MAI-JUIN 1963 - N° 3 ------ LA HOUILLE BLANCHE 255

Flow analysis with radio-isotopesAPPLIED TO A HYDRO-ELECTRIC PLANT

L'analyse de l'écoulement

au moyen d'isotopes radio-actifs

APPLICATION A UNE INSTALLATION HYDROÉLECTRIQUE

BY

Nu,s L. SVANTESSON,D. PHIL.

AND M. SUNDBERG-FALKENUARK,STATE HYDROLOGIST

SWEDISH INS'l'ITUTE OF METEOROLOGY AND HYDHOLOGY, STOCKHOLM

The relative dilution method combined willza radioactive '1 -emitting tracer, has been usedto determine the flow throngh a power plantwith low head and short water ways. 1'0 coun­teract the incomplete mixing measurementswere performed at 20 different points for each

of which tu'o determinations were made.The accuracy of the final value in. thisfir:;t trial was estimated at about 4 %. Il

is pointed out that the accuracy aitainablewith this mcthod is mainly a question of enlarg­

ing the number of measurements (injectionnumber multiplied by deteelor Ilul1lber). Theconcentration records made at the mouth of thedraft tube, using "in situ" scintillation detec­tors ilS probes, were also used to localize flowdisturbances, e.g. separation from the ceilingof the draft tube. The constrnctor of the powerplant is hereby given a possibility to check thequality of the conduit shape more exhausti­vely tlwn is possible in model studies. Ananalysis of flow parameters has been perform­ed for a straighi hypothetical tube giving thesame dispersion as the acillal water way.The qllality of the water way shape has beenexpressed as the friction coefficient of thistube. A trial has been made to calculate para­meters for small scale as weil as large scnleeddies.

Combinée avec llll indicateur à émissions y,la méthode de dilution relative a été employéepour déterminer l'écoulement au travers d'uneinstallation hydroélectrique de basse chuteet ayant de courts chemins d'eau. Afin de tenircompte d'erreurs possibles dues à un mélangeincomplet, les mesures ont porté sur une ving­taine de points différents pour chacun des­quels on fit deux déterminations. La précisionde la valeur définitive découlant de ce premieressai est évaluée à 4 %. Les auteurs font remar­quer que la précision obtenue avec cetteméthode croît avec le nombre des mesures(nombre des injections multiplié par le nom­bre des détecteurs). Les mesures de concentra­tion faites au débouché de l'aspirateur (pourlesquelles des détecteurs de scintillation pla­cés à l'endroit même ont été employés commecapteurs) ont été également employées poursituer les perturbations de l'écoulement, tellesque les phénomènes de décollement dZl plafondde l'aspirateur, par e:J.'emple. Cette méthodepermet au constructeur des groupes d'effecluerun contrôle de la qualité du profil de la con­dZlite plus complètement qZle cela ne seraitpossible par une étZlde SZlr modèle. Cette étZlileprésente également une analyse des paramètresd'écoulement pour un tube fictif rectiligne pré­sentant la même dispersion que la conduite« nature ». La qualité du profil de la conduitea été exprimée par le coefficient de frotte­ment du tube considéré et un essai de calcul aété fait en vue de déterminer des paramètresvalables tant pour les petits tourbillons quepour ceux de plus grande étendue.

Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1963015

256 LA HOUILLE BLANCHE N° 3 - MAI-JUIN 1963'

INTRODUCTION

In the relative dilution method the flow isdetermined from the expression :

(1)MT

Icdto

Q=where:Q means the flow;

1"1 the amount of tracer injected;c the concentration of the tracer at the time t

at the gauging section;'l'the time used by the tracer cloud to pass

the gauging section.

The relation above is valid only when themixing is complete Le. the integral has to stayconstant over the gauging section. As will beshown by the present investigation, however,the method can be applied even when this con­dition is not fulfilled as, for instance, in a powerplant with a low head of water and short waterways. Furthermore, the data collected can beused not only to evaluate the flow but also todetermine disturbances in the flow as weIl assome parameters of the turbulence.

Since the end of the nineteenth centurydilution methods combined with chemical orcolorimetric technique to determine the con­centration of the tracers, have been used forflo'w measurements in hydro-electric plantsand natural streams [1] . Two types ofdilution methods may be distinguished: oneusing a continuous injection of tracer andmeasuring the concentration in the homogenousregion, the other introducing the tracer instan­taneously and measuring the mean concentra­tion in the tracer wave. The latter is oftennamed the relative dilution method and hasbeen used since 1H26 [2, 3, 4, 5]. The relativedilution method was in IH58 combined ,vithradio isotopes and was called the total-countmethod [6]. This technique was used at theSwedish Meteorological and Hydrological Insti­tute for the first time in 1H58 in a flow measure­ment in a natural stream and in 1H5H also forturbine testing [7J. Lately the method [8J andits application to hydro-electric plants [9J hasb~en discussed in this journal.

1. - EVALUATION OF FLOW AND EFFICIENCY

l A 7" 'r _ _-Co 1 'L Idt (Ui - U) (Ci - c) (2 b)

Jr 1 0

where A is the area of the gauging section; U i

and Ci are the velocity and concentration at a

ing was m08t incomplete in the power plantinvestigated. Therefore the integral in thedenominator in expression (1) was replaced byits mean value from k individual measurementsat points distributed over the gauging section:

(2 a)MQ­ i 'LfCdt

This is an approximation which is good if thepoints are suitably chosen and if the correlationis low between the velocity and the tracer con­centration at each part of the gauging section."Then the gauging section is split up in a largenumber k of small areas Alk the flow can infact he written [18 ]

The power plant, equipped with Kaplan tur­bines, has a head of water of about 12 m. Duringthe investigation the flow was about 200 m 8/s.The shape of the water ways in. shown in Fi­gures land 2.

The radio isotope, a few tenths of a gram24Na2C08 or about 15 millicuries, was dissolvedin water and injected installLly in Olle point nearthe intake. Two scintillation detectors, combin­ed with recorders, measured the concentrationin situ in a section about 75 m downstream atthe mouth of the draft tube. This means a dis­tance between the point of injection and the gaug­ing section of only about seven diameters as cal­culated for a circulaI' tube of the same meanarea as that of the draft tube. 1\rIeasurementswere made in 10 difIerent verticals with twomeasuring points in each. As a rule, twoinjections were made for every vertical. Thustotally 40 recordings of the concentrations as afunction of time were made.

As mentioned above, the flmv is given by theexpression (1) when the mixing is complete.Preliminary investigations proved that the mix-

MAI-JUIN 1963 - N° 3 --- SVANTESSON AND SUNDBERG-FALIŒNMARK 257

o 5 IOmtJters~,---"~----"

Gougingsection

FIG. 1

Vertical section of the power plant, showing the water ways.

.._.. _1

section

: /J::c==~> 1/

1

right

-----"--"--"--"~ .. -- ..-- .. _-- .. -- .. --- .. ~!

-"--"--"--"---"---"-1

/'";1

/dt =:::;> :~t Gougi~

///: ..~ ..

.. Turbine \

( œ )

~ "-- /..~.~

---------- 1----- .. _- .. __ .. _- ..-- .._- .. '

o, 5,

FIG. 2

Horizontal section of the draft tube.

258 LA HOUILLE BLANCHE N° 3 - MAI-JUIN 1963'

right /eft

~Vet:ti~nr

70 9 8 7 6 5 3 2- 1nr

level (m .f ~detector+ f+

f~

12 EeO.87 .0.71 .1.09 • 1.83 .'.04- .0.85 ·0.88 ·'.361.35 1.25 1.11 1.33 7.43 0.90 - 1.0911 0.90 e7.14- ·'.94

0.77 7.26

10

S .'.54 .'.63 .'.6! e 1.82 .2.13 .0.85 e2.20 ·2.5~ 5U! 2.30 1.94 0.92 1.94 2.03 - 2.4$

7.58 .7.32 .2.238 7.54 2.75

7

~1

~1 Z 3 1- 5 6 7 8 9 10 11 12 13 14- 15 16 17 18(m)

FIG. 3

Variations of the relative concentration over the two partsof the gauging section at the downstream end of the draft tube.

Flow direction out from the paper.

poin t in one of the small areas ; li and (; theirmean values over the section:

no is the net counting rate of the detectorat the calibration;

n Ct) the counting rate at the time t;

eœAt a correction for the radioactive decaybetween the measurement and the cali­bration.

(4)

1.5 Relativeef'fïciency

1.0.0.5

p'i}=

9,8 AH Q

Depth(m)

FIG. 4

The relative efficiency as a fnnction of .the depth.

Orart tuberoof 8

6

E •

Oetecter3

S z

battom 0

0

The efficiency 'i}, the determination of whichwas the primary aim of this investigation, isobtained from the expression:

(3)

C=(LC;)/Je;

1 '1'Uo L - fn Ct) dt

lJ 0

Q=

li = ( L u;)1Je = QIA,

where:

'1'and C = Je dt.

oSince the velocity and the tracer concentration

at the various areas should be essentially un­correlated, the second tenu can be neglected andthe expression (1) replaced by expression (2 a).

At each measurement a certain volume u ofthe radioactive solution was injected. The cali­bration of the instrument was made in a largevolume Vo of water, in which a volume Uo fromthe same mother solution was uniformly dissolv­ed. The expression (2), transformed into suitablequantities, is then:

MAI-JUIN 1963 - N° 3 --- SVANTESSON AND SUNDBERG-FALKENMARK 259

where P is the electric power in kW and AH isthe head of water in meters and Q is measuredin ma/s. A combination of lhe expressions (3)and (4) gives:

(5)

which can also be looked upon as a mean of kdifferent efficiency values. The nUlnerical valuesof the individual integrals were evaluated fromthe recorder papers.

The variation of the relative concentrations

over the gauging section is shown in Figure 3.No trend is observed in horizontal direction. Inverlical direction however there is a pronouncedvariation with the depth. The standard devia­tion observed at the two instruments levels, nam­ed the E- and S-levels, was 15 and 20 % respec­tively. '1'0 determine the mean efficiency for thewhole gauging section a graphical procedurewas used (Fig. 4). The efficiency thus evaluatedcoincided, within the estimated errors, with thevalue determined from conventional cUITentmeter measurements. The accurracy in this firsttrial was estimated at about 4 % (standarddeviation of the mean value).

II. - FLOW DISTURBANCES AND TURBULENCE PARAMETERS

1. Theory of dispersion in water ways. ;LI' for which case the differential equation islransformed to:

where Xl = x - Uland c represents the meanconcentration over a tube section. K representsa virtual diffusion coefficient including turbulentdispersion effects due to variations of velocityover the tube section (radial diffusion) andsecondly also effects due to longitudinal diffu­sion. K can also be interpreted as a quantitythat reflects the wall friction in the tube. Taylorhas shown that for flow through hoth smoothand rough pipes the following formula holds:

The present data, primarily intented for thedetermination of the efficiency, can also be usedfor a more detailed study of the flow throughthe conduits of the power station. Already fromthe very shape of the individu al concentration­time distributions Cc-t CllI'Ves) some informationabout the turbulence generated in the waterways can he gained. '1'0 get a deeper knmvledgeabout the turbulence, it has seemed desirahleto make a comparison with the theOl'y availahleat present, which is concerned with dispersion instraight tubes and open water ways. SOlneexperiences from this field have also been used.

Turhulent dispersion of a tracer lnaterial inwater is described by the following expression,if effects due to molecular difIusion, densityvariations, chemical and biological factors canhe lleglected:

ocol

K= ~\iyaU

(7)

(8)

where k,J'-1fkz are diffusion coefficients in thedirections x, y, Z respectively; ÏÎ is the meanvelocity and c the concentration at a point(x, y, z). The x-axis represents the direction offlow.

Taylor [10] discusses this equation appliedto dispersion in a straight uniform tuhe. Hemakes use of a reference system, moving withthe mean velocity of the flow U in the direction

oc + _ ocu-ol OX

~ (k ..§!:) + ~( k ~)ox x ox oy 11 oy

o ( OC)+ Bi k z Bi (6)

\vhere a is the radius of the pipe, ~ a numericalcoefIicient, and y the friction coefficient. For astraight pipe the theoretical value of ~, borneout by experiment, is 7.14, but for curved pipesthe value is fOlllld to be greater.

In the case of a plane, instantaneous injection,the solution can be written [11, 12]

Mc (x, t) = e[- (x-UtV!4KtJ (9)A\i41CKl

where c (x, t) is the concentration at the time lat the distance x from the injection, A the crosssection.

260 LA HOUILLE BLANCHE ------ N° 3 - MAI-JUIN 1963'

The standard deviation 0";/) of the tracer cloudat the same time t is givenby:

Experimentally plane injections are oftenhard to attain. The experiment can thereforebe simplified by making the injection in a sin­gle point. In this case the dispersion initiallyoccurs in both radial and longitudinal direction.Through the re11ection l'rom the tube walls,however, a smoothing out of the concentration,variations in radial direction is rapidly attained.This leads to a concentration distribution,approximately corresponding to the one receivedfrom a plane injection. Until such completemixing in radial direction has been reached, i.e.until this approximation is good enough, themean value of the concentration C over the sec­tion can replace c (x, t) in the expression CH).Thomas [11 J has used the same approach totwo-dimensional dispersion in natural waterways. The lnean value of the concentrationIllight be still more improved through the in­creasing of the number of injections.

The concentration distribution in expression(9) is for a constant value of t a normal distri­bution, symmetrical around x = Ul. The maxi­mum has the value:

Taylor [10J proved that in the case of flowat a constant speed through a long uniformpipe the virtual diffusion coefficient (dispersioncoefficient) for sufficiently large times can bewritten as:

(14)VU12 /U2

instance, this factor is 10-2 the difference betweenthe maximum concentrations of the two distribu­tions is 0.5 %, and between the two correspond­ing times 1 %' The difference between 0"a:2 andO"t2U2 is then 6 % [Il J.

The pUI'pose of the present calculations was,as mentioned above, not only to determine the11ow, but also to try to evaluate parameterswhich could be used to describe the turbulencein the water ways of the plant. The statisticaltheor'y of turbulence [ 11, 13, 14J oU'ers sornepossibilities. In this theor'y it is generallyassumed that the motion can be separated intoa Inean 110w and a superposed turbulent flow, themean value of which is zero. The instantaneousvelocity in the direction of 110w in a certainpoint ï's then given by n = U + !lI, where U I isthe additional velocity l'rom the mean value Uin the same point (the turbulent fluctuation).

The intensity of the turbulence can be express­ed by the standard deviation of the instanta­neous velocities. In the isotropic case, wherethe mean values in the Huee coordinate (lirec­tions of the additional velocities squared areequal, the relative intensity of turbulence isgiven by:(10)

(11)

NIcma" = ----

Av4 nKt

or:

O",c2 X1l22

SIn 2(12)

'"K = lI'2JR ('t') d"Co

(15)

,.."here Xl/2 stands for the distance, over whichhaIt' the maximum concentration is exceeded.The concentration distribution for a given valueof x has a maximum which exceeds the maxi­mum given by the expression (10) and more thelarger the ratio K/Ux is. The maximum willpass the section at a time tma", which is earlierthan the time of arrivaI of the centre of mass,tcm = (x/U).

The standard deviation O"t of the c-t distribu­tion is given by

In this case the dispersion coefficient can bewritten as:

""vere n' is the difference between the local velo­city li. and the mean velocity U of the flow,u' = II - D, and ZZ'2 is the mean square velocitydeviation. R ("C) is the correlation between thevelocity in the direction x of a partic1e at oneinstant of time and that of the same particle adefinite time "C later. The integral is a para­meter which represents the time scale of theturbulence; for sufficiently large values of "C thefunction R ("C) falls to zero and. the integralbecomes finite:

2 = 2 Ktcm (1 + 6 K)O"t U2 Ux (13)

'"JR ("C) d"C - too

(16)

The ratio tmll,,/tcm as weIl as the ratio 0"",2/O"t2U2

is a function of' the factor K/Ux. \Vhen, for

MAI-JUIN 1963 - N° 3 -- SVANTjESSON AND SUNDBERG-FALKENMARK 261

The expressions (11) and (15) combine ta: which is consistent vlith the definition (16) of t~.

It fo11ows from (19) and (21) that:

. '"O'i = 2 l[12 tIR (-ç) d-ço

(18)(22)

Before the integral has reached its final value,the correlation function R (-ç) can be evaluatedas:

For t » to this becomes :

(19)(23)

which should be compared with the expression(18) in the form :

Thus, if the standard deviation O',e is lmownas a function of time as weIl as Il'2, the relationbetween the correlation coefficient R (-ç) and thetime -ç can be determined. As far as diffusionis concerned the length 2. Analysis and results.

(*) Ralinske and Pien used x as a variable. insteadof t by replacing t by x/V.

is analogous to a nllxmg length. If the turbu­lence is superposed on a mean velocity li, thelength scale can also be expressed by the para­meter:

In the fo11ovvillg calculations, an experimentalwork by Kalinske and Pien [15] will be usedto determine the function R (-ç). In a model theseauthors studied the dispersion of materia byturbulence in a straight open channel. The tra­cer was injected in one point at mid-depth.Theil' investigation showed that the distributionin vertical direction downstream the point ofinjection was a Gaussian, which is in agreementwith an expression of the same type as expres­sion (9). They were able to evaluate local valuesof the coefficient of dispersion from the rela­tion between the vertical tracer spread and thedistance from the section of injection. Thisdispersion coefficient was also evaluated in ver­tical sections near the point of injection, wherethe integral in equation (16) has not reachedthe final value t~. Using expression (19) theyshowed that the correlation coefficient R, as hadbeen pointed out ·by for instance Dryden [16],varied as (*) :

a) FLOW DISTUHBANCES:

Thus, during this time interval, the meanconcentration in the two levels varies in thesame way. From Figure 6 it is obvious, thatthe activity in the E-Ievel 10 minutes artel' theinjection still exceeds the natural backgroundcontrary ta the situation at the S-level. Thus,the taiIs must have considerably different lengthsat the two levels. Furthermore strong fluctua­tions in the concentration are visible in theE-taiIs in Figure 5. The priIllary data showedthat this was true even 10 minutes artel' injec­tion.

As already mentioned, a total of 40 measure­ments were performed over the gauging sectionin order ta compensate for the incomplete mix­ing in this power plant. An idea about the stateof mixing has already been given when Figure 3was discussed. From this it is evident, thatthere is a significal1t difference in the relativemean concentrations at the two instrumentlevels, the average of which are 1.16 for the E­and 1.82 for the S-level. This indicates a diffe­rent character of the flow observed by the twodetectors.

On the other hand, no trend can be observedin the concentration values at a single level.Nor do the times of arrivaI of the individualtracer clouds exhibit a certain trend. It shouldtherefore be justified to add the c-t curves overthe E- and S-levels respectively. This sum-curvefor the S-level is representated in Figure 5.The corresponding E-curve, normalized to themaximum of the S-curve, is shown in the samefigure. When compared, the shapes of the twocurves show tobe practically congruent fortimes less than about three minutes.

(21)

(20 a)

(20 b)L =lito

R = e-t/to

4

262 LA HOUILLE BLANCHE N° 3 - MAI-JUIN 1963'

Mean concentration as a function of the time after theinjection fOl' both E- and S-Iayer. The E-values have

been normalized on the maximum of the S-values.The carves represcnt fitted Taylor distributions.

Thus, on one hand the shape of the sum­curves for the hvo layers agreed and on theother hand a long-lasting fluctuating activitywas observed in the E-Ievel. From this it hasbeen concluded, that the E-deteclor simultane­ously registered the concentration in two differ­ent layers: a lower layer, the flow of \vhich wasmuch like the flow observed by the S-detectorand an upper layer with strong turbulence andslow water exchange with the Io-wer layer. Theposition of the boundary zone between the twolayers must be near the middle of the draft tube,as can be seen from the Yj-depth curve in Fi­gure 4. Another estimate Can be made from thedefinite Yj-value combined with the Yj-value forthe S-layer, under the assumption that theupper layer did not contain any activity; thi:>indicates a boundary to be about 1 m above the

Relativeconcentrnion

7000

6500

16000 1

11

5500 ~

"5000

4500

4000

3500

3000

250011

2000 l,1

11500 J

1

1000 1

1

1

5001

111

00 20 40

1

1

1

1

1

\1

1

1

1\

\\

\\

60

Notation: • S-{evelx E-Ievel

- Expression 9K·ZO m'YsU· 2,6m/,X·46 m

- - - Expression 9K- 5 m0U' 2,2m/sX'76 m

M fOO 120 fI,() 160 IBO 200 secTime

FIG. 5

E-detector. The boundary zone if extrapolatedupstream would touch the ceiling of the drafttube at an angle of 5-10degrees. Thus, it is afairly large part of the draft tube in this powerplant that is not effectively used.

It is also possible to study the influence onthe flo\V from the vertical wall, which dividesthe draft tube in hvo parts (Fig. 2). The clock­wise rotating turbine axis is displaced from theaxis of symmetry towards the right part ofthe draft tube. Differences in the flow throughthe two parts are evident already from the acti­vitY concentrations in Figure 6. The activity inthe E-Ievel of the left part of the draft tubeabout 10 minutes after the injection is consider­ably higher near the dividing wall than else­where. As might be expected from Figure 2, nosuch disturbance can be observed on the rightside of the wall. At this time, the activity in theS-level has decreased ta the bacl,ground level,due to the more active water exchange in thebottom layer. If, however, the sum curves forthe right and the left part in this level (Fig. 7)are considered it is clear that the activity levelfrom about H minutes after injection is higherin the left than in the right part. This indicatesthat the disturbance from the dividing wallobserved in the E-Ievel also occurs in the bottomlayer.

From this interpretation of the data it isobvious, that the shape of the ceiling as weIlas of the dividing wall generates intensive tur­bulence.

b) FLOW PAHAMETERS:

In the following an attempt \vill be Iuade tadescribe the flow in the power plant conduitsby evaluating some parameters from a fictitiouspipe.

From hydraulic considerations il can beexpected that the main turbulence is generatedwhere the aeeeler:::l1.ion of the flow passes into aretardation [17]. The geometrical dimensionsof the water way show, that this takes place inthe turbine 46 m upstream the gauging section.As a matter of fact, the point of injection waschosen as to let the tracer pass into the tnrbinewith a minimum of spread. Vnder these cir­cums tances, the spread of the tracer upstreamthe turbine can be neglected and the injectionthought of as taking place at the turbine. Theaverage time required for the transportation ofa tracer cloud from this place to the gaugingsection has, from the flow and the mean area ofthe water way been calculated at 18 sec.

When calculating the flow parameters thecondui t Oll the downstream side· of the turbine

MAI-JUIN 1963 - N° 3 --- SVANTESSON AND SUNDBERG-FALKENMARK 263

eS

right

rrelative

activity

110 9 8 7 6

leTt

Detector

Vertical

FIG. 6

Concentration of the radioactivity af the mouthof the draft tuLe 10 min:s after the injections.

Thc effect of the natura! radioactivity is included.

RelativeConcentration

4000

3500

3000

Notation: G {ef't port

" right port

2500

2000

~ "

FIG. 7

Mean concentrationas a function of the timeafter the injectionfor rightand left partof the draft tube.

o

ID 20 30 40 50 60 70 80 80 100 110 120 130 140 150 sec.Time

~

oi-~,---,-.L,-~--r~-r-~-'-~--'--~.....-r-r--r---.-r-,---,--y-~"''---'-'-r-r---.----,o

500

fOOO

1500

264 LA HOUILLE BLANCHE -----~ N° 3 - MAI-JUIN 1963

is thought of as replaced by a fictitious tube,the lenght of which agrees with the distancefrom an assumeù "point of injection" in theturbine to the gauging section.

Turbulence parameters can be determinedboth from the individual tracer clouds and fromthe sum cloud.

The turbulence in the pipe consists of eddiesof aIl dimensions from the scale of the pipedownwards, but the spread of an individualcloud can be expected to be eaused mainly byeddies of sizes up to that of the elouù itself(much laI'ger eddies only carrying the wholecloud along). With this fact in mind the disper­sion coefficient ohtained from individual tracerclouds can he described as corresponding to"small-scaie eddies":

(24)

Here lI'. = II lIcloud is the velocity deviationfrom the mean velocity lIcloud = Xli of the cloudand tO,8 is the eorresponding time scale [cf.

equations (16) and (17)]; X is the distance(= length of the fictitious pipe) and i the timeof travel. Owing to lm'ger eddies i anù lIcloud

will vary from one cloud to another. The dis­persion coefficient of the sum cloud, with meanvelocity D, can then he written [18J:

where 1I't = llcloud - D, the index 1 referring tolarge-scale eddies. Of course the tenus "smaIl­seale edùies" and "large-seale eddies" are hereused in a rather loose sense.

In order to detennine the turbulence para­meters for the small scale eddies, Ks and io,. therelative spreaù values of the individual tracerclouds have heen plotted against the thnes ofarrivaI (modal times) in Figure 8. (Ji was there­by eaJculated from the expression (12) by meansof the haU values of the c-i elIrVes. This means,that the peaks of the c-i curves were Teplacedby Gaussians, which gives the same result asthe expession (13) when the second tenu of the

~-----------------

250

Notation: G individual tracer clouds

0sum cloud

200

150

'100

50

\small seale eddies

1

00 ? 1• >-

10 20 • ...-0,.,.· 40 60 70 80 90 lOO,,!

1----.l..-.,..--0.,-=-L.~-·--2b,.LI--J'h....J~O---40rl-LI--5'~....JI---6b,.L1 --7,iJ-lI---ao'-l-LI--8,~--J/~_0--IOrl~.l.fO__,,..,';~

tmax

FIG. 8

Variance of x, as determined in the difTerent concentration distributions measuredin the S layer, as a function of the mode. The standard rleviation has been measllredin the scale of the distance from the turbine to the gauging section. The curve repre-

sents the expression 22. Concerning the dashed lines cf. the text.

MAI-JUIN 1963 - N° 3 -- SVANTiESSON AND SUNDBERG-FALIŒNMARK 265

1.0 ------

correction factor is neglected. The influence ofthe reflexion from the walls of the draft tube onthe spread is by this approach minimised. Thetwo parameters could then be evaluated by fitt­ing the spread function in the expression (22) to

The correlation R (c) of the small scale eddiesin the fictitious pipe describing the draft tube.

o

/0

o 10

tmax-Ia-?:

FIG. 9

2D JO

50 60

40

the individual tracer spread values. Because ofthe spread of these data this has been perform­ed by dividing the cluster of points in two parts,tHting a spread function to the means. Thetime scale tO• 8 then beeing lmown, the lengthscales, the velocity variance u? and the relativeintensity of the flow can be determined from theexpressions (20), (17) and ('14) respectively. Theparameter values are given in table l, whereasthe correlation function corresponding to thesmall scale eddies is shown in Figure 9.

Information concerning the large seale eddiescan be gained from a study of the sum curve.A fittîng of a Taylor distribution [expression(9)] to the peak of the experimental sum C1Uvedetermines the dispersion coefficient K and themean velocity U for the flow in the fictitiouspipe. The values thus ohtained are found iùtable 1 and the correspondence between the twocurves is fairly good, as can be ~een from Fi·gure 5. The conformity does apparently notinclude the "taiIs," i.e. the extended later partsof the c-t curves. These are therefore attributedto such boundarv effects in the conduits as notallowed for in the deduction of the theoreticalexpression. The K-value just determined lends

TABLE l

Flow parameters of a fictitious pipe simulating the flow through the power plant

PIPE LENGTHPARAMETER SYMBOL PIPE LENGHT=ACTUAL SPHEAD DISTANCE= = LENGTH OF WHOLE

46m water way=76 m

Mean velocity ........... U 2.6 mis 2.2 misReynolds number ........ lR- 2.7.107 2.3.107

Dispersion coefficient ofthe sum cloud = plane

1injection. .. ..... . K1

20 m2/s 5 m2/sFriction coefficient o ••• y 0.045 0.004Standard deviation of the 1

sum cloud ............. Cf;]] 24m

SMALL SCALE EDDIES LARGE SCALE EDDIES

Dispersion coefficient .... K 3 m2/s 17 m2/sTime scale .............. to 5 s 17 sLength scale ............ l 4 m 17 m

- •••••••••• 0·0L 14 m 44 m

Standard deviation ofcloud. ......... (j';]] 7 m -

Velocity variance ....... U'2 0.5 m2/s2 1.0 m2/s2

Relative intensity ........ Vu'230 0/0 40 0/0--

U

266 LA HOUILLE BLANCHE N° 3 - MAI-JUIN 1963'

itself to the determination of the friction coeffi­cient of the straight fictitious pipe [expression(8) J. The value calculated is about 25 times thevalue that could be expeeted for smooth flow ofthe same Reynolds number. The variance lI'/-'can be determined from the spread of the timesof arrivaI of the individual clouds (modal times).The time scale tO,1 can then he calculated fromthe expression (24) and the dispersion coeffi­cient KI from the expression (25). The valuesreceived are given in table 1.

Ii is also of interest to compare the standarddeviation 0"", with the largest dimension of thedraft tuhe, in this case about 18 m. For a singlecloud 0",,, is abou t 1/3 of this '\vidth, whereas forthe sum curve 0",,, is about the same. Ii can heseen from Figure 8, that a dispersion time of110 sec. would have heen required in the case ofa point injection ta reach a spread as large asthat of the snm curve if effects due ta wall reflec-

tion are neglected. This corresponds to a flowdistance of ahout 300 m. Even if the wall reflec­tion would have been ef1'ective in reducing thisdistance, it is apparent that the use of severalinjections made it possible to simulate in astretch of 46 m a mixing quite as complete asthat ohtained at a much larger distance after asingle point injection.

For a comparison with other power plants itmight be of interest to lmow the K-value of thesum CIIrVe and the friction coefficient for a ficti­tians pipe of the same length as the actualmeasuring distance, 76 m. These values are alsoshmvn in table 1.

The friction coefficient for this longer pipe isonly twice as large as it would have heen if theflow had been smooth. The fit in this case isnot as good as when the spread was looked uponas starting in the turbine as can be seen fromFigure 5.

III. - DISCUSSION

A discussion about the relative dilution me­thad is essentially a question concerning themixing in the ,vatel' ways. The resuIts of thepresent investigation indicate, that in a hori­zontal layer at the end of the draft tuhe themean concentration observed in a gauging pointis subject ta fluctuations of statistical nature.This is shown also by the smoothness of thesunI curves for each level. As a consequence,the average of the local mean concentrationsrepresents the mean value for the level in ques­tion. If the fluctuations are statistical, how­ever, the same average value '\'lOuld be obtainedby adding mean concentrations from severalmeasurements in one single point. This meansthat the time necessary for determining a singlefla\\' value can he considerably hrought downsince the detectors, in the experimental arrange­ment used, can be moved fI'am one vertical taanother only when the turbine is unloaded.

In the actnal plant, measurements performedin different levels give dif1'erent mean concentra­tions. This might OCCUl" even in other plants.If so, the representative mean concentration hasto he measured for a sufficient numher of levelsto guarantee knowledge about the depth depen­dence.

In the present investigation, where the mix­ing was most incomplete, the standard devia­tian of a single measnrement was between 15and 20 %' This includes, beside a statisticalerraI' in the counting rate, also errors caused

by the instrumentation. The total standarddeviation of the calculated mean of the 40 mea­surements was estimated ta 4 %'

Ii can he mentioned that a second investiga­Uon using the same technique was carried outrecently in another power plant, equipped withFrancis turbines and with a head of ahout70 meters. The distance between the injectionand the gauging section was about 20 diameters.In this case, the standard deviation was 6 %mainly due ta the much better mixing condi­tions but also ta the considerably improvedapparatus used. \Vith 9 injections and twodetectors the inaccuracy of the average washrought down ta 1.4 %' It might be added, thatno vertical variation in the concentration wasohserved. This meuns, that if the numher ofinjections is increased by a factor 2, the erraI'should he about 1 %'

The measurement of aIl the quantities deter­mining the Q-value in expression (3) can beimproved sa that the inaccuracy becomes of theorder of one pel' mille or less. The accuracyattainahle '\vith this method for measuring theflow would therefore remain mainly a questionof enlarging the number of measurements. Thiscan be made by increasing the numher of injec­tions or that of the detectors.

In this connection it can be mentioned thatthe flow also is determined from the ratio be­tween the volume of the effective water way andthe mean time of fla,,,. In the present investi-

MAI-JUIN 1963 - N° 3 ---. SVAN'DESSON AND SUNDBERG-FALIŒNMARK 267

gation this value agreed weIl ,vith the valuedetermined frOIn the dilution method used.

The technique of using radio isotopes used inthe present investigation gives, together withthe flow value, considerable information aboutthe character of the flow in the plant, whichmakes it possible to localize disturhances in thewater ways. The consiructor of the power plantis herehy given a possihility to check the qua­litY of the conduit shapes more exhaustivelythan is possible in model studies. It shouldhowever be observed, that in investigationswhere the primary aim is to localize such dis­turhances, the points of injection and the gaug­ing points should he chosen with special regardto this.

The analysis of flow parameters is performedby replacing the ,vatel' ways through the powerstation hy a straight tube, for which a disper­sion coefficient is determined. This has heenlIsed to get a measure of the qllality of thewater \Vay shapes, expressed as a friction coeffi­cient. It might he mentioned that this coeffi­cient in the plant in question, where the effi­ciency ,vas extremely low, was much laI'ger thanthat for smooth pipes. For the plant recentlyinvestigated the preliminary data ohtained incli-

cate that the friction coefficient was essentiallythe same as for a smooth pipe. The efficiency\Vas in this case extremely high.

From the spread of the individual tracerclouds as weIl as of the sllm cloud it has beenpossible to evaluate some turbulence parametersrepresenting smaII scale and large scale eddiesrespectively.

Acknowledgements

The authors ,,,ish to express their gratitudeto the Swedish Meteorological and HydrologicalInstitute, particularly to the chief of the Hydro­logical Department, Gunnar Nyhrant. The Swe­dish Power Board supported the part of thepresent stndy concerned with the determinationof the discharge. Thanks are also due to thepersonnel at the power station for their splendidassistance. \Ve are indehted to S.B. Nilsson,D. phil., Dep. of Mathematical Physics, The Ins­titute of Technology of Lund, for valuable dis­cussions on the theoretical part of this paper.The valuable collaboration in the experimentalpart hy Captain Claes Ljungdal and in the ca1cu­lation by Ml' Erik Nyherg is also announced.

REFERENCES

[10] TAYLOI\ (G.I.). - The dispersion of matter in tur­bulent ilow through a pipe. The scientific papersof Sir Geoffrey Ingram Taylor, pp. 466-488 Ed.by G l{ Batchelor Cambridge 1960.Also : Proc. Roy. Soc., A, vol. 223 (954), pp. 446­68.

[11] THOMAS (I.E.). - Dispersion in open channel flow.Dissertation. iVorthwestern University, Chicago,Ill., 1958.

r12] MOSETTI (F.). - Su alcuni criteri ed esperienze pel'misurazioni idrologiche mediante sostanze trac­cianti. L'Energia Eletlrica, 37 (1960), n° 9, Milano.

[13 J LIN (C.C.). - Statistical theory of turbulence,Princeton, New Jersey 1961.

[18] NILSSON (S.B.).

[1] l{OLUPAILA (S.). - Bibliography of hydrometry.Notre Dame,. Indiana 1961.

[2] BARBAGELATA (A.). - Il metodo ehimieo-elettricopel' la misura delle portate dei corsi d'acqLuLL'Elettrotecnica, 13 (1926), nO 5, Milano.

[3] BOTTANI (E.). - Teoria, errori ed alyprossimazionedeI metodo chimico-elettrico pel' la misura delleporta!te. L'EleUrotecnica, 13 (1926), nO 5, Milano.

[4J AASTAD (J.) et S:aGNEN (R.). - Ny metode for best­emmelse av vannfffringen i naturlige og kunstigevannl:ap. Teknisk Ukeblad 4·6 (1928), nO 29, Oslo.

[5] AASTAD (J.) et S'0GNEN (R.). - Discharge nwasure­ments bv means of a salt solution, "the rela­tive dilu"tion method". Association Internationaled'Hydrologie Scientifique, Assemblée générale deRome 195·~, vol. 3.

[6] HULL (D.E.). - Total-count technique: a llew prin­ciple in ilow measurement. lntcrn. JOZll'n. Appl.Rad. & Isotopes, 4· (958), n° 1/2, London.

[7] SVANTESSON (N.L.) et SUNDIlEI\G-FALKENMAI\K (M.). ­Spridningsproblem i sjüar oeh vattendrag. Tek­nisl, Tidskrift (1959): '12, Sto<'1dlOlm.

[8] ANDHl~ (H.). - Méthode chimique de dilution. Pro­cédé par intégration. La Houille Blanche, n° spé­cial B, 1960, Grenoble.

[9J \VOLF et Pl~I\EZ. - Quelques applications dc laméthode de dilution à des mesures de débit enusines hydroélectriques. La Houille Blanche,nO spécial A, 1961, Grenoble.

[.1 4J

[15]

[16]

[17]

TAYLOR (G.I.). - Diffusion by continuous move­mellts. Ref. nI'. 10, pp. 172-184.A]so: ['l'OC. London Math. Society, Sel'. 2, 200H21), pp. 1%-212.

K~LINSIΠ(A .A.) et PIEN (C.L.). - Eddy diffusion.Industl'ial ancl Engincering Clwmistry, 86, 220(944).

DUYDEN (H.L.). - Turbulence and diffusion. lndus­trial and Engineering Chemistry, 31, 416 (1939).

GERBER (H.). - \Vassermessungen nach dem Salz­verdünllungs-Verfahren. Schweizerische Bauzeit­ung, 65, p. 524 (1947), Zürich.

Private communication.

268 LA HOUILLE BLANCHE

RËSUMÉ

-----~ N° 3 - MAI-JUIN 1963'

Depuis la fin du XIX' siècle, la méthode de dilution a été utilisée pour la détermination desdébits des fleuves et des usines hydroélectriques. En 1958, des traceurs radioactifs furent utiliséset la méthode prit pour nom: méthode de comptage total.

Le débit est donné par l'expression:

Q=M

T.f cdto

où Q est le débit cherché;

M, la masse du traceur injecté;

c, la concentration du traceur à l'instant t;

T, le temps pendant lequel, le nuage passe dans la section de mesure.

De tels essais sont faits par l'Institut Météorologique et Hydrologique de Suède depuis 1958,et un des essais dans une usine équipée de turbines Kaplan. La chute était de 12 m et le débitd'environ 200 mals (voir fig. 1 et 2); le traceur utilisé était du 24Na2COa. Les injections, de 15 milli­curies chacune, se faisaient dans les pertuis d'entrée et les concentrations à la sortie du diffuseurétaient déterminées par deux compteurs à scintillations et leurs évolutions enregistrées; 10 verti­cales ont été explorées en deux points chacune; chaque point étant doublé, il y eut en tout40 injections.

A partir des courbes (concentration en fonction du temps) enregistrées, il a été possibleT

de déterminer en chaque point le .f cdt correspondant et, en faisant leur moyenne, de calculero

le débit et le rendement correspondant (fig. 3, 4, 5, 6, 7).L'écart-type des distributions gaussiennes des concentrations a été calculé. Il était de 4 %.

On a pu constater une différence très importante entre la moyenne des concentrations relativesobtenues dans les deux parties du diffuseur: respectivement 1,16 et 1,82. De même, les tempsde passage des nuages étaient très nettement différents.

Les valeurs des concentrations en des points d'un même plan horizontal ont lllontréun caractère statistique; leur moyenne représen tait donc bien la valeur moyenne de la concen­tration par palier horizontal.

Les caractéristiques hydrauliques (voir tabl. 1) ont été tirées d'expériences faites en rem­plaçant l'ensemble où s'opéraient les mesures par un tuyau circulaire fictif de même longueur etd'un diamètre moyen tenant compte de toutes les caractéristiques géométriques.

Selon un type de calcul donné par G.1. Taylor, les auteurs ont pu déterminer d'aprèsleur mesure de concentration les valeurs de divers paramètres caractérisant la turbulence del'écoulement (voir le tab!. 1 et les fig. 8 et 9).