Application of the finite volume method to processes in wood technology.

Horman, Izet ; Martinovic, Dunja

1. INTRODUCTION

The finite volume method for stress analysis is equally applicable to linear, isotropic, anisotropic, porous and non-linear materials. In what follows an outline of the method is given and some results illustrating the method's abilities are presented. More details about the method can be found in (Demirdzic, Martinovic, Horman, at al., 1993, 1995, 2000, 2001, 2003).

2. MATHEMATICAL MODEL AND NUMERICAL SOLUTION PROCEDURE

The behaviour of an arbitrary part of a solid, porous body of volume V bounded by the surface S at any instant of time t can be described by equations of momentum (1), thermal energy (2) and mass balance (3) (Demirdzic, at al., 2000)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

In order to close the system of Eqs. (1) to (3) or (1) and (2), the constitutive relations are used:

--for an elastic, porous, orthotropic material

* for Eq. (1) is

[[sigma].sub.ij] = [C.sub.ijki][[epsilon].sub.ki] - [[alpha].sub.ij][DELTA]T - ([[beta].sub.ij][DELTA]M) (4)

where [C.sub.ijkl] are the elastic constant tensor components (the nine non-zero orthotropic elastic constants) (Demirdzic, at al., 2000, Martinovic, at al., 2001). The terms in <> brackets are "active" only for M < [M.sub.h]([M.sub.h] is the moisture potential at the fibre saturation point).

* for Eqs. (2) and (3) are heat and mass flux vector

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

--for a thermo-elasto-plastic isotropic material (Demirdzic, at al., 1993)

* for Eqs. (1) and (2) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and constitutive relation (5) for [m.sub.j] = 0

In the case of elastic conditions, the expression within the brackets <> vanishes, and the constitutive relation (7) reduces to the Duhamel-Neumann form of Hooke's law.

By introducing corresponding constitutive relations into governing equations a closed system of 2 or 3 (generally nonlinear and coupled) equations with two or three unknown functions of spatial coordinates and time ([u.sub.i], T, M or [u.sub.i], T) is obtained. To complete the mathematical model, initial and boundary conditions have to be specified.

The solution domain is discretized by a finite number of contiguous hexahedral control volumes {CV) or cells of the volume V which are bounded by six cell faces of the area [S.sub.j] with calculation points P in the CV's centres (Figure 1).

The time domain is subdivided into a number of time intervals [delta]t.

Equations (1), (2) and (3) are integrated over time interval [delta]t and over each control volume resulting in a system of 5N (generally non-linear and coupled) algebraic equations of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [phi] stands for displacement components [u.sub.i] (i = 1,2,3) or temperature T or moisture potential M, n is the number of cell faces of a control volume (Fig. 1).

Systems of algebraic equations (8) are solved by an iterative procedure (Demirdzic & Muzaferija, 1995).

[FIGURE 1 OMITTED]

3. APPLICATION OF THE METHOD

The method described in the previous sections has been applied to a number of both isotropic and orthotropic body deformation problems, few of which will be presented in this paper.

3.1. Numerical analysis of a wood drying process

The beech-wood beams are exposed to the (uniform, unsteady) flow of the hot air in a laboratory dryer with an automatic control of the ambient air parameters. The temperature and moisture dependent physical properties of the wood are used (Martinovic, at al., 2001).

Equations (1) to (3) and the constitutive relations (4) to (6) are used. Fig. 2 shows the fields of moisture, stress [[sigma].sub.xx], displacements at t = 159 h and cross section shape of deformed wood sample, at the end of the drying process (t = 246 h).

[FIGURE 2 OMITTED]

The maximal normal stresses are in regions near the sample's surface, and the contraction of the wood sample is 6,5 % and 3,3 % (axis x and y) at the end of the drying process.

3.2. Numerical analysis process of wood heat treatment

The log is exposed to the (unsteady) flow of the steam during his thermal preparation in the veneers production (Horman, at al., 2003).

Equations (1) and (2) and the constitutive relations (7) and (5) (for [m.sub.j] = 0) are used. The problem is considered to be a 2D plane strain problem. Fig. 3 and Fig. 4 show temperature and circular stress histories at three reference sections.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Fig. 5 shows the temperature distribution at [phi] = const., in four time instants. The temperature gradients are the largest in the region near the log's surface and this is the region of the largest stress, where the residual stress occurs (Horman, at al., 2003).

4. CONCLUSIONS

The numerical method for stress analysis has been outlined and its applicability to the solution of transient problems involving various porous, orthotropic and elastoplastic materials has been demonstrated. The mathematical model and the numerical calculation employing the finite volume method presented enable the prediction of the distribution of deformation and stresses in wood during a drying process and wood steaming. The development of computer technology and numerical methods have made the research much easier and enabled obtaining information of what is happening inside a loaded product.

5. REFERENCES

Demirdzic, I.; Martinovic, D. (1993). Finite volume method for thermo-elasto-plastic stress analysis, Comput. Methods Appl. Mech. Engrg., 109, 331-349.

Demirdzic, I.; Muzaferija, S. (1995). Numerical method for coupled fluid flow, heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology, Comput. Methods Appl. Mech. Engrg., 125, 235-255.

Demirdzic, I.; Horman, I. & Martinovic, D. (2000). Finite volume analysis of stress and deformation in hygrothermo-elastic orthotropic body, Comput. Methods Appl. Mech. Engrg., 190, 1221-1232.

Horman, I.; Martinovic, D. & Bijelonja, I. (2003). Numerical Analysis of Process of Wood Heat Treatment, 4th Int. Symposium RIM, Bihac, 443-450.

Martinovic, D.; Horman, I. & Demirdzic, I. (2001). Numerical and Experimental Analysis of a Wood Drying Process, Wood Science and Technology, 35, 143-156.