Let's consider the next turboprop arrangement:
What I am looking for
I am looking for an equation that involves $\dot{W}_T$, $\dot{W}_C$ and $\dot{W}_P$ (i.e. the power done by each component). This needs to be done through an energy balance in the shaft.
What we know
We know that the shaft has an efficiency of $\eta_{\text{shaft}}$, and the gearbox has an efficiency of $\eta_{\text{box}}$.
My try (#1)
Since energy is derived from the turbine, we adopt the convention $\dot{W}_T>0$, therefore $\dot{W}_P, \dot{W}_C < 0$. In other words, energy is required to move the compressive devices.
Following the definition of mechanical efficiency: $$\boxed{\eta_{\text{shaft}}=\dfrac{-\dot{W_C}-\dot{W_P}}{\dot{W_T}}}$$
However, I don't know how to incorporate the efficiency $\eta_{\text{box}}$ into this equation. This led me to try and find another expression.
My try (#2)
Following an hydraulic analogy for the power, I thought that whatever power the turbine generates needs to be invested in:
- Increasing the temperature of the bearings of the main shaft (because the shaft is not ideal and has an efficiency of $\eta_{\text{shaft}}$).
- Moving the compressor.
- Increasing the temperature of the gearbox (because the box is not ideal and has an efficiency of $\eta_{\text{box}}$).
- Moving the propeller.
Following this thinking, I can intuitively find this next equation (which I don't know if is correct or not): $$\boxed{\eta_{\text{shaft}}\times \dot{W}_T = -\dot{W}_C-\dfrac{\dot{W}_P}{\eta_{\text{box}}}}$$
The thing is I don't know if I assigned the efficiency $\eta_{\text{box}}$ correctly or not. Is there an easier way (i.e. less intuitive/more mechanical way) to apply the energy equation? What if we had 2 or more shafts?