This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: L». EFIGURE P4.4O4 V: Volt + M4“ sintcoril 4.41 Water flows over the crest of a dam with speed Vas shown in
Fig. P441. Determine the speed if the magnitude of the normal ac
celeration at point (i) is to equal the acceleration of gravity, g. EFIGUFIE P4.41 4.42 Assume that the streamlines for the wingtip vortices from
an airplane (see Fig. P419 and Video V4.6) can be approximated
by circles of radius r and that thé speed is V = K/r, where K is a
constant. Determine the streamline acceleration, as, and the normal
acceleration, at", for this ﬂow. 4.43 A fluid ﬂows past a sphere with an upstream velocity of
V0 = 40 m/s as shown in Fig. P443. From a more advanced theory
it is found that the speed of the ﬂuid along the front part of the sphere
is V = gift, sin 0. Determine the streamwise and normal components
of acceleration at pointA if the radius of the sphere is a = 0.20 m. IFIGUFIE P4.43 *4.44 For ﬂow past a sphere as discussed in Problem 4.43, plot a graph
of the sueamwise acceleration, (2,, the normal acceleration, a,,, and the
magnitude of the acceleration as a function of 9 for 0 s 6 s 90° with
V0 = 50 ft/s anda = 0.1, 1.0, and 10 it. Repeat for V0 = 5 ft/s. At
what point is the acceleration a maximum; 3 minimum? ”34.45 The velocity components for steady flow through the nozzle
shown in Fig. P445 are L! = —Vo IN and 1) = Vo[1 + (yr/8)}, EFIGURE 134.45 183 Problems where V0 and t? are constants. Determine the ratio of the magnitude
of the acceleration at point (1) to that at point (2). *4.46 A ﬂuid ﬂows past a circular cylinder of radius a with an
upstream speed of V0 as shown in Fig. P446. A more advanced the»
ory indicates that if viscous effeéts are negligible, the velocity of the
fluid along the surface of the cylinder is given by V = 2% sin 6.
Determine the streamline and normal components of acceleration
on the surface of the cylinder as a function of V0, a, and 6 and plot
graphs of a, and an for 0 S 6 S 90° with V0 = 10 m/s and
a = 0.01, 0.10, 1.0, and 10.0 m. V
.I'
V
._°>. _'_/;
\ EFIGURE P4.46 4.47 Determine the x and y components of acceleration for the
ﬂow given in Problem 4.11. If c > 0, is the particle at point
x = x0 > 0 and y = 0 accelerating or decelerating? Explain.
Repeat if x0 < 0. <4.4§‘>When ﬂood gates in a channel are opened, water ﬂows
h'long the channel downstream of the gates with an increasing
speed given by V = 4(1 ~l— 0.1:) ft/s, for 0 S t S 20 s, where I
is in seconds. For t > ‘20 s the speed is a constant V = 12 ﬁle.
Consider a location in the curved channel where the radius of
curvature of the streamlines is 50 ft. For t = 10 5 determine (a)
the component of acceleration along the streamline, (b) the
component of acceleration normal to the streamline, and (c)
the net acceleration (magnitude and direction). Repeat for I: 3.0 s. @9 ater ﬂows steadily through the funnel shown in
igf”P4.49. Throughout most of the funnel the flow is approxi
.mately radial (along rays from O) with a velocity of V = c/rz,
where r is the radial coordinate and c is a constant. If the veloc—
ity is 0.4 m/s when 'r x 0.1 m, determine the acceleration at points A and B. FIGURE P4.49 4.50 Water flows though the slit at the bottom. of a two—
dimensional water trough as shown in Fig. P450. Throughout most
of the trough the ﬂow is approximately radial (along rays from O)
with a velocity of V = c/r, where r is the radial coordinate and c is
a constant. If the velocity is 0.04 m/s when r m 0.1 m, determine
the acceleration at points A and B. 4.62 In the region just downstream of a sluice gate, the water
may develop a reverse flow region as is indicated in Fig. P4.62
and Video VlO.9. The velocity proﬁle is assumed to consist of
two uniform regions, one with velocity V,l = 10 fps and the other
with Vb = 3 fps. Determine the net ﬂowrate of water across the
portion of the control surface at section (2) if the channel is 20 ft wide.
I. Sluice gate
Vb = 3 fits (1f EFIGUHE P4.62 4.63 At time t = 0 the valve on an initially empty (perfect vac
uum, p = 0) tank is opened and air rushes in. Ifthe tank has a vol
ume of V0 and the density of air within the tank increases as
p = pm(i — 8“"), where b is a censtant, determine the time rate of
change of mass within the tank. "{4.64 From calculus, one obtains the following formula (Leibnitz
rule) for the time derivative of an integral that contains time in both
the integrand and the limits of the integration: d 13(3) 116 [ix (it
—l ream] gateway: were? 11 Discuss how this formula is related to the time derivative of the
total amount of a property in a system and to theReynolds transport
theorem. 4.65 Water enters the bend of a river with the uniform velocity
proﬁle shown in Fig. P465. At the end of the bend there is a re
gion of separation or reverse ﬂow. The ﬁxed control volume ABCD
coincides with the system at time t = 0. Make a sketch to indicate
(a) the system at time t = 5 s and (b) the ﬂuid that has entered and
exited the control volume in that time period. _;_ Control votume EFIGUFIE 134.65 4.66 A layer of oil ﬂows down a vertical plate as“ shown in 'Fig. P466 with a velocity of V = (VD/hi) (231x — A?” where V0
and h are constants. (a) Show that the ﬂuid sticks to the plate and
that the shear stress at the edge of the layer (x = h) is zero. (b) De
termine the ﬂowrate across surface AB. Assume the width of the
plate is 1;. (Note: The velocity proﬁle for laminar ﬂow in a pipe has
a similar shape. See Video V6.13.) 185 Problems nix)
Plate BFIGUFIE P4.66 4.67 ater flows in the branching pipe shown in Fig. P467 with
uniform velocity at each inlet and outlet. The ﬁxed control volume
indicated coincides with the system at time t = 20 3. Make a sketch
to indicate (a) the boundary of the system at time t : 20.1 s, (b) the
fluid that left the control volume during that 0.15 interval, and (c)
the ﬂuid that entered the control volume during that time interval. ,, 7 Control volume
EFIGURE P4.67 4.68 TWO plates are pulled in opposite directions with speeds of
1.0 his as shown in Fig. P468. 'Ilre oil between the plates moves
with a velocity given by V = 10 yi ft/s, where y is in feet. The ﬁxed
control volume ABCD coincides with the system at time t = 0. Make
a sketch to indicate (a) the system at time t = 0.2 s and (h) the fluid
that has entered and exited the control volume in that time period. Control 1 it/s EFIGUFIE P4.68 4.69 Wateris squirted from a syringe with a speed of V = 5 m/s by
pushing in the plunger with a speed of V!J = 0.03 m/s as shown in
Fig. P469. The surface of the deforming control volume consists of
the sides and end of the cylinder and the end of the plunger. The sys
tem consists of the water in the syringe at r = 0 when the plunger
is at section (1) as shown. Make a sketch to indicate the control sur
face and the system when t = 0.5 s. 5.11 Air ﬂows steadily between two cross sections in a long,
straight section of 0.1m inside diameter pipe. The static tempera»
ture and pressure at each section are indicated in Fig. Pill. If the
average air velocity at section (1) is 205 m/ s, determine the average
air velocity at section (2). Section (1) . Section (2)
p1 = 77 kPa (abs) p2 = 45 kPa tabs}
T1=268K T2=240K
V1 = 205 mfs HFIGURE 135.11 5.12 A hydraulic jump (see Video V1010) is in place downstream
from a spillway as indicated in Fig. P112. Upstream of the jump,
the depth of the stream is 0.6 ft and the average stream velocity is
18 ft/s. Just downstream of the jump, the average stream velocity is
3.4 ft/s. Calculate the depth of the stream, h, just downstream of
the jump. 3 FIGURE P5.12 5.13 'An evaporative cooling tower (see Fig. P513) is used to cool
water from 110 to 80°F. Water enters the tower at a rate of
250,000 lbm/ hr. Dry air (no water vapor) flows into the tower at a
rate of 151,0001bm/hr. If the rate of wet air flow out of the tower
is 156,900 lbrn/ hr, determine the rate of water evaporation in
lbrn/iu‘ and the rate of cooled water ﬂow in lbrn/ hr. Wet alr
rh = 156,900 lbmihr 1‘ Warm water —)
th = 250.000 tbmlhr _ Dry air w»)—
m =151.000lbmihr EFIGUHE P5.13 Cooled
water 5.14 At cruise conditions, air flows into a jet engine at a steady
rate of 65 lbm/s. Fuel enters the engine at a steady rate of 0.60 lbm/s.
The average velocity of the exhaust gases is 1500 ft/ s relative to the
engine. If the engine exhaust effective crosssectional area is
3.5 ftz, estimate the density of the exhaust gases in lbrn/ft3. 247 Problems 5.15 Water at 0.1 111313 and alcohol {SG=0.8) at 0.3 msls are mixed
in a y—duct as shown in Fig. 5.15. What is the average density of the
mixture of alcohol and water? Water and
alcohol mix
—l>
Water 0 \
Q = 0.1 male \
Alcohol (so = 0.8}
Q a 0.3 mals
> FIGURE P5.15 5.16 Freshwater ﬂows steadily into an open 55gal drum initially
ﬁlled With seawater. The freshwater mixes thoroughly with the sea—
water and the mixture overﬂows out of the drum. If the freshwater
ﬂowrate is 10 galjmin, estimate the time in seconds required to de«
crease the difference between the density of the mixture and the
density of freshwater by 50%. Section 5.1.2 Fixed, Nondet‘orming Control Volume—
Nonum‘form Velocity Proﬁle 5.17 A water ‘et pump (see Fig. Pit?) involves a jet cross~sectional
area of 0.01 111 , and a jet velocity of 30 m/s. The jet is surrounded by
entrained water. The total cross—sectional area associated with the
jet and entrained streams is 0.075 m2. These two ﬂuid streams leave
the pump thoroughly mixed with an average velocity of 6 m/s
through a cross—sectional area of 0.075 ml. Determine the pumping
rate (i.e., the entrained fluid ﬂowrate) involved in liters/s. EFIGURE P5.‘i7 5.18 ’IWo rivers merge to form a larger river as shown in Fig. P5.18. At a location downstream from the junction (before the
two streams completely merge), the nonuniform velocity proﬁle is as shown and the depth is 6 ft. Determine the value of V. E FIGURE £35.18 248 Chapter 5 E Finite Control Volume Analysis 5.19 Various types of attachments can be used with the shop vac 5.23 An incompressible ﬂow velocity ﬁeld (water) is given as
shown in Video V5.2. Two such attachments are shown in Fig. P5.19 1 1 ma nozzle and a brush. The ﬂowrate is 1 ft3/s. (a) Determine the V = __ 9r + _ 36 m/s
average velocity through the nozzle entrance, V”. (b) Assume the air I‘ enters the brush attachment in a radial direction all around the brush
with a velocity proﬁle that varies linearly from 0 to Vb along the length
of the bristles as shown in the ﬁgure. Determine the value of Vb. h where r is in meters. (3) Calculate the mass ﬂowrate through the
cylindrical surface at r = 1 m from 7. = 0 to z = 1 In as shown in
Fig.P5.23a. (b) Show that mass is conserved in the annular control
voiurnefromrz lintorz 2mandz= Otoz = 1 masshown Q = 1 “3,5 in Fig. P5231). Z 1m Vn a Fl G U Fl E P5.19 (a) W ‘ EFEGUHEP5.23
5.20 An appropriate turbulent pipe flow velocity proﬁle is R _ m, 5.24 Flow of a viscous ﬂuid over a ﬂat plate surface results in the
V g ”a (4) i development of a region of reduced velocity adjacent to the wetted
R surface as depicted in Fig. P524. This region of reduced flow is h 2 t l' 1 . : l . z . d‘ , called a boundary layer. At the leading edge of the plate, the veloc—
:vﬂ de €622 nit‘ifielciad r1321: 12:3]er eht erﬁfﬁggims’. Re th 6133353131]: ity proﬁle may be considered uniformly distributed with a value U.
erage velocity, 5, to centerii n e velocity, u” for (a) n : 4, (b) n z 6, All along the outer edge of the boundary layer, the fluid velocity = 8, d = 1 .C th d'ff t 1 ‘ ﬁl _ component parallel to the plate surface is also U. If thexdirection
(GE—M ( ) n 0 ompare e l eren “3 oc1ty pro es velocity proﬁle at section (2) is @Ps shown in Fig. P521, at the entrance to a 3ft—wide channel 1 1,?
e velocity distribution is uniform with a velocity V. Further down i = (X)
stream the velocity proﬁle is given by u : 4y H 23:2, where u is in U 5
ft/3 andy is in ft Detel‘miﬂﬁ the “11116 0f V develop an expression for the volume flowrate through the edge of the boundary layer from the leading edge to a location downstream
at x where the boundary layer thickness is 8. Section (2)
Section {1} \ U Outer edge IFIGURE P5.21 5.22 A water flow situation is described by the velocity ﬁeld equation
v 2 (3x + 2)i + (2y e 4)} — 5zi< ft/s wherex, y, and z are in feet. (a) Determine the mass ﬂowrate through E F  G u n E P5.24 the rectangular area in the plane corresponding to z = 2 feet having comers at (x, y, z) x (0, 0, 2), (5, 0, 2), (5, 5, 2), and (0, 5, 2) as shown Section 5.1.2 Fixed, Nondeforming Control Volume—
in Fig P5.22a. (b) Show that mass is conserved in the control volume Unsteady Flow having corners at (x, y, z) = (0,0, 2), (5, O, 2), (5, 5, 2), (0, 5, 2), (0, O, 0), )1: . _ _ _
(5’ 0’ 0), (5, 5, a), and (0, 5, 0), as shown in Fig. pink (2; at standard conditions enters the compressor shown 1n Fig.
P525 at a rate of 10 ft3/s. It leaves the tank through a I.24n.diame— ter pipe with a density of 0.0035 slugs/ft3 and a uniform speed of
700 ft/s. (a) Determine the rate (slugs/s) at which the mass of air in
the tank is increasing or decreasing. (b) Determine the average time
rate of change of air density within the tank. Compressor Tank volume = 20113 10 rats. (a) (1;) 9.00233 slugsma
FIGURE 135.22  _EFIGURE 135.25 ...
View
Full Document
 Spring '09
 Smith

Click to edit the document details