CVEN-3313 Theoretical Fluid Mechanics MODULE#2 : Emptying of a Tank by an Orifice

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1

October 10,

2003


CVEN
-
3313 Theoretical Fluid Mechanics

MODULE#2 : Emptying of a Tank by an Orifice


1. OBJECTIVE

To
ap
ply the
principle of mass conservation

for

unsteady
f
low
of water
through an orifice in a
vertical tank.


2. THEORY

1)

For mass conser
vation, the discharge through
an

orifice is equal to the drainage rate from the
tank.
By
applying
Bernoulli’s equation

between points 1 and 2 in
Figure 1
, it can be shown
that :






Equatio
n
(1)


where,
v
2

= velocity at the orifice,
h

= elevation of water surface above the level of orifice
at time
t
,
g

= acceleration due to gravity.


2)

Theoretical discharge
Q
theore
tical

through the orifice is :






Equation
(2)


where,
a

= cross sectional area of the orifice.


3)

Due to frictional losses, the actual discharge
Q
actual

will be less than the theoretical discharge.
That is,



Equation
(3)


where,
C
d

=
discharge coefficient

for
an
orifice
,

which

accounts for
frictional energy losses.
C
d

can be
determined experimentally.


4)

The rate
at which the water surface in the tank decreases
is
-
, where
A

= cros
s sectional
area of
the
tank,
and

dh/dt

= change of water height
at the surface over

time. Since the
discharge
Q
actual

at the orifice must be equal to the rate
at which the water surface in the tank
decreases
,






Equation

(4)


Note

: There are two different cross sectional areas
A
1

(full section) and
A
2

(half section).
2

3. PROCEDURE

1)
Place

the module on the hydrostatic bench
, and attach the
3mm
-
diameter

orifice
plate
.

2) Cover the orifice, and

f
ill the tank with water

above the 16
-
inch line
.

3
)
Uncover the orifice, and let

the water
discharge
through the orifice.

4
)
Begin

timing when
h
=16
-
in
ches,

and record
the
time (
t=t
i
) when the water level
reaches

the
specified heights

(
e.g.,
h
=15
-
in
ch
, 14
, 13, etc
)
,

as indicated on the data table
.


For
h
<1
-
in
ch
, measure time
s

for

h
=0.75, 0.5, and 0.25
-
in
ch
.


4.
RESULTS &
ANALYSIS

Let Part A of the procedure be for 7<h<16
-
inch, and Part B be for 0<h<6
-
inch.

1)
Use
Bernoulli’s equation

to derive E
quation
(1). Refer to figure 3.11 in
the

textbook.


2)
Plot
h

vs.
t

for 0<h<16
-
inch.

W
hy
does the
change

in height get smaller
as time goes on?


3
)

Plot

vs.
t

for 0<h<16
-
in
ch
.
Explain why the graph is shaped as it is.



4
)

Derive an a
nalytical expression for water level

at the surface


as a function of time
t
, i.e.,
=f(t)
. Use equation (4).

It

will
be necessary
to integrate.


5
)

The

equation from
4
) should be of the form

w
here
m
is the slope of the line.
Fit a line to
the

graph

from 3)
for
Part A and
Part
B

and

determine the slope of
each Part
.
Use th
is

slope to find the discharge coefficient

for
each Part.


6
)

D
ata
was recorded
until the water leve
l
h(t)

reached

0.25
-
in
ch
. How
much time would have
elapsed
if
the experiment
w
as

continued until
h(t)

= 0?
Indicate this point
on
the

graph
s from
2) and 3
). Do the two graphs appear to agree?


7
)

If
A
2

is further reduced,
at some point
E
q
uation

(4) wil
l no longer
be
valid
, because Equation
(1) w
ill

no longer be valid
.


State why

this is the case
.

For this case, derive a new
Equation
(1)

for V
2

in terms of
A, a, g, and h.

Then
, derive a new
Equation (4).

Finally, show that the
new Equation (4) goes to

the original Equation (4) as a/A goes to zero.


5. DISSCUSION


1
)
Under ideal conditions, what do you think the maximum value of
C
d

would be
? State
qualitatively

why. What would you suggest
for making

C
d

closer to the maximum value
?


6
. MISCELLANEO
US

1)

Type a lab report, completely answering all questions in this hando
ut.

2)

Please specify all variables and units. Remember units in graphs and tables.

3)

You will be graded as much for neatness and presentation as you will for correctness.

4)

Lab Reports are du
e
October 24, 2003, by 3:00 p.m.

3



Definitions


h(t)


: water level
height
from orifice

center

(in)


t


: time for water surface to
decrease

from h=16
-
in
ch

to a specified height (sec).


A
1

: cross
-
sectional area of the tank for
P
art A

(= 23.22 in
2
)



A
2

: cross
-
sectional area of the tank for
P
art B

(= 10.93 in
2
)


a

: cross
-
sectional area of the orifice (= 0.01096 in
2
)


g

: gravity (= 32.2 ft/sec
2
)