MATHEMATICAL MODELING OF TWO-PHASE GAS-LIQUID MEDIUM FLOW IN METANTANK

Yu. V. Karaevajulieenergy@list.ru

The Branch - of Research Center of Power Engineering Problems of Institution the Russian Academy of Sciences the Kazan Scientific Centre RAS

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

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.

Simple models Complex models

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.

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.

Modeling of anaerobic digestion

MATHEMATICAL MODEL

Assumptions

1. movement of a two-phase gas-liquid medium is laminar, unsteady, axisymmetric, isothermal;

2. the carrier phase is a viscous incompressible Newtonian fluid;

3. dispersed phase consists of strong dominant size bubbles of ideal gas. The energy and other effects of the chaotic and internal motions of dispersed particles (gas bubbles) may be neglected. We can ignore the direct interaction and collisions of particles too; no processes of crushing, agglomeration and coagulation of particles.

4. the hypothesis of equality of pressure in phases is accepted.

Rheological equation of state for carrier phase

where Т – tensor of tension, Р – pressure, I – unite tensor, μ (I2) – effective

viscosity, I2 – second invariant of deformation speed ,

– vector of speed, index Т –symbol of transposition.

Tvgradvgrad 2

1D

DIT 22 IP

v

,exp **

,0

*

,0

1*

where - fluidity, - fluidity at and ;

- measure and limit of liquid structural stability.

,0 0

1 ,

)(

1,

2

2

22 I

II

where ρ1, ρ2 are the real density of liquid and gaseous phases; – vectors of

speed for liquid and gaseous phases;α1, α2 – volume concentrations for liquid and

gaseous phases; j1-2– intensity of mass turning from liquid to gaseous phase; j2-1 -

intensity of mass turning from gaseous phase to liquid one.

21 , vv

1222222

jvdivt

2111111

jvdivt

121

Mass conservation equations

022 vndivdt

ndParticle conservation equation

where n – number of particles per unit volume.

2121211

2

11

1111

2

112 2

11

22

11 Fvvjdiv

dt

vdT

gg 222

1111

2

11

221

2121211

2222

2

112 2

3

22

11 Fvvjdiv

dt

vdT

gg

21

2

1 222211

where g - vector of gravitational acceleration; T1 - reduced stress tensor for

the carrier phase; - strength of the interphase interaction.

Equations of motion

21F

, 2

1 T

s vgradvgrad D

,2211 vvv ,1 ss trI D

ID

IIT

s

1

sI

njvdiv

dt

dP

1

1

2122

222

3

12

31

3

4

where Р – pressure; I – unite tensor; μ – effective viscosity; Ds– strain rate tensor;

Т – symbol for transposition.

Reduced stress tensor for the carrier phase

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

-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

0.015

0.005

а) b) c)The initial distribution of velocity:

a) tangential; b) longitudinal; c) radial velocity

t=30 t=60 t=120

Evaluation of the tangential velocity component of the carrier phase v1φ on time

Evaluation of the tangential velocity component of the dispersed phase v2φ on time

-0.01

0

-0.02

-0.03

-0.04

-0.02

0

-0.04

-0.06

-0.08

t=30 t=60 t=100

Evaluation of the longitudinal velocity component of the carrier phase v1z on time

0.03

0.02

0.01

0

Evaluation of the longitudinal velocity component of the dispersed phase v2z on time

0.02

0.01

0

t=30 t=60 t=100

Evaluation of the radial velocity component of the carrier phase v1r on time

Evaluation of the radial velocity component of the dispersed phase v2r 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

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