Control Volume (Integral) Technique
Introduction: Techniques for solving flow problems
There are three different techniques for solving these types of problems:
- Control Volume Analysis
- This is just like in thermodynamics class
- One can calculate the gross properties (total power output,
total heat transfer, etc.)
- With this however, we do not care about the details inside the control
volume (In other words we can treat the control volume as a "black
Dimensional Analysis and Experiment
- In this technique, one solves differential equations of motion everywhere
(i.e. The Navier-Stokes equation).
- Here we solve for all of the details in the flow
- This method is used when methods one or two are not possible.
- One uses wind tunnels, models, etc. to employ this method.
Introduction to systems and control volumes
System: Consider a system of a fluid:
Conservation of mass states that the mass of a system
- Recall from thermo class, that a system is defined as
a volume of mass of fixed identity.
- Let m = the mass of the system
Let V = the velocity of the system
Let a = the acceleration of the system
- Now we can write three basic conservation laws which apply to this
system. Note: These conservation laws apply directly to a system.
This can be written as the following equation:
In this equation m = the mass of the system.
Conservation of linear momentum which is a restatement
Newton's Second Law.
- In equation form this is written as:
Where mV = the linear momentum of the system.
- Note that this is the same as Newton's second law, it is just written
a little differently.
- Using the above equation, we can obtain a form of the equation in terms
Conservation of Energy
- This equation is obtained through expansion using the product rule.
- We know that ,
by the conservation of mass found above.
- Using these ideas, we can then see that
Conservation of Angular Momentum
- For this, use the First Law of Thermodynamics in rate form to obtain
the following equation:
- Where E = the total energy of the system. In the above equation
is the rate of change of system energy.
- is the rate of heat added
to the system
- is the rate of work done
by the system .
Because work is done by the system, the negative sign is in the equation
for the first law of thermodynamics.
- Now, these conservation laws must always hold for a system.
We will not have time to study this, but see the text for details.
- One can think of the system approach as the Lagrangian description,
which if recalled is the description where we follow the individual chunks
of the fluid.
- However, the trouble with this is that we prefer to use the Eulerian
Description, where we define a control volume with fluid moving
- In other words, if we define some control volume, fixed in
space, the system will flow through it.
- Our goal here is to try to write the conservation laws above in a form
applicable to a control volume (Eulerian description)
rather than to a system (Lagrangian description).
- There is a way to do this, and it is called the Reynolds Transport
- Now our goal is to write all three of these conservation laws in terms
of a control volume. But, to save us some work, let's not derive it three
times, but just once for a general property B, and then use our
result for any of the three conservation laws:
- In other words, let's write R.T.T. for some arbitrary property B.
Let B = some arbitrary property (vector or scalar) and
per unit mass.
This way, we can later substitute B for mass, momentum, energy, etc.
- We are then able to make the following substitutions for :
- For conservation of mass, let B = m, and .
- For conservation of momentum, ,
- For conservation of energy, B = E, and .
- SEE the text for the derivation.
REYNOLDS TRANSPORT THEOREM (R.T.T.)
- This is for some property B, and is also for a fixed
Deforming the control volume is more complicated; see the text for details.
- The equation for Reynolds Transport Theorem is written as the following:
(This is equation 3.12 in the text.)
For a fixed control volume, the order of differentiation
and integration doesn't matter. So alternatively,
(This is equation 3.17 in the text. )
- Comments on the above equation:
- The total rate of change
of B following the system.
This is the term to which the conservation laws directly apply.
- The time rate of change
of B within the control volume. This is due to unsteadiness.
- The net flux of B out of
the control surface.
Due to convection, B changes because system moves to a new
part of the flow field, where conditions are different. A circle
was put around the integral
to emphasize that this is an integral over the entire control
surface. (This is not done in the text however).
- Sometimes and are
used to emphasize that these are volume
and area integrals. (Again this is not done in the text).