# Maximum/Lowest achievable expansion ratio (generally) in a turbine? [closed]

Let's think about a turboprop with a single shaft. The turbine moves the compressor and the propeller.

My question: what is the "limit" to the expansion ratio achieved by the turbine?

If we call $5$ the air that is exiting the turbine, and $4$ the air that enters the turbine, then the expansion (or compression) ratio is defined by: $$\pi_t\equiv \dfrac{p_{05}}{p_{04}}<0\Longleftarrow \left(\textit{Because air is expanding}\right)$$

But for a non-ideal, single-stage axial turbine this ratio can be related to $T_{04}$ (that is, the temperature at the burner exit) by: $$\pi_t=\bigg[1-\dfrac{W_t}{c_p\eta_tT_{04}}\bigg]^{\gamma/\left(\gamma-1\right)}\quad \text{where}\quad \gamma\approx 1,35>0$$ where $W_t$ is the specific energy extracted from the fluid by the turbine.

On the one hand, as $T_{04}$ increases, $\pi_t$ will increase as well.

But on the other hand, I think we are interested in $T_{04}$ as high as possible BUT $\pi_t$ as low as possible, which comes into direct contradiction right? (Please, correct me here if I'm wrong).

What is the compromise solution between $T_{04}$ and $\pi_t$? If we make $T_{04}$ too high (which is thermodinamically convenient) then $\pi_t$ will be higher than its optimal value, and the other way around.

EDIT:

$T_{04}$ has itself a structural/thermal limit (if it goes too high, it can ultimately melt and/or generate way too high pressure on the turbine blades until it breaks). I think this defines a maximum value for $\pi_t$, right? Although knowing the maximum $\pi_t$ I think doesn't make sense, because we want to minimize it.

## closed as off-topic by Firee, Federico♦, SMS von der Tann, kepler22b, PondlifeJun 9 '16 at 12:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about aviation, within the scope defined in the help center." – Firee, Federico, SMS von der Tann, kepler22b, Pondlife
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