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
(total power output, total heat transfer, etc.)__gross properties__ - 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
box.")

**Differential Analysis**- 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

**Dimensional Analysis and Experiment**- 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 __

- Recall from thermo class, that a
is defined as a volume of mass of fixed identity.__system__ - Let m = the mass of the system

Let= the velocity of the system__V__

Let= the acceleration of the system__a__

- Now we can write three basic conservation laws which apply to this system. Note: These conservation laws apply directly to a system.
states that the mass of a system is constant.__Conservation of mass__

This can be written as the following equation:

In this equation m = the mass of the system.

which is a restatement of__Conservation of linear momentum__

Newton's Second Law.- In equation form this is written as:

Where m__V__= 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
of acceleration:

- This equation is obtained through expansion using the product rule.
- We know that ,

and

by the conservation of mass found above. - Using these ideas, we can then see that

__Conservation of Energy__- 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
the system*to* - is the rate of work done
the system . Because work is done by the system, the negative sign is in the equation for the first law of thermodynamics.*by* - Now, these conservation laws must always hold
.__for a system__

__Conservation of Angular Momentum__

We will not have time to study this, but see the text for details.

- One can think of the system approach as the
, which if recalled is the description where we follow the individual chunks of the fluid.__Lagrangian description__ - However, the trouble with this is that we prefer to use the
, where we define a control volume with fluid moving through it.__Eulerian Description__

- In other words, if we define some control volume,
, the system will flow__fixed in space__it.__through__ - Our goal here is to try to write the conservation laws above in a form
applicable to a
(Eulerian description) rather than to a__control volume__(Lagrangian description).__system__ - There is a way to do this, and it is called the
__Reynolds Transport Theorem__(R.T.T.) - 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
, let B = m, and .*conservation of mass* - For
, , and .*conservation of momentum* - For
, B = E, and .*conservation of energy* - SEE the text for the derivation.

__Control Volume__:

__REYNOLDS TRANSPORT THEOREM (R.T.T.)__

- This is for some property B, and is also for a
control volume.__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__OR__

of differentiation and integration doesn't matter. So alternatively,*order*

(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, 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__convection__control surface. (This is not done in the text however).*entire*

- Sometimes and are
used to
that these are*emphasize**volume*

andintegrals. (Again this is not done in the text).*area*