### 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.