Isentropic efficiencies of steady-flow devices in thermodynamic systems

Most steady-flow devices operate under adiabatic conditions, and the ideal process for these devices is the isentropic process. The parameter that describes how efficiently a device approximates a corresponding isentropic device is called isentropic or adiabatic efficiency.^[13]

Isentropic efficiency of turbines:

${\displaystyle \eta _{\text{t}}={\frac {\text{actual turbine work}}{\text{isentropic turbine work}}}={\frac {W_{a}}{W_{s}}}\cong {\frac {h_{1}-h_{2a}}{h_{1}-h_{2s}}}.}$

Isentropic efficiency of compressors:

${\displaystyle \eta _{\text{c}}={\frac {\text{isentropic compressor work}}{\text{actual compressor work}}}={\frac {W_{s}}{W_{a}}}\cong {\frac {h_{2s}-h_{1}}{h_{2a}-h_{1}}}.}$

Isentropic efficiency of nozzles:

${\displaystyle \eta _{\text{n}}={\frac {\text{actual KE at nozzle exit}}{\text{isentropic KE at nozzle exit}}}={\frac {V_{2a}^{2}}{V_{2s}^{2}}}\cong {\frac {h_{1}-h_{2a}}{h_{1}-h_{2s}}}.}$

For all the above equations:

${\displaystyle h_{1}}$ is the specific enthalpy at the entrance state,

${\displaystyle h_{2a}}$ is the specific enthalpy at the exit state for the actual process,

${\displaystyle h_{2s}}$ is the specific enthalpy at the exit state for the isentropic process.

Isentropic devices in thermodynamic cycles

Note: The isentropic assumptions are only applicable with ideal cycles. Real cycles have inherent losses due to compressor and turbine inefficiencies and the second law of thermodynamics. Real systems are not truly isentropic, but isentropic behavior is an adequate approximation for many calculation purposes.

Derivation of the isentropic relations

For a closed system, the total change in energy of a system is the sum of the work done and the heat added:

${\displaystyle dU=\delta W+\delta Q.}$

The reversible work done on a system by changing the volume is

${\displaystyle \delta W=-p\,dV,}$

where ${\displaystyle p}$ is the pressure, and ${\displaystyle V}$ is the volume. The change in enthalpy (${\displaystyle H=U+pV}$) is given by

${\displaystyle dH=dU+p\,dV+V\,dp.}$

Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), ${\displaystyle \delta Q_{\text{rev}}=0}$, and so ${\displaystyle dS=\delta Q_{\text{rev}}/T=0}$ All reversible adiabatic processes are isentropic. This leads to two important observations:

${\displaystyle dU=\delta W+\delta Q=-p\,dV+0,}$

${\displaystyle dH=\delta W+\delta Q+p\,dV+V\,dp=-p\,dV+0+p\,dV+V\,dp=V\,dp.}$

Next, a great deal can be computed for isentropic processes of an ideal gas. For any transformation of an ideal gas, it is always true that

${\displaystyle dU=nC_{v}\,dT}$, and ${\displaystyle dH=nC_{p}\,dT.}$

Using the general results derived above for ${\displaystyle dU}$ and ${\displaystyle dH}$, then

${\displaystyle dU=nC_{v}\,dT=-p\,dV,}$

${\displaystyle dH=nC_{p}\,dT=V\,dp.}$

So for an ideal gas, the heat capacity ratio can be written as

${\displaystyle \gamma ={\frac {C_{p}}{C_{V}}}=-{\frac {dp/p}{dV/V}}.}$

For a calorically perfect gas ${\displaystyle \gamma }$ is constant. Hence on integrating the above equation, assuming a calorically perfect gas, we get

${\displaystyle pV^{\gamma }={\text{constant}},}$

that is,

${\displaystyle {\frac {p_{2}}{p_{1}}}=\left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }.}$

Using the equation of state for an ideal gas, ${\displaystyle pV=nRT}$,

${\displaystyle TV^{\gamma -1}={\text{constant}}.}$

(Proof: ${\displaystyle PV^{\gamma }={\text{constant}}\Rightarrow PV\,V^{\gamma -1}={\text{constant}}\Rightarrow nRT\,V^{\gamma -1}={\text{constant}}.}$ But nR = constant itself, so ${\displaystyle TV^{\gamma -1}={\text{constant}}}$.)

${\displaystyle {\frac {p^{\gamma -1}}{T^{\gamma }}}={\text{constant}}}$

also, for constant ${\displaystyle C_{p}=C_{v}+R}$ (per mole),

${\displaystyle {\frac {V}{T}}={\frac {nR}{p}}}$ and ${\displaystyle p={\frac {nRT}{V}}}$

${\displaystyle S_{2}-S_{1}=nC_{p}\ln \left({\frac {T_{2}}{T_{1}}}\right)-nR\ln \left({\frac {p_{2}}{p_{1}}}\right)}$

${\displaystyle {\frac {S_{2}-S_{1}}{n}}=C_{p}\ln \left({\frac {T_{2}}{T_{1}}}\right)-R\ln \left({\frac {T_{2}V_{1}}{T_{1}V_{2}}}\right)=C_{v}\ln \left({\frac {T_{2}}{T_{1}}}\right)+R\ln \left({\frac {V_{2}}{V_{1}}}\right)}$

Thus for isentropic processes with an ideal gas,

${\displaystyle T_{2}=T_{1}\left({\frac {V_{1}}{V_{2}}}\right)^{(R/C_{v})}}$ or ${\displaystyle V_{2}=V_{1}\left({\frac {T_{1}}{T_{2}}}\right)^{(C_{v}/R)}}$

Table of isentropic relations for an ideal gas

${\displaystyle {\frac {T_{2}}{T_{1}}}}$	${\displaystyle =}$	${\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}}$	${\displaystyle =}$	${\displaystyle \left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}}$	${\displaystyle =}$	${\displaystyle \left({\frac {\rho _{2}}{\rho _{1}}}\right)^{(\gamma -1)}}$
${\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)^{\frac {\gamma }{\gamma -1}}}$	${\displaystyle =}$	${\displaystyle {\frac {P_{2}}{P_{1}}}}$	${\displaystyle =}$	${\displaystyle \left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }}$	${\displaystyle =}$	${\displaystyle \left({\frac {\rho _{2}}{\rho _{1}}}\right)^{\gamma }}$
${\displaystyle \left({\frac {T_{1}}{T_{2}}}\right)^{\frac {1}{\gamma -1}}}$	${\displaystyle =}$	${\displaystyle \left({\frac {P_{1}}{P_{2}}}\right)^{\frac {1}{\gamma }}}$	${\displaystyle =}$	${\displaystyle {\frac {V_{2}}{V_{1}}}}$	${\displaystyle =}$	${\displaystyle {\frac {\rho _{1}}{\rho _{2}}}}$
${\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)^{\frac {1}{\gamma -1}}}$	${\displaystyle =}$	${\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{\frac {1}{\gamma }}}$	${\displaystyle =}$	${\displaystyle {\frac {V_{1}}{V_{2}}}}$	${\displaystyle =}$	${\displaystyle {\frac {\rho _{2}}{\rho _{1}}}}$

Derived from

${\displaystyle PV^{\gamma }={\text{constant}},}$

${\displaystyle PV=mR_{s}T,}$

${\displaystyle P=\rho R_{s}T,}$

where:

${\displaystyle P}$ = pressure,

${\displaystyle V}$ = volume,

${\displaystyle \gamma }$ = ratio of specific heats = ${\displaystyle C_{p}/C_{v}}$,

${\displaystyle T}$ = temperature,

${\displaystyle m}$ = mass,

${\displaystyle R_{s}}$ = gas constant for the specific gas = ${\displaystyle R/M}$,

${\displaystyle R}$ = universal gas constant,

${\displaystyle M}$ = molecular weight of the specific gas,

${\displaystyle \rho }$ = density,

${\displaystyle C_{p}}$ = specific heat at constant pressure,

${\displaystyle C_{v}}$ = specific heat at constant volume.