**3. Experimental research of 2D temperature distribution in logs subjected to freezing**

For the validation of the suggested above mathematical model, it is necessary to have experimentally obtained data about the temperature distribution in logs during their freezing. The logs subjected to freezing in our experimental research were with a diameter of 240 mm, length of 480 mm, and *u* > *u*fsp. This means that the logs contained the maximum possible amount of bound water for the separate wood species. They were produced from the sapwood of a freshly felled pine (*Pinus sylvestris* L.) and spruce (*Picea abies* L.) trunks. Before the experiments, four holes with diameters of 6 mm and different lengths were drilled in each log parallel to its axis until reaching the characteristic points of the log [22].

The coordinates of the characteristic points of the logs are given in **Figure 2**. These coordinates of the points allow the determination of the 2D temperature distribution in logs during their freezing. For refrigeration of the logs according to the suggested methodology by the authors [22], a horizontal freezer was used with adjustable temperature range from �1 to �30°C.

Sensors Pt100 with long metal casings were positioned in the four drilled holes of the logs. The automatic measurement and record of *t*m, ϕm, and *t* in the

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

#### **Figure 2.**

The multiplier 3.6�10<sup>6</sup> in the denominator of Eq. (37) ensures that the values of

Based on Eq. (33), it is possible to calculate the energy *<sup>Q</sup>*fr‐bw according to the

<sup>3</sup>*:*<sup>6</sup> � <sup>10</sup><sup>6</sup> � <sup>272</sup>*:*<sup>15</sup> � *<sup>T</sup>*w�fr�avg

.

� �*,* (34)

<sup>4</sup> *:* (37)

� �*,* (38)

(35)

exp 0*:*<sup>0567</sup> *<sup>T</sup>*w�fr�avg � <sup>272</sup>*:*<sup>15</sup> � � � � 1 þ *u*

*T r*ð Þ *; z; τ* d*S*w@*T*<sup>w</sup>�fre�avg≤*T r*ð Þ *; z; τ* ≤272*:*15K*,* (36)

, instead of in J�m�<sup>3</sup>

�

and the area of ¼ of the longitudinal section of the log subjected to freezing, *S*w,

*<sup>S</sup>*<sup>w</sup> <sup>¼</sup> *<sup>D</sup>* � *<sup>L</sup>*

Based on Eq. (33), it is possible to calculate also the energy *<sup>Q</sup>*Lat‐bw according to

where the specific heat capacity, which is formed by the release of the latent heat of the bound water, *c*Lat�bw, is calculated according to Eq. (21) and the density

For the validation of the suggested above mathematical model, it is necessary to have experimentally obtained data about the temperature distribution in logs during their freezing. The logs subjected to freezing in our experimental research were with a diameter of 240 mm, length of 480 mm, and *u* > *u*fsp. This means that the logs contained the maximum possible amount of bound water for the separate wood species. They were produced from the sapwood of a freshly felled pine (*Pinus sylvestris* L.) and spruce (*Picea abies* L.) trunks. Before the experiments, four holes with diameters of 6 mm and different lengths were drilled in each log parallel to its

The coordinates of the characteristic points of the logs are given in **Figure 2**. These coordinates of the points allow the determination of the 2D temperature distribution in logs during their freezing. For refrigeration of the logs according to the suggested methodology by the authors [22], a horizontal freezer was used with

Sensors Pt100 with long metal casings were positioned in the four drilled holes

of the logs. The automatic measurement and record of *t*m, ϕm, and *t* in the

**3. Experimental research of 2D temperature distribution in logs**

<sup>3</sup>*:*<sup>6</sup> � <sup>106</sup> � <sup>272</sup>*:*<sup>15</sup> � *<sup>T</sup>*<sup>w</sup>�fr�avg

*<sup>Q</sup>*fr‐bw <sup>¼</sup> *<sup>c</sup>*bw‐avg � *<sup>ρ</sup>*<sup>w</sup>

fsp � 0*:*12 � �

*<sup>Q</sup>*Lat‐bw <sup>¼</sup> *<sup>c</sup>*Lat‐bw � *<sup>ρ</sup>*<sup>w</sup>

axis until reaching the characteristic points of the log [22].

adjustable temperature range from �1 to �30°C.

*<sup>Q</sup>*<sup>w</sup> are obtained in kWh�m�<sup>3</sup>

*Low-temperature Technologies*

where according to [8, 9]

*<sup>c</sup>*bw�avg <sup>¼</sup> <sup>1</sup>*:*<sup>8938</sup> � <sup>10</sup><sup>4</sup> *<sup>u</sup>*272*:*<sup>15</sup>

*<sup>T</sup>*<sup>w</sup>�fr�avg <sup>¼</sup> <sup>1</sup>

@*T*w�fre�avg≤*T*≤272*:*15K*,*

*S*w

*S*w ðð

*ρ*<sup>w</sup> is calculated according to Eq. (9).

**subjected to freezing**

equation

is equal to

the equation

**120**

*Radial (left) and longitudinal (right) coordinates of four characteristic points for measurement of the temperature in logs subjected to refrigeration.*

#### **Figure 3.**

*Experimentally determined change in tm, ϕm, and t in four points of the studied logs P1 (above) and S1 (below) with D = 0.24 m and L = 0.48 m during their 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. **4.2 Computation of 2D temperature distribution in logs during their**

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

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

**Figure 4** presents, as an example, the calculated change in *t*<sup>m</sup>fr, log's surface

The comparison to each other of the analogical curves in **Figure 3**—above and

temperature *t*s, and *t* of four characteristic points of the studied pine log P1.

**Figure 4** shows good conformity between the calculated and experimentally determined changes in the very complicated temperature field of the pine log

*Experimentally determined and calculated change in tm, ts, and* t *in four points of the studied pine log P1*

**refrigeration**

in Eqs. (30) and (31).

during its refrigeration.

**Figure 4.**

**123**

*during its 30 h refrigeration.*

fields in the logs' longitudinal sections.

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

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 data logger with intervals of 5 min.
