Abstract This work presents the mathematical modeling of clarification kinetics of the sugarcane juice by ozonation in batch reactor considering the reaction occurs homogeneously and ozone saturation in the juice is guaranteed. The sugarcane juice color was considered proportional to the light absorbance at wavelength λ = 420 ηm, according to the International Unit specified by ICUMSA - International Commission for Uniform Methods of Sugar Analysis. The model equation coefficients were obtained numerically, using the MATLAB® software, based on the experimental absorbance as a function of time, obtained for three different temperatures of the juice. The obtained results were so much satisfactory and led to the conclusion that mathematical model represents well the kinetics of clarification in the proposed conditions.

RESUMO Neste estudo, grãos de sorgo foram hidratados para avaliar os efeitos do tempo e temperatura na umidade e modelar o processo com diferentes equações matemáticas. Para esse fim, as cultivares de sorgo BRS 310 e BRS 655 foram submetidas à hidratação nas temperaturas de 30 ºC, 40 ºC, 50 ºC e 60 ºC, por um período de 12 horas. O processo de hidratação foi modelado com as equações empíricas de Peleg e Page e com os modelos fenomenológicos de Difusão e Omoto-Jorge. O tempo e a temperatura afetaram significativamente a cinética de hidratação dos grãos de sorgo. Os modelos de Peleg e de Difusão apresentaram ajustes mais satisfatórios para o processo e, por isso, foram generalizados. O coeficiente de difusão efetivo de água nos grãos durante a hidratação variou de 2,02.10-11 a 6,34.10-11 m2.s-1 para a cultivar BRS 310 e de 2,76.10-11 a 4,38.10-11 m2.s-1 para a BRS 655 com energias de ativação de 31,21 e 11,52 kJ.mol-1, respectivamente.

ABSTRACT Sorghum grains were soaking to evaluate the effects of time and temperature on moisture and process modelling was maked with different mathematical equations. Then, sorghum BRS 310 and BRS 655 were hydrated at 30 °C, 40 °C, 50 °C and 60 °C, for 12 hours. The hydration kinetics modelling with empirical equations of Peleg and Page and phenomenological equations of Diffusion and Omoto-Jorge were considerad. The time and temperature affected significantly the hydration kinetics of grain sorghum. The Peleg and Diffusion models showed the best fitting for soaking and, therefore, these models were used to create generalized models. The effective diffusion coefficient of water during the grain sorghum hydration ranged from 2.02x10-11 to 6.34x10-11 m2.s-1 for BRS 310 and from 2.76x10-11 to 4.38x10-11 m2.s-1 for BRS 655 with activation energies of 11.52 and 31.21 kJ.mol-1, respectively.

Abstract This study aimed to assess the drying kinetics of barley grains and, using mathematical modeling, to evaluate the diffusivity during drying and after drying to evaluate the effect of high drying temperatures on the seed germination index for malt production. Three drying temperatures 40, 60 and 80°C were used. The last two, 60 and 80°C, reached moisture close to ideal for storage with 6 and 4 hours respectively; at 40°C, drying process took 12 hours and would still take additional hours to reach 13% moisture on a dry basis. Drying temperature influenced the germination power. The protein content was reduced according to increased drying time. The diffusivity coefficient was higher in conventional than in intermittent drying, probably due to heat loss in the intermittent process. It was observed that higher the temperature higher is the diffusivity. Ks values of the Omoto model ranged from 1.07 to 2.05E-4.

The hydration kinetics of transgenic corn types flint DKB 245PRO, semi-flint DKB 390PRO, and dent DKB 240PRO was studied at temperatures of 30, 40, 50, and 67 °C. The concentrated parameters model was used, and it fits the experimental data well for all three cultivars. The chemical composition of the corn kernels was also evaluated. The corn cultivar influenced the initial rate of absorption and the water equilibrium concentration, and the dent corn absorbed more water than the other cultivars at the four temperatures analyzed. The effect of hydration on the kernel texture was also studied, and it was observed that there was no significant difference in the deformation force required for all three corn types analyzed with longer hydration period.

The hydration kinetics of five barley cultivars was studied at six different temperatures ranging from 10 to 35 ºC for 32 hours applying the Peleg model. Response Surface was used to describe dynamic of the process and identify the hydration time for each cultivar. The activation energy (Ea), enthalpy (ΔH*), entropy (ΔS*), and Gibbs free energy (ΔG*) were estimated from the adjusted parameters and Arrhenius equation. Temperature had significant effect on the hydration of the five cultivars. At low temperatures, the stabilization time for hydration was faster. Peleg constants K1 and K2 decreased with increasing temperature. The cultivar BRS BRAU showed the lowest value of initial absorption rate (R0 = 0.149 kg.h-1) at 10 ºC, while the cultivar BRS BOREMA had the highest value of R0 (0.367 kg.h-1 at 35 ºC). The equilibrium moisture content (Me) increased with increasing temperature. The cultivars BRS CAUE and BRS BRAU showed the lowest values of Ea, ΔH*, ΔS* showed negative values, and ΔG* increased with increasing temperature, confirming the effect of temperature on hydration.

The aim of this work was to study the drying curves and equilibrium isotherms of extruded fish feed. The drying curves were determined at air temperatures of 50, 60, 70 and 80°C and airflow velocities of 1.5, 2.5 and 3.5 m/s). The equilibrium isotherms of relative humidity of air were obtained between 10 and 80% at 30, 40, 50, and 70°C. The experimental data were fitted for non-linear regression by using Statistica® to the models reported in the literature. The results showed that the drying curves and the equilibrium isotherms were significantly influenced by variations of the air temperature in a similar way to solid materials as described in the literature. The statistical results for models of Page and Peleg showed that the fitting of the experimental drying curve and isotherm data were satisfactory.

An industrial system for the production of white liquor of a pulp and paper industry, Klabin Paraná Papéis, formed by ten reactors was modeled, simulated, and analyzed. The developed model considered possible water losses by the evaporation and reaction, in addition to variations in the volumetric flow of lime mud across the reactors due to the composition variations. The model predictions agreed well with the process measurements at the plant and the results showed that the slaking reaction was nearly complete at the third causticizing reactor, while causticizing ends by the seventh reactor. Water loss due to slaking reaction and evaporation occurred more pronouncedly in the slaker reactor than in the final causticizing reactors; nevertheless, the lime mud flow remained nearly constant across the reactors.

Desenvolveu-se uma análise comparativa do estudo teórico-experimental efetuado por Hsu sobre a hidratação de grãos de soja Amsoy 71 a partir da execução de diversos ensaios de hidratação de soja CD 202 nas temperaturas de 10, 20, 30, 40 e 50 °C, medindo-se a umidade ao longo do tempo. Os resultados revelam que o valor da umidade de equilíbrio, Xeq, da soja CD 202 independe da temperatura e é 21% superior àquele encontrado por Hsu, sugerindo que a variedade de soja exerce grande influência sobre Xeq. O modelo de Hsu foi resolvido numericamente e teve seus parâmetros ajustados pelo método dos mínimos quadrados, apresentando desvios máximos de +/-10% em relação aos valores experimentais. A etapa limitante do processo de transferência de massa na hidratação corresponde à difusão de água no interior do grão, conduzindo ao surgimento de gradientes radiais de umidade que diminuem com o aumento da temperatura e do tempo. Independentemente da variedade de soja utilizada, a difusividade aumenta com a elevação da temperatura e do teor de umidade, porém os valores desta propriedade de transporte para soja Amsoy 71 foram superiores aos da CD 202, sendo muito próximos no início da hidratação a 20 °C e quase três vezes maior ao final da hidratação a 50 °C.

A comparative analysis of the theoretical-experimental study, developed by Hsu on the hydration of Amsoy 71 soybean grain, was performed through several soaking experiments using CD 202 soybean at 10, 20, 30, 40, and 50 °C, measuring moisture content over time. The results showed that CD 202 soybean equilibrium moisture content, Xeq, does not depend on temperature and is 21% higher than that found by Hsu, suggesting that soybean cultivar exerts great influence on Xeq. The Hsu model was numerically solved and its parameters were adjusted by the least squares method, with maximum deviations of +/- 10% relative to the experimental values. The limiting step in the mass transfer process during hydration corresponds to water diffusion inside the grain, leading to radial moisture gradients that decrease over time and with an increase in temperature. Regardless of the soybean cultivar, diffusivity increases as temperature or moisture content increases. However, the values of this transport property for Amsoy 71 were superior to those of CD 202, very close at the beginning of hydration at 20 °C and almost three times higher at the end of hydration at 50 °C.

Desenvolveu-se um modelo matemático para estimar a concentração de água em grãos de ervilha durante sua hidratação, o qual foi obtido a partir de um balanço de massa em regime transiente, considerando-se volume constante e geometria esférica. Este modelo é representado por uma equação diferencial ordinária de primeira ordem e possui dois parâmetros: o coeficiente de transferência de massa (Ks) e a concentração de equilíbrio (ρAeq). Ambos foram ajustados numericamente tomando-se como critério a minimização da soma dos resíduos quadráticos, utilizando-se dados experimentais de hidratação de grãos de ervilha a 20, 30, 40, 50 e 60 ºC. Em todas as temperaturas, o modelo representou as principais tendências do processo de hidratação com desvios inferiores a 5%. Foi observado que Ks apresentou um comportamento segundo a equação de Arrhenius frente à temperatura, Ks = 182,8 exp (-3521/T), enquanto ρAeq permaneceu praticamente constante, obtendo-se um valor médio de 0,53 g.mL-1. Utilizando-se a correlação de Ks e assumindo-se o valor de ρAeq médio, obteve-se um modelo generalizado, o qual apresentou para todos os dados um desvio ligeiramente superior a 7%. Adicionalmente, constatou-se que a densidade dos grãos de ervilha permaneceu praticamente constante ao longo da hidratação independentemente da temperatura.

A mathematical model for estimating the water concentration in pea grains during hydration was developed from a transient water mass balance inside a grain of pea considering a constant volume and spherical geometry. This model is represented by an ordinary first order differential equation and has two parameters: the mass transfer coefficient (Ks) and the equilibrium concentration (ρAeq). Both parameters have been numerically fitted using the minimization of the sum-of-squared residuals criterion and the hydration experimental data for pea grains at 20, 30, 40, 50, and 60 ºC. At all temperatures, the model represented the major trends of the hydration process with standard deviations smaller than 5%. For the model fitted parameters, it was observed that the mass transfer coefficient behaved according to Arrhenius's equation in relation to the temperature variation, Ks = 182.8.exp(-3521/T) while the equilibrium concentration remained practically constant showing an average value of ρAeq = 0.533 g/cm³. Using the Ks Arrhenius's correlation and assuming the average value of ρAeq, a generalized model was obtained, which resulted in a standard deviation slightly greater than 7% for all the data values. Additionally, it was observed that the pea grains density remains practically constant along the hydration process and does not depend on the temperature.

Este trabalho apresenta um novo modelo fenomenológico de parâmetros concentrados para a hidratação de grãos de soja, obtido a partir de um balanço de massa em regime transiente nos grãos, admitindo duas hipóteses básicas: 1º) variação do volume do grão diretamente proporcional à variação da sua massa ao longo da hidratação; e 2º) variação exponencial do coeficiente de transferência de massa aparente em função da concentração de água na soja. Este modelo possui apenas dois parâmetros, os quais foram estimados a partir do ajuste deste às medidas da umidade do grão ao longo do tempo, avaliadas em várias temperaturas. Os resultados revelam que o modelo proposto é adequado, enquanto o coeficiente de transferência de massa aparente depende da temperatura e apresenta variações significativas ao longo da hidratação.

A new first principle model with lumped parameters was developed for the hydration of soya beans. A transient mass balance was applied to the soya beans taking into consideration two basic hypotheses:1º) variation of the volume directly proportional to the variation of the mass of the grain during the process, 2º) Exponential variation of the apparent coefficient of mass transfer in function of water concentration in the soy. This model has got just two parameters, which were evaluated from adjustment of model to the measures of grain moisture along the time, at several temperatures. The results indicate that the proposed model was able to represent the hydration process, while the mass transfer coefficient depends on the temperature and it presents significant variations along the hydration.

Foi desenvolvido e testado um modelo dinâmico do reator de apagamento do sistema de caustificação da Klabin Paraná Papéis, responsável pela geração do licor branco utilizado na planta. O modelo contempla perdas de água por evaporação e por reação química e apresentou boa concordância com dados industriais de álcali ativo, álcali total titulável e temperatura, sem a necessidade de ajuste de nenhum parâmetro. Os resultados obtidos a partir de simulações revelam que o consumo de água pela reação de apagamento, bem como pela evaporação, exercem uma influência significativa sobre a vazão volumétrica na saída do reator, impondo uma diminuição de 4,6% sobre o teor de água na corrente de saída do reator em relação à alimentação.

A dynamic model of the slaker reactor was developed and validated for Klabin Paraná Papéis causticizing system, responsable for white liquor generation used by the plant. The model considered water losses by evaporation and chemical reaction. The model showed a good agreement with the industrial plant measures of active alkali, total titratable alkali and temperature, without the need of adjustment of any parameter. The simulated results showed that the water consumption by the slaking reaction and evaporation exerted significant influence on the volumetric flow rate of limed liquor, which imposed a decrease of 4.6% in the amount of water in reactor outlet.

Neste trabalho desenvolveu-se um modelo fenomenológico de parâmetros concentrados para a hidratação de grãos de soja, obtido a partir de um balanço de massa em regime transiente nos grãos, levando em consideração a variação do diâmetro destes ao longo do processo. Os resultados das simulações deste modelo foram comparados com as previsões de modelos empíricos encontrados em literatura. Tanto o modelo fenomenológico quanto os modelos empíricos foram validados frente a dados experimentais de hidratação de grãos de soja, obtidos neste trabalho nas temperaturas de 10, 20, 30, 42 e 49°C. Todos os modelos representaram as principais tendências do processo de hidratação, a partir do ajuste dos seus parâmetros para cada temperatura. Entretanto, o modelo fenomenológico foi o que apresentou menores desvios em relação aos dados experimentais. Adicionalmente, foi proposto um modelo fenomenológico generalizado.

A first principle model with lumped parameters was developed for the hydration of soya beans. A transient mass balance was applied to the soya beans taking into consideration the variation of the bean diameters during the process. Simulation results were compared to the empirical models available in the literature and with experimental data. Both the first principle and empirical models were validated by the experimental data that were obtained for the temperatures of 10, 15, 20, 30, 42 and 49°C. All models were able to represent the main data trends of the hydration process, after adjustment of their parameters for each temperature. However, the first principle model has the smallest deviations in relation to the experimental data. In addition, a generalized first principle model was proposed.