Question 12.5: Figure 12.17 shows the turbine and compressor maps for a sin......
Figure 12.17 shows the turbine and compressor maps for a single-spool no-load gas turbine engine. The engine is operating under the following conditions:
• Fuel-to-air ratio ( f ) = 0.024
• Mechanical efficiency (η_{m}) = 0.95%
Select any reasonable magnitude for the T_{t\ 4}/T_{t\ 2} ratio. Now select a point on the compressor map and determine its “conjugate” point on the turbine map.
NOTE
In solving this problem, do not use equation (12.41) to verify the actual magnitude of T_{t\ 4}/T_{t\ 2}, as this is the major requirement in the next example. This is where we will verify and subsequently modify this temperature ratio in a rather tedious procedure, as will be obviously clear.
{\frac{T_{t\ 4}}{T_{t\ 2}}}=\left[{\frac{1}{(1+f)\eta_{m}\eta_{T}\eta_{C}}}\right]\left({\frac{c_{p_{C}}}{c_{p_{T}}}}\right){\frac{\left[\left({\frac{p_{t\ 3}}{p_{t\ 2}}}\right)^{\frac{\gamma_{C}-1}{{\gamma_{C}}}}-1\right]}{\left[1-\left({\frac{p_{t\ 5}}{p_{t\ 4}}}\right)^{\frac{\gamma_{T}-1}{{\gamma_{T}}}}\right]}} (12.41)
Let us arbitrarily set the ratio (T_{t\ 4}/T_{t\ 2}) to 5.4. As part of the outer loop (in the last segment of the text), let us (also arbitrarily) consider the point C_{1} (shown on the compressor map in Fig. 12.17), which is associated with the following variables:
\pi_{C}=9.5
(\dot m)_{C2}=12.0\,\mathrm{kg/s}
N_{C2}=32,400\,{\mathrm{rpm}}
With this choice, we can now proceed to calculate the turbine corrected speed as follows:
{N}_{C4}=N_{C2}\sqrt{\frac{T_{t\ 2}}{T_{t\ 4}}}=14,000\;\mathrm{rpm}
Referring again to the same double-loop procedure at the end of the text, we are now looking for the point “T_{1},” on the turbine map, which corresponds to the previously chosen point “C_{1}” (on the compressor map). In doing so, note that the only turbine variable of which we are aware is the corrected speed computed earlier. The process is indeed iterative. In the following, only the first step of this process is presented, followed by the final result, for the purpose of brevity.
Step 1:
Let us pick the point on the turbine map where p_{t\ 4}/p_{t\ 5}=4.2 and \left(\dot m\right)_{C4}=3.4\,\mathrm{kg/s}. With this choice, we are now able to calculate the compressor corrected mass-flow rate as follows:
({\dot{m}})_{C2}=({\dot{m}})_{C4}\biggl({\frac{1}{1+f}}\biggr)\pi_{C}{\sqrt{\frac{T_{t\ 2}}{T_{t\ 4}}}}=13.6\ \mathrm{kg/s}
which is far from that associated with point “C_{1},” with the magnitude sought after being 12.0 kg/s.
Repeating this computational step, the final results are
{\frac{p_{t\ 4}}{p_{t\ 5}}}=4.12
(\dot m)_{C4}=2.8\;\mathrm{kg/s}
These two variables are sufficient to place the point “T_{1}” on the turbine map to correspond to the previously-selected point “C_{1}” on the compressor map.