Encyclopaedia Britannica > Volume 15, PLA-RAM

ill Airl:
N Motw
P N E U M
feet and O the area or fe&ion of the orifice, expreffed
in fuperficial or ^uare feet; and let the natural denfity
of the air be D. „ . , ^ . , •
Sin'-e the quantity of aerial matter contained in a
vefiel depends on the capacity of the veffel and the
denfity of the air jointly, we may exprefs the air which
would fill this veffel by the fymbol CD when the air
is in its ordinary ftate, and by C J when it has the
denfity /. In order to obtain the rate at which it fills,
we muft take the fluxion of this quantity C This
is C j ; for C is a conftant quantity, and i is a variable
or flowing quantity. r . . , ,
But we alfo obtain the rate of influx by our know¬
ledge of the velocity, and the area of the orifice, and
the"denfity. The velocity is V, or B^/H, at the firft
inftant; and when the air in the vefiel has acquired the
denfity <?, that is, at the end of the time /, the velocity
is vh^/ °r
D
a/V-J
VD
; X
—i
A T I C S.
the time in feconds of completely filling it will be
8" 1152"
Ti*666’ or 666
of a fquare inch, that is, if its fide is to
or 1,7297
or Bv^hl
V xj
The rate of influx therefore (which may be con¬
ceived as meafured by the little mafs of air which ^ ill
enter during the time / with tills velocity) will be
Bty/HOPy'D—f t , or B-v/HOv'D'/D—mul-
7d ~
tiplying the velocity by the orifice and by the denfity..
Here then we have two values of the rate of influx.
By ftating them as equal we have a fluxionary equation,
from which we may obtain the fluents, that is, the
time t in feconds neceffary for bringing the air in the
veffel to the denfity or the denfity J1 which will be
produced at the end of any time t. We have the equa¬
tion Sy'HOy'DV'D—Hence we derive
C >
Of this the fluent is
f “Bv'HO^D
C .
^=4v7HO\/D^' ^ +in which A is a con¬
ditional conilant quantity. The condition which de¬
termines it is, that t muft be nothing when <> is nothing,
that is, when a/D—^zrA^D; for this is evidently
the cafe at the beginning of the nlotion. Hence it
follows, that the conftant quantity is VD, and the
complete fluent, fuited to the cafe, is
C
miuftr ;d
by'ex
iiniplt n
iiflfiib 5.
qVHOv'D x v'D — a/D
The motion ce.afes when the air in the veffel has ac¬
quired the denfity of the external air; that is, when
J=D, or when t = X ^D, = •
1 herefore the time of completely filling the veflel is.
4VHO*
Let us illuftrate this by an example in numbers.
Suppofing then that air is 840 times lighter than
water, and the height of the homogeneous atmofphere
27720 feet, we have 4v'H = 666. Let us further
iuppofe the veffel to contain 8 cubic feet, which is
nearly a wine hogfhead, and that the hole by which the
air of the ordinary denfity, which we ftiall make = i,
enters is, an inch fquare, or of a fquare fpot. Then
Platt
ccccvr;
If the hole is only
of an
inch, the time of completely filling the hogfhead will
be 1737 very neatly, or fomething lefs than three
minutes.
If we make the experiment with a hole cut in a thin
plate, we fhall find the time greater nearly in the pro¬
portion of 63 to 100, for reafons obvious to all who
have ftudied Irydraulics. In like manner we can tell
the time neceffary for bringing the air in the velfel to £
of its ordinary denfity. The only variable part of our
fluent is the coefficient — or a/737. Let be
— I, then a/ 1—^=^1 = 4, and 1 — 1—1.— 4- ;
and the time is 864-" very nearly when the hole is
an inch wide.
Let us now fuppofe that the air in the veffel ABCD
(fig. 64.) is compreffed by a weight a£ting on the
cover AD, which is moveable down the veffel, and is ^
thus expelled into the external air. city 0f
The immediate effect of this external preffure is tow;th the
comprefs the air and give it another denfity. The additional
denfity D of the external air correfponds to its preffure iniPuJ^ °£
P. Let the additional preffure on the cover of the joying
veffel be p, and the denfity of the air in the veffel down tha
be cl. We fhall have P : —D : d; and therefore veffcU
d—n
/» = PX—g—. Then, becaufe the preffure which ex¬
pels the air is the difference between the force which
compreffes the air in the veffel and the force which
compreffes the external air, the expelling force is
p. And becaufe the quantities of motion are a& the
forces which finrilarly produce them, we fhall have
d-T>
where M and m exprefs
the quantities of matter expelled, V expreffes the velo¬
city with which air rufhes into a void, and v expreffes
the velocity fought. But becaufe the quantities of
aerial matter which iffue from the fame orifice in a mo¬
ment are as the denfities and velocities jointly, we fhall
have MV : mv — DVV : dvv, zrDV1 : dvz. There-
d—D
fore P : p—^—irDV1 : dv. Hence we deduce
P:PX-j)—= MV
*=vv/-
D_
D!
We may have another expreffion of the velocity with¬
out confidering the denfity. We had D: P-fpzr D : dc
therefore </=? + and d—D=D,
__D XP + p—DP
and
_P + ^-P
_ P
P+/’
d—D __D X Y+p—DP
d D X P+p
therefore-
P-pp P-f-p ^ v p
which is a very Ample and convenient expreffion. ^
Hitherto we have confidered the motion of air as The efftifl
produced by its weight only. Let us now confider the of the air’s-
cffedl of its elafticity. elafticity
Let ABCD (fig. 64.) be a veffel containing air 0fconluicre(I»
any denfity D. This air is in a ftate of compreffion ;
and if* the comprcffing force be removed, it will expand,
and its elafticity will diminifh along with its denfity.
Its