# What is the equation for the energy balance between turbine, compressor and propeller?

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?

• The shaft is generally pretty stiff and doesn't directly have moving parts. So I assume by shaft efficiency you are thinking about bearing losses? – OSUZorba May 24 '16 at 0:48
• Exactly OSUZorba. That's the energy lost due to friction in the bearings (unconveniently used to increase its temperature ) – Jose Lopez Garcia May 24 '16 at 1:21

$$\dot{W_t}=\frac{-\dot{W_c}}{\eta_{shaft}}+\frac{-\dot{W_p}}{\eta_{shaft}*\eta_{gearbox}}$$
As you add more efficiencies to a component, you would just continue to multiple the all $\eta$'s together to get the total efficiency of that component, since efficiencies are multiplicative and the associative property of multiplication.