Rate of Change of an Extensive Property

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The flow of fluid through a control volume transports mass, energy, or momentum into and out the control volume.  Before we can attempt to analyze the flow of a fluid, we must have a way of quantifying this transport and relating it to the rate of change of various fluid properties.

By Joseasorrentino (Own work) [CC-BY-SA-3.0], via Wikimedia Commons

The following derivation for the rate of change of an extensive property leads to the Reynolds Transport Theorem that is one of the most fundamental theorems in fluid dynamics.


A control volume is shown in the figure.  At time, t, there is a system of fluid particles that occupy the control volume.  At this instance in time, the extensive property B of the system of particles is the same as that for the control volume.

 

After a small increment in time, δt, the system of particles has moved relative to the control volume.  At t+δt, the extensive property B is equal to 

 


However, this is no longer equal to B for the control volume.  In terms of the control volume, 

The change of B in the system is

This equation can be rewritten as

Dividing by δt

 

The left hand side of the equation is simply the time rate of change of B in the system.  


The first term on the right hand side of the equation is the time rate of change of B for the material that is in the control volume. 

The second, third, and fourth terms give the rate of change of B due to material being transported into and out of the control volume.   If b is an intensive property obtained by dividing B by the mass, then the net rate at which b is transferred into and out of the control volume is given by

The Reynolds Transport Theorem is obtained by combining the previous three equations to yield


The Reynolds Transport Theorem states that the time rate of change of B for a system of fluid particles is equal to the time rate of change B in the control volume plus the net rate at which B is transported across the control surface.

It is an important theorem because it relates the time rate of change of a property for a fixed system of fluid particles to what is happening in a control volume that does not contain a fixed system of particles.  The Reynolds Transport Theorem will allow the fundamental of equations of Physics for a system of particles to be used to obtain the fundamental equations for a control volume.

 

Learning Objectives:

Learn how the Reynolds Transport Theorem is derived