THERMAL EFFECTS OF FLUIDS IN MOTION. 
323 
for periods of from 3 or 4 minutes up to nearly half an hour after the pressure had 
become sensibly uniform, depend on a complication of circumstances, which appear 
to consist of (1) the change of cooling effect due to the instantaneous change of 
pressure ; (2) a heating or cooling effect produced instantaneously by compression 
or expansion in all the air flowing towards and entering the plug, and conveyed 
through the plug to the issuing-stream ; and (3) heat or cold communicated by con- 
tact from the air on the high-pressure side, to the metals and boxwood, and con- 
ducted through them to the issuing stream. 
The first of these causes may be expected to influence the issuing stream instanta- 
neously on any change in the stopcock; and after fluctuations from other sources have 
ceased, it must leave a permanent effect in those cases in which the stopcock is per- 
manently changed. But after a certain interval the reverse agency of the second 
cause, much more considerable in amount, will begin to affect the issuing stream, 
will soon preponderate over the first, and (always on the supposition that this con- 
vection is uninfluenced by conduction of any of the materials) will affect it with all 
the variations, undiminished in amount, which the air entering the plug experiences, 
but behind time by a constant interval equal to the time occupied by as much air as 
is equal in thermal capacity to the cotton of the plug, in passing through the appa- 
ratus*; this, in the experiments with the stopcock shut, would be very exactly a 
* To prove this, we have only to investigate the convection of heat 
through a prismatic solid of porous material, when a fluid entering it 
with a varying temperature is forced through it in a continuous and 
uniform stream. Let A B be the porous body, of length a and trans- 
verse section S ; and let a fluid be pressed continuously through it in 
the direction from A to B, the temperature of this fluid as it enters 
at A being an arbitrary function F if) of the time. Then if v be the 
common temperature of the porous body and fluid passing through it, at a distance x from the end A, we have 
dv drv 6 dv ... 
dt ~~ ^ dx ‘ S dx’ 
if k be the conducting power of the porous solid for heat (the solid surrounding it being supposed to be an 
infinitely bad conductor, or the circumstances to be otherwise arranged, as is practicable in a variety of ways, 
so that there may be no lateral conduction of heat), a the thermal capacity of unity of its bulk, and 0 the 
thermal capacity of as much of the fluid as passes in the unit time. Now if, as is probably the case in the 
actual circumstances, conduction through the porous solid itself is insensible in its influence as compared with 
the convection of the fluid, this equation will become approximately 
dv Q dv 
<J ~dt~~^Tx 
which, in fact, expresses rigorously the effect of the second cause mentioned in the text if alone operative. 
If F denote any arbitrary function, and if 0 be supposed to be constant, the general integral of this equation is — 
(3) 
and if the arbitrary function be chosen to express by F (t) the given variation of temperature where the fluid 
enters the porous body, we have the particular solution of the proposed problem. We infer from it that, at 
any distance x in the porous body from the entrance, the temperature will follow the same law and extent of 
2 T 2 
