SCALING LAWS AND SMALL ORBITAL LAUNCH VEHICLES
I thought it would be good to start on some of the most basic issues related to small orbital rockets: their most basic flight principles and how the size of the launcher affects its flight characteristics.
BASIC FLIGHT FORCES
Like any vehicle flying in the air, a small orbital rocket launcher is controlled by four fundamental forces:
1. Weight caused by gravity
2. Thrust caused by the rocket motor
3. Lift caused by aerodynamic forces
4. Drag caused by aerodynamic forces
These are the same four forces that effect other air vehicles such as airplanes.
Now, if we simplify the flight of an orbital rocket, we need not concern ourselves with lift. A rocket without lifting surfaces which is flying vertically, or directly into the slipstream will not experience any lift. This will apply to most rocket designs.
The area of a rocket is affected by what is known as the Square Law. Knowing how area scales is important to understanding scaling effects on aerodynamic forces.
As the vehicle is scaled linearly, the area scales by the square of the linear distance.
Area = L^2
Thus, if you double the linear scale of the vehicle, its area will quadruple:
4 = 2^2
And if you halve the linear scale of the vehicle, its area will be reduced to a quarter:
1/4 = (1/2)^2
The most obvious effect derived from this is frontal area impinging into the airstream. If you scale the rocket by 1/2, then the frontal area is 1/4th.
The volume, weight and mass of a vehicle is affected by what is known as the Cube Law.
As the vehicle is scaled linearly, the volume (and thus the mass) scales by the
cube of the linear distance. If you think of a cube with any side of dimension, L,
Volume = L^3
If you double L, then the volume (as well as the mass) increases by eight:
8 = 2^3
If you halve L, then the volume (and mass) is 1/8th of the original.
1/8 = (1/2)^3
ROCKET THRUST AND SCALING
A common equation relating the thrust of a rocket motor to the relevant
Thrust = mdot * Ve + (pe - po)*Ae
mdot is the mass flow rate of exhaust gas from the rocket motor
Ve is the speed of the exhaust gas from the rocket motor
Pe is the exit pressure of the rocket motor
po is the chamber pressure of the rocket motor
Ae is the exit area of the rocket nozzle
It is necessary to break down the mdot part of this equation with this equation:
mdot = rho * Ve * Ae
rho is the density of the gas exhaust from the rocket motor
Ve is the exit velocity of the gas exhaust from the rocket motor
Ae is the nozzle exit area of the rocket motor
This allows us to relate mdot to Ve and Ae.
Combining these two equations by substitution, we get:
Thrust = rho * Ae * Ve^2 + (pe - Po) * Ae
We can rearrange to bring home the significant point:
Thrust = ((rho * Ve^2) + (pe - po)) * Ae
Thrust = k* Ae = k * L^2
The most important thing to realize from this equation is the scaling effects
on the thrust is proportional to the area of the nozzle or the square of the
linear scaling dimension. The implication of this is that as you scale a rocket
motor, the thrust from the rocket motor is proportional to the area of the throat,
or the square of the scaling.
AERODYNAMIC FORCES AND SCALING
Now that we've covered the most basic aspects of scaling, we can begin to
look at the four flight forces on the rocket to begin to understand the
effects of scaling on them.
The aerodynamic drag which is experienced by a rocket in motion is one of the
most significant changes. A commonly used equation to model the force of
drag on a rocket in motion is:
Force = 1/2 * rho * Cd * Area * Velocity^2
In this equation:
rho is the air density
Cd is the coefficient of drag
Area is the frontal area of the rocket
Velocity is the speed of the rocket in the air
Based on our earlier analysis, we can see that scaling a rocket by double will
quadruple the area, and therefore, the drag force will also quadruple. Scaling
the rocket by half will reduce the drag forces by one quarter.
DRAG AND ACCELERATION
Now that we know something about the effect of scaling on area, mass and drag,
we can begin to appreciate the problem with small scale rockets.
Isaac Newton showed the relationship between forces, mass and the accelerative
affects on that mass by those forces. His Second Law showed that the relationship
between forces, masses and accelerations is:
Force = Mass * Acceleration
Acceleration = Force / Mass
We should reiterate two important facts that are relevant here: scaling effects mass
by the cube of linear scaling but the force of drag is affected by the square of linear
scaling. Therefore, if we write these three equations together, we get:
Accleration = Force / Mass = (L^2)/(L^3)
If we simplify this, then we get:
Acceleration = 1/L
This means that Acceleration or Deceleration on the vehicle due to drag is
inversely proportional to linear scale. In other words, as we double the
scale of the vehicle, the acceleration (or deceleration) is affected
by 1/2. If we halve the scale of the vehicle, the deceleration due to
drag is doubled.
This is the key aspect that effects the design of small orbital launchers.
The summary is that the effects of aerodynamic forces are significantly
greater for small rocket launchers than for larger ones.