Abstract

Two-dimensional mutually connected mathematical models have been created, solved, and verified for the transient non-linear heat conduction in logs during their freezing and subsequent defrosting. The models reflect the influence of the internal sources of latent heat of both the free and bound water on the logs' freezing process and also the impact of the temperature on the fiber saturation point of 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 chapter presents solutions of the models with explicit form of the finite-difference method and their validation towards own experimental studies. Results from experimental and simulative investigation of 2D non-stationary temperature distribution in the longitudinal section of beech and pine logs with a diameter of 0.24 m and length of 0.48 m during their many hours freezing in a freezer and subsequent defrosting at room temperature are presented, visualized, and analyzed.

Keywords: logs, modeling, heat conduction, latent heat, freezing, defrosting, free water, bound water

## 1. Introduction

It is known that the duration of the thermal treatment of the frozen logs in the winter aimed at their plasticizing for the production of veneer and the very high energy consumption needed for this treatment depend on the degree of the logs' icing [1–10]. For example, for the defrosting and plasticizing of beech and oak logs with an initial temperature of ˜10°C and moisture content of 0.6 kg°kg˜<sup>1</sup> approximately 68 and 81 kWh°m˜3 thermal energy respectively [10] is needed.

In the specialized literature there are limited reports about the temperature distribution in subjected to defrosting frozen logs [8, 11–22] and there is very little information about research of the temperature distribution in logs during their freezing given by the authors only [23–25]. That is why the modeling and the multi-parameter study of the mutually connected freezing and defrosting processes of logs are of considerable scientific and practical interest.

Considerable contribution to the calculation of the non-stationary distribution of the temperature in frozen and non-frozen logs and to the duration of their heating at conductive boundary conditions has been made by Steinhagen [11, 12] and later

one-dimensional and two-dimensional models have been created and solved [13–17]. The thermal energy, which is needed for the melting of the ice, which has been formed from the freezing of the bound water in the wood, has not been taken into account in the models of cited references.

The models assume that the fiber saturation point is identical for all wood species (i.e., ufsp = 0.3 kg�kg�1 = const). However, it is known that there are significant differences between the fiber saturation points of the different wood species [1–8]. The indicated complications and incompleteness in these models have been overcome in the suggested by Deliiski [9, 18, 19] 2-dimensional mathematical model of the transient non-linear heat conduction in frozen logs during their heating and defrosting.

For different engineering calculations it is needed to be able to determine the icing degree of the wood materials depending on the temperature of the influencing on them gas or liquid medium and on the duration of their staying in this medium. Such calculations are carried out using mathematical models, which describe adequately the complex processes of the freezing of both the free and bound water in the wood.

The computer solutions of these models give the non-stationary distribution of the temperature in the materials during their cooling below temperatures, at which a freezing of the whole amount of the free water and a freezing of respective, depending on the temperature, part of the bound water in the wood occurs [24, 25].

This chapter presents the creation, numerical solving and validation of two mutually connected two-dimensional mathematical models of the transient nonlinear heat conduction in logs during their freezing and subsequent defrosting at convective boundary conditions.

The model of the freezing process takes into account for the first time 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 chapter also presents and visualizes the results from experimental and simulative investigation of the 2D non-stationary temperature distribution in the longitudinal section of beech and poplar logs with a diameter of 0.24 m, length of 0.48 m, and different moisture content during many hours their freezing in a freezer and subsequent defrosting at curvilinear changing temperature of the processing air medium.
