G.4 Adsorption and desorption

At each timestep and within each active computational cell, the WQ Module distributes the total FRP mass between dissolved FRP and adsorbed FRP if the phosphorus adsorption constituent model is used. Total FRP is conserved via Equation (G.5).

\[\begin{equation} \left[FRP\right]_{tot} = \left[FRPads\right] + \left[FRP\right] \tag{G.5} \end{equation}\]

\(\left[FRP\right]_{tot}\) is the total FRP available for distribution between dissolved and adsorbed states, and \(\left[FRPads\right]\) and \(\left[FRP\right]\) are the ambient adsorbed and dissolved FRP concentrations, respectively.

The distribution between adsorbed and dissolved FRP concentrations is computed within the WQ Module via either a linear or quadratic model. These offer alternative approaches to simulating the distribution of total FRP between adsorbed and free states.

G.4.1 Linear model

Equation (G.6) is the linear adsorption model, and the default.

\[\begin{equation} \left.\begin{aligned} \left[FRPads\right] &= \left(\frac{K_{ads-L}^{FRP} \times \left[SS\right]}{1 + K_{ads-L}^{FRP} \times \left[SS\right]}\right) \times \left[FRP\right]_{tot}\\ \\ \left[FRP\right] &= \left(\frac{1}{1 + K_{ads-L}^{FRP} \times \left[SS\right]}\right) \times \left[FRP\right]_{tot} \end{aligned}\right\} \tag{G.6} \end{equation}\]

\(\left[SS\right]\) is the sum of suspended sediment concentrations of sediment fractions that have been designated as able to adsorb FRP and \(K_{ads-L}^{FRP}\) is the linear sorption partitioning coefficient.

Figure G.3 provides an indicative representation of the general form of the linear relationship, for assumed total FRP (0.1 mg/L) and SS (5mg/L) concentrations.

Figure G.3: The change in distribution of dissolved and adsorbed FRP under the linear adsorption model. This figure has no slider bar

As a guide, equation (G.6) has that if \(K_{ads-L}^{FRP} \times \left[SS\right]\) has a value of 1, then an equal (50:50) proportioning of free and adsorbed FRP will result. This guide can be useful if ambient suspended sediment concentrations are known - even approximately - in advance.

G.4.2 Quadratic model

Equation (G.7) is the quadratic adsorption model.

\[\begin{equation} \left.\begin{aligned} \left[FRPads\right] &= \frac{1}{2} \times \left[\left(\left[FRP\right]_{Tot} + \frac{1}{K_{ads-Q}^{FRP}} + \left(\left[SS\right] \times Q^{FRP}_{max}\right)\right) - \text{C} \right] \\ \\ \left[FRP\right] &= \frac{1}{2} \times \left[\left(\left[FRP\right]_{Tot} - \frac{1}{K_{ads-Q}^{FRP}} - \left(\left[SS\right] \times Q^{FRP}_{max}\right)\right) + \text{C} \right] \\ \\ \text{C} &= \sqrt{\left(\left[FRP\right]_{Tot} + \frac{1}{K_{ads-Q}^{FRP}} - \left(\left[SS\right] \times Q^{FRP}_{max}\right)\right)^2 + \left(\frac{4.0\times \left[SS\right] \times Q^{FRP}_{max}}{K_{ads-Q}^{FRP}}\right)} \end{aligned}\right\} \tag{G.7} \end{equation}\]

\(\left[SS\right]\) is again the sum of suspended sediment concentrations of sediment fractions that have been designated as able to adsorb FRP, \(K_{ads-Q}^{FRP}\) is the ratio of adsorption and desorption rate coefficients and \(Q^{FRP}_{max}\) is the maximum adsorption capacity of suspended sediment, for FRP.

Figure G.4 provides an indicative representation of the general form of the quadratic relationship, again for assumed total FRP (0.1 mg/L) and SS (5mg/L) concentrations.

Figure G.4: The change in distribution of dissolved and adsorbed FRP under the quadratic adsorption model

The parameters used to prepare the above figures were deliberately chosen to illustrate the different behaviour that can result from these two models. This underscores the need to make careful decisions regarding the selection of an adsorption model.