Question 16.2: The following data refer to a gas turbine with a free power ......

For chocked free turbine $\dot{m}\sqrt{T_{04}}/P_{04}$ = constant, or $\dot{m}\sqrt{T_{04}}/P_{04}$ = 220. Compatibility of mass flow rate between two turbines

$\frac{\dot{m} \sqrt{T_{04}}} {P_{04}} =\frac{\dot{m}\sqrt{T_{03}}}{P_{03}} \times \sqrt{\frac{T_{04}}{T_{03}} } \times \frac{P_{03}}{P_{04}}$

where

$\frac{T_{04}}{T_{03}} =1-\eta_{\text{t}}\left(1-\frac{1}{\pi_{1}^{\text{G}_{\text{t}}}} \right)$

and $π_t$ is the pressure ratio of gas generator turbine and $G_{\text{t}}=(\gamma_{\text{t}}-1)/\gamma_{\text{t}}$ .
Define the calculated mass flow parameter of the free power turbine as $(\dot{m}\sqrt{T_{04}}/P_{04})_{\text{A}}$, which is calculated from Equation 1.

$\left(\frac{\dot{m}\sqrt{T_{04}}}{P_{04}} \right) _{\text{A}}=\frac{\dot{m}\sqrt{T_{03}}}{P_{03}} \times \left[1-\eta_{\text{t}}\left(1-\frac{1}{\pi_1^{\text{G}_{\text{t}}}} \right) \right] ^{0.5} \times \pi_{\text{t}} \quad \quad \quad (1)$

The calculated mass flow parameter of the free turbine $(\dot{m}\sqrt{T_{04}}/P_{04})_{\text{A}}$ is plotted together with the given choked value $\dot{m}\sqrt{T_{04}}/P_{04}$ as shown in Figure 16.13.
The intersection point gives the matching point data; namely,

$\pi_{\text{t}}=\frac{P_{03}}{P_{04}} =2.438 \quad \text{ and } \quad \frac{\dot{m}\sqrt{T_{03}}}{P_{03}} =98.9667$

Next, matching between the compressor and gas generator turbine may be done as follows:
Continuity equation

$\frac{\dot{m}\sqrt{T_{01}}}{P_{01}} =\frac{\dot{m}\sqrt{T_{03}}}{P_{03}} \sqrt{\frac{T_{01}}{T_{03}} }\frac{P_{03}}{P_{02}} \frac{P_{02}}{P_{01}} =\frac{\dot{m}\sqrt{T_{03}}}{P_{03}} \sqrt{\frac{T_{01}}{T_{03}} }\pi_{\text{b}}\pi_{\text{c}}=98.967 \times 0.97 \sqrt{\frac{T_{01}}{T_{03}} } \times \pi_{\text{c}} \\ \frac{\dot{m}\sqrt{T_{01}}}{P_{01}} =96 \sqrt{\frac{T_{01}}{T_{03}} } \times \pi_{\text{c}} \quad \quad \quad (2)$

Power balance

$\dot{m}\text{C}_{\text{P}_{\text{c}}}T_{01} \times \frac{1}{\eta_{\text{C}}} \left[\pi_{\text{c}}^{\text{G}_{\text{c}}}-1\right] =\dot{m}\text{C}_{\text{P}_{\text{t}}}T_{03} \eta_{\text{m}}\eta_{\text{t}}\left[1-\left(\frac{1}{\pi_{\text{t}}} \right) ^{\text{G}_{\text{t}}}\right]$

where $G_{\text{c}}=(\gamma_{\text{c}}-1)/\gamma_{\text{c}}$,thus

$\frac{T_{01}}{T_{03}} =\frac{\text{C}_{\text{P}_{\text{t}}}}{\text{C}_{\text{P}_{\text{c}}}} \eta_{\text{t}}\eta_{\text{m}}\left[1-\left(\frac{1}{\pi_{\text{t}}} \right)^{\text{G}_{\text{t}}} \right] \frac{\eta_{\text{C}}}{\pi_{\text{c}}^{\text{G}_{\text{c}}}-1} \\ \frac{T_{01}}{T_{03}} =0.19 \frac{\eta_{\text{C}}}{\pi_{\text{c}}^{\text{G}_{\text{c}}}-1} \quad \quad \quad (3)$

Equations (2) and (3) are used to calculate the mass flow parameter of the compressor that is tabulated in following table against the given input values for the compressor map:

Graphical solution of both values $\left(\left(\dot{m}\sqrt{T_{01}}/P_{10}\right) _{\text{calculated}} \text{ and }(\dot{m}\sqrt{T_{01}}/P_{01})_{\text{map}}\right)$ gives the following results for the compressor operating point:

$\pi_{\text{c}}=5.6, \quad \eta_{\text{c}}=0.84, \quad \frac{T_{01}}{T_{03}} =0.2506 \quad \text{and} \quad \frac{\dot{m}\sqrt{T_{01}}}{P_{01}} =269.5$

The turbine inlet temperature is then

$T_{03}=\frac{T_{01}}{0.2506} =\frac{288}{0.2506} =1149.24 \text{ K}$