2016-08-23

3. Free energy at the tip of the needle?

In the second part of this narrative we have studied the coaxial capacitor with a rough outer surface. Such a surface greatly reduces the input voltage, which can begin the process of ionization of the surrounding gas [1]. And if you do not avoid this restriction, but rather to use the effect fully? Obviously, the surface needs all the more to make uneven, to increase the curvature to the maximum. The most suitable form for this — the needle shape, and because we have the likeness of a plate, a brush consisting of such needles. But this design is not very practical, so it is best a brush roll to roll to the needles turned sticking out, and the second external plate of such a condenser to perform a conventional cylinder. Such a design is called high voltage diode, but to avoid confusion we shall call it And the diode and to define further concepts as shown in the figure to the right. We will try to calculate its parameters and to calculate efficiency for prostheses scheme with his participation. The operation of the circuit is reduced to a quick-charging-diode \(C\) from the voltage source \(U\) through the switch SW1, after which he opens, and the charge forms an ionic breeze that carries from the cathode to the anode. At some point key closes SW2 and closes the circuit through the load resistor \(R\). The figure to the right shows a fragment of a migration process, consisting of the charged cathode which come off negatively charged gas ions. Because the ion takes away from the cathode the charge, each palebisi will accelerate with lower speeds depending on the interelectrode distance. This process is similar to the current movement in the conductor with the difference that the charge carriers — ions will have a different speed.

Dependence of speed on voltage can be expressed from the following formula, derived generally from [1] and [2]. \[ V = {\mu \, U \over h} = {\mu \, Q \over h \, C} \qquad (3.1) \] where: \(V\) is the velocity of the flow of ions, \(\mu\) — the mobility of gas, \(U\) is the voltage between the cathode and the anode, \(h\) is the distance between the electrodes, \(Q\) — charge and-of the diode, \(C\) is its capacity. The calculation we make the assumption that \(h\) is equal to or less than the free path length of gas molecules. It is adjustable, for example, the gas pressure in the housing And diodes. Now we need to understand the qualitative picture of this effect, a more complex calculation, if desired, can be done later.

Therefore, the time required ion to fly the distance \(h\) are looking for this: \[ \tau = {h \over V} = {h^2 \, C \over \mu \, Q} \qquad (3.2) \] But we already know that each ion will have its speed, and hence the time of flight. Next, we will not need absolute time, and relative intervals between the collisions of the ions with the anode. They are so: \[ \Delta\tau_i = {h^2 \, C \over \mu} \left( {1 \over Q_0 - e\,(i+1)} - {1 \over Q_0 - e\,i} \right) \qquad (3.3) \] \[ i \in 0..N, \qquad N = {Q_0 \over e} \] where \(\Delta\tau_i\) the time between adjacent collisions of ions with the anode is \(i\)-th step, \(Q_0\) is the initial charge on the diode \(e\) is the charge of one electron. It is seen that with increasing index \(i\) from zero to \(N\), the charge on the diode is spent with \(Q_0\) to zero. Next we do the calculation on the assumption that the ions take the entire charge from the cathode, although in reality a small part of the charge will remain.

Remember the formula for the current \(I = {\Delta Q/\Delta t}\) and energy \(W = \sum I^2\,R \, \Delta t\). As we explore the time between collisions of neighboring charges, it is clear that \(\Delta Q\) it will be the elementary charge \(e\). Substituting all the relations get the energy flowing through the diode and through the load resistance \(R\): \[ W = R\sum_{i=0}^N {e^2 \over \Delta\tau_i} \qquad (3.4) \] Simplify the expression for \(\Delta\tau_i\) provided that \(e/Q_0\) is a very small unit: \[ \Delta\tau_i = {h^2 \, C \, e \over \mu\,(Q_0 - e\,i)^2 } = {h^2 \, C \, e \over \mu\,Q_0^2\,(1 - e\,i/Q_0)^2 } \qquad (3.5) \] Thus: \[ W = {\mu\,R\,e\,Q_0^2 \over h^2\,C} \sum_{i=0}^N (1 - e\,i/Q_0)^2 \qquad (3.6) \] the Expression under the sign of the sum is a number, the amount of which for sufficiently large \(N\) is: \[ \sum_{i=0}^N (1 - e\,i/Q_0)^2 = {Q_0 \over 3\,e} \qquad (3.7) \] Hence, the General formula for energy is: \[ W = {\mu\,R\,Q_0^3 \over 3\,h^2\,C} \qquad (3.8) \] the Energy expended in charging and diodes: \[ W_0 = {Q_0^2 \over 2\,C} \qquad (3.9) \] now compare these two energy: \[ K_{\eta2} = {W \over W_0} = \frac23 {\mu\,R\,Q_0 \over h^2} \qquad (3.10) \]

Let's try to find \(K_{\eta2}\) for some averages, and for this we find \(\mu\) from [2]. It can be seen that the ion mobility of the gas depends on the number of electrodes and distances between them, but for example, you can take some optimal value for the 60 electrodes: \(\mu = 2 \cdot 10^{-5}\,(m)\), which, by the way, an order of magnitude less than in [1], which gives a great "margin of safety" in our calculations. Express the initial charge as \(Q_0 = U_0\,C\), where \(U_0\) — charge voltage And the diode, and the remaining parameters take the following: \[ U_0 = 1.5 \cdot 10^{3}\,(B), \qquad C = 10^{-10}\,(f), \qquad R = 10^{6}\,(Om), \qquad h = 10^{-3}\,(m), \] then: \[ K_{\eta2} = \frac23 {\mu\,R\,C\,U_0 \over h^2} = 2. \]

Insights

The above example is taken for the air environment and other non-optimized parameters. From the formula (3.10) shows that more optimally to decrease the distance between the electrodes \(h\), than to increase the voltage \(U_0\). The limit could be the minimum voltage to start the ionization of the gas. \(K_{\eta2}\) can be increased due to the shape and number of electrodes gas fill And a diode, and the pressure in it. If you increase the speed of the ionic wind, an additional air pump, it is possible not only to some extent to increase the efficiency, but also to build the air engine.