Thermodynamics and Propulsion

11.2 Thermal and Propulsive Efficiency

It is often convenient to break the overall efficiency into two parts: thermal efficiency and propulsive efficiency, where

$$\eta_{\textrm{thermal}} = \frac{\textrm{rate of production of kinetic energy}}{\textrm{fu... ...left(\cfrac{\dot{m}_e u_e^2}{2}-\cfrac{\dot{m}_0 u_0^2}{2}\right)}{\dot{m}_f h}$$

$$\eta_{\textrm{propulsive}} =\frac{\textrm{propulsive power}}{\textrm{rate of production of p... ...frac{T u_0}{\left(\cfrac{\dot{m}_e u_e^2}{2}-\cfrac{\dot{m}_0 u_0^2}{2}\right)}$$
such that

$$\eta_{\textrm{overall}} = \eta_{\textrm{thermal}} \eta_{\textrm{propulsive}}$$

The thermal efficiency in this expression is the same as that which we used extensively in Part I. For an ideal Brayton cycle it is a function of the temperature ratio across the compressor,

$$\eta_{\textrm{thermal, ideal Brayton cycle}} = \frac{W_{\textrm{net}}}{Q_{\textrm{in}}} = 1 -\frac{T_1}{T_2}$$

Note that we can use our expression for thrust to rewrite the equation for propulsive efficiency in a more convenient form,

$$T \approx \dot{m}(u_e - u_0) \qquad \textrm{since $\dot{m}_e \approx \dot{m}_0 \approx \dot{m}$}$$

then

$$\eta_{\textrm{propulsive}} = \cfrac{\dot{m} u_0 (u_e-u_0)}{\cfrac... ...m}}{2}(u_e^2-u_0^2)} = \frac{2 u_0}{u_0 + u_e} = \cfrac{2}{1+\cfrac{u_e}{u_0}}.$$