Presented at the COMSOL Conference 2010 Paris. A Model of Gas Bubble Growth by Comsol Multiphysics. Laboratorio MUSP

Transcription

1 Presented at the COMSOL Conference 2010 Paris A Model of Gas Bubble Growth by Comsol Multiphysics Bruno Chinè 1,2, Michele Monno 1,3 1, Piacenza, Italy; 2 ITCR, Cartago, Costa Rica; 3 Politecnico di Milano, Italy bruno.chine@musp.it November 17-19, 19, 2010

2 Outline of the presentation Introduction Metal foams and bubbles growth Bubble growth model Simulations by Comsol Multiphysics Results Conclusions

3 Metal foams Uniform gas-liquid mixture (gas-metal or gas-alloy) in which the volume fraction of the liquid phase is small (10-20%: wet foam, <10% dry foam) D.J. Durian (UCLA):...a random packing of bubbles... or...a most unusual form of condensed matter... solidification solidified metal foam

4 Process and bubble growth Metal and foaming agent mixing mixing of the foaming agent powder to obtain a uniform distribution in the base metal powder powder cold compaction in order to break the oxide layer covering the aluminium particle Consolidation and extrusion extrusion of the pre-compacted billet in order to obtain a precursor material whose density is close to that of the base metal

5 Process and bubble growth Shaped mould chopping of the precursor material in small pieces placing inside a sealed split mould heating to a temperature a little above the solidus temperature of the alloy foaming agent decomposition and foam formation cooling and extraction

6 Foaming process Foaming is a complex phenomena: simultaneous mass, momentum and energy transfer mechanisms several physical phenomenon on interfaces, interface motion bubble dynamics, coarsening, rupture other aspects (drainage, mould filling, geometry) difficulty for experimental measurements (foams are hot, opaque,..) On the other hand: - mechanical properties of solids foams depend on relative density ρ/ρ s, ρ = foam density, ρ s = density of the solid metal matrix - to reduce defects of pieces is need to control foaming and mould filling process

7 A bubble growth model At the beginning, simplified models may be used to study metal foaming processes. transient bubble growth in a 2D region, circular symmetry isothermal, no mass diffusion: growth is only driven by a pressure difference, surface tension effects are considered gas follows the ideal gas law pv = nrt, liquid is incompressible gas and liquid are immiscible R Ω R(t) y Liquid liquid R(t) pg p G (t) Gas gas n pext p EXT (t) n θ x

8 R R 0 eq = = p G,0 p EXT,0 p G p EXT Comsol Multiphysics: t = 0, equilibrium t = t fin, equilibrium A bubble growth model t = time R(t) y pext p EXT (t) Liquid liquid Two Phase Flow, Level Set Application Mode Weakly-Compressible R Ω R(t) pg p G (t) θ ρ + t ( ρ ) = 0 u continuity u T ρ + ρ ( u ) u = [ pi + η( u + ( u) ) t 2η ( κ DV )( u ) I] + F + ρg + F ST momentum 3 Gas gas n n x φ + t u φ = γ [ ε φ φ(1 φ) φ ] φ level set

9 A bubble growth model equations for gas density - if= 0 : - if 0 : ρ ( t) = G ρ G,0 p (1 + η G,0 L t ) t = time R(t) y Liquid liquid pext = 0 p EXT (t) [ C AR( t)] C = p G R G,0 2η L 2 0 C, exp = [ C AR exp AR( t) C A( R0 At) 0 A = ρ G (t ) 2η L ] R Ω R(t) pg p G (t) Gas gas n θ x n

10 Simulations: properties and parameters Magnitude Symbol Value Universal gas J/(mol K) constant Gas molar mass M 2 g/mol Gas density ρ G ideal gas law Liquid density ρ L 10 kg/m 3 Gas viscosity η G 10-3 Pa s Liquid viscosity η L 10-1 Pa s Gas bulk viscosity κ DV 0 Pa s Surface tension coefficient Initial bubble radius Initial bubble pressure 0 N/m N/m 10-2 m R 0 p G,0 Ambient pressure p EXT 0 Pa Constant T 933 K temperature 0.2 Pa 1.2 Pa; 2.2 Pa ρ L 2 η L 2 ρ G,0 4x10 η G 10 Magnitude Symbol Value Max element size m of the mesh Time stepping - set by the solver Relative tolerance s Absolute tolerance s Interface thickness ε 10-4 m Reinitialization γ m/s 4 mesh :10 8x10 4 DOF triangle elements Direct solver PARDISO ( Comsol Multiphysics 3.5a) 3 2 step size 10 s, solution time 10 min ( f ( t fin ))

11 Results bubble growth R ( t = 1.4s) = m = R eq R eq = p G p EXT

12 pressure p ( t = 1.4s) = L p EXT Results bubble area pg ( t = 1.4s) = 0. 83Pa R eq equilibrium

13 Results pressure and velocity contours p G, 0 = 2.2 Pa, t = 0.2s, bubble is still expanding pg,0 R0 Req = = m 2 R( t = 0.2s) m R eq 0.45Pa < p( t = 0.2s) 0.73Pa

14 Conclusions A Comsol Multiphysics model simulates bubble growth with flow in gas and liquid regions. Gas pressure drives the expansion. A weakly-compressible model, coupled to a level set equation, allows to capture the interface. The gas is ideal and surface tension effects are present. The model takes into account moderate density and viscosity differences values for the fluids, but it could represent a basis for successive realistic simulations of foam expansions. In this sense, for a future work: - accurate and larger transient at the beginning of the growth phenomena, together with a denser mesh on interface will be required - mass diffusion and heat transfer will be added to the model.

15 References J. Banhart, Manufacture, characterization and application of cellular metals and metal foams, Progress in Materials Science, 46, (2001). S. Osher and J.A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulation, Journal of Computational Physics, 79, (1988). Comsol AB, Comsol Multiphysics-Chemical Engineering Module, User s Guide, Version 3.5 a (2008). J. Bruchon, A. Fortin, M. Bousmina and K. Benmoussa, Direct 2D simulation of small gas bubble clusters: From the expansion step to the equilibrium state, International Journal for Numerical Methods in Fluids, 54, (2007).

16 Many thanks for your attention. Thanks also to the organizers of