Chapter 6 Mathematical solution and validation
6.4 Model validation
The purpose of the model validation was to analyse predictions by comparison with experimental observations to gain an indication of the model performance. The experiments conducted and presented in Chapter 4 were used to determine the level of agreement. The predictions of relative permeability for different coating formulations (Equation 5.8) were compared with observed relative permeability. The relative permeability was calculated as the ratio of the WVTR measured for barrier dispersion coatings with different filler volume fraction and the WVTR of unfilled barrier dispersion coatings at the same coating thickness level (prepared with the same rod) (Equation 6.11).
where
ܴܲ௦: Relative permeability of analysed barrier dispersion coatings
ܹܸܴܶௗ: Normalised water vapour transmission rate of barrier dispersion coatings filled at different volume fraction of filler
ܴܲ௦ൌ
ܹܸܴܶௗ
ܹܸܴܶ௨ௗ: Normalised water vapour transmission rate of unfilled barrier dispersion coatings
The geometrical characterisation of fillers presented in Chapter 3 was used to define the distribution of the parameter of distribution of length, elongation, and thickness for all the selected filler types. The size of the coatings was based on the estimation of REV presented in the previous section.
A summary of all the system inputs required to run the model are presented in Tables 6.1 to 6.6.
Table 6.1 Summary of the system inputs related to coating geometry and properties
Parameter description Symbol Unit Value
Length ܬ ߤ݉ 12
Width ܯ ߤ݉ 12
Thickness ܭ ߤ݉ 5
Glass transition temperature of the coating ܶ ιܥ 2
Table 6.2 Condition for definition of grid of coating geometry
Parameter description Symbol Value
Initial grid division in ݔ-axis ݊ݔ 5 Initial grid division in ݕ-axis ݊ݕ 5 Initial grid division in ݖ-axis ݊ݖ 5
Table 6.3 Summary of the system inputs related to fillers
Parameter description Symbol Clay A Clay B Clay C
Average of the logarithm of the length ߤ௧ 0.733 0.727 0.570 Standard deviation of the logarithm of the length ߪ௧ 0.731 0.746 0.781 Average of the logarithm of the thickness ߤ௧௦௦ -2.055 -1.968 -2.222 Standard deviation of the logarithm of the thickness ߪ௧௦௦ 0.231 0.154 0.227 Location parameter of elongation ߤ௧ 0.275 0.303 0.268 Scale parameter of elongation ߪ௧ 0.206 0.225 0.193 Shape parameter of elongation ݇௧ 1.273 1.278 1.262
Maximum limit of length ܯܽݔ௧ 22.83 19.34 14.49
Maximum limit of thickness ܯܽݔ௧௦௦ 0.200 0.200 0.179 Maximum limit of elongation ܯܽݔ௧ 6.670 5.100 6.200
Minimum limit of length ܯ݅݊௧ 0.550 0.550 0.550
Minimum limit of thickness ܯ݅݊௧௦௦ 0.070 0.080 0.050 Minimum limit of elongation ܯ݅݊௧ 1.000 1.000 1.000
Table 6.4 Condition of exposure of the dispersion coating
Parameter description Symbol Unit Value
Top coating surface Temperature ߠ °C 23
Relative humidity ܴܪ % 50
Bottom coating surface Temperature ߠ °C 23
Relative humidity ܴܪ % 0
Table 6.5 Summary of the system inputs related to permeability parameters
Parameter description Symbol Unit Value
Oxygen Water vapour
Lennard-Jones temperature ߝ
݇ ܭ 93 809
Ratio of the collision diameter of the permeant and
nitrogen ቆ ߪ௫ ߪேమ ቇ ଶ 0.83 0.48
Table 6.6 Constants used for simulations
Parameter description Symbol Units Value
Calculation of partial pressure of air (Equation 5.14) ܥ଼ െͷǤͺͲͲʹʹͲ ൈ ͳͲଷ
Calculation of partial pressure of air (Equation 5.14) ܥଽ ͳǤ͵ͻͳͶͻͻ͵
Calculation of partial pressure of air (Equation 5.14) ܥଵ െͶǤͺͶͲʹ͵ͻ ൈ ͳͲିଶ Calculation of partial pressure of air (Equation 5.14) ܥଵଵ െͶǤͳͶͺ ൈ ͳͲିହ Calculation of partial pressure of air (Equation 5.14) ܥଵଶ െͳǤͶͶͷʹͲͻ͵ ൈ ͳͲି଼
Calculation of partial pressure of air (Equation 5.14) ܥଵଷ െǤͷͶͷͻ͵
Ideal gas constant ܴ ܬ ή ݈݉ିଵή ܭିଵ ͺǤ͵ͳͶ
6.4.1 Variation of the permeant concentration due to filler arrangement of barrier dispersion coatings
Figure 6.6 and Figure 6.7 show the concentration profile across the coating was affected by the distribution of fillers. Between large filler gaps, the concentration tended to change in vertical direction. In cases where parallel fillers created channels, the concentration mainly changed in a horizontal direction. It was also observed that the concentration gradients were relatively large through channels compared with other regions with no fillers around. This was due to the constriction effect around filler channels. The constriction can be divided into two components; the resistance that the slit generates itself and the resistance created when the permeant is crossing through the slit (Cussler, et al., 1988; Wakeham & Mason, 1979). If the concentration is analysed (for example at ݔ=13.6 ߤ݉ at the left boundary in Figure 6.7), it was possible to see that it changed from almost 0 ݈݉ ή ݉ିଷ at ݕ=0.5 ߤ݉ to 90 ݈݉ ή ݉ିଷ at ݕ=1 ߤ݉, in part because of the small gaps between fillers at ݕ=1 ߤ݉, and ݔ between 16 and 18.5 ߤ݉ and ݔ between 20.5 and 24 ߤ݉. The variation of permeant concentration and the constriction of the channels affected the permeant flowrate. In Figure 6.7 the permeant flowrates at three points
were measured. Those were 6.68 ൈ10 -9݈݉ ή ݉ିଶή ݏିଵ followed by 1.05ൈ10 -8݈݉ ή ݉ିଶή ݏିଵ and 7.98ൈ10 -5݈݉ ή ݉ିଶή ݏିଵ for points 2, 1 and 3 respectively. These results showed that as the channel narrows, the permeant flux reduces.
Figure 6.6 Profile of permeant concentration through a coating filled with Clay A at 10.3% of volume fraction
Figure 6.7 Profile of permeant concentration through a coating filled with Clay A at 10.3% filler volume fraction at
ݔ െ ݖ axis view of the geometry at ݕ=19.6 ߤ݉
6.4.2 Comparison against experimental data
Figure 6.8 shows the prediction of relative permeability as a function the volume fraction for the three selected fillers studied in previous chapters. As expected, the general trend was to reduce the relative permeability if the volume fraction of the fillers increases. The reduction of relative permeability was the most for the largest filler size (Clay A). The increase in volume fraction, particularly for large fillers, creates more intricate pathways. These pathways increased the tortuosity and constriction effects and, as a result, reduced the relative permeability. However, as the volume fraction increased, the filler size became less important in the reduction of relative permeability and the prediction tended to be similar for all selected fillers.
3 1
Figure 6.8 Prediction of relative permeability as a function of the volume fraction of fillers for all the selected fillers
(the error bars represent standard deviation)
For example, at 4.9% volume fraction, the variation of relative permeability reached 0.05 and for larger amount of fillers the variation becomes constant at between 0.02 and 0.03 for each formulation. This means that the chance of significantly different arrangements of fillers is lower due to the large number of fillers and, as a result, most of the coating is covered with fillers. The relationship between the variation of relative permeability and the volume fraction can be seen in Figure 6.9. This figure shows the ratio between the relative permeabilities and their average as a function of the ratio between the volume fractions and their average for simulation carried out at 4.9% of filler volume fraction. As well as the predictions presented in Figure 6.8, the trend of the variation of relative permeability is inversely proportional to the fillers volume fraction.
Figure 6.9 Comparison between the variation of relative and volume fraction of fillers for barrier dispersion coating
formulations with Clay A at 4.9% of volume fractions (%v/dv)
Figure 6.10 shows a comparison between observed and predicted relative permeability at different volume fractions. The predictions follow the same trend as the experimental results; however, underestimations of the relative permeability for all the evaluated volume fractions
0 0.2 0.4 0.6 0.8 1 0% 5% 10% 15% 20% Relativ e p erm eab ility
Volume fraction of fillers
Clay A Clay B Clay C 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 Relativ e p erm eab ility / relativ e p erm eab ility av erag e
and types of fillers were found. At low volume fraction, the variability of both predictions and observations were overlapped. The differences were larger when the volume fraction increased reaching almost twice the lower predictions than the observed relative permeability for volume fraction equal to 15%.
Figure 6.10 Predicted and observed relative permeability of water vapour (WV) and oxygen (O2) for coatings
prepared with Clays A, B, and C (the error bars represent standard deviation)
From this point, the comparisons between experimental data and predictions were carried out only for the water vapour relative permeability. In the case of the oxygen relative permeability larger variability was observed. Other factors that were not related to the coating formulation maybe associated to large experimental variability as discussed in Section 4.3 Chapter 4.