Agitation in metatanks of biogas plants as a rule is carried out mechanically (in 90% of plants). Currently, due to the necessity to process organic wastes with high concentration of solids, as well as frequent breakdown of agitation systems together with pumping systems within the stage of biological treatment of water in consequence of improper engineering based on the wrong assumption on Newtonian nature of a substrate. The first mathematical models of anaerobic digestion process appeared in 2000 (Masse, Droste). ADM1 seems to be the most well-known one due to its being more comprehensive model of anaerobic digestion. In scientific literature there are few models with anaerobic digestion process description. That could be explained by poor investigations of biogas production processes in bioreactors, as well as by the difficulty in modeling and solution of the problem. Furthermore, most of the models do not take into consideration hydrodynamic structure of the flow which influences these processes. The given work is devoted to the investigation of hydrodynamics of fermentable matter motion with its presentation as two-phase gas-liquid medium that ensures more accurate interpretation of the structure of velocity fields that influence biogas generation process quality. The proposed mathematical model is based on a system of equations of Heterogeneous-Continuum Mechanics. The motion description in this case is based on hypothesis of multispeed continuum. Kinetics of biogas production is found by the model proposed by Chen and Hashimoto. The mathematical model allows: -carrying out studies of substrates with different concentration of a solid (that is modeling of non-Newtonian qualities) - analyzing the peculiarities of fermentable matter motion in reactors that differ in construction; - considering different configurations of agitators; - studying and choosing those hydrodynamic regimes that ensure the best agitation of organic substrate.

1. MATHEMATICAL MODELING OF TWO-PHASE GAS-LIQUID MEDIUM FLOW IN METANTANK Yu. V. Karaeva [email_address] The Branch - of Research Center of Power Engineering Problems of Institution the Russian Academy of Sciences the Kazan Scientific Centre RAS

2. Agitation in metatanks of biogas plants as a rule is carried out mechanically (in 90% of plants).

3. Currently, due to the necessity to process organic wastes with high concentration of solids, as well as frequent breakdown of agitation systems together with pumping systems within the stage of biological treatment of water in consequence of improper engineering based on the wrong assumption on Newtonian nature of a substrate.

4. In scientific literature there are few models with anaerobic digestion process description. That could be explained by poor investigations of biogas production processes in bioreactors, as well as by the difficulty in modeling and solution of the problem. Furthermore, most of the models do not take into consideration hydrodynamic structure of the flow which influences these processes.

5. Modeling of anaerobic digestion 2000 Masse and Droste; 2002 Batstone, Keller, Angelidaki, Kalyuzhnyi, Pavlostathis, Rozzi, Sanders, Siegrist and Vavilin ( anaerobic digestion model №1 (ADM1)); 2002 Minott, Fleming; 2005 Vesvikar and Al-Dahhan; Blumensaat and Keller ; 2008 Wu and Chen ; 2009 Wu, Bibeau and Gebremedhin . 1950 Buswell and Mueller; 1978 Chen and Hashimoto; 1982 Hill; 1983 Hashimoto; 1993 Angelidaki and Ahring; 1994 Safley and Westerman; 1995 Toprak; 1999 Vartak , Angelidaki et al.; 2003 Keshtkar et al.; Yilmaz and Atalay . C omplex models S imple models

7. Rheological equation of state for carrier phase where Т – tensor of tension , Р – pressure , I – unite tensor , μ (I 2 ) – effective viscosity , I 2 – second invariant of deformation speed , – vector of speed , index Т – symbol of transposition . where - fluidity , - fluidity at and ; - measure and limit of liquid structural stability .

8. where ρ 1 , ρ 2 are the real density of liquid and gaseous phases ; – vectors of speed for liquid and gaseous phases ; α 1 , α 2 – volume concentrations for liquid and gaseous phases ; j 1-2 – intensity of mass turning from liquid to gaseous phase; j 2-1 - intensity of mass turning from gaseous phase to liquid one. M ass conservation e quation s Particle conservation e quation where n – number of particles per unit volume .

9. where g - vector of gravitational acceleration; T 1 - reduced stress tensor for the carrier phase ; - strength of the interphase interaction. Equation s of motion

10. where Р – pressure; I – unite tensor; μ – effective viscosity; D s – s train r ate t ensor ; Т – symbol for transposition . Reduced stress tensor for the carrier phase

11. , The problem was solved using the software package COMSOL in frame of which the custom application was created taking into account all the features of the task. A cylindrical reactor with a base radius equal to 0.5 m and a height of 1 m was considered. The numerical calculation was made for organic substrate which humidity W = 92%. To determine a rotation speed of the mixing device it is necessary to take account the fact that in anaerobic reactors speed must be low enough, since high rates can lead to unacceptable physical separation of individual groups of bacteria from each other, as well as particles of the substrate with which the bacteria have a close affinity. As a mechanical mixing device considered agitator with blades of rectangular shape, placed perpendicular to the bottom of the tank. The distribution of the velocity field at the moment after the mixing device stopped is a result of the corresponding stationary problem. To obtain the solution moving coordinate system that rotates by the agitator having a constant angular velocity was used. The homogeneous approximation of mixing medium was considered. The results were used as initial condition. Numerical results

12. -0.02 0 -0.04 -0.06 -0.08 0.02 0.01 0 0.002 0 -0.002 -0.004 -0.006 -0.008 -0.01 0.0 15 0.0 05 а) b ) c ) The initial distribution of velocity: a ) tangential; b) longitudinal ; c) radial velocity

13. t=30 t=60 t=120 Evaluation of the tangential velocity component of the carrier phase v 1 φ on time Evaluation of the tangential velocity component of the dispersed phase v 2 φ on time -0.01 0 -0.02 -0.03 -0.04 -0.02 0 -0.04 -0.06 -0.08

14. t= 3 0 t= 6 0 t=100 Evaluation of the longitudinal velocity component of the carrier phase v 1z on time 0.03 0.02 0.01 0 Evaluation of the longitudinal velocity component of the dispersed phase v 2 z on time 0.02 0.01 0

15. t=30 t=60 t=100 Evaluation of the radial velocity component of the carrier phase v 1r on time Evaluation of the radial velocity component of the dispersed phase v 2r on time 0.002 0 -0.002 -0.004 -0.006 -0.008 -0.01 0.002 0 -0.002 -0.004 -0.006 -0.008 -0.01

16. The mathematical model allows : -carrying out studies of substrates with different concentration of a solid (that is modeling of non-Newtonian qualities) - analyzing the peculiarities of fermentable matter motion in reactors that differ in construction; - considering different configurations of agitators; - studying and choosing those hydrodynamic regimes that ensure the best agitation of organic substrate. CONCLUSIONS