Theoretical models of vegetable drying by convection

The results of experiments are often used to model empirical phenomena. However, the term model is applied in various meanings. A model is usually treated as an abstract formal structure that can replace a material system considered as original, in respect to the aim of modeling. Certain formal structures may be treated as theoretical models of empirical phenomena. On the other hand, a material system can also be referred to as a model of an abstract system, e.g., a set of equations or a hypothesis. Such a material system, if it is a distinct empirical interpretation of the language of a given theory, is then called a real model. Both kinds of models are applied in drying technology, but the second one is more inventive. The mathematical structures are treated as empirical formulae or as theoretical models properly derived from true or legitimated promises of a given theory. The advantages of some mathematical theoretical models of drying processes versus empirical formulae are discussed. The creation of new mathematical theoretical models of convection drying kinetics of some shrinking solids is presented and analyzed. One of the above models was also hypothetically suggested for modeling the drying of cut vegetables in a fluidized-bed. Despite its initial acceptance due to peer empirical justification on cut carrots and celery, it still requires further theoretical analysis. Other models indicated here are theoretical models of vegetable drying in a tunnel drier. These models are created by deduction from laws of heat and mass transfer theory and its basic equations. XI Polish Drying Symposium, Poznań, Poland, 13–16 September 2005.     

Nonequilibrium Effects During Spontaneous Imbibition

Accurate models of multiphase flow in porous media and predictions of oil recovery require a thorough understanding of the physics of fluid flow. Current simulators assume, generally, that local capillary equilibrium is reached instantaneously during any flow mode. Consequently, capillary pressure and relative permeability curves are functions solely of water saturation. In the case of imbibition, the assumption of instantaneous local capillary equilibrium allows the balance equations to be cast in the form of a self-similar, diffusion-like problem. Li et al. [J. Petrol. Sci. Eng. 39(3) (2003), 309–326] analyzed oil production data from spontaneous countercurrent imbibition experiments and inferred that they observed the self-similar behavior expected from the mathematical equations. Others (Barenblatt et al. [Soc. Petrol. Eng. J. 8(4) (2002), 409–416]; Silin and Patzek [Transport in Porous Media 54 (2004), 297–322]) assert that local equilibirum is not reached in porous media during spontaneous imbibition and nonequilibirium effects should be taken into account. Simulations and definitive experiments are conducted at core scale in this work to reveal unequivocally nonequilbirium effects. Experimental in-situ saturation data obtained with a computerized tomography scanner illustrate significant deviation from the numerical local-equilibrium based results. The data indicates: (i) capillary imbibition is an inherently nonequilibrium process and (ii) the traditional, multi-phase, reservoir simulation equations may not well represent the true physics of the process.     

Systematic multiscale models for deep convection on mesoscales

This paper builds on recent developments of a unified asymptotic approach to meteorological modeling [ZAMM, 80: 765–777, 2000, SIAM Proc. App. Math. 116, 227–289, 2004], which was used successfully in the development of Systematic multiscale models for the tropics in Majda and Klein [J. Atmosph. Sci. 60: 393–408, 2003] and Majda and Biello [PNAS, 101: 4736–4741, 2004]. Biello and Majda [J. Atmosph. Sci. 62: 1694–1720, 2005]. Here we account for typical bulk microphysics parameterizations of moist processes within this framework. The key steps are careful nondimensionalization of the bulk microphysics equations and the choice of appropriate distinguished limits for the various nondimensional small parameters that appear. We are then in a position to study scale interactions in the atmosphere involving moist physics. We demonstrate this by developing two systematic multiscale models that are motivated by our interest in mesoscale organized convection. The emphasis here is on multiple length scales but common time scales. The first of these models describes the short-time evolution of slender, deep convective hot towers with horizontal scale ~ 1 km interacting with the linearized momentum balance on length and time scales of (10 km/3 min). We expect this model to describe how convective inhibition may be overcome near the surface, how the onset of deep convection triggers convective-scale gravity waves, and that it will also yield new insight into how such local convective events may conspire to create larger-scale strong storms. The second model addresses the next larger range of length and time scales (10 km, 100 km, and 20 min) and exhibits mathematical features that are strongly reminiscent of mesoscale organized convection. In both cases, the asymptotic analysis reveals how the stiffness of condensation/evaporation processes induces highly nonlinear dynamics. Besides providing new theoretical insights, the derived models may also serve as a theoretical devices for analyzing and interpreting the results of complex moist process model simulations, and they may stimulate the development of new, theoretically grounded sub-grid-scale parameterizations.     

Internal Heating Effect and Enhancement of Drying of Ceramics by Microwave Heating with Dynamic Control

The effectiveness of internal heating for enhancing the drying of molded ceramics is evaluated by both modeling and experiments. In the theoretical analysis, three dimensional drying-induced strain–stress are modeled, and the numerical solutions show that the internal heating generates lower internal stress than continuous convective heating or intermittent convective heating. Microwave drying is examined experimentally to study the effect of internal heating on the drying behavior of a wet sample of a kaolin slab. The drying behavior is compared among three modes: microwave heating, hot air heating and radiation heating. The transient behavior of temperatures in microwave drying is quite different from conventional drying by external heating. In particular, the temperature of the slab drops once in the progress of drying. This phenomenon cannot be predicted adequately by a simple model of one-dimensional heat conduction and moisture diffusion accompanied with an internal heat generation rate given as a linear function of the moisture content. It should be noted that the temperature behavior takes place due to the combined interactions with internal evaporation of moisture by rise in internal vapor pressure and shift of impedance or interference in the applicator. Microwave heating with a constant power above 100 W results in sample breakage due to the internal vapor pressure. However, if the power is dynamically controlled so as to maintain the temperature less than the boiling point of water, the drying succeeds without any crack generation until completion with a significantly faster drying rate than drying in convective heating or in the oven.     

Periodic Fluctuations of Flow and Porosity in Spouted Beds

The results of experimental investigations of bed flow hydrodynamics in spouted beds are compared with CFD simulations (Eulerian–Eulerian approach) for two different column geometries. The experimental results of bed porosity and fluctuation frequency of mass flow rate of grain in the fountain region are compared with the corresponding results of simulations. The simulation results confirmed the observations of Muir et al. (1990, Chem. Eng. Comm. 88: 153–171) and Yang and Keairns (1978, AlchE Symp. Ser. No. 176 74: 218) that fluctuations of bed flow in DTSB are caused by particle cluster formation in the loading region at the bottom of column. The solids cross into the jet and cover the column inlet and are carried upward periodically through a draft tube. Subsequent figures obtained from simulations, which show stages of particle cluster formation at the entrance of column, exactly match visual observations. The frequency of fluctuations of grain mass flow rate predicted in simulations (~5–6 Hz) is in the range of that experimentally determined. The fluctuating inflow of solids results in slug formation and explains the vertical variations of height and porosity of the fountain.     

Modeling of water transport in roof tiles by removal of moisture at isothermal conditions

The main objective of this article is to describe the drying process of ceramic roof tiles, shaped from red clay, using diffusion models. Samples of the product with initial moisture content of 0.24 (db) were placed inside an oven in the temperatures of 55.6, 69.7, 82.7 and 98.6°C; and the data of the drying kinetics were obtained. The analytical solutions of the diffusion equation for the parallelepiped with boundary conditions of the first and third kinds were used to describe the drying processes. The process parameters were determined using an optimization algorithm based on inverse method coupled to the analytical solutions. The analysis of the results makes it possible to affirm that the boundary condition of the third kind satisfactorily describes the drying processes. The values obtained for the convective mass transfer coefficient were between 8.25 × 10⁻⁷ and 1.64 × 10⁻⁶ m s⁻¹, and for the effective water diffusivity were between 9.21 × 10⁻⁹ and 1.80 × 10⁻⁸ m² s⁻¹.     

Effect of Rainfall-Induced Soil Seals on the Soil Water Regime: Drying Interval and Subsequent Wetting

The effects of rainfall-induced soil seals on drying processes and on infiltration following drying intervals are simulated for two different soils, a loam and a sandy loam. The simulated drying processes include water content redistribution without evaporation and under a constant evaporation rate of 5 mm day^–1. During evaporation, the water content at the seal surface decreases rapidly. A high water content gradient develops within the seal, which increases along the drying interval. It indicates that, at least during the first hours of drying, the seal layer fulfilled all the evaporation demand and therefore dries faster that an unsealed soil where the evaporation is supplied by a much deeper zone of the soil profile. This phenomenon is more accentuated in the loam than in the sandy loam soil. Considering the subsequent infiltration curves during rainfall following different drying intervals, the ponding time and the post-ponding infiltration rates increase when the antecedent drying period is longer, but no significant effect on the final infiltration is found following drying intervals of few days. Also, the water content at the sealed soil surface before rainfall seems to play a major role on infiltration. Very close infiltration curves were obtained after different drying intervals that ended with similar surface water content.     

A constrained theory for single crystal shape memory wires with application to restrained recovery

The theory of thin wires developed in Dret and Meunier (Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 337:143–147, 2003) is adapted to phase-transforming materials with large elastic moduli in the sense discussed in James and Rizzoni (J Elast 59:399–436, 2000). The result is a one-dimensional constitutive model for shape memory wires, characterized by a small number of material constants. The model is used to analyze self-accommodated and detwinned microstructures and to study superelasticity. It also turns out that the model successfully reproduces the behavior of shape memory wires in experiments of restrained recovery (Tsoi et al. in Mater Sci Eng A 368:299–310, 2004; Tsoi in 50:3535–3544, 2002; S̆ittner et al. in Mater Sci Eng A 286:298–311, 2000; vokoun in Smart Mater Struct 12:680–685, 2003; Zheng and Cui in Intermetallics 12:1305–1309, 2004; Zheng et al. in J Mater Sci Technol 20(4):390–394, 2004). In particular, the model is able to predict the shift to higher transformation temperatures on heating. The model also captures the effect of prestraining on the evolution of the recovery stress and of the martensite volume fraction.     

Relative permeability relations: A key factor for a drying model

In the modelling of heat, mass and momentum transfer phenomena which occur in a capillary porous medium during drying, the liquid and gas flows are usually described by the generalised Darcy laws. Nevertheless, the question of how to determine experimentally the relative permeability relations remains unanswered for most materials that consist of water and humid air, and as a result, arbitrary functions are used in the drying codes. In this paper, the emphasis is on deducing from both numerical and experimental studies a method for estimating pertinent relations for these key parameters. In the first part, the sensitivity of liquid velocity and, consequently, of drying kinetics in the variation of the relative permeabilities is investigated numerically by testing various forms. It is concluded that in order to predict a realistic liquid velocity behaviour, relative permeabilities can be linked to a measurable quantity: the capillary pressure. An estimation technique, based on simulations coupled with experimental measurements of capillary pressure, together with moisture content kinetics obtained for low or middle temperature convective drying, is deduced. In the second part, the proposed methodology is applied to pine wood. It is shown that the obtained relations provide closer representation of physical reality than those commonly used.     

Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography

A wetting–drying condition (WDC) for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real geometries. The WDC has been incorporated into a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting–drying condition based on steady‐state conditions lead to numerical errors in unsteady cases over configurations with strong variations on bed slope. A modification of the wetting–drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and efficient 2D river flow simulation tool. Comparisons of experimental and numerical results are shown for some of the applications. Copyright © 2004 John Wiley & Sons, Ltd.     

2-D hydro-viscoelastic model for convective drying of deformable and unsaturated porous material

The aim of this work was to simulate in two dimensions the spatio-temporal evolution of the moisture content, the temperature, the solid (dry matter) concentration, the dry product total porosity, the gas porosity, and the mechanical stress within a deformable and unsaturated product during convective drying. The material under study was an elongated cellulose–clay composite sample with a square section placed in hot air flow. Currently, this innovative composite is used in the processing of boxes devoted to the preservation of heritage and precious objects against fire damage and other degradation (moisture, insects, etc.). A comprehensive and rigorous hydrothermal model had been merged with a dynamic linear viscoelasticity model based on Bishop's effective stress theory, assuming that the stress tensor is the sum of solid, liquid, and gas stresses. The material viscoelastic properties were measured by means of stress relaxation tests for different water contents. The viscoelastic behaviour was described by a generalized Maxwell model whose parameters were correlated to the water content. The equations of our model were solved by means of the ‘COMSOL Multiphysics’ software. The hydrothermal part of the model was validated by comparison with experimental drying curves obtained in a laboratory hot-air dryer. The simulations of the spatio-temporal distributions of mechanical stress were performed and interpreted in terms of material potential damage. The sample shape was also predicted all over the drying process.     

Kinetics of Atmospheric Freeze-drying of Apple

We present investigations of the effect of Atmospheric Freeze-Drying kinetics on the quality (dehydration rate, shrinkage, color, and antioxidant properties) of apple cubes. The experimental data are compared with the result of convective and vacuum freeze-drying processes, and suitable operating parameters are determined. The experiments were carried out in an Internet controlled, fully automated heat-pump assisted drying system.     

On the isometric isomorphism of probabilistic metric spaces

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Flow-induced crystallization of high-density polyethylene: the effects of shear and uniaxial extension

The effects of shear, uniaxial extension and temperature on the flow-induced crystallization of two different types of high-density polyethylene (a metallocene and a ZN-HDPE) are examined using rheometry. Shear and uniaxial extension experiments were performed at temperatures below and well above the peak melting point of the polyethylenes in order to characterize their flow-induced crystallization behavior at rates relevant to processing (elongational rates up to 30 s^− 1 and shear rates 1 to 1,000 s^− 1 depending on the application). Generally, strain and strain rate found to enhance crystallization in both shear and elongation. In particular, extensional flow was found to be a much stronger stimulus for polymer crystallization compared to shear. At temperatures well above the melting peak point (up to 25°C), polymer crystallized under elongational flow, while there was no sign of crystallization under simple shear. A modified Kolmogorov crystallization model (Kolmogorov, Bull Akad Sci USSR, Class Sci, Math Nat 1:355–359, 1937) proposed by Tanner and Qi (Chem Eng Sci 64:4576–4579, 2009) was used to describe the crystallization kinetics under both shear and elongational flow at different temperatures.     

Drying Rate Measurements in Convection- and Diffusion-Driven Conditions on a Shaly Sandstone Using Nuclear Magnetic Resonance

Carbon Capture and Storage (CCS) is one of the solutions studied to reduce greenhouse gas accumulation in the atmosphere. Depleted oil and gas reservoirs have been studied for potential storage sites but also saline aquifers that have the advantages of much larger pore volume. In this latter case, injection of large volume of anhydrous carbon dioxide will lead to a strong water desaturation of the near wellbore region because of evaporation mechanisms. Even the capillary trapped water can be removed by thermodynamical transfer of water vapor in the CO₂ phase. The extension in time and space of the dry zone will be controlled by the drying rate induced by the gas flow. Consequences of drying may induce alteration of the injectivity by salt precipitation and/or alteration of the rock fabric itself, especially for shaly sandstones in the case of clay drying. The context of CCS has raised new interests in the understanding of drying kinetic where the water vapor is evacuated by gas convection. In this study, we investigated experimentally the drying rate evolution with time on a shaly sandstone sample in two conditions of drying: convective and diffusive. In convective conditions, air is injected at different flow rates through the porous media in conditions of drying representative of a CO₂ injection site at one million ton per year. In diffusive conditions, no flow is imposed and the water vapor escape by diffusion. Drying rates dynamics in both conditions were measured by Nuclear Magnetic Resonance (NMR) and compared. We varied the temperature and the salinity in diffusive-driven drying and the gas flow rate in convective-driven drying. The water distribution in the pore network and the water saturation profiles were monitored continuously using T₂ relaxation and 1D imaging NMR techniques. For the range of temperature and air flow rate used, we show that drying rates in the two drying conditions are similar but not identical. They both present different periods characteristic of the main mechanisms for water mass transfer. Drying rate has a power law dependence on the temperature, as predicted by thermodynamic, and drying rate was found proportional to the flow rate in convective drying. Presence of salt has a complex effect: an increase of the drying rate at early stage of drying followed by a strong decrease for the remaining time of drying.     

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

Polynomial Filtration Laws for Low Reynolds Number Flows Through Porous Media

In this study, we use the method of homogenization to develop a filtration law in porous media that includes the effects of inertia at finite Reynolds numbers. The result is much different than the empirically observed quadratic Forchheimer equation. First, the correction to Darcy’s law is initially cubic (not quadratic) for isotropic media. This is consistent with several other authors (Mei and Auriault, J Fluid Mech 222:647–663, 1991; Wodié and Levy, CR Acad Sci Paris t.312:157–161, 1991; Couland et al. J Fluid Mech 190:393–407, 1988; Rojas and Koplik, Phys Rev 58:4776–4782, 1988) who have solved the Navier–Stokes equations analytically and numerically. Second, the resulting filtration model is an infinite series polynomial in velocity, instead of a single corrective term to Darcy’s law. Although the model is only valid up to the local Reynolds number, at the most, of order 1, the findings are important from a fundamental perspective because it shows that the often-used quadratic Forchheimer equation is not a universal law for laminar flow, but rather an empirical one that is useful in a limited range of velocities. Moreover, as stated by Mei and Auriault (J Fluid Mech 222:647–663, 1991) and Barree and Conway (SPE Annual technical conference and exhibition, 2004), even if the quadratic model were valid at moderate Reynolds numbers in the laminar flow regime, then the permeability extrapolated on a Forchheimer plot would not be the intrinsic Darcy permeability. A major contribution of this study is that the coefficients of the polynomial law can be derived a priori, by solving sequential Stokes problems. In each case, the solution to the Stokes problem is used to calculate a coefficient in the polynomial, and the velocity field is an input of the forcing function, F, to subsequent problems. While numerical solutions must be utilized to compute each coefficient in the polynomial, these problems are much simpler and robust than solving the full Navier–Stokes equations.     

A methodology for coupling DNS and discretised population balance for modelling turbulent precipitation

In this paper, we present a methodology for simulating nanoparticle formation in a turbulent flow by coupling Direct Numerical Simulation (DNS) and population balance modelling. The population balance equation (PBE) is solved via a discretisation method employing a composite grid that provides sufficient detail over the wide range of particle sizes reached during the precipitation process. The coupled DNS/PBE approach captures accurately the strong interaction between the dynamics of turbulent mixing and particle formation processes. It also allows the calculation of the particle size distribution (PSD) of the product and enables an investigation on how it is controlled by turbulent mixing. Finally, it provides the statistics of kinetic processes and their timescales so that further analysis can be performed. The methodology is applied to the simulation of experiments of hydrodynamics and nanoparticle precipitation in a T-mixer (Schwertfirm et al., Int. J. of Heat and Fluid Flow 28, pp. 1429–1442; Schwarzer et al., Chem. Eng. Sci. 61, pp. 167–181), and the agreement with the experimental results is very good.