[AR] Re: Orions and PDEs (was Re: More MAX delays.)
- From: Norman Yarvin <yarvin@xxxxxxxxxxxx>
- To: arocket@xxxxxxxxxxxxx
- Date: Tue, 28 Jan 2020 15:16:31 -0500
On Mon, Jan 27, 2020 at 03:47:56PM -0600, Jim Davis wrote:
Looking up various cycles, I see the ideal Brayton cycle quoted as
having the usual efficiency formula (T2-T1)/T2, the maximum for any
thermodynamic cycle.
That's the efficiency for the Carnot cycle
Okay, I see what I missed. The formula can be used for the Brayton
cycle too, and T1 is the same (the cold temperature), but T2 is
different: for the Brayton cycle, instead of being the hottest
temperature, it's the temperature to which the air is compressed
(before fuel is injected and burned). That's with reference to a jet
engine, which is what the Brayton cycle better describes; there's no
identifiable T2 in a rocket engine, at least not as a physical
temperature measured at a given location. (Even the jet engine leaves
out one side of the theoretical Brayton cycle diagram, the side which
recycles the used air; rockets leave out the compression-of-intake-air
part too.)
[...]; the ideal Brayton cycle efficiency is
eff = 1- 1/PR^(Y-1/Y)
where
PR = pressure ratio
Y = ratio of specific heats
That formula needs a pair of parentheses added, I think:
eff = 1 - 1/PR^((Y-1)/Y)
and then it indeed applies to rocketry, where it's the usual
expression for the nozzle efficiency.
The ideal Humphrey cycle efficiency is
eff = 1- 1/PR^(Y-1)
For a given amount of heat addition the constant volume process
entails a smaller entropy rise than the constant pressure
process. This translates to a higher pressure after expansion in a
nozzle of a given length or the ability to use a longer nozzle to
expand to the same pressure.
The thing is, if the pressure were higher after expansion, wouldn't
that have to mean it was higher before expansion, too? And if you
pumped up your conventional engine to that same higher pressure,
wouldn't it have the same efficiency?
In other words, I'm still suspecting imprecision here in talking about
the pressure ratio. In the Brayton cycle it is a single quantity: the
pressure ratio by which the intake air is compressed is taken to be
the same as the pressure ratio by which the exhaust is expanded.
(Real jet engines don't have to obey this rule, but that's how the
Brayton cycle is defined.) But in the Humphrey cycle, the two
pressure ratios are different, so there's a question of which one that
formula uses.
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