The following thoughts are considerations one would need to think about to develop an autonomous flying machine, utilising EHD thrusters as its main propulsion system. It has been mathematically shown that even for low performance EHD thrusters, type Lifter V1, an autonomous condition can be possibly engineered.
High performance EHD thrusters are made up of modular shapes of high performance EHD thruster cells as described in my EHD Thrusters Collection. This project will go into all details required to fully develop a complete flying machine. Knowledge about various other sciences has to be gained in order to go from a simple cell to such a stable flying machine.
Electric propulsion involves the acceleration of ionised air, or other onboard gases, through the application of an electric field. It also represents the only practical method for rapid interplanetary travel currently known to mankind. While conventional rockets produce large amounts of thrust through the highly inefficient process of chemical combustion, such high thrust levels are only important for rapid planetary lift-off scenarios. A much more important characteristic for interplanetary travel is a quantity called specific impulse. Specific impulse is essentially an indication of how much a rocket's velocity can be increased by using a given quantity of fuel. Electrically propelled spacecraft generally possess very high values of specific impulse, allowing them to travel 10 to 100 times faster than chemical rockets with the same amount of fuel.
Formidable engineering challenges must still be addressed before the advantages of electric thrusters can be fully exploited. Most notably, thrust levels must be increased and high voltage power supply weights must be reduced. While extensive research was performed in the 1950's and 1960's on the use of electric propulsion for interplanetary spaceflight, many promising concepts had to be abandoned due to the technological limitations of the power conditioning systems in use at the time. To date no consistent effort has been made to re-evaluate these approaches in light of modern power processing technologies. In fact, research into the development of electric thrusters for interplanetary travel has been largely abandoned by NASA and the scientific community since the late 1960's. It was at that time that the major focus of electric propulsion shifted from interplanetary missions to low-power near-earth applications such as orbit stabilisation and directional control. This change arose when plans for large space power systems were abandoned in favour of the gradual development of large solar panels.
Many of the analytical problems that could not be solved with the computer technology available in the 1960's are now tractable using modern computer modelling techniques and hardware, whilst the technological limitations on hardware are diminishing and lend themselves to the possibility of constructing a fully autonomous EHD thruster.
As with all aerotype vehicles, the drag coefficient of the craft is of primary importance.
This diagram shows various drag coefficients for different shapes having the same frontal area. As shown, the airfoil shape has the lowest value.
The choice of the best shape depends mainly on how the machine is going to travel and how fast it will go. For low speed applications, such as method of transport (at a fraction of a Mach) that works well in all directions in the horizontal plane, the front-to-front airfoil shapes works best, with its rounded edges on both ends. For transonic/supersonic flights, a very thin shape with sharp edges as in the back to back design is the best.
The terminal velocity of the flying machine in air is given by:
v=sqrt(2F/pACd) ...F=accelerating force, p=density of air, A=cross sectional area in the direction of travel, Cd= drag coefficient for its shape
The physics of a frisbee has to date been only applied to toys, but in effect it stays aloft thanks to the same aerodynamic lift that keeps the most advanced jet flying in air, and has the advantage of gyroscopic stability.
Forward flight splits rushing air at the disk's leading edge: half goes over the curved upper side of the frisbee; half goes under the flat side. Because that edge is tipped up at an 'angle of attack', the disk deflects the lower airstream downward. As the Frisbee pushes down on the air, the air pushes upward on the Frisbee - a force known as aerodynamic lift. The upper airstream is also deflected downward. Like all viscous fluids, flowing air tends to follow curving surfaces, even when those surfaces bend away from the airstream. The inward bend of the upper airstream is accompanied by a substantial drop in air pressure just above the Frisbee, sucking it upward. In this way, forward motion is translated into vertical thrust.
Limits to the airstream's ability to follow a surface explain why a Frisbee flies so poorly upside down. When the upper airstream tries to follow the sharp curve of an inverted Frisbee's hand grip, its inertia breaks it away from the surface. A swirling air pocket forms behind the Frisbee and destroys the suction, raising the air resistance. Once this air resistance has sapped the inverted disk's forward momentum, it drops like a rock. Players can take advantage of this effect in a hard-to-catch throw called the hammer.
It is a true fact that frisbees do not fly if they are not rotating, but rotation by itself does not produce vertical thrust. This can be easily confirmed by noting that you can spin a frisbee in place, about a fixed axis, and it will not 'lift up'. You also can see that the frisbee is still spinning at nearly 'full speed' when it finally hits the ground, so you have another piece of evidence that shows that the spinning doesn't lift the frisbee. Rotation is crucial. Without it, even an upright Frisbee would flutter and tumble like a falling leaf, because the aerodynamic forces aren't perfectly centered. Indeed, the lift is often slightly stronger on the forward half of the Frisbee, and so that half usually rises, causing the Frisbee to flip over. A spinning Frisbee, though, can maintain its orientation for a long time because it has angular momentum, which dramatically changes the way it responds to aerodynamic twists, or torques. The careful design of the Frisbee places its lift almost perfectly at its center. The disk is thicker at its edges, maximizing its angular momentum when it spins. And the tiny ridges on the Frisbee's top surface introduce microscopic turbulence into the layer of air just above the label. Oddly enough, this turbulence helps to keep the upper airstream attached to the Frisbee, thereby allowing it to travel farther.
Rotating systems exhibit some behaviour that appears strange when we apply our intuition, developed for linear motion. The motion of the gyroscope shown here is an example. The silver colored rotor inside the gimbal rings is spinning rapidly about its axis. As we will see, it is the presence of the little weight, hung on one end of the axis of rotation of the spinning rotor, that causes the observed motion about the vertical axis. Our intuition might suggest that the weighted end of the rotor's axis would move down, not go around. Gyroscopic precession is a phenomenon occurring in rotating bodies in which an applied force is manifested 90 degrees later in the direction of rotation from where the force was applied.
If wp is the angular frequency of the axis of rotation, called the precessional frequency:
wp = (Frsina)/Iw
w is the angular velocity of rotation
wp is the precessional frequency
F is the external influence force acting of the gyro
r is the distance at which force F is acting from the spinning axis
I is the moment of inertia of the gyro
a is the angle which force F makes with the spinning axis
Although precession is not a dominant force in rotary-wing aerodynamics, it must be reckoned with, because turning rotor systems exhibit some of the characteristics of a gyro. The graphic shows how precession affects the rotor disk when force is applied at a given point:
A downward force applied to a clockwise rotating disk at any point A results in a downward change in disk attitude at point B at an angular distance of 90 degrees, and an upward force applied at Point A results in an upward change in disk attitude at point B. If the disk is rotating anticlockwise, the disk attitudes will result at an angular distance of -90 degrees, say point C. Since points B & C are 180 degrees from each other, that is, on opposite sides of the disk, a clockwise disk will react exactly the opposite way of an anticlockwise disk when an external force is applied.
This behaviour explains some of the fundamental effects occurring during various helicopter manouvers. For example: The helicopter behaves differently when rolling into a right turn than when rolling into a left turn. During the roll into a left turn, the pilot will have to correct for a nose down tendency in order to maintain altitude. This correction is required because precession causes a nose down tendency and because the tilted disk produces less vertical lift to counteract gravity. Conversely, during the roll into a right turn, precession will cause a nose up tendency while the tilted disk will produce less vertical lift. Pilot input required to maintain altitude is significantly different during a right turn than during a left turn, because gyroscopic precession acts in opposite directions for each.
Corona discharge in flying EHD thrusters
It was found that even for low gas velocities (<1 m/s), the current-voltage characteristics of the discharge for positive polarity are modified significantly, and the type of the discharge can be changed by the flow. When the gas velocity is increased from 0 to 4 m/s, the breakdown voltage increases by about 25%.
The time averaged discharge current increases for gas flows from 0 to 0.5 m/s, and next decreases again. This phenomenon is not observed for negative polarity of the corona discharge in this gas velocity range.
The air flow in the interval up to about 1 m/s has no effect on the corona onset voltage. Two negative resistance regimes can be observed in the current - voltage characteristics in flowing air. One is the spark discharge regime, and the second is the arc discharge. Between theses two modes of discharge a narrow range of current exists in which the discharge resistance is positive. The discharge current and the mode of discharge are however unstable in this regime and they can transit easily to each other. The positive resistance occurs for flow velocities lower than 2 m/s, and for positive polarity.
For positive polarity, the voltage at which the corona discharge transits into the spark discharge (breakdown voltage) also increases with the flow velocity increasing. For flow velocity of 4 m/s, this voltage increases by about 25% as compared to the still air.
The gas flow affects in different ways the corona characteristics at negative and positive polarity. At low interelectrode spacing, the negative corona current is conducted mainly by free electrons of high mobility, which cannot be easily removed by the flowing gas from the discharge space. In the case of positive corona, there are only heavy ions in the interelectrode space, which form a stable space charge. For low values of electrical Reynolds number for ions, which is of the order of magnitude of about 0.1 for the velocity of a few m/s, the ions can be removed by the flowing gas, reducing the space charge and increasing locally the electric field. That cause the increase in the discharge current.
Hydrogen, Helium or Vacuum Cells?
One option for cancelling out some of the mass of the vehicle is to use lighter than air gases, such as helium or hydrogen, within the thruster's cells. Another way is to eliminate lifting gas altogether and use vacuum lifting cells. Since vacuum has no mass, it provides even greater buoyancy than hydrogen. The only problem is that you're going to need extreme ultra-tech materials to maintain a vacuum cell under the external atmospheric pressure.
Air density at sea level is about 1.2 kg/m3, so each cubic metre of vacuum gives us 1.2 kgF (kg*g) of lift. This is only about 10% greater than hydrogen, and 20% greater than helium, but with increasing volume this small difference may result in a substantial gain in lifting capacity.
Assume a roughly cylindrical vacuum cell, say 100 m long and 10 m radius, that holds 31,400 m3 of vacuum. That generates 37,700 kgF of lift. Surface area of the cell is 6911 m2, so your vacuum cell structural material is limited to a maximum density of about 5.4 kg/m2.
The density of steel is 7800 kg/m3, so the maximum thickness of steel plate you can use is 0.7 millimetres (1/36"). Obviously this isn't going to be anywhere near strong enough! And it needs to be even thinner if you want any appreciable usable lift (as opposed to lift merely used to keep the vacuum cell itself aloft).
To get an idea of the strength required from an ultra-tech material, the vacuum cell has an air pressure equivalent to just over 10 tonnes on every square metre of its surface. This is like taking a sheet of that 0.7 millimetre thick steel, using it to bridge a 100 metre wide chasm, parking 500 cars on it (you'd have to stack them about 10-high to fit them on), and expecting it to not even *bend* appreciably - let alone collapse.
In order to gain an advantage over helium lifting gas, the structural material required to maintain a vacuum cell must have a mass of less than 0.2 kg per cubic metre of cell volume greater than that required for a helium cell of the same size. (The limit is 0.1 kg per cubic metre for an advantage over hydrogen, but it can easily be argued that safety considerations make hydrogen less desirable!)
I think that realistically you will need extreme ultra-tech materials before you can approach the required lightness, strength, and above all stiffness to maintain a usable vacuum cell. I would hazard the guess that such materials wouldn't be available for a long time, if ever. So, hydrogen cells seem to be more feasable than vacuum ones, and when considering safety, one would better go for Helium LTA's.
In EHD propulsion devices, dissipation of ions as they reach their target electrode can result in power losses of up to 10%. Once neutralised at the target electrode, a new ion cloud has to be generated to continue generating thrust. One way to avoid this power loss is to continously move the target electrode away from the ion cloud on its approach, kind of playing donkey-and-carrot with the ion cloud. Also, from the equation Thrust= id/k, we see that thrust is directly proportional to 'd', the path distance of the ion cloud. One can achieve higher thrust levels, by 'moving the carrot' for a longer distance than the nominal single stage air gap. This can be done electronically by lowering the voltage on the target electrode as the ion cloud approaches, whilst at the same time increasing the voltage on a farther target electrode. One such designed was discussed between myself and Evgenij and works as follows. The design consists of ONE ionising stage, and a multiple number of accelerating electrodes driven by sinewave hv sources to achieve the required effect. When they reach the last electrode, they are finally neutralised to reduce ion imbalance in the surrounding air. Since, each time that the ion cload approaches a target electrode the voltage on that electrode goes down to zero, the ions never get neutralized on their path and so a single ionisation stage may be enough. This also helps reducing power requirements for multiple ionisation stage. The frequency of field modulation should be such that during one period ions fly over the whole number of electrodes. This frequency depends on the ion cloud flight velocity and electrode separation. Ion flight speed in this configuration is quite low (a few m/s) and the frequency required for inter-electrode distances of 30mm, is in the order of 90Hz, which in practice should be tweaked upon current feedback and velocity of the EHD propulsion thruster. Self capacitance of the accelerating electrodes should be kept to minimum to reduce ion build up around them, which can neutralise part of the ion cloud as it passes over them.
Above simulated results show that properly timed hv signal variations to the accelerating electrodes result in continuous forward field experienced by positive and negative ion
clouds generated on first (corona) electrode. The ion cloud never reaches its target within its complete travel from the corona generator till the exit. This is done by timing the phase and frequency with respect to each electrode so that the voltage on each
electrode approaches zero as ion cloud is approaching it. The average field experienced by ion cloud on the flight is much lower compared to a single stage device. Integration of the field shows efficiency increase of about 600%! For more details on this design refer to Evgenij's AC surfing page.
When a fluid is in motion such as in EHD thrusters, it must move in such a way that mass is conserved. To see how mass conservation places restrictions on the velocity field, consider the steady flow of air through a duct (that is, the inlet and outlet flows do not vary with time). The inflow and outflow are one-dimensional, so that the velocity V and density \rho are constant over the area A (shown below).
|One-dimensional duct showing control volume.|
Now we apply the principle of mass conservation. Since there is no flow through the side walls of the duct, what mass comes in over A1 goes out of A2, (the flow is steady so that there is no mass accumulation). Over a short time interval \Delta t,
This is a statement of the principle of mass conservation for a steady, one-dimensional flow, with one inlet and one outlet. This equation is called the continuity equation for steady one-dimensional flow. For a steady flow through a control volume with many inlets and outlets, the net mass flow must be zero, where inflows are negative and outflows are positive.
A streamline is a line that is tangential to the instantaneous velocity direction (velocity is a vector, and it has a magnitude and a direction). To visualize this in a flow, we could imagine the motion of a small marked element of fluid. For example, we could mark a drop of water with fluorescent dye and illuminate it using a laser so that it fluoresces. If we took a short exposure photograph as the drop moves according to the local velocity field (where the exposure needs to be short compared to the time it takes for the velocity to change appreciably), we would see a short streak, with a length V \Delta t, and with a direction tangential to the instantaneous velocity direction. If we mark many drops of water in this way, the streamlines in the flow will become visible. Since the velocity at any point in the flow has a single value (the flow cannot go in more than one direction at the same time), streamlines cannot cross, except at points where the velocity
magnitude is zero, such as at a stagnation point.
There are other ways to make the flow visible. For example, we can trace out the path
followed by our fluorescent drop using a long-exposure photograph. This line is called a
pathline, and it is similar to what you see when you take a long-exposure photograph
of car lights on a freeway at night. It is possible for pathlines to cross, as you can
imagine from the freeway analogy: as a car changes lanes, the pathline traced out by its
lights might cross another pathline traced out by an adjoining vehicle at a different
Another way to visualize flow patterns is by streaklines. A streakline is the line traced out by all the particles that passed through a particular point at some earlier time. For instance, if we issued fluorescent dye continuously from a fixed point, the dye makes up a streakline as it passes downstream. To continue the freeway analogy, it is the line made up of the lights on all the vehicles that passed through the same toll booth. If they all follow the same path (a steady flow), a single line results, but if they follow different paths (unsteady flow), it is possible for the line to cross over itself. In unsteady flow, streamlines, pathlines and streaklines are all different, but in steady flow, streamlines, pathlines and streaklines are identical.
The Bernoulli equation states that,
Although these restrictions sound severe, the Bernoulli equation is very useful, partly
because it is very simple to use and partly because it can give great
insight into the balance between pressure, velocity and elevation.
How useful is Bernoulli's equation? How restrictive are the assumptions governing its use? Here we give some examples.
|One-dimensional duct showing control volume.|
When streamlines are parallel the pressure is constant across them, except for hydrostatic head differences (if the pressure was higher in the middle of the duct, for example, we would expect the streamlines to diverge, and vice versa). If we ignore gravity, then the pressures over the inlet and outlet areas are constant. Along a streamline on the centerline, the Bernoulli equation and the one-dimensional continuity equation give, respectively,
These two observations provide an intuitive guide for analyzing fluid flows, even when the
flow is not one-dimensional. For example, when fluid passes over a solid body, the
streamlines get closer together, the flow velocity increases, and the pressure
decreases. Airfoils are designed so that the flow over the top surface is faster than over
the bottom surface, and therefore the average pressure over the top surface is less than
the average pressure over the bottom surface, and a resultant force due to this pressure
difference is produced. This is the source of lift on an airfoil.
Lift is defined as the force acting on an airfoil due to its motion, in a direction normal to the direction of motion. Likewise, drag on an airfoil is defined as the force acting on an airfoil due to its motion, along the direction of motion.
The Autonomous EHD Hovercraft
Powered at 40kV, our first trial used a 23 gramme EHD hovercraft which lifted off, hovering at 1cm to 2cm above the surface of the test bench. Working out pressure from the force exerted over the lower skirt opening of 30cm by 35cm by approximately 1.5cm air cushion depth:
Pressure = F/A = 0.23N/(0.3*0.35) = 2.19Pa
This is not the maximum pressure, as a lot of air is flowing out through the air cushion gap of 1.5cm, so further extra payload was added in small increments, until the device just hovered. This happened at an extra payload of 15g, which together with the device own 23g weight, results in a 0.38N of force over the base area, giving a total pressure of 3.6Pa. Still however, leakages at the bottom could not be totally eliminated if the device is not sealed. Lowering the air leakage at the bottom to zero, is the only way we can calculate the maximum pressure generated by this EHD thruster. For this purpose, I modified the above device by closing the lower base with a square sheet of plastic. Then put the whole device on a scale with a contact surface area of 22 x 23 cm, and tightened the thruster against the table top. The scale was reset to zero, and the below readings show the value obtained by dividing the recorded weight by the contact area (g assumed 9.8). I got a maximum recorded value of 26g, equivalent to 5.04 Pa was recorded at 43kV @ 0.91 mA. Higher voltages resulted in arcing. Our second prototype, was a similar hover device having a base cross sectional area of 1m2. The lower edge was finished with a smooth soft rubber strip which touched the ground around the whole perimeter when the device was powered off. A lightweight version of HVPS03 (stripped off from all digital meters and extras) was mounted over inside the hover, in the space between the collector grid and ground. Surely enough, when switching the device on, it started to hover around at a clearance of 1mm or so, between the lower rubber edge and a flat granite top. Total device weight, including power supply and LiPo cells was 490 grammes.
The closed static pressure does not depend on the box shape and dimensions, but is entirely governed by electric pressure. We know that the pressure in a point to plane geometry can be assumed constant over the gap, and the force acting is given by Sigmond's equation:
where i= dc corona current, d= distance from wire to plane, μ= ion mobility
This equation is independent of actual path taken by the ion to travel across the gap.
Shown below is a proposed ion spreading mechanism within the gap, that results in an increasing cross section of ions proportional to the distance travelled from the corona wire. The ions slow down their radial velocity, thus becoming more populated at each increasing radial distance. The repulsion between them, being unipolar, results in rotation which spreads them out across the whole cylindrical volume.
Independant of the pressure variation within the air gap, the pressure transmitted below the grid, will be equal to that at its inner side, and is equal to the average value which we can calculate by knowing the force acting upon the inner trough surface area of Π*d*l:
Pelectric = F/A = (id/μ)/(Π*d*l) = (i/l)/(Π*μ) = J/(Π*μ) ... J= current density (A/m)
For a fully static pressure case we have:
Pstatic= Pelectric = J/(10*Π*μ) ..... where J = current density in mA/cm and mobility μ is 2E-4 m2/V-1s-1
For our effective corona length of 28cm, this gives a theoretical pressure of:
Pelectric= (0.91/28)/(10*Π*μ) = 5.17 Pa ..... just under 2% discrepancy with our experimental results.
The above plot is for gramme force of thrust vs current. This shows a very linear behaviour of thrust vs current. The lower non linear part is probably due to the fact that the inflating plastic at the base is not yet fully streched over the scale, however all points above 0.5mA show a clear linear curve. In this device, the corona wire was set straight from edge to edge over the whole cylinder of length 32cm, so maximum current was limited due to sparks occuring at the ends of the wire. So here our effective corona wire length is 32cm. For maximum current of 50uA/cm we would get 50uA*32cm= 1.6mA, so I have interpolated the curve to this point, which results in a 40gF, which over the scale area of 22*23cm, would represent a pressure of 7.7Pa. Therefore 7.7Pa would be a good approximation for the maximum practical pressure generated with such a device.
Bernoulli's equation leads to some interesting conclusions regarding the variation of pressure along a streamline. Consider a steady flow impinging on a perpendicular plate.
|Stagnation point flow.|
There is one streamline that divides the flow in half: above this streamline all the flow goes over the plate, and below this streamline all the flow goes under the plate. Along this dividing streamline, the fluid moves towards the plate. Since the flow cannot pass through the plate, the fluid must come to rest at the point where it meets the plate. In other words, it 'stagnates.' The fluid along the dividing, or 'stagnation streamline' slows down and eventually comes to rest without deflection at the stagnation point.
Bernoulli's equation along the stagnation streamline gives
The stagnation or total pressure, p0, is the
pressure measured at the point where the fluid comes to rest. It is the highest pressure
found anywhere in the flowfield, and it occurs at the stagnation point. It is the sum of
the static pressure
(p0), and the dynamic pressure
measured far upstream. It is called
the dynamic pressure because it arises from the motion of the fluid.
The dynamic pressure is not really a pressure at all: it is simply a convenient name for the quantity (half the density times the velocity squared), which represents the decrease in the pressure due to the velocity of the fluid.
We can also express the pressure anywhere in the flow in the form of a non-dimensional pressure coefficient Cp, where
At the stagnation point Cp = 1, which is its maximum value. In the freestream, far from
the plate, Cp = 0.
For short discharge times and weight critical applications, specific power is more important than specific energy. Unfortunately, as with most things, nothing is ideal in real life. Some energy sources are good at giving short powerful bursts of energy, others are good at giving long term moderate energy levels. The powerful but short term capacity energy sources are in general the lightest sources. A battery that can only be discharged relatively slowly will be inappropriate for a rapid discharge, of the order of a few minutes. Specific power (SP) relates the power achievable from the energy source per unit weight of the source, and this is what we are after. The following plot compares the discharge performance of some reserve batteries with generator systems, for specific energy (SE) and specific power (SP). There are clearly large differences between achievable power densities. Thermal batteries can deliver up to 5W/g of power, whilst silver-zinc can manage only 1W/g and lithium about 0.1W/g if cooling is used. Lead-acid batteries can only achieve about 0.45W/g.
Note that if the weight of fuel can be ignored the theoretical SP of a generator is fixed (its power/weight ratio). In this case, the SE increases proportionately with the time of use, i.e. the longer it runs the more energy it produces.