# 3D Modeling and Visualization of Non- Stationary Temperature Distribution during Heating of Frozen Wood

Abstract: A 3-dimensional mathematical model has been developed, solved, and verified for the transient non-linear heat conduction in frozen and non-frozen wood with prismatic shape at arbitrary initial and boundary conditions encountered in practice. The model takes into account for the first time the fiber saturation point of each wood species, u fsp , and the impact of the temperature on u fsp of frozen and non-frozen wood, which are then used to compute the current values of the thermal and physical characteristics in each separate volume point of the material subjected to defrosting. This paper presents solutions of the model with the explicit form of the finite-difference method. Results of simu- lation investigation of the impact of frozen bound water, as well as of bound and free water, on 3D temperature distribution in the volume of beech and oak prisms with dimensions 0.4 x 0.4 x 0.8 m during their defrosting at the temperature of the processing medium of 80 °C are presented, analyzed and visualized through color contour plots.

## Summary (2 min read)

### 3D modeliranje i vizualizacija nestacionarne

- Distribucije temperature tijekom zagrijavanja smrznutog drva Original scientific paper Izvorni znanstveni rad Received – prispjelo: 27.
- This paper presents solutions of the model with the explicit form of the finite-difference method.
- 3D mathematical model, frozen wood, finite difference method, temperature distribution, contour plots Sažetak Kreiran je i riješen 3D matematički model te provjeren za nelinearno provođenje topline u smrznutome i nesmrznutom drvu prizmatičnog oblika pri proizvoljnim početnim i rubnim uvjetima koji se susreću u praksi, also known as Keywords.
- Rad prikazuje rješenja modela s eksplicitnim oblikom metode konačnih razlika.

### 1. UVOD

- For the optimization of the control of the heating process of wood in veneer and plywood mills, it is necessary to know the temperature distribution at every moment of the process (Shubin, 1990; Trebula and Klement, 2003; Pervan, 2009).
- The heat energy, required for melting the ice, formed from bound water in the wood, has not been taken into account in these models.
- The models assume that the fiber saturation point is identical for all wood species (i.e. ufsp = 0.3 kg·kg-1 = const) and that the melting of the ice, formed from free water in the wood, occurs at 0 ºC.
- The complications and deficiencies indicated in these models have been overcome by a 2-dimensional mathematical model of the transient non-linear heat conduction in frozen and non-frozen logs suggested by Deliiski (2004, 2011).
- This paper also presents the results of simulation investigation of the impact of the frozen bound water and free water on 3D temperature distribution in the volume of beech and oak prisms with dimensions 0.4 x 0.4 x 0.8 m during their defrosting at the temperature of the processing medium of 80 °C.

### 2.1. 3D matematički model procesa odmrzavanja prizmatičnoga drvnog materijala

- Other equations quoted by the above authors present mathematical descriptions of wood density, r, and of its thermal conductivity, l, in different anatomical directions.
- This has been done using the method presented by Deliiski (2013) during the update of the mathematical description of l.

### 3.1. Computation of 3D temperature distribution in

- Izračun 3D raspodjele temperature u smrznutome drvnom materijalu tijekom njegova odmrzavanja Besides taking into account the stability condition for solving the 3D model, the value of the step tD is calculated so as to be divisible by the input value of INT, using the software package.
- It must be noted that there are no such almost horizontal sections in the change of wood temperature during defrosting of the ice formed only by bound water in the wood (Fig. 4).

### 3.2 Color visualization of 3D non-stationary

- Temperature distribution in prisms during defrosting 3.2.
- The results obtained by Visual Fortran for 3D temperature distribution in the volume of wooden prisms undergoing defrosting have been subjected to the following visualization with the help of the software Excel 2010.
- For the solution of the model, an explicit form of the finite-difference method is used, with the possibility of excluding any model simplifications.
- This paper was written as a part of the solution of the project “Modelling and Visualization of Wood Defrosting Processes in Technologies for Wood Thermal Treatment”, supported by the Scientific Research Sector of the University of Forestry in Sofia (Project 114/2011).

### 5. LITERATURA

- Modeling and Technologies for Steaming of Wood Materials in Autoclaves, also known as 3. Deliiski, N., 2003a.
- Steinhagen, H. P., 1991: Heat Transfer Computation for a Long, Frozen Log Heated in Agitated Water or Steam - A Practical Recipe. 21. Videlov, H., 2003: Drying and Thermal Treatment of Wood.

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