Lecture 15: Biological Reaction Kinetics (Cont.)

Lecture 15 Biological Reaction Kinetics (continued) We have addressed the "How?" and "How much?" of biological wastewater treatment. Need to also consider the "How fast?" (i.e. the kinetics). Biological reactions generally involve enzymes to catalyse the reactions:   E + S <=> ES -> E + P substrate enzyme products Example from V.T.H, pg. 528. sucrose + invertase -> glucose + fructose + invertase CH₂OH CH₂OH CH₂OH CH₂OH CH₂OH | | | | | OH OH OH O OH | | | || | H H H H H | | | | | OH OH OH OH OH | | | | | H H H H H | | | | | CH₂OH CH₂OH CH₂OH CH₂OH CH₂OH E + S <=> ES -> E + P Assume at start: [E] = [E]_0 [ES] = 0 [S] >> [E] d[ES]/dt = k₁[E][S] - k₋₁[ES] - k₃[ES] ≈ 0 for reaction at uniform rate Also [E] + [ES] = [E]_0 -> [E] = [E]_0 - [ES] k₁([E]_0 - [ES])[S] = (k₋₁ + k₃)[ES] k₁[E]_0[S] = (k₋₁ + k₃ + k₁[S])[ES] [ES] = k₁[E]_0[S] / (k₋₁ + k₃ + k₁[S]) [ES] = [E]_0[S] / ([S] + (k₋₁ + k₃)/k₁) [ES] = [E]_0[S] / ([S] + Km) Km = Michaelis-Menten constant or half-velocity const Rate of product formation, r r = d[P]/dt = k₃[ES] r = k₃[E]_0[S] / ([S] + Km) k₃[E]_0 is maximum rate of P formation, occurring when all E is in the form of ES d[P]/dt|max = k₃[ES]max = k₃[E]_0 = rmax r = rmax[S] / ([S] + Km) Goal of wastewater treatment is not to grow cells per se but for the microbiological culture to utilize substrate in the form of organic matter in wastewater. Substrate utilization rate is closely related to biological growth and follows Monod-type equation. Equation accounts for effect of substrate conc (S) as well as effect of biomass conc (x) Rate of substrate utilization,   rsu denotes substrate is being reduced) (<0 to \[r_{su}==\frac{kxs}{k_{s}+s}\] \(r_{5u}\) = rate of substrate utilization \([\frac{M}{\mathcal{P}T}]\) k M substrate maximum specific substrate. utilization rate Morganism T X Morganism = organism (biomass) conc L3 S= concentration of growth-limiting. M substrate substrate \[K_{s}=\] half-saturation constant [substrate \[=I_{sw}\] kx saturation KX/2 Note that specific growth rate µg [Mass new cells/Mass cells-time] is related to substrate utilization rate rsu [Mass substrate/time] by cell yield Y [Mass new cells/Mass substrate] and cell conc X [Mass cells/vol] and rsu = kxs / (Ks + S) k = µmax / Y -rsu = µg/Y * x k = max spec substrate utilization rate µmax = max spec growth rate Actual bacterial growth is generally less than µmax depending on substrate availability and endogenous respiration: rg = -Y * rsu - Ke * X rg = net biomass production (growth) rate [M biomass/L³-T] (g VSS/m³-d) Ke = endogenous decay coefficient [T⁻¹] Endogenous decay includes: Cell material used to generate energy for cell maintenance Cell death Predation by protozoa and other organisms Specific biomass growth rate (as opposed to cell growth   rate) µ = rg/x = -Y/x * rsu - Ke = -Y/x * kxs / (Ks + S) - Ke = -Y * ks / (Ks + S) - Ke