The First Order Kinetic (k - C*) Model
- Geoff Davis
- Mukta Sapkota
When a parcel of water carrying materials such as suspended solids, phosphorus, or nitrogen enters a treatment measure such as a pond or wetland, the water quality of the parcel begins to change. Several physical processes are involved, and the detailed behavior can be very complex. But the overall effect is that contaminant concentrations in the parcel tend to move by an exponential decay process towards an equilibrium value for that site at that time. This behavior can be described by the first order kinetic (or k-C*) model, in which C* is the equilibrium value or background concentration, and k is the exponential rate constant.
The process can be expressed algebraically as:
(Cout - C*) / (Cin - C*) = e -k/q
where
C* is the background concentration
Cin is the input concentration
Cout is the output concentration
k is the rate constant, and
q is the hydraulic loading (flow rate per surface area) of the treatment measure.
Thus, a higher k means a faster approach to equilibrium, and hence a higher treatment capacity (provided C* is less than Cin).
Tip Box
The rate constant k can be visualized as the hydraulic loading which gives an output concentration (above C*) which is e -1 (~0.37) times the inflow concentration (above C*) for a given situation.
The water quality performance of a treatment measure may depend upon the inflow rate. In particular, stormflow and baseflow may be handled very differently. Baseflows may be confined to a distinct low flow channel or pipe, while stormflows potentially occupy the whole area of the treatment measure. To allow for this, the package recognizes two separate background concentrations in treatment measures that do not consist of a permanent pool, thus allowing for a better description of the low flow operating conditions in these measures.
We redefine C* to be the event background concentration, which applies at higher flows when the extended detention storage is in use. The new parameter C** becomes the baseflow background concentration, which applies when flows are largely confined to a low flow channel. Where a permanent pool is present, only a single background concentration (C*) applies. The C** feature can be disabled by setting it to have the same value as C*.