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:

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


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.


Write both the integral and differential forms of the second law


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


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.