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A fellow engineer told me that there are greenhouses which exploit the stack effect, in order to cover some or all of their electrical energy needs. This is achieved by installing small electrical generators with fans mounted on the rotor, on or near a chimney, which has to be large. Due to pressure difference between the greenhouse and the environment, there is a flow of cold air, from outside, which is capable of rotating the fans and therefore producing electrical energy.

By reading the wiki article on the effect, I understood why the chimney has to be large, since $\Delta P$ is proportional to the height h:

$\Delta P = C\alpha h(\frac{1}{T_0}-\frac{1}{T_i})$

where

  • ΔP = available pressure difference [Pa]
  • C = 0.0342, the temperature gradient [K/m]
  • a = atmospheric pressure [Pa]
  • h = height or distance [m]
  • To = absolute outside temperature [K]
  • Ti = absolute inside temperature [K]

But can someone explain to me why the height is proportional to ΔP in a physical/mechanical way? or recommend an article/book/paper that explains it?


EDIT:

Something similar (if not exactly the same) is the solar updraft tower. Again, tower has to be as tall as possible, in order to maximize the power generation. But how is this mechanically explained? Is it because of greater pressure, the speed of the air increased, so fans rotate faster?

enter image description here

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  • $\begingroup$ Pressure difference being proportional to height is just the hydrostatic formula. $\endgroup$ – Deep Dec 16 '18 at 5:51
  • $\begingroup$ @Deep ok... so which other formulas matter in this case? $\endgroup$ – thece Dec 16 '18 at 11:18
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The chimney only has to be high enough to contain an extended column of warm/light air. Too short and the warm air might spill out into the ceiling of the greenhouse instead of being siphoned up the chimney. Every home fireplace relies on this effect to keep smoke from filling the room.

Pressure decreases with height because the pressure comes from the weight of the air above. The higher you go, the less air is above you. The same effect occurs in liquids.

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  • $\begingroup$ I understand both of your arguments. But which is the mechanism behind the correlation ~ Taller Chimney <=> More Energy Production ~ ? $\endgroup$ – thece Dec 16 '18 at 14:16
  • $\begingroup$ @thece More correctly, Greater density difference <=> More energy production. You must know that temperature decreases with altitude. $\endgroup$ – Deep Dec 17 '18 at 5:00
  • $\begingroup$ haha. that's right dear @Deep. Would you mind share with me some book/notes/lectures that analyze the above phenomenon? $\endgroup$ – thece Dec 17 '18 at 11:04
  • $\begingroup$ @thece Books on natural ventilation would give you the general theory. But I don't know of books specific to this topic. $\endgroup$ – Deep Dec 18 '18 at 8:08
  • $\begingroup$ See pages 3 and 12 of E.M. Clancy's paper from 2000 titled "Derivation of simple ventilation and thermal models for a naturally ventilated auditorium with high internal heat gains." $\endgroup$ – Richard H Downey Dec 18 '18 at 14:10

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