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THE GAS TURBINE
AN " INTERNAL COMBUSTION" PRIME-MOVER
BY
SANFORD A. MOSS, M.S. (UN1V.CAL.)
A THESIS PRESENTED TO THE FACULTY OF CORNELL UNIVERSITY
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
ITHACA, N, Y.
MAY, 1903
THE GAS TURBINE
AN " INTERNAL COMBUSTION" PRIME-MOVER
BY
SANFORD A. MOSS, M.S. (UNIV. CAL.)
A THESIS PRESENTED TO THE FACULTY OF CORNELL UNIVERSITY
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
. MAY, 1903
ANDRUS & CHURCH
ITHACA, N. Y.
CONTENTS.
PAGE
CHAPTER i. General Account of the Gas Turbine 5
CHAPTER 2. History of the Gas Turbine 13
CHAPTER 3. Thermodynamics of the Brayton Cycle 17
CHAPTER 4. Theory of Nozzles and Free Expansion 25
CHAPTER 5. Theory of a Rapidly Rotating Disc on a Flexible Shaft__ 36
CHAPTER 6. Experiments with a Gas Turbine. 48
THE GAS TURBINE.
AN "INTERNAL COMBUSTION" PRIME MOVER.
CHAPTER I.
GENERAI, ACCOUNT OF THE GAS TURBINE.*
The " Gas Turbine " is a form of heat engine, or mechanism
for changing the potential energy of. fuel into mechanical work.
As we shall have occasion to see in Chap. 3, the gas turbine
seems to offer greater theoretical possibilities than the steam en-
gine and boiler, or the Otto cycle gas engine, which are the heat
engines now most usual. Of course considerable experimental
perfecting would be necessary before the gas turbine could de-
mand serious attention for commercial uses. However, the theo-
retical possibilities seem to warrant the present investigation at
least.
Before discussing the gas turbine itself we shall consider an-
other form of heat engine based on the same thermodynamic
principles, but somewhat simpler in conception.
Consider a compressed air transmission plant, comprising an
air compressor and a motor like an ordinary steam engine. The
work done by the motor varies directly with the absolute temper-
ature of the air supplied to it. That is to say, the work done by
the air depends only on its volume when supplied to the motor,
and hence increasing this volume by raising the temperature in-
creases the work, regardless of the actual quantity or weight of
the air. For this reason compressed air is usually " reheated "
before use in a motor.
If the air is compressed without any gain or loss of heat, that
is, " adiabatically," the temperature will rise considerably. In
our theoretical discussion we will assume such compression.
Practically it is always convenient to cool the air somewhat dur-
ing compression. If the air supplied to the motor is at the
adiabatic temperature at which it theoretically leaves the com-
* The substance of this chapter was published as an advance portion of
this thesis in The Engineer (Cleveland), Vol. 40, No. 8, April 15, 1903.
6
pressor, the power from the motor will be theoretically the same
as that required for the compressor. By reheating the air suffi-
ciently, the power from the motor may be made to exceed the
power for the compressor by as much as we please. The excess
power represents work obtained from heat energy of the fuel used
in reheating.
In a transmission plant, compressor and motor are placed at a
distance from each other. If, however, they were at the same
place, the motor could drive the compressor, and the excess
power would be available for outside purposes. The apparatus
would then be a heat engine operating on the fuel used in heat-
ing the air between compressor and motor.
The air might be heated by passing it through a closed vessel
of some sort exposed to an outside flame, or by " internal com-
bustion " which we will discuss later. The entire connected
system of piping and vessels between the compressor and motor,
including the heating vessel, is everywhere at the same pressure,
which, of course, is the maximum compression pressure. The
air is therefore heated at constant pressure, and its volume in-
creases so that air leaves the heating chamber much faster than
it enters. The motor must of course use up the entire volume of
heated air as fast as it is supplied in order that the pressure may
remain constant.
The series of operations performed on a portion of working
substance as it goes through a heat engine is called its "cycle,"
the cycle used in the heat engine just described being named the
" Bray ton cycle." This cycle is distinguished by the fact that
while the heat used is being added to the air, or other working
substance which actuates the engine, this working substance is
kept at constant pressure. The same cycle or series of operations
maybe carried out by very different kinds of engines. The gas
turbine operates differently from the engine just described, but it
uses the same Brayton cycle. That is, the same series of opera-
tions are carried out, but in a different way.
Theoretically the Brayton cycle should be easier to execute
than the Otto cycle used in modern gas engines. The thermo-
dynamic efficiency, or theoretical fraction of the heat supply
which is given out as mechanical work, is the same for the Otto
as for the Brayton cycle, if the same compression pressure is
used. However, the maximum pressure is very much greater
with the Otto cycle, where the heat is added at constant volume
with consequent increase of pressure, than with the Brayton cycle,
where the pressure does not rise above the compression pressure.
The maximum temperature is also much greater for the Otto
cycle than for the Brayton, for the same compression pressure
and theoretical efficiency. Having lower maximum pressure and
temperature we should expect the Brayton cycle to be mofe
easily managed than the Otto for equal efficiency.
However, the method in which a cycle is carried out is as im-
portant as the form of cycle. Losses not taken account of by
theory always occur, and the magnitude of these depends only on
the mechanism of the heat engine which executes the cycle. We
must carry out the Brayton cycle in an actual engine without in-
curring greater losses than occur in an Otto gas engine in order
to secure better results. In all actual attempts to execute the
Brayton cycle in the past the losses have been so serious as to
more than outweigh the theoretical advantages. On paper, at
least, the gas turbine avoids these losses, as we shall see in
Chap. 3.
Let us next consider how the Brayton cycle is carried out by
the gas turbine, which, as has been stated, is the equivalent of
the compressor- motor combination already discussed. In the
case of the compressor-motor combination the compressed air was
supposed to be heated by being passed through a vessel exposed
to an outside flame. In the gas turbine there is substituted for
this internal combustion, used so successfully in the modern gas
engine. Since air is our original working substance and since it
is also requisite for the combustion of the fuel used in the heating,
we may use the air which is to operate the motor to support the
combustion of the fuel. That is to say, we will heat the air for
the motor by burning the fuel right in it. This is called " inter-
nal combustion," as distinguished from " external combustion,"
where the heat of the fuel is transferred through the walls of a
vessel to the working substance, as in the case of a steam boiler.
The air must be heated between the compressor and motor, when
it is under considerable pressure. A closed vessel or enlargement
of the pipe between the compressor and motor is provided, called
the " combustion chamber," in which the air and fuel combine
or burn, always, of course, while at the maximum compression
pressure. From this chamber the products of combustion pass to
the motor. Theoretically this could be an engine like a steam
engine, or any substitute for it, and in this particular case we will
use a turbine wheel similar to a steam turbine. The details of
this we will discuss later, first considering more fully the method
of operating the combustion chamber.
8
Heat engines using the Bray ton cycle and having a combustion
chamber with internal combustion, but using a motor similar to
an ordinary steam engine, instead of a turbine wheel as in our
case, have been built or planned in a multitude of forms, and the
type of combustion chamber proposed for any one of these is of
course suitable for the gas turbine. The relation of these engines
to the gas turbine we shall discuss more fully in Chap. 2.
In the Cayley engine, actually in operation in 1807, coal was
used as fuel. A supply sufficient for some time was placed on a
grate in a closed vessel and lighted, and- the air from the com-
pressor forced through it. When the fuel had all burned out, the
motor had to be shut down, the combustion chamber opened, and
a fresh fuel supply introduced. Of course a method could be
easily contrived for introducing coal into the combustion chamber
while the pressure is on, so that the motor could be operated con-
tinuously.
In other cases a gaseous fuel has been forced into the combus-
tion chamber against pressure within by means of a small gas
compressor, and the air forced in by a separate air compressor,
the two pipes meeting as they enter the combustion chamber.
After once being lighted a continuous jet of flame issues from the
meeting point of the pipes. The first gas turbine ever proposed,
which we will discuss later, was to operate in this way.
The most convenient fuel, however, for a gas turbine will be a
liquid of some kind, as, for instance, gasoline, kerosene, or dis-
tilled or crude petroleum. Many proposed Brayton cycle engines
with motors similar to steam engines have been planned for
liquid fuel. The original engine of this type was that of George
Brayton, actually in commercial use about 1872. It is due to
this engine that the name of Brayton is given to the cycle in
which heat is added at constant pressure. Such a cycle had been
proposed long before, however, and Brayton only invented a par-
ticular form of engine for executing the cycle.
The cylinder of Bray ton's motor was really his combustion
chamber. He forced the oil and air through passages at the en-
trance to the cylinder as the piston was advancing on the forward
stroke, and by means of a small flame, kept constantly lighted,
ignited and burned the mixture as it entered the cylinder. The
hot products of combustion, of course, at the maximum compres-
sion pressure, forced the motor piston ahead. In this case the
combustion was intermittent, the supply of air and oil being cut
off when the motor piston reached a certain point, and the com-
bustion resumed at the beginning of the next stoke.
9
In the case of the gas turbine air and oil are continuously
forced into one end of the combustion chamber, and, once being
lighted, the jet continues to burn. The pressure within the com-
bustion chamber and the air pipe leading to it is, of course,
always the maximum compression pressure.
"Combustion at constant pressure" occurs in all of the cases
we have considered, whether the fuel be solid, liquid, or gaseous.
The fuel being present, and air supplied as fast as necessary by
the compressor, combustion ensues exactly as if the fuel were
burned in the open air. The fact that the combustion takes place
in a closed vessel under pressure makes no difference whatever.
The products of the combustion are nitrogen and carbon dioxide,
together with oxygen from any air in excess of that required for
combustion. All of the heat of combustion of the fuel is emitted
just as when it is burned in the open air, and this goes to heat
the products of combustion, which are thereby expanded so that
they occupy a considerably increased volume. Now the products
of combustion have practically the same volume at any given
temperature and pressure as would the original air. Therefore
the volume and temperature of the products after combustion will
be the same as the original air would have if the entire heat of
combustion of the fuel were added to the air by passing it through
a heated vessel, without changing the chemical composition by
burning the fuel in it. In effect, then, internal combustion is
merely a means of adding a certain amount of heat to the air,
with consequent increase of volume, and is equivalent to passing
it through a heated pipe. The change of chemical composition
is merely incidental. Of course we always assume that the pro-
ducts of combustion are drawn off from the combustion chamber
as fast as they are formed, so that the pressure within it is always
the maximum compression pressure. Owing to the expansion
due to heating, the volume taken from the combustion chamber
and supplied to the motor is much greater than the volume of air
supplied by the compressor. As already stated, the power ob-
tained from the motor is greater than that required to drive the
compressor in the same ratio.
Next let us consider briefly the theory of the turbine wheel or
11 impulse wheel " as it should properly be called. Suppose we
have a vessel from which we may obtain a continuous supply of
liquid or gas under pressure. This might be a reservoir of water,
a steam boiler, or the combustion chamber of a gas turbine. By
using an engine similar to a steam engine we could obtain power
IO
from the pressure of the liquid or gas. Suppose, however, that
we connect an open nozzle with the vessel, .so that the water,
steam, or gas can escape into the air or into any region where the
pressure is lower. A jet will then issue from the nozzle with con-
siderable velocity. If the nozzle be properly shaped so as to
avoid friction losses, the kinetic energy of the escaping jet will be
exactly the same as the energy which could be obtained by ex-
panding the same liquid or gas from the higher pressure to the
lower in a pressure engine like a steam engine. That is to say,
the velocity with which the jet escapes represents exactly the
same power as could be obtained by using the liquid or gas in an
ordinary engine, We will give the proof of this, and consider
the whole matter in greater detail in Chap. 4.
By directing the jet upon the vanes or buckets of a properly ar-
ranged impulse wheel, the power of the jet may be taken from
it and applied to do useful work. Examples of this are shown
by the Pelton type of water wheel, and the De Laval steam tur-
bine.
Suppose, then, that we place at the farther end of the combus-
tion chamber a proper nozzle and direct the jet of gases issuing
from it upon a turbine wheel. The wheel could be of the type
used in the De Laval steam turbine with a single set of buckets,
or a number of wheels could be placed in series, as with the
Curtis or Parsons steam turbines. We will then obtain the same
power from the wheel as we would if we used a motor with a
piston like that of Brayton, Cayley, and others. As already
stated, this power exceeds the power required to drive the com-
pressor by as much as the heating expanded the products of com-
bustion. Hence, we may drive the compressor from the impulse
wheel, and have surplus power, which is the net income from the
heat energy of the fuel used.
Having examined the individual steps in the operation of the
gas turbine, we are prepared to consider the apparatus as a
whole. Fig. i is a diagrammatic sketch of the complete machine,
which is so labelled as to be sufficiently explicit. We will discuss
possible modifications of this fundamental arrangement in Chap.
3 and actual details of construction and operation in Chap. 6.
Oil and air are forced into the combustion chamber as indi-
cated, and begin to burn at the entrance. The flaming stream
passes along the combustion chamber, and combustion is com-
plete by the time the nozzle is reached. The same constant pres-
sure is, as previously stated, maintained throughout the system
II
from the compressor, through the pipes and combustion chamber
to the nozzle, since the nozzle is of such size as to allow the in-
creased volume of products of combustion to pass out of the com-
bustion chamber as fast as they are produced by the entrance of
FIG. i. Diagrammatic Plan of Fundamental Gas Turbine.
air and fuel. The diagram shows the arrangement for liquid
fluid, to which the gas turbine is perhaps best adapted. Gaseous
fuel, such as any kind of coal gas or producer gas, could be used
by replacing the oil pump by a gas compressor. Solid fuel could
be used by devising some means of continuously feeding it
against the pressure of the combustion chamber, or by having a
combustion chamber large enough to contain a supply of fuel for
some time, and forcing in air only.
The combustion chamber must, of course, be designed to with-
stand the high temperature of the burning gases and the pressure
of the compressor. The nozzle must be so designed that the
products of combustion have free expansion from the maximum
pressure to the external pressure, hence, as in the case of the
steam turbine, the channel must first converge and then diverge.
The proper shape for the nozzle we will consider in detail in
12
Chap. 3. The gases will be cooled considerably in expanding
through the nozzle, but are still hot at the exit. In all heat
engines only a portion of the heat of the fuel is utilized and the
rest must be thrown away. In the case of the gas turbine, the
kinetic energy of the jet issuing from the nozzle represents that
part of the heat energy of the fuel which can be utilized, while
the temperature of the jet represents that portion of the energy
which will be lost.
The gases issue from the nozzle with a velocity comparable
with that in the De Laval steam turbine, and the turbine must
therefore rotate at the same extremely high speed if we use a
single stage wheel as does De Laval, the velocity of the buckets
being approximately half that of the jet. The theory of a disc
rotating at such extraordinary speeds is given in Chap. 5. Some
form of speed-reducing mechanism must also be introduced, since
it is difficult to utilize power from the wheel at the original speed.
However the speed of the wheel could be reduced by using
several wheels in series, as in the Curtis or Parsons turbines.
The use of a true turbine, where expansion occurs within the
wheel itself, is also conceivable. Everything considered how-
ever, a single stage impulse wheel .seems best.
CHAPTER II.
HISTORY OF THE GAS TURBINE.
As is the case with many other supposed novelties, the gas tur-
bine was conceived in ancient times. What is probably the orig-
inal gas turbine was patented in England in 1791 by one John
Barber. The drawing of this, Fig. 2, and the following descrip-
FIG. 2. Barber's Gas Turbine, 1791.
i 4
tion are taken from the British Patent Reports, Volume XXIII,
1791-2, Patent No. 1833, which seems to be the only source of
information concerning the curious machine. It will be seen that
Barber probably understood thoroughly the principle of the gas
turbine, although the mechanical knowledge of the time would
hardly have sufficed for the construction of a working machine.
The vessels marked i are retorts for the production of the gas
to be used, by distillation of coal, wood, etc., by means of an ex-
ternal flame. These are in duplicate, so that one can always be in
use while the other is being emptied of coke and recharged. The
vessel above the retorts is a cooling and condensing chamber,
from which the gas is drawn by the pipe B. The apparatus up
to this point is merely for the production of the gaseous fuel to be
used, and is not an essential feature of the gas turbine. The
parts marked CC, DD, the two lower vessels of those marked 4,
and the elevated water tanks above the apparatus comprise two
peculiar hydraulic compressors, the details of which we need not
investigate. The triangular shaped vessel between the -tanks 4
and the wheels is the combustion chamber. By means of the
front compressor, the gas taken from B is compressed and dis-
charged from the front vessel 4 into the combustion chamber by
the pipe shown. By means of the rear compressor air is com-
pressed and discharged from the rear vessel 4 by a pipe not shown
into the rear side of the combustion chamber. The upper vessel
4 discharges water into the combustion chamber, which absorbs
some of the heat of combustion and reduces the maximum tem-
perature. This is not essential to the operation of the apparatus,
however, as the same result could, of course, be obtained by the
use of an excess of air.
The air and gas must burn within the combustion chamber,
and the products, issuing from the lower end, are directed upon
vanes of the wheel marked 8, which is thereby impelled to rotate.
Barber's description of the action within the combustion chamber
is not very explicit, and there is a possibility that he did not un-
derstand the gas turbine after all. It may be that the triangular
chamber was only intended as a mixing chamber, and that the
combustion was not to occur until the mixture had passed through
its outlet, since the flame would not strike back owing to the ve-
locity through the orifice. It might be, on the other hand, that
the air and gas were to be throttled before they entered the cham-
ber, so that atmospheric pressure existed within it. In either
event the heat would be added while the working substance was
at atmospheric pressure, instead of while it was at the maximum
pressure. The addition of heat would then be of no use what-
ever, and the wheel 8 would operate equally well whether the
gases were lighted or not. Theoretically the power required for
compression would then be equal to the power of the turbine
wheel ; and practically, owing to friction, etc., would be in excess ;
so that the machine would not operate at all.
In case, however, that Barber had the correct conception of the
matter and arranged to have the combustion take place in the tri-
angular chamber, where also the maximum compression pressure
existed, then the operation would be, as he states, as follows : A
pinion on the impulse wheel shaft operates the gear wheel 10, on
the shaft of which are the cams 9, which raise rods attached to
the walking beams 55, actuating the compressors by means of the
chains shown. A pinion on the same shaft as the wheel 10 ope-
rates the upper gear wheel. From the projecting shaft of this up-
per gear the difference between the compressor and motor power
can be taken off and applied to useful work.
The maximum compression pressure which Barber could obtain
was that due to the head of water from the tanks in the upper
part of the figure, which practically could not be made very great.
A modern gas turbine would use much greater pressures than Bar-
ber contemplated.
Nothing seems to have come of Barber's gas turbine and the
matter lay untouched for years. In the meantime a great number
of internal combustion Brayton cycle engines with reciprocating
engines as motors, were proposed as already stated, none attain-
ing success however. A number of these are described by Mr.
Charles Lucke in a paper presented to the American Society of
Mechanical Engineers, Dec. 190^, andjorming part of his Doc-
tor's thesis, presented to Columbia University in 1902. He dis-
tinguishes two types, the first type having intermittent, and the
second type continuous internal combustion at constant pressure.
The latter type of course involves a combustion chamber or re-
ceiver sufficiently large to prevent excessive fluctuations of
pressure owing to the intermittent demand of the motor. A heat
engine of this type is essentially equivalent to the gas turbine
with the single exception that a reciprocating motor is used
instead of an impulse wheel. Mr. Lucke mentions the possibility
and even the desirability of using a "gas-expansion turbine"
instead of a reciprocating motor in this type of heat engine, but
he does not go into details.
i6
A number of gas turbines have been proposed by various in-
ventors in which an explosive mixture is repeatedly admitted to
a chamber, exploded, and discharged with gradually decreasing
pressure upon the vanes of a turbine wheel. These might be
called " intermittent combustion" gas turbines. A gas turbine
of this type due to Leon Le Pontois was the subject of the Doc-
tor's thesis presented to Cornell University by Mr. W. O.
Amsler. All gas turbines of this character seem to be based
on the false idea that "explosion" is the only proper way
to conduct internal combustion, and -the inventors seem to
be ignorant of the possibility of continuous combustion under
constant pressure. For obvious reasons a continuous combustion
gas turbine is immeasurably superior to one with intermittent
combustion and variable pressure.
The writer began the study of the gas turbine in 1898, believ-
ing the idea original Until research disclosed Barber's patent.
A thesis for the degree M.S. entitled "Thermodynamics of the
Gas Turbine" was presented to the University of California in
1900, the substance of which is given in the next chapter.
CHAPTER III.
THERMODYNAMICS OF THE BRAYTON CYCI,E.
The heat engine used in the Gas Turbine, commonly called the
Brayton Cycle, as already stated, is distinguished by the fact that
heat is added to and taken from the working substance while it is
at constant pressure. The various operations performed on a
portion of working substance as it passes through a Brayton cycle
heat engine are as follows :
The working substance, initially separate portions of air and
fuel in the case of the gas turbine, is first compressed to the maxi-
mum pressure p v While at this pressure heat is added by some
means or other ; in the case of the gas turbine by "internal com-
bustion. ' ' The volume and temperature of the working substance
are thereby increased. The chemical composition may also be
changed but this is merely on incident which we have on occasion
to take account of at present. The working substance next ex-
pands adiabatically to atmospheric pressure. During addition of
heat and expansion, work is done which is eventually utilized by
the motor. In our case an impulse wheel is used, which is, how-
ever, as will be shown in Chap. 4, equivalent to a piston engine.
The working substance, now at atmospheric pressure but at a
considerable temperature, is next discarded, and a new lot taken
into the compressor at atmospheric temperature and pressure. In
order to close the thermodynamic cycle we may suppose that the
same working substance is used over again, enough heat having
been extracted at constant pressure to reduce the temperature
appropriately.
The mathematical treatment of an actual cycle such as the
above has never been attempted directly, and the best that can be
done is to discuss the matter on the assumption that the working
substance is a perfect gas throughout. The approximations that
this assumption involves in the case of the gas turbine are as fol-
lows : The working substance is initially separate portions of oil
and air. Since the work of bringing the oil to the maximum
pressure is insignificant, the actual work of compression is not
quite so great as we take it to be by assuming the working sub-
stance wholly gaseous. However, as we shall see later, in cases
where the proportion of oil is reduced sufficiently to give work-
able temperatures, the oil forms two or three per cent, by weight of
i8
the working substance. Hence but slight error is committed by
assuming that the work of pumping the oil is equal to the work of
compressing on equal weight of air.
In the next place the variation of volume due to addition of
heat by internal combustion is very irregular, and does not follow
the simple law of the perfect gas cycle, where the volume varies
directly with the heat added. Some irregularities, probably
slight owing to the small percentage of oil, arise from the fact
that some of the heat of combustion of the oil is absorbed by its
latent heat of vaporization. This heat causes increase of volume
but not of temperature. Other irregularities arise from the "ex-
pansion " due to combustion. When the volume of a perfectly
combustible mixture of gases is compared with volume of the
products of combustion ; both calculated by Avogadro's law, and
reduced to the same pressure and temperature, an expansion or
contraction is found to have occurred. It can be shown that ex-
pansion always occurs when a hydrocarbon C a H m burns, if m is
greater than ^. Therefore, since most of the hydrocarbons of
which oil is composed are quite complex, an expansion occurs
due to the change of chemical composition. This is independent
of the heating of the products of combustion due to the addition
of the heat of combustion. This is therefore another source of
increase of volume without change of temperature. Usually
there is a considerable amount of dilution in the mixtures used,
particularly on account of the nitrogen of the air, so that the ex-
pansion is not a very large percentage of the whole volume in-
volved, making this irregularity slight also.
The increase of volume due to the two causes referred to of
course represents work done due to liberation of the chemical
energy of the fuel It may be remarked that the work thus done
is probably never included as it ought to be in the tabulated
values of the heat of combustion of fuels. The " constant press-
ure calorimeter " of the Junkers' or other form, which is usually
used, takes no account of energy liberated by combustion but
causing volume variation merely without incidental change of
temperature.
The differences between the actual cycle and the assumed per-
fect gas cycle have thus far been found slight. There is, how-
ever, another difference of much greater magnitude. Recent re-
search has probably established the fact that the specific heat of
all gases increases with the temperature. Therefore the specific
heat of the products of combustion at the high temperatures
attained in the gas turbine is much greater than the constant
value assumed in the perfect gas cycle. It follows that the actual
temperatures attained in the gas turbine are much less than those
calculated on the perfect gas assumption. Since so much approxi-
mation is involved in the perfect gas assumption, we may as well
assume in addition that the working substance has throughout
the specific heat of air at atmospheric temperatures.
Let us consider the effect of the "imperfect" nature of the
actual gases upon the theoretical efficiency of the cycle. As is
well known, in the case of the Carnot cycle imperfection of the
working fluid has no effect UROII the theoretical efficiency. It
may also be shown that the theoretical efficiency is independent
of the nature of the working fluid for any cycle in which heat is
added and taken away along lines of the same constant specific
heat* It does not appear therefore as if the imperfection of the
working substance would have any marked effect on the theo-
retical efficiency in the case of our cycle. That is to say we will
make the hypothesis that the theoretical efficiency of the actual
gas turbine is approximately the same as that calculated on the
basis that the working substance is a perfect gas.
The algebraical and arithmetical calculations for a number of
cases of perfect gas Brayton cycles comprised the master's thesis
of the writer, entitled " Thermodynamics of the Gas Turbine,"
filed at the University of California. A brief abstract will be
given of the most important results.
The heat Q added by the combustion, per pound of working
substance, may be as a maximum the heat emitted by the com-
bustion of as much oil as is necessary to form a perfectly com-
bustible mixture of air and oil amounting to one pound. Ordi-
nary crude or refined petroleum is about 85 per cent, carbon and
15 per cent. Hydrogen. Calculation from this analysis gives
about 15 pounds of air as necessary to burn i pound of oil, with
emission of about 20,000 B.T U. Hence a perfect mixture of
air and oil weighing i pound emits 2O ' OC ^ 1250 B. T. U.
However in order to be sure that no oil will be unconsumed there
should be a slight excess of air, so that we will assume a round
figure of i ,000 as the greatest desirable value of Q. By decreas-
ing the proportion of oil to air we may make Q anything that we
please under 1,000. Of course this figure may need modification
if fuels other than oil are used.
*See Generalization of Carnot' s Cycle, Physical Review, Vol. 16, No. i,
Jan., 1903.
20
The temperature and volume after combustion are calculated
from the value of Q and the specific heat at constant pressure for
air, '238, assumed as constant as already explained. The final
temperature and volume are calculated by assuming adiabatic
expansion from maximum to atmospheric pressure. The heat q
which must be extracted to return the working substance to the
initial condition is found from the specific heat at constant
volume, as before. The heat transformed into mechanical work
is the difference Q q so that the efficiency of the cycle which
we will call e is given by e = 2.
The work done in compression is C and the work done by the
motor is M, so that the net work available for useful purposes is
C M. Of course C M = J (Q q} where /is the mechani-
cal equivalent of heat. A measure of the complication involved
in the necessary compression of the air before combustion will be
given by the ratio of the power required for compression, to the
net power, which we will call r.
Then r = --.
C M
In deciding the values of the various quantities to be used in a
particular Brayton Cycle, we may choose arbitrarily the maxi-
mum pressure p l and the heat added per pound of working sub-
stance Q. The latter, however, must never exceed 1000 as already
stated. When/j and Q are decided, everything else may be cal-
culated directly. On the other hand, we may assign arbitrary
values to any other two quantities whatever, and calculate the
values of p l and Q which must necessarily be used to .secure them.
In cases where the cycle of a heat engine is executed in a motor
involving a reciprocating piston, the maximum pressure and tem-
perature must be brought within reasonable limits, so that these
are made the arbitrarily determined variables. In the case where
ah "impulse wheel" is used as a motor for the Brayton Cycle,
giving a Gas Turbine ; the temperature at the end of expansion
and the linear velocity of the impulse wheel must be within con-
trollable limits. The peripheral velocity of a De Laval Steam
Turbine, a Pelton water wheel, or any similar impulse wheel,
should theoretically be one-half, and practically about three-
eighths of the velocity of the jet which drives it. The velocity of
the jet Fis such that the kinetic energy of a pound of working
F 2
substance, , is, as already stated, equal to the work which
21
would be done by the same pound in expanding behind a piston,
given by M. Hence the assignment of a value to the velocity of
the impulse wheel of a gas turbine amounts to assigning a value
to M. An extreme value for the velocity is 100,000 feet per
minute.
We shall next consider some numerical results for a number of
cases of the Bray ton Cycle, given in Table i. We will among
other things give the speed which the impulse wheel should have
in case the cycle is executed by using a gas turbine. Of course
all of the other values hold good for any form of engine whatever
in which the cycle is executed, whether a reciprocating piston
engine or whatnot. As previously stated, we will assume the
working substance as a perfect gas, so the temperatures we obtain
are very much higher than will occur in an actual case.
Case I is a theoretical Brayton Cycle in which compression and
expansion are both adiabatic. We will suppose a perfect ma-
chine with no losses.
Case II is a modification of the Brayton Cycle, in which the
compression is isothermal and a regenerator is used. We will
again assume no losses. We will find that the regenerator gives
a remarkable gain. The formulas show that the work done by
the motor is equal to the heat added by combustion. Hence we
obtain as useful work all of the heat of combustion except the
work required for compression. The less the compression pres-
sure the less this lost work and therefore the greater the efficiency.
This is not the case, however, if any losses occur. The theo-
retical efficiencies for Case II are remarkably high, but are of
course unattainable since we have assumed a perfect heat engine.
Case III is a recalculation of Case II in which we attempt to
find values for the efficiency which might be expected in an
actual case by assuming the various losses which will occur.
We will assume that the motor only gives 70% of the power
which it theoretically should, that the compressor requires 20%
more power than it should, and that the regenerator has an effi-
ciency of 60%. It appears that we can in no case use the full
value looo for the heat added per pound of working substance,
since the temperatures will be too high for a reciprocating engine
and the velocities too high for a Gas Turbine. An excess of air
is therefore used to reduce the heat added per pound.
The values given for the efficiency in this case are remarkably
high, and since they are results which might be expected in an
actual engine, they are worthy of attention. Of course experi-
22
merit will be necessary to determine just what values of tempera-
ture, etc., are practicable. As stated the values of the tempera-
tures given in Table i do not hold for the actual cycle.
TABLE i. BRAYTON CYCI,E CALCULATIONS.
1
IH
5
*7 bo
(4-4
o
'
3 cO
i,
i
1
o
'
oD
% *>
V&
w ! ||
8.J
S|S
"cd O *
O
gfc
frfr
0*
^ cC
IT* ^j
-M F be
VH ^* ^yQ
fi ^
K**
O S3
^ C
'O'o "S
w t/3
PH J_T
^j . ' t
t4H -j -*-*
S ^
""
co &c
g n W
g^H
S 8^
3 0>
H'S
<r
S 'S 5
.2
ili<r
{S
^S^
J:^
si
CO 2
o
'o'tS
s&^
* 5
o^^
if
H
as,
1"
CO
S-s
cS
C
to
^5
5
t!
^
B
if
s
S
A
X
4
^3
r
r
e
CASE I.
90
90
IOOO
250
O
O
4665
1505
2435
652
.22
.87
115,300
71,680
43
43
Adiabatic Compression.
Perfect Machine with
No Losses.
195
495
495
IOOO
250
IOOO
250
o
4860
1710
5190
2040
2008
546
1562
434
.27
1.07
34
1.38
131,2001 .54
83,740 .54
147,600 .64
98,300 .64
CASE II.
90
IOOO
o
9239
5039
;o8
159,100 .93
Isothermal Compression
with Regenerator.
Perfect Machine with
90
195
195
495
250
IOOO
250
IOOO
o
o
o
o
1965
7376
1498
6087
915
3176
448
1887
39
.1 1
.61
15
79,370
159,100
79,570
159,100
.72
91
.62
87
No Losses.
495
250
o
1176
126
1.03
79,570
49
CASE III.
56
333
2147
1200
74
75,4oo
.27
Iso. Comp. with Regen.
Actual Machine with
Assumed Losses.
Air Excess.
82
61
98
33
61
392
445
545
445
569
o
o
o
o
2 39 8
2867
3233
3179
3699
I2OO
1619
1619
2139
2139
.68
50
47
.40
36
84,800
86,485
99,960
78,155
96,775
30
35
.28
33
CASE IV.
Iso. Comp. with Regen.
79
625
37
1275
600
67
75,000
13
Assumed Losses.
47
638
.36
1680
IOOO
43
75,ooo
.15
Cooling Water.
The excess of air required to keep the temperature down in the
above case demands an increased size of compressor. However,
the temperature could be kept down by the use of water, which
can be introduced more readily. Case IV is therefore one in
which the working substance is a mixture of air and water, the
air being just sufficient for perfect combustion. The water be-
comes superheated steam at the same temperature as the products
of combustion and the mixture operates the motor. We suppose
isothermal compression of the necessary air, and a regenerator as
-23-
before. The regenerator can only utilize the sensible heat of the
steam however, and must reject the latent heat. This lowers the
efficiency, and therefore this case is not as favorable as Case III.
The compressor is much smaller in proportion however, as shown
by the small value of r. The assumed efficiencies of compressor
motor and regenerator were needlessly taken somewhat differently
in this case than in the preceding.
The efficiencies of Case III are so much higher than those of
Case IV that it probably will be best to use air in excess rather
than water, in order to give controllable temperatures.
As stated, we have attempted to take account of all possible
losses in calculating Case III "ley assuming dynamical efficiencies
for the various parts of the apparatus. It is to be noted that we
do not have any extra-thermodynamic or thermal losses in the gas
turbine, which prove so serious in heat engines with reciprocating
motors, such as the steam engine and Otto gas engine. The most
important source of loss not taken account of by theory in a gas
engine is the loss of heat through the cylinder walls to the jacket
water. There is no such loss in the case of the gas turbine, as
the maximum temperature occurs within a chamber where there
are no moving parts, so that cooling is unnecessary. By lining
the combustion chamber thickly, or covering it with a non-con-
ductor, the temperature at the outside, and consequently the
radiation loss, may be made very small, even though very high
temperatures exist within.
One source of dynamic loss in the gas turbine is the friction as
the gases pass through the nozzle and as they pass along the
buckets of the impulse wheel. The steam turbine is subject to
exactly similar losses, and the fact that they have not proven
serious in its development indicates that they will probably not
be serious in the case of the gas turbine. We have attempted to
take account of the loss in Case III by assuming an efficiency of
70% for the motor.
Another source of dynamic loss in the gas turbine is that occur-
ring in the compressor, since its "mechanical efficiency" is.
always less than unity. We have attempted to take account of
this loss in Case III by assuming that the compressor requires
20% more power than it theoretically should.
It is to be noted that losses in compressor and motor which are
comparatively small percentages of the powers of these machines
are much larger percentages of the net power, which is the differ-
ence between the two. Suppose, for instance, that the compres-
24
sor requires theoretically 10 and actually 12 horsepower, and that
the motor yields theoretically 50 but actually 35 horsepower.
The net power will then be theoretically 40 and actually only 23.
That is to say a 20% loss in the compressor and a 30% loss in the
motor cause a loss of 42^ % of the theoretical net power. It is
probably due to this great effect on the net power of compara-
tively small compressor and motor losses that the Brayton cycle
has never been really successful in a reciprocating engine, al-
though tried many times. A reciprocating motor necessarily has
a water jacket, and this adds possibly a 50% thermal loss to the
dynamic motor loss. The net power is therefore enormously re-
duced. In the gas turbine we escape the thermal loss and thus
have some possibility of obtaining reasonably high values of the
net power. This is shown by the net efficiencies of Case III,
Table i, which have been calculated with reasonable values of
the various losses taken into account, as already stated. These
efficiencies compare so favorably with those obtained in the mod-
ern steam and gas engine that it seems quite possible that after
sufficient experiment and development the gas turbine may trans-
form heat into work more efficientlv than either of these.
CHAPTER IV.
THEORY OF NOZZLES AND FREE EXPANSION.*
Suppose that a portion of fluid, liquid or gaseous, with a given
volume and temperature, exists in a region where a given pres-
sure is continuously maintained. The fluid then represents a
certain amount of stored or "potential" energy. Part of this
exists as heat energy due to the temperature of the portion, and
the balance is due to its presence in a region where the pressure
is continuously maintained. -If the volume leaves the region,
work will be done, the amount being given by the product of
pressure and volume. Hence the presence of the portion of fluid
in the region under pressure implies stored energy of this amount.
Suppose that in some way or other our portion of fluid reaches
a region of lower pressure, the change being made without add-
ing or extracting heat, that is " adiabatically." The potential
energy will then be found to be less than before. The difference
has left the gas and must be sought elsewhere.
There are two ways in which the portion of fluid may pass
from one pressure to another. In the first place, it may be
placed behind a piston and allowed to expand gradually, as when
steam expands in a steam engine. Then the difference between
the potential energies for the two pressures appears as the me-
chanical work done by the piston. This may be called "con-
strained expansion. ' ' In the second place, the portion of fluid may
expand from one pressure to the other by passing through a
" nozzle " or orifice in the wall of the vessel in which the higher
pressure exists, into the region where the lower pressure exists.
This is called "free expansion." The result of free expansion is
the reduction of the portion of fluid to the same final condition as
regards pressure, volume and temperature as in the case of con-
strained expansion, so that the loss of potential energy must be
the same.
Now each small portion of the fluid, while in the orifice, is
always passing from one point to another where the pressure is
slightly lower. At any instant, therefore, any small portion is
acted on by a force due to the difference between the pressures
behind it and before it. This force does work in increasing the
velocity of the portion of fluid considered, so that the mass ac-
*The substance of this chapter was published as an advance portion of
this thesis in Sibley Journal, Vol. 17, No. 6, March, 1903.
26
quires a gradually increasing stock of kinetic energy as it moves
through the orifice. The kinetic energy of the fluid as it finally
leaves the orifice, together with any energy dissipated in the
form of heat due to friction against the sides of the orifice, repre-
sents the difference in the potential energy of the fluid when at
the higher and when at the lower pressure. That is to say, in
free expansion we have kinetic energy corresponding to the work
done on the piston in constrained expansion.
In both cases, however, work is done representing the differ-
ence between the potential energies corresponding to the two
pressures. The kinetic energy due to free expansion is therefore
exactly the same as the amount of work which would be done on
an engine piston by the same fluid expanding between the same
pressures.
The kinetic energy of a jet of fluid issuing from an orifice after
free expansion may be taken from the fluid and applied to do use-
ful work by means of an " impulse wheel " having a number of
vanes or buckets on which the jet impinges. Any form of motor
in which a piston is moved by a fluid under pressure may there-
fore have substituted for it an impulse wheel driven by a jet of
the fluid issuing from a vessel under the same initial pressure. A
hydraulic pressure engine may therefore be replaced by a Pelton
water wheel, a steam engine by a De Laval steam turbine, and a
Brayton Gas Engine by a " gas turbine."
The kinetic energy of a jet of gas after free expansion is usually
enormous compared with the mass, so that very high velocities
are attained. The friction losses may therefore become serious if
the nozzle is not properly designed. On the other hand the
friction losses are comparatively small with proper precautions.
A requisite for minimizing friction loss is that the velocity in the
nozzle shall gradually and continuously increase, since sudden
variations cause shock. As the increase of velocity is caused by
the decrease of pressure from point to point, the pressure should
decrease gradually and continuously.
In the case of water, the volume does not vary with the
pressure, and hence the velocity at any point along a water nozzle
depends only on the cross-section at that point, regardless of the
pressure. Hence a continuous increase of velocity is secured by a
continuous decrease of section. That is to say a nozzle for water
should converge continuously from the point of greatest pressure
to the point of least pressure.
A nozzle for a gas must have a radically different shape in order
that the velocity may increase and the pressure decrease con-
tinuously. The volume of a given mass of gas increases as pres-
sure decreases and this increase of volume must be taken into ac-
count in conjunction with the desired increase of velocity in order
to find the nozzle shape. Now the nozzle must converge in order
to give an increase of velocity, and diverge in order to allow for a
change of volume due to decrease of pressure. If the velocity is
to increase faster than the volume increases, the nozzle will con-
verge on the whole. However, it is quite possible that the pressure
may decrease so that the consequent rate of increase of volume is
greater than the consequent rate of increase of velocity. Then
the nozzle must diverge in order to provide for the greater rate at
which the volume increases. \
As we will show presently a decrease of pressure down to about
one-half of the initial pressure causes a greater rate of increase of
velocity than of volume, requiring the nozzle to converge. If the
pressure is to decrease still further the volume begins to increase
more rapidly than the velocity and hence the nozzle must diverge.
That is to say when the final pressure is less than about one-half
of the initial pressure the nozzle first converges and then diverges
in order to allow for a continuous decrease of pressure.
If the nozzle is not shaped in this manner the pressure cannot
decrease continuously down to the final pressure. That is to say
a convergent nozzle is an abnormal shape for a gas, and gives a
very abnormal action which we will later briefly investigate. A
nozzle first convergent and then divergent is the normal shape for
a continuous decrease of pressure and nothing abnormal or
mysterious occurs when such a nozzle is used. It is often popu-
larly supposed that the free expansion of gases is very mysterious
owing to the irregular action in a convergent nozzle, whereas this
irregularity is due solely to the fact that a convergent nozzle is
proper only for non-expansible fluids such as water.
With this introduction we may proceed to treat our subject
mathematically. We will take the case of a perfect gas although,
as already stated, the gases used in a gas turbine are by no means
perfect. However the results will apply approximately. Our
results also hold approximately for the case of superheated steam,
now commonly used in steam turbines, and for compressed air
issuing from an orifice. The same methods would be used for the
case of saturated steam, but the equations would be somewhat
different. We will neglect friction altogether. The gas passes
through a nozzle from a region where the pressure is p^ pounds
per square foot to a region where the pressure is zero. Since the
28
pressure is to decrease continuous!}' from />, to O we may cut the
nozzle off at a point where the pressure has diminished to/ 3 and
thus obtain a nozzle suited for discharge into a region of pres-
sure /> 3 . Let the general conditions at any point whatever along
the nozzle be denoted by letters without subscripts, p being the
absolute pressure in pounds per square foot, v the corresponding
specific volume in cubic feet per pound, T the absolute Fahrenheit
temperature, Fthe velocity in feet per second, and A the nozzle
area in square feet.
Let the conditions at the entrance of the nozzle be denoted by
subscriptj, at the point of minimum diameter by subscript 2 , at
the point where any particular pressure (as for instance atmo-
spheric pressure) is reached, by subscript 8 , and at the point
where zero pressure would be reached if the nozzle continued so
far, by subscript 4 . Let M be the weight of gas passing through
in pounds per second. Let <: p and c v be the specific heats of the
gas and k their ratio. R is the gas equation constant, equal to
^, and / is the mechanical equivalent of heat. As is well
known R =J( c^ c v ).
Let us consider that portion of the nozzle between the entrance
and the general point where the pressure is p. The mathematical
statement of the fact that no energy is created within this space,
(neglecting the kinetic energy of the gas entering the nozzle) is
The neglect of the entering kinetic energy amounts to the as-
sumption that the entrance velocity is zero, so that the nozzle
area at the entrance must be infinite. Practically, however, the
entrance area may be comparatively small and the entrance
velocity be in the neighborhood of 100 to 200 feet per second, and
yet the kinetic energy of the entering gas will be quite negligible
as compared with the other terms of the expression.
The above expression reduces to
~r=P.v l -pv+Jc v (T l -T)
Since p l v l = R T pv=RT, and
this becomes
Hence we have as a general expression for the velocity
29
V-VigJc^Tt-T) (0
We will assume, as is usual in similar cases, that the gas ex
pands adiabatically in passing through the nozzle.
Hence the usual adiabatic relation holds,
T,
Therefore
This expression shows the relation between the pressure and
velocity at any point. It will be seen that the velocity always in-
creases as the pressure decreases.
We may express the relation between the nozzle area and the
pressure at any point by substituting for Fin (2). From the re-
lation A V = Mv and the adiabatic relation
, we have V-
v '
Then
r .
This gives us a means of investigating the variation of the area
A as the pressure/ varies. Differentiating with respect to/,
where B is a group of coefficients all of which are essentially posi-
tive. Suppose that / is equal to /j. Then since k is about 1.41,
will be positive. will continue positive, that is the noz-
dp dp
zle area will decrease as the pressure decreases, while the pressure
varies from/! to a value such that the parenthesis on the right
vanishes. That is to say the nozzle must converge while the
pressure varies from p l to a value / 2 such that
'-' (5)
Evidently - vanishes for/ / 2 and hence A has a minimum
dp
value for this pressure. As the pressure / still further decreases
is a negative and hence the nozzle diverges.
dp
30
That is to say a nozzle in which the pressure continuously de-
creases has a point of minimum area at the place where the pres-
sure is/> 2 and diverges beyond this point. If the shape is differ-
ent from this, the pressure cannot decrease continuously and an
irregular action occurs.
In the case of air and a number of other gases k= 1.41 and
the pressure at the throat or point of minimum area is / 2 =
527^1-
By substituting the value of the throat pressure in the adiabatic
relation between pressure and temperature, the throat temperature
is found to be
T 2 T
T+l l
By substituting this value for T 2 in the general expression for the
velocity (i) we have as the throat velocity
This may be shown to be the velocity of sound in the gas for the
conditions of pressure and temperature obtaining at the throat.
It is therefore the velocity at which a disturbance will be pro-
pagated along the stream flowing through the nozzle. Since
the velocity of motion beyond the throat is greater than the
velocity at which a disturbance can be propagated, it is impossible
for a disturbance to be propagated backward along the stream of
gas. Hence if the portion of the nozzle beyond the throat were
to be removed, so that the gas was discharged at a pressure p^
into a region of lower pressure /> 3 , the disturbance thus produced
could not in any way affect the flow of gas between the entrance
and the throat. That is to say if a nozzle is wholly convergent and
the region into which it discharges has a pressure less than / 2 the
discharge will nevertheless be at pressure p 2 and the gas will ex-
pand from / to /> 3 by diverging in the region beyond the end of
the nozzle, just as if the proper divergent portion of the nozzle
existed beyond the throat. This explanation of the well known
fact is in essence due to Professor Osborne Reynolds.
In order to compute the sizes necessary for the design of a
nozzle we proceed as follows : By substituting the value of p 9
given by (5) in the general expression (4) for the area we obtain
a value for the throat area which reduces to
\
A \ 2gk
For the case of air and those gases for which k= 1.41 the throat
area becomes
In order to most conveniently compute the area and velocity at
the end of the nozzle where it discharges into, a region at pressure
/ 3 we first find the final temperature and specific volume by sub-
stituting the value of p 3 in the general adiabatic relations, whence
We may then obtain the final velocity by substituting the value
of 7~ 3 in the general expression for the velocity (i), which gives
Also, v .
The final area may then be found by substituting the values thus
found for F 3 and v 3 in the general expression A V Mv.
By taking p = o in (4) it appears that the final area for dis-
charge into an absolute vacuum is infinite. This is du'e to the
fact that the specific volume is infinite for zero pressure. How-
ever, the area increases very rapidly for a very slight decrease in
pressure in this vicinity, and the final area for what is practical!}'
considered a vacuum is not particularly great. It may be re-
marked that the final temperature is zero for discharge into a
vacuum.
The theory thus far has been known in essence for some time,
although the present method of treatment is possibly new. We
have found, however, only the entrance, throat, and final diame-
ters of the nozzle. The curvature of the nozzle and the distance
between these points, is not determined. The following attempt
to complete the discussion is believed to be original.
We shall assume as a basis for procedure the manner in which
a particle of gas is to be accelerated as it passes through the noz-
zle. Evidently in order to avoid impact loss due to changes of
velocity the acceleration should proceed in some regular manner.
One possible assumption is uniform acceleration. That is to say
the force furnished by the gradually decreasing pressure which
serves to increase the velocity may be taken as constant. We
will presently work out the shape of a nozzle to secure this result.
However, this is not necessarily the shape giving a minimum
friction loss. For instance it is conceivable that the losses might
be less if the particles of gas were accelerated less rapidly as their
32-
velocity increased. Then the accelerating force would vary in-
versely as the velocity. Theoretical considerations may be dis-
covered which will give some absolute criterion as to the best
method of acceleration. However, the matter must probably be
decided by experiments with nozzles designed according to vari-
ous reasonable assumptions.
The assumption of uniform acceleration seems most reasonable
and therefore we will take it up here.
Let x be the variable distance from the nozzle entrance to any
point where the pressure is p. Then x^ will be the throat dis-
tance, x s the total length of the nozzle, and x the length for dis-
charge into a vacuum. We have by hypothesis
<Tx =
dt*
where C is a constant to be determined presently. Also
V=
dt
Let us substitute from these expressions in the identity
dt dt*
The substitution gives
F 2 = 2 C
dx
Integrating and inserting the obvious value o for the integration
constant,
V* = 2 Cx. (6)
We evaluate the constant C by assuming a value for x 9t the length
of the nozzle for discharge at pressure p s . This must be decided
by experience, being long enough to avoid losses due to too rapid
acceleration, and short enough to avoid excessive friction against
the nozzle walls. Having computed F 3 in the manner already
described we have, by substitution in the above expression,
c=YL
2*3
By substituting the value for V. A obtained from the general ex-
pression (2), we have as an equivalent expression for C,
Next, let us express the relation between the pressure at any
point and the distance to it. By substituting in (2) the value of
]/ from (6) we have
ZLEIWITh UNIFORM ACCELERATIO
FIG. 3. Nozzle Section with Pressure, Temperature and Velocity Curves.
34
-gjf* T, \ l _(P_
We have taken arbitrarily x 3 the length of nozzle for discharge
into a region of pressure p s . However the nozzle up to x^ will be
the same if we suppose that the nozzle is long enough to discharge
into a vacuum, and that the particular pressure p. A is reached at
the arbitrary intermediate point x^. The total length of the noz-
zle will then be x^ which is found by placing p = o in the above
expression. Then
We shall find it convenient to use x simply as an abbreviation.
Then the relation between x and p is
*-,Jx-(A^l (9)
Next, let us express the area at any point in terms of the dis-
tance to that point. By substituting the value of Ffrom (6) and
the value of p from (9) in (3), we have
A
do)
This is the equation giving the nozzle shape in order that the
gas passing through may be uniformly accelerated. Curve i, Fig.
3 shows the section of a nozzle for a particular case constructed by
this equation. The final pressure is zero, but by cutting off the
nozzle at the proper point the discharge pressure may have any
value that we please.
It must be remarked that the equations deduced give the areas
perpendicular to the stream tines, or lines of flow. When the
area is small compared with the length, these are practically par-
allel to the axis and the areas may be taken on normal plane sec-
tions of the nozzle. When the nozzle begins to diverge consider-
ably near the end the stream lines spread out and the areas given
by the equation can no longer be taken on plane surfaces.
If we express A in full by substituting the values of the con-
stants C and x t from (7) and (8) it will be found that A is directly
proportional to ^/and \/T r inversely proportional to p^ and also
involves the ratio ( $- ) . It follows, therefore, that a nozzle
-35-
constructed for a given initial pressure and temperature may have
these quantities varied and still be of the correct shape for uniform
acceleration. The quantity of gas passing through, M, will change
however, since - l is constant for a given shape: Also,
Pi
PA. 1 must be constant for a given shape, so that the final pres-
P^J
sure will vary directly with the initial pressure for a given length
of nozzle x a and a given shape.
By differentiating (10) and putting -- =o we obtain the fol-
% d x
lowing value for x tt the distance to the throat or point of minimum
area :
That is to say, the throat is at a constant fraction of the length
of a complete nozzle, regardless of conditions. This value of JC 2
substituted in (10) will give the throat area A 2 previously found.
It will be interesting to note the variation of pressure, tempera-
ture and velocity as we proceed from point to point along the noz-
zle. (6) gives the variation of velocity from point to point and
(9) the variation of pressure. By substituting in (9) the adiabatic
relation between pressure and temperature, we obtain as the tem-
perature variation
This shows that the temperature decreases uniformly for our case
of uniform acceleration.
'From these equations the values of pressure, temperature and
velocity for the nozzle of Curve i, Fig. 3 are plotted in Curves
2, 3, 4, each value being laid off opposite the point of Curve i, to
which it corresponds.
CHAPTER V.
THEORY OF A RAPIDLY ROTATING DISC ON A FI<F,XIBI,E SHAFT.
One of the possible methods of operating a gas turbine is the
use of an apparatus similar to that of the De Laval steam tur-
bine, involving a single wheel rotating at a very high speed.
The wheel must of course be very carefully balanced, so that the
center of gravity coincides as nearly as possible with the center
of the hole bored to fit the shaft on which the wheel is mounted.
Even with most careful workmanship, however, the center of
gravity is always far enough away from the bore center to give
rise to centrifugal forces of considerable magnitude at the high
rotative speeds necessary. The evil effect of this lack of balance
is obviated by Gustav De Laval's highly ingenious invention of
a flexible shaft. As the action of the flexible shaft is very com-
monly misunderstood, and as the full mathematical theory has
probably never been given, it will be here discussed as a matter
of incidental interest.
We will use the following notation throughout this chapter.
O = center of rotation.
B = center of bore, provided the shaft is initially straight. In
general, B is the point coinciding with O when there is no
load on the shaft.
C = center of gravity of disc.
a = distance between center of bore and center of gravity = BC.
This is the amount by which the shaft is out of balance. It
is always a small quantity.
K= force required to deflect the shaft unit distance. This is a
function of the dimensions and material of the shaft and
character of the supports. For ordinary bearings the shaft
is a beam fixed at the ends. If there is a swivel bearing at
one or both ends, the beam is supported merely.
M= weight of disc. The weight of .the shaft is assumed to be
neglegible.
w = angular velocity of disc in radians per second. This is the
angular velocity of B about O, and is supposed to be main-
tained constant by some external agency.
* = period which the disc would have if the shaft were set into
transverse vibration. Then it may be shown that \l/ V
\
.
M
37
where g is the gravitation constant. The time of a complete
vibration is then .
R = distance OC between center of gravity and center of rotation
for equilibrium. We will find later R =
L = twisting moment required to twist the shaft and disc through
one radian.
/ = moment of inertia of the disc about its center of gravity.
< = period which the disc would have if the shaft were set into
torsional vibration. Then it may be shown that <j> = \~JL
The time of a complete vibration is ?- rr .
<p
Fig. 4 represents the ordinary case of a shaft slightly out of
balance. The centrifugal force deflects the shaft until the inward
force due to the deflection is equal to the centrifugal force. We
neglect gravity for the present, and may suppose the shaft verti-
cal. The centrifugal force acting on the center of gravity is
r w 2 . This gives rise to a rotating force of the same magnitude
g
pulling on the bearings and tending to cause their vibration.
The shaft deflection is O B = r a. The condition that the force
due to the deflection is equal to the centrifugal force is
M rv ? = K(r a)
g
which reduces to
a
(i)
-0'
For ordinary shafts $ is large, so that for moderate speeds
is less than unity. Then the greater w the greater will be r.
That is to say, the greater the speed the greater will be the shaft
deflection, and the more violent the rotating pull on the bearings,
given by
MJ K
This is the theory of the well known difficulties occurring with
high speed machinery. The remedy is to make M the weight of
the rotating parts, as small as possible, K which gives the stiff-
- 3 8-
ness of the shaft, as great as possible, and a which gives the ec-
centricity of balancing, as small as possible.
If the angular velocity be made greater, the deflection will in-
crease, and finally, if the angular velocity w be made equal to ^,
the deflection will theoretically become infinity. This will
not actually occur, however, as the law of proportionality of
load and deflection does not hold except for small deflections, and
because vibrations of period ^ which may occur, will also have a
period o>. \j/ is called the "critical speed," and considerable dis-
turbance occurs however when o> reaches this value. No mathe-
matical account has ever been given of the exact action.
Suppose now that w is made to exceed the critical speed. Then
the denominator of the expression (i) becomes negative, and in
order to have a positive value for r we must make a negative also.
This is done by taking it in a direction opposite that of Fig. 4, so
that the center of gravity is inside of the center of bore, as in Fig.
5. Fig. 5 therefore represents the position of equilibrium when w
is greater than the critical speed if/. If we count a as essentially
positive in this position, the condition for equilibrium is
M R ^__
g
which reduces to
o_
(2)
a]
-39-
Here the greater the angular velocity to the less will be R. That
is to say, the center of gravity actually comes nearer to the center
of rotation as the speed increases. For any given rotative speed,
we secure this result by making tf = as small as possible,
M
which is accomplished by making K, which gives the stiffness of
the shaft, as small as possible. That is to say, the more flexible
the shaft the nearer the center of gravity comes to the center of
rotation. The center of gravity always rotates in a circle whose
radius is given by (2), however. The popular statement that " a
flexible shaft allows the disc to'rotate about its center of gravity "
is therefore erroneous. If the disc were to rotate about its .center
of gravity, the shaft would be deflected a distance a, so that there
would be an unbalanced deflection force a K.
The rotating force pulling on the bearings and tending to vi-
brate them, for the case of Fig. 5 is
g J _ g
K Mrf
This is diminished and the machine made to run with slight vi-
bration by making K, which gives the stiffness of the shaft, as
small as possible and Mthe weight of the disc, and o> the angu-
lar velocity, as great as possible. This is all expressed by the
statement that w -i- \f/ should be as great as possible. That is to
say, the working value of the speed should bear as great a ratio
as possible to the critical speed in order to reduce vibration of the
bearings.
The position of Fig. 4 is sometimes called rotation with the
"heavy side out" while that of Fig. 5 is rotation with the
" heavy side in." That is to say, if a mark be made during rota-
tion by touching the portion of the edge of the disc furthest from
the center of rotation, in order to find the place to cut away metal
in order to correct the balancing, then metal must be taken from
the side of disc nearest the mark if the speed is below the critical
speed, and from the side away from the mark if the speed is
above the critical speed.
The critical speed in radians per second we have found to be
t = \ K g. This reduces to [32.16 X 12 k = , 8? - 6 \~k
\| M 27T^J M ^-^r
revolutions per minute, where k is the load in pounds required to
deflect the shaft one inch and M is the total load in pounds sup-
posed concentrated at the shaft center.
4
As we have seen, Fig. 5 shows a position of equilibrium. We
have next to show that the equilibrum is stable. That is to say,
we must show that if any accidental cause displaces the center of
gravity from the position shown, it will return and not seek some
new position.* The balance of the chapter will be a tedious
proof of this fact.
In discussing the motion of the disc we will make use of the
well known fact that the center of gravity C moves as if all of
the forces acting on the disc were applied at it, the mass being
concentrated there also ; and that the disc rotates about the center
of gravity as if it w r ere fixed in space.
First we will consider the motion of the center of gravity only,
and will assume that the shaft is vertical. Suppose that the disc
is not rotating so that the B Fig. 5 will coincide with the point O.
Then the center of gravity C will assume a position C slightly
below O.
If the point B be displaced in any direction from O the elastic-
ity of the shaft will cause a force proportional to the displacement
tending to return B to O, We will suppose the disc displaced
without rotation, so that the center of gravity C will have an ex-
actly equal displacement from its position of rest C ' . Since the
center of gravity moves as if the displacement force acted on it,
we may consider that the center of gravity C is acted on by a force
proportional to its displacement from C '. The center of gravity
will then have a simple harmonic motion, and will vibrate back
and forth until the energy is absorbed by molecular friction. The
period of such a vibration is well known to be $ =& using the
notation already given.
The most general case of harmonic motion will be when two
displacements at different angles and at different times are given
to the disc. It will then vibrate in an ellipse with period \j/.
We may now apply any other forces to the shaft when it has
this vibration, and the disc will then have a displacement com-
pounded of that due to the vibration and that due to the new forces.
Suppose, for instance, we apply a pull to the center of gravity of
constant magnitude but of varying direction, rotating in fact with
* The discussion up to this point substantially as here given, was published
by Stodola in Zeitschrift des Vereines Deutscher Ingenieure, Vol. 47, Nos.
2 and 4, Jan. 10 and 24, 1903. However, the matter is given here since it
was prepared independently before the above publication. The balance of
this chapter is probably new. Stodola gives a treatment along totally dif-
ferent lines which does not seem satisfactory.
an angular velocity o>. So far as the shaft is concerned this force
will cause a rotating displacement, to be compounded with the
vibrating displacement. Such a rotating force is furnished by the
"centrifugal force," which is in equilibrium with the force due
to a rotating displacement R + a of the shaft center B. Hence
the disc may rotate in the position shown by Fig. 5, and at the
the same time have an elliptical vibration.
Conversely, if the disc be rotating in the position of Fig. 5, and
any accidental displacement be given to it, it will at once begin
to vibrate, in addition to the rotation, in the same way as it would
if the displacement were given to it when at rest.
The above reasoning is perfectly rigid, but nevertheless the step
is perhaps a large one to make at once. We will therefore at a
later period, for verification, form the differential equation of mo-
tion of the disc, and show that the integral gives a displacement
compounded of an elliptical vibration and a rotating displace-
ment R. Such a mathematical treatment hides the true nature
of the actions, however, and the method above used is much more
luminous.
It must be noted that if a displacement from the position of Fig.
5 be gived the disc during rotation, the resulting vibration will
always be parallel to the original displacement, and will not rotate
in direction. That is to say, if a displacement should happen to
be made in the direction of the instantaneous position of the radius
to the center of gravity, the vibration induced will not continue to
be radial, since then the direction of vibration would constantly
change, but it will be in a direction fixed in space parallel to the
initial direction.
If the elastic force resulting from an initially radial displace-
ment were always radial, it can be shown that the position of Fig.
5 would be unstable.
Any accidental displacement from the position of Fig. 5 will
therefore in general cause the point C to vibrate in an ellipse about
the equilibrium position, the axes of the ellipse always remaining
parallel to their original position in space. Owing to molecular
friction this vibration will soon die away, the size of the ellipse
gradually decreasing until the position of Fig. 5 is again reached.
Next let us consider the effect of the vibrations just considered
in causing rotation of the disc about the center of gravity. Let
Fig. 6 represent any instantaneous position of the points O, Z?and
C while C is making an elliptical vibration. The path of B will
be parallel since the vibrational displacements have involved no
42
rotation of the disc. The net force acting on the disc is then pro-
portional to O B. If the center of gravity is now fixed and a force
proportional to OB acts on B as shown, there will be a twisting
moment on the disc proportional to OB X CD, Fig. 6. This
twisting moment is resisted by the torsional elasticity of the shaft,
so that there is a tendency to produce torsional^ vibration.
Since all of the displacements are very small the shaft may be
considered as perfectly straight in computing the torsional elas-
ticity. Now, both O B and CD are very small quantities as com-
pared with the torsional elasticity, so that the angle through
which the shaft is rotated by the twisting moment OB x CD is
of the second order of small quantities and may be safely neglect-
ed. That is to say, a displacement of the disc has no appreciable
effect in causing additional rotation of the disc about its center of
gravity beyond that due to the constant angular velocity.
There is another possible displacement from the position of Fig.
5, consisting of a finite rotation of the disc so as to cause torsion
in the shaft, as shown in Fig. 7. If the angular velocity o> did
not exist, a torsional displacement would cause a harmonic tor-
sional oscillation of the disc about its center of grovity. The
FIG. 6.
Transverse
Vibration.
FIG. 7.
Torsional
Vibration.
FIG. 8.
Effect of
Gravity.
FIG. 9
Most General
Vibration.
twisting moment due to torsion of the shaft is a pure couple, so
that it causes rotation about the center of gravity, even though
the center of bore is at a different point. The period of the tor-
sional oscillation will be <f> = \f-* using the notation already
given. If now the disc be given a constant angular velocity w in
addition to the torsional vibration, the latter will persist just as
before, so far as rotation of the disc about its center of gravity is
concerned.
The force due to the net deflection, proportional to OB, Fig. 7,
will here also give a twisting moment about the center of gravity
proportional to OB x CD. This is of the second order of small
quantities and will be neglected as before.
-43-
The line B C, Fig. 7, will therefore rotate with a constant an-
gular velocity o>, and also oscillate back and forth so as to make
a variable angle a with the normal position B' C' . Since a varies
harmonically with period <, we will have a = c t cos <f> t. The in-
troduction of the constant angular velocity does not affect the har-
monic oscillation as already stated.
Next let us consider the effect of such a torsional oscillation on
the displacement of the center of gravity. Owing to the previ-
ously considered displacements without rotation, the points^ and
C were displaced equally beyond the displacement necessary to
balance the centrfugal force, SQ that the force acting on the center
of gravity could be taken as proportional to its own displacement.
Now we must consider that the force acting on the center of grav-
ity is proportional to the displacement of B beyond the position
necessary to balance the centrifugal force. This is not the dis-
placement of the center of gravity owing to the variable angle a.
The position of C without the vibration we are now considering
would be at a distance R from O, Fig. 7, near C ', and owing to
the addition of the vibration will be supposed to be C at any in-
stant. The vibration we are considering is the absolute vibration
of the center of gravity in space in addition to the constant rota-
tion.
Suppose we put OC' = = fi-\-v and C C' = rj and resolve
forces and accelerations along and perpendicular to the moving
axis O C' . v and rj will then give the departures from the equi-
librium position due to the vibration we are considering.
The force perpendicular to the axis is AT X B B' or K(-rj +
flsina), since we have CC' = t] = BB' a sin a. Since a will
always be small we take a instead of sin a. We have already re-
marked that a = r t cos <f> t, so that the force perpendicular to the
moving axis is K '(rj -f- ac^ cos <f> t}.
The total force in the direction of the moving axis is
- K X O C or K ( + a cos a).
Now a portion of this, K (R + ), is required to maintain equi-
librium against the centrifugal force, so that the remaining force,
K ' (y a -f- a cos a), is the force in the direction of the moving
axis tending to cause the vibration we are considering. Since a
is small we may for a second approximation put cos a i, so that
the force becomes Kv. The accelerations along and perpendic-
ular to a moving axis are respectively (Routh's Rigid Dynamics,
Vol. I, page 176),
d 2 2 i d ,
44-
^9_,.+L _)
Since w is constant we may reduce the final terms. Also substi-
tuting in terms of v, we have as the accelerations,
_-_.
Now the term R w 2 gives the acceleration due to the rotation in
the equilibrium position so that the acceleration due to the added
vibration alone is found by omitting this term. We have already
omitted the corresponding centrifugal force, K '(R + a).
Bquating the forces previously found along and perpendicular
to the moving axis to the forces required to produce the above ac-
celerations we have
,2 , . .-. d 2 77 o i d v
^ 2 (77 -fa^COS^O = ' 7?a> 2 -f 20)-.
at at
This is a pair of simultaneous linear differential equations whose
solution is complicated. The integration will be omitted and the
integrals only given. These will be found correct by substitution
in the equations.
v = 2 B to < sin < /
y = _ B O 2 - co 2 - < 2 ) cos <f> t
where
Since v and rj are displacements along and perpendicular to the
rotating axis O C' , and since the values above evidently repre-
sent a harmonic motion in an ellipse with axes in these directions,
it follows that any accidental torsional displacement will cause a
vibration of the center of gravity with period < in an ellipse whose
axes rotate with an angular velocity w, one of them always coin-
ciding with the radius vector to Ihe equilibrium position of the
center of gravity. This is in addition to the simultaneous tor-
sional oscillation of the disc and shaft. This oscillation and con-
sequently the vibration of the center of gravity induced by it will
of course rapidly die out owing to molecular friction, so that the
equilibrium position of Fig. 5 will be again reached.
-45-
We have always supposed a vertical shaft so that gravity
need not be considered. Let us now make the shaft hori-
zontal and take gravity into account. First taking all the
forces as acting on the center of gravity, we will have a down-
ward pull M and an equal upward pull due to a down-
ward deflection of the shaft, the displacement of the disc
M
being ^> The point C will therefore rotate in a circle of radius
R whose center is O Fig. 8, a distance below the axis of ro-
JK
tation O'.
Next let us consider if rotation occurs about the center of grav-
ity due to the action of the weight as shown in Fig 8. If the
center of gravity Cbe fixed there will be a force proportional to
the net deflection O B acting at B, tending to rotate the disc and
resisted by the torsional elasticity of the shaft. The twisting
moment will be proportional to O B X CD, Fig. 8, and will
therefore be a small quantity of the second order, so that the
torsional vibration induced will be negligible just as in the pre-
vious cases.
We have now analysed the matter completely and taken ac-
count of everything which can occur. The methods used have
been chosen so as to show most clearly the actual nature of the
various actions, and have perhaps been rather inelegant mathe-
matically. We will therefore go through with the matter again
from a strictly mathematical standpoint for verification of our re-
sults. This treatment is distinctly obscure so far as the nature
of the action is concerned however.
Fig. 9 shows the most general position of the various points.
Since we have a rotation with angular velocity w, induced by an
external source, we take as a reference line an axis rotating with
this velocity and therefore making an angle <o t with the X axis
at any instant. The most general displacement of the shaft
center B will be along and perpendicular to the rotating axis.
We will take as the amount of the net deflection at any instant
p O B, and suppose that it makes an angle 8 with the reference
axis. Such a displacement would leave B C parallel to the refer-
ence axis. We will also suppose a rotation of the disc and the
line B C and a twist of the shaft through an angle a. We will as
is usual first find the motion of the center of gravity by taking all
of the forces as acting on it. We will resolve forces along the X
and Faxes. At first sight it would seem simpler to resolve forces
- 4 6-
along and perpendicular to the moving axis. However, the re-
sulting equations turn out to be much more difficult to integrate.
The force equations are
M d 2 x f * *\
- - = K p cos (o> t 8)
g dt
Evidently
p cos (co / 8) = x + a cos (w t -f a)
p sin (w / 8) = jj/ + a sin (w /f + a) .
Hence, our force equations may be written
^.= - ^ { x + a cos (o, / + a) j
2 /* *\
^^ = ^ 2 J j/ -f a sin (w / + a) I g.
In order to integrate these we must find an expression for a.
This we do by forming the equation of rotation about the center
of gravity, assuming that it is fixed and that all of the forces act
to cause rotation about it'.
The force moment equation then is
We have
dt* dt*'
Now A' si n (a -f 8) is of the same order of magnitude as L a.
p and a are both small quantities so that their product is of the
second order of small quantities, and therefore negligible. Hence
the term Kp a sin (a -f- 8) is negligble as compared with L a.
This amounts to saying that the twisting moment A" X OB X
CD, Fig. 9 can never cause an appreciable torsional vibration of
the shaft.
Hence we have
This is a linear equation whose solution is well known to be
a = c v cos <j> t.
There is in general a second arbitrary constant which we may
take zero without loss of generality, beginning our count of time
from the instant when the vibration is at one extreme.
47-
It is to be remarked that a variable twisting moment would
really be required to drive the shaft at the constant angular
velocity w, owing to the variation of a and of the shaft displace-
ment perpendicular to the reference line. Both of these should
be balanced by a variable external twisting force in order that
there should be no angular Deceleration at the extremity of the
shaft. However, there are usually rotating bodies with consider-
able mass geared to the shaft, and the inertia of these serves to
prevent any appreciable angular acceleration.
Substituting our value of a in the force equations above, and
taking a as small so that we can put cos a=r i sin a = a, we have
5 -f ^ x if/ 2 a cos w t H- V 2 # *\ cos < t sin w /
d 2 v
5 + "fy ^ 2 # sin <o / i/r 2 a-c^ cos <f> t cos o> t g.
These are linear differential equations which are integrated by
well known methods which we need not consider here. The form
to which we reduce the integrals has been arranged to exhibit
the various individual vibrations already discussed.
The integrals are
x = R cos o> / -f c 3 cos ft cos O / + ft ) + <: 3 sin ft sin O t + ft)
-f- B \ ( i/f 2 to 2 < 2 ) sin co t cos <f> t + 2 co < cos co / sin ^> / !
<o t + ^ 2 sin ft cos (i/' / + ft) + ^ 3 cos ft sin ($ t -\- ft)
co 2 < 2 ) cos <o / cos < / 2 co c/> sin co t sin </)/[
3 K
the arbitrary constants of the initial displacements. These are
evidently the projections on the J^and Y axes of the following :
A rotation in a circle of radius R with a constant angular velocity
o>, an elliptica 1 vibration with axes fixed in space and angular
velocity ^, an elliptical vibration with axes along and perpendic-
ular to the moving reference axis and angular velocity <, and a
displacement vertically downward. These are the motions already
found otherwise the coefficients also agreeing. Hence, we con-
clude finally that the most general possible motion will consist of
vibrations about the position of Fig. 5, which will die out ow-
ing to molecular friction so that the actual position of Fig. 5 will
always be reached.
CHAPTER VI.
EXPERIMENTS WITH A GAS TURBINE.
This portion of the investigation of the gas turbine, really the
most important of all, has unfortunately turned cfUt in a very un-
satisfactory manner. So far as principles are concerned all of the
statements previously made were found correct, but no actual re-
sults were obtained. After many failures a gas turbine was
actually operated, but the net output was negative. That is to
say, the turbine wheel did not even yield enough power to com-
press the air required. This result is to be attributed solely to
the crudness of the apparatus used. The apparatus was arranged
very much like the diagram of Fig. 2, except that the air was
supplied by an independent source and the whole output of the
wheel measured by a brake.
Considerable difficulty was found in igniting the air and oil and
starting the continuous combustion under pressure. A successful
method of doing this was finally found however. Next trouble
occured in securing smokeless combustion, which was also over-
come. After this the combustion gave no further trouble, and
gases at a white hot temperature issued from the nozzle with per-
fect regularity. However, the nozzles first used were soon de-
stoyed. After many experiments a nozzle was built which lasted
a reasonable time. Permanent success was not attained, however.
The turbine wheel used was that from a very small DeL,aval
steam turbine and was about 5 inches mean diameter. The best
speed was found to be about 19,000 revolutions per minute, and at
this speed about 3 horsepower was developed. About 4 horse-
power was theoretically required to compress the air.
No experimental work was done in the way of varying the
pressure, the shape of the nozzle, and the shape of wheel buckets.
There is of course an almost infinite field for variation in these
particulars and the writer has no doubt that experiments in these
directions will ultimately be crowned with success.
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