Rules of Thumb for Mechanical Engineers. A manual of quick, accurate solutions to everyday mechanical engineering problems

410 Pages·1996·17.07 MB·English

by  J. Edward Pope (Auth.)

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1: Fluids, 1
Friction Factor and Darcy Equation ............................... 9
Losses in Pipe Fittings and Valves .................................. 10
Fluid Properties ........................................................ 2 Pipes in Series ................................................................. O1
Pipes in Parallel .............................................................. 10
Density, Specific Volume, Specific Weight,
Specific Gravity, and Pressure .................................... 2 Open-Channel Flow 11
Surface Tension .............................................................. 2 Frictionless Open-Channel Flow .................................... 11
Vapor Pressure ................................................................ 2 Laminar Open-Channel Flow ......................................... 12
Gas and Liquid Viscosity ................................................ 3 Turbulent Open-Channel Flow ....................................... 12
Bulk Modulus ................................................................. 3 Hydraulic Jump ............................................................... 12
Compressibility ............................................................... 3 Fluid Measurements ....... 13
Units and Dimensions ..................................................... 3 Pressure and Velocity Measurements ............................. 13
Fluid Statics .............................................................. 4 Flow Rate Measurement ................................................. 14
Hot-Wire and Thin-Film Anemornetry ........................... 14
Manometers and Pressure Measurements ....................... 4
Hydraulic Pressure on Surfaces ...................................... 4 Open-Channel Flow Measurements ............................... 15
Buoyancy ........................................................................ 5 Viscosity Measurements ................................................. 15
Basic Equations ........................................................ 5 Other Topics .............................................................. 16
Continuity Equation ........................................................ 5 Unsteady Flow, Surge, and Water Hammer .................... 16
Euler's Equation ............................................................. 5 Boundary Layer Concepts .............................................. 16
Bernoulli's Equation ....................................................... 6 Lift and Drag ................................................................... 16
Energy Equation ............................................................. 6 Oceanographic Flows ..................................................... 17
Momentum Equation ...................................................... 6
2: Heat Transfer, 18
Moment-of-Momentum Equation ................................... 6
Advanced Fluid Flow Concepts .............................. 7
Dimensional Analysis and Similitude ............................ 7 Introduction .............................................................. 19
Nondimensional Parameters ........................................... 7 Conduction ................................................................ 19
Equivalent Diameter and Hydraulic Radius ................... 8 Single Wall Conduction .................................................. 19
Pipe Flow ................................................................... 8 Composite Wall Conduction ........................................... 21
The Combined Heat Transfer Coefficient ....................... 22 Diesel Cycle: Another Power Cycle ............................... 63
Critical Radius of Insulation ........................................... 22 Gas Power Cycles with Regeneration ............................. 64
Convection ................................................................. 23
Dimensionless Numbers ................................................. 23
:4 Mechanical Seals, 66
Correlations ..................................................................... 24
Typical Convection Coefficient Values .......................... 26
Radiation ................................................................... 26 Basic Mechanical Seal Components ....................... 67
Emissivity ....................................................................... 27 Sealing Points ............................................................ 67
View Factors ................................................................... 27 Mechanical Seal Classifications .............................. 68
Radiation Shields ............................................................ 29 Basic Seal Designs ..................................................... 68
Finite Element Analysis ........................................... 29 Basic Seal Arrangements ......................................... 72
Boundary Conditions ...................................................... 29 Basic Design Principles ............................................ 74
2D Analysis .................................................................... 30 Materials of Construction ....................................... 77
Transient Analysis .......................................................... 30
Desirable Design Features ...................................... 79
Evaluating Results .......................................................... 31
Equipment Considerations ...................................... 80
Heat Exchanger Classification ................................ 33
Calculating Seal Chamber Pressure ....................... 81
Types of Heat Exchangers .............................................. 33
Seal Flush Ham ........................................................ 82
Shell-and-Tube Exchangers ............................................ 36
Integral Pumping Features ...................................... 85
Tube Arrangements and Baffles ..................................... 38
Seal System Heat Balance ........................................ 87
Shell Configurations ....................................................... 40
Flow Rate Calculation .............................................. 89
Miscellaneous Data ......................................................... 42
References ............................. 91
Flow Regimes and Pressure Drop in Two-Phase
Heat Transfer ........................................................ 42
Flow Regimes ................................................................. 42 :5 spmuP dna ,srosserpmoC 29
Flow Maps ...................................................................... 46
Estimating Pressure Drop ............................................... 48
Pump Fundamentals and Design ............................ 93
:3 ,scimanydomrehT 15 Pump and Head Terminology ......................................... 93
Pump Design Parameters and Formulas ......................... 93
Types of Pumps ............................................................... 94
Thermodynamic Essentials ...................................... 52 Centrifugal Pumps .......................................................... 95
Phases of a Pure Substance ............................................. 52 Net Positive Suction Head (NPSH) and Cavitation ........ 96
Thermodynamic Properties ............................................. 53 Pumping Hydrocarbons and Other Fluids ...................... 96
Determining Properties ................................................... 55 Recirculation ................................................................... 97
Types of Systems ............................................................ 56 Pumping Power and Efficiency ...................................... 97
Types of Processes .......................................................... 56 Specific Speed of Pumps ................................................ 97
The Zeroth Law of Thermodynamics ............................. 57 Pump Similitude ............................................................. 98
First Law of Thermodynamics ................................ 58~ ~ Performance Curves ........................................................ 98
Work ................................................................................ 58 Series and Parallel Pumping ........................................... 99
Heat ................................................................................. 58 Design Guidelines ........................................................... 100
First Law of Thermodynamics for Closed Systems ....... 58 Reciprocating Pumps ...................................................... 103
First Law of Thermodynamics for Open Systems .......... 58 Compressors ............................................................. 110
Second Law of Thermodynamics ............................ 59 Definitions ...................................................................... 110
Reversible Processes and Cycles .................................... 59 Performance Calculations for Reciprocating
Thermodynamic Temperature Scale ............................... 59 Compressors ............................................................... 111
Useful Expressions ......................................................... 59 Estimating Suction and Discharge Volume Bottle
Thermodynamic Cycles ........................................... 60 Sizes for Pulsation Control for Reciprocating
Basic Systems and Systems Integration ......................... 60 Compressors ............................................................... 114
Carnot Cycle ................................................................... 60 Compression Horsepower Determination ....................... 117
Rankine Cycle: A Vapor Power Cycle ............................ 61 Generalized Compressibility Factor ............................... 119
Reversed Rankine Cycle: A Vapor Refrigeration Cycle. 61 Centrifugal Compressor Performance Calculations ....... 120
Brayton Cycle: A Gas Turbine Cycle ............................. 62 Estimate HP Required to Compress Natural Gas ........... 123
Otto Cycle: A Power Cycle ............................................ 63 Estimate Engine Cooling Water Requirements .............. 124
vi
Estimate Fuel Requirements for Internal Combustion Lubricant Selection ......................................................... 162
Engines ....................................................................... 124 Lubricating Methods ....................................................... 163
References ................................................. 124 Relubrication ................................................................... 164
Cleaning, Preservation, and Storage ............................... 165
6: Drivers, 125 Mounting ................................................................... 166
Shafting ........................................................................... 166
Housings ......................................................................... 169
Motors: Efficiency .................................................... 126
Bearing Clearance ........................................................... 172
Motors: Starter Sizes ............................................... 127
Seals ................................................................................ 174
Motors: Service Factor ............ 127
Sleeve Bearings ......................................................... 175
Motors: Useful Equations ......................................... 128
References ................................................................. 177
Motors: Relative Costs ...... ....................................... 128
Motors: Overloading ................................................ 129
9: Piping and Pressure Vessels, 178
Steam Turbines: Steam Rate .................. 129
Steam Turbines: Efficiency ........................ 129
Gas Turbines: Fuel Rates ......................................... 130 Process Plant Pipe .................................................... 179
Gas Engines: Fuel Rates .......................................... 132 Definitions and Sizing .................................................... 179
Gas Expanders: Available Energy .......................... 132 Pipe Specifications .......................... . ............................... 187
Storing Pipe .................................................................... 188
7: Gears, 133 Calculations to Use ......................................................... 189
Transportation Pipe Lines 190
Steel Pipe Design ............................................................ 190
Ratios and Nomenclature ................ 134
Gas Pipe Lines ............................................................ 190
Spur and Helical Gear Design ................................. 134
Liquid Pipe Lines ........................................................ 192
Bevel Gear Design .................................................... 139 Pipe Line Condition Monitoring ............................. 195
Cylindrical Worm Gear Design ............. 141 Pig-based Monitoring Systems ....................................... 195
Materials .................................................................. 142 Coupons .......................................................................... 196
Summary of Gear Types .......................................... 143 Manual Investigation ...................................................... 196
Buying Gears and Gear Drives .......... 144 Cathodic Protection ........................................................ 197
References ...................................... 144 Pressure Vessels ........................................................ 206
Stress Analysis ................................................................ 206
8: Bearings, 145 Failures in Pressure Vessels ............................................ 207
Loadings ......................................................................... 208
Stress ............................................................................... 209
Types of Bearings ..................................................... 146
Procedure "1 General Vessel Formulas ........................... 213
Ball Bearings .................................................................. 146
Procedure 2: Stresses in Heads Due to Internal
Roller Bearings ............................................................... 147
Pressure ....................................................................... 215
Standardization ............................................................... 149
Joint Efficiencies (ASME Code) .................................... 217
Materials ......................................................................... 151
Properties of Heads ......................................................... 218
Rating and Life ......................................................... 152
Volumes and Surface Areas of Vessel Sections .............. 220
ABMA Definitions ......................................................... 152
Maximum Length of Unstiffened Shells ........................ 221
Fatigue Life ..................................................................... 153
Useful Formulas for Vessels ........................................... 222
Life Adjustment Factors ................................................. 154
Material Selection Guide ................................................ 224
Load and Speed Analysis ......................................... 156 References ....................................................................... 225
Equivalent Loads ............................................................ 156
Contact Stresses .............................................................. 157
10: Tribo (cid:12)9 I ogy, 226
Preloading ....................................................................... 157
Special Loads .................................................................. 158
Effects of Speed ........... ............................................ 159 Introduction .............................................................. 227
Lubrication ............................................................... 160 Contact Mechanics ................................................... 227
General ............................................................................ 160 Two-dimensional (Line) Hertz Contact of Cylinders ..... 227
Oils .................................................................................. 161 Three-dimensional (Point) Hertz Contact ....................... 229
Greases ............................................................................ 161 Effect of Friction on Contact Stress ................................ 232
vii
Yield and Shakedown Criteria for Contacts ................... 232 Mechanical Testing ... 284
Topography of Engineering Surfaces ..................... 233 Tensile Testing ................................................................ 284
Definition of Surface Roughness .................................... 233 Fatigue Testing ................................................................ 285
Contact of Rough Surfaces ............................................. 234 Hardness Testing ............................................................. 286
Life Factors ..................................................................... 234 Creep and Stress Rupture Testing ................................... 287
Friction ...................................................................... 235 Forming ..................................................................... 288
Wear ........................................................................... 235 Casting ....................................................................... 289
Lubrication 236 Case Studies .............................................................. 290
References ................................................................. 237 Failure Analysis .............................................................. 290
Corrosion ........................................................................ 291
:11 Vibration, 238 References ................................................................. 292
Vibration Definitions, Terminology, and :31 Stress dna Strain, 294
Symbols ................................................................. 239
. . . . . . . .
Solving the One Degree of Freedom System .......... 243 Fundamentals of Stress and Strain ......................... 295
Solving Multiple Degree of Freedom Systems ....... 245
Introduction ..................................................................... 295
Vibration Measurements and Instrumentation ..... 246 Definitions--Stress and Strain ....................................... 295
Table A: Spring Stiffness ......................................... 250 Equilibrium ..................................................................... 297
Table B: Natural Frequencies of Simple Systems.. 251 Compatibility .................................................................. 297
Table C: Longitudinal and Torsional Vibration of Saint-Venant's Principle .................................................. 297
Uniform Beams ..................................................... 252 Superposition .................................................................. 298
Table D: Bending (Transverse) Vibration of Plane Stress/Plane Strain ................................................ 298
Uniform Beams ..................................................... 253 Thermal Stresses ............................................................. 298
Stress Concentrations ............................................... 299
Table E: Natural Frequencies of Multiple DOF
Determination of Stress Concentration Factors .............. 300
Systems .................................................................. 254
Design Criteria for Structural Analysis ................. 305
Table F: Planetary Gear Mesh Frequencies .......... 255
General Guidelines for Effective Criteria ....................... 305
Table G: Rolling Element Bearing Frequencies
Strength Design Factors .................................................. 305
and Bearing Defect Frequencies ......................... 256
Beam Analysis ........................................................... 306
Table H: General Vibration Diagnostic
Limitations of General Beam Bending Equations .......... 307
Frequencies ........................................................... 257
Short Beams .................................................................... 307
References ................................................................. 258
Plastic Bending ............................................................... 307
Torsion ............................................................................ 308
:21 Materials, 259 Pressure Vessels ........................................................ 309
Thin-walled Cylinders .................................................... 309
Classes of Materials .................................................. 260 Thick-walled Cylinders .................................................. 309
Definitions ................................................................. 260 Press Fits Between Cylinders .................................. 310
Metals ........................................................................ 262 Rotating Equipment ................................................. 310
Steels ............................................................................... 262 Rotating Disks ................................................................ 310
Tool Steels ...................................................................... 264 Rotating Shafts ................................................................ 313
Cast Iron .......................................................................... 265 Flange Analysis ................................................. (cid:12)9 ....... 315
Stainless Steels ................................................................ 266 Flush Flanges .................................................................. 315
Superalloys ..................................................................... 268 Undercut Flanges ............................................................ 316
Aluminum Alloys ........................................................... 269 Mechanical Fasteners ............................................... 316
Joining ............................................................................. 270 Threaded Fasteners ......................................................... 317
Coatings .......................................................................... 273 Pins ................................................................................. 318
Corrosion ........................................................................ 276 Rivets .............................................................................. 318
Powder Metallurgy ......................................................... 279 Welded and Brazed Joints ....................................... 319
Polymers .................................................................... 281 Creep Rupture .......................................................... 320
Ceramics .................................................................... 284 Finite Element Analysis ........................................... 320
viii
Overview ......................................................................... 321 Strain Measurement ................................................. 362
The Elements .................................................................. 321 The Electrical Resistance Strain Gauge .......................... 363
Modeling Techniques ...................................................... 322 Electrical Resistance Strain Gauge Data Acquisition ..... 364
Advantages and Limitations of FEM .............................. 323 Liquid Level and Fluid Flow Measurement .......... 366
Centroids and Moments of Inertia for Common Liquid Level Measurement ............................................. 366
Shapes .......... 324 Fluid Flow Measurement ................................................ 368
Beams: Shear, Moment, and Deflection Formulas References ................................................................. 370
for Common End Conditions .............................. 325
References ................................................................. 328 :61 gnireenignE ,scimonocE 372
:41 Fatigue, 923
Time Value of Money: Concepts and Formulas .... 373
Simple Interest vs. Compound Interest ........................... 373
Introduction .............................................................. 330 Nominal Interest Rate vs. Effective Annual
Stages of Fatigue ....................................................... 330 Interest Rate ................................................................ 374
Design Approaches to Fatigue ................................. 331 Present Value of a Single Cash Flow To Be Received
Crack Initiation Analysis ......................................... 331 in the Future ................................................................ 374
Future Value of a Single Investment ............................... 375
Residual Stresses ............................................................ 332
The Importance of Cash Flow Diagrams ........................ 375
Notches ........................................................................... 332
Analyzing and Valuing Investments/Projects with
Real World Loadings ...................................................... 335
Multiple or Irregular Cash Flows ............................... 375
Temperature Interpolation .............................................. 337
Perpetuities ..................................................................... 376
Material Scatter ............................................................... 338
Future Value of a Periodic Series of Investments ........... 377
Estimating Fatigue Properties ......................................... 338
Annuities, Loans, and Leases ......................................... 377
Crack Propagation Analysis .................................... 338
Gradients (Payouts/Payments with Constant
K--The Stress Intensity Factor ...................................... 339
Growth Rates) ............................................................. 378
Crack Propagation Calculations ..................................... 342
Analyzing Complex Investments and
Creep Crack Growth ....................................................... 344
Cash Flow Problems ................................................... 379
Inspection Techniques .............................................. 345
Decision and Evaluation Criteria for Investments
Fluorescent Penetrant Inspection (FPI) .......................... 345
and Financial Projects .......................................... 380
Magnetic Particle Inspection (MPI) ................................ 345
Payback Method ............................................................. 380
Radiography .................................................................... 345
Accounting Rate of Return (gog) Method .................... 381
Ultrasonic Inspection ...................................................... 346
Internal Rate of Return (IRR) Method ............................ 382
Eddy-current Inspection .................................................. 347
Net Present Value (NPV) Method ................................... 383
Evaluation of Failed Parts ............................................... 347
Nonmetallic Materials .............................................. 348 Sensitivity Analysis ................................................... 384
Decision Tree Analysis of Investments and
Fatigue Testing .......................................................... 349
Financial Projects ................................................. 385
Liability Issues .......................................................... 350
Accounting Fundamentals ....................................... 389
References ................................................................. 350
References and Recommended Reading ................ 393
:51 Instrumentation, 352
,xidneppA 394
Introduction .............................................................. 353
Temperature Measurement ..................................... 354 Conversion Factors .................................................. 395
Fluid Temperature Measurement .................................... 354 Systems of Basic Units .............................................. 399
Surface Temperature Measurement. ............................... 358 Decimal Multiples and Fractions of SI units ......... 399
Common Temperature Sensors ....................................... 358 Temperature Conversion Equations ....................... 399
Pressure Measurement ..... 359
Total Pressure Measurement ........................................... 360 Index, 400
Static/Cavity Pressure Measurement .............................. 361
sdiulF
inabahB .P ,ytnahoM Ph.D., tnempoleveD ,reenignE nosillA enignE ynapmoC
Fluid Properties .......................................................... Nondimensional Parameters ............................................. 7
Density, Specific Volume, Specific Weight, Equivalent Diameter and Hydraulic Radius ..................... 8
Specific Gravity, and Pressure ...................................... Pipe Flow .................................................................... 8
Surface Tension ................................................................ Friction Factor and Darcy Equation ................................. 9
Vapor Pressure .................................................................. Losses in Pipe Fittings and Valves ................................... 10
Gas and Liquid Viscosity ................................................. Pipes in Series .................................................................. 10
Bulk Modulus ................................................................... Pipes in Parallel ................................................................ 10
Compressibility ................................................................ Open-Channel Flow ................................................... 11
Units and Dimensions ...................................................... Frictionless Open-Channel Flow ...................................... 11
Fluid Statics ................................................................ Laminar Open-Channel Flow ........................................... 12
Turbulent Open-Channel Flow ......................................... 12
Manometers and Pressure Measurements ........................
Hydraulic Jump ................................................................ 12
Hydraulic Pressure on Surfaces ........................................
Buoyancy.. ........................................................................ Fluid Measurements .................................................. 13
Pressure and Velocity Measurements ............................... 13
Basic Equations .......................................................... Flow Rate Measurement ................................................... 41
Continuity Equation .........................................................
Hot-Wire and Thin-Film Anemometry ............................ 14
Euler's Equation ...............................................................
Open-Channel Flow Measurements ................................. 15
Bemoulli's Equation .........................................................
Viscosity Measurements ................................................... 15
Energy Equation ...............................................................
Other Topics ............................................................... 16
Momentum Equation ........................................................
Unsteady Flow, Surge, and Water Hammer ..................... 16
Moment-of-Momentum Equation ....................................
Boundary Layer Concepts ................................................ 16
Advanced Fluid Flow Concepts ................................ Lift and Drag .................................................................... 16
Dimensional Analysis and Similitude .............................. Oceanographic Flows ....................................................... 17
2 Rules of Thumb for Mechanical Engineers
FLUID SEITREPORP
A fluid is defined as a "substance that deforms contin- ear relationship between the applied shear stress and the
uously when subjected to a shear stress" and is divided into resulting rate of deformation; but in a non-Newtonian
two categories: ideal and real. A fluid that has zero vis- fluid, the relationship is nonlinear. Gases and thin liquids
cosity, is incompressible, and has uniform velocity distri- are Newtonian, whereas thick, long-chained hydrocar-
bution is called an idealfluid. Realfluids are called either bons are non-Newtonian.
Newtonian or non-Newtonian. A Newtonian fluid has a lin-
,ytisneD cificepS ,emuloV Specific ,thgieW cificepS ,ytivarG dna erusserP
The density p is defined as mass per unit volume. In in- (cid:12)9 The specific gravity s of a liquid is the ratio of its
consistent systems it is defined as lbm/cft, and in consis- weight to the weight of an equal volume of water at stan-
tent systems it is defined as slugs/cft. The density of a gas dard temperature and pressure. The s of petroleum
can be found from the ideal gas law: products can be found from hydrometer readings using
API (American Petroleum Institute) scale.
p = p/RT (1)
where p is the absolute pressure, R is the gas constant, and (cid:12)9 The fluid pressure p at a point is the ratio of normal
T is the absolute temperature. force to area as the area approaches a small value. Its
The density of a liquid is usually given as follows: unit is usually lbs/sq, in. (psi). It is also often measured
as the equivalent height h of a fluid column, through
(cid:12)9 The specific volume sv is the reciprocal of density"
the relation:
~v = 1/p
(cid:12)9 The specific weight ), is the weight per unit volume"
Y=Pg
ecafruS noisneT
Near the free surface of a liquid, because the cohesive product of a surface tension coefficient and the length of
force between the liquid molecules is much greater than that the free surface. This is what forms a water droplet or a mer-
between an air molecule and a liquid molecule, there is a cury globule. It decreases with increase in temperature, and
resultant force acting towards the interior of the liquid. depends on the contacting gas at the free surface.
This force, called the surface tension, is proportional to the
ropaV erusserP
Molecules that escape a liquid surface cause the evapo- of the temperature and increases with it. Boiling occurs
ration process. The pressure exerted at the surface by these when the pressure above the liquid surface equals (or is less
free molecules is called the vapor pressure. Because this is than) the vapor pressure of the liquid. This phenomenon,
caused by the molecular activity which is a function of the which may sometimes occur in a fluid system network,
temperature, the vapor pressure of a liquid also is a function causing the fluid to locally vaporize, is called cavitation.
Fluids 3
saG dna diuqiL ytisocsiV
Viscosity is the property of a fluid that measures its re- The t.l above is often called the absolute or dynamic
sistance to flow. Cohesion is the main cause of this resis- viscosity. There is another form of the viscosity coefficient
tance. Because cohesion drops with temperature, so does called the kinematic viscosity v, that is, the ratio of viscosity
viscosity. The coefficient of viscosity is the proportional- to mass density:
ity constant in Newton's law of viscosity that states that the
shear stress x in the fluid is directly proportional to the ve- v=~p
locity gradient, as represented below:
Remember that in U.S. customary units, unit of mass den-
du sity p is slugs per cubic foot.
x = t~ ~dY (2)
kluB suludoM
A liquid's compressibility is measured in terms of its bulk The bulk modulus of elasticity K is its reciprocal:
modulus of elasticity. Compressibility is the percentage
change in unit volume per unit change in pressure: K= 1/C
K is expressed in units of pressure.
~Sv/v
C = -------
8p
ytilibisserpmoC
Compressibility of liquids is def'med above. However, for sVp = RT
a gas, the application of pressure can have a much greater
effect on the gas volume. The general relationship is gov- where p is the absolute pressure, sV is the specific volume,
erned by the perfect gas law: R is the gas constant, and T is the absolute temperature.
stinU dna snoisnemiD
One must always use a consistent set of units. Primary mass is ever referred to as being in Ibm (inconsistent sys-
units are mass, length, time, and temperature. A unit system tem), one must first convert it to slugs by dividing it by
is called consistent when unit force causes a unit mass to 32.174 before using it in any consistent equation.
achieve unit acceleration. In the U.S. system, this system is Because of the confusion between weight (lbf) and mass
represented by the (pound) force, the (slug) mass, the (foot) (Ibm) units in the U.S. inconsistent system, there is also a
length, and the (second) time. The slug mass is def'med as similar confusion between density and specific weight
the mass that accelerates to one ft/sec a when subjected to one units. It is, therefore, always better to resort to a consistent
pound force (lbf). Newton's second law, F = ma, establish- system for engineering calculations.
es this consistency between force and mass units. If the
4 Rules of Thumb for Mechanical Engineers
DIULF SCITATS
Fluid statics is the branch of fluid mechanics that deals there is no relative motion between fluid layers, there are
with cases in which there is no relative motion between fluid no shear stresses in the fluid under static equilibrium.
elements. In other words, the fluid may either be in rest or Hence, all free bodies in fluid statics have only normal forces
at constant velocity, but certainly not accelerating. Since on their surfaces.
, , , , ,
sretemonaM dna erusserP stnemerusaeM
Pressure is the same in all directions at a point in a sta- above expression, we neglected the vapor pressure for
tic fluid. However, if the fluid is in motion, pressure is de- mercury. But if we use any other fluid instead of mercury,
fined as the average of three mutually perpendicular nor- the vapor pressure may be significant. The equilibrium
mal compressive stresses at a point: equation may then be:
P = xP( + Py + pz)/3 aP = (0.0361)(s)(h) + Pv(144)
Pressure is measured either from the zero absolute pres- where 0.0361 is the water density in pounds per cubic
sure or from standard atmospheric pressure. If the reference inch, and s is the specific gravity of the fluid. The consis-
point is absolute pressure, the pressure is called the absolute tent equation for variation of pressure is
pressure, whereas if the reference point is standard atmos-
p=rh
pheric (14.7 psi), it is called the gage pressure. A barom-
eter is used to get the absolute pressure. One can make a
where p is in lb/ft ,2 T is the specific weight of the fluid in
simple barometer by filling a tube with mercury and in-
lb/ft ,3 and h is in feet. The above equation is the same as p
verting it into an open container filled with mercury. The
- ywsh, where wY is the specific weight of water (62.4
mercury column in the tube will now be supported only by
lb/ft )3 and s is the specific gravity of the fluid.
the atmospheric pressure applied to the exposed mercury
Manometers are devices used to determine differential
surface in the container. The equilibrium equation may be
pressure. A simple U-tube manometer (with fluid of spe-
written as:
cific weight 7) connected to two pressure points will have
a differential column of height h. The differential pressure
aP = 0.491(144)h
will then be Ap = 2P( - Pl) = 7h. Corrections must be
where h is the height of mercury column in inches, and 0.491 made if high-density fluids are present above the manome-
is the density of mercury in pounds per cubic inch. In the ter fluid.
ciluardyH erusserP no secafruS
For a horizontal area subjected to static fluid pressure, 1
Pavg = ~" (hi + h2) sin 0 (3)
the resultant force passes through the centroid of the area.
If the plane is inclined at an angle 0, then the local pressure However, the center of pressure will not be at average depth
will vary linearly with the depth. The average pressure but at the centroid of the triangular or trapezoidal pressure
occurs at the average depth: distribution, which is also known as the pressure prism.
Fluids 5
ycnayouB
The resultant force on a submerged body by the fluid The principles of buoyancy make it possible to determine
around it is called the buoyant force, and it always acts up- the volume, specific gravity, and specific weight of an un-
wards. If v is the volume of the fluid displaced by the sub- known odd-shaped object by just weighing it in two different
merged (wholly or partially) body, T is the fluid specific fluids of known specific weights lT and -2T This is possi-
weight, and tnayoubF is the buoyant force, then the relation ble by writing the two equilibrium equations:
between them may be written as:
'~ (4) W = F l + ")v l = F 2 + 2Tv (5)
Fbuoyan t -- V (cid:141)
CISAB SNOITAUQE
In derivations of any of the basic equations in fluids, the 3.1 st and 2nd Laws of Thermodynamics
concept of control volume is used. A control volume is an 4. Proper boundary conditions
arbitrary space that is defined to facilitate analysis of a flow
region. It should be remembered that all fluid flow situa- Apart from the above relations, other equations such as
tions obey the following rules: Newton's law of viscosity may enter into the derivation
process, based on the particular situation. For detailed pro-
.1 Newton's Laws of Motion
cedures, one should refer to a textbook on fluid mechanics.
2. The Law of Mass Conservation (Continuity Equation)
ytiunitnoC noitauqE
For a continuous flow system, the mass within the fluid Q is defined as Q = A.V, the continuity equation takes the
remains constant with time: dm/dt = .O If the flow discharge following useful form:
rh = p IAIVI = 2V2A2p (6)
s'reluE noitauqE
Under the assumptions of: (a) frictionless, (b) flow When p is either a function of pressure p or is constant, the
along a streamline, and (c) steady flow; Euler's equation Euler's equation can be integrated. The most useful rela-
takes the form: tionship, called Bernoulli's equation, is obtained by inte-
grating Euler's equation at constant density p.
-~+ g.dz + v.dv = 0 (7)