WINDAGE RESISTANCE OF STEAM-TURBINE WHEELS

A CRITICAL STUDY OF THE EXPERIMENTAL DATA WHICH HAVE BEEN

PUBLISHED AND OF EQUATIONS FOR REPRESENTING THEM
By E. Buckingham
I. General Considerations.The power dissipated in driving a
given wheel against the resistance of the surrounding medium,
depends on the speed of rotation and the mechanical properties
of the medium.
If the medium is an homogeneous fluid such as water, air, or
dry steam, its mechanical behavior is determined by its density,
viscosity, and compressibility. If the linear speeds involved are
all small compared with the speed of sound in the medium, the
energy of the acoustic waves generated is insignificant, the drag
on the wheel due to the generation of these waves is negligible,
and the medium acts sensibly as if they did not exist, i. e., as if
it were incompressible. Except possibly in extreme cases, it
can hardly be doubted that treating the medium as incompres-
sible is allowable to the degree of approximation needed in steam-
turbine calculations or justified by the accuracy of pubHshed
determinations of windage losses. The density and viscosity of
the medium in which the wheel rotates are therefore the only
ones of its physical properties which we need to take into accotmt,
so long as the medium is homogeneous.
If instead of confining our attention to a particular wheel and
casing we extend it to other wheels of various sizes wliich are,
together with their casings, geometrically similar to the wheel
and casing first considered, we see that the power dissipated may,
further, depend on the diameter of the wheel, this single magni-
tude sufiicing for the complete geometrical specification of the
system when the shape is once given. The shape itself may be
191
192
BtUletin
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Standards [Voi.io
specified by the values of the ratios of a number of lengths to
some one length, such as the diameter of the wheel. It is to be
understood, in what follows, that when we speak of wheels of the
same shape, or "geometrically similar wheels," the similarity is
to extend to the casings as well as to the wheels proper.
Let P be the power dissipated in driving a wheel of given
shape and of diameter Z>, at a speed of n revolutions per unit
time in an homogeneous medium of density
p
and viscosity /Lt.
Then since the power can depend only on the size of the wheel,
its speed of rotation, and the properties of the medium we have
P
=
f{p,n,D,ti)
This physical equation must consist of terms which are all of the
same dimensions, and this can be true only if the equation is of
the general form
P
=
l,Np''n^D'fi^ (i)
in which the N's are ptu-e numbers and the a,
/8, 7, 8, of each
term have a set of values which give that term the same dimen-
sions as P. There may be any number of terms, but for each
of them a dimensional equation,
[P]
=
[pn^Dy]
(2)
must be satisfied.
It is most convenient to use the ordinary m, I, t system of
fundamental units. If we do so and derive our other units from
them, we have the following dimensional equations for the quan-
tities which appear in equation (i) :
[P]
=
[mlH-^]
[p]
= [ml-^]
[n]
=
[t-^]
The exponents of each term in equation (i) must therefore be
such as to satisfy a dimensional equation
[ml^ir^] =[mH-^''1rH^mH-^t-^]
Buckinoham] Windage Losses
of
Steam Turbines
193
The units of m, I, and t being independent, this equation must be
homogeneous in each of them, so that we have
I
=
a + S 1 a=i-8
2
=
-Sa+y-B whence /3
=
3-a
3
=
/3 +B \ 7
=
5-28
as the relations which must exist between the exponents of each
term of equation (i).
If we substitute the foregoing values of a,
/9,
and
7,
in equation
(i) and take out the common factor {pn^D^), we have
'-'^^K^y
^'^
The number of terms, the coefficient A^ of each term, and the
exponent h with which the single variable
(

^
j
appears in that
term, all remain indeterminate, i. e., they may have any values
whatever without violating the dimensional requirements. We
may also write equation
(3)
in the simpler form
P-pn>D^<^-^)
(4)
where
^
is an unknown function of (

^y
For wheels of any given shape, the form of the fimction tp is
fixed, but for anyother shape the form of cp will or may be different.
It depends on the values of a number of ratios of lengths sufficient
to fix the shape of the wheel and casing, including the closeness
of the casing to the wheel as part of the "shape." These ratios
determine the values of the A^'s and S's of equation
(3).
The
variable
( V>,
)
has no dimensions, so that its numerical value in
any absolute units is fixed by the physical magnitudes of /x,
/>,
n,
and D, and is independent of the magnitudes of the fundamental
imits.
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Bulletin
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2. Remarks on the Function cp.

Equations
(3)
and
(4)
contain
all the information obtainable from the principle of dimensional
homogeneity, and further information must be sought elsewhere.
The resistance offered by a fluid to a solid body in steady motion
through it may be looked upon as a retarding drag due to trans-
verse communication of momentum between currents or layers
of the fluid which are moving over each other with different average
velocities and are maintained by the motion of the solid body.
In quiet stream-Hne motion, this transfer of momentum takes
place by intermixing on a molecular scale, i. e., by interdiffusion of
the different streams of fluid; and the coefficient of viscosity,
/I, is a measure of the activity of this molecular intermixing.
But whenever, by reason of high speed or of roughness or irregu-
larity of the sohd, the motion of the fluid becomes very turbulent,
the lateral transfer of momentum between different streams occurs
mainly by motions of relatively large masses of fluid in all sorts
of irregular cross-currents and eddies. For any given geometrical
arrangement of such a state of tiurbulent motion, this molar trans-
fer of momentum is evidently proportional to the masses involved,
or in other words, to the density of the fluid.
We must therefore expect that when the motion of the fluid
about the solid body is perfectly quiet and regular, which will
usually mean that all the motions are slow, the resistance encoun-
tered by the solid will be directly proportional to the viscosity
of the fluid. But if, on the other hand, the circumstances are
such that the motion of the fluid is very tiurbulent, it is to be
expected that density will play the determining part in the phe-
nomena of resistance, viscosity being of relatively small or even
vanishing importance. The substantial correctness of this general
qualitative reasoning is estabHshed by well-known facts relating
to skin friction, aeroplane and ship resistance, and the flow of
liquids and gases tlurough pipes. Let us now see what it leads
to in connection with equation
(3)
.
If we have to deal with a smooth disk at a low speed of rotation,
we must expect the retarding torque and therefore the power ab-
sorbed at any given speed to be directly proportional to the viscosity
of the medium about the disk. This requires that the second
member of equation
(3)
shall consist of only a single term and
Bucktngham] Wiudage Losscs
of
Steam Turbines
195
that its exponent shall be S =
i . The equation for the power thus
reduces to the very simple form
P
=
Nn^D'/JL
(5)
and the retarding torque is therefore independent of the density
and proportional to the speed of rotation, the viscosity of the
medium, and the cube of the diameter of the disk.
If the speed of rotation is increased, the stream-Hne motion
will become unstable and will break up, at or near a certain definite
critical speed, into a quite different, turbulent motion. At all
events, this is what we should expect from our knowledge of the
flow of fluids through pipes, and otir expectation is confirmed
by experiments on disks. After this abrupt change in the char-
acter of the fluid motion, the density of the fluid must appear
in any equation which is to describe the facts, and if the vis-
cosity fi appears at all it must be, in each term, with a smaller
exponent than that of
p
in the same term. In the Hmiting case,
the viscosity might be of altogether vanishing importance and
then equation
(3)
would necessarily have the form
P
=
Npn'D'
(6)
the retarding torque being proportional, for a given disk, to the
density and the square of the speed, while for disks of different
diameters but geometrically similar

^in regard to roughness as

well as general shapethe torque would be proportional to the
fifth power instead of the cube of the linear dimensions. In
practice we should expect, rather, that the relation would not
reach this very simple limiting form and that
fi would still appear,
though only in terms with small exponents.
If it is found that the dependence of P on
p,
n, D^ or fjL can be
represented with a degree of approximation sufficient for the
experimental accuracy by giving the independent variable a single
fixed exponent, it follows that the second member of equation
(3)
may be represented by a single term, the others being negli-
gible. Equation
(3)
then reduces to
p
=
Np'-^n^-^D'-^^/x'
(7)
/
1
96 Bulletin
of
the Bureau
of
Standards [Voi. 10
It appears that such an equation is adequate to the representa-
tion of the experimental data which are available

^the experi-
mental accuracy not having been high. Equations
(5)
and
(6)
correspond to the limiting values B=i and S =
o.
The value of A^ depends on the form of the wheel and casing.
For a bladeless disk running in the open, it depends on the pro-
file and the roughness. For a thin flat disk it depends on the
roughness alone; but the roughness is to be measured in terms of
the diameter, so that for a surface with granulations or irregulari-
ties of a certain general absolute size, a large disk is "smoother"
than a small one, in the present sense of the term smooth.
For smooth plane disks of small thickness, the critical speed, at
wliich stream line motion breaks up into turbulent motion and
the exponent B drops suddenly from nearly unity to a much
smaller value, is fairly definite and the change in the law of resist-
ance is sharp. For ordinary tm-bine wheels, consisting of a disk
and blades, the change is less abrupt and there is no definite
critical speed; for the motion about the blades must always be
turbulent at any speeds which are high enough to be of practical
interest. The smaller the ratio,
y^,
of blade length to disk diam-
eter, the more nearly the behavior of the wheel approaches that of
a simple disk, while with long blades, the additional resistance
due to the blades is so much greater than that which would be
encountered by the disk alone, that although the law of resist-
ance is different at high and at low speeds the change is gradual.
In an experimental study of the form of the function
(p we have
first to work with wheels of some one shape, preferably a simple
one, to start with. If we then find, as in fact we do for consider-
able ranges of the variables, that the dissipation P is proportional
to a fixed power of
p,
or of t^, or of Z^, or of fi, we know that equa-
tion
(7)
is adequate for this shape of wheel and our experiments
give us the value of the coefficient N for this shape as well as the
value of 8, no matter which of the four variables we may have
selected for independent variation during the experiments. There
are thus, in principle, four different modes of attacking the prob-
lem which must lead to the same result and may be used for
checking one another. In practice we can not always conven-
Buckingham] Windage Losses
of
Steam Turbines
197
iently vary the density and viscosity of the medium independ-
ently, so that there are often only three modes of attack which
are practicable.
The general procedure is obvious; keeping any three of the
variables at fixed values, we vary the fourth and observe the
corresponding values of P. If the phenomena can be described
by equation
(7),
the values of log P when plotted against the
logarithm of the independent variable will give points which lie
on a straight line, within the experimental errors. The slope of
this line is the exponent with which the independent variable
appears in equation
(7)
; it determines the value of S. The posi-
tion of the line, taken in connection with the fixed values of the
other three variables, determines the value of N. We may now
proceed to examine the experimental data which illustrate the
foregoing statements. Readers who are interested only in the
jSnal result of this somewhat laborious examination may proceed
at once to section
6,
page 214.
3.
The Relation
of
Power Dissipated to Speed
of
Rotation.

A. Disks Without Blades: As the simplest sort of wheel we may

take a flat bladeless disk rotating in the open air. We have
data on such disks from Stodola's
^
experiments on a smooth
but unmachined disk of thin boiler plate and from Odell's
^
experi-
ments on paper disks.
Stodola's disk had a diameter of
537
mm or 21.1 inches, and
he gives points on a plot of log P against log n only for
1 500, 1 800
and 2000 r.
p.
m. These three observations very nearly satisfy
the relation Pozn^-^^, The retarding torque was thus propor-
tional to the 1.92 power of the speed n, which is about what
might have been expected from the work of Reynolds,^ Froude,*
Zahm,^ and others.
Odell's experiments covered a wider range of speed and he
used foiu: disks with diameters of 15.0, 21.8, 26.8, and
47.1
inches,
recording the torque needed to drive each disk at measured
speeds. When log torqiie is plotted against log n it is found
that for any given disk, the points for all the higher speeds lie
close to a straight line. Upon decreasing the speed, a critical
speed, Wc,
is reached and the law of resistance changes abruptly.
198
BtUletin
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of
Standards [Vol. lo
The results, so far as they concern us at present are collected in
Table i.
TABLE 1
Odell's Experiments on Paper Disks Rotating in Air
A B C D
Diameter in inches, D= 15.0 21.8 26.8 47.1
Range of speeds, (r. p. m.)
j
113
12100
200
900
100
525
250
740
Critical speed, wc= 918 376 243 (100)
Value of p
for n>no 3.32 3.55 3.54 3.08
10-5?le>2 2.07 1.79 1.75
The values oi
Uc
and of the speed exponent yS =
3

3 were
fotmd by replotting Odell's observations; they do not differ
much from Odell's values. The value of
^
is, of cotu-se, obtained
by adding unity to the slope of the line on the plot of log torque
against log n.
Upon inspection of the logarithmic plot there is no doubt that
above the critical speed, /3 is very nearly constant, and there is
also no doubt that the values of
P
are distinctly different for the
different disks, the uncertainty in finding /3 from the published
observations being not over o.i. If the disks had all been
geometrically similar, the form of
^
in equation
(4)
would have
been the same for all and the exponent
^
would therefore have
had the same value. Since the values of
^
are not the same,
we are sure that the disks were sensibly dissimilar. They were,
in fact, made of four different kinds of paper and it could not be
expected that they should be either similarly rough or similarly
stiff.
When thin paper disks are driven at high speeds they become
covered with fluttering waves which must have an important
effect in increasing the resistance, so that it is not safe to draw
conclusions regarding rigid disks from results obtained with
flexible ones. Disk D made of "canvas-backed diagram paper"
was no doubt much the stiffest, the others being of two kinds
of drawing paper and of cartridge paper. It gave the value
)S =
3.i which is not far from the value 2.92 for Stodola's disk.
Buckingham] Windage Losses
of
Steam Turbines
199
Disks B and C gave nearly the same value of /S at the high speeds
and seem to have behaved as if geometrically similar; the wave
systems in them had similar effects on the resistance.
When two geometrically similar wheels or disks are compared,
any singular point on the curv^e P
=/
(n) , such as the critical speed,
must by equation
(4)
occur at the same value of
7^
for both.
In Odell's and Stodola's experiments the meditun (air) was always
approximately unchanged, so that the value of ncD^ should be
the same for geometrically similar disks. The values in the last
line of Table i show that this was nearly true for Odell's disks
B and C, which gave the same value of yS at speeds above the critical.
The value of
nc
given in the table for disk D was computed
from that observed for A which was most nearly like D ; it is
only a rough approximation, and being far below the lowest
speed used with this disk, it was not observed. The critical speed
for Stodola's disk, computed in the same way, would be in the
vicinity of 500 r.
p.
m., which is also below the lowest speed
recorded.
Disk C, of cartridge paper, shows the interesting phenomenon
of two critical speeds, the lower, not mentioned in Table i, being
at about 150 r.
p.
m. Below this, the torque is proportional to
n^-^ or, within the experimental errors, directly to n. We have
thus the limiting case described by equation
(5)
or by equation
(7)
with 8
=
1. At the first critical speed it appears that the
stream-line motion of the air broke up into turbulent motion,
the torque became proportional to n^-'' and the power dissipated
to n^-"^. For the range between the two critical speeds we there-
fore have /8
=
3

3
=
2.7,
^=0-3- At the second critical speed
of about
243
r.
p.
m. the disk suddenly began to flutter
*
and the
law of resistance changed to
Poc^^
-^
a new element being intro-
duced by the presence of waves on the disk. The middle range,
between the two critical speeds corresponds to Stodola's observa-
tions on his rigid disk. The value =
2.7
which is somewhat
lower than Stodola's /6
=
2.92 indicates that the cartridge paper,
as long as it stayed plane, was smoother than the boiler plate,
which is not smprising.
*
This is the writer's inference; Odell is not responsible for it.
200 Bulletin
of
the Bureau
of
Standards ivoi. lo
Disk B shows only the upper critical speed, the observations
below this point being very few and the lowest coming about
where the lower critical speed would probably have been foimd.
Since the upper critical speed and the phenomena at still higher
speeds depend on the stiffness of the disk, which has not appeared
at all in otu* equations, the close agreement of /3 and of nj)^,
for disks B and C, must be regarded as in some degree accidental.
It happened that the disks were similarly stiff or, at all events,
developed wave systems which had similar effects on the resistance.
The few observations on disk B below the upper critical speed
give Pccfi^-^. We can not say that this is different from
tte
value
yS =
2.7 obtained for disk C, the evidence not sufficing to decide
the point. But the difference, if real, is in the direction indicated
by what we can judge of the smoothness of the two disks as
described by Odell. For B was of a smoother thinner drawing
paper than A, i. e., probably a rather hard smooth-surfaced
paper, while the "cartridge paper" of disk C probably had a
somewhat fuzzy surface.
If we compute the critical speeds of the rigid disk D and of
Stodola's steel disk from the lower critical speed of disk C, we
get
49
and
244
r.
p.
m., respectively, values about half those
computed from disk A and probably nearer the truth.
In the case of disk A, which was relatively stiffer and rougher
than B and C, the observations below the single critical speed
observed are too scattered for us to detect any lower critical
speed, if one occiured, or to distinguish the effects of turbulence
of the air and fluttering of the paper. The effect of the flut-
tering at the high speed was less than with B and C as is shown
by the fact that the value
^
=
3.3
is nearer the values for rigid
disks, e.
g.,
Stodola's disk
{I3=2.g2),
disk D
{^
=
3.14)
and disk
C between its critical speeds (/3=2.7). The greater stiffness is
also shown by the larger value of ncD^; the stiffer disk requires
a higher speed to make it break into a flutter.
Odell's experiments were only preliminary and his own opinion
of their accuracy is indicated by his saying that P
=
const X n^D^
"agrees pretty well with the experimental results." But in view
of the very good agreement which they show with the predictions
of the general theory in all cases where the range of speed is
Buckinoham] Windage Losses
of
Steam Turbines 201
large enough, the present writer forms a higher estimate of the
interest and relative accuracy of these experimental results, even
though the absolute values of the power dissipated may be con-
siderably in error, and though we can evidently not safely pre-
dict the behavior of rigid disks from experiments on flexible ones.
B. Ordinary Single-Row Steam-Turbine Wheels: Besides the
bladeless disk already mentioned, Stodola experimented on five
single-row turbine wheels with various blade lengths and with
disk diameters, measured at the root of the blades, of 20 to
45
inches. The wheels were run. in air; all of them in the
open, and three of them also inclosed in casings. At all the
higher speeds the results, as Stodola shows them on a logarithmic
plot, satisfy the relation Pccn^ quite closely. For the ten series
shown, the value of 13 is from 2.82 to
2.97
with a mean of 2.89
and a mean residual of 0.04. The wheels all acted very nearly
like the bladeless disk as regards the variation of power with speed.
Lewicki

gives the results of a few observations on the windage
losses of a small Laval turbine at
14
000 to 20 000 r.
p.
m. The
wheel had a disk diameter of 180 mm or
7.09
inches, and the
blades were 20 mm or
0.79
inch long. Three observations in air
at atmospheric pressure and 30 C give P^n^"^^ nearly. Four
observations in sattu-ated steam at atmospheric pressure lie fairly
close to
Pocn'^*o
while three of them give almost exactly
Pocn^'O.
There is no doubt that /3 was nearly constant in each case, but
the observations are so few and the range of speeds is so small
that no great weight can be given to the numerical values of /8.
We have, finally, the results published by Holzwarth."^ Admit-
ting that for a given wheel running at a given speed in steam, the
power dissipated in windage is proportional to the density of the
steam, and admitting, fiuther, that for practical ptuposes the
density may be treated as proportional to the absolute pressure
p,
Holzwarth sets P=Kp, and gives the values of K deduced
from his experiments, by means of ctuves on a three-coordinate
diagram. Each curve show^s the relation oi K to n for a given
diameter, the diameters being 10, 20, 30, 40,
and
50 inches. The
lowest speed is
750
r.
p.
m. and the highest runs up to 4000 r.
p.
m.
for the small wheels. There is a separate diagram for each of
the five blade lengths
0.5,
i.o,
1.5, 2.0, and 2.5 inches. The wheels
202 Bulletin
of
the Bureau
of
Standards [Voi. lo
were inclosed but had somewhat more clearance than is usual in
practice. No information is given as to how nearly geometri-
cally similar the wheels were, except the mere statement of disk
diameter and blade length. The results appear to be the most
comprehensive we have; but all details are omitted so that w^e
have no means of estimating the accuracy of the experiments,
of the reduction of the observations, or of the construction and
reproduction of the diagrams as published. Though the curves
may be quite adequate to the practical end for which their author
intended them, it is evident at a glance that they can not all be
represented by a single equation containing only n, D, and / as
independent variables, and that readings of K from them are
liable to rather large percentage errors.
With the exception of a few curves for the largest diameters
and blade lengths, each curve shows two distinct forms and may
be divided, though not sharply, into a low-speed and a high-speed
part. The relation
K=f(n)
is different for the two regions and
the transition corresponds to the definite critical speed in the case
of a bladeless disk. The lo-inch curves fall almost entirely in
the low-speed range which is in general of little practical interest.
The results for this diameter are therefore omitted from further
consideration. For each of the other four diameters the writer
made readings of the value of the coefficient i^* for each of the
five blade lengths and at the various speeds shown on the dia-
grams. The value of log K was then plotted against that of log
n, and 20 series of points were thus obtained, each referring to
a fixed diameter and blade length but varying speed.
For the higher speeds, the points of each series lie fairly close
to straight fines, sometimes quite close. These lines were drawn
in by inspection and their slopes, which represent )8in the equation
P
=
const Xn^ were read off. For the small wheels with short
blades there is a fairly definite critical speed at which the rela-
tion P
=
f{n)
changes; but the points for the "low" speeds are
not exact or numerous enough to show what the relation is at
these speeds. For larger wheels or longer blades the transition
from "high" to "low" speeds when shown at all is gradual and
*Holzwarth used the symbol m for this coefficient.
BuckiTighami Windage Losses
of
Steam Turbines
203
not sharp, the lowest points on the logarithmic plot being above
the straight lines drawn to represent the high-speed points.
The values obtained for the speed exponent
/3, are from
2.3 to
3.6;
the mean is
3.0
and the mean residual is 0.3. The uncer-
tainty of each value, due to doubt as to just where the straight
line should be drav^m, is generally about 0.1 but may be 0.2.
For most of the series there is no doubt that Pocn^
is an approxi-
mate representation of Holzwarth's published ciuves for the higher
speeds. There is also no doubt that the values of
/3, are distinctly
different in different cases, but there is no evident systematic
variation of the values with either diameter or blade length.
To smn up the conclusions which may legitimately be drawn
from the data discussed in the foregoing section, we may say
that both for flat bladeless disks and for single-row tiu-bine wheels
of ordinary forms, running either inclosed or in the open and
in either air or steam, the power dissipated in windage by a
given wheel is very nearly proportional to a fixed power of the
speed of rotation throughout the range of speeds actually used
with the diameters in question, except possibly at the lowest
speeds, where the windage losses are economically insignificant.
It follows that only a single term need be used in the second mem-
ber of equation
(3)
, so that an equation of the form
(7)
or
p
=
Np'-^^n^-^D'-^^fi'
(7)
is adequate to representing the facts, over the range mentioned,
as closely as we know what the facts are.
The observed values of
3

8
=
/3 are grouped about a general
mean yS =
3
and as a first -approximation we have
P=Npn^D^
(8)
the coefficient N having a value determined by the shape of the
wheel, and the viscosity not appearing at all. In individual cases
the observed values of /3 vary from 2.5 to
3.5
without oiu* being
able to decide, from the rather meagre accoimts published,
whether the variations are genuine, i. e., not due to errors of
measurement, and if they are genuine, what causes them.
204
Bulletin
of
the Bureau
of
Standards \yoi.io
The best experiments we have, Stodola's, give y3
=
2.9 very
nearly, for all of his wheels and all conditions, and this value is
quite consistent with the results by other experimenters in similar
fields. It is probable that in general the equation
obtained from
(7)
by setting 38 =
2.9 will give a better approxi-
mation than the simpler equation above, in which S =
0.
At a speed of rotation of 20 000 r.
p.
m. the peripheral speed of
the tips of the blades of Lewicki's wheel was
755
ft. sec. The
speed of soimd in air at 30
C. is about
11 50 ft. sec, and in satu-
rated steam at one atmosphere it is about
1350
ft. sec. The linear
speeds which occurred in Lewicki's experiments were therefore by
no means all what would commonly be thought of as "small" in
comparison with the acoustic speed. Nevertheless, an equation
developed on the assumption that compressibility is negligible
appears to hold for these high speeds as well as for lower ones. It
would have been interesting to have data on the windage of
Lewicki's wheel at even higher speeds, for it seems probable that if
the speed had been run up to
30
000 the equation P
=
const X n^
would have failed completely. Experiments on the resistance of
projectiles
^
indicate that the effects of compressibility begin to
be sensible at about three-fourths of the acoustic speed and increase
rapidly for some distance beyond that point, so that within a range
of from three-fourths to one and one-half times the acoustic speed,
the law of resistance is rather complicated. The speeds in
Lewicki's experiments appear to have been nearly but not quite
high enough to necessitate the consideration of compressibility in
making up the theoretical equations. We may conclude that the
simple theory as given in section i is probably always sufficiently
exact when the peripheral speeds are not over one-half the acoustic
speed in the medium in question.
4.
The Relation
of
Power Dissipated to Diameter.The experi-
mental results discussed in section
3
suffice to show than an equa-
tion of the form
(7)
describes the facts, and they give us the
value of h in some specific cases. The same information might
be got from experiments on geometrically similar wheels, by
Bvckinoham] Windage Losses
of
Steam Turbines 205
making D the only independent variable and comparing the dis-
sipation for wheels of various diameters run at the same speed
in the same medium; but no experimental results on exactly
similar wheels have been published, so that an investigation by
this method can not be based on any published data. Neverthe-
less, as a check on the previous work it is interesting to compare
such imperfect data as we have with equation
(7)
and the values of
h already found in section
3.
A. Disks Without Blades: We have to select for comparison,
disks which gave nearly the same value of /S =
3

3, since disks
which do not satisfy this condition can not possibly be geomet-
rically similar. Odell's paper disks B and C gave the same value
of ^ above the fluttering point, but since the rigidity of the disk
is here an essential element which has not been allowed for in otir
theory, we must not expect equation
(7)
to hold at all exactly.
If we compare these disks at 500 r.
p.
m., the observed power
ratio is 4.1 as against
3.5,
computed from the diameter ratio
and the value /3
=
3.54
obtained from the experiments at varying
speed. The discrepancy of
15
per cent may be due to experi-
mental error but it seems quite as likely to be due to the fact
that the flexibility of the disks has not appeared in the equations.
We may next compare these two disks in the region just below
their upper critical speeds, where they were still behaving as if
rigid. From the writer's plot of Odell's observations he finds
for disk B P
=
2
.5
1 X
10-^^.470
for disk C P
=
2.28 X
10-10^2.654
The observations on B in this range are so few that the exponent
^
=
2.47
is very uncertain and the equation could not safely be
used for extrapolation, but it represents the actual observations
reasonably well. At n
=
200 r.
p.
m., which is within the range
of speeds for both disks, the power computed from these equations
is 1.21X10-* hp for B and 2.92X10-^ hp for C. Comparing
the power ratio with the diameter ratio we have
2.92
/26.8Y'
I.2I~V2I.8/
2764714-
2o6 BtUletin
of
the Bureau
of
Standards ivoi. lo
If we regard the disks as similar we have
5

28
=
4.26,
whence
3
S
=yS =
2.63,
which is sensibly identical with the value 2.65
obtained directly for disk C.
As a more severe test of the theory we may compare Odell's
stiffest disk D, of
47.1
inches diameter, with Stodola's boiler-plate
disk of 2 1. 1 inches diameter. At n
=
2ooo Stodola observed
P
=
0.147
hp and his speed exponent was /3
=
2.92; hence we have
for this disk
P
=
3.38Xio-iin2'92
From Odell's measurements on his disk D we have approximately
P
=
48.
Xio-iin^'os
Odell regarded his experiments on this disk, especially at the lower
speeds, as less accurate than his other work, so that it seems only
fair to make the comparison at one of the higher speeds within
the range used by Odell. Taking n
=
700,
we have by the above
equations; for Stodola's disk P =0.00679 hp, and for Odell's disk
D, P =0.278 hp. Comparing the power ratio with the diameter
ratio we have
0.278
0.00679
<m"'
=
40.9
Regarding the disks as similar, we have
5

28
=
4.62, whence
3

8
=
2.81 as compared with /3
=
2.92, obtained directly for
Stodola's disk.
The value /8
=
2.92 or
5

28
=
4.84
would give a power ratio of
48.7.
Hence if we use equation
(7),
with the values of N and 8
foimd for Stodola's disk, to compute the power for Odell's disk at
700 r.
p.
m. we get within 20 per cent of the observed value.
The equation given above for Odell's disk would give, at 2000
r.
p.
m. and on a diameter of 21.1 inches, P =0.1 12 hp while
Stodola's observed value at this speed was
0.147
^P-
^^ view of
all the circumstances and the long extrapolation from observa-
tions which Odell himself regarded with suspicion, the agreement
must be considered quite satisfactory.
Buckingham] Windage Losses
of
Steam Turbines 207
B. Single-Row Turbine Wheels: When we look for experi-
ments on geometrically similar turbine wheels we find only a
very few cases where we can assume that the wheels were even
roughly similar.
Oiie of Stodola's wheels had a disk diameter of
504
mm, or
19.84
inches and a blade length I
=
60 mm, so that the blade-
length ratio was ^=0.119.
This wheel was rtm in air at atmos-
pheric pressure and temperature, in a casing which allowed an
axial clearance of
4
mm at the edge of the blades. Lewicki,* as
already mentioned, made observations on a I^aval wheel which
had a disk diameter of 180 mm or
7.089
inches and a blade-length
ratio ^=0.111. These two wheels were therefore nearly similar in
respect to the important element
^.
From the small scale draw-
ing it appears that Lewicki's wheel had an axial clearance at the
blade edges of about 2 mm. For geometrical similarity, the clear-
ance on Stodola's wheel would therefore have had to be about
6 mm instead of the actual
4
mm. Lewicki's wheel thus had
relatively more clearance; and since it is known experimentally
that reducing the clearance round the blades reduces the windage,
we must expect that in making comparisons between the two
wheels, lycwicki's observed values will be somewhat larger than
values computed for the same speed and diameter from the data
obtained by Stodola.
lyCwicki's experiments on windage were only a subsidiary part
of a larger investigation and are not comprehensive enough to
furnish satisfactory values of the constants N and B.
We shall
therefore deduce an equation from Stodola's observations and
compare values computed from this equation with averages from
Lewicki's observations.
The speed exponent given by Stodola for this wheel is
2.87,
but it appears from the plot that a larger value might be taken.
We can probably do no better than to set yS =
2.9 and use equation
(9).
At 2100 r.
p.
m. the dissipation was foimd to be
0.704 hp.
2o8 BtUletin
of
the Bureau
of
Standards ivou lo
The disk diameter was
19.84
inches and the density of the air, if
we take the value given for the bladeless disk, was /o =
i . 1 2 kgm/m'
or 0.070 Ib./ft.^ From these values we get
AT/li^'i =
1.058 X IO-^^
whence
P = 1 .06 X
lo-iy-
WD* /-^Y'
(10)
in which
P
=
the horsepower dissipated.
/o =
the density of the medium in Ib./ft.^
n
=
the speed of rotation in r.
p.
m.
Z?=the disk diameter in inches.
/io
=the viscosity of the air during Stodola's experiments.
/x =
the viscosity of the medium in any other case for which we
wish to compute the value of P.
Since most of Lewicki's work was done at a speed of 20 000
r.
p.
m. we may confine our attention to this speed. If we then
set w
=
20 000 and D
=
7.089,
we have
P
=
38.o/.(^0'
(11)
This equation would give us the power dissipated by a wheel
geometrically similar to Stodola's wheel but of the same disk
diameter as Lewicki's, when running at 20 000 r.
p.
m. in any
homogeneous medium of density
p
and viscosity /x.
To find the correction for the lack of exact similarity, we first
make the computation for air because in this case the value of

will certainly be sensibly equal to unity. The mean of

Lewicki's three observations in air, which do not differ enough to
make it worth while to treat them separately, was P(obs)
=4.37
hp
at a mean density
p
=
0.0647
Ib./ft.^ Substituting in equation (i i)
4 37
we have P(calc) =3.24 hp. The ratio
-^^^^
=
1.35
indicates that
the combined effect of the relatively greater clearance and the
slightly smaller blade length of Lewicki's wheel, was to increase
the windage loss by about
35
per cent. We therefore modify
Buckingham] Windage Losses
of
Steam Turbines
209
equation
(11)
by introducing this correction factor and so obtain,
for further use, the equation
5^AP' <0-
(
To compare the values computed by equation
(12)
with I^ewicki's
observations in steam we must form an estimate of the value of

j
and it is fortunate that this appears with so small an exponent
since we know almost nothing about the viscosity of steam.
According to measurements by
h.
Meyer and O. Schumann (see
Landolt and Bomstein's tables) the viscosity of sattirated steam
at
100
is about 0.72 times that of air at room temperature. We
may assume as a sufficient approximation, under the circum-
stances, that jji is proportional to the square root of the absolute
temperature and is independent of the density. In applying
equation
(12)
to steam we therefore have
where / is the temperature of the steam on the Fahrenheit scale.
Equation (12)
then takes the form
p-./<i^r
..4)
applicable to dry steam of density
p
Ib./ft.^ and temperature t F.
We may first consider the
15
observations made by I^ewicki in
superheated steam, which was presumably dry and homogeneous
though at the lower temperatures this may not have been quite
true. The data which concern us are collected in Table 2, reduced
to English units.
2IO BtUlettn
of
the Bureau
of
Standards [Voi. w
TABLE 2
Lewicki's Observations in Superheated Steam; n
=
20 000 r.
p. m.
Pressure
[lb./i..]
Temperature
t
[F.]
Density
[lb./ft.3]
Power
P
[U.S.hp.]
P
P
P
pO-9
14.7 221 0.0368 3.02 82 59
14.7 253 0.0350 2.77 79 57
14.7 406 0.0286 2.33 81 57
14.7 462 0.0268 2.30 86 60
14.7 478 0.0263 2.00 76 53
14.7 531 0.0249 1.98 80 55
14.7 556 0.0243 1.87 77 53
14.7 574 0.0239 1.83 77 53
9.56 586 0.0153 1.08 70 46
5.69 462 0.0100 0.88 88 56
5.69 489 0.0097 0.81 84 52
5.68 259 0.0127 0.94 74 48
5.63 561 0.0089 0.60 69 42
5.39 295 0.0115 0.90 78 50
5.39
Means
590 0,0083 0.58 70 43
448 0.0202 78 52
Mean residuals 5.8% 3.5%
1
The values of the density are not very exact, partly because
for the first eight observations the pressure is merely stated to
have been atmospheric, and partly because the densities at high
superheats are not very accurately known.
In view of the unavoidable errors we can probably do no bet-
ter than to average the values and set P
=
'jSp, the mean density
being
p
=
o.o202 and the mean temperature 448 F. Under these
circumstances we therefore have P(obs)
=
1.58 hp. Equation
(14)
with these values of
p
and t gives us P(calc)
=
i.5i hp. The
agreement of the observed and calculated values to within
5
per
cent must be regarded as very satisfactory.
We may also represent the observations fairly well by setting
P(obs) =52 X/^*^^ while equation
(14)
gives us P(calc) =50.5 X/'^
The agreement is a trifle closer but there is no great difference.
The result of the comparison of Stodola's and I^ewicki's wheels
is to show that as nearly as we can tell, equation
(9)
represents
the facts and that dry steam is entirely comparable with air when
the proper physical constants are used.
Buckingham] Windage Losses
of
Steam Turbines 211
We may now consider Ivevvicki's five observations in saturated
steam, the results of which are shown in Table
3.
TABLE 3
Lewicki's Observations in Saturated Steam; n
=
20 000 r. p. m.
Pressure
[lb../m.21
Density
[lb./ft.s]
Power
[U. S. hp.]
P
P
P
14.7 0.03732 3.22 86.2 62.0
10.50 0.02727 2.69 98.3 68.9
8.56 0.02253 2.05 91.1 63.7
6.44 0.01726 1.66 96.0 64.0
5.69
Mean
0.01538 1.49 96.8 63.7
0.02395 93.7 64.5
Mean residuals 4.3% 2.8%
As for superheated steam, we can represent the results approxi-
mately by Poc/o or Pocp^'^, the second being in this case distinctly
the better, as is shown by the fact that the mean residual is only
2.8 per cent as compared with
4.3
per cent.
Not knowing how the viscosity varies with temperature when
the steam is kept saturated, we ignore the last factor of equa-
tion
(14)
which is certainly nearly unity, and we then have
P(calc) =49.7/0^'^ as compared with P(obs)
=
64.5/3*^ The observed
value is thus 1.30 times the calculated. Making the computa-
tion by the mean density we have P
=
93.7/5,
/9
=
0.02395
P(obs)=2.23 hp while equation
(14)
gives us P(calc)
=
1.73,
the ratio being now 1.29 in place of 1.30.
The cause of the discrepancy of 30
per cent is clear. To obtain
dry saturated steam is a difficult matter, requiring elaborate pre-
cautions, though this was not so well known at the date of Lewicki's
experiments. The increase of
30
per cent in the resistance in
passing from air or dry steam to saturated steam, was due to the
wetness of the steam. Not knowing how wet the steam may have
been, we have, perforce, used the density of the steam alone and
not the mean density of the mixture. The mean density would
have been larger and so would, if used in the computations,
have reduced the discrepancy between the observed and calcu-
212 Bulletin
of
the Bureau
of
Standards [Vol. lo
lated values. But it is most unlikely that the steam was so wet
that the whole
30
per cent could be accounted for in this way,
even supposing that the water remained completely suspended.
And under such conditions only a very small amount of water
remains in suspension as fog; most of it is deposited and that is
imdoubtedly what happened here.
Just
how wet the steam was
and why the deposition of water on the wheel should have
increased the resistance
30
per cent it is of course impossible to
say. But it is clear that the steam did not act like an homogene-
ous medium, so that equations developed for homogeneous media,
which, as we have seen, describe the facts satisfactorily for both
air and dry steam, are not strictly apphcable. All we can say
at present is that the resistance to the rotation of a wheel in steam
increases considerably if the steam changes from dry to wet; but
how the amount of the increase depends on the wetness or other
circumstances can only be decided by further experiments.
It remains to examine Holzwarth's results for wheels which
had the same ratio of blade height to disk diameter and so had at
least this one element of geometrical similarity. We have the
figiu-es given in Table
4,
for -^
=
0.05.
TABLE 4
Holzwarth's Results on Wheels with the same Blade Length Ratio
Disk diameter
D (ins.)
Blade
length
/ (ins.)
20
30
40
50
1.0
1.5
2.0
2.5
3.6
3.0
2.9
3.1
2.5
3.5
2.5
2.3
The values of
y3,
taken from the straight lines drawTi on the plot
of log K against log n vary considerably and are rather uncertain
so that the round values yS
=
3
and
7
=
5
have been adopted. The
values in the last column were got by averaging over all the
points which lay distinctly above any indication of a rapid change
in the exponent of n, omitting a few doubtful readings. If these
Buckinoham] Windage Losses
of
Steam Turbines
213
wheels were, together with their casings, all geometrically similar,
and if we had 8 =
0,
the values of
ttW
should all be the same;
it is seen that they are nearly so. The high value for the 30-inch
wheel is consistent with other readings which make it appear that
the 30-inch disks were much rougher than those of the other
diameters.
We have also the following figures for
^^
=0.025
TABLE 5
D 1
P
10iK
n'Ds
20
40
0.5
1.0
3.6
2.5
1.31
1.29
If the readings of K from the published curves really represent
the facts as Holzwarth observed them, the great difference in the
two values of
^
proves that these two wheels were far from geo-
metrically similar. The agreement of the values in the last
column is therefore little better than accidental, though it shows
that the facts may be represented, at least roughly, by the equation
P
=
Npn^D\
5.
The
Effect of
the Density
of
the Medium.All writers on the
subject appear to agree that when a given wheel runs at a given
speed, the power dissipated is directly proportional to the density
of the medium. Holzwarth says that this relation holds for
steam "within limits acctirate enough for practical pturposes,"
and I^asche, in an equation quoted by Stodola,* sets Pocpn^.
Stodola experimented on a multi-disk impulse turbine driven
in stagnant steam of densities of o.i to 1.7
kgm/m^, which
correspond, if the steam was dry satiu-ated, to pressm-es of 2.5 to
45
Ib/in^ absolute. His observations as plotted on a small scale
diagram indicate the existence of a linear relation between P
and
p,
and he sets
Pocp and uses tliis relation without further
question. Lewicki's observations as given in Tables 2 and
3
also indicate that the resistance is approximately proportional
to the density. We may therefore say that within the range
2
1
4
Bulletin
of
the Bureau
of
Standards [Voi. to
and the accuracy of the published experimental data on this
point, the windage of turbine wheels in a given medium is pro-
portional to the density of the medium, but the evidence is not
at all sufficient to show how nearly exact the relation is. We can,
in fact, only regard this evidence as roughly conformatory of that
presented in section
3,
which shows that when the medium is
homogeneous the exponent of
p
in equation
(7)
can not differ
much from imity.
6. Remarks on the Comparison
of
Dissimilar Wheels.

^The
considerations already set forth having shown that an equation
of the form
(7)
describes the behavior of wheels of any one shape,
it remains for us to find out if possible how the coefficient N
depends on the shape of the wheel and its casing.
Since N depends on shape and not on absolute size, any correct
expression for N must contain as variables only ratios of lengths.
An equation for P which can not be reduced to the form
(3),
or
practically to the form
(7),
with the N's satisfying the above
condition, is not a rational equation and can not have any general
validity, even approximate, though it may be satisfactory as an
empirical working formula over limited ranges of the variables.
In attacking the problem of finding a satisfactory expression
for N we are met at the outset by the obvious fact that the shape
of a turbine wheel and its casing, even if confined to general
conformity with commercial practice, may vary in a great many
ways. Thus N must be regarded as a function of a large number
of variables which are, at least within certain limits, all inde-
pendent. But while a complete solution of the problem of pre-
dicting the value of N from geometrical measiurements is thus
out of the question, we may nevertheless make some progress
if it is found that in practice one or a very few variables are of
so much importance that the effects of changing the others are
small or negligible.
It is evident, both a priori and from experiment, that two
very important geometrical elements to be considered are the
blade length and the closeness with which the casing surrounds
the blades, both measured in terms of the disk diameter. Other
important elements are the roughness of the disk; the width,
pitch, and angles of the blades; the number of rows of blades,
Buckinoham] Windage Losses
of
Steam Turbines
215
and whether they are shrouded or not. Of these most evidently
important data, the blade length ratio,
j^,
and the number of rows
of blades are often the only ones given in pubhshed accounts of
experiments, the others either not being given at all or having
been varied so unsystematically that they can not be used. To
start with, we shall of necessity confine our attention to wheels
with one row of blades, together with their Umiting form, bladeless
disks. As regards pitch, width, and angles of the blades, we
can only say that in practice different single-row steam-turbine
wheels are usually somewhere near similar in respect to these
points, as they are also, though perhaps less nearly, in respect
to roughness and profile of the disk.
Stodola's experiments
^
showed that the amount of clearance
between wheel and casing has a large influence on the windage
resistance, and he made a few measurements relating to this
piont. But since his results are not numerous and we have no
other satisfactory data on the effect of altering the casing which
incloses a given wheel, we are reduced to the expedient of elimi-
nating the effect of the casing by removing it altogether; in other
words, we must, at least in starting, use only data obtained from
wheels running in the open without any casing. We may then
hope to get an approximate expression for N in terms of the blade-
length ratio, the hope being founded on the expectation that this
will prove to be much more important in specifying the shape
of the wheel itself than all the other variable elements combined,
so long as these others remain within the Hmits which obtain in
practice.
We then have available for study, Odell's results on disks and
Stodola's results on one disk and five wheels. Tentatively, we
shall also use Holzwarth's results on inclosed wheels in steam,
the assumption being that the casings were similar. Since the
nature of these data does not warrant the use of any refinements
in analyzing them, we shall accept the equation
P^Npn^D^
(15)
as a sufficiently approximate description of the facts for wheels
of any given shape.
2 1
6
Bulletin
of
the Bureau
of
Standards ivoi. to
7.
Expressions
for
N in terms
of
Roughness and Blade Length.

If we assume that the differences of resistance of different single-

row wheels of the same disk diameter, when driven at the same
speed in the same medium, are expressible in terms of the rough-
ness of the disk and the blade length
/,
the coefficient A^ must have
the form
Af = >l+/(^)
(16)
in which
/
is some function which vanishes with its argument, and
A is the constant limiting value of Nwhen the blades are shortened
indefinitely. For simple flat wheels the "disk coefficient" A
will depend mainly on the superficial roughness of the metal of
the disk; but it will also, in general, depend on the profile of
the wheel and the nature of the shrouding over the ends of the
blades. For brevity we may include all these factors in the
single term ''roughness" since we have no data which would
enable us to separate them.
If, further, we admit that equation
(7)
is correct in having
only a single term in the second member, it follows that any gen-
eral equation for P which contains no other elements of shape than
roughness and blade length must necessarily have the form
p=p'-v-"d-v[a+/(^)]
(17)
or approximately
P
=
imW^\A+f(^
(18)
The three equations which the writer has seen given for com-
puting the power dissipated in windage, do not satisfy this require-
ment and so are not general, i. e., they can not safely be used
for extrapolation to values of the variables outside the limits of
the experiments from which the equations were deduced. These
tliree equations are the following:
Lasche is quoted by Stodola
^
as having deduced, from experi-
ments on wheels with i,
2, 3,
and
4
rows of blades, equations
which may, for comparison with
(18)
be written in the form
'4<5K]
pn^D^\B[-j.]D
(19)
Buckingham] Windage Losscs
of
Steam Turhiues 217
This equation does not profess to be valid except within rather
narrow limits, hence there is no occasion for criticising the fact that
it leads to an absurdity for bladeless disks or that it violates the
dimensional requirements.
Jude,
after an examination of the experimental results of
Odell, Lewicki, Stodola, and Holzwarth, gives an equation which
may be written
P =
fmm^A +b(^D~'^~'A (20)
The second or blade term of this equation violates the dimensional
conditions completely.
Stodola
^
gives an equation which may be written
P =fm^D'W +
^(^y^^]
(2
1)
This, too, violates the dimensional requirements but to a less
extent than equation
(20)
; it could therefore be used over a wider
range of diameters before involving excessive errors. Equation
(20)
can apply only over a limited range of speed as well as of
diameter, while equation
(21)
involves time correctly to the same
degree of approximation as equation
(15).
It has seemed to the writer worth while to attempt to repre-
sent the same data as were used by Stodola and
Jude, or such of
them as it appeared might legitimately be used, by an equation
which should be free from the defects just noticed. The very
simple form
P
=
pn'D'\A+B(^^'\
(22)
was therefore tested, to see if it was possible to find satisfactory
values of the constants A, B, an4 x. The disk coefficient. A,
must evidently depend on the roughness of the disk and we can
not expect all disks to give the same value. The "blade coeffi-
cient," B, should doubtless involve the width, pitch, angles, pro-
file, and form of shrouding of the blades, but it will, tentatively,
be treated as a constant, as will the exponent x. The question
is whether such an equation can be made to describe the observed
facts with reasonable accuracy and completeness.
2l8 Bulletin
of
the Bureau
of
Standards [Vol. lo
8. The Form
of
the Coefficient N
for
Stodola's Unenclosed Wheels.

We may first examine Stodola's data on his ^yq. wheels nmning

in the open air. He gives his observed value of the power absorbed
by each wheel, only for one speed, namely the highest used with
that wheel. The data which concern us here are collected in
Table 6.
TABLE 6
Stodola's Observations on Unenclosed Wheels in Air
I n m IV V VI vn vm
D
Disk
diameter
D
(ins)
Mali-
mum
speed
n
(r. p. m.)
Power
absorbed
P
(Up.)
lOiW
p
(calc)
P (obs)
-P (calc)
in per
cent
D 0.0311 34.84 1600 2.306 1.569 2.315 -
0.4
c 0.0364 26.50 2200 1.778 1.786 1.736 + 2.4
A 0. 0396 19.88 2200 0.536 2.319 0.446 +18.
0. 0476 45.47 980 2.895 2.264 3.000
-
3.5
B 0. 1190 19.84 2100 1.850 9.306 1.833 + 0.9
N is defined by P=Npn^D^ and the density of the air is assumed to have been
/)=o.o699 i^lf^
P (calc) =io-^>wj^i
+
593(^-^y]
If the observed values of P. are expressible by equation
(22),
i. e., by setting N
=
A +B(-j^)
,
we have
log (N-A)=logB+x\og-j^
(23)
and the values of A, B, and x may be found by trial. We assume
a value of A
,
and plot log {N

A) against log
y^.
If we can find a
value of A such that the points all lie on a straight line within the
observational errors, equation (22)
is satisfied, the slope of the line
is the value of x, and the position of the Hne gives the value of B.
By this method it was foimd that the observations on foiu* of
the five wheels are satisfactorily represented by writing equation
(22)
in the particular form
P= lor^^prvz>{i+
593(5)]
(24)
Buckingham] Windage Losses
of
Steam Turbines 219
and that no other values oi A,B, and x are sensibly better. Values
computed by equation
(24)
are given in column VII of Table 6
and the agreement of the results mth the observed values in column
V is shown by column VIII to be good except for wheel A. The
value for this wheel can evidently not be brought into accord
with those observed for the others by any simple expression for N
as a function of
y^.
It is possible that the discrepancy may be due
to an experimental error, but this seems very unlikely. Unless it
is merely a typographical error, it probably arises from some geo-
metrical peculiarity of this wheel which is not evident from the
description given. The writer has assumed that the blades of all
the wheels were shrouded, because that is the usual construction;
but if the blades of wheel A were open-ended, the relatively
greater windage would be easily understood.
However, there is no advantage to be gained from splitting
hairs in analysing so small a number of data by means of confess-
edly only roughly approximate assumptions, and we may be
content to say that equation
(24)
represents the results of Stodola's
experiments on single-row wheels of ordinary forms running in
the open air, quite as well as could be demanded of any equation
deduced from so inadequate data.
9.
Deduction
of
an Expression
for N from Holzwarth's Data.

Holzwarth's results and the manner in which they were treated

for finding the relation of P to n, have been described in section
3.
The values foimd for y3
=
3

8 are given in Table
7
as they were
read from the straight lines drawn on the plot of log K against
log n and they are imcertain by from o.i to 0.2, because the lines
were drawn merely by inspection.
TABLE 7
Values of K from Holzwarth's Diagrams
Blade length
(inches)
D=20
inches
Z?=30
inches
I>=40
inches
D=50
inches
0.5 3.61 3.39 2.51 3.24
1.0 3.62 3.00 2.51 3.30
1.5 2.28 S.OO 2.58 2.58
2.0 2. 83 .98 2.90 2.89
2.5 2.69 3.39 S.09 S.I4
220 Bulletin
of
the Bureau
of
Standards ivoi.zo
From the great variations in yS it appears either that the values
of Holzwarth's coefficient K read from his diagrams are erroneous
or that the wheels, even for the same blade-length ratio, were far
from geometrically similar. It would be a waste of time to attempt
to reconcile all the readings of K with an equation of the form
(22)
in which /3
=
3,
and I have therefore arbitrarily selected the eight
series which gave values of /3 {italicized in Table
7)
between 2.8
and
3.2,
i. e. values which we can not be sure are different from
3.0,
and I shall confine my analysis to these series.
For each of the selected eight series, the mean value of

tt^^t
was found by averaging over all the values of K for the higher
speeds, omitting a few points where the readings from the dia-
grams were obviously liable to exceptionally large errors. We
thus get eight separate values.
In the units we have been using we have, approximately,
P=o.s^pK
(25)
Since Holzwarth's diagrams are drawn for a constant value of
the density we may treat K as we have previously treated P,
which is proportional to it when
p
is constant. If we proceed in
this way, we find that by selecting an appropriate disk coefficient
A' for each diameter and plotting log I
3^^
~^) ^g^-iiist log
-^,
seven of the eight points may be made to fall very close to a single
straight line of which the slope is 2. The exception is the one
rather doubtful point for the 20-inch diameter. We thus find that
those of Holzwarth's results which are comparable with Stodola's
in giving y8
=
3,
approximately, may be represented by an equation
very like
(24)
which describes Stodola's results. The exponent of
( ^
j
is the same and the disk and blade coefficients are not very
different, though somewhat smaller for Holzwarth's wheels, in
accordance with the fact that the wheels were inclosed instead of
run in the open.
If we write
P
=
pn'D^A +
Bf-Jl
(26)
we have the values of A and B given in Table 8.
Buckingham] Wtudage LossBs
of
Steam Turbines 221
TABLE 8
Values of the Coefficients of Equation
(26)
Blade
coeffi-
cient
BXIOI8
Disk
diam-
eter!?
(inclies)
Disk
coeffi-
cient
AX10i
19.8-45.5 1.0
20 (0.54)
30 0.86
40 0.32
50 0.22
21.1 0.63
47.1 a5
0.44
593
(243)
414
414
414
Stodola's observations on wheels B, C, D, E, in open air
Holzwarth's observations. Wheels inclosed in steam
Stodola's bladeless disk run in open air
Odell's disk D in open air
Jttde; from observations by Lewicki, Odell, Stodola, and Holzwarth, for
wheels run in open air
The difference between the coefficients deduced from Stodola's
and from Holzwarth's observations would probably have been
greater if Holzwarth's values had all been obtained from experi-
ments in air. His results were obtained "from occasional experi-
ments carried on for some years

partly in air, partly in steam of

different density." Since he says nothing to the contrary, the
steam was doubtless saturated and probably by no means dry, so
,
that we may infer from Lewicki's observations, described in
section
4,
that the value of A^ or of the coefficients A and B would
have been smaller if the experiments had all been made in air or
perfectly dry steam.
On account of the varying density in Holzwarth's experiments,
some sort of reduction to a standard density must have been
made in order to get the values of K for drawing the diagrams.
Holzwarth sets P<^p and we must assume that he used this rela-
tion in the reduction. But if in equation
(7)
the value of yS is
very different from
3,
the exponent of
p
must differ by the same
amount from unity and a reduction of this sort is not permissible.
Although no detailed criticism is possible, it is evident that we
have here an additional ground for omitting from consideration
those series of Holzwarth's results which do not conform approxi-
mately to the equation P
=
const X n^.
The degree of approximation to which equation
(26)
with the
coefficients given in Table 8 reproduces the readings from Holz-
2764714
5
222 Bulletin
of
the Bureau
of
Standards [Vol. TO
warth's diagrams, may be seen from Table
9
in which are given
the values K(ohs) read from the diagrams, and the values X(calc)
computed from equations equivalent to
(26)
with the coefficients
in Table 8. The readings may be in error by one tmit at the low
values and several imits at the higher.
TABLE 9
Comparison of Observed and Computed Values of Holzwarth's
Coefficient K
speed in r. p. m. n= 1000 1250 1500 1750 ^r] 2500 3000 3500
D
(in.)
I
(in.)
5 6 8.5
5.9
20
20.1
28.5
28.8
40
41
85
86.9
123
124
161
172
236
244
10
9.4
31.5
31.9
44
45.8
70
65.2
150
138
211
197
288
272
446
388
14.5
14.1
45
47.7
62.5
68.3
109.5
97.3
224
206
334
294
24.5
27.5
94
93.1
131
133
205
190
375
402
560
574
45.5
47.5
160
161
238
231
344
328
527
695
(120)
75.5
293
256
20
30
40
50
2.0
1.0
1.5
2.0
2.0
2.5
2.0
2.5 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
10.5
6
12.5
8.5
13
12.2
26
25.7
34
36.8
50
50.8
65
72.3
13.5
11.6
19
16.7
20
23.8
46
50.3
63
71.8
86.5
99.2
118
141
For the smaller wheels the observed resistance at the lower
speeds is considerably larger than the calculated, and the law of
resistance is evidently not the same as at the high speeds. At
750
r.
p.
m. the observed K is larger than the calculated in every
instance. But if we omit the low speeds, where agreement is not
to be expected, and the highest speeds, where slight errors in the
drawing of the diagrams cause large errors in the values of K read
from them, the agreement of observed and calculated values is
faircertainly as good as would be demanded of any formula by
one who has studied the diagrams.
Buckingham] Windage Losses
of
Steam Turbines
223
10. Further Values
of
the Disk Coefficient.We have now
exhausted the few published data which are available for inves-
tigating the dependence of the coefficient N of equations
(7),
(8), (9),
etc., on the blade-length ratio, and there remain for
consideration only experiments on bladeless disks which will give
values of the disk coefficient for comparison v^th the values
obtained from experiments on wheels with blades.
Stodola's boiler-plate disk of 21.1-inch diameter absorbed
0.147
hp when running 2000 r.
p.
m. in air of density
p
=
o.o'j lb. per
cu. ft^. From these values we find A =0.63 X io~^*. The only
remaining experiments on a rigid disk, which gave ^8
=
3
nearly,
are those of Odell on his largest disk. From data already given in
section
4
we find for this disk ^4 =0.5 X
10-*
approximately. Both
these values are included in Table 8. The difference between the
values of A deduced from Stodola's observations on wheels and on
the disk need not cause any surprise, for we do not know anything
about the relative roughness in the two cases. The shrouding over
the blades and the difference of profile between wheels and disk
may accoimt for the difference, which appears to be genuine.
One more value of A is included in Table
8, namely,
A =0.44 X io~*, deduced by Jude from a general examination of
all the experiments already mentioned in this paper. It is appli-
cable according to its author to flattish disks in open air.
11. The Influence
of
Axial Clearance.For information on this
point we have only Stodola's comparative runs of three of his
wheels in the open air and in casings. The results are given in
Table 10, some of the data being repeated from Table 6.
TABLE 10
Stodola's Experiments on Varying Clearance
I n m TV V VI
I
D
D
inch
Axial
clearance 1016JV Ratio
00 2.32
A 0.0396 19.88 0.0079 1.28
2.26'
0.55
E 0.0476 45.47 0.0056
00
0.57
9.31
0.25
B 0.1190 19.84 0.0079 3.53 0.38
224
Bulletin
of
the Bureau
of
Standards [Voi.w
In column V of the table are given the values of A'' in the equa-
tion P
=
Npn^D^ for the axial clearances given in column IV, the
clearance
oo meaning that the wheel was run in the open air. In
column VI there is given for each wheel the ratio in which the
dissipation was reduced by the presence of the casing.
It appears from this table that the effect of a close casing is
much greater on the blades than on the disk. By comparing
wheels A and B we find that the presence of a casing, which left
at the blade edges an axial clearance of about 0.008 times the
disk diameter, reduced the resistance considerably more in the
case of B, which had blades three times as long as those of A, the
ratio of reduction being 0.38 for the long blades as against
0.55
for the short ones. If we compare wheels A and
E,
which did
not differ very widely as to blade-length ratio, we find that reduc-
ing the clearance from
0.0079 ^ ^o 0.0056 D, i. e., to two-thirds,
decreased the resistance, expressed as a fraction of the open-air
resistance, by about one-half, or from
0.55
to 0.25. We have
also already found in section
4,
when comparing wheel B with
Lewicki's wheel of about the same blade-length ratio, that increas-
ing the axial clearance by about one-half, or from 0.008 D to
0.012 jD, increased the resistance some
35
per cent.
These few isolated data are evidently not a sufficient ground for
any general quantitative statement about the effect of clearance
on windage, but they may be valuable as a basis for guesswork
in cases which happen to be nearly similar to the ones mentioned.
In this connection we may also note the results obtained by
Stodola
^
in comparing wheels run in the normal or forward direc-
tion with the same v/heels run backward. In open air the resist-
ance backward was in one example as much as
5.4
times the
resistance forward, though in other examples the ratio was not
so large. But inclosing the blades reduced the difference very
much, and the longer the blades and the greater the part of the
resistance due to the blades the greater is this effect of the casing,
so that with very small clearances the resistance when the wheel
is run backward is very little greater than when it is run forward.
With wheel B for which
y^
--=0.1
19,
when the axial clearance was
0.008 D, the resistance backward was only
13
per cent more than
the resistance forward.
Buckingham] Wiudage LossBs
of
Steam Turbines
225
12. Remarks on Further Experimental Results.In the foregoing
discussion the symbol P has everywhere denoted the power
required to drive a wheel against the resistance of the otherwise
stagnant medium surrounding it, and the experiments noted
have referred to this state of affairs. But the conditions of ordi-
nary practice are different, and it remains a question whether,
in designing a steam turbine, a windage correction based on even
completely satisfactory data of the kind considered could be
considered reliable. The only answer that can be made to this
question is that we do not know; and the best, because the only,
thing we can do at present is to compute windage corrections for
designing purposes as if the turbine were to be driven independ-
ently from without, acting merely as a brake, a condition which
occurs only in the case of marine turbines with reversing stages
or with cruising stages which are by-passed at full power.
Another pertinent question is : How much ought the computed
windage loss to be reduced when a part or all of the blades are
working in the ordinary manner, with steam from the nozzles
passing through them? The experiments of Lasche, quoted by
Stodola,^ and of Jasinsky
^^
are not sufficiently comprehensive to
tell us more than that the windage decreases as the admission
arc increases and fewer blades are idle. With rectangular nozzles
and a continuous steam belt, it will probably be not far wrong to
multiply the blade term of the computed resistance by the frac-
tion of the whole circumference which is not occupied by open
nozzles, i. e. by the fraction of the whole number of blades which
is idle at each instant.
The experiments we have considered having referred only to
disks or single-row wheels, one further question remains: How
does the windage, i. e. the value of N in our equations, depend
on the number of rows of blades ? Here again we have only the
most meager information. Experiments by Lasche, quoted by
Stodola,^ gave for wheels with from i to
4
rows but otherwise,
presumably, similar, resistances which stood in the following
relation
:
Number of rows of blades
1234
Relative resistance i 1.2 1.6 2.4
226 Bulletin
of
the Bureau
of
Standards [Voi.io
But the experiments were on wheels which were not closely
encased so that they do not give us much practical assistance.
The increase of resistance with number of rows of blades would
probably be very much less rapid if the wheels were run in cas-
ings with fine clearances.
13. Dynamically Similar Wheels.The one fact which emerges
clearly from the foregoing discussion is that the task of provid-
ing data for the development of a satisfactory general formula
for computing the windage losses of high-speed steam turbines,
has hardly been begun. Suggestions as to futtire lines of experi-
ment are therefore in place here.
Up to peripheral speeds of at least one-half the speed of sound,
or about 700
feet per second for steam, we are justified in treat-
ing the medium as incompressible and the resistance phenomena
may be described by an equation of the general form
(4)
. If we
in
introduce the "kinematic viscosity" v = -, the equation may be
written
P=/m'i?^9^^)
(27)
q)
being a function of which the form, though imknown, is fixed
by the shape of the wheel and casing.
Let us compare two geometrically similar wheels of diameters
D and Do
running in media of densities
p
and
Po
and kinematic
viscosities v and v^ at speeds of rotation which stand in the ratio
no
v^D/
Speeds thus related are
"
corresponding speeds."
At corresponding speeds
nD""
noDo^
and since the form of cp depends only on the shape, which is the
same for the two wheels, we have
n^j^K'd?)
Bttckinvkam} Wiudoge Losses
of
Steam Turbines
227
no matter what the shape of the wheels and the form of the func-
tion
(p may be. The ratio of the dissipation by the two wheels
at corresponding speeds is therefore, by equation
(27)
P pn^D'^
Po~Pono'Do'
or by equation
(28)
which defines corresponding speeds,
Any two geometrically similar wheels running at corresponding
speeds constitute a pair of ** dynamically similar" systems, and
the power dissipated by either may be determined from an experi-
ment on the other by means of equation
(29)
, if the diameters of
the wheels are measured and the densities and kinematic vis-
cosities of the media are known.
If the experiments are all made in the same medium so that
p
=
Po
and V = Vq, corresponding speeds are inversely as the squares
of the diameters and the powers dissipated at corresponding
speeds are inversely as the diameters.
If T and Tq are the torques required to drive the wheels at
corresponding speeds, since PccnT we have by (28)
and
(29)
Tr -^.1
At corresponding speeds in a given medium the torque is directly
as the diameter.
The shearing stress in the metal of the shaft, so long as the
shaft is not sensibly distorted, is proportional to the torque
divided by the cube of the shaft diameter, or with similar wheels
T
to
yp.
Hence the ratio of the shearing stresses, 5 and
So,
in. the
shafts of two dynamically similar wheels is
At corresponding speeds in a given medium the shearing stresses
are inversely as the squares of the diameters.
228 Bulletin
of
the Bureau
of
Standards [Vol. 10
The centrifugal stresses C and
Co in the metal of the two wheels,
if they are of the same density, will stand approximately in the
ratio.
r.-(rJ(&J
At corresponding speeds in a given medium, the centrifugal
stresses are inversely proportional to the squares of the diameters
if the wheels are of the same density.
14. The Use
of
Model Wheels in Studying Windage.

^To avoid
the difficulties of working with large wheels, it may be desirable
to utilize the results of model experiments as is done in determin-
ing ship resistance.
If we limit ourselves to tne use of a single medium, the prac-
ticable range of scale reduction is not great, unless the full-sized
original runs very slowly, and this case is not interesting because
we know that at low speeds the windage losses are too insignificant
to demand much attention. For dynamical similarity in a given
medium, a quarter-scale model must, by equation
(28)
run at 16
times as many revolutions per minute as the full-sized original;
and by equation
(32),
the centrifugal stresses in the wheel will
then be 16 times as great in the model as in the original; an
increase which would usually not be permissible. Furthermore,
the peripheral speed of the model will be
4
times that of the original
and may approach the acoustic speed so closely as to invalidate our
fundamental assumption that the medium behaves sensibly as if it
were
incompressible. There is evidently not much information
to be got from small scale models unless they can be nm in a med-
ium of much less kinematic viscosity than steam, so that the speed
of the model may be reduced, in accordance with equation
(28)
.
Water is such a medium. Using values from I^andolt and Bom-
stein's tables, and comparing water with air which is known to
act like dry steam, we have at
20
C and i atmosphere pressure.
Air Water
p=0. 001205
^=0.0001898
v=0. 1575
p=0.9982
;^
0.01012
y= 0.01014
Buckingham^ Windage Losses
of
Steam Turbines 229
Let it be desired to find the power dissipated by a wheel of diam-
eter Do
running no
revolutions per minute in air, by means of
experiments on a model of diameter D
=
^
running in water,
the temperatiu-e being
20
C in each case and the air being at
normal pressure. We then have to substitute in equations
(6)
to (10)
p
V

=828
=
0.0644
the uncertainty of

being about

3
per cent.
For any given scale ratio r, the ratio of corresponding speeds,
n for the model and no for the original will be, by equation
(28)
,
n
=
o.o644r2

3
per cent
(33)
/fro
The ratio of the powers dissipated at corresponding speeds will
be, by equation
(29)
,
p
p- =0.222 r 6 per cent
(34)
from which Po can be foimd if P has been measured in an experi-
ment on the model.
We must next consider whether the stresses in the wheel and
shaft of the model will rise too high when we make a convenient
reduction of diameter. Taking first the centrifugal stresses in
the wheel, we have by substitution in equation
(32)
^
=
0.00415 r^
(35)
Setting C
=
Co
and solving for r, we find that the stress of any
point in the model will not exceed that at the corresponding point
in the original until r>i5.7. As a 10 to i reduction will usually
be ample, there will be no difficulty regarding the bursting strength
of the model if it is made of the same material as the full-sized
wheel.
230
Bulletin
of
the Bureau
of
Sta^tdards [Vol. lo
For the ratio of the shearing stresses in the shaft, equation
(31)
gives us
5
So
= 3.43^^
(36)
A scale reduction r
=
10 would thus give us
343
times as great a
stress in the shaft of the model as in the original shaft. This, at
first sight, looks quite impracticable. But in fact turbine shafts
are generally made much stronger, for the sake of stiffness, than is
required by torsional strength; and, furthermore, the torque due
to windage when the wheel is driven light is only a few per cent
of the torque of the wheel running at full load. Hence the ratio
343
would probably not always be excessive. However, doubling the
diameter of the shaft of the model would have hardly any effect on
its windage resistance, and by such a small sacrifice of exact geo-
metrical similarity the use of a model in water might always be
made practicable, so far as the strength of the shaft is concerned.
To make the matter more concrete we may carry out the com-
putations for a few typical cases of wheels run in air in comparison
with models of 12 inches diameter run in water. We have the
values given in Table 1 1
.
TABLE 11
FuU-sized
diameter
Do
(inches)
Wheel
. speed
no
(r. p. m.)
Diam-
eter
ratio for
12-inch
model
Do
Speed of
model
n
(r. p. m.)
Power
ratio
P
Po
Stress
ratio in
shafts
S
So
36
60
84
120
3600
1800
900
450
3
5
7
10
2090
2900
2840
2900
0.67
1.11
1.55
2.22
31
86
168
343
It therefore appears that the investigation of such cases as are
commonly met with, by means of
12-inch models run in water,
would be quite practicable so far as the points already treated
are concerned, and only one further point remains to be considered.
This is the question of cavitation.
Buckingham] Windage Losses
of
Steam Turbines
231
If cavitation occurs in the water the similarity of the model
and the original will cease to exist, and equation
(29)
will not give
correct results. To avoid cavitation, the water surrounding the
model must be under pressure and the casing must be constructed
accordingly. At atmospheric pressure and high speeds cavita-
tion will certainly occur. If the pressure on the water is increased
the cavitation will decrease and the torque at a fixed speed will
change until the presstue has been increased so much that cavita-
tion has been eliminated. By observing the variation of torque
or power with the pressure at any given speed we have thus a
means for finding what pressure is needed in order to eliminate
cavitation and make sure that the model in water is comparable
with its original in air or steam. It remains to be seen whether
the pressures required would be impracticably high, but it appears
that the method of model experiments is worth trying.
15.
Summary.I. The power P required to drive a turbine
wheel of diameter D 2it n revolutions per unit time against the
resistance of a homogeneous medium of density
p
and viscosity /x,
when the peripheral speed does not exceed one-half that of soimd,
may be represented by an equation of the general form
in which the form of the unknown function cp depends solely on
the shape of the wheel and its casing.
II. Throughout the practical range of the experimental data,
equation (I) has the simpler form
p^Np'-^n^D^^li^ (II)
the abstract numerical constant N having a value which is deter-
mined solely by the shape of the wheel and casing. All the
reliable data which we have agree with equation (II) or with direct
deductions from it.
III. At low speeds the value of 3 is nearly unity and the resist-
ance is directly proportional to the viscosity of the medium. At
232
Bulletin
of
the Bureau
of
Standards [Voi.io
the speeds at which stationary turbines are usually run, 8 is a
small quantity and we have approximately
P
=
Nfm'D' (III)
A closer approximation is obtained by setting 8
=
o.i whence
IV. For wheels of ordinary shapes, running either in the open
or in casings with fairly large clearances, the "shape coefficient"
A^ may be expressed approximately by the equation
JV=A+b(^J (IV)
in which l/D is the ratio of the blade length to the disk diameter.
The disk coefficient A increases with roughness of the disk; the
blade coefficient B decreases when the clearance round the blades
is decreased, but no more definite statements are warranted.
V. (a) For designing purposes we may first compute the windage
loss by the following equation deduced from Stodola's results
:
P
=
io-^ >^D^[i+59o(0]
(V)
In this equation
P
=
the horsepower dissipated
p
=
the density of the medium, in pounds per cubic foot
w=the speed, in revolutions per minute
D
=
the diameter, in inches, at the root of the blades
/
=
the length of the blades, in inches
The equation is applicable to wheels of ordinary shapes with one
row of shrouded blades, running in the open or in casings which
leave large clearances, in a homogeneous medium such as air or
dry steam.
(b) Reducing the clearances, especially round the blades, reduces
the windage. In some cases the amount of this reduction may be
estimated from Table lo but no general quantitative statement
is possible. The reduction affects mainly the blade term.
(c) Open-ended blades have more resistance than shrouded
blades, to which equation (V) refers, but there are no data to
Buckingham] Windage Losses
of
Steam Turbines
233
show how much. When the radial clearance over the blade tips is
small, the presence or absence of shrouding will have Uttle effect.
(d) A wheel run backward experiences a greater resistance
than the same wheel run forward, i. e., in the normal direction.
When nmning in the open the difference may be considerable,
but if the clearances round the blades are small, the resistance
backward is not much greater than the resistance forward.
(e) Each additional row of blades increases the resistance.
When run in the open, a fotu--row wheel may have two and one-half
times the resistance of a one-row wheel, but there are no adequate
data published. With short blades and fine clearances the effect
of increasing the number of rows will be much less than that
indicated.
(/)
A wheel run in wet steam experiences more resistance
than in dry steam or air of the same density. In Lewicki's experi-
ments the increase was
30
per cent; we have no other informa-
tion on this point.
(g)
Equation (V) gives the power absorbed when the wheel is
driven from without. If the wheel is working in the usual manner
we may reduce the blade term in the ratio of the number of idle
blades to the whole number.
(h) The values given by equation (V) may be too large, espe-
cially for smooth wheels with short blades ; for the disk coefficient
used, viz.: A
= 10-^^,
is larger than the values deduced from
other experiments.
VI. The necessary indefiniteness of many of the statements
made under V is due to lack of experimental data. No formula
based only on our present knowledge can be trusted to give accurate
results, and no formula which has not the general form (II) should
be used for extrapolation beyond the limits of the experiments
from which it was deduced. Equation (V) , if used as indicated,
probably gives safe maximum values.
VII. The method of model experiments is applicable to the
study of the windage losses of steam turbines and might prove very
useful-
by
decreasing some of the difficulties encountered in making
systematic experiments on a large number of full-sized wheels.
Washington,
July 25, 191
3.
234
Bulletin
of
the Bureau
of
Standards vvoi. to
REFERENCES
1. Stodola, Die Dampfturbinen, 4th ed., 1910, pp.
120.
2. Odell, Engineering, 77, p. 30; Jan.
i,
1904.
3.
Osborne Reynolds, Scientific Papers,
2, p. 51;
or Phil. Trans.; March
15, 1883.
4. W. Froude, B. A. Report;
1874.
5.
Zahm, Phil. Mag.
(6), 8, p. 58; 1904
II.
6.
E. Lewicki, Zs. d. Ver. Deutsch, Ingen., 47, p. 492;
Apr.
4, 1903.
7.
Holzwarth, Power, 27, p. 50; Jan. 1907.
8. See Kranz, Lehrb. der Ballistik
1, p. 49,
also Mallock, Proc. Roy. Soc. London,
(A) 79, p. 262; March,
1907.
9. Jude, The Theory of the Steam Turbine, 2nd ed. 1910, p.
221.
10. Jasinsky, Zs. d. Ver. Deutsch. Ingen.,
63, pp. 492, 538; 1909
I.
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