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Full text of "On the losses in convergent nozzles"

*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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