ADM Models

ADToolbox exposes two anaerobic digestion models:

CLI command	Model	Parameter folder	Stoichiometric builder	ODE system
adtoolbox ADM adm1	ADM1	adm1	adm.build_adm1_stoichiometric_matrix	adm.adm1_ode_sys
adtoolbox ADM e-adm	e-ADM	e_adm	adm.build_e_adm_stoichiometric_matrix	adm.e_adm_ode_sys

The CLI and public API use only the names ADM1 and e-ADM. The preferred parameter format is one models.json file containing all models, keyed by model name.

Model Inputs

The canonical model input is a single JSON file with one top-level key per model:

{
  "adm1": {
    "species": [],
    "reactions": [],
    "model_parameters": {},
    "base_parameters": {},
    "initial_conditions": {},
    "inlet_conditions": {}
  },
  "e_adm": {
    "species": [],
    "reactions": [],
    "model_parameters": {},
    "base_parameters": {},
    "initial_conditions": {},
    "inlet_conditions": {}
  }
}

The first layer is always the model key. The second layer is the complete model payload used to instantiate adm.Model.

Top-level key	Public model	CLI command
adm1	ADM1	adtoolbox ADM adm1 --models-json models.json
e_adm	e-ADM	adtoolbox ADM e-adm --models-json models.json

The older six-file layout is still supported as a compatibility path, either by passing --parameters-dir or by passing each file explicitly.

Input file	ADM1 name	e-ADM name	Description
Model parameters	adm1_model_parameters.json	e_adm_model_parameters.json	Kinetic, equilibrium, inhibition, gas transfer, and yield parameters used in reaction rates and acid/base calculations.
Base parameters	adm1_base_parameters.json	e_adm_base_parameters.json	Reactor-level constants such as liquid volume, gas volume, influent flow, gas transfer constants, pressure, temperature, and gas constants.
Initial conditions	adm1_initial_conditions.json	e_adm_initial_conditions.json	Initial concentration of every model state. Keys must match the species list.
Inlet conditions	adm1_inlet_conditions.json	e_adm_inlet_conditions.json	Influent concentration of every model state, using the same species names with _in appended.
Reactions	adm1_reactions.json	e_adm_reactions.json	Ordered reaction names. This order defines the columns of the stoichiometric matrix and the reaction-rate vector.
Species	adm1_species.json	e_adm_species.json	Ordered state names. This order defines the rows of the stoichiometric matrix and the concentration vector.

The CLI also accepts:

Option	Applies to	Description
--models-json	ADM1, e-ADM	Preferred input. JSON file containing all models keyed by model name.
--parameters-dir	ADM1, e-ADM	Directory containing the six JSON files for the selected model.
--model-parameters, --base-parameters, --initial-conditions, --inlet-conditions, --reactions, --species	ADM1, e-ADM	Override individual JSON file paths.
--report	ADM1, e-ADM	Output mode. Use csv to write a report or dash to open the interactive app. If omitted, the model is solved without creating an output view.
--control-states	e-ADM	JSON object of states to hold constant. By default, e-ADM holds S_H_ion at 10^-6.5.

ADM1 Input Contract

Input object	Required shape	Required naming rule	How it is used
species	list of 38 strings	Names must match every key in initial_conditions; each inlet key must be species_name + "_in".	Defines the order of the state vector c.
reactions	list of 28 strings	Names must match the reaction names used in build_adm1_stoichiometric_matrix and adm1_ode_sys.	Defines the order of the reaction vector v and the columns of S.
initial_conditions	object with 38 numeric values	Keys are exactly the ADM1 species names.	Converted to y0, the initial state passed to solve_ivp.
inlet_conditions	object with 38 numeric values	Keys are exactly the ADM1 species names with _in appended.	Converted to c_in, used in the liquid and gas flow terms.
base_parameters	object with reactor constants	Must include q_in, V_liq, V_gas, R, T_op, T_base, P_atm, and K_W or equivalent water-equilibrium value used by the model.	Controls hydraulic dilution, gas headspace scaling, and gas partial pressures.
model_parameters	object with kinetic, yield, fraction, acid/base, inhibition, and gas-transfer constants	Must include all keys referenced by ADM1 rates and stoichiometry, such as k_dis, k_hyd_*, k_m_*, K_S_*, Y_*, f_*, C_*, N_*, K_a_*, K_pH_*, K_I_*, k_L_a, K_H_*, and k_p.	Controls reaction rates, stoichiometric coefficients, inhibition factors, acid/base rates, and gas transfer.

e-ADM Input Contract

Input object	Required shape	Required naming rule	How it is used
species	list of 49 strings	Names must match every key in initial_conditions; each inlet key must be species_name + "_in".	Defines the order of the state vector c.
reactions	list of 45 strings	Names must match the reaction names used in build_e_adm_stoichiometric_matrix and e_adm_ode_sys.	Defines the order of the reaction vector v and the columns of S.
initial_conditions	object with 49 numeric values	Keys are exactly the e-ADM species names.	Converted to y0, after applying any control_state overrides.
inlet_conditions	object with 49 numeric values	Keys are exactly the e-ADM species names with _in appended.	Converted to c_in, used in the liquid and gas flow terms.
base_parameters	object with reactor constants	Must include q_in, V_liq, V_gas, R, T_op, T_base, P_atm, and K_W.	Controls hydraulic dilution, gas headspace scaling, and gas partial pressures.
model_parameters	object with kinetic, yield, fraction, acid/base, inhibition, and gas-transfer constants	Must include all keys referenced by e-ADM rates and stoichiometry, including Y_su, Y_aa, Y_fa, Y_ac_et, Y_ac_lac, Y_pro_et, Y_pro_lac, Y_bu_et, Y_bu_lac, Y_va, Y_cap, Y_bu, Y_Me_ac, Y_Me_CO2, Y_ac_et_ox, and Y_pro_lac_ox.	Controls reaction rates, stoichiometric coefficients, inhibition factors, acid/base rates, and gas transfer.
control_state	optional object	Keys must be valid species names.	Overrides initial conditions and forces selected derivatives to zero. The CLI sets S_H_ion = 10^-6.5 by default for e-ADM.

Other Keyed Database Files

The same first-layer-key pattern should be used for other structured databases. For example, feeds can live in one feeds.json file:

{
  "food_waste": {
    "name": "food_waste",
    "carbohydrates": 10,
    "proteins": 20,
    "lipids": 20,
    "si": 30,
    "xi": 50,
    "tss": 80
  },
  "manure": {
    "name": "manure",
    "carbohydrates": 5,
    "proteins": 15,
    "lipids": 5,
    "si": 20,
    "xi": 55,
    "tss": 70
  }
}

That keeps the rule consistent: one file per database type, one top-level key per named item.

Model Outputs

Output	Source	Description
Concentration trajectories	Model.solve_model(...)	Returns a SciPy OdeResult. sol.t contains time points and sol.y contains species concentrations ordered exactly like model.species.
CSV report	--report csv or Model.csv_report(...)	Writes <model.name>_Report.csv. Rows are species and columns are simulated time points.
Interactive dashboard	--report dash or Model.dash_app(...)	Opens a Dash app with concentration plots and model tables.
Reaction fluxes	model.info["Fluxes"]	The most recent reaction-rate vector v, ordered exactly like model.reactions.
Stoichiometric matrix	model.s	Matrix S with shape number_of_species x number_of_reactions. Entry S[i, j] is the coefficient for species i in reaction j.

ADM1 has 38 species and 28 reactions. e-ADM has 49 species and 45 reactions.

Output Shapes

Output	ADM1 shape	e-ADM shape	Notes
sol.t	(n_time_points,)	(n_time_points,)	Time is in days in the default examples. The CLI uses np.linspace(0, 30, 10000).
sol.y	(38, n_time_points)	(49, n_time_points)	Rows follow model.species; columns follow sol.t.
model.s	(38, 28)	(49, 45)	Rows follow species; columns follow reactions.
model.info["Fluxes"]	(28, 1)	(45, 1)	Stores the most recent flux calculation from the ODE function.
CSV report	38 rows plus header	49 rows plus header	The first column is the species index. Remaining columns are simulated time points.

The solver returns a fallback solution filled with 1e10 if integration fails. That is a failure sentinel, not a physical concentration.

Mathematical Setup

The JSON files describe the state names, reaction names, and parameter values. The Python model converts them into a dynamic system by building a stoichiometric matrix and evaluating reaction rates at every time step.

Step	Formula	Meaning
Species vector	c(t) = [c_1(t), ..., c_n(t)]^T	Concentrations are ordered by the species JSON file.
Reaction-rate vector	v(c, theta) = [v_1, ..., v_m]^T	Fluxes are ordered by the reactions JSON file and are calculated from concentrations and parameters.
Stoichiometric matrix	S in R^(n x m)	Rows are species, columns are reactions. S[i, j] converts reaction j into the concentration change of species i.
Reaction contribution	r(c, theta) = S v(c, theta)	Net production or consumption of every species from biochemical, acid/base, and gas-transfer reactions.
Liquid flow contribution	(q_in / V_liq) * (c_in - c)	Continuous stirred tank dilution or influent loading for liquid-phase states.
Gas flow contribution	(q_gas / V_gas) * (c_gas,in - c_gas)	Headspace gas outflow for gas states.
Gas flow rate	q_gas = max(0, k_p * (P_gas - P_atm))	Gas leaves only when calculated gas pressure exceeds atmospheric pressure.
Final derivative	dc/dt = S v(c, theta) + flow terms	This is the vector returned to scipy.integrate.solve_ivp.

The implemented derivative can be written as:

c = current state vector
c_in = inlet state vector
S = stoichiometric matrix
v = reaction-rate vector

r = S @ v

for liquid states:
    dc_liq/dt = r_liq + (q_in / V_liq) * (c_in,liq - c_liq)

for gas states:
    dc_gas/dt = r_gas + (q_gas / V_gas) * (c_in,gas - c_gas)

The gas states are the final three species in both models:

S_gas_h2, S_gas_ch4, S_gas_co2

The gas pressure and gas outflow equations are:

p_gas_h2  = S_gas_h2  * R * T_op / 16
p_gas_ch4 = S_gas_ch4 * R * T_op / 64
p_gas_co2 = S_gas_co2 * R * T_op
p_gas_h2o = 0.0313 * exp(5290 * (1 / T_base - 1 / T_op))

P_gas = p_gas_h2 + p_gas_ch4 + p_gas_co2 + p_gas_h2o
q_gas = max(0, k_p * (P_gas - P_atm))

For any reaction j, the instantaneous contribution to species i is:

dc_i/dt from reaction j = S[i, j] * v_j

This is why reaction and species names are used to build S: the equations are much easier to audit than when the matrix is filled with positional numbers.

Reaction Rate Forms

Reaction type	Formula pattern	Used for
First-order conversion	v_j = k * X	Disintegration, hydrolysis, and decay reactions.
Single-substrate uptake	v_j = k_m * S_a / (K_S + S_a) * X * I	Sugar, amino acid, LCFA, acetate, hydrogen, ethanol, lactate, and VFA uptake reactions.
Competitive ADM1 C4 uptake	v_j = k_m * S_a / (K_S + S_a) * X_c4 * S_a / (S_va + S_bu + eps) * I	ADM1 valerate and butyrate uptake.
Two-substrate uptake	v_j = k_m * S_a * S_b / (K_a * S_a + K_b * S_b + S_a * S_b + eps) * X * I	e-ADM syntrophic reactions that require paired substrates.
Inhibition or limitation	I = product(I_pH, I_H2, I_NH3, I_IN, ...)	Multiplies uptake rates to suppress a reaction under limiting or inhibitory conditions.
Acid/base kinetic form	v_AB = k_AB * (S_ion * (K_a + S_H) - K_a * S_total)	Kinetic acid/base reactions when the state is not solved algebraically.
DAE acid/base update	S_ion = K_a / (K_a + S_H) * S_total	Algebraic speciation used when model.switch == "DAE".
Gas transfer	v_gas = k_La * (S_liq - H * p_gas)	Transfer between dissolved gas states and headspace gas states.

Acid/Base Equations

ADM1 and e-ADM both calculate kinetic acid/base rates, then set selected acid/base derivatives to zero in DAE mode. The kinetic form is:

v_AB,acid = k_A_B,acid * (S_ion * (K_a,acid + S_H) - K_a,acid * S_total)

For carbonate and inorganic nitrogen:

v_AB,co2 = k_A_B_co2 * (S_hco3_ion * (K_a_co2 + S_H) - K_a_co2 * S_IC)
v_AB,IN  = k_A_B_IN  * (S_nh3      * (K_a_IN  + S_H) - K_a_IN  * S_IN)

In DAE mode, ionized species are updated algebraically. e-ADM explicitly updates:

S_va_ion  = K_a_va  / (K_a_va  + S_H) * S_va
S_bu_ion  = K_a_bu  / (K_a_bu  + S_H) * S_bu
S_pro_ion = K_a_pro / (K_a_pro + S_H) * S_pro
S_cap_ion = K_a_cap / (K_a_cap + S_H) * S_cap
S_ac_ion  = K_a_ac  / (K_a_ac  + S_H) * S_ac
S_lac_ion = K_a_lac / (K_a_lac + S_H) * S_lac
S_hco3_ion = S_IC - S_co2

Hydrogen ion concentration is calculated from electroneutrality unless it is controlled:

S_H = -phi / 2 + 0.5 * sqrt(phi^2 + 4 * K_w)

When S_H_ion is included in control_state, the controlled value is used and dS_H_ion/dt = 0.

Inhibition Equations

The pH inhibition terms use Hill-type functions:

I_pH_x = K_pH_x^n_x / (S_H^n_x + K_pH_x^n_x)

Nitrogen limitation is a Monod term:

I_IN_lim = S_IN / (K_S_IN + S_IN + eps)

Hydrogen and ammonia inhibition terms use:

I_h2_x = 1 / (1 + S_h2 / (K_I_h2_x + eps))
I_nh3  = 1 / (1 + S_nh3 / (K_I_nh3 + eps))

ADM1 uses the same conceptual forms, with its original I_IN_lim = 1 / (1 + K_S_IN / S_IN) expression.

ADM1 Reactions

The ADM1 reactions follow the Batstone ADM1 structure: disintegration, hydrolysis, soluble-substrate uptake, biomass decay, acid/base speciation, and gas transfer.

ADM1 Stoichiometric Coefficients

ADM1 uses yield coefficients Y_*, product fractions f_*, carbon contents C_*, and nitrogen contents N_* to fill S. Nitrogen limitation sets selected yields to zero before S is built.

For the main carbon-balance terms:

Y_su = 0 if nitrogen_limited else Y_su
Y_aa = 0 if nitrogen_limited else Y_aa
Y_fa = 0 if nitrogen_limited else Y_fa

s_1  = -C_xc + f_sI_xc*C_sI + f_ch_xc*C_ch + f_pr_xc*C_pr + f_li_xc*C_li + f_xI_xc*C_xI
s_2  = -C_ch + C_su
s_3  = -C_pr + C_aa
s_4  = -C_li + (1 - f_fa_li)*C_su + f_fa_li*C_fa
s_5  = -C_su + (1 - Y_su)*(f_bu_su*C_bu + f_pro_su*C_pro + f_ac_su*C_ac) + Y_su*C_bac
s_6  = -C_aa + (1 - Y_aa)*(f_va_aa*C_va + f_bu_aa*C_bu + f_pro_aa*C_pro + f_ac_aa*C_ac) + Y_aa*C_bac
s_7  = -C_fa + (1 - Y_fa)*0.7*C_ac + Y_fa*C_bac
s_8  = -C_va + (1 - Y_c4)*0.54*C_pro + (1 - Y_c4)*0.31*C_ac + Y_c4*C_bac
s_9  = -C_bu + (1 - Y_c4)*0.8*C_ac + Y_c4*C_bac
s_10 = -C_pro + (1 - Y_pro)*0.57*C_ac + Y_pro*C_bac
s_11 = -C_ac + (1 - Y_ac)*C_ch4 + Y_ac*C_bac
s_12 = (1 - Y_h2)*C_ch4 + Y_h2*C_bac
s_13 = -C_bac + C_xc

These terms are inserted into the S_IC row with negative signs for conversion reactions and positive return through decay. For example:

S[S_IC, Uptake of sugars] = -s_5
S[S_IC, Uptake of acetate] = -s_11
S[S_IC, Decay of Xsu] = -s_13

Nitrogen stoichiometry follows the same pattern:

S[S_IN, Uptake of sugars] = -Y_su * N_bac
S[S_IN, Uptake of amino acids] = N_aa - Y_aa * N_bac
S[S_IN, Uptake of acetate] = -Y_ac * N_bac
S[S_IN, Decay of Xsu] = N_bac - N_xc

Reaction	Rate expression	Main conversion represented in S
Disintegration	k_dis * X_xc	Composite particulate matter X_xc is split into carbohydrates X_ch, proteins X_pr, lipids X_li, soluble inert S_I, particulate inert X_I, inorganic carbon S_IC, and inorganic nitrogen S_IN.
Hydrolysis carbohydrates	k_hyd_ch * X_ch	Particulate carbohydrates become soluble sugars S_su, with carbon correction through S_IC.
Hydrolysis of proteins	k_hyd_pr * X_pr	Particulate proteins become amino acids S_aa, with carbon correction through S_IC.
Hydrolysis of lipids	k_hyd_li * X_li	Lipids split into LCFA S_fa and sugars S_su, with carbon correction through S_IC.
Uptake of sugars	k_m_su * S_su / (K_S_su + S_su) * X_su * I5	Sugars are consumed to grow X_su and produce butyrate S_bu, propionate S_pro, acetate S_ac, hydrogen S_h2, inorganic carbon, and nitrogen demand.
Uptake of amino acids	k_m_aa * S_aa / (K_S_aa + S_aa) * X_aa * I6	Amino acids are consumed to grow X_aa and produce valerate S_va, butyrate, propionate, acetate, hydrogen, inorganic carbon, and inorganic nitrogen.
Uptake of LCFA	k_m_fa * S_fa / (K_S_fa + S_fa) * X_fa * I7	LCFA are consumed to grow X_fa and produce acetate, hydrogen, inorganic carbon, and nitrogen demand.
Uptake of valerate	k_m_c4 * S_va / (K_S_c4 + S_va) * X_c4 * S_va / (S_va + S_bu + eps) * I8	Valerate is consumed by X_c4, producing propionate, acetate, hydrogen, biomass, carbon, and nitrogen demand.
Uptake of butyrate	k_m_c4 * S_bu / (K_S_c4 + S_bu) * X_c4 * S_bu / (S_bu + S_va + eps) * I9	Butyrate is consumed by X_c4, producing acetate, hydrogen, biomass, carbon, and nitrogen demand.
Uptake of propionate	k_m_pr * S_pro / (K_S_pro + S_pro) * X_pro * I10	Propionate is consumed to grow X_pro and produce acetate, hydrogen, carbon, and nitrogen demand.
Uptake of acetate	k_m_ac * S_ac / (K_S_ac + S_ac) * X_ac * I11	Acetate is consumed by acetoclastic methanogens, producing methane S_ch4, X_ac, carbon correction, and nitrogen demand.
Uptake of Hydrogen	k_m_h2 * S_h2 / (K_S_h2 + S_h2) * X_h2 * I12	Hydrogen is consumed by hydrogenotrophic methanogens, producing methane S_ch4, X_h2, carbon correction, and nitrogen demand.
Decay of Xsu	k_dec_X_su * X_su	Sugar degraders decay back to composite particulate matter X_xc, with carbon and nitrogen returned through S_IC and S_IN.
Decay of Xaa	k_dec_X_aa * X_aa	Amino acid degraders decay to X_xc, with carbon and nitrogen returned.
Decay of Xfa	k_dec_X_fa * X_fa	LCFA degraders decay to X_xc, with carbon and nitrogen returned.
Decay of Xc4	k_dec_X_c4 * X_c4	C4 degraders decay to X_xc, with carbon and nitrogen returned.
Decay of Xpro	k_dec_X_pro * X_pro	Propionate degraders decay to X_xc, with carbon and nitrogen returned.
Decay of Xac	k_dec_X_ac * X_ac	Acetoclastic methanogens decay to X_xc, with carbon and nitrogen returned.
Decay of Xh2	k_dec_X_h2 * X_h2	Hydrogenotrophic methanogens decay to X_xc, with carbon and nitrogen returned.
Acid Base Equilibrium (Va)	k_A_B_va * (S_va_ion * (K_a_va + S_H) - K_a_va * S_va)	Valerate speciation between total valerate S_va and ionized valerate S_va_ion.
Acid Base Equilibrium (Bu)	k_A_B_bu * (S_bu_ion * (K_a_bu + S_H) - K_a_bu * S_bu)	Butyrate speciation between S_bu and S_bu_ion.
Acid Base Equilibrium (Pro)	k_A_B_pro * (S_pro_ion * (K_a_pro + S_H) - K_a_pro * S_pro)	Propionate speciation between S_pro and S_pro_ion.
Acid Base Equilibrium (Ac)	k_A_B_ac * (S_ac_ion * (K_a_ac + S_H) - K_a_ac * S_ac)	Acetate speciation between S_ac and S_ac_ion.
Acid Base Equilibrium (CO2)	k_A_B_co2 * (S_hco3_ion * (K_a_co2 + S_H) - K_a_co2 * S_IC)	Carbonate speciation through bicarbonate S_hco3_ion and inorganic carbon S_IC.
Acid Base Equilibrium (In)	k_A_B_IN * (S_nh3 * (K_a_IN + S_H) - K_a_IN * S_IN)	Nitrogen speciation between ammonia S_nh3, ammonium S_nh4_ion, and total inorganic nitrogen S_IN.
Gas Transfer H2	k_L_a * (S_h2 - 16 * K_H_h2 * p_gas_h2)	Dissolved hydrogen leaves the liquid and enters S_gas_h2 with V_liq / V_gas scaling.
Gas Transfer CH4	k_L_a * (S_ch4 - 64 * K_H_ch4 * p_gas_ch4)	Dissolved methane leaves the liquid and enters S_gas_ch4 with V_liq / V_gas scaling.
Gas Transfer CO2	k_L_a * (S_co2 - K_H_co2 * p_gas_co2)	Dissolved carbon dioxide leaves the liquid and enters S_gas_co2 with V_liq / V_gas scaling.

ADM1 Inhibition Factors

Factor	Formula pattern	Applied to
I5	I_pH_aa * I_IN_lim	Sugar uptake.
I6	I5	Amino acid uptake.
I7	I_pH_aa * I_IN_lim * I_h2_fa	LCFA uptake.
I8, I9	I_pH_aa * I_IN_lim * I_h2_c4	Valerate and butyrate uptake.
I10	I_pH_aa * I_IN_lim * I_h2_pro	Propionate uptake.
I11	I_pH_ac * I_IN_lim * I_nh3	Acetate uptake.
I12	I_pH_h2 * I_IN_lim	Hydrogen uptake.

e-ADM Reactions

e-ADM keeps the same dynamic structure but expands the soluble fermentation network. It separates ethanol-linked and lactate-linked routes, adds caproate, lactate, ethanol, explicit TSS/TDS disintegration, and two methanogenesis routes.

e-ADM Stoichiometric Coefficients

e-ADM uses the same matrix logic as ADM1, but most soluble conversions are written as explicit product fractions plus a carbon-balance residual. Selected yields are set to zero under nitrogen limitation.

For sugar, amino acid, and LCFA uptake:

Y_su = 0 if nitrogen_limited else Y_su
Y_aa = 0 if nitrogen_limited else Y_aa
Y_fa = 0 if nitrogen_limited else Y_fa

f_ac_su = 1 - f_pro_su - f_et_su - f_lac_su
f_ac_aa = 1 - f_pro_aa - f_et_aa - f_lac_aa
f_ac_fa = 1 - f_pro_fa - f_et_fa - f_lac_fa

f_IC_su = -(-C_su + (1-Y_su)*f_pro_su*C_pro + (1-Y_su)*f_et_su*C_et
              + (1-Y_su)*f_lac_su*C_lac + (1-Y_su)*f_ac_su*C_ac + Y_su*C_bac)

f_IC_aa = -(-C_aa + (1-Y_aa)*f_pro_aa*C_pro + (1-Y_aa)*f_et_aa*C_et
              + (1-Y_aa)*f_lac_aa*C_lac + (1-Y_aa)*f_ac_aa*C_ac + Y_aa*C_bac)

f_IC_fa = -(-C_fa + (1-Y_fa)*f_pro_fa*C_pro + (1-Y_fa)*f_et_fa*C_et
              + (1-Y_fa)*f_lac_fa*C_lac + (1-Y_fa)*f_ac_fa*C_ac + Y_fa*C_bac)

For chain elongation and VFA conversion, e-ADM uses paired substrate routes:

f_IC_ac_et  = -(-C_ac  + f_et_ac*C_et  + (1-f_et_ac-Y_ac_et)*f_bu_ac*C_bu  + Y_ac_et*C_bac)
f_IC_ac_lac = -(-C_ac  + f_lac_ac*C_lac + (1-f_lac_ac-Y_ac_lac)*f_bu_ac*C_bu + Y_ac_lac*C_bac)

f_IC_pro_et  = -(-C_pro + f_et_pro*C_et  + (1-f_et_pro-Y_pro_et)*f_va_pro*C_va  + Y_pro_et*C_bac)
f_IC_pro_lac = -(-C_pro + f_lac_pro*C_lac + (1-f_lac_pro-Y_pro_lac)*f_va_pro*C_va + Y_pro_lac*C_bac)

f_IC_bu_et  = -(-C_bu + f_et_bu*C_et  + (1-f_et_bu-Y_bu_et)*f_cap_bu*C_cap  + Y_bu_et*C_bac)
f_IC_bu_lac = -(-C_bu + f_lac_bu*C_lac + (1-f_lac_bu-Y_bu_lac)*f_cap_bu*C_cap + Y_bu_lac*C_bac)

Methanogenesis and oxidation use:

f_IC_Me_ach2 = -(f_ac_h2*C_ac + (1 - Y_Me_ac)*C_ch4 + Y_Me_ac*C_bac)
f_IC_et_ox   = -(-C_et  + (1-Y_ac_et_ox)*C_bac + Y_ac_et_ox*C_ac)
f_IC_lac_ox  = -(-C_lac + (1-Y_pro_lac_ox)*C_bac + Y_pro_lac_ox*C_pro)

The stoichiometric matrix then combines these residuals with explicit product coefficients. For example:

S[S_su, Uptake of sugars] = -1
S[S_pro, Uptake of sugars] = (1 - Y_su) * f_pro_su
S[S_et, Uptake of sugars] = (1 - Y_su) * f_et_su
S[S_lac, Uptake of sugars] = (1 - Y_su) * f_lac_su
S[S_ac, Uptake of sugars] = (1 - Y_su) * f_ac_su
S[S_IN, Uptake of sugars] = -Y_su * N_bac
S[S_IC, Uptake of sugars] = f_IC_su
S[X_su, Uptake of sugars] = Y_su

Reaction	Rate expression	Main conversion represented in S
TSS_Disintegration	k_dis_TSS * TSS	Total suspended solids are split into X_ch, X_pr, X_li, and inert particulate biomass X_I using feed fractions.
TDS_Disintegration	k_dis_TDS * TDS	Total dissolved solids are split into X_ch, X_pr, X_li, and soluble inert S_I using feed fractions.
Hydrolysis carbohydrates	k_hyd_ch * X_ch	Particulate carbohydrate is hydrolyzed to soluble sugar S_su.
Hydrolysis proteins	k_hyd_pr * X_pr	Particulate protein is hydrolyzed to amino acids S_aa.
Hydrolysis lipids	k_hyd_li * X_li	Particulate lipids are hydrolyzed to LCFA S_fa.
Uptake of sugars	k_m_su * S_su / (K_S_su + S_su) * X_su * I5	Sugars are consumed to grow X_su and form propionate, ethanol, lactate, acetate, carbon, and nitrogen demand.
Uptake of amino acids	k_m_aa * S_aa / (K_S_aa + S_aa) * X_aa * I6	Amino acids are consumed to grow X_aa and form propionate, ethanol, lactate, acetate, inorganic carbon, and inorganic nitrogen.
Uptake of LCFA	k_m_fa * S_fa / (K_S_fa + S_fa) * X_fa * I7	LCFA are consumed to grow X_fa and form propionate, ethanol, lactate, acetate, carbon, and nitrogen demand.
Uptake of acetate_et	k_m_ac * S_ac * S_et / (K_S_ac*S_ac + K_S_ac_et*S_et + S_ac*S_et + eps) * X_ac_et * I11	Acetate and ethanol are converted by X_ac_et, producing butyrate, hydrogen, carbon correction, and new biomass.
Uptake of acetate_lac	k_m_ac * S_ac * S_lac / (K_S_ac*S_ac + K_S_ac_lac*S_lac + S_ac*S_lac + eps) * X_ac_lac * I11	Acetate and lactate are converted by X_ac_lac, producing butyrate, hydrogen, carbon correction, and new biomass.
Uptake of propionate_et	k_m_pro * S_pro * S_et / (K_S_pro*S_pro + K_S_pro_et*S_et + S_pro*S_et + eps) * X_chain_et * I10	Propionate and ethanol are converted by chain-elongating biomass, producing valerate, hydrogen, carbon correction, and X_chain_et.
Uptake of propionate_lac	k_m_pro * S_pro * S_lac / (K_S_pro*S_pro + K_S_pro_lac*S_lac + S_pro*S_lac + eps) * X_chain_lac * I10	Propionate and lactate are converted by chain-elongating biomass, producing valerate, hydrogen, carbon correction, and X_chain_lac.
Uptake of butyrate_et	k_m_bu * S_bu * S_et / (K_S_bu*S_bu + K_S_bu_et*S_et + S_bu*S_et + eps) * X_chain_et * I14	Butyrate and ethanol are converted to caproate, hydrogen, carbon correction, and X_chain_et.
Uptake of butyrate_lac	k_m_bu * S_bu * S_lac / (K_S_bu*S_bu + K_S_bu_lac*S_lac + S_bu*S_lac + eps) * X_chain_lac * I14	Butyrate and lactate are converted to caproate, hydrogen, carbon correction, and X_chain_lac.
Uptake of valerate	k_m_va * S_va / (K_S_va + S_va) * X_VFA_deg * I15	Valerate is degraded to propionate and X_VFA_deg.
Uptake of caproate	k_m_cap * S_cap / (K_S_cap + S_cap) * X_VFA_deg * I13	Caproate is degraded to acetate and X_VFA_deg.
Uptake of butyrate	k_m_bu_deg * S_bu / (K_S_bu + S_bu) * X_VFA_deg * I13	Butyrate is degraded to acetate and X_VFA_deg.
Methanogenessis from acetate and h2	k_m_h2_Me_ac * S_gas_h2 * S_ac / (K_S_h2_Me_ac*S_gas_h2 + K_S_ac_Me*S_ac + S_ac*S_gas_h2 + eps) * X_Me_ac * I12	Acetate and gas-phase hydrogen are converted to dissolved methane S_ch4, methanogenic biomass X_Me_ac, carbon correction, and nitrogen demand.
Methanogenessis from CO2 and h2	k_m_h2_Me_CO2 * S_gas_h2 * S_gas_co2 / (K_S_h2_Me_CO2*S_gas_h2 + K_S_CO2_Me*S_gas_co2 + S_gas_co2*S_gas_h2 + eps) * X_Me_CO2 * I12	Gas-phase hydrogen and carbon dioxide are converted to gas-phase methane S_gas_ch4, X_Me_CO2, gas-phase carbon dioxide change, and nitrogen demand.
Uptake of ethanol	k_m_et * S_et / (K_S_et + S_et) * X_et * I16	Ethanol is oxidized to acetate, inorganic carbon, and X_et.
Uptake of lactate	k_m_lac * S_lac / (K_S_lac + S_lac) * X_lac * I16	Lactate is oxidized to propionate, inorganic carbon, and X_lac.
Decay of Xsu	k_dec_X_su * X_su	Sugar degraders decay to TSS, returning biomass carbon and nitrogen.
Decay of Xaa	k_dec_X_aa * X_aa	Amino acid degraders decay to TSS, returning biomass carbon and nitrogen.
Decay of Xfa	k_dec_X_fa * X_fa	LCFA degraders decay to TSS, returning biomass carbon and nitrogen.
Decay of X_ac_et	k_dec_X_ac * X_ac_et	Acetate-ethanol consumers decay to TSS, returning biomass carbon and nitrogen.
Decay of X_ac_lac	k_dec_X_ac * X_ac_lac	Acetate-lactate consumers decay to TSS, returning biomass carbon and nitrogen.
Decay of Xpro	Not currently assigned in e_adm_ode_sys	Listed in the reaction set but not given a rate in the current ODE block. If kept, it behaves as zero-flux unless populated elsewhere.
Decay of X_chain_et	k_dec_X_chain_et * X_chain_et	Ethanol-linked chain elongators decay to TSS, returning biomass carbon and nitrogen.
Decay of X_chain_lac	k_dec_X_chain_lac * X_chain_lac	Lactate-linked chain elongators decay to TSS, returning biomass carbon and nitrogen.
Decay of X_VFA_deg	k_dec_X_VFA_deg * X_VFA_deg	VFA degraders decay to TSS, returning biomass carbon and nitrogen.
Decay of X_Me_ac	k_dec_X_Me_ac * X_Me_ac	Acetate/hydrogen methanogens decay to TSS, returning biomass carbon and nitrogen.
Decay of X_Me_CO2	k_dec_X_Me_CO2 * X_Me_CO2	CO2/hydrogen methanogens decay to TSS, returning biomass carbon and nitrogen.
Decay of Xet	k_dec_X_et * X_et	Ethanol oxidizers decay. This rate is present in the ODE; the current stoichiometric builder should be checked if this flux is expected to affect states.
Decay of Xlac	k_dec_X_lac * X_lac	Lactate oxidizers decay. This rate is present in the ODE; the current stoichiometric builder should be checked if this flux is expected to affect states.
Acid Base Equilibrium (Va)	k_A_B_va * (S_va_ion * (K_a_va + S_H) - K_a_va * S_va)	Valerate speciation between S_va and S_va_ion.
Acid Base Equilibrium (Bu)	k_A_B_bu * (S_bu_ion * (K_a_bu + S_H) - K_a_bu * S_bu)	Butyrate speciation between S_bu and S_bu_ion.
Acid Base Equilibrium (Pro)	k_A_B_pro * (S_pro_ion * (K_a_pro + S_H) - K_a_pro * S_pro)	Propionate speciation between S_pro and S_pro_ion.
Acid Base Equilibrium (Cap)	k_A_B_cap * (S_cap_ion * (K_a_cap + S_H) - K_a_cap * S_cap)	Caproate speciation between S_cap and S_cap_ion.
Acid Base Equilibrium (Lac)	k_A_B_lac * (S_lac_ion * (K_a_lac + S_H) - K_a_lac * S_lac)	Lactate speciation between S_lac and S_lac_ion.
Acid Base Equilibrium (Ac)	k_A_B_ac * (S_ac_ion * (K_a_ac + S_H) - K_a_ac * S_ac)	Acetate speciation between S_ac and S_ac_ion.
Acid Base Equilibrium (CO2)	k_A_B_co2 * (S_hco3_ion * (K_a_co2 + S_H) - K_a_co2 * S_IC)	Carbonate speciation through bicarbonate S_hco3_ion and inorganic carbon S_IC.
Acid Base Equilibrium (In)	k_A_B_IN * (S_nh3 * (K_a_IN + S_H) - K_a_IN * S_IN)	Nitrogen speciation between ammonia S_nh3, ammonium S_nh4_ion, and total inorganic nitrogen S_IN.
Gas Transfer H2	max(0, k_L_a * (S_h2 - 16 * K_H_h2 * p_gas_h2))	Dissolved hydrogen leaves the liquid and enters S_gas_h2 with V_liq / V_gas scaling.
Gas Transfer CH4	max(0, k_L_a * (S_ch4 - 64 * K_H_ch4 * p_gas_ch4))	Dissolved methane leaves the liquid and enters S_gas_ch4 with V_liq / V_gas scaling.
Gas Transfer CO2	max(0, k_L_a * (S_co2 - K_H_co2 * p_gas_co2))	Dissolved carbon dioxide leaves the liquid and enters S_gas_co2 with V_liq / V_gas scaling.

e-ADM Inhibition Factors

Factor	Formula pattern	Applied to
I5	I_pH_aa * I_IN_lim	Sugar uptake.
I6	I5	Amino acid uptake.
I7	I_pH_aa * I_IN_lim * I_h2_fa	LCFA uptake.
I10	I_pH_pro * I_IN_lim * I_h2_pro	Propionate plus ethanol or lactate uptake.
I11	I_pH_ac * I_IN_lim * I_nh3	Acetate plus ethanol or lactate uptake.
I12	I_pH_h2 * I_IN_lim	Methanogenesis.
I13	I_pH_cap * I_IN_lim * I_h2_c4	Caproate uptake and direct butyrate degradation.
I14	I_pH_bu * I_IN_lim * I_h2_c4	Butyrate plus ethanol or lactate uptake.
I15	I_pH_va * I_IN_lim * I_h2_c4	Valerate uptake.
I16	I_IN_lim * I_nh3 * I_pH_aa * I_h2_oxidation	Ethanol and lactate oxidation.

Methane-Producing Reactions

Model	Methane-producing reactions
ADM1	Uptake of acetate, Uptake of Hydrogen
e-ADM	Methanogenessis from acetate and h2, Methanogenessis from CO2 and h2

Gas Transfer CH4 does not create methane. It moves methane between dissolved S_ch4 and gas-phase S_gas_ch4.

State Vectors

ADM1 Species

S_su, S_aa, S_fa, S_va, S_bu, S_pro, S_ac, S_h2, S_ch4, S_IC, S_IN, S_I, X_xc, X_ch, X_pr, X_li, X_su, X_aa, X_fa, X_c4, X_pro, X_ac, X_h2, X_I, S_cation, S_anion, S_H_ion, S_va_ion, S_bu_ion, S_pro_ion, S_ac_ion, S_hco3_ion, S_co2, S_nh3, S_nh4_ion, S_gas_h2, S_gas_ch4, S_gas_co2

e-ADM Species

S_su, S_aa, S_fa, S_va, S_bu, S_pro, S_et, S_lac, S_cap, S_ac, S_h2, S_ch4, S_IC, S_IN, S_I, TSS, TDS, X_ch, X_pr, X_li, X_su, X_aa, X_ac_et, X_ac_lac, X_fa, X_VFA_deg, X_et, X_lac, X_chain_et, X_chain_lac, X_Me_ac, X_Me_CO2, X_I, S_cation, S_anion, S_H_ion, S_va_ion, S_bu_ion, S_pro_ion, S_cap_ion, S_lac_ion, S_ac_ion, S_hco3_ion, S_co2, S_nh3, S_nh4_ion, S_gas_h2, S_gas_ch4, S_gas_co2

Minimal CLI Run

adtoolbox ADM adm1 --models-json /path/to/ADToolbox/models.json --report csv

adtoolbox ADM e-adm --models-json /path/to/ADToolbox/models.json --report csv