Conclusions

n(V′_{, y(x, t)) gives the number of daughter drops that are formed by the breakage of a drop with volume V}′_. Physically, the following boundary conditions should be prescribed onΓwall= Γstirrer∪ Γtank:

u= ustirrer onΓstirrer u= 0 on Γtank f (x, V, t) = 0 on Γwall

whereΓstirrerdescribes the stirrer, andΓtank the boundary of the tank. The initial conditions are chosen such that they are consistent with the boundary conditions. Further details about the modelling of stirred liquid-liquid sys- tems can be found in [52, 91, 92].

In practice it is often assumed that drops belong to some fixed size groups, the number of which is finite. Hence, discretizing (8.8) in respect to size, we obtain a continuity equation for the size group-i:

∂

∂tραi+ ∇ · ραiu= Si, (8.10)

whereSiis the rate of mass transfer into or out of the size group due to break-up and coalescence. It is obvious, that the sum of all particle volume fractions equals the volume fraction of the dispersed phase:

X

αi= α.

As before, we can rewrite the individual size-group volume fraction in terms of the total as: fiα = αi

and rewrite (8.10):

∂

∂tραfi+ ∇ · ραufi= Si, (8.11)

wherefiis the fraction of the dispersed phase volume fraction in group-i. This equation has the form of the trans- port equation of a scalar variablefiin the dispersed phase.

Due to one way coupling we can apply the coordinate transformation technique to calculate the velocity field u, which will be used later in the population balance equation (8.9). Numerical experiments in this direction we will also leave as a topic for future.

8.4 Conclusions

From the previous two sections we observe that the coordinate transformation concept and therefore the modified discrete projection framework are applicable for the numerical simulation of population balance and ’Reynolds- averaged’ turbulence models, where precise calculation of a flow field along moving boundaries is important. We showed that the modified discrete projection method is suitable for simulation of complex industrial models. We also noticed that for complex 3D simulations presented in this chapter the iterative solver demonstrates the same properties already observed for test models of the chapter 5. The particle tracing tool gives a realistic behaviour of the flow field in the stirred tank reactor. At the moment, theFeatflowgroup at TU Dortmund continues intensive research in the directions of turbulent flows [53] and populations balance modeling [4]. As future extensions one can choose simulation of such complex models by a general-purpose scheme (4.8)–(4.10), where convection and the Coriolis force term are written in a form of the cross-product operator w_{(u, ω, ·) × u.}

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Conclusion and outlook

In this thesis we proposed a new discrete projection method for the incompressible Stokes and Navier-Stokes equations with Coriolis force which includes new multigrid and preconditioning techniques for the arising sub- problems for pressure and velocity. In particular, the constructed multigrid method for the velocity matrix shows a robust convergence behaviour for a wide range ofω∆t values. Moreover, its explicit inversion does not require any additional memory or computational resources. The modified discrete pressure Poisson-like operator in a pro- jection step was deduced using pressure Schur complement preconditioning technique. It appears to be much more efficient both in accuracy in time and in convergence to the steady state solution than the standard one since con- vective as well as rotational parts were taken into account. The numerical results showed that the modified DPM is more efficient and robust with respect to the variation in problem parameters than the standard projection scheme.

Furthermore we analysed the accuracy of the modified projection scheme. It was proven that the proposed DPM for the Navier-Stokes equations with the Coriolis force (1.3) has the same order of accuracy as the classical projection scheme for the Navier-Stokes equations (1.1). Namely, the velocity is weakly first-order approximation and the pressure is weakly order1₂ approximation.

As the next step we introduced the rotational form of convection. By doing so, we extended the framework of the modified scheme to the general case, which made it possible to treat any terms written in a form of the cross-product operator w_{(u, ·)×u. Though we did not gain advantageous numerical behaviour of the rotation form of convection} with respect to those of the standard form, we showed that with the help of edge-oriented and_{∇ div-stabilization} techniques one can obtain sufficiently accurate results up to medium Reynolds numbers. As test models we took the lid-driven cavity and the flow around cylinder benchmark problems.

Finally, with the code for the modified discrete projection scheme we performed nonsteady simulations for two configurations of stirred tank reactor models. In the obtained flow field we injected virtual particles and observed their distribution and mixture. These tests showed that the proposed DPM can be successfully used for real-life models. We also showed possible applications of our DPM for turbulent flows in the stirred tank reactor, where prescription of boundary layers is of primary importance. Numerical analysis of this model by a general scheme with coriolis-convection operator w_{(u, ω, ·) × u we leave as a topic for the future.}

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Appendix A