CSTR WITH SOLIDS RECYCLE

Components:

- Ideal CSTR, Volume = V

- Ideal solids separator, volume = 0

Flows

- Inflow (Q) and treated outflow (Q-QW)

- Waste biomass (QW) (wasting from CSTR mixed contents) to maintain steady-state biomass concentration

- QW << Q

Characteristics:

Hydraulic residence time: τ or HRT

τ = V / Q

Solids residence time Θ or SRT = mass solids in CSTR/solids mass wasting rate

Θ = VX / (QW X) = V / QW

Influent: Q, So,

Xo = 0

CSTR: V, S, XB,

XD

Treated effluent:

Q-QW, S,

Xout = 0

Waste biomass:

QW, XB, XD, S

Ideal separator

Note that since QW << Q, Θ >> τ

General Mass Balance Expression for one reactive component, S:

QSO – (Q-QW)S - QWS + VrS = V dS/dt

At steady state condition for S:

QSO – (Q-QW)S - QWS + VrS = 0

Rearranging:

Q(SO-S) + VrS = 0

rS = (SO - S) / τ

For heterotrophic growth and decay in a CSTR, four components and two reactions are of interest: SUBSTRATE COD (SS), OXYGEN (SO), CELL-COD (XBH), and DEBRIS-COD (XD)

Components Rates

Process | Soluble COD SS (mg/L COD) | Dissolved O2, SO (mg/L O2) | Heterotrophic biomass, XBH (mg/L COD) | Debris, XD (mg/L COD) | ρj

---|---|---|---|---|---

Aerobic Heterotrophic Growth | -1/YH | - (1-YH)/YH (1) | 1 | na | μH XBH (2)

Decay and Lysis of Heterotrophs | 1-fD (3) | -1 fD (6) | -1 | fD (6) | bH XBH (4)

(1) heterotrophic cell yield = YH = g-heterotroph cell-COD/g-COD consumed

(2) heterotrophic growth rate (ρ1) = XBH μH = μ̂H (Ss / (Ks + Ss)) XBH where μH is specific growth rate (d-1)

(3) fD = g debris-COD produced/g biomass-COD decayed. Assumes direct exertion of decayed biomass COD

(4) heterotrophic decay rate (ρ2) = XBH bH where bH = specific decay rate (d-1)

net heterotrophic biomass growth rate = rXB = (μH - bH)XBH

debris generation rate = rD = fDbHXBH

COD consumption rate = rS = -(μH / YH) XBH (growth only)

O2 consumption rate = rO = -((1-YH)/YH μH + (1-fD)bH) XBH

Apply rate expressions to steady-state mass balances on CSTR with solids retention:

Viable heterotrophic cells:

0 – 0 - QWXBH + V XBH(μH – bH) = 0

simplify and note that Θ = V/QW

1/Θ = μH - bH (1)

for CSTR with biomass retention (recycling), the growth rate is a function of Θ, not τ.

Assuming Monod kinetics with COD the limiting substrate:

μH = μ̂H Ss / (Ks + Ss) (2)

Substituting (2) → (1):

Ss = Ks(1/Θ + bH) / (μ̂H - (1/Θ + bH)) (I)

Steady-state mass balance on COD substrate, S:

QSSO – QWSS – (Q-QW)SS – (μH/YH)XBHV = 0

(μH/YH)XBH = (SSO – SS)/τ

XBH = (1/τ) YH(SSO - SS) / μH (3)

Substituting (1) into (3) for μH:

XBH = (Θ/τ) YH(SSO - SS) / (1 + bHΘ) (II)

Heterotrophic cell concentration is proportional to Θ and inversely proportional to τ

Can substitute into (II) for SS from (I)

XBH = (Θ/τ) YH( SSO - [Ks(1+bHΘ)/(μ̂HΘ - (1+bHΘ))] ) / (1 + bHΘ) (IIa)

Steady-state mass balance of debris:

-QWXD + V fDbHXBHV = 0

XD = fDbHXBHΘ

Substituting for XBH from (II):

XD = fDbHΘ (Θ/τ) YH(SSO - SS) / (1+bHΘ) (III)

Total biomass, XT = XBH + XD

XT = (1 + fDbHΘ)(Θ/τ) YH(SSO - SS) / (1+bHΘ) (IV)

Define active fraction of biomass, fA = XBH / XT

fA = 1 / (1 + fDbHΘ)

Define observed yield as the net yield, considering biomass decay (different from growth yield) = YHobs

To maintain system in steady-state with respect to biomass components, net growth and accumulated debris must be wasted:

Net growth = YHobsQ(SSO – SS)

Wasted biomass = QWXT

Steady-state condition:

QWXT = YHobsQ(SSO – SS)

YHobs = QWXT / Q(SSO – SS)

Note that QW/Q = (V/Θ)/(V/τ) = τ/Θ

And substituting for XT from (IV):

YHobs = YH(1 + fDbHΘ) / (1 + bHΘ) (V)

Since fD < 1, (1 + fDbHΘ)/(1 + bHΘ) < 1 and YHobs always < YH

Waste biomass production:

WT = QWXT (rate of total biomass wasting)

OXYGEN CONSUMPTION

ro,1 = - (1-YH)/YH rXBH for heterotrophic aerobic growth

incorporate recycling of decayed biomass assuming direct conversion of decayed biodegradable fraction of biomass to COD and exertion of oxygen demand:

Components Rates

Process | Soluble COD SS (mg/L COD) | Dissolved O2, SO (mg/L O2) | Heterotrophic biomass, XBH (mg/L COD) | Debris, XD (mg/L COD) | ρj

---|---|---|---|---|---

Aerobic Heterotrophic Growth | -1/YH | - (1-YH)/YH (2) | 1 | na | μ̂H (4)XBH

Decay and Lysis of Heterotrophs | 1-fD | -1 | -1 | fD (6) | bHXBH (7)

For decay stoichiometry that converts recycled COD directly to oxygen consumption:

-XBH – (1-fD)SO + fDXD = 0

ro,2 = (1-fD)rXBH

combining

ro = ro,1 + ro,2

ro = -((1-YH)/YH μH + (1-fD)bH) XBH

substitute μH = 1/Θ + bH and for XBH

rO = (SSO - SS)/τ [1 - (YH(1+fDbHΘ)/(1+bHΘ))] (VIa)

but

(YH(1+fDbHΘ)/(1+bHΘ)) = YHobs

so

rO = (SSO - SS)/τ [1 - YHobs] (VIb)

multiply by V to get overall system oxygen uptake rate RO = rOV = rOQτ for COD oxidation and heterotrophic biomass decay producing additional oxygen demand

RO = Q(SSO – SS)(1 – Yobs) (mg/day)

NITROGEN: Two processes for removing ammonia:

1. Heterotrophic net growth requirement for nitrogen = XNcells is also a function of Θ:

XNcells = iNXB YHobs (SSO – SS) mg-N/l

XNcells = iNXB (YH(1+fDbHΘ)/(1+bHΘ)) * (SSO – SS) mg-N/l

Ammonia nitrogen uptake rate in cell synthesis, rNH:

rNH = XNcells / τ = iNXB YHobs (SSO – SS) / τ (mgN/l/d)

2. Nitrification (ammonia substrate is rate-determining component)

μA = μ̂A SNH / (KNH + SNH)

where μA is the autotrophic growth rate, SNH is ammonia nitrogen concentration, and KNH is the half-saturation constant for nitrifying bacteria for ammonia nitrogen.

And

SNH = KNH(1/Θ + bA) / (μ̂A - (1/Θ + bA))

where bA is the decay coefficient for nitrifying bacteria (autotrophs).

And

XBA = (Θ/τ) YA( S*NHO - [KNH(1+bAΘ)/(μ̂AΘ - (1+bAΘ))] ) / (1+bAΘ)

Where S*NHO = SNHO – NR and SNHO = influent ammonia nitrogen.

The rate of oxygen consumption for nitrification, rON:

rON = (4.57/τ)(SNHO – SNH) - iNXB YHobs (SSO – SS) (VII)

Now calculate the total rate of oxygen consumption for both COD oxidation (growth plus decay) and nitrification, rOT by combining VIb and VII:

rOT = (1/τ)[(SSO – SS)(1 – YHobs) + 4.57(SNHO – SNH) - iNXB YHobs (SSO – SS)]