Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin, Germany,

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1 Volker John On the numerical simulation of population balance systems Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin, Germany, Free University of Berlin, Department of Mathematics and Computer Science, Arnimallee 6, Berlin, Germany Numerical simulation of population balance systems Berlin, (44)

2 Outline of the talk 1 Modeling of a bulk precipitation by a population balance system 2 Finite element methods for convection dominated equations 3 Incompressible turbulent flows 4 A finite element projection based VMS method 5 Calcium carbonate precipitation in 2d/3d 6 Calcium carbonate precipitation in 3d/4d 7 Summary and outlook Numerical simulation of population balance systems Berlin, (44)

3 Modeling of a bulk precipitation by pop. bal. sys. 1 Modeling of a bulk precipitation by a population balance system calcium carbonate precipitation chemical reaction in a flow CaCl 2 + Na 2 CO 3 CaCO 3 +2NaCl main feature: precipitation starts if local concentration of CaCO 3 exceeds saturation concentration chemical mechanisms: nucleation of particles growth of particles reactive flows with particles particle size distribution (PSD) is of interest, not the behavior of individual particles modeling with population balance systems Numerical simulation of population balance systems Berlin, (44)

4 Modeling of a bulk precipitation by pop. bal. sys. 1 Momentum balance of the continuous phase momentum balance of the Navier Stokes equations ( ) t (ϱ v m ) + rk ϱ v k v m + π mk = Jim(φ, v v)f i dv x + ϱ g m Ω x ϱ density, v velocity, π stress tensor accumulation convection, diffusion exchange with the disperse phase body forces in applications: flows very often turbulent Numerical simulation of population balance systems Berlin, (44)

5 Modeling of a bulk precipitation by pop. bal. sys. 1 Mass and energy balances of the continuous phase system of convection diffusion reaction equations ) t (ϱ φ l ) + rk (ϱ v k φ l + j φ lk = J φ il (φ, v)f idv x + σ l (φ) Ω x φ l concentrations accumulation convection, diffusion exchange with disperse phase chemical reactions convection dominant reaction dominant Numerical simulation of population balance systems Berlin, (44)

6 Modeling of a bulk precipitation by pop. bal. sys. 1 Population balances of the disperse phase disperse distribute more or less evenly throughout a medium system of convection diffusion equations ( ) ) t f i + xj G ij (φ, v)f i + rk (v k f i + j f ik accumulation growth convection, diffusion = h i,br (f, φ, v)dv x Ω x + h i,agg (f, φ, v)dv x Ω x breakage agglomeration PSDs depend on time, space and properties of the particles (internal coordinates) equations are defined in a higher dimensional domain than the other equations convection dominant, often even no diffusion global integral kernels on right hand side Numerical simulation of population balance systems Berlin, (44)

7 Modeling of a bulk precipitation by pop. bal. sys. 1 Challenges simulation of reaction and convection dominated equations with the goal to obtain solutions with sharp layers and without spurious oscillations simulation of turbulent flows coupling of equations defined in domains with different dimension Numerical simulation of population balance systems Berlin, (44)

8 Finite element methods for convection dominated quations 2 Finite element methods for convection dominated equations comparison of 20 stabilized finite element methods, J., Schmeyer (2008, 2009) talk by its own FEM FCT schemes (Kuzmin (2005,2009)) clearly the best methods linear FEM FCT scheme (Kuzmin (2009)) has good ratio of accuracy and costs = method for population balance systems Numerical simulation of population balance systems Berlin, (44)

9 Incompressible turbulent flows 3 Incompressible turbulent flows Navier Stokes equations: fundamental equations of fluid dynamics Claude Louis Marie Henri Navier ( ), George Gabriel Stokes ( ) Numerical simulation of population balance systems Berlin, (44)

10 Incompressible turbulent flows 3 The incompressible Navier Stokes equations conservation laws conservation of linear momentum conservation of mass u t 2Re 1 D(u) + (uu T ) + p = f in (0, T ] Ω u = 0 in [0, T ] Ω u(0, x) = u 0 in Ω + boundary conditions given: Ω R d, d {2, 3}: domain T : final time u 0 : initial velocity boundary conditions parameter: Reynolds number Re to compute: velocity u, where D(u) = u + ut 2 is the velocity deformation tensor pressure p Numerical simulation of population balance systems Berlin, (44),

11 Incompressible turbulent flows 3 The incompressible Navier Stokes equations Reynolds number Re = LU ν L [m] characteristic length scale (diameter of a channel, diameter of a body in the flow) U [ms 1 ] characteristic velocity scale (inflow velocity) ν [m 2 s 1 ] kinematic viscosity (water: ν = 10 6 m 2 s 1 ) rough classification of flows: Re small: steady state flow field (if data do not depend on time) Re larger: laminar time dependent flow field Re very large: turbulent flows Numerical simulation of population balance systems Berlin, (44)

12 Incompressible turbulent flows 3 The incompressible Navier Stokes equations Reynolds number Re = LU ν L [m] characteristic length scale (diameter of a channel, diameter of a body in the flow) U [ms 1 ] characteristic velocity scale (inflow velocity) ν [m 2 s 1 ] kinematic viscosity (water: ν = 10 6 m 2 s 1 ) rough classification of flows: Re small: steady state flow field (if data do not depend on time) Re larger: laminar time dependent flow field Re very large: turbulent flows There is no exact definition of what is a turbulent flow! Numerical simulation of population balance systems Berlin, (44)

13 Incompressible turbulent flows 3 Characteristics of turbulent flows posses flow structures of very different size hurricane Katrina (2005) some large eddies (scales), many very small eddies (scales) Numerical simulation of population balance systems Berlin, (44)

14 Incompressible turbulent flows 3 Characteristics of turbulent flows Richardson energy cascade: energy is transported in the mean from large to smaller eddies smallest scales important for physics of the flow start of cascade: kinetic energy introduced into flow by productive mechanisms at largest scale inner cascade: transmitting energy to smaller and smaller scales by processes not depending on molecular viscosity end of cascade: molecular viscosity enforcing dissipation of kinetic energy at smallest scales Numerical simulation of population balance systems Berlin, (44)

15 Incompressible turbulent flows 3 Characteristics of turbulent flows Kolmogorov (1941): energy is dissipated from eddies of size (Kolmogorov scale) λ Re 3/4 Kolmogorov during a visit at the Akademie der Wissenschaften der DDR, mid of 1950-ies Numerical simulation of population balance systems Berlin, (44)

16 Incompressible turbulent flows 3 Impact on numerical simulations Galerkin method aims to simulate all persisting eddies, Direct Numerical Simulation (DNS) number of degrees of freedom Re 9/4 Ω = (0, 1) 3 = L = 1 approx 10 7 cubic mesh cells ( ) low order method (mesh width resolution of discretization) = λ 1/215 = Re 1290 applications: Reynolds numbers larger by orders of magnitude Direct Numerical Simulation not feasible! only resolved scales can be simulated Numerical simulation of population balance systems Berlin, (44)

17 Incompressible turbulent flows 3 The Kolmogorov energy spectrum energy of scales in wave number space (Fourier space) logarithmic axes resolved scales large scales resolved small scales unresolved scales, subgrid scales k wave number E(k) turbulent kinetic energy of modes with wave number k k 5/3 law of energy spectrum: E(k) ɛ 2/3 k 5/3 Numerical simulation of population balance systems Berlin, (44)

18 Incompressible turbulent flows 3 Summary DNS impossible (very) small scales important, have to be taken into account 3d simulations necessary literature P.A. Davidson, Turbulence, Oxford University Press, 2004 U. Frisch, Turbulence, Cambridge University Press, 1995 S.B. Pope, Turbulent Flows, Cambridge University Press, 2000 Numerical simulation of population balance systems Berlin, (44)

19 Incompressible turbulent flows 3 Summary DNS impossible (very) small scales important, have to be taken into account 3d simulations necessary literature P.A. Davidson, Turbulence, Oxford University Press, 2004 U. Frisch, Turbulence, Cambridge University Press, 1995 S.B. Pope, Turbulent Flows, Cambridge University Press, 2000 Impact on numerical simulations only large scales of a turbulent flows possible to simulate, two approaches Large Eddy Simulation (LES) Variational Multiscale (VMS) methods impact of the small scales has to be modeled Numerical simulation of population balance systems Berlin, (44)

20 A finite element projection based VMS method 4 A finite element projection based VMS method What is large? (traditional) Large Eddy Simulation (LES): large flow structures defined by an average in space two scale decomposition of scales: large, resolved scales small, unresolved, subgrid scales based on strong formulation of equation commutation errors boundary conditions for large scales references: Sagaut (2006), Berselli, Iliescu, Layton (2006), J. (2004) Numerical simulation of population balance systems Berlin, (44)

21 A finite element projection based VMS method 4 A finite element projection based VMS method Variational Multiscale (VMS) methods: large flow structures defined by projections based on ideas of Hughes (1995), Guermond (1999) often three scale decomposition of scales: resolved large scales resolved small scales small, unresolved, subgrid scales based on variational formulation of equation variety of realizations can be found in the literature Numerical simulation of population balance systems Berlin, (44)

22 A finite element projection based VMS method 4 A finite element projection based VMS method J., Kaya (2005), based on ideas from Layton (2002) (V h, Q h ) conform velocity pressure finite element spaces fulfilling the inf sup condition for all resolved scales L H finite dimensional space of symmetric tensor valued functions in L 2 (Ω) d d (large scale space) Numerical simulation of population balance systems Berlin, (44)

23 A finite element projection based VMS method 4 A finite element projection based VMS method J., Kaya (2005), based on ideas from Layton (2002) (V h, Q h ) conform velocity pressure finite element spaces fulfilling the inf sup condition for all resolved scales L H finite dimensional space of symmetric tensor valued functions in L 2 (Ω) d d (large scale space) find u h : [0, T ] V h, p h : (0, T ] Q h, G H : [0, T ] L H : (u h t, v h ) + (2Re 1 D(u h ), D(v h )) + ((u h )u h, v h ) (p h, v h ) + (ν T (D(u h ) G H ), D(v h )) = (f, v h ) v h V h (q h, u h ) = 0 q h Q h (D(u h ) G H, L H ) = 0 L H L H ν T (t, x) 0 turbulent viscosity, turbulence model G H = P L H D(u h ) L 2 projection nonlinear (in the viscosity) version of local projection stabilization (LPS) schemes for stabilizing convection dominated equations Numerical simulation of population balance systems Berlin, (44)

24 A finite element projection based VMS method 4 Properties three scale decomposition: (resolved) large scales resolved small scales unresolved small scales turbulence model acts directly only on the resolved small scales modeling the influence of unresolved small scales indirect influence onto large scales by coupling of resolved small and large scales parameters of the VMS method L H ν T (Smagorinsky type models) Numerical simulation of population balance systems Berlin, (44)

25 A finite element projection based VMS method 4 Properties three scale decomposition: (resolved) large scales resolved small scales unresolved small scales turbulence model acts directly only on the resolved small scales modeling the influence of unresolved small scales indirect influence onto large scales by coupling of resolved small and large scales parameters of the VMS method L H ν T (Smagorinsky type models) finite element error analysis: J., Kaya (2008); J., Kaya, Kindl (2008) similar approach with finite volume methods by Gravemeier (2006) Numerical simulation of population balance systems Berlin, (44)

26 A finite element projection based VMS method 4 How to choose the large scale space L H? standard bases for velocity pressure finite element spaces here: L H defined on the same grid: l L H j H 0 0 = span 0 0 0, lj H 0, l H j 0 l H j 0 0 lj H 0 lj H ,, lj H 0 0 lj H lj H 0 0, j = 1,..., n L two level method (for convection diffusion equations), J., Kaya, Layton (2006) Numerical simulation of population balance systems Berlin, (44)

27 A finite element projection based VMS method 4 How to choose the large scale space L H? coupled system A 11 A 12 A 13 B1 T G 11 G 12 G A 21 A 22 A 23 B2 T 0 G 22 0 G 24 G 25 0 A 31 A 32 A 33 B T G 33 0 G 35 G 36 B 1 B 2 B G M M G 21 G M G 31 0 G G M 0 0 M 0 G 52 G G M u h 1 u h 2 u h 3 p h g H 11 g H 12 g H 13 g H 22 g H 23 g H 33 = f h 1 f h 2 f h additional matrices Numerical simulation of population balance systems Berlin, (44)

28 A finite element projection based VMS method 4 How to choose the large scale space L H? condensation à 11 à 12 à 13 B T 1 à 21 à 22 à 23 B T 2 à 31 à 32 à 33 B T 3 B 1 B 2 B 3 0 u h 1 u h 2 u h 3 p h = f h 1 f h 2 f h 3 0 à 11 = A 11 G 11 M 1 G G 24 M 1 G G 36 M 1 G 63.. à 33 = A 33 G 36 M 1 G G 11 M 1 G G 24 M 1 G 42 goal: sparsity pattern of à αβ same like A αβ Numerical simulation of population balance systems Berlin, (44)

29 A finite element projection based VMS method 4 How to choose the large scale space L H? conditions on L H : support of each basis function of L H only one mesh cell basis of L H is L 2 orthogonal = discontinuous finite element spaces with bases of piecewise Legendre polynomials simulations found in the literature: J., Kaya (2005), J., Roland (2007), J., Kindl (2008) L H (K) = P 0 (K) for all mesh cells K L H (K) = P1 disc (K) for all mesh cells K Numerical simulation of population balance systems Berlin, (44)

30 A finite element projection based VMS method 4 How to choose the large scale space L H? conditions on L H : support of each basis function of L H only one mesh cell basis of L H is L 2 orthogonal = discontinuous finite element spaces with bases of piecewise Legendre polynomials simulations found in the literature: J., Kaya (2005), J., Roland (2007), J., Kindl (2008) L H (K) = P 0 (K) for all mesh cells K L H (K) = P1 disc (K) for all mesh cells K goal: method should determine local coarse space L H (K) a posteriori such that L H (K) is a small space where flow is strongly turbulent turbulence model has large influence L H (K) is a large space where flow is less turbulent turbulence model has little influence Numerical simulation of population balance systems Berlin, (44)

31 A finite element projection based VMS method 4 Adaptive large scale space assumption: local turbulence intensity reflected by size of local resolved small scales size of resolved small scales large = many unresolved scales can be expected size of resolved small scales small = little unresolved scales can be expected compute the deformation tensor of the large scales G H computation is not necessary for static L H additional matrices to assemble in comparison to static L H Numerical simulation of population balance systems Berlin, (44)

32 A finite element projection based VMS method 4 Adaptive large scale space assumption: local turbulence intensity reflected by size of local resolved small scales size of resolved small scales large = many unresolved scales can be expected size of resolved small scales small = little unresolved scales can be expected compute the deformation tensor of the large scales G H computation is not necessary for static L H additional matrices to assemble in comparison to static L H define indicator of the size of the resolved small scales in mesh cell K η K = GH D(u h ) L 2 (K) 1 L 2 (K) = GH D(u h ) L 2 (K) K 1/2, K T h size of the resolved small scales does not depend on size of mesh cell size of the mesh cell scales out Numerical simulation of population balance systems Berlin, (44)

33 A finite element projection based VMS method 4 Adaptive large scale space compare η K to some reference value similar to a posteriori error estimation and mesh refinement reference values mean value at current time η := time average of mean values η t := 1 no. of cells linear combination η t/2 := η + ηt 2 K T h η K 1 no. of time steps time steps η Numerical simulation of population balance systems Berlin, (44)

34 A finite element projection based VMS method 4 Adaptive large scale space local spaces (V h = Q 2 or V h = P bubble 2 ) L H (K) = 0 = P 00 (K) turbulence model influences locally all resolved scales L H (K) = P 0 (K) L H (K) = P 1 (K) L H (K) = P 2 (K) set ν T (K) = 0, locally no turbulence model Numerical simulation of population balance systems Berlin, (44)

35 A finite element projection based VMS method 4 Adaptive large scale space: procedure procedure: choose three values 0 C 1 C 2 C 3 choose a mean value η choose a frequency of updating the large scale space n update in every n update th step: compute η K and determine the local large scale space L H (K) = P2 disc (K), ν T (K) = 0 if η K C 1 η L H (K) = P1 disc (K) if C 1 η < η K C 2 η L H (K) = P 0 (K) if C 2 η < η K C 3 η L H (K) = P 00 (K) if C 3 η < η K first VMS method with adaptive large scale space Numerical simulation of population balance systems Berlin, (44)

36 A finite element projection based VMS method 4 Turbulent flow around a cylinder at Re = domain and coarse grid vortex street (iso surfaces of the velocity) statistically periodic flow Re = (mean inflow, diameter of cylinder, viscosity) Numerical simulation of population balance systems Berlin, (44)

37 A finite element projection based VMS method 4 Turbulent flow around a cylinder at Re = domain and coarse grid vortex street (iso surfaces of the velocity) statistically periodic flow Re = (mean inflow, diameter of cylinder, viscosity) Q 2 /P disc 1, no. of d.o.f.: velocity, pressure Crank Nicolson scheme with t = static Smagorinsky model with van Driest damping for ν T ν T = 0.01(2h K,min ) 2 D(u h ) F, h K,min shortest edge of K Numerical simulation of population balance systems Berlin, (44)

38 A finite element projection based VMS method 4 Turbulent flow around a cylinder at Re = characteristic values of the flow lift coefficient c l, c l temporal mean, c l,rms root mean squared drag coefficient c d Strouhal number St Numerical simulation of population balance systems Berlin, (44)

39 A finite element projection based VMS method 4 Turbulent flow around a cylinder at Re = characteristic values of the flow lift coefficient c l, c l temporal mean, c l,rms root mean squared drag coefficient c d Strouhal number St time averaged values and rms values (30 periods) C 1 C 2 C 3 mean n update c l c l,rms c d c d,rms St static large scale space VMS with L H = P VMS with L H = P1 disc large space adaptive method: results not much different η η η η t/ experimental results Numerical simulation of population balance systems Berlin, (44)

40 A finite element projection based VMS method 4 Turbulent flow around a cylinder at Re = over-prediction of c d in all simulations all other values in reference intervals or close to reference value notable difference in c l between static and adaptive large scale space good parameter choices similar to other flow problems large scale space (pictures for every 100-th time steps) Numerical simulation of population balance systems Berlin, (44)

41 A finite element projection based VMS method 4 Summary and outlook for large scale adaptive VMS method more details: J., Kindl (2010) first VMS method with adaptive large scale space size of the resolved small scales is used to determine large scale space large space adaptive VMS method is able to adapt large scale space to local intensity of the turbulence method can be extended to tetrahedral meshes and P bubble 2 /P disc 1 finite element (J., Kindl, Suciu (2009), in press) Numerical simulation of population balance systems Berlin, (44)

42 A finite element projection based VMS method 4 Summary and outlook for large scale adaptive VMS method more details: J., Kindl (2010) first VMS method with adaptive large scale space size of the resolved small scales is used to determine large scale space large space adaptive VMS method is able to adapt large scale space to local intensity of the turbulence method can be extended to tetrahedral meshes and P bubble 2 /P disc 1 finite element (J., Kindl, Suciu (2009), in press) often results with respect to time averaged references similar to method with fixed large scale space further studies of parameters of the method (C 1, C 2, C 3, n update ) necessary mathematical analysis for adaptive method not yet available Numerical simulation of population balance systems Berlin, (44)

43 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d J., Mitkova, Roland, Sundmacher, Tobiska, Voigt (2009); J., Roland (2010, in press) flow: 2d, incompressible, laminar chemical reaction: CaCl 2 + Na 2 CO 3 CaCO 3 +2NaCl PSD: one internal coordinate (diameter of particles) = 3d chemical processes: nucleation of particles growth of particles no back coupling of PSD and concentrations to flow Numerical simulation of population balance systems Berlin, (44)

44 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d dimensionless population balance system u t 1 u + (u )u + p = 0 in (0, T ] Ω Re u = 0 in [0, T ] Ω c A t DA l c c A + u c A + k R c Ac B = 0 in (0, T ] Ω u l u c B DB l c c B + u c B + k R c Ac B = 0 in (0, T ] Ω t u l u c C t DC c C + u c C Λ chem c Ac B u l +Λ nuc max 0, (c C 1) 5 f + u f + kgcc, c C csat t A CaCl 2 B Na 2 CO 3 C CaCO 3 + c C csat C, R 1 c C, d 2 d p,min pf d(d p) = 0 in (0, T ] Ω C, c C, «l u d p, f d p = 0 in (0, T ] Ω (d p,min, 1) Numerical simulation of population balance systems Berlin, (44)

45 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d discretization of Navier Stokes equations Crank Nicolson scheme (fully implicit) Galerkin FEM Q 2 /P1 disc finite element, inf sup stable discretization of convection diffusion reaction equations Crank Nicolson scheme linear FEM FCT Q 1 finite element explicit treatment of coupling terms with PSD Numerical simulation of population balance systems Berlin, (44)

46 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d discretization of Navier Stokes equations Crank Nicolson scheme (fully implicit) Galerkin FEM Q 2 /P1 disc finite element, inf sup stable discretization of convection diffusion reaction equations Crank Nicolson scheme linear FEM FCT Q 1 finite element explicit treatment of coupling terms with PSD solution of PSD equation expensive because of higher dimension What happens if cheap methods are used? studied methods: explicit Euler method with finite difference upwind stabilization implicit Euler method with finite difference upwind stabilization implicit linear FEM FCT method Numerical simulation of population balance systems Berlin, (44)

47 Calcium carbonate precipitation in 2d/3d 5 Coupling strategy discrete time t k 1. solve Navier Stokes equations independent of concentrations and PSD Numerical simulation of population balance systems Berlin, (44)

48 Calcium carbonate precipitation in 2d/3d 5 Coupling strategy discrete time t k 1. solve Navier Stokes equations independent of concentrations and PSD 2. solve equations for c CaCl2 and c Na2CO 3 use velocity field computed in step 1 independent of c CaCO3 and PSD nonlinear system, solved iteratively Numerical simulation of population balance systems Berlin, (44)

49 Calcium carbonate precipitation in 2d/3d 5 Coupling strategy discrete time t k 1. solve Navier Stokes equations independent of concentrations and PSD 2. solve equations for c CaCl2 and c Na2CO 3 use velocity field computed in step 1 independent of c CaCO3 and PSD nonlinear system, solved iteratively 3. solve equation for c CaCO3 use velocity field (step 1) and c CaCl2, c Na2CO 3 (step 2) use PSD and c CaCO3 from t k 1 in nucleation and in coupling term = linear equation Numerical simulation of population balance systems Berlin, (44)

50 Calcium carbonate precipitation in 2d/3d 5 Coupling strategy discrete time t k 1. solve Navier Stokes equations independent of concentrations and PSD 2. solve equations for c CaCl2 and c Na2CO 3 use velocity field computed in step 1 independent of c CaCO3 and PSD nonlinear system, solved iteratively 3. solve equation for c CaCO3 use velocity field (step 1) and c CaCl2, c Na2CO 3 (step 2) use PSD and c CaCO3 from t k 1 in nucleation and in coupling term = linear equation 4. solve equation for PSD f use velocity field (step 1) and c CaCO3 (step 3) Numerical simulation of population balance systems Berlin, (44)

51 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d volume fraction q 3 (t, d p ) := d 3 p f(t, d p ) e dp,max ed p,0 d3 p f(t, dp ) d( d p ) d p [m] diameter of particles, f(t, d p ) [1/m 4 ] PSD cumulative volume fraction median of volume fraction Q 3 (t, d p ) := e dp ed p,0 q 3 (t, d p ) d( d p ) d p,50 (t) := { d p : Q 3 (t, d p ) = 0.5} Numerical simulation of population balance systems Berlin, (44)

52 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d Numerical simulation of population balance systems Berlin, (44)

53 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d structured flow field Numerical simulation of population balance systems Berlin, (44)

54 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d structured flow field median of the volume fraction at the center of the outlet temporal mean values (64 intervals for internal coordinate) t = t = t = FWE UPW FDM 3.055e e e-6 BWE UPW FDM 4.030e e e-6 FEM FCT 4.246e e e-6 iso surfaces of PSD (100, 1000, 10000, particles) Numerical simulation of population balance systems Berlin, (44)

55 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d unstructured flow field Numerical simulation of population balance systems Berlin, (44)

56 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d unstructured flow field median of volume fraction at center of outlet temporal mean values (64 intervals for internal coordinate) t = t = t = FWE UPW FDM 1.125e e e-6 BWE UPW FDM 1.207e e e-6 FEM FCT 1.947e e e-5 Numerical simulation of population balance systems Berlin, (44)

57 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d academic test example at coupled 2d/3d problem: FEM FCT scheme more accurate than the other schemes conclusion: accurate method for PSD equation necessary for turbulent flows, e.g. linear FEM FCT scheme Numerical simulation of population balance systems Berlin, (44)

58 Calcium carbonate precipitation in 2d/3d 5 Calcium carbonate precipitation in 2d/3d academic test example at coupled 2d/3d problem: FEM FCT scheme more accurate than the other schemes conclusion: accurate method for PSD equation necessary for turbulent flows, e.g. linear FEM FCT scheme drawback: computing time (per time step, in seconds), unstructured flow field t = t = t = FWE UPW FDM BWE UPW FDM FEM FCT bottle neck: matrix assembling possible remedy: Group FEM method, Fletcher (1983) Numerical simulation of population balance systems Berlin, (44)

59 Calcium carbonate precipitation in 3d/4d 6 Calcium carbonate precipitation in 3d/4d setup simular to 2d/3d problem 3d/4d simulation, Re = 10000, turbulent flow turbulence model: finite element variational multiscale (VMS) method median of the volume fraction at the center of the outlet similar observations as in 2d/3d example with unstructured flow field Numerical simulation of population balance systems Berlin, (44)

60 Summary and outlook 7 Summary and outlook accurate and non oscillatory schemes necessary for discretizing all equations FEM: alternatives to FEM FCT? Discontinuous Galerkin methods? developing good schemes for simulating turbulent flow fields increasing the efficiency of the simulations parallelization of the code adaptive time stepping schemes current topics in the numerical simulation of population balance systems: precipitations in 3d/4d simulation of turbulent flows with droplets (clouds), E. Schmeyer simulation of the synthesis of urea, C. Suciu Numerical simulation of population balance systems Berlin, (44)