STATE CLAUSIUS INEQUALITY AND EXPLAIN ITS SIGNIFICANCE

We were discussing “Clausius theorem” and “Carnot cycle and its efficiency” in our previous post. Today we will see here the fundamental of Clausius inequality with the help of this post.

So let us see here the Clausius inequality

First we need to know the importance of Clausius inequality in the field of thermodynamics and after that we will see here the basic concept of Clausius inequality. We will also derive the expression for Clausius inequality in this post. 

Clausius inequality suggests us the criterion for reversibility and irreversibility of a thermodynamic cycle.

Let us consider the following figure, we can see here the curve between temperature (T) and Volume (V) here. There are few almost vertically drawn lines and these lines indicate the reversible adiabatic lines as shown and mentioned in figure. There is one more horizontal line drawn below represent the constant temperature line and having temperature T₂ as mentioned in figure.

Let us consider the cycle ABCD as shown in figure, let us consider one process AB is one general process which will be either reversible or irreversible process. Other processes of cycle such as process BC and AD are reversible adiabatic process and process CD is reversible isothermal process.

Now let us focus over the process AB and we will clearly observe here that addition of heat will not be done here at constant temperature but also heat will be added at different temperature during the process AB.

Let we have divided the cycle ABCD in to number of elementary cycles by drawing the straight lines at top as shown in figure and these straight lines represent the constant temperature lines.

Now let us consider one elementary cycle and we can say here that dQ heat is added here at temperature T and dQ₂ heat is rejected at temperature T₂ as shown in figure. Now we will see here the efficiency of this elementary cycle and it will be as mentioned here.

η = 1-(dQ₂/dQ)

As we know that efficiency of a general cycle will always be less than or equal to the efficiency of a reversible cycle and therefore we will have following equation as mentioned here

Or we can also say that for any process reversible or irreversible such as process AB in our example, we will have following equation

As we know that for a reversible process, we will have

dS = dQ_rev/T = dQ₂/T₂

For any process reversible or irreversible such as process AB in our example, we will have following equation

The above equation will be termed as Clausius inequality and it will suggest us the criterion for reversibility and irreversibility of a thermodynamic cycle.

Do you have any suggestions? Please write in comment box.

We will see another topic i.e. "Entropy change in reversible and irreversible processes" in our next post in the category of thermal engineering.

Reference:

Engineering thermodynamics by P. K. Nag

Image Courtesy: Google

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