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. 1. PR HEAD:: DUE TO PR OF LIQUID = p/w

2. 2. PE HEAD: DUE TO POSITION OF FLUID LEVEL = z

3. 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

1,A1,v1,

p1,z1

SECTION-1

SECTION-2

2,A2,v2

p2,z2

p1 /w + v12/2g + z1 = p2/w + v2

2/2g +z2

FLOW

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

VENTURIMETERA 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…..