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E

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

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

CONTROL S YSTEMS 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 ACKN CKNOWLE OWLEDG DGM MENTS ENTS ACRONYMS

xv xvii xvii

xix

COMM COMMON ON ELEC ELECTR TRIC ICAL AL DEFI DEFINIT NITIO IONS NS

xxii xxiiii

1. CSE CSE PE PE EXA EXAM M – GENE GENERA RAL L INF INFOR ORMA MATI TION ON 2. MISC MISCEL ELLA LANE NEOU OUS S REV REVIE IEW W MAT MATER ERIA IAL L

1

3

2.1 Mathematics 3 2.1.1 SI Pr Prefixes 3 2.1.2 Algebra 3 2.1.3 Trigonometry 6 2.1.4 Calculus 7 2.1. 2.1.66 Diff Differ eren enti tial al Equ quat atio ions ns 11 2.1.7 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. 2.2.22 He Heat at Ad Addi diti tion on and and Te Temp mper erat atur uree 15 2.2.3 2.3 Mollier SStteam Di Diagr agram 15 2.2.4 2.4 Psychro hrometric Ch Chart 15 2.2.5 2.2.5 Proper Propertie tiess of Water Water 17 2.2. 2.2.66 Pr Prop oper erti ties es of Satu Satura rate ted d Stea Steam m 18 2.2. 2.2.77 Prope ropert rtie iess of of Su Supe perh rhea eate ted d Ste Steam am 20 2.3 Statistics 22 2.4 2.4 Bool Boolea ean n Logi Logicc Ope pera rati tion onss 24 2.5 Conversion Fa Factors 26 2.6 2.6 Eq Equ uatio ations ns/L /Law awss/F /For orm mulas ulas 29 3. MEASUREMENT

35

Topic Hi Highlights 35 3.1 3.1 Te Temp mper erat atur uree Me Meas asur urem emen entt Se Sens nsor orss 35

V

3.1.1 3.1.2 3.1.3 3.1.4 3.1. 3.1.55 3.1.6

Thermocouple (T/C) 35 Resistanc Resistancee Temperatur Temperaturee Detector Detector (RTD) (RTD) 38 Thermistor 41 Temperature Switch 42 Temp Te mper erat atur uree Indic Indicat ator or (The (Therm rmom omet eter er)) 42 Thermowell 42

3.2 3.2 Press ressu ure Measu easure rem ment ent Sen Senso sors rs 45 3.2.1 Manometer 45 3.2.2 Bourdon Tube 46 3.2.3 2.3 Pressure Diaphragm 46 3.2. 3.2.44 Pr Pres essu sure re Tra Trans nsdu duce cer/ r/Tr Tran ansm smit itte terr 47 3.2.5 Diaphragm Se Seal 47 3.2.6 Pressure Pressure Sensor Sensor Installatio Installation n Details Details 47 3.3 3.3 Volu Volume metr tric ic Flow Flow Me Meas asur urem emen entt Sen Senso sors rs 49 3.3. 3.3.11 Se Sens nsor orss Based Based on Dif Diffe fere rent ntia iall Pres Pressu sure re (D/P (D/P Pro Produ duce cers rs)) 50 3.3. 3.3.22 Elec Electr tron onic ic Volu Volume metr tric ic Flow Flowme mete ters rs 62 3.3.3 3.3.3 Mass Mass Flowme Flowmeter terss 65 3.3. 3.3.44 Me Mech chan anic ical al Vol Volum umet etri ricc Flo Flowm wmet eter erss 67 3.3. 3.3.55 Op Open en Chan Channe nell Volu Volume metr tric ic Flow Flow Me Meas asur urem emen entt 68 3.3. 3.3.66 Flow lowmete meterr Sel Selec ecti tion on Gui Guide 71 3.4 3.4 Lev Level Measu easure reme ment nt Se Sens nso ors 72 3.4. 3.4.11 Infe Infere rent ntia iall Le Leve vell Me Meas asur urem emen entt Te Tech chni niqu ques es 72 3.4. 3.4.22 Visu Visual al Lev Level el Mea Measu sure reme ment nt Tec Techn hniq ique uess 78 3.4. 3.4.33 Elec Electr tric ical al Prop Proper erti ties es Leve Levell Measu Measure reme ment nt Tech Techni niqu ques es 80 3.4. 3.4.44 Floa Float/ t/Bu Buoy oyan ancy cy Le Leve vell Mea Measu sure reme ment nt Tec Techni hniqu ques es 82 3.4. 3.4.55 Time Time of Flig Flight ht Lev Level el Me Meas asur urem emen entt Tec Techni hniqu ques es 83 3.4. 3.4.66 Misc Miscel ella lane neou ouss Level Level Meas Measur urem emen entt (Switc (Switch) h) Tech Techni niqu ques es 85 3.5 Analytical Analytical Measurement Measurement Sensors Sensors 86 3.5. 3.5.11 Comb ombusti ustib ble Gas Gas Ana Anallyzer yzerss 87 3.5.2 Dew Point 88 3.5.3 Humidity SSeensors 89 3.5. 3.5.44 Elec Electr tric ical al Con Condu duct ctiv ivit ity y Anal Analyz yzer erss 91 3.5.5 pH/ORP An Analyzers 92 3.5. 3.5.66 Disso issollve ved d Oxy Oxyg gen Ana Anallyzer yzerss 93 3.5.7 Oxygen Co Content tent (in Ga Gas) 94 3.5.8 Turbidity An Analyzers 95 3.5. 3.5.99 TO TOC C (To (Tota tall Orga Organi nicc Car Carbo bon) n) Ana Analy lyze zers rs 97 3.5. 3.5.10 10 Ligh Lightt Wav Wavel elen engt gth h Typ Typee Ana Analy lyze zers rs 99 3.5. 3.5.11 11 Chro Chroma mato togr grap aphs hs and and Spe Spect ctro rome mete ters rs 99 3.5. 3.5.12 12 Cont Contin inuo uous us Emi Emiss ssio ion n Moni Monito tori ring ng Sys Syste tems ms (CE (CEMS MS)) 102 3.5.13 Vibrati ation An Analysis 104 104 4. SIGNA SIGNALS LS,, TRA TRANS NSMI MISS SSIO ION N AND AND NETW NETWOR ORKI KING NG

4.1 Signals/Transmission 107 4.1.1 Copper Cabling 107 4.1.2 1.2 Fiber O Opt ptiic Ca Cabling 109

VI

107 107

4.1.3 IEEE 802.11 & ISA 100.11a Wireless LAN Communication Protocols 111 4.1.4 Transducers 113 4.1.5 Intrinsic Safety (I.S.) 113 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) 148 5. FINAL CONTROL ELEMENTS

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 195 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 205 5.6.2 VFD Applications 206 5.6.3 Harmonics Associated with VFDs 206 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) 215 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 231 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

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) 240 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 246 6.4.1 Component Block Diagram 246 6.4.2 Basic Building Block 247 6.4.3 Elementary Block Diagrams 247 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 254 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) 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 300 8.2.3 Definitions 300 8.2.4 Layers of Protection 301 8.3 Determining PFD (Probability of Failure on Demand) 302 8.3.1 Basic Reliability Formulas 303 8.3.2 Architectures 304 9. CODES, STANDARDS AND REGULATIONS

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) 313 9.1.13 OSHA 313 9.2 NEC 315 9.2.1 Allowable Conduit Fill 9.2.2 Wiring Methods 315

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 335 9.4.5 Qualified Personnel 335 9.4.6 Label Requirements 336 9.5 Lightning Protection 336 9.5.1 Lightning Protection (NFPA 780; UL96 & 96A; LPI 17S; IEEE Std 487) 337 9.5.2 Facility Lightning Protection per NFPA 780 337

X

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) 358 12.3 RTD Resistance Tables 359 12.4 Thermocouple milliVolt Tables 369 12.5 Instrument Air Quality 375 12.6 Thevenin & Norton Equivalencies 376 12.6.1 Thevenin 376 12.6.2 Norton’s Theorem 380 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) 388 13.2.3 Step-Wave 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

−b ± b 2 − 4ac r1, r2 = a( x − r1 )( x − r2 ) = 0 2a

REVIEW MATERIAL – MATHEMATICS

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. -

This is an indication of an over-damped system.

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

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 x a =

( x ) a

b x a

b

a

x

= x ab

= a xb =

( ) a

x

b

Logarithms

If bx = y, then x=log by. Example: If y = 10 x , then x = log10 y OR If ex = y, then x = loge y = ln y log100 = log10 2 log 0.01 = 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           −1 2 0  0 −4 3   −1 − 0 2 − ( −4) 0 − 3   −1 6 −3   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 =  2 y − 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)  = 2 1 × − − 0 3 −1        0 4  (0 × 0) + (3 × ( −2)) + (( −1) × 0) (0 × 3) + (3 × ( −1)) + (( −1) × 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    5 2

w Let A −1 =   x

If A = 

y 



z 

8w + 3x 8y + 3z   1 0   = 0 1 5 w 2 x 5 y 2 z + +    

Then AA −1 = 

The resulting equations

8w + 3 x = 1 8 y + 3z = 0 5w + 2 x = 0

5 y + 2z = 1

Have the solution w=2; x=-5; y=-3; z=8

2

−3 

 −5

8

∴ A−1 = 



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  = 57.3 1⋅ radian =    π 

csc-1 1-sin

B u s e n e t p o H y

c

A

θ

b sinθ =

6

Adjacent

opposite a = hypotenuse c

e t i s o p p O

csc

c o t

sin

t a n

sin θ

1-cos

1-sec

cos

a

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 sinA = = adjacent b cos A

cot θ =

adjacent b cos A = = opposite a sin A

Trigonometric Identities sin2 x + cos2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = csc2 x sin2x = 2sinx • cosx 2tan x 1 − tan2 x cos 2 x = 2cos 2 x − 1 = cos2 x − sin2 x = 1 − 2sin2 x tan2 x =

sin( x + y ) = sin x cos y + cos x sin y 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’(x 0 )= Slope of the tangent line x 0

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 change in x

=

∆y ∆ x

The idea is to compute the rate of change as the limiting value of the ratio of the differences ∆y/∆x as ∆x 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 '   = g 2  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 =

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

y ' = 0( x + 2) − 1(4) 2

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

at (2,1); y ' =

−4 4 1 =− =− 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

1 d ln( x ) = dx x

d csc x = − csc x cot x dx d sec x = sec x tan x dx d cot x = − csc2 x dx

Inverse Trigonometric 1 −1 d d sin−1 x = csc −1 x = dx dx x 2 − 1 x x 2 − 1 1 −1 d d cos−1 x = sec −1 x = dx dx 1 − x 2 x x 2 − 1 1 −1 d d tan−1 x = cot −1 x = 2 1+ x 1+ x2 dx dx 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−a



b

a

udx

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   = v 2 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  kdx = kx + C  sin xdx = − cos x + C  sec xdx = tan x + C  sec x tan xdx = sec x + C 2

  tan xdx = − ln cos x + C  sec xdx = ln sec x + tan x + C e xdx = e x + C

dx

  x

a2 − x 2 dx

= sin−1

x 2 − a2

=

x +C a

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

  cos xdx = sin x + C  csc dx = − cot x + C  csc x cot xdx = − csc x + C n

2

dx = ln x + C x

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

2

dx 1 x = tan−1 + C 2 +x a a

1 x sec−1 + C a a

Examples:

 −8dx = −8x + C  3 x dx = x + C 2



10

3

(6 x 2 + 5 x − 3)dx =

6 x 3 5 x 2 + − 3 x + 2 = 2 x 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 (t)

(t)]

1

1 s

e at

1 s −a

sin at

a s2 + a2

cos at

s s2 + a2

t sin at

2as

( s 2 + a2 ) t cos at

s 2 − a2

(s eat sin bt

2

2

+ a2 )

2

b 2

(s − a) + b2 e at cos bt

s −a 2

(s − a) + b2 t n n! at

e ×

; n ∈ N t n n!

; n ∈ N

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.


Cutoff Frequency ) B d ( e d u t i n g a M

Slope: –20 dB/decade

Passband

) g e d ( e s a h P

Passband

Stopband

Stopband

Frequency (rad/sec)

Figure 2-3. Bode Plot Imaginary G

(contour) ΓF (s)

F(S) Plane Real G

Figure 2-4. Nyquist Plot

• A feedback control system is stable if, and only if, the contour Γ F(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. ∆U = 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); when a system absorbs heat from its surroundings, Q is indicated by a positive value (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 (∆T = 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 ( ∆Q=mc∆T). 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 diagram 8 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)

V o S p lu e c i m f ic e 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/ft

Weight lb/gal

3

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   ( d2 − d1 ) g g −  2 1

d = d1 + 

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   (1045.43 − 1543.2 ) = − 550 600  

∴ d = 1045.3 + 

1045.3 + ( −0.5 ) ( −497.77) = 1045.3 + 248.89 = 1294.19ps ia

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 Steam 10 Specific Volume V Pressure psig

Saturated Temperature ºF

0

3

ft /lb

Heat Content BTU/lb

Latent Heat of Vaporization (hfg) BTU/lb

Saturated Liquid (Vf)

Saturated Vapor (Vg)

Saturated Liquid (hf)

Saturated Vapor (hg)

212

0.0167

26.8

180

1150

970

1

215

0.0167

24.3

183

1151

967

2

218

0.0167

23.0

186

1153

965

3

222

0.0168

21.8

190

1154

963

4

224

0.0168

20.7

193

1155

961

5

227

0.0168

19.8

195

1156

959

6

230

0.0168

18.9

198

1157

958

7

232

0.0169

18.1

200

1158

956

8

235

0.0169

17.4

203

1158

955

9

237

0.0169

16.7

205

1159

953

10

239

0.0169

16.1

208

1160

952

11

242

0.0169

15.6

210

1161

950

12

244

0.0170

15.0

212

1161

949

13

246

0.0170

14.5

214

1162

947

14

248

0.0170

14.0

216

1163

946

15

250

0.0170

13.6

218

1164

945

16

252

0.0170

13.2

220

1164

943

17

254

0.0170

12.8

222

1165

942

18

255

0.0170

12.5

224

1165

941

19

257

0.0171

12.1

226

1166

940

20

259

0.0171

11.1

227

1166

939

25

267

0.0171

10.4

236

1169

933

30

274

0.0172

9.4

243

1171

926

35

281

0.0173

8.5

250

1173

923

40

287

0.0173

7.74

256

1175

919

45

292

0.0174

7.14

262

1177

914

50

298

0.0174

6.62

267

1178

911

55

302

0.0175

6.17

272

1179

907

60

307

0.0175

5.79

277

1181

903

65

312

0.0176

5.45

282

1182

900

70

316

0.0176

5.14

286

1183

897

75

320

0.0176

4.87

290

1184

893

80

324

0.0177

4.64

294

1185

890

85

327

0.0177

4.42

298

1186

888

90

331

0.0178

4.24

301

1189

887

95

334

0.0178

4.03

305

1189

884

100 105

338 341

0.0178 0.0179

3.88 3.72

308 312

1190 1191

882 877

110

343

0.0179

3.62

314

1191

877

115

347

0.0180

3.44

318

1192

872

120

350

0.0180

3.34

321

1193

872


19

125

353

0.0180

3.21

324

1193

867

130

355

0.0180

3.12

327

1194

867

135

358

0.0181

3.02

329

1194

864

140

361

0.0181

2.92

332

1195

862

145

363

0.0181

2.84

335

1196

860

150

366

0.0182

2.75

337

1196

858

155

368

0.0182

2.67

340

1196

854

160

370

0.0182

2.60

342

1196

854

165

373

0.0183

2.53

345

1197

852

170

375

0.0183

2.47

347

1197

850

175

378

0.0183

2.40

350

1198

848

180 185

380 382

0.0184 0.0184

2.34 2.29

352 355

1198 1199

846 844

190

384

0.0184

2.23

357

1199

842

200

388

0.0185

2.14

361

1199

838

210

392

0.0185

2.05

365

1200

835

220

396

0.0186

1.96

369

1200

831

230

399

0.0186

1.88

373

1201

828

240

403

0.0187

1.81

377

1201

824

250

406

0.0187

1.75

380

1201

821

260

410

0.0188

1.68

384

1201

817

270

413

0.0188

1.63

387

1202

814

280

416

0.0189

1.57

391

1202

811

290

419

0.0190

1.52

394

1202

807

300

421

0.0190

1.47

397

1202

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 Steam 12 Pressure

Sat. Temp psia psig . 15

0.3

Total Temperature ºF 350

400

500

600

700

800

900

1000

1100

1300

1500

213.03 V 31.939 33.963 37.985 41.986 45.978 49.964 53.946 57.926 61.905 69.858 77.807 hg 1216.2 1239.9 1287.3 1335.2 1383.8 1433.2 1483.4 1534.5 1586.5 1693.2 1803.4

20

5.3

227.96 V 23.900 25.428 28.457 31.466 34.465 37.458 40.447 43.435 46.420 52.388 58.352 hg 1215.4 1239.2 1286.9 1334.9 1383.5 1432.9 1483.2 1534.3 1586.3 1693.1 1803.3

30

15.3 250.34 V 15.859 16.892 18.929 20.945 22.951 24.952 26.949 28.943 30.936 34.918 38.896 hg 1213.6 1237.8 1286.0 1334.2 1383.0 1432.5 1482.8 1534.0 1586.1 1692.9 1803.2

40

25.3 267.25 V 11.838 12.624 14.165 15.685 17.195 18.699 20.199 21.697 23.194 26.183 29.168 hg 1211.7 1236.4 1285.0 1333.6 1382.5 1432.1 1482.5 1533.7 1585.8 1692.7 1803.0

50

35.3 281.02 V

9.424 10.062 11.306 12.529 13.741 14.947 16.150 17.350 18.549 20.942 23.332

hg 1209.9 1234.9 1284.1 1332.9 1382.0 1431.7 1482.2 1533.4 1585.6 1692.5 1802.9 60

45.3 292.71 V

7.815

8.354

9.400

10.425 11.438 12.446 13.450 14.452 15.452 17.448 19.441

hg 1208.0 1233.5 1283.2 1332.3 1381.5 1431.3 1481.8 1533.2 1585.3 1692.4 1802.8 70

55.3 302.93 V

6.664 7.133

8.039

8.922

9.793 10.659 11.522 12.382 13.240 14.952 16.661

hg 1206.0 1232.0 1282.2 1331.6 1381.0 1430.9 1481.5 1532.9 1585.1 1692.2 1802.6 80

65.3 312.04 V

5.801 6.218

7.018

7.794

8.560

9.319

10.075 10.829 11.581 13.081 14.577

90

hg 1204.0 1230.5 1281.3 1330.9 1380.5 1430.5 1481.1 1532.6 1584.9 1692.0 1802.5 75.3 320.28 V 5.128 5.505 6.223 6.917 7.600 8.277 8.950 9.621 10.290 11.625 12.956 hg 1202.0 1228.9 1280.3 1330.2 1380.0 1430.1 1480.8 1532.3 1584.6 1691.8 1802.4

100

85.3 327.82 V

4.590

4.935

5.588

6.216

6.833

7.443

8.050

8.655

9.258

10.460 11.659

hg 1199.9 1227.4 1279.3 1329.6 1379.5 1429.7 1480.4 1532.0 1584.4 1691.6 1802.2 120 105.3 341.27 V 3.7815 4.0786 4.6341 5.1637 5.6813 6.1928 6.7006 7.2060 7.7096 8.7130 9.7130 hg 1195.6 1224.1 1277.4 1328.2 1378.4 1428.8 1479.8 1531.4 1583.9 1691.3 1802.0 140 125.3 353.04 V

-

3.4661 3.9526 4.4119 4.8588 5.2995 5.7364 6.1709 6.6036 7.4652 8.3233

hg

-

1220.8 1275.3 1326.8 1377.4 1428.0 1479.1 1530.8 1583.4 1690.9 1801.7

160 145.3 363.55 V

-

3.0060 3.4413 3.8480 4.2420 4.6295 5.0132 5.3945 5.7741 6.5293 7.2811

hg

-

1217.4 1273.3 1325.4 1376.4 1427.2 1478.4 1530.3 1582.9 1690.5 1801.4

180 165.3 373.08 V

-

2.6474 3.0433 3.4093 3.7621 4.1084 4.4508 4.7907 5.1289 5.8014 6.4704

hg

-

1213.8 1271.2 1324.0 1375.3 1426.3 1477.7 1529.7 1582.4 1690.2 1801.2

200 185.3 381.80 V

-

2.3598 2.7247 3.0583 3.3783 3.6915 4.0008 4.3077 4.6128 5.2191 5.8219

hg

-

1210.1 1269.0 1322.6 1374.3 1425.5 1477.0 1529.1 1581.9 1689.8 1800.9

220 205.3 389.88 V

-

2.1240 2.4638 2.7710 3.0642 3.3504 3.6327 3.9125 4.1905 4.7426 5.2913

hg

-

1206.3 1266.9 1321.2 1373.2 1424.7 1476.3 1528.5 1581.4 1689.4 1800.6

240 225.3 397.39 V

-

1.9268 2.2462 2.5316 2.8024 3.0661 3.3259 3.5831 3.8385 4.3456 4.8492

hg

-

1202.4 1264.6 1319.7 1372.1 1423.8 1475.6 1527.9 1580.9 1689.1 1800.4

260 245.3 404.44 V

-

-

2.0619 2.3289 2.5808 2.8256 3.0663 3.3044 3.5408 4.0097 4.4750

hg

-

-

1262.4 1318.2 1371.1 1423.0 1474.9 1527.3 1580.4 1688.7 1800.1

280 265.3 411.07 V

-

-

1.9037 2.1551 2.3909 2.6194 2.8437 3.0655 3.2855 3.7217 4.1543

hg

-

-

1260.0 1316.8 1370.0 1422.1 1474.2 1526.8 1579.9 1688.4 1799.8

300 285.3 417.35 V

-

-

1.7665 2.0044 2.2263 2.4407 2.6509 2.8585 3.0643 3.4721 3.8764

hg

-

-

1257.7 1315.2 1368.9 1421.3 1473.6 1526.2 1579.4 1688.0 1799.6

320 305.3 423.31 V

-

-

1.6462 1.8725 2.0823 2.2843 2.4821 2.6774 2.8708 3.2538 3.6332

hg

-

-

1255.2 1313.7 1367.8 1420.5 1472.9 1525.6 1578.9 1687.6 1799.3


21

340 325.3 428.99 V

-

-

1.5399 1.7561 1.9552 2.1463 2.3333 2.5175 2.7000 3.0611 3.4186

hg

-

-

1252.8 1312.2 1366.7 1419.6 1472.2 1525.0 1578.4 1687.3 1799.3

360 345.3 434.41 V

-

-

1.4454 1.6525 1.8421 2.0237 2.2009 2.3755 2.5482 2.8898 3.2279

hg

-

-

1250.3 1310.6 1365.6 1418.7 1471.5 1542.4 1577.9 1686.9 1798.8

380 365.3 439.61 V

-

-

1.3606 1.5598 1.7410 1.9139 2.0825 2.2484 2.4124 2.7366 3.0572

hg

-

-

1247.7 1309.0 1364.5 1417.9 1470.8 1523.8 1577.4 1686.5 1798.5

400 385.3 444.60 V

-

-

1.2841 1.4763 1.6499 1.8151 1.9759 2.1339 2.2901 2.5987 2.9037

hg

-

-

1245.1 1307.4 1363.4 1417.0 1470.1 1523.3 1576.9 1686.2 1798.2

3

V = specific volume, ft /lb h g = total heat of steam, BTU/lb

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 ( ) 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 8

=

40 8

=5

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% 34.1%

0.1% –3σ

2.1%

2.1% 13.6% –2σ

–1σ

Mean

13.6% +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 Level

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

Density of Water

62.4 Lbs/Ft3

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

Unit Lbs Water @ 60°F

Table 2-10. Distance Factors Multiply Inch

By 2.54

To Obtain Centimeter

Centimeter

0.3937

Inch

Foot

0.3048

Meter

3.28083

Foot

Meter

Table 2-11. Volume Factors Multiply

By

To Obtain

Gallon

0.13368

Ft 3

Gallon

0.003754

M3

Gallon

3.7853

Liter

Liter

0.2642

Gallon

Liter

0.03531

Ft3

Liter

0.001

M3

Ft3

7.481

Gallon

Ff3

28.3205

Liter

Ft3

0.028317

M3

M3

35.3147

Ft3

M3

3.28083

Gallon

M3

1000

Liter

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 105

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

N/M2 or Pa

0.004018

Inches WC

N/M2 or Pa

0.00014508

PSI

* 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 Ft 3 Charles’s Law V1 T1

=

V 2 T 2

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 Ft 3 Gay-Lussac's Law P 1 T 1

=

P 2 T 2

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 Ft 3 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, mm Hg, etc.

∆ P = ρ g (∆h) ∆P = Hydrostatic pressure ρ = Mass density

g= Gravitation constant ∆h = 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 1 1 T1

=

P2V2 T 2

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


∆Pπ r 4 Q= 8 µ 

Q = Volumetric flow rate, in3/sec  = 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

ACFM = SCFM 

 P a

×

  519.67  T a


equivalent to

P 1V 1 T 1

=

P 2V 2 T 2

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 (µ JT) 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: V  ∂T  = (α T − 1)   ∂P H C P

µJT ≡ 

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 µ JT 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 µJT is:

Gas Temperature

sign of

P

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 t

=

 V  p     103 R  t  T  M

Substitute for Q: w =

MQ  p  103 R  T 

Q = k D ; k =

Mk f 103 R


Control-Systems-Engineer-Technical-Reference-Handbook-Chapter-2.pdf

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