Mathematical models for mixing in deep jet bioreactors: analysis

A mathematical model for single and multi step deep-jet bioreactors is presented. A stagewise approach based on macroscopic mechanistic model which divides the reactor into compartments with good quality of mixing and plug flow regions (macromixer), was used. For the mathematical representation of this model a system of differential equations, describing the concentration of tracer in structural elements based on mass balance, and the Runge-Kutta-Fehlberg numerical method of integration, was applied. The mixing time in a 300 dm³ tank was determined by conductivity method with NaCl as tracer.List of Symbols V _g dm³ total volume of liquid - V ₁; V ₆ dm³ volumes of ideally mixed compartments in the vessel - V ₂; V ₇ dm³ volumes of macromixer in the inner circulation flows - V ₃; V ₉ dm³ volumes of liquid phase in the pump - V ₄; V ₈ dm³ volumes of liquid phase in the pipe between the vessel and the pump - V ₅; V ₁₀ dm³ volumes of liquid phase in pipes between the pump and the air input system, including falling jet - F _E; F _E,1; F _E,2 dm³/s the inner volumetric circulation flow rates accross the macromixers - F _E,3; F _E,4 dm³/s exchanges volumetric flow rates between two ideally mixed compartments in the vessel - F _cir; F _1,cir; F _2,cir dm³/s external volumetric circulation flow rates (pumping capacity) - t _A s time interval of puls application - t _AA s time point of impuls application related to the free chosen point of simulation - t _end s end time of simulation - F _qu g²/dm⁶ sum of quadratic error - C _*,* kg/m³ concentration of the tracer in the indicated compartment - C ₀ kg/m³ concentration of the tracer before the injection - C _t kg/m³ concentration of the tracer at the indicated time - C kg/m³ theoretical concentration of full mixed tracer - i index of an arbitrary tank - C _sim kg/m³ calculated concentration of the tracer by numerical integration method