4. Numerical solution of the mathematical models of the logs' freezing and subsequent defrosting processes

For the numerical solution of the mutually connected mathematical models, a software package was prepared in Visual FORTRAN Professional developed by Microsoft. Using the package, computations were carried out for the determination of the 2D non-stationary change of t in the characteristic points of ¼ of the longitudinal sections of the studied logs, whose experimentally determined temperature fields are presented on Figures 3 and 4.

The models have been solved with the help of explicit schemes of the finite difference method in a way, analogous to the one used and described in [8–10, 18, 19]. For the computation of the temperature distribution in ¼ of the longitudinal section of the logs, which is mirror symmetrical towards the remaining 3/4 of the

same section, the models were solved with step Δr = Δz = 0.006 m along the coordinates r and z and with the same initial and boundary conditions, as they were during the experimental research.

During the solving of the models, mathematical descriptions of the thermo- <sup>293</sup>:<sup>15</sup> physical characteristics of beech wood with ufsp <sup>¼</sup> <sup>0</sup>:31 kg�kg�<sup>1</sup> , Kwr = 1.35, and <sup>293</sup>:<sup>15</sup> <sup>K</sup>wp = 2.40, and also of poplar sapwood with ufsp <sup>¼</sup> <sup>0</sup>:35 kg�kg�<sup>1</sup> , Kwr = 1.48, and Kwp = 2.88 [8, 10].

The mathematical models of the logs' freezing and subsequent defrosting processes have been solved with different values of the exponents Еfr and Еdfr in Eqs. (15)–(18). The calculated by the models change of the temperature in the four characteristic points of the longitudinal logs'sections with each of the used values of Еfr and Еdfr during the freezing and defrosting has been compared mathematically with the corresponding one experimentally determined change of t in the same points with an interval of 15 min. The aim of this comparison was to find that the values of Еfr and Еdfr, 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, σavg, has been used.

The average value of RSME, σavg, has been calculated according to the following equation:

$$
\sigma\_{\text{avg}} = \sqrt{\frac{\sum\_{n=1}^{N} \sum\_{p=1}^{P} \left( t\_{p,n}^{\text{calc}} - t\_{p,n}^{\text{exp}} \right)^2}{P \cdot (N-1)}},\tag{51}
$$

exp where t calc and tp,n are the calculated and experimentally established tempera- p,n tures in the characteristic points; p—number of the characteristic points in the log's longitudinal section: р = 1, 2, 3, 4, i.e., Р = 4 is inputted into Eq. (51); n—number of the moments of the freezing and subsequent defrosting processes: n = 1, 2, 3,,….., N = (τfr+τdfr)/(150Δτ) because of the circumstance that the comparison of the calculated values of t with experimentally determined values in the same points has been made with an interval of 15 min = 900 s = 150Δτ.

For the determination of σavg software program in the calculation environment of MS Excel has been prepared. At τfr+τdfr = 100 h for the beech log and at τfr+τdfr = 120 h for the poplar log, RSME has been calculated with the help of the program simultaneously for total (N + 1)�P = 1604 temperature–time points for the beech log and for total 1924 such points for the poplar log during their freezing and subsequent defrosting. It was determined that the minimum values of RSME overall for the studied 4 characteristic points are equal to σavg = 1.29°C for the beech log and to σavg = 1.50°C for the poplar log.

The minimum values of σavg were obtained with the following values of the exponents in Eqs. (15)–(18):


Figures 7 and 8 presents the calculated change in t<sup>m</sup>�fr and t<sup>m</sup>�dfr, which are represented unitedly as tm, and also in logs'surface temperature ts and t of 4 characteristic points of the studied beech and poplar logs respectively.

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

#### Figure 7.

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

#### Figure 8.

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

The comparison to each other of the analogical curves on Figures 5 and 7, and also on Figures 6 and 8 shows good conformity between the calculated and experimentally determined changes in the very complicated temperature fields of the studied logs during their freezing and subsequent defrosting.

During our wide simulations with the mathematical models, we observed good compliance between computed and experimentally established temperature fields of logs various wood species with different moisture content.

The overall RSME for the studied 4 characteristic points in the logs does not exceed 5% of the temperature ranges between the initial and the end temperatures of the logs subjected to freezing or subjected to subsequent defrosting.

#### 5. Conclusions

This chapter describes the creation, solution, and validation of two mutually connected 2D non-linear mathematical models for the transient heat conduction in subjected to freezing and subsequent defrosting logs with any u ≥ ufsp. The model of the freezing process takes into account the impact of the internal sources of latent heat of both the free and bound water on the temperature distribution. The both models reflect the impact of the temperature on the fiber saturation point of each wood species, with whose participation the current values of the thermo-physical characteristics in each separate volume point of the subjected to freezing and subsequent defrosting logs are computed.

The mechanism of the temperature distribution in the longitudinal section of the logs during their freezing and subsequent defrosting has been mathematically described by 2D equations of heat conduction. Boundary conditions for convective heat transfer have been implemented in the models. For the transformation of both models into discrete analogues, which are suitable for programming, an explicit form of the finite-difference method has been used.

For the numerical solution of the discrete analogues of the models a software package has been prepared using the programming language FORTRAN, which has been input in the calculation environment of Visual Fortran Professional.

A validation of the models towards own experimentally determined 2D temperature distribution in beech and poplar logs with a diameter of 0.24 m, length of 0.48 m during their separate 50 h freezing in a freezer and many hours subsequent defrosting at room temperature has been carried out. The influence of the curvilinear changing temperature of the air medium in the freezer until reaching of approximately ˜30°C and also of the air processing medium during the defrosting of logs has been investigated. The following minimum values of the average RSME total for the temperature change in four characteristic points in each of the logs have been obtained:


During our experimental research it has been determined that in situated on the logs' inner layers characteristic points the specific practically horizontal sections of retention of the temperature for many hours in the range from 0 to ˜1°С arise, while in these layers a complete freezing of the free water occurs. Analogous retention of the temperature in the range from ˜1 to 0°C arises during the logs' defrosting. The further the point is distanced from the logs'surfaces and the larger the amount of the free water in the wood is, that much more these sections with temperature retention are extended. Our simulations show that this phenomenon of the freezing and defrosting processes has been correctly reflected in the models (see Figures 7 and 8).

Good adequacy and precision of the models towards the results from wide own experimental studies allow for the carrying out of various calculations with the models, which are connected to the non-stationary temperature distribution in logs from different wood species during their freezing and subsequent defrosting.

The validation of the models with curvilinear change in the temperature of both the freezing and defrosting air mediums will allow us in the future to solve the models with curvilinear changing of the climate temperature [29] over many winter days and nights. The solution will allow for the calculation of the temperature distribution, icing degrees from both the free and bound water, and also different energy characteristics of logs for each desired moment of the their freezing and subsequent defrosting.

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

The mutually connected models of the freezing and defrosting processes can be applied for the development of scientifically based and energy saving optimized regimes for thermal treatment of frozen logs and also in the software for controllers used for model predictive automatic control [21, 22, 30] of this treatment.
