# CVEN-3313 Theoretical Fluid Mechanics MODULE#2 Emptying of a

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```					                                                                                  February 21, 2006

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

1. OBJECTIVE
To apply the principle of mass conservation for unsteady flow of water through an orifice in a
vertical tank.

2. THEORY
1) For mass conservation, 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 :

v 2 = 2 gh                              Equation (1)

where, v2 = 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 Qtheoretical through the orifice is :

Qtheoretical = av 2 = a 2 gh                     Equation (2)

where, a = cross sectional area of the orifice.

3) Due to frictional losses, the actual discharge Qactual will be less than the theoretical discharge.
That is,
Qactual = C d a 2 gh                             Equation (3)

where, Cd = discharge coefficient for an orifice, which accounts for frictional energy losses.
Cd can be determined experimentally.

dh
4) The rate at which the water surface in the tank decreases is - A        , where A = cross sectional
dt
area of the tank, and dh/dt = change of water height at the surface over time. Since the
discharge Qactual at the orifice must be equal to the rate at which the water surface in the tank
decreases,

dh
A      = !C d a 2 gh                          Equation (4)
dt

Note : There are two different cross sectional areas A1 (full section) and A2 (half section).
1
3. PROCEDURE
1) Place the module on the hydrostatic bench, and attach the 3mm-diameter orifice plate.
2) Cover the orifice, and fill 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-inches, and record the time (t=ti) when the water level reaches the
specified heights (e.g., h=15-inch, 14, 13, etc), as indicated on the data table.
For h<1-inch, measure times for h=0.75, 0.5, and 0.25-inch.

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 Equation (1). Refer to figure 3.11 in the textbook.

2) Plot h vs. t for 0<h<16-inch. Why does the change in height get smaller as time goes on?

3) Plot   h vs. t for 0<h<16-inch. Explain why the graph is shaped as it is.

4) Derive an analytical expression for water level at the surface   h as a function of time t, i.e.,
h =f(t). Use equation (4). It will be necessary to integrate.

5) The equation from 4) should be of the form h = mt + C where 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 this slope to find the discharge coefficient C d for each Part.

6) Data was recorded until the water level h(t) reached 0.25-inch. How much time would have
elapsed if the experiment was continued until h(t) = 0? Indicate this point on the graphs from
2) and 3). Do the two graphs appear to agree?

7) If A2 is further reduced, at some point Equation (4) will no longer be valid, because Equation
(1) will no longer be valid. State why this is the case. For this case, derive a new Equation
(1) for V2 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 Cd would be? State
qualitatively why. What would you suggest for making Cd closer to the maximum value?

6. MISCELLANEOUS
1) Type a lab report, completely answering all questions in this handout.
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 due March 10, 2006, by 5:00 p.m.

2
Definitions
h(t)    : water level height from orifice center (in)
t       : time for water surface to decrease from h=16-inch to a specified height (sec).
A1      : cross-sectional area of the tank for Part A (= 23.22 in2)
A2      : cross-sectional area of the tank for Part B (= 10.93 in2)
a       : cross-sectional area of the orifice (= 0.01096 in2)
g       : gravity (= 32.2 ft/sec2)

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