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}