Fluid Mechanics & Hydraulic Machines - 4

8/2/2019 Fluid Mechanics & Hydraulic Machines - 4

http://slidepdf.com/reader/full/fluid-mechanics-hydraulic-machines-4 1/32

MACHINES

Lecture 3: Fluid DynamicsEr. Deepak Kumar Mallick,

School of Civil Engineering.



• One-, two- and three dimensional Flow.

three directions.

When the velocity change in any of the direction is negligible then it can beassumed as two dimensional flow.

When the velocity change in two directions are negligible then it can be

assumed as one dimensional flow.



..• Steady & unsteady Flow.

For steady flow the values of all fluid properties (velocity, temperature,

densit etc. at an ixed oint are inde endent o time but are unction o

flow passage.

P

t, .

t, is the time.



…• Uniform and non-uniform flow.

If the flow parameters such as pressure, velocity, density, viscosity and

tem erature remain constant throu hout the low ield at an iven time.

P0

S

P, represents any one of the flow parameters.



..• Laminar and turbulent flow.

.

one lamina of fluid over another. This is generally seen in viscous fluid

flows in smooth pipe with low velocity.

In turbulent flow the fluid particles move in erratic or unpredictable path.

This t e o low is seen in rivers, canals and atmos here.



..• Compressible and Incompressible flow

and temperature is negligible.

In compressible fluid the density change is appreciable. The gases arereadily compressible fluid. They expand infinitely in the absence of pressure

and contract easil under ressure.



Different Types of Heads (Energies) of

a Liquid in Motion

Potential Head / Potential Energy: This is due to configuration or position

above some suitable datum line. It is denoted by “z”.

Velocity Head / Kinetic Energy: This is due to the velocity of the flowing liquid

an s measure as g, w ere s e ve oc y o e ow an g s

acceleration due to gravity.

Pressure Head / Pressure Energy: This is due to the pressure of liquid and is

measured as P/ρg, where P is the Pressure of the flow and ρg is weight

.



..• Total Head Total Energy: Total head of liquid particle in motion is the

sum of its potential head, kinetic head and pressure head.

Total Head, H = m of liquid

2V P

z



• The mass across a pipe at different section is constant.

=. . 1 1 1 2 2 2

i.e. A1V1 = A2V2 (Incompressible Fluid)



’ASSUMPTIONS:

• Steady flow of an ideal fluid.

• -

and length ds.

• P= Pressure on the element’s 1st face.

• P+dP= Pressure on the element’s other face.• V= Velocity of the fluid element.



..1. Net pressure force in the direction of flow is

= P.dA - (P+dP).dA = -dP.dA

.

= -ρ.g.dA.ds.cosθ = -ρ.g.dA.ds.(dz/ds) = -ρ.g.dA.dz

3. Mass of the fluid element is

=ρ.dA.ds4. The acceleration of the fluid element is

a = t = s x s t = . s

5. According to Newton’s second law

=

-dP.dA - ρ.g.dA.dz = ρ.dA.ds x V.(dV/ds) [dividing both sides by ρ.dA]

[Euler’s Equation for motion]V.dV g.dz 0



..

•

V.dV g.dz 0

1 dP VdV .dz cons tant ,

2P V

z constant 2

P V2 z cons an

g 2g

• ressure ea ressure energy per un t we g t + net c ea net c

energy per unit weight) + Potential Head (Potential energy per unit

weight) = Total or Hydrodynamic Head

[Bernoulli’s Equation]

2 2

1 1 2 21 2

P V P Vz z



..



’ASSUMPTIONS:

• The fluid is ideal, viscosity is zero.

• .

• The flow is irrotational.

• The fluid must be incompressible, even though the pressure varies the

density must remain constant.• Flow must be one-dimensional, along a streamline.

.

• The only forces acting on the fluid are the gravity forces and the pressure

forces.



..P1=Pressure at LL.

V1=Velocity of liquid at LL.

=1 .

A1=Area of Pipe at LL.

P2,V2,z2,A2 = Corresponding

values at MM.

W=Weight of the liquid between

LL and L’L’As the flow is continuous,

W = ρgA1dl1 = ρgA2dl2



..

’ ’ =

1 1 2 2

WA dl A dl

g

= P1A1dl1

Work done by pressure at MM in moving the liquid to M’M’ = -P2A2dl2

Total Work done by the pressure = P1A1dl1 - P2A2dl2

= 1 1 1- 2

1 2

W

(P P )

Loss of Potential Energy = W (z1-z2)

Gain in Kinetic Energy =

2 22 22 1

2 1

V V W

W (V V )2g 2g 2g



..Loss of Potential Energy+ Total Work done by the pressure = Gain in Kinetic Energy

- =W 2 2W 1 2 1g 2g

2 2

1 2 2 1P P V V 1 2

g g 2g 2g

[Bernoulli’s Equation]

2 2

1 1 2 21 2

P V P VZ Z

g 2g g 2g



’• Bernoulli s equation was derived with an assumption that the fluid is non-

viscous, frictionless.

• Real fluid flows offer resistance to flow there was alwa s some losses in

the fluid flow, hence the losses has to be taken into consideration.

Thus the Bernoulli s equation for real fluid flow is given as

2 2

1 1 2 21 2 LZ Z h

g 2g g 2g

hL = Loss of energy between point 1 and 2.



’ Venturimeter

A venturimeter is a device used for measuring the rate of a flow /

dischar e of a fluid flowin throu h a i e.

Types of Venturimeter

i) Horizontal Venturimeterii) Vertical Venturimeter

Parts of Venturimeteri) A short Converging Part

ii) Throat

iii) Diverging Part



Let, d1 = Diameter at inlet or at section (1)

P1 = Pressure at section (1)

=1

a1 = Area at section (1) = πd12/4

d2, P2, v2, a2 corresponding values at section (2).

Applying Bernoulli’s Equation,2 2

P v P v

=

1 2z z

g 2g g 2g

,

2 2

1 1 2 2P v P v

2 2

1 2 2 1P P v v

g g g g g g g



...P P

= Difference of pressure heads at sections 1 and 2 and equal to h.g

2 2

Applying equation of continuity at section 1 and 2, we have

h2g 2g

A1v1 = A2v2 => v1 = A2v2/A1

ence, 2

2 2

2 2 2 2

12 2 1 2

A v

Av v A A

2

1

22

2g 2g 2g A

A A

2 22 2 2 2

1 2 1 2

v g v g

A A A A



…A A A

2 2 22 2 2 2

1 2 1 2

Disc arge, Q A v A 2g Q 2gA A A A

A A

2 2

1 2

Q C h, where, C = Constant of Venturimeter 2gA A

The above e uation ives the theoretical dischar e value but the actual dischar e Q ac

theis less than the theoretical discharge (Q ) is by,

A Aact d

2 2

1 2

Actual Discharge, Q C x 2ghA A

, -

d

- .

0.96 to 0.98. Due to variation of C venturimeters are not suitable for very low velocities.

d

c ua sc arge

Theoretical DischarC act

thege Q



…Value of ‘h’ given by differential U-tube manometer.

CASE-I:

Let, the differential manometer contains a liquid which is heavier than the.

Let, Sh = Sp. gravity of the heavier liquid

S = S . ravit of the li uid flowin throu h the i e hS

x = Difference of heavier liquid column in U-tube 0

,

S

CASE-II:

Let, the differential manometer contains a liquid which is lighter than the

.

Let, Sl = Sp. gravity of the lighter liquid

S0 = Sp. gravity of the liquid flowing through the pipe lS

Then, h x 1

x = Difference of lighterr liquid column in U-tube 0



• Vertical and inclined venturimeters are employed for measuring discharge

on pipelines which are not horizontal. The same formula for discharge is

used for these cases as well.

CASE-III:

• Inclined venturimeter with differential U-tube manometer. Let the

1 2 h1 2

P P Sh z z x 1

CASE-IV:

0

• For Inclined venturimeter with differential U-tube manometer contains

lighter liquid then h is given by

1 2 l

1 20h z z x 1g g S



Orifice meter or orifice plate is a device (cheaper than venturimeter)

employed for measuring the discharge of fluid through a pipe. It also

works on the same principle of a venturimeter.

• .

• The diameter of the orifice varies between 0.4 to 0.8 times the diameter

of the pipe, but the value is choosen as 0.5.

• The differential manometer is at

a distance of 1.5 to 2 times the

pipe diameter upstream from

the orifice plate.

• A1= Area of the pipe at section

• V1=Velocity at section (1)

• P =Pressure at section (1)

• A2,V2,P2 are the correspondingvalues at section (2)



…• A l in Bernoulli’s e uation at section 1 & 2 we et

2 2

1 1 2 2

1 2

P v P vz z

g 2g g 2g

2 2

1 2 2 11 2

P P v vz z

2 2

2 1v vh 1 2P P

h z z differential head

2g 2g g g

2 2

2 1v v

h2g 2g

2

212 1

vv 2g h 2gh v

2



…• Section (2) is at vena contract and A2 is the area at vena contracta. If A0 is

the area of the orifice then,

A

A2=A0Cc

c c

0

ere = o-e c en o con rac onA

Using equation of continuity, we get

A1V1=A2V2 =>V2=A2V2/A1 => V2=A0CcV2/A1

Substituting the value of V1, we get

22 2 22 20 c 2 0

2 2 c 222

1 120

c

V 2gh V 1 C 2gh VA A

A1 C

1



…• The discharge, Q=A2V2=A0CcV2

2

0A

2 0 c2

20

2ghV A C

A

1

d c2

AC C

c

1A

c

1

1 C

A

• d = o-e c ent o sc arge

2

2A c

1

c d2

AC C

0

1

1

A



…2

20

c1

A1 C

A

0 0 10 d d2 2 22 2

1 0020 0

c11 1

Q A .C Q Q CA AAA A 11 1 C

AA A

d o-e c ent o sc arge o an or ce s muc sma er t an t at o a

venturimeter.



Difference Between Venturimeter and

Orifice meter1. The venturimeter can be used for measuring the flow rates of all

incompressible flows (gases with low pressure variation, as well as liquid),

where as the orifice meters are generally used for measuring the flow ratesof liquids.

2. Venturimeter is installed in pipelines only, the flow continues through the

.

converted to kinetic energy, and the jet discharges freely into the openatmosphere.

. n ventur meter, t e ow ve oc ty s measure y not ng t e pressure

difference between he inlet and the throat of the venturimeter, where as

in the orifice meter the discharge velocity is measured by using pitot tubeor by trajectory method.



Pitot Tube is one of the most accurate devices for velocity measurement. It

works on the principle that if the velocity of flow at point becomes zero, the

pressure there is increased due to conversion of kinetic energy into pressure.

• A l in Bernoulli's e uation

bwteen stagnation point (S) and

point (P).

‘ ’• 0= ressure a a c ressure

• V= Velocity at ‘P’ (Free flow velocity)

• P = Sta nantion ressure at ‘S’

• Δh = Dynamic pressure

= Difference between stagnation

pressure head (hs) and static

pressure head (h0).

0 s

0 s s 0

P PV V

h h V 2g h h V 2g hg 2g g 2g

Fluid Mechanics & Hydraulic Machines - 4

Documents

Transcript of Fluid Mechanics & Hydraulic Machines - 4