Calculation and measurement of a neutral air flow velocity impacting a high voltage capacitor with asymmetrical electrodes Open Access

The phenomenon of the force originating on the high voltage asymmetrical capacitors is known for more than 80 years. There even has been a growing interest in this so called Biefeld-Brown effect in the recent years.¹ However the detailed description of all interesting aspects of this phenomenon is missing. One very important aspect is the existence of the air flow between the electrodes. The hypothesis about the origin of this air flow was already proposed by the authors of this article in,² but the full mathematical description–especially the relation between the current and the velocity of the flow–was never made.

The main purpose of this article is to make such a description and compare the calculated values with the experimentally obtained results, which were measured on a model capacitor using the Particle Image Velocimetry (PIV) method.

This phenomenon, commonly called the Biefeld-Brown effect after its discoverers, is based on the motion of charged particles through an electric field between two asymmetrical electrodes interacting on the way with a neutral medium. High DC voltage (>10 kV) is applied to one of the electrodes (usually the smaller one), while the other is grounded. In our experimental setup we used an extremely thin wire (diameter of 0,1 mm) as the smaller electrode, while the larger consisted of an aluminium foil coated styrofoam box (see Fig. 1). The distance between electrodes was 30 mm. This system of two highly asymmetrical electrodes used for the experiments with the Biefeld-Brown effect is called an asymmetrical capacitor.

A respected researcher Thomas B. Bahder has in his report for the U.S. Army Research Laboratory in 2003³ considered one of two phenomenons–either ion wind or ion drift–as the cause of the generated force. He has attempted to prove or disprove one of them as a cause by computing the theoretical values of generated force using either of the effects. To quote him: “Ionic wind is a ballistic flow of charges from one electrode to the other. Clearly the force due to ionic wind is at least three orders of magnitude too small to account for the observed force on an asymmetric capacitor (in air). There is another type of classical transport: drift of charge carriers in an electric field. In the case of drift, the carriers do not have ballistic trajectories, instead they experience collisions on their paths between electrodes. However, due to the presence of an electric field, the carriers have a net motion toward the opposite electrode. This type of transport picture is more accurate (than ballistic ionic wind) for a capacitor whose gap contains air.”³ Further Bahder gives a macroscopic thermodynamic analysis of the effect stating: “Biefeld-Brown force, generated on an asymmetric capacitor, can be described by the thermodynamics of a fluid dielectric in an external electric field produced by charged conductors. The (partially ionized) air between capacitor electrodes is the fluid dielectric.”³ 

The authors of this paper reached similar conclusions to Bahder. Based on several experiments we see the ion drift as the most likely cause for the generated force. The high electric field around the charged thin wire electrode causes ionisation of the surrounding air. This produces a large number of charged particles, which move towards the grounded larger electrode and on the way they interact with the surrounding medium. As they are in the presence of the electric field between the electrodes, which drives them to the grounded electrode, their collisions with the neutral molecules of the surrounding air is returned back as momentum onto the capacitor. As these collisions happen approximately 10⁹-times a second, a continuous force in the direction of the smaller electrode is generated.

The presence of ionized particles of air generated around the thin wire electrode can be proven by observing corona discharge around the wire. The following image, Fig. 2, was taken using a corona visualizing camera (UVIRCO). Since the image was taken in darkness the outlines of the asymmetrical capacitor were added later to make the image more comprehensible.

The details of this phenomenon were already described in our previously published paper⁴ and as a detailed mathematical description will not be necessary here, it will not be included. Suffice it to say that a formula was derived, which describes the value of the mechanical force:

where I is the electric current flowing through the capacitor, d is the distance between the electrodes and k represents the ion mobility coefficient.

The mechanical force generated by the asymmetrical capacitor is relatively small (ranging in milinewtons). If we consider the presence of high voltage, the force is also difficult to effectively measure. Thus a suitable measuring technique had to be used. In previous dealings with this phenomenon we measured the force by setting the capacitor on a precise digital balance and after applying high voltage, we measured the change in weight of the device.

To protect the balance from the effects of the high voltage, the capacitor was set onto a grounded styrofoam stand which in turn stood on the balance (see Fig. 3).

Following this method of measurement, we obtained a set of values of generated force and current at several voltage settings ranging from 8 kV to 16 kV.

As was said in Basic Theory the phenomenon is based on the motion of charged particles through the air from one electrode to the other. The collisions of the charged particles with the neutral molecules are transferred back to the device through existing electric field as momentum (thus causing the mechanical force F_E). But the collisions also produce a flow of neutral air. In ideal conditions the air can continue on without any resistance and carry the opposite momentum away from the device.

However, we are not working in ideal conditions, in our setup the flow of neutral air partly carries its momentum back to the device thus creating a force F_D with an opposite direction to the force F_E we are trying to generate. That can be represented by a simple formula:

where F_E is the force generated by the motion of ions, F_D represents the force of the neutral air flow affecting the device and F is the resulting force, which we get to measure or use.

From that we can see that the force of the neutral air flow is reducing the resulting force. This fact is known to us and in our experiments we are trying to minimize F_D.

However in this case we will use F_D to ascertain the neutral air flow velocity v for the purposes of both the possibility of using the neutral air flow in itself and also the fact that the value of the velocity will be useful in proving the theory behind the phenomenon.

First we should define the interacting forces. The generated force is described by the formula (1). We can rewrite it like this:

where I is the current flowing between the electrodes, d is the distance between the electrodes and μ₊ is positive ion mobility. Since we are using a setup where the thin wire electrode is connected to positive high voltage and the larger electrode is grounded the main carriers of charge will be positively charged ions. That is why we can use the mobility of positive ions (in air μ₊ = 2.10⁻⁴ m².V⁻¹.s⁻¹).

To describe F_D, the force caused by the neutral air flow, we will use the drag equation used to calculate the drag force experienced by an object due to movement through a fully enclosing fluid:

where ρ is air mass density (in atmospheric pressure ρ = 1,29 kg.m⁻³), S is the surface of our larger electrode perpendicular to the air flow, v is the neutral air velocity, which we are trying to find and C_X is the drag coefficient (for our geometry we can safely assume C_X = 1).

If we combine the formulae (2)–(4) we will get the following:

Because the only unknown value in this formula is the air flow velocity, we can derive:

We have already defined the majority of the variables in this formula. The values for the current I and the measured force F can be obtained from the graph in Fig. 4. We have a choice, which set of values to select, but in this case we are interested in the air flow velocity at voltage U = 16 kV, which corresponds with the last (highest) set of values of current (I = 25 μA) and force (F = 3,4 mN).

The last unknown variable we need to obtain the value of is the surface S. On the device, which we are using to measure the force, this corresponds with the surface on the larger electrode closest to the thin wire electrode. As the larger electrode is 10 cm long (length is parallel with the thin wire electrode) and 1 cm thick the surface is easily obtained as S = 10⁻³ m.

Using all of this we will obtain the value of air flow velocity on our measuring device at the voltage of 16 kV:

There are two main problems with the measurement of the air flow velocity in our experimental setup. The first is the presence of high voltage in the range of tens of kilovolts on the electrodes. This makes it impossible to use any contact methods or any method, which requires an electrical device being present close to either of the electrodes.

The second reason is a rather low air flow velocity (<1 m.s⁻¹). This prevents us from using any of the rougher measuring techniques.

We have taken these two technical limitations into consideration and decided to use the Particle Image Velocimetry (PIV). It is an optical method used for flow visualisation (and measurement). The examined flow is seeded with the tracking particles (in our measurement we have used small droplets of olive oil). The particles are then illuminated using Nd-YAG laser and filmed using a fast digital camera. Since the particles are following the flow faithfully, with the use of appropriate software we are able to visualize the flow (see Fig. 5) and calculate its velocity.

We used this method to visualize the flow in the middle of the length of the capacitor in a plane perpendicular to the capacitor with voltage setting U = 16 kV, so we can compare the results with the calculated value.

The values of velocity measured by the PIV method on the face of the larger electrode are in the range 0,7–0,8 m.s⁻¹. This corresponds well with the calculated value of 0,7366 m.s⁻¹.

The purpose of this article was to describe the air flow velocity around the face of the larger electrode of a high voltage capacitor with asymmetrical electrodes and to compare values of the velocity with those measured by Particle Image Velocimetry.

This has been successfully achieved and there is a satisfactory agreement between the values of the air flow velocity measured by the PIV technique and the value obtained through the formula (6). This proves the formula right and viable to be used in further experimentation.

Because we have successfully compared on one hand a method directed strictly at measuring flow velocities regardless of origins and on the other hand a formula, which is partially based on several assumptions from electrohydrodynamics this paper has another contribution. It has given us yet another proof of the assumptions concerning the effect's principle and the ion origin of the generated force.

This work was supported by Centre for nanomaterials, advanced technologies and innovation CZ.1.05/2.1.00/01.0005 on the Technical University of Liberec.