I've been asked many times now whether I can design a backyard system for them. Of course the answer is yes, but since my time isn't generally free1, the issue is often dropped. These aquaponic system ideas are for those who aren't particular, or are just looking for a starting point for a system they wish to build.
I would like to start with a very simple aquaponic system in a single tank. With only one tank you have basically two options for growing plants. You can use water loving plants like watercress where you don't need any growing medium, or you can use rock where it is either piled on one side of the tank (for large tanks) or contained in baskets. Even if you use rock, you will need to select your plants carefully and ensure you water is very well oxygenated.
The only component necessary for this system is an air compressor to run one or more air lifts. The number of air lifts will depend on how large of a tank you use, and the vegetation and fish load. To give you an idea, I would suggest four air lifts in a one foot deep tank that is four foot wide and five foot long (20 cubic feet or 0.56 cubic meters) which holds roughly 150 gallons (560 liters).
You can use any fish you want in this system, but make sure you first 'clean' the fish (I will write an article on this subject soon), and they should be appropriate to the number of gallons of swim-able water. For smaller tanks I'd recommend mosquito fish since they are very cheap to obtain, they breed prolifically without any intervention, and feeding them only requires a few goldfish flakes every now and then (a medium pinch every three days or so will easily take care of thirty fish if the tank is not indoors).
If you use cuttings of plants and they are taken from a source that might harbor diseases that could affect your fish (ponds, other aquaponic systems, etc), you should isolate the plants (with an air lift or bubbler if necessary) for a minimum of 10 days at 80° temperatures. This will kill most diseases since they will not have a host (fish) to breed in.
If you choose to use a floating raft style system, I would suggest doing some research on rafts to choose a material that is non-toxic. Most foams are poor choices in an aquaponic system because they are often made with materials that leach into the water. Also avoid any fire retardants, particularly if you intend to consume anything from your system.
Finally we need to talk about water. You can safely ignore the pH of your aquaponic system's water as long as it is stable. If you are not sure if it is stable, or if your water doesn't have a natural pH buffer, then I would suggest burying eggshells in the rock (or hang a mesh bag of them in the water). The calcium in the shells will buffer your pH around an 8 (slightly alkaline). In addition to ensuring you have a pH buffer, you also may need to degas your water if you are using tap water (not from a well). Tap water often has additives such as chlorine which should be removed to protect your plants and fish. Removal is simply a matter of letting the water sit (preferably in the sun) for up to 48 hours, less if you use a bubbler (12 would probably be sufficient). Note: You should check with your water authority and ensure they do not add chloramines as a cleanser. If they do use chloramines, you will not be able to use the water without first removing them. Before you start thinking that rain water or reverse osmosis water would be best, let me caution you that both of these can be problematic. Rain water often does not contain necessary minerals, and you will need to add them. This includes calcium and potassium, but may also include others I am forgetting at the moment. Reverse osmosis water is even more dangerous because it is so clean that it can actually leach nasty chemicals from metals and plastics that come in contact with the water.
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