## Entropy

### The Clausius Inequality

- The first law is just a balance of energy.
- The second law states an inequality ie an irreversible process is less efficient than a reversible process.
- One of such important inequalities is that of the Clausius inequality in Thermodynamics according to which the cyclic integral of δQ / T is always less than or equal to zero.
- It is valid for all cycles, be it reversible or irreversible.
- The basis for the definition of a new property called entropy is formed by the Clausius inequality.
- For an internally reversible process, the cyclic integral of δQ / T is zero.
- If the cyclic integral of a quantity is zero, it means that it depends on the state only and not the process path. Hence, it is a property.
- For an irreversible process, the value of entropy change for a closed system is greater than the value of integral of dQ/T evaluated for that process.
- If the energy exchange takes place, δQ will be the energy transfer from the surroundings to the system.

**TEMPERATURE-ENTROPY PLOT **

Now dQ_{rev}, = TdS

Thus, area under the T -S plot on S axis will give the heat transfer in a reversible process.

**Fig.: Area under a reversible path on the T-s plot**

**THE INCREASE OF ENTROPY PRINCIPLE:**

Let us assume a cycle that is made up of two processes as shown below:

process 1-2, which is an arbitrary process (reversible or irreversible),

and process 2-1, which is internally reversible in nature, as shown in the Figure below,

**Fig.: Combination of reversible & irreversible process**

where the equality is for the internally reversible process and the inequality for the irreversible process.

It should be kept in mind that the entropy generation S_{gen} is always a positive quantity or zero. Its value is process-dependent, and thus it is not a property of the system.

For an isolated system,

dQ = 0 since no energy interaction occurs between the system and the surrounding.

Therefore, for an isolated system

dS_{iso} ≥ 0

For a reversible process,

dS_{iso} = 0

implies, S = constant

For an irreversible process

dS_{iso} > 0

**Entropy change of the system:**

Entropy charge of the system is summation entropy change due to internal irreversibility and entropy change due to external interaction

dS = (dS)_{Irr,} + (dS)_{EI}

**T-dS EQUATION**

........(1)

........(2)

Note:

Equation (1) & (2) are applicable for both reversible process as well as the irreversible process because it contents all properties.

**APPLICATIONS OF ENTROPY PRINCIPLE:**

For every irreversible process, there is an increase of entropy of the universe, and this entropy increase determines the extent of the irreversibility of the process. The higher the entropy increase of the universe; the higher will be the irreversibility of the process.

Some of the applications of the entropy principle are illustrated in the following.

(A) Heat transfer through a Finite Temperature Difference.

(B) Two fluids mixing with each other.

(C) Maximum Work that can be obtained from Two Finite Bodies at Temperatures T_{1} and T_{2} interacting in a reversible manner.

The sources of energy can be divided into two groups i.e., high grade energy (mechanical work, electrical energy, water power, wind power) and low grade energy (heat or thermal energy, heat derived from nuclear fission or combustion of fossil fuels). That part of the low grade energy which is available for, conversion is referred to as available energy, while the part which is not available is known as unavailable energy.

**Availability:** When a system is subjected to a process from its original state to dead state the maximum amount of useful work that can be achieved under ideal conditions is known as available energy or availability of the system.

W_{max} = AE = Q_{xy} – T_{0}(S_{y}-S_{x})

Unavailable Energy: UAE = T_{0}(S_{y}-S_{x})

where, S_{x} and S_{y} are the entropy at x and y, respectively.

The Available Energy (AE) is also known as exergy and the Unavailable Energy (UAE) as anergy. The energy which cannot be utilised for doing useful work is called unavailable energy. Irreversibility is equivalent to energy destroyed, hence also known as energy destruction consider the example given below.

**DECREASE IN AVAILABILITY WHEN HEAT TRANSFER THROUGH FINITE TEMPERATURE DIFFERENCE**

Consider a reversible heat engine operating between T_{1 }and T_{0}.

Q_{1} = T_{1}. Δs

W = AE= (T_{1} – T_{0})Δs

Let us now consider heat Q_{1} is transferred through a finite temperature difference.

Q_{1} = T_{1} Δs = T’_{1} . Δs’

Δs’ > Δs

Q_{2} = T_{0} Δs → Initial UAE

Q’_{2} = T_{0} Δs’ ⇒ Afterward UAE

Q’_{2} > Q_{2}

W’ =Q’_{1} – Q’_{2}

W’ = T’_{1} Δs’ – T_{0}Δs’

W’ = (T’_{1} – T_{0}) Δs’

W = (T_{1} – T_{0}) Δs

Hence increase in UAE and the shaded portion represent increase in UAE.

**SECOND LAW EFFICIENCY: **Second law efficiency is the ratio of the exergy recovered to the exergy spent. OR It is the ratio of actual work produced to the max work produced under reversible condition. Consider the case of heat engine.

The second law efficiency is measure of the performance of a device relative to its performance under reversible conditions.

**EXERGY OF A CLOSED SYSTEM : **Consider a piston cylinder device that contains a fluid of mass m at temperature T and pressure P. The system is then allowed to undergo a differential change of state during which volume changes by dV and heat is transferred from the system to surroundings.

For change of exergy from state 1 to state 2.

Φ_{1} = (E_{1} – E_{0}) + P_{0}(V_{1} – V_{0}) – T_{0}(S_{1} – S_{0})

Φ_{2} = (E_{2} – E_{0}) + P_{0}(V_{2} – V_{0}) – T_{0}(S_{2} – S_{0})

Φ_{1} – ϕ_{2} = Energy at state 1 – Energy at state 2

Φ_{1} – ϕ_{2 }= (E_{1} – E_{2}) + P_{0} (V_{1} – V_{2}) – T_{0}(S_{1} – S_{2})

**EXERGY OF OPEN SYSTEM: **A flowing fluid has an additional form of energy, called flow energy, which is the energy needed to maintain flow in Pipe.

W_{flow} = PV

Exergy associated with system is

Φ_{flowing fluid} = ϕ_{non flow fluid} + ϕ_{flow}

Φ_{flowing fluid} = (E_{1} – E_{0}) + P_{0}(V_{1} – V_{0}) – T_{0}(S_{1} – S_{0}) +P_{1}V_{1} – P_{0}V_{0}

Φ_{flowing}= U_{1}- U_{0} + (KE)_{1} - (KE)_{0} + (PE)_{1} – (PE)_{0} + P_{1}V_{1} – P_{0}V_{0} – T_{0} (S_{1} – S_{0})

Φ_{flowing fluid} = (U_{1} + P_{1}V_{1}) – (U_{0} + P_{0}V_{0}) – T_{0}(S_{1} – S_{0}) + (KE)_{1} – (KE)_{0} + (PE)_{1} – (PE)_{0}

Φ_{flowing} = (H_{1} – H_{0}) – T_{0} (S_{1} – S_{0}) + (KE)_{1} – (KE)_{0} + (PE)_{1} – (PE)_{2}

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