Chapter 3 – Essential Fluid Mechanics for Energy Conversion

CHE 3107-Renewable Energy Engineering I

Prof. Byung-Hwan Um Chemical Engineering Hankyong National University Spring, 2012

Chapter 3 – Essential Fluid Mechanics

for Energy Conversion

• If the control volumes are in form of streamlines and streamtubes, where the flow in these streamlines and streamtubes moves in its designated path without

crossing each other, the flow can be assumed to be inviscid or frictionless since the fluid molecules do not interact with their adjacent counterparts.

• Hence, no intermolecular friction, which is also referred to the viscous property, is being produced.

• The application of the conservation of mass and

momentum along streamlines and streamtubes produces

the well-known Bernoulli equation.

Objectives

At the end of this chapter, you should be able to :

 identify flow problems in which the Bernoulli equation is valid,

 understand the use of hydraulic and energy grade lines,

 analyse frictionless flow problems using the

Bernoulli equation.

• Application of the second Newton’s law of motion along streamlines of fluid flow leads to a very famous equation in Fluid Mechanics, i.e. the Bernoulli equation.

• There are four assumptions used to derive the equation and these four assumptions must always be remembered to

ensure that it is used correctly, i.e.

1. The flow is inviscid or frictionless, i.e. viscous effects are negligible which is valid for low viscosity fluids such as water and air,

2. The flow is steady, i.e. the flow pattern is fully developed and does not change with time,

3. The flow is incompressible, which is valid for all

liquids and low speed gas of Mach 0.3 or below since the change in gas density is less than 5%,

4. The flow considered is along the same streamline, as the variation of properties for fluid molecules

travelling in the same path can be simulated more accurately through conservation laws of physics.

3.1 Basic physical properties of fluids (2)

The conservation of mass principle can be expressed as

Where and are the total rates of mass flow into and out of the CV, and

dm

/dt is the rate of

change of mass within the CV.

CV

in out

m m dm

− = dt

 

m 

m 

The amount of mass flowing through a control surface per unit time is called the mass flow rate and is denoted

The dot over a symbol is used to indicate time rate of change.

Flow rate across the entire cross- sectional area of a pipe or duct is obtained by integration

While this expression for is exact, it is not always convenient for

engineering analyses.

c c

n c

A A

m ^ = ∫ δ m = ∫ ρ V dA

m 

m 

3.3 Mass continuity (2)

For steady flow, the total

amount of mass contained in CV is constant.

Total amount of mass entering must be equal to total amount of mass leaving

For incompressible flows,

in out

m = m

∑ ∑ ^{ }

n n n n

in out

V A = V A

∑ ∑

Bernoulli’s Equation

 Acceleration of Fluid Particles give Fluid Dynamics

 Newton’s Second Law is the Governing Equation

 Applied to an Idealized Flow and Assumes Inviscid Flow

 There are numerous assumptions

 “Most Used and Abused Equation”

Daniel Bernoulli (1700-1782)

Swiss mathematician, son of Johann Bernoulli, who showed that as the velocity of a fluid increases, the pressure decreases, a statement known as the Bernoulli principle. He won the annual prize of the French Academy ten times for work on vibrating strings, ocean tides, and the kinetic theory of gases. For one of these victories, he was ejected from his jealous father's house, as his father had also submitted an entry for the prize. His kinetic theory proposed that the properties of a gas could be explained by the motions of its particles.

The Bernoulli Equation is Listed in Michael Guillen's book "Five Equations that Changed the World: The Power and Poetry of

Mathematics"

Original Bernoulli Equation

p₁ + ½V₁² + gz₁ = p₂ + ½V₂² + gz₂ = constant ^(3.2) ρ ρ

Eq. (3.2) is the original Bernoulli equation, which is applicable for inviscid, steady and incompressible flows along a streamline.

It can be rewritten in form of pressure (SI unit: Pa) as follows,

p₁ + ½ρV₁² + ρgz₁ = p₂ + ½ρV₂² + ρgz₂ = constant (3.2.1)

or, in form of head (SI unit: m) such that

p₁ + V₁² + z₁ = p₂ + V₂² + z₂ = constant (3.2.2) ρg 2g ρg 2g

• These terms are typically added to the head-form of Bernoulli equation, thus Eq. (3.2.2) can be transformed to be

p₁ + V₁² + z₁ + h_s = p₂ + V₂² + z₂ + h_L (3.2.3) ρg 2g ρg 2g

where h_L represents head loss due to wall friction, etc. and h_s represents the head associated with shaft work from pumps and turbines.

The value for h_s should be positive for pumps and negative for turbines.

Eq. (3.2.3) is called in some Fluid Mechanics text as the one-dimensional energy equation since it also represents conservation of energy between two positions.

In addition, it is also known as the modified Bernoulli equation as it

accommodates losses and any fluid machine installed along the flow path.

3.4 Energy conservation in an ideal fluid: Bernoull’s Equation (2)

Physical Interpretation: Normal and Along a Streamline Basic Assumptions:

1) Steady 2) Inviscid

3) Incompressible (=constant ρ)

 A violation of one or more of the assumptions mean the equation is invalid.

 The “Real World” is never entirely all of the above.

 If the flow is nearly Steady, Incompressible, and Inviscid, it is possible to adequately model it.

The three terms that the equations model are: pressure, acceleration, and weight.

Pressure Acceleration Weight

The Bernoulli equation is a statement of the work-energy principle:

The work done on a particle by all forces acting on the particle is equal to the change of kinetic energy of the particle.

Physical Interpretation: Normal and Along a Streamline

As a fluid particle moves, pressure and gravity both do work on the particle:

p is the pressure work term, and γz is the work done by weight.

1/2ρV² is the kinetic energy of the particle.

Alternatively, the Bernoulli equation can be derived from the first and second laws of Thermodynamics (energy and entropy) instead of the Newton’s 2^nd Law with the appropriate restrictions.

Bernoulli’s Equation can be written in terms of heads:

. 2 tan

2

streamline a

on t cons g z

V

p + + =

γ

Pressure

Head Velocity Head

Elevation Term

Pressure Head: represents the height of a column of fluid that is needed to produce the pressure p.

Velocity Head: represents the vertical distance needed for the fluid to fall freely to reach V.

Elevation Term: related to the potential energy of the particle.

3.4 Energy conservation in an ideal fluid: Bernoull’s Equation (4)

Static, Stagnation, Dynamic, and Total Pressure: Bernoulli Equation

Static Pressure

Dynamic Pressure

Hydrostatic Pressure

Follow a Streamline from point 1 to 2

1 2 1 1

2 2

2

2 2

1 2

1 V z p V z

p + ρ +γ = + ρ +γ

Following a streamline:

0 0, no elevation 0, no elevation

2 1 1

2 2

1 V p

p = + ρ ^p₂ =γ^H

H > h

Note:

(^{H h})

V₁ = γ − In this way we obtain a measurement of the centerline flow with piezometer tube.

“Total Pressure = Dynamic Pressure + Static Pressure”

 Static pressure – representing the actual or thermodynamic pressure at a particular point in the streamline.

 Dynamic pressure – representing the kinetic energy for fluid molecules passing at the same point.

 Hydrostatic pressure – representing the potential energy for fluid molecules at the same point which changes with elevation.

Stagnation Point: Bernoulli Equation

Stagnation point: the point on a stationary body in every flow where V= 0 Stagnation Streamline: The streamline that terminates at the stagnation point.

Symmetric:

Axisymmetric:

If there are no elevation effects, the stagnation pressure is largest pressure obtainable along a streamline: all kinetic energy goes into a pressure rise:

2 V2

p₊ ρ

streamline a

on t cons p

z V

p _T tan

2

1 ₂ + = =

+ ρ γ

Total Pressure with Elevation:

Stagnation Flow I:

Stagnation Flow II:

3.4 Energy conservation in an ideal fluid: Bernoull’s Equation (6)

The Bernoulli Equation is a statement of the conservation

of ____________________ Mechanical Energy

1

2

p gz V C

ρ ^{+ + =}

2

"

2

p V

z C

γ ^{+ +} g ⁼

Pressure head

p γ ⁼

z =

p z γ ^{+ =}

2

2 V

g =

Elevation head Velocity head

Piezometric head

2

2

p V

z g

γ ^{+ + =} Total head

Energy Grade Line

Hydraulic Grade Line

Energy and Hydraulic Grade Lines

 Pressure head p/ρg – representing the height of a fluid column of density ρ required to generate the pressure p at its datum.

 Velocity head V²/2g – representing the height of a fluid mass initially at rest to free fall under the influence of gravity with no resistance or friction and accelerate to a velocity V.

 Elevation head z – representing the potential energy of a

fluid and is directly given by its height from the datum.

The combination of the pressure head p/ρg and the elevation head z from the piezometric head.

• Using different constant of the Bernoulli equation for positions (1) and (2) to represent any losses and addition or extraction of energy to and from the system, the Bernoulli equation can be modified to include new parameters.

• Here, energy may be added or extracted to or from the fluid system if fluid machines such as pumps and turbines are installed in between the travelling path, while factors such as wall friction, convergence and divergence of flow, bends and any fitting components such as valves, filters and faucets installed in between points (1) and (2) lead to energy losses.

Energy and Hydraulic Grade Lines

• In this section, we will look into all terms in the Bernoulli equations, Eq. (3.2.1) and Eq. (3.2.2), which can be termed based on their functionality.

• Since positions (1) and (2) are arbitrary, by omitting the subscripts, we can rewrite Eq. (3.2.1) as,

p + ½ ρ V

+ ρ gz

= p

Here, p_T is defined as total pressure which is always constant along the same streamline and each term in the left side can be defined as follows:

 Static pressure p – representing the actual or thermodynamic

pressure at a particular point in the streamline.

 Dynamic pressure ½ρV² – representing the kinetic energy for fluid molecules passing at the same point.

 Hydrostatic pressure ρgz – representing the potential energy for fluid molecules at the same point which changes with elevation.

• If the fluid has a certain velocity V travelling along one streamline with small elevation, the hydrostatic pressure is usually small and insignificant compared to the static pressure and the dynamic

pressure. The combination of the static pressure and the dynamic pressure forms the stagnation pressure p₀, or

p + ½ ρ V

= p

Energy and Hydraulic Grade Lines

• The stagnation pressure is usually used in gas flows as an equivalent term for total pressure for liquids and represents the pressure generated when a fluid is suddenly being stopped.

• Now, let us consider another form of the Bernoulli Equation, Eq. (3.3.2), which can be rewritten as,

p

+ V

+ z = H

ρ g 2g

• Here, H is defined as total head which is always constant along the same streamline and each term in the left side can be defined as follows:

• In this section, we will apply the Bernoulli equation to selected applications in Fluid Mechanics, namely the free jet flows.

• For the Bernoulli equation to be applicable, all the flows are steady, incompressible, frictionless and along the same streamline.

• Firstly, we are going to consider is the free jet formed by the flow out of a large tank through a nozzle at the bottom of the tank.

• Flow and the streamline under consideration are shown in Fig. 5.4. Here, using the Bernoulli equation, we can form a relation between point (1) and point (2) as follows:

p

+ ½ ρ V

+ ρ gz

= p

+ ½ ρ V

+ ρ gz

Applications of the Bernoulli Theorem (2)

• At point (1), the pressure is atmospheric (p₁ = p₀), or the gage pressure is zero, and the fluid is almost at rest (V₁ = 0).

At point (2), the exit pressure is also atmospheric (p₂ = p₀), and the fluid moves at a velocity V. By using point (2) as the datum where z₂ = 0 and the elevation of point (1) is h, the above relation can be reduced to

p

+ ½ρ(0)

+ ρ gh = p

+ ½ ρ V

+ ρ g (0) ρ gh = ½ ρ V

Hence we can formulate the velocity V to be

V = √ 2gh

• Notice that we can also obtain the similar relation by using the relation between point (3) and point (4). The pressure and the velocity for point (4) is similar to point (2). However, the pressure for point (3) is the hydrostatic pressure, i.e. p₃ = p₀ + ρg(h - ) and the velocity is also zero due to an assumption of a large tank. Hence, the relation becomes

ρgh = ½ρV ²

Hence we can formulate the velocity V to be

p₀ + ½ρ(0)² + ρgh = p₀ + ½ρV₅² + ρg ( -H ) V₅ = √ 2g ( h + H )

Applications of the Bernoulli Theorem (4)

where (h + H) is the vertical distance from point (1) to point (5).

• For a nozzle located at the side wall of the tank as in Fig. 5.5(b), we can also form a similar relation for the Bernoulli equation, i.e.

For a nozzle having a small diameter (d ∨ h), then we can conclude that

V

≅ V

≅ V

= V ≈ √ 2gh

i.e., the velocity V is only dependent on the depth of the centre of the nozzle from the free surface h. If the edge of the nozzle is sharp, as illustrated in Fig. 5.5, flow contraction will be occurred to the flow. This phenomenon is known as vena contracta, which is a result of the inability for the fluid to turn at the sharp corner 90°. This effect causes losses to the flow.

Applications of the Bernoulli Theorem (6)

In this section, we are going to use the Bernoulli Equation in the measurement of flow-rate. This can be accomplished by introducing an obstacle to the flow between two positions as depicted in Fig. 5.6.

• For this case, since the level for points (1) is equal to that for point (2), z₁= z₂. Then, using the Bernoulli equation, we can write the relation to be

p

+ ½ ρ V

= p

+ ½ ρ V

p

+ p

= ½ ρ (V

– V

)

• Hence, we can see that the pressure decrease with an increase in velocity due to flow contraction and increase after passing the neck structure at point (2) with a decrease in velocity as a result of flow expansion.

• Hence the minimum pressure can be generated at the neck. For low and medium values of flowrate, the minimum pressure can be calculated provided that we know the cross sectional area of both points.

Flow Measurement (2)

• From the continuity equation, if A and d is the cross sectional area and the diameter of the pipe, respectively, we can write

Q = A₁ V₁ = A₂ V₂ ( ¼πd₁²)V₁ = ( ¼πd₁²)V₂

V₁ = V₂ d₂ ²

d₁

By defining the diameter ratio as β = d₂/d₁, i.e. the ratio between the neck diameter and the pipe diameter, then the relation for velocity between these two points becomes

V₁

= β

Since the pressure at point (1), we can write the relation for pressure difference ∆p to be

∆p = p₁ – p₂ = ½ρ (V₂² – V₁²)

( )

• Putting Equation (5.18) into V₁ in this pressure difference relation gives ∆p = p₁ – p₂ = ½ρV₂² (1 –

β

Hence, the velocity at point (2) can be formulated as

V₂

=

and the formula for flow rate is

Q = A₂V₂ = A₂ ^(3.3.11)

• If the area at the neck becomes very small in such a way that the minimum pressure is projected lower than the vapour pressure of the liquid at that temperature, cavitation will occur at the neck where the liquid vapour starts to form at the neck and moves together with the flow as bubbles. This phenomenon is highlighted in Fig. 5.6 for high flowrate.

2 ∆p ρ (1 – β ⁴)

√

2 ∆p ρ (1 – β ⁴)

√

Flow Measurement (4)

• In addition, Eq. (3.3.11) presents a theoretical formula for flowrate.

However, the actual flow will be lower that the value calculated using this formula. This is due to flow contraction at the neck which generates losses as the vena contracta forms just downstream of the neck.

• Hence, Eq. (3.3.11) has to be modified to include the coefficient of discharge C_D to associate the losses into the flowrate formula.

Thus, the formula becomes

Q_actual = C_DQ_theory = C_DA₂ (3.3.12)

• The value of C_D depends on the fitting used to form the obstacle.

Typically, there are three types of obstacle as shown in Fig. 5.7.

2 ∆p ρ (1 – β ⁴)

√

Flow Measurement (6)

• Typically, C_D varies with the flowrate and can be correlated to be a function of Reynolds number Re (see Section 3.2, Unit 1 and Section 7.1, Unit 3).

• Hence, the graphs of C_D for orifice, nozzle and venturi meter are respectively given in Fig. 5.8, Fig. 5.9 and Fig. 5.10.

Flow Measurement (7)

• The second flow measuring apparatus which uses the Bernoulli equation is the Pitot tube. Fig. 5.11(a) illustrates the schematic representation of the Pitot tube, while Fig. 5.11(b) gives the actual layout of the Pitot tube where point (3) and point (4) are linked to manometers or other measuring apparatus.. Again, from Fig. 5.11(a), we can make the relation between point (1) and point (2) to be

p

+ ½ ρ V

+ ρ gz

= p

+ ½ ρ V

+ ρ gz

For this case, z₁ = z₂, V₁ = V, V₂ = 0, p₁= p₀ + ρgh dan p₂= p₀ + ρgH.

Hence, the above relation becomes

V = √2g (H - h) (3.3.13)

Summary

This chapter has summarized on the aspect below:

 you should be able to understand that the Bernoulli equation for frictionless flows and should be able to apply to relevant fluid flows that are steady, incompressible, inviscid and along a streamline.

 This knowledge is very helpful in analysing many simple flow problems such as the flow in pipes.