SL: The entropy "balance" equations

Entropy equations: Open/Closed Integrated/Differential

Closed, diff:

$nds - \frac{\delta Q}{T} = nds + \frac{\delta Q_{surr}}{T} \ge 0$

Closed, int:

$\Delta S - \int \frac{\delta Q}{T} \ge 0$

Open, diff:

$\left ( \frac{dS}{dt} \right )_{surr} + \left ( \frac{dS}{dt} \right )_{sys} = \left ( \frac{dS}{dt} \right )_{sys} + \sum_{out} \dot n_{out}s_{out} - \sum_{in} \dot n_{in}s_{in} + \frac{\dot Q_{surr}}{T_{surr}} \ge 0$

Open, SS, 1in/out:

$\Delta S + \frac{\dot Q_{surr}}{T_{surr}} \ge 0$

Note:

Often, we make an equality out of these expressions by calculating the (rate of) entropy generation, $S_G$, which must be greater than or equal to zero at all times.

Outcome:

Write both the integral and differential forms of the second law

Outcome:

Identify when the open and closed forms of the second law are applicable

Outcome:

Determine when each term in the second law is zero or negligible

Test Yourself:

Calculate the rate of entropy generation for a steady flow process where 1 mol/s of air at 600K and 1 atm is continuously mixed with 2 mol/s of air at 450K and 1 atm to yield a stream of air at 400K and 1 atm. Take air to be an ideal gas with $c_P = 7/2R$ and the surroundings to be at 300K.