Analysis on the influence of grain size and grain layer thickness on the sorption kinetics of grained wood at low relative humidity with the use of water vapour sorption experiments
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Abstract
Water vapour sorption (WVS) experiments on grained Norway spruce wood (Picea abies) at low relative humidities were carried out to test the influence of grain size and grain layer thickness on the sorption kinetics. Samples were compared under identical climatic conditions (i.e. humidity and temperature), and the kinetic behaviour was analysed with selected modelling approaches existing in the literature. Both, grain size and grain layer thickness influenced the initial kinetics, with the latter showing a larger impact. This confirms the notion of a transport limited initial mass increase with diffusion of water vapour/H_{2}O-molecules to the sorption sites being a possible candidate. In contrast, the long-time behaviour was only slightly affected, supporting the concept of a relaxation and reorganisation dominated long-time behaviour. An analysis on the WVS kinetics of cut and grained wood with comparable sample material has further shown a very similar behaviour, which allows to draw some conclusions for cut wood. Regarding the modelling approaches, the parallel exponential kinetics model provided the best fitting results as the predictive models could not properly capture the split-up for a variation in grain size or grain layer thickness.
Introduction
Water vapour sorption experiments are a frequently used method to get information on the transport of water vapour through the macrostructure of wood and its subsequent sorption behaviour (e.g. Christensen and Kelsey 1959; Wadsö 1993; Eitelberger et al. 2011). Samples were exposed to variable climate conditions and weighted continuously. The amount of bound water is measured directly without any further assumptions. Experimental investigations on the sorption kinetics of wood have been performed for various wood species (e.g. Zaihan et al. 2009), sample sizes (e.g. Eitelberger and Svensson 2012) and also for grained wood (e.g. Hill et al. 2010b). However, an investigation on the influence of grain size and grain layer thickness on the WVS behaviour as well as an appropriate comparison between cut and grained wood seems to be missing.
Range of diffusion coefficients for water vapour (\(D_{i}\)) and bound water (\(D_{i,\text{cw}}\)) in spruce wood in m^{2}/s at room temperature
Longitudinal (L) | Radial (R) | Tangential (T) | |
---|---|---|---|
\(D_{i}\) | \(2.6 \cdot 10^{-5}\) | \(1.9 \cdot 10^{-6}\) | \(1.3 \cdot 10^{-6}\) |
\(D_{i,\text{cw}}\) | \(10^{-12}\!\cdots 10^{-10}\) | \(0.5\cdot D_{L,\text{cw}}\) | \(0.3\cdot D_{L,\text{cw}}\) |
In the present study, the influence of grain size and grain layer thickness on the WVS behaviour of grained wood at low RH is investigated and analysed with the use of three different sorption kinetic models. These modelling approaches to describe and simulate the mass change of wood for a step change in RH are presented in “Existing models” section. The experimental set-ups are given in “Material and methods” section, including details on the performed simulations (e.g. geometry and used parameters). “Results and discussion” section represents the experimental results, showing the similarity between sliced and grained wood (“Comparison of sliced and grained wood” section), the effect of grain size (“Comparison of various grain sizes (monolayer)” section) as well as the influence of grain layer thickness (“Comparison of multilayer experiments” section). The objective of this study is thus to provide a detailed analysis on the WVS behaviour of grained wood and to get more information on the processes dominating the sorption kinetics at low relative humidity. Additionally, the differences between various modelling approaches and their application on grained wood will be discussed.
Existing models
- (1)
Relaxation dominated case
- (2)
Transport dominated case
- (3)
Mixed case, where transport and relaxation processes are comparable
Accordingly, the various modelling approaches in the literature can be separated into three cases. A similar classification can be found in Hill et al. (2011) for the diffusion behaviour of swelling polymers. In the following, three modelling approaches are presented, covering all of the before-mentioned cases. They are based on different ideas and were used frequently in recent years. Further, the given models range from a heuristic relationship (PEK model) to a semi-predictive (bound water diffusion model) and a predictive approach (coupled diffusion model).
Parallel exponential kinetics (PEK) model
Bound water diffusion model
Here, the coefficient c specifies the maximum moisture uptake before relaxation and d represents the additional amount available after relaxation and reorganisation processes took place. There are thus fast accessible sorption sites and sites which need a certain relaxation time \(\tau\) to be available (mixed case). This idea on a creation of new sorption sites seems reasonable for swelling materials and has already been mentioned by many authors (see e.g. Hartley et al. 1992). Regarding the non-Fickian behaviour, it was emphasised by Olek et al. (2011) the modified boundary condition with Eq. (8) yields more adequate results than the moisture-dependent diffusion coefficient.
Coupled diffusion model
The coefficients were adjusted to the experiments of Christensen (1965), where almost identical sorption kinetics were found for 20 μm and 1 mm thick samples. In both cases, a physical interpretation of the proportionality factor and its coefficients seems to be difficult and is still missing.
Material and methods
Sample preparation
Classification of sliced wood (A1, A4) and grained wood by minimal–maximal particle size and sample mass (at RH = 20%) for the experimental investigations
Sample mass (mg) | Particle size | Label |
---|---|---|
365 | 0.5 mm × 43 mm × 43 mm | A1 |
200 | 125–250 μm | A2 |
400 | 125–250 μm | A3 |
22 | 25 μm × 10 mm × 10 mm | A4 |
22 | 20–63 μm | A5 |
20 | 20–63 μm | B1 |
20 | 500–1000 μm | B2 |
200 | 20–63 μm | C1 |
200 | 125–250 μm | C2 |
200 | 500–1000 μm | C3 |
200 | 20–63 μm | C4 |
400 | 20–63 μm | C5 |
600 | 20–63 μm | C6 |
800 | 20–63 μm | C7 |
Measuring apparatus
A change of relative humidity at \({T} =25\,^{\circ }\mathrm {C}\) took about 1 min for the experiments with \(\Delta \hbox {RH} = 5\%\) and \(\Delta \hbox {RH} = 10\%\) with a deviation below 1%. A deviation less than ± 0.3% to the pre-set humidity values could be ensured except for \(\hbox {RH}\le 1\%\) (\(\pm \,0.1\%\)). The achievable humidity ranges from \(\hbox {RH} = 0.1\%\) to \(\hbox {RH} = 95\%\) for \({T} = 25\,^{\circ }\mathrm {C}\). Variations of temperature were below \(\pm \,0.1\,^{\circ }\mathrm {C}\). To achieve a stabilised climate for the samples, measurement cycles were executed in an 8 min interval.
Experimental set-up
Cleaned sample bowls were placed on the rotating plate and tared at \(\hbox {RH} = 20\%\) and \({T} = 25\,^{\circ }\mathrm {C}\). Sliced and grained samples with a weight according to Table 2 were poured in the sample bowls. The surface was carefully flattened with a spatula in order to avoid compression among the various grain sizes. Minimum sample mass was chosen at 20 mg to obtain a reasonable mass resolution. Sample masses were weighted out with an accuracy of ± 0.2 mg for approximately 20 mg and ± 1 mg for a mass \(\ge\) 200 mg. The used step size in RH was chosen to be \(\Delta\)RH = 5% in order to reduce the influence of the non-instantaneous step change on the sorption kinetics. Temperature was kept constant at \(25\,^{\circ }\mathrm {C}\) during the whole step. Only for experiments on the lower resolution scale, step size was chosen at \(\Delta \hbox {RH} = 10\%\) in order to increase the signal-to-noise (S/N) ratio by a larger amount of absolut water uptake. In the following, experimental set-ups for three different investigations on the sorption behaviour of grained wood is given.
Comparison of sliced and grained wood
To analyse the differences in the kinetic behaviour of sliced and grained wood, an experimental set-up with a sample thickness in the mm and μm-range was used. Care was taken regarding sample thickness and weight (i.e. amount of sorption sites per unit cross section) as water vapour has to be transported to and through the sample material. For the comparison in the mm-range, a slice of cut wood in longitudinal direction with 0.5 mm thickness and a mass of 365 mg (A1) was used. The sliced sample stood out of the sample bowl and was exposed to the surrounding RH on both sides. In contrast, grained samples were only exposed on the upper side (i.e. on one side) to RH, but had a higher accessibility based on the spherical shape. As the number of sorption sites is an important parameter for a diffusion process with a sink, grained wood samples with a mass of 200 mg (A2) and 400 mg (A3) were chosen. Grain size of 125–250 μm was used as pre-testings with other grain sizes have shown a similar behaviour (cf. “Comparison of multilayer experiments” section). The corresponding grain layer thickness can be estimated to 0.7 and 1.4 mm (Eq. 16). Cross-sectional areas between the cut and grained samples were comparable. For the comparison in the μm-range, sample mass was chosen to be small enough to avoid overlapping of the slices or grains but still as high to provide an acceptable S/N-ratio with the given sorption analyser. Hence, a sample mass of 22 mg was used. About 20 slices of the microtomed samples with 25 μm thickness (A4) were needed and only a few were partly overlapping. In contrast to the comparison of sliced and grained wood in the mm-range, the microtomed and grained samples were both exposed to RH in a similar manner. Therefore, a similar thickness appeared reasonable and grain size was chosen slightly larger as it offers a faster accessibility due to the spherical geometry. The grained sample was consequently chosen with a grain size of 20–63 μm and equal mass (A5). Further, a similar cross-sectional area of approximately 19 cm^{2} could be reached.
Comparison of grain sizes (monolayer)
To figure out possible differences in the WVS behaviour of various grain sizes, grained wood with a single grain layer (monolayer) and equal sample mass was used. Tests on monolayer experiments with five different grain size distributions were performed in advance to estimate the differences in their sorption kinetics. To point out the differences more clearly, an experiment with only two samples was performed. Thus, grained wood with a grain size of 20–63 μm (B1) and 500–1000 μm (B2) and a mass of 20 mg was used. With this set-up, a reduction of the measurement cycle time from 8 to 5 min could be achieved, corresponding to a faster time resolution of the measuring device. Similar to the comparison of sliced and grained wood in the μm-range, sample mass in this set-up is on the lower range limit of the sorption measuring system. Hence, the S/N-ratio is rather low and thus step size in RH was chosen to a larger value.
Comparison of multilayer experiments
In order to determine the effect of layering of grains on the sorption kinetics of grained wood, two experimental set-ups with multiple grain layers (multilayer) were used. To analyse the influence of grain sizes, multilayer experiments with three different grain sizes and equal mass were compared. Grain size distributions were chosen to 20–63 μm (C1), 125–250 μm (C2) and 500–1000 μm (C3) with a sample mass of 200 mg. As all samples do have the same amount of sorption sites, any additional effects caused by an external water vapour supply limitation were minimised. Other sample masses were tested in advance showing a similar behaviour. With these multilayer experiments possible differences could be distinguished more accurately than for the monolayer experiments, as the relative error decreases with increasing sample mass. According to Eq. 16, the given sample mass is equivalent to a grain layer thickness of \(0.67\,\hbox {mm}\) (C1), \(0.71\,\hbox {mm}\) (C2) and \(0.84\,\hbox {mm}\) (C3). For the determination of the influence of grain layer thickness (i.e. amount of sorption sites per unit cross section), grained wood with multiple grain layers and a single grain size of 20–63 μm was used. Sample mass was chosen to 200 mg (C4), 400 mg (C5), 600 mg (C6) and 800 mg (C7), respectively. These sample masses correspond to a grain layer thickness (cf. Eq. 16) of \(0.7\,\hbox {mm}\) (C4), \(1.3\,\hbox {mm}\) (C5), \(2.0\,\hbox {mm}\) (C6) and \(2.7\,\hbox {mm}\) (C7).
Error estimation
To determine the measurement error expected in the WVS experiments, three different sources have to be taken into consideration. For experiments with a sample mass in the order of 20 mg, the reproducibility of the balance (± 10 μg) becomes important. With a total mass increase of approximately 520 μg for a step change of \(0\rightarrow 10\%\) RH, this error contributes with 2% to the mass increase of these samples. As the results are represented in relative changes of sample mass (cf. Eq. 17), the errors arising from balance reproducibility decrease with increasing sample mass. The second source of error which is more dominant at the beginning of the sorption process (i.e. the first measuring values) is caused by local and global fluctuations in humidity. This error has been estimated (on the basis of previous experiments) to be in the order of ± 20 μg for fluctuations in the RH-step change between 0 and 10% RH and contributes thus with 4% to the absolute mass increase. Again, samples with a larger thickness and a higher weight are less affected as both, diffusion and the amount of water molecules in the vapour fluctuation restrict the resulting impact. The third source of error arises from random disturbances of the balance caused by temporary vibrations that are transferred over the floor/building. Even though this error could be rather large, it affects usually only a single measuring value and can thus be easily detected. For the following results error bars were chosen according to the maximum value of the three mentioned errors which is appropriate to the particular experiment. It has to be noted, in the case of grained wood there is a large amount of single grains in each sample bowl, and thus, an average value for the mass increase is measured.
Simulations
The simulations of the mentioned models in “Existing models” section were done with the software Wolfram Mathematica 10.4. For the PEK model, the parameters were determined by a nonlinear model fit. As no spatial variables are included in this approach, sample geometry has not to be taken into account. In the case of the bound water diffusion model, Eqs. (4), (5), (7) and (8) were used without performing an inverse analysis as given in Olek et al. (2011). The bound water diffusion coefficient (\(D_{b}\)), the surface emission coefficient (\(\sigma\)) and the time constant (\(\tau\)) were fitted to the experiment, while the remaining parameters were used as given for the lowest RH range in their work (Olek et al. 2011). For the coupled diffusion model, Eqs. (9), (10), (11), (12) and (13) were used, including all necessary parameters given in Frandsen et al. (2007) which were based on earlier experiments of sliced Klinki pine wood (Araucaria hunsteinii). No adjustment on the parameters was performed as this model serves to predict the WVS kinetics of wood.
Estimation of grain layer thickness
Results and discussion
Each step change starts at t = 0 and ends when the equilibrium conditions for all samples are fulfilled (\(t_{\rm max}\)). For a better comparability, time axis was chosen with a scale according to the relevant domain although equilibrium (Eq. 15) was usually achieved at later times. As the comparison of sliced and grained wood is mainly to indicate the similarity of their WVS behaviour, measured data were only fitted with the PEK model.
Comparison of sliced and grained wood
Comparison in the mm-range
Regarding the PEK model, all three samples could be fitted within the error bars. Both, the time constants for the fast (\(\tau _{1}\)) and slow process (\(\tau _{2}\)) show an increasing tendency with sample weight as given in Table 3. The difference at the beginning of mass increases between the two grained samples (with equal grain size but different layer thicknesses) in Fig. 5 is in contrast to the notion of both exponential processes in the PEK model being assigned to a mechanical origin (Eq. 2). More plausible would be the interpretation of a transport process at the beginning (fast process) and a relaxation process for long measuring times (slow process) similar to that suggested by Popescu et al. (2014).
Comparison in the μm-range
The results for the comparison of the WVS kinetics of sliced and grained wood in the μm-range are given in Fig. 5, for a step change in RH of \(0\rightarrow 10\%\) at \({T} = 25\,^{\circ }\mathrm {C}\). Both samples show a similar behaviour, even though the sorption measuring system was too slow to sufficiently resolve the mass increase at the beginning. With a weight of 22 mg, both samples are on the lower limit of the measuring device. Hence, the reproducibility error and the error due to fluctuations in RH are comparatively large. Comparing the apparent fast mass increase for the μm-samples with the markedly slower increase in the mm-samples supports the significance of a transport process at the beginning of the sorption kinetics in the low range of RH. The long-time behaviour shows within the measurement error an identical small mass increase for the microtomed and grained sample until equilibrium condition is achieved. Within the measuring accuracy, this increase seems to be independent of the used sample geometry. Consequently, also for the μm-range it seems as if there is no essential difference in the WVS behaviour of cut and grained wood, assuming that suited sample material is used.
Results for the fit parameters of the five experiments for the PEK model (\(\tau _{1}, \tau _{2}\)) and the bound water diffusion model (\(\tau , \sigma\), \(D_{b}\))
Sample mass | \(\tau _{1} \,(s)\) | \(\tau _{2}\) (\(10^{3}\) s) | \(\tau\) (\(10^{3}\) s) | \(\sigma\) (\(10^{-6}\) m/s) | \(D_{b}\)(\(10^{-9}\) m\(^{2}\)/s) |
---|---|---|---|---|---|
365 mg (0.5 mm) | 560 | 3.8 | |||
200 mg (125–250 μm) | 380 | 2.8 | |||
400 mg (125–250 μm) | 830 | 5.3* | |||
22 mg (25 μm) | 120 | 11.3* | |||
22 mg (20–63 μm) | 110 | 18* | |||
20 mg (20–63 μm) | 80 | 3.1* | 1.4 | 1.2 | 1.6 |
20 mg (500–1000 μm) | 130 | 1.7* | 1.4 | 1.2 | 1.6 |
200 mg (20–63 μm) | 280 | 3.8 | 3.3 | 3.0 | 1.6 |
200 mg (125–250 μm) | 320 | 4.2 | 3.3 | 3.0 | 1.6 |
200 mg (500–1000 μm) | 380 | 4.6 | 3.3 | 3.0 | 1.6 |
200 mg (20–63 μm) | 220 | 1.7 | 3.3 | 3.0 | 1.6 |
400 mg (20–63 μm) | 700 | 4.1 | 3.3 | 3.0 | 1.6 |
600 mg (20–63 μm) | 1190 | 5.9* | 3.3 | 3.0 | 1.6 |
800 mg (20–63 μm) | 1640 | 16.7* | 3.3 | 3.0 | 1.6 |
Comparison of various grain sizes (monolayer)
To point out the influence of grain size on the WVS behaviour of wood, the largest and smallest available grain sizes were used. Figure 6 shows the sorption kinetics for the 20–63 µm and 500–1000 μm grain size distributions for a step change in RH of \(0\rightarrow 10\%\) and \({T}=25\,^{\circ }\mathrm {C}\). The measurement error in the normalised representation is relatively large, as sample mass for approximately one monolayer of the smallest grain size is around 20 mg. A marked difference can be seen for the first measuring point (up to 10 min), where the smaller grains show a faster mass increase than the larger ones. The long-time behaviour of the two samples shows within the measuring accuracy a similar small mass increase. According to the difference in grain size, these results point towards a transport limited process at the beginning of the mass increase. Either the pathways through the macrostructure, the higher accessibility of the cell wall material or the total amount of sorption sites per unit cross section could serve as a possible candidate for limiting the short-time kinetics. It should be mentioned that even if there is only cell wall material without any enclosed lumen pathways for the smallest grain size (i.e. the kinetic is for example limited by diffusion and adsorption of water molecules in the cell wall), there must be a reason why the larger grain size takes more time at the beginning of the mass increase than the smaller one. The difference in the WVS kinetics is much less than expected for a diffusion process with a factor of 20 difference in sample thickness (see, e.g., the results for the bound water diffusion model in Fig. 6b). This indicates processes like the accessibility of binding sites or the total amount of sorption sites per unit cross section to be more relevant in this stage. The results are compatible with the early experiments of Christensen and Kelsey (1959), where a small difference between the sorption kinetics of the 20 μm microtomed slices and the 1 mm cut slice could be seen for the lowest step change in RH, with the difference being much smaller than expected for the given sample lengths.
Comparison of multilayer experiments
Comparison of grain size (multilayer)
In order to get a better S/N-ratio of the monolayer experiment and to test the influence of grain size for multilayer experiments, three different grain sizes with an equal sample mass were compared. The results are shown in Fig. 7 for a sample mass of 200 mg and a step change in RH of \(0\rightarrow 5\%\) at \({T}=25\,^{\circ }\mathrm {C}\). A small difference can be seen at the beginning of the mass increase (up to 20 min) with the larger grains showing a slightly slower increase than the smaller ones. For the long-time behaviour, a good congruence between the three grain size distributions is given until equilibrium condition is achieved. Similar results were also obtained with larger sample masses. Comparing with the monolayer results, it seems as if the influence of grain size (or cell wall accessibility) becomes less important for the multilayer experiments. Still, the grain layer thickness or the total amount of sorption sites per unit cross section seems to be of importance as the mass increase at the beginning of the multilayer experiments is markedly slower than in the monolayer case. The small differences in sheet thickness for the three grain sizes (“Experimental set-up” section) might thus be used to roughly estimate the differences in the WVS kinetics of grained wood with multiple grain layers. According to the similarity of the kinetic curves, it seems as if the given graining process has no significant impact on the WVS behaviour of wood. This includes any irregular graining losses among the various grain size distributions as also mechanical damage of the cell wall structure.
Comparison of grain layer thickness
As the grain layer thickness (i.e. sample thickness) of grained wood seems to be important for the WVS kinetics, various masses with one grain size distribution were compared. Figure 8 shows the sorption kinetics for the smallest grain size (20–63 μm) with a variation of sample thickness for a RH-step change of \(0\rightarrow 5\%\) at \(T=25\,^{\circ }\mathrm {C}\). An obvious difference between the four masses can be seen at the beginning, where thicker samples show a slower mass increase than thinner ones. Using a representation of the measured data over square root of time shows an approximately linear part for the mass increase at the beginning (Fig. 8d). This linear behaviour is characteristic for a diffusion process and is pointing towards diffusion being relevant for the short-time kinetics. However, the split-up seems to decrease with increasing grain layer thickness, indicating a deviation from a simple diffusion process. Similar results were also reported for cut wood (e.g. Wadsö 1993), where samples with various thickness do show a smaller split-up than expected for a diffusive process using Fick’s law (Krabbenhoft and Damkilde 2004). In contrast, the long-time behaviour shows only small differences between the four samples, with a steeper slope for an increasing sample thickness. This seems likely for the consecutive humidification of single grains (or layers), as expected for a diffusive process through the multilayers. Hence, the consecutive adsorption of water/\(\hbox {H}_{2}\hbox {O}\)-molecules of the grained samples initiates a successive starting of relaxation or reorganisation processes. The summation of all these processes thus leads to a delayed long-time mass increase for multilayer samples with a larger grain layer thickness. In a similar manner, the same mechanism should also hold for sliced wood with variable thickness. That is, the relaxation or reorganisation processes in the cell wall of wood are a local phenomenon initiated as soon as water vapour/\(\hbox {H}_{2}\hbox {O}\)-molecules reach the given position. Such concepts of a local treatment of relaxation have already been mentioned in the literature (e.g. Engelund et al. 2012 and references therein) and requires a local sink as in the case of the coupled diffusion model. A global treatment of all these successive processes (as in the PEK model) seems thus to be an approximation, which might be reasonable merely for samples with a small thickness in longitudinal direction. For larger sample geometries, a local treatment of the relaxation and reorganisation processes with a corresponding equation for the water vapour supply appears to be necessary.
Conclusion
The comparison of cut and grained wood showed a similar WVS behaviour in the mm and μm-range, proving there are no principal differences in their sorption kinetics. For the cut and grained samples, it could be seen that thicker samples were appreciably slower in their short-time kinetics (i.e. mass increase at the beginning) than thinner ones, indicating a transport limited initial phase. In the same manner, monolayer experiments of grained wood exhibit a faster mass increase for smaller grains. Thus, transport effects are significant even for very small sample sizes. For multilayer experiments with equal sample mass, the influence of grain size seems to be less pronounced. A variation of grain layer thickness with equal grain size caused, however, a marked split-up in the short-time kinetics. Additionally, thinner samples showed a slightly faster equilibration in the long-time range than the thicker ones, which seems obvious for a successive adsorption of water vapour/\(\hbox {H}_{2}\hbox {O}\)-molecules followed by a relaxation and reorganisation process. This trend holds even down to the smallest tested sample thickness (20–63 μm) and is pointing towards a connection between the two stages. The WVS kinetics at low RH seems thus to be limited by a diffusion-like process at the beginning of the sorption kinetics and by a relaxation and reorganisation limited process for the long-time behaviour. Consequently, the second stage might be used to get insights into structural changes and rearrangements of water molecules, whereas the first stage seems to provide information about the penetration and support of water vapour/\(\hbox {H}_{2}\hbox {O}\)-molecules to the sorption sites. Regarding the modelling approaches and their application to grained wood, a few general statements could be drawn:
PEK model: This study confirms the fast process of the PEK model being related to a transport limited process in the lower range of RH. The measurements indicate diffusion of water vapour/\(\hbox {H}_{2}\hbox {O}\)-molecules to the sorption sites being a possible candidate. Considering the long-time behaviour, the slow process in the PEK model was supported to be related to a relaxation and reorganisation limited process. Consequently, the independent treatment of both processes seems to be an approximation for small samples. A proper treatment of the transport processes and/or a coupled description appears to be advantageous and might avoid the exclusion of the first measurement points until target RH is reached.
Bound water diffusion model: The bound water diffusion model showed a too large split-up for the grain sizes (monolayer) and for the various grain layer thicknesses (multilayer). This seems to result as a consequence of both, treating the relaxation processes as a global phenomenon at the boundary and neglecting the water vapour transport inside and to the sample. Hence, a local treatment might provide better results when comparing similar samples with a variable thickness. For the long-time behaviour, this model leads to similar results as the PEK model, as both approaches use the same expression to treat relaxation and reorganisation processes.
Coupled diffusion model: This model serves as the most comprehensive approach of the tested models as it accounts for the transport through the macrostructure, inside the cell wall and includes also the sorption process. Still, with the given sorption term and its parameters it led to the largest deviations to the experiments and the split-up for a variation of grain layer thickness could not be captured properly. A modification of this term might thus be worthwhile to treat the mass increase for samples with a thickness below 1mm.
Notes
Acknowledgements
Open access funding provided by University of Innsbruck and Medical University of Innsbruck. This work has been founded by the Research Program Translational Research of the Standortagentur Tirol under the Project DigiPore3D and the Grant Doktoratsstipendium NEU aus der Nachwuchsförderung by the Universität Innsbruck, Vizerektorat für Forschung under the code 2015/2/TECH-26. Their financial support is gratefully acknowledged.
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