Proposals for air breathing hypersonic craft. II

(I.) In this thread on sci.astro I argued for using the boundary layer air at zero
relative velocity to the craft to eliminate the problem of the ram
drag created by ingesting and slowing down the surrounding air for
combustion:

From: Robert Clark
Date: Thurs, May 6 2004 9:12 pm
Email: rgregorycl...@yahoo.com (Robert Clark)
Groups: sci.astro, sci.space.policy, sci.physics, sci.mech.fluids,
sci.engr.mech
Subject: Proposals for air breathing hypersonic craft. I
http://groups.google.com/group/sci.astr ... abf58f307/

Another possibility would be to accelerate the fuel up to the same
velocity of the craft then eject this into the air flow. Call it
Accelerated Fuel Combustion (AFC). Then you would not have to slow down
the air inflow at all for combustion. The problem then would be to be
able to accelerate the fuel up to the maximum velocity of the craft to
reach orbit, about 7.5 to 8 km/sec.
DARPA and Johns Hopkins' Applied Physics Laboratory are already
investigating a partial version of this idea at lower velocities:

New Powerplant Key To Missile Demonstrator
By Stanley W. Kandebo/Aviation Week & Space Technology
September 3, 2002
"APL's dual combustion ramjet is yet another way to obtain hypersonic
speeds. In this powerplant, supersonic air ingested through one inlet
is slowed to subsonic speeds, mixed with a conventional hydrocarbon
fuel in a fuel-rich environment and ignited, as in a ramjet. To break
through the ramjet's operating speed limitations, though, the expanding
combustion products are then mixed with supersonic air entering through
a second inlet and are more completely burned in a supersonic
combustor. According to APL researchers, the DCR has an operating
threshold of about Mach 3, and a maximum operating speed of about Mach
6.5."
http://www.aviationnow.com/avnow/news/c ... rj0903.xml

I shall argue that the method of not slowing the incoming air at all
but accelerating the fuel up to the relative air speed will result in a
marked improvement in fuel efficiency. Specifically, the exponential
increases in fuel according to velocity given by the rocket equation
will no longer be needed.
Let X be the mass of the rocket with fuel, v the rocket velocity, r
the ratio of air to fuel in mass, and e the nominal exhaust velocity of
combusting still air with still fuel.
For this method to work I will assume that the force produced by the
combustion of the air with the fuel can be fully communicated to the
craft. To derive the thrust equation take the craft including the fuel
to be a closed system and the air to be outside the system and take the
rest frame to be the Earth, or likewise the still air.
Now if we did not combust the ejected fuel with the air then by
momentum conservation we would have:

(X + dX)(v + dv) + 0(-dX) = Xv

In the first term on the left we add dX to X because dX is negative
since the mass is decreasing as fuel is consumed. So the first term
represents the mass of the rocket less the ejected fuel times the
increased velocity of the rocket. In the second term we are multiplying
the velocity with respect to ground of the ejected fuel times the mass
of the fuel ejected. Since we are ejecting the fuel at a speed to stay
at zero relative velocity to air, i.e., to the ground, this velocity
here is 0. The negative sign in front of dX again is because dX is
negative so -dX is the positive mass of the fuel.
This equation expanded out is Xv + Xdv+ vdX + dXdv = Xv. So the change
in momentum is Xdv + vdX + dXdv = 0, and the rate of change of momentum
is:

0 = Xdv/dt + vdX/dt + (dXdv)/dt = Xdv/dt + vdX/dt , because the term
with two differentials dXdv vanishes as dt ---> 0.

Now when we do combust the fuel with the air, then the rate of change
in momentum of the system is the force on the craft due to the
combustion of the air and fuel. This is the thrust produced by this
combustion which equals mass flow rate, air + fuel, times the nominal
exhaust velocity of the combustion of still air and still fuel:

Xdv/dt + vdX/dt = -ed(rX+X)/dt = -e(r+1)dX/dt , where the minus sign
comes from dX being negative.

Let c = e(r+1). Then the equation becomes Xdv/dt + (c + v)dX/dt = 0,
which is equivalent to:

d[(c +v)X]/dt = 0

This has solution (c + v)X = constant. Let X0 be the initial mass and
v0 the initial speed of the rocket. Then the solution is (c +v)X = (c +
v0)X0.
Therefore X0/X = (c + v)/(c + v0), i.e., the mass ratio of the fully
fueled rocket to the empty rocket is just a linear function of ending
velocity.

(II.) There are a couple of problems with this idea. First, you have
to accelerate the fuel up to the velocity of the craft which can be up
to 8 km/sec to reach orbit. Secondly, we shall see communicating the
full thrust of the combustion to the craft is no easy matter. Actually
I think probably only some portion of this thrust will wind up being
applied to the craft, call it a fraction given by f. Then the
calculation will carry through similarly to as before so the final
equation will be (fc + v)X = (fc + v0)X0.
For accelerating the fuel, the DARPA/APL method is to combust a fuel
rich mixture first using subsonic combustion which results in
uncombusted fuel in the exhaust moving at the exhaust speed. However,
even if you used the highest exhaust speed for chemical rockets of 4500
m/s this still would not be fast enough.
So my suggested method is to use the idea of using high temperature
atomic hydrogen stored on board:

From: Robert Clark
Date: Tues, Jun 13 2006 3:44 am
Email: "Robert Clark" <rgregorycl...@yahoo.com>
Groups: sci.astro, sci.space.policy, sci.physics, sci.chem, sci.energy
Subject: Storing atomic hydrogen propellant at high temperature.
http://groups.google.com/group/sci.astr ... 2c95826eee

The Accelerated Fuel Combustion method since the fuel requirements are
so low would be ideal for the stored atomic hydrogen since the high
temperature, high pressure tanks to hold the atomic hydrogen could be
minimized in size. Atomic hydrogen propulsion can also have ISP up to
1600 sec, which means the exhaust velocity can be up to about 16,000
m/s.
How much fuel would be required? Hydrogen/LOX engines can have exhaust
speeds of 4500 m/s. However, this is by using liquid oxygen oxidizer
which results in high flame temperatures and using high pressures in
the combustion chamber. Using ambient oxygen from air that also
contains 80% nitrogen that does not contribute to the combustion and
using incoming air that is not compressed as with typical (sc)ramjet
methods would result in significant reduction in performance. Let's
suppose the exhaust speed for still air, fuel is 2000 m/s. At
stoichiometric mixture ratio of 8 to 1 oxygen to hydrogen and 5 times
the total mass of air as the mass of oxygen this gives r = 8*5 = 40 and
c = e(r + 1) = 2000*41 = 82,000 m/s.
If you wanted to reach 8000 m/s from 0 initial velocity the equation
would be X0/X = (c + v)/(c + v0) = (82,000 + 8,000)/82,000 = 1.098.
This means less than 10% of the empty rocket mass would have to be
carried as fuel. This compares to typical rocket fuel loads that are
several times larger than the mass of the empty rocket.

(III.) However, the key problem is how to communicate the thrust of the
combustion, which is taking place in still air, to the craft. The
problem is the fuel is being combusted in still air while the rocket is
moving away at up to 8000 m/s. So even if the combustion products are
moving at 4500 m/s they still can not catch up to the craft to impart
momentum to the vehicle.
A couple of proposed solutions. Both of these though require the
combustion to be pulsed. Pulsed combustion for hypersonic craft is
being researched with Pulsed Detonation Engine (PDE) propulsion for
craft:

PDE Faq.
http://www.innssi.com/pde01.htm

The idea is to carry out detonations many times a second to result in
smooth propulsion. The key distinction of PDE propulsion is that the
combustion is through detonations rather than through simple burning
(deflagration.) A benefit of this that the combustion can take place
orders of magnitude faster than with simple burning.
The technical problems with PDE though still have not been worked out.
I believe though am far from certain that the Accelerated Fuel
Combustion method will not require PDE to work.
(a.) First method to communicate thrust to rocket: use the accelerated
fuel to propel a plate rearward to be at the same speed of the fuel/air
mixture and at the front of it. When the fuel air is ignited since the
plate is still with regards to the fuel/air it receives the momentum of
the combustion products moving forwards. You see here it can only
receive a portion of the thrust produced since it does not receive the
momentum of the combustion products moving rearward. At best it could
receive 50% of the thrust produced.
For this to work this "pusher plate" if you will needs to be of a
light material, lighter in fact than the mass of the fuel/air
combusted. Then the momentum imparted to it will give it a velocity
higher than that of the exhaust gases to be a speed at least as high as
the speed of the rocket moving forward. Once it has received the
greatest momentum boost from the expanding combustion gases, it is
allowed to catch to the walls of the rocket or to a stop bumper towards
the front thereby transferring its momentum to the rocket.
(b.) Second method to communicate thrust to rocket: use in fact not
only a pusher plate but a full combustion chamber moving rearward at
the same velocity of the fuel/air and containing the fuel/air, with its
nozzle pointing rearward. As with the pusher plate it would need to be
made of a light material to wind up at a higher velocity moving forward
than the exhaust gases. To make it lighter you might only want it to
consist of a front plate and a rear nozzle connected by strong thin
rods to keep the volume of the chamber constant as the combuston gases
expand. The walls of the rocket would then serve as the walls of the
combustion chamber. This method has the advantage that more of the
thrust produced will be transmitted to the rocket.

Bob Clark