Revision as of 21:19, 27 October 2018

First Law of Thermodynamics

Standard Presentation

Following the detailed discussion of the laws of thermodynamics that can be found, for example, in Chapter I of [C67] 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,

Fundamental Law of Thermodynamics

<math>~ d\epsilon + PdV \, , </math>

[C67], Chapter II, Eq. (2)
[H87], §1.2, Eq. (1.2)
[KW94], §4.1, Eq. (4.1)
[HK94], §1.2, Eq. (1.10)
[BLRY07], §1.6.5, Eq. (1.124)

where, <math>~\epsilon</math> is the specific internal energy, <math>~P</math> is the pressure, and <math>~V</math><math>~= 1/</math><math>~\rho</math> 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, <math>~T</math>, and the specific entropy of the fluid, <math>~s</math>, via the expression,

[C67], Chapter I, Eq. (76) & Chapter II, Eq. (44)
[H87], §1.4, p. 16
[HK94], §1.2, Eq. (1.10)

If, in addition, it is understood that the specified changes are occurring over an interval of time <math>~dt</math>, then from this pair of expressions we derive what will henceforth be referred to as the,

Standard Form
of the First Law of Thermodyamics

<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math>

[T78], §3.4, Eq. (64)
[Shu92], Chapter 4, Eq. (4.27)
[HK94], §7.3.3, Eq. (7.162)

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,

Adiabatic Form
of the First Law of Thermodyamics

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math>

[C67], Chapter II, Eq. (13)
[T78], §3.4, Eq. (70)

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 <math>~\rho</math> and rearranging terms, we find that,

<math>~0</math>	<math>~=</math>	<math>~ \rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr) </math>
	<math>~=</math>	<math>~ \frac{d(\rho\epsilon)}{dt} - \epsilon \frac{d\rho}{dt} - \frac{P}{\rho} \frac{d\rho}{dt} </math>
	<math>~=</math>	<math>~ \frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon) \frac{1}{\rho}\frac{d\rho}{dt} </math>
	<math>~=</math>	<math>~ \frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon)\frac{d\ln\rho}{dt} \, , </math>

is an equally valid statement of the conservation of specific entropy in an adiabatic flow. In combination, first, with

Form B
of the Ideal Gas Equation

<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>

and, second, with the

Lagrangian Form
of the Continuity Equation

<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>

we may furthermore rewrite this expression as,

<math>~\frac{d(\rho\epsilon)}{dt}</math>	<math>~=</math>	<math>~ \gamma_g (\rho\epsilon)\frac{d\ln\rho}{dt} </math>
<math>~\Rightarrow ~~~ \frac{1}{\gamma_g} \frac{d\ln(\rho\epsilon)}{dt}</math>	<math>~=</math>	<math>~ \frac{d\ln\rho}{dt} </math>
<math>~\Rightarrow ~~~ \frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt}</math>	<math>~=</math>	<math>~ - \nabla\cdot\vec{v} \, . </math>

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

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

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,

<math>~\frac{\tau}{\rho}</math>	<math>~=</math>	<math>~ \epsilon^{1/\gamma_g} \cdot \rho^{1/\gamma_g - 1} </math>
<math>~\Rightarrow ~~~ \ln\biggl(\frac{\tau}{\rho}\biggr)</math>	<math>~=</math>	<math>~ \frac{1}{\gamma_g} \biggl[ \ln\epsilon - ( \gamma_g-1)\ln\rho \biggr] \, . </math>

Now, from the first law, we can write,

<math>~ds</math>	<math>~=</math>	<math>~\frac{1}{T} \biggl[ d\epsilon - \frac{P}{\rho} {d\ln\rho} \biggr] </math>
	<math>~=</math>	<math>~ c_V~ d\ln\epsilon - \frac{\Re}{\mu} ~{d\ln\rho} </math>
<math>~ \Rightarrow ~~~ \frac{ds}{c_P} </math>	<math>~=</math>	<math>~ \frac{c_V}{c_P}~ d\ln\epsilon - \frac{\Re/\mu}{c_P} ~{d\ln\rho} </math>
	<math>~=</math>	<math>~ \frac{1}{\gamma_g} \biggl[ d\ln\epsilon - (\gamma_g-1){d\ln\rho} \biggr] \, , </math>

which, upon integration, gives,

<math>~ \frac{1}{\gamma_g} \biggl[ \ln\epsilon - (\gamma_g-1)\ln\rho \biggr] + \mathrm{constant} \, . </math>

To within an additive constant, this is precisely the expression for the logarithm of the entropy tracer, as provided immediately above. Hence, we see that,

<math>~s = c_P \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant} \, ,</math>

that is, we see that the variable, <math>~\tau</math>, traces the fluid entropy just as <math>~\rho</math> traces the fluid mass.

We have found one other instance in the literature — although there are undoubtedly others — where the role of this entropy tracer previously has been identified. In chapter IX of [LL75] we find that, "apart from an unimportant additive constant," the specific entropy is,

<math>~c_P \ln \biggl(\frac{P^{1/\gamma_g}}{\rho} \biggr) \, .</math>

[LL75], §80, Eq. (80.12)

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 <math>~\rho</math>, we can now write,

<math>~\rho T ~\frac{ds}{dt}</math>	<math>~=</math>	<math>~ \frac{d(\rho\epsilon)}{dt} - \gamma_g (\rho\epsilon) ~\frac{d\ln\rho}{dt} </math>
<math>~\Rightarrow~~~ \frac{\rho T}{\gamma_g(\rho\epsilon)} ~\frac{ds}{dt}</math>	<math>~=</math>	<math>~ \frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt} - \frac{d\ln\rho}{dt} </math>
<math>~\Rightarrow~~~ \frac{1}{c_P} ~\frac{ds}{dt} </math>	<math>~=</math>	<math>~ \frac{d\ln(\tau/\rho)}{dt} </math>
<math>~\Rightarrow~~~ \frac{1}{c_P}\biggl( \frac{\tau}{\rho} \biggr) ~\frac{ds}{dt} </math>	<math>~=</math>	<math>~ \frac{d(\tau/\rho)}{dt} </math>
	<math>~=</math>	<math>~ \frac{1}{\rho} \biggl[ \frac{d\tau}{dt} - \frac{\tau}{\rho}\frac{d\rho}{dt}\biggr] </math>
	<math>~=</math>	<math>~ \frac{1}{\rho} \biggl[ \frac{d\tau}{dt} + \tau \nabla\cdot\vec{v} \biggr] </math>
<math>~\Rightarrow~~~ \biggl( \frac{\tau}{c_P} \biggr) ~\frac{ds}{dt} </math>	<math>~=</math>	<math>~ \frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) </math>

Now,

<math>~c_P</math>	<math>~=</math>	<math>~c_V \gamma_g</math>
	<math>~=</math>	<math>~\biggl(\frac{c_V}{\rho\epsilon}\biggr) \gamma_g \tau^{\gamma_g}</math>
	<math>~=</math>	<math>~\biggl(\frac{1}{\rho T}\biggr) \gamma_g \tau^{\gamma_g} \, .</math>

Hence,

<math>~ \frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) </math>

<math>~ \biggl( \frac{1}{\gamma_g \tau^{\gamma_g-1}} \biggr) ~\rho T~\frac{ds}{dt} \, . </math>

Marcello & Tohline (2012), §2.2, Eq. (31)

Notice, as well, that we can write,

<math>~ \frac{ds}{dt} </math>	<math>~=</math>	<math>~c_P~ \frac{d\ln(\tau/\rho)}{dt} </math>
	<math>~=</math>	<math>~c_V \biggl[ \frac{d\ln(\tau/\rho)^\gamma}{dt} \biggr] </math>
	<math>~=</math>	<math>~c_V \frac{d}{dt}\biggl[ \ln\biggl( \frac{\rho\epsilon}{\rho^{\gamma_g}}\biggr) \biggr] </math>
<math>~ \Rightarrow ~~~ \rho T ~\frac{ds}{dt} </math>	<math>~=</math>	<math>~\rho\epsilon ~\frac{d}{dt}\biggl[ \ln\biggl( \frac{\rho\epsilon}{\rho^{\gamma_g}}\biggr) \biggr] </math>
<math>~ \Rightarrow ~~~ \frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[ \ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr) \biggr] </math>	<math>~=</math>	<math>~ \rho T ~\frac{ds}{dt} \, . </math>

Specifically for the case, <math>~\gamma_g = \tfrac{5}{3}</math>, this gives,

<math>~ \frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[ \ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr) \biggr] </math>

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