Temperature and thrust of SABRE engine core

Calculating the temperature after combustion in SABRE core combustion
chamber involves calculating the energy inputs and using the specific
heat of the exhaust components.

Unfortunately, specific heats of gasses are not constant over the kind
of temperatures we're discussing. I'm using

http://www.engineeringtoolbox.com/nitrogen-d_977.html
http://www.engineeringtoolbox.com/wa...por-d_979.html

SABRE preheats its air and hydrogen inputs to 692.5K, so any energy
deriving from combustion heats the exhaust gas from that temperature to
a resulting temperature T. The simplest way to calculate T is to use the
average specific heats over the range 692.5 to T. Of course, to
determine that you already need to know T. Fortunately I do, and the
corresponding average specific heats in kJ/kgK, are

Nitrogen 1.248
Water Vapour 2.673

This can be checked by the reader, if they're so motivated, once T is
calculated below.

I'm taking Air to be, by mass, 1.29% argon, 23.14% oxygen, and the rest
nitrogen, so we'll also need the specific heat of Argon, which is
unvarying at 0.520.

We can do this calculation with any amount of air and hydrogen. For
simplicity, I'll do it with 1kg of air.

1kg of air contains 0.231 kg of Oxygen, which will burn 0.029 kg of
hydrogen to produce 0.260 kg of water vapour.

Water vapour's enthalpy of formation, into the gas phase, is 241.82
kJ/mole. The molecular weight is 18.02, so there are 14.46 moles, and
thus 3497.5 kJ released when the hydrogen and oxygen combine.

The exhaust consists of 0.7557 kg Nitrogen, 0.26 kg of water vapour, and
0.0129 kg of Argon, totaling 1.0268 kg. The average specific heat of the
three combined is obtained by taking the average specific heats of the
components, weighted according to their masses.

That is (0.7557 * 1.248 + 0.26 * 2.673 + 0.0129 * 0.52) / 1.0268

This comes to 1.6 kJ/kgK. Divide the 3497.5 kJ by 1.6kJ, and by the
total mass of the exhaust, and we get a temperature rise of 2125
degrees. Add that to the 692.5 we started with, and the total is 2817.

So the combustion chamber temperature for the SABRE core is 2817K. This
is perhaps surprisingly high, but it must be remembered that the input
air hydrogen were already very hot to begin with.

This value is of course the T used in calculating the average specific
heats.

It should be noted that the specific heat tables contained a caveat
about dissociation 1500K and we're well past that. The combustion
product will not really be just combination of nitrogen and steam, but
will contain a variety of other substances which will reduce the real
temperature.

That said, we can go a step further, and seek to calculate the exhaust
velocity, and thus the thrust.

http://en.wikipedia.org/wiki/De_lava...t_gas_velocity

The value of k, the isoentropic expansion factor, is a problem, but for
this purpose let's take it as 1.4.

Looking at the air breathing ascent data in

http://www.reactionengines.co.uk/dow...ory_output.xls

at 600 seconds after launch, there is a time when almost no air is
bypassing the core. At that point the ambient pressure is about 2207 Pa,
which is about 1/45 bar. The SABRE chamber pressure is 105 bar.

The average molecular weight of the exaust (i.e. the component molecular
weights weighted by the respective masses) is 24.64.

We now have all the numbers to fit into the exhaust velocity equation.
The result comes to 2462 m/s.

Going back to the spread sheet, and we see that the air passed into the
core in kg/s is 380.198 - 14.40845, or about 363, per nacelle The thrust
is then 893706N per nacelle, or about 1.79 MN. This somewhat shy of the
stated 2.09 MN, but it's in the right area.

But given the rate of air flow, we can calculate the amount of hydrogen
consumed, and the result is 21kg/sec, which is much less than the
61kg/sec given in the spread sheet as the rate of mass loss. I'll leave
the ramifications of that, and the high combustion temperature, until later.

Sylvia.

On 15/05/2012 12:21 AM, Sylvia Else wrote:

Going back to the spread sheet, and we see that the air passed into the
core in kg/s is 380.198 - 14.40845, or about 363, per nacelle The thrust
is then 893706N per nacelle, or about 1.79 MN. This somewhat shy of the
stated 2.09 MN, but it's in the right area.

Oops - for every kg into the nacelle, there's 1.0268 kg of exhaust, so
the thrust is 2 * 363 * 1.0268 * 2462 N, or about 1.84MN.

Sylvia.