PGE/FirstLawOfThermodynamics - Revision history

Jet53man: /* Example C */

2021-10-07T15:40:48Z

Example C

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	where, <math>\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math> is the Stefan-Boltzmann constant. In such a case, the right-hand-side of the equation describing the corresponding time-rate-of-change of the entropy of the ''radiation'' field, <math>s_\mathrm{rad}</math>, would necessarily contain the same two terms, but in both cases with ''opposite signs.'' That is, the entropy of the radiation field sees <math>\Lambda</math> as a ''source'' while it sees <math>\Gamma</math> as a "sink."		where, <math>\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math> is the Stefan-Boltzmann constant. In such a case, the right-hand-side of the equation describing the corresponding time-rate-of-change of the entropy of the ''radiation'' field, <math>s_\mathrm{rad}</math>, would necessarily contain the same two terms, but in both cases with ''opposite signs.'' That is, the entropy of the radiation field sees <math>\Lambda</math> as a ''source'' while it sees <math>\Gamma</math> as a "sink."

	[In addition to <math>~\Gamma</math> and <math>~\Lambda</math>, other terms involving spatial variations in the velocity field and in the radiation energy density also appear on the right-hand-side of the expression for <math>~ds_\mathrm{rad}/dt</math>. For simplicity, and because these other terms are not relevant to the principal point we are making, we have opted not to detail the entire expression for <math>~ds_\mathrm{rad}/dt</math> here. The additional terms and details can be found in, for example, ~~[http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006)]~~ or {{ MT2012 }}.]		[In addition to <math>~\Gamma</math> and <math>~\Lambda</math>, other terms involving spatial variations in the velocity field and in the radiation energy density also appear on the right-hand-side of the expression for <math>~ds_\mathrm{rad}/dt</math>. For simplicity, and because these other terms are not relevant to the principal point we are making, we have opted not to detail the entire expression for <math>~ds_\mathrm{rad}/dt</math> here. The additional terms and details can be found in, for example, {{ ZEUS-MP2006 }} or {{ MT2012 }}.]

	====First Elaboration====		====First Elaboration====

Jet53man: /* Example C */

2021-10-07T00:04:58Z

Example C

← Older revision		Revision as of 20:04, 6 October 2021
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	where, <math>\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math> is the Stefan-Boltzmann constant. In such a case, the right-hand-side of the equation describing the corresponding time-rate-of-change of the entropy of the ''radiation'' field, <math>s_\mathrm{rad}</math>, would necessarily contain the same two terms, but in both cases with ''opposite signs.'' That is, the entropy of the radiation field sees <math>\Lambda</math> as a ''source'' while it sees <math>\Gamma</math> as a "sink."		where, <math>\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math> is the Stefan-Boltzmann constant. In such a case, the right-hand-side of the equation describing the corresponding time-rate-of-change of the entropy of the ''radiation'' field, <math>s_\mathrm{rad}</math>, would necessarily contain the same two terms, but in both cases with ''opposite signs.'' That is, the entropy of the radiation field sees <math>\Lambda</math> as a ''source'' while it sees <math>\Gamma</math> as a "sink."

	[In addition to <math>~\Gamma</math> and <math>~\Lambda</math>, other terms involving spatial variations in the velocity field and in the radiation energy density also appear on the right-hand-side of the expression for <math>~ds_\mathrm{rad}/dt</math>. For simplicity, and because these other terms are not relevant to the principal point we are making, we have opted not to detail the entire expression for <math>~ds_\mathrm{rad}/dt</math> here. The additional terms and details can be found in, for example, [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006)] or ~~[http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello & J. E. Tohline (2012)]~~.]		[In addition to <math>~\Gamma</math> and <math>~\Lambda</math>, other terms involving spatial variations in the velocity field and in the radiation energy density also appear on the right-hand-side of the expression for <math>~ds_\mathrm{rad}/dt</math>. For simplicity, and because these other terms are not relevant to the principal point we are making, we have opted not to detail the entire expression for <math>~ds_\mathrm{rad}/dt</math> here. The additional terms and details can be found in, for example, [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006)] or {{ MT2012 }}.]

	====First Elaboration====		====First Elaboration====

Jet53man: /* Incorporation Into the First Law */

2021-10-07T00:03:25Z

Incorporation Into the First Law

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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	~~[https://ui.adsabs.harvard.edu/abs/2002ApJS..138..121M/abstract Motl et al. (2002)]~~ §4.1, Eq. (33)		{{ MTF2002 }}, §4.1, Eq. (33)
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Incorporation Into the First Law */

2021-10-07T00:02:32Z

Incorporation Into the First Law

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	</tr>		</tr>
	</table>		</table>
	~~[http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello & Tohline (2012)]~~, §2.2, Eq. (31)		{{ MT2012 }}, §2.2, Eq. (31)
	</div>		</div>

Jet53man: /* Left-Hand Side (LHS) */

2021-10-06T23:41:24Z

Left-Hand Side (LHS)

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	</table>		</table>

	In [[~~User:Tohline/~~Appendix/Ramblings/~~RadiationHydro~~#Optically_Thick_Regime\|optically thick environments]] where the radiation field is intermixed and in equilibrium with the fluid (gas), the time-rate-of-change in the entropy of the radiation field is characterized by the expression,		In [[Appendix/Ramblings/Radiation/RadHydro#Optically_Thick_Regime\|optically thick environments]] where the radiation field is intermixed and in equilibrium with the fluid (gas), the time-rate-of-change in the entropy of the radiation field is characterized by the expression,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* First Law of Thermodynamics */

2021-07-28T03:19:43Z

First Law of Thermodynamics

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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=First Law of Thermodynamics=		=First Law of Thermodynamics=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#Context\|<b>1<sup>st</sup> Law of<br />Thermodynamics</b>]]</font>
			\|}

	==Standard Presentation==		==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,		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,
			<br />
	<div align="center">		<div align="center">
			<br />
			<br />
	<span id="FundamentalLaw"><font color="#770000">'''Fundamental Law of Thermodynamics'''</font></span>		<span id="FundamentalLaw"><font color="#770000">'''Fundamental Law of Thermodynamics'''</font></span>

Jet53man: /* Standard Presentation */

2021-07-11T17:38:39Z

Standard Presentation

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	<span id="VariableDimensions"><table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">		<span id="VariableDimensions"><table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
	<div align="center">'''Variable Dimensions'''</div>		<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 ~~entropy,~~ {{ Template:Math/VAR_SpecificEntropy01 }}, has units of ~~specific~~ energy per Kelvin.		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>		</td></tr></table></span>

Jet53man: /* Standard Presentation */

2021-07-11T01:57:30Z

Standard Presentation

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	</div>		</div>

	<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">		<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 entropy, {{ Template:Math/VAR_SpecificEntropy01 }}, has units of specific energy per Kelvin.		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 entropy, {{ Template:Math/VAR_SpecificEntropy01 }}, has units of specific energy per Kelvin.
	</td></tr></table>		</td></tr></table></span>

Jet53man: /* Standard Presentation */

2021-07-11T01:55:01Z

Standard Presentation

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	[<b>[[Appendix/References#HK94\|<font color="red">HK94</font>]]</b>], §1.2, Eq. (1.10)		[<b>[[Appendix/References#HK94\|<font color="red">HK94</font>]]</b>], §1.2, Eq. (1.10)
	</div>		</div>

			<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">
			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 entropy, {{ Template:Math/VAR_SpecificEntropy01 }}, has units of specific energy per Kelvin.
			</td></tr></table>

174.64.8.247: /* Left-Hand Side (LHS) */

2021-07-05T17:14:59Z

Left-Hand Side (LHS)

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	</td>		</td>
	</table>		</table>
			In a [[Apps/SMS#Entropy_of_Radiation_Field\|separate discussion]] we identify other references where this expression for <math>s_\mathrm{rad}</math> can be found.

	==Right-Hand Side (RHS)==		==Right-Hand Side (RHS)==