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.

AREA SCALING

============

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.

MASS SCALING

============

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,

then:

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

parameters is:

Thrust = mdot * Ve + (pe - po)*Ae

Where:

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

Where:

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

or simplifying:

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

or:

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.

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