[Footnote: A paper lately read before the Institution of Mechanical
Engineers.]
By ARTHUR ACHARD, of Geneva.
But allowing that the figure of 22 H. P., assumed for this power (the
result in calculating the work with compressed air being 19 H. P.) may be
somewhat incorrect, it is unlikely that this error can be so large that its
correction could reduce the efficiency below 80 per cent. Messrs. Sautter
and Lemonnier, who construct a number of compressors, on being consulted
by the author, have written to say that they always confined themselves in
estimating the power stored in the compressed air, and had never measured
the gross power expended. Compressed air in passing along the pipe, assumed
to be horizontal, which conveys it from the place of production to the
place where it is to be used, experiences by friction a diminution of
pressure, which represents a reduction in the mechanical power stored up,
and consequently a loss of efficiency.
The loss of pressure in question can only be calculated conveniently on the
hypothesis that it is very small, and the general formula,
p1 - p 4L
------- = ---- f(u),
[Delta] D
[TEX: \frac{p_1 - p}{\Delta} = \frac{4L}{D}f(u)]
is employed for the purpose, where D is the diameter of the pipe, assumed
to be uniform, L the length of the pipe, p1 the pressure at the entrance, p
the pressure at the farther end, u the velocity at which the compressed air
travels, [Delta] its specific weight, and f(u) the friction per unit of
length. In proportion as the air loses pressure its speed increases, while
its specific weight diminishes; but the variations in pressure are assumed
to be so small that u and [Delta] may be considered constant. As regards
the quantity f(u), or the friction per unit of length, the natural law
which regulates it is not known, audit can only be expressed by some
empirical formula, which, while according sufficiently nearly with the
facts, is suited for calculation. For this purpose the binomial formula, au
+ bu², or the simple formula, b1 u², is generally adopted; a b and b1 being
coefficients deduced from experiment. The values, however, which are to
be given to these coefficients are not constant, for they vary with the
diameter of the pipe, and in particular, contrary to formerly received
ideas, they vary according to its internal surface. The uncertainty in this
respect is so great that it is not worth while, with a view to accuracy, to
relinquish the great convenience which the simple formula, b1 u², offers.
It would be better from this point of view to endeavor, as has been
suggested, to render this formula more exact by the substitution of a
fractional power in the place of the square, rather than to go through
the long calculations necessitated by the use of the binomial au + bu².
Accordingly, making use of the formula b1 u², the above equation becomes,
p1 - p 4L
------- = ---- b1 u²;
[Delta] D
[TEX: \frac{p_1 - p}{\Delta} = \frac{4L}{D} b_1 u^2]
or, introducing the discharge per second, Q, which is the usual figure
supplied, and which is connected with the velocity by the relation, Q =
([pi] D² u)/4, we have
p1 - p 64 b1
------- = --------- L Q².
[Delta] [pi]² D^5
[TEX: \frac{p_1 - p}{\Delta} = \frac{64 b_1}{\pi^2 D^5} L Q^2]
Generally the pressure, p1, at the entrance is known, and the pressure, p,
has to be found; it is then from p1 that the values of Q and [Delta] are
calculated. In experiments where p1 and p are measured directly, in order
to arrive at the value of the coefficient b1, Q and [Delta] would be
calculated for the mean pressure ½(p1 + p). The values given to the
coefficient b1 vary considerably, because, as stated above, it varies with
the diameter, and also with the nature of the material of the pipe. It
is generally admitted that it is independent of the pressure, and it is
probable that within certain limits of pressure this hypothesis is in
accordance with the truth.
D'Aubuisson gives for this case, in his _Traité d'Hydraulique_, a rather
complicated formula, containing a constant deduced from experiment, whose
value, according to a calculation made by the author, is approximately b1 =
0.0003. This constant was determined by taking the mean of experiments made
with tin tubes of 0.0235 meter (15/16 in.), 0.05 meter (2 in.), and 0.10
meter (4 in.) diameter; and it was erroneously assumed that it was correct
for all diameters and all substances.
M. Arson, engineer to the Paris Gas Company, published in 1867, in the
_Mémoires de la Société des Ingénieurs Civils de France_, the results of
some experiments on the loss of pressure in gas when passing through pipes.
He employed cast-iron pipes of the ordinary type. He has represented the
results of his experiments by the binomial formula, au + bu², and gives
values for the coefficients a and b, which diminish with an increase in
diameter, but would indicate greater losses of pressure than D'Aubuisson's
formula. M. Deviller, in his _Rapport sur les travaux de percement du
tunnel sous les Alpes_, states that the losses of pressure observed in the
air pipe at the Mont Cenis Tunnel confirm the correctness of D'Aubuisson's
formula; but his reasoning applies to too complicated a formula to be
absolutely convincing.
Quite recently M. E. Stockalper, engineer-in-chief at the northern end of
the St. Gothard Tunnel, has made some experiments on the air conduit of
this tunnel, the results of which he has kindly furnished to the author.
These lead to values for the coefficient b1 appreciably less than that
which is contained implicitly in D'Aubuisson's formula. As he experimented
on a rising pipe, it is necessary to introduce into the formula the
difference of level, h, between the two ends; it then becomes
p1 - p 64 b1
------- = --------- L Q² + h.
[Delta] [pi]² D^5
[TEX: \frac{p_1 - p}{\Delta} = \frac{64 b_1}{\pi^2 D^5} L Q^2 + h]
The following are the details of the experiments: First series of
experiments: Conduit consisting of cast or wrought iron pipes, joined by
means of flanges, bolts, and gutta percha rings. D = 0.20 m. (8 in.); L =
4,600 m. (15,100 ft,); h= 26.77 m. (87 ft. 10 in.). 1st experiment: Q =
0.1860 cubic meter (6.57 cubic feet), at a pressure of ½(p1 + p), and a
temperature of 22° Cent. (72° Fahr.); p1 = 5.60 atm., p =5.24 atm. Hence p1
- p = 0.36 atm.= 0.36 x 10,334 kilogrammes per square meter (2.116 lb. per
square foot), whence we obtain b1=0.0001697. D'Aubuisson's formula would
have given p1 - p = 0.626 atm.; and M. Arson's would have given p1 - p =
0.9316 atm. 2d experiment: Q = 0.1566 cubic meter (5.53 cubic feet), at a
pressure of ½(p1 + p), and a temperature of 22° Cent. (72° Fahr.); p1
= 4.35 atm., p = 4.13 atm. Hence p1 - p = 0.22 atm. = 0.22 X 10,334
kilogrammes per square meter (2,116 lb. per square foot); whence we obtain
b1 = 0.0001816. D'Aubuisson's formula would have given p1 - p = 0.347 atm;
and M. Arson's would have given p1 - p = 0.5382 atm. 3d experiment: Q =
0.1495 cubic meter (5.28 cubic feet) at a pressure of ½(p1 + p) and a
temperature 22° Cent. (72° Fahr.); p1 = 3.84 atm., p = 3.65 atm. Hence p1 -
p = 0.19 atm. = 0.19 X 10,334 kilogrammes per square meter (2.116 lb. per
square foot); whence we obtain B1 = 0.0001966. D'Aubuisson's formula would
have given p1 - p = 0.284 atm., and M. Arson's would have given p1 - p =
0.4329 atm. Second series of experiments: Conduit composed of wrought-iron
pipes, with joints as in the first experiments. D = 0.15 meter (6 in.), L
- 0.522 meters (1,712 ft.), h = 3.04 meters (10 ft.) 1st experiments: Q =
0.2005 cubic meter (7.08 cubic feet), at a pressure of ½(p1 + p), and a
temperature of 26.5° Cent. (80° Fahr.); p1 = 5.24 atm., p = 5.00 atm. Hence
p1 - p = 0.24 atm. =0.24 x 10,334 kilogrammes per square meter (2,116 lb.
per square foot); whence we obtain b1 = 0.3002275. 2nd experiment: Q =
0.1586 cubic meter (5.6 cubic feet), at a pressure of ½(p1 + p), and a
temperature of 26.5° Cent. (80° Fahr.); p1 = 3.650 atm., p = 3.545 atm.
Hence p1 - p = 0.105 atm. = 0.105 x 10,334 kilogrammes per square meter
(2,116 lb. per square foot); whence we obtain b1 = 0.0002255. It is clear
that these experiments give very small values for the coefficient. The
divergence from the results which D'Aubuisson's formula would give is due
to the fact that his formula was determined with very small pipes. It is
probable that the coefficients corresponding to diameters of 0.15 meter
(6 in.) and 0.20 meter (8 in.) for a substance as smooth as tin, would be
still smaller respectively than the figures obtained above.
The divergence from the results obtained by M. Arson's formula does not
arise from a difference in size, as this is taken into account. The author
considers that it may be attributed to the fact that the pipes for the St.
Gothard Tunnel were cast with much greater care than ordinary pipes, which
rendered their surface smoother, and also to the fact that flanged joints
produce much less irregularity in the internal surface than the ordinary
spigot and faucet joints.
Lastly, the difference in the methods of observation and the errors which
belong to them, must be taken into account. M. Stockalper, who experimented
on great pressures, used metallic gauges, which are instruments on whose
sensibility and correctness complete reliance cannot be placed; and
moreover the standard manometer with which they were compared was one of
the same kind. The author is not of opinion that the divergence is owing to
the fact that M. Stockalper made his observations on an air conduit, where
the pressure was much higher than in gas pipes. Indeed, it may be assumed
that gases and liquids act in the same manner; and, as will be [1]
explained later on, there is reason to believe that with the latter a rise
of pressure increases the losses of pressure instead of diminishing them.
[Transcribers note 1: corrected from 'as will we explained']
All the pipes for supplying compressed air in tunnels and in headings of
mines are left uncovered, and have flanged joints; which are advantages not
merely as regards prevention of leakage, but also for facility of laying
and of inspection. If a compressed air pipe had to be buried in the ground
the flanged joint would lose a part of its advantages; but, nevertheless,
the author considers that it would still be preferable to the ordinary
joint.
It only remains to refer to the motors fed with the compressed air.
This subject is still in its infancy from a practical point of view. In
proportion as the air becomes hot by compression, so it cools by expansion,
if the vessel containing it is impermeable to heat. Under these conditions
it gives out in expanding a power appreciably less than if it retained its
original temperature; besides which the fall of temperature may impede the
working of the machine by freezing the vapor of water contained in the air.
If it is desired to utilize to the utmost the force stored up in the
compressed air it is necessary to endeavor to supply heat to the air during
expansion so as to keep its temperature constant. It would be possible
to attain this object by the same means which prevent heating from
compression, namely, by the circulation and injection of water. It would
perhaps be necessary to employ a little larger quantity of water for
injection, as the water, instead of acting by virtue both of its heat of
vaporization and of its specific heat, can in this case act only by virtue
of the latter. These methods might be employed without difficulty for air
machines of some size. It would be more difficult to apply them to small
household machines, in which simplicity is an essential element; and we
must rest satisfied with imperfect methods, such as proximity to a stove,
or the immersion of the cylinder in a tank of water. Consequently loss of
power by cooling and by incomplete expansion cannot be avoided. The only
way to diminish the relative amount of this loss is to employ compressed
air at a pressure not exceeding three or four atmospheres.
The only real practical advance made in this matter is M. Mékarski's
compressed air engine for tramways. In this engine the air is made to pass
through a small boiler containing water at a temperature of about 120°
Cent. (248° Fahr.), before entering the cylinder of the engine. It must
be observed that in order to reduce the size of the reservoirs, which
are carried on the locomotive, the air inside them must be very highly
compressed; and that in going from the reservoir into the cylinder it
passes through a reducing valve or expander, which keeps the pressure of
admission at a definite figure, so that the locomotive can continue working
so long as the supply of air contained in the reservoir has not come down
to this limiting pressure. The air does not pass the expander until after
it has gone through the boiler already mentioned. Therefore, if the
temperature which it assumes in the boiler is 100° Cent. (212° Fahr.), and
if the limiting pressure is 5 atm., the gas which enters the engine will be
a mixture of air and water vapor at 100° Cent.; and of its total pressure
the vapor of water will contribute I atm. and the air 4 atm. Thus this
contrivance, by a small expenditure of fuel, enables the air to act
expansively without injurious cooling, and even reduces the consumption of
compressed air to an extent which compensates for part of the loss of power
arising from the preliminary expansion which the air experiences before its
admission into the engine. It is clear that this same contrivance, or what
amounts to the same thing, a direct injection of steam, at a sufficient
pressure, for the purpose of maintaining the expanding air at a constant
temperature, might be tried in a fixed engine worked by compressed air with
some chance of success.
Whatever method is adopted it would be advantageous that the losses of
pressure in the pipes connecting the compressors with the motors should be
reduced as much as possible, for in this case that loss would represent
a loss of efficiency. If, on the other hand, owing to defective means of
reheating, it is necessary to remain satisfied with a small amount of
expansion, the loss of pressure in the pipe is unimportant, and has only
the effect of transferring the limited expansion to a point a little lower
on the scale of pressures. If W is the net disposable force on the shaft
of the engine which works the compressor, v1 the volume of air at the
compressor, p1. given by the compressor, and at the temperature of the
surrounding air, and p0 the atmospheric pressure, the efficiency of the
compressor, assuming the air to expand according to Boyle's law, is given
by the well-known formula--
p1 v1 log (p1 / p0)
-------------------.
W
[TEX: \frac{p_1 v_1 \log \frac{p_1}{p_0}}{W}]
Let p2 be the value to which the pressure is reduced by the loss of
pressure at the end of the conduit, and v2 the volume which the air
occupies at this pressure and at the same temperature; the force stored
up in the air at the end of its course through the conduit is p2 v2
log(p2/p0); consequently, the efficiency of the conduit is
p2 v2 log(p2/p0)
----------------
p1 v1 log(p1/p0)
[TEX: \frac{p_2 v_2 \log\frac{p_2}{p_0}}{p_2 v_2 \log\frac{p_2}{p_0}}]
a fraction that may be reduced to the simple form
log(p2/p0)
----------,
log(p1/p0)
[TEX: \frac{\log\frac{p_2}{p_0}}{\log\frac{p_2}{p_0}}]
if there is no leakage during the passage of the air, because in that cause
p2 v2 = p1 v1. Lastly, if W1 is the net disposable force on the shaft of
the compressed air motor, the efficiency of this engine will be,
W1
----------------
p2 v2 log(p2/p0)
[TEX: \frac{W_1}{p_2 v_2 \log \frac{p_2}{p_0}}]
and the product of these three partial efficiencies is equal to W1/W, the
general efficiency of the transmission.
III. _Transmission by Pressure Water_.--As transmission of power by
compressed air has been specially applied to the driving of tunnels, so
transmission by pressure water has been specially resorted to for lifting
heavy loads, or for work of a similar nature, such as the operations
connected with the manufacture of Bessemer steel or of cast-iron pipes.
The author does not propose to treat of transmissions established for this
special purpose, and depending on the use of accumulators at high pressure,
as he has no fresh matter to impart on this subject, and as he believes
that the remarkable invention of Sir William Armstrong was described for
the first time, in the "Proceedings of the Institution of Mechanical
Engineers." His object is to refer to transmissions applicable to general
purposes.
The transmission of power by water may occur in another form. The motive
force to be transmitted may be employed for working pumps which raise the
water, not to a fictitious height in an accumulator, but to a real height
in a reservoir, with a channel from this reservoir to distribute the water
so raised among several motors arranged for utilizing the pressure. The
author is not aware that works have been carried out for this purpose.
However, in many towns a part of the water from the public mains serves to
supply small motors--consequently, if the water, instead of being brought
by a natural fall, has been previously lifted artificially, it might be
said that a transmission of power is here grafted on to the ordinary
distribution of water.
Unless a positive or negative force of gravity is introduced into the
problem, independently of the force to be transmitted, the receivers of
the water pressure must be assumed to be at the same level as the forcing
pumps, or more correctly, the water discharged from the receivers to be at
the same level as the surface of the water from which the pumps draw their
supply. In this case the general efficiency of transmission is the product
of three partial efficiencies, which correspond exactly to those mentioned
with regard to compressed air. The height of lift, contained in the
numerator of the fraction which expresses the efficiency of the pumps, is
not to be taken as the difference in level between the surface of the water
in the reservoir and the surface of the water whence the pumps draw their
supply; but as this difference in level, plus the loss of pressure in the
suction pipe, which is usually very short, and plus the loss in the channel
to the reservoir, which may be very long. A similar loss of initial
pressure affects the efficiency of the discharge channel. The reservoir, if
of sufficient capacity, may become an important store of power, while the
compressed air reservoir can only do so to a very limited extent.
Omitting the subject of the pumps, and passing on at once to the discharge
main, the author may first point out that the distinction between the
ascending and descending mains of the system is of no importance, for two
reasons: first, that nothing prevents the motors being supplied direct from
the first alone; and second, that the one is not always distinct from the
other. In fact, the reservoir may be connected by a single branch pipe with
the system which goes from the pumps to the motors; it may even be placed
at the extreme end of this system beyond the motors, provided always that
the supply pipe is taken into it at the bottom. The same formula may be
adopted for the loss of initial pressure in water pipes as for compressed
air pipes, viz.,
p1 - p 64 b1
------- = --------- L Q² ± h;
[Delta] [pi]² D^5
[TEX: \frac{p_1 - p}{\Delta} = \frac{64 b_1}{\pi^2 D^5} L Q^2 \pm h]
h being the difference of level between the two ends of the portion of
conduit of length, L, and the sign + or - being used according as the
conduit rises or falls. The specific weight, [delta], is constant, and the
quotients, p1/[delta] and p/[delta], represent the heights, z and z1, to
which the water could rise above the pipes, in vertical tubes branching
from it, at the beginning and end of the transit. The values assigned to
the coefficient b1 in France, are those determined by D'Arcy. For new
cast-iron pipes he gives b1 - 0.0002535 + 1/D 0.000000647; and recommends
that this value should be doubled, to allow for the rust and incrustation
which more or less form inside the pipes during use. The determination of
this coefficient has been made from experiments where the pressure has
not exceeded four atmospheres; within these limits the value of the
coefficient, as is generally admitted, is independent of the pressure. The
experiments made by M. Barret, on the pressure pipes of the accumulator at
the Marseilles docks, seem to indicate that the loss of pressure would be
greater for high pressures, everything else being equal. This pipe, having
a diameter of 0.127 m. (5 in.), was subjected to an initial pressure of 52
atmospheres. The author gives below the results obtained for a straight
length 320 m. (1050 ft) long; and has placed beside them the results which
D'Arcy's formula would give.
Loss of head, in meters or ft. respectively
per 100 meters or ft. run of pipes.
+-----------------^-------------------+
| |
Calculated loss.
+-----------^-----------+
| |
Velocity of flow Actual loss
per second. observed. Old pipes. New pipes.
Meters. Feet. Met. or Ft. Met. or Ft. Met. or Ft.
0.25 0.82 1.5 0.12 0.06
0.50 1.64 2.5 0.48 0.24
0.75 2.46 3.7 1.08 0.54
1.00 3.28 5.5 1.92 0.96
1.25 4.10 6.1 3.00 1.50
1.50 4.92 7.3 4.32 2.16
1.75 5.74 8.0 5.88 2.94
2.00 6.56 10.2 7.68 3.84
2.25 7.38 11.7 9.72 4.86
2.50 8.20 14.0 12.00 6.00
Moreover, these results would appear to indicate a different law from that
which is expressed by the formula b1 u², as is easy to see by representing
them graphically. It would be very desirable that fresh experiments should
be made on water pipes at high pressure, and of various diameters. Of
machines worked by water pressure the author proposes to refer only to two
which appear to him in every respect the most practical and advantageous.
One is the piston machine of M. Albert Schmid, engineer at Zurich. The
cylinder is oscillating, and the distribution is effected, without an
eccentric, by the relative motion of two spherical surfaces fitted one
against the other, and having the axis of oscillation for a common axis.
The convex surface, which is movable and forms part of the cylinder, serves
as a port face, and has two ports in it communicating with the two ends of
the cylinder. The concave surface, which is fixed and plays the part of a
slide valve, contains three openings, the two outer ones serving to admit
the pressure water, and the middle one to discharge the water after it has
exerted its pressure. The piston has no packing. Its surface of contact has
two circumferential grooves, which produce a sort of water packing acting
by adhesion. A small air chamber is connected with the inlet pipe, and
serves to deaden the shocks. This engine is often made with two cylinders,
having their cranks at right angles.
The other engine, which is much less used, is a turbine on Girard's system,
with a horizontal axis and partial admission, exactly resembling in
miniature those which work in the hydraulic factory of St. Maur, near
Paris. The water is introduced by means of a distributer, which is fitted
in the interior of the turbine chamber, and occupies a certain portion
of its circumference. This turbine has a lower efficiency than Schmid's
machine, and is less suitable for high pressures; but it possesses this
advantage over it, that by regulating the amount of opening of the
distributer, and consequently the quantity of water admitted, the force can
be altered without altering the velocity of rotation. As it admits of great
speeds, it could be usefully employed direct, without the interposition of
spur wheels or belts for driving magneto-electric machines employed for the
production of light, for electrotyping, etc.
In compressed air machines the losses of pressure due to incomplete
expansion, cooling, and waste spaces, play an important part. In water
pressure machines loss does not occur from these causes, on account of the
incompressibility of the liquid, but the frictions of the parts are the
principal causes of loss of power. It would be advisable to ascertain
whether, as regards this point, high or low pressures are the most
advantageous. Theoretical considerations would lead the author to imagine
that for a piston machine low pressures are preferable. In conclusion, the
following table gives the efficiencies of a Girard turbine, constructed by
Messrs. Escher Wyss & Co., of Zurich, and of a Schmid machine, as measured
by Professor Fliegnor, in 1871: