Thermodynamic Processes

ds=C_v\dfrac{dT}{T}+R\dfrac{d\nu}{\nu}

(2.34)

d\nu=\dfrac{\nu}{R}ds-C_v\dfrac{\nu}{RT}dT=\dfrac{\nu}{R}ds-\dfrac{C_v}{p}dT

(2.35)

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

ds=C_p\dfrac{dT}{T} - R \dfrac{dp}{p}

(2.36)

dp=-\dfrac{p}{R}ds+C_p\dfrac{p}{RT}dT=-\dfrac{p}{R}ds+C_p\rho dT

(2.37)

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

Temperature-entropy process trends — Figure 2.1:Isentropic process trends

Since $\nu$ decreases with temperature and pressure increases with temperature for an isentropic process, we can see from Eqn. (2.35) that $d\nu$ 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=\dfrac{\nu}{R}\left(ds-C_v\dfrac{dT}{T}\right) \Rightarrow \dfrac{dT}{ds}=\dfrac{T}{C_v}

(2.38)

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=\dfrac{p}{R}\left(C_p\dfrac{dT}{T}-ds\right) \Rightarrow \dfrac{dT}{ds}=\dfrac{T}{C_p}

(2.39)

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 $C_p > C_v$ .

2.5 Thermodynamic Processes