[AR] Re: thinking big once more
- From: Norman Yarvin <yarvin@xxxxxxxxxxxx>
- To: arocket@xxxxxxxxxxxxx
- Date: Fri, 30 Sep 2016 15:12:28 -0400
On Thu, Sep 29, 2016 at 11:30:09PM -0400, Henry Spencer wrote:
Uh, remember that thrust is thrust coefficient, times throat area, *times*
*chamber* *pressure*. So that's a 55% increase in thrust, not 3.5%.
The traditional reason for wanting high pressure in a first-stage engine
is not higher Isp, but getting more thrust out of an engine whose size is
constrained by packaging issues, e.g. the size of the vehicle base. This
is why the SSME's pressure was so high -- three of them had to fit in the
lee of a dense, compact orbiter during reentry. Similarly, the RD-170 was
constrained to fit within the maximum allowable cross-section of a Russian
railroad cargo, so a fully-assembled Zenit first stage could be moved by rail
without restrictions.
Historically, designs for big rockets tend to gain width faster than they gain
height, simply because they need the extra base area. Elon's megarocket would
have to be a good deal shorter and fatter without that alarmingly high chamber
pressure.
Well, that effect is a bit subtler than might be thought. When
packing engines together, the limit is not the size of the throat but
the size of the nozzle; and though for higher-pressure engines the
throat size is proportionately smaller, the expansion ratio is larger.
The advantage for higher-pressure engines is because although the
expansion ratio is larger, it's not larger in exact proportion to the
pressure ratio difference, but rather somewhat less than that. That's
because the exhaust cools as it expands, and cooler exhaust takes up
less volume, and higher-pressure engines expand the gas more. (They
start from about the same chamber temperature, since that temperature
is mainly governed by how much energy the fuel/oxidizer combo has.)
To put some numbers to this, the rule for adiabatic expansion (which
is pretty much what one gets in a rocket nozzle) is
P*V^gamma = constant.
So for expansion over a pressure ratio of P_exhaust/P_throat, the
volume change is the gamma'th root of that. So say you're comparing
two engines, one of which is three times the chamber pressure of the
other. Take the two chamber temperatures to be the same, and the exit
pressure to be the same (since that's mainly governed by the ambient
pressure that the engine has to work in). And let's use the value of
gamma for a diatomic gas (7/5). The 3x pressure ratio translates into
a 3^(5/7) = 2.19 volume ratio. So as a first approximation the
higher-pressure engine has an exit nozzle area of 2.19/3 = 73% of the
lower-pressure engine's. A noticeable advantage, but you _did_ have
to work for it.
(Translating volume changes into cross-sectional area changes, as
above, is permissible if you assume the same flow speed. Here the
flow speed is a bit different, since the higher-pressure engine has a
bit higher exhaust velocity, but not all that much different.
Multiply by the ratio of Isps to give the higher-pressure engine full
credit. But against that, you're probably not going to get exactly
diatomic gamma, but rather something worse from the point of view of
the higher-pressure engine.)
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