Isentropic relations - Revision history

Nian at 10:24, 1 April 2026

2026-04-01T10:24:24Z

← Older revision		Revision as of 10:24, 1 April 2026
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	[[Category:Compressible flow]]		<!--
	[[Category:Thermodynamics]]		-->[[Category:Compressible flow]]<!--
	<noinclude><!--		-->[[Category:Thermodynamics]]<!--
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	-->[[Category:Compressible flow:Topic]]<!--		-->[[Category:Compressible flow:Topic]]<!--
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Nian at 13:29, 30 March 2026

2026-03-30T13:29:21Z

← Older revision		Revision as of 13:29, 30 March 2026
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			--><noinclude><!--
	-->{{#vardefine:secno\|1}}<!--		-->{{#vardefine:secno\|1}}<!--
	~~--><noinclude><!--~~
	-->{{#vardefine:eqno\|9}}<!--		-->{{#vardefine:eqno\|9}}<!--
	--></noinclude><!--		--></noinclude><!--

Nian at 09:56, 30 March 2026

2026-03-30T09:56:40Z

← Older revision		Revision as of 09:56, 30 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
	[[Category:Thermodynamics]]		[[Category:Thermodynamics]]
	<noinclude>		<noinclude><!--
	[[Category:Compressible flow:Topic]]		-->[[Category:Compressible flow:Topic]]<!--
	</noinclude>		--></noinclude><!--

	<nomobile>		--><nomobile><!--
	__TOC__		__TOC__<!--
	</nomobile>		--></nomobile><!--

			-->{{#vardefine:secno\|1}}<!--
			--><noinclude><!--
			-->{{#vardefine:eqno\|9}}<!--
			--></noinclude><!--
			-->
	=== First law of thermodynamics ===		=== First law of thermodynamics ===

	~~{{#vardefine:secno\|1}}~~
	~~{{#vardefine:eqno\|9}}~~

	First law of thermodynamics:		First law of thermodynamics:

Nian at 09:48, 30 March 2026

2026-03-30T09:48:46Z

← Older revision		Revision as of 09:48, 30 March 2026
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	{{#vardefine:secno\|1}}		{{#vardefine:secno\|1}}
	{{#vardefine:eqno\|7}}		{{#vardefine:eqno\|9}}

	First law of thermodynamics:		First law of thermodynamics:

Nian at 06:41, 30 March 2026

2026-03-30T06:41:01Z

← Older revision		Revision as of 06:41, 30 March 2026
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	</noinclude>		</noinclude>

			<nomobile>
	__TOC__		__TOC__
			</nomobile>

	=== First law of thermodynamics ===		=== First law of thermodynamics ===

	{{#vardefine:secno\|1}}		{{#vardefine:secno\|1}}
	{{#vardefine:eqno\|5}}		{{#vardefine:eqno\|7}}

	First law of thermodynamics:		First law of thermodynamics:

Nian at 19:55, 29 March 2026

2026-03-29T19:55:06Z

← Older revision		Revision as of 19:55, 29 March 2026
Line 20:		Line 20:
	{{NumEqn\|<math>		{{NumEqn\|<math>
	de=Tds-pd\left(\frac{1}{\rho}\right)		de=Tds-pd\left(\frac{1}{\rho}\right)
	</math>\|eq-first-law-b}}		</math>\|label=eq-first-law-b}}

	Enthalpy is defined as: <math>h=e+p/\rho</math> and thus		Enthalpy is defined as: <math>h=e+p/\rho</math> and thus
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp		dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp
	</math>\|eq-dh}}		</math>\|label=eq-dh}}

	Eliminate <math>de</math> in {{EquationNote\|eq-first-law-b}} using {{EquationNote\|eq-dh}}		Eliminate <math>de</math> in {{EquationNote\|label=eq-first-law-b}} using {{EquationNote\|label=eq-dh}}

	{{NumEqn\|<math>		{{NumEqn\|<math>
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	ds=C_p\frac{dT}{T}-R\frac{dp}{p}		ds=C_p\frac{dT}{T}-R\frac{dp}{p}
	</math>\|eq-ds}}		</math>\|label=eq-ds}}

	Integrating {{EquationNote\|eq-ds}} gives		Integrating {{EquationNote\|eq-ds}} gives
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)
	</math>\|eq-ds-b}}		</math>\|label=eq-ds-b}}

	For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus		For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)
	</math>\|eq-ds-c}}		</math>\|label=eq-ds-c}}

	An alternative form of {{EquationNote\|eq-ds-c}} is obtained by using <math>de=C_v dT</math> in {{EquationNote\|eq-first-law-b}}, which gives		An alternative form of {{EquationNote\|label=eq-ds-c}} is obtained by using <math>de=C_v dT</math> in {{EquationNote\|label=eq-first-law-b}}, which gives

	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>\|eq-ds-d}}		</math>\|label=eq-ds-d}}

	Again, for a calorically perfect gas, we get		Again, for a calorically perfect gas, we get
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>\|eq-ds-e}}		</math>\|label=eq-ds-e}}

	=== Isentropic Relations ===		=== Isentropic Relations ===

	Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus {{EquationNote\|eq-ds-c}} reduces to		Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus {{EquationNote\|label=eq-ds-c}} reduces to

	{{NumEqn\|<math>		{{NumEqn\|<math>
Line 86:		Line 86:
	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}		\frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}
	</math>\|eq-isentropic-a}}		</math>\|label=eq-isentropic-a}}

	In the same way, {{EquationNote\|eq-ds-e}} gives		In the same way, {{EquationNote\|label=eq-ds-e}} gives

	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)}		\frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)}
	</math>\|eq-isentropic-b}}		</math>\|label=eq-isentropic-b}}

Nian at 23:20, 28 March 2026

2026-03-28T23:20:48Z

← Older revision		Revision as of 23:20, 28 March 2026
Line 8:		Line 8:

	=== First law of thermodynamics ===		=== First law of thermodynamics ===

			{{#vardefine:secno\|1}}
			{{#vardefine:eqno\|5}}

	First law of thermodynamics:		First law of thermodynamics:
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	ds=C_p\frac{dT}{T}-R\frac{dp}{p}		ds=C_p\frac{dT}{T}-R\frac{dp}{p}
	</math>}}		</math>\|eq-ds}}

	Integrating ~~Eqn. \ref~~{eq:ds} gives		Integrating {{EquationNote\|eq-ds}} gives

	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)
	</math>}}		</math>\|eq-ds-b}}

	For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus		For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)
	</math>}}		</math>\|eq-ds-c}}

	An alternative form of ~~Eqn. \ref~~{eq:ds:c} is obtained by using <math>de=C_v dT</math> ~~Eqn. \ref~~{eq~~:firstlaw:~~b}, which gives		An alternative form of {{EquationNote\|eq-ds-c}} is obtained by using <math>de=C_v dT</math> in {{EquationNote\|eq-first-law-b}}, which gives

	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>}}		</math>\|eq-ds-d}}

	Again, for a calorically perfect gas, we get		Again, for a calorically perfect gas, we get
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>}}		</math>\|eq-ds-e}}

	=== Isentropic Relations ===		=== Isentropic Relations ===

	Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus ~~Eqn. \ref~~{eq:ds:c} reduces to		Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus {{EquationNote\|eq-ds-c}} reduces to

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{C_p}{R}\ln\left(\frac{T_2}{T_1}\right)=\ln\left(\frac{p_2}{p_1}\right)		\frac{C_p}{R}\ln\left(\frac{T_2}{T_1}\right)=\ln\left(\frac{p_2}{p_1}\right)
	</math>		</math>}}

	<math display="block">		<math display="block">
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	</math>		</math>

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}		\frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}
	</math>		</math>\|eq-isentropic-a}}

	In the same way, ~~Eqn. \ref~~{eq:ds:e} gives		In the same way, {{EquationNote\|eq-ds-e}} gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)}		\frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)}
	</math>		</math>\|eq-isentropic-b}}


	Eqn. ~~\ref~~{eq:isentropic:a} and Eqn. ~~\ref~~{eq:isentropic:b} constitutes the isentropic relations		Eqn. {{EquationNote\|eq-isentropic-a}} and Eqn. {{EquationNote\|eq-isentropic-b}} constitutes the isentropic relations

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^{\gamma}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}		\frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^{\gamma}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}
	</math>		</math>}}

Nian at 22:02, 28 March 2026

2026-03-28T22:02:26Z

← Older revision		Revision as of 22:02, 28 March 2026
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	First law of thermodynamics:		First law of thermodynamics:

	{{NumEqn\|<math>de=\delta q - \delta w</math>\|1}}		{{NumEqn\|<math>de=\delta q - \delta w</math>}}

	For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math>		For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math>
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	de=Tds-pd\left(\frac{1}{\rho}\right)		de=Tds-pd\left(\frac{1}{\rho}\right)
	</math>\|eq-first-law-b\|2}}		</math>\|eq-first-law-b}}

	Enthalpy is defined as: <math>h=e+p/\rho</math> and thus		Enthalpy is defined as: <math>h=e+p/\rho</math> and thus
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp		dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp
	</math>\|eq-dh\|3}}		</math>\|eq-dh}}

	Eliminate <math>de</math> in {{EquationNote\|eq-first-law-b}} using {{EquationNote\|eq-dh}}		Eliminate <math>de</math> in {{EquationNote\|eq-first-law-b}} using {{EquationNote\|eq-dh}}
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	Tds=dh-\cancel{pd\left(\frac{1}{\rho}\right)}-\left(\frac{1}{\rho}\right)dp+\cancel{pd\left(\frac{1}{\rho}\right)}		Tds=dh-\cancel{pd\left(\frac{1}{\rho}\right)}-\left(\frac{1}{\rho}\right)dp+\cancel{pd\left(\frac{1}{\rho}\right)}
	</math>\|4}}		</math>}}

	{{NumEqn\|<math>		{{NumEqn\|<math>
	ds=\frac{dh}{T}-\frac{dp}{\rho T}		ds=\frac{dh}{T}-\frac{dp}{\rho T}
	</math>\|5}}		</math>}}

	Using <math>dh=C_p T</math> and the equation of state <math>p=\rho RT</math>, we get		Using <math>dh=C_p T</math> and the equation of state <math>p=\rho RT</math>, we get
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	ds=C_p\frac{dT}{T}-R\frac{dp}{p}		ds=C_p\frac{dT}{T}-R\frac{dp}{p}
	</math>\|6}}		</math>}}

	Integrating Eqn. \ref{eq:ds} gives		Integrating Eqn. \ref{eq:ds} gives
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)
	</math>\|7}}		</math>}}

	For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus		For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)
	</math>\|8}}		</math>}}

	An alternative form of Eqn. \ref{eq:ds:c} is obtained by using <math>de=C_v dT</math> Eqn. \ref{eq:firstlaw:b}, which gives		An alternative form of Eqn. \ref{eq:ds:c} is obtained by using <math>de=C_v dT</math> Eqn. \ref{eq:firstlaw:b}, which gives
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>\|9}}		</math>}}

	Again, for a calorically perfect gas, we get		Again, for a calorically perfect gas, we get
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>~~\|10~~}}		</math>}}

	=== Isentropic Relations ===		=== Isentropic Relations ===

Nian at 18:13, 28 March 2026

2026-03-28T18:13:27Z

← Older revision		Revision as of 18:13, 28 March 2026
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	de=Tds-pd\left(\frac{1}{\rho}\right)		de=Tds-pd\left(\frac{1}{\rho}\right)
	</math>\|2}}		</math>\|eq-first-law-b\|2}}

	Enthalpy is defined as: <math>h=e+p/\rho</math> and thus		Enthalpy is defined as: <math>h=e+p/\rho</math> and thus
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp		dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp
	</math>\|3}}		</math>\|eq-dh\|3}}

	Eliminate $de$ in ~~Eqn. \ref~~{eq~~:firstlaw:~~b} using ~~Eqn. \ref~~{eq:dh}		Eliminate <math>de</math> in {{EquationNote\|eq-first-law-b}} using {{EquationNote\|eq-dh}}

	{{NumEqn\|<math>		{{NumEqn\|<math>

Nian: /* First law of thermodynamics */

2026-03-27T15:49:55Z

First law of thermodynamics

← Older revision		Revision as of 15:49, 27 March 2026
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	For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math>		For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math>

	<math ~~display="block"~~>		{{NumEqn\|<math>
	de=Tds-pd\left(\frac{1}{\rho}\right)		de=Tds-pd\left(\frac{1}{\rho}\right)
	</math>		</math>\|2}}

	Enthalpy is defined as: <math>h=e+p/\rho</math> and thus		Enthalpy is defined as: <math>h=e+p/\rho</math> and thus

	<math ~~display="block"~~>		{{NumEqn\|<math>
	dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp		dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp
	</math>		</math>\|3}}

	Eliminate $de$ in Eqn. \ref{eq:firstlaw:b} using Eqn. \ref{eq:dh}		Eliminate $de$ in Eqn. \ref{eq:firstlaw:b} using Eqn. \ref{eq:dh}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	Tds=dh-\cancel{pd\left(\frac{1}{\rho}\right)}-\left(\frac{1}{\rho}\right)dp+\cancel{pd\left(\frac{1}{\rho}\right)}		Tds=dh-\cancel{pd\left(\frac{1}{\rho}\right)}-\left(\frac{1}{\rho}\right)dp+\cancel{pd\left(\frac{1}{\rho}\right)}
	</math>		</math>\|4}}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	ds=\frac{dh}{T}-\frac{dp}{\rho T}		ds=\frac{dh}{T}-\frac{dp}{\rho T}
	</math>		</math>\|5}}

	Using <math>dh=C_p T</math> and the equation of state <math>p=\rho RT</math>, we get		Using <math>dh=C_p T</math> and the equation of state <math>p=\rho RT</math>, we get

	<math ~~display="block"~~>		{{NumEqn\|<math>
	ds=C_p\frac{dT}{T}-R\frac{dp}{p}		ds=C_p\frac{dT}{T}-R\frac{dp}{p}
	</math>		</math>\|6}}

	Integrating Eqn. \ref{eq:ds} gives		Integrating Eqn. \ref{eq:ds} gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)
	</math>		</math>\|7}}

	For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus		For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus

	<math ~~display="block"~~>		{{NumEqn\|<math>
	s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)
	</math>		</math>\|8}}

	An alternative form of Eqn. \ref{eq:ds:c} is obtained by using <math>de=C_v dT</math> Eqn. \ref{eq:firstlaw:b}, which gives		An alternative form of Eqn. \ref{eq:ds:c} is obtained by using <math>de=C_v dT</math> Eqn. \ref{eq:firstlaw:b}, which gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>		</math>\|9}}

	Again, for a calorically perfect gas, we get		Again, for a calorically perfect gas, we get

	<math ~~display="block"~~>		{{NumEqn\|<math>
	s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>		</math>\|10}}

	=== Isentropic Relations ===		=== Isentropic Relations ===

← Older revision		Revision as of 19:55, 29 March 2026
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	de=Tds-pd\left(\frac{1}{\rho}\right)		de=Tds-pd\left(\frac{1}{\rho}\right)
	</math>\|eq-first-law-b}}		</math>\|label=eq-first-law-b}}

	Enthalpy is defined as: <math>h=e+p/\rho</math> and thus		Enthalpy is defined as: <math>h=e+p/\rho</math> and thus
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp		dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp
	</math>\|eq-dh}}		</math>\|label=eq-dh}}

	Eliminate <math>de</math> in {{EquationNote\|eq-first-law-b}} using {{EquationNote\|eq-dh}}		Eliminate <math>de</math> in {{EquationNote\|label=eq-first-law-b}} using {{EquationNote\|label=eq-dh}}

	{{NumEqn\|<math>		{{NumEqn\|<math>
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	ds=C_p\frac{dT}{T}-R\frac{dp}{p}		ds=C_p\frac{dT}{T}-R\frac{dp}{p}
	</math>\|eq-ds}}		</math>\|label=eq-ds}}

	Integrating {{EquationNote\|eq-ds}} gives		Integrating {{EquationNote\|eq-ds}} gives
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)
	</math>\|eq-ds-b}}		</math>\|label=eq-ds-b}}

	For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus		For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)		s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)
	</math>\|eq-ds-c}}		</math>\|label=eq-ds-c}}

	An alternative form of {{EquationNote\|eq-ds-c}} is obtained by using <math>de=C_v dT</math> in {{EquationNote\|eq-first-law-b}}, which gives		An alternative form of {{EquationNote\|label=eq-ds-c}} is obtained by using <math>de=C_v dT</math> in {{EquationNote\|label=eq-first-law-b}}, which gives

	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>\|eq-ds-d}}		</math>\|label=eq-ds-d}}

	Again, for a calorically perfect gas, we get		Again, for a calorically perfect gas, we get
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)		s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)
	</math>\|eq-ds-e}}		</math>\|label=eq-ds-e}}

	=== Isentropic Relations ===		=== Isentropic Relations ===

	Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus {{EquationNote\|eq-ds-c}} reduces to		Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus {{EquationNote\|label=eq-ds-c}} reduces to

	{{NumEqn\|<math>		{{NumEqn\|<math>
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	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}		\frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}
	</math>\|eq-isentropic-a}}		</math>\|label=eq-isentropic-a}}

	In the same way, {{EquationNote\|eq-ds-e}} gives		In the same way, {{EquationNote\|label=eq-ds-e}} gives

	{{NumEqn\|<math>		{{NumEqn\|<math>
	\frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)}		\frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)}
	</math>\|eq-isentropic-b}}		</math>\|label=eq-isentropic-b}}