# Fluid Mechanics for Power Generation

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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..
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.

Particle p at time t1
Particle p at time t2
Uniform Flow

r rotcev noitisop   1
2
r rotcev noitisop                     

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 Newton’s second law for each individual particle!
                     
Fparticle  mparticle a particle
Say particle p is at position r1(t1) and at position r2(t2) then,
                    
r2  r1
V particle  lim
t  0 t  t
2    1

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
Eularian Approach

Eulerian: focus attention on a fixed point in space.

              ˆ
ˆ j
x  xi  yˆ  zk

In general,
  
ˆ j       ˆ
V  V ( x , t )  ui  vˆ  wk

Velocitycomponents

where, u = u(x,y,z,t), v = v(x,y,z,t), w = w(x,y,z,t)
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.
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).
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 ……
Major Components of Coal Fired Steam
Generator
Schematic of typical coal pulverized system

A Inlet Duct;
B Bowl Orifice;
C Grinding Mill;
D Transfer Duct to Exhauster;
E Fan Exit Duct.
Velocity through various regions of the mill during
Cyclone-type classifier.

Axial and
velocity
components
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.
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.
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.
Polar Coordinates
Volume Rate of Flow (flow rate, discharge)
• Cross-sectional area oriented normal to velocity vector (simple case
where V . A).
Volume Rate of Flow in A General Control Volume


Q   V .ndA
ˆ
CS


Q   V cosdA
CS

 

m    V .n dA
           ˆ
CS
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.
In the Eulerian approach the velocity is a function of both
space and time; consequently,

ˆ                                        ˆ
V  u ( x, y, z , t )i  v( x, y, z , t ) ˆ  w( x, y, z , t )k
j
                                        
Velocitycomponents

x,y,z are f(t) since we must follow the total derivative approach
in evaluating du/dt.
Similarly for ay & az,

In vector notation this can be written concisely
Basic Control-Volume Approach
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 GDE’s 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.
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.
The fluid that was in the control volume at time t0 will be seen
at time t0 +dt as:       .
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    .
• 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
t 0.

III

II
I
• the same system occupies regions (II+III) 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.
The control volume may move as time passes.

III has left CV at time t0+dt
III

II
I is trying to enter CV at time t0

II           At time t0+dt

I                        VCV
At time t0
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 :
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

BCM  B t                 Bt
0 dt          0

BCM  BII   t0 dt
 BIII t
0 dt

 BI   t0
 BII   t0

0 dt
BCM  BII   t0 dt
 BIII t
0 dt

 BI       t0
 BII   t0
 B       I t dt
0
 BI   t0 dt

BCM  BI  BII  t                  BIII t                BI  BII  t  BI
0 dt                   0 dt                              0            t0 dt

BCM  BCV            t0 dt
 BCV                 BIII t             BI
t0                0 dt                t0 dt

The above mentioned change has occurred over a time dt, therefore
Time averaged change in BCM is

BCM   BCV           t0 dt
 BCV                     BIII t                BI
0 dt             t0 dt
                                                                   
t0

dt                      dt                                dt                      dt
For and infinitesimal time duration
BCM          BCV
       t0 dt
 BCV        
         BIII t dt         BI t dt
 lim                                  dt o             lim
t0
lim                                                  lim         0                0

dt o  dt    dt o

           dt                
            dt        dt o   dt

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

dBCM dBCV         
      BIII  BI
dt   dt
Conservation of Mass

• Let b=1, the B = mass of the system, m.

dmCM dmCV
      mout  min
      
dt   dt

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

dmCV
 mout  min  0
      
dt
Conservation of Momentum

• Let b=V, the B = momentum of the system, mV.

              
              
             
d mV           d mV                           
CM
         CV
 mV
      out    mV
       in
dt             dt

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

 

             
d mV                                   
CV
 mV
     out    mV
     in   F
dt
Conservation of Energy

• Let b=e, the B = Energy of the system, mV.

d meCM d meCV
          me out  mein
          
dt      dt

The rate of change of energy of a control mass should be equal
to difference of work and heat transfers.

d meCV
 me out  me in  Q  W
                   
dt
First Law for A Control Volume

• Conservation of mass:
dmCV
 mout  min  0
      
dt

• Conservation of energy:
dECV               
 Eout  Ein  Q  W
dt

        dECV          
Q  Ein        Eout  W
dt
Complex Flows in Power Generating Equipment

Separation, Vortices, and Turbulence
Classification of Flows in Power Generation
Pipe Flows
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
• 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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