Finite Columns Research Articles

Modelling of multicomponent reactive transport in finite columns — Application to gold recovery using iodide ligands

Existing semi-analytical solutions to the problem of reactive transport in porous media are either restricted to single aqueous species, to decoupled processes or to infinite/semi-infinite domains. Our approach addresses the problem of multicomponent reactive transport in finite columns, where coupling feedbacks emanating from the non-equilibrium thermodynamics of chemical species are taken into account. The main purpose of this research work is to investigate the in-situ leaching of precious metals—a technique that is expected to be relevant to the Earth's regolith consisting of fragmented and/or weathered rocks where permeability is sufficiently high for reactive transport. Producing precious metals from such areas may be beneficial especially when the grade is too low for conventional mining techniques to be applicable.The proposed solution method is validated experimentally using existing tests of column leaching using iodine-based lixiviants. The obtained results show good agreement with the experimental data. In addition, the results indicate that the rate of effluent gold concentration increases until its maximum when the flow velocity, concentration of iodine lixiviants, or specific surface increase. After the peak, the behaviour is more complex and the obtained results allow to optimise the amount of recovered gold.

A sensitivity analysis of DC resistivity prospecting on finite, homogeneous blocks and columns

Besides field applications to geophysical prospecting and subsurface hydrology, electrical resistivity tomography can be applied to finite-scale blocks in the laboratory to characterize the resistivity structure of the blocks and to monitor internal physical and chemical processes. This requires a fast and accurate calculation of the sensitivity matrix to perform a successful resistivity inversion for such blocks. However, the complex geometric shape and boundary and the finite size of the block limit the application of field-suitable sensitivity calculation methods to these blocks. As blocks and finite columns are often used in the laboratory experiments, this paper develops practical analytic expressions, based on the method of image charges, for the sensitivity matrix for these two types of homogenous bodies. The corresponding formulae for the electric potential distribution and theelectrode array coefficient are also presented. As a result of the theory, the effects of placing limits on the sum index in the electric-potential calculation can be analyzed, and a comparison of the theoretical and the numerically simulated electric potential is shown. The results demonstrate the correctness of the theory and indicate that even the addition of only one set of mirror current sources greatly reduces the effects of the block boundary on the electric-potential calculation. Finally, several interesting sensitivity distributions for cross-surface arrays on blocks, and for circular and vertical arrays on columns, are given. Although the formulae developed here are only valid for homogeneous blocks and columns, and an element of relatively small volume is required to permit a good approximation to the sensitivity, the theory is useful in the verification of numerically simulated results, in sensitivity-analysis for optimum probing-scheme design, and in successful resistivity inversion calculation for finite bodies.

Two software tools for plane-wave scattering on nonspherical particles in the German Aerospace Center’s virtual laboratory

We present the methodological background, the range of applicability, and the on-line usage of two software packages, MIESCHKA and CYL, which we have developed for light-scattering analysis on nonspherical particles. MIESCHKA solves Maxwell's equations in a rigorous way but is restricted to axisymmetric geometries, whereas CYL is an approximation for finite columns with nonspherical cross sections. We have established an easy on-line access to both of these programs through the Virtual Laboratory. Its generic software infrastructure was designed to simplify the web-based usage and to support the intercomparability of scientific software.

Estimating falling rate evaporation from finite soil columns

Estimating evaporation is very important in many fields including agriculture and geo-environmental engineering. For example evaporation has a definitive role in the performance of soil covers widely used in mining and landfill applications for environmental protection. The ability to estimate evaporation from different soils without performing detailed experiments or modeling could be very useful especially during the initial stages of design. Gardner and Hillel developed an equation for calculating evaporation from finite soil columns involving the soil moisture diffusivity. The capability of the equation to predict falling rate evaporation from finite coarse sand and fine sand was studied. The studied cases were also modeled using the finite element model SoilCover. The results from the model and the equation were compared to experimental data and found to agree reasonably. The agreement indicates that the analytical solution could be used to estimate falling rate evaporation from finite columns of coarse sand and fine sand. It can facilitate the estimate of actual evaporation from soils especially in the design of engineered covers. The necessary data required are soil–water characteristic curve, the hydraulic conductivity-suction function and initial water content. The analytical method relies heavily on the soil diffusivity; hence the concepts of diffusivity and its role in water movement in unsaturated soil are also discussed.

Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths.

We investigate the errors associated with the use of circular cylinders as surrogates for hexagonal columns in computing the optical properties of pristine ice crystals at infrared (8-12-microm) wavelengths. The equivalent circular cylinders are specified in terms of volume (V), projected area (A), and volume-to-area ratio that are equal to those of the hexagonal columns. We use the T-matrix method to compute the optical properties of the equivalent circular cylinders. We apply the finite-difference time-domain method to compute the optical properties of hexagonal ice columns smaller than 40 microm. For hexagonal columns larger than 40 microm we employ an improved geometric optics method and a stretched scattering potential technique developed in previous studies to calculate the phase function and the extinction (or absorption) efficiency, respectively. The differences between the results for circular cylinders and hexagonal columns are of the order of a few percent. Thus it is quite reasonable to use a circular cylinder geometry as a surrogate for pristine hexagonal ice columns for scattering calculations at infrared (8-12-microm) wavelengths. Although the pristine ice crystals can be approximated as circular cylinders in scattering calculations at infrared wavelengths, it is shown that optical properties of individual aggregates cannot be well approximated by those of individual finite columns or cylinders.

Predicting Multiple Steady States in Equilibrium Reactive Distillation. 1. Analysis of Nonhybrid Systems

During the last few years, multiple steady states (output multiplicities) have been discovered for reactive distillation processes, for example, for the production of fuel ethers and for some esterification processes. Using the transformation of Ung and Doherty (Chem. Eng. Sci. 1995e, 50 (1), 23−48), a method is presented to predict the existence of output multiplicities based on the reactive vapor−liquid equilibrium for the limiting case of reactive columns of infinite length operated at infinite internal flows. This graphical method rests upon the ∞/∞ analysis for azeotropic distillation columns by Bekiaris et al. (Ind. Eng. Chem. Res. 1993, 32 (9), 2023−2038). It is directly applicable to systems where the reactions take place in the entire column (“nonhybrid” columns). When all possible profiles and products are located through a bifurcation analysis, qualitative and quantitative predictions are obtained. The region of feed compositions leading to multiple steady states can be constructed graphically by applying a necessary and sufficient geometrical condition. The prediction results are shown to carry over to finite columns by application to the methyl tert-butyl ether process and are verified by simulation.

Oscillations of finite liquid columns

The resonance frequencies of cylindrical liquid bridges spanned between coaxial circular disks are investigated. First, the oscillations of infinite liquid columns are determined, which are periodic in time but damped in axial direction. For each frequency, an infinite set of such oscillations is obtained. Second, turning to finite columns linear combinations of these axially damped oscillations are formed, which satisfy the boundary conditions on the flow velocity at the supporting disks. The resulting infinite sums are truncated by restricting satisfaction of the boundary conditions to a finite number of circles. The fluiddynamic pressure onto the disks is calculated. The maxima at the resonance frequencies clearly decrease in sharpness and eventually vanish with increasing viscosity and decreasing surface tension of the liquid used. The explicit dependence of the resonance frequencies on the aspect ratio between length and diameter of the liquid columns and their viscosity and surface tension is given. Off-resonance, in particular at high frequencies, the length of the columns actually accelerated by the vibrating disk is found to correspond to 30 percent of their diameter.

Multiple steady states in multicomponent countercurrent mass-transfer processes

Steady-state multiplicity under idealized conditions in a single column, or in a column section with fixed entry conditions at both ends, depends critically on whether the model postulate infinite or finite column length or mass-transfer rates. Like the multiplicity in a column with infinite mass-transfer rates, that in an infinitely long column with finite mass-transfer rate is exclusively a matter of phase-equilibrium and flow-rate and hold-up ratios and is independent of mass-transfer effects. Such a multiplicity does not exist if both the column length and mass-transfer rates are finite, but the approach to the unique steady state may be exceedingly slow (“creep”), a phenomenon known from asymmetric dynamics. Mathematically exact multiplicity in finite columns with finite mass-transfer rates requires an unusual mass-transfer behavior as well as entry conditions different from those for multiplicity in infinitely long columns or in finite columns with infinite mass-transfer rates.

Capillary Pressure Techniques: Application to Exploration and Development Geology

Capillary pressure can be viewed as the pressure required to drive a fluid through a pore throat and displace the pore wetting fluid, with greater pressure being required as the pore throat becomes smaller. The size and distribution of pore throats within a host rock control its capillary pressure characteristics, which in turn control fluid behavior in the pore system. Mercury-injection capillary pressure curves, which measure capillarity, are obtained by injecting mercury into sample plugs to produce a plot of injection pressure vs. mercury saturation. The resulting capillary pressure curves can provide qualitative assessments of reservoir rock using such calculated values as pore-throat sorting, reservoir grade, and oil columns required to obtain 50 and 75% oil saturat ons. Regional capillary pressure maps of these calculated values identify a north-south trend of favorable reservoir rock along the state line between Montana and North Dakota in the third porosity (C) zone of the Ordovician Red River Formation. Oil migration and trapping can be significantly controlled by capillary pressure. At Bell Creek field, in southeastern Montana, multiple oil-water contacts in Cretaceous Muddy Sandstone barrier sands can be explained with the concept of a capillary pressure release valve. Semi-permeable siltstones can only trap finite columns with the excess oil being released updip to migrate into other reservoir quality sand bodies. Oil columns can be estimated from capillary pressure data by determining the buoyancy force of the oil floating on water and the converting from a mercury-air capillary system to an oil-water capillary system. Relative permeability data may also be calculated from capillary pressure data and can be used to estimate fractional water production at given water saturations. Tilted oil-water contacts can result from hydrodynamics, capillary pressure, or both, and capillary pressure data can help determine the type of tilting present. By understanding capillarity, we can obtain better insight on the nature of reservoir rock, oil accumulations, and permeability relationships.

Reliability approach to the random imperfection sensitivity of columns

Buckling of stochastically imperfect finite columns on a mixed quadratic-cubic elastic foundation is studied. The imperfection function is assumed to be a normally distributed random function of the space coordinate with given mean function and autocorrelation function. The problem is solved by the Monte-Carlo-Method. The Fourier coefficients of the initial imperfection function, expanded in terms of the buckling modes of the associated perfect column, are simulated. For each trial the buckling load is obtained numerically, and the results are used in constructing the reliability function at a specified load (defined as the probability of the buckling load exceeding this specified load). The study is a sequel to an earlier work of the author concerning the reliability of stochastically imperfect columns on a purely cubic nonlinear elastic foundation.

Boundary Conditions for Displacement Experiments through Short Laboratory Soil Columns

AbstractThis paper presents a discussion of the physical and mathematical significance of various boundary conditions applicable to one‐dimensional solute transport through relatively short laboratory soil columns. Based on mass balance considerations, it is shown that a first‐type or concentration‐type condition at the inlet boundary incorrectly predicts the volume‐averaged or resident concentration inside both semi‐infinite and finite systems. A third‐type or flux‐type inlet boundary condition preserves mass in semi‐infinite systems, but underpredicts effluent concentrations from finite columns unless a local transformation is used to convert volume‐averaged concentrations into flux‐averaged concentrations. This transformation leads to an expression for the effluent concentration that is identical to the solution for the semi‐infinite system using a concentration‐type boundary condition. For column Peclet numbers greater than about five, the resulting analytical expression for the effluent curve is shown to be nearly identical to the analytical solution for a finite system based on a flux‐type inlet boundary condition and a zero‐concentration gradient at the exit boundary. Both solutions correctly preserve mass in the system; other solutions of the convective‐dispersive transport equation are shown to be inappropriate for analyzing column effluent data.

Buckling of a Stochastically Imperfect Finite Column on a Nonlinear Elastic Foundation: A Reliability Study

The paper is concerned with the problem of buckling of finite columns with initial imperfections, resting on a “softening” nonlinear elastic foundation. The approach is probabilistic. The initial imperfections are assumed to be Gaussian random fields with given mean and autocorrelation functions, and the problem is solved by the Monte Carlo Method. For each realization of the initial imperfection function, the buckling load was found through transformation of the two-point nonlinear boundary-value problem into an initial-value problem and results were used in constructing the empirical reliability function at the specified load (relative number of columns with buckling loads exceeding this specified load). Numerous results are presented with regard to the influence of the parameters of the columns on their imperfection sensitivity.

Comment on ‘Analytical solution of the equation for transport of reactive solutes through soils’ by H. M. Selim and R. S. Mansell

Water Resources ResearchVolume 13, Issue 3 p. 701-701 Free Access Comment on ‘Analytical solution of the equation for transport of reactive solutes through soils’ by H. M. Selim and R. S. Mansell J.-Y. Parlange, J.-Y. ParlangeSearch for more papers by this authorJ. L. Starr, J. L. StarrSearch for more papers by this author J.-Y. Parlange, J.-Y. ParlangeSearch for more papers by this authorJ. L. Starr, J. L. StarrSearch for more papers by this author First published: June 1977 https://doi.org/10.1029/WR013i003p00701AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. References Brenner, H., The diffusion model of longitudinal mixing in beds of finite length, Numerical values, Chem. Eng. Sci., 17, 229– 243, 1962. Carslaw, H. S., J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, New York, 1960. Parlange, J.-Y., J. L. Starr, Linear dispersion in finite columns, Soil Sci. Soc. Amer. Proc., 39, 817– 819, 1975. Parlange, J.-Y., J. L. Starr, First and zeroth order kinetics and influence of boundary conditions on linear dispersion in finite columns, Soil Sci. Soc. Amer. Proc., 1977. Selim, H. M., R. S. Mansell, Analytical solution of the equation for transport of reactive solutes through soils, Water Resour. Res., 123, 528– 532, 1976. Volume13, Issue3June 1977Pages 701-701 ReferencesRelatedInformation

Minimum Reinforcement of Nonuniform Plates

Considerable attention has been given to the plastic design of plates and grillages of uniform depth for minimum reinforcement material consumption, but very little has been given to nonuniform depth structures. This paper attempts to fill some of this gap. It will be shown that the optimality conditions for plates of nonuniform thickness can be derived quite simply from the Marcal-Prager-Shield “associated structure” optimization theory. These conditions are applied to an axisymmetric plate or grillage with predefined thickness variation, a square simply supported flat plate with differing reinforcement depths and to the more practical problem of an infinite flat plate with drop panels, supported on finite columns. Some remarks about the practical application of the results close the paper.

Linear Dispersion in Finite Columns

AbstractThe effect of the finite length of a soil column on the miscible displacement of a solute is analyzed. The boundary conditions at both ends of the column affect the breakthrough curve, while only the boundary condition at the entrance of the column affects the profile within the column to any extent. The interaction between boundary conditions and column length is such that while the breakthrough curve is essentially that given by the standard solution for a semi‐infinite column, even for relatively short columns, the concentration profile is greatly affected by the finite length of the soil column.

Total potential energy of finite columns

Total potential energy of finite columns

Total potential energy of finite columns

The relation between enthalpy and total potential energy in a column of finite horizontal extent is investigated. The limitations of the quasistatic approximation are introduced a priori rather than a posteriori , i.e., cyclone-scale air masses are assumed to be in hydrostatic equilibrium, but any differential equation implying small-scale hydrostatic balance is deliberately avoided. It is found that the enthalpy and total potential energy are equal for columns over ocean, but in general, there is an orographic term which cannot be eliminated by redefining the zero-level of geopotential. There is also a term consequent of the departure from hydrostatic balance in the neighborhood of surface irregularities; this term is presumably negligible for columns of cyclone-scale lateral extent. DOI: 10.1111/j.2153-3490.1968.tb00407.x