There are no systematic approaches in currently available specialist literature regarding any optimization process for drying hardwood species (such as birch (
The pine and spruce wood drying simulation program TORKSIM (version 3.1 onwards), has significantly contributed towards the optimization of spruce and pine drying, both of these being common coniferous tree species in Estonia (Sova
Mathematical models which are usually validated on the basis of experimental data which can be presented in the same article are less universal than TORKSIM. Sometimes such models contain important simplifying assumptions. For example, Baronas
The main goal of this article was to apply the experimental and theoretical methodology which had been developed for the drying of pine sapwood, in order to now be able to determine the critical diffusion constant and corresponding critical drying air humidity for each hardwood species, something which is important in the optimization of hardwood drying.
The fundamental equation for the diffusion for non-stationary isothermal moisture transfer through the wood specimen represents Fick's second law in the one-dimensional case (Crank, 1956):
When wood drying is carried out within a narrow temperature range of 50–60°C, that is, in a special isothermal case, then the simplified Fick's second law can be used:
The solution to such a differential equation is a polynomial of the second degree. A drying plan, one which considers the dynamics of moisture content and temperature, provides a parabolic distribution of the moisture content perpendicular to the surface of a material. This was referred to by Luikov as a quasi-stationary drying regime (Luikov, 1966; Tamme
The local diffusion coefficient can be experimentally determined according to Fick's first law (Fick, 1855; Crank, 1956; Salin, 1990; Tamme, 2016):
During the convective drying of wood, heat is transferred from the surrounding air through the surface of the wood into its interior and, at the expense of the heat energy being transferred to it, moisture evaporates from the wood, i.e. the wood is dried. The main equations are as follows when it comes to describing the heat flow of dry air, which is transferred to wood, and the heat flow of moist air which leaves the wood (Salin, 1990):
The basic equation for the wood deformation calculation in the drying process is the following:
It should be kept in mind that the modulus of elasticity is not constant, but instead depends upon the wood moisture content and temperature (Salin, 1990).
Five out of eight measurement channels from the Scanntronik (Scanntronik, 2023) resistance meter were calibrated from a 10LogR resistance channel to the measurement channels for the wood moisture content (MC%), using a section-linear calibration model. In the section-linear calibration model (Tamme
The methodology for the individual calibration of the resistance-type wood moisture meter channels is described in more detail in Tamme's doctoral thesis (2023).
Formula (3) must be adapted to a suitable format in order to be able to process experimental data (i.e. MC%'s dependences on time
The drying experiments made use of three specimens of black alder, aspen, and birch which had been cut from the same materials, with a thickness of 35 mm, width of 150 mm, and length of 100 mm along the grain. The electrodes which are shown in the figure were placed on the first specimen. The second specimen in the drying experiments was intended for collecting drying curve data using the dry weight method. The third specimen was used for the individual calibration of electrodes by means of the slicing method (Tremblay
In all specimens listed here, moisture distribution was carefully obstructed in the longitudinal and tangential direction by means of special vapour barriers. Moisture could only leave the specimens in the radial direction. In this way a one-dimensional moisture gradient was guaranteed in the specimens (Tamme, 2023).
All of the experiments with the three hardwood species were carried out using the same drying plan (see Table 1). A FEUTRON climate chamber (Feutron, 2023) was used as a dryer. Only one tree species at a time was kept in the drying chamber. The drying time was 120 hours. A total of three drying sessions were conducted (one drying cycle for each hardwood species), following the plan shown in Table 1.
The same drying schedule for drying all hardwoods.
Time (h) | Surrounding air temp. (°C) | Wet bulb temp. (°C) | Surrounding air RH (%) |
---|---|---|---|
0 | 20 | 19.44 | 60 |
1 | 47 | 46.11 | 95 |
12 | 48 | 47.10 | 95 |
36 | 50 | 48.13 | 90 |
60 | 52 | 48.02 | 80 |
84 | 52 | 45.54 | 69 |
108 | 52 | 43.05 | 59 |
132 | 52 | 40.27 | 49 |
Sensors and collecting experimental data may be described as follows:
For a wood drying optimization system it is first necessary to develop reliable and accurate sensors to be able to record the average moisture, local moisture, and surface moisture levels in the wood, as well as a sensor to record the deformation of the wood surface. These sensors must simultaneously withstand temperatures of 50–80°C and high relative humidity levels of 95–100% RH which are characteristic of a convective kiln (Tamme
The methodology for the experiment is described in more detail, together with photos, in the final report for the EIC contract No. 16200 (Tamme
For the process of monitoring the drying process, the nine-channel data logger Al-memo® 2890-9 was used, which is manufactured by Ahlborn (Ahlborn, 2023), as well as the eight-channel data loggers, Thermofox and Gigamodule, which are produced by Scanntronik (Scanntronik, 2023).
In the doctoral thesis (Tamme, 2023), the following hypothesis was established for pine sapwood:
‘If the critical diffusion constant is exceeded, the drying process for the wood's surface layer will also affect the wood's inner layers.’
In the case of pine sapwood, the validity of this hypothesis was tested experimentally by means of a moistening pulse, along with calibrated and uncalibrated resistance sensors (Tamme
To test the same hypothesis for hardwood species as part of this particular study, a statistical analysis was carried out in the freeware environment R (R Core Team, 2023) (see Table 5). The purpose of testing the hypothesis was to determine the strength of the linear relationship using the correlation coefficient, R, in layers with a depth of 1 mm and 12 mm. The 12 mm layer also served to characterise the average moisture content of the wood according to the EN 13183-2:2005 (2005) standard and the recommendations from EDG (Welling, 2010) (see Figure 2).
For data processing and figure formatting, use was made of the spreadsheet program Excel, and the freeware programs R (R Core Team, 2023) and MatPlotLib v3.4.3.
A total of 15 calibration functions at five different depths were determined for each hardwood species. A total of 45 calibration functions were determined for the three tree species.
Section-linear calibration functions determined on the basis of Formula (7) at depths of 1 mm and 4 mm from the surface, using the example of alder wood. Independent x-value is electrical resistance of wood (unit 10LogR). Dependent y-value is the moisture content (MC) of wood.
Depth (mm) | AB | BC | CD |
---|---|---|---|
1 mm | y = −8.0269x+475.3 | y = −0.676x+61.97 | y = −0.41x+43.335 |
4 mm | y = −12.813x+714.47 | y = −1.474x+119.696 | y = −0.721x+69.18 |
For wood moisture content levels which were below FSP (< 30% MC), it was mathematically proven that the local moisture content of wood, as measured at a depth of a third of the material thickness from the surface, is numerically equal to the average wood moisture content level (Kretchetov, 1972). Even so, for wood moisture content levels which exceed FSP (> 30% MC), it is not clear whether the recommendation which is contained in the EN13183-2:2005 (2005) standard in regard to measuring the average moisture content at a depth of a third of the material thickness is in fact valid for every drying plan and every wood material thickness.
Figure 2 also presents an important result for the validation of the EN 13183-2:2005 (2005) standard for moisture content levels which exceed the FSP.
Using alder wood as an example, the validation result confirms that the recommendation which is contained in the EN 13183-2:2005 (2005) standard to measure the average moisture content of wood at a depth of a third of the thickness is also relevant for moisture content levels which exceed the FSP, even without the corresponding mathematical proof.
Within the wood surface layer (in this study, about 2.5 mm from the wood surface), two local diffusion coefficients can be found, i.e. the local diffusion coefficient in the first drying phase for the surface layer, and the local diffusion coefficient in the second drying phase for the surface layer. The surface layer's local diffusion coefficient in which the wood's moisture content remains >FSP (i.e. greater than 30%) was referred to as the critical diffusion coefficient because, numerically speaking, it can be several times higher than the surface layer's diffusion coefficient in the second drying phase: “The maximum value of the diffusion coefficient immediately before entering into the second drying phase was named the
The critical diffusion coefficient (Dcr) for the surface layer of three hardwood species and pine sapwood, along with the corresponding critical air humidity (RHcr) and the ratio of the diffusion coefficients for the first drying phase and the second drying phase in the surface layer (Dcr/D2ph), for different tree species.
Type of wood | Dcr (*10−4 mm2/s) | RHcr (%) | D2ph (*10−4 mm2/s | Dcr/D2ph |
---|---|---|---|---|
Black alder | 36.57 | 75.6 | 13.87 | 2.64 |
Aspen | 30.71 | 85.4 | 11.72 | 2.62 |
Birch | 16.35 | 85.4 | 6.92 | 2.36 |
*Pine sapwood | 27 | 81 | 18 | 1.5 |
The experimentally determined moment of time, for the separating line (SL) and the corresponding average moisture content in the wood at the point at which the wood's surface layer transitions from the first drying phase into the second drying phase, and figures related to that transition.
Type of wood | Separating line (SL) (h) of drying phases | Avg MC (%) | Related figures |
---|---|---|---|
Black alder | 61 | 55 | Figure. 9, line 1; Figure. 11a |
Aspen | 50 | 62 | Figure. 9, line 2; Figure. 11b |
Birch | 46 | 64 | Figure. 9, line 3; Figure. 11c |
*Pine sapwood | 94 | 60 | Figure. 11d |
In Figure 9, line 1 is the transition for the alder wood surface layer from the first drying phase to the second drying phase; line 2 is the transition for the aspen wood surface layer from the first drying phase to the second drying phase; line 3 is the transition for the birch wood surface layer from the first drying phase to the second drying phase; and line 4 is the average wood moisture content which corresponds to the FSP value (i.e. 30% MC) for all of those tree species which were part of the experiment.
An additional explanation is required for Figure 9, line 4. If the wood average moisture content becomes lower than the FSP's corresponding average moisture content value (Figure 9, line 4), maximum drying stresses will begin to develop throughout the material at every depth. Therefore, if AvgMC < 30%, there is a risk of breaking stress developing in the material's surface layer (Salin, 1990; Tamme
According to Table 5 there is a strong linear relationship between the surface layer (1 mm) and the inner layer which corresponds to the average moisture content (12 mm) throughout the drying process, which itself indicates a causal relationship, i.e. the moisture content in the surface layer significantly affects the moisture content in the inner layers throughout the drying process. Consequently, the hypothesis which has already been established for pine wood (Tamme, 2023) is also valid for hardwoods.
Hypothesis testing results for hardwood species. Independent x-value is moisture content (MC) in the depth level of 1 mm from the surface. Dependent y-value is the MC in the depth level of 12 mm from the surface.
Type of wood | Equation: 12mmMC (y) ~ 1mmMC (x) | R | R2 | p-value |
---|---|---|---|---|
Black alder | y = 0.951x + 17.303 | 0.989 | 0.977 | < 0.001 |
Aspen | y =1.275x + 11.8 | 0.979 | 0.958 | < 0.001 |
Birch | y =1.302x + 12.028 | 0.965 | 0.931 | < 0.001 |
*Pine sapwood | y =0.9377x + 57.898 | 0.992 | 0.984 | < 0.001 |
An explanation is required for the definition of the effective diffusion coefficient (EDC). Formula (2) represents Fick's second law, in which the partial derivative with respect to time is assumed to be approximately constant, as it occurs in the quasi-stationary drying phase (Luikov, 1966; Salin, 1990; Tamme
In Table 6, the given effective diffusion coefficient (EDC) was determined using the parabolas method (Kretchetov, 1972; Tamme
A comparison of the effective diffusion coefficient (EDC) (Deff) for different tree species under quasi-stationary drying conditions.
Type of wood | Deff (*10−4 mm2/s) |
---|---|
Black alder | 11.92 |
Aspen | 11.44 |
Birch | 4.18 |
*Pine sapwood | 9.58 |
Equations of experimentally determined parabolic moisture profiles for different tree species, and an R-squared approximation. Independent x-value is the depth level (mm) from the board surface. Dependent y-value is the moisture content (MC%) at the depth level.
Type of wood | Time (h) | Equation of parabolic MC profile | R2 |
---|---|---|---|
Black alder (Figure 10a) | 0 | --------- | ------ |
61 | y = − 0.0976x2 +3.0933x +30.2 | 0.9512 | |
96 | y = − 0.0655x2 +2.0568x+ 12.884 | 0.9964 | |
119 | y = − 0.047x2 +1.5694x + 6.4665 | 0.9974 | |
Aspen (Figure 10b) | 0 | ----------- | -------- |
50 | y = − 0.1197x2 +3.4894x+32.592 | 0.8463 | |
99.5 | y = − 0.0597x2 +1.9168x + 11.8 | 0.9916 | |
125 | y = − 0.0315x2 +1.18x +9.1838 | 0.993 | |
Birch (Figure 10c) | 0 | ----------- | -------- |
46 | y = − 0.1154x2 +3.9316x +23.5 | 0.9698 | |
94 | y = − 0.0599x2 +2.0472x +13.274 | 0.9981 | |
122 | y = − 0.0547x2 +1.9479x +10.274 | 0.9984 | |
*Pine sapwood (Figure 10d) | 22 | y = − 0.054x2 +1.5471x +91.484 | 0.87 |
92 | y = − 0.0781x2 +2.3962x +24.791 | 0.8978 | |
116 | y = − 0.0867x2 +2.2641x +18.533 | 0.8615 | |
140 | y = − 0.0625x2 + 1.8709x +11.191 | 0.9769 |
For different tree species the characteristic maximum values were analysed for the difference between drying air and wood surface temperatures (see Figures 11a–d), which occur simultaneously in the material's critical surface layer (i.e. accelerated diffusion). Based on Formula (5), the heat flux of moist air which comes from the wood's surface is proportional to the temperature difference, with the comparison benchmarks being the mass transfer coefficient of moisture and the heat capacitance of the drying air. Therefore, it can be concluded that the maximum mass flux of moisture from the surface of the wood is also in sync with the critical diffusion coefficient in the surface layer of the wood, at approximately 2.5 mm deep from the surface. The physical reason for the temperature difference between the wood surface and the surrounding air is the fact that the thermal energy which is required for the evaporation of water (i.e. vaporisation heat or, approximately, enthalpy) is taken from the wood's surface, due to which the wood's surface is also cooled at the same time.
When comparing Figures 3, 5, 7 and Figure 11a–d, it can be concluded that the separating line (which is marked as ‘SL’ in the figures) for the first and the second drying phases within the surface layer is located in approximately the same position on the time axis as the specific maximum levels of the air and surface temperature difference (Figure 11a–d). Since the Dcr was determined on the basis of Figures 3, 5, 7, the maximum figures for temperature differences are therefore in sync with the Dcr figure which was achieved for all of the tree species. The maximum figures for temperature differences also form additional proof of the validity of the definition of the critical diffusion coefficient (Dcr) for the hardwood species.
The final stage of this process involves a look at the comparative behaviour of surface layer deformations (see Figure 13), which are calculated on the basis of Figure 12, and also on the basis of readings from the displacement sensor which monitored the shrinkage of the surface layer. A relevant analysis was carried out on different tree species and under the same drying plan conditions. The total experimental deformation of the surface layer was calculated by means of the following formula:
The total surface deformation as represented by the theoretical Formula 6 can be considered as being a mathematical description of the wood's mechanical properties.
Formula 6 contains the components as additives when it comes to elastic deformation, viscoelastic deformation, and deformation which are all caused by mechanosorption (Salin, 1990). Figure 13 shows that the total deformation according to the aforementioned formula is linearly dependent upon time for pine sapwood and alder. In the case of birch and aspen, the dependence of the total deformation on time is rather exponential, which suggests that the wood modulus of elasticity (MOE) is not linearly dependent upon the wood moisture content. Another possibility is that the MOE retains a linear dependence upon the wood's moisture content, but the viscoelastic deformation dominates, which is indeed basically an exponent function (Salin, 1990).
The experimental and theoretical drying methodology which has been developed for pine sapwood under the EIC Grant No. 16200 project also worked surprisingly well in terms of the study of hardwood drying. The methodology clearly highlighted quite important differences between the process of drying hardwoods and that of drying pine sapwood. At the same time, the basic definitions were confirmed for the methodology, sometimes in an even more prominent form when compared to that for pine wood (see Table 3). For all species of hardwood which were used in the tests, the existence of a critical diffusion coefficient was confirmed in the surface layer, as well as the maximum values for the difference between the drying air temperature and that of the wood's surface which is something that occurs practically in sync with the Dcr. The effective diffusion coefficients found at the beginning of the quasi-stationary drying period were also highly comparable (see Table 6). When comparing the total deformations in the surface layer of pine wood and hardwoods (see Figure 13), the influence of the mechanical properties of different wood species was vividly highlighted in terms of the dynamics behind the total deformation. And, finally, with the help of statistical analysis, the basic hypothesis which had been set up for pine sapwood in the doctoral thesis (Tamme, 2023) in regard to the transfer of changes in the wood's surface layer to its inner layers (see Table 5) was proved also to be true for hardwood species.