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Unformatted text preview: 296 csnrrnr 7 I DIMENSIONAL ANALYSIS AND srururuos 7.7 The equation describing motion of ﬂuid in a pipe due to an
applied pressure gradient. when the ﬂow starts from rest. is (Clo 1 6p ('92:: l ﬂu
% = —— + . —
:31 par 1((31‘2 + far) Use the average velocity V, pressure drop Ap. pipe length L. and
diameter D to nondimensionalize this equation. Obtain the dimen
sionless groups that characterize this ﬂow. 7.8 An unsteady. two dimensional. compressible. inviscid ﬂow
can be described by the equation 631,9 + 2 data
Oil ('71 6x3
where [/1 is the stream function. u and v are the x and y com
ponents of velocity. respectively. c is the local speed of sound,
and r is the time. Using L as a characteristic length and CD {the
speed of sound at the stagnation point) to nondimensionalize this equation. obtain the dimensionless groups that characterize the
equation. (”2+U2)+(M2,CE) +(u2~r2)j——+2uv . =0 7.9 At very low speeds. the drag on an object is independent of
ﬂuid density. Thus the drag force. F. on a small sphere is a funcu
tion only of speed. V. fluid viscosity. it. and sphere diameter. D.
Use dimensional analysis to determine how the drag force F de—
pends on the speed V. 7.10 At relatively high speeds the drag on an object is indepen—
dent of ﬂuid viscosity. Thus the aerodynamic drag force. F, on an
automobile. is a function only of speed. V. air density p, and
vehicle size. characterized by its frontal area A. Use dimensional
analysis to determine how the drag force F depends on the speed V. 7.11 Experiments show that the pressure drop for ﬂow through
an oriﬁce plate of diameter d mounted in a length of pipe of diam~
eter D may be expressed as Ap = p. — p2 =j’lp. in? d. D).
You are asked to organize some experimental data. Obtain the re~
suiting dimensionless parameters. 7.12 The speed. V. of a freesurface wave in shallow liquid is a
function of depth. D. density. p. gravity. g. and surface tension. cr.
Use dimensional analysis to [ind the functional dependence of V
on the other variables. Express Vin the simplest form possible. 7.13 The wall shear stress, I... in a boundary layer depends on
distance from the leading edge of the body, at. the density. p. and
viscosity. it. of the ﬂuid. and the freestream speed of the ﬂow. U.
Obtain the dimensionless groups and express the functional
relationship among them. 7.14 The boundarylayer thickness. d. on a smooth ﬂat plate in
an incompressible ﬂow without pressure gradients depends on the
freestream speed. U. the ﬂuid density. p. the ﬂuid viscosity. ,u. and
the distance from the leading edge of the plate. A”. Express these
variables in dimensionless form. 7.15 If an object is light enough it can be supported on the sur—
face of a lltrid by surface tension. Tests are to be done to investi
gate this phenomenon. The weight. W. supportable in this way
depends on the object’s perimeter. p. and the fluids density. p.
surface tension rt. and gravity. g. Determine the dimensionless
parameters that characterize this problem. 7.16 The mean velocity. E. for turbulent ﬂow in a pipe or a
boundary layer may be correlated using the wall shear stress. r...
distance from the wall. y. and the ﬂuid properties. p and ,u. Use dimensional analysis to ﬁnd one dimensionless parameter containf
ing E and one containing _v that are suitable for organizing exper
imental data. Show that the result may be written ﬁ_ m
u_._f(v) where the = (“7pr is theﬁ‘r'cn'an velocity. 7.17 The speed, V. of a freesurface gravity wave in deep water
is a function of wavelength. I1. depth, D. density. p. and accelera.
tion of gravity. g. Use dimensional analysis to ﬁnd the functional:
dependence ol‘ V on the other variables. Express V in the simplest
form possible. 7.18 The torque. T. of a handheld automobile buffer is a tune
tion of rotational speed. 0). applied normal force. F. automobile
surface roughness. e, bufﬁng paste viscosity. u. and surface ten
sion. 0. Determine the dimensionless parameters that characterize
this problem. 7.19 Measurements of the liquid height upstream from an ob.
struction placed in an openchannel flow can be used to determine}.
volume flow rate. (Such obstructions. designed and calibrated to.
measure rate of openchannel ﬂow. are called weirs.) Assume the:
volume ﬂow rate. Q. over a weir is a function of upstream heights
11. gravity, 3. and channel width. I). Use dimensional analysis to:
ﬁnd the functional dependence of Q on the other variables. 7.20 Capillary waves are limited on a liquid free surface as a tie2
sult of surface tension. They have short wavelengths. The speed,
of a capillary wave depends on Surface tension. a. wavelength, i.
and liquid density. p. Use dimensional analysis to express waves:
speed as a function of these variables. 7.2] The loadcarrying capacity. W. of a journal bearing is;
known to depend on its diameter. D. length. i. and clearance. c. in:
addition to its angular speed, a), and lubricant viscosity. ,u. Deten
mine the dimensionless parameters that characterize this problem. 7.22 The time. I. for oil to drain out of a viscosity calibration
container depends on the ﬂuid viscosity, ,u. and density. p. the OtisI
lice diameter. d. and gravity. g. Use dimensional analysis to ﬁnd
the functional dependence of t on the other variables. ExpreSs tilt
the simplest possible form. 7.23 The power, 97°. used by a vacuum cleaner is to be correlated;
with the amount of suction provided (indicated by the pressure
drop. Ap. below the ambient room pressure). It also depends on
impeller diameter. D. and width. d. motor speed. to. air density.
p. and cleaner inlet and exit widths. d, and (1... respectively. De»
termine the dimensionless parameters that characterize this.
problem. 7.24 The power per unit crosssectional area. E. transmitted by a.
sound wave is a function of wave speed. V. medium density, ,0.
wave amplitude. r. and wave frequency. n. Determine, by dimen
sional analysis. the general form of the expression for E in tents.
of the other variables. 7.25 You are asked to ﬁnd a set of dimensionless parameters to
organize data from a laboratory experiment. in which a tank is.
drained through an oriﬁce from initial liquid level ha. The time,
1'. to drain the tank depends on tank diameter. D. oriﬁce diam
eter. d. acceleration of gravity. 5.). liquid density. p, and liquid
viscosity. ,u. How many dimensionless parameters will result?
How many repeating variables must be selected to determine the. dimensionless parameters? Obtain the TI parameter that contains
the viscosity. 7.26 The power. 9P. required to drive :1 fan is believed to depend
on ﬂuid density. p. volume ﬂow rate. Q. impeller diameter. D. and
angular velocity. to. Use dimensional analysis to determine the de
pendence of Cil’ on the other variables. 7.27 A continuous bell moving vertically through a bath of vis—
cous liquid drags a layer of liquid. of thickness 1‘]. along with it.
The volume ﬂow rate of liquid. Q. is assumed to depend on ti. p.
g. h. and V. where V is the belt speed. Apply dimensional analysis
to predict the form of dependence of Q on the other variables. 7.28 In a ﬂuid mechanics laboratory experiment a tank of water.
with diameter D. is drained from initial level It“. The smoothly
rounded drain hole has diameter (1‘. Assume the mass [low rate
from the tank is a function of h. D. d. g. p. and ,u. where g is the
acceleration of gravity and p and it are ﬂuid properties. Measured
data are to be correlated in dimensionless form. Determine the
number of dimensionless parameters that will result. Specify the
number of repeating parameters that must be selected to determine
the dimensionless parameters. Obtain the TI parameter that con tains the viscosity. 7.29 Small droplets of liquid are formed when a liquid jet breaks
up in spray and fuel injection processes. The resulting droplet
diameter. d. is thought to depend on liquid density. viscosity. and
surface tension. as well as jet speed. V. and diameter. D. How
many dimensionless ratios are required to characterize this pro:
cess? Determine these ratios. 7.30 The diameter. d. of the dots made by an ink jet printer de—
pends on the ink viscosity. Ju. density. p. and surface tension, 0. the
nozzle diameter. D. the distance. L. of the nozzle from the paper
surface. and the ink jet velocity. V. Use dimensional analysis to
ﬁnd the [1 parameters that characterize the ink jet’s behavior. 7.31 The sketch shows an air jet discharging vertically. Exper
iments show that a ball plaCed in the jet is suspended in a stable
position. The equilibrium height of the ball in the jet is found to
depend on D. d. V. ,0. p. and W. where W is the weight of the ball.
Dimensional analysis is suggested to correlate experimental data.
Find the I] parameters that characterize this phenomenon. P731 7.32 The terminal speed V of shipping boxes sliding down an in
cline on a layer of air (injected through numerous pinholes in the
incline surface) depends on the box mass. m. and base area. A.
gravity. g. the incline angle. t), the air viscosity. p. and the air
layer thickness, 5. Use dimensional analysis to ﬁnd the ll par
ameters that characterize this phenomenon. 7.33 The diameter, d. of bubbles produced by a bubblemaking
toy depends on the soapy water viscosity, p, density, p. and sur
face tension. cr. the ring diameter. D. and the pressure differential. PROBLEMS 297 Ap. generating the bubbles. Use dimensional analysis to ﬁnd the
Ti parameters that characterize this phenomenon. 7.34 A washing machine agitator is to be designed. The power. '91’.
required for the agitator is to be correlated with the amount of
water used (indicated by the depth, H, of the water). It also de
pends on the agitator diameter. D. height. [7. maximum angular
velocity. ton“... and frequency of oscillationsfl and water density.
p. and viscosity. p. Determine the dimensionless parameters that
characterize this problem. 7.35 The time. t. for a ﬂywheel. with moment of inertia. I. to
reach angular velocity, to, from rest, depends on the applied torque.
T. and the following ﬂywheel bearing properties: the oil viscosity.
,u. gap. (5. diameter. D. and length, L. Use dimensional analysis to
ﬁnd the Ti parameters that characterize this phenomenon. 7.36 A large tank of liquid under pressure is drained through a
stnoothly contoured nozzle of area A. The mass flow rate is
thought to depend on nozzle area. A. liquid density. p. difference
in height between the liquid surface and nozzle. h. tank gage pres
sure. Ap. and gravitational acceleration. g. Determine how many
independent Fl parameters can be formed for this problem. Find
the dimensionless parameters. State the functional relationship for
the mass flow rate in terms of the dimensionless parameters. 7.37 The ventilation in the clubhouse on a cruise ship is insufﬁ
cient to clear cigarette smoke (the ship is not yet completely
smokefree). Tests are to be done to see if a larger extractor fan
will work. The concentration of smoke. c (particles per cubic me
ter) depends on the number of smokers. N. the pressure drop pro
duced by the fan. Ap. the fan diameter. D. motor speed. to. the
particle and air densities, pp. and ,0, respectively. gravity. g. and
air viscosity. it. Determine the dimensionless parameters that
characterize this problem. 7.38 Spin plays an important role in the ﬂight trajectory of golf.
PingPong. and tennis balls. Therefore. it is important to know the
rate at which spin decreases for a ball in flight. The aerodynamic
torque, T. acting on a ball in flight. is thought to depend on ﬂight
speed. V. air density. p. air viscosity, ,tt. ball diameter. D, spin rate
(angular speed). to. and diameter of the dimples on the ball. at. De
termine the dimensionless parameters that result. 7.39 The power loss. 9)“. in ajoumal bearing depends on length.
l. diameter. D. and clearance. c. of the bearing. in addition to its
angular speed. a). The lubricant viscosity and mean pressure are
also important. Obtain the dimensionless parameters that charac—
terize this problem. Determine the functional form of the depen‘
dence of '3’ on these parameters. 7.40 The thrust of a marine propeller is to be measured during
"open—water” tests at a variety of angular speeds and forward
speeds ( "speeds of advance"). The thrust. F 1. is thought to depend
on water density, p, propeller diameter. D, speed of advance, V. ac—
celeration of gravity, g, angular speed. a). pressure in the liquid. 1).
and liquid viscosity. ,tt. Develop a set of dimensionless parameters
to characterize the performance of the propeller. (One of the result
ing parameters. gD/Vg. is known as the F made speed ofaduonce.) 7.41 In a fanassisted convection oven. the heat transfer rate to a
roast. Q (energy per unit time). is thought to depend on the
speciﬁc heat of air, c... temperature difference. E). a length scale.
L, air density. p. air viscosity. ,u, and air speed. V. How many
basic dimensions are included in these variables? Determine the ...
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 Fall '08
 ZOHAR

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