Total thrust pressure is the sum of momentum thrust pressure and exit pressure in a rocket motor. Confused yet? Well, it's rocket science. Skip to the bottom line: with only a few exceptions, the deciding dimension of the size of a liquid fuelled rocket motor is the diameter of the nozzle exit. It has to fit in under your booster, and it has to generate a certain amount of thrust to lift your booster off the ground, sea, or preceding stage of flight. As a booster designer, you also want fuel efficiency. Above all, you want a cheap ride to orbit (or else you probably wouldn't bother reading any of this.)
You want fuel efficiency? You therefore want a high expansion nozzle. The higher the expansion ratio, the more efficient the nozzle is at high altitude. There are two problems:
- On Earth's surface, atmospheric pressure tries to keep your rocket exhaust from expanding out of the nozzle. The worst case scenario is that your rocket exhaust quits expanding before it actually leaves the nozzle, separating from the walls and leaving part of the nozzle unused. This is especially annoying in a crosswind, which can cause your exhaust to separate only on one side of the nozzle (and this might happen anyway). This normally happens when a motor is first started, and is why most large boosters are bolted down until the engines are up and running.
- In the vacuum, it would be nice to have an infinite expansion ratio, but you will never have enough room in your design to do this. You therefore have to decide on an expansion ratio that will fit in your booster while also providing enough thrust. The length of the nozzle, and the skirt or truss around it to allow the stage to be carried by the booster's first stage, is generally the deciding factor, but can be mitigated in three ways. The Russians like multi-motor engines (look up the RD-0110 and RD-0124 for examples), but because not all of the base area is nozzle area, the required thrust pressure increases by at least 45.7% (assuming four motors instead of one.) The Americans like extendable nozzles (look up RL-10B-2), which slide the the bottom part of the nozzle up around the top part, much like a plastic lightsaber toy. Many providers (look up Transtage and Breeze-M) like to use donut shaped tanks (technical word is "toroidal") so that they can snuggle the upper, narrow portion of the motor up inside it. You can also see this in certain solid motors (look up Star 10).
As it relates to "big dumb boosters", I'll deal with the first stage only. As a cheap booster designer, you'll be willing to spend about the same amount of money on the first stage as the second stage, and the second stage will be smaller, and therefore somewhat more sophisticated (unless your first stage is much more reusable than your second stage, since the cost per flight is what really matters.)
I did a quick and dirty study on the effects of total thrust pressure scaling all by itself. I picked the typical parameters of a dumb booster that burns oxykerosene fuel at a mixture ratio of 2.5:1, is seven times as long as it is in diameter, uses common bulkhead propellant tanks and miraculous zero-length interstages (resulting from the desire for a quick study, not as a realistic consideration...this is why most launch vehicles, even ones matched to considerably narrower lines than their actual diameters, are considerably longer than 7 diameters...that and the fact that many of them use oxyhydrogen upper stages, which has a much lower average density.) Finally, its burnout mass is 10% of its full mass, and its payload mass is 1.5% of its liftoff mass. It lifts off with an accelleration of 12m/s2 (2.19m/s2 more than gravity, peppy for a big orbital booster, but dangerously pitiful for a model rocket.) After running it through a series of simple geometric formulae (such things as area = pi*radius squared and pressure = force/area), I came up with the following general rules about how the required total thrust pressure would scale using these assumptions:
Booster mass = 7.39 diameters cubed (diameter in m, mass in kg)
Payload mass = 0.348 diameters cubed (diameter in m, mass in kg)
Total thrust pressure required = 88.6 diameters (diameter in m, pressure in kPa)
0.1m (4 inches) diameter: 7.29kg total mass; 348g orbital payload. Requires 8.86kPa
0.2m (8 inches) diameter: 58.3kg total mass; 2.78kg orbital payload. Requires 17.72kPa
0.3m (12 inches) diameter: 197kg total mass; 9.40kg orbital payload. Requires 26.88kPa
As orbital concepts, these above are pretty fanciful. The closest real rockets are high power model rockets. They require about four times as much liftoff accelleration for safe operation, and therefore four times as much thrust pressure. Even the biggest, needing 108kPa, is fairly easy to meet.
New: 19 November 2008: Previously ignored OTRAG CRPU stage evaluated. The OTRAG system sought to achieve low cost orbital access buy building up multistage boosters from small common modules. Details at http://www.astronautix.com/lvs/otrag.htm. The CPRU has a loaded mass of 1500kg, diameter of 0.27m, thrust of 25kN at sea level and a length of 16m (producing a fineness ratio of ). The total thrust pressure achieved is 145.0kPa.
The vehicle failed primarily because of politics before orbit could be reached. It was being built in West Germany, and its neighbours were skittish about its potential use as a missile. This wasn't helped by the use of a Libyan test site and the theft of some CPRU hardware by the Libyans in 1983. Prime Aero (my shorthand for Boeing, Lockheed Martin, Orbital Sciences and their predecessors in the US) saw the OTRAG as a market threat. I've noticed throughout my study of history that when something happens to threaten the US military/industrial complex or energy industry, "conspiracy deception" happens. OTRAG was accused of being a cover for a joint German/South African nuclear cruise missile, and so appears to have fallen victim to this phenomenon. The primer on conspiracy deception is Nick Cook's The Hunt For Zero Point
Technically, OTRAG probably would have made orbit with only small payloads, but I'm sure, had it the chance, the company would probably have expanded to an Ascent Roadmap-like family. The CPRU, when clustered square, produces a 99.5kPa total thrust pressure, suggesting an orbital payload limit in the ballpark of 500kg.
0.5m (20 inches) diameter: 911kg total mass; 43.5kg orbital payload. Requires 44.3kPa
0.8m (31 inches) diameter: 3.73 tonnes total mass; 178kg orbital payload. Requires 70.9kPa
1.0m (39 inches) diameter: 7.29 tonnes total mass; 348kg orbital payload. Requires 88.6kPa
1.2m (47 inches) diameter: 12.6 tonnes total mass; 601kg orbital payload. Requires 106.3kPa
This one best matches the Orbital Pegasus booster, the one that is air dropped. It starts out at about 20m/s2, solid thrust, and masses about twice as much as this. Also, a diameter of 50in. It's actual total thrust pressure is 384kPa, hinting at an unnecessarily smart booster.
1.5m (59 inches) diameter: 24.6 tonnes total mass; 1175kg orbital payload. Requires 132.9kPa
1.8m (71 inches) diameter: 42.5 tonnes total mass; 2030kg orbital payload. Requires 159.5kPa
2.0m (79 inches) diameter: 85.3 tonnes total mass; 2785kg orbital payload. Requires 177.2kPa
The actual diameter of Taurus at 73 tonnes/1380kg payload is 2.36m. It's actual total thrust pressure is 315.5kPa, again hinting at an unncecessarily smart booster. Pegasus and Taurus are solid propellant with very high pressures, resulting in heavy stages and high drag. This is why their payload proportions are smaller. They're not "smart" in the same way as conventional boosters.
2.5m (98 inches) diameter: 117 tonnes total mass; 5438kg orbital payload. Requires 221.5kPa
This line pegs the Delta II very closely. Payload, diameter (base diameter is about 4m with the strap-ons, core diameter is 2.4m) and lift off mass are surprisingly close. Delta II also happens to be the best from a $/kg to orbit perspective in LEO On The Cheap (go to http://www.dunnspace.com/ to download this excellent Air Force University Press book by John R. London III.)
3.0m (118 inches) diameter: 197 tonnes total mass; 6396kg orbital payload. Requires 268.8kPa
This line is probably the limit of practical pressure-fed boosters. It is difficult to exceed 300kPa by much with pressure-fed liquid fuelled motors. With the fancy features to do so (Tridyne pressurization, composite wound tanks), turbopumps become competitive with the better educated dumb boosters. This opinion is based on a few pages of ASME B&PV Code Sec VIII/1 calculations to estimate tank mass proportions at various pressures and a few thousand pages of AFAL Isp Calculator output (also at http://www.dunnspace.com/) and the fact that conventional booster's $/kg payload cost bottoms out right here.
3.5m (138 inches) diameter: 313 tonnes total; 14.9 tonne orbital payload. Requires 310.1kPa
This line pegs the Atlas V non-strapped configurations almost perfectly, except that the payloads are smaller, at about 8 tonnes (and the actual core diameter is 3.81m.) This is probably a side-effect of the two motor arrangement, which is the least efficient way to pack multiple motors under the base of a booster (note: the Atlas V uses and RD-180 engine, which has two RD-190 class motors ganged to a single set of turbopumps. The RD-190 engine (used in Angara) has one such motor, while the RD-170 (Energia strap-on) and RD-171 (Zenit...real name for Sea Launch and Land Launch) have four such motors. American ganged engines include the MA-5 Boost (Atlas to version IIAS) and LR-91 (Titan II, III, and IV core stages). This line also pegs the Soyuz, which is an interesting study in total thrust pressure economics. The gracefully tapered shape of the Soyuz booster results from the relatively low total trust pressure of the four motor ganged engines used in the first and second stages (RD-107 used in the first stage, that is, the strap-ons and the very similar RD-108 engine used in the core second stage.)
4.0m (157 inches) diameter: 467 tonnes total; 22.3 tonnes orbital payload. Requires 354.4kPa
Here we have just passed the largest operational solid rocket motor: the RSRM. Translation for rocketry noobs: the Shuttle strap-on booster, the one that blew up the teacher in 1986. This line pegs almost every booster in the payload class: Ares I, Atlas V, Zenit, Proton, and Ariane 5. Ascent Roadmap's Lilmax is also looking for real estate on this line.
4.5m (177 inches) diameter: 664 tonnes total; 31.7 tonnes orbital payload. Requires 398.7kPa
5.0m (197 inches) diameter: 911 tonnes total; 43.5 tonnes orbital payload. Requires 443.0kPa
6.0m (236 inches) diameter: 1575 tonne total; 75.2 tonnes orbital payload. Requires 531.6kPa
An astute rocket fan will notice that both the total mass, payload mass have passed that of the 6.5m diameter Saturn IB (590 tonnes, 18.6 tonnes orbital payload). Ouchy, no? Actually, the Soyuz is about that diameter across its base. This shows that a wide tapered or squat launch vehicle is another approach to low cost space access by keeping the total thrust pressure requirement down.
7.0 m (276 inches) diameter: 2500 tonnes total, 119.4 tonnes orbital. Requires 620.2kPa
8.0 m (315 inches) diameter: 3732 tonnes total, 178.2 tonnes orbital. Requires 708.8kPa
There would be few matches past this point because the thrust pressure requirements are getting too high for any economical vehicle to match them. For example, the Saturn V (<140 tonnes orbital, 2800 tonnes total, 10.1m diameter) matches the ~7.2m line except for diameter, which lowers the total thrust pressure requirement by about 29% to 440kPa. Saturn V's actual total thrust pressure is 425kPa...pretty close for a quick and dirty study.
9.0 m (354 inches) diameter: 5314 tonnes total, 254 tonnes orbital. Requires 797.4kPa
This is Freezerburn's core diameter. It has no hope of meeting that total thrust pressure with clustered Lilmax modules. The result is that it will have a lot more base real estate than this quick and dirty study thinks such a vehicle should have, following the trend of many successful conventional boosters.
10.0m (394 inches) diameter: 7290 tonnes total, 348 tonnes orbital. Requires 886kPa
And that's as far as my quick and dirty total thrust pressure study went. It is clear that booster designers will have to take the best that today's technology can offer (staged combustion, 20+MPa chamber pressures, composite construction, preheated pressurization systems, etc.) and arrange them very creatively to launch 350 tonne payloads economically. These payloads will be asking for diameters of 10.0m or even more...we can expect their boosters to be even wider at ground level!!