where qA is the maximum specific rate of acetic acid uptake, hr ', SA is the acetic acid concentration in COD units, KA is a half-saturation coefficient for acetic acid, XP|. is the poly-P concentration in the biomass, expressed as a liquid phase P concentration, Kpp is a half-saturation coefficient for poly-P in the same units, and Xu.p is the concentration of PAO biomass in COD units. The stoichiometry of the reaction is such that each mg/L of acetic acid COD removed from the medium forms a mg/L of PHB COD in the biomass. Thus, the rate expression for PHB (Xm„) formation, rXpni), >s:

Furthermore, Yp units of soluble phosphate, Sp, are released for each unit of acetic acid stored as COD, increasing SP by an amount equal to the decrease in stored PolyP concentration. Thus:

The value of YP selected by Henze et al.?6 was 0.40 mg P/mg COD, reflecting the average stoichiometry for the process. The values of KSA and K,M. were chosen to be small to make the parenthetical terms in Eq. 3.82 serve as switching functions that change rapidly from one to zero, thereby turning the reaction on and off. The value chosen for KSA was 4.0 mg COD/L whereas the value chosen for KpF was 0.01 mg P/mg PAO COD.

Under aerobic conditions, the PAOs grow by using the stored PHB as a carbon and energy source. This is assumed to be their only substrate for growth, even though they are capable of growth on soluble substrates. Because little soluble substrate is likely to be present in the aerobic portion of a biological phosphorus removal process, Henze et al.'Sh ignored it to simplify the model. Furthermore, because the process can only occur under aerobic conditions, a switching function for oxygen was included to make the rate go to zero when oxygen is absent. Considering all of these factors, the rate of PAO growth can be described by:

where |lp is the maximum specific growth rate coefficient for PAOs, X,.MH is the stored PHB concentration in mg/L as COD, SP is the soluble phosphate concentration in mg/L as P, K,. is the half-saturation coefficient for soluble phosphate, S(, is the dissolved oxygen concentration, and K,, is the half-saturation coefficient for dissolved oxygen. It should be noted that the expression for the effect of PHB concentration on biomass growth is written in terms of the amount of PHB available per unit of biomass COD because the PHB is not free in the medium, but is stored in the biomass. As a result, K,,,IH has units of mg PHB COD/mg PAO COD. Because of biomass lysis, phosphate will continually be released to the medium. Consequently, S(. will never reach a zero concentration and phosphorus will always be available for growth. The values chosen for the half-saturation coefficients by Henze et al.s" were 0.01 mg PHB COD/mg PAO COD, 0.20 mg P/L, and 0.20 mg 0:/L. for K,m„ K,., and K<„ respectively.

The stoichiometry of the aerobic growth reaction on a COD basis is the same as that in Eq. 3.33, except that PHB is the growth substrate. Consequently, the relationship between rxlil., rxl.H,„ and rs<) (with all in COD units) will be the same as the relationship between rMt, rss, and rS() in Eq. 3.34, or:

where Y|.AO is the yield coefficient for PAOs growing on stored PHB. The value assumed for it in ASM No. 2 is 0.63 mg PAO COD/mg PHB COD. " It should be noted that the rates of PHB loss and oxygen consumption expressed by Eq. 3.86 are that associated only with PAO growth.

Storage of polyphosphate also occurs under aerobic conditions and the energy for it also comes from PHB utilization. The rate expression includes all of the parenthetical terms in Eq. 3.85. It has been observed, however, that storage of poly-P stops if its content in the PAOs becomes too high/" It is necessary to include a term that decreases the rate of Poly-P storage as the Poly-P concentration per unit of PAOs approaches a maximum value of K,.MAX. Considering these factors, the rate of PolyP storage, rx,.,., can be expressed as:

where qHp is the maximum specific rate of Poly-P storage, which has a typical value of 0.06 mg P/(mg PAO COD h) at 20°C. K„,p is the inhibition coefficient for PolyP storage, with an assumed value of 0.02 mg P/mg PAO COD. All other terms were defined following Eq. 3.85. Soluble phosphate is removed from the medium in direct proportion to the amount incorporated into Poly-P. Furthermore, PHB is lost and oxygen is utilized proportionally as well. The relationship between the rates is determined from the stoichiometrv as:

where Ypm, is the PHB requirement for poly-P storage, which has a typical value of 0.20 mg PHB COD/mg P/" The rates of PHB loss and oxygen consumption in this expression are those associated with only Poly-P storage. The total rates of each under aerobic conditions must be obtained by adding the expressions from Eqs. 3.86 and 3.88. Furthermore, Eq. 3.88 does not give the total rate of soluble phosphate loss since polyphosphate formation is not the only mechanism for removing soluble phosphate from the liquid. Rather, phosphorus is also a required nutrient for biomass synthesis. If i,> x„ is the mass of phosphorus incorporated into cell material per unit of PAO COD formed, the total rate of removal of soluble phosphorus by the PAOs will be:

where rxm, is given by Eq. 3.85. Cellular biomass contains about 2.5 percent phosphorus on a mass basis, so on a biomass COD basis, i,, XH has a value around 0.02 mg P/mg biomass COD. If heterotrophs and autotrophs are growing in the system, they will also consume soluble phosphate for incorporation into biomass with the same stoichiometry.

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