Wood chips
We analyze wood chips made of stem wood of predominantly Norway spruce (Picea abies) with possible small amounts of Baltic pine (Pinus sylvestris) from middle Sweden, i.e., conifer or softwood. The woodchips were produced for industrial purpose and in particular as fuel in district heating. It was, therefore, not possible to evaluate the permittivity and anisotropy of the wood before the it was cut into wood chips and used in the study. Figure 1 shows wood chips of stem wood of the type investigated here, where the left image is a top view and the right a side view. The side view is blurred since it is taken through transparent plastic used in the lab, in order to have the wood chips oriented as in the radio measurements (describe below). Notice that the individual wood chips have the shape of long rods in most cases. The length is in the range 1–10 cm. The short side of the individual wood chips is 10–20% of the length. The wood chips are not randomly oriented in all directions. Instead, they are oriented in such a way that the long side is predominantly horizontal (perpendicular to gravity) [8].
Experimental equipment
During the measurements the wood chips were placed in a plastic box of the type euro container (an industrial stacking container conforming to the VDA 4500 standard [12]). Four antenna housings were placed on each side of the box for radio transmission measurement (see Fig. 2). The box had the interior size of 26.8 × 36.8 × 31.5 cm^{3}. The antenna housings were made of steel and had the size of 14.0 × 14.0 × 14.0 cm^{3}. Each housing included one balanced Vivaldiantenna, which could be rotated to enable the measurement of the permittivity parallel and perpendicular to gravity (cf. [13]). The transmit and receive antennas were both in the parallel or perpendicular orientation, in the respective measurement cases. The system was used in transmission mode, i.e., the antenna on one side is transmitting a signal that propagates through the wood chips and is received by the opposite antenna.
The antennas were connected to a radar system called DiRP (digital radar processor) that is based on Msequence technology (cf. [14]), to generate the UWB signal. The sampling frequency was 25 GHz and the signal’s center frequency was 2.5 GHz in a 3.5 GHz wide band. In this frequency range the signal level after propagating through wood chips is high enough to be detectable for wood chips of both high and low moisture content. Higher frequencies would be more attenuated resulting in too low signal levels. The antenna size is practical in laboratory and industry applications. Lower frequency would require larger antennas.
A conventional cement mixer (Biltema 17686), Fig. 3 (left), was used to mix wood chips and water to get wood chips of moisture content. A scale (Dini Argeo IP67/IP68) was used to measure the weight of water and wood chips, Fig. 3 (right).
Experimental procedure
To characterize the wood chips, 100 pieces of wood chips were randomly selected and the dimensions were measured using a caliper. The wood chips were also put in a box of transparent plastic without any lid and photographed (see Fig. 1 above).
The moisturization process started by placing 8975 kg of relatively dry wood chips (starting chips) in a larger conditioning box. This amount was used in the successive moistening. We use the dry based moisture content,
$$M_{{\text{d}}} = \frac{{m_{\text{w}}  m_{\text{d} }}}{m_{\text{d} }},$$
(1)
where \(m_{{\text{w}}}\) and \(m_{{\text{d}}}\) is the weight of the wet and dry wood chips, respectively.
Initially, the moisture content of the starting chips was measured by taking samples (à 500 g) that were weighted before, denoted \(m_{{{\text{w1s}}}}\), and after drying, denoted \(m_{{{\text{d1s}}}}\). The drying was performed during 24 h in + 105 °C, which is the approved industrial drying procedure in Sweden [15]. The moisture content of the starting chips, M_{d1}, was determined using Eq. 1. The moisture content of the starting chips was M_{d1} = 2.23%. The dry weight, m_{d1}, of the entire amount of starting chips was determined from Eq. 1, using M_{d1} and the measured weight of the starting chips, \(m_{{{\text{w1}}}}\).
The moisture content of the wood chips was increased in a number of steps. In each step, n, the moisture content, M_{dn}, was determined from the measured weight, \(m_{{{\text{w}n}}}\), and the initial weight, \(m_{{{\text{w1}}}}\) using Eq. 1. The wood chips were the same in all steps.
The moistening process was performed by sprinkling 1 L of water on the wood chips during mixing by the cement mixer (Fig. 3a) for each moistening cycle. Altogether, 10 moistening cycles were performed. Each succeeding moistening cycle (except for the first one) was based on preceding moistened wood chips. Each moistening cycle took about 30 min and the whole moistening experiment was carried out for 8 h (one working day). In each cycle, following acts and measures were made:

1.
Wood chips and water were mixed in the cement mixer (Fig. 3a). Water was sprinkled over the wood chips,

2.
Wood chips were moved from the cement mixer to the conditioning box and mixed again,

3.
The weight, \(m_{{{\text{w}n}}}\), of the wood chips in the conditioning box was measured (Fig. 3b),

4.
Wood chips were moved from the conditioning box to the sample box and mixed again,

5.
The sample box was packed by shaking the box towards the floor three times and adding more wood chips until the box was filled. Approximately half of the amount of wood chips from the conditioning box was needed to fill the sample box,

6.
Radio transmission measurements with antennas in vertical and horizontal direction were performed on the sample box (Fig. 2a), and

7.
The weight, \(m^{\prime}_{{{\text{w}n}}}\), of the wood chips in the sample box was measured.
The mixing in both step 1 and 2 was made to achieve a homogeneous moisture distribution in the sample box.
A conventional error analysis gives the error in the measured moisture content
$$\Delta M_{{{\text{d}n}}} = \left( {\frac{{M_{{{\text{d}n}}} }}{{m_{{{\text{d1}}}} }} + \frac{1}{{m_{{{\text{d1}}}} }}} \right)\left( {\frac{{2m_{{{\text{d1}}}} }}{{m_{{{\text{ds}}}} }} + 1} \right)\Delta w + \frac{\Delta w}{{m_{{{\text{d1}}}} }},$$
(2)
where \(\Delta w\) is the error in the weight measuring.
Determining permittivity
We determined the complex permittivity of wood chips from transmitted radar pulses. We describe the main steps of the determination method. Details can be found elsewhere [8, 13]. We model the wave propagation using the complex refractive index, \(\tilde{n}\). The real and imaginary parts of the permittivity, \(\tilde{\varepsilon } = \varepsilon^{\prime} + {{i}}\varepsilon^{\prime\prime}\), are related to the complex refractive index \(\tilde{n} = n + {{i}}\kappa\) by
$$\varepsilon^{\prime} = n^{2}  \kappa^{2} ,$$
$$\varepsilon^{\prime\prime} = 2n\kappa ,$$
(3)
or \(\tilde{n} = \sqrt {\tilde{\varepsilon }}\). We determine \(\tilde{n}\) from measured radar pulses that have propagated through the wood chips. The time delay gives the real part, n, [8]. The imaginary part, κ, is obtained from the relative damping vs. frequency [13].
In a reference measurement with air in the test box, a reference pulse, \(y_{0} (t)\), is measured vs time, t. In a second measurement a pulse, \(y_{1} (t)\), is measured after it has been transmitted through the wood chips.
The refractive index, n, is determined from the difference in time delay between \(y_{1} (t)\) and \(y_{0} (t)\). In order to make the time estimate more robust, we use a technique in which the Fourier transform of a transfer function, h(t) was calculated as [16]
$$H(\nu ) = \frac{{Y_{1} (\nu )Y_{0}^{*} (\nu )}}{{Y_{0} (\nu )Y_{0}^{*} (\nu ) + \gamma }},$$
(4)
where \(Y_{1} (\nu )\) and \(Y_{0} (\nu )\) are the Fourier transforms of \(y_{1} (t)\) and \(y_{0} (t)\), respectively, \(\nu\) is the frequency, and * denotes the complex conjugate; h(t) is obtained by taking the inverse Fourier transform of \(H(\nu )\). The parameter γ is small and real valued and works as noise filtering. The time of the peak value of h(t) is the difference in time between \(y_{1} (t)\) and \(y_{0} (t)\), i.e., T_{1}T_{0}. The refractive index is determined as [8]
$$n = \frac{{c_{0} (T_{1}  T_{0} )}}{z} + 1.$$
(5)
We estimate the error in the refractive index as:
$$\Delta n = (n  1)\frac{\Delta z}{z} + \frac{{c_{0} \Delta (T_{0}  T_{1} )}}{z},$$
(6)
where z is the thickness of the box, and ∆z its error; ∆ (T_{1}T_{0}) is the error in the determined time difference.
The imaginary part, κ, is determined from the damping of \(y_{1} (t)\) relative to \(y_{0} (t)\) as described in [13]. The ratio of the signals’ Fourier transforms is analyzed,
$$\frac{{\left {Y_{1} (\nu )} \right}}{{\left {Y_{0} (\nu )} \right}} = \left {H(0)} \right\exp (  {{(2\pi \kappa z\nu )} \mathord{\left/ {\vphantom {{(2\pi \kappa z\nu )} {c_{0} }}} \right. \kern\nulldelimiterspace} {c_{0} }})\beta_{0} \beta_{1}^{  1} ,$$
(7)
where \(\beta_{0}\) and \(\beta_{1}\) are functions that that take into account the effects such as finite sample size and finite distance between the antennas [13]. These coefficients have small frequency dependence. We neglect all frequency dependence in H except that from the attenuation in the wood chips, which is given by the slope \(\alpha =  2\pi \kappa z/c_{0}\). Taking the natural logarithm of Eq. 7, we get that \(\ln \left( {\left {{{Y_{1} (\nu )} \mathord{\left/ {\vphantom {{Y_{1} (\nu )} {Y_{0} (\nu )}}} \right. \kern\nulldelimiterspace} {Y_{0} (\nu )}}} \right} \right)\) depends linearly on ν with the slope \(\alpha =  2\pi \kappa z/c_{0}\). Thus, we determine the slope, \(\alpha\) by the least square method and then calculate \(\kappa = {{  \alpha c_{0} } \mathord{\left/ {\vphantom {{  \alpha c_{0} } {(2\pi z)}}} \right. \kern\nulldelimiterspace} {(2\pi z)}}\). We estimate the error in \(\kappa\) as
$$\frac{\Delta \kappa }{\kappa } = \frac{\Delta \alpha }{\alpha } + \frac{\Delta z}{z},$$
(8)
where \(\Delta \alpha\) is the error in the fitted coefficient of a straight line by linear regression (see e.g., [17]) and \(\Delta z\) is the error in the thickness of the wood chips in the test box.
The error in the permittivity is obtained from putting the errors in Eq. 8 into Eq. 3,
$$\begin{gathered} \Delta \varepsilon^{\prime} = 2n\Delta n + 2k\Delta k, \hfill \\ \Delta \varepsilon^{\prime\prime} = 2n\Delta k + 2k\Delta n. \hfill \\ \end{gathered}$$
(9)
To describe the anisotropy of \(\tilde{\varepsilon }\), we the calculate the anisotropy ratios
$$\begin{gathered} k^{\prime} = {{\varepsilon^{\prime}_{{{\text{hor}}}} } \mathord{\left/ {\vphantom {{\varepsilon^{\prime}_{{{\text{hor}}}} } {\varepsilon^{\prime}_{{{\text{ver}}}} }}} \right. \kern\nulldelimiterspace} {\varepsilon^{\prime}_{{{\text{ver}}}} }}, \hfill \\ k^{\prime\prime} = {{\varepsilon^{\prime\prime}_{{{\text{hor}}}} } \mathord{\left/ {\vphantom {{\varepsilon^{\prime\prime}_{{{\text{hor}}}} } {\varepsilon^{\prime\prime}_{{{\text{ver}}}} }}} \right. \kern\nulldelimiterspace} {\varepsilon^{\prime\prime}_{{{\text{ver}}}} }}, \hfill \\ \end{gathered}$$
(10)
where \(\varepsilon^{\prime}_{{{\text{hor}}}}\)(\(\varepsilon^{\prime\prime}_{{{\text{hor}}}}\)) is the real (imaginary) part of \(\tilde{\varepsilon }\) for horizontal Efield, and \(\varepsilon^{\prime}_{{{\text{ver}}}}\)(\(\varepsilon^{\prime\prime}_{{{\text{ver}}}}\)) for vertical Efield, respectively.
Effective medium theory
Effective medium models (or dielectric mixing models) are used for calculating the complex dielectric permittivity, \(\tilde{\varepsilon } = \varepsilon^{\prime} + {{i}}\varepsilon^{\prime\prime}\), of inhomogeneous media of different materials, in which the length scale of the inhomogeneities is small compared to the wavelength of the electromagnetic radiation [9, 10]. Effective medium models like Bruggeman and Maxwell Garnett are derived from Maxwell’s equations for electromagnetic wave propagation and are quasistatic approximations. The models contain physical parameters and a comparison between modeled and experimentally determined permittivity can, hence, give insight into the physics of the inhomogeneous mixing. We use three different effective medium models that include inhomogeneities that give rise to anisotropic permittivity. All models have as input parameters the permittivity and volume fractions of the different materials (in our case air, wood and water) and the depolarization factors that model the effect of anisotropic inhomogeneities (in our case the elongation of the wood chips and the air and water in between them). We use literature values for the permittivity of wood at different moisture content in the models, together with volume fractions that we estimate from the measured masses.
The relaxations mechanism of wood with moisture is complex (see e.g., [18]). To describe the effect on the permittivity we use the concept of bounded and free water [19]. At moisture contents below the fiber saturation points the water molecules are adsorbed to the cell walls and do not interact as free water molecules with the Efield. Above the fiber saturation point, some water molecules are in the cell wall cavities or on the wood chips’ surfaces and interact with the Efield as in free water. The fiber saturation point is at moisture contents of around 30% [19].
Wood chips have anisotropic permittivity. There are two physical properties that may give anisotropic effective permittivity. The first is that wood has different permittivity in different directions, or more precisely, for the Efield 1) radial (to the annual rings); 2) tangential (to the annual rings), and 3) parallel to the fiber direction or tangential to the stem [20]. The second is that the wood chips that are in the shape of plates or needles are oriented by gravity such that the effective permittivity becomes different for the Efield parallel or perpendicular to gravity [8]. We use an isotropic equivalent permittivity in the models. In [21], different methods to describe an isotropic equivalent permittivity of wood were compared. It was found that the arithmetic mean of the permittivity of the three main axes differs only by a few percent from other physically motivated methods. We therefore use the arithmetic mean in the cases when we use an isotropic permittivity of wood, i.e., \(\tilde{\varepsilon }_{{\text{w}}} = 2\varepsilon_{{_{{\parallel \user2{,w}}} }} + \varepsilon_{{_{{ \bot \user2{,w}}} }}\) where \(\varepsilon_{{_{{\parallel \user2{,w}}} }}\) and \(\varepsilon_{{_{{ \bot \user2{,w}}} }}\) are the permittivity of the wood with the Efield parallel and perpendicular to the fibers, respectively.
The first model that we use is a multiphase Bruggeman (BGMPA) effective medium model for spheroidal inclusions [10, 22]. This model has the advantage of being symmetric in the constituents, i.e., the analytical expressions do not distinguish between a host material, like air, and a particle material, like wood. This feature is advantageous in our case, since the volume fractions of the constituents may vary with water content. The spheroidal inclusions—in contrast to spherical ones—give anisotropic effective permittivity. The BGMPA gives an effective complex permittivity for three directions, \(\tilde{\varepsilon }_{e,x}\), \(\tilde{\varepsilon }_{e,y}\), and \(\tilde{\varepsilon }_{e,z}\), which are obtained by solving
$$\sum\limits_{n = 1}^{N} {f_{n} \frac{{\tilde{\varepsilon }_{n}  \tilde{\varepsilon }_{e,j} }}{{\tilde{\varepsilon }_{e,j} + L_{n,j} \left( {\tilde{\varepsilon }_{n}  \tilde{\varepsilon }_{e,j} } \right)}}} = 0$$
(11)
for \(\tilde{\varepsilon }_{e,j}\) for \(j = x,y,z\). In Eq. 11, N is the number of different phases or materials, \(f_{n}\) is the volume fraction of material n, \(\tilde{\varepsilon }_{n}\) is the complex permittivity of material n, and \(L_{n,j}\) is the depolarization factor for inclusions of material n in direction j. Notice, that \(f_{1} + f_{2} + \ldots f_{N} = 1\) and that \(L_{n,x} + L_{n,y} + L_{n,z} = 1\) for each n. For isotropic inclusions, \(L_{n,x} + L_{n,y} + L_{n,z} = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern\nulldelimiterspace} 3}\). For infinitely long needles in the zdirection, \(L_{n,x} + L_{n,y} = 0\) and \(L_{n,z} = 1\). When solving Eq. 11 an equation of order N in \(\tilde{\varepsilon }_{e,j}\) has to be solved and consequently multiple roots are obtained. We solve the equations analytically for N = 2 and numerically for N = 3. Only one root is physically correct and we use the method in [23] to select the correct one. In the modeling below we use \(\tilde{\varepsilon }_{1}\) for the permittivity air, \(\tilde{\varepsilon }_{2}\) for wood, and \(\tilde{\varepsilon }_{3}\) for water, respectively.
The second model that we use is the Maxwell Garnett model for layered spheroidal inclusions (MGLSI) [24]. It is derived for a geometry of a spheroidal with a coating in a host material. It resembles the wood particles in air with the assumption of free water on the particles’ surface. It gives the effective complex permittivity [24]:
$$\begin{gathered} \tilde{\varepsilon }_{e,j} = \tilde{\varepsilon }_{1} + \frac{{\tilde{\varepsilon }_{1} \sigma_{j} }}{{1  L_{3,j} \sigma_{j} }}, \hfill \\ \sigma_{j} = \left( {f_{2} + f_{3} } \right)\left[ {\left( {\tilde{\varepsilon }_{3}  \tilde{\varepsilon }_{1} + \left( {\tilde{\varepsilon }_{3} + L_{3,j} (\tilde{\varepsilon }_{1}  \tilde{\varepsilon }_{3} )} \right)\frac{{(\tilde{\varepsilon }_{2}  \tilde{\varepsilon }_{3} )f_{2} /(f_{2} + f_{3} )}}{{\tilde{\varepsilon }_{3} + L_{2,j} (\tilde{\varepsilon }_{2}  \tilde{\varepsilon }_{3} )}}} \right)} \right. \hfill \\ \times \left. {\left( {\left( {\tilde{\varepsilon }_{1} + L_{3,j} (\tilde{\varepsilon }_{3}  \tilde{\varepsilon }_{1} ) + L_{3,j} (1  L_{3,j} } \right)(\tilde{\varepsilon }_{3}  \tilde{\varepsilon }_{1} )\frac{{(\tilde{\varepsilon }_{2}  \tilde{\varepsilon }_{3} )f_{2} /(f_{2} + f_{3} )}}{{\tilde{\varepsilon }_{3} + L_{2,j} (\tilde{\varepsilon }_{2}  \tilde{\varepsilon }_{3} )}}} \right)^{  1} } \right], \hfill \\ \end{gathered}$$
(12)
where \(\tilde{\varepsilon }_{1}\) is the complex permittivity of the host material (air), \(\tilde{\varepsilon }_{3}\) of the outer layer material (water) and \(\tilde{\varepsilon }_{2}\) of the core material (wood). \(f_{1}\), \(f_{2}\), and \(f_{3}\) are the volume fractions of materials 1, 2, and 3, respectively, and, as for the Bruggeman model, \(f_{1} + f_{2} + f_{3} = 1\). \(L_{j,n}\) are as before the depolarization factors of material n in the direction j (j = x, y, or z).
The third model is an anisotropic Maxwell Garnett model for spheroidal inclusions of multiple types and multiple depolarization factors (MGMPA) [25]. The inclusion can also have different orientation and distributions of orientation. Paz et al. [11] used the Maxwell Garnett model for sawdust with randomly oriented anisotropic particles, causing an isotropic effective permittivity. We use the model assuming that all inclusions are aligned to the coordinate system of the measurements system (which is given by the orientation of the antennas, the box and the wood chips’ orientation by gravity). The effective permittivity in the x,y, and zdirections is given by the diagonal elements of the matrix \({{{\upvarepsilon}}}_{e}\),
$$\begin{gathered} {{{\upvarepsilon}}}_{e} = \tilde{\varepsilon }_{1} {\mathbf{I}} + \frac{1}{3}\sum\limits_{n = 2}^{3} {\tilde{\varepsilon }_{1} f_{n} \left[ {\sum\limits_{j = 1}^{3} {{{{\upeta}}}_{n} ({\mathbf{I}} + {{{\upeta}}}_{n} L_{n,j} )^{  1} } } \right]} \hfill \\ \times \left\{ {{\mathbf{I}}  \frac{1}{3}\sum\limits_{n = 2}^{3} {f_{n} \left[ {\sum\limits_{j = 1}^{3} {L_{n,j} {{{\upeta}}}_{n} ({\mathbf{I}} + {{{\upeta}}}_{n} L_{n,j} )^{  1} } } \right]} } \right\}^{  1} , \hfill \\ \end{gathered}$$
(13)
where \(\tilde{\varepsilon }_{1}\) is the permittivity of the host material (air), \({\mathbf{I}}\) is the identity matrix and the matrix \({{{\upeta}}}_{n}\) is diagonal in our case with the diagonal elements
$${{{\upeta}}}_{n,jj} = \frac{{16f_{n} (\tilde{\varepsilon }_{n}  \tilde{\varepsilon }_{1} )}}{{3(\tilde{\varepsilon }_{1} + L_{n,j} (\tilde{\varepsilon }_{n}  \tilde{\varepsilon }_{1} ))}},$$
(14)
and j = 1,2,3 corresponds to the x, y, and zdirection, respectively. As before, \(L_{n,j}\) is the depolarization factor of material n in direction j and \(f_{n}\) is the volume fraction of material n.
Modeling wood chips moisture content
The effective medium models above require that the volume fractions of the different materials are known. From the experiments we get the mass of the wood chips in the sample box, \(m^{\prime}_{{\text{w}}}\), for different moisture contents, \(M_{{\text{d}}}\) (we omit the index n which refers to moistening cycle). The dry weight of the wood chips in the sample box, \(m^{\prime}_{{\text{d}}}\), can be calculated from \(M_{{\text{d}}}\) and \(m^{\prime}_{{\text{w}}}\) using Eq. 1. The weight of the water is then \(m^{\prime}_{{{\text{H}}_{{{2}}} {\text{O}}}} = m^{\prime}_{{\text{w}}}  m^{\prime}_{{\text{d}}}\). The volume of the sample box containing the wood chips is V_{sb}.
Below the fiber saturation point (which we take to be M_{d} = 30% [19]) there is no free water and the mixture consists of air and wood. The volume fraction of wood becomes
$$f_{2} = \frac{{m^{\prime}_{{\text{w}}} }}{{\rho_{{\text{w}}} V_{{{\text{sb}}}} }},$$
(15)
where \(\rho_{{\text{w}}}\) is the density of the wood, which depends on M_{d} [26]
$$\rho_{{\text{w}}} = \rho_{{{\text{H}}_{{\text{2}}} {\text{O}}}} G_{{\text{w}}} (1 + M_{{\text{d}}} ),$$
(16)
where G_{w} is the specific gravity of wood and \(\rho_{{{\text{H}}_{{{2}}} {\text{O}}}}\) = 1 g cm^{−3} is the density of water. The density \(\rho_{{\text{w}}}\) is an average of all the wood chips, similarly to the depolarization factors \(L_{n,j}\). The variation in \(\rho_{{\text{w}}}\) will be relatively small since all wood chips are of conifer wood.
The volume fraction of air is obtained as \(f_{1} = 1  f_{2}\) below the fiber saturation point (i.e., we set f_{3} = 0, in the effective medium model above, since it is the volume fraction of free water).
Above the fiber saturation point, we assume that the wood’s volume fraction, f_{2}, is the same as at the fiber saturation point (M_{d} = 30%); the mass of the free water, denoted \(m^{\prime}_{{{\text{H}}_{{\text{2}}} \user2{O,fr}}}\) is calculated as \(m^{\prime}_{{{\text{H}}_{{\text{2}}} \text{O,fr}}} = m^{\prime}_{{\text{w}}}  m^{\prime}_{{\text{w}}} (30\% )\) and the volume fraction for water becomes
$$f_{3} = \frac{{m^{\prime}_{{{\text{H}}_{{{2}}} \user2{O,fr}}} }}{{\rho_{{{\text{H}}_{{{2}}} {\text{O}}}} V_{{{\text{sb}}}} }}$$
(17)
and the volume fraction of air becomes \(f_{1} = 1  f_{2}  f_{3}\).