## Introduction

Over the last two decades the simulation of wood drying has been the subject of much research impetus, and it is clear that the field is truly interdisciplinary, with key articles found in top engineering, wood science, and mathematical journals. These publications cover a wide range of diverse topics, including the derivation of the physical and mechanical formulation, the development of analytical and numerical solutions, the determination of the physical and mechanical characterization of the medium being dried, and the process experiments carried out on both laboratory and industrial scales. Nowadays, as a result of ever-increasing computational power, numerical simulation has fast become a very powerful tool that drying practitioners can use to study the drying process at a fundamental level. Such knowledge has guided the introduction of many new and innovative drying operations into the industrial sector. Nevertheless, it is the opinion of the authors that a close interaction still must exist between drying theoreticians and practitioners, because of the complexity of a global strategy for wood-drying improvement, or kiln control, that must address a range of important issues. In particular, the final choice of the modeling approach must take into consideration an extensive list, which includes the choice of the most appropriate numerical model for the study, the knowledge of the important product properties needed for the simulations, and whether all of the required physical parameters are available and if not, whether it is possible to perform or propose measurements for the unknown properties.

The literature highlights several sets of macroscopic equations being used to model the drying process. The first fundamental difference between the underlying mathematical models lies in the number of state variables used to describe the process. Three possibilities exist, which are known as one-variable, two-variable, or three-variable models. A one-variable model typically uses moisture content as the primary variable or an equivalent variable, such as saturation or water potential. Two-equation models use both moisture content (or its equivalent) and temperature, *T*, or an equivalent variable such as enthalpy, in the formulation, and the most sophisticated three-equation models use moisture content (or its equivalent), *T* (or its equivalent), and gaseous pressure *P*_{g}, or an equivalent variable such as air density or intrinsic air density. Perré (1999) gives a critical review of the possibilities and limitations given by these different sets of equations. The use of moisture content alone is the basis of many correlations of lumber-drying rates (see Keey et al., 2000). Doe et al. (1996) found that at the relatively low temperatures (<40°C) used in drying eucalyptus hardwoods, both moisture content and temperature are needed to describe the movement of moisture. The importance of the role of gaseous pressure appears in the simulation of high-temperature seasoning, vacuum drying, or microwave drying (Perré and Degiovanni, 1990; Turner and Perré, 1996; Perré and Turner, 1999a, b). As a general rule, a one-variable model should be avoided, because it is not able to account for the very important coupling between heat and mass transfer that exists during drying. A two-variable model is appropriate for most of the drying conditions encountered in industry. Finally, the quite complex three-variable model should be reserved for processes during which the internal pressure has a significant impact on internal moisture transport, such as the processes with internal vaporization that include vacuum drying, high-temperature drying, and radiofrequency drying.

The second fundamental difference between the drying models lies in the number of spatial dimensions used to describe the process. One-dimensional (1-D) models that use the thickness of the sample as the only spatial dimension are fast running and allow the most important transport phenomena to be captured, including, for example, transfers and drying stress. For a refined approach, a 2-D model can be useful where the thickness and the width of the sample are used in the formulation to enable a better evaluation of the drying rate and stress development. Moreover, such models can account for the grain direction and the presence of both sapwood and heartwood in wood. Due to the large anisotropy ratios evident in wood, the length of the sample is absolutely required in the case of processes where internal vaporization is important. Finally, a 3-D model allows a comprehensive geometrical modeling to be investigated (Perré and Turner, 1999b).

Once the specific formulation has been decided, various approaches in the computational solution strategy can be adopted to simulate the drying process. For example, simple and efficient methods that use the assumption of constant physical parameters can be used to address specific problems, including kiln sizing and the global effect of the product size. Among these methods, the dimensionless drying curves proposed by Van Meel (1958) or the simple analytical solutions outlined by Crank (1975) have to be noted. However, the development of these methods fails to provide a complete understanding of the internal transfer mechanisms and the coupling that always exists between heat and mass transfer.

A second set of computational strategies can be used that try to be more realistic by using a more suitable set of simplifying assumptions. All models based on the concept of “drying front” belong to this category (Hadley, 1982; Rogers and Kaviany, 1992; Perré et al., 1999). Finally, all computational models based on a complete numerical solution of the nonlinear conservation laws constitute the third category. The first representative examples of this category appeared more than 20 years ago (Bramhall, 1979; Kawai, 1980). Thereafter, a more comprehensive physical formulation has been applied with success, at first in one-dimension (Stanish et al., 1986; Moyne, 1987; Ben Nasrallah and Perré, 1988; Ilic and Turner, 1989) and later in two-dimensions (Perré and Degiovanni, 1990; Fyhr and Rasmuson, 1996; Boukadida and Ben Nasrallah, 1995; Pang, 1996). Nowadays, as a result of the significant advancements in numerical techniques, including efficient inexact Newton iterative solvers and the use of unstructured meshes, together with the ever-increasing power of computers, the numerical simulations have become a real tool, able to deal with a comprehensive formulation with any geometrical formulation (Turner and Perré, 1996; Perré, 1999a, b). Before going further, one has to note that the vacuum drying of wood involves two particular and important features:

- 1The accelerated internal mass transfer due to the overpressure that can exist within the product;
- 2The effect of the high anisotropy ratio of wood permeability (from 100 to more than 10
^{4}), especially for wood with a high aspect ratio between the length and width of the sample.

No realistic simulation can be expected from a model that is not able to account for these two important features.

The objectives of Part II of this research are to use the extensive experimental data sets documented in Part I for the purposes of assessing the accuracy and predictive ability of two previously developed drying models. The first model is a comprehensive drying model known as *TransPore* (Turner and Perré, 1996; Perré and Turner, 1999a, b), which is able to capture the quite subtle and convoluted heat- and mass-transfer mechanisms that evolve throughout the drying process. The second drying model, which is known as *Front_2D* (Perré et al., 1999), uses a number of simplifying assumptions to reduce the complexity of the comprehensive model to a system that enables a semianalytical approach to be exploited for its solution. Although the physical formulation used in both of these models always had the capability of simulating any external pressure applied to the medium, this is the first time that they are actually used at an external pressure different from the atmospheric value. Specific differences in the two formulations for vacuum drying will be elucidated in the following sections of the text.