A
Accelerating systems
choked flow, 246
in constant area duct, 537–543
Aerocyclones, 369
separation efficiency, 372–375
velocity distribution, 370–371
Air flow
other pipe, 515
through Schedule 40 steel pipe, 512–514
Angular momentum, 119–120, 217, 219
Annular flow, 183
API gravity, 66
Apparent viscosity, 51, 56, 60
Archimedes number, 339, 408, 414–415
B
Barnea–Mizrahi
correlation, 416
Batch flux curve, see Thickening
Batch settling test, 418
Bathwater rule, 25
Bernoulli equation, 109, 111–113, 116–117, 407
Best efficiency point (BEP), see Efficiency; Pumps
Bethe–Zel’dovich–Thompson (BZT) fluids, 256
all flow regimes, 151
friction factor, 153
turbulent pipe flow, 153
unknown diameter, 163–165, 201
unknown driving force, 155–157, 198
Blake–Kozeny equation, 386–387
Boundary layer, 11, 142, 144, 336–337
Buckingham Π theorem, 25
Buckingham–Reiner equation, 152–153
Bulk modulus, 241
Buoyancy, 88
Burke–Plummer equation, 386
C
Capital cost of pipe (CCP), 185–186
Capital cost of pump stations (CCPS), 185–187
Carreau fluids
Carreau–Yasuda model, 64
Cavitation, see Pumps
correction factor, 365
disk-bowl centrifuge, 366
immiscible liquids separation, 366–369
radial acceleration, 363
radial terminal velocity, 364
Centrifuge, see Centrifugal separation
Choked flow, see Compressible flows
Choking, see Two-phase flow
Clarifier, 416
Clausius–Clapeyron equation, 443
Collection efficiency, see Efficiency
gas properties
generalized gas flow expressions, ideal gas
high-speed ideal gas problems, 259–260
HDI method, 320
ideal adiabatic flow, 242
ideal nozzle, 249
adiabatic flow, 244
friction factor, 242
frictionless adiabatic flow, 248–250
general fluid properties, 250
mass flow rate and mass flux, 241
Reynolds number, 242
Compressors
density, 228
efficiency, 230
energy balance equation, 228
high-pressure pump, 226
isentropic compression, 228–229
isothermal compression, 228
work of compression, 228
kinetic energy per unit mass, 22
law of conservation of dimensions, 21
numerical values, 22
principles, 22
convention for heat, 104
convention for work, 104
energy balance, 105
heat balance, 105
irreversible effects
constant density, 110
energy dissipation, 109
enthalpy, 109
head form of Bernoulli equation, 110
kinetic energy correction, 110–113
mechanical energy balance, 109
thermodynamics, 108
mechanical energy, 105
pressure energy, 104
shaft work, 104
thermal energy, 105
Conservation of linear momentum
conservation of angular momentum, 119–120
macroscopic conservation, 114
macroscopic momentum balance, 113
microscopic momentum balance, 122–125
moving boundary systems and relative motion, 120–121
one-dimensional flow in a tube, 114–116
Conservation of mass
macroscopic mass balance, 101–103
microscopic mass balance, 103–104
of angular momentum, 122
conservation law, 101
conservation of energy
convention for heat, 104
convention for work, 104
energy balance, 105
heat balance, 105
mechanical energy, 105
pressure energy, 104
shaft work, 104
thermal energy, 105
conservation of linear momentum
conservation of angular momentum, 119–120
macroscopic conservation, 114
macroscopic momentum balance, 113
microscopic momentum balance, 122–125
moving boundary systems and relative motion, 120–121
one-dimensional flow in a tube, 114–116
conservation of mass
macroscopic mass balance, 101–103
microscopic mass balance, 103–104
of dimensions, 103
of dollars, 1
of energy
compressible flow, 251
in pipe flow, 251
of mass
of momentum
principles, 101
system with inputs and outputs, 101–102
Constant density fluids
isochoric fluid, 82
cavitation and flashing liquids
definition, 311
liquid choking, 311
liquid critical pressure ratio, 311–313
pressure differential ratio factor, 313–314
characteristics
equal percentage, 302
globe-style valve bodies, 301–302
guidelines for applications, 303–304
linear flow, 302
modified parabolic, 303
trim cages, 302
compressible fluids
choked flow, 318
equation constant, incompressible fluids, 305–306
expansion factor, 318
flow coefficient, 307, 310, 319
flow control piping system, 300–301
universal gas sizing equation, 305
valve coefficients
liquid pressure recovery factor, 308–309
piping geometry factor, 308
valve style modifier, 309
valve–system interaction, 323–325
as variable resistance, 300
viscosity correction
unknown flow rate, 314
unknown pressure drop, 314–315
unknown valve size, 314
Control volume, 101
Coriolis flow meter, 284
Cost
Couette viscometer, see Cup-and-bob viscometer
Crane method, 193
Creeping flow, 333
average shear rate, 53
shear stress, 53
Taylor vortices, 55
Cycle time, see Filtration
Cyclone, see Cyclone separations
advantages, 370
disadvantages, 370
Cylinder drag, 336
Cylindrical coordinates
continuity equation, 497
momentum equation, 498
stress tensor components, 500
D
Dallavalle equation, 336, 338, 340, 362, 415
Darcy friction factor, 30
Diffusivity, 7
Dimensional analysis
accuracy and precision
definitions, 34
implied precision, 35
relative uncertainty, 36
reproducibility, 34
advantages, 24
bathwater rule, 25
Buckingham Π theorem, 25
dependent variable, 29
dimensionless variables, 28–29
dimensions
definition, 17
scientific and engineering systems, 18
Euler number, 30
friction factor, 30
pressure gradient, 30
problem solution, 29
Reynolds number, 30
uniqueness, 28
Dimensionless groups, 24, 34–35
Dimensions
engineering, 20
Dissipation of energy, 140
Dodge and Metzner equation, 150
Drag coefficient (CD)
boundary layer effects, 336–337
definition, 333
nonspherical particles, 343
Reynolds numbers, 335
spherical drops and bubbles, 349
Drops and bubbles, 349
E
Economic pipe diameter
best efficiency point (BEP), 219–220, 225
collection efficiency, 375
grade efficiency, 372, 374–375
Einstein equation, 431
Electromagnetic flow meter, 283
Ellis model, 63
Energy, see Conservation of energy
Energy balance, 138
Energy cost to power pumps (EC), 185, 187
Energy dissipation, 109, 140, 146, 433
Equivalent L/D method, 191–193
Ergun equation, 386
Euler turbine equation, 120
Expansion factor, see Compressible flows
External flows
correction factors
drops and bubbles, 349
particle shape effects, 341–349
drag coefficient
boundary layer effects, 336–337
cylinder drag, 336
definition, 333
Reynolds numbers, 35
falling particles
unknown diameter, 339
unknown viscosity, 340
non-Newtonian fluids
wall effects, 354
F
Falling particles
unknown diameter, 339
unknown viscosity, 340
Fanning friction factor, 12, 30, 141, 149
Fanno flow, 251
Fick’s law of diffusion, 7
Film flow, 182
constant flow, 396
plate-and-frame, 397
Fittings, see Valves and fittings
Flooding point, 390
Flow inclination number, 446
Flow index, 60
Flow measurement
control valves, 267
noninvasive techniques
Coriolis flow meter, 284
ultrasonic flow meter, 283–284
vortex-shedding flow meter, 282
orifice meter
concentric, eccentric and segmental orifice designs, 280
friction loss coefficient, 279–280
mass flow rate, 274
obstruction meter, 271
pressure tap locations, 273, 276
square-edged orifice, 273, 275
vena contracta, 272
pitot tube
conduit cross section, 267
stagnation point, 268
total flow rate, 268
volumetric flow rate, 267
venturi and nozzle meters
applicable range and accuracy, 271, 273
ASME flow, 269
discharge coefficient parameters, 271–272
discharge coefficient vs. Reynolds number, 270–271
obstruction meters, 269
unrecovered (friction) loss, 271, 274
Flow meters
Coriolis, 284
Doppler, 284
electromagnetic, 283
Flow work, 104
Fluid flows
baby and bathwater principle, 2
phenomenological laws
Fick’s law of diffusion, 7
Fourier’s law of heat conduction, 6
Newton’s law of viscosity, 7–10
Ohm’s law of electrical conductivity, 7
rate of transport, 4
transport coefficient, 5
turbulent flux, 5
generalized dimensionless correlation, 415–416
minimum bed voidage, 409
minimum fluidization velocity, 408
nonspherical particles, 409–410
schematic representation, 407–408
classification of materials
apparent viscosity, 51
inviscid (Pascalian) fluid, 50
linear elastic (Hookean) solid, 50
nonlinear elastic solid, 50
non-Newtonian fluid, 51
rheological properties, 49
shear modulus, 50
shear strain, 49
shear stress, 49
density, 66
fluid viscous (rheological) properties
cup and bob (Couette) viscometer, 52–55
simple shear test, 51
tube flow (Poiseuille) viscometer, 55–57
non-Newtonian fluid behavior
Newtonian fluid, 59
shear stress vs. shear rate, 57–58
structural viscosity models, 62–64
viscosity vs. shear rate, 57–59
pressure–enthalpy diagrams
carbon dioxide, 492
ethane, 494
ethylene, 495
nitrogen, 491
oxygen, 490
surface tension, 67
temperature dependence of viscosity, 64–65
Fluid–solid separations, 361
Fluid–solid two-phase pipe flows
approximate slurry flow regimes, 430
heterogeneous liquid–solid flows dimensionless groups, 433
dimensionless solids, 434
drag coefficient, 433
minimum deposit velocity, 431
pressure gradient and flow regimes, 431–432, 435
state diagram, suspension transport, 433–434
pneumatic solids transport
dense phase and dilute phase, 436
hydraulic conveying system, 435–436
pressure gradient–velocity relation, 436
saltation velocity, 437
vertical transport, 439
pseudohomogeneous flows, 430–431
Fluid statics
arbitrary region within fluid, 81
basic equation, 82
buoyancy, 88
constant density fluids
isochoric fluid, 82
ideal gas law
isothermal, 84
momentum balance, 81
moving systems
horizontally accelerating free surface, 86–87
vertical acceleration, 86
standard atmosphere, 85
static forces on solid boundaries, 88–89
Fluid viscous (rheological) properties
cup and bob (Couette) viscometer
average shear rate, 53
shear stress, 53
Taylor vortices, 55
simple shear test, 51
tube flow (Poiseuille) viscometer
momentum (force) balance, 55
pressure drop, 55
Free settling fluid–solid separations
centrifugal separation, 363–369
definition, 361
Fanning
homogeneous gas–liquid flow, 12, 30, 116, 166
inclined plane, 182
pipe flow, 116
pneumatic solids transport, 435–439
porous medium, 385
Friction loss; see also Energy dissipation; Loss coefficient
non-Newtonian fluids, 194, 196
rough tubes, 146
smooth pipe, 145
Froude number, 35
Fruit salad law, 1–2, 21–22; see also Conservation of dimensions
Fully turbulent friction factor, 153, 206, 243, 294
G
Galileo number, 339
Gas–liquid two-phase pipe flow
flow regimes
fluid property correction factors, 441
horizontal and vertical flow, 440–441
mechanistic models, 442
pressure gradient, 442
homogeneous gas–liquid models
acceleration, 442
average specific volume, 443
definition, 442
homogeneous direct integration method, 447
for ideal gas, 443
maximum (choked) mass flux, 444
no-slip/equilibrium density, 444
Reynolds number, 444
total pressure gradient equation, 443
two-phase multiplier method, 445
separated flow models, 447–449
slip and holdup
acceleration term, 452
coefficients, 451
empirical equation, 451
general correlation for slip, 450
Hughmark correlation, 451
pressure drop, 452
Gas pressure-adjustment coefficient, 316
Generalized gas flow expressions, ideal gas
governing equations
BZT fluids, 256
continuity, 251
dimensionless equations, 253
dimensionless thermal expansion coefficient, 252
energy, 251
impulse function, 254
polytropic exponent, 254
high-speed ideal gas problems, 259–260
Generalized viscosity plot, 483
Grade efficiency, 372; see also Efficiency
Gravitational systems, 19
Gravity settling, 361–363, 410
H
Hagen–Poiseuille equation, 141, 146, 150, 181
Hedstrom number, 35, 153, 157, 190
Hindered settling, 361–362, 410–411
Holdup, see Two-phase flow
Homogeneous direct integration (HDI) method, 296–297, 318–320, 447
Homogeneous gas–liquid flow, see Two-phase flow
Hookean solid, 50
Hydraulic diameter, 115, 184–185, 384–385
I
Ideal gas
generalized gas flow expressions
high-speed ideal gas problems, 259–260
homogeneous gas–liquid models, 443
isothermal, 84
SRV, 295
Immiscible liquids, 282, 366–369
Impeller, see Pumps
Inclined plane, 123
friction factor, 182
Internal flow applications
economical diameter
TC, 187
total system cost, 185
friction loss in valves and fittings
Crane method, 193
energy loss per unit mass of fluid, 191
equivalent L/D method, 191–193
loss coefficient, 191
3-K (Darby) method, 191, 193–196
2-K (Hooper) method, 193
noncircular conduits
hydraulic diameter, 181
non-Newtonian fluids
Bingham plastic, 196
effective viscosity, 194
power law model, 196
unknown driving force, 197–198
Interstitial velocity, 383, 407, 428
exponent, steam values, 487
Isothermal, 239
compression, 228
ideal gas law, 84
Isotropic stress, 89
K
Kinematic viscosity, 8
Kinetic energy correction
kinetic energy correction factor, 110–111
sudden expansion, 112
L
Laminar flow, 137
Laplace equation, 388
Leibnitz’ rule, 165
Liquid pressure recovery factor, 308–309
Loading point, 390
Lockhart–Martinelli method, 447–448; see also Two-phase flow
Loss coefficient; see also Energy dissipation; Friction factor; Friction loss
fittings, 191
general, 116
pipe, 116
M
Mach number, 243–244, 252, 254–257, 259
Macroscopic momentum balance, 113
Material classification, 49–51
Maximum allowable working pressure (MAWP), 291
Mechanistic models, 442
Memory materials, 51
Meter model, 64
Microscopic conservation equations
continuity equation, 497
momentum equation
in cylindrical coordinates, 498
in rectangular coordinates, 497–498
in spherical coordinates, 498–499
stress tensor components
cylindrical coordinates, 500
rectangular coordinates, 499
spherical coordinates, 500
Microscopic momentum balance
components, 122
constitutive equations, 122
flow down an inclined plane, 123–125
Navier–Stokes equations, 122
non-Newtonian fluids, 123, 125
Molerus method, see Two-phase flow
Momentum; see also Conservation principles, of momentum
balance, 139
microscopic balance, 122–123, 139, 181, 333
Moody diagram, 146–149, 158, 188–189
Moving boundary systems, 120
N
Navier–Stokes equations, 122, 341
Net positive suction head (NPSH), 223–224; see also Pumps
Networks, see Pipe flow
all flow regimes, 149
Fanning friction factor, 141, 149
internal flow applications, economical diameter, 188–189
turbulent flow
boundary layer, 142
friction loss in rough pipe, 146, 148
friction loss in rough tubes, 146–147
friction loss in smooth pipe, 145
turbulent momentum flux, 142–143
unknown diameter, 153, 162–165, 188–189, 197, 199–200
unknown driving force, 154–155, 197–198
unknown flow rate, 158, 198–199
Newton’s law of viscosity
dynamic viscosity, 8
kinematic viscosity, 8
momentum flux and shear stress, 8
Newtonian and non-Newtonian fluids, 9
transport expression, 7
Noncircular conduits
Non-ideal gas, 106–108, 240, 244, 249, 252
Non-Newtonian fluids, 51
behavior
Newtonian fluid, 59
shear stress vs. shear rate, 57–58
structural viscosity models, 62–64
viscosity vs. shear rate, 57–59
fluid–solid flow, 439
internal flow applications
Bingham plastic, 196
effective viscosity, 194
power law model, 196
wall effects, 354
Non-spherical particles, 341–344, 409–410
O
Obstruction meters, 269, 271, 274, 280
Omega method, see Two-phase flow
Orifice meter
concentric, eccentric and segmental orifice designs, 280
friction loss coefficient, 279–280
mass flow rate, 274
obstruction meter, 271
pressure tap locations, 273, 276
square-edged orifice, 273, 275
vena contracta, 272
P
Packed columns
flooding, 390
friction factor, 390
Particles
in Newtonian fluid
in power law fluids, 149, 350, 353
Permeability, 386–387, 394, 398
Fick’s law of diffusion, 7
Fourier’s law of heat conduction, 6
Newton’s law of viscosity
dynamic viscosity, 8
kinematic viscosity, 8
momentum flux and shear stress, 8
Newtonian and non-Newtonian fluids, 9
transport expression, 7
Ohm’s law of electrical conductivity, 7
rate of transport, 4
transport coefficient, 5
turbulent flux, 5
Physical properties
of gases, 486
of ordinary water and common liquids
EE units, 473
of SAE oils and lubricants, 474
Pipe dimensions, 26
Pipe flow
air in schedule pipe, 164
Bernoulli equation, 154
compressible flows
adiabatic flow, 244
friction factor, 242
frictionless adiabatic flow, 248–250
general fluid properties, 250
mass flow rate and mass flux, 241
Reynolds number, 242
continuity, 140
dimensional analysis, 24–30, 34–37, 141
friction factor, 145–146, 149, 153, 166, 168
laminar flow, 137
momentum balance, 139
Newtonian fluids
Fanning friction factor, 149
non-Newtonian fluids
schedule number, 89
tube flow (Poiseuille) viscometer, 165–166
turbulent drag reduction
dimensionless Deborah number, 167–168
friction factor vs. Reynolds number, 167
friction loss, 170
generalized friction factor, 167–168
polyacrylamide solutions, 166
turbulent flow, 137
unknown driving force, 154–157
wall shear rate, 165
wall stress, 144, 150, 355, 438
water in schedule pipe, 164, 507–515
Plain-end pipe, 506
Plate-and-frame filter, 397
Pneumatic solids transport, see Two-phase flow
Poiseuille flow viscometer, 55–57, 165–166
Polyacrylamide solutions, 62, 166
Porous media
filtration
constant flow operation, 396
constant pressure operation, 395–396
plate-and-frame filter, 397
friction factor, 385
interstitial velocity, 383
multidimensional flow, 387–389
pile of solid particles concept, 383
porosity, 383
porous medium friction factor, 385
pressure drop, 386–387, 389–390
specific surface area, 384, 390
superficial velocity, 383, 385
Positive displacement pumps, 217
Power-law fluids
all flow regimes, 151
drag coefficient, 350
turbulent flow, 150
unknown driving force, 155, 198
unknown flow rate, 159–161, 199
unknown particle diameter, 352–354
unknown particle velocity, 351–353
unknown pipe diameter, 162–163, 200–201
viscosity function, 350
Prandtl mixing length, 145
Pressure–enthalpy diagrams
carbon dioxide, 492
ethane, 494
ethylene, 495
nitrogen, 491
oxygen, 490
Pseudoplastic fluid, 351
BEP (see Efficiency)
head-capacity range charts, 517–536
positive displacement, 217
required head
suction lift, 224
suction specific speed, 225–227
vapor lock and cavitation, 223
Q
Quality, see Two-phase flow
R
Rectangular coordinates
continuity equation, 497
stress tensor components, 499
Reynolds number, 30
Bingham plastic, 153
cyclones, 375
Newtonian fluid, 31, 142, 146, 150
non-circular conduits, 183–185
power law fluid, 150–151, 159, 190, 198–200, 350
swarm, 415
two-phase flow, 433, 444, 449, 451
unknown driving force, 154–155, 157, 197–198
Reynolds, Osborne, 137
Reynolds stresses, 122, 143–144, 146
Rheological properties, 29, 49, 51–57
Richardson–Zaki equation, 413
Rietema and Bradley geometries, 377
Roughness, 188
equivalent wall roughness, 148–149
S
Safety relief valves (SRV)
discharge coefficient, 298–300
fluid models
homogeneous direct integration method, 296–297
ideal gases, 295
incompressible/isochoric fluids, 294–295
nonequilibrium (flashing) flows, 297–298
sizing
fluid property model, 293
spring-loaded safety valve, 291–292
Scaling, 22
Schedule number, see Pipe flow
Schedule 40 steel pipe
Sedimentation; see also Fluidization
Dallavalle equation, 415
definition, 410
generalized dimensionless correlation, 415–416
single-particle terminal velocity, 414
suspension velocity, 414
Shear modulus, 50
Shear rate, see Fluid properties
Shear stress
tensor, 122
two-phase flow, 442
Shear-thickening, 60
Similitude, 17
Specific speed, see Pumps
Speed of sound, 240–241, 243, 255, 283, 318, 444, 446
Spherical coordinates
continuity equation, 497
stress tensor components, 500
Sphericity factor (ψ), 385, 409
Standard atmosphere, 85
Standard steel pipe dimensions and capacities, 501–505
Statics
Stokes drag, 336
Stokes flow, 333–334, 338, 340, 349, 355–356, 362, 365, 368, 373
Stress, tensor, 79–81, 122, 499–500
Suction lift, see Pumps
Suction specific speed, see Pumps
Superficial velocity, 383, 385, 407–408, 415, 428
Surface tension, 26, 28–29, 67
T
Temperature dependence
density, 66
Tensor
anisotropic, 80
Terminal velocity, 338–339, 341, 344–349, 363, 407, 409, 412, 415
Thermal diffusion coefficient, 6
batch settling curve, 418
batch settling test, 418
definition, 416
mass ratio of solids to fluid (R), 417
middle zone, 416
solids volume fraction (φ), 417
stable steady-state operation, 418, 420
thickener area, 420
underflow rate (Qu), 420
3-K method, see Darby 3-K method
Toms effect, 166
Total annual cost (TC), 187
Transport coefficient
of heat, 11
turbulent models, 12
Transport laws, see Phenomenological laws
Tube flow
momentum balance, 141
Tube flow (Poiseuille) viscometer, 165–166
momentum (force) balance, 55
pressure drop, 55
Turbulent flows, 137
Bingham plastics, 153
Newtonian fluids, 8, 29, 31, 51, 138, 150, 153
boundary layer, 142
friction loss in rough pipe, 146, 148
friction loss in rough tubes, 146–147
friction loss in smooth pipe, 145
Prandtl mixing length, 143–144
2-K method (see Hooper 2-K method)
Turbulent macroscopic (convective) transport models
Fanning friction factor, 12
force, 12
heat flux, 11
mass flux, 11
momentum flux, 12
2-K method, see Hooper 2-K method
choking, 318
definitions
gas–liquid (G-L) flows, 428
inlet velocity, 429
in-situ holdup, 429
mass flow rate, 428
mass flow ratio, 429
mass flux, 428
phase velocity, 428
relative (slip) velocity and slip ratio, 428–429
solid–liquid (S-L) mixtures, 427–428
volume flow rate, 428
volume flux, 428
fluid–solid two-phase pipe flows
approximate slurry flow regimes, 430
heterogeneous liquid–solid flows, 431–435
pneumatic solids transport, 435–439
pseudohomogeneous flows, 430–431
frozen flow, 443
gas–liquid two-phase pipe flow
homogeneous gas–liquid models, 442–447
separated flow models, 447–449
Lockhart–Martinelli method, 447–448
Molerus method, 435
numerical solutions, 447
pneumatic solids transport, 435–439
pseudo-single-phase fluid, 427
separated flow models, 447–449
two-phase multiplier, 445, 447–448
volume flux, 428
Typical pump head capacity range charts, 517–536
U
Ultrasonic flow meter, 283–284
Units
Unknown diameter
compressible flow, 259
falling particles, 339
Newtonian fluid, 162–165, 188, 199–200
power law fluid, 162–163, 200–201, 352–353
Unknown driving force
compressible flow, 259
Newtonian fluid, 154–155, 197–198
Unknown flow rate
compressible flow, 259
orifice, 281
V
Valves and fittings, friction loss, 191–196; see also Control valves
Valve–system interaction, 323–324
Vapor lock, 223
Venturi and nozzle meters
applicable range and accuracy, 271, 273
ASME flow, 269
discharge coefficient parameters, 271–272
discharge coefficient vs. Reynolds number, 270–271
obstruction meters, 269
unrecovered (friction) loss, 271, 274
Viscoelastic fluid, 9, 51, 350
Viscometer
tube flow (Poiseuille), 55–57, 165–166
Viscoplastic fluid, 59
Viscosity
nomograph, 480
generalized plot, 483
nomograph, 469
of water, 470
and liquid petroleum products, 464
von Karman equation, 144–146, 150
W
Wall correction factor (Kw), 340
Water flow
other pipe, 515
through Schedule 40 steel pipe, 508–511
Wetted perimeter, 115, 126, 203
Weymouth equation, 242
Y
Yasuda model, 64