Selection of numerical method for solving ordinary differential equation systems for a high-speed model of hydrocarbons steam cracking

Relevance. Caused by the need to increase production of light olefins. The use of advanced process control systems and Real–Time Optimization makes it possible to increase the efficiency of steam cracking plants, but requires a high-speed mathematical model of the process. Aim. To select a method for numerical solution of systems of ordinary differential equations, which provides the highest speed when calculating the reaction coil of a steam cracking furnace. Reducing the time spent on calculating each scenario will allow the proposed model to be used for real-time process optimization tasks. Object. Mathematical model of ethane steam cracking, numerical methods for ordinary differential equations systems solution. Methods. System analysis, mathematical modeling. To solve the ordinary differential equations systems, various explicit numerical methods were used, differing in approach to integration step determination. Results. The authors have developed and tested a steady-state model of ethane steam cracking. The developed model was used to compare the calculation time required for solving ordinary differential equations systems using different numerical methods. It was demonstrated, that the use of an adaptive integration step reduces calculation time by more than 20 times (from more than 11 hours to 34 minutes) while maintaining the accuracy of calculations. This is due to different reaction rates through the length of the reaction coil – in areas of high temperatures and high concentrations of reagents, a reduction in the integration step is required to obtain the desired accuracy. And in low reaction rates areas an increase in the step and reduction in the total calculated iterations are acceptable

Femoral histology and growth patterns of the ceratopsian dinosaur Psittacosaurus sibiricus from the Early Cretaceous of Western Siberia

The early ceratopsian dinosaur Psittacosaurus sibiricus from the Early Cretaceous of Western Siberia, Russia, is one of the most advanced and largest (up to 2.5 m) members of the genus. Here we present a description of ontogenetic changes in the long-bone histology of this species. Analysis of a growth series of femora demonstrates significant histological maturation during ontogeny, expressed by the progressive appearance of signs of bone remodeling (erosion bays, secondary bone formation), decreasing of vascularity, changing of the orientation of vascular canals from reticular to longitudinal, and appearance of parallel-fibred bone in the outer part cortex. These ontogenetic changes in the longbone histology of P. sibiricus are generally similar to those of another relatively advanced species, P. lujiatunensis from China. The basal P. mongoliensis from Mongolia shows less mature long-bone histology during late ontogeny (e.g., the late appearance of signs of remodeling and the predominance of reticular vascularization at later stages). We suggest that the earlier achievement of histological maturity is an evolutionary trend of the genus Psittacosauru