An experimental determination of the critical diffusion coefficient and critical relative humidity (RH) of drying air when optimizing the drying of three hardwood species (birch, aspen, and black alder)

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): (1) ∂u∂t=∂∂x(Dt(T, u)∂u∂x), {{\partial u} \over {\partial t}} = {\partial \over {\partial x}}\left( {{D_t}\left( {T,\,u} \right){{\partial u} \over {\partial x}}} \right), where D_t is the diffusion coefficient perpendicular to the wood (m² · s⁻¹).

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: (2) Deff∂2u∂x2=const, {D_{eff}}{{{\partial ^2}u} \over {\partial {x^2}}} = const, where D_eff is the effective diffusion coefficient perpendicular to wood (m² ·s⁻¹).

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 et al., 2011).

The local diffusion coefficient can be experimentally determined according to Fick's first law (Fick, 1855; Crank, 1956; Salin, 1990; Tamme, 2016): (3) F=−D∂u∂x, F = - D{{\partial u} \over {\partial x}}, where F is mass flux (kg/m²s); D is the diffusion coefficient (m²/s); u is mass concentration (kg/m³); and x is the coordinate (m).

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): (4) ∅1=αS(T0−T), {\emptyset _1} = \alpha S\left( {{T_0} - T} \right), (5) ∅2=βcp S(T0−T), {\emptyset _2} = \beta {c_p}\,S\left( {{T_0} - T} \right), where Ø₁ and Ø₂ is the heat flow (W); α is the heat transfer coefficient (W/m² °C); ß is the mass transfer coefficient (m²/s); c_p is the specific heat of humid air which is in equilibrium with the wood's surface (J/°C m³); S is the surface area of the specimen (m²); T₀ is the surrounding air temperature (°C); and T is the wood's surface temperature (°C).

The basic equation for the wood deformation calculation in the drying process is the following: (6) ∂ε∂t=1E∂σ∂t+∂εv∂t+(α+mσ)∂ρb∂t, {{\partial \varepsilon } \over {\partial t}} = {1 \over E}{{\partial \sigma } \over {\partial t}} + {{\partial {\varepsilon _v}} \over {\partial t}} + \left( {\alpha + m\sigma } \right){{\partial {\rho _b}} \over {\partial t}}, where ε is total deformation; σ is tensile stress (Pa); E is the modulus of elasticity (Pa); ε_v is the viscoelastic strain; a is the unrestricted shrinkage coefficient (m³ kg⁻¹); m is the mechano-sorptive creep coefficient (m³ kg⁻¹ Pa); ρ_b is the content of bound water (kg m³); and t is time (s).

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 et al., 2021a; Tamme, 2023), the following relationship holds true for each segment of the calibration function: (7) y−y1y2−y1=x−x1x2−x1, {{y - {y_1}} \over {{y_2} - {y_1}}} = {{x - {x_1}} \over {{x_2} - {x_1}}}, where points A(x₁; y₁) and B(x₂; y₂) are the coordinates for the segment's endpoints.

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 t and coordinate x) in order to experimentally determine the local diffusion coefficient (Tamme et al., 2021a). In principle, the diffusion coefficient (DC) can be given according to Formula (3) as the ratio of the mass flux to the gradient: (8) D=−MassfluxGradient, D = - {{Massflux} \over {Gradient}}, (9) Massflux=−Δ(MC%)mdryΔt*S, Massflux = - \Delta \left( {MC\% } \right){{{m_{dry}}} \over {\Delta t*S}}, (10) Gradient = Δ(MC%)ρwood,dryΔx, Gradient\, = \,\Delta \left( {MC\% } \right){{{\rho _{wood,dry}}} \over {\Delta x}}, where D is the diffusion coefficient (DC) (m²/s); while Δ(MC%) is the finite increment of the wood's MC% on the time axis for the mass flux and, in the material thickness (x-axis), for the gradient (MC%); m_dry is the wood dry mass (kg); S is the specimen surface area (m²); Δt is the time increment (s); ρ_{wood, dry} is the wood's dry density (kg/m³); and Δx is the x coordinate's increment (m).

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 et al., 2000; Aboltins & Kic, 2019; Tamme et al., 2021a; Tamme, 2023).

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.

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 et al., 2021a).

The methodology for the experiment is described in more detail, together with photos, in the final report for the EIC contract No. 16200 (Tamme et al., 2021b) and doctoral thesis (Tamme, 2023). The basic scheme for the experiment is shown in the photograph in Figure 1a and Figure 1b.

Figure 1.

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 et al., 2021a).

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).

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.

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 critical diffusion coefficient.” (Tamme et al., 2021b; Tamme et al., 2021a; Tamme, 2023).

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.

Figure 9.

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 et al., 2021b; Tamme, 2023). In the discussion section of Tamme's doctoral thesis, he explained that, although tensile stresses also occur in the surface layer only when the surface layer transitions to a situation in which the appropriate value falls beneath the FSP figure, apparently those stresses which are developed only in the surface layer are not numerically sufficient to create such a maximum tensile stress which would lead to the formation of drying cracks.

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.

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 et al., 2011). It is important to note that, according to Formula (2), the effective diffusion coefficient is the same at all depths from the wood surface, i.e. it no longer depends on the coordinate, as with the local diffusion coefficient which is determined based on Fick's first law (Formula (3)).

In Table 6, the given effective diffusion coefficient (EDC) was determined using the parabolas method (Kretchetov, 1972; Tamme et al., 2011). The principle behind the parabolas method involves the parabolic moisture profiles (see Figures 10a–d and Table 7) containing all of the necessary information for determining the mass flux and gradient, and consequently also for determining the effective diffusion coefficient. According to Formula (2), in Fick's second law the mass flux becomes approximately constant, i.e. a quasi-stationary drying period occurs. In Figure 10a–d, the last two moisture profiles have almost parallel curves, which is a visual sign of the quasi-stationary state (Kretchetov, 1972; Tamme et al., 2011).

Figure 10.

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.

Figure 11.

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: (11) Relative shrinkage =deformation = (l0−l)/L (mm /mm), {\rm{Relative}}\,{\rm{shrinkage}}\, = {\rm{deformation}}\, = \,\left( {{{\rm{l}}_0} - {\rm{l}}} \right)/{\rm{L}}\,\left( {{\mathop{\rm mm}\nolimits} \,/{\rm{mm}}} \right), where l is the reading from the displacement sensor at time (t), while l₀ is the reading from the displacement sensor at the beginning of the surface layer's drying shrinkage process, and L is the length of the displacement sensor's base. In all tests, L=120 mm.

Figure 12.

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).

Figure 13.