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

DOFPro Team

Introduction

The Second Law of Thermodynamics or entropy balances

  • closed systems
  • open transient systems
  • open steady state systems

Mathematical Statement of the Second Law

\(dW_{\text{lost}} \geq 0\)


or

\(\Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \geq 0\)


Remember:

\(\Delta S_{\text{system}} \text{ or } \Delta S_{\text{surroundings}}\)


can be less than zero. But


\(\Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \geq 0\)

Entropy Balance for Closed System


\(\Delta S_{\text{system}} + \Delta S_{\text{surroundings}} = S_G \geq 0\)


Defining heat transferred from the surroundings into the system as positive, i.e., using the same convention for entropy as for heat with regard to the system, then


\(\Delta S_{\text{surroundings}} = - \sum\limits_j \dfrac{Q_j}{T_{\sigma j}}\)


\(\boxed{\Delta S_{\text{system}} - \sum\limits_j \dfrac{Q_j}{T_{\sigma j}} = S_G \geq 0}\)


\(Q_j\) is heat transferred into system from reservoir \(j\).


\(T_{\sigma j}\) is temperature of reservoir \(j\).

Entropy Balance for Open Transient System


\(\dfrac{d(m \hat{S})_\mathrm{cv}}{dt} + \Delta (\dot{m} \hat{S})_{\text{fs}} + \dfrac{d S_{\text{surr}}}{dt} = \dot{S}_G \geq 0\)


or \(\ \ \dfrac{d(n \hat{S})_\mathrm{cv}}{dt} + \Delta (\dot{n} \hat{S})_{\text{fs}} + \dfrac{d S_{\text{surr}}}{dt} = \dot{S}_G \geq 0\)


Defining heat transferred from the surroundings into the system as positive, i.e., using the same convention for entropy as for heat with regard to the system, then


\(\dfrac{d S_{\mathrm{surr}}}{dt} = - \sum\limits_j \dfrac{\dot{Q}_j}{T_{\sigma j}}\)

Entropy Balance for Open Transient System (cont.)


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


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


\(\dot{Q}_j\) is heat transferred into system from reservoir \(j\).


\(T_{\sigma j}\) is temperature of reservoir \(j\).

Entropy Balance for Open Steady-State System


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


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


\(\dot{Q}_j\) is heat transferred into system from reservoir \(j\).


\(T_{\sigma j}\) is temperature of reservoir \(j\).


The ΔS-sentials of Calculating Entropy Changes video explained the methods we have available for calculating changes in entropy.

The Takeaways

  1. The Second Law of Thermodynamics for a closed system is \(\Delta S_{\text{system}} - \sum\limits_j \dfrac{Q_j}{T_{\sigma j}} = S_G \geq 0\).
  2. The Second Law of Thermodynamics for a open transient system is \(\dfrac{d(m \hat{S})_\mathrm{cv}}{dt} + \Delta (\dot{m} \hat{S})_{\text{fs}} - \sum\limits_j \dfrac{\dot{Q}_j}{T_{\sigma j}} = \dot{S}_G \geq 0\).
  3. The Second Law of Thermodynamics for a open steady-state system is \(\Delta (\dot{m} \hat{S})_{\text{fs}} - \sum\limits_j \dfrac{\dot{Q}_j}{T_{\sigma j}} = \dot{S}_G \geq 0\).






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The DOFPro Team