Numerical Simulation of Physical Foaming Processes 7th OpenFOAM® Workshop June 26, 2012 Florian Gruber and Manfred Piesche Institute of Mechanical Process Engineering University of Stuttgart Good afternoon everybody and welcome to my talk. I‘m here today to present a modeling approach for the simulation of physical foaming processes Fittingly realized with OpenFOAM.

Introduction Modeling physical foaming processes with OpenFOAM Variety of applications for foamed products Plastics processing industry  thermal insulation  packaging industry Food technology Growing demand for suitable modeling approaches *1 Foamed products in general have become quite commonly used for various industrial and everyday life applications. Some examples include the plastics processing industry, where thermoplastic foams based on Polystyrene or Polyethylene are used for thermal insulation or as packagin materials. Further applications of foaming materials include for example the food or cosmetics industry. The global polymeric foams market has reached approximately 20 million metric tons by 2010, at an average annual growth rate of more than 3.5% during 2000-2010. Today the majority of foaming agents used are unreactive inert gases like, for example, nitrogen, Helium or carbon dioxide. Now with a growing market for foamed products there is also increasing demand for suitable models that can be used to gather valuable information like resulting foam shapes and foam properties without costly experiments- and I’m here to present one of those modeling approaches. *2 *1 © jpdschoolofdesign.blogspot.de *3 *2 © colourbox.com *3 © thermo-soft.at

Outline Physical Foaming Numerical Approach 1D Foam Density Model 3D FVM Model Simulation Examples Batch Foaming Process Continuous Foaming Process Summary and Outlook Now here is a short overview with the outline of my talk First I will shed some light on the general process of physical foaming Then I will present the modeling approach, consisting of a 1D Foam density model and a 3D Finite-volume-based model I will give some simulation examples, specifically a batch foaming process and a continuous foaming process and conclude my talk with a short summary

General Process of Physical Foaming Source Phases Two-Phase Mixture Single Phase Solution Thermodynamic Instability Cellular Foam Gas + Liquid Here you can see the most commonly used process of physical foaming: In contrast to foaming by chemical reaction, physical foaming processes are based on the thermodynamic state of the involved materials under different process conditions. A blowing gas is injected into a liquid matrix at elevated pressure or low temperatures and completely or partially dissolved. Ideally, the result is a nearly-homogeneous single phase solution of liquid and dissolved gaseous components. Then a thermodynamic instability is induced. The most common method is to lower the pressure, but it is also possible to increase the temperature to induce degassing. This instability leads to nucleation of small microbubbles which continue to grow, until a steady-state condition and a cellular foam is established. Elevated Pressure Low Pressure

Simulation of Physical Foaming Currently predominant modeling approaches: Micro-Scale Modeling Macro-Scale Modeling Bubble growth on cell level Differential equations describing motion of bubble surface Mostly finite-element based ALE (Arbitrary Lagrangian-Eularian) methods Complex models required for transient material properties The currently available modeling approaches for physical foaming can be roughly divided in aspects of the foaming process. There exists a number of micro-scale models that describe the dynamics of the growth of a single gas bubble in the foam matrix. Macro-Scale modeling of physical foaming processes are scarce, but can be mostly related to modeling foam extrusion processes. General approach is often usage of finite-element based arbitrary-lagrangian methods that picture the foam interface with the underlying mesh. Taliadorou E., Georgiou G. and Mitsoulis E.: Numerical simulation of the extrusion of strongly compressible Newtonian liquids . Rheologica Acta 47 (2008) 49-62

Numerical Approach 1D foam density model 3D FVM model ρFoam time Necessary to calculate temporal evolution of ρFoam Requires information on process conditions and material data Based on compressibleInterFoam Volume-of-Fluid method to calculate transient foam - air interface Custom material properties models 1D: Used to calculate temporal evolution of foam density Requires Information on process conditions like Concentration of dissolved gases Pressure dynamics Initial temperature

Numerical Approach 1D foam density model 3D FVM model ρFoam time Necessary to calculate temporal evolution of ρFoam Requires information on process conditions and material data Based on compressibleInterFoam Volume-of-Fluid method to calculate transient foam - air interface Custom material properties models Requires Information on process conditions concentration of dissolved gases pressure dynamics initial temperature

1D Foam Density Model Model assumptions: Ideal gas law valid Phase interface in thermodynamic equilibrium Henry‘s law valid System of coupled differential equations (mass, momentum and energy conservation) Equibiaxial extensional flow in a spherical shell Ideal gas law valid Henry‘s law valid Phase interface in thermodynamic equilibrium Die Einzelblase besitzt den Radius rB und ist von einer endlichen Kugelschale mit dem äußeren Radius rS umgeben. Auf dem Rand der Kugelschale herrscht der Umgebungsdruck pinf. In der Flüssigkeitsschale sind beliebig viele, flüchtige Komponenten i gelöst. Zum Zeitpunkt t=0 befindet sich ausschließlich Schleppmittel in der Blase. Pg ist der Blaseninnendruck, tg die Temperatur. Für Zeiten t>0 beginnen die flüchtigen Komponenten in die Blase zu diffundieren. Dadurch und durch die Absenkung des Umgebungsdrucks wächst die Blase und damit der Schaum an. Es werden folgende Annahmen getroffen: ... Desweiteren wird aufgrund der geringen Konzentrationen bei der Restentgasung die Gültigkeit des Henryschen Gesetzes vorausgesetzt. Die Gleichgewichtskonzentration an der Phasengrenze ist damit über den Henry-Koeffizienten HW direkt mit dem Partialdruck der flüchtigen Komponente i in der Blase verknüpft. Demnach befindet sich die Phasengrenze zu jeder Zeit im thermodynamischen Gleichgewicht. [1] Equation of motion for bubble surface: [1] Nonnenmacher, S.: Numerische und experimentelle Untersuchungen zur Restengasung in statischen Entgasungsapparaten. VDI-Fortschrittsberichte (2003) 3, Nr. 793

Numerical Approach 1D foam density model 3D FVM model ρFoam time Necessary to calculate temporal evolution of ρFoam Requires information on process conditions and material data Based on compressibleInterFoam Volume-of-Fluid method to calculate transient foam - air interface Custom material properties models Requires Information on process conditions concentration of dissolved gases pressure dynamics initial temperature

3D FVM Model Reduction to two-phase model: air phase Surrounding air phase: Constant material properties Pseudo-homogeneous foam phase: Averaged material properties based on amount of gaseous blowing agent: Viscosity ηFoam Density ρFoam Thermal conductivity λFoam air phase liquid phase gas bubbles foam phase Now even with the processing power available today, it is impossible to resolve each single gas bubble of the foam. As a result, a necessary simplification has to take place. The foam matrix composed of the liquid phase and the gas bubbles is approximated as one single phase This effectively reduces the simulation to two phases one surrounding air phase with constant properties one pseudo-homogeneous foam phase with averaged material properties depending on the volume fraction of dissolved gas. This also means that sound approaches for parameters like thermal conductivity, heat capacity or foam rheology are required

3D FVM Model Cell density 𝛒 modeled with mixture law: ρ= α F ρF+ 1−αF ρ Air Foam density 𝛒 𝐅 modeled as a function of mass fraction of gaseous components xG: ρ F = 1 x G ρ G + 1− x G ρ L x G = 𝑚 𝐺 𝑚 𝐺 + 𝑚 𝐿 The average density in each cell of the calculation domain is modeled with mixture law according to volume-fractions of pure components: We use a residence time based formulation for the calculation of the foam density. So Rho F is defined as a function of the local cell values for the residence time tr and the local pressure. The residence time is considered to be the time that went by since the foam has experienced the pressure drop leading to nucleation and subsequent bubble growth. The connection to the calculated density values from the bubble growth model is established with the amount of blowing agent that has already changed to a gaseous state. Density of blowing gas fraction: simple pressure dependence (ideal gas law), based on a reference density at atmospheric pressure: Blowing gas density ρ g : linear pressure dependence ρ G (p) = ρ G,ref ∙ p p atm Liquid density ρ L assumed to be constant

3D FVM Model Link between 1D model and 3D model:  local residence time t R : scalar variable expressing time after pressure drop Process Conditions Density model Density-time-relationship p time ρFoam time xG time pF The average density in each cell of the domain is modeled with mixture law according to volume-fractions of pure components: We use a residence time based formulation for the calculation of the foam density. So Rho F is defined as a function of the local cell values for the residence time tr and the local pressure. The residence time is considered to be the time that went by since the foam has experienced the pressure drop leading to nucleation and subsequent bubble growth. The connection to the calculated density values from the bubble growth model is established with the amount of blowing agent that has already changed to a gaseous state. Density of blowing gas fraction: simple pressure dependence (ideal gas law), based on a reference density at atmospheric pressure: tR time ρ F t R ,p = 1 x G ( t R ) ρ G (p) + 1− x G ( t R ) ρ L tR [s] 1

3D FVM Model Transport of phase fraction: 𝑡 𝑖=0 Calculate phase fraction αF Solve mass balance 𝑡 𝑖+1 = 𝑡 𝑖 +∆𝑡 Explicit Calculation of the phase fraction alphaI Transport of phase fraction: 𝜕(𝜌𝐹𝛼𝐹) 𝜕𝑡 +𝛻∙ 𝜌𝐹 𝑢 𝛼𝐹 +𝛻∙ 𝜌𝐹 𝑢 𝑟 𝛼𝐹 1−𝛼𝐹 =0 Mass balance: 𝜕𝜌 𝜕𝑡 +𝛻∙ 𝜌 𝑢 =0

Required number of corrector steps 3D FVM Model 𝑡 𝑖=0 Calculate phase fraction αF Solve mass balance 𝑡 𝑖+1 = 𝑡 𝑖 +∆𝑡 Required number of corrector steps Calculate ρFoam based on local residence time and pressure Solve momentum balance 𝜕𝜌 𝑢 𝜕𝑡 +𝛻∙ 𝜌 𝑢 ∙ 𝑢 =−𝛻𝑝+𝜌 𝑔 +𝛻∙ 𝜇∙(𝛻 𝑢 + 𝛻 𝑢 𝑇 +𝜎𝜅𝛻𝛼 Momentum balance:

Required number of corrector steps 3D FVM Model 𝑡 𝑖=0 Calculate phase fraction αF Solve mass balance 𝑡 𝑖+1 = 𝑡 𝑖 +∆𝑡 Required number of corrector steps Calculate ρFoam based on local residence time and pressure Solve momentum balance Solve energy balance and scalar transport equations Explicit Calculation of the phase fraction alphaI Energy balance: 𝜕(𝜌𝑇) 𝜕𝑡 +𝛻∙ 𝜌 𝑢 −𝛻∙ 𝜆 𝑐 𝑝 𝛻𝑇 = 𝑞 𝑐 𝑝 𝜙 𝑝 =0 𝑓𝑜𝑟 𝑝> 𝑝 𝐹 Scalar transport equation for tR: 𝜕(𝜌 𝑡 𝑅 ) 𝜕𝑡 +𝛻∙ 𝜌 𝑢 ∙ 𝑡 𝑅 =𝜌 𝜙 𝑝 𝜙 𝑝 =1 𝑓𝑜𝑟 𝑝≤ 𝑝 𝐹

Required number of corrector steps 3D FVM Model 𝑡 𝑖=0 Calculate phase fraction αF Solve mass balance 𝑡 𝑖+1 = 𝑡 𝑖 +∆𝑡 Required number of corrector steps Calculate ρFoam based on local residence time and pressure Solve momentum balance Solve energy balance and scalar transport equations 𝑡>= 𝑡 𝑒𝑛𝑑 ? End simulation Convergence? no yes yes no

Simulation Examples Batch foaming process Continuous foam extrusion

Simulation Examples Batch foaming process Continuous foam extrusion

Batch Foaming Process High-viscosity silicone oil foamed with Helium and Nitrogen Pressure chamber designed for reproducible foaming experiments Used to verify time-density-relationship from bubble growth model three dimensional foam expansion calculated with FVM model Initial conditions from image analysis bubble radius r0 gas fraction xG,0 This is an experimental setup where Anlage zum kontrollierten, druckgetriebenen physikalischen Schäumen von mit Treibmittel beaufschlagtem Silikonöl pressure chamber Known phase equilibrium silicone oil – He and silicone oil – N2 Image Analysis to accurately measure the initial conditions Bildanalyse der Silikonöl-Treibmittelmischung Standzylinderversuche zur Ermittlung des initialen Gasanteils Used material silicone oil Korasilon with a viscosity of 100 Pas Treibmittel Helium, Stickstoff als Reinstoffe und Mischung Residence time tR Pressure signal

Batch Foaming Process Foam rheology η F = A (1+B γ) C Shear-thinning Carreau-type model Model parameters dependent on foam density η F = A (1+B γ) C ρF= 850 kg/m3 Foam viscosity µF [Pas] ρF= 400 kg/m3 ρF= 100 kg/m3 Local shear rate 𝛾 [1/s]

Batch Foaming Process Simulation example: foam expansion Foam mixture: 0.62 kg Oil / 0.03 g Helium Pressure reduction: 4 bar  0.2 bar over 9.5 s

Batch Foaming Process Free foam expansion: simulation vs. experiment time after pressure drop pressure signal foam experiment So at first, I want to show some results 3D-VoF-simulation density dynamics from bubble growth model

23.04.2017 Batch Foaming Process Experiment to visualize transient flow of expanding foam Flexible installation of flow obstacles Used to verify simulation results with shear- and density-dependent rheology model In the following study, the experimental setup was slightly modified in order to visualize the transient flow of expanding foam around obstacles. Different geometric shapes were examined like this cylinder in the centre of the pressure tank, or different setups of lateral positioned cuboids. The main goal was to verify the the used rheology and density models under transient flow conditions with easily Die eingebauten Strömungshindernisse können in ihrer Form und Position variiert werden und sollen der Strömungsquerschnitt unterschiedlich verengen. Visualisierung des instationären Fließverhaltens der Schaumphase unter gegebenen Druck- und Geometrierandbedingungen Strömungshindernisse verengen den Strömungsquerschnitt flexibler Einbau (Form und Position)

Batch Foaming Process Example 1: Centered cylinder t = 0 s t = 7,2 s

Batch Foaming Process Example 2: Lateral cube t = 0 s t = 7,1 s

Batch Foaming Process Solution sensitivity with regard to different rheology models: t = 8,8 s t = 11,8 s Current Carreau type viscosity model Standard Newtonian viscosity model

Simulation Examples Batch foaming process Continuous foam extrusion

Continuous Foaming Process Polystyrene foam extrusion Slit die Now the second experimental setup I‘d like to present is the continuous extrusion of polystyrene foam. Here, polystyrene melt is loaded with a mixture of carbon dioxide and ethanol under high pressure. When the melt All experimental research was conducted with a twin screw extruder at pilot plant scale. Polystyrene Foam

Continuous Foaming Process Pressure profile in slit die as boundary condition to calculate ρ F with 1D model Additional scalar transport equation solved for local residence time tR Locally strongly varying foam density Local residence time tR [s] Foam density 𝛒 𝐅 [𝐤𝐠/ 𝐦 𝟑 ] When modeling a continuous foaming process, the local foam density can vary greatly depending on the age of the foam with respect to the initial pressure drop which induces foaming. As boundary condition for the calculation of the foam density, knowledge of the dynamics of the pressure drop occuring in the slit die is required. The local residence time of the foam

Continuous Foaming Process Simulation results: Polystyrene foam extrusion Current model suitable for foam extrusion processes with dynamic change in density Realistic foam shape with adequate assumptions regarding material data experiment Solution is highly dependent on the underlying density and rheology model. So, with the right assumptions regarding initial and process conditions and material data, it is possible to generate realistic foam shapes. simulation

Summary and Outlook Simulation model for physical foaming processes Combination of 1D and 3D models to evaluate transient foam growth Experimentally verified solutions for given pressure conditions Stable solution for processes with dynamic decrease in foam density Work in progress Two-phase viscoelastic rheology model Further enhanced thermal model Successful combination of 1-D and 3-D models suited for simulation of pressure and diffusion driven foaming processes Now the next sentence is true for every simulation, but I feel that the highly dynamic nature of foaming processes and the associated changes in material properties require even more attention regarding the underlying material data The simulation is suitable for foaming processes with highly dynamic density changes, which typically pose very difficult for ALE methods as the mesh gets distorted very much in short timespans. This can often lead to stability problems or requires time-consuming re-meshing of the simulation domain Still work in progress is

Thank you for your attention! Questions? This concludes my speech, thank you very much for your attention. I‘ll be happy to answer your questions

1D Foam Density Model Phase 1: Bubble Foam Phase 2: Polyhedral Foam Process of bubble growth divided in two successive model phases Assumed initial condition: gaseous components present as bubble nuclei Phase 1: Bubble Foam Phase 2: Polyhedral Foam As adequate means to calculate the density development of a closed cell foam in response to an occurring pressure drop, a 1D foam density model is used. The process of bubble growth divided in two consecutive phases. As starting condition, it is assumed that a certain fraction of the gaseous components are present as bubble nuclei During the first At a Grundlage des Simulationsmodells ist die Unterteilung der strömungsmechanischen Vorgänge in zwei zeitlich aufeinanderfolgende Modellphasen: In der 1. Modellphase des Kugelschaums sind die Blasen monodispers, kugelförmig in einer regelmäßigen kfz-Struktur angeordnet. Ausgangspunkt des Schaumwachstums sind dabei dispergierte Schleppmittelblasen. Mit zunehmendem Schaumvolumen beginnen sich beim Schaumwachstum (infolge von Wechselwirkungen zwischen den Blasen) polyederförmige Blasenformen auszubilden. In der zweiten Modellphase wird daher postuliert, daß die Blasen die idealisierte Form von Pentagon-Dodekaederen besitzen. Mit der Blasenform verändert sich dabei die Verteilung der Flüssigkeit zwischen und um die Blasen. Für beide Wachstumsphasen gilt, daß der Schaum geschlossenzellig ist. Circular bubbles, kfz-structure Pentagon-dodecahedral bubbles Nonnenmacher, S.: Numerische und experimentelle Untersuchungen zur Restengasung in statischen Entgasungsapparaten. VDI-Fortschrittsberichte (2003) 3, Nr. 793

Continuous Foaming Process Extended rheology model: Plasticizing effect of dissolved components  modeled with equivalent temperature increase XCE : volume fraction of dissolved ethanol XCC: volume fraction of dissolved carbon dioxide T eqv =T+ C 1 ∙ X CE − C 2 ∙ X CE 2 X CE + C 3 + C 4 ∙ X CC − C 5 ∙ X CC 2 1+ C 6 ∙ X CE 2 Temperature dependence of foam viscosity  Williams-Landel-Ferry (WLF) equation η T eqv η T ref = a T =exp⁡ − K 1 ∙( T eqv − T ref ) K 2 + T eqv − T ref Modeling of polymer foam requires additional modifications regarding the used rheology model η F = η 0 ∙ a T 1+ a T ∙ γ γ c 2 n−1 2 Modified carreau-type model: η F =f( γ , ρ F , T, X i )