P.J.A.M. Kerkhof

A modified Maxwell-Stefan model for transport through inert membranes : the binary friction model

This paper focuses mainly on the development of a model for permeation through inert membranes, as encountered in many cases in ultrafiltration and in gas permeation through inert porous plugs. The ultrafiltration model is made up of a boundary layer transport model and a porous membrane model in series, which are connected by an equilibrium relation. The boundary layer model is developed with the Vieth approximation for turbulent diffusivity. For the internal membrane transport, a modification of the Maxwell-Stefan-Lightfoot equation is derived (the binary friction model), which in a natural way includes both interspecies (diffusive) and species-wall forces. Application for the partial separation of PEG-3400 from aqueous solution shows that membrane friction coefficients can simply be estimated from membrane resistance measurements and mixture viscosity data. The only adjustable parameter to be determined is the distribution coefficient between the free solution and the membrane pores. The differences between the Lightfoot approach and the dusty gas model (DGM) are shown to stem from errors in the drivations of the latter, thus invalidating the dusty gas approach in the normal region in which viscous friction effects become important. For gases, the binary friction model is developed to include Knudsen and viscous wall friction terms as well as intermolecular diffusion. It is shown to give excellent coverage of the He-Ar diffusion data of Evans et al. (J. Appl. Phys., 33 (1962) 2682; 34 (1963) 2020), with wall friction coefficients derived directly from Knudsen coefficients and gas viscosity data. The apparent success of the DGM in describing the same phenomena is shown to be caused by the relatively small importance of the wall friction forces at elevated pressures, and by the correct transition to Knudsen flow at low pressures. In addition, it is shown that diffusive slip phenomena in capillaries can be described well by the binary friction model.

THEORETICAL AND PRACTICAL ASPECTS OF AROMA RETENTION IN SPRAY DRYING AND FREEZE DRYING

Abstract For many food products the presence of volatile aroma components is a prime quality feature. Upon drying part of these components may be lost, leading to unbalanced flavour patterns in the reconstituted product. The mechanisms of aroma loss in most drying processes are well understood by now. For homogeneously dissolved aroma components this can be described succesfully by Thijssens selective diffusion theory. Somewhat more complicated is the loss of aroma components which are present in the form of a dispersion. An extensive overview is given of theory and experimental work on aroma loss in slab drying, spray drying and freeze drying. Practical processing rules following from the basic insights are discussed. It is the intention to present a broad coverage of the subject.

Drying kinetics: A comparison of diffusion coefficients from moisture concentration profiles and drying curves

The mathematical formulation of mass transfer in drying processes is often based on the diffusion equation. In principle the diffusion coefficient as a function of moisture content has to be determined experimentally. The most direct approach is to derive the diffusion coefficient from experi- mental moisture concentration profiles in the material during drying. In this work, the diffusion coefficient determined in this way is called the actual diffusion coefficient. Very often, however, an indirect method is used based on drying curves (average moisture content of a sample vs time). Since the diffusion coefficient is determined indirectly from the macroscopic behaviour of the sample, this coefficient will be called the apparent diffusion coefficient. A comparison of diffusion coefficients as a function of moisture content using both methods shows that for porous materials such as clays, the apparent diffusion coefficient depends on experimental conditions. This can be explained by the fact that the apparent diffusion coefficient is in agreement with the actual diffusion coefficient over a limited range of moisture content only. For materials such as clays, drying curves are not suited to derive diffusion coefficients as a function of moisture content in a satisfactory and consistent way.

Fundamental aspects of sludge filtration and expression

Abstract The filtration and expression behaviour of sewage sludge is discussed. Due to the increase of costs for controlled dumping and transport and more severe environmental legislation the need for decreased sludge volumes is rising. Filtration and expression are the cheapest dewatering operations and it is therefore desirable to remove the maximal feasible amount of water by mechanical dewatering. High dry solids contents of 35–40 wt% can already be reached at pressures of 300–400 kPa and optimal flocculation conditions; however at pressures of 6–10 MPa dry solids contents of 60 wt% can be reached. Further the modelling of the dewatering is discussed; model and experiment show acceptable agreement.

THE ROLE OF THEORETICAL AND MATHEMATICAL MODELLING EN SCALE-UP

Abstract 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.

The influence of shrinkage on drying behaviour of clays

Abstract A correct description of the evolution of moisture concentration profiles in shrinking materials is complicated by the influence of shrinkage on mass transfer. Shrinkage has to be accounted for in the diffusion equation. Simulations have been performed for three types of shrinkage: isotropic, unidirectional and no shrinkage. Since many materials show isotropic shrinkage behaviour, unidirectional and no shrinkage should be considered as approximations of the actual material behaviour. Various numerical simulations show that for clays the influence of the type of shrinkage on drying curves is small.

Isothermal moisture transport in partially saturated porous media

ABSTRACT The prediction of a hydrodynamic model for the isothermal transport of liquid in partially saturated porous media is compared with experimentally obtained values of water transport in clay. The transport obeys the diffusion equation, The diffusion coefficient is described as a function of porosity, permeability and pore size distribution. The comparison indicates that the model needs some refinement.

Molecular Dynamics Simulation of Self-Diffusion and Maxwell-Stefan Diffusion Coefficients in Liquid Mixtures of Methanol and Water

Abstract Self-diffusion coefficients and Maxwell-Stefan diffusion coefficients in liquids have been determined by the equilibrium molecular dynamics calculation of the appropriate Green-Kubo equation. Simulations of water, methanol and mixtures of water and methanol have been carried out to calculate the diffusion coefficients at 300 K. In order to study the influence of the force field on the calculated self-diffusion coefficients of the pure liquids, two different force fields for each component have been used. The Van Leeuwen/Smit force field calculated the self-diffusion of methanol accurately. The SPC/E force field gave the best, but moderate, results for water. In mixtures of water and methanol the self-diffusion coefficients of both components were more accurate at high mole fractions of methanol. This can be explained by the better performance of the methanol force field. The Maxwell-Stefan diffusion coefficients in the mixtures of methanol and water agreed fairly well with the experimental values...

Some modeling aspects of (batch) fluid-bed drying of life-science products

Some specific properties of life-science products, and consequences for fluid-bed drying are discussed. The sorption isotherm and maximum temperature are related to limits in process conditions, and capacity. The occurrence of stickyness is combined with the sorption isotherm and the psychrometric chart, resulting in an extended stickyness diagram, from which operating limits for air temperature and humidity can easily be deduced. Models for the air-side mass transfer are considered. The expanded-bed model, with possible bypass, is then combined with a model for concentration- and temperature-dependent diffusion inside the particles. Also a model thermal degradation reaction is included to illustrate possible quality changes in the product. Simulations show the differences between laboratory and large-scale operations, and illustrate the effect of some process conditions on drying rate, product temperature and quality. For the often implicitly used time-averaged air temperature and humidity to calculate particle drying a theoretical basis is provided. It is also shown that for quality aspects this assumption probably is too much simplified.

Molecular Dynamics Simulation of the Maxwell-Stefan Diffusion Coefficients in Lennard-Jones Liquid Mixtures

Abstract Maxwell-Stefan (MS) diffusion coefficients in multicomponent liquids have been determined by the equilibrium molecular dynamics calculation of the appropriate Green-Kubo equation. Simulations were performed for systems of 300 LJ particles at various compositions. The unary system was divided into three components by attaching a colour label to the particles, which plays no role in the dynamics. The binary system argon + krypton was divided into three species by attaching a colour label to the particles of argon. The ternary system consisted of argon, krypton and neon. The results of the calculation of the MS diffusion coefficients in the unary and binary systems agreed well with the literature values. The MS diffusion coefficients of the unary system did not differ significantly from the self-diffusion coefficient. The MS diffusion coefficients of the ternary system behaved as expected from other physical properties.