Entropy Balances Reference Page
How Much Entropy Can You Balance on the Head of a Pin (or in an Open Steady-State System)?

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Intro to Entropy Balances

The Second Law of Thermodynamics can be written as a balance equation for entropy, similar in form to mass and energy balances.

Entropy balances allow us to determine whether a process is thermodynamically possible and to quantify the irreversibility of real processes.

Entropy balances can be written for closed systems, open steady-state systems, and open transient systems.

How Much Entropy Can You Balance on the Head of a Pin (or in an Open Steady-State System)?

This video introduces entropy balance equations and shows how they apply to closed systems, steady-state flow systems, and transient systems.

Visuals

Examples and Definitions

Definitions

Entropy Balance for a Closed System
The entropy balance for a closed system is

\[ \Delta S_{\text{system}} - \sum\limits_j \dfrac{Q_j}{T_{\sigma j}} = S_G \geq 0 \]

where \(S_G\) is the entropy generated by irreversibilities within the system.

Entropy Balance for an Open Steady-State System
For a steady-state control volume,

\[ \Delta (\dot{m} \hat{S})_{\text{fs}} - \sum\limits_j \dfrac{\dot{Q}_j}{T_{\sigma j}} = \dot{S}_G \geq 0 \]

Entropy Balance for an Open Transient System
For a general control volume,

\[ \dfrac{d(m \hat{S})_\mathrm{cv}}{dt} + \Delta (\dot{m} \hat{S})_{\text{fs}} - \sum\limits_j \dfrac{\dot{Q}_j}{T_{\sigma j}} = \dot{S}_G \geq 0 \]

Entropy Generation, \(S_G\)
Entropy produced within a system due to irreversibilities such as friction, mixing, chemical reactions, heat transfer across finite temperature differences, or turbulence.

The Second Law requires

\[ S_G \ge 0 \]

Transient System
A system in which process variables change with time.
Steady State System
A system in which process variables do not change with time, although they may vary with position. (What electrical engineers call a DC steady state.)
Closed or Batch System
Throw stuff in. Wait. Pull products out.
Open or Continuous System
Continually throw stuff in and pull stuff out.