## Reynolds Transport Theorem

**Introduction: **It can be noticed that the same derivations carried for the density can be carried for other intensive properties such as specific entropy, specific enthalpy. Suppose that g is intensive property (which can be a scalar or a vector) undergoes change with time. The change of accumulative property will be then

This theorem named after Reynolds, Osborne, (1842-1912) which is actually a three dimensional generalization of Leibniz integral rule1. To make the previous derivation clearer, the Reynolds Transport Theorem will be reproofed and discussed. The ideas are the similar but extended somewhat.

Leibniz integral rule2 is a one dimensional and it is defined as

Initially, a proof will be provided and the physical meaning will be explained. Assume that there is a function that satisfies the following:

Notice that lower boundary of the integral is missing and is only the upper limit of the function is present. For its derivative of equation

Differentiating (chain rule d uv = u dv v du) by part of left hand side of the Leibniz integral rule (it can be shown which are identical) is

**REYNOLDS TRANSPORT THEOREM:** The terms 2 and 4 in above equation are actually (the x2 is treated as a different variable)

The first term (1) in equation:

The same can be said for the third term (3). Thus this explanation is a proof the Leibniz rule. The above “proof” is mathematical in nature and physical explanation is also provided. Suppose that a fluid is flowing in a conduit. The intensive property, f is investigated or the accumulative property, F. The interesting information that commonly needed is the change of the accumulative property, F, with time. The change with time is

For one dimensional situation the change with time is

If two limiting points (for the one dimensional) are moving with a different coordinate system, the mass will be different and it will not be a system. This limiting condition is the control volume for which some of the mass will leave or enter. Since the change is very short (differential), the flow in (or out) will be the velocity of fluid minus the boundary at x_{1}, U_{rn} = U_{1} − U_{b}. The same can be said for the other side. The accumulative flow of the property in, F, is then