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# Heat Entropy

A brief introduction to heat entropy and isentropic processes
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### Key Facts

Gyroscopic Couple: The rate of change of angular momentum ($\inline&space;\tau$) = $\inline&space;I\omega\Omega$ (In the limit).
• $\inline&space;I$ = Moment of Inertia.
• $\inline&space;\omega$ = Angular velocity
• $\inline&space;\Omega$ = Angular velocity of precession.

## Overview

Key facts

The change in entropy for a system undergoing a reversible process is given by:

$\delta&space;S&space;=&space;\frac{\delta&space;Q}{T}$

where $\inline&space;T$ is the temperature of the system, and $\inline&space;\delta&space;Q$ the infinitely small amount of heat which is absorbed by the system in a reversible way.

Isentropic processes are processes which are assumed to proceed without changes in the entropy of the system.

It can be shown that any reversible adiabatic process is an isentropic process.

In order to define the heat entropy, consider the curve of state $\inline&space;1-2$ diagramed in Figure 1.

##### MISSING IMAGE!

This curve of state is plotted on a graph with the absolute temperature $\inline&space;T$ on the ordinate, such that the area under the curve represents the heat $\inline&space;Q$ supplied or rejected. The quantity along the abscissa is a property, or function of state, called entropy, and denoted by the letter $\inline&space;S$.

As entropy is a function of state, changes in its value depend only on the initial and final state, and not on the process which causes the change. The change in entropy for a system undergoing a reversible process (for a more detailed discussion on reversible processes also see Reversible Processes ) can be written as:

$\delta&space;S&space;=&space;\frac{\delta&space;Q}{T}$

where $\inline&space;T$ is the temperature of the system, and $\inline&space;\delta&space;Q$ the infinitely small amount of heat which is absorbed by the system in a reversible way.

## Isentropic Processes

Isentropic processes are processes which are assumed to proceed without changes in the entropy of the system. Therefore, during isentropic processes, the entropy is assumed to remain constant.

It can be shown that any reversible adiabatic process is an isentropic process. However, it should be noted that it is possible to have adiabatic processes which are not reversible, and also isentropic processes which are not adiabatic. Consider, for example, a rotary compressor. The processes involved are adiabatic, since no heat goes in or out the machine. Nevertheless, the entropy does increase due to heat produced internally by friction. If, however, just enough heat is removed by a cooling system to counteract the heat produced by friction, then the entropy will remain constant, and the process will become isentropic, but no longer adiabatic.