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Cost of a T-Space CXV launch

Posted by: Andy Hill - Tue Aug 23, 2005 8:03 am
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Cost of a T-Space CXV launch 
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Post    Posted on: Thu Sep 27, 2007 2:48 pm
From the numbers listed in the previous post – taken in their more precise versions I get the following surfaces:

Code:
LOX-tank       1. stage   32.66 m^2
Propane-tank   1. stage   21.77 m^2
LOX-tank       2. stage   15.21 m^2
Propane-tank   2. stage    9.43 m^2 – 23.22 m^2
Connection                 1.45 m^2


In total these are 80.52 m^2 to 94.31 m^2– without the nozzles, engines and the container for the payload. The container will be replaced by the vehicle in the case of the CXV and so can be neglected.

The surface(s) of the first stage is/are 54.43 m^2 – of the second stage 26.09 m^2 to 39.88 m^2. Using the aluminium-weight applied in the SpaceX-thread I get the following weights for these surfaces:

Code:
           1 mm         1 cm
1. stage   146.961 kg   1469.61 kg
2. stage    70.443 kg    704.43 kg
         - 107.676 kg – 1076.76 kg
Total      217.404 kg   2174.04 kg
        - 254.637 kg – 2546.37 kg


These weights are the equivalents to the hardware weights of the tanks and stages in the Lunar Soyuz thread and will be subtracted from the total QuickReach-weight later. After subtraction of the weights of the payload, the container of the payload and the propellants the weight of the engines and nozzles is available – and the calculation announced can start.



Dipl.-Volkswirt (bdvb) Augustin (Political Economist)


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Post    Posted on: Tue Oct 09, 2007 5:00 pm
In between I recognized a few imprecisions that had essential impacts on the results and corrected several things.

The thickness applied now is 1 mm. The LOX-tank of the first stage is calculated round now instead of edged. The volume of the connection between the secodn stage and ist nozzle is subtracted from the volume of the propane-tank of the first stage. Another connection has been assumed inside the LOX-tank of the first stage and the two data Airlaunch LLC list in the QuickReach Fact Sheet are applied – meaning another correction of the numbers got using the ruler.

The first correction was required because I originally had rounded up the total length of the QuickReach applying a ruler to the graphic and had forgot it. The result were too large weights. By an experiment I recognized that taking into account the precise geometry of tanks – round instead of edged – would correct that. I applied the ruler again and was reminded that I had rounded up the original result. Next I applied the correct numbers got by this second measurment by a ruler. Again a too large weight resulted. The numbers listed in the QuickReach Fact Sheet removed the problem.

Because of this the question is if more data have to be corrected. I used the ruler to measure the central length as well as the diameter of the QuickReach. Next I calculated the ratio between the diamter and the central length measured. This ratio I multiplied by the real length of the QuickReach listed in the Fact Sheet – and got a too large diameter. The excess was 10 cm which caused the too large weight..

So the question is if the lengths of the tanks and stages are too large also. The only proper way to check that is to calculate the ratio of the two real values listed in the Facht Sheet, compare it to the ratio of the measured values and then to measure the diameter again.

It turned out that the ratio to be got by measurements is NOT equal to the ratio of the real numbers.It’s larger and thus results in a too large diameter. It simply is required to select one of the two measured numbers and to ignore the other. I select the measured length here and ignore the measured diamter. If the graphic would show too large lengths also then this would cause too large weights which still fit into the real weight – and this would be a safety margin.

The calculation is described in the appendix.

The resulting numbers are:

Code:
           geometry   length   diameter   surface   thickness   volume   amount

                                          first stage

LOX-tank   cylinder   4.17     2.1        34.45     0.001       14.45    1
           sph.segm   0.42     2.1         5.50     0.001        1.07    2
conn.      cylinder   4.69     0.11       1.66      0.001        0.05     1
total                 4.59     2.1        31.60     0.001       16.54    1

fl-tank    cylinder   3.51     2.1        23.17     0.001       10.11    1
conn       cylinder   0.89   0.53        1.93     0.001        0.2     1
nozzle     cone       2.22     1.78        6.7      0.001        0,01    1
total                                                            9.9     1



                                          second stage

LOX-tank   sphere     2.22     2.22       13.85     0.001        4.85    1
fl-tank    cylinder   1.43     2.1        16.36     0.001        4.95     1


The total of the surface of the LOX-tank of the first stage will wonder – the reason for that total being less than the sum resulting from the table is that the calculated volume of the spherical segment includes a bottom that doesn’t exist – the area of that bottom has been subtracted to get the total.

From the cylindrcal numbers about the fuel tank of the first stage the volume of other tanks extended into it have been subtracted. The tank fuel-tank of the second stage has been calcukated as cylinder because the round portion of it extending into the fuel-tank of the first stage seems to fit nerly into ist opposite side which then would result in a cylinder.

The volume of the cylindrical connection between the tanks of the second stage and the nozzle of the second stage reduce the volume of the propane tank of the first stage – as well as the nozzle of the scond stage does that. It was required to take that into account also.

Regarding the first stage I recognized the problem that no connection pipe from the propane tank to the engine can be seen. The graphic shows vaguely something that might indicate a pipe going along the stage outside – but this is far from sure and appears a bit strange to me – and so I assume a pipe passing the LOX-tank. The diameter I got by using the ruler and calculating a ratio between the real diameter and a correct diameter in the graphic. By multiplying the ratio with the measured value I got the diameter to be applied

One or two diameters are larger than the diameter Airlaunch LLC listed – this is an error causing a safety margin.

Applying the density of propane whonos applied much earlier in this thread the resulting weights are

Code:
           material   density   weight

                      first stage

LOX-tank   aluminum   2700            85.32
conn.      aluminum   2700          4.48
oxidizer   LOX        1141         18843.20 

fl-tank    aluminum   2700            62.57
fuel       propane     507.21    5014.98



                      second stage

LOX-tank   aluminum   2700      37.41
oxidizer   LOX        1141          5516.96

fl-tank    aluminum   2700            44.18
conn.      aluminum   2700         5.20
nozzle     aluminum   2700        18.09
fuel       propane     507.21    2503.89


Unfortunately whonos has deleted the text of his post I am referring to but I had copied the numbers into an Excel file for afew reasons. He applied 1.92 kg/gallon. According to Wikipedia a gallon is 3.785411784 litres from this a density of 507.21034052764495752940784948959 kg/m^3 is resulting.

The numbers are valid applying the connection pipe inside the LOX-tank – if that pipe is left away the amount of oxygen is higher while the hardware weight is smaller. In difference the total weight of the QuickReach is smaller by much less than 100 kg.

What’s left is the weight of the conical container for the payload. At a length of 3.3 m, a diameter of 2.1 m, athickness of 0.001 m, material aluminum and a bottom plate I get a surface of 21.78237786 m^2 and a weight of 58.81242023 kg. To this the weight of the cylinder connecting it to the portion of the tank has to be added – 36.54 kg at a length of a bit more than 1 m and a surface of 13.53 m^2. These are 94.35 kg in total

The total weight got this way is 32213.52932 kg – then 445.1206772 kg are left to the total weight. If the pipe passing the LOX-tank of the first stage is left away then the weight got is 32209.05431 kg – now 449.5956926 kg are left. This is the weight left for engines.

Before going on the question needs to be considered if the weights left include the payload of 1000 pounds = 453.59237 kg. This would mean that the QuickReach described by the Fact Sheet includes a payload – and that the drop tests are done including a payload. This appears strange and unlikely to me. It would mean that a way had to be found to reduce the weight calculated by a few hundred kg.A short check showed me that four to eight of the pipes and connections are required I included already. So I think that the weight left does NOT include the payload – it will have to be added before the calculation of the QuickReach2.

To check if the weight(s) left are the two engines let’s think of the weights of engines applied in the Lunar Siyuz-thread – they are all close to the weight of the engine of the Block DM which is 230 kg. Two of them would weigh 460 kg – by around 11 kg more than the larger value but by around 15 kg more than the smaller value. Alternatively www.bernd-leitenberger.de say that the engines of the Falcon 1 include a pump weighing 150 kg – two such pumps would fit into both the weights left.

Next the weight needs to be split between the two engines. The nozzels appear to be of very similar size and weight. But the connection of the engine of the first stage is larger and thus will be a bit heavier. I don’t think that it would be of help to do more measurements by a ruler and calculations because the weight will be determined by the inner parts of the connections to a significant degree. So the distribution will be a bit arbitrary. Let’s subtract the weight of a nozzle from the both values and then assign 2/3 of the remainder to the engine of the first stage and 1/3 to that of the second stage. Then the nozzles can be added again to get the weight of each engine. The weight of the nozzle of the engine of the second stage must be subtracted from the weight of the second stage.

So the weights to be applied to calculate a QuickReach2 carrying a CXV weighing 3,600 kg are

Code:
comp.     weight

first stage

tanks       152.36
engine      327.13
prop.     23858.18



second stage

tanks        86.79
engine      172.61
prop.      8038.93
total    8298.33



vehicle     548.94


So now the method applied in the Lunar Siyuz-thread can be applied. But it will be simplified – and in difference to that thread a stages rocket is to be calculated here.

The data listed above result in the following:

Code:
1.stage                      2.stage
          weights   ratio   share      weights   ratio   share
                            prop.                        prop.

vehicle    548.94   0.06     1409.73    548.94    0.69    5542.75
tank       152.36   0.02      391.29     86.79    0.11     876.29
engine     327.13   0.03      777.56    172.61    0.20    1619.90
cargo     8298.33   0.89    21279.61      0.00    0.00       0.00


The ratio between the CXV weighing 3600 kg and the payload container including the payload is 6.56. Applying this to the values above the following results are got:

Code:
          2.stage               1.stage
          weight     geometry   weight      geometry
vehicle   36349.78                9245.14
engine     1619.90                 777.56
ratio                     5.23
cargo
cargo         0.00              111315.52
total     37969.68              121338.21
ratio         5.30                   5.17
orig.
tank       4645.26                2023.18
resul.
total     42614.94              123361.40

LOX       29245.76               97430.88
volume                  25.63                   85.39
diametr                  2.10                       2.10
length                   7.40                   24.65
surface                104.57                  332.23
thick                      0.001                   0.001
tank        282.34                    928.90

propane   13273.30               25930.52
volume                  26.17                   51.12
diametr                  2.10                       2.10
length                   7.56                   14,76
surface                106.62                  201.68
thick                   0.001                   0.001
tank        287.87                    544.55

2 tanks     570.22                1441.56

stage       730.65                1744.33

total     43345.59   125105.73


Then the QuickReach2 plus the CXV weighs 172051.32 kg. Some numbers in the above table(s) may wonder and be incorrect – but this weight I nonetheless consider to be a lower margin. What’s interesting really is the comparison to the earlier calculation which resulted in a total weight of between 300000 kg and 400000 kg. I also neglected 74 kg of the CXV here which has to do with the Lunar Siyuz-thread as well as with the unpreventable imprecisions – but I recognized by experimenting that such a weight does have marginal effects on the results only.

The data about the total amounts of propellant are

Code:
prop.     weight   costs

LOX       126676.64   10134.13 at 0.08 $/kg
propane   39203.82   20385.99 at 0.52 $/kg


An empty QuickReach2 then would weigh 2570.86 kg.This weight of aluminum would cost 6816.31 at an aluminum-price of 2651.375 $/mt. Then to $ 5 mio $ 4962663.57 are left. Remember the results for the Merlins got in the SpaceX-thread up to now – it’s no problem to keep the costs within the margins of DARPA’s program. The QuickReach-engines might be less complex and cheaper than the Merlins. They are NOT reusable at present – but it might be very interesting to make them reusable unless the ready QuickReach costs less than $ 100000. The reusability might drop the flight costs down to less than between $ 25.000 and $ 10.000 perhaps.

But what's essential here is that the QuickReach2 easyly can cost less and even much less than $ 10 mio!

During the calculations I kept the diameter of the QuicjReach at 2.1 m. I experimented in short with a diameter of 4.1 m. This saves around 1100 kg 700 kg of which are for the empty QuickReach.

Appendix

Calculation of QuickReach

1. length of the complete QuickReach measured in the graphic using a ruler
2. length of the cylindrical part of the LOX-tank measured in the graphic using a ruler
3. ratio of the two lengths calculated
4. real length of the complete QuickReach taken from the Airlaunch Fact Sheet available under www.airlaunchllc.com
5. ratio multiplied by the real length.
6. steps 1 to 5 for all tanks, two connection pipes, one nozzle and the cone containing the payload.
7. real diameter of the complete QuickReach taken from the Airlaunch Fact Sheet available under www.airlaunchllc.com
8. surface of the cylindrical part of the LOX-tank calculated
9. volume of the cylindrcial part of the LOX-tank calculated
10. steps 1 to 5 and 7 to 9 repeated for the spherical segments of the LOX-tank.
11. result of step 10 two times added to the result of step 9
12. surface of the LOX-tank got in steps 8 and its repetition for the spherical parts added on
13. thickness of 1 mm assumed and multiplied by the surface calculated
14. density of aluminum multiplied by the result of step 13
15. repetition of steps 8, 9 and 12 to 14 for all tanks
16. result of step 13 subtracted from the result of step 9
17. result of step 17 multiplied by the density of LOX.
18. steps 16 and 17 repeated for all tanks.
19. diameter of the connection pipe to the engines of the second stage measured in the graphic using a ruler
20. steps 1 to 5, 8 and 9, 12 to 14, 16 repeated using the result of step 19
21. step 20 repeated for the nozzle of the second stage.
22. subtraction of the results of steps 19 to 21 from the LOX-amount resulting in step 17
23. adding on all weights per stage to the weight of the stage – without and including the propellant(s)
24. repition of steps 1 to 5, 7 and 8, 13 and 14 repeated for the cone futurely containing the payload
25. adding the result of step 24 to the result of step 23
26. subtraction of the result of step 25 from the real weight of the QuickReach taken from the QuickReach Fact Sheet available under www.airlaunchllc.com
27. subtraction of the nozzle of the second stage from the result as the nozzle of the first stage.
28. Multiplication of the result by ratios distributing the result of step 27 over two stages
29. adding the nozzle of the second stage to both results

calculation completed!

Calculation of the QuickReach2 including the CXV

A. starting at the second stage
1. multiplication of the vehicle share of the QuickReach by the CXV/(cone + 1000 pounds)-ratio
2. addition of the original engine share
3. calculation of the ratio to the original joined share
4. multiplication of the result with the original tank share
5. addition of the result of step 4 to the result of step 2
6. multiplication of the result of step 3 by the orginal LOX-amount
7. calculation of the required volume by division by LOX-density
8. calculation of tank-length by application of a selected diameter
9. calculation of the surface by application of the length and the selected diameter
10. multiplication of the surface by a selected thickness and the density of aluminum to get the weight
11. epetition of steps 6 to 10 for propane
12. addition of the weights as the launch weight of the scond stage

B. first stage
like second stage but including additional steps:

1.2 calculation of the ratio between the second stage and the original second stage
1.3 multiplication of the share of the original second stage by the result of step 1.2
2.1 addition of the result of step 1.3 to the result of step 2
13. addition of all weights including the CXV
14. addition of all LOX-amounts
15. addition of all propane-amounts.


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