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1

Effect of dispersal in single-species discrete diffusion systems with source-sink patches

Elbetch Bilel

Mathematica Applicanda

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2023

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Vol. 51, no. 1

51--97

EN

A multi-patch source-sink model with and without intraspecific competition in the sink patches is considered. First, we study the dynamics of the model when the matrix of migration is irreducible and reducible. We show that, there is a threshold number of source patches such that the population potentially becomes extinct below the threshold and established above the threshold. Next, used the theory of singular perturbation and theorem of Tikhonov, in the case of perfect mixing, i.e. when the diffusion rate tends to infinity, we calculate the equilibrium of the model and we give a good approximations of the solutions in this case. Second, we determine, in some particular cases, the conditions under which fragmentation and the existence of sink patches can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities of the source patches. Finally, we study the effect of the rapid growth of the population in source patches and the rapid death of the population in sink patches on the dynamics of the total equilibrium population and on the coexistence of the populations in the patches.

PL

Rozważany jest model wielogatunkowy enklaw wykluczanych ekologicznie z i bez konkurencji wewnątrzgatunkowej w enklawach zanikających. Najpierw badamy dynamikę modelu, gdy macierz migracji jest nieredukowalna i edukowalna. Pokazujemy, że istnieje progowa liczba enklaw, która powoduje, że populacja potencjalnie wymiera poniżej progu i jest ustablilizowana powyżej pewnego pozomu. Następnie, korzystając z teorii perturbacji pojedynczych i twierdzenia Tichonowa, w przypadku doskonałego mieszania, tj. gdy współczynnik dyfuzji dąży do nieskończoności, obliczamy równowagę modelu i podajemy dobre przybliżenia rozwiązań w tym przypadku. Po drugie, określamy, w pewnych szczególnych przypadkach, warunki, w których fragmentacja i istnienie enklaw może prowadzić do tego, że całkowita populacja w równowadze jest większa lub mniejsza niż suma populacji enklaw. Wreszcie, badamy wpływ szybkiego wzrostu populacji źródłowej i szybkiego wymierania populacji populacji zlewowej na dynamikę całkowitej populacji równowagi i na koegzystencję gatunków.

2

A sensitivity analysis of kinetic characterizations in continuous flotation circuits under moderate deviations with respect to perfect mixing

Vinnett Luis, Pino-Muñoz Catalina A., Yianatos Juan, Díaz Francisco, Henríquez Felipe

Physicochemical Problems of Mineral Processing

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2022

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Vol. 58, iss. 5

art. no. 152420

EN

This paper studies the effect of moderate deviations with respect to perfect mixing on the estimated kinetic parameters in industrial flotation banks. Radioactive tracer tests and mass balance surveys were performed to characterize the mixing regimes and Cu kinetic responses. For three models (Single Rate Constant, Rectangular and Gamma), two approaches to incorporate the residence time distributions (RTD) in the kinetic characterizations of rougher banks were compared: (i) RTDs measured from the radioactive tracer tests; and (ii) pure perfect mixing in each flotation machine. The measured RTDs did not present significant bypass in the evaluated banks. In all cases, comparable model fitting was obtained with both RTD approaches, which indicates that the kinetic models add sufficient flexibility to compensate for moderate biases in the mixing regime. The studied kinetic models showed non-significant differences in the estimated maximum recoveries (R∞), mean (kmean) and median (k50) rate constants when comparing the process modelling from measured RTDs and pure perfect mixing. However, the Gamma model was more sensitive to the RTD assumption in terms of the shapes of the flotation rate distributions. From the results, kinetic characterizations focused only on model fitting, or on R∞ and kmean (or k50) estimations have low sensitivity to the assumption of perfect mixing when the RTDs present moderate deviations with respect to this regime. Special attention must be paid when characterizing floating components as the perfect mixing assumption may bias the shapes of the flotation rate distributions.

3

Modelling the gas flow in permeate channel in membrane gas separation process

Szwast M.

Chemical and Process Engineering

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2018

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Vol. 39, nr 3

271–-280

EN

This paper analyses the real behaviour of the fluid in the channels of a three-end membrane module. The commonly accepted mathematical model of membrane separation of gas mixtures in such modules assumes a plug flow of fluid through the feed channel and perfect mixing in the permeate channel. This article discusses the admissibility of accepting such an assumption regarding the fluid behaviour in the permeate channel. Throughout analysis of the values of the Péclet number criterion, it has been demonstrated that in the industrial processes of membrane gas separation, the necessary conditions for the perfect mixing in the permeate channel are not met. Then, CFD simulations were performed in order to establish the real fluid behaviour in this channel. It was proved that in the permeate channel the fluid movement corresponds to the plug flow, with the concentration differences at both ends of the module being insignificant. In view of the observations made, the admissibility of concentration stability assumptions in the mathematical models for the permeate channel was discussed.

4

Badanie stopnia zmieszania w krystalizatorze zbiornikowym o działaniu ciągłym

Machej K., Piotrowski K., Słuszniak K.

Inżynieria i Aparatura Chemiczna

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2001

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Nr 2

5-9

PL

Zaproponowano metodę pomiarową, pozwalającą w prosty i wystarczająco dokładny sposób określić wielkość odchylenia stopnia zmieszania w układzie od stanu zmieszania idealnego, odpowiadającego aparatowi MSMPR. Wykorzystując przedstawioną metodę określono następnie wpływ wybranych warunków mieszania (tj. typu mieszadła, liczby jego obrotów, położenia mieszadła względem dna) na stopień zmieszania roztworu lub zawiesiny wewnątrz krystalizatora.

EN

A measurement method enabling one to determine in a simple and sufficiently precise way a deviation of the actual degree of mixing from the ideal one, corresponding to MSMPR conditions, was proposed. Taking advantage of this method, an influence of selected mixing conditions (i.e. type of a stirrer, rotational speed and a position with regard to the bottom) on the degree of mixing of solution or suspension inside a crystallizer was determined.