Editing PGE/FirstLawOfThermodynamics (section)

=First Law of Thermodynamics=
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<font size="-1">[[H_BookTiledMenu#Context|<b>1<sup>st</sup> Law of<br />Thermodynamics</b>]]</font>
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==Standard Presentation==
Following the detailed discussion of the laws of thermodynamics that can be found, for example, in Chapter I of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] we know that, for an infinitesimal quasi-statical change of state, the change <math>dQ</math> in the total heat content <math>Q</math> of a fluid element is given by the,
&nbsp;<br />
<div align="center">
&nbsp;<br />
&nbsp;<br />
<span id="FundamentalLaw"><font color="#770000">'''Fundamental Law of Thermodynamics'''</font></span>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>dQ</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
d\epsilon + PdV \, ,
</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (2)<br />
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.2, Eq. (1.2)<br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;4.1, Eq. (4.1)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.2, Eq. (1.10)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;1.6.5, Eq. (1.124)
</div>
where, {{ Template:Math/VAR_SpecificInternalEnergy01 }} is the specific internal energy, {{ Template:Math/VAR_Pressure01 }} is the pressure, and {{ Template:Math/VAR_SpecificVolume01 }} <math>= 1/</math>{{ Template:Math/VAR_Density01 }} is the specific volume of the fluid element.  Generally, the change in the total heat content can be rewritten in terms of the gas temperature, {{ Template:Math/VAR_Temperature01 }}, and the specific entropy of the fluid, {{ Template:Math/VAR_SpecificEntropy01 }}, via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dQ</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~T ds \, .</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter I, Eq. (76) &amp; Chapter II, Eq. (44)<br />
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.4, p. 16<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.2, Eq. (1.10)
</div>

<span id="VariableDimensions"><table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
<div align="center">'''Variable Dimensions'''</div>
Each of the terms in these two expressions has units of specific energy, that is, energy per unit mass.  Specifically, these are the units for the two variables, <math>Q</math> and {{ Template:Math/VAR_SpecificInternalEnergy01 }}, while the product of {{ Template:Math/VAR_Pressure01 }} (energy per unit volume) and {{ Template:Math/VAR_SpecificVolume01 }} (volume per unit mass) gives specific energy.  It should be clear as well that {{ Template:Math/VAR_SpecificEntropy01 }} has units of specific energy per Kelvin; given that {{ Template:Math/VAR_SpecificEntropy01 }} is usually referred to in the literature as "specific entropy," we conclude that entropy, itself, has units of energy per Kelvin.
</td></tr></table></span>


If, in addition, it is understood that the specified changes are occurring over an interval of time d{{ Template:Math/VAR_Time01 }}, then from this pair of expressions we derive what will henceforth be referred to as the,

<div align="center">
<span id="Standard Form"><font color="#770000">'''Standard Form'''</font></span><br />
of the First Law of Thermodyamics

{{ Template:Math/EQ_FirstLaw01 }}

[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, Eq. (64)<br />
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Chapter 4, Eq. (4.27)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;7.3.3, Eq. (7.162)
</div>

If the state changes occur in such a way that no heat seeps into or leaks out of the fluid element, then <math>ds/dt = 0</math> and the changes are said to have been made ''adiabatically.''  For an adiabatically evolving system, therefore, the ''First Law'' assumes what henceforth will be referred to as the,

<div align="center">
<span id="Standard Form"><font color="#770000">'''Adiabatic Form'''</font></span><br />
of the First Law of Thermodyamics

{{ Template:Math/EQ_FirstLaw02 }}

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (13)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, Eq. (70)
</div>
Clearly this form of the ''First Law'' also may be viewed as a statement of ''specific entropy conservation.''

==Entropy Tracer==

===Initial Recognition===
Multiplying the ''Adiabatic Form of the First Law of Thermodynamics'' through by {{ Template:Math/VAR_Density01 }} and rearranging terms, we find that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d(\rho\epsilon)}{dt} - \epsilon \frac{d\rho}{dt} -  \frac{P}{\rho} \frac{d\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon) \frac{1}{\rho}\frac{d\rho}{dt} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon)\frac{d\ln\rho}{dt} \, ,
</math>
  </td>
</tr>
</table>
is an equally valid statement of the conservation of specific entropy in an adiabatic flow.  In combination, first, with
 
<div align="center">
<span id="IdealGasB"><font color="#770000">'''Form B'''</font></span><br />
of the Ideal Gas Equation

{{ Template:Math/EQ_EOSideal02 }}
</div>

and, second, with the
<div align="center">
<span id="Continuity"><font color="#770000">'''Lagrangian Form'''</font></span><br />
of the Continuity Equation

{{ Template:Math/EQ_Continuity01 }}
</div>

we may furthermore rewrite this expression as,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d(\rho\epsilon)}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\gamma_g (\rho\epsilon)\frac{d\ln\rho}{dt} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{1}{\gamma_g} \frac{d\ln(\rho\epsilon)}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d\ln\rho}{dt} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \nabla\cdot\vec{v} \, . 
</math>
  </td>
</tr>
</table>

This relation has the classic form of a conservation law.  It certifies that, within the context of adiabatic flows, the ''entropy tracer,''

<div align="center">
<math>\tau \equiv (\rho\epsilon)^{1/\gamma_g} = \biggl[ \frac{P}{(\gamma_g - 1)} \biggr]^{1/\gamma_g} \, ,</math>
</div>

is the volume density of a conserved quantity.  In this case, that conserved quantity is the entropy of each fluid element.

===Substantiation===
To further substantiate this claim, we note that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\tau}{\rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\epsilon^{1/\gamma_g} \cdot \rho^{1/\gamma_g - 1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \ln\biggl(\frac{\tau}{\rho}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\gamma_g} \biggl[ \ln\epsilon - ( \gamma_g-1)\ln\rho \biggr] \, .
</math>
  </td>
</tr>
</table>

Now, from the first law, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>ds</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{T} \biggl[
d\epsilon - \frac{P}{\rho} {d\ln\rho}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_V~ d\ln\epsilon - \frac{\Re}{\mu} ~{d\ln\rho}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \frac{ds}{c_P}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{c_V}{c_P}~ d\ln\epsilon - \frac{\Re/\mu}{c_P} ~{d\ln\rho}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\gamma_g} \biggl[  d\ln\epsilon - (\gamma_g-1){d\ln\rho} \biggr] \, ,
</math>
  </td>
</tr>
</table>
which, upon integration, gives,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{s}{c_P}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\gamma_g} \biggl[  \ln\epsilon - (\gamma_g-1)\ln\rho \biggr] + \mathrm{constant} \, .
</math>
  </td>
</tr>
</table>

To within an additive constant, the right-hand side of this relation is precisely the expression for the logarithm of the entropy tracer, as provided immediately above.  Hence, we see that,
<div align="center">
<math>s = c_P \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant}  \, ,</math>
</div>
that is, we see that the variable, <span title="Entropy tracer"><math>\tau</math></span>, traces the fluid entropy just as {{ Template:Math/VAR_Density01 }} traces the fluid mass.

<span id="EntropyLL75">We have found</span> one other instance in the literature &#8212; although there are undoubtedly others &#8212; where the role of this ''entropy tracer'' previously has been identified.  In chapter IX of [<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>] we find that, "apart from an unimportant additive constant," the specific entropy is,
<div align="center" id="LL75">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>s</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_P \ln \biggl(\frac{P^{1/\gamma_g}}{\rho} \biggr) \, .</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], &sect;80, Eq. (80.12)
</div>

Given that <math>\tau \propto P^{1/\gamma_g}</math>, this is clearly the same expression as we have derived for the specific entropy of the fluid.

===Incorporation Into the First Law===
Multiplying the ''Standard Form of the First Law of Thermodynamics'' through by {{ Template:Math/VAR_Density01 }}, we can now write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho T ~\frac{ds}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d(\rho\epsilon)}{dt} - \gamma_g (\rho\epsilon) ~\frac{d\ln\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{\rho T}{\gamma_g(\rho\epsilon)} ~\frac{ds}{dt}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt} - \frac{d\ln\rho}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ 
\frac{1}{c_P} ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d\ln(\tau/\rho)}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ 
\frac{1}{c_P}\biggl( \frac{\tau}{\rho} \biggr) ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d(\tau/\rho)}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\rho} \biggl[ \frac{d\tau}{dt} - \frac{\tau}{\rho}\frac{d\rho}{dt}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\rho} \biggl[ \frac{d\tau}{dt} + \tau \nabla\cdot\vec{v} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ 
\biggl( \frac{\tau}{c_P} \biggr) ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) 
</math>
  </td>
</tr>
<tr>
<td align="center" colspan="3">
{{ MTF2002 }}, &sect;4.1, Eq. (33)
</td>
</tr>
</table>

Now,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>c_P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_V \gamma_g</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{c_V}{\rho\epsilon}\biggr) \gamma_g \tau^{\gamma_g}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{1}{\rho T}\biggr) \gamma_g \tau^{\gamma_g} \, .</math>
  </td>
</tr>
</table>
</div>

Hence,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{\gamma_g \tau^{\gamma_g-1}} \biggr) ~\rho T~\frac{ds}{dt} \, .
</math>
  </td>
</tr>
</table>
{{ MT2012 }}, &sect;2.2, Eq. (31)
</div>

Notice, as well, that we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> 
\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_P~
\frac{d\ln(\tau/\rho)}{dt}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_V \biggl[
\frac{d\ln(\tau/\rho)^\gamma}{dt}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_V \frac{d}{dt}\biggl[
\ln\biggl( \frac{\rho\epsilon}{\rho^{\gamma_g}}\biggr)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\rho T ~\frac{ds}{dt} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho\epsilon ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{\rho\epsilon}{\rho^{\gamma_g}}\biggr)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr)
\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho T ~\frac{ds}{dt} \, .
</math>
  </td>
</tr>
</table>

Specifically for the case, <math>\gamma_g = \tfrac{5}{3}</math>, this gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> 
\frac{3}{2} ~P ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{P}{\rho^{5/3}}\biggr)
\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho T ~\frac{ds}{dt} \, .
</math>
  </td>
</tr>
</table>

[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Chapter 9, Eq. (9.26)
</div>
It is fair to say, therefore, that in this specific case [<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>] also recognized the relevance of and the conservative nature of, what we have referred to as, the ''entropy tracer.''