**2. Mathematical model of the 2D temperature distribution in logs subjected to refrigeration**

#### **2.1 Mechanism of the temperature distribution in logs during their refrigeration**

The mechanism of the temperature distribution in logs during their refrigeration can be described by the equation heat conduction [2, 5, 8–10].

When the length of the logs is less than their diameter by at least 3–4 times, for the calculation of the change in the temperature in the longitudinal sections of the logs (i.e., along the coordinates *r* and *z* of these sections) during their refrigeration in the air medium, the following 2D mathematical model can be used [23]:

$$\begin{split} \omega\_{\text{w}\sigma}\rho\_{\text{w}}\frac{\partial T(r,z,\tau)}{\partial\tau} &= \lambda\_{\text{wr}} \left[ \frac{\partial^2 T(r,z,\tau)}{\partial r^2} + \frac{1}{r} \frac{\partial T(r,z,\tau)}{\partial r} \right] + \frac{\partial \lambda\_{\text{wr}}}{\partial T} \left[ \frac{\partial T(r,z,\tau)}{\partial r} \right]^2 \\ &+ \lambda\_{\text{wp}} \frac{\partial^2 T(r,z,\tau)}{\partial z^2} + \frac{\partial \lambda\_{\text{wp}}}{\partial T} \left[ \frac{\partial T(r,z,\tau)}{\partial z} \right]^2 + q\_{\text{v}} \end{split} \tag{1}$$

with an initial condition

$$T(r, z, \mathbf{0}) = T\_{\mathbf{w}\mathbf{0}} \tag{2}$$

and boundary conditions for convective heat transfer:

• along the radial coordinate *r* on the logs' frontal surface during the cooling process

$$\frac{\partial T(r,\mathbf{0},\boldsymbol{\tau})}{\partial r} = -\frac{a\_{\mathrm{wp}\cdot\mathrm{fr}}(r,\mathbf{0},\boldsymbol{\tau})}{\lambda\_{\mathrm{wp}}(r,\mathbf{0},\boldsymbol{\tau})}[T(r,\mathbf{0},\boldsymbol{\tau}) - T\_{\mathrm{m}\cdot\mathrm{fr}}(\boldsymbol{\tau})],\tag{3}$$

• along the longitudinal coordinate *z* on the logs' cylindrical surface during the cooling

$$\frac{\partial T(\mathbf{0}, z, \tau)}{\partial \mathbf{z}} = -\frac{a\_{\text{wr-fr}}(\mathbf{0}, z, \tau)}{\lambda\_{\text{wr}}(\mathbf{0}, z, \tau)} [T(\mathbf{0}, z, \tau) - T\_{\text{m-fr}}(\tau)].\tag{4}$$

Equations (1)–(4) represent a common form of a mathematical model of the logs' freezing process.

#### **2.2 Mathematical description of the heat sources in logs during their freezing**

The volume internal heat source in the logs, *q*v, reflects in Eq. (1) the influence of the latent heat of water in the wood on the logs' freezing process. In the available literature for hydrothermal treatment of frozen wood materials, no information can be found on the approaches for quantitative determination of the heat source *q*v.

That is why as a methodology for the determination of *q*<sup>v</sup> during the freezing of logs, in [23], a perspective is used, which is already applied for determination of the volume heat source, *q*vM, during the process of solidification of melted metal [25–28]. According to this methodology, the following equations for determination of the volume internal sources of latent heat separately of the free and the bound water in wood during the logs' freezing process have been obtained [23]:

$$q\_{\text{v-fw}} = K\_{\text{y}-\text{fw}} \rho\_{\text{w}} L\_{\text{cr-ice}} \frac{\partial \varphi\_{\text{ice-fw}}}{\partial \pi},\tag{5}$$

$$q\_{\text{v-bw}} = K\_{\text{y}-\text{bw}} \rho\_{\text{w}} L\_{\text{cr-ice}} \frac{\partial \mu\_{\text{ice-bw}}}{\partial \pi},\tag{6}$$

where *<sup>L</sup>*cr�ice = 3.34�10<sup>5</sup> <sup>J</sup>�kg�<sup>1</sup> [2, 5, 22–24] and

$$K\_{\rm \nu - fw} = \frac{\rho\_{\rm w} - (1 - \Psi\_{\rm ice-fw}) \cdot \rho\_{\rm wUfsp}}{\Psi\_{\rm ice-fw} \cdot \rho\_{\rm w}},\tag{7}$$

$$K\_{\rm \nu \text{--bw}} = \frac{\rho\_{\rm wUfsp} - (1 - \Psi\_{\rm ice\text{-}bw}) \cdot \rho\_{\rm wUnfw}}{\Psi\_{\rm ice\text{-}bw} \cdot \rho\_{\rm w}}.\tag{8}$$

A numerical approach and an algorithm for the computation of <sup>Ψ</sup>ice‐fw and <sup>Ψ</sup>ice‐bw are given in [23]. The difference *<sup>ρ</sup>*<sup>w</sup> � *<sup>ρ</sup>*wUfsp in the right-hand part of Eq. (7) reflects the entire mass of free water (in kg), which is contained in 1 m<sup>3</sup> of the logs. The wood densities *ρ*<sup>w</sup> and *ρ*wUfsp, which participate in Eqs. (7) and (8), are determined above the hygroscopic range according to the equations below [1–19, 29]:

$$
\rho\_{\mathbf{w}} = \rho\_{\mathbf{b}} \cdot (\mathbf{1} + \mathbf{u}),
\tag{9}
$$

$$
\rho\_{\text{wUfsp}} = \rho\_{\text{b}} \cdot \left(\mathbf{1} + \mathfrak{u}\_{\text{fsp}}\right). \tag{10}
$$

The density of the wood *ρ*wUnfw in Eq. (8) is determined according to the following equation in relation to the present entirely liquid quantity of nonfrozen water in the wood, *u*nfw, corresponding to the current wood temperature *Т* < 272.15 K [2, 10]:

$$\rho\_{\text{wUnfw}} = \rho\_{\text{b}} \cdot \frac{1 + u\_{\text{nfw}}}{1 - \frac{S\_{\text{v}}}{100} \left( u\_{\text{fsp}}^{272.15} - u\_{\text{nfw}} \right)},\tag{11}$$

where

$$
u\_{\rm fsp}^{272.15} = 
u\_{\rm fsp}^{293.15} + 0.021,\tag{12}$$

$$u\_{\rm nfw} = 0.12 + \left( u\_{\rm fsp}^{272.15} - 0.12 \right) \cdot \exp \left[ 0.0567 (T - 272.15) \right] \uplus 213.15 \text{K} \cdot \text{s} \, 272.15 \text{K} \, \text{s} \tag{13}$$

information about 2D temperature distribution in logs subjected to freezing was given by the authors [22–24]. In the available literature for hydrothermal treatment of frozen wood materials, there is no information at all about the quantitative determination of the energy characteristics of both the free and bound water during

The aim of the present work is to suggest a methodology for mathematical modeling and research of two mutually connected problems: 2D nonstationary temperature distribution in logs subjected to refrigeration and change in two important energy characteristics of the bound water in logs during its freezing thermal energy of the gradual phase transition of bound water from liquid into solid state and latent thermal energy of the bound water released in the logs

**2. Mathematical model of the 2D temperature distribution in logs**

The mechanism of the temperature distribution in logs during their refrigeration

When the length of the logs is less than their diameter by at least 3–4 times, for the calculation of the change in the temperature in the longitudinal sections

> 1 *r :*

• along the radial coordinate *r* on the logs' frontal surface during the cooling

• along the longitudinal coordinate *z* on the logs' cylindrical surface during the

Equations (1)–(4) represent a common form of a mathematical model of the

*∂λ*wp *∂T*

*<sup>∂</sup>T r*ð Þ *; <sup>z</sup>; <sup>τ</sup> ∂r*

þ *∂λ*wr *∂T*

*T r*ð Þ¼ *; z;* 0 *T*w0 (2)

*<sup>λ</sup>*wpð Þ *<sup>r</sup>;* <sup>0</sup>*; <sup>τ</sup>* ½ � *T r*ð Þ� *;* <sup>0</sup>*; <sup>τ</sup> <sup>T</sup>*<sup>m</sup>‐frð Þ*<sup>τ</sup> ,* (3)

*<sup>λ</sup>*wrð Þ <sup>0</sup>*; <sup>z</sup>; <sup>τ</sup>* ½ � *<sup>T</sup>*ð Þ� <sup>0</sup>*; <sup>z</sup>; <sup>τ</sup> <sup>T</sup>*<sup>m</sup>‐frð Þ*<sup>τ</sup> :* (4)

*<sup>∂</sup>T r*ð Þ *; <sup>z</sup>; <sup>τ</sup> ∂z* <sup>2</sup>

*<sup>∂</sup>T r*ð Þ *; <sup>z</sup>; <sup>τ</sup> ∂r* <sup>2</sup>

(1)

þ *q*<sup>v</sup>

**2.1 Mechanism of the temperature distribution in logs during their**

of the logs (i.e., along the coordinates *r* and *z* of these sections) during their refrigeration in the air medium, the following 2D mathematical model

> *T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ

> > *T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ

can be described by the equation heat conduction [2, 5, 8–10].

*∂*2

*∂*2

and boundary conditions for convective heat transfer:

*<sup>∂</sup><sup>r</sup>* ¼ � *<sup>α</sup>*wp‐frð Þ *<sup>r</sup>;* <sup>0</sup>*; <sup>τ</sup>*

*<sup>∂</sup><sup>z</sup>* ¼ � *<sup>α</sup>*wr‐frð Þ <sup>0</sup>*; <sup>z</sup>; <sup>τ</sup>*

þ *λ*wp

their freezing in the wood or in other capillary porous materials.

at temperatures below �1°C.

*Low-temperature Technologies*

**refrigeration**

can be used [23]:

*<sup>∂</sup>T r*ð Þ *; <sup>z</sup>; <sup>τ</sup>*

with an initial condition

*<sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>λ</sup>*wr

*<sup>∂</sup>T r*ð Þ *;* <sup>0</sup>*; <sup>τ</sup>*

*<sup>∂</sup>T*ð Þ <sup>0</sup>*; <sup>z</sup>; <sup>τ</sup>*

*c*we*ρ*<sup>w</sup>

process

cooling

logs' freezing process.

**114**

**subjected to refrigeration**
