An approximation method for the effectiveness factor in porous catalysts

An efficient method for computing approximate value of the effectiveness factor is presented. The method is developed for an arbitrary rate expression and for three representative catalyst shapes, namely, an infinite slab, an infinite cylinder and a sphere. In the method, two new asymptotic expansions for small and large Thiele moduli are developed and joined together at an intermediate value of the Thiele modulus, which can be calculated by a simple equation. The approximation is highly accurate for small and large Thiele moduli, and it has small or negligible errors for intermediate values of the Thiele modulus. As demonstrated with examples, the approximation method is fast, straightforward and is a dependable alternative to numerical methods for computing exact values of the effectiveness factor.     

Effectiveness factor for spatial gradient sensing in living cells

We consider the steady-state pattern of messenger molecules produced in the membrane of a cell perceiving and responding to an extracellular gradient of chemoattractant, which directs cell movement towards the chemoattractant source. Specifically, we analyze the undesirable effect of lateral diffusion in blurring the intracellular messenger profile. The concept of an effectiveness factor, akin to the analysis of reactions in porous catalysts, is applied to the spatial gradient sensing problem, with the distinction that slow, not fast, diffusion is required for effective gradient sensing. Analytical effectiveness factor expressions are derived for ideal geometries and then generalized to arbitrary cell shapes. In the case of mouse fibroblasts responding to gradients of platelet-derived growth factor, we conclude that the cell morphology and orientation with respect to the gradient can dictate whether messenger diffusion obliterates gradient sensing or has very little effect. The analysis outlined here allows the effect of intracellular messenger diffusion on spatial gradient sensing to be quantified for individual cells.     

Approximate residence time distribution of fully develop laminar flow in a straight rectangular channel

For microfluidic applications the residence time distribution (RTD) of laminar flow in rectangular channels is of interest. The exact velocity profile for this type of flow consists of an infinite series and does not allow analytical evaluation of the RTD curve. In this paper we adopt a simpler binomial product profile which was proposed in literature and serves as good approximation. This allows us to determine in an analytical manner approximate expressions for the diffusion-free RTD of fully developed laminar flow in a straight rectangular channel of arbitrary aspect ratio. Since the evaluation of this RTD is computationally elaborate because it involves the Gauss hypergeometric function, we fit it by an empirical model which is suitable for engineering applications. We find that for a Newtonian fluid there is a narrowing of the RTD as the aspect ratio decreases from unity (square channel) to zero (parallel plates). We investigate the range of applicability of the diffusion-free RTD and show that it is a good estimation for liquids in a certain range of Reynolds numbers. The actual limits of this range depend on the Schmidt number and on the aspect ratio and length-to-hydraulic-diameter ratio of the channel.     

A packing algorithm for particles of arbitrary shapes

Particle packing is a subject of both academic and industrial importance, and a number of packing algorithms have been proposed and widely used. However, most of these packing algorithms can only deal with spheres and a few regular shapes. If applied to arbitrarily shaped particles, they would have difficulties in at least one of three aspects. First, arbitrary shapes are notoriously difficult to model mathematically. Secondly, efficient algorithms for collision and overlap detection of arbitrary shapes are difficult to derive and more difficult to implement. Thirdly, the packing program would be very computationally expensive for routine and practical use. This paper describes a new, digital approach to particle packing, which can avoid many of the difficulties suffered by conventional methods. The key innovation is digitisation of both particle shapes and the packing space. Thus, a particle is now just a coherent collection of pixels or voxels, regardless of its shape, moving on a square lattice, onto which the packing space is mapped. Using the digital approach, it is easy to pack particles of any shapes and sizes into a container of any geometry, generally requiring no more than an ordinary PC. Although, as yet, the packing algorithm does not involve physical forces explicitly, it can simulate some physical phenomena such as size segregation. The ability to pack particles in their real shapes, rather than approximated as spheres, opens up many new industrial and academic opportunities, some of which are discussed. Examples of packing density predictions of particles subject to various effects, including vibration and rotation, are given.     

Note on the rate of diffusional mass transfer at the surface of a nonspherical solid particle

The effect that the change in the geometric form of a nonspherical particle exerts on the mass transfer at its surface is discussed. The specific model cases of an infinite cylinder and a general ellipsoid are analyzed and expressions for the variation of their dimensions with time derived. The problem of shape stability is pointed out.     

Conditional and Marginal Mutual Information in Gaussian and Hyperbolic Decay Time Series

We consider the amount of available information about an arbitrary future state of a Gaussian stochastic process. We derive an infinite series for the marginal mutual information in terms of the autocorrelation function. We derive an infinite series for the newly available information for prediction, the conditional mutual information, in terms of the moving average parameters, and directly characterize predictability in terms of sensitivity to random shocks. We apply our results to long memory, or more generally, hyperbolic decay models, and give information‐theoretic characterizations of the transition from persistence to anti‐persistence, stationary long memory to nonstationarity, and a stationary regime where the mutual information is not summable.     

Shape normalization of catalyst pellet

A shape normalization, which is applicable in the entire range of Thiele modulus φ, is developed. A shape normalization established here for small φ and the shape normalization already established here for large φ are used in developing the normalization for all φ. This normalization brings the η - φ curves for all pellet shapes to a single curve corresponding to infinite slab geometry for all φ. The effectiveness factor for any shape of catalyst is simply the effectiveness factor for an infinite slab when the Thiele modulus for the slab is properly defined in terms of the characteristic pellet length and the reaction kinetics. The shape normalization is shown to give negligible error for any pellet configuration and first order reaction, and is postulated to hold for general kinetics and any pellet configuration, by proper definition of the Thiele modulus.     

Spatial Grid for Estimation of Thermal Fields in Glass Molds

Methods for imposition of a multidimensional grid on a set of molds of arbitrary shapes are considered.     

Population balance modeling in Simulink: PCSS

In this work we develop, demonstrate, and distribute the code for a new Simulink block that models the dynamic evolution of the population density function for a physical system which can be modeled by a population balance equation. The name of the block is PCSS, for population balance modeling using the conservation element/solution element method in Simulink. The block interfaces with an auxiliary user-defined function that allows the user to specify arbitrary expressions for growth and generation/loss terms, as well as an arbitrary number of inputs, outputs and auxiliary states for the block. The versatility of the block allows a wide variety of physical systems to be modeled, and the implementation in Simulink facilitates rapid model development and permits the use of pre-existing MATLAB/Simulink packages for system identification and control.     

FINITE ELEMENT SOLUTION OF THE MAXWELL EQUATIONS FOR ABSORPTION AND SCATTERING OF ELECTROMAGNETIC RADIATION BY A COATED DIELECTRIC PARTICLE

The problem of absorption and scattering of electromagnetic radiation by particles can be solved analytically for only the simplest cases, but established numerical methods can allow a straightforward extension to particles with arbitrary inhomogeneities, shapes, and nonlinear response. In this paper a recently developed finite element technique for the scattering and absorption by a homogeneous sphere is extended to the problem of a coated dielectric sphere of arbitrary size parameter. Numerical results showing good agreement with analytical solutions for the size parameter of 5.93 and various core radii are presented. Results obtained suggest that finite element methods have promise for analytically intractable scattering/absorption problems and show that the Debye amplitude formulation of the problem offers some advantages in a numerical scheme.     

Dispersion in electrochemical cells with radial flow between parallel electrodes. I. A dispersive plug flow mathematical model

The conventional relations for dispersion in tubes and ducts are inappropriate for radial flows such as are found in the disc-stack and pump cells. A first-order mathematical model is therefore developed for radial flow between infinite parallel planes. The shapes of the predicted responses to pulses of injected material are shown to be in qualitative agreement with experimental curves, however the form of the mathematical expression is inappropriate for routine data analysis. Nevertheless simple relationships are derived which enable the dispersion coefficient and mean residence time of an electrogenerated species to be determined from the first and second moments of the response and a knowledge of the geometry of the system. In Part II experimental data are analysed in detail with the aid of the model; however, it is clear that an improved model is of a three-phase flow, a slow phase creeping along either plane with a faster ‘core’ flow in between.     

Safe operating conditions for semibatch processes involving consecutive reactions with autocatalytic behavior

When strongly exothermic chemical processes are dealt with, safe and productive operating conditions may be difficult to identify; particularly if the reaction scheme is complex and the desired product is an intermediate. In this work, a new procedure for building boundary and temperature diagrams is developed to account for arbitrary kinetic expressions and multiple reactions. Such a procedure has been validated by comparison with literature experimental data involving autocatalytic consecutive reactions.     

BIASES OF ESTIMATORS IN MULTIVARIATE NON-GAUSSIAN AUTOREGRESSIONS

Abstract. Expressions for the bias of the least-squares and modified Yule-Walker estimators in a correctly specified multivariate autoregression of arbitrary order are obtained without assuming that the innovations are Gaussian. Instead, the innovations are assumed to form a martingale difference sequence which is stationary up to sixth order and which has finite sixth moments. The errors in the expressions are shown to be O( n ^-3/2), as the sample size n under some moment conditions. The expressions obtained are the same in the Gaussian and non-Gaussian cases.     

Improved Equation of the Continuous Particle Size Distribution for Dense Packing

The Furnas model describes the discrete particle size distribution for densest packing. Using a model that considers a continuous particle size distribution for the densest packing to be a mixture of infinite Furnas discrete particle size groups, an equation for the cumulative particle size distribution providing the densest packing was derived. Monosize particles with different shapes have a different packing pore fraction. One parameter in the equation is the pore fraction of packed monosize particles; the particle size distribution for achieving densest packing is a function of this pore fraction. A reduced form of this equation is also presented as a working equation. The equation derived here is compared to the modified Andreasen equation for dense packing. An equation and the correlated graph for calculating theoretically the geometric mean particle size and an equation for calculating the specific surface area of the particle size distribution of the improved equation are also derived.     

Effect of Thermal Asymmetry on Laminar Forced Convection Heat Transfer in a Porous Annular Channel

The effect of thermal asymmetry on laminar forced convection heat transfer in an annular porous channel with a Darcy dissipation of fluid kinetic energy was investigated numerically. The cylindrical surfaces making the channel boundaries were kept at constant but different temperatures. The thermal asymmetry thus imposed on the system results in an asymmetric temperature field and different heat fluxes across the channel boundaries. Depending on the Darcy, Péclet and Reynolds numbers, the thermal asymmetry may lead to a reversal of the heat flux along the channel at least at one of the channel walls. The corresponding Nusselt number becomes zero and subsequently experiences a discontinuity, thereby jumping from infinite negative to infinite positive, or vice versa. This feature is observed in the region of thermal development. In the fully developed heat transfer region, the Nusselt numbers can be positive or negative for the same inlet conditions, depending on the heat source strength. In the case of a plug flow, the analytical expressions for the Nusselt numbers have been obtained.     

A GENERAL APPROACH FOR ANALYZING THE ARBITRARY MOTION OF A CIRCULAR DISK IN A STOKES FLOW

A concise analytic method is developed to investigate the arbitrary motion of a circular disk through an unbounded fluid satisfying Stokes equation. Four elementary motions are considered within the same mathematical framework: broadside translation, edgewise translation, in-plane rotation and out-of-plane rotation of a disk. Stokes equations are reduced to a set of dual integral expressions relating the velocity and traction in the plane of the disk. The dual integral equations are solved exactly for each motion and lead, in turn, to closed-form analytical expressions for the velocity and pressure fields. Although many of these results have been previously reported, the approach described here unifies the analysis of the four different motions and presents a straightforward solution technique.     

Computation of asymmetric signal extraction filters and mean squared error for ARIMA component models

Abstract.  Standard signal extraction results for both stationary and nonstationary time series are expressed as linear filters applied to the observed series. Computation of the filter weights, and of the corresponding frequency response function, is relevant for studying properties of the filter and of the resulting signal extraction estimates. Methods for doing such computations for symmetric, doubly infinite filters are well established. This study develops an algorithm for computing filter weights for asymmetric, semi-infinite signal extraction filters, including the important case of the concurrent filter (for signal extraction at the current time point). The setting is where the time series components being estimated follow autoregressive integrated moving-average (ARIMA) models. The algorithm provides expressions for the asymmetric signal extraction filters as rational polynomial functions of the backshift operator. The filter weights are then readily generated by simple expansion of these expressions, and the corresponding frequency response function is directly evaluated. Recursive expressions are also developed that relate the weights for filters that use successively increasing amounts of data. The results for the filter weights are then used to develop methods for computing mean squared error results for the asymmetric signal extraction estimates.     

Perturbation theory and one-fluid corresponding states theories for fluid mixtures

A family of perturbation theories is defined which gives rise rationally, in zeroth order, to an infinite number of sets of corresponding states theories of the one-fluid type, for solutions of molecules whose potentials are conformal. It is argued that the optimum zeroth order member of this family yields the highly successful one-fluid van der Waals (vdWl) theory. This places that theory on a firm theoretical foundation and enables it to be systematically extended. The theory is derived to the second-order terms and preliminary numerical results are presented for the case of hard-sphere mixtures. Suggestions are made for empirical evaluation of some of the exact expressions.