Métodos numéricos para resolução de equações diferenciais ordinárias lineares baseados em interpolação por spline

In this work we have elaborated a spline-based method of solution of inicial value problems involving ordinary differential equations, with emphasis on linear equations. The method can be seen as an alternative for the traditional solvers such as Runge-Kutta, and avoids root calculations in the linear time invariant case. The method is then applied on a central problem of control theory, namely, the step response problem for linear EDOs with possibly varying coefficients, where root calculations do not apply. We have implemented an efficient algorithm which uses exclusively matrix-vector operations. The working interval (till the settling time) was determined through a calculation of the least stable mode using a modified power method. Several variants of the method have been compared by simulation. For general linear problems with fine grid, the proposed method compares favorably with the Euler method. In the time invariant case, where the alternative is root calculation, we have indications that the proposed method is competitive for equations of sifficiently high order.

Degradação fotocatalítica oxidativa do fenol utilizando carvão obtido da pirólise de diferentes biomassas

The modern industrial progress has been contaminating water with phenolic compounds. These are toxic and carcinogenic substances and it is essential to reduce its concentration in water to a tolerable one, determined by CONAMA, in order to protect the living organisms. In this context, this work focuses on the treatment and characterization of catalysts derived from the bio-coal, by-product of biomass pyrolysis (avelós and wood dust) as well as its evaluation in the phenol photocatalytic degradation reaction. Assays were carried out in a slurry bed reactor, which enables instantaneous measurements of temperature, pH and dissolved oxygen. The experiments were performed in the following operating conditions: temperature of 50 °C, oxygen flow equals to 410 mL min-1 , volume of reagent solution equals to 3.2 L, 400 W UV lamp, at 1 atm pressure, with a 2 hours run. The parameters evaluated were the pH (3.0, 6.9 and 10.7), initial concentration of commercial phenol (250, 500 and 1000 ppm), catalyst concentration (0, 1, 2, and 3 g L-1 ), nature of the catalyst (activated avelós carbon washed with dichloromethane, CAADCM, and CMADCM, activated dust wood carbon washed with dichloromethane). The results of XRF, XRD and BET confirmed the presence of iron and potassium in satisfactory amounts to the CAADCM catalyst and on a reduced amount to CMADCM catalyst, and also the surface area increase of the materials after a chemical and physical activation. The phenol degradation curves indicate that pH has a significant effect on the phenol conversion, showing better results for lowers pH. The optimum concentration of catalyst is observed equals to 1 g L-1 , and the increase of the initial phenol concentration exerts a negative influence in the reaction execution. It was also observed positive effect of the presence of iron and potassium in the catalyst structure: betters conversions were observed for tests conducted with the catalyst CAADCM compared to CMADCM catalyst under the same conditions. The higher conversion was achieved for the test carried out at acid pH (3.0) with an initial concentration of phenol at 250 ppm catalyst in the presence of CAADCM at 1 g L-1 . The liquid samples taken every 15 minutes were analyzed by liquid chromatography identifying and quantifying hydroquinone, p-benzoquinone, catechol and maleic acid. Finally, a reaction mechanism is proposed, cogitating the phenol is transformed into the homogeneous phase and the others react on the catalyst surface. Applying the model of Langmuir-Hinshelwood along with a mass balance it was obtained a system of differential equations that were solved using the Runge-Kutta 4th order method associated with a optimization routine called SWARM (particle swarm) aiming to minimize the least square objective function for obtaining the kinetic and adsorption parameters. Related to the kinetic rate constant, it was obtained a magnitude of 10-3 for the phenol degradation, 10-4 to 10-2 for forming the acids, 10-6 to 10-9 for the mineralization of quinones (hydroquinone, p-benzoquinone and catechol), 10-3 to 10-2 for the mineralization of acids.

Análise da biorremediação de compostos monoaromáticos em água através da pseudomonas aeruginosa

The monoaromatic compounds are toxic substances present in petroleum derivades and used broadly in the chemical and petrochemical industries. Those compounds are continuously released into the environment, contaminating the soil and water sources, leading to the possible unfeasibility of those hydrous resources due to their highly carcinogenic and mutagenic potentiality, since even in low concentrations, the BTEX may cause serious health issues. Therefore, it is extremely important to develop and search for new methodologies that assist and enable the treatment of BTEX-contaminated matrix. The bioremediation consists on the utilization of microbial groups capable of degrading hydrocarbons, promoting mineralization, or in other words, the permanent destruction of residues, eliminating the risks of future contaminations. This work investigated the biodegradation kinetics of water-soluble monoaromatic compounds (benzene, toluene and ethylbenzene), based on the evaluation of its consummation by the Pseudomonas aeruginosa bacteria, for concentrations varying from 40 to 200 mg/L. To do so, the performances of Monod kinetic model for microbial growth were evaluated and the material balance equations for a batch operation were discretized and numerically solved by the fourth order Runge-Kutta method. The kinetic parameters obtained using the method of least squares as statistical criteria were coherent when compared to those obtained from the literature. They also showed that, the microorganism has greater affinity for ethylbenzene. That way, it was possible to observe that Monod model can predict the experimental data for the individual biodegradation of the BTEX substrates and it can be applied to the optimization of the biodegradation processes of toxic compounds for different types of bioreactors and for different operational conditions.

Análise da biorremediação de compostos monoaromáticos em água através da pseudomonas aeruginosa

The monoaromatic compounds are toxic substances present in petroleum derivades and used broadly in the chemical and petrochemical industries. Those compounds are continuously released into the environment, contaminating the soil and water sources, leading to the possible unfeasibility of those hydrous resources due to their highly carcinogenic and mutagenic potentiality, since even in low concentrations, the BTEX may cause serious health issues. Therefore, it is extremely important to develop and search for new methodologies that assist and enable the treatment of BTEX-contaminated matrix. The bioremediation consists on the utilization of microbial groups capable of degrading hydrocarbons, promoting mineralization, or in other words, the permanent destruction of residues, eliminating the risks of future contaminations. This work investigated the biodegradation kinetics of water-soluble monoaromatic compounds (benzene, toluene and ethylbenzene), based on the evaluation of its consummation by the Pseudomonas aeruginosa bacteria, for concentrations varying from 40 to 200 mg/L. To do so, the performances of Monod kinetic model for microbial growth were evaluated and the material balance equations for a batch operation were discretized and numerically solved by the fourth order Runge-Kutta method. The kinetic parameters obtained using the method of least squares as statistical criteria were coherent when compared to those obtained from the literature. They also showed that, the microorganism has greater affinity for ethylbenzene. That way, it was possible to observe that Monod model can predict the experimental data for the individual biodegradation of the BTEX substrates and it can be applied to the optimization of the biodegradation processes of toxic compounds for different types of bioreactors and for different operational conditions.