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Study Group Problem 1: Forces on a diffuser lifting screw
Industry:
Sugar Cane Processing
Industrial Representative:
Richard Loubser, Sugar Milling Research Institute NPC,
c/o University of KwaZulu-Natal, Durban.
Moderator:
Student Moderators:
Problem Statement:
In South Africa, the diffuser is a popular method for extracting sugar from sugar cane. It is a counter current washing machine with shredded cane added at one end and imbibition added at the other end of a chain (ladder) conveyor system. The water percolates through the cane bed and is pumped forward and reintroduced to the cane bed closer to the end where the cane is added. This is how the counter current effect is achieved.
The system relies on the free percolation of the water or juice through the cane bed. It was found during the development of the cane diffuser, however, that fine particles tend to form a layer in the bed which retards the free flow of the juice through the bed.
To ensure that the flow through the bed is restored, lifting screws are used. These are arranged across the diffuser bed as shown in Figure 1.
Figure 1. Lifting screws in a diffuser
Most lifting screws rotate at 40 revs per minute and are of a similar design regardless of make of diffuser.
As the cane passes the screws, they churn up the cane to disrupt any layers of fine particles that may have formed on the cane bed. Adjacent screws turn in opposite directions to enhance the effect of the screws.
The mounting of the lifting screw is designed with a shear pin that fails to protect the screw itself from failure. Given the design of the screw, it is possible to determine the strength of the pin.
Many factories are operating with lifting screws that are not working because the shear pin has failed. There is very little in the literature to guide the design of the lifting screw to handle the forces experienced in churning up the bed.
It is tempting to replace the pin with stronger components. This, however, has disastrous consequences. It would be far more acceptable if the lifting screws could be designed according to the maximum forces that would be experienced.
It would be useful if a method for estimating the load on the lifting screw could be determined. A typical cane preparation is shown in Figure 2. It consists of long strands with lengths in the order of 7 cm and thickness of a few millimetres. In between the long strands are shorter, thinner strands and then there is the fines and pith component. To churn this up, it is required that the bed is torn apart as it passes the lifting screws.
Figure 2. Prepared cane
Anecdotal evidence indicates that the effort required by the lifting screws reduces with increasing the amount of juice held up in the bed. Does the lubricating and buoyancy effect outweigh the mass effect? Is there a balance point?
The desired output from this study would be a way of estimating the worst-case design load for the lifting screw. The model would need to estimate how factors such as strand length, packing density and juice hold-up would interact and influence the design load.
Supporting Material
First-day Presentation
Report-back Presentation
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Study Group Problem 2: Deep-sea mining
A dream to come true, or irreversible damage about to hit Coastal-Marine Tourism & Biodiversity and threaten Climate Change Resilience
Industry: Tourism Sector (Coastal & Marine Tourism), Robotics
Industry Representative: Dr Lombuso Precious Shabalala,
Department of Applied Management, UNISA
Student Moderators:
Problem Statement:
The sea or ocean hosts a unique, largely unexplored biodiversity that is incredibly fragile. The impacts of deep-sea mining could be irreversible, resulting in habitat destruction, species extinction, climate disruption, and disruption of oceanic food chains, as well as an impact on tourism activities. Fast-advancing technologies have now granted access to even the deepest parts of the ocean, fuelling interest in commercial exploration of deep-sea resources, such as metals and rare earth elements. Deep-sea mining (DSM) activities are currently planned in the economic zones of several national jurisdictions, particularly Pacific Island States, and in areas beyond national jurisdiction (Kaikkonen & van Putten, 2021). Ma et al. (2022) define Deep-sea mining as the utilisation of hydrodynamic or mechanical methods to transport mineral ores from the seabed to the ocean surface and then transport the ores to land-based processing plants by ship. The scholars further note that, despite the vast amount of research and development that has made significant progress, there are still many obstacles to its industrial development activities, which include concerns about environmental pollution and sustainable development issues, regardless of the employment of technological advancements such as ore exploration, robotics, and hydrodynamic lifting (See Figure 1).
Figure 1: Schematic diagram of the mining system. Source: Yao et al., 2025.
Additionally, these activities have garnered attention from governments, companies, and scientific research institutions. Figure 2 presents the multifaceted impacts of plume generation. A deep-sea plume is a cloud of fine sediment suspended in the water, primarily caused by human activities like deep-sea mining or natural events like storms. These plumes can spread over large areas, reduce water clarity, and potentially smother or poison marine life by burying them or releasing toxic compounds.
Figure 2: Schematic of plume impacts. Source: Yao et al., 2025.
Coastal and Marine tourism are affected by deep-sea mining. For instance, a study conducted in Fiji found that pristine coral reefs possess a tremendous potential for contributing to tourism and economic development (Folkersen et al., 2018). In addition, the study revealed that its tourism economy relies heavily on diving and coastal activities. Unfortunately, despite the importance of coral reefs to the Fijian tourism sector, the Fijian Government has granted exploration licenses to mining companies to assess the viability of deep-sea mining (DSM). Understanding divers’ perceptions of coral reefs and environmental issues is, therefore, paramount to sustaining the tourism sector. There is concern that DSM may negatively impact reef-related tourism due to tourists’ perception that DSM activities degrade Fiji's coral reefs. A study by Kaikkonen and van Putten (2021) confirms that economic development and human activities in the ocean are accelerating, with maritime activities at the forefront of numerous government, research, and industry initiatives aimed at expanding the ‘Blue Economy’. The deep sea is being promoted as the new frontier for resource extraction (Kaikkonen & van Putten, 2021).Tourism is one of the top sectors that contributes to Gross Domestic Product and creates jobs in South Africa and globally. A call to halt deep-sea mining in Africa (Greenpeace, 2025) before it begins has already garnered over 4,822,507 signatures (on petitions) opposing deep-sea mining. These individuals argue that the deep sea is a treasure trove of biodiversity and home to countless wonders and possibilities. It is also one of our best allies against climate change. Kaikkonen and van Putten (2021) concur regarding the deep sea as the largest ecosystem on the planet.
The problem to investigate
It is understood that oceans produce approximately 50% of the oxygen we breathe, absorb around 25% of global CO₂ emissions, and capture approximately 90% of the excess heat generated by those emissions (Greenpeace Africa, 2025). This suggests that oceans are our first line of defence against climate change. Destabilising such could lead to a disaster against humanity.
- Currently, there is no guarantee of a sustainable mining system that optimises the conservation of life forms (biodiversity) and tourism activities, including climate change resilience.
- Secondly, there is no assurance that benefits yielded from the blue economy will reach local communities and contribute to genuine sustainable development.
Building on the previously presented Figures 1 and 2, this problem seeks to develop:
- Mathematical models for the creation and spread of plumes in deep-sea mining.
- A mathematical model that supports a sustainable deep-sea mining system or method, which, upon its implementation, could optimise the conservation of life forms and tourism activities and contribute to climate change resilience.
References
Folkersen, M.V., Fleming, C.M. and Hasan, S., 2018. Deep-sea mining's future effects on Fiji's tourism industry: A contingent behaviour study. Marine Policy, 96, pp.81-89.
Greenpeace, 2025. Stop deep sea mining before it starts. Available from: https://www.greenpeace.org/international/act/stop-deep-sea-mining/ https://www.greenpeace.org/africa/en/blog/58872/deep-sea-mining-africa-cannot-stay-still/ . Access Date:10 Nov 2025.
Kaikkonen, L. and van Putten, I., 2021. We may not know much about the deep sea, but do we care about mining it?. People and Nature, 3(4), pp.843-860.
Ma, W., Zhang, K., Du, Y., Liu, X. and Shen, Y., 2022. Status of Sustainable Development of Deep-Sea Mining Activities. Journal of Marine Science and Engineering, 10(10), p.1508.
Yao, W., Tian, C., Teng, Y., Diao, F., Du, X., Gu, P. and Zhou, W., 2025. Development of deep-sea mining and its environmental impacts: a review. Frontiers in Marine Science, 12, p.1598584.
Additional Material
Greenpeace Africa. URGENT petition telling African Govts to deliver a firm NO’ to deep sea mining before it starts. https://www.facebook.com/share/p/1FZTrFPksA/. Access Date: 11 Nov 2025.
PHYS.ORG. Deep-sea mining risks disrupting the marine food web, study warns. Available from:https://phys.org/news/2025-11-deep-sea-disrupting-marine-food.html. Access Date: 11 Nov 2025.
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Study Group Problem 3: Mathematical models for Tornadoes
Industry: Meteorology
Industry Representative: Puseletso Mofokeng, South African Weather Service
Moderator:
Student Moderator(s):
Problem Statement
The sudden development of tornadoes with their localised but severe devastating destruction of property and environment remains a mayor challenge for meteorologists, disaster management practitioners and the public, to name a few. Collaterally, injuries with potential or real loss of human and animal lives form part of the challenge. In South Africa, the problem further extends to inconsistent reports of tornadic events which are sometimes unreliable. Moreover, centuries of lack of traceable raw data of tornadic events, followed by their scarcity decades after the advent of satellite and RADAR technology, has slowed down research and effective operational decision-making.
Regardless of this compounded challenge, the forecasting desk of the national weather authority (i.e. the South African Weather Service) only once issued a tornado warning at very short lead-time to the public on the afternoon of 30 December 2017. Then, about 5-minute updates of RADAR images were used and a meagre lead-time of less than 5-minutes was achieved. This time difference was worked out by comparing the time of issue of the tornado warning (i.e. 16:42 SAST) and time from the call log (16:46 SAST) of the residents of Protea Glen (Soweto), in the City of Johannesburg. The resident made a call at the instance of tornado touchdown.
Thereafter, a study of 34 tornadoes which occurred across South Africa was conducted between June 2016 to June 2021. This bulk study used images of meteorological satellite updating on 15-minute intervals. It was then deduced that slightly over 75% of tornadoes developed during the cross-intersection of a thunderstorm with fast velocities of lifted horizontal winds of another thunderstorm. The formation mechanisms of these tornadoes differed with the majority of those which occur across the south-eastern region (i.e. east of Rocky Mountains) of the United States of America (USA). Such USA tornadoes typically occur in association within the occluded fronts.
Approximations of the increase (decrease) in the rate of change of the vorticity advection with the Equation 1 below:
However, occluded fronts are every rare in South Africa. Scrapping through historical records shows that their occurrences are decadal; they occur under exceptionally deep cut-off low pressure systems. Nonetheless, the deduction was that pseudo-occluded fronts occasionally occur and seem to be the cause of majority of tornadoes in South Africa. Pseudo-occluded fronts appear to be favourable for tornado development in South Africa when one thunderstorm deviated from the mean wind flow (i.e., steering winds) onto the strong outflow (i.e., gust front) of adjacent or approaching storm. The storm deviation could be the result of blocking significant topography (i.e., notably uneven features of land surface) or the influence of strong gust fronts change the steering winds of adjacent newly developed cells towards the outflow of the approaching storms. Such resulting quantum theory is illustrated in Figure 1 with a schematic model of two thunderstorms. One of these idealised thunderstorms deviate (due to significant topography) to temporally fixed point and onto the strong gust in front of an approaching storm.
Figure 1 – Schematic Tornado Model: Tornadogenesis resulting from the cross-intersection of one thunderstorm with the gust front of another storm (circled) in proximity (e.g. within 1 km) . Such situations typically result when one storm (S2) deviate onto strong gust front of the approaching storm (S1).
Upon contact with lifted moist fluids (i.e., thunderstorm), fast velocities of horizontal dry fluids (i.e., winds) are assumed to undergo a degree of detour.
Figure 2: Thunderstorm activities from meteorological satellite along with the associated critical vectors.
Their sustenance into moist fluids typically experience counter-detour from pre-existing flows, resulting in downward spiral (oscillatory motion) of the lower sector of the moist fluids (i.e. drop of the portion of cloud base) (see Figure 2).Additionally, this portion of the cloud base do so due to the accumulation of cold temperatures from their advection by the winds (vectors). In optimal conditions the downward spiral of moist fluid typically develops into a funnel cloud or a tornado. The rapid decrease of radii towards the land surface is accepted to exponentially increase the speed of rotation within a tornado (see Figure 3).
Figure 3: A strong tornado and its enhanced-Fujita (EF) rating.
Theoretical results of further study suggest geographic locations of at least two thunderstorms could be used in the idealised straight lines together with approximation procedure to determine the potential of passing through a temporally fixed point (i.e. common geographic location). Such temporally fixed point seems important to find the subregion of less than 50 km across which a tornadic development is possible. This background and partitioning of constituent factors suggest the differential vector calculus as the best approach to develop mathematical tornado models.
The Mathematics in Industry Study Group is requested to consider the scenario below and answer the questions below:
A short-lived tornado hit oThongathi in KwaZulu-Natal on the late afternoon of 3rd June 2024. The phenomenon left considerable damage to property and reports have it that 16 residents lost their lives. Further, local businesses were halted and by extension local economy was negatively impacted. Images of meteorological satellite and RADAR show moments before, during and after the occurrence of a tornado.
Figure 4: Focus on Storm B and C. The 15-minute updates of top-view meteorological satellite images (© EUMETSAT). These storms were approximated with bottom-view RADAR images, as supplied by the South African Weather Service.
Figure 5: Evidence of impact across subregions of oThongathi, in KwaZulu-Natal.
Retrospectively, and given the two positions for each of these thunderstorms at 16:00 SAST and 16:15 SAST, coordinates were collected as follows,
Storm_B = {(-29.286, 30.761), (-29.450, 30.849)}
Storm_C = {(-29.264, 31.061), (-29.394, 30.947)}
Assuming straight propagation of these storms, attempt to address the following research questions:
- Where could the theoretical temporally fixed point (i.e., common geographic position) be for each storm to pass through?
- Which of these two thunderstorms (Storm_B or Storm_C) will reach the idealized temporally fixed point, first?
- Upon cross-intersection with fast velocities (e.g., constant of 20 m/s) of the supporting storm, the horizontal speed of the main storm will be retarded. As influenced by the strong outflow of the supporting storm, detect the speed of the downward spiral (i.e., rotation of cloud) around the z-component with decreasing radii at lower height?
- Could we use these constituent factors to develop a functional mathematical model for tornado genesis?
An alternative post-analysis study could be conducted with the questions above. A tornado that hit open field on the afternoon of 6 November 2025 could be used. For the time intervals 16:15 and 16:30, coordinates of storms labelled S1 and S2 are respectively given as:
S1 = {(-28.460, 29.112), (-28.422, 29.145)}
S2 = {(-28.359, 27.953), (-28.298, 28.203)}
Key words: Moist fluids, thunderstorms, dry fluids, gust front, vorticity, fluid spin
References
- Boyer, C. H., and Dahl, J. M. (2020). The mechanisms responsible for large near-surface vertical vorticity within simulated supercells and quasi-linear storms. Monthly Weather Review, 4281–4297. doi:10.1175/mwr-d-20-0082.1
- Elsom, D. M. (1993). The thunderstorm gust front as a trigger for tornado formation: the long Stratton tornado of 14 December 1989. Journal of Meteorology, 18(175), 3-12.
- Jiang, Q., Dawson II, D. T., Li, F. & Chavas, D. R. (2025). Classifying synoptic patterns driving tornadic storms and associated spatial trends in the United States. Climate and Atmospheric Science, 8(7), 1-10. doi:10.1038/s41612-025-00897-1
- Lekoloane, L. E., Bopape, M.-J. M., Rambuwani, T. G., Ndarana, T., Landman, S., Mofokeng, P., Gijben, M., and Mohale, N. (2021). A dynamic and thermodynamic analysis of the 11 December 2017 tornadic supercell in the Highveld of South Africa. Weather and Climate Dynamics, 373-393. doi:10.5194/wcd-2-373-2021
- Mofokeng, S. P. (2024). Study of the influence gust fronts and topographical features in the development of severe thunderstorms across South Africa. Dissertation, University of the Witwatersrand.
- Sibolla, B. H., Van Zyl, T., and Coetzee, S. (2020). Determining real-time patterns of lightning strikes from sensor observations. Journal of Geovisiualization and Spatial Analyis, 5(4), 1-18. doi:10.1007/s41651-020-00070-7 2021
- Stewart, J. (1998). Multivariable calculus: concepts and contexts. Carlifonia: Brooks/Cole Publishing.
- Trafalis, T. B., Adrianto, I., Richman, M. B., and Lakshmivarahan, S. (2013). Machine-learning classifiers for imbalanced tornado data. Computational Management Science. doi:10.1007/s10287-013-0174-6
Supporting Materials
First-day Presentation
Report-Back Presentation
