Abstract:
Performance of an electrohydrodynamic fluid accelerator device may be improved and adverse events such as sparking or arcing may be reduced based, amongst other things, on electrode geometries and/or positional interrelationships of the electrodes. For example, in a class of EHD devices that employ a longitudinally elongated corona discharge electrode (often, but not necessarily, a wire), a plurality of generally planar, collector electrodes may be positioned so as to present respective leading surfaces toward the corona discharge electrode. The generally planar collector electrodes may be oriented so that their major surfaces are generally orthogonal to the longitudinal extent of the corona discharge electrode. In such EHD devices, a high intensity electric field can be established in the “gap” between the corona discharge electrode and leading surfaces of the collector electrodes.

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) 
       [0001]    The present application claims the benefit of U.S. Provisional Application No. 61/094,028, filed Sep. 3, 2008. 
     
    
     BACKGROUND 
       [0002]    1. Field 
         [0003]    The present application relates to the thermal management, and more particularly, to micro-scale cooling devices that use electrohydrodynamic (EHD, also known as electro-fluid-dynamic, EFD) technology to generate ions and electrical fields to control the movement of fluids, such as air, as part of a thermal management solution to dissipate heat. 
         [0004]    2. Related Art 
         [0005]    In general, electrohydrodynamic (EHD) technology uses corona discharge principles to move fluids (e.g., air molecules). Basic principles of EHD fluid flow are reasonably well understood by persons of skill in the art. Accordingly, a brief illustration corona discharge principles in a simple two electrode system sets the stage for the more detailed description that follows. 
         [0006]    With reference to the illustration in  FIG. 1 , corona discharge principles include applying a high intensity electric field between a first electrode  10  (often termed the “corona electrode,” the “corona discharge electrode,” the “emitter electrode” or just as the “emitter”) and a second electrode  20 . Fluid molecules, such as surrounding air molecules, near the corona discharge region  18  become ionized and form a stream  14  of ions  16  that accelerate toward second electrode  20 , colliding with neutral fluid molecules  22 . During these collisions, momentum is imparted from the stream  16  of ions  14  to the neutral fluid molecules  22 , inducing a corresponding movement of fluid molecules  22  in a desired fluid flow direction, denoted by arrow  2 , toward second electrode  20 . Second electrode  20  is variously referred to as the “accelerating”, “attracting”, “collector” or “target” electrode. While stream  14  of ions  16  are attracted to, and neutralized by, second electrode  20 , neutral fluid molecules  22  move past second electrode  20  at a certain velocity. The movement of fluid produced by corona discharge principles has been variously referred to as “electric,” “corona” or “ionic” wind and has been defined as the movement of gas induced by the repulsion of ions from the vicinity of a high voltage discharge electrode. 
         [0007]    Devices built using the principle of the ionic movement of a fluid are variously referred to in the literature as ionic wind machines, electric wind machines, corona wind pumps, electrostatic air accelerators, electro-fluid-dynamics (EFD) devices, electrostatic fluid accelerators (EFA), electrohydrodynamic (EHD) thrusters and EHD gas pumps. Some aspects of the technology have also been exploited in devices referred to as electrostatic air cleaners or electrostatic precipitators. 
         [0008]    In the present application, embodiments of the devices illustrated and described herein are referred to as electrohydrodynamic fluid accelerator devices, also referred to in an abbreviated manner herein as “EHD devices”, and are utilized as a component in a thermal management solution to dissipate heat generated by an electronic circuit. 
       SUMMARY 
       [0009]    It has been discovered that performance of an electrohydrodynamic fluid accelerator device may be improved and adverse events such as sparking or arcing may be reduced based, amongst other things, on electrode geometries and/or positional interrelationships of the electrodes. For example, in a class of EHD devices that employ a longitudinally elongated corona discharge electrode (often, but not necessarily, a wire), a plurality of generally planar, collector electrodes may be positioned so as to present respective leading surfaces toward the corona discharge electrode. The generally planar collector electrodes may be oriented so that their major surfaces are generally orthogonal to the longitudinal extent of the corona discharge electrode. In such EHD devices, a high intensity electric field can be established in the “gap” between the corona discharge electrode and leading surfaces of the collector electrodes. 
         [0010]    In general, it can be desirable to decrease the gap distance so as to increase intensity of the ionizing electric field near the corona discharge electrode. However, at the same time, it can be desirable to “spread” that field over a comparatively large surface of the collector electrodes. In this way, EHD device designs seek to maximize intensity of the ionizing electric field while minimizing arcing between corona discharge and collector electrodes. In the EHD device configuration previously introduced, the respective leading surfaces of the collector electrodes present comparatively large surfaces. In an attempt to spread the field evenly over such leading surfaces, designs have been proposed in which such leading surface present a circular profile such that all points on the surface are equidistant from a corona discharge electrode positioned at a distance equal to the radius of the circular profile. 
         [0011]    While such configurations have some intuitive appeal, it has been discovered that performance may be improved in configurations in which a curved leading surface presents a nearest point at a minimum distance, D min , from the corona discharge electrode and in which additional points in either direction away therefrom along the curved leading surface are at increasing distance, d(θ)&gt;D min . In some configurations, the presented curvature is non-circular, such as in the case of certain parabolic, elliptical, caternary or other curved profiles in which the increasing distance constraint is maintained. In some configurations, curvature presented by the leading surface of a collector electrode may even be circular as long as electrode positioning is such that minimum distance, D min , is less than the radius of circular curvature. 
         [0012]    Building on the foregoing, we present a variety of embodiments. In some embodiments, collector electrodes of the EHD device are themselves thermally coupled to a heat source such that at least some surfaces thereof act as fins of a heat exchanger. In some embodiments, the EHD device motivates flow of a fluid (typically air) past a heat exchanger that is distinct from the collector electrodes. In some embodiments, multiple EHD device instances are ganged and/or staged so as to increase volume of flow, pressure or both. These and other embodiments will be understood with reference to the description that follows and with respect to the appended claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0013]    The description of illustrative embodiments will be understood when read in connection with the accompanying drawings. Drawings are not necessarily to scale; emphasis has instead been placed upon illustrating the structural and fabrication principles of the described embodiments. 
           [0014]      FIG. 1  is a graphical depiction of certain basic principles of corona-induced electrohydrodynamic (EHD) fluid flow. 
           [0015]      FIG. 2  is a side view of an EHD device in accordance with some embodiments of the present invention. 
           [0016]      FIG. 3A  is a partial cross-sectional perspective view, in accordance with some embodiments of the present invention, of a collector electrode geometry that may be employed by the EHD device illustrated in  FIG. 2 . 
           [0017]      FIG. 3B  is a simplified cross-sectional view of the corona discharge electrode and a single collector electrode in accord with the leading surface profile illustrated  FIG. 3A . 
           [0018]      FIG. 4  is a side view of an EHD device configuration in accordance with some embodiments of the present invention and in which multiple assemblies are ganged to increase total volume of fluid flow. 
           [0019]      FIG. 5  is a schematic block diagram illustrating a configuration in which an EHD device in accordance with some embodiments of the present invention may be used. 
           [0020]      FIG. 6  is a schematic block diagram illustrating another configuration in which EHD devices in accordance with some embodiments of the present invention may be used. 
           [0021]      FIG. 7  is a perspective view of an EHD device in accordance with some embodiments of the present invention and in which collector electrodes are thermally coupled to a heat source. 
           [0022]      FIG. 8A  is a schematic block diagram illustrating the components of a multi-stage EHD device. 
           [0023]      FIG. 8B  is a block diagram illustrating distance relationships between two stages of the multi-stage EHD device of  FIG. 8A , used in some embodiments to determine an effective distance between successive stages in the multi-stage EHD device. 
           [0024]      FIG. 8C  is a block diagram representing current relationships between two stages of the multi-stage EHD device of  FIG. 8A  used in some embodiments to determine an effective distance between successive stages in the multi-stage EHD device. 
           [0025]      FIG. 9  is a schematic block diagram illustrating an operating configuration for some multi-stage EHD device embodiments in accordance with the present invention. 
           [0026]      FIG. 10A  depicts an embodiment of a multi-stage EHD device in accord with  FIG. 9 . 
           [0027]      FIG. 10B  depicts a further embodiment of a multi-stage EHD device in accord with  FIG. 9  and in which may be suitable for use as a direct replacement for a conventional rotary fan ventilated heat exchanger design such as employed in some laptop computer designs. 
           [0028]      FIG. 11  is a schematic block diagram illustrating an operating configuration for some multi-stage EHD device embodiments in accordance with the present invention and in which a temperature sensor feedback system is provided. 
           [0029]      FIG. 12  is a schematic block diagram illustrating a multi-stage EHD device in accordance with some embodiments of the present invention and in which collector electrodes are thermally coupled to the heat source. 
           [0030]      FIGS. 13A ,  13 B and  13 C illustrate simulation plots of electric field characteristics for a variety of illustrative geometries and operational pairings of corona discharge and collector electrodes. In particular,  FIG. 13A  is a simulation plot of electric field strength and equipotential lines for a representative configuration in which a collector electrode having a generally circular curved leading surface profile is paired with a corona discharge electrode positioned equidistant from all points along the curved leading surface of the collector electrode.  FIG. 13B  is a simulation plot of equipotential lines for an alternative configuration of a collector electrode having a generally circular curved leading surface profile paired with a corona discharge electrode of differing geometry. Finally,  FIG. 13C  is a simulation plot of equipotential lines for a configuration in which a collector electrode presents a generally parabolic leading surface profile. 
           [0031]      FIG. 14A  is a side view of an EHD device in accordance with some embodiments of the present invention. 
           [0032]      FIG. 14B  is a cross-sectional view of the EHD device illustrated in  FIG. 14A . 
           [0033]      FIGS. 15 ,  16 ,  17  and  18  illustrate a variety of electrode configurations and geometries for EHD devices in accordance with some embodiments of the present invention and in which points that are at increasing angular distance along curved leading surfaces of respective collector electrodes are at increasing distance from a corona discharge electrode. In particular, 
           [0034]      FIG. 15  illustrates a generally parabolic leading surface profile for which points at increasing angular distance are at increasing positional distance from a corona discharge electrode. 
           [0035]      FIG. 16  illustrates a generally elliptical leading surface profile for which points at increasing angular distance are at increasing positional distance from a corona discharge electrode. 
           [0036]      FIG. 17  illustrates a generally curved leading surface profile displaced from a corona discharge electrode so that points at increasing angular distance along a collector electrode are at increasing distance from the corona discharge electrode. 
           [0037]      FIG. 18  illustrates a circular leading surface profile having a radius of curvature substantially greater than a minimum distance between corona discharge and collector electrodes. 
       
    
    
       [0038]    The use of the same reference symbols in different drawings indicates similar or identical items. 
       DETAILED DESCRIPTION 
       [0039]    Geometries and/or positional interrelationships of the electrodes of an electrohydrodynamic fluid accelerator device (hereafter an EHD device) have been studied together with the electric fields that may be established during device operation. Based, in part, on insights developed, candidate geometries and/or positional interrelationships have been adapted to generally improve performance and/or reduce incidence of adverse events in such EHD devices. For example, in some embodiments in accordance with the present invention, specific geometries and/or positional interrelationships of corona discharge and collector electrodes are employed in which points that are at increasing angular distance, θ, from a dominant fluid flow direction along curved leading surfaces of respective collector electrodes are at increasing distance, d(θ), from a corona discharge electrode. 
         [0040]    In general, a variety of curved leading surface profiles are envisioned for collector electrodes, together with a variety of positional interrelationships between corona discharge and collector electrodes of an EHD device. For concreteness of description, we focus on certain illustrative embodiments and certain illustrative surface profiles and positional interrelationships. For example, in much of the description herein, plural planar collector electrodes are arranged in a parallel, spaced-apart array proximate to a corona discharge wire that is displaced from curved leading surfaces of the respective collector electrodes. Likewise, in some embodiments, planar portions of the collector electrodes are oriented generally orthogonally to the longitudinal extent of the corona discharge wire. In some embodiment, collector electrodes of an EHD device act as heat exchanger fins and are thermally coupled to a heat source, typically an electrical assembly or semiconductor integrated circuit, and typically by one or more heat pipes. In some embodiments, an EHD device may motivate fluid flow past a separate heat exchanger. In some embodiments, multiple EHD device instances are staged, ganged, or staged and ganged to provide a composite EHD device. 
         [0041]    Of course, these embodiments are merely illustrative and, notwithstanding the particular context in which any particular embodiment is introduced, persons of ordinary skill in the art having benefit of the present description will appreciate a wide range of design variations and exploitations for the developed techniques and configurations. Moreover, reference to particular materials; dimensions, electrical field strengths; exciting voltages, currents and/or waveforms; packaging or form factors, thermal conditions, loads or heat transfer conditions and/or system designs or applications is merely illustrative. In view of the foregoing and without limitation on the range of designs encompassed within the scope of the appended claims, we now describe certain illustrative embodiments 
       Electrohydrodynamic (EHD) Fluid Acceleration, Generally 
       [0042]    Basic principals of electrohydrodynamic (EHD) fluid flow are well understood in the art and, in this regard, an article by Jewell-Larsen, N. et al., entitled “Modeling of corona-induced electrohydrodynamic flow with COMSOL multiphysics” (in the  Proceedings of the ESA Annual Meeting on Electrostatics  2008) (hereafter, “the Jewell-Larsen Modeling article”), provides a useful summary. Likewise, U.S. Pat. No. 6,504,308, filed Oct. 14, 1999, naming Krichtafovitch et al. and entitled “Electrostatic Fluid Accellerator” describes certain electrode and high voltage power supply configurations useful in some EHD devices. U.S. Pat. No. 6,504,308, together with sections I (Introduction), II (Background), and III (Numerical Modeling) of the Jewell-Larsen Modeling article are hereby incorporated by reference herein for all that they teach. 
         [0043]    Note that the simple illustration of corona-induced electrohydrodynamic fluid flow shown in  FIG. 1  (which has been adapted from the Jewell-Larsen Modeling article and discussed above) includes shapes for first electrode  10  and second electrode  20  that are particular to the simple illustration thereof. Likewise, the electrode configurations illustrated in U.S. Pat. No. 6,504,308 and aspects of the power supply design are particular thereto. Accordingly, such illustrations, while generally useful for context, are not intended to limit the range of possible electrode or high voltage power supply designs in any particular embodiment of the present invention. 
       An Illustrative EHD Device 
       [0044]      FIG. 2  is a front, side view of an EHD device in accordance with some embodiments of the present invention. EHD device  100  will be understood relative to a three-dimensional coordinate system  101  in which the x-y plane respectively designates the width and depth of device  100  and the z direction (into the page) designates the height, h, of device  100 . In  FIG. 2 , device  100  motivates flow of a fluid in the y direction; that is, fluid is drawn into the first, or front, surface of device  100  shown in  FIG. 2  and typically exits a rear surface, opposite the first surface, not shown in  FIG. 2 . 
         [0045]    In the configuration illustrated, EHD device  100  comprises first and second opposing frame members  104  and  106  that function to hold, or support, corona discharge electrode  110  and collector electrode array  120 . Frame members  104  and  106  may be fabricated of a dielectric material in order to provide electrical isolation from other components of EHD device  100 . Corona discharge electrode  110  in device  100  has a small radius of curvature and, in some embodiments, may take the form of a wire or rod. Other shapes for corona discharge electrode  110  are also possible; for example, corona discharge electrode  110  may take the shape of barbed wire, a band, blade or place that, in some embodiments, may present a knife- or serrated-edge. In some embodiments, a cross-section such as illustrated in  FIG. 1  for electrode  10  may be employed. Typically, a small radius of curvature or sharp point tends to facilitate ion production at an appropriate point when high voltage is applied. 
         [0046]    In general, corona discharge electrode  110  may be fabricated in a wide range of materials. For example, in some embodiments, compositions such as described in U.S. Pat. No. 7,157,704, filed Dec. 2, 2003, entitled “Corona Discharge Electrode and Method of Operating the Same” and naming Krichtafovitch et al. as inventors may be employed. U.S. Pat. No. 7,157,704 is incorporated herein for the limited purpose of describing materials for some corona discharge electrodes that may be employed in some embodiments. In general, a high voltage power supply (not specifically shown) creates the electric field between corona discharge electrode  110  and collector electrode array  120 . 
         [0047]    In the embodiment of  FIG. 2 , frame members  104  and  106  include a pair of curved recesses  108 , generally conformal with an end portion of corona discharge electrode  110 . Each opposing end of corona discharge electrode  110  passes through a respective recess  108  and is attached to an interior portion (not shown) of a respective frame member. Recess  108  provides a transition region for corona discharge electrode  110  to pass through from its positioning proximate to collector electrode array  120  and one of frame members  104  and  106 . The transition region eliminates the sharp points that may occur at an abrupt junction between corona discharge electrode  110  and its respective frame member, thereby reducing arcing and other undesirable effects in the surrounding high electric field created during operation of EHD device  100 . 
         [0048]    With continued reference to  FIG. 2 , collector electrode array  120  comprises a plurality of substantially parallel unit structures  130  attached to a pair of parallel and substantially flat, spaced apart support members  132 . Each unit structure  130  functions as a collector electrode and may generally have greater depth (in the y direction) than width (in the x direction). Unit structures  130  may be fabricated of any suitable metal material, such as aluminum or copper. The number of, and distance between, unit structures  130  in collector array  120  may vary according to device specifications. Unit structures  130  are generally planar and present a curved leading surface exposed toward corona discharge electrode  110 . In some embodiments, unit structures  130  include a generally rectangular extent in the direction of fluid flow (the y direction), although, more generally, may be formed in other shapes. 
         [0049]      FIG. 3A  illustrates a partial cross-sectional perspective view, in accordance with some embodiments of the present invention, of a collector electrode array that may be employed by the EHD device illustrated in  FIG. 2 . More specifically,  FIG. 3A  illustrates several adjacent unit structures  130  of collector electrode array  120 . For simplicity of description (and generality with respect to alternative EHD device configurations), such unit structures  130  are hereafter referred to as collector electrodes  130 , although persons of ordinary skill in the art will immediately recognize that, in some configurations, additional structures (such as support members  132  illustrated in  FIGS. 2 ,  3 A and  3 B) may be electrically conductive and act as part of an overall “collector electrode.” In view of the foregoing, we now turn to the generally curved leading surface(s)  136  of collector electrodes  130 . 
         [0050]    Fluid flow through collector electrode array  120  is generally in the direction of arrow  102 . In the embodiment shown in  FIG. 3A , collector electrodes  130  are substantially rectangular in shape, having a leading edge  138  disposed closest to corona discharge electrode  110  and a trailing edge  137  opposite to leading edge  138 . Leading edge  138  includes a contoured or curved surface  136 . As will be understood by persons of ordinary skill in the art, consistent with principles of high voltage design, curved leading surface  136  is intended to present generally curvaceous surface contours toward corona discharge electrode  110  and any sharp exposed edges are merely an artifact of the illustration and cross section of  FIG. 3A . Corona electrode  110  is shown positioned a distance, d, above collector electrodes  130 . Distance d may sometimes be referred to as the “gas gap” or “air gap.” 
         [0051]    EHD device  100  may be constructed in a variety of sizes, and thus is suitable for a variety of thermal management applications involving the cooling of electronic circuits. In one exemplary implementation of EHD device  100 , frame sections  104  and  106  and collector electrodes  130  of collector array  120  have a height of approximately 3 mm, corona discharge electrode  110  may be a bare or coated tungsten wire having a diameter of about 12.5 μm, collector electrodes  130  in collector array  120  have a width (thickness) of about 0.25 mm and are spaced approximately 3 mm apart on center, and the distance, d, between corona discharge electrode  110  and collector electrodes  130  is approximately 1.6 mm. The voltage applied across the air gap between corona discharge electrode  110  and collector electrodes  130  may be in the range of 1.5 kV to 4 kV. 
         [0052]      FIG. 3B  is a simplified cross-sectional view of corona discharge electrode  110  and a single collector electrode  130  instance in accord with the curved leading surface profiles illustrated  FIG. 3A . In particular,  FIG. 3B  illustrates a side elevation view of collector electrode  130 . In operation, when an electric field is created between corona discharge electrode  110  and collector electrode  130 , ions generally flow in the directions of the electric field lines represented by the arrows. Curved leading surface  136  may provide certain enhancements to the operation of EHD device  100 . For example, as demonstrated by 3D electrical field simulations performed on a corona discharge electrode and collector electrode configuration similar to that illustrated in  FIG. 3B , utilizing a generally curved leading surface  136  for instances of collector electrode  130  may allow for a shorter distance, d, between corona electrode  110  and collector electrode  130 , while at the same time increasing ion production and assisting in preventing sparks and arcing. In addition, utilizing curved surface  136  for collector electrode  130  may provide electrical separation between adjacent corona discharge electrodes in some embodiment described elsewhere herein that gang multiple EHD device instances. 
       Simulation Analysis 
       [0053]      FIGS. 13A ,  13 B and  13 C illustrate simulation plots of electric field characteristics for a variety of illustrative geometries and operational pairings of corona discharge and collector electrodes. In particular,  FIG. 13A  is a simulation plot of electric field strength and equal potential surfaces for a representative configuration in which a collector electrode having a generally circular curved leading surface profile is paired with a corona discharge wire electrode positioned equidistant from all points along the curved leading surface of the collector electrode. 
         [0054]    Simulation plot  340  illustrates the direction and distribution of the electric field using conventional symbology in which the length and direction of an arrow emanating from a given point code strength and direction of the field at that point. Equipotential surfaces are also illustrated. More specifically, simulation plot  340  illustrates electric field characteristics when a voltage is applied between a wire-type corona discharge electrode  344  and a collector electrode  347  having circular profile  348  which distributes the electric field over a larger surface. Modeled collector electrode  347  is shown spanning a distance between two frame or support sections  353 . 
         [0055]    In general, the amplitude of the electric field at a particular location is represented by shading and length of arrows. For example, arrows (e.g., arrows  350  and  351 ) emanating from points close to corona discharge electrode  344  indicate the highest electric field strength. Arrows emanating from points further from corona discharge electrode  344  (e.g., arrow  352 ) indicate points where the electric field of lesser strength, while arrows emanating from points much further from corona discharge electrode  344  (e.g., arrow  354 ) indicate points where the electric field of significantly lower strength. The ion movement follows the electric field lines represented by the arrows in simulation plot  340 . It will be appreciated that contoured collector electrode  347  tends to increases the travel distance of ions following the steeped electric field gradients, which in the illustration of  FIG. 13A  provides a net flow direction generally aligned with arrow  351 . 
         [0056]    This increased distance can increase the efficiency of the momentum exchange between the ions and air molecules and may be an advantage of some curved leading surface designs. However, it should be noted that, in the results illustrated in  FIG. 13A , electric field lines tend to collect (and a large gradient exists) at the uppermost surfaces of modeled collector electrode  347  where it joins support sections  353 . The large gradient suggests an increased likelihood of arcing between corona discharge electrode  344  and the top portion of support sections  353 . 
         [0057]    Like the collector electrode illustrated in  FIG. 3B , modeled collector electrode  347  presents a curved leading surface; however, the unlike collector electrode  130  and unlike the configuration illustrated in  FIG. 3B , the modeled configuration positions modeled corona discharge electrode  344  and modeled collector electrode  347  so that points along the curved leading surface of modeled collector electrode  347  are equidistant from corona discharge electrode  344 . 
         [0058]      FIG. 13B  presents a modeled configuration in which the wire-type corona discharge electrode is replaced with a blade type configuration. For simplicity,  FIG. 13B  illustrates equal potential contours but omits the corresponding electric field lines. In simulation plot  360 , the gradient of the electric field at a particular location is shown by the closeness of the equal potential contour lines. Examples of equal potential contour lines are lines  361  and  372 . The more closely spaced contour lines indicate a steeper (or larger) electric field gradient. In  FIG. 13B , simulation plot  360  is based on a blade-shaped corona discharge electrode  364  paired with a collector electrode  367  having a circular curved surface  368 . As before, collector electrode  367  is shown spanning a distance between two frame or support sections  363 . Area  370  represents the gradient of the electric field at surface  368  of collector electrode  367 . The fact that area  370  is substantially not uniform in width (i.e., distance between plot lines) indicates that the electric field gradient is non-uniform at surface  368 . In addition, the electric field has a large gradient as it approaches each of support sections  363 , as indicated by contour line  372 . The large gradient at support sections  363  suggests a higher likelihood of arcing between corona discharge electrode  364  and the top portion of support sections  363 . 
         [0059]    In contrast,  FIG. 13C  is a simulation plot of equal potential contours for a configuration in which a collector electrode presents a generally parabolic leading surface profile. Corona discharge electrode  384  presents a bulbous leading profile and small radius of curvature trailing cross section. As before, the collector electrode (here collector electrode  387 ) spans a distance between two frame or support sections (here support sections  383 ), though with a curved profile that, unlike that illustrated in  FIGS. 13A and 13B , places a center portion of collector electrode  387  at a minimum distance from corona discharge electrode  384  and presents portions of the generally parabolic leading surface profile that extend outward toward support sections  383  at increasing distances therefrom. 
         [0060]    Area  390  represents the gradient of the electric field at surface  388  of collector electrode  387 . It can be seen that, in contrast to plot  360 , area  390  is more uniform in width, indicating that the electric field gradient is substantially more uniform across surface  388 . In addition, it can be seen from simulation plot  380  that a comparatively smaller gradient exists at the top of support sections  383 , as indicated by contour line  392 , in comparison to that shown in plot  360 . The lower gradient at support sections  383  suggests a lower likelihood of arcing between corona discharge electrode  384  and the top portion of support sections  383 . Thus, in some embodiments of the EHD devices illustrated herein, it may be desirable to provide a generally curved surface of a collector electrode, such as that provided by parabolic curved surface  388 , which may provide increasing inter-electrode distance at portions of the collector electrode extending away from a minimum distance point (here at the central axis). 
         [0061]    With continued reference to  FIGS. 13B and 13C , plots  360  and  380  also illustrate an operational characteristic of the corona discharge electrode. The shape of corona discharge electrode  384  in plot  380  is different from the shape of corona discharge electrode  364  in plot  360 . While this difference in shape has no (or at least negligible) effect on the electric field gradient at the surface of the collector electrode, as discussed above, the shape of the corona discharge electrode may either adversely or positively effect the operation of the EHD device. In plot  260 , blade-shaped corona discharge electrode  364  has the same thickness from top to bottom. This profile forms two regions  374  and  375  of high electric field gradient, indicated by the closeness of contour lines on both top and bottom ends of electrode  364 . Both of these regions are two corona discharge regions. Top corona region  374  reduces EHD device efficiency because the ion movement direction away from collector electrode  367  is against the air flow direction. In contrast, corona discharge electrode  384  has a “teardrop” shape; specifically, top region  384   a  of electrode  384  has a larger radius of curvature than the bottom end facing the collector electrode. Plot  380  shows that this shape reduces the electric field gradient at top region  384   a , as indicated by contour line  394 . The teardrop shape, therefore, limits the corona discharge from top end  384   a  of corona discharge electrode  384 , and improves device efficiency. 
       Exemplary Curved Leading Surface Profiles for Collector Electrodes 
       [0062]    Consistent with the foregoing simulations,  FIGS. 15 ,  16 ,  17  and  18  illustrate a variety of electrode configurations and geometries for EHD devices in accordance with some embodiments of the present invention and in which points that are at increasing angular distance from a dominant fluid flow direction along curved leading surfaces of respective collector electrodes are at increasing distance from a corona discharge electrode. In particular,  FIG. 15  illustrates a generally parabolic leading surface profile  1501  for a collector electrode (such as collector electrode  130 , recall  FIG. 3B ) in which points  1502 ,  1503  at increasing angular distances (θ 1 , θ 2 ) from an axis  1511  are at increasing distances (d(θ 1 ), d(θ 2 )) from a corona discharge electrode  1504 . A nearest point along generally parabolic leading surface profile  1501  is at a minimum distance D min  from corona discharge electrode  1504 . 
         [0063]      FIG. 16  illustrates a generally elliptical leading surface profile in which points at increasing angular distance are at increasing distance from a corona discharge electrode. As before, a leading surface profile  1601  for a collector electrode (such as collector electrode  130 ) exhibits points  1602 ,  1603  at increasing angular distance (θ 1 , θ 2 ) from an axis  1611  that are at increasing distances (θ 1 ), d(θ 2 )) from a corona discharge electrode  1604 . A nearest point along generally parabolic leading surface profile  1601  is at a minimum distance D min  from corona discharge electrode  1604 . 
         [0064]      FIG. 17  illustrates a generally curved leading surface profile  1704  for a collector electrode displaced from a corona discharge electrode so that points  1702 ,  1703  at increasing angular distance (θ 1 , θ 2 ) from an axis are at increasing distances (d(θ 1 ), d(θ 2 )) from corona discharge electrode  1704 . In general any of a variety of curved surface profiles (e.g., parabolic, elliptical, circular, caternary, etc.) may be employed.  FIG. 18  illustrates, in particular, a circular leading surface profile  1801  for a collector electrode having a radius of curvature, R, substantially greater than a minimum distance, D min , between corona discharge and a nearest point on the collector electrode. As before, the leading surface profile  1801  exhibits points  1802 ,  1803  at increasing angular distances (θ 1 , θ 2 ) from an axis that are at increasing distances (d(θ 1 ), d(θ 2 )) from the corona discharge electrode (here corona discharge electrode  1804 ). 
       Additional Embodiments and Design Variations 
       [0065]      FIG. 14A  is a side view of an EHD device  300 , which is a variation of corona discharge electrode  110  and collector electrode array  120  of EHD device  100  previously described with reference to  FIG. 2 . EHD device  300  has a width in the x-direction greater than that shown for EHD device  100 . EHD device  300  comprises first and second opposing frame members  304  and  306  that function to hold, or support, corona discharge electrode  310  and collector electrode array  320 . Collector array  320  comprises a plurality of substantially parallel unit structures  330  attached to a pair of parallel and substantially flat, spaced apart support members  332 . As before, individual unit structures  330  may be understood to constitute collector electrodes. Also as before, individual collector electrodes exhibit curved leading surfaces such as described above. 
         [0066]    During operation of the EHD device  300 , corona discharge electrode  310  may lose some of its initial tension and may sag downward toward collector electrode array  320 . Support member  324  provides a resting place for corona discharge electrode  310  and maintains a substantially equal distance of corona discharge electrode  310  from unit structures  330  during operation of EHD device  300 . Support member  324  may be made of a dielectric material in order to provide the necessary electrical isolation from other components of EHD device  300 .  FIG. 14B  is a cross-sectional view of the EHD device illustrated in  FIG. 14A  and illustrate a curved or rounded profile for support member  324 . The curved or rounded profile of support member  324  tapers from a narrow profile proximate to corona discharge electrode  310  to a broader profile proximate to collector electrode array  320 . This profile may function to decrease the electric field gradient in the vicinity of support member  324 , thereby decreasing the risk of arcing proximate to support member  324 . In a variation of the embodiment shown in  FIG. 14B , curved or rounded support member  324  need not extend so far as to contact corona discharge electrode  324 ; instead, the extent of support member  324  may be such that it provides a gap between the support member and corona discharge electrode  324 , thereby providing a lower resting place for corona discharge electrode  324  should it sag during operation of EHD device. 
         [0067]      FIG. 4  is a side view of an EHD device configuration in accordance with some embodiments of the present invention and in which multiple assemblies are ganged to increase total volume of fluid flow. A first, or front, surface of EHD device  400  is situated in a three-dimensional coordinate system  101  in which the x-y plane respectively designates the width and depth of device  400  and the z direction designates the height, h, of device  400 . In  FIG. 4 , device  400  motivates flow of a fluid in the y direction; that is, fluid is drawn into the first, or front, surface of device  400  shown in  FIG. 4  and exits a rear surface, opposite the first surface, not shown in  FIG. 4 . 
         [0068]    EHD device  400  comprises a plurality of corona discharge electrodes and associated collector electrode arrays of the type described with respect to EHD device  100  of  FIG. 2 , assembled in a single housing or frame. The presentation in  FIG. 4  of EHD device  400  as having three EHD device  100  assemblies is solely for the sake of illustration, and is not intended to be limiting in any way. EHD device  400  comprises first and second opposing frame members  404  and  406  that function to hold, or support, corona discharge electrodes  110 ,  112  and  114  and associated collector electrode arrays  120 ,  122  and  124 . Each frame member  104  and  106  comprises a plurality of recesses  108 . An opposing end of each corona discharge electrode  110 ,  112  and  114  passes through a recess  108  in each of the opposing frame members and is attached to an interior portion (not shown) of a respective frame member. 
         [0069]    Corona discharge electrodes  110 ,  112  and  114  in device  400  have small radii of curvature and, in some embodiments, may take the form of wire or rods. Other shapes are also possible; for example, corona discharge electrodes  110 ,  112  and  114  may take the shape of barbed wire, a band, blade or, in some embodiments, may present a knife- or serrated-edge. In some embodiments, a cross-section such as illustrated in  FIG. 1  for electrode  10  (or as illustrated with reference to the simulation plot of  FIG. 13C ) may be employed. As noted with respect to EHD device  100  of  FIG. 2 , corona discharge electrodes  110 ,  112  and  114  may likewise be fabricated in a wide range of materials. A high voltage power supply that creates the electric field between corona discharge electrodes  110 ,  112  and  114  and respective collector electrode arrays  120 ,  122  and  124  will be understood, but is not separately shown in  FIG. 4 . 
         [0070]    With continued reference to  FIG. 4 , each collector electrode array  120 ,  122  and  124  comprises a plurality of substantially parallel unit structures  130  attached to a pair of parallel and substantially flat, spaced apart support members  132 . Support members  132  are disposed within openings  134  in frame sections  404  and  406 . Each unit structure  130  functions as a collector electrode and may generally have greater depth (in the y direction) than width (in the x direction). Unit structures  130  may be fabricated of any suitable metal material; aluminum, for example, may be chosen for its low cost, good electrical and thermal conductivity, and for its ease in forming a desired shape. Unit structures  130  may be substantially rectangular in shape, or may be formed in other shapes. As before, individual unit structures  330  may be understood to constitute collector electrodes. Also as before, individual collector electrodes exhibit any of a variety of curved leader surfaces such as described above. 
         [0071]    In one exemplary implementation of EHD device  400 , frame sections  404  and  406  may have a height, h, of approximately 9 mm, and the distance  412  between adjacent corona discharge electrodes  112  and  114  may be approximately 4 mm. 
         [0072]      FIG. 5  schematically illustrates an environment in which devices such as EHD device  100  of  FIG. 2  and EHD device  400  of  FIG. 4  may operate. EHD device  400  or EHD device  100  may be positioned proximate to an electronic circuit  510  (or other heat source) that generates heat during its operation. Note that electronic circuit  510  may also be in thermal contact with a heat sink or other thermal management device (not shown) positioned above, below or adjacent to electronic circuit  510 . As defined herein, a heat sink is an object that absorbs and dissipates heat from another object using either direct or radiant thermal contact. In operation, high voltage power supply  530  is operated to create a voltage difference between corona discharge electrodes  110 ,  112  and  114  ( FIG. 4 ) and collector electrode arrays  120 ,  122  and  124 , generating an ion stream (as described with respect to  FIG. 1 ) that moves ambient air toward collector electrode arrays  120 ,  122  and  124 . The resulting air movement over and around electronic circuit  510  dissipates heat in the air above and around electronic circuit  510 . Note that the position of power supply  530  relative to EHD device  400  and electronic circuit  510  may vary from that shown in  FIG. 5 . Also, note that while  FIG. 5  illustrates flow toward electronic circuit, in some configurations flow may also (or alternatively) be drawn past an electronic circuit or heat sink. 
         [0073]      FIG. 6  is a schematic block diagram illustrating a second environment in which devices such as EHD device  100  of  FIG. 2  and EHD device  400  of  FIG. 4  may operate. Electronic device  600  comprises a substantially rectangular housing  616 , or case, having a cover  610  that includes a display device  612 . A portion of the front surface  621  of housing  620  has been cut away to reveal interior  622 . Housing  620  of electronic device  600  may also comprise a top surface not shown in  FIG. 6  that supports one or more input devices that may include, for example, a keyboard, touchpad and tracking device. Electronic device  600  further comprises electronic circuit  660  which generates heat in operation. A thermal management solution comprises heat pipes  644  that draw heat from electronic circuit  660  to heat sink devices  642 . Further shown in interior  622  of electronic device  600  is a pair of EHD devices  620  which may represent EHD device  100  of  FIG. 2 , EHD device  400  of  FIG. 4  or a variation of either device as described above. Each one of a pair of EHD devices  620  is powered by high voltage power supply  630  and is positioned proximate to one of the heat sinks  642 . Electronic device  600  may also comprise many other circuits, depending on its intended use; to simplify illustration of this second embodiment, other components that may occupy interior area  622  of housing  620  have been omitted from  FIG. 6 . 
         [0074]    With continued reference to  FIG. 6 , in operation, high voltage power supply  630  is operated to create a voltage difference between the corona discharge electrodes and collector electrode arrays disposed in each EHD device  620 , generating an ion stream (as described with respect to  FIG. 1 ) that moves ambient air toward the collector electrode arrays. The moving air leaves EHD device  620  in the direction of arrows  602 , traveling through the protrusions of heat sinks  642  and through an exhaust grill or opening (not shown) in the rear surface  618  of housing  620 , thereby dissipating heat accumulating in the air above and around heat sinks  642 . Note that the position of power supply  630  relative to EHD devices  620  and electronic circuit  660  may vary from that shown in  FIG. 6 . Note also that other flow topologies may be supported. In a variation of the operating environment shown in  FIG. 6 , each EHD device  620  may be supplied with voltage from its own dedicated power supply. 
         [0075]      FIG. 7  is a schematic diagram of a second embodiment of an EHD device. EHD device  700  comprises corona discharge electrodes  714  supported in frame  712 . Collector electrode array  720  comprises individual unit structures  722  that function both as collector electrodes and components of a heat sink. The leading edge of each individual unit structure  722  closest to corona discharge electrode array  710  exhibits a curved leading surface, such as curved surface  136  illustrated in  FIG. 3A  or any of the curved leading surfaces described with reference to  FIGS. 15-18 . A high voltage power supply (not shown) provides electrical energy to corona discharge electrode  714  and to collector electrode array  720 . In operation, an ion stream generated by corona discharge electrode  714 , according to the corona discharge principles illustrated in  FIG. 1 , produces fluid movement in the direction of arrow  702  between unit structures  722  of collector electrode array  720 . Collector electrode array  720  may be positioned proximate to a heat pipe  740  that transfers heat from an electronic circuit or other heat source not specifically shown in  FIG. 7 . The components of EHD device  700  may be assembled on surface  704  which may be a thermally conductive substrate for transferring heat from heat pipe  740  to collector electrode array  720 . EHD device  700  and surface  704  may be fabricated in a wide range of sizes, according to the specifications of a particular thermal management application. When corona discharge electrode  714  is fabricated as described for EHD device  400  of  FIG. 4 , EHD device  700  may have a height (z direction) in the range of 0.5 mm to 30 mm, and thus may be suitable for use in an electronic device having a thin form factor. Note that the scale of the individual components shown in  FIG. 7  is solely for illustration purposes; each component may have height, width and depth dimensions that are different from the relative dimensions shown in the Figure. 
       Embodiments of EHD Devices with Stages 
       [0076]      FIG. 8A  is a schematic block diagram illustrating the components of a multi-stage EHD device  800 . Individual EHD devices  810   a ,  810   b  through  810   n , each of which is referred to individually herein as EHD device stage  810   n , are disposed in parallel and orthogonal to a desired air flow direction  802 . High voltage power supply  830  is connected to each individual EHD device stage  810   n  by way of electrical conductors  836 . Individual EHD device stage  810   n  may comprise any one EHD device  100  of  FIG. 2 , EHD device  400  of  FIG. 4 , or EHD device  700  of  FIG. 7 , or variations of those devices as described above. In a variation of multi-stage EHD device  800 , high voltage power supply  830  may comprise a set of individual power supplies, each one coupled to an individual EHD device stage  810 . In operation, each individual EHD device stage  810   n  may be operated simultaneously and synchronously with the others in order to produce increased volume and pressure of air flow in the direction of arrow  802 , thereby sequentially accelerating a fluid through the multiple stages. Synchronous operation of EHD device stages  810   n  is defined herein to mean that a single power supply, or multiple synchronized and phase-controlled power supplies, provide high voltage power to each EHD device stage  810   n  such that both the phase and amplitude of the electric power applied to the same type of electrodes in each stage (i.e., the corona discharge electrodes or the collector electrodes) are aligned in time. 
         [0077]    U.S. Pat. No. 6,727,657, entitled “Electrostatic Fluid Accelerator for and a Method of Controlling a Fluid Flow” (hereafter, “the &#39;657 patent”) provides a discussion of the configuration and operation of multi-stage EHD devices. U.S. Pat. No. 6,727,657 is incorporated by reference herein in its entirety for all that it teaches. In particular, the &#39;657 patent discloses that increasing electrode density, defined as stages-per-unit-length, and eliminating or significantly decreasing stray currents between neighboring stages is accomplished, in part, by powering neighboring EHD stages with substantially the same voltage waveform, i.e., the potentials on the neighboring electrodes have the same or very similar alternating components so as to eliminate or reduce any a.c. differential voltage between stages. Operating in such a synchronous manner between stages, electrical potential differences between neighboring electrodes of adjacent EHD stages remains constant and any resultant stray current from one electrode to another is minimized or completely avoided. 
         [0078]    Synchronization among the EHD device stages may be implemented by different means, but most easily by powering neighboring EHD device stages with respective synchronous and syn-phased voltages from one or more power supplies, or with power supplies synchronized to provide similar amplitude a.c. components of the respective applied voltages. This may be achieved with the same power supply connected to neighboring EHD stages or with different, preferably matched power supplies that produce a synchronous and syn-phased a.c. component of the applied voltage. In one embodiment, a suitable power supply may comprise a plurality of converters for transforming a primary power to high voltage power, with each converter being independently connected to a respective one of the EHD device stages for providing high voltage power thereto. A suitable power supply may further comprise a controller connected to the converters for synchronizing the alternating components of the high voltage power provided by each converter. 
         [0079]      FIG. 8B  schematically illustrates a first embodiment for computing a suitable inter-stage distance. As applied to  FIG. 8B  herein, the &#39;657 patent discloses that the distance  850  between corona discharge electrode  812  and collector electrode  814  of EHD device stage  810   a  may be comparable to the distance  860  between the trailing edge of collector electrode  814  and corona discharge electrode  816  of subsequent EHD device stage  810   b ; that is, the closest distance between elements of adjacent stages is not much greater than the distance between electrodes within the same stage. In one embodiment, the inter-stage distance  860  between collector electrode  814  and corona discharge electrode  816  of the adjacent stage may be between 1.2 and 2.0 times that of the intra-stage spacing distance  850  between corona discharge electrode  812  and collector electrode  814  (or, similarly, the distance between corona discharge electrode  816  and collector electrode  818 ) within the same stage. For illustration purposes only, in  FIG. 8B , distance  860  is shown as being approximately 1.4 times distance  850 . Because of this consistent spacing, capacitance between EHD device stage electrode pair  812  and  814  and between inter-stage corona electrode pair  812  and  816  are of the same order. 
         [0080]    The &#39;657 patent further explains that the closest spacing of electrodes of adjacent EHD device stages may be approximated as follows. A typical EHD device operates efficiently over a rather narrow voltage range. The voltage V c  applied between the corona discharge and collecting electrodes of the same stage should exceed the so-called corona onset voltage V onset  for proper operation. That is, when voltage V c  is less than V onset , no corona discharge occurs and no air movement is generated. At the same time V c  should not exceed the dielectric breakdown voltage V b  so as to avoid arcing. Depending on electrode geometry and other conditions, V b  may be more than twice as much as V onset . For typical electrode configurations, the V b /V onset  ratio is about 1.4-1.8 such that any particular corona discharge electrode should not be situated at a distance from a neighboring collecting electrode where it may generate a “back corona.” Therefore, the normalized distance “aNn” between the closest electrodes of neighboring stages may be at least 1.2 times greater than the normalized distance “aNc” between the corona discharge and the collecting electrodes of the same stage but not more than 2 times greater than distance “aNc.” That is, electrodes of neighboring stages may be spaced so as to ensure that a voltage difference between the electrodes is less than the corona onset voltage between any electrodes of the neighboring stages. Thus, the voltage frequency and phase control discussed above allow neighboring EHD stages to be closely spaced at a distance of from 1 to 2 times an inter-electrode distance within a stage. 
         [0081]      FIG. 8C  schematically illustrates a second embodiment for computing a suitable inter-stage distance between upstream EHD device stage  810   a  and downstream EHD device stage  810   b  positioned in the same plane as upstream EHD device stage  810   a , designated respectively in the mathematical notation that follows as the “A” unit and the “B” unit. In  FIG. 8C , arrows  870 ,  880  and  890  indicate ionic current flows; specifically, arrows  870  and  890  indicate current flows I AF  and I BF  to collector electrodes in the respective A and B units, and arrow  880  indicates a current flow I BR  to the collector electrode of unit A upstream of unit B. A suitable inter-stage distance between upstream EHD device stage  810   a  and downstream EHD device stage  810   b  may be computed as a distance that satisfies the relationship 
         [0000]    
       
         
           
             
               
                 I 
                 BF 
               
               
                 I 
                 BE 
               
             
             &gt; 
             Z 
           
         
       
     
         [0000]    where Z should be greater than 1, and for high efficiency operation, Z should be greater than 100. 
         [0082]      FIG. 9  is a block diagram of a multi-stage EHD device  900 , of the type illustrated in  FIG. 8 , utilizing individual EHD devices  400  of  FIG. 4 , designated in the Figure as EHD device stages  400   a ,  400   b  and  400   c . Multi-stage EHD device  900  is implemented in the operating environment of the electronic device  600  of  FIG. 6 , as shown by the reference to components  660 ,  644  and  642  of  FIG. 6 . Heat generated by the operation of electronic circuit  660  is drawn through heat pipes  644  to heat sink device  642 . EHD device stages  400   a ,  400   b  and  400   c  are disposed in a parallel arrangement and orthogonal to air flow direction  902  toward heat sink  642 . in operation, high voltage power supply  930  produces a voltage differential between respective pairs of corona discharge electrodes and collector electrodes in each EHD device stage  400   a ,  400   b  and  400   c , via conductors  936 , as described above with respect to  FIG. 8 , in order to generate an ion stream (as described with respect to  FIG. 1 ) that moves ambient air toward the collector electrode arrays in each EHD device stage. The moving air exits EHD device stage  400   c  in the direction of arrow  902 , traveling through the protrusions of heat sink  642  and through an exhaust grill or opening (not shown) in the rear surface of electronic device  600  ( FIG. 6 ), thereby dissipating heat accumulating in the air above and around heat sink  642 . Note that the position of power supply  930  relative to EHD device stages  400   a ,  400   b  and  400   c  and electronic circuit  660  may vary from that shown in  FIG. 9 . Note also that each EHD device stage  400   a ,  400   b  and  400   c  may be supplied with voltage from its own dedicated power supply, each of which may be operated as disclosed in the &#39;657 patent. 
         [0083]      FIG. 10A  depicts an illustration of one embodiment of device  900  of  FIG. 9 . Multi-stage EHD device  1000  comprises three EHD device stages  1010  which accelerate a fluid such as air through duct  1020  into heat sink  1030 . Heat sink  1030  receives heated air from heat pipe  1040  that draws the heated air from an electronic circuit not shown in the Figure. In  FIG. 10B , multi-stage EHD device  1000  is shown positioned within curved frame  1050 . In one implementation of multi-stage EHD device  1000 , curved frame  1050  resides in the interior of a laptop computer and has the dimensions of approximately 55 mm in width, 50 mm in depth, and 4 mm in height. The volume encompassed by curved frame  1050  is sufficient to hold a low-profile axial fan for use in pushing air toward heat sink  1030  in order to dissipate heat generated by an electronic circuit not shown in the Figure. In  FIG. 10B , multi-stage EHD device  1000  is shown as a direct replacement for the fan. 
         [0084]      FIG. 11  is a block diagram of multi-stage EHD device  900  of  FIG. 9  including one embodiment of a temperature sensor feedback system comprising temperature sensors  1105 ,  1107  and  1109 . In the embodiment illustrated in  FIG. 11 , temperature sensor  1107  positioned internally in heat sink  642  sends temperature data along electrical connection  1108  to a control unit in power supply  930 ; temperature sensor  1105  positioned at the exhaust end of heat sink  642  sends temperature data along electrical connection  1106  to the control unit in power supply  930 ; and temperature sensor  1109  positioned in electronic circuit  660  sends temperature data along electrical connection  1110  to the control unit in power supply  930 . In response to the temperature data received, the control unit of power supply  930  may adjust the operation of EHD device stages  400   a ,  400   b  and  400   c  via electrical connectors  936 . In one embodiment, the control unit of power supply  930  may also adjust the operation of electronic circuit  660  by sending control signals to circuit  1114  along electrical connector  1112  in response to received temperature data. 
         [0085]      FIG. 12  is a block diagram of a multi-stage EHD device  1200 , of the type illustrated in  FIG. 8 , utilizing individual EHD devices of the type illustrated in  FIG. 7  in which the collector electrode arrays  1220  and  1222  comprise individual unit structures that function both as collector electrodes and heat sink fins. Each collector electrode array  1220  and  1222  is paired with a respective corona discharge electrode array  1210  and  1212 . Heat generated by the operation of electronic circuit  1260  is drawn through heat pipes  1244  which divide into two portions that direct heated air separately to each EHD device stage. The EHD device stages are placed in series and disposed in a parallel arrangement and orthogonal to air flow direction  1202 . Note that the scale of the individual components shown in  FIG. 12  is solely for illustration purposes; each component may have height, width and depth dimensions that are different from the relative dimensions shown in the Figure. In addition, the number of stages may be more or less than shown in  FIG. 12 . 
         [0086]    With continued reference to  FIG. 12 , in operation, high voltage power supply  1230  produces a voltage differential between respective pairs of corona discharge electrodes and collector electrodes in each EHD device stage, via conductors  1236 , as described above with respect to  FIG. 8 , in order to generate an ion stream (as described with respect to  FIG. 1 ) that moves ambient air toward collector electrode arrays  1220  and  1222  in each EHD device stage. The moving air dissipates heat that collects within collector electrode arrays  1220  and  1222  in each EHD device stage as it moves in the direction of arrow  1202 . Note that the position of power supply  1230  relative to the EHD device stages and electronic circuit  1260  may vary from that shown in  FIG. 12 . In a variation of the embodiment shown in  FIG. 12 , each EHD device stage may be supplied with voltage from its own dedicated power supply, each of which may be operated as disclosed in the &#39;657 patent. 
         [0087]    While the techniques and implementations of the EHD devices discussed herein have been described with reference to exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the appended claims. In addition, many modifications may be made to adapt a particular situation or material to the teachings without departing from the essential scope thereof. Therefore, the particular embodiments, implementations and techniques disclosed herein, some of which indicate the best mode contemplated for carrying out these embodiments, implementations and techniques, are not intended to limit the scope of the appended claims.