1. 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.

2. 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
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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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:

15. 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

16. 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

17. Simulation Examples
Batch foaming process
Continuous foam extrusion

18. Simulation Examples
Batch foaming process
Continuous foam extrusion

19. 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

20. 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]

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

22. 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.
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)

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

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

26. 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

27. Simulation Examples
Batch foaming process
Continuous foam extrusion

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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 C 4 ∙ X CC − C 5 ∙ X CC 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 a T ∙ γ γ c n−1 2
Modified carreau-type model:
η F =f( γ , ρ F , T, X i )