SOME QUESTIONS IN THE THEORY OF AN ION-CONVECTION GENERATOR

V. D. ESKOV

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)

At the present time, in connection with the rapid development of nuclear physics and research on outer space, interest has increased in electrostatic generators and, in particular, in ion-convection-type generators as sources of high voltage.

The ion-convection generator (ICG), despite its low efficiency, possesses a number of advantages that may prove important when considering the question of its practical application. Its operating principle is similar to that of a Van de Graaff generator, but gas or liquid is used as the charge carrier. Like other electrostatic generators, the ICG consists of a charge source (ionizer), the space in which the charges are transported (the working section), and a charge receiver (collector).

Ionization of the gas may be produced either by means of a corona discharge [1] or by means of radioactive radiation [3], with subsequent separation of electrons. The collector may be made in the form of a system of metal tubes or grids placed inside a spherical shell in order to satisfy the conditions of a Faraday cylinder and to avoid corona discharge when high voltages are obtained. The most important part is the working section, the theoretical consideration of whose properties is the task of our article.

Theory of Charge Transport

The apparatus shown in Fig. 1 consists of a cylindrical tube with internal radius \(\vartheta = R\) and length \(z = l\), made of a semiconducting material with a nonlinear characteristic. Inside it, a gas with dielectric constant \(\varepsilon\) is driven through, moving with mean velocity \(v\), the latter being assumed constant in time. At the section \(z = 0\), ions with mobility \(k\) are introduced into the gas stream; the collector at \(z = l\) removes them into the load connected between the collector and the ionizer. The charges are distributed in the volume of the ICG with volume density \(\rho\), and their motion constitutes a current with density \(\delta\). The potential of the collector, which depends on the current density \(\delta\) and the conductance of the external load \(G\), is taken to be \(U_{\text{н}}\); in the system there exists an electric field with intensity \(\mathbf{E}\).

If the steady-state operating mode of the ICG is regarded as a quasistatic process, then our problem reduces to the simultaneous solution of two equations: Poisson’s equation \(\nabla^2 U=-\rho/\varepsilon\) and the principle of continuity of the electric current \(\operatorname{div}\vec{\delta}=0\). Owing to the axial symmetry of our setup, in these expressions \(\partial^2 U/\partial \varphi^2=0\) and \(\partial \delta_\varphi/\partial \varphi=0\); hence we have:

\[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial U}{\partial r}\right) + \frac{\partial^2 U}{\partial z^2} = -\frac{\rho}{\varepsilon}, \tag{1} \]

\[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\delta_r\right) + \frac{\partial \delta_z}{\partial z} = 0, \tag{2} \]

where \(\delta_r\) and \(\delta_z\) are the radial and axial components of the current density.

Taking into account that, according to [1—1],

\[ \delta_r=-k\rho\frac{\partial U}{\partial r}, \tag{3a} \]

\[ \delta_z=\rho\left(v-k\frac{\partial U}{\partial z}\right), \tag{3b} \]

from (1) and (2) we obtain:

\[ -\frac{\partial \rho}{\partial r}\cdot \frac{\partial U}{\partial r} + \frac{\rho^2}{\varepsilon} + \left(\frac{v}{k}-\frac{\partial U}{\partial z}\right) \frac{\partial \rho}{\partial z} = 0. \tag{4} \]

If we consider an isolated gas cloud having the shape of a cylinder of infinitely great length, which expands under the action of repulsive forces of the charges of one sign contained in it, then it turns out that, for \(\rho=\mathrm{const}\) at the time \(t=0\), the volume charge density will decrease with time to the same degree at all points of the space occupied by the gas.

Fig. 1. Schematic diagram of the ICG.
1 — collector, 2 — ionizer, 3 — load.

Since in our setup \(l\gg R\), in the first approximation we may assume that \(\rho\) does not depend on \(r\), i.e.

\[ \frac{\partial \rho}{\partial r}=0. \]

The desire to obtain, all other conditions being equal, a higher voltage at the generator output requires a uniform distribution of the potential along its walls. The presence on the lateral surface of the working part of the ICG of a semiconducting layer with a nonlinear volt-ampere characteristic makes it possible to achieve this. Calculations show that, in order to satisfy the boundary conditions \(U=0\) at \(z=0\) and \(U=U_{\mathrm{n}}\) at \(z=l\), the axial component of the field strength \(E_z\) must depend on \(r\) and \(z\), but this dependence proves to be appreciable only over sections whose length is of the order of the radius near the ionizer and the collector. Therefore, in the first approximation one may take

\[ \frac{\partial U}{\partial z}=\mathrm{const} \]

throughout the entire volume of the charge-transfer space.

Taking into account the approximations adopted above, from equations (1) and (4) we obtain, respectively:

\[ \frac{1}{r}\frac{\partial}{\partial r}(rE_r)=\frac{\rho}{\varepsilon}, \tag{5} \]

where \(E_r=-\dfrac{\partial U}{\partial r}\) is the radial component of the electric-field intensity, and

\[ \frac{\rho^2}{\varepsilon}+\left(\frac{v}{k}+E_z\right)\frac{\partial \rho}{\partial z}=0. \tag{6} \]

Integrating (5) with respect to \(r\) over the limits from 0 to \(r\), we obtain

\[ E_r=\frac{\rho r}{2\varepsilon}. \]

Integration of (6) with respect to \(z\) over the limits from 0 to \(z\) gives

\[ \rho=\rho_0\left[1+\frac{\rho_0 z}{\varepsilon\left(\dfrac{v}{k}+E_z\right)}\right]^{-1}, \tag{7} \]

where \(\rho_0\) is the volume charge density at \(z=0\), which is determined by the capabilities of the ionizer.

Hence, returning to equation (3b), we find

\[ \delta_z=k\rho_0\left(\frac{v}{k}+E_z\right) \left[1+\frac{\rho_0 z}{\varepsilon\left(\dfrac{v}{k}+E_z\right)}\right]^{-1}. \tag{8} \]

It is also possible to calculate the retarding pressure of the ions \(P_1\), determined, according to [2], by the formula

\[ \operatorname{grad} p_1=\frac{dp_1}{dz}=\rho E_z, \tag{9} \]

which is a consequence of the conversion of mechanical energy into electrical energy and, when calculating the expended power, is added to the hydrodynamic losses in the system, since \(E_z\) is directed opposite to the gas flow. Integrating equation (9) with respect to \(z\) over the limits from 0 to \(l\), we obtain

\[ p_1=\varepsilon E_z\left(\frac{v}{k}+E_z\right) \ln\left[1+\frac{\rho_0 l}{\varepsilon\left(\dfrac{v}{k}+E_z\right)}\right]. \tag{10} \]

In expressions (7), (8), (10), the unknown quantity is \(E_z\), which can be found from the boundary conditions. At \(z=l\), by Ohm’s law we have:

\[ U_{\mathrm{н}}G=I_{\mathrm{н}}, \tag{11} \]

where \(I_{\mathrm{н}}\) is the current in the load.

Since, within the limits of our approximations, \(\delta_z\) does not depend on \(r\), then

\[ I_{\mathrm{н}}=\delta_l S, \tag{12} \]

where \(\delta_l\) is the axial component of the current density at \(z=l\), \(S=\pi R^2\) is the cross-sectional area of the working part of the ICG.

In addition,

\[ U_{\mathrm{n}}=-E_z l . \tag{13} \]

Solving equations (11), (12), (13), and (8) jointly with respect to \(E_z\), we obtain

\[ E_z=-\frac{v(2a+b+1)}{2k(a+1)} \left[ 1-\sqrt{1-\frac{4a(a+1)}{(2a+b+1)^2}} \right], \tag{14} \]

where \(a\) and \(b\) are dimensionless quantities equal, respectively, to

\[ a=\frac{k\rho_0 S}{lG}, \qquad b=\frac{k\rho_0 l}{\varepsilon v}. \]

If we now introduce the notation \(c=-\dfrac{kE_z}{v}\), where \(c=f(a,b)\) is likewise a dimensionless quantity, then, according to (12), (13), and (10), the following expressions describing the operating process of the generator can be written:

\[ U_{\mathrm{n}}=\frac{vl}{k}\,c, \tag{15} \]

\[ I_{\mathrm{n}}=\rho_0 vS\,\frac{(1-c)^2}{1-c+b}, \tag{16} \]

\[ p_1=\frac{\varepsilon v^2}{k^2}\,c(1-c)\ln\left(1+\frac{b}{1-c}\right). \]

Then the power at the generator output, equal to the product of \(U_{\mathrm{n}}\) and \(I_{\mathrm{n}}\), is determined by the formula

\[ P_{\mathrm{n}}=\frac{\rho_0 v^2}{k}\,lS\,\frac{c(1-c)^2}{1-c+b}. \tag{17} \]

The efficiency of the installation can be found from the expression

\[ \eta=\frac{P_{\mathrm{n}}}{(p_1+p_2)Q}; \tag{18} \]

here \(Q=vS\) is the gas flow rate, and \(p_2\) is the hydrodynamic pressure loss in the generator, which according to the known formula is equal to

\[ p_2=\frac{\lambda}{4}v^2\gamma\,\frac{l}{R}, \]

where \(\gamma\) is the gas density,
\(\lambda\) is the coefficient of resistance to the gas flow.
As a result of substitution, we obtain from (18):

\[ \frac{1}{\eta}= \frac{1}{b}\left(1+\frac{b}{1-c}\right) \left[ \ln\left(1+\frac{b}{1-c}\right) +\frac{m}{4c(1-c)}\cdot\frac{l}{R} \right], \tag{19} \]

where

\[ m=\frac{\lambda\gamma k^2}{\varepsilon} \]

is a dimensionless quantity depending on the properties of the gas.

Operating Regimes and Specific Characteristics of the ICG

Let us examine expression (14). For \(a=0\) we also have \(c=0\), hence \(U_{\mathrm{n}}=0\) and \(G\to\infty\). This is the short-circuit regime. From (16) we find

\[ I_{\mathrm{sc}}=\rho_0 vS\left(1+\frac{k\rho_0 l}{\varepsilon v}\right)^{-1}. \]

For \(a \to \infty\) we obtain \(c=1\), then \(I_{\mathrm{n}}=0\) and \(G=0\). This is the no-load operation of the generator. From (15) we obtain

\[ U_{xx}=\frac{vl}{k}. \]

Let us now examine expression (18). It reaches a maximum at \(c \simeq \frac{1}{2}\), if \(b<0.05\), and at \(c \simeq \frac{1}{3}\), if \(b>0.8\). The corresponding expressions for the optimal output power are

\[ P_{01}=\frac{\rho_0 v^2}{4k}\cdot\frac{lS}{2b+1}, \]

\[ P_{02}=\frac{4\rho_0 v^2}{9k}\cdot\frac{lS}{3b+2}. \]

In this case the value of the ionic back pressure is found to be

\[ p_{11}=\frac{\varepsilon v^2}{4k^2}\ln(2b+1), \]

\[ p_{12}=\frac{2\varepsilon v^2}{9k^2}\ln\left(\frac{3}{2}b+1\right). \]

In regime 1, corresponding to \(b\leq 0.03\), the value of the output power is limited mainly by the capabilities of the ionizer. The maximum specific power (per unit volume) is equal to

\[ P_{\mathrm{sp}1}\simeq\frac{\rho_0 v^2}{4k}. \tag{20a} \]

In the regime corresponding to \(b\geq 15\), the value of the power at the generator output is limited by the opposing field and by the resulting redistribution of the space charge. The maximum specific power here is equal to

\[ P_{\mathrm{sp}2}\simeq\frac{\varepsilon v^3}{7lk^2}. \tag{20б} \]

Expressions (20a) and (20б) are similar to the corresponding expressions in [2], where it is shown that the maximum power in the second regime is always greater than in the first, while in the interval between regimes 1 and 2 a continuous increase in the optimal power is observed. The efficiency of the installation at maximum power in regime 1 is equal to

\[ \eta_1 \simeq \frac{1}{2+\dfrac{ml}{bR}} = \frac{1}{2+\dfrac{\chi k v}{\rho_0 R}}, \]

and in regime 2

\[ \eta_2 \simeq \frac{2}{3\left[\ln\left(\frac{3}{2}b\right)+\dfrac{ml}{R}\right]}. \]

An analytical comparison of the last two equations shows that the efficiency at maximum output power in regime 2 is greater than in regime 1 as long as \(m>1\), and precisely this case is of technical significance.

Conclusions

The theory described above is suitable for both gases and liquids as charge carriers, although, of course, the assumptions made in it require experimental verification. The theory makes it possible to design the working part of an ICG in accordance with specified output parameters and with the values of the permissible \(E_z\) and \(E_r\), which are determined by the electric strength of the medium. Further development of this question will help ion-convection generators demonstrate in practice their unquestionable advantages: the possibility of direct conversion of the kinetic energy of gases and liquids into electrical energy, the absence of moving parts, ease of maintenance, low requirements on manufacturing precision, compactness, relatively low weight, and the possibility of obtaining large powers in a small volume.

Literature

1. N. A. Kaptsov. Electrical Phenomena in Gases and Vacuum. Gostekhizdat, 1947.
2. O. M. Stuetz­er. Ion Transport High Voltage Generators. “Rev. Sc. Instrum.,” 1961, 32, 1.
3. M. C. Goudrine. Wind Energy Convertors Provide High Power. “Electronics,” 1960, 33, 33.