Modeling of Transport Phenomena in Metal Foaming

Transcription

1 Modelin of Transport Phenomena in Metal Foamin B. Chinè *1,3, M. Monno 2,3 1 Instituto Tecnolòico de Costa Rica, Cartao, Costa Rica, 2 Politecnico di Milano, Dipartimento di Meccanica, Milano, Italy, 3 Laboratorio MUSP, Macchine Utensili e Sistemi di Produzione, Piacenza, Italy *Correspondin author: ITCR, School of Materials Science and Enineerin, P.O. Box , Cartao, Costa Rica, bchine@itcr.ac.cr; bruno.chine@musp.it Abstract: Metal foams are interestin materials which include as voids in the material structure with the real possibility to modify ad hoc their physical properties. Durin the foamin process of a metal, simultaneous mass, momentum and enery transport mechanisms arise. In this work we propose a model considerin mass transfer phenomena coupled to the rowth and motion of a as bubble in the liquid metal. The diffusion of the as in the liquid is studied by applyin the Fick's law and convective transport. The equilibrium concentration at the as-liquid interface is modeled by the Sievert's law with surface tension effects included. The numerical results of the simulation show that the computational model, usin the phase field method for capturin the phase interface, can be effective. The computations simulate satisfactorily mass transfer, bubble expansion, interface movement and fluid flow. In this way other physical mechanisms of foamin could be included in a future more comprehensive model. Keywords: metal foams, multi-phase flow, transport phenomena, as diffusion. 1. Introduction Metal foams [1,2] are interestin materials with many potential applications in enineerin. Foamed metals or alloys include as voids in the material structure with the real possibility to modify ad hoc their physical properties. Therefore, a wide rane of possibilities arise in the automotive, aerospace, nautical, railway, buildin, civil enineerin and medical industries. It is possible to manufacture foamed component via the powder metallurical route, i.e. startin from a compacted mixture (precursor) of metal (or metal alloy) and blowin aent powders, such as Al and titanium hydride (Ti 2 ). Remeltin the precursor leads to in situ as evolution within the metal as the blowin aents release 2 as. With this process, closed molds can be filled with foam, and structural foam parts of complex shape can be manufactured. Shaped sandwich panels with two dense face sheets and a cellular core can also be made [1]. Durin the foamin process of a metal, simultaneous mass, momentum and enery transport mechanisms arise between different phases, the solid one (mold walls and the solidified metal), the liquid metal (Al) and the as ( 2 as bubbles and the surrounded air). Furthermore, other physical phenomena like complex interface processes, bubble motion, coexistence, coalescence and collapse of bubbles, have place. Experimental works carried out by observation techniques cannot be sometimes applied owin to the specific properties of liquid metals, since they are hot, opaque and very reactive with oxyen. ence the computational techniques could represent a useful tool to investiate these mechanisms, althouh the computational work is very challenin, because the phenomena are not independent amon them and are simultaneous. The ultimate aim of our underway researches is to simulate the expansion of a metal foam in a mold, by takin all these mechanisms into account. Followin our previous efforts to simulate the foamin process of a metal [3], we develop a model, by COMSOL Multiphysics version 4.3b, which considers mass transfer phenomena coupled to the rowth and movement of a hydroen as bubble in liquid aluminium. The purpose is to model the expansion of 2 as in metal foam, resultin from the diffusion of the as towards the bubble. In the previous work this expansion was modeled by means of a specific expansion rate [3]. The structure of the paper is the followin. The description of the physical and mathematical model is iven in Section 2, while Section 3 deals with the use of Comsol Multiphisics. Finally the computational results are presented in Section 4 and the conclusions in Section 5.

2 2. Model description and equations 2.1 Physical model We consider that under the precursor heatin, the foamin aent starts to release 2 as in the melted aluminium. The as dissolves in the aluminium and insoluble as diffuse towards existin bubbles or nuclei, which causes them to inflate. The as in the bubble is considered as an ideal as and the equilibrium concentration at the as-liquid interface is iven by the Sievert's law. Because of surface tension effects, the thermodynamical equilibrium at the as-liquid interface between the hydroen partial pressure p in the as bubble and the dissolved hydroen in the liquid aluminium is expressed by the Gibbs-Thomson equation [4,5]. Fi. 1 shows an isolated circular 2 as bubble surrounded concentrically by a shell of liquid Al. In this 2D model ravity is absent and the system is considered isothermal. The melted metal is considered as an incompressible Newtonian liquid of constant viscosity. At the boundary of the system with external radius R ext the pressure is fixed at atmospheric pressure p ATM. We define r the radial coordinate, t the time, R=R(t) the time-dependent bubble radius, p the pressure of hydroen in the bubble, σ the interfacial surface tension, V the molar volume and C the molar concentration. Followin the work of Atwood et al. [4] the hydroen concentration at the interface between the as bubble and the melt is: C V exp R( t) T * L S where is the universal ideal as constant, (1) L S the hydroen solubility in liquid aluminium and T the absolute temperature. The hydroen solubility is computed by the equation: 2760 los L 2.796, ml/100 STP (2) T iven in ml of hydroen dissolved in 100 of aluminium under standard conditions of temperature and pressure. Fiure 1. Schematic of a hydroen as bubble expandin in a 2D liquid aluminum. In Eq. 1 the exponential term accounts for the effects of curvature accordin to the Gibbs- Thompson equation [4,5]. By assumin a supersaturated aluminium melt of initial * hydroen concentration C C, hydroen will diffuse towards the bubble and the chane of the bubble radius will depend on the concentration radient of C at the interface. Therefore, by applyin a mass balance at the interface we may relate the chane of R(t) to the concentration radient of hydroen. 2.2 Mathematical model Eq. 3 ives the expression used by Atwood et al. [4] to compute the chane of R(t): C R ext R(t) Gas Liquid * dr( t) C C D dt r (3) r R( t) where C is the hydroen concentration in the C as bubble, r denotes the r R(t) concentration radient at the interface and D is the diffusion coefficient of hydroen in liquid aluminium. We consider only 2 as in the bubble and express C by the ideal as equation: y p (t) σ p L n p ATM x

3 C where the 2 pressure is: p p (4) T pl (5) R(t) with p L denotin the pressure in the liquid aluminum. Equations 1 to 5 complete the mathematical model of the mass transfer mechanism coupled to the rowth and movement of a 2 as bubble in liquid Al. The transport of the diluted 2 in aluminium is modeled by the Fick s law of diffusion and takin the convection of species into account. In the model we use the laminar equations of fluid dynamics and the phase field method in order to capture the interface as-liquid. We assume the liquid to be an incompressible Newtonian fluid and represent the compressibility of the as throuh a weaklycompressible model, valid for as flows with low Mach numbers (approximately Ma 0. 3 ). The continuity and momentum equations modeled in Comsol Multiphysics are the followin [6]: ( u) 0 (6) t u T ( u ) u [ pi ( u ( u) ) t 2 ( u) I] Fst F F 3 (7) ext The evolution of the phase field function in the fixed computational reion is iven by the Cahn illiard equation: u t 2 where ψ is called help variable, iven by: fext ( 1) (8) (9) In Eq. 8 the parameter ε is a measure of the interfacial thickness, γ is the mobility and λ represents the mixin enery density. λ and ε are related to the surface tension coefficient σ throuh the expression: 2 2 (10) 3 In the momentum transport equation, the scalar manitudes ρ and μ are the fluid density and the dynamic viscosity, respectively. On the other hand u is the fluid velocity, I is the identity tensor and the term F st accounts for the surface tension force actin at the interface between the two fluids. F st is implemented as a body force linked to the chemical potential G of the multiphase system throuh the equation: where f f F st G (11) is a defined source of free enery. The force arisin due to a specific source of free enery is represented by F ext and is computed accordin to: f F ext (12) Finally F models other body forces of the system. The transport mechanism of hydroen to the bubble is modeled by the convection diffusion equation available in the Transport of Diluted Species User Interface of COMSOL Multiphysics [7], iven by Eq. 13: C t DC C R u (13) where R is a reaction rate expression for the same species. We assume that the initial concentration of hydroen in aluminium is * C, 0 C and fix the same value C, 0 on the external boundary of the shell r R ext. In the model R is set equal to zero.

4 3. Solution with COMSOL Multiphysics To simulate a hydroen as bubble of radius 0.01 m expandin in liquid Al, we use the values of fluid properties iven in Table 1, while other parameters of the model are shown in Table 2. The hydroen concentration in the liquid on the as-liquid interface is iven by Eq. 1, while the initial hydroen concentration C, 0 in the bulk liquid Al is set in the simulations. The same value of C, 0 is assined to the shell boundary. The diffusion coefficient D is also selected and set in the numerical experiments. Because the expansion of a metal foam obtained via the powder metallurical route is of the order of a few minutes when the metal is completely melted, we use larer values for the hydroen diffusion coefficient, which normally is very small (close to 10-7 m 2 /s). The model equations are numerically solved with Comsol Multiphysics 4.3b, usin the CFD Module [6] and the Chemical Reaction Enineerin Module [7]. Eq.1 modelin the transport of hydroen in aluminium is implemented only in the shell, conversely the other equations are also solved in the as phase. The ordinary differential equation modelin the chane of the bubble radius R(t), as a function of the concentration radient of C at the as-liquid interface (Eq.3), is coupled to the former equations. To compute this equation we add the Global ODEs and DAEs User Interface in order to evaluate the time dependent radius and modify the density of 2 as in the bubble. The as density variation causes the interface to move forward. The computational domain is obtained by meshin the circular reion with nearly 7x10 4 trianle elements. To solve the resultin system of equations we have used the BDF time dependent solver and the MUMPS direct solver for the interation in space. The computational model was run in a PC with Intel Xenon CPU X5660, 6 core, 2.80 Gz, 48 GB RAM, 64bit and Windows 7 Operative System. The solution time is approximately of s for a simulated foamin process of 0.14 s. 4. Results and discussion Fi. 2 shows the hydroen concentration field in the shell at time t= 0.14 s, while the Table 1: Fluid properties used in the simulations of 2 as bubble rowth in liquid Al. Manitude Symbol Value Gas density ρ G Ideal as Liquid density ρ L 2.4x10 3 k/m 3 Gas viscosity μ G 10-3 Pa s Liquid viscosity μ L 10-1 Pa s Surface tension coefficient σ 0.9 N/m Gas bubble radius R m Liquid shell R ext 5x10-2 m radius Ambient pressure p EXT KPa Constant temperature Universal ideal constant Molar concentration of 2 in the bubble ydroen solubility in liquid aluminium ydroen concentration in the liquid on the as-liquid interface Initial hydroen concentration in the shell and on the boundary Diffusion coefficient of hydroen in aluminium T 933 K J/mol K C iven by Eq. 4 L S iven by Eq. 2 * C iven by Eq. 1 C set in the,0 simulations D set in the simulations Table 2: Model parameters used in the diffuse interface model. Manitude Symbol Value Max element size - 6x10-4 m of the mesh Interface ε 3x10-4 m thickness Mobility tunin parameter (*) χ 1 m s/k (*) This parameter determines the time scale of the Cahn- illiard diffusion and therefore overns the diffusion related time scale of the interface [6].

5 Fiure 2. Values of hydroen concentration in the shell of aluminium melt (C,0 = 15xC *, D = 10-3 m 2 /s). from the peripheral reion to the bubble. As the convection diffusion equation was solved only in the shell, the central reion is void. For the same time, Fi. 4 ives the position of the as liquid interface, from which it is possible to et the fast expansion of the bubble. This is confirmed by the plot of Fi. 5 which shows the chane of the bubble time dependent radius, iven by Eq. 3. The sinificant motion acquired by the two fluids is depicted in Fi. 6. ere it is evident the characteristic radial pattern of the expansion flow, where the manitude of the liquid velocity decreases with increasin distance from the as liquid interface. Finally, Fi. 7 ives the pressure field in the system. Fiure 3. Values of the diffusive flux manitude of hydroen in the shell of aluminium melt (C,0 = 15x C *, D = 10-3 m 2 /s). Fiure 5. Plot of the as bubble time dependent radius (C,0 = 15xC *, D = 10-3 m 2 /s). Fiure 4. Position of the 2 as-liquid aluminium interface after t = 0.14 s (C,0 = 15xC *, D = 10-3 m 2 /s). diffusive flux manitude of hydroen is iven in Fi. 3. We observe the usual concentric pattern indicatin the radial transport of the hydroen Fiure 6. Plot of velocity field and streamlines in 2 bubble and aluminium shell; position of the as-liquid interface is iven by the contours (C,0 = 15xC *, D = 10-3 m 2 /s).

6 Fiure 7. Plot of the pressure field in 2 bubble and aluminium shell after t =0.14 s (C,0 = 15xC *, D = 10-3 m 2 /s). 5. Conclusions We have proposed a computational model considerin mass transfer phenomena coupled to the rowth and motion of as bubbles in the liquid metal. The diffusion of the as in the liquid has been simulated by applyin the Fick's law and convective transport. The equilibrium concentration at the as-liquid interface has been modelled by the Sievert's law with surface tension effects included. The numerical results of the simulation has shown that the computational model, usin the phase field method for capturin the phase interface, can be effective. The computations simulate satisfactorily mass transfer, bubble expansion, interface movement and fluid flow. In this way other physical mechanisms of foamin could be included in a future more comprehensive model. situ observation and modellin, Acta Materialia, 48, , (2000). [5] Y. Sun and C. Beckermann, Phase-field modelin of bubble rowth and flow in a ele- Shaw cell, International Journal of eat and Mass Transfer, 53, (2010). [6] Comsol AB, Comsol Multiphysics-CFD Module, User s Guide, Version 4.3b, (2013). [7] Comsol AB, Comsol Multiphysics-Chemical Reaction Enineerin Module, User s Guide, Version 4.3b, (2013). 7. Acknowledements One author (B. Chinè) ratefully acknowledes the financial aid provided by the Vicerrectoria de Docencia and Vicerrectoria de Investiaciòn y Extensiòn of the Instituto Tecnolòico de Costa Rica. 6. References [1] J. Banhart, Manufacture, characterization and application of cellular metals and metal foams, Proress in Materials Science, 46, (2001). [2] J. Banhart, Liht-metal foams-istory of innovation and technoloical challenes, Advanced Enineerin Materials, 15, doi: /adem (2012). [3] B. Chinè and M. Monno, Multiphysics modelin of a metal foam, Proceedins of 2012 European Comsol Conference, Milan, (2012). [4] R.C. Atwood, S. Sridhar, W. Zhan W. and P.D. Lee, Diffusion-controlled rowth of hydroen pores in aluminium-silicon castins: in