Fluid Mechanics Machinery Chapter02 Fluid Flow Sub Teacher

Fluid Mechanics & Machinery Chapter-02 Fluid Flow Sub. Teacher: Ms. Nilesha U. Patil Mr. C. V. Chimote

Program : Mechanical Engineering Course: Fluid Mechanics & Machinery CO-Apply continuity equation, Bernoulli’s theorem in different situations

Faculty of Engineering and Technical Studies Program : Mechanical Engineering Course: Fluid Mechanics & Machinery CO-Apply continuity equation, Bernoulli’s theorem in different situations

Introduction § Discusses the analysis of fluid in motion: fluid dynamics. § When a fluid flows through pipes and channel or around bodies such as aircraft and ships, the shape of the boundaries, the externally applied forces and the fluid properties cause the velocities of the fluid particles to vary from point to point throughout the flow field. § The motion of fluids can be predicted using the fundamental laws of physics together with the physical properties of the fluid. § The geometry of the motion of fluid particles in space and time is known as the kinematics of the fluid motion. § A fluid motion may be specified by either tracing the motion of a particle through the field of flow or examining the motion of all particles as they pass a fixed point in space. 4

Objectives Comprehend the concepts necessary to analyse fluids in motion. Identify differences between steady/unsteady, uniform/non-uniform and compressible/incompressible flow. Appreciate the Continuity principle through Conservation of Mass and Control Volumes. Derive the Bernoulli (energy) equation. Familiarise with the momentum equation for a fluid flow. 5

Types of flow Steady Flow- The flow in which the velocity of fluid is constant at any point is called as steady flow. Unsteady Flow-When the flow is unsteady , the fluid velocity differs between any two points. 6

Uniform Flow, Steady Flow uniform flow: flow velocity is the same magnitude and direction at every point in the fluid. 7

Uniform Flow, Steady Flow (cont. ) Steady uniform flow: § Conditions: do not change with position in the stream or with time. § Example: the flow of water in a pipe of constant diameter at constant velocity. Steady non-uniform flow: § Conditions: change from point to point in the stream but do not change with time. § Example: flow in a tapering pipe with constant velocity at the inlet-velocity will change as you move along the length of the pipe toward the exit. Unsteady uniform flow: § At a given instant in time the conditions at every point are the same, but will change with time. § Example: a pipe of constant diameter connected to a pumping at a constant rate which is then switched off. Unsteady non-uniform flow: § Every condition of the flow may change from point to point and with time at every point. § Example: waves in a channel. 8

Laminar and Turbulent Flow Laminar flow § all the particles proceed along smooth parallel paths and all particles on any path will follow it without deviation. § Hence all particles have a velocity only in the direction of flow. Typical particles path Figure 3. 1 a: Laminar flow 9

Turbulent Flow § The particles move in an irregular manner through the flow field. § Each particle has superimposed on its mean velocity fluctuating velocity components both transverse to and in the direction of the net flow. Particle paths Transition Flow Figure 3. 1 b: Turbulent flow § exists between laminar and turbulent flow. § In this region, the flow is very unpredictable and often changeable back and forth between laminar and turbulent states. § Modern experimentation has demonstrated that this type of flow may comprise short ‘burst’ of turbulence embedded in a laminar flow. 10

Laminar Flow- The flow in which the fluid particles move in the same direction is called as Laminar Flow. Turbulent Flow- The flow in which the fluid particles move randomly in any direction is called as Turbulent Flow. 11

Relative Motion Observer Boat moving Boat stationary Flow pattern moves along channel with boat changes with time UNSTEADY Fluid moving past boat pattern stationary relative to boat does not change with time STEADY Figure 3. 2: Relative motion 12

Compressible or Incompressible § All fluids are compressible - even water - their density will change as pressure changes. § Under steady conditions, and provided that the changes in pressure are small, it is usually possible to simplify analysis of the flow by assuming it is incompressible and has constant density. § As you will appreciate, liquids are quite difficult to compress - so under most steady conditions they are treated as incompressible. 13

Compressible or Incompressible • Compressible Flow- When the density of flow varies with pressure it is termed as compressible flow. • Incompressible Flow- When the density of flow does not varies with pressure it is termed as incompressible flow. Viscous/Non-viscous flow- When the viscosity of the fluid is zero, it is termed as non-viscous otherwise it is viscous. • 14

One, Two or Three-dimensional Flow § In general, all fluids flow three-dimensionally, with pressures and velocities and other flow properties varying in all directions. § In many cases the greatest changes only occur in two directions or even only in one. § In these cases changes in the other direction can be effectively ignored making analysis much more simple. 15

§ Flow is one dimensional if the flow parameters (such as velocity, pressure, depth etc. ) at a given instant in time only vary in the direction of flow and not across the cross-section. The flow may be unsteady, in this case the parameter vary in time but still not across the cross-section. § An example of one-dimensional flow is the flow in a pipe. Note that since flow must be zero at the pipe wall - yet non-zero in the center - there is a difference of parameters across the crosssection. § Should this be treated as two-dimensional flow? Possibly - but it is only necessary if very high accuracy is required. A correction factor is then usually applied. 16

§ Flow is two-dimensional if it can be assumed that the flow parameters vary in the direction of flow and in one direction at right angles to this direction. § Streamlines in two-dimensional flow are curved lines on a plane and are the same on all parallel planes. § An example is flow over a weir for which typical streamlines can be seen in the figure below. Over the majority of the length of the weir the flow is the same - only at the two ends does it change slightly. Here correction factors may be applied. 17

Mass and volume flow rate Mass flow rate mass flow rate = m = time = mass of fluid time taken to collect the fluid mass flow rate 18

Discharge or Rate of flow It is defined as the volume of liquid flowing per second. The amount of fluid flow per unit second is called rate of flow or discharge. It is represented by ‘Q’ Q=V/T Q=a x v Its S. I. unit is m 3/sec or litre/sec or kilolitre/sec 19

Volume flow rate - Discharge volume of fluid discharge = Q = time = mass of fluid density x time ( density = mass volume ) & = mass fluid rate density m = 20

The Fundamental Equations of Fluid Dynamics 1. The law of conservation of matter § stipulates that matter can be neither created nor destroyed, though it may be transformed (e. g. by a chemical process). § Since this study of the mechanics of fluids excludes chemical activity from consideration, the law reduces to the principle of conservation of mass. 2. The law of conservation of energy § states that energy may be neither created nor destroyed. § Energy can be transformed from one guise to another (e. g. potential energy can be transformed into kinetic energy), but none is actually lost. § Engineers sometimes loosely refer to ‘energy losses’ due to friction, but in fact the friction transforms some energy into heat, so none is really ‘lost’. 21

3. The law of conservation of momentum § states that a body in motion cannot gain or lose momentum unless some external force is applied. § The classical statement of this law is Newton's Second Law of Motion, i. e. force = rate of change of momentum 22

Continuity Equation (Principle of Conservation of Mass) • Matter cannot be created nor destroyed - (it is simply changed in to a different form of matter). • This principle is known as the conservation of mass and we use it in the analysis of flowing fluids. Continuity equation is based upon , principle of conservation of mass. For a fluid flowing through the pipe at all cross section, the quantity of fluid flowing per second is constant. Or in the other words, for a steady and incompressible flow, rate of flow of liquid remains constant at different sections. Inflow CONTROL VOLUME Outflow Control surface Figure 3. 10: A control volume 23

For any control volume the principle of conservation of mass says Mass entering = Mass leaving + Increase of mass in the control per unit time volume per unit time For steady flow: § § (there is no increase in the mass within the control volume) Mass entering per unit time = Mass leaving per unit time Mass entering per unit time at end 1 = Mass leaving per unit time at end 2 Figure 3. 11: A streamtube section 24

§ flow is incompressible, the density of the fluid is constant throughout the fluid continum. Mass flow, m, entering may be calculated by taking the product (density of fluid, ) (volume of fluid entering per second Q) § Mass flow is therefore represented by the product Q, hence Q (entering) = Q (leaving) § But since flow is incompressible, the density is constant, so Q (entering) = Q (leaving) (3. 5 a) § This is the ‘continuity equation’ for steady incompressible flow. 1 A 1 V 1= 2 A 2 V 2 But 1= 2 § A 1 V 1=A 2 V 2 § 25

§ If the velocity of flow across the entry to the control volume is measured, and that the velocity is constant at V 1 m/s. Then, if the cross-sectional area of the streamtube at entry is A 1, Q (entering) = V 1 A 1 § § Thus, if the velocity of flow leaving the volume is V 2 and the area of the streamtube at exit is A 2, then Q (leaving) = V 2 A 2 § § Therefore, the continuity equation may also be written as Q 1= Q 2= Q 3 a 1 v 1= a 2 v 2 =a 3 v 3 flow , V 1 A 1 = V 2 A 2 (3. 5 b) Q rate of flow in , a=area of V= velocity of flow 26

Energy Possessed by flowing Fluid • Energy • Three forms: • Kinematic Energy • Potential Energy • Pressure Energy 27

Energy Possessed by flowing Fluid • Kinematic Energy: Ability of mass to do work by virtue of its velocity. • Potential Energy: It is by virtue of the position of the liquid with respect to some datum level • Pressure Energy: Energy possessed by a liquid particle by virtue of its existing pressure. 28

Application of Continuity Equation § We can apply the principle of continuity to pipes with cross sections which change along their length. § A liquid is flowing from left to right and the pipe is narrowing in the same direction. By the continuity principle, the mass flow rate must be the same at each section - the mass going into the pipe is equal to the mass going out of the pipe. So we can write: 1 A 1 V 1 = 2 A 2 V 2 § As we are considering a liquid, usually water, which is not very compressible, the density changes very little so we can say 1 = 2 =. This also says that the volume flow rate is constant or that § Discharge at section 1 = Discharge at section 2 Q 1 = Q 2 A 1 V 1 = A 2 V 2 or V 2 = A 1 V 1 A 2 Figure 3. 12: Pipe with a contraction 29

§ As the area of the circular pipe is a function of the diameter we can reduce the calculation further, pd 12/4 A 1 V 1= V 2 A = 2 pd 2 /4 2 V( = ) 2 d 12 d 22 d 12 V 1 2 d 2 2 V 1 (3. 6) Another example is a diffuser, a pipe which expands or diverges as in the figure below 30

The continuity principle can also be used to determine the velocities in pipes coming from a junction. Total mass flow into the junction = Total mass flow out of the junction 1 Q 1 = 2 Q 2 + 3 Q 3 When the flow is incompressible (e. g. water) 1 = 2 = Q 1 = Q 2 + Q 3 A 1 V 1 = A 2 V 2 + A 3 V 3 (3. 7) 31

Work and Energy (Principle Of Conservation Of Energy) § § friction: negligible sum of kinetic energy and gravitational potential energy is constant. Recall : Kinetic energy = ½ m. V 2 Gravitational potential energy = mgh (m: mass, V: velocity, h: height above the datum). 32

To apply this to a falling body we have an initial velocity of zero, and it falls through a height of h. § § We Initial kinetic energy = 0 Initial potential energy = mgh Final kinetic energy = ½ m. V 2 Final potential energy = 0 know that, § kinetic energy + potential energy = constant { Initial kinetic Energy } +{ Initial potential Energy mgh = ½ m. V 2 } ={ Final Kinetic Energy } +{ Final Potential Energy } or 33

Energy Possessed by flowing fluid 1. 2. 3. Kinetic Energy Potential Energy Pressure Energy

Bernoulli’s theorem This theorem states that whenever there is a continuous flow of liquid the total energy at every section remains the same provided that there is no loss or addition of the energy. Z= potential energy V 2/2 g=kinetic energy P/w=pressure energy 35

Bernoulli's Equation We see that from applying equal pressure or zero velocities we get the two equations from the section above. They are both just special cases of Bernoulli's equation. Bernoulli's are: equation has some restrictions in its applicability, they § Flow is steady; § Density is constant (which also means the fluid is incompressible); § Friction losses are negligible. § The equation relates the states at two points along a single streamline, (not conditions on two different streamlines). 36

Figure 3. 19 : A contracting expanding pipe 37

Limitations of Bernoulli’s equation 1. 2. 3. 4. It is applicable to ideal incompressible flow. The heat transfer into or out of fluid should be zero The temperature remains constant so that internal energy does not change. The effect of presence of any mechanical device between two sections is ignored. 38

Assumption for Bernoullis Equation Copyright © ODL Jan 2005 Open University Malaysia 39 39

Modifications of Bernoulli Equation • In practice, the total energy of a streamline does not remain constant. Energy is ‘lost’ through friction, and external energy may be either : § added by means of a pump or § extracted by a turbine. • Consider a streamline between two points 1 and 2. If the energy head lost through friction is denoted by Hf and the external energy head added (say by a pump) is or extracted (by a turbine) HE, then Bernoulli's equation may be rewritten as : ± HE = H 2 + Hf (3. 11) or (3. 12) HE = energy head added/loss due to external source such as pump/turbines This equation is really a restatement of the First Law of Thermodynamics for an incompressible fluid. 40

The Power Equation In the case of work done over a fluid the power input into the flow is : P = g. QHE (3. 13) where Q = discharge, HE = head added / loss If p = efficiency of the pump, the power input required, Pin = (3. 14) 41

3. 4 Application of Bernoulli Equation • The Bernoulli equation can be applied to a great many situations not just the pipe flow we have been considering up to now. • In the following sections we will see some examples of its application to flow measurement from tanks, within pipes as well as in open channels. Objectives • Acknowledge practical uses of the Bernoulli and momentum equation in the analysis of flow • Understand how the momentum equation and principle of conservation of momentum is used to predict forces induced by flowing fluids • Apply Bernoulli and Momentum Equations to solve fluid mechanics problems 42

• Thank you 43