conservation of mass

Practice Problems on Conservation of Mass

C. Wassgren, Purdue University of 23 Last Updated: 2010 Sep 08

COM_01

Construct from first principles an equation for the conservation of mass governing the planar flow (in the xy plane) of a compressible liquid lying on a flat horizontal plane. The depth, h(x,t), is a function of position, x, and time, t. Assume that the velocity of the fluid in the positive x-direction, u(x,t), is independent of y. Also assume that the wavelength of the wave is much greater than the wave amplitude so that the horizontal velocities are much greater than the vertical velocities.

Answer(s):

0h uht x

If incompressible: 0h

uht x

x y

h(x,t) u(x,t)

free surface

liquid



COM_02

Consider the flow of an incompressible fluid between two parallel plates separated by a distance 2H. If the velocity profile is given by:

2

2

1H

yuu c

where uc is the centerline velocity, determine the average velocity of the flow, u . Assume the depth into the page is w.

Answer(s):

23 cu u

y H

H

u u



COM_03

An incompressible flow in a pipe has a velocity profile given by:

2

2

1)(R

ruru c

where uc is the centerline velocity and R is the pipe radius. Determine the average velocity in the pipe. Answer(s):

12 cu u

u r R



COM_04

Water enters a cylindrical tank through two pipes at volumetric flow rates of Q1 and Q2. If the level in the tank remains constant, calculate the average velocity of the flow leaving the tank through a pipe with an area, A3. Answer(s):

1 23

3

Q QV

A

h=constant

Q2

Q1

?3 V A3



COM_05

Water enters a cylindrical tank with diameter, D, through two pipes at volumetric flow rates of Q1 and Q2 and leaves through a pipe with area, A3, with an average velocity, 3V . The level in the tank, h, does not remain constant. Determine the time rate of change of the level in the tank. Answer(s):

2 1 3 32

4

Q Q V Adh

dt D

hconstant

Q2

Q1

3V A3

D



COM_06 A spherical balloon is filled through an area, A1, with air flowing at velocity, V1, and constant density, 1. The radius of the balloon, R(t), can change with time, t. The average density within the balloon at any given time is b(t). Determine the relationship between the rate of change of the density within the balloon and the rest of the variables. Answer(s):

21 1 1

3

4

43

bb

dRV A Rd dt

dt R

1, V1, A1

R(t)

b



COM_07

A box with a hole of area, A, moves to the right with velocity, ubox, through an incompressible fluid as shown in the figure. If the fluid has a velocity of ufluid which is at an angle, , to the vertical, determine how long it will take to fill the box with fluid. Assume the box volume is Vbox and that it is initially empty. Answer(s):

CV

box fluid sin

VT

u u A

ufluid

ubox

area, A

volume, Vbox



COM_08

Determine the rate at which fluid mass collects inside the room shown below in terms of , V1, A1, V2, A2, Vc, R, and . Assume the fluid moving through the system is incompressible. Answer(s):

2CV1 1 2 2 2

cos cdM

V A V A V Rdt

A2

V2

V1

A1

room

2R

2

2

1R

rVV c



COM_09

Water enters a rigid, sealed, cylindrical tank at a steady rate of 100 L/hr and forces gasoline (with a specific gravity of 0.68) out as is indicated in the drawing. The tank has a total volume of 1000 L. What is the time rate of change of the mass of gasoline contained in the tank? Answer(s):

gasgas H20 gas H20 H20

dMQ SG Q

dt

dMgas/dt = -68 kg/hr = 0.019 kg/s

water

gasoline



COM_10

In order to avoid getting wet, is it better to walk or run in the rain? Assume that the rain falls at an angle from the vertical. Clearly state your assumptions and provide justification for your conclusion. Answer(s): So, what is the best approach to minimize getting wet when it’s raining? It depends. If the wind (and, hence, the rain) is always in your face (the first case considered in this solution) then it’s best to run as fast as possible. However, if the wind is at your back, then if

top

side

tanA

A

you are better off traveling at the horizontal speed of the rain, i.e. U = Vsin. Otherwise you should still run.



COM_11

The (symmetric) V-shaped container shown in the figure has width, b, into the page and is filled from the inlet pipe at volume flow rate, Q. Derive expressions for: a. the rate of change of the surface height, dh/dt b. the time required for the surface to rise from h1 to h2.

Answer(s):

tan

2

dhQ

dt hb

2 22 1

2 1 tan

b h ht t

Q

h(t)

Q



COM_12

An incompressible fluid is being squeezed outward between two large circular disks by the uniform downward motion of the upper disk moving at velocity, V0. Assuming one-dimensional radial outflow, derive an expression for the radial flow at any given radius, V(r). Answer(s):

0

1

2

rV V

h

V0

r

V(r) h(t)

fixed circular disk



COM_13

An air bellows may be modeled as a deforming wedge-shaped volume as shown in the figure below. Let b be the width of the bellows into the paper. a. Determine the outlet mass flow rate for the conditions shown.

b. Determine the mass flow rate, with uncertainty, for the following conditions. L = 30 1 mm b = 25 1 mm = /6 0.1 rad d/dt = -/3 0.2 rad/s air = 1.2 kg/m3 (no uncertainty)

Answer(s):

2 2out sec

dm L b

dt

5out 3.8*10 kg/sm

out23.6%mu

L

air out



COM_14 The motion of a hydraulic cylinder is cushioned at the end of its stroke by a piston that enters a hole as shown. The cavity and cylinder are filled with hydraulic fluid of uniform density, .

a. Obtain an expression for the average velocity, Vout, at which hydraulic fluid escapes from the cylindrical hole

assuming that the cylinder moves at a constant velocity, Vcyl.

b. Determine the velocity, Vout, with relative uncertainty, for the following conditions.

= 900 5 kg/m3 L = 100 0.1 mm R = 15 0.1 mm r = 10 0.1 mm Vcyl = 100 1 mm/s

Answer(s):

out cyl 2

1

1V V

Rr

out 80.0 3.6 mm/sV

R

r

Vcyl

Vout

x

L

hydraulic fluid of density,



COM_15

Calculate the mass flux through the control surface shown below. Assume a unit depth into the page. Answer(s):

m VD

V

D

control surface



COM_16

A new jet engine is being tested in a wind tunnel. Air, with properties characteristic of U.S. Standard Atmosphere at 6000 m enters the engine at a velocity of 275 m/s through a circular intake port of radius 0.5 m. Fuel enters the engine at a mass flow rate of 2.5 kg/s. If the gas leaves the engine with an average velocity of 300 m/s through an exit port of radius 0.4 m, calculate the density of the exhaust gas. Answer(s):

inlet inlet fueloutlet inlet

outlet outlet outlet outlet

U A m

U A U A

outlet = 0.961 kg/m3



COM_17

The spray system shown in the figure below is used to manufacture an aluminum film. Plot the thickness of the film, h, as a function of distance along the belt, x. Answer(s):

S S

F

Vh x

R

R

L

R

S, VS

x

h

F

The conveyor has a width, w, into the page.



COM_19

A sanding operation injects 105 particles/s into the air in a room as shown in the figure. The amount of dust in the room is maintained at a constant level by a ventilating fan that draws clean air into the room at section 1 and expels dusty air at section 2. Determine the concentration of particles (particles/m3) in the exhaust air for steady state conditions.

Answer(s): C2 = 5*105 part/m3

A2=0.1 m2

A1

V2=2 m/s

V1

sander

105 part/s



COM_20

Air at standard conditions enters the compressor shown in the figure below at a rate of 10 ft3/s. The air leaves the tank through a 1.2 in. diameter pipe with a density of 0.0035 slug/ft3 and a uniform speed of 700 ft/s. Determine the average time rate of change of air density within the tank. Answer(s):

i i o o ot

t

d Q U A

dt V

4 32.28*10 slug ft std

dt

1.2 in. 700 ft/s

0.0035 slug/ft3 10 ft3/s

0.00238 slug/ft3

tank volume = 20 ft3



COM_21

An incompressible fluid flows past an impermeable flat plate of width b into the page as shown in the figure. The velocity profile at the inlet is uniform with u = U0 while the exit velocity has the cubic profile:

3

0

3

2

u

U

where = y/

Compute the volume flow rate, Q, across the top surface of the control volume. Answer(s): Not yet available.

Q

y



COM_22

A hydraulic accumulator is designed to reduce pressure pulsations in a machine tool hydraulic system. For the instant shown, determine the rate at which the accumulator gains or loses hydraulic oil. Answer(s): Not yet available.

5.75 gpm 4.35 ft/s

1.25 in.



COM_23

Consider the process of blowing up a spherical balloon. Measurements indicate that the “surface tension” of the balloon material is k (assumed constant here with units of force per unit length). Assuming that an air compressor used to blow up the balloon can deliver a constant mass flow rate of air of m , plot the balloon radius, Rb, as a function of time assuming Rb(t) = 0. Answer(s):

242

3 b bR R t

atmb b

RTR R

k

3

atm

atm

RTmt t

k

0

1

2

3

4

5

6

0 100 200 300 400 500 600 700 800

dimensionless time, t'

dim

en

sio

nle

ss

ba

lloo

n r

ad

ius

, R' b



COM_24

Consider a simple roof with the geometry shown below. Assume that on each side of the house there is a gutter running the length of the house. The gutter has a width of w. During a rainstorm in which rain falls straight down at a rate of r (in depth per hour, e.g. 1 in. per hour), estimate the rate at which the depth in the gutter increases if the gutter exit is clogged. Assume that no water accumulates on the roof, i.e. the water landing on the roof immediately drains into the gutter. If L = 100 ft, W = 40 ft, w = 3.5 in., h = 3.5 in., and r = 1 in./hr estimate how long it will take before the water in the gutter spills out. Answer(s): tfill = 3.0 min

W

L

w h

conservation of mass

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