## Project Type

individual

## Authors' Class Standing

Jacob Engle. Junior

## Lead Presenter's Name

Jacob Engle

## Faculty Mentor Name

Dr. Jeremy Riousset

## Abstract

The ability to protect modern infrastructure as effectively as possible from lightning strikes has become essential with the development of complex electronic, communication, and power systems (Riousset, 2010). In order to determine the most effective geometry of a lightning rod, one must first understand the physical difference between their current designs. Benjamin Franklin’s original theory of sharp tipped rods suggests an increase of local electric field, while Moore et al.’s (2000) studies of rounded tips evince an increased probability of strike (Moore et al., 2000; Gibson et al., 2009). The beginning of the connection process between the descending lightning channel and the upward connecting leader, there is the formation of a “precursor” plasma discharge around the rod in the form of an ionization front (Golde, 1977). In this analysis, the plasma discharge is produced between two electrodes with a high potential difference, resulting in ionization of the neutral gas particle and creating a current in a gas medium. This process, when done at low current and low temperature can create corona and “glow” discharges, which can be observed as a luminescent emission. The Cartesian geometry known as Paschen, or Townsend, theory is particularly well suited to model experimental laboratory scenario, however, it is limited in its applicability to lightning rods. Franklin’s sharp tip and Moore et al.’s (2000) rounded tip fundamentally differ in the radius of curvature of the upper end of the rod. As a first approximation, the rod can be modelled as an equipotential conducting sphere above the ground. Hence, we expand the classic Cartesian geometry into spherical geometries. In a spherical case, a small radius effectively represents a sharp tip rod, while larger, centimeterscale radius represents a rounded, or blunt tip. Experimental investigations of lightning-like discharge are limited in size. They are typically either a few meters in height, or span along the ground to allow the discharge to develop over a large distance. Yet, neither scenarios account for the change in neutral charge density, which conditions the reduced electric field, and therefore hardly reproduce the condition of discharge as it would occur under normal atmospheric conditions (Raiser, 1991). In this work we explore the effects of shifting from the classical parallel plate analysis to spherical geometries more adapted for studies of lightning rods. Utilizing Townsend’s equation for corona discharge, we estimate a critical radius and minimum breakdown voltage that allows ionization of the air around an electrode in air. Additionally, we explore the influence of the gas in which the discharge develops. We use BOLSIG+, a numerical solver for the Boltzmann equation, to calculate Townsend coefficients for CO2-rich atmospheric conditions (Haagler and Pitchford, 2005). This allows us to expand the scope of this study to other planetary bodies such as Mars. We solve the problem both numerically and analytically to present simplified formulas per each geometry and gas mixture. The development of a numerical framework will ultimately let us test the influence of additional parameters such as background ionization, initiation criterion, and charge conservation on the values of the critical radius and minimum breakdown voltage.

## Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?

No

Numerical and analytical studies of critical radius in spherical and cylindrical geometries for corona discharge in air and CO2 -rich atmospheres

The ability to protect modern infrastructure as effectively as possible from lightning strikes has become essential with the development of complex electronic, communication, and power systems (Riousset, 2010). In order to determine the most effective geometry of a lightning rod, one must first understand the physical difference between their current designs. Benjamin Franklin’s original theory of sharp tipped rods suggests an increase of local electric field, while Moore et al.’s (2000) studies of rounded tips evince an increased probability of strike (Moore et al., 2000; Gibson et al., 2009). The beginning of the connection process between the descending lightning channel and the upward connecting leader, there is the formation of a “precursor” plasma discharge around the rod in the form of an ionization front (Golde, 1977). In this analysis, the plasma discharge is produced between two electrodes with a high potential difference, resulting in ionization of the neutral gas particle and creating a current in a gas medium. This process, when done at low current and low temperature can create corona and “glow” discharges, which can be observed as a luminescent emission. The Cartesian geometry known as Paschen, or Townsend, theory is particularly well suited to model experimental laboratory scenario, however, it is limited in its applicability to lightning rods. Franklin’s sharp tip and Moore et al.’s (2000) rounded tip fundamentally differ in the radius of curvature of the upper end of the rod. As a first approximation, the rod can be modelled as an equipotential conducting sphere above the ground. Hence, we expand the classic Cartesian geometry into spherical geometries. In a spherical case, a small radius effectively represents a sharp tip rod, while larger, centimeterscale radius represents a rounded, or blunt tip. Experimental investigations of lightning-like discharge are limited in size. They are typically either a few meters in height, or span along the ground to allow the discharge to develop over a large distance. Yet, neither scenarios account for the change in neutral charge density, which conditions the reduced electric field, and therefore hardly reproduce the condition of discharge as it would occur under normal atmospheric conditions (Raiser, 1991). In this work we explore the effects of shifting from the classical parallel plate analysis to spherical geometries more adapted for studies of lightning rods. Utilizing Townsend’s equation for corona discharge, we estimate a critical radius and minimum breakdown voltage that allows ionization of the air around an electrode in air. Additionally, we explore the influence of the gas in which the discharge develops. We use BOLSIG+, a numerical solver for the Boltzmann equation, to calculate Townsend coefficients for CO2-rich atmospheric conditions (Haagler and Pitchford, 2005). This allows us to expand the scope of this study to other planetary bodies such as Mars. We solve the problem both numerically and analytically to present simplified formulas per each geometry and gas mixture. The development of a numerical framework will ultimately let us test the influence of additional parameters such as background ionization, initiation criterion, and charge conservation on the values of the critical radius and minimum breakdown voltage.