Download control volume approach and continuity principle

Some Definitions
Chapter 5
CONTROL VOLUME
APPROACH AND
CONTINUITY PRINCIPLE
Discharge or Volume Flow Rate:
For variable velocity distribution:
Mass Flow Rate:
For variable velocity distribution:
Fluid Mechanics, Spring Term 2011
System, Control Volume, and Control Surface
(cross section of area must be perpendicular to velocity)
Selection of Control Volume
• 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.
unsteady flow
steady 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 b) may be preferable.
Plan for this lecture:
Change of Mass in a Control Volume
• 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.
Skipping some fairly obvious derivation on p. 148:
• 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.
Dividing by a small !t gives the rate form:
Continuity principle
(note that this is the conservation equation for mass)
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)
Reynolds Transport Theorem
Do read the derivation on pp. 151-153 in the book. Itʼs a bit
lengthy, though…
Essentially, it boils down to this:
a)
b)
c)
Examples: Velocity (V ), Energy per unit mass (e )
For any extensive property B and intensive property b:
a) The total amount of B in a control volume changes
b) …if that property is created inside the system (e.g., we
may “create” thermal energy from internal chemical
energy), or
c) …if there is a net flux of B out of the control volume
(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).
A
M
V
The fluid of mass m moves a distance
in time !t.
(mass crossing an area per
unit time)
Property Transport across Control Surface
and Sign Convention for Control Surface
Flow rate for any extensive property B:
A
Letʼs look at the mass flow rate again in detail:
B
V
Total amount of B in the colored volume:
Net flow rate of B out of the volume:
If velocity is constant across area 1 and across area 2:
We choose the surface normal to point outward from the
control volume so that
In a similar manner, we can write for the convective
transport of any extensive property B out of the control
volume:
Back to our equation for the total change in B:
We just derived an equation for
Now letʼs look at
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)
Turning things around we can now write:
b)
a)
Conservation of Mass
c)
No mass is created or destroyed!
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.
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.)
General Form of the Continuity Equation
Now letʼs see what happens when we substitute mass for
the property B in the Reynolds Transport Theorem.
Notice that b = M / M = 1:
Notice that this equation still has the same meaning as
the one we started out with:
Conservation of mass
First term: Either time-change in density or in volume.
Second term: Net flow out of the volume (across cs).
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.
The continuity equation we derived is an integral equation:
We start with the first term. Recall that our continuity
equation came from this basic idea:
We now write this equation in differential form. We will
show that the above is equivalent to
From the way we developed the equation, the derivative
is the full time derivative in moving along with cv.
which can also be written as:
Letʼs assume that the control volume does not change:
The entire enclosing surface cs is fixed in space and time.
We would like to have both integrals as volume integrals.
To get this, we apply Gaussʼ Theorem to the 2nd term:
Then, for the first term:
Moreover, since we move along with cv and that velocity
is zero:
Description of Gaussʼ Theorem:
for any vector r . We let
to get
Illustration of Gaussʼ Theorem:
cv
A positive divergence of the vector field means
There is a net flow of material
out of a an infinitesimal
volume
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.
Substituting into the continuity equation gives:
Moreover, since
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.
An incompressible material cannot change density.
Hence,
From continuity (
Continuity equation for
incompressible materials:
and
or
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
Example 5.5: Flow into tank through orifice with A=0.0025m2.
Flow out at rate of 0.003m3/s. What is the rate of change of
mass in the tank?
Example 5.10: Incompressible flow into and out of a tank.
Velocity in pipe 4 has
following distribution:
We want to know an
average over the
control volume, so
any of the integral
forms of the continuity
equation will do.
Find Vmax
This is very similar to the last problem. The difference is
that V is not constant across pipe 4, so we canʼt use
This is the simplest form of the equation which can be used
here. Notice:
and
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.
Use
instead.