Editing Entropy (section)

=== Entropy balance equation for open systems ===
[[File:First law open system.svg|thumb|upright=1.4|During [[Steady-state (chemical engineering)|steady-state]] continuous operation, an entropy balance applied to an open system accounts for system entropy changes related to heat flow and mass flow across the system boundary.]]
In [[chemical engineering]], the principles of thermodynamics are commonly applied to "[[Open system (systems theory)|open systems]]", i.e. those in which heat, [[work (thermodynamics)|work]], and [[mass]] flow across the system boundary.  In general, flow of heat <math display="inline">\dot{Q}</math>, flow of shaft work <math display="inline"> \dot{W}_\mathsf{S} </math> and pressure-volume work <math display="inline">P \dot{V}</math> across the system boundaries cause changes in the entropy of the system. Heat transfer entails entropy transfer <math display="inline">\dot{Q}/T</math>, where <math display="inline">T</math> is the absolute [[thermodynamic temperature]] of the system at the point of the heat flow. If there are mass flows across the system boundaries, they also influence the total entropy of the system. This account, in terms of heat and work, is valid only for cases in which the work and heat transfers are by paths physically distinct from the paths of entry and exit of matter from the system.<ref>{{cite book|author=Late Nobel Laureate Max Born|title=Natural Philosophy of Cause and Chance|url=https://books.google.com/books?id=er85jgEACAAJ|date=8 August 2015|publisher=BiblioLife|isbn=978-1-298-49740-6|pages=44, 146–147}}</ref><ref>{{cite book|last1=Haase|first1=R.|title=Thermodynamics|date=1971|publisher=Academic Press|location=New York|isbn=978-0-12-245601-5|pages=1–97}}</ref>

To derive a generalised entropy balanced equation, we start with the general balance equation for the change in any [[extensive quantity]] <math display="inline">\theta</math> in a [[thermodynamic system]], a quantity that may be either conserved, such as energy, or non-conserved, such as entropy. The basic generic balance expression states that <math display="inline">\mathrm{d} \theta / \mathrm{d} t</math>, i.e. the rate of change of <math display="inline">\theta</math> in the system, equals the rate at which <math display="inline">\theta</math> enters the system at the boundaries, minus the rate at which <math display="inline">\theta</math> leaves the system across the system boundaries, plus the rate at which <math display="inline">\theta</math> is generated within the system. For an open thermodynamic system in which heat and work are transferred by paths separate from the paths for transfer of matter, using this generic balance equation, with respect to the rate of change with time <math display="inline">t</math> of the extensive quantity entropy <math display="inline">S</math>, the entropy balance equation is:<ref name="Pokrovskii 2020">{{Cite book|url=|title= Thermodynamics of Complex Systems: Principles and applications. |last= Pokrovskii |first=Vladimir|language=English | publisher= IOP Publishing, Bristol, UK.|year=2020|isbn=|pages=|bibcode= 2020tcsp.book.....P }}</ref><ref>{{Cite book|last=Sandler|first=Stanley, I.|title=Chemical and Engineering Thermodynamics|publisher=John Wiley & Sons|year=1989|isbn=978-0-471-83050-4}}</ref><ref group="note" name=overdot>The overdots represent derivatives of the quantities with respect to time.</ref><math display="block">\frac{\mathrm{d} S}{\mathrm{d} t} = \sum_{k=1}^K{\dot{M}_k \hat{S}_k  + \frac{\dot{Q}}{T} + \dot{S}_\mathsf{gen}}</math>where <math display="inline">\sum_{k=1}^K{\dot{M}_k \hat{S}_k}</math> is the net rate of entropy flow due to the flows of mass <math display="inline">\dot{M}_k </math> into and out of the system with entropy per unit mass <math display="inline">\hat{S}_k</math>, <math display="inline">\dot{Q} / T</math> is the rate of entropy flow due to the flow of heat across the system boundary and <math display="inline">\dot{S}_\mathsf{gen}</math> is the rate of [[entropy production|entropy generation]] within the system, e.g. by [[chemical reaction]]s, [[phase transition]]s, internal heat transfer or [[Friction|frictional effects]] such as [[viscosity]].

In case of multiple heat flows the term <math display="inline">\dot{Q}/T</math> is replaced by <math display="inline">\sum_j{\dot{Q}_j/T_j}</math>, where <math display="inline">\dot{Q}_j</math> is the heat flow through <math display="inline">j</math>-th port into the system and <math display="inline">T_j</math> is the temperature at the <math display="inline">j</math>-th port.

The nomenclature "entropy balance" is misleading and often deemed inappropriate because entropy is not a conserved quantity. In other words, the term <math display="inline">\dot{S}_\mathsf{gen}</math> is never a known quantity but always a derived one based on the expression above. Therefore, the open system version of the second law is more appropriately described as the "entropy generation equation" since it specifies that:<math display="block">\dot{S}_\mathsf{gen} \ge 0</math>with zero for reversible process and positive values for irreversible one.