The Petersen matrix is a comprehensive description of systems of
biochemical reactions used to model
reactors for
pollution control
Pollution is the introduction of contaminants into the natural environment that cause adverse change. Pollution can take the form of any substance (solid, liquid, or gas) or energy (such as radioactivity, heat, sound, or light). Pollutants, the ...
(engineered
decomposition
Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is e ...
) as well as in
environmental systems
In science and engineering, a system is the part of the universe that is being studied, while the environment is the remainder of the universe that lies outside the boundaries of the system. It is also known as the surroundings or neighborhood, ...
. It has got as many columns as the number of relevant involved components (
chemicals
A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent elements by physical separation methods, i.e., wit ...
,
pollutants
A pollutant or novel entity is a substance or energy introduced into the environment that has undesired effects, or adversely affects the usefulness of a resource. These can be both naturally forming (i.e. minerals or extracted compounds like oi ...
,
biomass
Biomass is plant-based material used as a fuel for heat or electricity production. It can be in the form of wood, wood residues, energy crops, agricultural residues, and waste from industry, farms, and households. Some people use the terms bi ...
es,
gases
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
) and as many rows as the number of involved
processes (biochemical reactions and physical degradation). One further column is added to host the description of the
kinetics
Kinetics ( grc, κίνησις, , kinesis, ''movement'' or ''to move'') may refer to:
Science and medicine
* Kinetics (physics), the study of motion and its causes
** Rigid body kinetics, the study of the motion of rigid bodies
* Chemical ki ...
of each transformation (
rate equation
In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reacti ...
).
Matrix structure
The mass conservation principle for each process is expressed in the rows of the matrix. If all components are included (none omitted) then the mass conservation principle states that, for each process:
:
where
is the density rate of each component. This can also be seen as the process
stoichiometric relation.
Moreover, the rate of variation of each component for all processes simultaneous effect can be easily assessed by summing the columns:
:
where
are the reaction rates of each process.
Example
A system of a third
order reaction followed by a
Michaelis–Menten enzyme reaction.
:
+ 2B -> S
:
+ S <=> _fk_r] ES -> _\mathrm + P
where the reagents A and B combine forming the substrate S (S = AB
2), which with the help of enzyme E is transformed into the product P.
Production rates for each substance is:
:
Therefore, the Petersen matrix reads as
The Petersen matrix can be used to write the system's rate equation
:
References
{{Reflist
Biodegradation
Biodegradable waste management
Chemical processes