,
A new methodology for early BMP assessment using a mathematical model
St ´ephane Mottelet 1 Sabrina Gu ´erin 2 Sam Azimi 2 Jean Bernier 2 Vincent Rocher 2 Andr ´e Pauss 1 Thierry Ribeiro 3
1
Universit ´e de Technologie de Compi `egne, FRANCE
2
SIAAP Direction D ´eveloppement et Prospective, Colombes, FRANCE
3
UniLaSalle Beauvais, FRANCE
,
Context
Besides the classical measure of BMP can need as long as 15 days, the dynamics of the
production can be identified during the first few days if rigorous test methodology is respected and a biologically inspired mathematical model is used. This approach can provide accurate predictions of BMP value with shortened experimentation duration.
But finding mathematical models that are simple enough to be used for process control and
prevision is particularly important. In this study we have used the modified AM2 model, calibrated and then validated with 90 different experiments as triplicates of 30 different substrate/inoculum mixes of sludge coming from the Paris’ conurbation WWTP.
The obtained model allows a good prediction after only 4 days with an acceptable error and at the same time gives the possibility to understand the influence of the initial proportion of substrates on the production profile.
Experimental protocol
I 500 ml reactors,I/S ratio=3 I CO
2trapping
I Mean flow measurement by ≈ 10ml throttles I Full compliance with experts
recommendations [1]
I 36 batchs in triplicates
I VSS, TSS, COD, BOD measurements I BMP obtained after 20 days
AMPTS
Modified AM2 model
Reaction scheme [2, 3, 4] :
S
0−−−−→
r0S
1, (Hydrolysis)
S
1−−−−→
r1Y
X1X
1+ (1 − Y
X1)S
2+ k
4CO
2, (Acidification) S
2−−−−→
r2Y
X2X
2+ (1 − Y
X2)CH
4+ k
5CO
2, (Methanogenesis) S
0: insoluble organic molecules, S
1: simple compounds (fatty acids, peptides, amino acids, . . .), S
2: volatile fatty acids
Differential equations system :
I Perfectly mixed batch reactor, states of the sytem : S
0, S
1, S
2, X
1, X
2S
00= −r
0, S
10= r
0− r
1, S
20= (1 − Y
X1)r
1− r
2, X
10= Y
X1r
1, X
20= Y
X2r
2, CH
40= (1 − Y
X2)r
2I initial conditions : S
0(0) = S
00, S
1(0) = S
10, X
1(0) = X
10, X
2(0) = X
20. I reaction rates :
r
0= µ
0S
0, r
1= µ
max1S
1X
1S
1+ K
S1, r
2= µ
max2S
2X
2S
2+ K
S2+ S
22/K
I· I Parameters θ = (Y
X1, Y
X2, µ
0, µ
max1, µ
max2, K
S1, K
S2, K
I| {z }
θc : kinetic parameters
, X
10, X
20, S
00, S
10, S
20| {z }
θb : batch parameters
)
Identification of parameters
I Goals
Obtain a mathematical model allowing to reproduce the methane rate of all experiences,
without necessarily uniquely describe state variables (X
1, X
2, S
1, S
2, S
0) and being able to use this model to predict the BMP from new data measured after only 4 days
I Available measurements For each batch #i we have
I (t
ki)
k=1...mithe times of throttle switchs
I (D
ki)
k=2...mithe mean CH
4flow rate measured at t = t
ki, k = 1 . . .
I Simulations
For θ = (θ
c, θ
b) we can simulate the mean flow of CH
4: d
ki(θ
c, θ
b) =
CH
4(t
ki) − CH
4(t
ki −1)
/(t
ki− t
ki −1), The function
J
i(θ
c, θ
b, T ) = X
k=2
tki ≤T
(t
ki− t
ki −1)(D
ki− d
ki(θ
c, θ
b))
2evaluates the misfit between measurements of batch #i and the simulation with parameters θ = (θ
c, θ
b) at horizon T
I Learning phase T = ∞, batchs #1 . . .#69
Minimize with respect to ξ = (θ
c, θ
b1. . . , θ
b69) ∈ R
353ξ ˆ = arg min
ξ
J (ξ ) = X
i=1...69
J
i(θ
c, θ
bi, ∞) + λkξ k
2,
We only keep θ ˆ
c∈ R
8which is used for the prediction.
Optimization is done with interior points method (fminc, MATLAB) and computation time is small despite problem size (computer with 20 processors Xeon E5-2660-v2).
I Prediction/validation phase T = 4 days, batchs #70 . . .#108 Independently minimize with respect to θ
bi∈ R
5θ ˆ
bi= arg min
θb
J
i( θ ˆ
c, θ
b, 4), i = 70 . . . 108
Results
I Learning on batchs #1 to #69 Kinetic parameters θ ˆ
cY
X10, 58 Y
X20, 12 µ
00, 29 µ
max13, 63 µ
max22, 67 K
S11, 02 K
S23, 45
K
I1, 44
0 200 400true BMP (NmL)600 800 1000 1200 14000 200 400 600 800 1000 1200
predicted BMP (NmL)
Model-3
I Prediction/validation on batchs #70 to #108 at T = 4 days
0 200 400 600 800 1000 1200 1400
true BMP (NmL) 0
200 400 600 800 1000 1200
predicted BMP (NmL)
Model-3
I Prediction at T = 4 days, floated sludge
0 5 10 15
t (days)
0 100 200 300 400
flow (NmL/d)
Model-3 - batch n°92 (Manip 20140305, cell=11)
0 5 10 15
t (days)
0 500 1000
Vol (NmL))
Model-3 - batch n°92 (Manip 20140305, cell=11)
0 5 10 15
t (days) 0
5 10
g.COD/L
Model-3 - batch n°92 (Manip 20140305, cell=11)
S0
S1
S2
0 5 10 15
t (days) 0
5 10
g.COD/L
Model-3 - batch n°92 (Manip 20140305, cell=11)
X1
X2
I Prediction at T = 4 days, thickened sludge
0 5 10 15
t (days)
0 100 200 300 400
flow (NmL/d)
Model-3 - batch n°106 (Manip 20140407, cell=13)
0 5 10 15
t (days)
0 500 1000
Vol (NmL))
Model-3 - batch n°106 (Manip 20140407, cell=13)
0 5 10 15
t (days) 0
5 10
g.COD/L
Model-3 - batch n°106 (Manip 20140407, cell=13)
S0
S1
S2
0 5 10 15
t (days) 0
5 10
g.COD/L
Model-3 - batch n°106 (Manip 20140407, cell=13)
X1
X2
Trends and conclusions
I Results :
I Well fitted kinetics in learning phase and good prediction of BMP at 4 days I Ratios of S 0 , S 1 , S 2 seem to be interpretable
I Planned improvements :
I Theoretical study of identifiability of parameters in learning phase I BOD and VSS measurements should be taken into account
I Coupling between triplicates has to be considered
I Confidence intervals should be computed for the predicted BMP
I Actual model should be simplified and compared with other models
References
[1] C. Holliger et al. (2016), Towards a standardization of biomethane potential tests, Water Science &Technology
[2] O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi, J. P. Steyer (2001), Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnology and bioengineering
[3] R. Fekih Salem, N. Abdellatif, T. Sari, and J. Harmand (2012), On a three step model of anaerobic digestion including the hydrolysis of particulate matter, MATHMOD 2012
[4] A. Donoso-Bravo, S. P ´erez-Elvira and F. Fdz-Polanco (2014), Simplified mechanistic model for the two-stage anaerobic degradation of sewage sludge, Environmental Technology
