MAIN CONCEPTS

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All physical laws are stated in terms of various physical parameters, such as velocity, acceleration, mass or temperature. Let B represent any of these fluid parameters and b represents the amount of that parameter per unit mass. That is,

where m is the mass of the portion of the fluid of interest. The parameter B is termed the extensive property and the parameter b is termed the intensive property. The amount of an extensive property that a system possesses at a given instant, , can be determined by adding up the amount associated with each fluid particle in the system. For infinitesimal fluid particles of size  and mass , this summation takes the form of an integration over all the particles in the system and can be written as

In our analysis, we are usually more interested in the time rate of change of an extensive property of a fluid system. This is expressed as

Similarly, we written the time rate of change of an extensive property within a control volume, , as

where the limits of integration, denoted by cv, cover the control volume of interest.

Now we know about the extensive property, we’ll move swiftly along to derive a simple version of the Reynolds transport theorem relating system concepts to control volume concepts by looking at one-dimensional flow through a fixed control volume as shown below.

We consider the control volume to be the stationary volume within the pipe or duct between sections (1) and (2). The system is that fluid occupying the control volume at some initial time t. After a small interval of , the system has moved slightly to the right.

The fluid particles at section (2) of the control surface at time t has moved a distance  to the right where  is the velocity of the fluid as it passes section (2). Similarly, the fluid initially at (1) has moved a distance  with velocity . Notice that  and  need not be equal by the simple reasoning that the area of each control surface is different.

By looking at the various regions in the diagram, we see that the outflow from the control volume from time  to  is denoted as volume II, the inflow as volume I, and the control volume itself as CV.

If B is the extensive parameter of the system, then the value of it for the system at time t is

since the system and the fluid within the control volume coincides at this time. Its value at time is

Thus, the change in the amount of B in the system in  divided by this time interval is given by

Making use of the fact , we get

In the limit , the left side of the equation gives us the time rate of change of B for the system as we denote it as . We use the material derivative to emphasize that this is the time rate of change of that quantity associated with a given fluid particle.

The first term on the right side of the equation is equal to the time rate of change of the amount of B within the control volume

The third term on the right hand side represents the rate at which the extensive parameter B flows from the control volume, across the control surface. This can be seen from the fact that the amount of B within region II, the outflow region, is its amount per unit volume, , times the volume , and so

where  are across section (2). The rate at which this property flows from the control volume, , is given by

By a similar argument

Combining all these equations, we see that the relationship between the time rate of change of B for the system and that for the control volume is given by

This version of the Reynolds transport theorem is valid under the assumptions of a fixed control volume with one inlet and one outlet having uniform properties (density, velocity, and the parameter b) across the inlet and outlet with the velocity normal to sections (1) and (2). We later extend this equation to accommodate multiple inlets and outlets.