Thermodynamic processes - Revision history

Nian at 13:30, 30 March 2026

2026-03-30T13:30:26Z

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Nian at 09:53, 30 March 2026

2026-03-30T09:53:33Z

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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	ds=C_v\dfrac{dT}{T}+R\dfrac{d\nu}{\nu}		ds=C_v\dfrac{dT}{T}+R\dfrac{d\nu}{\nu}

Nian at 09:49, 30 March 2026

2026-03-30T09:49:14Z

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Nian at 09:44, 30 March 2026

2026-03-30T09:44:10Z

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	{{NumEqn\|<math>		{{NumEqn\|<math>
	ds=C_v\dfrac{dT}{T}+R\dfrac{d\nu}{\nu}		ds=C_v\dfrac{dT}{T}+R\dfrac{d\nu}{\nu}
	</math>}}		</math>\|label=eq_process_ds_a}}

	{{NumEqn\|<math>		{{NumEqn\|<math>
	d\nu=\dfrac{\nu}{R}ds-C_v\dfrac{\nu}{RT}dT=\dfrac{\nu}{R}ds-\dfrac{C_v}{p}dT		d\nu=\dfrac{\nu}{R}ds-C_v\dfrac{\nu}{RT}dT=\dfrac{\nu}{R}ds-\dfrac{C_v}{p}dT
	</math>}}		</math>\|label=eq_process_dnu}}

	for an isentropic process (<math>ds=0</math>), <math>d\nu < 0</math> for positive values of <math>dT</math>.		for an isentropic process (<math>ds=0</math>), <math>d\nu < 0</math> for positive values of <math>dT</math>.

	<math ~~display="block"~~>		{{NumEqn\|<math>
	ds=C_p\dfrac{dT}{T} - R \dfrac{dp}{p}		ds=C_p\dfrac{dT}{T} - R \dfrac{dp}{p}
	</math>		</math>\|label=eq_process_ds_b}}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	dp=-\dfrac{p}{R}ds+C_p\dfrac{p}{RT}dT=-\dfrac{p}{R}ds+C_p\rho dT		dp=-\dfrac{p}{R}ds+C_p\dfrac{p}{RT}dT=-\dfrac{p}{R}ds+C_p\rho dT
	</math>		</math>\|label=eq_process_dp}}

	for an isentropic process (<math>ds=0</math>), <math>dp > 0</math> for positive values of <math>dT</math>.		for an isentropic process (<math>ds=0</math>), <math>dp > 0</math> for positive values of <math>dT</math>.
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	Since <math>\nu</math> decreases with temperature and pressure increases with temperature for an isentropic process, we can see from ~~Eqn.~\ref~~{~~eqn:process:dnu~~} that <math>d\nu</math> will be greater at lower temperatures and thus isochores (lines of constant specific volume) will be closely spaced at low temperatures and more sparse at higher temperatures. For an isochore <math>dv=0</math> which implies		Since <math>\nu</math> decreases with temperature and pressure increases with temperature for an isentropic process, we can see from {{EquationNote\|label=eq_process_dnu}} that <math>d\nu</math> will be greater at lower temperatures and thus isochores (lines of constant specific volume) will be closely spaced at low temperatures and more sparse at higher temperatures. For an isochore <math>dv=0</math> which implies

	<math ~~display="block"~~>		{{NumEqn\|<math>
	0=\dfrac{\nu}{R}\left(ds-C_v\dfrac{dT}{T}\right) \Rightarrow \dfrac{dT}{ds}=\dfrac{T}{C_v}		0=\dfrac{\nu}{R}\left(ds-C_v\dfrac{dT}{T}\right) \Rightarrow \dfrac{dT}{ds}=\dfrac{T}{C_v}
	</math>		</math>}}

	and thus we can see that the slope of an isochore in a <math>T-s</math>-diagram is positive and that the slope increases with temperature.		and thus we can see that the slope of an isochore in a <math>T-s</math>-diagram is positive and that the slope increases with temperature.
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	In analogy, we can see that an isobar (<math>dp=0</math>) leads to the following relation		In analogy, we can see that an isobar (<math>dp=0</math>) leads to the following relation

	<math ~~display="block"~~>		{{NumEqn\|<math>
	0=\dfrac{p}{R}\left(C_p\dfrac{dT}{T}-ds\right) \Rightarrow \dfrac{dT}{ds}=\dfrac{T}{C_p}		0=\dfrac{p}{R}\left(C_p\dfrac{dT}{T}-ds\right) \Rightarrow \dfrac{dT}{ds}=\dfrac{T}{C_p}
	</math>		</math>}}

	and consequently isobars will also have a positive slope that increases with temperature in a <math>T-s</math>-diagram. Moreover, isobars are less steep than ischores as <math>C_p > C_v</math>.		and consequently isobars will also have a positive slope that increases with temperature in a <math>T-s</math>-diagram. Moreover, isobars are less steep than ischores as <math>C_p > C_v</math>.

Nian at 08:36, 30 March 2026

2026-03-30T08:36:44Z

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	<math ~~display="block"~~>		{{#vardefine:secno\|1}}
			{{#vardefine:eqno\|7}}

			{{NumEqn\|<math>
	ds=C_v\dfrac{dT}{T}+R\dfrac{d\nu}{\nu}		ds=C_v\dfrac{dT}{T}+R\dfrac{d\nu}{\nu}
	</math>		</math>}}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	d\nu=\dfrac{\nu}{R}ds-C_v\dfrac{\nu}{RT}dT=\dfrac{\nu}{R}ds-\dfrac{C_v}{p}dT		d\nu=\dfrac{\nu}{R}ds-C_v\dfrac{\nu}{RT}dT=\dfrac{\nu}{R}ds-\dfrac{C_v}{p}dT
	</math>		</math>}}

	for an isentropic process (<math>ds=0</math>), <math>d\nu < 0</math> for positive values of <math>dT</math>.		for an isentropic process (<math>ds=0</math>), <math>d\nu < 0</math> for positive values of <math>dT</math>.

Nian at 19:17, 20 March 2026

2026-03-20T19:17:12Z

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			[[Category:Compressible flow]]
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	~~[[Category:Compressible flow]]~~
	[[Category:Compressible flow:Topic]]		[[Category:Compressible flow:Topic]]
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Nian at 19:15, 20 March 2026

2026-03-20T19:15:29Z

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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
	[[Category:Compressible flow:Topic]]		[[Category:Compressible flow:Topic]]
	[[Category:Thermodynamics]]		[[Category:Thermodynamics]]
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Nian at 19:02, 20 March 2026

2026-03-20T19:02:41Z

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Nian at 18:45, 20 March 2026

2026-03-20T18:45:08Z

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Nian at 15:45, 20 March 2026

2026-03-20T15:45:33Z

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