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5.4: Ion Drag Anemometer

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    The example of this section is intended to illustrate how charged particle trajectories can be computed using the approximations introduced in Sec. 5.3. A pair of electrodes is embedded in the wall bounding a fluid moving uniformly to the right, as shown in Fig. 5.4.1. A potential \(V\), applied to the right electrode gives an electric field intensity which terminates on the left electrode. In the neighborhood of the coordinate origin this field can be approximated as azimuthal. Thus, the imposed velocity and electric field intensity distributions are

    \[ \overrightarrow{v} = U [\sin \, \theta \overrightarrow{i}_r + \cos \, \theta \overrightarrow{i}_{\theta}] \label{1} \]

    \[ \overrightarrow{E} = - \frac{V}{\pi r } \overrightarrow{i}_{\theta} \label{2} \]

    Fig. 5.4.1. Electrodes embedded in a smooth wall have the potential difference \(V\). Ions enter from the left, entrained in the uniform velocity \(U\). With a positive \(V\), the left electrode intercepts some of the ions from the flow.

    Fluid flow is represented as inviscid, and hence uniform right up to the electrode surfaces. Positive ions, present in the stream entering from the left, are sampled by the electrodes. The flux of ions to the left electrode caused by applying a positive voltage \(V\) to the right electrode is to be computed with a view toward obtaining the associated current i as a way of measuring the gas velocity.\(^1\)

    It follows from Eqs. 5.3.7 that

    \[ A_E =\frac{V}{\pi} \, \ln \, (\frac{r}{a}) \label{3} \]

    \[ A_v = -Ur \, \cos \, \theta \label{4} \]

    The characteristic lines, along which the charge density is constant, are given by Eq. 5.3.13, which in view of Eqs. \ref{3} and \ref{4} becomes

    \[ -Ur \, cos \, \theta + \frac{bV}{\pi} \, \ln \, (\frac{r}{a}) = \text{constant} \label{5} \]

    The constant is evaluated by fixing attention on the characteristic line entering at an altitude \(h\) over the left edge of the left electrode. Thus, at \(r \, sin \, \theta = -c, \, r \, cos \, \theta = h\) and \(r = \sqrt{h^2 + c^2}\).Then, Equation \ref{5} becomes

    \[ V \, \ln \Bigg [ \frac{\sqrt{h^2 + 1}}{r} \Bigg ] = h - r \, cos \, \theta \label{6} \]

    where normalization of \((V,h,r)\) is introduced:

    \[ \underline{V} = \frac{bV}{\pi U c}, \, \underline{h} = \frac{h}{c}, \, \underline{r} = \frac{r}{c} \nonumber \]

    The quantity on the right is the distance downward (toward the electrode) measured from the initial altitude, \(\underline{h}\), of a particle. Hence, the particle trajectories can be simply plotted by specifying the normalized voltage \(\underline{V}\) and \(\underline{h}\) for the trajectory of interest. With compass in hand,a graphical construction of a trajectory is obtained by picking a normalized radial coordinate \(\underline{r}\), computing the left-hand side of Equation \ref{6},and finding the azimuthal angle \(\theta\) at which the distance downward from the initial height \(\underline{h}\), is as computed.

    Fig. 5.4.1. Vertical and horizontal distances have been normalized to \(c\), with the left electrode then extending from \(1 \rightarrow a/c\). In this sketch, \(\underline{V}=0.5\).

    Typical plots are shown in Fig. 5.4.2. Concern is with positive ions only so that characteristic lines emanating from the wall to the right of the origin enter the volume of interest where there is no source of charge. Hence, the constant charge density to be associated with those lines is zero.On lines entering from the left, the charge density is a constant determined by conditions to the left.

    The point \((r,\theta) = (0.5,0)\) shown in Fig. 5.4.2 is one of zero force. Setting the \(r\) and \(\theta\) components of \(v + bE\) to zero shows that this critical point is at \(\underline{r} = \underline{V}\) and \(\theta = 0\). At this point, characteristic lines entering from the left split into those that remain in the stream and those that reach the plane \(\theta = - \pi/2\).

    The characteristic line passing through the critical point is found by evaluating Equation \ref{5} at \(\underline{r} = \underline{V}, \, \theta = 0\):

    \[ -r \, cos \, \theta + V \, ln \, (r \frac{c}{a}) = -V + V \, ln \, (V \frac{c}{a}) \label{7} \]

    The position \(\underline{r} = \underline{r}^{*}\) on the surface \(\theta = - \pi/2\) where this critical characteristic line impinges then follows by evaluating Equation \ref{7} with \(\theta = - \pi/2\):

    \[ r^{*} = \frac{V}{e} \label{8} \]

    Thus, the critical characteristic line impinges on the electrode if \(\underline{r}^{*} > (a/c)\),i.e., if

    \[ \underline{V} > (\frac{a}{c}) e \label{9} \]

    For lesser values of \(\underline{V}\), all of the electrode surface collects particles entering from the left, and the total current \(i\) is the integral of \(-\rho bE_{\theta}\) over the entire electrode surface:

    \[ i = w \int_a^c \frac{\rho b V}{\pi} \frac{dr}{r} = (\rho U c w) \underline{V} \, ln(\frac{c}{a}) \label{10} \]

    This dependence of \(i\) on \(V\) is presented graphically in Fig. 5.4.3, valid so long as \(\underline{V} < \frac{a}{c} e\).

    If \(V\) is increased beyond this value, only that portion of the electrode to the left of \(\underline{r} = \underline{r}^{*}\) collects particles. The rest intercepts characteristic lines carrying no charge because they originate on the boundary \(\theta = \pi/2\) to the right. Thus, the current is

    \[ i = w \int_{r^{*{(c \underline{V}/e)}}}^c \frac{\rho b V}{\pi} \frac{dr}{r} = \rho U c w \, \underline{V}(1- ln \underline{V}) \label{11} \]

    Fig. 5.4.1 as function of normalized voltage \(\underline{V} = bV/ \pi U c\).

    With \(V\) beyond the value e, all of the characteristic lines reaching the electrode surface originate to the right where there is no source of particles. For voltages greater than this, the electric field diverts the particles completely before they can reach the electrode, and \(i = 0\). The current dependence given by Equation \ref{11} is also summarized in Fig. 5.4.3.

    It should be clear from the \(i-V\) characteristic summarized by Fig. 5.4.3 that there are many ways in which practical use could be made of the charged particle collection process. The peak current is a measure of \(U\), while the voltage at which the curve peaks, or cuts off, gives a measure of either the velocity or the mobility.

    1. K. J. Nygaard, "Anemometric Characteristics of a Wire-to-"Plane" Electrical Discharge," Rev. Sci.Instr. 36, 1771 (1965).

    5.4: Ion Drag Anemometer is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James R. Melcher via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.