The second law can be written as an entropy balance that separates entropy transferred across a boundary from entropy produced internally by irreversibility. With specific entropy $s=S/m$, a fixed control volume obeys

$$\boxed{\frac{dS_{CV}}{dt}=\sum_j\frac{\dot Q_j}{T_j}+\sum_{in}\dot m s-\sum_{out}\dot m s+\dot S_{gen}},$$

where $T_j$ is the absolute boundary temperature at which heat $\dot Q_j$ crosses and

$$\boxed{\dot S_{gen}\ge0}.$$

The terms have different physical meanings:

• $dS_{CV}/dt$ is entropy accumulation inside the control volume;
• $\dot Q_j/T_j$ is entropy transferred with heat;
• $\dot m s$ is entropy transported by flowing mass;
• $\dot S_{gen}$ is entropy created inside by irreversibility.

Common sources of entropy generation include friction, viscous dissipation, finite-temperature-difference heat transfer, mixing, electrical resistance, shocks, and throttling. A reversible process has $\dot S_{gen}=0$.

Steady adiabatic flow

At steady state,

$$\frac{dS_{CV}}{dt}=0.$$

For an adiabatic one-inlet/one-outlet device with equal mass flow rate,

$$0=\dot m s_1-\dot m s_2+\dot S_{gen},$$

so

$$\boxed{\dot S_{gen}=\dot m(s_2-s_1)\ge0}.$$

An adiabatic device is therefore not automatically isentropic. It is isentropic only in the internally reversible limit, when $\dot S_{gen}=0$.

Worked example

A steady adiabatic turbine carries

$$\dot m=2.0,\mathrm{kg/s}$$

with

$$s_1=6.50,\mathrm{kJ/(kg,K)},\qquad s_2=6.65,\mathrm{kJ/(kg,K)}.$$

Then

$$\dot S_{gen}=\dot m(s_2-s_1)$$

$$=(2.0)(6.65-6.50)=\boxed{0.30,\mathrm{kW/K}}.$$

The positive entropy generation measures irreversibility inside the turbine even though no heat crosses its boundary.

Closed-system and isolated-system limits

For a closed system, the mass-flow terms vanish:

$$\boxed{\frac{dS}{dt}=\sum_j\frac{\dot Q_j}{T_j}+\dot S_{gen}}.$$

For an isolated system, heat and mass transfer are both absent, giving

$$\boxed{\frac{dS}{dt}=\dot S_{gen}\ge0}.$$

The same balance therefore unifies closed and open systems: entropy can cross boundaries with heat and matter, while irreversibility produces additional entropy internally.