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.     

Drying of Transformer Board

The internal insulation in shell type power transforms is usually ensured by the stacking of plates of board impregnated with oil. A board is in equilibrium with the atmosphere at a moisture content varying from seven to eight percent by weight, but, when the insulating function is required, the water content must be lower than 0.5 %. The aim of the present work is the understanding and the modelling of transformer boards drying.

The first part of this work presents experimental studies showing that the drying of transform board follows two regimes :

- a fast regime in the fim period

- a slow regime, up to 99 percent of dryness.

The effect of heating and the influence of the board thickness on the drying rate also studied. In the second part of this paper, we presnt a model capable to represent the observed kinetics of transformer board drying. This model is based on the diffusion of water vapour in the gaseous phases combined with the transfer of water vapor from the fibres to the -us phase. The due of the average global transfer coefficient can be deduced from the slope of the curve 1 f(drying time), where x     

MODELISATION PRINCIPLES FOR DRYING OF GELS

ABSTRACT

When one wants to understand and model the drying of high water content gels, one needs to gather all thermal physical parameters and to derive a physical model based on independently measured parameters. We chose to work experimentally and theoretically on water diffusion and drying of spheres of polyacrylamide gel.

We have demonstrated the strong influence of vitreous transition on the relationship of water diffusivity with moisture content. Values of the diffusion coefficient and temperature shift factor for diffusion have been proposed.

Convective and micro-wave drying kinetics show the presence of a drying phase purely specific to radiation absorption. Finally, we have proposed a mathematical model containing independent physical parameters, which' not only predicts mass fluxes but also analyses critical physical phenomena.     

DEVELOPMENT AND VALIDATION OF A PROCESS SIMULATOR FOR DRYING SUBBITUMINOUS COAL

Drying subbituminous coal has never been practiced commercially. The commercial dryers built to date have been designed for drying surface moisture in conjunction with upstream coal preparation facilities. This type of drying is mainly controlled by input energy and the basis of the design is an energy balance. In drying inherent moisture from subbituminous coal, the thermal conductivity of the coal and the diffusion of molecular water within coal particles impose limitations on the process conditions. Energy input and solids residence time in the dryer have to be controlled properly for simultaneously balancing the heat and mass transfer within the coal particles. Improper control of either parameter can cause fires and explosions during the key steps of the drying process—drying and cooling

In parallel to the Anaconda coal drying pilot plant program, a cross-flow, fluid-bed coal drying/cooling process simulator was developed for: (1) understanding the drying phenomena on an individual particle basis; (2) analyzing potential risks and safety limits, and (3) designing the Anaconda pilot plant program

The development of the process simulator was based on both first principles and laboratory data and can be divided into two phases:

1 Development of a semi-mechanistic drying model for Powder River Basin subbituminous coal employing an analytical solution of the diffusion equation

2.Formulation of a fluid-bed cross-bed cross-flow dryer/cooler simulator employing simultaneous heat and mass transfer

This model was validated against process variables data taken on a 4 tph pilot plant. An operable range, or process envelope, has been developed through the pilot plant experience and the process simulation study. Based on the model predictions, an uncertainly range was defined in the design recommendations of a pioneer coal drying plant in scale-up.     

Vacuum Drying of Oak Wood

Vacuum drying, j,e drying under absolute gas pressure of about 10⊃ Pa. is an efficient means of reducing the process period and of producing good quality wood. We will examine here continuous vacuum drying where the plank surfaces are kept at a constant temperature, which remains above the boiling point, and moisture flowing to the surface is extracted from the kiln.

We have carried out an experimental study of oak drying under such conditions. The drying rate and moisture content profile of the sample (40 mm thick) are recorded during the whole drying period.

A model of continuous drying is established from general conservation equations with the main approximation that the air is rapidly extracted. The two constitutive equations of the model which describe temperature and water content fields are of a diffusive type and coupled through coefficients. The adequate boundary equation is not a convective one, but expresses a hygroscopic equilibrium between the vapour in the chamber and the wood surface. The mass diffusive coefficient can be adjusted to the drying rates through capillary pressure and bound water diffusion functions. The wood heterogeneity (seasonal growth) is the main factor of discrepancy in these functions. The simulated drying rates correspond with the experimental ones.     

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.     

A MATHEMATICAL MODEL FOR PARALLEL FLOW DRYER FOR SLABS WITH HIGH THERMAL DIFFUSIVITY

An analytical model for the process is developed. The thermal diffusivity of the drying slabs is assumed infinite and the moisture diffusivity constant during the entire drying process.

With specified initial and boundary conditions, the mathematical model yields a two-part solution for the diffusion equation. The first part is valid for the initial drying during which the surface moisture content exceeds the value of fiber saturation. This part of the solution is used until the surface moisture content drops to the fiber saturation value. The moisture profile at the end of this period is used as the initial condition for the second period of drying which takes place under hygroscopic conditions.

Two simplifying assumptions are adapted for the hygroscopic region: 1. The dependence between the surface temperature and the moisture content is linear. 2. Constant (average) absorption heat is used during this second drying period.

For both parts of the solution, the surface moisture gradient is proportional to the local temperature difference between the drying air and the slab surface. This temperature difference can be expressed by means of a water mass balance equation for the part of the dryer between the slab in-feed and the point considered and by using the thermodynamic properties of the humid air.     

A Simulation of Wet Pocket Lumber Drying

In general, wood containing wet pockets is difficult to dry and to ensure uniformity of moisture content at the end of the drying process. Large variations of final moisture content and severe case hardening are common problems associated with the drying of wet wood. In order to devise optimal strategies for drying wood containing wet pockets, it is necessary to understand its complex moisture movement mechanisms and therefore predict drying times and final moisture content. Sub-alpine fir dimension lumber was used in this research because of its inherent issues related to wet pockets.

A two-dimensional mathematical drying model for wood containing wet pockets was developed. An effective diffusion coefficient (D_eff) was utilized in the model and heat and mass transfer equations were solved using a control volume approach. The difficulties involved in the simulation of the drying process of wet pocket lumber are due to the differences in moisture content and physical properties between wet and normal wood. Thus, an adjustable D_eff based on the moisture content (for both below and above fiber saturation point) was used during the simulation.

Four drying runs involving green unsorted sub-alpine fir lumber were carried out in a 3-ft laboratory kiln and in an 8-ft pilot kiln. The results of the simulations were in agreement with the results obtained through the drying experiments.     

PROCESSES RELATED TO DRYING: PART II USE OF THE SAME MODEL TO SOLVE TRANSFERS BOTH IN SATURATED AND UNSATURATED POROUS MEDIA

From the mathematical formulation presented in part I, a numerical code is developed to simulate heat and: mass transfers in porous media. The aim of this· tool is to understand and to improve each process related to drying. The association of a comprehensive set of equations with a efficient 2-D computer code allows us to predict the comportment of several porous media even if submitted to severe drying conditions. A few runs have been selected with special attention paid to the effect of internal gaseous pressure:

Convective drying of softwood at high temperature illustrate the typical two-dimensional transfers that occur in an anisotropic medium.

Microwave drying of light concrete pinpoints liquid expulsion of water which is driven by the pressure due to internal heating.

Finally, appropriate physical behaviours of a bed of glass spheres allows one to deal with simple processes for which full saturation occurs.     

THE INFLUENCE OF PRESSURE AND TEMPERATURE ON THE KINETICS OF VACUUM DRYING OF KETOPROFEN

This paper explores the influence of temperature and pressure on drying kinetics of 2-(3-benzoylphenil propionic acid) ketoprofen, in a vacuum dryer on laboratory scale, Experimentally determined relations between moisture content and drying rate vs time, were approximated with an exponential model. Model parameters were correlated with drying conditions (temperature, pressure) and defined by functions of their potentions.

From an energy balance of the process, a mathematical model for simulating dependence of sample temperature vs drying time, and moisture content of material, has been developed.

Simulation of the drying kinetics and sample temperature, by use of those functional dependencies shows good agreement with experimental results.     

CONVECTIVE DRYING OF PAPER CALCULATED WITH A NEW MODEL OF THE PAPER STRUCTURE

Paper drying models are based on the assumption of a rigid solid matrix. Because of the interaction of water molecules and paper fibre the dimensions of the sheet are changing, especially the thickness. These changes influence the moisture transport within the material. A model of the paper structure has been developed taking into account different kinds of pores in the paper.

By the assistance of this model the dependence between water content and thickness or transport parameters as permeability and thermal conductivity can be predicted. The only informations which are needed are sorption isotherms, apparent paper density and permeability of the single phase flow. Further informations as the pore size distribution are useful but not necessary.

The transport parameters are generated in the suggested way for wood pulp. It is shown that in the case of convective drying, the agreement between experimental results and model simulations is good.     

RELATIONSHIP BETWEEN A SOLAR DRYING MODEL OF RED PEPPER AND THE KINETICS OF PURE WATER EVAPORATION (I)

Drying of red pepper under solar radiation was investigated, and a simple model related to water evaporation was developed. Drying experiments at constant laboratory conditions were undertaken where solar radiation was simulated by a 1000 W lamp.

In this first part of the work, water evaporation under radiation is studied and laboratory experiments are presented with two objectives: to verify Penman's model of evaporation under radiation, and to validate the laboratory experiments. Modifying Penman's model of evaporation by introducing two drying conductances as a function of water content, allows the development of a drying model under Eolar radiation.

In the second part of this paper, the model is validated by applying it to red pepper open air solar drying experiments.     

A New Approach to Contact Drying Modelling

This paper deals with contact drying modelling. The more general models of granular material contact dryers are based on assumptions proposed by Schlunder and al. (1.21. In this paper, a sensitivity study of the models is presented in order to find and to explain the influence of the different arameters.

A new approach, based on the same Schlunder's assumptions, but with less restrictive hypothesis, which can be applied to drying in the presence of an inert gas as well as vacuum drying and which take into account phenomena which were ignored untill now such as the local grain dehydration kinetics or the vapour diffusion inside the bed has been developped.

This new model has been compared uith some experimental results found in the litterature. A good agreement between the calculated and the experimental results has been observed. Moreover, this model is able to justify some assumptions made by Schlunder, which have not been so untill now.     

HEAT AND MASS TRANSFER DURING SAPWOOD DRYING ABOVE THE FIBRE SATURATION POINT

Pine sapwood was dried in an air convection kiln at temperatures between 60-80 °C. Temperature and weight measurements were used to calculate the position of the evaporation front beneath the surface. It was assumed that the drying during a first regime is controlled by the heat transfer to the evaporation front until irreducible saturation occurs. Comparisons were made with CT-scanned density pictures of the dry shell formation during initial stages of drying of boards.

The results indicate a receding evaporation front behaviour for sapwood above approximately 40-50% MC when the moisture flux is heat transfer controlled. After that we finally reach a period where bound water diffusion is assumed to control the drying rate.

The heat transfer from the circulating air to the evaporation front controls the migration flux. In many industrial kilns the heating coils therefore have too small heat transfer rates for batches of thin boards and boards with high sapwood content.     

Rapid Method of Determination of Sorption Isotherms and Water Apparent Diffusivity

In the context of a research on the quality of pasta made from soft wheat, reaction kinetics had to be measured at definite temperatures and water contents. It was chosen to realize such experiments in a pilot dryer. To be able to fix temperature and water content of product for a definite period of time, it was first necessary to determine sorption curves of pasta and a drying model including the values of apparent diffusivity of water at elevated temperatures.

This paper presents a method to meet these objectives. It consists in realizing a sequence of plateaux of air relative humidities of decreasing values, at constant temperature, in the dryer. Air-product equilibrium at the end of each plateau would have been too long to attain, therefore equilibrium product water content was calcultated using an exponential model. Apparent water diffusivity was determined by adjustement of a diffusive model, taking in account that the air humidity change from one plateau to the next one was a linear ramp and not a step-wise change.

The average relative repeatability of equilibrium water Content is 0.1% water/dry basis. There is more dispersion on apparent diffusivity.     

RELATIONSHIP BETWEEN A SOLAR DRYING MODEL OF RED PEPPER AND THE KINETICS OF PURE WATER EVAPORATION (II)

In part one, a simple drying model of red pepper related to water evaporation was developed. In this second part the drying model is applied by means of related experiments.

Both laboratory and open air drying experiments were carried out to validate the model and simulation results are presented.     

Adhesion of Plasticized Polyvinyl Butyral) to Glass

A study of safety glass provides a good example of the interplay among the many physical properties involved in “adhesion”, and the relationship between adhesion and performance. This work demonstrates the value of applying known fundamentals to practical problems.

An idealized model of a windshield fracture event is described in terms of interactions among mechanical responses of the interlayer, the fracture characteristics of the glass and the high speed, low angle peel behavior.

Data on the surface energies of glass, polyvinyl butyral) and water show that at thermodynamic equilibrium a stable system comprising glass, water and polyvinyl butyral) phases, an aqueous phase must lie between the glass and PVB.

The potassium salts are shown to be effective because they are deliquescent and give solutions at equilibrium with the water in the PVB at water contents of ∼0.40% or higher. The greater the amount of salt at the interface and the higher the water content of the sheeting during lamination, the thicker the interfacial layer and delamination occurs more readily. This relationship is quantified using a modified form of the Stefan equation.

Data on diffusion of water and salt are shown to be consistent with the amount of salt at the interface required for the observed performance (∼ 3 mg KAc/m²).

Data on electrical resistivity of the interface correlate with peel force and provide convincing support for the hypothesis.     

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

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.