User:AKAF/Efficiency

The performance of a launch system is complex. Normally craft are designed to maximise range ($$R$$), orbital radius ($$R$$) or payload mass fraction ($$\Gamma$$) for a given engine and fuel. This results in tradeoffs between the efficiency of the engine (takeoff fuel weight) and the complexity of the engine (takeoff dry weight), which can be expressed by the following:


 * $$\Pi_e+\Pi_f+\frac{1}{\Gamma}=1$$

Where :
 * $$\Pi_e=\frac{m_{empty}}{m_{initial}}$$ is the empty mass fraction, and represents the weight of the superstructure, tankage and engine.
 * $$\Pi_f=\frac{m_{fuel}}{m_{initial}}$$ is the fuel mass fraction, and represents the weight of fuel, oxidiser and any other materials which are consumed during the launch.
 * $$\Gamma=\frac{m_{initial}}{m_{payload}}$$ is initial mass ratio, and is the inverse of the payload mass fraction. This represents how much payload the vehicle can deliver to a destination.

Obviously, changing the engine will change all three of the above values. Scramjets decrease $$\Pi_f$$ and increase $$\Pi_e$$. It can be difficult to decide whether this will result in an increased $$\Gamma$$ (which would be an increased payload delivered to a destination for a constant vehicle takeoff weight.

Additionally, the drag of the new configuration must be considered. The drag of the total configuration can be considered as the sum of the vehicle drag ($$D$$) and the engine installation drag ($$D_e$$). The installation drag traditionally results from the pylons and the coupled flow due to the engine jet, and is a function of the throttle setting. Thus it is often written as:

$$D_e=\phi_eF$$ Where:
 * $$\phi_e$$ is the loss coefficient
 * $$F$$ is the thrust of the engine

For an engine strongly integrated into the aerodynamic body, it may be more convenient to think of ($$D_e$$) as the difference in drag from a known base configuration.

The overall engine efficiency can be represented as a value between 0 and 1 ($$\eta_0$$), in terms of the specific impulse of the engine:


 * $$\eta_0=\frac{g_0V_0}{h_{PR}}\cdot I_{sp}=\frac{\mbox{Thrust Power}}{\mbox{Chemical energy rate}}$$

Where:
 * $$g_0$$ is the acceleration due to gravity at ground level
 * $$V_0$$ is the vehicle speed
 * $$I_{sp}$$ is the specific impulse
 * $$h_{PR}$$ is fuel heat of reaction

Specific impulse is often used as the unit of efficiency for rockets, since in the case of the rocket, there is a direct relation between specific impulse, specific fuel consumption and exhaust velocity. This direct relation is not generally present for airbreathing engines, and so specific impulse is less used in the literature. Note that for an airbreathing engine, both $$\eta_0$$ and $$I_{sp}$$ are a function of velocity.

The specific impulse of a rocket engine is independent of velocity, and common values are between 200 and 600 seconds (450s for the space shuttle main engines). The specific impulse of a scramjet varies with velocity, reducing at higher speeds, starting at about 1200s, although values in the literature vary.

For the simple case of a single stage vehicle, the fuel mass fraction can be expressed as:
 * $$\Pi_f=1-exp\left[-\frac{\left(\frac{V_{initial}^2}{2}-\frac{V_i^2}{2}\right)+\int{g}\,dr}{\eta_0h_{PR}\left(1-\frac{D+D_e}{F}\right)}\right]$$

Where this can be expressed for single stage transfer to orbit as:


 * $$\Pi_f=1-exp\left[-\frac{g_0r_0\left(1-\frac{1}{2}\frac{r_0}{r}\right)}{\eta_0h_{PR}\left(1-\frac{D+D_e}{F}\right)}\right]$$

or for level atmospheric flight from air launch (missile flight):


 * $$\Pi_f=1-exp\left[-\frac{g_0R}{\eta_0h_{PR}\left(1-\phi_e\right)\frac{C_L}{C_D}}\right]$$

Where $$R$$ is the range, and the calculation can be expressed in the form of the Breguet range formula:


 * $$\Pi_f=1-e^{-BR}$$
 * $$B=\frac{g_0}{\eta_0h_{PR}\left(1-\phi_e\right)\frac{C_L}{C_D}}$$

Where:
 * $${C_L}$$ is the lift coefficient
 * $${C_D}$$ is the drag coefficient

This extremely simple formulation, used for the purposes of discussion assumes:
 * Single stage vehicle
 * No aerodynamic lift for the transatmospheric lifter

However they are true generally for all engines.