Thermodynamic laws and efficiencies of turbomachines

First law (steady flow energy equation)

For one inlet and one outlet, neglecting potential energy, per unit mass:

$$q - w = (h_2 - h_1) + \frac{C_2^2 - C_1^2}{2}$$

For an adiabatic machine $q = 0$. Using stagnation enthalpy $h_0 = h + C^2/2$:

• Turbine: $w = h_{01} - h_{02}$
• Compressor: $w_{in} = h_{02} - h_{01}$

For a perfect gas, $h_0 = c_p T_0$, so $w = c_p (T_{01} - T_{02})$ for the turbine.

Second law

For an adiabatic machine $s_2 \ge s_1$. The ideal (reversible) machine is isentropic, $s_2 = s_1$. Irreversibility (friction, shock, tip leakage) raises the entropy, so the turbine delivers less work and the compressor needs more work than the isentropic value.

 T                            T          2 (actual)
 |  1                         |    2s   /
 |  |\                        |    |   /
 |  | \                       |    |  /
 |  |  \                      |    | /
 |  |   2 (actual)            |    |/
 |  2s                        |    1
 +------------ s              +------------ s
    turbine                      compressor

In both cases 2s lies vertically above or below 1 (same entropy), and the actual state 2 lies to its right at a higher temperature.

Isentropic efficiencies

$$\eta_t = \frac{h_{01} - h_{02}}{h_{01} - h_{02s}} \approx \frac{T_{01} - T_{02}}{T_{01} - T_{02s}}, \qquad \eta_c = \frac{h_{02s} - h_{01}}{h_{02} - h_{01}} \approx \frac{T_{02s} - T_{01}}{T_{02} - T_{01}}$$

Polytropic (small-stage) efficiency

It is the isentropic efficiency of an infinitesimal stage, assumed constant through the machine. For a compressor with pressure ratio $r$ and $\gamma$:

$$\eta_{p} = \frac{\frac{\gamma - 1}{\gamma}\ln r}{\ln (T_{02}/T_{01})}, \qquad \eta_c = \frac{r^{(\gamma-1)/\gamma} - 1}{r^{(\gamma-1)/(\gamma \eta_p)} - 1}$$

For a turbine expanding by ratio $r$:

$$\eta_t = \frac{1 - r^{-\eta_p (\gamma-1)/\gamma}}{1 - r^{-(\gamma-1)/\gamma}}$$

For the same $\eta_p$, the compressor $\eta_c$ is less than $\eta_p$ (preheat effect: the losses in each stage heat the air entering the next stage, which then needs more work), while the turbine $\eta_t$ is greater than $\eta_p$ (reheat effect: the losses in each stage are partly recovered as work in the following stages). The difference grows with the pressure ratio.