chapter 1 hydrolic machine

8/11/2019 Chapter 1 Hydrolic Machine

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Introductionunif orm flow: flow velocity is the same magnitude and direction at every point in the

fluid.

non-uniform: If at a given instant, the velocity is not the same at every point theflow. (In practice, by this definition, every fluid that flows near a

solid boundary will be non-uniform - as the fluid at the boundarymust take the speed of the boundary, usually zero. However if thesize and shape of the of the cross-section of the stream of fluid isconstant the flow is considered uniform.)

steady: A steady flow is one in which the conditions (velocity, pressure andcross-section) may differ from point to point but DO NOT change

with time.

unsteady: If at any point in the fluid, the conditions change with time, the flowis described as unsteady. (In practice there is always slight variationsin velocity and pressure, but if the average values are constant, theflow is considered steady.)



Turbulent Flow

the particles move in an irregular manner through the flow field.

Each particle has superimposed on its mean velocity fluctuating velocitycomponents both transverse to and in the direction of the net flow.

Fig 2 Turbulent Flow

Transition 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.

Particlepaths



Mass flow rate

mass flow rate =

time =

Volume flow rate – Discharge

Discharge/volume flow rate =

=

mass of fluidtime taken to collect the fluid

mass

mass flow rate

volume of fluid

time

mass fluid rate

density

m

r =



Basic principle of flow1. The law of conservation of matter

stipulates that matter can be neither created nor destroyed, though it may betransformed (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 energycan be transformed into kinetic energy), but none is actually lost.

Engineers sometimes loosely refer to ‘energy losses’ due to friction, but infact the friction transforms some energy into heat, so none is really ‘lost’.



3. The law of conservation of momentum

states that a body in motion cannot gain or losemomentum unless some external force is applied.

The classical statement of this law is Newton'sSecond Law of Motion, i.e

force = rate of change of momentum



Continuity (Principle of Conservation of Mass )

• Matter cannot be created nor destroyed - (it is simply changed in to adifferent form of matter).

• This principle is known as the conservation of mass and we use it in

the analysis of flowing fluids. • The principle is applied to fixed volumes, known as control volumes

or surfaces

CONTROL

VOLUME

Control surface

OutflowInflow

Figure 3.10: A control volume



For any control volume the principle of conservation of mass says

Mass entering = Mass leaving + Increase of mass in the control. per unit time 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

Figure 3.11: A streamtube section

Mass entering per unit time atend 1 = Mass leaving per unittime at end 2



flow is incompressible, the density of the fluid is constant throughout the fluidcontinum. Mass flow, m, entering may be calculated by taking the product

(density of fluid, r ) (volume of fluid entering per second Q)

Mass flow is therefore represented by the product r Q, hence

r Q (entering) = r Q (leaving)

But since flow is incompressible, the density is constant, so

Q (entering) = Q (leaving)

This is the ‘continuity equation ’ for steady incompressible flow.

If the velocity of flow across the entry to the control volume is measured, and thatthe velocity is constant at V 1 m/s. Then, if the cross-sectional area of the streamtubeat entry is A1,

Q (entering) = V 1 A1

Thus, if the velocity of flow leaving the volume is V 2 and the area of the streamtubeat exit is A2, then

Q (leaving) = V 2 A2

Therefore, the continuity equation may also be written as

V 1 A1 = V 2 A2



Application of Continuity Equation

We can apply the principle of continuity to pipes with cross sections whichchange along their length.

A liquid is flowing from left to right and the pipe is narrowing in the samedirection. By the continuity principle, the mass flow rate must be thesame at each section - the mass going into the pipe is equal to the massgoing out of the pipe. So we can write:

r 1 A1V 1= r 2 A2V 2

As we are considering a liquid, usually water,which is not very compressible, the densitychanges very little so we can say r 1 = r 2 = r .This also says that the volume flow rate is

constant or that Discharge at section 1 = Discharge at section

2

Q1 = Q2

A1V 1 = A2V 2 or V 2 =

Fig Pipe with contraction

A1V 1

A2



The continuity principle can also be used to determine thevelocities in pipes coming from a junction.

Fig: A pipe with junction

Total mass flow into the junction = Total mass flow out of thejunction

r 1Q1 = r 2Q2 + r 3Q3

When the flow is incompressible (e.g. water) r 1 = r 2 = r

Q1 = Q2 + Q3

A1V 1 = A2V 2 + A3V 3



Work and Energy

(Principle Of Conservation Of Energy)

friction: negligible

sum of kinetic energy and gravitational potential energy is

constant. Recall :

Kinetic energy = ½ mV 2

Gravitational potential energy = mgh

(m: mass, V : velocity, h: height above the datum).



To apply this to a falling body we have an initial velocity of zero, andit falls through a height of h.

Initial kinetic energy = 0

Initial potential energy = mgh

Final kinetic energy = ½ mV 2

Final potential energy = 0We know that,

kinetic energy + potential energy = constant

mgh = ½ mV 2 or ghV 2



continuous jet of liquid

a continuous jet of water coming from a pipe with velocity V 1. One particle of the liquid with mass m travels with the jet and falls from

height z 1 to z 2. The velocity also changes from V 1 to V 2. The jet is traveling in air where the

pressure is everywhere atmospheric so there is no force due to pressure

acting on the fluid. The only force which is acting is that due to gravity. The sum of the kinetic

and potential energies remains constant (as we neglect energy losses due tofriction) so :

mgz 1 + mV 12 = mgz 2 + mV 2

2

As m is constant this becomes :

V 12 + gz 1 = V 22 + gz 2

Figure: Thetrajectory of ajet of water



Flow from a reservoir



It is based on the law of conservation of momentum or on the momentum

principle, which states that the net force acting on a fluid mass is equal to the

change in momentum of flow per unit time in that direction. The force acting on

a fluid mass‘m’

is given by theNewton’s

second law of motion,

chapter 1 hydrolic machine

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