Chapter 6: Bernoulli and energy equations Fall 2009 University of Palestine College of Engineering &...

Chapter 6: Bernoulli and energy equations

Fall 2009

University of PalestineCollege of Engineering & Urban PlanningApplied Civil Engineering

Lecturer:Eng. Eman Al.Swaity

Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 2

OBJECTIVES

EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations

Derive the Bernoulli (energy) equation.

Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow.

understand the use of hydraulic and energy grade lines.

Apply Bernoulli Equation to solve fluid mechanics problems (e.g. flow measurement).

Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 3EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations

Let us first derive the Bernoulli equation, which is one of the most well-known equations of motion in fluid mechanics, and yet is often misused. It is thus important to understand its limitations, and the assumptions made in the derivation.

The assumptions can be summarized as follows: Inviscid flow (ideal fluid, frictionless) Steady flow (unsteady Bernoulli equation will not be discussed

in this course) Applied along a streamline Constant density (incompressible flow) No shaft work or heat transfer

The Bernoulli Equation


The Bernoulli Equation Limitations



The Bernoulli equation is an approximate relation between pressure, velocity, and elevation and is valid in regions of steady, incompressible flow where net frictional forces are negligible.Equation is useful in flow regions outside of boundary layers and wakes, where the fluid motion is governed by the combined effects of pressure and gravity forces.


Acceleration of a Fluid Particle

Describe the motion of a particle in terms of its distance s along a streamline together with the radius of curvature along the streamline.The velocity of a particle along a streamline is V = V(s, t) = ds/dt The acceleration can be decomposed into two components: streamwise acceleration as along the streamline and normal acceleration an in the direction normal to the streamline, which is given as an = V2/R.


Acceleration of a Fluid Particle

Note that streamwise acceleration is due to a change in speed along a streamline, and normal acceleration is due to a change in direction. The time rate change of velocity is the acceleration

In steady flow, the acceleration in the s direction becomes

(Proof on Blackboard)


Derivation of the Bernoulli Equation

Applying Newton’s second law in the s-direction on a particle moving along a streamline in a steady flow field gives

The force balance in s direction gives

where

and



Therefore,

Integrating steady flow along a streamline

Steady, Incompressible flow

This is the famous Bernoulli equation.



This is the Bernoulli equation, consisting of three energy heads



A head corresponds to energy per unit weight of flow and has dimensions of length.Piezometric head = pressure head + elevation head, which is the level registered by a piezometer connected to that point in a pipeline.Total head = piezometric head + velocity head.

Applying the Bernoulli equation to any two points on the same streamline, we have



Without the consideration of any losses, two points on the same streamline satisfy

where P/ as flow energy, V2/2 as kinetic energy, and gz as potential energy, all per unit mass. The Bernoulli equation can be viewed as an expression of mechanical energy balanceWas first stated in words by the Swiss mathematician Daniel Bernoulli (1700–1782) in a text written in 1738.

2 21 1 2 2

1 21 22 2

P V P Vz z

g g g g


Example 1


Example 2


Example 2

-29.9 kpa


Example 3


The Energy Equation

The energy equation is more general than the Bernoulli equation, because it allows for (1) friction, (2) heat transfer, (3) shaft work, and (4) viscous work (another frictional effect).

Where Ws is the shaft work , hL, called the head loss,In the absence of these two terms, the energy equation is identical to the Bernoulli equationWe must remember however that the Bernoulli equation is a momentum equation applicable to a streamline and the energy equation above is applied between two sections of a flow


The Energy Equation


Example 1


Example 2


Example 2

waterof 97.12

1081.9*2

64.70000

22

2

2

222

1

211

mH

H

Hhzg

V

g

pz

g

V

g

p

P

P

PL


Example 3

Pump draws water from reservoir (A) and lifts it to a higher reservoir (B), as shown below, the head loss from A to the pump = 4v2/2g and the head loss from the pump to B = 7 v2/2g. compute the pressure head the pump must deliver. The pressure head at the inlet of pump = -6m.

A

B


Example 3

mH

HV

Hhzg

V

g

pz

g

V

g

p

mV

V

VV

hzg

V

g

pz

g

V

g

p

P

P

PL

L

7281.9*2

26.61150

81.9*2

115400400

22

26.681.9*2

510

81.9*2

40

81.9*26400

22

2

2

3

233

1

211

2

22

2

222

1

211

12

3


HGL and EGL

It is often convenient to plot mechanical energy graphically using heights.

P/g is the pressure head; it represents the height of a fluid column that produces the static pressure P.V2/2g is the velocity head; it represents the elevation needed for a fluid to reach the velocity V during frictionless free fall.z is the elevation head; it represents the potential energy of the fluid.H is the total head.


HGL and EGL

Hydraulic Grade Line (HGL)

Energy Grade Line (EGL) (or total head)

PHGL z

g

2

2

P VEGL z

g g


HGL and EGL


Something to know about HGL and EGL:

For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide with the free surface of the liquid, since the velocity is zero and the static pressure (gage) is zero.The EGL is always a distance V2/2g above the HGL. In an idealized Bernoulli-type flow, EGL is horizontal and its height remains constant. This would also be the case for HGL when the flow velocity is constant .For open-channel flow, the HGL coincides with the free surface of the liquid, and the EGL is a distance V2/2g above the free surface.



At a pipe exit, the pressure head is zero (atmospheric pressure) and thus the HGL coincides with the pipe outlet.The mechanical energy loss due to frictional effects (conversion to thermal energy) causes the EGL and HGL to slope downward in the direction of flow. A steep jump occurs in EGL and HGL whenever mechanical energy is added to the fluid. Likewise, a steep drop occurs in EGL and HGL whenever mechanical energy is removed from the fluid.



The pressure (gage) of a fluid is zero at locations where the HGL intersects the fluid. The pressure in a flow section that lies above the HGL is negative, and the pressure in a section that lies below the HGL is positive.


Examples on HGL and EGL:


Static, Dynamic, and Stagnation Pressures

P is the static pressure; it represents the actual thermodynamic pressure of the fluid. This is the same as the pressure used in thermodynamics and property tables.V2/2 is the dynamic pressure; it represents the pressure rise when the fluid in motion.gz is the hydrostatic pressure, depends on the reference level selected.

The Bernoulli equation


Static, Dynamic, and Stagnation PressuresThe sum of the static, dynamic, and hydrostatic pressures is called the total pressure (a constant along a streamline).The sum of the static and dynamic pressures is called the stagnation pressure,

The fluid velocity at that location can be calculated from


APPLICATIONS OF BERNOULLI & MOMENTUM EQUATION

1)Pitot tube.2)Changes of pressure in a tapering pipe:3)Orifice and vena contracta.4)Venturi, nozzle and orifice meters.5)Force on a pipe nozzle.6)Force due to a two-dimensional jet hitting an

inclined plane.7)Flow past a pipe bend.


PITOT TUBE


PITOT TUBE IN THE PIPE-(method1)

Two piezometers

for ideal flow

To account for real fluid effects, the equation can be modified into

where is the coefficient of velocity to be determined empirically.



Using a static pressure taping in the pipe

1

water

mercurymercurywater hh



Example

mh

hh

water

water

mercurymercurywater

45.211025

136002.0

1

95.0 if

pipe in the water of velocity actual theDetermine

kg/m 1025

200mh shown, figure In the3

water

mercury

VC

Sol.

smV

ghCV

actual

Vactual

/ 6.645.281.9295.0

2



Using combined Pitot static tube



Using combined Pitot static tube. In which the inner tube isused to measure the impact pressure while the outer sheathhas holes in its surface to measure the static pressure

The total pressure is know as the stagnation pressure (or total pressure)


CHANGES OF PRESSURE IN ATAPERINGPIPE.

Changes of velocity in a tapering pipe were determined by using the continuity equation.

Changes of velocity will accompanied by a changed in pressure, modified by any changed in elevation or energy loss, which can be determined by the use of Bernoulli's equation.


CHANGES OF PRESSURE IN ATAPERINGPIPE-Example

Find:¢ the pressure difference across the 2m length ignoring any losses ofenergy.¢ the difference in level that would be shown on a mercury manometer connected across this length.


Orifice and vena contracta

We are to consider the flow from a tank through a hole in the side close to the base

Looking at the streamlines you can see how they contract after the orifice to a minimum cross section where they all become parallel, at this point, the velocity and pressure are uniform across the jet. This convergence is called the vena contracta (from the Latin 'contracted vein'). It is necessary to know the amount of contraction to allow us to calculate the flow.


Venturi , nozzle and orifice meters

The Venturi, nozzle and orifice-meters are three similar types of devices for measuring discharge in a pipe.


A Venturi meter in laboratory


Venturi Meter


Venturi Meter

This is also the theoreticaldischarge in terms ofmanometer readings


Venturi Meter

22

21

21

21

22

2

21

2

22

22

12

2

21

22

21

22

21

2221

2

222

1

211

.2

11

2

)(2 )(2

)(2 22

22

22

AA

AAghQ

AAg

Qh

A

Q

A

QhgVVhg

VVhgg

V

g

Vh

g

V

g

V

g

p

g

p

zg

V

g

pz

g

V

g

p

22

21

21.2AA

AAghCQ dactual

Horizontal Venturimeter


Venturi Meter Example 1

mh

h

AA

AAghCQ dactual

756.0

00786.003141.0

00786.003141.081.9296.01000/30

.2

22

22

21

21



mh

hh

mh

water

water

mercurymercurywater

mercury

15.311000

1360025.0

1

25.0


Venturi Meter


Venturi Meter

smQ

Q

AA

AAghCQ dactual

/ 138.0

0314.01257.0

0314.01257.0*92.081.9298.0

.2

3

22

22

21

21

mh

hh

mh

oil

oil

mercurymercuryoil

mercury

92.01700

1360005.0

1

05.0


PIPE ORIFICES

¢ A similar effect as the venturi meter can be achieved by insertingan orifice plate¢ The orifice plate has an opening in it smaller than the internal pipediameter

Chapter 6: Bernoulli and energy equations Fall 2009 University of Palestine College of Engineering &...

Documents

Transcript of Chapter 6: Bernoulli and energy equations Fall 2009 University of Palestine College of Engineering &...