Submerged Discharges of Dense Effluent

R79-35

SU BMERGED DISCHARGES

OF

DENSE EFFLUENT

by

Siu Shing Tong

and

Keith D. Stolzenbaeh

RALPH M. PARSONSLABORATORY

FOR WATERRESOURCESAND HYDRODYNAMICS

Report No. 243

Pre pa red under the support of the

Marine Environmental Assessment

Division

and the

Sea Grant Program Office

of the

National Oceanie and Atmospherie Administration

U.S. Department of Commerce

and the

Strategie Petroleum Reserve Office

U.S. Department of Energy

MIT report, no. 243.

Tong, S.S., and K.D. Stolzenbach.

Submerged discharges of dense effluent.

1979

.

~~:l

NAAM

vakgroep Kustwaterbouwkunde.

NAAM Datum uitlening Datum terugontvangsl

I

Gl

"'

;;

cl

'/'J-/

₉

R 9-35

Bibliotheek '.

(

SUBMERGED DISCHARGES OF DEN SE EFFLUENT

by Bibliotheel(

afd. CivieleTecnniek

T.H

.

Stevinweg ~ '::Detft.

Siu Shing Tong and

Keith D. Sto1zenbach

Ra1ph M. Parsons Laboratory for

Water Resources and Hydrodynamics Department of Civi1 Engineering Massachusetts Institute of Techno1ogy

Report No. 243

Prepared under the support of the Marine Environmental Assessment Division

and the

Sea Grant Program Office

Nationa1 Oceanic and Atmospheric Administration U.S. Department of Commerce

and the

Strategie Petroleum Reserve Office U.S. Department of Energy

ABSTRACT

An analytical and experimental study of discharge of dense fluid is conducted. Of interest is the near field dilution obtained by a multiport diffuser structure located on the bottom of the water body and consisting of vertical ports extending above the bottom and spaced equally along a line perpendicular to a steady receiving water current.

Experiments are performed in a laboratory flume in which a single dense jet with varying discharge and cross flow condition are studied. Some two dimensional two layer flows obtained by multiport discharges are also studied.

On the bases of the laboratory investigations, the near field is separated into four regions; the jet region; impact region; upstream region and downstream region. Empirical equations for dilution and geometry of the jet based on dimensional analysis are obtained for the jet region. A previously developed two dimensional theory for the water depth in a channel below free overfalls is applied to the impact region to provide a boundary condition for the analysis of the up-stream wedge region. The observed behavior of the upstream wedge is found to correspond to frictional two-layer flow theory.

Application of the results to an actual design is possible if the effluent flow rate and density, receiving water depth and current

ranges are known. A design procedure is developed for choosing the port diameter, discharge velocity, port spacing and port height to meet a specified level of near field mixing.

ACKNOWLEDGEMENTS

This study was supported by the Strategie Petroleum Reserve Office of the U.S~ Department of Energy and the Marine Environmenta1 Assessment Division and the U.S. Sea Grant Office of the Nationa1

Oceanic and Atmospheric Administration. (OSP #84669, #84830, and #86400). The experiments and analyses were performed by Mr. Siu Shing Tong

Research Assistant under the supervision of Di. Keith D. Sto1zenbach, Associate Professor of Civi1 Engineering. The assistance of Mr.

W.R. De1ap1ane, Jr. of the U.S. Department of Energy and Dr. Dai1 Brown and Captain Charles A. Burroughs of the Marine Environmenta1 Assessment Division is gratefu11y acknowledged. Thanks are a1so due to Dr. E. Eric Adams for wi11ingness to share his know1edge and insight into various aspects of the study, to Mssrs. Mark Pape, Edward McCaffrey, Bob Rar1eman, and Roy Mi11ey for assistance in the construction of the experimenta1 apparatus, and to Zigrida Garnis and Caro1e Solomon for their ro1e in the preparation of the report.

ABSTRACT ACKNOWLEDGEMENTS TABLE OF CONTENTS I INTRODUCTION TABLE OF CONTENTS 3 4 6 1.1 Background

1.2 Scope and Objectives of this Study 1.3 Summary of the Study

6 7 7 11 EXPERIMENT INVESTIGATION 10 2.1 Introduction 2.2 Description of Apparatus 10 10 2.2.1 Model Set-up 2.2.2 Instrumentatiön 10 13 2.3 Experimenta1 Procedure

2.4 Presentation and Discussion of Data 2.4.1

2.4.2

15

17

Summary of Run Parameters Discussion of Resu1ts

17

17

111 THEORETICAL ANALYSIS 25

3.1 Introduction

3.2 Dominant Flow Region 3.3 Dense Jet Regions

25 26 28 3.3.1 3.3.2 3.3.3 3.3.4 General Characteristic Dimensiona1 Ana1ysis

Review of Previous Investigation Ana1ysis of Data 28 28 33 36 3.4 Impact Region 3.4.1 3.4.2 3.4.3 45 Review of Previous Investigations

Theoretica1 Model of this Study Discussion 45

46

49 3.5 Upstream Region 3.5.1 3.5.2 51 Review of Predictive Mode1s

Ana1ysis of Data

51 55

3.6 Do,~stream Region

Page 58

IV

DESIGN APPLICATION AND CASE STUDY 63

4.1 Design Application ₆₃

4.1.1 4.1. 2 4.1.3

Statement of the Problem

Application of Experimental and Theoretical Analysis Summary of Design Procedure

63 65

73

4.2 Case Study: Offshore Brine Disposal 74

4.2.1 4.2.2 4.2.3 Background Site Information Design Considerations 74 74

75

V CONCLUSIONS AND RECOMMENDATION FOR FUTURE STUDY

79

5.1 Summary and Conclusions

5.2 Recommendation for Future l-lork

79

82 BIBLIOGRAPHY

84

LIST OF SYMBOLS 87 APPENDIX 92 5

CHAPTER 1 INTRODUCTION

1.1 Background

BibliottiéeR

afd. Civiele TectinTel<

IK

Stevinweg 1 - Delft.

Liquid eff1uents with densities greater than water may resu1t from a number of industria1 processès. Among these are ~he cases of

desa1ination p1ants, solution mining of salt, and brines associated with the extraction of oi1 and gas. In many cases, because of

economic or technica1 considerations, these materials are disposed of into the nearby natura1 water bodies such as rivers, 1akes and oceans. The quantities and properties of the eff1uents vary for different activities. Brine from an oi1 field may have a sa1inity of 100 ppt and a pumping rate of 0.02 m3/sec. Effluent from a desa1ination plant of 10 mi11ion gallons per day capacity may have a concentration of 60 ppt and flow rate of 0.6 m3/sec. Solution from salt mining may be .

essentia11y saturated with sa1inities ón the order of 250 ppt. In addition to the excess sa1inities mentioned above, a dense effluent may contain potentia11y toxic substances. Metals common1y found in brines are lead, zinc, copper, arsenic, cadmium and nickel. The concentration of these may be as high as 10 ppM. Typica1 va1ues of copper range from 40 to 380 ppb.

Because of the potentia11y high concentrations of salt and other toxic substances mentioned above, it is c1ear that protection of the marine environment must at a minimum invo1ve a mechanism for reducing the concentrations from effluent va lues to be10w critica1 levels.

Where treatment of effluent streams is not practical, such concentration reduction must be accomp1ished by rapid mixing of the effluent with the

receiving water by the use of a properly designed discharge structure.

1.2 Scope and Objectives of this Study

This study is motivated by the need, as mentioned in the previous section, to achieve a desired level of mixing of a dense effluent with a receiving water body. The basic scope and objectives of this investigation are as follows:

(a) Of interest is the near field dilution obtained by a multi-port diffuser structure located on the bottom of a coasta1 receiving water body and consisting of vertical ports extending above the bottom and spaced equally a10ng a line perpendicular to the receiving water current as shown in Figure 1.1. This type of discharge is likely to be employed where existing structures are not availab1e for mounting near surface discharges. The effluent concentration distribution associated with such discharges also exhibits intermediate field

(lateral spreading) and a transient far field (natural diffusion and advection) regions as shown in Figure 1.1. However, for this study the focus wi1l be on the near field region.

(b) This study will assume that for a given site the effluent flow rate and density and the receiving water depth, and current ranges are known. The objective of this study is to develop, by experimental and theoretical investigations, a design procedure for choosing the nozz1e diameter, discharge velocity, port spacing and port height above the bottom to meet a specificed level of near field mixing and associated reduction of effluent constituènt concentration.

1.3 Summary of the Study

In Chapter 11, a purely experimental investigation of dense effluents 7

v

-NEAR FIELD (SIDE VIEW)

----

---

---_.

_

INTERMEDIATE FIELD (TOPVIEW)

--

-

--

-

-

-

--

-

-

-

----

-

-

--

-

-

- -

--FAR FIELD (TOPVIEW) ~ DIFFUSER

U./

a

Figure 1.1 Schematic Views of Regions of a Discharge of

Dense Effluent from a Submerged l1u1tiport Diffuser 8

is discussed. In Chapter 111, the results of the experiments are analyzed and compared with various theoretical modeis. Af ter the complete flow pattern associated with nearfield mixing is weIl described by models selected in this way, the roodels are synthesized in Chapter IV to produce a set of design recommendations for submerged discharges of dense effluents. A case study is used to illustrate the application of these principles. Finally, in Chapter V, some conclusions and suggestions for further study are presented.

CHAPTER 2

EXPERIMENTAL INVESTIGATION

2.1 Introduction

This chapter describes the laboratory equipment and procedures used in the experimental part of the investigation. Two different set-ups were used: (1) a single jet to study the jet region behavior of a multi-port diffuser (2) a multi-port diffuser to produce uniform two dimensional flow for examining two-dimensional characteristics.

The experimental apparatus and procedures are described in

Section 2.2 and 2.3. Experimental parameters and results are presented in Section 2.4.

2.2 Description of Apparatus 2.2.1 Model set-up

(a) Basin

All experiments were performed in a metal tank 17 meters long, 1 meter deep, and 1.5 meter wide as shown in Figure 2.1. A false floor was constructed 20 cm off the bottom. A movable side wall was built to reduce the channel width to 60 cm. Both the false floor and side wall were made of 2 cm thick latex paint coated plywood. The water level was controlled by a system of two weirs downstream. The first one was a vertical piece of plywood extending 60 cm above the false floor perforated with holes which could be pluged when the channel flow was smalle The second weir was a metal sheet inclined at a variable angle for the purpose of maintaining the water level downstream of the plywood weir.

r

SM

·1

Diffu ...

metal flume """"\

D,.fltl

113ecm weir...J

I

2 cm plywood wal!

:l

•

.",.

1~"'"

~ - - --- -)et _. ~ Drain 11 M TOP VIEW rota~e" gla. wal!

,

~.

I

-I-" IM I-"

A.J

~60C~ 20 cm Section AA SIDE VIEW carr_ t ~ ~. i

•

•

T

_{90 cm}

1

concrete block

I·

60c:f S. ctfon BB

(b) Single jet nozzle and multi-port diffuser structure The single jet nozzles were made out of copper tubings. For some desired sizes which were not available in commercial tubing, plastic pipette tips were cut to get the right sizes and placed on top of a copper tubing. In order to make the jet angle easily adjustable, a 3 inch PVC balI valve was converted into a rotatabie connection

between the nozzles and the salt water supply hose. One end of the valve was sealed. The balI inside the valve was rotated 90 degrees and a

copper tubing connection was threaded into the balI to hold the nozzles. The handle of the valve was removed and became the inlet of salt water.

The multi-port diffuser was constructed by connecting a piece of 1 cm diameter plexiglass tubing horizontallyon top of a vertical nozzle. Equally spaeed holes were drilled on top of the plexiglass tubing and the ends sealed.

(c) Channel water supply

The channel flow water was supplied from a BOOOcubic meters reservoir through a centrifugal pump. In all cases the water had been stored for 24 hours to keep the channel water temperature close to room temperature. Channel flow rate was measured by a series of six parallel rotameters. Downstream at the end of the tank the channel flow was drained by gravity either to the city sewer or back to the reservoir.

(d) Dense water supply

Hot water was supplied by a steam heat exchanger and mixed with cold water controlled by a mixing valve. The mixed water was around 60°C and was discharged into a one cubic meter constant head

tank. Concentrated salt water with specific gravity of 1.2 was stored in a one cubic meter container and pumped up to a temporary storage which had an automatic water level control feedback to the pump. Salt water flowed down by gravity into a small constant he ad tank. The overflow from this head tank was directed back to the salt water container.

The flow rates of hot water and salt water were measured by two rotameters. Small diameter tubing was connected upstream of the hot water rotameter to bleed off the air bubbles. Hot and salt water merged af ter the rotameters and passed through 10 meters of insulated hose into the experimental flume. Small amounts of concentrated dye was injected into the hose before it reached the flume. A T-connection was instalIed for the purpose of sampling and flushing out the left-over fluid in the hose. Figure 2-2 shows the arrangement of the dense water supply system components.

2.2.2 Instrumentation (a) Flow meters

Flow rates were measured by different sizes of Brooks rotameters.

(b) Initial specific gravity measurement

The initial specific gravity was measured by a Troemner balance with an accuracy of lppt.

(c) Temperature measurement systems

(i) Initial discharge temperature was measured by a Yellow Spring Instrument model 401 thermistor probe inserted inside the PVC

valve with a United System M468A digital readout. Time constant of the probe was 9 seconds.

,_.

.t:-pump salt wat.r r. s.rvoir drain automatic water level control hot wat.r h.ad tank bl .. d rotam.t." • hot wat. I

r eo-ci

h.at uchanQ.r dy •

Figure 2.2: Dense Water Supply System

samplinQ ta p

(ii) Channel flow and dense layer temperatures upstream and downstream from the dense jet were measured by a Yellow Spring

Instrument model 700 thermistor probe with a United System M636 scanner, a model 251 digital readout and a model 691 digital printer. The time

constant of the 700 probe was one second. A single probe was mounted on the upstream screen for ambient temperature measurement. Thirteen thermistors were grouped together with 2 cm separation to measure the vertical temperature distribution in the dense layer.

(iii) Temperatures in the discharge plume were measured by a fast responding (0.1 sec.) temperature system consisting of a Fenwal #K486C thermistor mounted on a motor driven point gauge.

Horizontal position of the probe was measured by a potentiometric voltage divider. Temperature distribution and position along a horizontal

profile was recorded on an Omnigraphic x-y recorder.

2.3 Experimental procedure

a) The channel flow was first set up by adjusting the valves at the rotameters. The flow distribution was recorded by muIt i-exposure Polaroid photography. Some or all of the holes in the plywood weir would be plugged if the cross flow was low.

b) The left-over salt water in the hose was flushed out and the initial specific gravity was measured with the Troemner balance.

c) Dye injection into the discharge flow was started. The flow was permitted to developed for 30 to 60 minutes until the dense fluid distribution was observed to be stabIe. The upstream wedge tip location was checked frequently. Crystals of potassium permanganate were dropped

into the channel to indicate the flow direction on the bottom. This information was recorded by color pictures and by hand sketches.

d) Ambient and downstream fixed probe (YSI 700 thermistor) were scanned every 5 minutes. Initial temperature reading from the YSI 401 probe was recorded for all temperature scans. For large size nozzles, the fast response Fenwal K496C thermistor was used to locate the

maximum temperature along the plume. Three tb six cross-sections of the plume perpendicular to the channel flow were taken. In each cross section, the carriage traveled horizontally and the point gauge was set at

various heights to find the point of maximum temperature difference. For thin dense.layers the vertical temperature was measured by a single YSI 700 probe mounted on a point gauge. Detailed temperature distribution we re obtained by moving the probe vertically in small increments. In each increment, ten to thirty readings were recorded to get the average temperature. The probe was moved very slowly so as not to disturb the layer. Those measurements were taken away from the region of jet impact (the place where the plume first hit the floor) to avoid three dimensional effects. The visual interface of dense and fresh layers were also recorded for comparison.

e) The initial specific gravity was checked again af ter all data had been taken.

2.4 Presentation and Discussion of Data 2.4.1 Summary of run parameters

The parameters relevant to each test are defined schematically in Figures 2.3 and 2.4. Tables 2.1 and 2.2 give the value of all para-meters for all tests. In addition, more detailed results are presented for each test in Appendix I.

2.4.2 Discussion of results

The experimental runs were divided into distinct groups, each with a particular discharge set up and with a distinct purpose. The general characteristics and results from each group of tests is summarized in the follow!ng:

(a) Tests No. 4 to 16 were single jet experiments with large nozzle diameters. Three different orientations of nozzles instalied perpendicular to cross flow were studied. A fast response thermistor probe was used to determine the jet centerline trajectories. The temperature distribution was found to be relatively insensitive to the orientation of the nozzle. For high cross flow the interface between the salt and fresh flows was indistinct. The round jets were distorted

immediately af ter exiting from out of the nozzles apparently by the wake formation on the backside of the jet. In some of the cases the nozzle

was found to be submerged by the dense bot tom layer and a stationary

wedge formed upstream of the nozzle.

(b) Test No. 17 ~s a single jet test with a relatively low Froude number. The nozzle diameter was also reduced. A thiner dense

layer was observed. The cross flow was set to be high enough to keep the

upstream wedge away from the nozzle.

... (XI

u.

ZJ H h Z

_--",

....

,

... Wa

t.r

Surface

XI

"

,

_,

_,

\ \ \

,

\ \ \ \ \ \ Impact point

Figure 2.3: Schematic of Single Jet Parameters

hl

h2

h _{Wo ter Su rfo ce} ----+ H

ua

z

~ 1.0 /

/

/

/

/

/

I

I

I

I

I

J_

/~

D

I

.

X

h2

XI

y

N

o

D.

U.

g~

u

a

H

z.

J J _J

2

a

J

Test No.

(cm)

(cm/sec)

(cm/sec')

(cm/sec)

(degree)

(cm)

(cm)

4

.794

108

32.8

6.66

90°

64

3

6

.794

108

29.7

6.66

60°

64

6

7

.794

108

34.4

7.04

45°

64

4

8

.794

100

33.9

7.05

45°

64

4

9

.794

100

39.9

6.93

90°

64

3

10

.794

100

32.2

7.07

60°

64

6

11

.794

100

30.4

5.53

60°

64

6

12

.794

100

35.3

2.29

60°

62

6

13

.794

100

34.7

0.97

60°

62

6

14

.794

100

33.8

2.64

90°

62

6

15

.794

100

35.3

0.98

90°

62

6

16

.794

100

33.8

2.49

90°

62

6

17

.366

43.4

37.2

4.70

90°

63

6

18

.318

57.7

28.4

3.87

90°

63

7.6

19

.318

57.7

28.4

4.33

90°

63

7.6

20

.318

57.7

32.3

3.32

90°

63

7.6

.

21

.318

57.7

33.3

2.85

90°

62

7.6

22

.318

57.7

33.3

5.29

90°

64

7.6

N

I-'

Dj

U. _{g~ U R- H}

Zj

J _J

₂

a

0

Test No.

(cm)

(cm/sec)

(cm/sec

·

)

(cm/sec)

(cm)

(cm)

(cm)

25

0.132

41.8

33.2

4.18

7.62

63

10

26

0.132

45.4

48.6

4.18

7.62

63

10

27

0.132

45.4

48.6

4.85

7.62

63

10

28

0.132

45.4

48.6

4.51

7.62

63

10

29

0.132

45.4

48.6

4.21

7.62

63

10

(c) Tests No. 18 to 22 were a series of vertical, small diameter single jet experiments with different cross flows. Complicated

flow patterns were ohserved downstream from the point where the jet hit the hottom of the channel. An ohlique hydraulic jump with or without normal hydraulic jump in the middle was generally formed. A short

distance af ter the jump (40 cm to 80 cm), the hottom flow reversed its direction.

(d) Tests No. 25 to 29 were multiport diffuser experiments designed to provide uniform two dimensional flow patterns. The ports were àll directed vertically. The hydraulic jump downstream from

the jet impact point was of the normal, rather than ohlique type. It was ohserved that the wedge dep th decreased dramatically in the upstream direction. The net dilutions for these tests were found to he higher than those in single jet tests.

Figures 2.5 and 2.6 show a series of discharge conditions which do not correspond directly to any of the quantitative experimental runs hut which show the general configuration of the dense jet and

the formation of the hottom layer, particularly at low cross flow ratios.

co r-I ...:t 00 11 0 0 _or; ~ 11 11 ..c: 1-4 1-4 ~ :> :> "r"l ) Ul ~ al ...., 23

v

=

.02

r

v

r

=

.04

N ~

v

=

.06 r

v =

r .08

Figure 2.6: Photograph of Dense Jets with F. J 10

CHAPTER 3 THEORETICAL ANALYSIS

3.1 Introduction

In this chapter the experimental data are analyzed in the context of the theoretical models for different regions of the flow as defined in Section 3.2. Each region is then treated separately in Section 3.3 to 3.6.

Because of the complexity of the general problem defined in Chapter 1, only the following simplified case is treated:

(a) all flows are steady

(b) the receiving water current is perpendicular to the diffuser (c) the diffuser is very long and only the center portion away from the end is being considered

(d) the water is sufficiently deep that the jet will not hit the surface

(e) the bottom is horizontal

(f) the receiving water density is constant

(g) the nozzle spacing is assumed to be long enough to avoid inter-action between jets, but at the same time close enough to neglect three dimensional effects for the dense layer~ it is also assumed that the spacing between nozzles is constant

(h) the jets are turbulent and directed vertically

(i) the density difference is small enough to apply Boussinesq approximation.

3.2 Dominant Flow Regions

The dense fluid flow is divided into four regions. Figure 3-1 shows the location of each region.

a) Dense jet region

Immediately af ter coming out of the diffuser port, the dense fluid forms a turbulent jet. The dense jet reaches a maximum height above the bot tom and then falls back. Ambient water is entrained as a result of turbulent dilution. The jet is deflected by the

receiving water current. b) Impact region

In the impact region the falling jet is diverted upstream and downstream by the presence of the bottom, forming a dense layer.

c) Upstream region

The part of the fluid that goes upstream against the current as a result of its excess density will eventually form a steady wedge of finite length. There will be no net flow across a vertical section in the wedge when it reaches a steady state.

d) Downstream region

In the downstream region, an abrupt transient between the high velocity flow in the impact region and low velocity far downstream may occur in the form of hydraulic jump. Additional dilution may occur at this point. A two layer stratified flow is generally maintained for some distance downstream.

26

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Ifd. Civiele Techniel</ •

CD

CROSS FLOW

N -...J

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\

_..,.,

..,.,-..,.,

,...

,...

,...

..,.,

®

---.

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___.

®

__...

__.

...

<D

DE NSE JET

REGION

®

IMPACT REGION

®

UPSTREAM

REGION

@

DOWNSTREAM

REGION

3.3 Dense Jet Region

3.3.1 General characteristic

The schematic of a round dense turbulent jet is shown in Figure 3-2. _{A flow,}

_Q.,

_{of dense effluent issues from a nozz1e with}

J.

diameter D. at uniform velocity U.. The jet width increases a10ng

J . J

the jet trajectory due to entrainment of ambient f1uid. The trajectory is def1ected as the entrained f1uid has momentum in x-direction.

The upward motion is dece1erated by entraining f1uid with no vertica1 momentum, and by the gravity force acting downward at the jet. The jet reaches its maximum height z at a distance x. The velocity vector

p p

of the fa11ing jet form an ang1e

e

with the horizontal bottom. 3.3.2 Dimensiona1 ana1ysis

For the simp1ified case defined in Section 3.1, the independent variables reduce to the fo110wing:

reduced initia1 gravitationa1 acce1eration nozz1e diameter

initia1 discharge velocity

velocity of uniform ambient current x x coordinate

y y coordinate z z coordinate

Any dependent variab1e ~ (velocity, minimum dilution, etc.) in the jet must be a function of the above independent variables.

~ = f(g~, D., U., U , x, y, z)

J J J a (3.1)

Dimensiona1 analysis indicates that there are five independent dimensionless

..

z

\

Ua

..,..--<,

'"

<,

<,

<,

-,

-,

-,

\

\

\

uJ~

---- -

--

'-- Vs

---..tJ

\

...

X

--

... N \.C)

g

~

/

/

/

/

/

I

I

- Xp

...

--l-____________

Xb

XI

groups in this problem. (Only leng th and time appear in the independent variables). One classical set of parameters are as follows:

*

(x

.s.

4>

D.'

D.' J J

U.

_2)=f( J D. r-t="D J yg:uj J (3.2)

where 4>* is a dimensionless dependent quantity, and

U.

__~J_

=

F.

=

discharge densimetric Froude number Ig!D. J

J J

(3.3)

u

_1!

=

U

=

cross fLow ratio

U. r

J

(3.4)

However, List and Imberger [21], Wright [29,30] and others have

demonstrated the advantage of considering the kinematic mass flux Q _{D. U.,}2 J J momentum flux M = Q uj and buoyancy flux B = gjQ, instead of gj, Dj and U., which gives

J

f(B,M, Q, U , x, y, z)

a (3.5)

Furthermore, various length scales can be defined by combining the above variables. 2

B g.U.D.

1

= ---

=

J J J buoyancy leng th scale

b U 3 U 3 a a (3.6) 1 m

Ml/2

D.U. = -_ = _J__J_

U

U

a a

momentum length scale _(3.7)

IQ

= ~

=

Dj volumetrie length scale _(3.8)

which again gives

(3.9)

Up to this point the two different approaches are equivalent. Eq. 3.2 can be obtained by expressing Froude number and cross flow ratio in terms

U. J 1 3/2 m F. J (3.10)

li'D.

J 1 1 1/2

Q

b Ua .IQ -=-= U U. 1 r J m (3.11)

However, the second way of grouping allows the simp1ification of

neg1ecting the initia1 volume

flux Q.

This approach is justified by

the high di1ution ratio expected in such jets. By doing that Equations

(3.5) and (3.9) become:

~ =feB, M, U , x, y, z)

a (3.12)

(3.13)

~ =

f(lb, 1m, x, y, z)

In Eq. 3.13, on1y the unit of 1ength appears in the variables.

The dimension1ess equation is now reduced to: M1/2U 2

~*(x*,y*,z*) = f( a )

B (3.14)

~*(x*,y*,z*) _(3.15)

The dimension1ess position veetors x*,y* and z* ean be chosen as

(x Ua/M1/2, y Ua/M1/2, z Ua/M1/2) and (x/1 ,y/1 ,z/l ) for Eq. 3.14 and mmm

amd 3.15 respectively. To express them in physical variables Ua, gj' and 3.6 obtains D., dividing

Eq. 3.7

J by

Eq.

1

Ml/2

u

2

U2

m a a -~ = --B-- = gjDj (3.16) which gives

u

2 f(F~.U 2) <jl*(~' _:j_, 2.) = _f(g~D.)a = D. D. D. _J r J J J J J (3.17)

It implies that dependent dimensionless variabIe

~*

can be

characterized by one parameter~ which may be used to define a new densi-metric Froude number.

U 2

a

=

g!D.

J J

(3.18)

This condition wil1 be true at points in the jet where the total volume flux is significantly greater then the initial volume flux.

(This will in genera1be true at distance much greater than IQ from the jet).

To define the jet induced dilution. however. it is necessary to express the volume flux Q at any point in the jet in dimensionless term:

x

(3.19)

or

(3.20)

The net dilution may then be obtained by dividing by the initial

mass flux,

Q.

S = net di1ution _(3.21)

or

(3.23)

Thus the need to define di1ution, S, has added another parameter, U , but r in a slight1y 1ess general formu1ation than Eq. 3.2.

3.3.3 Review of previous investigations

In the past, most research attention has been focused on the prob1em of positive1y bouyant jets. A theoretica1 model of a dense jet was most recent1y proposed by Chu, V.H. [7]. Anderson, J.L., Parker, F.C. and Benedict, B.A. [3] have performed a set of experiments and compared the resu1ts with modified integra1 bouyant jet mode1s. Ho11ey, F.H. and

Grace, J.C. [16] deduced a set of empirica1 equations for this prob1em

based on the experiments they performed. The fo11owing are a short

summary of these investigations:

(A) Chu's model: His model is based on the fo11owing

assump-tions:

(i) The hor1zonta1 ve1oc1ty components of the jet are

approx1mate1y equa1 to the cross flow.

(1i) The pressure drag is negligib1e compa red wit]1the

contr1bution from turbulent entra1nment.

(iii) The jet entrainment is based on a machanistic model which describes the formation of the vortex pair in the jet. The dependent quantities of the jet are found to be:

(a) the ascending jet;

(_3_)1/3 1 D. 2

₊

_{(~)2 x]1/3.} z = [- -(

J)

_x 4a2 2

U F :-

U

r r aj (3.24)

SU

r (6a)2/3

U 2

1 r (-"2( 2 2) D. F . J aJ U x2

+

_..!. x]2/3 D. J (3.25)

where z vertical coordinate of the jet trajeetory x horizontal coordinate of the jet trajectory

a empirica1 entrainment coefficient (b) at the maximum height:

(3.26)

(c) for the descending jet

2z 3_ (2z _ z)3 = _3_ p p 4a2 l D. [__ ( J)

2 U F ~

r aj (3.27)

SU

r U - _E x

+

F

.2]2/3

D. aj J (3.28)

(d) at the impact point, z

=

0:

2 D.

U U.

xI

=

(1

+

17)

F_aj _]_ '"_U 3.6~ _g r

(3.29)

(60.)2/3 [4 F :]2/3 U aj r

(3.30)

di =-dx '"0.5 (F_aJ

:)-2/3

(3.31)

The entrainment coefficient Cl was found to be a constant approximate1y

equa1 to 0.5. To re late the characteristic di1ution S with the minimum di1ution Sm' Chu fitted experimental data in buoyant jets to obtain,

S = 3.54 S

m

(3.32)

Combining with Eq. (3.30) and a.

=

0.5, the above gives the minimum dilution at the impact point,

(3.33)

(B) Anderson, Parker and Benedict performed a series of dense jet experiments with various cross flow ratios Ur and densimetric Froude numbers F.. Integral jet models were used to compare the experimental

J

resu1ts, obtaining reasonable results for jet trajectories and dilutions using constant entrainment coefficient.

(C) Ho11ey and Grace empirically studied the effects of the independent variables, U , U., Dj and ~p on the maximum height, lateral

a J

spreading and downstream dilution of dense jets. They proposed the following empirical equations.

(3.34)

S

m (3.35)

Pincince, A.B. and List, E.J. [20] had pointed out that scat ter in Ho11ey and Grace's experiment was very large and the formu1a was obvious1y incor-rect because, for F. equa1 to 0.5/U , the di1ution became infinitely large.

J r

3.3.4 Ana1ysis of data

The dimensionless jet parameters as defined in Section 3.3.2 are 1isted in Tab1e 3.1 for this study and in Table 3.2 for previous

studies. The results are p10tted on Fig. 3.3 to 3.7 using the relationship obtained from dimensional analysis.

Fig. 3.3 shows the minimum di1ution at the impact point. fit curve for the data _{in this study is:}

2 (F ~)0.67

~=-I

U

_r aJ

The best

(3.36)

which has exactly the same form as Chu's model with a 30% higher constant. The experimenta1 data obtained by others are found to have slight1y higher va1ues but to be in general agreement with the data from this study.

Fig. 3.4 shows the minimum di1ution along the x- axis. Again, the shape of Chu's theoretica1 curve and the data are simi1ar. For low

W -....J

F.

u

D./U

F2

Smr

_XrU/Dj

e

J

r

J

r

aj

Test No.

_(U;//giD;)

(Ua/U;)

(1 )

_m

(U 2/g!D.)

_a

(~T

/~T)

(Xr/1m)

(degree)

J_.l,

max

0

4

21.2

0.062

12.8

1.73

48

4.69

45

9

17.7

0.069

11.5

1.49

40

4.17

45

14

19.3

0.026

30.5

0.26

26

0.721

75

15

18.8

0.010

79.4

0.034

20

0.213

80

16

19.3

0.025

30.5

0.25

26

0.721

75

17

11.7

0.108

3.39

1.60

26

4.42

45

18

19.2

0.067

4.75

1.65

35

4.84

45

19

19.2

0.075

4.24

2.07

50

7.07

30

20

18.0

0.058

5.48

1.09

50

2.74

55

21

17.7

0.049

6.49

0.75

50

2.16

50

I

22

17.7

0.092

3.46

2.65

--

8.24

25

,

lAl co

F.

u

D./U

_Faj

2

Smr

XrUr/Dj

J

r

J

r

Test No.

_(U;//g;D;)

(U/U;)

(1 )

_m

(Ua2/giD

i)

([lT

_max

hT)₀

(Xr/1m)

APB 15

10

0.05

0.63

0.25

18

1.25

APB 9

10

0.10

0.31

1.0

25

4.00

APB 10

10

0.20

0.16

4.0

40

11.0

APB 16

20

0.05

0.63

1.0

30

3.0

APB 11

20

0.10

0.31

4.0

55

13.0

HG 619

25

0.012

41.5

0.09

41*

0.29**

HG 627

25

0.024

20.8

0.36

54*

3.36**

HG 635

25

0.036

13.9

0.81

71*

3.24**

HG 643

33

0.038

13.2

1.57

98*

15.2**

HG 659

25

0.057

8.77

2.03

115*

5.7**

RG 667

24

0.086

5.81

4.26

224*

34.4**

~-- ---- ~

Tab1e 3.2

Dimension1ess jet parameters obtained by others.

**

Anderson, Park and Benedict

Ho11y and Grace

Based on empirica1 equation derived by Ro11y and Grace

interpreted from di1ution plots, impact location based on empirica1 equation derived by

Rolly and Grace

APB

RG

o

This study

•

Anderson, Parker and Benedict

x

Holly and Grace

Chu' s Model x x •

x

2

=-.

~

10 F 2 aj U 2 a

=

g~D. ] ]

Figure 3.3: Minimum Dilution at Impact Point

Test F

7

Symbols No. aJ

•

4

1.7

0

₁₈

_1.7

•

17

1.6 S U ₀ ·9

1.5

m r A ₁₄

_.23

•

15

Chu's

2U

D r j

Figure 3.4: Minimum Dilution Along the x-axis

2

value of F ., the Chu's model predicts higher dilution than observed. The aJ

2

Chu's model and data agree very well when Faj approximately equal 0.23.

For F : around 1.6, the prediction is lower than observation. aJ

Fig. 3.5 shows the impact distances normalized by D./U. Once again

J r

the form of the best fit curve is the same as Chu's model. The constant

is around thirty percent lower. For F . equal to 0.034, the data deviates2

aJ

from the straight line. In this run, the cross flow ratio was very low

and the jet trajectory cannot be easi1y determined. This deviation may

a1so be caused by breakdown of previous assumptions for 10w va1ue of 2

F .. Until further studies are performed in this region, the re1ationship

aJ

of impact distance to F : is assumed to be:

aJ

XI 2

-D U = 2.8 Faj

. r

J (3.37)

Fig. 3.6 shows the maximum jet height norma1ized by Dj/Ur•

jet height zp was found to be different from Chu's model. The best fi-t

The maximum

curve for the data in this study is: z

t

Ur = 1.7 Faj (3.38)

J

The observed impact ang1es are p10tted on Fig.

3.7.

The scatter in

this plot is 1arger due to experimental difficu1ty in measuring the

impact ang1e. Impact angles are determined by ang1e measurement on

photographs in increments of 5 degrees. According to Chu's model,

the curve shou1d have a slope equa1 to

-2/3.

A best fit curve with

this slope is found to be,

tan

el

= 1.3(F :)-2/3

aJ (3.39)

and the curve shows good agreement with the data. The run with F :

aJ equa1 to 0.034 is neg1ected in this fitting for the reason mentioned before.

10 o Impact Distance Best Fit Chu's Model

XI

- U = D. r J 2

=

2.8 F . aJ .I~ __ L-~_L-~-LLLL- __ L-~_L-~-LLLL- __ L-~_L-~-L~ .01 . I 10 F 2 aj I U 2 a

=

g~D. J J Fig. 3.5 Impact Distance

10

o

Maximum Jet Height

Best Fit Chu's Model .IL-__~L-_L~_J-L~~ ~ __~~~~~~--_J--~~_L~LLU

.01

. I I

U 2

a = gjDj 10 . 2

F .

aJ

Fig. 3.6 Maximum Jet Height

10 -- --- _ o . This Study Bes~ Fit Chu's Model U 2 F ~ a aj = g

!D.

J J

Figure 3.7: Impact Angles

3.4 Impact Region

When the falling dense jet hits the bottom, the fluid spreads as a dense bottom layer. Most of the fluid from the jet will be diverted in the downstream direction immediately. A portion of the jet flow will go upstream into a region of zero net flow and then will be reentrained by the jet. To understand the upstream region, it is necessary to predict the heights and velocities of the dense fluid when it leaves the impact region. Previous studies of buoyant jets impinging on flat surfaces are reviewed in the next section. In Section 3.42, a theoretical model based on a free overfall analysis is formulated. The theory and

the experimental data will be compared in Section 3.4.3. 3.4.1 Review of previous investigations

No studies of the impact region of d.ensejets have been performed. Jirka, G. [17] had studied the impinging region of a two dimensicnal

buoyant jet in shallow water. He drew a control volume over the whole

impingement region and related the energy loss to the two velocity

heads upstream and downstream. The two different head loss coefficients

were dependent on the angle of the impinging jet and were approximated

by experimentally determined head loss coefficients for flow in smooth

pipe bends.

Lee, J.H. [22] similarly treated a three dimensional round buoyant

jet impinging the water surface by radial hydraulic jump. He neglected

the external force and assumed a very abrupt hydraulic jump with negligible

jump length. This approach is applicable only in an unconfined region

and with no return flow. For this study the return flow from upstream

and lateral interaction with other impact regions make this approach

inapplicable.

3.4.2

Theoretical model of this study

A theoretical model analogous to the basin flow under an overfa11 is adopted here to ana1yze this region. The part of f1uid f10wing upstream after impact is said to be stopped by static pressure somewhere upstream. This sets up a circu1ation upstream of the impact point. A model of two dimensiona1 flow is first deve10ped to gain some insight of the prob1em. Three control volumes are drawn to separate this complex flow phenomenon into manageab1e sub-regions.

(a) Fig. 3.8 shows the first sub-region, the control volume I. Inside this control volume the flow is treated 1ike a uniform jet hitting an inc1ined p1ain as shown in Fig. 3.9. Both pressure difference and energy 10ss are assumed to be zero. The net densimetric gravity effect (static pressure of hb and h2 with g')is said to be neg1igib1e compared to the velocity head. This simp1ification is not abso1ute1y necessary but wil.! help simp1ify the resu1ting formu1ation. The energy conservation assumption leads to:

(3.40)

Momentum equation in x direction can be written for control volume I in Fig. 3.8 with the resu1t of Eq.

3.40:

(3.41)

Plus the mass conservation equation

(3.42)

Figure 3.8: Control Volume I

Figure 3.9: Jet Hitting an lnclined Plain 47

immediately gives the relationship of q2 and qb as a function of the impact angle

q2 1

+

cos

e

·

- =

-":'-_....:....:..~-=-qb 1 - cos

e

(3.43)

(b) The second control volume is drawn outside of the first one.

(Fig. 3.10). The previous simplification of negligible net gravity

force is again used. Pressure inside the re-entraining zone is assumed

to be the same as outside.

The re-entrained fluid qe is assumed to have negligible momentum.

With these assumptions and Eq. 3.40 the momentum equation in x direction

can be written as:

where qt js the total flow input to this region. The flow input to

this region contains mostly entrained ambient fluid having a velocity

U. It is therefore reasonable to say that: a

(3.45)

The mass conservation equation for this control volume gives:

(3.46)

By combining equations 3.44, 3.45, and 3.46, the following expressions

are obtained:

u

= U (2 cos

e

)

a i 1

+

cos

e

(3.47) U2

=

(1

+

cos e) U 2 cos

e

a (3.48) 48

(e) To find the height of h3, the third control volume is extended upstream to the point where most of the f1uid has been returned (Fig. 3.11). The gravitationa1 forces are put back to the x momentum equation as

h3 is much higher than h2•

(3.49)

where gi is the redueed gravity at impact =

Bj/SI

governing the depth upstream of impact point is obtained: h

(2)-2 =

H (3.50)

Define the channe1 densimetric

2

2

2

q q

SI

F a a

aI = 'H3 = q ~H3

gI J

Froude number at impact F

aI

(3.51)

Eq. 3.50 can be rewritten as:

(3.52)

3.4.3 Discussion

The above theoretica1 model gives an explicit expression of h3/H as a function of jet impact ang1e 8, channe1 densjmetric Froude number

at impact-Fa ,and flow ratïo q2/qa' The jet flow at impact q2 ean

I .

easi1y be obtained by setting:

h Q _j[. Q = _lL D2 U

w ere j

=

4 4 j j

Figure 3.10: Control Volume 11

Figure 3.11: Control Volume 111

A more complicated expression can be deduced if it is desired to account for the net reduced gravity force in the first and second control volume. The simplified model is believed to be adequate specially when the density difference is small and the angle of impact is large.

Since the layer depth h3 is thin and close to the impacting jet, the height is hard to measure. Verification of this ·model will be

included in the discussion of the next region.where the predicted height

h3 will be used as the downstream control of the upstream wedge.

Estimated h3 from photographs are found to agree with the model quite

well.

3.5 Upstream Region

The behavior of the upstream wedge is one of the major concerns of

this study. A prediction of wedge height is necessary to prevent

submer-gence of the diffuser port by the dense bottom layer. Section 3.5.1

discusses the possible governing mechanisms for this region. The theories

will then be compared with experimental results in Section 3.5.2 and the

choice of theory will be made based on the observed behavior of the

upstream wedge.

3.5.1 Review of predictive models

There are two major but completely different theories describing

the intrusion of a wedge of heavy fluid upstream against a current. The

first is the gravity current analysis proposed by Von Kármán, where

excess gravity force from the heavier fluid is balanced by hydrodynamic

pressure drag on the layer. The second theory proposes that the excess

gravity force is balanced by frictional force at the interface between 51

the two fluids.

(a) Gravity current model:

Benjamin T.B. [5] modified Von Kármán's derivation and obtained an expression re1ating the velocity to the depths and densities of the two f1uids. He proposed that a stagnation point existed at the tip of the wedge and that the dynamic pressure everywhere'in the heavier f1uid was constant. To simp1ify the prob1em, the free surface e1evation changes are rep1aced by statie pressure difference. With Benjamin's assumption the fo11owing re1ationship ho1ds

U 2

a

PA - Po

=

~p

=

p 2 _(3.54)

where PA is the pressure inside the wedge and Po is the pressure far upstream.

App1ying an integra1 momentum ba1ance for the control volume as shown in Fig. 3.12 yie1ds,

2 1 2 pUa H

+

~gH _(3.55) combining F 2 a with Eq. U 2S a g!H

=

J

3.54 and using mass conservation, Eq. 3.55 becomes: h3/H (h3/H-1)(h3/H_2)

h3/H-1 (3.56)

which gives the re1ationship between 1ayer depth h3 and di1ution S. A schematic of gravity current and the resu1t of Eq. 3.56 are shown in

h3 1

Fig. 3.12 and 3.13. Benjamin has proven that the case of

lH

>

2

is impossib1e un1ess energy is added to the system. For some va1ues of

2

F_a there are two possib1e solutions to Eq. 3.56. This can be exp1ained by the formation of hydraulic jump.

6

I I IH I I

Fig. 3.12 Schematic of Gravity Current

Fig. 3.13 Height of Dense Layer vs. Froude Number F 2 based on Gravity Current Model. a

(b) Friction control:

Von Kármán and Benjamin viewed this wedge as a dynamic flow phenomenon taking place in a short distance. The frictional force at the interface was thus assurned to be negligible. On the other hand, almost all analyses dealing with salt water wedge intrusion in estuaries have

observed gradually varying flow and the heavier fluid was said to be balanced by interfacial friction.

The analysis of a frictional wedge is based on the two layer strati-fied flow equation [Ref. 17, p. 109] applicable to the case with the bottom layer velocity equal to zero.

Ti (__!_

+

1.)

dh2 = Pg hl h dx 2 !lp __q_l_~

'h 3

g 1

(3.57)

p

where H

=

hl

+

hand Ti is the interfacial shear stress that may be parameterized by:

T.

1.

(3.58)

with fi being an interfacial friction factor. (See Fig. 3.14) Combining the above and rearranging yields,

2

___a=x~=

(h/H)[Fa (l-h/H)3] a(h/H) F 2 a

(3.59)

where x

=

f. x 1.

- 8lI

(3.60) 54

Integrating Eq. 3.57 with x yields

1 (l __1_) (h)2

+

_1_ (.!!)3 _ 1_1_ (.!!)4

+

_1_ (h)5

+

c (3.61)

x =

2

F 2 H F 2 H 4 F 2 H 5F 2 H

a a a a

where c is a integration constant which depends on the specification of a boundary condition at x =

O.

Similar to single layer open channel flow, the height of the layer is said to be critical depth when Eq. 3.59 is equal to zero. (i.e. ah2/ax

=

00). Hence, the critical depth h is given

c as h .c -= H 1 - F 2/3a (3.62)

The critical depth, h , is the maximum possible depth achievable by

c

the stagnant wedge and will be observed whenever the downstream layer

depth is equal to or greater than h. Therefore, taking the location of

c

critical depth as the origin, the constant c in Eq. 3.61 can be found by

letting

x

=

0 and h

=

h

=

(1 - F 2/3)H. After the value c is known, the

c a

wedge length measured form

x

=

0 can then be obtained by letting h go to

zero. The schematic of stagnant wedge and normalized plot of h/h vs

c

x/L are shown in Fig. 3.14 and 3.15. It is found that the shape of the

curve is relatively insensitive to F 2

a

If the downstream layer depth is controlled by some means to be

small than the critical depth h , the wedge will start at that height c

following the same curve in Fig. 3.15.

3.5.2 Analysis of data

model.

Fig. 3.16 shows the comparison of data with the gravity current 2

The maximum value of F in Eq. 3.56 is equal to 0.278, which

a

indicates that for F 2 greater than this value, there will be no wedge

a

o

Fig. 3.14 Schematic of Friction Controlled Upstream loledge

1.0

0.8

06

0.4

0.2

o

X/L

Fig. 3.15 h/h vs x/L for a Friction Controlled Upstream Wedge c

Symbol Test No.

•

29

•

28

•

26

0

21

•

D.

20

0 ₁₉ C\I

QBr-•

'V

18

0

•

LL

₀

₁₄

~~Q6L

:.

- Theory VI :::> Ol ... ...__ I 0 A'V

o

o

0

.

1

0

.

2

0.3 0.4

0

.

5

h / H

upstream. Looking at the data it is c1ear that the wedge is not behaving as predicted by this model.

To compare the data with friction control theory, initia1 depths h3 are obtained from the impact ana1ysis. These resu1ts are then checked to find whether h3 is greater or 1ess than hc. The experimenta1 data are 1isted in Tab1e 3.3 and compared with the theory in Fig. 3.17.

The resu1ts of this ana1ysis definite1y indicates that the upstream wedge is controlled by interfacia1- friction and that the ana1ysis of Benjamin is not app1icab1e to this case of a steady arrested wedge.

3.6 Downstream Region

In this region the dense f1uid is f10wing away in the same direction as the current forming a co-f1owing two layer flow. The densimetric Froude numbers for each layer may be defined as:

2 UI

=-,-g hl

U 2

F 2

2

2 = g'h2 (3.64) F 2 I (3.63)

where UI and hl are the velo city and depth of the upper 1ayer. The value of g' is based on ave rage dilution of the dense layer measured from the bottom to the visua1 interface or to where the gradient of the temperature distribution if largest. Simi1ar to single layer open channel flow, if FI2

+

F22 > 1, the

flow is supercritical and the flow upstream will not be controlled by downstream condition (Yih[32]).

In the experimental study loca1 hydrau1ic jumps were observed downstream of the impact region implying no downstream inf1uence on the impact zone. Additional di1ution caused by mixing at the jump resulted in increased

VI \0

F 2

_h/R

_{h /R}

critica1

(h/R)x=O

h/hc

x/L

Test No.

a

c

control

18

0.29

0.033

0.34

No

0.0326

0.10

·

0.97

19

0.52

0.029

0.031

·

No

0.029

0.19

0.94

0.14

0.96

20

0.27

0.062

0.35

No

0.062

0.16

0.91

21

0.19

0.057

0.42

No

0.057

0.15

0.94

0.13

0.96

26

0.62

0.093

0.147

No

0.093

0.45

0.37

0.40

0.40

28

0.76

0.10

0.087

Yes

0.087

0.49

0.50

0.37

0.67

29

0.65

0.056

0.15

No

0.056

0.38

0.70

0.24

0.82

---

-

-Tab1e 3.3

Upstream wedge data

Note:

F

_a2

based on observed di1ution upstream of impact point

. _.

hc

based on friction theory and observed di1ution upstream of impact point

hx=O

based on impact model with observed ang1e of impact and di1ution

L

1ength of the stagnant wedge from critica1 depth to zero depth

h/hc

based on observed temperature distribution

0-o

1.0

u

•

<,

•

•

_•

Symbol Test No.

• 29 • 28 • 26

o

21

A

20

o

19 "IJ 18 Theory

0

.

2

0

.

8

o~---~---~

~

~

~

~

o

0

.

4

0.6

_1.0

X/l

value of Fl F 2 and F 2

1 2

and F2 as the value of g' decreased. The observed value of

2 2

(Fl

+

F2 ) downstream from the jump are listed in Table 3.4 and

for all of the experimental cases. With the exceptions of Test No. 12 to 15, the sum of two Fraude numbers were greater than one, indicating that downstream control was not criticàl to the observed fe?ture of the jet, impact and upstream regions.

(j')Q)(D r+ ...-. (1)0.0'" < . -_.0 ö' :J-._ «::r. èIi _. (1), co

*"

~

~-f " (I) 0°

92..5-;t

ëö

~ ~ ~ 0'1 N

-I

Test No

F.

U S

u

1(cm/sec)

u

2(cm/sec)

h1(cm)

h2(cm)

F 2

F 2

~F2

J

r

1

2

4

21.2

.062

110

7.63

3.66

41

23

5.13

2.27

7.4

,

6

22.2

.062

120

7.63

3.97

40

24

5.77

2.5

8.3

7

20.6

.065

120

7.63

3.97

40

24

5.77

2.5

8.3

8

19.3

.075

75

7.02

3.66

48

16

2.51

2.13

4.6

9

17.7

.069

80

7.02

3.97

48

16

2.67

2.6

5.3

10

19.7

.071

80

7.32

3.36

46

18

3.05

1.

79

4.8

11

20.4

.055

70

6.41

2.96

42

19

2.23

1.05

3.3

12

18.8

.023

40

2.32

2.65

49

12

0.14

0.78

0.92

I

.010

32

0.79

1.65

46

15

I

0.2

13

19.1

_0.014

_0.19

14

19.3

.026

34

2.59

1.92

47

14

0.16

0.29

0.45

15

18.8

.010

35

0.76

2.75

51

10

0.013

0.86

0.87

16

19.3

.025

44

2.30

3.05

50

11

0.15

1.22

1.4

17

11.7

.108

170

5.34

1.29

53

10

2.46

0.76

3.2

18

19.2

.067

60

4.02

1.31

59.5

3.5

0.57

1.03

1.6

19

19.2

.075

60

4.43

1.83

60.5

2.5

0.69

2.83

3.5

20

18.0

.

058

100

3.44

1.90

59

4

0.49

2.23

2.7

21

17.7

.049

80

2.99

1.22

58

5

0.37

0.72

1.1

22

17.7

.092

160

5.56

2.30

58.7

5.3

2.53

4.80

7.3

25

-

_-

240

4.66

0.33

56

7

2.79

0.12

2.9

26

-

_-

135

4.75

0.19

55

8

1.14

0.01

1.2

27

-

-

250

5.52

0.22

51

12

3.07

0.02

3.1

29

-

_-

150

4.98

0.16

53

10

1.44

.01

1.5

CHAPTER 4

DESIGN APPLICATION AND CASE STUOY

In this chapter the previous analyses of different regions of dense discharges wi11 be brought together and a suggested design procedure wi11 be presented in Section 4.1. Section 4.2 contains a case study

invö1ving the offshore disposa1 of saturated brine.

4.1 pesign App1ication

4.1.1 Statement of the prob1em

The main purpose of this study is to determine, for a given site and discharge specification, the sets of designs which can achieve a

certain requirement. Tab1e 4.1 1ists all the design parameters being considered. By virtue of the number of variables invo1ved, some of the variables have to be considered as fixed by externa1 considerations.

The fo11owing are the assumptions used in this study to define a design

-procedure.

(a) Site and discharge specification:

The depth of water H, the range of ambient current

magnitude Ua' the excess discharge density in terms of gj and the tota1 amount of effluent to be disposed in unit tireeQT are assumed to be known.

(b) Required performance:

For a given minimum ambient current Ua min' the average concentration of the dense jet before it hits the ground must be smaller than a given va1ue, CR. A1so the jet shou1d not hit the water surface. Le. Z < H.

Pm

Total amount of flow per unit time of the dense effluent Initial reduced gravity of the dense effluent

H:

_{Total depth of the receiving water body}

U :

a Ambient current, usually given in a certain range

[u - U ]

a min a max D.: Port diameter

J

Z.: Height of the ports J

N: Number of ports

t: Spacing between ports

L:

Diffuser length,

L

=

(N-l)t

CR:

Maximum allowed average concentration at impact point

Z : Maximum allowed height of the jet center line Pm

Table 4.1 Glossary of design parameters

Cc) Fixed design variables:

In current engineering practice, the maximum discharge velocity is of ten fixed by the power of the avai1ab1e pumps.

first fixed to reduce the number of independent variables.

Thus, U. is J Sensitivity studies are later performed to determine the inf1uence of the choice of

U

j on the design.

Cd) Design variables to be se1ected:

The remaining variables to be chosen to meet the specified conditions (a) and Cc) and the performance requirements eb) above are the diffuser port diameter. Dj' the port height. Zj' the number of ports, Nt and the port spacing, t.

4.1.2 App1ication of experimenta1 and theoretical analysis In this section, the experimental resu1ts and theoretica1 analyses of previous chapters wi1l be used to attack the design problem defined above. A short summary of design procedure wi11 be presented in the next section (4.1.3).

(a) Design of port diameter:

The average excess concentration of the dense jet at a point can be re1ated to the initia1 discharge density using the volumetrie di1ution, S. For the case of smal1 density difference as expected in the ~pact region:

C=l~

S

Pa

where 6p is the initia1 density of the dense effluent Pj minus the ambient (4.1)

water density Pa' Combining Eq. 4.1 with 3.36 yie1ds:

u

C

=

/!.P

_!_

(F ~)-0.67

~ P 2 aj

(4.2)

where C is the maximum concentration inside the jet at impact point. mI

It was found in this study that the dense jet maximum concentration was around twice as much as the ave rage concentration. With this

information the average concentration of the jet at impact point cl can he expressed as:

u

C

=

/!.P

_!.

(F 2) -0.67

I p 4 aj (4.3) also

=

_i (F 2)

0.67

SI U aj r

(4.4)

By rearranging the terms and letting Cl equal the required maximum average concentration CR' Ua equal the lower limit of the cross flow range Ua min, and U. equal

a

fixed reasonahle value, the design nozzle diameter can he

J determined hy 3 3

U

.

CR

U

j

]1/2

D. = 8 [_..:::;a...;m=l.=n~~...J..__ J

2 (/!.p)5

g P

(4.5)

(h) Readjustment of dis charge velocity:

At this point U. and D. are known. The initial choice

J J

of discharge velocity may have to he readjusted hecause of insufficient pumping capacity to overcome the head loss at the exit. Also the height of the jet is of concern. Eq. 3.38 gives the maximum height of the jet center line as

z

-..E.

U

=

1.7 F. D. r aj J

(4.6)

66

or, in dimensional form U.D.1/2

Zp = 1 7 J J (4.7)

• g ~ 1/2

J

The total depth H should be considerably higher than Z. A P maximum jet height, Z , mayalso be imposed for other reasons.

Pm

(c) Choice of diffuser length and nozzle height:

The criteria on choosing nozzle height Zj is based on the impact and upstream region analysis described in Section 3.4 and 3.5. There are two conditions determining the design,

(i) If .h3 _-> h ,_c h

c

The value of h3 is given in Eq. 3.50. Multiplied by

(H/Dj)2

it becomes, h 2 (~)2 U

-4/3

F

813

D F l4/3f (2)2 _{= 'Ir fl}

₊

8'1r

.=t u/

13

Dj R. r J j 2 ( 2 cos

e

2 where fl = ) 1

+

cos

e

f2 = 1 - cos

e

2 cos

e

(4.8)

h3/Dj depends on three independent variables,

R.

/Dj'

Ur and Fj. In order to reduce the number of independent variables, the critical cross flow ratio Urc (which gives the highest value of h/Dj for a given Fj) were

by trial and error for given value of Fj. is out

found Thus, unless U

r

c

of the range of given Ur' the design Zj should be based on the critical