Numerical simulation of polyurethane foaming processes on bubble scale

Transcription

1 Numerical simulation of polyurethane foaming processes on bubble scale 7th OpenFOAM Workshop Darmstadt, June 2012 Stephanie Geier and Manfred Piesche Institute of Mechanical Process Engineering University of Stuttgart

2 Outline Motivation Foaming process Modeling approach Phase change modeling Bubble-bubble interaction Examples Conclusion and outlook Stephanie Geier 2

3 Motivation Polyurethane foaming process Mixing of polyol and isocyanate Foaming and mold filling due to reaction progress Local foam structure? Foam properties, e.g. thermal conductivity and impact strength depend on local foam structure Stephanie Geier 3

4 Foaming process Gelling reaction Increasing viscosity (urethane links) Blowing reaction Increasing viscosity (urea links) Density reduction - chemical blowing of the foam (CO 2 ) Evaporation of physical blowing agent Density reduction physical blowing of the foam (e.g. pentanes) Stephanie Geier 4

5 Modeling approach Assumptions and simplifications Foam is a two-phase system Gas bubbles Liquid reacting polymer phase Isothermal Gas and liquid phase are incompressible Constant viscosities Numerical approach Volume-of-fluid (VOF) based on solver interphasechangefoam Stephanie Geier 5

6 Modeling approach- Governing equations Continuity equation u = m g 1 ρ g 1 ρ l (1) Momentum balance ρu t + ρu u = p + ρg + μ u + ςκ α + f (2) Volume fraction balance α t + uα + u r α 1 α = m g ρ l (3) source terms describing phase change additional body forces Stephanie Geier 6

7 Modeling approach Phase change Phenomenological approach Density evolution known from mold filling simulations or experiments ρ foam ρ foam m g t 1 t 1 +Δt t Volumetric gas creation rate accounting for phase change m g = α m g V int Δt (4) V int - total volume of liquid in phase interface cells Stephanie Geier 7

8 Modeling approach Bubble-bubble interaction Repulsive forces between neighboring bubbles expressed through disjoining pressure π [1] π = k π d max d d < d max (5) 0 d d max Conversion to body force f π = π α (6) π π [1] C. Körner et al.: Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming. J. Stat. Phys., 121 (2005), Stephanie Geier 8

9 Modeling approach Determination of disjoining pressure α [1] marker [1] Volume fraction field Bubble marker field π [N/m²] Phase interface region (blue) and region of disjoining pressure (red) Stephanie Geier Disjoining pressure field in phase interface region 9

10 Examples Rising bubble Effect of disjoining pressure implementation No disjoining pressure t = 0 s t = 2,5 s t = 5 s t = 6,25 s t = 6,75 s t = 7 s t = 7,25 s t = 10 s Stephanie Geier 10

11 Examples Rising bubble Effect of disjoining pressure implementation Disjoining pressure included t = 0 s t = 2,5 s t = 5 s t = 6,25 s t = 6,75 s t = 7 s t = 7,25 s t = 10 s Stephanie Geier 11

12 Examples Bubbles in confined geometry ρ foam [kg/m³] Boundary and initial conditions: Solid walls: base and sides repulsive forces 53 bubbles randomly distributed Initial bubble diameter: 16 μm Foam density (from experiments) ,5 5 7,5 1012,51517,5 t [s] ρ g = 2 kg/m³ ρ l = 1100 kg/m³ ρ foam,init = 1095 kg/m³ Stephanie Geier 12

13 Examples Bubbles in confined geometry ρ foam [kg/m³] Bubbles growing in a confined geometry t = 0 s t = 2 s t = 4 s t = 6 s t = 8 s t = 10 s t = 12 s t = 14,25 s Stephanie Geier ,5 5 7, , ,5 t [s] experiment simulation 13

14 Examples Bubbles in confined geometry Deforming and rearranging bubbles t = 9 s t = 9,5 s t = 10 s t = 12 s t = 12,75 s t = 13,5 s t = 14,25 s Stephanie Geier 14

15 Conclusion and outlook Model for polyurethane foaming processes on bubble scale Phenomenological phase change model Bubble-bubble interaction u Work in progress: Foams with lower density Extension to time-varying polymer viscosity appropriate boundary conditions accounting for varying flow conditions during foaming process u Stephanie Geier 15

16 Numerical simulation of polyurethane foaming processes on bubble scale 7th OpenFOAM Workshop Darmstadt, June 2012 Stephanie Geier and Manfred Piesche Institute of Mechanical Process Engineering University of Stuttgart Thank you very much for your attention.