3.2. DISCRETIZATION OF THE LAYER DOMAIN

The concentration profile is calculated by Finite Element Approximations. The packaging mass which is taken for the diffusion expressions can be treated as an ‘infinite’ surface of thickness ‘d’ (i.e. infinite in two directions -y, z directions- and of layer thickness ‘d’ in the third -x direction-). Mathematically expressed, we can establish that inside the bulk the partial pressure gradients of a migrant ‘M’ in the x-direction are an order of magnitude higher than the gradients in the x,y-directions at any time and in any layer:

		(3.15)
The above facts are evident because for a step dy or dz we have:		(3.16)
		(3.17)
		(3.18)

for any layer ‘U’.

Using the generalized mass balance eq. 3.11 over one layer element in the packaging wall, we can relate the diffusion of the migrant M in each layer. The scheme of the grid-point distribution applied for calculating the concentration distribution in each layer is presented in Fig. 3.2.


Fig.3.2: Description of a layer ‘domain’. The grid-point distribution is chosen with variable step lengths in the diffusion direction x as well as in the time direction.

The functions of the mass balance (eqs. 3.11-3.12) are singular at the interface of the different layers and at the beginning of the diffusion process (times around 0). Therefore the grid-point distribution must be chosen with variable step lengths (see Fig. 3.2). The generation of adaptive meshes allow the achievement of a desired resolution in localized regions and decreases by orders of magnitude the calculation time. Grid points are added in regions of high gradients to generate a denser mesh in that region and substracted from regions where the solution is decaying or flattening out.

The Fig. 3.2 illustrates the discretization of the governing partial differential eqs. 3.11-3.12 describing the diffusion processes occurring inside the layer. The layers can be divided into a series of N mesh planes each having IxJ elements. Applying FEA, the position of the mesh planes is moved along the time-axis allowing the calculation of the concentrations of both species (migrant and stimulant) at each location for every x, t grid points of each layer.

Discretization: Let us choose ‘n’ for satisfying following conditions:

		(3.19)
		(3.20)

where

.

e.g.

describes with arbitrary units the length of a desired thickness inside one layer.

After computing a series expansion of the above equations with respect to the variable ‘k’, we obtain:

		(3.21)
Taking the natural logarithm, we can solve the inequality 3.22 for ‘n’ and round the obtained value to the nearest integers towards minus infinity:		(3.22)
		(3.23)

The partial pressure profile for the migrant may now be expressed by substituting finite-difference approximations. Taylor series expansions of the second derivates are computed, with respect to the variable x, up to the order three. The series data structure represents an expression as a truncated series in one indeterminate node, expanded about the particular point (i ,j) (see Fig. 3.2). The detailed presentation of the Finite Element Analysis rests upon the scope of this help.