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Index
Half-title page
Title page
Copyright page
Dedication
Contents
List of Figures
Preface
1 Philosophy and Approach
1.1 Introduction
1.2 Elements of environmental modeling
1.3 Education vs. training
1.4 Principles ensuring education
1.4.1 Education and expansion of thought processes
1.4.2 Challenge, persistence, critical analysis, and growth
1.4.3 Learning through reading, discussion, problems, and critical assessment
1.4.4 Increase fundamental and general understanding
1.5 Conclusion
1.6 Problems
2 Thoughts on Use of Data
2.1 Introduction
2.2 On numbers in elementary education
2.3 What’s the answer?
2.4 Given the process, what’s the answer?
2.5 Given the answer, what’s the process?
2.6 Given an answer, is it useful?
2.6.1 Given the process, is the proposed answer useful?
2.6.2 Given the data, is the answer useful?
2.7 Conclusion
2.8 Problems
3 Models as a Framework for Study of Data
3.1 Introduction
3.2 Models
3.2.1 Physical models
3.2.2 Conceptual models
3.2.3 Mathematical models
3.3 Definitions
3.4 Elements of a deterministic model
3.4.1 Conceptual model of the mathematical model
3.5 Modeling principles
3.6 Approaches to modeling
3.7 Summary of the modeling procedure
3.8 Quantifying the utility of a model
3.8.1 How uncertainty gets introduced
3.8.2 Reducing uncertainty
3.8.3 Verification and validation development
3.8.4 Verification
3.8.5 Validation
3.8.6 Uncertainty quantification
3.8.7 Calibration
3.8.8 Final thoughts on validation
3.9 Conclusion
3.10 Problems
4 Length and Time Scales
4.1 Introduction
4.2 Scales of observation
4.3 Continuum scale, averaging, and the rev concept
4.4 Reflections on measurement of properties of a continuum
4.5 Reflections on a general governing equation
4.6 Length scales
4.7 Time scales
4.8 Problems
5 Mechanisms of Change
5.1 Introduction
5.2 Elements of body sources
5.2.1 Sources of total mass
5.2.2 Sources of chemical species
5.2.3 Sources of momentum
5.2.4 Sources of energy
5.3 Elements of surface sources and transport
5.3.1 Velocity
5.3.2 Diffusion and dispersion processes
5.3.3 Mass, momentum, and energy diffusion
5.3.4 Mass, momentum, and energy dispersion
5.3.5 Convection, advection, and flux
5.3.6 Waves
5.4 Conclusion
5.5 Problems
6 Dimensional Analysis
6.1 Introduction
6.2 Parameter selection
6.2.1 Problem identification
6.2.2 Modeling objectives
6.3 Dimensional analysis in general
6.4 Dimensional analysis of solid–liquid problem
6.4.1 Well-mixed solid–liquid problem
6.4.2 Solid–liquid problem without mixing
6.5 Dimensional analysis of sedimentation
6.6 Dimensional analysis of pipe flow
6.7 Dimensional analysis of porous media flow
6.8 Some important dimensionless numbers
6.8.1 Reynolds number
6.8.2 Prandtl number
6.8.3 Schmidt number
6.8.4 Peclet number
6.8.5 Biot number
6.8.6 Froude number
6.8.7 Rossby number
6.8.8 Bond and capillary numbers
6.9 Conclusion
6.10 Problems
7 Mathematical Instruments of Change
7.1 Introduction
7.2 First order discrete time models
7.2.1 Malthusian model
7.2.2 Logistic model
7.2.3 Gompertz model
7.2.4 Harvest models
7.3 Stability for discrete models
7.3.1 Stability theorem
7.3.2 Stability for Malthusian models
7.3.3 Stability for logistic models
7.4 Other contexts
7.4.1 Finance
7.4.2 Roots of equations
7.5 Conclusion
7.6 Problems
8 Derivatives and Scales
8.1 Continuous change in time and space
8.2 Preliminary concepts for continuous modeling
8.3 Scales in model formulation
8.4 Time derivatives
8.4.1 General time derivative
8.4.2 Eulerian approach
8.4.3 Lagrangian approach
8.5 Comments on time derivatives
8.6 Dot product or inner product
8.7 Cross product
8.8 Gradient
8.9 Comment on coordinate systems
8.10 Normal direction to a surface
8.11 Normal velocity of a surface
8.11.1 Case I: fixed CV with w = 0 at every point on the boundary
8.11.2 Case II: the CV is moving with w ≠ 0 at some boundary locations
8.12 Divergence
8.12.1 Divergence in cylindrical coordinates
8.13 Conclusion
8.14 Problems
9 Integral Theorems and Volume Kinematics
9.1 Introduction
9.2 Divergence theorem
9.3 Gradient form of the divergence theorem
9.4 Digression
9.5 General transport theorem
9.5.1 Kinematics of a general control volume
9.6 Reynolds transport theorem
9.6.1 Kinematics of a material volume
9.7 Transport theorem for a fixed volume
9.7.1 Kinematics of a fixed volume
9.8 Conclusion
9.9 Problems
10 Mass Conservation
10.1 Introduction
10.2 Mathematical tools
10.3 Integral forms of conservation of mass
10.3.1 Conservation of mass for a general volume
10.3.2 Conservation of mass for a material volume
10.3.3 Conservation of mass for a fixed volume
10.4 Point form of conservation of mass
10.4.1 Point conservation of mass from a general volume
10.4.2 Point conservation of mass from a material volume
10.4.3 Point conservation of mass from a fixed volume
10.5 Total mass conservation in a stirred tank
10.5.1 Differential mass conservation in a CSTR
10.5.2 Discrete mass conservation in a CSTR
10.6 Hydrologic routing
10.6.1 Hydrologic routing for a reservoir
10.6.2 Hydrologic routing for a channel
10.7 Conclusion
10.8 Problems
11 Species Mass Conservation
11.1 Introduction
11.2 General species mass conservation principle
11.2.1 Conservation of species mass for a general volume
11.2.2 Conservation of species mass for a material volume
11.2.3 Conservation of species mass for a fixed volume
11.3 Point form of mass conservation for a chemical species
11.4 Conservation of moles
11.5 On the velocity of a multispecies solution
11.5.1 Barycentric or mass average velocity
11.5.2 Molar average velocity
11.6 Introduction to the diffusion/dispersion vector
11.7 Approximate form of the diffusion/dispersion vector
11.7.1 Mass dispersion vector
11.7.2 Molar dispersion vector
11.8 Species mass conservation in a stirred tank
11.8.1 General solution for species mass fraction
11.8.2 Constant inflow and outflow
11.8.3 Discrete solution for species mass fraction
11.9 Advection–Dispersion equation
11.10 Solution of the advection–dispersion equation
11.10.1 One-dimensional analytic solution
11.10.2 Superposition for linear ADE solution
11.11 Conclusion
11.12 Problems
12 Statement of Conservation of Momentum
12.1 Introduction
12.2 Elements of the momentum conservation equation
12.3 Second order tensor
12.3.1 Mathematical relations
12.3.2 Physical meaning of two example tensors
12.4 Integral momentum conservation equations
12.4.1 Momentum conservation for a general volume
12.4.2 Momentum conservation for a material volume
12.4.3 Momentum conservation for a fixed volume
12.5 Point form of the momentum conservation equation
12.6 Viscous stress tensor
12.7 Navier–Stokes equation
12.8 Bernoulli equation
12.9 Microscale mechanical energy balance
12.10 System-scale mechanical energy balance
12.11 Conclusion
12.12 Problems
12.13 Appendix: angular momentum equation
13 Conservation of Total Energy
13.1 Introduction
13.2 Statement of the total energy equation
13.3 Point forms of total energy conservation
13.4 Thermal energy balance
13.5 Internal energy equation in terms of temperature
13.5.1 Introduction of heat capacity
13.5.2 Introduction of Fourier’s law
13.6 Conclusion
13.7 Problems
14 Mixed-scale Modeling
14.1 Introduction
14.2 Starting point
14.3 Equations for hydraulic routing
14.3.1 Mass conservation for hydraulic routing
14.3.2 Momentum conservation for hydraulic routing
14.3.3 Special form of the momentum equation
14.4 Friction slope parameterization
14.5 Implementation of hydraulic routing techniques
14.5.1 Unsteady, non-uniform flow (full dynamic equation) with with β = 1
14.5.2 Steady, non-uniform flow (quasi-steady dynamic wave approximation)
14.5.3 Steady, non-uniform flow (diffusion wave approximation)
14.5.4 Steady, uniform flow rate Q (normal flow)
14.5.5 Kinematic wave approximation
14.6 Data support
14.7 Mechanical energy equation for hydraulic routing
14.8 Shallow-water equations
14.8.1 Mass conservation for shallow-water flow
14.8.2 Momentum conservation for shallow-water flow
14.9 Conclusion
14.10 Problems
15 Porous Media and Groundwater Systems
15.1 Introduction
15.2 Considerations for porous media flow and transport
15.3 Fluid mass conservation in a porous medium
15.4 Macroscale fluid mass conservation equation
15.5 Matrix deformation and material compressibility
15.6 Darcy’s law
15.7 Flow equation in terms of pressure
15.8 Flow equation in terms of head
15.9 Constant density flow in a deformable, homogeneous porous medium
15.10 Steady-state flow with constant ρ[sup(w)] and K
15.11 Summary of main points for flow equations
15.12 Porous medium-scale convective–dispersive equation
15.13 Conclusion
15.14 Problems
15.15 Appendix: anisotropic hydraulic conductivity
16 Advection–Dispersion Equation Solution
16.1 Introduction
16.2 Simplified advection–dispersion equation
16.3 One-dimensional formulation
16.4 Discrete form of the advection–dispersion equation
16.4.1 Dimensionless form of the advection–dispersion equation
16.4.2 Discrete approximations of derivatives
16.4.3 One-dimensional discrete advection–dispersion equation
16.5 Two-dimensional formulation
16.5.1 Construction of a difference equation
16.5.2 Steady-state case
16.6 Conclusion
16.7 Problems
17 Stability Revisited
17.1 Introduction
17.2 Lake/CSTR simulation
17.2.1 Discrete form and stability
17.3 First example of lake operation
17.4 Second example of lake operation
17.5 Third example of lake operation
17.6 Conclusion
17.7 Problems
References
Index
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