# Why this decay term in Anaerobic Digestion Model?

I am reading about the AM2 model, which appears to be a simplified version of the ADM1 model for modeling anaerobic processes.

The author of this doctoral thesis extended the simpler AM2 model by introducing hydrolysis. Given a substrate state variable $$S_1(t)$$, he then proceeds as follows:

$$\frac{dS1}{dt}=D_{in}(S_{1in}-S)-k_1\mu_1X_1+k_7\left(k_{dis}X_c-k_{dec,X_1}X_1-k_{dec,X_2}X_2\right)+k_8\left((k_{hyd,ch}X_{ch}-f_{ch,X_c}k_{dis}X_c)+(k_{hyd,pr}X_{ch}-f_{pr,X_c}k_{dis}X_c)+(k_{hyd,li}X_{li}-f_{li,X_c}k_{dis}X_c)\right)$$

The variables $$X_1,X_2$$ describe the chemical oxygen demand of acidogenic and methanophobic bacteria. The variables $$X_{ch},X_{li},X_ch$$ describe the chemical oxygen demand of the hydrolized composite $$X_c$$.

I do not understand the underbraced parts:

$$\frac{dS1}{dt}=D_{in}(S_{1in}-S)-k_1\mu_1X_1+k_7\left(k_{dis}X_c-k_{dec,X_1}X_1\underbrace{-k_{dec,X_2}X_2}\right)+k_8\left((k_{hyd,ch}X_{ch}-f_{ch,X_c}k_{dis}X_c)+(k_{hyd,pr}X_{ch}\underbrace{-f_{pr,X_c}k_{dis}X_c})+(k_{hyd,li}X_{li}\underbrace{-f_{li,X_c}k_{dis}X_c})\right)$$

Unfortunately I have not been able to reach the author. Why would we substract the underbraced terms? Imagine a scenario, where $$k_8=0, k_{dis}=0$$. In that case, the substrate concentration $$S_1$$ could become negative just because of dying bacteria (the $$k_{dec}$$ terms). That does not make any sense to me.

Does it make sense to you?

• I would start be asking simple questions. Does your imagined scenario make sense (biologically and/or practically)? Are there organic substrates with these characteristics (a zero disintegration parameter ({k_dis}) and a zero yield coefficient for proteins, lipids, and carbohydrates)? And if so, why would you feed them into your anaerobic processor? Perhaps my lack of expertise in the area is just preventing me from understanding why the scenario you described is problematic. May 12, 2022 at 15:22
• Thank you for the answer. The continued substraction of available substrate from for example dead biomass (decayed bacteria) strikes me as peculiar. You are right, I will more deeply investigate these basoc questions May 13, 2022 at 7:18