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SCALE EFFECTS ON THE FORMATION OF VORTICES AT INTAKE STRUCTURES

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

FERHAT ARAL GÜRBÜZDAL

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

CIVIL ENGINEERING

SEPTEMBER 2009

Approval of the thesis:

SCALE EFFECTS ON THE FORMATION OF VORTICES AT INTAKE STRUCTURES

submitted by FERHAT ARAL GÜRBÜZDAL in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering Department, Middle East Technical University by, Prof. Dr. Canan Özgen Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Güney Özcebe Head of Department, Civil Engineering Prof. Dr. Mustafa Göğüş Supervisor, Civil Engineering Dept., METU Examining Committee Members Prof. Dr. Nevzat Yıldırım Civil Engineering Dept., Gazi University Prof. Dr. Mustafa Göğüş Civil Engineering Dept., METU Assoc. Prof. Dr. İsmail Aydın Civil Engineering Dept., METU Assoc. Prof. Dr. M. Ali Kökpınar TAKK Dept., State Hydraulic Works Inst. Dr. Mete Köken Civil Engineering Dept., METU Date: 10.09.2009

iii

I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Ferhat Aral GÜRBÜZDAL

iv

ABSTRACT

SCALE EFFECTS ON THE FORMATION OF VORTICES AT INTAKE

STRUCTURES

Gürbüzdal, Ferhat Aral

M.Sc., Department of Civil Engineering

Supervisor: Prof. Dr. Mustafa Göğüş

September 2009, 50 pages

In the present study, possible scale effects on the formation of air-entraining

vortices at horizontal intakes are studied experimentally. Basic dimensionless

parameters that govern the onset of vortices at a horizontal intake in a model

and a prototype are stated by dimensional analysis. Series of experiments are

conducted on four intake pipes of different diameters located in a large

reservoir.

The relationship of critical submergence ratio with other dimensionless

parameters is considered for a given Froude number and it is found out that

the critical submergence ratio is affected by model length scale ratio and its

natural result of side-wall clearance ratio and Reynolds number differences

between model and prototypes. It is observed that, side-wall clearance ratio is

not effective on the critical submergence ratio after it exceeds about 6. In

addition to this, Reynolds number limit, beyond which viscous forces do not

affect the vortex flow, is found out to be increasing with the increase in Froude

number.

v

An empirical relationship, which gives the critical submergence ratio as a

function of Froude number, side-wall clearance ratio and Reynolds number is

obtained by using data collected in the experiments.

Keywords: Intake Structure, Air-Entraining Vortex, Critical Submergence,

Model Scale Effects, Side-Wall Clearance.

vi

ÖZ

SU ALMA YAPILARI ÖNÜNDEKİ VORTEKSLERİN OLUŞUMUNA

ÖLÇEK ETKİLERİ

Gürbüzdal, Ferhat Aral

Yüksek Lisans, İnşaat Mühendisliği Bölümü

Tez Yöneticisi: Prof. Dr. Mustafa GÖĞÜŞ

Eylül 2009, 50 sayfa

Bu çalışmada yatay ağızlı su alma yapılarında oluşan hava girişli vorteksler

üzerindeki olası model ölçeği etkileri incelendi. Model ve prototiplerdeki yatay

ağızlı su alma yapılarında oluşan vortexin başlangıcını etkileyen boyutsuz temel

parametreler boyut analizi ile ortaya konuldu. Büyük bir rezervuara

yerleştirilmiş dört farklı çaptaki su alma ağzı ile deneyler gerçekleştirildi.

Model ve prototipler üzerinde sabit Froude sayısı altında kritik batıklık oranının

diğer boyutsuz parametrelerle ilişkisi incelendiğinde, kritik batıklık oranının

model büyüklüğü oranı ve bunun doğal bir neticesi olan yan duvar yakınlık

oranı ile Reynolds sayısındaki farklılıklardan etkilendiği görüldü. Yan duvar

yakınlığının yaklaşık 6’ dan büyük olduğu durumda etkisiz olduğu gözlendi.

Buna ek olarak viskoz kuvvetlerin, vorteks akışını etkilemediği Reyolds sayısı

limitinin artan Froude sayısı ile yükseldiği tespit edildi.

Deneylerde elde edilen veriler sonucunda kritik batıklık oranını Froude sayısı,

yan duvar yakınlığı ve Reynolds sayısının bir fonksiyonu cinsinden ifade eden

deneysel bir bağıntı bulundu.

Anahtar Kelimeler: Su Alma Yapısı, Hava Girişli Vorteks, Kritik Batıklık, Model

Ölçeği Etkileri, Yan Duvar Yakınlığı.

vii

To my Granddady...

viii

TABLE OF CONTENTS

ABSTRACT…………………………………………………………………………………………………….. iv

ÖZ…………………………………………………………………………………………………………………. vi

DEDICATION…………………………………………………………………………………………………. vii

TABLE OF CONTENTS…………………………………………………………………………………… viii

LIST OF TABLES……………………………………………………………………………………………. x

LIST OF FIGURES…………………………………………………………………………………………. xi

LIST OF SYMBOLS………………………………………………………………………………………… xii

ABBREVIATIONS..………………………………………………………………………………………… xiv

CHAPTER

1. INTRODUCTION…………………………………………………………………………………. 1

1.1. Introductory Remarks on the Intake Vortex……………………………. 1

1.2. Scope of the Study……………………………………………………………………. 6

2. LITERATURE REVIEW…………………………………………………………………………. 7

3. MODELLING OF AIR-ENTRAINING VORTICES…………………………………… 17

3.1. Introductory Remarks………………………………………………………………. 17

3.2. Dimensionless Parameters……………………………………………………….. 17

3.2.1. Influence of Kolf Number……………………………………………… 19

3.2.2. Influence of Weber Number…………………………………………. 20

3.2.3. Influence of Reynolds Number……………………………………… 20

3.2.4. Influence of Froude Number…………………………………………. 20

3.2.5. Influence of Model Length Scale…………………………………… 21

4. EXPERIMENTAL EQUIPMENT AND PROCEDURE………………………………… 23

4.1. Experimental Equipment…………………………………………………………… 23

4.2. Experimental Procedure……………………………………………………………. 24

4.3. Observations……………………………………………………………………………… 26

5. RESULTS AND DISCUSSIONS……………………………………………………………. 28

5.1. Introduction………………………………………………………………………………. 28

5.2. The Relation between Dimensionless Parameters and

Submergence Scale Ratio…………………………………………………………..

29

5.2.1. The Relation between Sc / Di and b / Di ……………………… 31

5.2.2. The Relation between Sc / Di and Re ………………………… 32

ix

5.2.3. The Relation between (Sc / Di)r and Lr ………………………… 34

5.2.4. The Relation between (Sc / Di)r and Rer ……………………… 36

5.3. Verification of Results……………………………………………………………….. 38

5.4. Curve Fitting…………………………………………………………………………….. 40

6. CONCLUSIONS…………………………………………………………………………………… 42

REFERENCES..………………………………………………………………………………………………. 44

APPENDICES

A. EXPERIMENTAL RESULTS..................................................... 46

x

LIST OF TABLES

TABLES

Table 5.1. Summary of Experimental Study..................................... 28

Table 5.2a. Relation between Dimensionless Parameters and 29

Submergence Scale Ratio for 0,65 ≤ Fr ≤ 0,90....................

Table 5.2b. Relation between Dimensionless Parameters and

Submergence Scale Ratio for 1,00 ≤ Fr ≤ 3,00....................

30

Table 5.3. Comparison between the present study and study 39

of Gordon (1970)..........................................................

Table A.1. Experimental Results of Pipe 1........................................ 47

Table A.2. Experimental Results of Pipe 2........................................ 48

Table A.3 . Experimental Results of Pipe 3........................................ 49

Table A.4 . Experimental Results of Pipe 4........................................ 50

xi

LIST OF FIGURES

FIGURES

Figure 1.1. Vorticity Sources (Durgin & Hecker, 1978)...................... 2

Figure 1.2. Directional and Structural Classification of Vortices

(Knauss, 1987) ...........................................................

3

Figure 1.3. Relative Comparison of Eddy-Dimple-Vortex Tail.............. 4

Figure 1.4. ARL Vortex Type Classification (Knauss, 1987)................. 5

Figure 2.1. Critical Spherical Sink Surface Approach

(Yıldırım and Kocabaş, 1995).........................................

16

Figure 3.1. Modelling Elements for Experimental Intake.................... 18

Figure 4.1. Experimental Equipment............................................... 25

Figure 4.2. Swirl Pattern During Experiments................................... 27

Figure 5.1. Relation between b / Di and Sc / Di

for 0,65 ≤ Fr ≤ 1,50......................................................

32

Figure 5.2a. Relation between Re and Sc / Di

for 0,65 ≤ Fr ≤ 0,90......................................................

33

Figure 5.2b. Relation between Re and Sc / Di

for 1,00 ≤ Fr ≤ 1,50......................................................

34

Figure 5.3a. Relation between Lr and (Sc / Di)r

for 0,65 ≤ Fr ≤ 0,90......................................................

35

Figure 5.3b. Relation between Lr and (Sc / Di)r

for 1,00 ≤ Fr ≤ 1,50......................................................

36

Figure 5.4a Relation between Rer and (Sc / Di)r

for 0,65 ≤ Fr ≤ 0,90......................................................

37

Figure 5.4b. Relation between Rer and (Sc / Di)r

for 1,00 ≤ Fr ≤ 1,50......................................................

37

Figure 5.5. Data Fit Plot................................................................ 41

xii

LIST OF SYMBOLS

a Intake gate height

B Headrace channel width

b Side-wall clearance

b1 Horizontal distance from the center of the intake to the right-side-wall

of the reservoir

b2 Horizontal distance from the center of the intake to the left-side-wall of

the reservoir

c Vertical distance of the intake to the bottom of the reservoir

Cd Discharge coefficient of the intake in a uniform canal flow

Di Intake diameter

Fr Intake Froude number

g Gravity acceleration

H Vertical intake submergence

K Viscous correction factor

Ko Kolf number

K1 Viscosity and circulation correction factor

k Constant = 6 x 10-5

L Distance from the headrace entrance to intake center line

Lr Model length scale ratio

Nν Ratio of intake Froude number to intake Reynolds number

NΓ Circulation number

NΓ* Submergence circulation number

Qi Intake discharge

R Correlation coefficient

Re Intake Reynolds number

ReR Radial Reynolds number

r0 Shadow radius of vortex at the bottom of canal

Sc Critical submergence at horizontal intakes

U∞ Velocity of uniform canal flow at the upstream of intake

Vi Intake velocity

xiii

x Regression variable

y Regression variable

z Regression variable

We Intake Weber number

α Approach flow angle

Γ Circulation

μ Viscosity

ν Kinematic viscosity

ρ Density of the fluid

σ Surface tension

xiv

ABBREVIATIONS

ABS Absolute

ARL Alden Research Laboratory

kW Kilowatt

SI International System of Units

1

CHAPTER 1

INTRODUCTION

1.1. Introductory Remarks on the Intake Vortex

Modern society is facing a serious challenge to meet increasing water demands

for power generation, irrigation, domestic and industrial supply from seas,

lakes, rivers or simply from reservoirs through intakes. Similar to many other

civil engineering structures, design of intakes is an optimization process which

has two dependency; minimizing cost together with maximizing the ease of

operation or satisfying the design requirements. It is the susceptibility of the

intakes to the formation of air-entraining vortices that determines the design

criteria. The position of an intake should be so arranged that under the most

critical scenario, that is to say operating when the reservoir is at dead or at

minimum storage level, water level should be well above the intake so as no

vortices occur. On the other hand, the intake must be located close to the

water surface so as to reduce the cost of construction.

Distance between the free surface and the intake is known as submergence.

When this submergence drops to a critical level which is known as “critical

submergence”, air-entraining vortex starts to occur at the free surface. For an

effective intake, submergence should be large enough to prevent inducing air-

entraining vortices extending from the free surface down to the intake

entrance, which could lead to serious problems such as increased head losses,

disturbed flow pattern before the intake and hence reduced flow rate by air

ingestion through the opening formed at the core region of the free surface

vortex. Further, this air ingestion can cause mechanical damage like cavitation,

vibrations, loss of pump and hydraulic turbine efficiency and operational

difficulties by the suction of floating debris pulled down by free surface

vortices.

2

Among the many vorticity sources in literature, Durgin & Hecker (1978)

defined three fundamental types as shown in Figure1.1. Vortices are mainly

triggered by: the eccentric orientation of the intake relative to a symmetric

approach flow, the viscous induced velocity gradients with the flow boundary

itself a vorticity source and the tendency of obstructions to form rotational

wakes.

Figure 1.1. Vorticity Sources (Durgin & Hecker, 1978)

In the literature and practice, many intake arrangements can be seen. In order

to present a clear classification, two distinctions may be proposed. The first

one is a distinction related to intake direction. The second one is a structural

distinction that considers whether an intake is located in the floor or on the

walls of the basin or is projecting into the reservoir or the sump. Figure 1.2

illustrates this classification of intakes based on directional and structural

distinctions.

Classification of the intake vortices according to their strength can be done by

using some visual techniques or by measuring some quantities directly or

indirectly related to the strength of vortices. For the latter case, changes in

intake discharge coefficient, the magnitude of inlet pipe flow swirl or the

determination of air ingestion can be used. However even for an air core

vortex, if small compared to the inlet, it may not produce effects that can be

measured reliably and the other one is that, correlation between vortices and

selected dependent parameter may be weak and variable with other

parameters. The most obvious way of determining the type and hence the

severity of a vortex is by visual observation. Before stating these vortex types,

various words such as swirl, eddy, dimple and vortex tail should be defined.

3

Figure 1.2. Directional and Structural Classification of Vortices (Knauss, 1987)

Eddy, dimple and vortex tail are all used to describe the appearance of water

surface and also to denote the degree of vortex air core development. A

quantitative distinction has not been made between an eddy and dimple or a

dimple and vortex tail. However, in a qualitative sense, the depression of an air

cavity downward from the water surface is greater for a vortex tail than for a

dimple, and the depression of a dimple is greater than an eddy as seen in the

Figure 1.3.

4

The depression of an eddy or swirl is very slight and is observed only by

reflection of light from the water surface, whereas, a dimple and a vortex tail

are more readily seen due to obvious depression by naked eye.

Figure 1.3. Relative Comparison of Eddy-Dimple-Vortex Tail

A visual classification of vortices consists of following stages (See Figure 1.4):

1) A weakly developed vortex with no air core and only a small eddy on the

water surface indicates presence of the vortex.

2) A coherent surface swirl turns into a small depression on the free surface.

3) Type 3 vortex is the one with a tail which is non-air-entraining, and air

bubbles are not dragged from the tail. Dye placed in the vortex tail is

carried downward into the intake forming a filament which reveals location

of the vortex axis.

4) In type 4, vortex is so strong that it can ingest more buoyant particles such

as floating trash but not air.

5) A partially developed air-entraining vortex does not have a continuous air

core. The air core extends only part way down from the water surface and

end with a vortex tail. Occasionally, small bubbles may be dragged from

the vortex tail, travel down the longitudinal axis of the vortex, and enter

the intake.

5

Figure 1.4. ARL Vortex Type Classification (Knauss, 1987)

6

6) A fully developed air-entraining vortex has a continuous air core extending

from the water surface into the intake. Near the water surface the air core

has a funnel shape and below the water surface a rope-like appereance.

1.2. Scope of the Study

The scope of the study is to scrutinize possible scale effects on the formation of

air-entraining vortices at horizontal intakes by conducting experiments and

comparing experimental results with the general theory of vortex subject and

previous data.

In the present work, a hydraulic model is constructed to study the onset of air-

entraining vortices (Type 6) at a horizontal intake. Chapter 2 deals with the

literature review in which previous studies of several investigators interested in

vortices will be summerized by noting their method for the determinance of

possible scale effects, limits proposed beyond which no scale effects are

present and formulas, if any, relating critical submergence to remaining

independent variables. In Chapter 3, the basic non-dimensional parameters

that govern the flow in air-entraining vortices at a horizontal intake are derived

and relation between critical submergence ratio and other non-dimensional

variables is stated. Chapter 4 is devoted to the description of experimental

setup, observations during experiments and procedure for the data taking.

Results of measurements in terms of set of curves, a practical formula for the

critical submergence of intakes, and comparison of the present study with

previous studies are all included in Chapter 5. Final comments and concluding

remarks are given in Chapter 6.

7

CHAPTER 2

LITERATURE REVIEW

Comprehensive analytical and numerical solutions to the vortex problem are

now available and these apparently produce realistic descriptions of the vortex

flow field despite of some simplifying assumptions. Applicational difficulty of

these analytical models to the real cases other than their mathematical

complexity stems mainly from the requirement of boundary conditions of the

flow field need to be specified. An additional difficulty in applying these

analytical models is that the effects of local asymmetrical geometric features

on the flow cannot be analytically included. Therefore, due to the unique

nature of every hydraulic project and complexity of vortex flow, physical model

studies were mostly prefered by previous investigators dealed with vortices.

Iversen (1953) studied the effects of sump boundaries on the pump efficiency

and also on the critical submergence for vertically downward sump intake.

According to the experiments undertaken, side-wall clearances in the range of

Di / 4 to Di / 2 and bottom clearance of Di / 2 do not affect normal pump

performance together with critical submergence, where Di is the intake

diameter.

Anwar (1965,1967 and 1968) studied both experimentally and theoretically on

a steady vortex with an air core forming at the entrance of an outlet pipe

discharging from a cylindrical tank. It was concluded from these experiments

that radial flow and thus the full development of a vortex can be prevented by

roughening the floor and solid boundaries since radial flow at the boundary

supplies the energy necessary to maintain an open vortex. In addition to this,

intake performance can be improved by using a floating raft on the surface

immediately above the intake and by baffle walls to dissipate the energy by

providing excessive roughness. Anwar also reported that, performance of a

pump depends very much on the side-wall and floor clearances.

8

Experiments were undertaken by Zielinski and Villemonte (1968) in the vortex

tank with five different orifice diameters using oil and water. As a conclusion,

physical effects of viscosity on vortex-orifice flow noted, as viscosity increases

circulation decreases from inlet to outlet due to the increase in viscous shear.

Consequently as circulation decreases, the dropdown decreases; as the

dropdown decreases air core decreases, thereby increasing the area of the jet;

as the area of the jet increases, the coefficient of discharge increases, thus in

order to maintain a constant discharge, head decreases. The study also

resulted in that when the Reynolds number, Re = ViDi / ν, is greater than 104,

the effects of viscosity on the discharge coefficient can be neglected. Here Vi is

the velocity at the intake and ν is the kinematic viscosity.

As one of the mostly used design practice investigator Gordon (1970) stated

that the factors affecting the vortex formation are: the geometry of the

approach flow to the intake; the velocity at the intake; the size of the intake;

and the critical submergence (Sc). Based on a study of 29 existing

hydroelectric intakes, a relation between these terms was proposed as

c

i

S 1.72FrD

= [2.1]

for symmetrical flow conditions and,

c

i

S 2.29FrD

= [2.2]

for non-symmetrical flow conditions, where all variables are in SI unit system

and Sc is measured from top of the intake.

Reddy and Pickford (1972) considered the flow boundary within the intake

region as the largest factor contributing to the vortex formation and being a

free surface phenomenon Reynolds number can be eliminated from the field of

the vortex formation. It was concluded that when vortex prevention devices

are not used Sc / Di = Fr (otherwise Sc / Di = 1 + Fr) will give vortex-free

operation either in hydroelectric practice or pump sump design.

9

Dagget and Keulegan (1974) investigated the effects of viscosity and surface

tension on the incipient condition for the vortex formation, the vortex shape,

the vortex size and the efficiency of the outlet under vortex action.

Experiments were conducted in two cylindrical testing tanks. Various

combinations of vane angle settings and outlet diameters were used for water-

glycerine and various grades of oil mixtures each having different surface

tensions and kinematic viscosities in both tanks. A number of flow rates was

used for each combination of liquid, vane angle and outlet diameters. By

plotting coefficient of discharge versus Reynolds number at costant values of

circulation number, NΓ = ΓDi / Qi, it was determined that effect of viscosity

becomes negligible for values of Re > 5 x 104, where Re = Qi / Diν, Γ =

circulation and Qi = intake discharge. From the comparison of the flow

conditions between different liquids used, it was concluded that surface tension

does not affect the vortex flow significantly when radial Reynolds number, ReR

> 3 x 103, where ReR = Qi / Hν and H = vertical intake submergence. The

study resulted in a relation for critical depth ratio such that:

(H / Di)c = 35 x 10-3 NΓRe ; Re < 5 x 104

(H / Di)c = 150NΓ ; Re ≥ 5 x 104

where (H / Di)c, as the smallest of H / Di for which an air core does not form.

Durgin and Hecker (1978) presented a general method to investigate potential

scale effects on the free surface vortices such that estimation to prototype

operating conditions can be made. Sources of vorticity together with indicators

of vortex severity were mentioned and vortex type classification based on

visual observation was made. In order to project the vortex severity of the

model to the prototype, existance of exact geometric and Froude similitude

was stated to be ensured and secondary effects of the Reynolds number must

be taken into consideration. In the projection technique, based on the idea to

get higher Reynolds numbers other than Froude similitude implies, models

operating at different water temperatures to have different viscosities ,and for

additional data points, flow rates both above and below those indicated by

Froude scaling were used by writers. In the application of method, Froude

number ratio versus Reynolds number produced, when increasing the model

flow rate at a given water temperature was plotted and recording the vortex

severity indicators (such as air ingestion, swirl and coefficient of discharge) at

10

a given operating point lines of constant vortex severity were produced. The

prototype performance is predicted by noting the vortex severity indicated at

the corresponding Reynolds number and properly scaled Froude number that is

Frr = 1.

Investigation conducted by Anwar et al. (1978) on the onset of air-entraining

vortices at a horizontal intake showed that flow conditions in an air-entraining

vortex is not affected by surface tension and the viscosity of the test fluid when

radial Reynolds number and Weber number are larger than 3 x 104 and 104,

respectively. It was shown that bellmouth entry do not improve critical

submergence heads as compared with the simple pipe intake. In the case of an

intake with and without bellmouth and mounted flush with side wall of the

flume, it was noted that the boundary wall reduces circulation and thus the

critical submergence to a point that the water surface almost reaches the

intake lip before air-entraining vortices occure.

Jain et al. (1978) carried out an investigation in two geometrically similar

cylindrical tanks each installed with a vertical pipe intake centrally located in

the tank bottom and radial flow was ensured by setting adjustable guide vanes

in the radial direction. Experiments were performed by three different intake

diameters and by using liquids such as water, water-Cepol solution and water-

isoamyl alcohol solution to get different viscosities and surface tensions

for each tank. The critical submergence was found to be practically

independent of the viscosity as well as the surface tension within the range of

experiments (2.5 x 103 ≤ Re = ViDi / ν ≤ 6.5 x 105; 1.2 x 102 ≤ We = ρVi2Di / σ

≤ 3.4 x 104, where ρ = density of the fluid and σ = surface tension). By plotting

Sc / Di versus Fr for experimental data it was seen that data can be

represented by

0.50c

i

S 0.47FrD

= [2.3]

As a second part of the previous study, Jain et al. (1978) presented a more

detailed experimental study to establish the conditions of similarity for the

onset of air-entraining vortices at vertical pipe intake. A very same

11

experimental set up and test liquids were used. Similar to the previous study,

influence of surface tension on the critical submergence for vertically

downward pipe intake was found to be negligible when Weber number, We =

ρVi2Di / σ ≥ 120. The following relation of critical submergence ratio with

viscous, gravity and circulation effects was obtained by plotting Sc / Di versus

Fr on a double log paper:

0.50c1

i

SK FrD

= [2.4]

in which correction factor K1 = f(Nν, NΓ)

Here Nν = g1/2 Di 3/2 / ν is a ratio of Froude number to Reynolds number and

used as a viscous parameter; NΓ = ΓSc / Qi is a circulation parameter. After

some manipulations, final relation was presented as:

0.42 0.50c

i

SK 5.6N FrD Γ= [2.5]

in which correction factor K = f(Nν) and attains a value of unity for Nν ≥ 5 x

104.

A significant conclusion of the experiments showed that the limit of Reynolds

number at which viscous effects become negligble is dependent on the Froude

number; the higher the Froude number, the greater is the limit of Reynolds

number for freedom from viscous influences. Completing the study, Jain et al.

propsed a method from which the prototype critical submergence can be

readily determined. The method considers that for a geometrically similar

model run at the same Froude number as that in the prototype, the only

distortion introduced is due to the change in Reynolds number, the effect of

which is taken care of by the viscous correction factor K. For such case model

and prototype critical submergence has a relation as

c c

i im p

KS KSD D

⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ [2.6]

12

from which

c

i p m

pc

i m

SD K

KSD

⎛ ⎞⎜ ⎟⎝ ⎠ =⎛ ⎞⎜ ⎟⎝ ⎠

[2.7]

Km and Kp being the viscous correction factors evaluated in the model and

prototype respectively and they were evaluated graphically. Since prototype

Reynolds number is usually so high as to be above the corresponding limit of

zero influence (Kp = 1), equation 2.7 becomes

c

i

c

i

S for zero viscous influenceD KS at any Reynolds numberD

= [2.8]

Anwar and Amphlett (1980) conducted experiments with a vertically inverted

pipe arrangement to measure variables such as intake height, side and back

wall clearance and circulation intensity which were thought to be responsible

for the formation of air-entraining vortices and results of this study was

compared with the results of horizontal intake arrangement. Three different

pipe diameters were tested with and without bellmouthed entry, and different

guide vane settings were used to get various swirl intensities. It was shown

that bellmouth entry did not improve critical submergence heads as compared

with the simple pipe intake. By plotting circulation number, Γr0 / Qi, versus

radial Reynolds number, ReR = Qi / νH, it was observed that circulation number

decreases rapidly when ReR increases from 1 to 3x104 and it is almost

independent when ReR > 4 x 104, where r0 is the shadow radius of the vortex

reflected at the bottom of the canal. Circulation number plotted against

coefficient of discharge indicated a dependence on H / Di and Di whereas

independence on b / Di, where b = side-wall clearance. These plots showed

that a model for which the circulation number is almost independent on viscous

effects can be designed and coefficient of discharge is now a function of Di and

H. Moreover, the submergence height H / Di becomes independent of wall

clearance b / Di > 8 for a high value of circulation and of b / Di > 4 for a non-

swirl flow.

13

Padmanabhan and Hecker (1984) conducted experiments using one full-sized

and two reduced-scale models of a pump sump to geometric scales of 1 : 2

and 1 : 4 to determine scale effects on free surface vortexing. It was found

that there occurs no significant scale effects on modeling free surface vortexing

in 1 : 2 and 1 : 4 models operated according to Froude similitude. This is due

to the fact that, their comparisons between full-sized and reduced-scale models

were based on vortex types indicated in ARL vortex classification chart (See

Figure 1.4.) instead of on the critical submergence for air core vortices. In

addition to this, it was stated that full-scale inlet losses were well predicted by

the reduced-scale losses when model pipe Reynolds numbers, Re = ViDi / ν,

are above 1 x 105. On the other hand, some scale effects were observed for Re

< 1 x 105 since higher loss coefficients were indicated by the reduced-scale

models.

Odgaard (1986) considered a Rankine vortex model as the basis for an

equation of critical submergence at intakes. By solving equations of motion in

the vicinity of the vertical axis that is steady, axi-symmetric and laminar, the

critical submergence was related as

22 i iVH VH 0.9 0.0043

g gΓσ

= − +ρ ν ν

[2.9]

or, in terms of dimensionless parameters

*

4 5 / 22 1/ 2 1 2 2

i i

H H1.0Fr Re We 0.00337Fr N ReD D

−Γ

⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ [2.10]

in which NΓ = circulation number = ΓDi / Qi or in terms of submergence NΓ* =

ΓH / Qi.

From the relation of ( ) 3iN Fr 4 / gDΓ = Π Γ , equation 2.10 without a surface

tension term can be reduced to

3i i

H 0.074 ReD gD

Γ= [2.11]

14

It was also noted that intake velocity, Vi, should be replaced by Vi + kΓ where

constant k = 6 x 10-5, in order to use presented formulations for turbulent flow

conditions.

Gulliver and Rindels (1987) presented an experimental investigation to predict

the formation of weak, free surface vortices at vertical intakes with a headrace

channel. In order to predetermine the flow approach angle to the headrace and

to find the effect of it on the vortex formation, guide vanes were used. After

the critical submergence at which persistent dye core vortices form was

measured over a range of intake Froude numbers, the lineer regression yielded

an equation as follows:

2 / 3 * 3c

i

S 42.5 Fr 40ND 3 Γ= + + [2.12]

where NΓ* = tanα / [1 + (L / B)tanα], L = distance from the headrace entrance

to intake center line, B = headrace channel width and α = approach flow angle.

Hite and Mih (1994) worked theoretically to determine closed-form equations

for axial, radial and tangential velocities as well as the water surface profile of

vortices at hydraulic intakes. These equations were found to be agreed with

experimental measurements and are applicable to vortex motion in general.

Yıldırım and Kocabaş (1995) investigated, both analytically and experimentally,

the critical submergence for vertically oriented intake in a horizontal

rectangular open flume. Investigators tried to solve the vortex problem

analytically with the potential flow solution for the combination of a point sink

and a uniform canal flow. According to this approach, the critical submergence

was considered to be equal to the radius of an imaginary spherical sink surface

which is, at this stage, called as critical spherical sink surface. As it can be seen

in Figure 2.1, this spherical sink surface was thought to have the same center

and discharge with the intake entrance. By carrying out dimensional analysis

together with the theoretical approach of Rankine half-body of revolution, final

relation of critical submergence was obtained as

15

1/ 2

c id

i

S V1 CD U2 2 ∞

⎛ ⎞= ⎜ ⎟

⎝ ⎠ [2.13]

where Vi = average intake velocity, Cd = discharge coefficient and U∞ = velocity

of uniform canal flow. All parameters can be seen in Figure 2.1. This final

relation was compared with experimental measurements and the agreement

was reported to be good.

A large-scale physical model study was undertaken by Jiming et al. (2000) for

the determinance of the minimum submergence before double entrance

pressure intakes. Based on the comparison made between single entrance and

double entrance pressure intakes with the same model scales and under the

same operating conditions, it was observed that besides having better flow

patterns, double entrance pressure intakes also would not induce air-entraining

vortex whereas air-entraining vortices were found before single entrance

intakes. As a consequence of the experimental study, it was stated that double

entrance intakes for large projects can be designed safely using suggested

formula as

cS 2.39Fr 0.01a

= − [2.14]

for symmetrical flow conditions and,

cS 3.17Fr 0.01a

= − [2.15]

for non-symmetrical flow conditions, where a = intake gate height.

16

Figure 2.1. Critical Spherical Sink Surface Approach

(Yıldırım and Kocabaş, 1995)

Yıldırım et al. (2000) searched for the flow-boundary effects on critical

submergence of intake pipe and hence a better prediction of critical

submergence. Experiments were performed on a horizontal intake pipe sited in

a dead-end canal flow. It was observed that as the distance between the intake

pipe and the dead-end gets smaller than Sc, deviation between theoretical and

experimental results increases. It was reported that potential flow solution still

gives acceptable results when this distance is smaller than Sc, however it

overpredicts by about 80% when the distance between the intake pipe and the

dead-end becomes much smaller than Sc.

Yıldırım and Kocabaş (2002) tried a critical spherical sink surface which has

radius of Sc / √2 and obtained good agreement between theoretical and

experimental results. Especially when the distance of the impervious dead-end

wall to the intake center is smaller than or equal to the diameter of the intake.

Results were obtained to be better comparing to a critical spherical sink surface

which has radius of Sc.

17

CHAPTER 3

MODELLING OF AIR-ENTRAINING VORTICES

3.1. Introductory Remarks

The formation of air-entraining vortices is the result of the complex interaction

between the geometry of the intake medium and the approach channel, the

flow velocity and the liquid properties such as surface tension and viscosity.

The flow near an intake is quite complex and is hence not readily suitable to

theoretical solution, except possibly in the case of idealised shape of the intake

medium. Design of intakes to be free of objectionable air-entraining vortices is,

therefore, based on physical-model studies.

3.2. Dimensionless Parameters

Based on the dimensional analysis theory, all dimensionless parameters can be

listed under the title of three major modelling elements. These are:

• Properties of the liquid: Density of the fluid (ρ), viscosity of the fluid (μ),

surface tension of the fluid (σ).

• Flow properties: Velocity at the intake pipe (Vi), circulation imposed to flow

(Γ), and gravitational acceleration (g).

• Geometric properties of the flow medium: Diameter of the intake pipe (Di),

distance between the intake center and reservoir bottom (c), right-side-wall

distance of the reservoir to the intake center (b1) and left-side-wall distance

of the reservoir to the intake center (b2).

Considering an intake of type shown in Figure 3.1, the critical submergence,

Sc, (defined as one which is sufficient to prevent formation of air-entraining

vortices) may be expressed by

( )c 1 i i 1 2S f , , ,g,V, ,D ,c,b ,b= ρ μ σ Γ [3.1]

18

Sc

Figure 3.1. Modelling Elements for Experimental Intake

By carrying out dimensional analysis procedure, following relation is obtained

between dimensionless variables.

c 1 22 O

i i i i

S b b cf , , ,Re,Fr,We,KD D D D

⎛ ⎞= ⎜ ⎟

⎝ ⎠ [3.2]

where

ρ= =

μi iVDRe Intake Reynolds number

19

= = i

i

VFr Intake Froude numbergD

ρ= =

σ

2i iV DW e Intake Weber number

Γ

= =Oi i

K Intake Kolf numberVD

In the experimental setup, bottom of the intake was so arranged that bottom

clearance, c, is always equal to Di / 2 and hence c / Di ratio becomes 0.5.

Therefore c / Di ratio can be dropped from Equation 3.2. Moreover, intake pipe

is placed mid-way between left and right side-walls so that b1 is equal to b2

and accordingly there remains only one dimensionless geometric parameter

which is b / Di. After these modifications Equation 3.2 becomes

c2 O

i i

S bf ,Re,Fr,We,KD D

⎛ ⎞= ⎜ ⎟

⎝ ⎠ [3.3]

In a geometrically similar model, dimensionless geometric parameters would

be the same in the model and prototype (However, in the present study, four

different scaled intake pipes were used for the same reservoir assuming that

the reservoir is sufficiently large). Equality of Sc / Di in the model and

prototype, in general, require equality of Fr, Re, We and KO between model and

prototype which would be imposible to achieve. The following discussion is,

therefore, devoted to an analysis of the relative importance of these four

parameters, which would enable the designer to decide upon the suitable

modelling criteria for his study. Based on the study about these parameters in

the literature, sequence of the influences of these parameters to be

investigated will be from minor importance to major importance in order to

simplfy the relation obtained in equation 3.2 each time if it is possible.

3.2.1. Influence of Kolf Number

Circulation is dependent on the characteristics of the approach flow, the

geometry of the intake chamber and the discharge. Since all geometric

20

parameters of the intake pipe, intake medium and approach flow channel are

included in Equation 3.1 and no imposed circulation is applied, the parameter Γ

could be deleted from this equation. Therefore, effect of the dimensionless

parameter of Kolf number can be neglected on the formation of air-entraining

vortices. Then Equation 3.2 becomes

c2

i i

S bf ,Re,Fr,WeD D

⎛ ⎞= ⎜ ⎟

⎝ ⎠ [3.4]

3.2.2. Influence of Weber Number

Weber number is basically effective in shallow-dimple like vortices. As it was

stated in the researches of Dagget& Keulegan (1974), Jain et al. (1978),

Gulliver & Rindels (1987) and others, surface tension effect can be neglected in

practice of air-entraining vortices. Therefore Equation 3.3 becomes

c2

i i

S bf ,Re,FrD D

⎛ ⎞= ⎜ ⎟

⎝ ⎠ [3.5]

3.2.3. Influence of Reynolds Number

In literature, almost all researchers considered the effect of viscosity on the

formation of vortices and determined limits of Reynolds number ,beyond which

viscous forces are negligible, according to their own experimental results. Due

to this fact that, influence of Reynolds number for this study was taken into

account.

3.2.4. Influence of Froude Number

Practically all studies carried out till now, with exception of Yıldırım & Kocabaş

(1995, 2000, 2002), have indicated Froude number to be one of the important

parameters influencing the critical submergence. This is understandable

because it is a free surface phenomenon and affected by gravity. It is therefore

customary to base the model study of vortex formation on Froude number

21

similarity. Corrections to model results are made to account for the distortion

likely to be introduced by the non-constancy of Reynolds numbers in the model

and the prototype if any distortion is present to the this non-constancy.

Therefore, final relation between dimensionless numbers was found to be as in

Equation 3.5.

3.2.5. Influence of Model Length Scale

Equation 3.5 gives the relationship between Sc / Di and other related

independent dimensionless parameters involved in the formation of vortices at

intake structures. This relationship is valid for a prototype intake structure and

its hydraulic model having a certain length scale Lr. In order to have a

complete similarity between the prototype and model structure, all of the

corresponding dimensionless terms given in Equation 3.5 must be equal to

each other. However, it is known that to satisfy this condition for the equality

of both Reynolds and Froude numbers, is not practically possible. Therefore in

the modelling of the intake structures, the equality of Froude numbers is

demanded. Neglecting the Reynolds number and some other terms given in

Equation 3.3 for mentioned reasons results in errors at certain percentages on

the values of Sc / Di to be calculated for the prototype. This deviation between

the calculated Sc / Di value for the prototype and the one corresponding to the

model is due to the length scale used in the modelling. Since dimensional

analysis and related model studies aim to predict prototype values of Sc / Di by

performing experimental study and considering the possible length scale

effects, Equation 3.5 can be expressed in the final form of:

⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

cr r

i ir r

S bf ,Re ,FrD D

[3.6]

In model studies based on the equation given above Frr = 1, Rer ≠ 1, (Sc / Di)r

≠ 1 and due to the fact that the central distance of each pipe to side-wall

boudaries is the same in the experimental setup (b / Di)r can be expressed as

follows:

22

⎛ ⎞⎜ ⎟⎛ ⎞ ⎝ ⎠= = ⋅ =⎜ ⎟ ⎛ ⎞⎝ ⎠⎜ ⎟⎝ ⎠

i i pm m

i p i m rr

i p

bD (D )(b)b 1

D (b) (D ) LbD

[3.7]

Substituting Equation 3.7 into Equation 3.6, one can get

( )⎛ ⎞=⎜ ⎟

⎝ ⎠c

r r ri r

S f L ,Re ,FrD

[3.8]

23

CHAPTER 4

EXPERIMENTAL EQUIPMENT AND PROCEDURE

4.1. Experimental Equipment

An experimental reservoir was built to investigate scale effects on the

formation of air-entraining vortices at horizontal intakes. The reservoir, shown

in Figs. 4.1, is in 2.20 m length, 1.40 m width and 2.00 m depth and consisting

of dead-volume section, active reservoir section and intake section. A diaphram

slab was constructed so as to leave a 40 cm space from rear-wall of the

reservoir and to create a dead-volume by dividing reservoir into two part in the

horizontal plane. Flow was supplied from a large elevated-constant head tank

to an inflow pipe ,in 80 cm diameter, which enters the reservoir below the

bottom level of diaphram slab so that water fills dead-volume section first. This

enables entrance of water being still, while space provided allows water to rise

and fill the active reservoir calmly. Screens in two rows were attached over the

diaphram slab and fastened to the reservoir side walls. These screens allows to

dissipate the energy of water and move uniformly without any circulation to

the intake. Dead-end was created by means of plexiglass panels attached to

each other with screws and waterproofed with silicone paste. A small draining

pipe was located at the face of dead-end to enhance water-level adjustment.

Water levels were read from a milimetric paper sticked to dead-end face. For

the ease of observations, part of dead-end, in 103,5 cm length and 63,5 cm

width, was extended from the reservoir front plane. In order to provide space

for the attachment of intake pipes square hole, at equal distance to the

plexiglass side walls, was opened to the face of extended dead-end. Plexiglass

intake pipes ,in 19,72 cm, 15,23 cm, 9,28 cm, and 6,12 cm diameters, were

installed to the extended dead-end for each set of experiments in this order. A

steel plate was placed in front of the intake pipe entrance to the same level

with the bottom of intake pipe (i.e. c = Di / 2). Intake pipes were connected to

a 7,5 kW centrifugal pump which conveys water to a steel pipe in 10,4 cm

diameter. Water falls freely from steel pipe to an open channel which ends with

24

a weir. Discharged water from this weir and from outlet pipe during reservoir

emptying process was conveyed by canals to the sump below laboratory and

water in the sump is pumped to elevated water tank.

4.2. Experimental Procedure

After installation of each intake pipe and levelling of steel plate accordingly,

set-up becomes ready for each set of experiments. In each experiment,

discharge and water surface elevation at which air-entraining vortex initiates,

were measured. The data are given in Appendix-A.

Before different set of experiments conducted at different times, first the water

was pumped from a sump to an elevated tank in order to achieve a constant

head. Then valve on the inflow pipe was opened and reservoir was filled to a

level that is very much higher than the critical submergence. By starting the

pump operation and adjusting the valve on the pumping line, predetermined

discharge amount was obtained. At this stage since discharged water from the

reservoir cannot be recirculated to the reservoir again, in order to achieve a

constant reservoir water surface level, valves on the supply pipe and pumping

line were opened or closed at certain amount while observing the water surface

level (This proccess of obtaining constant water level took 30 minutes to 1,5

hours for different experiments). By the time, since adjustment is mainly done

by valve on supply pipe, discharge of intake pipe is affected slightly. After

constant water level was reached, observation of water surface for possible

swirls, surface depressions and air-entraining vortices was started. Water level

was decreased for a small amount carefully by opening the valve of small drain

pipe. Valve was then closed and reservoir surface was observed. These steps

were continued untill an air-entraining vortex was resulted in (This proccess of

air-enraining vortex initiation took 30 minutes to 1 hour). At that point

discharge and water surface level were measured. By changing the opening of

valve at supply pipe, various values of discharge and hence Froude numbers

were obtained and steps untill to get air-entraining vortex were repeated.

25

Figure

4.1

. Exp

erim

enta

l Equip

men

t

26

There are two important notes related to experimental studies. First, ten

minutes of observation time was selected throughout the experiments before

further decreasing the water level. This means that after each reduction of

water level by the use of drain pipe, water surface is observed for possible air-

entarining before further reduction of water level. Second, since, as already

mentioned, discharged water cannot be recirculated, flow through supply pipe

is increased after each reduction of reservoir water level since the decreasing

submergence of the supply pipe causes an increase in discharge coefficient of it

and hence the discharge of it. Although it seems that, each reduction of water

level requires re-fixing the water surface at a constant level, after couple of

fixing proccesses water level does not change during ten minutes of

observation time. Moreover, further fixing attempts require much less time

after first fixing of water level is completed.

4.3. Observations

All the vortices that occured in the experiments were of an intermittent type,

where the vortices formed and then dissipated. In many instances there was

the same pattern of vortex development where a swirl existed near an intake

pipe entrance and organized vortex action would develop in the swirl.

Rotational velocity increased at the center of the swirl, a dimple formed,

increased in size, formed a vortex tail, and then further increased in size, an

air core vortex occured.

Almost for all experimental data, swirl pattern shown in Figure 4.2 resulted in

and could be observed by naked eye. This swirl pattern is an indication that the

plexiglass side-walls of dead-end penetrated through reservoir are inherently

sources of swirl generation. At high submergences, when compared to the

critical submergence at that discharge, swirls on the water surface formed and

disappeared instantly. When water surface was reduced, duration and intensity

of these swirls increased. Continuing to decrease water surface level, swirls

turned into small dimples. Further decrease in water level caused deeper

dimples and if strong enough these dimples extended, increased in diameter

and formed vortex tails. Very near to the critical submergence, air core vortices

occured but air core was not continuous, only small bubbles entered to intake.

27

Finally, when the critical submergence was reached, continuous air core

vortices formed together with air ingestion. After this point on, decreasing

water levels caused stronger air-entraining vortices and hence increase in air

ingestion amount.

Figure 4.2. Swirl Pattern During Experiments

28

CHAPTER 5

RESULTS AND DISCUSSIONS

5.1. Introduction

In this chapter the data collected in the present study are reported and

comparison between four sets of experiments for different intake pipe

diameters is made based on Froude number equality. According to this

comparison, possible scale effects due to the Reynolds number and side-wall

clearance parameter, b / Di, are interpreted. Moreover, results of this study are

verified by using formulation proposed by Gordon (1970). Finally, curve fitting

will be achieved by regression analysis. Table 5.1 is a summary of the

experimental study examined in the present study. Detailed results of the

experimental study are given in Appendix A.

Table 5.1. Summary of Experimental Study

Intake Pipe

Range of Qi (lt/s)

Range of Fr

Range of Re

Range of We Di (cm) b/Di

Number of observations

1 21,62

~ 38,85

0,51 ~

0,92

1,39E+05 ~

2,50E+05

1,35E+03 ~

4,38E+0319,72 1,597 10

2 14,37

~ 34,68

0,65 ~

1,56

1,20E+05 ~

2,89E+05

1,30E+03 ~

7,57E+0315,23 2,068 11

3 4,12

~ 19,63

0,64 ~

3,04

5,63E+04 ~

2,68E+05

4,72E+02 ~

1,07E+049,28 3,394 10

4 1,43

~ 9,19

0,63 ~

4,03

2,96E+04 ~

1,90E+05

1,98E+02 ~

8,19E+036,12 5,147 10

29

5.2. The Relation between Dimensionless Parameters and

Submergence Scale Ratio

The relation between dimensionless parameters and (Sc / Di)r is summarized in

Table 5.2a and Table 5.2b.

Table 5.2a. Relation between Dimensionless Parameters and Submergence

Scale Ratio for 0,65 ≤ Fr ≤ 0,90

Fr Pipe Di (cm) Sc/Di* Re b/Di Lr** (Sc/Di)r** Scale Effect***(%)

0,65

1 19,72 1,003 1,776E+05 1,597 1,000 1,000 0,00 2 15,23 0,818 1,205E+05 2,068 0,772 0,816 18,44 3 9,28 0,428 5,732E+04 3,394 0,471 0,427 57,33 4 6,12 0,287 3,070E+04 5,147 0,310 0,286 71,39

0,70

1 19,72 1,011 1,912E+05 1,597 1,000 1,000 0,00 2 15,23 0,876 1,298E+05 2,068 0,772 0,866 13,35 3 9,28 0,452 6,173E+04 3,394 0,471 0,447 55,29 4 6,12 0,312 3,306E+04 5,147 0,310 0,309 69,14

0,75

1 19,72 1,108 2,049E+05 1,597 1,000 1,000 0,00 2 15,23 0,934 1,391E+05 2,068 0,772 0,843 15,70 3 9,28 0,476 6,614E+04 3,394 0,471 0,430 57,04 4 6,12 0,338 3,542E+04 5,147 0,310 0,305 69,49

0,80

1 19,72 1,132 2,186E+05 1,597 1,000 1,000 0,00 2 15,23 0,968 1,483E+05 2,068 0,772 0,855 14,49 3 9,28 0,5 7,055E+04 3,394 0,471 0,442 55,83 4 6,12 0,364 3,778E+04 5,147 0,310 0,322 67,84

0,85

1 19,72 1,153 2,322E+05 1,597 1,000 1,000 0,00 2 15,23 0,992 1,576E+05 2,068 0,772 0,860 13,96 3 9,28 0,524 7,496E+04 3,394 0,471 0,454 54,55 4 6,12 0,389 4,015E+04 5,147 0,310 0,337 66,26

0,90

1 19,72 1,22 2,459E+05 1,597 1,000 1,000 0,00 2 15,23 1,057 1,669E+05 2,068 0,772 0,866 13,36 3 9,28 0,548 7,937E+04 3,394 0,471 0,449 55,08 4 6,12 0,415 4,251E+04 5,147 0,310 0,340 65,98

* Sc / Di values were interpolated from the experimental results of each intake

pipe for the corresponding Froude numbers.

** Lr = (Di)m / (Di)p and (Sc/Di)r = (Sc/Di)m / (Sc/Di)p

*** Scale effect = 100 x (1,00 - (Sc/Di)r)

30

Table 5.2b. Relation between Dimensionless Parameters and Submergence

Scale Ratio for 1,00 ≤ Fr ≤ 3,00

Fr Pipe Di (cm) Sc/Di* Re b/Di Lr** (Sc/Di)r** Scale Effect***(%)

1,00 2 15,23 1,147 1,854E+05 2,068 1,000 1,000 0,00 3 9,28 0,597 8,819E+04 3,394 0,609 0,520 47,95 4 6,12 0,466 4,723E+04 5,147 0,402 0,406 59,37

1,10 2 15,23 1,221 2,040E+05 2,068 1,000 1,000 0,00 3 9,28 0,696 9,701E+04 3,394 0,609 0,570 43,00 4 6,12 0,518 5,195E+04 5,147 0,402 0,424 57,58

1,20 2 15,23 1,282 2,225E+05 2,068 1,000 1,000 0,00 3 9,28 0,795 1,058E+05 3,394 0,609 0,620 37,99 4 6,12 0,573 5,668E+04 5,147 0,402 0,447 55,30

1,30 2 15,23 1,351 2,410E+05 2,068 1,000 1,000 0,00 3 9,28 0,922 1,146E+05 3,394 0,609 0,682 31,75 4 6,12 0,637 6,140E+04 5,147 0,402 0,472 52,85

1,40 2 15,23 1,413 2,596E+05 2,068 1,000 1,000 0,00 3 9,28 1,054 1,235E+05 3,394 0,609 0,746 25,41 4 6,12 0,701 6,612E+04 5,147 0,402 0,496 50,39

1,50 2 15,23 1,521 2,781E+05 2,068 1,000 1,000 0,00 3 9,28 1,154 1,323E+05 3,394 0,609 0,759 24,13 4 6,12 0,765 7,085E+04 5,147 0,402 0,503 49,70

1,75 3 9,28 1,312 8,819E+04 3,394 1,000 1,000 0,00 4 6,12 0,907 4,723E+04 5,147 0,659 0,691 30,87

2,00 3 9,28 1,469 9,701E+04 3,394 1,000 1,000 0,00 4 6,12 1,173 5,195E+04 5,147 0,659 0,799 20,15

2,25 3 9,28 1,586 1,058E+05 3,394 1,000 1,000 0,00 4 6,12 1,443 5,668E+04 5,147 0,659 0,910 9,02

2,50 3 9,28 1,736 1,146E+05 3,394 1,000 1,000 0,00 4 6,12 1,613 6,140E+04 5,147 0,659 0,929 7,09

2,75 3 9,28 1,963 1,235E+05 3,394 1,000 1,000 0,00 4 6,12 1,707 6,612E+04 5,147 0,659 0,870 13,04

3,00 3 9,28 2,103 1,323E+05 3,394 1,000 1,000 0,00 4 6,12 1,913 7,085E+04 5,147 0,659 0,910 9,03



** Lr = (Di)m / (Di)p and (Sc/Di)r = (Sc/Di)m / (Sc/Di)p

*** Scale effect = 100 x (1,00 - (Sc/Di)r)

31

Due to the experimental equipment limitations, the relationship between

dimensionless parameters and submergence scale ratio including four intake

pipes was achieved only within the range of 0,65 ≤ Fr < 0,92. Between ranges

of 0,92 ≤ Fr < 1.56 and 1,56 ≤ Fr ≤ 3,04, this relationship is available for pipes

2-3-4 and 3-4, respectively. For each range, the largest intake diameter was

assumed to be as a prototype and Lr and (Sc / Di)r were determined

accordingly.

5.2.1. The Relation between Sc / Di and b / Di

In literature, some researchers like Gordon (1970) eliminated the effect of b /

Di on Sc / Di by working on prototypes whereas some like Anwar and Amphlett

(1980) worked with different b / Di ratios and concluded that b / Di is not

effective when b / Di > 4 for non-swirl flow. However, in the present study

since b / Di values are not that large and differs in each pipe, there occurs a

direct scale effect of this parameter on Sc / Di.

The effect of b / Di on Sc / Di can be observed for constant Froude numbers and

varying Reynolds numbers in Figure 5.1 for 0,65 ≤ Fr ≤ 1,50. For a given

Froude number, Sc / Di values rapidly decrease as b / Di increases up to the

value of b / Di ≅ 3,50, then the decreasing rate of Sc / Di with b / Di decreases

for b / Di values greater than 3,50. From Figure 5.1 it may also be concluded

that Sc / Di values approach to almost constant values for a given Froude

number. In other words, it can be stated that as b / Di gets larger, Sc / Di

becomes independent of b / Di and function of only Froude number and

Reynolds number.

32

Figure 5.1. Relation between b / Di and Sc / Di for 0,65 ≤ Fr ≤ 1,50

5.2.2. The Relation between Sc / Di and Re

In literature, there are some Reynolds number limits beyond which it is

reported that Reynolds number effects on Sc / Di can be neglected. It is hard to

give such a limit for the present study.

In Figure 5.2a one may say that there are inflection points on the curves of Sc /

Di versus Re of about 1,60 x 105. Up to this value of Re, Sc / Di increases

rapidly with increasing Re for the Froude number range between 0,65 and

0,90. The increasing rate of Sc / Di with increasing Re for Re > 1,60 x 105

decreases. From the general trend of these curves one may conclude that at

Reynolds numbers much larger than 2,60 x 105, the variation of Sc / Di with Re

gets smaller.

In Figure 5.2b, which covers the data of experiments having Froude numbers

greater than 1,00, the above mentioned inflection points can be seen only on

the curves of Froude numbers of 1,30, 1,40 and 1,50. As the Froude number

0,000

0,200

0,400

0,600

0,800

1,000

1,200

1,400

1,600

0,00 1,00 2,00 3,00 4,00 5,00 6,00

S c/D

i

b/Di

Fr=0,65Fr=0,70Fr=0,75Fr=0,80Fr=0,85Fr=0,90Fr=1,00Fr=1,10Fr=1,20Fr=1,30Fr=1,40Fr=1,50

Di=19

,72 cm

Di=15,23 cm

Di=9,28 cm

Di=6,12

cm

Di=15

,23 cm

33

increases, the Reynolds number at which these inflection points are observed,

also increases. Even after these inflection points, the slopes of Sc / Di versus Re

curves are much steeper than those of Fr < 1,00 given in Figure 5.2a.

Therefore it is quite difficult to say something about the limit values of

Reynolds numbers after which the effect of Re is negligable on the values of Sc

/ Di for experiments of Fr ≥ 1,00. Consequently, it can be concluded from

Figure 5.2a and Figure 5.2b that, limit of Re beyond which viscous forces do

not affect the vortex flow, increases with the increase in Froude number as

stated in Jain et al. (1978).

Figure 5.2a. Relation between Re and Sc / Di for 0,65 ≤ Fr ≤ 0,90

0,200

0,400

0,600

0,800

1,000

1,200

1,400

1,000E+04 6,000E+04 1,100E+05 1,600E+05 2,100E+05 2,600E+05

S c/D

i

Re

Fr=0,65Fr=0,70Fr=0,75Fr=0,80Fr=0,85Fr=0,90

Di=6,12

cm

Di=9,28 cm

Di=19,72 cm

34

Figure 5.2b. Relation between Re and Sc / Di for 1,00 ≤ Fr ≤ 1,50

5.2.3. The Relation between (Sc / Di)r and Lr

For model length scale ratio equals to unity it means that there is a complete

similarity between the model and the prototype. Since Lr = 1 is not satisfied for

practical reasons, for a given Froude number there is direct scale effect of

Reynolds number and b / Di on the values of Sc / Di due to deviation of Lr from

unity. In order to see how this deviation can be represented by Lr only, the

relationship between Lr and (Sc / Di)r is presented in Figures 5.3a and 5.3b.

This relationship has been obtained based on Froude similitude since the

Froude number is the main dimensionless parameter that affects the vortex

flow.

Figure 5.3a clearly shows that as Lr decreases (Sc / Di)r rapidly decreases for all

of the Froude numbers tested. For a given Lr, (Sc / Di)r increases slightly with

increasing Fr, for the tests having the Froude numbers up to 0,90. The model

of Di = 6,12 cm (Lr=0,310) has (Sc / Di)r values varying between 0,286 and

0,340 which result in scale effects varying between 71,39 % and 65,98 %,

0,200

0,400

0,600

0,800

1,000

1,200

1,400

1,600

1,000E+04 6,000E+04 1,100E+05 1,600E+05 2,100E+05 2,600E+05 3,100E+05

S c/D

i

Re

Fr=1,00

Fr=1,10

Fr=1,20

Fr=1,30

Fr=1,40

Fr=1,50

Di=6,12 cm

Di=9,28 cm

Di=15,23 cm

35

respectively. When Lr value gets larger, such as Lr = 0,772, the corresponding

scale effects get much smaller; 18,44 % and 13,36 %.

For tests of Fr ≥ 1,00 (Figure 5.3b), the effect of Fr on the variation of (Sc / Di)r

for a given Lr is more significant than those tests descibed in Figure 5.3a. The

model of minimum length scale, Lr = 0,402, has (Sc / Di)r values of 0,406 and

0,503 corresponding to the scale effects of 59,37 % and 49,70 %, respectively.

From the above discussions it can be concluded that smaller the model length

scale, much larger is the scale effect.

Figure 5.3a. Relation between Lr and (Sc / Di)r for 0,65 ≤ Fr ≤ 0,90

0,200

0,300

0,400

0,500

0,600

0,700

0,800

0,900

1,000

1,100

0,200 0,300 0,400 0,500 0,600 0,700 0,800 0,900 1,000 1,100

(Sc

/Di) r

Lr

Fr=0,65Fr=0,70Fr=0,75Fr=0,80Fr=0,85Fr=0,90

Di=19,72 cm (Prototype )

Di=15,23 cm

Di=9,28 cm

Di=6,12 cm

36

Figure 5.3b. Relation between Lr and (Sc / Di)r for 1,00 ≤ Fr ≤ 1,50

5.2.4. The Relation between (Sc / Di)r and Rer

In order to see the effect of Rer on the variation of (Sc / Di)r, Figures 5.4a and

5.4b were plotted for the tests of Fr ≤ 0,90 and Fr ≥ 1,00. The general trends of

the curves given in these figures are the same as those presented in Figures

5.3a and 5.3b. The strong dependence of (Sc / Di)r on Rer is clearly seen in

these figures. It is obvious that for selected prototypes Frr = 1, Rer = 1 and (Sc

/ Di)r =1. As Rer decreases, which means that Lr is decreasing, (Sc / Di)r attains

much smaller values resulting in very large scale effects.

0,200

0,300

0,400

0,500

0,600

0,700

0,800

0,900

1,000

1,100

0,300 0,400 0,500 0,600 0,700 0,800 0,900 1,000 1,100

(Sc

/ Di) r

Lr

Fr=1,00Fr=1,10Fr=1,20Fr=1,30Fr=1,40Fr=1,50

Di=15,23 cm

Di=6,12 cm

37

Figure 5.4a. Relation between Rer and (Sc / Di)r for 0,65 ≤ Fr ≤ 0,90

Figure 5.4b. Relation between Rer and (Sc / Di)r for 1,00 ≤ Fr ≤ 1,50

0,200

0,300

0,400

0,500

0,600

0,700

0,800

0,900

1,000

1,100

0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1,10

(Sc

/Di) r

Rer

Fr=0,65Fr=0,70Fr=0,75Fr=0,80Fr=0,85Fr=0,90

Di=6,12 cm (Lr=0,310)

Di=9,28 cm (Lr=0,471)

Di=15,23 cm (Lr=0,772)

Di=19,72 cm (Lr=1 ⇒Prototype )

0,200

0,300

0,400

0,500

0,600

0,700

0,800

0,900

1,000

1,100

0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1,10

(Sc

/ Di) r

Rer

Fr=1,00

Fr=1,10

Fr=1,20

Fr=1,30

Fr=1,40

Fr=1,50

Di=6,12 cm (Lr=0,310)

Di=9,28 cm (Lr=0,471)

Di=15,23 cm (Lr=1 ⇒Prototype )

38

5.3. Verification of Results

Experimental results have been tested with formulation (Equation 2.1)

reported by Gordon (1970). One should note that formulation proposed by

Gordon was generated based on prototype observations of hydropower intakes

for 0,2 ≤ Fr ≤ 2. Due to this fact, the comparison of the present study with the

mentioned one was performed up to Fr = 2 and the related parameters were

given in table Table 5.3.

Equation 2.1 implies that the critical submergence ratio Sc / Di is dependent

only on Froude number. Operational conditions of prototypes lead to large

enough Reynolds number and b / Di. Therefore it is an expected result that Sc /

Di is dependent only on Froude number as the equation 3.5 indicates. Table 5.3

shows that in each group of the constant Froude numbers, the first pipe, which

has the maximum intake diameter, considered as the prototype, yields the

minimum scale effect. On the other hand, the other pipes result in much higher

scale effect as the pipe diameter decreases. This situation implies that as the

model length scale gets smaller, the effect of length scale on the value Sc / Di

rapidly increases.

39

Table 5.3. Comparison between the present study and study of Gordon (1970)

Fr Pipe Di (cm) Sc/Di* Vi (m/s) Sc (m) Sc/DiG** Scale

Effect***(%)

0,65

1 19,72 1,003 0,904 0,198 1,120 10,42 2 15,23 0,818 0,795 0,125 1,120 26,95 3 9,28 0,428 0,620 0,040 1,120 61,78 4 6,12 0,287 0,504 0,018 1,120 74,37

0,70

1 19,72 1,011 0,974 0,199 1,206 16,16 2 15,23 0,876 0,856 0,133 1,206 27,35 3 9,28 0,452 0,668 0,042 1,206 62,52 4 6,12 0,312 0,542 0,019 1,206 74,13

0,75

1 19,72 1,108 1,043 0,218 1,292 14,24 2 15,23 0,934 0,917 0,142 1,292 27,71 3 9,28 0,476 0,716 0,044 1,292 63,16 4 6,12 0,338 0,581 0,021 1,292 73,84

0,80

1 19,72 1,132 1,113 0,223 1,378 17,86 2 15,23 0,968 0,978 0,147 1,378 29,76 3 9,28 0,5 0,763 0,046 1,378 63,72 4 6,12 0,364 0,620 0,022 1,378 73,59

0,85

1 19,72 1,153 1,182 0,227 1,464 21,26 2 15,23 0,992 1,039 0,151 1,464 32,25 3 9,28 0,524 0,811 0,049 1,464 64,21 4 6,12 0,389 0,659 0,024 1,464 73,43

0,90

1 19,72 1,22 1,252 0,241 1,550 21,31 2 15,23 1,057 1,100 0,161 1,550 31,82 3 9,28 0,548 0,859 0,051 1,550 64,65 4 6,12 0,415 0,697 0,025 1,550 73,23

1,00 2 15,23 1,147 1,222 0,175 1,723 33,42 3 9,28 0,597 0,954 0,055 1,723 65,34 4 6,12 0,466 0,775 0,029 1,723 72,95

1,10 2 15,23 1,221 1,345 0,186 1,895 35,56 3 9,28 0,696 1,050 0,065 1,895 63,27 4 6,12 0,518 0,852 0,032 1,895 72,66

1,20 2 15,23 1,282 1,467 0,195 2,067 37,98 3 9,28 0,795 1,145 0,074 2,067 61,54 4 6,12 0,573 0,930 0,035 2,067 72,28

1,30 2 15,23 1,351 1,589 0,206 2,239 39,67 3 9,28 0,922 1,240 0,086 2,239 58,83 4 6,12 0,637 1,007 0,039 2,239 71,56

1,40 2 15,23 1,413 1,711 0,215 2,412 41,41 3 9,28 1,054 1,336 0,098 2,412 56,30 4 6,12 0,701 1,085 0,043 2,412 70,93

1,50 2 15,23 1,521 1,833 0,232 2,584 41,14 3 9,28 1,154 1,431 0,107 2,584 55,34 4 6,12 0,765 1,162 0,047 2,584 70,39

1,75 3 9,28 1,312 1,670 0,122 3,015 56,48 4 6,12 0,907 1,356 0,056 3,015 69,91

2,00 3 9,28 1,469 1,908 0,136 3,445 57,36 4 6,12 1,173 1,550 0,072 3,445 65,95

40



** Sc / DiG values were calculated based on equation 2.9.

*** Scale effect = 100 x ABS(Sc/Di - Sc / DiG) / Sc / Di

G

Another outcome of Table 5.3 is that, Sc / Di values calculated in the present

sudy are smaller than the predicted Sc / Di values based on formulation of

Gordon. Side-wall proximity and smaller Reynolds numbers than the prototype

conditions are the main reasons of this fact.

5.4. Curve Fitting

In part 3.2.2, the relation of the critical submergence ratio, Sc / Di with

dimensionless parameters was given in Equation 3.5. Analysis of the

experimental results indicated that side-wall clearance b / Di and the Reynolds

number affects the critical submergence ratio Sc / Di in the present study as

stated in part 5.2.1 and 5.2.2, respectively. Therefore, in order to get a

practical formula for the critical submergence of intakes with different scale

ratios, regression analysis was performed with a computer program named

DataFit (Oakdale Engineering, 2009). In order to apply regression analysis

following equation is defined to the program.

⎛ ⎞= ⎜ ⎟

⎝ ⎠

yx zc

i i

S bFr ReD D

[5.1]

Based on regression analysis, regression variables x, y, and z are obtained as

0,865, -0,565 and 0,0424, respectively. The correlation coefficient, R2, of the

analysis is found as 0,950 and the agreement of regression results with

experimental data can be seen in Figure 5.5. Analyzed values of x, y and z are

valid in the experimental limits of Fr, b / Di and Re parameters.

41

R² =

0,9

497

0,20

0

0,70

0

1,20

0

1,70

0

2,20

0

2,70

0 0,20

00,

700

1,20

01,

700

2,20

02,

700

(Sc/Di)measured

(Sc

/Di) c

alcu

late

d

Figure

5.5

. D

ata

Fit

Plo

t

42

CHAPTER 6

CONCLUSIONS

In the present study the effects of Froude number, side-wall clearance and

Reynolds number on the formation of air-entraining vortices at horizontal

intakes are investigated by conducting experiments and comparing

experimental results with the general theory of vortex subject and previous

data. An empirical relationship for the critical submergence accounting the

possible scale effects between model and prototype conditions is obtained.

Also, experimental results are compared with the study of Gordon (1970). By

this experimental study, the followings have been discerned:

1. For constant Froude numbers, effect of b / Di on Sc / Di decreases as b /

Di increases although Re decreases with the increasing b / Di. Also it can

be interpreted that Sc / Di becomes almost independent to b / Di for b /

Di > 6.

2. Reynolds number limit, beyond which viscous forces do not affect the

vortex flow, increases with the increase in Froude number as stated in

Jain et al. (1978).

3. As the model length scale gets smaller, the effect of length scale on the

value of (Sc / Di)r rapidly increases. For a given Lr, (Sc / Di)r increases

slightly with increasing Fr for the tests having the Froude numbers up to

0,90. For tests of Fr ≥ 1,00, the effect of Fr on the variation of (Sc / Di)r

for a given Lr is more significant than those tests of Fr ≤ 0,90. These

deviaitons are due to Reynolds number differences between model and

prototype conditions and also due to differences of side-wall clearences.

4. Based on regression analysis the dependence of Sc / Di to Froude

number, b / Di and Reynolds number is formulated as in equation 5.1

43

with R2 = 0,950. Equation 5.1 is valid for 0,51 ≤ Fr ≤ 4,03, 1,597 ≤ b /

Di ≤ 5,147 and 2,96 x 104 ≤ Re ≤ 2,89 x 105.

44

REFERENCES

Anwar, H.O. (1965), “Flow in a Free Vortex”, Water Power 1965(4), 153-161.

Anwar, H.O. (1967), “Vortices at Low Head Intakes”, Water Power 1967(11), 455-457.

Anwar, H.O. (1968), “Prevention of Vortices at Intakes”, Water Power 1968(10), 393-401.

Anwar, H.O., Weller, J.A. and Amphlett, M.B. (1978), “Similarity of Free-Vortex at Horizontal Intake”, J. Hydraulic Res. 1978(2), 95-105.

Anwar, H.O. and Amphlett, M.B. (1980), “Vortices at Vertically Inverted Intake”, J. Hydraulic Res. 1980(2), 123-134.

Daggett, L.L. and Keulegan, G.H. (1974), “Similitude in Free-Surface Vortex Formations”, J. Hydraulic Div., ASCE, HY11, 1565-1581.

Durgin, W.W. and Hecker, G.E. (1978), “The Modelling of Vortices at Intake Structures”, Proc. IAHR-ASME-ASCE Joint Symposium on Design and Operation of Fluid Machinery, CSU Fort Collins, June 1978, Vols. I and III.

Gordon, J.L. (1970), “Vortices at Intakes”, Water Power 1970(4), 137-138.

Gulliver, J.S. and Rindels, A.J. (1987), “Weak Vortices at Vertical Intakes”, J. Hydraulic Div., ASCE, HY9, 1101-1116.

Hite, J.E. and Mih, W.C. (1994), “Velocity of Air-Core Vortices at Hydraulic Intakes”, J. Hydraulic Div., ASCE, HY3, 284-297.

Iversen, H.W. (1953), “Studies of Submergence Requirements of High- Specific Speed Pumps”, ASME, Vol. 75, 635-641.

Jain, A.K., Kittur, G.R.R., and Ramachandra, J.G. (1978), “Air Entrainment in Radial Flow Towards Intakes”, J. Hydraulic Div., ASCE, HY9, 1323-1329.

45

Jain, A.K., Kittur, G.R.R., and Ramachandra, J.G. (1978), “Vortex Formation at Vertical Pipe Intakes”, J. Hydraulic Div., ASCE, HY10, 1429-1448.

Jiming, M., Yuanbo, L. and Jitang, H. (2000), “Minimum Submergence Before Double-Entrance Pressure Intakes”, J. Hydraulic Div., ASCE, HY10, 1429-1448.

Knauss, J. (1987), “Swirling Flow Problems at Intakes”, A.A Balkema, Rotterdam.

Oakdale Engineering web site, http://www.oakdaleengr.com/download.htm, last accessed on 07.03.2009.

Odgaard, A.J. (1986), “Free-Surface Air Core Vortex”, J. Hydraulic Div., ASCE, HY7, 610-620.

Padmanabhan, M. and Hecker, G.E. (1984), “Scale Effects in Pump Sump Models”, J. Hydraulic Engng., ASCE, 110, HY11, 1540-1556.

Reddy, Y.R. and Pickford, J.A. (1972), “Vortices at Intakes in Conventional Sumps”, Water Power 1972(3), 108-109.

Yıldırım, N. and Kocabaş, F. (1995), “Critical Submergence for Intakes in Open Channel Flow”, J. Hydraulic Engng., ASCE, 121, HY12, 900-905.

Yıldırım, N., Kocabaş, F. and Gülcan, S.C. (2000), “Flow-Boundary Effects on Critical Submergence of Intake Pipe”, J. Hydraulic Engng., ASCE, 126, HY4, 288-297.

Yıldırım, N. and Kocabaş, F. (2002), “Prediction of Critical Submergence for an Intake Pipe”, J. Hydraulic Res. 2002(4), 507-518.

Zielinski, P.B. and Villemonte, J.R. (1968), “Effect of Viscosity on Vortex-Orifice Flow”, J. Hydraulic Div., ASCE, HY3, 745-751.

46

APPENDIX A

EXPERIMENTAL RESULTS

This appendix provides detailed data for the experimental results of each

intake pipe used during experiments.

47

Table A.1. Experimental Results of Pipe 1

PIPE 1

Di= 19,72 cm

c=Di/2 9,86 cm

ν= 1,004E‐06 m2/s

ρ= 9,982E+02 kg/m3 at 20°Cσ= 7,280E‐02 N/m

Qi(l/s) Sc'(cm)* Sc

''(cm)** Sc'/Di Sc''/Di Vi(m/s) Fr Re We

1 21,62 14,98 24,84 0,760 1,260 0,708 0,509 1,39E+05 1,35E+032 23,68 15,78 25,64 0,800 1,300 0,775 0,558 1,52E+05 1,63E+033 25,45 17,78 27,64 0,902 1,402 0,833 0,599 1,64E+05 1,88E+034 27,57 19,78 29,64 1,003 1,503 0,903 0,649 1,77E+05 2,20E+035 30,15 19,98 29,84 1,013 1,513 0,987 0,710 1,94E+05 2,63E+036 30,98 21,38 31,24 1,084 1,584 1,014 0,729 1,99E+05 2,78E+037 32,13 21,98 31,84 1,115 1,615 1,052 0,756 2,07E+05 2,99E+038 34,90 22,48 32,34 1,140 1,640 1,143 0,822 2,24E+05 3,53E+039 36,33 22,78 32,64 1,155 1,655 1,190 0,855 2,34E+05 3,83E+0310 38,85 24,48 34,34 1,241 1,741 1,272 0,915 2,50E+05 4,37E+03

* Sc' is from summit point of the intake

** Sc'' is from intake center of the intake

48


PIPE 2

Di= 15,23 cm

c=Di/2 7,615 cm

ν= 1,004E‐06 m2/s

ρ= 9,982E+02 kg/m3 at 20°Cσ= 7,280E‐02 N/m

Qi(l/s) Sc'(cm)* Sc



11 34,68 24,67 32,285 1,620 2,120 1,904 1,558 2,89E+05 7,57E+03



49


PIPE 3

Di= 9,28 cm

c=Di/2 4,64 cm

ν= 1,004E‐06 m2/s

ρ= 9,982E+02 kg/m3 at 20°Cσ= 7,280E‐02 N/m

Qi(l/s) Sc'(cm)* Sc





50


PIPE 4

Di= 6,12 cm

c=Di/2 3,06 cm

ν= 1,004E‐06 m2/s

ρ= 9,982E+02 kg/m3 at 20°Cσ= 7,280E‐02 N/m

Qi(l/s) Sc'(cm)* Sc





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