page_title: Vaneless ion wind generator
section_title: Simplified analytical model
hierarchical_section_title: Vaneless ion wind generator / Design and construction / Theoretical operation / Simplified analytical model
caption_attribution_description: English: Two droplets acted upon by wind and a uniform electric field, whose equal charges repel one other.
context_page_description: A vaneless ion wind generator or power fence is a device that generates electrical energy by using the wind to move charged particles across an electric field.
  Ion wind generators are not commercially available, though working prototypes and proofs of concept have been created. Several prototypes exist in the Netherlands, one of which resides in Delft University of Technology, whose researchers developed some of the underlying technology. Ion wind generators are currently experimental, while conventional wind turbines are the most common form of wind energy generation. But ion wind generators, which have no moving parts, could be used in urban settings where wind turbines are impractical due to vibrational noise, moving shadows, and danger posed to birds.
context_section_description: Researchers from Delft University of Technology devised an equation to model the behavior of the water droplets as they move through the air in order to optimize the system mathematically and run computer simulations. For the purposes of the model, a simple electrode configuration and uniform electric field is assumed, wherein the electric force exerted on the particles will be directly opposite that of the wind.
  Each particle is acted upon by the force of gravity,
  where is the mass of the iᵗʰ droplet and  is the gravitational acceleration of Earth. The model assumes  is constant and does not take evaporation into account. The atmosphere also exerts a force in the form of buoyancy as the droplets fall,
  where  is the volume of the droplet and is the air density. The droplets are also acted upon by the wind,
  where  is the drag coefficient, is the wind speed, and is the droplet speed. The equation may be simplified in cases of laminar flow, which can be expressed using the Reynolds Number (Re), which is used in fluid mechanics to determine flow patterns. The flow is considered laminar when the Reynolds Number is less than 1, 
   
  where  is the viscosity of air. When the flow is indeed laminar, the drag force can be calculated using Stoke's law, 
   
  where  is the Cunningham slip correction factor, which is assumed to be 1 for particles greater than 1μm in diameter.
  The electric force acting on the droplets is affected by both the external electric field () of the device's electrodes,
  where  is the charge of the iᵗʰ droplet, and the electric fields of other charged droplets,
  where  is the distance between droplet i and droplet j. The sum of these forces represents the researchers' full equation,
  where  is the total force exerted on the iᵗʰ droplet and  is the acceleration of the iᵗʰ droplet. The work done on the iᵗʰ droplet can be calculated using the previous equation,
  where  is the droplet's displacement. The researchers use this to calculate the potential energy difference for the droplet. The sum of work done on each droplet yields the total energy generated from the wind.
caption: Two droplets acted upon by wind and a uniform electric field, whose equal charges repel one other[4]
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