SA Graduate Modelling Camp Problems

Presenter

Professor Neville Fowkes, University of Western Australia, Perth, Australia.

Problem statement

Even specially fitted well designed masks such as N95 leak, so that infected wearers can spread covid. Such “good” masks are uncomfortable to wear and may even cause breathing problems. On the other hand cloth masks are comfortable to wear but are not so good in terms of preventing covid spread. Can one quantify mask efficiency and comfort, and design a best mask?

At MISG 2021 a study group produced a simple flow model for determining the through flow and leakage flow from a mask due to a sneeze or cough, and identified relevant design parameters. The aim at this Graduate Modelling Camp is to extend the model to quantify the response to continual coughing, and also to determine what proportion of virus carrying particles escape from the mask and how that depends on design. Optimal design issues may also be addressed.

Public transport and covid

If time permits we may also address the public safety question:

Under what circumstances should wearing a mask in a bus be mandatory?

Supporting Material

On respiratory droplets and face masks

Air permeability of woven fabrics

Mask Design for ÍøÒ×ÌåÓý

An overview of filtration efficiency through the masks

Presentation

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Presenter

Professor David P Mason, University of the Witwatersrand, Johannesburg

Problem statement

One of the most important flow regimes in gas liquid two-phase flow is annular flow in a tube. The gas phase moves in the core of the tube surrounded by a thin liquid film on the wall of the tube. In a typical flow regime the film surface is composed of small ripples and larger waves. There exist states where the interface can take the form of a train of what are known as disturbance waves. These are large amplitude waves having a regular height, much greater than the minimum film thickness, and large regular separations which are two or three orders of magnitude greater than their height.

The Study Group will work through the paper by Hall Taylor et al (1) which attempts to provide a basic understanding of disturbance waves by modelling them mathematically as moving packets of inviscid liquid. The effect of inertia dominates over the effect of gravity which can be neglected and the Reynolds numbers of the gas and liquid phases based on the tube radius and the film thickness are so high that the effects of viscosity can be neglected. The effects of gas compressibility are also ignored.  

Supporting material

1. Hall Taylor N S, Hewitt I J, Ockendon J R and Witelski T P. A new model for disturbance waves, International J of Multiphase Flow, 66 (2014), 38-45.
2. A proposal concerning laminar wakes behind bluff bodies at large Reynolds number
3. A new model for disturbance waves

First day Presentation

Graduate Modelling Camp Problem 2 Disturbance waves Presentation

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Presenter

Dr Matthews M. Sejeso, University of the Witwatersrand, Johannesburg

Problem statement

A brewing company in North America has established itself in the market with the concept of beer blending. This concept consists of mixing different types of ready-to-drink beers in order to create uniquely new flavours of beer. One famous example of beer blending is the “Black and Tan”, which is obtained by layering a pale ale and dark beer. In order to achieve a perfect blending, the company considers a large set of beers as raw materials. Each raw material has multiple attributes such as brightness, colour, thickness, coarseness, and flavour. Each of these attributes is graded on a numeric scale. Thus, a raw material is “finger printed” for different attributes, that is, raw material is given a numeric score. On the other hand, each final blend of beer is also “figure printed” against the same attributes.

The objective is then to find a combination of raw materials to produce the blends to match their attributes score requirements, within the availability of the given raw materials and based on the weekly demand. In the event that the attributes score requirements cannot be matched, it is important to find the combination of raw materials that achieves the closest match to the attributes score requirements. Usually, some of the attributes of a blend may be more critical in achieving the blend’s quality than others. Therefore, in order to achieve the closest match of a blend, the decision maker would prefer the scores of the critical attributes not to be violated even if that implies some violations to the scores of the less critical attributes.

The aim of this project is to develop a decision support tool that is able to recommend lowest cost option for recipes for beer blends based on the price, the availability and the attributes of raw materials. The solution should ensure the attributes score requirements are matched whenever possible. When it is not possible to match these scores, the closest match solution should be found.

Some data are attached for testing the solution. The “Raw_Materials.xls” file is a set of raw materials with numeric scores of the attributes, their availability as well as their costs. The file “Some_Testing_Scenarios.xlsx” presents three test scenarios which can be used to assess the decision support tool on a small scale. Finally the file “Blends.xls” is a set of beer blends with the score requirement of each of their attributes as well as the amount of demand that needs to be achieved using the raw materials in the file “Raw_Materials.xls”. This will be used to assess the efficiency of the decision support tool.

Supporting Material

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Presenter:

Erick Mubai, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg

Problem statement

Sound waves are used to detect structures such as salt domes and seeps at the seafloor. These structures might indicate the presence of oil and gas since they can trap and prevent oil and gas from escaping.

The layers of the seafloor are examined using seismic reflection and refraction. A sound pulse is sent from a ship using an airgun array and the reflected/refracted sound from the seafloor is detected by an array of hydrophones. The detected sound can be used to create three-dimensional image of the seafloor and its layer.

Airguns rapidly release compressed air, forming a bubble. This bubble formation produces a loud sound which can penetrate the seafloor as much as 20-30 km below.

The time it takes the sound to return to the surface of the sea can be used to determine the depth of the seafloor and the thickness of the layers in the seafloor.

The sound waves are refracted as the pass-through different layers of the seafloor. The refracted sound can be used to determine density of the layers.

Possible questions:

1. How can the refracted sound be used to calculate the densities of the seafloor layers?
2. How can the refracted sound be separated from the reflected sound?

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Presenter:

Dr Adewunmi Gideon Fareo, University of the Witwatersrand, Johannesburg

Problem statement

Close to 50% of the world’s population is living in urban areas or cities. According to a united nations report, it is predicted that by the year 2050, 66% of the global population will live in urban areas, resulting in higher levels of densification. The continuous increase in urban inhabitants has led to the spread of urban areas into nearby farmlands causing farm reclamations and deforestation activities. When vegetative spaces, like farmlands are turned into urban structures, there is often an associated rise in temperatures of the environment, a phenomenon first noticed in London in the 19^th century, and referred to as the Urban Heat Island effect (UHI).

The Urban Heat Island is a phenomenon where the temperature of urban areas and the inner cities is higher than the surrounding less dense outskirts.

It has been recognized that the temperature increase is primarily due to the absorption of radiation during daytime by dense materials like concrete which we use for building houses; dark materials like bitumen which we use for road construction and the roofs of buildings that make up the cities and urban areas. This heat from solar radiation is stored until the evening when it is released into the atmosphere by convection. The combination of these materials causes the inner city areas and urban areas to be several degrees warmer than the surrounding less dense areas.

The surrounding less dense areas or nearby countryside has a lot of vegetation and less of concrete, bitumen, etc. Therefore there are much less dense materials to absorb heat.

There is a body of knowledge that suggests that the installation of vegetation on roofs of buildings (otherwise known as green roofs) can mitigate the urban heat island through shading, insulation of the soil layer and evapotranspiration.

In this study, we will investigate whether green roofs can indeed help reduce Urban Heat Island. Mathematical modelling techniques will be used to estimate the energy stored in a green roof and in concrete. We will work through the paper by Aguareles et.al (1) which provides a good theoretical introduction as well as sound mathematical modelling of green roofs. Susca et.al (2) and Fitchett et.al (3) provide great introductions to the subject of green roofs.

References

1. Maria Aguareles, Marc Calvo-Schwarzwalder, Francesc Font, Timothy G. Myers,A mathematical model for the energy stored in green roofs, Applied Mathematical Modelling, 115 (2023), 513-540.
2. Susca, T., Gaffin, S.R., Dell'Osso, G.R., Positive effects of vegetation: Urban heat island and green roofs. Environmental pollution, 159 (2011), 2119-2126.
3. Fitchett, A., Govender, P., Vallabh, P., An exploration of green roofs for indoor andexterior temperature regulation in the South African interior. Environment Development and Sustainability 22, 5025–5044 (2020).

Supporting material

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