The modified Gompertz model in BMP tests: lag, rate and potential

The modified Gompertz model describes cumulative methane production with a sigmoidal curve: B(t) = P·exp{−exp[(Rm·e/P)·(λ − t) + 1]}, where P is the final potential, Rm is the maximum production rate (NmL/g VS/d), λ is the lag time (d) and e is the base of the natural logarithm. It is used when, after the test begins, production starts with a delay and then accelerates and fades, that is, when the first-order model does not fit the data.

When a BMP curve has an S shape

• the inoculum requires adaptation to the substrate (e.g. fats, substrates with inhibitors),
• the substrate requires hydrolysis before methane is produced (lignocellulosic materials),
• a low I/S causes temporary acidification and a delay,
• two stages of degradation occur in the sample.

Parameters

Calculation example

Let us assume P = 300, Rm = 25, λ = 2 d. The coefficient Rm·e/P = 25 × 2.718 / 300 ≈ 0.2265 d⁻¹.

(Calculation for t = 10: exp(−0.812) ≈ 0.444, exp(−0.444) ≈ 0.641, B = 300 × 0.641 ≈ 192.) The maximum rate occurs at the inflection point, where B = P/e ≈ 110 NmL/g VS; it follows from the equation that this happens around t = λ + P/(Rm·e) ≈ 2 + 4.4 ≈ 6.4 d.

Fitting step by step

1. Enter the data (t, cumulative B) into a tool with non-linear regression (a spreadsheet with a solver, R, Python).
2. Set the starting values: P ≈ the last point, Rm ≈ the greatest slope, λ ≈ the point where the tangent crosses the time axis.
3. Fit the parameters; check convergence.
4. Assess RMSE, residuals and confidence intervals.
5. Compare with the first-order model (e.g. the AIC criterion); choose the simpler one if the difference is small.

Pitfalls

• Too many parameters. Three parameters with a short curve of few points give unreliable estimates.
• P as BMP. If the test was ended prematurely, P will be an extrapolation; give it with a comment.
• Interpretation of λ. The lag results from the inoculum and I/S; it is not a characteristic of the material.
• Comparison between laboratories. Kinetic parameters show greater scatter than B₀/P.
• No mechanism. The model is empirical; it does not explain why the lag appeared.

Use in practice

The Gompertz model is useful for describing curves in publications and reports, for assessing the effect of pretreatment on the lag, and also for detecting problems: a lengthened λ for a known substrate compared with previous series may indicate a weaker inoculum or an inhibitor. It should not be used to design a digester directly, because continuous conditions differ from batch ones.

Documentation

In the report give: the form of the equation, the fitting software, the values of P, Rm, λ with uncertainty, the goodness-of-fit measures and the range of data used for the fit. If you use a two-phase or another model, name it explicitly. This allows the reader to reproduce the result.

Choosing a model — a short cheat sheet

Additional example: the effect of pretreatment

Let us assume that before pretreatment a substrate has P = 280, Rm = 12, λ = 4 d, and after pretreatment P = 300, Rm = 22, λ = 1.5 d. The increase in P of about 7% is small, but the maximum rate has almost doubled and the lag has shortened by 2.5 d. For a plant with a short HRT this is a significant change: a larger part of the potential will be used in the digester. If pretreatment were assessed only by the final BMP, the benefit would be underestimated.

Information on data limitations

Gompertz fitting requires dense sampling around the inflection point, that is, when production grows fastest. With measurements every few days, the parameter Rm will be estimated with high uncertainty. In a study programme it is worth planning more frequent measurements in the first two weeks.

Differences relative to other sigmoidal models

The literature also uses the logistic and Richards models, which have a similar shape but different parameterisations. Changing the model changes the values of λ and Rm, so comparing parameters is possible only within a single model. When choosing, be guided by the fit and simplicity, not by the number of parameters.