SA Graduate Modelling Camp Problems

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

Problem Statement

The demand for parking spaces in various urban and suburban environments often presents significant planning challenges. This problem encompasses facilities, allocation, and conflicts, which are central issues for traffic managers in a broader context. The main issues arise either from scarcity (limited parking availability compared to the high demand) or management (inefficient use of existing parking facilities). Parking allocation is increasingly recognized as a critical land use issue, particularly in densely populated areas. The rise in vehicle ownership among residents, employees, and visitors, combined with constrained parking spaces, exacerbates the difficulty of efficient parking space allocation (PSA). Often, solutions to these problems are attempted without a deep understanding of their nature and the most effective resolution methods.

The lack of parking near key activity centres becomes more acute with the increase in vehicle ownership, especially in areas with high urban density. Considering the various rules, regulations, and restrictions that govern parking in these areas, it is vital to identify optimal methods for allocating parking space, taking into account specific constraints and requirements.

While many have approached this issue from an administrative and managerial perspective, there is a growing belief that mathematical modelling and optimization techniques can offer significant insights and solutions. The challenge lies in developing a model that effectively balances land use in competitive, policy-driven environments. It is essential to investigate the major constraints and parameters that traffic managers deem critical for optimal parking. The ultimate objective is to devise practical solutions that aid in the efficient planning, design, and allocation of parking spaces and facilities in these complex environments.

Presenter: Professor Neville Fowkes, University of Western Australia, Perth, Australia

Problem statement

Platinum is a noble metal noted for its extreme resistance to tarnishing and corrosion, and it is also rare; as such it is much valued in the fashion industry. Platinum has a very high melting point (3 215°C), good electrical properties and, together with the other members of the platinum group of metals (PGM), are much valued for their catalytic properties. They are in great demand and short supply; platinum is about 30 times rarer than gold. South Africa holds more than 80% of the world's platinum reserves, with more than 70% coming from the Bushveld Complex in the Transvaal in S Africa.

The ore is mined, crushed and a rotation process is used to remove most of the gangue (useless minerals) to produce a concentrate which, after the addition of flux, is then smelted in a furnace, see Figure 1. Smelting is a density separation process carried out at temperatures high enough to melt and chemically convert the in-feed. The metallic sulphides released during smelting separate out from the remaining gangue (oxides) because of their higher density (SG>4.5) compared with that of the in-feed and SG<3.5).

In the situation of interest a combustion lance is inserted into the furnace and used to raise the temperature to the very high levels (1,600 °C) required for smelting, see Figure 1. The lance inputs air, coal and oxygen, together with the other ingredients required to start and sustain the smelting process. The air jet assists in mixing the reacting components in the slag layer. The lance is thus the main initiator and controller of the process. Metal sulphide droplets are formed in the slag layer and gravitate down to the liquid matte below, leaving behind the gangue which oat to the surface and are later discarded. The liquid matte layer largely contains liquid copper, iron and nickel sulphides, but small quantities of PMG are also present. Further chemical processing is needed to extract the PMGs. Typically the matte to slag volume ratio in the furnace is 1 to 8 with the matte depth small (typically 20 cm) compared with a slag depth of 1-3.5 m. The furnace operates continuously with in- feed added to the top of the furnace and with slag and matte tapped at from different locations along the side of the furnace. Ideally this in-feed and tapping process would be continuous and designed to ensure a steady state operation, however in practice the in-feed and tapping processes are intermittent and produce rapid changes in depth of the slag and matte layers, potentially causing an heating imbalance within the furnace; explosions can result. Also there can be variations in the `quality' of the in-feed which will effect the temperature levels in the furnace. There are other instabilities that can effect the smooth operation of the furnace.

Figure 1: The PGM Furnace: The furnace is a cylinder container of radius 2 m and height about 20 m. A combustion lance is inserted from the top of the furnace and into the concentrate. This raises the temperature above to about 1,600°C; above the melting point of the in-feed and sufficient to cause smelting . Metallic sulphide droplets are formed in the in-feed and drip down into the matte layer formed under the slag. The furnace is run continuously with in-feed added and matte and slag extracted at appropriate time intervals.

Our aim is to set up a model/s for determining the temperature variations in the furnace under steady state and transient conditions. In particular we would like to determine the effect of quality variations and intermittent in-feed on temperature levels within the furnace. It would be nice if we can determine `an optimal' operational in-feed scheme for the furnace. We may also be able to identify possible instabilities.

Background needed

Modelling. The Diffusion Equation. Fourier series. ODE and PDE integration using MATLAB or MATHEMATICA or equivalent.

Supporting material

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MISG2024 Grad Mod Camp Prob 2 First Day Presentation

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MISG2024 Grad Mod Camp Prob 2 Report Back Presentation

Presenter: Dr Mathibele Nchabeleng, University of Pretoria

Problem statement

Integral equations are equations in which some unknown function to be determined appears under one or several integral signs [1, 2]. The name integral equation was given by du Bois-Reymond in 1888. Integral equations arise in several  elds of science; for example, in queuing theory , medicine, acoustics, heat and mass transfer, economics [3]. There are many types of integral equations. The classification of integral equations depends mainly on the limits of integration and the kernel of the equation. More details about integral equations and their origins can be found in [4, 5]. 

Read the full problem description here: MISG 2024 Graduate Modelling Camp Problem 3

Presenter:  Mr Kendall Born, University of the Witwatersrand, Johannesburg.

Problem Statement

In structural engineering, the analysis of beams is crucial for understanding the behaviour of various materials under different loading conditions. We will introduce three essential types of models for beams:  the Euler-Bernoulli beam, the Timoshenko beam, and the voussoir beam.

Euler-Bernoulli beam

The Euler-Bernoulli beam theory provides a fundamental understanding of how beams deform under applied loads. Given specific material properties, geometry, and boundary conditions, the study group is tasked with solving the mathematical model that predicts the deflection or fracturing along the beam. The limitations of this theory will be explored and the study group will identify scenarios where it may not accurately represent real-world behaviour.

Timoshenko beam

The Timoshenko beam theory extends the Euler-Bernoulli model by considering the effects of shear deformation and rotational inertia. The study group will analyse situations where these additional factors become significant, leading to deviations from the Euler-Bernoulli predictions. This theory is particularly good at describing the behaviour of beams that are subject to high-frequency excitation.

Voussoir beam

The voussoir beam, also known as the arch beam, introduces curvature and non-uniform loading conditions while accounting for discontinuities in the beam. Discontinuities may be caused by many things, but of particular importance are fractures. The study group will investigate   the unique characteristics of voussoir beams, considering the geometric intricacies and material properties that influence their behaviour.

The goal of this modelling camp problem is to equip members of the study group with the knowledge of different model types for beams. The study group should understand the strengths and limitations of each beam theory and be able to follow along the modelling procedure used in problems that involve rock specimens that fracture or wind turbine blades that vibrate.

References

Supporting material

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Presenter:  Professor D P Mason, University of the Witwatersrand, Johannesburg

Problem Statement

Some mathematical models for wind turbines will be investigated. There are two types of wind turbines, the horizontal axis wind turbine (HAWT) and the vertical axis wind turbine (VAWT). In the HAWT the rotor shaft is aligned parallel to the wind and the main components, the generator and gear box, are located high above ground level. In the VAWT the rotor shaft is set transverse to the wind and the generator and gear box are at the base of the wind turbine at ground level. The VAWT does not need to be pointed into the wind.

The actuator disc model for a HAWT will be described and used to derive the Betz limit for the maximum efficiency of a HAWT. The derivation of a conserved quantity and similarity solution for the turbulent wake behind a non-rotating HAWT will be investigated. The finite boundary of the wake will be determined.

The stress and strain in a rotating blade of a HAWT will be investigated using the plane stress model of a thin two- dimensional rotating beam. The maximum normal and shear stress in the rotating beam will be calculated and compared with the tensile strength of the beam.

In both the HAWT and the VAWT the blades experience lift and drag. The role of circulation in the generation of lift in a symmetric blade modelled as a thin flat plate at a small angle of attack to the oncoming wind will be investigated.

First day Presentation

MISG 2024 Graduate Modelling Camp Problem 5 First day presentation

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MISG 2024 Modelling Camp Problem 5 - Wind turbines - Report-back Presentation

Presenter:  Dr Erick Mubai, University of the Witwatersrand, Johannesburg          

Problem Statement

Rogue waves have crest heights that are significantly larger than the average wave in the sea and pose a danger to ships travelling in the sea. Reports suggest that these waves are highly unpredictable and can appear and disappear in a short period. Folklore stories about these waves have existed for a long time, but many people treated their existence with skepticism. The first rogue wave was recorded in 1995 when it hit the Draupner natural gas platform situated 160 km from the southern tip of Norway. This caught the attention of oceanographers and a wide range of researchers.

Various models have been proposed in the literature to explain the rogue waves. Some research suggests that rogue waves might be caused by the nonlinear instability of smaller sea waves that grow to the height of rogue waves by extracting the energy of neighbouring small waves. This phenomenon can be modelled using the nonlinear Schrödinger equation.

This problem aims to investigate the initial conditions of the sea that are responsible for the creation of rogue waves. An understanding of how these waves are created might assist in the development of warning criteria for the danger of rogue waves.

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