Understanding Rocket Equations

How do we arrive at the equations describing the relation between mass flow and thrust?

We are going to look at how we can explain we arrive at the equation that says that mass flow dm/dt is defined as:

dm/dt = F/(Isp * g₀)
dm/dt = F/vₑ

Where g₀ is the acceleration of gravity on earth, measured to 9.80665 m/s². F is the thrust of the rocket engine. Isp is the specific impulse of the engine. Basically a measure of propellant efficiency.

The exhaust velocity vₑ is another way of looking at the efficiency of the engine. It is basically how fast each particle in the exhaust from the rocket engines moves.

We will explain this by starting with Newton’s second law, usually expressed this way:

F = am

It says the force F working on an object with mass m causes an acceleration of a. However purposes we will look at another definition which is more practical.

This requires us to look at momentum first. Momentum is defined as:

p = mv

This is used when two objects of different size and velocity collide and you want to check what their new velocities should be. We can do that due to the conservation of momentum.

Anyway momentum provides us with an alternative definition of Newton’s second law:

F =  Δp/ Δt

One way to think about this is that if you throw a snowball while standing on the ice, applying some constant force F to it for a time period equal to Δt, then the snowball will "experience" a change in momentum of Δp, but so will you. However the outcomes will be different because you have vastly different mass.

You cannot measure the force F directly, but you can figure it out by measuring:

1. How long time Δt the snowball was in your hand getting accelerated, with some unknown accelerations a.
2. What the velocity v was when the snowball exited your hand.
3. Weigh the snowball before it is thrown, to determine its mass m

Since the snowball started with velocity 0, the change in velocity Δv = v - 0 = v. Since the snowball weighed the same before and after the throw, we get that Δp = m Δv. Thus we could write this as:

F =  Δp/ Δt = m Δv / Δt = m(Δv / Δt) = ma

But this does not explain the mass flow equation. Do do that we need to do a little thought experiment. Imagine you are throwing snowballs of varying mass extremely rapidly in succession. Basically there is a flow of snowball mass emanating from your body, propelling you across the ice sheet.

Imagine that the velocity of each snowball is the same vₑ every time. That is the case with the exhaust from a rocket engine. The only thing varying slightly is the size of the snowballs.

That would mean the Δp will be fluctuating, and consequently the F would fluctuate too. But the fluctuations of Δp will not be caused by a change in velocity since it is the same. Hence there is a Δm fluctuations in mass. Thus we will instead write:

F =  Δp/ Δt = Δm v / Δt = v(Δm / Δt)
Δm / Δt = F/v

Which is basically telling us that the mass of the snow Δm I am throwing in time period Δt gives a mass flow of Δm / Δt equal to the force F I am applying to the snow balls I throw, divided by the velocity v they have acquired when they exit my hand.

If we shift this snowball analogy over to the rocket engine, which we are actually interested in, we can attempt to explain what is going on.

When the fuel and oxidizer reactants react in the combustion chamber they produce products with high kinetic energy. When they exit the nozzle it is the same as the snow ball being thrown out. Except these particles are so small it does not make sense to count each one of them. Instead we deal with them as a mass flow.

So for some period Δt a certain amount of propellant with a mass of Δm will have escaped. Each particle will have had a velocity we call the exhaust velocity vₑ. As we shrink the time interval Δt close to zero we get the equation:

dm/dt = F/vₑ

How is Mass Flow Useful to Know?

If like me you are writing a simulation of a rocket launch, you want to know at every time step you simulate how much mass the rocket has. That will be influenced by how much fuel has been consumed. To determine the amount of fuel consumed in a time step of length Δt, you need to know the mass flow.

Usually you will have been provided with the Isp, thrust Force and propellant mass of the rocket. Isp and Force of an engine should be easy to measure on a test stand. They can just measure the time it takes the empty the propellant tanks and the force generated in that period to get the mass flow. From there they can calculate the Isp.