Entropy Balance for Open Systems | Thermodynamics - Mechanical Engineering PDF Download

Entropy Balance for Open Systems

As with energy balance for open systems, once can extend the equation 4.24 to a generalized entropy balance equation that may be written for the system shown in fig. 3.6.  There is, however, an important point of departure from the first law. Unlike energy, entropy is not a conserved quantity for real world processes in which both mechanical and thermal irreversibilities are inevitable. Hence for such processes are attended by a positive entropy generation rate,   This conclusion may be expressed in mathematical terms as follows:

                 ..(4.25)

As with eqn. 4.24, for eqn. 4.25 too, the left side reduces to zero if the processes occurring in the open system are totally reversible; that is, both with respect to the system as well to the surroundings. Or else there is a net entropy generation.
A process is said to be internally reversible if all the processes occurring within the system are mechanically reversible, that is they are not subject to dissipative forces.  External reversibility, on the other hand, signifies that all heat transfer between the system and surrounding occur under infinitesimal gradients and are therefore thermally reversible. In principle reversible heat transfer is possible if the surroundings have heat reservoirs with temperatures equal to those of the control surface or if reversible Carnot engines operate between the control-surface temperatures and the heat-reservoir temperatures.

We next expand eqn. 4.25 to a further level of detail.  Let there be heat transfer at the rate   at a particular part of the control surface of the open system where the surrounding temperature is given by 

Thus: 

                     ..(4.26)

Here, j runs over all the heat reservoirs associated with the system. The negative sign is used for the entropy term for the surroundings as heat transfer terms by convention are associated with the system. 

Putting eqn. 4.26 in eqn. 4.25 one obtains:
 

                                  ..(4.27)

For steady flows through the control volume eqn. 4.27 reduces to:

                         ..(4.28)

Further, for the simplest case of one inlet and exit, and one surrounding temperature:

                        ..(4.29)