Delta V Calculation question

[ChainsDD]

Ok, So I understand how to calculate Delta V for a particular stage, and for the rocket as a whole. Where I get confused is this, if I have a stage where I have a combination of different fuel types. For example, I have a rocket similar to the Space Shuttle, with a liquid fueled engine and 2 SRBs. To calculate the total Delta V for that stage, would I calculate the Delta V for the liquid engine, then the SRBs and add them together, keeping their fuels separate, or would I find the average Isp for both types of engine and the total fuel?

Things really get confusing for me when it comes to the mass and calculating the Delta-Vs separately. I need to use the dry/total mass of the entire ship when doing the calculations, but should I use the dry or the total mass for the SRBs when calculating the liquid engine?

I'm not concerned about possible efficiency issues, just curious about the calculations.

Edited October 17, 2013 by ChainsDD

[Specialist290]

Before we begin, I'm going to assume a "best-case" scenario where the solid-rocket boosters run out of fuel before the liquid engines do, and are dropped as soon as they're empty. The other way around is possible, but generally less efficient for a number of reasons.

The easiest way to do this is to treat both phases of the burn as two separate "stages": On one stage you're burning all of your SRBs and part of your liquid fuel, and on the second, after you've dropped the boosters, you're burning the rest of the fuel.

The determining factor on the delta-v of this first "stage" is going to be the time it takes for your SRBs to finish their burn. Since you can't throttle a solid rocket, this is going to remain more or less static. (In reality, of course, changing values of Isp as your rocket rises through the atmosphere are going to make this a little wobbly.)

Once you know that, you'll need to figure out the amount of fuel burnt by your liquid engines in that time. We'll call this parameter "delta-m" or "dm" (for change in propellant mass) for now. You can use:

dm = (F * t) / (g0 * Isp)

F = average thrust in kilonewtons provided by liquid engine for the duration of the burn (which will be full thrust if the throttle stays at max; otherwise, it gets tricky)

t = burn time

g0 = standard surface gravity (9.8 m/s)

The value you get should be the mass of the liquid fuel (including oxidizer) burnt.

You'll take this value and subtract both it and the propellant mass of the boosters from the craft's full mass to find the remaining mass of your craft. When you run the rocket equation, you'll plug the full mass into the rocket equation for m0, and the remaining mass for m1.

Now we need to find the Isp for the combined system, which gets tricky when you have engines of different specific impulses. You'll need to find the total rate of propellant mass flow (mDot) to do this. You can find the mass flow rate for each engine or booster using the following equation:

mDot = Fe / (g0 * Isp)

Fe = The thrust provided by that engine or booster

Isp = The specific impulse of that engine or booster

Then you add all the mDot values together and invert the equation to find the average specific impulse for the entire stage:

Isp = Ft / (g0 * mDot)

Ft = Total thrust of all engines

Plug the values you've derived for m0, m1, and Isp into the rocket equation. That should give you the delta-v for your "first stage." Mark that down.

Now subtract the dry mass of your boosters from the remaining mass (to represent dropping your empty boosters). That is your new m0 for the "second stage." Subtracting the remaining liquid fuel in the stage to find your dry mass will give you your new m1. Again, solve the rocket equation with those values. Record that as well.

Now add the two delta-v values you've gotten to find the total delta-v for both the liquid engine and the boosters.

Hope this helps

[ChainsDD]

That looks good to me :-) Thanks! Now to figure out how to code it...

[Oddible]

Or you can just grab a mod like Kerbal Engineer (my fave as I like the way the info is displayed and there is no temptation to use it for gameplay, it is just information), or MechJeb. (Fix to make them work with .22 is here, if they haven't posted an update already).