Radial Porosity Profile Research Articles

Radial porosity profiles are a powerful tool for tracing locomotor maturation in developing limb bones.

Radial porosity profiles (RPP) are a new quantitative osteohistological parameter designed to capture the dynamic changes in the primary porosity of limb bones through ontogeny, providing insights into skeletal growth and functional development of extant and extinct vertebrates. Previous work hypothesized that RPP channelization-the intraskeletal alignment of RPPs across different bones resulting from similar cortical compaction patterns-indicates increasing locomotor performance of the developing limbs. By investigating RPPs in ontogenetic series of pheasants, pigeons and ducks representing distinct locomotor developmental strategies, we test this hypothesis here and show that RPPs are indeed powerful osteohistological correlates of locomotor ontogeny. Qualitative and quantitative analyses reveal strong association between RPP channelization and fledging, the most drastic locomotor transition in the life history of volant birds. The channelization signal is less clear in precocial leg function; however, when additional intraskeletal and intercohort RPP characteristics are considered, patterns related to leg precocity can also be identified. Thus, we demonstrate that RPPs can be used in future by palaeobiologists to generate breakthroughs in the study of the ontogeny and evolution of flight in fossil birds and pterosaurs. With further baseline data collection from modern terrestrial vertebrates, RPPs could also test hypotheses regarding ontogenetic postural shifts in dinosaurs and other terrestrial archosaurs.

Area of the Intersection between a Sphere and a Cylindrical Plane

A proper understanding of the porous structure of packed beds of spheres is imperative in the analysis and design of the processes involving fluid flow and heat and mass transfer. The radial variation in porosity is of specific interest. When the positions and sizes of the spheres are known, the radial variation in porosity can be determined using volume-based, area-based, or line-based approaches. Here, the focus is on the area-based methods which employ the intersections between the spheres and selected cylindrical planes to determine the radial variation in porosity, focusing specifically on the calculation of the area of the curved elliptic intersection between a sphere and a cylindrical plane. Using geometrical considerations, analytical integral expressions have been derived based on the axial direction, angular direction, or the radial direction as independent variables. The integral expressions cannot be integrated analytically and have been evaluated using approximations or numerical integration. However, only indirect validation of the calculation of the intersection area has been provided by comparing the radial porosity profiles obtained with experimental data. This study provides direct validation of the calculated area through refined numerical integration of the primary integral expressions and the evaluation of the area employing computer-aided design software.

Radial porosity profiles: a new bone histological method for comparative developmental analysis of diametric limb bone growth.

In fossil tetrapods, limb bone histology is considered the most reliable tool not only for inferring skeletal maturity—a crucial assessment in palaeobiological and evolutionary studies—but also for evaluating the growth dynamics within the ontogenetic window represented by the primary bone cortex. Due to its complex relationship with bone growth and functional maturation, primary cortical vascularity is an indispensable osteohistological character for reconstructing growth dynamics, especially in the context of various developmental strategies along the precocial–altricial spectrum. Using this concept as our working hypothesis, we developed a new quantitative osteohistological parameter, radial porosity profile (RPP), that captures relative cortical porosity changes in limb bones as trajectories. We built a proof-of-concept RPP dataset on extant birds, then added fossil paravian dinosaurs and performed a set of trajectory-grouping analyses to identify potential RPP categories and evaluate them in the context of our ontogeny—developmental strategy working hypothesis. We found that RPPs, indeed, reflect important developmental features within and across elements, specimens and taxa, supporting their analytical power. Our RPPs also revealed unexpected potential osteohistological correlates of growth and functional development of limb bones. The diverse potential applications of RPPs open up new research directions in the evolution of locomotor ontogeny.

Estimation of the radial distribution of axial velocities in fixed beds of spherical packing

The vessel wall of densely packed beds exert a strong influence on particle distribution that generates a radial porosity profile and, in turn, a radial profile of axial velocity. A direct consequence is the appearance of an uneven flow distribution on the bed cross-section and radial variations of transport properties. A widespread alternative for estimating such a velocity profile is to solve the so-called extended Brinkman equation, using an available radial porosity profile. To this end, it is employed a local version of an expression quantifying the drag force on the packing (usually the Ergun equation) and an effective viscosity to account for the radial transport of axial momentum. It is shown that the existing expressions to evaluate the effective viscosity are only suitable for a restricted range of Reynolds numbers. Supported on the expressions from averaging the microscopic conservation equations, it is reassessed the local evaluation of the drag force and the estimation of the effective viscosity. The use of the extended Brinkman equation with such changes allows a satisfactory description of the radial velocity profile for mono-sized spherical packing, as arisen from the comparison with available experimental results and with an important number of results from pore-scale simulation.

Enhancing the X-ray contrast of polymeric biochromatography particles for three-dimensional imaging

A major limitation of the three-dimensional imaging of polymeric biochromatography particle packings using X-ray computed tomography is that the particles have a low density and a high porosity, making them almost undistinguishable from the surrounding liquid phase. Additionally, the employed media are typically composed of materials with low atomic numbers, which exhibit low X-ray absorption. We report an improvement of packed column reconstruction using micro X-ray computed tomography. A simple, inexpensive, and fast method to increase the contrast factor of highly porous polymer-based chromatographic particles was developed by applying a modified pore-blocking method. This approach relies on the selective filling of the porous chromatographic particles with a hydrophilic phase while a hydrophobic phase occupies the void spaces between the particles. The hydrophilic phase contains a dissolved X-ray absorbing radiocontrast agent. No chemical modifications of the chromatographic beads or columns were necessary. The developed method can be applied in-situ in a previously packed column and can be used for media with different organic backbones. We show the applicability of this method by carrying out the first 3D-reconstruction of packed micro columns with an inner diameter of 760 μm. The micro column contained agarose- and methacrylate-based particles commonly used in preparative biochromatography with mean diameters of 40 and 65 μm, respectively. Based on the obtained high-resolution 3D-reconstructions, we exemplarily computed packing properties such as global extraparticle porosity and radial porosity profiles, and visualized the presence of void spaces using 3D image analysis.

What happens in a catalytic fixed-bed reactor for n-butane oxidation to maleic anhydride? Insights from spatial profile measurements and particle resolved CFD simulations

Industrial catalytic reactors are made of steel, operate at high temperatures and pressures and contain hazardous chemicals. What happens inside remains hidden. Reactor optimization requires costly trial and error or is based on simplified mathematical models employing more or less accurate transport correlations and reaction kinetics. In the present work a pilot-scale fixed-bed reactor was developed for measuring concentration and temperature profiles for n-butane oxidation to maleic anhydride on vanadyl pyrophosphate catalyst pellets under industrially-relevant conditions. The reactor was equipped with five heating zones. The reactor was modeled by particle-resolved computational fluid dynamics. The catalyst bed was created by discrete element simulation and the result was validated by comparison with experimental radial porosity profiles. Catalytic chemistry was included by a kinetic model of intrinsic reaction rates. Transport resistances and packing deviations were lumped in reaction rate multipliers determined by fitting the model to profiles measured at a uniform reactor wall temperature. Simulation results reveal strong inhomogeneities inside the bed. A hot-spot develops at uniform wall temperature. At this hot-spot temperature differences of 40 K exist on one and the same pellet with negative impact on maleic anhydride selectivity and catalyst lifetime. An optimized wall temperature profile was derived by combining knowledge from the experimental profiles at uniform wall temperature and the CFD results. A gradual increasing temperature was predicted by the model to eliminate the hot-spot and increase integral maleic anhydride selectivity at constant n-butane conversion. This prediction was confirmed by experiment. At 80% n-butane conversion the maleic anhydride selectivity could be improved by 2%. Facing the scale of the process, this improvement translates into significant n-butane savings, reduced COx emissions and increased revenue.

Transient Heat Transfer in a Packed-Bed Elliptic Cylindrical Reactor: A Finite-Volume Approach

This work aims to develop a transient three-dimensional mathematical model using the elliptic cylindrical coordinate system, to predict heat transfer in a elliptic cylindrical packed fixed bed reactor. The model considers variable thermo physical properties and a parabolic temperature profile at the fluid inlet. The governing equation is solved using the finite volume method. Results of temperature profile along the reactor are presented and discussed at different moments.It was verified that the maximum heat transfer rate inside the reactor occurs near the extreme region close to minor semi-axis of the ellipse; the higher temperatures at the reactor surface are also in this region, along the entire height of the bed; the steady-state regime is reached at t = 4.5 s of process, presenting after this time interval,small axial temperature gradients and high radial gradients along of the reactor bed; the parabolic temperature profile give to the bed a predominance of radial temperature gradients, and the radial porosity profile favours a higher heat transfer rate at reactor surface.

A DNS study of flow and heat transfer through slender fixed-bed reactors randomly packed with spherical particles

A fully resolved direct numerical simulation of flow and heat transfer is presented for slender randomly packed bed reactors. The flow and temperature field are solved over a non-body fitted, non-conformal Cartesian computational domain. The coupling between fluid and solid (both spherical particles and cylindrical wall) is enforced by a second order accurate, sharp interface immersed boundary method (IBM). The present numerical technique neither requires any challenging volumetric mesh generation process nor demands manipulation of the geometry near the particle-particle and particle-wall contact points. Conjugate heat transfer has been considered where the temperature field is calculated both inside the solid particles and in the fluid. A discrete element method (DEM) is used to generate the random packings of spherical particles in the cylindrical column, and a methodology is proposed to calculate the radial porosity profile of the bed.The column-to-particle diameter ratio (N) is varied from 4 to 8, and two separate cases have been considered where N→∞. The particle Reynolds number (Red) is varied from 1 to 500. The numerically obtained pressure drop and overall wall-to-bed heat transfer coefficient for different simulation cases are critically compared with empirical correlations and a good agreement is reported. Moreover, based on the current numerical results, correlations are proposed for the pressure drop and the wall-to-bed heat transfer coefficient. The effect of the column-to-particle diameter ratio (N) on both the flow and heat transfer, as-well-as the effect of the solid to fluid thermal conductivity ratio on the conjugate heat transfer are discussed. Furthermore, the fully resolved accurate numerical simulations have helped to elucidate the detailed pore-scale flow and heat transfer feature in the packed beds.

Influence of the porosity profile and sets of Ergun constants on the main hydrodynamic parameters in the trickle-bed reactors

The Eulerian multiphase code has been applied for the comparative simulation of the experimental results obtained for varying flow rates of both fluid phases and for systems of various physicochemical properties used in the laboratory trickle-bed reactor (TBR). Therefore six different radial porosity profiles and three sets of the Ergun constants used in the relations defining the interphase momentum exchange have been implemented into the CFD code. The mean relative error of the pressure drop and of the averaged holdup as well as the standard deviation determined with respect to the experiments have been used as the criterion for the validation of the theoretical model. These values explicitly confirmed the necessity of applying the radial porosity profile in the beds of low values of the D/dp ratio i.e. for laboratory equipment in order to obtain accurate values of the pressure drop and the averaged holdup. They also enabled to select the optimum radial porosity correlation and the optimum Ergun constants in the relations of interphase momentum exchange which in consequence could be recommended for the modeling of the hydrodynamics in the laboratory TBR by means of the CFD code.

Numerical Generation of a Fixed Bed Structure

Abstract A numerical algorithm is presented for the filling process of a cylindrical column with equilateral cylinders. The process is based on simplified mechanics - the elements are added one by one until the mechanical equilibrium is reached. The final structure is examined with respect to the global and local porosity distribution. Oscillating radial porosity profile is obtained in accordance with experimental data.

Geometrical and topological measures for hydrodynamic dispersion in confined sphere packings at low column-to-particle diameter ratios

At low column-to-particle diameter (or aspect) ratio (dc/dp) the kinetic column performance is dominated by the transcolumn disorder that arises from the morphological gradient between the more homogeneous, looser packed wall region and the random, dense core. For a systematic analysis of this morphology–dispersion relation we computer-generated a set of confined sphere packings varying three parameters: aspect ratio (dc/dp=10–30), bed porosity (ɛ=0.40–0.46), and packing homogeneity. Plate height curves were received from simulation of hydrodynamic dispersion in the packings over a wide range of reduced velocities (v=0.5–500). Geometrical measures derived from radial porosity and velocity profiles were insufficient as morphological descriptors of the plate height data. After Voronoi tessellation of the packings, topological information was obtained from the statistical moments of the free Voronoi volume (Vfree) distributions. The radial profile of the standard deviation of the Vfree distributions in the form of an integral measure was identified as a quantitative scalar measure for the transcolumn disorder. The first morphology–dispersion correlation for confined sphere packings deepens our understanding of how the packing microstructure determines the kinetic column performance.

A simple method for determining sphere packed bed radial porosity

Abstract A new and simple method is established to examine the radial porosity profile of mono-sized spheres in packed bed cylinders based on the sphere center coordinates. The method is derived from geometrical and analytical techniques and uses arc lengths to calculate the radial porosity profile at any given axial position, a given interval axial position, or the total axial position, thus evaluating the local, interval, or axially averaged radial porosity profile, respectively. The new method and analytical functions are simple and straightforward and provide an accurate representation of the radial porosity profile given the sphere center coordinates. The simple method is used to calculate the radial porosity profile for the fixed packing of identical spheres in cylindrical containers with D/d ≥ 1.0. The evaluated results for the radial porosity profile are benchmarked with existing analytical equations and available experimental and numerical data, respectively, for mono-sized spheres in cylindrical containers. A concise FORTRAN program for the new arc length-based technique is presented for this simple method for calculating radial porosity profiles.

Morphological analysis of physically reconstructed capillary hybrid silica monoliths and correlation with separation efficiency

We report an experimental study on the structural (especially radial) heterogeneity of eleven 100 μm i.d. capillary tetramethoxysilane–methyltrimethoxysilane hybrid silica monoliths with different pore and skeleton sizes, which were imaged by an optimized confocal laser scanning microscopy method. This method allows the optical sectioning of the monoliths, which is a prerequisite for quantitative morphological image analysis. Both radial porosity profiles and chord length distributions were calculated in the macropore domain for each column from at least 100 complete cross-sectional views along the column axis. The statistical approach visualized radial heterogeneities on different length scales in the monolithic structures. Chord length distributions followed a simplified k-gamma function, and a structural parameter obtained from this function is introduced to provide a scalar measure of column heterogeneity. It enables the comparison of monoliths with different pore sizes and helps to establish correlations between the microscopic properties of a column, eddy dispersion, and its separation efficiency.

Evaluation of Structural Properties of Cylindrical Packed Beds Using Numerical Simulations and Tomographic Experiments

This contribution is focused on the analysis of the structure of packed beds of spherical particles at relatively low aspect ratios (i.e., particle to tube diameter ratio) as those arising in multitubular fixed bed reactors. On one hand, the computed tomography (CT) technique is employed to evaluate the position of each particle in the packing and from this information local properties such as particle center distribution and radial porosity profile were obtained. On the other hand, results from a previously developed algorithm to simulate packings were compared with those from our CT data and from literature sources. The agreement was very satisfactory.

CFD modeling of radial spreading of flow in trickle-bed reactors due to mechanical and capillary dispersion

CFD simulations of trickle-bed reactors are presented with radial spreading of the liquid due to mechanical and capillary dispersion. Simulations are performed with various particle sizes and the significance of the dispersion mechanisms at the industrially relevant particle size range is analyzed. The effect of the bed porosity distribution and particle size to the simulation results is also discussed. The choice of the radial porosity profile is found to have a significant impact to the simulation results, especially when the column to particle diameter ratio, D/ d p , is small, in which case the wall flow is important. The dependence of the standard deviation of porosity on the sample size is determined experimentally. Introducing just random variation of porosity to the model is found to describe inadequately the dispersive flow behavior. The presented hydrodynamic model with proper capillary and mechanical dispersion terms succeeds in capturing the features of the two independent physical phenomena. Separate models are presented for each dispersion mechanisms and it is shown that they both can have a significant contribution to the overall dispersion of liquid flowing through a packed bed. The hydrodynamic model is validated against the experimental dispersion profiles from Herskowitz and Smith [1978. Liquid distribution in trickle-bed reactors. A.I.Ch.E Journal 24, 739–454], Boyer et al. [2005. Study of liquid spreading from a point source in trickle-bed via gamma-ray tomography and CFD simulation. Chemical Engineering Science 60, 6279–6288] and Ravindra et al. [1997. Liquid flow texture in trickle-bed reactors: an experimental study. Industrial & Engineering Chemistry Research, 36, 5133–5145]. The extent of liquid dispersion predicted by the presented hydrodynamic model is in excellent agreement with the experiments.

Comparison and Extension of Dancoff Factors for Pebble-Bed Reactors

A method is developed for the computation of infinite-medium Dancoff factors for spherical kernels with a stochastic packing as used in high-temperature reactors. The method is used to compute Dancoff factors that are then compared to those obtained by four preexisting methods from the literature. The older methods assume either infinitesimally small kernels with a random distribution, or finite kernels with some assumptions. In the new method, the infinite-medium Dancoff factor is calculated rigorously by numerically integrating the Dancoff factor of two adjacent finite-size kernels over their surfaces and relative positions. It turns out that for practical pebble-bed fuel designs, as currently envisioned, all four methods give results accurate (i.e., compatible with one another) within 2%, but that larger deviations are observed for extreme cases (either at high or low dilution).A Monte Carlo program, named INTRAPEB, was written to calculate the average value and the space dependency of the Dancoff factor of one single fuel pebble, as well as the angular distribution of neutrons escaping the pebble. For the Dancoff factor, the analytical results from literature agree very well with the new computational approach. However, for a cubic packing of particles, as is usually modeled in MCNP calculations, a larger Dancoff factor is found for the pebble. The angular distribution of neutrons escaping from the moderator zone of a pebble is much more forwardly peaked than the cosine angular distribution assumed in the derivation of the analytical methods. If the previously developed analytical methods need improvement, this could be achieved by using a more forwardly peaked neutron distribution.A second program, named PEBDAN, was written to calculate the average value and the space dependency of the interpebble Dancoff factor (the probability that a neutron escaping from the fuel zone of a pebble crosses a fuel particle in another pebble) and the pebble-pebble Dancoff factor (the probability that a neutron escaping from the fuel zone of a pebble crosses the fuel zone of another pebble). In this program, the coordinates of the pebbles in a randomly packed bed are determined, after which the Dancoff factors are calculated using a Monte Carlo ray-tracing method. The packing distribution obtained in this code reproduces the principal features of experimental data; however, the predicted radial porosity profile of the packed bed displays less-pronounced oscillations (i.e., lower peaks) and a slightly larger average porosity value. Because of the larger first-flight escape probability of neutrons, the Dancoff factors drop several tens of percent along the inner and outer reflector of the core.

A cellular automata model for liquid distribution in trickle bed reactors

A cellular automata model for liquid distribution studies in trickle bed reactors is presented. It is a potential tool for describing non-uniform distribution of gas and liquid in a trickle bed. This non-uniformity may arise from a wide range of potential sources, such as improper distribution of the feed, random or radial porosity variation, wall effect, partial wetting of the catalyst, and gas–liquid surface tension related effects. Axial and radial dispersion of the liquid flow are inherently included in the model, since the fundamental model probability parameters are directly related to the dispersion coefficients. The present model is extremely fast due to simple single-event modeling, and it is well suited for parallelization. Three examples of the model performance are shown. In the first a liquid jet spreading from a point source is followed, and in the second, effect of radial porosity profile to wall flow is examined. The third example illustrates the potential of the model to predict pulsing flow regime.

Oxygen Distribution in Packed-Bed Membrane Reactors for Partial Oxidations: Effect of the Radial Porosity Profiles on the Product Selectivity

A two-dimensional, pseudohomogeneous reactor model was presented to describe the radial and axial concentration profiles in a packed-bed membrane reactor and the local velocity field while accounting for the influences due to the distributive membrane flow and the radial porosity profile. The effect of the radial porosity profile on the model results has been evaluated for cases with a low tube-to-particle diameter ratio (dt/dp) with insufficient radial transport resulting in strong radial oxygen concentration profiles. From these calculations, it was concluded that the overall effect of the porosity profile on the conversion-selectivity plot is unexpectedly small. If the implementation of a radial porosity profile is not required by other considerations, e.g., for an accurate description of temperature effects, then the effect of the porosity profile can safely be neglected. If a radial porosity profile is implemented, however, the effect of the porosity profile on the dispersion coefficient should be taken into account to achieve a proper description of the radial transport.

CFD of multiphase flow in packed‐bed reactors: II. Results and applications

AbstractNumerical simulations of multiphase flow using the k‐fluid CFD model described in Part I of this issue are presented for packed beds at various operating conditions. Both steady‐state and unsteady‐state (e.g., periodic operation) feed conditions were studied numerically. Predictions of the k‐fluid CFD model are comparable with the experimental data in the literature for liquid upflow in a cylindrical packed bed. In addition to the mean porosity and the longitudinally averaged radial porosity profile, the variance of the porosity distribution is needed for predicting the probability density function of the sectional flow velocity. In the trickling flow regime, the k‐fluid CFD model provides reasonable predictions of the global liquid saturation and the pressure gradient. Relevant applications of the k‐fluid CFD model are identified in quantifying the relationship between bed structure and flow distribution in various‐scale packed beds. The combined flow‐reaction modeling scheme is proposed through the “mixing‐cell” network concept, in which the k‐fluid CFD simulation can provide the information on sectional flow distribution.

Boundary element method in modelling of momentum transport and liquid-to-wall mass transfer in a packed-bed reactor

In this paper, the boundary element method has been applied to solve highly non-linear differential equations describing hydrodynamics and liquid/solid mass transfer at the wall of a packed-bed reactor with single-phase liquid flow. It is based on the boundary integral element concepts where both the dependent variable and its gradient become the primary variables. Two radial porosity profiles were used to take into account the non-uniformity character of the bed. It was shown that the computed velocity profiles have the same shape as the porosity profiles. The mass transfer experimental measurements, obtained for four different Schmidt numbers, are in good agreement with the computed predictions when uniform velocity profiles are used.