Introduction

Solid wood can be compressed in radial direction under certain conditions, when wood material is softened above its glass transition temperature (T g) which is highly dependent on its moisture content [1–3]. Rather than compressing solid wood throughout, the densification can be targeted on the surface only [4–6]. Therefore, the process can be short, only few minutes, and less energy is needed for the modification. The surface densification process has several phases. First, a relatively dry wood specimen is heated only on one surface in contact with a hot plate and compressed by an unheated plate during a process phase called closing time. After the unheated loading plate has contacted with the stops of the device, the press remains closed and the heating continues during the holding time phase. Furthermore, a cooling system is turned on during the phase of cooling time, while the load is maintained until the temperature is below a fixed value. It is well known that wood panel boards (e.g. particle board) suffer immediately spring back after pressing, if the press is opened when wood material is in soften state. Cooling phase decreases the immediately spring back after the press is opened. However, some spring back still occurs [7, 8] because of the stored internal stresses. An important feature of the process is that the compression in radial direction does not produce major fractures [4] of the cell walls above the glass transition point. The surface densification affects both the hygro-thermal behaviour of wood and its mechanical properties. The result of this process can be an environmental friendly product, since no toxic chemicals needed and only heat and moisture, showing several potentialities in different sectors of the wood industry [9].

In the last few years, the experimental research on wood surface densification has pointed out the influence of various process parameters (wood species, initial moisture content, press temperature, closing time, holding time and compression ratio) on the development of density profiles in the treated samples [4, 5]. This knowledge is very useful to obtain optimal properties for the treated wood. Due to the large amount of experiments required to check the influence of the process parameters for wood surface densification, the experimental approach may be rather time consuming and laborious. For this reason, numerical modelling can significantly help to optimize the experimental work. To numerically simulate the full process of wood surface densification, an accurate hygro-thermal model is required for the description of simultaneous transport of heat and moisture inside the wood. In the existing literature, several numerical models for the hygro-thermal behaviour of wood-based composites under hot-pressing processes are available [10–14]. In addition, a large amount of scientific papers are available on the numerical modelling of drying and high-temperature treatment of solid wood as, among others [15–20]. These earlier models provide precious suggestions for the hygro-thermal modelling of surface densification in solid wood.

The importance of temperature and moisture content levels coupled with heat and mass transfer in wood is pointed out in an earlier study [21]. In the same study, the use of computational modelling for a realistic prediction of the quantities inside the wood is strongly suggested. Multi-physics is a primary tool to be used for solving the coupled problems occurring in wood at high temperatures, and the exploitation of material properties assessed by micromechanics and multiscale modelling can improve the understanding of these phenomena in future [22].

In the present work, a three-dimensional FEM model for simulation of hygro-thermal phenomena characterizing wood surface densification is used which is a specialization of the multiphase model previously proposed in [17] by adding the modelling of convection used in [11, 12] and [14–16]. Employed mathematical model is based on the conservation equations for energy, air and moisture content, taking into account both the hygroscopic (or bound) water and the water vapour phase, while the free water is neglected as moisture contents remain below the fibre saturation point. Flow of bound water through the solid occurs only by molecular diffusion while air and steam transport includes both diffusion and convection. System of three coupled PDEs describing the process is solved through the commercial FEM code Abaqus [23], where a special finite element is programmed into the user subroutine Uel [24].

The presented model is used to numerically simulate the distribution of moisture content and temperature in wood samples tested under surface densification within a previous study [4]. The correspondence between the available peaks of density measured during the experiments and the numerical peaks of moisture content and temperature is discussed. The proposed computational approach is particularly suitable as the first step for a 3D full hygro-thermo-mechanical FEM simulation of wood surface densification also when other mechanical loads are applied.

A variant of previous multiphase models for wood at high temperatures

The wood densification process involves the simultaneous transfer of heat and mass. Thus, it is necessary to analyse moisture, air and heat transport throughout the material. The existing literature on this topic has been re-visited and a model has been selected which is a specialization of the one at proposed in [17] for high-temperature treatment of wood. This model has been slightly modified and then integrated with the approaches presented in [11, 12] and [14–16] for the modelling of convection. In the following, the matrix and vector forms of the tensors involved in the partial differential equations governing the problem will be used.

In this work, the following assumptions were done:

  1. 1.

    The values of bound water diffusion and thermal conduction parameters in radial and tangential directions are assumed to be equal. Since wood exhibits a larger anisotropy between the cross sectional plane and the longitudinal direction (compared to the anisotropy in the cross sectional plane), different diffusion parameters are used in the longitudinal direction [25]. However, the material parameters for diffusive and convective fluxes for vapour and air are assumed to be the same in all directions [17].

  2. 2.

    The earlywood and latewood rings are not considered as different materials [25].

  3. 3.

    Since the positions of wood piths were located very far from the cut samples in order to have the growth ring as parallel as possible, a rectangular coordinate system (with tangential, radial and longitudinal directions along the X-Y-Z axis) is used in the numerical simulations.

  4. 4.

    At each point of the material, local equilibrium between bound water and vapour water is assumed, with a sorption isotherm that couples the moisture contents in the solid and the vapour phase.

  5. 5.

    Ideal gas behaviour is considered for the gaseous phases.

According to an earlier study [17], the mass balance equations for water and dry air are described by Eqs. 1 and 2, respectively:

$$ \frac{{\partial \varrho_{b} }}{\partial t} + \varepsilon \frac{{\partial \varrho_{v} }}{\partial t} + \nabla \cdot ({\mathbf{J}}_{b} + {\mathbf{J}}_{v} ) = 0 $$
(1)
$$ \varepsilon \frac{{\partial \varrho_{a} }}{\partial t} + \nabla \cdot {\mathbf{J}}_{a} = 0 $$
(2)

where J b , J v and J a represent the flux vectors of the bound water (b), water vapour (v), and air (a) respectively, \( \varrho_{b} \) is the bound water concentration with the dry wood (d) volume as reference, \( \varrho_{v} \) and \( \varrho_{a} \) are the concentrations of the gaseous phases referred to the lumens volume, so porosity \( \varepsilon = 1 - \varrho_{d} /1500 \) is the volume of voids with respect to the one of dry wood whose density is \( \varrho_{d} \). Finally t is the time, ∇ collects the derivatives with respect to the coordinates x, y and z in space and (·) indicates the dot product.

Let \( M \) be the moisture content per unit mass of dry wood, Eq. 1 becomes:

$$ \varrho_{d} \frac{\partial M}{\partial t} + \nabla \cdot ({\mathbf{J}}_{b} + {\mathbf{J}}_{v} ) = 0 $$
(3)

By exploiting Fick’s law, the flux of bound water becomes \( {\mathbf{J}}_{b} = - \varrho_{d} {\mathbf{D}}_{b} \nabla M_{b} \) where \( M_{b} = \varrho_{b} /\varrho_{d} = M - \varepsilon \varrho_{v} /\varrho_{d} \) and D b represents the diffusion matrix of bound water. The diffusion coefficient in the radial and tangential directions is calculated as in [17] and its value is multiplied by an amplification factor of 2.5 when referring to the longitudinal direction.

As pointed out in [11] and [12], during high-temperature processes and in particular in hot-pressing cases, the diffusion of air and steam is a minor phenomenon compared to convection. According to [11], if the velocity of the gaseous mixture is expressed by means of the Darcy law, the diffusive and convective fluxes for vapour and air are defined as

$$ {\mathbf{J}}_{v}^{d} = - \frac{{m_{v} }}{RT}{\mathbf{D}}_{\text{eff}} \nabla P_{v} , \quad {\mathbf{J}}_{v}^{c} = - \frac{{m_{v} }}{{RT\eta_{g} }}{\mathbf{K}}_{g} P_{v} \nabla P_{g} $$
(4)
$$ {\mathbf{J}}_{a}^{d} = - \frac{{m_{a} }}{RT}{\mathbf{D}}_{eff} \nabla P_{a} , \quad {\mathbf{J}}_{a}^{c} = - \frac{{m_{a} }}{{RT\eta_{g} }}{\mathbf{K}}_{g} P_{a} \nabla P_{g} $$
(5)

where \( {\mathbf{D}}_{\text{eff}} \) represents the effective diffusivity matrix and \( {\mathbf{K}}_{g} = k_{g} {\mathbf{I}} \) the relative permeability matrix for the gas flow, \( {\mathbf{I}} \) being the identity matrix and \( k_{g} \) a relative permeability coefficient. \( {\mathbf{D}}_{\text{eff}} =\varvec{\delta}D_{va} \) is obtained from the so-called inter-diffusion coefficient of an air–vapour mixture \( D_{va} \) used in [16] where δ = δ I is a reduction matrix accounting for the porous structure and the tortuous path of wood, δ being the attenuation factor.

Furthermore, in Eqs. (4) and (5) m a and P a , m v and P v represent the molar mass and the partial pressure for the dry air and the vapour, respectively; R and T are the ideal gas constants and the temperature; \( P_{g} = P_{v} + P_{a} \) and \( \eta_{g} \) represent the total pressure and the dynamic viscosity of the air–vapour mixture, respectively. Viscosity \( \eta_{g} \) is taken from [26], while the partial vapour pressure P v is related to the so-called saturation partial pressure of water vapour P sv by means of the following sorption isotherm used in [17] as a function of moisture content and temperature:

$$ P_{v } = P_{sv} \left( T \right)\psi \left( {M,T} \right) $$
(6)

where \( P_{sv} = { \exp }\left( {25.5058 - \frac{5204.9}{T}} \right) \) and

$$ \psi = { \exp }\left[ {\left( {17.884 - 0.1423T + 23.63 \times 10^{ - 5} T^{2} } \right)\left( {1.0237 - 67.4 \times 10^{ - 5} T} \right)^{92M} } \right] $$

The temperature dependence of the sorption isotherm in wood is an important phenomenon in particular in the presence of high temperatures (see [27–29] for more details).

To complete the formulation of the problem, the conservation of energy is expressed in the form used in [17]:

$$ \varrho c_{p} \frac{\partial T}{\partial t} = \nabla \cdot \left( {{\mathbf{K}}\nabla T + h_{b} {\mathbf{J}}_{b} + h_{v} {\mathbf{J}}_{v} + h_{a} {\mathbf{J}}_{a} } \right) $$
(7)

where K is the thermal conductivity matrix, \( \varrho c_{p} = \varrho_{d} (c_{pd} + M_{b} c_{pw} ) + \varepsilon \varrho c_{pg} \) is the total heat capacity, \( c_{pd} \) and \( c_{pw} \) being the specific heats of the dry wood and water and \( \varrho c_{pg} = \varrho_{v} c_{pv} + \varvec{ }\varrho_{a} c_{pa} \) refers to the gas mixture depending of the specific heats \( c_{pa} \) and \( c_{pv} \) of the two gas phases. Finally \( h_{b} \), \( h_{v} \), \( h_{a} \) are the specific enthalpies of air, steam and bound water. The thermal conductivity K is calculated by the classical approach described in [30], starting from the value in the transversal direction. For the longitudinal direction amplification by 2.5 is considered. Standard expressions for the enthalpies of bound water and the different gaseous phases are taken from [16, 17]. Finally for the specific heats, the values proposed in [31] are used for steam and air, while the expressions reported in [17] are adopted for dry wood and water.

Introducing also the hypothesis of ideal gas behaviour for air and steam, after some manipulations, Eqs. (2) and (3) can be rewritten as

$$ \varepsilon \frac{{\partial \varrho_{a} }}{\partial t} = \nabla \cdot \left( {{\mathbf{D}}_{AA} \nabla \varrho_{a} + {\mathbf{D}}_{AM} \nabla M + {\mathbf{D}}_{AT} \nabla T} \right) $$
(8)
$$ \varrho_{d} \frac{\partial M}{\partial t} = \nabla \cdot \left( {{\mathbf{D}}_{MA} \nabla \varrho_{a} + {\mathbf{D}}_{MM} \nabla M + {\mathbf{D}}_{MT} \nabla T} \right) $$
(9)

where the various matrices are:

$$ {\mathbf{D}}_{AA} = \frac{{P_{a} }}{{\eta_{g} }}{\mathbf{K}}_{g} + {\mathbf{D}}_{\text{eff}} ,\quad \quad {\mathbf{D}}_{AM} = \frac{{m_{a} }}{RT} \frac{{P_{a} }}{{\eta_{g} }}P_{sv} \frac{\partial \psi }{\partial M}{\mathbf{K}}_{g} ,\quad \quad {\mathbf{D}}_{MA} = \frac{{m_{v} P_{v} }}{{m_{a} \eta_{g} }}{\mathbf{I}}, $$
$$ {\mathbf{D}}_{AT} = \frac{{m_{a} }}{RT} \frac{{P_{a} }}{{\eta_{g} }}{\mathbf{K}}_{g} \left( {\varrho_{a} \frac{R}{{m_{a} }} + \frac{{\partial P_{sv} }}{\partial T}\psi + \frac{\partial \psi }{\partial T}P_{sv} } \right) + \frac{{\varrho_{a} }}{T}{\mathbf{D}}_{\text{eff}} , $$
$$ {\mathbf{D}}_{MMv} = \frac{{m_{v} }}{RT}\left( { \frac{{P_{v} }}{{\eta_{g} }}{\mathbf{K}}_{g} + {\mathbf{D}}_{\text{eff}} } \right)P_{sv} \frac{\partial \psi }{\partial M}, \quad{\mathbf{D}}_{MMb} = \varrho_{d} \left( {1 - \frac{\varepsilon }{{\varrho_{d} }}\frac{{m_{v} }}{RT}P_{sv} \frac{\partial \psi }{\partial M}} \right){\mathbf{D}}_{b} , $$
(10)
$$ {\mathbf{D}}_{MTb} = \frac{{\varepsilon m_{v} }}{RT}\left\{ {\frac{{P_{v} }}{T} - \left( {\frac{{\partial P_{sv} }}{\partial T}\psi + \frac{\partial \psi }{\partial T}P_{sv} } \right)} \right\}{\mathbf{D}}_{b} , $$
$$ {\mathbf{D}}_{MTv} = \frac{{m_{v} }}{RT}\left\{ {\frac{{P_{v} }}{{\eta_{g} }}{\mathbf{K}}_{g} \frac{{\varrho_{a} R}}{{m_{a} }} + \left( {\frac{{P_{v} }}{{\eta_{g} }}{\mathbf{K}}_{g} + {\mathbf{D}}_{eff} } \right)\left( {\frac{{\partial P_{sv} }}{\partial T}\psi + \frac{\partial \psi }{\partial T}P_{sv} } \right)} \right\}, $$
$$ {\mathbf{D}}_{MM} = {\mathbf{D}}_{MMv} + {\mathbf{D}}_{MMb} , \quad {\mathbf{D}}_{MT} = {\mathbf{D}}_{MTv} + {\mathbf{D}}_{MTb} $$

By using the same notation, the energy balance is

$$ \varrho c_{p} \frac{\partial T}{\partial t} = \nabla \cdot \left( {{\mathbf{D}}_{TA} \nabla \varrho_{a} + {\mathbf{D}}_{TM} \nabla M + {\mathbf{D}}_{TT} \nabla T} \right) $$
(11)

with the matrices \( {\mathbf{D}}_{TA} = h_{a} {\mathbf{D}}_{AA} + h_{v} {\mathbf{D}}_{MA} \) , \( {\mathbf{D}}_{TM } = h_{a} {\mathbf{D}}_{AM} + h_{v} {\mathbf{D}}_{MMv} + h_{b} {\mathbf{D}}_{MMb}, \quad {\mathbf{D}}_{TT } = {\mathbf{K}} + h_{a} {\mathbf{D}}_{AT} + h_{v} {\mathbf{D}}_{MTv} + h_{b} {\mathbf{D}}_{MTb}\).

The direct consequence of Eq. 6 is that sorption phenomena (see [32] and [28] for a systematic description) are not considered. To include the simulation of sorption, a reformulation of the model and further computational work are needed.

Boundary conditions

Since the hot plate is assumed to be impervious to gas, no gas fluxes are imposed, while a thermal resistance for the contact is assumed as in [12] and [14] that imply the following condition

$$ - {\mathbf{K}}\nabla T \cdot {\mathbf{n}} = h_{Tp} (T - T_{\text{plate}} ) $$
(12)

where \( h_{Tp} \) represents the heat transfer coefficient at the contact between the wood sample and the hot plate and \( {\mathbf{n}} \) is the outside normal to the surface.

According to [14], the following boundary conditions are imposed for the faces in contact with the ambient air:

$$ - \left( {{\mathbf{D}}_{AA} \nabla \varrho_{a} + {\mathbf{D}}_{AM} \nabla M + {\mathbf{D}}_{AT} \nabla T} \right) \cdot {\mathbf{n}} = h_{p} \frac{{\varrho_{a} }}{{\eta_{\text{g}} }}(P_{\text{g}} - P_{\text{amb}} ) $$
(13)
$$ - \left( {{\mathbf{D}}_{MA} \nabla \varrho_{a} + {\mathbf{D}}_{MM} \nabla M + {\mathbf{D}}_{MT} \nabla T} \right) \cdot {\mathbf{n}} = h_{p} \frac{{\varrho_{v} }}{{\eta_{\text{g}} }}(P_{\text{g}} - P_{\text{amb}} ) $$
(14)
$$ - {\mathbf{K}}\nabla T \cdot {\mathbf{n}} = h_{T} \left( {T - T_{\text{amb}} } \right) $$
(15)

where the subscript ‘amb’ refers to the quantities related to the ambient.

The values used for the different material parameters described above are summarized in Table 1.

Table 1 Material parameters for the wood surface densification modelling
Full size table

FEM implementation in Abaqus code

In order to use the model in the FEM commercial code Abaqus, a standard eight-node isoparametric brick element with the same interpolation for all the unknown variables is implemented in the user subroutine Uel [24]. A weak form of the problem presented in the previous section is obtained multiplying Eqs. (8), (9) and (11) by the three weight functions w A , w M and w T and then integrating over the volume V. Introducing a linearization of the functions at the time t starting from the corresponding values at the time t + ∆t and by applying the product rule and the divergence theorem, the following equations are obtained:

$$ \begin{gathered} \frac{1}{\Updelta t}\int\limits_{V} {w_{A}\varepsilon \left({\varrho_{a , t + \Updelta t} - \varrho_{a , t}}\right)} + \int\limits_{V} {{\boldsymbol{\nabla}} w_{A} \cdot\left({{\mathbf{D}}_{AA} {\boldsymbol{\nabla}} \varrho_{a} + {\mathbf{D}}_{AM} {\boldsymbol{\nabla}} M + {\mathbf{D}}_{AT}{\boldsymbol{\nabla}} T} \right)} \hfill \\ - \int\limits_{S}{w_{A} {\mathbf{n}} \cdot \left({{\mathbf{D}}_{AA}{\boldsymbol{\nabla}} \varrho_{a} + {\mathbf{D}}_{AM}{\boldsymbol{\nabla}} M + {\mathbf{D}}_{AT} {\boldsymbol{\nabla}}T} \right)} = 0,\quad \forall w_{A} \hfill \\ \end{gathered} $$
$$ \begin{gathered} \frac{1}{\Updelta t}\int\limits_{V} {w_{M}\varrho_{d} \left({M_{, t + \Updelta t} - M_{, t}} \right)}+\int\limits_{V} {{{\boldsymbol{\nabla}} w_{M}} {\cdot}\left({{{\mathbf{D}}_{MA}} {{\boldsymbol{\nabla}} \varrho_{a}} +{{\mathbf{D}}_{MM}} {{\boldsymbol{\nabla}} {M}} + {{\mathbf{D}}_{MT}} {{\boldsymbol{\nabla}} {T}}} \right)} \hfill \\ - \int\limits_{S} {w_{M} {\mathbf{n}} \cdot\left({{{\mathbf{D}}_{MA}} {{\boldsymbol{\nabla}}}{\varrho_{a}} + {{\mathbf{D}}_{MM}} {{\boldsymbol{\nabla}} {M}} + {{\mathbf{D}}_{MT}} {{\boldsymbol{\nabla}} {T}}} \right)} = 0,\quad \forall w_{M} \hfill \\ \end{gathered} $$
(16)
$$ \begin{gathered} \frac{1}{\Updelta t}\int\limits_{V} {w_{T} \varrho c_{p} \left({T_{, t + \Updelta t} - T_{, t}} \right)} + \int\limits_{V} {{{\boldsymbol{\nabla}} w_{T}} {\cdot} \left({{{\mathbf{D}}_{TA}} {{\boldsymbol{\nabla}} \varrho_{a}} + {{\mathbf{D}}_{TM}} {{\boldsymbol{\nabla}} {M}} + {{\mathbf{D}}_{TT}} {{\boldsymbol{\nabla}} {T}}} \right)} \hfill \\ - \mathop \int \limits_{S}^{{}} w_{T} {\mathbf{n}} {\cdot} \left({{{\mathbf{D}}_{TA}} {{\boldsymbol{\nabla}} \varrho_{a}} + {{\mathbf{D}}_{TM}} {{\boldsymbol{\nabla}} {M}} + {{\mathbf{D}}_{TT}} {{\boldsymbol{\nabla}} {T}}} \right) = 0,\quad\forall w_{T} \hfill \\\end{gathered} $$

in which the differentiations are equally distributed between the unknowns and the test functions w. Finally, S indicates the boundary surface.

Experimental tests

In this paper, some of the tests presented in [4] are simulated. The tested wood material was the clear sapwood with regular growth rings of kiln dried Scots pine (Pinus sylvestris L.) from Southern Finland. An initial moisture content MC0 = 9.6 % was reached in the specimens after 2 months conditioning in relative humidity of 35 %. The initial value for density was \( \varrho_{\text{d}} = 5 30{\text{ kg}}/{\text{m}}^{ 3} \) (RH 65 %, 20 °C). More details are given in [4]. The dimensions of the specimens numerically analysed in the present work are 140 mm (longitudinal), 50 mm (tangential) and 16 mm (radial).

The surface densification was obtained by using an electrically heated plate incorporated into a press tool fitted to a Zwick 1475 testing machine combined with an MTS Premium Elite controller (see Fig. 1). The used device included a water cooling system to enable the heated surface of the press tool to be cooled from the maximum reached temperature to below 100 °C within ~30 s. The plate temperature was monitored by a PT-100 sensor below the stainless steel surface. Mechanical stops forced the specimens to be compressed to a target final thickness (15 mm), when compression ratio of 6 % was reached. Earlier studies have been shown that the material properties can be enhanced rather much even with 1-mm compression, especially hardness [5, 6]. Density profiles were measured at intervals of 0.1 mm through the thickness, using an ATR Density Profilometer DPM201 (1995) using a gamma ray. More details are available in [4]. Some of the results from [4] in terms of density profiles of vertical path of the specimen are reported in the next section.

Fig. 1
figure 1

Device for wood surface densification

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Numerical results and discussion

In the following, the hygro-thermal responses of the specimens with dimensions, initial conditions and loading rate described in the previous section are simulated under different process parameters. The studied tests are simulated following some assumptions:

  • The hygro-thermal analysis is conducted on the undeformed mesh, so the mechanical effects due to the compression rate during closing time are not taken into account. The reason of this choice is that the proposed analysis has to be considered as a first hygro-thermal step to be used within a sequential hygro-thermo-mechanical analysis.

  • The dry wood density during the simulation is assumed to be constant and equal to the initial density (\( \varrho_{\text{d}} = 5 30{\text{ kg}}/{\text{m}}^{ 3} \)).

For all the considered cases, the initial air density is calculated following [25] starting from the vapour pressure value in equilibrium with MC0 at the temperature T 0 = 20 °C, according to the sorption isotherm exploited in this work.

Case a: influence of the holding time

The experiments presented in [4] for a value of initial moisture MC0 = 9.6 %, heating temperature T = 150 °C, closing time = 0.5 min, two different values of holding time (1 min and 10 min) and cooling time = 0.5 min, were simulated. The test with holding time 1 min is taken as reference test for the other cases studied in this work.

The experimental results do not show relevant differences between the reached profiles of density (see Fig. 2). A slightly higher peak of density closer to the hot surface occurs for the case with holding time = 1 min, while the density peak is lower and slightly shifted further away from the surface for the case with holding time = 10 min. These results can be explained by analysing the numerical profiles of temperature and moisture content (Fig. 4) along the vertical path drawn in Fig. 3 and by the curves of T and MC during time (Fig. 5):

Fig. 2
figure 2

Case a: influence of the holding time. Experimental results. Density profiles along the vertical path at the end of the experimental process

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Fig. 3
figure 3

Case a: influence of the holding time. Numerical results for the case with holding time 1 min. Distribution of moisture content on the undeformed mesh at t = 120 s (end of cooling time) and vertical path in the middle of the specimen used for the calculation of temperature and MC profiles (arrow from the surface in contact with the press to the heated surface)

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  • End of holding time For the test with holding time 10 min, the higher temperature reached at the end of the holding time makes the wood surface drier compared to the case with holding time 1 min and this causes a bigger reduction of the moisture content on the surface as well as a higher peak of MC shifted away from the surface itself (Fig. 4). On the contrary, the peak of moisture content for the case with holding time = 1 min occurs close to the heated surface. This behaviour is explained by the plots of T and MC (Fig. 5). As described in [13] for hot-pressed wood composites, also in the present case the sudden increase of temperature on the surface in contact with the hot press at the end of the closing time (30 s) produces a sudden decrease of MC. The gradient of vapour pressure drives the vapour from the surface into the specimen where MC increases. The increase of moisture content on the internal surfaces continues until the end of holding time with a higher growing rate for the case with holding time 10 min. This phenomenon produces the shift of the MC peaks towards the middle of the specimen. At the end of holding time, the peak of MC for the case with holding time 1 min occurs almost in correspondence of the final peak of density while for the case with holding time 10 min, the peak of MC appears shifted into the specimen rather far from the position of the final peak of density.

    Fig. 4
    figure 4

    Case a: influence of the holding time. Numerical values of temperature and moisture content along the vertical path of Fig. 3 at the end of holding time (t = 90 s and t = 630 s) and at the end of cooling time (t = 120 s and t = 660 s). Top: T profiles. Bottom: MC profiles

    Full size image
    Fig. 5
    figure 5

    Case a: influence of the holding time. Numerical curves of temperature and moisture content during time on the heated and middle surface of the specimen until the end of cooling time (t = 660 s). Top: T curves. Bottom: MC curves

    Full size image

  • End of cooling time During the cooling time there is a slight increase of MC on the external surface and a slight decrease on the internal surface with lower peaks of MC that are located almost in the same positions as the density peaks reached at the end of the densification process.

The above results point out that the response in terms of MC during holding time 1 min has a significant influence on the final position of the density peak. On the other hand, the specimen was compressed only 1 mm. In the case of holding time 10 min, the higher peak of MC far from the surface and very close to the middle of the specimen suggests that, in the presence of a larger compression, the final density peak can be more shifted inside the specimen and closer to the reached peak of MC.

Case b: influence of the heating temperature

The test presented in [4] for the case of closing time = 0.5 min, holding time = 1 min and cooling time = 0.5 min with heating temperature T = 200 °C was simulated and compared with the numerical results of the reference test (Case a, closing time = 0.5 min, holding time 1 min, cooling time = 0.5 min, T = 150 °C).

The experimental profiles of density obtained in [4] show a clear and higher peak of density close to the surface for the case with T = 150 °C, while the density peak is lower and slightly shifted further away from the surface for the case with T = 200 °C (see Fig. 6). This behaviour is explained in Fig. 7 by the numerical profiles of temperature and moisture content on the vertical path of Fig. 3:

Fig. 6
figure 6

Case b: influence of the heating temperature. Experimental results. Density profiles along the vertical path at the end of the experimental process

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Fig. 7
figure 7

Case b: influence of the heating temperature. Numerical values of temperature and moisture content along the vertical path of Fig. 3 at the end of holding time (t = 90 s) and at the end of cooling time (t = 120 s). Top: T profiles. Bottom: MC profiles

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  • End of holding time In the case with T = 200 °C, the high temperature reached at the end of the holding time (Fig. 7, top) makes the wood surface drier compared to the case with T = 150 °C and a higher peak of MC occurs. In both the compared cases, the peaks of MC are slightly shifted from the external surface into the specimen and remain close to the heated surface. The plots of T and MC during time (Fig. 8) explain the reason of this behaviour, in fact the moisture content on the middle surface shows a low-rate increase until about 1 min from the beginning of the test and then a low-rate decrease.

    Fig. 8
    figure 8

    Case b: influence of the heating temperature. Numerical curves of temperature and moisture content during time on the heated and middle surface of the specimen until the end of cooling time (t = 120 s). Top: T curves. Bottom: MC curves

    Full size image

  • End of cooling time In both the compared cases, there is a shifting of the MC peaks deeper into the specimen (Fig. 7), but the values of the peaks are reduced due to the decrease of temperature (and increase of moisture content) on the surface (Fig. 8).

The obtained results confirm that the peaks of MC reached at the end of a short holding time influence the final position of the peaks of density. The effect of higher temperatures produces a shift of the MC peak from the heated surface which is also in agreement with the shift of density peak obtained from the experiments (Fig. 6). This MC shift is less relevant than the one obtained in Case (a) for holding time 10 min and T = 150 °C but larger than the shift for holding time 1 min and T = 150 °C. This result also suggests that, in the presence of a larger compression (more than 1 mm), the final density peak for T = 200 °C and holding time = 1 min can be closer to the reached MC.

Some notes on future extension of the model

The future extension of the model for a full hygro-thermo-mechanical simulation of the surface densification process in solid wood requires a study of the coupling between hygro-thermal phenomena and mechanical response. For the hot-press cases, in [10] a three-dimensional heat and mass transfer model was combined with a one-dimensional rheological model. Further features can be exploited by referring to other mechanical approaches used for treatment at high temperatures or modelling of wood under compression. Orthotropic-viscoelastic-mechanosorptive models for wood with mechanical variables depending on hygro-thermal quantities were developed mainly for drying cases, for example in [33]. Interesting results on the nonlinear mechanical behaviour of wood under compression can be found in [34] where a material point method (MPM) was used. Special Abaqus user subroutines for inelastic behaviour of wood under compression are developed in [35].

The model proposed in the present paper can be further developed in Abaqus FEM code by transferring the hygro-thermal results to a mechanical analysis that includes a suitable model for wood under temperature up to 200 °C. A full model can be able to evaluate the reached density and to deeply understand the phenomenon of spring back. This further step is under development.

Conclusion and future work

In this study, the hygro-thermal behaviour of wood during wood surface densification is simulated by using a numerical approach. In this way, it is possible to numerically evaluate the distribution of moisture and temperature inside the material during the treatment, taking into account the phases of bound water and water vapour as well as the dry air involved in the process.

The FEM strategy used to solve the governing equations of the problem is implemented in the Uel subroutine of Abaqus FEM code. The numerical profiles of temperature and moisture content explain the reasons of the differences between the experimental density profiles obtained within a previous study with different values of holding time and heating temperature. The presented model does not reproduce experimental results but it gives indications about the near-correspondence of density peaks and the numerical peaks of moisture content and temperature. Further experimental work is needed to verify the model. The experimental measurement for temperature is rather straightforward, but dynamic moisture content measurement will be extremely challenging. The presented method can be considered as a first hygro-thermal step for a full hygro-thermo-mechanical analysis of wood under surface densification.