Transcript 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 7th OpenFOAM.

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
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Introduction
Modeling physical foaming processes with OpenFOAM

Variety of applications for foamed products
 Plastics processing industry
 thermal insulation
 packaging industry
*1
 Food technology

Growing demand for suitable modeling
approaches
*2
*1 © jpdschoolofdesign.blogspot.de
*3
*2 © colourbox.com
*3 © thermo-soft.at
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Outline
 Physical Foaming
 Numerical Approach
 1D Foam Density Model
 3D FVM Model
 Simulation Examples
 Batch Foaming Process
 Continuous Foaming Process
 Summary and Outlook
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General Process of Physical Foaming
Source
Phases
Two-Phase
Mixture
Single Phase
Solution
Thermodynamic
Instability
Cellular Foam
Gas
+
Liquid
Elevated Pressure
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Low Pressure
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Simulation of Physical Foaming
Currently predominant modeling approaches:
Micro-Scale Modeling

Bubble growth on cell level

Differential equations describing
motion of bubble surface
Macro-Scale Modeling

Mostly finite-element based ALE
(Arbitrary Lagrangian-Eularian) methods

Complex models required for transient
material properties
Taliadorou E., Georgiou G. and Mitsoulis E.: Numerical simulation of
the extrusion of strongly compressible Newtonian liquids . Rheologica
Acta 47 (2008) 49-62
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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
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
Based on compressibleInterFoam

Volume-of-Fluid method to calculate
transient foam - air interface

Custom material properties models
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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
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
Based on compressibleInterFoam

Volume-of-Fluid method to calculate
transient foam - air interface

Custom material properties models
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1D Foam Density Model
 Model assumptions:
 Ideal gas law valid
 Phase interface in
thermodynamic equilibrium
 Henry‘s law valid
GL ,i 
pi
,
HW,i (i , T)
i 
 System of coupled differential equations
(mass, momentum and energy conservation)
 Equation of motion for bubble surface:
[1]
rS
 1 1  2 d 2 rB 1  rB 4
  drB  2 
rB
2σ
1


τ rr  τ θθ  dr
ρ   rB


4

3

p


p

2



g

2
4



dt
2  rS
rS
rB
r
 rB rS 
rB
  dt  
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[1] Nonnenmacher, S.: Numerische und experimentelle Untersuchungen zur Restengasung
in statischen Entgasungsapparaten. VDI-Fortschrittsberichte (2003) 3, Nr. 793
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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
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
Based on compressibleInterFoam

Volume-of-Fluid method to calculate
transient foam - air interface

Custom material properties models
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3D FVM Model
Reduction to two-phase model:
air phase
liquid phase
Surrounding air phase:
Constant material properties
gas bubbles
Pseudo-homogeneous foam phase:
Averaged material properties based
on amount of gaseous blowing agent:
air phase
foam phase
• Viscosity ηFoam
• Density ρFoam
• Thermal conductivity λFoam
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3D FVM Model
Cell density 𝛒 modeled with mixture law:
ρ = αF ρF + 1 − αF ρAir
Foam density 𝛒𝐅 modeled as a function of
mass fraction of gaseous components xG:
1
ρF =
xG 1 − xG
+
ρG
ρL
xG =
𝑚𝐺
𝑚𝐺 + 𝑚𝐿
 Blowing gas density ρg : linear pressure dependence
ρG (p) = ρG,ref ∙
p
patm
 Liquid density ρL assumed to be constant
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3D FVM Model
Link between 1D model and 3D model:
 local residence time t R :
Process Conditions
scalar variable expressing time after pressure drop
Density-time-relationship
Density model
xG
ρFoam
p
pF
time
time
tR
time
1
ρF t R , p =
xG (t R ) 1 − xG (t R )
+
ρL
ρG (p)
time
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tR [s]
1
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3D FVM Model
𝑡𝑖=0
𝑡𝑖+1 = 𝑡𝑖 + ∆𝑡


• Calculate phase fraction αF
• Solve mass balance
Transport of phase fraction:
Mass balance:
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𝜕(𝜌𝐹𝛼𝐹 )
+ 𝛻 ∙ 𝜌𝐹 𝑢𝛼𝐹 + 𝛻 ∙ 𝜌𝐹 𝑢𝑟 𝛼𝐹 1 − 𝛼𝐹
𝜕𝑡
=0
𝜕𝜌
+ 𝛻 ∙ 𝜌𝑢 = 0
𝜕𝑡
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3D FVM Model
𝑡𝑖=0
𝑡𝑖+1 = 𝑡𝑖 + ∆𝑡
• Calculate phase fraction αF
• Solve mass balance
Required number
of corrector steps
• Calculate ρFoam based on local
residence time and pressure
• Solve momentum balance

Momentum balance:
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𝜕𝜌𝑢
+ 𝛻 ∙ 𝜌𝑢 ∙ 𝑢 = −𝛻𝑝 + 𝜌𝑔 + 𝛻 ∙ 𝜇 ∙ (𝛻𝑢 + 𝛻𝑢
𝜕𝑡
𝑇
+ 𝜎𝜅𝛻𝛼
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3D FVM Model
𝑡𝑖=0
𝑡𝑖+1 = 𝑡𝑖 + ∆𝑡
• Calculate phase fraction αF
• Solve mass balance
• Calculate ρFoam based on local
residence time and pressure
• Solve momentum balance


Energy balance:
Scalar transport equation for tR:
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Required number
of corrector steps
Solve energy balance and
scalar transport equations
𝜕(𝜌𝑇)
𝜆
𝑞
+ 𝛻 ∙ 𝜌𝑢 − 𝛻 ∙
𝛻𝑇 =
𝜕𝑡
𝑐𝑝
𝑐𝑝
𝜕(𝜌𝑡𝑅 )
+ 𝛻 ∙ 𝜌𝑢 ∙ 𝑡𝑅 = 𝜌𝜙𝑝
𝜕𝑡
𝜙𝑝 = 0 𝑓𝑜𝑟 𝑝 > 𝑝𝐹
𝜙𝑝 = 1 𝑓𝑜𝑟 𝑝 ≤ 𝑝𝐹
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3D FVM Model
𝑡𝑖=0
• Calculate phase fraction αF
𝑡𝑖+1 = 𝑡𝑖 + ∆𝑡
Required number
of corrector steps
• Solve mass balance
• Calculate ρFoam based on local
residence time and pressure
Solve energy balance and
scalar transport equations
• Solve momentum balance
Convergence?
no
yes
𝑡 >= 𝑡𝑒𝑛𝑑 ?
End simulation
yes
no
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Simulation Examples
Batch foaming process
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Continuous foam extrusion
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Simulation Examples
Batch foaming process
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Continuous foam extrusion
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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
Residence time tR
Pressure signal
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 bubble radius r0
 gas fraction xG,0
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Batch Foaming Process
Foam rheology
 Shear-thinning Carreau-type model
 Model parameters dependent on foam density
A
Foam viscosity µF [Pas]
ηF = (1+Bγ)C
ρF= 850 kg/m3
ρF= 400 kg/m3
ρF= 100 kg/m3
Local shear rate 𝛾 [1/s]
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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
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Batch Foaming Process
Free foam expansion:
simulation vs. experiment
time after
pressure drop
pressure signal
foam experiment

3D-VoF-simulation

density dynamics from
bubble growth model
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Batch Foaming Process
Experiment to visualize transient flow of expanding foam
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
Flexible installation of flow obstacles

Used to verify simulation results with shearand density-dependent rheology model
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Batch Foaming Process
Example 1:
t=0s
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Centered cylinder
t = 7,2 s
t = 8,3 s
t = 8,9 s
t = 11,2 s
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Batch Foaming Process
Example 2:
t=0s
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Lateral cube
t = 7,1 s
t = 8,2 s
t = 8,8 s
t = 11,8 s
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Batch Foaming Process
Solution sensitivity with regard to different rheology models:
t = 8,8 s
Current Carreau type
viscosity model
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t = 11,8 s
Standard Newtonian
viscosity model
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Simulation Examples
Batch foaming process
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Continuous foam extrusion
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Continuous Foaming Process
Polystyrene foam extrusion
Slit die
Polystyrene
Foam
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Continuous Foaming Process

Pressure profile in slit die as boundary
condition to calculate ρF with 1D model

Additional scalar transport equation solved
for local residence time tR

Locally strongly varying foam density
Local residence time tR [s]
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Foam density 𝛒𝐅 [𝐤𝐠/𝐦𝟑 ]
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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
simulation
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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
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Thank you for your attention!
Questions?
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1D Foam Density Model

Process of bubble growth divided in two successive model phases

Assumed initial condition: gaseous components present as bubble nuclei
Phase 1: Bubble Foam
Circular bubbles, kfz-structure
Phase 2: Polyhedral Foam
Pentagon-dodecahedral bubbles
Nonnenmacher, S.: Numerische und experimentelle Untersuchungen zur Restengasung in statischen Entgasungsapparaten. VDIFortschrittsberichte (2003) 3, Nr. 793
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Continuous Foaming Process
Extended rheology model:
Plasticizing effect of dissolved components
 modeled with equivalent temperature increase
Teqv = T + C1 ∙ XCE −
2
X CE
2
X CC
C2 ∙
C4 ∙ X CC − C5 ∙
+
2
XCE + C3
1 + C6 ∙ XCE
XCE : volume fraction of dissolved
ethanol
XCC: volume fraction of dissolved
carbon dioxide
Temperature dependence of foam viscosity
 Williams-Landel-Ferry (WLF) equation
ηTeqv
ηTref
−K1 ∙ (Teqv − Tref )
= aT = exp
K 2 + Teqv − Tref
ηF =
 Modified carreau-type model:
ηF = f(γ, ρF , T, X i )
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η0 ∙ aT
1 + aT ∙
γ
γc
2
n−1
2
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