Townsend Equation Research Articles

Realizability of the rapid distortion theory spectrum: The mechanism behind the Kelvin–Townsend equations

In the context of homogeneous turbulence, we prove that the Rapid Distortion Theory (RDT) model for the spectral tensor preserves the symmetry, positive semidefiniteness, and integrability properties required in Cramér’s characterization of the spectral tensor of a continuous homogeneous random process. From this, a statistically valid correlation tensor is obtained that returns a Reynolds stress tensor model that satisfies realizability conditions. The number of hypotheses used is kept to a minimum, which allows a flexible use of the model in the applications. The Kelvin–Townsend equations allow us to construct the solution and prove its properties by means of a factorization approach. Since the RDT spectral tensor model is a system of transport equations plus an algebraic restriction due to incompressibility, we deal with the existence, uniqueness, and persistence of solutions in a specific set of functions by using DiPerna–Lions renormalization techniques.

Electrostatic charging and precipitation of nanoparticles in technical nitrogen: Highly efficient diffusion charging by hot free electrons

Electrostatic charging and deposition of a liquid nano-aerosol was studied in dry air and in technical (3.6 % O2) nitrogen. The experiments have shown that electronic charging of aerosols can be important in technical scale electrostatic precipitators (ESPs). Already at operation voltages just slightly above the corona onset voltage, the contribution of the free electrons to the overall current is estimated to be around 50 %. Due to the high temperature of free electrons, diffusion charging by free electrons allows to reach exceptionally high particle charge and extremely high precipitation efficiency.A strongly simplified theoretical model was developed, which gives a good prediction of particle charge based on averaged values of particle diameter, current density, electric field strength, electron temperature and residence time. The ion mobilities were determined by fitting the current-voltage characteristics with a modified Townsend (Monrolin et al. 2018) equation and are significantly higher than the values typically used to describe diffusion charging in air. This may be ascribed to the very dry gas phase and the short average lifespan of the ions under ESP conditions, which is in the order of 1 ms.From the practical point of view, electronic charging might be relevant in a number of technical applications, including high temperature ESPs, ESP applications in dry and oxygen-free gases and pulsed corona systems.

Entropy Generation through the Interaction of Laminar Boundary-Layer Flows: Sensitivity to Initial Conditions

A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to the interaction of a longitudinal vortex with a laminar boundary-layer flow. These three-dimensional equations are cast into a Lorenz-format system of equations for the spectral velocity component solutions. Tsallis-form empirical entropic indices are obtained from the solutions of the modified Lorenz equations. These solutions are sensitive to the initial conditions applied to the time-dependent coupled, non-linear differential equations for the spectral velocity components. Eighteen sets of initial conditions for these solutions are examined. The empirical entropic indices yield corresponding intermittency exponents which then yield the entropy generation rates for each set of initial conditions. The flow environment consists of the flow of hydrogen gas with impurities at a given temperature and pressure in the interaction of a longitudinal vortex with a laminar boundary layer flow. Results are presented that indicate a strong correlation of predicted entropy generation rates and the corresponding applied initial conditions. These initial conditions may be ascribed to the turbulence levels in the boundary layer, thus indicating a source for the subsequent entropy generation rates by the interactive instabilities.

An analytical comparison of gas gain in spherical, cylindrical and hemispherical low-pressure proportional counters intended for use in experimental microdosimetry

Traditionally experimental microdosimetry has employed low pressure single cavity spherical Tissue Equivalent Proportional Counters (TEPCs). Multi-Element Tissue Equivalent Proportional Counters (METEPCs) with numerous cylindrical cavities have been constructed in order to increase sensitivity per unit volume; however existing METEPC designs are prohibitively complex and sensitive to motion and audible noise. This work proposes a novel hemispherical element with a wire-less anode ball as a solution to these issues. The gas gain characteristics of this hemispherical METEPC element were analyzed first for a single hemispherical TEPC to evaluate performance relative to current cylindrical and spherical counter designs that have been demonstrated experimentally to perform very well. This gain analysis evaluated relative avalanche size and the uniformity in maximum gain for electrons originating throughout the gas cavities of each of the three counters. Radial gas gain distributions for each counter were determined using both theoretical potential distributions as well as analytical equipotential distributions generated with ANSYS Maxwell (V. 14.0) to solve the Townsend equation. It was found that the hemispherical counter exhibits completely uniform gas gain for electrons approaching the anode from all directions and its avalanche region occupies only 3.5×10−3% of the entire gas cavity volume, whereas in the cylindrical and spherical counters the avalanche occupies 0.6% and 0.12% of the total respective gas cavity volumes. These analytical gas gain results are promising, suggesting that the hemisphere should exhibit uniform signal amplification throughout the gas cavity and if the recommended follow-up experimental work demonstrates the hemispherical counter works as anticipated it will be ready to be incorporated into an METEPC design.

Spectral Entropy, Empirical Entropy and Empirical Exergy for Deterministic Boundary-Layer Structures

A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to a laminar three-dimensional boundary-layer flow in a methane fired combustion channel flow environment. The objective of this study is to explore the applicability of a set of low dimensional, coupled, nonlinear differential equations for the prediction of possible deterministic ordered structures within a specific boundary-layer environment. Four increasing channel pressures are considered. The equations are cast into a Lorenz-type system of equations, which yields the low-dimensional set of equations. The solutions indicate the presence of several organized flow structures. Singular value decomposition of the nonlinear time series solutions indicate that nearly ninety-eight percent of the fluctuating directed kinetic energy is contained within the first four empirical modes of the decomposition. The empirical entropy computed from these results indicates that these four lowest modes are largely coherent structures with lower entropy rates. Four regions are observed: low-entropy structures over the first four modes; steep increase in entropy over three modes; steady, high entropy over seven modes; and an increase to maximum entropy over the last two modes. A measure, called the empirical exergy, characterizes the extent of directed kinetic energy produced in the nonlinear solution of the deterministic equations used to model the flow environment. The effect of increasing pressure is to produce more distinct ordered structures within the nonlinear time series solutions.

Optimal Generation of 254nm ultraviolet radiation

The science of the application of 254nm UV from mercury doped glow discharge tubes has been a major topic since Johann Ritter discovered UV via its chemical inducing reactions in 1801 and Niels Finsen's 1860 work on UV therapy in treating rickets. In 1857 Siemens AG patented UV254nm creation via filamentary discharge, subsequently widely used for ozone production. By 1932 the Coblentz Congress had defined the three regions of the UV action spectrum. This paper presents the science of a new design for a sterilizer module fabricated from extruded, recycled aluminium. This novel design achieves better than D10 performance using six UV tubes per module driven by three electronic ballasts drawing a total current of only 1.26 amps at 240V single phase. This module delivers more than 45,000 microwatts per square centimetre of 254nm UV which sterilises one litre per second in a module with a dwell time of 1.6 seconds in a design with less than 0.5 bar pressure drop across each module. This system takes the electrical efficiency of 254 UV generation from less than 25% to more than 92% as measured by an NPL-traceable calibration against a current industry standard. Since 254nm UV generating tubes are also the basis of fluorescent lighting this new work on optimising the generation of 254nm UV also has application worldwide to improved efficiency of fluorescent tube electrical lighting, because we have shown that most of the fluorescent lamps operating today (particularly the T8 1 diameter) are running at less than 25% efficiency as opposed to the over 92% which is possible with the methods we describe. The work reported here shows that the Townsend equation for electron transport in glow discharge plasmas is not adequate since it does not address either plasma diameter or plasma drive frequency both of which fundamentally alter the electron energy transfer efficiencies to mercury atoms in the plasma.

Ordered Regions within a Nonlinear Time Series Solution of a Lorenz Form of the Townsend Equations for a Boundary-Layer Flow

A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to a laminar three-dimensional boundary-layer flow. These equations are cast into a Lorenz-type system of equations. The initial system of Lorenz equations yields the generation of masked output signals containing internal ordered regions. The self-synchronizing property of the Lorenz system of equations is then exploited by considering the initial Lorenz system as a transmitter system providing chaotic masked information signals to a series of identical Lorenz receiver systems. The output signal from each successive receiver system indicates the growing recovery of ordered regions in the chaotic output signal. Finally, the three-dimensional graph of the output velocity wave vector signal from the fourth receiver system and the spectral entropy rates for the output axial velocity wave vector indicate the presence of ordered regions which are characterized as axially-directed spiral vortices.

Zonal approach to centrifugal, elliptic and hyperbolic instabilities in Stuart vortices with external rotation

The stability analysis of a street of Stuart vortices in a rotating frame is performed by integrating the Kelvin–Townsend equations along the mean flow trajectories, using the geometrical optics technique (Lifschitz & Hameiri 1991) for short-wave perturbations. A parallel is drawn between the formulations of this zonal approach and that of rapid distortion theory, better known to the turbulence community. The results presented confirm those obtained by the standard stability analysis based on normal-mode decomposition: depending on the rotation parameter and the oblique mode considered, three unstable zones are identified, related to the centrifugal, elliptic and hyperbolic instabilities, as observed for Taylor–Green cells (Sipp et al. 1999). Anticyclonic rotation is shown to destabilize Stuart vortices through a combination of the elliptical and centrifugal instability mechanisms, depending on the ratio of its rate to the structure core vorticity. Available stability criteria are discussed in the general case of two-dimensional rotating flows, in relation to their streamline topology and the values of the local Rossby number or vorticity.

Streamwise inhomogeneity of decaying grid turbulence

Accounting for the streamwise inhomogeneity of turbulence in Kolmogorov’s equation for the third-order moment of the velocity increment between two points allows compliance with two important classical results. These are retrieved when the separation between the points either exceeds the integral length scale or becomes comparable to the Kolmogorov length scale. In the context of decaying grid turbulence, the results correspond to the mean turbulent energy equation and the Batchelor–Townsend equation for the decay of the mean-squared vorticity. Analogous results are obtained when the streamwise inhomogeneity is included in Yaglom’s equation. The form of the inhomogeneous term is illustrated and discussed in the context of measurements in a turbulent grid flow.

An improved method to estimate ionic size for calculating polyatomic ion mobility

A method is described as an alternative to the shadowgraphic method developed by Lin et al. for estimating ionic radii (RI) of protonated molecular ions of polynuclear aromatic hydrocarbons (PAH). These investigators had applied the estimated RI values to calculate the gas phase ion mobility values by means of the Langevin equation. The new method assumes that a neutral molecule and its positive molecular ion can be treated as effective hard spheres of the same radius RI which can be estimated from total molecular volume or total molecular surface area values computed using an algorithm previously described. These RI values are then used to estimate collision cross-sectional areas. The gas phase ion mobilities were calculated utilizing the Einstein–Nernst–Townsend equation as well as the Langevin equation. The ion mobility values so calculated compare reasonably well with experimental values for the 21 PAH previously used by Lin et al.

A study of electrostatic precipitation processes — calculation of electric fields due to a single unipolar source

A predictive model of the electrical conditions in an electrostatic precipitator under steady unipolar operating conditions has been developed to facilitate calculation of electric fields in a precipitator. Electrode geometries are not restricted provided that charging electrode possesses twofold mirror symmetry; the procedure adopted is easily extended to more general electrode designs. The model assumes a conservative current density field and presumes the availability of the Laplacian potential. A procedure has been designed to modify this potential to account for the effects of space charge. The model predicts accumulation of charge density near the charging electrode and the onset of saturation at high operating currents. The analysis of the results yields a voltage-ampere relationship which is comparable with the form of the Townsend equation for corona currents.

Electrical ionization and breakdown in gaseous ethylene derivatives and in isoprene

Townsend primary and secondary ionization coefficients have been measured and sparking voltages deduced for di- and tri-methyl ethylene and for isoprene over a pressure range of 2 to 200 Torr and a parallel plate gap distance up to 1 cm yielding derived sparking voltages in excess of 500 000 V. The Townsend equation for ionization is found to be closely obeyed by the gases and vapours and the Townsend constants have been derived. A detailed analysis of the experimental results shows that progressive methyl substitution around the ethylene double bond influeneces the electrical characteristics less markedly than straight chain substitution, in line with further electrical data taken from other sources on hydrocarbons.

Electrical ionization and breakdown of mono- and cyclo- olefin vapours

Townsend primary and secondary ionization coefficients have been measured and spark voltages deduced for pentene-1, hexene-1 and their cyclo-derivatives up to their respective saturation vapour pressure, corresponding to 48  E/po  2000 V cm–1 torr–1 at 0°C and voltages up to 500 kV. The Townsend equation α/po = A exp (— Bpo/E) is found to bo oboyed over an α/po range of 1000 to 1 and the constants A and B are deduced. This allows the cyclo-olefins to bo correctly categorized in relation to their straight chain counterparts. Giant avalanches in the region of exp (αd) = 108 have again been measured. A short historical survey of this discovery is given together with a. novel application.

Electrical ionization and breakdown of paraffin vapours

Townsend primary and secondary utilization coefficients have been measured and sparking voltages deduced for n-pentane, n-hexane and n-heptane vapours up to their respective saturation vapour pressures, corresponding to 54 ≤ E/ p 0 ≤ 2000 and voltages up to 500 kV. The Townsend equation α /p 0 = A exp ( −Bp 0/E) is found to be obeyed progressively near the Paschen minimum over an α/p 0 range of 300 to 1 and the constants A and B together with the sparking voltages increases systematically with increase in molecular weight. Again large electron avalanches exceeding a multiplication factor of 108 have been observed. The sparking voltage of n-hexane vapour at p 0 d = 10,000 Torr cm at 0°C is 1.75 times that of air.

Ionization and breakdown in ethylene, propylene and butene-1

Townsend primary and secondary ionization coefficients have been measured and sparking voltages deduced for ethylene, propylene and butene-1 over the pressure range 1 to 400 torr, corresponding to 36 ≲ E/p 0 ≲ 2000 and voltages up to 500 kV. Townsend's equation, α/p 0 = A exp (— Bp 0/E), is found to be obeyed over the region 36 ≲ E/p 0 ≲ 400 corresponding to a range of α/p 0 of ratio 1000 : 1, and the constants A and B are found to increase systematically with increase in molecular weight of the gases tested. Large electron avalanches exceeding exp (αd) = 108 have been measured, The primary and secondary ionization coefficients decrease, and the sparking voltage increases, with increases in molecular weight.

Prebreakdown current growth and the ionization coefficients in hydrogen

The prebreakdown current growth between gold-plated parallel plane electrodes in hydrogens is measured for 76 Td<E/N<576Td and analysed by fitting the Townsend equation and the Lucas equation. The results suggest that the current growth for E/N<or approximately=310Td, where the photon ( delta p) and positive ion ( gamma i) secondary actions are appreciable, is represented better by the Lucas equation. At higher E/N, the growth becomes equally well represented by both equations because the gamma i action tends to prevail over the delta p.

Ionization coefficients and sparking voltages from methane to butane

Townsend primary (α/p 0)T and secondary ionization coefficients have been measured and sparking voltages have been deduced for methane, ethane, propane and butane over the pressure range 1 to 400 torr corresponding to 26 ≲ E/p 0≲2000 and voltages up to 500 kV. Townsend's equation α/p 0 = A exp ( - Bp o/E) is found to be obeyed over the region 30≲E /p 0≲670 corresponding to a range of α/p 0 of ratio 1000: 1 and the constants A and B increase systematically with increase in molecular weight of the gases tested. Large electron avalanches have been measured exceeding exp (ad) = 108. The secondary ionization coefficient decreases, whilst the sparking voltage increases, with increase in molecular weight.

Hypothetical mechanism of the helium detector : Operation in the positive and linear modes

On the basis of the Townsend equation for the gas amplified current and of the Dutton—Powell equation for the enhancement of the primary ionization coefficient in gas mixtures due to the Penning effect, a mathematical model of the chromatographic helium detector is proposed. Because both calculations of the relative current increase for E/ p = 4, 5 and 6 V/em·torr and actual measurements lead to linear calibration graphs for the sample calibrations at the parts per million level, a simplified and linearized relationship was developed, although further experimental confirmation is needed. The suggestion is made that the derived equation will make possible measurements of the 0 number of metastable states produced per electron per centimetre of drift in an electric field, and the cross-sections for the Penning effect. The rate constants for Penning ionization by triplet metastable atoms of helium for hydrogen sulphide and ethylene were evaluated but the results are of a relative character. The shortcomings of the model are considered: the lack of explanation of negative peaks, the role of specially added admixtures to the carrier gas, and no account of recombination phenomena are the most important. A brief scheme of necessary investigations is outlined.

The mechanism of breakdown of ambient air in long uniform-field gaps

Measurements of pre-breakdown ionization currents in uniform-field gaps in filtered atmospheric air have been undertaken for values of the parameter pd (pressure × gap length) up to 10 500 Torr cm, corresponding to a maximum breakdown voltage of 350 kV. Over the whole range investigated, the spatial growth of ionization in the gaps was found to be in accordance with the generalized Townsend equation, the main source of secondary electrons being due to the photoelectric effect at the cathode. For given values of Es/p, where Es is the breakdown field stress, there is good agreement between the values of ds (sparking distance) obtained by experiment and those predicted by the Townsend breakdown criterion.

The spatial growth of ionization currents in nitrogen at voltages up to 500 kV

Accurate measurements of the spatial growth of pre-breakdown ionization currents in nitrogen under uniform field conditions were made for values of E/N in the range from 8.48*10-16 to 15.2*10-16 v cm2 and 0.825*1019<or=N<or=8.25*1019 cm-3 (E is the electric field and N the gas number density). Each set of measured currents at a constant value of E/N and N was found to satisfy the generalized Townsend equation, indicating that the spatial growth of current resulted from the action of primary and secondary ionization processes. Analysis of the measurements gave values of the primary ionization coefficient alpha and the generalized secondary ionization coefficient omega / alpha . It was found that alpha /N and omega / alpha are dependent on E/N but not on N. A quantitative analysis of the observed increase of omega / alpha with E/N led to the conclusion that the predominant secondary ionization process was that of photoelectric emission from the cathode. That cathode processes are significant was confirmed by the fact that a spark between clean electrodes led to a considerable reduction in the value of omega / alpha .