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CONTROL SYSTEMS ENGINEER TECHNICAL REFERENCE HANDBOOK BY CHUCK CORNELL, PE, CAP, PME

International Society of Automation P.O. Box 12277 Research Triangle Park, NC 27709

CONTENTS PREFACE

xiii

ABOUT THE AUTHOR xv ACKNOWLEDGMENTS ACRONYMS

xvii

xix

COMMON ELECTRICAL DEFINITIONS

xxiii

1. CSE PE EXAM – GENERAL INFORMATION 2. MISCELLANEOUS REVIEW MATERIAL

1

3

2.1 Mathematics 3 2.1.1 SI Prefixes 3 2.1.2 Algebra 3 2.1.3 Trigonometry 6 2.1.4 Calculus 7 2.1.6 Differential Equations 11 2.1.7 Laplace Transforms 11 2.1.8 Bode Plot 11 2.1.9 Nyquist Plot 12 2.2 Thermodynamics 14 2.2.1 Terminology 14 2.2.2 Heat Addition and Temperature 15 2.2.3 Mollier Steam Diagram 15 2.2.4 Psychrometric Chart 15 2.2.5 Properties of Water 17 2.2.6 Properties of Saturated Steam 18 2.2.7 Properties of Superheated Steam 20 2.3 Statistics 22 2.4 Boolean Logic Operations 2.5 Conversion Factors

24

26

2.6 Equations/Laws/Formulas 29 3. MEASUREMENT

35

Topic Highlights

35

3.1 Temperature Measurement Sensors

35

V

3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6

Thermocouple (T/C) 35 Resistance Temperature Detector (RTD) 38 Thermistor 41 Temperature Switch 42 Temperature Indicator (Thermometer) 42 Thermowell 42

3.2 Pressure Measurement Sensors 45 3.2.1 Manometer 45 3.2.2 Bourdon Tube 46 3.2.3 Pressure Diaphragm 46 3.2.4 Pressure Transducer/Transmitter 47 3.2.5 Diaphragm Seal 47 3.2.6 Pressure Sensor Installation Details 47 3.3 Volumetric Flow Measurement Sensors 49 3.3.1 Sensors Based on Differential Pressure (D/P Producers) 3.3.2 Electronic Volumetric Flowmeters 62 3.3.3 Mass Flowmeters 65 3.3.4 Mechanical Volumetric Flowmeters 67 3.3.5 Open Channel Volumetric Flow Measurement 68 3.3.6 Flowmeter Selection Guide 71

50

3.4 Level Measurement Sensors 72 3.4.1 Inferential Level Measurement Techniques 72 3.4.2 Visual Level Measurement Techniques 78 3.4.3 Electrical Properties Level Measurement Techniques 80 3.4.4 Float/Buoyancy Level Measurement Techniques 82 3.4.5 Time of Flight Level Measurement Techniques 83 3.4.6 Miscellaneous Level Measurement (Switch) Techniques 85 3.5 Analytical Measurement Sensors 86 3.5.1 Combustible Gas Analyzers 87 3.5.2 Dew Point 88 3.5.3 Humidity Sensors 89 3.5.4 Electrical Conductivity Analyzers 91 3.5.5 pH/ORP Analyzers 92 3.5.6 Dissolved Oxygen Analyzers 93 3.5.7 Oxygen Content (in Gas) 94 3.5.8 Turbidity Analyzers 95 3.5.9 TOC (Total Organic Carbon) Analyzers 97 3.5.10 Light Wavelength Type Analyzers 99 3.5.11 Chromatographs and Spectrometers 99 3.5.12 Continuous Emission Monitoring Systems (CEMS) 3.5.13 Vibration Analysis 104 4. SIGNALS, TRANSMISSION AND NETWORKING 4.1 Signals/Transmission 107 4.1.1 Copper Cabling 107 4.1.2 Fiber Optic Cabling 109

VI

107

102

4.1.3 IEEE 802.11 & ISA 100.11a Wireless LAN Communication Protocols 4.1.4 Transducers 113 4.1.5 Intrinsic Safety (I.S.) 113

111

4.2 Networking 115 4.2.1 OSI Model 115 4.2.3 Protocol Stack 116 4.2.4 Network Hardware 116 4.2.5 Network Topology 119 4.2.6 Buses/Protocols 121 4.3 Circuit Calculations 134 4.3.1 DC Circuits 134 4.3.2 AC Circuits 140 4.3.3 Voltage Drop 147 4.3.4 Cable Sizing 148 4.3.5 Electrical Formulas for Calculating Amps, HP, KW & KVA (Table 4-13) 5. FINAL CONTROL ELEMENTS

148

149

5.1 Control Valves 149 5.1.1 Selection Guide 149 5.1.2 Control Valve Inherent Flow Characteristics 149 5.1.3 Control Valve Shutoff (Seat Leakage) Classifications 151 5.1.4 Control Valve Choked Flow/Cavitation/Flashing 151 5.1.5 Control Valve Noise 152 5.1.6 Control Valve Plug Guiding 156 5.1.7 Control Valve Packing 157 5.1.8 Control Valve Bonnets 159 5.1.9 Control Valve Body Styles 159 5.1.10 Common Valve Trim Material Temperature Limits 165 5.1.11 Control Valve Installation 166 5.2 Actuators 166 5.2.1 Failure State 166 5.2.2 Action 167 5.2.3 Valve Positioner 168 5.2.4 Actuator Types 168 5.2.5 Actuator Selection 170 5.3 Control Valve Sizing 171 5.4 Pressure Regulators 175 5.4.1 Pressure Reducing Regulator 175 5.4.2 Back Pressure Regulator 177 5.4.3 Vacuum Regulators & Breakers 177 5.4.4 Regulator Droop 178 5.4.5 Regulator Hunting 178 5.4.6 Pressure Regulator Sizing 178 5.5 Motors 181 5.5.1 Types of Motors

181

VII

5.5.2 5.5.3 5.5.4 5.5.5 5.5.6 5.5.7 5.5.8 5.5.9

Motor Enclosure Types 194 Nameplate Voltage Ratings of Standard Induction Motors Motor Speed 195 Motor NEMA Designations 197 Life Expectancy of Electric Motors 198 Motor Positioning 199 Example Motor Elementary Diagrams 200 Motor Feeder Sizing Table 204

5.6 Variable Frequency Drives (VFDs) 205 5.6.1 Types of Variable Frequency AC Drives 5.6.2 VFD Applications 206 5.6.3 Harmonics Associated with VFDs 206

205

5.7 Pressure Safety Devices 207 5.7.1 Terminology 207 5.7.2 Types of Pressure Relief/Safety Valves 208 5.7.3 Tank Venting 209 5.7.4 Types of Rupture Disks 210 5.7.5 Rupture Disk Accessories 211 5.7.6 Rupture Disk Performance 212 5.7.7 Pressure Relief Device Sizing Contingencies 212 5.7.8 Pressure Levels as a % of MAWP 214 5.7.9 Pressure Relief Valve Sizing (per ASME/API RP520) 5.7.10 Rupture Disk Sizing (per ASME/API RP520) 217 5.7.11 Selection of Pressure Relief/Safety Devices 219 5.8 Relays/Switches 219 5.8.1 Relays 219 5.8.2 Switches 220 6. CONTROL SYSTEMS

223

6.1 Documentation 223 6.1.1 Instrumentation Identification Letters 223 6.1.2 Instrumentation Line Symbols 224 6.1.3 Instrumentation Location Symbols 225 6.1.4 Binary Logic Diagrams 226 6.1.5 Functional Diagram/Symbols 226 6.1.6 Loop Diagram 229 6.2 Control System - Controller Actions 6.2.1 Terminology 231 6.2.2 PID Control 232 6.2.3 Cascade Control 233 6.2.4 Feedforward Control 234 6.2.5 Ratio Control 235 6.2.6 Split Range Control 235 6.2.7 Override Control 236 6.2.8 Block Diagram Basics 236

VIII

231

215

195

6.3 Controller (Loop) Tuning 238 6.3.1 Process Loop Types: Application of P, I, D 238 6.3.2 Tuning Map – Gain (Proportional) & Reset (Integral) 6.3.3 Loop Dynamic Response 240 6.3.4 Loop Tuning Parameters 242 6.3.5 Manual Loop Tuning 242 6.3.6 Closed Loop Tuning 243 6.3.7 Open Loop Tuning 244 6.3.8 Tuning Rules of Thumb 246 6.4 Function Block Diagram Reduction Algebra 6.4.1 Component Block Diagram 246 6.4.2 Basic Building Block 247 6.4.3 Elementary Block Diagrams 247

240

246

6.5 Alarm Management 250 6.5.1 Good Guidelines for Alarm Management 250 6.5.2 Characteristics of Good System Alarms 251 6.5.3 Alarm Terms 251 6.5.4 Alarm Review Methodology 253 6.6 Types of Control System Programming 6.6.1 Ladder Diagram 254 6.6.2 Function Block Diagrams 264 6.6.3 Structured Text 268 6.7 Batch Control 272 6.7.1 Automation Pyramid 273 6.7.2 Physical Model 273 6.7.3 Procedural Model 276 6.7.4 Recipes 277 6.7.5 Sequential Function Chart (SFC)

254

278

6.8 Advanced Control Techniques 279 6.8.1 Fuzzy Logic 279 6.8.2 Model Predictive Control 280 6.8.3 Artificial Neural Networks 280 6.9 Example Process Controls 281 6.9.1 Boiler Control 281 6.9.2 Distillation Column Control 283 6.9.3 Burner Combustion Control 284 7. ISA-95 285 7.1 ISA-95 Hierarchy Model

285

7.2 ISA-88 Physical Model As It Pertains to ISA-95 286 7.3 Levels 4–3 Information Exchange

286

IX

8. HAZARDOUS AREAS AND SAFETY INSTRUMENTED SYSTEMS

287

8.1 Hazardous Areas 287 8.1.1 NEC Articles 500–504 287 8.1.2 NEC Article 505 (Class I, Zone 0, Zone 1 and Zone 2 Locations)

296

8.2 Safety Instrumented Systems (SIS) 299 8.2.1 Safety Integrity Level (SIL) 299 8.2.2 Relationship between a Safety Instrumented Function and Other Functions 8.2.3 Definitions 300 8.2.4 Layers of Protection 301 8.3 Determining PFD (Probability of Failure on Demand) 8.3.1 Basic Reliability Formulas 303 8.3.2 Architectures 304 9. CODES, STANDARDS AND REGULATIONS

302

307

9.1 Standards Listings 307 9.1.1 ISA 307 9.1.2 ASME 310 9.1.3 API 311 9.1.4 NFPA 311 9.1.5 IEC 312 9.1.6 CSA 312 9.1.7 UL 312 9.1.8 FM 312 9.1.9 CE 313 9.1.10 ANSI 313 9.1.11 IEEE 313 9.1.12 AICHE (American Institute of Chemical Engineers) 9.1.13 OSHA 313 9.2 NEC 315 9.2.1 Allowable Conduit Fill 9.2.2 Wiring Methods 315

313

315

9.3 NEMA/IEC-IP Enclosure Classifications 328 9.3.1 NEMA Designations (Non-Hazardous) 328 9.3.2 NEMA Designations (Hazardous) 329 9.3.3 IEC-IP (Ingress Protection) 329 9.4 NFPA 70E Electrical Safety in the Workplace: 330 9.4.1 Shock Hazard Analysis 331 9.4.2 Arc Flash Hazard Analysis 333 9.4.3 Protective Clothing Characteristics 334 9.4.4 Personal Electrical Shock Protection Equipment 9.4.5 Qualified Personnel 335 9.4.6 Label Requirements 336

335

9.5 Lightning Protection 336 9.5.1 Lightning Protection (NFPA 780; UL96 & 96A; LPI 17S; IEEE Std 487) 9.5.2 Facility Lightning Protection per NFPA 780 337

X

337

300

10. SAMPLE PROBLEMS

341

11. SAMPLE PROBLEMS - SOLUTIONS

351

12. MISCELLANEOUS TABLES/INFORMATION

357

12.1 Viscosity Equivalency Nomograph 357 12.2 Copper Resistance Table (Table 12-1) 12.3 RTD Resistance Tables

359

12.4 Thermocouple milliVolt Tables 12.5 Instrument Air Quality

358

369

375

12.6 Thevenin & Norton Equivalencies 12.6.1 Thevenin 376 12.6.2 Norton’s Theorem 380

376

13. UNINTERRUPTIBLE POWER SUPPLY (UPS)

387

13.1 UPS Topologies 387 13.1.1 Single-Conversion 387 13.1.2 Double-Conversion 387 13.2 Inverter Technologies 388 13.2.1 Ferro-resonant 388 13.2.2 PWM (Pulse Width Modulation) 13.2.3 Step-Wave 388

388

13.3 Mechanical Flywheel 388 14. RECOMMENDED RESOURCES

391

XI

PREFACE The information in this book was prepared so that it can serve a dual purpose. The first purpose is to provide a study aid for the Control Systems Engineering Professional Engineering Exam that is presided over by the NCEES. The second purpose is to provide a technical reference for future use by the instrumentation / automation professional. Where the author cites any references to commercially available products, it is for reference only and is by no means an endorsement by the author of any commercially available product. Sample problems presented in this book are not meant to influence the reader on specific problems that may be on the exam, but rather to reinforce the technical material that has been presented to the reader.

XIII

2. MISCELLANEOUS REVIEW MATERIAL 2.1 MATHEMATICS This section only intends to provide a high-level overview of the various math concentrations, not specific in-depth coverage.

2.1.1 SI Prefixes The SI prefixes are derived from a Greek, Latin, Italian and Danish names that precedes an SI unit of measure. This name indicates a decade multiplier or divider.

Table 2-1. SI Prefixes Prefix

Symbol

Value

Exa

E

1018

Peta

P

1015

Tera

T

1012

Giga

G

109

Mega

M

106

Kilo

k

103

Hecto

h

102

Deca

da

101

Deci

d

10-1

Centi

c

10-2

Milli

m

10-3

Micro

µ

10-6

Nano

n

10-9

Pico

p

10-12

Femto

f

10-15

Atto

a

10-18

cgs units (centimeter, gram, second) mks units (meter, kilogram, second)

2.1.2 Algebra The part of mathematics in which letters and other symbols are used to represent numbers and quantities in formulae and equations. Quadratic Equations ax 2 + bx + c = 0 r1, r2 =

−b ± b 2 − 4ac 2a

REVIEW MATERIAL – MATHEMATICS

a( x − r1)( x − r2 ) = 0

3

The variables r1 and r2 can be real or imaginary depending on the coefficients a, b and c. •

If b2–4ac > 0, then there are two different REAL roots. -

•

If b2–4ac = 0, then there are two identical REAL roots. -

•

This is an indication of an over-damped system.

This is an indication of a critically-damped system.

If b2–4ac < 0, then there are two complex conjugate roots with the following: Real part:

−b 2a

Imaginary part: ± c −  b  a  2a  

-

2



This is an indication of an under-damped system.

Exponentiation x a x b = x a+b 1 x −a = a x 1 xa = a x b x a = x ab

( ) b

x a = a xb =

( x) a

b

Logarithms If bx = y, then x=logby. Example: If y = 10 x , then x = log10 y OR If e x = y , then x = loge y = ln y log100 = log10 2 log0.01 = log10 − 2

Constants: loga 1 = 0 loga a = 1

4


Other Identities: logb y a = (a)logb y a a x = b( x log b ) loga y = (logb y )(loga b) logb xy = logb x + logb y log( x = jy ) = log( x 2 + y 2 ) + j (log e) × tan−1

y x

Antilog: The antilog function is the inverse of the log function: antilog 2 = 102

antilog− 2 = 0.01

Matrix Mathematics A matrix is a rectangular array of numbers. Addition and Subtraction: Matrices MUST be of the same size in order for addition/subtraction to work.  0 1 2   6 5 4  0 + 6 1 + 5 2 + 4   6 6 6  9 8 7  + 3 4 5  = 9 + 3 8 + 4 7 + 5  = 12 12 12         0 − 3   −1 6 −3   −1 2 0  0 −4 3   −1 − 0 2 − ( −4)  0 3 6  − 9 −4 −3  =  0 − 9 3 − ( −4) 6 − ( −3) =  −9 7 9           −3 x   4 6   1 7   2y 0  +  −3 1 =  −5 1       First simplify the left side of the equation:  −3 x   4 6   −3 + 4 x + 6   1 7   2y 0  +  −3 1 =  2y − 3 0 + 1 =  −5 1         ∴ x + 6 = 7 so x = 1 and 2y − 3 = −5 so y = −1

Multiplication: Matrix multiplication is not commutative: the order in which matrices are multiplied is important. To multiply matrices, their ranks2 must be compatible. In the matrix example shown below a (2x3) matrix is multiplied by a (3x2). Their product is a (2x2) matrix. To check for rank compatibility simply write the ranks as (M x N) x (N x Q). The matrices may be multiplied together ONLY if the (N) values are equal. If the (N) values are indeed equal then the resultant matrix will have a rank of (M x Q).

2.

Rank of a matrix is defined as, the maximum number of linearly independent column vectors in the matrix, OR the maximum number of linearly independent row vectors in the matrix.


5

Example: Multiply the ROWS of Matrix A by the COLUMNS of Matrix B. 0 3  1 0 −2    (1× 0) + (0 × ( −2)) + (( −2) × 0) (1× 3) + (0 × ( −1)) + (( −2) × 4)  0 3 −1 ×  −2 −1 = (0 × 0) + (3 × ( −2)) + (( −1) × 0) (0 × 3) + (3 × ( −1)) + (( −1) × 4) =   0 4     Matrix A Matrix B (0 + 0 + 0) (3 − 0 − 8)   0 −5  (0 − 6 + 0) (0 − 3 − 4) =  −6 −7     

Division: There is NO such operation as matrix division. You MUST multiply by a reciprocal. Not all matrices may be inverted because there is no inverse of zero and you cannot divide by zero. 8 3  If A =   5 2 

w Let A−1 =  x

y z 

8w + 3 x 8 y + 3z   1 Then AA−1 =  = 5w + 2 x 5 y + 2z  0 8w + 3 x = 1 The resulting equations 5w + 2 x = 0

0 1 8 y + 3z = 0 5 y + 2z = 1

Have the solution w=2; x=-5; y=-3; z=8  2 −3  ∴ A −1 =    −5 8 

2.1.3 Trigonometry The branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles. r  180  (360 ) =   2π r  π 





 180   1⋅ radian =   = 57.3  π 

csc-1 1-sin

c H

A

θ

e

Adjacent

b sinθ =

6

s nu ote p y

opposite a = hypotenuse c

Opposite

B

csc

sin

c ot t an

sin θ 1-cos cos

a

1-sec sec

C

csc θ =

hypotenuse c 1 = = opposite a sin A


cos θ =

adjacent b = hypotenuse c

sec θ =

hypotenuse c 1 = = adjacent b cos A

tanθ =

opposite a sin A = = adjacent b cos A

cot θ =

adjacent b cos A = = opposite a sin A

Trigonometric Identities sin2x + cos2 x = 1 1 + tan2 x = sec 2x 1 + cot 2x = csc 2x sin2x = 2sinx • cosx 2 tan x 1 − tan2 x cos 2 x = 2cos2 x − 1 = cos2 x − sin2 x = 1 − 2sin2 x sin( x + y ) = sin x cos y + cos x sin y tan2 x =

sin( x − y ) = sin x cos y − cos x sin y cos( x + y ) = cos x cos y − sin x sin y cos( x − y ) = cos x cos y − sin x sin y 2sin x sin y = cos( x − y ) − cos( x + y ) 2cos x cos y = cos( x − y ) + cos( x + y ) 2sin x cos y = sin( x + y ) + sin( x − y )

2.1.4 Calculus A form of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of very small differences. The two main types are differential calculus and integral calculus. Differential Calculus Differentiation is the procedure used to take the derivative of a function. The derivative is a measure of how a function changes as its input changes, as indicated by a tangent line to a curve on the graph (Figure 2-1). For commonly used derivative formulas, reference Table 2-2.

f(x) f’(x0)= Slope of the tangent line x0

Figure 2-1. Derivative


7

The derivative method is used to compute the rate at which a dependent output y changes with respect to a change in the independent input x. This rate of change is called the derivative of y with respect to x (i.e. the dependence of y upon x means that y is a function of x). If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point. This functional relationship is often denoted y = ƒ(x), where ƒ denotes the function. The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x) = m x + b, for real numbers m and c, and the slope m is given by: ‘m =

change in y Δy = change in x Δx

The idea is to compute the rate of change as the limiting value of the ratio of the differences Δy/Δx as Δx becomes: dy dx

Differentiation Rules Constant Rule: if ƒ(x) is constant, then f′ = 0 Sum Rule: for all functions ƒ and g and all real numbers a and b. (af + bg)′ = af′ +bg′ Product Rule: for all functions ƒ and g. (fg)′ = f′g + fg′ Quotient Rule: for all functions ƒ and g where g ≠ 0. '

 f  f ' g − fg '   = g2 g 

Power Rule: if f′(x)=x′′, then f′(x) = nx(n-1) Chain Rule: If f(x) = h(g(x)), then F′(x) = h′(g(x)) * g′(x) Examples: Find

dy 3x + 5 if y= dx 2x − 3

dy 3(2x − 3) − 2(3x + 5) = = dx (2x − 3)2 6x − 9 − 6x − 10 (2x + 3)2

8

=

−19 (2x + 3)2

if y =

Find f '(3) if f ( x ) = x 2 − 8 x + 3 f '( x ) = 2 x − 8 f '(3) = (2)(3) − 8 = −2

4 find y ' at (2,1) x+2

y ' = 0( x + 2) − 1(4) at (2,1); y ' =

4 1 −4 =− =− (2 + 2)2 16 4


Alternate solution to the above example (right): y = 4 ( x + 2)

−1

 y ' = −4 ( x + 2 )

−1−1

=

−4

( x + 2)

2

 y ' ( 2,1) ) =

−4

( 2 + 2)

2

=−1

4

Table 2-2. Table of Derivatives Power of x d d C=0 x =1 dx dx

d n x = nx ( n −1) dx

Exponential / Logarithmic d x d x e = ex b = b x ln(b ) dx dx Trigonometric d sin x = cos x dx d cos x = − sin x dx d tan x = sec 2 x dx

d 1 ln( x ) = dx x

d csc x = − csc x cot x dx d sec x = sec x tan x dx d cot x = − csc 2 x dx

Inverse Trigonometric d d 1 −1 sin−1 x = csc −1 x = 2 dx dx x −1 x x2 −1 d d −1 1 cos−1 x = sec −1 x = 2 dx dx x x2 − 1 1− x d d −1 1 tan−1 x = cot −1 x = dx dx 1 + x2 1 + x2 Hyperbolic d sinh x = cosh x dx d cosh x = sinh x dx d tanh x = 1 − tanh2 x dx

d csc hx = −(coth x csc hx ) dx d sec hx = −(tanh x sec hx ) dx d coth x = 1 − coth2 x dx

Integral Calculus A common application of integration is to find the average value of a function. For a function u, the average value from x = a to x = b is: u=

1 b udx b − a a

It is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b (Figure 2-2). For commonly used integral formulas, reference Table 2-3. The term integral may also refer to the notion of the antiderivative, a function F whose derivative is the given function ƒ.


9

y = f(x) area =

a



b

a

f ( x )dx

b

Figure 2-2. Average Value of a Function

Product Rule:

d (uv ) = udv + vdu

Quotient Rule:

 u  vdu − udv d  = v2 v 

Chain Rule:

(u × v )' = (u '× v )v '

Table 2-3. Table of Integrals

 f ( x)dx = F( x) + C  kf ( x)dx = k  f ( x)dx

 [f ( x) ± g( x)]dx =  f ( x)dx ±  g( x)dx x ( n−1) + C; n ≠ 1 n +1

 kdx = kx + C

n  x dx =

 sin xdx = − cos x + C  sec xdx = tan x + C  sec x tan xdx = sec x + C

 cos xdx = sin x + C  csc dx = − cot x + C  csc x cot xdx = − csc x + C

 e dx = e



2

x

x

+C

 tan xdx = − ln cos x + C  sec xdx = ln sec x + tan x + C dx



2

x

a −x dx 2

= sin−1

2

x −a

2

=

x +C a

2

dx = ln x + C x

 cot xdx = ln sin x + C  csc xdx = − ln sec x + tan x + C a

2

1 dx x = tan−1 + C 2 +x a a

1 x sec −1 + C a a

Examples:

 −8dx = −8x + C  3x dx = x + C 2

 (6x

10

3

2

+ 5 x − 3)dx =

6x3 5x 2 + − 3 x + 2 = 2x 3 + 2.5 x 2 − 3 x + C 3 2


2.1.6 Differential Equations A differential equation is a mathematical expression combining a function and one or more of its derivatives. First Order (Linear): A first order differential equation is an equation that contains a first, but no higher, derivative of an unknown function. It can be written as a sum of products of multipliers of the function and its derivatives. If the multipliers are scalar then the differential equation is said to have constant coefficients. If the function or one of its derivatives is raised to some power the equation is said to be non-linear. The form for the first order linear equation is y’ + p(x)y = q(x) where p and q are continuous functions of x. Second Order (Linear): A second order differential equation is an equation that involves the second derivative of an unknown function, but no derivative of a higher order: y” + p(x)y’ + q(x) = r(x) where p, q and r are continuous functions of x. Homogeneous: The sum of the derivatives terms is equal to zero. Non-Homogeneous: The sum of the derivative terms is equal to non-zero.

2.1.7 Laplace Transforms The Laplace transform3 is an important tool for solving systems of linear differential equations with constant coefficients. The strategy is to transform the more difficult differential equations into simple algebra problems where solutions are more easily obtained. This book will not attempt to teach Laplace transforms, but rather just provide a table of commonly used transforms (Table 2-4).

2.1.8 Bode Plot A Bode plot (Figure 2-3)4 is a graphical representation of a system, used to evaluate its stability and performance. It is a combination of two graphs (plots) on log (logarithmic) paper and consists of a Bode phase plot and a Bode magnitude (gain) plot. They are both drawn as functions of frequency where each cycle represents a factor of ten in frequency. The Bode magnitude plot is a graph where the frequency is plotted along the x-axis and the resultant gain (represented as decibels – dB) at that frequency is plotted along the y-axis. The Bode phase plot is a graph where the frequency is again plotted along the x-axis and the phase shift of that frequency is plotted along the y-axis. A “passband” is the range of frequencies or wavelengths that can pass through a circuit without being attenuated. A “stopband” is a band of frequencies, between specified limits, through which a circuit does not allow signals to pass. 3. 4.

Named for French mathematical astronomer Pierre-Simon Laplace. Image derived from Wikipedia.

REVIEW MATERIAL – BODE & NYQUIST PLOTS

11

Table 2-4. Laplace Transforms ƒ(t)

ℒ[ƒ(t)]

1

1 s

e at

1 s −a

sinat

a s + a2 2

cos at

s s + a2 2

t sin at

2as

(s t cos at

+ a2

2

+ a2

e at ×

tn ;n∈N n!

)

2

2

+ b2

s −a

(s − a ) tn ;n∈N n!

2

b

(s − a) e at cos bt

)

s 2 − a2

(s e at sin bt

2

2

+ b2

1 s n +1 1

(s − a )

n +1

2.1.9 Nyquist Plot A Nyquist5 plot (Figure 2-4) exhibits a relationship to the Bode plots of the system. If the Bode phase plot is plotted as the angle θ, and the Bode magnitude plot is plotted as the distance r, then the Nyquist plot of a system is the polar representation of the Bode plot.6 Nyquist Stability Criteria This is a test for system stability. The criteria states that the number of unstable closed-loop poles (zeroes) is equal to the number of unstable open-loop poles (zeroes) plus the number of encirclements of the origin of the Nyquist plot of the complex function D(s) (aka the Argument Principle7).

5. 6. 7.

12

Named after Harry Theodore Nyquist of Bell Labs. Image derived from www.math.uic.edu. Developed by Augustin Louis Cauchy.


Phase (deg)

Magnitude (dB)

Cutoff Frequency Slope: –20 dB/decade

Passband

Stopband

Passband

Stopband

Frequency (rad/sec) Figure 2-3. Bode Plot Imaginary G

(contour) Γ F(s)

F(S) Plane Real G

Figure 2-4. Nyquist Plot

•

A feedback control system is stable if, and only if, the contour ΓF(s) in the F(s) plane does not encircle the (-1, 0) point when P (number of poles) is 0 (i.e. the point –(1, 0) is used because that is where a unit circle drawn with its center at the origin (0,0) crosses the realaxis which means that point is -180° from the origin). Therefore, the feedback control system is stable when the unit circle crossing point is at a frequency lower than -180°.

•

A feedback control system is unstable when the same unit circle crossing point on the real-axis is at a frequency higher than -180°.


13

2.2 THERMODYNAMICS Thermodynamics is the study of the effects of work, heat, and energy on a system. There are three laws: First Law (Law of Conservation): Energy can be changed from one form to another, but it cannot be created or destroyed. ΔU = Q – W (U = internal energy; Q = heat added to system; W = work done by system). Second Law (Law of Entropy): Energy spontaneously disperses from being localized to becoming spread out if it is not hindered from doing so (i.e., heat is transferred from high temperature to low temperature regions). Third Law: This law is an extension of the second law: as temperature approaches absolute zero, the entropy of a system approaches a constant.

2.2.1 Terminology Exothermic: A type of chemical reaction that releases energy in the form of heat, light, or sound. This type of reaction may occur spontaneously. Endothermic: A type of chemical reaction that must absorb energy in order to proceed. This type of reaction does not occur spontaneously because work must be done in order to get this reaction to occur. Entropy (s): Entropy is a measure of how much heat must be rejected to a lower temperature receiver at a given pressure and temperature (measured in BTU/lbm–°R). Heat Q released by a system into its surroundings is indicated by a negative quantity (Q°°0). An example of this heat rejection is to place a glass of hot liquid into a colder environment, this results in a flow of heat from the glass to the environment’s surrounding atmosphere until an equilibrium is reached. s=

Q T

Q = heat content of the system

T = Temperature of the system (°R)

Enthalpy (h) (inherent heat): Enthalpy is measured in British thermal units per pound (mass), or BTU/lbm, and represents the total energy content of the system (the enthalpy SI unit is J/kg). It expresses the internal energy and flow work, or the total potential energy and kinetic energy contained within a substance. h = U + PV (U = internal energy; P = pressure; V = volume)

Adiabatic process: A thermodynamic process in which there is no transfer of heat between the process and the surrounding environment. An adiabatic process is generally obtained by surrounding the entire system with a strong insulating material or by carrying out the process so quickly that there is no time for significant heat transfer to take place. Isothermal process: A thermodynamic process in which no temperature change occurs (ΔT = 0). Note that heat transfer can occur without causing a change in temperature of the working fluid.

14

REVIEW MATERIAL – THERMODYNAMICS

2.2.2 Heat Addition and Temperature When heat is added to a material, one of two things will occur: the material will change temperature or the material will change state. When a substance is below the temperature at a given pressure required to change state, the addition of sensible heat will raise the temperature of the substance. Sensible heat applied to a pot of water will raise its temperature until it boils. Once the substance reaches the necessary temperature at a given pressure to change state, the addition of latent heat causes the substance to change state. Adding latent heat to the boiling water does not get the water any hotter, but changes the liquid (water) into a gas (steam). One can state that a certain amount of heat is required to raise the temperature of a substance one degree. This energy is called the specific heat capacity (ΔQ=mcΔT). The specific heat capacity of a substance depends upon the volume and pressure of the material, except for water, the specific heat capacity is 1 BTU/lbm-°F (1kcal/kg-°C in SI units) and remains constant. This means that if we add 1 BTU of heat to 1 lbm of water, the temperature will rise 1°F. The specific heat value and the specific heat capacity value for water are equal.

2.2.3 Mollier Steam Diagram A Mollier diagram (Figure 2-5) can be used to determine enthalpy versus entropy of water and steam with the pressure identified on the y-axis in a log scale, and enthalpy identified on the xaxis. Other properties identified on the Mollier diagram are constant temperature, density and entropy lines. The Mollier diagram8 is useful when analyzing the performance of adiabatic steady-flow processes.9 How To Read A Mollier Diagram Example: Superheated steam at 700psi and 680°F is expanded at constant entropy to 140psi. 1. Locate point 1 at the intersection of the 700psi and 680°F line – then read h (h = 1333 BTU/lbm). 2. Follow the entropy line downward vertically to the 140psi line and read h (h = 1178 BTU/lbm) ∴ h = 1178 – 1333 = –155 BTU/lbm.

2.2.4 Psychrometric Chart A psychrometric chart (Figure 2-6) is graphical representation of the thermodynamic properties of moist air. The chart is used to determine the state of an air-water vapor mixture when at least two properties are known. How to Use a Psychrometric Chart Example: Assume dry bulb temperature = 78°F and wet bulb temperature = 65°F. 8. 9.

Mollier diagrams are named after Richard Mollier, a professor at Dresden University who pioneered the graphical display of the relationship of temperature, pressure, enthalpy, entropy and volume of steam and moist air. Reference www.chemicalogic.com for a blank example of this Mollier Diagram.


15

v = specific volume

Entrpoy s=1.0

s=1.1

s=1.2

s=1.3

s=1.4

s=1.5

s=1.6 s=1.7

s=1.8

s=1.9

s=2.0

s=2.1

s=2.2

s=2.3

s=2.4

s=2.5

s=2.6

s=2.7

s=2.8

Figure 2-5. Mollier Diagram (based upon the Scientific IAPWS-95 formulation)

ific ec Sp me lu Vo

Intersection Point

Dewpoint

78°F

Figure 2-6. Psychrometric Chart (Linric Company)

16


•

First locate 78°F on the DB Temperature Scale at the bottom of the chart.

•

Then locate 65°F WB on the saturation curve scale.

•

Extend a vertical line from the 78°F DB point and a diagonal line from the 65°F WB point to intersect the vertical DB line.

•

The point of intersection of the two lines indicates the condition of the given air. As a result: -

Enthalpy of air = 30 BTU/lb Specific volume of air = 13.7 ft3/lb Dewpoint = 57.5°F

2.2.5 Properties of Water Table 2-5. Properties of Water Specific Temperature Saturation Of Water °F Pressure psia Volume ft3/lb

Weight Density lb/ft3

Weight lb/gal

32

0.08859

0.016022

62.414

8.3436

40

0.12163

0.016019

62.426

8.3451

50

0.17796

0.016023

62.410

8.3430

60

0.25611

0.016033

62.371

8.3378

70

0.36292

0.016050

62.305

8.3290

80

0.50683

0.016072

62.220

8.3176

90

0.69813

0.016092

62.116

8.3037

100

0.94924

0.016130

61.996

8.2877

110

1.2750

0.016165

61.862

8.2698

120

1.6927

0.016204

61.7132

8.2498

130

2.2230

0.016247

61.550

8.2280

140

2.8892

0.016293

61.376

8.2048

150

3.7184

0.016343

61.188

8.1797

160

4.7414

0.016395

60.994

8.1537

170

5.9926

0.016451

60.787

8.1260

180

7.5110

0.016510

60.569

8.0969

190

9.340

0.016572

60.343

8.0667

200

11.526

0.016637

60.107

8.0351

210

14.123

0.016705

59.862

8.0024

212

14.696

0.016719

59.812

7.9957

220

17.186

0.016775

59.613

7.9690

240

24.968

0.016926

59.081

7.8979

260

35.427

0.017098

58.517

7.8226

280

49.200

0.017264

57.924

7.7433

300

67.005

0.01745

57.307

7.6608


17

350

134.604

0.01799

55.586

7.4308

400

247.259

0.01864

53.684

7.1717

450

422.55

0.01943

51.467

6.8801

500

680.86

0.02043

48.948

6.5433

550

1045.43

0.02176

45.956

6.1434

600

1543.2

0.02364

42.301

5.6548

650

2208.4

0.02674

37.397

4.9993

700

3094.3

0.03662

27.307

3.6505

Saturation Pressure: P = 10

  B   A−     (C +T )  

A, B and C are the values of the Antoine constants A, B and C for the temperature T from NIST. Density: ρ = P/RT (P in Pascal; T in Kelvin) Specific Volume: 1/ρ Linear Interpolation in Tables Linear interpolation is a convenient way to fill in holes in tabular data. The formula for linear interpolation is:  g − g1  d = d1 +   ( d2 − d1 )  g2 − g1 

Example: Find the saturation pressure for water at 575°F. From Table 2-5: d1 = 1045.43 psia d2 = 1543.2 psia g = 575°F g1 = 550°F f2 = 600°F  575 − 550  ∴ d = 1045.3 +   (1045.43 − 1543.2 ) =  550 − 600  1045.3 + ( −0.5 ) ( −497.77) = 1045.3 + 248.89 = 1294.19psia

2.2.6 Properties of Saturated Steam Saturated steam occurs when steam and water are in equilibrium. Once the water’s boiling point is reached, the water’s temperature ceases to rise and stays the same until all the water is vaporized. As the water goes from a liquid state to a vapor state it receives energy in the form of “latent heat of vaporization.” As long as there is some liquid water left, the steam’s temperature is the same as the liquid water’s temperature. This type of steam is called saturated steam. Industries normally use saturated steam for heating, cooking, drying and other processes.

18


Table 2-6 lists the properties of saturated steam at different pressures. Table 2-6. Properties of Saturated Steam10 Saturated Pressure Temperature psig ºF 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120

212 215 218 222 224 227 230 232 235 237 239 242 244 246 248 250 252 254 255 257 259 267 274 281 287 292 298 302 307 312 316 320 324 327 331 334 338 341 343 347 350

Specific Volume V ft3/lb

Heat Content BTU/lb

Saturated Liquid (Vf)

Saturated Vapor (Vg)

Saturated Liquid (hf)

Saturated Vapor (hg)

0.0167 0.0167 0.0167 0.0168 0.0168 0.0168 0.0168 0.0169 0.0169 0.0169 0.0169 0.0169 0.0170 0.0170 0.0170 0.0170 0.0170 0.0170 0.0170 0.0171 0.0171 0.0171 0.0172 0.0173 0.0173 0.0174 0.0174 0.0175 0.0175 0.0176 0.0176 0.0176 0.0177 0.0177 0.0178 0.0178 0.0178 0.0179 0.0179 0.0180 0.0180

26.8 24.3 23.0 21.8 20.7 19.8 18.9 18.1 17.4 16.7 16.1 15.6 15.0 14.5 14.0 13.6 13.2 12.8 12.5 12.1 11.1 10.4 9.4 8.5 7.74 7.14 6.62 6.17 5.79 5.45 5.14 4.87 4.64 4.42 4.24 4.03 3.88 3.72 3.62 3.44 3.34

180 183 186 190 193 195 198 200 203 205 208 210 212 214 216 218 220 222 224 226 227 236 243 250 256 262 267 272 277 282 286 290 294 298 301 305 308 312 314 318 321

1150 1151 1153 1154 1155 1156 1157 1158 1158 1159 1160 1161 1161 1162 1163 1164 1164 1165 1165 1166 1166 1169 1171 1173 1175 1177 1178 1179 1181 1182 1183 1184 1185 1186 1189 1189 1190 1191 1191 1192 1193


Latent Heat of Vaporization (hfg) BTU/lb 970 967 965 963 961 959 958 956 955 953 952 950 949 947 946 945 943 942 941 940 939 933 926 923 919 914 911 907 903 900 897 893 890 888 887 884 882 877 877 872 872

19

125 130 135 140 145 150 155 160 165 170 175 180 185 190 200 210 220 230 240 250 260 270 280 290 300

353 355 358 361 363 366 368 370 373 375 378 380 382 384 388 392 396 399 403 406 410 413 416 419 421

0.0180 0.0180 0.0181 0.0181 0.0181 0.0182 0.0182 0.0182 0.0183 0.0183 0.0183 0.0184 0.0184 0.0184 0.0185 0.0185 0.0186 0.0186 0.0187 0.0187 0.0188 0.0188 0.0189 0.0190 0.0190

3.21 3.12 3.02 2.92 2.84 2.75 2.67 2.60 2.53 2.47 2.40 2.34 2.29 2.23 2.14 2.05 1.96 1.88 1.81 1.75 1.68 1.63 1.57 1.52 1.47

324 327 329 332 335 337 340 342 345 347 350 352 355 357 361 365 369 373 377 380 384 387 391 394 397

1193 1194 1194 1195 1196 1196 1196 1196 1197 1197 1198 1198 1199 1199 1199 1200 1200 1201 1201 1201 1201 1202 1202 1202 1202

867 867 864 862 860 858 854 854 852 850 848 846 844 842 838 835 831 828 824 821 817 814 811 807 805

2.2.7 Properties of Superheated Steam As opposed to saturated steam, when all the water is vaporized any subsequent addition of heat will raise the steam’s temperature. Steam heated beyond the saturated steam level is called superheated steam. Superheated steam is used almost exclusively for turbines. Turbines have a number of stages. The exhaust steam from the first stage is directed to a second stage on the same shaft, and so on. This means that saturated steam would get wetter and wetter as it went through the successive stages. This is due to the fact that saturated steam has a greater volume of water in it as the pressure gets lower. Not only would this situation promote water hammer11, but the water particles would cause severe erosion within the turbine. There is a good reason why superheated steam is not as suitable for process heating as is saturated steam. Superheated steam has to cool to saturation temperature before it can condense to release its enthalpy of evaporation. The amount of heat given up by superheated steam as it cools to saturation temperature is relatively small in comparison to its enthalpy of evaporation. Thus, if the steam has a large degree of superheat, it may take a relatively long time to cool, during which time the steam will release very little energy. 10. There are many free online tools available to calculate these values based upon the pressure in psig, www.spiraxsarco.com is just one example of a website that makes one of these calculation tools available free of charge. 11. Water hammer (aka fluid hammer) is a pressure surge that occurs when the fluid in motion is suddenly stopped or forced to change directions abruptly. This pressure wave then propagates throughout the equipment causing noise and vibration.

20


Table 2-7 lists the properties of superheated steam at various pressures. Table 2-7. Properties of Superheated Steam12 Pressure

Sat. Temp psia psig . 15

0.3

20

5.3

30

15.3

40

25.3

50

35.3

60

45.3

70

55.3

80

65.3

90

75.3

100

85.3

120 105.3 140 125.3 160 145.3 180 165.3 200 185.3 220 205.3 240 225.3 260 245.3 280 265.3 300 285.3 320 305.3

213.03 V hg 227.96 V hg 250.34 V hg 267.25 V hg 281.02 V hg 292.71 V hg 302.93 V hg 312.04 V hg 320.28 V hg 327.82 V hg 341.27 V hg 353.04 V hg 363.55 V hg 373.08 V hg 381.80 V hg 389.88 V hg 397.39 V hg 404.44 V hg 411.07 V hg 417.35 V hg 423.31 V hg

Total Temperature ºF 350

400

500

600

700

800

900

1000

1100

1300

1500

31.939 1216.2 23.900 1215.4 15.859 1213.6 11.838 1211.7 9.424 1209.9 7.815 1208.0 6.664 1206.0 5.801 1204.0 5.128 1202.0 4.590 1199.9 3.7815 1195.6 -

33.963 1239.9 25.428 1239.2 16.892 1237.8 12.624 1236.4 10.062 1234.9 8.354 1233.5 7.133 1232.0 6.218 1230.5 5.505 1228.9 4.935 1227.4 4.0786 1224.1 3.4661 1220.8 3.0060 1217.4 2.6474 1213.8 2.3598 1210.1 2.1240 1206.3 1.9268 1202.4 -

37.985 1287.3 28.457 1286.9 18.929 1286.0 14.165 1285.0 11.306 1284.1 9.400 1283.2 8.039 1282.2 7.018 1281.3 6.223 1280.3 5.588 1279.3 4.6341 1277.4 3.9526 1275.3 3.4413 1273.3 3.0433 1271.2 2.7247 1269.0 2.4638 1266.9 2.2462 1264.6 2.0619 1262.4 1.9037 1260.0 1.7665 1257.7 1.6462 1255.2

41.986 1335.2 31.466 1334.9 20.945 1334.2 15.685 1333.6 12.529 1332.9 10.425 1332.3 8.922 1331.6 7.794 1330.9 6.917 1330.2 6.216 1329.6 5.1637 1328.2 4.4119 1326.8 3.8480 1325.4 3.4093 1324.0 3.0583 1322.6 2.7710 1321.2 2.5316 1319.7 2.3289 1318.2 2.1551 1316.8 2.0044 1315.2 1.8725 1313.7

45.978 1383.8 34.465 1383.5 22.951 1383.0 17.195 1382.5 13.741 1382.0 11.438 1381.5 9.793 1381.0 8.560 1380.5 7.600 1380.0 6.833 1379.5 5.6813 1378.4 4.8588 1377.4 4.2420 1376.4 3.7621 1375.3 3.3783 1374.3 3.0642 1373.2 2.8024 1372.1 2.5808 1371.1 2.3909 1370.0 2.2263 1368.9 2.0823 1367.8

49.964 1433.2 37.458 1432.9 24.952 1432.5 18.699 1432.1 14.947 1431.7 12.446 1431.3 10.659 1430.9 9.319 1430.5 8.277 1430.1 7.443 1429.7 6.1928 1428.8 5.2995 1428.0 4.6295 1427.2 4.1084 1426.3 3.6915 1425.5 3.3504 1424.7 3.0661 1423.8 2.8256 1423.0 2.6194 1422.1 2.4407 1421.3 2.2843 1420.5

53.946 1483.4 40.447 1483.2 26.949 1482.8 20.199 1482.5 16.150 1482.2 13.450 1481.8 11.522 1481.5 10.075 1481.1 8.950 1480.8 8.050 1480.4 6.7006 1479.8 5.7364 1479.1 5.0132 1478.4 4.4508 1477.7 4.0008 1477.0 3.6327 1476.3 3.3259 1475.6 3.0663 1474.9 2.8437 1474.2 2.6509 1473.6 2.4821 1472.9

57.926 1534.5 43.435 1534.3 28.943 1534.0 21.697 1533.7 17.350 1533.4 14.452 1533.2 12.382 1532.9 10.829 1532.6 9.621 1532.3 8.655 1532.0 7.2060 1531.4 6.1709 1530.8 5.3945 1530.3 4.7907 1529.7 4.3077 1529.1 3.9125 1528.5 3.5831 1527.9 3.3044 1527.3 3.0655 1526.8 2.8585 1526.2 2.6774 1525.6

61.905 1586.5 46.420 1586.3 30.936 1586.1 23.194 1585.8 18.549 1585.6 15.452 1585.3 13.240 1585.1 11.581 1584.9 10.290 1584.6 9.258 1584.4 7.7096 1583.9 6.6036 1583.4 5.7741 1582.9 5.1289 1582.4 4.6128 1581.9 4.1905 1581.4 3.8385 1580.9 3.5408 1580.4 3.2855 1579.9 3.0643 1579.4 2.8708 1578.9

69.858 1693.2 52.388 1693.1 34.918 1692.9 26.183 1692.7 20.942 1692.5 17.448 1692.4 14.952 1692.2 13.081 1692.0 11.625 1691.8 10.460 1691.6 8.7130 1691.3 7.4652 1690.9 6.5293 1690.5 5.8014 1690.2 5.2191 1689.8 4.7426 1689.4 4.3456 1689.1 4.0097 1688.7 3.7217 1688.4 3.4721 1688.0 3.2538 1687.6

77.807 1803.4 58.352 1803.3 38.896 1803.2 29.168 1803.0 23.332 1802.9 19.441 1802.8 16.661 1802.6 14.577 1802.5 12.956 1802.4 11.659 1802.2 9.7130 1802.0 8.3233 1801.7 7.2811 1801.4 6.4704 1801.2 5.8219 1800.9 5.2913 1800.6 4.8492 1800.4 4.4750 1800.1 4.1543 1799.8 3.8764 1799.6 3.6332 1799.3


21

340 325.3 428.99 V hg 360 345.3 434.41 V hg 380 365.3 439.61 V hg 400 385.3 444.60 V hg 3 V = specific volume, ft /lb hg = total heat of steam, BTU/lb

-

1.5399 1252.8 1.4454 1250.3 1.3606 1247.7 1.2841 1245.1

1.7561 1312.2 1.6525 1310.6 1.5598 1309.0 1.4763 1307.4

1.9552 1366.7 1.8421 1365.6 1.7410 1364.5 1.6499 1363.4

2.1463 1419.6 2.0237 1418.7 1.9139 1417.9 1.8151 1417.0

2.3333 1472.2 2.2009 1471.5 2.0825 1470.8 1.9759 1470.1

2.5175 1525.0 2.3755 1542.4 2.2484 1523.8 2.1339 1523.3

2.7000 1578.4 2.5482 1577.9 2.4124 1577.4 2.2901 1576.9

3.0611 1687.3 2.8898 1686.9 2.7366 1686.5 2.5987 1686.2

3.4186 1799.3 3.2279 1798.8 3.0572 1798.5 2.9037 1798.2

2.3 STATISTICS Degrees of Freedom (df) df is the number of values that are free to vary in the final calculation of a statistic: df = n – 1 where n = number of samples. Standard Deviation (σ) is a measure of the spread of the data about the mean value. Reference Figure 2-7 for the normal distribution curve between standard deviations.

σ=

Σ(x − x ) n −1

2

x = sample value; x = mean value; n = # of samples; n − 1 = df

Example: Consider a population consisting of the following values: 2, 4, 4, 4, 5, 5, 7, 9 There are eight data points in total, with a mean (or average) value of 5: 2 + 4 + 4 + 4 + 5 + 5 + 5 + 7 + 9 40 = =5 8 8

To calculate the population standard deviation, we compute the difference of each data point from the mean, and square the result: (2 – 5)2 = (–3)2 = 9 (4 – 5)2 = (–1)2 = 1 (4 – 5)2 = (–1)2 = 1

12. There are many free online tools available to calculate these values based upon the pressure and superheat temperature, www.spiraxsarco.com is just one example of a website that makes one of these calculation tools available free of charge.

22

REVIEW MATERIAL – STATISTICS

(4 – 5)2 = (–1)2 = 1 (5 – 5)2 = (–0)2 = 0 (5 – 5)2 = (–0)2 = 0 (7 – 5)2 = (–2)2 = 4 (9 – 5)2 = (–4)2 = 16 Next we average these values and take the square root, which gives the standard deviation: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 8 −1

32 = 2.138 7

34.1%

0.1% –3σ

34.1%

2.1%

2.1% 13.6% –2σ

–1σ

13.6% Mean

+1σ

+2σ

0.1% +3σ

Figure 2-7. Percentages in Normal Distribution between Standard Deviations

Six Sigma Six Sigma comes from the notion that if one has six standard deviations between the process mean and the nearest specification limit, there will be practically no items that fail to meet specifications. To achieve Six Sigma, a process must not produce more than 3.4 defects per one million opportunities. 1.5 Sigma Shift It has been shown that in the long term, processes usually do not perform as well as they do in the short term. As a result of this performance, the number of sigmas that will fit between the process mean and the nearest specification limit is likely to drop over time, as compared to an initial short-term study. To account for this real-life increase in process variation over time, an empirically-based 1.5 sigma shift is introduced into the calculation. According to this concept, a process that fits Six Sigmas between the process mean and the nearest specification limit in a short-term study will, in the long term, only fit 4.5 sigmas. Either the process mean will move over time, or the long-term standard deviation of the process will be greater than that observed in the short term, or possibly both. Therefore, the widely accepted definition of a Six Sigma process is one that produces 3.4 Defective Parts per Million Opportunities (DPMO).

REVIEW MATERIAL – STATISTICS

23

This definition is based on the fact that a process that is normally distributed will have 3.4 defective parts per million beyond a point that is 4.5 standard deviations above or below the mean. So the 3.4 DPMO of a “Six Sigma” process corresponds in fact to 4.5 sigmas, that is, 6 sigmas minus the 1.5 sigma shift introduced to account for long-term variation. This is done to prevent underestimation of the defect levels likely to be encountered in real-life operation. Table 2-8 gives long-term DPMO values corresponding to various short-term sigma levels. Table 2-8. Long-Term DPMO Values ΣLevel

DPMO

% Defective

% Yield

1

691,462

69%

31%

2

308,538

31%

69%

3

66,807

6.7%

93.3%

4

6,210

0.62%

99.38%

5

233

0.023%

99.977%

6

3.4

0.00034%

99.99966%

2.4 BOOLEAN LOGIC OPERATIONS13 AND Gate: All inputs must be true for output to be true In1

In2

Out

0

0

0

0

1

0

1

0

0

1

1

1

In1 In2

Out

NAND Gate: All inputs must be false for output to be true In1

In2

Out

0

0

1

0

1

1

1

0

1

1

1

0

In1 In2

Out

13. Developed by George Boole (1815-1864).

24

REVIEW MATERIAL – BOOLEAN LOGIC

OR Gate: Any input can be true for output to be true In1

In2

Out

0

0

0

0

1

1

1

0

1

1

1

1

In1

Out

In2

NOR Gate: If any input is true the output will be false In1

In2

Out

0

0

1

0

1

0

1

0

0

1

1

0

In1

Out

In2

XOR Gate: All the inputs must be different for the output to be true In1

In2

Out

0

0

0

0

1

1

1

0

1

1

1

0

In1

Out

In2

S-R Flip-Flop: Latch Circuit S

R

Q

Q

0

0

Keep output state

Keep output state

0

1

0

1

1

0

1

0

1

1

Unstable condition

Unstable condition

S

NAND

Q

S R

NAND

SET

Q

Q

R

CLR

Q

Equivalent Circuit

REVIEW MATERIAL – BOOLEAN LOGIC

25

2.5 CONVERSION FACTORS Table 2-9. Common Conversion Factors Unit

=

Gallon

8.34

Unit Lbs Water @ 60°F

3

Density of Water

62.4 Lbs/Ft

Density of Air

0.07649 Lbs/Ft3

SG Water @ 60°F

1

MW of Air

29

SG of Liquid

MW of Liquid ÷ 18.02

SG of Gas

MW of Gas ÷ 29

Table 2-10. Distance Factors Multiply Inch

By 2.54

To Obtain Centimeter

Centimeter

0.3937

Inch

Foot

0.3048

Meter

Meter

3.28083

Foot

Table 2-11. Volume Factors Multiply

By

To Obtain

Gallon

0.13368

Ft3

Gallon

0.003754

M3

Gallon

3.7853

Liter

Liter

0.2642

Gallon

Liter

0.03531

Ft3

Liter

0.001

M3

Ft3

7.481

Gallon

3

28.3205

Liter

3

Ft

0.028317

M3

M3

35.3147

Ft3

M

3

3.28083

Gallon

M

3

1000

Liter

Ff

Table 2-12. Mass Factors

26

Multiply

By

To Obtain

Pound

0.4536

Kilogram

Kilogram

2.2046

Pound

REVIEW MATERIAL – CONVERSION FACTORS

Table 2-13. Force Factors Multiply

By

To Obtain

Newton

0.22481

Pound-Force

Pound-Force

4.4482

Newton

Table 2-14. Energy Factors Multiply

By

To Obtain

BTU

778.17

Ft-Lbf

BTU

1.055

KJoules

BTU/Hr

0.293

Watt

HP

0.7457

Kilowatt

HP

2545

BTU/Hr

Table 2-15. Temperature Factors Unit

Use Equation

To Obtain

°F

(°F – 32)*1.8

°C

°F

(°F + 459.67)/1.8

°K

°F

(°F + 459.67)

°R

°C

(°C × 1.8) + 32

°F

°C

°C + 273.15

°K

°C

(°C × 1.8) + 32 + 459.67

°R

°K

(°K × 1.8) – 459.67

°F

°K

°K - 273.15

°C

°K

°K × 1.8

°R

°R

°R – 459.67

°F

°R

(°R – 32 – 459.67)/1.8

°C

°R

°R/1.8

°K


27

Table 2-16. Pressure Factors Multiply

By

To Obtain

Atmosphere

1.01295

Bar

Atmosphere

29.9213

Inches Hg

Atmosphere

760

mm Hg

Atmosphere

406.86

Inches WC *

Atmosphere

14.696

PSI

Atmosphere

1.01295 x 10

N/M2 or Pa

Bar

0.9872

Atm

Bar

29.54

Inches Hg

Bar

750.2838

mm Hg

Bar

401.65

Inches WC

Inches WC

0.03612

PSI

Inches WC

0.07354

Inches Hg

Inches WC

1868.1

mm Hg

Inches WC

248.9

N/M2 or Pa

Inches WC

0.001868

Micron or mtorr

PSI

27.68

Inches WC

PSI

2.036

Inches Hg

PSI

51.71

mm Hg

PSI

0.068046

Atm

PSI

0.068948

Bar

PSI

6892.7

N/M2 or Pa

Micron or mtorr

0.0005353

Inches WC

0.004018

Inches WC

0.00014508

PSI

2

N/M or Pa 2

N/M or Pa

5

* WC indicates water column

Table 2-17. Viscosity Multiply

By

To Obtain

cSt

0.999g/cm3

cP

cP

1/0.999g/cm3

cSt

Kinematic viscosity (stoke) = Absolute viscosity (poise)/S.G. Dynamic viscosity (cP) = 0.001 Pa-s

28


2.6 EQUATIONS/LAWS/FORMULAS Pressure P=

F A

F = Force applied A = Area Boyle’s Law PV 1 1 = P2V2

Boyle’s law states that at constant temperature, the absolute pressure and the volume of a gas are inversely proportional. The law can also be stated in a slightly different manner: that the product of absolute pressure and volume is always constant. P = Pressure in PSIA V = Volume in Ft3 Charles’s Law V1 V2 = T1 T2

OR

V1T2 = V2T1

These expressions may be combined into the form of PV/T = constant for a fixed mass of gas. Charles’s law states that at constant pressure, the volume of a given mass of an ideal gas increases or decreases by the same factor as its temperature on the absolute temperature scale (i.e., the gas expands as the temperature increases, the temperature is the average of molecular motion, therefore, the molecular motion will increase with a corresponding temperature increase, thus causing the gas to expand.). T = Temperature in °R (Note the absolute temperature scale) V = Volume in Ft3 Gay-Lussac's Law P1 P2 = T1 T2

OR

PT 1 2 = P2T1

The pressure of a fixed mass and fixed volume of a gas is directly proportional to the gas's temperature. T = Temperature in °R P = Pressure in PSIA

REVIEW MATERIAL – EQUATIONS/LAWS

29

Ideal Gas Law (for compressible fluids) PV = RT

R = Gas Constant (Value = 1544/MW) P = Pressure in PSIA V = Volume in Ft3 T = Temperature in °R MW = Molecular Weight Pascal’s Law Pascal’s Law states that a change in the pressure of an enclosed incompressible fluid is conveyed undiminished to every part of the fluid and to the surfaces of its container. Note: this is the principle used in the pressure factor table (Table 2-16) to convert between pressure and inches-WC, mmHg, etc.

ΔP = ρg (Δh) ΔP = Hydrostatic pressure ρ = Mass density g= Gravitation constant Δh = Difference in elevation between the two points within the fluid column Bernoulli’s Equation The Bernoulli equation states that as the speed of a moving fluid increases, the pressure within the fluid decreases: PV PV 1 1 = 2 2 T1 T2

P + ½ ρv2 + ρgh = Constant P = Pressure in PSIA ρ = Mass Density g = Gravitation constant h = Height above reference level v = Velocity This form of the Bernoulli equation ignores viscous effects. If the flow rate is high, or the flowing material has a very low viscosity, Bernoulli’s equation should not be used. For example, liquid flowing in a pipe may be more accurately described with Poiseuille’s equation14 to account for flow rate, viscosity and pipe diameter. Poiseuille’s equation states:

14. Also known as the Hagen-Poiseuille Law that states that the volume flow of an incompressible fluid through a circular tube is equal to π/8 times the pressure differences between the ends of the tube, times the fourth power of the tube's radius divided by the product of the tube's length and the dynamic viscosity of the fluid.

30


Q=

ΔPπ r 4 8μ

Q = Volumetric flow rate, in3/sec μ  = Tube length, inches r = Tube radius, inches μ = Viscosity, lb•sec/in2 Volumetric Flow Rate Q = AV

Q(gpm) = 3.12 A(sq in) x V(ft/sec)

Q = Volumetric Flow Rate A = Cross Sectional Area of the Pipe V = Velocity of the Fluid Darcy’s Formula (general formula for pressure drop) h=

fLV 2 2Dg

h = Pressure drop in feet of fluid L = Length of pipe (feet) V = Velocity of the fluid (ft/sec) g = Acceleration of gravity (32.2 ft/sec2) D = Pipe ID (feet) f = The Darcy-Weisbach friction factor f = 16 ÷ Re (Re = Reynolds Number) Velocity of Exiting Fluid

h A

V = 2gh

Q = A 2gh

V = Velocity of the fluid (ft/sec) g = Gravitation constant (32.2 ft/sec2) h = Height above reference level (in feet) A = Area of opening (in sq ft) (the smaller the area, the greater the fluid velocity) Q = Volumetric flow rate (ft3/sec) Convert Actual Cubic Feet per Minute (ACFM) to Standard Cubic Feet per Minute (SCFM)  14.7 Ta  × ACFM = SCFM   519.67   Pa


equivalent to

P1V1 P2V2 = T1 T2

31

Pa = Actual pressure (PSIA) Ts = Standard temperature (519.67°R) NOTE: °R =60°F+459.67 (convert from °F to °R) Ta = Actual temperature (°R) Joule-Thomson (Kelvin) Effect/Coefficient When the pressure of a non-ideal (real) gas changes from high to low (such as through a valve), a change of temperature occurs, proportional to the pressure difference across the restriction. The Joule-Thomson coefficient (μJT) is the change of temperature per unit change of pressure. The rate of change of temperature T with respect to pressure P in a Joule-Thomson process (that is, at constant enthalpy H) is the Joule-Thomson (Kelvin) coefficient. This coefficient can be expressed in terms of the gas's volume V, its heat capacity at constant pressure Cp, and its coefficient of thermal expansion α as:  ∂T 

V

μJT ≡  (αT − 1)  =  ∂P H CP

Wh

Where: V = Volume of gas Cp = The gas’ heat capacity at constant pressure α = The gas’ coefficient of thermal expansion H = Enthalpy constant ∂ T = Rate of change of temperature ∂ P = Rate of change of pressure The value of μJT is typically expressed in °C/bar (SI units: °K/Pa) Table 2-18 defines when the Joule-Thomson effect cools or warms a real gas: Table 2-18. Joule-Thomson Effect sign of ∂ P

µJT is:

Gas Temperature

sign of ∂ T

∴The gas

< Inversion Temperature (1)

Positive

Negative (2)

Negative

COOLS

> Inversion Temperature (1)

Negative

Negative (2)

Positive

WARMS

(1) Inversion Temperature: The critical temperature below which a non-ideal (real) gas that is expanded (with a constant enthalpy) will experience a temperature decrease. (2) When a gas expands the pressure is always lower; therefore ∂ P is always negative.

Mass Flow – Gas Equations Substitute Q for V/t: w=

32

m M V  p  = 3    t 10 R  t   T 

Substitute for Q: w=

MQ  p  103 R  T 

Q = k D;k =

Mk f 103 R


p

Simplified: w = k D   T  w = Mass flow rate (kg/sec) Q = Volume flow rate (m3/sec) p = Absolute pressure (pascal) T = Absolute temperature (Kelvin) M = MW (g/mol) R = Universal gas constant = 8.314 J ÷ (°K x mol) D = Flowmeter D/P (pascal) k = Mass flow proportionality constant kf = Flowmeter proportionality constant M = ρ AV M = Mass flow rate (lbs/sec) A = Cross sectional area (ft2) ρ = Fluid density (lbs/ft3) V = Velocity (ft/sec) Density will vary in inverse proportion to temperature, and in direct proportion to pressure. Surface Area Formulas •

Sphere: 4π r 2

•

Right Circular Cone: π r 2 + π rs

•

Right Circular Cylinder: 2π rh + 2π r 2

•

Pyramid: Area of Base + Area of the (4) Triangular Sides

Volume Formulas •

Sphere: 4 3 πr 3

•

Right Circular Cone: 1 2 πr h 3

•

Right Circular Cylinder: π r 2h

•

Pyramid: 1 A • h (A = Area of base) 3


33