The key phrase here appears to be "for launcher-like values"
For the sort of launchers that can carry a man, m_I/m_f is greater than e, and
log curve flattens out to such an extent that given a fixed m_f (a reasonable
approximation of payload), exponentially greater quantities of fuel are
necessary for linear increases in delta v.
It is precisely this flat wimpyness of the log curve that makes improvement in
Isp (and staging) so attractive for these launchers.
People like Zach and I don't build launchers that can carry men though, we
build launchers that men can carry.
At our scales, minimum gauge effects driven by aerodynamic loads, fixed sizes
of many components, and cube square effects cause our m_i/m_f to be rather low.
For these launchers, the log curve is much steeper, and unlike in the previous
case a small increase in propellant yields great results (1). In this case
optimization of mass fraction is the better choice (2).
Pedant's note (1): in both cases it takes an exponential increase in mass
fraction to to cause a linear increase in delta v. It's just that in the second
case we are looking at a different and unloved part of the function.
Pedant's note 2: at the scale of man-carried launchers, aerodynamics plays a
role larger than either part of the Tsiolkovsky's equation.
-------- Original message --------
From: Zachary Martinez <znm3m8@xxxxxxx>
Date: 09/05/2016 15:05 (GMT-08:00)
To: arocket@xxxxxxxxxxxxx
Subject: [AR] Re: Issues with operating at low chamber pressure
It really does depend on the vehicle almost exclusively. You will have to
explore the options and pick the one with the most delta v because it's not
like you are ever given a choice between 10% Isp and 10% mass fraction in a
design. It also really depends on the mass fraction. For example at the scale I
am working at ~15-30% propellant mass fraction, a relatively small reduction in
dry mass can lead to almost a doubling of propellant while this same choice
(400psi chamber pressure vs 200psi chamber pressure) only leads to about a 20%
reduction in Isp. The logic obviously doesn't apply to orbital launch vehicles
so the advice that Isp is better than mass fraction or the other way around is
kinda irrelevant because these decisions 10% this vs the same 10% that rarely
come up in design. Just my 2 cents
Zach Martinez
Sent from my iPhone
On Sep 5, 2016, at 16:37, Norman Yarvin <yarvin@xxxxxxxxxxxx> wrote:
On Mon, Sep 05, 2016 at 07:10:08AM -0400, William Claybaugh wrote:
Norm:
A logarithm is an exponential function.
Not when you're talking about an "exponential effect", it isn't. An
exponential effect is a big effect that grows strongly as you increase
the factor in question; a logarithmic effect is a wimpy one that gets
ever-wimpier as the factor in question increases. In particular,
here, a 10% increase in Isp buys you a 10% increase in delta-v,
whereas a 10% increase in m_initial/m_final buys you less, at least
for launcher-like values of that ratio. (If, say, it starts out as
10, a 10% increase buys you about 4% more delta-v; or if it starts out
as 5, a 10% increase buys you 5.6% more delta-v.)
