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**The Bernoulli Equation**

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**SEQUENCE OF CHAPTER Introduction Objectives**

Frictionless Flow Along Streamlines Energy and Hydraulic Grade Lines Application of Bernoulli Theorem Flow Measurements Summary

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Introduction 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.

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**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.

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**Frictionless Flow Along Streamlines**

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,

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**Frictionless Flow Along Streamlines**

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.

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**Frictionless Flow Along Streamlines**

Original Bernoulli Equation p1 + ½V12 + gz1 = p2 + ½V22 + gz2 = constant (7) Eq. (7) is the original Bernoulli equation, which is applicable forinviscid, steady and incompressible flows along a streamline. It can be rewritten in form of pressure (SI unit: Pa) as follows, p1 + ½V12 + gz1 = p2 + ½V22 + gz2 = constant (8) or, in form of head (SI unit: m) such that p1 + V12 + z1 = p2 + V22 + z2 = constant (9) g 2g g 2g

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**Frictionless Flow Along Streamlines**

Fig. 5.2 gives several examples in which the Bernoulli equation can be applied or not.

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**Frictionless Flow Along Streamlines**

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.

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**Frictionless Flow Along Streamlines**

These terms are typically added to the head-form of Bernoulli equation, thus Eq. (9) can be transformed to be p1 + V12 + z1 + h3 = p2 + V22 + z2 + hL (10) g 2g g 2g where hL represents head loss due to wall friction, etc. and hs represents the head associated with shaft work from pumps and turbines. The value for hs should be positive for pumps and negative for turbines. Eq. (10) 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.

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**Frictionless Flow Along Streamlines**

Usually, the rating of pumps and turbines are given in terms of power, i.e. horsepower (hp) for the British unit and watt (W) for the SI unit. If the mass flowrate mL or the volumetric flowrate Q is known, the corresponding head can be converted into power due to shaft work Ws using the following equation: Ws = mghs = gQhs (11) Consequently, power loss in the system WL can be calculated using a similar form of equation. WL = mghL = gQhL (12) ˙

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**Energy and Hydraulic Grade Lines**

In this section, we will look into all terms in the Bernoulli equations, Eq. (8) and Eq. (9), which can be termed based on their functionality. Since positions (1) and (2) are arbitrary, by omitting the subscripts, we can rewrite Eq. (8) as, p + ½V2 + gz1 = pr (13) Here, pT 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:

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**Energy and Hydraulic Grade Lines**

Static pressure p – representing the actual or thermodynamic pressure at a particular point in the streamline. Dynamic pressure ½V² – representing the kinetic energy for fluid molecules passing at the same point. Hydrostatic pressure gz – 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 + ½V 2 = p0 (14)

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**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. (5.9), which can be rewritten as, p1 + V 2 + z = H (15) 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:

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**Energy and Hydraulic Grade Lines**

Pressure head p/g – representing the height of a fluid column of density 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/g and the elevation head z from the piezometric head.

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**Applications of the Bernoulli Theorem**

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.

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**Applications of the Bernoulli Theorem**

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: p1 + ½V12 + gz1 = p2 + ½V22 + gz2

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**Applications of the Bernoulli Theorem**

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 p0 + ½(0)2 + gh = p0 + ½V 2 + g (0) gh = ½V 2 Hence we can formulate the velocity V to be V = 2gh (16)

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**Applications of the Bernoulli Theorem**

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 + g(h - ) and the velocity is also zero due to an assumption of a large tank. Hence, the relation becomes [ p0 + g(h - )] + ½(0)2 + g = p0 + ½V 2 + g (0) gh = ½V 2 Hence we can formulate the velocity V to be p1 + ½V12 + gz1 = p5 + ½V5 2 + gz5 p0 + ½(0)2 + gh = p0 + ½V52 + g ( -H ) V5 = 2g ( h + H )

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**Applications of the Bernoulli Theorem**

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. V1 = 2g(h – d/2), V2 = 2gh, V3 = 2g (h + d/2), For a nozzle having a small diameter (d h), then we can conclude that V1 V2 V3 = V 2gh

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**Applications of the Bernoulli Theorem**

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.

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Flow Measurement 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.

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**Flow Measurement p2 + p1 = ½ (V22 – V12) (17)**

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 p1 + ½V12 = p2 + ½V2 2 p2 + p1 = ½ (V22 – V12) (17) 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.

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**( ) Flow Measurement ( ¼d12)V1 = ( ¼d12)V2 V1 = V2 d2 2 d1**

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 ( ¼d12)V1 = ( ¼d12)V2 V1 = V d2 2 d1 By defining the diameter ratio as = 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 = 2V1 (18) Since the pressure at point (1), we can write the relation for pressure difference p to be p = p1 – p2 = ½ (V22 – V12) ( )

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Flow Measurement Putting Equation (18) into V1 in this pressure difference relation gives p = p1 – p2 = ½V22 (1 – 4) (19) Hence, the velocity at point (2) can be formulated as V2 = (20) and the formula for flow rate is Q = A2V2 = A2 (21) 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 (1 – 4) 2 p (1 – 4)

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** Flow Measurement Qactual = CDQtheory = CDA2 (22)**

In addition, Eq. (21) 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. (21) has to be modified to include the coefficient of discharge CD to associate the losses into the flowrate formula. Thus, the formula becomes Qactual = CDQtheory = CDA2 (22) The value of CD depends on the fitting used to form the obstacle. Typically, there are three types of obstacle as shown in Fig. 5.7. 2 p (1 – 4)

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Flow Measurement

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Flow Measurement Typically, CD varies with the flowrate and can be correlated to be a function of Reynolds number Re Hence, the graphs of CD for orifice, nozzle and venturi meter are respectively given in Fig. 5.8, Fig. 5.9 and Fig. 5.10.

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Flow Measurement 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

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**Flow Measurement p1 + ½V12 + gz1 = p2 + ½V22 + gz2**

For this case, z1 = z2, V1 = V, V2 = 0, p1= p0 + gh and p2= p0 + gH. Hence, the above relation becomes ( p0 + gh )+ ½V 2 = (p0 + gH) + ½(0)2 V = 2g (H - h) (23)

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**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 as will be discussed in Unit 3..

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Example 1 Velocity measurement by a Pitot tube

Example 1 Velocity measurement by a Pitot tube

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