Table Of Content,It* .....
RULES OF THUMB
FOR MECHANICAL ENGINEERS
A manual of quick, accurate solutions
to everyday mechanical engineering problems
J. EDWARD POPE, EDITOR
RULES OF THUMB FOR
MECHANICAL ENGINEERS
Copyright (cid:14)9 1997 by Gulf Publishing Company,
Houston, Texas. All rights reserved. Printed in the
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Gulf Publishing Company
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Library of Congress Cataloging-in-Publication Data
Rules of thumb for mechanical engineers : a manual of
quick, accurate solutions to everyday mechanical
engineering problems / J. Edward Pope, editor ; in
collaboration with Andrew Brewington ... [et al.].
p. cm.
Includes bibliographical references and index.
ISBN-13:978-0-88415-790-8 ISBN-10:0-88415-790-3 (acid-free paper)
1. Mechanical engineering~Handbooks, manuals,
etc. I. Pope, J. Edward, 1956- . II. Brewington,
Andrew.
TJ151.R84 1996
621---dc20 96-35973
CIP
Printed on acid-free paper (,,~).
ISBN- 13: 978-0-88415-790-8
ISBN-10:0-88415-790-3 (acid-free paper)
Fluids
Bhabani P. Mohanty, Ph.D., DevelopmentE ngineer, Allison Engine Company
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 ................................................... 14
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 PROPERTIES
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-
Density, Specific Volume, Specific Weight, Specific Gravity, and Pressure
The density p is defined as mass per unit volume. In in- (cid:12)9T he 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)9T he 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)9T he specific volume vs is the reciprocal of density"
the relation:
v~ = 1/p
(cid:12)9T he specific weight ), is the weight per unit volume"
Y=Pg
Surface Tension
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
Vapor Pressure
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
Gas and Liquid Viscosity
Viscosity is the property of a fluid that measures its re- The l.t 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 Yd~ (2)
Bulk Modulus
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
Compressibility
Compressibility of liquids is def'med above. However, for pVs = 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, Vs is the specific volume,
erned by the perfect gas law: R is the gas constant, and T is the absolute temperature.
Units and Dimensions
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/seca 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
FLUID STATICS
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.
, , , , ,
Manometers and Pressure Measurements
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 = (Px + Py + pz)/3 Pa = [(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 Yw 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
Pa = 0.491(144)h
will then be Ap = (P2 - 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.
Hydraulic Pressure on Surfaces
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
Buoyancy
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 Tl and T2- This is possi-
weight, and Fbuoyant is the buoyant force, then the relation ble by writing the two equilibrium equations:
between them may be written as:
Fbuoyan t -- V • '~ (4) W = F l + v)"l = F 2 + vT2 (5)
BASIC EQUATIONS
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)
Continuity Equation
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 = p2A2V2 (6)
Euler's Equation
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)
6 Rules of Thumb for Mechanical Engineers
Bernoulli's Equation
Bemoulli's equation can be thought of as a special form namic head, and the p/pg term is called the static head. All
of energy balance equation, and it is obtained by integrat- these terms represent energy per unit weight. The equation
ing Euler's equation defined above. characterizes the specific kinetic energy at a given point
within the flow cross-section. While the above form is
V 2
convenient for liquid problems, the following form is more
z = constant (8)
+ ~ + p
2g pg convenient for gas flow problems:
The constant of integration above remains the same along
pv 2
a streamline in steady, frictionless, incompressible flow. The Tz +- (cid:1)89 + p = constant (9)
term z is called the potential head, the term v2/2g is the dy-
Energy Equation
The energy equation for steady flow through a control where qhe~t is heat added per unit mass and W~haeti s the shaft
volume is: work per unit mass of fluid.
V2l Pl
qheat + Pl + gz + Ul +---+ gz
191 T Wshaft P2
=
+--+ u2 (10)
2
Momentum Equation
The linear momentum equation states that the resultant
F = d (mv)
force F acting on a fluid control volume is equal to the rate
of change of linear momentum inside the control volume plus
the net exchange of linear momentum from the control
boundary. Newton's second law is used to derive its form:
Moment-of-Momentum Equation
The moment-of-momentum equation is obtained by tak- normal to the plane containing these two basis vectors and
ing the vector cross-product of F detailed above and the po- obeying the cork-screw convention. This equation is of great
sition vector r of any point on the line of action, i.e., r x F. value in certain fluid flow problems, such as in turboma-
Remember that the vector product of these two vectors is chineries. The equations outlined in this section constitute
also a vector whose magnitude is Fr sin0 and direction is the fundamental governing equations of flow.
Fluids 7
ADVANCEDF LUID FLOW CONCEPTS
Often in fluid mechanics, we come across certain terms, Table 1
such as Reynolds number, Prandtl number, or Mach num- Dimensions of Selected Physical Variables
ber, that we have come to accept as they are. But these are
extremely useful in unifying the fundamental theories in this Physical Variable Symbol Dimension
field, and they have been obtained through a mathematical Force F MET- 2
analysis of various forces acting on the fluids. The math- Discharge Q L3T- 1
Pressure p ML-1T- 2
ematical analysis is done through Buckingham's Pi Theo-
Acceleration a LT- 2
rem. This theorem states that, in a physical system de-
Density p ML -3
scribed by n quantities in which there are m dimensions, Specific weight 7 ML-2T-2
these n quantities can be rearranged into (n-m) nondimen- Dynamic viscosity 11 ML-1T- 1
Kinematic viscosity v L2T- 1
sional parameters. Table 1 gives dimensions of some phys-
Surface tension 0 MT -2
ical variables used in fluid mechanics in terms of basic mass Bulk modulus of elasticity K ML-1T -2
(M), length (L), and time (T) dimensions. Gravity g LT- 2
Dimensional Analysis and Similitude
Most of these nondimensional parameters in fluid me- These nondimensional parameters allow us to make
chanics are basically ratios of a pair of fluid forces. These studies on scaled models and yet draw conclusions on the
forces can be any combination of gravity, pressure, viscous, prototypes. This is primarily because we are dealing with
elastic, inertial, and surface tension forces. The flow sys- the ratio of forces rather than the forces themselves. The
tem variables from which these parameters are obtained are: model and the prototype are dynamically similar if (a)
velocity V, the density p, pressure drop Ap, gravity g, vis- they are geometrically similar and (b) the ratio of pertinent
cosity Ix, surface tension G, bulk modulus of elasticity K, forces are also the same on both.
and a few linear dimensions of 1.
,, , , , ,, , ,
Nondimensional Parameters
The following five nondimensional parameters are of between the two) through a critical value. For example, for
great value in fluid mechanics. the case of flow of fluids in a pipe, a fluid is considered tur-
bulent if R is greater than 2,000. Otherwise, it is taken to
Reynolds Number be laminar. A turbulent flow is characterized by random
movement of fluid particles.
Reynolds number is the ratio of inertial forces to viscous
forces: Froude Number
Froude number is the ratio of inertial force to weight:
R= p V1 (12)
V
This is particularly important in pipe flows and aircraft
model studies. The Reynolds number also characterizes dif- This number is useful in the design of spillways, weirs, chan-
ferent flow regimes (laminar, turbulent, and the transition nel flows, and ship design.
8 Rules of Thumb for Mechanical Engineers
Weber Number where c is the speed of sound in the fluid medium, k is the
ratio of specific heats, and T is the absolute temperature. This
Weber number is the ratio of inertial forces to surface ten-
parameter is very important in applications where velocities
sion forces.
are near or above the local sonic velocity. Examples are fluid
machineries, aircraft flight, and gas turbine engines.
W = VElp
t~ (14/
Pressure Coefficient
This parameter is significant in gas-liquid interfaces where
surface tension plays a major role. Pressure coefficient is the ratio of pressure forces to in-
ertial forces:
Mach Number
Ap = All
Mach number is the ratio of inertial forces to elastic forces: Cp = pV2/2 pV2/2g (16)
V V
M = = - (15)
-c- 4kRT This coefficient is important in most fluid flow situations.
Equivalent Oiameter and Hydraulic Radius
The equivalent diameter (Deq) is defined as four times If a pipe is not flowing full, care should be taken to com-
the hydraulic radius (rh). These two quantities are widely pute the wetted perimeter. This is discussed later in the sec-
used in open-channel flow situations. If A is the cross-sec- tion for open channels. The hydraulic radii for some com-
tional area of the channel and P is the wetted perimeter of mon channel configurations are given in Table 2.
the channel, then:
A Table 2
rh (17)
----P- Hydraulic Radii for Common Channel Configurations
Note that for a circular pipe flowing full of fluid,
Cross-Section rh
D~q = 4rh = 4(~:D2 / 4) = D Circular pipe of diameter D DI4
xD Annular section of inside dia d and outside dia D (D - d)/4
Square duct with each side a a/4
and for a square duct of sides and flowing full, Rectangular duct with sides a and b aJ4
Elliptical duct with axes a and b (~)/Kia + b)
4a 2
Semicircle of diameter D D/4
D~q =4r h = =a
4a Shallow fiat layer of depth h h
PIPE FLOW
In internal flow of fluids in a pipe or a duct, considera- layer of fluid at the wall must have zero velocity, with pro-
tion must be given to the presence of frictional forces act- gressively increasing values away from the wall, and reach-
ing on the fluid. When the fluid flows inside the duct, the ing maximum at the centerline. The distribution is parabolic.
