simulations of reactive settling of activated sludge with a...

INTRODUCTION GOVERNING MODEL IBV PROBLEM NUMERICAL SCHEME AND EXAMPLES CONCLUSIONS AND REFERENCES

Simulations of reactive settling of activated sludge with areduced biokinetic model

Raimund Burger1, Julio Careaga1, Stefan Diehl2, Camilo Mejıas1,Ingmar Nopens3 & Peter Vanrolleghem4

1CI2MA & Departamento de Ingenierıa MatematicaUniversidad de Concepcion, Chile

2Center of Mathematical Sciences, Lund University, Sweden3BIOMATH, Ghent University, Belgium4modelEAU, Universite Laval, Canada

4th French-Chilean Workshop on Bioprocess Modeling,Santiago, Chile, September 22, 2016

R. Burger, J. Careaga, S. Diehl, C. Mejıas, I. Nopens & P. Vanrolleghem Simulations of reactive settling of activated sludge with a reduced biokinetic model


IntroductionScope

• Benchmark simulations of entire WWTPs are today performedwith 1D SST simulation models.

• “Burger-Diehl model” (BD model): hindered settling, volumetricbulk flows, compression of the biomass at high concentrationsand dispersion near the feed inlet can be included flexibly.



This contribution

• First step towards extending the advances made by Burger et al.(2011) to the numerical treatment of non-reactive settling to thereactive case

• Focus on a reduced-order problem for a sequencing batchreactor

• Constitutive assumptions that determine its mathematicalnature:

(i) hindered settling of the flocculated particles,(ii) compression of the flocculated particles at high

concentrations when a network is formed,(iii) reaction terms containing nonlinear growth rate kinetics and

a constant decay rate of biomass.

• Treatment leads to a working numerical scheme



Governing model

Assumptions

• 1D batch sedimentation of suspended particles in closedcylinder, height L, depth z measured from suspension surface

• particulate microorganisms (biomass) may be active or inert(inert matter) with concentrations pX and (1− p)X respectively

• biomass flocculated into large particles: X

• constitutive assumption: each particle settles with a velocity

v = v (X,Xz) , Xz := ∂X/∂z.

• active biomass consumes substrate, soluble in water;concentration S, concentration NO3 and concentration N2

• spatial movement of substrate is captured by a single diffusioncoefficient dS.



System equations:

∂X

∂t= − ∂

∂z(v(X,Xz)X) + (µ(SNO3

, SS)− (1− fP)b)pX, (XE)

∂(pX)

∂t= − ∂

∂z

(v(X,Xz)pX

)+(µ(SNO3

, SS)− b)pX, (pXE)

∂SNO3

∂t= dS

∂2SNO3

∂z2− 1− Y

2.86Yµ(SNO3

, SS)pX,

∂SS

∂t= dS

∂2SS

∂z2−(µ(SNO3

, SS)

Y− (1− fP)b

)pX,

∂SN2

∂t= dS

∂2SN2

∂z2+

1− Y2.86Y

µ(SNO3, SS)pX,

• flux function:

v(X,Xz)X = fb(X)− ∂D(X)

∂z

• Y : dim’less yield factor, b [s−1]: constant decay rate,fP: proportion parameter



Initial-boundary conditions:

X(z, 0) = X0, p(z, 0) = p0,

SNO3(z, 0) = S0

NO3, SS(z, 0) = S0

S , SN2(z, 0) = 0,

v(X,Xz)X|z=0 = v(X,Xz)X|z=B = 0,

(SNO3)z(0, t) = (SNO3

)z(B, t) = 0,

(SS)z(0, t) = (SS)z(B, t) = 0,

(SN2)z(0, t) = (SN2

)z(B, t) = 0.

• each particle consists of percentage p0 of active biomass (Xa),remaining is inert matter (Xi)



Constitutive functions

• growth rate function:

µ(SNO3, SS) := µmax

SNO3

KNO3+ SNO3

SS

KS + SS

µmax: maximum specific growth rate, KS, KNO3: saturation

constants

• particle velocity takes into account both hindered settling andcompression (Burger, Diehl & Nopens 2011):

• fb(X) := Xvhs(X) : batch settling flux function,

D(X) :=

0 for X < Xc,∫ X

Xc

ρsvhs(s)σ′e(s)

g∆ρds for X > Xc.

vhs(X): hindered settling velocity, σe: effective solid stress, ρs:density of solids, ∆ρ: solid-fluid density difference, Xc: criticalconcentration above which particles form a network



Numerical scheme and examplesNumerical scheme

• can use numerical method for non-reactive model (Burger et al.2012, 2013)

• numerical updates use the idea given by Diehl (1997) andJeppsson & Diehl (1996)

• divide (0, L) into N cells, ∆z := L/N . Let tn, n = 0, 1, . . . denotethe discrete time points; ∆t: time step that should satisfy

∆t ≤ 1

max{k1, k2},

with

k1 :=

max0≤X≤Xmax

∣∣f ′b(X)∣∣

∆z+

2 max0≤X≤Xmax

dcomp(X)

∆z2+ max

{µmax − (1 − fP)b, (1 − fP)b

},

k2 :=2dS

∆z2+µmaxXmax

Ymax

{(1 − Y )

2.86KNO3

,1

KS

}.



• Xnj , Pnj : approx concentration in cell j at time tn

• convective numerical flux: Godunov flux

Gnj+1/2 :=

min

Xn

j≤X≤Xn

j+1

fb(X), if Xnj ≤ Xn

j+1,

maxXn

j≥X≥Xn

j+1

fb(X), if Xnj > Xn

j+1.

• total flux in (XE) between cells j and j + 1 is approximated by

Fnj+1/2 := Gnj+1/2 −D(Xn

j+1)−D(Xnj )

∆z.

• the corresponding flux of (pXE) is Pnj+1/2Fnj+1/2,

Pnj+1/2 =

{Pnj+1, if Fnj+1/2 ≤ 0,Pnj , if Fnj+1/2 > 0.

• numerical updates of SNO3, SS and SN2

are then straightforwardfor the corresponding equations.



• boundary conditions: Fn1/2 = Fn

N+1/2 = Pn1/2 = Pn

N+1/2 = 0

• Explicit Euler temporal approximation, j = 1, . . . , N :

Xn+1j = X

nj + ∆t

(−Fn

j+1/2 − Fnj−1/2

∆z+(µ(S

nNO3,j

, SnS,j) − (1 − fP)b

)P

nj X

nj

)If Xn+1

j > 0 :

Pn+1j =

1

Xn+1j

[P

nj X

nj + ∆t

(−Pn

j+1/2Fnj+1/2 − Pn

j−1/2Fnj−1/2

∆z+(µ(S

nNO3,j

, SnS,j) − b

)P

nj X

nj

)]If Xn+1

j = 0 : Pn+1j = P

nj

Sn+1NO3,j

= SnNO3,j

+ ∆t

(dSSNO3,j+1 − 2SNO3,j + SNO3,j−1

∆z2−

1 − Y

2.86Yµ(SNO3,j , SS,j)P

nj X

nj

),

Sn+1S,j = S

nS,j + ∆t

(dSSS,j+1 − 2SS,j + SS,j−1

∆z2−(µ(SNO3,j , SS,j)

Y− (1 − fP)b

)P

nj X

nj

)S

n+1N2,j

= SnN2,j

+ ∆t

(dSSN2,j+1 − 2SN2,j + SN2,j−1

∆z2+

1 − Y

2.86Yµ(SNO3,j , SS,j)P

nj X

nj

)



Examples 1–2: Kynch test• For all examples:

vhs(X) =v0

1 + (X/X)q, v0 = 1.76 × 10−3 m s−1, X = 3.87 kg m−3,

q = 3.58, Xc = 5 kg m−3,

σe(X) =

{0 for X ≤ Xc,α(X −Xc) for X > Xc,

α = 0.1m s−2,

L = B = 1 m, N = 100, p0 = 5/7 ≈ 0.7143

and initial values

S0S = 9.00 × 10−4 kg m−3 and S0

NO3= 6.00 × 10−3 kg m−3,

• Examples 1–2: X0 = 3.5 kg m−3

• all other parameters take standard values in this type of process.



Example 1: Kynch test dS = 1.00× 10−6 m2 s−1.

Total solids Active biomass Inert matter

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

X(z,t)[kg/m

3]

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

Xa(z,t)[kg/m

3]

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

Xi(z,t)[kg/m

3]

NO3 Substrate S Substrate N2 Substrate

0

0.5

1

0

1

2

0

2

4

6

x 10−3

z [m]t [h]

SNO

3(z,t)[kg/m

3]

0

0.5

1

0

1

20

0.1

0.2

0.3

0.4

z [m]t [h]

SS(z,t)[kg/m

3]

0

0.5

1

0

1

20

2

4

6

8

x 10−3

z [m]t [h]

SN

2(z,t)[kg/m

3]

1

*SNO3plot has been rotated 180 grade with respect the zt-plane.*



Example 2: Kynch test dS = 9.00× 10−6 m2 s−1.


0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

X(z,t)[kg/m

3]

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

Xa(z,t)[kg/m

3]

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

Xi(z,t)[kg/m

3]


0

0.5

1

0

1

2

0

2

4

6

x 10−3

z [m]t [h]

SNO

3(z,t)[kg/m

3]

0

0.5

1

0

1

20

0.1

0.2

0.3

0.4

z [m]t [h]

SS(z,t)[kg/m

3]

0

0.5

1

0

1

20

2

4

6

8

x 10−3

z [m]t [h]

SN

2(z,t)[kg/m

3]

1




Examples 3–4: Diehl test

• Initially suspension is located above clear liquid.

• For Example 3

X(z, 0) =

{7 kg/m3 for 0 < z < 0.5m,0 for 0.5m < z < 1m,

and for Example 4

X(z, 0) =

{14 kg/m3 for 0 < z < 0.25m,0 for 0.25m < z < 1m,

• All parameters as in Example 1.



Example 3: Diehl test X(z, 0) = 7 kg/m3 over 0.5 m.


0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

X(z,t)[kg/m

3]

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

Xa(z,t)[kg/m

3]

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

Xi(z,t)[kg/m

3]


0

0.5

1

0

1

2

0

2

4

6

x 10−3

z [m]t [h]

SNO

3(z,t)[kg/m

3]

0

0.5

1

0

1

20

0.1

0.2

0.3

0.4

z [m]t [h]

SS(z,t)[kg/m

3]

0

0.5

1

0

1

20

2

4

6

8

x 10−3

z [m]t [h]

SN

2(z,t)[kg/m

3]

1




Example 4: Diehl test X(z, 0) = 14 kg/m3 over 0.25 m.


0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

X(z,t)[kg/m

3]

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

Xa(z,t)[kg/m

3]

0

0.5

1

0

1

20

5

10

15

20

z [m]t [h]

Xi(z,t)[kg/m

3]


0

0.5

1

0

1

2

0

2

4

6

x 10−3

z [m]t [h]

SNO

3(z,t)[kg/m

3]

0

0.5

1

0

1

20

0.1

0.2

0.3

0.4

z [m]t [h]

SS(z,t)[kg/m

3]

0

0.5

1

0

1

20

2

4

6

8

x 10−3

z [m]t [h]

SN

2(z,t)[kg/m

3]

1




normalized inventory

Measure of the time-dependent normalized total mass of nitrate, theso-called normalized inventory:

INO3(t) :=

1

S0NO3

B

∫ B

0

SNO3(z, t) dz ≈ 1

S0NO3

N

N∑j=1

SnNO3,j=: I∆

NO3(t).

0 0.5 1 1.5 20.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I∆ NO

3(t)

t [h]

Example 1 (KT)

Example 2 (KT)

Example 3 (DT)

Example 4 (DT)



Example 5: Overcompressed test (Numerical test)

• A highly compressed body of sludge near the bottom of thecolumn.

• For Example 5

X(z, 0) =

{0 for 0 m < z ≤ 0.5 m,20 kg m−3 for 0.5 m < z ≤ 1 m

• All parameters as in Example 1.



Example 5: Overcompressed test X(z, 0) = 20 kg/m3 below 0.5 m.


0

0.5

1

0

1

20

10

20

30

z [m]t [h]

X(z,t)[kg/m

3]

0

0.5

1

0

1

20

10

20

30

z [m]t [h]

Xa(z,t)[kg/m

3]

0

0.5

1

0

1

20

10

20

30

z [m]t [h]

Xi(z,t)[kg/m

3]


0

0.5

1

0

1

2

0

2

4

6

x 10−3

z [m]t [h]

SNO

3(z,t)[kg/m

3]

0

0.5

1

0

1

20

0.2

0.4

0.6

0.8

z [m]t [h]

SS(z,t)[kg/m

3]

0

0.5

1

0

1

20

2

4

6

8

x 10−3

z [m]t [h]

SN

2(z,t)[kg/m

3]

1




accuracy

Relative error with a reference solution for N = 3200, fixed time point:

erelN :=‖Xa −Xref

a ‖L1

‖X0a‖L1

+‖Xi −Xref

i ‖L1

‖X0i ‖L1

+‖SNO3

− SrefNO3‖L1

‖S0NO3‖L1

+‖SN2

− SrefN2‖L1

‖SendN2‖L1

+‖SS − Sref

S ‖L1

‖S0S‖L1

Example 1 (Kynch Test) Example 3 (Diehl Test)t = 4 min t = 30 min t = 6 min t = 30 min

N erelN θ erelN θ N erelN θ erelN θ20 0.097 — 0.740 — 20 0.204 — 0.224 —50 0.033 1.176 0.302 0.978 50 0.110 0.674 0.069 1.285

100 0.018 0.853 0.148 1.026 100 0.066 0.745 0.029 1.231200 0.008 1.194 0.072 1.045 200 0.038 0.800 0.016 0.861400 0.004 1.141 0.034 1.092 400 0.021 0.885 0.010 0.737800 0.002 0.878 0.014 1.224 800 0.010 1.032 0.005 0.936

Table : Examples 1 and 3: approximate relative errors erelN ,

convergence rates θ (calculated between neighboring values of N ).



Conclusions and references

• model correctly reproduces expected settling, compression andreaction behaviour

• numerical method is robust with respect to scenarios,parameters, and choices of (monotone) numerical flux

• open issue: well-posedness and convergence analysis,extension to continuously operated equipment, improvement ofefficiency (e.g., semi-implicit time stepping)



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R. Burger, S. Diehl, S. Farras, and I. Nopens. On reliable and unreliable numerical methods for thesimulation of secondary settling tanks in wastewater treatment. Computers & Chemical Eng., 41:93–105,2012.

R. Burger, S. Diehl, S. Farras, I. Nopens, and E. Torfs. A consistent modelling methodology for secondarysettling tanks: a reliable numerical method. Water Sci. Tech., 68:192–208, 2013.

R. Burger, S. Diehl, and I. Nopens. A consistent modelling methodology for secondary settling tanks inwastewater treatment. Water Res., 45(6):2247–2260, 2011.

R. Burger, K.H. Karlsen, and J.D. Towers. A model of continuous sedimentation of flocculated suspensions inclarifier-thickener units. SIAM J. Appl. Math., 65:882–940, 2005.

Diehl, S. (1997). Continuous sedimentation of multi-component particles. Mathematical Methods in theApplied Sciences, 20, 1345–1364.

Jeppsson, U., & Diehl, S. (1996). On the modelling of the dynamic propagation of biological components inthe secondary clarifier. Water Science and Technology, 34 (5–6), 85–92.

A. A. Kazmi and H. Furumai. Field investigations on reactive settling in an intermittent aeration sequencingbatch reactor activated sludge process. Water Sci. Tech., 41(1):127–135, 2000.

A. A. Kazmi and N. Furumai. A simple settling model for batch activated sludge process. Water Sci. Tech., 42(3-4):9–16, 2000.

J. Keller and Z. Yuan. Combined hydraulic and biological modelling and full-scale validation of SBR process.Water Sci. Tech., 45(6):219–228, 2002.


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