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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Bring mathematical principles to bear on engineering problems with this updated text
The evolution of industrial processes has resulted in greater emphasis upon analytical and numerical problem solving. Process improvement through experimentation is impractical and consequently engineers must rely upon computational and technical analysis. Furthermore, the ease with which time-series data can be collected and processed has made harmonic signal interpretation routine. Thus, the ability of engineers to analyze, model, compute, and interpret process phenomena is crucial to professional practice.
Problem Solving in Engineering meets these needs with a foundational introduction to mathematical techniques in applied sciences and engineering. Incorporating examples from a range of scientific fields, it communicates principles that can be adapted to many hardware-software combinations. Now fully updated to reflect the latest research and applications, it remains an essential tool for engineers and applied scientists everywhere.
Readers of the second edition will also find:
- Extensive time devoted to problem formulation
- Detailed discussion of integro-differential equations and the processing and analysis of time-series data
- The use of vorticity transport for the solution of momentum, heat, and mass transfer problems in two dimensions
- Examples and problems drawn from aviation, telegraphy, structural failures, railroad operation, chemical processes, automatic process control, seismology, neutron diffusion, gravitation, and quantum theory
- Many additional narrative-type exercises written to appeal to students who find problems in context better suited to their learning style
- Solutions manual available for qualified instructors
Problem Solving in Engineering is ideal for advanced undergraduate, graduate students, and technical professionals in the physical sciences, specifically chemical, civil, biochemical, electrical, and mechanical engineering, as well as physics, chemistry, and biology.
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
About the Companion Website
1 Problem Formulation, Models, and Solution Strategies
1.1 Introduction
1.2 Algebraic Equations: Force Resolution and Vapor–Liquid Equilibria (VLE)
1.3 Macroscopic Balances—Lumped-Parameter Models
1.4 Force Balances—Newton's Second Law of Motion
1.5 Distributed Parameter Models—Microscopic Balances
1.6 Using the Equations of Change Directly
1.7 Discretizing a Problem that is Continuous in Time or Space
1.8 A Contrast: Deterministic Models and Stochastic Processes
1.9 Problems with Integer-Valued Variables
1.10 Empiricisms and Data Interpretation
1.11 Conclusion
Problems
References
2 Algebraic Equations
2.1 Introduction
2.2 Elementary Methods
2.3 Simultaneous Linear Algebraic Equations
2.4 Simultaneous Nonlinear Algebraic Equations
2.5 Algebraic Equations with Constraints
2.6 Conclusion
Problems
References
3 Vectors and Tensors
3.1 Introduction
3.2 Elementary Operations
3.3 Review of Some Basic Mechanics
3.4 Other Important Vector Operations
3.5 Green's Theorem
3.6 Stokes' Theorem
3.7 Conclusion
Problems
References
4 Numerical Quadrature
4.1 Introduction
4.2 Trapezoid Rule
4.3 Simpson's Rule
4.4 Newton–Cotes Formulae
4.5 Roundoff and Truncation Errors
4.6 Romberg Integration
4.7 Adaptive Integration Schemes
4.8 Gaussian Quadrature and the Gauss–Kronrod Procedure
4.9 Integrating Discrete Data
4.10 Multiple Integrals (Cubature)
4.11 Monte Carlo Methods
4.12 Conclusion
Problems
References
5 Analytic Solution of Ordinary Differential Equations
5.1 Some Introductory Examples
5.2 First-Order Ordinary Differential Equations
5.3 Nonlinear First-Order Ordinary Differential Equations
5.4 Higher-Order Linear ODEs with Constant Coefficients
5.5 Higher-Order Equations with Variable Coefficients
5.6 Bessel's Equation and Bessel Functions
5.7 Power Series Solutions of Ordinary Differential Equations
5.8 Regular Perturbation
5.9 Linearization
5.10 Frequency Response for Model Development
5.11 Conclusion
Problems
References
6 Numerical Solution of Ordinary Differential Equations
6.1 An Illustrative Example
6.2 The Euler Method
6.3 Runge–Kutta Methods
6.4 Simultaneous Ordinary Differential Equations
6.5 Limitations of Fixed Step-Size Algorithms
6.6 Richardson Extrapolation
6.7 Multistep Methods
6.8 Split Boundary Conditions
6.9 Finite-Difference Methods
6.10 Stiff Differential Equations
6.11 BDF (Backward Differentiation Formula) Methods
6.12 Bulirsch–Stoer Method
6.13 Phase Space
6.14 Summary
Problems
References
Note
7 Analytic Solution of Partial Differential Equations
7.1 Introduction
7.2 Classification of Partial Differential Equations and Boundary Conditions
7.3 Fourier Series
7.4 The Product Method (Separation of Variables)
7.5 Parabolic Equations
7.6 Elliptic Equations
7.7 Application to Hyperbolic Equations
7.8 Applications of the Laplace Transform
7.9 Approximate Solution Techniques
7.10 The Cauchy–Riemann Equations, Conformal Mapping, and Solutions for the Laplace Equation
7.11 Conclusion
Problems
References
Note
8 Numerical Solution of Partial Differential Equations
8.1 Introduction
8.2 Finite Difference Approximations for Derivatives
8.3 Boundary Conditions
8.4 Elliptic Partial Differential Equations
8.5 Parabolic Partial Differential Equations
8.6 Hyperbolic Partial Differential Equations
8.7 Problems with Moving Boundaries
8.8 Elementary Problems with Convective Transport
8.9 A Numerical Procedure for Two-Dimensional Flow and Transport Problems
8.10 MacCormack's Method
8.11 Adaptive Grids
8.12 Conclusion
Problems
References
9 Integro-Differential Equations
9.1 Introduction
9.2 An Example of Three-Mode Control
9.3 Population Problems with Hereditary Influences
9.4 An Elementary Solution Strategy
9.5 VIM: The Variational Iteration Method
9.6 Integro-Differential Equations and the Spread of Infectious Disease
9.7 Examples Drawn from Population Balances
9.8 Conclusion
Problems
References
Note
10 Time-Series Data and the Fourier Transform
10.1 Introduction
10.2 A Nineteenth-Century Idea
10.3 The Autocorrelation Coefficient
10.4 A Fourier Transform Pair
10.5 The Fast Fourier Transform
10.6 Aliasing and Leakage
10.7 Smoothing Data by Filtering
10.8 Modulation (Beats)
10.9 Some Important Examples
10.10 Conclusion and Some Final Thoughts
Problems
References
Note
11 An Introduction to the Calculus of Variations and the Finite Element Method
11.1 Some Preliminaries
11.2 Notation for the Calculus of Variations
11.3 Brachistochrone Problem
11.4 Other Examples
11.5 The Rayleigh–Ritz Method and Sturm–Liouville Problems
11.6 Contemporary COV Analyses of Old Structural Problems
11.7 Systems with Surface Tension
11.8 Less Familiar COV Applications
11.9 The Connection Between COV and the Finite Element Method
11.10 Conclusion
Problems
References
Note
Index
End User License Agreement
List of Tables
Chapter 7
Table 7.1 Illustration of Infinite Series Convergence for Small
t
‘s.
Chapter 8
Table 8.1 Explicit Computation with Unstable Parametric Choice(s).
Table 8.2 Explicit Computation with Stable Parametric Choice(s).
List of Illustrations
Chapter 1
Figure 1.1 A DC circuit with five resistors.
Figure 1.2 A desalination process consisting of an evaporator and a crystall...
Figure 1.3 Annular volume element extracted from conductor for the thermal e...
Figure 1.4 Displacement of a suspended weight attached to the ceiling with a...
Figure 1.5 Comparison of a distributed parameter model (filled circles) with...
Figure 1.6 Closed-loop response for an underdamped, second-order system with...
Figure 1.7 System response obtained with PID control. Notice how rapidly the...
Figure 1.8 Comparison of actual control valve operation with the linearized ...
Figure 1.9 The growth of a population of animals (per acre) as described by ...
Figure 1.10 Ant travel simulated as a random walk, with no “learning” on the...
Figure 1.11 Friction factors measured experimentally for flow through a poly...
Figure 1.12 A recording of a human voice speaking the word, “integration.”
Figure P1.18 Temperature history of the plate during radiant heating.
Figure P1.19 Simulated loop output for PI control of a second-order system w...
Chapter 2
Figure 2.1 Results of the preplanned search in which the appropriate quadran...
Figure 2.2 Contour plot of the results obtained through application of the G...
Figure 2.3 Contour plot for the function,
F
. The actual solution for this pr...
Figure 2.4 Progress of the Rosenbrock search method applied to the problem i...
Figure P2.20 Comparison of data with regression line obtained with
LinReg
. N...
Figure P2.21 SRB performance from J. A. Roach memo (MSFC/NASA-EL-42) of July...
Figure P2.22 Measured generator output as a function of engine speed for an ...
Chapter 3
Figure 3.1 Shear stress created by fluid motion past a solid surface. For th...
Figure 3.2 A force of 400 N applied in the
x–y
plane at an angle of 25...
Figure 3.3 Effect of a crosswind upon a light aircraft flying due east at 12...
Figure 3.4 Effect of a quartering wind upon a light aircraft flying east at ...
Figure 3.5 A structural member (or beam) lying against a vertical wall. The ...
Figure 3.6 A cylindrical object of radius,
R
, and weight,
W
, has rolled up a...
Figure 3.7 Coplanar forces acting upon a rigid body at different locations....
Figure 3.8 Surface in the
x–y
plane with for .
Figure 3.9 Illustration of coplanar vectors
A
and
B
forming an angle of 30°....
Figure 3.10 Vector field for
F
= (
y
3
,0).
Figure P3.7 Two-dimensional object in the
x–y
plane.
Figure P3.22 Scalar field for problem 3.22.
Chapter 4
Figure 4.1 Behavior of
x
2
exp(−
x
2
)
. The value of the integral (4...
Figure 4.2 Nonlinear discrete data for numerical quadrature. Note that the o...
Figure 4.3 Cubic spline curve fit for the nonlinear data set. We now have a ...
Figure 4.4 Progress of the Monte Carlo integration of the double integral, e...
Figure P4.10 Plot of the function, .
Figure P4.23 Volume–pressure data for ethylene from York and White (1940).
Figure P4.24 Velocity measurements for air discharged through an axisymmetri...
Chapter 5
Figure 5.1 Cooling process for 500 mL water, initially heated to incipient b...
Figure 5.2 Comparison of the solutions for the three ODEs :
y′
= −2 −
Figure 5.3 Local behavior of
y
(
x
) for eq. (5.66) as estimated through the co...
Figure 5.4 Behavior of frictionless pendulum with a starting position of 90°...
Figure 5.5 Characteristic behavior of a slightly underdamped second-order sy...
Figure 5.6 Response of negative feedback control loop to a unit step change ...
Figure 5.7 Bessel functions
J
0
(
r
) and
Y
0
(
r
) for
r
from 0 to 10.
Figure 5.8 Comparison of power series truncated at
n
= 7 and
n
= 59 with the...
Figure 5.9 Comparison of the approximate solution obtained with regular pert...
Figure 5.10 Phase-plane portrait of the nonlinear, second-order system revea...
Figure 5.11 Response of a second-order system to a sinusoidal forcing functi...
Figure 5.12 Phase lag for first-, second-, and third-order systems,
all
with...
Figure 5.13 Phase lag for a “black-box” system.
Figure P5.5 Suspension behavior for two cases, underdamped with
ζ
= 0.1...
Figure P5.12 Exact solution for steady conduction in a slab of material for ...
Figure P5.20 Numerical solution for the nonlinear ODE, .
Chapter 6
Figure 6.1 Comparison of the analytic and numerical (Euler method) solutions...
Figure 6.2 Effect of the damping coefficient,
ζ
, upon the dynamic behav...
Figure 6.3 Concentrations of reactants for series reactions carried out in a...
Figure 6.4 Oscillations in the cellular activity of CDK1 (
C
), APC (
A
), and P...
Figure 6.5 A measure of the local error for the equation,
dy/dx = xy
...
Figure 6.6 Behavior of the step size,
h
, using the Runge–Kutta Fehlberg sche...
Figure 6.7 Charge on the capacitor as computed with the analytic solution (f...
Figure 6.8 Solution of the Blasius equation for flow in the boundary layer o...
Figure 6.9 Comparison of the analytic solution (solid curve) for steady-stat...
Figure 6.10 Computed results for Gear's example (a chemical kinetics problem...
Figure 6.11 The system trajectory for this sinusoid is a closed path, or lim...
Figure 6.12 A two-dimensional (
Y-Z
plane) portrait of a
strange attractor
fr...
Figure 6.13 Polar plot of a solution to the Kepler problem with
f(r) = − ku
...
Figure 6.14 Numerical solution of the linear ODE,
u″ + u = c1/ exp(c2θ)
...
Figure 6.15 Numerical solution of a nonlinear Binet equation,
u″ + u = c1/u1
...
Figure 6.16 An inward-directed spiral obtained from eq. (6.109),
u″ + u = c1
...
Figure 6.17 Phase-space portrait of the behavior of the ODE,
u″ + u = 0.275u
...
Figure 6.18 Phase plane for the LaSalle model with
y
(0) = −½. Note that the ...
Figure P6.16 A few trajectories plotted for the LaSalle model.
Figure P6.22 Approximation using Legendre polynomial,
P
4
.
Figure P6.24 Phase plane for Duffing's equation with
r
= −1/6,
K
= 1/4, and
Figure P6.25 Poincaré section for the Duffing equation.
Figure P6.27 Numerical solution of Emden's equation with
n
= 2.
Chapter 7
Figure 7.1 A triangular wave on the interval (
−π ≤ x ≤ + π
...
Figure 7.2 Representation of the triangular wave with Fourier's series techn...
Figure 7.3 Reconstruction of triangular function using 2, 5, 10, 20, and 40 ...
Figure 7.4 Computation of the initial temperature distribution in the slab u...
Figure 7.5 Concentration distributions for diffusion in a plane sheet of thi...
Figure 7.6 Solution for heat transfer to a long, solid cylinder when the sur...
Figure 7.7 Solution for a sphere, initially at a uniform temperature,
T
i
. At...
Figure 7.8 Typical temperature distribution in slab at
t
= 0.10 given an ini...
Figure 7.9 Temperature distribution in a two-dimensional slab with the top m...
Figure 7.10 Temperature distribution in a two-dimensional slab with the righ...
Figure 7.11 Concentration distribution in a rectangular slab where the top i...
Figure 7.12 Temperature distribution in a slab with symmetric (left–right, a...
Figure 7.13 The temperature should vary across the bottom from 100° at
x
= 0...
Figure 7.14 Temperature contours in a two-dimensional slab with constant pro...
Figure 7.15 Temperature distribution in a circular cylinder with ends mainta...
Figure 7.16 Solution for steady conduction in a squat cylinder with the flat...
Figure 7.17 Temperature distribution in a flat disk with half of the edge ma...
Figure 7.18 String displacement for
t
‘s of 1, 2, 3, 4, 5, and 6 from solutio...
Figure 7.19 Comparison of the exact numerical solution with the collocation ...
Figure 7.20 Comparison of the numerical solution (solid curve) of the nonlin...
Figure 7.21 Numerical solution for the elliptic partial differential equatio...
Figure 7.22 Exact numerical solution for non-Newtonian flow through a rectan...
Figure 7.23 Legendre polynomials,
P
0
through
P
4
, on the interval −1 to 1.
Figure 7.24 Solution for the elliptic partial differential equation, .
Figure 7.25 Plot of the stream function for the complex potential given by e...
Figure 7.26 A partial construction of streamlines for a potential flow enter...
Figure P7.1 Fourier series approximation for
f
(
x
) with
n
= 50 for −2 <
x
< +...
Figure P7.3 Comparison of
f
(
x
) represented by a truncated Fourier sine serie...
Figure P7.27 Computed numerical solutions for
t
‘s of 0.02. 0.05, 0.08, and 0...
Chapter 8
Figure 8.1 Velocity distribution in a rectangular duct computed with the Gau...
Figure 8.2 Centerline velocity as a function of the number of iterations for...
Figure 8.3 Number of iterations required to achieve
ε
= 2 × 10
−7
...
Figure 8.4 Isotherms computed for a mild steel slab with the left-hand side ...
Figure 8.5 Approach of the node
T
(30,90) to the correct temperature, 3.1225,...
Figure 8.6 Computed Nusselt number as a function of
z/R
. The Reynolds number...
Figure 8.7 Concentration of contaminant in the interior of the porous spheri...
Figure 8.8 Development of
U
(
x,y
) in a square domain for time (variable
TT
in...
Figure 8.9 Cube of material with four vertical sides maintained at different...
Figure 8.10 Evolution of the temperature distribution
on the top surface
of ...
Figure 8.11 Propagation of a finite impulse in the
x
-direction due to a cons...
Figure 8.12 Advection of a unit step in the
x
-direction due to the constant ...
Figure 8.13 Analytic solution for the wave eq. (8.104) for specific
x
-positi...
Figure 8.14 Computed string displacement for times of 0.06, 0.36, 0.72, 1.08...
Figure 8.15 Propagation of an initial disturbance centered at
x
= 25 m, corr...
Figure 8.16 Change in the temperature distribution near the interface betwee...
Figure 8.17 Movement of the interface over the first 16 s of heat transfer. ...
Figure 8.18 Dispersion model results for a flow reactor with
Pe
=
VL/D
= 6,4...
Figure 8.19 Heat transfer to fully developed laminar flow between parallel w...
Figure 8.20 Transport of the tracer plume between concentric cylinders. The ...
Figure 8.21 Computed streamlines for recirculating flow in a square cavity. ...
Figure 8.22 Flow in a deep cavity for which
H
= 2 W and Re = 2,000; this is ...
Figure 8.23 Isotherms for the deep cavity with the heated panel (upper porti...
Figure 8.24 Streamlines illustrating the effects of buoyancy upon solution o...
Figure 8.25 Development of the convection rolls in a two-dimensional Rayleig...
Figure 8.26 Application of vorticity transport to buoyancy-driven flow in a ...
Figure 8.27 Thermal plumes behind wedge with Pr = 1, Re = 80, and
t
= 10 s (...
Figure 8.28 Contours of
v
x
(
x,y
) for Re = 900 and
t
= 18 s. The vertical heig...
Figure 8.29 Thermal plume rising from a heated rectangular duct in free conv...
Figure 8.30 Solutions for the linearized “inviscid” Burgers' equation using ...
Figure 8.31 Solutions obtained with MacCormack's method for the nonlinear Bu...
Figure 8.32 Half of a duct with increasing flow area. Fluid enters at the le...
Figure 8.33 Contours plotted for
ψ
(
x,y
) obtained from the solution of e...
Figure 8.34 An object placed in a two-dimensional flow field. This is an ins...
Figure 8.35 Results from the numerical solution of transient heat transfer i...
Figure 8.36 Velocity distribution in a square microchannel (18 μm on each si...
Figure 8.37 Temperature distribution in a two-dimensional slab with spatiall...
Figure P8.14 Note that the initial exponential distribution of displacement ...
Figure P8.21 Temperature distribution in the three-layer device when the
bot
...
Figure P8.22 Porous medium with an impermeable obstruction placed in the mig...
Figure P8.32 Dimensionless temperature as a function of radial position,
r/R
Figure P8.35 Surface and center temperatures for a spherical specimen with
R
Figure P8.36 Typical velocity distribution for start-up flow through a duct ...
Figure P8.39 Temperature of the calorimeter liquid in dimensionless form. No...
Figure P8.40 Streamlines and isotherms for natural convection inside a horiz...
Figure P8.45 Vortex shedding from impeller “blade” at Re = 240. This image c...
Figure P8.46 Startup flow across the top of a square (
L/D
= 1) cavity; compu...
Chapter 9
Figure 9.1 Solutions for the IDE, eq. (9.1), with increasing contributions f...
Figure 9.2 Phase-space portrait of the dynamic behavior of the predator–prey...
Figure 9.3 Illustration of the response of eq. (9.12) to sinusoidal error (t...
Figure 9.4 Solutions of the ordinary differential eq. (9.18), with increasin...
Figure 9.5 Phase-plane portrait for two populations in conflict with heredit...
Figure 9.6 Comparison of the analytic solution (filled gray circles) of the ...
Figure 9.7 Numerical results for IDE eq. (9.31) with different contributions...
Figure 9.8 Increase in the number of infected individuals for
β
's of 0....
Figure 9.9 Solutions of eq. (9.78) for a fixed time using values for
K
rangi...
Figure 9.10 Emergence of the traveling wave from the initial block of infect...
Figure 9.11 Floc formation in Couette device in which the outer cylinder is ...
Figure 9.12 (a and b). Batch flocculation of colloidal kaolin in a stirred 1...
Figure 9.13 Change in the number density function for the growth-only case. ...
Figure 9.14 Change in the number density function (growth only) using a grow...
Figure 9.15 Growth process with breakage imposed at a threshold size (corres...
Figure 9.16 (a) Case 1: Dynamic behavior from the discretized model with the...
Figure P9.12 Phase-plane portrait of two populations in conflict; notice how...
Figure P9.15 Droplet (number) density function for problem 9.15.
Chapter 10
Figure 10.1 A scatter plot of the minimum and maximum (average) temperatures...
Figure 10.2 Line plot of temperature data from the Bradford Station with an ...
Figure 10.3 The Dow Jones Industrial Average from 1896 to 2024. These data w...
Figure 10.4 A synthetic signal constructed from four sinusoids with radian f...
Figure 10.5 Application of Schuster's test to the data shown in Figure 10.4....
Figure 10.6 Autocorrelation coefficient for synthetic signal comprising four...
Figure 10.7 Periodogram for
x(t) = cos(ω1t) + 1.25 sin(ω2t) + Z
...
Figure 10.8 Computed spectrum for data of Figure 10.4. The important radian ...
Figure 10.9 Power spectrum for the signal produced by four sinusoids, eq. (1...
Figure 10.10 Spectrum from a modified version of Brigham's BASIC DFT (radix-...
Figure 10.11 Synthetic signal consisting of 11 sinusoids with a random contr...
Figure 10.12 The computed spectrum has captured all 11 sinusoidal contributi...
Figure 10.13 A plot illustrating sin(
ωt
) and 0.75 cos(4
ωt
), with
ω
...
Figure 10.14 (a–c) Comparison of spectra obtained by DFT for time-series dat...
Figure 10.15 (a–c) Comparison of spectra computed by DFT for record lengths ...
Figure 10.16 Application of the cosine window to a step change in the time-s...
Figure 10.17 The initial square wave (rectangular window) in shown in gray a...
Figure 10.18 Spectra obtained from the original time-series data (a), and fr...
Figure 10.19 Comparison of spectra for three cases: B) moderate noise and no...
Figure 10.20 Original noisy data (dotted), filtered once (continuous gray), ...
Figure 10.21 Comparison of the spectra for the original, noisy data (top, a)...
Figure 10.22 Effects of a
single
application of the digital triangular filte...
Figure 10.23 DFT computed for the simple sum of four periodic functions with...
Figure 10.24 DFT computed for two
products
added together, the cosine pair a...
Figure 10.25 An oscillating function with frequency,
ω
, for which both ...
Figure 10.26 Time-series data for lateral acceleration measured with
Physics
...
Figure 10.27 Spectrum,
S
(
ω
), computed for lateral acceleration experien...
Figure 10.28 G-forces recorded on the middle span of a beam bridge over the ...
Figure 10.29 Models of interferometer strain for event GW150914 detected and...
Figure 10.30 Computed correlation coefficient, γ(
τ
), for modeled event ...
Figure 10.31 USGS data from the Loma Prieta earthquake in 1989. The large-am...
Figure 10.32 Computed spectrum for USGS data from Loma Prieta earthquake of ...
Figure 10.33 Recorded velocity for a turbulent, recirculating flow in a box....
Figure 10.34 Characteristic shapes of air bubbles in water produced by jet a...
Figure 10.35 Spectrum for the recorded acoustic noise at the top of the colu...
Figure 10.36 Spectrum of measured accelerations resulting from car coupling ...
Figure P10.5 Slosh amplitude for cryogenic liquids in tanks.
Figure P10.6 Data of the type obtained from a piezoelectric sensor mounted o...
Figure P10.8 Simulated locomotive bearing box vibrations, with Δ
t
= 0.00025 ...
Figure P10.9 Spectrum for torsional motions of the center span. The frequenc...
Figure P10.10 (a–d) Time-series data for four spoken words, “example” and “e...
Figure P10.12 (a) Time-series data recorded with an accelerometer positioned...
Figure P10.13 Accelerometer data for Bridge 55, a beam bridge with curvature...
Figure P10.19 Three spectra with two distinct frequencies, 0.985 and 0.375 r...
Figure P10.20 Comparison of spectra for a single data segment versus the ave...
Chapter 11
Figure 11.1 Graph of for
0 ≤
x
≤ 4.5
....
Figure 11.2 Potential and kinetic energies of an object launched with an ini...
Figure 11.3 Illustration of the “family” of comparison functions. If
ε
...
Figure 11.4 Solution for the brachistochrone problem for a particle falling ...
Figure 11.5
φ
(
x
) for illustration of the Ritz method. The reader should...
Figure 11.6 Calculated values for the integral eq. (11.54); the minimum occu...
Figure 11.7 Optimal column shape if both ends are hinged (profile shown in h...
Figure 11.8 Typical behavior of the batch chemical process in which “B” is t...
Figure 11.9 (a and b) Flow around a rectangular obstruction at low Reynolds ...
Guide
Cover
Title Page
Copyright
Preface
About the Companion Website
Table of Contents
Begin Reading
Index
End User License Agreement
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Problem Solving in Engineering
Analytical Mathematics and Numerical Analysis
Larry A. Glasgow
Department of Chemical Engineering,Kansas State University,Manhattan, KS, USA
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Preface
This book, Problem Solving in Engineering: Analytical Mathematics and Numerical Analysis, is intended to help students acquire key competencies crucial to their ability to solve important practical problems. The text can be used in a formal classroom setting (as I have employed it at Kansas State University) or for self-study, and both students and practicing professionals in STEM fields may use it effectively.
The Programme for International Student Assessment (PISA) has determined that “…students in the United States have particular weaknesses in performing mathematic tasks with higher cognitive demands, such as taking real-world situations, translating them into mathematical terms, and interpreting mathematical aspects in real-world problems.” It has been suggested that a factor contributing to this crisis is the lack of integration between science and mathematics in secondary education. It seems that the focus in the U.S. system is often based on content coverage (possibly as a consequence of our emphasis upon standardized testing) and not on the contextual connection between math and real, physical phenomena. For students in the applied sciences—who are often visual and tactile learners—the latter is absolutely critical.
It should also be noted that the life experiences of contemporary adolescents are vastly different from those of previous generations. Although it is easy to attribute problems in U.S. secondary education to PC gaming, social media, and smartphones, this is too conveniently simplistic. Indeed, smartphones can be used to great advantage in STEM education, as we demonstrate in Chapter 10. That said, many aspects of contemporary life have conspired to deprive students of opportunities for somatic learning. Rites of passage for teenagers 60 years ago (e.g., adjusting the carburetor or ignition timing on an automobile) simply no longer exist. It is up to educators to step into this void with creative, contextual learning experiences. This is important because of a characteristic noted among many of the most successful engineering professionals: They typically have an unbridled curiosity about how and why machines and processes work the way they do. In applied mathematics, we must do our very best to nurture this mindset by demonstrating the connections between theory and important physical phenomena.
With these observations in mind, I have revised this book to emphasize how applied mathematics can be used by engineers to design processes, analyze their dynamic behavior, and interpret their performance in industrial settings. We note that the success of our efforts in this area has profound implications for economics, safety, productivity, and environmental stewardship. Accordingly, many new detailed examples and narrative-type problems have been included in this revision; these additions emphasize the essential connections between math and the physical world around us. I have drawn from diverse fields, including navigation, aerospace, signal processing, fluid flow, heat and mass transfer, chemical reactor operations, the response engineered structures to their environment, process optimization, machinery monitoring, and others. The intent, of course, is to find ways to appeal to the interests and talents of a broad range of engineering students and practicing professionals and to demonstrate to all users of this book why the topics we study are worthy of their attention.
Larry A. Glasgow
Department of Chemical Engineering,
Kansas State University,
Manhattan, KS, USA
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/glasgow/problemsolvingengineering
This website include:
Solutions Manual
Ancillary Materials and Problems
1Problem Formulation, Models, and Solution Strategies
1.1 Introduction
Many engineering textbooks describe the steps one should follow when solving engineering problems, and the list given by Eide et al. (2008) is appropriate for our purposes:
Recognize the problem and develop a physical understanding of it
Accumulate pertinent data if they are available
Select an appropriate principle or theory as the basis for the model
Apply the simplifications necessary to make the model tractable
Solve the problem with the best available tools
Verify the results using experience and physical insight.
Most of our work here will be focused upon 4), 5), and 6) since the models we will be working with—for the most part—have already been formulated for us and the data needed are provided in the problem statements. However, we should remember a physical understanding of a particular problem, 1), is a key element of a successful process; we cannot evaluate the results of our solution or our computational procedure without it. We should also observe that there is one aspect of the philosophy of engineering problem solving that does not appear in this list: Always begin with the simplest possible model and the most direct solution procedure. You are less likely to make a mistake, and additional complexity can always be added should you find it necessary. An inelegant solution procedure that provides the required result is always better than an elegant one that fails.
In this context, we note that many of the numerical methods described in this text were developed with PBCC™ (the Power BASIC Console Compiler). This has not been done with the expectation that students of this material will learn to write their own programs. A few may be so inclined, but for most it is more practical to utilize readily available commercial software. Nevertheless, there are important reasons for providing these examples: i) BASIC codes are pretty transparent—the logic is clear even to students with no background in high-level languages, ii) the author has found that PBCC generates very fast executables; although this is of little consequence for, say, root finding with Newton–Raphson, it may be critically important if one is trying to solve partial differential equations, and iii) both the structure and the intent of these example programs are obvious, so the solution strategies can be adapted to many different hardware/software combinations.
This touches on an old debate in engineering education: Should engineering students be taught computer programming? Proponents of this idea argue that programming in a high-level language fosters a systematic approach to problem solving. In the era of centralized computing prior to 1980 (in centralized computing users submitted their programs on punched cards to a mainframe computer and then waited for the printed output to be returned) this meant learning FORTRAN, a high-level language designed specifically for scientific computations. Software available for technical calculations included IBM's Scientific Subroutine Package (SSP) and, after 1971, SAS (the Statistical Analysis System). The former had capabilities for common operations like solving algebraic equations, performing numerical quadrature, and solving sets of ordinary differential equations. The SSP routines had to be “called” by a main program written by the user; furthermore, an engineer with an atypical computational problem would need to write their own code from scratch. This situation persisted into the beginning of the personal computing revolution when IBM introduced the PC based upon Intel's 8088 processor. Technical professionals immediately discovered the benefits of accessible computing power; unfortunately, software available for personal computers was limited to the disk operating system (DOS), a BASIC interpreter, and rudimentary spreadsheets. By necessity, engineers developed BASIC programs for many technical computations despite the speed limitations of the line-by-line interpreter. This changed rapidly, though, with the availability of the 8087 numeric coprocessor and Microsoft's publication of high-level language compilers for both FORTRAN and BASIC. By 1990 the computing power of PCs had evolved to the point where engineers could solve important problems without ever leaving their desk. It is no exaggeration to say that the decentralization of computing power greatly facilitated progress in the technical professions.
It is possible—even likely—that at some point in the near future an AI (artificial intelligence) system for engineering problem solving will exist. We can assume that it will make calculations for routine engineering problems autonomously with a near-zero probability of error. However, this does not mean that skilled engineering problem solvers will disappear. What the incorporation of automation in industrial settings has demonstrated is that when some job functions are eliminated, others are created. We should anticipate a future where highly capable engineering professionals will work in concert with AI to produce solutions that are practical, efficient, and economically optimal.
Exactly what are we trying to accomplish in this course?
We want to model a system, process, or phenomenon so that future behavior can be predicted.
We want to select an appropriate solution procedure so the model can be used as a tool for process design, improvement, or control. In the case of the latter, we are particularly concerned with problems that may result from transient operations or process excursions. In these instances the fidelity of the model and the accuracy of the solution procedure may have significant implications for safety.
We want to interpret data that have already been collected; i.e., we need to develop a framework that allows us to understand what has transpired and confidently apply that to analogous situations.
To accomplish these objectives we will review some mathematical techniques that can be used to solve important problems in engineering and the applied sciences. We will focus upon problem types that are crucial to the analysis and simulation of real, physical phenomena. As noted previously, sometimes our objective will be to predict future behavior of a system and sometimes it will be to interpret behavior that has already occurred. We want to stress that the author and the readers are collaborators in this effort, and whether this text is being used in a formal setting, or for self-study, the ultimate goal is the same: We want to be able deal with problems that arise in the applied sciences and do so efficiently. And—this is important—we do not want to rely upon calculation software unless we know something about the method(s) being employed. Too often real problems can have alternative solutions, so it is essential that the analyst be able to exercise some judgment based upon understanding of the problem and of the algorithm that has been selected.
We will redirect our discussion momentarily to underscore a common difficulty faced by engineers engaged in problem solving—and one not described above. Before you commit the time required to solve a particular problem you should ask the following question: Do I have enough information to formulate a model and to find an appropriate solution? Though the answer is usually obvious in instructional examples, it is less likely to be so in real-world problems. All engineers (and managers) want to make efficient use of the time spent solving problems, so allocating effort to determine whether a particular problem as posed is solvable prior to commencing the actual work is always justified. Let us illustrate this with two examples beginning with an elementary DC circuit illustrated in Figure 1.1.
Assume a voltage is applied and the resulting current flow in the circuit is measured. The resistors are parallel–series–parallel, so we write the appropriate equations beginning with Ohm's law:
Figure 1.1 A DC circuit with five resistors.
Figure 1.2 A desalination process consisting of an evaporator and a crystallizer.
Now, suppose the following information is available: V = 75 V, I = 2.19 amps, R1 = 8 Ω, R2 = 15 Ω, and R3 = 10 Ω. Can we determine R4 and R5? We cannot since we have two unknowns:
or
So what we can calculate is the sum of the inverses—we must know something about one of these resistors if we are to go farther. For example, if we know R4 = R5, one can then show that R4 = 38 Ω. In this simple problem the lack of information necessary for solution was apparent. But that will not always be the case, more complex problems may be missing subtle elements that render solution impossible; e.g., many engineering students have experienced this in the analysis of forces in trusses or solving material balances for chemical processes with recycle streams. Let's look at an example of the latter.
Material balance problems for chemical processes produce sets of algebraic equations, and it is often unclear whether or not a solution can be obtained with the information provided. For example, consider a desalination process in which a salt solution is fed (F) to an evaporator to produce salt crystals (C) and water (W) shown in Figure 1.2.
The process operates at steady state, so we first picture an envelope that encloses both the evaporator and the crystallizer (the entire system); since the mass coming in must exit, the overall balance is:
Next, we write a balance on the evaporator:
and a balance on the crystallizer,
You can see that if we add eqs. (1.3) and (1.4) together, we obtain eq. (1.2), so of course only two of these equations are independent. The brine produced by the crystallizer is recycled (R) to the evaporator. There are two components in this process, salt and water, so we can also develop component balances: Let xF and yF be the mass fractions of salt and water, respectively, in the feed stream, F; now suppose xF = 40% and F = 2,500 kg/hr. The only place salt exits the process is in the stream from the crystallizer, C. If C contains only salt crystals (and more on this momentarily) then:
We can also write salt balances for both the evaporator and the crystallizer:
It is reasonable to assume that both the mass flow rate of the feed, F, and the salt concentration will be specified, i.e., F and xF are known. This means that both C and W can be immediately determined. Which quantities are unknown, and do we have sufficient information to calculate them? The four unknowns are E, R, xE, and xR; but we have only two remaining equations, the salt balances on the evaporator and the crystallizer—there is insufficient information to complete the solution for this problem. Now, assume xE = 50% and xR = 45% and demonstrate that R = 10,000 kg/hr: F is provided, so 0.4F = C = 1,000 kg/hr. Since F = W + C, W = 1,500 kg/hr. Next we perform a salt balance on the evaporator: 0.45R + 0.4F = 0.5E and as noted previously, F + R = W + E and we eliminate E using the evaporator salt balance: 1,000 + R = 0.9R + 2,000 or 0.1R = 1,000, and R = 10,000 kg/hr. Finally, we observe that E = 2,500 + 10,000 − 1,500 = 11,000 kg/hr. There are two concluding observations we must make: i) There are just two components, salt and water. We should exercise care when using both component balances since the mass fractions in a given stream must sum to 1, e.g., for the recycle stream xR + yR = 1. Thus, if one added the two component balances on the crystallizer, you would simply obtain eq. (1.4), E = C + R. ii) As a practical matter it is virtually impossible to eliminate all of the water from C (the crystallizer product stream). Could you work this problem if C contained both salt (say 85%) and water (15%)? See if you can verify for the revised problem R = 8235.3 kg/hr and E = 9411.8 kg/hr.
Many of the problems we will be solving come from transient or equilibrium balances where something (force, momentum, mass, energy, charge, etc.) may be subject to a conservation principle. In their simplest forms such balances might be written as [Rate in] – [Rate out] = [Accumulation] or ∑F = 0. Often they will involve forces, fluxes, and the couplings between driving force–flux pairs. Examples of the latter are Newton's, Fourier's, Fick's, and Ohm's laws:
where τyx is the shear stress (acting upon a y-plane due to fluid motion in the x-direction), qy is the flux of thermal energy in the y-direction, NAz is the molar flux of species “A” in the z-direction, and Jz is the current density. Note that these four fluxes are linearly related to the velocity, the temperature, the concentration, and the potential, respectively. Each driving force–flux pair has, under ideal conditions, a constant of proportionality (the viscosity, μ, the thermal conductivity, k, the diffusivity, DAB, and the electrical conductivity, σ); these constants are molecular properties of the medium that can be determined from first principles if the right conditions are met. Unfortunately, it is also possible for viscosity to depend upon velocity, for thermal conductivity to depend upon temperature, for diffusivity to depend upon concentration, etc. In such cases the driving force–flux relationships are no longer linear as indicated by the above set of eq. (1.7).
Example A Microscopic Thermal Energy Balance
Let us illustrate how one of the driving force–flux couplings from eq. (1.7) is incorporated into a balance; we will formulate a microscopic balance in which Fourier's law will be used to describe the thermal energy flux. Consider heat transfer occurring in an electrical conductor, perhaps a copper wire. The conductor is carrying an electric current, so thermal energy will be produced in the interior by dissipation (I2R heating) and thermal energy will be lost to the surroundings at the wire's surface. We will construct a thermal energy balance upon a volume element shown in Figure 1.3, an annular region extracted from the interior of the wire of length L that extends from r to r + Δr.
We express the balance verbally in the form:
Figure 1.3 Annular volume element extracted from conductor for the thermal energy balance. The thickness of the annular shell is Δr and the length is L.
Since the temperature in the conductor may vary continuously with both position and time, the result of this balance will be a partial differential equation. We can rewrite the balance eq. (1.8) introducing the appropriate symbols:
Now we divide by 2πLΔr, take the limit as Δr → 0, apply the definition of the first derivative, and substitute Fourier's law for qr (we also divide by r):
Note that we have assumed that the volumetric rate of thermal energy production, Pe, is a constant; this cannot be strictly correct since the resistance of copper wire (e.g., AWG 12) is 1.65 ohms/1,000 ft at 25 °C, but increases to 3.08 ohms/1,000 ft at 250 °C. In our model we neglected the temperature dependence of the conductor's resistance; this would probably be acceptable if the temperature change in the wire is modest. For steady-state conditions, the solution for eq. (1.10) is simply
If T is finite at the center then clearly C1 = 0. One question that arises in such problems concerns the speed of approach to steady state: When might we use eq. (1.11) and when must we proceed with the solution for eq. (1.10)? We can illustrate this concern using 2 AWG bare copper wire (d = 0.6544 cm) with a constant surface temperature of 30 °C (this is an example of a Dirichlet boundary condition). We set °C/s and let the wire have a uniform initial temperature of 30 °C. Because copper has a very large thermal diffusivity, = 1.14 cm2/s, the approach to steady state should be quick:
t
(s)
0.005
0.01
0.02
0.05
0.075
0.100
0.175
T
center (°C)
59.67
86.87
124.51
162.43
168.01
169.20
169.52
As we anticipated, the steady-state condition is attained very rapidly suggesting that for many similar situations eq. (1.11) could be used to find T(r).
The example above is a microscopic balance, i.e., we are modeling a distributed parameter system. We will also have occasion to use macroscopic balances for lumped-parameter systems in which the field (or dependent) variable does not vary with position. For the electrical wire with dissipation discussed above, this would mean that the temperature would not vary in the r-direction. This is clearly not valid for the case we just examined where T(r = R) was forcibly maintained at 30 °C by removing heat at the surface. We will discuss the circumstances under which the temperature might be (nearly) independent of position a little later.
1.1.1 Rationale for Modeling and Some Unwanted Complications
In modern industrial production applied scientists and engineers constantly struggle to meet product specifications, satisfy regulatory constraints, increase output, and maximize the return for investors and stakeholders. A reality of modern industrial operations is that economic survival is often predicated upon continuous process improvement. And because of the scale of industrial processes, even incremental improvements can be very significant to the bottom line. In the early twentieth-century process tweaking was carried out empirically by trial and error; this usually worked since margins were wide, there was less global competition, and product specifications were often loose. Since there was little automatic process control, skilled operators quickly learned through experience how to make adjustments to improve production. That era has passed, and now operational decisions and control strategies are often based upon models or process simulations. As Hanna and Sandall (1995) point out, contemporary economic reality dictates that modeling and simulation be favored over labor-intensive experimental investigations. In this introduction we will examine a few of the possible ways models can be formulated, and we will look at some examples illustrating the underlying principles that are key to modeling and simulation.
Before we do that, however, we need to recognize that a model—however complex—is merely a representation of reality. Though we may understand the governing physical principles thoroughly our mathematical formulation will never be in perfect fidelity with the “real” world. This is exactly what Himmelblau and Bischoff (1968) referred to when they noted, “…..the conceptual representation of a real process cannot completely encompass all of the details of the process….” Nearly always in real processes there are random events, stochastic elements, or nonlinear couplings that simply cannot be anticipated. Nowhere does this become more apparent than in the examination of engineering or industrial catastrophes; the actual cause is almost always due to a chain or cascade of events many of which are quite improbable taken individually. In cases of this kind, the number of state variables may be very large such that no mathematical model—at least none that can be realistically solved—will account for every contingency. And even in relatively simple systems quite unexpected behavior can occur, such as a sudden jump to a new state, or the appearance of an aperiodic oscillation. Examples of real systems where such behaviors are observed include the driven pendulum, the Belousov–Zhabotinsky chemical reaction, and Rayleigh–Bénard buoyancy-driven instability. Real systems are always dissipative, i.e., they include “frictional” processes that lead to decay. Where we get into trouble is in situations that include both dissipation and at least one mechanism that acts to sustain the dynamic behavior. In such cases the behavior of the system may evolve into something much more complicated, unexpected, and possibly even dangerous.
There is an area of mathematics that emerged in the twentieth century (the foundation was established by Henri Poincaré) that can provide some qualitative indications of system behavior in some of these cases; though what has become popularly known as catastrophe theory is beyond the scope of our discussions, it may be worthwhile to describe a few of its features. In catastrophe theory, we concern ourselves with systems whose normal behavior is smooth—i.e., that possess a stable equilibrium, but that may exhibit abrupt discontinuities (become unstable) at instants in time. Saunders (1980) points out that catastrophe theory applies to systems governed by sets (even very large sets) of differential equations, to systems for which a variational principle exists, and to many situations described by partial differential equations. In typical applications, the number of state variables may indeed be very large, but the number of control variables may be quite small. Let us explain what we mean by control variable with an example: Suppose we wished to study the flow of water through a cylindrical tube. We impose a particular pressure gradient (or head, Δp) and then measure the resulting flow rate. The head is the control variable and the flow rate through the tube is established in response to Δp. If the number of control variables is less than or equal to four, then there are only seven elementary types of catastrophes. The beauty of catastrophe theory is that it makes it possible to predict the qualitative behavior of a system, even for cases in which underlying differential equations are unknown or hopelessly complicated. An excellent review of this field with numerous familiar examples (including biochemical reactions, population dynamics, orbital stability, neural activity in the brain, the buckling of structures, and hydrodynamic instability) has been provided by Thompson (1982). His book is a wonderful starting point for students interested in system instabilities. If you would like to explore catastrophe theory with the objective of gaining greater understanding, please consider the team exercise, problem 1.21, in which a catastrophe machine is constructed and tested.
The principal fact we wish to emphasize as we conclude this introduction is that every model is an idealization and when we rely upon a mathematical analysis (or a process simulation) it is prudent to keep its limitations in mind. We would do well to remember the statistician George E. P. Box's admonition, “…..essentially, all models are wrong but some are useful.” In the modern practice of applied science we must add a corollary: Not only can models be useful, sometimes they are absolutely necessary even when they are wrong in some limited sense.
Let us now look at just a few examples of how problems of the types we wish to solve are actually developed. We will begin with two very different situations—force resolution in statics, and equilibrium for four chemical species partitioned between gas and liquid phases; both problems require solution of sets of algebraic equations.
1.2 Algebraic Equations: Force Resolution and Vapor–Liquid Equilibria (VLE)
Imagine two steel rods of unequal length attached to the underneath side of a horizontal beam; the longer rod (L = 16 m) is attached at point “A” and the shorter rod (L = 9.24 m) at point “B.” The free ends of the two rods are pinned together at point “C” to form a 30-60-90 triangle, and a 1,000 lb. weight is suspended from “C.” Our interest is in the tension created in each of the two rods. The system is at equilibrium, so the 1,000 lbf weight must be balanced by the vertical components of the tensions in the rods; therefore
It is also necessary that the horizontal components of the tensions cancel, that is, there can be no net force acting in the horizontally (∑Fhoriz = 0):
Consequently, TAC = 0.577TBC and by eliminating TAC from eq. (1.12) we obtain
so that TBC = 866.1 lbf and TAC = 499.7 lbf. In this elementary example from statics the simultaneous algebraic equations were formulated by noting that the resultant forces must be zero at equilibrium.
Problems in vapor–liquid equilibria require solution of mass balances but in cases where the temperature (T) is unknown (as in this instance), a trial-and-error process can be employed. We will assume that we have a vapor consisting of an equimolar mixture of light hydrocarbons, ethane (1), propylene (2), propane (3), and isobutane (4). The vapor-phase mole fractions are all ¼, i.e., y1 = y2 = y3 = y4 = 0.25. The constant total pressure is 14.7 psia (1.013 bars), and the vapor phase is cooled slightly until the first drop of liquid is formed (this temperature is the dew point). Our objective is to find the temperature, T, at which this occurs, and the composition of the liquid that forms (in equilibrium with the vapor). We will solve this problem two different ways and then compare the results.
First, we will use the Antoine equation to get the vapor pressures of all four species as functions of temperature:
The necessary constants will be obtained from Lange's Handbook of Chemistry (Revised Tenth Edition, 1961).
A
B
C
Ethane
6.80266
656.4
256
Propylene
6.8196
785
247
Propane
6.82973
813.2
248
Isobutane
6.74808
882.8
240
Keep in mind that T must be °C and p
