Problems calculating Effective Exhaust Velocity

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Swartz55

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So, I really have no idea where to post this, so I'm going to drop it here and it can be moved to wherever it is appropriate.

I'm trying to calculate the effective exhaust velocity of an engine which burns aluminum oxide and hydrogen gas at a bare-bones minimum temperature of 973.15 degrees Kelvin (700 C, the ignition temperature) The formula I have for specific impulse is Tc x Rgas, where Tc is combustion chamber temperature and Rgas is the specific exhaust velocity. It's a simplified version of the whole equation, and you can see where I got it here. First off, I believe my problem is in my equation for the Universal Gas Constant. The only one I've found that I know how to use is 8.3144621x(Tc/M) where M is the atomic weight. I've seen where the Universal Gas Constant is listed as 8.3144621x(J/mol K), but I don't know what the J, mol or K stand for, and have yet to find a definition.

When I use the weight in amu's, I get a number for the Isp ranging in the thousands (which I doubt), and when I use the weight in kg (1.66053982x10^-27) I get numbers to the power of -32, 40, 60 and so on. Another main issue is that I have been unable to get the same number twice. I know I'm doing something horribly wrong, but I don't know what. Here is everything that I know that I use for my calculations:

Exhaust total weight (amu): 43.98853
Exhaust average weight (amu): 14.66284
R: 8.3144621(Tc/m)
Tc: 973.15 degrees Kelvin
Ve (effective exhaust velocity): Rgas x Tc
Rgas: R/MM (total exhaust gas weight)

I posted this on the Physics Forums, but with over 30 reads and no replies, I assumed that nobody there has any idea how to help. This is for my school Science project, and I need an estimate of the Isp of an ALICE (Aluminum Ice) powered rocket to compare with conventional motel rockets. Plus, I like doing this level of math. Any help would be appreciated. Thanks!
 
OK, just one thing I have to point out. I only have Honors Chem under my belt (taking AP right now) and I know that J/mol K is Joules divided by (Moles*Kelvin). Just a quick thing that googling will solve. :p


On the other hand, why are you trying to calculate this stuff?


Thirdly, do you happen to do anything with rocketry? Just asking. :p



Sorry I cannot answer your question.
 
Swartz55 said:
So, I really have no idea where to post this, so I'm going to drop it here and it can be moved to wherever it is appropriate.

I'm trying to calculate the effective exhaust velocity of an engine which burns aluminum oxide and hydrogen gas at a bare-bones minimum temperature of 973.15 degrees Kelvin (700 C, the ignition temperature) The formula I have for specific impulse is Tc x Rgas, where Tc is combustion chamber temperature and Rgas is the specific exhaust velocity. It's a simplified version of the whole equation, and you can see where I got it here. First off, I believe my problem is in my equation for the Universal Gas Constant. The only one I've found that I know how to use is 8.3144621x(Tc/M) where M is the atomic weight. I've seen where the Universal Gas Constant is listed as 8.3144621x(J/mol K), but I don't know what the J, mol or K stand for, and have yet to find a definition.

When I use the weight in amu's, I get a number for the Isp ranging in the thousands (which I doubt), and when I use the weight in kg (1.66053982x10^-27) I get numbers to the power of -32, 40, 60 and so on. Another main issue is that I have been unable to get the same number twice. I know I'm doing something horribly wrong, but I don't know what. Here is everything that I know that I use for my calculations:

Exhaust total weight (amu): 43.98853
Exhaust average weight (amu): 14.66284
R: 8.3144621(Tc/m)
Tc: 973.15 degrees Kelvin
Ve (effective exhaust velocity): Rgas x Tc
Rgas: R/MM (total exhaust gas weight)

I posted this on the Physics Forums, but with over 30 reads and no replies, I assumed that nobody there has any idea how to help. This is for my school Science project, and I need an estimate of the Isp of an ALICE (Aluminum Ice) powered rocket to compare with conventional motel rockets. Plus, I like doing this level of math. Any help would be appreciated. Thanks!
Click to expand...

I’ve used these formulas before, but to tell you the truth, I don’t understand some of the parameters you’re giving values for. (e.g.; Why is total weight in amu?) Also, I’d suggest that the exhaust products of ALICE are so inideal that such formulas aren’t going to be accurate. Since this is essentially a back-of-the-envelope computation, would suggest you simplify by doing the following:

Suppose you have a given mass of propellant, M, which releases energy of reaction, E. That energy is enough to accelerate the mass to a certain speed, which is an upper bound for your effective exhaust velocity. So

E = .5*M*(Ve)^2 (Under the ridiculous assumption that all of it goes into Ve. Hence the upper bound)

Ve(Upper bound) = SQRT{2*E/M}

In practice, expect 60%-70% of that number, but really in practice… you get what you get.

Where do you get the reaction energy? Grab a Handbook of Chemistry and physics. Get the heats of formation for water (as ice), and Al2O3.

Problems arise because a very high percent (almost all, really) of the exhaust is Al2O3, which is a ceramic. If you assume this is all vaporized, you have very little energy left over to accelerate that gas.
If you assume it's virtually all crystallized, it interferes with your nozzle. (You need a large expanding section to allow the stuff to crystallize, and to harness the heat of crystallization by expanding the hydrogen - which should carry the
Al2O3 along with it... if there's enough of it. Similar problems occur with Zn/S propellants. Essentially, too much of the reaction energy is released outside the combustion chamber.)

Not to burst you bubble, but as a rocket propellant, ALICE is pretty bad.

Regards,
-Larry
 
Swartz55 said:
So, I really have no idea where to post this, so I'm going to drop it here and it can be moved to wherever it is appropriate.

I'm trying to calculate the effective exhaust velocity of an engine which burns aluminum oxide and hydrogen gas at a bare-bones minimum temperature of 973.15 degrees Kelvin (700 C, the ignition temperature) The formula I have for specific impulse is Tc x Rgas, where Tc is combustion chamber temperature and Rgas is the specific exhaust velocity. It's a simplified version of the whole equation, and you can see where I got it here. First off, I believe my problem is in my equation for the Universal Gas Constant. The only one I've found that I know how to use is 8.3144621x(Tc/M) where M is the atomic weight. I've seen where the Universal Gas Constant is listed as 8.3144621x(J/mol K), but I don't know what the J, mol or K stand for, and have yet to find a definition.

When I use the weight in amu's, I get a number for the Isp ranging in the thousands (which I doubt), and when I use the weight in kg (1.66053982x10^-27) I get numbers to the power of -32, 40, 60 and so on. Another main issue is that I have been unable to get the same number twice. I know I'm doing something horribly wrong, but I don't know what. Here is everything that I know that I use for my calculations:

Exhaust total weight (amu): 43.98853
Exhaust average weight (amu): 14.66284
R: 8.3144621(Tc/m)
Tc: 973.15 degrees Kelvin
Ve (effective exhaust velocity): Rgas x Tc
Rgas: R/MM (total exhaust gas weight)

I posted this on the Physics Forums, but with over 30 reads and no replies, I assumed that nobody there has any idea how to help. This is for my school Science project, and I need an estimate of the Isp of an ALICE (Aluminum Ice) powered rocket to compare with conventional motel rockets. Plus, I like doing this level of math. Any help would be appreciated. Thanks!
Click to expand...

Do a quick unit check and you'll find that Tc*Rgas is units of m^2/s^2. Isp is a function of sqrt(Tc*Rgas), specifically Isp=sqrt(Tc*Rgas)/g0, where g0 is acceleration due to gravity at Earth sea level, and Rgas=Runiv/(molar mass of exhaust). I'll assume the following about your chemical reaction:
Al2O3+3H2->3H2O+2Al
Or to get 1 mole of product...
0.2Al2O3+0.6H2->0.6H2O+0.4Al
So the molar mass of products is (0.6*18+0.4*27)=21.6 g/mol.

Of course, you should really use a chemical equilibrium solver like GasEQ (free) to get the equilibrium composition, since that depends on things like temperature and pressure. Of course, this means you'd have to select a chamber pressure and assume perfect expansion in the nozzle. Even this isn't a perfect equation for Isp, but it's pretty close and can certainly be used for comparing Isp's.
 
Last edited:
Holy smokes. This hobby is truly where nerds go to die. Looks like I am one of them, but I will die without knowing who any of this means. Thank you Open Rocket. (Before anyone becomes overly sensitive, I will admit to my nerdiness with pride.)
 
Okay, thanks for the replies! So I guess I have some explaining to do. I'm doing this for my school science fair, where my friends and I are going to compare homemade rockets to store-bought ones, specifically one powered by ALICE and one by a sugar fuel. I'm the only one in my group of three that even bothers trying these equations, but I am no expert so I expected to be totally wrong about a lot.

When I said burns aluminum oxide and hydrogen gas, I actually meant that's what the exhaust is. The exhaust is Al2O3 and H1, with a total molar mass (I believe) of 43.98853 amu's and an average molar mass of 14.66284 amu's. Well, that's what I calculated, but we can all agree that I am a newbie, so I'm probably wrong. It's actually a mixture of aluminum powder 80nm thick and water. We got the idea from a Purdue experiment on a rocket with ALICE. I was listing the weight in amu's because I had no real directive to list them in any other format, and based on my (horrific) calculations, the weights in amu's made numbers more reasonable. Faulty logic, but it seemed that amu's was the correct unit.

So I'm a bit confused as to which equation(s) I should use to find the Isp. We were assuming the temperature at ignition (and would do further calculations once we measured how hot the fuel actually burns) and the simplified equation I linked above (here) eliminated the need for pressure and all that. And again, the Isp doesn't really need to be that great, because we're comparing it to a model rocket with an Isp of 10.

And so, to correctly use the Universal Gas Constant, I would need to do 8.3144621 J/ mol K? Would I find the Joules used in the burn by looking at the heats of formation for water and AL2O3? And then from there, how would I use that? Thank you all for the help, and for persevering with this newbie!
 
If it's any consolation, I really didn't understand the values I gave either. I found all of this information myself, and I didn't really have any idea where to go to get the right information. That's why I'm here!
 
Not sure if you're planning to get to this stage, but don't pour 80 nm aluminum powder into water and stir it up. You're gonna have a really bad time. Look up Kittell's paper on the topic, if it ever got published. ALICE is a mess.
 
Ah. Your chemical reaction, then is the following:
2Al+3H2O->Al2O3+3H2
or
0.5Al+0.75H2O->0.25Al2O3+0.75H2

Again, you should use a chemical equilibrium solver to determine the equilibrium composition, as it won't just be stoichiometric aluminum oxide and hydrogen. (By the way, hydrogen as a product will typically be H2, as H by itself isn't terribly stable--it wants to react!) Anyway, once you find your products, you can find the mass of the products. The reason I use the fractional coefficients in the products is so that it would be easy to compute the mass of 1 mole of products. The amu can easily be converted to useful masses. For example, H2 has a molecular mass of 2 amu, which is also 2 g/mol. Al2O3 has an molecular mass of 102 amu, or 102 g/mol. So, our total products in the stoichiometric case have a mass of (0.25*102+0.75*2), or 78 g/mol. This is independent of state. Combustion at different pressures will give you different products (including oddballs like Al, H2O, OH-, H+, and other odd radicals involving Al, O, and H). Technically, you should account for these as they do influence your result, but stoichiometry will keep you in an order of magnitude.

If you are assuming perfect expansion into a vacuum, you can get away with using the equation I proposed earlier:

Isp=sqrt(Tc*Rgas)/g0
(I realize I used Tc/Rgas previously. That is incorrect. What I wanted to get across is that Isp=f(sqrt(T/MM)). Editing previous post for clarity.)

Where Rgas=(8.3144621 J/(mol K) / 78 g/mol) and g0=9.81 m/s^2.
You may find it helpful to convert the molecular mass to kg/mol (0.078 kg/mol), as Joules are actually kg-m^2/s^2.

You'll find that the performance isn't great at such low temperatures.
 
Last edited:
aerostadt said:
Joe, It looks like you left out the specific heat ratio "gamma". This is another item that the investigator needs to get from a thermochemical run.
Click to expand...

You're right, though, actually it should be gamma/(gamma-1). Dimensional analysis doesn't help you catch omitted nondimensional parameters!

Isp=sqrt(gamma*Tc*Rgas/(gamma-1))/g0

GasEQ will also give you the gamma value of the products of your reaction, so one-stop shopping.
 
So if I use GasEq and plug all those numbers into Isp= sqrt(gamma*Tc*Rgas/gamma-1))/g0, then I can find out the atrocious Isp of ALICE? Sweet! I'm hoping this science project will go far. We're the only kids who are actually building our own rocket and making our own rocket fuel for it, but we're not the only ones doing rockets. I think we're in good standing, though.

So Joules are kilograms minus meters per second squared? Would I just use the rocket's maximum acceleration, or the instantaneous acceleration at a specific point? I do believe I understand everything else, so thanks so much for the help! If my science project goes anywhere, I'll be sure to acknowledge you kind folks.
 
Also, I think we planned on testing 40nm aluminum powder to see if that would work. Would that still be a bad idea to pour into water? How would you recommend I mix it? I certainly don't want any explosions while I'm mixing! And I do know you have to store ALICE and -30 C minimum. I'll look up that paper you referenced, too.
 
You're going to find that the theoretical Isp of ALICE is actually pretty decent. But that does not include the fact that a large percentage of the output material is liquid/solid phase. GasEQ is a pretty crappy solver for that. CEA is a little better and you can run it from the Internet machine. That will give you a sense of the loss in Isp due to condensed phase effects by giving you an amount of condensed phase product. (It will be high.)

But I digress. How you do the calculations is not the problem. Mixing nAl and water is. When you put the nAl into the water, what will happen? What effect do you think decreasing the particle size will have on this event?

Based on your answers, you'll understand why (a) you'll want to do this remotely, and (2) [sic] the finished propellant will either barely burn or not burn at all. The original ALICE motor was supposed to have a K motor's worth of total impulse. It ended up taking a Mongoose 98 to 1300 feet. (This was after they set the specialized $30,000 mixer on fire trying to figure out how to make the stuff safely.)

Note that I'm not disparaging the fantastic work that was done by my labmates over several years—I'm just pointing out that they conclusively proved it's of dubious usefulness for propulsion applications in its current state.

In a high school lab, you're probably going to end up hurting yourself playing with nanosize energetic materials unless you have a very experienced and well-trained supervisor. It's not worth it. I made motors in high school too, using composite propellants—it can be done, with good supervision and training and modern propellants, and you'll have way more fun than you will cursing at a bubbling, exotherming puddle of explosive water/aluminum goo.
 
Swartz55 said:
So Joules are kilograms minus meters per second squared? Would I just use the rocket's maximum acceleration, or the instantaneous acceleration at a specific point? I do believe I understand everything else, so thanks so much for the help! If my science project goes anywhere, I'll be sure to acknowledge you kind folks.
Click to expand...

kg times m^2/s^2. Sometimes you'll see it as a dash. Habit of notation...

Basically energy (Joules) is force dotted over an length.
 
Last edited:
daveyfire said:
You're going to find that the theoretical Isp of ALICE is actually pretty decent. But that does not include the fact that a large percentage of the output material is liquid/solid phase. GasEQ is a pretty crappy solver for that. CEA is a little better and you can run it from the Internet machine. That will give you a sense of the loss in Isp due to condensed phase effects by giving you an amount of condensed phase product. (It will be high.)
Click to expand...

I totally didn't think about that. Good point.
 
Thanks for the warning. I didn't realize it was that dangerous. In their research paper, at least the part I read, they stated that they machine mixed the ALICE but never really said how or why. I'll talk to my lab group to come up with something else. At least now I know how to calculate Isp! Thanks for the help, guys.

P.S. Would you have any other ideas on what kind of rocket fuel would be good and safe to make? We're already making a sugar rocket, too.
 
So, I started to use GasEq to try and find all the information I need to calculate the Isp of an Estes C6-5, just to test my equations and make sure I know how to do this. The Isp is already given for the engine, so I wanted to practice. But my main issue is that I don't believe GasEQ has Potassium, which is important because the engine uses potassium. Any suggestions?
 
What are the intervals CEARUN wants? Like it says low value, high value, interval.
 
Swartz55 said:
What are the intervals CEARUN wants? Like it says low value, high value, interval.
Click to expand...

I think you can make your calculations a lot simpler from a thrust-time curve the obtaining of which would add enormously to your science project. If you have a high speed video camera you can construct a static test stand pretty cheaply.

Don't be afraid of the simple integral if you haven't studied calculus. Nakka gives a very straight forward way of finding the integral graphically.

Richard Nakka's Experimental Rocketry Web Site

https://www.nakka-rocketry.net/impcalc.html
 
You know, if all else fails you can still estimate Isp from the thrust curve and the mass of the motor.

F=Isp*mdot*g0
or
Isp=(Average thrust)/(propellant mass/burn time)/(9.81 m/s^2 or 32.2 ft/s^2). That should be approximately correct assuming constant F and mdot.
 
jmattingly13 said:
You're right, though, actually it should be gamma/(gamma-1). Dimensional analysis doesn't help you catch omitted nondimensional parameters!

Isp=sqrt(gamma*Tc*Rgas/(gamma-1))/g0

GasEQ will also give you the gamma value of the products of your reaction, so one-stop shopping.
Click to expand...

Yes. This works if:

1) You are expanding your exhaust gas bto absolute zero. Otherwise, you have to account for the residual energy in the gas at the exit temperature.

2) If gamma makes sense. (It doesn't maker sense where phase changes pertain.) That is to say, the exhaust gas has to act like a gas.
 
Sounds like the OP is worried about this kind of thing, and will never actually get around to launching a Rocket.
That's pretty sad.
 
Larry Curcio said:
Yes. This works if:

1) You are expanding your exhaust gas bto absolute zero. Otherwise, you have to account for the residual energy in the gas at the exit temperature.

2) If gamma makes sense. (It doesn't maker sense where phase changes pertain.) That is to say, the exhaust gas has to act like a gas.
Click to expand...

1. Technically correct, though the assumption usually makes more sense when you claim to expand to vacuum pressure since nozzles are sized for expansion pressure and not temperatures.
2. Due to the temperatures (and pressures) involved, I don't know of any propulsion systems whose exhausts are anything but gases travelling through the most of the combustion chamber and all of the nozzle. So while gamma might change by a few percent as the reaction progresses downstream, I'd say gamma should generally make sense. If you have counter-examples, please let me know--I would find those interesting.
 
jmattingly13 said:
1. Technically correct, though the assumption usually makes more sense when you claim to expand to vacuum pressure since nozzles are sized for expansion pressure and not temperatures.
2. Due to the temperatures (and pressures) involved, I don't know of any propulsion systems whose exhausts are anything but gases travelling through the most of the combustion chamber and all of the nozzle. So while gamma might change by a few percent as the reaction progresses downstream, I'd say gamma should generally make sense. If you have counter-examples, please let me know--I would find those interesting.
Click to expand...

Alas it's ALICE.
Almost all of the exhaust is Al2O3. In the combustion chamber, this is in equilibrium between solid and gas.

That said, I read an article in which one of the students, after a test, said the measured Isp was close to the theoretical figure. I would expect that the theory looked much like these calculations.
 
After doing a bit of digging around, but not too much, the ignition temperature of that propellant should be sufficient to also gasify the Al2O3. If the products of the reaction were not primarily gases, I don't think it would make much sense to stick a nozzle on the back of the motor. These equations are predicated on a gaseous state of matter.
 
Although Al2O3 will be gas in the combustion chamber, it may not necessarily be a gas in the divergent section of the nozzle after the temperature of the combustion products has dropped substantially. There are at least two ways to handle this 1) make a Computational Fluid Dynamics (CFD) run that includes two phase flow for the particles with perhaps some special provisions 2) making hand calculations that use an equivalent gamma (specific heat ratio) assuming a homogeneous gas that allows for the fact that some of the combustion products are particles (Al2O3) and some is gas (H2). Say, for example, air has a gamma=1.4, but quite often AP-composite solid propellant combustion products may be modeled for engineering purposes with a gamma=1.14 neighborhood.
 
aerostadt said:
Although Al2O3 will be gas in the combustion chamber, it may not necessarily be a gas in the divergent section of the nozzle after the temperature of the combustion products has dropped substantially. There are at least two ways to handle this 1) make a Computational Fluid Dynamics (CFD) run that includes two phase flow for the particles with perhaps some special provisions 2) making hand calculations that use an equivalent gamma (specific heat ratio) assuming a homogeneous gas that allows for the fact that some of the combustion products are particles (Al2O3) and some is gas (H2). Say, for example, air has a gamma=1.4, but quite often AP-composite solid propellant combustion products may be modeled for engineering purposes with a gamma=1.14 neighborhood.
Click to expand...

Another concern is that Al2O3's heat of crystallization is considerable. Since much of the crystallization takes place in the expanding section, you have a highly exothermic reaction going on there. The equations are predicated on adiabatic expansion. (i.e.; no heat comes in or goes out.) Even if the expanding section is designed to take advantage of this reaction, I don't believe the equations account for it.

Same problem as ZnS propellants. People see a blinding flash in the exhaust, and they assume this is from unburned propellant being ejected. Actually, most of it is due to ZnS crystallizing and releasing the resulting heat as light.
 

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