Table of Contents.
I.)Introduction.
II.)Lightweight propellant tanks.
III.)Kerosene fuel and engines for the X-33/Venture star.
IVa.)Aerodynamic lift applied to ascent to orbit.
b.)Estimation of fuel saving using lift.
V.)Kerosene fueled VentureStar payload to orbit.
I.) A debate among those questing for the Holy Grail of a reusable,
single-stage-to-orbit vehicle is whether it should be powered by hydrogen or a
dense hydrocarbon such as kerosene. Most concepts for such a vehicle centered
on hydrogen, since a hydrogen/LOX combination provides a higher Isp. However,
some have argued that dense fuels should be used since they take up less
volume (equivalently more fuel mass can be carried in the same sized tank) so
they incur less air drag and also since the largest hydrocarbon engines
produce greater thrust they can get to the desired altitude more quickly so
they also incur lower gravity drag loss.
Another key fact is that for dense fuels the ratio of propellant mass to tank
mass is higher, i.e., you need less tank mass for the same mass of propellant.
This fact is explored in this report:
Single Stage To Orbit Mass Budgets Derived From Propellant Density and
Specific Impulse.
John C. Whitehead
32nd AIAA/ASME/SAE/ASEE Joint Propulsion ConferenceLake Buena Vista, FLJuly
1-3, 1996
http://www.osti.gov/bridge/servlets/pur ... 379977.pdf
Whitehead notes that the propellant mass to tank mass ratio for kerosene/LOX
is typically around 100 to 1, while for liquid hydrogen/LOX it's about 35 to
1, which would result in a significantly greater dry mass for the
hydrogen-fueled case just in tank weight alone. Based on calculations such as
these Whitehead concludes the best option for a SSTO would be to use
kerosene/LOX.
The case for the X-33/VentureStar is even worse because the unusual shape of
the tanks requires them to use more tank mass than a comparably sized
cylindrical tank. This is discussed here:
Space Access Update #91 2/7/00.
The Last Five Years: NASA Gets Handed The Ball, And Drops It.
"...part of L-M X-33's weight growth was the "multi-
lobed" propellant tanks growing considerably heavier than promised.
Neither Rockwell nor McDonnell-Douglas bid these; both used proven
circular-section tanks. X-33's graphite-epoxy "multi-lobed" liquid
hydrogen tanks have ended up over twice as heavy relative to the
weight of propellant carried as the Shuttle's 70's vintage aluminum
circular-section tanks - yet an X-33 tank still split open in test
last fall. Going over to aluminum will make the problem worse; X-
33's aluminum multi-lobed liquid oxygen tank is nearly four times as
heavy relative to the weight of propellant carried as Shuttle's
aluminum circular-section equivalent."
http://www.space-access.org/updates/sau91.html
The X-33's twin liquid hydrogen tanks had a weight of 4,600 pounds each, and
the liquid oxygen tank a weight of 6,000 pounds, for total of 15,200 pounds
for the tanks:
Marshall Space Flight Center
Lockheed Martin Skunk Works
Sept. 28, 1999
X-33 Program in the Midst of Final Testing and Validation of Key Components.
http://www.xs4all.nl/~carlkop/x33.html
The weight of the propellant carried by the X-33 was supposed to be 210,000
lb. So the propellant to tank mass ratio for the X-33 was only about 14 to
1(!). This would be a severe problem for the full-scale VentureStar. Its gross
lift off weight was supposed to be 2,186,000 lbs with a fuel weight of
1,929,000 lbs:
X-33 Advanced Technology Demonstrator.
http://teacherlink.ed.usu.edu/tlnasa/Ot ... trator.PDF
So the VentureStar would have a dry mass of 257,000 lbs. Since the same design
would be used for the VentureStar tanks as those of the X-33, the propellant
mass to tank mass ratio would also be 14 to 1, so the tank mass would be
138,000 lbs. But this means the empty tank mass alone would be over half of
the vehicle's dry weight (!)
It would have been extremely difficult for the VentureStar to have made orbit
with such a large weight penalty from the start. From all accounts the weight
problem with the tanks drove other problems such as the need to add larger
wings, increasing the weight problem further. NASA wound up canceling the
program when Lockheed couldn't deliver the working liquid hydrogen tanks even
at this excessive weight. However, rather than canceling the program I believe
the better course would have been to open up competition for coming up with
alternative, creative solutions for reducing the weight of the tanks. This
would also have resolved some of the problems with the vehicles weight growth.
II.) I have proposed one possibility for lightweighting the X-33 tanks on this
forum:
http://www.bautforum.com/space-explorat ... ost1495726
The idea would be to achieve the same lightweight tanks as cylindrical ones by
using multiple, small diameter, aluminum cylindrical tanks. You could get the
same volume by using varying lengths and diameters of the multiple cylinders
to fill up the volume taken up by the tanks. The cylinders would not have to
be especially small. In fact they could be at centimeter to millimeter
diameters, so would be of commonly used sizes for aluminum tubes and pipes.
The weight of the tanks could be brought down to the usual 35 to 1 ratio for
propellant to tank mass. Then the mass of the tanks on the X-33 would be
210,000 lbs/35 = 6,000 lbs, saving 9,200 lbs off the vehicle dry weight. This
would allow the hydrogen-fueled X-33 to achieve its original Mach 15 maximum
velocity.
The same idea applied to the full-scale hydrogen-fueled VentureStar would
allow it to significantly increase its payload carrying capacity. At a 35 to 1
ratio of propellant mass to tank mass, the 1,929,000 lbs propellant mass would
require a mass of 1,929,000/35 = 55,000 lbs for the tanks, a saving of 83,000
lbs off the original tank mass. This could go to extra payload, so from 45,000
lbs max payload to 128,000 lbs max payload.
An analogous possibility might be to use a honeycombed structure for the
entire internal makeup of the tank. The X-33's carbon composite tank was to
have a honeycombed structure for the skin alone. Using a honeycomb structure
throughout the interior might result in a lighter tank in the same way as does
multiple cylinders throughout the interior.
Still another method might be to model the tanks standing vertically as
conical but with a flat front and back, and rounded sides. Then the problem
with the front and back naturally trying to balloon out to a circular cross
section might be solved by having supporting flat panels at regular intervals
within the interior. The X-33 composite tanks did have support arches to help
prevent the tanks from ballooning but these only went partially the way
through into the interior. You might get stronger a result by having these
panels go all the way through to the other side.
These would partition the tanks into portions. This could still work if you
had separate fuel lines, pressurizing gas lines, etc. for each of these
partitions and each got used in turn sequentially. A preliminary calculation
based on the deflection of flat plates under pressure shows with the tank made
of aluminum alloy and allowing deflection of the flat front and back to be
only of millimeters that the support panels might add only 10% to 20% to the
weight of the tanks, while getting similar propellant mass to tank mass ratio
as cylindrical tank. See this page for an online calculator of the deflection
of flat plates:
eFunda: Plate Calculator -- Simply supported rectangular plate with uniformly
distributed loading.
http://www.efunda.com/formulae/solid_me ... niform.cfm
Note you might not need to have a partitioned tank, with separate fuel lines,
etc., if the panels had openings to allow the fuel to pass through. These
would look analogous to the wing spars in aircraft wings that allow fuel to
pass through. You might have the panels be in a honeycomb form for high
strength at lightweight that still allowed the fuel to flow through the tank.
Or you might have separate beams with a spaces between them instead of solid
panels that allowed the fuel to pass through between the beams.
Another method is also related to the current design of having a honeycombed
skin for the composite hydrogen tanks. Supposed we filled these honeycombed
cells with fluid. It is known that pressurized tanks can provide great
compressive strength. This is in fact used to provide some of the structural
strength for the X-33 that would otherwise have to be provided by heavy
strengthening members. This idea would be to apply fluid filled honycombed
cells. However, what we need for our pressurized propellant tanks is *tensile
strength*.
A possible way tensile strength could be provided would be to use the
Poisson's ratio of the honeycombed cells:
Poisson's ratio.
http://en.wikipedia.org/wiki/Poisson%27s_ratio
Poisson's ratio refers to the tendency of a material stretched in one
direction to shrink in length in an orthogonal direction. Most isotropic solid
materials have Poisson's ratio of about .3. However, the usual hexagonal
honeycombed structure, not being isotropic, can have Poisson's ratios in the
range of +1. This is mentioned in this article about non-standard honeycombed
structures that can even have negative Poisson ratios:
Chiral honeycomb.
http://silver.neep.wisc.edu/~lakes/PoissonChiral.html
However, note that from the formula for the volumetric change in the Wikipedia
Poisson's ratio page, a stretching of a material with a +1 Poisson's ratio
implies a *decrease* in volume; actually this is true for any case where the
Poisson's ratio is greater than +.5. Then fluid filled honeycombed cells would
resist the stretching of tensile strain by the resistance to volume
compression. This would be present with both gases and liquids. Gases are
lighter. However, they are highly compressible and it might take too large an
internal pressure in the cells to provide sufficient resistance, and so also
too thick cell walls to hold this pressure. Liquids are heavier but they are
highly non-compressible so could provide strong resistance to the volume
compression and thereby to the tensile strain.
Then for liquid hydrogen tanks we might use liquid hydrogen filled cells
within the skin of the tanks. Hydrogen is rather light compared to other
liquids at a density of only about 72 kg/m^3. This then could provide high
tensile strength at a much lower weight than typical solid wall tanks.
Kerosene and liquid oxygen would be used in the honeyombed cells for their
corresponding tanks, to keep the storage temperatures comparable. These are
heavier liquids than liquid hydrogen, approximately in the density range of
liquid water. Still these liquid filled honeycombed cells would provide much
lighter tanks than comparable solid wall tanks.
III.) Any of these methods might allow you to reduce the weight of the tanks
to be similar to that of cylindrical tanks and thus raise the payload to over
100,000 lbs. This would be for keeping the original hydrogen/LOX propellant.
However, in keeping with the analyses that show dense propellants would be
more appropriate for a SSTO vehicle I'll show that replacing the
hydrogen-fueled engines of the X-33/VentureStar with kerosene ones would allow
the X-33 to actually now become an *orbital* craft instead of just suborbital,
and the payload capacity of the VentureStar would increase to be comparable to
that proposed for Ares V.
The volume of the X-33 liquid hydrogen tanks was 29,000 gallons each and the
liquid oxygen tank, 20,000 gallons, for a total of 78,000 gallons volume for
propellant. This is 78,000gal*3.8 L/gal = 296,000 liters, 296 cubic meters.
How much mass of kerosene/LOX could we fit here if we used these as our
propellants? Typically the oxidizer to fuel ratio for kerosene/LOX engines is
in the range of 2.5 to 2.7 to 1. I'll take the O/F ratio as 2.7 to 1. The
density of kerosene is about 806 kg/m^3 and we can take the density of liquid
oxygen to be 1160 kg/m^3 when densified by subcooling:
Liquid Oxygen Propellant Densification Unit Ground Tested With a Large-Scale
Flight-Weight Tank for the X-33 Reusable Launch Vehicle.
http://www.grc.nasa.gov/WWW/RT/RT2001/5 ... omsik.html
These requirements of the propellants' total volume and densities, result in a
total propellant mass of 307,000 kg, with 83,000 kg in kerosene and 224,000 kg
in LOX. Kerosene/LOX tanks weigh typically 1/100th the propellant mass, so the
tank mass would be 3,070 kg. The current X-33 LH/LOX tanks weighed 15,200 lbs,
or 6,900 kg. So the empty weight of the X-33 is reduced from 63,000 lbs,
28,600 kg, to 28,600kg - 6,900kg + 3070kg = 24,800 kg.
How about the engines? The X-33 is to be reusable so you want to use reusable
kerosene engines. The RS-84 might be ideal when it is completed for the
full-scale VentureStar, but it turns out it's a bit too heavy for the X-33.
It would have a weight of about 15,000 lbs, 6,800 kg:
RS-84.
http://www.astronautix.com/engines/rs84.htm
about the weight of the two aerospike engines currently on the X-33:
Bringing launch costs down to earth.
"Three federally funded projects are underway to develop new rocket engines
that can make it more affordable to send payloads into orbit."
http://www.memagazine.org/backissues/me ... aunch.html
With 307,000 kg kerosene/LOX fuel and 24,800 kg dry weight, the mass ratio
would be 13.4. According to the Astronautix page, the sea level Isp of the
RS-84 would be 301 s, and the vacuum 335 s. Take the average Isp as 320s. The
total Isp for a rocket to orbit including gravity and air drag losses is
usually taken to be about 9,200 m/s. Then an average exhaust velocity of 3200
m/s and mass ratio of 13.4 would give a total delta-v of 8,300 m/s. Even if
you add on the 462 m/s additional velocity you can get for free by launching
at the equator this would not be enough for orbit.
So for the X-33 I'll look at the cases of the lighter for its thrust NK-33,
used as a trio. Note that though not designed to be a reusable engine to make,
say, 100 flights, all liquid fuel rocket engines undergo extensive static
firings during testing so the NK-33 probably could make 5 to 10 flights before
needing to be replaced.The NK-33 is almost legendary for its thrust to weight
ratio of 136. According to the Astronautix page its weight is 1,222 kg , with
a sea level Isp of 297 sec and a vacuum Isp of 331:
NK-33.
http://www.astronautix.com/engines/nk33.htm
I'll take the average Isp as 315 s. With three NK-33 engines the mass of the
X-33 becomes 21,700 kg, and the mass ratio becomes 15.15. Then with an average
Isp of 315 s, the total delta-v would be 8561 m/s and if you add on the 462
m/s additional equatorial velocity it's 9,023 m/s. Still slightly below the
delta-v typically given for orbit of 9,200 m/s.
However, it should be noted that the extra delta-v required beyond the 7,800
m/s orbital velocity is highly dependent on the vehicle and trajectory. Here's
a page that gives the gravity loss and air drag loss for some orbital rockets:
Drag: Loss in Ascent, Gain in Descent, and What It Means for Scalability.
Thursday 2008.01.10 by gravityloss
* Ariane A-44L: Gravity Loss: 1576 m/s Drag Loss: 135 m/s
* Atlas I: Gravity Loss: 1395 m/s Drag Loss: 110 m/s
* Delta 7925: Gravity Loss: 1150 m/s Drag Loss: 136 m/s
* Shuttle: Gravity Loss: 1222 m/s Drag Loss: 107 m/s
* Saturn V: Gravity Loss: 1534 m/s Drag Loss: 40 m/s (!!)
* Titan IV/Centaur: Gravity Loss: 1442 m/s Drag Loss: 156 m/s
http://gravityloss.wordpress.com/2008/0 ... alability/
Note that the gravity loss for the Delta 7925 is particularly small. As a
general principle the gravity loss can be minimized if you have a high thrust
vehicle that rapidly develops high vertical velocity sufficient to reach the
altitude for orbit. For then it can more quickly apply the horizontal thrust
required to achieve the 7,800 m/s orbital, tangential, velocity, there being
no gravity loss over the horizontal thrust portion. Note then the liftoff
thrust to liftoff weight ratio for the Delta 7925 is the relatively large 1.4;
in comparison for the Saturn V it was only 1.14. And note now also that the
X-33 with three NK-33 engines has total mass of 328,700 kg and total thrust of
4,530,600 N giving a liftoff thrust to liftoff weight ratio of 1.4. Then this
reconfigured X-33 would likely have comparable gravity drag loss as the Delta
7925. When you take into account the high thrust means it would rapidly reach
high altitude, implying the Isp would quickly get close to the vacuum Isp, the
average Isp over the trajectory is most likely closer to the 331 s vacuum Isp
than just 315 s, giving an actually higher achieved delta-v.
IV.a) It's capability of reaching orbit and possibly even with a small payload
could be increased with another additional factor. Among the questors for a
SSTO vehicle, the idea to use wings for a horizontal landing has been derided
because of the view they were just dead weight on ascent and you need to save
as much weight off the empty weight of the vehicle as possible to achieve
orbit. However, the key fact is that wings or a lifting body shape can reduce
the total delta-v for orbit by using aerodynamic lift to supply the force to
raise the vehicle to a large portion of the altitude of orbit rather than this
force being entirely supplied by the thrust of the engines. This fact is
discussed on page 4 of this report:
AIAA 2000-1045
A Multidisciplinary Performance Analysis of a Lifting-Body
Single-Stage-to-Orbit Vehicle.
Paul V. Tartabini, Roger A. Lepsch, J. J. Korte, Kathryn E. Wurster
38th Aerospace Sciences Meeting & Exhibit.
10-13 January 2000 / Reno, NV
"One feature of the VentureStar design that could
be exploited during ascent was its lifting body shape.
By flying a lilting trajectory, it was possible to significantly
decrease the amount of gravity losses, thereby
improving vehicle performance and payload capability.
Yet increasing the amount of lift during ascent generally
required flight at higher angles-of-attack and resulted
in greater stress on the vehicle structure. Accordingly,
the nominal trajectory was constrained to keep the parameter
q-_ below a 1500 psf-deg structural design limit
to ensure that the aerodynamic loads did not exceed
the structural capability of the vehicle. The effect of this
trajectory constraint on vehicle performance is shown
in Fig. 3. There was a substantial benefit associated with
using lift during ascent since flying a non-lifting trajectory
resulted in a payload penalty of over 1000 lbs compared
to the nominal case."
http://ntrs.nasa.gov/archive/nasa/casi. ... 025539.pdf
Then the gravity losses could be further reduced by flying a lifting
trajectory, which would also increase the payload capability by a small
percentage.
The trajectory I'll use to illustrate this will first be straight-line at an
angle up to some high altitude that still allows aerodynamic lift to operate.
At the end of this portion the vehicle will have some horizontal and vertical
component to its velocity. We'll have the vertical component be sufficient to
allow the vehicle to reach 100 km, altitude. The usual way to estimate this
vertical velocity is by using the relation between kinetic energy and
potential energy. It gives the speed of v = sqrt(2gh) to reach an altitude of
h meters. At 100,000 m, v is 1,400 m/s.
Now to have orbital velocity you need 7,800 m/s tangential, i.e., horizontal
velocity. If you were able to fly at a straight-line at a constant angle to
reach 7,800 m/s horizontal velocity and 1,400 m/s vertical velocity and such
that the air drag was kept at the usual low 100 to 150 m/s then you would only
need sqrt(7800^2 + 1400^2) = 7,925 m/s additional delta-v to reach orbit. Then
the total delta-v to orbit might only be in 8,100 m/s range. Note this is
significantly less than the 9,200 m/s delta-v typically needed for orbit,
including gravity and air drag.
The problem is with usual rocket propulsion to orbit not using lift the thrust
vector has to be more or less along the center-line of the rocket otherwise
the rocket would tumble. You can gimbal the engines only for a short time to
change the rocket's attitude but the engines have to be then re-directed along
the center line. However, the center line has to be more or less pointing into
the airstream, i.e., pointing in the same direction as the velocity vector, to
reduce aerodynamic stress and drag on the vehicle. But the rocket thrust
having to counter act gravity means a large portion of the thrust has to be in
the vertical component which means the thrust vector has to be nearly vertical
at least for the early part of the trip when the gross mass is high. Then the
thrust vector couldn't be along the center line of a nearly horizontally
traveling rocket at least during the early part of the trip.
However, using lift you are able to get this large upwards vertical component
for the force on the rocket to allow it to travel along this straight-line. A
problem now though is that at an altitude short of that of space, the air
density will not be enough for aerodynamic lift. Therefore we will use lift
for the first portion of the trajectory, traveling in a straight-line at an
angle. Then after that, with sufficient vertical velocity component attained
to coast to 100 km altitude, we will supply only horizontal thrust during the
second portion to reach the 7,800 m/s horizontal velocity component required
for orbital velocity.
IV.b) How much fuel could we save using a lifting straight-line portion of the
trajectory? I'll give an example calculation that illustrates the fuel savings
from using aerodynamic lift during ascent. First note that just as for
aircraft fuel savings are best at a high L/D ratio. However, the hypersonic
lift /drag ratio of the X-33/VentureStar is rather poor, only around 1.2,
barely better than the space shuttle:
AIAA-99-4162
X-33 Hypersonic Aerodynamic Characteristics.
Kelly J, Murphy, Robert J, Nowak, Richard A, Thompson, Brian R, Hollis
NASA Langley Research Center
Ramadas K. Prabhu
Lockheed Martin Engineering &Sciences Company
http://ntrs.nasa.gov/archive/nasa/casi. ... 091447.pdf
This explains the low increase in payload, about 1,000 lbs., less than .5% of
the vehicle dry weight, by using a lifting trajectory for the VentureStar.
However, some lifting body designs can have a lift/drag ratio of from 6 to 8
at hypersonic speeds:
Waverider Design.
http://www.aerospaceweb.org/design/wave ... ider.shtml
The L/D is usually optimized for a specific speed range but we can imagine
"morphing" wings that allow a good L/D ratio over a wide speed range. For
instance note on the "Waverider Design" web page the vehicles optimized for
the highest hypersonic speeds have a long, slender shape, compared to those
for the slower hypersonic speeds. Then for an orbital craft we could have
telescoping sides of the vehicle that would be extended when full of fuel at
the slower speeds, and retracted, producing a slimmer vehicle, when most of
the fuel is burned off and the vehicle is flying faster. Note that a good L/D
ratio at the highest hypersonic speeds means the vehicle will experience less
aerothermal heating on return.
Then we can imagine a second generation lifting trajectory vehicle having this
high L/D ratio over a wide speed range. So in the example I'll take the
supersonic/hypersonic L/D ratio as 5, and for lack of a another vehicle I'll
use the reconfigured kerosene-fueled X-33's thrust and weight values.
Here's the calculation for constant L/D at a constant angle θ (theta).
I'll regard the straight-line path as my X-axis and the perpendicular to this
as the Y-axis. Note this means my axes look like they are at an angle to the
usual horizontal and vertical axes, but it makes the calculation easier. Call
the thrust T, and the mass, M. Then the force component along the
straight-line path, our X-axis, is Fx = T - gMsin(θ) - D and the force
component along the Y-axis is Fy = L - gMcos(θ). We'll set L =
gMcos(θ). Then the force along the straight-line is Fx = T -
gMsin(θ) - gMcos(θ)/(L/D). As with the calculation in the horizontal
case, divide this by M to get the acceleration along this line, and integrate
to get the velocity. The result is V(t) = Ve*ln(M0/Mf) -g*tsin(θ) -
g*tcos(θ)/(L/D), with M0 the initial mass, and Mf, the mass at time t, a
la the rocket equation. If you make the angle θ (theta) be shallow, the
g*tsin(θ) term will be smaller
than the usual gravity drag loss of g*t and the (L/D) divisor will make the
cosine term smaller as well.
I'll assume the straight-line path is used for a time when the altitude is
high enough to use the vacuum Ve of 331s*9.8 m/s^2 = 3244 m/s. According to
the Astronautix page, 3 NK-33's would have a total vacuum thrust of 4,914,000
N and for an Isp of 331s, the propellant flow rate would be
4,914,000/(331x9.8) = 1,515 kg/sec. I'll use the formula: V(t) = Ve*ln(M0/Mf)
- g*tsin(θ) - g*tcos(θ)/(L/D) , to calculate the velocity along the
inclined straight-line path. There are a couple of key facts in this formula.
First note that it includes *both* the gravity and air drag. Secondly, note
that though using aerodynamic lift generates additional, large, induced drag,
this is covered by the fact that the L/D ratio includes this induced drag,
since it involves the *total* drag.
I'll take the time along the straight-line path as 100 sec. Then Mf =
328,700kg -100s*(1,515 kg/s) = 177,200 kg. After trying some examples an angle
of 30º provides a good savings over just using the usual non-lifting
trajectory. Then V(t) = 3244*ln(328,700/177,200) - 9.8*100(sin(30º) +
cos(30º)/5) = 1,345 m/s. Then the vertical component of this velocity is Vy =
1,135*sin(30º) = 672.3 m/s and the horizontal, Vx = 1,135*cos(30º) = 1,164.5
m/s.
To compare this to a usual rocket trajectory I'll calculate how much fuel
would be needed to first make a vertical trip to reach a vertical speed of
672.3 m/s subject to gravity and air drag, and then to apply horizontal thrust
to reach a 1,164.5 m/s horizontal speed.
The air drag for a usual rocket is in the range of 100 m/s to 200 m/s. I'll
take the air drag loss as 100 m/s for this vertical portion. Then the equation
for the velocity along this vertical part including the gravity loss and the
air drag loss would be V(t) = 3244*ln(M0/Mf) - 9.8*t - 100 m/s, where M0
=328,700 kg and Mf = 328,700 - t(1,515). You want to find the t so that this
velocity matches the vertical component in the inclined case of 672.3 m/s.
Plugging in different values of t, gives for t = 85 sec, V(85) = 680 m/s.
Now to find the horizontal velocity burn. Since this is horizontal there is no
gravity loss, and I'll assume this part is at very high altitude so has
negligible air drag loss. Then the velocity formula is V(t) = 3244*ln(M0/Mf).
Note in this case M0 = 328,700 - 85*1,515 = 199,925 kg, which is the total
mass left after you burned off the propellant during the vertical portion, and
so Mf = 199,925 - t*1,515. Trying different values of t gives for t = 40,
V(40) = 1,171.5 m/s.
Then doing it this way results in a total of 125 sec of fuel burn, 25 percent
higher than in the aerodynamic lift case, specifically 25s*1,515 kg/s = 37,875
kg more. Or viewed the other way, the aerodynamic lift case requires 20% less
fuel over this portion of the trip than the usual non-lift trajectory. With a
307,000 kg total fuel load, this corresponds to a 12.3% reduction in the total
fuel that would actually be needed. Or keeping the same fuel load, a factor
1/.877 = 1.14 larger dry mass could be lofted, which could be used for greater
payload. For a reconfigured X-33 dry mass of 21,700 kg, this means 3,038 kg
extra payload. Remember though this is for our imagined new X-33 lifting shape
that is able to keep a high L/D ratio of 5 at hypersonic speed, not for the
current X-33 shape which only has a hypersonic L/D of 1.2.
With the possibility of using morphing lifting body or wings with high
hypersonic L/D ratio allowing a large reduction in fuel requirements to orbit,
this may be something that could be tested by amateurs or by the "new space"
launch companies.
V.) Now for the calculation of the payload the VentureStar could carry using
kerosene/LOX engines. The propellant mass of the VentureStar was 1,929,000
lbs. compared to the X-33's 210,000 lbs., i.e., 9.2 times more. Then its
propellant tank volume would also be 9.2 times higher, and the kerosene/LOX
they could contain would also be 9.2 times higher, or to 9.2*307,000 =
2,824,400 kg.
We saw the VentureStar dry mass was 257,000 lbs, 116,818 kg, with half of this
as just the mass of the LH2/LOX tanks, at 138,000 lbs, 62,727 kg. However,
going to kerosene/LOX propellant means the tanks would only have to be 1/100th
the mass of the propellant so only 28,244 kg. Then the dry mass would be
reduced to 82,335 kg. We need kerosene/LOX engines now. I suggest the RS-84 be
completed and used for the purpose. You would need seven of them to lift the
heavier propellant load. They weigh about the same as the aerospike engines on
the current version of the VentureStar so you wouldn't gain any weight savings
here.
To calculate how much we could lift to orbit I'll take the average Isp of the
RS-84 as 320. Then if we took the payload as 125,000 kg the total liftoff mass
would be 2,824,400 + 82,335 + 125,000 = 3,031,735 kg, and the ending dry mass
would be 207,335 kg, for a mass ratio of 14.6. Then the total delta-v would be
3200ln(14.6) = 8,580 m/s. Adding on the 462 m/s equatorial speed brings this
to 9042 m/s. With the reduction in gravity drag using a lifting trajectory
this would suffice for orbit.
Bob Clark
I.)Introduction.
II.)Lightweight propellant tanks.
III.)Kerosene fuel and engines for the X-33/Venture star.
IVa.)Aerodynamic lift applied to ascent to orbit.
b.)Estimation of fuel saving using lift.
V.)Kerosene fueled VentureStar payload to orbit.
I.) A debate among those questing for the Holy Grail of a reusable,
single-stage-to-orbit vehicle is whether it should be powered by hydrogen or a
dense hydrocarbon such as kerosene. Most concepts for such a vehicle centered
on hydrogen, since a hydrogen/LOX combination provides a higher Isp. However,
some have argued that dense fuels should be used since they take up less
volume (equivalently more fuel mass can be carried in the same sized tank) so
they incur less air drag and also since the largest hydrocarbon engines
produce greater thrust they can get to the desired altitude more quickly so
they also incur lower gravity drag loss.
Another key fact is that for dense fuels the ratio of propellant mass to tank
mass is higher, i.e., you need less tank mass for the same mass of propellant.
This fact is explored in this report:
Single Stage To Orbit Mass Budgets Derived From Propellant Density and
Specific Impulse.
John C. Whitehead
32nd AIAA/ASME/SAE/ASEE Joint Propulsion ConferenceLake Buena Vista, FLJuly
1-3, 1996
http://www.osti.gov/bridge/servlets/pur ... 379977.pdf
Whitehead notes that the propellant mass to tank mass ratio for kerosene/LOX
is typically around 100 to 1, while for liquid hydrogen/LOX it's about 35 to
1, which would result in a significantly greater dry mass for the
hydrogen-fueled case just in tank weight alone. Based on calculations such as
these Whitehead concludes the best option for a SSTO would be to use
kerosene/LOX.
The case for the X-33/VentureStar is even worse because the unusual shape of
the tanks requires them to use more tank mass than a comparably sized
cylindrical tank. This is discussed here:
Space Access Update #91 2/7/00.
The Last Five Years: NASA Gets Handed The Ball, And Drops It.
"...part of L-M X-33's weight growth was the "multi-
lobed" propellant tanks growing considerably heavier than promised.
Neither Rockwell nor McDonnell-Douglas bid these; both used proven
circular-section tanks. X-33's graphite-epoxy "multi-lobed" liquid
hydrogen tanks have ended up over twice as heavy relative to the
weight of propellant carried as the Shuttle's 70's vintage aluminum
circular-section tanks - yet an X-33 tank still split open in test
last fall. Going over to aluminum will make the problem worse; X-
33's aluminum multi-lobed liquid oxygen tank is nearly four times as
heavy relative to the weight of propellant carried as Shuttle's
aluminum circular-section equivalent."
http://www.space-access.org/updates/sau91.html
The X-33's twin liquid hydrogen tanks had a weight of 4,600 pounds each, and
the liquid oxygen tank a weight of 6,000 pounds, for total of 15,200 pounds
for the tanks:
Marshall Space Flight Center
Lockheed Martin Skunk Works
Sept. 28, 1999
X-33 Program in the Midst of Final Testing and Validation of Key Components.
http://www.xs4all.nl/~carlkop/x33.html
The weight of the propellant carried by the X-33 was supposed to be 210,000
lb. So the propellant to tank mass ratio for the X-33 was only about 14 to
1(!). This would be a severe problem for the full-scale VentureStar. Its gross
lift off weight was supposed to be 2,186,000 lbs with a fuel weight of
1,929,000 lbs:
X-33 Advanced Technology Demonstrator.
http://teacherlink.ed.usu.edu/tlnasa/Ot ... trator.PDF
So the VentureStar would have a dry mass of 257,000 lbs. Since the same design
would be used for the VentureStar tanks as those of the X-33, the propellant
mass to tank mass ratio would also be 14 to 1, so the tank mass would be
138,000 lbs. But this means the empty tank mass alone would be over half of
the vehicle's dry weight (!)
It would have been extremely difficult for the VentureStar to have made orbit
with such a large weight penalty from the start. From all accounts the weight
problem with the tanks drove other problems such as the need to add larger
wings, increasing the weight problem further. NASA wound up canceling the
program when Lockheed couldn't deliver the working liquid hydrogen tanks even
at this excessive weight. However, rather than canceling the program I believe
the better course would have been to open up competition for coming up with
alternative, creative solutions for reducing the weight of the tanks. This
would also have resolved some of the problems with the vehicles weight growth.
II.) I have proposed one possibility for lightweighting the X-33 tanks on this
forum:
http://www.bautforum.com/space-explorat ... ost1495726
The idea would be to achieve the same lightweight tanks as cylindrical ones by
using multiple, small diameter, aluminum cylindrical tanks. You could get the
same volume by using varying lengths and diameters of the multiple cylinders
to fill up the volume taken up by the tanks. The cylinders would not have to
be especially small. In fact they could be at centimeter to millimeter
diameters, so would be of commonly used sizes for aluminum tubes and pipes.
The weight of the tanks could be brought down to the usual 35 to 1 ratio for
propellant to tank mass. Then the mass of the tanks on the X-33 would be
210,000 lbs/35 = 6,000 lbs, saving 9,200 lbs off the vehicle dry weight. This
would allow the hydrogen-fueled X-33 to achieve its original Mach 15 maximum
velocity.
The same idea applied to the full-scale hydrogen-fueled VentureStar would
allow it to significantly increase its payload carrying capacity. At a 35 to 1
ratio of propellant mass to tank mass, the 1,929,000 lbs propellant mass would
require a mass of 1,929,000/35 = 55,000 lbs for the tanks, a saving of 83,000
lbs off the original tank mass. This could go to extra payload, so from 45,000
lbs max payload to 128,000 lbs max payload.
An analogous possibility might be to use a honeycombed structure for the
entire internal makeup of the tank. The X-33's carbon composite tank was to
have a honeycombed structure for the skin alone. Using a honeycomb structure
throughout the interior might result in a lighter tank in the same way as does
multiple cylinders throughout the interior.
Still another method might be to model the tanks standing vertically as
conical but with a flat front and back, and rounded sides. Then the problem
with the front and back naturally trying to balloon out to a circular cross
section might be solved by having supporting flat panels at regular intervals
within the interior. The X-33 composite tanks did have support arches to help
prevent the tanks from ballooning but these only went partially the way
through into the interior. You might get stronger a result by having these
panels go all the way through to the other side.
These would partition the tanks into portions. This could still work if you
had separate fuel lines, pressurizing gas lines, etc. for each of these
partitions and each got used in turn sequentially. A preliminary calculation
based on the deflection of flat plates under pressure shows with the tank made
of aluminum alloy and allowing deflection of the flat front and back to be
only of millimeters that the support panels might add only 10% to 20% to the
weight of the tanks, while getting similar propellant mass to tank mass ratio
as cylindrical tank. See this page for an online calculator of the deflection
of flat plates:
eFunda: Plate Calculator -- Simply supported rectangular plate with uniformly
distributed loading.
http://www.efunda.com/formulae/solid_me ... niform.cfm
Note you might not need to have a partitioned tank, with separate fuel lines,
etc., if the panels had openings to allow the fuel to pass through. These
would look analogous to the wing spars in aircraft wings that allow fuel to
pass through. You might have the panels be in a honeycomb form for high
strength at lightweight that still allowed the fuel to flow through the tank.
Or you might have separate beams with a spaces between them instead of solid
panels that allowed the fuel to pass through between the beams.
Another method is also related to the current design of having a honeycombed
skin for the composite hydrogen tanks. Supposed we filled these honeycombed
cells with fluid. It is known that pressurized tanks can provide great
compressive strength. This is in fact used to provide some of the structural
strength for the X-33 that would otherwise have to be provided by heavy
strengthening members. This idea would be to apply fluid filled honycombed
cells. However, what we need for our pressurized propellant tanks is *tensile
strength*.
A possible way tensile strength could be provided would be to use the
Poisson's ratio of the honeycombed cells:
Poisson's ratio.
http://en.wikipedia.org/wiki/Poisson%27s_ratio
Poisson's ratio refers to the tendency of a material stretched in one
direction to shrink in length in an orthogonal direction. Most isotropic solid
materials have Poisson's ratio of about .3. However, the usual hexagonal
honeycombed structure, not being isotropic, can have Poisson's ratios in the
range of +1. This is mentioned in this article about non-standard honeycombed
structures that can even have negative Poisson ratios:
Chiral honeycomb.
http://silver.neep.wisc.edu/~lakes/PoissonChiral.html
However, note that from the formula for the volumetric change in the Wikipedia
Poisson's ratio page, a stretching of a material with a +1 Poisson's ratio
implies a *decrease* in volume; actually this is true for any case where the
Poisson's ratio is greater than +.5. Then fluid filled honeycombed cells would
resist the stretching of tensile strain by the resistance to volume
compression. This would be present with both gases and liquids. Gases are
lighter. However, they are highly compressible and it might take too large an
internal pressure in the cells to provide sufficient resistance, and so also
too thick cell walls to hold this pressure. Liquids are heavier but they are
highly non-compressible so could provide strong resistance to the volume
compression and thereby to the tensile strain.
Then for liquid hydrogen tanks we might use liquid hydrogen filled cells
within the skin of the tanks. Hydrogen is rather light compared to other
liquids at a density of only about 72 kg/m^3. This then could provide high
tensile strength at a much lower weight than typical solid wall tanks.
Kerosene and liquid oxygen would be used in the honeyombed cells for their
corresponding tanks, to keep the storage temperatures comparable. These are
heavier liquids than liquid hydrogen, approximately in the density range of
liquid water. Still these liquid filled honeycombed cells would provide much
lighter tanks than comparable solid wall tanks.
III.) Any of these methods might allow you to reduce the weight of the tanks
to be similar to that of cylindrical tanks and thus raise the payload to over
100,000 lbs. This would be for keeping the original hydrogen/LOX propellant.
However, in keeping with the analyses that show dense propellants would be
more appropriate for a SSTO vehicle I'll show that replacing the
hydrogen-fueled engines of the X-33/VentureStar with kerosene ones would allow
the X-33 to actually now become an *orbital* craft instead of just suborbital,
and the payload capacity of the VentureStar would increase to be comparable to
that proposed for Ares V.
The volume of the X-33 liquid hydrogen tanks was 29,000 gallons each and the
liquid oxygen tank, 20,000 gallons, for a total of 78,000 gallons volume for
propellant. This is 78,000gal*3.8 L/gal = 296,000 liters, 296 cubic meters.
How much mass of kerosene/LOX could we fit here if we used these as our
propellants? Typically the oxidizer to fuel ratio for kerosene/LOX engines is
in the range of 2.5 to 2.7 to 1. I'll take the O/F ratio as 2.7 to 1. The
density of kerosene is about 806 kg/m^3 and we can take the density of liquid
oxygen to be 1160 kg/m^3 when densified by subcooling:
Liquid Oxygen Propellant Densification Unit Ground Tested With a Large-Scale
Flight-Weight Tank for the X-33 Reusable Launch Vehicle.
http://www.grc.nasa.gov/WWW/RT/RT2001/5 ... omsik.html
These requirements of the propellants' total volume and densities, result in a
total propellant mass of 307,000 kg, with 83,000 kg in kerosene and 224,000 kg
in LOX. Kerosene/LOX tanks weigh typically 1/100th the propellant mass, so the
tank mass would be 3,070 kg. The current X-33 LH/LOX tanks weighed 15,200 lbs,
or 6,900 kg. So the empty weight of the X-33 is reduced from 63,000 lbs,
28,600 kg, to 28,600kg - 6,900kg + 3070kg = 24,800 kg.
How about the engines? The X-33 is to be reusable so you want to use reusable
kerosene engines. The RS-84 might be ideal when it is completed for the
full-scale VentureStar, but it turns out it's a bit too heavy for the X-33.
It would have a weight of about 15,000 lbs, 6,800 kg:
RS-84.
http://www.astronautix.com/engines/rs84.htm
about the weight of the two aerospike engines currently on the X-33:
Bringing launch costs down to earth.
"Three federally funded projects are underway to develop new rocket engines
that can make it more affordable to send payloads into orbit."
http://www.memagazine.org/backissues/me ... aunch.html
With 307,000 kg kerosene/LOX fuel and 24,800 kg dry weight, the mass ratio
would be 13.4. According to the Astronautix page, the sea level Isp of the
RS-84 would be 301 s, and the vacuum 335 s. Take the average Isp as 320s. The
total Isp for a rocket to orbit including gravity and air drag losses is
usually taken to be about 9,200 m/s. Then an average exhaust velocity of 3200
m/s and mass ratio of 13.4 would give a total delta-v of 8,300 m/s. Even if
you add on the 462 m/s additional velocity you can get for free by launching
at the equator this would not be enough for orbit.
So for the X-33 I'll look at the cases of the lighter for its thrust NK-33,
used as a trio. Note that though not designed to be a reusable engine to make,
say, 100 flights, all liquid fuel rocket engines undergo extensive static
firings during testing so the NK-33 probably could make 5 to 10 flights before
needing to be replaced.The NK-33 is almost legendary for its thrust to weight
ratio of 136. According to the Astronautix page its weight is 1,222 kg , with
a sea level Isp of 297 sec and a vacuum Isp of 331:
NK-33.
http://www.astronautix.com/engines/nk33.htm
I'll take the average Isp as 315 s. With three NK-33 engines the mass of the
X-33 becomes 21,700 kg, and the mass ratio becomes 15.15. Then with an average
Isp of 315 s, the total delta-v would be 8561 m/s and if you add on the 462
m/s additional equatorial velocity it's 9,023 m/s. Still slightly below the
delta-v typically given for orbit of 9,200 m/s.
However, it should be noted that the extra delta-v required beyond the 7,800
m/s orbital velocity is highly dependent on the vehicle and trajectory. Here's
a page that gives the gravity loss and air drag loss for some orbital rockets:
Drag: Loss in Ascent, Gain in Descent, and What It Means for Scalability.
Thursday 2008.01.10 by gravityloss
* Ariane A-44L: Gravity Loss: 1576 m/s Drag Loss: 135 m/s
* Atlas I: Gravity Loss: 1395 m/s Drag Loss: 110 m/s
* Delta 7925: Gravity Loss: 1150 m/s Drag Loss: 136 m/s
* Shuttle: Gravity Loss: 1222 m/s Drag Loss: 107 m/s
* Saturn V: Gravity Loss: 1534 m/s Drag Loss: 40 m/s (!!)
* Titan IV/Centaur: Gravity Loss: 1442 m/s Drag Loss: 156 m/s
http://gravityloss.wordpress.com/2008/0 ... alability/
Note that the gravity loss for the Delta 7925 is particularly small. As a
general principle the gravity loss can be minimized if you have a high thrust
vehicle that rapidly develops high vertical velocity sufficient to reach the
altitude for orbit. For then it can more quickly apply the horizontal thrust
required to achieve the 7,800 m/s orbital, tangential, velocity, there being
no gravity loss over the horizontal thrust portion. Note then the liftoff
thrust to liftoff weight ratio for the Delta 7925 is the relatively large 1.4;
in comparison for the Saturn V it was only 1.14. And note now also that the
X-33 with three NK-33 engines has total mass of 328,700 kg and total thrust of
4,530,600 N giving a liftoff thrust to liftoff weight ratio of 1.4. Then this
reconfigured X-33 would likely have comparable gravity drag loss as the Delta
7925. When you take into account the high thrust means it would rapidly reach
high altitude, implying the Isp would quickly get close to the vacuum Isp, the
average Isp over the trajectory is most likely closer to the 331 s vacuum Isp
than just 315 s, giving an actually higher achieved delta-v.
IV.a) It's capability of reaching orbit and possibly even with a small payload
could be increased with another additional factor. Among the questors for a
SSTO vehicle, the idea to use wings for a horizontal landing has been derided
because of the view they were just dead weight on ascent and you need to save
as much weight off the empty weight of the vehicle as possible to achieve
orbit. However, the key fact is that wings or a lifting body shape can reduce
the total delta-v for orbit by using aerodynamic lift to supply the force to
raise the vehicle to a large portion of the altitude of orbit rather than this
force being entirely supplied by the thrust of the engines. This fact is
discussed on page 4 of this report:
AIAA 2000-1045
A Multidisciplinary Performance Analysis of a Lifting-Body
Single-Stage-to-Orbit Vehicle.
Paul V. Tartabini, Roger A. Lepsch, J. J. Korte, Kathryn E. Wurster
38th Aerospace Sciences Meeting & Exhibit.
10-13 January 2000 / Reno, NV
"One feature of the VentureStar design that could
be exploited during ascent was its lifting body shape.
By flying a lilting trajectory, it was possible to significantly
decrease the amount of gravity losses, thereby
improving vehicle performance and payload capability.
Yet increasing the amount of lift during ascent generally
required flight at higher angles-of-attack and resulted
in greater stress on the vehicle structure. Accordingly,
the nominal trajectory was constrained to keep the parameter
q-_ below a 1500 psf-deg structural design limit
to ensure that the aerodynamic loads did not exceed
the structural capability of the vehicle. The effect of this
trajectory constraint on vehicle performance is shown
in Fig. 3. There was a substantial benefit associated with
using lift during ascent since flying a non-lifting trajectory
resulted in a payload penalty of over 1000 lbs compared
to the nominal case."
http://ntrs.nasa.gov/archive/nasa/casi. ... 025539.pdf
Then the gravity losses could be further reduced by flying a lifting
trajectory, which would also increase the payload capability by a small
percentage.
The trajectory I'll use to illustrate this will first be straight-line at an
angle up to some high altitude that still allows aerodynamic lift to operate.
At the end of this portion the vehicle will have some horizontal and vertical
component to its velocity. We'll have the vertical component be sufficient to
allow the vehicle to reach 100 km, altitude. The usual way to estimate this
vertical velocity is by using the relation between kinetic energy and
potential energy. It gives the speed of v = sqrt(2gh) to reach an altitude of
h meters. At 100,000 m, v is 1,400 m/s.
Now to have orbital velocity you need 7,800 m/s tangential, i.e., horizontal
velocity. If you were able to fly at a straight-line at a constant angle to
reach 7,800 m/s horizontal velocity and 1,400 m/s vertical velocity and such
that the air drag was kept at the usual low 100 to 150 m/s then you would only
need sqrt(7800^2 + 1400^2) = 7,925 m/s additional delta-v to reach orbit. Then
the total delta-v to orbit might only be in 8,100 m/s range. Note this is
significantly less than the 9,200 m/s delta-v typically needed for orbit,
including gravity and air drag.
The problem is with usual rocket propulsion to orbit not using lift the thrust
vector has to be more or less along the center-line of the rocket otherwise
the rocket would tumble. You can gimbal the engines only for a short time to
change the rocket's attitude but the engines have to be then re-directed along
the center line. However, the center line has to be more or less pointing into
the airstream, i.e., pointing in the same direction as the velocity vector, to
reduce aerodynamic stress and drag on the vehicle. But the rocket thrust
having to counter act gravity means a large portion of the thrust has to be in
the vertical component which means the thrust vector has to be nearly vertical
at least for the early part of the trip when the gross mass is high. Then the
thrust vector couldn't be along the center line of a nearly horizontally
traveling rocket at least during the early part of the trip.
However, using lift you are able to get this large upwards vertical component
for the force on the rocket to allow it to travel along this straight-line. A
problem now though is that at an altitude short of that of space, the air
density will not be enough for aerodynamic lift. Therefore we will use lift
for the first portion of the trajectory, traveling in a straight-line at an
angle. Then after that, with sufficient vertical velocity component attained
to coast to 100 km altitude, we will supply only horizontal thrust during the
second portion to reach the 7,800 m/s horizontal velocity component required
for orbital velocity.
IV.b) How much fuel could we save using a lifting straight-line portion of the
trajectory? I'll give an example calculation that illustrates the fuel savings
from using aerodynamic lift during ascent. First note that just as for
aircraft fuel savings are best at a high L/D ratio. However, the hypersonic
lift /drag ratio of the X-33/VentureStar is rather poor, only around 1.2,
barely better than the space shuttle:
AIAA-99-4162
X-33 Hypersonic Aerodynamic Characteristics.
Kelly J, Murphy, Robert J, Nowak, Richard A, Thompson, Brian R, Hollis
NASA Langley Research Center
Ramadas K. Prabhu
Lockheed Martin Engineering &Sciences Company
http://ntrs.nasa.gov/archive/nasa/casi. ... 091447.pdf
This explains the low increase in payload, about 1,000 lbs., less than .5% of
the vehicle dry weight, by using a lifting trajectory for the VentureStar.
However, some lifting body designs can have a lift/drag ratio of from 6 to 8
at hypersonic speeds:
Waverider Design.
http://www.aerospaceweb.org/design/wave ... ider.shtml
The L/D is usually optimized for a specific speed range but we can imagine
"morphing" wings that allow a good L/D ratio over a wide speed range. For
instance note on the "Waverider Design" web page the vehicles optimized for
the highest hypersonic speeds have a long, slender shape, compared to those
for the slower hypersonic speeds. Then for an orbital craft we could have
telescoping sides of the vehicle that would be extended when full of fuel at
the slower speeds, and retracted, producing a slimmer vehicle, when most of
the fuel is burned off and the vehicle is flying faster. Note that a good L/D
ratio at the highest hypersonic speeds means the vehicle will experience less
aerothermal heating on return.
Then we can imagine a second generation lifting trajectory vehicle having this
high L/D ratio over a wide speed range. So in the example I'll take the
supersonic/hypersonic L/D ratio as 5, and for lack of a another vehicle I'll
use the reconfigured kerosene-fueled X-33's thrust and weight values.
Here's the calculation for constant L/D at a constant angle θ (theta).
I'll regard the straight-line path as my X-axis and the perpendicular to this
as the Y-axis. Note this means my axes look like they are at an angle to the
usual horizontal and vertical axes, but it makes the calculation easier. Call
the thrust T, and the mass, M. Then the force component along the
straight-line path, our X-axis, is Fx = T - gMsin(θ) - D and the force
component along the Y-axis is Fy = L - gMcos(θ). We'll set L =
gMcos(θ). Then the force along the straight-line is Fx = T -
gMsin(θ) - gMcos(θ)/(L/D). As with the calculation in the horizontal
case, divide this by M to get the acceleration along this line, and integrate
to get the velocity. The result is V(t) = Ve*ln(M0/Mf) -g*tsin(θ) -
g*tcos(θ)/(L/D), with M0 the initial mass, and Mf, the mass at time t, a
la the rocket equation. If you make the angle θ (theta) be shallow, the
g*tsin(θ) term will be smaller
than the usual gravity drag loss of g*t and the (L/D) divisor will make the
cosine term smaller as well.
I'll assume the straight-line path is used for a time when the altitude is
high enough to use the vacuum Ve of 331s*9.8 m/s^2 = 3244 m/s. According to
the Astronautix page, 3 NK-33's would have a total vacuum thrust of 4,914,000
N and for an Isp of 331s, the propellant flow rate would be
4,914,000/(331x9.8) = 1,515 kg/sec. I'll use the formula: V(t) = Ve*ln(M0/Mf)
- g*tsin(θ) - g*tcos(θ)/(L/D) , to calculate the velocity along the
inclined straight-line path. There are a couple of key facts in this formula.
First note that it includes *both* the gravity and air drag. Secondly, note
that though using aerodynamic lift generates additional, large, induced drag,
this is covered by the fact that the L/D ratio includes this induced drag,
since it involves the *total* drag.
I'll take the time along the straight-line path as 100 sec. Then Mf =
328,700kg -100s*(1,515 kg/s) = 177,200 kg. After trying some examples an angle
of 30º provides a good savings over just using the usual non-lifting
trajectory. Then V(t) = 3244*ln(328,700/177,200) - 9.8*100(sin(30º) +
cos(30º)/5) = 1,345 m/s. Then the vertical component of this velocity is Vy =
1,135*sin(30º) = 672.3 m/s and the horizontal, Vx = 1,135*cos(30º) = 1,164.5
m/s.
To compare this to a usual rocket trajectory I'll calculate how much fuel
would be needed to first make a vertical trip to reach a vertical speed of
672.3 m/s subject to gravity and air drag, and then to apply horizontal thrust
to reach a 1,164.5 m/s horizontal speed.
The air drag for a usual rocket is in the range of 100 m/s to 200 m/s. I'll
take the air drag loss as 100 m/s for this vertical portion. Then the equation
for the velocity along this vertical part including the gravity loss and the
air drag loss would be V(t) = 3244*ln(M0/Mf) - 9.8*t - 100 m/s, where M0
=328,700 kg and Mf = 328,700 - t(1,515). You want to find the t so that this
velocity matches the vertical component in the inclined case of 672.3 m/s.
Plugging in different values of t, gives for t = 85 sec, V(85) = 680 m/s.
Now to find the horizontal velocity burn. Since this is horizontal there is no
gravity loss, and I'll assume this part is at very high altitude so has
negligible air drag loss. Then the velocity formula is V(t) = 3244*ln(M0/Mf).
Note in this case M0 = 328,700 - 85*1,515 = 199,925 kg, which is the total
mass left after you burned off the propellant during the vertical portion, and
so Mf = 199,925 - t*1,515. Trying different values of t gives for t = 40,
V(40) = 1,171.5 m/s.
Then doing it this way results in a total of 125 sec of fuel burn, 25 percent
higher than in the aerodynamic lift case, specifically 25s*1,515 kg/s = 37,875
kg more. Or viewed the other way, the aerodynamic lift case requires 20% less
fuel over this portion of the trip than the usual non-lift trajectory. With a
307,000 kg total fuel load, this corresponds to a 12.3% reduction in the total
fuel that would actually be needed. Or keeping the same fuel load, a factor
1/.877 = 1.14 larger dry mass could be lofted, which could be used for greater
payload. For a reconfigured X-33 dry mass of 21,700 kg, this means 3,038 kg
extra payload. Remember though this is for our imagined new X-33 lifting shape
that is able to keep a high L/D ratio of 5 at hypersonic speed, not for the
current X-33 shape which only has a hypersonic L/D of 1.2.
With the possibility of using morphing lifting body or wings with high
hypersonic L/D ratio allowing a large reduction in fuel requirements to orbit,
this may be something that could be tested by amateurs or by the "new space"
launch companies.
V.) Now for the calculation of the payload the VentureStar could carry using
kerosene/LOX engines. The propellant mass of the VentureStar was 1,929,000
lbs. compared to the X-33's 210,000 lbs., i.e., 9.2 times more. Then its
propellant tank volume would also be 9.2 times higher, and the kerosene/LOX
they could contain would also be 9.2 times higher, or to 9.2*307,000 =
2,824,400 kg.
We saw the VentureStar dry mass was 257,000 lbs, 116,818 kg, with half of this
as just the mass of the LH2/LOX tanks, at 138,000 lbs, 62,727 kg. However,
going to kerosene/LOX propellant means the tanks would only have to be 1/100th
the mass of the propellant so only 28,244 kg. Then the dry mass would be
reduced to 82,335 kg. We need kerosene/LOX engines now. I suggest the RS-84 be
completed and used for the purpose. You would need seven of them to lift the
heavier propellant load. They weigh about the same as the aerospike engines on
the current version of the VentureStar so you wouldn't gain any weight savings
here.
To calculate how much we could lift to orbit I'll take the average Isp of the
RS-84 as 320. Then if we took the payload as 125,000 kg the total liftoff mass
would be 2,824,400 + 82,335 + 125,000 = 3,031,735 kg, and the ending dry mass
would be 207,335 kg, for a mass ratio of 14.6. Then the total delta-v would be
3200ln(14.6) = 8,580 m/s. Adding on the 462 m/s equatorial speed brings this
to 9042 m/s. With the reduction in gravity drag using a lifting trajectory
this would suffice for orbit.
Bob Clark
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