Kinetics of Decay Stoichiometry for decay of viable cells producing debris and (eventually, soluble substrate: Biomass-COD  fDDebris-COD + (1-fD)SS In usual notation: (-1)XB + (1-fD)SS + fDXD = 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: iNXBBiomass-COD  iNXDfDdebris-COD + (iNXB - iNXDfD)NH4-N (-1)iNXB XB + iNXDfD XD + (iNXB - iNXDfD)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 cytoplasmDecay 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) BDXDXBbXfr1r (3) where rXD = rate of formation of debris-COD (mg-COD/L/hr) from (3): rXD = fDbXB assuming all biodegradable particulate COD from decay becomes soluble COD: BNXDDNXBSNHDSDXDXBbX)ifi(r)f1(rfr1r (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 + fDXD = 0 (1a) where XS = biodegradable particulate matter (mg-COD/L) (-1)iNXB XB + iNXDfD XD + (iNXB - iNXDfD)XNS = 0 (2a) where XNS = particulate organic nitrogen Then: BNXDDNXBXNSDXSDXDXBbX)ifi(r)f1(rfr1r (3b) and (-1)XS + SS = 0 (4) For the hydrolysis rate, rH: )XX(r)XX(r1r1r)X/X(K(X/XXkrSNSSNHSNSXNSSXSBHSXBHSBHHH (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)note that kH, like , is a function of the substrate for hydrolysis enzymes, which is particulate COD, XS. Actually, the hydrolysis rate has been found to be a function of the ratio of the fraction of particulate COD normalized by the heterotrophic biomass (autotrophs do not make appreciable hydrolysis enzymes for particulate organics), as well as the biomass concentration. Also note that with hydrolysis, another compartment, XS, is added to the COD degradation process reactions. DECAY CYCLE Biomass COD, XB Biomass N, iNXB XB Debris COD, XD Debris N, iNXDXD rXD = fDbXB rXB = bXB: Particulate organic COD, XS Particulate organic N, XNS = iNXBXS rXS = (1-fD)bXB Soluble COD, SS Soluble NH4-N,