Aeronautical Engineering Chapter 3 Homework When immersed in a uniform stream

Solution 3.71

From Prob. 3.50, recall that the essential data was

Problem 3.72*

When immersed in a uniform stream, a thick elliptical cylinder creates a broad downstream

wake, as idealized in Fig. P3.72. The pressure at the upstream and downstream sections are

approximately equal, and the fluid is water at 20 8 C. If U0 = 4 m/s and L = 80 cm, estimate the

drag force on the cylinder per unit width into the paper. Also compute the dimensionless drag

coefficient CD = 2F/(



Uo2bL).

Solution 3.72

This is a ‘numerical’ version of the “analytical” body-drag Prob. 3.44. The student still must

Problem 3.73

A pump in a tank of water at 20°C directs a jet at 45 ft/s and 200 gal/min against a vane, as

shown in Fig. P3.73. Compute the force F to hold the cart stationary if the jet follows (a) path A

or (b) path B. The tank holds 550 gal of water at this instant.

Solution 3.73

The CV encloses the tank and passes through jet B.

(a) For jet path A, no momentum flux crosses the CV, therefore F = 0 Ans. (a)

Problem 3.74

Water at 20°C flows down a vertical 6-cm-diameter tube at 300 gal/min, as in the Fig P3.74..

The flow then turns horizontally and exits through a 90° radial duct segment 1 cm thick, as

shown. If the radial outflow is uniform and steady, estimate the forces (Fx, Fy, Fz) required to

support this system against fluid momentum changes.

Solution 3.74

First convert 300 gal/min = 0.01893 m3/s, hence the mass flow is



Q = 18.9 kg/s. The vertical–

tube velocity (down) is Vtube = 0.01893/[(



/4)(0.06)2] = −6.69 k m/s. The exit tube area is

(



/2)Rh = (



/2)(0.15)(0.01) = 0.002356 m2, hence

Vexit = Q/Aexit = 0.01893/0.002356 = 8.03 m/s. Now estimate the force components:

Problem 3.75*

A jet of liquid of density ρ and area A strikes a block and splits into two jets, as in Fig. P3.75.

Assume the same velocity V for all three jets. The upper jet exits at an angle θ and area αA . The

lower jet is turned 90° downward. Neglecting fluid weight, (a) derive a formula for the forces

(Fx , Fy ) required to support the block against fluid momentum changes. (b) Show that Fy = 0

only if α ≥ 0.5. ( c ) Find the values of α and θ for which both Fx and Fy are zero.

Solution 3.75

(a) Set up the x– and y-momentum relations:

( cos ) ( )

xx

F F m V m V where m AV of the inlet jet

  

 = = − − − =

xy

Problem 3.76

A two-dimensional sheet of water, 10 cm thick and moving at 7 m/s, strikes a fixed wall inclined

at 20° with respect to the jet direction. Assuming frictionless flow, find (a) the normal force on

the wall per meter of depth, and the widths of the sheet deflected (b) upstream, and

(c) downstream along the wall.

Solution 3.76

(a) The force normal to the wall is due to the jet’s momentum,

Problem 3.77

Water at 20°C flows steadily through a reducing pipe bend, as in Fig. P3.77. Known conditions are

p1 = 350 kPa, D1 = 25 cm, V1 = 2.2 m/s, p2 = 120 kPa, and D2 = 8 cm. Neglecting bend and water

weight, estimate the total force which must be resisted by the flange bolts.

Solution 3.77

First establish the mass flow and exit velocity:

Problem 3.78

A fluid jet of diameter D1 enters a cascade of moving blades at absolute velocity V1 and angle



1 and it leaves at absolute velocity V1 and angle



2 as in Fig. P3.78. The blades move at

velocity u. Derive a formula for the power P delivered to the blades as a function of these

parameters.

Solution 3.78

Let the CV enclose the blades and move upward at speed u, so that the flow appears steady in

that frame, as shown at right. The relative velocity Vo may be eliminated by the law of cosines:

Problem 3.79

The Saturn V rocket in the chapter opener photo was powered by five F-1 engines, each of which

burned 3,945 lbm of liquid oxygen and 1738 lbm of kerosene per second. The exit velocity of

burned gases was approximately 8,500 ft/s. In the spirit of Prob. P3.34, neglecting external

pressure forces, estimate the total thrust of the rocket, in lbf.

Solution 3.79

We are given the inlet mass flow,

so we don’t need the (huge) exit area,

Problem 3.80

A river of width b and depth h1 passes over a submerged obstacle, or “drowned weir,” in

Fig. P3.80, emerging at a new flow condition ( V2 , h2 ). Neglect atmospheric pressure, and

assume that the water pressure is hydrostatic at both sections 1 and 2. Derive an expression for

the force exerted by the river on the obstacle in terms of V1 , h1 , h2 , b , ρ , and g . Neglect water

friction on the river bottom.

Solution 3.80

The CV encloses (1) and (2) and cuts through the gate along the bottom, as shown. The volume

flow and horizontal force relations give

RP-1

Ve

CV

Problem 3.81

Torricelli’s idealization of efflux from a hole in the side of a tank is

2,V gh

as shown in

Fig. P3.81. The cylindrical tank weighs 150 N when empty and contains water at 20°C. The

tank bottom is on very smooth ice (static friction coefficient



 0.01). For what water depth h

will the tank just begin to move to the right?

Solution 3.81

The hole diameter is 9 cm. The CV encloses the tank as shown. The coefficient of static friction

is



= 0.01. The x-momentum equation becomes

Problem 3.82*

The model car in Fig. P3.82 weighs 17 N and is to be accelerated from rest by a 1–cm-diameter

water jet moving at 75 m/s. Neglecting air drag and wheel friction, estimate the velocity of the

car after it has moved forward 1 m.

Solution 3.82

Problem 3.83

Gasoline at 20°C is flowing at V1 = 12 m/s in a 5-cm-diameter pipe when it encounters a 1-m

length of uniform radial wall suction. At the end of this suction region, the average fluid velocity

has dropped to V2 = 10 m/s. If p1 = 120 kPa, estimate p2 if the wall friction losses are neglected.

Solution 3.83

The CV cuts through sections 1 and 2 and the inside of the walls. We compute the mass flow at

each section, taking



 680 kg/m3 for gasoline

Problem 3.84

Air at 20°C and 1 atm flows in a 25–cm-diameter duct at 15 m/s, as in Fig. P3.84. The exit is

choked by a 90° cone, as shown. Estimate the force of the airflow on the cone.

Solution 3.84

The CV encloses the cone, as shown. We need to know exit velocity. The exit area is

approximated as a ring of diameter 40.7 cm and thickness 1 cm:

Problem 3.85

The thin-plate orifice in Fig. P3.85 causes a large pressure drop. For 20°C water flow at

500 gal/min, with pipe D = 10 cm and orifice d = 6 cm, p1 − p2  145 kPa. If the wall friction is

negligible, estimate the force of the water on the orifice plate.

Solution 3.85

The CV is inside the pipe walls, cutting through the orifice plate, as shown. At least to one-

Problem 3.86

For the water-jet pump of Prob. 3.36, add the following data: p1 = p2 = 25 lbf/in2, and the

distance between sections 1 and 3 is 80 in. If the average wall shear stress between sections

1 and 3 is 7 lbf/ft2, estimate the pressure p3. Why is it higher than p1?

Problem 3.36

The jet pump in Fig. P3.36 injects water at U1 = 40 m/s through a 3-in pipe and entrains a

secondary flow of water U2 = 3 m/s in the annular region around the small pipe. The two flows

Solution 3.86

Problem 3.87

A vane turns a water jet through an angle



, as shown in Fig. P3.87. Neglect friction on the

vane walls. (a) What is the angle



for the support force to be in pure compression?

(b) Calculate this force if the water velocity is 22 ft/s and the jet cross-section is 4 in2.

Solution 3.87

(a) From the solution to Example 3.8, the support will be in pure compression (aligned with F) if

the vane angle is twice the support angle.

Problem 3.88

The boat in Fig. P3.88 is jet-propelled by a pump which develops a volume flow rate Q and

ejects water out the stern at velocity Vj. If the boat drag force is F = kV2, where k is a constant,

develop a formula for the steady forward speed V of the boat.

Solution 3.88

Let the CV move to the left at boat speed V and enclose the boat and the pump’s inlet and exit.

Then the momentum relation is

Problem 3.89

Consider Fig. P3.36 as a general problem for analysis of a mixing ejector pump. If all conditions

(p,



, V) are known at sections 1 and 2 and if the wall friction is negligible, derive formulas for

estimating (a) V3 and (b) p3.

Solution 3.89

Use the CV in Prob. 3.86 but use symbols throughout. For volume flow,

Problem 3.90

As shown in Fig. P3.90, a liquid column of height h is confined in a vertical tube of cross-

sectional area A by a stopper. At t = 0 the stopper is suddenly removed, exposing the bottom of

the liquid to atmospheric pressure. Using a control volume analysis of mass and vertical

momentum, derive the differential equation for the downward motion V(t) of the liquid. Assume

one-dimensional, incompressible, frictionless flow.

Solution 3.90

Let the CV enclose the cylindrical blob of liquid. With density, area, and the blob volume

constant, mass conservation requires that V = V(t) only. The CV accelerates downward at blob

Problem 3.91

Extend Prob. P3.90 to include a linear (laminar) average wall shear stress resistance of the form

τ ≈ cV , where c is a constant. Find the differential equation for dV/dt and then solve for V (t),

assuming for simplicity that the wall area remains constant.

Solution 3.91

The downward momentum relation from Prob. 3.90 above now becomes

Problem 3.92*

A more involved version of Prob. 3.90 is the elbow-shaped tube in Fig. P3.92, with constant cross-

sectional area A and diameter

,Dh

L. Assume incompressible flow, neglect friction, and

derive a differential equation for dV/dt when the stopper is opened. Hint: Combine two control

volumes, one for each leg of the tube.

Solution 3.92

Use two CV’s, one for the vertical blob and one for the horizontal blob, connected as shown by

pressure.

Problem 3.93

According to Torricelli’s theorem, the velocity of a fluid draining from a hole in a tank is

V  (2gh)1/2, where h is the depth of water above the hole, as in Fig. P3.93. Let the hole have

area Ao and the cylindrical tank have cross-section area Ab >>Ao . Derive a formula for the time

to drain the tank completely from an initial depth ho.

Solution 3.93

Problem 3.94

A water jet 3 inches in diameter strikes a concrete (SG = 2.3) slab which rests freely on a level

floor. If the slab is 1 ft wide into the paper, calculate the jet velocity which will just begin to tip

the slab over.

Solution 3.94

For water let



= 1.94 slug/ft3. Find the water force and then take moments about the lower left

corner of the slab, point B. A control volume around the water flow yields

Problem 3.95

A tall water tank discharges through a well-rounded orifice, as in Fig. P3.95. Use the Torricelli

formula of Prob. P3.81 to estimate the exit velocity. (a) If, at this instant, the force F required to

hold the plate is 40 N, what is the depth h ? (b) If the tank surface is dropping at the rate of

2.5 cm/s, what is the tank diameter D ?