Aquaponics · Hydroponics · Carbonate system · v5
CO₂ / Alkalinity / Buffer capacity
Measurement-based calculation – coupled and decoupled RAS/Aquaponics systems
⚠ Model limitations & error propagation
This tool is an academic demonstration of relationships, not a precision instrument. Critical pH leverage effect: ±0.1 pH units (typical uncalibrated field use) produce ±23 % CO₂ error. With test strips (±0.5), the error exceeds 100 % – the calculation becomes meaningless. Other non-corrected sources: ammonium buffering, organic acids (UASB effluent, acetate/propionate), ionic strength >3 mS/cm, temperature gradients. Phosphate correction of TA is implemented only for ortho-PO₄ – polyphosphates and organically bound P are missing. No substitute for TA titration, ICP-OES/AAS and regular pH calibration.
This tool is an academic demonstration of relationships, not a precision instrument. Critical pH leverage effect: ±0.1 pH units (typical uncalibrated field use) produce ±23 % CO₂ error. With test strips (±0.5), the error exceeds 100 % – the calculation becomes meaningless. Other non-corrected sources: ammonium buffering, organic acids (UASB effluent, acetate/propionate), ionic strength >3 mS/cm, temperature gradients. Phosphate correction of TA is implemented only for ortho-PO₄ – polyphosphates and organically bound P are missing. No substitute for TA titration, ICP-OES/AAS and regular pH calibration.
Measurement input ▾
M1 – Carbonate system
Alkalinity (input unit)
Input converted internally to mmol/L HCO₃⁻ · °dH = drinking water analysis from public utilities
TA Total alkalinity
mmol/L · titration to pH 4.3
pH (electrode)
Daily calibration? Buffer pH 4 + pH 7
Water temperature
°C · at pH measurement point
pH accuracy ±
Dominates CO₂ error propagation
TA accuracy ±
mmol/L
PO₄ correction of titration alkalinity
PO₄ input mode
Only ortho-PO₄ corrected · polyphosphates missing
[PO₄-total] measured
mg/L as P
Feed amount
kg/day
P content of feed
% P (declaration, typically 1.0–1.5 %)
P retention fish
% · Tilapia ≈30, trout ≈35, carp ≈28
M2 & M3 – Fish / System
NH₄-N / day
g/d · from biofilter calculator
System volume
L · open
Fish biomass
kg · open · 0 = no fish
Fish species / Preset
Sets R_ref, T_ref, Q10
R_ref (CO₂ basis)
mg CO₂/kg·h at T_ref · resting metabolism
T_ref
°C · reference temperature for R_ref
Q10
– · species-specific
Dark period
h · 0–24 · 0 = no dark period
M4 – Decoupled system
System mode
CO₂ fish loop (probe)
mg/L · before stripper
CO₂ plant tank (probe)
mg/L · after stripper
Vol. fish loop
L
Vol. plant tank
L
M1 · Carbonate system & error propagation
CO₂ dissolved
–mg/L
HCO₃⁻ effective
–mmol/L
TA measured–
TA unit → mmol/L–
PO₄ correction of TA–
TA_carbonate (corr.)–
CO₂ (mmol/L)–
CO₃²⁻–
Buffer capacity β–
pKa₁ at T–
HCO₃⁻ as mg/L CaCO₃–
HCO₃⁻ as mg/L CaO–
HCO₃⁻ as mg/L MgO–
–
Gaussian error propagation
–
M2 · Alkalinity drift / Nitrification
TA loss / day
–mmol/L·d
TA depleted in
–d
TA current / critical limit–
H⁺ prod. / day–
ΔpH/day (via β)–
KHCO₃ / NaHCO₃ / K₂CO₃–
Alk. equivalent as mg/L CaCO₃·d–
M3 · CO₂ accumulation at night
ΔpH night
–ΔpH
CO₂ morning
–mg/L
Q10 factor (R(T)/R_ref)–
CO₂ rate at system temp.–
Nighttime ΔCO₂–
pH dark drop / morning–
Reference table: TA_carbonate × pH → CO₂ (mg/L) at 25 °C ▾
white <15 · yellow 15–25 · orange 25–40 · red >40 mg/L CO₂ · — = outside practical range
Formulas, assumptions & explicit model limits ▾
Units HCO₃⁻
1 mmol/L HCO₃⁻ = 50.044 mg/L CaCO₃ = 28.04 mg/L CaO = 20.16 mg/L MgO = 2.804 °dH
1 °dH = 0.3566 mmol/L HCO₃⁻ = 17.85 mg/L CaCO₃ = 10.00 mg/L CaO
1 °dH = 0.3566 mmol/L HCO₃⁻ = 17.85 mg/L CaCO₃ = 10.00 mg/L CaO
Carbonate system (Harned & Davis 1943)
pKa₁(T°C) = 6.352 − 0.00521·(T−25) [valid 5–40 °C, deviation <0.005]
HCO₃⁻ ≈ TA_corrected [valid pH 5.5–9; CO₃²⁻ negligible at pH <9]
CO₂(aq) = HCO₃⁻ / 10^(pH−pKa₁) · 44.01 [mg/L]
β ≈ ln(10) · [HCO₃⁻]·[CO₂] / ([HCO₃⁻]+[CO₂]) [mol/L·pH, approximation]
HCO₃⁻ ≈ TA_corrected [valid pH 5.5–9; CO₃²⁻ negligible at pH <9]
CO₂(aq) = HCO₃⁻ / 10^(pH−pKa₁) · 44.01 [mg/L]
β ≈ ln(10) · [HCO₃⁻]·[CO₂] / ([HCO₃⁻]+[CO₂]) [mol/L·pH, approximation]
PO₄ correction
f_HPO4(pH) = Ka₂ / ([H⁺]+Ka₂) with Ka₂ = 10⁻⁷·²⁰ (pKa₂ phosphoric acid)
TA_corr = TA − [PO₄_total mmol/L] · f_HPO4(pH)
⚠ Polyphosphates, pyrophosphate, organic P not included. NH₄⁺ buffering (pKa 9.25) missing.
TA_corr = TA − [PO₄_total mmol/L] · f_HPO4(pH)
⚠ Polyphosphates, pyrophosphate, organic P not included. NH₄⁺ buffering (pKa 9.25) missing.
Nitrification & alkalinity drift (M2)
2NH₄⁺ + 4O₂ → 2NO₃⁻ + 4H⁺ + 2H₂O
TA loss [mmol/L·d] = g_NH₄N · 1000 / (14.01 · V_L)
ΔpH/d ≈ −TA loss / β [approximation for small Δ, not valid for pH jumps]
TA loss [mmol/L·d] = g_NH₄N · 1000 / (14.01 · V_L)
ΔpH/d ≈ −TA loss / β [approximation for small Δ, not valid for pH jumps]
CO₂ rate fish (M3)
R(T) = R_ref · Q10^((T−T_ref)/10)
CO₂ rate_vol [mg/L·h] = R(T) · fish_kg / V_L
⚠ Resting metabolism. Feeding peaks: +50–150 %. Nighttime reduction not modeled.
CO₂ rate_vol [mg/L·h] = R(T) · fish_kg / V_L
⚠ Resting metabolism. Feeding peaks: +50–150 %. Nighttime reduction not modeled.
CO₂ error propagation (Gaussian)
δCO₂_pH = CO₂ · ln(10) · δpH ← dominant term
δCO₂_TA = CO₂ · δTA / HCO₃⁻
δCO₂_total = √(δCO₂_pH² + δCO₂_TA²)
δCO₂_TA = CO₂ · δTA / HCO₃⁻
δCO₂_total = √(δCO₂_pH² + δCO₂_TA²)
Model limits (complete)
- Ionic strength >3 mS/cm: Davies correction missing → pKa deviation
- Organic acids from UASB effluent (acetate, propionate): produce uncaptured TA
- NH₄⁺/NH₃ buffering (pKa 9.25): relevant at pH >8
- CO₂ supersaturation due to biological peaks: equilibrium model underestimates
- Q10 model: feeding peaks, diurnal activity not captured
- Decoupling (M4): purely measurement-based – stripper geometry/type not modeled
- Roux et al. 2025: M4 based on general decoupling principle
T_ref
Where to get T_ref?
Preliminarily: there is no single universal source.
In practice, there are three approaches:
- Use the preset — T_ref and R_ref come from the preset and are consistent. As long as the system temperature does not deviate extremely from the preset T_ref, this is sufficient.
- Own measurement — Measure O₂ consumption (mg O₂/kg·h), convert with RQ ≈ 0.85: CO₂ = O₂ consumption × (44/32) × RQ. This gives R_ref at the current system temperature – then set T_ref = system temperature, Q10 then no longer changes anything.
- Fish species not in preset — Search literature for the species, looking for resting metabolism CO₂ or O₂ consumption at a defined temperature. Worst case: take values for the closest related species and treat Q10 as an uncertainty parameter.
Conclusion: If you have no literature values, your own O₂ measurement at system temperature + T_ref = system temperature) is the most precise solution – then the Q10 extrapolation is completely eliminated.
Reference values used:
Tilapia: Colt & Tchobanoglous (1981), Brett & Groves (1979)
Carp: Schreckenbach et al. (2001)
Trout: Ultsch et al. (1980), Randall & Daxboeck (1984)
Catfish and perch: estimate
Literature (among others)
- Harned, H.S. & Davis, R. (1943). Ionization constant of carbonic acid, 0–50 °C. J. Am. Chem. Soc. 65:2030–2037.
- Stumm, W. & Morgan, J.J. (1996). Aquatic Chemistry, 3rd ed. Wiley. (Chap. 3)
- Timmons, M.B. & Ebeling, J.M. (2013). Recirculating Aquaculture, 3rd ed. (Chap. 5)
- Roux, N. et al. (2025). Decoupled aquaponic systems. Local copy / Public Domain
- Masser, M.P. et al. (1999). Recirculating aquaculture tank production systems. SRAC Publ. 452.
- FAO: FISHERIES AND AQUACULTURE TECHNICAL PAPER 589 by UNO: Local copy, caution: 56 MB / Public Domain
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