In solution salt becomes Na (sodium) and Cl Chloride. When passed over a weak electrical field the CL is broken into two atoms of chlorine (Cl-)
You may want to re-phrase this. Simply dissolving NaCl in water does not break it chemically. Chloride is the anion Cl− of Chlorine, which is an element. Cl cannot break into two atoms because as part of the NaCl compound there is only one Cl- per one molecule of NaCl.
Chlorine as a substance is a two-atom molecule Cl2, and is gaseous at standard conditions. I hope your chlorinating system does not produce too much of it ;-)