CONTROL VOLUME APPROACH AND CONTINUITY PRINCIPLE
Introduction: 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.
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.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.
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:
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)
Flow rate for any extensive property B:
Total amount of B in the colored volume:
Net flow rate of B out of the volume:
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.
Turning things around us can now write:
Notice that V in the last term has to be the velocity relative to the control surface.