On time discretizations for the simulation of the settling ... Burger.pdf · f(C) = Cv hs(C), including hindered settling velocity function v hs(C) (the Vesilind function is used)

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RESULTS AND DISCUSSION

On time discretizations for the simulation of thesettling-compression process

Raimund Burger1, Stefan Diehl2 & Camilo Mejıas1

1CI2MA & Departamento de Ingenierıa MatematicaUniversidad de Concepcion, Concepcion, Chile

2Center of Mathematical Sciences, Lund University, Sweden

9th IWA Symposium on Systems Analysis and IntegratedAssessment (Watermatex 2015)

Gold Coast, Australia, 14–17 June 2015R. Burger, S. Diehl and C. Mejıas Time discretizations for settling-compression

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GOVERNING PDE

SOLUTION BEHAVIOUR

Introduction

Scope

Benchmark simulations of entire WWTPs are today performedwith 1-d 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.

For long-time simulations of entire WWTPs, need fine resolutionin space and time, which implies long computational times. Onthe other hand, need short computational times.

Efficient numerical methods: high rate of reduction of numericalerror per computational (central processing unit; CPU) time.

R. Burger, S. Diehl and C. Mejıas Time discretizations for settling-compression

INTRODUCTION

METHODS

NUMERICAL TESTS


SCOPE

THIS CONTRIBUTION

GOVERNING PDE

SOLUTION BEHAVIOUR

Introduction

Scope






INTRODUCTION

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NUMERICAL TESTS


SCOPE

THIS CONTRIBUTION

GOVERNING PDE

SOLUTION BEHAVIOUR

Introduction

Scope






INTRODUCTION

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SCOPE

THIS CONTRIBUTION

GOVERNING PDE

SOLUTION BEHAVIOUR

Introduction

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GOVERNING PDE

SOLUTION BEHAVIOUR

This contribution

Simulation model by B. et al. (2013) is based on amethod-of-lines formulation of the underlying nonlinear PDE=⇒ system of time-dependent ODEs, one for each layer.

Simulations of PDEs are stable and reliable if the fully discretemethod satisfies a CFL (Courant-Friedrichs-Lewy) condition=⇒ maximal time step ∆t for each given layer thickness ∆z.

Only hindered settling and bulk flows:CFL cond. =⇒ may choose ∆t ∼ ∆z =⇒ fast simulations.

With compression or dispersion and standard ODE solvers:CFL cond. =⇒ ∆t ∼ ∆z2 =⇒ very small ∆t when ∆z small.

We investigate different time-integration methods (one new),with respect to efficiency and implementation complexity.


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GOVERNING PDE

SOLUTION BEHAVIOUR

Governing PDE

Model PDE for batch sedimentation in a cylindrical vessel:

∂C∂t

= −∂f (C)

∂z+

∂

∂z

(d(C)

∂C∂z

), 0 < z < 1 m, t > 0.

(PDE)

f (C) = Cvhs(C), including hindered settling velocity functionvhs(C) (the Vesilind function is used).

The compression function satisfies d(C) = Kvhs(C)σ′e(C), here:

σ′e(C) =

{0 for C < Cc,α(C − Cc) for C > Cc,

α = 0.5 m2/s2, Cc = 6 kg/m3.

Here σ′e is the derivative of the effective solid stress function σe.


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SOLUTION BEHAVIOUR

Governing PDE


∂C∂t

= −∂f (C)

∂z+

∂

∂z

(d(C)

∂C∂z

), 0 < z < 1 m, t > 0.

(PDE)



σ′e(C) =


α = 0.5 m2/s2, Cc = 6 kg/m3.



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SOLUTION BEHAVIOUR

Governing PDE


∂C∂t

= −∂f (C)

∂z+

∂

∂z

(d(C)

∂C∂z

), 0 < z < 1 m, t > 0.

(PDE)



σ′e(C) =


α = 0.5 m2/s2, Cc = 6 kg/m3.



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SOLUTION BEHAVIOUR

Solution behaviour: batch settling of initially homogeneoussuspension (Kynch test) C0 = 5 kg/m3, Cc = 6 kg/m3

he

igh

t [m

]

time [h]

10

9

8

7

6

6

5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10 0.5 1 1.5


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DISCRETIZATION

EXPLICIT EULER METHOD

SEMI-IMPLICIT (SI) METHOD

LINEARLY IMPLICIT (LI) METHOD

MethodsDiscretization in space and time

Rewrite (PDE) as

∂C∂t

=∂

∂z

(−f (C) +

∂D(C)

∂z

), D(C) :=

∫ C

0d(s) ds.

N layers of thickness ∆z = 1 mN , Cj (t): concentration of layer j .

Method-of-lines formulation (system of N ODEs):

dC1

dt=−G3/2 + J3/2

∆z, (ML1)

dCj

dt= −

Gj+1/2 −Gj−1/2

∆z+

Jj+1/2 − Jj−1/2

∆z, j = 2 :N − 1,

(ML2)dCN

dt=

1∆z(GN−1/2 − JN−1/2). (ML3)


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Rewrite (PDE) as

∂C∂t

=∂

∂z

(−f (C) +

∂D(C)

∂z

), D(C) :=

∫ C

0d(s) ds.



dC1

dt=−G3/2 + J3/2

∆z, (ML1)

dCj

dt= −

Gj+1/2 −Gj−1/2

∆z+

Jj+1/2 − Jj−1/2

∆z, j = 2 :N − 1,

(ML2)dCN

dt=

1∆z(GN−1/2 − JN−1/2). (ML3)


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DISCRETIZATION





Rewrite (PDE) as

∂C∂t

=∂

∂z

(−f (C) +

∂D(C)

∂z

), D(C) :=

∫ C

0d(s) ds.



dC1

dt=−G3/2 + J3/2

∆z, (ML1)

dCj

dt= −

Gj+1/2 −Gj−1/2

∆z+

Jj+1/2 − Jj−1/2

∆z, j = 2 :N − 1,

(ML2)dCN

dt=

1∆z(GN−1/2 − JN−1/2). (ML3)


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DISCRETIZATION




Gj+1/2: numerical convective flux (bulk flows/hindered settling)between layer j and layer j + 1, e.g. the Godunov flux:

Gj+1/2 =

{minCj≤C≤Cj+1 f (C) if Cj ≤ Cj+1,maxCj≥C≥Cj+1 f (C) if Cj > Cj+1.

Jj+1/2: compressive numerical flux:

Jj+1/2 =D(Cj+1)− D(Cj )

∆z=⇒

∂2D(C)

∂z2 ≈Jj+1/2 − Jj−1/2

∆z=

D(Cj+1)− 2D(Cj ) + D(Cj−1)

∆z2 .

Create fully discrete scheme by time discretization of(ML1)–(ML3).

No gain by high-order ODE (Diehl et al., 2015).

Wish to advance the numerical solution from t = tn totn+1 = tn + ∆t . Quantities are evaluated either at tn or at tn+1.


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Gj+1/2 =



Jj+1/2 =D(Cj+1)− D(Cj )

∆z=⇒

∂2D(C)

∂z2 ≈Jj+1/2 − Jj−1/2

∆z=

D(Cj+1)− 2D(Cj ) + D(Cj−1)

∆z2 .





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Gj+1/2 =



Jj+1/2 =D(Cj+1)− D(Cj )

∆z=⇒

∂2D(C)

∂z2 ≈Jj+1/2 − Jj−1/2

∆z=

D(Cj+1)− 2D(Cj ) + D(Cj−1)

∆z2 .





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DISCRETIZATION





Gj+1/2 =



Jj+1/2 =D(Cj+1)− D(Cj )

∆z=⇒

∂2D(C)

∂z2 ≈Jj+1/2 − Jj−1/2

∆z=

D(Cj+1)− 2D(Cj ) + D(Cj−1)

∆z2 .





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DISCRETIZATION





Gj+1/2 =



Jj+1/2 =D(Cj+1)− D(Cj )

∆z=⇒

∂2D(C)

∂z2 ≈Jj+1/2 − Jj−1/2

∆z=

D(Cj+1)− 2D(Cj ) + D(Cj−1)

∆z2 .





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Explicit Euler method

Standard method used for authors’ previous simulations.

CFL condition for fully discrete scheme:

∆t ≤ k2∆z2.

=⇒ if ∆z is replaced by ∆x/2, CPU time increases 23 = 8 times

Very easy to implement: all terms in RHS of (ML1)–(ML3) areevaluated at t = tn.

Fully discrete version of (ML2) is

Cn+1j − Cn

j

∆t= −

Gnj+1/2 −Gn

j−1/2

∆z+

Jnj+1/2 − Jn

j−1/2

∆z, j = 2 :N − 1,

with analogous formulas for boundary updates (ML1), (ML3).


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∆t ≤ k2∆z2.




Cn+1j − Cn

j

∆t= −

Gnj+1/2 −Gn

j−1/2

∆z+

Jnj+1/2 − Jn

j−1/2

∆z, j = 2 :N − 1,



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DISCRETIZATION







∆t ≤ k2∆z2.




Cn+1j − Cn

j

∆t= −

Gnj+1/2 −Gn

j−1/2

∆z+

Jnj+1/2 − Jn

j−1/2

∆z, j = 2 :N − 1,



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DISCRETIZATION







∆t ≤ k2∆z2.




Cn+1j − Cn

j

∆t= −

Gnj+1/2 −Gn

j−1/2

∆z+

Jnj+1/2 − Jn

j−1/2

∆z, j = 2 :N − 1,



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Semi-implicit (SI) method

We call the present semi-implicit, nonlinearly implicit methodsimply “semi-implicit.”

CFL condition, fully discrete scheme:

∆t ≤ k1∆z

To advance from tn to tn+1 we must solve the nonlinear system

Cn+1j − Cn

j

∆t= −

Gnj+1/2 −Gn

j−1/2

∆z+

Jn+1j+1/2 − Jn+1

j−1/2

∆z

= −Gn

j+1/2 + Gnj−1/2

∆z+

D(Cn+1j+1 )− 2D(Cn+1

j ) + D(Cn+1j−1 )

∆z2 ,

j = 2 :N − 1, plus boundary updates.

Nonlinear system is solved e.g. by Newton-Raphson method.


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DISCRETIZATION







∆t ≤ k1∆z


Cn+1j − Cn

j

∆t= −

Gnj+1/2 −Gn

j−1/2

∆z+

Jn+1j+1/2 − Jn+1

j−1/2

∆z

= −Gn

j+1/2 + Gnj−1/2

∆z+

D(Cn+1j+1 )− 2D(Cn+1

j ) + D(Cn+1j−1 )

∆z2 ,




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DISCRETIZATION







∆t ≤ k1∆z


Cn+1j − Cn

j

∆t= −

Gnj+1/2 −Gn

j−1/2

∆z+

Jn+1j+1/2 − Jn+1

j−1/2

∆z

= −Gn

j+1/2 + Gnj−1/2

∆z+

D(Cn+1j+1 )− 2D(Cn+1

j ) + D(Cn+1j−1 )

∆z2 ,




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DISCRETIZATION







∆t ≤ k1∆z


Cn+1j − Cn

j

∆t= −

Gnj+1/2 −Gn

j−1/2

∆z+

Jn+1j+1/2 − Jn+1

j−1/2

∆z

= −Gn

j+1/2 + Gnj−1/2

∆z+

D(Cn+1j+1 )− 2D(Cn+1

j ) + D(Cn+1j−1 )

∆z2 ,




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DISCRETIZATION




The linearly implicit (LI) method

Method goes back to Berger, Brezis and Rogers (1979).Practical purpose: one benefits from CFL condition ∆t ≤ k1∆zbut avoids solution of nonlinear systems.

Basic idea: now to replace D(Cnj ) by ξqn

j so that

Cn+1j − Cn

j

∆t=

Gnj−1/2 −Gn

j+1/2

∆z+ ξ

qn+1j−1 − 2qn+1

j + qn+1j+1

∆z2 ,

via a simple update formula for qnj to be executed first.

A stable implicit Euler time step means that

qn+1j − qn

j

∆t= ξ

qn+1j−1 − 2qn+1

j + qn+1j+1

∆z2 .

This is a linear system of equations, which gives qn+1j at the next

time step. Then Cnj can be updated explicitly.


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DISCRETIZATION







j so that

Cn+1j − Cn

j

∆t=

Gnj−1/2 −Gn

j+1/2

∆z+ ξ

qn+1j−1 − 2qn+1

j + qn+1j+1

∆z2 ,



qn+1j − qn

j

∆t= ξ

qn+1j−1 − 2qn+1

j + qn+1j+1

∆z2 .




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NUMERICAL TESTS


DISCRETIZATION







j so that

Cn+1j − Cn

j

∆t=

Gnj−1/2 −Gn

j+1/2

∆z+ ξ

qn+1j−1 − 2qn+1

j + qn+1j+1

∆z2 ,



qn+1j − qn

j

∆t= ξ

qn+1j−1 − 2qn+1

j + qn+1j+1

∆z2 .




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DISCRETIZATION




Complete description of LI method (for update tn → tn+1):

(1) For j = 1 :N, set

qnj =

D(Cnj )

ξ, ξ = γ max

0≤C≤Cmax

d(C), γ > 1.

(2) Solve the following linear system for qn+11 , . . . ,qn+1

N :

qn+11 − qn

1

∆t= −ξ

qn+11 − qn+1

2∆z2 ,

qn+1j − qn

j

∆t= ξ

qn+1j−1 − 2qn+1

j + qn+1j+1

∆z2 , j = 2 :N − 1,

qn+1N − qn

N

∆t= ξ∆t

qn+1N−1 − qn+1

N

∆z2 .


INTRODUCTION

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DISCRETIZATION




Complete description of LI method (for update tn → tn+1):

(1) For j = 1 :N, set

qnj =

D(Cnj )

ξ, ξ = γ max

0≤C≤Cmax

d(C), γ > 1.

(2) Solve the following linear system for qn+11 , . . . ,qn+1

N :

qn+11 − qn

1

∆t= −ξ

qn+11 − qn+1

2∆z2 ,

qn+1j − qn

j

∆t= ξ

qn+1j−1 − 2qn+1

j + qn+1j+1

∆z2 , j = 2 :N − 1,

qn+1N − qn

N

∆t= ξ∆t

qn+1N−1 − qn+1

N

∆z2 .


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DISCRETIZATION




(3) Calculate Cn+1j from

Cn+11 − Cn

1

∆t= −

Gn3/2

∆z+

qn+11 − qn

1

∆t,

Cn+1j − Cn

j

∆t=

Gnj−1/2 −Gn

j+1/2

∆z+

qn+1j − qn

j

∆t, j = 2 :N − 1,

Cn+1N − Cn

N

∆t=

GnN−1/2

∆z+

qn+1N − qn

N

∆t.

This involves in each time step the numerical solution of a linearsystem with tridiagonal coefficient matrix.

The CFL condition of the scheme is

∆t ≤ k3∆z, where k3 = k3(γ) = const. ·(

1− 1γ

)(see B. et al., in preparation).


INTRODUCTION

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DISCRETIZATION





Cn+11 − Cn

1

∆t= −

Gn3/2

∆z+

qn+11 − qn

1

∆t,

Cn+1j − Cn

j

∆t=

Gnj−1/2 −Gn

j+1/2

∆z+

qn+1j − qn

j

∆t, j = 2 :N − 1,

Cn+1N − Cn

N

∆t=

GnN−1/2

∆z+

qn+1N − qn

N

∆t.




1− 1γ



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DISCRETIZATION





Cn+11 − Cn

1

∆t= −

Gn3/2

∆z+

qn+11 − qn

1

∆t,

Cn+1j − Cn

j

∆t=

Gnj−1/2 −Gn

j+1/2

∆z+

qn+1j − qn

j

∆t, j = 2 :N − 1,

Cn+1N − Cn

N

∆t=

GnN−1/2

∆z+

qn+1N − qn

N

∆t.




1− 1γ



INTRODUCTION

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NUMERICAL TESTS


KYNCH TEST

DIEHL TEST

Numerical tests

Batch settling: conventional Kynch test (KT) and Diehl test (DT;Diehl, 2007): concentrated suspension initially above clearliquid, separated by a membrane.

Measure performance of methods in terms of numerical errorand CPU time, assess choice of γ for LI method.

Error of a simulated concentration CN with N layers is calculatedby comparing with reference soln Cref (obtained by Euler’smethod, N = 2430):

EN =

∫ 1.5 h0

∫ 1 m0 |CN(z, t)− Cref(z, t)|dz dt∫ 1.5 h0

∫ 1 m0 Cref(z, t) dz dt


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KYNCH TEST

DIEHL TEST

Numerical tests




EN =

∫ 1.5 h0




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KYNCH TEST

DIEHL TEST

Numerical tests




EN =

∫ 1.5 h0




INTRODUCTION

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KYNCH TEST

DIEHL TEST

Simulations of KTLI method, C0 = 5 kg/m3, N = 90, test various values of γ:

00.2

0.40.6

0.81

0

0.5

1

1.50

2

4

6

8

10

12

z [m]t [h]

C(z,t) [kg/m

3]

γ = 1.5

00.2

0.40.6

0.81

0

0.5

1

1.50

2

4

6

8

10

12

z [m]t [h]

C(z,t) [kg/m

3]

γ = 3

00.2

0.40.6

0.81

0

0.5

1

1.50

2

4

6

8

10

12

z [m]t [h]

C(z,t) [kg/m

3]

γ = 5


INTRODUCTION

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KYNCH TEST

DIEHL TEST

Efficiency curves for LI method (recall ∆t∆z ≤ const ·

(1− 1

γ

)).

Competing goals for choice of γ: small errors versus short CPU times.

computational time [s]10-2 10-1 100 101 102

rela

tive

L1 err

or

0

0.02

0.04

0.06

0.08

0.1

0.12N=10N=30N=90N=270N=810.=10

.=5

.=4

.=3

.=2.=1.5

.=1.1

Suitable choice: γ = 3 (among tested values)R. Burger, S. Diehl and C. Mejıas Time discretizations for settling-compression

INTRODUCTION

METHODS

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KYNCH TEST

DIEHL TEST

Simulations of DT

LI method, N = 90, γ = 3, C0 =

{10 kg/m3 for z < 0.4 m,0 for z > 0.4 m

:

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

0

2

4

6

8

10

12

t [h]z [m]

C(z

,t)

[kg

/m3]

heig

ht [m

]

time [h]

9

8

76

6

7

8

9

1

6

1.5

2.5

2

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


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EFFICIENCY ANALYSIS

REFERENCES

Results and discussionEfficiency analysisEfficiency curves: KT (left), DT (right)

computational time [s]10-2 10-1 100 101 102 103

rela

tive

L1 e

rro

r

10-3

10-2

10-1For each method: N = 10, 30, 90, 270, 810

SILIode15sEuler

computational time [s]10-2 10-1 100 101 102

rela

tive L

1 e

rror

10-2

10-1

For each method: N = 10, 30, 90, 270, 810

SILIode15sEuler


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EFFICIENCY ANALYSIS

REFERENCES

Conclusions

Explicit Euler method Semi-implicit (SI)method

Linearly implicit (LI)method

(+) Implementationeasy; numerical ap-proximate solutionsprovably converge to thePDE solution as; robustmethod.

(+) Under the assump-tion that NR iterationsfind a solution each timepoint, convergence canbe proved. Most effi-cient of the investigatedmethods.

(+) Implementationeasy. The ingredientin addition to the Eulermethod is basicallythat a linear system ofequations is solved ateach time step. Robustmethod.

(-) Inefficient. (-) Implementation morecomplex than Euler/LI.At each time step, anonlinear algebraic sys-tem is solved (e.g. byNR iterations). Conver-gence of NR not guaran-teed (but observed). Tol-erance parameter has tobe set.

(±) Convergence proofin preparation. Secondmost efficient for N ≈100 for batch sedimen-tation. The efficiencycan be adjusted to someextent by a parameter.Fastest method for agiven N ≥ 30, but leastaccurate.


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EFFICIENCY ANALYSIS

REFERENCES

ReferencesBerger, A.E., Brezis, H. & Rogers, J.C.W. 1979 A numerical method for

solving the problem ut − ∆f (u) = 0. RAIRO Anal. Numer. 13, 297–312.

Burger, R., Diehl, S., Faras, S. & Nopens, I. 2012 On reliable and unreliablenumerical methods for the simulation of secondary settling tanks inwastewater treatment. Computers & Chemical Eng. 41, 93–105.

Burger, R., Diehl, S., Faras, S., Nopens, I. & Torfs, E. 2013 A consistentmodelling methodology for secondary settling tanks: A reliablenumerical method. Water Sci. Tech. 68, 192–208.

Burger, R., Diehl, S. & Mejıas, C. (in preparation) A linearly implicit numericalscheme for strongly degenerate diffusion equations.

Diehl, S. 2007 Estimation of the batch-settling flux function for an idealsuspension from only two experiments. Chem. Eng. Sci. 62,4589–4601.

Diehl, S., Faras, S. & Mauritsson, G. 2015 Fast reliable simulations ofsecondary settling tanks in wastewater treatment with semi-implicit timediscretization. Comput. Math. Applic., in press.

Kynch, G.J. 1952 A theory of sedimentation. Trans. Farad. Soc. 48, 166–176.R. Burger, S. Diehl and C. Mejıas Time discretizations for settling-compression

INTRODUCTION

METHODS

NUMERICAL TESTS


EFFICIENCY ANALYSIS

REFERENCES









INTRODUCTION

METHODS

NUMERICAL TESTS


EFFICIENCY ANALYSIS

REFERENCES









INTRODUCTION

METHODS

NUMERICAL TESTS


EFFICIENCY ANALYSIS

REFERENCES









INTRODUCTION

METHODS

NUMERICAL TESTS


EFFICIENCY ANALYSIS

REFERENCES









INTRODUCTION

METHODS

NUMERICAL TESTS


EFFICIENCY ANALYSIS

REFERENCES









INTRODUCTION

METHODS

NUMERICAL TESTS


EFFICIENCY ANALYSIS

REFERENCES









Documents

On time discretizations for the simulation of the settling ... Burger.pdf · f(C) = Cv hs(C), including hindered settling velocity function v hs(C) (the Vesilind function is used)