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.

     

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.     

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.     

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.     

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.     

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.     

DIFFUSION LAG TIME FOR MULTIPLE AND PERIODIC LAMINATES

The permeability and lag time for a heterogeneous diffusion system, in which the diffusivity and partition coefficient for the diffusant are dependent on position, are formulated in terms of a linear asymptotic analysis. A repeated integration of the diffusion equation is used to obtain the time dependence of the total solute release into the receiver, Q(t). The asymptotic form of Q(t) is linear in time. The slope, and intercept of this linear asymptote with the time axis, respectively, give formulas for the steady-state permeability and lag time. These formulas are then applied to diffusion systems of multiple laminates, consisting of a series of different homogeneous slabs. Thus, for the first time, a concise treatment of diffusion in multiple laminates is obtained. The formulas are also applied to periodic laminates, consisting of a series of identical slabs, but with position-dependent diffusivity and partition coefficient. We found that the lag time can be well approximated by (nh))²/(6D_eff), where n and h are, respectively, the number and thickness of individual lamella, and D_eff is an effective diffusivity, for which a relation in terms of the local property distributions is obtained. This approximation becomes more accurate with increasing number of lamellae. At n = 5, the relative error is already within 4%. Finally a procedure is discussed for readily obtaining the lag time for periodic structures consisting of a serial repetition of a multi-laminate.     

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.     

MOISTURE DIFFUSION IN GELATIN SLABS BY MODELING DRYING KINETICS

ABSTRACT

The aim of this work is to improve the knowledge concerning moisture mobility in gelatin slabs, through the evaluation of moisture diffusivity (function of temperature and moisture content) by modeling of drying kinetics

The first part of this work is devoted to the determination of the gel characteristics needed for a good modeling. Drying experiments were then carried out on a pilot drier according to a central composite design with 3 factors (temperature, moisture content and pH).

Moisture diffusivity was finally calculated by fitting a model to the drying experiments. The chosea model (a conventional diffusive model with variable diffusion coefficient, solved by finite difference calculations in a solid-related frame of coordinates) did not perfectly fit the experimental results. It especially under-estimates moisture gradients at the surface of the gel     

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.     

NONDESTRUCTIVE MEASUREMENT OF MOISTURE DIFFUSION COEFFICIENT IN WOOD DRYING

The computer tomography(CT image of wood was analyzed to estimate the moisture content from CT number. Comparing moisture contents, densities and CT numbers, moisture distributions were nondestructively determined by CT numbers detected with the scanner.

By periodically measuring CT images of the same cross section of specimens in drying, change of moisture distribution was computed. The coefficients of moisture movement at given positions inside the specimen were directly calculated from Fick's Law with known values of moisture content change, distance and drying time.

The diffiion coeficient was described as a function of various driving forces of moisture content, partial vapor pressure and chemical potential of moisture in wood. The results were comparable to data in the literature. Furthermore, it is shown that diffusion coefficients based on moisture content varies as a curve with the maximum value at about 15% m.c. during drying, and that the moisture content is actually adaptable to the driving force for moisture movement in wood because of capability over wide moisture content range.     

离子扩散型抗菌陶瓷中银离子的扩散

研究了离子扩散法制造抗菌陶瓷过程中Ag+在釉层中的扩散行为.用扫描电镜和电子探针分析了扩散过程中Ag的分布.用Fick定律分析扩散过程,提出用表层Ag+质量分数和表观扩散系数来描述实际的扩散过程.用余误差函数近似法确定扩散温度下不同成分釉层中Ag+的表观扩散系数.对于所研究的釉层,表观扩散系数为1.6×10-11～1.3×10-10 cm2/s.确定了上述2个参数与釉层成分、熔盐比例、扩散温度及扩散时间的关系.利用这些关系建立了优化制造条件的基础.     

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.     

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.     

Kinetics and Mass Transfer during Atmospheric Freeze Drying of Red Pepper

Drying is applied for moisture removal to allow safe and extended storage. Red pepper (Capsicum annum) samples were heat pump dried in fluidized bed at different air temperatures. A slightly modified solution of the diffusion equation was used to describe the kinetics and drying rates of red pepper. The model well described the low- and medium-temperature drying processes. The determined effective mass diffusivities varied from 0.7831 to 4.0201 × 10^-9 m²/s and increased consistently with drying air temperature. The mass diffusivity was correlated to temperature by linear regression with coefficient of determination equal to 0.999 and negligible standard error.     

MASS TRANSFER AND DIFFUSION COEFFICIENT DETERMINATION IN THE WET AND DRY SALTING OF MEAT

Dehydrated salted meat is widely used in Brazil as a very important source of animal protein. The main objective of this kind of processing is water removal. initially by osmotic pressure changes and then by drying, resulting in a product with intermediate moisture levels.

In this work, mass transfer and salt diffusion in pieces of meat submitted to wet and dry salting were studied. Slabs of beef m. trapezius with an infinite plate geometry were salted in a NaCl saturated solution or in a dry salt bed, at two temperatures (10 and 20°C) and different time exposures (120 min and 96 hours). Equilibration studies were extended up to six days.

It was observed that water loss increased with salt uptake, for increasing periods of times. At 20°C the moisture loss was higher than it was at 10°C in both salting processes. On the other hand, the kinetics of salt uptake and moisture loss were of greater importance in the process of dry salting than in that of wet salting.

The salt diffusion coefficient for wet salting was 0.26 × 10^-10m²/s at20°C and 0.25 × 10^-10 m²/s at 10°C and for the dry salting the values were 19.37 × 10^-10 m²/s at 20°C and 17.21 × 10^-10 m²/s at 10°C.     

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).     

Estimation of the Effect of Shape and Temperature on Drying Kinetics Using MLP

The kinetics of drying is described in the article using artificial neural networks (multilayer perceptron MLP). Drying curves for vegetables are possible to obtain theoretically on the basis of the equations of mass transfer in a porous material. A key role in these equations is played by the effective coefficient of water diffusion in the form of liquid, vapor or jointly as liquid and vapor. The diffusion coefficient which depends both on moisture content in the material and temperature should be determined experimentally. The drying kinetic curve in this article is treated as a time series dependent on the state of material prior to drying and on the constant K characterizing process parameters such as drying temperature and describing the material, e.g., its shape and moisture content. Constant K characterizes the analyzed material from the drying point of view because it contains a diffusion coefficient and depends on the shape of material. The following materials were analyzed: beetroot and potato dried in the form of cubes, slices, chips, and strips. Experiments were carried out in a laboratory oven dryer at process temperature 50, 60, 70, 90, and 106°C.     

A METHOD OF DETERMINATION OF THE DIFFUSION COEFFICIENT BASED ON KERNEL MOISTURE CONTENT AND ITS TEMPERATURE

Mathematical analysis of convection drying in the falling rate period was used to determine the overall coefficient of diffusion as a function of temperature and moisture content. The mass diffusion equation and its series solution was adopted for a single wheat kernel, assuming that it can be represented by a sphere as a model system. The results of the experimental data and the calculations are presented in the graphical form.