9 of the academic manual may result in penalties as outlined.
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Learning Objectives
Skills in Mathematical Modelling and Analysis
As part of this module, throughout the core learning activities, workshops and courseworks,
we aim to equip you with the essential skills in mathematical techniques, analytical and
theoretical modelling and critical analysis of results.
The learning objectives associated with this assessment are listed in more detail below.
Detailed Learning Objectives
Model 1 Apply a technique: Determining eigenvalues and eigenvectors.
Apply a technique: Double and triple integrals.
Apply a technique: Differential vector calculus.
Apply a technique: Statistical decision tests.
Apply a technique: Least Square methods.
Show an understanding of: conservative fields and potential functions.
Show an understanding of: the implications of eigenvalues and eigenvectors.
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.
Legend Techniques Understanding Communication Computational
Modelling
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Model 1: Thermal Energy Storage [100%]
Figure 1. Diagram of a thermal energy storage battery.
Problem Description
Global commitment to action against climate change is intensifying, with the goal of reaching
net zero emissions by 2050 in order to maintain global temperature warming to 1.5 ℃ from
pre-industrial levels. This unprecedented transformation cannot be solved by a single solution
– a variety of technologies optimal for different needs are likely going to contribute to
addressing such a complex problem.
One of the answers is transitioning to renewable sources of energy – of which wind and solar
energy are some of the most common and cheapest in terms of initial investment. However, it
is well-known that solar and wind energy are not available always and there is often a
mismatch of supply and demand for the energy – it is not needed as much during the peak
production hours, while there is not enough supply during peak usage time. For this reason, a
necessary part of a transition to renewable energy is the supply of sufficient energy storage,
which would allow the excess energy to be stored until needed.
Chemical batteries are a common energy storage solution, and attractive due to their very
high efficiencies that can reach upwards of 95%. However, they are not always the best
choice, especially given their environmental impact and relatively high cost and short life-span.
This is why different energy needs should be met by different solutions.
In the UK, 17% of all carbon emissions are due to residential use, primarily heating. Globally,
this figure stands at around 15% and can go higher if industrial manufacturing heating needs
are included. One possible solution for this is to store energy directly as heat for later use,
through thermal energy storage.
Thermal energy batteries (Figure 1) work similarly to chemical batteries but instead of storing
electric energy, they store thermal energy. Initially, they are heated using the excess electrical
energy produced at peak times using resistance heaters, and the energy is stored in an
insulated vessel, full of a material that can store a lot of heat. This energy can be preserved
in this manner for months, and used when needed through heat exchangers.
In this coursework, we will consider some aspects of such thermal energy storage batteries to
run some scenarios of their operation and feasibility for potential use.
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Question 1 [15 marks]
We know that the divergence is a measure of the flux or flow. In the context of heat energy
transfer, the amount of heat flow per unit time, , through a region of space is given by
= div

,
where is the heat energy flux, measured per unit cross-sectional area per unit time. The
variables are given in more detail in Table 1.
Table 1. Model variables.
Variable Description Units
Heat flow per unit time Js
1 = W
Heat energy flux, measured as heat flow per unit
cross-sectional area per unit time
Js
1 m 2 = Wm 2
Region in space (Volume), defined by the volume of
the thermal energy battery
m3
Find for a thermal energy storage battery, in which case is defined as the volume of a
cylinder of radius 10 m and height 10 m, if it is known that
= [3
2 ,
3 + 2 ] = (3
2 ) + (
3 + 2 )
Question 2 [15 marks]
Characterise the field . Start by plotting the field in MATLAB, then determine its curl
curl
both analytically and numerically using the inbuilt MATLAB function curl. Discuss what
deductions you can make based on your findings about the field .
Remember to include your MATLAB code in your solution.
Question 3 [10 marks]
Determine if the field is conservative. Then find an expression for the potential function
such that
= ,
where is a constant, known as the thermal conductivity (measured in units of WK
1m 1
) of
the material used to store the heat in the thermal energy battery.
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Question 4 [15 marks]
One commercial decision aimed at keeping production costs low in the development of thermal
energy storage batteries is to use sand as the storage medium. This is an attractive choice
since there are additional sustainability considerations if waste sand from the mining industry
is used, for example.
However, knowledge of the thermal properties of the sand used in the battery is needed in
order to design optimal operating conditions, such as minimum and maximum temperatures,
charging duration, storage period for maximum efficiency. Hence, taking measurements of
material properties such as the thermal conductivity (WK
1m 1
) is essential.
Table 2 lists measurements of thermal conductivity, taken at 25 ℃, for two different sand types.
The sand is not homogeneous, therefore in order to have a better measurement, 4 samples
are taken for each type.
Perform appropriate statistical testing to determine if the thermal conductivity of the
two sand types is statistically different.
Note that as part of your answer you should choose an appropriate level of significance for
your test.
Table 2. Thermal conductivity data.
Thermal conductivity (

)
Sample Type 1 Type 2
1 0.253 0.237
2 0.256 0.239
3 0.256 0.239
4 0.257 0.236
Question 5 [15 marks]
Further investigation of the thermal properties of sand indicate that at sand temperatures
higher than 100 ℃, there is a relationship between the thermal conductivity, , and the
temperature, .
The data in Table 3 have been found experimentally. Using the method of Least Squares
Estimation, develop a linear model that matches these data to obtain an expression for as
a function of temperature. Solve the model in MATLAB.
Remember to take into account your decision from Question 4 to inform the model.
Table 3. Experimental data of thermal conductivity at different temperatures.
Thermal conductivity (

)
Temperature (℃) Type 1 Type 2
100 1.9 1.5
150 1.5 1.25
200 1.4 1.1
250 1.05 0.85
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Question 6 [20 marks]
At the start we discussed how thermal storage solutions are not the only possible technology
of choice. Competitive options are also hydrogen storage and chemical batteries.
Market research indicates that the probability to choose one of the three technologies or switch
to another after each year is given by the probability matrix :
From T1 From T2 From T3
= [
0.8 0.1 0
0.2 0.6 0.1
0 0.3 0.9
]
To T1
To T2
To T3
where the three technologies are
T1: Thermal storage T2: Hydrogen storage T3: Chemical batteries
Note each element in the matrix is a probability.
Given this, the adoption of each technology after, for example, two years can be found by
= for any initial state . Note represents a relative proportion of adoption of each
of the three technologies, therefore its elements should add up to 1.
a) [5 marks] Find the eigenvalues and eigenvectors of this matrix analytically (without the
use of in-built software tools).
b) [5 marks] Use your knowledge of eigenvalues, eigenvectors and the characteristic
equation
= ,
and the help of computational modelling, to determine if the funding distribution will
reach a steady-state and if so, what that steady state funding allocation is. What are
the implications for adoption of thermal storage batteries
c) [10 marks] Consider what would happen to the long-term adoption of the relative share
of the three main technologies in the event that the probabilities of some of the
elements in are changed as outlined below.
(1,1) is given by the probability density function
∫ ∫ 0.02
(

10)

(

5
)
0
20

0
20
(2,3) is given by a Poisson distribution process, which represents the probability
that 4 events occur, when the expected number of events occurring over the period
is 6.
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Summary and Reflection [10 marks]
Question 7 [10 marks]
As a way to summarise the analysis performed so far, consider the relationship between
thermal conductivity and temperature.
Reflect on how this relationship affects the rate at which heat flows out of the thermal heat
battery. Discuss how this can be incorporated into your model and what its effect would be on
the model results.
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