Entropy production

The importance of avoiding irreversible processes (hence reducing the entropy production) was recognized as early as 1824 by Carnot.

[1] In 1865 Rudolf Clausius expanded his previous work from 1854[2] on the concept of "unkompensierte Verwandlungen" (uncompensated transformations), which, in our modern nomenclature, would be called the entropy production.

In our formulation we assume that heat and mass transfer and volume changes take place only separately at well-defined regions of the system boundary.

represents the entropy production rates due to internal processes.

refers to the fact that the entropy is produced due to irreversible processes.

the enthalpy flows into the system due to the matter that flows into the system (Hmk its molar enthalpy, hk the specific enthalpy (i.e. enthalpy per unit mass)), and dVk/dt are the rates of change of the volume of the system due to a moving boundary at position k while pk is the pressure behind that boundary; P represents all other forms of power application (such as electrical).

Some important irreversible processes are: The expression for the rate of entropy production in the first two cases will be derived in separate sections.

This leads to a significant simplification of the first and second law: and The summation is over the (two) places where heat is added or removed.

is the heat removed at ambient temperature Ta, and P is the power delivered by the engine.

The COP is then given by the Carnot coefficient of performance In both cases we find a contribution

Hence: No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.

In this Section we will calculate the entropy of mixing when two ideal gases diffuse into each other.

The volume Va contains amount of substance na of an ideal gas a and Vb contains amount of substance nb of gas b.

In the final state the temperature is the same as initially but the two gases now both take the volume Vt.

The relation of the entropy of an amount of substance n of an ideal gas is where CV is the molar heat capacity at constant volume and R is the molar gas constant.

The result is Introducing the concentration x = na/nt = Va/Vt we arrive at the well-known expression The Joule expansion is similar to the mixing described above.

It takes place in an adiabatic system consisting of a gas and two rigid vessels a and b of equal volume, connected by a valve.

The volume, taken by the gas, is doubled while the internal energy of the system is constant (adiabatic and no work done).

Assuming that the gas is ideal, the molar internal energy is given by Um = CVT.

As CV is constant, constant U means constant T. The molar entropy of an ideal gas, as function of the molar volume Vm and T, is given by The system consisting of the two vessels and the gas is closed and adiabatic, so the entropy production during the process is equal to the increase of the entropy of the gas.

So, doubling the volume with T constant gives that the molar entropy produced is The Joule expansion provides an opportunity to explain the entropy production in statistical mechanical (i.e., microscopic) terms.

If the gas has amount of substance n, the number of molecules is equal to n⋅NA, where NA is the Avogadro constant.

This gives change in molar entropy of So, in an irreversible process, the number of microscopic possibilities to realize the macroscopic state is increased by a certain factor.

In this section we derive the basic inequalities and stability conditions for closed systems.

In most cases T is the temperature of the surroundings with which the system is in good thermal contact.

Substitution in the second law, and using that T is constant, gives With the Helmholtz free energy, defined as we get If P = 0 this is the mathematical formulation of the general property that the free energy of systems with fixed temperature and volume tends to a minimum.

If the process inside the system is completely reversible the equality sign holds.

Hence the maximum work, that can be extracted from the system, is equal to the free energy of the initial state minus the free energy of the final state.

As p is constant the first laws gives Combining with the second law, and using that T is constant, gives With the Gibbs free energy, defined as we get In homogeneous systems the temperature and pressure are well-defined and all internal processes are reversible.

by setting Finally, the entropy production reads [8] A recent application of this formula is demonstrated in neuroscience, where it has been shown that entropy production of multivariate Ornstein-Uhlenbeck processes correlates with consciousness levels in the human brain.

Rudolf Clausius
Fig. 1 General representation of an inhomogeneous system that consists of a number of subsystems. The interaction of the system with the surroundings is through exchange of heat and other forms of energy, flow of matter, and changes of shape. The internal interactions between the various subsystems are of a similar nature and lead to entropy production.
Fig.2 a : Schematic diagram of a heat engine. A heating power enters the engine at the high temperature T H , and is released at ambient temperature T a . A power P is produced and the entropy production rate is .
b : Schematic diagram of a refrigerator. is the cooling power at the low temperature T L , and is released at ambient temperature. The power P is supplied and is the entropy production rate. The arrows define the positive directions of the flows of heat and power in the two cases. They are positive under normal operating conditions.