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MULTILAYERS: 1D DESORPTION SIMULATION STANDARD FINITE FORMULATION FOR MULTI GEOMETRIES (v. 1.05)

Notations: (dimensionless equations are detailed here)

NAME DESCRIPTION
_F or layer 0
food or food simulant
_Pi
layer i of plastics or polymer (layer 1 is in contact with F)

_FP

ratio in F and in P
M
migrant or diffusant
0
initial
slab
sheet or planar packaging (most common)
cylindrical
cylinder (L_FP gives the internal radius)
spherical
sphere (L_FP gives the internal radius)

List of input parameters (geometry, thermodynamics, transport, initial concentration, contact time)

NAME PHYSICAL PROPERTY / UNITS / MAIN ASSUMPTION / SAFE VALUE
l_Pi

thickness of the layer i [ SI unit = m)] - ONE SINGLE SIDE CONTACT

No safe value (must be measured)

L_FP

mass dilution factor [SI unit: kgF/kgP)] - NO SIDE EFFECT (1D TRANSPORT)
= mass of F / overall mass of P

No safe value (must be measured)   >>3D»1D approximations

rho_

density [SI unit: (kg/m³) - since ratios of density are used, other units are also possible] - NB: k_ and rho ARE USED AS k_/rho_
If your concentrations are expressed in mass/mass (usual case), you must assign realistic densities to achieve accurate results.
If your concentrations are expressed in mass/volume, let all rho values to 1 and interpret L_FP as a volume dilution factor instead. Do not mix different units for concentrations.

No safe value (must be measured)

k_F

k_Pi

phenomenological coefficient (Henry like coefficients, such that the partition coefficient between i and j is K_ij = kj /ki [SI unit: (kgM/kgj)/(kgM/kgi)]
DESORPTION EQUILIBRIUM OBEYS TO HENRY's LAW

This formulation converts concentrations into dimensionless partial pressures, pi=ki·Ci, which are continuous between materials. The equilibrium condition corresponds to uniform pi values.

Safe choice = k_Pi>1000.k_F (almost complete extraction at equilibrium)   >>How to handle uncertainty on k values

D_Pi

diffusion coefficient in Pi [SI unit: m²/s] - THE FOOD/SIMULANT DOES NOT INTERACT WITH THE POLYMER

Safety value = must be [over]estimated or measured at a temperature higher than the conditions of use.
For an overestimate, use: Piringer Calculator

Bi

dimensionless mass Biot number [no unit] - EXTERNAL MASS TRANSFER RESISTANCE DUE TO FOOD TEXTURE/ABSENCE OF STIRRING/MASS TRANSFER IN FOOD
Bi=h.D_iref·rho_iref/(k_iref·l_iref) with iref layer with maximal transport resistance [(k_i·l_i)/(D_i·rho_i)]
NB: The internal solver use a dimensionless formulation based on the layer with a maximum transport resistance.

Safety value = 1000 (almost no external mass transfer resistance)

C0_Pi

initial concentration in Pi [SI unit: kgM/kgP] - ASSUMES AN HOMOEGENOUS AND UNIFORM INITIAL CONCENTRATION IN THE LAYER _Pi

Safety value = 5000 ppm   >>maximum recommended concentrations

t

contact time [SI unit s] - NO REACTION AND NO MASS LOSSES DURING DESORPTION

Default value = 10 days (use the maximum shelf life combined with the maximum storage temperature)

PARAMETER FORM (see above for details, only layers with non-zero properties are considered and in the presented order: _P1=layer in contact with food)
Advanced users:  >> How to take a decision with different sources of uncertainty
Optimized tools:   >> FV MULTILAYER simulation (faster and more reliable)   >>Optimized FV MONOLAYER simulation

 parameter _F _P1 _P2 _P3 _P4 _P5 _P6 _P7 _P8 unit l - µm mm cm m L_FP - - - - - - - - m³·m-³ rho kg·m-3 or g·cm-3 k variable D - m2.s-1 cm2.s-1 Bi - - - - - - - - none C0 - ppm kgM/kgP t - - - - - - - - years months days hours minutes seconds

<< The solution is calculated using a 2nd degree Finite Element (FE) technique and integrated over time by a variable order implicit scheme.
As a result, the calculations are accurate but may require several minutes for very complex multilayer structures. To prevent the congestion of the server, all calculations longer than 5 to 10 minutes are removed
from the server. To prevent this behavior, the alternative consists in using the new code based on an optimized Finite Volume (FV) method: multilayer FV code.
All calculations are absolutely stable, conservative and performed in few secondes with a satisfactory accuracy. Since the calculations are performed with a weak formulation (FV) of the diffusion problem, the solution
is however expected not to be as accurate in time as with a strong one (FE). Your choice should driven by your needs and the reliability of calculations.