DRYING OF CARROTS. I. DRYING MODELS.

Three models of different complexity are proposed to describe the falling rate period of the carrot drying process with shrinkage. A moving or fixed boundary problem as well as a constant or local moisture and temperature dependent effective diffusivity are considered. The moving boundary problem is solved by an explicit finite difference method. Heat transfer coefficient and effective diffusivity identification were carried out. The results of the heat transfer coefficient show a good agreement with other sources. Using experimental data and the models. describing the heat and mass transfer three different expressions for the effective diffusivity are established. Two of them are only temperature dependent considering or not particle shrinkage. The third one takes into account temperature and local moisture as well as shrinkage.

Drying of foods is a complicated process involving simultaneous coupled heat and mass transfer phenomena which occur inside the material being dried (Chiang and Petersen, 1987). Several models are found in the literature, representing mass and energy transfer which take place during food drying (King, 1968; Sokhansanj and Gustafson, 1980). Usually, approximate solutions are obtained with these     

DRYING OF CARROTS. I. DRYING MODELS.

Three models of different complexity are proposed to describe the falling rate period of the carrot drying process with shrinkage. A moving or fixed boundary problem as well as a constant or local moisture and temperature dependent effective diffusivity are considered. The moving boundary problem is solved by an explicit finite difference method. Heat transfer coefficient and effective diffusivity identification were carried out. The results of the heat transfer coefficient show a good agreement with other sources. Using experimental data and the models. describing the heat and mass transfer three different expressions for the effective diffusivity are established. Two of them are only temperature dependent considering or not particle shrinkage. The third one takes into account temperature and local moisture as well as shrinkage.

Drying of foods is a complicated process involving simultaneous coupled heat and mass transfer phenomena which occur inside the material being dried (Chiang and Petersen, 1987). Several models are found in the literature, representing mass and energy transfer which take place during food drying (King, 1968; Sokhansanj and Gustafson, 1980). Usually, approximate solutions are obtained with these     

DRYING OF CARROTS IN A FLUIDIZED BED. I. EFFECTS OF DRYING CONDITIONS AND MODELLING

Drying curves were determined in a mechanically agitated fluidized bed dryer, at temperatures between 70°C and 160°C, air velocities between 1.1 m/s and 2.2 m/s and stirring rates between 30 rpm and 70 rpm for batch drying of 3 kg lots of carrot slices, measuring the moisture content and shrinking of the particles in time. This was complemented by a study of the rate and degree of swelling of dried carrot particles in water between 20 and 75°C. Drying kinetics were modeled by Fick's second law, for which an optimal agreement with the experimental data was obtained when the effective diffusivity (D_e) was determined by a correlation based on the air velocity (v), the air temperature (T) and the dimensional moisture content of the carrot particles (X/X_o). Loss of carotenes is minimized when dehydration is carried out at about 130°C with a drying time below 12 min.     

DRYING OF CARROTS IN A FLUIDIZED BED. I. EFFECTS OF DRYING CONDITIONS AND MODELLING

ABSTRACT

Drying curves were determined in a mechanically agitated fluidized bed dryer, at temperatures between 70°C and 160°C, air velocities between 1.1 m/s and 2.2 m/s and stirring rates between 30 rpm and 70 rpm for batch drying of 3 kg lots of carrot slices, measuring the moisture content and shrinking of the particles in time. This was complemented by a study of the rate and degree of swelling of dried carrot particles in water between 20 and 75°C. Drying kinetics were modeled by Fick's second law, for which an optimal agreement with the experimental data was obtained when the effective diffusivity (D _e ) was determined by a correlation based on the air velocity (v), the air temperature (T) and the dimensional moisture content of the carrot particles (X/X _o ). Loss of carotenes is minimized when dehydration is carried out at about 130°C with a drying time below 12 min.     

SIMULATION MODELS OF MULTI-CYLINDER PAPER DRYING

ABSTRACT

The literature on simulation models for multi-cylinder paper dryers is surveyed, 20 such models being reviewed. Models for intraweb heat and mass transfer are treated extensively, a separate chapter being devoted to them. In some of the models the specific physical transfer phenomena are modelled in detail, whereas in other models a more macroscopic approach is adopted. Normally the latter approach requires the incorporation of a large number of empirical constants into the model to describe the complicated heat and mass transfer processes involved and to obtain a reasonable agreement between the model and the measured data. In order to develop the simulation models further without use of an inordinately large number of empirical constants, the different physical phenomena which occur in the dryer must be modelled in greater detail, bnsed both on fundnmental theory and on empirical correlations.     

SIMULATION MODELS OF MULTI-CYLINDER PAPER DRYING

The literature on simulation models for multi-cylinder paper dryers is surveyed, 20 such models being reviewed. Models for intraweb heat and mass transfer are treated extensively, a separate chapter being devoted to them. In some of the models the specific physical transfer phenomena are modelled in detail, whereas in other models a more macroscopic approach is adopted. Normally the latter approach requires the incorporation of a large number of empirical constants into the model to describe the complicated heat and mass transfer processes involved and to obtain a reasonable agreement between the model and the measured data. In order to develop the simulation models further without use of an inordinately large number of empirical constants, the different physical phenomena which occur in the dryer must be modelled in greater detail, bnsed both on fundnmental theory and on empirical correlations.     

CLASSIFICATION OF DRYING MODELS FOR POROUS SOLIDS

Models to describe drying pmesses are necessary for engineering design and optimization. Many resenrch studies in the past century have had the objective of developing mathematical models to describe drying processes in porous solids. This review identifies key characteristics of drying models including controlling process resistances, internal mechanisms of moisture movement, and methods of model cufficient determination, model solution, and model validation. Similarities and differences between previous work are noted, and strategies for future drying model development are discussed.     

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.     

TECHNICAL AND ECONOMICAL EVALUATION OF SOLAR DRYING

Precondition of the successful application of Solar drying is the economy mainly influenced by the savings and the costs. Components of savings and costs are related to the design of the solar dryer and the drying technology to be performed. Analysis of relations between the technical solutions and the economy is presented with regard to some often neglected or undervaluated effects, i.e. savings obtained by the absolute cleanness of solar energy, quality of the dried product and, the energy effectiveness of the drying process itself. Conclusions of- fer some contributions of principle to the selection of the system of solar dryer to be used, to design of solar dryers and, to direction of the drying technology 131.     

TECHNICAL AND ECONOMICAL EVALUATION OF SOLAR DRYING

Precondition of the successful application of Solar drying is the economy mainly influenced by the savings and the costs. Components of savings and costs are related to the design of the solar dryer and the drying technology to be performed. Analysis of relations between the technical solutions and the economy is presented with regard to some often neglected or undervaluated effects, i.e. savings obtained by the absolute cleanness of solar energy, quality of the dried product and, the energy effectiveness of the drying process itself. Conclusions of- fer some contributions of principle to the selection of the system of solar dryer to be used, to design of solar dryers and, to direction of the drying technology 131.     

COMPARISON OF DIFFERENT DRYING KINETICS MODELS FOR SINGLE PARTICLES

Different kinetic drying models applied to non-shrinking materials are scrutinised and compared. Rigorous mechanistic models are difficult to apply because of the large number of unknown parameters; hence simpler alternatives have been applied instead. This paper focuses on the two most commonly used kinetic models, namely; the diffusion model and the characteristic drying curves. These were compared by plotting the predicted drying curves (moisture content versus time) in the same diagram as a reference curve. The response to different changes in drying conditions and sensibility to extrapolated conditions were tested. The parameters studied were the temperature, velocity and humidity of the gas and the size and initial moisture content of the panicles. The comparison was based on two approaches; one theoretical where the reference drying curves were generated by an rigorous drying model; and one experimental where the reference drying curves were measured using a thin-layer kinetics rig. The materials under study were softwood, ceramic clay, silica gel and purolit. The results were promising in the sense that the characteristic drying curves (CDCs) and diffusion model could both predict the response to most changes in external conditions. The predictions for drying above the boiling point were however less accurate than the ones below it, probably due to internal overpressure effects.     

COMPARISON OF DIFFERENT DRYING KINETICS MODELS FOR SINGLE PARTICLES

ABSTRACT

Different kinetic drying models applied to non-shrinking materials are scrutinised and compared. Rigorous mechanistic models are difficult to apply because of the large number of unknown parameters; hence simpler alternatives have been applied instead. This paper focuses on the two most commonly used kinetic models, namely; the diffusion model and the characteristic drying curves. These were compared by plotting the predicted drying curves (moisture content versus time) in the same diagram as a reference curve. The response to different changes in drying conditions and sensibility to extrapolated conditions were tested. The parameters studied were the temperature, velocity and humidity of the gas and the size and initial moisture content of the panicles. The comparison was based on two approaches; one theoretical where the reference drying curves were generated by an rigorous drying model; and one experimental where the reference drying curves were measured using a thin-layer kinetics rig. The materials under study were softwood, ceramic clay, silica gel and purolit. The results were promising in the sense that the characteristic drying curves (CDCs) and diffusion model could both predict the response to most changes in external conditions. The predictions for drying above the boiling point were however less accurate than the ones below it, probably due to internal overpressure effects.     

EVALUATION AND MODELING OF TWO-STAGE OSMO-CONVECTIVE DRYING OF APPLE SLICES

Two-stage drying kinetics of cylindrical pieces of apples were evaluated by subjecting test samples first to various osmotic treatments and then to convective air drying to complete the drying process. Osmotic drying was carried out with cut apple cylinders of three different sizes (12, 17 and 20 mm diameter), all with a length to diameter ratio of 1 : 1, in a well agitated large tank containing the osmotic solution at the desired temperature. Solution to fruit volume ratio was kept greater than 60. After the osmotic treatment, apple slices were further dried in a cabinet drier at an average temperature 58°C. A central composite rotatable design (CCRD) with five levels of sucrose concentrations (34-63°Brix) and five temperatures (34-66°C) was used for osmotic treatment. Half-drying time and solids gain time were used as measures of rate of drying and associated diffusion coefficients for moisture loss and solids gain were evaluated. Half-drying time decreased with an increase in temperature or concentration, or a decrease in sample size. Diffusion coefficients were lower for smaller samples, and were higher for migration of moisture as compared to solids. For a given level of moisture removal, air drying times were shorter than osmotic drying times. Composite models were developed to describe the effect of process variables and particle size on the drying behavior of apple slices.     

EVALUATION AND MODELING OF TWO-STAGE OSMO-CONVECTIVE DRYING OF APPLE SLICES

ABSTRACT

Two-stage drying kinetics of cylindrical pieces of apples were evaluated by subjecting test samples first to various osmotic treatments and then to convective air drying to complete the drying process. Osmotic drying was carried out with cut apple cylinders of three different sizes (12, 17 and 20 mm diameter), all with a length to diameter ratio of 1 : 1, in a well agitated large tank containing the osmotic solution at the desired temperature. Solution to fruit volume ratio was kept greater than 60. After the osmotic treatment, apple slices were further dried in a cabinet drier at an average temperature 58°C. A central composite rotatable design (CCRD) with five levels of sucrose concentrations (34–63°Brix) and five temperatures (34–66°C) was used for osmotic treatment. Half-drying time and solids gain time were used as measures of rate of drying and associated diffusion coefficients for moisture loss and solids gain were evaluated. Half-drying time decreased with an increase in temperature or concentration, or a decrease in sample size. Diffusion coefficients were lower for smaller samples, and were higher for migration of moisture as compared to solids. For a given level of moisture removal, air drying times were shorter than osmotic drying times. Composite models were developed to describe the effect of process variables and particle size on the drying behavior of apple slices.     

EVALUATION OF LAGRANGIAN PARTICLE DISPERSION MODELS IN TURBULENT FLOWS

This work evaluates the performance of Lagrangian turbulent particle dispersion models based on the Langevin equation. A family of Langevin models, extensively reported in the open literature, decompose the fluctuating fluid velocity seen by the particle in two components, one correlated with the previous time step and a second one randomly sampled from a Wiener process, i.e., the closure is at the level of the fluid velocity seen by the particle. We will call those models generically the “standard model.” On the other hand, the model proposed by Minier and Peirano (2001 Minier , J. P. and Peirano , E. ( 2001 ). The PDF approach to turbulent polydisperesed two-phase flows , Phys. Rep. , 352 , 1 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) is considered; this approach is based on the probability density function (PDF) and performs the closure at the level of the acceleration of the fluid seen by the particle. The formulation of a Langevin equation model for the increments of fluid velocity seen by the particle allows capturing some underlying physics of particle dispersion in general turbulent flows while keeping simple the mathematical manipulation of the stochastic model, avoiding some pitfalls, and simplifying the derivation of macroscopic relations. The performance of the previous dispersion models is evaluated in the configurations of grid-generated turbulence (Snyder and Lumley, 1971 Snyder , W. H. and Lumley , J. L. ( 1971 ). Some measurements of particle-velocity autocorrelation functions in a turbulent flow , J. Fluid Mech. , 48 , 41 – 71 .[Crossref], [Web of Science ®] , [Google Scholar]; Wells and Stock, 1983 Wells , M. R. and Stock , D. E. ( 1983 ). The effect of crossing trajectories on the dispersion of particles in a turbulent flow , J. Fluid Mech. , 136 , 31 – 62 .[Crossref], [Web of Science ®] , [Google Scholar]), simple shear flow (Hyland et al., 1999), and confined axisymmetric jet flow laden with solids (Hishida and Maeda, 1987 Hishida , K. and Maeda , M. ( 1987 ). Turbulent characteristics of gas-solids two-phase confined jet: Effect of particle density , Jpn. J. Multiph. Flow , 1 , 56 – 69 . [Google Scholar]).     

THE ESTIMATION OF RANDOM COEFFICIENT AUTOGRESSIVE MODELS. II

Abstract. In Nicholls and Quinn (1980) a procedure was proposed for the determination of strongly consistent estimates of random coefficient autoregressive models. These estimates are used here as starting values in a Newton-Raphson algorithm which is employed to obtain the maximum likelihood estimates of a class of random coefficient autoregressions. The maximum likelihood estimates are shown to be strongly consistent and to satisfy a central limit theorem. The problem of testing for the randomness of the coefficients is also briefly discussed. The results of a number of simulations are reported which illustrate the theoretical results obtained.     

EVALUATION OF LAGRANGIAN PARTICLE DISPERSION MODELS IN TURBULENT FLOWS

This work evaluates the performance of Lagrangian turbulent particle dispersion models based on the Langevin equation. A family of Langevin models, extensively reported in the open literature, decompose the fluctuating fluid velocity seen by the particle in two components, one correlated with the previous time step and a second one randomly sampled from a Wiener process, i.e., the closure is at the level of the fluid velocity seen by the particle. We will call those models generically the “standard model.” On the other hand, the model proposed by Minier and Peirano (2001) is considered; this approach is based on the probability density function (PDF) and performs the closure at the level of the acceleration of the fluid seen by the particle. The formulation of a Langevin equation model for the increments of fluid velocity seen by the particle allows capturing some underlying physics of particle dispersion in general turbulent flows while keeping simple the mathematical manipulation of the stochastic model, avoiding some pitfalls, and simplifying the derivation of macroscopic relations. The performance of the previous dispersion models is evaluated in the configurations of grid-generated turbulence (Snyder and Lumley, 1971; Wells and Stock, 1983), simple shear flow (Hyland et al., 1999), and confined axisymmetric jet flow laden with solids (Hishida and Maeda, 1987).     

DRYING OF CLAY. II RHEOLOGICAL MODELISATION AND SIMULATION OF PHYSICAL PHENOMENA

The present study proposes the development of a complete mathematical modelling transfer phenomena involving at the same time heat, mass and momentum transfer during the drying process of clay. Clay is a generic example of colloid materials forming particulate gels. That can be considered as bi-constituent, homogeneous, isotropic, and highly deformable. The model was numerically solved by the finite difference method and validated by comparison of the numerical results with a previous set of experiments data. The simulation has allowed the determination of spatio-temporal evolution within the solid of different variables: temperature fields, moisture contents, displacement, deformation and stresses. The parametric sensibility has been analyzed in the case of thermophysical properties and the external heat transfer coefficient. Various values of external conditions have been analyzed.     

DRYING OF CLAY. II RHEOLOGICAL MODELISATION AND SIMULATION OF PHYSICAL PHENOMENA

ABSTRACT

The present study proposes the development of a complete mathematical modelling transfer phenomena involving at the same time heat, mass and momentum transfer during the drying process of clay. Clay is a generic example of colloid materials forming particulate gels. That can be considered as bi-constituent, homogeneous, isotropic, and highly deformable. The model was numerically solved by the finite difference method and validated by comparison of the numerical results with a previous set of experiments data. The simulation has allowed the determination of spatio-temporal evolution within the solid of different variables: temperature fields, moisture contents, displacement, deformation and stresses. The parametric sensibility has been analyzed in the case of thermophysical properties and the external heat transfer coefficient. Various values of external conditions have been analyzed.