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Reducing the cost for space launches is an accepted necessity. The question is, how.  In view of the incredible sizes of rocket hardware (most of which is being thrown away during launch) the engineering approach so far has always been to optimize systems, make the rocket smaller, increase engine power, use better and lighter-weight materials etc. with the ultimate goal of building an SSTO (Single Stage To Orbit) system. The recently approved US X33 Venture Star vehicle follows along that line. Critics point to the exuberant development cost of such projects and their uncertain outcome. Fact is that the so-called high-tech approach, which is best exemplified in the Space Shuttle, has failed to produce any cost reduction over throw-away systems so far - bearing in mind that current throw-away systems are by no means cheap!
 

This verdict shall be subject to a mathematical screening analysis. Can BDB’s be built?

We want our BDB to fly into LEO. Useing kerosine and LOX, for ex, it will have to be a several-stage (number variable) rocket with a pressure-feed system. To fly into orbit, a total velocity requirement of apprx. 9000 m/s will be needed.

Flight of a rocket is mathematically governed by the so-called Tsiolkowsky formula (after Russian school teacher and rocket pioneer Konstantin Tsiolkowski). This formula states that the terminal velocity of a rocket (velocity at burn-out = VB)

is a function of the gas exhaust velocity c and the natural log of the so-called mass ratio (i.e. the total weight of the fully fueld rocket M0 incl. payload divided by the rockets empty / dry weight).

The rocket's take-off weight (or the weight of any stage when it starts its burn) is obviously a sum of several components, such as its structural weight, the fuel weight and the payload it has to carry:

It would be convenient to express the stage structural weight (= dry weight without payload) as a percentage of the fully fueld stage, or:

Putting theese formulas together and with a bit of arithmatic, we will get:

whereby

This formula is good for any stage. For lower stages, the payload is obviously the total mass of all the upper stages including the actual space-going payload.

To do calculations, we must know the following:

• the total velocity requirement (for ex. 9000 m/sec)
• the number of stages
• the velocity gain per stage
• the exaust velocity of each stage
• the percentage of structural weight in a fully fueled stage

For our screening calculations we will further assume (only for reasons of simplicity) that all stages will give an equal incremental velocity gain.

As BDB concepts advocate cheap and simple-to-handle fuels, we will assume a LOX/kerosine mixture. This gives us a c of 3500 m/s for all stages except the lowest one. Here, an average c of 3000 m/s seems reasonable. Actually, at sea level pressure, c will be only 2800 m/s (with a 68 bar = 1000 psi rocket chamber pressure), but it increases as the vehicle climbs into the upper atmosphere. The reader is also encouraged to play with different numbers. Tanks will obviously be much lighter if the pressure were only 30 or 40 bar. Exaust velocities will then be lower, but not much if operated in a vacuum (i.e. for upper stages, pressure becomes less important).

Tank pressure should be apprx. 10 bar higher than chamber pressure to prevent combustion instabilities.

So far, all figures are on pretty secure grounds. The big joker in the calculation then is obviously which percentage of structural weight is achievable. A percentage of 10 (structural weight 10 % of the fully fueld stage) is usually considered pretty good. A percentage of 20 is not very good, obviously, but would it work ? What about 25% etc. What do "real" rockets or typical every-day pressure vessles such as a camping gas tank have ?

Let's do some screening calculations for various percentages and put them into a graphical relationship. The following data and graphical representation gives the overall mass-to-payload ratio for various structural weight percentages and gives the overall take-off-mass-to-payload ratio for a 2, 3 and 4-stage vehicle (graph), respectively (Delta-V = 9000 m/s, C1 = 3000 m/s, all other stages C = 3500 m/s). Similar calculations/graphs could be done for higher Delta-Vs and/or C's)

 

So what does this show us ?

If we could build rocket stages with a 10% structural weight, then a 2-stage vehicle would have an overall mass ratio of arount 38 - in other words, to launch a 1-ton payload, the rocket's take-off weight would be 38 tons. For a 3-stage vehicle, take-off weight would be around 20 tons etc. These figures can be taken directly from the graph. We can also see that there are mass ratios where the rocket will no longer fly. For ex, with a 20% mass ratio, a 2-stage rocket could no longer fly. This is what the Big-Dumb-Boosters opponents claim (and it is obvious). A 3-stage-rocket, however, would again fly with an acceptable overall mass ratio.

It may also be noted from the graph that in the 10%-range the benefit (in fuel saved) in the transgression from a two-stage to a 3 stage vehicle really does not warrant the cost and complexity of the extra stage. LOX and kerosine are both cheap and not worth conserving (in a rocket application !).

So which percentages are realistic ?

Please note line A. This is the Shuttle's SRB (Solid Rocket Booster, clearly visible in picture) percentage (ca. 13,5%). The SRBs are in essence pressure-fed systems (the whole rocket motor case is a pressure vessle). The SRB motor case is not weight optimised ! It is made from steel with a good overpressure safety margin. Even with this non-optimized steel technology, a 2-stage vehicle with an overall mass ratio of 60 would fly. Add a 3^rd stage, and you would be in a comfortable 25:1 payload range. Again, the SRB case is not weight optimised. The Ariane V SRB case, which is also made of steel, but in an improved manufacturing process, is in the 10% range (line C). A camping gas tank made from aluminum (as have become available a few years ago) is in that range, too! Much could be gained if the cases were made of Titanium - or even better: carbon or glass fiber. Then, a percentage of 7 does not seem unrealistic (line B). Glass fiber should not be considered a high-tech or high-cost material. Much could also be gained in building spherical rather than cylindrical tanks.

All in all, there can be no doubt that BDB’s would fly. Usually, arguments against run around added overweight, for ex. the "wet" weight of the engines or a 1% fuel residue etc. If we have a 15% dry weight and a 1% fuel residue, then calculate with 16%. The overall result will not change that much!

Screening calculations show also, that the fuel expense of a BDB is quite acceptalbe. If we spent only 30 kg of a Kerosine/LOX mixture per kg payload as our only or main operational cost at a current market price of , say, $ 7,- then space flight would be so cheap to stand a real chance to replace trips to Mallorca for a vaccation. Clearly, there is other cost, both operational and hardware. Boeing has done studies in the 70ies to come up with a reasonable cost estimate for large pressure tanks made from steel using standard procedures, on a per-kg tank weight basis. Prices were around the $ 7,- range [quote]. Thus, if 15 % of the 30 kg weight expense were tank weight, then our per kg payload expense (fuel and tank) in a total throw-away rocket would still be under $ 40,- - again not expensive. There will of course also be other operational cost, such as for the engine etc. Still, no justification for the prices we see in rocketry today.

Calculations also show, that BDB’s are indeed inferior to optimized systems. A Saturn V, for ex. has an overall mass ratio of 20. So does the Space Shuttle which is even more impressive considering that the Shuttle’s first stage (i.e. the SRB’s) are, in fact, pressure-feed systems. Current believe and textbook opinion is that the bigger the rocket, the more advantageous is a turbo pump systems, as the tanks would become too heavy. Interestingly, this relation, while true of course, must be put into relative proportions. The Space Shuttle SRBs show that turbo pumps are a relative advantage, but not a pre-requisite for large rockets.

So what exactly is dumb ? Usually, BDB’s are associated with cheap materials and simple manufacturing techniques. As always in engineering, once you have a working concept, the temptation to optimize becomes almost overwhelming. BDB’s are no different. When they were thought up, the only materials available were steel, stainless steel and aluminium (being somewhat more difficult to process). Today, we have even lighter-weight materials, namely carbon and glass fiber composites. As a rule of thumb, 1 kg of carbon fiber will replace 5 kg of steel. Thus, if the SRB caseing were made of carbon fiber (why not ?  - they are being refilled), our percentage would theoretically not be 13 but 2.5! Fiber glass is heavier, but still the gain is substantial. A percentage of 5 seems possible. Immagine a BDB with a 5% structural weight. Turbo pumps would be no longer needed! It is true that a fiber glass / epoxy composite is substantially more expensive than steel, but not unreasonably so, in particular if you plan on re-usability. While usully considered high-tech substances, they are really not. Lots of people use them to build boats and other stuff. Fiber glass processing is today no more high tech than welding VA steel. Plus a lot of processing time and cost is gained because there is no need for special bending and shapeing techniques or special machinery. Fiber glass is usually woven around a mandrell which is later remooved. Fiber glass, therefor, fullfills the requirement of a BDB construction material.

As this article's heading states, the above is ment to be a screening calculation. Many more detailed calculations should follow, but as a resume, it is safe to say that we can build BDB's - and today we can build even smaller and smarter Big Dumb Boosters than in the 60ies..

What are we waiting for?