Air always contains a little water vapour, in arid lands the air is usually too hot to allow the water carried to condense. This occurs under hot desert conditions. In cold deserts any water present is only available as ice and the air is almost devoid of water.
To extract water from hot desert air the air must be cooled to blow the dew point. If the air is cooled to below freezing water temperatures then ice will be deposited.
A possible simple way of cooling the air is to compress it, allow the heat of compression to dissipate then allow the air to expand into an insulated cavity.
If air is forced through a nozzle containing a porous material such as sharp sand and then allowed to expand into an insulated chamber, it will cool because of adiabatic expansion.
A simple device that will do this is shown in the diagram below. The pump needs to deliver air to the chamber so that for a cooling rate of (Power Output of motor/R (the gas constant)) degrees a second in suitable units (such as MKS, FPS or SI).
The amount of water extracted would depend on the absolute humidity and the rate of mass flow of the air.
Mass (Kg) of water extracted per second= power of pump (Watt)
/(latent heat of vaporisation of water/Kg)
Volume of air/second = Absolute Humidity (mass (Kg) /volume (L))/Power of Pump.