**1. Introduction**

It is known that the duration and the energy consumption of the thermal treatment of frozen logs in winter, aimed at their plasticizing for the production of veneer, depend on the degree of the logs' icing [1–10].

In the accessible specialized literature, there are limited reports about the temperature distribution in frozen logs subjected to defrosting [8, 11–21], and there is very little information about the research of the temperature distribution in logs during their freezing given by the authors only [22–24]. That is why the modeling and the multiparameter study of the freezing process of logs are of considerable scientific and practical interest.

For different engineering, technological, and energy calculations, it is necessary to be able to determine the nonstationary temperature field in logs depending on the temperature of the gaseous or liquid medium influencing them and on the duration of their staying in this medium. Such calculations are carried out using mathematical models, which describe adequately the complex processes of freezing both the free and bound water in the wood. In the specialized literature,

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

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

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

*<sup>∂</sup>ψ*ice‐fw

*<sup>∂</sup>ψ*ice‐bw

*ρ*<sup>w</sup> ¼ *ρ*<sup>b</sup> � ð Þ 1 þ *u ,* (9) *<sup>ρ</sup>*wUfsp <sup>¼</sup> *<sup>ρ</sup>*<sup>b</sup> � <sup>1</sup> <sup>þ</sup> *<sup>u</sup>*fsp *:* (10)

fsp � *<sup>u</sup>*nfw *,* (11)

fsp þ 0*:*021*,* (12)

(13)

*<sup>∂</sup><sup>τ</sup> ,* (5)

*<sup>∂</sup><sup>τ</sup> ,* (6)

*,* (7)

*:* (8)

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

*Modeling of the Energy for Bound Water Freezing in Logs Subjected to Refrigeration*

water in wood during the logs' freezing process have been obtained [23]:

*<sup>q</sup>*<sup>v</sup>‐fw <sup>¼</sup> *<sup>K</sup>ψ*�fw*ρ*w*L*cr‐ice

*<sup>q</sup>*<sup>v</sup>‐bw <sup>¼</sup> *<sup>K</sup>ψ*�bw*ρ*w*L*cr‐ice

*<sup>K</sup>ψ*�fw <sup>¼</sup> *<sup>ρ</sup>*<sup>w</sup> � ð Þ� <sup>1</sup> � <sup>Ψ</sup>ice‐fw *<sup>ρ</sup>*wUfsp

*<sup>K</sup>ψ*�bw <sup>¼</sup> *<sup>ρ</sup>*wUfsp � ð Þ� <sup>1</sup> � <sup>Ψ</sup>ice‐bw *<sup>ρ</sup>*wUnfw

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

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*

> *<sup>ρ</sup>*wUnfw <sup>¼</sup> *<sup>ρ</sup>*<sup>b</sup> � <sup>1</sup> <sup>þ</sup> *<sup>u</sup>*nfw <sup>1</sup> � *<sup>S</sup>*<sup>v</sup>

> > fsp <sup>¼</sup> *<sup>u</sup>*<sup>293</sup>*:*<sup>15</sup>

*u*<sup>272</sup>*:*<sup>15</sup>

<sup>100</sup> *u*<sup>272</sup>*:*<sup>15</sup>

fsp � <sup>0</sup>*:*<sup>12</sup> � exp 0½ � *:*0567ð Þ *<sup>T</sup>* � <sup>272</sup>*:*<sup>15</sup> @213*:*15K≤*T*≤272*:*15K*:*

<sup>Ψ</sup>ice‐fw � *<sup>ρ</sup>*<sup>w</sup>

<sup>Ψ</sup>ice‐bw � *<sup>ρ</sup>*<sup>w</sup>

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

of the heat source *q*v.

*DOI: http://dx.doi.org/10.5772/intechopen.83772*

below [1–19, 29]:

< 272.15 K [2, 10]:

where

**115**

*<sup>u</sup>*nfw <sup>¼</sup> <sup>0</sup>*:*<sup>12</sup> <sup>þ</sup> *<sup>u</sup>*<sup>272</sup>*:*<sup>15</sup>

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 at temperatures below �1°C.
