Centrifugal settling of flocculated suspensions: A sensitivity analysis of parametric model functions

(1)

Centrifugal settling of flocculated suspensions:

A sensitivity analysis

of parametric model functions

Rodrigo Garcés Muñoz

Advisor : Dr. Stefan Berres

Co-Advisor : Dr. Raimund Bürger

Dr. Walter Gómez

a

_{Universidad Católica de Temuco,}

b

_{Universidad de Concepción,}

c

_{Universidad de la Frontera}

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Centrifugal settling of flocculated suspensions: A sensitivity analysis of parametric model functions

Outline

1 Introduction

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Centrifugal settling of flocculated suspensions: A sensitivity analysis of parametric model functions Introduction

(4)

Governed equation

We consider the following parabolic-hyperbolic equation:

Centrifugation-Model

∂

t

φ +

1 r

∂

r

−w

2 _r

g

f (φ)

=

∂

r

(∂

r

A(φ)),

(r , t) ∈ Q

T

,

(1)

φ(r , 0)

=

φ

0 (r ),

r ∈ [R

0 ,

R],

(2)

(f (φ) + ∂

r

A(φ))(r

b

,

t)

=

0,

t ∈ [0, T ],

r

b

∈ {R

0 ,

R},(3)

φ(r , t) solid volumetric concentration.

φ

0 (r ) initial concentration.

(5)

Model

In this case f and A

Describe the rheology of the suspended material.

Are based on model specifications and constitutive

assumptions (finite numbers of parameters).

De [2]

A(φ) :=

Z

φ

0 a(s) ds,

a(φ) :=

−f (φ)σ

0 e

(φ)

∆%gφ

∆%

solid-fluid density difference.

g gravity.

σ

e

effective solid stress function.

f density flux function.

(6)

Model

Richardson & Zaki [1]

f (φ) =

(

ν

∞

φ (1 − φ)

C

for

0 < φ ≤ φ

max

,

0 for

φ ≤

0 and φ ≥ φ

max

,

Tiller & Leu [2]

σ

e

(φ) =

(

0 for

φ ≤ φ

_c

,

σ

0 (

_φ

φ

c

)

k

_{− 1}

_for

_{φ > φ}

c

,

C > 1 R&Z exponent.

ν

∞

<

0 settling velocity of a single particle in an unbounded fluid.

k > 0,

σ

0 ≥ 0,

φ

c

critical concentration (gel point).

It holds a ≡ 0 in [0, φ

c

]

(Hyperbolic pde).

Strongly degenerate parabolic equation.

(7)

Solution structure

0.06 0.1 0.2 0.3 _0.4 0.2 0 0.6 0.8 0 0.1 0.2 0.3 0.4 t [s] r [m] φ [⋅]

(8)

Profiles

0.1 0.060 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 r [m] φ [⋅]

(9)

Resolution strategy

Direct Problem

Due to diffusion term and flux function nonlinearity,

solutions of the direct are in general discontinuos.

Must be characterized as weak solutions of the problem.

To ensure uniqueness, such solutions must be defined as

entropy solutions in the Kružkov [5] sense.

(10)

Parameter identification

Cost function

J(φ) =

1

2 Z

L

0 |φ(r , t) − ˆ

φ(r , t)|

2 dr

ˆ

φ(r , t) observation profile.

φ(r , t) solution for a fixed t.

Optimization Problem

min

p

J(p)

s.a

u

direct problem solution

(4)

Inverse Problem

(11)

Resolution strategy

Inverse Problem

Discretize cost function and the restriction.

Nonlinear optimization algorithms.

The solution of the discretize version depends of the

resolution scheme chosen.

(12)

Resolution strategy

Inverse Problem

To solve the Inverse Problem, we discretize (i.e. cost

function and the restriction).

Nonlinear optimization algorithms.

The solution of the discretize version depends of the

resolution scheme chosen.

(13)

Resolution strategy

Direct Problem

Implicit method (faster, stable).

Engquist-Osher for the numerical flux.

Inverse Problem

Quasi-Newton method.

BFGS for the Hessian actualization.

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Centrifugal settling of flocculated suspensions: A sensitivity analysis of parametric model functions Preliminary Results

One-parameter cost function: Intermediate data

C

σ

0 k

φ

C

(15)

One-parameter cost function: Intermediate data

C

σ

0 k

φ

C

1

2

3

4

5

6

7

8

9

10

0

0.002

0.004

0.006

0.008

0.01 C

J(C)

(16)

One-parameter cost function: Intermediate data

C

σ

0 k

φ

C

2

3

4

5

6

7

8

0

0.5

1

1.5

2

2.5

3

3.5 x 10

−3

σ

0

J(σ

0

)

(17)

One-parameter cost function: Intermediate data

C

σ

0 k

φ

C

2

4

6

8

10

12

14

16

18

20

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07 k

J(k)

(18)

One-parameter cost function: Intermediate data

C

σ

0 k

φ

C

0.06

0

0.08

0.1

0.12

0.14

0.01

0.02

0.03

0.04 φ

c

J(φ

_c

)

(19)

One-parameter cost function: Stationary data

C

σ

0 k

φ

C

(20)

One-parameter cost function: Stationary data

C

σ

0 k

φ

C

2

4

6

8

10

0

0.5

1

1.5

2

2.5 x 10

−3

C

J(C)

(21)

One-parameter cost function: Stationary data

C

σ

0 k

φ

C

2

3

4

5

6

7

8

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035 σ

0

J(σ

0

)

(22)

One-parameter cost function: Stationary data

C

σ

0 k

φ

C

2

4

6

8

10

12

14

16

18

20

0

0.05

0.1

0.15

0.2

0.25

0.3 k

J(k)

(23)

One-parameter cost function: Stationary data

C

σ

0 k

φ

C

0.06

0

0.08

0.1

0.12

0.14

0.05

0.1

0.15

0.2

0.25 φ

c

J(φ

_c

)

(24)

Two-parameter cost function: Intermediate data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

(25)

Two-parameter cost function: Intermediate data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

4

5

6

7

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0

0.005

0.01

0.015

0.02

0.025

0.03 σ

0

φ

c

J(σ

₀

,φ

_c

)

(26)

Two-parameter cost function: Intermediate data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

8

9

10

0.08 _0.1

0.12 _0.14

0

0.01

0.02

0.03

0.04 k

k

φ

_c

J(k,φ

c

)

(27)

Two-parameter cost function: Intermediate data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

2

4

6

8

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0

0.005

0.01

0.015

0.02

0.025

0.03 φ

c k

C

J(C,φ

c

)

(28)

Two-parameter cost function: Intermediate data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

4

5

6

7

8

8.5

9

9.5

10

0

2

4

6

8 x 10

−3

k

σ

0

J(σ

₀

,k)

(29)

Two-parameter cost function: Intermediate data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

2

4

6

8

8 _8.5

9 _9.5

10

0

0.005

0.01

0.015 C

k

J(C,k)

(30)

Two-parameter cost function: Intermediate data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

2

3

4

5

6

7

8

4

6

7

0

2

4

6

8 x 10

−3

C

σ

0

J(C,σ

0

)

(31)

Two-parameter cost function: Stationary data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

(32)

Two-parameter cost function: Stationary data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

4

5

6

7

0.08

0.09

0.1

0.11

0

0.02

0.04

0.06

0.08

0.1 σ

0

φ

c

J(φ

c

,σ

0

)

(33)

Two-parameter cost function: Stationary data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

8

8.5

9

9.5

10

0.08

0.09

0.1

0.11

0

0.05

0.1

0.15 k

φ

c

J(φ

_c

,k)

(34)

Two-parameter cost function: Stationary data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

2

4

6

8

0.08

0.085

0.09

0.095

0.1

0.105

0.11

0

0.02

0.04

0.06

0.08 C

φ

c

J(C,φ

c

)

(35)

Two-parameter cost function: Stationary data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

4

5

6

7

8

8.5

9

9.5

10

0

0.02

0.04

0.06

0.08 k

σ

0

J(σ

0

,k)

(36)

Two-parameter cost function: Stationary data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

2

4

6

8

8.5

9

9.5

10

0

0.01

0.02

0.03

0.04

0.05

0.06 k

C

J(C,k)

(37)

Two-parameter cost function: Stationary data

(σ

0 , φ

C

)

(k , φ

C

)

(C, φ

C

)

(σ

0 ,

k )

(C, k )

(C, σ

0 )

2

4

6

8

3.5

4

4.5

5

5.5

6

6.5

7

7.5

0

2

4

6

8 x 10

−3

σ

0

C

J(C,σ

0

)

(38)

Bibliografía

A. Coronel and F. James and M.Sepúlveda (2003):

Numerical identification of parameters for a model of

sedimentation processes. Inverse Problems, vol.

19, pág.

951-972.

S. Berres and R. Bürger and A. Coronel and M. Sepúlveda

(2005): Numerical identification of parameters for a strongly

degenerate convection-diffusion problem modelling

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Math., vol.

52, pág. 311-337.

J. Bonnans and J. C. Gilbert and C. Lemaréchal and C.

Sagastizábal (2003):

Numerical optimization. Theoretical and practical,

Universitext. Berlin: Springer.

(39)

Bibliografía

J.F. Richardson and W.N. Zaki (1954): The sedimentation

of uniform spheres under conditions of viscuos flow Chem.

Eng. Sci., vol.

3, pág. 65-73.

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Centrifugal settling of flocculated suspensions: A sensitivity analysis of parametric model functions