Aquaponics - 2D nitrogen cycle

The nitrogen cycle

Fish excrete ammonia (total ammonia nitrogen, TAN), roughly 3 % of the feed mass. Nitrifying bacteria living on the biofilter surface turn TAN into nitrite (ammonia oxidisers) and then nitrite into nitrate (nitrite oxidisers); plants take nitrate up. Each step is a Monod rate: flux = A r_max x f(T) f(pH) c/(K + c), with A the biofilter area, x the active biofilm fraction, c the concentration and K the half-saturation constant. Concentrations follow V dc/dt = production − consumption − exchange.

Cycling and toxicity

A new system starts with almost no bacteria (x = 0.02 by default). The two biofilm fractions grow logistically, the nitrite oxidisers lag behind, which is why an ammonia peak is followed by a nitrite peak before nitrate rises: the classic fishless cycle of several weeks. Unionised NH₃ is the toxic form; its share is 1/(1 + 10^{pK_a − pH}) with pK_a = 0.09018 + 2729.92/T (Emerson et al., T in kelvin), so warm, alkaline water is far more dangerous. Nitrification speeds up by about 2× per 10 °C (Q₁₀ = 2) and slows below pH 6.5.

Honest simplifications: a lumped (well-mixed) model with one tank volume; fixed kinetic constants chosen in the typical literature range, not fitted to one farm; no oxygen limit, no solids or organic matter mineralisation, no denitrification or micronutrient limits; plant growth is a visual height tied to nitrate uptake; fish number on screen is only a rough drawing of the biomass; thresholds drawn (0.02 mg/L NH₃, 0.5 mg/L nitrite) are general guidelines for warm-water fish.

• Pick Fishless cycling and run it fast: ammonia peaks, then nitrite, then nitrate.
• Pick Overfed fish: unionised ammonia crosses the white line and the fish turn red.
• Pick Too few plants and watch nitrate climb without a ceiling.
• Lower the temperature to 12 °C and see how slowly the bacteria work.