Thermodynamic processes: Difference between revisions

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[[Category:Compressible flow]]
[[Category:Compressible flow]]
[[Category:Thermodynamics]]
[[Category:Thermodynamics]]<!--
--><noinclude><!--
-->[[Category:Compressible flow:Topic]]<!--
--></noinclude><!--


__TOC__
--><nomobile><!--
 
-->__TOC__<!--
=== Specific Heat Relations ===
--></nomobile><!--
 
--><noinclude><!--
For thermally perfect and calorically perfect gases
-->{{#vardefine:secno|1}}<!--
 
-->{{#vardefine:eqno|23}}<!--
<math display="block">
--></noinclude><!--
\begin{aligned}
-->
&C_p=\frac{dh}{dT}\\
{{NumEqn|<math>
&C_v=\frac{de}{dT}
\end{aligned}
</math>
 
From the definition of enthalpy and the equation of state <math>p=\rho RT</math>
 
<math display="block">
h=e+\frac{p}{\rho}=e+RT
</math>
 
Differentiate Eqn. \ref{eq:enthalpy} with respect to temperature gives
 
<math display="block">
\frac{dh}{dT}=\frac{de}{dT}+\frac{d(RT)}{dT}
</math>
 
Inserting the specific heats gives
 
<math display="block">
C_p=C_v+R
</math>
 
Dividing Eqn. \ref{eq:specificheat:b} by <math>C_v</math> gives
 
<math display="block">
\frac{C_p}{C_v}=1+\frac{R}{C_v}
</math>
 
Introducing the ratio of specific heats defined as
 
<math display="block">
\gamma=\frac{C_p}{C_v}
</math>
 
Now, inserting Eqn. \ref{eq:gamma} in Eqn. \ref{eq:specificheat:c} gives
 
<math display="block">
C_v=\frac{R}{\gamma-1}
</math>
 
In the same way, dividing Eqn. \ref{eq:specificheat:b} with <math>C_p</math> gives
 
<math display="block">
1=\frac{C_v}{C_p}+\frac{R}{C_p}=\frac{1}{\gamma}+\frac{R}{C_p}
</math>
 
and thus
 
<math display="block">
C_p=\frac{\gamma R}{\gamma-1}
</math>
 
=== Isentropic Relations ===
 
First law of thermodynamics:
 
<math display="block">
de=\delta q - \delta w
</math>
 
For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math>
 
<math display="block">
de=Tds-pd\left(\frac{1}{\rho}\right)
</math>
 
Enthalpy is defined as: <math>h=e+p/\rho</math> and thus
 
<math display="block">
dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp
</math>
 
Eliminate $de$ in Eqn. \ref{eq:firstlaw:b} using Eqn. \ref{eq:dh}
 
<math display="block">
Tds=dh-\cancel{pd\left(\frac{1}{\rho}\right)}-\left(\frac{1}{\rho}\right)dp+\cancel{pd\left(\frac{1}{\rho}\right)}
</math>
 
<math display="block">
ds=\frac{dh}{T}-\frac{dp}{\rho T}
</math>
 
Using <math>dh=C_p T</math> and the equation of state <math>p=\rho RT</math>, we get
 
<math display="block">
ds=C_p\frac{dT}{T}-R\frac{dp}{p}
</math>
 
Integrating Eqn. \ref{eq:ds} gives
 
<math display="block">
s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)
</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
 
<math display="block">
s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)
</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
 
<math display="block">
s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)
</math>
 
Again, for a calorically perfect gas, we get
 
<math display="block">
s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)
</math>
 
=== Isentropic Relations ===
 
Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus Eqn. \ref{eq:ds:c} reduces to
 
<math display="block">
\frac{C_p}{R}\ln\left(\frac{T_2}{T_1}\right)=\ln\left(\frac{p_2}{p_1}\right)
</math>
 
<math display="block">
\frac{C_p}{R}=\frac{\gamma}{\gamma-1}
</math>
 
<math display="block">
\frac{\gamma}{\gamma-1}\ln\left(\frac{T_2}{T_1}\right)=\ln\left(\frac{p_2}{p_1}\right)\Rightarrow
</math>
 
<math display="block">
\frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}
</math>
 
In the same way, Eqn. \ref{eq:ds:e} gives
 
<math display="block">
\frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)}
</math>
 
Eqn. \ref{eq:isentropic:a} and Eqn. \ref{eq:isentropic:b} constitutes the isentropic relations
 
<math display="block">
\frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^{\gamma}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}
</math>
 
=== Flow Processes ===
 
<math display="block">
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}}


<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>|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>.  

Latest revision as of 13:30, 30 March 2026

ds=CvdTT+Rdνν(Eq. 1.24)
dν=νRds−CvνRTdT=νRds−CvpdT(Eq. 1.25)

for an isentropic process (ds=0), dν<0 for positive values of dT.

ds=CpdTT−Rdpp(Eq. 1.26)
dp=−pRds+CppRTdT=−pRds+CpρdT(Eq. 1.27)

for an isentropic process (ds=0), dp>0 for positive values of dT.


Since ν decreases with temperature and pressure increases with temperature for an isentropic process, we can see from (Eq. 1.25) that dν 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 dv=0 which implies

0=νR(ds−CvdTT)⇒dTds=TCv(Eq. 1.28)

and thus we can see that the slope of an isochore in a T−s-diagram is positive and that the slope increases with temperature.

In analogy, we can see that an isobar (dp=0) leads to the following relation

0=pR(CpdTT−ds)⇒dTds=TCp(Eq. 1.29)

and consequently isobars will also have a positive slope that increases with temperature in a T−s-diagram. Moreover, isobars are less steep than ischores as Cp>Cv.


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