Abstract

This document presents a study on the contribution of the electrostatic force in the generation of thrust using an asymmetric electrodes configuration subjected to a high potential differential. The study uses computational models to provide an optimized configuration which generates 296.99% higher levels of electrostatic force when compared to a baseline configuration. Experimental testing of a system having comparable geometries and potential differentials to the computational models are conducted. The computational models are then compared to empirical results from the experimental tests. Results show a decreased in thrust with an increase of electrostatic force down to 61.58%. The discrepancy between the computational and experimental results is investigated by conducting a third experiment having an experimental set-up with multiple independent structural supports. The use of multiple independent supports shows and increase of internal forces of 69.23%. It is then concluded that the optimized experimental set-up increases the electrostatic repulsive forces between electrodes having common electric charges, but it does not translate into an increase in thrust. Instead, the increase in electrostatic force using the electrodes configurations used in this study shows a deleterious effect in thrust levels generated using an asymmetric capacitor configuration.

1. Introduction

This document presents a study on the contribution of the electrostatic force in the generation of thrust using an asymmetric electrodes configuration. The study combines results from computational models results and empirical data to establish a correlation between the mechanisms responsible for generating thrust in asymmetrical electrodes subjected to a high potential differential. Since its discovery in 1928 ¹ by Biefeld-Brown, the subject of propulsion thru the generation of an electrohydrodynamic (EHD) flow from a corona discharge has remained a subject of high interest and extensive research in the fields of physics and engineering ¹.

Understanding of the mechanisms and physics which govern EHD propulsion create new opportunities for its further application in green energy thrusters. The historical progression of EHD research has been obscure, from its erroneous link to creating an anti-gravity phenomenon, to its research deemed classified by the US Army, and the proliferation of unfounded ideas by conspiracy theorists ². Over the last three decades, EHD research has become increasingly more scholastic, with sound physics principles and mathematical models which have corrected the path toward EHD’s meaningful discoveries.

The primary focus of this work is replicating a previous computational model, which capture the EHD phenomenon, to then attempting validate the model with empirical data from experimental tests. From the literature review conducted by the authors, the work by Martins and Pinheiro ³ was selected to replicate their computational model. A deep dive into the role of electrostatic force in the EHD phenomenon is presented to evaluate the conclusions drawn by Martins and Pinheiro ³:

“The generated ion wind is a reaction to the electrostatic thrust mechanism and not the cause of the thrusting force as it is usually conceived. The main thrust force on the electrodes is electrostatic, not hydrodynamic”.

2. Methodology

Using the computational procedure set forth by Martins and Pinherio ³ and using the COMSOL 5.4 Multiphysics software, a numerical computational model is generated for a case where a pair of asymmetric electrodes are subjected to a direct current (DC) high potential differential.

The geometry selected are the cross sections of a pair of asymmetric cylindrical electrodes (i.e., wire and ground electrode) with a pre-defined distance apart and nitrogen as the fluid medium (Figure 1) ⁴.

Once the computational model is successfully duplicated (named “Baseline”), different arrangement and number of electrodes are modeled to determine an electrodes configuration that provides large electrostatic forces. If electrostatic forces are the main contributor of EHD propulsion, this “optimized” model should provide higher thrust when compared to the baseline model.

After gathering the computational results, experimental testing is conducted using a comparable “Baseline” test set-up and two “Optimized” test set-ups. Details on the testing configurations are provide in later sections of this document. Figure 2 shows a flow diagram of the methodology used.

To generate the computational model, the following COMSOL 5.4 modules are used:

1. Steady State incompressible Navier-Stokes: used to compute the fluid dynamic equations.

2. The Electrostatic Module is used to calculate the electric potential distribution and the electrostatic forces.

3. The PDE module (using coefficient form) is used to calculate the results of the charge transport equation.

Continuing with the procedure used by Martins an Pinheiro ³, there are two distinctive zones during corona discharges: the ionization zone and the drift zone each one with their additional parameters in the numerical model. Variables and their corresponding parameters used in the simulations are presented in the Appendix section.

For the formulation of the computational model, all nomenclature and their corresponding values can be found in the Appendix section of this document. The model assumes that ionization only occurs at a prescribed distance from the top electrode (anode). Using Kaptsov hypothesis ³ and Peek’s empirical formula ³, the electric field strength Ep for a smooth cylindrical electrode of radius r_c is calculated:

(1)

the ionization zone boundary can be calculated using:

(2)

The voltage (Vi) at the boundary of the ionization zone is calculated by:

(3)

The mobility µ_i of an ion can be calculated as the ratio of the velocity v achieved by an ion by the electric filed E where it travels:

(4)

Using Moseley ³ approach with a pressure of 1 atm and gas temperature of 300 K:

(5)

The ion diffusion coefficient Di can be approximated by the Einstein relation:

(6)

Outside the ionization zone, the model assumes that only elastic collisions occur where no additional ions are generated. On this outside zone named “drift” or transport zone, the electric field vector is given by:

(7)

Using Gauss’s Law:

(8)

The electric potential V is obtained by solving the Poisson equation. The total volume ionic current density Ji created by the space charge drift is given by:

is calculated using:

(9)

Where μ_i is the mobility of ions in the nitrogen gas subject to an electric field, u is the gas (nitrogen neutrals) velocity and Di is the ion diffusion coefficient. The current density satisfies the charge conservation (continuity) equation:

(10)

When applied to a DC problem, in steady state conditions, it simplifies to:

(11)

The hydrodynamic mass continuity equation for the nitrogen neutrals is given by ⁵:

(12)

If the nitrogen fluid density ρ_f is constant, like in incompressible fluids, then it reduces to ⁶:

(13)

Approximating nitrogen as incompressible and it must satisfy the Navier-Stokes’ equation ⁷:

(14)

Since the discharge is DC, the electrical force density on the nitrogen ions that is transferred to the neutral gas is:

(15)

Using the current density definition into the current continuity, the charge transport equation is obtained:

(16)

Since the fluid assumed incompressible the equation reduces to:

(17)

Figure 3 shows the surface electric potential distribution with a maximum value of 28,000 volts at the surface of the top electrode. Figure 4 shows the electrostatic forces generated at the top and bottom electrode represented by the profiles of the Maxwell Surface Stress Tensor. Table 1 shows the electrostatic forces in the vertical (y-axis) direction calculated in the computational model.

Following the successful simulation using the electrodes configuration presented by Martins and Pinheiro ³, changes on the number of electrodes and distances between electrodes were modeled to increase the asymmetric electrostatic force between them. The configuration which showed generating the largest electrostatic force consist of three top electrodes positioned in a triangular arrangement with one bottom electrode (Figure 5). This configuration required adding a dielectric element between the lowermost top electrode and the bottom electrode. The dielectric element serves as a barrier to prevent breakdown between these elements since their proximity exceeds the dielectric strength of the medium (Nitrogen). The dielectric material used is poly-tetrafluoro ethylene with a dielectric constant of 2.99 X 10⁶ V/m. Figure 5 shows the optimized electrodes configuration and dielectric element and the corresponding Maxwell Surface Stress Tensors. The electrodes have the same diameters to the simulation shown in Figure 4. Table 2 shows the coordinate location of the electrodes and the dielectric element used in the computational simulation. Table 3 shows the forces along the y-axis of the optimized electrodes and dielectric configuration. Table 4 shows a comparison between the two computational models. The model with the optimized electrodes and dielectric element shows an increase in the total electrostatic force of 296.99%.

As opposed to the computational models, which were conducted using Nitrogen as medium, the experimental tests were conducted using atmospheric air as medium. A correction factor of 0.788 Mol N₂ to 1 .000 Mol to account for the difference of the molecular mass between the two fluids.

For all cases, the top electrode material was copper wire of 2.5 X 10-4 m radius, and the bottom electrode was aluminum foil with a thickness of 2.5 X 10-5 m formed around a foam core with a round top edge. The total height h of the aluminum foil formed onto the foam core is 2.54 X 10-2 m. The distance between the top and bottom electrodes is per the distances used in the computational models as nitrogen and air have comparable dielectric strengths ⁸. The length of the electrodes was 0.305 m.

A total of three configurations were tested:

Baseline: one top (anode) and one bottom (cathode) electrode (shown in Figure 5). Using this configuration, upward force measurements were recorded and compared to the computational results shown in Table 1.

Case 1: Three top electrodes (anodes) and one bottom (cathode) electrode (shown in Figure 8) with the center anode supported to a secondary structure. This configuration allows recording the repulsive electrostatic forces between the anodes.

Case 2: Three top electrodes (anodes) and one bottom (cathode) electrode (shown in Figure 9). This configuration measures the thrust (upward force) of the three electrodes together obviating the electrostatic forces between the electrodes.

For all cases, upward force measurements were recorded to evaluate the contribution of the electrostatic forces. The results are compared with the computational results shown in Table 4.

Figure 6 shows a sketch of the cross section of the electrode configuration of the experimental set-up for the base configuration. The electrodes are connected to a high voltage power supply. The high voltage supply used was an AHVAC30KVR5MABT 30KV, 0.5 milli-amps from Analog Technologies. The high voltage terminal of the power supply was connected to the top electrode (anode) and the ground was connected to the bottom electrode (cathode). The electrodes were fixed to a single structure which was placed onto a scale. The scale has an accuracy of 0.01 grams and a maximum capacity of 3,000 grams. For the experimental set-up, the scale captures vertical changes in the weight of the electrodes and structure resulting from thrust generation. Figure 6 and Figure 7 show the experimental set-up used for the baseline configuration.

The voltage input was increased from 10 KV to 30 KV in 1 KV intervals. For each data point, reading from the scale were recorded and the thrust in Newtons was calculated by multiplying the recorded grams by gravity (9.81 m/s²)/1,000.

Similar to the configuration used in the computational models, the experimental set-up consisted of three top electrodes (anodes) and one bottom electrode (cathode). As shown in Figure 4, anode #3 has a dielectric material placed between itself and the bottom electrode (cathode). The dielectric material used in the experimental set-up was poly-tetrafluoro ethylene. The voltage input was increased from 10 KV to 30 KV in 1 KV intervals. For each data point, reading from the scale was recorded and the upward force in Newtons was then calculated by multiplying the grams by gravity (9.81 m/s2)/1,000.

Two cases were experimentally evaluated for the optimized geometry configuration. For both cases, the anodes A, B, C are located at the top of the setup and cathode D is located at the bottom of the setup. A dielectric is placed at the bottom of anode C to prevent exceeding the dielectric strength of air between anode C and cathode D.

Case 1: Anode C and the dielectric barrier supported by a secondary Structure 2. Figure 8 shows the FBD for Case 1.

Case 2: All Anodes and Cathode are supported by a single Structure 3. Figure 9 shows the FBD for Case 2.

For all experiments, the weight of the structures m_stucture1*g and m_stucture2*g were not included as the scale was tared to zero prior to conducting each experiment. The thrust and/or contribution of the electrostatic forces is measured as the net force in the upward direction registered by the scale. For Case 1, the equilibrium condition on the y-axis can be determined by:

(18)

(19)

(20)

(21)

For Case 2, the equilibrium condition on the y-axis can be determined by:

(22)

(23)

Where:

is the sum of forces applied to the structure 1 in the y-direction.

is the electrostatic force in the y-direction between electrodes A and C.

is the electrostatic force in the y-direction between electrodes A and D.

is the electrostatic force in the y-direction between electrodes B and C.

is the electrostatic force in the y-direction between electrodes B and D.

is the electrostatic force in the y-direction between electrodes D and A.

is the electrostatic force in the y-direction between electrodes D and B.

is the electrostatic force in the y-direction between electrodes D and C.

is the force applied to the scale by the structure in the y-direction.

is the vertical thrust generated by EHD propulsion in the y-direction.

is the sum of forces applied to the structure 2 in the y-axis.

is the electrostatic force in the y-direction between electrodes C and A.

is the electrostatic force in the y-direction between electrodes C and B.

is the electrostatic force in the y-direction between electrodes C and D.

is the right side reaction force of the table onto the Structure 2 in the y-direction.

is the left side reaction force of the table onto the Structure 2 in the y-direction.

is the sum of forces applied to the structure 3 in the y-direction.

3. Results

For each case, tests were conducted increasing the voltage input from 10 KV to 30 KV in 1 KV increments. Each datapoint consists of one reading from the scale at a corresponding voltage. The data from the scale was recorded in grams and was then multiplied by gravity (9.81 m/s2) and divided by 1 gram per 1,000 Kg to obtain the upward force in Newtons.

The results of the baseline configuration correspond to force upward resulting from thrust generated by the electrodes since a single structure holds all electrodes together and single point of contact exist with the scale.

The results of Case 1 for the optimized configuration include thrust generated by the system and the repulsive electrostatic forces between the two top anodes and one bottom anode. The repulsive force is transferred from Structure 2 to Structure 1 by the reaction forces between Support 1 and Support 2 with the table.

Results for the Case 2 for the optimized configuration correspond to force upward solei from the upward thrust generated by the electrodes as the configuration has a single point of support with the scale.

For the baseline, the test was limited to 26 KV since breakdown was observed between the top and bottom electrode. The maximum force recorded at this voltage was 2.86 X 10^-0.2 Newtons Table 5.

The maximum upward force recorded for Case 1 was 6.29 X 10^-2 Newtons at 30 KV. Also, the upward force recorded at 26 KV was 4.84 X 10^-2 Newtons.

The maximum upward force recorded for Case 2 was 2.55 X 10^-2 Newtons at 30 KV of input voltage and the upward force recorded at 26 KV input was 1.77 X 10^-2 Newtons.

Figure 10 shows the experimental results of the thrust recorded for the baseline configuration, optimized configuration with supports, and optimized configuration without supports.

4. Conclusions

The experimental results show the Case 1 of the optimized configuration generated the largest thrust of all experiments conducted with a recoded value of 6.29 X10^-2 at a 30 KV input. Figure 10 shows this configuration also recorded a value of 5.55 X10^-2 Newtons at 28 KV input. The computational model for an input voltage of 28 KV shows an upward force 1.88X10-1 N/m (Table 4). Applying the correction factor (0.788 Mol N₂ to 1 .000 Mol of air) and length of the electrode (0.305 m) used in the test set-up, the computational model yields an upward force of 7.28 X 10^-2 Newtons. The computational model shows a discrepancy of 31% when compared to the experimental results.

Among the sources of the discrepancy between the computational and experimental results is the fact that nitrogen gas at atmospheric conditions only forms positive ions ⁹. The plasma chemistry of air is complex, and the exited states may have hundreds of reactions ⁵. Air having a composition made from mainly oxygen (which is electronegative) and nitrogen (which is electropositive) ⁵ requires further considerations in applying a correction factor when comparing the computational and experimental results. Additionally, the computational model only accounts for ions created in the ionization zone but omits ions created in the transport regions resulting from secondary ionization ¹⁰.

When comparing the three experimental tests conducted, Case 1 shows the highest upward force and Case 2 shows the lowest upward force, with the Baseline values falling consistently between them for the voltages applied (Figure 10). Comparison of the upward forces generated by the three experimental results at 26 KV, provides evidence of the contribution of the electrostatic force in the generation of EHD propulsion in air at atmospheric conditions (Table 5).

Comparing the upward force of the Baseline to Case 1, it shows in increase of upward force of 69.23%. Comparing the Baseline to Case 2, shows a decrease in upward force of 61.58%. Case 1 and Case 2 have the same electrode geometry and distances from each other but showed the largest difference in upward force with a percentage value of 253.11%.

It can be concluded that the Case 1 and Case 2 configurations produced larger electrostatic forces between the electrodes when compared to the Baseline. This is consistent with the optimized computational model using three anodes and one cathode. However, the increase in electrostatic force does not translate into an increase of EHD propulsion. The large increase in upward force observed in Case 1 is attributed to the repulsive electrostatic forces between electrodes A+ B+ and C+ (Figure 8) which are translated to the additional supports (Support1 and Support2), thus generating the increase in upward force. As results of Case 2 includes upward force generated by EHD propulsion and the repulsive forces between the electrodes, the increase in upward force is attributed to Newton’s second law and not to EHD propulsion.

The decrease in upward force observed when comparing Case 2 to the Baseline, shows that an increase in the electrostatic forces between asymmetrical electrodes may be deleterious to generating EHD propulsion. For the electrodes configuration used in this study, electrostatic force is not the main mechanism for generation of EHD propulsion.

Further investigation is needed to determine if, under a specific configuration, electrostatic forces may contribute to increase the EHD thrust generated by asymmetrical electrodes configuration. A deeper dive into the properties of plasma chemistry of air and the creation and/or destruction of ions occurring during EHD propulsion may contribute to create a computational model that can more accurately weigh the contribution of electrostatic forces in EHD propulsion.

References

Appendix