Transcript derived along a fluid flow streamline is often called the

FLUID DYNAMICS
BERNOULLI’S EQUATION
BY
GP CAPT NC CHATTOPADHYAY
Daniel Bernoulli

(Groningen, 8 February 1700 –
Basel, 8 March 1782) was a
Dutch-Swiss mathematician
and was one of the many
prominent mathematicians in
the Bernoulli family. He is
particularly remembered for his
applications of mathematics to
mechanics, especially fluid
mechanics, and for his
pioneering work in probability
and statistics. Bernoulli's work
is still studied at length by
many schools of science
throughout the world.
INTRODUCTION
•A statement of the conservation of energy in a form
useful for solving problems involving fluids.
• For a non-viscous, incompressible fluid in steady flow,
the sum of pressure, potential and kinetic energies per
unit volume is constant at any point
•A special form of the Euler’s equation derived along a
fluid flow streamline is often called the Bernoulli Equation
AVAILABLE ENERGY HEADS

1. PR HEAD: DUE TO PR OF LIQUID = p/w

2. PE HEAD: DUE TO POSITION OF FLUID
LEVEL = z

3. VELOCITY HEAD: DUE TO VELOCITY i.e
KINETIC ENERGY HEAD
= v2/2g
STATEMENT
FOR A STEADY,STREAMLINE FLOW
OF AN IDEAL, INCOMRESSIBLE
FLUID, THE SUM OF KINETIC,
POTENTIAL AND PR ENERGY IS
CONSTANT
EXPLAINATION
SECTION-1
1,A1,v1,
p1,z1
SECTION-2
FLOW
2,A2,v2
p2,z2
p1 /w + v12/2g + z1 = p2/w + v22/2g +z2
DERIVATION


AS DERIVED IN THE CLASS
ALSO, PL REFER TO RECOMMENDED
TEXT BOOKS
NUMERICALS


1. DIA OF A PIPE CHANGES FROM 200mm AT A SECTION 5m
ABOVE DATUM TO 50mM AT A SECTION 3m ABOVE DATUM.
PRESSURE OF WATER IS 500kPa AT INLET WITH A VELOCITY
1m/s. DETERMINE PR AND VELOCITY AT EXIT.
2. BRINE OF S.G 1.15 IS DRAINING FROM BOTTOM OF A LARGE
OPEN TANK. THE DRAIN PIPE ENDS 10 m BELOW THE FREE
SURFACE. CONSIDERING THE FLOW AS STEADY AND ALONG
STREAMLINE CALCULATE THE DISCHARGE VELOCITY. (NEGLECT
FRICTION)
ASSIGNMENT

1. PRACTISE DERIVATION OF BERNOULLI’S EQUATION

2. SOLVE.

A 5m LONG PIPE IS INCLINED AT 150 TO THE HORIZONTAL.
SMALLER END OF PIPE IS AT LOWER LEVEL AND IS OF 80mm
DIA WHILE THE LARGER SECTION IS OF 240mm DIA. IF THE
INLET VELOCITY IS 1m/S, FIND EXIT VELOCITY AND PR
DIFFERENCE BETWEEN TWO SECTIONS
EULER’S EQUATION OF MOTION


AS DERIVED ON THE BOARD
BERNOULLI’S EQUATION WILL BE
ESTABLISHED FROM ABOVE
ASSUMPTIONS








FLOW IS STEADY
FLOW IS INCOMPRESSIBLE
FLOW IS ALONG STREAMLINE (1 D)
FLOW IS INVISID
NO HEAT OR WORK TRANSFER
NO ENERGY LOSS TO ENVIRONMENT
VELOCITY IS UNIFORM (Um)
ONLY FORCES ARE DUE TO PR AND GRAVITY
LIMITATIONS




VELOCITY MAY NOT BE UNIFORM IN A REAL FLOW
VISCOUS AND FRICTIONAL FORCES EXIST IN A REAL
FLOW
CENTRIFUGAL FORCE MAY ALSO BE PRESENT IN A
FLOW THROUGH CURVED PATH
HEAT TRANSFER ALSO OCURS DUE TO CONVERSION
OF KINETIC ENERGY INTO HEAT
Application of
Bernoulli’s Principle
MAJOR APPLICATIONS


MEASUREMENT OF FLOW VELOCITY
MEASUREMENT OF FLOW DISCHARGE
PITOT TUBE (WITH AOAI)
PITOT TUBE
Stagnation pressure = static
pressure + dynamic pressure
Which can also be written
Solving that for velocity we get:
USE OF PITOT SYSTEM
PITOT STATIC SYSTEM
DISCHARGE MEASUREMENT

VENTURIMETER

ORIFICEMETER

ROTAMETER
VENTURIMETER
A venturi can be used to measure the volumetric flow rate Q.
Since
then
ORIFICE METER
ROTAMETER
A rotameter is a device that measures
the flow rate of liquid or gas in a closed
tube.
It belongs to a class of meters called
variable area meters, which measure
flow rate by allowing the cross-sectional
area the fluid travels through to vary,
causing some measurable effect.
SO,WHAT DO U DO ?



CONCENTRATE ON THE BOARD FOR THE
DERIVATION
GO THROUGH THE TOPIC COVERED SO
FAR AND ……
HAVE PATIENCE TILL NEXT FM CLASS
ON THE BOARD PL…..
TIME TO ENJOY…….
EID MUBARK….
SEE U ALL ON THE FIRST DAY… FIRST
SHOW…. AFTER BREAK…..