Chapter 8

PAGE Basic Fluid Mechanics

Dr. V. K. Sarda

Chapter 8

Fluid Dynamics-I[Equation of motion and Energy Equation]

There are a few instances (of the order of ten) when Science had the greatest impact on peoples life. Fluid Dynamics provided at least one of them. The list of greatest impacts could include, for example,

Telecommunications (Electromagnetic theory) Nuclear energy (Quantum mechanics) Airplanes (Fluid mechanics)

Fluid dynamics is the key to our understanding of some of the most important phenomena in our physical world: ocean currents, floods and weather systems.

1 Introduction

Fluid dynamics is the branch of fluid mechanics which deals with fluid in motion as a consequence of pressure and such agents.A fluid motion can be completely analyzed by the application of following fundamental equations: The continuity equation

The energy equation

The momentum equationThese equations are based on three fundamental laws, namely:

The law of conservation of mass flow

The law of conservation of energy

The law of conservation of momentum

Continuity equation has already been dealt in previous chapter. In this chapter we will deal with the remaining two.

2 Two Types of Forces in Fluids

(i) Long-range forces (decrease slowly with distance).

Examples:

(1) Gravity,

(2) Electromagnetic forces (when the fluid contains charged particles, like plasma),

(3) Inertial forces (when the fluid dynamics is considered from a non-inertial

frame).

These forces can penetrate into the interior of the fluid. They are called volume or body forces.

A body force is characterized by the force density per unit mass.The force acting on the fluid particle with volume surrounding a point x at any instant t =

(ii) Short-range forces (decrease rapidly with distance). These refer to local dynamic stresses developing within the fluid itself as it moves; specifically, the forces acting on a given element of fluid by the surrounding fluid.

They are negligible, unless there is a direct mechanical contact between the interacting fluid elements.

Example: Two marbles do not interact, unless they are in contact, i.e. within the distance of intermolecular forces.

If we consider a fluid, we can think of it as consisting of two parts (with some imaginary boundary in between). These two parts are in a direct mechanical contact, and so should act on each other with some short-range force.

How to describe such a force? It is determined not only by the point x and instant t, but also by the orientation of the surface, passing through that point, i.e. by the unit normal vector . Consider Fig. 1.

Fig. 1 Force on a surafce

Here, force is exerted by the fluid element which points to, on the fluid element which points away from.

This force is proportional to the surface area,.The force per unit area, , is called stress.

3. Energy possessed by a fluid bodyA fluid body possesses the following energies, namely:

potential energy, pressure energy,

kinetic energy. 3.1 Potential Energy or Datum Energy: This is the energy possessed by a fluid body by virtue of its position or location in space. Consider W Newtons of a fluid at a height of zm above a datum plane. The potential energy of W Newtons of the fluid = Wz (N-m or joule) Thus the potential energy per Newton of the fluid = z (N-m/N) Or we say, the potential head = Z m.

(1)

3.2 Pressure energy: This is the energy possessed by a fluid body by virtue of the pressure at which it is maintained.

Fig. 2 Liquid chamberFig. 2 (a) shows a large chamber containing a liquid at a pressure intensity p. If now a piezometer tube be fitted to the chamber as shown in Fig. 2 (b), we know the liquid will rise in the tube by a height h metres. Now consider the fluid particles at the surface in tube. Let the weight of these particles be W Newton. Obviously these particles now have a potential energy of Wh Newton metre or joule. When these particles were inside the chamber, they had only pressure energy. Hence as these particles left the chamber and reached the surface in the tube, the pressure energy is converted into potential energy. Therefore, pressure energy of W Newton of the fluid in the chamber = Wh Newton metre or joule

Also, we know p = h,

where is the specific weight of the fluid.Pressure energy of W Newton of the fluid in the chamber = Newton metre or joule

Pressure energy per Newton of the fluid in the chamber = Newton metre per Newton

Or we say the pressure head = m..

(2)3.3 Kinetic energy. This is the energy possessed by a fluid body by virtue of its motion. Suppose W Newton of a fluid be moving at a velocity of V metres per second. Kinetic energy of W Newton of the fluid = N-m or joule. Therefore, kinetic energy per Newton of the fluid body = N-m/N Or we say the kinetic head is m.

(3)Thus we find that if a fluid body of weight W Newton be at a height of Z metres above a datum, and at a pressure intensity of p Newton / metre2, and at a velocity of v m/s, then the total energy of W Newton of the fluid body;= N-m

:. Total energy per Newton of the fluid body = Nm/N

Or we say the total energy head = m

(4)4. Newtons Laws

Newtons laws are relations between motions of bodies and the forces acting on them.

First law: a body at rest remains at rest, and a body in motion remains in motion at the same velocity in a straight path when the net force acting on it is zero.

Second law: the acceleration of a body is proportional to the net force acting on it and is inversely proportional to its mass. i.e.

the net force Fx acting on a fluid element in the direction of x is equal to mass m of the fluid element multiplied by the acceleration ax in the x -direction. i.e.

(5)Third law: when a body exerts a force on a second body, the second body exerts an equal and opposite force on the first.In a fluid flow, the following forces are present: 1. , gravity force.

2. , the pressure force.

3. , force due to viscosity.4. , force due to turbulence. 5. , force due to compressibility.

So that:

(6)(i) If the force due to compressibility, Fc is negligible, the resulting net force:

(7)These equations of motions are called Reynold's equations of motion.

(ii) For flow, where Ft is negligible, the resulting equations of motion are known as Navier-Stokes Equations.

(8)(iii) If the flow is assumed to be ideal, viscous force is zero and equation of motions are known as Euler's equation of motion.

(9)5. Euler's equation of motionIn this equation of motion, only forces due to gravity and pressure are taken into consideration.

Consider a stream-line in which flow is taking place in S-direction as shown in Fig. 3. Consider a cylindrical element of cross-section dA and length dS. The forces acting on the cylindrical element are:

(i) Pressure force in the direction of flow.

Fig. 3 Stream tube element

I

(ii) Pressure force , opposite to the direction of flow.(iii) Weight of element

Let is the angle between the direction of flow and the line of action of the weight of element.The resultant force on the fluid element in the direction of S must be equal to the mass of fluid element acceleration in the direction S.

--=

(10)Where as is the acceleration in the direction of S,

Now as = ==

Where v is a function of s and t.

If the flow is steady, = 0.

=

Substituting this in (10) and simplifying the equation, we get

-=

EMBED Equation.3

Or, -=

+ + = 0

Also,

EMBED Equation.3 + + = 0

EMBED Equation.3 + + = 0

(11)

Total change in energy per unit mass is equal to zero. (11) is known as Euler's equation of motion.6. Bernoulli's equation from Euler's equationBernoulli's equation is obtained by integrating the Euler's equation of motion (11) as:

+ + = constantFor incompressible flow, =constant

+ gz +=constant

Or, + z +=constant

(12)

Which is Bernoullis equation.

= pressure energy per unit weight of fluid or pressure head

= kinetic energy per unit weight or kinetic head

z = potential energy per unit weight or potential head

.4. Bernoulli's theorem This theorem is a form of the well known principle of conservation of energy. The theorem states that:

In a steady continuous flow of a frictionless incompressible fluid, the sum of the potential head, the pressure head and the kinetic head is the same at all points.4.1 Assupmtions

(i) Flow is ideal i.e. viscosity is zero

(ii) Flow is steady

(iii) Flow is incompressible

(iv) Flow is irrotational

4.2 Proof of Bernoulli's theorem. Consider the case of water flowing though a smooth pipe (Fig. 4). Such a situation is depicted in the figure below.

Fig. 4 Flow through pipe

We examine a fluid section of mass m traveling to the right as shown in the schematic above. The net work done in moving the fluid:

=

(13)where P denotes a force and an x a displacement. The second term picked up its negative sign because the force and displacement are in opposite directions.

Pressure is the force exerted over the cross-sectional area, or p =

Rewriting this as F = PA and substituting into (13) we find that:

(12)

The displaced fluid volume is the cross-sectional area A times the thickness x. This volume remains constant for an incompressible fluid, so

(14)

Using 13 and 14, we have

(15)

Since work has been done, there has been a change in the mechanical energy of the fluid segment (Fig. 5).

Fig. 5 Change in mechanical energy

The energy change between the initial and final positions is:

=

Or,

EMBED Equation.3 -

(16)

Here, the kinetic energy,, where m is the fluid mass and v is the speed of the fluid. The potential energy U = mgh where g is the acceleration of gravity, and h is average fluid height.

The work-energy theorem says that the net work done is equal to the change in the system energy.

(17)Substitution of Eq.(15) and Eq.(16) into Eq.(17) yields:

= -

(18)Dividing Eq.(18) by the fluid volume, V gives:

= -

(19)

Reorganize Eq.(19),

(20)Finally, note that Eq.(20) is true for any two positions. Therefore,

(21)Equation (21) is commonly referred to as Bernoulli's equation. Keep in mind that this expression was restricted to incompressible fluids and smooth fluid flows.5. Applications of Bernoulli theorem (Flow Measurement)The Bernoulli equation can be applied to several commonly occurring situations in which useful relations involving pressures, velocities and elevations may be obtained.A very important application in engineering is fluid flow measurement.Measurement of flow rate: Venturi meter

Orifice meterMeasurement of velocity: Pitot-static tube

5.1 Measurement of flow rate:

Basic principle: Increase in velocity causes a decrease in pressure. Fluid is accelerated by forcing it to flow through a constriction, thereby increasing kinetic energy and decreasing pressure energy. The flow rate is determined by measuring the pressure difference between the meter inlet and a point of reduced pressure. Desirable characteristics of flow meters:

Reliable, repeatable calibration

Introduction of small energy loss into the system

Inexpensive

Minimum space requirements

5.2 Generalized flow obstruction in a pipe

Fig. 6 Obstruction in a pipe

For Fig. 6, Continuity equation between (1) and (2)

EMBED Equation.3 Bernoulli equation between (1) and (2)

+ = 0

EMBED Equation.3 =

In above equation, frictional losses have not been taken into account

To account for frictional losses we use a discharge coefficient, Cd:

=

The volumetric flow rate can be easily calculated as,

5.3 The Venturi Meter

This device consists of a conical contraction, a short cylindrical throat and a conical expansion as shown in Fig. 7. The fluid is accelerated by being passed through the converging cone. The velocity at the throat is assumed to be constant and an average velocity is used. The venturi tube is a reliable flow measuring device that causes little pressure drop. It is used widely particularly for large liquid and gas flows.

Fig. 7 Venturi meter

In the venturi meter the fluid is accelerated through a converging cone of angle15-20oand the pressure difference between the upstream side of the cone and the throat is measured and provides a signal for the rate of flow.

The fluid slows down in a cone with smaller angle (5 - 7o) where most of the kinetic energy is converted back to pressure energy. Because of the cone and the gradual reduction in the area there is no "Vena Contracta". The flow area is at a minimum at the throat.

= As = =

Where h is the difference in level between two piezometers at 1 and 2 in venturimeter.

Hence,

==

=

Where the discharge coefficient, Cd = f(Re), can be found in Figures available in textbooks.

High pressure and energy recovery makes the venturi meter suitable where only small pressure heads are available.

A discharge coefficientCd= 0.975can be indicated as standard, but the value varies noticeably at low values of the Reynold number.

The pressure recovery is much better for the venturi meter than for the orifice plate.

The venturi tube is suitable for clean, dirty and viscous liquid and some slurry services.

Pressure loss islow Typical accuracy is1%of full range

Required upstream pipe length5 to 20diameters

Viscosity effect ishigh Relative cost ismediumIt may be noted that there will be no change in the result whether the venturimeter is horizontal, vertical or inclined5.4 The Orifice Plate

This type of meter consists of a thin flat plate with a circular hole drilled in its center as shown in Fig. 8. It is very simple, inexpensive and easy to install, but it can cause significant pressure drops.

The orifice meter consists of a flat orifice plate with a circular hole drilled in it.The orifice diameter is kept generally 0.5 times the diameter of the pipe, though it may vary from 0.4 to 0.8 times the pipe diameter.

Fig. 8 Orifice meterThere is a pressure tap upstream from the orifice plate and another just downstream. There are in general three methods of placing the taps. The coefficient of the meter depends upon the position of taps. Flange location - Tap location1 inchupstream and1 inchdownstream from face of orifice

"Vena Contracta" location - Tap location 1 pipe diameter (actual inside) upstream and0.3 to 0.8pipe diameter downstream from face of orifice

Pipe location - Tap location2.5times nominal pipe diameter upstream and 8 times nominal pipe diameter downstream from face of orifice

=

The discharge coefficient, Cd, varies considerably with changes in area ratio and theReynolds number. A discharge coefficientCd= 0.60may be taken as standard, but the value varies noticeably at low values of the Reynolds number.Discharge Coefficient- Cd

Diameter Ratiod = D2/ D1Reynolds Number Re

104105106107

0.20.600.5950.5940.594

0.40.610.6030.5980.598

0.50.620.6080.6030.603

0.60.630.610.6080.608

0.70.640.6140.6090.609

The pressure recovery is limited for an orifice plate and the permanent pressure loss depends primarily on the area ratio. For an area ratio of 0.5, the head loss is about 70 - 75% of the orifice differential.

The orifice meter is recommended for clean and dirty liquids and some slurry services.

The pressure loss ismedium Typical accuracy is2 to 4%of full scale

The required upstream diameter is10 to 30 The viscosity effect ishigh The relative cost islow5.5 The Nozzle or nozzle meterThe nozzle meter uses a contoured nozzle as shown in Fig. 9 . The resulting flow pattern for the nozzle meter is closer to ideal.

Fig. 9 Nozzle meterDischarge Coefficient- cd

Diameter Ratiod = D2/ D1Reynolds Number -Re

104105106107

0.20.9680.9880.9940.995

0.40.9570.9840.9930.995

0.60.950.9810.9920.995

0.80.940.9780.9910.995

The flow nozzle is recommended for both clean and dirty liquids

The relative pressure loss ismedium Typical accuracy is1-2%of full range

Required upstream pipe length is10 to 30diameters

The viscosity effecthigh The relative cost ismedium5.6 RotameterAlso known as variable-area meter is shown in Fig. 10.

It consists of a vertical transparent conical tube in which there is a rotor or float having a sharp circular upper edge. The rotor has grooves on its head which ensure that as liquid flows past, it causes the rotor to rotate about its axis. The rotor is heavier than the liquid and hence it will sink to the bottom of the tube when the liquid is at rest. But as the liquid begins to flow through the meter, it lifts the rotor until it reaches a steady level corresponding to the discharge. This rate of flow of liquid can then be read from graduations engraved on the tube by prior calibration, the sharp edge of the float serving as a pointer. The rotating motion of the float helps to keep it steady. In this condition of equilibrium, the hydrostatic and dynamic thrusts of the liquid on the under side of the rotor will be equal to the hydrostatic thrust on the upper side, plus the apparent weight of the rotor.

Fig. 10 Rotameter5.7 Elbow Meter (or Pipe-bend Meter)

An elbow meter (or pipe-bend meter) consists of a simple 90o pipe bend provided with two pressure taps, one each at the inside and the outside of the bend, as shown in Fig. 11.

Fig. 11 Elbowmeter

Its operation is based on the fact that as liquid flows round a pipe bend its pressure increases with the radius, due to approximately free vortex conditions being developed in the bend. As such a pressure difference is produced on the inside and outside of the bend which is used as a measure of the discharge. The pressure taps are connected to a differential manometer to measure the differential pressure head h. The discharge Q may then be computed as:

where CD is the coefficient of discharge of the elbow meter and A is its cross-sectional area. The coefficient of discharge CD depends mainly on the ratio R/c (where R is the radius of the axis of the bend and c is the radius of the pipe), and its value can be obtained by calibration. The main advantage associated with an elbow meter is that it entails no additions or alterations to an existing pipe system, except for the drilling of pressure taps, and if suitably calibrated it can be used for precision measurements. ..5.7 Pitot static TubeApitottubeis a pressure measurementinstrument used to measurefluidflow velocity. The pitot tube was invented by theFrenchengineerHenri Pitotin the early 1700sand was modified to its modern form in the mid 1800s by French scientist Henry Darcy. It is widely used to determine the airspeed of anaircraftand to measure air and gas velocities in industrial applications.

The basic pitot tube consists of a tube pointing directly into the fluid flow as shown in Fig. 10.

(a) Simple Pitot tube

(b) Pitot tube in flow

Fig. 10 Pitot tubeAs this tube contains fluid, a pressure can be measured; the moving fluid is brought to rest (stagnates) as there is no outlet to allow flow to continue. This pressure is thestagnation pressureof the fluid, also known as the total pressure or (particularly in aviation) the pitot pressure.

The measured stagnation pressure cannot of itself be used to determine the fluid velocity.However,Bernoullis equationat (1) and (2) in Fig. 10:

(i)

Since, Z1 = Z2 and V2 = 0, (i) can be simplified as:

(ii)Stagnation Pressure = Static Pressure + Dynamic PressureThedynamic pressure, then, is the difference between the stagnation pressure and thestatic pressure. The static pressure is generally measured using thestatic portson the side of the tube as shown in Fig. 10 and Fig. 11.

Fig. 11 Static Pressure tubeNow

EMBED Equation.3 and

So that (ii) becomes:

= = y

EMBED Equation.3 Which is theoretical velocity at a point in a flow.If is the coefficient of pitot tube, then actual velocity at any point will be:

(iii)

Or, Instead of static ports, a pitot-static tube (Fig. 12) (also called aPrandtltube) may be employed, which has a second tube coaxial with the pitot tube with holes on the sides, outside the direct airflow, to measure the static pressure.

Fig. 12 Pitot static tube

The disadvantages of the Pitot tube:

Do not give the averaged velocity;

Its readings for gases are extremely small;Summary The mechanical energy equation (or generalized Bernoulli equation) is an expression of the energy balance equation for steady flow and constant-density fluids.

The mechanical energy equation can be applied with negligible error to almost all steady flows of liquids and for steady flows of gases at low velocities.

A special case of the mechanical energy equation, the Bernoulli equation, can be derived if we assume frictionless flow and absence of shaft work.

A large number of devices for the measurement of fluid velocity and flow rate are based on the conservation of energy. The Bernoulli equation can be conveniently used to make the appropriate calculations.

PAGE 15

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