# What is the right Model Rocket Performance Figure of Merit?

### Help Support The Rocketry Forum:

#### TXWalker

##### Active Member
I've been pondering a thought experiment on what is the best figure of merit for model rocketry. In the military we have "bang for the buck", in aerospace we have "$/Kg payload", so what is the right metric for model rocketry? Since we are not intentionally trying to blow things up or deliver a payload to orbit I propose the objective is "up". Or how high can a particular rocket go on a given impulse. Let's start with the rocket altitude equation, greatly simplified for perfect trajectory, zero drag and constant gravity. Boost Phase Coast phase: h = deltaV^2/2g I would propose that the following are design parameters and should be left to the optimization of designer: drag, mass initial and final, and burn time tb. You'll note that Isp shows up in both terms of the boost equation with gravity as a constant. If we divide hb/Isp we get our figure of merit for model rocketry. In other words how much altitude can you generate for a given design. If you prefer Isp is directly proportional to cost so roughly speaking Altitude/$.

Taking this new figure of merit I compared the performance of the various high altitude record holders by impulse. The assumption being that these designs have all been fully optimized for mass, drag, and motor selection. I have grouped them by total impulse class as well as single vs staged design. I looked primarily at the lower impulse classes trying to avoid the complications of going super sonic. The first thing that jumps out is the altitude does not double with doubling impulse as the first order prediction from the rocket equation. For example doubling the impulse from F to G only results in 30% more altitude. Perhaps less surprising the two stages designs are slightly more efficient than the single stage designs. This is likely due to the increased burn time achieved by dividing the same total impulse into two ore more motors.

General trends when optimizing for altitude.
1) Smaller impulse rockets are more "efficient" than larger impulse rockets. ie Altitude/$2) When selecting a motor you want as long a burn time as possible for a given impulse. 3) It is more efficient to divide the total impulse between multiple motors. ie 2-stages better than one. #### Steve Shannon ##### Well-Known Member TRF Supporter The metric that makes the most sense to me is “specific impulse” (ISP) as a material property of propellant. Most simply explained it’s total impulse per unit of weight which results in a unit of time. For example, a pound of APCP may have 220 pound seconds total impulse, so its ISP is 220 seconds. #### TXWalker ##### Active Member The metric that makes the most sense to me is “specific impulse” (ISP) as a material property of propellant. Most simply explained it’s total impulse per unit of weight which results in a unit of time. For example, a pound of APCP may have 220 pound seconds total impulse, so its ISP is 220 seconds. Steve, Thanks for chiming in. Interesting discussion. No doubt that ISP is important. However it only addresses part of the question; "The How". IMO what is missing is "The Why" Why do you need 220 ISP? What work are you going to do with all that energy? This is the reason I added altitude to the equation. I assume the goal is to have the rocket fly to a given altitude. Altitude/ISP is a measure of how efficiently you achieve a given altitude. The figure of merit encompasses more than the specific motor and propellant efficiency. Including altitude encompasses much much more; how light you can build a given airframe, your ability to maintain a straight trajectory, what is the drag coefficient of your specific design, and etc. . #### Alan15578 ##### Well-Known Member I've been pondering a thought experiment on what is the best figure of merit for model rocketry. In the military we have "bang for the buck", in aerospace we have "$/Kg payload", so what is the right metric for model rocketry? Since we are not intentionally trying to blow things up or deliver a payload to orbit I propose the objective is "up". Or how high can a particular rocket go on a given impulse. Let's start with the rocket altitude equation, greatly simplified for perfect trajectory, zero drag and constant gravity.

Boost Phase

View attachment 432758
Coast phase:

h = deltaV^2/2g

I would propose that the following are design parameters and should be left to the optimization of designer: drag, mass initial and final, and burn time tb. You'll note that Isp shows up in both terms of the boost equation with gravity as a constant. If we divide hb/Isp we get our figure of merit for model rocketry. In other words how much altitude can you generate for a given design. If you prefer Isp is directly proportional to cost so roughly speaking Altitude/$. Taking this new figure of merit I compared the performance of the various high altitude record holders by impulse. The assumption being that these designs have all been fully optimized for mass, drag, and motor selection. I have grouped them by total impulse class as well as single vs staged design. I looked primarily at the lower impulse classes trying to avoid the complications of going super sonic. The first thing that jumps out is the altitude does not double with doubling impulse as the first order prediction from the rocket equation. For example doubling the impulse from F to G only results in 30% more altitude. Perhaps less surprising the two stages designs are slightly more efficient than the single stage designs. This is likely due to the increased burn time achieved by dividing the same total impulse into two ore more motors. View attachment 432759 General trends when optimizing for altitude. 1) Smaller impulse rockets are more "efficient" than larger impulse rockets. ie Altitude/$
2) When selecting a motor you want as long a burn time as possible for a given impulse.
3) It is more efficient to divide the total impulse between multiple motors. ie 2-stages better than one.
Ya know, the NAR used to have a competition event that was altitude per N-s. But people quickly figured out that it just 1/2A altitude. The 1/4As were a bit funky. A slightly better metric would have been payload mass times altitude/total impulse.

#### TXWalker

##### Active Member
That makes a lot of sense. The physics favors smaller motors on this metric.

#### GalantVR41062

##### Celebrate Recovery
TRF Supporter
You could look at mass fraction, and altitude achieved.

#### boatgeek

##### Well-Known Member
I’m with heada. It’s what you like so the metric is smiles. Perhaps smiles per launch but that disadvantages hangar queens.

#### manixFan

##### Not a rocket scientist
Of course one thing that accounts for the lack of linear increase in altitude vs impulse is motor diameter and mass fraction.

For example, for H impulse class, I can get a full H impulse in 29mm - the AT H268R (320NS). But in a 29mm I motor, I can only get 381 NS. To get to a full I motor, I have to step up to a 38mm motor. And of course increasing the diameter has a dramatic impact on potential altitude. Just running a super quick sim shows the H268R gets about 30% less altitude in a 38mm rocket vs a 29mm.

I've run many sims comparing which motor type gives me the most altitude per NS – a super light rocket with a long burn motor with lower overall impulse or a heavier rocket with fast burn motor with higher impulse. (The two motors I've used the most are the L265 (2644.6Ns and 9.9 seconds) and the L935 (3146.8Ns and 3.4 seconds). Both fit the 54mm 6XL case. It's an interesting exercise.

I like the metric of foot per NS, it eliminates all other variables, but it advantages smaller diameter motors. Lately I've been thinking about different metrics, such as foot/$within a specific impulse. What motor in the J motor class can I go the highest for the least amount of money? But that becomes very difficult when you mix in SU vs reloads and the cost of the motor hardware. No matter what metric there are many caveats. And hey, who's to tell me my idea of fun!? Tony #### AeroAggie ##### Well-Known Member TRF Supporter Smiles per unit of recovery distance. If you're not chasing records where numbers actually count, the only thing that matters is how much fun you had building and firing it, and how far you had to chase it. #### Walter Longburn ##### Well-Known Member TRF Supporter Maybe feet of altitude per foot of rocket... just an idea I haven't studied out. #### boatgeek ##### Well-Known Member A factor in the number of smiles is the flame length to rocket length ratio. Over 1 is a universally popular flight. #### TXWalker ##### Active Member You could look at mass fraction, and altitude achieved. Excellent suggestion adding mass to the figure of merit. At first I considered mass purely a design feature however this doesn't appear to be true. How about we consider "work" as the desired outcome. Classically work is defined as Force x Distance. In our case F= mass x gravity and distance is the peak altitude achieved. Since gravity is essentially constant over the altitudes we are dealing with our new FOM can be expressed as; FOM = m(initial) x h(altitude)/ Total impulse Unfortunately I don't have any real data on rocket mass for comparison. I guess I could do it by simulation in OR. Take the same design and vary to total impulse and measure the altitude. #### TXWalker ##### Active Member Of course one thing that accounts for the lack of linear increase in altitude vs impulse is motor diameter and mass fraction. For example, for H impulse class, I can get a full H impulse in 29mm - the AT H268R (320NS). But in a 29mm I motor, I can only get 381 NS. To get to a full I motor, I have to step up to a 38mm motor. And of course increasing the diameter has a dramatic impact on potential altitude. Just running a super quick sim shows the H268R gets about 30% less altitude in a 38mm rocket vs a 29mm. I've run many sims comparing which motor type gives me the most altitude per NS – a super light rocket with a long burn motor with lower overall impulse or a heavier rocket with fast burn motor with higher impulse. (The two motors I've used the most are the L265 (2644.6Ns and 9.9 seconds) and the L935 (3146.8Ns and 3.4 seconds). Both fit the 54mm 6XL case. It's an interesting exercise. I like the metric of foot per NS, it eliminates all other variables, but it advantages smaller diameter motors. Lately I've been thinking about different metrics, such as foot/$ within a specific impulse. What motor in the J motor class can I go the highest for the least amount of money? But that becomes very difficult when you mix in SU vs reloads and the cost of the motor hardware. No matter what metric there are many caveats.

And hey, who's to tell me my idea of fun!?

Tony
I would argue that airframe diameter is simply a design constraint. Do doubt a lower drag design will be more efficient for the same impulse.

Longer burn time is key to achieving higher boost altitude. Note tb in the boost phase of the rocketry equation above. That is why all the altitude junkies are so excited about the new AT long burn motors. ;-)

#### manixFan

##### Not a rocket scientist
I would argue that airframe diameter is simply a design constraint. Do doubt a lower drag design will be more efficient for the same impulse.

Longer burn time is key to achieving higher boost altitude. Note tb in the boost phase of the rocketry equation above. That is why all the altitude junkies are so excited about the new AT long burn motors. ;-)
"Do doubt a lower drag design will be more efficient for the same impulse."

That must be a typo - you must have meant 'no doubt'.

In theory longer burn times at slower velocities are better, but in practice the much lower ISP of typical longer burning propellants is hard to overcome. The two motors I gave as examples show this - the long burn has 15% lower total impulse in the same motor case. That's pretty typical. I saw the demo flight of the new AT long burn H13ST motor at Airfest and it did not go very well. Likely just a poor rocket design. But to achieve maximum altitude with such a low thrust requires an impossibly light rocket design. (Well maybe not impossible but very challenging.) I ran some quick sims in OpenRocket and from a practical standpoint it's much easier to get the same altitude with an AT H125W than with the H13ST.

My observation that motor impulse is tied to diameter was simply a response to your comment "The first thing that jumps out is the altitude does not double with doubling impulse as the first order prediction from the rocket equation." Your statement seems to ignores that fact that in order to double impulse (especially as the class increases) the motor diameter increases as well.

I've been on an altitude kick for quite a while and the AT motors are certainly intriguing. But based on the sims I've run, at least on the H13, it looks like it will be quite a challenge to build a rocket that matches the thrust profile of that motor for optimum altitude. I've flown a fair number of MD all carbon fiber rockets at BALLS slowly working my way up to extreme optimization for a particular motor. It will be very interesting to see what types of rockets builders come up with for the H13.

Tony

#### Funkworks

##### Well-Known Member
A FoM in interesting when a goal is universal or agreed on by a community.

In model rocketry, there are many different goals that can conflict.

So maybe there could be one FoM for those who want to go fast, one for those who want to go high, one for this and one for that.

Specs for an F1 racer are different than those of a monster truck yet they’re both automobiles.

#### manixFan

##### Not a rocket scientist
A FoM in interesting when a goal is universal or agreed on by a community.

In model rocketry, there are many different goals that can conflict.

So maybe there could be one FoM for those who want to go fast, one for those who want to go high, one for this and one for that.

Specs for an F1 racer are different than those of a monster truck yet they’re both automobiles.
I agree completely that there are many FoMs for any hobby or sport. But in staying with the spirit of the OP, he is looking for a pure performance measurement. So for at least this discussion, it makes sense that we limit the topic of discussion to that very narrow scope.

However, I must admit, after working on a number of altitude seeking 38mm and 54mm MD CF rockets, I am ready to try something else. I'm getting tired of hauling my tower out every time I want to launch and the challenges of working in very tight quarters of short MD rockets. At Airfest I noticed that I really liked a more 'scale look' to flights - moderate speed with longer burns and lots of flame and smoke, much like classic White Lightning on a larger rocket. So when my current builds run their course, I'll be ready for a different FoM, much like some of the others mentioned above. I also need to up my painting game, which is yet another metric.

Tony