BERNOULLI AND ENERGY EQUATIONS Conservation of Mass General equation 흏 휌푢 흏 휌푣 흏 휌푤 흏 휌 + + + = 0 흏풙 흏풚 흏풛 흏풕

흏 휌 steady flow ( ) incompressible 흏풕 one dimensional flow, in X- fluids : direction: 흏푢 흏푣 흏푤 흏푢 + + = 0 = 0 흏풙 흏풚 흏풛 흏풙

two-dimensional flow: 흏푢 흏푣 + = 0 2 흏풙 흏풚 The Bernoulli Equation

• 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. • Bernoulli equation: An approximate relation between pressure, velocity, and elevation, and is valid in regions of steady, incompressible flow where net frictional forces are negligible.

Derivation of the Bernoulli Equation

- 퐹푠 = 푚푎푠 푑푉 - 푃푑퐴 − 푃 + 푑푝 푑퐴 − 푊푠푖푛휃 = 푚푉 푑푠 m = 휌V = 휌dA ds is the mass, W = mg = 휌g dA ds is the weight of the fluid particle, and sin 휃 = dz/ds. 푑푧 푑푉 −푑푃푑퐴 − 휌푔푑퐴푑푠 = 휌푑퐴푑푠푉 푑푠 푑푠 Canceling dA from each term and simplifying, −푑푃 − 휌푔푑푧 = 휌푉푑푉 Noting that V dV =0.5 d (V 2) and dividing each term by 휌 gives 푑푝 푑(푉2) + 푔푑푧 + = 0 Integrating, 휌 2 푑푝 푉2 - Steady flow: + 푔푧 + = 푐표푛푠푡푎푛푡 휌 2 - Steady, incompressible flow: 풑 (푽ퟐ) + 품풛 + = 풄풐풏풔풕풂풏풕 흆 ퟐ - Steady, incompressible flow: 푃 푉2 푃 푉2 1 + 푔푧 + 1 = 2 + 푔푧 + 2 4 휌 1 2 휌 2 2 The sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow when compressibility and frictional effects are negligible.

The incompressible Bernoulli equation is derived assuming incompressible flow, and thus it should not be used for flows with significant compressibility effects. 5 • The Bernoulli equation can be viewed as the “conservation of mechanical energy principle.” • This is equivalent to the general conservation of energy principle for systems that do not involve any conversion of mechanical energy and thermal energy to each other, and thus the mechanical energy and thermal energy are conserved separately. • The Bernoulli equation states that during steady, incompressible flow with negligible The Bernoulli equation friction, the various forms of mechanical states that the sum of the energy are converted to each other, but their kinetic, potential, and flow sum remains constant. energies of a fluid particle is • There is no dissipation of mechanical energy constant along a streamline during such flows since there is no friction that during steady flow. converts mechanical energy to sensible thermal (internal) energy. • The Bernoulli equation is commonly used in practice since a variety of practical fluid flow problems can be analyzed to reasonable 6 accuracy with it. Limitations on the Use of the Bernoulli Equation 1. Steady flow The Bernoulli equation is applicable to steady flow. 2. Frictionless flow Every flow involves some friction, no matter how small, and frictional effects may or may not be negligible. 3. No shaft work The Bernoulli equation is not applicable in a flow section that involves a pump, turbine, fan, or any other machine or impeller since such devices destroy the streamlines and carry out energy interactions with the fluid particles. When these devices exist, the energy equation should be used instead. 4. Incompressible flow Density is taken constant in the derivation of the Bernoulli equation. The flow is incompressible for liquids and also by gases at Mach numbers less than about 0.3. 5. No heat transfer The density of a gas is inversely proportional to temperature, and thus the Bernoulli equation should not be used for flow sections that involve significant temperature change such as heating or cooling sections. 6. Flow along a streamline Strictly speaking, the Bernoulli equation is applicable along a streamline. However, when a region of the flow is irrotational and there is negligibly small vorticity in the flow field,7 the Bernoulli equation becomes applicable across streamlines as well. Frictional effects, heat transfer, and components that disturb the streamlined structure of flow make the Bernoulli equation invalid. It should not be used in any of the flows shown here.

When the flow is irrotational, the Bernoulli equation becomes applicable between any two points along the flow (not just on the same streamline). 8 Static, Dynamic, and Stagnation Pressures

The kinetic and potential energies of the fluid can be converted to flow energy (and vice versa) during flow, causing the pressure to change. Multiplying the Bernoulli equation by the density gives

P is the static pressure: It does not incorporate any dynamic effects; it represents the actual thermodynamic pressure of the fluid. This is the same as the pressure used in thermodynamics and property tables. V 2/2 is the dynamic pressure: It represents the pressure rise when the fluid in motion is brought to a stop isentropically. gz is the hydrostatic pressure: It is not pressure in a real sense since its value depends on the reference level selected; it accounts for the elevation effects, i.e., fluid weight on pressure. (Be careful of the sign— unlike hydrostatic pressure gh which increases with fluid depth h, the hydrostatic pressure term gz decreases with fluid depth.) Total pressure: The sum of the static, dynamic, and hydrostatic pressures. Therefore, the Bernoulli equation states that the total pressure along a streamline is constant. 9 Stagnation pressure: The sum of the static and dynamic pressures. It represents the pressure at a point where the fluid is brought to a complete stop isentropically.

Close-up of a Pitot-static probe, showing the stagnation pressure hole The static, dynamic, and and two of the five static circumferential stagnation pressures measured pressure holes. using piezometer tubes.

10 Hydraulic Grade Line (HGL) and Energy Grade Line (EGL) It is often convenient to represent the level of mechanical energy graphically using heights to facilitate visualization of the various terms of the Bernoulli equation. Dividing each term of the Bernoulli equation by g gives

P /g is the pressure head; it represents the height of a fluid column that produces the static pressure P. V 2/2g 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.

An alternative form of the Bernoulli equation is expressed in terms of heads as: The sum of the pressure, velocity, and elevation heads is constant along a streamline.

11 Hydraulic grade line (HGL), P/g + z The line that represents the sum of the static pressure and the elevation heads. Energy grade line (EGL), P /g + V 2/2g + z The line that represents the total head of the fluid. Dynamic head, V 2/2g The difference between the heights of EGL and HGL.

The hydraulic grade line (HGL) and the energy grade line (EGL) for free discharge from a reservoir through a horizontal pipe with a diffuser.

12 HGL and EGL

Hydraulic Grade Line (HGL) P HGL z g

Energy Grade Line (EGL) (or total head)

PV2 EGL   z gg2

Examples on HGL and EGL: Notes on HGL and EGL • For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide with the free surface of the liquid. • The EGL is always a distance V2/2g above the HGL. These two curves approach each other as the velocity decreases, and they diverge as the velocity increases. • In an idealized Bernoulli-type flow, EGL is horizontal and its height remains 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. The slope is a measure of the head loss in the pipe. A component, such as a valve, that generates significant frictional effects causes a sudden drop in both EGL and HGL at that location. • A steep jump/drop occurs in EGL and HGL whenever mechanical energy is added or removed to or from the fluid (pump, turbine). • The (gage) pressure 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. 17 In an idealized Bernoulli-type flow, EGL is horizontal and its height A steep jump occurs in EGL and HGL remains constant. But this is not whenever mechanical energy is added to the case for HGL when the flow the fluid by a pump, and a steep drop velocity varies along the flow. occurs whenever mechanical energy is removed from the fluid by a turbine.

The gage pressure of a fluid is zero at locations where the HGL intersects the fluid, and the pressure is negative 18 (vacuum) in a flow section that lies above the HGL. Example: Water Discharge Example: from a Large Tank Spraying Water into the Air

19 Example: Siphoning Out Gasoline from a Fuel Tank

20 Example: Velocity Measurement by a Pitot Tube

21 22 23 Example Example Example Example

-29.9 kpa Example Example Example Example

p V 2 p V 2 1  1  z  2  2  z  h  H g 2g 1 g 2g 2 L P

7.642 0  0  0  0  10  H 2*9.81 P

H P 12.97 m of water 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 PITOT TUBE IN THE PIPE

Using combined Pitot static tube. In which the inner tube is used to measure the impact pressure while the outer sheath has holes in its surface to measure the static pressure

The total pressure is know as the stagnation pressure (or total pressure) 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. • The performance of an orifice is related with three constants, namely, coefficient of contraction (Cc), coefficient of velocity (Cv), coefficient of discharge (Cd). • Coefficient of contraction; ratio Cc, of area Aactual, of the smallest section of the discharging flow to area A of the small hole as shown in Figure :

• 퐴푎푐푡푢푎푙 = 퐶푐퐴표푟푓푖푐푒

• Coefficient of velocity;. Ratio Cv, of actual velocity u to 2푔퐻 is called the coefficient of velocity:

• 푢 = 퐶푣푢퐵 = 퐶푣 2푔퐻 • Coefficient of discharge; the coefficient of discharge is the ratio of actual rate of discharge (Q) to ideal rate of discharge Qideal at no-friction and no- contraction condition:

• 푄푎푐푡푢푎푙 = 퐶푐퐶푣퐴표푟푓푖푐푒푢퐵 = 퐶푐퐶푣퐴표푟푓푖푐푒 2푔퐻

p1 • Furthermore, setting 퐶푐퐶푣 = 퐶 : A  C A (1) actual c orfice • 푄푎푐푡푢푎푙 = 퐶푑퐴표푟푓푖푐푒 2푔퐻 x

y p2 vena contractor

u2 A standard sharp-edged orifice

(2)

(b) (a) Orifice and vena contracta

Flow from an orifice at the side of a tank under a change head

The theoretical flow velocity is 푢 = 2푔퐻 Assume that dQ of water flows out in time dt with the water level falling by –dH (Figure . Then

푑푄 = 퐶푑퐴표푟푓푖푐푒 2푔퐻푑푡 = −푑퐻퐴tank 퐴 푑퐻 푑푡 = − tank 퐶푑퐴표푟푓푖푐푒 2푔퐻 푡2 퐴 퐻2 푑퐻 푑푡 = − tank 푡1 퐶푑퐴표푟푓푖푐푒 2푔 퐻1 퐻

where Atank is the cross-sectional area of the tank. The time needed for the water level to descend

from H1 to H2 is 2퐴tank 푡2 − 푡1 = 퐻1 − 퐻2 41 퐶푑퐴표푟푓푖푐푒 2푔 Flow from an orifice at the side of a tank where the descending velocity of the head is constant

• Assume that the bottom has a small hole of area a, through which water flows , then 2 • 푑푄 = 퐶푑퐴표푟푓푖푐푒 2푔퐻푑푡 = −푑퐻퐴tank = −푑퐻휋푟 • Whenever the descending velocity of the water level (-dH/dt=u) is constant, the above equation becomes: 푑퐻 퐶 퐴 2푔퐻 • 푢 = − = 푑 표푟푓푖푐푒 푑푡 휋푟2 2 휋푢 • 퐻 = − 푟4 퐶푑퐴표푟푓푖푐푒 2푔 • 퐻 ∝ 푟4

42

Venturi , nozzle and orifice meters

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

Horizontal Venturi Meter

2 2 p1 V1 p2 V2   z1    z2 g 2g g 2g Horizontal Venturimeter p p V 2 V 2 1  2  2  1 g g 2g 2g V 2 V 2 h  2  1  2g(h)  V 2 V 2 2g 2g 2 1 2 2 2 2 Q Q 2g(h)  V2 V1  2g(h)  2  2 A2 A1 Q2  1 1  h      2 2  2g  A2 A1    A .A  A .A  Q  2gh 1 2  Q  C  2gh 1 2   A2  A2  actual d  2 2   1 2   A1  A2  Venturi Meter Example

 A .A  Q  C  2gh 1 2  actual d  2 2   A1  A2   0.031410.00786  30 /1000  0.96 29.81 h   2 2   0.03141  0.00786  h  0.756m Venturi Meter Example

hmercury  0.25m     mercury  hwater  hmercury  1  water  13600  hwater  0.25 1  3.15m  1000  Venturi Meter Venturi Meter Example Venturi Meter

hmercury  0.05m     mercury  hoil  hmercury  1  oil  13600  hoil  0.05 1  0.92m  700   A .A  Q  C  2gh 1 2  actual d  2 2   A1  A2   0.12570.0314  Q  0.98 29.810.92 *   2 2   0.1257  0.0314  Q  0.138 m3 / s