Entropy

Entropy: 

It is defined as; 

Measure of irreversibilities or randomness of the system.

Clausius Inequality:

The clausius inequality is developed from the Kelvin-Planck statement of the 2nd Law and can be expressed as:

The inequality of clausius is a consequence of the 2nd Law of thermodynamics.

Where; 

• Q is the heat transfer to or from the system.
• T is the absolute temperature at the boundary.
• The symbol ∮ is the cyclic integral.

For total reversible and for internally Reversible:

A special case of Internally Reversible Isothermal Heat Transfer Process:

In Isothermal Process, temperature doesn't change (so it is treated as constant)

Entropy change will be positive or negative when:

• For heat transfered  ↝  Entropy is positive
• For heat rejected  ↝  Entropy is negative

Increase of Entropy Principle:

The total Entropy is given by:

Now, for the the process 1 to 2 and 2 to 1 is given by:

Entropy Change of a Pure Substance:

The Entropy change for pure substance is given by:

ΔS = m Δs

ΔS = m ( s2 - s1 )

Isentropic Process:

Entropy of a fixed mass can be changed by two ways:

1. By Heat transfer 
2. By Irreversibilities 

It means that for an adiabatic internally reversible process, the Entropy will not change. Such a process is called Isentropic Process.

Q. A Reversible adiabatic process is always Isentropic but an Isentropic process doesn't mean that the process is adiabaticlly reversible. Explain? 

If heat change is negative that is heat is given out but the losses always increase Entropy and if they are equal with opposite sign , so total Entropy is zero.

Q. What is the area under the internally reversible process curve on a T-S diagram indicate?

Now the differential area is equal to the product of temperature and differential Entropy.

Mollier Diagram:

Diagram between enthalpy and Entropy ( h-s ) is called Mollier Diagram.

T ds Relation:

The differential form of conservation of energy equation for closed stationary system containing a simple compressible substance can be expressed for an internally reversible process as:

This equation is called 1st TdS equation or Gibb's Equation.

Furthermore;

The above equations (3) and (4) are important as they relate the entropic changes in terms of changes in their properties.

Entropy changes of Solids and Liquids:

Since there is no effect of pressure on liquids and solids so there is no change in volume.

dV ⋍ 0

Cp = Cv = C 

So, equation (3) becomes:

Entropy Change of Gases:

For Gases:

1. Using Constant Specific Heat method:

2. Using Variable Specific Heat method:

When  the temperature change during a process is larger, then the constant specific heat assumption cannot be used. For this purpose, we define new function S° called reference Entropy.

Isentropic Process of Ideal Gases:

1. By using Constant Specific Heat method:

2. Using Variable Specific Heat method:

Steady Flow Reversible Work:

Energy balance for a steady flow devices undergoing an internally reversible process can be expressed in a differential form as:

If,

d k.e.  ⋍ 0

d p.e.  ⋍ 0

So above equation becomes:

Q. Why steam is superheated before sending it to the turbine?

Since, reversible steady flow work is proportional to the specific volume, therefore superheating is done to kept specific volume as high as possible so as work output.

Q. Why cooling is done during compression?

Since, reversible steady flow work is proportional to the specific volume, therefore cooling is done to kept specific volume as low as possible so as work input.

Q. What is the area to the left on an internally reversible process curve on a PV Diagram?

Now the differential area is equal to the product of Specific Volume and differential pressure.

Area to the left of process curve is steady flow reversible work.

Minimizing the Compressor Work:

It can be minimized by maintaining the temperature of the gas as low as possible by cooling during the compression process.

1. Isentropic compression ↝ Adiabatic therefore no cooling, high work input 
2. Polytropic compression ↝ Moderate cooling 
3. Isothermal compression ↝ maximum cooling, °•° least work input.

Thermal Efficiency:

When the Compressor is operated at Isothermal Process,  it's work input is minimum, so the Isothermal efficiencies compares the actual work input to the Isothermal efficiencies compares the actual work input to the Isothermal compressor work input.

Q.  Out of the above compression process which one will have least work input?

Since, area to the left of the process curve on a PV Diagram indicates steady flow reversible work. So, the Isothermal Process has the least area to the left of the curve and Isentropic has the maximum work.

Steady Flow Reversible Work for:

1. Isentropic Process:

As we know for an isentropic process:

Now, formula for work done:

From the very first equation:

2. Polytropic Process:

As we know for a Polytropic Process:

Now, formula for work done is:

3. Isothermal Process:

As we know for an Isothermal Process:

Now, formula for work done is:

Multi-stage Compression:

Multi stage compression is in between cooling by water or oil by water jack.

Now the formula for the input compressor work is given by:

The Px value that gives the minimum value if work input cab be found by differentiating  Px. The above expression and setting the resultant expression equals to zero.

After simplifying:

Isentropic Efficiencies of Steady Flow Devices:

The isentropic efficiency is the measure of the derivation of actual processes from the corresponding idealized ones. In Isentropic efficiency, the ideal processes are Isentropic processes.

1. Isentropic Efficiency of Turbines:

Now the Isentropic efficiency is given by:

2. Isentropic Efficiency of Compressor:

Now the Isentropic efficiency is given by:

3. Isentropic Efficiency of Pump:

Now the Isentropic efficiency is given by:

4. Isentropic Efficiency of Nozzles:

Now the Isentropic efficiency is given by:

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