Thesis (M.Sc. (Statistics )) -- University of Limpopo, 2021 / Temperature extremes have a crucial impact on agricultural, economic, health and energy sectors due to the occurrence of climate extreme events such as heat waves and cold waves. Limpopo province is among the hottest provinces of South Africa and experiences little rainfall which affect the water availabil ity, food production and biodiversity. In the Limpopo province, temperature extremes are expected to become more frequent as a result of climate change. The aim of this study was to model temperature extremes in the Limpopo province of South Africa using extreme value theory (EVT). The stationarity of the data was tested using augmented Dickey-Fuller (ADF), Phillips-Peron (PP) and Kwiatkowski-Phillips-Schmit-Shin (KPSS). Four candidate parent distri butions: normal, log-normal, gamma and Weibull distributions, were fitted to the average monthly maximum and minimum daily temperatures. Prior to the selection of the parent distributions, the data set at each station was subjected to normality test using the Shapiro-Wilk (SW) and Jarque-Bera (JB) tests. The stationarity and normality tests revealed that the maximum and minimum temperature data series at all the stations are neither stationary nor normally distributed. Akaike information criterion (AIC) and Bayesian information cri terion (BIC) were used to select the best fitting distribution at a particular site. The findings revealed that both maximum and minimum temperatures series at all the stations belong to the Weibull domain of attraction. The findings from the Mann-Kendall (M-K) test and time series plots trend analyses showed that there is a monotonic downward and upward long-term trend in minimum and maximum temperature data, respectively. Two fundamental approaches of EVT, block maxima and peaks-over-threshold (POT), were used in this dissertation. The generalised extreme value (GEV), generalised Pareto (GP) and Poisson point process distributions were fitted to the data set for each station. In order to account for climate change impact, non-stationary models were considered with Seasonal Oscillation Index (SOI) as covariates of the parameters of the GEV distribution. The findings revealed that both the maximum and minimum temperature data can be modelled by the Weibull family of distribution. The EVT return level analysis findings of above 400C for maximum temperature suggests impending heat waves and droughts in the Limpopo province. The bivariate conditional extremes ap proach with a time-varying threshold was used. The findings revealed both significant positive and negative extremal dependence in some pairs of meteo rological stations. The findings of this study play an important role in revealing information useful to meteorologists, climatologists, agriculturalists and plan ners in the energy sector where temperature extremes play an important role. The scientific contribution of this study was to reduce the risk and impact of temperature extremes on agricultural, energy and health sectors in the Limpopo province. An understanding of temperature extremes will help gov ernment and other stakeholders to formulate mitigation strategies that will minimise the negative impact resulting from temperature extremes in the Limpopo province. Among the major contributions of the study was the use of a pe nalised cubic smoothing spline to perform a nonlinear detrending of the tem perature data, before fitting bivariate time-varying threshold excess models based on Laplace margins, to capture the climate change effects in the data. Future studies may consider exploring the use of extreme value copulas, as well as spatio-temporal dependence between temperature extremes using the conditional extremes model of Heffernan and Tawn (2004). / DST National Research Foundation (NRF)

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There is significant interest by the food industry in applying milder processing conditions. A major area of research within predictive modelling has been the search for models which accurately predict the effect of combining multiple processes or hurdles. For a mild process, which has temperature as the major microbial injury step, the effect of the other combined hurdles in inhibiting growth of the injured organisms must be understood. The latter means that the inoculum size dependency of the time to growth must also be fully understood. This essentially links injury steps with the potential for growth. Herein, we have been developing the use of optical density (O.D) for obtaining growth rates and lag times using multiple inocula rather than using the traditional methods which use one single inoculum. All analyses were performed in the Bioscreen analyser which measures O.D. The time to detection (TTD) was defined as the time needed for each inoculum to reach an O.D=0.2 and O.D was related to microbial numbers with simple calibration curves. Several primary models were used to predict growth curves from O.D data and it was shown that the classic logistic, the Baranyi and the 3-phase linear model (3-PLM) were the most capable primary models of those examined while the modified Gompertz and modified logistic could not reproduce TTD data. Using the Malthusian approximation of the logistic model the effect of mild temperature shifts was studied. The data obtained showed that for mild temperature shifts, growth rates quickly changed to the new environment without the induction of lags. The growth of Listeria monocytogenes, Salmonella Typhimurium and Escherichia coli was studied at 30⁰C and/or 37⁰C, in different NaCl concentrations, pH and their combinations. The classical 3-parameter logistic with lag model was rearranged to provide the theoretical foundation for the observed TTD and accurate growth rates and lag times could be estimated. As the conditions became more unfavourable, the lag time increased while the growth rate decreased. Also, the growth rate was found to be independent from the inoculum size; the inoculum size affected only the TTD. The Minimum Inhibitory Concentration (MICNaCl and MICpH) was calculated using the Lambert and Pearson model (LPM) and also the Growth/No Growth (G/NG) interface was determined using combinations of NaCl and pH. These data were transformed in rate to detection (RTD) and fitted with a response surface model (RSM) which was subsequently compared with the Extended LPM (ELPM). The LPM and the ELPM could analyse results from individual and combined inhibitors, respectively. Following a mild thermal process a lag due to thermal injury was also induced, the magnitude of which was dependent on the organism and environmental conditions; the observed distribution of the lags appeared, in general, to follow the Log-normal distribution. After the lag period due to injury, growth recommenced at the rate dictated by the growth environment present. Traditional growth curves were constructed and compared with the data obtained from the Bioscreen under the same conditions. From the results obtained, it can be suggested that the increased lag times and growth rates obtained from the traditional plate counts compared with the values obtained from the Bioscreen microbiological analyser, might be an artifact of the plating method or may be due to the use of the modified Gompertz to study the growth. In conclusion, O.D can be used to accurately determine growth parameters, to give a better understanding and quantify the G/NG interface and to examine a wealth of phenomena such as fluctuating temperatures and mild thermal treatments. The comparison between the traditional growth curves against the data obtained from the Bioscreen showed that the TTD method is a rapid, more accurate and cheaper method than the traditional plate count method which in combination with the models developed herein can offer new possibilities both to the research and the food industry.

Die Temperaturverhältnisse beim Ziehen und Wärmebehandeln von Draht werden mit mathematisch-analytischen Methoden auf der Grundlage der FOURIERschen Wärmeleitungsgleichung eingehend untersucht. Insbesondere wird unter den spezifischen Wärmeübergangsbedingungen zwischen Draht und Ziehdüse sowie zwischen Draht und Ziehtrommel deren thermische Wechselwirkung analysiert. Ein Näherungsverfahren zur Berechnung der Drahttemperaturen in Zugfolgen unter Berücksichtigung des Ziehdüseneinflusses wird angegeben und mit einem Beispiel zum Nassziehen stark verzinkten Stahldrahts illustriert. Aus geschwindigkeitsabhängig gemessenen Änderungen des Drahtdurchmessers werden unter thermoelastischer Ziehringdurchmesserkorrektur Schmierfilmdicken bestimmt. Diffusionsgleichungen werden analysiert und ein Zusammenhang zur Reaktionskinetik wird hergestellt. Ein neues reaktionskinetisches Werkstoffmodell, das insbesondere auch im Falle stärker anisothermer Verhältnisse, also bei Kurzzeitwärmebehandlung anwendbar ist, wird vorgestellt.

The horizontal twin-roll casting (TRC) process is an energy saving and cost-efficient method for producing near-net-shape sheets of castable metals for light-weight production. In order to investigate the TRC process numerically, a code is generated in OpenFOAM and the commercial software STAR-CCM+ is used. Both are validated with the Stefan problem, the gallium melting test case, and a continuous casting experiment for magnesium AZ31. Different solidification models are tested that are similar to solution domain definitions and solid-fraction temperature relations. The comparison with temperature measurements of the MgF GmbH Freiberg pilot plant and the final microstructure exhibits good correlation. Sensitivity studies are carried out for thermophysical properties of AZ31 as well as pilot plant parameters. Furthermore, the rolls are incorporated into the simulation to determine the effect of a location-dependent heat-transfer coefficient. Finally, the results are compared to a second pilot plant situated at the Helmholtz-Centre Geesthacht in order to explore differences and similarities. / Das horizontales Gießwalzen ist eine energiesparende und kostengünstige Methode zur Erzeugung von Flachprodukten, die im Leichtbau verwendet werden. Um dieses Verfahren numerisch zu untersuchen wurde ein Programmcode in OpenFOAM entwickelt und die kommerzielle Software STAR-CCM+ verwendet, wobei beide mit dem Stefan Problem, dem Schmelzen von Gallium und Messdaten des Stranggusses von Magnesium AZ31 validiert wurden. Verschiedene Erstarrungsmodelle werden ebenso getestet wie Variationen des Simulationsbereiches und Feststoff-Temperatur-Verläufe. Vergleiche mit Temperaturmessdaten der Pilotanlage MgF GmbH Freiberg und der finalen Mikrostruktur zeigen gute Übereinstimmungen. Sensitivitätsanalysen werden durchgeführt, um die Einflüsse von thermophysikalischen Eigenschaften und Anlagenparametern abzuschätzen. Des Weiteren werden die Walzen in die Simulation mit einbezogen, um den Effekt eines lokal veränderlichen Wärmeübergangskoeffizienten zu beurteilen. Schließlich werden die Ergebnisse mit denen einer zweiten Pilotanlage am Helmholtz-Zentrum Geesthacht verglichen. / Le laminage de coulée continue horizontal possède une faible consommation d’énergie et est bon marché pour la production des feuilles de métaux coulables utilisés dans la construction légère. Afin d’examiner ce processus numériquement, un code est généré dans OpenFOAM et le logiciel commercial STAR-CCM+ est utilisé, tous les deux sont validés en utilisant le problème de Stefan, la fusion du gallium et la coulée continue verticale de magnésium AZ31. Plusieurs modèles de solidification sont testés, ainsi que la variation du domaine de simulation, et des rélations entre la teneur en matière solide et la température. Des comparaisons avec des résultats de mesures de la température à l’installation pilote de MgF GmbH Freiberg ainsi que la microstructure donnent des bons résultats. Des analyses de sensibilité sont effectuées afin d’évaluer l’influence des propriétés thermophysiques et des paramètres de l’installation. De plus, les cylindres sont intégrés dans la simulation pour estimer l’impact du coefficient de transfert de chaleur dépendant du lieu. Finalement, les résultats sont comparés avec ceux du Helmholtz-Centre Geesthacht.