Wheeler Equation Research Articles

Evaluating porous-layer open-tubular capillaries as vapor preconcentrators in a microanalytical system

Measuring environmental concentrations of organic vapors with microfabricated chemical sensors or sensor arrays often requires a means to enrich collected samples prior to detection. With such an application in mind, a preliminary evaluation is described of two porous-layer open tubular (PLOT) capillary traps as vapor preconcentrators for a series of vapors. Short (1-cm) sections of commercial PLOT-Q and PLOT-S capillary having wall coatings of styrene–divinylbenzene copolymer and vinylpyridine–divinylbenzene copolymer, respectively, are fitted with a metal sleeve for rapid thermal desorption of preconcentrated vapor samples, and tested using a downstream 97-MHz polyisobutylene-coated surface acoustic wave (SAW) sensor. Calibrated responses to vapors of 2-butanone (MEK), trichloroethylene (TCE), toluene, and m-xylene are collected with and without preconcentration. Dimethylmethylphosphonate could not be efficiently desorbed from either PLOT trap. For the remaining vapors, increases in sensitivity of 3–9-fold are achieved by preconcentrating and analyzing just 1 ml of sample air. Calculated limits of detection (LOD) range from 1–8 ppm. Differences in sensitivities are observed between the PLOT-Q and PLOT-S sampling trains for MEK and TCE. A theoretical model of penetration yields limiting values of flow rate and trap dimensions. Measured 10%-breakthrough times at 1 ml/min ranged from ∼1 to 6 min and, for PLOT-Q, are ≥ modeled values obtained using the modified Wheeler equation. The implications of the results for the design and operation of microanalytical systems for vapor analytes are discussed.

Gauge invariant perturbations of black holes. II. Kerr space–time

We cast the perturbed Bianchi identities into a form involving only tetrad and coordinate-gauge invariant field quantities. From the perturbed Bianchi identities we then derive a system of gauge invariant wavelike gravitational perturbation equations. These are the Teukolsky equation, a gauge invariant analog of the Regge–Wheeler equation, and a new gravitational perturbation equation. The Bianchi identities also provide the transformations between the perturbation equations. In particular, the transformation between the Teukolsky equation and the analog of the Regge–Wheeler equation arises in this way. This work extends our previous analysis of the perturbations of the Schwarzschild black hole to the Kerr case.

Adsorption properties of vapor-protective fabrics containing activated carbon

Abstract Barrier properties of air-permeable protective fabrics incorporating microporous activated carbon are evaluated using simulants of toxic chemical agents, both in dry atmospheres and at high humidity. Performance of an adsorbent fabric is dependent on kinetic factors as well as adsorption capacity, with the overall rate limited by vapor phase mass transfer over a wide range of environmental conditions. For dynamic adsorption, the characteristic sigmoid breakthrough curve of effluent vapor concentration against time is best described using a modified Wheeler equation including an adjustable parameter ( a ) which compensates for axial dispersion in the fabric. For powdered carbon impregnated in a polyurethane foam fabric, the calculated dynamic adsorption capacity ( A d ) is ca 10 percent lower than the equilibrium adsorption capacity ( A e ) obtained at the same vapor concentration for each carbon and adsorbent fabric. An additional 10 percent decrease in A d and A d for the supported carbon relative to dry powdered carbon is attributed to partial blocking of the pore volume by components of the acrylic binder formulation used in the fabric process. Adequate protection is attained even in the presence of faults (e.g. holes, tears and open seams) provided that the smallest dimension of the fault does not exceed a minimum size determined by the net rate of adsorption. This concept is demonstrated by experiments with fabrics possessing known fault arrays.

Gauge invariant perturbations of black holes. I. Schwarzschild space–time

We cast the perturbed Bianchi identities, in the Schwarzschild background, into a form involving only tetrad and coordinate gauge invariant Newman–Penrose field quantities. These quantities, which arise naturally in our approach, are gauge invariant quantities of spin-weight ±2, ±1, and 0. Some of the integrability conditions for the Bianchi identities then provide a system of six gauge invariant perturbation wave equations for the spin-weighted quantities. These wave equations are, respectively, the (spin-weight ±2) Bardeen–Press equations, two new (spin-weight ±1) gravitational wave equations, and two (spin-weight 0) Regge–Wheeler equations. Other integrability conditions provide the transformation identities that relate the field quantities to each other, and hence relate the various perturbation wave equations to one another. In particular, this method provides an alternative derivation of the transformations between the Bardeen–Press and Regge–Wheeler equations. The integrability conditions also allow us to relate the Bardeen–Press quantities of opposite spin-weight, and we investigate how this relationship compares with the Teukolsky–Starobinsky identities. Finally, we give a derivation of the gauge invariant Zerilli equation, and show how it is related to the fundamental equations mentioned above.

Asymptotic behaviour of quasi-normal modes of Schwarzschild black holes

The behaviour of the Regge--Wheeler equation in the vicinities of transition points is studied and an approximate analytic procedure is developed for the determination of highly damped quasi-normal modes of a Schwarzschild black hole. It is shown that the leading term in the expansion of the imaginary part of the modes is exactly , and the second and third terms of the expansion are proportional to and , respectively, where , l is the angular harmonic index, and the index n, , characterizes the highly damped modes for a given l. This expansion is found to be in qualitative agreement with the numerical formula given by Nollert.

Adsorption Prediction of Binary Mixtures on Adsorbents Used in Respirator Cartridges and Air-Sampling Monitors

A mathematical method was developed for predicting the performance of respirator cartridges and air-sampling monitors in the presence of binary mixtures. This method uses only data for pure-component adsorption equilibrium to calculate the breakthrough curves of the two components in a binary vapor mixture. The Ideal Adsorbed Solution Theory was used to calculate the adsorption equilibria for the components in a binary mixture. The adsorption capacities were then used in expressions similar to the modified Wheeler equation for pure compounds to determine the breakthrough curves for the binary mixture. This technique applies to a wide variety of binary mixture/adsorbent systems. In the present study it was successfully applied to the adsorption of four binary mixtures of acetone and m–xylene on activated carbon, three binary mixtures of acetone and styrene on activated carbon, and a binary mixture of carbon dioxide and water vapor on a molecular sieve. The predicted breakthrough curves were in agreement wi...

Estimating Service Lives of Organic Vapor Cartridges

Procedures were developed for estimating service lives of air-purifying organic vapor respirator cartridges, including methods for untested compounds and use conditions (concentration, temperature, and airflow rate). Correlations of adsorption capacities and adsorption-rate coefficients based on equilibrium and breakthrough curve data were reviewed. These correlations were combined using the Reaction Kinetic (modified Wheeler) equation to estimate breakthrough times. The proposed estimation procedure showed 95% confidence intervals of up to ±50% when no breakthrough data were assumed. It is shown how even limited breakthrough curve data for one vapor/carbon combination can be used to substantially improve accuracy of the estimation. Only dry conditions (below 50% relative humidity) have been considered so far.

Estimating Service Lives of Organic Vapor Cartridges

Procedures were developed for estimating service lives of air-purifying organic vapor respirator cartridges, including methods for untested compounds and use conditions (concentration, temperature, and airflow rate). Correlations of adsorption capacities and adsorption-rate coefficients based on equilibrium and breakthrough curve data were reviewed. These correlations were combined using the Reaction Kinetic (modified Wheeler) equation to estimate breakthrough times. The proposed estimation procedure showed 95% confidence intervals of up to ±50% when no breakthrough data were assumed. It is shown how even limited breakthrough curve data for one vapor/carbon combination can be used to substantially improve accuracy of the estimation. Only dry conditions (below 50% relative humidity) have been considered so far.

Effects of adsorbed water vapor on the adsorption rate constant and the kinetic adsorption capacity of the Wheeler kinetic model.

A recent trend in occupational safety and health has focused on the use of respiratory protective equipment to supplant engineering controls as the primary means of protecting workers from toxic substances. Respirator adsorbent cartridges have been demonstrated to have a finite capacity to adsorb toxic vapors. The knowledge of when this limit is approached or has been exceeded is crucial to the user. The Wheeler kinetic breakthrough model has been shown to describe accurately organic vapor penetration through beds of activated carbon. The model, however, does not account for competitive adsorption of water vapor or other organic vapors. The investigations reported here demonstrate the effect of adsorbed water vapor on the kinetic adsorption parameters (kinetic rate constant and kinetic saturation capacity) of the Wheeler equation. Adsorbent beds were equilibrated at varying concentrations of water vapor and then challenged with carbon tetrachloride vapor-laden air. Dry carbon had an initial rate constant of 62.5 s-1 and a kinetic adsorption capacity of 0.36 g of adsorbed CCl4/gram (g/g) of adsorbent. These parameters decreased in proportion to the amount of water vapor adsorbed, with the minimum predicted values occurring at 100% relative humidity. The minimum experimental value for the kinetic rate constant was 17.6 s-1, a decrease of 73% from the dry carbon values. The minimum predicted value for the kinetic adsorption capacity was 0.16 g/g, a decrease of 45%.

On the numerical solution of the Hill–Wheeler equation

A particular numerical method is discussed for solving the Hill–Wheeler equation, which is an integral equation that arises in the generator coordinate method analysis of problems in nuclear physics. The method is applicable to scattering problems with finite range potentials and exploits prior knowledge of the asymptotic form of the solution to convert the Hill–Wheeler equation into a Fredholm integral equation of the first kind. The resulting Fredholm equation is ill-posed, which often results in numerically unstable solutions. The method of regularization is used to produce numerically stable, approximate solutions. Optimizing the choice of regularization parameter is discussed and calculations are presented for an example problem.

Organic vapor (OV) respiratory cartridge testing--potential Jonas model applicability.

More efficient methods of characterizing organic vapor (OV) respirator performance against the wide range of compounds and environmental conditions found in industrial settings are needed to improve respiratory protection in the workplace. One method, based on Jonas' kinetic model, has potential application for evaluating OV cartridges and more efficiently utilizing available research, testing and certification resources. Jonas has shown that the kinetic adsorption capacity and gas adsorption rate of a small packed charcoal sorbent bed is described by the linear relationship between gas breakthrough time and sorbent weight as predicted by the modified Wheeler equation. This research is based on the hypothesis that characterizing sorbent performance against a reference compound will allow prediction of breakthrough times, and therefore prediction of cartridge life, for other compounds under similar conditions. Baseline data on dried cartridges at dry conditions are presented that demonstrate that the linear relationship between breakthrough time and and sorbent weight that Jonas showed for small sorbent columns also holds when commercially manufactured respiratory cartridges are stacked in series to resemble a packed column of varying bed length and sorbent weight. This suggests the potential applicability of the Jonas model for characterizing and evaluating commercially available OV respirator cartridges.

On solving directly the Griffin–Hill–Wheeler equation for scattering problems

We investigate a method for solving the Griffin–Hill–Wheeler equation, in which the generator coordinate amplitudes are approximated by regular functions. The method is applied to elastic α–α scattering and the results are compared with those yielded by other methods. Particular attention is paid to the calculation of physical wave functions free of spurious components.

The generator coordinate method for molecular wavefunctions: A moment method and a simple intrinsic function

An extension of the generator coordinate method to the description of the electronic structure of molecules is presented. An exact formal solution to the Hill–Wheeler equation is obtained for a certain class of intrinsic functions, namely, those whose Hamiltonian and overlap kernels are of degenerate form. Since the exact kernels are used in the Hill–Wheeler equation, the variational principle is retained. The formal solution is represented by the set of moments of the generator coordinate with respect to the weighting function. The features of the method are illustrated by application to the hydrogen molecule. A simple trial intrinsic function and a PPP Hamiltonian are used to describe the π-electron structure of three linear conjugated polyenes (1,3-butadiene; 1,3,5-hexatriene; and 1,3,5,7-octatetraene). A significant part of the apparent ground state correlation energy is recovered for each molecule and π-electron excited state energies are also calculated. These results are compared with PPP CI calculations and the limitations of this simple trial intrinsic function are discussed.

Illustration of the generator coordinate method in terms of model problems

Numerical aspects of the generator coordinate method are discussed in terms of three simple examples: a) the linear harmonic oscillator, b) the He atom and c) the H atom with trial generating functions: a) infinite square-well solutions, b) screened Slater-type orbitals and c) a Gaussian function, respectively. By numerical integration of the Hill Wheeler equation, straightforward and accurate results are obtained for the lower-lying eigenstates of the three cases studied. Some features of the method and the numerical solution are commented upon.

Respirator cartridge efficiency studies: VII. Effect of relative humidity and temperature.

Cartridge service life is not appreciably affected at relative humidities below 50%. However, performance is severely compromised when the humidity exceeds 65%. Experimental values are compared from those calculated from the adsorption isotherm, Mecklenburg and modified Wheeler equations. Calculations indicate that temperature does not play a significant role in service life predictions.

Accurate analytical determination of quasi-static microstrip line parameters

In the design of microstrip components, the analytical equations of Wheeler have an advantage in handling-speed and simplicity over alternative quasi-static analyses involving complex computation by numerical methods. Wheeler equations can produce excessive errors, however, unless care is taken particularly in the choice of changeover point between equations for narrow or wide strips. This paper shows that for alumina-type substrates with 8 < er < 12, the Wheeler analysis and synthesis equations produce results within 1% of those given by selected numerical methods, when the changeover points are correctly chosen. Additional new and modified formulae are presented for the direct calculation of microstrip effective permittivity from either W/h or Z0, also to an absolute accuracy of less than 1%. These complete an accurate set of analytical equations for quasi-static analysis or synthesis of microstrip lines.

Dependence of the kinetics of the decomposition of cyclohexene on the structure of the catalyst

1. With the transformation of cyclohexene over aluminosilicate catalysts varying in structure taken as example, it has been shown that the kinetic method of assessing catalyst activity from rate constants that take retardation by diffusion into account (ks) is applicable to such transformations. 2. It was shown that the value of the dimensionless parameter h is a characteristic of the extent of diffusional retardation, being proportional to the ratio of kinetic and diffusio rates, 3. It was shown that the value of the reaction rate constant ko does not depend on the surface area. 4. Wheeler's equation was transformed into a simple form of relationship between the constants kv ko, and ks and the value of the effective (accessible) surface, i.e. the physical properties of the catalyst.

The linear stability of the Schwarzschild solution to gravitational perturbations

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge--Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of nonlinear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.