3.1 Experimental research of the 2D temperature distribution in beech and poplar logs during their freezing and subsequent defrosting

For the validation of the suggested above mathematical models, it is necessary to have experimentally obtained data about the 2D temperature distribution in logs during their freezing and subsequent defrosting. That is why we carried out such experiments.

The logs subjected to freezing and subsequent defrosting in our experimental research were with a diameter of 240 mm, length of 480 mm, and u > ufsp. This means, that the logs contained the maximum possible amount of bound water. They were produced from the sap-wood of a freshly felled beech (Fagus sylvatica L.) and poplar (Populus nigra 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 of the characteristic points of the log [23].

The coordinates of the characteristic points of the logs are given on Figure 4. These coordinates of the points allow for the determination of the 2D temperature distribution in logs during their freezing and subsequent defrosting.

For the freezing of the logs according to the suggested methodology by the authors [23], a horizontal freezer was used with adjustable temperature range from ˜1 to ˜30°C.

Sensors Pt100 with long metal casings were positioned in the drilled four holes of the logs. The automatic measurement and record of tm, φm, and t in the characteristic points of the logs during the experiments was realized by Data Logger type HygroLog NT3 produced by the Swiss firm ROTRONIC AG (http:/www.rotronic.c om). The Data Logger has software HW4 for graphical presentation of the experimentally obtained data. After reaching of about ˜28°C in the log's center during the freezing, the freezer was switch off. Then its lid was opened and a defrosting of the log at room temperature was carried out.

In Figures 5 and 6 the change in the temperature of the processing air medium, tm, and in its humidity, φm, and also in the temperature in 4 characteristic points of a beech and of a poplar log respectively during their separately 50 and 70 h is presented. The record of all data was made automatically by Data Logger with

#### Figure 4.

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

#### Figure 5.

Experimentally determined change in tm, φm, and t in four points of the studied beech log during its 50 h freezing and 50 h subsequent defrosting.

#### Figure 6.

Experimentally determined change in tm, φm, and t in four points of the studied poplar log during its 50 h freezing and 70 h subsequent defrosting.

intervals of 15 min. The left coordinate axis on the figures is graduated at % of φm, and the left one is graduated at °C of t.

The initial temperature, tw0, basic density, ρb, moisture content, u, duration of the freezing, τfr, and duration of the subsequent defrosting, τdfr, of the logs during the experiments were as follow:


Numerical Solution to Two-Dimensional Freezing and Subsequent Defrosting of Logs DOI: http://dx.doi.org/10.5772/intechopen.84706

### 3.2 Mathematical description of the air medium temperature during logs' freezing and subsequent defrosting

The change in the shown on Figures 5 and 6 freezing and defrosting air medium temperatures, Тm�fr and Тm�dfr, with very high accuracy (correlation 0.98 and Root Square Mean Error (RSME), σ < 1.0°C) has been approximated with the help of the software package Table Curve 2D (http://www.sigmaplot.co.uk/products/table curve2d/tablecurve2d.php) by the following equations:

• during the freezing of the beech log:

$$T\_{\rm m-fr} = \frac{a\_{\rm fr} + c\_{\rm fr} \tau^{0.5}}{1 + b\_{\rm fr} \tau^{0.5}},\tag{48}$$

whose coefficients are equal to:

аfr = 294.3352069, bfr = 0.010648218, cfr = 2.468350514;

• during the freezing of the poplar log:

$$T\_{\rm m-fr} = \frac{a\_{\rm fr} + c\_{\rm fr}\tau + e\_{\rm fr}\tau^2 + g\_{\rm fr}\tau^3}{1 + b\_{\rm fr}\tau + d\_{\rm fr}\tau^2 + f\_{\rm fr}\tau^3 + h\_{\rm fr}\tau^4},\tag{49}$$

whose coefficients are equal to: аfr = 301.8210985, bfr = 0.0004515197, <sup>c</sup>fr = 0.111484207, dfr = �6.6585073�10�<sup>9</sup> , efr = �1.6653�10�<sup>6</sup> , ffr = 2.52712�10�14, <sup>g</sup>fr = 6.46801�10�12, hfr = 2.94924�10�<sup>21</sup> ;

• during the defrosting of both the beech and poplar logs:

$$T\_{\rm m-dfr} = \frac{a\_{\rm dfr} + c\_{\rm dfr} \tau^{0.5}}{1 + b\_{\rm dfr} \tau^{0.5}},\tag{50}$$

whose coefficients are equal to as follows:

аdfr = 297.1420433, bdfr = �0.00237763, cdfr = �0.70526837 for the beech log; аdfr = 296.3637194, bdfr = �0.00236425, cdfr = �0.69281743 for the poplar log. Eqs. (48) and (49) were used for the solving of Eqs. (3) and (7) and Eq. (50) was used for the solving of Eqs. (4) and (8) of the model.
