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EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Migration Modelling for Multilayer
Materials
Migration Modelling for Multilayer
Materials
Peter MerceaFABESFABES GmbH - Schragenhofstraße 35, 80992 Munich / Germany
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Topics :
�Basic aspects of modelling migration from multilayer materials
�Finite Difference (FD) Method to solve the Mass Transport Equation
�Performances and Validation of the FD NUMERIC Algorithm
�Input Menus and Results computed by MIGRATEST©EXP
�Case Examples: - „Invisible set-off“
- Functional barrier
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
CFtCCFtFt
Layer 1
Food
Layer 2
Layer 3
Multi-Layer Packaging
To be compared with the legal migration limits !
Migration from a Multi-layer Packging into a Food
MigrantCpo (mg/kg)
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Basic Aspects of Migration from Multi-Layer Materials (MLM)
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
When k >2 No analytical solution of the Mass
Transport Equation
To solve Fick‘s Equation one can use:
Numerical methods
Variational methods
Monte-Carlo methods
Finite Element Analysis (FEA)
Finite Volume Method (FVM)
Boundary Element Method (BEM)
Finite Difference Method (FDM)
SFPP3-INRA
2
2
x
cD
t
c kk
k
∂
∂=
∂
∂Fick‘s
2nd Equation
Mass Transport Equation in a Multi-Layer in contact with a Food1.
Basic Aspects of Migration from Multi-Layer Materials (MLM)
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Discretisation scheme
FD Discretisation scheme
only the boundary is discretised
Numerical Methods to Solve a Differential Equation over a Domain
Finite Element Method, FEM
Finite Volume Method, FVM
Boundary Element Method, BEM
Finite Difference Method, FD
SFPP3-INRA
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Constant mesh:
each Layer divided in equally thick slabs
hi = dpi /M
M=10
Finite Differences Method to Calculate Migration from Multi-Layers in Food
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Adaptive mesh:
each Layer divided in M slabs of varying thickness.Higher mesh density close to the boundaries.M=22
Finite Differences Method to Calculate Migration from Multi-Layers in Food
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
V.Tosa, K. Kovacs, P. Mercea and O. Piringer,
2008 AIP Conference Proceedings, 1048, 802.
- Write equations to approximate the derivatives of the
diffusion equation, at each mesh point, with appropriate
discrete formulas.
- Solve the equations and calculate the migrant flux.
2
2
x
cD
t
c kk
k
∂
∂=
∂
∂
Finite Differences Method to Calculate Migration from Multi-Layers in Food
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Expert Software for the Calculation of Diffusion/Migration from a Multilayer Material into a Contact Medium
Dr. Valer Tosa © 2006 updated 2008, 2009 & 2010
One-dimension - Plane Geometry
FD Method - Relatively simple and suited to quantify the migration
from MLM’s
Finite Differences Method to Calculate Migration from Multi-Layers in Food
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
- free choice of diffusion coefficients Dpi in the MLM and DF in the Contact Medium
- partitioning, Kpi,pk at each interface i,k of the MLM – Contact Medium system
ρiMLM made of 2 to 10 homogenous Layers = const.
- constant diffusion coefficients in each layer Dpi = const.
Solving the Diffusion/Migration Equations for:
liquid, viscous or solid
ρF = const.
Contact Medium
homogenous
one sided or two sidedContact between the MLM and Medium
dpi = const. - constant thickness
Main Features of the Expert Software NUMERIC
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Migrant concentration profiles in each layer of the MLM and in the Contact Medium
Main results computed by:
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Main results computed by:
Time dependent mean concentrations in each layer of the MLM and in the Contact Medium
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Performances of the Finite Difference (FD) algorithm:
Solves in MIGRATEST©EXP the
diffusion/migration equations for:
- a single Migrant with Molecular Weight, Mw, from 2 to 3,500 g/mol
- Migrant initially distributed in one, some or all Layers of the MLM
- initial concentration/s, C p,o‘s, of Migrant from 1 to 500,000 mg/kg
- thickness, dp, of each Layer from 1 to 100,000 µm
- density, ρρρρ, of each Layer from 0.1 to 20 g/cm³
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
- Migrant diffusion coefficients, Dp‘s, defined for each Layer- D‘s, from 10-2 cm²/s (very well mixed liquid) to 10-25 cm²/s (diffusion in solids)
- Migrant partitioning coefficients between Layers, Kpi,pj
- Kpi,pj‘s from 0.01 (extraction) to 1,000,000 (virtually a barrier)
Solves in MIGRATEST©EXP the
diffusion/migration equations for:
Performances of the Finite Difference (FD) algorithm:
- Migrant partitioning coefficients between MLM and Food, Kp,f - Kp,f from 0.01 (extraction) to 1,000,000 (virtually a barrier)
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Black BoxWarning for non-experts:
Do not open !!!
Diffusion Coeffs.,Dpi and DF
Partition Coeffs.,KPiPj and KPF
INPUT DATA
Contact area, A
Volume of food, VF
Thicknesses, dpi
Densities, ρρρρP and ρρρρF
Migration time, t
Initial conc., Cpo
START
RESULTS
Are the calculations with NUMERIC correct or not ? This is the question !
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Validation of the Finite Difference (FD) algorithm:
Validation of the FD algorithm isnecessary in order to assure the
user of the software MIGRATEST©EXP that the results
computed are correct:
- Validation by comparison with data calculated with other methods oralgorithms
- Validation by mathematical analysis of the computed results
- Validation by comparison between input data and calculated results
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Test 1 Comparison with results calculated with analytical algorithms
ρi = ρj = const Di = Dj = const
all ki,j, = kjF = 1
DF = 10-5 cm²/s - liquid medium/food
Cp,i (0) = Cp,j (0) = 1000 mg/kg
CF (0) = 0 mg/kg
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
CF,t = 8.9621 mg/kg
Relative difference of less than 0.1% between the numerical and analytical results !
Test 1 Comparison with results calculated with an analytical algorithm
CF,t = 8.962 mg/kg
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Comparison with results calculated with an analytical algorithm
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Test 2 Continuity of concentration profiles at interfaces
ρi = ρj = const
all ki,j, = kjF = 1
DF = 10-5 cm²/s - liquid medium/food
Cp,1 (0) = Cp,2 (0) = Cp,4 (0) = Cp,5 (0) = 0 mg/kg
CF (0) = 0 mg/kg
Di = Dj
Cp,3 (0) = 1000 mg/kg
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Test 2 Continuity of concentration profiles at interfaces
Relative error
less that 0.5%
for a mesh of
200 or larger
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Test 3 Values of migrant concentration jump at interfaces versus input partition coefficient
Partition values defined as input values
at each interface
all Ki,j, and KjF = 1
Di = Dj
Concentration jumps at interface calculated by NUMERIC
Cp,3 (0) = 1000 mg/kg
Relative deviation between input and calculated data
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Cp,3 (0) = 1000 mg/kg
100 200 300 4000,02
0,03
0,04
0,05
0,06
0,07
0,08
Rela
tive
De
via
tio
n (
%)
Number of mesh points
Relative deviation between the ration Ci / Cj and the partitioning Kij
Test 3 Values migrant concentration jump at interfaces versus input partition coefficient
Ki,j, = 20
Partition defined as input value
at the interface ijfrom 210 µm
Concentration values calculated by NUMERIC at interface ij and different time points:
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Test 4 – Overall mass conservation in the MLM – Contact Medium System
0 10 20 300
1
2
3
4
1
0-6 δ
qr
Time (hours)
Rela
tive t
ota
l m
ass
devia
tion (m
g)
Cp,3 (0) = 1000 mg/kg
Mass conservation assumption
∑=
+=n
i
titFt qqq1
,,
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Conclusions Conclusions
very good accuracy & results
fast computation with a regular PC
FD model used in
suited for plane geometry !
Development of a user-friendly PC program
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Main Features & Menus of the Software:
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus– Migration from the MLM into a Contact Medium
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Values
of mean
concentrations
and amounts
For 12 time intervals
in each layer of the MLM
Graphs
of concentration
profiles&
mean concentrations
Results of Computations
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Values
of mean
concentrations
and amounts
For 12 time intervals
in each layer of the MLM and the CM
Graphs
of concentration
profiles&
mean concentrations
Results of Computations
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Layer 1
Layer 2
Layer 1
Layer 2
Contact Medium
Time dependent mean
Migrant Concentration
in each Layer
of the MLM and the CM
Time dependent mean
Migrant Concentration
in each Layer
of the MLM and the CM
Contact MediumTime dependent mean
Migrant Concentration
in each Layer
of the MLM and the CM
Contact MediumContact Medium
Results of Computations – 2nd Step
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Case ExamplesCase Examples1. Process of „Invisible set-off“ during Manufacturing & Storage of a Multilayer
Material
Lamination or extrusion of the Multi-Layer Film
Tw
isti
ng
to a
bobbin
Zoom in
The Multi-Layer Material manufactured as a bobbin and stored for a certain time.
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
1st Winding 2nd Winding 3rd Winding
Migrant, CpoMigrant, Cpo
Process of „Invisible set-off“ during Manufacturing & Storage of a Multilayer
Material
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
1st Winding 2nd Winding 3rd Winding
Migrant, CpoMigrant, Cpo
Contact between windings
Process of „Invisible set-off“ during Manufacturing & Storage of a Multilayer
Material
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Menus
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Diffusion from Layer 1
Results of Computations – without „invisible set-off“
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Results of Computations – with „invisible set-off“
Diffusion from Layer 1 Contamination of Contact Layer due to migration from the next multi-layer
Diffusion from Layer 1
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Results of Computations – Migration without considering the „invisible set-off“
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Results of Computations – Migration with considering the „invisible set-off“
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Case ExampleCase Example2. Functional Barrier in a Multi-layer Packaging Material
2. Estimation of Migration through a Functional Barrier
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Data
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Input Data - Migration from the MLM into a Food for 4 years @20 °C
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Migration from the MLM into a Food for 4 years @20 °C – without PET layer
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Migration from the MLM into a Food for 4 years @20 °C – with barrier effect of PET
EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010
Migration from the MLM into a Food for 30 days @20 °C –with PET barrier layer but with „invisible set-off“ from the printing ink layer