Paper Pulp Dewatering

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1 Paper Pulp Dewatering Dr. Stefan Rief Flow and Transport in Industrial Porous Media November 12-16, 2007 Utrecht University

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 Page 2

3 Introduction and Motivation Examples of Porous Media Paper Diapers Clothes Air and oil filters Soil and sand Dewatering felts Foams Sintered metal Characterizing Property of Porous Media At least two distinct length scales Macro scale (=level of observation) Second length scale introduced by microstructure -> micro scale 10m 1mm Paper Machine Page 3 Tomography of a Dewatering Felt

4 Introduction and Motivation 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 Microscopic details of the flow field, temperature distribution or deformation are not of interest Upscaling Find simpler macroscopic description of phenomena (upscaling) Porous Medium Homogenous Medium Page 4

5 Introduction and Motivation Macroscopic description of flow in porous media (experiments) Sand Column Experiment (1856) Q: flow rate D: diameter of sand column l: length of sand column h 1 -h 2 : measure of pressure drop Darcy s Law Henry Darcy ( ) A = πd 2 /4: cross-sectional area K: permeability Page 5

6 Introduction and Motivation Darcy s Law in differential form, where v: effective velocity p: effective pressure µ: dynamic viscosity : permeability tensor Applicability and extensions of Darcy s 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): Page 6

7 Introduction and Motivation Theoretical Derivations of Macroscopic Descriptions Tube models of porous media based on Hagen-Poiseuille flow Overlapping continua descriptions Volume averaging of the Navier-Stokesequations using an REV Two-scale homogenization of the Navier-Stokes equations Important Questions How good is the macroscopic description? Relation between the microproblem and its macroscopic counterpart? Convergence? Page 7

8 Derivation and Properties of NS2P Periodic Porous Medium Periodicity cell Y=]0,1[ 2 : Porous Fluid Part Ω ε Y f fluid part Y s 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 Page 8

9 Derivation and Properties of NS2P Stationary, incompressible Navier-Stokes equations: Stationary, incompressible Navier-Stokes equations (dimensionless) Transformation of variables by characteristic quantities. Reynolds number, Froude number Page 9

10 Derivation and Properties of NS2P Scaling of the Reynolds number and Froude number: and. Scaled Navier-Stokes equations with boundary conditions: S ε : Union of obstacle boundaries (inner boundary) Γ 1 : periodic part of the outer boundary of the medium Γ 2 : no-slip part of the outer boundary of the medium Page 10

11 Derivation and Properties of NS2P Formal two-scale Analysis Series expansion:,. Equations of 0 th order: Navier-Stokes system with two pressures Micro Problem Macro Problem Page 11

12 Derivation and Properties of NS2P 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 -ε and prolongation operator ~ auf S ε. Page 12

13 Derivation and Properties of NS2P Relation of NS2P to Darcy s Law Using, γ<1 yields a Stokes system with two pressures (S2P) Consider cell problems Due to linearity of S2P, we have, hence (Darcy s Law), where and. Page 13

14 Splitting-Ansatz Numerical Solution of NS2P (Micro Problem) (Permeability function), (Macro Problem) Page 14

15 Numerical Solution of NS2P Splitting approach Properties of the permeability function around zero (Marušić-Paloka, Mikelić): - Monotony: - Ellipticity: - Taylor expandable: - Taylor coefficients determined by Stokes problems Micro problem I: Computation of Taylor coefficients Micro problem II: Discrete computation of Page 15

16 Numerical Solution of NS2P 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 L 2 -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 Page 16

17 Numerical Solution of NS2P Navier-Stokes solver on the periodicity cell (micro problem II) Formulation as least-squares problem (Glowinski), where Minimization of J using a CG method until ξ=0 3 Stokes problems in each iteration step Finite element approximation properties remains unchanged Page 17

18 Numerical Solution of NS2P Quasilinear, elliptic solver on Ω (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 Linear systems remain the same for each problem Application of the direct solver SuperLU 3.0 Inversion of the matrix is done only once -> fast (, but memory demanding) Page 18

19 Numerical Solution of NS2P Solution of the full Navier-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 Linear combinations of cell problem solutions yields a Poisson problem (2D) for p 0 Validation of the solution by comparison with solutions of the splitting approach Page 19

20 Numerical Results Discrete computation of the permeability function in case of a square obstacle Page 20

21 Numerical Results Recirculation zones create anisotropy in the permeability function: Streamfunctions for α=(1,0) T und α=(1,1) T Page 21

22 Numerical Results Darcy s Law vs. permeability function: Circular obstacle, µ= Taylor coefficients vs. fitted discrete Data Squared obstacle, µ=0.02 Page 22

23 Numerical Results Macro problem with quadratic micro and macro obstacle: f=(0.25,0) T Change of flow pattern Effective flow rate decreases by 30% in the nonlinear case Page 23

24 Remarks Numerical Results 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 Algorithm to solve the full 4d problem provides quite similar results (see thesis) Page 24

25 Modeling and Simulation of the Pressing Section of a Paper Machine Heimbach GmbH & Co., Düren, Germany Main Products: Paper Machine Clothings Filters Idea: Better Understanding of Dewatering Virtual Design of Felts Need: Computer Tool to Simulate Press Nips of Paper Machines Page 25

26 Modeling and Simulation of the Pressing Section of a Paper Machine Fiber Suspension Forming Section Pressing Section Drying Section Paper Page 26

27 Modeling and Simulation of the Pressing Section of a Paper Machine Paper Machine Press Nip Dimensions 10m width 3mm height 100mm Press zone 1.0m roll diameter v s up to 2000m/min Press force kN Page 27

28 Modeling and Simulation of the Pressing Section of a Paper Machine Roll Press Multi-Layered Felt Roll Press Paper What is the Importance of Press Nips? Mechanical Dewatering is 10 Times Cheaper than Thermal Drying! Page 28

29 Modeling and Simulation of the Pressing Section of a Paper Machine Deformation model Paper: visco-elastic-plastic Felt: visco-elastic Flow model Two-phase Darcy s Law Richards assumption: air infinitely mobile Pressure-saturation relation Nonlinear filtration laws in the fully saturated zone: Page 29

30 Modeling and Simulation of the Pressing Section of a Paper Machine 1. Deformation Input: Press force Iteration to achieve force balance Runge-Kutta method of 4 th order Solution of the model equations 2. Flow Linearization + relaxation FE discretization SuperLU 3.0 to solve linear systems Porosity, solid velocity v s, grid Page 30

31 Modeling and Simulation of the Pressing Section of a Paper Machine 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 Nonlinear filtration laws from 3dsimulations of a virtual felt using ITWM s ParPac and GeoDict Virtual Felt Page 31

32 Modeling and Simulation of the Pressing Section of a Paper Machine Paper 3 felt layers Roll surface Page 32

33 Modeling and Simulation of the Pressing Section of a Paper Machine Darcy s Law Nonlinear Filtration Laws Page 33

34 Modeling and Simulation of the Pressing Section of a Paper Machine Dry solids content of paper 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 Page 34

35 Modeling and Simulation of the Pressing Section of a Paper Machine Real World Experiment Measurements at STFI Sweden Paper Machine Speed 1200 m/min Batt Press Force 800 kn/m Base Weave Two Configurations: Batt A) Felt Belt B) Felt Turned Over Page 35

36 Modeling and Simulation of the Pressing Section of a Paper Machine Water Content Page 36 Felt Turned Over

37 Modeling and Simulation of the Pressing Section of a Paper Machine Saturation Page 37 Felt Turned Over

38 Modeling and Simulation of the Pressing Section of a Paper Machine Dryness Profile of the Paper Layer 43% 41% Page 38 Felt Turned Over

39 Summary 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 2. Consideration of nonlinear filtration laws 3. Proposal of a numerical solution algorithm 4. Presentation of numerical results and discussion Page 39

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