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Hi Steve & All, Jan 24, =
2015
Haldane was just putting the gullible reader on because angels, =
being spirits, would be weightless (or massless if you wish) so the =
wings would be used variously for ornament, supporting harps, modesty or =
maneuvering but not for lift.=20
But as you say understanding flight can be complicated, as =
illustrated in a poem (Amer. Sci. 53(1): p. 78, 1965.
Philosophers gravely expound
Metaphysical concepts profound.
While they argue all night
On the meaning of "Flight,"
The Wrights get a plane off the ground.
I leave it to evolution to iterate a useful balance between wing =
dimensions and body weight and approached the question of flight =
duration by calculating rate of fuel consumption (g tallow to keep one =
gram airborne for one second =3D k) and assumed no additional energy for =
forward motion; i.e. fighting gravity with a suitable angle of attack =
would provide both at little additional cost. Then used differential =
equations to allow for decrease in weight kept aloft as a function of =
time and integration to calculate the time required to burn a specified =
amount of tallow. I used 3 g initial and 1 g final (2 g tallow used) but =
any initial and final weights could be substituted by replacing 3 and 1 =
(2 equations up) with the other numbers.
The full text, with the two typos corrected as noted and with spurious =
carriage returns thrown in is as follows--
START OF PASTE\\\\\\\\\\\\\\\\\\\\\\\\\
Hi Chris & All, Jan 7, 2002
Your post about 70 hours of non-stop flight got me wondering about =
energy
equivalents of fat reserves so I cobbled together an estimate of the =
upper limit.
This is unfamiliar territory so don't bet your life savings that this =
upper imit
is correct. Assumptions are 100% efficiency and zero energy consumed by =
life
processes; obviously both wrong but I don't know probable values of =
efficiency
and metabolic consumption. No energy is reserved for air resistance or =
forward
motion, the idea being that forward motion is a practical way to climb =
against
gravity and maintain height.
GIVEN, ASSUMED:
Acceleration due to gravity (a); a =3D 980.665 cm/sec^2,
One gram.cm =3D 2.3427 x 10^-8 kg.cal,
Tallow heat of combustion =3D 9.5 kg.cal/g
Model bird weighs 3 grams, 2 grams being tallow.
GRAMS TALLOW TO KEEP ONE GRAM WEIGHT AIRBORNE FOR ONE SECOND:
A body at uniform acceleration a, will move in time t, a distance =
equal to s,
i.e.
s =3D 1/2 at^2
So instantaneous ds/dt, to overcome gravity, is
ds/dt=3D at cm/sec =3D 980.665 cm/sec
So the work (energy) required to keep one gram weight airborne for one =
second is
980.665 gram.cm which is equal to
980.665 x 2.3427 x 10^-8 kg.cal =3D 2.297 x 10^-5 kg.cal .
And the grams tallow required to keep one gram weight airborne for one=20
second,
call this k, is
2.297 x 10^-5 kg.cal/ 9.5 kg.cal/gram=3D 2.4183 x 10^-6 grams =
tallow/grams
weight.seconds
TIME THAT MODEL BIRD CAN REMAIN AIRBORNE; 3 GRAMS WEIGHT INITIAL, 1 GRAM =
WEIGHT FINAL, 2 GRAMS TALLOW USED:
So if w is instantaneous bird weight, the loss in weight per second will =
be
dw/dt=3D kw and conversely
dt/dw=3D 1/kw and dt=3D 1/k x dw/w
and T, the seconds to burn 2 g tallow, is the integral (int) of 1/k x =
dw/w, i.e.
T=3D 1/k x int(dw/w)
=3D 1/k x (ln w + c) and for initial and final values of 3 and 1 grams =
is
=3D1/k x [(ln 3 + c) - (ln 1 + c)] COMMENT =3D>+
=3D 1.0986/k
=3D 4.5428 X 10^5 seconds
=3D ~126 hours
Note that (ln 30 - ln 10) is also equal to 1.0986, so as long as =
proportions of
initial and final weight remain the same, the same upper limit for =
airborne=20
time will apply. COMMENT ln 20 THREE LINES UP CHANGED TO ln 10
This also explains why it is so difficult to work off that extra =
serving=20
of rich gravy.
Yours truly, Dave Webster, Kentville
END OF PASTE\\\\\\\\\\\\\\\
----- Original Message -----=20
From: Stephen Shaw=20
To: naturens@chebucto.ns.ca=20
Sent: Saturday, January 24, 2015 6:46 PM
Subject: Re: [NatureNS] Semipalmated Sandpiper migratory route
Hi Dave et al,=20
This is perhaps splitting hairs because the Sandpiper (28g) and =
Blackpoll Warbler (European, probably ~10-11g) are both quite small =
birds and therefore may be nearly equivalent, but any calculation needs =
to be framed in terms of what mass has to be kept aloft. It would take =
far more fuel consumption per hour to keep a heavy crow (450g) in the =
air compared to that for a small light warbler, if they have similarly =
efficient lift-generating wings (doubtful). It is usually framed in =
terms of body mass being proportional to the cube of the average linear =
dimension (LD^3), while lift generation is proportional to the square of =
the LD (LD^2) =97 so you need proportionally more wing lift area as the =
body mass increases until it finally becomes infeasible to fly at all =
=97 from memory the Great Bustard was reckoned to be the most massive =
bird that could still manage to fly. =20
I may have used this example before: biologist J.B.S. Haldane (an =
atheist) once penned a mischievous essay on the impossibility of the =
existence of angels, at least in their depiction in medieval =
manuscripts, because to power wings that size (a big LD^2) would require =
a breastbone extending down to the ground to carry the enormous muscles =
required to flap them (humungous LD^3), not so illustrated in the =
manuscripts.
Albatrosses don=92t count much in this because an engineer C. =
Pennycuick (sp?) in the 1960s calculated that one species he looked at =
mostly used the updraft from wave crests to glide along on a sinuous =
path using lift energy derived from that, somewhat analogous to the =
larger scale thermals used by some migrating raptors and storks.=20
Understanding flight is complicated.
Steve (Hfx)=20
=20
On Jan 24, 2015, at 3:33 PM, David & Alison Webster =
<dwebster@glinx.com> wrote:
Hi Angus & All, Jan 24, 2015
In an e-mail of Jan 7, 2002 I estimated maximum flight duration =
based on energy content of tallow and with initial conditions of 3 g =
total weight of which 2 g is tallow to be 126 hours (what I call weight =
would usually now be called mass). In scanning this quickly I noticed =
two typos but whether these introduced error I don't know.
At that time Richard dug out an example which showed this simple =
model to considerably underestimate actual endurance--
START OF PASTE\\\\\\\\\\\\\\\
Hi Richard, Elizabeth & All, Jan 9, 2002
<SNIP>
> -- "A typical Blackpoll Warbler at the end of its
> breeding season weighs about 11 grams, equivalent to the weight of =
4
> pennies. In preparing for its transatlantic trek, it may =
accumulate enough
> fat reserves to increas