Bernoulli Chapter 3

2/21/2014 9:57 AM FLUIDS 1

Chapter 3: The Bernoulli Equation

Newton’s Second Law F=ma along a streamline F=ma normal to a streamline Physical interpretations Static, Stagnation, Dynamic and Total Pressure Examples of use of the Bernoulli Equation The energy line and the hydraulic grade line Restrictions of use of the Bernoulli Equation

Newton’s 2nd Law

The net force acting on the fluid particle must equal its mass times its acceleration

For inviscid fluid, we are assuming that the fluid motion is governed by pressure and gravity forces only

Newton’s 2nd Law (cont.)

Streamlines – lines that are tangent to the velocity vectors throughout the flow field

Along the streamline,

Normal to the streamline,

Note : For steady, inviscid, incompressible flow, the pressure variation across streamline is merely hydrostatic (because of gravity alone), even though the fluid is in motion

s

VVas

R

Van

2

F=ma along a streamline The equation of motion along the streamline

direction is

◦ A change in fluid particle speed is accomplished by the appropriate combination of pressure and particle weight along the streamline

s

VV

s

p

sin

F=ma along a streamline (cont.)

Rearranging and integrating the equation for inviscid, incompressible flow gives

zVp 2

2

1Constant along a streamline

Applicable to steady flows only

Example 1

Some animals have learned to take advantage of the Bernoulli effect. For example, a typical prairie dog burrow contains two entrances – a flat front door and a mounded back door. When the wind blows with velocity Vo across the front door, the average velocity across the back door is greater than Vo because of the mound. Assume the air velocity across the back door is 1.07Vo. For a wind velocity of 6 m/s, what pressure difference, p1-p2, is generated to provide a fresh air flow within the burrow.

F=ma normal to a streamline

For steady, inviscid, incompressible flow

zdnR

Vp

2

Constant across the streamline

F=ma normal to a streamline (cont.)

When the fluid travels along a curved path, a net force directed towards the center of curvature is required, due to either gravity or pressure or both.

When the streamlines are straight, the centrifugal effect is negligible and the pressure variation across the streamline is due to gravity alone even though the fluid is in motion.

Physical Interpretation

An equivalent form of the Bernoulli Equation

Hzg

Vp

2

2

Constant along a streamline

Elevation head -related to potential energy of the particle

Velocity head -vertical distance needed for the fluid to fall freely (neglecting friction) if it is to reach V from rest

Pressure head -height of the column of fluid that is needed to produce the pressure p

Total head

Pressure head + Elevation head = Piezometric head

Static, stagnation, Dynamic and Total Pressure

TpzVp 2

2

1Constant along a streamline

Dynamic pressure

Hydrostatic pressure

Total pressure

Static pressure Actual thermodynamic pressure

Static, stagnation, Dynamic and Total Pressure (cont.)

For 2 points at the same height with V2=0

2

1122

1Vpp

Figure 1 : Measurement of

static and stagnation

pressures

Static, stagnation, Dynamic and Total Pressure (cont.)

Then, p2 is called the stagnation pressure

The pressure at stagnation point, p2, is greater than the static pressure, p1

There is a stagnation point on any stationary body that is placed into a flowing fluid

Example 2

Air is drawn into a small open circuit wind tunnel as shown. Atmospheric pressure is 98.7 kPa (abs) and the temperature is 27C. If viscous effects are negligible, determine the pressure at the stagnation point on the nose of the airplane. Also determine the manometer reading, h, for the manometer attached to the static pressure tap within the test section of the wind tunnel if the air velocity within the test section is 60 m/s. Elevation of (1),(2) and (3) is assume the same.

Pitot Static Tube

Fluid speed can be calculated if we know the values of the static and stagnation pressures in a fluid.

Figure 2 : The Pitot static tube

43

14

2

21

3

2 ppV

ppp

Vpp

Pitot Static Tube

Examples of use of the Bernoulli Equation

Free Jets

Assumptions ◦ z1=h, z2=0 ◦ Reservoir is large, V1=0 ◦ Reservoir is open to atmosphere, p1=0 gage ◦ Fluid leaves as a free jet, p2=0

◦ Once outside nozzle, the stream continues as a free jet, p5=0

Figure 3 : Vertical flow from a tank


Free Jets (cont.)

Figure 4 : Vertical flow from a tank

ghh

22v2

Hhg 2v5

Example 3

For the system in the figure, h= 36 ft and the diameter of the side opening is 2 in. Find the

(a) Jet velocity in units of ft/s

(b) Volume flow rate in units of gallon per min. (gpm)

Solution: Example 3

P1 + ½ ρV12 + γZ1 = P2 + ½ ρV2

2 + γz2 P1 = 0 V1 = 0 Z1 = h Z2 = 0 P2 = 0 d1 = 2 in = 2/12 = 0.167 ft

Note that: γ = ρg, thus: γ/ρ = g

γ Z1 = ½ ρV22

a)→ V = (2gh)½ = (2 x 32.2 ft/s x 36 ft)½ = 48.15 ft/s

0 0 0 0

Solution: Example 3

Example 4

A smooth plastic, 10-m long garden hose with an inside diameter of 15 mm is used to drain a wading pool as shown. If viscous effects are neglected, what is the flowrate from the pool?

Solution: Example 4


Confined Flows ◦ In many cases, fluid is confined and its pressure cannot be prescribed a priori – need to use the concept of conservation of mass

Figure 5 : Steady flow into and out of a tank


Confined Flows (cont.) ◦ In such case, mass is conserved, i.e. inflow rate must equal to the outflow rate

◦ In general, following Bernoulli, an increase in velocity (could be due to reduction of flow area) is accompanied by a decrease in pressure

◦ For flows of liquids, this may result in cavitation, a potentially dangerous situation that results when liquid pressure is reduced to vapor pressure and the liquid “boils”.

ible)incompress(ifor 211222111 VAVAVAVA

AVmAVQ ,

Cavitation

• Cavitation is the formation

and then immediate implosion

of cavities in a liquid – i.e.

small liquid-free zones

("bubbles") – that are the

consequence of forces acting

upon the liquid.

• It usually occurs when a

liquid is subjected to rapid

changes of pressure that cause

the formation of cavities where

the pressure is relatively low.

Example 5

Water is siphoned from the tank as shown. The water barometer indicates a reading of 30.2 ft. Determine the maximum value of h allowed without cavitation occurring. Note that the pressure of the vapor in the closed end of the barometer equals the vapor pressure.

Solution: Example 5

Example 6

Water flows from a large tank as shown. Atmospheric pressure is 14.5 psia and the vapor pressure is 2.88 psia. If viscous effects are neglected, at what height, h, will cavitation begin?

Solution: Example 6


Free Jets (cont.)

– If exit of tank is not

smooth, well

contoured nozzle, the

diameter of the jet will

be less than the

diameter of the hole –

vena contracta effect

– Contraction coef.,

Cc=Aj/Ah

hole jet

Figure 3 : Typical flow patterns and

contraction coef. for various round exit

configurations


Flowrate measurement ◦ Assumptions – steady, inviscid and incompressible

Figure 6 : Typical devices for measuring

flowrate in pipes


Flowrate measurement (cont.) ◦ Between points (1) and (2)

2

12

212

2211

2

221

2

2

121

1

1

)(2

hence

and

AA

ppAQ

VAVAQ

VpVp


Flowrate measurement (cont.) ◦ The actual measured flowrate, Qactual will be smaller than this theoretical results because of the assumptions made in deriving the Bernoulli Equation

◦ Other flowmeters based on Bernoulli equation are used to measure flowrates in open channels such as flumes and irrigation ditches.

Orifice – Nozzle – Venturi Flow Meter

The Energy Line and the Hydraulic Grade Line

Energy line is a line that represents the total head available to the fluid

Under the assumptions of the Bernoulli equation, the energy line is horizontal

If the fluid velocity changes along the streamline, the hydraulic grade line will not be horizontal

The Energy Line and the Hydraulic Grade Line (cont.)

Figure 7 : Representation of the

energy line and the hydraulic

grade line

Measures the sum of

the pressure head and

the elevation head. The

sum is called

piezometric head


Figure 8 : Representation of the energy line and the hydraulic grade line for flow from a tank


The distance from the pipe to HGL in Fig. 7 indicates the pressure within the pipe. ◦ If the pipe lies below HGL the pressure within the pipe is positive

◦ If the pipe lies above HGL the pressure is negative

Example 7

Draw the energy line and the hydraulic grade line for the flow of Example 6.

Restrictions of use of the Bernoulli Equation

Assumptions involved in deriving the Bernoulli equation ◦ Fluid is incompressible – ok with liquids

◦ Flow is steady

◦ Inviscid flow In the absence of viscous effects, the total energy of the

system remains constant

◦ There are no mechanical devices in the system between the two points along the streamline to which the equation is applied

Bernoulli Chapter 3

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