Verification of Bernoulli’s Theorem

Bernoulli’s equation states that in a steady, irrotational flow of an ideal incompressible fluid, the total energy at any point/ section is constant.

Object

To verify the Bernoulli's Theorem experimentally.

Theory

The principle of conversation of energy gives rise to Bernoulli's theorem which states that in a pipe flow

if an incompressible ideal fluid flows through a closed passage, the total head, i.e. the sum of datum head ‘z’, pressure head (p/w) and velocity head (v2/2g) will be the constant at all points, i.e.

H= (p/w) +z + (v2/2g)

The validity of above theorem is subject to the following conditions

(i) the flow being steady,

(ii) fluid being frictionless, i.e. non-viscous

(iii) flow being irrotational, and,

(iv) no external work on the flow system is done by any external machine

Experimental setup

The apparatus consist of diverging duct (e.g. 38mm width) issuing out of a water container and discharging into another. peizometric connections are made at small equal intervals to show pressure heads at different sections. The discharge is measured by collecting a known volume of water in time t.

The water is supplied to apparatus from the laboratory storage tank which can be regulated by inlet valve. Another valve is fitted at the end of duct to get sufficiently low pressure at the central tube. A graph is attached to vertical board to take the piezometric readings. Alternatively, a scale may be used for same purpose.

Procedure1. Open the inlet valve to obtain a steady flow. Collect

the water in tank for certain time ‘t’. (col. 2,3)

2. Consider the base of the apparatus as base datum; measure the height of water level in the tube above base, (p/w) +z (col. 9)Note: (p/w) is the pressure head and ‘z’ is the datum head and the sum of these two quantities is called piezometric head. By proper adjustment of outlet valve, sufficiently deep curve can be obtained.

3. Note down the width ‘b’, and depth ‘d’ of the duct at various cross sectional (col. 5)

Formula used

• Q = A1v1=A2v2 (continuity equation)• E = z+(p/w)+(v2 /2g) …. ( total energy of flow) • E1= E2 = E3 =… = constant (Bernoulli equation)

where, z = datum head (p/w) = pressure head (v2/2g) = velocity head

Observation TableTable - 1Discharge Measurement

S. No.

Area of Collecting Tank

(A)

Initial Readin

g(h1)

Final Reading (h2)

Depth of Water

Collected(h2- h1)

Volume of Water Collecte

d(A*V)

Time of Collectio

n(t)

Discharge

(Q)

Table - 2

S.No Tube No.

Depth of Conduit

Velocity

(v)

Velocity Head

(v2/2g)

Piezometric Head (H)

Total Energy

(E)

Remarks

123

123

Total Energy

Presentation of Results

1. Mark on the graph the ordinate of (p/w+z) on a suitably chosen horizontal scale which is called Hydraulic Gradient Line (H.G.L.).

2. Mark velocity head (v2/2g) over (p/w+z) curve. It is called Total Energy Line (T.E.L.).

3. Show that the resulting curve is a mildly sloping straight-line adding to it the loss of head, result in a horizontal line. Thus, it proves the Bernoulli's theorem.

Precautions

1. Discharge should be kept constant throughout one set of observation.

2. Depth of passage should be very carefully observed.

Viva-voce

1. What is meant by incompressible and ideal fluid?

2. What difference does it make to the Bernoulli's theorem if the flowing medium is not ideal fluid but real one?

3. Is the theorem applicable to gases and vapors too?

4. Discuss this point ?

5. Are the datum, pressure and velocity energies inter- convertible ?

6. Give example of each ?

7. What is meant by irrotational flow?

8. If a prime mover or any other machine is installed between any two flow sections, how is it accounted for in Bernoulli's theorem?