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.     

TWO-STAGE MOISTURE DIFFUSION IN WOOD WITH CONSTANT TRANSPORT COEFFICIENTS

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.     

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.     

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.     

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.

     

DIFFUSION MODELS OF PAPAYA AND MANGO GLACe' DRYING

The objective of this research was to develop diffusion models for papaya and mango glace' drying. Effective diffusion coefficients of papaya and mango glace' were evaluated by regression analysis of the experimental data to drying kinetic equation. Models 1 and 2 were developed by assuming that effective diffusion coefficients were constant and varied proportionally with the moisture ratio. Model 3, which the Arrhenius factor was a second-degree polynomial function of moisture content, was developed by assuming that the value of effective diffusion coefficient was constant over a short time interval. Model 4, which was similar to Model 3, was developed by considering the effect of volume shrinkage during drying. Four diffusion models were compared and it was found that the predicted values of moisture contents calculated by using Models 1 and 2 were close to experimental values during the early period of drying. Models 3 and 4 were able to have better predictions particularly towards the final period of drying. However, Model 4 was complicated. Therefore, Model 3 was recommended for calculating drying curves of papaya and mango glace' drying.     

DIFFUSION OF MOISTURE IN DRYING OF SUGAR CANE FIBERS AND BUNDLES

Sugar cane fibers and arrangements of fibers in cylindrical bundles were dried in a thermoanalyzer and their diffusive coefficients were calculated using the slope method. The effect of temperature, moisture content as well as structural changes were analyzed. Diffusion coefficients changed nanlineariy with moisture content and followed an Arrhenius-like functionality with temperature. The analysis of these effects suggested a liquid diffusion transport mechanism of moisture transfer inside sugar cane fibers and bundles.     

DIFFUSION OF MOISTURE IN DRYING OF SUGAR CANE FIBERS AND BUNDLES

ABSTRACT

Sugar cane fibers and arrangements of fibers in cylindrical bundles were dried in a thermoanalyzer and their diffusive coefficients were calculated using the slope method. The effect of temperature, moisture content as well as structural changes were analyzed. Diffusion coefficients changed nanlineariy with moisture content and followed an Arrhenius-like functionality with temperature. The analysis of these effects suggested a liquid diffusion transport mechanism of moisture transfer inside sugar cane fibers and bundles.     

THEORETICAL ANALYSIS OF THE DIFFUSION PROCESS INSIDE PROLATE SPHEROIDAL SOLIDS

A numerical solution of the diffusion equation to describe solute transport inside ellipsoids, assuming a constant difiusion coefficient and a corrective boundary condition, is presented. The difiusion equation in a prolate spheroidal coordinate system was used for a hidimensional case. The finite volume method was employed to discretize the basic equation, utilizing a uniform grid size. The equation was solved irteratively using the Gauss-Seidel method. The effect of Biot number and the aspect ratio of the body on diffusion rate and concentration during the process is presented. Plots are presented for Biot numbers from 0.05 to infinity and aspect ratios of 1.1, 2.0 and 5.0. To investigate the effect of the aspect ratio, different results changing the aspect ratio for Bi= 1.0 are shown. The results show that the model is consistent and it may be used to solve other cases such as those that include cylinder and sphere geometry, and/ or those with variable diffusion coefficients with small modifications.     

THEORETICAL ANALYSIS OF THE DIFFUSION PROCESS INSIDE PROLATE SPHEROIDAL SOLIDS

ABSTRACT

A numerical solution of the diffusion equation to describe solute transport inside ellipsoids, assuming a constant difiusion coefficient and a corrective boundary condition, is presented. The difiusion equation in a prolate spheroidal coordinate system was used for a hidimensional case. The finite volume method was employed to discretize the basic equation, utilizing a uniform grid size. The equation was solved irteratively using the Gauss-Seidel method. The effect of Biot number and the aspect ratio of the body on diffusion rate and concentration during the process is presented. Plots are presented for Biot numbers from 0.05 to infinity and aspect ratios of 1.1, 2.0 and 5.0. To investigate the effect of the aspect ratio, different results changing the aspect ratio for Bi= 1.0 are shown. The results show that the model is consistent and it may be used to solve other cases such as those that include cylinder and sphere geometry, and/ or those with variable diffusion coefficients with small modifications.     

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).     

A DIFFUSION MODEL WITH A MOISTURE DEPENDENT DIFFUSION COEFFICIENT FOR PARBOILED RICE

A mathematical model describing moisture migration by diffusion in a solid sphere with variable diffusion coefficient is proposed. An analytical expression for dependence of the diffusion coefficient with moisture content was derived based on the assumption that the activation energy for diffusion varies linearly with the desorption energy.

The expression for moisture dependence of diffusion coefficient was used to simulate drying of parboiled rice in the temperature range 50-90°C. The mathematical model shows good agreement between observed and predicted drying rate curves.     

DIFFUSION COEFFICIENTS FOR PREDICTING ROUGH RICE RE-WETTING BEHAVIOR

Thin-layer re-wetting experiments were conducted with medium grain rough rice in the temperature range of 17.8 to 45^°C and for relative humidities between 56 and 89.3%, with initial moisture contents in the range of 10.26 to 12.71% dry-basis to determine the diffusion coefficient of rough rice. Recent efforts to characterize the re-wetting characteristics of rough rice are summarized. New equation for temperature dependent liquid diffusion coefficients for medium grain rough rice are presented. The diffusion coefficient in re-wetting was lower than the drying. The results presented here, over typically five day re-wetting, will be useful in studying the longer term moisture transfer process occurring during ventilated storage.     

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

ABSTRACT

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).     

DRYING GRANULAR POROUS MEDIA: GRAVITATIONAL EFFECTS IN THE ISENTHALPIC REGIME AND THE ROLE OF DIFFUSION MODELS

In this paper we have examined the influence of gravity on the moisture transport process during the isenthalpic drying period, and we have considered the use of diffusion models both to predict saturation pro- files and to extract apparent diffusivities from experi- mental data. For granular or unconsolidated porous media, the one-dimensional moisture transport process can be characterized by two dimensionless groups that account for capillary forces, gravitational forces and viscous forces. Detailed numerical solutions of the saturation transport equation indicate under what circumstances the diffusion model can be used with confidence, and under what circumstances the diffusion model can be used to predict saturation profiles even though it is an incorrect representation of the moisture transport process. In addition to exploring the predictive capabilities of the diffusion model, we have     

DRYING GRANULAR POROUS MEDIA: GRAVITATIONAL EFFECTS IN THE ISENTHALPIC REGIME AND THE ROLE OF DIFFUSION MODELS

In this paper we have examined the influence of gravity on the moisture transport process during the isenthalpic drying period, and we have considered the use of diffusion models both to predict saturation pro- files and to extract apparent diffusivities from experi- mental data. For granular or unconsolidated porous media, the one-dimensional moisture transport process can be characterized by two dimensionless groups that account for capillary forces, gravitational forces and viscous forces. Detailed numerical solutions of the saturation transport equation indicate under what circumstances the diffusion model can be used with confidence, and under what circumstances the diffusion model can be used to predict saturation profiles even though it is an incorrect representation of the moisture transport process. In addition to exploring the predictive capabilities of the diffusion model, we have     

DIFFUSION COEFFICIENTS FOR PREDICTING ROUGH RICE RE-WETTING BEHAVIOR

ABSTRACT

Thin-layer re-wetting experiments were conducted with medium grain rough rice in the temperature range of 17.8 to 45^°C and for relative humidities between 56 and 89.3%, with initial moisture contents in the range of 10.26 to 12.71% dry-basis to determine the diffusion coefficient of rough rice. Recent efforts to characterize the re-wetting characteristics of rough rice are summarized. New equation for temperature dependent liquid diffusion coefficients for medium grain rough rice are presented. The diffusion coefficient in re-wetting was lower than the drying. The results presented here, over typically five day re-wetting, will be useful in studying the longer term moisture transfer process occurring during ventilated storage.     

Modeling of Moisture Diffusion in Microwave Drying of Hardwood

A one-dimensional mathematical model was developed to predict temperature and moisture content profiles in red maple (Acer rubrum L.) and white oak (Quercus alba) during microwave drying. The model was solved using the finite element analysis with MATLAB software. The predictions for temperature and moisture content agreed favorably well with the experimental data. The diffusion coefficients of the red maple and the white oak in microwave drying conditions were calculated and analyzed. Equations of the diffusion coefficient in longitudinal and transverse directions based on input microwave power level are presented in this article. In microwave drying of hardwood, the red maple was heated more efficiently than the white oak because of higher absorbing efficiency of the microwave power.     

Inverse Finite Element Analysis of Technological Processes of Heat and Mass Transport in Agricultural and Forest Products

Due to complexity of agricultural and forest products, the mathematical model coefficients are often dubious, as experimental determination of their values leads to erroneous results. To solve this problem an inverse finite element analysis software was developed to identify coefficient values of the heat and mass transport model and to predict and visualize the processes. The model reflected 3D structure of investigated systems comprising the heat conduction and moisture diffusion in heterogeneous, anisotropic, and irregularly shaped products represented by wood and cereal grain kernels. Test cases used to validate the software covered identification of the thermal conductivity, convective heat transfer coefficient, diffusion coefficient, equilibrium moisture content, and convective mass transfer coefficient in pine and beech wood, and also in corn. Implementation of the proposed optimization algorithm and improvement of the software functionality resulted in more effective and accurate identification of the coefficient values, demonstrated by increased accuracy and reliability of predicting the heat conduction and water diffusion processes.     

Modeling of Moisture Diffusion in Microwave Drying of Hardwood

A one-dimensional mathematical model was developed to predict temperature and moisture content profiles in red maple (Acer rubrum L.) and white oak (Quercus alba) during microwave drying. The model was solved using the finite element analysis with MATLAB software. The predictions for temperature and moisture content agreed favorably well with the experimental data. The diffusion coefficients of the red maple and the white oak in microwave drying conditions were calculated and analyzed. Equations of the diffusion coefficient in longitudinal and transverse directions based on input microwave power level are presented in this article. In microwave drying of hardwood, the red maple was heated more efficiently than the white oak because of higher absorbing efficiency of the microwave power.