Predicting Size Distribution of Dispersed Phase Systems
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1 Predicting Size Distribution of Dispersed Phase Systems Evren Bayraktar Supervisors: Prof. S.Turek (Dortmund), Dr. Ing. Otto Mierka, Dr. Ing. Frank Platte , Sulzer Page 1
2 Outline Introduction Theory Coupling PBE-CFD Diffusive Transport Modeling of Coalescence Phenomena Modeling of Breakage Phenomena Discretization of internal (size) coordinate Results & Discussion Simple Pipe Problem Complex Geometry Problem : Static Mixer High Quality CFD Data Future Plans Page 2
3 Introduction Evolution of the size distribution of the secondary phase IN Time Spatial coordinate Internal coordinate WITH Hydrodynamic quantities Physical qualities Page 3
4 Introduction The effort results in an inevitable coupling of CFD and PBE. No analytical solution for practical problems! No commercial code sufficiently good for every case! Accurate CFD calculations for complex geometries! High computational costs! Lack of unified framework of breakage kernel! Robust, fast and accurate prediction of mixing properties! Page 4
5 Theory: C-D Problem A scalar non-linear time dependent non-homogenous convection-diffusion problem. Non-homogeneity: Source and Sink terms due to Coalescence Breakage Robustness and fastness depends on the numerical schemes. Accuracy depends on modeling of source and sink terms. Page 5
6 Theory: PBEs PBEs are introduced into C-D problem An integro-differential equation which has to be treated very carefully! which has to be closed with proper kernels to obtain accurate results. which has to be discretized with proper techniques to obtain robust and fast solvers. Page 6
7 Theory: Kernels Coalescence kernel Breakage kernel Most of the models shows the similar trends. Equlibrium BSD is independent of choosen model. (Buwa&Ranade, 2002) Page 7
8 Theory: Breakage kernel Luo&Svendsen, 1996 Lehr et. al., 2001 Bazan et. al., 1999 Theoretical model: Lehr et. al., 2001, NO emprical parameters Phenomenological model: Martinez-Bazan et. al., 1999 Kg = 0.25 β = 8.2; by Batchelor,1956. Page 8
9 Theory: Breakage kernels Page 9
10 Theory: Tsouris & Tavlarides Geometric parameter k 8 : ratio of impeller volume to tank volume. Empirical parameter C 1 : found by LSF to measured data. The breakage frequency is too much problem specific! Page 10
11 Theory: Tsouris & Tavlarides Є min : energy required to create minimum and largest size bubble. Є max : energy required to create two equal sized bubble. Є d : surface energy of a bubble with size d. Unequal breakage is preferred. The formulation is general. Page 11
12 Theory: Lehr et al., 2001 A theoretical kernel which does not contain any empirical parameter. NOT problem specific. DSD function can represent extremes. Page 12
13 Theory: Lehr et al., 2002 Work with non-dimesional parameters for wide range of operating conditions. Charecteristic time, length are introduced. Page 13
14 Theory: Lehr et al., 2002 α = 0.13 ε = m 2 s 3 A: α = 0.20 ε = m 2 s 3 B: α = 0.08 ε = m 2 s 3 Page 14
15 Theory: Martinez et al., 1999 Breakup frequency (s -1 ) f*(d*) Ref 4 mm ε = 10 m 2 s -3 d ε = 4100 mmm 2 s -3 Ref 3 mm ε = 1000 m 2 s -3 d = 1 mm Ref 1 mm d = 0.5 mm Ref 50 mm Piecewise Breakup Daughter Frequency Constant Size Distribution Approximation and Velocity A phenomenological model with two emprical parameter. NOT problem specific. DSD function is I-type, unequal breakage is preferable. 1 Breakup velocity, u b (m s - 1 ) d d* d* (m) x 10-3 Page 15
16 Theory: Size Discretization To solve the integro-differential PBE for practical applications, the integrals should be discretized with a proper discretization technique. Method of Moments (MOM), McGraw and Saunders, 1984 Quadrature Method of Moments (QMOM), Marchsio et al., 2003 Direct Quadrature Method of Moments (DQMOM), Marchsio et al., 2005 Parallel Parent Daughter Classes (PPDC), Bove et al., 2005 Method of Classes (MC), Kumar and Ramkrishna, 1996 Cell Averaged Technique (CAT), Kostoglou, 2007 IF THE DISCRETIZATION IS FINE ENOUGH, THE RESULT SHOULD BE INDEPENDENT OF THE CHOSEN METHOD!? Page 16
17 Theory: D/Q/MOM All is based on formulating a tracer equation for moments or abscissas and weights. MOM: the key is to formulate the problem in terms of the moments in closed forms. MOM can be used when the source and sink terms directly computed. Which is possible only for growth terms. MOM is extended to QMOM for breakage and coalescence. DQMOM avoids the PD algorithm which is used in QMOM and tracks the abscissas and weights. DQMOM QMOM MOM Page 17
18 Theory: D/Q/MOM Reformulation of the problem in terms of moments Page 18
19 Theory: D/Q/MOM Page 19
20 Theory: MC & PPDC MC formulation of the problem yields, Page 20
21 Theory: MC & PPDC What s a good value for discretization constant q? Lehr et al., Page 21
22 Theory: MC & PPDC Generally, initial condition of our problems is a dirac function or superposition of few. Bove et al., 2005 Page 22
23 Theory: MC & PPDC The key is product difference algorithm. (Gordon, 1968; Bove et al., 2005) i. BSD Construction with M parent of matrix classes P. at t n. ii. Compute Generating the a new vector expanded α with entries BSD at of tp. n+1. iii. Calculate Assembling the a first symmetric 2M moments tridiagonal of the matrix new expanded J with entries BSD. of α. iv. Obtain Eigenvalues the reduced of J are BSD the abscissas with M classes of reduced by using PSD PD and algorithm. the number v. Repeat densities this can until be tfound n+1 Tby the first entries of corresponding eigenvectors. max. Page 23
24 Theory: MC & PPDC N 0 = 1, C = 1, v 0 = 1 N 0 = 1 & C = 1 Page 24
25 Theory: MC & PPDC Comparison of MC and PPDC for the following case: α = 0.18, ε = , water-air mixture x # relative density iad Error (m %Err % -1 (m ) -3 ) d43 d32 (m) (m) ppdc2 ppdc3 ppdc4 Reference Volume Interfacial Sauter Reference Number Average Mean solution: Area Density solution: Density Diameter Method vs PPDC5 Time vs of vs Time Classes Time MC MC PPDC2 PPDC2 PPDC3 PPDC3 PPDC4 PPDC4 MC MC PPDC2 PPDC2 PPDC3 PPDC3 PPDC2 PPDC4 PPDC4 PPDC3 PPDC fractional 2 time order moment 2.5 (s) moment order time (s) Page 25
26 Theory: PPDC with LMM & MB Simulation Convergence of coalescence of different models and breakage with 2,3, dominant and 4 Parent cases. classes. # density (m-3) d 32 (m) x Sauter Number Mean LMM Density vs Diameter MB vs d 32 Time vs Time LMM4C LMM2 MB4C LMM3 LMM4B MB4B MB3 MB2 MB LMM2 2 LMM3 LMM4 MB2 LMM4C 1 MB3 MB4C MB4 LMM4B MB4B time (s) Page 26
27 Theory: CFD-PBE Non-stationary two way coupled, CFD - PBE. X X X X Non-stationary one way coupled, CFD - PBE. Stationary one way coupled, CFD - PBE. X X Page 27
28 Theory: PPDC and Convection Suggested by Bove (Bove, 2005) at t = t n for a known f(d,x,t) with m parent class, i. Calculate the source and sink terms due to coalescence and breakage at every x i according to PPDC. ii. With these calculated terms obtain the source and sink terms for first 2m order of moments. iii. Convect the first 2m moments with these source and sink terms, solve 2m C-D problem. iv. Obtain the reduced PSD by product difference algorithm at every x i which is f(d,x,t n+1 ). Page 28
29 Results: MC Page 29
30 Results: PPDC number densities 1.10E9 2.00E8 5.40E6 # density E8 1.50E8 4.21E6 5.70E8 1.01E8 3.02E6 # density E8 5.10E7 1.84E6 # density E7 1.40E6 6.50E5 Page 30
31 Results: PPDC abcis abcis abcis abcis mm 4.10 Page 31
32 Results: PPDC moments 1.15E9 6.80E5 1.95E-2 0 th moment 9.36E8 5.42E5 1.85E E8 5.42E5 1.75E-2 1 st moment 4.15E8 2.68E5 1.65E-2 4 th moment 9.80E8 1.30E5 1.55E-2 Page 32
33 Results: PPDC vs MC d 32 PPDC d 32 MC mm Page 33
34 Results: Static Mixer Page 34
35 Results: Static Mixer xy plane at z= m, distribution just after the mixer. yz plane is located at x = m, midplane in x. d 32 = 10.55E-6 m Page 35
36 Results: Static Mixer , Your Name Sulzer Dortmund xy plane at z= m. d32 =10.63E-6 m. Page 36
37 Results: Static Mixer xy plane at z= m, outlet. d32 = 10.76E-6 m iad (1/m) α α d (m) d (m) iad (1/m) Void Fraction vs Diameter iad vs Diameter d (m) Page 37
38 Results: Static Mixer d 32 distribution 10.5E-6< d 32 <11.2E-6, d 32 = 10.8E-6 (all in meters) Dispersed phase is mixed homogenously in spatial space. Nevertheless, previous graphs has shown that mixing has a bimodal distribution. Page 38
39 Results: Static Mixer x sauter mean diameter d 32 (m) time(sec) Changes in the cutplanes are too small. After the first flow through ( sec), d32 = 10.8E-6 m. Page 39
40 Results: Static Mixer Calculations for three different initial condition. d 32 = 760E-6 m in equilibrium. d 32 = 10.8E-6 m in equilibrium (recent). Page 40
41 Results: Static Mixer eps (m 2 s -3 ) eps (m 2 s -3 ) d E-6 m d E-6 m Page 41
42 Discussion: Mesh Deformation (2D) Page 42
43 Discussion: Fictitious Boundary Page 43
44 Plans PLANS NEAR FUTURE FUTURE High Quality CFD Multifluid models PBE_LMM_MC/PPDC Parallelization PBE_MB_MC/PPDC Hybrid Models Page 44
45 Page 45
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