Second law analysis of a control volume

It was shown that the change in entropy of a system is given by

(24.12)

Where the equality sign holds good for the reversible processes and the inequality sign holds for all irreversible processes. This can be expressed as

(24.13)

Where represents entropy generation in the system and it cannot take a negative value.

for an irreversible process.

Consider a control volume through which material flows (Figure 24.2)

Figure 24.2

Let us identify a system such that its mass remains constant during a given process. At time t, the system constitutes both the mass inside the control volume and the mass about the enter the control volume during a small interval dt (that is, the mass occupying the region A). At time (t+dt), the system is the mass inside the control volume as well as the mass in region B. During the time interval dt, the system undergoes a change in configuration and receives energy as heat from the surroundings.

At time t, the entropy of the system

(24.14)

At time , the entropy of the system

(24.15)

Then,

(24.16)

or,

(24.17)

or,

(24.18)

This can be rearranged as

(24.19)

In words:

The rate of accumulation of entropy = Rate of inflow of entropy

• Rate of outflow of entropy + Rate of generation of entropy