Fluids in Motion

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Fluids in Motion

• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• IIT Delhi

An Unique Option for Many Power Generation
Devices..
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Velocity and Flow Visualization

• Primary dependent variable is fluid velocity
  vector V V ( r ) where r is the position
  vector.
• If V is known then pressure and forces can be
  determined.
• Consideration of the velocity field alone is
  referred to as flow field kinematics in
  distinction from flow field dynamics (force
  considerations).
• Fluid mechanics and especially flow kinematics is
  a geometric subject and if one has a good
  understanding of the flow geometry then one knows
  a great deal about the solution to a fluid
  mechanics problem.

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Flow Past A Turbine Blade
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Velocity Lagrangian and Eulerian Viewpoints
There are two approaches to analyzing the
velocity field Lagrangian and Eulerian
Lagrangian keep track of individual fluids
particles. Apply Newtons second law for each
individual particle!
Say particle p is at position r1(t1) and at
position r2(t2) then,
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Of course the motion of one particle is
insufficient to describe the flow field. So the
motion of all particles must be considered
simultaneously which would be a very difficult
task. Also, spatial gradients are not given
directly. Thus, the Lagrangian approach is only
used in special
circumstances.
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Eularian Approach
Eulerian focus attention on a fixed point in
space.
In general,
where, u u(x,y,z,t), v v(x,y,z,t), w
w(x,y,z,t)
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This approach is by far the most useful since we
are usually interested in the flow field in some
region and not the history of individual
particles.
This is similar to description of A Control
Volume. We need to apply newton Second law to a
Control Volume.
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Eularian Velocity

• Velocity vector can be expressed in any
  coordinate system e.g., polar or spherical
  coordinates.
• Recall that such coordinates are called
  orthogonal curvilinear coordinates.
• The coordinate system is selected such that it is
  convenient for describing the problem at hand
  (boundary geometry or streamlines).

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Fluid Dynamics of Coal Preparation Supply

• BY
• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• I I T Delhi

Aerodynamics a means of Transportation
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Major Components of Coal Fired Steam Generator
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Schematic of typical coal pulverized system
A Inlet Duct B Bowl Orifice C Grinding
Mill D Transfer Duct to Exhauster E Fan Exit
Duct.
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Velocity through various regions of the mill
during steady operation
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Cyclone-type classifier.
Axial and radial gas velocity components
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Centrifugal Classifiers

• The same principles that govern the design of
  gas-solid separators, e.g. cyclones, apply to the
  design of classifiers.
• Solid separator types have been used
  preferentially as classifiers in mill circuits
• centrifugal cyclone-type and gas path deflection,
  or
• louver-type classifiers.
• The distributions of the radial and axial gas
  velocity in an experimental cyclone precipitator
  are shown in Figures.
• The flow pattern is further characterized by
  theoretical distributions of the tangential
  velocity and pressure, the paths of elements of
  fluid per unit time, and by the streamlines in
  the exit tube of the cyclone.

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Particle Size Distribution--Pulverized-Coal
Classifiers

• The pulverized-coal classifier has the task of
  making a clean cut in the pulverized-coal size
  distribution
• returning the oversize particles to the mill for
  further grinding
• but allowing the "ready to burn" pulverized coal
  to be transported to the burner.
• The mill's performance, its safety and also the
  efficiency of combustion depend on a sufficiently
  selective operation of the mill classifier.

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Mill Pressure Drop

• The pressure loss coefficients for the
  pulverized-coal system elements are not well
  established.
• The load performance is very sensitive to small
  variations in pressure loss coefficient.

Correlation of pressure loss coefficient with
Reynolds number through the mill section of an
exhauster-type mill.
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Polar Coordinates
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Volume Rate of Flow (flow rate, discharge)

• Cross-sectional area oriented normal to velocity
  vector (simple case where V . A).

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Volume Rate of Flow in A General Control Volume
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Acceleration

• The acceleration of a fluid particle is the rate
  of change of its velocity.
• In the Lagrangian approach the velocity of a
  fluid particle is a function of time only since
  we have described its motion in terms of its
  position vector.

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In the Eulerian approach the velocity is a
function of both space and time consequently,
x,y,z are f(t) since we must follow the total
derivative approach in evaluating du/dt.
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Similarly for ay az,
In vector notation this can be written concisely
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Basic Control-Volume Approach
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Control Volume

• In fluid mechanics we are usually interested in a
  region of space, i.e, control volume and not
  particular systems.
• Therefore, we need to transform GDEs from a
  system to a control volume.
• This is accomplished through the use of Reynolds
  Transport Theorem.
• Actually derived in thermodynamics for CV forms
  of continuity and 1st and 2nd laws.

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Flowing Fluid Through A CV

• A typical control volume for flow in an
  funnel-shaped pipe is bounded by the pipe wall
  and the broken lines.
• At time t0, all the fluid (control mass) is
  inside the control volume.

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• The fluid that was in the control volume at time
  t0 will be seen at time t0 dt as           .

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The control volume at time t0 dt        .


The control mass at time t0 dt        .
The differences between the fluid (control mass)
and the control volume at time t0 dt        .
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• Consider a system and a control volume (C.V.) as
  follows
• the system occupies region I and C.V. (region II)
  at time t0.
• Fluid particles of region I are trying to enter
  C.V. (II) at time t0.

III
II

• the same system occupies regions (IIIII) at t0
  dt
• Fluid particles of I will enter CV-II in a time
  dt.
• Few more fluid particles which belong to CV II
  at t0 will occupy III at time t0 dt.

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The control volume may move as time passes.
III has left CV at time t0dt
I is trying to enter CV at time t0
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Reynolds' Transport Theorem

• Consider a fluid scalar property b which is the
  amount of this property per unit mass of fluid.
• For example, b might be a thermodynamic property,
  such as the internal energy unit mass, or the
  electric charge per unit mass of fluid.
• The laws of physics are expressed as applying to
  a fixed mass of material.
• But most of the real devices are control volumes.
• The total amount of the property b inside the
  material volume M , designated by B, may be found
  by integrating the property per unit volume, M
  ,over the material volume

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Conservation of B

• total rate of change of any extensive property B
  of a system(C.M.) occupying a control volume C.V.
  at time t is equal to the sum of
• a) the temporal rate of change of B within the
  C.V.
• b) the net flux of B through the control surface
  C.S. that surrounds the C.V.
• The change of property B of system (C.M.) during
  Dt is

add and subtract
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The above mentioned change has occurred over a
time dt, therefore Time averaged change in BCM is
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For and infinitesimal time duration

• The rate of change of property B of the system.

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Conservation of Mass

• Let b1, the B mass of the system, m.

The rate of change of mass in a control mass
should be zero.
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Conservation of Momentum

• Let bV, the B momentum of the system, mV.

The rate of change of momentum for a control mass
should be equal to resultant external force.
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Conservation of Energy

• Let be, the B Energy of the system, mV.

The rate of change of energy of a control mass
should be equal to difference of work and heat
transfers.
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First Law for A Control Volume

• Conservation of mass

• Conservation of energy

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Complex Flows in Power Generating Equipment

• Separation, Vortices, and Turbulence

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Classification of Flows in Power Generation
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Pipe Flows
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Turbulent Flow

• Turbulent flow fuller profile due to turbulent
  mixing extremely complex fluid motion that defies
  closed form analysis.
• Turbulent flow is the most important area of
  power generation fluid flows.
• The most important nondimensional number for
  describing fluid motion is the Reynolds number

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• Internal vs. External Flows
• Internal flows completely wall bounded
• Usually requires viscous analysis, except near
  entrance.
• External flows unbounded i.e., at some
  distance from body or wall flow is uniform.
• External Flow exhibits flow-field regions such
  that both inviscid and viscous analysis can be
  used depending on the body shape and Re.

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