TWO-STAGE MOISTURE DIFFUSION IN WOOD WITH CONSTANT TRANSPORT COEFFICIENTS

ABSTRACT

To determine whether transport coefficients in desorption curves for northern red oak are constant, four sets of desorption data were compared against previously established mathematical conditions for infinite-series solution of the non-steady-state diffusion equation. For each data set, when moisture fraction in wood is above a certain value, designated as the first stage, these conditions are satisfied, with the diffusion and surface emission coefficients being positive and finite; below that value, designated as the second stage, these conditions are still satisfied, with the diffusion coefficient taking a smaller positive value but the surface emission coefficient becoming negative and finite. Mathematically, these two pairs of transport coefficients can be used to predict the whole diffusion curve that describes the variation of moisture fraction with time. However, the negative surface emission coefficient in the second stage of the desorption process implies that the moisture gradient has cut the surface at a point below the equilibrium moisture content, which is physically impossible. Alternatively, the second stage can be considered as a new stage with moisture fraction values normalized with respect to the lowest value in the first stage, which is also the starting point of the second stage. The transport coefficents are obtained in the same manner as in the first stage and are found to be positive and finite. The two pairs of transport coefficients can describe the diffusion curve with high accuracy. However, the assumption used in the second stage is that the initial moisture content in wood is uniform, which again is physically impossible. We therefore conclude that the transport coefficients for northern red oak are not constant. Although the two-stage approaches presented in this study can predict the diffusion curves accurately, their physical interaction is difficult to justify.     

SOLUTIONS OF DIFFUSION EQUATION WITH CONSTANT DIFFUSION AND SURFACE EMISSION COEFFICIENTS

Closed-form solutions of the non-steady state diffusion equation with constant transport coefficients are presented. The diffusion coefficient is assumed to be finite, but the surface emission coefficient can be either finite or infinite. Mathematical conditions are established for the transport coefficients to be constant. When these conditions are met, the transport coefficients can then be easily evaluated. Diffusion test data can be compared against these conditions to determine whether or not the transport coefficients are constant. Desorption test data of northern red oak indicate that initial moisture content in wood and equilibrium moisture content in the environment are closely related to the constancy of the transport coefficients.     

SOLUTIONS OF DIFFUSION EQUATION WITH CONSTANT DIFFUSION AND SURFACE EMISSION COEFFICIENTS

ABSTRACT

Closed-form solutions of the non-steady state diffusion equation with constant transport coefficients are presented. The diffusion coefficient is assumed to be finite, but the surface emission coefficient can be either finite or infinite. Mathematical conditions are established for the transport coefficients to be constant. When these conditions are met, the transport coefficients can then be easily evaluated. Diffusion test data can be compared against these conditions to determine whether or not the transport coefficients are constant. Desorption test data of northern red oak indicate that initial moisture content in wood and equilibrium moisture content in the environment are closely related to the constancy of the transport coefficients.     

Mathematical Relationship Between Surface Emission and Diffusion Coefficients

ABSTRACT

This naver analvzes the surface emission coefficient corresponding to anv diffusion coefficient expressed as an exponential function of the concentration diffusing substance in capillary porous solids. Theoretical equations for surface emission coefficient for both sorption and desorption are presented. Procedures to derive the diffusion coefficient and to verify the corresponding surface emission coefficient are made using experimental sorption data of aspen (Populur sp.) wood. It is of interest to note that the theoretical models for diffusion and surface emission coefficients can be established and/or evaluated using the same set of experimental dara.     

A STUDY ON THE SEPARATION OF DIFFUSION AND SURFACE EMISSION COEFFICIENTS IN WOOD

ABSTRACT

For a more accurate simulation of the moisture sorption process in wood, the precise separation of surface emission and diffusion coefficients is essential. In this paper, a non-linear simplex technique that was developed to obtain the optimum pair of the surface emission coefficient he, and the diffusion coefficient D from a single sorption curve is presented and tested with experimental data. An easy to use computer program was employed to carry out the evaluation. Data analysis showed that the new method resulted in a more statistically accurate calculation of b_c and D than currently existing methods.     

PARAMETERS INFLUENCING DRYING BEHAVIOR OF RUBBER WOOD (HEVEA BRAZILLIENSIS) AS DETERMINED FROM DESORPTION EXPERIMENT

In this study, the drying properties of rubber wood, which are the basic parameters for kiln scheduling design, were determined from desorption experiment. Equilibrium moisture content expression was developed. The diffusion coefficients at different drying environments were evaluated. It is more appropriate to determine the diffusion coefficients by the optimum scheme in comparison to other schemes; the logarithmic, square-root and half-fraction of evaporable moisture schemes. Finally, the diffusion coefficient of rubber wood was described by, instead of wood moisture content, the drying temperature and relative humidity, which are the parameters controlling the drying kiln operation.     

PARAMETERS INFLUENCING DRYING BEHAVIOR OF RUBBER WOOD (HEVEA BRAZILLIENSIS) AS DETERMINED FROM DESORPTION EXPERIMENT

ABSTRACT

In this study, the drying properties of rubber wood, which are the basic parameters for kiln scheduling design, were determined from desorption experiment. Equilibrium moisture content expression was developed. The diffusion coefficients at different drying environments were evaluated. It is more appropriate to determine the diffusion coefficients by the optimum scheme in comparison to other schemes; the logarithmic, square-root and half-fraction of evaporable moisture schemes. Finally, the diffusion coefficient of rubber wood was described by, instead of wood moisture content, the drying temperature and relative humidity, which are the parameters controlling the drying kiln operation.     

Moisture Diffusion Coefficients for Modeling the First and Second Drying Sections of Green Bricks

A model has been developed that describes the dependence of the moisture diffusion coefficient on the water fraction. Until the end of shrinkage has been achieved, the moisture diffusion coefficient is proportional to the second power of the water fraction. Due to shrinkage, the relevant capillary spaces available for water transport become smaller. Consequently, the moisture diffusion coefficient decreases continually. After the end of shrinkage, the flow resistance to the water moving toward the surface increases sharply due to penetrating air. This leads to a steep drop of the moisture diffusion coefficient by several powers of ten. Measurements were carried out with specimens of defined geometry to determine the moisture diffusion coefficient. On the basis of a specified limiting value, the model is capable of calculating the moisture diffusion for all initially specified raw materials moistures. The moisture can also be determined if the degree of drying shrinkage is known. Using the determined moisture diffusion coefficient, the first and the second drying section can be located. Drying tests were carried out in a laboratory dryer and the experimental results obtained were compared to the simulation results. The simulation results are in good agreement with the experimental results.     

MATHEMATICAL RELATIONSHIP BETWEEN DESORPTION AND SORPTION SOLUTIONS

This paper presents a theoretical equation relating the dimensionless times of Newman's solutions of the diffusion equation for desorption and sorption. The derived equation provides a theoretical proof of the linear relationship between dimensionless time and the inverse of transport ratio for a given fraction of diffusing substance as observed numerically in the literature. Following the same approach proposed earlier by the writer, a theoretical expression for the diffusion coefficient is derived as a function of time and the inverse of desorption rate when the fraction of diffusing substance is 0.5. Evaluation of the diffusion coefficient from an experimental desorption curve is discussed.     

Mathematical Relationship Between Surface Emission and Diffusion Coefficients

This naver analvzes the surface emission coefficient corresponding to anv diffusion coefficient expressed as an exponential function of the concentration diffusing substance in capillary porous solids. Theoretical equations for surface emission coefficient for both sorption and desorption are presented. Procedures to derive the diffusion coefficient and to verify the corresponding surface emission coefficient are made using experimental sorption data of aspen (Populur sp.) wood. It is of interest to note that the theoretical models for diffusion and surface emission coefficients can be established and/or evaluated using the same set of experimental dara.     

MATHEMATICAL RELATIONSHIP BETWEEN DESORPTION AND SORPTION SOLUTIONS

ABSTRACT

This paper presents a theoretical equation relating the dimensionless times of Newman's solutions of the diffusion equation for desorption and sorption. The derived equation provides a theoretical proof of the linear relationship between dimensionless time and the inverse of transport ratio for a given fraction of diffusing substance as observed numerically in the literature. Following the same approach proposed earlier by the writer, a theoretical expression for the diffusion coefficient is derived as a function of time and the inverse of desorption rate when the fraction of diffusing substance is 0.5. Evaluation of the diffusion coefficient from an experimental desorption curve is discussed.     

Effects of material parameters on the diffusion and sorption properties of wood‐flour/polypropylene composites

Composites of wood in a thermoplastic matrix (wood–plastic composites) are considered a low maintenance solution to using wood in outdoor applications. Knowledge of moisture uptake and transport properties would be useful in estimating moisture‐related effects such as fungal attack and loss of mechanical strength. Our objectives were to determine how material parameters and their interactions affect the moisture uptake and transport properties of injection‐molded composites of wood‐flour and polypropylene and to compare two different methods of measuring moisture uptake and transport. A two‐level, full‐factorial design was used to investigate the effects and interactions of wood‐flour content, wood‐flour particle size, coupling agent, and surface removal on moisture uptake and transport of the composites. Sorption and diffusion experiments were performed at 20°C and 65 or 85% relative humidity as well as in water, and diffusion coefficients were determined. The wood‐flour content had the largest influence of all parameters on moisture uptake and transport properties. Many significant interactions between the variables were also found. The interaction between wood‐flour content and surface treatment was often the largest. The diffusion coefficients derived from the diffusion experiments were different from those derived from the sorption experiments, suggesting that different mechanisms occur. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 103: 752–763, 2007     

Isothermal Vapour and Liquid Transport Inside Clay During Drying

A model is presented to describe the moisture transport inside a partially saturated porous material. The transport is caused by vapour diffusion and liquid diffusion. The evaporation inside the porous material is described with a mass transfer coefficient and a specific evaporating surface. Predictions of the model for moisture profiles are compared to experimentally oblained profiles found in the literature. The model needs further extension in the form of incorporating sorption isotherms.     

Isothermal Vapour and Liquid Transport Inside Clay During Drying

A model is presented to describe the moisture transport inside a partially saturated porous material. The transport is caused by vapour diffusion and liquid diffusion. The evaporation inside the porous material is described with a mass transfer coefficient and a specific evaporating surface. Predictions of the model for moisture profiles are compared to experimentally obtained profiles found in the literature. The model needs further extension in the form of incorporating sorption isotherms.     

Isothermal Vapour and Liquid Transport Inside Clay During Drying

ABSTRACT

A model is presented to describe the moisture transport inside a partially saturated porous material. The transport is caused by vapour diffusion and liquid diffusion. The evaporation inside the porous material is described with a mass transfer coefficient and a specific evaporating surface. Predictions of the model for moisture profiles are compared to experimentally oblained profiles found in the literature. The model needs further extension in the form of incorporating sorption isotherms.     

AN INVERSE MOISTURE DIFFUSION ALGORITHM FOR THE DETERMINATION OF DIFFUSION COEFFICIENT

The finite difference approximation is applied to estimate the moisture-dependent diffusion coefficient by utilizing test data of isothermal moisture desorption in northern red oak (Quercus rubra). The test data contain moisture distributions at discrete locations across the thickness of specimens, which coincides with the radial direction of northern red oak, and at specified times. Also, the rate of moisture variation at each specified time and location must be known or correctly estimated. The functional form of the diffusion coefficient as well as the boundary conditions at the surfaces are not known a priori. The resulting system of finite difference equations defines an inverse problem, whose solution may be sensitive to small changes of input data. Results indicate that the diffusion coefficient increases with increasing moisture content below the fiber saturation point, which defines the upper limit applied by the diffusion theory.     

Isothermal Vapour and Liquid Transport Inside Clay During Drying

ABSTRACT

A model is presented to describe the moisture transport inside a partially saturated porous material. The transport is caused by vapour diffusion and liquid diffusion. The evaporation inside the porous material is described with a mass transfer coefficient and a specific evaporating surface. Predictions of the model for moisture profiles are compared to experimentally obtained profiles found in the literature. The model needs further extension in the form of incorporating sorption isotherms.     

AN INVERSE MOISTURE DIFFUSION ALGORITHM FOR THE DETERMINATION OF DIFFUSION COEFFICIENT

The finite difference approximation is applied to estimate the moisture-dependent diffusion coefficient by utilizing test data of isothermal moisture desorption in northern red oak (Quercus rubra). The test data contain moisture distributions at discrete locations across the thickness of specimens, which coincides with the radial direction of northern red oak, and at specified times. Also, the rate of moisture variation at each specified time and location must be known or correctly estimated. The functional form of the diffusion coefficient as well as the boundary conditions at the surfaces are not known a priori. The resulting system of finite difference equations defines an inverse problem, whose solution may be sensitive to small changes of input data. Results indicate that the diffusion coefficient increases with increasing moisture content below the fiber saturation point, which defines the upper limit applied by the diffusion theory.

     

Vacuum Drying of Oak Wood

ABSTRACT

Vacuum drying, j,e drying under absolute gas pressure of about 10? Pa. is an efficient means of reducing the process period and of producing good quality wood. We will examine here continuous vacuum drying where the plank surfaces are kept at a constant temperature, which remains above the boiling point, and moisture flowing to the surface is extracted from the kiln.

We have carried out an experimental study of oak drying under such conditions. The drying rate and moisture content profile of the sample (40 mm thick) are recorded during the whole drying period.

A model of continuous drying is established from general conservation equations with the main approximation that the air is rapidly extracted. The two constitutive equations of the model which describe temperature and water content fields are of a diffusive type and coupled through coefficients. The adequate boundary equation is not a convective one, but expresses a hygroscopic equilibrium between the vapour in the chamber and the wood surface. The mass diffusive coefficient can be adjusted to the drying rates through capillary pressure and bound water diffusion functions. The wood heterogeneity (seasonal growth) is the main factor of discrepancy in these functions. The simulated drying rates correspond with the experimental ones.     

Variability in Transport Properties Regarding Drying Behavior for Blackbutt Timber in New South Wales

Variability is a key issue in the processing of many biological materials, in this case the drying of hardwood timber. This article reports the measurements of variability of the diffusion coefficient (a transport property), the initial moisture content, and the basic density that are relevant to the drying of blackbutt, Eucalyptus pilularis Sm, from northern New South Wales in Australia. The diffusion coefficient was quantified using a mathematical model solving Fick's second law of diffusion for mass transfer, and Fourier's law for heat transfer. The initial moisture content and the basic density were measured using experimental procedures. Specifically, within-tree and between-tree variations are reported. The coefficients of variation of the initial moisture contents and final moisture contents are 0.24 and 0.19, respectively, for within-tree variability. A similar result was found for the amount of between-tree variability. Compensating differences in the diffusion coefficients of the timber boards were a significant reason for the small dispersion of final moisture contents, despite the large variation in initial moisture contents.

An analysis of variance showed that some timber properties were affected by the board positions within trees and between trees. Circumferential and radial effects were significant for the within-tree variability of most transport properties. Moreover, principal components analysis suggested that timber boards with low densities have high initial moisture contents and high diffusion coefficients. A potential reason is that if there is less wood material per unit volume (lower density), then there is more space to be occupied by water (higher initial moisture content), and there is also less resistance to the diffusive transport of moisture (higher diffusion coefficients).