Da Silva, C., Peces, M., Faundez, M., Hansen, H., Campos, J. L., Dosta, J., et al. (2022). Gamma distribution function to understand anaerobic digestion kinetics: Kinetic constants are not constant. Chemosphere, 306, 135579.
Abstract: The Gamma model is a novel approach to characterise the complex degradation dynamics taking place during anaerobic digestion. This three parameters model results from combining the first-order kinetic model and the Gamma distribution function. In contrast to conventional models, where the kinetic constant is considered invariant, the Gamma model allows analysing the variability of the kinetic constant using a probability density function. The kinetic constant of mono-digestion and co-digestion batch tests of different wastes were modelled using the Gamma model and two common first-order models: one-step one-fraction model and one-step twofraction model. The Gamma distribution function approximates three distinct probability density functions, i.e. exponential, log-normal, and delta Dirac. Specifically, (i) cattle paunch and pig manure approximated a lognormal distribution; (ii) cattle manure and microalgae approximated an exponential distribution, and (iii) primary sludge and cellulose approximated a delta Dirac distribution. The Gamma model was able to characterise two distinct waste activated sludge, one approximated to a log-normal distribution and the other to an exponential distribution. The same cellulose was tested with two different inocula; in both tests, the Gamma distribution function approximated a delta Dirac function but with a different kinetic value. The potential and consistency of Gamma model were also evident when analysing pig manure and microalgae co-digestion batch tests since (i) the mean k of the co-digestion tests were within the values of the mono-digestion tests, and (ii) the profile of the density function transitioned from log-normal to exponential distribution as the percentage of microalgae in the mixture increased.
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Fierro, R., & Leiva, V. (2017). A stochastic methodology for risk assessment of a large earthquake when a long time has elapsed. Stoch. Environ. Res. Risk Assess., 31(9), 2327–2336.
Abstract: We propose a stochastic methodology for risk assessment of a large earthquake when a long time has elapsed from the last large seismic event. We state an approximate probability distribution for the occurrence time of the next large earthquake, by knowing that the last large seismic event occurred a long time ago. We prove that, under reasonable conditions, such a distribution is exponential with a rate depending on the asymptotic slope of the cumulative intensity function corresponding to a nonhomogeneous Poisson process. As it is not possible to obtain an empirical cumulative distribution function of the waiting time for the next large earthquake, an estimator of its cumulative distribution function based on existing data is derived. We conduct a simulation study for detecting scenario in which the proposed methodology would perform well. Finally, a real-world data analysis is carried out to illustrate its potential applications, including a homogeneity test for the times between earthquakes.
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