Control Volume (Integral) Technique 2

Control Volume (Integral) Technique 2

1. Control Volume:

   • 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 through it.
   • 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 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 (*picture) per unit mass. This way, we can later substitute B for mass, momentum, energy, etc.
               • This way, we can later substitute B for mass, momentum, energy, etc.
                 • For conservation of mass, let B = m, and
                   • For conservation of momentum,
                     • For conservation of energy B = E, and

2. REYNOLDS TRANSPORT THEOREM (R.T.T.)

   • This is for some property B, and is also for a fixed control volume. Deforming the control volume is more complicated
     • The equation for Reynolds Transport Theorem is written as the following:
       • For a fixed control volume, the order of differentiation and integration doesn't matter. So alternatively,
         • LONG comments about RTT:
           • 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.
                 • are used to emphasize that these are volume and area integrals.

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