Fluid Mechanics 2010-2011 Fluid Kinematics Part 2

7/29/2019 Fluid Mechanics 2010-2011 Fluid Kinematics Part 2

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FLUID KINEMATICS

PN. SALMIE SUHANA CHE ABDULLAH

[email protected]


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Control Volume and System Representative

System a collection of matter of fixed identity (always the sameatoms or fluid particles) which may move, flow and interact withsurrounding. All the particles that involve in the system are identifiable .

System can change shape or speed as forces act on it.The mass of the system does not change (no mass crosses its boundary)

Also called as a closed system

Control volume

volume in space through which fluid may flow.Allows mass to flow in or out across its boundaries, which are called thecontrol surface (surface that encloses the control volume.)

Also called as a open system


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Control volume and System are identical before the spraying process.(t=0)

When some contents of the fire extinguisher are discharge, the system

approach considers the discharged mass as part of the system.

The mass of the system remains constant.

The control volume approach is not concerned at all with the sprayed mass.

Mass of control volume decreases, volume remains constant.


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Typical Control Volume

Fixed control volume

Fixed or moving control volume

Deforming control volume


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Fixed control volume

The control volume consists of the inside of the pipe between(1) and (2) The fixed control surfaces (the surface of the control volume)consist of the inside surface of the pipe

Fluid can flow across the ends of the control surface.


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Fixed or moving control volume

Example: the rectangular volume surrounding the jet engine Air is continually passing through the engine. The system that was in the engine at t = t1 is well past the engine att = t2 . At this later time other air (a different system) is within the engine.

The control volume is stationary if the jet itself is stationary. If thejet is moving then the control volume itself is moving.


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Deforming control volume

Example: deflating balloon Surface is the inner surface of the balloon As time increase, the volume decrease in size


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The Reynolds Transport Theorem

In fluid mechanics, it is usually more convenient to work with controlvolumes, and thus there is a need to relate the changes in a controlvolume to the changes in a system.

Reynolds transport theorem (RTT) provides :

- the link between the system and control volume approaches.

- a way to relate what is happening to the system and what ishappening in the control volume.

- a means for determining the rate of change of some quantityof interest (e.g. mass, momentum etc) following the motion.


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When describing a system, there are physical properties like mass,energy, momentum that need consideration. Let B be the property ofinterest.

We can write

where,B : fluid parameter of the system which is proportional to amount ofmassm : mass of the portion of fluid of interestb : the amount of B per unit mass (independent to the mass)

The parameter B is termed an extensive propertyand the parameter bis termed an intensive property.


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What is an extensive property?

An extensive property describes a specific part of the fluid e.g. the mass is different for different volumes of the same fluid Do depend on the quantity of the sample

What is an intensive property?

In simple terms, an intensive property is the extensive property per unitmass e.g. the density is the same for different volumes of the same fluid Do not depend on the size of the sample of matter and can be used to

identify substances.

Extensiveproperty

Intensiveproperty


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Examples of extensive property and intensive property

Extensive Property Intensive Property

B b (B = mb)m (Mass) 1 mv2 (K.E) 1/2v 2

mv(Momentum) v


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Derivation of the Reynolds Transport Theorem

Control volume and system for flow through a variable area pipe


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At time t,Control Volume (CV) and System (SYS) coincide

At t+t,CV : fixed and SYS : Move slightly

fluid particles at section (1): Move a distance dl1 = V1t fluid particles at section (2): Move a distance dl2 = V2t : Volume of Inflow (entering CV ) : Volume of Outflow (leaving CV )

At time t,SYS = CV

At time t+t,CV-+


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At time t,the system consists of the fluid in section CV;(SYS=CV)

At time t+t,

Thus, time rate of change in B can be;

When using B: extensive fluid property,


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At initial time t, making the fact that

We get

The time rate ofchange ofB for thesystem

Time rate ofchange of theamount ofBwithin thecontrol volume


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Time rate of change of the amount of B in control volume

For the extensive parameter B f lows from the control volume, across thesurface,

Since,


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Thus, the rate at which this property f lows from the CV,

Similarly, for inflow of B into control volume, the rate of inflow ofproperty B is given by


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=

Net flowrate of B leaving CV (Outflow) across thecontrol surface between II and CV (CS out )

=

Net flowrate of B entering CV (Inflow) across thecontrol surface between I and CV (CS in )


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Combining all these equations, we see that the relationship

between the time rate of change ofB for the system and that forthe control volume is given by

moving fixed out in


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Restrictions for the above Equation:

Fixed control volume One inlet and one outlet Normal velocity to section (1) and (2)


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Example 6

Consider the flow from the fire extinguisher shown below. Letthe extensive property of interest be the system mass (B = m, thesystem mass, so that b = 1).

Write the appropriate form of the Reynolds Transport Theorem

for this f low.


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Solution

At t = 0 (fig. a),Control volume : the fire extinguisherSystem : fluid within fire extinguisher

No inlet no fluid f lows into the control volume

A1 = 0

At t > 0 (fig. b),there is outlet, A2

Thus,


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DBsys = Bcv + outAout Vout bout - inAin Vin bin

Dt t

Multiple inlet and outlet control volume


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General expression of Reynolds Transport Theorem


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Example 7


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Solution


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Relationship between Material Derivative and Reynolds Transport Theorem


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Comparison with the definition of Material Derivative

Time rate of change of a property offluid particle

Time rate of change of a propertyat a local space

Change of a property due to the fluid motionConvective effect

Localpart

Convectivepart

Lagrangian concept

Eulerian concept: Unsteady effect


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Time rate of change of an extensive B ofa systemLagrangian concept

Time rate of change ofB within a control volume

Eulerian concept

Net flowrate of B across the entire control surfacemotion of a f luid

General expression of Reynolds Transport Theorem


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Conclusion

Reynolds Transport Theorem and Material Derivative bothrepresent methods to transform from fundamentally

Lagrangian concepts to Eulerian interpretations of thoseconcept.


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Selection of a Control Volume

Any volume in space can be considered as a control volume.None are wrong, but some are much better than others.

Fig(a), (b), (c) illustrates three possible control volumes associatedwith flow through pipe.CV(a) is better than CV(b) because point (1) lies on control surface.CV(a) is better than CV(c) because the flow is normal to the inletand exit portions of control volume.


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Quiz 3


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solution

Fluid Mechanics 2010-2011 Fluid Kinematics Part 2

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