Thermodynamic processes: Difference between revisions
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[[Category:Compressible flow]] | [[Category:Compressible flow]] | ||
[[Category: | [[Category:Thermodynamics]]<!-- | ||
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<math | --></nomobile><!-- | ||
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{{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}} | ||
<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 | {{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 | {{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 | 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 | {{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 | {{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>. | ||
