**4.2 Computation of 2D temperature distribution in logs during their refrigeration**

The mathematical model of the logs' freezing process has been solved with different values of the exponent *Е*fr in Еqs. (30) and (31). The calculated by the model change of the temperature in four characteristic points of the longitudinal logs'sections with each of the used values of *Е*fr during the freezing has been compared mathematically with the corresponding experimentally determined change of *t* in the same points with an interval of 5 min. The aim of this comparison was to find the value of *Е*fr, which ensures the best qualitative and quantitative compliance between the calculated and experimentally determined temperature fields in the logs' longitudinal sections.

As a criterion of the best compliance between the compared values of the temperature total for the four characteristic points, the minimum value of RSME, *σ*avg, has been used. For the determination of RSME, a software program in the calculation environment of MS Excel has been prepared. With the help of the program, RSME simultaneously for a total of 1440 temperature-time points during a separate 30 h refrigeration of the logs has been calculated. During the simulations the same initial and boundary conditions have been used as during the experiments. It was determined that the minimum values of RSME overall for the studied four characteristic points are *σ*avg = 1.67°C for P1 and *σ*avg = 1.54°C for S1. The minimum values of *σ*avg were obtained with the values of *Е*fr = 0.52 for P1 and *Е*fr = 0.48 for S1 in Eqs. (30) and (31).

**Figure 4** presents, as an example, the calculated change in *t*<sup>m</sup>fr, log's surface temperature *t*s, and *t* of four characteristic points of the studied pine log P1.

The comparison to each other of the analogical curves in **Figure 3**—above and **Figure 4** shows good conformity between the calculated and experimentally determined changes in the very complicated temperature field of the pine log during its refrigeration.

**Figure 4.**

*Experimentally determined and calculated change in tm, ts, and* t *in four points of the studied pine log P1 during its 30 h refrigeration.*

characteristic points of the logs during the experiments were realized by data logger-type HygroLog NT3 produced by ROTRONIC AG (http:/www.rotronic. com). The data logger has software HW4 for graphical presentation of the data. In **Figure 3**, the change in the temperature of the processing air medium, *t*m, and in its humidity, *ϕ*m, and also in the temperature in 4 characteristic points of pine log named below as P1 with *<sup>u</sup>* = 0.33 kg�kg�<sup>1</sup> and *<sup>ρ</sup>*<sup>b</sup> = 470 kg�m�<sup>3</sup> and spruce log named as S1 with *<sup>u</sup>* = 0.36 kg�kg�<sup>1</sup> and *<sup>ρ</sup>*<sup>b</sup> = 479 kg�m�<sup>3</sup> during their separate 30 h refrigeration is presented. The record of all data was made automatically by the

**4. Numerical solution of the mathematical model of the logs'**

For numerical solution of the mathematical model, a software package was prepared in Visual Fortran Professional developed by Microsoft. Using the

package, computations were carried out for the calculation of the 2D nonstationary change of *t* in the characteristic points of ¼ of the longitudinal sections of the studied logs, whose experimentally determined temperature fields are presented

The model has been solved with the help of explicit schemes of the finite difference method in a way analogous to the one used and described in [9, 10, 33]. For the computation of the temperature distribution in ¼ of the longitudinal section of the logs, which is symmetrical towards the remaining ¾ of the same section, the model was solved with step Δ*r* = Δ*z* = 0.006 m along the coordinates *r* and *z* and

with the same initial and boundary conditions, as they were during the

The interval between the time levels, Δτ (i.e., the step along the time coordinate), has been determined by the software package according to the condition of stability for explicit schemes of the finite difference method [10],

During the solving of the model, the mathematical descriptions of the

**4.1 Mathematical description of** *T* **in the freezer during logs' refrigeration**

*<sup>Т</sup>*<sup>m</sup>�fr <sup>¼</sup> *<sup>a</sup>*fr <sup>þ</sup> *<sup>c</sup>*fr*τ*<sup>0</sup>*:*<sup>5</sup>

whose coefficients are *a*fr = 309.7863391, *b*fr = 0.007125039, *c*fr = 1.321533597, and *<sup>d</sup>*fr <sup>=</sup> �2.769�10�<sup>6</sup> for log P1 and to *<sup>a</sup>*fr = 305.6335660, *<sup>b</sup>*fr = 0.005833651, *<sup>c</sup>*fr = 1.061216339, and *<sup>d</sup>*fr <sup>=</sup> �2.275�10�<sup>6</sup> for log S1. Equation. (39) and its coefficients were introduced in the software for solving Eqs. (3) and (4) of the model.

1 þ *b*fr*τ*<sup>0</sup>*:*<sup>5</sup> þ *d*fr*τ*

The curvilinear change in the freezing air medium temperature, *T*<sup>m</sup>�fr, which is shown in **Figure 1**, with high accuracy (correlation 0.98 for the both studied logs and root square mean error (RSME) σ = 1.28°C for P1 and σ = 1.22°C for S1) has been approximated with the help of the software package TableCurve 2D [34] by

fsp <sup>¼</sup> <sup>0</sup>*:*30 kg�kg�<sup>1</sup> and volume

fsp <sup>¼</sup> <sup>0</sup>*:*32 kg�kg�<sup>1</sup> and

*,* (39)

thermophysical characteristics of pine wood with *u*<sup>293</sup>*:*<sup>15</sup>

shrinkage *S*<sup>v</sup> = 11.8% and also of spruce sapwood with *u*<sup>293</sup>*:*<sup>15</sup>

data logger with intervals of 5 min.

**freezing process**

*Low-temperature Technologies*

experimental research.

and in our case it was equal to 6 s.

*S*<sup>v</sup> = 11.4% were used [8, 10].

the following equation:

**122**

in **Figure 3**.

#### *Low-temperature Technologies*

During our wide simulations with the mathematical model, we observed good compliance between computed and experimentally established temperature fields during refrigeration of logs from different wood species with different moisture content. The overall RSME for the studied four characteristic points in the logs does not exceed 5% of the temperature ranging between the initial and the end temperatures of the logs subjected to refrigeration.

<sup>Ψ</sup>icebw*=*d*<sup>T</sup>* from 0 to maximal values, equal to 0.0056 K<sup>1</sup> for P1 and 0.0057 K<sup>1</sup>

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

These maximal values remain practically unchanged until reaching 10.00th h for P1 and 10.75th h for S1, when the crystallization of the whole amount of the free water in the logs ends and the freezing of the bound water even in the logs' center

The further decrease of the temperature in the freezer (refer to **Figures 3** and **4**) causes a gradual crystallization of the bound water in the logs. Because of that the derivatives Ψicebw*=*d*T* decrease slowly, and at the end of the 30 h freezing, they

**Figure 6** presents according to Eqs. (34) and (38) the calculated change of

It can be seen that the change of the energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw is happening according to complex curvilinear dependences on the freezing time. The change of

<sup>Ψ</sup>icebwavg, and the change of *<sup>Q</sup>*Lat‐bw is similar to that of the derivative <sup>Ψ</sup>icebw*=*d*T*. At the beginning of the logs' refrigeration process, when Ψicebwavg = 0, both energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw are also equal to 0. After that *<sup>Q</sup>*fr‐bw increases gradually from 0 to 6.224 kWh<sup>m</sup><sup>3</sup> for P1 and 6.925 kWh<sup>m</sup><sup>3</sup> for S1 at the end of the 30 h freezing (see **Figure 6**, left). The larger value of *<sup>Q</sup>*fr‐bw for S1 is caused from the larger amount of the frozen bound water in S1 in comparison with P1 at the end of

As it was mentioned above, the fiber saturation point of the pine wood is equal

**Figure 5**, left, the relative icing degree, Ψicebwavg = 0.487 for P1 and Ψicebwavg = 0.498 for S1 at the end of the 30 h freezing. This means that the amount of the crystallized bound water is equal to 0.487 0.30 = 0.146 kgkg<sup>1</sup> in P1 and it is equal to 0.498 0.32 = 0.159 kgkg<sup>1</sup> in S1. Because of this not only the value of *<sup>Q</sup>*fr‐bw for S1 is larger than that of P1, but also the maximal value of *<sup>Q</sup>*Lat‐bw = 0.431 kWh<sup>m</sup><sup>3</sup> for S1 is larger than the maximal value of *<sup>Q</sup>*Lat‐bw= 0.405 kWh<sup>m</sup><sup>3</sup> for P1. After reaching the maximal values, the energy *<sup>Q</sup>*Lat‐bw decreases gradually and at the end of the 30 h freezing obtains a value of 0.232 kWh<sup>m</sup><sup>3</sup> for P1 and of 0.268

*Change in <sup>Q</sup>*fr‐bw *(left) and <sup>Q</sup>*Lat‐bw *(right) during refrigeration of the studied logs.*

, and of the spruce wood, it is 0.32 kgkg<sup>1</sup> [8, 10]. According to

the thermal energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw during the 30 h refrigeration of the

*<sup>Q</sup>*fr‐bw, depending on the freezing time, is similar to that of the icing degree

reach the value of 0.0013 K<sup>1</sup> for P1 and 0.0014 K<sup>1</sup> for S1.

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

**4.4 Change of the thermal energies** *Q* **fr-bw and** *Q***Lat-bw**

for S1.

starts.

studied logs.

the freezing process.

to 0.30 kgkg<sup>1</sup>

kWh<sup>m</sup><sup>3</sup> for S1.

**Figure 6.**

**125**

### **4.3 Change of the relative icing degree of logs Ψice-bw-avg and its derivative**

The calculation of the average mass icing degree of the logs caused from the freezing the bound water, Ψice�bw�avg, is carried out according to the following equation, which was obtained in [23] using Eq. (24):

$$\Psi\_{\rm ice-bw-zwg} = \frac{1}{S\_{\rm w}} \left\{ \iint \mathbf{1} - \frac{0.12 + \left( u\_{\rm fsp}^{272.15} - 0.12 \right) \cdot \exp\left\{ 0.0567 [T(r, z, \tau) - 272.15] \right\}}{u\_{\rm fsp}^{293.15} - 0.001 [T(r, z, \tau) - 293.15]]} d\mathbf{S}\_{\rm w} \right\},\tag{40}$$
 
$$\text{ @}\, T\_{\rm w-fps-zwg} \in T(r, z, \tau) \circ 272.15 \text{K}. \tag{40}$$

**Figure 5** presents the calculated change of the logs' icing degree Ψice�bw�avg and the derivative Ψice�bw*=*d*T* according to Eqs. (40) and (25), respectively, during the 30 h freezing process of the studied pine and spruce logs. The graphs show that the change of these variables is happening according to complex dependences on the freezing time.

It can be seen that the values of Ψice�bw�avg increase gradually from 0 to 0.487 for P1 and to 0.498 for S1 at the end of the 30 h freezing (**Figure 5**—left). These values of Ψice�bw�avg mean that 1–0.487 = 0.513 relative parts (i.e., 51.3%) of the bound water in P1 and 1–0.498 = 0.502 relative parts (i.e., 50.2%) of the bound water in S1 remains in a liquid state in the cell walls of the wood at the end of the 30th h of the logs' freezing process when the calculated average logs' mass temperatures are equal to �27.59°C for P1 and �26.82°C for S1.

It can be pointed that during the first 2.42 h for P1 and 2.75 h for S1 of the freezing process the whole amount of the bound water in the logs is in a liquid state and because of that the icing degree <sup>Ψ</sup>ice�bw�avg= 0 and also <sup>Ψ</sup>ice�bw*=*d*T*=0K�<sup>1</sup> .

From 2.42th h for P1 and from 2.75th h for S1 the crystallization of the bound water in the peripheral layers of the logs starts. This causes а jump in the change of

#### **Figure 5.**

*Change in the relative icing degrees Ψice*�*bw*�*avg (left) and their derivatives Ψice*�*bw=dT (right) during the refrigeration of the studied logs.*

*Modeling of the Energy for Bound Water Freezing in Logs Subjected to Refrigeration DOI: http://dx.doi.org/10.5772/intechopen.83772*

<sup>Ψ</sup>icebw*=*d*<sup>T</sup>* from 0 to maximal values, equal to 0.0056 K<sup>1</sup> for P1 and 0.0057 K<sup>1</sup> for S1.

These maximal values remain practically unchanged until reaching 10.00th h for P1 and 10.75th h for S1, when the crystallization of the whole amount of the free water in the logs ends and the freezing of the bound water even in the logs' center starts.

The further decrease of the temperature in the freezer (refer to **Figures 3** and **4**) causes a gradual crystallization of the bound water in the logs. Because of that the derivatives Ψicebw*=*d*T* decrease slowly, and at the end of the 30 h freezing, they reach the value of 0.0013 K<sup>1</sup> for P1 and 0.0014 K<sup>1</sup> for S1.

## **4.4 Change of the thermal energies** *Q* **fr-bw and** *Q***Lat-bw**

**Figure 6** presents according to Eqs. (34) and (38) the calculated change of the thermal energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw during the 30 h refrigeration of the studied logs.

It can be seen that the change of the energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw is happening according to complex curvilinear dependences on the freezing time. The change of *<sup>Q</sup>*fr‐bw, depending on the freezing time, is similar to that of the icing degree <sup>Ψ</sup>icebwavg, and the change of *<sup>Q</sup>*Lat‐bw is similar to that of the derivative <sup>Ψ</sup>icebw*=*d*T*.

At the beginning of the logs' refrigeration process, when Ψicebwavg = 0, both energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw are also equal to 0. After that *<sup>Q</sup>*fr‐bw increases gradually from 0 to 6.224 kWh<sup>m</sup><sup>3</sup> for P1 and 6.925 kWh<sup>m</sup><sup>3</sup> for S1 at the end of the 30 h freezing (see **Figure 6**, left). The larger value of *<sup>Q</sup>*fr‐bw for S1 is caused from the larger amount of the frozen bound water in S1 in comparison with P1 at the end of the freezing process.

As it was mentioned above, the fiber saturation point of the pine wood is equal to 0.30 kgkg<sup>1</sup> , and of the spruce wood, it is 0.32 kgkg<sup>1</sup> [8, 10]. According to **Figure 5**, left, the relative icing degree, Ψicebwavg = 0.487 for P1 and Ψicebwavg = 0.498 for S1 at the end of the 30 h freezing. This means that the amount of the crystallized bound water is equal to 0.487 0.30 = 0.146 kgkg<sup>1</sup> in P1 and it is equal to 0.498 0.32 = 0.159 kgkg<sup>1</sup> in S1. Because of this not only the value of *<sup>Q</sup>*fr‐bw for S1 is larger than that of P1, but also the maximal value of *<sup>Q</sup>*Lat‐bw = 0.431 kWh<sup>m</sup><sup>3</sup> for S1 is larger than the maximal value of *<sup>Q</sup>*Lat‐bw= 0.405 kWh<sup>m</sup><sup>3</sup> for P1. After reaching the maximal values, the energy *<sup>Q</sup>*Lat‐bw decreases gradually and at the end of the 30 h freezing obtains a value of 0.232 kWh<sup>m</sup><sup>3</sup> for P1 and of 0.268 kWh<sup>m</sup><sup>3</sup> for S1.

**Figure 6.** *Change in <sup>Q</sup>*fr‐bw *(left) and <sup>Q</sup>*Lat‐bw *(right) during refrigeration of the studied logs.*

During our wide simulations with the mathematical model, we observed good compliance between computed and experimentally established temperature fields during refrigeration of logs from different wood species with different moisture content. The overall RSME for the studied four characteristic points in the logs does

not exceed 5% of the temperature ranging between the initial and the end

**4.3 Change of the relative icing degree of logs Ψice-bw-avg and its derivative**

fsp � 0*:*12 � �

**Figure 5** presents the calculated change of the logs' icing degree Ψice�bw�avg and the derivative Ψice�bw*=*d*T* according to Eqs. (40) and (25), respectively, during the 30 h freezing process of the studied pine and spruce logs. The graphs show that the change of these variables is happening according to complex dependences on the

It can be seen that the values of Ψice�bw�avg increase gradually from 0 to 0.487 for P1 and to 0.498 for S1 at the end of the 30 h freezing (**Figure 5**—left). These values of Ψice�bw�avg mean that 1–0.487 = 0.513 relative parts (i.e., 51.3%) of the bound water in P1 and 1–0.498 = 0.502 relative parts (i.e., 50.2%) of the bound water in S1 remains in a liquid state in the cell walls of the wood at the end of the 30th h of the logs' freezing process when the calculated average logs' mass temper-

It can be pointed that during the first 2.42 h for P1 and 2.75 h for S1 of the freezing process the whole amount of the bound water in the logs is in a liquid state and because of that the icing degree <sup>Ψ</sup>ice�bw�avg= 0 and also <sup>Ψ</sup>ice�bw*=*d*T*=0K�<sup>1</sup>

*Change in the relative icing degrees Ψice*�*bw*�*avg (left) and their derivatives Ψice*�*bw=dT (right) during the*

From 2.42th h for P1 and from 2.75th h for S1 the crystallization of the bound water in the peripheral layers of the logs starts. This causes а jump in the change of

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

� exp 0f *:*0567½*T r*ð Þ� *; z; τ* 272*:*15�g

9 = ;

(40)

.

fsp � <sup>0</sup>*:*001½*T r*ð Þ� *; <sup>z</sup>; <sup>τ</sup>* <sup>293</sup>*:*15Þ� <sup>d</sup>*S*<sup>w</sup>

The calculation of the average mass icing degree of the logs caused from the freezing the bound water, Ψice�bw�avg, is carried out according to the following

temperatures of the logs subjected to refrigeration.

equation, which was obtained in [23] using Eq. (24):

@*T*<sup>w</sup>�fre�avg≤*T r*ð Þ *; z; τ* ≤272*:*15K*:*

atures are equal to �27.59°C for P1 and �26.82°C for S1.

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

<sup>Ψ</sup>ice�bw�avg <sup>¼</sup> <sup>1</sup>

freezing time.

**Figure 5.**

**124**

*refrigeration of the studied logs.*

*S*w *:* ðð

*Low-temperature Technologies*

*S*w 1 �

8 < :

completed as an example for the case of a pine log P1 and a spruce log S1 with a diameter of 0.24 m and length of 0.48 m subjected to 30 h refrigeration in a freezer at approximately 30°C. Practically, *<sup>Q</sup>*fr‐bw‐total represents the energy needed for crystallization of that part of the bound water, which amount

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

The logs subjected to refrigeration were with the following combinations of their initial temperature *t*w0, basic density ρb, and moisture content *u*: *t*w0 = 25.2°C,

It has been determined that the values of the energies *<sup>Q</sup>*fr‐bw,*Q*Lat‐bw, and *<sup>Q</sup>*fr‐bw‐total of the studied logs change according to complex relationships depending on the freezing time and after 30 h freezing of the logs reach the following values:

Good adequacy and precision of the model toward the results from wide own experimental studies allow the carrying out of various calculations with the model, which are connected to the nonstationary temperature distribution and energy characteristics of logs from different wood species during their refrigeration. The mathematical model, after its connection with other our model of the logs'

defrosting process [9, 10], could be input into the software of programmable controllers for optimized model-based automatic control [8, 20, 21, 35] of thermal

<sup>K</sup><sup>1</sup>

α heat transfer coefficients between log's surfaces and the surrounding air

<sup>K</sup><sup>1</sup>

, or length, m

*S* shrinkage, %, or aria of ¼ of log's longitudinal section, m<sup>2</sup>

The approach for the computation of the thermal energies of the bound water in logs during their refrigeration could be used for the creation of analogous models for the computation of the temperature distribution and the energy required for the refrigeration of different capillary porous materials (fruits, vegetables, meet, meet

treatment of frozen logs in the production of veneer.

, *<sup>Q</sup>*Lat‐bw = 0.232 kWh<sup>m</sup><sup>3</sup>

, and *<sup>u</sup>* = 0.33 kgkg<sup>1</sup> for log P1 and *<sup>t</sup>*w0 = 23.5°C, <sup>ρ</sup><sup>b</sup> = 479 kg<sup>m</sup><sup>3</sup>

, *<sup>Q</sup>*Lat‐bw = 0.268 kWh<sup>m</sup><sup>3</sup>

,

, and *<sup>Q</sup>*fr‐bw‐total = 5.992 kWh<sup>m</sup><sup>3</sup>

, and

depends on *u*fsp, *t*<sup>m</sup>fr, and *τ*fr.

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

and *<sup>u</sup>* = 0.36 kgkg<sup>1</sup> for log S1.

for P1 and *<sup>Q</sup>*fr‐bw = 6.925 kWh<sup>m</sup><sup>3</sup>

*<sup>Q</sup>*fr‐bw‐total = 6.657 kWh<sup>m</sup><sup>3</sup> for S1.

*<sup>Q</sup>*fr‐bw = 6.224 kWh<sup>m</sup><sup>3</sup>

<sup>ρ</sup><sup>b</sup> = 470 kg<sup>m</sup><sup>3</sup>

**6. Conclusions**

products, etc.).

**Nomenclature**

*D* diameter, m *E* exponent, — *<sup>L</sup>* latent heat, Jkg<sup>1</sup>

*R* radius, m

**127**

*T* temperature, K *t* temperature, °C

*<sup>c</sup>* specific heat capacity, Jkg<sup>1</sup>

*<sup>q</sup>* internal heat source, W<sup>m</sup><sup>3</sup> *<sup>Q</sup>* thermal energy, kWh<sup>m</sup><sup>3</sup>

*r* radial coordinate: *0 ≤ r ≤ R*, m

medium, W<sup>m</sup><sup>2</sup>

<sup>λ</sup> thermal conductivity, W<sup>m</sup><sup>1</sup>

*<sup>u</sup>* moisture content, kgkg<sup>1</sup> = %/100 *z* longitudinal coordinate: 0 *≤ z ≤ L/*2, m

<sup>K</sup><sup>1</sup>

**Figure 7.** *Change in <sup>Q</sup>*fr‐bw‐total *during the refrigeration of the studied logs.*

**Figure 7** presents the change of the thermal energy *<sup>Q</sup>*fr‐bw‐total during the 30 h refrigeration of the studied logs, which is calculated according to Eq. (32). The change of this energy is similar to that of the energy *<sup>Q</sup>*fr‐bw(see **Figure 6**, left). At the end of the 30 h freezing, the energy *<sup>Q</sup>*fr‐bw‐total reaches the following values: 5.992 kWh�m�<sup>3</sup> for P1 and 6.657 kWh�m�<sup>3</sup> for S1.

#### **5. Discussions**

This paper presents 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 phase transition of the bound water in 1 m<sup>3</sup> wood from liquid into solid state, *<sup>Q</sup>*fr‐bw, and latent thermal energy of the bound water, *<sup>Q</sup>*Lat‐bw, which is released in 1 m<sup>3</sup> of the logs during water crystallization.

Mathematical descriptions and an approach for computing of the energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw during the freezing of the bound water at temperatures below �1°C have been carried out. These descriptions are introduced in our own 2D nonlinear mathematical model of the 2D heat distribution in logs during their refrigeration at convective boundary conditions. The model was transformed in a form suitable for programming with the help of explicit schemes of the finite difference method, which excludes the necessity of any simplifications of the model.

A software program for numerical solution of the mathematical model and computation of 2D nonstationary change of the temperature in logs subjected to refrigeration and of the thermal energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw has been prepared in Fortran, which has been input in the calculation environment of Visual Fortran Professional developed by Microsoft.

With the help of the program, computations for the determination of the energies *<sup>Q</sup>*fr‐bw and *<sup>Q</sup>*Lat‐bw and their difference, *<sup>Q</sup>*fr‐bw‐total <sup>¼</sup> *<sup>Q</sup>*fr‐bw � *<sup>Q</sup>*Lat‐bw, have been *Modeling of the Energy for Bound Water Freezing in Logs Subjected to Refrigeration DOI: http://dx.doi.org/10.5772/intechopen.83772*

completed as an example for the case of a pine log P1 and a spruce log S1 with a diameter of 0.24 m and length of 0.48 m subjected to 30 h refrigeration in a freezer at approximately 30°C. Practically, *<sup>Q</sup>*fr‐bw‐total represents the energy needed for crystallization of that part of the bound water, which amount depends on *u*fsp, *t*<sup>m</sup>fr, and *τ*fr.

The logs subjected to refrigeration were with the following combinations of their initial temperature *t*w0, basic density ρb, and moisture content *u*: *t*w0 = 25.2°C, <sup>ρ</sup><sup>b</sup> = 470 kg<sup>m</sup><sup>3</sup> , and *<sup>u</sup>* = 0.33 kgkg<sup>1</sup> for log P1 and *<sup>t</sup>*w0 = 23.5°C, <sup>ρ</sup><sup>b</sup> = 479 kg<sup>m</sup><sup>3</sup> , and *<sup>u</sup>* = 0.36 kgkg<sup>1</sup> for log S1.

It has been determined that the values of the energies *<sup>Q</sup>*fr‐bw,*Q*Lat‐bw, and *<sup>Q</sup>*fr‐bw‐total of the studied logs change according to complex relationships depending on the freezing time and after 30 h freezing of the logs reach the following values: *<sup>Q</sup>*fr‐bw = 6.224 kWh<sup>m</sup><sup>3</sup> , *<sup>Q</sup>*Lat‐bw = 0.232 kWh<sup>m</sup><sup>3</sup> , and *<sup>Q</sup>*fr‐bw‐total = 5.992 kWh<sup>m</sup><sup>3</sup> for P1 and *<sup>Q</sup>*fr‐bw = 6.925 kWh<sup>m</sup><sup>3</sup> , *<sup>Q</sup>*Lat‐bw = 0.268 kWh<sup>m</sup><sup>3</sup> , and *<sup>Q</sup>*fr‐bw‐total = 6.657 kWh<sup>m</sup><sup>3</sup> for S1.
