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The Brazilian National Program for Biofuel Production has been encouraging diversification of feedstock for biofuel production. One of the most promising alternatives is the use of microalgae biomass for biofuel production. The cultivation of microalgae is conducted in aquatic systems, therefore microalgae oil production does not compete with agricultural land. Microalgae have greater photosynthetic efficiency than higher plants and are efficient fixing CO2. The challenge is to reduce production costs, which can be minimized by increasing productivity and oil biomass. Aiming to increase the production of microalgae biomass, mixotrophic cultivation, with the addition of glycerol has been shown to be very promising. During the production of biodiesel from microalgae there is availability of glycerol as a side product of the transesterification reaction, which could be used as organic carbon source for microalgae mixotrophic growth, resulting in increased biomass productivity. In this paper, to study the effect of glycerol in experimental conditions, the batch culture of the diatom Phaeodactylum tricornutum was performed in a 2-liter flask in a temperature and light intensity controlled room. During 16 days of cultivation, the number of cells per ml was counted periodically in a Neubauer chamber. The calculation of dry biomass in the control experiment (without glycerol) was performed every two days by vacuum filtration. In the dry biomass mixotrophic experiment with glycerol concentration of 1.5 M, the number of cells was assessed similarly in the 10th and 14th days of cultivation. Through a volume element methodology, a mathematical model was written to calculate the microalgae growth rate. Was used an equation that describes the influence of irradiation and concentration of nutrients in the growth of microalgae. A simulation time of 16 days was used in the computations, with initial concentration of 0.1 g l-1. In order to compare simulation data with experimental data, we calculated the dry weight in 8 points in the course of sixteen days. In this way, it was possible to assess graphically biomass concentration versus time through the experiments and by numerical simulation. It was identified that the simulation results were consistent with the experiments and that the addition of glycerol greatly influenced the growth of microalgae. In the present analysis, the glycerol added increased 30% in biomass.

Summary A dynamic mathematical model is developed to estimate microalgae growth in medium‐ and large‐scale compact tubular photobioreactors (PBRs). Besides cell and other chemical species concentrations, temperature and local solar irradiation are important variables to be assessed. Three different experiments were conducted to adjust and validate the mathematical model for which a methodology based on the coefficient of determination is introduced. The first experiment was performed in a 100 L prototype PBR, the second in a column 78.5 L air‐lift PBR, and the third in a 12,000 L compact tubular PBR. Initially, cell growth numerical simulation curves were directly compared with data from the first experiment, which resulted in a coefficient of determination R2 = 0.4043, showing that model adjustment was needed. As a result, 3 adjustment parameters were defined: (i) local solar irradiation (Ψ1); (ii) medium CO2 concentration (Ψ2); and (iii) nutrient concentration (Ψ3). Then, the first experiment data set was used to solve an inverse problem of parameter estimation, obtaining Ψ1 = 1.05, Ψ2 = 0.95, and Ψ3 = 0.18, which resulted in R2 = 0.98584. Next, cell growth numerical simulation curves were compared with measured data from the second and third experiments, obtaining R2 = 0.9862 and R2 = 0.82969, respectively. With the experimentally validated model, a 29 day (or 696 hours) simulation of microalgae cultivation was conducted to calculate the 12,000 L PBR microalgae‐derived oil production, which allowed for the projection of the microalgae species Acutodesmus obliquus oil productivity as approximately 2300 L ha−1 yr−1, considering 11.4% microalgae dry biomass lipid content. Such low production demonstrates that achieving an economically viable process for microalgae‐derived biofuels will require more technological advances and the development of highly optimized processes.

A mathematical and computational modeling of a photobioreactor for the determination of the transient temperature behavior in compact tubular microalgae photobioreactors is presented. The model combines theoretical concepts of thermodynamics with classical theoretical and empirical correlations of Fluid Mechanics and Heat Transfer. The physical domain is discretized with the Volume Element Model (VEM) through which the physical system (reactor pipes) is divided into lumped volumes, such that only one time dependent ordinary differential equation, ODE, results for temperature, based on the first law of thermodynamics. The energetic interactions between the volumes are established through heat transfer empirical correlations for convection, conduction and radiation. Within this context, the main goal of this study is to present a numerical methodology to calculate the mixture (algae + water + nutrients) temperature inside the compact photobioreactor. A pilot plant is under construction, in the Center of Research and Development for Self-Sustainable Energy (NPDEAS), located at UFPR, and the experimental data obtained from this research unit will be used to validate the present numerical solution. Temperature is one of the most important parameters to be controlled in microalgae growth. Microalgae that are cultivated outside their growth temperature range may have a low growth rate or die. For this reason a numerical simulation of the system based on the operating conditions and environmental factors is desirable, in order to predict the transient algae growth temperature distribution along the reactor pipes. The VEM creates an “artificial” spatial dependence in the system or process under analysis by dividing the space (physical domain) into smaller sub domains, namely Volume Elements (VE). Each VE interacts with its neighbors by exchanging energy and/or mass. Thus, each VE is treated as a control volume from classical thermodynamics, i.e., with uniform properties and exchanging mass and energy with its neighbors. The problem is then formulated with the energy equation applied to the fluid VE and to the wall VE. These equations form a system of time dependent ODE’s, which are not dependent on space, therefore eliminating the need for the solution of a system of partial differential equations, PDE’s, depend on time and space, as is the case of traditional numerical methods (e.g., finite element, finite volume and finite differences). The resulting ODE’s were solved using a fourth order Runge-Kutta method with adaptive time step.

A presente pesquisa propõe uma ferramenta educacional inovadora para o ensino do comportamento gráfico de funções matemáticas. Nessa primeira etapa, o desenvolvimento da ferramenta foi realizado na plataforma do software Geogebra, tendo como objetivo principal encaixar intervalos de funções matemáticas em um desenho. Os resultados obtidos apontam para uma evolução intelectual significativa dos alunos, tanto no que se refere à compreensão do impacto dos coeficientes das funções na construção do seu gráfico, quanto no foco e na motivação para completar os desenhos. Esses achados evidenciam o potencial da ferramenta como recurso pedagógico para o ensino de matemática.