Small particles in air are the most dangerous thing in your life also. Science knows just a little of the all negative health effects as they go directly into your bloodstream from lungs (the most smallest particles).
Industry makes all kind of poisons, which end up into us.
As we grow spirulina, they will also end into it and we eat them.
We should have first a waterfilter, which will take away dust and particles from the air and after that, the air would be directed to the the spirulina.
If the water is in fact lye, made from wood ash: it will also serve as filter from microbe contamination, as most of them from the air, will stick into water and die couse of the high pH in lye: 12-13.
Then you change the lye from time to time.
So first the air is aerated through lye, and then aerated to the algae.
Algae is very good to grow inside home, so the air is cleaner. But if you intend to eat the algae, you should use this kind of simple system.
I will (when I got money for new aquarium pump) make this filter and post pictures also. It would be good to do also experiment, does it affect to the growth of algae in positive way.