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Typical Mass balance calculation

source("R/functions.R")
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  • inf = influent
  • eff = effluent
  • WAS = waste activated sludge
  • AS = activated sludge
  • WW = wet weight
  • DW = dry weight
  • OM = organic matter
  • OM = DW -> inorganic matter assumed negligible
  • COD = Chemical oxygen demand
  • Vp = total plant volume
  • Xp = total plant biomass

Define Constants

# Nominal plant volume
Vp <- 1
# Hydrolyic retension time
HRT <- 1  # days
# Solids retension time
SRT <- 30  # days
# Assumed yield
Y_assumed <- 0.28

# unit conversion factors
um3_per_cm3 <- 1e+12
g_per_kg <- 1000
cm3_per_m3 <- 1e+06

Dry weather influent

# COD to organic matter ratio
COD_per_OM_inf <- 1.42  # kg-COD kg-OM^-1 [2]
# mean influent COD
COD_inf <- 0.582  # kg-COD m⁻³        [1] 
# mean influent cell concentration
cells_inf <- 1e+08  # cells cm⁻³        [1, 4] 
# influent flow
flow_inf <- Vp * HRT  # V day⁻¹

cells_inf <- cells_inf * cm3_per_m3  # cells m^-3 
OM_inf <- COD_inf/COD_per_OM_inf  # kg-OM m⁻³

# calculation of cell fraction of organic matter cell mass = 95 fg cell⁻¹
# [3] cell mass = 9.5e-13 g cell⁻¹

cell_volume <- 0.285  # µm³-cell cell⁻¹ [1] 
cell_density <- 1.1  # g-cell cm⁻³     [1, 3] 
OM_per_cell_WW <- 0.22  # g-OM g-cell⁻¹   [1, 3] 

cell_OM <- calculate_cell_OM(cell_volume, cell_density, OM_per_cell_WW)
cell_OM_inf <- cells_inf * cell_OM
# g-OM m⁻³ cell m⁻³ * g-OM cell⁻¹
cell_OM_inf <- cell_OM_inf/g_per_kg
# kg-OM m⁻³
cell_OM_percent <- round(cell_OM_inf/OM_inf * 100, 1)

Waste activated sludge

# mean AS cell concentration
cells_AS <- 1e+10  # cells cm⁻³  
cells_AS <- cells_AS * cm3_per_m3  # cells m⁻³ 
# WAS OM & COD
OM_AS <- 5  # kg-OM  m⁻³

COD_per_VSS_AS <- 1.42  # kg-COD kg-VSS⁻¹
VSS_per_OM_AS <- 0.75  # kg-VSS kg-OM⁻¹
COD_AS <- OM_AS * VSS_per_OM_AS * COD_per_VSS_AS  # kg-COD m⁻³

# WAS thickening
WAS_thick <- 3
OM_WAS <- OM_AS * WAS_thick  # kg-OM  m⁻³
COD_WAS <- COD_AS * WAS_thick  # kg-COD m⁻³

cells_WAS <- cells_AS * WAS_thick

# WAS flow
flow_WAS <- Vp * (1/SRT)/WAS_thick  # V day⁻¹

Dry weather effluent

# effluent flow
flow_eff <- Vp - flow_WAS  # V day⁻¹
# effluent OM & COD
OM_eff <- 20  #  g-OM m⁻³
OM_eff <- OM_eff/g_per_kg  # kg-OM m⁻³
COD_eff <- OM_eff * COD_per_OM_inf  # kg-COD m⁻³
# mean effleunt cell concentration
cells_eff <- 1e+07  # cells cm⁻³  [5] measured
# cells_eff <- 1.0e7 # cells cm⁻³ [4] measured
cells_eff <- cells_eff * cm3_per_m3  # cells m⁻³

Mass balance

# Cell budget
ncells_in <- cells_inf * flow_inf
ncells_eff <- cells_eff * flow_eff
ncells_AS <- cells_AS * Vp
ncells_WAS <- cells_WAS * flow_WAS
ncells_out <- ncells_eff + ncells_WAS

pprint <- function(x) {
    paste0(round(x * 100, 1), "%")
}
# % of out cells in effluent
pprint(ncells_eff/ncells_out)
## [1] "2.9%"

# % of out cells in waste
pprint(ncells_WAS/ncells_out)
## [1] "97.1%"

# fraction of out cells coming with WW
pprint(ncells_in/ncells_out)
## [1] "29.1%"

# Daily effluent fraction of AS
pprint(ncells_eff/ncells_AS)
## [1] "0.1%"

# Daily waste fraction of AS
pprint(ncells_WAS/ncells_AS)
## [1] "3.3%"

References

[1]: Vollertsen et al. 2001. Comparison of methods for determination of microbial biomass in wastewater

[2]: Henze et al. 2002.

[3]: Loferer-Krössbacher et al. 1998. Determination of bacterial cell dry mass by transmission electron microscopy and densitometric image analysis.

[4]: Snaidy. 2009. PhD Thesis. Detection and Enumeration of E. coli and Campylobacter in wastewater and treated water by CARD-FISH.

[5]: Morgan-Sagastume 2008

[6]: Frølund 1996