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Lunar Siyuz flight

Posted by: Ekkehard Augustin - Sat Oct 08, 2005 11:50 am
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Lunar Siyuz flight 
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Post    Posted on: Sun Aug 27, 2006 9:45 am
Contents of this post:

Overapproximations
1. Overapproximations done by intent
2. Inherent overapproximations not intended
Checks



Overapproximations

In principle they are no way overestimations since I am not estimating something in this thread but by intent using too high numbers but it’s easier here to talk if I call it overestimations.

Doing the required checks I found out that there were unintended inherent overestimations also which I list as a separate group.



1. Overapproximations done by intent

Too large a weight used for Apollo even after detection of this error

Numbers are rounded up when they increased the amount of propellant – except for those cases where other reasons prevented that.



2. Inherent overapproximations not intended

Weight of boosters and propellant tanks added to the weight of the vehicles – Apollo and CXV

Weight of the Block DM’s engine kept included when the Block DM-weight was multiplied to get the weight of ist larger replacements.

As turned out applying Apollo results in 50% too much propellant requirements



Checks

A portion of the weight left out of calculation beginning at the first step where the amount of propellant were considered. This resulted in an amount of propellant too low by a bit more than a third – wrong calculation

Unrecognized change from a slightly less to a slightly higher number – errored calculation

The CXV is equivalent to the Apollo CM + the Apollo SM reduced by the weight of the tanks – both taken as empty – too inexact and unprecise application of informations, systematical error.

The required amount of propellant has been calculated from the Apollo CSM directly which appears to be incomplete or wrong to me now

Apollo used without its booster

Safety margin applied before finishing the pseudo-systematical calculations

Consequences of differences in propellant-density aren’t considered.

Required adjustments of weights and numbers of boosters probable overlooked because of not developing and thus not applying a system and a concept of calculation(s) – may be some required calculations have not been done

Numbers by error left out of calculation for the first variant of the landing trip

Amount of propellant applied for the lander of the lower boundary was wrong

Regarding the upper boundary for the ladning trip carrying the lander from Earth allways there have occurred one or a few calculation errors resulting in too low tank capacities

Not taken into account that even an unparachuted Falcon is decelerated by the atmosphere on Earth – according increase of required propelllant left away regarding the Moon

Previous calculations don’t include the increase of the Falcon caused if the parachutes left away and propellant is required to get a safe landing on Earth

Assumed amount of cerosne a Falcon V requires not checked against other rockets

The differences in densities aren’t included yet – this is not that severe an error or mistake because it would be an additional aspect simply

A redundancy hasn’t been recognized properly



Apart of all these points I found additional steps possible that might reduce the costs further – it might turn out that they are needed really. But this has to be tested yet. I also might find a way to get an idea about the costs of vehicles of larger passenger capacities.

Improvements required and possible may result in findings that perhaps should be made a part of the checks or their results. But I don't think I should add them because they have to do with the earlier posts less and less.


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


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Post    Posted on: Sat Sep 30, 2006 3:25 pm
Contents

Clear system of calculations going to be applied
Details of the vehicles and tanks
Sections of phases
The tank of the existing vehicle
Appendix
Existing vehicle flown
Phases
Case of the existing vehicle
Phase Launch
Phase Orbital Insertion
Purpose(s) of the existing vehicle
Adjustment of the tank of the existing vehicle



In between the system of calculations is ready – I am going to explain it here in several steps.

Clear system of calculations going to be applied

The calculations were done beginning with Trans Lunar Injection and were based on expendables then. This complicated the calculations and prevented a clear system or concept. So I take the data about the launch of the expendables as the data valid at return now. From there I calculate back to Trans Lunar Injection - this way the calculations are (more) transparent and clear and the concept and system can be kept for all calculations including future ones for other planets to be done in other threads. I seem to remember that I read somewhere that such backwards calculations are the usually applied method. May be it was about spaceflight but may be also that it was an Economics science book.

This system means that 10% safety margins, absolute amount margins and the like are allowed only after finishing all calculations. Safety margins allowed are roundings – to standard numbers of decimals where possible - , linear calculations where they tend to result in numbers threatening favourable results and cumulations of such margins by iterations of a calculation (which is a kind of exponentiality...).

Doing so I soon recognized that the number of iterations isn’t foreseeable and that to do those number of iterations takes very lot of time. Both of this would be a very bad property of a system that is meant to be applied by others here also.

Because of this I decided to return to the application of a system of linear equations. Each equation of that system is valid for what I call here a phase. This is a particular part of the flight towards or from the Moon or any planet. It is a section of the flight path also – but this of no meaning here.



Details of the vehicles and tanks

In difference to before I broke down the vehicles and tanks to the details I needed and could get any data about. This improved the chance to find errors, problems and mistakes and showed up the corrections and solutions more quickly.

More important however – the details provide more flexibility regarding the data to be applied. For example several different alternative tanks (stages) can be applied now.

The details also reduced the amount of propellant to be looked for by the system of equations as well as the portion of the amount of the propellant for the existing vehicle the system requires to look for that amount. Most of the details allow direct calculations of smaller portions of propellant.



Sections of phases

The calculation for each phase is divided into several sections. These sections are required because of the differences between the existing and the potential vehicles and tanks. There are sections for

    the existing vehicle
    the potential vehicle
    a correction factor
    adjustment(s) of details of the existing vehicle
    the standard of the potential tank
    adjustment of the propellant of the existing vehicle
    the parameters to calculate those portions of the potential propellant amount that can be calculated directly
    the parameters required to do the indirect calculations by the equation system
    the ratios of the details of the potential vehicle of the existing vehicle
    the propellant portions of the potential vehicle that can be calculated directly
    the sums of the directly calculated amounts of propellant
    the equations of the system of equations including hte amount of propellant resulting from the system


These sections I am going to explain now. There is an Appendix where the mathematics can be seen in small letters – the terms, the equations, the algebra, modifications, parameters and the like.



The tank of the existing vehicle

It turned out that first of all the number of the weight of the tank of the existing vehicle has to adjusted because the potential vehicle might use another propellant than the existing vehicle – and the weight has an effect on the amount of propellant required. This different propellant might have a different density. If so then the volume of that different propellant would be larger or smaller than that of the existing vehicle’s propellant. At the same weight of propellant then the tank might be heavier or lighter than the the tank of the existing vehicle.

To handle this I calculate the volume of the propellant of the existing vehicle by dividing its weight by its correct density. This volume I subtracted from the volume of the stage and then calculated the weight per unit of volume. Next I separated the weights of the top and the bottom of the stage – the remainder had to be adjusted. To do so I divided the propellant of the existing vehicle by the density of the potential vehicle. From that new volume the new length of the tank and thus the stage can be calculated. This gets me the new difference between the new volume of the propellant and the new volume fo the stage around the propellant. By multiplication with the weight per volume the adjsuted weight is got.

The data to be used are inserted into the section of the existing vehicle while the adjustment is done in the section of the adjustment(s) of details of the existing vehicle.



The effect on the propellant I will consider later.



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

Appendix: sections of the existing vehicle and the adjustment(s) of details of the existing vehicle

Existing vehicle flown

evw weight of existing vehicle
eew weight of engine of existing vehicle
tew weight of tank of existing vehicle - the stage somehow
pew weight of propellant of existing vehicle
cew weight of what the existing vehicle has to carry with it
(lander, propellant for lander, corrections etc.)
---
die diameter of tank of existing vehicle
hie height of tank of existing vehicle
dye density of propellant of existing vehicle
eiv Isp of the existing vehicle – either of its propellant or of its engine


Phases

Case of the Existing vehicle

Phase Launch

to be launched: evw, eew, tew, cew OF THIS PHASE
required propellant: pew

cew consists of: eew-s of following phases
tew-s of following phases
pew-s of following phases
cew-s of following phases

There are also die-s and hie-s for each phase.



Phase Orbital Insertion

to be inserted: like phase launch but OF THIS PHASE
required propellant: like phase launch but

cew consists of: like phase launch but

like phase launch



Purpose(s) of the existing Vehicle

trips from Earth to another planet:

- round trips 2 phases
- trips into orbits 4 phases
- landing trips 4 phases

A round trip does a launch out of an earthian orbit, flies around the other planet, flies back to Earth and does an orbital insertion there.

A trip into an orbit does a launch out of an earthian orbit, does an orbital insertion into an orbit around the other planet, does a launch out of the orbit of that other planet and finally does an orbital insertion into an orbit around Earth.

This requires a distinction between the launch out of the earthian orbit and the launch out of the orbit of the other planet as well as between the orbital insertion into the orbit of the other planet and the orbital insertion into an earthian orbit.

This holds too for a landing trip because the lander only will go down.

The launch phase out of the earthian orbit is called Trans Planetary Injection or TPI now, the orbital insertion into the orbit around the other planet Planetary Orbital Insertion or POI, the launch phase out of the orbit around the other planet now is the Trans Earth Injection or TEI and the orbital insertion into an earthian orbit now is Earthian Orbital Insertion or EOI.

At this point the cew needs to reconsidered a bit. Phases are distiguished here now - and there may be payloads in an earlier phase that are not there no more in a later phase and there may be a new payload then. But also the payload of the earlier phase may still be there in a later phase. It urgently must be avoided to add on payloads that aren't there together or to apply one and the same payload as if it were two payloads. It looks as if this doesn't fit into the logic of the tanks, propellants, engines etc.

It might of help to classify the payloads:

I. Payloads that are carried at TPI and will return to Earth at EOI might be cew(EOI) or a part of it.
2. Payloads that are carried at TPI, arrive at the other planet but will not return to Earth might be cewp(POI).
3. Payloads that are not carried at TPI but are carried at TEI seem to require split calculations with EOI and TEI left zero in the first part while TPI and POI left zero in the second part. The payloads will be cewp(EOI) then in the second part.

Then there are:

TPI: evw, eew(TPI), tew(TPI), cew(TPI)
pew(TPI)
POI: evw, eew(POI), tew(POI), cew(POI), cewp(POI)
pew(POI)
TEI: evw, eew(TEI), tew(TEI), cew(TEI)
pew(TEI)
EOI: evw, eew(EOI), tew(EOI), cew(EOI)
pew(EOI)



Adjustment of the tank of the existing vehicle

In the following (...) is to be replaced by (EOI), (TEI), (POI) or (TPI) – it didn’t make sense to repeat the calculations for each phase.

The surface of the stage is known while I have no informations about the dimensions of the tank. So the volume and weight of the tank have to be calculated. The volume of the tank can be calculated by the amount of propellamt and the density of the propellant - but there are no informations about the diameter and the length of the tank. The only way out seems to be that the weight of the stage is required which is the capacity

ect(...) (capacity of tank of the existing vehicle)

in principle. The volume of the tank is

ect(...)/dye(...)

while the volume of the stage according to the general equation

V = Pi * r^2 * h

is

Pi * (die(...)/2)^2 * hie(...)
.

To apply the volume and the weight correct I next subtract the volume got by the propellant density from the total volume got:

Pi * (die(...)/2)^2 * hie(...) - (ect(...)/dye(...))

This is the volume weighing tew.

So the division of the two results in the weight per m^3. To get the weight of the reduced stage I next separate the stage and this its tank into top plus bottom and the side. Then the side only will be reduced.

Since the tank is smaller than the stage and is going to be reduced in length here the weight of the top and the bottom has to be removed now to be handled separately. But there is the problem, that the diameter and length of the tank aren't available. And I don't have such numbers about other tanks that might be applied here as far as I remember. So I use a function relating the length to the diameter:

dte is the tank-diameter and thus known in this case.

ect(...)/dye(...) = Pi * (dte(...)/2)^2 * hte(...) (hte = length of tank existing)
(ect(...)/dye(...))/(Pi * (dte(...)/2)^2) = hte(...) =
(ect(...)/dye(...))/(Pi * (dte(...)^2/4)) =
(ect(...)/dye(...))/((Pi * dte(...)^2)/4) =
(4 * ect(...))/(Pi * dte(...)^2 * dye(...)) = hte(...).

Then hie - hte is the remainder of the length of the stage by which the stage is longer than the tank. The volume to which this remainder belongs is the bottom-volume plus the top-volume. These two multiplied by the weight per m^3 is an additional constant for the direct calculation of propellant.

This also is to be subtracted from the weight of the stage and the difference volume calculated above. The remainder of the difference volume then will be reduced by the ratio of the densities:

top + bottom = Pi * (die(...)/2)^2 * (hie(...) – hte(...))

side =
stage - (top + bottom) =
Pi * (die(...)/2)^2 * hie(...) - (ect(...)/dye(...)) - Pi * (die(...)/2)^2 * (hie(...) – hte(...))
Pi * (die(...)/2)^2 * hie(...) - (ect(...)/dye(...)) - Pi * (die(...)/2)^2 * hie(...) + Pi * (die(...)/2)^2 * hte(...) =
Pi * (die(...)/2)^2 * hie(...) - Pi * (die(...)/2)^2 * hie(...) + Pi * (die(...)/2)^2 * hte(...) - (ect(...)/dye(...)) =
Pi * (die(...)/2)^2 * hte(...) - (ect(...)/dye(...))

Then

(4 * ect(...))/(Pi * dte(...)^2 * dyp(...)) = hte2(...)

is the length of the side if the propellant of the existing vehicle had the density dyp – dyp is the density of the propellant of the potential vehicle which will be listed later when the section of that vehicle is required to a larger degree.

From this

Pi * (die(...)/2)^2 * hte2(...) - (ect(...)/dyp(...))

can be calculated as the adjusted side which then has to multiplied by the weight per m^3 yet to get the adjusted weight of the tank


I increased the size of the letters according to a useful and helpful hint I got from Sigurd - Thank You Very Much, Sigurd.

End of Appendix


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Post    Posted on: Sat Oct 07, 2006 8:00 am
Conetnts

The effect of the reduction of the required volume on the propellant
Applying the existing vehicle
Appendix
Adjustment(s) of details of the existing vehicle: ist of parameters
Calculation of the adjusted amount of propellant – unavailable number required
Existing Vehicle: Initial Calculations
Total Weight of Existing Vehicle
Shares of the Weights




The effect of the reduction of the required volume on the propellant

If the required volume of the tank is reduced then the weight of the tank is reduced while an increase of the volume menas an increase of ist weight. Since the tank has to be propellad together with the vehicle, the top and bottom, the engines , all the cargo etc. a reduced tank weight menas a reduction of the required amount of propellant while a heavier tank means an increase of the required amount of propellant.

To calculate the adjusted amount og propellant I simply calculate the amount of propellant per kilogram of the original combination of vehicle, engine(s), top(s), bottom(s), side(s) and cargos etc. and multiply it by the new weight got because of the adjusted tank volume.

I tested if that works and that seems to be the case. But I couldn’t derive that up to now mathematically because I seem to kno anumber I couldn’t find informations about up to now. I also couldn’t find a away to derive it from data available to me.

This means that the adjusted amount of propellant involves a safety margin if the adjsuted volume of the tank is smaller than the original volume – while in the case that the adjusted volume is larger than the original one the adjusted amount will be too low in comparison to the adjusted requirements. But it seems that this second case can be hndledby another section of calculation per phase.

So it is required to look if the density of the propellant of the potential vehicle is larger than that of the existing or if it is smaller.

Of course there is no problem if it is equal.

In the Appendix the formular can be seen that would require the number I at presnet can’t get.



Applying the existing vehicle

Having done the adjustements of the tank and the required propellant to the density of the propellant of the potential vehicle the details of the existing vehicle can be applied now.

The details are

    the vehicle itself,
    the engine of the vehicle,
    the top and the bottom of the tank,
    all cargos to be carried including
    the tanks and propellants required later during the flight.


All of these cause a requirement of a share of the total amount of propellant Regarding the potential vehicle this is valid too and so the shares of propellant the details of the potential vehicle will require can be calculated directly from the details of the existing vehicle – provided that the according detail of the potential vehicle is known...

To get the shares of propellant that are caused by the particular detail I add on the weights of all the details to a total and then divide the weight of each particular detail by the total.

This has to be done per phase of flight.

I will go on later.



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



Appendix

Adjustment(s) of details of the existing vehicle: list of parameters

ect existing vehicle - capacity of tank
hte length of tank existing
dte diameter of tank existing

(hte2 adjusted length of tank adjusted: is a symbol for a formular only and thus not used again futurely.)



Calculation of the adjusted amount of propellant – unavailable number required

pew as the total propellant of the existing vehicle is composed as

pew = pewt + pewr

with pewt being the propellant required because of the tank tew and pewr the propellant for the vehicle, the engine(s) etc.

Because tew and pew are going to be adjusted a number is added here:

pew1 = pewt1 + pewr

pewr is not adjusted and doesn’t need a number.

Calling the non-tank weights to be propelled erw andd the total of the two etw

etw1 = tew1 + erw

is valid.

Then

pewt1 = tew1/etw1 * pew
= tew1/(tew1 + erw) * pew

and

pewr = erw/etw1 * pew
= erw/(tew1 + erw) * pew

are got.

Taking into account that the tank to be propelled might jave been used before arriving in the orbit already I now apply the complete capacity the tank might have:

Because of the previous post then

tew1 = Pi * (die/2)^2 * hie - (ect(...)/dye(...))

is to be applied:

pewt1 = ((Pi * (die/2)^2 * hie - (ect(...)/dye(...)))/((Pi * (die/2)^2 * hie - (ect(...)/dye(...))) + erw)) * pew
pewr = (erw/((Pi * (die/2)^2 * hie - (ect(...)/dye(...))) + erw)) * pew.

The propellant per unit of weight of hardware is

pew/(tew1 + erw) = pew/((Pi * (die/2)^2 * hie - (ect(...)/dye(...))) + erw)

To get the new weight of tank a number has to be subtratced from tew1 that is looked for here. Calling the new weight of the tank tew2 and the number looked for b

tew2 = tew1 - b = (Pi * (die/2)^2 * hie - (ect(...)/dye(...))) – b

is to be solved.

This requires a second equation because tew2 and b both aren’t known and thus two numbers are looked for.

The propellant required to carry the new weight of tank is

pewt2 = tew2 * (pew/(tew1 + erw)) =
(tew1 - b) * (pew/(tew1 + erw)) =
((Pi * (die/2)^2 * hie - (ect(...)/dye(...))) - b) * (pew/((Pi * (die/2)^2 * hie - (ect(...)/dye(...))) + erw)

Then the new total amount of propellant pew2 is

pew2 = pewt2 + pewr =
((Pi * (die/2)^2 * hie - (ect(...)/dye(...))) - b) * (pew/((Pi * (die/2)^2 * hie - (ect(...)/dye(...))) + erw)) +
(erw/((Pi * (die/2)^2 * hie - (ect(...)/dye(...))) + erw)) * pew

The adjusted tank has an adjusted capacity

ect2

which is not known yet.but would have to replace pewc1 which was known in contrary to pewc2:

(Pi * (die/2)^2 * hie - (ect(...)/dye(...))) - b =
(Pi * (die/2)^2 * hie2 - (ect2(...)/dyp(...)))

The adjustment of the tank to the new propellant requirements caused by the initial tank adjustment menas a new length of the tank – which is hie2 above.

The equation can be solved for b:

b = (Pi * (die/2)^2 * hie - (ect(...)/dye(...))) –
(Pi * (die/2)^2 * hie2 - (ect2(...)/dyp(...))) =
Pi * (die/2)^2 * hie - (ect(...)/dye(...)) -
Pi * (die/2)^2 * hie2 + (ect2(...)/dyp(...)) =
Pi * (die/2)^2 * (hie - hie2) + (ect2(...) * dye(...) - ect(...) * dyp(...))/(dye(...) * dyp(...))

So an additional unknown parameter has been introduced – hie2. I may have overlooked something or still not found an existing way but at present I had to give up at this point. May be I’lll find a solution later but none is in sight yet.



Existing Vehicle: Initial Calculations

The tank of the potential vehicle Y and ist propellant X are to be got by linear calculations - to do linear calculations ratios are required. These can consist of constants konwn only. But as the chapter Phases showed there are constants going thruogh more than one phase while others do not and so disappear in a few phases. Because of this some constants have to be added on before ratios can be established and used.

The constants this has to be done for are the propellants pew2(...), engines eew(...), and tanks tew2(...) – but those for the future phases only because the ones used in the actual phase are going to be consumed or expended and so don’t require propellants in future phases to be transported.

All the adjustments done are to be applied now – pew and tew are replaced by pew2 and tew2.

Propellants pew2(...)

EOI: cargo = 0
TEI: cargo = pew2(EOI)
POI cargo = pew2(EOI) + pew2(TEI)
TPI cargo = pew2(EOI) + pew2(TEI) + pew2(POI)



Tanks tew(...)

EOI: cargo = 0
TEI: cargo = tew2(EOI)
POI: cargo = tew2(EOI) + tew2(TEI)
TPI: cargo = tew2(EOI) + tew2(TEI) + tew2(POI)



Engines eew(...)

EOI: cargo = 0
TEI: cargo = eew(EOI)
POI: cargo = eew(EOI) + eew(TEI)
TPI: cargo = eew(EOI) + eew(TEI) + eew(POI)

There are also real cargos that don’t have to do with the propulsion of the existing vehicle that need to be added:

Real cargos cew(...)

EOI: cargo = cew(EOI)
TEI: cargo = cew(EOI)
POI: cargo = cew(EOI) + cewp(POI)
TPI: cargo = cew(EOI) + cewp(POI)

All of these need to be added on per phase to one complete cargo:

EOI: cargo = 0 + 0 + 0 + cew(EOI)
TEI: cargo = pew2(EOI) + tew2(EOI) + eew(EOI) + cew(EOI)
POI: cargo = pew2(EOI) + pew2(TEI) +
tew2(EOI) + tew2(TEI) +
eew(EOI) + eew(TEI) +
cew(EOI) + cewp(POI)
TPI: cargo = pew2(EOI) + pew2(TEI) + pew2(POI) +
tew2(EOI) + tew2(TEI) + tew2(POI)
eew(EOI) + eew(TEI) + eew(POI)
cew(EOI) + cewp(POI)



Total Weight of Existing Vehicle

The total weight of the existiung vehicle is

evw + eew + tew2 + cew = etw existing total weight

here.

They vary from phase to phase:

EOI: evw + eew(EOI) + tew2(EOI) + cew(EOI) = etw(EOI)
TEI: evw + eew(TEI) + tew2(TEI) + cew(TEI) = etw(TEI)
POI: evw + eew(POI) + tew2(POI) + cew(POI) = etw(POI)
TPI: evw + eew(TPI) + tew2(TPI) + cew(TPI) = etw(TPI)



Shares of the weights

To get the amount of propellant required for the potential vehicle, its engine and its payload I break off known parts of etw and set them into relation to etw:

vehicle itself

evw/etw =
evw/(evw + eew + tew2 + cew) = evs existing-vehicle-share

Again the phases have an impact:

EOI: evw/etw(EOI) =
evw/(evw + eew(EOI) + tew2(EOI) + cew(EOI)) = evs(EOI)

TEI: evw/etw(TEI) =
evw/(evw + eew(TEI) + tew2(TEI) + cew(TEI)) = evs(TEI)

POI: evw/etw(POI) =
evw/(evw + eew(POI) + tew2(POI) + cew(POI)) = evs(POI)

TPI: evw/etw(TPI) =
evw/(evw + eew(TPI) + tew2(TPI) + cew(TPI)) = evs(TPI)



engine of the vehicle

eew/etw =
eew/(evw + eew + tew2 + cew) = ees existing-engine-share

The impact of the phases has to be considered:

EOI: eew(EOI)/etw(EOI) =
eew(EOI)/(evw + eew(EOI) + tew2(EOI) + cew(EOI)) = ees(EOI)

TEI: eew(TEI)/etw(TEI) =
eew(TEI)/(evw + eew(TEI) + tew2(TEI) + cew(TEI)) = ees(TEI)

POI: eew(POI)/etw(POI) =
eew(POI)/(evw + eew(POI) + tew2(POI) + cew(POI)) = ees(POI)

TPI: eew(TPI)/etw(TPI) =
eew(TPI)/(evw + eew(TPI) + tew2(TPI) + cew(TPI)) = ees(TPI)



cargos of the vehicle

cew(EOI)

EOI: cew(EOI)/(evw + eew(EOI) + tew2(EOI) + cew(EOI))
TEI: cew(EOI)/(evw + eew(TEI) + tew2(TEI) + cew(TEI))
POI: cew(EOI)/(evw + eew(POI) + tew2(POI) + cew(POI))
TPI: cew(EOI)/(evw + eew(TPI) + tew2(TPI) + cew(TPI))

cewp(POI)

POI: cewp(POI)/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) +
pew2(EOI) + pew2(TEI) + cewp(POI))
TPI: cewp(POI)/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))

The share of the sum of the top and the bottom isn’t listed here. The reason is that they have been calculated in the previous post:

top + bottom = Pi * (die(...)/2)^2 * (hie(...) – hte(...))

and the shares are

((Pi * (die(...)/2)^2 * (hie(...) – hte(...))) * (pew/((Pi * (die(...)/2)^2 * hie(...) - (ect(...)/dye(...))) + erw(...))))/(evw + eew(...) + tew2(...) + cew(...))

Because

erw = evw + eew + cew

the share is

((Pi * (die(...)/2)^2 * (hie(...) – hte(...))) * (pew/((Pi * (die(...)/2)^2 * hie(...) - (ect(...)/dye(...))) + evw + eew(...) + tew2(...) + cew(...))))/(evw + eew(...) + tew2(...) + cew(...))
=
((pew(...) * (Pi * (die(...)/2)^2 * (hie(...) – hte(...))))/((Pi * (die(...)/2)^2 * hie(...) - (ect(...)/dye(...))) + evw + eew(...) + tew2(...) + cew(...)))/(evw + eew(...) + tew2(...) + cew(...))
=
(pew(...) * (Pi * (die(...)/2)^2 * (hie(...) – hte(...))))/(((Pi * (die(...)/2)^2 * hie(...) - (ect(...)/dye(...))) + evw + eew(...) + tew2(...) + cew(...)) * (evw + eew(...) + tew2(...) + cew(...)))

I was thinking about abandoning some parenthises by modifications but that seems to result in too lang a caller of the fraction.

Calling top + bottom n for non-side this share could be called ecs.




End of Appendix


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Post    Posted on: Sat Oct 14, 2006 5:20 pm
Contents

Calculation of the Propellant per Detail of the Existing Vehicle
The Potential Vehicle
Existing and Potential Vehicle – Ratios.
First Partial Step towards the Requirements of the Potential Vehicle
Potential Tank of the Potential Vehicle
Standard Tank and the Impact of Forgotten Economies of Scale
Isps
Correction Factor
Portion of Propellant already calculated
Appendix
Propellants for the Details of the Existing Vehicle
The Potential Vehicle
Case of the Potential Vehicle
Phase Launch
Both Vehicles: Ratios
Tank: Top and Bottom
Known Portion of the Propellant of the Potential Vehicle


Calculation of the Propellant per Detail of the Existing Vehicle

The shares I talked about in the previous post simply are to be multiplied by the propellant the existing vehicle consumed in the particular phases. These are the propellant amounts to applied during the future steps of calculation.

The Potential Vehicle

To be able to calculate the propellant requirements of the potential vehicle the details of that potential vehicle are required – and they have to be the details according to the details of the existing vehicle. But in difference two details aren’t known yet – these are the reasons to do all theses calculations but can’t be calculated yet in this post.

Existing and Potential Vehicle – Ratios

The availability of according details of both the existing and the potential vehicle allows to calculate some terms the application of which to the amounts of propellant per detail of the existing vehicle result in the amounts of the propellant of the existing vehicle the particular details of the potential vehicle require.

Each weight of a detail of the potential vehicle can be divided by the weight of the according detail of the existing vehicle – resulting in the ratio between the two weights.

First Partial Step towards the Requirements of the Potential Vehicle

The ratios got are to be multiplied with the amounts of propellant got per dteail of the existing vehicle. These are the amounts of propellant the details of the potential vehicle need – but these are amounts of the propellant used by the existing vehicle.

Also the amount of propellant the tank of the potential vehicle needs isn’t considered yet.

Potential Tank of the Potential Vehicle

The calculations per phase have a particular section for the – potential – tank of the potential vehicle. The first reason is that there never was a reusable vehicle yet that has flown to the Moon and back that used a particular tank. Such a combination only is under consideation by Space Adventures and the Russians.

The second reason to do so may be considered to be more essential and more interesting. In Space Adventure’s concept the tank and the vehicle are launched separated from each other and usually the vehicle doesn’t use the tank to be applied in Space Adventure’s concept. Consequently the concept of calculation here should enable to apply several alternative tanks for one and the same potential vehicle.

Since the potential tank already has been said to be reusable earlier during this thread it will be the same in all phases. Because on the other hand the required volume isn’t known yet a standard tank is to be applied at this point of the calculations.

Standard Tank and the Impact of Forgotten Economies of Scale

The circumstance that the tank can be applied already without knowing the required volume is a bit puzzling I suppose and I discovered a way to do it because of an interesting aspect of economic nature. When I tested three or two other concepts of calculations prior to posting again I ended up in a concept similar to this one but significantly simpler.

When that simpler concept was ready I tried one or two initial numbers valid for the round trip and got reasonable results that were significantly above those got the non-concepzional way. I thought them to be correct and tested the orbital trip. I wondered taht those resulkt weren’t that much above the non-conceptional ones and detected an error. I corrected that error... – and got astonishing results now: they were between 30-times and 40-times the non-conceptional ones. They were unbelievable and unreasonable.

I investigated the results – and found that I had repeated a mistake of economical nature I has done the conceptional way also (simply without the effects). There was the standard tank and I simply applied the numbers got from www.bernd-leitenberger.de or www.astronautix.com . Thes included the engine which I already had subtracted and applied as a separate detail – like here. The remainder of the weight I kept together – but hadn’t taken into account what that meant: It menat that I applied several tanks that were separated from each other by their top plates and -walls and their bottom plates and -walls. And all these top-plates and –walls and bottom-plates and –walls were involved twice between two tanks because each tank had them.

But since one whole and complete tank is to applied here all these tops and bottoms aren’t required but only the one bottom and one top. The error was that I didn’t take into accoiunt the economies of scale of the possibla larger tank that I had already to assume to be larger because of the non-conceptional earlier calculations.

The correction means that up to now the tanks – or more correct stages – are cylindrical and that la potential tank larger than a standard tank here simply menas that the potential tank is higher but has the same diameter as the standard tank.

The logical consequence is that the weight of the tops and bottoms are constants here that can be applied without knowing the required volume that determines the height of the tank and thus its side-walls that are left out of consideration here.

This would have to be left away for spherical tanks or the weight of spherical tanks would have to be corrected for the weight of the tops and bottoms but this is of no menaing in this concept because of aspects to be talked about later,

Isps

Last but not least the Isps are ruling the required amount of propellant. Like alreday done the non-conceptional way the ratio of Isps is applied here to the amounts already calculated up to now. Formerly these were the Isps of the propellants. But this concept applies the engines as a seperated detail. For this reason it seems to be reasonable to at least allow the replacement of the Isps of the propellants by the Isps of the engines. This may be more correct because the engines have a reducing impact on the Isps according to at least one of my sources of informations

Correction Factor

The correction factor is empirical and will be explained later. When I tested the earlier concepts already mentioned I detected that an adjustment is required which might be due to unavailable data, partially insufficient degree of detailing and the like.

Portion of Propellant already calculated

At this point all the ratios calculated up to now can be multiplied with all amounts of propellant got per detail of the existing vehicle, multiplied by the Isps-ratio and the correction factor. The sume of these then is the amount of propellant already got for the potential vehicle.

What’s left yet is the unknown voulme and side of the tank and the unknown propellant required because of that tank and those portions of the total propellant at least that are required at the Moon and at return to the Earth.



In the next post the search for those unknowns will start.





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



Appendix

Propellants for the Details of the Existing Vehicle

From the shares calculated the propellants per detail follow by multiplication by the total propellant per phase:

vehicle itself

evs * pew2 = evp existing-vehicle-propellant

EOI: evs(EOI) * pew2(EOI) = evp(EOI)
TEI: evs(TEI) * pew2(TEI) = evp(TEI)
POI: evs(POI) * pew2(POI) = evp(POI)
TPI: evs(TPI) * pew2(TPI) = evp(TPI)



engine of the vehicle

ees * pew = eep existing-engine-propellant

EOI: ees(EOI) * pew2(EOI) = eep(EOI)
TEI: ees(TEI) * pew2(TEI) = eep(TEI)
POI: ees(POI) * pew2(POI) = eep(POI)
TPI: ees(TPI) * pew2(TPI) = eep(TPI)



cargos of the vehicle

cew(EOI)

EOI: (cew(EOI)/(evw + eew(EOI) + tew(EOI) + cew(EOI))) * pew2(EOI)
TEI: (cew(EOI)/(evw + eew(TEI) + tew(TEI) + cew(TEI))) * pew2(TEI)
POI: (cew(EOI)/(evw + eew(POI) + tew(POI) + cew(POI))) * pew2(POI)
TPI: (cew(EOI)/(evw + eew(TPI) + tew(TPI) + cew(TPI))) * pew2(TPI)

cewp(POI)

POI: (cewp(POI)/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI)
+ pew2(EOI) + pew2(TEI) + cewp(POI))) * pew2(POI)
TPI: (cewp(POI)/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) * pew2(TPI)

The Potential Vehicle

pvw weight of potential vehicle
epw weight of engine of the potential vehicle
Y weight of tank of existing vehicle - the stage somehow
UNKNOWN = LOOKED FOR
X weight of propellant of potential vehicle
UNKNOWN = LOOKED FOR
cpw weight of what the potential vehicle has to carry with it
(see existing vehicle)
---

piv Isp of the potential vehicle – either of its propellant or of its engine

Case of the Potential Vehicle

Phase Launch

to be launched: pvw, epw, Y, cpw OF THIS PHASE
required propellant: X

all further according to the existing vehicle - except for epw and cpw: the potential vehicle is a reusable one. This means that there is one engine for all phases and there is one tank only that is carried through each of the phases unchanged. So eew and Y are no parts of cpw.

TPI: pvw, epw, Y, cpw(TPI)
X(TPI)
POI: Pvw, epw, Y, cpw(POI), cpwp(POI)
X(POI)
TEI: pvw, epw, Y, cpw(TEI)
X(TEI)
EOI: evw, epw, Y, cpw(EOI)
X(EOI)

Both Vehicles: Ratios

Ratios are available for all values of the potential vehicle already known. There are

vehicles themselves

pvw/evw = bvr both-vehicles-ratio



engines of the vehicles

epw/eew = ber both-engines-ratios

EOI: epw(EOI)/eew(EOI) = ber(EOI)
TEI: epw(TEI)/eew(TEI) = ber(TEI)
POI: epw(POI)/eew(POI) = ber(POI)
TPI: epw(TPI)/eew(TPI) = ber(TPI)



cargos of the vehicles (ratio left unnamed)

cpw(EOI)/cew(EOI)
cpwp(POI)/cewp(POI)



The cargo ratios aren’t given a name nor they are calculated as a compund constant here because they can’t exist phase-wise due to their nature..

Also know is the ratio between the Isps which may differe phase by phase because in the case of the existing vehicle the propellants may differe phase by phase:

eiv(...)/piv = bir(...)

Tank: Top and Bottom

The way is like for the tank of the existing vehicle. The only exception is that no adjustment calculations for density are required.

This potntial tank or tank of the potential vehicle has a capacity

pct (capacity of the tank of the potential vehicle)

.

pct is a weight of a propellant of the density

dyp (dfensity of the propellant of the potential vehicle)

.

As Ass een here no phases are applied here – because the complete tank has to return back to Earth because of reuasbility and future reuse for the same purpose and so does not change from phase to phase like the tank of the existing vehilce that is expendable.

Then the volume of pct is

V = pct/dyp

.

The volume of the stage the tank is part of according to the formular

V = Pi * r^2 * h

is

Pi * (dip/2)^2 * hip

.

Then the volume of the stage will differ from the volume of the tank by

Pi * (dip/2)^2 * hip – pct/dyp

.

This is the volume in which that portion of weight of the fueled stage is loctaed in that is not the weight of the propellant. This portion of the weight is

tpw (weight of the tnak of the potential vehicle)

.
Then

tpw/(Pi * (dip/2)^2 * hip – pct/dyp)

is the weight per unit of volume (m^3) of the empty stage.

If the diameter of the tank is

dtp (diameter of the tank of the potential vehicle)

then the height of the tank is

htp (height of the tank of the potential vehicle)

with

htp =
V/(Pi * (dtp/2)^2) =
V/(Pi * (dtp^2/4)) =
V/((Pi * dtp^2)/4) =
(V * 4)/(Pi * dtp^2) =
((pct/dyp) * 4)/(Pi * dtp^2)

.

Then the difference

hip – htp

is the height of top + bottom of the tank/stage of the potential vehicle.

The volume of the top + bottom then is

Pi * (dip/2)^2 * (hip – htp)

.

The weight of the top plus the bottom then is

(Pi * (dip/2)^2 * (hip – htp) * tpw)/((Pi * (dip/2)^2 * hip – pct/dyp))

The ratio to be applied then is this formular divided by the weight of the adjusted tank of the existing vehicle tew2:

((Pi * (dip/2)^2 * (hip – htp) * tpw)/((Pi * (dip/2)^2 * hip – pct/dyp)))/tew2(...)

.

Insertzing the formular of tew2 seems to result in aformular of a complexity that is of no use here.

Known Portion of the Propellant of the Potential Vehicle

Applying the already known amounts and ratios the follwing portion of propellant for the potential vehicle is already known:

sum(...) =

bvr * bir(...) * cor(...) vehicles themselves
+
ber(...) * bir(...)* cor(...) engines of the vehicles in EOI
+
cpw(EOI)/cew(EOI) * bir(...)* cor(...) cargos of the vehicles
+
(((Pi * (dip/2)^2 * (hip – htp) * tpw)/((Pi * (dip/2)^2 * hip – pct/dyp)))/tew2(...)) * bir(...) * cor(...)

For POI

cpw(EOI)/cew(EOI)

is to be replaced by

cpwp(POI)/cewp(POI)

.

Summing these up for all phases results in

tsm =

bvr * bir(EOI) * cor(EOI) +
bvr * bir(TEI) * cor(TEI) +
bvr * bir(POI) * cor(POI) +
bvr * bir(TPI) * cor(TPI)
+
ber(EOI) * bir(EOI)* cor(EOI) +
ber(TEI) * bir(TEI)* cor(TEI) +
ber(POI) * bir(POI)* cor(POI) +
ber(TPI) * bir(TPI)* cor(TPI)
+
cpw(EOI)/cew(EOI) * bir(EOI)* cor(EOI) +
cpwp(POI)/cewp(POI) * bir(POI)* cor(POI)
+
(((Pi * (dip/2)^2 * (hip – htp) * tpw)/((Pi * (dip/2)^2 * hip – pct/dyp)))/tew2(EOI)) * bir(EOI) * cor(EOI) +
(((Pi * (dip/2)^2 * (hip – htp) * tpw)/((Pi * (dip/2)^2 * hip – pct/dyp)))/tew2(TEI)) * bir(TEI) * cor(TEI) +
(((Pi * (dip/2)^2 * (hip – htp) * tpw)/((Pi * (dip/2)^2 * hip – pct/dyp)))/tew2(EOI)) * bir(POI) * cor(POI) +
(((Pi * (dip/2)^2 * (hip – htp) * tpw)/((Pi * (dip/2)^2 * hip – pct/dyp)))/tew2(TEI)) * bir(TPI) * cor(TPI)

=

sum(EOI) + sum(TEI) + sum(POI) + sum(TPI)


End of Appendix


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Post    Posted on: Sat Oct 21, 2006 3:42 pm
Contents
Calculating the Tank of the Potential Vehicle
Alternative tanks...
Appendix
Additional Share: Cylindrical Tank
Amounts of Propellants resulting from the additional Shares
Conclusion to the really required cylindrical Tank
Propellant required for the Tank of the Potential Vehicle
Weight of the complete cylindrical Tank of the Potential Vehicle
Both Vehicles: Ratio of the Tank Weights
Resulting Amount of Propellant for the cylindrical Tank
Deriving the spherical alternative


Calculating the Tank of the Potential Vehicle

In difference to all the details calculated in the previous post the weight of the tank of the potential vehicle can’t be calculated directly yet because the total amount of propellant required isn’t known yet.

But the calculation can be prepared already because there are some data available that tell a bit about the data looked for – the share of propellant the tank of the exisitng vehicle needs, the ratios and numbers to translate the propellant of the existing vehicle into the propellant of the potential vehicle etc.

Alternative tanks...

One kind of data telling something like that are to be mentioned in particular here: the data about any exisitng tank. Since this thread is based on a concept which applies a tank to a vehicle that never before has been using that tank free choice of the tank for the potential vehicle must be provided here – the tank is not bound to the potential vehicle and alternative tanks can be tried.

So it is possible to consider several alternaitve tanks für one and the same potential vehicle.

Since these tanks are well-defined regarding their volume and weight they can’t be applied directly in calculations – but they can be used as standard tanks. This standard consists of properties that partially don’t change with volume or weight of the tank – and these constant properties can be and must be used here.

Spherical Tank

The calcukation of a spherical tank according to the cylindrical tank is possible here but results in equations and formulars that involve a cuberoot. It turned out that it takes to much time to isolate the required phase-related variables – propellant and tank – so that the equations for them can be solved.

So the spherical tank is going to be derived from the results for the cylindrical tank. This necessaryly leads to amounts and volumes higher than required. I already explained that a constant is missing by which the correct values can be got. But the descriotion of what needs to be done - in the Appendix – enables interested readers to approximate the correct values. Here the focus is on the alternative or even comparison to cylindrical tanks.



Now the approach is ready for the calculation of the propellant in the next post.



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


Appendix

Additional Share: Cylindrical Tank

tank(s)

tew2/etw2 =
tew2/(evw + eew + tew2 + cew) = ets existing-tank-share

The phases have in impact here too:

TPI: (tew2(TPI) + tew2(POI) + tew2(TEI) + tew2(EOI))/etw(TPI) =
(tew2(TPI) + tew2(POI) + tew2(TEI) + tew2(EOI))/(evw + eew(TPI) + tew2(TPI) + cew(TPI)) = ets(TPI)

POI: (tew2(POI) + tew2(TEI) + tew2(EOI))/etw(POI) =
(tew2(POI) + tew2(TEI) + tew2(EOI))/(evw + eew(POI) + tew2(POI) + cew(POI)) = ets(POI)

TEI: (tew2(TEI) + tew2(EOI))/etw(TEI) =
(tew2(TEI) + tew2(EOI))/(evw + eew(TEI) + tew2(TEI) + cew(TEI)) = ets(TEI)

EOI: tew2(EOI)/etw(EOI) =
tew2(EOI)/(evw + eew(EOI) + tew2(EOI) + cew(EOI)) = ets(EOI)

pew2(TPI) is left out of consideration here because that would be an error – that propellant only is subject to looking for the shares the tanks etc. need of it.. And so that propellant mustn’t be included here – while the other pew(...) are carried only and thus need a share of pew2(TPI).



propellant(s)

EOI: 0/etw(EOI) =
0/(evw + eew(EOI) + tew2(EOI) + cew(EOI)) =
0

TEI: pew2(EOI)/etw(TEI)
pew2(EOI)/(evw + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew(EOI))

POI: (pew2(TEI) + pew2(EOI)/etw(POI)
(pew2(TEI) + pew2(EOI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + cew(TEI) + pew2(TEI)
+ cewp(POI)) =
(pew2(TEI) + pew2(EOI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI)
+ cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))

TPI: (pew2(POI) + pew2(TEI) + pew2(EOI))/etw(TPI)
(pew2(POI) + pew2(TEI) + pew2(EOI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + cew(POI)
+ pew2(POI)) =
(pew2(POI) + pew2(TEI) + pew2(EOI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI)
+ tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))

Again pew2(TPI) is left out of consideration here because that would be an error here also – that propellant only is subject to looking for the shares the propellants to be carried. need of it.. And so that propellant mustn’t be included here – while the other pew(...) are carried only and thus need a share of pew2(TPI).



Amounts of Propellants resulting from the additional Shares

The amounts of propellants per tank or portion of propellant now can be got like those for the existing vehicle itself, its engine or its cargo(s):



tank(s)

ets(TPI) * pew2(TPI)
ets(POI) * pew2(POI)
ets(TEI) * pew2(TEI)
ets(EOI) * pew2(EOI)



propellant(s)

EOI: 0 * pew2(EOI)

TEI: (pew2(EOI)/(evw + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) *
pew2(TEI)

POI: ((pew2(TEI) + pew2(EOI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + cew(TEI) +
Pew2(TEI) + cewp(POI))) * pew2(POI) =
((pew2(TEI) + pew2(EOI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI)
+ cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) * pew2(POI)

TPI: ((pew2(POI) + pew2(TEI) + pew2(EOI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) +
cew(POI) + pew2(POI))) * pew2(TPI) =
((pew2(POI) + pew2(TEI) + pew2(EOI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI)
+ tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) *
pew2(TPI)

Conclusion to the really required cylindrical Tank

The sums of propellants that can be and have been calculated directly are parts of X. Those sums had to be claculated per phase and only then can be added to get a total sum over all phases. So obviuosly some parts of X can be got separated and split only and then have to be added on. This menas that the tank too can be calculated in portions only.

To get those parts of X that are needed to carry thge tank Y and those parts of X required for future phases the propellants for the shares of the tanks and propellants of the existing vehicle must be used – which too are available per phase only. So obviously the complete X and Y can be calculated per phase only – at least I didn’t find another way although such a way might exist. The serach for it simply became more and more nebulous and vague the further I treid to go the way towards avoiding splits.

Going the way of splitting X and Y the phase EOI is catching attention because of the following:

EOI: 0 * pew2(EOI)

It means that no propellant for any future phase is to be carried in that phase. So the only amounts of propellants required for that phase are those that are calculated directly plus the propellant required to carry Y. All the other phases require propellant to carry propellant for future phases.

So obviously EOI is the key-phase to get X and Y.

Propellant required for the Tank of the Potential Vehicle

Weight of the complete cylindrical Tank of the Potential Vehicle

But Y is NOT 0 because the tank is reusable and contains the propellant to be consumed in the phase EOI. Y has to have the capacity of all the portions and parts of X of all the phases. The weight of that capacity is the weight of the surface of the tank. To calculate that weight the weight per unit of volume – m^3 here – of any tank can be calculated and used. There might be lighter and heavier tanks but this can be considered later like other numbers also.

The volume the weight per unit has to be multiplied by can be concluded from X – which means the total X here but NOT the portions per phase or the sum(...) separatedly. This conclusion is possible only by applying the density of X which results in the volume of X and then to calculate the surface from the volume:

X/dyp = V

X = X(EOI) + X(TEI) + X(POI) + X(TPI)

and so

X/dyp = (X(EOI) + X(TEI) + X(POI) + X(TPI))/dyp = V

It’s obvious that all of phases need to be calculated to get the volume and thus Y while in parallel Y needs to be known to calculate each phase. So both X and Y must be calculated in parallel – this is possible only via a system of equations. Since there are four phases and four X(...) exactly four equations are required, These I am going to derive now..

A portion of each X(...) is known alreday from the direct calculations as sum(...). This means

X(EOI) = sum(EOI) + Z(EOI)
X(TEI) = sum(TEI) + Z(TEI)
X(POI) = sum(POI) + Z(POI)
X(TPI) = sum(TPI) + Z(TPI)

where Z(...) simply is the remainder of X(...) that couldn’t be calculated directly. Z(EOI) is the propellant for Y only. So the calculation of Y now is

X/dyp =
(X(EOI) + X(TEI) + X(POI) + X(TPI))/dyp =
(sum(EOI) + Z(EOI) + sum(TEI) + Z(TEI) + sum(POI) + Z(POI) + sum(TPI) + Z(TPI))/dyp =
(Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + sum(EOI) + sum(TEI) + sum(POI) + sum(TPI))/dyp =
V

V can be the volume of a cylinder here. The general formular for the volume of a cylinder is

V = Pi * r^2 * h.

r and h here are

r = dtp/2 and
h = htp

while

V = vtp volume of tank of the potential vehicle
.

Then

vtp =
Pi * (dtp/2)^2 * htp =
Pi * (dtp^2)/4) * htp =
((Pi * dtp^2)/4) * htp =
(Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + sum(EOI) + sum(TEI) + sum(POI) + sum(TPI))/dyp =
vtp
.

Both the dtp and htp are required to calculate the area of the tank and at least one of the two variables dtp and htp has to be given externally because it can’t be calculated here. That externally given variable should be dtp – in the raw and chaotic calculations of earlier posts I took the Block DM multiple and usually cylindrical stages are connected bottom circular area on top circular area.

Then htp can be calculated here

Modification

htp =
vtp/((Pi * dtp^2)/4) =
((Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + sum(EOI) + sum(TEI) + sum(POI) + sum(TPI))/dyp)/((Pi * dtp^2)/4) =
(4 * (Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + sum(EOI) + sum(TEI) + sum(POI) + sum(TPI)))/(dyp * (Pi * dtp^2)) =
(4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * (sum(EOI) + sum(TEI) + sum(POI) + sum(TPI)))/(dyp * Pi * dtp^2)

For shortness let’s set

sum(EOI) + sum(TEI) + sum(POI) + sum(TPI) = tsm total sum

again - meaning

(4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(dyp * Pi * dtp^2)

This is the volume of the propellant – around that is the volume between the tank and the side walls of the stage which contains all the materials and things that make up the weight of the tank. This is the

volume along tank length =

Pi * (dip/2)^2 * htp

.

The volume of the tank is

Pi * (dtp/2)^2 * htp

And because of this

volume around the propellant =

Pi * (dip/2)^2 * hip - Pi * (dtp/2)^2 * htp =

Pi * ((dip/2)^2 - (dtp/2)^2) * htp

iu to be applied:

Pi * ((dip/2)^2 - (dtp/2)^2) * ((4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(dyp * Pi * dtp^2))

.

This has to be multiplied by the weight per unit of volume whichz has been got in the previous post as

tpw/(Pi * (dip/2)^2 * hip – pct/dyp)

Then the weight of the tank is

Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(dyp * Pi * dtp^2) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))

Both Vehicles: Ratio of the Tank Weights

Like for the details the propellants are calculated for already the ratio between the tank of the potential vehicle and the tank of the existng vehicle is required. The weight of the tank of the potential vehicle simply has to be divided by the tank weight of the existing vehicle – but this has to be done phase by phase since the weight of the tank of the existing vehicle changes from phase to phase because the tank for each phase is expended when it’s empty.

The ratios are

EOI: (Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 *
tsm)/(dyp * Pi * dtp^2) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp)))/tew2(EOI) =
Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 *
tsm)/(dyp * Pi * dtp^2 * tew2(EOI)) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))

TEI: (Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 *
tsm)/(dyp * Pi * dtp^2) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp)))/(tew2(EOI) + tew2(TEI)) =
Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 *
tsm)/(dyp * Pi * dtp^2 * (tew2(EOI) + tew2(TEI))) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))

POI: (Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 *
tsm)/(dyp * Pi * dtp^2) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp)))/(tew2(EOI) + tew2(TEI) + tew2(POI)) =
Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 *
tsm)/(dyp * Pi * dtp^2 * (tew2(EOI) + tew2(TEI) + tew2(POI))) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))

TPI: (Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 *
tsm)/(dyp * Pi * dtp^2) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp)))/(tew2(EOI) + tew2(TEI) + tew2(POI) +
tew2(TPI)) =
Pi * ((dip/2)^2 - (dtp/2)^2) * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 *
tsm)/(dyp * Pi * dtp^2 * (tew2(EOI) + tew2(TEI) + tew2(POI) + tew2(TPI))) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))

The four formulars include some terms that are constant. These terms can be shortened and should be shortened to get improved overview:

tew2(EOI) = twe(EOI)
(tew2(EOI) + tew2(TEI)) = twe(TEI)
(tew2(EOI) + tew2(TEI) + tew2(POI)) = twe(POI)
(tew2(EOI) + tew2(TEI) + tew2(POI) + tew2(TPI)) = twe(TPI)
dyp * Pi * dtp^2 = fdca
Pi * ((dip/2)^2 - (dtp/2)^2) = fcad

The shortened formulars are

EOI: fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(EOI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))

TEI: fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(TEI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))

POI: fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(POI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))

TPI: fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/( fdca * twe(TPI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))

Resulting Amount of Propellant for the cylindrical Tank

What’s left is to multiply the ratios got by the propellant amounts got for the expendable tanks of the existing vehicle, the ratio of the isps and the correction factor:

EOI: (fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(EOI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(EOI) * pew2(EOI) * bir(EOI) * cor(EOI)

TEI: (fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(TEI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(TEI) * pew2(TEI) * bir(TEI) * cor(TEI)

POI: (fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(POI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(POI) * pew2(POI) * bir(POI) * cor(POI)

TPI: (fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/( fdca * twe(TPI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(TPI) * pew2(TPI) * bir(TPI) * cor(TPI)

Deriving the spherical alternative

When the complete amount of propellant required X is calculated later the volume V(X) this amount requires can be calculated as

X/dyp = V(X) = (X(TPI) + X(POI) + X(TEI) + X(EOI))/dyp

The volume of a sphere is

V = (4/3) * Pi * r^3

and so

V(X) = (4/3) * Pi * ras^3

as the spherical volume of X with ras being the spherical radius ( = dis/2 with dis being the spherical diameter).

Because of

X/dyp = V(X)

X/dyp = (4/3) * Pi * ras^3

is valid.

1. Modification

(X/dyp)/Pi = (4/3) * ras^3
= X/(dyp * Pi)

2. Modification

(X/(dyp * Pi))/(4/3) = ras^3
= (X/(dyp * Pi)) * (3/4)
= (X * 3)/((dyp * Pi) * 4)
= (X * 3)/(dyp * Pi * 4)

3. Modification

ras = cuberoot((X * 3)/(dyp * Pi * 4)

Using the difference between the diameter of the potential cylindrical tank – dtp -.and the diameter of the volume it is located in – dip - the thickness of a spherical stage around the spherical volume of X is available. It looks a bit dissatisfying that the thickness can’t be determined externally here but such a link improves comparability in my eyes.

The difference of the diameters would involve the thickness at one point plus the thickness at the opposite point which would be wrong here and so the difference has to be divided by 2:

(dip – dtp)/2

.

Then

ras2 =
ras + (dip – dtp)/2 =
cuberoot((X * 3)/(dyp * Pi * 4) + (dip – dtp)/2

has to be applied to get the volume around the spherical volume of X:

(4/3) * Pi * ras2^3 =
(4/3) * Pi * (ras + (dip – dtp)/2)^3 =
(4/3) * Pi * (cuberoot((X * 3)/(dyp * Pi * 4) + (dip – dtp)/2)^3

.

It is not required to calculate the polynomes and the weight can be got as

((4/3) * Pi * (ras + (dip – dtp)/2)^3) * tpw/(Pi * (dip/2)^2 * hip – pct/dyp)

.

This now is an adjustment very similar to the adjustement of the tank of the existing vehilce to the density of the propellant of the potential vehicle done three posts earlier – which means that the same problem occurrs: an additional constant is required that isn’t available. Because of the adjustment here is done according to the earlier one: X is divided by the total weight of the potential vehicle, its engine, its tank calculated already, its cargo(s) and the propellants it carries for future phases.

Because of two essential aspects next not only the new required amount of propellant is to be calculated but the calculations have to go at least one step further than in the case of the adjustment to another density:

1. The replacement of the cylindrical tank by ist spherical alternative might change the propellant requirements significantly
2. If that change is significant really then another adjustment of the tank volume and the required propellant will be necessary.

First the same has to be done like in the case of the adjustment to another density -the calculation of the new propellant amount X2. The original hardware-weight is

pvw + epq + cpw(EOI) + cpwp(POI) + (X/pct) * (tpw – weight of top + bottom) + weight of top plus bottom

with

(X/pct) * (tpw – weight of top + bottom) + weight of top plus bottom

being the weight of the cylindrical tank of the potential vehicle.

The required amunt of propellant in the spherical case then is

X2 = (X/(pvw + epq + cpw(EOI) + cpwp(POI) + (X/pct) * (tpw – weight of top + bottom) + weight of top plus
bottom)) * (pvw + epq + cpw(EOI) + cpwp(POI) + ((4/3) * Pi * (ras + (dip – dtp)/2)^3) * tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))

Now the same is repeated what has been applied to calculate the spherical case from the cylindrical case:

ras3 = cuberoot((X2 * 3)/(dyp * Pi * 4)

volume of spherical stage: (4/3) * Pi * (ras3 + (dip – dtp)/2)^3

weight of spherical stage: ((4/3) * Pi * (ras3 + (dip – dtp)/2)^3) * tpw/(Pi * (dip/2)^2 * hip – pct/dyp)

X3 = (X2/(pvw + epq + cpw(EOI) + cpwp(POI) + (X2/pct) * (tpw – weight of top + bottom) + weight of top
plus bottom)) * (pvw + epq + cpw(EOI) + cpwp(POI) + ((4/3) * Pi * (ras + (dip – dtp)/2)^3) * tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))

„weight of top plus bottom“ has been calculated in the previous post without introducing a symbol for it – I didn’t want to insert the formular got because I think that this would reduce the overview regarding the resulating formular(s).

Of course – in difference to the formulars above X2 and X3 must be calculated by calculated as sums of X2(...) or X3(...). Since these are calculations of propellants this will be done later.

It may be that in some cases results more iterations of the procedure described result in significant redutions of the propellant requirements than in other cases. The larger the amount of propellant got the more iterations are worth to be done.

Please note here: At present I didn’t fin a way to avoid iterations.




End of Appendix


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Post    Posted on: Sat Oct 28, 2006 6:41 pm
Contents

Getting the total amount of Propellant: Linear Algebra
Resulting Tank
Costs and Safety Margins
Applying the Formulars and Equations: Excel
Appendix
Propellant for the Propellants: Ratio
Propellant for Tanks and Propellants together
Propellant if the Tank is spherical
Preliminaryly resulting Costs got by the Calculations


Getting the total amount of Propellant: Linear Algebra

Like in the previous post the ratio between the amount of propellant the potential vehicle needs and the amount of propellant the existing vehicle needs can be calculated as formular but not as a number yet since the amount in the potential case isn’t known yet.

But the two unknown amounts – propellant for the tank and propellant to carry the propellant for future phases – can be added to the already directly calculated and thus known amounts. The result is the complete propellant the potential vehicle needs – this result still is not known yet.

Now there four phases considered up to now and the amount of propellant is divided so that in each of the four phases sufficient propellant is available. And so for each of these four phases one share of the unknown amount of propellant is taken into account. Each share-amount depends on the same total amount that is not known yet. This menas that there are four equations giving the amount of propellant for one of the four phases – and there are four unknowns: the share-amounts. Also all the unknowns are linear – they are not set to square, cube or the like nor to roots.

Thus there are four linear eqations and four unknowns – which menas that there is a linear equation system by which all four unknowns can be calculated. This is done in the Appendix.

Resulting Tank

Summing up the results not only tells the total propellant required but by division by the density of the propellant also tells the volume of the tank from which the tank weight can be calculated.

The case of a spherical tank also requires calculation per phase because the amounts for future phases may be changing also – menaing that here also summing up is required. This hadn’t been taken into account in the previous post yet because the final look into the propellant amounts is required for that.

Costs and Safety Margins

Now the costs can be calculated by applying the money invested into the vehicle(s) and the tank and the price and transportation cosst of the propellant. Then the number of passengers per flight can be applied.

This may require more detailed consideration but this will be done later and is limited because it is a matter of accounting merely.

But the costs got this way are preliminary only because most of this is derived from vehicles that necessaryly either aren’t lunar or not reusable. Because of this a safety margin must be applied. The safety margin can be

a percentage,
rounding up to the next quarter of 10^i (it’s not known yet if 1 = 5, i = 6 etc.)
rounding up to a half of 10^i
rounding up to the next value estimated by an expert.

There might be combinations.

The spherical tank is of particular meaning here because a decision criterion is to be mentioned here. I have in ind apherical tank built in the orbit. May be that this increases the construction costs and thus the investment – but on the other hand it reduces the weight and thus the propellant costs. The in-orbit construction is economical as long as ithe sume of ist depreciation per flight and the other dosts per flight ar less than or equal to the costs in the cylindrical case!

Applying the Formulars and Equations: Excel

The large number of data, shares and ratios to be applied menas a significant chance of errors and major mistakes. To avoid that all I am working on an Excel spreadsheet that is ready to be applied. But it requires improvements and may be enhancements. It will include tables of alternative vehicles etc. and the opportunity to add more vehicles etc. And trips can be formed because they are required to do the calculations.



In the next post something will have to be said about a few parameters and some test calculations will be considered that are done by the Excel-spreadsheet.



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



Appendix


Propellant for the Propellants: Ratio

Since the weights don’t have to be calculated here like for the tanks the unknown ratios are given directly

EOI: 1 – no propellant to be carried in both the existing and the potential case which means that the
multiplication by this ratio mustn’t change or alter values. That’s guaranteed by 1

TEI: X(EOI)/pew2(EOI) =
(Z(EOI) + sum(EOI))/pew2(EOI)

POI: (X(TEI) + X(EOI))/(pew2(TEI) + pew2(EOI)) =
(Z(TEI) + sum(TEI) + Z(EOI) + sum(EOI))/(pew2(TEI) + pew2(EOI)) =
(Z(TEI) + Z(EOI) + sum(TEI) + sum(EOI))/(pew2(TEI) + pew2(EOI))

TPI: (X(POI) + X(TEI) + X(EOI))/(pew2(EOI) + pew2(TEI) + pew2(POI)) =
(Z(POI) + sum(POI) + Z(TEI) + sum(TEI) + Z(EOI) + sum(EOI))/(pew2(POI) + pew2(TEI) + pew2(EOI)) =
(Z(POI) + Z(TEI) + Z(EOI) + sum(POI) + sum(TEI) + sum(EOI)/(pew2(POI) + pew2(TEI) + pew2(EOI))

The resulting but still unknown propellants then are

EOI: 1 * 0 * pew2(EOI) * bir(EOI)=
0 * pew2(EOI) * bir(EOI)=
0

TEI: ((Z(EOI) + sum(EOI))/pew2(EOI)) * ((pew2(EOI)/(evw + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI)
+ cew(EOI) + pew2(EOI))) * pew2(TEI)) * bir(TEI) =
((Z(EOI) + sum(EOI))/(evw + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) *
pew2(TEI) * bir(TEI) =
(((Z(EOI) + sum(EOI)) * pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) +
cew(EOI) + pew2(EOI))) =
((Z(EOI) * pew2(TEI) * bir(TEI) + sum(EOI) * pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) =
Z(EOI) * ((pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) +
((sum(EOI) * pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI)))

POI: ((Z(TEI) + Z(EOI) + sum(TEI) + sum(EOI))/(pew2(TEI) + pew2(EOI))) * (((pew2(TEI) +
pew2(EOI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) +
pew2(EOI) + pew2(TEI) + cewp(POI))) * pew2(POI) * bir(POI)) =
((Z(TEI) + Z(EOI) + sum(TEI) + sum(EOI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) +
eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) * pew2(POI) * bir(POI) =
(((Z(TEI) + Z(EOI) + sum(TEI) + sum(EOI)) * pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) +
eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) =
((Z(TEI) * pew2(POI) * bir(POI) + Z(EOI) * pew2(POI) * bir(POI) + (sum(TEI) + sum(EOI)) * pew2(POI) *
bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) +
pew2(EOI) + pew2(TEI) + cewp(POI))) =
Z(TEI) * ((pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) +
Z(EOI) * ((pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) +
(((sum(TEI) + sum(EOI)) * pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) +
eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI)))

TPI: ((Z(POI) + Z(TEI) + Z(EOI) + sum(POI) + sum(TEI) + sum(EOI)/(pew2(POI) + pew2(TEI) + pew2(EOI)))
* (((pew2(POI) + pew2(TEI) + pew2(EOI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) +
eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) * pew2(TPI) * bir(TPI)) =
((Z(POI) + Z(TEI) + Z(EOI) + sum(POI) + sum(TEI) + sum(EOI))/(evw + eew(TPI) + tew2(TPI) + eew(POI)
+ tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) +
cewp(POI) + pew2(POI))) * pew2(TPI) * bir(TPI) =
(((Z(POI) + Z(TEI) + Z(EOI) + sum(POI) + sum(TEI) + sum(EOI)) * pew2(TPI) * bir(TPI))/(evw +
eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) +
cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) =
((Z(POI) * pew2(TPI) * bir(TPI) + Z(TEI) * pew2(TPI) * bir(TPI) + Z(EOI) * pew2(TPI) * bir(TPI) +
(sum(POI) + sum(TEI) + sum(EOI)) * pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) +
tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) +
cewp(POI) + pew2(POI))) =
Z(POI) * ((pew2(TPI) * bir(TPI))/ /(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
Z(TEI) * ((pew2(TPI) * bir(TPI))/ /(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
Z(EOI) * ((pew2(TPI) * bir(TPI))/ /(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
(((sum(POI) + sum(TEI) + sum(EOI)) * pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) +
tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) +
cewp(POI) + pew2(POI)))

Propellant for Tanks and Propellants together

EOI: (fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(EOI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(EOI) * pew2(EOI) * bir + 0 =
(fcad * 4 * Z(EOI) + fcad * 4 * Z(TEI) + fcad * 4 * Z(POI) + fcad * 4 * Z(TPI) + fcad * 4 * tsm)/(fdca *
twe(EOI)) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))) * ets(EOI) * pew2(EOI) * bir =
((Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * fcad * 4)/(fdca * twe(EOI)) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))) * ets(EOI) * pew2(EOI) * bir =
(Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * ((fcad * 4)/(fdca * twe(EOI)) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))) * ets(EOI) * pew2(EOI) * bir

TEI: (fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(TEI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(TEI) * pew2(TEI) * bir +
Z(EOI) * ((pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) +
((sum(EOI) * pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) =
(fcad * 4 * Z(EOI) + fcad * 4 * Z(TEI) + fcad * 4 * Z(POI) + fcad * 4 * Z(TPI) + fcad * 4 * tsm)/(fdca *
twe(TEI)) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))) * ets(TEI) * pew2(TEI) * bir +
Z(EOI) * ((pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) +
((sum(EOI) * pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) =
(Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * ((fcad * 4)/(fdca * twe(TEI)) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))) * ets(TEI) * pew2(TEI) * bir +
Z(EOI) * ((pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI))) +
((sum(EOI) * pew2(TEI) * bir(TEI))/(evw + eew(TEI) + tew2(TEI) +
tew2(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI)))

POI: (fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/(fdca * twe(POI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(POI) * pew2(POI) * bir +
Z(TEI) * ((pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) +
Z(EOI) * ((pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) +
(((sum(TEI) + sum(EOI)) * pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) +
eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) =
(fcad * 4 * Z(EOI) + fcad * 4 * Z(TEI) + fcad * 4 * Z(POI) + fcad * 4 * Z(TPI) + fcad * 4 * tsm)/(fdca *
twe(POI)) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))) * ets(POI) * pew2(POI) * bir +
Z(TEI) * ((pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) +
Z(EOI) * ((pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) +
(((sum(TEI) + sum(EOI)) * pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) +
eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) =
(Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * ((fcad * 4)/(fdca * twe(POI)) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))) * ets(POI) * pew2(POI) * bir +
Z(TEI) * ((pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) +
Z(EOI) * ((pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) +
tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI))) +
(((sum(TEI) + sum(EOI)) * pew2(POI) * bir(POI))/(evw + eew(POI) + tew2(POI) + eew(TEI) + tew2(TEI) +
eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI)))

TPI: (fcad * (4 * Z(EOI) + 4 * Z(TEI) + 4 * Z(POI) + 4 * Z(TPI) + 4 * tsm)/( fdca * twe(TPI)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(TPI) * pew2(TPI) * bir +
Z(POI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
Z(TEI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
Z(EOI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
(((sum(POI) + sum(TEI) + sum(EOI)) * pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) +
tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) +
cewp(POI) + pew2(POI))) =
(fcad * 4 * Z(EOI) + fcad * 4 * Z(TEI) + fcad * 4 * Z(POI) + fcad * 4 * Z(TPI) + fcad * 4 * tsm)/(fdca *
twe(TPI)) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))) * ets(TPI) * pew2(TPI) * bir +
Z(POI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
Z(TEI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
Z(EOI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
(((sum(POI) + sum(TEI) + sum(EOI)) * pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) +
tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) +
cewp(POI) + pew2(POI))) =
(Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * ((fcad * 4)/(fdca * twe(TPI)) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))) * ets(TPI) * pew2(TPI) * bir +
Z(POI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
Z(TEI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
Z(EOI) * ((pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) + tew2(POI) + eew(TEI) +
tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) + cewp(POI) + pew2(POI))) +
(((sum(POI) + sum(TEI) + sum(EOI)) * pew2(TPI) * bir(TPI))/(evw + eew(TPI) + tew2(TPI) + eew(POI) +
tew2(POI) + eew(TEI) + tew2(TEI) + eew(EOI) + tew2(EOI) + cew(EOI) + pew2(EOI) + pew2(TEI) +
cewp(POI) + pew2(POI)))

To shorten all this the following constants are introduced:

a(...)

EOI: ((fcad * 4)/(fdca * twe(EOI)) * weight per unit of volume) * ets(EOI) * pew(EOI) * bir = a(EOI)
TEI: ((fcad * 4)/(fdca * twe(TEI)) * weight per unit of volume) * ets(TEI) * pew(TEI) * bir = a(TEI)
POI: ((fcad * 4)/(fdca * twe(POI)) * weight per unit of volume) * ets(POI) * pew(POI) * bir = a(POI)
TPI: ((fcad * 4)/(fdca * twe(TPI)) * weight per unit of volume) * ets(TPI) * pew(TPI) * bir = a(TPI)

c(...)

EOI: -
TEI: (pew(TEI) * bir(TEI))/(evw + eew(TEI) + tew(TEI) + eew(EOI) + tew(EOI) + cew(EOI) + pew(EOI)) =
c(TEI)
POI: (pew(POI) * bir(POI))/(evw + eew(POI) + tew(POI) + eew(TEI) + tew(TEI) + eew(EOI) + tew(EOI) +
cew(EOI) + pew(EOI) + pew(TEI) + cewp(POI)) = c(POI)
TPI: (pew(TPI) * bir(TPI))/(evw + eew(TPI) + tew(TPI) + eew(POI) + tew(POI) + eew(TEI) + tew(TEI) +
eew(EOI) + tew(EOI) + cew(EOI) + pew(EOI) + pew(TEI) + cewp(POI) + pew(POI)) = c(TPI)

Using the constants the equations look like this:

EOI: ((Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * a(EOI)) =
(Z(EOI) * a(EOI) + Z(TEI) * a(EOI) + Z(POI) * a(EOI) + Z(TPI) * a(EOI) + tsm * a(EOI)) =
Z(EOI) * a(EOI) + Z(TEI) * a(EOI) + Z(POI) * a(EOI) + Z(TPI) * a(EOI) + tsm * a(EOI) =
Z(EOI)

TEI: ((Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * a(TEI)) +
Z(EOI) * c(TEI) +
sum(EOI) * c(TEI) =
(Z(EOI) * a(TEI) + Z(TEI) * a(TEI) + Z(POI) * a(TEI) + Z(TPI) * a(TEI) + tsm * a(TEI)) +
Z(EOI) * c(TEI) +
sum(EOI) * c(TEI) =
Z(EOI) * a(TEI) + Z(TEI) * a(TEI) + Z(POI) * a(TEI) + Z(TPI) * a(TEI) + tsm * a(TEI) +
Z(EOI) * c(TEI) +
sum(EOI) * c(TEI) =
Z(EOI) * (a(TEI) + c(TEI)) + Z(TEI) * a(TEI) + Z(POI) * a(TEI) + Z(TPI) * a(TEI) + tsm * a(TEI)
+ sum(EOI) * c(TEI) =
Z(TEI)

POI: ((Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * a(POI)) +
Z(TEI) * c(POI) +
Z(EOI) * c(POI) +
(sum(TEI) + sum(EOI)) * c(POI) =
(Z(EOI) * a(POI) + Z(TEI) * a(POI) + Z(POI) * a(POI) + Z(TPI) * a(POI) + tsm * a(POI)) +
Z(TEI) * c(POI) +
Z(EOI) * c(POI) +
(sum(TEI) + sum(EOI)) * c(POI) =
Z(EOI) * a(POI) + Z(TEI) * a(POI) + Z(POI) * a(POI) + Z(TPI) * a(POI) + tsm * a(POI) +
Z(TEI) * c(POI) +
Z(EOI) * c(POI) +
(sum(TEI) + sum(EOI)) * c(POI) =
Z(EOI) * (a(POI) + c(POI)) + Z(TEI) * (a(POI) + c(POI)) + Z(POI) * a(POI) + Z(TPI) * a(POI) + tsm *
a(POI) + (sum(TEI) + sum(EOI)) * c(POI) =
Z(POI)

TPI: ((Z(EOI) + Z(TEI) + Z(POI) + Z(TPI) + tsm) * a(TPI)) +
Z(POI) * c(TPI) +
Z(TEI) * c(TPI) +
Z(EOI) * c(TPI) +
(sum(POI) + sum(TEI) + sum(EOI)) * c(TPI) =
(Z(EOI) * a(TPI) + Z(TEI) * a(TPI) + Z(POI) * a(TPI) + Z(TPI) * a(TPI) + tsm * a(TPI)) +
Z(POI) * c(TPI) +
Z(TEI) * c(TPI) +
Z(EOI) * c(TPI) +
(sum(POI) + sum(TEI) + sum(EOI)) * c(TPI) =
Z(EOI) * a(TPI) + Z(TEI) * a(TPI) + Z(POI) * a(TPI) + Z(TPI) * a(TPI) + tsm * a(TPI) +
Z(POI) * c(TPI) +
Z(TEI) * c(TPI) +
Z(EOI) * c(TPI) +
(sum(POI) + sum(TEI) + sum(EOI)) * c(TPI) =
Z(EOI) * (a(TPI) + c(TPI)) + Z(TEI) * (a(TPI) + c(TPI)) + Z(POI) * (a(TPI) + c(TPI)) + Z(TPI) * a(TPI) +
tsm * a(TPI) + (sum(POI) + sum(TEI) + sum(EOI)) * c(TPI) =
Z(TPI)

It can be seen that more shortenings are possible:

EOI: tsm * a(EOI) = d(EOI)
TEI: tsm * a(TEI) + sum(EOI) * c(TEI) = d(TEI)
POI: tsm * a(POI) + (sum(TEI) + sum(EOI)) * c(POI) = d(POI)
TPI: tsm * a(TPI) + (sum(POI) + sum(TEI) + sum(EOI)) * c(TPI) = d(TPI)



EOI: -
TEI: a(TEI) + c(TEI) = e(TEI)
POI: a(POI) + c(POI) = e(POI)
TPI: a(TPI) + c(TPI) = e(TPI)

The equations then finally are

EOI: Z(EOI) * a(EOI) + Z(TEI) * a(EOI) + Z(POI) * a(EOI) + Z(TPI) * a(EOI) + d(EOI) = Z(EOI)
TEI: Z(EOI) * e(TEI) + Z(TEI) * a(TEI) + Z(POI) * a(TEI) + Z(TPI) * a(TEI) + d(TEI) = Z(TEI)
POI: Z(EOI) * e(POI) + Z(TEI) * e(POI) + Z(POI) * a(POI) + Z(TPI) * a(POI) + d(POI) = Z(POI)
TPI: Z(EOI) * e(TPI) + Z(TEI) * e(TPI) + Z(POI) * e(TPI) + Z(TPI) * a(TPI) + d(TPI) = Z(TPI)

The solution for Z(EOI) then is

1. Modification

Z(TEI) * a(EOI) + Z(POI) * a(EOI) + Z(TPI) * a(EOI) + d(EOI) = Z(EOI) - Z(EOI) * a(EOI) =
Z(EOI) * (1 - a(EOI))

2. Modification

(Z(TEI) * a(EOI) + Z(POI) * a(EOI) + Z(TPI) * a(EOI) + d(EOI))/(1 - a(EOI)) = Z(EOI) =
Z(TEI) * a(EOI)/(1 - a(EOI)) + Z(POI) * a(EOI)/(1 - a(EOI)) + Z(TPI) * a(EOI)/(1 - a(EOI)) + d(EOI)/(1 - a(EOI)) = Z(EOI)

Calling a(EOI)/(1 - a(EOI)) f(EOI) and d(EOI)/(1 - a(EOI)) g(EOI) the equation is

Z(TEI) * f(EOI) + Z(POI) * f(EOI) + Z(TPI) * f(EOI) + g(EOI) = Z(EOI)



This can be inserted into the three remaining equations:

TEI: (Z(TEI) * f(EOI) + Z(POI) * f(EOI) + Z(TPI) * f(EOI) + g(EOI)) * e(TEI) + Z(TEI) * a(TEI) + Z(POI) *
a(TEI) + Z(TPI) * a(TEI) + d(TEI) = Z(TEI) =
Z(TEI) * f(EOI) * e(TEI) + Z(POI) * f(EOI) * e(TEI) + Z(TPI) * f(EOI) * e(TEI) + g(EOI) * e(TEI) + Z(TEI)
* a(TEI) + Z(POI) * a(TEI) + Z(TPI) * a(TEI) + d(TEI) = Z(TEI) =
Z(TEI) * (f(EOI) * e(TEI) + a(TEI)) + Z(POI) * (f(EOI) * e(TEI) + a(TEI)) + Z(TPI) * (f(EOI) * e(TEI) +
a(TEI)) + g(EOI) * e(TEI) + d(TEI) = Z(TEI)

POI: (Z(TEI) * f(EOI) + Z(POI) * f(EOI) + Z(TPI) * f(EOI) + g(EOI)) * e(POI) + Z(TEI) * e(POI) + Z(POI) *
a(POI) + Z(TPI) * a(POI) + d(POI) = Z(POI) =
Z(TEI) * f(EOI) * e(POI) + Z(POI) * f(EOI) * e(POI) + Z(TPI) * f(EOI) * e(POI) + g(EOI) * e(POI) +
Z(TEI) * e(POI) + Z(POI) * a(POI) + Z(TPI) * a(POI) + d(POI) = Z(POI) =
Z(TEI) * (f(EOI) * e(TEI) + e(POI)) + Z(POI) * (f(EOI) * e(POI) + a(POI)) + Z(TPI) * (f(EOI) * e(POI) +
a(POI)) + g(EOI) * e(POI) + d(POI) = Z(POI)

TPI: (Z(TEI) * f(EOI) + Z(POI) * f(EOI) + Z(TPI) * f(EOI) + g(EOI)) * e(TPI) + Z(TEI) * e(TPI) + Z(POI) *
e(TPI) + Z(TPI) * a(TPI) + d(TPI) = Z(TPI) =
Z(TEI) * f(EOI) * e(TPI) + Z(POI) * f(EOI) * e(TPI) + Z(TPI) * f(EOI) * e(TPI) + g(EOI) * e(TPI) + Z(TEI)
* e(TPI) + Z(POI) * e(TPI) + Z(TPI) * a(TPI) + d(TPI) = Z(TPI) =
Z(TEI) * (f(EOI) * e(TEI) + e(TPI)) + Z(POI) * (f(EOI) * e(TPI) + e(TPI)) + Z(TPI) * (f(EOI) * e(TPI) +
a(TPI)) + g(EOI) * e(TPI) + d(TPI) = Z(TPI)

This allows the solution for Z(TEI):

1. Modification

Z(POI) * (f(EOI) * e(TEI) + a(TEI)) + Z(TPI) * (f(EOI) * e(TEI) + a(TEI)) + g(EOI) * e(TEI) + d(TEI) =
Z(TEI) - Z(TEI) * (f(EOI) * e(TEI) + a(TEI)) =
Z(TEI) * (1 - (f(EOI) * e(TEI) + a(TEI))) =
Z(TEI) * (1 - f(EOI) * e(TEI) - a(TEI))

2. Modification

(Z(POI) * (f(EOI) * e(TEI) + a(TEI)) + Z(TPI) * (f(EOI) * e(TEI) + a(TEI)) + g(EOI) * e(TEI) + d(TEI))/(1 –
f(EOI) * e(TEI) - a(TEI)) = Z(TEI) =
Z(POI) * (f(EOI) * e(TEI) + a(TEI))/(1 - f(EOI) * e(TEI) - a(TEI)) + Z(TPI) * (f(EOI) * e(TEI) + a(TEI))/(1 –
f(EOI) * e(TEI) - a(TEI)) + (g(EOI) * e(TEI))/(1 - f(EOI) * e(TEI) - a(TEI)) + d(TEI)/(1 - f(EOI) * e(TEI) –
a(TEI)) = Z(TEI)

Calling (f(EOI) * e(TEI) + a(TEI))/(1 - f(EOI) * e(TEI) - a(TEI)) h(TEI) and
(g(EOI) * e(TEI))/(1 - f(EOI) * e(TEI) - a(TEI)) + d(TEI)/(1 - f(EOI) * e(TEI) - a(TEI)) i(TEI) the equation is

Z(POI) * h(TEI) + Z(TPI) * h(TEI) + i(TEI) = Z(TEI)

To get the remaining required solutions this is to be inserted again:

POI: (Z(POI) * h(TEI) + Z(TPI) * h(TEI) + i(TEI)) * (f(EOI) * e(TEI) + e(POI)) + Z(POI) * (f(EOI) * e(POI) +
a(POI)) + Z(TPI) * (f(EOI) * e(POI) + a(POI)) + g(EOI) * e(POI) + d(POI) = Z(POI) =
Z(POI) * h(TEI) * (f(EOI) * e(TEI) + e(POI)) + Z(TPI) * h(TEI) * (f(EOI) * e(TEI) + e(POI)) + i(TEI) *
(f(EOI) * e(TEI) + e(POI)) + Z(POI) * (f(EOI) * e(POI) + a(POI)) + Z(TPI) * (f(EOI) * e(POI) + a(POI)) +
g(EOI) * e(POI) + d(POI) = Z(POI) =
Z(POI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI)) + Z(POI) * (f(EOI) * e(POI) + a(POI)) + Z(TPI) *
(h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI)) + Z(TPI) * (f(EOI) * e(POI) + a(POI)) + i(TEI) * f(EOI) *
e(TEI) + i(TEI) * e(POI) + g(EOI) * e(POI) + d(POI) = Z(POI) =
Z(POI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI) + f(EOI) * e(POI) + a(POI)) + Z(TPI) * (h(TEI) *
f(EOI) * e(TEI) + h(TEI) * e(POI) + f(EOI) * e(POI) + a(POI)) + i(TEI) * f(EOI) * e(TEI) + i(TEI) * e(POI)
+ g(EOI) * e(POI) + d(POI) = Z(POI)

TPI: (Z(POI) * h(TEI) + Z(TPI) * h(TEI) + i(TEI)) * (f(EOI) * e(TEI) + e(TPI)) + Z(POI) * (f(EOI) * e(TPI) +
e(TPI)) + Z(TPI) * (f(EOI) * e(TPI) + a(TPI)) + g(EOI) * e(TPI) + d(TPI) = Z(TPI) =
Z(POI) * h(TEI) * (f(EOI) * e(TEI) + e(TPI)) + Z(TPI) * h(TEI) * (f(EOI) * e(TEI) + e(TPI)) + i(TEI) *
(f(EOI) * e(TEI) + e(TPI)) + Z(POI) * (f(EOI) * e(TPI) + e(TPI)) + Z(TPI) * (f(EOI) * e(TPI) + a(TPI)) +
g(EOI) * e(TPI) + d(TPI) = Z(TPI) =
Z(POI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI)) + Z(POI) * (f(EOI) * e(TPI) + e(TPI)) + Z(TPI) *
(h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI)) + Z(TPI) * (f(EOI) * e(TPI) + a(TPI)) + i(TEI) * f(EOI) *
e(TEI) + i(TEI) * e(TPI) + g(EOI) * e(TPI) + d(TPI) = Z(TPI) =
Z(POI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + e(TPI)) + Z(TPI) * (h(TEI) *
f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + a(TPI)) + i(TEI) * f(EOI) * e(TEI) + i(TEI) * e(TPI) +
g(EOI) * e(TPI) + d(TPI) = Z(TPI)

From this the solution for T(POI) follows:

1. Modification

Z(TPI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI) + f(EOI) * e(POI) + a(POI)) + i(TEI) * f(EOI) * e(TEI) + i(TEI) * e(POI) + g(EOI) * e(POI) + d(POI) = Z(POI) - Z(POI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI) + f(EOI) * e(POI) + a(POI)) =
Z(POI) * (1 - (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI) + f(EOI) * e(POI) + a(POI))) =
Z(POI) * (1 - h(TEI) * f(EOI) * e(TEI) - h(TEI) * e(POI) - f(EOI) * e(POI) - a(POI))

2. Modification

(Z(TPI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI) + f(EOI) * e(POI) + a(POI)) + i(TEI) * f(EOI) * e(TEI) +
i(TEI) * e(POI) + g(EOI) * e(POI) + d(POI))/(1 - h(TEI) * f(EOI) * e(TEI) - h(TEI) * e(POI) - f(EOI) * e(POI) –
a(POI)) = Z(POI) =
Z(TPI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI) + f(EOI) * e(POI) + a(POI))/(1 - h(TEI) * f(EOI) * e(TEI)
- h(TEI) * e(POI) - f(EOI) * e(POI) - a(POI)) + (i(TEI) * f(EOI) * e(TEI) + i(TEI) * e(POI) + g(EOI) * e(POI) +
d(POI))/(1 - h(TEI) * f(EOI) * e(TEI) - h(TEI) * e(POI) - f(EOI) * e(POI) - a(POI)) = Z(POI)

Calling (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(POI) + f(EOI) * e(POI) + a(POI))/(1 - h(TEI) * f(EOI) * e(TEI) - h(TEI) * e(POI) - f(EOI) * e(POI) - a(POI)) j(POI) and
(i(TEI) * f(EOI) * e(TEI) + i(TEI) * e(POI) + g(EOI) * e(POI) + d(POI))/(1 - h(TEI) * f(EOI) * e(TEI) - h(TEI) * e(POI) - f(EOI) * e(POI) - a(POI)) k(POI)

the equatiom is

Z(TPI) * j(POI) + k(POI) = Z(POI)

Inserting this into the only remaining equation to be solved the solution for Z(TPI) is got:

TPI: (Z(TPI) * j(POI) + k(POI)) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + e(TPI)) +
Z(TPI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + a(TPI)) + i(TEI) * f(EOI) * e(TEI)
+ i(TEI) * e(TPI) + g(EOI) * e(TPI) + d(TPI) = Z(TPI) =
Z(TPI) * j(POI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + e(TPI)) + k(POI) *
(h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + e(TPI)) + Z(TPI) * (h(TEI) * f(EOI) *
e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + a(TPI)) + i(TEI) * f(EOI) * e(TEI) + i(TEI) * e(TPI) + g(EOI)
* e(TPI) + d(TPI) = Z(TPI) =
Z(TPI) * (j(POI) * h(TEI) * f(EOI) * e(TEI) + j(POI) * h(TEI) * e(TPI) + j(POI) * f(EOI) * e(TPI) + j(POI) *
e(TPI)) + Z(TPI) * (h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + a(TPI)) + k(POI) *
(h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + e(TPI)) + i(TEI) * f(EOI) * e(TEI) + i(TEI)
* e(TPI) + g(EOI) * e(TPI) + d(TPI) = Z(TPI) =
Z(TPI) * (j(POI) * h(TEI) * f(EOI) * e(TEI) + j(POI) * h(TEI) * e(TPI) + j(POI) * f(EOI) * e(TPI) + j(POI) *
e(TPI) + h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + a(TPI)) + k(POI) * h(TEI) * f(EOI)
* e(TEI) + k(POI) * h(TEI) * e(TPI) + k(POI) * f(EOI) * e(TPI) + k(POI) * e(TPI) + i(TEI) * f(EOI) * e(TEI)
+ i(TEI) * e(TPI) + g(EOI) * e(TPI) + d(TPI) = Z(TPI)

1. Modification

k(POI) * h(TEI) * f(EOI) * e(TEI) + k(POI) * h(TEI) * e(TPI) + k(POI) * f(EOI) * e(TPI) + k(POI) * e(TPI) + i(TEI) * f(EOI) * e(TEI) + i(TEI) * e(TPI) + g(EOI) * e(TPI) + d(TPI) = Z(TPI) - Z(TPI) * (j(POI) * h(TEI) * f(EOI) * e(TEI) + j(POI) * h(TEI) * e(TPI) + j(POI) * f(EOI) * e(TPI) + j(POI) * e(TPI) + h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + a(TPI)) =
Z(TPI) * (1 - (j(POI) * h(TEI) * f(EOI) * e(TEI) + j(POI) * h(TEI) * e(TPI) + j(POI) * f(EOI) * e(TPI) + j(POI) * e(TPI) + h(TEI) * f(EOI) * e(TEI) + h(TEI) * e(TPI) + f(EOI) * e(TPI) + a(TPI))) =
Z(TPI) * (1 - j(POI) * h(TEI) * f(EOI) * e(TEI) - j(POI) * h(TEI) * e(TPI) - j(POI) * f(EOI) * e(TPI) - j(POI) * e(TPI) - h(TEI) * f(EOI) * e(TEI) - h(TEI) * e(TPI) - f(EOI) * e(TPI) - a(TPI))

2. Modification

(k(POI) * h(TEI) * f(EOI) * e(TEI) +
k(POI) * h(TEI) * e(TPI) +
k(POI) * f(EOI) * e(TPI) +
k(POI) * e(TPI) +
i(TEI) * f(EOI) * e(TEI) +
i(TEI) * e(TPI) +
g(EOI) * e(TPI) +
d(TPI))/(1 –
j(POI) * h(TEI) * f(EOI) * e(TEI) –
j(POI) * h(TEI) * e(TPI) –
j(POI) * f(EOI) * e(TPI) –
j(POI) * e(TPI) –
h(TEI) * f(EOI) * e(TEI) –
h(TEI) * e(TPI) –
f(EOI) * e(TPI) –
a(TPI)) =
Z(TPI)

So the amounts looked for are known now:

Z(TPI)
Z(POI)
Z(TEI)
Z(EOI)

Because of this the total propellant is known also now:

X =
X(TPI) + X(POI) + X(TEI) + X(EOI) =
Z(TPI) + Z(POI) + Z(TEI) + Z(EOI) + sum(TPI) + sum(POI) + sum(TEI) + sum(EOI)

Now for some reasons of practice

evw,
eiv

might be changed between phases because it might be that no vehicle has done all the phases. But this doesn’t change the formulars but simply turns some constants phase-related that weren’t before.

Propellant if the Tank is spherical

Like described in the previous post the weight of the spherical tank can be calculated using the now known total amount of propellant required. – but in difference to that post that now known amount of propellant has to be applied phase per phase (like said there.

The reason is that the propellant is effected phase-wise – meaning that the direct application described in the previous post might lead to too large amounts.

Of course the weight of the tank can be calculated directly from the total amounts X, X2 and X3 – but X2 and X3 cannot be calculated from the weights of the tanks to be carried directly!

The reason is that parts of the total amounts X2 and X3 are carried fro future phases – and these parts are effected also which results in additional changes of the required amounts X2 and X3.

Because of this not only the weight of the cylindrical tank has to be subtracted from the totl weight of the potential vehicle, ist engine, ist cargo(s) and ist cylindrical tank and then the weight of the spherical tank added – but the amount of propellant required for future phases

X(EOI) to X(POI) ,
X2(EOI) to X2(POI) and
X3(EOI) to X3(POI)

have to be added also.

To get the new required amounts per phase

X2(EOI) to X2(TPI) and
X3(EOI) to X3)TPI)

The original

X(EOI) to X(TPI) and
X2(EOI) to X2(TPI)

have to be divided by the original

total weight of potential vehicle including engine, tank for X and cargo + X(POI) + X(TEI) + X(EOI) or
total weight of potential vehicle including engine, tank for X2 and cargo + X2(POI) + X2(TEI) + X2(EOI) in TPI,
total weight of potential vehicle including engine, tank for X and cargo + X(TEI) + X(EOI) or
total weight of potential vehicle including engine, tank for X2 and cargo + X2(TEI) + X2(EOI) in POI,
total weight of potential vehicle including engine, tank for X and cargo + X(EOI) or
total weight of potential vehicle including engine, tank for X2 and cargo + X2(EOI) in TEI and
total weight of potential vehicle including engine, tank for X or X2 and cargo in EOI.

The results have to be multiplied in the order first EOI, then TEI, POI and last TPI by

total weight of potential vehicle including engine, tank for X2 and cargo + X(POI) + X(TEI) + X(EOI) or
total weight of potential vehicle including engine, tank for X3 and cargo + X2(POI) + X2(TEI) + X2(EOI) in TPI,
total weight of potential vehicle including engine, tank for X2 and cargo + X(TEI) + X(EOI) or
total weight of potential vehicle including engine, tank for X3 and cargo + X2(TEI) + X2(EOI) in POI,
total weight of potential vehicle including engine, tank for X2 and cargo + X(EOI) or
total weight of potential vehicle including engine, tank for X3 and cargo + X2(EOI) in TEI and
total weight of potential vehicle including engine, tank for X2 or X3 and cargo in EOI.

The order mentioned above is the only possible order because

total weight of potential vehicle including engine, tank for X2 or X3 and cargo in EOI

is the only weight available already while all the other weights result from this weight.

After these calculations the sums

X2 = X2(EOI) + X2(TEI) + X2(POI) + X2(TPI) and
X3 = X3(EOI) + X3(TEI) + X3(POI) + X3(TPI)

can be calculated and used to get the weight of the according spherical tank(s) – like described earlier.

Preliminaryly resulting Costs got by the Calculations

Now all these calculations are done only to get preliminary costs but not the correct propellants – thos must be calculated the engineering way.

And please note urgently – the result are preliminary costs which means that these costs might be going to become modified a way that might be non-calculational.

What has to be done here is relatively short and simple:

1. X * price of the propellant = propellant costs per flight in the cylindrical case
X3 * price of the propellant = propellant costs per flight in the spherical case
2. (X/transportation capacity) * transportation costs = transportations costs per flight in the cylindrical case
(X3/transportation capacity) * transportation costs = transportations costs per flight in the spherical case
3. Result of 1. + result of 2. As variable costs per flight
4. Multiplication of the result of 3. by a percentage the costs according to the passengers wouldn’t „feel“.
5. Result of 3. + result of 4. = required revenue per flight
6. (Investemnt into the vehicle + investment into the tank + investment into a lander)/result of 5. = required number of flights
7. price per flight/number of passengers per flight = price each passenger has to pay
8. (result of 3 + result of 5.)/number of passengers = initial price to cover lost-in-space costs

The transportation capacity is the capacity of an orbital tanker or an interplanetary tanker here – regarding those trips that include a landing such a capacity has to be considered two times

The transportation costs are the costs of the flight of the orbital tanker – and these too are to be considered two times for landing trips.

Landing trips have to include the investment into the lander additionally.

The landing trips might be considered later seperately and additionally but of course really will be explicitly included into the Excel spreadsheet already announced earlier.


End of Appendix


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Post    Posted on: Sat Nov 04, 2006 4:33 pm
Contents

The diameter and length of the tanks: What to do without data?
Tests: The Correction Factor
Apollo-Spyuz result

The diameter and length of the tanks: What to do without data?

The tank containing the propellant is a part within a stage here.

To calculate the tank weight I used the constants dte and hte in the case of the existing vehicle and dtp and htp in the case of the tank of the potential vehicle.

But these data seem to be available nowhere – at least I didn’t find no data.

So what to do?

One possibility is to apply a maximum diameter as well as a minimum diameter and thus calculate an upper boundary and a lower boundary. The maxima and minima are got by looking at the resulting heights because these will be higher than the stage height begiining with a certain diameter.

But the parameters dte, hte, dtp and htp shouldn’t be used that way –this can be done better by the saftey margins to be applied after the calculations, by the correction factor and other ways.

No data are available – but images, drawings and descriptions are available from which diamters can be supposed.

However there is another intersting way available.

The minimum amount of propellant is got by the formular(s) devloped if the dimater of the tank of the potential vehicle is the minimum possible and the dimater of the tank of the existing vehicle is the maximum possible. This combination I applied in the next chapter.



Tests: The Correction Factor

Now let’s test the formular(s) got. The test is intended to look if reasonable results are got that are compatible with the informations available.

Since the amount of propellant is subject to what has been done in the previous posts this amount needs to be checked.

The best test is to look if the data about an existing vehicle results in these data themselves or close to them. This means to apply the same existing and flown vehicle as existing vehicle as well as as potential vehicle. To do so I apply isolated phases which means that all phases except one are applied as if they didn’t exist (all data set to zero)

Another test is to apply one existing vehicle to calculate the data for another existing vehicle.

Here I apply Apollo and Soyuz. Of course Soyuz hasn’t flown to the Moon yet but data regarding such a flight have been published and are calculated by the Russians.



Code:
test                   correction                      non-          corrected
                       factor                       corrected     amount
                       required                     amount

Apollo-Apollo in TPI   0,75551109         80537,48228   106600
Soyuz-Soyuz in TPI    0,79036535         12053,0715    15250
Apollo-Soyuz in TPI    2,72531585148294   41561,0667    15250


(Excel spreadsheet applied)



What the results are meaning?

In the test Apollo against Apollo the non-corrected amount is an amount that is by a bit less than a quarter less than the complete capacity of a Satrun IV B. This result is reasonable and fits into the informations because the Saturn IV B carried more than the propellant required for TPI (TLI in that case) – before TPI it has been used to get into the orbit.

This means that the Apollo-Apollo correction factor mustn’t be applied. But it can be used to find the proper values of the parameters dte and hte – the non-corrected result is a bit below the data to be found under www.astronautix.com and www.bernd-leitenberger.de . This seems to be justified by a look onto the Soyuz-Soyuz correction factor and the propellant amounts of that test. The Block-DM hasn’t been used to get into the orbit as far as I understand the informations and the formular(s) result in an amount below the capacity.

The test Apllo against Soyuz results in an propellant-amount that requires 2,75... Block-DMs – but the informations available say that only one Block-DM is required. So the Apollo-Soyuz correction factor needs to be applied urgently because the results would be far to high.

There is an inherent safety margin here – without the Apllo-Soyuz correction factor the resulting am,ounts of propellant for the phase TPI are too high. This can be applied for the CXV also – leaving the correction factor at any value between 1 and the correction factor got means a safety margin. In the case of the Soyuz-Soyuz test also a safety margin can be applied – in that case by using the correction factor instead of not using it.



Apollo-Spyuz result

The reason of the correction factor in the Apollo-Soyuz-case might be cause by the circumstance that I don’t have no informations that allow to take into account the different geometery of the tanks. While The tank of the Saturn IV B is cylindrical as far as I know the tank of the Block-DM is a toroidal regarding the cerosene while the LOC-tank is asphere. This can be seen by drawings and picture and Bernd Leitenbergers describes it so. I didn’t find no informations about the diameters.

But from the images and drawings it seems as if a cylinder of the diamter of the complete Block-DM is a sufficient approach – in that case the above wouldn’t be the explanation of the correction factor.



More tests are required but those are not needed for the round-trip which will be calculated in the next post.



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


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Post    Posted on: Sat Nov 11, 2006 7:48 pm
Contents

Tank Diameters used
Never yet a Vehicle doing EOI
Propellant Amounts and Stage/Tank Weights
Prices Resulting
Excel Spreadsheet
Appendix
Data about the vehicles applied

Tank Diameters used

To calculate the round-trip the systematical way by the formular(s) using the Excel-spreadsheet I looked for those diameters of the tanks at which I get the same amount of propellant in TPI that had been inserted. It turned out that the stage diameter is identical with the result – which on the one hand seems to be not that surprising but on the other hand I am doubting a bit if this fits into the verbal descrptions under www.bernd-leitenberger.de .

I still apply the Isps of the propellants in this post.

The data about the vehicle applied are listed in the Appendix.

The modified diameters result in a correction factor of 3.443842893at checking Apollo against Soyuz in TPI.

Of course the application of a tank diameter that is identical with the stage diameter results in a tank weight that’s the same for each amount of propellant the Excel spreadsheet used calculates. This is quite inacceptable and unreasonable.

So I use other diamters but my informations are limited to verbal general descirptions. So I am forced to base assumed data on these informations. Under www.bernd-leitenberger.de it is said that the connections of the tanks are relative heavy and so should be kept short. I am not sure if this can be applied to get an idea about the diameter but I also remeber to have read something about thicknesses of 0.79 millimeters and variations of thicknesse from millimeters up to 13 centimeters. (perhaps I find the information again).

For these reasons I apply tank diameter of 6.46 m now – for the tank(s) of the Saturn IV B. Then the sum of thickness and distnce to the outer wall of the stage would be 7 centimeters.

Regarding the Block-DM the tanks seem to be exposed to space directly merely – but there is a structure that is looking as if would have to be increased if the tanks are increased – but it’s looking lighter than in the case of the Saturn IV B where the tanks aren’t directly exposed to space. Because of this I apply a sum of thickness and distnace to the outer wall of 4 centimeters only and thus get a tank diameter of 3.62 meters now.

Using these the new Apollo-Soyuz correction factor is 3,49844972 – which is only slightly less now and the weight of the tank also only slightly is increased. The Soyuz-Soyuz correction factor is 0,99719954 now

Never yet a Vehicle doing EOI

Next there is the problem that there never was a manned vehicle yet that entered an earthian orbit and was kept there for resue.

It seems that in principle for a given vehicle the same amount of propellant should be required to insert it into an earthian orbit when it returns from the Moon as is required to launch it out of that orbit towards the Moon.

But if I would apply Soyuz in both TPI and EOI I think I would get an error – the tank and thus the weight to be applied in TPI would have to be larger than the data available.

Because of this I apply Apollo/Saturn IV B in TPI and Soyuz/Block-DM in EOI only. Apollo/Saturn IV B had more than four times the weight Soyuz/Block-DM have.

Propellant Amounts and Stage/Tank Weights

If the correction factor is NOT applied the resulting amount of propellant required is 61,633.0349 kg. These require a cylindrical stage of 2,151.34326 which would mena an increase of a bit more than 81 kg only for an increase of propellant by a bit more than 46,383 kg – if that might be reasonable??? In the case of a reusable tank kept in space allways?? In the spherical case the required amount of propellant would be 40,414.7235 kg in a stage of 92.1552932 kg. This spherical stage is that much lighter because I left away the weight of the top and the bottom. Because of this I also calculate a spherical stage inclduing the weight of top plus bottom which weighs 2,164.63477 kg and contains required 61,769.9935 kg of propellant.

Up to it seems that spherical tanks have no advantages if the diameters of the tnaks are as assumed here –let’s try later if that is changed in the optimal case.

If the Apollo-Soyuz correction factor is applied in TPI and the Soyuz-Soyuz correction factor in EOI. Now the amount of propellant in the cylindrical case is 38,215.554 kg – indicating that my former unsystematical calculations were too low by around 15,000 kg to 18,000 kg – and a stage of 2,110.27535 kg. The spherical numbers are 25,071.6037 propellant and 67.1626396 kg stage while spherical tank plus top + bottom results in 38,354.3255 kg propellant and 2,131.84616 kg stage.

Prices Resulting

Based on all this the Excel-spreadsheet calculates the following costs applying an investment of $ 400 mio into the CXV-like vehicle, $ 20 mio as investment into a Block-DM used as standard here with the standards being ist capacity and weight, transport capacity into LEO of 3,600 kg propellant at costs of $ 10 mio and flight costs inot LEO for five passenegers together of $ 20 mio:

cylindrical tank

without safety margins

$ 12,617,915.7 depreciation 10% of variable costs
$ 138,797,073 flight costs
$ 27,759,414.5 per passenger

35,6729884 required number of flights for complete depreciation
179 required number of passengers

lost-in-space-costs

$ 55 mio maximum
3 required flights for coverage

$ 193,797,073 initial price per flight
$ 38,759,414.5 initial price per passenger

safety margin applied

next full quarter above calculated price $ 150 mio
safety margin of 10 % applied $ 152,676,780

rounding up to next full $ 10 mio $ 160 mio
180 per passenger $ 32 mio

lost-in-space-costs

$ 55 mio maximum
4 required flights for coverage

$ 215 mio initial price per flight
$ 43 mio initial price per passenger

The safety margin applied adds $ 22 mio to the calculated flight costs. This should be sufficient to cover propellant requirements higher than the calculated ones.

The other two alterantives in short:

spherical tank

Without safety margin(s) $ 98,625,604.16 per flight and $ 19,725,120.83 per passenger at 48,28045161 flights and 242 passengers. The numbers are higher because the price is less, thus 10 % are less dollars and complete depreciation takes more time.

The initial price without the safety margin is $ 153,625,604.2 per flight and $ 30,725,120.83 per passenger.

Including the in this case valid safety margin of the difference to the next full quarter above the calculated price and the 10 % safety margin again above that the prices are $ 109,000,000 per flight, $ 21,800,000 per passenger while the according initial prices are $ 164,000,000 and $32,800,000.

spherical tank + top ülus bottom

The results here don’t differ that much from the spherical case – they are very slightly above the cylindrical cas at calculation and equal to it after applying the safety margin(s).

For comparison I also applied a smallest diameter of the tank of the potential vehicle of 1.775 meters and a maximum diameter of the tank of the Saturn IV B of 3.69 meters. The diameter of the tank of the potential vehicle has to be applied for Soyuz also because in both cases the Block-DM is the standard.

The results will be listed in the next psot.

I applied the formerly used investment into the Block-DM here. But I found out that I also had calculated a smaller investment of $ 4 mio while www.astronautix.com is reporting $ 4 mio only. This I also will aplly and post the results somehow.

As can be seen by this post an intersting lower boundary as well as an interesting upper bopundary can be found

Excel Spreadsheet

The calculations above in principle are a first demonstartion of what it calculates all only – I am NOT going to list that much data in this thread in the future – because the I will make available the spreadsheet at the message board.

The spreadsheet will include the follwoing:

1 list of existing vehicles per phase
1 list of potential vehicles
1 list of tanks/stages for the potential vehicle
1 list of landers
1 list of trips
the possibility to add more vehicles, stages and trips
the possibility to insert alternative prices, investments and lost-in-space-costs
the possibility to insert percentages for depreciations and safety margins
the possibility to maintain the correction factors and tank diameters

Up to now there is no sufficient oversight to hanlde and use the spreadsheet properly - because of this I am developing a form in which all is organized so that oversight is kept and handling is simplified.

I have in mind to keep you informed about progresses.



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



Appendix

Data about the vehicles applied

Code:
              exisitng vehicles                             potential vehicle

              Apollo                    Soyuz               CXV-like vehicle

vehicle       Apollo CM       5806      capsule        7250   vehicle         3600

engine        J-2             1438      engine           230   engine           230
stage         Saturn IV B    11862      Block-DM        2070   Block-DM-like   looked for
stage-wgt                    13300      stage       2300   stage           looked for

cargo                        23323                     0                      0
weight to                    42429                 9550                   looked for
be
accelerated

amount                    80791.58              15250                   looked for
of. prop.
capacity                    106600                  15250                   looked for
of stage

diameter                      6.6               3.7                   3.7
height                       17.8               6.2                   6.2
propellant    LOX/LH2                  LOX/cerosene           LOX/cerosene
density                      0.28              1.02                   1.02
Isp prop.                    3830              2945                   2945


The cargo of Apollo are the engine, Apollo SM and propellants for POI(LOI) and TEI – for Soyuz these aren’t available at present.

End of Appendix


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Post    Posted on: Sun Nov 19, 2006 10:54 am
Contents

Applying the announced diameters
Applying alternative Block-DM investments
Prices at alternative tankers, orbital flight prices and numbers of passengers
Variable Costs compared to earlier posts
Apollo 14 used: Numbers to be recalculated
Excel-spreadsheet
Short Remark regarding numbers of the previous post




Applying the announced diameters

As annunced in the previous post the diameters are going to be applied now that result in the smallest amount of propellant that can be calculated.

The dieamter of the tank of the Saturn IV B is set to 3.59 m now and the diameter of the tank of the Block DM is set to 1.775 m.

The correction factor in TPI – Apollo against Soyuz - then is 0,77900672 while the correction factor for EOI is the result of the check of Soyuz against Soyuz in TPI at the diameter chosen for the tank of the Block DM - 0,225918738.

It turns out that the results are negative propellant amounts and negative tank weights without applying the correction factors – which is no wonder n´because the correction factors are less than 1

Even the application of the correction factors gets negative results – meaning that the diameters applied are too low.

This means that correction factors below 1 indicate or are warnings that the diameters might be chosen too small and that increased attention should be paid to the results as well as to data and informations available.

Of course it is possible to apply te correction factors got by the calculation of the trip considered instead of those got by tests and checks. But these correction factors are tricky to be interpreted – they include the situation that can’t be tested yet...!

For now it seems that the calculation need to be left to the diameters applied in the previous post.



Applying alternative Block-DM investments

In the previous post I applied a required investment into one Block DM of $ 20 mio. But www.astronautix .com lists $ 4 mio only – so let’s see what numbers this results in:

For a cylindrical tank the required number of flights is 21,76111967 or 109 passengers – which in both cases is more than 50% of the numbers got in the previous post. The fact that the investment is by 75% less has a limited effect because the investment into the CXV-like vehicle is 20 times that of the investment into the Block-DM in the previous post and 100 times the investment into that stage applied here.

If the spherical tank is considered 31,03805844 flights and 156 passengers are required until complete depreciation of the investment. The comments are according to the cylindrical tank.

A spherical tank including top and bottom is too similar to the cylindrical tank as that it would be worth to be mentioned here.

But what’s interesting here is the circumstance that in all cases the number of required passengers is much less than the number of potential customers Space Adventures have identified for their lunar Soyuz-round-trip which was 1,000. ...

Prices at alternative tankers, orbital flight prices and numbers of passengers

Up to now I only was calculating numbers based on an expendable QuickReach2 for the CXV and a tanker. If a reusable QuickReach2 is applied like in the earlier posts the results are changing significantly. At an orbital CXV price of $ 125,000 per flight that is applied for the tanker also the flight price calculated is

$ 2,535,607.15 with a cylindrical tank and $ 1,710,014.44 with a spherical tank,
$ 57,535,607.15 initially (lost-in-space costs) cylindrical and $ 56,710,014.44 initially spherical,
$ 2,800,000 at cylindrical tank including safety margin and $ 1,900,000 at spherical tank including safety margin and $ 57,800,000 initial with safety margin at a cylindrical tank and $ 56,900,000 initial including safety margin at a spherical tank.

At five passengers the price per passenger then is

$ 507,121.43 (cyl.) and $ 342,002.89 (sph.),
initial $ 11,507,121.43 (cyl.) and $ 11,342,002,89 (sph.),
with safety margin $ 560,000 (cyl.) and $ 380,000 (sph.) and
initial with safety margin $ 11,560,000 (cyl.) and $ 11,380,000 (sph.)

But correction factors are NOT applied here



[size=18]Variable Costs compared to earlier posts[/size]

In the posts before I developed the formular and the Excel-spreadsheet I used the variable costs. The number of flights to depreciate the vehicle and the tank by is a free choice of the entrepreneur, owner etc. and thus the prices might be much less than those listed above. To get the numbers above I applied a depreciation of 10% of the variable costs – an entrepreneur might apply 1% instead – or even less. In space this might make sense. I also suppose that vehicles kept in space will be capable to do flights until very long after complete depreciation.

This all are good and reasonable reasons to suppose that the prices might be reduced to a level that depends on the variable costs solely and only. Becaus of this I now consider the variable costs per passenger and compare them to earlier results.

At an expendable QuickReach meaning CXV-flight costs of $ 20 mio and tanker-flight costs of $ 10 mio the variable costs per flight are

$ 191,242,936.1 in the cylindrical case and $ 132,289,390,5 in the spherical case.

If the QuickReach2 is reusable and the flight costs of the CXV as well as of the tanker are $ 125,000 variable costs of

$ 2,305,097.41 cylindrical and $ 1,554,558.59.

At five passengers the variable costs per passenger then are

$ 38,248,587,22 at a cylindrical tank and $ 26,457,875.1 at a spherical tank in the case of the expendable QuickReach2 ,

$ 461,019.49 cylindrical and $ 310,911.72 spherical in the case of a reusable QuickReach2.

A Falcon 9 S9 would be used if no reusable QuickReach is available and the flight costs of a tanker are too high above the Falcon-flight costs. So if a Falcon is used for propellant delivery the costs of a CXV as taxi into the orbit will be $ 20 mio then. This is applied now.

The flight costs of a Falcon 9 S9 are $ 78 mio at a capacity of 24,750 kg.

The variable costs per flight are

$ 214,277,504.7 cylindrical and $ 147,393,883.1 spherical

which result in variable costs per passenger of

$ 42,855,500.94 and $ 29,478,776.62.

For comparison the variable costs per passenger at five passengers and a reusable QuckReach2 got eralier (and listed as price) were $ 170,000 – by $ 140,911.72 too low in comparison to the result for the spherical case listed here and by 291,019.49 too low compared to the cylindrical case.

In the case of an expendable QuickReach the earlier variable costs were $ 15.6 nio – too low by $ 10,857,875.1 and $ 22,648,587.22.

For the Falcon I seem not to have calculated numbers earlier.

In so far it seems that my ealier calculations included errors having significant impacts. It turns out that it is justified not only but required really that I developed the formular(s) and apply them by an Excel-spreadsheet.

Please note urgently: the correction factors are NOT applied here.

But the numbers got now mustn’t be considered to be the final ones for a round-trip yet.



Apollo 14 used: Numbers to be recalculated

As can be seen by looks to those earlier posts in which I considered the trip including a landing without lunar refuelling and carrying back the lander instead of leaving it on the Moon the amount of properllant I applied here was that required by Apollo 14...

Apollo 14 used that amount to carry a lander through TPI that weighed around 15,000 kg.

These 15,000 kg of the lander I left away for now. The reason is that I have to consider them first instaed of applying them simply.

Because of this I will recalculate the round-trip later another time and check if the results are changing more than a bit.



Excel-spreadsheet

Progresses are focussed on controls to selct or insert data at present – may be that results will be handled by VBA-macros but I am not sure about that.

Alternative numbers for investments into the stages/tanks are selectable now. To enable this for the flight-costs of tankers and taxis into LEO also lists I will add:

1 list of taxis
1 list of tankers.

I am not sure yet if there will be separated lists for tankers into lunar orbit.

By the way – the amounts of propellant will be shown per phase.



Short Remark regarding numbers of the previous post

I am allways keeping the principle to calculate down from a heavier existing vehicle to a potential lighter vehicle. Because of this I should check if the tank to be calculated fits into this.

The financially CXV-like vehicle weighs 3,600 kg – the tank in the previous post weighs below 2,500 kg. So the sum of 6,100 kg is well below the weight of the Soyuz + Block Dm as well as well below the weight of Apollo + Saturn IV B. Since the amount of propellant for TPI both must be kept out of consideration because it is consumed in that phase and is in EOI + TPI for the potential vehicle less than for Apollo + Saturn IV B in TPI alone this seems to be no problem.

It also turns out that the propellant got for EOI is much less than the capacity of one Block DM.

All in all the numbers fit into the principle I am applying.



Next a systematical calculation for landers is needed – but it will be much simpler than the formular(s) developed up to now. I also will calculate the prices and variable costs for the round-trip at application of the correction factor(s).



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


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Post    Posted on: Sat Nov 25, 2006 6:52 pm
Contents

Applying the Correction Factor(s)
Engine Isps instead of Propellant Isps
What has been illustrated up to now
Landers
How to allow for safer calculation linearily up?
Excel spredscheet
Appendix




Applying the Correction Factor(s)

If the correction factor is applied with the Falcon 9 S9 as taxi and tanker-carrier and maximum investment costs into the Block DM as standard unit the prices are ranging from variable costs of $ 28,041,482.6 to $ 35,000,000 with a safaty margin at five passengers. The initial prices are between $ 41,845,630.9 (not variable costs) and $ 46,000,000 then. The mnimum got are variable costs of $ 19,772,592.58 in the spherical case.

If a reusable QuickReach2 is applied at flight costs of $ 125,000 and the known investment costs into the Block DM as standard unit then the minimum are $ 201,993.154 variable costs in the spherical case and maximum $ 360,000 including safety margin in the cylindrical case. The inital prices are ranging from $ 11,222,192.5 to $ 11,360,000.

Obviously the minimum is very close what I got earlier this year as well as to the $ 200,000 Prof. Collins assumes to be a realistic price in the year 2030.

At this point it looks more interesting to me to redo the calculation(s) later by applying the total weight the Satrun IV B had to accelerate with the amount of propellant applied here according to the informations.



Engine Isps instead of Propellant Isps

Up to now I allways applied the ratio of the Isps of the propellants. But like I mentioned some posts earlier already it might be more correct to apply the Isps of the engines instaed.

Here the results got at five passengers, reusable QuickReach2 with flight costs of $ 125,000 and investments of $ 4 mio into a Block DM as standard unit: $ 294,219.954 minimum in the spherical case (variable costs) and $ 540,000 maximum in the cylindrical case including safety margin. The initial prices are between $ 11,323,641.9 minimum spherical and $ 11,540,000 including safety margin cylindrical.

Since these are without correction factor here the alternatives – the correction factor Apollo-Soyuz in TPI then is 3.25831812. The Soyuz-Soyuz factor still is slightly below 1 and thus shouldn’t be applied.

Now the numbers are between a minimum of $ 195,736.7234 variable costs in the spherical case and $ 360,000 in the cylindrical case inlcuding safety margin. The initial prices are between $ 11,215,310.4 spherical (not variable costs) and $ 11,360,000 cylindrical including safety margin.

Obviuosly the engine Isp results in smaller numbers yet – the lowest price got is less than $ 200,000. This makes it even more interesting to apply the weight including the Apollo LM accelerated by the Saturn IV B during the Apollo 14 TLI.



What has been illustrated up to now

In between the correction factor has been left away as well as included. On the other hand the propellant Isp has been replaced by the engine Isp in this post. 4 combinations of the two are possible. Additionally alternative diameters of tanks could be applied – which shouldn’t be done but might be required because there are no numbers about them.

Nect three alternative amounts of investment into the Block DM as standard unit are available.

This shows up the amount of alternative results possible – so each one has a good chance to apply what she or he personally considers to be closest to what she/he supposes to be realistic, correct etc.

But the possibilities and opportunities are still going to be growing.

Landers

In between at least two lunar landers have been tested data about are available – Armadillo Aerospace’s Pixel and Microspace’s lander. In the according General Forums I asked for the data and I also got some from the homepages of the two companies.

These data can be applied directly.

This is not the case regarding the Apollo LM I used for the non-systematical calculations earlier this year.

The earlier application of the Apollo LM required the calculation of a reusable LM form the data about the existing Apollo LM. That calculation needs to be redone now – the systematical way.

The redoing of the calculation will occur analogous to the way the formular(s) are got the Excel spreadsheet is using. But there is one problem – there seem to be no data available publicly about those parts of the Apollo LM that are neither engine nor tanks nor volume around propellant nor propellant. So those must be calculated. This is done in the appendix.

The formular got can be applied to each expendable sample return mission also if required.

The weight I get applying the formular derived in the appendix is 1616.03344. Because I up to now allways calculated down from above only to be on the safe side this number means that no potential lander heavier than that number can be calculated.

How to allow for safer calculations linearily up?

This would mean that the comparability to the earlier calculations of landings couldn’t be kept – it’s looking as if I had to give up the safety. I remebr that I was forced to do so earlier also but want to do this a better and safer way now.

There are data about a lot of rockets and vehicles of different weights. These can be used to find out what safety margins must be included into calculations linearily up.

I will do right that but it doesn’t mean new formulars etc. ...



Excel spreadsheet

The calculation introduced will be included in the Excel spreadsheet – and of course the apllication of the result to get data about potential reusable landers also.

It is now possible to switch between propellant Isp and engine Isp. Such a switch will be possibnle regarding the correction factor as well. This maens that the correction factor is going to become a component of the trip design.

At present I have in mind that the spreadsheet will consist of two levels of multipages. The upper level will distinguish between the calculation of a trip, the design of trips, the opportunity to add more vehicles etc. while the lower level will list results structured – one page for prices and variable costs, a second page for the composition of variable costs, a third page for depreciations, are fourth for investments and so on. Something like that.

At present the focus is on the prices etc.



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



Appendix

It seems that the data about the LM are listed as the descent stage without the ascent stage but the ascent stage including the carrier itself – the crew cabin with all required there. So if the total weight of the existing lander is called

letw lander existing total weight

and the two stages

ldsw lander descent stage weight
lasw lander ascent stage weight

then

letw = ldsw + lasw

is valid.

Now ldsw is unstructure while the ascent stage is structured as follows:

lasw = lecw + asw

with

lecw lander existing carrier weight
asw ascent stage weight

.

To get lecw the propellants can be used. If the propellants are

dpw descent propellant weight
apw ascent propellant weight

.

Since the data about the descent stage ldsw are withour the carrier and the ascent stage it can be used to get the weight required to contain the descent propellant aa

ldsw/dpw

The multiplication of this by the ascent propellant weight tells the weight required to contain the ascent propellant. This only must be subtracted from lasw

lecw = lasw - ldsw/dpw * apw


End of Appendix


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Post    Posted on: Sat Dec 02, 2006 4:10 pm
Contents

Insights by Comparisons
Investments and Landers
Depreciations as an Instrument to reduce required time
Handling calculations linearly up
Excel spreadsheet




Insights by Comparisons

Let’s have a look onto a few values got or applied to get some experiences and thus insights.

At frist the ratio of the propellant Isps were applied – that ratio is 1,300509 according to the Excel spreadsheet. At that ratio the correction factor was 3.443842893. The ratio of the engine Isps is a bit less - 1,211243. The correction factor at that ratio os less also - 3.25831812

In other words the correction factor is a bit less if the engine Isps are used instaed of the propellant Isps. This might be a sign that it is more correct to apply the engine Isps.

I will continue to apply the propellant Isps also to keep comparability to the results got unsystematically – but the engine Isps deliver results that are more correct.

And up to now it could be seen that these more correct results lead to less variable costs than the propellant Isps – and too lower prices also to some degree.

The amount of propellant required was 61,633.0349 kg without correction factor and 38,215.554 kg including the correction factor in the cylindrical case at the application of the ratio of the propellant Isps. The according amounts got by the ratio of the engine Isps are 58,026.8238 and 36,787.122 kg. Obviously the required amounts are less at the engine Isps – by 3,500 kg and 1,500 kg.

Obviously the difference is larger if the propellant Isps are applied – which is due to the differences considered already. But the larger difference is 2.3333 the smaller one.

This might be of maening regarding the lander(s). I calculated the weight of carrier of the Apollo LM to be around 1,600 kg – very similar to the smaller difference. And the larger difference is very similar to the weight applied for the financially CXV-like vehicle.

In the Apollo program weight savings were of meaning for the success of that program. Because of this it will be interesting which differences will be got if the landing trips are considered.



Investments and Landers

The arlier unsystematical calculations included a case where the lander(s) was left on the Moon. The lander is permanently installed and stationed there. To do so the lander was carried there. If that’s not done it couldn’t be used there. But it needs to be carried there once only – the costs of its transportation thus are an investment simply.

To get the correct results this investment must be calculated and inserted into calculkations automatically by the Excel spreadsheet.

The reusable tank and the CXV-like vehicle also must be installed – in the earthian orbit.

All the installation investment need to be added to the hardware investments – but to do this properly the installation of the lander and the tanker must be included which will be possible when the lander and the tanker are incorporated into the calculation(s) and the Excel spreadsheet completely.

But this will have no impact on the prices because the depreciations are calculated as percentage of variables costs here. What is impacted is the time required until the investments are depreciated completely at a given percentage. This time will be increased – more flights or passengers will be required.



Depreciations as an Instrument to reduce required time

Since the depreications are a percentage of variable costs here instead of a percentage or fraction of the investment percentages above 100% might make sense here.

The larger the percentage of variable costs applied the shorter the time until complete depreciation of the investment(s) – less flights or passengers are required.

Because of this a strategy similar to Virgin Galactic’s one may be designed using the Excel spreadsheet. The price Space Adventures will set is known. Because of this a price will be competitive that is significantly below Space Adventures‘ price but high in comparison to prices got in this thread. In that case the initial price will include a surplus of millions and tens of millions that can accelerate the depreciation by months or years – reducing the time and the number of flights and passengers until complete depreciation.

It might be possible that the investments can be depreicated faster than in the suborbital case. The reason simply is the high price forced by the expendability of Soyuz applied by Space Adventures.

If this is done by the Excel spreadsheet then the percentage of variable costs applied might be considered to be an average of two or more different percentages that are applied at two or more different initial prices – Founders, Pioneers and more groups could be defined who would pay higher prices than the standard customer to be flown after those initial groups are flown in total – like Virgin Galactic’s suborbital price policy.



Handling calculations linearly up

I found informations about one group of sample return missions that can be applied directly. Using these informations I will experiment what I get if the propellants etc. are calculated linearly up from this group of missions the landers of which are lighter than the landing Apollo LM. These landers apply the same propellant as the Apollo LM.

There are also a few informations about landers that used other propellants than Apollo LM – so the complete calculations are required that the Excel spreadsheet already does for the round trip. These landers remained on the Moon but didn’t return nothing to Earth. So these can be used for the landing only.

Obviousl the data available to investigate calculations linearly up are limited and landing missions to other planets may have to be incorporated also – but I think that what can be done without those other missions will be sufficient. The way to adjust date of reusable vehicles landing on Earth can be used as assistance if required and the Apollo Lm was used as lower boundary earlier in this thread – an upper boundary similar to the earlier one would provide required safety.

What’s unclear yet is if the tanks really should or must be calculated like for the round trip. It seems that Apollo LM had several spherical tanks. So the question arises if the tanks of the potential larger lander of 3,600 kg weight would be sphercial or cylindrical and what diameter they would have in the cylindrical case.

It might be possible to ignore the tank size simply or/and to add spherical tanks of the size already applöied. It might be possible alternatively to consider the case of a lunar Falcon as the cylindrical case.

In principle the already applied tank weight per unit of amount of propellant might overcome the question.



Excel spreadsheet

The spreadsheet has been extended to calculate the weight of the carrier separated from the stages and two lander lists for landing and launch – that’s an analog to the lists of existing vehicles.

The spreadsheet will have to handle the significant difference between landing trips that they might apply an already existing or being developed reusable lunar lander or a potential lander to be calculated using the extension to calculate such a potential lander from an exsiting expendable lander

Calculations of the investments into installation in orbit and at the Moon (or any other planet) will be added.



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


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Post    Posted on: Sat Dec 09, 2006 4:22 pm
Contents

Approach for Landers heavier than Existing Landers
Excel Spreadsheet


Approach for Landers heavier than Existing Landers

Under www.bernd-leitenberger.de there are usable data about the landers

    Luna 9 and 13
    Surveyor
    Ranger
    Luna 15 etc
    Apollo LM




The first three required some thinking and raw ideas what portion of the weight might be tank and stage and what the lander itself – here a check and perhaps some corrections may be needed.

Applying the data for landing unmodified the required amounts of propellant for the weights of the first three are linearly decreasing if the landers are ordered by weight withourt tanks/stages. This is changing if the last two landers are considered – here the required amounts are increasing but by decreasing rate(s).

This calls for a modification for the first three landers. Such a modification is reasonable and plausible for good reasons: the propellant are differing between them and in comparison to the last two which have identical propellants. The modification will be done by the propellant Isps while the engine Isps will not be applied because the engine of the particular vehicle doesn’t change and no calculation of one vehicle from another is done. The propellant to which the modification will be done is the propellant the potential lander will use.

For LOX/cerosene it turns out that the situation changes only slightly – the first part of the curvature is below the unmodified one while the second part is slightly above the unmodified one.

There is one problem regarding the modification – I didn’t find the data to calculate from the actual propellant to LOX/cerosene. The propellant of Luna 9 and 13 is a technical one according to Bernd Leitenberger and no Isp is listed on his website.

Now I up to now applied the weight of the empty lander without the tanks and stages only. The reason is that the weight and volume of tanks is unknown as long as the amount of propellant is unknown because the size of the tank and thus the stage depends on that volume.

Obviously that approach has to be given up regarding the landers. Applying the total weight of the landers including their tanks/stages results in a curvature with increasing slope – only at the beginning there is a now very short decreasing slope which is very slight only.

So it seems that this is a reasonable base to get a function by statistical regression.

In principle this procedure must be applied to the ascent-components too – but there data available to me about only two ascent-components which is too few. So no more than a simple comparison is possible.

If the weights of the larger ascent-component are divided by the weights of the smaller ascent-component each – carrier, stage/tank, propellant – it turns out that the weight of the stage/tank is increased by a larger factor than the amount of propellant while the weight of the lander itself without stage/tank and propellant is increased by an even larger factor than that of the stage/tank.

This means at least that the shape of the curvature of the function to be looked for is quite similar to that for the landing components. Of course the function got for the landing components have to be appled to the ascent-components also but an adjustment by the factor(s) is possible.

Please note – I don’t have in mind to apply technical links or connections here, I don’t have in mind something like what I did when I derived the formualr(s) for the trip from Earth to the Moon and back. I am simply and only looking for a method to include a saftey margin to be applied if a potential vehicle has to be calculated linearly up from a lighter vehicle. What I am doing here is statistics – but to get a safety margin or factor only.



Excel Spreadsheet

As far as possible and as far as it makes sense this statistics will be included into the Excel spreadsheet so that ist users will be enabled to add date, replace data etc. The spreadsheet will do the statistical regression and apply resulting function(s)



I suppose that I will have to do so regarding the CXV-like vehicle(s) later also.



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


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Post    Posted on: Sun Dec 17, 2006 12:37 pm
Contents

Connections/Links looked at
Excel-spreadsheet


Connections/Links looked at

In between I did some statistical regressions. There are at least three types of regressions – linear, exponential and polynomial.

The polynomial regression I consider to be ruled out of itself since in general tthe ratio between weight of the hardware and required amount of propellant will not go up and down repeatedly and to get a safety margin negative slopes lead to the opposite.

I had a look at one exponential regression – but the Coefficient of determination of below 40% was too low.

But the linear regressions all had Coefficients of determination of 97.3 %to 98.7% - except for one of neraly 68,4%.

Since all the regressions are linear I consequently got linear functions only. The independant variable in all cases is the weight of the hardware while the required amount of propellant is the dependent variable.

There three different relations modelled by the regressions and functions:

1. lander without stages/engines to required amount of propellant
2. lander without stages/engines to stage
3. lander plus stage(s) to required amount of propellant

It is the second kind where the Coefficient of determination is at 68,4% and a bit low – but it is needed to include the stage/tank properly I think.

For each of the regressions the stamdard-error is available which tells the bias of the results of the functions from the actual amounts and is an average error. This error can be added to the slope of the function(s) to improve the saftey margin if required.

Of course these funtions are valid within the range of the data only they are based on – I will apply them outside that range though because I need them and have no better alternative up to now. I continue to be out on too high amounts

Interestingly the functions got for actual propellants and for propellant calculated into LOX/cerosne are very close to each other – except for the third relation modelled where the function for the actual propellants is clearly above the data and the function for LOX/cerosne. Might be a very good safety margin but doesn’t fit into the formular(s) because these calculate propellant of existing vehicles into the propellant of a potential vehicle that might apply a different propellant. This propellant should be used for the regressions. Alternatively the propellants could be standardized to the propellant of the existing vehicle used in the formular(s) and applied prior to the recalcultion (done by the ratio of Isps).

I will have a look onto multiple regression perhaps because that might be a better model to take into account the dependies between lander, stage and propellant in parallel – but I am not that sure and have doubts.

The first to kinds of functions will be applied to the results per phase of the formular(s) for the lander while the third may be applied to the total result. I will further think about it.

I have to check the data applied another time because I found more required details under www.bernd-leitenberger.de , had to conclude some weights and amounts which is an additional source of possible errors.

The formular(s) for the lander require a check too.



Excel-spreadsheet

There was little progress last week. Besides little time I did the regressions and thought about how to incorprate them.

There will be a few additional tables for the data required. I intend to enable you to add landers and even to replace the data available to the optimal degree possible

The spreadsheet also will include diagrams showing the dta and the linear functions the regressions lead to.



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


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Post    Posted on: Sun Dec 24, 2006 7:24 am
Contents

Calculation of/for the lander
Landers: Tests with existing ones
Descent Stages of Landers
Fourth possible tanker
Excel-spreadsheet
Appendix
Propellant for Tanks and Propellants together – modified
Diameters and Heights of Tanks – getting Ideas
Apollo LM AS - Launcher
Luna 15 – Launcher
Data for Statistical Regression
Apollo LM DS
Luna 15
Ranger
Surveyor
Luna 9/13




Calculation of/for the lander

Like I said in the previous post I checked the formular(s) for the lander. Doing that it seemed to me that the modifications caused by the lower number of phases and thus equations are significant enough to require a section in the Appendix of this post. That section is constrained to the modifications.

Initially the formulars are applied as they are to get the weight of the tank/stage and an amount of propellant.



Landers: Tests with existing ones

Like for the vehicles going from Earth to the Moon first of all the formulars need to be tested to get correction factors. Of course here again this can be done per phase only but not for the two phases together.

But here the calculations are not to be done as calculation from an existing lander to itself only – the more interesting test is the calculation from a lighter lander to a heavier one.

One problem needs to be handled somehow – I didn’t find no data about diameters and heights of tanks. So I did have to use other data, images and graphics or drawings to get an idea about them. These and their results are described in the appendix.

The two different heights of the lander both result in too low amounts of propellant – less than 2,100 kg while the actual amount was 2,358 kg. Becasue of this the resulting correction factor of below 1 must be applied – which had been avoided regarding the vehicle going from Earth to Moon. It has been avoided in that former case because it might have lead to amounts larger than the capacity of the tank(s) while here the amount without correction factor is below the capacity.

Applying the lower height the required correction factor is 0.88262903986429177268871925360475.

The height is contradicting the fact that the tanks of the launcher´-part of the Apollo LM are spherical. But first of all there are four of them, second the formular(s9 are based on cylinders and third there is the correction factor.

The next lander to be applied is the much smaller Luna 15. It had 324.5 kg of NTO/ADMH available for ist return to Earth. Here I apply the version with the lower tank height. The resulting amount of propellant is between 200 kg and 201 kg. This calls for a correction factor of 0.619414483821263482280431432973806.

So now I test the calculation linear down from the Apollo LM AS the launcher part of Luna 15. Without the correction factor the result is nearly 80 kg which is too few – this seems to mean that I should think about the earlier calculations linear down also but first of all I have nearly no real data here and may be assuming wrong ones and second I nonetheless will do something like this for the other vehicles later.

Applying the correction factor the result is at 90 kg – obviously another dorrection factor is required – it’s 0.279112681510015408320493066255778 and needs to be multiplied by that of Apollo LM AS to Apollo LM AS. The resulting factor is 0.246352958133010650231124807395994.

This might be indicating that at this large difference of weights between the two landers the overhead of parts the weight of which don’t vary thta much with amount of propellant have an impact. These might be the nozzle, the engine, instruments etc.

But now let’s calculate linearly up from Luna 15 to Apollo LM AS. The amount of propellant got without correction factor is 4488.7866 kg – this way too high because 2358 is correct. If the Luna 15-correction factor is applied this number is increased to 7246.82215. So the Luna 15-factor mustn‘t be applied but the factor should be 4488.7866 kg/2358 kg = 1,90364147582697201017811704834606 simply.

This seems to mean that calculations linearly up might be safe while the oppsite is not safe – but this conclusion isn’t possible here. Two different sets of data are tried only and these two are the only ones available.No statistical regeression can be done, no lander/launchers lighter than Luna 15 are availabe.



Descent Stages of Landers

So this is no proper base yet. There are more data about the descent parts. These should be looked at and this will be done in the next post - I will list the results only instead of en explanation like that above.

Here the results of the statistical regressions only. In the appendix it is explained which data are applied and which of them are derived which way from available ones.

The functions are

    272.9296839 kg + 1.734654412 * carrier-weight = amount of propellant with a standrad-error of
    291.9371047 kg and 0.14243892 for the relation between carrier and propellant. This function fits into the data to around 98.01%.

    135.5051273 kg + 1.827245988 * carrier weight = amount of propellant if the propellant is set to LOX/cerosene as standard – but this standard can’t be applied in general but will have to be set per trip. The standard errors are 392.2368303 kg and 0.171189759. This function fits into the data at 98.27%.

    267.1013157 kg + 0.389875022 * stage weight = amount of propellant is the function for the relation between stage-weight and propellant with standard-errors of 313.6971513 kg and 0.153055856 – fitting into the data to 68.38%.

    -14.15174436 kg + 1.222398736 * weight of complete lander is the relation between lander weight and propellant having stndard errors of 346.7108494 kg and 0.114208164. This fits into the data to 97.45%.


I left away the functions for standard-propellants for the last two functions because they depend on the propellant used – in eralir posts I applied to alternative propellants: LOX/cerosene and LOX/LH2.



Fourth possible tanker

By the way – the most recent edition of Wirtschaftswoche includes an article reporting that a California-based company called Launchpoint Technologies is planning to launch microsats up to 100 kilograms weight by a railgun. They will apply a circle of superconducting eletromagnets of 5 km diameter. The velocity reached will be around 27500 km/h. Then a laser will disconnect the satellite from the carrier and shoot it into orbit via a ramp.

The Pentagon is going to further that method during the next two years and check ist feasability.

What’s interesting for this thread is that one launch is calculated to cost $ 50,000 - $ 500 per kg. One launch of a Falcon 9 S9 will cost $ 78 mio and carry 24,750 kg - $ 3,152 per kg. So it might be interesting to add this method avialable perhaps in the nearby future to the list of tankers. Of course it cannot be added to the list of orbital taxies because the CXV for example weighs 36 times such a microsat. But the delivery of propellant in small portions below 100 kg are possible – they simply would require more than 380 launches. Given a well-developed schedule this might fit into the requirements – the delivery could occur continuesly.



Excel-spreadsheet

Since a lander and ist required propellant are carried by the vehicles going from Earth to Moon and back but not used for TPI, POI, TEI or EOI they simply are cargo for them. The Excel spreadsheet will include them into the cargo automatically – you only need to select a lander.

The propellant costs and the transportation costs of that propellant will be calculated separately. This is done to keep the cost structure transparent but it is required also because the lander might use other propellants than the vahicle travelling between Earth and Moon.

Because of the fourth tanker the tankers also will be calculated separately. This too has an additional reason – there are tankers into LEO and other tankers into lunar orbit. Also there might be tankers from the Monn into LEO. All these tankrs might be different or use different propellants.

And since there are different propellants in general and different prices are to be found for each propellant there will be a list of propellants.



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


Appendix

Propellant for Tanks and Propellants together - modified

...
...
...

Launch: (fcad * (4 * Z(Launch) + 4 * Z(Landing) + 4 * tsm)/(fdca * twe(Launch)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(Launch) * pew2(Launch) * bir + 0 =
(fcad * 4 * Z(Launch) + fcad * 4 * Z(Landing) + fcad * 4 * tsm)/(fdca *
twe(Launch)) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))) * ets(Launch) * pew2(Launch) * bir =
((Z(Launch) + Z(Landing) + tsm) * fcad * 4)/(fdca * twe(Launch)) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))) * ets(Launch) * pew2(Launch) * bir =
(Z(Launch) + Z(Landing) + tsm) * ((fcad * 4)/(fdca * twe(Launch)) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))) * ets(Launch) * pew2(Launch) * bir

Landing: (fcad * (4 * Z(Launch) + 4 * Z(Landing) + 4 * tsm)/(fdca * twe(Landing)) * (tpw/(Pi *
(dip/2)^2 * hip – pct/dyp))) * ets(Landing) * pew2(Landing) * bir +
Z(Launch) * ((pew2(Landing) * bir(Landing))/(evw + eew(Landing) + tew2(Landing) +
tew2(Launch) + tew2(Launch) + cew(Launch) + pew2(Launch))) +
((sum(Launch) * pew2(Landing) * bir(Landing))/(evw + eew(Landing) + tew2(Landing) +
tew2(Launch) + tew2(Launch) + cew(Launch) + pew2(Launch))) =
(fcad * 4 * Z(Launch) + fcad * 4 * Z(Landing) + fcad * 4 * tsm)/(fdca *
twe(Landing)) * (tpw/(Pi * (dip/2)^2 * hip – pct/dyp))) * ets(Landing) * pew2(Landing) * bir +
Z(Launch) * ((pew2(Landing) * bir(Landing))/(evw + eew(Landing) + tew2(Landing) +
tew2(Launch) + tew2(Launch) + cew(Launch) + pew2(Launch))) +
((sum(Launch) * pew2(Landing) * bir(Landing))/(evw + eew(Landing) + tew2(Landing) +
tew2(Launch) + tew2(Launch) + cew(Launch) + pew2(Launch))) =
(Z(Launch) + Z(Landing) + tsm) * ((fcad * 4)/(fdca * twe(Landing)) * (tpw/(Pi * (dip/2)^2 * hip –
pct/dyp))) * ets(Landing) * pew2(Landing) * bir +
Z(Launch) * ((pew2(Landing) * bir(Landing))/(evw + eew(Landing) + tew2(Landing) +
tew2(Launch) + tew2(Launch) + cew(Launch) + pew2(Launch))) +
((sum(Launch) * pew2(Landing) * bir(Landing))/(evw + eew(Landing) + tew2(Landing) +
tew2(Launch) + tew2(Launch) + cew(Launch) + pew2(Launch)))



To shorten all this the following constants are introduced:

a(...)

Launch: ((fcad * 4)/(fdca * twe(Launch)) * weight per unit of volume) * ets(Launch) * pew(Launch) * bir = a(Launch)
Landing: ((fcad * 4)/(fdca * twe(Landing)) * weight per unit of volume) * ets(Landing) * pew(Landing) * bir = a(Landing)

c(...)

Launch: -
Landing: (pew(Landing) * bir(Landing))/(evw + eew(Landing) + tew(Landing) + eew(Launch) + tew(Launch) + cew(Launch) + pew(Launch)) =
c(Landing)

Using the constants the equations look like this:

Launch: ((Z(Launch) + Z(Landing) + tsm) * a(Launch)) =
(Z(Launch) * a(Launch) + Z(Landing) * a(Launch) + tsm * a(Launch)) =
Z(Launch) * a(Launch) + Z(Landing) * a(Launch) + tsm * a(Launch) =
Z(Launch)

Landing: ((Z(Launch) + Z(Landing) + tsm) * a(Landing)) +
Z(Launch) * c(Landing) +
sum(Launch) * c(Landing) =
(Z(Launch) * a(Landing) + Z(Landing) * a(Landing) + tsm * a(Landing)) +
Z(Launch) * c(Landing) +
sum(Launch) * c(Landing) =
Z(Launch) * a(Landing) + Z(Landing) * a(Landing) + tsm * a(Landing) +
Z(Launch) * c(Landing) +
sum(Launch) * c(Landing) =
Z(Launch) * (a(Landing) + c(Landing)) + Z(Landing) * a(Landing) + tsm * a(Landing)
+ sum(Launch) * c(Landing) =
Z(Landing)

It can be seen that more shortenings are possible:

Launch: tsm * a(Launch) = d(Launch)
Landing: tsm * a(Landing) + sum(Launch) * c(Landing) = d(Landing)



Launch: -
Landing: a(Landing) + c(Landing) = e(Landing)

The equations then finally are

Launch: Z(Launch) * a(Launch) + Z(Landing) * a(Launch) + d(Launch) = Z(Launch)
Landing: Z(Launch) * e(Landing) + Z(Landing) * a(Landing) + d(Landing) = Z(Landing)

The solutions then are

1. Modification

Launch: Z(Landing) * a(Launch) + d(Launch) = Z(Launch) - Z(Launch) * a(Launch) = Z(Launch) * (1 -
a(Launch))
Landing: Z(Launch) * e(Landing) + d(Landing) = Z(Landing) - Z(Landing) * a(Landing) = Z(Landing) * (1 –
a(Landing))

2. Modification

Solution for Z(Launch):

Launch: (Z(Landing) * a(Launch) + d(Launch))/(1 - a(Launch)) = Z(Launch)

Z(Launch) inserted:

Landing: ((Z(Landing) * a(Launch) + d(Launch))/(1 - a(Launch))) * e(Landing) + d(Landing) =
Z(Landing) - Z(Landing) * a(Landing) =
Z(Landing) * (1 - a(Landing)) =
(((Z(Landing) * a(Launch) + d(Launch)) * e(Landing))/(1 - a(Launch)) + d(Landing) =
((Z(Landing) * a(Launch) * e(Landing) + d(Launch) * e(Landing))/(1 - a(Launch)) + d(Landing) =
(Z(Landing) * a(Launch) * e(Landing))/(1 - a(Launch)) + (d(Launch) * e(Landing))/(1 - a(Launch)) +
d(Landing)

3. Modification

Landing: (d(Launch) * e(Landing))/(1 - a(Launch)) + d(Landing) =
Z(Landing) * (1 - a(Landing)) - (Z(Landing) * a(Launch) * e(Landing))/(1 - a(Launch)) =
Z(Landing) * (1 - a(Landing) - (a(Launch) * e(Landing))/(1 - a(Launch)))

4. Modification

Solution for Z(Landing):

Landing: ((d(Launch) * e(Landing))/(1 - a(Launch)) + d(Landing))/(1 - a(Landing) - (a(Launch) *
e(Landing))/(1 - a(Launch)))



Diameters and Heights of Tanks – getting Ideas

Apollo LM AS - Launcher

Looking on images and drawings it seems that one tank of the launcher is around half as large as one tank of the Lander – Apollo LM DS. There seem to be four tanks – two for the oxydizer and two for the fuel. Their heights need to be summed up while the diameter of only one is to be applied..

The diameter of one tank of the Apollo LM DS seems to be the distance bteween the lower ends of the landing legs divided by 6.5. Since thdiameter by landing legs is listed as 9.37 m it seems to me that the Theorem of Pythagoras must be applied to get the distance between two legs – I am not referring to the diameter. The triangle made up by the two ends of the diameter and the on leg as the end of a distance between two legs has two equal sides and one angle of 90° So the two sides making up the angle of 90° are of equal length. Consequently The square of the diameter simply needs to be divided by two and then the aquareroot is to be taken. Then the distance got is 6.63 m. The diameter searched for then is 1.02 m

The amount of propellant is 2.358 kg while the density is 1.34 g/cm^3 – this means a volume of 1.76 m^3.

There are two different numbers about the height of the launcher - 3.76 m vs. 3.54 m while the diamter is given nearly identical at 4.2 m vs. 4.27 m.

The data calculated in the previous post I corrected for the engine. The crew part or carrier then weighs 1535.66712 kg, the tank of the ascent stage 540.332884 kg



Luna 15 - Launcher

The diameter of the lander itself – descent part – is 0.58 m while the engine of the launcher part has a diameter of 0.55 m. The capsule seems to have a larger diameter than the remainder – it is listed as 0.8 m.

The amount of propellant was 324.5 kg NTO/UDMH the density of which www.bernd-leitenberger.de lists as 1.02 g/cm^3. This means a volume of 0.32 m^3. But the diameter of the tank can’t have been larger than 0.58 m.

I don’t know how the launcher and the descent part were mounted and docked to each other – it may have been like the Apollo LM or they may have been one within the other. There may be more thinkable alternatives also. But the more I think about the more I prefer to suppose that it were like the Apollo LM.

In that case the diameters of both stages will have been quite similar while the height of the launcher will have been a bit larger than that of the descent part because it will have included the payload carrier too.

So I will apply a diameter of 0.58 for the stage. The complete height of the vehicle was 3.81 m.

The diameter of the tank I cannot „estimate like for the Apollo LM AS because the images don’t allow for that. The only alternatives possible are to suppose a distance of a few centimeters between the walls of the tank and the outer walls of the stage or to suppose that the realtion between the length of the engine and the length of the tank is similar. Then the diameter of 0.40 m or 0.55 m.

The weights of the launcher-part of Luna 15 listed are an amount of propellant of 324.5 kg, a mass at launch from the Moon of 457 kg, an engine mass of 41.8 kg and a capsule mass of 39 kg. So the total mass of the empty launcher seems to be 132.5 kg. Subtracting the engine and the capsule 51.7 kg are left as mass of stage/tank.



Data for Statistical Regression

Apollo LM DS

The data are completely available in the web.
Luna 15

The launch mass of Luna 15 at Earth is 5720 kg. On the Moon 1888 kg are left. So the propellant consumed at landing was 3832 kg. The weight arriving on the Moon should be the sum of the launcher with ist propellant and the descent stage. Since the weight for launch is 457 kg the empty descent stage will have weighed 1431.

Ranger

The total weight listed is 368.7 kg. Of these 12.4 kg are propellant, 10.4 kg engine and 43 kg structure and mechanics. So the lander itself weighs 302.9 kg.while the stage weighs 53.4 because here I don’t subtract the engine.



Surveyor

The available data about Surveyor say that the engine weighs 625 kg including an amount of propellant of 564.1 kg. I calculated a carrier-weight of 152.7 kg and a stage-weight of 76.4 kg – but at present I unfortunately can’t reconstruct how I calculated that. So I will look for the way and explain the data got later. Perhaps I applied a relation got for a nother lander.



Luna 9/13

The data available are the weight of the empty bus of 138.2 kg and the total weight of the bus of 863 kg. The amount of propellant is listed as 725 kg but from the other two numbers 724.8 seem to be more precise. The lander as carrier plus stage is said to weigh 100 kg. I got 66 kg for the carrier and 34 kg for the stage – but here too I at present have forgotten how I got these numbers. May be that I applied a relation from another lander – I will look for the way of calculation




End of Appendix


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Post    Posted on: Sat Dec 30, 2006 3:56 pm
Contents

Remark about the tanker to be added
Clarification
Tests and Checks for calculating up landing parts of existing landers
What to do with the launchers?
Excel spreadsheet
Appendix
How the missing data about Luna 9 and Surveyor are derived
How diameters of lunar landers are supposed
Apollo LM DS
Luna 15 landing stage
Ranger
Surveyor
Luna 9



Remark about the tanker to be added

Please note that this tanker might be available in four years if it turns out to be feasable and working. In that case it would provide a significant cost drop towards the costs via a reusable QuickReach2. The drop would be down from $ 3,000 per kg to $ 500 per kg. And the website of Launchpoint Technologies links to an article saying that at 3,000 launchers per year the costs would drop down to $ 189 per kg – for a reusable QuickReach2 I get $ 34.7222... per kg which $ 189 is five to six times of while $ 3,000 are 15 to 16 times the $ 189!

But the reusable QuickReach2 I assume to be available in and short before 2030 while the system of Launchpoint Technologies will be available in the nearby future – two decades earlier. It is closer to becoming reality and the low costs are calculated and listed by the company while the reusable QuickReach2 is potential only and a calculation and thought of mine.

Next the application of the system to lunar flights like calculated in this thread might generate the number of launches required to drop the costs down to $ 189 per kg. A few posts earlier I got a required amount of propellant of aroung 38,000 kg if the correction factor was applied. At 100 kg per launch these would mean 380 launches. Since one round trip seems to require one week 52 launches might be done per year if there are 360 lunar private passengers per year. 52 flights times 380 launches of 100 kg are 19760 launches requiring 6 to 7 systems.



Clarification

In the previous post I said something about the application of correction factors that seems to me a bit unclear now.

The criterion for applying a correction factor or applying it not is and was if a test results in an amount of propellant that is larger than the capicity of the tank or stage of the other existing vehicle applied in the test as potential vehicle.

Because of this the correction factor was NOT applied if the resulting amount was BELOW the capacity. There is one reason for it – it occurred for Soyuz and the CXV-like vehicle is lighter than a Soyuz and so should require less propellant than the capacity of one Block DM if there is a test for the phase(s) where the Block DM is valid.

But the tests with Luna 15 and Apollo-LM meant that the result below the capacity was wrong. But there is one weak point in that: For Luna 15 at least I have no data about tank capacities and no safe data about diameters and heights – I only have data about the propellants themselves.

In so far the criterion above can’t be applied for the landers. And the low correction factors might be results of wrong capacities and wrong dimaters and heights.

On the other hand it sounds reasonable to me though that the complete capacity of a tank should be applied to improve safety margins by adding another one. I will think about it more.



Tests and Checks for calculating up landing parts of existing landers

First I apply Luna 15 to calculate Apollo LM DS.

There is one problem with the landing part of Luna 15 – instead of an Isp a close range of Isps is given. I calculate the averag – which is 2466.5 – and apply it.

Next an additional number must be incorporated – both the landing part of Luna 15 and the Apollo LM DS carried a payload which is the launching part.

Additional numbers required will be found in the Excel spreadsheet once it’s ready and available at this board.

Without a correction factor – correction factor = 1 – the result is 738.4970232 kg of propellant which is too low by a factor of more than 11.

The correction factor from the Luna 15 landing part to the Luna 15 landing part must be applied which is 0.127341152658618859300461688115341. Applying it to the test of the Luna 15 landing part against the Apollo-LM DS the result now is 5799.358713 kg required amount of propellant – which is much better but still too few.

So it seems that this calculation linearly up leads to an insufficient amount of propellant. Of course this looks quite similar to what happened in the case of the launcher parts – but here now I can apply the functions got via regressional analysis. (by the way – I am not sure but I suspect that I by error applied the Luna 15 launcher tank as potnential tank for the Apollo-LM AS as potential vehicle – in so far the tests in the previous post will have to be redone but I am not that sure if that is of any value.)

The complete lander to be landed by the Apollo-LM DS weighs 4547 kg – Apollo-LM carrier, Apollo-LM As and Ass quired propellant for ascent. The following results are got:

Code:
function with non-standardized propellants   8160.403295264 kg   very close to the correct number
plus standard error of the fix part          8452.340399964 kg   already above the correct value
incl. standard error of the variable part    9100.010169204 kg   more than 10% above the correct number

function with N2O4/ADMH as standard
fix part 131.2705921 var. part 1.77014455    8180.11786095 kg   very close to the correct number – above it
plus std. error of fix part 379.9794294      8560.09729035 kg   more above the correct value
incl. std error of var. part 0.165840079     9314.172129563   more than 10% above the correct value




I originally had in mind to apply other functions also and to test the landing parts of the remaining landers also. But at present this seems to be a bit much to me – and this post would become a bit long then.

But for the functions without applying the stndard errors I already have tested which amounts will be got -they will be below the correct number for Luna 15, Surveyor and Luna 9 while they will be above for Apollo-LM DS as listed and a multiple of the correct number for Ranger.

This means that the complete group of functions without and including stndard errors will be calculated and that result will be selected that is above the correct number and clostest to the 10% safety margin if that margin is the highest one – and so on according to the safety margin system I already apply for the round trip.



What to do with the launchers?

Since no such funtions can be found for the launchers there is no way out than to adjust the existing functions to them. The data available for landing and launch per vehicle involve a ratio around 0.25. So the precise ratio will be applied to the functions and/or their results – I will think about what’s best or most proper.



Excel spreadsheet

The Excel spreadsheet will continue to apply correction factors – so it will be possible to use numbers that are NOT got via the functions. But the functions can be applied by switching on an option button (I may decide to use something else but in pribciple it will be like that.)

Regarding the list of existing landers that don’t need and must not be calculated the use of the function(s) and formular(s) will be blocked. But more essential – the list will include one lander that actually is a potential one. This will be the Falcon-derived lander.

And because of the clarification above all correction factors will be selectable. May be that numbers will cause a warning.



In so far all seems to be ready for now to calculate landing trips.



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




Appendix

How the missing data about Luna 9 and Surveyor are derived

To derive the missing data I thougfht to be forced to suppose an analagy of shares on the weight of the hardware.

The empty Apollo LM Descent Stage has a share of nearly a third (30.3781963 %) of the sum of the weight of the empty Apollo LM DS plus the Apollo LM AS inclduing AS-propellant

The Luna 15 launcher has a share of nearly a fourth (24.3085106 %) of the sum of the empty landing part plus the launcher including the launcher-propellant.

Luna 9 and Surveyor had no launchers - so the share required to launch back to Earth must be considered to be at least partially reqplaced by scientific instruments and the like.

Because of the 25%-share of the Luna15-launcher I supposed or assumed Luna 9 to have had 75% of the weight it would have had if it had had a launcher part to return to Earth. This applies to the Lunas only because they are from the same agency and belong to the same program.

Because of the replacement of launcher-weight by instrument-weight I applied a share of two thirds for the carrier and a share of one third for the descent stage of Luna 9 and Surveyor – carrier plus descent stage making up the complete lander.

This I applied to the listed Luna 9-weight of 100 kg. I removed the weight of the engine because I found the information that the engine has been expended before landing when the propellant was consumed completely.

Regarding Surveyor I applied the shares to the result of a calculation described below. Here nothing has been said about expending the engine. The engine including propellant weighed 625 kg. 564.1 kg of this were propellant – so an engine weight of 60.9 kg reached the lunar surface. Surveqor weighed 290 kg on the lunar surfcae. From this I subtracted the 60.9 kg and got 229,1 kg for Surveyor-landing stage plus Surveyor-carrier and the instruments carried.

I didn’t find another way to get numbers. There is another way – but that way is as speculative and arbitrary as this one. It‘s quite weak and very insufficiently based on informations.

But there was no better way yet. If someone knows data I am missing he should insert them into the Excel spreadsheet once it’s ready and available. The replacment will have an impact on the percentage of determination, the standard errors and the function the statistical regression results in.



How diameters of lunar landers are supposed

The methods and ways are humble and weak – I am not content but didn’t find better ways or data yet. There might be some or a few in the sources listed under www.bernd-leitenberger.de but I found time to look at one of them only up to now. This was a NASA-archive where the data needed are NOT available. So it will require time to find them if they are available elsewhere. Perhaps one or the other of you has access to better data.

But these data nonetheless aren’t applied to get weights and amounts for potential landers and the launchers directly – they are applied only to check calculations linearly up or even down.



Apollo LM DS

Using the data listed in the previous post a diameter of 1.02 m results in a height of 7.456948879
m.



Luna 15 landing stage

I had a look into the previous post and applied a diameter of 0.55 m resulting in a height of 6,121102056
m at the given amount of propellant.



Ranger

In one of the drawings the Retro Motor seems to have a diameter of around one third of the space between the two sollar panels. This drawing is about Block II while I am using Block III here. The spaces between the solar panels are quite similar – 152 cm and 154 cm according to the data I found under www.bernd-leitenberger.de . So the diamater of the Motor can be assumed to be 50 cm.

It’s looking as if the tank and the engine are integrated into each other. I had some problems to make up my mind what to suppose. I finally thought that the geometry of the tank around the engine will have benn abit complex and so applied a simple geometry instead – a cylinder of 0.5 m diamter resulting in a height of 0,062527407 m.

The spherical diameter is got via the volume of the propellant calculated from the amount of propellant and a density of 1.01 g/cm^3

Surveyor

From one of the drawings it seems that the engine plus tank was spherical. Again I seem to be forced to suppose that the engine without the nozzle was surrounded by the tank. If this were not the case and the sphere were a tank in total then the spherical diameter oif that tank were 1.498926817 m If that were not the case the geometery of the tank again seems to be complex. For the simple geometery applied by the formulars I apply the spherical diameter and get a cylindrical height of 0.23856161 m.

The spherical diameter is got via the volume of the propellant which I calculated from the amount and a density of 1.34 g/cm^3 – which is NOT the density of the propellant actualy consumed but the density of HTPB/aluminum/ammoniaperchlorate. The catual propellant was acrylnitrid/aluminum/ammoniaperchlorate.



Luna 9

About Luna 9 there are no diamaters or heights available regarding the lander itself. So I did as for the previous two landers – I calculated a spherical diameter from the amount of propellant and ist density which is 1.47 g/cm^3 – But again there seem to be no data about the propellant because it is a particular one – nitric acid with NTO, a bit of water and what’s called „Jod“ in German. The information is from www.bernd-leitenberger.de where this micture is called a technical one and no density is listed. Since nitirc acid and NTO make up 90% of the mixture I suppose the density to be half way between nitirc acid and NTO which are very close to each other. This number then is runded down.

Then a spherical diameter of 1.58002374 m is got and the according cylindrical height is 0.251468588 m is got.


End of Appendix


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