Experiment No. 3

Objective:(i) To evaluate the performance parameters, namely, discharge coefficient, velocity coefficient and contraction coefficient for a free jet flowing through an orifice or a nozzle.

(ii) To appreciate the limitations of Toricelli’s Theorem which states that the velocity of efflux from a tank open to the atmosphere is same as that of a rigid body falling from a height equal to the level in the tank above the vent.

(iii) To determine the time of emptying of the tank.

Theoretical Background: Consider the flow taking place from a round opening(opening or nozzle) provided in the side of a large tank as shown in the figure below. If one assumes that the area of the tank is much larger than that of the opening then, applying Bernoulli’s equation between the surface and the exit (points 1 and 2 respectively), we have,

Patmρg

+H=Patmρg

+ V2

2g(1 )

Thus,

V= (2gH )12 (2 )

By applying Bernoulli’s equation we are tacitly assuming that there are no losses between points (1) and (2). In reality the exit velocity, V, is less than that given by equation(2) and this loss is characterized by the velocity coefficient, C v, which is the ratio of the actual velocity at the exit to the theoretical estimate given in the

equation(2). If the exit is well rounded the losses are minimum and C v is close to 1.

Another effect comes to play if the opening is not well rounded. The flow in the tank occurs over a much larger area than the opening, thus the streamlines are converging as they approach the exit. If the exit is not provided with a long straight section, the streamlines will continue to converge after the exit, thus reducing the effective area of the opening. The minimum area thus formed is called the vena-contracta. The ratio of the area at the vena-contracta, Avc, to the area of the opening, A, is called coefficient of contraction, C c. The value of C c varies from 0.6 to 1 depending on the type of opening.

The rate of efflux from the opening may now be estimated as,

Q=C cC v (2 gH )1 /2 A(3)

We can define a discharge coefficient, Cd, as the ratio of the actual discharge to the theoretically expected discharge. Thus,

Cd=C cC v (2 gH )

12 A

(2 gH )12 A

=CcC v (4 )

The value of Cd may be estimated experimentally in two ways. The first method is to directly determine the actual flow rate for a head, H, by either the volumetric or the gravimetric method and then calculate Cd. The second method is to let the tank drain from an initial height H 1to a final height H 2 and find the elapsed time ∆ t. Then Cd may be estimated as follows: (Note that the total drop in the height should be small compared to H 1for this method to be accurate.)

Q=−A tdHdt

(5 )

Where, At is the cross sectional area of the tank.

Also,

Q=Cd A (2gH )12 (6 )

Thus,

Cddt=−( A tA ) (2gH )−12 dH (7 )

Cd=2( A tA ) (2g )−12 (√H 1−√H 2 )

∆ t(8 )

The two estimates of Cd may be compared and equation(4) can also be verified in the experiment.

Experimental setup: The rig consists of a constant head supply tank with an opening on the side (orifice or nozzle), a volumetric tank at the outlet, a sump and a pump. A schematic diagram of the set-up is shown in figure below.

Observations:

a) Cross sectional area of the volumetric tank, a= 29x46.5 cm2

b) Cross sectional area of the supply tank, At= 42x42 c m2

READINGS FOR ORIFICE:

c) Supply tank head, H=41 cm

d) Diameter of the opening, D=0.98 cm

e) Time taken for 0.1 m rise in the volumetric tank level, ∆t(s):

1)96 s

2)98 s

3)97 s

Average ∆ t=97 s

f) Volume flow rate, Q=0.1a/∆t = 56.998 cm3

s

g) Diameter of vena-contracta, Dvc =0.79 cm

h) Contraction coefficient, C c= (D vc /D)2=0.6498

i) Observation Table 1

Sr. No. Horizontal Distance, x(cm)

Vertical distance, z(cm)

z th=x2/4H

(cm)C v=

x

2√zH

1 0 0 0 -2 5 0.2 0.1524 0.8733 10 0.8 0.6097 0.8734 15 1.7 1.3719 0.8985 20 3 2.4390 0.9016 25 4.5 3.8109 0.9207 30 6.3 5.4878 0.9338 35 8.6 7.4951 0.9319 40 10.9 9.7561 0.94610 45 13.6 12.3475 0.95211 50 16.8 15.2439 0.952

Average C v=0.918

READINGS FOR NOZZLE:

c) Supply tank head, H=41 cm

d) Diameter of the opening, D=0.98 cm

e) Time taken for 0.1 m rise in the volumetric tank level, ∆t(s):

1)65 s

2)66 s

3)65 s

Average ∆ t=65.33 s

f) Volume flow rate, Q=0.1a/∆t = 84.629 cm3

s

g) Contraction coefficient, C c= (D vc /D)2≈1

h) Observation Table 1

Sr. No. Horizontal Distance, x(cm)

Vertical distance, z(cm)

z th=x2/4H C v=x /(2√ zH )

1 0 0 0 -2 5 0.3 0.1524 0.7123 10 0.8 0.6097 0.8734 15 1.7 1.3719 0.8985 20 2.9 2.4390 0.9176 25 4.5 3.8109 0.9207 30 6.4 5.4878 0.9258 35 8.6 7.4951 0.9319 40 11.1 9.7561 0.93710 45 13.9 12.3475 0.94211 50 17.3 15.2439 0.938

Average C v=0.899

For calculation ofCd:

Serial No. Level of water in the tank, Hi(cm)

Time elapsed, t 1 (s)

Theoretical estimate(Cd=1),t 2(s)

Cd=t 2/t 1

1 41 0 0 -2 36 42.76 42.56 0.9953 31 89.81 88.2 0.9824 26 141.59 137.69 0.9725 21 197.39 192.22 0.9736 16 258.83 253.73 0.980

7 11 337.72 325.88 0.9698 6 429.89 417.47 0.9719 1 594.07 570.53 0.960

Average Cd=0.975

Graphs Plotted:

X: Horizontal distance(cm) ;Z: Vertical distance (cm)

0 10 20 30 40 50 60

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

theoretical & experiment trajectories for ori-fice

experimentalTheoretical

0 10 20 30 40 50 60

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Comparision of theoretical and exper-imental trajectories of nozzle

ExperimentalTheoretical

X: Height of water in the tank(cm); Y: time of emptying(s)

0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

600

700

time of emptying v/s height of water

theoreticalexperimental

Results

PART I

Orifice

Velocity coefficient = 0.918

Contraction coefficient =0.6498

Discharge coefficient =0.5965

Nozzle

Velocity coefficient =0.899

PART II

Theoretical time to empty the tank=570.53

Actual time taken to empty the tank=594.07

PART III

Time taken to empty the tank=

Conclusions

PART I

Nozzle has coefficients closer to 1 as compared to orifice.

This is because the nozzle gradually converges to the given size, while in an orifice there is a sudden contraction for the fluid.

In an orifice the sudden contraction of liquid forces the jet to further contract after leaving the orifice. This results in lesser flow rates as compared to the theoretical values.

The same phenomenon occurs in a nozzle also. But since the nozzle itself gradually converges, the contraction of liquid after exiting the nozzle is much less as compared to an orifice.

This deviation is accounted for by Contraction coefficient.

PART II

Time taken to empty the tank comes out to be more than that calculated experimentally. According to the torricelli’s theorem the area of the tank is very large as compared to that of the openings (orifice and nozzle) , thus assuming that the height of the water head does not change, but actually there is change in the height of the water in the tank i.e. it decreases thus the velocity at the opening goes on decreasing as compared to the one which that would have been there if no change in height would have taken place.

Sources of error:

1) Tip of the needle must be well placed at the top point of the flow stream .

2) The water flowing through the notch must not flow from above the v-notch.

3) The valve for the volumetric tank must be properly closed.

4) Reading of the time must be taken carefully .