J-2X Doghouse: The Rocket Equation! Wahoo!

J-2X Doghouse: The Rocket Equation!  Wahoo!

Welcome back to the J-2X Doghouse.  We’ve talked before about what a rocket actually is and we’ve talked about different kinds of rocket engine cycles and where J-2X fits in that family.  This time, in response to a couple of early requests on this blog, I’d like to talk about rocket engine performance characteristics and how they relate to successfully getting off the planet and into orbit.  Because this comes down to a matter of equations – and therefore at least half of the reading audience will click on their Facebook icon as soon as they see any equations – let me start with an old joke:

Q:  What do engineers use for birth control?
A:  Their personalities.

Of course you know that joke can’t be entirely true if you’ve read any of the articles profiling the J-2X office.  Just about everyone in the office have children.  But somewhere, deep down, in order for that joke to have lasted so long, there must be a tiny sliver of truth.  Okay, yes, I admit it, here it is: People who become engineers do so for a whole variety of reasons but typically share in common an aptitude towards mathematics and a desire to know how things work. 

For me, this whole “future-engineer” notion translated to an appreciation of physics and the representation of the real world, to some approximation, in equations.  Seriously, just think about that for a moment.  You can pick up a pencil, draw a simple sketch, apply some fundamental laws, and, boom, you’ve got a prediction right there on your paper for how the real world will function.  Now that’s darn exciting!  At least it is to me.  But, okay, word of advice:  Showing this level of enthusiasm regarding physics and neato equations is generally NOT good fodder for a first date.  Trust me.

However, since I’ve not had a “first date” in over a quarter of a century, I am now going to explain the derivation of the foundational equation for all rocketry:  The Rocket Equation.  Approximately 99.7% of the world’s population at large does not know this … and, yes, that is a 100% unverified, made-up statistic.  Regardless, today you will join an elite, exclusive, and fashionably eccentric club. 

[Warning:  Some of the mathematics gets a bit heavy here, but some of y’all asked for it.]

First, we start with a simple drawing.  Please note that my wife is the artist in the family; I “draw” in PowerPoint.  Sorry.

What you have is a thing, a blob, at time equal to t0 with a mass of M moving at a velocity of v.  At this point, don’t think of the blob as a rocket.  It’s just a thing in an imaginary space where there is no gravity, no friction, no environmental impacts at all.

We then go to the next step in time, time = t0 + dt, where dt is a small increment.

Our blob has ejected from itself a small piece of mass, dm, in the opposite direction from which it was moving.  The small mass has a velocity of vdm in the opposite direction of the original blob.  The blob, by the way, has a mass now diminished by dm and a velocity that has changed by some increment dv.

Do you want to play the gray blob at home?  Okay, do this.  On a smooth floor, tile perhaps, sit in a rolling chair holding a basketball.  Throw the basketball.  You, still in the chair, will roll in the opposite direction from the flight of the basketball.  Your initial velocity, v, was zero.  Your initial mass, M, included you, the chair, and the basketball.  Your new velocity is zero + dv.  Your new mass is now minus that of the basketball, dm, flying in the other direction.  Ta-da!

Now, how do we turn this simple concept and simplistic drawings into rocket science?  Simple, we call up the work of our friend Sir Issac Newton (1642 – 1727).  Newton’s First Law of Motion states: “An object at rest tends to stay at rest and that an object in uniform motion tends to stay in uniform motion unless acted upon by a net external force.”  I told you that there were no external forces acting on our blob and I drew the box around the whole thing, blob and small piece together.  Combining the philosophy of this First Law with the mathematics of the Second Law results in a simple conclusion that in absence of any external forces, momentum is conserved.  So, the total momentum in the first drawing is the same as the total momentum in the second drawing with momentum defined as mass × velocity.

The second term in the right-hand box is negative since the velocity of the small piece is in the opposite direction as the original movement of the blob.  Okay, now multiply this all out and eliminate redundant terms and – thanks to some niceties of differential calculus – eliminate second order terms to yield the following:

The dm = -dM switch-a-roo is possible since the incremental change in mass of the blob over the time period dt is exactly the negative of the piece ejected.

Rather than talking in terms of absolute velocity of the blob and the small piece, let’s instead talk in terms of ve, the “ejection velocity” of the small piece.  So, this is a relative speed.  Then the equation becomes:

Believe it or not, that’s it.  In its most rudimentary, simplified form, that the Rocket Equation expressed over a very small time increment dt.  If you add up a bunch of these very small time increments, or in other words integrate over a measureable time period [Oh no, integral calculus buried in a blog!  Call the blog police!], you get the following:

What does this say?  Equations always say something or they’re worthless.  It says that the change in velocity of a blob is equal to the relative ejection velocity of small pieces flung away from the blob times the natural log of the ratio of initial mass, M0, to final mass of the blob MF.  The natural log thing got in there thanks to the rules of integral calculus.  You’ll have to trust me on that one.

Now, what does this have to do with rockets?  Well, how about rather than ejecting small discrete chunks of mass we think about what a rocket engine does, which is spew out a continuous stream of mass in the form of high-speed hot gases.  The whole derivation above holds for that case with one modification.  The hot gases ejecting from the nozzle create a pressure field at the point of ejection.  That pressure field creates a force acting on the system.  Because there is a force involved now, momentum is not conserved.  The derivation is a bit more complicated, but it’s not too bad.  The result looks like this:

Where ueq is called the “equivalent exhaust velocity” and is defined as:

The first portion of that last equation deals with the pressure field.  Basically, it is the exhaust pressure at the end of the nozzle, Pe, minus the ambient pressure outside, Pa, times the exit area of the nozzle, Ae.  Force equals pressure acting over an area.  Simple.  The “m-dot” term is the mass flow of the hot gases out of the nozzle.

We’re almost there.  Really.  Hold on. 

Next, I want to define thrust.  We could have started this way by drawing a control volume around a rocket, but I like starting with the blob.  Thrust, T, is the force that the engine imparts on the vehicle.  So, it includes the pressure field aspect and it includes the aspect of ejecting hot gases at high speeds.  Here it is:

If you put a rocket engine on the test stand, fire it, and measure how hard it pushes against the stand, this is what you are measuring.  Sometimes you will see a rocket engine specification that will talk about “vacuum thrust” or “sea level thrust.”  The difference between those can be found in the ambient pressure term, Pa, in the equation above.  In a vacuum, Pa = 0.  At sea level, Pa = 14.7 pounds-force per square inch.  Note that depending on your system of measurements, there could be a “g-factor” conversion lurking in the mass flux term so be careful.

Substituting back into the Rocket Equation, we get this:

I have now cleverly introduced the concept of specific impulse, Isp, which is thrust divided by mass flowrate.  When talking about rocket engines, we typically describe this parameter as being analogous to gas mileage so that people can understand, but here you can see that it’s an integral part of the basic physics of the acceleration of a rocket vehicle.  (Again, beware of hidden g-factor conversions.)

Note that earlier I said that we didn’t have any gravity in our hypothetical situation and that we didn’t have any friction.  I can now add these things into our system a simplistic manner to facilitate the final discussion and to present the final equation:

That, right there, believe it or not, tells you 90% of the whole story about how rockets get into orbit and even how they go from there into the rest of the cosmos.  What do you need to get into and stay in orbit?  A lot of velocity.  And this equation tells you all about it.  Listen carefully to rocket scientists talking about “delta-v” in movies or the news or in documentaries.  “Delta-v” is everything.  You need so much delta-v to get to orbit.  You need so much delta-v to change orbits.  You need so much delta-v to get out of orbit and head towards the moon or anywhere else.  Whenever you hear this, they are referring to the Rocket Equation.

Let’s break it down by starting with the last two terms on the right-hand side.  These are loss terms and that’s why they are negative.  First, as long as you are gaining altitude, you are fighting against gravity.  If we had some way to dial down gravity, we could launch rockets more efficiently because this term would lessen (we’d also all float away).  That’s intuitive.  It takes energy to lift something.  Second, as long as you have friction caused by drag against the atmosphere, you’ve got losses.  Compared to a vacuum, especially at high speeds, our atmosphere is like soup to a launch vehicle and you need energy to overcome it.  Thus, both of these loss terms tell you that for greatest efficiency, it’s best to get up high, out of the atmosphere, and level out to stop fighting gravity as fast as you possibly can.  And that’s exactly how we launch rockets.  It’s not an accident or a whim.  It’s physics.

Now, the first term on the right-hand side in that final equation has two pieces.  First, there is specific impulse, Isp.  That is a measurement of how efficiently the rocket engine produces thrust.  For a given amount of propellant the engine produces this much thrust.  Second, there is the ratio of masses within the natural logarithm.  What this says is that the lower the final mass of the vehicle is relative to the starting mass, the more velocity can be gained.  So, what you want is very low final, burn-out mass as compared to where you started.  When you get to the end, you don’t want much by way of leftover, unused propellants, and you want as little superfluous structure as possible.  Remember, part of your final mass is your payload, which is the satellite or your capsule filled with astronauts.  That’s the important stuff.  It’s this notion of discarding unnecessary and heavy stuff along the way that results in the reason why rockets typically have multiple stages.  As you go along, you toss off heavy structures that you no longer need:  The more stuff that you can shed, the less that you have to carry along, and the more velocity that you can pick up.  Again, it’s intuitive.

If you’ve made it this far and if you’ve grasped the basic concepts of the physics involved, you truly know more about rocketry than almost anybody you’ll meet.  And it’s amazing how intuitive it all ends up once you’ve plowed through the mathematics.  Good equations are those that can tell a good story.  The Rocket Equation is one such equation.

My recommendation is that you print this out and take it along with you on your next date.  Really, you’ll be a big hit!  (Or not.)


11 thoughts on “J-2X Doghouse: The Rocket Equation! Wahoo!”

  1. well explained.. thanks. a difficult subject to cover, too.

    I’d be interested in (say) a comparison using our newly learned equations of how a J-2X compares with a cluster of RL-10 engines. presumably performance is better because the J-2X has less mass than the cluster of RL-10s? or does the J-2X have better ISP or thrust, too?

  2. @Starsilk

    Excellent question. The answer is actually hidden in the assumptions behind the derivation above. One assumption that I’ve made is that the thrust level is great enough to overcome your loss terms. If that is not true – if you are not throwing overboard propellants at a great enough rate and in an efficient manner – then your delta-v could go negative. Put more simply, if you don’t have enough thrust, you fall out of the sky during launch. That is why very low-thrust propulsion systems can be used for spacecraft once you’ve gotten out of the atmosphere and out of the gravity well.

    So, J-2X puts out approximately 290,000 pounds-force thrust (vacuum conditions). An RL10-A4-3 puts out approximately 24,000 pounds thrust (vacuum conditions). To make an equivalent thrust cluster of RL10s to substitute for one J-2X, I would need a cluster of twelve engines. If I have a vehicle configuration with, say, three J-2X engines, then an equivalent cluster of RL10s would consist of 36 engines for that vehicle. That’s not truly a practical design.

    The RL10 actually has a higher specific impulse than J-2X owing to its engine cycle: closed-expander and it is an excellent engine for certain missions. But if you are looking at a larger, heavy-lift configuration where you need the thrust of a J-2X (or multiple J-2Xs), then the clustering considerations make the RL10 choice untenable. And, yes, it all comes back to the rocket equation and your losses at a given point in your mission trajectory versus your engine performance needs. It is not that one engine is better than the other. It is that one fits better within a given mission.

  3. very interesting… thank you.

    I guess a fairer comparison then would be with something like an NK-43 (vacuum version of the NK-33). it (supposedly) puts out 390Klbf in a vacuum, but the ISP is a lot lower being kerolox (346s vacuum, versus J2-X 448s?).

    just trying to get a handle on where J2-X fits in the landscape of available engines.. obviously there are also some ‘ownership’ issues which might make using Russian engines a problem on some vehicles.

  4. @starsilk: I will skillfully sidestep the quagmire of rocket engine and international corporate politics by simply saying that J-2X was conceived to fill a particular mission. Now, that’s not the only mission that it can fulfill, of course, but there were baseline requirements and the design is intended to meet those.

    With the full nozzle extension, J-2X has a high specific impulse, nearly that of hydrogen engines using staged combustion or an expander cycle. No kerosene engine can come close to hydrogen in terms of upper stage propulsion performance (ROCKET EQUATION!).

    J-2X has a pretty hefty thrust level: more than the heritage J-2, less than SSME, much more than RL10, much less than RS-68 or any of a number of other booster engines. Actually, Vulcain is in the same thrust class, though it has a different mission profile.

    J-2X is intended to be started at altitude and then, additionally, have a second start on orbit for a tranfer trajectory to the moon or Mars or wherever we’re going next. SSME was not intended for an altitude start (though that could probably be overcome) and not for a second start (far more difficult to overcome). RL10, on the other hand, does have multiple re-start capability.

    And, finally, J-2X has been designed to a stringent set of standards and subject to a barrage of modern analyses like no engine before for NASA specifically because it was within our requirement set to make it “human rated,” meaning both that it is analytically safer and that we better understand our margins than at similar points in other engine program lifecycles.

    Bottom line: Rocket engines are expensive, specialized machines designed for specialized, difficult missions. If you try to make one that makes everyone happy in terms of performance and flexiblity, everyone ultimately will be miserable since you could never afford it. Make one stupidly simple and (relatively) cheap, and again everyone ultimately will be miserable since you’ve taken out any flexibility and possibily compromised on your margins and safety. Finding a balance between the extremes is an uneasy and inexact science.

  5. thanks. I appreciate the comparison.

    it’s certainly interesting to be reminded of the amount of margin analysis etc that has gone into the production of this new engine – that’s something that often gets ‘forgotten’ when comparing with for example the ex-Soviet engines which are basically ‘gifts from the gods’ with no real proof of ‘man rating’.. lost (if it ever existed) over the last 30 years.

  6. Thanks for the theory. Been too long married to remember what dating was all about. Engineers are sometimes concerned with how efficient a process is. So, how is thermodynamic efficiency calculated for a rocket engine? That is, how would you define work for a rocket engine and what percentage of the chemical energy generated by combustion is converted into work?

  7. @ Steve: Thermodynamic efficiency of a rocket engine is typically done as a comparison to an ideal chemical equilibrium reaction and ideal one-dimensional flow calculation. There are two primary performance characteristics for an engine. The first is called c-star and the second is called the thrust coefficient.

    C-star, also known as the “characteristic velocity,” is a measure of the efficiency and effectiveness of the injection and combustion of propellants in the combustion chamber. Mathematically, it is defined as: c-star = P0*A-star/m-dot. P0 is the stagnation pressure in the combustion chamber. A-star is the nozzle throat area, your choke point. And m-dot is the overboard mass flow. In essence, c-star is a measure of how much pressure you generate for a given flowrate of propellants. You can use ideal thermodynamics and chemical equilibrium calculations to tell you what you ought to be able to get out of a particular propellant combination. Our measure of true efficiency for a given design is then defined as the ratio of the as-tested, measured c-star to the theoretical value. This is called c-star-efficiency. Typically, we can see c-star-efficiency values for modern injectors/combustion chambers in the 98% to 99.5% range. If you define your efficiency for the overall engine so as to include engine cycle impacts, these numbers are reduced.

    The second factor is thrust coefficient or Cf. This is a measure of the efficiency of the expansion of your combustion products downstream of the throat. Mathematically, it is defined as: Cf = T/P0/A-star. Where “T” is thrust. So, it tells you how much thrust you produce for a given chamber pressure and throat size. Ideal thermodynamics can tell you what this ought to be for a given propellant combination and for a given nozzle expansion ratio. And, again, we take the measured value off of the test stand and compare that to the ideal value to yield a thrust-coefficient-efficiency value. Here again the typical numbers are in the upper nineties percent, ~99%, though it can get a bit more complex when taking into consideration multi-dimensional considerations and boundary layer effects.

    Note that: c-star * Cf = specific impulse.

  8. Thanks for the explanation of efficiency. It looks like you have pushed the envelope for engine performance about as far as is reasonably possible. Reducing the cost of manufacturing these engines might still be a big challenge; if your design works, possibly a goal for follow-on projects.

  9. You mentioned the barrage of modern analysis..with that level of detail, have you discovered new variables that impact the basic (or complicated) equations? Or perhaps a new twist on an old idea?
    I’m just wondering how the J-2X project has pushed the boundaries of your field of science 🙂
    Have you developed new analysis techniques that haven’t been used before?

  10. @Aaron: What I’ve presented here, in the blog, is basic textbook stuff. By the way, my favorite text on the topic (simply because it’s the one I had in college) is “Mechanics and Thermodynamics of Propulsion” by Hill and Peterson. The analysis shown here is entirely generic and based on first principles. So, no, our modern analyses are not going to impact this much with regards to the engine.

    Mostly our modern analysis techniques deal with the internal workings of the engine. At the system level, we have steady-state and transient engine system modeling tools that allow us to determine extreme potential operating conditions based upon statistical variations in boundary conditions and hardware performance. We have design tools that can then take these conditions, apply three-dimensional stress analysis and thermal analysis to produce the necessary design characteristics at key points. We use computational fluid dynamics almost anywhere we have complex flow fields and translate these fluid results into structural loads conditions, potential acoustic responses from the hardware, leading to, in some cases, analyses regarding fatigue and fracture control.

    Truly, we have analysis and design tools today that could only be dreamed of just ten or fifteen years ago. There have been times when occasionally we have even gotten ourselves into a bind by doing “too much” analysis (also referred to “paralysis by analysis”), but that is usually a temporary state overcome by good, old-fashioned validation in testing at the component or subcomponent level. We will be doing even more of this type of analysis validation and anchoring with our development testing of Engine 10001 and with the PowerPack Assembly. That is how we evolve from the point of an engineer having a good idea with regards to analysis to the point of defining an accepted analysis and design practice.

    The goal of all of this is to better understand the engine. The more you understand, the more likely will be your success. Also, the more that you understand, the more able you will be to recover from the inevitable setbacks that will occur during development testing. No matter how smart we think we are, rocket engine hot fire is the ultimate experience of humbling education and yet also the ultimate proof that we eventually came to the right design to enable our larger mission.

  11. About the rocket engines, is NASA, JPL or the Air Force using any form of scram jet tech in space travel? Why? And, or, why not?
    I was reading your rocket science equation, and right off the bat I began quesioning where is he applying gravity or G a Gravitational constant? Also, where in the equation is the force being appplied on the force moving upwards? Or, better asked, there is force on the rocket moving upwards besides the force of gravity, and that is also besides the force of P pressure that surrounds the totale veichle, sorta like at deep ocean levels. Physics is interesting to me as is rocet science and C.E.V.’s Civilian enviromental vehicles, which is like a space suttle yet more like a space suttle conneted to a luner orbiter and a payload area of the third stage. It is launched from buillding the puzzle pieces from a platform in gravitational orbit. Or, put your space suit on and travel up a elevator cable to gravitation orbit. Heliophysics is fascinating as well. I always wanted to capute a bolt of lightning to power America for one full year, yet the X class flares have so much power inthem that you could power America for a million years… that is an X class flare that is three of four times the size of our sun.

    When Einstien developed energy equals mass times the speed of light squared, it has mutipule text forms to be applied to, in the big bang theory how does that equation hold up to how matter or mass was made in our Universe in the first place? Something had to start mass and the electrical fields everywhere or gravitons, etc. What if there were mutipule Univereses, and like what the solar sun does in an X class flare, where it joins a huge magnetic loop to another sun spot, what if the same applied in muti universes in electric fields, where there is this less then of what the make up or matter of our Universe is?

    How can anyone conclude that the Universe that we are in is continual and never ending? What if there was like sheets of universes, sorta like papers piled on each other, in cross sections, and even know they are piled upon each other, in between there is another form of space where there is a force and a pressure applied to each sheet of paper acting as another universe?

    Unlock the basic fundemetal laws of teh Universe adn the very foundation of all apperatuses would be efficent, effective and less costly, and would work in comparraion to the Know Universe and Universes. I wish I could take a bar magnet and spin it on my finger. If I was to have multipule bar magnets spinning on one axle or rod, which a bar magnet is reqtangular shaped, it’s flux lines are unique. There is a lobe of flux lines or an egg looking type of flus lines. Take other magnets and place them flatly, and slithly away from the magnets on the spindal, and could youhave magnets pushing magnets around a shaft that is connected to aan alternator or generator?

    On Earth we are already harnessing the energy from the sun. Can we get closer to teh sun to harness the energy from any class of solar flare? I wish I could plug a giant electrical cord into the sun to power earth, yet I believe in micro sense thta already takes place as it is of 46% solar radiation enter earth’s atmosphere. If we had solar collectors in gravitational orbit to collect solar emitances from the sun, or even set monster sized solar cells on the face of the mooon that we always see, then somehow find a way to store that emf up there for electrical uses when we are on the sun, and or beam it back to Earth with a lazer once collected, thenn I think we would be in business.

    g=Xfsquared “My own equations of a hydroelectrics dam that works under the suface of a lake or ocean. Build a geotech dome, or half of a shpere under the surface of a water table, a half of a globe sorta speak, round so al p is equal, or all pressures are equal, without any flat sides, build it above a pump that would pump the water out of the geotech dome, then dig a hole to crawl into teh dome next to it to dig another hole for a turbine that would be implanted in the ocean or lake’s bottom floor level. Build a pump in the orignal tunnel that was made for the men to crawl in, and build that pump to pump out the water from the geotech dome so the workers could build the hydroelectrics dam under water. The atmospheric pressures and the ocean level pressures times the angular pressures of water going into the hole where the tubine is at would help the efficency of the under water pump X to pump the water out at abyssal plan surface level or then carry it to sea suface level.

    What are the fundamental laws of Newton and how do those apply to multi universes also including thermodynamics of heat transfer which heat is an engine or energy, manmade or otherwise. How does that inner play with a theory of mulitidemational Universes? And if there are, then thermodynamics had to be one reasoning point as to how mass was created in the Universe along with whatever is inbetween the universes…sorta like an atom, an atom has protons nuetrons and electrons, yet they are not connected. They have a field between them, at least between the electrons and the inner circle core of the protons and neutrons. There is a fundamental law there that would apply to creation of mass in our universe of mutidementional universes.

    Respectfully yours in science indeed,
    Patrick & Terri

Comments are closed.