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
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
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
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
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
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
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?
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
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
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
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
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
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
Numerical Solution of NS2P
Splitting Splitting Splitting
Splitting----AnsatzAnsatzAnsatzAnsatz
(Micro Problem)
(Permeability function) ,
Splitting approach Splitting approach Splitting approach Splitting approach
• Properties of the permeability function around zero (Marušić-Paloka, Mikelić): - Monotony:
- Ellipticity:
- Taylor expandable:
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
3d-Modeling and Simulation of the Pressing Section of a Paper
Machine
Paper
3 felt layers
Nonlinear NonlinearNonlinear
Nonlinear Filtration Filtration Filtration LawsFiltration LawsLawsLaws Darcy
DarcyDarcy
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
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
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
Saturation Saturation Saturation Saturation
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
• 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