CONTROL VOLUME APPROACH AND CONTINUITY PRINCIPLE

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Chapter 5 CONTROL VOLUME APPROACH AND CONTINUITY
PRINCIPLE
Fluid Mechanics, Spring Term 2007
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Some Definitions
Discharge or Volume Flow Rate
For variable velocity distribution
Mass Flow Rate
For variable velocity distribution
(cross section of area must be perpendicular to
velocity)
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System, Control Volume, and Control Surface

• A fluid system is a continuous mass of fluid
  that always contains the same fluid particles.
  By definition, the mass of a system is constant.
• A control volume is some selected volume in
  space which can deform, move and rotate. Mass
  (i.e., particles) can flow into or out of the
  control volume.
• A control surface is the surface that encloses
  the control volume.

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Selection of Control Volume
steady flow
unsteady flow
You choose the control volume that is most
convenient for the problem you want to
solve. Steady flow problems are often easier than
unsteady ones, so control volume a) may be
preferable.
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Plan for this lecture

• Use the concept of the control volume to derive
  a mathematical description for how fluid
  properties change with time.
• This gives us a relation between the Eulerian
  and the Lagrangian description.
• In this chapter, the fluid property we deal
  with is mass.
• However, the concept is much more general, and
  we will apply it to other properties later as
  well.

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Change of Mass in a Control Volume
Skipping some fairly obvious derivation on p. 148
Dividing by a small Dt gives the rate form
Continuity principle
(note that this is the conservation equation for
mass)
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Intensive and Extensive Properties
Extensive Properties Proportional to mass of
system. Examples Mass (M ), Momentum (MV ),
Energy (E ) Intensive Properties Independent
of mass of system. (Obtained by dividing
extensive properties by mass) Examples Velocity
(V ), Energy per unit mass (e )
For any extensive property B and intensive
property b
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Reynolds Transport Theorem
Do read the derivation on pp. 151-153 in the
book. Its a bit lengthy, though Essentially,
it boils down to this
a) b) c)

• The total amount of B in a control volume changes
• if that property is created inside the system
  (e.g., we may create thermal energy from
  internal chemical energy), or
• if there is a net flux of B out of the control
  volume

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(from previous slide)
a) b) c)
The real physics happens in the
system Concepts such as mass conservation
apply to a set of particles where the particles
remain the same. But we want to know what
happens in the control volume in which particles
usually flow in or out. Derive more useful
expressions for terms a) and c). We start with
term c).
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Lets look at the mass flow rate again in detail
M
A
V
The fluid of mass m moves a distance in
time Dt.
(mass crossing an area per unit time)
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Flow rate for any extensive property B
B
A
V
Total amount of B in the colored volume
Net flow rate of B out of the volume
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Property Transport across Control Surface and
Sign Convention for Control Surface
If velocity is constant across areas 1 and 2
We choose the surface normal to point outward
from the control volume so that
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In a similar manner, we can write for the
convective transport of any extensive property B
out of the control volume
The total amount of property B in the volume
changes if there is a net flux of B out of the
volume, where the net flux may be written as the
concentration of B (i.e., b B / M ) times the
mass flow rate.
(integral form)
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Back to our equation for the total change in B
We just derived an equation for Now lets look
at
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Turning things around we can now write
b) a) c)
Notice that term a) is quite complicated It
includes time changes in b (fairly obvious), in
density (because b is just B referenced to mass),
and in the volume (if we increase the control
volume, we probably include more of the total
property B ) Notice also that V in the last term
has to be the velocity relative to the control
surface.
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Conservation of Mass
No mass is created or destroyed! This is
intuitive and thus the book does not stress it,
but this is one of the most fundamental equations
of physics. (Note that Einstein would probably
disagree Mass can be converted to energy and
vice versa, , but as long as we
restrict ourselves to non-relativistic problems,
the equation on top is true.)
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General Form of the Continuity Equation
Now lets see what happens when we substitute
mass for the property B in the Reynolds Transport
Theorem. Notice that b M / M 1
First term Either time-change in density or in
volume. Second term Net flow out of the volume
(across cs).
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Notice that this equation still has the same
meaning as the one we started out with
Conservation of mass
But the equation on top contains the correction
we need when we apply it to a control volume with
moving boundaries and flow across the boundaries.
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The continuity equation we derived is an integral
equation
We now write this equation in differential form.
We will show that the above is equivalent to
which can also be written as
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We start with the first term. Recall that our
continuity equation came from this basic idea
From the way we developed the equation, the
derivative
is the full time derivative in moving along with
cv.
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Lets assume that the control volume does not
change The entire enclosing surface cs is fixed
in space and time.
Then, for the first term
Moreover, since we move along with cv and that
velocity is zero
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We would like to have both integrals as volume
integrals. To get this, we apply Gauss Theorem
to the 2nd term
for any vector r . We let to get
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Description of Gauss Theorem
A positive divergence of the vector field means
There is a net flow of material out of a an
infinitesimal volume
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Illustration of Gauss Theorem
cv
When summing all the divergences over the volume
cv, only the net divergence across cs
contributes. All the velocities on the inside of
the volume cancel each other.
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Substituting into the continuity equation gives
But now both integrals are over the same volume.
Since the equation is true for any volume cv, it
follows that the equation is true for the
integrands alone
This is the differential form of the continuity
equation.
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Moreover, since
and
or
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An incompressible material cannot change
density. Hence,
From continuity (
)
Continuity equation for incompressible materials
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So we have several different equations that
describe conservation of mass. Which one do we
use? Answer Which ever is most convenient for a
given problem. Example 5.13 A flow is given
by Is this flow incompressible? If we use an
integral equation, we find out whether the flow
on average is incompressible over the volume. If
we use the differential equation, we can check
whether the flow is incompressible everywhere
since
Incompressible everywhere
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Example 5.5 Flow into tank through orifice with
A0.0025m2. Flow out at rate of 0.003m3/s. What
is the rate of change of mass in the tank?
We want to know an average over the control
volume, so any of the integral forms of the
continuity equation will do.
This is the simplest form of the equation which
can be used here. Notice
and
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Example 5.10 Incompressible flow into and out of
a tank.
Velocity in pipe 4 has following distribution
Find Vmax
This is very similar to the last problem. The
difference is that V is not constant across pipe
4, so we cant use
Use
instead.
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A related concept Cavitation
Fluid flow can cause pressure drops large enough
to result in boiling. Vapor bubbles and pockets
form such that the fluid in its liquid phase is
no longer continuous.
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