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VERIFICATION OF BERNOULLIS EQUATION

AIM OF THE EXPERIMENT:

To verify modified bernoullis equation and to determine thehead loss in the pipe.

THEORY:

The dynamics equation of fluid motion is obtained byapplying Newtons second law of motion to fluid element as a freebody.The fluid is assumed to be incompressible and non-viscous.

The continuity equation is based upon the principle ofconservation of mass and it states that if no fluid is added or

removed from the pipe of any length, then the mass passing acrossdifferent sections shall be same

i.e 1 A1 V1 = 2 A2 V2or

AV = ConstantA fluid element in motion possesses intermolecular energy,

kinetic energy and potential energy.In a fluid flow, the energy perunit weight is termed as head. Thus ,There are three types of heads

causing the fluid flow are interpreted as pressure head (pressureenergy or flow work per unit weight),velocity head(kinetic energyper unit weight) and potential or datum head (potential energy perunit weight).

Pressure head,= P/g m

Velocity head,= V2/2g m

Potential/datum head = Z mEulers equation along a steamline is derived by applying

Newtons second law of motion to a fluid element moving along astreamline and is given as

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i.e. vdv/ds= -1/ dp/ds-gdz/ds

integrating the eulers equation,we getv/t + Vv/s = -1/ p/s-gdz/ds

vdv/ds = -1/dp/ds ds - gdz/ds ds

i.e v2/2 + dp/ + gz = constantfor incompressible flow,

p/ + V2/2 + gz = constant

p/g + V2

/2g + z = constant

BERNOULLIS EQUATION: Bernoullis equation states that in an idealincompressible fluid when the flow is steady and continuous; thesum of pressure head, kinetic head and potential head is constantalong a stream line.

PROOF OF BERNOULLIS THEOREMConsider an ideal incompressible liquid through a non-uniform

pipe as shown in fig. Let us consider two section L-L and M-M andassume that the pipe is running full and there is continuity of flowbetween the two sections.

Let the liquid between the two sections L-L and M-M move toL-L and M-M through very small lengths dl1 & dl2.thismovement of liquid between L-L and M-M is equivalent to movement

of the liquid L-L and L-L and M-M and M-M;the remaining liquidbetween L-L and M-M being unaffected.Let,W=weight of liquid between L-L and L-L.As the flow is continues,

W=wA1.dl1=w.A2dl2

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or

A1.dl1=W/w similarly , A2dl2=W/wor

A1dl1= A2dl2

Workdone bypressure at L-L,in moving the liquid to L-L=force*distance=p1. A1dl1

Similarly,W.D by pressure at M-M in moving liquid to M-M=- p2. A2dl2 .(the ve sign indicates that dirn of p2

is opposite to p1).. Total workdone by pressure

=P1.A1dl1- P2. A2dl2

=W/w(P1-P2)Loss potential energy is =W(Z1-Z2)Gain in Kinetic Energy =W[V22/2g - V12/2g] =W/2g[V22- V12 ]Also loss of potential energy + workdone by pressure = gain in

Kinetic energy.. W(Z1Z2)+ W/w(P1-P2) = W/2g(V22-V12)

.. (Z1 Z2)+ (P1-P2)/w = (V22-V12)/2g

i.e. P1/w-P2/w + Z1Z2 = V22/2g-V12 /2g.. P1/w + Z1 + V12 /2g = P2/w + Z2 + V22 /2g

-:BERNOULLIS EQUATION:-

ASSUMPTION:-1. The liquid is ideal and incompressible .2. The flow is steady and continuous .

3. The flow is along the stream line i.e one dimesional.4. The velocity is uniform over the section and is equalto mean velocity.

5. The only forces acting on the fluid are the gravityforces and pressure forces.

EXPERIMENTAL SET UP RESOURCES:-

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Bernoullis apparatus :- sump tank with monoblock centrifugalpump H.P for fluid flow through the apparatus ;constant headsupply and delivery tanks , venturi tappings with venturi tube havingspecified inlet and outlet diameters made of transparent acrylic

material;piezometer panel consisting of transparent tubes mountedon a graduated scale for measuring piezometric head;digital stopwatch.

PRECAUTIONS:-1.Check all connections and electrical supply systems.2.Operate the inlet and outlet vavels gently and continuously.3.Ensure before taking reading the flow is steady and continuous.

4.Remove the air bubbles,if any from the monometer limbs usingthe vent cock on the monometer.5.After removing the air bubbles, put some back pressure (if thereading on monometer goes beyond the reading scale)using pressurevalve.6.Control the flow by flow valve to maintain different set ofconstant head in the supply and delivery tank.

PROCEDURE:-1. Ensure that the inlet and outlet valve is open for the pump .2. Switch on the electric power supply and fill the sump tank.3. Adujust the flow control valves and discharge valves to maintain

constant water level in the tanks ,allowing the water to flowthrough the apparatus and fill the piezometric tubes.

4. Remove the air bubbles , if any from monometer and piezometertubes.

5. At the time of measuring the discharge , close the outlet valveand measure the time required for increase in water levelindicater scale provide for discharge tank.

6. Note down the readings in observation table and ensure that allthe precaution and experimental assumption or requirement.

7. Repeat the procedure for different heads and floe rates.

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OBSERVATION TABLETUBE NO. PIEZOMETRIC

HEAD cmSECTIONDIAMETER

ACTUALDISCHARGEcm3/sec

RESULT TABLE

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TUBENO.

SUPPLYHEADH1

DELIVERYHEAD H2

LOSSOFHEADhL=

H1- H2

PIEZOMETRICHEAD(P/g + Z) cm

VELOCITYHEAD(V2/2g)cm

TOTALHEAD[(P/g +Z) +

V2/2g]

CONCLUSION:-