ENTROPY

Entropy

  1. Entropy is a thermodynamic property that measures the degree of randomization or disorder at the microscopic level. The natural state of affairs is for entropy to be produced by all processes.
  2. A macroscopic feature which is associated with entropy production is a loss of ability to do useful work. Energy is degraded to a less useful form, and it is sometimes said that there is a decrease in the availability of energy.
  3. Entropy is an extensive thermodynamic property. In other words, the entropy of a complex system is the sum of the entropies of its parts.
  4. The notion that entropy can be produced, but never destroyed, is the second law of thermodynamics

Reversible and Irreversible Processes

Processes can be classed as reversible or irreversible. The concept of a reversible process is an important one which directly relates to our ability to recognize, evaluate, and reduce irreversibilities in practical engineering processes.

Consider an isolated system. The second law says that any process that would reduce the entropy of the isolated system is impossible. Suppose a process takes place within the isolated system in what we shall call the forward direction. If the change in state of the system is such that the entropy increases for the forward process, then for the backward process (that is, for the reverse change in state) the entropy would decrease. The backward process is therefore impossible, and we therefore say that the forward process is irreversible.

If a process occurs, however, in which the entropy is unchanged by the forward process, then it would also be unchanged by the reverse process. Such a process could go in either direction without contradicting the second law. Processes of this latter type are called reversible.

The key idea of a reversible process is that it does not produce any entropy.

Entropy is produced in irreversible processes. All real processes (with the possible exception of superconducting current flows) are in some measure irreversible, though many processes can be analyzed quite adequately by assuming that they are reversible. Some processes that are clearly irreversible include: mixing of two gases, spontaneous combustion, friction, and the transfer of energy as heat from a body at high temperature to a body at low temperature.

Recognition of the irreversibilities in a real process is especially important in engineering. Irreversibility, or departure from the ideal condition of reversibility, reflects an increase in the amount of disorganized energy at the expense of organized energy. The organized energy (such as that of a raised weight) is easily put to practical use; disorganized energy (such as the random motions of the molecules in a gas) requires “straightening out” before it can be used effectively. Further, since we are always somewhat uncertain about the microscopic state, this straightening can never be perfect. Consequently the engineer is constantly striving to reduce irreversibilities in systems, in order to obtain better performance.


Examples of Reversible and Irreversible Processes

Processes that are usually idealized as reversible include:

  • Frictionless movement
  • Restrained compression or expansion
  • Energy transfer as heat due to infinitesimal temperature nonuniformity
  • Electric current flow through a zero resistance
  • Restrained chemical reaction
  • Mixing of two samples of the same substance at the same state.

Processes that are irreversible include:

  • Movement with friction
  • Unrestrained expansion
  • Energy transfer as heat due to large temperature non uniformities
  • Electric current flow through a non zero resistance
  • Spontaneous chemical reaction
  • Mixing of matter of different composition or state.

Entropy Changes in an Ideal Gas

Many aerospace applications involve flow of gases (e.g., air) and we thus examine the entropy relations for ideal gas behavior. The starting point is form (a) of the combined first and second law,

clip_image002

For an ideal gas, clip_image004. Thus

clip_image005

Using the equation of state for an ideal gas (clip_image006 ), we can write the entropy change as an expression with only exact differentials:

clip_image007

(5..2)

We can think of Equation (5.2) as relating the fractional change in temperature to the fractional change of volume, with scale factors clip_image008and clip_image009; if the volume increases without a proportionate decrease in temperature (as in the case of an adiabatic free expansion), then clip_image010increases. Integrating Equation (5.2) between two states “1” and “2”:

clip_image011

For a perfect gas with constant specific heats

clip_image012

In non-dimensional form (using clip_image014)

clip_image015

(5..3)

Equation 5.3 is in terms of specific quantities. For clip_image016moles of gas,

clip_image017

This expression gives entropy change in terms of temperature and volume. We can develop an alternative form in terms of pressure and volume, which allows us to examine an assumption we have used. The ideal gas equation of state can be written as

clip_image019

Taking differentials of both sides yields

clip_image020

Using the above equation in Eq. (5.2), and making use of the relations clip_image021; clip_image023, we find

clip_image024

or

clip_image025

Integrating between two states 1 and 2

clip_image026

(5..4)

Using both sides of (5.4) as exponents we obtain

clip_image027

(5..5)

Equation (5.5) describes a general process. For the specific situation in which clip_image028, i.e., the entropy is constant, we recover the expression clip_image029. It was stated that this expression applied to a reversible, adiabatic process. We now see, through use of the second law, a deeper meaning to the expression, and to the concept of a reversible adiabatic process, in that both are characteristics of a constant entropy, or isentropic, process.

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