Chapter 5 The Reynolds transport theorem is used to relate

5.39: PROBLEM DEFINITION

Situation:

InFig5.11in§5.2ofEFM10e,

a. the CV is passing through the system.

b. the system is passing through the CV.

SOLUTION

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5.40: PROBLEM DEFINITION

Situation:

Reynolds transport theorem.

Find:

The purpose of Reynolds transport theorem.

SOLUTION

The Reynolds transport theorem is used to relate Lagrangian equations to their

42

5.41: PROBLEM DEFINITION

Situation:

Mass is ﬂowing into and out of a tank.

Vi=10m/s,Ai=0.10 m2,ρi=3.00 kg/m3.

Vo=5m/s,Ao=0.20 m2,ρo=2.00 kg/m3.

Find:

Select the statement(s) that are true.

SOLUTION

Mass ﬂow out

Mass ﬂow in

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5.42: PROBLEM DEFINITION

Situation:

A piston in a cylinder is moving up and control consists of volume in cylinder.

Find:

Indicate which of the statements are true.

SOLUTION

a) True, there is no ﬂow entering or leaving across the control surface.

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5.43: PROBLEM DEFINITION

Situation:

Two ﬂow cases: a closed tank is ﬁlled with a ﬂuid and a pipe contracts.

V=12ft/s,A=1.5ft

2,ρ=2slug/ft3..

V1=1ft/s,A1=2ft

2,ρ1=2slug/ft3.

V2=2ft/s,A2=1ft

2,ρ2=2slug/ft3.

Find:

(a) Value of b.

(b) Value of dBsys/dt.

(c) Value of PbρV·A

(d) Value of d/dt Rcv bρdV

SOLUTION

Case (a) Case (b)

1) b=1 1) B=1

45

5.44: PROBLEM DEFINITION

Situation:

The law of conservation of mass for a closed system requires that the mass of the

system is:

a. constant

b. zero

SOLUTION

46

5.45: PROBLEM DEFINITION

Situation:

Consider the simpliﬁed form of the continuity equation, Eq. 5.29 of EFM10e. An

engineer is using this equation to ﬁnd the QCof a creek at the conﬂuence with a

large river, because she has automatic electronic measurements of the river discharge

upstream (QRu) and downstream (QRd)of the creek conﬂuence.

Find:

a. Which of the 3 terms on the left-hand side of Eq. 5.29 will the engineer assume

is zero? Why?

b. Sketch the creek and the river and sketch the CV you would select to solve this

problem.

SOLUTION

a. The engineer will assume that the d

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5.46: PROBLEM DEFINITION

Situation:

Pipe ﬂows full with water.

Find:

Isitpossibleforthevolumeﬂow rate into the pipe to be diﬀerent than the ﬂow

rate out of the pipe?

SOLUTION

Application of the continuity equation to a control volume passing through the inlet

section and outlet section shows

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5.47: PROBLEM DEFINITION

Situation:

Air ﬂows in a tube.

Find:

Is it possible for the mass ﬂow rate into the tube to be diﬀerent than the ﬂow rate

out of the tube?

Air is pumped into one end of a tube at a certain mass ﬂow rate. Is it necessary that

thesamemassﬂow rate of air comes out the other end of the tube?

Application of the continuity equation over a control surface that includes the inlet

49

5.48: PROBLEM DEFINITION

Situation:

Tire develops a leak.

Find:

How do air and density change with time?

How is air density related to tire pressure?

Assumptions:

Constant temperature.

SOLUTION

If an automobile tire develops a leak, how does the mass of air and density change

inside the tire with time?

Assuming the temperature remains constant, how is the change in density related to

the tire pressure?

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5.49: PROBLEM DEFINITION

Situation:

Two pipes are connected in series.

D1=2D2,V1=4m/s.

Find:

Velocity in smaller pipe ( m/s).

SOLUTION

Use continuity equation for discharge. Q=AV which is valid since density is con–

stant.

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5.50: PROBLEM DEFINITION

Situation:

The level in the tank is inﬂuenced by the motion of pistons A and B moving left.

VA=2VB,DA=3in,DB=6in.

Find:

Determine whether the water level is rising, falling or staying the same.

Sketch:

PLAN

Apply the continuity equation. Select a control volume as shown above. Assume it

is coincident with and moves with the water surface.

SOLUTION

Continuity equation

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5.51: PROBLEM DEFINITION

Situation:

Two round plates move together. At the instant shown, the plate spacing is h.

Air ﬂows across section A with a speed V0.

Find:

An expression for the radial component of convective acceleration at section A.

Sketch:

Assumptions:

Assume V0is constant across section A.

Assume the air has constant density.

PLAN

Apply the continuity equation to the control volume deﬁned in the problem sketch.

SOLUTION

Continuity equation

Convective acceleration

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5.52: PROBLEM DEFINITION

QA=0.04tm3/s,QB=0.006t2m3/s.

Aexit =0.01 m2,t=1s.

Find:

Velo city at the exit, Vexit.

Acceleration at the exit, aexit.

Assumptions:

Incompressible ﬂow.

PLAN

Apply the continuity equation.

SOLUTION

Since the ﬂow is incompressible, the unsteady term is zero. Continuity equation

The acceleration along a pathline at the (s→x)exitis

∂t +V∂V

∂x

Since Vvaries with time, but not with position, there is no convective acceleration

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5.53: PROBLEM DEFINITION

D=0.1m,h=0.6cm.

Find:

(a) Expression for acceleration at point A.

(b) Value of acceleration at point A.

(c) Velocity in the pipe.

PLAN

Apply the ﬂow rate equation.

SOLUTION

a)

Flow rate equation

Evaluate convective acceleration along a radial pathline (s→r)

b)

Vpipe =Q

Apipe

=(0.380 m3/s)

π

4(0.1m)

2

Vpipe =48.4m/s

c)

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5.54: PROBLEM DEFINITION

Situation:

Air ﬂow downward through a pipe and then outward between two parallel disks.

Q=Q0(t/t0),r=20cm.

D=10cm,h=1cm.

t0=1s,Q0=0.1m

3/s.

Find:

(a) At t=2 s, acceleration at point A: a2.

(b) At t=3 s, acceleration at point A: a3.

SOLUTION

Local acceleration

From solution to Problem 5.53 in EFM10e

At t=3s, Q=0.3m3/s

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5.55: PROBLEM DEFINITION

Situation:

Water ﬂows into a tank through a pipe on the side and then out a pipe on the

bottom of the tank.

Aout =Ain =0.0025 m2,Atank =0.1m

2,

At h=1m,dh/dt =0.1m,V =√2gh.

Find:

Velocity in the inlet: Vin.

Assumptions:

Incompressible ﬂow.

PLAN

Apply the continuity equation. Let the control surface surround the liquid in the

tank and let it follow the liquid surface at the top.

SOLUTION

Continuity equation

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5.56: PROBLEM DEFINITION

Situation:

A bicycle tire is inﬂated with air.The density of the air in the inﬂated tire is 0.4

lbm/ft3.

V=0.035 ft3,Qin =0.8ft

3/min.

Find:

Time needed to inﬂate the tire.

Properties:

ρin =0.075 lb/ft3,ρ

CV =0.04 lb/ft3.

PLAN

Apply the continuity equation. Select a control volume surrounding the air within

tire.

SOLUTION

Continuity equation

or

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5.57: PROBLEM DEFINITION

Situation:

A cylinder falls in a tube containing a liquid.

VC=4 ft/s,Dtube =8in,Dcylinder =6in.

Find:

Mean velocity of the liquid in between the cylinder and the wall.

Sketch:

PLAN

Apply continuity equation and let the c.s. be ﬁxedexceptatthebottomofthe

cylinder where the c.s. follows the cylinder as it moves down. The top of the control

volume is stationary with resepct to the wall.

SOLUTION

Continuity equation

59

5.58: PROBLEM DEFINITION

Situation:

A round tank is being ﬁlled with water.

Vp=10ft/s,DT=4ft,Dp=1ft.

Find:

Rate at which the water surface is rising.

Sketch:

PLAN

Apply the continuity equation and let the c.s. move up with the water surface in the

tank.

SOLUTION

Continuity equation

but dh/dt =VRso

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