On the losses in convergent nozzles online

*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-