Fluid dynamic

Fluid Dynamics:

(ii) Hydrodynamics: Different forms of energy in a flowing

liquid, head, Bernoulli's equation and its application, Energy

line and Hydraulic Gradient Line, and Energy Equation

Dr. Mohsin Siddique

Assistant Professor

1

Fluid Mechanics

Forms of Energy

2

� (1). Kinetic Energy: Energy due to motion of body. A body of mass, m, when moving with velocity, V, posses kinetic energy,

� (2). Potential Energy: Energy due to elevation of body above an arbitrary datum

� (3). Pressure Energy: Energy due to pressure above datum, most usually its pressure above atmospheric

2

2

1mVKE =

mgZPE =

m andV are mass and velocity of body

Z is elevation of body from arbitrary datumm is the mass of body

hγ=PrE !!!

Forms of Energy

3

� (4). Internal Energy: It is the energy that is associated with the molecular, or internal state of matter; it may be stored in many forms, including thermal, nuclear, chemical and electrostatic.

HEAD

4

� Head: Energy per unit weight is called head

� Kinetic head: Kinetic energy per unit weight

� Potential head: Potential energy per unit weigh

� Pressure head: Pressure energy per unit weight

g

VmgmV

Weight

KE

2/

2

1head Kinetic

22 =

== mgWeight =Q

( ) ZmgmgZWeight

PE=== /head Potential

γ

P

Weight==

PrEhead Pressure

TOTAL HEAD

5

� TOTAL HEAD

= Kinetic Head + Potential Head + Pressure Head

g

VPZ

2HHead Total

2

++==γ

g

V

2

2

γ

PZ

Bernoulli’s Equation

6

� It states that the sum of kinetic, potential and pressure heads of a fluid particle is constant along a streamline during steady flow when compressibility and frictional effects are negligible.

� i.e. , For an ideal fluid, Total head of fluid particle remains constant during a steady-incompressible flow.

� Or total head along a streamline is constant during steady flow when compressibility and frictional effects are negligible.

21

22

22

12

11

2

22

2Head Total

HH

g

VPZ

g

VPZ

consttg

VPZ

=

++=++

=++=

γγ

γ

1

2

Pipe

Derivation of Bernoulli’s Equation

7

� Consider motion of flow fluid particle in steady flow field as shown in fig.

� Applying Newton’s 2nd Law in s-direction on a particle moving along a streamline give

� Where F is resultant force in s-direction, m is the mass and as is the acceleration along s-direction.

ss maF =

Assumption:Fluid is ideal and incompressibleFlow is steadyFlow is along streamlineVelocity is uniform across the section and is equal to mean velocityOnly gravity and pressure forces are acting

ds

dVV

dtds

dsdV

dsdt

dsdV

dt

dVas ====

Eq(1)

Eq(2)

Fig. Forces acting on particle along streamline


8

W=weight of fluidWsin( )= component acting along s-directiondA= Area of flowds=length between sections along pipe

θ

( ) θsinWdAdpPPdAFs −+−=

Substituting values from Eq(2) and Eq(3) to Eq(1)

Eq(3)

( )ds

dVmVWdAdpPPdA =−+− θsin

ds

dVdAdsV

ds

dzgdAdsdpdA ρρ =−−

( )gdAdsmgW ρ==

ds

dz=θsin

Cancelling dA and simplifying

VdVgdzdp ρρ =−−

Note that 2

2

1dVVdV =

2

2

1dVgdzdp ρρ =−−

Eq(4)

Eq(5)

Fig. Forces acting on particle along streamline


9

� Dividing eq (5) by

� Integrating

� Assuming incompressible and steady flow

� Dividing each equation by g

ρ

02

1 2 =++ dVgdzdp

ρ

conttdVgdzdp

=

++∫

2

2

1

ρ

conttVgzP

=++ 2

2

1

ρ

conttg

Vz

g

P=++

2

2

ρ

� Hence Eq (9) for stead-incompressible fluid assuming no frictional losses can be written as

Eq (6)

Eq (7)

Eq (8)

Eq (9)

( ) ( )21

22

22

12

11

Head TotalHead Total

22

=

++=++g

VPZ

g

VPZ

γγ

Above Eq(10) is general form of Bernoulli’s Equation

Eq (10)

Energy Line and Hydraulic Grade line

10

� Static Pressure :

� Dynamic pressure :

� Hydrostatic Pressure:

� Stagnation Pressure: Static pressure + dynamic Pressure

Hg

Vz

P=++

2

2

γ

Head TotalheadVelocity headElevation head Pressure =++

P

gZρ

2/2Vρ

conttV

gzP =++2

2

ρρ

Multiplying with unit weight,γ,

stagPV

P =+2

2

ρ


11

� Measurement of Heads

� Piezometer: It measures pressure head ( ).

� Pitot tube: It measures sum of pressure and velocity heads i.e.,

g

VP

2

2

+γ

γ/P

What about measurement of elevation head !!


12

� Energy line: It is line joining the total heads along a pipe line.

� HGL: It is line joining pressure head along a pipe line.


13

Energy Equation for steady flow of any fluid

14

� Let’s consider the energy of system (Es) and energy of control volume(Ecv) defined within a stream tube as shown in figure. Therefore,

� Because the flow is steady, conditions within the control volume does not change so

� Hence

in

CV

out

CVCVs EEEE ∆−∆+∆=∆

in

CV

out

CVs EEE ∆−∆=∆

0=∆ CVE

Eq(1)

Eq(2)

Figure: Forces/energies in fluid flowing in streamt ube

Energy Equation for steady flow of any

fluid

15

� Now, let’s apply the first law of thermodynamics to the fluid system which states ” For steady flow, the external work done on any system plus the thermal energy transferred into or out of the system is equal to the change of energy of system”

( )

( ) in

CV

out

CV

s

EEshaftworklowwork

Eshaftworklowwork

∆−∆=++

∆=++

=+

ferredheat transf

ferredheat transf

energy of changeferredheat transdone work External

� Flow work: When the pressure forces acting on the boundaries move, in present case when p1A1 and p2A2 at the end sections move

through ∆s1 and ∆s2, external work is done. It is referred to as flow work.

msAsApp

mg

sAp

sAp

sApsAp

∆=∆=∆

−∆=

∆−∆=∆−∆=

222111

2

2

1

1

222

2

2111

1

1111111

workFlow

workFlow

ρργγ

γγ

γγ

Q Steady flow

Eq(3)

Eq(4)

Eq(5)


fluid

16

� Shaft work: Work done by machine, if any, between section 1 and 2

( ) ( ) mm

m

hmghsA

thdt

dsAtime

weight

energy

time

weight

∆=∆=

∆

==

111

111

Shaft work

Shaft work

γ

γ

� Where, hm is the energy added to the flow by the machine per unit weight of flowing fluid. Note: if the machine is pump, which adds energy to the fluid, hm is positive and if the machine is turbine, which remove energy from fluid, hm is -ve

� HeatTransferred: The heat transferred from an external source

into the fluid system over time interval ∆t is

( ) ( ) HH

H

QmgQsA

tQdt

dsA

∆=∆=

∆

=

111

111

ferredHeat trans

ferredHeat trans

γ

γ

� Where, QH is the amount of energy put into the flow by the external heat source per unit weight of flowing fluid. If the heat flow is out of the fluid, the value QH is –ve and vice versa

Eq(6)

Eq(7)


fluid

17

� Change in Energy: For steady flow during time interval ∆t, the weight of fluid entering the control volume at section 1 and leaving at section 2 are both equal to g∆m . Thus the energy (Potential+Kinetic+Internal) carried by g∆m is;

( )

( )

++∆=∆

++=∆

++

=∆

++∆=∆

++=∆

++

=∆

2

2

22

2

2

222222

2

22

222

1

2

11

1

2

111111

2

11

111

2

22

2

22

Ig

VzmgE

Ig

VzdsAtI

g

Vz

dt

dsAE

Ig

VzmgE

Ig

VzdsAtI

g

Vz

dt

dsAE

out

CV

out

CV

in

CV

in

CV

α

αγαγ

α

αγαγ

α is kinetic energy correction factor and ~ 1

Eq(8)

Eq(9)


fluid

18

� Substituting all values from Eqs. (5),(6), (7), (8), & (9) in Eq(4)

( ) in

CV

out

CV EEshaftworklowwork ∆−∆=++ ferredheat transf

( ) ( )

++∆−

++∆=∆+∆+

−∆ 1

2

112

2

22

2

2

1

1

22I

g

VzmgI

g

VzmgQmghmg

ppmg Hm αα

γγ

++−

++=++

− 1

2

112

2

22

2

2

1

1

22I

g

VzI

g

VzQh

ppHm αα

γγ

+++=++

++− 2

2

22

2

21

2

11

1

1

22I

g

Vz

pQhI

g

Vz

pHm α

γα

γ

This is general form of energy equation, which applies to liquids, gases, vapors

and to ideal fluids as well as real fluids with friction, both incompressible and compressible. The only restriction is that its for steady flow.

Eq(10)

Energy Equation for steady flow of

incompressible fluid

19

� For incompressible fluids

� Substituting in Eq(10), we get

( )12

2

22

2

2

11

1

22II

g

Vz

pQh

g

Vz

pHm −+

++=++

+−

γγ

γγγ == 21

( ) Hm QIIg

Vz

ph

g

Vz

p−−+

++=+

+− 12

2

22

2

2

11

1

22 γγ

Lm hg

Vz

ph

g

Vz

p+

++=+

+−

22

2

22

2

2

11

1

γγ Eq(11)

( ) HL QIIh −−= 12Q

� Where hL=(I2-I1)-QH= head loss. It equal to is gain in internal energy minus any heat added by external source.

� Hm is head removed/added by machines. It can also be referred to head loss due to pipe fitting, contraction, expansion and bends etc in pipes.

Energy Equation for steady flow of

incompressible fluid

20

� In the absence of machine, pipe fitting etc, Eq(11) can be written as

� When the head loss is caused only by wall or pipe friction, hL

becomes hf, where hf is head loss due to friction

Lhg

Vz

p

g

Vz

p+

++=

++

22

22

22

21

11

γγEq(12)

Power

21

� Rate of work done is termed as power

Power=Energy/time

Power=(Energy/weight)(weight/time)

� If H is total head=total energy/weight and γQ is the weight flow rate

then above equation can be written as

Power=(H)(γQ)= γQH

In BG:

Power in (horsepower)=(H)(γQ)/550

In SI:

Power in (Kilowatts)=(H)(γQ)/1000

1 horsepower=550ft.lb/s

Reading Assignment

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� Kinetic energy correction factor

� Limitation of Bernoulli’s Equation

� Application of hydraulic grade line and energy line

NUMERICALS

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� 5.2.1

24

� 5.2.3

25

� 5.3.2

26

� 5.3.4

27

� 5.3.6

28

� 5.9.6

Momentum and Forces in Fluid Flow

29

� We have all seen moving fluids exerting forces. The lift force on an aircraft is exerted by the air moving over the wing. A jet of water from a hose exerts a force on whatever it hits.

� In fluid mechanics the analysis of motion is performed in the same way as in solid mechanics - by use of Newton’s laws of motion.

� i.e., F = ma which is used in the analysis of solid mechanics to relate applied force to acceleration.

� In fluid mechanics it is not clear what mass of moving fluid we should use so we use a different form of the equation.

( )dt

mdma sV

F ==∑

Momentum and Forces in Fluid Flow

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� Newton’s 2nd Law can be written:

� The Rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force.

� The symbols F and V represent vectors and so the change in momentum must be in the same direction as force.

It is also termed as impulse momentum principle

( )dt

md sVF =∑

=

=∑

mV

F Sum of all external forces on a body of fluid or system s

Momentum of fluid body in direction s

( )smddt VF =∑

Impact of a Jet on a Plane

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Impact of a Jet on a Plane

32

Thank you

� Questions….

� Feel free to contact:

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Engineering

Fluid dynamic