*B 77 515
_^^__ ^^^^H
i
On the Losses in
Convergent Nozzles
"By Professor A. L. MELLANBY, D.SC., and
WM. KERR, A.R.T.C.
A PAPER READ BEFORE THE NORTH EAST COAST
INSTITUTION OF ENGINEERS AND SHIPBUILDERS, ON
THE i8TH FEBRUARY, 1921, AND REPRINTED BY ORDER
OF THE COUNCIL.
NEWCASTLE-UPON-TYNE :
PUBLISHED BY THE NORTH EAST COAST INSTITUTION
OF ENGINEERS AND SHIPBUILDERS, KOLBEC HALL.
LONDON :
E. ftf F. N. SPON, LIMITED, 57, HAYMARKET, S.W. i,
On the Losses in
Convergent Nozzles
'By Professor A. I.. MELLANBY, D.SC., and
WM. KERR, A.R.T.C.
A PAPER READ BEFORE THE NORTH EAST COAST
INSTITUTION OF ENGINEERS AND SHIPBUILDERS, ON
THE' 1 8-TH FEBRUARY, 1921, AND REPRINTED BY ORDEPs.
OF THE COUNCIL.
NEWCASTLE-UPON-TYNE s
PUBLISHED BY THE NORTH EAST COAST INSTITUTION
OF ENGINEERS AND SHIPBUILDERS, BOLBEC HALL.
LONDON :
E. & F. N. SPON, LIMITED, 57, HAYMARKET, S.W. i.
1921.
-re <-?
ON THE LOSSES IN CONVERGENT NOZZLES.
BY PROFESSOR A. L. MELLANBY, D.Sc., Associate Member, AND
WM. KERB., A.R.T.C.
[READ IN NEWCASTLE-UPON-TYNE, ISxn FEBRUARY, 1921.]
Introductory. The action of a fluid in a nozzle is enveloped
in such experimental and theoretical difficulties that only a
modicum of useful fact is to be expected from even the most
strenuous inquiry. The attack 011 the problem* however, must
continue to be made until such time as the mathematician
demonstrates the sum of a nozzle's peculiarities with all due
rigour. Examination of the remarkable series of intractable
equations laid down, in mathematical physics, for the action of
moving fluids will show very clearly that that time is not yet
reached. It seems necessary, therefore, to 1 be content with what
the meagre experimental processes reveal and to* endeavour
slowly to extend the field of fact, so that, ultimately, it may be
possible to have a fairly clear view of what is a highly important
operation.
It is with the idea of assisting this gradual enlightenment
that the series of papers on nozzle flow by the present authors has
been entered upon. The matter herein submitted represents the
third section of the series and, while it is to some extent in direct
continuation with its predecessors, and though a certain amount
of reference backwards is unavoidable, it is hoped with but little
repetition to make it almost self-contained.
One main difficulty in all experimental investigations on this
subject arises out of the necessity to confine attention mainly to
elementary nozzle forms of rather minute dimensions. The
enormous steam capacity of even quite small nozzles prohibits
extensive examination of practical forms of any reasonable size.
While this may detract from the direct application, in practice,
of any results so obtained, it should be remembered that' the
use of the simpler types probably eliminates several disturbing
4 ON THE LOSSES IN CONVERGENT NOZZLES.
factors from a problem which is, at best, rather complex.
Beyond this, their use is desirable where the intention is to
achieve some decision as to the causes of loss in expansion since,
thereby, the necessary internal examination of the jet is simpli-
fied to the maximum extent.
The Authors' experiments and analyses had this underlying
intention and, consequently, they felt justified in using the
customary small circular nozzles. While such have been fre-
quently and thoroughly examined for complete effects, it does
not seem that much has been done with a view to determine the
losses in detail. It will be obvious that this kind of investiga-
tion must be made before reasonably definite knowledge can be
claimed as to the real action in jet expansion, or before a search-
ing study of actual types can be entered upon, complicated as
these are by the requirements of constructional forms.
Again, the use of ordinary saturated steam in such investiga-
tions, introduces, as is now well known, the upsetting condition
of supersaturated now. While the effect of this in creating
excessive discharge is well understood, the degree of its occur-
rence can hardly be accepted as definite, nor can it be held to
be exactly calculable; and, hence, steam initially wet or dry
must be considered unsuitable for accurate experiment. Since
a similar objection applies to the use of those lower values of
superheat, which would compel some part of the expansion to
take place in the wet or supersaturated fields, it appears neces-
sary to employ superheat of sufficient magnitude to prevent the
steam condition from passing through the limiting dry state.
These main points outline the Authors' scheme, which
amounts to an attempt at the study o<f nozzle expansion and
losses in the simplest forms by the internal examination of
superheated steam jets. The first paper* was of a general nature
and gave a method of analysing the losses througii the medium
of search-tube experiments, and it is by the use of that method
that the points to be herein dealt with are developed. The
succeeding paperf g*ave an account of the experiments and a
statement and descriptive treatment of the results. It is now
proposed to carry out a closer investigation of the results there
given for the convergent nozzle types used.
* " Steam Action in Simple Nozzle Forms," British Association, Section
" G," August 26th, 1920. Engineering, September 3rd, 1920.
f " Pressure Flow Experiments on Steam Nozzles," Proc. Inst. Eng.
& Shipbuilders in Scotland, November 16th, 1920.
ON THE LOSSES IN CONVERGENT NOZZLES.
As, in its scope, the present matter encircles one of the
outstanding peculiarities of nozzle action it is, perhaps, desirable
to give, first of all, some consideration to this particular point
in its various aspects.
(i)
of the
SECTION I. REVIEW OF THE PROBLEM.
The Anomaly of the Velocity Co-efficients. In the course
developments of this subject O'f nozzle expansion anomalies
NO. 1. RATEAU
2. LOSCHGE
3.-'
4. GIBSON
5. MELLANBY & KERR.
STRAIGHT CONVERGENT NOZZLE SAT. STEAM.
VENTURI METER
STRAIGHT CONVERGENT NOZZLE.
,, CONV. PAR
ZOELLY TYPE NOZZLE
SUP. STEAM.
AIR.
HIGH SUP. (WITH SEARCH
TUBE EFFECT).
BY FLOW RATIO WITH
STRAIGHT TYPE.
60 -8S -tfo <)$
FIG. 1. VARIOUS DISCHARGE CO-EFFICIENTS.
have frequently been disclosed, but only in certain cases eluci-
dated. Outstanding in the latter achievements are the physical
interpretations of critical pressure ratio, and of excessive steam
discharge ; the former by Osborne- Reynolds in 1886, and the
latter by Mr. H. M. Martin in Engineering in 1912. These
two solutions are rather remarkable for their inherent simplicity
and apparent adequacy, and are now universally accepted in
their respective applications.
Several points still present, however, certain elements of
mystery. It is unnecessary here to enumerate these in their full
variety as, in the course of the present discussion, contact is
made with one only ; although that is probably the chief of them.
6 ON THE LOSSES IN CONVERGENT NOZZLES.
The particular point may be briefly expressed as the fall away
in the standard of performance of convergent type nozzles with
restriction of the range o>f expansion. The fact has been thor-
oughly demonstrated by experiment, and is usually exhibited by
the form of the curve showing the variation of the co-efficient
of velocity or of discharge. This curve, on a base of pressure
ratio of operation, or jet speed developed by the expansion,
always shows continuous reduction of the co-efficient with limi-
tation of the range ; thus apparently indicating higher propor-
tionate energy losses for the lower fluid speeds.
In illustration, Fig. 1 gives a few such curves and, while it
indicates differences between the various experimental results, it
shows very clearly the peculiarity referred to. The curves for
saturated steam flow in straight convergent nozzles show the
feature of excessive discharge by rising to values above unity
towards the critical range ; but, even with this effect, the curve
forms are of the same general type as for superheated steam,
where such an influence is absent.
In bringing the various values to a common base of pressure
ratio, some approximation has been indulged in as, in their
original forms, several of the results were otherwise shown. The
introduction of Professor Gibson's curve* derived from experi-
ments on an air venturi meter demonstrates that the effect is
common to the expansion of different fluids. Although the
venturi form is apparently convergent-divergent it is only oper-
ating as a\ convergent type within the expansion ranges to which
this discussion applies, and from which the co-efficients were
obtained.
Fig. 1 shows co'-efficients of discharge. The velocity
co-efficients are directly comparable with these but, in general,
slightly higher. The nozzle efficiency may be taken as given by
the square of the co-efficient- of velocity ; and, since the higher
ratios show the lower co-efficients and correspond to the lower
speeds of flow, the efficiency is apparently poo-rer with the less
rapid motion.
Such a result is contrary to any pre-conceived ideas of the
matter, as it would seem only natural to expect the best effi-
ciencies at the lowest speeds. The anomaly so presented has
been frequently remarked upon, and has- created a feeling of
* " Measurement of Air flow by Venturi Meter/' Proc. Inst. Mech. Eny.,
October, 1919.
ON THE LOSSES IN CONVERGENT NOZZLES. 7
perplexity that shows itself in the widely varied explanations
that have at times been advanced.
(ii) Various Hypotheses. The vagaries o>f nozzle flow have
very frequently been credited to probable heat conduction effects
through the nozzle walls and, owing to the little that is known
about these and the almost insurmountable difficulty of examin-
ation, the idea of charging them with all the anomalous features
that have been shown to exist has, perhaps, been too readily
entertained .
So far as the present question is concerned, Professor Gibson
has dealt, in a fairly conclusive fashion, with this particular
conception by showing that, if these effects are appreciable, it
would be natural to expert definitely modified influences by
radical change of conditions as regards nozzle dimensions, or
temperature relations within and without the nozzle. In both
these respects his experiments on air flow are in contrast with
the usual steam nozzle tests; and, since he obtains co-officient
ranges very similar in nature to those found for steam in dis-
similar conditions, it is justifiable to suppose that the heat
conduction influences are practically negligible.
One interesting, but rather speculative, theory on nozzle
expansion has been brought forward by Dr. Stewart.* It
involves an extension of Boltzmann's hypothesis, in the kinetic
theory of gases, which postulates the equal division of molecular
energy between the various degrees of freedom O'f a molecule
The justification for this assumption is fairly substantial, as
the values of the adiabatic index thereby defined for gases of
different molecular structure are in excellent agreement with the
known figures. The general result is that the smaller the
number of atoms in the molecule and, consequently, the fewer
the degrees of freedom, the higher the adiabatic index.
Dr. Stewart's developments o>f this matter further assumes
that very rapid expansion of a gas is equi valent to a reduction of
the degrees of freedom, by the " locking up " of the molecular
energy of rotation. Hence an increase in the index of the law
PV" = constant ensues, and the gas gives a discharge in
excess of that shown by the ordinary theory. Accepting the
premises the result follows in rigorous sequence.
* "The Theory of the Flow of Gases through Nozzles," Proc. In$t Mech,
Eng., December, 1914.
8 ON THE LOSSES IN CONVERGENT NOZZLES.
This would seem susceptible of easy verification, since it
means that all polyatomic gases could give high mass flow.
Stewart has obtained such for air, but no other recent experi-
ments with air show the effect. Besides this, the same result
should be given by superheated steam, and it is doubtful if this
ha,s ever beeoi obtained in any reasonably careful experiments,
Stewart's high discharge co-efficients are, moreover, deduced
from a curve of flow which is not very rational in form. Practi-
cally all nozzle experiments agree that, when the critical range
is exceeded, a straight line graph very closely connects the flow
and the absolute pressure of supply. The experimental curve
leading to the stated result does not satisfy this condition, and
this result cannot be held conclusive even in the matter that is
the main contention of the theory.
It is, therefore, hardly necessary to consider the extension
of the idea to the possible explanation of the fall in the velocity
co-efficient, since that would make further serious calls on the
imagination in connection with the variability of the degree of
" locking up " with speed of action. Professor Gibson shows that
his air co-efficients are brought fairly level on such assumption,
but the purely presumptive nature of a basis of this kind elimin-
ates all value in the deductions made therefrom.
In practically all nozzle calculations and theoretical consider-
ations, the assumption is tacitly made that there is uniform
pressure and velocity across a flow section. Professor Sir J. B.
Henderson* has pointed out that the inertia effects during the
rapid convergence to the throat would tend to set up pressure
variations across the throat section, with an inverse velocity
range in keeping therewith. This effect would result in a wave
flow beyond this point; but since the mass flow and, consequently,
the co-efficients deduced therefrom arise from the distribution
of values across either the throat or outlet sections, it is only
necessary to consider the effect of such distribution. At the time
this aspect of the matter was brought forward attention was
principally directed to the difficulty of excessive discharge. It
is, however, easy to see that, although such an effect is quite
probable, any occurrence of the kind can explain neither large
flows nor the point under discussion.
* " Theory and Experiment in the Flow of Steam through Nozzles/'
Proc. Inst. Mech. Eng., February, 1913.
ON THE LOSSES IN CONVERGENT NOZZLES.
In a convergent nozzle working" at the critical pressure, the
maximum discharge, theoretically and actually, can only occur
if there is uniformity of pressure across the throat. Any other
pressure value above or below the critical, over even a minute
portion of this area, would entail smaller now quantities, since
such values, with their corresponding velocities and volumes,
require larger areas than when at the critical pressure condition.
With a back' pressure higher than the critical, assumption of
variability of pressure over the outlet section would create dis-
crepancy with the theoretical in different directions, depending
on whether the variation were above or below the set back
pressure. If below, the flow would be too great, and the co-
c >K +C
FIG. 2. DISTRIBUTION OP VELOCITY.
efficient too high, which is contrary to the actual finding; if
above, it would be low, which agrees better with the observed
facts. A diminution of the co-efficient from this cause would,
however, demand an increasing pressure discrepancy with
decreasing pressure range and, since the inertia effects on which
the changes presumably depend are naturally the more severe
at the higher speeds, this application of the argument would
result in a finding in direct conflict with the premises. Besides
this, actual pressure determinations! in nozzles seem to show
t" Pressure Flow Experiments on Steam Nozzles," Proc. Inst. Eng.
& Shipbuilders in Scotland, November 16th, 1920.
E 2
10
ON THE LOSSES IN CONVERGENT NOZZLES.
that the tendency in convergent types, working above the critical
value, is towards slight over-expansion rather than under-
expansion.
Alteirnatively to the point of view just treated, it is possible
to conceive a non-uniform distribution of velocity across a section
independent of pressure variation. This might be considered
due to boundary and viscosity effects, and is actually known
from experiment, to exist in slower pipe flow. If, again, the
idea of no slipping at the boundary wall usual in mathematical
physics is assumed, it is possible to make a rough quantitative
examination in the following way.
97
l-o
r h e 170 1 -f-Q Vzfoc 'i hy
VT
(Q
FIG. 3. COMPARISON OF CO-EFFICIENTS.
Taking a circular nozzle with internal search tube as in the
case of the Authors' experiments, and assuming the curve of
velocity indicated in Fig. 2 to be given by :
U = l/ l f)X 2m
and noting that u o for x = c or + r there results :
The area of the figure is :
ON THE LOSSES IN CONVERGENT NOZZLES. 11
If the theoretical velocity is u t the co-efficient of velocity is :
J = s/ &2_Y
Or, if /! is supposed practically equal to u t :
2m
This represents a co-efficient falling with the speed in much
the same way as a convergent nozzle coefficient as is shown
in Fig. 3. In this, m is considered proportional to speed, and
the curve No. 5 in Fig. 1 is put on a similar base for comparison.
Since m is quite large the motion envisaged in the problem,
and roughly illustrated in Fig. 2, is practically equivalent to
that in which the whole central mass moves forward with uniform
speed, but this speed rises to its full value from zero at the
boundary through a thin film of fluid. As higher m values are
required at the higher speeds it follows that this boundary film
would become thinner as the speed increased a fact readily
understood, and usually adopted in explanation of heat trans-
mission phenomena.
With the velocity not uniform across the section the energy
co-efficient c e - is not equal to c v 2 . The value of c e is easily
obtained as above for c v , and is :
8m 2
C ~
(2m + 1) (4m + 1)
and :
>T=V~
Ce V (2i
8m 2
\m + 1) (4m + 1)
This also is shown in Fig. 3, where \/ c e is plotted, and lies
definitely above the c v curve.
The idea so developed is simpler than any of those previously
discussed, is less conjectural and seems superficially more
adequate. Rational as the initial conception is, however, the
adequacy of the development here given, as a possible explana-
tion of the falling co-efficients in nozzle expansion, is much more
apparent than real.
The failure hinges on the neglect of the ratio uju in the
expression :
_u l f 2m \
^-*/A2m + U
as the co-efficients against which the factor in brackets has been
compared are outlet co*-efficients only. These outlet co-efficients
rise with increasing speed but, if a convergent nozzle with n
12 ON THE LOSSES IN CONVERGENT NOZZLES.
fairly long parallel tail piece be considered, it will be found that
the velocity co-efficients fall with the increasing speed along the
tail. It is, therefore, evident that both factors in the expression
for c v are operative, and that the ratio ujut is probably the more
important, since c v may be either an increasing or decreasing
function of the speed, depending on whether growth of outlet
speed by change of the ratio of expansion or internal development
of speed is considered.
This argument does not detract in any way from the possi-
bility of some such variation of velocity over a section ; it only
shows that the non-uniformity of the velocity can provide no
explanation of the peculiarity in nozzle coefficients.
The foregoing treatment of these several points of view
demonstrates that not one of a fair variety of conceptions can
serve to explain the feature under discussion. The more ingenious
seem improbable and entirely lack real experimental support,
while the more probable fail somewhere in application.
It would seem necessary, therefore, to make some internal
examination that would enlighten the growth of loss along the
expansion, and this has been the Authors' view and effort. A
preliminary outlook on how the losses might operate 1 to cause the
effect can be given in a simple way.
The ordinary idea of nozzle loss is that it is mainly due to
frictional effects that increase rapidly with the speed. Since
the energy of expansion is proportional to the square of the speed
developed, and the loss would seem dependent if of a frictional
type on a higher power it would appear necessary that the
efficiency should drop as the expansion is extended. Of course
the increased pressure range creates a greater mass flow, which
might influence the matter slightly on the assumption of a
frictional loss dependent alone on speed. The nature of the
relationship can be readily shown as follows :
Let e = total energy loss per sec. in the nozzle
2/0 =- actual velocity of outlet
rj = nozzle efficiency.
Then :
Theoretical energy per Ib. fluid = - - (a = constant). And,
since the energy loss per Ib. fluid is e/G, the efficiency is given
by:-
11 ~ ~ aQu 3 '
ON THE LOSSES IN CONVERGENT NOZZLES. 13
The flow, in terms of the velocity, area, and specific volume at
outlet, is :
G_ AQ^Q
~~ V '
Vo
From these it follows that :
1
1 + *^? (b = constant).
W
The efficiency therefore falls as the value of V ^o 3 rises,
and this occurs so long as Y e increases at a more rapid rate
than w 3 . Since V itself increases with u , e need, only be
dependent on some power of u less than 3 in order that ^ should
diminish as u increases. If e represented a purely frictional
loss, it would seem certain that Y e would increase at a greater
rate than u Q 3 .
That f] does not diminish with increasing u is definite proof
that e is not solely a loss of this nature. Such a loss may be
involved in it, but this must be accompanied by another effect
either of constant magnitude, or increasing only with a low power
of the speed, but sufficiently important definitely to counteract
the natural influences of the normal frictional loss. In such case
the efficiency would be written :
UQ
ming, for argument,
then :
Assuming, for argument, that ^ varied as u 0) and e t as u s ,
"n = z-rr-
u 2
Obviously, this is a rising or falling function of u depending
on the relative magnitudes of the two terms in the denominator.
It can hardly be anticipated that the results will be of such a
degree of simplicity; but if the loss is separable into two effects
of this order the problem that has been discussed is effectively
elucidated.
(iii.) Value of a Solution. The importance of this whole
matter is perhaps less obvious than real ; and that importance is
at once scientific and practical ; the former because of the value
of a sound knowledge of the losses accompanying fluid expansion,
and the latter on account of the influence of that knowledge on
turbine theory and design.
14
ON THE LOSSES IN CONVERGENT NOZZLES.
If two loss effects exist and can even imperfectly be
separated, it follows that some indication must be obtained as to
the order of frictional resistance at high fluid speeds, and whether
these retain their characteristic dependence on the square of the
speed as is well established for moderate values. Besides this,
and second effect must be of importance as a matter of principle,
since it has not been customary to consider the existence of such
in appreciable degree.
In impulse turbine design there is no very rational rule as to
the best number of stages in any given case. Complete working
knowledge would require, as a minimum, nozzle and blading
efficiencies on a base of, say, theoretical speed, together with the
Cast. fcQ
V
Cos*
fV-
. - 'I 10
FIG. 4. EFFICIENCY CURVE FORMS.
best blade-steam speed ratios. This last will, however, not vary
greatly and it might be supposed eliminated by the possession of
blading efficiencies for the best ratios. The correct energy allot-
ment per stage would then be that at which the product of the
nozzle and blading efficiencies is a maximum, and this will, of
necessity, depend on the forms of these curves.
The convergent nozzle is the type most generally used in
modern practice, and it will have been observed from the atten-
tion given to the co-efficients that there is direct experimental
evidence that a velocity of efflux closely agreeing with that of
sound represents the best condition. Is this due to some particu-
lar virtue in this high speed ? Is the characteristic fact repro-
duced in the action on the blading or is it peculiar to the nozzle
ON THE LOSSES IN CONVERGENT NOZZLES. 15
alone? Is it, in brief, an effect of the speed or something
peculiar to the expansion? The problem of the best staging
hinges on these questions, since the form of the blading efficiency
curve depends on the answers.
The necessity to deal with rational curve forms may bear
some emphasis. Thus Fig. 4 shows roughly the three possible
combinations of forms that could be used as results of more or
less accurate deductions from test figures, where nozzle and blade
effects cannot be definitely separated. The dotted curves marked
''wheel efficiencies" represent a change of form due to the
necessary consideration of mechanical losses. The stage efficiency
curves give entirely different interpretations of the best condi-
tions, and cannot be all sound. Case (a) requires the highest
speed, case (b) a figure somewhat lower, while case (c) is content
with very moderate values indeed.
All turbine designers, in their designs, classify themselves
consciously or unconsciously in one or other of these three
cases, and a consideration of actual conditions will show that all
three are represented. The protagonists of cases (a) and (b) will
probably be astonished that anyone should be influenced by case
(c), while the adherents of this last may be considerably surprised
at the mere existence of the two former. Case (a) might be
supposed to include those who clamour at the door of Metallurgy
for better materials that they may be empowered to use higher
blade speeds, case (b) those who are obviously content that practi-
cally the best conditions have been reached, while in case (c)
must be grouped those guilty of multiplication of stages. Of the
three different bases of judgment the last involves an irrational
form.
The tendency in high power land turbine practice is certainly
towards cases (a) and (b). Whether this arises from an apprecia-
tion of the detail facts or from actual experience of power units
is not so clear, since at one time not so far distant such
turbines as the Rateau were built with an exceptionally large
number of stages. In marine impulse turbines in certain
instances there exists a distinct tendency towards excessive
staging, but while this may partly arise from the influence of
some .such conditions as are' embodied in case (c), Fig. 4, it is
also affected by the somewhat lower blade speeds supposed allow-
able in marine applications. It cannot, however, be altogether
16 ON THE LOSSES IN CONVERGENT NOZZLES.
due to this as, then, there need not be the noticeable discrepancy
between the speeds in the low pressure and high pressure units.
In the above, consideration has been given to the impulse
type only but, since in the reaction type the blading fulfils the
double function of nozzle and blade, the question of the expan-
sion losses is also important, although opinion may not vary
over such a wide range.
In turbine testing the nozzle and blading effects are inextric-
ably mixed and it would seem essential to achieve a rational and
adequate comprehension of nozzle action before attempting the
problem of blading, since that can only be carried out on actual
power units. Treating the matter scientifically, therefore, it is
necessary to obtain some insight into the nozzle losses that
produce the admittedly peculiar form of the co-efficient curve,
and this is the Authors' purpose in carrying out the experimental
analysis that follows.
SECTION II. EXPERIMENTAL ANALYSIS.
(i) Explanatory of Method. The bases from which the suc-
ceeding work is developed are fully treated in the Authors'
previous Papers, but it is desirable to give here a resume of
certain features so 1 that a suitable starting point may be estab-
lished.
The experimental method simply consists in passing a search-
tube along the axis of an expanding jet, thereby obtaining full
information as to the manner in which the pressure falls. In
conjunction with this, accurate measurement of the mass flow is
made. These observations are carried out on any given nozzle
under fixed initial conditions of supply, which are arranged so
that the expansion is confined to the superheat field. The
various quantities so determined by experiment serve to define
the reheating effect on the fluid due to the losses in the expansion
through the medium of the following equation derived else-
where.*
* 4- r n
*" Steam Action in Simple Nozzle Forms," British Association, Section
" G," August 26th, 1920. Engineering, September 3rd, 1920.
ON THE LOSSES IN CONVERGENT NOZZLES. 17
in which :
P! = pressure of supply Ib. per square inch.
Y x = specific volume cubic feet per Ib.
G = mass flow Ib. per second.
A flow area* square inches.
p
r = D = pressure ratio at any point where the pressure
*i
is P.
k reheating or loss factor.
n index of the law (PV W = constant) for the reversible
adiabatic in the field O'f expansion.
= 1'3 for superheated steam.
Clearly P 1? V 15 G and r, are direct results of experimental
observations. To make k determinate, A must be defined by
some means. In certain cases this is simply a matter 4 O'f
measurement; in others, however, it is less definite, and special
consideration must be given to it. An example of this point
will ensue in due course.
Either side of equation (1) expresses the value of what has
been termed the jet function since certain important quantities
that outline the jet conditions are embodied therein. Which
side is used in any particular instance depends upon which
variable is being 1 considered. Thus to write the jet function
Fas:
provides a form in which the flow area of the jet is the essential
variable, and it is seen that this is simply inversely proportional
to F. If the function is written :
= --
i ............ (p)
k + r~^~
a form is obtained in which the loss factor is the important
quantity considered. The relationship between k and F is not
very direct but, with F and r known, k can always be computed.
Again it is easy to' show that : :
is proportional to the kinetic energy of the jet; while :
( V
\\-k-r " )
18 ON THE LOSSES IN CONVERGENT NOZZLES.
is similarly proportional to the velocity. Also the factor:
n l
k + r n
r
is in direct proportion with the specific volume of the fluid.
These meanings fo>r the detail factors are rather useful, and
besides the fundamental importance o<f equation (1), it is prob-
ably advisable to emphasize the following :
n 1
If Volume Factor = - = m,
r
then :
Jet Flow Area varies as ^
Actual Jet Energy varies as (Fm) 2 .
Actual Flow Velocity varies as (Fm).
The absolute values of the various quantities so represented
by mere ratios can be readily obtained at any time, since all
are referable to the initial conditions, thus :
Energy -ft.-lbs. per Ib. = p lVl l - k - r
Velocity ft. per sec. = |(|^-) PI V, } * (l - * -
Specific Volume
From the) forms of these it follows that the loss o-f energy
per Ib. is :
and this energy being dissipated in heat causes the specific
volume to be :
Is*
r
in excess of what it wo>uld be theoretically.
These considerations show that if by experiment, or other-
wise the jet function is established for any point on the jet
by means of relation (2) then, by means of experimental values
of r and equation (3), it becomes possible to determine the
velocity and volume conditions at the point considered, and the
total expansion loss that has accrued up to that point.
ON THE LOSSES IN CONVERGENT NOZZLES. 19
Strictly speaking 1 , A; is a reheating effect rather than a loss
effect but, under the conditions of single nozzle testing, the
difference between these is entirely negligible, and need not
be considered. Consequently, the terms reheat and loss are
used indiscriminately in specifying the quantity that k repre-
sents.
By the method of deduction of equation (1) the expansion is
tacitly assumed as of the " f notional adiabatic " type. Thus
no heat is supposed to enter or leave the stuff. To cover for
such an effect as a heat exchange k would require to be made
a composite factor, as :
k = /(\ + k,
where k represents the reheating effect of the energy loss, and
& 2 the equivalent value of the heat exchange per Ib. While k
must always be positive, k 2 might be either positive or negative.
It has already been shown, however, that this effect, if existent
at all, must be practically negligible.
There is one other point of view affecting the consideration
of k in the general expression which might be of some moment.
This is the modification imposed on the jet function if some of
the energy of body movement is really disposed in an eddying
action. These eddies might be conceived either as " fringes "
of swirling 1 fluid at a free surface, or as molar currents within
the fluid body. For present purposes an eddy might, in fact,
be defined as any mass motion which utilizes some energy of
expansion, but contributes nothing to the forward flow. It
represents some ineffective form of kinetic energy which will
ultimately be dissipated as heat ; but, while it exists, the k of
the energy factor and that of the volume factor are necessarily
unequal since, to satisfy the jet function conditions, the velocity
of forward motion only must be used. Hence writing :
k 11 + r n
it follows that, if such an eddying motion exists, k 1 > Jc 11 , and
the F value is thereby reduced.
This point reappears towards the close of the present discus-
sion.
The various co-efficients employed in nozzle work may be
readily obtained from the general expression. Thus, the co-
20 ON THE LOSSES IN CONVERGENT NOZZLES.
efficient of discharge dealing with outlet or throat sections and
using suffix t for theoretical, and a foi j actual, values is :
/ n ~ l \ I n l \ 8
~ n \ / 1 7. n \
(4)
(5)
+ r / \ 1 r n >
The co-efficient of velocity is given directly by :
/I -Jc- r~*~\
clearly a figure somewhat in excess of c d This excess represents
the effect of the increased volume in diminishing the flow
quantity. The nozzle efficiency is:
M-l
1 __ Jc-r n
n-l
(6)
1-r
FIG. 5. CONVERGENT NOZZLE FORMS.
If there is a variation of velocity over a flow section it has already
been shown that rj > c v 2 , but owing
the matter this has to be neglected,
appear desirable to remember that :
to the indefiiiiteness of
Nevertheless, it would
and that, therefore, the direct deduction of an efficiency from a
flow co-efficient give too low a result.
Various other points can be deduced from consideration of the
jet function but, as these do not greatly enter into the present
treatment, reference may simply be made to the previous papers.
ON THE LOSSES IN CONVERGENT NOZZLES.
21
(ii) Experimental Data. In the close numerical considera-
tion to be given to the convergent nozzle data it is desirable to
cut this down to the minimum. In the actual tests two convergent
nozzles were used of forms as shown in Fig. 5. In each case two
initial conditions of working were employed, viz. : about 75 Ibs.
per square inch, abs., and at temperatures of about 560 and
400 Fahr. The former temperature represented the main tests,
and the latter were additional for more complete definition of
the pressure curves. In view of the very small difference of the
results for the two conditions, showing that initial values of
supply have little influence on the nozzle action, attention may
be concentrated almost solely on the higher temperature figures.
Table I. gives the supply, mass flow and jet function values
for the two forms of nozzle. The function values included there-
in are for definite flow areas only, i.e\., for 1 plane cross sections in
the parallel regions, as it is not permissible to employ such
sections gratuitously in the convergent portions.
TABLE 1.
NOZZLE.
Initial
Pressure
Ib. per
Initial
Tempera-
ture.
Steam
Flow
Duration
of
Test.
Ratio :
Nozzle
Outlet
Press.
Values of F.
sq. in.
(abs).
Fahr.
Ib.
Bee.
"Throat,."
At
Outlet.
Init. Press.
Simple |
[76-0
559
46
1301
580
2193
Convergent I
Nozzle
[76-0
J76-0
565
568
46
46
1304
1314
574
593
2195
2181
"a"
; 76-0
569
43
1311
710
2044
\
76-0
567
35
1321
843
1650
Convergent j
Parallel
76-4
76-4
558
566
59
49
1742
1460
498
495
2088
2077
2100
2088
Nozzle I
76-4
565
45
1355
582
2055
2066
6"
76-4
565
40
1301
717
1903
1914
76-4
567
33
1349
852
1515
1524
The five different sets o<f figures in each case are obtained by
variation of the back pressure and the flow and function values
are, therefore, obtained in terms of the pressure ratio of operation
of the nozzle.
In conjunction with these, fall of pressure curves are obtained
for each condition. These are given in Figs. 6 and 7, wherein
the pressure at any position on the jet axis is given by its ratio
with the supply pressure. The forms of the curves beyond the
22
ON THE LOSSES IN CONVERGENT NOZZLES.
nozzle outlet are not included, as these have no bearing on the
present matter. They show, however, several interesting
features, and will be found complete in the preceding paper.
The expansion curves in Figs. 6 and 7 show themselves as
very smooth lines, and it was fairly characteristic O'f convergent
types that pressure fluctuations within the nozzle length were not
indicated to any serious extent. This type, therefore, presents
itself as one very suitable for detailed investigation ; the same
cannot be said, however, of divergent types generally, and the
task of analysing results from these is much more difficult.
10
\
\
\
7
FIG. 6. PRESSURE RATIO CURVES FOR IMOZZLE "a."
(iii) Reduction of Data. If the function values at the outlet
section are plotted against the nozzle pressure ratios, both as
given in Table I., the curves termed " flow curves " in Fig. 8 are
obtajned. In this Fig. is also included a theoretical curve for F,
which represents the value the function should have for any
specified ratio, if expansion took place without loss. The flow
curves are so named because the direct ratio of any value shown
by them to the corresponding theoretical magnitude gives di-
rectly the co-efficient of discharge as defined by equation (4).
ON THE LOSSES IN CONVERGENT NOZZLES.
23
From the same sets of values the co-efficient of velocity may
be derived according to the relation in equation (5), and dis-
charge and velocity co<-efficients for both nozzles are shown in
Fig. 9. It is obvious from this diagram that a difference of
sensible amount exists between these co-efficients, and it is also
noticeable that the difference is much greater in the case of " b "
where, of course, the total losses are much higher, and the re-
heating effect with its influence on the outflow volume more
considerable.
-<o -8
Jef" 'in
I-Q
FIG. 7. PRESSURE RATIO CURVES FOR NOZZLE " b."
Fig. 8 shows that the experimental points lie on quite a
smooth curve; and that the form of this curve is rational is
shown by the nature of the co-efficient lines in Fig. 9, where the
tendency is well in line with that established by other experi-
ments and as illustrated in Fig. 1. Admittedly the experimental
points for the determination of the flow curves might have been
somewhat more numerous, but the correctness of form of the
co-efficient curves justifies the acceptance of the flow curves as
accurate, especially in view of the manner in which they
sweep through the actual points.
24
ON THE LOSSES IN CONVERGENT NOZZLES.
Now, examination of the pressure ratio curves in Figs. 6 and
7, and the relative positions of the points on the flow curves in
Fig. 8, will disclose the fact that the three most extreme ranges
of expansion in each case represent results very similar in nature
and extent. It follows that it is allowable to omit from further
consideration the second and third sets in each case ; and, conse-
quently, in order to cover the full range of action dealt
with experimentally, it is sufficient if only the cases repre-
51'
FIG. 8. F CURVES.
sented by the points marked (i), (ii), and (iii) on each flow
curve are considered in the detailed investigations.
The point has now been reached at which three different
outlet conditions for each nozzle are completely established.
These value represent variations in total effects, due to differ-
ON THE LOSSES IN CONVERGENT NOZZLES.
25
ences in the ranges of operation. It now becomes necessary to
pass to the discussion of the probable internal conditions for any
and all of these cases.
Considering- nozzle ' ' a" only f or the instant, the problem
takes the form of a determination o>f the F curves which
represent the internal conditions along- the jet, the data possessed
being- the pressure values along- the jet (Fig-. 6), and the outlet
F values (i), (ii), and (iii) o<n flow curve " a." These last are,
therefore, simply the terminal points of F curves yet to be
drawn.
?8
4:
c
.v
f
U
<?o
-7
'Tf
?
-Y\
Fia. 9. VELOCITY AND DISCHARGE CO-EFFICIENTS.
Since by equation (2)
F = -718
the problem takes the form of a determination of A values
along the jet since V^ P x and G are known. It might appear
that this was only a matter of taking- various positions and
calculating A from the nozzle sectional areas at these places.
The values of F co>uld then be calculated and, as r is known,
they could be directly plotted in Fig-. 8. This is, however,
incorrect.
E 3
26
ON THE LOSSES IN CONVERGENT NOZZLES.
The assumption is probably justifiable that where the bound-
ary form is nearly parallel to the jet axis as at outlet the
area to flow is sensibly the same as the nozzle cross section. But,
where the curvature is changing- rapidly the pressure is certainly
not uniform over a plane cross section, and the area for any
given pressure may be something quite different. Since this
investigation depends largely on pressure values it is essential to
consider flow areas in consonance therewith. It is quite probable
FIG. 10.
that by examination O'f imaginary stream lines, simply as
dependent on the boundary form, a very close approximation to
the required areas could be obtained from the curves orthogonal
to these lines. However, a simple and eminently practicable
method although involving slight trial and error presents
itself in the following way.
It wo'iild seem quite rational to assume that, since the
boundary form is well defined and exercises the same influence
ON THE LOSSES IN CONVERGENT NOZZLES. 27
o<n the flow lines under all conditions, for any one position on the
jet the flow area is always the same despite changes in pressure
and quantity. This assumption is actually made in deialing
with the o<utlet flow area but seems there to require no special
justification.
In Fig-. 10, therefore, showing- three imaginary ratio curves
for different conditions, any vertical line may be drawn, and
it is to be taken that the flow area is constant for the three
ratios so* obtained by intersection. If the line at the outlet edge
is also drawn the six points marked a to f are obtained. If,
say, F (a) etc., is written for the values of the jet functions
corresponding to points (<?), etc., and it is remembered that G
is constant along any one ratio line, then :
Ffc) A^ Ffc) = A_ FO) _ A_
and, therefore:
_ F(V) - Fffl _ F(Q -
FO) .... (7)
The correct forms of the F curves on this, idea are most
readily obtained by trial and error, using equations (7) as a
check. For nozzle "a" the three curves shown in Fig. 8 are
thus obtained, and are seen to be definitely distinct from each
other near their terminal values, but tend to merge into a
common line at the highest ratios. The curves actually pre-
sented in Fig. 8 are not in exact agreement with (7), but they
are nearly sp, and any attempt to change the form either way
increases the disagreement. The disparity probably only re-
flects the normal errors in the pressure ratio figures.
It will be noticed that these curves are not continued beyond
a ratio of 0'9. They seem, however, to be converging together
and are there very close to the theoretical. The above method
of derivation is not employed to cany the curves further because
that method ceases to be at all .sensitive when transition is made
from the steep portions of the pressure ratio curves to the flat
top parts. Consequently, for F values required in this high
pressure region the assumption is made that the curves fall away
to zero at r=l'0 in much the same fashion as does the theoretical
curve. This serves, therefore, to provide complete F curves for
nozzle "a."
ON THE LOSSES IN CONVERGENT NOZZLES.
When the convergent parallel nozzle " b " is considered it is,
of course, to be noted that the throat functions are known as
well as the outlet values, and both are given in Table I. When
these throat values are plotted, the interesting 1 result appears
that each practically lies on the corresponding' F curve of nozzle
"a." When the fact that the entry curvatures are identical! is
taken into account, this would seem to be quite appropriate, and
tends to justify the curves obtained by the above tentative pro>-
cess. Again, since the tail area is practically constant a
straight line joining the throat and outlet values for each case
in nozzle " I " should be a correct representation of the form of
the F curve for this region. It could, therefore, be taken that
the curves for the convergent-parallel nozzle are formed of these
10
03.
FIG. 11. COMPARISON OF AREAS GIVEN BY F VALUES WITH THE ACTUAL AREAS.
straight lengths, and the curved portions which they cut off
from the corresponding curves "a."
This would all appear quite consistent. Nevertheless, diffi-
culties are shown when the flow areas represented by the F
curves so established for the convergence are compared with the
nozzle cross sections. This comparison is roughly made in Fig.
11, and it is seen that, from a position midway along the con-
vergence, F gives area values increasingly smaller than those
calculated from plane sections of the nozzle. There is noi a priori
reason why the real flow areas should not be smaller than such
plane sections, as will be understood if the probable curvature
ON THE LOSSES IN CONVERGENT NOZZLES. 29
of the equal pressure lines be taken into account; but it would
seem unlikely that the difference should be so great. Now,
when examination is made of the point where the two curves
in Fig. 11 begin to separate it is found that this is where
pressure ratios of Q'9 to 0'95 are reached in all cases. In other
words, it represents the position at which the change of curva-
ture takes place in the pressure ratio curves, and hence the
length within which the difference is shown corresponds to the
largely indeterminate portion of the F curves. As relatively
small changes in the lie of the F curves at the high pressure
ratios would mean considerable change in F and, consequently,
definite modification of the curve in Fig. 11, it would be
possible to get better agreement by shearing down the F curves
somewhat. It is, however, very undesirable to make casual
alterations of this kind 011 curves which have all been derived
from a single assumption. Moreover, since the curves have been
obtained for the purpose of having specific values with which to
work, it would be better to reserve the matter of correct form
until the further analysis shows whether more definite evidence
can be adduced.
The F curves in Fig. 8 may, therefore, be used for the
purpose of determining the forms of the loss curves throrughoiut
the varioois expansion ranges and, in this respect, it is fortunate
that these curves are fairly definite for the later stages O'f the
expansions and particularly so for those portions effected in the
parallel tail piece of nozzle " 6."
In passing from the F values to the losses which these values
denote, it is necessary to realize that the resultant figures suffer
immediately and definitely from the influence of the experi-
mental errors. While these errors are small in so far as their
effect on the F values are concerned, this doies not, hold when the
differences between the theoretical and actual F figures are
dealt with, and it is on these differences that the losses depend.
This is a fault inherent in the usual direct measurement of flow,
in that judgment requires to be based on small differences greatly
affected by experimental accuracy. Besides this, further errors
or irregularities are naturally introduced by the extensive
reduction of data entailed. On the whole, it would seem
permissible to indulge in a slight smoothing o>f curves where
such is advantageous ; and it is desirable at all times, in dealing
with the actual losses, to recollect that relative agreement only
can be looked for.
80
ON THE LOSSES IN CONVERGENT NOZZLES.
With the F curves fixed as in Fig. 8 the losses for different
r values can be determined by means of the relation given by
equation (3) ante. . The calculation is tedious if directly
attempted, but is facilitated by means of calculating charts
based on (3).
/O
FIG. 12. Loss CTTRVES.
The investigation results in Fig. 12 in which the k values
are shown on a base of pressure ratio. It will be seen that the
different nozzles give quite consistent results, and the change
from " entrance " to " tail " effects is definitely indicated by
ON THE LOSSES IN CONVERGENT NOZZLES.
31
the conspicuous change of slope. It is, however, quite im-
possible to derive much from a direct consideration of Fig 1 . 12 as
the loss forms when so charted are divorced entirely from
the nozzle forms which cause them. It is necessary to relate the
FIG. 13. EXPANSION LINES.
K values to position on the jet or some, such variable and one
of the main functions of the pressure ratio curves 'is to enable
this to be done.
As the total heat-entropy diagram is the usual medium for
the presentation O'f expansion effects, it is probably of interest
32
ON THE LOSSES IN CONVERGENT NOZZLES.
to show the corresponding forms when these co-ordinates are
used, although such a demonstration has no special application
in the present study. This is carried out, approximately, in
Fig. 13, and it will be appreciated therefrom that no simple
assumption as to the form of the expansion line on this type cf
diagram can hope to represent the nature of the expansion
adequately; while it is also obvious that expansion lines shown
on this basis are like Fig. 12 quite useless for detail examina-
tion unless so-und knowledge can be obtained of the pressure
distribution alo'ng the nozzle.
0/5-0
0076-
0050
FIG. 14. Loss CURVES. NOZZLE "b" CASE (i).
The step may now be taken to relate the losses to the develop-
ments within the nozzle and this, at last, leads to the discussion
of the important matter, viz. : the nature of the losses.
(iv) The Boundary Loss. Consideration of Fig. 12 will
show that the convergent parallel nozzle loss effects embody the
same type of result as is given by the purely convergent form,
in addition to the influence of the straight tail piece. It is,
therefore, the more general of the two and may be dealt with
first. It may be emphasized here that, on account of this, of the
ON THE LOSSES IN CONVERGENT NOZZLES. 38
steadiness of its action, and its ability to show variations of
critical pressure, this type of nozzle oilers the best prospects for
research in nozzle phenomena. It is also much superior, for
this purpose, to divergent types where throat arid tail conditions
are very complex and present many uncertainties.
The loss curve in Fig. 12 has been used in conjunction with
the ratio curve in Fig. 7 to give in Fig. 14 the loss on a base
of length along the jet axis for nozzle " b " case (i). The unit
of length is simultaneously altered to feet, as this is desirable
for various reasons. Similar curves for the other cases will be
disclosed in due order.
It will be noticed that the points showing the loss along the
parallel portion lie nearly oil a straight line, and that such a
line has been drawn in. The reason for this is twofold; firstly,
because the discrepancy from the straight form is slight and may
be due to unavoidable errors in reduction and, secondly,
because the form of the pressure line as obtained by experiment
is closely in keeping with such a result. This last may be
demonstrated, generally, as follows :
From the constancy of area in a parallel length F must be
constant along it. Assume that the final value of k (say &J at
outlet is known, then it can be shown* that the lowest
possible value of the outlet pressure ratio, i.e., critical or true
throat pressure, i&> given by :
(8)
' | JU *
-l
where : a = .
n
Also the pressure ratio r at the commencement o>f the parallel
portion, on assumption of a loss k at that place, is given by :
(9)
Now, with a straight line giving the loss, at any distance x
along the tail of length I the value of the loss k is : -
Jc = (ki -
= nix
* " Steam Action in Simple Nozzle Forms," British Association, Section
" G," August 26th, 1920. Engineering, September 3rd, 1920.
ON THE LOSSES IN CONVERGENT NOZZLES,
At all points along the tail :
and, when the value of k is substituted, this takes the form :
*2 + zA(& ,m,F,r)+/ 2 (& ,m,F,r) = . . . (10)
With assumption of any value of r, between the known limits
of r and r^ given by equations (8) and (9), this quadratic in x
serves to determine the position at which the chosen r will exist.
-so -76-
\fcfucs of */l.
'0
FIG. 15. THEORETICAL AND ACTUAL PRESSURE EATIO CURVES IN A PARALLEL
TAIL.
Direct computation from (10) would be exceptionally laborious,
but the calculating charts used for other purposes give results
quite easily. Thus, choosing &! = 0'02 and & = 0'004 and working
with ratios #/Z, the curve shown full in Fig. 15 is obtained.
The dotted curve is a reproduction oi the actual tail portion of
the pressure ratio curve for nozzle " b," case (1) and the agree-
ment will be seen to be quite good so far as form is concerned.
The difference in values is, od: course, due to' the arbitrary loss
assumptions made for the plotting of the theoretical curve.
ON THE LOSSES IN CONVERGENT NOZZLES.
35
This result is interesting" as showing that the rapid fall of
the pressure line towards the outlet of such a nozzle is actually
required by theory, and it would seem to justify the simplifica-
tion made in drawing the straight line to represent the tail loss
in Fig. 14. It follows, therefore, that any consideration of the
nature of this loss must give a curve approximately parallel
to this line, and reasonably straight.
The complete curve in Fig. 14 gives a clear idea of the
growth of the loss as the expansion proceeds, and the rather
peculiar form is worthy of remark. The seemingly natural
tendency in the tail length is set bodily upwards by a rapid rate
of loss before the " throat." The total value of this " shear "
o/e
oog
JL.
00(0
OOU.
"76
05
Og
PIG. 16. Loss CURVES. NOZZLES "b" CASE (ii).
in the curve is not very great, but it certainly cannot be due to
the same causes that govern the following form. It is quite
clear, in fact, that the losses must be of at least two distinct
types, one probably due to a f rictional effect at the swept surface,
and the other a loss incurred during the rapid expansion in the
entrance form.
The curves for the other two cases of " b " are given in
Figs. 16 and 17 and exactly similar effects are shown. The
influence of the range O'f expansion is demonstrated in the
much smaller real slopes of the straight portions, and it is clear
that there is a definite speed effect.
36
ON THE LOSSES IN CONVERGENT NOZZLES.
It is, therefore, desirable to discover whether consideration
of friction would give loss curves reasonably straight in the
tail length and respectively parallel to the three different cases
with steady values o<f any constant entailed. The necessary
development is as follows.
The frictional resistance to the flow of a fluid over a
boundary surface may be approximately written as dependent
on the density, some power of the fluid speed, and the extent of
the surface. This neglects certain refinements that dimensional
theory introduces into the law of resistance, but it is not possible
OOfc
005
001*.
003
002
001
03
~ oc ph.
FIG. 17. Loss CURVES. NOZZLE " b " CASE (iii).
that the data available herein are of such a degree of excellence
as to necessitate consideration of these refinements. Hence the
resistance on an element of boundary Ida; may be written :
JD _ 1 m jj ( n < = constant \
' V ' \ / = perimeter (nozzle + search tube/
and, consequently, the energy loss per second on this element
could be stated as :
(11)
Now, in the nozzle a flow of G lb. per second takes place and it
has been shown that the loss per lb. is represented by :
ON THE LOSSES IN CONVERGENT NOZZLES. 37
where b is a constant. Hence :
rfE =
and .'. G^P 1 V 1 ^ = ~-w Idx . . . ... (12)
Again, by the equation of continuity :
and, therefore, by substitution in (12) :
i<fifc = an Idx ........... (13)
/ U / ^
again : u = [fydtY^)* (l - k - r n )
where k is the total losS at the ratio r. The loss being considered
is not necessarily the same thing as the total loss and, conse-
quently, it is desirable to distinguish them. The suffix t will
serve to represent the total loss, and 5 that resulting from the
effect under discussion. Hence substituting for u in (13) :
A
It is advisable to choose a value for the index m and test
whether such a value is suitable. Taking the usual 2 which
is the most generally applicable figure, this gives :
dk. = c(l Tc t - r ~" r )^ dx
and, therefore:
" 1X |<fe . . . . (14)
In this, c is some constant and the integration over any part
of the nozzle length gives the loss due to boundary friction
within this length. The main simplification in the work has
been due to the choice of 2 for m, and the resulting form of (14)
is distinctly significant of the correctness of this. It is a well-
known fact in nozzle flow that, with change of initial condi-
tions of supply but the same overall pressure ratio the dis-
charge is related to these conditions by :
'FT
v, '
with fair accuracy. This means that :
?-**
is practically a constant for this particular ratio. Hence Tc
38 ON THE LOSSES IN CONVERGENT NOZZLES.
should also be constant for this value of r, and independent of
the initial conditions. It is obvious that (14) gives a loss
exactly in line with this finding- and is, therefore, a highly
probable relation. The absolute validity of expression (15) will
depend, however, on whether any difference that exists between
s and Tc t is influenced by initial conditions or not.
The integrations required by (14) have been carried out
graphically for all three cases, and the resulting curves repre-
senting boundary losses are shown in Figs. 14, 16 and 17. In
all three the same value of the constant e has been employed,
and the straightness of the curves, and tjie distinct parallelism
with the actual tail loss lines are very noticeable. Besides this,
the average constant used does not greatly differ from those
that would be demanded for the best agreement in the separate
cases. Thus c = 0'0051 represents the average multiplying all the
integrals, while the best values fitting the separate curves
would be:
6 (i) - c = Q'00502
b (ii) - c = Q'00513
b (iii) - c = Q'00528
In the actual calculations I, A and dx have been taken in
feet units, but from the form of '(14) it is clear that c is a
dimensionless quantity, and will be unaffected by the length
units employed so long as these are not mixed.
The curve for k s in Fig. 14 shows that the assumption made
of a straight line loss curve is not quite correct as it indicates
a certain concavity upwards which is, of course, due to the
increasing speed along the tail. The curvature is, however,
not great, and the curve may be taken as agreeing with the
experimental data as closely as could be expected. Again, if
the slight variations in the values of the constant c have any
real significance they are probably due to the neglect of the
special function involving viscosity, etc., that is demanded by
a rigorous law of resistance.
On the whole, then, it may be taken that equation (14) is
a fairly suitable and adequate representation of the losses in
a straight tail piece, and it may also be allowed to give the
surface loss in a convergent part where perhaps the develop-
ment that has been given is not quite so exact. But in a con-
vergence this particular loss is small, and the approximation
must be close.
ON THE LOSSES IN CONVERGENT NOZZLES
39
The k s curves that have been derived admit of the elimina-
tion of the boundary loss from the total loss effects. This leaves
the part already noticed as occurring in the entrance, but as
nozzle " a " shows this to a greater extent than nozzle " ~b " it
will be preferable to deal with the former in the subsequent
work. Equation (14) with the constant obtained for nozzle " b ''
may, therefore, be used to obtain the net entrance effects in
nozzle " a." The total and boundary losses for " a " are
shown in Fig. 18, and the differences of these giving the
* /""\
Ok (J"J.
006*
OOtf
oo3
-ooz
009
02
03 '01
Je/- ,
Od
FIG. 18. Loss CURVES. NOZZLE "a."
entrance or ~k e values are sketched in Fig. 19. It now becomes
necessary to deal with these last.
(v) The Convergence Loss. The k e curves in Fig. 19 must
be accepted as merely hinting at the forms, and only the final
values can be assumed accurate. The consideration already
given to the manner in which the F curves are obtained has
demonstrated that, at least, certain portions of these are open
to question, and this detracts from the value of the loss curve
forms resulting therefrom. The end values of the losses shown
in Fig. 19 must, however, be fairly reasonable, and it is probable
that the top portions of the curves are also near the truth.
40
ON THE LOSSES IN CONVERGENT V ^7
Examination of the total values will show clearly that these
depend somewhat on the range of expansion, but are not pro-
portional to the available energy ; they do not fall away so
quickly as this. Hence it follows at once that in this effect is
contained the reason for the characteristic form O'f convergent
nozzle co-efficients considered at length in the first part of this
paper. The drop in these co-efficients is entirely dependent on
some occurrence in the convergence which is relatively the more
important as the expansion range becomes more limited, and
which exists in addition to the normal f Fictional effects.
004
FIG. 19. CONVERGENCE Loss CURVES. NOZZLE "a."
The fall of the total k e values indicates then some effect of
the expansion ; and the slowness of the fall would point to the
influence of some fairly steady factor. Now practically the only
steady influence is that of the entrance form a fixed material
restriction to the direction of flow at all times. Consequently,
it would seem desirable to examine a combined effect of, say,
speed and curvature.
In considering how the entrance form could affect the flow
adversely, it will be realized that the rapid convergence thereby
ON THE LOSSES IN CONVERGENT NOZZLES. 41
imposed on the fluid must cause a correspondingly rapid de-
formation of all elements of fluid engaged in the flow. This
again, acquires further importance when viewed in conjunction
with the growing velocity. Although the convergence of the
stream lines is most rapid in the initial stages, it is to be realized
that in that position it is operative on relatively slow moving fluid
and is, therefore, probably causing less rapid deformation than
occurs at a distance within the nozzle inlet. Now, when it is
considered how fluid viscosity causes loss when deformation of
shape takes place at a fair rate an idea is obtained as to how a
reheating effect may ensue with maximum rapidity at about the
position indicated by the curves in Fig. 19.
This point of view gives then an indication of how a combined
speed and convergence effect might produce results of the kind
obtained, and might even give agreement with the actual curve
forms. However, it also charges the loss against viscosity and
tacitly assumes stream line flow; but the idea is sufficiently
plausible to justify some quantitative consideration, which,
owing to the inherent difficulties of accurate examination, might
be carried out roughly as follows.
If a flowing viscous fluid be subjected to deformation of form,
stresses are established which depend on the rates of distortion,
and oil viscosity. It is shown in hydrodynamics* that the direct
and shear stresses on planes parallel to the co-ordinate planes are
given by :
2 f du dv dw \ . n du
(16)
2 / du dv
2 ( du dv
In these (16) gives the normal stresses, while (17) defines the
shear stresses; u, v and w are the component velocities; /* is the
co-efficient of viscosity ; p is the fluid pressure. The stresses are
clearly alike and equal to the fluid pressure for the perfect non-
* Lamb's Hydrodynamics, p. 512.
42 ON THE LOSSES IN CONVERGENT NOZZLES.
viscous fluid. This equality is upset by viscosity, which estab-
lishes differences depending on the rates of deformation in the
different directions. The equations are very similar to those
which give the stresses in elastic solids but, in that case, the
partial derivatives denote strains, whereas here they measure
rates of strain.
The product of the stresses and the rates of strain give the
rate at which the stresses are doing work on the fluid. This
results in : *
Aw , dv*. du dw\* dv
' +
The terms containing viscosity as a factor represent a dissipa-
tion of energy in the form of heat at the expressed rate per unit
of volume, i.e., a reheating effect. These terms also represent
what is called the " Dissipation Function."
The general expressions stated above are too formidable for
full treatment and drastic simplification is necessary. It must
be understood that only a rough measure of the order of the
effect is required ; and, needless to say, if sufficient detail know-
ledge of the motion were available to deal adequately with such
involved equations, there would be little necessity at this date
for experiments on elementary nozzles.
Simplification may be introduced by assuming that the direc-
tions of the principal stresses in the fluid are along and perpen-
dicular to the direction of flow. This eliminates equations (17)
and for ease of working (16) may be written:
2
= -P Q
2
(16 1 )
, . , ~ du . dv . dw
m which :- 8 = , + , + , = rf - + dy + g- -
From the simple equation of continuity :
GV
T. . ....... (18)
* Lamb's Hydrodynamics, p. 518.
ON THE LOSSES IN CONVERGENT NOZZLES. 43
Now this represents the actual flow velocity, but it may be
taken as an approximation to the velocity in the x direction
(axis of jet) since it becomes more- nearly equal to this as the
speed increases.
Then from (18):-
, 7T7 , A
= A V ~ "A 5 "
du _ u^ dV u^ d A _
/. ~dx,~V~dx~~ ~K~dx^'' &l .... (19)
Now, the expanding fluid is suffering alteration of volume
at the rate dVjdt as it flows. Hence the rate of volume strain
might be w ritten :
u dV
V' fa
But this rate is also:
8 = e l + e 2 + e s
/. from (19),
^ + * = s-^|^
A ax
and by symmetry it may be taken that :
u dA.
~~ e * ~ 21 tf* . ....... (20)
With the stress equations expressed as in (16 1 ) the Dissipa-
tion Function becomes :
and, by substitution from (19) and (20) and reduction :
, = 4 ./ ldV_l 1 bdA^
3^ V Vax A dx) .... (21)
In this it will be seen that V and A appear only in ratio
form, and hence it is permissible to use the most ready repre-
sentation of these quantities. On page 13 it was shown
that if :-
* + r ^~ = m, then V a m
r
and also that Aoc.^. Using these proportional relations (21)
becomes :
. 4
-m!ti te ..... (22)
Now, from the nature of the dissipation function (22) represents-
44 ON THE LOSSES IN CONVERGENT NOZZLES.
a rate of energy loss per unit volume ; hence in a volume element
Kdx there is a loss rate of :
But through this element there pass G Ib. per sec., therefore
the loss per Ib. is :
*.
The loss per Ib. throughout this work has been given by : -
&?! Vi k
and, therefore, for the type of loss considered:
4
/x .fldm.l-bdF
^7^7=7 U ~ ( T T -VS r~
Vi G \m dx F dx
Now :
G = Y?, Y = Y! iw, ti = ^2.^Pi Vi Wl - * - r"^
Also (l k t r~n~ j may be written Fm.
Wherefore, by substitution and simplification :
where ^ is a constant.
This obviously gives a loss dependent on the rate oi' con-
vergence, on the range of expansion and on the viscosity. It
is also, however, influenced by initial conditions, and in this
respect it is peculiar that the experimental results for the lower
conditions of initial superheat for both convergent nozzles used
show somewhat lower values of h. The differences vary a good
deal and are obviously affected by the experimental errors, but
the variation is distinctly in the direction required by (23).
Numerical agreement cannot, however, be claimed, and it is
sufficient to note the fact.
The investigation required by (23) has been carried out and
the integral curve for " " case (1) is shown in Fig. 20. As the
loss under discussion receives various considerations it is thought
desirable to limit the diagrams by showing curve forms for ease
(i) only and to give the values of the constants as indicating the
agreement with the other cases. Wherever this is done it may
be taken that the curves for cases (ii) and (iii) are of entirely
similar form to that for case (i).
Since the initial conditions are practically constant for all
three cases the factors outside the integral sign in (23) may be
taken as one constant. The values obtained for this are:
ON THE LOSSES IN CONVERGENT NOZZLES.
45
a(iii)
-43-5x10
,, =42-9 x 10
=40-7 x 10
-6
Considering the nature of the investigation, and the fact
that each of the actual, figures dealt with represents the residue,
after a series of reductive and deductive operations, it will be
allowed that these constants are reasonably steady.
005
OOf
PIG. 20.
There is, however, no agreement between the nature of the
loss curve obtained from the investigation and that established
in Fig. 19 mainly from the F curves fixed in Fig. 8. This new
curve demands a considerable rate of loss in the early stages of
the expansion, which would require a fair departure of the F
curve from the theoretical at the higher ratios. If this were
given, the loss curve resulting from (23) would be toned down
somewhat but would nowise come to agreement with the prev-
iously established form, which would now seem to give too much
importance to the expansion and too little to the convergence.
The double curvature in Fig. 20 when considered along with
the area curves in Fig. 11 also seems to show that the earlier F
values are too high.
46
ON THE LOSSES IN CONVERGENT NOZZLES.
These points need not be elaborated further os the whole
question o>f the validity of the basis of working is raised by
examination o*f the above constants. It is clear that even taking
into account the very serious simplifications made in the hydro-
dynamic theory the constant should have some rough agreement
with the co-efficient of viscosity. When the value of this,
implicity embodied in the constant, is enumerated, it appears
that /^ is of the order of '0003 (engineer's units of mass, ft. and
sees.). Now this is many hundred times greater than any
possible value of the co-efficient and, consequently, viscosity
would not seem to be the direct cause of loss, and the idea of
005
OOif
FIG. 21.
pure stream line motion in the convergence fails. No doubt, a
viscosity loss exists, but it would appear and this might be a
natural opinion apart from analysis two small to account for
the values actually obtained and indeed negligible in comparison.
With the collapse of this conception, the confession is forced
that a complete treatment of the convergence effect is impossible
from the data in hand, since values for one convergent form only
are available. It is desirable, however, to examine the matter
further to obtain, if possible, the best view of this loss, giving
reasonable curve forms and agreement with the total values.
Suppose that the rate of growth of loss is proportional to the
rate of growth of some power of the speed the constant involved
ON THE LOSSES IN CONVERGENT NOZZLES.
in such a relation being of course a function of the convergence
but constant for a given nozzle. Then
fc, = (constant) x (Fm) c . . . . ., . . (24)
It is known from the nature of the loss that c cannot be a large
number but, for purposes of clearness, both 2 and 1 may be tried.
AVith c = 2 the curve in Fig. 21 is obtained. The agreement
with the original form of loss curve is exceptionally close but the
constants are definitely out, thus :
a (i) - - constant = 0'0364
a (ii) - 0*0465
a (iii) - 0'0642
005
OOif
O03
Fio. 22.
This demonstrates two facts, viz. : that the loss is not depen-
dent on the square of the speed, and that the loss curves
originally established in Fig. 19 are of a wrong form, since they
agree with a condition that shows a wide departure from the
main facts.
Taking c = 1, Fig. 22 is obtained. Here the curve is of
the same foT-m as the original, but is definitely detached froira
it; still, in view of the frequently emphasized probability that
the original occupies too* late a position, owing to the F values
being too high in the early stages, the disagreement is somewhat
in favour of the new curve. When the constants are enumerated
a remarkable closeness is found, thus:
48 ON THE LOSSES IN CONVERGENT NOZZLES.
a (i) - constant = '0123
a (ii) '0123
a-(iii) '0122
The exactness of this agreement is probably a coincidence,
but it is significant of a correct readirxg of the effect. The
corresponding 1 values for nozzle " b " give approximately similar
constants, but not of quite the same steady nature.
On the whole, therefore, this last and very simple considera-
tion of a rather difficult question gives by far the best results.
It is seen that for a given nozzle having 1 fixed entrance foam,
and working under invariable initial conditions, there is a
convergence loss which, measured in energy units per Ib. of
fluid passed, is proportional to the first power of the speed.
Accepting the curve form obtained this finding applies through-
out the convergence and not merely to the total effect.
The quantity (Fw) has been used in place of the speed
because of its direct proportionality therewith in the cases con-
sidered. If dependent on absolute value o<f the speed the Tc e
loss is obviously influenced by initial conditions which as has
been noted previously is in keeping with the general tendency
o>f the experimental results for the different temperatures used.
Why the loss should vary only with this low power O'f the
speed is rather difficult to understand, but the following explana-
tion might be advanced. It has also the merit of rationalizing
the loss curve actually derived from the above consideration .
On page 19 a short discussion was given of the influence of
a supposed " eddying " action of the fluid on the value of the
jet function. This showed that, if, up to any point at which
this eddying or perhaps cross currents existed, the reheating
effect were k 11 and the loss in the " forward " energy added to
this gave F then :
F =
n-l
n
r
Now in the early stages of the expansion the F values that
have been used lie very close to the theoretical and, therefore,
envisage very low values of k 11 . If, however, k 1 is not low
owing to the presence of energetic eddies, then F must be
considerably reduced in a relative sense and the loss curve
obtained from such revised F values would be translated towards
the commencement O'f the expansion and, therefore, approach
ON THE LOSSES IN CONVERGENT NOZZLES. 49
more closely to the curve calculated on a basis of loss varying
with speed. At the same time this calculated curve would move
along to meet it and close agreement of position could thus be
obtained by modification of the F values to suit eddying. Again
this would also mean a better agreement of actual nozzle areas
with those indicated by F, the discrepancy on the previous
basis having been shown in Fig. 11.
This idea then tends greatly to modify, if not to eliminate
the outstanding disagreement of curve forms. Furthermore, the
severity O'f the occurrence could be attributed to the acute-ness
or general unsuitability of the entrance form. Thus a curved
axis and a sharp form would be more objectionable in its effect
than a straight axis and a slow curvature; and hence practical
nozzle forms might be more faulty in this respect than the
elementary types.
The necessary change in the F values, is most extensive in
the earlier stages, hence this supposed eddying would be most
violent near the commencement. If, however, the no-zzle
boundary shape is really effective from the beginning the eddies
must be considered an internal rather than a surface effect.
Also, they must be supposed relatively unimportant by the time
the convergence is completed, and therefore naturally accom-
pany high rates of deformation.
There still remains the question as to what conditions would
lead this kind o<f occurrence to show itself as a loss per Ib.
that varies with speed, and this might be dealt with in the
following manner.
The mass disturbed from smooth forward flow might be sup-
posed proportional to some function of the convergence, and to
the density. It is also very probable that the dimensions of
the nozzle have an influence, but this may be taken as embodied
in the unknown function. The rate at which energy is being
potentially lost per second at any position along the convergent
length may then be written proportional to the product of this
function, the density, and the rate at which energy is being
liberated per Ib., i.e.
ff . du
f(c,x)pu Tx
where f (c, x) represents the at present indeterminate effect of
the convergent form in producing agitation. But this rate of
50 ON THE LOSSES IN CONVERGENT NOZZLES.
loss is also proportional to :
Q^
J dx
and since :
GrGC p u
it follows that :
Owing to the fixed nature of the convergence in any one
nozzle, and to the similarity of the velocity curve forms for
different overall pressure ratios, this would represent a loss
gTOwing, at first, rather more rapidly than the fluid velocity
but, for total values probably in close direct proportion to the
final speed. That is, for a particular nozzle there is a conver-
gence loss by diffusive agitation of the flowing mass which is
approximately dependent on the speed attained.
It is also clear that such a loss would depend somewhat on
the initial conditions, since it is the absolute value of the
velocity that is concerned in the matter. Thus, the total value
of the loss might be roughly expressed as :
*. oc (P^Oi/fo x) (Fm) c
for the sake of comparison of overall effects. In this, (Fm) c
represents the " velocity factor " at the termination of the
convergent part. The differential equation form is, however,
the more correct, since k is, at all times, an integral quantity,
so that :
(25)
would give a presumably general form. In (25) f (c, x) depends
on the convergence form, and probably also includes an effect
of the absolute dimensions ; (Fm) is mainly dependent on the
range of expansion, while Pj and Vj represent initial conditions.
(vi) Main Points. The essential matter of the rather
lengthy investigation may advantageously be summarized, as the
many details involved are somewhat confusing.
The losses in a nozzle are referred to by a factor Jc which, in
continued product with the initial pressure and volume condi-
tions, gives the energy loss per Ib. If this factor is strictly
constant for any given ratio of expansion, independent of supply
conditions, then the flow per square inch of area is accurately
proportional to :
VT
ON THE LOSSES IN CONVERGENT NOZZLES. 51
Any dependence of k on P x and V x upsets this proportionality,
but the discrepancy will be the smaller if only a portion of the
total Jc is thus affected. General experimental work shows that
mass flow is closely proportional to the above factor, but not
exactly so, the disagreement being of much the same order as
experimental error.
It has been shown that in the expansion of a fluid through
a nozzle there are at least two distinct types of loss. One, an
effect of friction, seems demonstrated with fair conclusiveness ;
the other is apparently an inherent feature of the convergent
portion of the expansion.
The frictional effect termed the boundary loss is due to
the sweeping of the confining surfaces by the high speed fluid.
The frictional drag thereby established shows its characteristic
dependence 011 the square of the speed even up to such a high
figure as the speed of sound. The loss factor k in this case
appears to be entirely independent of the initial conditions, and
is given by integration along the axis, the integrand being a
function of the hydraulic mean depth and of a speed factor
that varies only with the ratio of expansion. The constant
involved will, of course, cover the influence of the surface
finish.
This boundary loss may conveniently be considered as due
to a scattering of flowing molecules resulting from impingement
on the irregularities of the surface, with consequent retrans-
formation of their flow motion into heat motion.
The second type called the convergence loss is caused by
the rapid convergence in the entrance. It is shown that this
effect is too great to arise from the work done against the de-
formative stresses created by viscous action ; and it is indicated
that the cause probably lies in ineffectual mass motions within
the streaming body of fluid. The loss factor is shown to be
largely dependent on the velocity developed in the convergent
portion of the jet, but it must also be influenced by the actual
nozzle form; and, therefore, any expression giving the loss must
contain some function of the convergence. Since with a fixed
convergence the velocity is primarily important the loss is
affected by initial conditions of supply and, therefore, departure
from the practical rule that flow is proportional to the square
root of PI/VJ. must be expected. This departure will be the
greater in those cases where this particular loss is a large propor-
tion of the whole, e.g., in the purely convergent types, and this
is in rough agreement with the experimental evidence available.
52 ON THE LOSSES IN CONVERGENT NOZZLES.
This convergence loss is, then, attributable to the establish-
ment of molar currents in the main stream by and in the conver-
gence. The damping o<ut of these as the jet narrows to a
straight path would cause a reheating effect by retransforniation
of kinetic energy.
GENERAL CONCLUSIONS.
It will be seen that the separation of frictional and
convergence effects gives at once, through their different
natures, a definite reason why the nozzle co-efficiente
should fall as the expansion range is narrowed. The latter
effect, measured per lb., is not reduced so rapidly as the avail-
able energy and has, therefo-re, an increasingly important
influence as the pressure ratio of action rises. This demonstrates
the validity of the idea simply treated on page
The rate of fall of the co-efficient whether of discharge or
velocity will depend on the relative magnitudes of the two
losses. With any convergent type having a fixed inlet form the
fall should be less rapid as the straight tail length is increased,
owing to the ready manner in which the frictional effects acquire
a dominant influence. With nozzles of equal tail length, how-
ever, the more objectionable convergence will cause the more
rapid fall. Thus considering the convergent types used in actual
turbine work the forms at inlet are such as would compel belief
that their convergence loss is more severe than that of the simple
circular straight types, since there is rectangular formation,
curvature of axis, and distinct differences in curvature between
the two pairs of opposite sides. In this connection it is significant
that the very scanty data obtainable on practical nozzles confirm
this finding, the curves derived by Christlein, for instance, hav-
ing an extremely rapid fall.
Again, the size of the nozzle would presumably have an
influence on this loss, as the disturbing currents might then be.
supposed more extensive. If so, the slope downwards of the
co-efficient curves would be more noticeable for the larger
nozzles. There are practically no large scale steam experiments,
but reference ,to Fig. 1 will show that Professor Gibson's curve
for air displays this feature. Now these experiments were made
with diameters of the order of 1J inches, that is, several times the
size of experimental steam nozzles, and it would, therefore, seem
that there is a definite dimensional influence.
ON THE LOSSES IN CONVERGENT NOZZLES. 53
The effect of dimensions on the convergence loss will be
opposite to that on the boundary loss. It must be kept clearly
in view that the frictional occurrences are apparently inversely
dependent on the hydraulic mean depth and, therefore, small
scale experiments are most suitable for the examination of these.
Thus, in the Authors' experiments the introduction of an |-inch
diameter search tube into a J-inch bore nozzle increases the sur-
face by 50 per cent, and reduces the flow by 25 per cent., i.e.,
the search tube doubles the effect and therefore, makes it the
more readily determinate. For the same reason efficiencies
obtained from such small nozzles are not directly applicable to
larger sizes and, therefore, to say that the experiments are faulty
because in practice better values are apparently shown is no
sound charge. Application to the practical forms can only
rationally be made when the constant factors in the different
loss effects are known.
It may be taken, then, that the total loss to any point x along
the jet within the boundary form is given by :
(c, j>)^(Fm)|<fo . (26)
This expression is too involved to be of much practical value, but
considerable simplification is possible with only moderate loss of
accuracy. It may be permitted to charge the frictional loss
against the tail piece only, and relate this to the average velocity
factor therein ; while the entrance loss may be made directly
dependent on the speed established in the convergence thus :
* =-^(Fm) 2 +ft(P 1 V 1 ) i (Fm) .... (27)
where
a constant for surface.
L length of parallel portion.
h hydraulic mean depth of parallel portion.
(Fm) average velocity factor in parallel portion.
& factor varying with different convergence forms.
(Fm) c velocity factor at end of convergence.
Although (Fm) contains k as a term the influence of this is
not great, and (Fm) might be taken as equal to the theoretical :
,
1-r
Actually, however, it will be found that by using a suitable chart
for calculation rather closer approximations to the real values
can be obtained.
54 ON THE LOSSES IN CONVERGENT NOZZLES.
Taking the actual nozzles used, the constants are of the
following order:
6 = -0005.
The constant a is for a moderately well finished bored out brass
nozzle, while b covers the convergence effect for a small nozzle
having a straight axis and a radius of entry nearly equal to the
diameter.
Under reasonable assumptions it would be possible to obtain
a fair amount of information from other published data, but this
need not be entered into at the moment.
For practical application further experimental investigations
seem necessary, and a few final remarks on the best forms for
these might be- given.
The boundary loss will be most readily obtained from exam-
ination of the action in straight parallel lengths. The value of
the jet function F is constant along such a part, and the growth
of loss is clesely represented by a straight line. For these reasons
pressure readings are only necessary at entrance to and outlet
from the parallel portion. Again, this frictional loss is most
clearly shown by flow experiments on small size nozzles.
Consequently, for the determination of frictional constants
a few long parallel nozzles might be used of small bore and
having different surfaces ; provision also being made for the
observation of, at least, two pressure values.
Examination of the convergence loss requires determination
of the conditions in a clearly defined area at the termination of
the entry curve. Any chosen convergence must then be fitted
with a very short parallel outlet; although with the more
awkward forms rather greater lengths might be necessary to
ensure fairly stabilized conditions. There are effects both of
curvature and size, and it is probable that experiments with air
flow might give a more ready means of examination.
For the convergence effects, then, a few standardized en-
trance forms might be used each with short parallel outlet, each
form, say, in triplicate differing only in linear dimensions. Mass
flow determination accompanying pressure reading at outlet
would then allow of establishment of the general variation of this
loss, and any odd form that may be used at times in practice
could be reasonably covered by interpolation from the known
standard results.
DISCUSSION THE LOSSES IX CONVERGENT NOZZLES. 55
DISCUSSION.
Professor G. G. STONEY, F.R.S., Vice-President, congratu- Prof . 8t one y
lated Professor Mellanby on his Paper and on his method of
analysing- the nozzle losses by the help of the factors K and F.
Professor Mellanby had also been wise to use superheated steam,
and thus avoid the complications due to having condensation
taking place in his nozzles, more especially as there was much
uncertainty as to the point between the saturation line and the
Wilson line at which condensation really began.
The nozzles upon which the experiments had been carried
out were small, but were comparable with Parsons' blading.
They had a mean hydraulic depth of O031 inch while the usual
inch Parsons' blade had a hydraulic depth of 0*04 inch, and
except in the low-pressure part of the very largest Parsons'
turbine the mean hydraulic depth rarely exceeded T V inch.
The results of the energy efficiency of nozzles agreed
generally not only with the results which Professor Mellanby
quoted, but also with those obtained by Christlein and Josse in
Germany. Considerable doubt had been thrown on the relia-
bility of the published results of the tests of Christlein and Josse,
and there was- a strong suspicion that they were " cooked," as
with them the efficiencies of modern Parsons 7 turbines would be
impossible. He was afraid that Professor Mellanby 's results
were also unreliable.
Professor Stoney confined himself in his remarks entirely to
the efficiency of the nozzle as the discharge co-efficient was not
really of interest to turbine engineers and to nozzle " a "
which was the practical form.
In Parsons' blading, which, of course, was really all nozzles,
an efficiency of well over 90 per cent, and more probably 93 per
cent, or 94 per cent., had been attained with steam velocities of
about 150 feet per second, giving an expansion ratio of about
0'99. Recent tests of a larger Parsons' turbine showed that the
efficiency between the thermal units in the steam from the
B.E.A.M.A. Tables and the B.H.P. on the shaft was 83 per
cent., assuming the alternator efficiency to be 96 per cent., and
allowing for terminal loss. The steam velocities varied from
about 300 feet per second in the high-pressure part to 700 feet
per second in the low-pressure part giving an expansion ratio* of
from 0-98to0'81.
Reference to Fig. 9 of the Paper for the velocity co-efficient
of nozzle " a," showed that the nozzle efficiency with about 0*99
E 5
56 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
Prof, storey expansion ratio would be about 88 per cent, or below the-
efficiency actually shown by Parsons' Wading which had been
shown by experiment to be about 93 per cen-t. In the case of
the large Parsons' turbine to which Professor Stoney had
referred earlier, the efficiency would vary from about 92 per
cent, to 88 per cent, of the nozzles according to* Professor
Mellanby. Professor Stoney asked how the brake efficiency for
the whole turbine of 83 per cent, could possibly be obtained with
such low nozzle efficiencies.
The accuracy of Professor Mellanby 's efficiencies depended
on the pressure at the outlet of the nozzle not only being uniform
across the cross section o>f the nozzle, but also shown truly by
the search tube. Also, it depended on the effective area of the
nozzle being the same as the actual area, and that there was no
vena contracta. Both these assumptions seemed to Professor
Stoney exceedingly doubtful. Had Professor Mellanby ascer-
tained by test that there was no vena contracta. if the nozzle was
used for water? An incompressible fluid such as water had the
same properties in a jet as a compressible fluid such as steam for
small amounts of expansion. Such discharge curves as were
shown in Fig. 1 would be accounted for by the existence o>f a
vena contracta at small amounts of expansion which would be
reduced when the expansion was larger. It was possible that a
better shaped nozzle might show different results by Professor
Mellanby 's method.
Professor Mellanby, so far as Professor Stoney could see, did
not raise the question of pressure distribution in his Paper, but
he made various observations on the question of whether there
was any vena contracta or not. It was quite evident, however,
from the remarks that he made on page 16, with reference to
the design of turbines, that he assumed there was no vena
contracta, and that he placed considerable reliance, at all
events, upon the relative efficiencies of his nozzles which showed
that the efficiency was a maximum at about the critical velocity
and low for small expansion ratio. As Professor Stoney had
already shown, there was considerable doubt as to the reliability
of these assumptions.
Professor Stoney could not at all 'agree with Professor
Mellanby 's remarks on page 15. Large diameters and high
blade speeds had to be adopted in large turbines both to reduce
the cost of manufacture and also to reduce the leaving losses.
The danger of going to too high blade speeds was well shown by
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 57
the numerous recent breakdowns in America of turbines with ?
high blade speeds, and English designers had been most wise in
keeping to moderate blade speeds, which of course meant that
the steam velo'city was somewhat below the critical. This was
especially the case in marine work where not only might a
breakdown cause immense loss of life and property, but also
there was considerably mo-re risk of a breakdown due to a
wheel bursting, owing to the fact that excessive speed might
easily be attained if the propeller came out of the water in a
heavy sea and the governor failed to act promptly. Marine
designers had, theiefore, been most wise in keeping to moderate
blade speeds, and there was at present no reliable evidence
that the efficiency was reduced thereby.
Professor Stoney wished, therefore, to point out that the
highest efficiency of a turbine yet recorded was that of that
Parsons' turbine to which he had just referred, and that in that
turbine the steam speeds varied from about J to f of those usual
in large impulse turbines, which were not much removed from
the critical.
Mr. A. JUDE said that it was rather difficult to see quite what Mr. Jude
the Authors wished discussed. Certain graphically presented
facts of experiment were given, the accuracy of which there was
no reason to doubt, so far asi jthey w^ent, as many of them were
(he thought admittedly) pretty well known. A very complicated
and original mathematical edifice, however, was built upon these
experimental results, part of which left him with very mixed
feelings difficult to express much the same as was felt at
having to put up with proofs by " induction " while other
parts were more satisfying as they did at any rate sweep away
certain refuges commonly sought by the destitute for instance,
those relating to viscosity and friction.
To begin at the end as with a novel, we turn to the last page
first to see what becomes of everybody, he was disappointed at
the Author's confession that the " intractable equations'' of
their predecessors had not been displaced by anything much
more tractable, and that, after all said and done, we were left
to the plain arithmetical values of the things that entered into
practical service as given by specific experiments.
It was impossible to enter into the complete mentality of the
Paper in a few hours, it having taken the Authors many years
to produce, so he was compelled to confine himself to a few
58 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
broad points and to take the side of the prosecution, whatever
he might believe outside court.
The Authors discussed at some length the disputed
phenomenon of " excess discharge " in the vicinity of the
critical drop of pressure. Personally, Mr. Jude had an open
mind ; but it was the easiest thing imaginable to make a nozzle
give an excess discharge over a very large range of ratios. When
excess (or any other) discharge had been spoken of, it had always
been taken for granted that the conventional thermo-clynamic
formula was an absolute and precise law of nature. It was
certainly pretty close to it and served well in everyday affairs,
but after all "pv n = c" whether n were simple or compound,
was really only a mathematical convention. All mathematical
expressions had characteristic peculiarities, and while they
might represent phenomena truly over a large range it did not
always follow that they did so through all the algebraic trans-
formations, critical values and limits of which they were capable
as pure mathematics.
It really mattered little what the datum for comparison was
in a co-efficient of discharge, so long as it was a convenient one,
universal and amenable to the everyday tables of the properties
of the gas in question. If desired, the simple hydraulic formula
might be adopted over the whole scale as a datum. It simply
meant that co-efficients would have certain values under that
convention, which .values would be used in a practical applica-
tion based on the same convention. So it would not matter
whether the co-efficient for a certain ratio were O9, I'O, or any
value it might happen to> come.
Leaving a weakness in the mathematical side of the excess-
discharge controversy, Mr. Jude turned to look at a practical
or observational point, which not only applied to the so-called
critical region, but to the whole supercritical one as well. How
was it known that there was not already a tendency to get
" excess discharge " over the whole scale after the losses had
been given their proper value?
Mr. Jude explained himself. Suppose that for a certain
convergent nozzle the co'-efficient at r = 0'8 was 0'9'; the Authors
then said that the loss was O'l, being mad up of two or more
separate losses, say, for argument 0'06, 0*03, 0*01. Mr. Jude's
point was that the losses might really be, say, 0'06, 0'04, 0'04,
and that the convergent nozzle devoid of internal losses, were it
but found, might actually show an excess discharge. Taking*
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
59
nozzle " fl 1 .," Fig. 23, and giving it any supercritical drop of pres- Mr.
sure; it liad an apparent co-efficient of discharge of, say, 0'9,
and passed 0'9G pounds of gas per second, with p lt p 3 as source
and sink pressures. If a divergent tail like Fig. 236 were
added it passed ever so much more than O9G, probably nearly
3G ; that is, its co-efficient of discharge referred to the throat
and to the same reservoir pressures was 3. Everyone knew that
this arose from the pressure p 2 at the throat falling below the
sink pressure, and this effect took place to a more or less propor-
tional degree as the divergent tail was cut down. It would,
therefore, be seen how easy it was to make a nozzle give
" excess discharge."
FIG. 23.
Now, in the case of the sharp-edged nozzle, the venacontracta
was an accepted and in fact unavoidable phenomenon ; and if
such a thing as a stable vena contracta were possible without)
shaped retaining walls., why not a vena expanda as well ? If the
Authors' pressure curves in Fig. 6 were examined, the third one
particularly, it would be seen that the pressure always sank
below the reservoir pressure, apart from the wave formation.
It was not very conspicuous in that figure, but had their gauges*
been more sensitive and their chart enlarged, the record would
have been better. It would be found that the mean pressure
persisted for some considerable distance at below the datum
60 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
Mr. jude pressure. Dr. Morley had pointed this out in his 1916 I.M.E.
Paper. Mr. Jude had himself given an example of it with
small drops of air pressure, equal to 10 inches of water, in his
1920 I.M.E. Paper, and he had some very ancient notes of it.
The jet having emerged into the free space, something-
became O'f it as it did not keep on for ever. To; save a lot of
trouble it was said, too glibly as a rule, that the kinetic energy
was transformed into heat. No doubt it was in time if " heat "
were an elastic enough term ; but there were, in relation to the
time of things in a nozzle, long intermediate stages. One of
them was ." sound " which was another name for the struggle
between potential and kinetic energy. It always had seemed
to him that the first natural effort to equilibrium was to convert
" velocity " into "pressure," not heat, with the result that
static waves were formed. Under Mr. Martin's telephone
system the jet from the nozzle "a," Fig 23, desired to be supplied
with a divergent enclosure like Fig. 236 as its kinetic energy
became dissipated, and if the desire were granted its pressure
line would be as X in Fig. 23c. As the desire was not satisfied,
the jet tried to adopt the same tactics as it did in a sharp-edged
entry and provided itself with a makeshift cone in which for
some little distance a semblance to integrity was maintained.
Losses in the convergence intervened as well, with the result
that the pressure line was as Y, Fig. 23c. Possibly, with a
divergent tail in substance, the co-efficient of discharge at a
certain r would be, say, 2; without it, its co-efficient appeared
to be, say, 1'02 (with a better conoidal nozzle than the Authors').
The Authors had shown (Figs. 5, 9, second Paper) that at and
below the theoretical critical ratio, the drop in the compound
nozzle at the throat was much greater than in the simple
convergent.
If there really were the least tendency to a divergent tail
effect, then a nozzle of unity efficiency was a physical impossi-
bility. It might show unity, but unity was merely the result of
an addition and subtraction of gains and losses, and the Authors'
ks might have quite incorrect relative values. He inten-
tionally laboured the point because the effect of a measureable
divergence on a convergence was not only a very con-
spicuous one, not a matter o<f an odd one per cent., but anything
up to 300 per cent, or so, and this property of the compound
nozzle had been entirely obscured by the Authors' attention
being confined to the outlet area quite (Correctly from some of
their points of view, but not necessarily altogether so.
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
61
The point underlying the remarks he had just made was that Mr.
both from the ordinary mathematical and the observational
point of view there might be nothing very remarkable in a so-
called '* excess discharge."
In the case of steam, the effects of supersaturation had yet
to be proved of more than relatively trivial moment, although
he agreed with the Authors in keeping it out of their steam
conditions.
While on the subject of pressure curves he referred to the
convergent losses with which the Authors dealt towards the
end of the Paper. It was an experimental fact that the
efficiency of the nozzles formed by the Parsons type of blading
was very high. Acting purely as no-zzles and with superheated
steam it was somewhere about 95 per cent, (energy efficiency).
FIG. 24.
As he had said elsewhere, these nozzles contained quite a lot of
unavoidable mechanical imperfections not present with the
elementary form used by the Authors, others and himself in
these little diversions ; but in the turbine the convergence only
occurred parallel to one plane and not about a centre. The
Authors very rightly drew attention in the beginning to the
possibility of the pressures not being uniform over the various
planes of the nozzle, but they themselves were guilty later on
of assuming that they were uniform when they said that the
axial Pitot tube gave " full information "'(page 16) as to the
manner in which the pressure fell. As a matter of general fact
the pressure was a long way from uniform, but in support of
their conclusion that the convergent losses were quite a big slice
of the wholei, Mr. Jude drew Fig. 24, which gave an idea of what
the isobars really were like. He had taken the 1 pressures over
62 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
the flat surfaces of large rectangular nozzles of various forms
including Parsons-type blade passages,* and lie found the same
characteristic of the centre being left behind, as it were. He
thus was able to confirm by direct experiment the form of the
isobars which the Authors had assumed by way of argument in
their second Paper.
There was thus a very rapid fighting for place in the
entrance, which fighting must obviously be more severe in the
circular than in the rectangular nozzle.
As Professor Stoney had just pointed out, the convergence of
the Parsons' blade passages was very much less than that of the
experimental nozzle " a " ; but the Authors appeared to claim
the discovery of a possible large and over-riding convergent loss.
There was nothing new in this ; indeed it must be obvious from
what was known to happen as the convergence widened and
ultimately became a " hole in a thin plate." Mr. Jude had
given two examples in his I.M.E. Paper of 1920 (one a cone, the
other a rectangular skew). In both cases a maximum co-
efficient was given with an included angle of between 15 and
25, beyond that falling rapidly with the angle. This angle
agreed very closely with the Parsons' blade entry, so far as
curves could be substituted for straight lines.
The Authors, after the criticism of their previous Papers and
in their interim reply to the present, insisted that they did not
wish their experiments to be taken as a guide to turbine practice
but as an attempt to establish a method of analysing the losses
and to exhibit their general trend. He was therefore somewhat
taken aback at their remarks on page 15, as to the conscious or
unconscious wickedness of turbine designers. They certainly
did all sorts of both things, being, he hoped, true engineers who
coated their arithmetic with a good thick layer of judgment, but
they did not use Fig. 4, a, b, or c, in either way for multi-
stage turbines.
An outsider reading their remarks about staging would think
that the rational rules for staging grew out of the values for the
efficiency of nozzles and blading. That was only a little bit of
the story. From that point of view, the number of stages for
a given diameter would be exactly the same for a 1 H.P. as for
a 100,000 H.P. turbine.
Again, it was not altogether a matter of the graduation of
overall losses. Each size had to be a commercial proposition or
* Jude, Theory of Steam Turbine, page 41 1 .
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
63
it was no use designing" it at all. Moreover, not only a com- Mr - Jude
mercial pro-position from the maker's sordid point of view, but
from that of the man who had to run it.
The Authors said that there was direct experimental evidence
that a velocity of efflux closely agreeing 1 with that of sound
represented the best condition. If they got this from the curve
O'f co-efficient of discharge the definite evidence was weak
because the most effective jet energy was not necessarily as the
square of the discharge. If they got it from a certain German-
9S
80
7S-L
>re
900
10
Load
FIG. 25.
Swiss turbine catalogue, then they were merely quoting what in
America was called a " stunt." However, a land turbine
worked under variable load at constant speed and thus offered
one way of getting an idea of the effect of altering the ratio of
blade to steam speed ; and the efficiencies obtainable under those
conditions coupled with other information of a general
character, such for instance as the efficiency of the reaction
turbine, dissolved all pretence of the critical velocity being the
best. That was not to say it was the worst; there was nothing 1
in it that could be said to establish the dictum.
64 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
Mr. Jude gave an example in Fig. 25 of land turbines work-
ing under variable load, having* a full load steam speed of
roughly 1,100 feet per second. The efficiency of the turbine
proper for the purpose in view was the ratio of the work done
to the available energy between the steam chest (not the stop
valve) and the exhaust. These were Rateau-type pressures
turbines working wholly on low-pressure steam, the H.P.
stages running idle. The figures were not from a single turbine
but from quite a large number off the same drawings. The
average speed ratio at full load was about 0'43, neglecting
carry-over.
It should be understood that this efficiency was not just that
of the blading but covered the whole of the turbine losses,
friction, leakage, pump, etc. Although he did not guarantee
the light-load efficiency and the slope of this curve to two or
three per cent. since the chest pressures were taken on an
ordinary vacuum gauge the general indication in this case
was that the efficiency of the blading was higher with the
smaller drops of pressure than with the higher drops, and so
high in any case that there was very little left for nozzle
research to accomplish. Of course, the big loss so far as actual
steam consumption rate went w r as in the drop across the governor
valve, but that was not curable by nozzle design. Partial
admission could do very little because the no-load consumption
was practically the same whatever was done.
The curve was not to be taken as a law ; he could give
examples where the slope was in the other direction, but what
general law there was about the whole thing was that the real
working efficiency tended to keep a constant only collapsing
toward the no-load zero at very light loads. It was not
affected so very much by the speed ratio itself because it was a
point in design to conserve the carry-over velocities from stage
to stage as much as possible ; indeed the very life of the Parsons'
turbine hung on it. This was one reason why the Authors in*
Fig. 4 were off the track altogether. With perfect carry-over
it would not matter much how many or how few stages there
were, if the mechanical losses were ignored, for the only
residual loss was that in the final leaving velocity and this only
affected the whole some one or two per cent, between the extreme
commercial possibilities of design. So, the approximation to
sound velocity was merely fortuitous. The factors that settled
the steam speed were the necessary mechanical proportions of
the whole turbine, and the environment of engineering and
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 6f)
commercial factors. Pattern-makers, for instance, were a very Mr * Jude
expensive luxury !
Early practice with the so-called Curtis stage in Curtis-
Rateau turbines was to make it big with a big sub-critical drop.
The practical difficulties attending the manufacture; of the
necessary nozzles and getting what was wanted and the up-
setting- of pretty calculations by actual running conditions had
led to the modern! practice of elimination or reducing the drop
to not more than about " sound " conditions a point on which
he thought the Authors would agree.
The Authore' mathematics was rather horrible to read, and
they seemed to have gone out of their way to make it difficult by
adopting several indefinite factors indefinite because it was
necessary to keep digging and referring back to find out what
they were. " Loss factor " apparently was not a " factor " at
all; then there were other curious names "volume factor,''
" jet flow area," " velocity factor." He could forgive them,
however, as he w r as not guiltless: himself ; but when the reader
was given a very limited time to get through rather a large
meal, frequent definition and clearly dimensioned symbols
were more than helpful.
He thought that the final " boil-clowns " should have had
some direct application to practical service; e.g., (27) would be
better expressed as a ratio or the complement of an efficiency.
Form (27) it was thus seen that after all, the convergent loss
per cent in a simple convergent nozzle was adjudged to be
constant
approximately / ~n-~\* Surely they now could find that serial
v 1 r
n
constant without much trouble ! He was rather interested in this
because the results of his own " series " experiments (I.M.E.
1920) demanded that the co-efficient of discharge for all nozzles
should be about zero for r=l, a point on which several people*
seemed to disagree with him.
The Authors were to be complimented on the persevering
way in which they were attacking their problem. It was only
to be regretted that the tools and data at their command, both
physical and mathematical had proved inadequate, but like
everything else these things took time to grow up.
Mr. A. H. FITT congratulated the Authors on tackling a Mr. Fitt
very difficult problem and carrying out a great number of
valuable and painstaking experiments.
66 DISCUSSION THE LOSSES* IN CONVERGENT NOZZLES.
The search tube had been generally adopted for ascertaining
the condition, etc., of steam in a nozzle. It had defects and
losses due to its use certainly occurred in the nozzle. The
Authors had not indicated what they considered to be the
magnitude of these losses. Further, it would seem probable
that a vena contracta occurred at the entrance to the nozzles,
and this the Authors had apparently ignored.
The co-efficient of friction was certainly more important
than that of discharge. For impulse turbines the discharge
co-efficient could be ignored. It was of importance only in
the high-pressure or first-stage nozzle as this determined the
steam consumption of the turbine, but as many disturbing
influences, such as superisaturation and errors of manufacture
entered into the question, a theoretical discharge co-efficient was
of doubtful value and generally speaking it could be ignored.
Wfth cast nozzles, Mr. Fitt's difficulty had been to get the
nozzles finished to anything like one or two per cent, as regards
area. In other words, the efficiency O'f the nozzle was the
important thing". With built-up OT machined nozzles greater
accuracy was obtainable.
Mr. Fitt's chief point of criticism was that the experimental
nozzles used by the Authors were circular and straight, and
nothing like those used in actual turbine practice, which made
him inclined to think that the whole theory was upset when it
was considered that a rectangular nozzle had a greater boundary
surface than the circular for the same area. Further, the axis
of the practical nozzle was never straight, the steam being
turned through a considerable angle, and this meant another
loss. Again, with a velocity compounded wheel the nozzle was
convergent divergent, and this introduced further complica-
tions. The circular nozzle was taboo because it gave a circular
jet of .steam thus not filling the blades, and allo'wing expansion
in these, with consequent waste of energy and lowering O'f the
efficiency. The rectangular shape was, therefore, obligatory.
The circular nozzle did not represent practical design and Mr.
Fitt thought that this largely detracted from Prof. Mellanby's
results-.
With regard to Fig. 4 representing different forms of
efficiency curves for impulse turbines, Mr. Fitt said that he had,
never used curves o>f this description when designing machines,
and did not place much value on them, for one had to design
a turbine not as a technical and highly scientific instrument, but
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 67
as a commercial proposition which must give a certain output Mr - F ^
and be designed to give the best steam consumption with the
minimum cost of manufacture. The higher the speed the
smaller and lighter the machines could be built, with a
onsequent reduction in cost.
Professor Stoney had referred to wheel bursts which had
occurred in America and to a minor degree in this country.
Mr. Fitt had had some experience with wheels and had investi-
gated as far as povssiMe the cause of the breaks. In America
speeds up to about 900 feet per second had been used, and this
was in excess oif practice in this coomtry, wheT'e mean speeds
varied from about -500-700 feet per second. The cause of some
of the fractures at any rate had been definitely traced to defects
in the steel, and there was no question that, with the advance in
metallurgy, steels could be and were produced which would
safely withstand the stresses at these high speeds.
He suggested to the Authors that they might now tackle the
problem of determining the losses in rectangular nozzles
which could be machined out quite easily. The data obtained
would be immensely valuable in actual turbine design, which
was iioit the case with the circular noizzle, because it did not
represent the actual problem designers were up against every
day.
VOTE OF THANKS.
The PRESIDENT (Mr. A. Ernest Doxford, M.A.) moved a The president
vote of thanks to the Authors, which was carried by
acclamation.
CORRESPONDENCE.
Professor Gr. G. STONEY, Vice-Pre\sident, said that since the Prof, stoney
meeting Mr. Dempster Smith had tested at the College of
Technology, Manchester, with water, two no'zzlies, marked A
and B, in Figure 26. A was one with a good stream line
form and gave no perceptible vena conlracta and a smooth jet.
It had a coefficient of discharge of 0*99, B was one with the same
proportions as Professor Mellanby's nozzle, "a" and it gave
a rather rough jet showing that the convergence was too rapid
toi give a smooth jet and its coefficient of discharge was 0'92.
It could be seen that there was a vena contracta but the
amount of it could not be measured accurately on account o>f
68
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
Prof. Stoney
Mr. Cook
the roughness oif the jet; it seemed, however, to be between
^ inch and ^ inch. This confirmed Professor Stoney's
impression that if Professor Mellanby had experimented with
better shaped nozzles he would not have obtained low efficiency
at small ratios of expansion.
NOZZLE A
NOZZLE B
FIG. 26.
Mr. S. S. COOK, Member, said that the Authors had made a
courageous attack upon a difficult and important subject,
although by methods that would hardly command general appro-
bation. An important part of their conclusion was that the
losses were of two kinds, friction losses and eddying OT conver-
gence losses. It hardly required so much ingenious and
laborious analysis to arrive at this obvious conception which.,
Mr. Cook thought the Authors might well have adopted as the
starting point of their enquiry.
Five or six pages had been devoted to an elaborate mathe-
matical investigation of the effects of viscosity when a very
simple calculation would have sufficed to show that viscosity
effects were of too low an order to have any influence on the
present problems.
DISCUSSION - THE LOSSES IN CONVERGENT NOZZLES. 69
The experiments of the Authors had apparently been carried Mr. cook
out on nozzles of symmetrical form, which permitted a search
tube to be moved along- the axis, so that the same aperture in the
tube was in use all the time. Bearing in mind the difficulties
usually met with when using search tubes such, for instance,
as the effect of any slight irregularities in the edge of the hole
Mr. Cook was afraid this method could not be so easily applied
to an actual turbine nozzle which had as a rule a curved axis,
and was not symmetrical. In such a nozzle, too, the losses due
to convergence might be expected to be considerably greater
than for the symmetrical nozzles employed by the Authors.
In formula (1) page 16, which apparently assumed the steam
to be in the state of a perfect gas, the value of k was dependent
upon the ratio o<f expansion as well as upon the efficiency of
the nozzle. Would it not be preferable to adopt in its place
some co-efficient not directly dependent upon the pressure ratio,
but only so in so far as this affected the efficiency? The right
hand side o>f the formula might be written
r(l-r X ) 4 , n-1
- where A. = r
77 being the average hydraulic efficiency and n = 1-3 as before.
Mr. J. G. STEWART, D.Sc., said that he had been particu- Dr ste wart
larly interested in the Author's separation of the losses into
boundary and convergent parts. They had mentioned the
hypothesis of a lag in the partitioning of energy in the molecule
which Dr. Stewart had put forward in 1917 to explain ex-
cessive flows in nozzles. Besides the behaviour of gases in
nozzles he had collected a few other experimental facts which
similarly supported the hypothesis and published these in a
Paper in the Philosophical Magazine in the following year ; but
at the time he had received no evidence from the behaviour of
sound to support it. Subsequently the late Lord Eayleigh had
drawn his attention to a paragraph in one of his (Lord
Rayleigh's) Papers in which he had considered the same
hypothesis as an explanation of the rate of dissipation of sound
energy in the air. While the evidence was thus in favour of
the hypothesis of " molecular locking " as Prof. Mellanby and
Mr. Kerr had termed it, it did not necesssarily account for the
whole of the convergent loss ; and Dr. Stewart was particularly
curious to know, in connexion with the design of gas flow
70
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
Dr. Stewart whether any part of the loss was a function of the viscosity of
the gas.
Dr. walker Dr. WILLIAM J. WALKER said that he agreed with the
Authors' conclusion that nozzle losses required to be divided
into two distinct types. Whether matters would be simplified,
however, by separating these into " frictional " and " conver-
gence " losses as denned in the Paper remained to be determined
by experiment. It was more t-han likely that there was so
close a dependence between the two as to render their separation
impracticable. Possibly it might be found when more direct
1-19
0-93
2 4.6 8 10 12 11- 16
DIFFERENCE IN HEAD (INCHES OF WATER).
FIG. 27.
experimental evidence was available, that the separation was
best carried out in terms of v and v 2 losses (i.e., viscosity
and inertia losses). That the relation between these was
directly affected by convergence appeared, from experimental
evidence, to be highly probable.
Although discharge co-efficients above unity had not, in
general, been obtained for elastic fluids under low pressure
heads, Dr. Walker had recently obtained the results shown in
Fig. 27 in the measurement of air flow through a Yenturi
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 71
meter, 3 inch pipe diameter and 1 inch throat diameter, Dr
approximately. These results were entirely unexpected, but
frequent repetition of the tests confirmed the general trend of
the curve to be as shown. Discharges had been measured by the
fall of a calibrated bell-tank container. This result, however,
was not more paradoxical than similar results for water flow,
for it was well known that curves similar to Fig. 27 and also
curves of opposite tendency (i.e., image curves of the type
shown in Fig. 27) might be obtained in hydraulic experiments.
Thus values for C rf as high as 1'36* and as low as 0'75f had. been
obtained at low velocities of flow. In a recent PaperJ Dr.
Walker had advanced an explanation for this based on the fact
that resistance to fluid motion might be written ~R = av + b v 2
where " a " and " b " were constants, of which it appeared
11 a " might be positive or negative according to conditions of
flow. The nature and value of the constant " a" appeared to be
directly affected by the geometrical outline of the convergent
and possibly also of the divergent portion (if any) of the nozzle.
It appeared likely, at least, that the agreement between Dr.
Walker's curves of either type as theoretically deduced (and
given in the Paper above mentioned) and those actually obtained
in practice, was some evidence that the practical solution of the
whole problem might be more readily found by exploring the
variations in the constants " a " and " b " under different
convergences and velocities of flow.
Mr. M. G. S. SWALLOW, Member, and Mr. N. A. LAMB,
Member, in a joint contribution, said that the Authors had
presented a Paper which raised their admiration of the pains-
taking research work which was being carried out at the
Glasgow Technical College. They naturally hesitated to
criticize research of such magnitude, and wished to assure the
Authors that they fully appreciated the value of the results
obtained ; but they wished to criticize one aspect which the
Paper forced iipon the reader. Although a very large amount
of research work had been done during recent years on this
particular subject little reference was made to work carried out
by previous investigators. The only exceptions were the papers
read before the Institution of Mechanical Engineers, Mr. H.
* Coker, Canadian Soc. Civil Engineers, March 1902.
+ Herschel, Trans. Amer. Soc. G. K., vol. xvii., p. 228.
t Phil. Mag., Feb. 1921, "Fluid Discharges as Affected by Resistance to
Flow."
E 6
72 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
^' Martin's articles in Engineering, and the reference on Fig.
1 to Rateau and Loschge's tests.
Apart from Bateau and Loschge a large number of tests had
been carried oait during! recent years by Continental workers of
which the following were instances :
On page 14 the Authors stated that there was direct
experimental evidence that a velocity of efflux closely agreeing
with that of sound represented the best condition. This was an
effect which was pointed out by Josse and Christlein in their
Papers published in 1911 and 1912, and formed the subject
matter of their British Patents Nos. 7043/11, 8844/12 and
9975/12. Although the results of the experiments had at the
time of publication been severely criticized, the Authors' tests
appeared to confirm the Jesse results and the reduction in
stages in recent impulse turbine designs gave considerable
evidence of the correctness of their results.
With reference to the Authors' very interesting investiga-
tion into* the question of boundary and convergence loss, these
had already been dealt with in tests carried out by Prof. Stodola
and the results published in January, 1919. Whilst the
Authors' investigation was an excellent one and the publication
in the English language of these results would be a great
advantage, a considerable amount of time and effort might have
been spared, if the Swiss experiments had been more widely
known. The Stodola experiments formed an attempt to work
out the conditions in a nozzle from the three-dimensional point
of view in contrast with the study o>f the conditions along the
Tines O'f flow, as those latter, according to him " are sufficiently
well known." Prof. Stodola apparently did not hesitate to
employ saturated steam and although the phenomenon of super-
saturation was not a new one, yet the present Authors did not
feel justified in including tests which might introduce it.
Both the present Authors and Stodola assumed the distribu-
tion of the velocity across the jet to be similar :
Authors' formula :
Stodola's w = n(l--)
\ ' /
where w and r referred to the velocity at the centre and the
radius of the section respectively, and w and r the velocity and
distance of the point in question from the axis of the nozzle.
Close approximation to actual results was obtained with a value
of n = 25 for the particular nozzle,
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 16
Stodola remarked that it must not be assumed from the level
form of the curve in the centre that friction was absent but
that the type of the frictional losses in the centre of the jet was
different from the friction of the walls. To the former he gave
the name of turbulence, and directly anticipated the present
Authors' division of the K into the two types of boundary loss
and convergence loss.
The theory of the so-called turbulence was plainly and
straightforwardly developed from the consideration of art
element of the jet in the form of a cylinder A, B, C, D.
In the course of the motion of this cylinder the shape was not
preserved, but certain parts were intruded and extruded. The
idea of " Work of Turbulence " was developed in addition to
the " Work of Friction." The turbulence was denned as the/
quantity of momentum which passed through each unit of
surface of the cylinder, and it appeared that this increased pro-
portionately to the distance from the axis of the nozzle.
FIG. 28.
For the sake of simplicity the original expression for the
turbulence was altered by considering it as analogous to fluid
friction. If df was the surface of an element of the jet then
the turbulence was
r> Jf d w
R = y . df' -
a r
y was termed the turbulence co'-efficient and was comparable
with the coefficient introduced into the Authors' equations Nos.
16, 17, etc. As an average value for speeds of the order of
700 m/sec. Stodola estimated y to be about 7xlO~ 4 ICQ. sec. m~ 2 ,
that is of thei same dimensions as the viscosity co-efficient. In
Ibs. sec.-ft.~ 2 this value became 0'000143, which was comparable
with the Authors' figure of '0003.
Stodolia also remarked that this was very much larger than
the viscosity co-efficient, but as he recognized its true nature
he made no further comments and utilized it in some very inter-
esting calculations regarding the velocity variations. The
74 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
MI. swallow further development of the eddying (or turbulence) effect hinted
and Mr. Lamb
at by the Authors under their reference to " re-heating " was
expanded by Prof. Stodola at some length.
Mr. Swallow and Mr. Lamb finally quoted two of the three
paragraphs of Prof. Stodola 's summary of his report, namely :
(1) The velocity of flow of a jet issuing from the mouth of a nozzle is mainly
constant across the section and only decreases rapidly to zero in the immediate
neighbourhood of the walls. The equalization of the kinetic energy is effected by
the turbulence, the intensity of which is approximately proportional to the distance
from the axis.
(2) Numerical values for the turbulence are deducted from the tests of Diebold
and Triipel. It appears from these that it may be considered as a sort of friction
which depends on the radial change of speed, analogous to the viscosity, but which
is very much greater than the latter.
Mr. Anderson Mp j) 3 ANDERSON gaid tnat the me thod of analysis
developed by the Authors promised to give a very useful and
convenient means of determining the absolute losses in a nozzle
as well as of comparing the performances of different nozzles.
Having in his possession a small square convergent-parallel
nozzle O'f dimensions similar to those of the found nozzle li 1) " in
Fig. 5, he had thought that it might be of interest, both to the
Authors and to the members of the Institution, to apply this
new method of analysis to the square nozzle inasmuch as it
approximated more closely than nozzle "b" to the shape O'f
practical nozzles.
The conditions of the test had been arranged to be as nearly
as possible identical with those of case (i) nozzle " b." The pro-
cedure also had been similar; search tube readings gave the
pressure ratio curve and the flow quantity had been carefully
measured. The loss curve ~Kt was shown plotted (Fig. 29) on a
base of distance along the nozzle. The portion of this curve
occurring before the throat was left dotted as no attempt was
made to compare this with the convergence loss suggested by the
Authors. The end value of this convergence loss, that is, the
value at the throat, was probably fairly definite and it was to
be noted that the square nozzle gave less loss up to this point.
The radius of convergence wa.s precisely the same as that of the
round nozzle, but being of square section the cross-sectional
area was greater. Perhaps the greater area might have some
influence in reducing the eddy-flow to which the Authors attri-
buted the main portion of the convergence lo>ss. The subse-
quent portion of the graph was practically a straight line and
parallel with that of the round nozzle. The rate; of loss was
DISCUSSION THE LOSSES IN CONVEKGENT NOZZLES.
75
thus the .same in the two nozzles and hence the co-efficient Mr. Anderson
0*0051 as found for the round nozzle would also be applicable to
this case.
When this test had been completed the parallel portion of
the nozzle was turned away to leave a pure convergent portion
similar to nozzle "a."
This nozzle had then been tested under the conditions of case
(1). It had been expected that the total loss would be roughly
of the order of 0'004 as indicated by the loss curve of the first
experiment, but when calculated out it had proved to be 0*006
020
-o/s
QIC
00$
i'
^'
O 'O2. <*>. -06 'O$ lo
NOZZLE LENGTH IN FEET. Loss CURVE FOR SQUARE CONVERGENT- PARALLEL.
FIG. 29.
or 50 per cent. more. The nozzle had been carefully examined,
and the experiment repeated but the loss remained unaltered.
If the convergence loss was dependent o<n the speed developed one
would certainly expect more loss in the convergence of a pure
convergent nozzle than in the convergence of a convergent
parallel one, but the difference would not be of the order indi-
cated, especially as the figure 0*004 made an allowance for an
increase, the loss up to this point in the convergent parallel
being 0*0035. It would appear, therefore, that the conver-
gence loss could be very seriously modified by the presence or
absence of a tail piece. This probably did not apply to circular
76 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
Mr. Anderson nozzles where the convergence was the same in all planes pass-
ing through the nozzle axis but might be typical of nozzles such
as square nozzles where the convergence was not uniform
around the nozzle in a radial direction. The presence of a tail
piece in the latter type might serve to steady the flow in the
convergent portion and thus reduce the loss. That the tail
piece did steady the flow had been demonstrated by allowing
water to pass through the nozzles. The jet issuing from the
pure convergent nozzle had a decided rotation about its axis,
while that from the convergent-parallel was absolutely devoid
of rotation. It seemed reasonable to assume that any rotatioit
of the jet would tend to increase the eddy-flow and thus give
rise to the high losses observed.
AUTHORS' REPLY.
The Authors In the introductory part of the Paper we suggested that the
subject of nozzle experiment was o>ne of great difficulty, and it
is, therefore, gratifying to find that so many contributors have
been attracted to the discussion. Especially is this the case
when the majority of our critics are in a position to obtain
the best practical information and are, therefore, enabled
to correct and, ultimately, to apply the facts we have obtained
from small scale experiments. In a subject which has been
attacked from so many different angles the arguments advanced
are so varied that, in order to avoid encroachment upon valu-
able space, we shall only touch lightly upon or altogether
neglect some of the minor side issues raised. In connexion
with many of these it may be remarked that a further reading
of the Paper will provide the necessary illumination, and we
can hardly be expected in a reply to criticism to recapitulate
points that it has been considered necessary to deal with at some
length in the papers preceding the one now under review.
Mr. Stanley S. Cook's remarks form a model expression of
opinion by the clearness and brevity with which he presents his
critical attitude. It is at once admitted that we might have
started with the assumption of the two kinds of loss w r hich we
have actually established by argument. But such a procedure
would have been liable to severe criticism, since it would not
have permitted the rigorous exclusion of alternative ideas that
are shown in the Paper to be quite inadequate. Everyone
familiar with this subject will admit that many unsupported
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 77
opinions actually achieve a certain measure of respect, while the The Author
perpetual references to viscosity in connexion with fluid expan-
sion rendered it necessary to combat this point; very definitely.
Moreover, if Mr. Cook refers to some of the other contributions
lie will realize that the comparatively unimportant effects which
we attribute to viscosity do not meet with unqualified approval.
We would also suggest that although the analysis shows the
unimportance O'f this effect it demonstrates the certain influence
of convergence form by the steadiness of the constants obtained.
The difficulty of using a search tube in general cases is remarked
upon by Mr. Cook, but in the examination of ultimate causes of
loss we are justified in using the tube in simple forms. At the
end of the Paper we indicate the method we would propose for
arriving 1 at figures applicable to practical nozzles, and in that
we see no* great difficulty. In view of the findings of the
Paper the necessity to read pressures at all points of any given
form vanishes. The indefinite effect of errors in the hole of the
search tube is often remarked upon. It is only necessary herein
to consider the influence of a slight error in the pressure reading
on say the overall loss in the tail in nozzle (b) to realize that, in
most cases, this may be neglected. Mr. Cook criticizes the
fundamental equation (1) and suggests an alternative .method to
the use of k by the introduction of the " average hydraulic
efficiency " as a factor modifying (n l)jn. We would again
point out that our problem is one of examination of detail
losses in which an average hydraulic efficiency can have no
place. This will be clearer if figure 13 is studied, and it is
remembered that ^ varies with the slope of an expansion line.
Prof. Gr. Gr. Stoney refers to the results of Josse and
Christlein, which he considers unreliable or " cooked," and he
is afraid that ours are also unreliable. In the attitude Prof.
Stoney takes up we have a repetition of the view generally
taken of the efficiencies we have given. They are too low !
Criticism on this point is based on an entirely wrong under-
standing of the problem, and this blind acceptance of an
efficiency value as an unvarying figure does much to justify the
lengthy examination we have given to causes of loss. It is an
enlightening process that is obviously necessary.
In dealing with this matter we may bring together all the
points that are centred therein. Thus, Mr. Jude in his excep-
tionally interesting contribution is essentially in line with Prof.
Stoney in emphasizing the high efficiency of Parsons' blading
78 DISCUSSION -- THE LOSSES IN CONVERGENT NOZZLES.
4\thors ( or no>zzles). Aguin, in a supplementary note, Prof. Stoney
adduces results obtained by Mr. Dempster Smith for water flow
through two convergent types. All the stated facts merely
snow that there are good and bad convergences. We have not
denied it ; nor have we claimed that the entirely arbitrary
circular curve of entry in our experimental types possesses any
special merit. Indeed, the results show it to be rather objection-
able from the point of view of efficient action. But these
experiments constitute a study of loss effects, not a process of
continuous refining until the minimum value is reached which
is the method by which the Parsons' blading section has been
evolved. Now we are quite willing 1 to believe that the conver-
gence loss in a re-action blade passage is less severe than has
been demonstrated in these experiments for a circular nozzle.
Prof. Stoney's remarks concerning the relative hydraulic mean
depths are beside the point, as the whole question in this
application centres on the convergence loss, the steam speeds
being too small to create controlling frictional effects. On the
supposition that the loss factor is TO per cent, of what we have
obtained for the circular entrance, we get (for nozzle action
only)
0085
-(7w?5*
and with r = -95 :
'0085
Since Prof. Stoney and Mr. Jude quote figures that are appar-
ently uncertain between 90 per cent, and 95 per cent., agreement
is quite clearly possible. Prof. Stoney refers to pressure ratios
of 0*99, but special consideration must be given to such small
ranges of expansion, and it could hardly be expected that
extreme extrapolation of results from experiments with large
pressure drops could g*ive immediately serviceable guidance.
The reference to the high overall efficiency of a large Parsons'
set is in no sense critical of our results, as there is no difficulty
in understanding the possibility of such performance from
figures as above. Thus at a ratio of 0'85 the " nozzle "
efficiency of a Parsons' blade might reach
1-^95.57,
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 79
We are thus willing to admit on the basis of the data under The Authors
review that, when employed simply as nozzles, Parsons' blad-
ing might have efficiency values within 92-95 per cent, for most
of the ranges employed in reaction turbines. There are, of
course, other sources of loss in the turbine unit, but we are not
in a position to cover these. The figure of 83 per cent, on the
brake may then be quite possible with high-power units, but we
have unfortunately met no similar figures in our experi-
ence. There are, however, forms of turbine nozzles other than
that of the Parsons' type and inspection will show that in
many cases the convergence losses may be particularly severe.
We anticipate that when this point is fully realized the atten-
tion of designers will be directed to a feature which has often
been neglected.
Both Prof. Stoney and Mr. Jude bring heavy artillery to
bear on figure 4 and its context. We hardly expected this to
meet with a smooth reception by experts engaged in turbine
manufacture, where the manifold influences of finance and con-
struction entirely hide secondary factors that are rather more
subtle. We simply postulated that, if nozzle and blading
efficiency curves had certain forms, certain findings regarding
the best staging followed, if the number of stages were
unaffected by other considerations. There is a difficulty in
allowing this point o>f view, when one is prejudiced by the limi-
tations of practical design, but it nevertheless remains that
there are turbines constructed wherein the design is greatly
influenced by such curves as (c) Fig. 4. If these are wrong
and we contend that they are so then the designers are allow-
ing themselves to be misled by faulty forms, and we are
justified in pointing out the possibility.
Mr. Jude's lengthy disquisition on the subject does not help
us much, and his figure 25 is inapplicable. An impulse turbine
under variable load does not provide a full range variation of
pressure ratio per stage, since there is only a moderate influence
on intermediate stages with extreme effects on those at both
terminals. Besides this, such a process of working involves
modifications of speed ratio. This has important effects
apart from those of absolute steam speeds, and, therefore, the
issue is entirely confused. Mr. Jude states that the effective
jet energy is not necessarily as the square of the discharge,
a point to which we had already directed attention. He also
imagines that we may have been influenced by some data that
80 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
The Authors ht> designates a kt stunt." This could hardly be so, as we do
not know the data referred to.
The point we have raised regarding staging is so susceptible
to misconstruction, and has been so deliberately misconstrued
that we do not propose to say more regarding it and leave it to
our critics to re-examine the matter from the proper attitude,
with the request that they negiect for the time being* the troubles
of a pattern-shop, the purchasing power of clients and the
absorbing power of shareholders.
Prof. Stoney refers to the pressure not being uniform across
a plane section and to the possibility of a vena qontracta. The
pressure is certainly not uniform across the section in a curved
entrance form (see Mr. Jude's Fig. 24 and o>ur preceding- Paper),
but in a parallel tail-piece it must be very nearly so 1 . The
ability of a search tube to give the true pressure depends on the
" lie " of the exploring hole relative to the stream. In our
case this hole was bored straight through both walls and, there-
fore, the two sides would tend to balance any error. The ques-
tion of a vena contracta is one previously raised by Mr. Martin.
We consider it very unlikely that in high speed-flow a vena
contracta of any effective magnitude can exist in any well
rounded form, since experiment shows a relatively very large
growth in the discharge when the change is made from sharp
entrance to even the smallest curve. Thus in the previous Paper
we presented results showing how quickly the discharge
increased with radii of -$ inch and of -j^ inch, and any con-
traction that can exist with a radius of J inch must be so small
that the assumption of a completely filled area is certainly
justifiable. A contraction! shown in water flow as remarked
upon in connection with Mr. Dempster Smith's experiment
may hold for very small steam expansions as Prof. Stoney
suggests, but can hardly be supposed representative of the high
ranges we have employed in conjunction with full entrance
curves.
Mr. Jude considers the question of excessive discharge, and
states that it is the easiest thing imaginable to get this. He
apparently labours under the belief that this is only a matter
of manipulating the standard of comparison. The point is,
however, that, on a certain definite theoretical process, excessive
discharge shows itself under particular conditions. But on this
same basis it is not so very easy to get it. To those who wish
to observe it we would recommend the use of very short parallel
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 81
nozzles with a supply condition of a few degrees of superheat The Authors
where it will be found that, with accurate tests, excessive dis-
charge is a feature of expansion in the debateable supersaturated
or wet field. Mr. Jude is unconcerned about the value of the
discharge coefficient, and is willing to accept any reasonable
basis of judgment whether is gives 0'9, I'O, 11, etc. This is
a typical designer's attitude; but we fail to see why Mr. Jude
has brought it into this discussion. In large range nozzle
expansion excessive discharge is well defined and quite well
understood, and there is no obvious necessity for attacking the
usual mathematical " convention '' in cases where the dis-
charge cannot be and never is excessive.
Mr. Jude's detail development in the attempt to be quite
clear is rather difficult to grasp. He seems to convey the idea
that losses which are apparently 0-06 + 0'03 + 001 may really be
0'06 + 0'04 + 0'04, and hence we may have a real action exceeding
the theoretical in any case. This is a deliberate confusion of
facts already sufficiently complicated; and unnecessary in view
of the defined limits of excessive discharge. We might as well
say that the losses are 0'5 + 0'25 + 0'25 and that the flow itself
is merely something to be grateful for, or a bounty of Mature
conferred on the investigator for his industry. In his follow-
ing treatment by means of Figs. 23(a), (b) and (e) he deals at
length with the pressure values as influenced by form and this
question we have considered in previous work. Mr. Jude's
remarks on this are well made and, generally, we are in agree-
ment. We would, however, point out that we have, in the
present work, dealt with the actual pressure values at any given
poinl in order to arrive at losses, and are not primarily con-
cerned with modification of values by form. Again, our critic's
demonstration of the variability of the co'-eflicient, with and
without a divergent tail, is hardly valid, since ordinary theory
is quite competent to give sound ideas and rational values. Mr.
Jude declares that he labours the whole point because it is not
merely a matter of a few per cent., but anything up to 300
per cent, or so. Herein he again obscures the issue as we have
very clearly confined our attention to conditions at definite
areas unaffected by the subsequent mode of dispersion of the jet.
We are glad to have Fig. 24 in illustration of the pressure
conditions in the entry, but Mr. Jude considers us guilty in
assuming that the axial Pitot tube gives " full information."
If he will reconsider the discussion regarding flow areas he
will understand better the meaning it was intended to convey.
82 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
Th Authors After his powerful attack on the question of " staging "
w r hich we have already referred to, Mr. Jude submits that the
mathematics is rather " horrible," and that the final " boil-
downs " should be otherwise given. We regret the difficulty
of reading the former, but are glad to receive his " forgiveness,"
even although it is only on the score of similarity of criminal
activities. The " boil-downs " can be expressed as a ratio or
complement of efficiency as required, but are sufficiently simple
to allow of adjustment to suit personal taste. It is wrong,
however, to suppose that the convergence loss per cent,
is " adjudged to be approximately constant." We have:
and therefore :
Fractional Loss =
^
l-r )
which is not constant, although b may be nearly so.
Mr. Jude's many points are well worthy of study, but to
deal with them in greater detail would lead to a reply of
inordinate length. We are, however, gratified by the attention
he has given the Paper and hope that he will accord some of
the matter therein a more careful consideration.
Dr. Stewart's note regarding his theory of unequal parti-
tioning of molecular energy is welcome in view of the
additional support he has been able to obtain for it. The fact
that Lord Eayleigh gave the hypothesis some attention is of
importance; but while the general idea is distinctly ingenious'
it is a pity that Dr. Stewart did not demonstrate it more con-
clusively for the case of nozzle expansion. We can only remark
that in very extensive tests with superheated steam we are
unable to find any real evidence of its applicability.
Dr. Walker seemingly agrees with our division of the losses
into two parts, although apparently differing from our findings
as to the general nature of the separate items. He says
" inertia and viscosity," we say "* friction and mass diffusion."
Moreover, Dr. Walker considers so close a dependence exists
that separation is impracticable; we consider the essential
feature of our work to lie in the definiteness of the separation.
It is obvious, therefore, that there is real disagreement despite
the form of his opening sentence. Dr. Walker writes the
resistance to fluid motion as
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 83
which is clearly a law applicable to every stag*e of the expan- The Authors
sion, and gives a loss depending on the second and third powers
of the speed. Now the view we have taken is roughly that
there are two laws of loss, and that one gives a dependence on
the first power and the other on the third, while each law applies
mainly to one portion of the expansion. Dr. Walker's expres-
sion is significant of an attempt to explain complete results ;
and the fact that a constant A in an equation for resistance may
vary in sign does not help in any way to make the attempt
appear successful.
The calibration curve for a venturi meter which is given by
Dr. Walker is rather surprising, and presents no agreement
with the full range curve obtained by Prof. Gibson and shown
in our Fig. 1. It should always be realized in working with
fluid floAv under minute pressure heads that the discharge is
extremely sensitive to pressure variation, and it is probable thai
no ordinary method of pressure measurement will serve. Thus,
owing to the rapid way in which the flow varies with the
pressure near the limit r = l, the very smallest discrepancy
between actual and real pressures will greatly upset the co-
efficient. Mr. Jude has made a most careful examination at
approach to the limiting pressure ratio, and we do not think his
results are in any way corroboratory of Dr. Walker's curve.
That this type of curve and its " image " can both be obtained
is rather perplexing, but while such a difficulty exists we shouM
incline to believe the more natural result, and accept the
t image " as the more truthful representation.
Mr. Anderson's contribution is very welcome. It will be
seen that his square nozzle gives resiilts closely in line with the.
round nozzle (6) although the convergence loss would appear
smaller. It is difficult to see why it should be so as we should
rather expect the reverse. Either Mr. Anderson's argument
holds, or we are here faced with the influence of experimental
error. It has been emphasized in the Paper that in deriving
losses from flow measurements the margin of error is rather
high, and we may have a case in this comparison of the sum-
mation of such errors. But Mr. Anderson's most interesting
point lies in the changed result for the convergence effect when
the tail is eliminated. The tail has obviously had a steadying
influence on the entrance expansion and its disappearance allows
the awkward convergence form to do its worst. From Mr.
Anderson's figures it would seem, therefore, that tlie tail
84 DISCUSSION THE LOSSES IN CONVERGENT NOZZLES.
The Authors influences the convergence loss, and it will be remembered that
no such effect was obtained from the round nozzles. Hence,
questions of symmetry are apparently important.
Mr. Fitt is concerned about the losses due to the search tube.
In a large degree those are due to the changed relation between
surface and area arising from the employment o>f a central tube.
This effect is of course allowed for in the examination^ of loss.
The real drawback is due to a possible discrepancy between
actual pressure and recorded pressure, but we think the margin
negligible when working with large ranges of expansion such
as adopted in this work. Moreover, since the main check is by
overall losses, slight pressure errors would not seriously affect
the determination of these. It is readily admitted that, in
practice, the efficiency of the no<zzle is the main matter, but
knowledge of the discharge co-efficient is also useful in the
investigation of the losses, and must be considered. The
difficulty of finishing actual nozzles to a closely calculated area,
thus rendering the discharge coefficient valueless, is well
recognized. Mr. Fitt raises the question about the great differ-
ence between these experimental nozzles and the practical forms,
but since we must get down to fundamentals in such a study as
this the simpler forms have to be used. If he will refer to Mr.
Anderson's contribution, he will see there is not much difference
between simple round and simple square types ; while at the end
of the Paper we give an outline of the method we would propose
to deal with the complex practical types.
Apparently Mr. Fitt, like Prof. Stoney and Mr. Jude, ha^
no use for Fig. 4. We must point out that this is a root matter
not necessarily the main one, but still of some value. If Mr.
Fitt ever makes a predetermination of a performance he must
use some facts akin to those shown, not in this simple form
perhaps, but still derivable therefrom. We have said that
designers are 1 consciously or unconsciously influenced by these
forms, and the trend of the discussion has demonstrated that
they find a difficulty in discerning anything through the
luxuriant foliage of " costs and construction." Mr. Fitt's
remarks on steel defects are interesting, as there are
undoubtedly cases of failure due to these.
Mr. Swallow and Mr. Lamb introduce very pertinent matter
and we are much indebted to them. The general result we have
pictured in Fig. 1 has been established by many different
experiments, and we were only concerned with a few cases to
DISCUSSION THE LOSSES IN CONVERGENT NOZZLES. 85
illustrate it. Hence, we took tlie most readily available, and The Aufchors
the neglect of Jesse and Christlein is not deliberate. Their
results are well known, and while they have been alternately
discredited and supported their nature must be accepted what-
ever belief is held regarding numerical values. Our main
purpose was to deal with the causes of loss that would give the
curve forms mentioned and, therefore, extensive reference to
all cases of that form was unnecessary. We are very glad to
observe that Mr. Swallow and Mr. Lamb consider that this
result has had an influence on some designs, and we would
commend their attitude to some of our other critics. It is when
we come to the reference to Dr. Stodola's work that we reach
the main theme of their contribution. Their method of stating
this may create the idea that we have been influenced by Dr.
Stodola's investigation, and followed it without acknowledg-
ment. We must earnestly repudiate such a suggestion, as we
had no idea that such a treatment had been given by this
eminent authority. Within recent years Continental litera-
ture has been most difficult of access, and our own technical
journals do not give very much guidance as to what is being
done abroad.
The similarity between our expression for variation of
velocity across a section and that used by Stodola, according to
Messrs. Swallow and Lamb, is a coincidence, but these expres-
sions are used for widely different purposes. We limit our dis-
cussion to the effect on the velocity coefficient of different values
of 2m (Stodola's n) and we have shown that variations in u 19
(Stodola's iv ) are of primary importance. The formula is
presumably the basis of Stodola's work and in this connexion
several points arise.
The writers say " to the former he gives the name of turbu-
lence and directly anticipates the Authors' division of K into
the two types of boundary and convergence loss." This is not
very clear as the turbulent action so indicated is not merely a
convergence loss, but must if deduced from the stated equation
be inherent in all parts of the expansion. Now it hardly
seems feasible to deduce a turbulence effect from an equation of
this order no matter how it is handled ; and, moreover, our
treatment of the action in a parallel tail shows this to be
explainable oil an ordinary frictional basis, i.e., our separation
of losses results in two types mainly referable to two different
forms of expansion. It is, of course, quite possible that both
86 DISCUSSION -- IHE LOSSES IN CONVERGENT NOZZLES.
Authors types appear in all forms of expansion but, if so, the separation
effected in our cases of convergent nozzles demonstrates that
they do change entirely in their relative magnitudes.
The definition of turbulence quoted by Messrs. Swallow and
Lamb is merely a definition of viscosity on the kinetic, theory,
and this is further shown by the equation
which is the mathematical expression of viscosity modified by a
new constant. We are, therefore, asked to believe that a
treatment along orthodox lines applicable to stream line motion
can account for an effect entirely incommensurate therewith.
The writers also remark on the similarity of the findings
regarding the incompatibility of the actual loss co-efficient and
viscosity effects. With this disclosed we take the step of dis-
carding all modes of treatment based on stream-line motion
orthodox or artificially modified since such a process is equi-
valent to applying a method to a type of action which is not
envisaged in the fundamentals of that method. In the same
way the ordinary treatment failed to account for certain observa-
tions regarding pendulum motion when employed by such an
authority in hydrodynamics as Sir Gabriel Stokes.
While there is an apparent similarity of view between the
work referred to by Messrs. Swallow & Lamb and the ideas
we have advanced, there is in reality a basic disagreement.
So far as we can understand the method of attack indicated it
constitutes an extension of a purely molecular action (to cover
what we conceive to be a molar effect) by the artificial enlarge-
ment of a definite physical constant.
We are extremely glad to have had this contribution, and
the writers would increase our obligation if they would favour
us with a complete reference to the work in question. It is
obviously of importance to anyone dealing with this problem.
The discussion has entailed a very lengthy reply, but we
are grateful to the many contributors who have given the
Paper their attention. Where the reply appears meagre on any
important point, it is because we believe that further considera-
tion of our published work on this subject will assist an under-
standing of our ideas.
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