Entropy production

Entropy production is the amount of entropy which is produced during heat process to evaluate the efficiency of the process.

Short history

is produced in irreversible processes. The importance of avoiding irreversible processes was recognized as early as 1824 by Carnot. In 1865 Rudolf Clausius expanded his previous work from 1854 on the concept of "unkompensierte Verwandlungen", which, in our modern nomenclature, would be called the entropy production. In the same article in which he introduced the name entropy, Clausius gives the expression for the entropy production for a cyclical process in a closed system, which he denotes by N, in equation which reads
Here S is the entropy in the final state and S₀ the entropy in the initial state; S₀-S is the entropy difference for the backwards part of the process. The integral is to be taken from the initial state to the final state, giving the entropy difference for the forwards part of the process. From the context, it is clear that if the process is reversible and in case of an irreversible process.

First and second law

The laws of thermodynamics apply to well-defined systems. Fig. 1 is a general representation of a thermodynamic system. We consider systems which, in general, are inhomogeneous. Heat and mass are transferred across the boundaries, and the boundaries are moving. 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. The expression, given here, are not the most general formulations of the first and second law. E.g. kinetic energy and potential energy terms are missing and exchange of matter by diffusion is excluded.
The rate of entropy production, denoted by, is a key element of the second law of thermodynamics for open inhomogeneous systems which reads
Here S is the entropy of the system; T_k is the temperature at which the heat enters the system at heat flow rate ; represents the entropy flow into the system at position k, due to matter flowing into the system and specific entropy ; represents the entropy production rates due to internal processes. The subscript 'i' in refers to the fact that the entropy is produced due to irreversible processes. The entropy-production rate of every process in nature is always positive or zero. This is an essential aspect of the second law.
The Σ's indicate the algebraic sum of the respective contributions if there are more heat flows, matter flows, and internal processes.
In order to demonstrate the impact of the second law, and the role of entropy production, it has to be combined with the first law which reads
with U the internal energy of the system; the enthalpy flows into the system due to the matter that flows into the system, and dV_k/dt are the rates of change of the volume of the system due to a moving boundary at position k while p_k is the pressure behind that boundary; P represents all other forms of power application.
The first and second law have been formulated in terms of time derivatives of U and S rather than in terms of total differentials dU and dS where it is tacitly assumed that dt > 0. So, the formulation in terms of time derivatives is more elegant. An even bigger advantage of this formulation is, however, that it emphasizes that heat flow rate and power are the basic thermodynamic properties and that heat and work are derived quantities being the time integrals of the heat flow rate and the power respectively.

Examples of irreversible processes

Entropy is produced in irreversible processes. Some important irreversible processes are:

heat flow through a thermal resistor
fluid flow through a flow resistance such as in the Joule expansion or the Joule–Thomson effect
heat transfer
Joule heating
friction between solid surfaces
fluid viscosity within a system.

The expression for the rate of entropy production in the first two cases will be derived in separate sections.

Performance of heat engines and refrigerators

Most heat engines and refrigerators are closed cyclic machines. In the steady state the internal energy and the entropy of the machines after one cycle are the same as at the start of the cycle. Hence, on average, dU/dt = 0 and dS/dt = 0 since U and S are functions of state. Furthermore, they are closed systems and the volume is fixed. This leads to a significant simplification of the first and second law:
and
The summation is over the places where heat is added or removed.

Engines

For a heat engine the first and second law obtain the form
and
Here is the heat supplied at the high temperature T_H, is the heat removed at ambient temperature T_a, and P is the power delivered by the engine. Eliminating gives
The efficiency is defined by
If the performance of the engine is at its maximum and the efficiency is equal to the Carnot efficiency

Refrigerators

For refrigerators holds
and
Here P is the power, supplied to produce the cooling power at the low temperature T_L. Eliminating now gives
The coefficient of performance of refrigerators is defined by
If the performance of the cooler is at its maximum. The COP is then given by the Carnot coefficient of performance

Power dissipation

In both cases we find a contribution which reduces the system performance. This product of ambient temperature and the entropy production rate is called the dissipated power.

Equivalence with other formulations

It is interesting to investigate how the above mathematical formulation of the second law relates with other well-known formulations of the second law.
We first look at a heat engine, assuming that. In other words: the heat flow rate is completely converted into power. In this case the second law would reduce to
Since and this would result in which violates the condition that the entropy production is always positive. Hence: No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work. This is the Kelvin statement of the second law.
Now look at the case of the refrigerator and assume that the input power is zero. In other words: heat is transported from a low temperature to a high temperature without doing work on the system. The first law with would give
and the second law then yields
or
Since and this would result in which again violates the condition that the entropy production is always positive. Hence: No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature. This is the Clausius statement of the second law.

Expressions for the entropy production

Heat flow

In case of a heat flow rate from T₁ to T₂ the rate of entropy production is given by
If the heat flow is in a bar with length L, cross-sectional area A, and thermal conductivity κ, and the temperature difference is small
the entropy production rate is

Flow of mass

In case of a volume flow rate from a pressure p₁ to p₂
For small pressure drops and defining the flow conductance C by we get
The dependences of on and on are quadratic.
This is typical for expressions of the entropy production rates in general. They guarantee that the entropy production is positive.

Entropy of mixing

In this Section we will calculate the entropy of mixing when two ideal gases diffuse into each other. Consider a volume V_t divided in two volumes V_a and V_b so that. The volume V_a contains amount of substance n_a of an ideal gas a and V_b contains amount of substance n_b of gas b. The total amount of substance is. The temperature and pressure in the two volumes is the same. The entropy at the start is given by
When the division between the two gases is removed the two gases expand, comparable to a Joule–Thomson expansion. In the final state the temperature is the same as initially but the two gases now both take the volume V_t. The relation of the entropy of an amount of substance n of an ideal gas is
where C_V is the molar heat capacity at constant volume and R is the molar gas constant.
The system is an adiabatic closed system, so the entropy increase during the mixing of the two gases is equal to the entropy production. It is given by
As the initial and final temperature are the same, the temperature terms cancel, leaving only the volume terms. The result is
Introducing the concentration x = n_a/n_t = V_a/V_t we arrive at the well-known expression

Joule expansion

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. Initially, the valve is closed. Vessel a contains the gas while the other vessel b is empty. When the valve is opened, the gas flows from into b until the pressures in the two vessels are equal. The volume, taken by the gas, is doubled while the internal energy of the system is constant. Assuming that the gas is ideal, the molar internal energy is given by. As C_V is constant, constant U means constant T. The molar entropy of an ideal gas, as function of the molar volume V_m 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