Univ., Diss.--Frankfurt/Main, 2000.

A two-step model order reduction method is investigated in order to overcome problems of certain one-step methods. Not only optimal combinations of one-step reductions are considered but also the selection of a suitable intermediate dimension (ID) is described. Several automated selection methods are presented and their application tested on a gear box model. The implementation is realized using a Matlab-based Software MORPACK. Several recommendations are given towards the selection of a suitable ID, and problems in Model Order Reduction (MOR) combinations are pointed out. A pseudo two-step is suggested to reduce the full system without any modal information. A new node selection approach is proposed to enhance the SEREP approximation of the system’s response for small reduced representations. / Mehrschrittverfahren der Modellreduktion werden untersucht, um spezielle Probleme konventioneller Einschrittverfahren zu lösen. Eine optimale Kombination von strukturmechanischen Reduktionsverfahren und die Auswahl einer geeigneten Zwischendimension wird untersucht. Dafür werden automatische Verfahren in Matlab implementiert, in die Software MORPACK integriert und anhand des Finite Elemente Modells eines Getriebegehäuses ausgewertet. Zur Auswahl der Zwischendimension werden Empfehlungen genannt und auf Probleme bei der Kombinationen bestimmter Reduktionsverfahren hingewiesen. Ein Pseudo- Zweischrittverfahren wird vorgestellt, welches eine Reduktion ohne Kenntnis der modalen Größen bei ähnlicher Genauigkeit im Vergleich zu modalen Unterraumverfahren durchführt. Für kleine Reduktionsdimensionen wird ein Knotenauswahlverfahren vorgeschlagen, um die Approximation des Frequenzganges durch die System Equivalent Reduction Expansion Process (SEREP)-Reduktion zu verbessern.

Slags plays a significant role at high temperature processes. The estimation of the slag viscosity is vital for the safe run of e.g. entrained flow gasifiers. One opportunity of determination is rotational viscometry. This technique is disadvantageous in view of elevated temperatures, applied materials and the necessary time. Additionally, the viscosity can be predicted by the help of viscosity models, where viscosity is a function of slag composition and temperature. Due to changing slag properties within the technical processes, the calculated viscosities can hugely differ from measured ones. In this work, the viscosities of 42 slags where measured up to 100 Pa s and temperatures up to 1700 °C. Oxidizing and reducing conditions were applied. Additionally, selected slag samples were quenched at defined temperatures to qualitatively and quantitatively determine the formed minerals by X-ray diffraction (XRD). Differential temperature analysis (DTA) was applied to find the onset of crystallization for the complementation of investigations. The Einstein-Roscoe equation was chosen to improve the classic viscosity models. Reducing atmosphere decreased viscosity and the number of formed minerals was increased. Slags show a shear-thinning behavior above ca. 10 vol.-% of solid mineral matter. Also, Newtonian behavior was observed up to 60 vol.-%. To overcome problems with the kinetic cooling behavior of the slags, a viscosity approximation method was applied afterwards. This can result in optimized viscosity predictions when several preconditions are fulfilled.

A two-step model order reduction method is investigated in order to overcome problems of certain one-step methods. Not only optimal combinations of one-step reductions are considered but also the selection of a suitable intermediate dimension (ID) is described. Several automated selection methods are presented and their application tested on a gear box model. The implementation is realized using a Matlab-based Software MORPACK. Several recommendations are given towards the selection of a suitable ID, and problems in Model Order Reduction (MOR) combinations are pointed out. A pseudo two-step is suggested to reduce the full system without any modal information. A new node selection approach is proposed to enhance the SEREP approximation of the system’s response for small reduced representations.:Contents Kurzfassung..........................................................................................iv Abstract.................................................................................................iv Nomenclature........................................................................................ix 1 Introduction........................................................................................1 1.1 Motivation........................................................................................1 1.2 Objectives........................................................................................1 1.3 Outline of the Thesis........................................................................2 2 Theoretical Background.......................................................................3 2.1 Finite Element Method......................................................................3 2.1.1 Modal Analysis...............................................................................4 2.1.2 Frequency Response Function.......................................................4 2.2 Model Order Reduction.....................................................................5 2.3 Physical Subspace Reduction Methods.............................................7 2.3.1 Guyan Reduction...........................................................................7 2.3.2 Improved Reduced System Method...............................................8 2.4 Modal Subspace Reduction Methods...............................................10 2.4.1 Modal Reduction...........................................................................11 2.4.2 Exact Modal Reduction..................................................................11 2.4.3 System Equivalent Reduction Expansion Process.........................13 2.5 Krylov Subspace Reduction Methods...............................................14 2.6 Hybrid Subspace Reduction Methods..............................................15 2.6.1 Component Mode Synthesis........................................................16 2.6.2 Hybrid Exact Modal Reduction......................................................19 2.7 Model Correlation Methods.............................................................21 2.7.1 Normalized Relative Frequency Difference...................................21 2.7.2 Modified Modal Assurance Criterion.............................................22 2.7.3 Pseudo-Orthogonality Check.......................................................22 2.7.4 Comparison of Frequency Response Function.............................23 3 Selection of Active Degrees of Freedom............................................25 3.1 Non-Iterative Methods...................................................................26 3.1.1 Modal Kinetic Energy and Variants..............................................26 3.1.2 Driving Point Residue and Variants..............................................27 3.1.3 Eigenvector Component Product..................................................28 3.2 Iterative Reduction Methods...........................................................29 3.2.1 Effective Independence Distribution.............................................29 3.2.2 Mass-Weighted Effective Independence.......................................32 3.2.3 Variance Based Selection Method.................................................33 3.2.4 Singular Value Decomposition Based Selection Method................34 3.2.5 Stiffness-to-Mass Ratio Selection Method.....................................34 3.3 Iterative Expansion Methods...........................................................35 3.3.1 Modal-Geometrical Selection Criterion...........................................36 3.3.2 Triaxial Effective Independence Expansion...................................36 3.4 Measure of Goodness for Selected Active Set..................................39 3.4.1 Determinant and Rank of the Fisher Information Matrix................39 3.4.2 Condition Number of the Partitioned Modal Matrix........................40 3.4.3 Measured Energy per Mode..........................................................40 3.4.4 Root Mean Square Error of Pseudo-Orthogonality Check.............41 3.4.5 Eigenvalue Comparison................................................................41 4 Two-Step Reduction in MORPACK.......................................................42 4.1 Structure of MORPACK.....................................................................42 4.2 Selection of an Intermediate Dimension.........................................43 4.2.1 Intermediate Dimension Requirements........................................44 4.2.2 Implemented Selection Methods..................................................45 4.2.3 Recommended Selection of an Intermediate Dimension...............48 4.3 Combination of Reduction Methods.................................................49 4.3.1 Overview of All Candidates..........................................................50 4.3.2 Combinations with Modal Information.........................................54 4.3.3 Combinations without Modal Information....................................54 5 Applications........................................................................................57 5.1 Gear Box Model...............................................................................57 5.2 Selection of Additional Active Nodes................................................58 5.3 Optimal Intermediate Dimension......................................................64 5.4 Two-Step Model Order Reduction Results........................................66 5.5 Comparison to One-Step Model Order Reduction Methods..............70 5.6 Comparison to One-Step Hybrid Model Order Reduction Methods...72 5.7 Proposal of a New Approach for Additional Node Selection..............73 6 Summary and Conclusions...................................................................77 7 Zusammenfassung und Ausblick..........................................................79 Bibliography............................................................................................81 List of Tables..........................................................................................86 List of Figures.........................................................................................88 A Appendix.............................................................................................89 A.1 Results of Two-Step Model Order Reduction.....................................89 A.2 Data CD............................................................................................96 / Mehrschrittverfahren der Modellreduktion werden untersucht, um spezielle Probleme konventioneller Einschrittverfahren zu lösen. Eine optimale Kombination von strukturmechanischen Reduktionsverfahren und die Auswahl einer geeigneten Zwischendimension wird untersucht. Dafür werden automatische Verfahren in Matlab implementiert, in die Software MORPACK integriert und anhand des Finite Elemente Modells eines Getriebegehäuses ausgewertet. Zur Auswahl der Zwischendimension werden Empfehlungen genannt und auf Probleme bei der Kombinationen bestimmter Reduktionsverfahren hingewiesen. Ein Pseudo- Zweischrittverfahren wird vorgestellt, welches eine Reduktion ohne Kenntnis der modalen Größen bei ähnlicher Genauigkeit im Vergleich zu modalen Unterraumverfahren durchführt. Für kleine Reduktionsdimensionen wird ein Knotenauswahlverfahren vorgeschlagen, um die Approximation des Frequenzganges durch die System Equivalent Reduction Expansion Process (SEREP)-Reduktion zu verbessern.:Contents Kurzfassung..........................................................................................iv Abstract.................................................................................................iv Nomenclature........................................................................................ix 1 Introduction........................................................................................1 1.1 Motivation........................................................................................1 1.2 Objectives........................................................................................1 1.3 Outline of the Thesis........................................................................2 2 Theoretical Background.......................................................................3 2.1 Finite Element Method......................................................................3 2.1.1 Modal Analysis...............................................................................4 2.1.2 Frequency Response Function.......................................................4 2.2 Model Order Reduction.....................................................................5 2.3 Physical Subspace Reduction Methods.............................................7 2.3.1 Guyan Reduction...........................................................................7 2.3.2 Improved Reduced System Method...............................................8 2.4 Modal Subspace Reduction Methods...............................................10 2.4.1 Modal Reduction...........................................................................11 2.4.2 Exact Modal Reduction..................................................................11 2.4.3 System Equivalent Reduction Expansion Process.........................13 2.5 Krylov Subspace Reduction Methods...............................................14 2.6 Hybrid Subspace Reduction Methods..............................................15 2.6.1 Component Mode Synthesis........................................................16 2.6.2 Hybrid Exact Modal Reduction......................................................19 2.7 Model Correlation Methods.............................................................21 2.7.1 Normalized Relative Frequency Difference...................................21 2.7.2 Modified Modal Assurance Criterion.............................................22 2.7.3 Pseudo-Orthogonality Check.......................................................22 2.7.4 Comparison of Frequency Response Function.............................23 3 Selection of Active Degrees of Freedom............................................25 3.1 Non-Iterative Methods...................................................................26 3.1.1 Modal Kinetic Energy and Variants..............................................26 3.1.2 Driving Point Residue and Variants..............................................27 3.1.3 Eigenvector Component Product..................................................28 3.2 Iterative Reduction Methods...........................................................29 3.2.1 Effective Independence Distribution.............................................29 3.2.2 Mass-Weighted Effective Independence.......................................32 3.2.3 Variance Based Selection Method.................................................33 3.2.4 Singular Value Decomposition Based Selection Method................34 3.2.5 Stiffness-to-Mass Ratio Selection Method.....................................34 3.3 Iterative Expansion Methods...........................................................35 3.3.1 Modal-Geometrical Selection Criterion...........................................36 3.3.2 Triaxial Effective Independence Expansion...................................36 3.4 Measure of Goodness for Selected Active Set..................................39 3.4.1 Determinant and Rank of the Fisher Information Matrix................39 3.4.2 Condition Number of the Partitioned Modal Matrix........................40 3.4.3 Measured Energy per Mode..........................................................40 3.4.4 Root Mean Square Error of Pseudo-Orthogonality Check.............41 3.4.5 Eigenvalue Comparison................................................................41 4 Two-Step Reduction in MORPACK.......................................................42 4.1 Structure of MORPACK.....................................................................42 4.2 Selection of an Intermediate Dimension.........................................43 4.2.1 Intermediate Dimension Requirements........................................44 4.2.2 Implemented Selection Methods..................................................45 4.2.3 Recommended Selection of an Intermediate Dimension...............48 4.3 Combination of Reduction Methods.................................................49 4.3.1 Overview of All Candidates..........................................................50 4.3.2 Combinations with Modal Information.........................................54 4.3.3 Combinations without Modal Information....................................54 5 Applications........................................................................................57 5.1 Gear Box Model...............................................................................57 5.2 Selection of Additional Active Nodes................................................58 5.3 Optimal Intermediate Dimension......................................................64 5.4 Two-Step Model Order Reduction Results........................................66 5.5 Comparison to One-Step Model Order Reduction Methods..............70 5.6 Comparison to One-Step Hybrid Model Order Reduction Methods...72 5.7 Proposal of a New Approach for Additional Node Selection..............73 6 Summary and Conclusions...................................................................77 7 Zusammenfassung und Ausblick..........................................................79 Bibliography............................................................................................81 List of Tables..........................................................................................86 List of Figures.........................................................................................88 A Appendix.............................................................................................89 A.1 Results of Two-Step Model Order Reduction.....................................89 A.2 Data CD............................................................................................96

Slags plays a significant role at high temperature processes. The estimation of the slag viscosity is vital for the safe run of e.g. entrained flow gasifiers. One opportunity of determination is rotational viscometry. This technique is disadvantageous in view of elevated temperatures, applied materials and the necessary time. Additionally, the viscosity can be predicted by the help of viscosity models, where viscosity is a function of slag composition and temperature. Due to changing slag properties within the technical processes, the calculated viscosities can hugely differ from measured ones. In this work, the viscosities of 42 slags where measured up to 100 Pa s and temperatures up to 1700 °C. Oxidizing and reducing conditions were applied. Additionally, selected slag samples were quenched at defined temperatures to qualitatively and quantitatively determine the formed minerals by X-ray diffraction (XRD). Differential temperature analysis (DTA) was applied to find the onset of crystallization for the complementation of investigations. The Einstein-Roscoe equation was chosen to improve the classic viscosity models. Reducing atmosphere decreased viscosity and the number of formed minerals was increased. Slags show a shear-thinning behavior above ca. 10 vol.-% of solid mineral matter. Also, Newtonian behavior was observed up to 60 vol.-%. To overcome problems with the kinetic cooling behavior of the slags, a viscosity approximation method was applied afterwards. This can result in optimized viscosity predictions when several preconditions are fulfilled.:List of Tables ............................................................................................................ vi List of Figures ........................................................................................................ viii Symbols and Abbreviations .................................................................................. xviii 1. Introduction and Aim ....................................................................................... 1 2. General Overview of Slag ............................................................................... 2 2.1 Viscosity ...................................................................................................... 2 2.1.1 Viscosity Introduction ........................................................................... 2 2.1.2 Flow behavior of fluids ......................................................................... 3 2.2 Slag Definition and Phase Diagrams ........................................................... 4 2.3 Solid Slag Structure .................................................................................... 5 2.4 Liquid Slag Structure ................................................................................. 10 2.5 Basicity and B/A-ratio ................................................................................ 11 2.6 Slag Components...................................................................................... 13 2.6.1 Silicon dioxide .................................................................................... 13 2.6.2 Aluminum oxide ................................................................................. 13 2.6.3 Calcium oxide .................................................................................... 15 2.6.4 Iron oxide ........................................................................................... 16 2.6.5 Magnesium Oxide .............................................................................. 18 2.6.6 Potassium Oxide ................................................................................ 19 2.6.7 Sodium Oxide .................................................................................... 20 2.6.8 Titanium Oxide ................................................................................... 21 2.6.9 Phosphorous ...................................................................................... 22 2.6.10 Sulfur .............................................................................................. 22 2.7 Summary of Last Chapters ........................................................................ 23 3. Slag Viscosity Toolbox .................................................................................. 25 3.1 Slag Viscosity Predictor............................................................................. 25 3.2 Slag Viscosity Database............................................................................ 26 3.3 Prediction Quality of Viscosity Models ....................................................... 27 4. Classic Slag Viscosity Modelling ................................................................... 30 4.1 Selected Classic Viscosity Models ............................................................ 31 4.1.1 S2 ....................................................................................................... 32 4.1.2 Watt-Fereday ..................................................................................... 32 4.1.3 Bomkamp ........................................................................................... 32 4.1.4 Shaw .................................................................................................. 32 4.1.5 Lakatos .............................................................................................. 33 4.1.6 Urbain ................................................................................................ 33 4.1.7 Riboud ............................................................................................... 33 4.1.8 Streeter .............................................................................................. 34 4.1.9 Kalmanovitch-Frank ........................................................................... 34 4.1.10 BBHLW .......................................................................................... 34 4.1.11 Duchesne ....................................................................................... 34 4.1.12 ANNliq ............................................................................................ 35 4.2 Need of Improvement in Viscosity Literature ............................................. 35 4.3 Summary of Last Chapters ........................................................................ 36 5. Advanced Slag Viscosity Modelling .............................................................. 37 5.1 Crystallization ............................................................................................ 37 5.1.1 Nucleation .......................................................................................... 38 5.1.2 Crystallization Rate ............................................................................ 39 5.1.3 Crystallization Measurement Methods ............................................... 39 5.2 Slag Properties Changes During Crystallization ........................................ 40 5.2.1 Slag Density ....................................................................................... 40 5.2.2 Solid Volume Fraction ........................................................................ 46 5.2.3 Estimation of Slag Composition During Cooling ................................. 46 5.3 Viscosity Depending on Particles and Shear Rate..................................... 47 5.3.1 Einstein-Roscoe Equation .................................................................. 48 5.3.2 Improved Modelling Approach by Modified Einstein-Roscoe .............. 49 5.4 Summary of Last Chapters ........................................................................ 50 6. Experimental Procedures ............................................................................. 52 6.1 Viscosity Measurements ........................................................................... 52 6.1.1 Estimating Parameter Ranges of Viscosity Measurements ................ 53 6.1.2 Viscosity Measurement Procedure ..................................................... 54 6.2 Thermal Analysis of Slags ......................................................................... 55 6.2.1 Experimental Conditions of DTA ........................................................ 55 6.3 Phase Determination ................................................................................. 55 6.3.1 Quench Experiment Processing ......................................................... 56 6.3.2 Phase Determination on XRD Results ............................................... 56 6.4 Summary of Last Chapters ........................................................................ 57 7. Results and Discussion ................................................................................ 58 7.1 Selected Slag Samples ............................................................................. 58 7.1.1 Slag Sample Composition Before Viscosity Measurements ............... 58 7.1.2 Slag Sample Composition After Viscosity Measurements .................. 59 7.2 General Results of Viscosity Measurements ............................................. 60 7.2.1 Viscosity under Air Atmosphere ......................................................... 63 7.2.2 Viscosity under Reducing Atmospheres ............................................. 65 7.2.3 Viscosity under Constant Partial Oxygen Pressure ............................ 66 7.2.4 Summary of Last Chapter .................................................................. 68 7.3 Mineral Formation ..................................................................................... 69 7.3.1 General Results on Primarily Mineral Formation ................................ 69 7.3.2 Influences on Primarily Mineral Formation ......................................... 70 7.3.3 Mineral Formation over Wide Temperature Ranges ........................... 71 7.3.4 Summary of Last Chapter .................................................................. 77 7.4 Results Obtained by DTA .......................................................................... 78 7.4.1 Comparing Results obtained by DTA and Quenching ........................ 80 7.4.2 Summary of Last Chapter .................................................................. 82 7.5 Shear Rate Influence on Slag Viscosity ..................................................... 82 7.5.1 Shear Rate Influence under Oxidizing Atmospheres .......................... 83 7.5.2 Shear Rate Influence under Reducing Atmospheres .......................... 87 7.5.3 Shear Rate Influence under Constant Atmospheres .......................... 91 7.5.4 Summary of chapter ........................................................................... 92 7.6 Atmospheric Influence on Viscosity ........................................................... 93 7.6.1 Summary of Last Chapter .................................................................. 95 7.7 Cooling Rate Influence on Slag Viscosity .................................................. 95 7.7.1 Summary of Last Chapter .................................................................. 97 8. Advanced Viscosity Modelling Approach ...................................................... 99 8.1 Prediction Quality of Classical Viscosity Models ........................................ 99 8.1.1 Selecting the Best Viscosity Model for Newtonian Flow ..................... 99 8.1.2 Summary of Last Chapter ................................................................ 103 8.2 Predicting Liquidus Temperature ............................................................. 103 8.2.1 Comparing Liquidus Calculations and Quenching Experiments ....... 103 8.2.2 Comparing DTA Results and Liquidus Calculations ......................... 105 8.2.3 Summary of Last Chapter ................................................................ 107 8.3 Predicting Liquid Slag Composition ......................................................... 108 8.3.1 Results of Slag Composition Calculations at Oxidizing Conditions ... 108 8.3.2 Results of Slag Composition Calculations at Reducing Conditions ... 110 8.3.3 Summary of Last Chapter ................................................................ 111 8.4 Modelling Approach ................................................................................ 112 8.4.1 Development of Datasets for Advanced Viscosity Modeling ............. 113 8.4.2 Summary of Last Chapter ................................................................ 116 8.5 Results of Advanced Slag Viscosity Modelling Approach ........................ 116 8.5.1 Summary of Last Chapter ................................................................ 121 9. Summary .................................................................................................... 123 10. Appendix: Information on Classic Viscosity Modelling ................................. 126 10.1 Backgrounds of Applied Viscosity Models............................................ 126 10.2 Viscosity Model of the BCURA (S2) ..................................................... 129 10.3 Watt-Fereday ....................................................................................... 130 10.4 Bomkamp ............................................................................................ 130 10.5 Shaw ................................................................................................... 131 10.6 Lakatos Model ..................................................................................... 132 10.7 Urbain Model ....................................................................................... 133 10.8 Riboud Model ...................................................................................... 134 10.9 Streeter Model ..................................................................................... 136 10.10 Kalmanovitch-Frank Model .................................................................. 137 10.11 BBHLW Model ..................................................................................... 137 10.12 Duchesne Model .................................................................................. 139 10.13 ANNliq Model ...................................................................................... 141 11. Appendix: Settings of Equilibrium Calculations ........................................... 143 12. Appendix: Parameters of Einstein-Roscoe Equation ................................... 153 13. Appendix: Ash and Slag Sample Preparation ............................................. 155 14. Appendix: Experimental Procedures: Viscometer ....................................... 159 14.1 General Viscometer Description .......................................................... 159 14.2 Temperature Calibration ...................................................................... 160 14.3 Viscometer Calibration ......................................................................... 160 14.4 Accuracy and Reproducibility of HT-Viscosity Measurements .............. 161 14.5 Influence of Inductive Heating .............................................................. 163 14.6 Influence of Measurement System Materials ....................................... 164 15. Appendix: Experimental Procedures: Quenching Furnace .......................... 167 16. Appendix: Slag Sample Parameters and Composition ................................ 168 17. Appendix: Slag Viscosity Measurements Results ....................................... 175 18. Appendix: Viscosities at Different Cooling Rates ........................................ 182 19. Appendix: Slag Viscosity Modelling: AALE Calculations ............................. 187 20. Appendix: Advanced Viscosity Modelling: a-factors .................................... 193 21. Appendix: Slag Mineral Phase Investigations and Modelling ...................... 197 22. Appendix: Results of DTA Measurements on Slags .................................... 207 23. Appendix: Advanced Slag Viscosity Modelling Approach ............................ 211 References ........................................................................................................... 228