User:ThermoGroup6/sandbox

KERS Flywheel
The energy of a flywheel can be described by this general energy equation, assuming the flywheel is the system:
 * $$ E_{in}-E_{out}= \Delta E_{system}$$.

Where:
 * $$E_{in}$$ is the energy into the flywheel.
 * $$E_{out}$$ is the energy out of the flywheel.
 * $$\Delta E_{system}$$ is the change in energy of the flywheel.

An assumption is made that during braking there is no change in the potential energy, enthalpy of the flywheel, pressure or volume of the flywheel, so only kinetic energy will be considered. As the car is braking, no energy is dispersed by the flywheel, and the only energy into the flywheel is the initial kinetic energy of the car. The equation can be simplified to:
 * $$\frac{mv^2} {2} = \Delta E_{fly}$$.

Where:
 * $$m$$ is the mass of the car.
 * $$v$$ is the initial velocity of the car just before braking.

The flywheel collects a percentage of the initial kinetic energy of the car, and this percentage can be represented by $$\eta_{fly}$$. The flywheel stores the energy as rotational kinetic energy. Because the energy is kept as kinetic energy and not transformed into another type of energy this process is efficient. The flywheel can only store so much energy, however, and this is limited by its maximum amount of rotational kinetic energy. This is determined based upon the inertia of the flywheel and its angular velocity. As the car sits idle, little rotational kinetic energy is lost over time so the initial amount of energy in the flywheel can be assumed to equal the final amount of energy distributed by the flywheel. The amount of kinetic energy distributed by the flywheel is therefore:
 * $$KE_{fly}=\frac{\eta_{fly} mv^2} {2} $$

Regenerative Brakes
Regenerative braking has a similar energy equation to the equation for the mechanical flywheel. Regenerative braking is a two-step process involving the motor/generator and the battery. The initial kinetic energy is transformed into electrical energy by the generator and is then converted into chemical energy by the battery. This process is less efficient than the flywheel. The efficiency of the generator can be represented by:
 * $$ \eta_{gen}=\frac{W_{out}}{W_{in}}$$

Where:
 * $$W_{in}$$ is the work into the generator.
 * $$W_{out}$$ is the work produced by the generator.

The only work into the generator is the initial kinetic energy of the car and the only work produced by the generator is the electrical energy. Rearranging this equation to solve for the power produced by the generator gives this equation:
 * $$P_{gen}= \frac{\eta_{gen} mv^2}{2 \Delta t} $$

Where:
 * $$\Delta t$$ is the amount of time the car brakes.
 * $$m$$ is the mass of the car.
 * $$v$$ is the initial velocity of the car just before braking.

The efficiency of the battery can be described as:
 * $$ \eta_{batt}=\frac{P_{out}} {P_{in}} $$.

Where:
 * $$P_{in}=P_{gen}$$
 * $$P_{out}=W_{out} \Delta t$$

The work out of the battery represents the amount of energy produced by the regenerative brakes. This can be represented by:
 * $$ E_{out}=\frac{\eta_{batt} \eta_{gen} mv^2}{2}$$