Macroscopic X-ray computed tomography aided numerical modelling of moisture flow in sawn timber

The drying schedule (wet-bulb temperature, \({T}_{\mathrm{w},\mathrm{s}}\), and dry-bulb temperature, \({T}_{\mathrm{d},\mathrm{s}}\)) used in the experiment is presented in Fig. 2a. This schedule was provided by a sawmill outside Skellefteå in Northern Sweden. The figure also presents the recorded wet-bulb temperature, \({T}_{\mathrm{w},\mathrm{a}}\), and dry-bulb temperature, \({T}_{\mathrm{d},\mathrm{a}}\), provided by the integrated sensors in the climate chamber, and it gives an indication of the onset of each drying regime (1–5). The schedule consisted of a sequence of heating, capillary, transition, diffusion, and conditioning regimes.

Within the heating regime, the \({T}_{\mathrm{d},\mathrm{s}}\) and the \({T}_{\mathrm{w},\mathrm{s}}\) were raised to 66 °C and 63 °C, respectively, in 14 h. In the capillary regime, \({T}_{\mathrm{d},\mathrm{s}}\) was slowly increased to 67.5 °C and the \({T}_{\mathrm{w},\mathrm{s}}\) was kept constant for 32 h to remove capillary (free) water from the lumen above FSP. The transition regime was used to bring \({T}_{\mathrm{d},\mathrm{s}}\) up to 70 °C and \({T}_{\mathrm{w},\mathrm{s}}\) up to 64.5 °C in a period of 14 h to achieve a smooth transition between capillary-driven and diffusion-driven drying. In the diffusion regime, \({T}_{\mathrm{d},\mathrm{s}}\) was kept constant and \({T}_{\mathrm{w},\mathrm{s}}\) was gradually lowered to 60 °C over a period of 28 h to lower the relative humidity inside the kiln and to remove bound water from the cell walls below FSP. The conditioning regime was characterized by a constant \({T}_{\mathrm{d},\mathrm{s}}\) and an increase in \({T}_{\mathrm{w},\mathrm{s}}\) to 67.5 °C to raise the relative humidity in the kiln in 7.65 h, in order to elevate the MC directly beneath the board surface. In the experimental validation presented in this paper, the data collected from the transition regime onwards are used. Figure 2a also shows the recorded temperature inside the timber board, \({T}_{{\mathrm{A}}_{1}}\), obtained with a TGP-4017 Tinytag plus 2 (Intab, Stenkullen, Sweden) in combination with a PB-5006-1m5 Allround Tinytag sensor (− 40 °C to +125 °C,  ± 0.2 °C between 0 °C and +70 °C). It can be seen that \({T}_{{\mathrm{A}}_{1}}\) was generally lower than \({T}_{\mathrm{d},\mathrm{a}}\). The position of the sensor within the cross section of the timber board is shown in Fig. 2b. The air velocity, \({v}_{\mathrm{a}}\), within the chamber was set to a constant value of 5 m s⁻¹.

The medical CT scanner used to scan the timber consisted of an X-ray tube and four rows of detectors, both of which were aligned at opposite positions and could rotate around the gantry where the board was located to produce a fan beam. For each full rotation, ca. 1000 projections were collected and used to reconstruct a two-dimensional image through a back-projection algorithm (Hsieh 2015). The scan parameters were set to 110 kVp and 88 mAs, and the total scanning time was 0.8 s. The reconstruction process resulted in images with a pixel resolution of 0.586 × 0.586 mm² and a 5 mm voxel depth. Since only single scans were considered, the reconstruction was two-dimensional. The location of this image was at the centre of the board in the lengthwise direction.

The image collection was done at regular time intervals. A one-hour (h) time interval was adopted during the heating, capillary, transition, and diffusion regimes, and a 15 min time interval during the conditioning regime. Each image was stored in the medical image format DICOM. The raw images contained a CT number (CTN) in Hounsfield units at each voxel point. The CTN was determined according to:where \(\mu\) is the linear attenuation coefficient of the material, and \({\mu }_{\mathrm{w}}\) and \({\mu }_{\mathrm{a}}\) are the attenuation coefficients of water and air, respectively (Kalender 2011; Hsieh 2015). There may be further processing of the CT images by the CT scanner that can affect the CTNs, such as filtering, but this is part of the proprietary software and unknown to the user of the scanner. The CTNs were converted to density in kg m⁻³ by the metadata of the DICOM files based on an internal law that gives the linear relationship between these two variables as:where \(\rho\) is the density of the scanned wood. Equation (2) is applicable to all ranges of MC. The accuracy of the scans is ensured by a calibration of the machine using phantoms of known density, such as water and air. If necessary, a corrected calibration function can be included in the density calculation. The calibration for the current CT scans was shown to be well within the range of accepted values. The existence of a linear relationship between the CTN and wood density was proven by, among others, Lindgren (1991) and has been widely applied since then.

The image-processing algorithm presented by Hansson and Cherepanova (2012) was adopted to estimate the MC at each voxel point, \({w}_{\mathrm{v}}\), using the wet-density, \({\rho }_{\mathrm{w}}\), and dry-density, \({\rho }_{0}\), images and the expressions:where \({\beta }_{\mathrm{max}}\) is a coefficient indicating the dimensional change due to shrinkage, \({V}_{\mathrm{w}}\) is the wet volume, and \({V}_{0}\) is the dry volume (Hansson and Cherepanova 2012). The proportionality between the grey scale values of a CT image and the MC of wood was established in the early 1990s by for example Lindgren (1991). The temperature-dependent fibre saturation point, \({w}_{\mathrm{f}}\), was estimated with Eq. (5) established by Siau (1995). The Hough transform for circles was used to determine the pith location.

A challenge in the estimation of the MC is the shrinkage of the timber during drying. This distorts the cross-section of the timber and prevents the one-on-one comparison between density images. The bidirectional elastic registration algorithm published by Arganda-Carreras et al. (2006) was therefore integrated into the image-processing algorithm and used to transform the shape of the dry-density image to the shape of the wet-density image. The mapped density values at each pixel were corrected by using polygon clipping (Hansson and Fjellner 2013).

A manual threshold segmentation method was adopted to define the geometric boundary of the specimen. This segmentation resulted in a two-dimensional binary image, \({B}_{0}(x,y)\), using the dry-density image, \({P}_{0}\), and a threshold value, \({\tau }_{\mathrm{B}}\). The threshold value was based on the two density modes (air and wood) found in the histogram of \({P}_{0}\) (Prewitt and Mendelsohn 1966). The value used for \({\tau }_{\mathrm{B}}\) was 250 kg m⁻³. The segmentation procedure was implemented in the MATLAB^® software (MathWorks, Natick, USA).

The dense latewood of Norway spruce generally has a higher MC than the less dense earlywood (Fromm et al. 2001). These differences in density and MC between latewood and earlywood were also detected in both the density images obtained through CT scanning (Fig. 3) and in the MC images created with the image-processing algorithm (Fig. 4). However, the numerical model assumed a homogenized cross section which made no distinction between earlywood and latewood. To be able to use the density and MC data in the development of the numerical model and in the execution of the experimental validation, the density and MC were therefore smoothed. This smoothing also improved the numerical convergence and meant that parts of the data were emphasized. For example, the cross-sectional variation in density between sapwood and heartwood became more distinct, as well as the moisture gradient within the cross section of the board. An averaging scheme was chosen that used a two-dimensional Gaussian kernel, \(G\), which represents a normal distribution based on a standard deviation, \(\sigma\) (Russ 1995). To deal with smoothing at the boundary of the board, the surface detection (see subsection 2.2.2) preceded the smoothing. The smoothing was executed in the MATLAB^® software using statement ‘imgaussfilt’.

The oven-dry density image, \({P}_{0}\), of specimen A₁ presented in Fig. 3a was determined by exposing the specimen to a temperature of 103 ± 2 °C until the difference in mass between two successive weighings (interval of 2 h) was less than 0.1% (CEN 2003). Figure 3a also shows the actual thickness (154.2 mm) and width (52.2 mm) of the specimen at the beginning of the transition regime, and the location of the pith (80.4 mm, 54.6 mm) determined with the image-processing algorithm.

In Fig. 3b and c, the oven-dry density is given along a row of pixels centered along the width and thickness of the specimen’s cross section, respectively. The plots give both the original data (exp) and the smoothed data (gauss). Based on the smoothed data, the mean dry-density of the specimen was determined to be 349 kg m⁻³ with minimum and maximum dry-density values of 256 kg m⁻³ and 391 kg m⁻³, respectively.

Figure 3d gives the green state (wet) density image obtained at the beginning of drying (\(t\) = 0). The border between heartwood and sapwood was derived from the difference in color between heartwood and sapwood in the green state. This difference is due to the rapid increase in MC between the border and the surface of the board. The percentage of heartwood in the specimen was 54%.

Figure 4a and b show a moisture vs distance plot at the beginning of the transition regime along the thickness and width of the specimens, respectively. The figures show the original data (exp), the smoothed data (gauss), and an indication of the temperature-dependent FSP (27%) based on the equation presented by Siau (1995). In the case of the smoothed data, most of the cross section was below FSP at the start of the transition regime, indicating that the moisture flow was driven only by diffusion. Figure 4c and d present smoothed moisture vs distance plots at different stages of the drying process starting at the transition regime. The onset of the transition regime is indicated as \(t\) = 0. The plots show that the MC distribution was unsymmetric and differed in magnitude between the two edges.

The balance equation and constitutive relationship used to describe the non-linear transient moisture flow in wood are given by Eqs. 9 and 10, respectively:where \({c}_{w}\) is the moisture capacity, \({\rho }_{w}\) is the density of water, \({\varvec{q}}\) is the moisture flux vector, \({\varvec{D}}\) is a moisture-, dry-density- and temperature-dependent diffusion matrix, \({\rho }_{0}\) is the dry density, \(w\) is the unit-less MC, \(T\) is the temperature and \(\nabla\) defines gradients. The symbol \(\left( {\overline{ \bullet } } \right)\) in Eq. (8) refers to the orthogonal local coordinate system for wood. The moisture capacity is an indicator of how well the wood retains moisture. For simplicity, it was assumed that the balance equation is divided by \({c}_{w}\) and \({\rho }_{w}\) and that the inverses of \({c}_{w}\) and \({\rho }_{w}\) are implicitly included in the diffusion coefficient. In general, \(w\) is dependent on the concentration of water inside the wood, \(c\), and the dry density in the sense that \(w=c/ {\rho }_{0}\). In the current model this is not incorporated, which means instead that \(w\) is assumed to be based on a constant dry density.

The Neumann boundary condition that describes the flux normal to the exchange surface \({q}_{n}\) is defined as:where, \(s\) is a density-, moisture-, and temperature-dependent surface emission coefficient and \({w}_{\infty }\) is the unit-less MC of the ambient air. Further information concerning the derivation of the FE formulation and how to iteratively solve the non-linear system of equations is given by Florisson et al. (2020) and Johannesson (2019).

The meshed geometry of board A₁ is shown in Fig. 5a and b. Figure 5a shows the initial moisture content distribution, while Fig. 5b shows the dry-density distribution. These are discussed in further detail in subsection 2.3.4. The model is assumed to be 1 mm thick, creating a more-or-less two-dimensional scenario.

Figure 5c presents the dry-density CT image with an indication of the absolute thickness (154.2 mm) and width (52.2 mm) of the specimen at the beginning of the transition regime. The figure also shows the local (l, r, t) coordinate system associated with the wood’s fiber direction, and the global (x, y, z) coordinate systems. The estimated pith location (80.4 mm, 54.6 mm in the global domain) obtained with the image-processing algorithm (see subsection 2.2.1) was used to define the annual ring pattern in the cross-section of the board using user-subroutine ORIENT.

To mesh the board, a geometric meshing technique was used. The surface geometry was first obtained from the CT image corresponding to the beginning of the transition regime. The threshold segmentation method presented in subsection 2.2.2 was used to identify the voxels located at the surface of the board in relation to the global coordinate system. These data were then imported into the FE program as individual points and interpolated linearly. A partition, indicated by the black solid line in Fig. 5a, was made to create a structured 1 mm hexahedral mesh of type DC3D8 in the central region of the specimen, and an unstructured hexahedral mesh at the edges, in order to benefit the numerical analysis and to enable the more efficient extraction of results at certain locations.

The expressions used to determine the diffusion coefficient, \(D\), and the surface emission coefficient, \(s\), are:where \(\mathrm{r}\) and \(\mathrm{t}\) indicate the radial and tangential material directions, respectively. Both material parameters were assumed to be dependent on oven-dry density, \({\rho }_{0}\), moisture content, \(w\), and temperature, \(T\) (Avramidis and Siau 1987; Sehlstedt-Persson 2001; Yeo et al. 2008). The coefficients \({D}_{\mathrm{w}}\), \({D}_{\mathrm{T}}\), \({s}_{\mathrm{w}}\) and \({s}_{\mathrm{T}}\) were determined on the basis of experimental data previously collected by Florisson et al. (2021) for different types of pine spp. from for example Avramidis and Siau (1987), Rosenkilde (2002) and Yeo et al. (2008). The data are listed in Table 1. The linear relationship between the diffusion coefficient and dry density, which results in larger coefficients for smaller densities, is based on experimental data published by Sehlstedt-Persson (2001). Due to a lack of data, the same relationship was used to describe the dependence of the surface emission coefficient on the density.

In Fig. 6a and c, the diffusion and surface emission coefficient, respectively, based on the coefficients presented in Table 1 and used in the numerical validation, are given for the temperature ranges and a mean density of 349 kg m⁻³ obtained through the experiment. The assumed relationship between the material parameters and density is shown in Fig. 6b and d for a relative humidity of 85% and a temperature of 60 °C. The experimental data (exp) for heartwood and sapwood specimens of Norway spruce in Fig. 6b are taken from Sehlstedt-Persson (2001). The values lie between 3.24e-6 and 1.08e-6 m² h⁻¹ given by Eq. (10).

Figure 5a presents the MC distribution at the beginning of the simulation, i.e. at the beginning of the transition regime. Figure 5b shows the dry-density distribution needed to define the material parameters. These initial conditions were generated by the FE program based on the smoothed MC and dry-density data described in subsections 2.2.4 and 2.2.5, respectively. The FE program used a distance weighting algorithm as a deterministic method for multivariable interpolation of the imported data to the nodes that make up the FE mesh. The default settings for mapping set by the FE program were used. The change in temperature inside the sawn timber was included in the simulation by assigning the measured temperature \({T}_{{\mathrm{A}}_{1}}\) (see Fig. 2) to the mesh nodes that lie within the partition that encloses the inner region of the cross-section. This created a constant temperature field within each time increment used to determine the size of the material parameters.

The climate surrounding the board during drying was simulated as the MC of the ambient air (see Fig. 7), \({w}_{\infty }\), presented in Eq. (9). The value of \({w}_{\infty }\) within each time increment was determined in two steps, where the first step was to determine the RH, \(\phi\), based on the recorded \({T}_{\mathrm{d},\mathrm{a}}\) and \({T}_{\mathrm{w},\mathrm{a}}\), and the method presented in ASHRAE Handbook ASHRAE (2017). This method was implemented in MATLAB^®. The equations used to determine \(\phi\) were:where \({p}_{\mathrm{v},\mathrm{w}/\mathrm{d}}\) is the vapor pressure determined with the Antoine formula based on either the recorded dry-bulb temperature inside the climate chamber, \({T}_{\mathrm{d},\mathrm{a}}\), or on the recorded wet-bulb temperature, \({T}_{\mathrm{w},\mathrm{a}}\). \({p}_{\mathrm{a}}\) is the atmospheric pressure and had a value of 101.325 kPa. \(\omega\) is the degree of saturation and \({\omega }_{\mathrm{w}}\) is the degree of saturation corresponding to the wet-bulb temperature. \({T}_{\mathrm{d},\mathrm{a}}\) and \({T}_{\mathrm{w},\mathrm{a}}\) in Eq. (14) are expressed in Kelvin.

The second step was to estimate \({w}_{\infty }\) using Simpson’s formula (Simpson 1971), \(\phi\) and \({T}_{\mathrm{d},\mathrm{a}}\). The procedure was executed inside user-subroutine FILM. The recorded temperature \({T}_{\mathrm{d},\mathrm{a}}\) was assigned to the mesh nodes that lie outside the partition, close to the timber surface (see Fig. 5). Simpson’s formula was originally created for Sitka spruce (Picea sitchensis) and is based on an average of the adsorption and desorption curves.