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I know this is a silly question, but shouldn't the high speed rotating blades (in case of an axial air compressor or an axial air fan for example) block the air from flowing? What makes air flow between blades?

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    $\begingroup$ They do partially block the flow. When designing a high performance compressor or fan, the number of blades is chosen as a trade off between not doing enough work on the air, and causing too much blockage. For example it isn't just a coincidence that almost all large jet engines, designed by different companies, have close to 25 blades in the fan (find some pictures and count them yourself!) $\endgroup$ – alephzero Jan 5 '19 at 19:09
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The blades are set at an angle which causes them to "dig in" to the air and propel it through the fan disc as the disc rotates. In this manner, the blades add kinetic energy to the air by performing work on it.

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The blades are basically small helicopter wings. They are set to a small angle to the shaft so they create "lift", but because they can't move up with the lift they move the air.

However they can block the air in some cases. When they stall. Stall is when the smooth flow of air around a blade gets turbulent and the blade/wing loses its aerodynamics properties.

In early era of jet engines stalling inside a jet engine was a serious problem. It has been taken care of by digital automatic control of thrust and fuel regimen independent of pilot input.

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