Spectral analysis and limit behaviours in a spring-mass system

We consider a model for a damped spring-mass system that is a strongly damped wave equation with dynamic boundary conditions. In a previous paper we showed that for some values of the parameters of the model, the large time behaviour of the solutions is the same as for a classical spring-mass damper ODE. Here we use spectral analysis to show that for other values of the parameters, still of physical relevance and related to the effect of the spring inner viscosity, the limit behaviours are very different from that classical ODE

Regulation of mass and function of pancreatic β-cells: identification of anti-apoptotic peptides and role of GLP-1

Résumé La masse de cellules β sécrétrices d'insuline est un tissu dynamique qui s'adapte aux variations de la demande métabolique pour assurer une normoglycémie. Cette adaptation se fait par un changement de sécrétion d'insuline et de la masse totale des cellules β. Une perte complète ou partielle des cellules β conduit respectivement à un diabète de type 1 et de type 2. Les mécanismes qui régulent la masse de cellules β et maintiennent leur phénotype differencié sont encore peu connus. Leur identification est nécessaire pour comprendre le développement du diabète et développer des stratégies de traitement. La greffe d'îlots est une approche thérapeutique prometteuse pour le diabète de type 1, mais est limitée par une perte précoce des cellules β due à une apoptose induite par des cytokines. Afin d'améliorer la survie des cellules β lors de la greffe d'îlots, le premier but était de trouver des peptides pouvant bloquer l'apoptose induite par FasL et TNF-α. Pour ce faire, deux librairies de phages ont été criblées pour sélectionner des peptides se liant au Fas DD ou au TNFRl DD. Nous avons identifié six peptides différents. Cependant, aucun d'entre eux n'était capable de protéger les cellules de l'apoptose induite par FasL ou TNF-α. Deuxièmement, le GLP-1 est une hormone qui stimule la sécrétion d'insuline, et est impliquée dans la prolifération des cellules β, la différentiation, et inhibe l'apoptose. Nous avons fait l'hypothèse que le GLP-1 joue un rôle crucial dans le contrôle de la masse et de la fonction des cellules β. Afin de l'évaluer, une analyse par puce à ADN a été réalisée en comparant des cellules βTC-Tet traitées avec du GLP-1 à des cellules non-traitées. 376 gènes régulés ont été identifiés, dont RGS2, CREM, ICERI et DUSP14, augmentés significativement par le GLP-1. Nous avons confirmé que le GLP-1 augmente l'expression de ces gènes, aussi bien au niveau des transcripts que des protéines. De plus, nous avons montré que le GLP-1 induit leur expression par activation de la voie cAMP/PKA, et nécessite l'entrée de calcium extracellulaire. D'après leur fonction biologique, nous avons ensuite supposé que ces gènes pourraient agir comme régulateurs négatifs de la signalisation du GLP-l, et donc freiner son effet proliférateur. Pour vérifier notre hypothèse, des siRNAs contre ces gènes ont été développés, et leurs effets sur la prolifération des cellules β seront évalués ultérieurement. Abstract The pancreatic β-cell mass is a dynamic tissue which adapts to variations in metabolic demand in order to ensure normoglycemia. This adaptation occurs through a change in both insulin secretion and the total mass of ,β-cells. An absolute or relative loss of β-cells leads to type 1 and type 2 diabetes, respectively. The mechanisms that regulate the pancreatic β-cell mass and maintain the fully differentiated phenotype of the insulin-secreting β-cells are only poorly defined. Their identification is required to understand the progression of diabetes, but also to design strategies for the treatment of diabetes. Islet transplantation is a promising therapeutic approach for type 1 diabetes, but it is still limited by an early graft loss due to cytokine-induced apoptosis. In order to improve β-cell survival during islet transplantation, our first goal was to find novel blockers of FasL- and TNF-α-mediated cell death in the form of peptides. To that end, we screened two phage display libraries to select Fas DD- or TNFR1 DD-binding peptides. We identified six different small peptides. However, none of these peptides was able to prevent cells from FasL- or TNF-α-mediated apoptosis. Secondly, GLP-1 is a hormone that has been shown to stimulate insulin secretion and to be involved in β-cell proliferation, differentiation and inhibition of apoptosis. We hypothesized that GLP-1 plays a crucial role to control mass and function of β-cells. To evaluate this hypothesis, we performed a cDNA microarray analysis with GLP-1-treated βTC-Tet cells compared to untreated cells. We found 376 regulated genes, among these, RGS2, CREM, ICERI and DUSP14, which were significantly upregulated by GLP-1. We confirmed that both their mRNA and protein levels were strongly and rapidly increased after GLP-1 treatment. Moreover, we found that GLP-1 activates their expression mainly through the activation of the cAMP/PKA signaling pathway, and requires extracellular calcium entry. According to their biological function, we then hypothesized that these genes might act as negative regulators of the GLP-1 signaling. In particular, they might brake the effects of GLP-1 on β-cell proliferation. To verify this hypothesis, siRNAs against these genes were developed. The effect of these siRNAs on GLP-1-induced β-cell proliferation will be evaluated later.

Quantum transport phenomena and shallow impurity states in CdSb

This work is dedicated to investigation of the energy spectrum of one of the most anisotropic narrow-gap semiconductors, CdSb. At the beginning of the present studies even the model of its energy band structure was not clear. Measurements of galvanomagnetic effects in wide temperature range (1.6 - 300 K) and in magnetic fields up to 30 T were chosen for clarifying of the energy spectrum in the intentionally undoped CdSb single crystals and doped with shallow impurities (In, Ag). Detection of the Shubnikov - de Haas oscillations allowed estimating the fundamental energy spectrum parameters. The shapes of the Fermi surfaces of electrons (sphere) and holes (ellipsoid), the number of the equivalent extremums for valence band (2) and their positions in the Brillouin zone were determined for the first time in this work. Also anisotropy coefficients, components of the tensor of effective masses of carriers, effective masses of density of states, nonparabolicity of the conduction and valence bands, g-factor and its anisotropy for n- and p-CdSb were estimated for the first time during these studies. All the results obtained are compared with the cyclotron resonance data and the corresponding theoretical calculations for p-CdSb. This is basic information for the analyses of the complex transport properties of CdSb and for working out the energy spectrum model of the shallow energy levels of defects and impurities in this semiconductor. It was found out existence of different mechanisms of hopping conductivity in the presence of metal - insulator transition induced by magnetic field in n- and p-CdSb. Quite unusual feature opened in CdSb is that different types of hopping conductivity may take place in the same crystal depending on temperature, magnetic field or even orientation of crystal in magnetic field. Transport properties of undoped p-CdSb samples show that the anisotropy of the resistivity in weak and strong magnetic fields is determined completely by the anisotropy of the effective mass of the holes. Temperature and magnetic field dependence of the Hall coefficient and magnetoresistance is attributed to presence of two groups of holes with different concentrations and mobilities. The analysis demonstrates that below Tcr ~ 20 K and down to ~ 6 - 7 K the low-mobile carriers are itinerant holes with energy E2 ≈ 6 meV. The high-mobile carriers, at all temperatures T < Tcr, are holes activated thermally from a deeper acceptor band to itinerant states of a shallower acceptor band with energy E1 ≈ 3 meV. Analysis of temperature dependences of mobilities confirms the existence of the heavy-hole band or a non-equivalent maximum and two equivalent maxima of the light-hole valence band. Galvanomagnetic effects in n-CdSb reveal the existence of two groups of carriers. These are the electrons of a single minimum in isotropic conduction band and the itinerant electrons of the narrow impurity band, having at low temperatures the energies above the bottom of the conduction band. It is found that above this impurity band exists second impurity band of only localized states and the energy of both impurity bands depend on temperature so that they sink into the band gap when temperature is increased. The bands are splitted by the spin, and in strong magnetic fields the energy difference between them decreases and redistribution of the electrons between the two impurity bands takes place. Mobility of the conduction band carriers demonstrates that scattering in n-CdSb at low temperatures is strongly anisotropic. This is because of domination from scattering on the neutral impurity centers and increasing of the contribution to mobility from scattering by acoustic phonons when temperature increases. Metallic conductivity in zero or weak magnetic field is changed to activated conductivity with increasing of magnetic field. This exhibits a metal-insulator transition (MIT) induced by the magnetic field due to shift of the Fermi level from the interval of extended states to that of the localized states of the electron spectrum near the edge of the conduction band. The Mott variablerange hopping conductivity is observed in the low- and high-field intervals on the insulating side of the MIT. The results yield information about the density of states, the localization radius of the resonant impurity band with completely localized states and about the donor band. In high magnetic fields this band is separated from the conduction band and lies below the resonant impurity bands.

Volcanic debris avalanche deposits of the upper Maronne valley (Cantal Volcano, France): evidence for contrasted formation and transport mechanisms

The deposits of two volcanic debris avalanches (VDA I and II) that occur in the upper Maronne valley, northwest sector of Cantal Volcano, France, were studied to establish their mechanisms of formation, transport and deposition. These two volcanic debris avalanches that clearly differ with regard to their structures, textures and extensions, exemplify the wide spectrum of events associated with large-scale sector collapse. VDA I is voluminous (similar to1 km(3) in the upper Maronne valley) and widespread. The deposits comprise two distinct facies: the block facies that forms the intermediate and upper part of the unit and the mixed facies that crops out essentially at the base of the unit. The block facies consists of more or less brecciated lava, block-and-ash-flow breccia and pumice-flow tuff megablocks set in breccias resulting from block disaggregation. Mixing and differential movements are almost absent in this part of the VDA. The mixed facies consists of breccias rich in fine particles that originate from block disagregation, as well as being picked up from the substratum during movement. Mixing and differential movements are predominant in this zone. Analysis of fractures on lava megablocks suggests that shear stress during the initial sliding is the principal cause of fracture. These data strongly indicate that VDA I is purely gravitational and argue for a model in which the initial sliding mass transforms into a flow due to differential in situ fragmentation caused by the shear stress. VDA II is restricted to low-topography areas. Its volume, in the studied area, is about 0.3 km(3). The deposits consist of brecciated, rounded blocks and megablocks set in a fine-grained matrix composed essentially of volcanic glass. This unit is stratified, with a massive layer that contains all the megablocks at the base and in the intermediate part, and in the upper part a normally graded layer that contains only blocks <1 m in size. The different lithologies present are totally mixed. These observations suggest that VDA II may be of the Bezymianny-type and that it underwent a flow transformation from a turbulent to a stratified flow consisting of a basal hyperconcentrated laminar body overlain by a dilute layer. (C) 2000 Elsevier Science B.V. All rights reserved.

Flow phenomena, heat and mass transfer in microchannel reactors

The studies of flow phenomena, heat and mass transfer in microchannel reactors are beneficial to estimate and evaluate the ability of microchannel reactors to be operated for a given process reaction such as Fischer-Tropsch synthesis. The flow phenomena, for example, the flow regimes and flow patterns in microchannel reactors for both single phase and multiphase flow are affected by the configuration of the flow channel. The reviews of the previous works about the analysis of related parameters that affect the flow phenomena are shown in this report. In order to predict the phenomena of Fischer-Tropsch synthesis in microchannel reactors, the 3-dimensional computational fluid dynamic simulation with commercial software package FLUENT was done to study the flow phenomena and heat transfer for gas phase Fischer-Tropsch products flow in rectangular microchannel with hydraulic diameter 500 ¿m and length 15 cm. Numerical solution with slip boundary condition was used in the simulation and the flowphenomena and heat transfer were determined.

Particle transport method for convection-diffusion-reaction problems

Convective transport, both pure and combined with diffusion and reaction, can be observed in a wide range of physical and industrial applications, such as heat and mass transfer, crystal growth or biomechanics. The numerical approximation of this class of problemscan present substantial difficulties clue to regions of high gradients (steep fronts) of the solution, where generation of spurious oscillations or smearing should be precluded. This work is devoted to the development of an efficient numerical technique to deal with pure linear convection and convection-dominated problems in the frame-work of convection-diffusion-reaction systems. The particle transport method, developed in this study, is based on using rneshless numerical particles which carry out the solution along the characteristics defining the convective transport. The resolution of steep fronts of the solution is controlled by a special spacial adaptivity procedure. The serni-Lagrangian particle transport method uses an Eulerian fixed grid to represent the solution. In the case of convection-diffusion-reaction problems, the method is combined with diffusion and reaction solvers within an operator splitting approach. To transfer the solution from the particle set onto the grid, a fast monotone projection technique is designed. Our numerical results confirm that the method has a spacial accuracy of the second order and can be faster than typical grid-based methods of the same order; for pure linear convection problems the method demonstrates optimal linear complexity. The method works on structured and unstructured meshes, demonstrating a high-resolution property in the regions of steep fronts of the solution. Moreover, the particle transport method can be successfully used for the numerical simulation of the real-life problems in, for example, chemical engineering.

Modeling of non-isothermalvapor membrane separation with thermodynamic models and generalized mass transfer equations

A rigorous unit operation model is developed for vapor membrane separation. The new model is able to describe temperature, pressure, and concentration dependent permeation as wellreal fluid effects in vapor and gas separation with hydrocarbon selective rubbery polymeric membranes. The permeation through the membrane is described by a separate treatment of sorption and diffusion within the membrane. The chemical engineering thermodynamics is used to describe the equilibrium sorption of vapors and gases in rubbery membranes with equation of state models for polymeric systems. Also a new modification of the UNIFAC model is proposed for this purpose. Various thermodynamic models are extensively compared in order to verify the models' ability to predict and correlate experimental vapor-liquid equilibrium data. The penetrant transport through the selective layer of the membrane is described with the generalized Maxwell-Stefan equations, which are able to account for thebulk flux contribution as well as the diffusive coupling effect. A method is described to compute and correlate binary penetrant¿membrane diffusion coefficients from the experimental permeability coefficients at different temperatures and pressures. A fluid flow model for spiral-wound modules is derived from the conservation equation of mass, momentum, and energy. The conservation equations are presented in a discretized form by using the control volume approach. A combination of the permeation model and the fluid flow model yields the desired rigorous model for vapor membrane separation. The model is implemented into an inhouse process simulator and so vapor membrane separation may be evaluated as an integralpart of a process flowsheet.

Dynamic Model of a Small Once-Through Boiler

In this study, a model for the unsteady dynamic behaviour of a once-through counter flow boiler that uses an organic working fluid is presented. The boiler is a compact waste-heat boiler without a furnace and it has a preheater, a vaporiser and a superheater. The relative lengths of the boiler parts vary with the operating conditions since they are all parts of a single tube. The present research is a part of a study on the unsteady dynamics of an organic Rankine cycle power plant and it will be a part of a dynamic process model. The boiler model is presented using a selected example case that uses toluene as the process fluid and flue gas from natural gas combustion as the heat source. The dynamic behaviour of the boiler means transition from the steady initial state towards another steady state that corresponds to the changed process conditions. The solution method chosen was to find such a pressure of the process fluid that the mass of the process fluid in the boiler equals the mass calculated using the mass flows into and out of the boiler during a time step, using the finite difference method. A special method of fast calculation of the thermal properties has been used, because most of the calculation time is spent in calculating the fluid properties. The boiler was divided into elements. The values of the thermodynamic properties and mass flows were calculated in the nodes that connect the elements. Dynamic behaviour was limited to the process fluid and tube wall, and the heat source was regarded as to be steady. The elements that connect the preheater to thevaporiser and the vaporiser to the superheater were treated in a special way that takes into account a flexible change from one part to the other. The model consists of the calculation of the steady state initial distribution of the variables in the nodes, and the calculation of these nodal values in a dynamic state. The initial state of the boiler was received from a steady process model that isnot a part of the boiler model. The known boundary values that may vary during the dynamic calculation were the inlet temperature and mass flow rates of both the heat source and the process fluid. A brief examination of the oscillation around a steady state, the so-called Ledinegg instability, was done. This examination showed that the pressure drop in the boiler is a third degree polynomial of the mass flow rate, and the stability criterion is a second degree polynomial of the enthalpy change in the preheater. The numerical examination showed that oscillations did not exist in the example case. The dynamic boiler model was analysed for linear and step changes of the entering fluid temperatures and flow rates.The problem for verifying the correctness of the achieved results was that there was no possibility o compare them with measurements. This is why the only way was to determine whether the obtained results were intuitively reasonable and the results changed logically when the boundary conditions were changed. The numerical stability was checked in a test run in which there was no change in input values. The differences compared with the initial values were so small that the effects of numerical oscillations were negligible. The heat source side tests showed that the model gives results that are logical in the directions of the changes, and the order of magnitude of the timescale of changes is also as expected. The results of the tests on the process fluid side showed that the model gives reasonable results both on the temperature changes that cause small alterations in the process state and on mass flow rate changes causing very great alterations. The test runs showed that the dynamic model has no problems in calculating cases in which temperature of the entering heat source suddenly goes below that of the tube wall or the process fluid.

Mass transfer and particleinteractions in granular systems and suspension flows

The purpose of this study was to investigate some important features of granular flows and suspension flows by computational simulation methods. Granular materials have been considered as an independent state ofmatter because of their complex behaviors. They sometimes behave like a solid, sometimes like a fluid, and sometimes can contain both phases in equilibrium. The computer simulation of dense shear granular flows of monodisperse, spherical particles shows that the collisional model of contacts yields the coexistence of solid and fluid phases while the frictional model represents a uniform flow of fluid phase. However, a comparison between the stress signals from the simulations and experiments revealed that the collisional model would result a proper match with the experimental evidences. Although the effect of gravity is found to beimportant in sedimentation of solid part, the stick-slip behavior associated with the collisional model looks more similar to that of experiments. The mathematical formulations based on the kinetic theory have been derived for the moderatesolid volume fractions with the assumption of the homogeneity of flow. In orderto make some simulations which can provide such an ideal flow, the simulation of unbounded granular shear flows was performed. Therefore, the homogeneous flow properties could be achieved in the moderate solid volume fractions. A new algorithm, namely the nonequilibrium approach was introduced to show the features of self-diffusion in the granular flows. Using this algorithm a one way flow can beextracted from the entire flow, which not only provides a straightforward calculation of self-diffusion coefficient but also can qualitatively determine the deviation of self-diffusion from the linear law at some regions nearby the wall inbounded flows. Anyhow, the average lateral self-diffusion coefficient, which was calculated by the aforementioned method, showed a desirable agreement with thepredictions of kinetic theory formulation. In the continuation of computer simulation of shear granular flows, some numerical and theoretical investigations were carried out on mass transfer and particle interactions in particulate flows. In this context, the boundary element method and its combination with the spectral method using the special capabilities of wavelets have been introduced as theefficient numerical methods to solve the governing equations of mass transfer in particulate flows. A theoretical formulation of fluid dispersivity in suspension flows revealed that the fluid dispersivity depends upon the fluid properties and particle parameters as well as the fluid-particle and particle-particle interactions.

Dynamic drag modeling of submerged aquatic vegetation canopy flows

Vegetation has a profound effect on flow and sediment transport processes in natural rivers, by increasing both skin friction and form drag. The increase in drag introduces a drag discontinuity between the in-canopy flow and the flow above, which leads to the development of an inflection point in the velocity profile, resembling a free shear layer. Therefore, drag acts as the primary driver for the entire canopy system. Most current numerical hydraulic models which incorporate vegetation rely either on simple, static plant forms, or canopy-scaled drag terms. However, it is suggested that these are insufficient as vegetation canopies represent complex, dynamic, porous blockages within the flow, which are subject to spatially and temporally dynamic drag forces. Here we present a dynamic drag methodology within a CFD framework. Preliminary results for a benchmark cylinder case highlight the accuracy of the method, and suggest its applicability to more complex cases.

Steady-state global optimization of metabolic non-linear dynamic models through recasting into power-law canonical models

Background: Design of newly engineered microbial strains for biotechnological purposes would greatly benefit from the development of realistic mathematical models for the processes to be optimized. Such models can then be analyzed and, with the development and application of appropriate optimization techniques, one could identify the modifications that need to be made to the organism in order to achieve the desired biotechnological goal. As appropriate models to perform such an analysis are necessarily non-linear and typically non-convex, finding their global optimum is a challenging task. Canonical modeling techniques, such as Generalized Mass Action (GMA) models based on the power-law formalism, offer a possible solution to this problem because they have a mathematical structure that enables the development of specific algorithms for global optimization. Results: Based on the GMA canonical representation, we have developed in previous works a highly efficient optimization algorithm and a set of related strategies for understanding the evolution of adaptive responses in cellular metabolism. Here, we explore the possibility of recasting kinetic non-linear models into an equivalent GMA model, so that global optimization on the recast GMA model can be performed. With this technique, optimization is greatly facilitated and the results are transposable to the original non-linear problem. This procedure is straightforward for a particular class of non-linear models known as Saturable and Cooperative (SC) models that extend the power-law formalism to deal with saturation and cooperativity. Conclusions: Our results show that recasting non-linear kinetic models into GMA models is indeed an appropriate strategy that helps overcoming some of the numerical difficulties that arise during the global optimization task.

DLL4 promotes continuous adult intestinal lacteal regeneration and dietary fat transport.

The small intestine is a dynamic and complex organ that is characterized by constant epithelium turnover and crosstalk among various cell types and the microbiota. Lymphatic capillaries of the small intestine, called lacteals, play key roles in dietary fat absorption and the gut immune response; however, little is known about the molecular regulation of lacteal function. Here, we performed a high-resolution analysis of the small intestinal stroma and determined that lacteals reside in a permanent regenerative, proliferative state that is distinct from embryonic lymphangiogenesis or quiescent lymphatic vessels observed in other tissues. We further demonstrated that this continuous regeneration process is mediated by Notch signaling and that the expression of the Notch ligand delta-like 4 (DLL4) in lacteals requires activation of VEGFR3 and VEGFR2. Moreover, genetic inactivation of Dll4 in lymphatic endothelial cells led to lacteal regression and impaired dietary fat uptake. We propose that such a slow lymphatic regeneration mode is necessary to match a unique need of intestinal lymphatic vessels for both continuous maintenance, due to the constant exposure to dietary fat and mechanical strain, and efficient uptake of fat and immune cells. Our work reveals how lymphatic vessel responses are shaped by tissue specialization and uncover a role for continuous DLL4 signaling in the function of adult lymphatic vasculature.

Auger-type granular fertylizer distributor: matemathical model and dynamic simulation

The objective of this study was to model mathematically and to simulate the dynamic behavior of an auger-type fertilizer applicator (AFA) in order to use the variable-rate application (VRA) and reduce the coefficient of variation (CV) of the application, proposing an angular speed controller θ' for the motor drive shaft. The input model was θ' and the response was the fertilizer mass flow, due to the construction, density of fertilizer, fill factor and the end position of the auger. The model was used to simulate a control system in open loop, with an electric drive for AFA using an armature voltage (V A) controller. By introducing a sinusoidal excitation signal in V A with amplitude and delay phase optimized and varying θ' during an operation cycle, it is obtained a reduction of 29.8% in the CV (constant V A) to 11.4%. The development of the mathematical model was a first step towards the introduction of electric drive systems and closed loop control for the implementation of AFA with low CV in VRA.

A multibody dynamic model of the cross-country ski-skating technique

The objective of this thesis is the development of a multibody dynamic model matching the observed movements of the lower limb of a skier performing the skating technique in cross-country style. During the construction of this model, the formulation of the equation of motion was made using the Euler - Lagrange approach with multipliers applied to a multibody system in three dimensions. The description of the lower limb of the skate skier and the ski was completed by employing three bodies, one representing the ski, and two representing the natural movements of the leg of the skier. The resultant system has 13 joint constraints due to the interconnection of the bodies, and four prescribed kinematic constraints to account for the movements of the leg, leaving the amount of degrees of freedom equal to one. The push-off force exerted by the skate skier was taken directly from measurements made on-site in the ski tunnel at the Vuokatti facilities (Finland) and was input into the model as a continuous function. Then, the resultant velocities and movement of the ski, center of mass of the skier, and variation of the skating angle were studied to understand the response of the model to the variation of important parameters of the skate technique. This allowed a comparison of the model results with the real movement of the skier. Further developments can be made to this model to better approximate the results to the real movement of the leg. One can achieve this by changing the constraints to include the behavior of the real leg joints and muscle actuation. As mentioned in the introduction of this thesis, a multibody dynamic model can be used to provide relevant information to ski designers and to obtain optimized results of the given variables, which athletes can use to improve their performance.

Fluid dynamic modelling and simulation of circulating fluidized bed reactors: importance of the interface turbulence transfer

The main objective of this work is to analyze the importance of the gas-solid interface transfer of the kinetic energy of the turbulent motion on the accuracy of prediction of the fluid dynamic of Circulating Fluidized Bed (CFB) reactors. CFB reactors are used in a variety of industrial applications related to combustion, incineration and catalytic cracking. In this work a two-dimensional fluid dynamic model for gas-particle flow has been used to compute the porosity, the pressure, and the velocity fields of both phases in 2-D axisymmetrical cylindrical co-ordinates. The fluid dynamic model is based on the two fluid model approach in which both phases are considered to be continuous and fully interpenetrating. CFB processes are essentially turbulent. The model of effective stress on each phase is that of a Newtonian fluid, where the effective gas viscosity was calculated from the standard k-epsilon turbulence model and the transport coefficients of the particulate phase were calculated from the kinetic theory of granular flow (KTGF). This work shows that the turbulence transfer between the phases is very important for a better representation of the fluid dynamics of CFB reactors, especially for systems with internal recirculation and high gradients of particle concentration. Two systems with different characteristics were analyzed. The results were compared with experimental data available in the literature. The results were obtained by using a computer code developed by the authors. The finite volume method with collocated grid, the hybrid interpolation scheme, the false time step strategy and SIMPLEC (Semi-Implicit Method for Pressure Linked Equations - Consistent) algorithm were used to obtain the numerical solution.