NORTHERN ILLINOIS UNIVERSITY - Department of Mechanical Engineering
MEE 390 EXPERIMENTAL METHODS IN MECHANICAL ENGINEERING
©1990-2001 M. Kostic

Lab: Experimental verification of Bernoulli equation

Objective: To verify experimentally the validity of Bernoulli’s equation for fluid flow

Apparatus:
Bernoulli (Venturi-type) flow apparatus with column-manometers and Pitot tube;
3-5 gallon bucket; and
stopwatch.

Theory: Bernoulli's law indicates that, if an inviscid fluid is flowing along a pipe of varying cross section, then the pressure is lower at constrictions where the velocity is higher, and higher where the pipe opens out and the fluid stagnates. Many people find this situation paradoxical when they first encounter it (higher velocity, lower pressure). The well-known Bernoulli equation is derived under the following assumptions:

  1. fluid is incompressible ( density r is constant );
  2. flow is steady:
  3. flow is frictionless (t = 0);
  4. along a streamline;

Then, it is expressed with the following equation:

           

Where (in SI units):      

p = fluid static pressure at the cross section in N/m2.

r = density of the flowing fluid in kg/m3

g = acceleration due to gravity in m/s2 (its value is 9.81 m/s2 = 9810 mm/s2)

v = mean velocity of fluid flow at the cross section in m/s

z = elevation head of the center of the cross section with respect to a datum z=0

h* = total (stagnation) head in m

The terms on the left-hand-side of the above equation represent the pressure head (h), velocity head (hv ), and elevation head (z), respectively. The sum of these terms is known as the total head (h*). According to the Bernoulli’s theorem of fluid flow through a pipe, the total head h* at any cross section is constant (based on the assumptions given above). In a real flow due to friction and other imperfections, as well as measurement uncertainties, the results will deviate from the theoretical ones.

In our experimental setup, the centerline of all the cross sections we are considering lie on the same horizontal plane (which we may choose as the datum, z=0), and thus, all the ‘z’ values are zeros so that the above equation reduces to:

                   (This is the total head at a cross section).

For our experiment, we denote the pressure head as hi and the total head as h*i, where ‘i’ represents the cross section we are referring to.

Procedure:

1.         Check if the drain valve is open and keep it wide open and check whether the outlet pipe goes to the drain. Initiate flow through the Venturi test section by opening inlet valve(s).

2.         Check that all manometer tubing are properly connected to the corresponding pressure taps and are air-bubble free. If needed flush the air-bubbles by slowly closing the exit valve and draining the water (and the air-bubbles) through the manometer tubing.

3. Adjust both (inlet and outlet) valves so that you get the maximum difference in levels between tapping point #7 and #8.

4.         Wait for some time for the level in manometer tube #8 or #(*) to stabilize (it takes some time for it to reach steady state).

5.         After the steady state is achieved, redirect the water outlet hose into a container whose capacity is known (10 litters, for example) and record the time taken for the water to fill it up. Take at least 3 measurements and record the timings in order to calculate (average) flow rate.

6.         Gently push (slide) the Pitot (total head measuring) tube, connected to manometer #8, so that its end reaches the cross section of the Venturi tube at #1, for example. Wait for some time and note down the readings from manometer #8 (or *) and #1. The reading shown by manometer #8(or *) is the sum of the pressure and velocity heads, i.e. the total (or stagnation) head (h*), because the Pitot tube is held against the flow of fluid forcing it to a stop (zero velocity). The reading in manometer #1 measures just the pressure head (h) because it is connected to the Venturi tube pressure tap, which does not obstruct the flow, thus measuring the flow static pressure.

7.         Repeat step 6 for other cross sections (3, 5, and 7, for example).

OBSERVATIONS:

VOLUME FLOW RATE

Trial No.

Volume
Vol
(ml)

Time
t
(s)

Flow rate
Qi=Vol/t
(ml/s)

1

 

 

 

2

 

 

 

3

 

 

 

Average flow rate Qav [ cm3 / sec ]
Qav [ mm3 / sec ]

 

NOTE: 1 ml = 1 cm3 = 1000 mm3

MANOMETER READINGS AND CALCULATIONS

Cross Section

Using Bernoulli equation

Using Continuity equation

Difference


i


h*=h8


hi

ViB =
=Ö [2*g*(h* - hi )]

Ai =
=p Di2 / 4

ViC =
Qav / Ai


(ViB-ViC)/ ViC

 

mm

mm

mm/s

mm2

mm/s

%

1

 

 

 

507

 

 

2

 

 

 

380

 

 

3

 

 

 

302

 

 

4

 

 

 

263

 

 

5

 

 

 

238

 

 

6

 

 

 

214

 

 

7

 

 

 

199

 

 

NOTE: Di=25.4, 22.0, 19.6, 18.3, 17.4, 16.5, and 15.9 mm; for i=1,2,3,4,5,6, and 7.

NOTE: Your previous Lab assignment and Lab report are due before the demonstration of the next Lab. It is the best for you if you do your lab experiments right after the demonstration while TA is still in the Lab. Also, you have to perform the uncertainty analysis for every experimental lab and include it in your lab report.