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ENGF0004 Mathematical Modelling and Analysis II

2023/2024

Coursework 2

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

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.

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 V is given by

where q 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.

Find  for a thermal energy storage battery, in which case V is defined as the volume of a cylinder of radius 10 m and height 10 m, if it is known that

Question 2 [15 marks]

Characterise the field q. Start by plotting the field in MATLAB, then determine its curl

curl q

both analytically and numerically using the inbuilt MATLAB function curl. Discuss what deductions you can make based on your findings about the field q.

Remember to include your MATLAB code in your solution.

Question 3 [10 marks]

Determine if the field q is conservative. Then find an expression for the potential function T such that

where k 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.

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 k (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.

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, k, and the temperature, T.

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 k 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.

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 A:

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 y2 = y0 for any initial state y0. Note y0 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.

• (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.ENGF0004

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.