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First Order Rate Model

The parameters that describe biogas production are:

The volume of the container: V in m3

The Retention time R is the time the feed material is in the container, for a batch plant.

For a continuous digester:  

, where v m3 is the volume fed each day.  

\[R = \frac{V}{v}\]

The digestible matter in the slurry can be measured from the dry weight in the feed material and the volatile solids (VS) proportion or the Chemical Oxygen Demand (COD). The initial concentration (S0) of digestible matter in the slurry is then measured as     kg.VS m-3 or kg.COD m-3. As biogas is produced, the concentration of the digestible matter (S) reduces.  

The first assumption is that the gas produced is proportional to the mass of feed material (substrate) digested:

\[\begin{gathered} G = C\,V\left( {S_0 - S} \right) \hfill \\ g = C\,v\left( {S_0 - S} \right) \hfill \\ \end{gathered} \]

The total (cumulative) gas production G m3 is measured for a batch plant, while the daily gas production g m3 d-1 is used for continuous plants.

C m3 kg.VS-1 or m3 kg.COD-1 is defined as the biogas potential and is assumed to be constant for a feed material.

The second assumption is that the rate of gas production (how fast biogas is produced) is proportional to the concentration of the substrate:

\[\frac{{dS}}{{dt}} = - k\,S\]

k d-1 is defined as the rate constant.

For a batch plant, this gives:

\[S = S_0 \exp \left\{ { - k\left( {t - t_0 } \right)} \right\}\]
\[G = C\,V\,S_0 \left[ {1 - \exp \left\{ { - k\left( {t - t_0 } \right)} \right\}} \right]\]

For a continuous plant, mass in - mass out = mass converted per day:

\[\begin{gathered} v\,S_0 - v\,S = k\,S\,V \hfill \\ S_0 - S = k\,S\,R \hfill \\ g = C\,V\,S_0 \frac{k}{{1 + k\,R}} \hfill \\ \end {gathered} \]

Values for C and k have been determined for cattle dung and food wastes, as well as examples of their use with the equations, based on properties of the materials.