Elmer Tory | Mount Allison University (original) (raw)

Papers by Elmer Tory

Industrial Engineering Chemistry Fundamentals, 1965

Journal of Colloid and Interface Science, 1966

The Canadian Journal of Chemical Engineering, 1978

ABSTRACT In a slurry of radioactive particles, radiation count and solids concentration are relat... more ABSTRACT In a slurry of radioactive particles, radiation count and solids concentration are related by a Fredholm equation of the first kind. Two iterative methods are described and conditions for their convergence are given. Two hypothetical examples illustrate computational difficulties and their solution. Experiments with slurries of radioactive praseodymium oxalate demonstrate the feasibility of the method. Solutions are proposed for various practical problems which arise.On a utilisé une équation de Fredholm de premier type, pour établir une relation entre le décompte des radiations et la concentration en matières solides dans une suspension de particules radioactives. On décrit deux měthodes d'itération dont on présente les conditions de convergence. On illustre les difficultés de calcul et la façon de les résoudre au moyen de deux exemples hypothétiques. On démontre le caractère pratique de la méthode en faisant des expériences sur des suspensions d'oxalate de praséodymium radioactif; on propose aussi des solutions dans le cas de divers problèmes qui se présentent dans la pratique.

Canadian Journal of Chemistry, 1953

ABSTRACT Cotton linters dewaxed with benzene and alcohol possess a slightly expanded structure at... more ABSTRACT Cotton linters dewaxed with benzene and alcohol possess a slightly expanded structure attributed to the swelling effect of the alcohol. Storage causes a partial collapse of the linters especially so when moisture is present. Wetting with water followed by rigorous drying produces a marked reduction in accessibility, but with each additional wetting–drying cycle accessibility of the dried linters increases slightly as measured by reaction with thallous ethylate in ether, a nitration mixture, and in hydrogen–deuterium exchange. Differences in nitration found for limited reaction times are obliterated when these reaction times are extended. Increasing accessibility due to repeated wetting and drying is accompanied by lower water sorption and smaller heats of wetting. This anomaly is due to the fact that cellulose samples obtained by alternately wetting and drying dewaxed linters, when stored with a desiccant, compete for the limited amount of water present and adsorb moisture in proportion to their accessibility. Upon further exposure to water the sample of least accessibility, having adsorbed less water, can now adsorb to a greater extent than do the linters of somewhat greater accessibility.The evidence indicates that the difference in accessibility occasioned by repeated wetting and drying is of a physical rather than chemical nature.

The Canadian Journal of Chemical Engineering, 1973

Industrial Engineering Chemistry, 1965

Journal of Mathematical Analysis and Applications, 1982

The Journal of Chemical Physics, 1979

The radial distribution function of hard rods is discussed and several point not explained in pre... more The radial distribution function of hard rods is discussed and several point not explained in previous paper 1,2 are explained. (AIP)

Journal of Applied Mechanics, 1981

Chemical Engineering Science, 1969

Chemical Engineering Science, 1972

Chemical Engineering Journal, 2000

Molecular collisions with very small particles induce Brownian motion. Consequently, such particl... more Molecular collisions with very small particles induce Brownian motion. Consequently, such particles exhibit classical diffusion during their sedimentation. However, identical particles too large to be affected by Brownian motion also change their relative positions. This phenomenon is called hydrodynamic diffusion. Long before this term was coined, the variability of individual particle trajectories had been recognized and a stochastic model had been formulated. In general, stochastic and diffusion approaches are formally equivalent. The convective and diffusive terms in a diffusion equation correspond formally to the drift and diffusion terms of a Fokker-Planck equation (FPE). This FPE can be cast in the form of a stochastic differential equation (SDE) that is much easier to solve numerically. The solution of the associated SDE, via a large number of stochastic paths, yields the solution of the original equation. The three-parameter Markov model, formulated a decade before hydrodynamic diffusion became fashionable, describes one-dimensional sedimentation as a simple SDE for the velocity process {V(t)}. It predicts correctly that the steady-state distribution of particle velocities is Gaussian and that the autocorrelation of velocities decays exponentially. The corresponding position process {X(t)} is not Markov, but the bivariate process {X(t), V (t)} is both Gaussian and Markov. The SDE pair yields continuous velocities and sample paths. The other approach does not use the diffusion process corresponding to the FPE for the three-parameter model; rather, it uses an analogy to Fickian diffusion of molecules. By focusing on velocity rather than position, the stochastic model has several advantages. It subsumes Kynch's theory as a first approximation, but corresponds to the reality that particle velocities are, in fact, continuous. It also profits from powerful theorems about stochastic processes in general and Markov processes in particular. It allows transient phenomena to be modeled by using parameters determined from the steady-state. It is very simple and efficient to simulate, but the three parameters must be determined experimentally or computationally. Relevant data are still sparse, but recent experimental and computational work is beginning to determine values of the three parameters and even the additional two parameters needed to simulate three-dimensional motion. If the dependence of the parameters on solids concentration is known, this model can simulate the sedimentation of the entire slurry, including the packed bed and the slurry-supernate interface. Simulations using half a million particles are already feasible with a desktop computer.

Powder Technology, 1995

Transit times (the times for individual particles to traverse the distance between two fixed heig... more Transit times (the times for individual particles to traverse the distance between two fixed heights) have been widely used to estimate the mean of the instantaneous velocity. Individual particles in a dilute dispersion have velocities that are approximately normally distributed. Also, the velocity of any particle varies continuously, but erratically, with time. These features, which complicate the analysis of transit times, are accurately described by the three-parameter Markov model. We have applied a fourth-order stochastic Runge-Kutta method to the differential equations for this model to obtain position and velocity at discrete times. Crossing times and velocities are determined by interpolation. We have used simulations to clarify the concepts of crossing velocities, first-crossing velocities, 'calming sections', and transit times. Means calculated from total distance/total time agree closely with the mean chosen for the simulation. This agreement, which persists far beyond any variability yet encountered in experimental work, validates the use of transit times to estimate mean velocity.

Lecture Notes in Applied and Computational Mechanics, 2003

ABSTRACT The behaviour of sedimenting monodisperse suspensions is usually deduced from the flux p... more ABSTRACT The behaviour of sedimenting monodisperse suspensions is usually deduced from the flux plot, but this approach is not available for polydisperse suspensions. Also, the diversity of velocities, even in monodisperse suspensions, produces a hydrodynamic diffusion that is not taken into account in Kynch’s theory. The rapid improvement in computing power has made simulation an attractive method. Sedimentation of suspensions with many species can be handled easily, and stochastic effects can be included, if desired. We show that two sources of difficulty, generation of a concentration gradient and control of fluctuations in concentration, can be overcome by choosing the controlling concentration as that immediately below the test sphere.

This paper reviews some recent advances in mathematical models for the sedimentation of polydispe... more This paper reviews some recent advances in mathematical models for the sedimentation of polydisperse suspensions. Several early models relate the settling velocity to the solids concentration for a monodisperse suspension. Batchelor's theory for dilute suspensions predicts the settling velocity in the presence of other spheres that differ in size or density. However, this theory is based on the questionable assumption

Siam Journal on Applied Mathematics, Jul 27, 2006

We show how existing models for the sedimentation of monodisperse flocculated suspensions and of ... more We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression" or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convectiondiffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.

Nonlinearity, Mar 1, 2011

The well-known kinematic sedimentation model by Kynch states that the settling velocity of small ... more The well-known kinematic sedimentation model by Kynch states that the settling velocity of small equal-sized particles in a viscous fluid is a function of the local solids volume fraction. This assumption converts the one-dimensional solids continuity equation into a scalar, nonlinear conservation law with a nonconvex and local flux. This work deals with a modification of this model, and is based on the assumption that either the solids phase velocity or the solid-fluid relative velocity at a given position and time depends on the concentration in a neighbourhood via convolution with a symmetric kernel function with finite support. This assumption is justified by theoretical arguments arising from stochastic sedimentation models, and leads to a conservation law with a nonlocal flux. The alternatives of velocities for which the nonlocality assumption can be stated lead to different algebraic expressions for the factor that multiplies the nonlocal flux term. In all cases, solutions are in general discontinuous and need to be defined as entropy solutions. An entropy solution concept is introduced, jump conditions are derived and uniqueness of entropy solutions is shown. Existence of entropy solutions is established by proving convergence of a difference-quadrature scheme. It turns out that only for the assumption of nonlocality for the relative velocity it is ensured that solutions of the nonlocal equation assume physically relevant solution values between zero and one. Numerical examples illustrate the behaviour of entropy solutions of the nonlocal equation.