Zoltan Lozong explains along this amazing study how one could extract exergy from diephoretic process.
In the tutorial about electrostatic forces in dielectrics we introduced the dielectrophoretic force that creates a pressuredifference at the edge of a flat capacitor when merged into a liquid dielectric, and pushes up a liquid column between the plates (fig. 1a).
Fig. 1.
We have derived the formula for the force that pushes the liquid column upwards, and it is:
The pressure increase at the bottom edge of the capacitor can be calculated from this as:
The pressure p in our discussion actually means a pressuredifference Δ p and not an absolute pressure.
Two questions may arise naturally in connection with this setup. The first question is whether the liquid would pour out from the elevated liquid column at the front and back edges when they are left open. The other question is whether a jet of liquid would be ejected upwards into the air (like a fountain) when the height of the capacitor is lower than the calculated height of the liquid column between the plates. If this would be possible then a very efficient pump and fountain could be constructed since the pumping effect does not require any current flow but only static high voltage. By covering the plates with a very good solid insulator layer, and using a liquid dielectric of high permittivity and high resistance, the current could be minimized and the power consumption reduced to an insignificant value compared to the power generated by the pumping effect.
The answer to one of these questions answers both questions because the same principle determines both these phenomena. If the law of energy conservation has to remain valid, then the liquid may not flow out either at the open side edges or at the top of the capacitor since then a closed loop movement of the liquid would be established  producing energy  while theoretically (using very high resistance dielectrics) requiring no electric current and no input power to maintain the pumping effect. This explanation of the classical science seems to be supported by the fact that the electric field is the same at all four edges of the capacitor. Thus the same force is supposed to push the liquid dielectric into the capacitor at all sides when it reaches into the inhomogeneous Efield region of the edges where the pumping forces appear. According to this interpretation the liquid would flow into the inhomogeneous Efield region outside the plates, but it would stop there, held back by the dielectrophoretic forces, and form an arced convex liquid surface (fig. 1b). In practice this might not be fully realized, since at the edges that are not fully surrounded by the dielectric (above the liquid surface) some little force component will be missing that would otherwise be generated by the weak Efield farther away from the edges if the whole capacitor were surrounded with the liquid. Thus the dielectrophoretic pressure at the bottom edge (where there is a deep space below, filled with dielectric) will be slightly stronger than that at the edges above the liquid surface, and a weak leakage might appear. However the pressuredifference will be minimal and cannot provide an efficient pumping effect. So as a first step approximation we can assume that the law of energy conservation is satisfied in this case, and that no significant pumping effect can be achieved that would pump the liquid through the capacitor, since the inward sucking forces are nearly the same at all four edges, and the retarding force at the outlet will cancel the upward pumping force at the bottom input.
However we may try to go around this difficulty by not expecting the liquid to flow out at the edges, where the retarding forces are strong (sealing the edges), but through a hole drilled into the middle of the grounded plate, as shown in fig. 2.
Fig. 2.
Now the question is how much retarding pressuredifference will be generated at the hole outlet by the dielectrophoretic forces, and whether this retarding force will be less than the pumping force at the bottom edge. This is a difficult question to answer without measurements since the shape of the Efield at the hole is fairly complex as shown in fig.3a. But we have good reason to suspect that the forces will be less intense here because the Efield is less intense outside the hole (the plate is grounded) compared to the Efield intensity at the edges of the capacitor. Naturally the indirect method for calculating the forces is not applicable in this case since that can yield a correct result only if the law of energy conservation is valid. Thus that method would not allow any possibility of leakage from the capacitor.
Fig.3.
The fig. 3a. shows a slightly inhomogeneous and arced Efield shape that can still produce significant retarding force. Let’s try to find some arrangement that could minimize the curvature and inhomogeneousness of the Efield at the hole. One such possibility is shown in fig. 3b, when a parallel shielding plate is placed in front of the hole and connected to the ground together with the right electrode of the condenser. The shielding plate s ensures a more uniform Efield distribution, and this arrangement might produce less dielectrophoretic resistive force in the way of the flowing liquid. All this is written in conditional mode, since without numerical analysis or exact measurements we cannot know the magnitude of the forces at the hole.
However our endeavor to theoretically violate the law of energy conservation is not completely hopeless, and we will do the impossible and prove the invalidity of energy conservation for the complete system in some special arrangements, based on (and assuming) the local validity of energy conservation for each single component of the system.
In order to find the possible components of such a FE system, we should derive the formulas of the dielectrophoretic forces for each element through mathematical analysis, firmly based on the assumed validity of energy conservation for each component. In the case of a rectangular flat capacitor this has been already done (see above), and now let’s see if there is any difference if we use discshaped electrodes instead of rectangular plates (fig. 4a).
Fig. 4.
The flat disc capacitor is fully merged into the liquid dielectric but there is a flexible, movable wall w between the electrodes (perpendicular to the plates) that prevents the liquid from entering into the center part of the capacitor which is filled with air. Let us calculate the force that is squeezing this insulating wall towards the center, and also the pressure within the capacitor caused by the dielectrophoretic forces at the edges. The capacitance of the condenser is:
The electric energy in the condenser is:
The electric energy change of the capacitor per wall movement, when the radius of the wall is changed is:
The force upon the wall is:
The pressure of the liquid inside the capacitor is:
This is the same formula as that of the rectangular flat capacitor and both produce the same dielectrophoretic pressure.
Now let’s do the same calculation for the coaxial capacitor shown in fig. 4b.
In this case the force is directed towards the increase of x, thus:
This is an interesting result since the pressuredifference of the coaxial capacitor is not the same as that of the rectangular or disc shaped flat capacitors using the same dielectric, voltage, and distance between electrodes. This difference can provide a firm ground for the establishment of the asymmetry in pressures and the consequent violation of the energy conservation.
Before analyzing this possibility, let us calculate the force and pressure in the case of a semicylindrical capacitor (fig. 5).
Fig. 5.
The energy stored in the capacitor is:
The torque is calculated as:
(1)
Supposing that the torque is caused by a homogeneous pressure, it is calculated as:
Equating this expression with the formula (1) we get the pressure:
The total force developed by this pressure upon the capacitor is:
The examined capacitor is shown in fig. 6. The capacitance is:
The energy content of the condenser is:
We want to calculate how much pressure would be required to create the derived torque. This is done by assuming a constant pressure over the examined half circlering area, and integrating the elementary torques to get the resultant torque M (fig. 6b)
Fig. 6.
By equating this torque resulting from the pressuredifference with the above formula (2) we get the pressure:
Supposing that this pressure is the same over the whole surface area of the circlering between the bottom edges of the hemispherical capacitor, the force upon the condenser is:
These formulas for the hemispherical capacitor have been derived using a torque around an axis, and a doubt may arise about its correctness. Therefore the same formulas have been derived with another method, assuming that the dielectric rises into the capacitor through the whole open bottom surface area simultaneously, and develops a torque around a single point at the center. The calculation can be found in the appendix.
The following table summarizes the derived formulas of the pressures and forces at the open edges of the different capacitor types:
pressure 
force 

Rectangular flat capacitor 

Discshaped flat capacitor 

Cylindrical coaxial capacitor 

Semicylindrical capacitor 

Hemispherical capacitor 
Pressuredifferences p_{d} = p_{r} and p_{x} = p_{c}, thus we have only 3 different pressuredifference for the 3 different electrode shapes. The following diagram compares these pressuredifferences for changing r_{1} (radius of the smaller electrode) with fixed r_{2} = 5mm and uses the following parameters: l = r_{2} – r_{1} ; U=10 kV ; e =56e _{0} (the permittivity of glycerin).
Diagram 1.
The pressuredifference ratios of flatcylindrical and flatspherical condensers are:
The curves nicely illustrate that when r_{1} approaches r_{2} all 3 pressuredifferences will become nearly identical (right side). However when r_{1} is much smaller than r_{2}, there will be big differences between the pressuredifferences. For example when r_{1} is 10 times smaller then r_{2} then p_{f}=1.4p_{c} and p_{f} =3.7p_{s}. When r_{1} is 100 times smaller than r_{2} then p_{f} =2.35p_{c} and p_{f} =33.67p_{s}.
With a clever arrangement we can exploit these pressuredifferences to create an efficient electrostatic pump that violates the law of energy conservation. A flat capacitor’s edge should be merged into the liquid dielectric, since that gives the highest pressuredifference; and a smooth transition should be made to a coaxial cylindrical capacitor, that in turn should end in a semispherical capacitor where the fluid leaves the condenser, since the retarding pressuredifference is the lowest for that shape (fig. 7).
Fig. 7.
At the curved region t where the flat disc capacitor merges into a cylindrical coaxial capacitor, the Efield lines are straight and thus do not retard the flow of the liquid. There will be a dielectrophoretic force towards the center of curvature O that is perpendicular to the movement of the fluid, tending to push it against the wall of the upper electrode, but not retarding the movement of the dielectric. The only retarding pressuredifference appears at the top opening, and that pressuredifference of the semispherical condenser has just been calculated to be less than the pressuredifference of the flat capacitor at the bottom. Thus the resultant pressuredifference that this pump can create is calculated as:
If the output ending would not have the semispherical shape, but is left to be the natural ending of the coaxial capacitor (fig. 7b), then the resultant pumping pressuredifference would be:
The dependence of these pressuredifferences from the radius of the inner electrode of the cylindrical part r_{1} and voltage is shown in diagram 2.
Diagram 2.
The semispherical outlet shows a better performance than the simple cylindrical coaxial capacitor ending. Other possible pump shapes are shown in fig. 7c. and fig. 8, which have basically the same performance as the one shown in fig. 7a. How high can these pumps elevate the liquid column? The formulas for spherical and cylindrical endings are:
Fig. 8.
After deriving the formulas, and supposing that there is no major flaw in the derivation, we have arrived to a weird contradiction:
It seems that there is no possibility to satisfy the law of energy conservation simultaneously for the whole pump and also for each basic capacitor shape component. The above conclusions assumed that the liquid dielectric is a very good insulator and it does not allow any current flow. In practice there is always some leakage current that can be minimized by carefully choosing the liquid dielectric and optionally covering the electrodes with a very good solid insulator layer. But even if there still would be significant leakage current, the law of energy conservation would be still violated, since the input power (caused by the unwanted current) is fully transformed into heat and into accumulation of electrostatic charge (and energy) on some bodies. Thus the input power is not consumed for the propulsion of the liquid, but for producing heat, thus the kinetic energy represents an excess energy that still violates the law of energy conservation.
How much energy can we generate with the pump? Knowing the produced pressuredifference it is easy to calculate the power:
Where p – is the pressuredifference; dV  is the elementary volume; dt – time difference; S – surface area; v – velocity of the liquid.
The pressuredifference p and the surface area through which the liquid flows out are determined by the size and other parameters of the pump. However, the velocity v of the liquid dielectric is not definitely determined. This velocity would be limited only by the length of the accelerating path and the frictional resistance of the fluid path. Since the pressuredifference does not depend on the velocity of the liquid but is constant, the possibility naturally offers itself to artificially create an external additional pressuredifference by connecting an external pump in series with the dielectrophoretic pump, and this way further increase the velocity of the dielectric. The external pressuredifference can be generated by any conventional means and the increased velocity increases the excess energy. To calculate the increase of excess energy let us assume that the velocity of the fluid through a pipe is linearly proportional with the pressuredifference v=Kp. The input power consumed by the external conventional pump is then:
The output power is:
The excess power is:
If there would be no external pump increasing the velocity then the excess energy would be , thus the external pump increases the excess energy by the factor of:
If the external pump provides 9 times higher pressure than the electrostatic pump then the excess energy generated by the electrostatic pump will be 10 times greater than without external pump. The COP of the whole system is:
When the external pressuredifference is much higher than that of the dielectrophoretic pump then the COP approaches unity. Naturally the best is to use dielectrophoretic pumps also as external pumps, thus they would not consume significant energy, and the COP could be maintained at high value.
Fig. 9.
Using this method we assume that there is a flexible frustum of a cone between the electrodes with vacuum (or air) inside, and that the space below the cone is filled with the liquid dielectric that is squeezing the cone as in fig. 10a.
Fig. 10.
The capacitance of a full spherical condenser is:
The capacitance of a sphere segment is proportional to its outer surface area (or in other words, with the space angle viewed from the center of the capacitor), thus the partial capacitances are calculated as follows.
The outer surface area of the conical sphere segment C_{1} (fig. 10a) is . The total surface area of the sphere is . The partial capacitance C_{1} can be calculated when the capacitance of the full spherical capacitor is multiplied with the quotient of these two surface areas:
The electric energy of the capacitor is:
The energy change per angle displacement:
Thus the torque around the center point should be:
Let’s now calculate the torque on the frustum of a cone if the liquid pressure is constant everywhere on that surface area (fig. 10b):
Equating this formula with (3) we get the wanted pressure difference:
This formula is the same as derived above with a different method, thus it is correct.
NOTE: The above figures have been made manually based on rough estimations, thus the Efield shapes are not based on numerical analysis or exact analytical calculations. The real Efield shape might be somewhat different than what is shown in the drawings. These illustrations serve only for the purpose of aiding the understanding of the presentation.
Created by Zoltan Losonc (feprinciples@on.mailshell.com) on 4 July 2003. Last updated on 14 July 2003. Linguistic proofreading done by Steven Dufresne.