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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-

_^^__ ^^^^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-

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