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

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 in **Figure 3**.

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 experimental research.

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], and in our case it was equal to 6 s.

During the solving of the model, the mathematical descriptions of the thermophysical characteristics of pine wood with *u*<sup>293</sup>*:*<sup>15</sup> fsp <sup>¼</sup> <sup>0</sup>*:*30 kg�kg�<sup>1</sup> and volume shrinkage *S*<sup>v</sup> = 11.8% and also of spruce sapwood with *u*<sup>293</sup>*:*<sup>15</sup> fsp <sup>¼</sup> <sup>0</sup>*:*32 kg�kg�<sup>1</sup> and *S*<sup>v</sup> = 11.4% were used [8, 10].

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

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 the following equation:

$$T\_{\rm m-fr} = \frac{a\_{\rm fr} + c\_{\rm fr} \tau^{0.5}}{1 + b\_{\rm fr} \tau^{0.5} + d\_{\rm fr} \tau},\tag{39}$$

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.

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