A Two-Fluid/DQMOM Methodology For Condensation In Bubbly Flow

Transcription

1 A Two-Fluid/DQMOM Methodology For Condensation In Bubbly Flow Gothenburg Region OpenFOAM User Group Meeting, November 11, 2015 Klas Jareteg Chalmers University of Technology Division of Nuclear Engineering, Department of Applied Physics Gothenburg, Sweden

2 Contact: Klas Jareteg Webpage: Linkedin: Blog: (Coming soon!) Slides and some code and cases: GitHub: Disclaimer: DISCLAIMER: This offering is not approved or endorsed by OpenCFD Limited, the producer of the OpenFOAM software and owner of the OPENFOAM and OpenCFD trade marks Following the trademark policy DISCLAIMER: The ideas and code in this presentation and all appended files are distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE 2

3 Background and scope Background: Water used as a coolant for Light Water Reactors (LWRs) Coolant properties important for multiphysics aspects Bubbly flows encountered for low vapor fractions due to boiling Existing code base in foam-extend for LWR multiphysics of single phase flow Aim: Formulate and implement a two fluid framework combined with a population balance equation (PBE) to simulate subcooled boiling 3

4 Background and scope Background: Water used as a coolant for Light Water Reactors (LWRs) Coolant properties important for multiphysics aspects Bubbly flows encountered for low vapor fractions due to boiling Existing code base in foam-extend for LWR multiphysics of single phase flow Aim: Formulate and implement a two fluid framework combined with a population balance equation (PBE) to simulate subcooled boiling 3

5 Model Model PBE: DQMOM/MUSIG Results Void and size distributions Computational efficiency 4

6 Population Balance Method Problem: Two fluid methodology treating vapor and liquid phases as interpenetrating continua Potential remedy: PBM Retrieve bubble size distribution Size-dependent formulations of particle forces and condensation Number density Bubble size Population balance equation: ( ) n(x, r, t) x(x, r, t) + x n(x, r, t) t t + r (U(x, r, t)n(x, r, t)) = S(x, r, t) 5

7 Population Balance Method Problem: Two fluid methodology treating vapor and liquid phases as interpenetrating continua Potential remedy: PBM Retrieve bubble size distribution Size-dependent formulations of particle forces and condensation Number density Bubble size Population balance equation: ( ) n(x, r, t) x(x, r, t) + x n(x, r, t) t t + r (U(x, r, t)n(x, r, t)) = S(x, r, t) 5

8 Population Balance Method Problem: Two fluid methodology treating vapor and liquid phases as interpenetrating continua Potential remedy: PBM Retrieve bubble size distribution Size-dependent formulations of particle forces and condensation Number density Bubble size Population balance equation: ( ) n(x, r, t) x(x, r, t) + x n(x, r, t) t t + r (U(x, r, t)n(x, r, t)) = S(x, r, t) 5

9 MUSIG Static classes Fixed classes, a priori determined Discrete size intervals, limiting the resolution, and no continuous size change captured Condensation occuring between the classes Number density Bubble size Transport equation for size j: α g,j ρ g t + (α g,j jρ g U g ) = S j Implementation in foam-extend-31 Each bubble size represented by a volscalarfield Point list with field iterated until convergence for each time step Kernels and condensation computed as sums of contributions 6

10 MUSIG Static classes Fixed classes, a priori determined Discrete size intervals, limiting the resolution, and no continuous size change captured Condensation occuring between the classes Number density Bubble size Transport equation for size j: α g,j ρ g t + (α g,j jρ g U g ) = S j Implementation in foam-extend-31 Each bubble size represented by a volscalarfield Point list with field iterated until convergence for each time step Kernels and condensation computed as sums of contributions 6

11 DQMOM Dynamic sizes Dynamic bubbles sizes, based on a quadrature approximation n(ξ; x, t) N w j (x, t)δ(ξ ξ j (x, t)) i=1 Sizes and weights solved for using transport equations Number density Bubble size Proposed condensation as: ξ(r, t) t = C(ξ, r, t) Implementation in foam-extend-31 Each abscissa and weight represented by a volscalarfield Kernels and condensation computed as sums of contributions Additional algorithmic details required to achieve stable systems 7

12 DQMOM Dynamic sizes Dynamic bubbles sizes, based on a quadrature approximation n(ξ; x, t) N w j (x, t)δ(ξ ξ j (x, t)) i=1 Sizes and weights solved for using transport equations Number density Bubble size Proposed condensation as: ξ(r, t) t = C(ξ, r, t) Implementation in foam-extend-31 Each abscissa and weight represented by a volscalarfield Kernels and condensation computed as sums of contributions Additional algorithmic details required to achieve stable systems 7

13 DQMOM Dynamic sizes Dynamic bubbles sizes, based on a quadrature approximation n(ξ; x, t) N w j (x, t)δ(ξ ξ j (x, t)) i=1 Sizes and weights solved for using transport equations Number density Bubble size Proposed condensation as: ξ(r, t) t = C(ξ, r, t) Implementation in foam-extend-31 Each abscissa and weight represented by a volscalarfield Kernels and condensation computed as sums of contributions Additional algorithmic details required to achieve stable systems 7

14 Remarks DQMOM vs MUSIG: Continuous condensation of bubbles allowed in DQMOM Arbitrarily small sizes in DQMOM Both methodologies requiring an iterative approach Coupling to two fluid equations: Continuity equations not solved in two fluid solver Vapor fraction distribution computed by DQMOM/MUSIG Pressure equation consistency with DQMOM/MUSIG Remarks on code structure: Code structured as classes with inheritance eg: pbemethod inherited by DQMOM and MUSIG Also twophasesolver implemented as a class hierarchy (in parts based on twophaseeulerfoam Inheritence increase development speed: new variation of one of the solver inherits the previous and overrides the base class Small additional cost from vptr due to rather large methods 8

15 Remarks DQMOM vs MUSIG: Continuous condensation of bubbles allowed in DQMOM Arbitrarily small sizes in DQMOM Both methodologies requiring an iterative approach Coupling to two fluid equations: Continuity equations not solved in two fluid solver Vapor fraction distribution computed by DQMOM/MUSIG Pressure equation consistency with DQMOM/MUSIG Remarks on code structure: Code structured as classes with inheritance eg: pbemethod inherited by DQMOM and MUSIG Also twophasesolver implemented as a class hierarchy (in parts based on twophaseeulerfoam Inheritence increase development speed: new variation of one of the solver inherits the previous and overrides the base class Small additional cost from vptr due to rather large methods 8

16 Remarks DQMOM vs MUSIG: Continuous condensation of bubbles allowed in DQMOM Arbitrarily small sizes in DQMOM Both methodologies requiring an iterative approach Coupling to two fluid equations: Continuity equations not solved in two fluid solver Vapor fraction distribution computed by DQMOM/MUSIG Pressure equation consistency with DQMOM/MUSIG Remarks on code structure: Code structured as classes with inheritance eg: pbemethod inherited by DQMOM and MUSIG Also twophasesolver implemented as a class hierarchy (in parts based on twophaseeulerfoam Inheritence increase development speed: new variation of one of the solver inherits the previous and overrides the base class Small additional cost from vptr due to rather large methods 8

17 Results Model PBE: DQMOM/MUSIG Results Void and size distributions Computational efficiency 9

18 System description and boundary conditions Wall Outlet Symmetry Geometry: Domain size: 50 cm 10 cm Mesh size: cells Inlet conditions: U g,max = (0, 0, 03) U l,max = (0, 0, 01) Wall conditions: U g : slip U l : noslip Inlet 10

19 Example: Inlet vapor bubbles I Axial void fraction at symmetry line Subcooled liquid (3 K, 01 MPa) water with vapor bubbles Void DQMOM, 2 DQMOM, 3 DQMOM, 4 DQMOM, 5 MUSIG, Axial position Bubble mean diameter 7 mm (normal distribution in size) Rapid decrease in void fraction 11

20 Example: Inlet vapor bubbles II Sauter mean diameter Convergence in abscissas, differeing first at z=01 m 10 2 Bubble size shrinkage dampened at low void fraction MUSIG decreasing notably slower ds [m] DQMOM, 2 DQMOM, 3 DQMOM, 4 DQMOM, 5 MUSIG, Axial position [m]

21 Example: Inlet vapor bubbles III 10 3 Particle size distribution (z=0025) 10 2 Weight Size [m] 0010 MUSIG,

22 Example: Inlet vapor bubbles IV 10 3 Particle size distribution (z=0025) 10 2 Weight Size [m] 0010 MUSIG, 30 DQMOM,

23 Example: Inlet vapor bubbles V 10 3 Particle size distribution (z=0025) 10 2 Weight Size [m] 0010 MUSIG, 30 DQMOM, 2 DQMOM,

24 Example: Inlet vapor bubbles VI 10 3 Particle size distribution (z=005m) 10 2 Weight Size [m] 0010 MUSIG, 30 DQMOM, 2 DQMOM,

25 Example: Inlet vapor bubbles VII Particle size distribution (z=01m) Weight Size [m] 0010 MUSIG, 30 DQMOM, 2 DQMOM,

26 MUSIG vs DQMOM: Computational time Solver N Time [au] DQMOM MUSIG MUSIG convergence slower for more classes (iterations) Computational cost primarily associated with cell wise computation of condensation (and optionally aggregation/breakage) 18

27 Summary and outlook Two fluid solver combined with DQMOM for subcooled boiling DQMOM allowing dynamic sizes for bubbles, ability to capture continuous bubble shrinkage DQMOM shown to be computationally less expensive than MUSIG for comparable precision 19

28 20

29 Extra material 21

30 Case 2: Wall boiling I 10 1 Horizontal void fraction, mid-elevation Void DQMOM, 2 DQMOM, 3 DQMOM, 4 DQMOM, 5 MUSIG, Horizontal position 0020 Vapor bubbles introduced at the wall, parabolic profile in α Void fraction increases away form the wall DQMOM predicting lower void fraction approach the centre

31 Case 2: Wall boiling II Horizontal Sauter mean diameter, mid-elevation Faster decrease in bubble size for DQMOM 10 2 Convergence in DQMOM for increasing N ds[m] DQMOM, 2 DQMOM, 3 DQMOM, 4 DQMOM, 5 MUSIG, Horizontal position