Kinetics of Decay

Stoichiometry for decay of viable cells producing debris and (eventually, soluble substrate:

Biomass-COD  fD Debris-COD + (1-fD) SS

In usual notation:

(-1)XB + (1-fD)SS + fD XD = 0 (1)

Where:

XB = biomass concentration (mg-COD/L)

XD = debris concentration (mg-COD/L)

SS = soluble COD (mg/L)

fD = fraction of decayed biomass that is nonbiodegradable debris (g-debris COD formed/g-biomass-COD decayed)

Assumption:

Simplification: all biodegradable particulate COD (1-fD) becomes soluble substrate COD eventually (hydrolysis is 100% effective)

Reaction stoichiometry for nitrogen in decay assuming nitrogen from decaying cells either becomes debris or soluble ammonia:

iNXB Biomass-COD  iNXD fD debris-COD + (iNXB - iNXD fD) NH4-N

(-1)iNXB XB + iNXD fD XD + (iNXB - iNXD fD) NH4-N = 0 (2)

where:

iNXB = 14 g-N/113 g-cells/1.42 g-COD/g-cells = 0.087 g-N/g-cell COD

iNXD < (0.6) since debris are thought to contain less N than cytoplasm

Decay rate expressions:

Define biomass decay coefficient, b, for a first-order rate expression:

rXB = -bXB

where

rXB = rate of decay of biomass (mg-COD/L-hr)

XB = biomass concentration (mg-COD/L)

b = decay coefficient (hr-1)

rXB / -1 = rXD / fD = bXB (3)

where rXD = rate of formation of debris-COD (mg-COD/L/hr)

from (3):

rXD = fD bXB

assuming all biodegradable particulate COD from decay becomes soluble COD:

rXB / -1 = rXD / fD = rS / (1-fD) = rSNH / (iNXB - fD iNXD) = bXB (3a)

Note that the general rate equation relationship is the same regardless of whether the organisms are heterotrophs or autotrophs.

A difference between the decay rate and growth rate expressions is that b is assumed to be constant for all process conditions (unlike the growth rate coefficient, which is a function of substrate and other nutrient concentrations).

Hydrolysis

Modify (1):

(-1)XB – (1-fD)XS + fD XD = 0 (1a)

where XS = biodegradable particulate matter (mg-COD/L)

(-1)iNXB XB + iNXD fD XD + (iNXB - iNXD fD) XNS = 0 (2a)

where XNS = particulate organic nitrogen

Then:

rXB / -1 = rXD / fD = rXS / (1-fD) = rXNS / (iNXB - fD iNXD) = bXB (3b)

and

(-1)XS + SS = 0 (4)

For the hydrolysis rate, rH:

rH = kH XBH (XS / XBH) / (KX + (XS / XBH)) = rXS / -1 = rS / 1 = rXNS / -(XNS / XS) = rSNH / (XNS / XS) (5)

where:

kH is a pseudo-first-order rate coefficient (hr-1)

XBH is heterotrophic biomass (mg-COD/L)

KX = the half saturation coefficient for the switching function for the concentration of hydrolysis enzyme substrate (mg-particulate-COD/mg-heterotrophic biomass-COD)