# Applied Dimensional Analysis and Modeling

**By**

- Thomas Szirtes, Ph.D, P.E., Thomas Szirtes and Associates, Inc. Toronto, Ontario Canada

Applied Dimensional Analysis and Modeling provides the full mathematical background and step-by-step procedures for employing dimensional analyses, along with a wide range of applications to problems in engineering and applied science, such as fluid dynamics, heat flow, electromagnetics, astronomy and economics. This new edition offers additional worked-out examples in mechanics, physics, geometry, hydrodynamics, and biometry.

View full description### Audience

Upper Undergraduate and First-year Graduate students in Mechanical, Civil, and Aerospace Engineering, students in Materials Engineering, Solid Mechanics, Engineering Mechanics, Professional Engineers in Mechanical, Civil and Aerospace Engineering, Manufacturers of primary structural materials, particularly structural metals like steel and aluminum, Manufacturing Engineers in the Aerospace, Aeronautical and Automotive industries

### Book information

- Published: November 2006
- Imprint: BUTTERWORTH HEINEMANN
- ISBN: 978-0-12-370620-1

### Table of Contents

Chapter 1: Mathematical Preliminaries by PĂˇl RĂłzsa 1.1 Matrices and Determinants 1.2 Operations with Matrices1.3 The Rank of a Matrix 1.4 Systems of Linear Equations 1.4.1 Homogeneous Case 1.4.2 Nonhomogeneous Case 1.5 List of Selected Publications Dealing with Linear Algebra and Matrices Chapter 2: Formats and Classification2.1 Formats for Physical Relations 2.1.1 Numeric Format 2.1.2 Symbolic Format 2.1.3 Mixed Format 2.2 Classification of Physical Quantities 2.2.1 Variability 2.2.2 Dimensionality Chapter 3: Dimensional Systems 3.1 General Statements 3.1.1 Monodimensional System 3.1.2 Omnidimensional System 3.1.3 Multidimensional System 3.2 Classification 3.3 The SI 3.3.1 Preliminary Remarks 3.3.2 Structure (a) Fundamental Dimensions (b) Derived Dimensionless Units (c) Derived Dimensional Units withSpecific Names (d) Derived Dimensional Units withoutSpecific Names (e) Non-SI Units Permanently Permitted tobe Used with SI(f) Non-SI Units Temporarily Permitted tobe Used with SI (g) Prohibited Units 3.3.3 Prefixes 3.3.4 Rules of Etiquette in Writing Dimensions3.3.4.1 Problems 3.4 Other Than SI Dimensional Systems 3.4.1 Metric, Mass-based Systems (a) CGS System (b) SI (for reference only) 3.4.2 Metric, Force-based System 3.4.3 American/British Force (Engineering) System 3.4.4 American/British Mass (Scientific) System 3.5 A Note on the Classification of Dimensional Systems Chapter 4: Transformation of Dimensions 4.1 Numerical Equivalences 4.2 Technique 4.3 Examples 4.4 Problems Chapter 5: Arithmetic of Dimensions Chapter 6: Dimensional Homogeneity 6.1 Equations 6.2 Graphs 6.3 Problems Chapter 7: Structure of Physical Relations 7.1 Monomial Power Form 7.2 The Dimensional Matrix 7.3 Generating Products of Variables of DesiredDimension 7.4 Number of Independent Sets of Products of GivenDimension (I) 7.5 Completeness of the Set of Products of Variables 7.6 Special Case: Matrix A is Singular 7.7 Number of Independent Sets of Products of GivenDimension (II); Buckinghamâ€™s Theorem 7.8 Selectable and Nonselectable Dimensions in aProduct of Variables 7.9 Minimum Number of Independent Products ofVariables of Given Dimension 7.10 Constancy of the Sole Dimensionless Product 7.11 Number of Dimensions Equals or Exceeds theNumber of Variables 7.11.1 Number of Dimensions Equals the Numberof Variables 7.11.2 Number of Dimensions Exceeds the Numberof Variables 7.12 Problems Chapter 8: Systematic Determination of Complete Setof Products of Variables 8.1 Dimensional Set; Derivation of Products of Variablesof a Given Dimension 8.2 Checking the Results 8.3 The Fundamental Formula Chapter 9: Transformations 9.1 Theorems Related to Some Specific Transformations 9.2 Transformation between Systems of DifferentD Matrices 9.3 Transformation between Dimensional Sets 9.4 Independence of Dimensionless Products of theDimensional System Used Chapter 10: Number of Sets of Dimensionless Productsof Variables 10.1 Distinct and Equivalent Sets 10.2 Changes in a Dimensional Set Not Affecting theDimensionless Variables 10.3 Prohibited Changes in a Dimensional Set 10.3.1 Duplications 10.4 Number of Distinct Sets 10.5 Exceptions 10.5.1 Dimensionally Irrelevant Variable 10.5.2 In Matrix C, One Row is a Multiple ofAnother Row 10.6 Problems Chapter 11: Relevancy of Variables 11.1 Dimensional Irrelevancy 11.1.1 Condition 11.1.2 Adding a Dimensionally Irrelevant Variableto a Set of Relevant Variables 11.1.3 The Cascading Effect 11.2 Physical Irrelevancy 11.2.1 Condition 11.2.2 Techniques to Identify a PhysicallyIrrelevant Variable Common Sense Existence of Dimensional Irrelevancy Heuristic Reasoning Tests Combined with Deft Interpretationof Results 11.3 Problems Chapter 12: Economy of Graphical Presentation 12.1 Number of Curves and Charts 12.2 Problems Chapter 13: Forms of Dimensionless Relations 13.1 General Classification 13.2 Monomial is Mandatory 13.3 Monomial is Impossibleâ€”Proven 13.4 Monomial is Impossibleâ€”Not Proven 13.5 Reconstructions 13.5.1 Determination of Exponents of Monomials The Measurement Method The Analytic Method The Heuristic Reasoning Method 13.5.2 Determination of Some Nonmonomials13.6 Problems Chapter 14: Sequence of Variables in theDimensional Set 14.1 Dimensionless Physical Variable is Present 14.2 Physical Variables of Identical Dimensions are Present 14.3 Independent and Dependent Variables 14.4 Problems Chapter 15: Alternate Dimensions Chapter 16: Methods of Reducing the Number ofDimensionless Variables 16.1 Reduction of the Number of Physical Variables 16.2 Fusion of Dimensionless Variables16.3 Increasing the Number of Dimensions 16.3.1 Dimension Splitting 16.3.2 Importation of New Dimensions 16.3.3 Using Both Mass and Force Dimensions 16.4 Problems Chapter 17: Dimensional Modeling 17.1 Introductory Remarks 17.2 Homology 17.3 Specific Similarities 17.3.1 Geometric Similarity 17.3.2 Kinematic Similarity 17.3.3 Dynamic Similarity 17.3.4 Thermal (or Thermic) Similarity 17.4 Dimensional Similarity 17.4.1 Scale Factors 17.4.2 Model Law 17.4.3 Categories and Relations CategoriesRelations 17.4.4 Modeling Data Table 17.5 Scale Effects 17.6 Problems Chapter 18: Forty-three Additional ApplicationsReferencesNumerical OrderAlphabetical Order (by Authorsâ€™ Surname)Appendices