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Hey! I've always been in love with atmospheric flight, but it was not until I played KSP that I found a fondness for spaceflight. I've always had a love/hate relationship with maths, i.e, I love the practical science/engineering/business applications of it, but it costs me horrors to do anything beyond basic equations. Anyway, a few weeks ago, as I was browsing the Internet, I came across a small PDF booklet that piqued my curiosity. It was titled "HOW to DESIGN, BUILD and TEST SMALL LIQUID-FUEL ROCKET ENGINES." I gave it a quick reading, skipping over most of the maths, and realized that the apparent complexity in the design of a rocket engine stems not from the engine itself, which is a relatively simple machine, but from the fact that a flight engine has to fit a very harsh set of criteria: It needs extreme levels of both thrust and efficiency. It has to be extremely lightweight, and the tanks and piping have to be lightweight too. Cost is usually not an issue, or is pretty low in the priority list. That set of criteria produces the awesome beasts we know and love, but in the process also makes them extremely complex and costly machines. (Think turbopumps, regenerative cooling, exotic materials and building techniques, cutting-edge avionics and software, ultra-precise machining, etc) I realized, that, were one to have a different set of priorities, one could take the design of rocket engines out of the realm of the true rocket engine engineers (usually teams of specialists in aerodynamics, chemistry, thermodynamics, stress analysis, avionics, and the list goes on and on) and into the hands of a single hobbyist with barely high-school math skills like me. Enthusiastic, I gave the book a more thorough reading, and found out that it was more of a "How To" guide (Insert X value into Equation 4, take it from table B, and so on), and less of a true rocket engine design book. Given the fact that I actually want to learn design instead of just blindly following along a guide, I decided upon complementing it with other bibliography, mainly "Rocket Propulsion Elements", a monster of a book at 700 pages, and filled to the brim with complex math, which, nevertheless, has managed to solve (with considerable effort and headache on my part ) all the doubts such as Why is X done in Y way?, where does this precomputed value we're told to use come from?, etc. left in the wake of the smaller book. Hence I started the design process, and am currently in the phase of producing CAD drawings for manufacturing and assembly (i.e, I'm almost done) I've decided to share the process with you in order to: Give back to this awesome community at least a tiny bit of which it has given me over the years of playing KSP. Fully review the design process from start to finish as I write this, in search of errors. Learn even more as I search for ways to explain complex concepts in forms that are simpler to grasp than mere maths. Without further ado, let's dive in! I started the design process by listing a set of criteria for the engine to meet, in order for it to be a realistic, doable project for myself. Things I want or need: Simple. Safe (Well, as safe as a controlled explosion can be anyway) Cheap to build and operate Things I do not want or need: Extreme high performance. Or any performance at all. As long as it makes a supersonic flame and lots of noise, I'm happy. Lightweight. Expensive/Hard to find/Toxic propellants. Regenerative cooling (Arguably the hardest part in the design of any rocket engine) Expensive/exotic materials. Complex/extremely precise machining of parts. Gimbaling Given that different design criteria, the project becomes a lot simpler indeed! After outlining my requirements, I made the three most basic decisions that will drive the rest of the design process. Propellants to be used. How will the propellants be fed to the engine Thrust level to be achieved. After careful consideration, and a dive into Elements of Rocket Propulsion, and some Wikipedia to check chemical properties, I settled upon Gaseous Oxygen and Methyl Alcohol as propellants. The oxidizer, gaseous oxygen (GOX) is cheap, easy to find, non toxic, non cryogenic (does not require cryogenic valves, piping, engine pre-chilling, etc), has a slightly higher performance than liquid oxygen, and it also comes pre-pressurized (No pump required). It has a big drawback, in that the required tanks and pressure regulation devices are large and heavy (think high pressure storage of a gas which uses up a large volume), and, while that would be an instantaneous No-No for an engine to be used in a flight rocket, it was unimportant for my intended use. The fuel, methyl alcohol, also known as methylene or wood spirits, was chosen because, while it is more expensive than gasoline or kerosene, it burns at lower pressures and temperatures than those, therefore making the unspoken requirement "The engine should not melt/explode" a bit easier to comply with, and it can be bought at any hardware store. Methyl alcohol is toxic, but only upon ingestion and it's not horribly toxic or carcinogenic like other propellants or oxidizers such as hydrazine, aniline, red fuming nitric acid, dinitrogen tetroxide, etc) I also decided to use a pressure fed design, as, in keeping with the simplicity premise, I want to avoid turbopumps, gas generators, and all that sort of things that make complexity, cost, and the number of potential failure points to increase. The thrust level I decided upon was 100 newtons (10.2 Kgf or around 22 lbf). It's pretty darn puny for a rocket engine, but it's a nice round number, and should still be an interesting challenge, which should be achievable without: Needing huge chamber pressures/temperatures. Having a large fuel consumption. Rocket engines are inefficient machines by nature, and I don't want to go broke after the first few minutes of operation, With that decided, it is time to determine the basic operating parameters of the engine, such as mass flow, chamber pressure, etc. that will then be used to determine the materials and physical dimensions of the engine. That is already done, but I have to review it and convert from my scribbled design notes to a good quality post. Until then, I leave you with this render of the combustion chamber / nozzle assembly as a teaser of things to come. Dec 04 2015: In the last installment, I decided upon the propellant combination and thrust target. Today, I will determine the most basic operating parameters of the engine, and, upon those, calculate some other parameters which must be known in order to start calculating the basic physical dimensions of the thrust chamber and nozzle. Also, I keep all the parameters that I determine/calculate, in a large table, that is kept handy and lets me have all the data that I might possibly need, ready at a quick glance. This is the table so far, and from now on all results will be added to it, and data for any calculations, sourced from it too. ENGINE MASTER DATA TABLE Parameter or Dimension Value Metric Imperial Propellants GOX/Methanol Thrust 10.2kgf / 100N 22.5 lbf Now, in order to get started with the physical design, we have to know a few parameters: Chamber pressure (Combustion pressure) Combustion Temperature. Mixture Ratio (Proportion of oxidizer to fuel) Approximate ISP (This is mostly a rough number dependent upon the propellant combination, and will be later adjusted to account for engine geometry losses) These parameters can be calculated, but designing an engine from scratch, with no reference numbers, is a daunting task. Fortunately there are huge amounts of precalculated data on the subject, made available by either government or private organizations, and we can easily source them from tables. As indicated on the table, these parameters are determined for expansion to 14.7 PSI, which is sea level atmospheric pressure. That is good enough for me, because this engine will not be used at very high altitude or a vacuum. (I live at 2700 ft above sea level) Also, not indicated on the table (one of the things the book assumes you to know/realize) is the fact that these pressures, temperatures, and ISP's, are based upon a stoichiometric mixture ratio (there is just enough oxidizer to burn all the fuel). Any other ratio will result in lower pressures and temperatures, and less performance, which makes sense, because you are either low on oxidizer, having unburned fuel go through the engine, and then burn with the outside atmospheric oxygen without producing useful thrust, or you have an excess of oxidizer going through the engine, and, given that there is not an unlimited amount of space inside the engine, any excess in oxidizer means a corresponding lack of fuel. (There is an exception to that if running fuel-rich reduces the molecular weight of your exhaust, such as in hydrogen/oxygen engines, but that is honestly beyond the scope of this discussion) Now we are ready to determine a few other rough parameters, most importantly, the engine mass flow rate. The engine mass flow rate will let us know the mass of propellants required for operating the engine at the desired thrust level. The formula for engine mass flow rate is: To understand why that does even make sense (It took me awhile to realize why it did, and I was very confused before that) you have to take two things into account: Mass conservation principle. No matter what is chemically happening inside the engine as propellants are burned, the same amount of mass that enters, will leave. Unless you somehow create a nuclear reaction, in which case some mass will be converted to energy, but that is a very, very, very unlikely outcome Specific Impulse (ISP) is just a fancy way of saying "Hey, this engine could develop X amount of thrust if it burned 1 unit of mass per unit of time" Thus, given that we know our thrust target and also know our rough ISP, we can proceed to calculate the amount of mass entering and exiting the engine per second. I'm pretty sure this engine will consume less than 0.1 kg of propellant per second, but let's find out the exact value. 100 Newtons are 10.1971621 kgf. Therefore our engine has a thrust of 10197 grams. Ah, metric system, how can I not love you 10197 / 248 = 41,116935483870967741935483870968 grams/sec. So, when the engine consumes 41.12 grams of propellant per second, it will emit 41.12 grams of exhaust gasses per second, and produce the 100 newtons of thrust. (In theory). Now based on that total, we will determine which part of the propellant mass is fuel and which part is oxidizer. (This will be used later in the design of the injectors, and fuel system and is better determined now than then) Given the oxidizer/fuel ratio of 1.2, as per table 1, we then can determine the mass flow ratios to be as follows: Oxidizer flow = Emfr * R / (R+1) Oxidizer flow =0.403*1.2/(1.2+ 1) Oxidizer flow = 0,2198181818181818 newtons / sec oxidizer Fuel Flow = Emfr * R /(R+1) = Fuel Flow = 0.403/(1.2+1) Fuel Flow = 0,1831818181818182 newtons/sec fuel I know I'm not supposed to use newtons as a mass unit, and later realized that mistake, but the results are the same whether I use pounds or grams , only expressed in the pertinent unit. I have no clue why this is so, and if anyone could explain, I'd be grateful. Now with these parameters calculated, we can dive into the meat and potatoes of the project, and start calculating the physical dimensions of the engine, and also you'll get to see me suffer through some more complicated maths, but that will have to wait for the next installment. Until then, this is the engine data table, with all the data that we have determined or calculated so far. ENGINE MASTER DATA TABLE Parameter Value Metric Imperial Propellants GOX/Methanol Thrust 10.2 kgf 22.5 lbf Chamber Pressure 2.068 Mpa 300 psi Mix ratio 1.2 ISP 248 s Total Mass Flow 41.1 gr/s 0.0906 lb/s Mass Flow (Oxidizer) 22.42 gr/s 0.049428 lb/s Mass Flow (Fuel) 18.68 gr/s 0.041182 lb/s If you have any insight, questions, or even better, have found an error, please let me know Dec 12 2015: Hey! After determining operating parameters, today we are going to determine some gas values, that we will then use to determine chamber dimensions, nozzle outlet diameter, expansion ratios, throat diameter, etc. Let's get started! The idea behind a DeLaval nozzle (That's how a rocket nozzle is called) is to transform a high pressure, high temperature, low velocity gas, like the combustion products, into a low pressure, relatively low temperature, and crazy-high speed gas. (Remember that momentum = mass times velocity, and given that gasses tend to be very light, in order to produce useful thrust, velocity has to be extremely high) Velocities of 2 km/s are not unheard of for small hobby engines This image shows the profile of the gases in a DeLaval Nozzle: Notice that the gasses after the throat are supersonic, and that is done in order to prevent pressure perturbations from travelling upstream (any pressure perturbations travel at the speed of sound) This is critical, because otherwise the nozzle would behave as a Venturi tube, and produce an exhaust of similar pressure and velocity as given in the inlet, which would be useless for us. Now, you'd think that calculating a diameter that will produce a desired Mach speed would be easy, but it turns out that the local sound speed (Mach number) of any gas is affected by pressure, temperature, and density... And guess what, a nozzle varies pressure and temperature along its whole length! Now the math starts to pick up in complexity! First, we have to determine the temperature of the gas in the nozzle throat (Tthroat). That is because, as explained above, the gas temperature at the nozzle throat is less than in the combustion chamber due to loss of some thermal energy during the acceleration of the gas to local speed of sound (Mach number = 1) at the throat. Gamma (the Y shaped Greek letter) is the ratio of gas specific heats, a dimensionless value (much like the Mach number), which relates to the heat capacity at a given volume for a gas. For the products of hydrocarbons and gaseous oxygen combustion, Gamma equals 1.2 Tgas = 1 / (1 + ((1.2-1)/2)) Tgas = 0.90909090909 of the Chamber temp. Tgas = 0.90909 * 3155 º K Tgas = 2868.18 º K or 2595 ºc The chamber (combustion) temperature is determined for this propellant combination from Table 1. Now, we have to determine the gas pressure at the nozzle throat.The pressure at the nozzle throat is less than in the combustion chamber due to acceleration of the gas to the local speed of sound (Mach number =1) at the throat, as given by So, Pgas = 300 psi * (1+((1.2-1)/2)) ^-(1.2 / (1.2 – 1)) I just rage-quitted there, and cheated with Wolfram Alpha. (which, by the way, is a wonderful free online tool that I recommend you check out) So, pressure at the throat is 169.34 psi. Quite a large drop, about half of the initial value, which is to be expected in this kind of nozzle. So far, so good. Also, the gas will have to be expanded to atmospheric pressure before exiting the engine. This is important for future calculations. Given that I live at 2700 ft above sea level, I just asked Wolfram Alpha which was the atmospheric pressure at that altitude. Unfortunately, time is in short availability for me right now. So, I give you the current status of our calculations. ENGINE MASTER DATA TABLE Parameter Value Metric Imperial Propellants GOX/Methanol Thrust 10.2kg 22.5 lbf Chamber Pressure 2.068 Mpa 300 psi Maximum Reaction Temperature 3155ºK 5679 ºR Mix ratio 1.2 ISP 248 s Expansion pressure 918 Mb 13.31 psi Total Mass Flow 41.1 gr/s 0.0906 lb/s Mass Flow (Oxidizer) 22.42 gr/s 0.049428 lb/s Mass Flow (Fuel) 18.68 gr/s 0.041182 lb/s Gamma 1.2 Throat Gas Temperature 2868.18ºK 5679 ºR Throat Gas Pressure 1.168 Mpa 169.34 psi Please join me in the next installment, when we determine Mach numbers and finally some physical dimensions! Until then, if you find any errors or have comments/suggestions, please do let me know. Thanks. Dec 18 2015: Hey! Real life has been hell these days! Fortunately, now I've had time to review another part of the design. Onward! Now that the gas parameters, such as temperature and pressure at the throat have been determined, and we know the mass flow of the engine, we can proceed to calculate throat area, and from that, derive throat diameter (The first physical dimension) Throat area is given by: Where R is the universal gas constant, M is the molecular weight of the exhaust gasses, and Gc is the universal gravitation constant. Athroat = ((Mflow/Pthroat) * ((R * Tthroat ) / (Gamma * gravitational constant)) ^1/2 Athroat = (0.0906 lb/sec / 169.34 psi) * ((64.388 foot-pound/pound/degree Rankine * 5679 degrees Rankine)/ (1.2 * 32.2 foot/sec^2 )) ^1/2 Athroat = (0.0906 lb/sec / 169.34 psi) * ((64.388 foot-pound/pound/degree Rankine * 5679 degrees Rankine )/ (1.2 * 32.2 foot/sec ^2 )) ^1/2 Athroat = (0.0906/169.34 psi) * ((64.388 *5679)/ (1.2*32.2)) ^1/2 Athroat = (0.0906/169.34 psi) * (365659.452 / 38.64) ^1/2 Athroat = (0.0906/169.34) * (365659.452 / 38.64) ^1/2 Athroat = (0.0906/169.34) * (365659.452 / 38.64) ^1/2 Athroat = 0.0520461 square inches, or 33.5781 square mm Given this area, we can proceed to determine diameter, by simple geometry of circles. Dthroat= 4*33.5781 / 3.14159265 Dthroat= 4*33.5781 / 3.14159265 Dthroat= 6.53858 mm I'm sorry guys, I really wanted to push out more content today but an unexpected work issue has arisen (yet again) *Sigh*.. I'll have a fuller update ASAP. Sorry for the really crappy little update, but hey, progress is progress! PS: Almost forgot, this is our Data Table now: ENGINE MASTER DATA TABLE Parameter Value Metric Imperial Propellants GOX/Methanol Thrust 10.2kg 22.5 lbf Chamber Pressure 2.068 Mpa 300 psi Maximum Reaction Temperature 3155ºK 5679 ºR Mix ratio 1.2 ISP 248 s Expansion pressure 918 Mb 13.31 psi Total Mass Flow 41.1 gr/s 0.0906 lb/s Mass Flow (Oxidizer) 22.42 gr/s 0.049428 lb/s Mass Flow (Fuel) 18.68 gr/s 0.041182 lb/s Gamma 1.2 Throat Gas Temperature 2868.18ºK 5679 ºR Throat Gas Pressure 1.168 Mpa 169.34 psi Throat Area 33.5781 mm2 0.0520461“2 Throat Diameter 6.53858 mm 0,2574244 “ Feb 18 2015: Hello guys! Sorry I left all of you hanging in there, but I've been having all kinds of Real Life Stuff™ going on! I can't promise updates will be regular anymore, but this project is in no way shelved or anything. In the last installment, we had determined the gas pressure, temperature, and throat area of the nozzle. Now, with that data on hand, we can proceed to calculate the best bell end diameter that will provide expansion to the desired pressure and prevent the engine from running under or overexpanded. (Don't worry, I'll explain these terms in a second) In order for us to understand why expanding to a predetermined pressure is important, you have to go back to the definition of a DeLaval nozzle that I posted some paragraphs above. "The idea behind a DeLaval nozzle [...] is to transform a high pressure, high temperature, low velocity gas, like the combustion products, into a low pressure, relatively low temperature, and crazy-high speed gas." So the nozzle does useful work (accelerating a gas) by taking energy from its heat while reducing its pressure. I never even thought this would be significant, I mean, the larger the expansion, the better the performance you extract from the engine, right? But as a thought experiment, I decided to imagine the "perfect" engine. This perfect engine would have an infinitely large exhaust nozzle, it would drop the exhaust pressure to zero and the exhaust temperature to absolute zero, and thereby convert all the available heat from the exhaust gasses into kinetic energy. Exhaust velocity would NOT be infinite, because there's only a limited amount of heat energy to begin with, and, given the infinitely large nozzle would also be infinitely heavy, that would render our perfect engine useless, but hey, this is only a thought experiment in a perfect vacuum... And then it hit me, that in fact a real engine would not operate in a perfect vacuum, where the ideal exhaust pressure is zero, but it would operate inside an atmosphere, where the ideal expansion is to atmospheric pressure. To better understand why this is so: Imagine you are sitting with your engine at sea level. Therefore, the pressure of the engine exterior is 1 atmosphere, or 14.7 psi. Now imagine you had 300 psi in the combustion chamber, and your hypothetical nozzle had been designed to reduce pressure at the exit to 100 psi. So, what happens when you start said engine? Your nozzle works as expected, and it reduces exhaust pressure to 100 psi, with a proportional temperature drop. Then, once the gasses leave the nozzle, what happens? They immediately proceed to expand to 14.7 psi, further cooling in the process. Therefore your nozzle is underexpanded, and it is wasting gas energy (Remember, any gas that expands outside the engine is useless for thrust, much like excess fuel would be (There is an exception to that if running fuel-rich reduces the molecular weight of your exhaust, such as in hydrogen/oxygen burning engines, but that is honestly beyond the scope of this discussion)). Now to the opposite end of the spectrum: Imagine you take the same engine and change the nozzle for one that goes to, say, 0.5 psi. As the gasses go further down the nozzle, their pressure will decrease, until it matches that of the atmosphere. At said point, they stop expanding, because the atmospheric pressure exerts a force equal and opposite to that of the inner gas pressure, and the exhaust will form a column that is "pinched" by the atmosphere and will exit the bell without expanding any further. This seems like it would be good enough, right? You get a slightly larger and heavier nozzle, but for that price, you make absolutely sure that you're expanding the gas as much as it can expand, and getting all the thermal and pressure energy you can get out of it. The exhaust is as cool as it can get, it's at ambient pressure, and you've extracted all the velocity you can extract. Then who cares if the nozzle is a bit too large? Well, in an ideal world that would be OK, but in the real world, having parts of the nozzle not filled with exhaust is a bad, bad idea. The best that can happen is that the gas "sticks" to the nozzle walls after its expansion is done, you get vacuum "bubbles", Mach diamonds, turbulence, etc. in the exhaust and you lose thrust. (That happens with mild overexpansions) and the worst that can happen is the flame flopping around like crazy and banging the nozzle walls randomly until the vibration, noise, and mechanical stress of the turbulent gas flow cause the engine to experience R.U.D. (Rapid Unscheduled Disassembly) Real rockets have a problem with that. Especially first stages! First stages have to go from sea level to almost a vacuum! So how do they avoid gross underexpansion or overexpansion? Well, by compromising, and using a nozzle that is designed to work halfway between sea level and vacuum. So upon start up they are overexpanded, and as they climb they reach their design altitude (perfectly expanded), and then past that they become underexpanded. Example overexpanded nozzle. You can see the telltale Mach disks. And my favorite underexpanded one, Saturn V going uphill You can check out the expansion of exhaust gasses in this video of the Mars Climate Orbiter launch. Check out how big that plume gets as the atmosphere gets thinner and thinner. That was when I came across what I thought was the simplest engine design calculation so far: With said constant already being helpfully provided by the author. But alas, I'm always curious, and I dived into Rocket Propulsion Elements, to find out why relative gas expansion was so simple. Oh, how I was to repent. Turns out, said constant is only valid for sea level. For expansion to a different pressure, you need either a new constant, or you need to do math of the kind that gives you chills. Nevertheless, once I was in, i had no choice but to press forward (Just kidding, I had fun learning about it) These equations will be used to calculate the Mach speed of the exhaust gasses, and once we have that, find an exhaust area that will yield exhaust pressure equal to the local atmospheric pressure for that Mach number. Once again, Wolfram Alpha proves to be an invaluable tool for the hobbyst rocket engineer who wants to save time and headache. An exhaust velocity of Mach 2.62 sounds incredibly high, but actually, is pretty much on the lowest end of what you will get with a rocket engine. Now that we know the area of the nozzle end, we can use simple circle geometry to calculate a diameter (It's the same formula we already used to derive nozzle throat diameter from nozzle throat area) Dexhaust=0.515609 inches or 13.0965 mm. Therefore our Master Data table now looks like this: ENGINE MASTER DATA TABLE Parameter Value Metric Imperial Propellants GOX/Methanol Thrust 10.2kg 22.5 lbf Chamber Pressure 2.068 Mpa 300 psi Maximum Reaction Temperature 3155ºK 5679 ºR Mix ratio 1.2 ISP 248 s Expansion pressure 918 Mb 13.31 psi Total Mass Flow 41.1 gr/s 0.0906 lb/s Mass Flow (Oxidizer) 22.42 gr/s 0.049428 lb/s Mass Flow (Fuel) 18.68 gr/s 0.041182 lb/s Gamma 1.2 Throat Gas Temperature 2868.18ºK 5679 ºR Throat Gas Pressure 1.168 Mpa 169.34 psi Throat Area 33.5781 mm2 0.0520461 “2 Throat Diameter 6.53858 mm 0,2574244 “ Exhaust gas velocity (Mach) 2.62167 Nozzle exit area 134.71 mm2 0.2088 “2 Nozzle exit diameter 13.0965 mm 0.515609 “ Join me in the next installment, where we'll calculate the combustion chamber parameters, and we will be then ready to begin sketching the innards of the chamber + nozzle. Until then, thanks for your time & patience in dealing with my ramblings, and as always, if you find a mistake, please DO let me know. I happen to dislike explosions if I have to pay for the exploding stuff. Mar 3 2015: Hi! Finally found a bit of free time! Real Life keeps me busy, and usually at the end of the day I'm too knackered to do anything other than crawl into bed .... But enough of my whining! You're here for the possible explosions rocket engine design theory. Given that we now know the throat diameter, and exit diameter, one would think that it's already time to calculate nozzle inlet diameter, but, a quick bit of thinking reveals that the nozzle inlet and chamber outlet are one and the same, so we'll kill two birds with a single stone, and calculate chamber dimensions which we can then use to derive nozzle inlet diameter. We will start by calculating the volume of the chamber, and, knowing that volume, we can make an educated guess about length/diameter ratio, and calculate exact values from there. What would a good volume be? A good volume would be one that ensures adequate mixing, evaporation, and complete combustion of propellants by the time they reach the nozzle inlet. That is so, because the nozzle is designed to work with a specific inlet pressure and temperature. Any propellant that goes past the nozzle inlet, will probably burn in the nozzle, which is a bad idea because temperatures at the throat are already pretty critical (despite being at lower temperatures, the throat is the area with less dissipation surface available, and therefore more susceptible to heat damage) and also it would throw off our pressure and temperature ratios for all the points along the nozzle, and if the chamber is too big, the gasses will have time to cool before they enter the inlet, thus reducing engine performance. So, in resume: Chamber too big: Colder inlet temperatures, performance wasted. Heavier engine. Somewhat easier cooling due to lowered gas temps at the nozzle. Risk of combustion instability. Chamber too small: Dangerously hotter nozzle, performance wasted. Lighter engine. Calculating the aerothermochemodynamics of complex hydrocarbons reacting while changing their state, pressure, mixture ratios, temperature, movement speed, and several other variables, in order to ensure complete combustion, is an awful, hellish nightmare. Trust me, I have looked at it. But turns out, there's a cheat for that. Even Real Life Rocket Scientists™ happen to use it for preliminary designs. It's called "characteristic chamber length" and is defined as the length that a chamber of the same volume should have if it were a straight tube and had no converging nozzle section. Characteristic chamber length, L* or L star, is determined experimentally for different propellant combinations, throat diameter, and combustion pressures, and it can be sourced from tables. For an hydrocarbon burning engine like mine, L* is between 50 to 70 inches. The variation is to account for injector design (propellant mixing) I decided to go with 60 inches. Vchamber = 60 * 0.0520461 cubic inches, therefore Vchamber = 3.122766 in3 or 51.173 cm3 To derive chamber length from volume, we also have to know a diameter. A good diameter for combustion chambers is around 5 times throat diameter. Dc = 5 Dthroat Dc = 5 * 6.53858 Dc = 32.6929 mm – 1.287122 inches This is for the cylindrical portion of the chamber. For a small chamber, we can just assume the convergent segment to be 1/10th of the chamber volume, and be done with it. For the chamber area, i just went with my trusty ally, Wolfram Alpha. Lc = Vc / (1.1 * Ac) Lc = 3.122766 in3 / (1.1 * 1.3012 in2) Lc = 3.122766 / (1.1 * 1.3012) Lc = 3.122766 / 1.43132 = 2.18174 inches - 55,416196 mm And thus, our Engine Master Data Table is beginning to fill with physical dimensions. ENGINE MASTER DATA TABLE Parameter Value Metric Imperial Propellants GOX/Methanol Thrust 10.2kg 22.5 lbf Chamber Pressure 2.068 Mpa 300 psi Maximum Reaction Temperature 3155ºK 5679 ºR Mix ratio 1.2 ISP 248 s Expansion pressure 918 Mb 13.31 psi Total Mass Flow 41.1 gr/s 0.0906 lb/s Mass Flow (Oxidizer) 22.42 gr/s 0.049428 lb/s Mass Flow (Fuel) 18.68 gr/s 0.041182 lb/s Gamma 1.2 Throat Gas Temperature 2868.18ºK 5679 ºR Throat Gas Pressure 1.168 Mpa 169.34 psi Throat Area 33.5781 mm2 0.0520461 “2 Throat Diameter 6.53858 mm 0,2574244 “ Exhaust gas velocity (Mach) 2.62167 Nozzle exit area 134.71 mm2 0.2088 “2 Nozzle exit diameter 13.0965 mm 0.515609 “ Chamber Volume 51.173 Cm3 3.122766 “3 Chamber Diameter 32.6929 mm 1.287122 ” Chamber Area 839.5 mm2 1.3012 “2 Chamber Length (including Convergent Segment) 55,416196 mm 2.18174” Please join me next time, were we'll calculate chamber walls, dabble in safety margins, and make a first crude sketch of the engine (Spoiler: It does end up looking like a rocket engine) Until then, if you happen to find any errors, or have feedback, please do so. Thanks Apr 30 2016: Wow! It's been a long time! Sorry for the delay guys... Real life has been absolutely hectic, work issues, study issues, family issues, you name it you got it! Despite the long time between updates this project is not dead at all and I've been itching to show some of the progress I've made. So, without further ado, let's dive in! In the last installment, we had finished determining chamber and nozzle dimensions, but these are the inside ones only, and now we will calculate wall thickness. Every point in the chamber and nozzle has to be strong enough to resist the pressures involved, otherwise the engine will explode. I've decided that, in order to simplify the design, I will simply use a constant wall thickness, suited for the highest pressure area. This is really overkill for parts of the nozzle where the pressure will be lower, and makes the engine significantly heavier, but greatly simplifies both design and machining. Thus I shall design a vessel that can contain 300 psi with an adequate safety margin. Given that the nozzle will be automatically overbuilt, due to its lower operating pressure, I will treat the chamber as a pipe and thus greatly simplify calculation. The equation for the stress on the wall of a tube is: where S is the stress on the pipe wall, P is Pressure, D is Diameter and Tw is the wall thickness. Thus, if we replace S with the ultimate strength of our material, we can calculate the minimum wall thickness. I choose copper, given that it has excellent thermal conductivity, is easy to machine, and is cheap. The ultimate strength of copper is around 10.000 psi, but I will use a conservative 8000 psi in this calculation. S= P * D / 2Tw Tw = P * D / 2S Tw = 300 psi * 1.287122 inch / 16000 Tw = 300 * 1.287122 /16000 Tw = 0.0241335375 inch or 0.61299185 mm Of course this is the absolute minimum value, and while going with 2 mm wall thickness should be more than enough, there are other things to consider, machinability being a top priority since I don't want this project to be unnecessarily hard to machine (Machining a nozzle with walls of that thickness, in copper, will be very hard to do without deforming it) Therefore, I will make an educated guess and use a 5 mm wall thickness, which should be easy to obtain. That also gives me an 815% safety margin. This baby may melt, but an explosion is now an extremely unlikely outcome. (Thankfully) Obviously this just made the engine a lot heavier, but, then again, I don't care about weight. Now that we know all our dimensions, we need to determine our half angles, or the angles of the lines that join inlet, throat, and outlet, thus conforming the nozzle walls. For this small engine, adding a bell shape would give me major machining headaches, and produce only a minor performance improvement. Based on a simpler geometry proposal by @A Fuzzy Velociraptor, I decided to go with 15º and 40º half angles, jointed by rounded unions. I proceeded to fire up my favorite CAD software and did a quick sketch. (All dimensions in mm) I don't know about you, but to me, that definitely looks like a rocket engine. What do you guys think? Next up: We will tackle the issue of cooling. Hopefully tomorrow. No promises. ENGINE MASTER DATA TABLE Parameter Value Metric Imperial Propellants GOX/Metanol Thrust 10.2kg 22.5 lbf Chamber Pressure 2.068 Mpa 300 psi Maximum Reaction Temperature 3155ºK 5679 ºR Mix ratio 1.2 ISP 248 s Expansion pressure 918 Mb 13.31 psi Total Mass Flow 41.1 gr/s 0.0906 lb/s Mass Flow (Oxidizer) 22.42 gr/s 0.049428 lb/s Mass Flow (Fuel) 18.68 gr/s 0.041182 lb/s Gamma 1.2 Throat Gas Temperature 2868.18ºK 5679 ºR Throat Gas Pressure 1.168 Mpa 169.34 psi Throat Area 33.5781 mm2 0.0520461 “2 Throat Diameter 6.53858 mm 0,2574244 “ Exhaust gas velocity (Mach) 2.62167 Nozzle exit area 134.71 mm2 0.2088 “2 Nozzle exit diameter 13.0965 mm 0.515609 “ Chamber Volume 51.173 Cm3 3.122766 “3 Chamber Diameter 32.6929 mm 1.287122 ” Chamber Area 839.5 mm2 1.3012 “2 Chamber Length + Convergent segment 55,416196 mm 2.18174” Chamber Wall thickness 5 mm 0,19685” Nozzle Half-Angle 15º Nozzle inlet Half-angle 40º May 06 2016: Did I say tomorrow? I totally meant in a week or so! Let's get started on cooling, shall we? In order to understand the cooling needs, we first have to understand how the heat flows through a rocket engine. Most of the heat of combustion is either used up accelerating the gasses, or leaves with the exhaust, while a part of it is transferred to the chamber wall, propellant injectors, and nozzle. Heating is a problem because it can debilitate the metals of the chamber to the point at which they cannot resist the chamber pressure anymore, causing deformations which are usually followed by RUD. Therefore, we can devise of several methods to keep the temperatures within reason. No cooling at all: Use the thermal mass of the engine as a heat sink, then radiate the heat away while the engine is off. Pros: Simplest method - Cons: Run time very constrained. Passive cooling: Use either the engine nozzle or chamber walls exposed to the atmosphere as radiators. Pros: Very reliable - Cons: Complex design, a large run time requires more radiating surface than may be available, and thus, the run time is still limited without adding heavy radiator vanes. Active cooling: Use a cooling fluid circulated against the walls to get heat out of the engine. Pros: Unlimited run time. Potential to be extremely lightweight, if regenerative cooling is used (Regenerative cooling means that propellant doubles as cooling fluid) Cons: Complex design. I shall use Active cooling for this engine, for the following reasons: Safety I: If I design the engine for unlimited run time, the chance of destroying it in a 5 second initial run is extremely low. Safety II: The cooling jacket doubles as a shrapnel shield, and protects the test stand equipment from a possible explosion. It should be an interesting and educative challenge, but not a hardcore one like regenerative cooling. Active cooling works like this: (in this example, the cooling fluid is water) Small hobby rocket engines have an average heat transfer from the hot gasses to the chamber walls of about 0.5 Kw/cm2/sec, or 3Btu/sq inch./sec. Therefore, and assuming a perfect wall conductivity, this is the amount of heat that has to be removed from each square cm of the engine. Now in order to know the total heat transfer per unit time, we have to determine the inner surface area. In order to simplify calculation, I will ignore fillets and treat the engine as a cylinder for the chamber, a truncated cone for the nozzle's convergent section, and another truncated cone for the divergent section. Atotal= Achamber + Anozzle convergent + Anozzle divergent The formula for the surface area of a cylinder is: I shall modify this formula, because I do not want the total area, I only want the area of the side wall + top (the injector plate) The bottom area is shared with the convergent section of the nozzle and there is no material there to absorb heat. Therefore, Achamber = 2 * 3.14159265359 * 16.345 ^ 2 + 2 * 3.14159265359 * 16.345 * 40 So, the area of the chamber inner side walls plus injector plate inner side: Achamber = 5786 sq milimeters. The lateral area of a truncated cone, is as given by: Thus, for the convergent segment of our nozzle, Anozzle c = 3.14159265359 * (16.345 + 3.408) * Sqrt ( (16.345 - 3.408)^2 + 15.838) We use lateral area because the "bottom" of the truncated cone is the chamber radius and is not in contact with the walls, and the "top" is the throat radius, and, as such, also not in contact with walls. Therefore, Anozzle c = 840 sq mm And now, the same for Anozzle d Atotal= Achamber + Anozzle c + Anozzle d Atotal= 5786+ 840 + 145 Atotal= 5786+ 840 + 145 = 6771 square mm, or 67.71 square cm, or 10.5 square inches. The total heat transfer, "Q", is equal to the heat transfer rate "q" times the surface area of the inner walls. Therefore Q = qA Q = 0.5Kw/cm2/sec * 67.71 cm2 And thus, the total heat transfer of the engine is 33.85 Kw, or about 45 horsepower... (For the Imperial guys, about 31.5 BTU/sec) Join me next time, when we will attempt to find out exactly how much water flow does it take to get these insane amounts of heat out of the engine! If such a small engine produces these amounts of heat, my respect for the guys and gals that work on the real deal with regenerative cooling has multiplied hundredfold. May 16 2016 A few days ago, we calculated the amount of waste heat that the engine would output when working, and now we need to devise a means to get said heat out of the engine, in order to keep the operating temperatures as low as possible. Injector cooling is not an issue, as they are cooled by the inflow of propellant. Injector plate and chamber, however, are. For the sake of simplicity, I will stick to using water as coolant. Therefore, the system now has a few defined constraints: The coolant fluid must not boil. I will use water as coolant, for its high specific heat, and availability The system must be more capable than strictly needed. I don't care about mass and therefore I will have ample safety margins. Coolant flow speed of 10 m/sec or around 30 fps The coolant shall enter near the nozzle, flow all the way around the chamber, and leave near the injector plate. The amount of water mass flow (mass/sec) needed can be calculated, given the desired temperature rise and the heat input to the fluid, as given by: This is a simplified equation that only will work for water. For other cooling fluids, you need to factor in specific heat capacity. A good ΔT could be 20 ºC, that way water entering the cooling system at ambient temperature, about 20 ºC, would leave at 40 ºC, and thus a 60 ºC margin would remain before its boiling point. (68 to 108 ºF, 42.22ºF ΔT,) Wm = 31.5 / 40 Wm = 0.7875 pounds/sec, or 357 grams/second of coolant fluid. Another cool thing about using water is that, given a density (σ) of 1kg/lt, we now also know that the engine will need 0.357 liters of water per second in order to operate. (That is around 21.5 liters per minute, or 1290 liters per hour.) Now we have to calculate a pipe of such area as to obtain the desired water flow at the desired flow velocity (10 m/s should be more than enough to prevent boiling for this engine). To simplify calculation, I will treat water as a perfectly incompressible fluid. To obtain the desired mass flow at the desired velocity, the cooling jacket must have an area Ajacket, given by: The cooling jacket will therefore be like a ring around the outside of the chamber walls, with cross-sectional area Ajacket , as given by: where D2 is the inner diameter of the outer jacket and D1 is the outer diameter of the combustion chamber, given by: D1 = Dc +2Tw Where Dc is Chamber inner diameter and Tw is the wall thickness. Now we substitute and solve as this: And thus: D2 = sqrt(4mw/(Vw ^ density ^pi) + D1 ^2) D2 = sqrt(4*0.357kg /(10 m/s ^ 1 kg/lt ^3.14159265359) + 42.69 ^2 ) So, 44.33 mm is the inner diameter of the cooling jacket. I will just round it up to 46 mm for ease of machinability. That will increase coolant consumption without significantly improving cooling, but I don't care about that. Please join me in the next installment, when we finish up the coolant jacket design, including yet again safety margins, and some weird math! Until then, I leave you our ENGINE MASTER DATA TABLE Parameter Value Metric Imperial Propellants GOX/Metanol Thrust 10.2kg 22.5 lbf Chamber Pressure 2068 kpa 300 psi Maximum Reaction Temperature 3155ºK 5679 ºR Mix ratio 1.2 ISP 248 s Expansion pressure 918 Mb 13.31 psi Total Mass Flow 41.1 gr/s 0.0906 lb/s Mass Flow (Oxidizer) 22.42 gr/s 0.049428 lb/s Mass Flow (Fuel) 18.68 gr/s 0.041182 lb/s Gamma 1.2 Throat Gas Temperature 2868.18ºK 5679 ºR Throat Gas Pressure 1168 kpa 169.34 psi Throat Area 33.5781 mm2 0.0520461 “2 Throat Diameter 6.53858 mm 0,2574244 “ Exhaust gas velocity (Mach) 2.62167 Nozzle exit area 134.71 mm2 0.2088 “2 Nozzle exit diameter 13.0965 mm 0.515609 “ Chamber Volume 51.173 cm3 3.122766 “3 Chamber Diameter 32.6929 mm 1.287122 ” Chamber Area 839.5 mm2 1.3012 “2 Chamber Length + Convergent segment 55,416196 mm 2.18174 ” Chamber Wall thickness 5 mm 0,19685” Nozzle Half-Angle 15º Nozzle inlet Half-angle 40º Average wall heat transfer 0.5 kw/sec/cm2 3 Btu/sec/“2 Total inner surface area 67.71 cm2 10.5“2 Total heat transfer 33.85 kw/sec Coolant fluid Water Coolant fluid ΔT 20º C 42.22º F Coolant mass flow 357 grams/sec 0.7875 lb/sec Coolant flow volume 0.357 liters/sec 12.07 fl oz/sec Coolant density 1kg/lt 62.43 lb/ft3 Coolant flow velocity 10 m/s 32.81 ft/sec Coolant jacket inner diameter 46 mm 1.811” June 29 2016 Man, time sure flies when you're having fun horribly busy! On with the show! In the last installment, we had almost finished the cooling jacket, but some dimensions still have to be known, such as jacket inlet/outlet diameters, and jacket wall thickness. I shall use a single outlet, and two offset inlets, in order to produce a swirling motion of the coolant that should help prevent hot spots. In order to avoid pressure variations, and to keep flow speed constant, I shall keep a constant area between inlets, jacket, and outlet. The jacket has to withstand the coolant pressure, but it also doubles as shrapnel shield in case of engine RUD, and thus I will simply go for an overkill 5 mm wall thickness for the jacket, which gives us an outer diameter of 56 mm. The area of the inlets equals 1/2 of the area between the chamber outside wall and the jacket inner wall. This, as given by the area of a circle formula, equals 5221 mm2 for the jacket, and 4499 mm2 for the chamber. Thus, the coolant flow passage area is 722 mm2. and the outlet is 3.032 cm in diameter, while the inlets are half that. I'll just round it to 30 and 15 mm, for ease of machining. I'm starting to feel that the extra area I've added is counterproductive, as the design might be wasteful of water. Although better safe than sorry. I'll stick to those dimensions, and if there's excessive cooling I can simply reduce flow. And with that, the cooling design is done. Next up: Injectors! Oh boy! ENGINE MASTER DATA TABLE Parameter Value Metric Imperial Propellants GOX/Metanol Thrust 10.2kg 22.5 lbf Chamber Pressure 2068 kpa 300 psi Maximum Reaction Temperature 3155ºK 5679 ºR Mix ratio 1.2 ISP 248 s Expansion pressure 918 Mb 13.31 psi Total Mass Flow 41.1 gr/s 0.0906 lb/s Mass Flow (Oxidizer) 22.42 gr/s 0.049428 lb/s Mass Flow (Fuel) 18.68 gr/s 0.041182 lb/s Gamma 1.2 Throat Gas Temperature 2868.18ºK 5679 ºR Throat Gas Pressure 1168 kpa 169.34 psi Throat Area 33.5781 mm2 0.0520461 “2 Throat Diameter 6.53858 mm 0,2574244 “ Exhaust gas velocity (Mach) 2.62167 Nozzle exit area 134.71 mm2 0.2088 “2 Nozzle exit diameter 13.0965 mm 0.515609 “ Chamber Volume 51.173 cm3 3.122766 “3 Chamber Diameter 32.6929 mm 1.287122 ” Chamber Area 839.5 mm2 1.3012 “2 Chamber Length + Convergent segment 55,416196 mm 2.18174 ” Chamber Wall thickness 5 mm 0,19685” Nozzle Half-Angle 15º Nozzle inlet Half-angle 40º Average wall heat transfer 0.5 kw/sec/cm2 3 Btu/sec/“2 Total inner surface area 67.71 cm2 10.5“2 Total heat transfer 33.85 kw/sec Coolant fluid Water Coolant fluid ΔT 20º C 42.22º F Coolant mass flow 357 grams/sec 0.7875 lb/sec Coolant flow volume 0.357 liters/sec 12.07 fl oz/sec Coolant density 1kg/lt 62.43 lb/ft3 Coolant flow velocity 10 m/s 32.81 ft/sec Coolant jacket inner diameter 46 mm 1.811” Coolant flow passage area 722 mm2 1.119”2 Coolant inlets diameter 15 mm 0.5906” Coolant outlet diameter 30 mm 1.181” Mar 10 2017: Not abandoned! It may take me a long time, but this project will be finished come hell or high water! It's been a long time, so I'd recommend that you read from the beginning as a refresher. With that said, let's proceed. So, where was I? Ah, yes, injectors, injectors. The function of an injector is to take high pressure propellants from the feed lines, meter the appropriate amount of each (much like a carburetor), and inject them into the chamber in such a way that they can properly and efficiently burn. There are several kinds of injectors, impinging, showerhead, hollow post, pintle, etc. For this design, I shall use an impinging design. It's easy to design and build, and, while it has several disadvantages (Less efficient, very hard to throttle, small variations in shape cause big mixture irregularities, etc), these disadvantages are irrelevant to the type of engine that I'm designing. There are several "eyeballed" parameters. 100 PSI pressure drop. This should be enough to help prevent instability without requiring structural reinforcement. 20 m/s injection velocity. I was unable to find data on how an injection velocity is chosen for different propellants, however, this value is mid of the range for small hydrocarbon/oxygen engines We can now proceed to determine injector hole area, based on the physical characteristics of the propellants. Ethanol can for all practical purposes be considered incompressible. Thus, the injection area that satisfies the mass flow and injection characteristics is given by Where m is the propellant flow mass, c is the discharge coefficient, δ the density, and Δp the pressure drop. A typical discharge coefficient for round hole, small size injectors with a larger fuel manifold behind is about 0.7 The density of ethanol is about 0.75 g/cm3 at ambient pressure, and almost does not change with pressure. Pressure drop will be 100 psi. And also the bibliography I'm using (For those of you crazy cool enough to attempt a similar project) Title Author Editor DESIGN OF LIQUID PROPELLANT ROCKET ENGINES Dieter K. Huzel and David H. Huang Rocketdyne Division, North American Aviation HOW to DESIGN, BUILD and TEST SMALL LIQUID-FUEL ROCKET ENGINES Leroy J. Krzycki ROCKETLAB / CHINA LAKE, CALIFORNIA MECHANICS AND THERMODYNAMICS OF PROPULSION Philip G. Hill and Carl R. Peterson Addison-Wesley Publishing Company Ignition!: An informal history of liquid rocket propellants John D. Clark Rocket Propulsion Elements 7th Edition GEORGE P. SUTTON and OSCAR BIBLARZ JOHN WILEY & SONS, INC If you have any insight, questions, or even better, have found an error, please let me know
I love putting the RE-M3 "Mainsail" Liquid Engine in my rockets, but since it's the heaviest are there any best liquid fuel engines for first stages out there? (no mods please) I use the 'Skipper' Liquid fuel engine for my second stage.
Just playing Kerbal for a few days and are starting to get the hang of it to put stuff into orbit arround Kerbal. The problem is getting back because the rocket engines wont start to get out of orbit. In my case i am using an LV-909 as i saw in a youtube video from Scott Manley where he uses the same rocket to get back to Kerbal. Have checked everything but the engine wont start only a puff kind of sound and thats it. There is enough electricity and fuel and oxidizer is at 100%. Also tested other kind of rockets but nothing works, what am i doing wrong -.-