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.     

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.     

DETERMINATION OF MOISTURE DIFFUSIVITY AND BEHAVIOR OF TOMATO CONCENTRATE DROPLETS DURING DRYING IN AIR

ABSTRACT

The drying mechanism and diffusion coefficient of water in spherical droplets (1.73 – 2.08 mm diameter) of tomato concentrates were successfully interpreted and modelled by using Fick's law. Solids content of the initial concentrate (5–15% w/w), and drying temperature (60^° – 100^° C) were varied but the drying air was kept at constant velocity and humidity.

The effective moisture diffusivity was estimated from the drying rate curves and expressed by an Arrhenius relation. Further, it was observed that case hardening has a large effect on the diffusion process causing the effective diffusional distance and the rate of moisture accumulation in the hardened crust to vary with the moisture content, according to a sorption controlled mechanism.     

Vacuum dehydration kinetics of onion slices

Onion slices were dried in a single layer of thickness varying from 1 to 5 mm in the temperature range of 50-70 °C in a laboratory scale vacuum dryer. The effect of pretreatment, drying temperature and slice thickness on the drying kinetics of onion slices was studied. Four thin layer drying models namely Lewis model, logarithmic model, Page model and Fick's law model were applied on the experimental moisture loss data with respect to time to predict the drying pattern properly on the basis of coefficient of determination and standard error. The Page model showed better fit to the experimental data compared to other models. Effective moisture diffusivity of the slice was measured using Fick's second law of unsteady state diffusion. The diffusivity values were found ranging from 1.32E−10 to 1.09E−09 m²/s for untreated and 1.32E−10 to 1.09E−01 m²/s for treated onion slices. Effective moisture diffusivity showed increasing trend with increase in temperature and thickness.     

Drying and Heat Transfer Characteristics for a Novel Fluidized Bed Dryer

Abstract

The use of a fluidized bed dryer with a lateral air flow and mechanical agitation to the drying of sludge from a wastewater treatment plant was investigated. Experimental curves of moisture content vs. drying time, as well as heat transfer coefficients and the size characteristics of the products, were determined at temperatures between 80°C and 110°C, a stirring rate of 55 rpm and air velocity of 0.9 m/s for 3 kg sludge batches with initial moisture contents of 0.55 and 0.65 (d.b.). Experimental drying kinetics were compared with values derived from three models based on Fick's second law, namely: the constant diffusivity model, the simplified variable diffusivity model, and the modified quasi-stationary model.     

Effective Diffusivity Determination Considering Shrinkage by Means of Explicit Finite Difference Method

Drying data of salted shark (Carcharhinus limbatus) muscle pieces samples, dried in a convective dryer, by using three different air conditions and two different air velocities, were treated by Fick's second law. The shrinkage during drying process was considered as a linear function of sample moisture. The Fick's second law equation was numerically solved by the explicit finite difference method to obtain effective diffusivity, considering shrinkage and with and without simplification of moisture content profile. The experimental data fitted very well independently of the simplification introduced on the moisture content profile. The best fit was obtained by considering volumetric average of sample moisture content. The effective diffusivity values calculated considering the shrinkage varied from 0.72 to 2.20 × 10^?10 m²/s with the mean relative deviation modulus from 1.02 to 6.51%. The activation energy expressed in function of air temperature varied from 3.42 to 19.23 kJ/mol.     

Drying Characteristics and Moisture Diffusivity of Distillers' Spent Grains Dried in Superheated Steam

Distillers' spent grain pellets were prepared from material with an initial moisture content of 25% (wb). These pellets were dried in pairs using superheated steam at 120°C in two orientations, horizontal and vertical. The drying characteristics, modeled by the Page equation, showed that there was a significant difference between orientations. The overall moisture diffusivity was calculated using a finite cylinder model based on Fick's law of diffusion accounting for a change in dimensions over the course of drying. The overall diffusivity values ranged from 4.08 × 10^?10 to 1.48 × 10^?8 m²/s.     

A New Correlation for the Estimation of Moisture Diffusivity in Corn Kernels from Drying Kinetics

ABSTRACT

The modelisation of cereal drying kinetics is more, and more often performed by the diffusion equation (Fick's law) Eor homogeneous materials rather than lumped or semi-empirical equations. The heterogeneity of the material is accounted for by the use of an effective diffusivity the prediction of which is essential for the determination of the functional relationship between diffusion coefficient and arain moisture content and temperature. For this dependence, the method of resolution may be analytical or numerical. In this study, the estimation of diffusion coefficient is based on drying curves obtained with arains in the ranae of initial moisture content of 18-509 and a temperature of 50:120??. The important result corresponds to the finding that the moisture diffusivity is a function not only of variabies of the state i.e. moisture content and temperature, but also of the initial moisture content as in the following : 'D = P4g X, exp(8 X ). AD and B are both dependent on temperature     

Determination of Moisture Diffusivity by Thermo-Gravimetric Analysis under Non-Isothermal Condition

Abstract:

In drying of solids, the diffusion model based on Fick's second law is usually applied to interpret the moisture migration within the solid. Then the temperature dependence of the moisture diffusivity, generally described by an Arrhenius-type equation, is obtained through the drying kinetics. In this article, a nonisothermal (linearly increasing temperature) procedure was used to determine the moisture diffusivity as a function of temperature with the complex optimization method, and the result was accessed by comparison with a classical isothermal procedure. All the experiments were conducted in a thermogravimetric analyzer (TGA) for accurately recording the mass loss from the sample and easily programming the heating profile.     

From Laboratory Experiments to Design of a Conveyor-Belt Dryer via Mathematical Modeling

Abstract

A conveyor-belt dryer for picrite has been modeled mathematically in this work. The necessary parameters for the system of equations were obtained from regression analysis of thin-layer drying data. The convective drying experiments were carried out at temperatures of 40, 60, 80, and 100°C and air velocities of 0.5 and 1.5 m/sec. To analyze the drying behavior, the drying curves were fitted to different semi-theoretical drying kinetics models such as those of Lewis, Page, Henderson and Pabis, Wang and Singh, and the decay models. The decay function (for second order reactions) gives better results and describes the thin layer drying curves quite well. The effective diffusivity was also determined from the integrated Fick's second law equation and correlated with temperature using an Arrhenius-type model. External heat and mass transfer coefficients were refitted to the empirical correlation using dimensionless numbers (J _h , J _D  = m · Re ⁿ ) and their new coefficients were optimized as a function of temperature. The internal mass transfer coefficient was also correlated as a function of moisture content, air temperature, and velocity.     

Enthalpy-Entropy Compensation for Water Loss of Vegetable Tissues during Air Drying

Transition state theory was used to study enthalpy-entropy compensation for water loss during air drying of potato and apple slices. Slices of either potato or apple of 4-mm thickness, 40 mm diameter and air drying temperatures of 323, 333, 343, and 353 K were employed in the experiments. Moisture content and internal potato and apple slice temperatures were recorded during the drying runs. Water loss during drying was described by the unsteady-state Fick's equation and moisture diffusivity evolution was established by applying the method of the slopes. Thus, the experimental drying curve was compared to the theoretical diffusion curve, and the slope of the two curves were estimated at the same moisture content to in order give the corresponding value of diffusivity. During drying, the moisture diffusivity reached a maximum value as the water content of potato and apple slice was around 1 kg water/kg dry solid, regardless of the air temperature. The isokinetic temperature was found to be 320.2 and 312.8 K for potato and apple tissues, respectively. These values were greater than the experimental harmonic mean temperature, which was found to be 307.4 and 308.3 K for potato and apple tissues, respectively. Thus, it was concluded that the water loss process is enthalpy controlled.     

Drying characteristics of sweet cherry

The effects of alkali emulsion of ethyl oleate and air temperature (60, 70 and 75 °C) on the drying characteristics of sweet cherry were studied using a hot air dryer at a constant air velocity of 2.0 m/s. It was observed that both the alkali emulsion of ethyl oleate and air temperature affected the drying time. The drying times of pre-treated samples were 19.5-22.6% shorter than those of control samples. Five semi-theoretical thin-layer models, namely, Lewis, Henderson and Pabis, Logarithmic, Page, Wang and Singh models were used for the modeling of the drying kinetics. The fit quality obtained with each model was evaluated using statistical tests. After comparing experimentally obtained values with the calculated values from the models, it was concluded that Page model represents the drying characteristics better than the other models. The effective moisture diffusivity was determined by using Fick's second law and was observed to lie between 5.683 × 10⁻¹⁰ and 1.544 × 10⁻⁹ m²/s for the pre-treated and control samples. Rehydration ratio was significantly affected by pre-treatment and air temperature. It was found to increase proportionally with the increase in air drying temperature.     

DETERMINATION AND INTERPRETATION OF THE CRITICAL MOISTURE CONTENT (C.M.C.) AND THE INTERNAL MOISTURE CONTENT PROFILE DURING THE CONSTANT DRYING RATE PERIOD.

Abstract

In this study, the critical moisture content ( CMC. ) and the internal moisture content profiles during the constant rate period were determined experimentally for several conditions of convective drying of plaster. The experimental CMC values obtained allow us to validate the theoretical model of prediction of this parameter presented by KEEY [1] and SUZUKI [2] for drying rates higher than 3×10^?4 kg/(m².s) and to determine the equivalent moisture difliisivity at the external surface of plaster which was found to be equal to 3.9 × 10^?9 m²/s On the other hand, expressions of the internal humidity profiles were determined without solving the second FICK's law and were found to be same that the ones obtained by KEEY [1] for thick products and KRJSCHER and KROLL [3] for thin products by solving this law. The moisture diffusivity at the surface previously obtained was used in these equations to predict the internal humidity profiles during the constant rate period Comparison between theoretical predictions and experimental humidity profiles obtained on plaster shows a quite good agreement.     

Air Drying Kinetics of Potato Cylinders

ABSTRACT

An experimental air tunnel dryer was used to investigate the kinetics of moisture transport in potato cylinders (Solanum tuberasum). Acoordingly, the experimental results, represnting only falling-rate drying behaviour and hence. dehydration completely controlled by internal mass transfer, were interpreted on the basis of Fick's diffusional model for non-stedy state radial diffusion. The effects of air velocity and temperature on the drying rate were studied. with he temperature being the principal controlling factor. Analysis of the drying curves by the method of slopes resulted in a variable effective moisture diffusivity. Shrinkage as a function of moisture content under various drying conditions was investigated. The volumetric shrinkage of the samples was affected mainly by air velocity. whilst air temperature had a negligible effect. Good agreement was obtained between the experimental apparent density data and the predicted correlation.     

Drying Characteristics of Oil Palm Kernels

ABSTRACT

The drying characteristics of a single oil palm kernel suspended in a drying tunnel is studied. Luikov's unsteady state simultaneous heat and mass transpon equations are proposed as the governing equations. The model is simulated using the explicit and modified implicit Crank Nicholson finite difference algorithms. The desorption isotherms of the kernel warj determined using an environmental chamber and the data generated was found to fit the Hasley's and Henderson's equations well. The mass diffusion coefficient was determined by using a distributed system parameter identification technique where a weighted least square criterion between model and experimental average moisture content and temperature profiles is optimized using the Lagrangian algorithm within the constraints of the governing equations. The simulated parameter estimated drying characteristics were found to be in good agreement with experimental values.     

A Method of Determination of Concentration Dependent Effective Moisture Diffusivity

ABSTRACT

A method based on Fourier series solution to Fick's diffusion equation has been proposed to evaluate effective diffusivity (D) as a function of moisture content in agricultural materials undergoing shrinkage during drying process. The shrinkage kinetics of the particulate was used to correlate its instantaneous size (spherical equivalent diameter) as a function of material moisture content A computer program was used to evaluate D based on shrinkage kinetics and experimental drying data and relate it to moisture content. The method was used to obtain moisture diffusivity data for thin layer drying of grape and corn.     

An Effective Moisture Diffusivity Model Deduced from Experiment and Numerical Solution of Mass Transfer Equations for a Shrinkable Drying Slab of Microalgae Spirulina

From experimental data, Spirulina effective moisture diffusivity was analytically estimated by considering two diffusion regions and the product shrinkage. Then, the moisture diffusivity was deduced from the numerical solutions of mass transfer equations by minimizing the difference between experimental and simulated drying curves and by taking into account the slab thickness variation. The range of moisture diffusivity used for simulations was estimated from minimal and maximal values of experimental effective diffusivities and calculation started with the mean value of experimental effective diffusivities. Identified effective diffusivities ranged from 1.79 × 10^?10 to 6.73 × 10^?10 m²/s. These diffusivities increased strongly with drying temperature and decreased slightly with moisture content. A suitable model correlating effective diffusivity, temperature, and moisture content was then established. Effective diffusivities given by this model were very close to experimental ones with a relative difference ranging from 0.5 to 24%.     

Thin-Layer Drying of Flax Fiber: I. Analysis of Modeling Using Fick's Second Law of Diffusion

Thin-layer drying of moist flax fiber was performed at four temperatures of 30, 50, 70, and 100°C with a constant absolute humidity of 0.0065 kg water per kg dry air. The coefficients of diffusion of the fiber at different drying conditions were estimated by modeling the drying process using the one- to five-term solutions of the second Fick's law of diffusion. The models underestimated the drying process during the initial stages of drying and overestimated this process during the final stages. The estimated coefficient of diffusions ranged from 5.11 × 10^?9 to 1.92 × 10^?8 m²/s and linearly increased with the drying air temperature.     

Experimental modelling of infrared drying of industrial grape by-products

The thin-layer infrared drying behaviour of industrial grape by-products was experimentally investigated in the temperature range from 100 to 160 °C. The drying rate was found to increase with temperature, thus reducing the total drying time. In particular, as drying temperature was raised from 100 °C up to 160 °C, the time period needed to reduce the moisture content of the sample from 204.32% down to 38.89% by weight (dry basis) decreased from 60.5 to 21 min.Using a non-linear regression (Marquart's method) together with a multiple regression analysis, a mathematical model for the thin-layer infrared drying process of wet grape residues was proposed. The values for the diffusivity coefficients at each temperature were obtained using Fick's second law of diffusion. They varied from 11.013 × 10^?9 to 26.050 × 10^?9 m²/s along the temperature range. The temperature dependence of the effective diffusivity coefficient was expressed by an Arrhenius type relationship. Activation energy for the moisture diffusion was determined as 19.27 kJ/mol.