Kinetics of growth
It is often convenient to state stoichiometric expressions with cells as a reference, since many rates are a function of growth processes.
Notation:
In general, S designates soluble components, X, particulate.
S\_S = soluble COD
S\_O = dissolved oxygen
S\_NH = ammonia nitrogen (NH4-N)
S\_NO = nitrate nitrogen (NO3-N)
S\_NS = soluble organic nitrogen
etc.
X\_BH = heterotrophic viable cells
X\_BA = autotrophic viable cells
X\_S = biodegradable particulate organic matter
X\_D = debris organic matter (nonbiodegradable)
X\_NS = particulate organic nitrogen (biodegradable)
etc.
Reaction: Heterotrophic growth, reference is viable bacteria cells (as COD)
-1/Y\_H S\_S - ((1 - Y\_H)/Y\_H) S\_O + X\_BH = 0
since nitrogen uptake in synthesis is growth-determined, it can “come along for the ride:”
-1/Y\_H S\_S - ((1 - Y\_H)/Y\_H) S\_O - i\_NXB S\_NH + X\_BH = 0
associated rate relationship:
r\_S / (-1/Y\_H) = r\_O / (-(1 - Y\_H)/Y\_H) = r\_BH = overall growth reaction rate, r\_j for j = 1 → n
where r\_S = rate of COD substrate consumption, r\_O = rate of oxygen consumption and r\_BH = heterotrophic growth rate.
If r\_BH is defined along with values for Y\_H and i\_NXB, then all other rates are defined as above.
Bacteria grow by fission, so the population growth rate is a function of the number of viable cells. Define the growth rate for any bacteria, r\_B
r\_B = μX\_B
where
μ = specific growth rate (hr-1, d-1, etc.)
X\_B = viable cell concentration (mg-COD/L = g-COD/m3)
Note that units of rate expressions for wastewater process analysis are always concentration/time. If μ is constant, this is first order kinetics:
r\_B = μX\_B (1)
Note that yield links substrate removal with growth:
r\_S = -μX\_B / Y
In ~1948 French microbiologist Jacques Monod found that in continuous cultures of bacteria μ was not constant. In fact it varied in non-linear proportion to a rate limiting substrate (often the electron donor).
μ = μ̂ S\_S / (K\_S + S\_S)
where μ̂ = the maximum specific growth rate and K\_S = the half-saturation (or half-velocity) constant.
Note that Monod kinetic expression developed for single culture growing on single limiting substrate. We will apply to mixed populations (e.g., heterotrophs) growing on compound mixture (COD)
Regions of Monod growth
First order:
If S\_S << K\_S, denominator → K\_S and
μ ≅ μ̂ S\_S / K\_S
growth rate is linearly proportional to rate limiting substrate concentration
Zero-order:
If S\_S >> K\_S, denominator → S\_S and
μ ≅ μ̂
Complimentary Nutrients:
COD, N, P, O2, etc.
Allowing simultaneous substrate limitation of growth by two or more components.
Interactive model uses a series of “switching functions” where 0 < SF < 1
μ = μ̂ (S\_S/(K\_S + S\_S)) (S\_O/(K\_O + S\_O)) (S\_NH/(K\_NH + S\_NH))
where
0 < S\_i/(K\_i + S\_i) < 1
Relation between a nutrient’s concentration and the half-saturation constant is key to determining value of each switching function.
Interactive model provides a smooth function for the specific growth rate which makes computational modeling easier.