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<h1>Typical Mass balance calculation</h1>
<pre><code class="r">source("R/functions.R")
</code></pre>
<pre><code>## Loading required package: phyloseq
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</code></pre>
<ul>
<li>inf = influent</li>
<li>eff = effluent</li>
<li>WAS = waste activated sludge</li>
<li>AS = activated sludge</li>
<li>WW = wet weight</li>
<li>DW = dry weight</li>
<li>OM = organic matter</li>
<li>OM = DW -> inorganic matter assumed negligible</li>
<li>COD = Chemical oxygen demand</li>
<li>Vp = total plant volume</li>
<li>Xp = total plant biomass</li>
</ul>
<h2>Define Constants</h2>
<pre><code class="r"># 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
</code></pre>
<h2>Dry weather influent</h2>
<pre><code class="r"># 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)
</code></pre>
<h2>Waste activated sludge</h2>
<pre><code class="r"># 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
</code></pre>
<h2>Dry weather effluent</h2>
<pre><code class="r"># 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
</code></pre>
<h2>Mass balance</h2>
<pre><code class="r"># 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)
</code></pre>
<pre><code>## [1] "2.9%"
</code></pre>
<pre><code class="r"># % of out cells in waste
pprint(ncells_WAS/ncells_out)
</code></pre>
<pre><code>## [1] "97.1%"
</code></pre>
<pre><code class="r"># fraction of out cells coming with WW
pprint(ncells_in/ncells_out)
</code></pre>
<pre><code>## [1] "29.1%"
</code></pre>
<pre><code class="r"># Daily effluent fraction of AS
pprint(ncells_eff/ncells_AS)
</code></pre>
<pre><code>## [1] "0.1%"
</code></pre>
<pre><code class="r"># Daily waste fraction of AS
pprint(ncells_WAS/ncells_AS)
</code></pre>
<pre><code>## [1] "3.3%"
</code></pre>
<h2>References</h2>
<p>[1]: Vollertsen et al. 2001. Comparison of methods for determination of microbial biomass in wastewater</p>
<p>[2]: Henze et al. 2002. </p>
<p>[3]: Loferer-Krössbacher et al. 1998. Determination of bacterial cell dry mass by transmission electron microscopy and densitometric image analysis.</p>
<p>[4]: Snaidy. 2009. PhD Thesis. Detection and Enumeration of E. coli and Campylobacter in wastewater and treated water by CARD-FISH.</p>
<p>[5]: Morgan-Sagastume 2008</p>
<p>[6]: Frølund 1996</p>
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