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# 1st Order Kinetic Model k-C* - SRG

The 1st Order Kinetic Model k-C* filter model describes the decay or reduction in inflow concentration within a treatment facility such as a grass filter strip. The effectiveness of the treatment is a function of the inflow concentration, the ‘background concentration’ of the treatment facility (ie some measure of the ability of the treatment to reduce concentration), the inflow, and the treatment area. The k-C* model is the fundamental model used in the Universal Stormwater Treatment Model (USTM) of music (eWater 2009).

## Scale

The model operates as a direct concentration reduction facility, where the filtration process is defined to act discretely in a single time-step, and is applied at the scale of the functional unit (FU) to which the filter is applied. The operation of a set or train of filters should be represented by an equivalent single filter area for the FU.

## Principal developer

Cooperative Research Centre for Catchment Hydrology.

## Version

Source v2.10. Based on music v 2.1, Feb 18, 2005.

Source has two sets of k and C* parameters: one for quickflow and one for slowflow.

music version 4, Oct 13, 2009.

## Availability/conditions

The 1st Order Kinetic Model k-C* model is automatically installed with Source.

## Flow phase

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 behaviour 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 behaviour 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 model process can be expressed as:

where:

C* is the background concentration (mg/L)

Cin is the input concentration (mg/L)

Cout is the output concentration (mg/L)

k is the rate constant (m/s)

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).

The rate constant k can be visualised 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.

 Note that the k-C* approach is applicable only for event operation as the k parameter lumps the influence of a number of predominantly physical factors in removal of stormwater pollutants.

In the long term, model performance is sensitive to the value of C*, as this is the value that the outflow concentration tends towards.

There is no routing of flow through the k-C* filter.

The water quality performance of a treatment measure may depend upon the inflow rate. In particular, storm flow 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 model recognises 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.

C* is redefined to be the event background concentration, which applies at higher flows (quick flow) when the extended detention storage is in use. The new parameter C*-slowflow 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*-slowflow feature can be disabled by setting it to have the same value as C*.

## Input data

Within Source, the input flow and concentration are provided by the rainfall-runoff and constituent generation model within a FU. The model time-step is determined by the models that have been selected, although it should be noted that the k-C* model is considered a lumped/event model for filtration of constituent loads.

The k-C* model also requires an input value of treatment area (m2) over which the filtration process acts. This is taken as the area of the FU over which the filter is applied.

In Source a transformation layer has been implemented that allows k-C* parameters to be entered in the usual units (k in m/yr; C* in mg/L), which are then transformed internally to Source consistent (ie. SI) units.

## Parameter settings

Model parameters are summarised in Table 1. Typical values (eWater 2009) for these parameters are shown in Table 2.

###### Table 1. 1st Order Kinetic Model k-C* parameters

Parameter

Description

Units

Default

Range

C*

The background concentration of quickflow

mg/L

1

0 - ∞

C*-slowflow

The background concentration of baseflow

mg/L

1

0 - ∞

Surface area - quick flow

Surface area of filter model for surface flow component

m2

60000

0 - ∞

Surface area - slow flow

Surface area of filter model for the baseflow component

m2

60000

0 - ∞

Cin

Input concentration

mg/L

60 000

0 - ∞

k

Areal decay rate constant

m/yr

1000

0 - ∞

###### Table 2. Typical values for 1st Order Kinetic Model k-C* parameters

Treatment Measure

Total Suspended Solids (TSS)

Total Phosphorus
(TP)

Total Nitrogen
(TN)

k
(m/yr)

C*
(mg/L)

k
(m/yr)

C*
(mg/L)

k
(m/yr)

C*
(mg/L)

Sedimentation Basins

15,000

95

12,000

0.22

1,000

1.7

Ponds

300

40

200

0.20

50

1.3

Swales

15,000

95

12,000

0.22

1,000

1.7

## Output data

The k-C* filter outputs a constituent load, based on the output (= input) flow and output concentration.

## Configuration

Only a single filter model can be applied to a FU or sub-catchment.

## Reference list

eWater 2009, music v4 by eWater User Manual, eWater CRC, Canberra.

## Bibiography

eWater 2009, music v4 by eWater User Manual, eWater CRC, Canberra.

Wong, THF, Duncan, HP, Fletcher, TD & Jenkins, GA 2001, ‘A unified approach to modelling urban stormwater treatment’, Proceedings of the 2nd South Pacific Stormwater Conference, New Zealand Water and Wastewater Association, Auckland, New Zealand, pp. 319-327.