The Models:
- The \(NKC^*\) Model: A generalized tank-in-series outflow model represented as
| Variable | Definition | Units |
|---|---|---|
| \(Q\) | Flow | \(\mathrm{m^{3}\ y^{-1}}\) |
| \(A\) | Area | \(\mathrm{m^{2}}\) |
| \(HLR\) | Hydraulic Looading Rate \(HLR = Q/A\) | \(\mathrm{m\ y^{-1}}\) |
| \(C_{in}\),\(C_{out}\) | Inflow/Outflow Concentration | \(\mathrm{mg\ m^{-3}\ or \ \mu g\ L^{-1}}\) |
| \(C^*\) | Background concentration | \(\mathrm{mg\ m^{-3}\ or \ \mu g\ L^{-1}}\) |
| \(k\) | areal net settling (removal velocity) | \(\mathrm{m\ y^{-1}}\) |
| \(N\) | Number of tanks in series | unitless |
When \(N = 1\) it indicates a fully mixed tank (conservative), meanwhile \(N = 3 - 5\) is the common STA design range.
Using equation above, here are some useful limited cases and/or simplifications of the model.
- \(N = 1\) (single Continuously Stirred Tank Reactor [CSTR])
\[C_{out} = C^{*} + \frac{C_{in} - C^{*}}{1 + \frac{k}{N \cdot HLR}}\]
- \(N \to\infty\) (plug-flow limit)
\[C_{out} = C^{*} + (C_{in} - C^{*})e^{-k / HLR}\]
Using the tank in series equation, here is the inverse relationship
Solve for required HLR (given target \(C_{out}\))
Rearrange the \(NKC^*\) formula:
\[\bigg( 1 + \frac{k}{N \cdot HLR} \bigg)^N = \frac{C_{in} - C^*}{C_{out} - C^{*}} \] \[HLR = \frac{k/N}{\big( \frac{C_{in} - C^*}{C_{out} - C^{*}} \big)^{1/N} -1}\] Then area for a specified flow \(Q\) is:
\[A = \frac{Q}{HLR}\] Domain constraints: you need \(C_{out} > C^{*}\) and \(C_{in} > C^{*}\). If \(C_{out} \le C^{*}\), the outflow is exceeding the assumed background concentration.
Solve for k (given observed \(C_{in}\),\(C_{out}\) and \(HLR\))
\[k = N \cdot HLR \bigg[ \bigg( \frac{C_{in} - C^{*}}{C_{out} - C^{*}} \bigg)^{1/N} -1 \bigg]\] This is the equivalent settling velocity (\(m \ yr^{-1}\)) implied by observed performance (at the chosen \(N\) and $C^{8}).
- \(N = 1\) (single Continuously Stirred Tank Reactor [CSTR])
\[k = HLR \cdot \bigg( \frac{C_{in}-C^{*}}{C_{out}-C^{*}} -1 \bigg)\]
- \(N \to\infty\) (plug-flow limit)
\[k = HLR \cdot ln\bigg( \frac{C_{in}-C^{*}}{C_{out}-C^{*}} \bigg)\]
This is often the method to estimate the upper bounds of \(k\) from observed annual data.
- DMSTA Model: Conceptually represented as
| Symbol | Description | Units |
|---|---|---|
| State Variables | ||
| M | Water column P storage | \(\mathrm{mg\ m^{-2}}\) |
| S | Biomass P storage | \(\mathrm{mg\ m^{-2}}\) |
| Z | Mean water depth | \(m\) |
| Driving Variables | ||
| L | External P load | \(\mathrm{mg\ m^{-2}\ yr^{-1}}\) |
| Q | Outflow rate | \(\mathrm{m\ yr^{-1}}\) |
| Parameter Variables | ||
| K1 | Maximum uptake rate | \(\mathrm{m^{3}\ mg^{-1}\ yr^{-1}}\) |
| K2 | Recycle rate | \(\mathrm{m^{2}\ mg^{-1}\ yr^{-1}}\) |
| K3 | Burial rate | \(\mathrm{yr^{-1}}\) |
| Compartment | Equation |
|---|---|
| Storage: | \(K1 \cdot C = K2 \cdot S + K3\) |
| Overall: | \(L - Q \cdot C = K3 \cdot S\) |
| Solution for C: | \(C = \frac{(K2 \cdot L + K3^{2})}{K3 \cdot K1 + Q \cdot K2}\) |
| Solution for S: | \(S = \frac{(K1 \cdot C - K3)}{K2}\) |
Whereas the steady-state DMSTA solution is equivalent to a simple \(KC^{*}\) model with \(C^{*} = C_0\) and \(K\) is interpreted as a net settling rate (\(m \ yr^{-1}\)). Therefore, for “equivalent design model” purposes \(C^{*} \equiv C_{0}\) and \(k \equiv K\). This means that the DMSTAs calibrated (\(K\) and \(C_{0}\)) can be subsituted directly into the \(NKC^{*}\) model.
Where \[C = \frac{(L + K \cdot C_{0})}{(K + Q)}\]
\[S = 1000 \cdot \frac{C-C_0}{C1 - C_0}\]
Where,
\[K = \frac{K2 \cdot K1}{K2}\]
Therefore, given (\(K\),\(C_0\),\(C_1\)), (\(K1\), \(K2\), \(K3\)) can be estimated:
\[K3 = K \cdot \frac{C_{1} - C_{0}}{1000}\]
\[K1 = \frac{K3}{C_0}\]
\[K_2 = \frac{K3 \cdot K1}{K}\]
Given (\(K1\), \(K2\), \(K3\)), (\(K\),\(C_0\),\(C_1\)) can be estimated:
\[C_{0} = \frac{K_3}{K_1}\]
\[K = \frac{K_1 \cdot K_3}{K_2}\]
\[ C_1 = \frac{1000 \cdot K_{2} + K_{3}}{K_{1}}\]
Helpful R Code
To see these equation in practice here are some simple R functions and examples.
# Forward NKC* (tanks-in-series) prediction
nkc_out <- function(Cin, k, HLR, N = 1, Cstar) {
Cstar + (Cin - Cstar) / (1 + k / (N * HLR))^N
}
# Solve for HLR given target Cout
nkc_hlr <- function(Cin, Cout, k, N = 1, Cstar) {
r <- (Cin - Cstar) / (Cout - Cstar)
(k / N) / (r^(1/N) - 1)
}
# Solve for k given observed Cin/Cout and HLR
nkc_k <- function(Cin, Cout, HLR, N = 1, Cstar) {
if (any(Cout <= Cstar))
stop("Cout must be greater than Cstar")
r <- (Cin - Cstar) / (Cout - Cstar)
N * HLR * (r^(1/N) - 1)
}
# DMSTA parameter translations
dmsta_KC_to_K123 <- function(K, C0, C1) {
K3 <- K * (C1 - C0) / 1000
K1 <- K3 / C0
K2 <- K3 * K1 / K
list(K1 = K1, K2 = K2, K3 = K3)
}
dmsta_K123_to_KC <- function(K1, K2, K3) {
C0 <- K3 / K1
K <- (K1 * K3) / K2
C1 <- (1000 * K2 + K3) / K1
list(K = K, C0 = C0, C1 = C1)
}
# Estimate k from plug and flow
estimate_k_plug <- function(
Cin, Cout,
HLR,
Cstar
) {
if (any(Cout <= Cstar))
stop("Cout must be greater than Cstar")
HLR * log( (Cin - Cstar) / (Cout - Cstar) )
}## Example Data
Cin <- 122
Cout <- 17
k <- 66.5
HLR <- 9.5
Cstar <- 2
K <- 16.8
C0 <- 2
C1 <- 22
K1 <- 0.168
K2 <- 0.00336
K3 <- 0.336
# Estimated Cout given Cin, k, & HLR
nkc_out(Cin, k, HLR, N = 1, Cstar)
#> [1] 17
# Estimated HLR given Cin, Cout & k
nkc_hlr(Cin,Cout,k,N = 1,Cstar)
#> [1] 9.5
# vs more tanks
nkc_hlr(Cin,Cout,k,N = 3,Cstar)
#> [1] 22.16667
## Estimate k
nkc_k(Cin,Cout,HLR,1,Cstar)
#> [1] 66.5
# estimate model parameters
dmsta_KC_to_K123(K,C0,C1)
#> $K1
#> [1] 0.168
#>
#> $K2
#> [1] 0.00336
#>
#> $K3
#> [1] 0.336
dmsta_K123_to_KC(K1,K2,K3)
#> $K
#> [1] 16.8
#>
#> $C0
#> [1] 2
#>
#> $C1
#> [1] 22