First-order kinetics in the BMP test: the rate constant k, B₀ and the t90 time

The first-order model describes cumulative methane yield with the equation B(t) = B₀·(1 − e^(−k·t)), where B₀ is the final potential (NmL CH₄/g VS) and k is the rate constant (1/d). From the model one reads the time needed to reach 90% of the potential (t90 = ln 10 / k ≈ 2.3/k). It is the simplest way of describing a BMP curve and the basis of many design calculations, but it has limitations: it does not account for the lag phase and assumes homogeneous, single-stage degradation.

Origin of the model

The assumption: the rate of methane production is proportional to the amount of material not yet degraded: dB/dt = k·(B₀ − B). The solution gives the exponential function above. In process models (e.g. ADM1) the hydrolysis of polymers is also described by first-order kinetics, so the constant k from BMP is an approximate "hydrolysis constant" under the test conditions.

Interpretation of the parameters

The values of k for substrates in the literature span a wide range (from fractions to several 1/d) and depend on the method; they should not be treated as material constants.

Fitting

1. Non-linear regression (least squares): fitting B₀ and k to the points (t, B) — preferred.
2. Linearisation: ln(1 − B/B₀) = −k·t; requires a known B₀ and is sensitive to errors when B is close to B₀.
3. Assessing the fit: R², RMSE, residual analysis (whether the residuals have a systematic shape).
4. Parameter uncertainty: confidence intervals from the fit or from bootstrapping.

Calculation example

Let us assume B₀ = 320 NmL/g VS and k = 0.10 d⁻¹.

• t = 5 d: B = 320·(1 − e^(−0.5)) = 320 × 0.3935 ≈ 126 NmL/g,
• t = 10 d: B = 320·(1 − e^(−1)) = 320 × 0.632 ≈ 202,
• t = 20 d: B = 320·(1 − e^(−2)) = 320 × 0.8647 ≈ 277,
• t90 = 2.303 / 0.10 ≈ 23 d.

For k = 0.05 d⁻¹, t90 ≈ 46 d. If the HRT of the digester is 40 days, then with a rigid residence time (as in a batch test) about 98% of the potential will be used in the first case and about 86% in the second (1 − e^(−2) ≈ 0.865). This is a simplification, because a fully mixed digester has a residence time distribution rather than a rigid time; in a CSTR digester the degraded fraction is k·HRT/(1 + k·HRT): for k = 0.10 and HRT of 40 d it is 4/5 = 80%, for k = 0.05 it is 2/3 ≈ 67%. The difference between the "single bottle" approach and the CSTR is considerable and explains part of the discrepancy between BMP and full scale.

Limitations of the model

• Lag phase. When production is delayed at the start, the first-order model will fit poorly; a model with a lag or the Gompertz model is used.
• Two-stage curves. Substrates containing an easily and a hardly degradable fraction (e.g. silage with fibre) require a two-fraction model (the sum of two exponentials).
• Inhibition. Toxicity and acidification effects do not follow from first-order kinetics.
• Dependence on conditions. k changes with I/S, temperature and particle size.

Good practice

1. Fit the model to the whole curve, not just the end point.
2. Give B₀, k and their uncertainties as well as the fitting method.
3. Do not compare k between series without checking that the inoculum and I/S were comparable.
4. Use k for approximate selection of HRT, but confirm with continuous tests.

Application in planning

For substrates with a similar B₀, a faster k means the possibility of a shorter HRT without loss of yield. This is an argument for pretreatment (size reduction, hydrolysis), which usually increases k and not necessarily B₀. If after pretreatment k doubles at the same B₀, a shorter HRT at the same performance can free up digester volume or increase throughput.

Quick checklist

• [ ] Cumulative, not daily, data are the input to the fit.
• [ ] All points were included, also those from the start of the test.
• [ ] It was checked whether there is a lag phase.
• [ ] B₀, k, uncertainties and R² are given.
• [ ] The tool used is recorded.