DMSTA2  Phosphorus Cycling Model 06/22/05
One CSTR at Steady-State
Unit Area Storage & Fluxes
Concs in mg/m3
Fluxes in mg/m2-yr      
Storage in mg/m2 Water Column
L Mass = M
Q  C
Conc = C =  M / Z
     
FC FZ  K1  S  C K2 S2
FZ  =  Depth Multiplier      
FC = Conc Multiplier Biomass P Storage
S
     
K3 S
State Variables:
M Water Column P Storage mg/m2
S Temporary P Storage in Biota, etc. mg/m2
Z Water Column Mean Depth m
Driving Variables:
L P Load, Including Atmos. Deposition mg/m2-yr
Q Outflow m/yr
Parameter Values:
K1 Maximum Uptake Rate m3 / mg - yr
K2 Recycle Rate m2 / mg-yr
K3 Burial Rate 1/yr
Steady-State Mass Balances 1 CSTR,  FZ = 1, & C << C2:
Storage: K1  C    =    K2 S  +  K3
Overall: L    -  Q C    =    K3 S
Solution for C: C     =      ( K2  L  +  K32 ) /  (  K3  K1  +  Q  K2 )  
Solution for S: S     =     (  K1  C  -  K3  )  /  K2
For Parameter-Estimation Purposes, Model Coefficients Are Re-expressed as Follows:
K Net Settling Rate at Steady State m/yr Fit to Concentration Time Series Data
C0 Water Column Conc at S = 0 mg/m2 ppb Estimated from Storage vs. Conc Relationship
C1 Water Column Conc at S = 1000 mg/m2 ppb Fit to Concentration Time Series Data
Steady-State Solution:
C  =   (   L   +     K   C0 )   /    (   K   +  Q )
S  =   1000    (  C   -   C0 )   /   (  C1  -   C0 ) used to calibrate C0 & C1
The steady-state solution is equivalent to the familiar K/C* model with C* = C0 :
L
C
Q  C
                 K  ( C  -  C0 )
where,
K  =  K2 K1 / K2
Translations of Parameter Sets:
K3  =  K  (  C1  -  C0 ) / 1000 K   =   K1  K3  /  K2
K1  =  K3 /  C0 C0  =  K3  /  K1
K2  =  K3  K1  /  K C1  =  (1000 K2  +  K3 ) / K1
Calibrations: 
EMERG emergent marsh (cattail etc.) on previously farmed or otherwise disturbed soils
PEW preexistent wetlend; former wetland with vegetation established prior to construction; calcitic waters
SAV cell managed for SAV or other favorable community; calcitic waters
PSTA periphyton treatment area (peat removed or capped)
RESERV lake or reservoir, generally depths > 150 cm
Parameter EMG_3 PEW_3 SAV_3 PSTA_3 RES_3
C0 WC Conc at 0 g/m2 Storage ppb 3 3 3 3 3
C1 WC Conc at 1 g/m2 Storage ppb 22 22 22 22 150
C2 Half-Saturation Conc for P Uptake ppb 300 300 300 300 0
K Net Settling Rate at Steady State m/yr 16.8 34.9 52.5 23.6 5.0
Z1 Depth at Maximum Uptake cm 40 40 40 0 40
Z2 Transition Depth cm 100 100 100 100 100
Z3 Depth at Minimum Uptake cm 200 200 200 200 400
FZ3 Relative Uptake Rate at Z >= Z3  - 0.2 0.2 0.2 0.2 0.2
K1 Uptake Rate m3 / mg-yr 0.1064 0.2210 0.3325 0.1495 0.2450
K2 Recycle Rate m2 / mg-yr 0.0020 0.0042 0.0063 0.0028 0.0360
K3 Burial Rate 1/yr 0.3192 0.6631 0.9975 0.4484 0.7350
Depth Effects:
Parameters inferred from STA operating experience & calibrations
Performance increases between 0 and Z1  (topographic effects, effective treatment area, short-circuiting)
Marsh performance starts to deteriorate at 30-day mean depth > Z2 (uprooting, light-limitation, short-circuiting, etc.)
Reservoir performance starts to deteriorate at depths > Z2 (stratification/anoxia, turbulence, light limitation)
Between Z2 and Z3 all calibrations converge to effective K = 1 m/yr (K1, K2, K3 adjusted accordingly)
These transitions are ignored if input Z1-Z3 values = 0 
STA's typically operated at static depth of 40 cm, &  40- 80 cm during inflow events
Z1 Saturated Uptake Depth daily cm
Z2 Lower Penalty Depth 30-day cm
Z3 Upper Penalty Depth 30-day cm
Concentration Multiplier for Gross Uptake Rate:
Reflects saturation of uptake rate at high concentrations
FC = C2 /  ( C  +  C2 )
C2 = half-sat. conc for P uptake = 300 ppb
Calibrated to marsh datasets in high conc range
Not applied to reservoirs.
Conc
ppb FC
0 1.00
50 0.86
100 0.75
150 0.67
200 0.60
250 0.55
300 0.50
350 0.46
400 0.43
450 0.40
500 0.38
Steady-State Solutions of P Cycling Model for Each Calibration
Mean Depth = 60 cm Hydraulic Load = 10 m/yr
Storage (mg/m2)             Storage Response Time (yrs)
Net P Removal  (g/m2-yr)  Net Settling Rate (m/yr)
6/22/2005