Moisture Diffusivity in Food and Biological Materials

The spatial distributions of water content and temperature inside moist food or biological materials (native or manufactured) during drying are important in interpretation and prediction of quality changes. The dynamics of drying in modern devices is not simple—for instance, the process of intermittent drying—thus, a liquid diffusion model is expected to be the simplest and hopefully the very effective. In this article, some controversial aspects of the concept of the effective Fickian liquid moisture diffusivity are described and discussed. These discussions are intended to stimulate more interest in working toward a better rationalization of the liquid diffusion concept. It is suggested that such a parameter may even be called a “liquid depletion coefficient” instead to avoid confusion at the fundamental level. An improved alternative model from the effective liquid diffusion model is possible and some new ideas have been explored.     

An Artificial Neural Network Model for Prediction of Drying Rates

Abstract

Drying rate data were generated for training of an ANN model using a liquid diffusion model for potato slices of different thicknesses using air at different velocities, humidities and temperatures. Moisture content and temperature dependence of the liquid diffusivity as well as the heat of wetting for bound moisture were included in the diffusion model making it a highly nonlinear system. An ANN model was developed for rapid prediction of the drying rates using the Page equation fitted to the drying rate curves. The ANN model is verified to provide accurate interpolation of the drying rates and times within the ranges of parameters investigated.     

Drying with Chemical Reaction in Cocoa Beans

Desirable flavor qualities of cocoa are dependent on how the cocoa beans are fermented, dried, and roasted. During fermentation and drying, polyphenols such as leucocyanidin and apecatechin are oxidized by polyphenols oxidase to form o-quinone, which later react nonenzymatically with a hydroquinone in a condensation reaction to form browning products and moisture. The objective of this article is to model the cocoa beans drying together with the browning reaction. A Luikov drying model for the moisture and a simple Fick's law diffusion model combined with first-order reactions for both the enzymatic oxidation and nonenzymatic condensation reactions were constructed. Both models were used to identify moisture diffusivity coefficient and total polyphenols diffusivity in cocoa beans from experimental drying and polyphenols degradation data and published kinetic data of the reactions. The theoretical drying model fitted the experimental cocoa bean drying curves with low mean square of residuals. The polyphenols diffusion and reaction model also fitted the experimental polyphenols degradation curves with minimum mean residual squares. The rate of polyphenols degradation in the cocoa beans increases at higher temperature and higher relative humidity. This is because the increasing reaction rate of polyphenols oxidation reaction as well as higher moisture diffusion at higher relative humidity and temperature. The effective moisture diffusivity in cocoa beans is estimated to be between 8.194 × 10^?9 and 8.542 × 10^?9 m²·s^?1, which is of the same order of magnitude as published data. The effective total polyphenols diffusivity is estimated to be between 8.333 × 10^?12 to 1.000 × 10^?11 m²·s^?1 with minimum mean residual squares. It is three orders of magnitude less than the estimated moisture diffusivity because of the larger polyphenols molecules. The estimated polyphenols diffusivity is very close to those published in the literature for sorption and ultrafiltration processes.     

An Artificial Neural Network Model for Prediction of Drying Rates

Drying rate data were generated for training of an ANN model using a liquid diffusion model for potato slices of different thicknesses using air at different velocities, humidities and temperatures. Moisture content and temperature dependence of the liquid diffusivity as well as the heat of wetting for bound moisture were included in the diffusion model making it a highly nonlinear system. An ANN model was developed for rapid prediction of the drying rates using the Page equation fitted to the drying rate curves. The ANN model is verified to provide accurate interpolation of the drying rates and times within the ranges of parameters investigated.     

Drying with Chemical Reaction in Cocoa Beans

Desirable flavor qualities of cocoa are dependent on how the cocoa beans are fermented, dried, and roasted. During fermentation and drying, polyphenols such as leucocyanidin and apecatechin are oxidized by polyphenols oxidase to form o-quinone, which later react nonenzymatically with a hydroquinone in a condensation reaction to form browning products and moisture. The objective of this article is to model the cocoa beans drying together with the browning reaction. A Luikov drying model for the moisture and a simple Fick's law diffusion model combined with first-order reactions for both the enzymatic oxidation and nonenzymatic condensation reactions were constructed. Both models were used to identify moisture diffusivity coefficient and total polyphenols diffusivity in cocoa beans from experimental drying and polyphenols degradation data and published kinetic data of the reactions. The theoretical drying model fitted the experimental cocoa bean drying curves with low mean square of residuals. The polyphenols diffusion and reaction model also fitted the experimental polyphenols degradation curves with minimum mean residual squares. The rate of polyphenols degradation in the cocoa beans increases at higher temperature and higher relative humidity. This is because the increasing reaction rate of polyphenols oxidation reaction as well as higher moisture diffusion at higher relative humidity and temperature. The effective moisture diffusivity in cocoa beans is estimated to be between 8.194 × 10^-9 and 8.542 × 10^-9 m²·s^-1, which is of the same order of magnitude as published data. The effective total polyphenols diffusivity is estimated to be between 8.333 × 10^-12 to 1.000 × 10^-11 m²·s^-1 with minimum mean residual squares. It is three orders of magnitude less than the estimated moisture diffusivity because of the larger polyphenols molecules. The estimated polyphenols diffusivity is very close to those published in the literature for sorption and ultrafiltration processes.     

干燥过程热质传递的简化模型

对多孔介质体积平均理论进行简化,得到简化的热质传递干燥模型,用一个理论式表达模型中有效扩散系数。对香蕉片千燥的研究表明,模型的预测值与实验结果相     

ESTIMATION OF FUNCTIONS IN DRYING EQUATIONS

In drying problem, particularly for drying foodstuff, modelling is very difficult. Many physical effects have to be taken into account for mass transfer ; then mass transfer coefficient varies

In different models unknown functions must be estimated. It is particularly the case in simple models of drying using average values of water content, where the mass transfer varies versus mean water content in falling rate period. On the other hand in the “diffusion model” we have the same problem concerning the diffusion coefficient which must be also estimated

The method we propose in this paper for these two models : simple and “diffusion model” of drying consists from measurements of temperature and water content of the product to search a numerical approach of the unknown function. This method uses optimization techniques on computer and least squares criterion between model values and experimental data

Results are given for the “diffusion model” applied to shelled corn drying to find the diffusion coefficient and for a simple 11107 del applied to plum drying to find the mass transfer coefficient.     

Convective drying of osmotically dehydrated apples described by three-dimensional numerical solution of the diffusion equation with analysis of water effective diffusivity spatial distribution

Abstract

This study presents two liquid diffusion models to represent the convective drying of apple, osmotically dehydrated in sucrose solution, cut into parallelepiped-shaped pieces. Model 1 considered water diffusivity and the volume of the slices with constant values. Model 2 considered water effective diffusivity and the dimensions of the slices as variable. The numerical solution of the three-dimensional diffusion equation in Cartesian coordinates was obtained through the finite volume method, with a fully implicit formulation and boundary condition of the third kind. Process parameters were estimated by an optimizer using experimental data. A spatial distribution analysis was carried out for water effective diffusivity and moisture content in the apple slices. The results showed that the concentration of the osmotic solution used in the pretreatment influenced the drying process and that the mathematical model that considered a variable diffusivity and shrinkage was more suitable to describe the experimental data.     

Air drying characteristics and moisture diffusivity of carrots

The effects of air temperature on drying kinetics of carrot cubes were investigated. Convective drying characteristics of carrot cubes in a spout-fluidized bed were evaluated through the effect of air temperature on drying kinetics. Drying was carried out at 60, 70, 80 and 90 °C and the falling drying rate data were used to calculate the effective diffusion coefficients from the Fick's equation. Four mathematical models available in the literature were fitted to the experimental data. The Two-term model is given better prediction than the Henderson and Pabis, Page and Lewis model and satisfactorily described drying characteristics of carrot cubes.     

Drying of Carbohydrate and Protein Solutions

Drying behavior of a single suspended droplet of various protein and carbohydrate solutions was investigated experimentally. For small molecular weight carbohydrates such as sucrose and glucose, the constant-rate period was observed, which was followed by the falling rate period characterized as the “regular regime”. During the falling rate period. the drying rate was very low due to a sharp decrease of the water diffusion coefficient with decreasing water concentration No remarkable constant rate periods were observed for large-molecular carbohydrates and proteins. This is because the water diffusion coefficient is low even at high water concentrations. The drying curves or bovine serum albumin, skimmilk and model-food liquid solutions were found to be predicted by a computer simulation with the concentration dependent water diffusion coefficient of maltodextrin of dextrose equivalent =11. The experimental investigations on the ethanol and enzyme retention during drying indicated that the ethanol is lost during the costant-rate period and the enzyme inactivation occurs during the constant-rate period (regular regime).     

MICROSCOPIC DETERMINATION OF TRANSPORT PARAHETERS IN DRYING POROUS MEDIA

ABSTRACT

The flow and distribution of liquid and vapor in the pore space of drying porous media are represented by a simple network model that incorporates the microscopic mechanisms. A method akin to Monte Carlo and molecular dynamics approaches is described for calculating from any such model the macroscopic, ‘volume- averaged’, or effective transport parameters: relative permeabilities for pressure-driven flow, effective diffusivities for molecular diffusion, and so on.

The method is to map representative small samples of pore space onto regular networks that have equivalent mean coordination and are made up of biconical pore segments intersecting in pore bodies, all dimensions being drawn from measured or estimated distributions. Evaporation and meniscus curvature, vapor movement by diffusion away from menisci and viscous flow toward them are described by one-dimensional local approximations. The resulting large set of ordinary differential and algebraic equations is solved by computer. Solutions from several such realizations are averaged to determine relative permeability to liquid, capillary pressure, and effective diffusivity of vapor as functions of liquid saturation and drying history. The results are to be used for interpreting, interpolating, and extrapolating experimental measurements of the same quantities.     

On the Applicability of Normalization for Drying Kinetics

A model to determine the requirements of normalization has been derived by means of equivalence of two models. The first model uses the Clausius-Clapeyron equation to recognize the influence of gas temperature on the drying rate according to normalization. The second approach applies the Arrhenius diffusion model to identify the influence of gas temperature on the drying rate. It was found that normalization works well when the activation energy for the effective diffusion coefficient has the “right” value. In this study the normalization works well for polypropylene and fails for polyamide 6.     

Drying with Internal Vaporisation : Introducing the Concept of Identity Drying Card (IDC)

Different drying configurations (convective drying with moist air and superheated steam, microwave drying and vacuum drying) on different materials (isotropic and anisotropic) were experimentally studied in order to model and visualise the evolution of internal pressure and temperature. To be able to do so, in addition to measuring the average moisture, a method which can determine internal-local pressure and temperature simultanously by using specially designed sensors was developed. In combination with the experiments, the numerical code TRANSPORE has been used to simulate drying processes. A less comprehensive but more comprehensible analytical model was also provided to facilitate the better understanding of internal phenomena. Based on the results gained from measurement and numerical analysis, the dynamic distribution and development of local temperature and pressure inside seasoned medium are coupled together by a temperature-pressure graph, which is herewith called “Identity Drying Card” (IDC), a new concept initialid in the paper. By using IDC, the internal profile of temperature and pressure, the dominant transport properties (penncability and difisivity), the mechanism of transport (diffusion, convection or both) and the phase transitions during drying can be visualised. More specifically, the amount of dry air, the moisture content in the hygroscopic rcgion or the danger due to internal mechanical loads of handled materials can be figured out with the aid of IDC.     

An Analytical Model for Moisture Diffusion in Solid Objects During Drying

ABSTRACT

An analytical model is proposed to determine the moisture diffusivities in geometrical solid objects (namely, infinite slab, infinite cylinder, sphere) subjected to drying applications. The analysis for moisture diffusion is carried out on the basis of two imponant criteria: 0.l?Bi?100 and Bi>100. New correlations for the mots of the characteristic equations corresponding these objects are developed. For a drying application, using the similarity between cooling and drying profiles, drying coefficient and lag factor are obtained. The model is then verified using available data from literature.Results show that the presented model can be used to determine the moisture diffusion coefficients in such solid objects in a simple and accurate manner in a variety of drying applications.     

THE ROLE OF THEORETICAL AND MATHEMATICAL MODELLING EN SCALE-UP

Drying processes are complex because of the non-linearity of physical phenomena and of the distributed parameter character on many levels of consideration. At different scales the relative importance of phenomena will differ. A discussion of the level structure is presented and the concept of “critical grain size” is introduced. As a general criterion for the importance of panicle diffusion resistance a modified Biot-number is defined. Theoretical models are treated for diffusional mass transfer, both rigorous and short-cut models. The integration of these models with approximating relations or models for specific drying processes is demonstrated for spray drying of foods and fluidized bed drying of bioproducts. Also a theoretical treatment of several quality aspects is given.     

Finite Element Simulation of Drying of Longan Fruit

A two-dimensional finite element model has been developed to simulate moisture diffusion in longan fruit during drying and moisture diffusivities of the components of longan fruit determined experimentally are used in this simulation. Shrinkage of the flesh of longan during drying is also taken into account. The finite element model is programmed in Compaq Visual FORTRAN version 6.5. This finite element model satisfactorily predicts the moisture diffusion during drying. Moisture contents in the different components in the longan fruit during drying are simulated. Moisture content profiles of the longan fruit are also predicted. Knowledge gained from this study will be useful in the understanding of the transport processes in the different components of the longan fruit.     

PARAMETER ESTIMATION OF FICK'S LAW DRYING EQUATION

Distributed parameter drying models such as the Fick's law diffusion model, unlike the lumped parameter model of van Meel whose parameters can be easily estimated by regression, suffer from the difficulty in estimating the parameters of the models quantitatively with accuracy. In the past they were estimated by visual inspection of the theoretical drying curves which fit the experimental drying curve best In this work, a quantitative parameter estimation technique originally suggested by Chavent, is developed by minimizing the integrated squares of error between theoretical and experimental curves over the drying lime (the criterion) subjected to the constraints that the theoretical curve is governed by the constant diffusivity Fick's taw diffusion equation (the constraint). Although the estimation of Fick's law constant diffusivity can be done by using the analytical solution developed by Crank, the use of the Fick's law model here is simply to demonstrate the utility of the proposed technique which can be used in more complex distributed models. The optimization problem is to solve for the adjoint equation for which the value of the Fick's law diffusivity minimizes the criterion. The Lagrangian derivative is solved by using a discrete derivative of the criterion. The theoretical curves are generated by using simple explicit (FSE) and modified Crank-Nicholson (FCR) algorithms The drying of oil palm kernels are used as a case study. Ii is found that the estimated diffusivities of moisture in oil palm kernels range from 0 5 to 5.0 × 10^-10 m²sol;s which are comparable with published data. It is also found that the estimated diffusivity is dependent on the initial moisture content.     

PARAMETER ESTIMATION OF FICK'S LAW DRYING EQUATION

ABSTRACT

Distributed parameter drying models such as the Fick's law diffusion model, unlike the lumped parameter model of van Meel whose parameters can be easily estimated by regression, suffer from the difficulty in estimating the parameters of the models quantitatively with accuracy. In the past they were estimated by visual inspection of the theoretical drying curves which fit the experimental drying curve best In this work, a quantitative parameter estimation technique originally suggested by Chavent, is developed by minimizing the integrated squares of error between theoretical and experimental curves over the drying lime (the criterion) subjected to the constraints that the theoretical curve is governed by the constant diffusivity Fick's taw diffusion equation (the constraint). Although the estimation of Fick's law constant diffusivity can be done by using the analytical solution developed by Crank, the use of the Fick's law model here is simply to demonstrate the utility of the proposed technique which can be used in more complex distributed models. The optimization problem is to solve for the adjoint equation for which the value of the Fick's law diffusivity minimizes the criterion. The Lagrangian derivative is solved by using a discrete derivative of the criterion. The theoretical curves are generated by using simple explicit (FSE) and modified Crank-Nicholson (FCR) algorithms The drying of oil palm kernels are used as a case study. Ii is found that the estimated diffusivities of moisture in oil palm kernels range from 0 5 to 5.0 × 10^-10 m²sol;s which are comparable with published data. It is also found that the estimated diffusivity is dependent on the initial moisture content.