Approximate Analytical Equations Research Articles

Determination of Rate Constants of Consecutive First Order Reactions Occurring on Chromatographic Columns

Abstract Approximate analytical equations were developed for the products of consecutive reactions occurring on a chromatographic column. The equations were tested both by numerically evaluated chromatograms and the acid catalyzed hydrolysis of phthalic dichloride in a liquid chromatographic reactor.

Approximate analytic equations for small shapiro step Josephson junctions

AbstractObservation of Shapiro step is important for evaluation of the Josephson junction. This theoretical characteristic is given by a nonlinear differential equation with the normalized frequency ω based on the current driven resistively shunted junction (RSJ) model as a parameter [14]. Although this model provides good agreement with the experimental results, the number of operations for the numerical solution of the equation is excessive. Hence, discussions of the experimental results by means of theoretical values are difficult. As a result, a proposal of a simple approximate equation which can describe the theoretical characteristics only with ω is considered extremely useful. In the high temperature superconducting junction extensively studied recently, the IcRn product is large and ω < 0.1 is expected. For ω < 0.1, the current amplitude of Shapiro step decreases excessively. In practice, Shapiro steps are often not observed at a high‐temperature superconducting junction. This is considered to be caused by the ω < 0.1 measurement condition. It is important for evaluation of the junction to investigate this problem. No detailed analysis has been carried out for the region with ω < 0.1. On the other hand, since ω < 0.1 is the ideal operating condition of the Josephson mixer, understanding of the characteristics is important from the application point of view. In this paper, understanding of the theoretical characteristics for ω < 0.1, the derivation of the approximate equation and the condition in which Shapiro step cannot be observed are investigated. The analysis results and the experimental values are compared, and the effectiveness of the analysis is presented.

Scanning Electrochemical Microscopy. 27. Application of a Simplified Treatment of an Irreversible Homogeneous Reaction following Electron Transfer to the Oxidative Dimerization of 4-Nitrophenolate in Acetonitrile

Scanning electrochemical microscopy (SECM) was employed to study the rate of the second-order homogeneous reaction of a species electrogenerated at an ultramicroelectrode tip. Oxidation of 4-nitrophenolate in acetonitrile to the phenoxy1 radical took place at a gold ultramicroelectrode (5-pm diameter) held in close proximity to a larger substrate electrode that detected the tip-generated species. A radical dimerization rate constant of 1.2 (*0.3) X 108 M-' s-I was determined. Approximate analytical equations for the treatment of irreversible kinetics following electron transfer were developed to simplify the analysis of feedback and collection data for both first- and second-order reactions. The effect of possible interference by the reference material added to allow precise determination of the tipsubstrate spacing (benzoquinone or ferrocene) was also investigated.

Tip-surface capillary interactions

The capillary adhesion force between the tip of a particle and a flat surface is calculated by numerical solution of the equations which are developed aa well aa by an approximate analytical equation which is suggested. Reeulte are presented for spherical, paraboloidal, and conical tips. They are compared with the classical approximate equation for the capillary adheeion force between a sphere and a flat surface, which predicta that the force is independent of the position of the liquid-fluid interface. The present analysis shows that the situations for which the classical equation can be UBBd are rather limited. For quantitative analysis of experimental data, such aa the measurement of thin film profilea using atomic force microscopy, accurate knowledge of the geometry of the tip of the particle and numerical solution of the equations are usually required.

An Analytical Criterion for Buckling Strength of Built-up Compression Members

In current design practice, North American and European specifications differ significantly regarding the rules governing the design of stitched built-up struts. The British and German specifications treat such members essentially like battened columns and apply more stringent provisions to limit their shear flexibility. Consequently, built-up struts designed according to these specifications must be closely stitched. Furthermore, the strength of built-up members are reduced due to shear flexibility. The American and Canadian specifications (except for recent AISC-LRFD), on the other hand, do not treat stitched built-up struts in a way similar to battened columns. They only require that minimum individual slenderness ratio between stitches not exceed the governing slenderness ratio of the built-up strut as a whole. Thus, in strength calculation, the detrimental effect of shear flexibility is not taken into account. Recently, AISC-LRFD Specification introduced an empirical equation to modify the overall slenderness ratio of built-up struts, including stitched members, in order to take into account the detrimental effect of shear flexibility on their strength. The criterion is based on experimental study by Zandonini on stitched double channel struts and verification by Astaneh and Goel on double angle bracing members. In 1952, Bleich proposed a similar but analytical criterion to modify the overall slenderness ratio of battened columns. The equation has the limitation of being strictly applicable to battened columns because of simplifications made in the derivation process. Another limitation of Bleichs equation is that it was derived strictly for hinged-end members. This paper presents a generalized version of the analytical equation proposed by Bleich. Thus, it does not have the limitations of Bleichs equation. Therefore, the proposed analytical equation is applicable to all built-up struts with general end conditions. The equation is verified both analytically and experimentally. For analytical verification, a parametric study is performed in which, for different over-all and individual slenderness ratios, the buckling load is calculated according to the proposed analytical equation, Bleichs approximate analytical equation, and the LRFD empirical equation. The results are compared and discussed. For experimental verification, test results from another study by the authors are compared with the calculated buckling load according to the proposed analytical equation and the LRFD empirical equation. All comparisons verify the validity of the proposed analytical equation.

A propagation-delay expression and its application to the optimization of polysilicon emitter ECL processes

A sensitivity analysis is used to compute an analytical expression for the propagation delay of an emitter-coupled logic (ECL) gate in terms of the electrical parameters of the circuit. These electrical parameters are in turn related to the processing parameters through approximate analytical equations and computer simulation programs. In this way, a simple means is evolved of optimizing an ECL process for maximum circuit speed. The accuracy of the propagation-delay expression and the related equations for the electrical parameters are verified by comparison with experimental results on 6- mu m geometry circuits. By means of the propagation-delay expression, it is shown that the optimum propagation delay at 6- mu m geometry is 320 ps for a conventional process and 160 ps for a self-aligned process. On scaling to a geometry of 0.5 mu m, a propagation delay of 20 ps is predicted for a fully optimized self-aligned process.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Mass transfer in ozone absorption. An approximate analytical equation is derived for predicting the enhancement of mass transfer by decomposition and ozanation reactions

A mathematical model has been formulated for the absorption of ozone accompanied by a first-order decomposition of ozone and the second-order, ozonation reaction between ozone and a liquid reactant in gas-liquid contactors. On the basis of the film model, an analytical approximate equation has been derived to predict the enhancement of mass transfer by the decomposition and ozonation reactions. Methods for estimation of chemical mass-transfer coefficients for ozone-absorption processes have been suggested. By combining a knowledge of physical mass-transfer coefficients from empirical correlations and enhancement factors from theoretical predictions, the chemical mass-transfer coefficient for an ozone-absorption process can be estimated as illustrated in the paper. 38 references, 6 figures.

On creep fracture by void growth

The growth of void-like creep damage is analysed. Voids can grow by mechanisms controlled by grain-boundary diffusion, by surface diffusion, by power-law creep, and by any combination of two of these Approximate analytical equations for the growth rate by each mechanism, under multiaxial stress states, are developed, and related to a number of published analyses. The growth equations are integrated to give times and strains to fracture under constant (multiaxial) stress, constant load and two simple load histories. Both quasi-uniform and non-uniform distributions of voids are considered. The formulation can be related to the continuum damage method of Kachanov and suggests modifications to this method. It leads to a new prescription for extrapolating creep data. A unified picture of void growth appears possible, embracing most aspects of previously published models.

Approximate analytical equations for rate curves in the case of biporous sorbents and linear sorption isotherms

1. Approximate analytical expressions were obtained for the rate curves in the case of biporous sorbents, linear sorption isotherms, and granules of the sorbent as spheres, cylinders with impervious faces, and platelets (prisms or cylinders with impervious sides). 2. The obtained equations can be used to solve both the direct problem of calculating the rate curves from the known transfer coefficients and the reverse problem of determining the transfer coefficients from the experimental rate curves.