[AR] Re: Mass fraction vs Isp
- From: Henry Spencer <hspencer@xxxxxxxxxxxxx>
- To: arocket <arocket@xxxxxxxxxxxxx>
- Date: Wed, 27 Jan 2021 00:15:12 -0500 (EST)
On Tue, 26 Jan 2021, Dr Edward Jones wrote:
In flights to orbit, won't a one percent better mass fraction be more
effective than a one percent improvement in Isp?
A meaningful answer requires defining what you mean by "effective".
If you measure effectiveness by delta-V, a 1% boost in Isp multiplies
delta-V by 1.01. Whereas a 1% boost to a mass ratio r multiplies delta-V
by (1 + ln(1.01)/ln(r)), and since ln(1.01) is pretty close to 0.01, this
is better only if ln(r) was pretty poor to begin with. If ln(r) was some
nice number like 2 or 3, then you get half to a third as much improvement.
And isn't one a linear improvement, while the other is exponential?
No, one is linear and the other is *logarithmic* -- the *inverse* of
exponential.
If "effective" refers to available payload for a constant delta-V, that
gets more complicated but the bottom line tends to be the same -- other
things being equal, improvements come more easily with Isp increases.
However, other things aren't equal. In particular, big increases in Isp
have a bad habit of coming with mass-ratio penalties, e.g. the bigger and
heavier tanks and poorer engine T/W of LH2, which make the net effect on
the bottom line harder to predict. Also, the delta-V required for flights
to orbit is not in fact constant -- it too is a (weak) function of Isp,
penalizing high Isp a little (on the steep end of the curve), which
complicates things further.
(My take? I'd say that in recent decades there has been too much
obsession with high Isp, and not enough attention to high mass ratio in
general and high engine T/W in particular, and I think the latter need to
come back into style. It's gotten bad enough that True Believers in
parametric models have been heard to claim that mass numbers for real
flown stages must be wrong because their mass ratios couldn't possibly be
that high.)
Henry
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