• No results found

Paper Pulp Dewatering

N/A
N/A
Protected

Academic year: 2021

Share "Paper Pulp Dewatering"

Copied!
39
0
0

Loading.... (view fulltext now)

Full text

(1)

Dr. Stefan Rief

stefan.rief@itwm.fraunhofer.de

Flow and Transport in Industrial Porous Media Flow and Transport in Industrial Porous MediaFlow and Transport in Industrial Porous Media Flow and Transport in Industrial Porous Media

(2)

Overview

• Introduction and Motivation

• Derivation and Properties of the Navier-Stokes System with Two Pressures (NS2P)

• Numerical Solution of NS2P • Numerical Results

• Modeling and Simulation of the Pressing Section of a Paper Machine • Summary

(3)

Examples of Porous Media Examples of Porous MediaExamples of Porous Media Examples of Porous Media • Paper

• Diapers • Clothes

• Air and oil filters

• Soil and sand • Dewatering felts • Foams

• Sintered metal Characterizing Property of Porous Media Characterizing Property of Porous MediaCharacterizing Property of Porous Media Characterizing Property of Porous Media • At least two distinct length scales

• Macro scale (=level of observation) • Second length scale introduced by

microstructure -> micro scale

Paper Machine 10m

(4)

Introduction and Motivation

Problem of Direct Numerical Simulation of Porous Media Problem of Direct Numerical Simulation of Porous Media Problem of Direct Numerical Simulation of Porous Media Problem of Direct Numerical Simulation of Porous Media • Unknown exact microstructure

• Required high resolution of the discretization (!!!!) ->

unrealistic demands on memory and computational time Solution

SolutionSolution Solution

• Microscopic details of the flow field,

temperature distribution or deformation are not of interest

• Find simpler macroscopic description

of phenomena (upscaling) Homogenous Medium

Upscaling

(5)

Macroscopic description of flow in porous media (experiments) Macroscopic description of flow in porous media (experiments) Macroscopic description of flow in porous media (experiments) Macroscopic description of flow in porous media (experiments)

Henry Darcy (1803-1858)

Sand Column Experiment (1856)

Q: flow rate

D: diameter of sand column l: length of sand column

h1-h2: measure of pressure drop

(6)

Introduction and Motivation

Darcy DarcyDarcy

Darcy‘‘‘‘s Law in differential forms Law in differential forms Law in differential forms Law in differential form

Applicability and extensions of Darcy Applicability and extensions of DarcyApplicability and extensions of Darcy

Applicability and extensions of Darcy‘‘‘‘s Laws Laws Laws Law • Darcy‘s Law is valid for slow flows

• Dupuit (1863): Pressure drops increase faster as predicted by Darcy‘s Law • Extension of Darcy‘s Law by Forchheimer (1901):

v: effective velocity p: effective pressure µ: dynamic viscosity

: permeability tensor , where

(7)

Theoretical Derivations of Macroscopic Descriptions Theoretical Derivations of Macroscopic Descriptions Theoretical Derivations of Macroscopic Descriptions Theoretical Derivations of Macroscopic Descriptions

• Tube models of porous media based on Hagen-Poiseuille flow • Overlapping continua descriptions

• Volume averaging of the Navier-Stokes-equations using an REV

• Two-scale homogenization of the

Important Questions Important QuestionsImportant Questions Important Questions • How good is the macroscopic

description?

• Relation between the micro-problem and its macroscopic counterpart?

(8)

Page 8

Derivation and Properties of NS2P

Periodic Porous Medium Periodic Porous MediumPeriodic Porous Medium Periodic Porous Medium • Periodicity cell YYY=]0,1[Y 2:

Y Y Y Yffff–––– fluid part Y Y Y

Yssss–––– solid part (obstacle)

• Construction of the periodic porous medium by translation and scaling of the periodicity cell

• 0 < ε << 1: characteristic length of the microstructure

(9)

Stationary, incompressible Navier Stationary, incompressible Navier Stationary, incompressible Navier

Stationary, incompressible Navier----Stokes equations:Stokes equations:Stokes equations:Stokes equations:

Stationary, incompressible Navier Stationary, incompressible Navier Stationary, incompressible Navier

Stationary, incompressible Navier----Stokes equations (dimensionless)Stokes equations (dimensionless)Stokes equations (dimensionless)Stokes equations (dimensionless) • Transformation of variables by characteristic quantities

. • Reynolds number , Froude number

(10)

Derivation and Properties of NS2P

Scaling of the Reynolds number and Froude number: Scaling of the Reynolds number and Froude number:Scaling of the Reynolds number and Froude number: Scaling of the Reynolds number and Froude number:

and . Scaled Navier

Scaled NavierScaled Navier

Scaled Navier----Stokes equations with boundary conditions:Stokes equations with boundary conditions:Stokes equations with boundary conditions:Stokes equations with boundary conditions:

• Sε: Union of obstacle boundaries (inner boundary)

• Γ1: periodic part of the outer boundary of the medium • Γ : no-slip part of the outer boundary of the medium

(11)

Formal two Formal two Formal two

Formal two----scale Analysisscale Analysisscale Analysisscale Analysis • Series expansion:

, . Equations of 0

Equations of 0 Equations of 0

Equations of 0thththth order: Navierorder: Navierorder: Navierorder: Navier----Stokes system with two pressuresStokes system with two pressuresStokes system with two pressuresStokes system with two pressures

(12)

Remarks Remarks Remarks Remarks

• Sanchez-Palencia, Lions formally derived NS2P in 1980 • Marušić-Paloka, Mikelić prove in 2000:

- Existence and uniqueness of the solution

- Convergence of and , i.e.

and ,

where

and prolongation operator ~ auf Sε.

Derivation and Properties of NS2P

(13)

Relation of NS2P to Darcy Relation of NS2P to Darcy Relation of NS2P to Darcy

Relation of NS2P to Darcy‘‘‘‘s Laws Laws Laws Law

• Using , γ<1 yields a Stokes system with two pressures (S2P) • Consider cell problems

(14)

Numerical Solution of NS2P

Splitting Splitting Splitting

Splitting----AnsatzAnsatzAnsatzAnsatz

(Micro Problem)

(Permeability function) ,

(15)

Splitting approach Splitting approach Splitting approach Splitting approach

• Properties of the permeability function around zero (Marušić-Paloka, Mikelić): - Monotony:

- Ellipticity:

- Taylor expandable:

(16)

Page 16

Numerical Solution of NS2P

Stokes solver on the periodicity cell (micro problem I) Stokes solver on the periodicity cell (micro problem I) Stokes solver on the periodicity cell (micro problem I) Stokes solver on the periodicity cell (micro problem I) • mixed finite element discretization on squared grids

• bi-quadratic (bilinear) approximation of the velocity (pressure) -> quadratic (linear) convergence in L2-norm

• Application of an Augmented Lagrangian Uzawa-CG Method (Fortin, Glowinski) to the discrete system:

- Reformulation as a constrained optimization problem

- Reformulation as saddle point problem using an augmented Lagrangian functional

- Application and interpretation of the Uzawa algorithm as Gradient method - Use of a CG method

(17)

Navier Navier Navier

Navier----Stokes solver on the periodicity cell (micro problem II)Stokes solver on the periodicity cell (micro problem II)Stokes solver on the periodicity cell (micro problem II)Stokes solver on the periodicity cell (micro problem II) • Formulation as least-squares problem (Glowinski)

, where

(18)

Numerical Solution of NS2P

Quasilinear Quasilinear Quasilinear

Quasilinear, elliptic solver on , elliptic solver on , elliptic solver on Ω, elliptic solver on ΩΩΩ (macro problem)(macro problem)(macro problem)(macro problem)

• Least-squares CG-method for linearization • Sequence of Poisson problems

• Biquadratic finite element discretization on squared grids • Approximation of the pressure is of second order

Remarks on the solvers Remarks on the solvers Remarks on the solvers Remarks on the solvers

• Linear systems remain the same for each problem • Application of the direct solver SuperLU 3.0

(19)

Solution of the full Navier Solution of the full Navier Solution of the full Navier

Solution of the full Navier----Stokes system with two pressures (4d)Stokes system with two pressures (4d)Stokes system with two pressures (4d)Stokes system with two pressures (4d)

• Application of a variational formulation similar to a variational formulation of the Navier-Stokes equations

• Use of a least-squares method

• Solve linear S2P problems (4d): Micro problems have y-dependent RHS (!!!) • Extension of cell problem idea: use FE basis functions as right hand sides

(20)

Numerical Results

Discrete computation of the permeability function in case of a s Discrete computation of the permeability function in case of a s Discrete computation of the permeability function in case of a s

(21)

Recirculation zones create anisotropy in the permeability functi Recirculation zones create anisotropy in the permeability functi Recirculation zones create anisotropy in the permeability functi

Recirculation zones create anisotropy in the permeability function: on: on: on: Streamfunctions

Streamfunctions Streamfunctions

(22)

Numerical Results

Darcy DarcyDarcy

Darcy‘‘‘‘s Law vs. permeability function:s Law vs. permeability function:s Law vs. permeability function:s Law vs. permeability function: Circular obstacle, µ=0.0002

Taylor coefficients vs. fitted Taylor coefficients vs. fitted Taylor coefficients vs. fitted Taylor coefficients vs. fitted discrete Data

discrete Data discrete Data discrete Data

(23)

Macro problem with quadratic micro and macro obstacle: f=(0.25,0 Macro problem with quadratic micro and macro obstacle: f=(0.25,0 Macro problem with quadratic micro and macro obstacle: f=(0.25,0 Macro problem with quadratic micro and macro obstacle: f=(0.25,0))))TTTT

• Change of flow pattern

• Effective flow rate decreases by 30% in the nonlinear case

(24)

Numerical Results

Remarks Remarks Remarks Remarks

• Recirculation zones block the flow and produce higher pressure drops

• NS2P provides quite complex extensions of Darcy‘s Law -> Extensions by just one term might be questionable!

• Taylor coefficients differ significantly from fitted coefficients due to small convergence radius of the Taylor series

• Effective macro flow rate decreases and flow patterns change

(25)

Heimbach GmbH & Co., D

Heimbach GmbH & Co., D

Heimbach GmbH & Co., D

Heimbach GmbH & Co., Dü

ü

ü

üren, Germany

ren, Germany

ren, Germany

ren, Germany

Main Products: Main Products: Main Products: Main Products:

• Paper Machine Clothings • Filters

Idea: Idea: Idea: Idea:

• Better Understanding of Dewatering • Virtual Design of Felts

(26)

Modeling and Simulation of the Pressing Section of a Paper

Machine

Fiber Suspension Fiber SuspensionFiber Suspension Fiber Suspension Paper Paper Paper Paper Forming Section Forming SectionForming Section Forming Section Pressing Section Pressing Section Pressing Section Pressing Section Drying Section Drying Section Drying Section Drying Section

(27)

Paper Machine Paper MachinePaper Machine

Paper Machine DimensionsDimensionsDimensionsDimensions • 10m width • 3mm height • 100mm Press zone • 1.0m roll diameter • vs up to 2000m/min • Press force 200-Press Nip Press Nip Press Nip Press Nip

(28)

Modeling and Simulation of the Pressing Section of a Paper

Machine

What is the

Importance of Press

Nips?

Mechanical

Dewatering is

10

10

10

10

Times Cheaper

Times Cheaper

Times Cheaper

Times Cheaper

than

Thermal Drying!

Roll Press

Roll Press

Paper

Multi-Layered

Felt

(29)

Deformation model Deformation model Deformation model Deformation model • Paper: visco-elastic-plastic • Felt: visco-elastic Flow model Flow modelFlow model Flow model • Two-phase Darcy‘s Law

• Richards assumption: air infinitely mobile

• Pressure-saturation relation • Nonlinear filtration laws in the

(30)

Modeling and Simulation of the Pressing Section of a Paper

Machine

1. Deformation 1. Deformation 1. Deformation 1. Deformation • Input: Press force

• Iteration to achieve force balance • Runge-Kutta method of 4th order

Solution of the model equations Solution of the model equations Solution of the model equations Solution of the model equations

2. Flow 2. Flow 2. Flow 2. Flow • Linearization + relaxation • FE discretization

• SuperLU 3.0 to solve linear systems

(31)

Model parameters Model parameters Model parameters Model parameters

• Machine parameters supplied by producer • Deformation parameters from measurements • Pressure-saturation relation from measurements • Use of heuristics to provide data for individual

layers

(32)

3d-Modeling and Simulation of the Pressing Section of a Paper

Machine

Paper

3 felt layers

(33)

Nonlinear NonlinearNonlinear

Nonlinear Filtration Filtration Filtration LawsFiltration LawsLawsLaws Darcy

DarcyDarcy

(34)

Modeling and Simulation of the Pressing Section of a Paper

Machine

Remarks Remarks Remarks Remarks

• Reasonable simulation results

• Nonlinear filtration laws increase fluid pressure significantly

• Complete coupling of flow and

deformation seems to be a necessary model extension at very high machine speeds

(35)

Real World Experiment Real World Experiment Real World Experiment Real World Experiment

• Measurements at STFI Sweden • Machine Speed 1200 m/min • Press Force 800 kN/m • Two Configurations: Paper Batt Base Weave Batt

(36)

Page 36

Modeling and Simulation of the Pressing Section of a Paper

Machine

Water Content Water Content Water Content Water Content

Felt Turned Over Felt Turned Over Felt Turned Over Felt Turned Over

(37)

Saturation Saturation Saturation Saturation

(38)

Page 38

Modeling and Simulation of the Pressing Section of a Paper

Machine

Dryness Profile of the Paper Layer Dryness Profile of the Paper Layer Dryness Profile of the Paper Layer Dryness Profile of the Paper Layer

43%

41%

Felt Turned Over Felt Turned Over Felt Turned Over Felt Turned Over

(39)

• Introduction to Porous Media and Upscaling Methods • Derivation and Properties of NS2P

• Proposition of two numerical solution approaches for NS2P: 1. Splitting approach into micro and macro problems

2. Solution of the full system in four dimensions • Presentation of numerical results and consequences

• Modeling and Simulation of the pressing section of a paper machine: 1. Two dimensional model

References

Related documents

• The design should follow the overall European Athletics design using the typography, colour, wireframe and/or pattern. • The event logo must be placed on all invitations

Dinamika konflik keagamaan acapkali melibatkan pemeluk agama dalam jumlah besar cendrung (Konflik Komunal) mempunyai dampak sosial politik lebih luas dan

The aim of this study was to investigate the effect of wave conditions and surfer ability on the physiological, ride and performance characteristics of surfing measured through

4 Entry (X) Visa Applicants arrived on Entry (X) visa are requested to furnish the following documents in 2 sets during the time of registration:.. a) Original valid Passport

TEMCO FIREPLACE PRODUCTS DIRECT VENT FIREPLACES INSTALLATION AND STARTUP CHECKLIST.

Human Resources &amp; Affirmative Action Southwest Tennessee Community College 737 Union Ave Memphis, TN 38103 Phone: (901) 333-5828 pthomas@southwest.tn.edu President-Elect

Based on the background, the author aimed to determine the relationship between nurse knowledge and nurse attitudes with the implementation of the Patient Safety

A discrete, non-local and non-reflecting boundary condition is specified at an artificial external boundary by the DNL method, yielding an equivalent problem that is solved in a