45197995 Book of Design Water System

WATER AND WASTEWATER SYSTEMS ANALYSIS

DEVELOPMENTS IN WATER SCIENCE, 34

OTHER TITLES IN THlS SERIES (Volumes 1-3 are out of print)

4 J.J. FRIED GROUNDWATER POLLUTION

5 N. RAJARATNAM TURBULENT JETS

6 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS

7 GROUNDWATER HYDRAULICS

B J.BALEK HYDROLOGY AND WATER RESOURCES IN TROPICAL AFRICA

9 RESERVOIR CAPACITY AND YIELD

10 0. KOVACS SEEPAGE HYDRAULICS

11 W.H. GRAF AND C.H. MORTIMER (EDITORS) HYDRODYNAMICS OF LAKES: PROCEEDINGS OF A SYMPOSIUM 12-13 OCTOBER 1978, LAUSANNE, SWITZERLAND

12 W. BACK AND D.A. STEPHENSON (EDITORS) CONTEMPORARY HYDROGEOLOGY: THE GEORGE BURKE MAXEY MEMORIAL VOLUME

13 M.A. MARl f l0ANDJ.N. LUTHIN SEEPAGE AND GROUNDWATER

14 D. STEPHENSON STORMWATER HYDROLOGY AND DRAINAGE

15 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS (completely revised edition of Vol. 6 in the series)

16 W. BACK AND R. LkTOLLE (EDITORS] SYMPOSIUM ON GEOCHEMISTRY OF GROUNDWATER

17 TIME SERIES METHODS IN HYDROSCIENCES

18 J.BALEK HYDROLOGY AND WATER RESOURCES IN TROPICAL REGIONS

19 D. STEPHENSON PIPEFLOW ANALYSIS

20 I. ZAVOIANU MORPHOMETRY OF DRAINAGE BASINS

21 M.M.A. SHAHIN HYDROLOGY OF THE NILE BASIN

V. HALEK AND J. SVEC

T.A. McMAHON AND R.G. MElN

A.H. EL-SHAARAWI (EDITOR) IN COLLABORATION WITH S.R. ESTERBY

22 H.C. RlGGS STREAMFLOW CHARACTERISTICS

23 M. NEGULESCU MUNICIPAL WASTEWATER TREATMENT

24 L.G. EVERETT GROUNDWATER MONITORING HANDBOOK FOR COAL AND OIL SHALE DEVELOPMENT

25 W. KINZELBACH GROUNDWATER MODELLING: AN INTRODUCTION WITH SAMPLE PROGRAMS IN BASIC

26 KINEMATIC HYDROLOGY AND MODELLING

D. STEPHENSON AND M.E. MEADOWS

27 STATISTICAL ASPECTS OF WATER QUALITY MONITORING - PROCEEDINGS OF THE WORKSHOP HELD AT THE CANADIAN CENTRE FOR ISLAND WATERS, OCTOBER 1985

A.H. EL-SHAARAWI A N D R.E. KWIATKOWSKI IEDITORS)

28 M.JERMAR WATER RESOURCES AND WATER MANAGEMENT

29 G.W. ANNANDALE RESERVOIR SEDIMENTATION

30 D.CLARKE MICROCOMPUTER PROGRAMS FOR GROUNDWATER

31 R.H. FRENCH HYDRAULIC PROCESSES ON ALLUVIAL FANS

32 ANALYSIS OF WATER RESOURCE SYSTEMS

WATER MANAGEMENT IN RESERVOIRS

L. VOTRUBA. Z. KOS. K. NACHAZEL. A. PATERA ANDV. ZEMAN

33 L. VOTRUBA AND V. BROZA

DAVD STEPHENSON Water Systems Research Group, University of the Witwatersrand, 1 Jan Smuts Avenue, Johannesburg, South Africa

ELSEVl ER

Amsterdam - Oxford - New York - Tokyo 1988

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, 1000 AE Amsterdam, The Netherlands

Distributors for the United States and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, NY 10017, U S A .

ISBN 0-444-42945-X (Vol. 34) ISBN 0-444-41 669-2 (Series)

0 Elsevier Science Publishers B.V., 1988

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publish- ers B.V./Physical Sciences & Engineering Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands.

Special regulations for readers in the USA - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher.

No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

Printed in The Netherlands

V

PREFACE

A systematic approach to decision making i n water resources p lann ing

i s presented w i th pa r t i cu la r reference to wastewater re-use.

Various methods of system simulat ion and optimization are appl ied in a

number of case studies. Methods of analysis and numerical methods

(Chapter 2, 4 ) are described as well as the basis of pol lut ion and water

qua l i t y (Chapter 1 , 3 ) . The economics of desal ination are also discussed

(Chapter 7 ) .

The author has considerable experience in p lann ing water pur i f i ca t ion

and recycl ing systems i n an a r i d area, Southern Afr ica. Water i s a t a

premium for mining and indus t r ia l development and considerable money is

spent on water treatment o r use of poor qua l i t y water. Careful management

and d is t r ibu t ion of water resources can i n these circumstances save a lot

of money. The general theory of optimization subject to qua l i t y constraints

i s presented i n Chapter 6.

The examples studied range from regional supplies (Chapter 10) to

internal re-circulat ion (Chapter 8). Groundwater and a r t i f i c i a l recharge

are considered (Chapter 9 ) and stormwater qua l i t y (Chapter 5 ) and

sewerage systems (Chapter 1 1 , 12) are also covered. Computer appl icat ions

exist throughout and a number of simulat ion and optimization programs in

BASIC are presented.

Chapter 13 i s on an often ignored subject, the necessity fo r scient i f ic

sampling procedures i n monitoring water qua l i t y . I t was wri t ten b y

Professor Tom Sanders of Colorado State Universi ty.

The theory and case studies should prove of value in many aspects of

p lann ing use of water resources w i th qua l i t y constraints. Wastewater

re-use and conservation therefore are promoted b y the approach adopted.

v i

CONTENTS

CHAPTER 1 . WATER QUALITY IN INDUSTRIAL SYSTEMS

Geochemical source of p o l l u t i o n Effect o f evaporat ion on concentrat ions Effects of poor q u a l i t y water Scal ing

Pred ic t ion of sca l ing a n d corrosion Prevent ion of s c a l i n g Calcium carbonate s c a l i n g Sulphate sca l ing Add i t i ves fo r the prevent ion of sca l ing

Fou l ing Control o f f o u l i n g

O i l emulsion breakdown Corrosion

Types of corrosion Corrosion prevent ion

Potable water s tandards A g r i c u l t u r e a n d i r r i g a t i o n

CHAPTER 2. MATHEMAT I CAL MODELLING O F WATER QUAL I TY

Mass Balances Mixed a n d p l u g f low systems Systems a n a l y s i s Terminal concentrat ion in a water c i r c u i t App l ica t ion to a mine water c i r c u i t Computer s imulat ion model

Mathematical b a s i s of model

CHAPTER 3. NON CONSERVATIVE PARAMETERS

In t roduc t ion Basic mass ba lance equat ion Oxygen balance in r i v e r s

Coupled equations f o r DO a n d BOD A n a l y t i c a l so lu t ion

Ca l ib ra t ion of a moving BOD model Oxygen balance Fie1 d measurements

CHAPTER 4. NUMERICAL METHODS

Simulat ion of H y d r a u l i c Systems Two-step method Demonstration of numer ica l inaccuracy I m p l i c i t f i n i t e d i f ference schemes Comments on f i n i t e d i f ference methods

d i f f e r e n t i a l equat ions The Eu ler method The modi f ied Eu ler method Runge-Kutta methods Mul t i s tep methods

Boundaries f o r numer ica l methods

Numerical methods f o r the so lut ion of s i n g l e

F i n i t e elements

1 2 2 3 3 3 5 6 6 8 9 10 10 13 14 15 17

20 21 24 24 26 31 31

35 35 37 37 39 40 40 45

51 52 52 55 56

57 57 59 60 61 62 62

v i i

CHAPTER 5. MASS BALANCE O F STORMWATER POLLUTANTS

Introduction Catchmen t descript ion Qual i ty Observations

Fa1 lout measurement Relationship between total pol lutant load and runoff volume Chemical constituents

on Hil lbrow catchment

on Montgomery Park catchment

Mass balance for event of 18 January 1985

Mass balance for event of 7 March 1983

Conclusions

CHAPTER 6. OPTIMUM ALLOCATION O F WATER RESOURCES SUBJECT TO QUAL I TY CONSTRA I NTS

I n t roduc t ion The system Solution method

Discussion Linear Programming Solution The I inear programming technique with separable

programming appl ied Sensi t iv i ty study for various acceptable TDS values

CHAPTER 7. ECONOM I CS OF DESALINATION OF WASTEWATERS

I n t roduc t ion Alternatives for optimal reuse of waste water Selection of optimum desalination methods Relevant desl ination methods

Indus t r ia l wastewater treatment Reverse osmosis Membrane descript ion E I ect rod i a I ysi s Ion exchange

Capital costs Indirect capi ta l costs Running costs Labour costs Membrane replacement

Cost analysis

Conc I us ions

CHAPTER 8. COMPUTER ANALYSIS JUST I F I ES DESAL I NAT ION

I n t roduct ion Application of optimization of water supply Systems Analysis

General optimization problem Program appl icat ion Optimization of mine water system Result of analysis Appendix 8.1

MlNSlM Program fo r simulat ing flow and TDS in closed systems. Tape o r disc management MlNSlM l i s t of symbols

64 64 66 66

67 67

72

73 77

79 80 82 85 85

91 95

99 99 101 103 1 04 104 105 105 105 107 107 108 108 108 108 1 1 1

115 116 118 121 122 123 123 128 128 128 128 129

v i i i

A p p e n d i x 8.2 MINOP p r o g r a m f o r o p t i m i z i n g d i s t r i b u t i o n MINOP l i s t o f symbo ls

136 136 136

CHAPTER 9. INTEGER PROGRAMMING PLANNING OF TREATED WASTEWATER CONVEYANCE FOR A R T I F I C I A L RECHARGE OF AN AQUIFER

I n t r o d u c t i o n Cost a n a l y s i s Ma themat i ca l f o rmu ta t i o n Resu l t s Summary a n d conc lus ions

141 146 149 151 153

CHAPTER 10. OPTIMAL PLANNING OF REGIONAL WASTEWATER TREATMENT

I n t r o d u c t i o n The mathemat i ca l model O p t i m i z a t i o n method

CHAPTER 1 1 . SIMULATION OF SEWER FLOW

I n t r o d u c t i o n H y d r a u l i c a n a l y s i s F low measurements

H i g h e r income r e s i d e n t i a l L o w income r e s i d e n t ia I Apar tmen t b u i l d i n g s Commercial a r e a s I n d u s t r i a l

Conc lus ions A p p e n d i x

P r o g r a m SEWSIM E f fec t o f l oca l p e a k s R o u t i n g e f fec t Non-Ci rcu l a r C o n d u i t s I nf low components D a t a P r o g r a m o u t p u t Sample d a t a f i l e

CHAPTER 12. SEWERAGE SYSTEMS MANAGEMENT

L e a r n i n g S i m u l a t i o n P r o g r a m O p t i m i z a t i o n

O p t i m a l Cont ro l a s a L i n e a r P r o g r a m m i n g Prob lem Sewer Ma in tenance D a t a P r o c e s s i n g in J o h a n n e s b u r g

A p p l i c a t i o n to J o h a n n e s b u r g ' s System Process ing of Sewer M a i n t e n a n c e D a t a

CHAPTER 13. WATER QUAL I TY MON I TORlNG NETWORKS

Necess i ty f o r Ne tworks M o n i t o r i n g System Framework F a c t o r s in Ne twork Des ign Se lec t ion of Water Q u a l i t y V a r i a b l e s t o Measure S a m p l i n g S t a t i o n L o c a t i o n S a m p l i n g F r e q u e n c y D iscuss ion

155 158 162

166 167 167 169 170 171 171 172 172 174 174 1 74 175 175 1 76 177 177 186

190 192 193 195 197 198

204 205 205 206 207 21 1 215

ix

AUTHOR INDEX

SUBJECT INDEX

21 7

21 9

This Page Intentionally Left Blank

1

CHAPTER 1

WATER QUALITY IN INDUSTRIAL SYSTEMS

GEOCHEMICAL SOURCE OF POLLUTANTS

Many of the chemicals ? n solution in water o r ig ina te from the

surroundings. Minerals which form rocks may be dissolved by water in a

sui table environment. Acidic waters i n pa r t i cu la r are known to dissolve

certain chemicals in the rock, Exposure to air, which contains oxygen,

assists the reaction, I ron sulphide i s one such chemical which can be

oxidized to sulphate. Bacteria a re also thought to p l a y an important p a r t

in the leaching of sulphides.

The so lub i l i t y of chemicals i s also dependent on temperature and the

total dissolved sol ids i n the water amongst other factors. In many cases

the ra te of dissolut ion i s slow. I t may take years to dissolve a l l the

sulphide from a rock sample. On the other hand chlorides dissolve very

rap i d I y . When a chemical compound dissolves i n water the ions appear as

posi t ively charged metal o r cations and negat ively charged anions.

Solubi l i ty depends on other charges present and i s expressed as a

so lub i l i t y K. In pa r t i cu la r water i s ionized as follows: (Brownlow, 1979).

H 0 = H+ + OH- The log of the hydrogen H+ ion concentration i s termed the pH:

pH = -log(H+)

2

Water wi th a pH below 7 i s acidic. I t may be rendered so by many

factors. For example absorption of carbon dioxide C02 from the a i r forms

carbonic acid which could reduce the pH as low as 3.0. O n the other hand

water d ra in ing from limestone or s i l icate minerals may have a pH greater

than 7 (Pel let ier, 1964).

The process of leaching sulphide from minerals i s self st imulat ing as

sulphuric acid i s formed i n the process. On exposure of sulphide bear ing

horizons ferrous sal ts oxidize to the fe r r i c state and sulphide i s oxidized

to sulphate:

4FeS2 + 1502 + 10H20 + 4FeO(OH) +' 8H2S04

I n mining environments, bacter ia th r ive in the acid mine water and

fu r ther promote the oxidat ion of Fe and S. The bacter ia th iobaci l lus

ferro-oxidans oxidize both Fe and S whereas thiobaci l lus thio-oxidans

oxidizes only S (Mrost and Lloyd, 198Oj.

EFFECT OF EVAPORATION ON CONCENTRAT IONS

The ra te of concentration of total dissolved sa l ts b y evaporation may be

predicted for any ambient condit ions using psychrometric relat ionships. In

addi t ion to evaporation in cooling towers, evaporation of water takes place

i n indus t r ia l systems and pa r t i cu la r l y vent i lated systems. Thus i f the

dry-bulb temperature of a i r i s 31OC and the a i r i s conveyed in a t 38%

re la t i ve humidity, i t w i l l increase in water content b y 5 g water

vapour/kg of a i r . For an a i r density of 1.2 kg/m3 and a vent i la t ion ra te

of 700 m’/s, the amount of water absorbed by the a i r w i l l be 5 l i t r es per

second (Barenbrug, 1965).

The loss of water by evaporation leaved behind sa l ts which may have

entered the system in d i lu te solut ion i n i t i a l l y . There i s therefore a

concentrat ing effect, and the ra te of increase in concentration i n time

depends on the volume of storage i n the system. Thus i f there i s a 12

hour retention and the flow ra te of service water i s 100 l i t r es per second,

the volume i n the system w i l l be 100 x 3600 x 24/1000 = 8640m’. I f the

evaporation loss i s 10 l i t r es a second which i s 864 m3/day the i n i t i a l ra te

of concentration w i l l be 10 percent of the i n i t i a l concentration per day.

The sa l t concentration would increase indef in i te ly unless the water was

replaced. The ra te of concentration i s usua l ly offset b y the fact that there

i s a source of purer water used for make-up bu t even th is adds to the

total load unless there i s a blowdown (Porges, 1971).

The concentration of total dissolved sol ids a t equ i l ib r ium w i l l be a

function of the evaporation ra te as well as the pumping discharge ra te

and ra te a t which sal ts a re introduced as a resu l t of make-up water and

leaching of chemicals from the environment (Van Staden, 1970).

EFFECTS OF POOR QUAL I TY WATER

High total dissolved sa l ts concentration in water gives r i se to a number

of problems. The nature of the problems varies, bu t i n a l l cases the

economic consequences of poor qua l i t y water are severe. High chlor ide

concentration i n mine water i s suspected to be one of the causes of

corrosion of pipework and equipment. Sulphates and carbonates i n the

waters give r i se to scal ing and blockages. Scal ing i s common i n heat

exchange equipment. I n many systems there may be scal ing in some areas

and corrosion in others. Plant has f requent ly to be replaced af ter only a

few years i n service i n many systems because of these effects. The recent

development of mechanized equipment has given r i se to new fears of the

3

consequences of poor qua l i t y water. Many of these machines are designed

to operate hyd rau l i ca l l y using oi l- in-water emulsions. Emulsion s tab i l i t y

and the hydrau l i c c i r cu i t s could be already affected b y poor qua l i t y

water.

SCAL I NG

Scaling i s the phenomenon of chemical deposition on submerged surfaces.

The deposits are due to c rys ta l l i za t ion o r precipi tat ion. Scal ing takes

place because of the dissolved sa l t concentration exceeds i t s saturat ion

l im i t and is usual ly a resul t of an excess of chemicals in solution which

could be caused by evaporation loss of water, leaching of chemicals from

surroundings, o r a change i n temperature.

The scal ing i s also a function of other parameters such as pH, total

dissolved solids concentration, a l ka l i n i t y , time and flow velocity.

The chemicals most frequently causing scale are calcium carbonate

(CaCO ) and calcium sulphate (CaS04). Calcium carbonate i s pa r t i cu la r l y

insoluble whilst so lub i l i t y of both salts i s h igh l y dependent on

temperature. Figure 1 . 1 i l l us t ra tes the effects of temperature on the

so lub i l i t y of these salts. T h e so lub i l i t y i s inf luenced b y chlorides and

other ions i n solution. Other chemicals, pa r t i cu la r l y oxides of magnesium

(e.g. Mg(OH)2), i ron, aluminium and s i l i ca are also sometimes found i n

scales (Betz, 1980).

3

Predict ion of Scal ing and Corrosion

T h e factors affect ing the equ i l ib r ium of calcium carbonate i n solut ion

have a complex interdependence. Langel ier (1954) developed an equation to

predict the tendency of calcium carbonate to form a scale. Ryzner (1944)

preferred to express the equation in terms of pH. However there any many

inf luencing effects and such formulae can only offer a guide to the l i ke l y

behaviour of the water, pa r t i cu la r l y i n respect to corrosion.

Prevention of Scal ing

One way of prevent ing scal ing or corrosion would be to desal inate the

water. Possible methods of desal inat ing mine water are ion exchange,

reverse osmosis, e lectrodialysis and thermal procedures. Although these are

expensive their possibi I i ties are being re-assessed.

4

Scale prevention i s cur ren t ly normally undertaken by pH adjustment

and, where necessary, a control led bleed (waste) from the system to

prevent excessive concentration of the dissolved sal ts. This treatment may

be supplemented with the use of scale and corrosion preventat ive

addit ives.

3000

2800

2600

2400

2200

zoo0

1800

1600

-I - 1400

1200 E

1000

800

600

400

200

100

50

0 0 20 40 6 0 80 100 120 140 160

TEMPERATURE O C

Fig . 1 . 1 Effect of temperature on so lub i l i t y of sca l ing sal ts

5

Calcium carbonate scal ing

The factors af fect ing the equ i l ib r ium of calcium carbonate in solut ion

have a complex interdependence. In addi t ion to temperature, the presence

Procedure : glven temp. OC TDS mgll Ca mgll Alkallnlty proceed 1-2-3-4-5

PHS Ryzner Stablllty Index RSI=lpH,-pH Calclum Carbonate Sal lng Ilkely If LSI>O and RSlc6 Colrorlon Ilkely If RSI>O

Fig. 1.2 Langelier Saturation Index Chart fo r Carbonate Scaling

6

of other dissolved solids, especial ly total a l k a l i n i t y and pH, affect the

tendency to form scale. A sudden reduction in pressure such as a t a

nozzle can induce scal ing, and suspended matter i n the water may serve

as nuclei for scale formation. Langel ier developed an equation to predict

the tendency of calcium carbonate to from scale. The Langel ier Saturation

Index i s LSI = pH - pHs, where pHs i s the saturat ion pH. I f the LSI i s

posit ive, there is a tendency to scale and i f i t i s negative, calcium

carbonate tends to dissolve. The pHs i s calculated from the equation:

pHs = pK(C,T) + pCa + pAPk

where K(C,T) i s a function of the temperature and total dissolved sol ids,

and represents the second dissociat ion constant and so lub i l i t y constant

which can be computed from thermo-dynamic considerations. pCa i s the

negative logarithm of the calcium content, and pAPk i s the negat ive

logarithm of the equivalent concentration of the a l k a l i n i t y . The LSI can be

computed read i l y from F igure 1.2.

Ryzner proposed a di f ferent arrangement of the terms i n the Langel ier

equation. The Ryzner S tab i l i t y Index ( R S I ) i s :

RSI = ZpHs - pH

I f the RSI i s less than 6, scal ing tendency increases, and i f i t i s

greater than 8, corrosion i s in fact more l i ke l y .

There are many other effects inf luencing scal ing and corrosion,

however, and such formulae can only provide pre l im inary guides

pa r t i cu la r l y i n relat ion to corrosion.

Sulphate scal ing

The so lub i l i t y of var ious forms of calcium sulphate i s higher than that

of calcium carbonate bu t i t i s also h igh l y dependent on temperature.

Calcium sulphate occurs i n three di f ferent c rys ta l l i ne forms: dihydrate,

CaS04.2H20 (gypsum), herni-hydrite CaS04.)H20 (p las te r of p a r i s ) and

anhydr i te, CaS04.

The so lub i l i t y of the hernihydrite and anhydr i te decreases w i th

temperature (F igure 1 . 1 1. The so lub i l i t y increases w i th chlor ide

concentration and i s affected by total dissolved solids.

Addi t ives for the prevention of scale

I n most systems, especial ly once-through systems, dernineral izat ion o r

softening the water wi th resin o r Zeolite i s not economically jus t i f iab le . In

some cases chemical inh ib i to rs a re used to prevent the formation of scale.

7

These agents control deposits by prevent ing c rys ta l growth, even in a

supersaturated solution. The basic mechanisms of scal ing and deposit

control are:

( 1 1 Control of in te rpar t i c le a t t rac t i ve forces e.g. dispersants.

( i i ) Control of part icle-to surface forces, e.g. surfactants o r

wetting agents. These involve electrostat ic forces. They act

non-stoichiometrically and hence low concentrations are

possible. They are used more for prevent ing foul ing than

scal ing . ( i i i ) Control of precipi tat ion rate, e.g. flocculants. These are h igh

molecu l a r weight pol ymers.

( i v ) Retardation of crystal growth, e.g. polyphosphates.

Some of the reagents used are l is ted below:

Polyphosphates: Applied i n rates from 0,5 to 5 mg/O. Absorbed onto the

surfaces of growing c rys ta ls and in incipient c rys ta l nuclei. They increase

the apparent so lub i l i t y of scale forming salts. These are successful for

carbonates and hydroxides but not for sulphates.

Organic Phosphates: Simi l a r to polyphosphates but they are more stable i n

cooling tower systems. Phosphonic acids have proved pa r t i cu la r l y

successfu I .

Phosphate: React w i th calcium to form insoluble calcium phosphate which

precipitates out. For th is reason i t s use has la rge ly been replaced by

d i spersan ts.

Polymers (especial ly polyacrylates): Absorbed onto surfaces of c rys ta l

growths. Effect ively dispersants as they maintain small par t i c les of

crystals i n suspension. Low molecular weight polymers have recently been

developed for th is purpose.

The reagents may be used on combination, o r even together wi th ac id to

reduce pH. Carbon dioxide can be added to closed systems to reduce pH.

Ferr ic chloride is also used.

Dispersants o r sequestrants are sometimes used to prevent scale

formation of i ron hydroxide o r oxide in par t i cu la r . Chelants o r complexing

agents are used to isolate and inh ib i t scale formers. I t should be noted

8

that c rys ta l inh ib i to rs a re not effect ive in e l im ina t ing foulants enter ing

the systems as suspended sol ids. Instead, agglomeration of these sol ids

must be prevented b y dispersants such as phosphonates and

I i gno-su I phates.

The deposit of phosphates i n closed systems due to addi t ives can be a

problem. The durat ion of effectiveness of addi t ives i s also unknown. H igh

velocit ies and turbulence can affect low-concentration dispersants i n

par t i cu la r .

The effects of chemical addi t ives on the rest of the system w i l l also

requ i re consideration. Deposits may block pipes o r machines. Suspensions

may erode high-velocity jets. Reactions w i th other chemicals may aggravate

total dissolved sol ids problems. There may also be an effect on sett lers

and demineral izat ion plants.

FOUL I NG

Besides chemical precipi tates there are many substances in suspension

which can settle out o r block pipework and machinery. The deposits may

mater ia l ize i n the form of f i lms b r idg ing openings o r bu i l d ing up in

cavi t ies where water velocit ies a re slow. The mater ia l deposited may be:

Sediment from ore or the atmosphere transported in suspension

Floc created b y chemical treatment

I ron oxide ( r u s t )

Chemicals used for scale o r corrosion inh ib i t ion which subsequently

cause deposits

Oi ls

Foam from chemical reactions o r aeration

Bacteriological slime collected or accumulated i n the system

The tendency to sett le i s a function of pa r t i c l e size, shape, densi ty,

water velocity and apperture bore. Turbulence due to f lowing water w i l l

maintain some par t i c les in continuous suspension al though the concentration

w i l l b e highest near the bed i n the case of par t i c les denser than water.

Once part ic les sett le out they may st ick to the surface, or to other

part ic les. Al ternat ively they may migrate along the bed. Under some

conditions the bed mater ia l may move as dunes w i th par t i c les being picked

up by the flow upstream of the dune crest and deposited downstream. The

resu l t ing r ipp led surface can aggravate f r i c t ion loss in conduits. In

9

addit ion to the reduction i n cross sectional-area, the capaci ty of the

conduit i s reduced due to the higher d rag on the perimeter.

Deposits may block f ine pores or ori f ices completely. The gaps between

f i l t e r media part ic les rap id l y block thus requ i r i ng backwashing. I n

machines with f ine je ts o r screens simi lar blockages are possible.

Deposits may remain i n flocculated blanket form, o r consolidate w i th

time and increasing deposits pressing down from above. They may st ick to

the surface due to chemical bonding.

Biological matter such as bacter ia l slime o r fungi can build up w i th in

a water system provided nutr ients such as nitrogen, phosphorous and

sometimes carbon and s i l i ca , are present. They may be anaerobic (not

requ i r ing oxygen for growth) o r aerobic. Some bacter ia th r ive on i ron o r

sulphate and cause deteriorat ion.

Control of fou l ing

Deposits i n machinery and pipe systems can be prevented o r reduced by

control l ing par t i c le at t ract ion forces, prevent ing sett l ing by turbulence,

i ns ta l l i ng set t l ing basins or keeping the par t i c les out of the system (e.g.

closed c i rcu i ts ) . Dispersants are used to control part icle-to-part icle and

par t i cI e- to-surface forces. They neutra I i ze electrostat ic a t t ract ion charges

or create repel l ing charges. One problem with these i s that i f there are

sedimentation basins i n the system they may hinder se t t l ing there.

High concentrations of dispersants may in fact be used for desludging

systems. Surface wett ing agents a re sometimes used to prevent deposition of

o i l and grease.

Biological foul ing may be control led by disinfection. Shock dosing

treatment appears more effect ive and economic than continuous dosing.

Chlorine i s widely used as a biocide to combat bio-matter. I t i s an

ox id iz ing agent and reduces the pH when dissolved i n water by forming

hypochlorous acid and hydrochlor ic acid. A free residual chlorine content

of less than 1 mg/t i s usual ly suff icient i f contact per iod i s an hour o r

more. Hypochlorite i s also used occasionaly.

Non-oxidizing biocides act by surface ac t iva t ing o r b y causing surface

lesions i n the metabolism. Into th is category f a l l quarternary ammonia

compounds (quats ) .

10

0 I L EMULS ION BREAKDOWN

Emulsions of o i l in water are used for d r i v i n g prototype machinery

amongst other things. The emulsions consist of o i l dispersed i n water in

the form of minute droplets. The emulsion i s stabi l ized by electr ical

charges on emulsifying agents.

Chemicals (such as polymers of opposite charge p o l a r i t y ) break

emulsions by neut ra l i z ing repuls ive charges between par t i c les

(coagulat ion), p rec ip i ta t ing or Crys ta l l i z ing out emulsi fy ing agents o r

a l te r ing the emulsi fy ing f i lm so that i t can read i l y be broken. Cations

and cationic polymers are pa r t i cu la r l y effect ive in separat ing d i l u te

oi I-in-water emulsions. Once charges have been neutral ized, o i I droplets

and suspended sol ids w i l l be absorbed on the surface of floc o r w i l l break

out and f loat on top thereby destroying the emulsion propert ies.

Although i t i s desirable to maintain the emulsion oi l - in-water

suspension whi lst in service, af ter the emulsion i s discharged to waste i t

may be desirable to separate the o i l and the water. This should take

place a t control led locations to prevent subsequent slime and cak ing i n

machinery fur ther in the cycle. Acid and aluminium sulphate (alum) have

been used to break oi l- in-water emulsions. The ac id lowers the pH to

about 3 and alum coagulates the o i l b y neut ra l i z ing the charges. Lime i s

added to raise the pH again and the aluminium i s precipi tated as

aluminium hydroxide. Cationic polymers are preferred and often used i n

double a i r f lotat ion (DAF) un i ts which col lects the o i l on the surface.

CORROS I ON

Corrosion i s the at tack and degradation of metal by chemical o r

electrochemical act ion. Pipework and machinery a re subject to corrosion

due p r imar i l y to h igh l y sal ine or ac id ic water. The destruction may be

general o r i n isolated points. I t may reduce the l i f e of p ipe and steelwork

b y many years.

I ron corrodes i n water as follows: I t replaces the hydrogen ion in

water since i t i s less noble i.e. i t i s less cathodic:

Fe + 2H20 = Fe(OHl2 + H2

I n the presence of oxygen, which i s usua l ly i n solut ion in water, the

ferrous oxide i s oxidized fu r ther to fe r r i c hydroxide, Fe(OHl3. This i s

insoluble, but i s ul t imately changed to fe r r i c oxide, Fe203. The reaction

manifests as p i t s i n the i ron surface, a form of oxygen corrosion. (F igure

1.3) .

1 1

TABLE 1.1 Nernst sca le of s tandard e q u i l i b r i u m poten t ia ls re la ted

to the s tandard hydrogen electrode a t 25OC

(Metal immersed in a normal so lut ion of one of i t s s a l t s )

Metal Electrode react ions Equi I i br iurn potent ia l

(vo l t s )

Potassi urn

Calcium

Sod i um

Magnesium

Al urn in iurn

Manganese

Z inc

C hrom i urn

I ron

Coba I t

Nickel

T i n

Lead

Hydrogen

Copper

Copper

S i l ver

P I a t inum

Gold

Gold

K = K+ + e-

Ca = Ca + 2e-

Na = Na+ + e-

Mg = Mg + 2e-

~t = AI+++ + 3e-

Mn = Mn + 2e-

Zn = Zn + 2e-

C r = C r + 3e-

Fe = Fe + 2e-

Co = Co + 2e-

Ni = Ni + 2e-

S n = Sn + 2e-

Pb = Pb++ + 2e-

H2 = 2H+ + 2e-

c u = cu + 2e-

cu = cu+ + e-

Ag = A g + + e-

Pt = Pt + 2e-

Au = A u + 3e-

AU = AU+ + e-

++

++

++ ++

+++ ++ ++ ++ ++

++

++ +++

- 2.922

- 2.87

- 2.712

- 2.34

- 1.67

- 1.05

- 0.762

- 0.71

- 0.440

- 0.277

- 0.250

- 0.136

- 0.126

- 0.000 b y convention

+ 0.345 + 0.522

+ 0.800

+ 1.2 approx.

+ 1.42

+ 1.68

area Cathodic area

a- Iron

Fig . 1.3 Corrosion ce l l on the surface of i r o n in water

12

1

P o t e n t i a l o f M e t a l

E h r e l a t i v e 0 t o h y d r o g e n

- 1

L \

\ \ \ - \

\ %

\ \

\

% %

\ \

- O x i d a t i o n - C o r r o s i o n - % \

- C o r r o s i o n - - Immuni ty due t o - low i r o n p o t e n t i a l

Fig . 1.4 A simpl i f ied form of the Pourbaix Diagram for i ron corrosion

I f the, or some of the, i ron oxides are present as protective layers

they may be eroded by f lowing water especial ly i f sediment i s present.

Cavitat ion can also erode the surface layers. The metal i s thereby exposed

and corrosion i s accelerated.

The equ i l ib r ium between i ron and var ious compounds i n the presence of

water was studied by Pourbaix. He presented h i s resul ts in a diagram

(F igure 1.4) which shows three zones:

A corrosion zone for low pH or h igh electr ical potent ia l re la t i ve to

l i qu id solution.

A corrosion inh ib i t ion zone fo r h igh pH due to passivat ion b y a f i lm

found on the surface

A cathodic protection zone for low i ron potent ia l re la t i ve to a standard

elect rode.

13

Hydrogen is used as a reference electrode i n the diagram. The

potential of the i ron w i l l depend on the reference system. Table 1 . 1 gives

the equi l ibr ium potent ia ls of metals immersed in a normal solution of one

of i t s salts, re la t i ve to the standard hydrogen electrode a t 25OC. There

are many texts on factors af fect ing corrosion e.g. Uh l ig (1963)

Types of Corrosion

There are many ways i n which corrosion can occur in the presence of

water. Corrosion i s commonly an electro-chemical phenomenon which occurs

a t an anode when electrons flow from an anode to a cathode, leaving a

posi t ively charged anode to react wi th oxygen. The cathode does not

corrode. Ways i n which the electrons migrate for corrosion to occur, are

described below (Uhl ig, 1963).

Galvanic Corrosion:

When electr ical ly dissimi lar metals are i n contact in or through an

electrolyte, a potential difference i s established. The more act ive (anodic)

metal corrodes, as i t i s least noble.

Selective Lea china:

One element of an al loy can be corroded more r a p i d l y than another.

Pitting :

A shell of permeable magnetite o r fe r r i c hydroxide may form over an

i ron surface. Salts may concentrate under the shell and the resu l t ing

env ironmen t becomes increasing I y corrosive.

Stress Corrosion :

Metals i n stress may exh ib i t abnormal corrosive propert ies in a

corrosive environment. Once a crack i s formed i t r a p i d l y deteriorates due

to salt bui ld-up s imi la r to p i t t ing . Chlorides and amonia appear to be the

chief aggressors i n th is type of corrosion. Welding may also induce l ines

of corrosion unless stress rel ieved.

Ac id Corrosion :

Acids, o r even carbon dioxide i n solution, can increase the hydrogen

ion concentration. This resul ts in general loss of metal by corrosion. Some

chelants, e.g. NTA (n i t r i l o t r i ace t i c acid) may also become corrosive as

they concentrate.

14

Bacter ia l Corrosion :

Bacteria can cause biochemical act ion which resu l ts i n corrosion. This

type of corrosion i s often encountered in su lphur ic condit ions.

E lec t r i ca l Corrosion

Electr ic currents, d.c. i n pa r t i cu la r , may cause severe corrosion. I f an

anode i s formed where the current leaves the conductor, corrosion occurs

there.

Reagent Corrosion :

Certain scale prevent ing agents such as acids and chelants and

complexing agents can promote corrosion

T h e effectiveness of a l te rna t ive corrosion prevention methods depends on

the preva i l ing circumstances and system to be protected. In small closed

cool ing systems re la t i ve l y h igh concentrations of chemical dosage are

possible. I n la rge c i rcu i ts and cool ing systems, i n order to be economic,

the treatment dosage must be less; sometimes l i t t l e more than p H and

concentration control (by bleeding off and rep lac ing w i th fresh water) can

be accomp I i shed.

I n ch i l led water c i r cu i t s the c i rcu la t ing water may be consumed by

human beings. I n these circumstances i t i s imperative that any treatment

used i s non-toxic. This requirement has the effect of severely l im i t i ng the

number of chemical corrosion inh ib i to rs which can be considered.

Corrosion prevention

Corrosion can be reduced by changing the character ist ics of the water

o r coating the metal. Metal i s sometimes i n fact coated na tu ra l l y by scale.

A uniform deposit of calcium carbonate can be created by dosing the water

w i th lime, soda ash o r caustic soda. The deposit i s f requent ly non uni form

o r unstable, and cannot be re l ied upon for 100 percent protection.

Deaeration of water w i l l also reduce i t s cor ros iv i ty . I n closed systems,

vacuum deaeration if feasible. Oxygen and carbon dioxide which a i d

corrosion are thus removed to some extent. Sodium sulphi te can be used to

remove oxygen i n the water. The reaction w i th w i th oxygen forms sodium

sulphate:

2Na2S03 + O2 = 2Na2S04

15

Corrosion inh ib i to rs are ava i lab le commercially. One type of i nh ib i t o r

passivates the surface by forming a protective oxide f i lm such as

magnetite (Fe304). Other inh ib i to rs react chemical ly to form insoluble

precipitates. Into the la t te r category fa1 I zinc, calcium carbonate, calcium

phosphate and ortho- and polyphosphates.

Other inhibi tors act by absorbing or by passivat ing. The la t te r form a

protective f i lm and include chromate, n i t ra te , molybdate and tungstate.

Silicates also appear to work on s imi la r pr inciples.

I n general the corrosion ra te i s dependent on conduct iv i ty. pH and

oxygen. I t increases with conduct iv i ty up to a l im i t , whereas i t i s most

s igni f icant when the pH drops below 4 (see Fig. 1 . 5 ) . Oxygen content

increases corrosion rate, especial ly a t h igher temperatures.

Chromates are pa r t i cu la r l y effect ive corrosion inhibi tors. Concentrations

up to 300 mi l l igrams per l i t r e in open c i rcu i ts and 2000 mi l l igrams per

I i t r e i n closed c i rcu i ts are used. I t i s therefore costly, and i t s toxic i ty i s

a deterrent. Addit ives of zinc and phophate reduce the chromate

requ i remen t s.

To overcome the toxic i ty problem of chromates, sui table ortho-phosphate

and polyphosphate mixtures have been developed. To prevent p rec ip i ta t ion

of calcium orthophosphate at orthophosphate concentrations above 5 to 7

mil l igrams per I i t r e , an inh ib i to r such as phosphonate can be added.

Simultaneous passivat ion of the anodic areas and precipi tat ion of calcium

salts a t the cathodic zone to form a protective layer i s thereby possible

(referred to as dianodic protection, a p ropr ie t ry name), (Betz, 1980).

Fi lming amines such as octadecylamine act d i f ferent ly. They form a

physical bar r ie r , often monomolecular i n nature.

POTABLE WATER STANDARDS

Although indus t r ia l water i s not often intended for human consumption

the qua l i t y should be adequate to ensure no harm i f i t i s consumed. I t

should be non-toxic, and i f drunk in l imited quant i t ies showld show no i l l

effects. The upper l im i ts to dissolved sal ts fo r potable water a re d i f f i cu l t

to f i x . They depend on the amount consumed and i t should be born in

mind that men could d r i nk up to 2 l i t res a shi f t .

Microbiological matter in the water can be more concern than dissolved

salts. After disinfection, normally wi th chlorine, bacter ia and viruses are

not normally present i n mine service water, bu t regu la r checks should be

made. Toxic substances include heavy metals, concentrated f luorides,

nitrates, some algae, organic phosphates and some poly-electrolytes ( the

16

la t te r two are used in t rea t ing water sometimes)

Highly mineral ized water possesses l axa t i ve properties. I t may also

affect the sweating process, blood pressure o r the cardio-vascular system.

Often human perception (taste, smell o r colour) w i l l i den t i f y the

poss ib i l i t y of unsafe water. Phenols, chlor ine and organic matter are

easi ly detected b y taste.

Suggested l i s t of l im i t s to cer ta in substances fo r po tab i l i t y i s g iven in

Table 1.2. Table 1.3 indicates the maximum al lowable concentrations of

other toxic substances.

Fig. 1.5 The effect of pH on the corrosion rate.

TABLE 1.2 Recommended potable water standards.

Substance Concentrat ion

mg/e

A I k y I ben zenesu I fona t e ( ABS ) , tast e-produc i ng

Arsenic (As)

Chloride ( C 1 1 , taste-producing

Carbon chloroform extract ( C C E ) , taste-producing

possi b I y toxic

Cyanide (CN)

I ron (Fe), taste- and colour-producing

Manganese (Mn), taste- and colour-producing

N i t ra te (NO ) , producing methemoglobinemia

P heno I s , Sulphate (SO)&), taste-producing and laxa t ive

Total dissolved solids, laxa t ive

Zinc (Zn) , taste producing

3 t as t e-p rod uc i n g a nd tox i c

0.5

0.1

250.0

0.2

0.01

0.3

0.05

45.0

0.001

250.0

500.0

5.0

17

TABLE 1.3 Toxic concentrations in water

Substance Concentration,

mg/t

Arsenic (As)

Barium (Ba)

Cadmium (Cd)

Chromium (hexavalent, C r

Cyanide ( C N )

Lead (Pb)

Selenium (Se)

Si lver (Ag)

6 + )

0.5

1 .o 0.01

0.05

0.02

0.05

0.01

0.05

AGRICULTURE AND IRRIGATION

I r r i ga t i on i s a major consumptive use of water. Crops cannot tolerate

h igh sa l t loads and yields deteriorate unless remedial act ion i s taken. The

fol lowing table shows levels of sal ts which affect crops.

TABLE 1.4 Water Quali ty which affect crops

Lower l im i t Upper l im i t

T DS mg/P 500

Root abstraction :

Chloride mg/P 150

Leaf water abstraction

(spr ink l ing)

Chloride mg/P 100

Nitrates mg/P 5

2000

350

1000

30

Rapid assessment of TDS i s often possible by measuring conduct iv i ty.

The conductivi ty in mS/m i s approximately equal to the TDS ( to ta l

dissolved sol ids concentrat ion) i n m g / t div ided b y 6.5.

F ig 1.6 shows the decrease in y ie ld for some crops as a function of

soi l moisture salinity.Some crops are more resistant than others to sa l ts

due to their p u r i f y i n g a b i l i t y . For instance, vegetables are more resistant

than f r u i t , but a re less prof i table.

There i s also the gradua l deteriorat ion in soi l to contend with. Salt

bu i l ds up due to evaporation and t ransp i ra t ion of water. The sal ts can b e

leached out by appl icat ion of excessive water, but , for instance, a t least

25% more water would be required to ensure good sol id condit ions w i th the

TDS levels of 800 mg/e. More i r r i ga t i on equipment i s also required to cope

with the higher flows.

The a l te rna t ive i s to change the cropping pattern. Crops requ i r i ng less

water o r adaptable to sal ine water would have to be developed.

\ Lucerne\ I

I I \ \ I

Conductivi ty of groundwater (mS/m)

Fig. 1.6 Crop y ie ld as a function of water qua l i t y

19

REFERENCES

Barenbrug, A.W.T., 1965. Psychrometry and psychrometr ic char ts .

Betz. 1980. Handbook of I n d u s t r i a l Water Condi t ion ing, 8 th Ed., Betz,

Brownlow, A.H., 1979. Geochemisty, Prent ice Ha l l , N.J. 498 pp. Langel ier , W.F. 1954. Journal America1 Water Works Assn., 46, 461. Mrost, M. and L loyd , P.J., 1980. Bac ter ia l Ox ida t ion of Wi twatersrand

Slimes, I .A.E.A. Johannesburg. Pel le t ier , R.A. 1964. Minera l Resources of South - Centra l Af r ica. Oxford

Un ivers i ty Press. Cape Town. 277 pp. Porges, J. 1971. Handbook of Heating, V e n t i l a t i n g a n d A i r Condi t ion ing.

6 th Ed., Newnes-Butterworths, London. Ryzner, J.W. A p r i l 1944. A new index f o r determin ing the amount o f

calcium carbonate scale formed b y water. JAWWA, 36, 472-473. Uh l ig , H.H. 1963. Corrosion a n d Corrosion Control. John Wiley a n d Sons,

N.Y. Van Staden, C.M.V.H., 1970. Steps Taken b y the M i n i n g I n d u s t r y to

Prevent and Overcome Water Pol lu t ion. Water f o r the Future Convention, Pretor ia .

Transvaal and O.F.S. Chamber of Mines. Johannesburg.

Trevose, 440 pp.

CHAPTER 2

MATHEMAT I CAL MODELL ING O F WATER QUAL I TY

A f ie ld to which many systems concepts can be appl ied i s that of water

qua l i t y deteriorat ion i n indus t r ia l systems. Cooling and washing systems

are examples where qua l i t y w i l l deteriorate i n time. I t i s not easy to

predict the ra te of bui ld-up of dissolved sal ts o r the equ i l ib r ium

concentrations i n water re t i cu la t ion systems, even w i th an understanding

of the or ig ins and methods of concentration of salts. This i s because of

the complex nature of indus t r ia l water recirculat ion systems. One way of

accounting for a l l these effects i n a real system appears to be b y

modell ing the system on a computer.

Once a model i s produced and val idated, i t may be used to improve the

operation of ex is t ing service water re t i cu la t ion systems and for opt imizing

the design of new systems.

The bui ld-up of impur i t ies i n water can be simulated mathematical ly

together wi th the water rec i rcu la t ion cycle. The flows of water i n conduits

o r i n vapour form i n the a i r in and out of the system can be calculated.

The processes of evaporation, condensation, po l lu t ion and make-up can a l l

be modelled.

MASS BALANCES

For the purposes of mathematical simulat ion o f water systems, the

system must b e described in terms of equations. One-stage systems can be

described i n terms of a mass balance equation which can be solved

ana ly t i ca l l y . I n other more complex s i tuat ions i t i s necessary to express

the equations in f i n i t e difference form and solve them numerical ly.

Different types of models and the assumptions therein are described below.

Parameters whereby pol lut ion i s measured may ei ther be conservative o r

non-conservative. I n a conservative system input to any p a r t of the system

equals outflow. Thus, i f the parameter studied i s water flow then

evaporation w i l l be neglected in a conservative model. S imi la r ly i f the

parameter i s a chemical compound i t i s assumed there i s no reaction,

deposition or solut ion i n a conservative model.

The model may be steady-state or t ime-varying. Dur ing the start-up

per iod of a mine as concentrations bui ld up the system i s said to be

unsteady. After a whi le the system may reach equ i l ib r ium. That is, in the

case of sal ts in solut ion, the increases i n mass of dissolved sol ids in the

21

system due to leaching or evaporation equals the loss by pumping or

deposition.

MIXED AND PLUG FLOW SYSTEMS

I n a plug-flow system, the water i s assumed to t ravel through the

pipes and dra ins at a certain rate, conveying impurit ies a t that rate. The

sal ts content at any point can therefore be affected in a series of steps

as water wi th di f ferent concentrations a r r i ves a t that point. I n a

completely mixed system, the concentration of sal ts w i l l be the same a t

every point. An input i s assumed to spread instantaneously through the

systems so that the concentration increases by the mass of sa l t input

div ided by the total volume of water in the system. This s impl i f ied

mechanism is often satisfactory to describe systems which exh ib i t gradual

rates of change i n concentrations. Real systems w i l l probably be between

p lug flow and completely mixed, as there w i l l be di f fusion and mix ing due

to tubulence and cross connections. I n general salts a re conveyed by

advection ( la te ra l t ransport) and dispersion.

Examples

The simplest i l l us t ra t ion of the use of the mass balance equations i s for

a steady-state system. Q i s flow ra te i n e/s or MP/d, C i s the

concentration i n mg/e. Inf low of water and of sal ts per un i t time equals

outflow rate:

F ig 2.1 Point Node

Flow Balance

Mass Balance

a, + a2 = a3 ale, + a2c2 = a3

= alcl+a2c2

Q 1 +Q2

3 .*.C

22

e.g. i f Q1 = 5 MP/d, Q2

then C3 = 200 mg/t

and the total mass of sa l t discharged per day

= 10 Me/d (water flow ra te )

C 1 = 400 mg/P, C2 = 100 mg/P (sa l t concentrat ion)

= Q3C3 = 15 x 200 = 3000 kg/d ( 2 . 4 )

A completely mixed system can be described by d i f fe ren t ia l equations:

Subscript i refers to inf low, e to exi t , s to i n i t i a l conditions:

Volume S

Conc. C

Fig . 2 .2 Mixed flow node

d (SC) Q iC i = Q C + - e dt

dC = Q C + Sx for constant S

SdC .*. dt = Qi C i-QeC

F ig . 2 .3 Diffuse node

Integrat ing and eva lua t ing the constant of integrat ion from the fact

that C = C at t = 0 :

T = S en QiCi-QsCs

1 - ( Q i C i - Q e C

Qe

( 2 . 8 )

Q i C i Q i C i / Q e - C s o r C = -- 1

Qe e ( Qe t /S

23

(2.9)

e.g. at t = 0, C = Cs, and at t = m , or Be = - o r S = 0,

c = (ai/ae)ci Observe that i f Q . does not equal Qe, there must be internal gains or

losses, e.g. due to evaporation.

The previous example could be studied numerical ly. Although th i s

requires specific numbers, i t i s often the only p rac t ica l way of so lv ing

more complex problems.

Assume S = 1000 m3, Q . = lm’/s = Q e’ Choose ~t = 100 5. The choice of ~t can affect the speed of solut ion,

the accuracy of resul ts and the numerical s tab i l i t y of the computations. I t

must be determined by t r i a l , from experience or from theoretical

Cs = 0, Ci = 500 mg/P.

considerations. c -c 2 1

NOW Q.C. - Q C = 5- I I At

.’. C2 = C t 8 . ( C . -C ) = C1 + 0.1(500-C1) 1 5 1 1 1

The computations can be set out in tabular form as follows:

c2 500-C1 xo.l c 1

t

(2.10)

(2.11)

0

100

200

300 0

0

0

1000

0 500 50 50

50 450 45 95

95 405 40 135

135 365 37 172

326 1 74 17 343 mg/t

Equation (2.9) would indicate C = 316 mg/e a t t = lOOOs, which i s

comparable w i t h the resul t indicated by the numerical solution of 343

m d e .

A p lot of C versus t i s cal led a pol lutograph, and CQ versus t a

loadograph. The numerical computations for change of pol lutant load i n the

above table are very s imi la r to flood rou t ing calculat ions assuming for

example the Muskingum method.

Reservoirs are general ly assumed to be completely mixed, whereas r i v e r s

24

are sometimes assumed to be p lug flow. I n fact i n both there i s a degree

of mix ing due to:

a ) Molecular di f fusion due to Brownian movement (neg l ig ib le in most

hydrau l i c systems).

b ) Turbulent mixing, due to eddies i n the stream.

c ) Short c i r cu i t i ng o r t rack ing e.g. in reservoirs where a t rack i s

made across the water body b y the f low. The stagnant water i n

corners i s cal led dead water

d ) Wind mix ing

e ) Thermal mix ing and inversion (e.g. Henderson-Sel lers, 1979).

The degree of mix ing can affect concentrations so much that monitoring

systems need to account fo r i t (Sanders, 1983).

SYSTEMS ANALYSIS

A more sophisticated approach than the simulat ion method described

above i s the use of systems analysis and optimization techniques, w i t h the

assistance of computers i f necessary. The methods al low an optimum design

to be selected from numerous al ternat ives (Thomann, 1374).

T h e a l te rna t ive standard engineering approach i s to select the best

option from a few selected designs. The la t te r approach i s tedious where

there are many al ternat ives.

T h e design optimization approach involves the creation of a general

conf igurat ion i n which the numerical value of independent var iables has

not been f ixed. A n overa l l economic objective i s defined and the system i s

described i n terms of equations or constraints.

TERMINAL CONCENTRATION I N A WATER C I R C U I T .

The total dissolved sol ids concentration i n a closed water recirculat ion

system w i l l bu i l d up due to evaporation and adsorption or leaching. The

concentrat ing effect w i l l continue indef in i te ly unless saturat ion occurs, o r

water i s replaced. Make-up water w i l l replace pol luted water and the

re la t i ve proport ion of ra te of replacement to water i n c i rcu la t ion w i l l

control the equ i l ib r ium dissolved sol ids concentration. Computation of the

equ i l ib r ium concentration i s performed as follows:

F I ~ ~ : ai = aD + a e ( 2 . 1 2 )

25

a. Plug flow

c I c,

t

C

t

b. C o m p l e t e l y m i x e d s y s t e m

t

c. D i f f u s e sys tea

Fig. 2.4 Comparison of p lug f l o w a n d mixed systems.

26

Salts : QiTi + QiTe - - QPTP

(2.13)

where Qi i s the water input ra te (.e.g. l i t r es per' second or megali tres

per day ) ,

Q i s the discharge pumping ra te

Q i s the evaporation ra te

T. i s the concentration of sal ts i n the replacement o r makeup water

P

stream.

T i s the concentration b u i l d up due to leaching, expressed i n e terms of the incoming water flow ra te here.

Tp i s the concentration in the pumped water which i s the same as

the c i rcu la t ing water for a mixed flow system.

I f Q i s i n megali tres a day and T i n mg/t then QT has the un i t s of

ki lograms of sa l t per day. Solving fo r T the sa l t concentration i n the

system, P'

(2.14)

Thus for no leaching ( T = 0) and an evaporation ra te equal to 50% of

the pumping rate, T = 1 . 5 T i i.e. the equ i l ib r ium sa l t concentration w i l l

be 150% of that of the make-up water. P

APPLICATION TO A MINE WATER CIRCUIT

South Afr ican gold mines use near ly 2000 m i l l i on l i t r es of water a day

underground (Holton and Stephenson, 1983). The water i s used p r i m a r i l y

for dust control and cooling. Owing to the great depths i.e. often over

3000 metres below surface, rock temperatures can reach 65°C. The most

ef f ic ient method of cool ing i s b y means of spray ing ch i l led water onto the

rock. The water i s also used fo r ore moving and to a l imi ted extent for

hydrau l i c emulsions in machinery.

The geological formations i n which gold i s mined are i n the Orange

Free State and Transvaal which suffer water shortages. Water i s i n fact

imported from adjoining catchments such as i n Natal for domestic d r i n k i n g

purposes. The cost of water i s therefore h igh and re-use of water i s

encouraged both to conserve water and to minimize the discharge of

pol luted wastewaters into surface streams. The water requirements of the

gold mines are therefore la rge ly met b y recycl ing and only approximately

10% of the requirement i s made up from surface water resources. Some

27

mines also have surplus underground water from in f i l t r a t i on and th i s i s

used where possible.

# Pure water

Remaining concent ra ted S a l i n e water

Fig. 2.5 Model of sa l t bu i ldup due to evaporation

The qua l i t y of surface water i s good and the total dissolved sol ids

content i s typ ica l l y less than 500 mg per l i t re . The qua l i t y of ground

water where there is any, i s also general ly good as the water or ig inates

largely from dolomitic aqui fers in the upper strata. Although the water i s

hard and contains magnesium and calcium carbonates, the dissolved sa l t s

concentration is ra re l y above lOOOmg per l i t r e . I n the Orange Free State

on the other hand the na tura l water i s known to contain h igh

concentrations of chlorides.

, Q i

Average s a l i n i t y I

Fig. 2.6 Model of sa l t equ i l ib r ium due to pumping

28

Despite the general p u r i t y of make-up water, concentrations of dissolved

and organic sal ts underground can t yp i ca l l y va ry from 3000 to 10000 mg

per I i t re. This water can therefore only be used for l imited purposes. Care

has to be taken to ensure that i t i s not used for d r i nk ing , in cer ta in

machines and w i th heat exchange apparatus. In many mines there i s

scal ing and fou l ing of machinery and pipework because of the poor qua l i t y

of water and i n other mines there i s corrosion of pipework and other

metal-work.

Reasons for the deteriorat ion in mine service water can be a t t r ibu ted

p r imar i l y to leaching from the mined ore. In add i t ion certain po l lu tan ts

a re brought from the source of the water and there i s a secondary

concentration effect due to the evaporative loss of water underground and

i n cooling towers. The chemical dosing of water for pur i f i ca t ion purposes

and for corrosion and scal ing inh ib i t ion can be ru led out as a major

contr ibutor owing to the small dosage rates re la t i ve to the increase in

dissolved solids i n the water.

Evaporation

Groundwater

Fig. 2.7 Mine water re t i cu la t ion system

After iden t i f y ing possible sources of chemical sa l ts appear ing i n the

water, laboratory and s i te test ing were performed to evaluate the possible

ra te of appearance of sa l ts in the water. The complicated nature of the

geochemical environment make% an exact quant i ta t i ve estimation of leaching

i n any pa r t i cu la r circumstance impossible. Nevertheless re la t i ve rates of

leaching can be assessed with these methods. Once p i l o t tests had been

29

conducted to ident i fy the prime effects of water qua1 i t y deteriorat ion, these

parameters were studied i n more detai l . I t appeared that the fineness of

the crushed ore or ig ina t ing from b las t ing or d r i l l i n g was a pr ime

parameter i n affect ing the rate of geochemical leaching. The composition of

the reef i.e. the ore bear ing stratum, also i s a factor. The contact time

with the water affects the amount of leaching from any pa r t i cu la r mass of

ore. Temperature affects the chemical reaction, as does the pH of the

water, and i n isolated cases possibly the presence of b io logical matter, in

pa r t i cu la r thio-baci l lus ferro-oxidans and thio-oxidans. The presence of

a i r appeared i n a l l cases suff icient to saturate the water w i th oxygen and

therefore was not a l im i t ing factor.

Conditions unless otherwise stated 209 fine, 3OoC, a i r bubbled through 2e

100 -

0 0 - E

E f 60

s, - .- > - "

P

.- a

c 4 0 -

/' 5-409 fine crushed ore

d0VS

Fig. 2.8 Laboratory leaching tests on crushed ore

Laboratory tests were performed by immersing samples ,o f ore crushed to

various finenesses in one to two l i t res of water. Temperature was

controlled by means of a bath and a i r was bubbled through the samples to

agitate and provide suff icient oxygen. Datum tests wi th pure d i s t i l l ed

water were performed simultaneously. Tests were run for longer than a

month and conduct iv i ty and various dissolved sal t parameters were

measured regu la r ly as well as pH and temperature.

30

The ra te of leaching of a typical batch of samples i s indicated in Fig.

2.8. I t w i l l be observed that the leaching rates were most r a p i d du r ing

the f i r s t day and then gradua l ly decelerated as the soluble chemicals in

the ore were depleted. Confirmatory tests w i th i n i t i a l water concentrations

a t various levels indicated that saturat ion of the water was not the cause

for the reduction i n leaching rate, The effects of d i f ferent passes of

crushed ore, d i f ferent sizes of part ic les and temperature, presence of a i r

and agi tat ion were studied in di f ferent samples.

The increase i n total dissolved sol ids in the water var ied from 5 to 30

grams of dissolved sol ids per k i logram of crushed ore.

The fol lowing inorganic sal ts were detected in the water samples

anal ysed : sulphates, chlorides, carbonates, n i t rates, calc i urn magnesi urn,

sodium as well as other elements i n the re la t i ve order indicated. The

concentration of sulphates in mi l l igrams per I i t r e (mg/P) of SO4 was

typ ica l l y one ha l f of the total dissolved sol ids concentration i n mg/e. This

can be a t t r ibu ted to the h igh sulphide concentration in the ore (up to 8

per cent sulphur by mass). In the presence of oxygen and water some

sulphide forms in pa r t i cu la r were oxidized to sulphates. The i ron from the

reaction was often precipi tated as i ron oxide and the chemical reaction

which i s well known i n both coal mining and gold mining po l lu t ion

problems is indicated below:

+ O2 + H20 4 FeO(OH) + H2S04 (2.15)

The pH of the solut ion remained between 6 and 8 for the f i r s t week in

most cases. By the end of the second or t h i r d week the pH often dropped

to below four. As the pH dropped an accelerat ion i n the leaching rate, as

indicated by an increase i n conduct iv i ty and total dissolved, sol ids, was

evidenced. The presence of bacter ia was noticed i n isolated samples a f te r

a month of test ing, but not i n a l l samples in which the pH dropped or the

ra te of leaching was noted to be pa r t i cu la r l y high.

I t i s therefore concluded that the leaching reaction i s p r imar i l y a

geo-chemical reaction and biological reaction can be said to be small in

the environment studied.

The appl icat ion of the laboratory resul ts to the f ie ld condit ions i s

par t i cu la r l y complicated. I t i s not only the total mass of f ine ore

generated by mining operations which i s of importance, but also the

exposed surface of the f ine which settles out r a p i d l y in the hor izontal

d ra ins tak ing water back to the shaft. Only the surface layer appears to

leach a t a h igh ra te and th is may exp la in the re la t i ve l y low leaching

r a t e underground as compared w i th the maximums measured i n the

laboratory. Tests i n the f i e ld could only indicate increase i n total

31

dissolved solids of the order of 100 to 300 mg/e per cycle as the water

ran from the workings back to the shaft.

I t was therefore not possible to insert the complete chemical process i n

equation form into the computer model of the system. Empir ical

relat ionships were therefore used and these w i l l have to be ver i f ied for

each mine and each ore mined

COMPUTER SIMULATION MODEL

The rates of use of water underground va ry considerably dur ing the

day and are highest dur ing the d r i l l i n g and ore moving shi f ts. Water i s

often stored i n the cascade dams underground or in surface dams a t

various stages. Fluctuation i n water qua l i t y i s therefore d i f f i cu l t to

predict unless the volumes of a l l the storage dams as well as the flow

rates i n the various conduits can b e modelled. External flows such as

evaporation, water removed with the ore, seepage and intermittent make-up

addit ions also affect internal volume, flow rates and qua l i t y . The most

logical method of s imulat ing the process was with a d i g i t a l computer

model. This was adapted to a micro computer wi th considerable success.

A general simulat ion program was developed for s imulat ing specif ic

models of mine water systems. Models are constructed i n general form fo r

pa r t i cu la r mines. The sizes of dams, the posit ions and the capacit ies of

conduits and the usage hydrographs can then be specified by the user.

The operating relat ionships which, for example, define the sa l t leaching

rate, c r i t e r i a for adding make-up water, s ta r t ing pumps etc. a re

programmed as pa r t of the model source code. The computer program w i l l

then be used to simulate the model. Flow rates, volumes and dissolved

sal ts concentrations are displayed at specified time in te rva ls as output.

Mathematical Basis of Model

The computer model was prepared in a modular structured fashion fo r

easy updat ing and modification. The va ry ing volumes and sa l t

concentrations are described i n models by means of f i rst-order o rd inary

di f ferent ia l equations. Al ternat ive methods of solv ing the equations

numerical ly are b u i l t into the simulat ion program and the methods can be

selected to suit the pa r t i cu la r equations. I n many cases a fast a lgor i thm

is sui table while i n other cases a more accurate algor i thm i s required to

solve the equations wi th suff icient accuracy on a numerical basis.

I n a mine water system many processes occur simultaneously and the

32

net effect i s either to increase or decrease the dissolved sa l t concentration

of flows and water volumes w i th time. The sa l t concentration of water in a

storage element, such as dams, depends on the mass of sal ts and the

volume of water i n storage, and the sa l t concentrat ion of any inf law and

outflow. Denote Q1 and Q2 as the inf low and outflow to a dam, and denote

C l and C 2 as the corresponding sa l t concentrations. I f M i s the mass of

dissolved salt and V the dam volume a t a cer ta in time t, then the ra te of

change of water volume and sa l t mass w i th time i s

and - dM = Qi.Cl - Q 2 . C 2 d t

( 2 . 1 6 )

( 2 . 1 7 )

I f perfect mix ing i s assumed to occur i n the dam then the sa l t

concentration of the outflow is: M

c2 = 0 (2 .18 )

A mine water model bas ica l l y consists of equations (2 .16 ) and ( 2 . 1 7 ) fo r

each storage element wherein the volume and sal t mass change w i th time.

Other relat ionships govern flow rates and changes i n sa l t concentrations of

flows between storage elements.

Start ing wi th known o r assumed i n i t i a l values fo r a l l M, V and C the

d i f fe ren t ia l equations are numerical ly solved using Euler and Runge-Kutta

methods. Values of M, V and C are determined a t each i terat ion

time-increment du r ing the simulat ion and can be displayed as output. The

s tab i l i t y and accuracy of the solution depends very much on the time step

and numerical method selected.

Considerable ef for t has to be expended i n ga ther ing data for the model

i t i s found. Owing to the unpredictable changes i n min ing patterns as the

character ist ics of the Reef change, the water re t i cu la t ion pattern i s

cont inual ly being extended or altered. The conduits and dams constructed

therefore form a complex storage d is t r ibu t ion system which i s often not

monitored as i t i s designed to operate automatical ly. Flow rates, stored

volumes and times of makeup were therefore often d i f f i c u l t to ascertain.

The model can thus oe used to predict the water qua l i t y a t any time a t

any point in th is system for a l te rna t ive operat ing conditions. F ig , 2 . 9

indicates a typ ica l var ia t ion in flow ra te a t the workings underground

and Fig. 2.10 indicates the water qua l i t y var ia t ion in the water pumped

to the surface a t the same gold mine over a per iod of a week.

33

12h00 24h00 l2hOO

Fig. 2.9 Flowrate from Coldwell to undergrund (M4/d)

The i n i t i a l conditions i n s ta r t ing up and runn ing the model could be

var ied to an extent. That i s the i n i t i a l water qua l i t y could be var ied

assuming that di f ferent make-up quant i t ies of surface water could be used

to replace poor qua l i t y water i n the surface storage dams over a weekend

when mining ac t iv i t ies were minimal. By comparing a l te rna t ive management

pol icies i n th is manner i t i s possible to reach a minimum cost procedure

for maintaining the water qua l i t y a t a certain selected level. The ra te of

usage underground was assumed f ixed by the mining operation and

therefore only storage dam capacit ies and make-up ra te could be var ied i n

th is way.

I t i s also possible that mining methods could be var ied to affect the

water qua l i t y . I t was recognised that the contact time between f ine ore in

suspension and i n the re tu rn water systems had an important bear ing on

the rate of deteriorat ion i n the water qua l i t y . Alternate methods of

returning the water were therefore investigated i n order to optimize the

water qua l i t y . In th is manner the effects of po l lu t ion can be minimized,

therefore requ i r ing less surface make-up water and reducing mining costs

i n el iminat ing to a large extent scal ing and erosion.

12000

10000

8000

5000

4000

2000

TueS Yed Thur F r i S a t Sun Hon Tues

Fig. 2.10 Salt concentration i n the settlers

34

REFERENCES

Henderson-Sellers, B. 1979. Reservoirs, McMi l lan , 1 2 8 ' p . Holton, M.C. and Stephenson, D . , 1983. A computer model of c i r c u l a t i n g

service water i n South Afr ican gold mines. I n t . J . M ine Water, 2 ( 2 ) p 33-42.

Qua l i ty . Water Resources Publ ics. 328 p .

McGraw H i l l , 286 p .

Sanders, T.G. ( E d . ) , 1983. Design of Networks for Monitor ing Water

Thomann, R . V . , 1974. Systems Analysis a n d Water Q u a l i t y Management.

35.

CHAPTER 3

NON CONSERVATIVE PARAMETERS

INTRODUCTION

Mass balances are not always possible. Many constituents in s t i l l

waters change concentration na tu ra l l y . Some react chemically to resul t i n

di f ferent salts. I f a l l the sal ts before and af ter reaction are soluble the

total concentration of dissolved salt i n mg/e i n the water remains the

same. Sometimes oxygen i s taken out of the water to release hydrogen gas

which i s more vo la t i le and escapes.

Oxygen i n water i s the cause of many changes. For instance ammonia i s

oxidized to n i t r i tes , and these in tu rn a re oxidized to ni t rates. The

ni t rates cannot be el iminated except by chemical replacement, absorption

or biochemically, as i s now done in some waste water treatment processes.

Absorption of oxygen and other chemicals i n water may occur due to

biological matter i n water. Decay i s general ly approximated b y a f i r s t

order equation

- _ - KC at

BASIC MASS BALANCE EQUATION

The one-dimensional balance equation al lowing for dispersion, decay

and sources or sinks is derived below

Source

I SdtAdx

direct i o n

Decay Kc Adxdt

Fig. 3.1 Mass balance

36

Net increase i n mass of C i n element in time d t i s

dC.Adx = d t (SAdx - KC Adx - C - dx - Q as dx+ a x ( A € a s ) d x j a x aa ax a x

For a uniform channel A = constant and t = constant and Q = constant

. ac a 2 c .. - + k C + v a t s- E , , I - ~ = O (3.3)

1 ) i s ra te of increase in concentration of po l lu tan t

2) i s decay ra te

3 ) i s advection

4) i s d i f fusion

5 ) i s source

E i s the turbulent di f fusion coefficient. I t i s s im i la r to the kinematic

viscosity which represents t ransfer of momentum between layers i n a

f r i c t ion model, e.g.

T = p E - against a wal l ,

where

du dy

where U,= v ' ( T / P ) = shear velocity

and k i s the von Karman constant, 0.4.

(3.4)

(3.5)

But i t i s not that simple i n channels as not on ly molecular di f fusion

but macro turbulence, t rack ing , dead water, s t ra t i f i ca t ion etc. complicate

the action, therefore one needs to ca l ib ra te models.

Elder (Deiniger, 1973) suggests E = A h J(ghS) (3.7)

where h = depth and A = coefficient (averag ing 0.07).

Normally di f fusion is neg l ig ib le in r i vers , except estuaries.

Thus one gets the Streeter-Phelps equation

ac - _ - - v - _ KC a t a x (omit t ing sources)

(3.8)

(3.9)

(3.10)

K ranges from 0.01 per day i n laboratory condit ions (as found b y Arnold,

1980) wi th publ ished f igures for r i ve rs averaging 0.1 per day.

37

t

or

C

X

Fig. 3.2 Decay curves

OXYGEN BALANCE I N R I V E R S

Oxygen concentration in a r i v e r i s measured in terms of DO (dissolved

oxygen). Shortage of oxygen i s measured as a chemical oxygen demand

(COD) o r a biochemical oxygen demand (BOD). The long term BOO i s about

1.45 x BOD5 where BOD5 i s the BOD as measured in a laboratory over 5

days, a standard test (AWWA, 1965).

Coupled equations for DO and BOD

I f DO concentration i s designated C and BOD i s L then

a2c ac - _ - E 7 - v~ - K , L + K,(C - C) * S c a t ax ( 3 . 1 1 )

38

J

Olcygen

Dissolved Oxygen Dernond Sag Curve

Dissolved Oxygen

P - . L . _ - I n - : - A

0 Carbonaceous plus

Lriilcui ruin1 R e oxygenot ion Curve

Deoxygenalion Curve

Distance Downstream

E f f luen t Outtall

or Time

Fig. 3 . 3 The dissolved oxygen sag curve

,

Fig. 3.4 Carbonaceous and nitrogenous oxygen demand curves

39

where C = saturation conc. of oxygen

and - = E 7 - v- - K,L, t S aL a 2 L aL a t ax ax L

(3.12)

These simultaneous equations can be solved a t points along a r i v e r and

over time increments, K, = K~~~~ e (T-20) i.e. i t i s a function of

temperature.

charac t e r i s t i C

A t n - l

Fig. 3.5 Solution g r i d

As an example of the solution of these two equations a two-step exp l i c i t

method can be employed (Deininger 1973, p 122). One can get pseudo

di f fusion where A x f v A t (3.13)

unless a careful numerical procedure i s used.

Where the r i v e r i s depleted of oxygen, the BOD equation must be

replaced by

KIL = K ( C - C) - Sc (3.14)

i.e. the quant i ty of oxygen consumed i s equal to the quant i t y of oxygen

introduced in the same time (Thomann, 1972).

2 s

Ana I y t ica I solution

d C d t 1 2 s

I f - = - K L + K (C - C )

and oxygen def ic i t D = C - C

_ - 2: - KIL - K2D

Integrat ing gives K I L O -K,t -K,t -K, t

K z - K i (e -e 1 + Doe D = -

One can also evaluate K, and K2 a t t C (Deininger 1972 p 126).

(3.15)

(3.16)

(3.17)

(3.18)

40

CALIBRATION OF A MOVING BOD MODEL

As an appl icat ion of the ca l ib ra t ion of a r i v e r oxygen model, the K l i p

r i v e r in South Afr ica was analyzed. The K l ip r i v e r has discharging into i t

ef f luent from major municipal sewage works and runoff from an

underdeveloped township. The stream i s also h igh l y mineral ized and flows

through reed beds. Measurements of BOD and DO over summer and winter

show r a p i d na tura l self aeration. The waters a re eventual ly recycled w i th

other sources. A numerical model predicts d a i l y var ia t ions i n BOD and DO.

The decay coefficient and sources and sinks were f i t t ed by l inear

programming opt im iza t ion.

The K l i p r i v e r r ises in the watershed of the Witwatersrand. O n i t s

banks are three major municipal sewage works and a la rge resident ia l

area. Separate san i ta ry sewers are provided general ly but a tendency to

l i t t e r i ng resul ts in h igh l y pol luted surface r u n o f f .

The populat ion of the area is near ly 2 mi l l ion. Of a total water

consumption w i th in the watershed of the K l i p r i v e r of approximately 500

mi l l ion l i t res per day, near ly 50 percent i s returned to the K l i p r i v e r v i a

sewage pur i f i ca t ion works or separate storm sewers (untreated) i .e. 2m3/s.

The base flow of the r i v e r i n the reaches studied amounts to only lm’/s.

OXYGEN BALANCE

The dissolved oxygen content (DO) of water i s a useful indicator of i t s

a b i l i t y to support l i fe. A lower level of 4 mg/t i s regarded as the l im i t

for f ish l i f e i n the area studied.

The ra te of which dissolved oxygen reduces the biochemical oxygen

demand is dependent on the level of free oxygen concentration. The upper

l im i t i s the saturat ion concentration, C s , estimated to be

CS (mg/e) = 14.6 - 0.41T + 0.008T‘ - 0.000778T’ (3.19)

where T i s i n “C

The DO in a pol luted stream var ies along the length i n accordance with

the ra te of takeup and the ra te of re-oxygenation (F ig . 3.3). I n add i t ion

to biodegradation of carbonaceous organic matter, oxygen i s required for

n i t r i f i ca t i on , ox id iz ing inorganic chemicals and p lan t respirat ion. W i t h a

h igh sulphur concentration i n the waters, due to mining ac t i v i t y , the

oxygen requirement i s f a i r l y h igh. This i s counterbalanced to some extent

by the h igh llime content, as the waters o r ig ina te from a dolomitic area.

Temperature and sludge deposits i n winter also inf luence the oxygen

demand.

41

Owing to deposits of sludge in the slow moving stream velocity ( less

than 0.2 m/s) du r ing winter months, BOD was observed to increase. After

the summer ra ins the deposists were scoured out and a more r a p i d

reoxygenation was observed. The sludge arose p r imar i l y from organic

matter, while benthal deposits were considered re la t i ve ly inact ive (Velz,

1970).

There are two pr imary biochemical oxygen abstractors; carbon and

nitrogen. The BOD removal curve typ ica l l y exh ib i ts an i n i t i a l hump due to

carbon and a subsequent hump due to ni t rogen (Fig. 3.4). The decay

equation smooths the curve out.

The coupled d i f fe ren t ia l equations describing the var ia t ion of BOD and

DO are 3.11 and 3.12 rewri t ten i n the form

(3.20)

(3.21)

A method of evaluat ing the coefficients K, and K2, and the source and

sink term, S and P, so that the equations represented the real r i v e r

system, would be to f i nd values for these parameters that would lead to

the minimum total difference between the concentrations predicted b y

equations 3.20 and 3.21 and the actual concentrations observed in the

f ie ld. L inear programming may be used for minimisation of an objective

function subject to certain constraints provided the system is l inear. In

the above case, the objective function would be

Minimise { z I Predicted BOD - observed BOD I + Z I Predicted DO - observed DO I } (3.22)

subject to the constraints formed by the system of equations and to the

constraint that the er ro r p lus the predicted value must equal the observed

value.

I n other words the ca l ib ra t ion of the model can be car r ied out by

minimising the sum of the absolute errors.

Another method would be by .means of least squares f i t t i n g techniques.

This has been attempted elsewhere, using the data from the sampling

survey on 21 March, 1979 (McPherson and Sharland, 1979) equations 3.20

and 3.21. The I inear ' programming method has been used be Kleinecke

(1971 for estimating geohydrologic parameters of groundwater basins.

O L - . ; ' . ' - . . . . I . . . . . 1 . . . . . . 1 1 . . . . . I . . . . . . . . a . A l . . . . . I I 1 . . . . . I . . , . . . . . . . . I . . . . . I ,

6h00 l2hOO 18h00 24h00 06h00 06h00 12h00 18h00 24h00 06h00 06h00 12h00 18h00 24h00 06h00

F i g . 3.6 Results of Simulation using minimum e r r o r ca l ib ra t ion parameters

43

1-1 I

Fig. 3.7 x - t g r i d

1+1 x

Considering the concentration-space g r i d in Fig. 3.7 i t can be shown

how the above coupled equations 3.20 and 3.21. can be formulated for

l inear programming evaluation of the parameters as follows:

The BOD concentration a t a point P can be wri t ten in terms of imp l ic i t

f i n i t e differences as:

1 - ( L 2At i ,n + L i+ l , n - Li ,n- l + Li+l,n-l

For I inear programming purposes two requirements must be met:

( i ) a l l terms must be l inear

( i i ) a l I var iables must be non-negative

I n the above f i n i t e difference form these condit ions are not satisf ied.

F i r s t l y the term - K + L . + L. + Li,n-l) i s not

l inear since both the K1 and the L. are unknowns. Secondly the net 1,n

source/sink term may be either posi t ive o r negative.

To overcome these problems the prediced L. are replaced by the known

observed values b. and the source/sink term i s sp l i t into an input term 1,n

+ S and an output term - T where one of S and T w i l l be posi t ive and the

other zero. The equation then becomes

1-1 , n-1 1 I n /4 .(Li-, ,n t , i

1,n

44

+ L . )/2 ( L . 1 ,n + Li+l,n)/2 - (Li,n-l i+1 ,n-1

(L i+ l ,n + Li+l,n-l - L. i,n - Li ,n- l 1 - -UAt - - 2 Ax

+ A t S i - At T i (3.24)

This can be rewri t ten as

1 U At 1 UAt (- 1 - UAt -)-L. 1 UL!) L i ,n - l (2 + d + L i+ l ,n - l (Z - ~ x ) - i,n 2 2Ax i + ~ , n ‘Z + 2AX

K1 iAt - - * (bi+ l ,n + bi+l,n-l + b i ,n + b i ,n- l 1

+ AtSi - AtTi = O (3.25)

I n addi t ion another set of equations can be wri t ten in terms of the

er ro r by which the predicted value of L . d i f fe rs from the actual value

of L . . 1,n

1 ,n

L . + Ui,n - Vi,n = b. (3.26) ‘,n i,n

Again the requirements that the var iab le must be posi t ive necessitates

the sp l i t t i ng of the er ro r into a posi t ive e r ro r U o r a negative er ro r -V,

one of which w i l l be zero in the solution.

S imi la r ly a set of equations can be wri t ten for equation 3.22. This

includes a reaeration term which i s also non-l inear unless the observed

values are substi tuted for the predicted values.

These equations are given below

1 UAt - ci+l,n ( - + -1

2 26x

+ A t Pi - At R i = o C. + M. - N. = d.

1,n 1,n 1,n 1,n

(3.27)

(3.28)

45

Equations 3.27 and 3.28 can be writ ten for a l l points i, except the last

point, along a study reach for which observed data i s ava i lab le over a

period of time. The observed values b and di,n a t each value of n may

have to be inerpolated from observations taken a t other times. Only

equations 3.26 and 3.28 can be wri t ten for the point furthest downstream.

The objective function now becomes

i ,n

+ N. (3.29) + V . + Mi,n 1,n {; ; (Ui,n 1,n

Minim i se

subject to the constraints given b y equations 3.27 and 3.28.

F I ELD MEASUREMENTS

The length of stream modelled was 6 km. I t was d iv ided into four

reaches and two sets of samples were taken as representative, one set i n

mid winter and one i n mid summer. Samples were taken every hour fo r 24

hours of each section, which was probably a b i t sparse. DO was measured

with a portable meter. The samples were tested for 5-day and 20-day BOD,

COD and pH, conduct iv i ty, ammonia, n i t rate, n i t r i t e , chloride, a l k a l i n i t y

and suspended sol ids were determined. Photosynthetic oxygen release was

estimated from l i gh t and dark bott le tests, and time of passage and

dispersion were determined with f luorescin dye,

Various methods were employed to ca l ib ra te the simulat ion model :

l inear programming was used to minimize the absolute value of the

differences between observed and simulat ion concentrations of BOD and DO.

The method i s described elsewhere. I n order to render the equations

I inear, the theoretical concentrations were approximated b y observed

values whenever products of two unknowns appeared in the equations. This

may have been the resul t of often apparent ly h i g h decay rates and

unaccounted for sources along some of the reaches. The methods are being

extended to non-l inear equations (McPherson and Sharland, 1979) wi th

encouraging results.

The input parameters for the plots given in Table 3.2 were derived by

t r i a l and error f i t s i n the model.

Even then, there appeared inexpl icably h i g h BOD or COD sources along

the r i v e r reaches. These were a t t r ibu ted to benthic deposits o r runoff from

adjacent sewage i r r i ga t i on works, and seepage from the indus t r ia l and

other townships to the north.

The accuracy of the BOD measurements a t the levels observed (5 to 10

mg/4) i s questionable, due to the complex way of determining i t . Various

46

researchers have proposed TOC ( total organic carbon) or COD (chemical

oxygen demand) as indicators of oxygen demand. Due to the h igh inert

f ract ion of COD, the change i n COD may be a more appropr iate parameter

than COD, and th is in fact gave better resul ts than the BOD model.

The sampling frequency of 1 hour was ra ther coarse. Once resul ts were

plotted i t was real ized that pol lut ion loading var ied rap id l y . This was

more l i ke ly due to surface runoff than to the ef f luent from the sewage

works.

The decay ra te of the COD was estimated to be up to 3,O per day,

which is h igh i n comparison w i th laboratory resul ts and other publ ished

data. This may be due to h igh turbulence, o r the h igh sa l i n i t y of the

water promoting reactions.

Photosynthesis was noticeable only on very overgrown reaches. A value

of 3 mg/P/day was typical .

Oxygen sinks were found to be large in winter (up to 75 mg/P/day) but

neg l ig ib le i n summer ( the ra iny season).

The dissolved oxygen content was found to be suff ic ient to support l i f e

(above 3 mg/P) a t a l l stages.

Typical results are included as Tables 3.1 to 3.3 and F igure 3.6.

REFERENCES

American Water Works Association, 1%5. Standard Methods fo r the Examination of Water and Wastewater.

Arnold, R.W., 1980. Modell ing Water qua l i t y in the upper KI ip r i ve r . MSc(Eng) Dissertation, Universi ty of the Witwatersrand.

Deininger, R.A., 1973. Models for Environmental Pol lut ion Control. Ann Arbor.

Kleinecke, D., 1971. Use of l i near programming for est imating geohydrologic parameters of groundwater basins. Water Resources Research, 7 ( 2 ) , p 367-374.

McPherson, D.R. and Sharland, P.J., 1979. River Qual i ty Tests.

Thomann, R.V., 1972. Systems Analysis and Water Qua l i t y Management.

Velz, C.J., 1970. Applied Stream Sanitat ion. Wiley Interscience, N.Y.

Undergraduate project, Universi ty of the W i twatersrand.

McGraw H i l l , N.Y.

TABLE 3.1 Results of ca l i b ra t i on using da ta of 21 March 1979 (end of summer)

Parameter

Dispersion coeff ic ien

Decay coeff icient

Reaeration coeff ic ien

BOD source/sink ( 1 )

DO source/sink (2 )

Photosynthetic DO (2

Iner t source/sink ( 1

Notes ( 1 )

(2)

jyrnbo

E

K1

K2

S

R1

p 1

Uni ts

COD Ca l ib ra t ion

?each 1

10.4

0.05

2.0

-1 80

8

3

-1 50

Value

3each 2

0.4

1.3

2.7

330

10

0

175

leach 3

10.0

3.0

3.0

100

0

0

80

BOD Cal ib ra t ion

Value

teach 1

10.0

3.0

2.0

75

30

2

Reach Z

0.4

2.0

2.7

30

10

5

Reach :

10.0

4.0

7 .o

-50

20

5

Not app l i cab le

-1)

A negat ive va lue indicates a source (pos i t i ve be ing a s ink ) .

A pos i t i ve va lue indicates a source (negat ive be ing a s ink ) .

Method of determination

Reaches 1 and 2 - Tracer studies Reach 3 - Ca l ib ra t ion

Ca I i b r a t ion

Reaches 1 and 2 - Formula Reach 2 - Ca l ib ra t ion

Ca l ib ra t ion

Ca l ib ra t ion

Bot t I e tests

Ca l ib ra t ion

The iner t f rac t ion of the input COD was taken as 60%

The BOD5/BOD20 r a t i o was taken as 0.69.

48

TABLE 3.2 Results of model f i t t ed to COD da ta of 18 July 1978 (mid-w i n ter

Parameter

D i spers i on

coefficient

Decay

coefficient

Reaera t ion

coefficient

BOD source/

sink ( 1 )

DO source/sinb

( 2 )

Photosynthesis

( 2 ) DO

~ n e r t source/

( 1 ) sink

ymbol

E

K 1

K2

S

R1

p1

- each

1

10.4 --

0.1

2.0

-32

- 1 3

3

-48

-

Value

- each

2

0.4

1.3

2.7

175

13

5

20

-

- each

3

10.0

1 .o

5.0

50

- 1 5

5

50

-

Method of

Det erm i n a t ion

Assumed same as

for March survey

Model f i t t i n g

Reaches 1 and

2 - formula

Reach 3 - model

f i t t i n g

Model f i t t i n g

Model f i t t i n g

Bottle tests

Model f i t t i n g

Notes: ( 1 ) A negative value indicates a source (pos i t i ve being a s ink )

( 2 ) A posi t ive va lue indicates a source (negat ive being a s ink )

The inert f ract ion of the COD was taken as 60%

49

TABLE 3 .3 Program output (.I D . I . LISIIng (lor mol

K l l p R l v w S l m l a t l m

ENn I 1.7s bn Vn = 23.40 kruleay DFFN - l0.00kruld.y aa* . 0.050 1/*y KFm . 2.00 I /&Y S I N t . -190.0 np / l /daY 91RE . 8.0 np/ l /daY

S I N t I . -150.0 np/l/dmy Po0 - 3.0 np/l/dmy

5.30 bn 20.03 krulday 10.00 krulday 3.m I/&y 1.00 I/&Y

100.0 -/1/dmy 0.0 np/ l /daY 0.0 np/ l /daY

m.0 np/i/ea~

lncrl Fncllm: 0.00 0.00 - O l u m l Varlal lm of Rolos,mlhesla used IFW VE IRO-ll

aLn - 0.007 days

O a X I l l * 0.203 bn OELXIII - 0.241 bn OaXl3l - 0,174 km

NO of pace 1n1ew.I~ NX - 27 No of Ilm Inl.walsNT 123

IbI 1yplc.l Lonplludlnal Oulpul

ULIPRIVER SIMULATION - Run 364 Tlm. aO.00 hrs

DI.l.nc. Sirnulaled Ownsirearn 9 o o w

statim m E

0.0 27.0 2.9 0.250 35.6 3.0 0.500 15.0 3 .2 0,750 52.3 3.3 I .ooo 5a.A 1.5 .. _._ 1.250 62.7 3.5 1.500 24.3 3.6 1.7% 62.4 3.5

SIa l lm M F

2 . W 50.6 3.4 2 . m 54.6 3.1 2.503 49.7 i.1 1.750 44.0 3.4 3 . W 17.6 3.5 3.250 30.9 1.7

Slal lm M G

3.500 U.6 3.9 3.750 24.0 1.9 4.m 23.3 4.3 4.250 22.9 4.1 4.503 22.9 4.1 4.750 23.0 4.2 5.m 20.2 4.2 5.254 23.3 4.2

Slmllm No. HI41

Ov..r".d moo

27.0 2.8

57.1 3.6

22.0 b.0

11.9 4.0

0 0 0 0 0

0 0 0 0 0

0 0

0 0

0 0

0 0

0 0

50

TABLE 3 . 3 Contd.

(c l Typical Time Variat ion Output

KLlPRlVER SIMULATION - RUN

eOp a n d 00 VARIATION = I T N TIME AT STATION F

TIME SIMULATED OBSERVED

N0.5

0.0 1.04 2.09 2.49 3.06 5.00 6.04 7.07 7.92 8.%

10.00 11.04 12.05 12.92 13.96 15.00 16.04 17.09 17.92 18.96 20.00 21.36 22.09 22.92 23.94 25.0

m a ,

57.56 3.38 59.69 3.45 57.55 3.52 53.24 3.65 52.68 3.91 52.24 4.21 52.55 4.47 M . 5 5 4.56 49.75 4.71 49.32 4.76 47.41 4.73 51.56 4.65 55.93 4.48 61.39 4.19 66.30 4.04 69.45 3.76 70.40 3.65 69.30 3.53 53.32 1.54 53.95 3.42 60.98 3.43 49.00 3.48 49.12 3.50 51.33 3.46 54.72 3.47 54.91 3.46

Kn

55.0 59.0 56.00 54.00 59.00 59.00 60.97 60.02 55.00 61.23 57.62 58.80 60.95 60.17 60.24 56.00 56.90 56.36 60.46 66.55 57.07 39.95 47.13 53.00 37.50 27.70

00

3.52 3.51 4.00 4.02 4.26 4.50 4.50 4.57 4.62 4.62 2.37 4.25 4.13 3.66 3.81 3.78 3.56 3.34 3.33 3.52 3.60 3.60 3.62 3.60 3.35 3.39

ox o n O N on 0%

XO SO NO NO LO S O

*O SO X

no NO

0 0 0 0 ON 0 0

no

0 . . 0.

0 . 0 .

0 . . . . . 0 .

a .

0 . 0 .

0 . . . . . . I .

0 . 0 . ..

6 . 0 . 0 .

. . . . .

5 10 15 20 25 30 35 44 4 2 50 55 60 6: ?$ 7: 80 0 .

I

0 I 5 10 15 20

51

CHAPTER 4

NUMER I CAL METHODS

SIMULATION OF HYDRAULIC SYSTEMS

Simulation of systems described b y d i f fe ren t ia l equations can be done in

a number of ways:

F in i te elements

Characterist ics

Fini te difference - Imp l ic i t - Four point 0-0

Fin i te difference - Exp l i c i t - Four point 7 'V

Leap frog

Dif fusive

Backward centred

Lax - Wendroff = d i f fusive/ leap frog

Exp l ic i t schemes are simple but not as accurate o r stable as imp l ic i t

schemes. Problems which manifest w i th exp l i c i t schemes include numerical

ins tab i l i t y and numerical dif fusion. I ns tab i l i t y can occur i f the time step

i s too great. The accepted s tab i l i t y c r i te r ion for d i f fus ive schemes i s

(Deininger, 1973);

Ax2/At 2 2 E (4 .1

or At 5 A x2 /2 E ( 4 . 2 )

There i s an addi t ional problem, that of numerical d i f fusion i.e.

spreading of the pol lut ion gradient due to successive calculat ions using

concentrations a t adjacent points. From a second order Taylor expansion

the maximum numerical d i f fusion i s E max = A x2/8At. (Deininger 1973)

Using the previous expression for A t , we get the pseudo di f fusion cannot

be less than €/4.

P

52

Two-step method

The water q u a l i t y equat ion i n c l u d i n g the d i f f u s i o n term can b e so lved

in two steps to ensure correct advect ion a n d d i f fus ion . Thus s t a r t i n g w i t h

a‘c ac a t ax _ _ EaxZ - v - kC

c . - C i d l

ci+l - 2c i + ci-l

use aC = Ax Ax

a 2 c - a x 7 - A X2

(4.3)

(4.4)

(4.5)

then C . = c . + EAt C i - l ,n+Ci+ l ,n -2Ci ,n - vAt ‘i+l,n-‘i,n - k C . i ,n+ l i,n 1,n

(4.6) A x 2 A X

The f i r s t a n d last two terms on the r i g h t h a n d side i n the above

equat ion for advect ion a n d decay can be used to get the f i r s t

approx imat ion to C . a n d then the d i f f u s i o n term. I, n+ l

F ig . 4.1 Basic r e c t a n g u l a r x- t g r i d

Demonstrat ion of numer ica l inaccuracy

The convection term in the water q u a l i t y equat ion w i l l be used to

i I lus t ra te problems a n d inaccurac ies due to a n incorrect numer ica l scheme.

Neglecting the d i f f u s i o n and decay term, we h a v e

= C. - v ~ t ‘i+l,n - ‘ i,n A X (4.7) ‘i , n+l 1,n

53

We should have a wave of concentration move downstream a t a ra te v ,

unattenuated or changed i n concentration.

C i ,n

i - 1 i i t 1 i t 2 A X

F ig . 4.2 Theoretical advection

I f Ax = vAt

then using a forward difference exp l i c i t method

- c. ) - (‘i+l,n i ,n ‘i , n+l 1,n = c. = 1 - (0-1)

= 2 which i s wrong, i t should be 0

i.e. dont use a forward difference

Instead use a backward difference ac/ax = ( C i - Ci-l)/Ax

ac/ax = (Ci+l - C i ) / ~ x

Then ‘i,n+l = c . t,n - (‘i,n - ‘i-l,n 1 = 1 - (1-0) = 0, correct.

on the other hand i f we use A X = 2vAt,

(4.7b)

= 1 - 0-0 = 1, also wrong. 2

I f we continued with th i s scheme, the value of C osci l lates (see below)

1

0

0.5

Fig. 4.3 Osci l lat ing scheme

54

O n the other h a n d i f one uses a backward d i f ference w i t h A x = 2vAt

numer ica l d i f fus ion occurs as ind ica ted below.

1 -

Fig . 4.4 Numerical d i f f u s i o n

I f At > Ax/v we get numer ica l i n s t a b i l i t y , e.g.

i f At = 2 Ax/v,

C. = Ci,n - * ( C . i,n - ‘i-1,n 1

Ax

1 - 2 (1-0) = -1.

Cont inu ing so, an o s c i l l a t i n g curve occurs:

I \

(4 .7d)

F i g . 4.5 I n s t a b i l i t y

55

Imp l ic i t f i n i t e dif ference schemes

i-1

Fig. 4.6 Impl ic i t scheme

i X i +1

(4.10)

z c, - vAt ) . A l l values a-t n+ l a re becomes C.

unknown and a set of i equations i s establ ished for i unknowns. The ('i,n+l - 'i-1 ,n+l ~ , n + l i ,n

method i s uncondit ional ly stable but solut ion of the i simultaneous

equations can be lengthy, especial ly for non l inear systems, e.g. i f we

use the hydrodynamic equation wi th the term v - this i s non-l inear ax

since v . ( V i , n + l - ' i - l ,n+ l ) - i s parabol ic. I , n+l

A x (4.11)

V . (4.12) So rather use 'i,n ('i,n+l - I-1,ntl) which i s l inear.

A X

Methods of solution of i equations include (Fr ied, 1975)

i ) Direct methods e.g. mat r i x methods and Gauss el imination.

i i ) I te ra t i ve method - i.e. assume reasonable values fo r a l l C 's and

i terate the equations subst i tut ing assumed values on the r i g h t hand

side un t i l the left hand side agrees with assumed values. This

only converges i f At < A X / V .

. . . I I I ) Relaxation methods (Timoshenko, 1951).

i v ) Al ternat ing direct ion imp l ic i t procedure (Fr ied, 1975), i.e. compute

56

der i v i t i ve w i th respect to x imp l i c i t l y and y exp l i c i t l y and then

vice versa (s tab le ) .

One also gets combined exp l i c i t / imp l ic i t methods for more accuracy

(e.9. McDonnel and O'Conner, 1977) .

Comments on finite difference methods

Exp l i c i t method:

1 .

2.

3 .

4 .

This must be designed to be stable i.e. any er ro rs due to 2nd order

terms in the Taylor expansion (we took jus t the f i r s t o rder ) must decay

d u r i n g comp u t ion.

The time in te rva l must threfore be smaller than for imp l ic i t method.

For exp l i c i t hydrodynamic equation, using Four ier series i t may be

shown to be stable i f 2 Jgy = wave celer i ty i.e. speed of

computation greater than speed of a disturbance i n the system.

-

I t must be accurate. Check with a few space and time in te rva ls and

against an ana ly t i ca l solut ion i f there i s one.

I t shohld minimize numerical d i f fusion

One can use va ry ing gr ids where greater accuracy i s required:

Fig. 4 . 7 Varying g r i d spacing (zooming)

57

NUMER I CAL METHODS FOR THE SOLUT ION OF SINGLE D I FFERENT I AL EQUAT IONS

Numerical solutions appear i n the form of a tabulat ion of the values of

the functions of various values of the independent time var iab le and not

as a functional re lat ionship. Numerical methods have the a b i l i t y to solve

prac t ica l l y any equation but they have the disadvantage that the en t i re

table must be recomputed i f the i n i t i a l condit ions are changed.

I f a function f ( t ) can be represented b y a power series i n a cer ta in

interval then i t can be represented b y the Taylor series expanded about a

point t = to, i.e. about the i n i t i a l value:

(4.13) I II I1

2! 3!

Y ( t )=y (tO)+Y ( to ) (t- tO)+y ( to ) (t- t0)2+y ( to ) (t- t0)3+ . . . - -

Lett ing n represent the previous step a t time to and n+l represent the

next step at t +h, the series can be wri t ten as: 0

I l l + Yn+l=~n+hyn l+h2 - yn I I + c y n *..

2 6

Consider the examp I e prob lem

(4.14)

(4.15)

wi th i n i t i a l conditions

Y(0) = 1 (4.16)

This i s a l inear time var ian t 1st order d i f fe ren t ia l equation. The

ana ly t i ca l solution to the problem, y = 2e-t-1 w i l l be used to compare the

numerical results of some of the methods and to i l l us t ra te the er ro r a t any

step.

t

The Euler Method

The Euler method i s the simplest but least accurate of a l l the methods

discussed. To obtain an exact numerical solut ion to the example problem I I I l l I V (4.151, a l l the der ivat ives y , y , y ... must be evaluated and

substi tuted into the Taylor series (4.14). Knowing the i n i t i a l values of y n '

yn , yn ..., Y,+~ could be evaluated a f te r a time increment h. The

values of a l l the der ivat ives could then be calculated a t n+l , and y n+2

could be evaluated a f te r the next time increment and so on. Der ivat ives of

a r b i t r a r y functions cannot easi ly be formulated in computer programs. The

der ivat ives y l ' , Y l I I , etc. are easy to evaluate fo r the example (4.14)

I I I

58

but th is i s not general ly the case. The Euler method truncates the Taylor

series by excluding the terms a f te r the f i r s t der iva t ive and el iminates the

problem of hav ing to evaluaate the second and subsequent der ivat ives.

Then

yn+l=yn+hynl+O(h') e r ro r (4.17)

Neglecting h'yn"/2 and the subsequent terms in (4.14) resul ts i n a

t runcat ion er ro r of order h' which i s denoted O(h*). This i s the l oca l

e r ro r and resul ts from one step only, i.e. from n to n+l. I t can be shown

that the g loba l e r ro r accumulated over many steps becomes O(h), i.e. an

er ro r of order h.

Substi tut ing the example (4.15) into the Euler algor i thm (4.17) gives:

Yn+l=Yn+h. (Yn+tn) (4.18)

The i n i t i a l condit ion y(O)=l means that y=O a t t=O. Choosing the time

increment h=0.02 and le t t ing the step number n=O a t t=O, the values for y

can be evaluated a t successive time increments as follows:

y =y +h(yo+tO) = 1+0.02(1+0) 1 0 y =y +h

2 1 y =y +h

3 2

y4

y5 etc.

+t ) = 1.0200+0.02( Y l 1 y +t ) = 1.0408+0.02(

2 2

= 1.0200

.0200+0.02) = 1.0408

.040+0.04) = 1.0624

= 1 .ow0

= 1 .lo81

Anal y t i c a1

solution

. c

solution

. c

Fig. 4.8 The Euler method

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

t The numerical solution af ter 5 steps i s y(0.10)=1 .lo81 whereas y=2e -t-1

gives the exact ana ly t i ca l solut ion as y(0.10)=1.1103. Hence the absolute

global e r ro r i s 0.0022, i.e. two-decimal-place accuracy. Since the global

59

e r r o r of the Euler method i s p ropor t iona l to h, i.e. O(h), the step size h

must be reduced a t least 22-fold to g a i n four-decimal accuracy, i.e. h

<0.004. Th is would increase the computational e f fo r t 22-fold. F ig . 4.8 shows

how the slope a t the beg inn ing of the i n t e r v a l yn l i s used to determine

the funct ion va lue a t the end of the i te ra t ion in the Eu ler method.

The slope a t the beg inn ing of the i n t e r v a l i s a lways wrong unless the

solut ion i s a s t r a i g h t l ine. Thus the simple E u l e r method su f fe rs from the

d isadvantage of lack of accuracy, r e q u i r i n g a n extremely smal l step size.

The Modi f ied Eu ler Method

F i g 4.8 and the subsequent discussion suggest how the Eu ler method can

be improved w i th l i t t l e add i t iona l computational e f for t . The ar i thmet ic

average of the slopes a t the beg inn ing and the end of the i n t e r v a l i s used

(on ly the slope a t the beg inn ing i s used in the Eu ler method).

1 1 yn+l = Yn + h'n +',+I

2 (4.24)

I The Eu ler a lgor i thm must f i r s t be used to p red ic t yn+l so tha t y

can be estimated. A p p l y i n g the same example (4.15) as before a n d

subs t i tu t ing y1 = x+t i n t o (4.24) g ives

n+l

Y n + l - -yn+h(Yn+tn) + (~,+~+t,+l) (4.25)

Subs t i tu t ing the Eu ler equat ion (4.18) for Yn+l g ives

2

'n+l = yn+h('n +t n 1 + (yn+h(Yn+tn) + tn+l 1 2

Using h=0.02 and the i n i t i a l condit ions: y = l , t =O 0 0

= 1 + 0.02 (1+0) + (1+0.02(1+0)+0.02) 2

= 1.0204

(4.26)

(4.27)

(4.28)

(4.29)

60

y = 2 1.0204 + 0.02(1 .0204+0.02)+(1.0204+0.02(1.0204+0.02)+0.04)

2

= 1.0416

(4.30)

(4.31)

y5= 1.1104 c f a n a l y t i c a l so lu t ion 1.1103

The answer agrees to w i t h i n 1 in the f o u r t h decimal p lace. Near ly twice

as much work was done a s in the Eu ler method b u t c e r t a i n l y not the 22

times more that would have been needed w i t h tha t method to a t t a i n four

decimal p lace accuracy. I t can be shown that the loca l a n d g loba l e r r o r s

of the Modif ied Euler method are O ( h 3 ) a n d O ( h 2 ) respect ive ly . The

Modi f ied Euler a n d the simple Eu ler methods a r e of ten re fe r red to as

second and f i r s t o rder methods respect ive ly .

Runge-Kutta Methods

The Fourth-Order Runge- Kut ta methods a r e amongst those which p r o v i d e

the greatest accuracy p e r u n i t o f computat ional e f for t . The development of

the method i s a l g e b r a i c a l l y complicated and i s g iven completely in Stummel

a n d Hainer (1978) w h i l e Gerald (1980) der ives the Second-Order

Runge-Kutta a lgor i thm a n d e x p l a i n s the p r i n c i p l e s beh ind the methods. Al I

the Runge-Kutta methods use the simple Eu ler method as a f i r s t estimate.

Improved estimates a r e then made u s i n g prev ious estimates a n d d i f fe ren t

t ime-values w i t h i n the t ime i n t e r v a l h. A weighted average of a l l the

estimates i s used to ca lcu la te yn+l. The Fourth-Order Runge-Kutta methods

a r e the most widely used because of t h e i r power a n d s i m p l i c i t y . The

fo l low ing i s a p a r t i c u l a r Fourth-Order method which i s commonly used a n d

which i s inc luded in the s imulat ion program:

=y + I ( kl +2k2+2k3+k4) Y n + l n 6

k2 = h f ( tn+ ih , yn+gkl )

k 3 = hf(t,+ih,yn+$k2)

k 4 = hf(tn+l,yn+k3)

(4.32)

(4.33)

(4.34)

(4.35)

(4.36)

61

Again the problem given i n (4.14) above i s solved as an example:

dy/dt=f(t,y)=t+y,y(O)=l. This time y(0.1) i s calculated in one step (h=0.1)

whereas ~ ( 0 . 1 ) was calculated i n f i ve time increments (h=0.02) using the

simple and modified Euler methods above.

kl =h(tn+yn)

=o. 1 (0+1 = 0.10000

k 2 =0.1 (0.05+1 .05) = 0.11000

k 3 =0.1 (0.05+1 .055) = 0.11050

k4 =0.1(0.10+1.1105 = 0.12105 1 6

y(0.1)=1.000+ -(0.10000+2x0.11000+2x0.11050+0.12105)

=1.11034

(4.37)

(4.38)

(4.39)

(4.40)

(4.41

(4.42)

This agrees to f i ve decimals w i th the ana ly t i ca l resul t and i l lus t ra tes

a fur ther gain i n accuracy wi th less effort than required b y the previous

Euler methods. I t s computationally more ef f ic ient than the modified Euler

method because, whi le four evaluations of the function a re required fo r

each step rather than two, the steps can be many-fold la rger for the same

accuracy. The simple E u l e r method would have required of the order of 220

steps to achieve five-decimal accuracy in y(O.1) but each step involves

only one evaluation of the function. The eff iciency of the Euler and

Runge-Kutta methods can be roughly compared b y ca lcu la t ing the number of

function evaluations required for the same order of accuracy. In th is

pa r t i cu la r example the Runge-Kutta method is approximately 50 times more

eff icient than the simple Euler method (220/4). The local e r ro r term fo r the

Fourth-Order Runge-Kutta a lgor i th (7.35) i s O(h ) and the global e r ro r

would be about O(h 1.

5

4

Mult istep Methods

The simple Euler, Modified Euler and Runge-Kutta methods are ca l led

single step methods because they use only the information from the last

step computed. I n th is they have the a b i l i t y to perform the next step w i th

a dif ferent step size and are ideal for beginning the solut ion where only

the i n i t i a l conditions are avai lable. The p r inc ip le behind a mult istep

method i s to u t i l i ze the past values of y and/or y l to construct a

polynomial that approximates the der iva t ive function and to extrapolate

th is into the next time interval . Most mult istep methods have the

disadvantage that they use a constant step size h to make the construction

of the polynomial easier. Another disadvantage of mult istep methods is that

62

several past points are required whereas only the i n i t i a l condit ions are

ava i lab le a t the start . The s ta r t ing values are general ly calculated from

the i n i t i a l condit ions using a single-step method such as a Runge-Kutta

method.

F I N I TE ELEMENTS

A n imp1 i c i t method invo lv ing mass balance across element boundaries

(Connor and Brebbia, 1976) i s popular i n f i xed systems but has not gained

much popu lar i t y i n hyd rau l i c systems owing to changing boundaries

necessitating i te ra t i ve methods. The steps are as follows:

Div ide body into elements ( 2 or 3- dimensional)

Define the nodal unknowns

The flow across an external boundary can be approximated as a

ma themat ical function.

Fig. 4.9 F in i te elements

One sets up equations g i v i n g balance for each element and solve

simu I taneousl y .

Boundaries for numerical methods

Conditions on a boundary may be either constant potent ia l (o r head o r

water level o r cencentrat ion) i.e. flow across boundary, stream1 ines (no

flow across) or mixed (same funct ion).

One can use pseudo points e.g.

hi-, = h. for no flow

h.

Simi lar schemes may be used fo r concentration def in i t ion.

-h. = h. - hi+, for flow perpendicular to boundary. 1-1 I

63

REFERENCES

Connor, J.J. and B r e b b i a , C.A. 1976. F i n i t e e lements f o r f lu id f l o w .

D e i n i n g e r , R.A., 1973. Models f o r E n v i r o n m e n t a l P o l l u t i o n Con t ro l . A n n

F r i e d , J.J., 1975. Groundwate r P o l l u t i o n , E l s e v i e r . Gera ld , C.F., 1980. A p p l i e d Numer i ca l A n a l y s i s ; 2 n d Ed. A d d i s o n VJesley. McDonel l , D.M., O 'Conner , B.A., 1977. H y d r a u l i c B e h a v i o u r of E s t u a r i e s .

Sturnel, F . a n d H a i n e r , K., 1978. I n t r o d u c t i o n to Numer i ca l A n a l y s i s ;

Timoshenko, S. and Goodier, J . M . , 1951. Theory of E l a s t i c i t y , McGraw H i l l .

Newnes-Bu t t e rwor ths .

A r b o r Science.

Macmi I Ian.

Sco t t i sh Academic P ress L t d .

64

CHAPTER 5

MASS BALANCE OF STORMWATER POLLUTANTS

I NTRODUCT ION

Pollut ion loadings from two catchments i n Johannesburg (Green et a l . ,

1986) were investigated. One, Montgomery Park, i s a suburban catchment

and the other, Hi l lbrow, a densely b u i l t up c i t y area. A comparison of

stormwater runoff and d r y weather flows from both catchments narrowed

down sources of po l lu tan ts and assisted in understanding the washoff

process. I t i s reported that non-point source po l lu t ion i s responsible fo r

70% of the load i n u rban runoff (Wanielista, 1979), and i t i s la rge ly th i s

type of contr ibut ion which i s detected here. Bradford (1977) attempts to

re la te pol lutant loads to landuse, and th i s paper contr ibutes to h i s

hypothesis. The unpred ic tab i l i t y of runoff qua1 i t y indicated by Simpson

and Kemp (1982) i s borne out though.

CATCHMENT DESCR I PT ION

Fig. 5.1 Montgomery Park Catchment

65

The Montgomery Park catchment i s si tuated 6 km north-west of

Johannesburg and measures 10.53 km2 (1053 ha) . The populat ion i s

estimated a t 15000. The developed area i s 75% of the total and the

remainder includes parks, a cemetery and undeveloped area. The

development i s housing and some commercial and l i gh t industry. There i s a

sol id waste t i p in the catchment f r o m which seepage occurs. The catchment

i s f a i r l y h i l l y , slopes rang ing from 0.02 m/m to 0.15 m/m. The main

drainage system comprises na tura l and a r t i f i c i a l channels (see F igure

5.1). Rainfal l over the catchment i s recorded a t f i ve locations b y

autographic r a i n gauges. Runoff i s measured a t a gauging stat ion a t the

catchment outlet i n which the measuring element i s a Crump weir w i th a

bubble type recorder. Electr ical conduct iv i ty of the water was recorded

continuously since March 1983.

The Hil lbrow catchment measures 67.2 ha and i s a f u l l y developed

urban area comprising high-r ise bui ld ings, some h igh density housing and

a school and is i l l us t ra ted i n Figure 2. The populat ion i s estimated a t

12000. There are four raingauges and a streamgauge for t h i s catchment.

Both catchments have separate stormwater drainage systems i .e.

seoarate from waste sewerage systems.

Fig. 5.2 Hil lbrow Catchment

66

QUALITY OBSERVATIONS

Fa1 lout measurement

An attempt was made to assess the level of TDS occurr ing as

atmospheric fa l lou t on the Montgomery Park catchment. After a per iod of 28

days without any r a i n f a l l , the raingauges in the catchment were "washed

down" w i th d i s t i l l ed water, th is water being collected in a sample bottle.

I t was found that the TDS wi th in the funnels averaged 9.5 mg. Since th i s

was deposited onto a funnel area of 0.020 m2 i t was deduced that the

equivalent fa l lou t loading on the Montgomery Park catchment was 4.75

kg/ha over 28 days. I f washout was omitted th i s would represent 62

kg/ha/annum. Atmospheric fa l lou t was collected i n a funnel wi th an area

of 0.72 mz a t a location near the Hi l lbrow catchment over a per iod of 18

days w i th no r a i n f a l l . I t was found that the TDS w i th in the funnel in th i s

case was 188 mg resu l t ing in an atmospheric loading ra te of 48

kg/ha/annum.

Stormwater runoff qua1 i t y data were collected for selected storms and

analyzed to determine whether relat ionships could be establ ished and to

obtain the proport ions of the di f ferent constituents.

Certain researchers have observed a correlat ion between the number of

d r y days preceding a storm and the level of po l lu t ion of the resu l t ing

runoff (e.g. Sartor et a l . , 1974; Colwi l l et al. , 1984) whi le others

maintain that no such relat ionship exists (e.g. Whipple et al. , 1977;

Bedient, 1980).

A n attempt was made to see whether the peak concentration of TDS

could be related to the number of antecedent d r y days. A regression

ana lys is was performed on a l l the data ava i l ab le and the best f i t resulted

from a l inear relat ionship, viz.

C =568 + 68N

where C i s the peak

antecedent d r y days w

corresponding i s 0.12

length of time between

I n a fu r ther tes

P

P

(5.1 1 TDS concentration in mg/l and N i s the number of

th a maximum value of 5. The correlat ion coeff icient

which i s poor. There i s an increase i n TDS w i th

storms i t appears.

, TDS was correlated w i th antecedent moisture

condit ion classes proposed by Terstr iep and Stal l (1974) used and the

fol lowing relat ionship emerged.

Cp = 1020 - 163 AMC

wi th a correlat ion coefficient of 0.29. C i s the peak TDS concentration in

mg/l and AMC i s the antecedent moisture condit ion class.

(5.2)

P

67

Relationship between Total Pol lutant Load and Runoff Volume

A regression analysis was performed on the po l lu tan t load - f low

volume data to determine whether any def in i te relat ionship could be

established between these parameters. I n a l l cases the best f i t was

obtained from l inear approximations wi th reasonably h igh correlat ion

coefficients.

Considering data from Montgomery Park alone resul ts in the equation

W = 3395 + 23V (5.3)

wi th a correlat ion coefficient of 0.84. W i s -the mass of transported

dissolved solids i n k g and V i s the volume of runoff in m’.

With the inclusion of the data from Hi l lbrow, the equation becomes

W = 1186 + = 0.27V (5.4)

wi th a correlat ion coefficient of 0.90.

Treated separately, the relat ionship between pol lut ion load and low

flow volume i s

W = 3.24 + 0.55V ( 5 . 5 )

with a coefficient of correlat ion of 0.97

Chem ica I Const i tuen ts

The resul ts of the chemical analyses of the “grab“ samples collected i n

both the Hil lbrow and the Montgomery Park catchments are l is ted i n Tables

1 to 5 .

These results were analyzed to quant i fy the presence of n i t rates,

chlorides and bicarbonates as i t was considered that these were the major

anions present in the water. The highest anion concentration was

bicarbonate, followed by sulphates dur ing storm runoff and chlor ide in d r y

weather conditions. Sulphates are predominant i n Johannesburg and could

be wind blown from neighbouring mine waste t ips which have h i g h

sulphate concentrations. Sulphates also reach concentrations over 300 mg/l

i n water supplies for the area.

T h e proport ion of n i t rates, sulphates, chlorides and bicarbonates to the

total dissolved salts i s much lower in storm runoff than i n the d r y

weather flow analyzed. I n the la t te r case 68.6% of the TDS consists of

these anions whereas this proport ion is as low as 38.8% i n the storm

runoff (averaged over both catchments) ind ica t ing probable washoff of

constituents that do not normally apear in the d r y weather flow.

68

TABLE 5.1

&gl*

Y r h

1811

18/2

18/3

18/U

18/5

1016

1811

18/8

18/9

18/10

13/11

18/12

R

Results o f chemical analyses on r a i n f a l l a n d runof f samples f o r H i l lb row on 03/01/85

T l l pM O D M Y C L l V l t Y

taWn

.s/*

i ~ n 2 i 6.35 46.10

14132 6.35 35.50

lh138 6.15 2h.50

14144 6.35 13.40

14hM 6.15 8.88

14h52 5.05 8.23

1hh54 6.20 6.29

lhh57 5.60 7.03

15Mo 5.65 6.14

15hO4 5.10 5.95

15hO9 5.85 5.92

15hl6 5.10 6.58

*/A b.07 P.ZO

-

OM

-

6.20

6.30

6.05

6.00

5.55

5.85

5.45

5.90

5.55

-

t O . l

<o. 1

to. 1

a. 1

go.1

0.3

0 .1

0.2

0.2

0.2

0.2

0.3

6U

57

a3

17

13

14

14

12

11

10

11

10

~~

conduct1 V l t Y

ma/.

14.31

13.12

11.81

9.69

9.91

10.88

13.37

15.43

6.60

I l l

taman

-

2Ohl 1

20hl4

20118

2ohz3

.?Oh26

2013 1

20150

2lho1

*/A

T D I

.PI 1

138

112

134

102

100

126

120

170

18

1010

242

160

770

512

232

110

102

0.2

0.3

0.8

4.1

8.6

5.3

15.0

12.9

12

10

10

10

11

13

16

21

4

5.1

4.1

3.0

3 .0

5.1

5.1

1.1

8.2

3.8

36

31

36

24

10

24

10

27

6 2.7

63 I

TABLE 5 . 2 Results o f chemical analyses on r a i n f a l l a n d runof f samDles f o r H i l lb row on 18/01/85

U I I C N t .

346

265

182

104

69

65

55

65

u 60

49

50

18

64

84

380

130

2Oh

\ u4 92

56

8

(1

Y1 41

37.0

30.0

12.3

10.3

8.2

7 .1

6.1

10.2

4.1

6 .1

6.1

5 .1

8.2

122

90

85

S6

20

14

12

7

10

7

7

1

6

69

LwIe

mark

50111

sol12

so113

Ylll

RFl/l*

TABLE 5.3 Resu l t s o f chemica l a n a l y s e s o n r a i n f a l l and r u n o f f samples f o r Montgomery P a r k on 07/03/83

T i n on COnaWtlvltY roa auspewd ~ ~ t n t m Su1ph.t. chiorlam c. c.rbmet.

taken so1 id.

mSlm -1 I -1 1 9 1 I -1 I -1 1 -1 1

*** 6.25 13.26 104 95 eo. 1 10 6.3 30

5.85 10.31 86 200 0.1 16 5.2 20

6.00 12.91 112 450 2.2 13 6. 3 31

6.20 22.30 166 44 1.5 25 16.0 16

1.25 10.86 52 0.4 .. .. 21

T O I

-11

104

1625

314

544

446

320

262

620

TABLE 5.4 Resu l t s o f chemica l a n a l y s e s on d r y wea the r f l o w samp les f rom Montgomery P a r k

8usp.na.a soilas

-1 I

10

4

12

12

10

24

14

la

210.0

91.0

10.0

11.8

30.4

b . 0

0.1

2.0

100

15

16

64

kz

18

25

40

34

480

45

101

86

29

10

120

15

456

175

202

1@3

not aow

not row

not aow

70

TABLE 5.5 Results of chemical analyses on d r y weather flow samples from Hi l lbrow

As one would expect, the concentration of suspended sol ids i n d r y

weather flow i s much lower than i n the storm runoff , ind ica t ing a h igher

transport rate of sediments as well as possible erosion du r ing storms. For

the samples analyzed, the suspended sol ids i n the d r y weather flow

averaged only 27 mg/l compared with an average of 236 mg/l for the storm

flows.

Comparing Tables 4 and 5 ( d r y weather f lows) w i t h Tables 1, 2 and 3

reveals that the TDS concentrations are considerably higher in d r y weather

flows than i n storm flows. The average TDS for the d r y weather flow

samples i s 644 mg/l whi le average values of TDS for the three runof f

events are 125 mg/l, 113 mg/l and 117 mg/l, ind ica t ing that the d r y

weather flow has about f i ve times as h igh a concentration as stormwater

runoff. The base load of TDS from Montgomery Park appears to be la rge ly

from a refuse t ip , which averages 160000 kg/annum o r 150 kg/ha/annum

averaged over the catchment (Ba l l , 1984).

I t was mentioned that samples of runoff were obtained on the r i s i n g

l imb of the hydrograph of 18 January 1985 in Hi l lbrow, making i t possible

to detect a f lushing effect at the s ta r t of the runoff . The h igh TDS

concentrations a t the ear ly stages of the runoff , v iz. 346 mg/l and 265

mg/l, followed b y a time-dependent decrease in TDS concentration to f i na l

levels of about 60 mg/l indicate a " f i r s t f lush" effect in accordance w i th

the f ind ings of many others (e.g. Cordery, 1977; Helsel et al. , 1979).

The proport ions of n i t rJ tes are also much higher in the d r y weather

flow than in the stormwater runoff. In the case of the d r y weather flow

sampled in August and October 1982 (see Table 5. 4) the levels of n i t r a t e

71

-.18

-.18

-.14

mar

(mpnl 18

18

14

12

10

8

8

4

2

160-

140-

120-

100-

10

14

12

10

8

PH

6.5

0.0

5.5

5.0

21

Fig . 5.3 Plot of p o l l u t a n t concentrat ion VS. time fo r r a i n f a l l - r u n o f f event on H i l lb row on 03/01/85

45 -

4 0 - 6.1

35- 8.t

30- 5.e

25 - 5.t

20 -

15-

10-

5-

- Nllrole --.-. Conducllvlly

---- Suop. Solids cc CMorldoo TO9 - Flowale \ -.-

HW+I+WI Sulohale - pn

14h25 30 35 40 45 50 55 15hoo 5 10 15 T h

Fig . 5.4 Plot of p o l l u t a n t concentrat ion vs. time fo r r a i n f a l l - r u n o f f event on H i l lb row on 18/01/85

72

are so h igh as to suggest possible blockage of a san i ta ry sewer w i th the

resu l t ing overflow enter ing the stream. I t was observed for a l l three

runoff events that the n i t ra te concentrations increased over the dura t ion of

each hydrograph, reaching the i r maximum on the recession limbs of the

respective hydrographs. A possible explanation for t h i s phenomenon i s that

l i gh t i ng ac t i v i t y w i l l increase the n i t ra te levels in the r a i n f a l l du r ing the

course of the storm, resu l t ing i n increasing n i t r a t e concentrations i n the

runoff w i th time. There were however large differences in magnitudes of

these concentrations between events. The i n i t i a l and f i na l n i t ra te

concentrations from the Hi l lbrow catchment were 0.2 mg/l and 12.9 mg/l in

the runoff on 3 January 1985 whi le the maximum n i t r a t e concentration i n

the runoff on 18 January 1985 d i d not exceed 0.3 mg/l. A maximum n i t ra te

concentration of 2.2 mg/l was recorded i n the runoff from the Montgomery

Park catchment on 7 March 1983. The recommended n i t ra te l im i t i n domestic

water i s 6.0 mg/l wi th an upper l im i t of 10.0 mg/l (SABS, 1984).

There does not apear to be any de f in i te time-related decrease o r

increase in the levels of the other consti tuents i n the runoff . For example

sulphate concentrations increase with time i n the runoff from Hi l lbrow on 3

January 1985 whi le the converse i s t rue for the runof f on 18 January 1985

from the same catchment.

Plots of pol lutant concentrations w i th time for the Hi l lbrow events are

presented i n Figures 5.3 and 5.4.

Mass Balance for event of 18 January 1985 on H i l lb row Catchment

A r a i n f a l l depth of 6 mm was measured for t h i s event and the TDS

concentration in the r a i n f a l l was 18 mg/l (see Table 2). This can also be

expressed as a r a i n f a l l loading ra te of 0.18 kg/ha/mm of r a i n or 1.08

kg/ha i n total. For a catchment size of 67.2 ha th i s depth of r a i n f a l l

corresponds to 4030 rn3 of r a i n f a l l over the catchment mass of 73 k g of

pol lutants.

For th is event a runoff volume of 475 m’ and a total load of 121 k g of

pol lutant were estimated. There was thus a net washoff of 48 k g of

pol lutant from the catchment. Expressing the po l lu tan t load i n the runoff

i n terms of catchment area and r a i n f a l l gives 0.30 kg/ha/mm o r 1.8 kg/ha

total.

The sources of these po l lu tan ts have not been ident i f ied, bu t i n a

densely developed area l i k e Hi l lbrow, the most l i k e l y sources are washoff

of deposits from wind and motor vehicles and soluble f ract ions of l i t t e r

which is usua l ly present.

73

Since the runoff was only 12% of the r a i n f a l l and the catchment s t i l l

experienced a net washoff of pol lutants, w i th 66% more pol lutant being

washed o f f than was deposited by the r a i n f a l l , i t i s conceivable that t h i s

washoff may reach even higher percentages for events where the proport ion

of runoff to r a i n f a l l i s greater. Such events would resul t from storms

haaving a greater depth of h igher intensity ra in fa l l . I t i s also possible

that input dur ing one storm is stored and released af ter loss of moisture,

to be washed off du r ing a subsequent storm.

A plot of hydrograph, pol lutograph and TDS var ia t ion wi th time for

th is event i s presented i n Figure 5.5.

Mass Balance for event of 7 March 1983 on Montgomery Park Catchment

On 7 March 1983 a total depth of 14 mm of r a i n f a l l was recorded on

the Montgomery Park catchment. This event was preceded b y a time per iod

exceeding f ive days of no ra in , so i t i s not surpr is ing that the TDS

concentration of the r a i n f a l l i s much higher than that measured in

Hil lbrow on 18 October 1985 when only two d r y days had passed. The

measured TDS of the r a i n f a l l was 52 mg/l (see Table 5.31, resu l t ing in a

r a i n f a l l loading ra te of 0.52 kg/ha/mm. The total mass of soluble

pol lutants deposited on this 10.53 km’ catchment was thus 7666 k g in

147420 m3 of ra in fa l l .

A runoff volume of 5508 m3 wi th a corresponding cumulative runoff load

of 1479 k g of dissolved pol lutants was measured. I n terms of r a i n f a l l t h i s

pol lutant load can be expressed as 0.10 kg/ha/mm. The runoff volume

represents only 4% of the r a i n f a l l and the TDS washed of f 19% of that

deposited by the ra in fa l l . I n th is case the catchment therefore experienced

a net ga in of 6187 k g of pol lutant, o r 81% of that deposited. This

corresponds to a net gain of 5.87 kg/ha or 0.42 kg/ha/mm of rain-borne

pol lutant i.e. net deposition of pol lutant occurred in the per i -urban

catchment while net washoff occurred i n the densely developed catchment.

Since there is a deposit ( loss of matter) from r a i n as indicated b y the

Montgomery Park catchment i t can be expected that a s imi la r deposit would

occur in Hil lbrow, so the l i t t e r load must be higher.

Once again i t i s d i f f i cu l t to attempt to ident i fy the sources of

pol lutants washed of f th is catchment. Referr ing to Tables 5.2 and 5.3 i t

w i l l be seen that n i t ra te levels in the runoff are higher fo r th is catchment

than for the Hi l lbrow catchment on 18 January 1985, s ign i f y ing the

possible washoff of decaying vegetation, animal faeces and garden

fert i l izers. This seems a reasonable deduction as the Montgomery Park

74

--.----. r--- L*'

,/-•

-

lEhoO TIME 14hOO 16h00

Fig. 5.5 Hydrograph, pol lutograph and TDS for Hi l lbrow for event on 18/01/85

I / - - '

1,20 -

E 0,90 - " E W I- U d 0,60 - s

LL s

0,30 -

15h00 I

17h00 19hOO 2lhoo 23h0 TIME

Fig. 5.6 Hydrograph, pol lutograph and TDS for Montgomery Park for event on 07/03/83

75

catchment consists of predominantly suburban resident ia l developments w i th

gardens. Another source in Montgomery Park could be leachate from the

ground (either previously deposited by r a i n seeping in o r from soi l

minerals) . I t i s noted that the proport ion of sulphates and carbonates i n

runoff i s s imi lar to the ra in , but chlorides increase.

I t appears that sulphates and chlorides are unaffected b y the two

dif ferent land-uses, the respective levels being of the same order for both

catchments which also indicates they may be air-borne into the catchment.

I t has also been observed that there are ( i l l e g a l ) discharges of i ndus t r i a l

wastes into the separate stormwater system in Hi l lbrow.

The hydrograph, pol lutograph and TDS var ia t ion w i th time for t h i s

event are i l l us t ra ted i n Figure 5.6.

I n the mass balance of pol lutants outl ined above i t was found possible

i n both the Hil lbrow and the Montgomery Park catchments to relate the

pol lutant load i n the runoff to the load in the r a i n f a l l causing that

runoff. To determine whether the catchment has experienced a net loss o r

gain of pol lutants i t i s also necessary to know the TDS concentration of

the r a i n f a l l as well as the runoff. In the present project r a i n f a l l qua l i t y

was only analyzed fo r three events, TDS levels i n the r a i n f a l l being 18

mg/l (H i l lb row) , 52 mg/l (Montgomery Park ) and 78 mg/l (H i l lb row) . A TDS

concentration of 118 mg/l i n r a i n f a l l was observed by Madisha (1983) a t a

location near the Hi l lbrow catchment.

Assuming a r a i n f a l l loading ra te of 0.52 kg/ha/mm for Montgomery Park

and an average r a i n f a l l loading ra te of 0.71 kg/ha/mm for Hi l lbrow, the

total weight of dissolved sol ids deposited on the two catchments was

computed for twelve ra in fa l l - runof f events fo r which both discharge and

electr ical conductivi ty data were avai lable. These resul ts a re presented in

Table 5.6.

I t can be deduced from Table 5.6 that the average pol lut ion load of

runoff expressed i n terms of r a i n f a l l i s 0.40 kg/ha/mm of r a i n f a l l for

Montgomery Park and 1.54 kg/ha/mm of r a i n f a l l for Hi l lbrow. This f i nd ing

i s i n accordance with the f ind ings of other researchers (e.g. Pol ls and

Lanyon, 1980; Mikalsen, 1984), v iz. that i n general the level of po l lu t ion

of stormwater i s higher from commercial and downtown land-use

developments than from resident ia l developments.

Another interesting deduction from Table 6 i s that more pol lutant was

deposited on the Montgomery Park catchment that was washed of f for f i ve

out of the seven events while th is was only the case for two out of f i ve

events in the Hi l lbrow catchment. The higher percentage imperviousness in

the Hil lbrow catchment i s possibly the reason fo r th is phenomenon.

76

TABLE 5.6

U l p h t o f

dap0alt.d

106

I h a l

7666

7118

9309

25100

30116

13141

36607

48

669

96

48

73

w a t i o n

and

data

U i p h t 01

106 In

rumrr

tho1

1479

7356

13086

23451

15872

7680

26391

247

217

01

193

121

mntwuw P a ~ h

01 /03 /03

W/I2/01

12/12/83

21/01/85

30/10/85

3 I / 10185

0111 1/85

n l l l b -

13/09/011

16/09/04

H)/lO/04

21/10/04

18/01/85

Comparison of po l lu t ion loads i n r a i n f a l l and runoff w i th r a i n f a l l depths

- l a i n f a l l

dapth

(-1

-

14

11

17

46

55

24

67

1

14

2

1

6

-

Ratio O f

runofr

I M d to

r a i n f a l l

laad

0.19

1.03

1.41

0.93

0.53

0.59

0.72

5.15

0.32

0.ou

4.02

1.66

?Oi I U t l O l

load in

wnorr

I h g / h . / r l

0.10

0.54

0.73

0.40

0.27

0.30

0.37

3.60

0.21

0.60

2.07

0.30

Avarmpa 10s for ni I IbIOr - 71 -/I

Having established relat ionships between depth o r r a i n f a l I and amount

of pol lutant washed of f a catchment, annual po l lu tan t loads can be

compu ted . Considering the Hi l lbrow catchment for example and assuming a mean

annual precipi tat ion of 763 mm (Adamson, 1981), the total mass of

po l lu tan ts washed of f t h i s catchment w i l l be of the order of 80000 k g per

annum or 1190 kg/ha/annum. For the Montgomery Park catchment the

amount of annual po l lu tan t loading w i l l be approximately 320 000 k g or

305 kg/ha/annum.

Assuming an average d r y weather flow of 0.0015 m’/s or 130 m’/day i n

Hi l lbrow and 310 d r y days per annum resul ts i n an annual d r y weather

flow volume of approximately 40300 m’. This resu l ts in an annual dry

weather pol lutant load of approximately 22100 k g or 330 kg/ha/annum. The

average d r y weather flow i n Montgomery Park i s about 0.004 m’/s so the

annual d r y weather flow off t h i s catchment i s approximately 110000 m’

which corresponds to a total po l lu tan t load of 60500 k g o r 57

kg/ha/annum. Therefore i t can be deduced that the annual po l lu tan t load

due to direct stormwater runoff i s about 3.6 times that due to d r y weather

77

flow for the Hi l lbrow catchment and about 5.3 times that due to d r y

weather flow for the Montgomery Park catchment.

The pol lutant loading rates derived from the di f ferent sources are

summarized i n Table 5.7.

T A B L E 5.7 Summary of dissolved loads i n kg/ha/mm

CONCLUSIONS

Despite the l imited monitoring, the fo l lowing tentat ive conclusions can

be drawn.

The total dissolved pol lut ion load i n stormwater and surface drainage

from Hil lbrow, a densely populated c i t y area i s about 15000 kg/ha/annum

which i s about 3 times as great from a suburban catchment, Montgomery

Park. The major i ty (70%-80%) occurs du r ing storm runoff in both cases.

Only about 430 kg/ha/annum fa l l s o r i s washed out of the atmosphere. The

major i ty i s therefore l i t t e r and from vehicles in the case of Hi l lbrow, and

decaying vegetable matter o r leachate from Montgomery Park.

There i s a net ga in of pol lutants from Hi l lbrow but in Montgomery Park

the total washoff i s about the same order as the total deposited from the

atmosphere. As a large proport ion of r a i n seeps into the ground, i t could

store TDS to be released i n future runoff. There i s a net ga in of n i t r a t e

in Montgomery Park however.

Dry weather concentrations are higher in both catchments, due to

seepage from a pol luted l and f i l l in the case of Montgomery Park, (Bal l ,

1984) and i l lega l waste discharge in Hil lbrow. Concentrations in storm

runoff increase with number of previous d r y days, s ign i f y ing that street

sweeping would reduce loads.

78

The m a j o r i t y o f d isso lved s a l t s i s washed o f f d u r i n g the r i s i n g l imb of

the storms except n i t r a t e s which e x h i b i t a lag. Release from the ground o r

a l t e r n a t i v l e y the in f luence of atmospheric l i g h t i n g cou ld be the cause of

th is . Before pred ic t ion b y model l ing can be under taken, in tens ive f u r t h e r

inves t iga t ion w i l l be requi red.

REFERENCES

Adamson, P.T., 1981. Southern Af r i can Storm R a i n f a l l . D i rectorate of Water Af fa i rs , Department o f Environment A f fa i rs , Technical Report TR 102.

B a l l , J.M., 1984. Degradat ion of g round a n d sur face water q u a l i t y in r e l a t i o n to a s a n i t a r y l a n d f i l l . MSc(Eng) Disser ta t ion, U n i v e r s i t y o f the W i twatersrand.

Bedient, P.B., Lambert, J.L. a n d Spr inger , N.K., 1980. Stormwater p o l l u t a n t load-runoff re la t ionships. Jnl. Water Po l lu t ion Control Fed.,

Bradford, W.J., 1977. Urban stormwater p o l l u t a n t loadings: a s t a t i s t i c a l summary th rough 1972. Jn l . Water Po l lu t ion Control Fed., 49, 613-622.

Co lw i l l , D.M., Peters, C.J. a n d Per ry , R., 1984. Water q u a l i t y o f motorway runof f . Transpor t a n d Road Research Labora tory , Dept. o f the Environment a n d Dept. o f Transpor t , TRRL Supplementary Report No. 823.

Cordery, I., 1977. Q u a l i t y charac ter is t i cs of u r b a n stormwater in Sydney, Aus t ra l ia . Water Resources Research, 13, 197-202

Green, I.R.A., Stephenson, D. a n d Lambourne, J.J., 1986. Stormwater p o l l u t i o n ana lys is . Urban Hydro logy a n d Dra inage Research Contract, Water Research Commission Report No. 115/10/86.

Helsel, D.R., Kim, J.I., Gr izzard, T.J., Randa l l , C.W. a n d Hoehn, R.C., 1979. L a n d use in f luences on meta ls in storm dra inage. Jnl. Water Po l lu t ion Control Fed., 51, 709-717.

Madisha, J.L., 1983. Inves t iga t ion pro ject on u r b a n stormwater p o l l u t i o n in Braamfontein. Department o f C i v i l Engineer ing, Un ivers i ty o f the W i twatersrand.

Mikalsen, K.T., 1984. Assessment of water q u a l i t y changes r e s u l t i n g from urban iza t ion , a g r i c u l t u r e and commercial fo res t ry in the s ta te of Georgia, U.S.A. Proceedings of the T h i r d I n t . Conf. "Urban Storm Drainage," Goteborg, Sweden, 801-810.

Pol Is, I . and Lanyon, R., 1980. Po l lu tan t concentrat ions from hogeneous l a n d uses. Jnl. Environmental Eng. Div., ASCE, 106, 69-80.

Sar tor , J.D., Boyd, G.B. a n d Agardy, F.J., 1974. Water p o l l u t i o n aspects of street sur face contaminants. Jnl. Water Po l lu t ion Control Fed., 46,

Simpson, D.E. a n d Kemp, P.H., 1982. Q u a l i t y a n d q u a n t i t y o f stormwater runof f from a commercial land-use catchment in Nata l , South Af r i ca . Water Sci. Tech., 14, 323-38.

South Af r i can Bureau of Standards (SABS), 1984. Speci f icat ion f o r water fo r domestic supplies. SABS 241.

Stephenson, D. and Green, I.R.A., 1987. Mass ba lance of stormwater po l lu tan ts . Water S.A.

Terst r iep, M.L. a n d Sta l l , J.B., 1974. The I l l i n o i s u r b a n d r a i n a g e area s imulator , ILLUDAS. I l l i n o i s State Water Survey, Urbana, B u l l e t i n 58.

Waniel ista, M.P., 1979. Stormwater Management Quant i t y a n d Q u a l i t y . Ann Arbor Science Pub l ishers Inc. Mich igan.

Whipple, W., Hunter, J.V. a n d Yu, S.L., 1977. Ef fects of storm f requency on p o l l u t i o n from u r b a n runof f . Jnl. Water Po l lu t ion Control Fed., 49,

52, 2396-2404.

458-467.

2243-2248.

79

CHAPTER 6

OPT I MUM ALLOCAT I ON OF WATER RESOURCES SUBJECT TO QUAL I TY CONSTRA I NTS

I NTROOUCT ION

We are reaching an age of compromise. The 1960's ha i led the era of

economic benefit cost analysis, the 1970's saw the appearance of the

environmentalists and ideal ists and the 1980's appear to be producing more

rea l i s t i c planners. Mu l t ip le objectives inc lud ing economic, sociological,

po l i t i ca l and environmental w i l l be considered bu t hopeful ly i n the correct

perspectives. High ideals can only resul t i n slow-down of growth and th is

may have detrimental effects on development of underdeveloped countries.

The watering down of engineering projects to meet h igh ideals can also

stagnate the engineering industry and lose va luab le b ra in power to other

profess ions . Water resources are regarded b y many as a never diminishing asset. O n

account of annual replenishment i t i s assumed the resource cannot be

mined. This i s a fa l lacy , for apart from over-exploitat ion and drainage

basin deteriorat ion, the nature of the resource can be altered. As more

and more usage occurs so there w i l l be greater waste water discharges

and poorer qua l i t y water i n our r i vers . New growth can only be met from

these r i ve rs o r from water fu r ther a f ie ld i f surface waters are to be re l ied

on. We can often not af ford the l uxu ry of pure mountain waters piped from

many hundreds of kilometres away. I t w i l l be necessary to p u r i f y waste

water to acceptable standards i n some cases. The cost of demineral izat ion

and nutr ient removal i s pa r t i cu la r l y high. This cost may not be warranted

for a l l uses. I n many countries potable water i s transported separately o r

obtained from containers while poorer qua l i t y water i s used for general

domestic and indus t r ia l purposes. Although separate piped water supplies

of dif ferent qua l i t y water w i l l be expensive there may be some areas

which are predominately h igh density resident ia l and could j us t i f y h igh

qua l i t y water. Other areas requ i r i ng lower qua l i t y could receive separate

supplies. This i s pa r t i cu la r l y the case i n min ing areas in South Afr ica

where these studies were in i t iated.

Para l le l studies are invest igat ing the cost of demineral izat ion and h igh

qua l i t y pur i f i ca t ion but that i s only one of the options. The others are to

seek fresh surface or groundwater resources fu r ther away, to make do w i th

poorer qua l i t y of local resources o r to al locate i n an optimal manner as

indicated here.

80

Methods of research could ei ther adopt the global systems approached or

a more simpl ist ic but perhaps easier understood methodology. There are a

number of sophisticated techniques for opt imizat ion of l inear and nonl inear

water systems subject to var ious constraints. The use of computers i s

read i l y , in fact sometimes too read i l y , adopted b y eager students. Whereas

these methods may form ideal subject matter fo r dissertat ions the output

from a computer program i s not easy to exp la in to regional p lanners and

pol i t ic ians. Simple graphical d isp lays or tabu la r resul ts a re much easier

to describe and present. Simple hand calculat ions often enable the analyst

to follow the al ternat ives and bear in mind marg ina l costs o r mu l t ip le

objectives. By fol lowing the en t i re p lann ing process through the analyst i s

also able to c l a r i f y a l te rna t ive objectives and al locate p r io r i t ies . I t i s

t h i s approach which i s adopted i n the s impl ist ic study below (Stephenson,

1982).

THE SYSTEM

Consider the t ransportat ion problem depicted i n Fig. 6.1. A number of

sources of water are ava i lab le (A , B and C ) and they each have l imi ted

resources indicated as 10, 20 and 15 megali tres per day (Ml /day) ,

respectively. T h e total a v a i l a b i l i t y may exceed the requirements of demand

of users W, X, and Y though, which requ i re 8, 12 and 16 Ml/day in this

example. The cost of t ransport along each route i s indicated i n Fig. 6.1,

and again i n the transportat ion tableau Table 6.2. As f a r as i t has been

described the system i s a simple t ransportat ion example w h i c h could easi ly

be optimized, i.e. the flow along each route to resul t i n a minimum total

transportat ion system could be der ived re la t i ve l y easi ly.

F ig . 6.1 Supply requirements and al ternat ives

81

The situation i s complicated by the Fact that the ind iv idua l consumers

have certain water qua l i t y requirements. The measurements of the relevant

impurity, e.g. TDS ( to ta l dissolved sol ids) i s in mg/l and the requirements

of W, X and Y are that the TDS shal l not exceed 10, 11 and 8 mg/l

respectively. Note that Ml/day mul t ip l ied by mg/l gives kg/day of salts, a

mass flow rate. The TDS of the source waters from A, B and C are 6, 1 1

and 8 mg/l respectively.

The lower l im i t on TDS may be achieved b y selecting correct sources,

blending dif ferent sources or, i f economic, p u r i f y i n g pa r t o r a l l of any of

the resources. The la t te r option, namely pur i f icat ion, could be handled by

assuming any source i s pu r i f i ed and adding the cost to the conveyance

cost.

Often the relat ionship between cost of pur i f i ca t ion and ra te of flow i s

nonl inear such as w i th desal ination be reverse osmosis and the system

becomes more complex. I n such case, separable programming methods are

possible (Stephenson, 1978). Al ternat ively a gradient method may be

employed to seek an optimum. I f only pa r t ( a va r iab le p a r t ) of a source

need be pur i f ied, the descript ive equations are more numerous bu t l inear

programming methods may be employed to optimize the system.

I n the present example (Fig.6.1) the resource and demand constraints

may also be writ ten as l inear constraints:

Demand

aAW + aBW + aCW = 8

Q~~ + aBX + aCX = 12

aAy + aBy + aCY = 16

The qua l i t y constraints may be writ ten:

6QAW + l l Q B W + 8QCW 110 x 8

6QAx + l l Q B x + 8QCX 511 x 12,

6QAy + l l Q B y + 8QBx 5 8 x 16

Provided there is a feasible solution the set of m + 2n constraints could

be analyzed by l inear programming methods. m i s the number of sources

and n the number of demand points. Al ternat ively the a v a i l a b i l i t y and

82

demand constraints could be considered i n a t ransportat ion mat r ix and the

qua1 i t y constraints handled separately using the p r inc ip le of decomposition

of l inear programmes (Dantzig, 1963; Stephenson, 1969). A t h i r d method i s

described below. This i s based on the transportat ion method w i th

addi t ional constraints on the distr ibut ions. The resu l t ing advantages over

the l inear programming al ternat ives are s imp l ic i t y , rap id i t y , no necessity

fo r computers and more interaction between the water resources planner

and the system.

I t i s assumed the reader i s f am i l i a r w i th l inear programming and

transportat ion programming techniques.

SOLUTION METHOD

The data are arranged in a tableau s imi la r to a transportat ion tableau

(Table 6 . 1 ) . Each demand i s represented b y a row, inc lud ing a row

label led "slack" since resources exceed demand here. A column represents

each source and there i s an addi t ional column label led "a r t i f i c i a l slack"

since the i n i t i a l assignment may not sat isfy qua l i t y constraints without i t .

I n fact, as there are three addi t ional constraints, one would expect up to

three addi t ional var iab les in the f i na l programmme. The cost coeff icients

of the a r t i f i c i a l slack flow var iab les should be very large, bu t not

necessarily so for sa l t mass flow slacks since they are of the type. I t

i s not necessary to assign a r t i f i c i a l cost coefficients in the fol lowing

method.

TABLE 6.1 Transportat ion mat r ix w i th shuf f l ing to el iminate a r t i f i c i a l

slack

X

Y

83

An i n i t i a l assignment i s made i n Table 6.1 using the Northwest corner

ru le (Loomba, 1964). At each assignment, two types of constraint must be

satisf ied - water flow and sal t balance. Thus i n most blocks, e.g. AW,

flow l im i ts the number, but i n block CY, TDS balance l im i ts flow to 2.25

Ml/day ( a flow of 6 would otherwise have been assigned to th is block).

The f i r s t step af ter making the i n i t i a l assignment should be to evacuate

flows from the a r t i f i c i a l slack column. Note each re-assignment must

satisfy flow constraints and TDS l imits. Observe that the water flows i n

Ml/day are wr i t ten i n the bottom left of each block followed b y / and the

TDS flow in kg/day. Thus 8/48 indicates 8 Ml/day a t a source TDS of 6

mg/l resul t ing in TDS flow of 48 kg/day. I t i s re la t i ve ly easy to check a t

each corner of a closed c i r cu i t whether water o r TDS l im i t the

re-allocation, and select the lowest permissible flow al location. I n Table

6.1, two re-al locations are necessary to evacuate block LY. a feasible

(non opt imal) solution results.

Now the optimization proceeds as for any transportat ion exercise, except

for the addi t ional constraint on each re-al location. After ca lcu la t ing

column and row cost coefficient and comparing implied costs i n each

vacant block wi th actual costs i t i s decided to re-al locate to block BY.

Although flow consideration would l im i t the al locat ion to 1 Ml/day, qua l i t y

constraints l imi t i t to 0.4 Ml/day. Then a l l the slack i n TDS for row Y i s

eliminated. Table 6 .2 results.

TABLE 6.2 Transportat ion mat r ix step two

04

I t w i l l be observed that there i s more than one possible cost coefficient

f o r some blocks, depending on which block i s used as a p ivo t . This arises

because the number of occupied blocks is greater than n + m - 1 where n

and m are the number of rows and columns i n the tableau excluding the

a r t i f i c i a l slack column. Each possible combination should be investigated.

Where the p ivo t sequence AW, AX, B X , B Y , BS, CY i s used, the maximum

difference between implied cost and actual cost coeff icient appears i n block

C X , and i s 8 vs 4 (Table 6.3). I t w i l l be found that by proceeding f i r s t

around the closed pa th CX-AX-AY-CY-CX and then CX-EX-BY-CY-CX that f i r s t

1.4 and then 0.9 ml/day can be al located to block CX without v io la t i ng

flow and qua l i t y constraints.

TABLE 6. 3 Transportat ion mat r ix optimized

Dunand :

Subsequent calculat ions w i l l reveal there is no fu r ther cost

improvement, i.e. no more implied costs exceed actual costs once new cost

coefficients are ca I cu I a ted . The optimum p lan , which satisf ies qua l i t y requirements, i s indicated in

Fig. 6.2.

85

D i scussionl

A l l water users do not require the same h igh qua l i t y water. Where

poorer qua l i t y i s tolerable, al location of a l te rna t ive sources may be

considered. Overal l economy of pur i f i ca t ion and d is t r ibu t ion result.

Fig. 6.2 Optimum al location subject to constraints

A technique for al locat ing water resources b y a form of t ransportat ion

programming has been demonstrated with an example. The technique i s

simple and not computer orientated. The resu l t ing d is t r ibu t ion system i s

depicted and can read i l y be updated as qua l i t ies of the sources vary .

The method is therefore of use for management and operation of water

d is t r ibu t ion systems as well as design. I n fact even more so, since

construction costs are not as a ru le easi ly l inear ized whereas pumping

costs are general ly proport ional to the rate of flow.

L I NEAR PROGRAMM I NG SOLUT ION

The previous sections high1 ighted the shortcomings of the t ransportat ion

and transportat ion extended techniques. The techniques requ i re that

onerous simp1 i f y ing assumptions be made about parameters. They thus of fer

an approach which do not model the d is t r ibu t ion comprehensively enough.

The most severe shortcomings are:

a ) Qua1 i t y constraints cannot be considered i n Transportat ion

Programming.

b ) Optimization of the amount of water to be desal inated and hence

blended cannot be achieved in ei ther of the Transportat ion techniques.

86

c ) Non-Linear cost functions have to be approximated by l inear functions

in both techniques.

Linear programming techniques provide means whereby a l I these

shortcomings can be overcome. I n l inear programming both qua l i t y and

quant i t y constraints, and any other l inear constraints, can be manipulated

to y ie ld an optimum solution. However, nei ther a non-l inear objective

function nor constraints can be used unless they are converted to a l inear

o r piecewise l inear form. This can be achieved by using l inear

programming in conjunction w i th separable programming.

This section describes how the d is t r ibu t ion problem i s transformed into

a representative mathematical model sui table for ana lys is using l inear

programming (Grosman, 1981).

The sets of data required are summarized below;

Sources : -

Water Board

Groundwater

Wastewater

Desalinated wastewater

Demands:-

Transfer

System 1

System 2

System 3

Waste (Slack)

Qua1 i ty

500 mg/P

600 mg/e

1750 mg/l

175 mg/P

1750 mg/e

700 mg/e

700 mg/e

700 mg/e

1750 mg/P

Quant i ty

100.0 Me/d

11.5 Me/d

Var iable

Var iable

7.0 Me/d

9.5 Me/d

0.7 Me/d

0.5 Me/d

unused

The water board supply i s assumed to be 100 Me/d. I n re la t ion to the

other sources, th is i s h igh , and consequently only a port ion thereof may

b e used. The port ion to be used w i l l be optimized.

Wastewater y ie lds 10 Me/d, of which a va r iab le port ion i s desal inated

to y ie ld an improved qua1 i t y ava i lab le from the desal inated wastewater.

This var iab le port ion i s an unknown and hence i t should be optimized.

The recovery ra t ion of feed flow to product flow in a desal ination p lan t i s

0,69 for th is case. Expressing the above i n mathematical terms,

U + D /0.69 = 10

U + 1.45 D = 10 (6.10)

where U = Used MSW i n Me/d

D = Desalinated Wastewater in MP/d

87

(c/m’ 1

WB R

GROUND WATER F

WASTE WATER u

DESAL. WASTE D WATER

-_

Two new v a r i a b l e s (U a n d D) a n d a new c o n s t r a i n t Eq. 6.10 a r e

i n t roduced to c a t e r f o r t he d e s a l i n a t i o n of a v a r i a b l e pe rcen tage o f waste.

U was assumed to b e 7.5 Mk‘/d, hence f rom Eq. 6.10, D was 1.73 Me/d. The

magn i tude of the s l a c k a l l o c a t i o n to waste (W) w i l l consequent ly v a r y . The

v a r i a t i o n i s acco rd ing to Eq. 6.11 where the sources a r e b a l a n c e d a g a i n s t

the demands:

W + 7.0 + 9.5 + 0.7 + 0.5 = 100 + 11.5 + U + D

but D = (10 - U ) 0.69 from Eq. 6.10 (6.11)

hence W = 93.8 + U + (10 - U ) 0.69

W = 100.7 + 0.31 U (6.12)

From a T r a n s p o r t a t i o n Extended a n a l y s i s U was 7.5 Me/d, a n d the 100

M t / d was o n l y 1 MP/d, hence W was 4.03 MP/d. However, U v a r i e s now

w i t h a maximum v a l u e of 10 Me/d, ( t h a t i s w i thou t d e s a l i n a t i o n ) .

Therefore from Eq. 6.12

W L 103.8 Me/d (6.13)

The acceptable q u a l i t y assumed f o r the System 1 (S), System 2 ( V ) a n d

System 3 ( M ) i s s t i l l 700 mg/Q. The nex t sect ion rev iews a n a n a l y s i s of a

r a n g e o f acceptable q u a l i t i e s .

The o n l y u n r e a l i s t i c assumpt ion necessary in t h i s a n a l y s i s , u s i n g

l i n e a r p rog ramming , i s t h a t the to ta l costs a r e l i n e a r l y r e l a t e d to feed

f low. The cost coe f f i c i en ts a r e summar ized below in T a b l e 6.4.

Abbrev ia t i ons f o r the sources a n d demands a r e a l so i nd i ca ted .

w l O / S I v l M 0.0 0.0 24.0 32.5 39.5

5.0 2.5 3.0 11.5 14.0

7.5 6.0 0.0 1 .o 0.0

44.0 43.0 38.0 46.0 51 .O

TABLE 6.4 Cost coe f f i c i en ts (c/m’) used in l i n e a r programme

DEMANDS

I WASTE I TRANSFER I SYSTEM 1 I SYSTEM 2 1 SYSTEM 3

cn W u K 3

SI

88

The fo rmu la t i on of the mathemat ica l model beg ins b y exp ress ing the

ob jec t i ve func t i on in terms of the express ion

n z = c c . x Hence

J J (6.14)

Z = O . O ( R W ) + O.O(RB) + 24.O(RS) + 32.5(RV) + 39.5(RM) + 5.0(FW) +

2.5(FB) + 3.0(FS) + 11.5(FV) + 14.O(FM) + 7.5(UW) + 6.0(UB) + O.O(US)

+ l.O(UV) + O.O(UM) + 44.O(DW) + 43.O(DB) + 38.O(DS) + 46.O(DV) + 51.5(DM) + O.O(U) + O.O(D)

(6.15)

where the terms in b racke ts represent the f l ow of water f rom a spec i f i c

source ( f i r s t l e t t e r ) to a spec i f i c demand (second l e t t e r ) a n d cons t i t u te the

unknowns. The term U and D represent the y i e l d s o f the wastewater a n d

Desal inated wastewater respec t i ve l y . Consequently they have zero cost

coef f ic ients and a r e a l so unknown.

The object ive func t i on 2 must be min imized subject to the f o l l o w i n g

l i n e a r const ra in ts .

Source Constra in ts : -

RW + RB + RS t R V t RM 5 100.0

FW + FB + FS + FV + FM = 11.5

UW + UB + US + UV + UM = U

DW + DB + DS + DV + DM = D

Demand Constra in ts : -

RW + FW + UW + DW 103.8 ( f rom 6.14)

RB + FB + UB + DB = 7.0

RS + FS + US + DS = 9.5

RV + FV + UV + DV = 0.7

RM + FM + UM + DM = 0.5

Q u a l i t y Constra in ts : -

500 R W + 600 + FW + 1750 UW + 175 DW I 103.8 (1750)

500 RB + 600 + FB + 1750 UB + 175 DB 5 7.0 (1750)

500 RS + 600 + FS + 1750 US + 175 DS 5 9.5 ( 700)

500 RV + 600 + FV + 1750 UV + 175 DV 5 0.7 ( 700)

500 RM + 600 + FM + 1750 UM + 175 DM 5 0.5 ( 700)

(6.16)

(6.17)

(6.18)

(6.19)

(6.20)

(6.21)

(6.22)

(6.23)

(6.24)

(6.25)

(6.26)

(6.27)

(6.28)

(6.29)

89

B I end i ng Constraints:-

U + 1.45 D = 10 ( from 6.10) (6.30)

Non-nega t i v i ty constra i n ts:-

A l l unknowns 0 (6.31 )

I t i s necessary to convert Eq. 6.18 and 6.19 to a form with no

unknowns on the r i g h t hand side. Hence:

UW + UB + US + UV + UM - U = 0 (6 .32 )

DW + DB + DS + DV + DM - D = 0 (6 .33)

The set of constraints (Eq. 6.16, 6.17 and 6.20 to 6.33 can be

optimized, subject to the objective function of Eq. 6.15, using the manual

Simplex Technique. Since the resul tant mat r ix i s ra ther large, a computer

programme to solve I inear programming problems b y the Simplex Technique

i s preferred. The fol lowing i s noted :

a ) Diagrammatical ly the solutions i s as indicated i n Fig. 6 . 3

b ) The water obtained from the water board i s not required, and i s

consequently al located to waste as slack. This i s interpreted to mean

that i s i s not necessary to obtain water from the water board.

c ) Most of the ground water i s allocated to system 1 and most of the

wastewater i s al located to the transfer.

11 .5MP/d /

I WASTE J 1750mg/L /f

Fig. 6.3 Optimal solution to the d is t r ibu t ion problems using l inear programming

90

The amount of water drawn from the desal inated wastewater source i s

zero. This implies that no desal ination i s required. The importance of

being able to optimize th is var iab le i s enhanced by comparing the

change in the values of the objective function.

The value of the objective function is now $1043/day, a decrease of

$453/day compared w i th the Transportat ion Programming Extended

Solution, solely as a resul t of a l low ing an unknown va r iab le

desal inat ion f ract ion to be opt imized.

The optimum solution was reached, since a l l the shadow values of the

unknowns were negative or zero. The lowest increase in the objective

function would occur i f a un i t of water was al located from the

groundwater to waste. The cost would increase by on ly 1 c/m' o r

$lO/day/unit of water. The highest increase would be $336/day/unit of

water, i f water was al located from the desal inated wastewater to the

transfer. Table 6.5 summarizes the shadow values in ascending order.

TABLE 6.5 Shadow values of empty cel ls using l inear programming.

SOURCE DEMAND SHADOW VALUE ($ /day /un i t -

Groundwater

Water Board

Desal. waste

Water Board

Desal. W w

Water Board

Desa I. WW

Water Board

Desa I. WW

Desa I. WW

Waste

Transfer

System 2

System 2

System 3

System 1

System 1

System 3

Waste

Transfer

10

15

225

238

24 2

244

257

280

331

336

g ) The assumption that cost i s l inear ly related to feed flow i s c lear ly not

correct. Hence the resul ts obtained are not a ' t r ue ' optimum. I n i t i a l l y ,

the expected feed flow rates from each source to each demand were

assumed, and hence a cost was derived. The inaccuracies of the

assumptions (and consequently of the cost coeff icients) are borne out

by the comparisons i n Table 6.6. I t may, however, be argued that the

overestimates cancel the underestimates.

91

TABLE 6.6 Errors incurred by using l inear cost functions instead of

non-l inear cost functions.

SOURCE DEMAND ASSUMED FLOWS OPT I MUM FLOWS PERCENTAGE AND CORRESPONDING FROM FIG. 6.3 DIFFERENCE COSTS AND CORRESPONDING IN COSTS

COSTS

FLOW COST FLOW COST

(Ml/d) (c/m’) (Ml/d) (c/m’) (c/m3 1

Ground- water System 1 3.0 3.0 8.7 1.6 47%

water Waste 2.0 7.5 3.8 5.2 31 % Waste-

Waste- water Transfer 2.0 6.0 5.3 4.5 25%

Ground- water Transfer 5.0 2.5 1.7 3.1 -24%

THE LINEAR PROGRAMM ING TECHN IQUE WITH SEPARABLE PROGRAMM I NG

APPL I ED

The assumption of a l inear cost function, and hence of objective

function, has been made throughout the discussion on transportat ion

programming, transportat ion programming extended and I inear

programming, and the appl icat ion of these techniques. I t has consistently

been mentioned that the assumption i s onerous b y the nature of the

cost-flow graphs and that in fact the technique of separable programming

could be employed to avoid th is assumption. Separable Programming

approximates non-l inear functions by piecewise l inear approximations. The

accuracy depends on the deviat ion of the l inear approximation from the

curve. For non convex separable functions, as i n th is case, the technique

does not guarantee a global optimum.

This section employs the separable programming technique, i n

conjunction wi th l inear programming, to provide an optimal solution. The

mathematical model of the system i s described in the next section, except

for the objective function.

92

T h e model i s thus expressed as below:-

Minimize the objective function Z subject to the l inear o r piecewise l inear

constraints

Z = CRW f CRB t CRS f CRV f CRM

+ CFW f CFB + CFS f CFV f CFM

+ CUW f CUB + CUS f CUV + CUM

f CDW t CDB t CDS + CDV + CDM (6.34)

where CXY represents the total cost of supp ly ing water from source X to

demand Y i n dol lars/day, and represents a functional equation of

separable programming. DJXY represents the J th increment D o r S for the

XY combination.

CRW = 0

CRB = 0

CRS = 117 DORS + 107 D l R S + 223 D2RS + 565 D3RS f 1658 D4RS

CRV = 124 DORV + 108 D l R V t 227 D2RV + 565 D3RV f 1670 D4RV

CRM = 117 DORM + 107 D l R M + 223 DZRM + 565 D3RM f 1658 D4RM

CFW = 37 DOFW + 25 D l F W f 52 D2FW f 113 D3FW + 351 D4FW

CFB = 13 DOFB + 19 D l F B + 4 0 D2FB + 80 D3FB + 156 D4FB

CFS = 42 DOFS f 20 D l F S + 24 D2FS + 29 D3FS + 89 D4FS

CFV = 4 9 DOFV t 22 D l F V t 26 D2FV + 30 D3FV + 101 D4FV

CFM = 42 DOFM + 20 D l F M + 24 D2FM f 29 D3FM + 89 D4FM

CUW = 6 2 DOUW + 38 DlUW + 64 D2UW f 131 D3UW + 407 D4UW

CUB = 42 DOUB + 34 D l U B + 5 7 DZUB + 109 D3UB + 236 D4UB

cus = 0

CUV = 4 DOUV + 1 D l U V f 2 D2UV f 3 D3UV + 6 D4UV

CUM = 0

CDW = 251 DODW + 229 DlDW + 583 D2DW + 1594 D2DW + 4595 D4DW

CDB = 231 DODB + 227 D l D B + 574 DZDB + 1572 D3DB + 4424 D4DB

CDS = 189 DODS + 193 D lDS + 517 D2DS + 1464 D3DS + 4287 D4DS

CDV = 193 DODV + 194 D l D V + 519 DZDV + 1466 D3DV + 4194 D4DV

CDM = 189 DODM + 193 D l D M + 517 D2DM f 1464 D3DM + 4287 D4DM

subject to source constraints 6.16, 6.17, 6.32 and 6.33,

demand constraints 6.20 to 6.24,

qua1 i ty constraints 6.25 to 6.29,

b I end i ng constraints 6.30,

U + 1.45 D = 10

Non-nega t i v i ty constraints,

adjacent constraints for separable variables:-

(6.35)

(6 .30)

93

I f any DJXY is non-zero, a l l the preceding DJXY values must take on

the value 1 , and a l l the succeeding DJXY values must take the value 0.

where

RS = 0.3 DORS + 0.6 DlRS + 1.6 D2RS + 4.5 D3RS + 13 D4RS

RV = 0.3 DORV t 0.6 DlRV + 1.6 DZRV + 4.5 D3RV + 13 D4RV

RM = 0.3 DORM + 0.6 DlRM + 1.6 DZRM + 4.5 D3RM + 13 D4RN

FW = 0.3 DOFW + 0.6 DlFW + 1.6 DZFW + 4.5 D3FW + 13 D4FW

FB = 0.3 DOFB + 0.6 D l F B + 1.6 D2FB + 4.5 D3FB + 13 D4FB

FS = 0.3 DOFS + 0.6 DlFS + 1.6 D2FS + 4.5 D3FS + 13 D4FS

FV = 0.3 DOFV + 0.6 D lFV + 1.6 DZFV .t 4.5 D3FV + 13 D4FV

FM = 0.3 DOFM + 0.6 DlFM + 1.6 DPFM + 4.5 D3FM + 13 D4FM

UW = 0.3 DOUW + 0.6 DlUW + 1.6 DZUW + 4.5 D3UW + 13 D4UW

UB = 0.3 DOUB + 0.6 DlUB + 1.6 DZUB + 4.5 D3UB + 13 D4UB

UV = 0.3 DOUV + 0.6 DlUV + 1.6 DZUV + 4.5 D3UV + 13 D4UV

DW = 0.3 DODW + 0.6 DlDW + 1.6 DZDW + 4.5 D3DW + 13 D4DW

DB = 0.3 DODB t 0.6 DlDB + 1.6 DZDB + 4.5 D3DB + 13 D4DB

DS = 0.3 DODS + 0.6 DlDS + 1.6 DZDS + 4.5 D3DS + 13 D4DS

DV = 0.3 DODV + 0.6 DlDV + 1.6 DZDV + 4.5 D3DV + 13 D4DV

DM = 0.3 DODM + 0.6 DlDM + 1.6 DZDM + 4.5 D3DM + 13 D4DM

(6.36)

and where XY represents the unknown quant i ty of water i n MP/d, suppl ied

from source X to demand Y. These set of equations i n 6.36 represents the

g r i d equa t ions of sepa rab I e programm i ng . Substi tut ing the set of equations of 6.35 in to 6.34 and re ta in ing the set

of equations of 6.36 as independent equations, the system i s solved using

l inear programming i n conjunction wi th separable programming. The

solution was obtained using the IBM Mathematical Programming System

Extended/370 (MPSX/370) Software Package.

I n order to ensure the solution obtained i s close to a global optimum

(as opposed to a local optimum) i t i s necessary to complete two computer

runs. The f i r s t , wi th the control programme, the second w i th the l i ne

XSETLB = -1 af ter the l ine BCDOUT in the control programme. Separable

programming for non-convex separable functions, as in this case, does not

guarantee a global optimum. The reader i s referred to the MPSX/370 IBM

Program Reference Manual (1976) for fur ther information. The sal ient resul ts

and conclusions apear below:

a ) Diagrammatically the solution i s indicated below i n Fig. 6.4.

b ) The 100 MP/d taken from the water board i s i n fact not used. I f

lOMP/d i s assumed to have been taken from the board, i t i s a l l

returned. Hence the supply from the board i s also optimized.

94

Fig. 6.4 Optimal solut ion using l inear programming in conjunction w i th

separable programming

Most of the ground water i s al located to system 1 and most of the

wastewater i s transferred.

The amount of water drawn from desalinated wastewater i s zero. Th is

implies that no desal ination is required. The main reason for th is i s

the abundance of f a i r l y good qua l i t y ground water w i th a TDS of 600

mg/e. I n th i s case the acceptable standard of 700 mg/e i s only jus t

above the qua l i t y of the groundwater supply.

The value of the objective function i s now $691/day, a decrease of

$352/day compared w i th the l inear programming solution. This comes

about solely as a resul t of the introduct ion of a more representative

cost function, using separable programming.

Since the objective function i s of the non-convex type, the solut ion

obtained i s not necessari ly the global optimum. When us ing the

95

XSETLB=-1 command (searching from lower bound to upper bound) the

value of the objective function was $714/d. When delet ing th is command

(searching from upper bound to lower bound) the value was reduced to

$691/d. Fig. 6.5 indicates the $691/d i s s t i l l a local optimum. The

global optimum i s around $650/d.

g ) There is no dif ference i n al location between th is solution, and the

solution using l inear programming (without a non-l inear cost funct ion).

h) The accuracy of the solution is w i th in 5% - 10% of the t rue global

optimum. I t would be improved by a refinement of the g r i d and

functional equations w i th in the v i c in i t y of the current al locations.

Sensi t iv i ty Study fo r var ious acceptable TDS values

The case studies presented were based on an acceptable Total Dissolved

Solids (TDS) of 700 mg/P for water used on the System 1 , System 2 and

System 3. I t i s necessary to examine the optimal solutions for var ious

possible acceptable TDS values i n order to establ ish the best qua l i t y . This

would lead to the establishment of an optimum TDS value, in terms of total

combined costs, required a t the demand zones mentioned above.

The range of acceptable TDS values examined here i s between 500 mg/O

and 1200 m g / P i n discrete steps of 50 mg/O from 600 mg/P upwards, and i n

closer steps between 500 mg/O and 600 mg/e. The only modif icat ion

required to the mathematical model i s the subst i tut ion of the relevant TDS

values for 700 mg/P i n Eq. 6.27 to Eq. 6.29. Results are also given fo r

the ex is t ing system, ignor ing a l l qua l i t y aspects.

Table 6.7 presents the optimal al location of water of water from each

source to each demand, the amount of wastewater used and the amount of

desalinated wastewater. I t also indicates whether the solut ion i s a global

o r local optimum, and shows the value of the objective function fo r

various selected TDS values.

Fig. 6.5 indicates graph ica l l y the var ia t ion i n the total cost of

procuring, desal inat ing and d i s t r i bu t i ng fo r various acceptable TDS

values.

From Table 6.7 and Fig. 6.5 the fol lowing conclusions are apparent:

a ) The graph i s characterized by two parts: one with acceptable qua l i t ies

greater than 600 mg/e and the other wi th qua l i t ies less than 600 mg/O.

The former has a f l a t slope of 0.2, ind ica t ing small cost decreases fo r

large TDS increases. The la t te r has a slope of 8.5, d isp lay ing the

reverse tendency.

96

T A B L E 6.7 Comparison of optimal solut ions fo r var ious TDS values

RW

R B

RS

RV

RM

FW

FB

FS

F V

F M

uw U B

us uv UM

DW

D B

DS

DV

DM

U

D

4.339 3.409 7.000 0.651

3.318 4.522 3.800

7,000

0.700

2.059

8.382

0.618

0.441

3.326

4.941

0.800 1.730

9.500 8.674

0.700 0.639

0.500 0.457

3.800 3.800

6.200 5.270

0.826

0.061

0.043

2.661

7.848

0.578

0.413

8.139

3;591

7.022

0.517

0.370

7.209

7.265

0.535

0.382

6.196

0.467

0.326

6.278

6.349

1.651

0.122

0.087

2.478

0.183

0.130

3.304

0.243

0.174

8.800

0.700

0.500

2.235

0.165

0.118

6.349

2.518

1.118

0.082

0.059

8.175

1.259

10.000 10.000 10.000 10 .ooo 10.000

TYPE GLOBAL GLOBAL GLOBAL L O C A L GLOBAL GLOBAL L O C A L GLOBAL

OBJ. F N . 1573 1213 698 691 633 61 2 64 9 359 ( R / d )

97

LLL 1 6 u

7

I --1

1 FIG. 6.5 V A R I A T I O N I N T O T A L COST FOR

V A R I O U S A C C E P T A B L E TOS V A L U E S

800 900 1000 1100 1200

T O T A L D I S S O L V E D S O L I D S ( m g / e )

The abrupt change in slope a t 600 mg/e i s a resu l t of the necessity fo r

procur ing Desalinated wastewater fo r lower acceptable TDS values. As

water of improved qua l i t y i s required, the amount of water requ i r i ng

desal ination incrases, and consequently the al locat ions to System 1 , 2

and 3 increase.

The previous argument val idates the selection of a qua l i t y w i th in the

reach, s l i gh t l y to the r i g h t of 600 mg/t. The exact qua l i t y would be

determined af ter a g raph was drawn.

Each successive increase i n acceptable TDS should cause a decrease i n

the total least-cost.

Not a l l the successive TDS increases manifest a decreased cost for the

previous TDS. This occurs as a resul t of the non-convex separable

objective function. However, on inspection of F ig . 6.5, i t i s evident

that 700 mg/4, 1000 mg/e and 1100 mg/e are in fact local optima, since

they do not follow the expected trend. Consequently the al locat ions

produced i n Table 6.7 for these TDS values are also not true optima.

These problems may be overcome be performing a sens i t i v i t y analysis,

o r possibly b y rev is ing the g r i d and funct ional equations.

The absolute lowest cost occurs when a l l qua l i t y aspects are ignored

completely. The resu l t ing cost i s $359/d, a decrease of $34/d in

comparison w i th the solut ion using transportat ion programming a lone.

Trends of increases, decreases and changes i n al locat ions as the TDS

var ies are evident from Table 6.6. Two typ ica l forms are:

1 ) An increase in al locat ion from 500 mg/Q to 600 mg/e and a

decrease thereafter - Groundwater to System 1 .

2) No allocation u n t i l af ter 600 mg/e and a steady increase

thereafter - Wastewater to System 2 .

Most of the groundwater i s al located to System 1, and most of the

wastewater t ransferred a t low TDS values discharaged to waste.

REFERENCES

Dantzig, G.B., 1963. L inear Programming and Extensions. Princeton Univ.

Grosman, D.D., 1981. Optimum al location of mine service water subject to

Lcomba, N.P., 1964. L inear Programming. McGraw-Hill, NY. Stephenson, D., 1969. Optimum al locat ion of water resources b y

mathematical programming. J . Hydrol. 9, 20-33. Stephenson, D., 1978. Optimum p lann ing of regional waste water treatment.

I n : Modelling the Water Qual i ty of the Hydrological Cycle (Proc. Baden Symp., September 1978), 351-360. IAHS Publ. No. 125.

Stephenson, D., 1982. Optimum al location of water resources subject to qua l i t y constraints. Proc. Exeter Symp. IAHS, Publ ic. 135, 299-305

Press, Princeton.

qua l i t y constraints. C iv i l Eng. in S.A.

99

CHAPTER 7

ECONOMICS OF DESALINATION OF WASTEWATERS

I NTRODUCT ION

Municipal wastewater i s general ly treated to remove suspended matter

and to neutral ise the biological ac t i v i t y . I t i s disinfected and rendered

innocuous before being discharged into streams. The mineral content of the

water i s , however, not affected noticeably b y treatment ei ther a t the

wastewater treatment works and a t the water pur i f i ca t ion works. Thus not

only i s there a mineral bui ld-up in the water due to i ndus t r i a l po l lu t ion

and to some extent domestic pol lut ion, but also th is mineral content i s

contr ibuted to by na tura l sources. Thus in addi t ion to the ni t rates,

phosphates and other nutr ient minerals coming from the resident ia l type

areas, we also have na tura l minerals such as calcium and sulphate being

contr ibuted to the system from stormwater runoff. The total mass of

dissolved solids thus discharged b y wastewater works into the r i v e r s

averages mi l l ions of tons per day.

The magnitude of the problem of removing the dissolved minerals i n the

water i s enormous. There are many options open, however, for optimum

reuse of th is wastewater. Some of the possibi l i t ies are suggested below:

ALTERNATIVES FOR OPTIMAL REUSE OF WASTE WATER

Present pol icy for many affected water supplies i s ef fect ively to d i l u te

p a r t l y treated and returned wastewaters w i th fresh water from r i ve rs and

other upstream sources. Provided that the water qua l i t y i s at an

acceptable l imi t , for example 500 mg/e per l i t r e total dissolved sol ids

according to world heal th organisation standards, then there i s l i t t l e

concern. I n order to achieve this di lut ion, i t may be necessary i n fu tu re

to discharge some of the wastewater downstream where other users w i l l

have s imi la r o r more concentrated problems. Al ternat ive to th is i s the use

of fur ther sources of fresh water (Stephenson and Corbetis, 1984) .

I t may be more prudent to adopt more operat ing intensive schemes and

less cap i ta l intensive schemes i n the l igh t of economic r i s k s involved i n

capi ta l intensive water supply schemes. ,In pa r t i cu la r where conjunctive

use i s thought of then h igh cap i ta l cost schemes should be used on a

steady base load supply basis whereas operat ing intensive schemes would

general ly be reserved for times of drought i n surface resources which are

100

,001 -

Clarl- flcatloi

REVERSE OSMOSIS

Grit

Sand

Silt

Clay

Collolc

500

50

10

1

0.1

0,Ol

0.00

S A N D FILTERS

E V A P O R A T I O N

0001 I I 1 10 100 1000 10000 too

Fig. 7.1 Selection of pur i f i ca t ion method based on water qua l i t y

101

cap i ta l intensive. Demineralisation and desal ination processes are more

operative intensive than surface resources development. Desalination

procedures i.e. those sui table for desal ination of sea water are often

h igh ly operating intensive as they use large amounts of power, for

example d i s t i l l a t i on processes. On the other hand the total dissolved sol ids

content of sea water i s near ly 35 000 mg/t per l i t r e , whereas we are

ta l k ing of dissolved sol ids contents of less than 1 000 mg/P per I i t r e i n

wastewaters for a r t i f i c i a l recharge of ground water aquifers.

The poss ib i l i t y of l imi ted treatment before discharging into aqui fers i s

now under consideration. I t i s possible that by t r i ck l i ng the water

through the aqui fers there w i l l b e na tura l aeration which would reduce the

biological oxygen demand as well as provide a degree of f i l t r a t i on and of

great interest, na tura l ion exchange resu l t ing i n demineral isat ion o r

neutral isat ion of some of the dissolved sol ids content. Al ternat ively, the

wastewaters could be discharged to lower levels of the acqui fer thereby

l i f t i n g the fresher waters which have seeped there by na tura l means such

as from rainwater and in f i l t r a t i on from surface streams.

Not only w i l l th is type of a r t i f i c i a l recharge have the advantage of

reducing pumping costs b y keeping a h igh water table, but i t may also

solve the problem of dewatering of dolomitic compartments which is l inked

to geotechnical problems. Previous dewatering exercises have resulted i n

collapse and d i re consequences i n resident ia l areas and a t mining

development so part ies concerned would be nervous about dewatering even

i f only intermittently to supply i n times of drought.

Other possibi l i t ies include the local recycl ing of wastewater from

pa r t i cu la r areas to other selected areas. In th is way there i s a

poss ib i l i t y of minimal treatment i f water i s used for successive lower

qua l i t y - requ i r ing uses. Yet another poss ib i l i t y i s the recycl ing wi th fresh

water pumped from r i ve rs instead of the na tura l recycl ing. In this way

pumping costs and p ip ing costs, as well as storage costs, would be saved

fo r the water would be recycled and not have to be pumped.

SELECT ION OF OPT I MUM DESAL I NAT ION METHODS

Although h igh desal ination costs are a deterrent to the general use of

desal ination fo r water supply, an optimised system may i n fact be

considerably more economic than may f i r s t appear. The location, scale,

type and adaptab i l i t y of a desal ination or demineral isat ion p lan t can a l l

be put to use i n reducing total water costs. Thus the location of in-house

desal ination p lan t may avoid the necessity of disposing of eff luents into

102

col lect ing sewers, then through municipal wastewater treatment works.

Distr ibut ion and pumping costs are thereby p a r t l y reduced, which offset

desal ination costs.

Studies for optimum location of desal ination p lan ts have been conducted

w i th the assistance of computer simulat ion programmes. The depict ion of

the ret iculat ion system and a l te rna t ive locations fo r desal ination p lan ts

p lus analysis have proved that such p lan ts can be economically instal led.

There may be savings in pumping costs i n pa r t i cu la r for h igh heads i f

p lan ts are underground. There is also the saving in purchase of raw

water and p ip ing costs. Desalination has also been shown to give a

considerably better water qua l i t y than the use of r i v e r water. This has

fu r ther imp1 ications i n reducing costs of corrosion and deter iorat ion to

pipework due to other chemical ac t i v i t ies such as scal ing.

The method of desal ination may also depend to a large extent on the

cost of energy. Whereas desal ination methods such as evaporation require

large amounts of energy and are therefore normal ly undertaken using coal

f i r ed boi lers, low energy consumption systems such as reverse osmosis

frequently use electr ic i ty from the g r i d systems. There are many industr ies

which have some form of heat exchange or heat generation. Thus

oil-from-coal systems generate large amounts of surp lus heat and may be

more sui table fo r evaporation type methods. Some industr ies requ i re

cooling. Some mines c h i l l water before sending i t underground and have

even contemplated the d is t r ibu t ion of ice underground. I n such cases

freezing desal ination may prove most v iable. The vapour compression

method i s also receiv ing close attention in te rna t iona l l y a t the moment.

The scale of the desal inat ion p lan t and the qua l i t y of the raw water

have a considerable effect on the optimum method of desal ination. Scale o r

size of p lan t w i l l inf luence the operat ing costs and these can be expected

to reduce the la rger p lan t and cap i ta l costs such as housing w i l l reduce

per k i l o l i t r e of water treated the la rger the p lan t .

For low total dissolved sol ids contents membrane-type processes such as

reverse osmosis and electrodialysis have proved most economical. For very

low concentrations ion exchange is economical. There are many problems

associated with the membrane type processes in pa r t i cu la r where there i s a

h igh sulphate content. I n such cases seeded systems have often prevented

the crystal isat ion of the sulphates on the membrane and kept the total

dissolved sol ids in suspension. The number of stages in such p lan ts cai

also effect the f i na l water qua l i t y . The cost per ton of dissolved sol id

removed may be a minimum for one pa r t i cu la r method whereas the cost pel

k i l o l i t r e treated, i f one i s not concerned w i th the amount of sal ts removed

103

may be cheaper for another system. Figures 7.7 and 7.8 compare the costs

on these basis for d i f ferent methods.

Mult i-stage demineral isat ion may also be sui table for pa r t i cu la r

appl icat ions. Thus i f a h igh water qua l i t y i s required, mult i-stage

methods are usual ly the most eff icient and the f i na l eff luent can be

brought to a low total dissolved sol ids concentration, fo r example, less

than 10 mi l l igrams per I i t r e , most economically through successive stages.

I t i s possible that succession stages employ di f ferent techniques e.g.

reverse osmosis coupled w i th ion exchange.

The dispossl of the br ine is also a problem when i t comes to

demineralisation of waste waters. Whereas the volume of concentration of

the br ine i s of l i t t l e concern i n the case of desal ination p lan ts on the

sea, th is i s not the case where the br ine has to be disposed of. I t i s

most desirable that the b r ine be brought to a very h igh concentration and

even possibly to sol ids form before disposal in land. This minimises the

cost of transport and the cost of disposal sites i f i t i s stored. I n such

cases the br ine may have to be concentrated through successive stages of

the p lan t . These concentration stages would b e designed d i f fe ren t ly to the

eff luent pur i f i ca t ion stages and may work on a di f ferent process again.

Heat exchange p lays an important p a r t in the operat ing costs of many

processes. Thus i f evaporation techniques are used then the ef f luent which

is at a h igh temperature can be used to heat the incoming stream to b r i n g

i t to near ly evaporation temperature such as i n mult i-stage f lash

evaporation methods. I n the case of freezing processes the ice product

could be used to cool the incoming stream of water to near freezing

temperature. This would be possible i f the f i na l product temperature were

immaterial but i t may not be wise i f cold water i s a product as well as

pure water.

I t i s thus evident that desal ination and demineral isat ion cannot easi ly

be bought i n the form of a package p lan t . The economics and

prac t icab i l i t y of the demineral isat ion process can only be selected when

considering the p lan t and a l l factors as a whole. The ent i re process must

be designed i n conjunction wi th the p lan t of the factory i f optimum

desalination costs are to be achieved. I n th is case effect ive costs less

than 50 cents per k i l o l i t r e , comparable wi th raw water, can be achieved.

RELEVANT DESALINATION METHODS

The potential for desal ination i s in te rna t iona l l y recognized and as a

result there was an increase i n water desal ination capacity of 40% dur ing

104

the last 5 years (65% of which are mult i-stage f lash d i s t i l l a t i on seawater

p lan ts and 25% reverse osmosis seawater and brack ish water p lan ts ) .

Membrane methods appear promising fo r fu tu re development. Reverse

osmosis was restr icted to small size p lan ts fo r b rack ish water u n t i l 10

year ago and now moving to la rge scale plants. Some examples are the

250000 m’/d p lan t in Jeddah for sea water treatment and the 400000 m’/day

p lan t i n Yuma (USA) , for desal ination of drainage.

Membrane methods are competitive w i th thermal processes, comparing

favourably i n energy consumption as well as suscept ib i l i ty to corrosion

and scal ing. Where b r ine disposal i s a problem however ( f o r example

i n land ) the cost of addi t ional fac i l i t i es may detract from the membrane

processes.

I ndus t r i a l Wastewater treatment

The increased energy costs du r ing the last decade have directed

research and development work for a l I desal inat ion methods towards

reducing energy consumption. Different methods of energy recovery have

been investigated and the i r app l i cab i l i t y depends on the costs and the

size of the p lan t (Binnies, 1981; Larson, 1 9 7 9 ) .

Some examples are:

1 ) RO p lan t energy can be recovered by i ns ta l l i ng a turbine on l ine i n

the (h igh pressure) b r i ne stream.

2 ) Underground ins ta l la t ion of RO p lan t can be j us t i f i ed based on

u t i l i sa t ion of the stat ic pressure instead of h igh pressure pumps.

This can be appl icable to the mining industry for fresh water production

underground. The energy consumption costs w i l I normal ly be high.

I n the USA i t has been suggested that advanced treatment methods

(demineral izat ion) for domestic and municipal wastewater i s the best

a l te rna t ive for solv ing the problems of water supply. This i s appl ied iri

Denver (USA) where the RO for demineral izat ion i s included i n a s ingle

p lan t , and is now being contemplated elsewhere.

Reverse Osmosis

T h e na tura l phenomenon of osmosis occurs when sal t water and fresh

water are separated by a semi-permeable membrane and fresh water flows

through the membrane to d i l u te the sal ine water. This water flow stops

when equ i l ib r ium i s establ ished and the pressure dif ference between the

two solutions i s cal led osmotic pressure, the magnitude of which depends

105

on the sal ine solution concentration. I f however, pressure is exerted in

the sal t solution greater than osmotic, fresh water diffuses through the

membrane free of sal t (DSS, 1980; Ludwig, 1980).

Membrane Description

The cellulose acetate membrane which i s cur ren t ly in general use i s

approximately 1 0 0 ~ thick, of which only one layer i s act ive and is

approximately 0,2p thick (2000A) on top of the membrane surface w i th the

rest act ing as physical support for the exerted pressure. This th in layer

acts as a f i l t e r to re ta in the ions such as Na' and Cl-.

E I ectrod i a I ysi s

I n the Electrodialysis process, water flows between al ternately placed

cation and anion permeable membranes.

A direct electr ic current i s the d r i v i n g force for the ion migrat ion

through the membranes. A series of a l te rna t ive cation and anion membranes

with a p las t i c spacer between is assembled into membrane stacks. Several

hundred membranes and their separating spacers are usua l ly assembled

between a s ingle set of electrodes to form a membrane stack.

The ion selection membranes are basical ly ion exchange resins i n sheet

form with selectivi t ies greater than 90%. Normally Electrodialysis systems

consists of one to s ix stages, wi th removal per stage va ry ing from 30 to

60% (normally 50%).

Energy consumption is based on Faraday 's Law, according to which for

100 mg/e removal of dissolved ionised sol ids from 5m3 of water, 200

amper-hours ( D C ) are required with voltages 1 - 2 V. Therefore about 0,3

kWh is needed for 5m' of water treated i n addi t ion to which 2kWh i s

required for pumping.

Several hundred Electrodialysis p lan ts have been instal led

internat ional ly for process water treatment o r portable water from feed

waters normally of less than 3000 mg/t sa l in i t y .

Reverse po la r i t y (electrodialysis reversal) conf igurat ion has been

introduced commercially to reduce polar izat ion and scal ing i n the

membranes.

Ion Exchange

The ion exchange process has been used for many years fo r softening

106

' 1 10 100 1000 10 000 100 1 Product Solinity mg/l

F ig . 7 .2 Suitable feed sal ini t ies and product sa l in i ty for various desalination processes. Recovery rat ios also indicated

iter

ter

107

of water and demineral izat ion for var ious indus t r ia l uses. I t i s normally

restr icted to waters of not more than 1000 mg/4 total dissolved solids.

The process i s based on the character ist ic of some neutral minerals

cal led zeolites which were found to exchange ions sui table fo r the

softening of water, l i ke exchange of Mg' and Ca for Na . Ar t i f i c i a l ion

exchange resins have superior exchange character ist ics and be de f in i t ion

are insoluable sol ids containing f i xed cations o r anions capable of

reversible exchange with mobile ions of the opposite s ign in solutions.

Resins normally absorb Na+ ions and other cations and release H+ or

absorbs CP- and other anions and release OH-; these are ca l led ac id

resins and base resins respectively. The ions release H and OH- in the

solution which combine to form H20.

++ +

+

Ion exchange i s un l i ke ly to prove economical fo r water treatment of

sa l i n i t y higher than 1000 mg/4. However, i t can be used in conjunction

w i th a membrane process.

COST ANALYSIS

The cost of desal ination techniques i s often expressed i n terms of

cents/m' of water produced. This approach can be misleading as i t f a i l s

to take into account a l l the var iables af fect ing the cost structure. Before

proceeding with cost estimation the fol lowing parameters have to be f ixed:

1 ) Product requirement, p lan t load factor and recovery.

The var iables affect the quant i ty and qua l i t y of the f ina l product, the

relat ionship between product water quant i ty, feed water quant i t y and

the plant operational eff iciency. Values assumed for the cost estimate

i n th is paper are; recovery ra t i o (Rc) 70% - 65% and p lan t load factor

90%.

2 ) Rate of interest which for present purposes i s taken as 10% wi th a

redemp t ion period of 20 years.

3) Plant l i fe i s taken a t 30 years.

The capi ta l and runn ing costs are affected by the above parameters.

Capi ta l Costs

Capital costs include C iv i l Engineering and p lan t as well as s i te

development (roads, electr ic i ty and water etc. 1 . They also include

equipment and controls as well as ins ta l la t ion of intake and b r ine

d i sposa I .

108

Ind i rec t Capi ta l Costs

10 - 12% long term interest ra te i s included du r ing p lan t construction

and also labour costs which amount to 5 - 6% of the total cap i ta l costs.

Running Costs

Running costs are general ly d i rec t l y proport ional to product throughput

and include energy costs, chemical costs, labour for operation and

ma i n tenance, membrane rep lacemen t , operat ing and maintenance costs.

For membrane p lan ts i t i s reasonable to assume e lec t r i c i t y cost wi th

100% load factor of 2c/kWh. Chemical cost and treatment costs va ry wi th

feed water character ist ics, the process used and the p l a n t ' s recovery

ra t io .

Labour Costs

These depend on the requirements o f the p lan t w i th respect to operation

and control. I t a l so depends on the r e l i a b i l i t y of the p lan t , as th is

affects i t s maintenance labour cost. Costs given here are for p lan ts up to

10 000m3/day capacity (medium size).

Membrane Replacement

For Electrodialysis ( E D ) 20% of cap i ta l cost i s spent on membranes

which on the average have a 7 year l i fe. For reverse osmosis t rea t ing

brack ish water the l i f e of the membrane is only 3 years.

I n Figure 7.3 the cap i ta l cost fo r both Electrodialysis and Reverse

Osmosis include the si te development costs, equipment costs and indirect

cap i ta l costs. Figure 7.4 shows the operat ing costs ( runn ing costs)

inc lud ing labour, energy and costs for both membrane processes. This data

i s based upon a feed water w i t h a sa l i n i t y in the range of 1 000 to 3 000

mg/e total dissolved sol ids.

The cap i ta l cost fo r the reverse osmosis process var ies from

approximately $700/m3/day for a 120rn3/day feed capaci ty p lan t to

$170/m3/day for a 10 000m3/day feed capaci ty p lan t , based upon 3 stages.

The cost al lows for di f ferent equipment designs and manufacturer 's pr ices

and si te costs which are dependent on the site.

The cap i ta l cost for Electrodialysis depends la rge ly on the sa l i n i t y of

the feedwater hence the number of stages involved in the process as well

109

DESRLINRTION PROCESSES

CFTXTRL L RUNNING co- ro11 nLcmtornmysxs L IINLRSL osnos~s

c

8

Fig

n \

I

Fig. 7.4 Running cost versus P l a n t size (Feed) for ED & RO

ED process ( -.-.d

) RO process (

I10

D E S A L I N A T I O N PROCESSES

ENERGY Rf O U I R M W S (hl4hk) FOR CLCCTRODIRLYSIS h REVERSE OSMOSIS

Fig. 7.5 Energy Demand versus Feed Salinity for ED & RO

Fig. 7.6 Energy Demand versus Plant size (Prod) for ED & RO

ED process (-.-.- )

RO. p roccsd )

111

as the plant size. The costs vary from $760/m3/day for two 4-stage p lan t

of 120m3/day feed capaci ty to a $200/m3/day for a 2 stage p lan t of

10000m3/day feed capacity. I t can be seen i n Figure 7.3 that cap i ta l costs

for Electrodialysis are general ly higher than other methods.

I n runn ing costs however electrodialysis i s cheaper than reverse

osmosis, va ry ing from 25c to 7c as opposed to 45c - 1Oc for reverse

osmosis from the same p lan t capacity (F igure 7.5). The costs i n Figures

7.3 and 7.4 account for d i f ferent water character ist ics and p lan t design

factors.

The water temperature, pH, tu rb id i t y , suspended matter etc. can

influence energy and pretreatment costs. Also recovery ra t i o (Rc) var ies

between 70 and 85% for Electrodialysis and 65 and 85% for reverse osmosis

which can affect the energy requirement.

The to ta l theoretical cost of the water var ies from 40c to 20c per m3

for dif ferent size p lan ts for Electrodialysis and 52c to 22c per m3 i n the

case of reverse osmosis (1983 f igures) .

I n Figure 7.5, i t can be seen that Electrodialysis i s less

energy-consuming for a feed sa l in i ty range of 500 - 3 000 mg/Q. This i s

due to the fact that energy demand is d i rec t l y proport ional to removal for

the Electrodialysis. The energy costs account for d i f ferent p lan t sizes,

assumed of the order of 1 000m3/day,

Figure 7.6 shows the var ia t ion in energy requirements for d i f ferent size

plants. Energy for reverse osmosis changes with recovery va ry ing from 70%

to 85%. The electrodialysis curve assumes a recovery r a t i o of 75%.

CONCLUSIONS

I t i s evident that there are many var iables af fect ing the costs of

demineralization. From the point of view of the app l i cab i l i t y of a

pa r t i cu la r process and the optimum process for a pa r t i cu la r appl icat ion

the fol lowing factors are relevant:

Load factor - relates cap i ta l to runn ing costs.

Local i ty - affects power cost, construction and labour costs.

Capital cost and interest rate.

Power and other operating costs.

TDS of raw water.

Qua l i t y requirements for eff luent

Method of b r ine disposal.

112

I f the costs a re expressed as cents per k i l o l i t r e , one pa r t i cu la r method

may prove most economic, whereas i f they a re expressed as cents per k g

of dissolved sol ids removed, another method may be optimal.

Methods re la t i ve ly insensit ive to the TDS of the incoming steam include

the thermal methods. In the case of evaporation methods, however, scal ing

i s d i rec t l y related to the TDS. There i s a tendency to use thermal methods

for h igh TDS waters as the cost per kg of sa l t removed i s low. O n the

other hand the cost of membrane and ion exchange processes are more

related to the amounts of sa l t removed and they a re therefore lowest fo r

low TDS waters. Figure 7.7 and 7.8 depict the costs of d i f ferent method

plotted ( a ) versusflow ra te and (b ) versus r a t e of TDS removal.

I t appears that membrane type processes are most economical for

indus t r ia l wastwaters (provided adequate pre-treatment i s p rac t i ca l ) .

Recent advances i n membranes now include those capable of passing h igh

rates a t low pressures provided only l imi ted sol ids removed i s required.

Using present day prices i t i s feasible that costs w i l l be competitive w i th

bu l k water supplies from a fa r and i n addi t ion there is the incentive of

reduced ret iculat ion costs, and the poss ib i l i t i es of use of the eff luents for

specific appl icat ions or fo r ground water recharge.

REFERENCES

Binnie and Partners, 1981. Desalination methods and costs, Water Systems

DSS Engineers Inc., 1980. Data Collection and Analysis of Commercial

Larson T.J. and Leitner G., 1979. Desalination of sea water and brack ish

Ludwig, L., 1980. Reverse osmosis i n the desal inat ion of b rack ish water

Stephenson, D. and Corbetis, S.S., 1984. Economics of Desalination of Wastewaters for the W i twatersrand. Water We1 I Assn. Conf. Johannesburg.

Research Programme, Universi ty of the Witwatersrand.

Membrane Desalination Plants.

water, A cost update. Desalination, 30, p525-539.

and sea water, Desalination, 36, p 153-178.

113

\ \

114

Fig. 7.8 Desalination costs per kg of dissolved solids removed as a function of feed s a l i n i t y , f l o w r a t e a n d method

115

CHAPTER 8

COMPUTER ANALYSIS JUSTIFIES DESALINATION

INTRODUCTION

Industry uses water for cleaning, a i r condit ioning, d i l u t i on and

transport amongst other purposes. Water may be recycled and deteriorates

in qua l i t y due to contact w i th contaminants and discharges into the system

(Holton and Stephenson, 1983). The total dissolved sol ids (TDS)

concentration i n some waters can exceed 10000 mg/l.

The poor water qua l i t y can lead to corrosion and deteriorat ion of

pipework and machinery. I n a pa r t i cu la r system, corrosion of the

refr igerat ion p lan t was extremely severe. Other problems associated w i th

the poor water qua l i t y are the heal th hazard, and surface ef f luent

discharge regulat ions.

The problem has been aggravated by deteriorat ion in surface water

qua l i t y over the last few years.

The approach adopted in th is study, namely to optimize the use of

resources, can be appl ied on a much wider scale than herein described

(Stephenson, 1986). The computer program developed for the purpose of

re-distr ibut ing water i n the most economical way to meet qua l i t y

constraints i s universal. I t could be apppl ied on a regional basis o r on a

smaller scale for indus t r ia l water systems. The bas is for j u s t i f y i n g h igh

operating-cost sources such as desal ination l ies in the fact that other

costs may thereby be saved, namely pumping costs o r storage dam costs.

Less water may be required i n total, as the water may be of a higher

qua l i t y . Eff luent discharge problems may be reduced as the volume of

effluent (b r ine) i s considerably less i f desal ination i s performed.

There are al ternat ive ways of j us t i f y i ng h igh operating-cost systems

such as desal ination in comparison with more conventional sources such as

r i v e r water. They l i e i n conjunctive use - namely use of capi ta l - intensive

schemes to supply the base load, and resort ing to standy low capital-cost

sources when there is a short fa l l i n surface resources e.g. du r ing a

drought. Al ternat ively desal ination could proceed on a regu la r small scale,

d ischarging into a reservoir (e.g. aqu i fe r ) which could be tapped i n times

of shortage elsewhere.

The f i t t i n g i n of a desal ination un i t wi th a system i s thus more l i ke l y

to jus t i f y desal ination than a straight comparison of u n i t costs of water

from a p lan t o r from more conventional sources. The optimization of

116

multi-source systems requires careful ana lys is and this i s where the

computer i s helpful . There are various ways i n which computer systems

analysis can be used. Simulation of the water d is t r ibu t ion system w i l l

reveal bu i lup of TDS over time. A simultaneous solut ion of the mass

balance equation a t nodes w i l l g ive steady state TDS loads, and

optimization can produce the best (general ly the most economic) p lan .

The type of desal ination p lan t most sui table for any given system w i l l

also depend on the circumstances (Stephenson and Corbetis, 1983). The TDS

values mentioned here could be described as requ i r i ng desal inat ion o r

demineral izat ion (usua l ly confined to lower TDS water) . Membrane type

processes, e.g. reverse osmosis, are often most suitable, a l though

freeezing desal ination i s now receiv ing attention.

Various attempts a t optimizing complex systems invo lv ing water qua l i t y

have been reported (R ina ld i e t a l , 1984). I n general, the problem i s

non-l inear so l inear programming packages (Loucks et a l , 1967) are of

l i t t l e use. Search methods (Smeers and Tyteca, 1981) must general ly be

used. The technique described i n th is paper i s one such method which uses

known characterist ics of the system to speed the search. The method and

the cost of desal ination are not explained i n de ta i l here although such

studies have been reported (Abulnour et a l , 1983).

APPL ICAT ION OF OPTIMIZATION OF WATER SUPPLY

The options studied, in view of the poor qua l i t y of the water on an

indus t r ia l system were:

1 . Accommodate the poor qua l i t y at the expense of h igher maintenance

costs of pipework and machinery.

2. Prevent deteriorat ion of the water a t the source of pol lut ion.

3. Remove the TDS i n the water by means of a desal inat ion p lan t .

4. Use greater quant i t ies of fresh water.

5. Re-distribute the water i n a way which maintains higher qua l i t y a t key

points.

Al ternat ive 1 , namely cont inuing w i th the present poor qua l i t y water,

was qu ick ly ru led out by assessing the cost of regu la r replacement of

pipework and machinery. Research was also in progress to reduce leaching

a t the source of pol lut ion but no posi t ive recommendations could be made.

The most economical combination of a l ternat ives 3-5 was investigated us ing

the computer program developed for the purpose.

117

No.3 Shaft

Flsrure water

Mlm u r v l c a water

- F a d rate;

----- MIM wwklngs u r d water Sludge llmr -. -

6 nrsure water

L- - C b r water return Ilne . . . . . . . . . Carcmd. rvstem overflow

chamber water system dam

Fig. 8.1 Section through a typical mine

The water d is t r ibu t ion system is often complex (Stephenson, 1983) and

not accurately documented. I n general, water i s piped from one working to

another and i t i s collected i n d ra ins and pumped to waste o r recycled

after set t l ing to remove suspended part ic les. The water may be used for

dust suppression and cooling. Minimum flows a re required bu t the

distr ibut ion pattern and the amount recirculated o r pumped can be varied.

118

Water from di f ferent locations i s often blended. Water i s cooled b y spray

evaporation o r draught towers before re f r igera t ion in an indirect

heat-exchange p lan t i n cool ing systems. The evaporation in the warm areas

as well as i n evaporation dams aggravates the TDS concentration i n the

water. Fig. 8.1 i l l us t ra tes a section through a typ ica l mine workings w i th

water taken from a fresh water source a t the surface, returned from

underground for cooling a t surface cooling towers, and p a r t l y replaced b y

fresh water before recycl ing.

SYSTEMS ANALYSIS

One shaft i n the complex d is t r ibu t ion system depicted in Fig. 8.1 can

be reduced to the flow diagram of Fig. 8.2, reproduced b y the computer

program described here. The hyd rau l i c p r i nc ip le used for ana lyz ing the

system i s the cont inui ty equation, that is, net inf low minus outflow a t any

node must equal zero. I t i s possible on th i s bas is to make up any

d is t r ibu t ion system out of a number of closed loops. I t may be necessary

to close the system by means of a dummy node i.e. any loose input to and

outflow from the system can be taken from o r to the dummy node. Such a

dummy node i s handled d i f fe ren t ly from other nodes as no qua l i t y

constraint or mass TDS balance appl ies to i t .

Fig. 8.2 Graphic of system analyzed

119

I t may be shown that the minimum number of closed loops in a network

of pipes and channels i s (Smith et a l , 1981):

L = P - N + l

where P i s the number of conduits connecting N nodes. Addit ional loops

may be defined by including pieces of other loops and the selection of the

best loops can speed ana lys is i f manual hydrau l i c ana lys is i s performed.

A n algori thm for f i nd ing loops i n a system described only in terms of

nodes comprised pa r t of the program. For p rac t ica l purposes i t i s easiest

to ident i fy conduits as f lowing from one node to another and not as

separating adjacent loops, i.e. loops are not ident i f ied at the data input

stage. The def in i t ion of loops a t data input stage i s also irksome when i t

comes to revis ing, adding to o r subtract ing conduits. Dur ing revis ion of

flows i n the network, however, flows should always balance a t nodes. The

simplest way of ensuring th i s i s to adjust flows by adding a flow around

closed loops.

MAIN PROGRAM SUB PROGS ~

IDENTIFY SYSTEM READ CONDUIT DATA IDENTlfY NODE CONS SEEK ML LOOPS

I HYDRAULIC NETWORK ANALYSIS FOR CONDUITS

I . I I TRYMTERNATNEDESM I I I I PUNT LOCATIONS AND-SCALE I

I I

SOLVE SIMULTANEOUS MASS W C E TDS EOS AT ALL NODES XPT 0

ADJUST FLOW IN LOOPS IN BEST WAY UNTIL AU TDS's WITHIN LIMITS

RETC COST

Fig. 8.3 Program flow diagram

1 20

A computational a lgor i thm for selecting the minimum number of closed

loops was developed. The algor i thm can include unclosed branches by

insert ing dummy conduits and nodes. Special hand l ing of these conduits

was required as only flow balance, not TDS balance occurs a t some dummy

nodes. For instance, evaporation can be represented as a negative flow fo r

a dummy node w i th zero TDS. A flow diagram for the procedure i s given in

Fig. 8.3. To ensure that each pipe i s included in a loop, the procedure

s ta r ts w i th each pipe in tu rn i n the system. I t proceeds in the posi t ive

flow direct ion (from top node to bottom node) from one pipe to another

leading from i t . Where there i s more than one pipe ex i t i ng from a node, a

new loop i s created for each branch. At each step a check i s made fo r

loop closure and then any pipes i n the series not in a closed loop are

dropped. Whenever a closed loop i s formed a check i s made, pipe b y pipe,

w i th other loops to ensure no dupl icat ion. Loops which have negative f lows

are ignored. A negative flow from node zero (wh ich has zero TDS) i s

prescribed to represent evaporation since node 0 i s always at zero TDS.

START WITH W C H PIPE IN TURN GO FROM TOP NODE TO BOTTOM OF SUCCESSIVE PIPES

Where there Is a branch-off store plpedata up to branch for another loop CHECK FOR LOOP CLOSURE ELIMINATE REDUNDANT PIPES CHECK FOR LOOP DUPLICATION BY COMPARING PIPE FOR PIPE AND ELIMINATE DUPLICATES

GO BACK TO CHECK FOR BRANCH LOOPS

Fig. 8.4 Loop seeking algor i thm

121

Once a l l the loops are identi f ied, an increment in flow i s added to one

loop a t a time to determine the corresponding change in TDS a t each node

and the corresponding cost increase. The node w i th the worst TDS, in

comparison w i th target TDS, i s ident i f ied and the improvement in TDS a t

the node per un i t increment i s established. This i s repeated fo r each loop

and the one w i th the maximum of d(TDS)/dC i s selected where C i s the cost

of circulat ion. The maximum necessary increase in flow i s made around the

loop consistent wi th qua l i t y constraints. The procedure i s repeated u n t i l

a l l TDS's are w i th in specified l imits.

I t i s necessary to recalculate TDS's a t each node in the system each

time an increment i s made to flows around a loop. This i s done b y

i te ra t ing for each node in succession the mass balance equation

T ' = z Q ( T + P)/YQ

where Q i s the flow from an upstream node to the node a t which T I i s to

be determined. T i s the TDS a t the upstream node, and P i s the p ickup

(o r decrease i n the case of desal ination) of TDS along the conduit. The

summation i s over a l l conduits leading to the node.

General optimization problem

The problem to be solved can be described as follows: given minimum

flow requirements and maximum TDS requirements a t certain points in a

network, as well as qua l i t y of raw water avai lable, ra te of deteriorat ion

in water a t certain points and the network layout, what amount of water

should be purchased from the raw water source and what level of

desal ination should be performed? At the same time the program yields

flows i n each conduit and qua l i t y (TDS) at each node. Costs i n the

objective function to be minimized include cost of raw water, cost of

transport per I i t r e along any conduit, e.g. pumping and cost of

desal ination in c/ l for a l te rna t ive levels of treatment (mg/l removal).

A t r i a l and er ro r process i s required to establ ish the best posit ion of

the desal ination p lan t o r plants, and the best level of treatment

(proport ion of TDS removed ) . The costs were assumed to be l inear ly proport ional to flow ra te

although non-linear objective functions could be handled b y the program.

Capital costs of conduits were not included as the conduits already exist

and can hangle larger flows than w i l l normally occur (they are based on

emergency condit ions). Flows can be control led b y valves i n the case of

g rav i t y lines and pumping power in the case of pumping lines.

122

PROGRAM APPL I CAT I ON

The program was run on a 128 kb x 16 b i t micro computer which

handles up to 100 conduits. I t i s often run in conjunction w i th a

simulat ion and graphics program. The simulat ion program al lows for time

var ia t ion in drawoffs fo r meeting specif ic water demands, and for storage

which can f luctuate over a period. I t does not, therefore, assume

cont inui ty of flow a t nodes and i s useful fo r more ref ined studies than the

optimization program such as time va ry ing qua l i t y (see Fig. 8 . 5 ) . The

graphics system i s useful for v isual izat ion of the system, al though i t

requires co-ordinates of nodes as input. A common data f i l e can be used

for a l l programs and th i s f i l e can be amended (conduits altered, added o r

subtracted) as ana lys is proceeds.

The optimization program commences with minimum flow data. Flows can

only be increased to reduce TDS a t any point. The location of desal inat ion

p lan ts must be selected by t r i a l , as well as the level of desal ination,

e.g. i f s l i p stream desal inat ion i s resorted to, i t i s equivalent to p a r t

removal. Although i t would be a simple matter to automat ical ly invest igate

desal ination p lan ts along successive conduits, the number of physical

constraints l im i t ing desal inat ion p lan ts to specif ic locations o r levels does

not warrant automatic reposit ioning of p lants. Br ine disposal, space

requirements and access are of pa r t i cu la r concern i n mines.

0 ,DAY 1 2 3 4 5 6 /

Fig. 8.5 Deterioration of TDS a t a strategic point as indicated by

simulat ion

123

OPTIMIZATION OF MINE WATER SYSTEM

Mines suffer from poor water qua l i t y due to poor r i v e r qua l i t y , la rge

concentrations of chlor ine i n ground water, and a severe shortage of water

which necessitates extensive recycl ing. Baker-Duly (1985) reported on a

pa r t i cu la r mine's method of solv ing the problems using a simulat ion

program.

The pa r t i cu la r mine presented as an example of the appl icat ion of the

current optimization program suffered the var ious problems out1 ined above.

The simpl i f ied system analyzed here i s depicted i n Fig. 8.2, whereas the

rea l mine involved 3 shafts wi th cross flows from one shaft to another. I n

the system analyzed there i s one main shaft from the surface down to

2140m below surface, and two sub-shafts down to 2430 and 2620m below

surface, respectively. There are sett lers for removing suspended sol ids a t

the lowest level and a t the bottom of the main shaft. Water from the

sett lers i s diverted into sumps and thence pumped to higher levels. Par t

of the water i s recycled with ' f resh ' water purchased from a regional

water board. Pr io r to the study 23 I /s was thus purchased, the makeup

from groundwater total led 22 I /s and evaporation amounted to 16 I /s

resul t ing i n a discharge to waste on the surface of 29 I /s. Owing to the

h igh heads, cost of pumping to waste was h igh and the purchase pr ice of

fresh water was also high. The qua l i t y of ' f resh ' water was in fact poor,

making marginal improvement to the system for the pr ice paid. As water

qua l i t y was of pa r t i cu la r concern at the re f r igera t ion plants, fresh water

was directed s t ra igh t to them. Unfortunately, water from cooling dams was

also sent to the re f r igera t ion plants, and th is water was of pa r t i cu la r l y

poor qua l i t y . That was because the dams received warm water from the

mine workings where the water had come into contact w i th ore, w i th a l l

i t s contamination, and evaporation a t the workings and the spray ing on

the dams had concentrated the dissolved sol ids even more by evaporation.

The fact that water i n the spray dams was of pa r t i cu la r l y poor qua l i t y

and that i t had to be improved considerably before re-use, indicated the

most appropriate posit ion fo r a desal ination p lan t .

RESULT OF ANALYSIS

Table 8.1 indicates flows and TDS's a t a l l points in the system p r i o r

to the analysis and Tables 8.2 and 8.3 present the optimum flows and

TDS's resul t ing from the analysis for the best posit ion of the desal ination

p lan t .

1 24

Table 8.1 was obtained from the simulat ion program, where water

qua l i t y was a r b i t r a r i l y assumed to s ta r t at 500 m g / P before increasing

over l i t t l e more than a week to equ i l ib r ium TDS's of over 4000 mg/l a t

p I aces.

TABLE 8.1 Water Flows and TDS without desal inat ion

Cost, $/a = 1.056825E+6

~~ ~

Pipe Node 1 Node 2 Water Flow Increase TDS Cost Equilibrium TDS number (W (will ( C P U (WlU

1 0 1 23 800 80 i91 2 1 3 23 0 0 193 3 3 6 8 0 0 183 4 6 I 8 0 0 3296 6 0 11 -8 0 0 4133 6 0 I -8 0 0 3296 I 2 4 36 0 0 2419 8 4 6 61 0 0 2298 9 0 6 18 1800 0 2298 10 5 8 69 400 0 8180 11 8 7 62 0 6 8295 12 8 9 67 0 0 3174 13 9 2 67 0 20 8188 14 I 10 36 0 0 8138 16 10 12 40 200 0 8826 16 I 11 26 0 0 4183 17 11 12 18 200 0 3826 18 0 10 4 1800 0 3138 19 12 8 60 0 10 3180 20 12 0 8 0 60 0 21 2 0 21 0 16 0 22 3 4 16 0 0 2419

Table 8.2 summarizes the TDS's, wi th a desal inat ion p lan t between

nodes 8 and 7, removing 1000 mg/l. The highest TDS in the system is 1334

mg/l and no addi t ional f lows above minimum were required. The net system

operat ing cost assuming a desal inat ion cost of 100 c /k l , would be $2.9

mil l ion/y. I n fact a maximum TDS of 2000 mg/l anywhere in the mine was

specified for th is r u n whereas a maximum of 1334 mg/l resulted without

any re-distr ibut ion or increase of flows.

For the r u n w i t h no desal ination i t was found that i t was impossible to

achieve a maximum TDS as low as 1334 mg/l everywhere pu re l y b y

rec i rcu la t ing more fresh water. This i s p a r t l y because the TDS of the raw

water was re la t i ve ly h i g h bu t even with better qua l i t y raw water ( i f

ava i lab le ) considerable geochemical deter iorat ion occurred underground, so

125

there were l imi ts to what could be achieved. Anyway, even w i th a

maximum TDS of 1603 mg/l, the cost of c i rcu la t ing water was $6.5 mi l l ion /y

which exceeds the desal ination a l te rna t ive for better water. On the other

hand, i f the TDS l im i t was relaxed to 2000 mg/l, the cost of the

recirculat ion solution became comparable wi th the desal inat ion solution,

namely $2.5 mi 1 I ion/year.

I t should be noted that the costs only include pumping, raw water

purchases and desal ination. Brine disposal i s not costed as the b r ine i s

used to pump to the metal lurgical p lan t where i t i s used. The cost of

maintenance, especial ly replacement of corroded pipes, i s omitted as the

constraints (maximum TDS) were set to eliminate corrosion problems. The

desal ination system cost i s more than jus t i f ied in savings i n replacements

and i t was only a question of whether to use more raw water o r to

desalinate.

TABLE 8.2 New Water Flows and TDS w i th desal ination

Max TDS 1344 mg/l, Cost, $/a = 2.912175E+6

~~

Pipe Node1 Node2 WatcrFlow lncmrrTD8 Cod BpuilibriumTD6 number (Ud (=g/l) (CW) cwn, 1 0 1 23 800 30 797

3 2s 0 0 798 2 1 3 3 6 8 0 0 783 4 6 7 8 0 0 448 6 0 11 -8 0 0 843

7 -8 0 0 448 6 0 7 2 4 36 0 0 1179 8 4 6 61 0 0 1339 9 0 6 18 1800 0 1839

10 6 8 69 400 0 1844 11 8 7 62 -1000 100 448 12 8 9 67 0 0 1342 1 s 9 2 67 0 20 1348 14 7 10 36 0 0 681 16 10 12 40 200 0 799 16 7 11 26 0 0 643 17 11 12 18 200 0 799

10 4 1800 0 681 18 0 19 12 8 60 0 10 1344 20 12 0 8 0 60 0 21 2 0 21 0 16 0 22 3 4 16 0 0 1179

126

One factor omitted a t th is stage, however, i s the requirement for

addi t ional cooling of the recirculated water, i f less raw water i s used.

The raw water i s general ly a t a lower temperature (2lOC) than the water

returned from the mine workings (28OC). The add i t iona l cost of operat ing a

desal ination p lan t underground where access i s l imited, i s also not

proper ly evaluated a t th is stage.

The most economical s i t i ng of the desal inat ion p lan t i s underground as

th is reduces pumping cost to the surface. I t should be a t the point of

highest TDS concentration as th i s resul ts i n minimum p lan t capaci ty per

un i t removed.

The example i l l us t ra tes the use of the program fac i l i ta ted r a p i d

comparison of a l te rna t ive water management systems on a cost and qua l i t y

basis. I n general i t demonstrates that, despite the fact that desal inat ion

costs quoted are often i n excess of raw water costs, there are often

addi t ional factors favour ing desal ination i n indus t r ia l systems, namely:

cleaner water, less corrosion and blocking, less eff luent, greater

conservation of na tu ra l resources, less pumping costs, and lower water

consumption.

TABLE 8.3 Optimum Flows to reduce TDS to 1603 mg/e

Best obtainable without desal ination, Cost, $/a = 6.544681E+6

Pipe Node 1 Node 2 Water Flow Increase TDS Cost Equilibrium TDS number (116) ( W l ) (c/W (mg/l)

1 0 1 176 800 30 800 2 1 3 176 0 0 799 3 3 6 118 0 0 798 4 6 7 118 0 0 1239 6 0 11 -8 0 0 1303 6 0 7 -8 0 0 1233 7 2 4 36 0 0 1106 8 4 6 94 0 0 1216 9 0 6 18 1800 0 1216

10 6 8 112 400 0 1604 11 8 7 113 0 6 1233 12 8 9 212 0 0 1609 13 9 2 212 0 20 1603 14 7 10 78 0 0 1489 16 10 12 194 200 0 1697 16 I 11 146 0 0 1303 17 11 12 138 200 0 1697 18 0 10 116 1800 0 1489 19 12 8 213 0 10 1604 20 12 0 118 0 60 0 21 2 0 176 0 16 0 22 3 4 68 0 0 1106

127

REFERENCES

A b u l n o u r , A.M., Sorour , FA.H., Hammouda, F. and A b d a l Dayem, A.M., 1983. Squeez ing d e s a l t e d w a t e r cos ts b y p r o p e r cho ice o f t h e d e s a l t i n g techno logy a n d w a t e r management. D e s a l i n a t i o n , 44, 189-198.

Baker -Du ly , H.L.G., 1985. O p t i m i z a t i o n o f w a t e r r e t i c u l a t i o n sys tems a t L o r a i n e Go ld M i n e L i m i t e d , Proc. M i n e V e n t i l a t i o n Soc ie ty o f South A f r i c a Conf.

Ho l ton , M.C. and Stephenson, D., 1383. A compu te r model o f c i r c u l a t i n g s e r v i c e w a t e r in South A f r i c a n g o l d mines . Intl. J. M i n e Water, 2 ( 2 )

Loucks , D.P., Reve l le , C.S. and Lynn, W . R . , 1967. L i n e a r p r o g r a m m i n g mode ls f o r w a t e r p o l l u t i o n c o n t r o l . Management Science, 14, 8166-8181.

R i n a l d i , S . , Soncini-Sessa, R., S teh fes t , H. and Tamura , H., 1979. M o d e l l i n g and Con t ro l o f R i v e r Q u a l i t y , McGraw H i l l , New Y o r k .

Smeers, Y . and Tyteca, D., 1981. On the o p t i m a l l o c a t i o n o f was te w a t e r t rea tment p l a n t s , I n : J. Th i sse and J. Z o l l e r (Eds . ) , L o c a t i o n and A n a l y s i s o f P u b l i c F a c i l i t i e s , N o r t h H o l l a n d , Amsterdam.

Smi th , A.A., H i l t o n , E. and Lew is , R.W., 1981. C i v i l E n g i n e e r i n g Systems A n a l y s i s and Des ign .

Stephenson, D., 1983. D i s t r i b u t i o n o f w a t e r in g o l d m ines in S.A., I n t l . M i n e Water J . , 2 ( 2 ) 21-30.

Stephenson, D., 1986. Computer a n a l y s i s j u s t i f i e s d e s a l i n a t i o n . Desa I i n a t ion , 58, 155-167.

Stephenson, D. a n d Corbe t i s , S . , 1984. Economics o f d e s a l i n a t i o n o f was te wa te rs f rom the W i t w a t e r s r a n d , Proc. I n t l . Conf. o n Water Resources and D e s a l i n a t i o n . Johannesburg , South A f r i c a , Water S u p p l y Improvement Assn.

33-42.

128

APPENDIX 8.1

MlNSlM PROGRAM FOR SIMULATING FLOW AND TDS IN CLOSED SYSTEMS

The program i s for s imulat ing flow and TDS changes in water

re t i cu la t ion systems. I t i s based on nodes and l inks , and calculates TDS

concentration a t a l l nodes, and p lo ts i t a t any selected node. Volume

var ia t ions at nodes are permitted although zero volume nodes can also be

specified. The program w i l l also draw a p i c tu re of the system a t any

selected scale o r v iewing angle. The program i s i n BASIC 3.0 for a HP

9816 micro computer.

To account for TDS bui ld-up along a route, one may specify the

increase in TDS along the route in mg/l.

I f desal ination i s done, a negative increment i n TDS i s inserted.

When evaporation increases the TDS concentration a t any node one

specifies a negative flow to that node from another node such as node '0 '

a t the node of o r i g in and zero increase i n TDS along the l i n k route.

The volume a t each node i s set i n i t i a l l y a t a specif ied value. I f th is

value i s zero or negative, a no-volume node i s assumed and inf low must

equal outflow. I t i s therefore not possible to specify flow var ia t ions

du r ing the day from a zero volume node. From other posi t ive volume

nodes, the flow can be specif ied over so many hours a day. Then in order

not to cause extreme volumes, the average flows into and out of a l l nodes

should balance du r ing each day, not the peaks which are specif ied in the

data. i.e. Q.Tin should = 0 over 24 hours.

A maximum of 5 pipes o r l i nks are permitted to each node.

When input t ing data make sure the ' top ' ( I ) node has been defined

before reading i n data on the 'bottom' ( J ) node. I f necessary use a

dummy pipe from ' 0 ' to the node i n question to define i t s co-ordinates.

Node 0 need not be so defined as no pipes from i t are plotted. Pipes to

node 0 are plotted bu t i t i s not wise to have pipes to 0 as they w i l l

redefine the TDS a t node 0, which could otherwise be used as a s ink for

evaporation and a source for f issures which would then increase i n TDS b y

a given f igure.

Costs are calculated i n Dol lars per annum i f pr ices i n c/k l a re input

in data.

Tape or Disc Management

The programme MlNSlM can be copied onto new tapes. Data f i l es

129

cannot. They have to be typed i n l ine by l ine a f te r the fo l lowing i s set

up on the tape.

CREATE "DATMIN", 100,88. This creates a f i l e of 100 records ( fo r 100

pipes) each 88 bytes long permit t ing 11 x 8 byte numbers per record. To

erase a previous data f i l e purge i t and re-create i t . Note a data f i l e

might have to be closed manually a f te r a bomb out. Al ternat ively whi le

executing the program i t pauses and asks whether graphics o r re-write of

data i s required. A new tape could be inserted a t t h i s stage.

MlNSlM L is t of Symbols

A1 A2 B1-9 C c2 E (

F ( ) G ( G1 GO H( H1 I ( J ( K ( I ,MI L ( 1 L1 MO M M1 M2 M3 M5 N NO N1 N2

N3 N4 P ( 1 PO Q ( 91 R( S ( 1 s1 T1 T2 T3 T4

angle of v iewing plane from X axis, degrees angle of viewing plane from 2 axis, degrees dummy input for a l terat ions pr ice c/kP total cost/rands per annum i n i t i a l volume, m3. Use negative or zero value to s ign i fy constant outflow over 24h. Must then balance flows over 24h not a t peaks. new mg/P a t node pol lutant concentration mg/P a t node G counter fo r p lots used in calculat ion of TDS a t nodes head, m not used size of device top node of p ipe bottom of p ipe number of pipes connecting into node (up to 5 permitted) length, m distance to device pipe counter

number of nodes connecting p ipe counter counter for i n i t i a l flow calculat ions pipe no. of device node counter device per p ipe node a t which TDS i s to be plotted 1 = TDS concentration p lo t required 2 = volume a t node, m3 0 = o ld data, 1 = new, 2 = revised 1 = graphic display, 0 = none, 2 = record data and stop input po l lu tan t concentration a t node 2, mg/e into p ipe in i t i a I concentration flow P / s Z Q design flow i n pipe P / s volume m3 S coun'ter for p lots durat ion of simuln. days simulat ion in te rva l , hours drawoff hours/day (1st hours of day) time in te rva ls for drawoff per iod

Pipe

1 30

T5 device type; 1 = f lange, 2 = valve, 3 = tank, 4 = arrow, 5 = square no. i terat ions per day i terat ion counter day counter day counter screen X co-ordinate screen Z co-ordinate X min on screen X max on screen Z min on screen Z max on screen X co-ordinate Y co-ordinate Z co-ordinate

Data Input

The computer asks for the fo l lowing data to be typed in in te rac t ive ly :

L l L2 L3 L4

L5 L6 L7

L B et seq

System name Simulation period, days Time increment, days Period in hours per day du r ing which drawoff occurs. The balance of the time water may flow to r e f i l l reservoirs. I n i t i a l TDS of the en t i re systems, mg/P Node no. a t which a p lo t of TDS versus time i s required Type 0, 1 o r 2 depending on whether the o ld data f i le , a new one or a revis ion of the o ld one i s required Type the fol lowing separated b y commas, wi th one l i ne per pipe or conduit;

Top (upstream) end node no. Bottom (downstream) end node no. x-co-ordinate of bottom node (hor izon ta l l y from a datum) y-co-ordinate of bottom node ( v e r t i c a l l y ) z-co-ordinate of bottom node ( i n to screen) Volume of storage a t bottom node, m’ Design flow in P / s ~ n p u t pol lut ion, mg/e Cost along route, cents per u n i t of flow (ke) Type a row of n ine zeros to end th is da ta Later the program may c a l l for add i t iona l graphics data per pipe : Pipe no. (counted from the top of the l i s t ) Position (distance from top end of p ipe) to a device Device type ( 1 = f lange, 2 = valve, 3 = tank, 4 = arrow, 5 = square) Size, m to draw i t Cost per u n i t size

The data w i l l then be f i l ed for subsequent re-use b y the second program MINOP. Examples of output, graphics and input are given in the char, ter.

131

Program listing

1 I RE-STORE"MINS1M' 10 I MINE WATER RETIC SIMULATION I"M1NSIM" 1 1 GRAPHICS OFF 12 DUMP DEVICE I S 707,EXPANDED 1 4 PRINTER I S 707 2 0 ASSIGN OPath l TO "DATMIN" I CREATE "FILNAM",100,88 ag"DATM1N" 21 D ISP "SYSTEM NAME"i 22 INPUT NS 2 4 PRINT NE 2 5 INTEGER N ,NO ,N1 ,N2 ,N3 ,N4 ,M ,M0 ,M1 ,M2 ,M5 ,S I ,I( 99 ) , J ( 99 ) ,K( 99.5 )

2 7 D IM X ( 9 9 ) ,Y (99 ) .Z( 9 9 ) ,U( 9 9 ) ,W(99) ,P( 9 9 ) , H ( 9 9 ) ,6( 9 9 ) ,S (99 ) 30 DIM E( 9 9 ) ,R( 99) ,P( 99) ,L( 9 9 ) .F( 99) ,C( 99 ) 31 D ISP "DURN OF SIMULATION,DAYS"I 3 2 INPUT 1 1 33 D ISP "TIME 1NCREHENT.HOURS"t 3 4 INPUT 1 2 3 5 D ISP "FLO OVER HOURS/DAY"i 36 INPUT T3 37 DISP " I N I T I A L TDS,mQ/l"( 3 8 INPUT P0 39 DISP "PLOT TDS AT N0DE"i 4 0 INPUT N1 4 2 DEG 4 3 D ISP "OLD OR NEW OR REV D A T A ( 0 / 1 / Z ) " i 45 INPUT N3 61 NZ-1 62 T4-T3/T2 I I T S / d 66 17 -24 /12 70 G(O )=0 80 X(0)-0 90 Y(0j-0 1 0 0 Z(0)-0 110 E ( 0 ) - 0 111 FOR N-1 TO 9 9 112 S ( N ) = 0 113 E ( N ) - 0 114 F ( N ) = 0 115 X ( N ) = 0 116 Y(N)=0 117 Z ( N ) = 0 118 C ( N ) - 0 119 NEXT N 120 c 2 - 0 122 G( 1 )=P0 125 M1=0 130 FOR M=1 TO 9 9 140 I F N3C>1 THEN 190 145 I NEW PIPE DATA 150 DISP "N1 ,N2,X2,YZ,ZZ,U2,l/s,tmg/l,c/"i 1 6 0 INPUT I ( M ) , J (M) , X ( J ( M ) ) ,Y( J ( M ) ) ,Z( J ( M ) ) ,E( J ( M ) ) , R ( M ) ,P(M) ,C( M ) 170 OUTPUT @Path1 .M i I ( M ) , J ( M ) , X ( J ( M ) ) .Y( J ( M ) ) ,Z( J (M) ) ,E( J ( M ) ) , R ( M ) ,P(M) ,C(M) 1 8 0 GOT0 2 10 185 I OLD PIPE DATA 190 ENTER @Path1 , M i I ( M ) , J ( M ) , X ( J ( M ) ) , Y ( J ( M ) ) , Z ( J ( M ) ) ,E( J ( M ) ) ,R(M),P(M),C(M) 2 1 0 I F I ( M ) + J ( M ) = 0 THEN 228 2 1 2 S ( J ( M ) ) = E ( J ( M ) ) 2 1 8 G ( J ( M ) ) - P 0 2 2 0 C2=CZtC( M )rR( M )*315 225 M l = M l + l

132

226 NEXT M 2 2 8 I F N3<2 THEN 256 2 3 0 FOR M0=l TO 99 I REV P I P E DATA 231 DISP "P IPE N o . " i 2 3 2 INPUT M 2 3 3 C2=CZ-C( M ) * R ( M ) * 3 1 5 234 DISP "N1 , N 2 , X 2 , Y 2 , ~ 2 , U 2 , l / s , t m g / l , c / " i 2 3 5 INPUT I ( M ) , J (M) , X ( J ( M ) ) , Y ( J ( M ) ) ,Z( J ( M ) ) ,E( J ( M ) ) , R ( M ) , P ( M ) , C ( M ) 2 4 5 OUTPUT B P a t h l , M i I ( M ) , J ( M ) , X I J ( M ) 1 , Y ( J ( M ) ) ,Z( J ( M ) ) ,E( J ( M ) ) , R ( M ) ,P (M) ,C(M) 246 I F I ( M ) t J ( M ) = 0 THEN 256 ? 4 8 I F MCMl THEN 251 2 4 9 Ml=MI+l 251 S ( J ( M ) )=E( J ( M ) ) 2 5 3 G( J ( M ) )-P0 2 5 4 CZ=C2+C(M)*R(M)*315 2 5 5 NEXT M0 256 FOR M - l TO M 1 2 5 7 L (M)=SQR( (X( J(M))-X(I(M)) ) ^ 2 + ( Y ( J ( M ) ) - Y ( I ( M ) ) ) * 2 + ( 2 ( J(M))-Z( I ( M ) ) ) ^ 2 1 2 5 8 NEXT M 262 PRINT "N1 N2 X2 Y2 22 U2 Q t m g c / " 263 FOR M=1 TO M 1 264 PRINT USING 2651 I ( M ) , J ( M ) , X ( J ( M ) ) , V ( J ( M 1 ) ,Z( J ( M ) ) ,E( J ( M ) ) , R ( M ) . P ( M ) , C ( M ) 265 IMAGE 20,2D ,5D, 4D,5D ,4D ,3D, 4D,3D 267 NEXT M 2 6 8 D ISP "LAYOUT GRAPHICS(0=NO,l=YES,2=RECORD DATA & STOP ) " I

2 6 9 INPUT N4 2 7 0 I F N4<2 THEN 280 271 ASSIGN O P a t h l TO "DATMIN" 272 FOR M=1 TO 99 2 7 4 OUTPUT @Path1 , M i I ( M ) , J (M) ,XC J ( M ) ) , Y ( J ( M ) ) ,Z( J ( M ) ) ,E( J ( M ) ) , R ( M ) ,P(M) ,C(M 1 276 I F I ( M ) t J ( M ) = 0 THEN 2910 2 7 8 NEXT M 286 I F N4-1 THEN 1570 7 0 6 ALPHA OFF 7 0 5 G I N I T 7 0 7 GRAPHICS ON 7 0 8 I F N2>1 THEN 8 5 0 710 WINDOW - . 5 , T I . -200 ,2*G( I )t3000 71 1 C L I P 0 ,T1 ,0 ,2*Gc 1 )+3006 7 2 0 AXES 1 ,100 730 C L I P OFF 7 4 0 MOUE T1-1,10 750 LABEL " O A Y " 7 6 0 FOR T0=1 TO T I

7 8 0 LC\BEL UALO(T0) 7 9 0 NEXT T0 800 FOR G I - 0 TO 2 * G ( l ) + 2 5 0 0 STEP 500 8 1 0 MOUE -.5,G1 8 2 0 LABEL UALS(G1 ) 8 3 0 NEXT 61 8 3 5 MOUE .5,2*G( 1 )+600 8 4 0 LABEL "TDSmg/l NODE"LUALB(N1 ) 8 4 5 60TO 890 8 5 0 WINDOW - .5,T l ,-10,E(NI ) t l U 851 C L I P 0 , T l .-10,E(N1 )+ I0

855 C L I P OFF

770 MOVE TO-.5,-200

854 AXES 1 , l

133

857 MOUE T l -1 ,0 868 LABEL "DAY" 860 FOR T0=l TO T1 862 MOUE T0-.5,-10 864 LABEL UALS(T0 ) 866 NEXT 1 0 868 FOR S l - 0 TO E ( N 1 ) t l O STEP 10 870 MOUE -.5,S1 872 LABEL UCILL(SI ) 874 NEXT S1 876 MOUE B,E(Nl ) + I 0 878 LABEL "UOLn3,NODE"bUALO(Nl) 890 I END OF LABELING 900 FOR N=1 TO M l t l I NODES CONS 905 M2=0 906 FOR M0=l TO 5 907 K ( N ,M0 )-0 908 I (K(N,M0) )=0 909 J(K(N,M0))=0 910 NEXT MO 915 FOR M0=1 TO M 1 1 PIPES 920 I F I (M0)<>N THEN 940 925 MZ-M2tl 927 K ( N ,M2 )-M0 930 GOTO 960 940 I F J(M0)<>N THEN 968 945 M2=M2+1 950 K(N,MZ)=MB 960 I F M2-5 THEN 970 968 NEXT M0 970 NEXT N 975 I F NZ>l THEN 990 980 MOUE 0,G(N1 )

985 GOTO 1090 990 MOUE 0,S(Nl )

1090 FOR T=1 TO T l ! DAYS 1100 FOR T9=1 TO T7 1105 I F T9>T4 THEN 1160 1110 FOR M=1 TO M1 1145 Q ( M ) = R ( M ) 1155 NEXT M 1157 GOTO 1200 1160 I F E ( I ( M ) ) < = 0 THEN 1200 1195 Q(M)-0

1210 FOR N-1 TO M 1 + 1 1220 I F E ( N ) > 0 THEN 1340 1225 Gl=Q 1228 Q1=.001 1230 FOR M0-1 TO 5 I 2 3 5 I F J (K(N,MB))ON THEN 1280 1236 I F G( I ( K ( N , M 0 ) ) )+P(K(N ,M0) )<=0 THEN 1255 1240 G1=( G( I ( K ( N ,M0) ) ) t P ( K ( N , M I ) ) )*Q(K(N .M0) )+G1 1255 01 =Ol + Q ( K ( N ,M0 ) )

1280 NEXT MO 1282 G l = G l / Q l 1285 GOTO 1440 1340 G l m 0 1 S>0 1345 G0=G( N )*S ( N )

1200 TDS a STORAGE ITNS

134

1350 FOR MO-1 TO 5 1360 I F J(K(N,M0))<>N THEN 1390 1370 G1-61+(6( I ( K ( N,M0 ) ) )+P(K( N ,M0 ) ) )*Q( K ( N ,M0 ) )

1375 S(N)=S(N)tQ(K(N,MB) )*T2*3.6 1380 GOT0 1420 1390 I F I (K(N,MB))<>N THEN 1420

1412 S(N)=S(N)-Q(K(N,MQ) )*T2*3.6 1428 NEXT M 0 1430 61-( 61 *T2*3.6+60 ) /S(N) 1440 F(N)=Gl 1472 NEXT N 1474 FOR N-1 TO M l + l 1476 G(N)=F(N) 1488 NEXT N 1492 I F N2>1 THEN 1488 1485 DRAW T- l tTg*T2/24 ,6(N l )

1486 60TO 1490 1488 DRAW T- l tT9*T2/24 ,S(N l )

1490 NEXT T9 1492 NEXT T 1494 DUMP GRAPHICS 8707 1496 GOT0 2920 1570 DISP "XMIN ,XMAX ,ZMIN ,ZMAX ,XANGL ,ZANGL' I

1590 GINIT 1592 GRAPHICS ON 1595 DEG 1600 WINDOW UQ,U9,W0,W9 1601 FOR M=1 TO M 1 ! NODES 1602 U(M)=X(M)*COS(Al ) t Y ( M ) + S I N ( A l ) 1604 W ( M )-Z( M )+COS( A2 )t( Y ( M )tCOS( A1 )-X( M )*SIN( A 1 ) )*SIN( A2 ) 1608 NEXT M 2170 FOR M=1 TO M l ! PIPES 2190 PEN 1 2195 I F I ( M ) - 0 THEN 2225 2196 I F J(M)-0 THEN 2230 2200 MOUE U( I ( M ) ) ,W( I ( M ) ) 2210 DRAW U ( J ( M ) ) , W ( J ( M ) ) 2220 LABEL Uc\L$( J ( M ) ) 2223 GOT0 2230 2225 MOUE U( J ( M ) ) , W ( J ( M ) ) 2228 LABEL UALO( J ( M ) ) 2230 NEXT M 2235 FOR N0=l TO 3 2240 FOR N=1 TO 100 2241 I F N>Ml THEN 2415 2242 I F NO-1 THEN 2256 2243 I F N0=3 THEN 2269 2246 M5-N I ARROWS 2247 I F I ( N ) = 0 THEN 2410 2248 L l = L ( N ) / 2 2249 I F J (N) -0 THEN 2410 2250 T5=4 2251 Hl-L1/10 2254 C0-0 2255 GOTO 2320 2256 M5=N ! TANKS 2258 L l = L ( N ) 2260 1513

i 400 GI=GI-G(N')*Q(K(N ,MQ ) )

1580 INPUT ~ 0 , ~ 9 , ~ 0 , ~ 9 , ~ 1 ,AZ

135

2262 H l = E ( J ( N ) ) / 2 5 2264 C0=0 2266 GOT0 2320 2268 ALPHA ON 2270 DISP "PIPEn ,X ,TYPE ,SIZE .COST/' 1

2280 INPUT M5,Ll ,T5,H1 ,C0 2320 I F M5-0 THEN 2420 2340 X5=X( I ( M 5 ) )+L l /L (M5 ) * ( X ( J(M5 1 ) -X( I(%))) 2350 Y5=Y( I ( M 5 ) )+Ll /L(M5 )*( Y ( J( M5 ) )-Y( I (M5 ) ) ) 2360 Z5=Z( I ( M 5 ) )+L1 /L(M5 )+(Z( J(M5) )-Z( I ( M 5 ) ) ) 2370 US=XS*COS(AI ) tY5*SIN(A1 ) 2380 W5=Z5*COS(A2 )t(YS*CDS(Al )-XS*SSN(Al ) ) * S I N ( h Z ) 2390 ON T5 60TO 2460,2490,2540,2590,2850 2400 I l=FLANGE,Z=UALUE,3=TANK,4=~RROW,S=S~UARE 2410 NEXT N 2415 NEXT N0 2420 MOUE U0,WB 2430 C2=1NT(C2 ) 2440 LABEL " R/s="LUALO(C2) 2445 DUMP GRAPHICS 2450 60T0 700 2460 MOUE US ,W5tHI /2 2470 DRAW US ,W5-H1/2 2480 GOT0 2410 2490 HOVE U5-H1/2,WStH1/2 2500 DRAW U5tHI 12 ,WS-HI / 2 2510 MOUE U5+Hl/2,WS+H1/2 2520 ORAW U5-H1/2,U5-H1/2 2530 GOTO 2410 2540 MOUE U5-HI / 2 ,WStWI 2550 DRAW U5-Hl /2 ,W5 2560 DRAW U S H 1 / 2 ,WE 2570 DRAW U5+Hl/Z,W5+Hl 2580 60TO 2410 2590 I F U ( J ( M S ) ) < > U ( I ( M S ) ) THEN 2601 2591 IF W ( J ( f l S ) ) > W ( I ( M 5 ) ) THEN 2594 2592 U8=270 2593 GOTO 2608 2594 U8=90 2595 GOT0 2608 2601 2602 I F U8>=0 THEN 2606 2603 I F W ( J(M5) ) < W ( I(M5) 1 THEN 2608 2604 GOTO 2607 2606 I F U(J(MS))>W(I (MS)) THEN 2608 2607 U8=U8t180 2608 UG=U5-Hl*COS(U8-45) 2610 W6=W5-Ht*SIN(U8-45) 2620 U7-U5-Hl*COS(U8t45) 2630 W7-W5-Hl*SIN(UEt45) 2810 MOUE U6,W6 2820 DRAW U5,W5 2830 DRAW U7,W7 2840 6QTO 2410 2850 MOUE US-H1/2,WStHl 2860 DRAW U5-H1/2 ,W5

2880 DRAW U5tH1/2,W5tHI 2890 DRAW U5-H1/2,W5tHl 2900 60TO 2410 2920 END

UE=ATN( (U( J( M 5 ) )-W( I( M5 ) ) ) / (U( J ( M 5 ) )-U( I( M5 ) ) ) )

2a70 DRAW u 5 + ~ 1 12 ,w5

136

APPENDIX 8.2 MINOP program fo r opt imizing d i s t r i bu t i on

MlNOP List of Symbols

pr ice c/ke cost total cost dummy TDS max. TDS desired, o r G of node w i th m a x . T D S increment in TDS max. increment i n TDS total TDS - mg/s in to node total flow in to node TDS G ( I ) - H ( I ) max. TDS top node bottom node pipe no. connecting to node (up to 5 permitted) number of p ipe connecting number loops best loop no. branches in loop posi t ive loop pipes out node loop counter number loops and begin number loop number connecting pipes to node p ipe number p ipe counter number of nodes

number of p ipes in loop number of connecting pipes from node dummy pipes out node number of p ipes in loop reduction i n no. pipes in loop, or, p ipe to node w i th max. TDS pipe number i n loop begin pipe for loops name node counter no. nodes pipe no. input TDS, mg/e

dQ/dC dQ co-ord. not used in MINOP

flow e/s

0 ,

I 1

137

Notes on program

The program i s in BASIC for an HP 9816 series 200 micro computer. The data fi le i s obtained from the MlNSlM program in appendix 8.1.

138

100 110 120 130 140 150 160 170 180 190 2 0 0 216 2 2 0 2 3 0 2 4 0 250 260 2 70 280 2 9 0 300 310 320 330 3 4 0 3 5 0 3 6 0 370 3 8 0 390 400 4 1 0 420 430 440 450 460 470 480 4 90 500 51 0 5 2 0 530 5 4 0 5 5 0 5 6 0 5 7 0 580

Program MINOP listing 101 RE-STORE"MIN0P" 20 I "MINOP " OPTIMZS FLOS I N NETWORK SUBJECT TO TOS L I M I T S 3a PRINTER IS 707 4 0 A S S I G N @Path1 TO "DATMIN" 5 0 I D I S P "SYSTEM NAME"! 60!INPUT NO 70 IPRINT NC 8 0

S.M6(99),M7,M8,M9,N,Nl

DIM Q( 9 9 ) ,G(99) ,P( 9 9 ) ,C( 99 ) , L ( 9 9 ) ,H( 9 9 ) 90 INTEGER I ( 9 9 ) , J ( 9 9 ) , K ( 9 9 , 5 ) ,K1 (99 ,9 ) ,L l ,L2 ,L3,M,M0,Ml ,M2(50 ,50 ) ,M3(99 ) .M4 ,M

M I 1 0 1 NO.PIPES G ( 0 ) = . 1 N l = l DISP "MAX TDS DES1RED"i INPUT 6 0 FOR M=l TO 99

ENTER @ P a t h 1 ,M; I ( M ) , J (M) ,X ,Y ,Z ,E ,Q( M ) ,P( M ) ,C( M ) H( .I( M ) )=G0 I F I ( M ) t J ( M ) = 0 THEN 2 5 0 I F I(M);=Nl THEN 2 1 0 N l = I ( M ) If J (M)<=N l THEN 2 3 0 N1 =.Jc M ) M I - M l t l

NEXT M H(0 ) -100000 FOR M0=l T O M 1

DISP "ANY CHANGES? PIPENo,TOPn,BOTn,FLOI/s,POLmg/l , c /h1 (O's=none ) " I

INPUT M , I ( M ) , J ( M ) , Q C M 1 ,P( M ) ,C( M ) I F I ( M ) + J ( M ) = 0 THEN 3 2 0 I F M > M l THEN M l = M l t l

NEXT MO FOR N-0 TO N l I NODES

G( N )=G0 M3( N )=0 L( N )=0 FOR M=l T O M l I P I P E S FROM NODE

I F I ( M ) < > N THEN 400 M3( N )-M3( N )t 1 K l ( N ,M3( N ) )=M

NEXT M FOR M0=1 TO M1I PIPES TO NODE

I F J(MQ)<>N THEN 450 L( N )=L( N )+ 1 K ( N ~ L ( N ) )=NO

NEXT M0 NEXT N G( 0 1-0 L1=01LOOFS FOR.M9=1 TO M1 IBEGINPIPE FOR LOOPS

L 0 - L l t l ITRY LOOP L8=0 MG(L0)= l INO.P IPES I N LOOP M2(L0,1 )=M91PIFES I N LOOPI LE=LBIPOS LOOP FOR L3=1 TO MlIBRANCH ROUTINE

L8=0 L4=M3( J ( M 2 ( L0 ,M6( L0 ) 1 ) ) FOR M5=1 TO L 4 IPIPES OUT NODE

139

5 90 600 610 620 630 640 650 660 670 680 690 700 71 0 720 730 740 750 760 770 780 7 90 800 81 0 820 830 840 850 860 870 880 890 900 91 0 920 930 940 950 960 970 980 990 1000 1010 1020

I F M5-1 THEN 670 L6=LG+lIANOTHER POS LOOP FROM BRANCH M6(L6 )=M6(L0) FOR M7=1 TO M6(L6)-1

NEXT M7 M2( L6 ,M6( L6 ) )=K 1 ( J ( M2( L 0 ,M6( LO )- 1 ) ) ,M5 )

GOTO 690 M6(L0)=M6(L0)+1 IN0 PIPES I N LOOP M2( L 0 ,M6( L 0 ) )=K 1 ( J ( M 9 ( L0 ,M6( L0 1- 1 ) ) ,M5 ) I NEXT PIPE

MZ(L6 ,M7)42(LB,M7 )ICOPIES PIPES I N PREU LOOP

NEXT M51 CHEK LOOP CLOSURE FOR M5-2 TO M6(L0)

FOR M7=1 TO M5 IF I( MZ(L0 ,M7) ) < > J ( M Z ( L 0 ,M5) ) THEN 800 L1 =L1 t 1 M6(L1 )4lS+I-M71SHUFFLE UP PIPES FOR M8=l TO M 6 ( L l )

NEXT ME L6=1 GOTO 840

M Z ( L I ,M8 )-M2( L 0 ,M8+M7- 1 )

NEXT M7 NEXT M5 GOTO 1000

I F L l ; = l THEN 1000 FOR LZ=1 TO L1-1

I CHEK DUP LOOP

M=O FOR M7=1 TO M6(Ll )

M=Mt 1 ME-1 I F MZiL l ,M)OM2(L2,M8) THEN 980 M8=M6+ 1 M=Mt 1 IF M:=MG(LI) THEN 950 M= 1 I F M8:=M6(L7) THEN 9G8

GOTO 1000 Ll=Ll- l lREMOUE OUP LOOF

NEXT M7 NEXT L2 I F L8<-0 THEN 1030 I F L6 ‘=L0 THEN 1040 L0=L0+ 1

NEXl L3 1030 1040 NEXT M9 I 0 5 0 FOR L:=l T O L I 1060 FOR M=I TO M6(L21 IQ701PRINT L:,MZ(L2,W) 1080 NEXT M 1090 NEXT L2 1100 GOSU8 1120 1110 GOTO 1300

1130 G5-0

1150 G l = . l 1160 G’L=.I 1170 FOR O=l TO L ( N ) 1180

1120 FOR L4=1 TO Mi

1140 FOR N=i TO N l

Gl=Gl t Q ( K ( N,O ) * ( G : I ( k‘: N , O ) ) ) t F ( K ( N ,O ) ) )

140

1190 62=62tQ(K(N,O)) 1200 NEXT D 1210 63=6(N) 1220 6( N )=GI 162 1230 1240 I F 64<65 THEN 1260 1250 6 5 4 4 1260 NEXT N 1270 I F 65<.00I THEN 12901 MAX FC 1280 NEXT L 4 1290 RETURN 1300 FOR M=1 TO N1 IITNS 1310 RI=BIDQ/DC 1320 R3=0IDQ 1330 FOR L 2 = l TO Ll lEEST LOOP 1340 C 1 = 0 1350 H1=0 1360 M0=6 1370 FOR M8=1 TO MSfL2 ) 1380 1390 I F Q(M2(LZ3M8))<0 THEN 1540 INEXT LOOP 1400 M0=M8 14 10 1420 I F J(M2(L2,M8))=0 THEN 1470 !NEXT PIPE 1430 I F G( J ( M 2 ( L2 ,M8 ) ) )-H( J( M 2 ( L Z ,M8 ) ) )<=HI THEN 1470 1440 1450 1460 60=G(J(MZ(LZ,M8))) 1470 NEXT ME 1480 I F H l i = l THEN 1540 1490 GOSUE 1120 1500 I F (GQ-G(J(M7) ) ) /C I~ '=R1 THEN 1540 1510 Rl=(G0-G( J(M7) ) ) / C l 1520 1536 L3=L2 1540 FOR M8=1 TO MQ 1550 1560 NEXT M8 1570 NEXT L t 158@ FOR M7=1 TO M6(L3i 1590 Q ( M ~ ~ L ~ , M ~ ) ) I Q ( M ~ ( L ~ , M ~ ) ) t R 3 1600 NEXT M7 1610 NEXT M 1620 C2=0 1630 PRINT "Pn N l N2 1 /s tTDSng1 c / L 1 TDS2' 1640 FOR M-1 TO M1 1650 PRINT USING 1660;M,I(M) , J ( M ) , Q ( M ) ,P(M) , C ( M ) ,G( J ( M ) ) 1660 IMAGE 2D ,4D ,4D ,4D ,6D ,50 ,SO 1670 C2=C2tC(f l)*Q(M)*315 1680 NEXT M 1690 C2=INT(C2 ) 1706 PRINT "COST ,R/a=" i C Z 1710 ASSIGN 0Pathl TO 1720 EN0

64=AES( 6( N )-63 )/G( N )

C1 =C1 tC(MZ(L2 ,M8 ) )

Q( M2( L2 ,M8 ) )=Q( M2( L2 ,M8 ) ) t 1

HI =G( J( M 2 ( L2 ,M8 ) 1 )-H( J ( M2( L2 ,M8 ) ) ) M7=MZ(L2,MB)IPIPE TO NODEWITH MAX TDS

R3=(G0-H( J(M7) ) ) / t G Q - G ( J (M7) ) )

Q( MZ(L2 ,ME) )=Q(MZ(L? ,Ma) ) - 1

141

CHAPTER 9

INTEGER PROGRAMM I NG PLANN I NG OF TREATED WASTEWATER CONVEYANCE FOR

ARTIFICIAL RECHARGE OF AN AQUIFER

INTRODUCTION

The internat ional growth i n water demand over the last few decades

has been persistently high. This ra te of growth i s l i ke l y to continue as a

large proport ion of the populat ion i s increasing r a p i d l y in standard of

l i v i ng . The a v a i l a b i l i t y of new sources of water has tended to l a g behind

demand. Even i f new sources were ava i lab le the cost of p rov id ing to meet

any possible drought extreme i n developing areas could be high, on

account of the un re l i ab i l i t y of r i v e r flow. Surface water can on ly be used

to i t s fu l lest extent i f a l te rna t ive sources are ava i lab le to meet essential

demands i n times of drought. For th is reason the wor ld i s now looking to

groundwater and wastewater to meet short fa l IS.

A scheme is investigated here to supply from groundwater on ly a t the

ra te a t which i t can be na tu ra l l y replenished. Separate studies are being

conducted on te r t ia ry treatment of wastewater bu t the idea of using the

wastewater to a r t i f i c i a l l y recharge groundwater w i th wastewater i s only

now receiv ing consideration. Such research i s a long term project and

cannot be expected to re1 ieve current droughts, which however precipi tated

research into al ternat ive sources of water.

I t i s proposed to use groundwater i n conjunction w i th surface water

resources i n such a way that deficiencies in surface water can be

supplemented b y groundwater, resu l t ing i n a h igher overal l a v a i l a b i l i t y .

Surface water resources can then be u t i l i zed to a greater degree since

groundwater reserves can be drawn on in times of short fa l ls in surface

r i ve rs (Pal ing, 1984). The ra te of recharge w i l l also be re la t i ve l y slow

owing to l imi ted sui table wastewater being ava i lab le , the poss ib i l i t y of

na tura l pur i f i ca t ion and the l imi ted permeabi l i ty of the soi l . The

hydraul ics of the recharge process should be investigated w i th s i te tests.

The case study analyzed i s the Witwatersrand area, a h igh growth ra te

conurbation based o r ig ina l l y on mining. Groundwater constitutes a t present

only one percent of the average da i l y supply to the Rand Water Board of

2400 Ml/d to the Witwatersrand area. Pr iva te ly owned boreholes for

farming and gardening purposes are however in common use as a

I

4

Fig. 9.1 Dolomite deposits in the Witwatersrand area

143

supplement to the formal supply. Fol lowing a series of low r a i n f a l l years

i n the ear ly 1980's water restr ict ions were introduced, several emergency

programmes in i t iated, and the study of ava i lab le groundwater sources

intensif ied . The main source of groundwater i s i n weathered zones, cav i t ies and

fissures of dolomite deposits, outcrops of which occur i n a wide c i rc le

around Johannesburg (F igure 9.1 1. These deposits, approximately one

kilometer thick, d ip gently away from the centre at a slope of some ten

degrees. To the South dolomite over lays the quartz i tes and small pebble

conglomerates of Black Reef Series, the Ventersdorp and W i twatersrand

geological System, the la t te r renowned for i t s aur i ferous reefs, which were

a t one time extensively mined on the Witwatersrand. The shales and

quartzites of the Pretor ia Series over ly these series, and vert ical

intersections by syenite dykes create a number of v i r t u a l l y independent

groundwater compartments. Since some of the resu l t ing compartments have

been sanctioned for dewatering in order to al low safer mining, the

groundwater levels i n these compartments have dropped considerably. Other

compartments have not been dewatered because of the danger of sinkholes

forming on the surface. The outcrops of dolomite a re general ly covered by

a l l u v i a l deposits of va ry ing depth.

Published studies on a r t i f i c i a l recharge by i n f i l t r a t i on w i t h p a r t i a l l y

treated wastewater are almost ent i re ly conducted in pr imary aqui fers of

considerable depth, under la in by an impermeable layer. Important aspects

of the performance of these schemes are the permanence of the i n f i l t r a t i on

ra te and the reduction i n contaminants by bacteriological processes and b y

adhesion to soi I part icles. Successful projects are reported from the U.S.A.

(Bouwer et a l . , 1980), Israel ( lde lov i tch and Michai l , 1984) and Austral ia

(Mathew et a l . , 1982). Careful selection of the i n f i l t r a t i on s i te and

extensive testing of the p u r i f y i n g capacit ies of the combined pr imary and

secondary aqui fer i n the Witwatersrand area w i l l have to precede a

decision on the qua l i t y of the i n f i l t r a t i on water.

An act ively operated groundwater reservoir wi l I create a f luc tua t ing

groundwater table and increased groundwater flow velocities. Therefore an

increased ac t i v i t y of solut ion and erosion processes i n the dolomite may be

expected, in some cases followed by the occurence of sinkholes and

subsidence. Bui l t -up areas located over some of the dolomitic areas leave

certain compartments unsui table for the envisaged scheme. Simulation of

groundwater movements and monitoring of the na tu ra l freshwater/wastewater

interfaces w i l l have to be done.

Ar t i f i c ia l recharge not only increases the ava i lab le amount of

F i g . 9.2 Sewage works and recharge si tes

145

groundwater, i t also enables, by i t s more stable supply, to develop

schemes for groundwater use in conjunction w i th surface waters. A

s impl ist ic model (Pal ing, 1985) indicates that by conjunctive use of surface

and groundwater the minimum guaranteed d ra f t can be increased b y 10% of

the total supply compared w i th the s i tuat ion in which each source supplies

ind iv idua l l y .

Sewage treatment in the Witwatersrand are i s s t i l l mainly organized on

a municipal basis. Johannebsurg operates three works South of the

watershed which div ides the town in two d is t inc t areas as f a r as

wastewater collection i s concerned, w i th a combined eff luent output of

approximately 500 Ml/d (megali tres per day) . Sewage from surrounding

municipal i t ies i s processed i n works w i th capacit ies as small as 6 Ml/d.

East of Johannesburg a potential a r t i f i c i a l recharge scheme could involve

the works located in the municipal i t ies of Germiston, Boksburg, Benoni,

Brakpan and Springs (F igure 9.2 and Table 9.1).

Table 9.1 Sewage works and Recharge site.

No. Name Discharge (Ml/d) Elevation (m)

average 1984

1

2

3

4

5

6

7

8

9

10

Davey ton

Mc Comb

Jan Smuts

Rynf ie ld

Ancor

Benon i

Mapleton

Dekema

Rondebul t

Vlakp laas

8

6

1 1

1 1

30

15

-- 56

39

44

1600

1580

1610

1620

1580

1640

1580

1530

1560

1520

I t has been histor ical pract ice to discharge the eff luent in streams

that form t r ibu tar ies of the' Vaal River upstream of the intake of the

local Rand Water Board. Indirect reuse in th i s form has been pract ised i n

the Witwatersrand area since 1923 and takes place af ter al leged self

pur i f i ca t ion in streams and reedbeds, and af ter d i l u t i on wi th fresh r i v e r

water. Intensive monitoring of the streams by the Rand Water Board in

146

recent years has indicated a considerable contamination of these streams,

mainly b y indus t r ia l discharge and leaching of the numerous goldmine

dumps.

Dolomite compartments sui table fo r a r t i f i c i a l recharge and abstract ion

are limited. Dewatered compartments in the gold min ing area and

compartments w i th bu i l t -up areas on top have been mentioned. Another

impediment could be the percolat ion from contaminated streams.

Eff luent from establ ished sewage water works would have to be

transported to the dolomite compartment by pipel ine. Canals might have

been a cheaper al ternat ive, bu t closed pipes are prefered for san i ta ry

reasons. For sewage works i n close prox imi ty to each other in comparison

w i th the distance to any seepage area combined conduits would probably

be most economic whereas other sources may jus t i f y seperate conduits. The

problem of minimizing total conveyance cost could be considered as a

network to be optimized. In the Witwatersrand si tuat ion c lear ly d is t inc t

clusters are discernable, which enabled the complexity of the network

model to be reduced to a minimum. Although relocation of ex is t ing sewage

treatment works cannot be jus t i f ied , fu tu re new works may be more

economically si ted over the seepage areas. This would use the advantage

of economy of scale, bu t may increase conveyance cost due to the greater

peak to average flow r a t i o for raw sewage.

Integer network programming has found several appl icat ions in the

f i e ld of sewage conveyance. Wanielista and Bauer (1972) appl ied i t to the

central izat ion of wastewater treatment fac i l i t i es i n the Econ River bas in

Flor ida, Leighton and Shoemaker (1984) used i t i n a s imi la r fashion fo r

the regional izat ion of wastewater col lect ion and treatment systems in Long

Is land, New York.

COST ANALYSIS

Pipe diameters were based on maximum permissible flow velocit ies decided

fo r p rac t ica l reasons. The next la rger commercially ava i lab le p ipe size

was selected in each case. Based on a cost per metre in f igure 9.3 and

w i th the length of the p ipe l ine section known, the total costs fo r supply

and construction may be calculated.

For purposes of est imating pumping heads, the f r i c t i on head was

calculated using the Darcy equation. The costs fo r pumps was estimated

w i th the relat ionships i n Table 9.2. The equations were based on data for

the range Q = 10 - 50 Ml/d. For intermediate heads the costs a re

interpolated.

147

c)

w

0

V

-0

m

u

.r .

0

0

N

0 0

w

0

0

Ln

0 0

OD

0 0

0

0 0

N

- c

0

E-

D 0 0

N

0 0

E

0

D

E

0 0

z D

D

N

D

D

z D

0

m

0 0

W

0 0

0

0 0

N

148

TABLE 9.2 Pump costs

Head (m 1

Cost ( Q i n Ml/d) ( $ 1

20 207.5 * Q + 4625 40 217.5 * Q + 4725

60 230.0 * a + 5800

The costs fo r the pump motors were based on the shaft power,

increased by a 20% safety margin (see Table 9.3).

TABLE 9.3 Motor costs

Motor Power Voltage costs

(kW) ( V ) ($/kW

0 - 250 400 60

250 - 1900 1000 - 3300 70

1900 - 6570 6600 - 11000 80

The number of instal led pump sets depends on the flow to be handled

and the cont inui ty of th is flow volume. For the present calculat ions one

basic and one back-up set of pumps was he ld to be suff ic ient f o r f lows

under 50 Ml/d and two basic and one back-up sets fo r b igger flows.

F ina l l y the costs for pipel ine, pumps and motors are added up. Th is

procedure i s repeated for each of the 113 var iab les per c luster.

149

MATHEMAT I CAL FORMULAT ION

I n the papers by Wanielista and Bauer (1972) and Leighton and

Shoemaker (1984) regional wastewater conveyance to central ized treatment

p lan ts i s dealt wi th i n one comprehensive computer model. For the

W i twatersrand area eff luent conveyance to recharge sites can successfully

be broken down i n a number of subsystems. A network of the

Witwatersrand was developed that could incorporate a l l the subsystems. In

pa r t i cu la r the most complex subsystem comprises the s i x treatment works i n

the Benoni, Brakpan and Springs municipal i t ies (node 1 to 6 ) and a

proposed in f i l t r a t i on area at Mapleton (node 7) . The network wi th i t s

possible flow direct ions i s presented in f i gu re 9.4.

I t i s the subsystems which could possibly supply the Mapleton aqui fer

which are considered here. At design stage i t i s unclear whether eff luent

should be conveyed from node 3 to node 5 or the other way round in order

to minimize the costs. Both options are therefore left for possible selection

in the optimization procedure. The same considerations apply to the link

between nodes 3 and 6. The introduction of two possible flow direct ions

excludes the appl icat ion of a dynamic programming approach as used by

Smith et a l . , (1983). On the other hand integer programming can be

appl ied to solve th is problem i n the fo l lowing way.

Rynfield 0--------- - - - - - - - - - O D a v e y t o n

,'; \, -0' i

Maple1 :on

MC Comb

Fig. 9.4 Network wi th possible flow direct ions

150

Each p lan t has a certain design discharge. Every p ipe l ine section

o r ig ina t i ng from a p lan t must a t least be able to convey this discharge.

As most l i nks are supposed to convey ef f luent from other p lan ts as wel l , a

series of flow rates can be defined as a resul t of d i f ferent combinations of

discharge volumes. By representing each flow ra te in each p ipe l ine section

b y a seperate integer decision var iab le the network optimization can then

be formulated as an integer programming problem. The number of var iab les

depends on the number of network nodes as well as on the number of l i n k s

w i th the i r specific direct ion. In the present example the network can b e

described by 113 var iables.

The flow ra te dictates under certain condit ions mentioned below the

required pipe diameter and pump capacit ies. Subsequently the costs

connected with these requirements can be determined. The objective

function can be expressed as the sum of the products of the integer

var iab les and the related cost factors, o r

Objective function = CCi * X i

The cost factors C i are calculated for each flow volume and each

pipel ine section as indicated i n the cost analysis. A separate program was

developed to calculate thse cost factors, based on the input of ( 1 ) the

maximum flow volume from each node, ( 2 ) the elevation of each node, ( 3 )

the length of each l i nk , ( 4 ) the maximum permissible flow velocity and ( 5 )

the effect ive p ipe roughness. Provision was made to el iminate certain l i nks

o r supply nodes i n order to maintain the f l e x i b i i t y required to adjust the

network for other c lusters of treatment plants. The cost calculat ion

program was extended i n such a way that the format of the output

program makes i t immediately sui table for submission fo r optimization,

using the I .B.M. mixed integer programming package MPSX-370/MIP.

The constraint mat r i x i s based on two simple p r inc ip les which exploi t

the essential feature of decision var iables:

a ) Y - C X i 5 zero

This expression i s used to describe the relat ionship between di f ferent

var iables. I f Y = 1 a t least one X. must be present. Interact ions of t h i s

type are mainly of a progressive nature, but some regressive steps had to

be included e.g. i f l ink 4-6 conveys Q1 + Q4, the l i nks 2-3, 2-5 and 2-7

may convey only Q2.

b) L X. = zero o r .un i t y .

For the r i g h t hand side to equal one, on ly one va r iab le assumes the

value one. I f the r i g h t hand side i s set a t zero, a l l var iab les must be

zero. I f a l l possible flows from a pa r t i cu la r node are grouped together

and then set a t one, only one flow w i l l be selected.

151

The combination of these two expressions guarantee unique paths

through the network and together wi th the objective function the pa th w i th

the lowest costs w i l l be found.

RESULTS

Optimal eff luent network configurations were computed for a potent ia l

a r t i f i c i a l recharge scheme car r ied around Mapleton. A visual inspection

appeared to indicate the minimum total p ipe length was as estimated in

f igure 9.5a. l is ing the flow volumes in table 9.1 and a maximum flow

velocity of 2.2 m/s the total construction costs for th is selected

configuration would be $13.4 mi l l ion. Under the same condit ions the

Optimized configuration shown in f igure 9.5b would cost $11.2 mi l l ion .

The relat ion between flow ra te and pipe diameter i s Q = v ( ~ / 4 ) D 2 .

Since the pipe diameter i s thus inversely proport ional to the square root

of the flow velocity, a reduction of the maximum velocity to 1.2 m/s

resulted i n a minimum construction cost of $17.5 m i l l i on ( f i gu re 9 . 5 ~ ) .

The or ig ina l problem could be expanded s l i gh t l y by the inclusion of

sources 8 , 9 and 10. Figure 9.6a shows the si tuat ion in which two clusters

supply i nd i v idua l l y to one recharge site. Each cluster i s optimized

separately, resu l t ing in a cost of $11.2 m i l l i on for the northern c luster as

indicated previously, and $5.1 mi l l ion for the western cluster, b r i ng ing

the total to $16.3 mi l l ion.

Fig. 9.5 Network optimization. Node 1 to 6 represent sewage works and node 7 the i n f i l t r a t i on site. See also Table 9.1 and f igure 9.1

152

Fig. 9.6 Linkage of networks. Node 1 to 6 and node 8 to 10 represent sewage works and node 7 the i n f i l t r a t i o n site. See also Table 9.1 and f i gu re 9.1

An a l te rna t ive arrangement which may save cost i s to supply eff luent

from the northern c luster v i a node 9 (see f igure 9.6b). T h e change in

input data for the northern c luster involves f i v e new values for the length

of the l i nks between node 2, 3, 4, 5 , 6 on the one hand and node 9 on

the other, resu l t ing i n a cost of $ 1 1 . 1 mil l ion. The fact that the

conf igurat ion of the optimal network remains the same as i n f igure 9.6a i s

coincidental.

The cost optimization of the western c luster i s performed a f te r

increasing the flow v i a node 9, resu l t ing i n an optimized cost of $6.1

mi l l ion for th is cluster. The total cost thus amounts to $17.2 mi l l ion .

Hence the lay-out of the sewage works i n th is case i s such that a l inkage

of the two clusters does not resul t i n any fu r ther cost reduction.

+' nodes 1 , 2

nodes 1 , 2 , 4 , 5 , 6 - supply points node 7 - demand point node 3 - locat ion of booster station node 3' - alternative s i t e for

booster stati a,

Fig . 9.7 Optimum location fo r a booster stat ion

153

SUMMARY AND CONCLUS IONS

Integer programming proved to be a useful tool fo r the evaluat ion of a

least-cost configuration of a pipel ine network. The support program was

designed to sui t a network wi th seven nodes and ra ther complex

interrelat ions. I t takes approximately ha l f a second on an I.B.M. 3083

computer to calculate the cost factors and to compile a program sui table

fo r integer optimization. For a cluster of s ix supply nodes and one demand

node optimization resul ts were obtained i n less than 30 seconds using the

I .B.M. mixed integer programming package MPSX-37O/MIP. Complicated

cases can be handled b y subd iv id ing a system into subsystems and

optimizing each cluster i nd i v idua l l y .

The f l ex ib i l i t y of the program enables one to introduce changes w i th

minimal effort and to compare the resu l t ing alternatives.

This feature may be i l lus t ra ted b y the fo l loh ing example of an

economic optimization of the location of a booster stat ion ( f i gu re 9.7).

By vary ing the distance between node 3 and the other f i xed nodes a

sequential search w i l l resul t i n the optimum solution. The "range" f a c i l i t y

of MPSX-370/MIP can for each run g ive a sens i t i v i t y analysis and thus

prevent the search from becoming a random process. The general

app l i cab i l i t y i s accompanied by the disadvantage that no provis ions are

made for introducing the costs for obstacles l i k e roads and r i vers .

Computer analysis of the most economic p ipe l ine conf igurat ion fo r

sewage eff luent conveyance to an a r t i f i c i a l groundwater recharge si te

resulted i n a cost reduction of 16% over an optimum solution based on

visual inspection. A lower flow velocity increased the total construction

costs as the influence of an increased pipe diameter strongly outweighs the

reduced cost i n pumping equipment. Depending on the lay-out of the

sewage works a fur ther reduction in the combined costs could possibly be

at ta ined by integrat ion of ind iv idua l clusters.

REFERENCES

Bouwer, H., Rice, R.C., Lance., J.C., and Gilbert, R.G., 1980. Rapid- inf i l t rat ion Research a t Flushing Meadows Project, Arizona. J. Water Polut. Control Fed., Vol. 52, No. 10, p 2457.

Idelovitch, E. and Michai l , M., 1984. Soil-aquifer Treatment - A New Approach to an O l d Method of Wastewater Reuse. J. Water Pol lut. Control Fed., V o l . 56, No. 8, p 936.

Leighton, J.P. and Shoemaker, C.A., 1984. An Integer Programming Analysis of the Regionalization of Large Wastewater Treatment and Collection Systems. Water Resources Research, Vol. 20, No. 6, p 671.

154

Mathew, K., Newman, P.W.G. and Ho, G.E., 1982. Groundwater Recharge w i t h Secondary Sewage E f f l uen t . A u s t r a l i a n Water Resources Counci l , Technical Paper No. 71, Canberra.

P a l i n g , W.A.J., 1984. Opt im iza t i on of Conjunct ive Use of Groundwater and Sur face Water Resources in the Vaal Basin. Proceedings o f the H a r a r e Symposium, IAHS Pub l . No. 144.

P a l i n g , W.A.J., 1985. Economic Opt imizat ion of A l te rna te Water Resources for the Wi twatersrand. Water Systems Research Programme, U n i v e r s i t y o f the Wi twatersrand, Report No. 4/1985.

P a l i n g , W.A.J. and Stephenson, D., 1985. I n tege r Programming of Treated Wastewater Conveyance f o r A r t i f i c i a l Recharge o f a n Aqu i fe r . J. I n t . SOC. Ecol. Mode l l i ng ( 7 ) .

Smith, A.A., Hinton, E. a n d Lewis, R.W., 1983. C i v i l Eng ineer ing Systems, Ana lys i s and Design, John Wiley G. Sons.

Wanie l is ta , M.P. and Bauer, C.S., 1972. C e n t r a l i z a t i o n of Waste Treatment Fac i l i t i es , J. Water Po l l u t . Control Fed., V o l . 44, No. 12, p 2229.

155

CHAPTER 10

OPTIMAL PLANNING OF REGIONAL WASTEWATER TREATMENT

I NTRODUCT ION

The Witwatersrand o r 'White Waters Ridge' i s an elevated r i dge in the

heart of South Afr ica formed b y the gold-bearing quartz i te rock s t ra ta

emerging above the surface of the surrounding country. I t i s where gold

was o r ig ina l l y discovered in 1886 and this sparked the growth of

secondary industry and te r t i a ry commercial development. There are now

about four mi l l ion inhabi tants of the area. The present water consumption

which i s supplied by the Rand Water Board, averages 1700 Ml/day and i s

increasing at a rate of 6 per cent a year.

The Witwatersrand runs i n an east-west direct ion and forms a na tu ra l

watershed (F ig . 10.1). I t i s the source of the number of streams and

o r ig ina l l y there were many spr ings along the r idge, hence the name 'White

Waters Ridge'. The water supply to Johannesburg was o r i g i n a l l y pumped

from the ground as the water resources of the area were not p len t i f u l (on

account of the fact that the elevated r idge was an o r ig in of streams

ra ther than a r i v e r basin). The streams to the nor th form t r ibu tar ies of

the Limpopo River, an ephemeral r i v e r which flows eastwards to the Ind ian

Ocean. The streams to the south flow in to the Vaal River, a t r i bu ta ry of

the Orange River, which flows westwards to the At lan t ic Ocean.

I t i s from the Vaal River that the Witwatersrand draws most of i t s

water. The Vaal Barrage, 50 km south of the Witwatersrand, was

constructed in 1923, followed by the Vaal Dam in 1933. The combined

re l iab le sustained y ie ld of these sources i s 3300 Ml/day and the Rand

Water Board has r i gh ts to 2400 Ml/day. Some water must also be passed on

to downstream users. The Vaal, o r 'murky ' r i v e r as i t i s t ranslated i s a

f lashy r i v e r and carr ies much s i l t (170 mg/l on the average) beyond the

Vaal Dam. O n the other hand i t has re la t i ve ly l i t t l e dissolved sal ts

(average 100 mg/ l ) o r i g ina t i ng mainly from sal ts leached from farmlands.

The Vaal River i s now being supplemented b y water diverted from the

re la t i ve ly untapped Tugela River, 300 km away. Plans were considered for

tapping the Orange River a t i t s source and d i ve r t i ng these waters to the

Vaal basin. These diversion schemes are expensive.

O n the other hand wastewater treatment technology i s now advancing

rap id l y and the cost of treatment i s appear ing more at t ract ive. Since only

about 50 per cent of water i s used consumptively on the Witwatersrand,

157

there is scope for water reclamation and recycl ing. In fact t h i s i s now

happening indirect ly. Most water returned to the sewers on the

Witwatersrand f inds i t s way, af ter treatment, to streams which discharge

to the Vaal Barrage. The Rand Water Board (1977) pumps 1000 Ml/day from

the Barrage. Most of the balance of i t s supply i s taken d i rec t l y from the

Vaal Dam through a pipel ine, which may be supplemented i n the fu tu re b y

a canal leading from the dam wal l to the Zuikerbosch pumping stat ion.

Some of the eff luent from the Witwatersrand which f inds i t s way to the

Vaal Barrage and i s not returned to the Witwatersrand b y the Rand Water

Board, i s consumed by communities fu r ther downstream. The qua l i t y of the

eff luents entering the Vaal Barrage thus affects the cost of treatment

before fur ther use of the water i s possible.

A number of possible schemes to re-use the wastewaters of the

Witwatersrand are possible:

1 .

2 .

3 .

4 .

5.

6.

7.

Par t ia l treatment of waste water treatment p lan ts on the Witwatersrand,

and return of the eff luent to the Vaal Barrage v i a streams, where

further pur i f i ca t ion occurs. The eff luents a re d i lu ted b y re la t i ve l y pure

r i ve r water then, af ter re-treatment, pumped back to the Witwatersrand

and/or passed dOWnStream of the Barrage.

Convey wastewaters from the Witwatersrand to the banks of the

Barrage, and pu r i f y them at a combined works before re-cycl ing.

Reclaim eff luent on the Witwatersrand to a sub-standard and re-cycle i t

i n a separate d is t r ibu t ion system for non-hygienic purposes.

Reclaim to a h igh standard and re-cycle together w i th the water

pumped from the Vaal River.

Ins ta l l a low-capital, h igh operating-cost, reclamation f a c i l i t y on the

Witwatersrand and maintain th is as a standby in case of na tu ra l r i v e r

droughts. Draw from the Vaal River a much h igher d ra f t than could

re l i ab l y be drawn, and use the reclamation p lan t when short fa l ls

occur.

Ins ta l l low-capacity reclamation fac i l i t i es on the Witwatersrand and

discharge the pu r i f i ed eff luent into storage dams; ei ther on the surface

o r underground. Draw on the Vaal River to a h i g h degree as for ( 5 )

and use the stored eff luent when shor t fa l l s occur in the Vaal River.

Pass p a r t l y treated eff luents downstream of the Vaal Barrage in

constructed condu i t s.

Past p lann ing and construction of waste water treatment fac i l i t i es has

been on a local municipal basis, whereas bu lk water supply was the

158

responsibi l i ty of the regional Rand Water Board. The establishment of a

regional waste water au thor i ty i s now being contemplated. This w i l l enable

comprehensive p lann ing a t least cost to be achieved. Regional treatment

fac i l i t i es and combined sewage out fa l l s can be planned w i th consequent

savings i n cost due to scale. The location of treatment fac i l i t i es can be

selected to resul t i n least overal l cost i.e. of sewers, wastewater

treatment, and water supply.

I t i s wi th these ideas i n mind that a project to study the water supply

and waste water system of the Witwatersrand was embarked on.

THE MATHEMAT I CAL MODEL

The system of dams, streams, wastewater treatment plants, potable

water works and conduits between the Witwatersrand and the Vaal River i s

growing more complex over the years. As pointed out, a regional p lann ing

author i ty responsible for overa l l p lann ing i s desireable. Such a body

should have a t i t s disposal data assimi lat ion fac i l i t i es and systems

analysis models to fac i l i t a te planning. A sui te of computer programs for

opt imizing p lann ing of wastewater treatment works, ou t fa l l sewers and

water works should be a t hand.

The system i s described later by equations and constraints which are

in some cases not l inear , and nonl inear programming methods are needed

to a r r i v e a t a least-cost p lan. A s impl ist ic mathematical model of p a r t of

the system i s assembled below, and methods of solut ion are out l ined later.

The chapter goes on to describe a method of l i n k i n g neighbour ing

basins b y a master program, which could also consider var ious time

horizons. The set of constraints developed below i s fo r stat ic condit ions in

a pa r t i cu la r basin. S ta t i s t i ca l l y averaged values of flows, water qua l i t ies

and consumptions are taken. To al low fo r var ia t ions by p robab i l i t y

d is t r ibu t ions would be theoret ical ly possible but would increase

computat iona I time man yfo Id.

Consider the s impl ist ic system i n F igure 10.2. The diagram embodies

the fol lowing concepts: the water requirements of a major consumer such as

the Witwatersrand ( 3 ) could be met from surface resources ( 1 1 , p a r t i a l l y

treated wastewaters returned to the r i v e r a t (2 ) o r reclaimed waste water

from (5) . The wastewaters from (3) could be treated a t (4) followed b y

te r t i a ry treatment a t ( 5 ) o r discharged into the r i v e r a t ( 6 ) a f te r l imi ted

treatment. Reclaimed water i s assumed to be c i rcu la ted in the same

d is t r ibu t ion systems as r i v e r water. The problem i s to determine what each

f low should be and what standard of treatment i s desirable.

159

Fig.

SUE-SKTEM I

'. \ \

10.2 Water c i rculat ion diagram

Each source, consumer, treatment p lan t o r junct ion i s referred to as a

node. The var iables are the flow rates between the nodes numbered i n the

diagram and the respective pol lutant concentrations, designated Q. . and

Pi-j respectively for flow from node i to node j. The flows are a l l

expressed i n Ml/day.

The pol lut ion load may be conservative such as total dissolved sol ids

(TDS) or a non-conservative var iab le such as biochemical oxygen demand

(BOD). I n the former case there must be a mass balance of po l lu tan t i n

the system and i n the la t te r case the effect of the r i v e r and barrages i n

d i l u t i n g the pol lutant must be assessed. BOD i s considered i n the present

model. Values of BOD are expressed i n mg/l and the product of flow in

Ml/day and pol lut ion in mg/l i s proport ional to the total load in tons/day

discharged by a stream or conduit.

I -J

The object of the study is to minimize the cost of the system. I t i s

convenient to convert a l l costs to a common basis, say annual interest and

redemption on cap i ta l cost p lus runn ing costs, a l l expressed in cents per

k i l o l i t r e . Cost coefficients, o r rates, are therefore required for each

var iable. Some of the cost coefficients, for instance for conveyance of

water i n closed conduits, a re nonl inear and must be approximated b y the

anticipated incremental costs. Although nonlinear objective functions are

theoretical ly feasible, i t i s necessary to s impl i fy the model as f a r as

possible since there are fu r ther nonl inear constraints as w i l l be revealed

further on. I f there are ex is t ing conduits these w i l l effect ively have zero

cap i ta l cost. But i f the fu tu re flow along a route i s l i ke l y to exceed

exist ing capacity, i t i s only the incremental cost of the new conduits and

works which need to be considered. Allowance for peak factors i s made in

siz ing the various works.

160

Associated w i th each flow ra te Q . . i s a conveyance cost coeff icient, I-J

The var iab le port ion of the cost component which which i s designated Ci .

i s a function of flow only i s thus

‘1 ‘1-2 + ‘2’2-3 ’ ‘3‘3-6 + ‘4‘4-3 (10.1)

Note that Q6-7 i s f i xed so i t s cost i s not va r iab le i.e. need not be

considered. The cost of conveyance i n na tura l channels i s zero.

Pur i f icat ion costs comprise a component proport ional to flow Q . . and a

component proport ional to pol lutant load Q. .P. . o r proport ional to

pol lutant load removed, Q . .(P. .-P. . ) where subscript 2 refers to

condit ions af ter treatment. The la t te r component, i .e. cost proport ional to

load removed, i s d i f f i cu l t to establ ish and i n fact i t i s often assumed that

works are designed to produce an eff luent of reasonable standard w i th

treatment costs proport ional to flow rate. Vie w i l l consider the general case

and designate the coefficients of Q P and Q2-6 P2-3 as C5 and C6

respectively. Since Q3-6 + Q3-4 i s a constant, the f low-proport ional cost

can b e omitted.

I-J

I - ] I-J

I-J I-J l -J/2

3-6 3-4

The equations or constraints descr ib ing the system are formulated next.

H a l l (1977) formulated the system wi th s imi la r constraints, but a t that

time was unaware of a simple method of solution.

For flow balance a t the var ious nodes:

At some source nodes such as ( 1 1 , the y ie ld may be l imi ted bu t i n our

case we consider unl imited augmentation possible, a t a cost. At consumer

nodes the supply must be suff ic ient :

42-3 + Q4-3 = a1 (10.2)

‘6-7 = a2 (10.3)

At consumer nodes the wastewater output i s known:

‘3-4 + ‘3-6 = a3 At treatment p lan ts and other nodes the flows must balance:

Q3-4 - 44-2 - Q4-3 = 0

‘1-2 + ‘4-2 - ‘2-3 - ‘2-6 =

‘2-6 + ‘3-6 - ‘6-7 - ‘6-8 =

Note that although the va r iab le Q6-7 could be el iminated using equation

(10.3) i t complicates the cost function and any later changes to the

constants which may be desired. The number of var iab les i s nevertheless

minimized in simple cases b y subs t i tu t ing Q and Q3-6 for Q3-9 and Qg-6. I t w i l l be observed that the so ca l led constraints

(10.2)-(10.7) a re i n fact a l l equations. They could equa l ly well be

constraints of the less-than o r greater-than type, in which case slack

var iab les would be introduced to form equations. The waste, Q6-8, i s in

fact a slack var iab le , bu t w i th more meaning than a pure ly algebraic

(10.4)

(10.5)

(10.6)

(10.7)

for Q4-5 and 4-3

161

slack var iable. I n equation (4) the waste water output i s given as a

constant. This constant i s in fact a function of the consumption a2, bu t i t

i s easier to insert a constant.

There are other forms of constraints on the flow var iables which could

be incorporated. For instance basin character ist ics may l im i t the amount of

waste water C13-6 which could (economically) be discharged beyond the

barrage (10.2). O r i f the supply C12-3 was considered as two components;

one through ex is t ing conduits and one through new conduits, there would

be a l im i t on the capacity of the exist ing conduits.

The next set of constraints appl ies to the pol lutants. Certain levels of

pol lut ion may be known:

PI-2 = a4 (10.8)

‘4-2 5 a5 (10.9)

‘2-3 5 a6 (10.10

‘6-7 5 a7 (10.11

’4-2 - ’4-2/2 = a8

In fact i t i s assumed for the BOD study that P1-2 = 0.

There may be tolerable l imi ts on certain levels of pol lut ion:

There i s an extent of na tura l pur i f i ca t ion i n r i vers :

(10.12

P2-3 - P2-3/2 = ag (10.13)

I n the case of TDS as the pol lutant there would be neg l ig ib le reduction

of P a t waste water treatment p lan ts and the TDS a f te r reclamation p lan ts

could be taken as zero. i n the case of BOD, i t can be assumed P1-2 = 0

and P5-3 = 0, and there i s some reduction i n P a t 4 and 9:

‘3-4 - ‘4-2 = (10.14)

(10.15) ‘3-4 - ‘9-6 = all A mass balance of po l lu tan ts must be maintained at nodes:

( f o r f l o w 2 - 6)

(10.16)

(10.17)

Q3-4P3-4+Q3-6P3-4-Q2-3P2-3-Q4-3p5-3 = al 2 (10.18)

Note P2-6 equals P 2-3, P4-5 equals P4-2 and P3-9 equals P3-4 so these

substi tut ions are made for s impl i f icat ion.

I t i s impl ic i t i n the optimization program that a l l var iab les are

non-negative and real so these constraints are not stated exp l i c i t l y .

The general model then is to minimize (10.1) subject to constraints

(10.2)-(10.17). A method of solution i s out l ined in the next section.

162

OPTIMIZATION METHOD

I t w i l l be observed that the objective funct ion and constraints are

l inear except fo r constraints (16)-(18). These constraints involve

two-dimensional products of var iables. There are techniques for convert ing

these products to separate functions, fo l lowing which a technique known as

separable programming may be employed to optimize the system.

Hadley (1964) proposed a simple method of transforming a product QP

b y introducing two new var iab les M and N, such that

M = ( Q + P)/2, N = ( Q - P)/2 (10.19)

QP = M‘ - NZ (10.20)

Q = M + N , P = M - N (10.21)

then

and

M and N are unrestr icted i n s ign but th is i s permitted in the del ta method

of separable programming (IBM, 1976).

The separable programming algor i thm i s based on the fact that a

separable function can be approximated b y piecewise I inear functions. The

polygonal approximation i s represented by a set of special var iables, so

any value of M can be represented as follows:

M = Mo + GIDl + G2D2 + ... G k k D +... GRDR (10.22)

where the Ds represent in te rva ls of M and the special var iab les GI, ..., GR

are defined as follows:

fo r M in in te rva l k,

G , = G 2 = G k-1 = 1 (10.23)

O ( G k ( 1 ( 10.24)

Gk+l = Gk+2,..GR = 0 (10.25)

i.e. M comprises a set of integral in te rva ls D up to k - 1 p lus a f ract ion

of in te rva l k.

Note that

M’ = Ma + G E (10.26)

where each in te rva l Ek corresponds to an in te rva l Dk. Thus for the

approximations i n Fig. The value of M

can be confined to a known range.

+... G E +... GRER 0 1 1 k k

10.3, Mo = 0, k = 4 and Gk = 0.3.

Although there i s a poss ib i l i t y of a t ta in ing a local optimum th i s chance

i s reduced i f the problem i s solved w i th the special var iab les set i n i t i a l l y

a t their upper bound, and then with them set a t the i r lower bounds, to

ve r i f y the results. I t w i l l be observed that 2R var iab les are introduced

in to the model for each product in the o r ig ina l constraints. I t i s therefore

desirable to s impl i fy the o r ig ina l system as much as possible to minimize

163

the number of products.

The fo l low ing s imp l i f i ca t ions a r e introduced. The system i s subd iv ided

i n t o two subsystems (see F ig. 10.21, which w i l l b e l i n k e d v i a a master

program accord ing to Dantz ig 's decomposition p r i n c i p l e (Dantzig, 1963) a n d

app l ied b y Stephenson (1970) to l ink r i v e r bas ins. Only sub-system 1 i s

considered here a n d there w i l l be shadow va lues imposed o n the cost

coeff icients of Q P a n d Q2-6P2-3 b y the master program (C7 a n d C8).

(10.31, (10.71, (10.11 1, (10.13) a n d (10.17) a r e thereby e l iminated.

E l i m i n a t i n g Pg-6 us ing (10.151, cost coef f ic ients become C o g = C3 - C7,C f5 = C5 + C7 a n d C I 6 = C6 + C8.

Cer ta in v a r i a b l e s , namely Plm2 a n d P5-3 a r e zero so cons t ra in t (10.8)

i s el iminated, a n d b y subs t i tu t ing from (10.12) a n d (10.4) i n t o (10.16)

a n d (10.18) respect ive ly , the number of products i s reduced. Now the

problem is :

3-6 9-6

a l 1

subject to:

0.28)

0.29)

0.30)

0.31)

0.32)

(10.33)

(10.34)

( 10.35)

( 10.36

(10.37)

(10.38)

Put

'4-2~4-2 = M t - ~ ' 1 (10.39)

Q2-3P2-3 = M i - N: (10.40)

Q2-6P2-3 = M - N: (10.41)

= M Z - N t (10.42)

Then (10.37) a n d (10.38) can be rep laced b y equat ions (10.43)-(10.52): Q3-4p3-4

164

(10.43

(10.44

(10.45

(10.46

(10.47)

(10.48)

(10.49)

(10.50)

(10.51)

(10.52)

Each M2 and NZ comprises composite polygonal functions according to

equation (10.22) and i t i s necessary to define the in te rva ls i n the

computer data input. The objective function must also be re-writ ten as

min c1a1-2 + c2a2-3 + c * ~ Q ~ - ~ + c4a4-3 + c ; (a3P3-& - M: + N:)

t /

(10.53)

b

Fig. 10.3 Polygonal approximation of a separable funct ion

T h e problem may thus be solved by s t ra igh t fo rward techniques, and

once the procedures a re adequately programmed, the process may be

expanded, using the decomposition pr inc ip le , to consider var ious time

horizons. The problem may also be considered i n more de ta i l than out l ined

here. Using decomposition pr inc ip les i t would be possible to incorporate

sub-programs fo r i nd i v idua l waste water treatment works (e.g. C I R I A ,

1975). This would ensure optimization of each component. Dynamic

programming methods could be employed to study po l lu t ion along stream

reaches and optimize the soacing and standard of waste water pu r i f i ca t i on

165

works d ischarg ing in to the stream. Where m u l t i p l e decisions a r e requi red,

fo r instance f o r a l t e r n a t i v e p u r i f i c a t i o n p l a n t locations, o r sewer ou t fa l l s ,

mixed integer programming could be employed. Computer s imulat ions of the

system as a lso requ i red to study extreme condi t ions a n d cost sens i t i v i t ies

to supplement the shadow values produced b y the opt imizat ion. There a r e

other sophis t icated techniques fo r op t im iz ing non l inear waste water systems

(Chiang and L a u r i a , 1977; Pra t ish thananda a n d Bishop, 1977) b u t the

above-formulated problem s imp l i f ied to a neat solut ion.

To incorporate a l l the concepts in a l a r g e number-crunching program

w o u l d not be r e a l i s t i c though, a n d i n t e r a c t i v e programming i s

recommended, i.e. human in tervent ion a t each step. The p l a n n i n g process

could be continued as d a t a a re updated b y successive i t e r a t i o n of the

master program and sub-programs.

REFERENCES

Chiang, C.H. and L a u r i a , D.T., 1977. Heur is t i c a lgor i thm f o r waste water p lann ing . Proc. Amer. SOC. Civ. Engrs 103 No. EE5, 863-876.

CIRIA, 1975. Cost e f fect ive sewage treatment - the c rea t ion of a n op t im iz ing model. C l R l A Report 46, London.

Dantz ig , G.B., 1963. L i n e a r Programming and Extensions: Pr inceton Un ivers i ty Press, Princeton, New Jersey, USA.

Hadley, G., 1964. Nonl inear and Dynamic Programming, pp. 448-465: Addi son-Wesley , Reading.

H a l l , G.C., 1977. A method fo r op t im iz ing the c i r c u l a t i o n of water in urban regions. Nat l . Ins t . Water Res. Counci l f o r Scient. a n d Indust . Res. Pretor ia , R.S.A.

IBM, 1976 Mathematical Programming System Extended/370, Program Reference Manual, 2nd edi t ion, pp. 230-251.

Prat ishthananda, S . a n d Bishop, A.B., 1977. A non l inear m u l t i l e v e l model fo r reg iona l water resources p l a n n i n g . Waf. Resour. Bu l l . 13 No. 3, 61 1-625.

Rand Water Board, 1977. Annual Report. Stephenson, D., 1970. Optimum design of complex water resource projects.

Stephenson, D. 1978. Optimal p l a n n i n g of reg iona l wastewater treatment. Proc. Amer. SOC. Civ. Engrs 96, no. HY6, 1229-1246.

Proc. IAHS Symposium. Model I ing the Water Qua1 i t y o f the Hydro log ica l Cycle. Baden, 125. 351-360.

166

CHAPTER 11

SIMULATION OF SEWER FLOW

I NTRODUCT ION

With the development of suburban areas w i th in c i t ies to the i r l im i ts i t

i s becoming necessary to consider subdiv is ion and more intense resident ia l

densities i n suburbs which were previously on ly sparsely populated. The

effect of more intense development on the services, such as sewerage, for

an area must be considered before such increase in density i s permitted. A

study of the consequences of increased loading i s d i f f i cu l t , p a r t i c u l a r l y as

the effect may cause a chain reaction down the length of the sewer

system. A method of iden t i f y ing possible problem areas and methods of

passing the increased flow was sought. A model f o r r a p i d l y selecting a

pa r t i cu la r area o r fo l lowing the flows through a system was developed.

There is frequently an appreciable time-lag as hydrographs flow down

the system as well as attenuation due to la te ra l dispersion of the

hydrograph, i.e. rout ing. I n order to al low fo r these effects a computer

simulat ion program seemed to be the most logical approach. The program

can draw data from ex is t ing land use inventories wherein data concerning

a l l stands w i th in the municipal area (i.e. f loor areas, number of rooms

and land usage type) are retained. I n add i t ion da ta f i les containing

engineering data ( i .e. sewer lengths, slopes, connections, diameters, drops

and condit ion) are compiled. The la t te r da ta may be used fo r other

projects such as data re t r ieva l for d rawing sections, establ ishing depths

of connections and management of the sewerage system a t a la te r stage. As

there are about 135 000 stands w i th in the Johannesburg area a systematic

and eff icient way of stor ing the data was required fo r th is appl icat ion.

The process of computerization also enables engineers to estimate

sewage design flows effect ively. Whereas i t was previously necessary to

design sewerage systems for the estimated peak flows based on averages

over ten hours of the day, i t i s now possible to d iv ide the flow into

di f ferent components. Stormwater ingress, i n f i l t r a t i on through jo in t s i n

manholes, leaks from san i ta ry f i t t i ngs and sewage discharges may be

accounted for separately. The actual time d is t r ibu t ion of sewage discharges

may also be considered. This w i l l affect the hydrograph lags where

successive hydrographs contr ibute to an ou t fa l I . Considerable data

therefore had to be gathered in order to provide the subdiv is ion fo r the

analysis and the development of contr ibutory hydrographs.

167

Considerable work has been done in South Afr ica, in pa r t i cu la r b y

Shaw (1963) and Crabtree (1976), on establ ishing contr ibutor hydrographs.

The actual lagging of hydrographs and consideration of rou t ing effects and

probabi l i t ies of d i f ferent connections discharging simultaneously have not

been tackled on a real scale. The ana lys is of flow in storm d ra ins has

probably received more attention (Stephenson 1981) due to the ease of

synthesizing inf low hydrographs and the la rger scale of storm sewer

systems (Stephenson and Hine, 1986).

HYDRAULIC ANALYSIS

In employing a computer simulat ion method allowance for time lag,

backwater effects, rou t ing and probab i l i t y effects could be included.

However, i f a l I these components were considered in one program,

computational time would be increased considerably. The hydrau l i c

equations were therefore s impl i f ied by omit t ing some of these effects and

time lag rou t ing was employed. Peaks due to ind iv idua l lavatory flushes

and so on can also be shown using probab i l i t y theory to be r a p i d l y

attenuated (Chan and Wang, 1980).

One computer model simulates the flow down sewers and accounts for

time lag as well as di f ferent types and times of inf low. However as the

ind iv idua l lengths of sewer number 135 000, i t i s often not convenient o r

in fact appropr iate to analyse the flow in each sewer. Another program

therefore exists for abstract ing data from relevant areas which require

analysis and even for condensing the da ta so that a number of lengths of

sewer could be considered together to speed ana lys is (Constantinides, 1982).

I t i s possible to ident i fy the type of inf low i n each case in order to

b u i l d the correct contr ibutor hydrograph. One of the objects of the

program i s to study the l ag and at tenuat ing effect on hydrographs. The

times of day a t which the flows s ta r t and increase and subsequently

decrease are therefore important and f i e ld measurements were required.

FLOW MEASUREMENTS

Measurements were made in manholes as near as possible to the source

of sewage in order to minimize the time lag and to avoid attenuation due

to rou t ing down the sewers. The flow depths in the sewers wer gauged a t

manholes over a period of weeks and the resu l t ing hydrographs plotted.

The var iat ions from week to week were s l i gh t and s imi la r weeks were

averaged for compil i ng the hydrographs. The observations taken a t n igh t

168

i n d r y weather were assumed to indicate leakage p lus i n f i l t r a t i on . By

comparing these readings a t the end of summer and the end of winter in

d r y periods i t was possible to estimate the re la t i ve proport ions of

i n f i l t r a t i on and leakage from the plumbing systems. Observations made

du r ing and af ter summer storms indicated storm inf low to the system.

Ra in fa l l i n Johannesburg i s normal ly restr icted to the summer season when

h i g h intensity storms of short durat ion occur du r ing the afternoons. The

sewers were assumed to flow unsurcharged du r ing storm flow and

surcharged conditions were discarded as they would have been d i f f i c u l t to

use for est imating actual con t r ibu t ing flows.

The peak flow ra te du r ing the day for h igher income housing averages

1.17 I/min per house excluding leakage, i n f i l t r a t i on and stormwater

ingress. The amount of leakage from the plumbing system in to the sewerage

system i s estimated to be 0.06 I/min per house, over 24 hours a day

throughout the year. The in f i l t r a t i on in leaking sewers i s estimated to be

0.05 I/min per metre of sewer per metre diameter. This i s greater for o lder

sewers in poor soil. The addi t ional flow du r ing and a f te r storms i s

estimated to average 1 % of the prec ip i ta t ion over the catchment. This w i l l

va ry widely depending on the methods of con t ro l l ing stormwater inf low into

gul l ies. This flow i s also associated w i th the one-year recurrence in te rva l

storm over n hours which i s estimated to be 2 mm/h for general analysis.

The flow suggested to peak flow design is 0.05 mm/h which i s 1 % of the

two-year storm of 5 mm/h. The effect ive cont r ibu t ing area i s about 50 m

width of catchment per metre of sewer.

Up to 5%, and i n isolated cases even more, of the r a i n f a l l over an

assumed 50 m wide s t r i p over a l l sewers was found to enter sewers i n

some cases. The actual sewer flows often increased b y over 50% - even to

69% dur ing and a f te r a storm.

I n the case of the f l a t areas a special g lass f i b re flume was used in

the invert of a manhole. This has a curved bottom and a hump which

made i t possible to measure re la t i ve l y small flows from a block of 273

f lats. This el iminates the problem of ascertaining ex is t ing sewer gradients.

Another advantage i s the fact that the resu l ts a re not affected b y the

existence of s i l t and so on. Where possible, a conventional flume was used

to gauge flows. The normal method of recording a t these flumes was on a

c i r cu la r chart w i th an integrated total flow read on a meter; the charts

a re changed a t weekly intervals. To g ive continuous flow rates a Fisher

Porter meter was instal led for several weeks to g ive accurate data a t 15

minute intervals.

169

The input hydrographs were reproduced b y the computer using a

Fourier series type of curve f i t .

Higher income resident ia l

An area of approximately 1220 ha with well establ ished medium-sized

houses and newer houses including town houses was selected fo r the

development of the hydrograph taken in a higher income resident ia l area

(Fig. 1 1 . 1 ) . Equivalent house un i ts were based on a 150 m2 f loor area

normalized by ra i s ing that to the power of 0.8 to al low for the reduction

in flow per un i t of f loor area as houses and f l a t s increase in size. Hotel

and servant accommodation is allowed for a t the ra te of one house

equivalent un i t for every three rooms. Shops, offices, schools and churches

are allowed for a t the ra te of one house equivalent per 300 m2 of f loor

area.

'"t

Metered Calculated (November 1981)

Daily Iota1 kl 58 5 58 7 Average Its 67 8 68 0 Peak 11s 1164 1156 Minimum 11s 24 0 24 0

- Calculated flow .-. 0 . Melered llow

I I

0 3 6 9 12 15 18 21 24 Time of day h

Fig. 1 1 . 1 Comparison of calculated and metered f lows in higher income resident ia l areas

Minimum o r base flow was assumed to be due to i n f i l t r a t i on into sewers

and manholes or to leakage w i th in the bui ld ings. This was shown to be

equal to 41% of the average, o r 20% of the maximum flow in d r y weather.

A t 5.15 a.m. the flow rises r a p i d l y to w i th in 80% of maximum flow,

which occurs a t 8 a.m. This flow pattern suggests that h igher income

residents in Johannesburg get up a t 5 a.m. onwards. Generally, production

workers start work at 7 a.m. and off ice workers a t 8 a.m. Tra f f i c

congestion demands an ear ly start for those t rave l l i ng the 12 km into the

c i t y .

170

The evening peak occurs a t about 7 p.m., ind ica t ing preparat ion of

meals, ablut ions and so on, and ac t i v i t y ceases a t midnight. Maximum

flow per un i t was found to average 1.17 I/min.

Low income residential

Detailed land use da ta were not ava i l ab le for the study of the low

income resident ia l area (Fig. 11 .2 ) . Detai ls of houses and f l a t s were

abstracted from construction drawings and checked on si te before the sewer

data were used. The area chosen embraced most of the newer sections of

Lenasia - a town 25 km south of Johannesburg - and the sewage flow was

monitored with a flume. A time lag of one hour was al lowed fo r when

comparing the hydrograph measured w i th the hydrograph a t the point of

o r ig in .

Daily Iota1 kl Average. 11s Peak. Us Minimum. Us

15

l 1

5 a E

Monday ,Tuesday Wednesday Average Calculaled 461 462 460 461 463 531 5 32 5 30 531 54 14 77 1532 1450 1486 14 7 0.1 5 0.28 0.28 0.23 0.3

- Calculaled l b w -- - Monday 30 January 1984 . . . Tuesday 31 January 1984 ........... Wednesday 1 February 1984

. 0 3 6 9 12 15 18 21 24

Time of day h

Fig . 11.2. Comparison of calculated and metered f lows i n low income resident ia l areas

Rapid increases i n flow occurred a t 5.30 a.m., peaking a t 7.15 a.m. A

smaller peak a t 10 a.m. indicated greater a c t i v i t y i n the home a f te r the

departure of the working populat ion than i n h igher income areas,

pa r t i cu la r l y on Mondays - the t rad i t iona l washing day. A greater

proport ion of the fami ly could remain a t home, which i s also indicated b y

the smaller evening peak a t 8 p.m.

Maximum flow per house un i t amounted to 0.46 I/min. The low minimum

flows metered are due to the recent construction of a l l sewers and

bu i Idings.

171

Apartment bu i Id ings

A large complex known as Helderberg i n Berea, Johannesburg, was

selected for the study of an apartment ( f l a t s ) area (Fig. 11.3). I t

comprises 273 f l a t s wi th a total of 585 rooms. Metering was car r ied out

close to the bu i l d ing so the time lag was minimal.

Points of interest of the hydrograph include the low level of

i n f i l t r a t i on and leakage, the sharp r i se and subsequent f a l l i n the

morning peak, the secondary morning peak a t 10 a.m. each day except on

the Thursday and the d is t inc t i ve peaks a t 7.30 p.m. and 9.30 p.m. which

may have something to do with the television v iewing of f lat-dwellers.

Inf low starts later than i n other resident ia l areas, possibly due to the

greater proximity of these f l a t s to places of work. Maximum flows appear

to be higher than i n other types of development, i.e. 2.05 I/min per un i t .

Monday Tuesday Wednesday Thursday Average Calculated 241 9 2374 2367 240

8 66 7 45 8 66 8 31 027 86 041 0 46 052 0 4 2 0 3

Daily lolal hI 238 2295 Average 11s Peak 11s Minimum l ls 0 27

2 74 2 66 2 82 276 275 2 8

Calculated (low - 10 - --- Monday 21 November 1983 -.- Tuesday 22 N o v e m k 1983

G

Time 01 day h

Fig. 11.3 Comparison of calculated and metered f lows i n a f l a t area

Commercial areas

For the study of a commerical area (Fig. 11.4) a port ion of the central

business d is t r i c t of Johannesburg was selected. House equivalent un i ts were

obtained on the basis of 300 mz of f loor area and amounted to 3026 uni ts,

indicat ing a total f loor area of 90.8 ha. The development i s exclusively

commerc i a I.

A r a p i d increase in flows occurred a t 6 a.m. The peak morning flow

occurred at 1 1 a.m. Flows then decl ined u n t i l a r a p i d increase occurred a t

around 3 p.m., resu l t ing in the peak d a i l y flow a t 4 p.m. These

phenomena are indicat ive of normal off ice hours. The afternoon peak must

b e due to the use of lavator ies and washing jus t before staf f lef t work.

172

The base f lows are h igh re la t i ve to h igher income resident ia l areas,

which can be a t t r ibu ted to a h i g h leakage rate.

20-23 April 1982 5-8 July 1982 Average Calculalecj Daily average total kl 4955 6 47185 4837 0 4849 8 Average 11s 57 4 54 6 56 0 56 1 Peak 11s 90 5 94 7 92 6 93 2 Minimum 11s 31 9 30 0 30 9 30 3

150 I - Calculated llnw ...... Weekday tlow 20-23 Aprll 1982 - .. . . . . Weekday l l w 5-8 July 1981

I I

0 3 6 9 12 15 18 21 24 Time 01 day h

Fig . 11.4 Comparison of calculated and metered flows in a commercial area

Industrial

Several indus t r ia l areas were investigated i n de ta i l to establ ish a

reasonably rea l iab le method of s imulat ing flows. Types of indus t ry were

general ly mixed bu t d i d not include any heavy indus t ry . I n i t i a l l y un i ts

were establ ished based on 100 mn of f loor area which showed la rge

discrepancies in flows due to the predominance of h i g h o r low water usage

by ind iv idua l f i rms which bore no re la t ion to f loor area. Var iat ion of the

f loor area per un i t d i d not therefore g i ve consistent resu l ts from one area

to another. I t was found that actual water supply gave the best indicat ion

of eff luent discharge. Water meter readings a re normal ly taken every three

months i n Johannesburg and stored i n computer f i les. I t was possible to

extract meter flows over a three-month per iod and then base un i ts on an

average d a i l y flow of 800 I/day. An indus t r ia l area o f 160 ha was selected

for t h i s study (F ig . 5 ) . Major industr ies included yeast manufacture which

has a very h igh water usage.

CONCLUSIONS

The composite hydrographs prepared from sewer flow measurements (F ig .

1 1 . l - 11 .5 ) indicate va ry ing peak times and the importance of assessing

i nd i v idua l hydrographs i s thereby emphasized. Out-of-phase peaks w i I I not

be cumulative and as a resul t sewer capaci t ies need not be the sum of

peaks. The time lag of i nd i v idua l contr ibutor hydrographs also adds to

173

300

the attenuation effect.

The computer simulat ion program i s used for the p lann ing and design

of extensions to the san i ta ry sewer col lect ion network. The program i s

used to ident i fy bottlenecks, study the effects of re-zoning or subdivision,

p lan new faci l i t ies, size temporary diversion works, p lan a l te rna t ive

routes, size t runk sewers and estimate loads a t ou t fa l l works.

The program i s l inked to a land use inventory for assessing

contr ibutions and to p lo t t ing program for d rawing sewer longi tudinal

sections. I t i s proposed to estimate future f lows using a land use

classif icat ion established in the computer data bank.

- Calculaled llow - - - Monday 5 July 1982 - .- Tuesday 6 J U I ~ 1982 . . . Wednesday 7 July 1982 ....... Thursaay 8 July 1982

-

Monday Tuesday Wednesday Thursdav Average Calculated Dailylolal k l 10317 11157 10565 10423 10615 10647 Average I/s 1194 129 1 122 3 1206 1228 1232 Peak lls 216 5 208 5 208 5 2165 2125 2132 Minimum 11s 60 2 69 9 75 0 67 4 68 1 74 5

,7 E F E R E N C ES

Chan, W.Y.W. and Wang, L.K. , 1980. Re-evaluating Hunter 's model for residential water demand, J. Am. VJat. Wks Ass.

Constantinides, C.A., 1982. Comparison of time lag and kinematic flow i n conduits. Water Systems Research Programme, Universi ty of the Witwatersrand.

Crabtree, P.R., 1976. Flow and in f i l t r a t i on gauging in sewers. National Bu i ld ing Research Inst i tute, Concil for Scienti f ic and Indus t r ia l Research. Pretoria.

Shaw, V.A., 1963. The development of contr ibutor hydrographs for san i ta ry sewers and the i r use in sewer designs. Civ. Engr. S. A f r . 5, No. 9, 246-252.

Stephenson, D., 1981. Stormwater hydrology and drainage, Elsevier, p 276. Stephenson, D. and Hine, A.E., 1986. Simulation of sewer flow. Municipal

Engineer, 3. 107-112.

174

APPENDIX 11.1

PROGRAM SEWS I M

This i s a micro-computer (HP9816) orientated version of a program to

store data and simulate flows down san i ta ry sewage networks.

Contr ibutor hydrograph character ist ics a re programmed for var ious

types of development e.g. resident ia l upper class ( type 1 1 , lower class

( type 2) , indus t r ia l ( t ype 3 ) and commercial ( t ype 4). The actual peak

flows per un i t (P/min/100m2) must be . inserted in the data. Equivalent

number of house un i ts (HE) or 100m’ i n the case of non-residential, i s

required for each pipe, and peak in P/min/HE for each section.

Hydrographs are accumulated with time l a g proceeding down a l l sewers.

Time lags are based on f u l l p ipe velocity as hydrodynamic ana lys is would

be too time consuming and not worth the ef for t .

Hydrographs over any per iod of time e.g. 24h (s ta r t i ng a t midn igh t )

for any time in te rva l (e.g. l h ) for any number of selected pipes (say 10

maximum) are tabulated, and plot ted i f required.

Also indicated are maximum flow for every pipe, i t s capaci ty and

overflow volume if not adequate. A summary of p ipe lengths and inf low

areas i s tabulated a t the end.

Each sewer i s ident i f ied b y a number. The numbering system fo r sewers

can be selected such that the f i r s t two d ig i t s of a 6-digi t number indicate

the ou t fa l l region, the second two the suburb and the last two the actual

p ipe which i s assumed the same as the top end manhole.

Effect of Local Peaks (Probab i l i t y and Routing)

The design flow from a house connection for sewer design i s t yp i ca l l y

1,5P/min. The actual peak discharge i s considerably higher but when the

effect of a number of houses i s accumulated the above f i gu re i s

reasonable. A proof that the instantaneous peaks can be neglected follows.

A typ ica l toi let f lush occurs a t a ra te of 204 in 7 sec i.e. 3 P / s .

Frequency of f lushing a t peak periods i s once every 2,5 minutes = 1/150s

per house. Therefore af ter 7 houses mean flow/house assuming only 1 house

flushes a t a time, i s 1,2P/min - which i s a normal design flow. i.e. a f te r

10 houses or so the flushes average to g ive a normal flow design f igure.

The probab i l i t y of any two houses f lush ing simultaneously i s (7/150)’ =

1/400, and of 3 houses 1/8000 etc. i.e. remote so the coincidence of a peak

from each house together i s remote. In any case those peaks are r a p i d l y

175

attenuated by the rou t ing effect described below.

Routing effect (Graphs from Stormwater Hydrology and Drainage Stephenson,

1981 ) .

I n order to shorten runn ing time, the program does not include

hydrau l i c rou t ing effects. The fol lowing section i s proof of the fact that

rou t ing has a neg l ig ib le effect on peak flows. Routing i s the spreading of

a wave and corresponding reduction in peak due to hydrodynamic forces.

I t i s superimposed on the time lag effect.

Consider the depth corresponding to a flow of 3P/s for 7s in a 150mm

dia. d ra in at a slope of 1/100:

From a chart f u l l flow Qf = 2OP/s.

Therefore Q/Qf = 0.15.

Therefore re la t i ve depth a t 3t/s from the chart i s

y/D = 0.25

Now for a reduction i n depth from 0.25D to 0.125D (i.e. ha l f o r i g ina l

depth) from the chart

Ey 0 15*x

0.003'7 ~0.013a100 - = 15m

.'. x = t2m

i.e. depth halves i n 12m of sewer pipe.

Therefore the rou t ing effect i s very r a p i d to s ta r t w i th i f the flow i s

very low.

On the other hand the rou t ing effect on a hydrograph from 100 houses

is calculated below:

Flow ra te q = 1.5P/min x 100/60s = 3e/s as well, bu t Q(volume) i s now

0.003 x 8h x 3600. i.e. depth w i l l have over 12rn x 3600 x 0/7 = 50 km

i.e. negl ig ib le rout ing over the f i r s t km or so and b y then the number of

contr ibut ing houses w i l l f a r exceed 100.

Non-Circular Conduits

Sewers are sometimes non-circular e.g. rectangular culverts o r egg

176

shaped. Then the 'diameter' in the data may be replaced by 4R where R

i s the hydrau l i c rad ius , equal to A/P and A is the cross sectional area,

and P the wetted perimeter, a t f u l l flow. This procedure resu l ts i n the

correct time lag i n the computations, which i s taken for the f u l l flow. The

flow capacity of the conduit w i l l however be incorrect ly indicated i n the

results. The actual discharge capacity is:

A Qe - De2

Q = A v =

4

i.e. the indicated flow Q i n the computer p r in tou t should be mul t ip l ied b y

the true cross sectional area d iv ided by ( nDe'/4) where De i s the

equivalent diameter 4A/P i.e. t rue capaci ty Q = (P2/4nA)Qe.

Inf low Components

Inf low to each sewer i s assumed to comprise four components; sewage flow

from connections, stormwater ingress, a function of sewer length, steady

groundwater i n f i l t r a t i on which is a function of s e w e r length, and leakage.

Each parameter can be supplied i n the data and u n t i l more accurate data

i s avai lable, the fo l lowing f igures are suggested:

Sewage Inf low : Peak net inf low ra te of 1.0 l i t r es per minute per house

equivalent i s average in middle class resident ia l areas.

Stormwater : From gul leys, manholes and leaks, mm/h. About 1 % of

precipi tat ion rate. e.g. 1 % x 10mm/h = O.lmm/h.

I n f i l t r a t i o n : 0.15 l i t r es per minute per metre of sewer per metre diameter.

Increase for o ld sewers.

Leaks : from cisterns, d r i pp ing taps etc. 0.15 l i t r es per minute per house

equivalent. Increase for older and la rger propert ies.

Inf low d is t r ibu t ion assumed i s a series of s in waves. The re la t i ve

peaks of each of the 3 s in waves for any type hydrograph are designated

Q 1 , Q2 and 43 i n the program. The time i n hours when each of those

peaks occur are T11, T21 and T31 and the time a t which the s in waves

(posi t ive ha l f on l y ) s ta r t a re T10, T20 and T30 respectively ( a l l i n

hours). The f i r s t wave should start a t T10. These values are b u i l t into

177

the program and the program must be edited to change them.

DATA

Data can be stored and edited in a separate f i le, named a t the time of

establishment and ident i f ied when SEWSIM i s run. Data requirements are

ident i f ied i n the data form. As the data f i l e i s appended to the main

program at the time of runn ing using the "GET" statement, l i ne numbers

should be equal to and greater than 2000 to avo id ob l i te ra t ing program

lines. The data f i l e should end with END. Data can be i n free format w i th

commas separating numbers.

Three items of data are not used i n SEVJSIM. These are sewer depth,

drop and ground level. They are intended for a da ta logging and p lo t t i ng

program later. Zeros may be inserted a t th is stage.

Program Output

The program sorts the pipe data in to order and ident i f ies the lowest

manhole(s). I f there i s more than one unconnected (downstream) manhole

the data should be checked. A picture i s drawn of the system which can

be copied using DUMP GRAPHICS. Then press CONTINUE for the program to

route the flows through the system and tabulate maximum flow etc. in each

pipe.

Hydrographs are also tabulated for nominated pipes. Note that the

hydrographs are tabulated a t the times corresponding to when they would

reach the ex i t of the system (the lowest manhole) and to get the actual

time a t which the tabulated flow occurs subtract the l ag time of the

correspond i n g p i pe from the tabu I at ed t i me.

The hydrographs are also plotted on the screen a t the correct times. To

plot a hydrograph DUMP GRAPHICS then/or continue.

F ina l l y a table summarizing total p ipe length and house un i t s for each

type of development i s given.

'LO! Fi'E-Sl'OHE"SEWSI~1" 12 ! D. STEPHENSON . W 1'1'8 7 16-2560 - 1 b I 03.87 15 N d = T 0 2 16 DlJMP O E V I C E I S Nd 18 PRINTER I S Nd 20 D I M M ( 4 0 0 ) . M d ( 4 0 0 ) . D ( 4 0 0 ) .S(408) . X ( 4 0 0 ) .Hq(400) . Y ( 4 0 0 ) .H(40GI). I t (400) .Tx (400) . T I (400) . Q ( 4 0 0 ) . Q c ( 4 0 0 ) . Q w ( 4 0 0 ) ,G!s (400) ,Qi (400) .Ql (400) . Q m ( 4 0 0 ) .Qv(4P)0) .Jd(40B) 3v1 DJM I21 (99) . Q 2 ( 9 9 ) , Q 3 ( 9 9 ) . T 1 8 ( 9 9 ) .T11 (99) . T 2 0 ( 9 9 ) . T 2 1 (99) . T 3 0 ( 9 9 ) ,T31 (99) . M h ( 9 9) , Jh (99) ,at (29.99) ,He(99) .G1 (400) 31 COM NSCZ03 40 I N P U T "NAME '?".NS 60 READ N s . T s . T i . N h , O l l !NO.SECNS. S I M L N h .T I N C h .Nhydqphs.GLmBOTMH 70 FOR Jn=l TO Nh ! HYDROGHAPHS 80 HEAD Mh (Jn) !PIPE NO OF HYGPHS 90 NEXT Jn 100 J2=B 110 FDR K = l TO Ns ' SECT DATA I20 READ Mb,Itk,Np,Am,Fw,Fs,Fi,Fl ! HOTM MH, ZONE TYPE,Npipes,MANNINljn.I-/MIN/HEQ ,mm/h STORM,INFIl/MIN/M/M.LK/MIN/HEQ 130 J2=J2+Np 140 M d ( J 2 ) = M b 150 FOR J=J2-Np+l TO J2 !PIPE DATA 160 READ M ( J ) , D ( J ) . X (J) ,S (J ) ,Hq(J) ,Y(J) ,H(J) ,G1 (;I) !NO. ,DIClmm,I..m,Sm/m.H~USE EQU VS( l00m2) ,DETTHm,DROP BOTMENDmm,GLmMH 180 I F J < = J Z - N p + l THEN 200 190 M d ( J - l ) = M ( J ) 200 I t . ( J ) = I t k 205 Q v ( J ) = K 260 T x ( J ) =X (J ) +Am*4". (2/3) / ( D (J) / 1000) ." ( 2 / 3 ) /S (J )". 5 ! L..AG, 5

270 Qc (J ) =. 785/Am/4 . " (213) * (D (J ) / 1000) ,*% (8/3) +S (J ) 280 Qw (J) =Fw+Hq ( J ) /h0/ 1000 !PEAK M3/S INFLOW 290 Q s ( J ) = F s * X ( J ) * l 0 0 / 1 0 0 0 / ~ 6 0 0 ! DO. STORM 300 Qi (J) = F i + X (,I) +D (J) / 1 0 0 0 / 6 0 / 1000 ! DO. I N F I L T N 310 Q1 (J)=Fl+Hq(J) /60/1000 ! DO. L E A K S 320 NEXT J 330 0 (K) =J2-Np+l ! TEMP. TOPMH FOR PLAN P1.OT

5+ 1000 ! CAPAC I TY , L./s

340 9rn (F:) =Np 350 NEXT K 4 0 0 T40=12 410 T41-17 420 91(1)=.7 430 92(1)= .5 440 93(1)=.6 450 T 1 0 ( 1 ) = 5 4 6 0 T l l ( i ) = 9 470 T 2 0 ( 1 ) = 7 4 8 0 T 2 1 ( 1 ) = 1 3 490 T 3 0 ( 1 ) = 1 4 500 T 3 1 ( 1 ) =20 510 91(2)=.8 520 92(2)=.7 530 c-13(2)=.6 540 T 1 0 ( 2 ) = 6 550 T11 (2)=8 560 T 2 0 ( 2 ) = 7 570 T Z 1 ( 2 ) =13 580 T 3 0 ( 2 ) = 1 6 590 T 3 1 ( 2 ) = 2 0 4500 01(3)=.4 6llb 02(3)=.S 620 93 (3) =. 45 630 T 1 0 ( 3 ) = 6 640 T 1 I (3) =I0 650 T 2 0 ( 3 ) = 4 660 T 2 1 (3)=13 670 T 3 0 ( 3 ) = 1 3 675 T 3 1 ( 3 ) = 1 6 680 01 (4)=.3 690 02(4)=.7 700 03(4)=.45 710 T 1 @ ( 4 ) = 6 720 1 1 1 (4)=10

740 T 2 1 ( 4 ) = 1 3

--. /d T 2 0 ( 4 ) = 6

!STORM START h !PEAK h ! RESID4UPCLAS. 1 ST PEAK=l !2ND PEAK !3RD PEW !START h 1ST PEW MUST BE 1st HG TO START 'PEAK h 1ST PEAK

!START h ZND PEAK !PEAK h 2ND PEAK

!START h 3RD PEAK !PEAK h 3RD PEAK

!RESIDLPOOR

! INDUST

!COMMERCIAL

d

W

i

w

t);

B

61 t.4

4

m

IXI

2

z

W

iii I

I

!- L

w

a + r.l

a-

1030 NEXT E 1032 G I N I T 1033 GRAPHICS ON ! DRAW LAYOUT 1034 GCLEAR 1035 WINDOW -1000 .T l (N l )+1000 ,0 ,Ns+ l 1039 FOR E=l TO Ns 1040 FOR J=B(E) TO O m ( K ) + O ( # ) - l 1041 MOVE T1 ( N l ) - T l ( J ) . K 1042 DRAW T1 ( N l ) - T l ( J d ( J ) ) - 2 0 . K 1043 L l = T l ( N l ) - T l (J)-20 1044 C S I Z E 3..4 1045 MOVE L1 .K 1 0 4 6 ! D I R l+PT/b 1047 LABEL M ( J ) 1 0 4 8 L D I R 0

1050 Jmm=Om ( K ) +O (K ) - 1 1051 I F J d ( J m m ) = 0 THEN 1055 1052 MOVE T1 ( N l ) - T l ( Jd (Jmm) ) -10.K 1053 DRAW 11 (N1 ) - T 1 ( J d ( J m m ) ) ,G!v(Jd(Jmm)) 1 0 5 5 NEXT K 1060 PAUSE 1070 FOR J=1 TO J2 1071 (;7m(J)=0 ! MAX. FLOW 1072 Q v ( J ) = 0 !SPILL VOL. 1073 NEXT J 1074 T i = 0 !COUNTER FOR TH 1075 FOR T h = T i TO T s STEP T i ! T I M E h AT BOTTM 1076 FOR J=1 TO 52 I N I T I C I L I Z E FLOWS 1077 R ( J ) = 0 1078 NEXT J 1080 FOR J=1 TO 52 J 090 TSTh-TI (J ) /3h00 !PIPE T IME FOR REACHNG E X I T AT T h 1092 I F T : : . T 1 0 ( I t ( J ) ) THEN 1100 1 0 9 3 T=T+24 1100 IF T > = T l B ( I t ( J ) ) THEN 1110 !HYOROGRAPH ORDINATE PER PIPE

104.9 NEXT J

1290 NEXT J 5 1300 NEXT J 1310 FOR J=1 TO J Z 1320 I F R(J)<=Qm(J) THEN 1340 1330 Qm(J)=Q(J) 1340 I F Q(J)<Pc(J)/l000 THEN 1360 1350 RV (J) =Qv (J) + ( Q (J) -Qc (J) / 1000) +3600+Ti 1360 NEXT J 1370 T i = T J + l 1380 FOR Jn=l TO Nh !HYDRAPH POTNTS 13Y0 R t (Jn, T j )=a (Jh (Jn) ) +lB00 1400 NEXT Jn 1.410 NEXT T h 1420 PRINT "SEWER NETWORK ANALYSIS , " 1430 PRINT N8 1440 PRINT 'I P I P E D I A SLOPE HUNITS LAGh MAXLs QCAP XFIJL.1. OFL.Om3" 1450 IMAGE DDDDDD.DDDDD.D. DDDD.DDDDD.DDD. DD.DDDI)D.T)I)DDD.onnI>. D.DDDD. D 146R FnR J=l TO 52 1470 T 1 ( J ) = T l (J)/360cI 1400 Qm (J ) =Qm (J) +1800 1490 Pf =am (J) /Qc (J ) +100 1500 PRINT USIND 1450:M(J) ,D(J) . S ( J ) .Hq(J) .T1 ( . 7 ) .Qm(,l) . R c ( , l ) . P f . R v ( J ) 1510 NEXT 3 1520 IMAGE DDDDDDDD.# 1525 IMAGE DDDDD.DD.# 1526 IMAGE DDDDD.DD 1527 IMAGE DDDDDDDD 1530 PRINT 1540 PRINT "SELECTED HYDROORAPHS AT":Ti ; "h 1NTVL.S STCIHTTNG AT -T l FKIR PTPF 1545 PRINT USING 1520;Ti 1550 FOR Jn=l. 7'0 Nh-1 1551 PRINT USING 1520;M(Jh(Jn)) 1552 NEXT Jn 1553 PRINT USING 1527:M(Jh(Nh) ) 1560 FOR T k = l TO T i 1565 PRINT USING 15'20:Tk 1570 FOR Jn=l TO Nh-1 1571 PRINT I.JSING 1525:Qt (-7n.Tk) 1572 NEXT Jn 1573 PRINT IJSING 1526:Qt (Nh.Tk)

1575 NEXT TIC 1380 GCLECIR 1600 FOR Jn=l TO Nh 1602 QCLEAR !PLOT HYDROORCIPHS ON SCRN- DUMP GRAPHICS &/OR CONT 1603 C S I Z E 4 1605 O m i = Q m ( J h ( J n ) ) 1610 WINDOW -3,24.5,-0.Qmj*l. 1 1620 CIXES 1,1,0,0,12,10 1630 MOVE l , Q m ( J h ( J n ) ) 1700 Qm (Jh (Jn ) ) = I N T (Qm ( Jh (Jn ) ) * 1000) / 1000 1701 LABEL Qm (Jh (Jn) ) ; "L/s PIPE" : Mh (Jn) 1710 MOVE 6,0 171 1 LCIBEL. NS 1720 MOVE 22.0 1721 LABEL "h 24" 1724 MOVE -2.10 1725 LhBEL 10 1730 FOR T=2 TO T i 1740 T h = ( T ) * T i - T l (Jh(Jn)) 1750 MOVE T h - T i .Qt ( J n , T - l ) 1755 DRAW Th,Qt ( J n . T ) 1760 NEXT T 1765 MOVE -T1 (Jh(Jn)) .Q t (Jn,T-1) 1766 DRAW T i - T l ( J h ( J n ) ) , Q t ! J n , l ) 1770 PAUSE !TYPE CONT (El-) TO DO NEXT t i Y B 13R DUMPFH TO DRAW EX 1780 NEXT Jn 1790 FOR 1=1 TO 6 1800 He(I)=0 1810 NEXT I 1820 FOR J=1 TO 52 1830 H e ( 1 t ( J ) ) = H e ( I t (J ) ) + H q ( J )

1850 NEXT J 1855 PRINT

1840 X l e X 1 + X ( J )

SEWSIM DATA FORM

2 W !

2010

2020

2030

2040

NAME -- DATA .......... NO. OF SECTIONS, SIMULATION DURATION h, TIME INCREMENT.h, NO. HYDROGRAPHS REQUIRED, G.L.(M) BOTM M i .

DATA ........................ .................................... .................. P I P E NO. OF HYDROGRAPHS

DATA .......................................................... SECTION DATA: ( 1 L INE PRECEEDING EACH SECN.)

BOTTOM MH NO., ZONE TYPE (1-41, NO. PIPES I N SECN, MANNINGN, SW/min/HE, STORM Chnm/h, INFILTN L/mm/m/m, EAKG L/mi /HE.

DATA ........................................................................................................................... PIPE DATA: ( I L PER P IPE)

P IPE NO.(=TOP MH NO.), -m, LENGTH m , SLOPE m/m, HE=HDWE EaUIVS( lWm’) , (DEPTH TOP MHm, DROP BOTM mm, G L m

DATA ............................................................................................................................

Give each d a t a f i l e a name - c a l l e d up when SEWSIM is RUN.

Store in ASCII form e.g. SAVE “FILENAME”

187

SAMPLE DATA FILE

SEWDAT (ASC I I FORMAT)

7 0 0 0 DATA 4 , 2 4 , 1 . 4 , 5 0 2 0 0 1 DATA lil,1'21,211,.311., 11.3,1 ,2, . 0 1 J , 1 ,. 1 , 0 , 0 ZQCI? W T A 111 ,100 ,100 , . 0 1 0 0 , 1 0 0 , 1 . 3 0 , 4 9 2061.3 DATA 1 1 2 , 3 0 0 , 3 0 0 , . 6 1 1 0 0 , l ~ 0 , 1 , 0 , 4 8

288'3 DATA 121 ,150 ,1B0 , . 0 1 5 , 1 0 0 , 1 , 0 , 4 7 2806 DATA 1 2 2 , 1 5 0 , 3 0 0 ,.004,75,1,C!Il46 2807 D(?ITA 123,200,i00,.802,95,1.5,30,45 2 0 0 8 DATA 1 2 2 , 3 , 1 , . 0 2 , 1 , 0 , a . . 1 2089 DATA 211,150,100,.81,100,1,0,48 2 0 1 0 DATA 1 2 3 , 4 , 1 , . 0 2 , 1 , 0 , 0 , 0 2 0 1 1 DATA 311,100,iQ0,.002,100,1,0,47 100pI0 END

2804 DATA 112,2,9, .02, I ,0, . 1 , . 0

: YF'B2'mX. 78B VOLUME LABEL: B982.5 F I L E NAME PRO TYPE FiEC/FILE HYTE/HEC ADDPESS

SEWDAT ASCII 3 256 50 BEWSIM PHDG 73 _I L 256 5 3

Note SAME _ - _ _ "SEWDAT", don't STORE ,I...

188

END P I P E WITH NO D.S .P IPE 2 112

SEWER NETWORK hNALYSIS FOR

PIPE DIA SLOPE HUNITS LAGh MAXLs PCAP %FULL 1 1 1 100 .0100 100 . I 2 2 4 37.3 112 308 .0100 100 .07 10 84 12.1 121 150 .0150 100 .44 2 12 14.0 122 150 .0840 75 .40 4 6 6 9 . 3 I23 200 .0020 95 .I6 7 10 73.3 211 150 .a100 100 .45 2 10 16.7 311 100 .00?0 100 .31 2 2 112.5

OFLOm3 0.0 0.0 0.0 0.0 0.0 0.0 0.0

SELECTED HYDROGRAPHS AT I h INTULS STfiRTING AT -TL FOR P I P E ... I 1

3 4 5 6 7 8 9 10 1 1 1: 13 1 3 15 16 17 16 19 20 21 22 23 24

1 L.

1 1 1 * 29 .03

0.00 0.0Q 0.00

. 7 9 1.25 1.56 I .67 1.57 1 .29

. 9 6

.97 ! . 1 8 1.34 1 . 4 0 1.36 1.21 1.17 1.07 .89 .73 .53

. a0

121 .03 .03 . 03 .03 -03 .03 .68

1.45 1.69 1 .?O .96

1 . 1 1 1.18 1.18 1.10

. 9 4

.94 1.03 1.00 1.01 1 .OO

. 6 4

.56

.20

21 1 .I7 .I7 .I7 .I7 .17 .17 .67 1.03 1.32 I .51 1.58 1.52 1.36 1.33 1.51 1.65 I .56 1 .:7

. b l

.51

.37

.23

. I 7

.I7

31 1 0.00 0.00 0.00 0.00 0.00 0.0Q .32 . 7 b 1.10 1.36 1.50 1 .5l 1 . 41 1 . 21 I . 4 7 1 .69 I . 4 3

* 76 .34 .08

0 .Q0 0.08 0.08 0.00

189

-

I 1.684 L /s PIPE 121

1.688 L/s PIPE 31 1

h 24

190

CHAPTER 12

SEWERAGE SYSTEMS MANAGEMENT

The in terest in sewerage systems has increased in recent years a s

management of e x i s t i n g systems i s improved in order to cope w i t h

inc reas ing f lows a n d t o improve catchment water balances. The design,

s imulat ion a n d management of such systems i s the subject of much research

(Yen, 1987). Dual systems pose p a r t i c u l a r problems in o l d e r ,a reas a s

p o l l u t i o n of waterways i s becoming more of a problem. Rehab i l i ta t ion of

o l d systems, i n c l u d i n g r e - l i n i n g to increase throughput i s a lso t a k i n g

p lace (Adams a n d Zukovs, 1987).

The operat ion of a l a r g e u r b a n sewer system was opt imized b y S c h i l l i n g

a n d Petersen (1987) u s i n g I i n e a r programming. The storm/waste combined

sewer system in Brenner, West Germany, comprises sewer pipes,

pumpstations, s torage ponds and a wastewater treatment p lan t . Unless

adequate ly contro l led, the system i s l i a b l e to f lood l o w l y i n g suburbs w i t h

severe economic consequences. The opt imizat ion model was run in

conjunct ion w i t h a catchment s imu la t ion model. The reason f o r t h i s was

tha t the opt imizat ion model was, o f necessity, a s i m p l i f i e d model assuming

l i n e a r const ra in ts . Conduit s torage therefore, a complex func t ion of f low

r a t e as g iven b y the St. Venant f low equations, could not e a s i l y b e

inc luded in a l i n e a r model.

A r a i n da ta co l lect ion network was coupled to a catchment model on a

r e a l t ime bas is to p r e d i c t f low r a t e s (Fuchs e t al., 1987).

LEARNING SIMULATION PROGRAM

The program used a n i t e r a t i v e l e a r n i n g process t o opt imize operat ion o f

the system. That is, successive r u n s used prev ious resu l ts to improve o n

the opera t ing r u l e u s i n g a r t i f i c i a l in te l l igence.

The sewer system s tud ied was designed to t rea t lower f lows whereas

over f low in storms r a n t o r i v e r s a n d lakes. Inc reas ing p o l l u t i o n awareness

forced the system to be improved. At the same t ime a s reduc ing overf lows,

p a r a l l e l ob ject ives were to reduce pumping energy costs a n d a v o i d s t reet

f lood i ng . The problem was set up to min imize a cost func t ion wi thout v i o l a t i n g

const ra in ts . A formal system ( re fe r red to a s a p roduc t ion system) i s

estab l ished w i t h three components:

191

A working memory w i th a l l data

A ru le base

A n interpreter to choose and apply productions

Improved control i s achieved by a l te r i ng the r u l e base or add ing new

ones. For each unsatisfactory production a l i s t i s created.

A meta production systems was fu r ther added. Meta productions do not

affect the working memory but can change the content of the r u l e base.

The meta system is evaluated by the control interpreter. A simple example

demonstates the technique:

Stormflow could be stored i n a detention bas in freely, whi le street

f looding would be an unsatisfactory state. This ru le could be described by

a meta function as flows:

( W E > 1.0) + (pump too L O W ) (Value = - 1 )

Whenever the water level in the sewers i s higher than manhole level which

may cause street f looding th is r u l e i s appl ied. At any selected time i f the

meta production r u l e i s appl icable the decision PUMP = OFF i s counter

ru led i.e. the corresponding productions are decreased b y 1 .

The facts in the working memory at the time may have been

W E = 0.4 where W E = water elevation

R I = 10 where R I = r a i n f a l l intensity.

Another s i tuat ion may also have been stored in the experience memory,

e.g.

WE = low

R I = 10

I t i s possible better productions could have been applied.

The total l i s t i n memory may now be

R I W E

-3 L O W 9

-1 L O W 10

- - Va I ue

-6 L O W a -10 L O W 1 1

A new production i s created tak ing the condit ion p a r t of the o l d one. A

second condition is added of the form

N I op x

where op i s ei ther < o r > and x i s the median value of R I in the l i s t of

experience memory, weighted with the level of punishment.

Thus s ta r t ing wi th the o ld production

( W E = L O W ) + (pump = OFF)

192

the new production w i l l switch pump on because the systems knows th is i s

connected w i t h h igh r a i n f a l l intensi ty.

Hence the new production i s

(WE = L O W ) (N I > 9.75) + (pump = ON)

The new production i s assigned a va lua t ion of 0 and stored in the r u l e

base.

I f the s i tuat ion WE = 0.4

R I = 10

occurs again the last ru les w i l l both app ly , but the la t te r r u l e i s chosen

as hav ing lowest evaluat ion level. Street f looding i s thus avoided.

OPT I M I ZAT ION

The same problem was simpl i f ied into a l i near system for direct

optimization a t discrete times. Sewers were lumped into three

subcatchments.

Wser

Hbsssliisc

I2

-1

I1

Fig . 12.1 Process var iab les for the s impl i f ied systems

193

A ra in fa l l / runof f model was used to compute inf low hydrographs. The

remaining system consists of two of f - l ine ponds and two t runk sewers w i th

backwater effects from the pumps. The system can be described b y 18

var iables (F ig . 12.11, namely:

- inf low I 1 into the pump sump of the downstream pumping stat ion, inf low

12 halfway up the upstream stat ion, and 13 in to the sump of the

upstream stat ion.

- the pumping rates PR3 into the upstream pond, P2 from the upstream

into the downstream system, PRl in to the downstream pond, and PKA to

the treatment p lan t .

- recycled flow from the ponds to the system ( R R l and RR3, respect ively) ,

- the stored sewage i n the t runks (V12 and V3, respectively) and i n the

ponds (R1 and R3, respect ively) ,

- overflow PO1 into the Weser estuary, 01 into the downstream creek

Wasserlose, and 03 into the upstream creek Krimpelfleet,

- flood volumes which cannot be handled b y the system (F12 and F3,

respectively 1. The simpl i f ied model was ver i f ied. This was done through a detai led

and phys ica l l y precise model.

Optimal Control as a L inear Programming Problem

The task i n the operation of the Bremen combined sewer system were

drainage ( i .e. minimization of f looding) and environmental protection (i.e.

minimization of combined sewer f low) whi le keeping the cost of operations

as low as possible. Since i t i s impossible to achieve perfect f lood

protection and no overflow simultaneously p r i o r i t i es have to be specified.

They include:

1 . minimum flooding (F12, F3)

2. minimum overflow into the creeks (01, 03),

3. minimum overlfow into the estuary (Po l ) ,

4. minimum pumping into the ponds (PRl, PR3),

5. minimum use of the ponds ( R l , R3)

Unit costs c are specified for every cubic metre flooded, cubic metre

overflow. etc. Using the technique of l inear programming the operational

optimization problem was formulated as

n z

t= l min cv3tV3t + cR3tR3t + cv12tV12t + c r l tRl t + cRR3tRR3t + cP2tP2t

+ cPR3tPR3t + cF3tF3t + co3t03t + cRRltRRlt + cPKAtPKAt

+ cPRltPRlt + cF12tF12t + cPOltPOlt + col tOl t

1 94

TABLE 12.1 Opt imal Control S t r a t e g y f o r M a j o r Storm 0708

t

C

_--- ----

1 2 3 4 6 6 7 8 9 10 11 12

PM PRl Po1

0.0 1.0 10

1630 0 0 3800 1277 0 3600 9414 691 3600 Be68 9000 3600 6346 6346 3600 2946 2946 3600 1373 1373 3800 603 603

3600 0 0 3600 0 0 3600 0 0

,----------------

.----------------

$600 a 0

93

0.26

1730 1730 1730 1730 1730 1730 1730 1730 1730 1730 1262 193 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

----- -----

1280 1280 1260

0 1260 1260 1260 1260

14268 1277

14268 10000 'I4268 10000

ill88 10000 14268 10000 14268 10000 14268 10000 14264 10000 13814 10000

13088 10000 13049 10000

1260 1260 1260 1260 1260 1260

13 14 16

11037 10000 10108 10000 9125 10000

3600 0 0

3600 0 0 3800 0 0 3600 0 0 3800 0 0 3600 0 0

3800 0 0

16 17 18 19

air4 10000 7163 10000 6182 10000 5201 10000

0 0 0 0 0 0 0 0 0

1260 1260 1260 1260 1280 1260 171 171 171 171

0 0 0 20 21 22 23

4220 10000 3231 10000 1612 9549

3800 0 0 3600 0 0 3600 0 0

0 0 0 0 0 0 0 0 451 -~~~

531 6560 531 6490' 531 4420 531 2350 531 260 531 0 531 0

3600 0 0 3600 0 0 3800 0 0 3600 0 0 3600 0 0 1610 0 0 1530 0 0

0 0 989 0 0 2070 0 0 2070

24 25 28 27 28 29

0 0 2070 0 0 2070 0 0 280 0 0 0

171 171 171

t R3

c 0 . 3

1 4944 2 9600 3 9600 4 9600 5 9600 6 9600 7 9600 8 9600 9 9600 10 9600 11 9600 12 9600 13 6704 14 7615 15 6626 16 5437 17 4848 18 3269 19 2170 20 1081 21 0 22 0

. - - -__ - - - - -

.----------

PR3 F3

1.0 1001

4944 0 6588 5710 6588 270 6032 0 3656 0 2047 0 2011 0 1577 0 997 0 396 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

------------ --__----_

1932 12283 9972 12715 8410 5080 3519 2752

13558 6118 6032 4916 3307 3271 2837 2257 1656 812 171 171 171 171 171 171 171 171 171 171 171

3483 1762 752

0 0 0

6588 6032 3656

._

275 205 194 191

0 0 0

2047 2011 1577 997 396 0

0 0 0

2147 189 189 189

1701 1386 1170 1170 1170 1170 1170 1170 1170

0 896 1069 1089

0 0 0 0

i 89 189 189 ias 189 189

1089 1089 1089 1089 1089 1081

0

0 0 0

~~~

189 189 189 189 189

0 1170 1170 1170

0 0 0

195

subject to the capacity constraints

V3 5 1730 m3

R3 5 9600 rn’

V12 5 14268 m’

R1 5 10000 m3

P2 5 0.70 m’/s

PR3 5 3.66 m’/s

PKA 5 2.00 m’/s

PRI 5 8.20 m’/s

PO1 5 5.00 m’/s

and the dynamic constraints for each of the four storage un i ts

V3t+l - V3t - RR3t + P2t + PR3t + F3t = 13t

R3t+l - R3t - PR3t + RR3t + 03t = o V12t+l - V12t - P2t-2 - RRlt + PKAt + PRlt + F12t = I l t + 12t-1

R l t+ l - R l t - PRlt + RRlt + POlt + Olt = o

The flow time from the inf low si te 12 to the downstream pump was

taken as one time step ( i .e. 30 min) and the flow time between the two

pumping stations as two time steps.

The problem was solved with standard software. A typical resul t i s

presented i n Table 12.1 fo r a 210 min storm and inf low forecasts of one

time step only (0 to 30 min from actual time). The table includes un i t

costs c of the objective function. Sensi t iv i ty analyses showed that these

could be specified qui te a r b i t r a r i l y , provided that un i t costs of d i f ferent

orders of magnitude are al located to objectives of d i f ferent p r i o r i t y .

SEWER MAINTENANCE DATA PROCESSING IN JOHANNESBUG

Johannesburg has near ly four thousand kilometres of sewerage to

operate and maintain on a continuous basis. Many of the areas are prone

to abuse and blockage and the nature of the topography and cl imate make

maintenance a h igh cost i n the system. That is, intense storms often resul t

in ingress into sewers and th is may b r i n g surface debr is and other

foreign matter which block the sewers. There i s also unauthorised access

in many places i t i s suspected, as a r t i c les obviously not from the

sanitat ion system are often found in sewers. Despite the h i g h ra te of

1%

growth i n Johannesburg many of the sewers are o l d and some are of poor

qua l i t y requ i r i ng regu la r maintenance and replacement and repairs.

While a t f i r s t i t may appear that ready a v a i l a b i l i t y of labour in South

Afr ica should fac i l i t a te c leaning and a t the same time the maintenance

should provide labour opportunit ies, the management of such a system

obviously imposes severe problems a t h igher levels. Maintenance of logs

fo r iden t i f y ing trouble spots i n the system would be of great va lue to

managers and sewerage engineers. This type of data i s useful fo r

budgeting for repa i r work such as to manholes and even pipes requ i r i ng

replacement as well as minor items such as manhole l i ds and stepirons and

benching i n manholes. There i s also much to be gained from ana lys is of

maintenance data in the way of types of blockage. For instance local i t ies

where foreign objects a re frequently encountered can be narrowed down

and the inhabi tants of that township made aware of the troubles caused b y

such pol lut ion. Where sand i s frequently found in sewers i t may point to

roads requ i r i ng surfacing as stormwater can reach sewers b y unexpected

ways.

Overflows and l i f t i n g of manhole l i ds in cer ta in areas may point to

inadequate sewer capacit ies. Al ternat ively they may indicate corroded

sewer l i n ings o r roots which block the sewers. Here aga in ident i f i ca t ion of

frequency and loca l i t y of such inadequacies indicates where maintenance i s

most urgent ly required.

The human management side i s also very complex. The ou t l y ing depots

where such maintenance takes place employ some s i x hundred people, which

are general ly organized into gangs a t each depot. The supervisors report

to managers who take messages and transmit the teams to problem points.

Even managers and frequently supervisors a re not h igh l y t ra ined and the

type of logs they keep are often d i f f i c u l t to process. However the

computerization of the log keeping on an experimental bas is a t one of the

depots has proved sat isfactory and w i th in the capab i l i t ies of the ex is t ing

type of staf f . Terminals connected to the mun ic ipa l i t y ' s main computer a t

head off ice are used and once basic keyboard s k i l l s have been picked up

then spread sheet type data logging has proved possible and i n fact of

great advantage to the engineers a t head of f ice concerned w i th p lann ing

and the engineers concerned w i th budget ing and maintenance and design.

Although Johannesburg's obvious solut ion i s through i t s mainframe

computer wi th ou t ly ing terminals, in fact many smaller mun ic ipa l i t ies may

resort to mini o r even micro computers to handle the i r system. The la t te r

would be popular wi th the smaller mun ic ipa l i t ies where one stat ion only i s

maintained.

197

The use of micro computers also enables micro graphics to be used to

ident i fy trouble areas. A screen map can h igh l i gh t zones w i th frequent

blockages. With the advent of the computers many f ie lds i n the C iv i l

Engineering f ie ld have been opened up to the benefi ts which can accrue in

both the design and constructional areas and the management and

administrat ion areas. Due to i n i t i a l costs and a na tura l reluctance to

adopt new methods progress i s sometimes slow but i t can usua l ly be sa id

that while computers do not necessarily save money they can de f in i te ly

give better resul ts a t the end of the day.

AppI icat ion to Johannesburg's system

Thus i t was with th is intention of g i v ing an improved service that

Johannesburg has persevered with computerization of many of i t s functions.

This chapter outl ines the progress made in the provis ion and maintenance

of sewerage ret iculat ion.

The analysis of sewer systems to ident i fy potent ia l overloading b y

sewers has already been established and has been used to analyse

townships for exist ing and future flows. In some cases the effect of

subdivision of stands has been assessed and accurate estimates of costs

given for addi t ional sewerage work (Stephenson and Hine, 1982 and 1985).

Sewer ret iculat ions need regu la r planned cleansing i f serious f looding

and subsequent danger to heal th i s to be avoided. I f regu la r cleansing of

pub l i c sewers i s well organized many of the blockages which occur can be

avoided. Maintainance of p r i va te l y owned sewers i s not the responsibi l i ty

o f the sewerage author i ty bu t i n Johannesburg i t i s the o f f i c ia l po l i cy to

unblock these sewers i f asked to do so b y the owner. In many cases the

owner i s the local author i ty so that there i s a vested interest to ensure

that these are well maintained so as to reduce the number of blockages.

Conventional systems have been used to record the work car r ied out

using cards etc. which has been successful but time consuming. I t was

considered that records of cleansing work and the clear ing of blockages

could be more effect ively done by computer and that re t r ieva l of records

and planning of work would be made easier.

Consequently the Maintainance Data System has been establ ished and i s

being applied where Sewer Data has been establ ished g i v i n g sizes and

lengths of sewers together w i th a unique manhole numbering system.

198

Data i s compiled by depot administrat ive staf f on Forms wich have a

numerical format sui table for input to the computer. Detai ls are abstracted

from work reports da i l y .

Forms used in the f i e ld g ive township and street names which become

numerical township codes and manhole numbers before being entered into

the computer. Incorporated i n the cleansing report i s an inspection of each

manhole and sewer length inc lud ing the measurement of the depth of flow.

Processing of Sewer Maintenance Data

The processing of sewer maintenance da ta has reached an advanced

stage using the programs and techniques described below.

The workforce is d iv ided in to gangs which work on ei ther c leaning of

sewers o r c lear ing of blockages.

The cleaning of sewers i s recorded b y the gang leaders i n the f i e ld

each day and manhole numbers are obtained from keyplans showing the

sewer network.

On the fol lowing working day information i s abstracted b y depot

administrat ive staf f and inserted i n a numerical format. (Table 12.1)

TABLE 12.1 Cleansing of Sewers

nEcono OF SYSTEMATIC CLEANINO L SEWERMANHOLE CONDITION

199

Program UPDATE i s then used to provide de ta i l s of sewer diameter,

length and slope which i s added to the data f i le . These de ta i l s a re

obtained from the Sewer Data F i le (Table 12.2)

A MERGE program i s used to add new data to a master maintenance

f i l e (Table 12.3).

Program MAINTENANCE produces a report of the sewers cleaned between

given dates as required and pr in ted out according to each township.

Program GANGS produces a report of work car r ied out by each gang

between given dates.

This report could form the basis for a bonus scheme (Table 12 .4 ) .

Blockages are recorded as reported with the time and date recorded.

The detai ls of actual clearance g i v ing the time started and completed

appear on work report sheets and enable data to be completed. "Private"

blockages which are w i th in stand boundaries are denoted b y a stand

number and "Main" blockages which occur in pub l ic sewers are denoted b y

a manhole reference number.

Reports which are found to b e problems i n the water re t i cu la t ion o r

storm water system are given a code which enables the computer to ignore

that report apart from showing how many of the reports have been referred

elsewhere for act ion (Table 12.5) .

Program BLOCKMACRO produces a report of blockages in townships

between given dates. The length of time taken to c lear the blockage and

possible cause is shown. The time which elapsed between the report and

completion of clearance i s also calculated to help ident i fy administrat ion

problems, lack of staf f etc. The severity of a blockage i s also shown b y

ind ica t ing the number of houses flooded as the resul t of a "main" blockage

and for a "pr ivate" blockage i f the house or ya rd i s flooded (Table 12.6) .

A macro program produces a report of a l l the work each gang has

done in unblocking sewers between given dates. Numbers of blockages and

total time spent i s shown (Table 12.7).

A program produces a report of a l l the stands and sewer lengths where

there has been more than one blockage in a given time period. This

information can be very useful in ident i f y ing possible defects and

overloading of pub l i c sewers and also when answering queries about

repeated blockages on p r i va te stands.

TABLE 12.2 Example of Updated Sewer D a t a F i l e h) 0 0

35032913086120202 51.01.5 35032913186120203 41.01.0 350329 13286 120203 ti 1.0 1.0 35032913386120203 &l. 01.5 350329 13986 120202 5 1.02.0 35032914086120202 51.02.5 35032914186120202 51.03.0 35132915086120103 01.01.0 35132915106120103 61.01.0 35132915286120103 41.02.0 35132915306120103 41.01.5 35132915486120102 51.02.0 35132915586120102 51.02.0 35132915686120102 51.01.5 35132315786120102 SO. 50.5

TABLE 12.3 Sewer Maintenance Records

ISEUER PlCIINlEWNCE R M R D S

REOUESTED CTLIRT WITEX- 861201 REOUESTED END DRTE I- 861204

TOUNSHIP sso - uwsm

1

2 2

3 5 1 2

6 1

5 4

1

l4RN GRNG tcIhlti SEWER NO No SIZE W R S LENGTH ~ ~~~-

329130 2 S 1.0 IS.% 3-2131 S 0 1.0 70--% 3--91;2 3 A 1.0 57.79 z2913 1 A 1.0 60.01 229139 2 S 1.0 A9.22 ‘329100 2 5 1.0 75-68 3,3101 2 S 1.0 A9.10

!saNn BKTS

1.S 1.0 1.0 1-s 2.0 2. S J. 0

DEBRIS

RUB R f f i MET YO00 GLRSS 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

1

1 3 1 2 2

1 1 3 5 6 2 4 3 6 3 2 2 1 1

152 152 152 152 152 152 152 152 152 152

1 152 152 152 152 152

15.54 100.0 70.26 100.0 57.79 100.0 60.01 00.0 49.22 80.0 75.68 00.0 49.10 110.0 76.90 70.0 70.73 70.0 64.00 60.0 62.97 60.0 65.40 66.0 62.97 46.0 63.03 79.2 63.00 79.2

ROOTS FRT 0 0 0 0 0 0 0 0 0 0 0 0 0 0

MRNMOLE CONDITION SEUER CONDITION

BEN 1NV U W SLBB COV FRR STEP COP FLOu RS X JNTS PIPE OIL DEPTH DF

0 0 0 0 0 0 1 0 19. 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1z.

0 0 0 0 0 0 0 0 12. 0 6 0 0 0 0 0 0 0 0 1 6. 0 0 0 0 0 0 0 0 0 0 6 19. 0 6 0 0 0 0 0 0 0 0 6 22. 0 6 0

0 0 0 1 0 0 0 0 0 0 0 29.

TOTnLSI 7.0 S77.60 12.S 1 2 1 0 1 0 0 1 0 0 0 0 0 1 1 ISEUER PlCIINlENRNCE RECORDS

6 1 6

c

0 N

86 TV9 oo '0

vv 'VSt 00 '0 vs '68 I 00 '0 H18N31 +OOOI

1UlOl :tj313wu1a

00 '0 00 '0

00 '0 00 '0 00 '0 00 '0 666/00S 662/002

do U3NU313 Stj3M3S

86 'SVV s 'L

vv 'f6Z ST I 1x3 UISUN31 9s '68 f 0 '* U I SUN31 66f/001 StlllOH 3WUN do SHUN31 NU313 d IHSNMOl

- : s7u101 - f SS 'OSZ

3a03NMOI

N 0 N

TABLE 12.6 Output f i l e

1SEULR OLOCKAGC R C C O R D S

R L P U E S I E D S T A C T DATE:- 8 7 C l U S *L:UC:TED CND D A T i :- 1175107

TOYNSnXP 5 2 - B O S l O W T 40. OF STANDS - 1 4 1 8 AREA(WECTARES> - 106.1

BLOCKAGL PRIVATE: nA1M: ILOCKAGE CLEARED M O U I S FRO@! D C P O R T F D STAND SLYER S T A R T F I N I S H J O B I E P O R T l N G TO GANG GANG MOUSES 11110 MOUSE M O U l l D A T E NO no DATF T l R f DATE T I R E T I R C CORPLETIOM SUMDAT NO S I Z E CAUSE L F F E C T E D FLOODED FLOODED

R A I N : C I I V A T L :

I ~ 7 ~ 1 ~ s 0 z z ~ i m 8 7 r i . r ~ i z :40 87i t ios i 4 : o o 1.20 6 0 5 1 4 2 I! n

Z O n P L A I h T : R E F i R R E O TO b A T E R S R A N C M - 0 EC’PLAIhTS R L F E R R L O 1 0 R O A D S AND YORKS - D

I S E Y i l l BLOCKAGE R C C O R D S

R L C U E S l r O S l 8 R T DATE:- 117OlOf D L i U C S T L D E k J C A T i :- 6 7 0 1 0 7

l O Y N S H I P 157 - ELDORAEO PAPK 30 . OF S T A W S - 19n9 AREACMECTLRES) - 128.6

@LOCKAGE P R I V A T E : RAIN: BLOCKAGE CLELRED M O U R S cnon MA1N: PRIVATE:

niui D A T L ID L’J DATL T I B E D A T E T I N E T I M E COf iPLETIOH . SUMDAT N O S l Z E CAUSE AFFECTED FLOODED FLOFDED J O B R E P O R T I N G TO GANG GAYG nousEs T A ~ D HOUSE i E P O P T l L S T A G O SEYER START F l t i i s n

? C 7 u l u b 4 J l o I I C I ? ~ I:DD 8 7 ~ 1 0 6 a:so 0.30 2 0 2 3 2 0 1 1

REFERENCES

20 3

Adams, B.J. and Zukovs, G., 1987. P r o b a b i l i s t i c models f o r combined sewer systems r e h a b i l i t a t i o n s ana lys i s . In Beck (1987).

Beck, M.B. (Ed.) 1987. Systems Ana lys i s in Water Q u a l i t y Management. IAWPRC Conf. London, Pergamon

Fuchs, L., Mu l l e r , D. a n d Neumann, A., 1987. L e a r n i n g p roduc t i on systems fo r the contro l o f u r b a n sewer systems. In Beck (1987).

S c h i l l i n g , W. a n d Petersen, S.O., 1987. Real t ime operat ion o f urban d r a i n a g e systems, v a l i d i t y a n d s e n s i t i v i t y o f op t im iza t i on techniques. In Beck (1987).

Stephenson, D. a n d Hine, A.E., 1982. Computer a n a l y s i s o f Johannesburg Sewers. Proc. Instn. Munic. Engrs. S.A. IMESAF, 7 (4 ) A p r i l . p13-23

Stephenson, D. and Hine, A.E. 1985. Sewer Flow Modules f o r Va r ious types o f development in Johannesburg. Proc.. I ns t . Munic. Engrs. S.A. (10)

Stephenson, D. a n d Hine, A.E., 1987. Maintenance p rog ram f o r

Yen, B.C. (Ed . ) 1987. Proc. 4 th I n t l . Conf. U rban Storm Water Hydro logy

Oct. p31-41.

Johannesburg Sewerage Systems.

and Dra inage, Lausanne.

TABLE 12.7 Outout f i l e

SLYER BLOCKAGE RECORDS

l E O U E S T E D S T l R l DATE:- 070105 SEOUESTED END DATE :- 870107

€ A M : 1 SIZE: 4

NO. OF NO. OF TOTAL T o Y N s n I P P R I Y I T E n A I N NO. O f PRIVATE- MAIN- TOThL-

lOYNCODE l A M E BLOCKbCLS BLOCKAGES BLOCKAGES JOB T I M E JOB T I M E JOE T I M E 155 E L D O R l D O PARK2 2 0 2 b.25 0.00 4.25 177 E L D O R A D O PARK4 4 0 4 7.25 0.00 7.25 3 4 2 K L I P S P R U I T YES 1 0 1 0.25 O.OG 0.25

TOTALS:- 7 0 7 12.15 0.00 12.15

204

CHAPTER 13

WATER QUALITY MON I TOR I NG NETWORKS

by Thomas G. Sanders, Colorado State U

NECESS I TY FOR NETWORKS

i ve i Y

Environmental legis lat ion and general water qua l i t y awareness have

been responsible for recent increased monitor ing and sampling of water i n

streams. Such monitoring and test ing can be expensive and a scient i f ic

approach to minimizing costs whi lst maximizing benefi ts i s desirable.

The assumption that a water qua l i t y monitor ing network can detect

trends i n water qua l i t y , check compliance w i th stream standards, and

measure ambient water qua l i t y , etc., i s incorporated into legis lat ion for

water qua l i t y management i n the United States. The legal view of water

qua1 i t y monitoring envisages conclusive information being generated to

ac t ive ly guide government's water qua l i t y management efforts. When

implemented, however, water qua l i t y monitor ing i s viewed more from a

technical feasibi I i t y stand-point. That is, the problems involved in

obtaining conclusive information w i t h the ava i l ab le resources force many

compromises and ha l f measures, the consequences of which are often not

f u l l y understood.

Monitoring performed by government agencies is, i n many cases,

conducted over large geographic areas (def ined b y po l i t i ca l and not

necessarily hydrologic boundaries) covering many k i lometres of streams.

Simply col lect ing samples in such a s i tuat ion often becomes a major

problem; so major, i n fact, that i t becomes an end i n i tsel f . In many

cases, l i t t l e thought i s given to the representativeness of the water

samples o r types of da ta ana lys is techniques to be used or even the

ul t imate use of the data. Consequently, the major i t y of resources a re

devoted to col lect ing data as i t i s the most immediate problem.

By using most resources to phys ica l l y col lect water samples, l i t t l e

resources are lef t to consider the representativeness of the sample i n time

and space, data analysis o r data use. A balanced (col lect ion versus use)

monitoring system should therefore be developed so the en t i re monitor ing

system should be examined and designed simultaneously ( a systems

approach).

The purpose of t h i s chapter i s to review the monitor ing system and

then del ineate the impacts that such a systems approach of monitor ing w i l l

have on network design b y considering the water qua l i t y var iab les to be

205

monitored, the sampling location and sampling frequency.

MONITORING SYSTEM FRAMEWORK

Before a monitoring network can be designed the goals of the

monitoring program should be delineated, and specif ic objectives applied.

I n addit ion, the decisions to be made based upon information from the

network and the subsequent actions should also be well developed p r i o r to

the collection o r a s ingle b i t of data.

The actual operation of a monitoring system can be categorized into

f i ve major functions:

1 . Sample Collection

2. Laboratory Analysis

3. Data Handl ing

4. Data Analysis

5. I nformation Ut i I izat ion

These f i ve functions serve as the feedback loop from in-stream water

qua l i t y conditions of water qua l i t y management decision making. A

management agency i s constantly making decisions (e.g. re la t i ve to s i te

approvals, regulat ions, pol lut ion abatement, etc.) that affect water

qua1 i t y . Without a monitoring feedback loop accurately documenting the

effects of those decisions, the management's past success and fu tu re

direct ion are uncertain.

Monitoring network design i s an over r id ing ac t i v i t y (covering the f i ve

operational functions l is ted above) that should care fu l l y integrate sample

collection (e.g. location and frequency) w i th the type of data ana lys is

used to obtain the information required and ac tua l l y u t i l i zed in decision

making. Thus, the design of water qua l i t y monitor ing networks must take

into account the decision making process, the type and level of s ta t i s t i ca l

analysis appl ied to the data, and ul t imate use of the data collected.

FACTORS I N NETWORK DESIGN

Monitoring network design, as a planning/design type function which

guides monitoring operations, can i tsel f be broken down into three major

componen ts:

1. Selection of Water Qua l i t y Variables to Monitor

2. Sampling Station Location

206

3. Sampling Frequency

The term water qua l i t y va r iab le i s used instead of water qua l i t y

parameter because water qua l i t y i s a random va r iab le and can be defined

b y stat ist ical parameters such as the mean and standard deviat ion. I n

addi t ion, the term parameter i s most often used to define constants of

determinist ic equations o r models and i t can lead to confusion b y

ident i f y ing i t as a random var iable.

Each of these factors i n network design effects a l l the monitor ing

system's operational functions I isted previously and vice versa. The degree

of impact, however, depends upon the purpose and goals of the monitor ing

system.

SELECT ION OF WATER QUALITY VARIABLES TO MEASURE

The selection of the water qua l i t y va r iab le to be sampled w i l l depend

to a la rge extent on the objectives of the sampling network and the

background o r frame of reference of the i nd i v idua ls responsible fo r

developing the objectives of the monitoring network. When a sampl i ng

network has i t s p r imary objective to monitor compliance w i th stream

stndards, the var iab les sampled are the ones specif ied i n the legis lat ion,

fo r example, dissolved oxygen (DO) . DO i s sampled because stream

standards specify a minimum level which should not be violated. Dissolved

oxygen and other var iab les deemed most important and included in stream

standard legis lat ion were those related to water supply, col iform bacter ia,

biochemical oxygen demand (BOD), temperature, t u rb id i t y , and suspended

and dissolved solids, because most i nd i v idua ls enter ing the f i e ld of water

qua l i t y management du r ing the last few decades have a background in

san i ta ry engineering.

Since ind iv idua ls other than besides san i ta ry (environmental) engineers

became interested i n water qua l i t y , the number of water qua l i t y var iab les

which should be sampled rout inely has increased. This compounding

syndrome cannot and should not be the major va r iab le selection mode for a

permanent, rout ine sampling program, bu t instead can be eas i l y

accommodated i n the much discussed synoptic surveys. The increasing

popu lar i t y of synoptic surveys w i th sampling agencies i s probably due to

the fact that the surveys are in fact an appl icat ion of a systems approach

to water qua l i t y monitoring. Unl ike the permanent, rout ine sampling

programs, the objectives, the use of the data, the sampling locations, the

sampling frequency, the var iab les to be sampled as well as the da ta

207

analysis procedures and decisions to be made should be developed

completely before the survey i s undertaken.

Both sampling location and sampling frequency can be developed

independently of the water qua l i t y va r iab le to be analyzed, as both

location and frequency are specified for the col lect ion of the water sample

( the analyses are made la te r ) . However, both c r i t e r i a a re affected by the

water qua l i t y var iab le being monitored. For example, sampling once a

week at a s ingle point i n a r i v e r may be more than adequate for

monitoring the re la t i ve ly stable r i v e r temperature, but may be ha rd l y

adequate for monitoring r a p i d l y va ry ing coliform bacter ia concentrations.

Therefore, before a water qua l i t y monitoring network can be designed i n a

systematic fashion, the var iables to be monitored should be specif ied so

that their na tura l and/or man-made var ia t ion i n time and space can be

considered when designing the monitoring network. In addi t ion to

considering water qua1 i t y variables, the i r respective un i ts should be

delineated. Network design d i f fe rs i f a d a i l y mean ( f low weighted)

concentration i s needed as opposed to an instantaneous g rab sample

concentration, the former being a resul t of several samples wi th flow

measurements spaced du r ing a 24-hour period, whi le the la t te r comprises

only a single sample (general ly i n the daytime, between 8.00 a.m. and

4.30 p.m.1.

I n rea l i t y , the specif icat ion of the water qua l i t y va r iab le to be

monitored p r i o r to i n i t i a t i n g network design would be ideal. In practice,

however, network design i s specified and one must know o r determine what

water qua l i t y var iables can be accurately monitored w i th the ex is t ing

network.

SAMPL I NG STAT ION LOCAT ION

The location of a permanent sampling stat ion i n a water’ qua l i t y

monitoring network i s probably the most c r i t i ca l aspect of the network

design, but a l l too often never proper ly addressed. Expediency and cost

comprises lead i n many cases to sampling from br idges o r near ex is t ing

r i v e r gauging stations. Whether the s ingle g rab sample from the br idge o r

the gauging stat ion i s t r u l y representative of the water mass being

sampled i s not known, bu t general ly i s assumed to be b y both the

collectors and users of the water qua l i t y data. Using r i v e r stage for

est imating discharge, measurement anywhere i n the la te ra l transect would

indicate exact ly the r i v e r discharge. However, t h i s does not necessari ly

follow when measuring water qua l i t y var iab le concentrations. In fact

208

Fig, 13.1 Macrolocation of Sampling Stations Within a River Basin Using the Percent Areal Coverage a s the Cr i te r ia Specifying Locat ion

209

research indicates the opposite, that ra re l y w i l l a s ingle sample be

indicat ive of the average water qua l i t y i n a r i v e r cross section.

Sampling locations fo r a permanent water qua l i t y network can be

classi f ied into two levels of design: macrolocation and microlocation, the

former being a function of the specific objectives of the network and the

la t te r being independent of the objectives bu t a function of the

representativeness of the water sample to be collected.

The macrolocation w i th in a r i v e r basin usua l ly i s determined b y

po l i t i ca l boundaries, areas of major pol lut ion loads, populat ion centres,

etc. Macrolocation can be specified, as well, according to percent areal

coverage using basin centroids (Sanders et a l , 1986). This methodology

locates sampling points in a systematic fashion maximizing information of

the ent i re basin wi th a few strategical ly located stations. F igure 13.1 i s

an example of locating sampling stat ions using bas in centroids and

sub-basin centroids w i th percent areal coverage as the c r i te r ia .

The procedure for locating sampling stat ions i s derived b y determining

the centroid of a r i v e r system. Each cont r ibu t ing exter ior t r i bu ta ry ( t h i s

i s a stream without defined t r ibu tar ies ) i s given the value of one; an

in te r io r stream resu l t ing from the intersection of two exter ior t r ibu tar ies

would have a value of two. Continuing downstream in the same manner, as

streams intersect, the resul tant downstream stretch of r i v e r would have a

value equal to the sum of the values of the preceeding intersecting

stream. At the mouth of the r i ve r , the value of the f i na l r i v e r section w i l l

be equal to the number of contr ibut ing exter ior t r ibutar ies, 22 in F igure

13.1. D iv id ing the value of the f i na l stretch of the r i v e r b y two, the

value of the centroid of the basin, 1 1 i s calculated. The section of r i v e r

hav ing a value equal to that of the centroid d iv ides the bas in into two

sections and i s the location of the sampling stat ion w i th highest order

( the assumption i s made that there exists a sampling stat ion a t the mouth

of the r i v e r bas in ) . I n many cases, when app ly ing th is procedure to a

r i v e r basin, there i s usua l ly not a stream hav ing a value equal to that

of the centroid. When th i s occurs, the stream segment hav ing a value

closest to the centroid i s chosen. The next order of sampling locations i s

determined by f ind ing the centroid value of the two equal sections above

and below the i n i t i a l r i v e r basin centroid. The procedure i s continued

u n t i l a percentage of areal coverage i s attained.

The percentage of area coverage specified by the monitoring agency i s

defined as the number of sampling stat ions d iv ided b y the magnitude of

the basin. I n t r i ns i c in th i s objective procedure i s the concept of a

sampling stat ion hierarchy that orders the importance of each sampling

210

stat ion in the basin (Sharp, 1973). This provides a rea l i s t i c methodology

i n which a ra t iona l implementation progam can proceed: the most important

stat ions (highest order) a re b u i l t f i r s t and as the resources become

avai lable, addi t ional stat ions can be bu i l t . As each succeeding h ie rarchy

of stat ions are establ ished the percentage of a rea l coverage i s increased.

Having establ ished the macrolocations w i th in a r i v e r basin, the

microlocation specifies the r i v e r reach to be sampled whi le the

microlocation specifies the point in the reach to be sampled. This point i s

the location of a zone in the r i v e r reach where complete mix ing exists and

only one sample i s required from the la te ra l transect in order to ob ta in a

representative ( i n space) sample. Being a function of the distance

downstream from the nearest ou t fa l l , the zone of complete mix ing can be

estimated using var ious methodologies.

Given the assumptions that a point source po l lu tan t d is t r ibu t ion in a

stream approximates a Gaussian d is t r ibu t ion , and that boundaries can be

modelled using image theory, the fo l lowing equation can pred ic t the

distance downstream in a s t ra igh t , uniform channel from a point source

po l lu tan t to a zone of complete mix ing (Sanders et al. , 1977).

(J 2u - Y

L Y - 2oy (13.1)

where L i s the mix ing distance fo r complete la te ra l mixing, a y i s

distance from source to farthest la te ra l boundary, u i s mean stream

velocity and D i s the la te ra l turbulent d i f fus ion coefficient.

Estimates of D can be made using equation 13.2

Y

Y

Y

D = 0.23 du' (13.2) Y

where d i s depth of flow u* i s shear velocity g i s accelerat ion f low

due to g rav i t y R i s hydrau l i c rad ius S i s slope o r the hyd rau l i c gradient

(Sanders et al., 1977).

Unfortunately, there may not exist in a given r i v e r reach any points

of complete mix ing due in p a r t to the random nature of the aforementioned

mix ing distance, i napp l i cab i l i t y of the assumptions used in the

determination of the m ix ing distance, o r more often than not, not enough

r i v e r length o r turbulence to assure complete m ix ing w i th in the specif ied

r i v e r reach. On the other hand f i e ld ver i f i ca t ion of a completely mixed

zone p r i o r to locating a permanent sampling stat ion can be easi ly done b y

col lect ing mul t ip le samples in the cross section and ana lyz ing the da ta

using a we1 I-known one- o r two-way ana lys is of var iance techniques.

21 1

I f there i s not a completely mixed zone in the r i v e r reach to be

sampled, there are three al ternat ives:

( 1 ) Sample anyway a t a s ingle point and assume i t i s representative ( t h i s

i s a general approach adopted today);

( 2 ) Don't sample the r i v e r reach a t a l l , because the data which would be

obtained does not represent the ex is t ing r i v e r qua l i t y , b u t only the

qua l i t y of the sample volume collected. In other words, the data i s

useless;

( 3 ) Sample a t several po in ts in the la te ra l transect col lect ing a composite

mean, which would be representative of the water qua l i t y in the r i v e r

a t that point in time and space.

I f the sample i s not representative of the water mass, the frequency of

sampling as well as the mode of data analysis, interpretat ion and

presentation and the rea l i s t i c use of the data for objective decision

making becomes inconsequential. I n spi te of t h i s fact , c r i t e r i a to establ ish

stat ion locations for representative sampling have received re la t i ve l y l i t t l e

attention from many inst i tut ions and agencies responsible for water qua l i t y

monitoring.

SAMPLING FREQUENCY

Once sampling stat ions have been located to ensure samples collected

are representative i n space, sampling frequency should be specif ied so

that the samples are representative in time.

Sampling frequency a t each permanent sampling stat ion w i th in a r i v e r

basin i s a very important parameter which must be considered i n the

design of a water qua l i t y monitoring network. A la rge port ion of the costs

of operating a monitoring network i s d i rec t l y related to the frequency of

sampling. However, the r e l i a b i l i t y and u t i l i t y of water qua l i t y data

derived from a monitoring network i s l ikewise related to the frequency of

sampling. Addressing th i s anomaly Quimpo (1968) summarized the

signif icance of sampling frequency and stated that:

On the one hand, b y sampling too often, the information

obtained is redundant and thus expensive, and on the other

hand, sampling too infrequent ly bypasses some information

necessitating an extended period of observation.

Signif icant as sampling frequency i s to detecting stream standards

v io la t ion , maintaining eff I uent standards, and estimating temporal changes

i n ambient water qua l i t y , very l i t t l e quant i ta t i ve c r i t e r i a which designate

appropriate sampling frequencies have been appl ied to the design of water

21 2

qua l i t y monitoring networks. In many cases, professional judgment and

cost constraints provide the basis for sampling frequencies. A l l too often,

frequencies are the same a t each stat ion and based upon rou t ing

capabi l i t ies, once-a-month, once-a-week, etc. and al though possibly the

on ly p rac t ica l means to implement a sampl i n g program considering the

s ta t i s t i ca l background of data collectors, there do exist many quant i tat ive,

s ta t i s t i ca l l y meaningful procedures to specify sampling frequencies a t each

stat ion (Sanders and Adrian, 1978). The methods include speci fy ing

frequencies as functions of the cyc l i c var ia t ions of the water qua l i t y

var iab le (Nyquist frequency), the drainage bas in area and the r a t i o of

maximum to minimum flow (Pomeroy and Orlob, 19671, the confidence

in te rva l of the annual mean (Ward et a l , 1976; Lof t i s and Ward, 1978),

the number of data per year for hypotheses (Sanders and Ward, 1978), and

the power of a test measuring water qua l i t y intervent ion (Lettenmaier,

1975).

A l l of the aforementioned procedures can be app l ied to the design of a

water qua l i t y monitoring network w i th each requ i r i ng a di f ferent level of

stat ist ical sophist icat ion insofar as data requirements as well as

assumptions app I y . One of the simplest approaches i s to assume that the water qua l i t y

var iab le concentrations are random, independent and ident ica l l y

d is t r ibu ted ( i i d ) and determine the number of samples per year as a

function of an al lowable (specif ied) confidence in te rva l of the mean annual

concentration ( t h i s i s analogous to the procedure for determining how many

analyses of a water sample should be made to determine a reasonable

estimate of the mean water qua l i t y va r iab le concentrat ion).

n = [ a izS ] (13.3)

where n i s the number of equal ly spaced samples collected per year, ta I2

i s a constant which i s a function of the level of s igni f icance and the

number of samples, S i s the standard deviat ion of the water qua l i t y

concentrations and R i s specif ied hal f -width of the confidence in te rva l of

the annual mean.

Using the same assumption, that the water qua l i t y va r iab le i s i id, the

number of samples per year can be specif ied as a function of the data

ana lys is procedure as well. For example, i f annual means were to be

tested for s igni f icant changes us ing the dif ference in means, then to detect

an assumed level of change, the number of samples can be specified.

A more sophist icated procedure, representing a h igher level of

21 3

0.9

0.8

0.7

0.6

R 0.5

0.4

0.3

0.2

0. I

R vs. Number of Somples per Yeor

I Wore 2 Conn. at Thompsonville 3 Deerfield 4 Conn. ot Montopue City 5 Millers 6 Conn.ot Vernon 7 Westfield 8 Conn. ot Turners Falls

I 1 I I I 10 20 30 40 50

Number of Somples per Yeor

Fig 13.2 A p l o t n u m b e r o f s a m p l e s per y e a r of the expected h a l f - w i d t h of t h e c o n f i d e n c e i n t e r v a l of m e a n log f l o w , R , v e r s u s n u m b e r of S a m p l e s for S e v e r a l R i v e r s in t h e C o n n e c t i c u t R i v e r B a s i n

214

stat ist ical analysis, would be to recognize that water qua l i t y ver iab les

may not be i i d , bu t h igh l y dependent, not iden t ica l l y distr ibuted, hav ing

seasonal var iat ion, and determine sampling frequency as a function of the

v a r i a b i l i t y of the water qua l i t y var iab le time series a f te r trend and

per iodic components have been removed. Unfortunately, other than mean

d a i l y discharge, data bases of water qua l i t y va r iab le of suf f ic ient

number, r e l i a b i l i t y and length are general ly not ava i lab le for appl icat ion

of th is procedure.

Once a uniform sampling frequency c r i te r ion i s selected i t can be

u t i l i zed to objectively d is t r ibu te sampling frequencies w i th in a water

qua l i t y monitoring network. For example, the expected ha l f -w id th of the

confidence in te rva l of the annual mean ( fo r speci fy ing sampling

frequencies) approach can be appl ied basin-wide in a consistent fashion

b y specifying equa l i t y of these expected hal f -widths a t each sampling

stat ion. Thus, stat ions where water qua l i t y var ies tremendously w i l l be

sampled more frequent ly than stat ions where the water qua l i t y var ies

l i t t le . With reference to F igure 13.2 which i s a plot of the expected

ha l f -w id th of the confidence in te rva l of mean log r i v e r flow versus the

number of samples per year, the number of samples collected a t each

stat ion w i th in the r i v e r bas in fo r a given R a re determined b y drawing a

horizontal l ine through R and reading the number of samples on the

abscissa ax is below the intersections on the horizontal l i ne w i th each

curve. Figure 13.2 may also be used i n an i te ra t i ve fashion to specify

sampling frequencies a t each stat ion when a total number of samples from

the basin i s specified. For example, i f on ly N samples per year were

collected and analyzed, a value of R i s assumed and a l ine i s drawn

hor izontal ly; the number of samples specif ied by the intersection of the

curves are summed and compared to N. I f the sum were not equal to N

then another estimate of R would be made u n t i l the sum of a l l the samples

i s equal to N.

I t should be noted that the expected ha l f -w id th of the annual mean i s

not the only s ta t i s t i c that can be used to specify sampling frequencies;

the expected hal f -width d iv ided b y the mean i s a measure of re la t i ve e r ro r

and may be more appropr iate when assigning sampling frequencies in a

bas in where water qua l i t y var ies tremendously from r i v e r to r i ve r .

When developing sampling frequencies, one must keep i n mind two very

important cycles which can have immense impact on water qua l i t y

concentrations, the d iu rna l cycle and the weekly cycle. The effect of the

d iu rna l cycle (which i s a function of the rotat ion of the ear th ) can be

el iminated b y sampling in equal time in te rva ls fo r a 24-hour per iod and

215

the effect of the weekly cycle (which i s a function of mans' a c t i v i t y ) can

be eliminated by specifying that sampling in te rva ls for a network cannot

be mult iples of seven, and occasional sampling on weekends would be

necessary.

Perhaps the major impact between network design in terms of var iables

to b e monitored, sampling location, and sampling frequency and the

operational monitoring functions i s in the area of data ana lys is and,

consequently, ult imate value of the monitoring network information. Any

sampling program that i s to generate conclusive resul ts from observing a

stochastic process (water qua l i t y concentrations) must be well planned and

s ta t i s t i ca l l y designed. S ta t i s t i ca l l y designed implies that the sampling i s

planned ( i n proper locations and numbers) so that the stat ist ical analysis

techniques chosen w i l l be able to y ie ld quant i ta t i ve information. Thus, the

data analysis techniques ( level and type of s ta t i s t i cs ) to be used must be

defined i n order to know how to compute proper sampling frequencies,

locations, etc.

D I SCUSS I ON

The above section has pointed out many problems due to not designing

a monitoring system i n a systems context. Perhaps the major concern i s

that a l l aspects of a monitoring program should match i n terms of

accuracy. For example, i t would not be wise to use time series ana lys is

on nonrepresentative, g rab sample data. The system would be prov id ing

excessive accuracy i n one segment compared to the accuracy in another

segment . I n a s imi lar manner, i t may be unrea l i s t i c to encourage use of more

sophisticated sample collection and laboratory ana lys is techniques i f the

data i s not to receive a thorough stat ist ical analysis.

I t i s d i f f i cu l t to test hypotheses, make decisions and in i t i a te action

using water qua l i t y data which are collected only in the daytime and not

flow weighted, several times a year, from locations which are not

completely mixed and using lab analyses procedures which may have more

var ia t ion in their resul ts when analyzing the same sample than the

ambiant var ia t ion of the water qua l i t y va r iab le in the r i ve r .

Perhaps an even la rger concern to those in monitoring network design

i s the use of water qua l i t y standards that general ly ignore stat ist ics.

This lowers the value of any information from a compliance viewpoint, to

that of spot checks. Incorporat ing water qua l i t y means and var ia t ion into

standards would great ly fac i l i ta te incorporat ing more stat ist ics into

21 6

monitoring. This would have the effect of t y i n g network design to da ta use

in a much more concrete, s ta t i s t i ca l manner than i s now possible. I t would

also encourage use of the system approach to network design as there

would be a s ta t i s t i ca l thread moving through the en t i re monitor ing

operat ion.

REFERENCES

Lettenmaier, D.P., 1975. Design of Monitor ing Systems for Detection of Trends i n Stream Quali ty. Technical Report No. 39, Charles W. Ha r r i s Hydraul ics Laboratory, Universi ty of Washington, Seattle.

Lof t is , J.C. and Ward, R.C., 1978. Stat ist ical Tradeoffs i n Monitor ing Network Design, presented a t AWRA Symposium Establishment of Water Qual i ty Monitoring Programs. San Francisco, Cal i fornia.

Pomeroy, R.D. and Orlob, G.T., 1967. Problems of Sett ing Standards of Survei l lance fo r Water Qual i ty Control. Ca l i fo rn ia State Water Qua l i t y Control Board Publ icat ion No. 65, Sacramento, Cal i fornia.

Quimpo, R.G., 1968. Stochastic Analysis of Da i l y River Flows. Journal of Hydraul ics, ASCE. 94(HY1) p43-47.

Sanders, T.G., Adr ian, D.D. and Joyce, J.M., 1977. M ix ing Length fo r Representative Water Qua l i t y Sampling. Journal Water Pol lut ion Control Federation. 49 p2467-2478.

Sanders. T.G. and Ward, R.C., 1978. Relat ing Stream Standards to Regulatory Water Qual i ty Monitor ing Practices. Presented a t the AWRA Symposium “Establishment of Water Qual i ty Monitor ing Programs, San Francisco, Ca I i fo rn ia.

Sanders, T.G. and Adrian, D.D., 1978. Sampling Frequency fo r River Quali ty Monitoring. Water Resources Research. 14(4) p 569-576.

Sanders, T.G., Ward, R.L. Lof t is, J.G. Steel, T.D, Adr ian, D.D. and Yevjevich, V., 1986. Design of Networks fo r Monitoring Water Qua l i t y , 2nd Edit ion, Water Resources Publications, Colorado.

Sharp, W.E., 1973. A Topological ly Optimum River Sampling Plan for South Carol ina. Water Resources Research Ins t i tu te Report No. 36, Clemson Universi ty , Clemson , South Carol ina.

Ward, R.C., Neilsen, K.S. and Bundgaard-Nielsen, M., 1976. Design of Monitoring Systems for Water Qua l i t y Management. Contr ibution for the Water Quali ty Inst i tute, Danish Academy of Technical Science, No. 3, Horshdm, Denmark.

21 7

AUTHOR INDEX

Abulnour, A.M. 116 Adarns, B.J. 190 Adarnson, P.T. 76 Adr ian, D.D. 209,212 Agardy, F.J. 66 American Water Works Associat ion 37 Arnold, R.W. 36

Baker-Duly, H.L.G. 123 B a l l , J.M. 70, 77 Barenbrug, A.W.T. 2 Bauer, C.S. 143, 146, 149 Beck, M.B. 202 Bedient, P.B. 66 Betz, 3 Bishop, A.B. 165 Boyd, G.B. 66 Bradford, W. J . 64 Brebbia, C.A. 62 Brownlow, A.H. 1 Bungaard-Nielsen, M. 210 Chan, W.Y.W. 167 Chiang, C.H. 165 CIRIA. 164 Co lw i l l , D.M. 66 Connor, J.J. 62 Corbetis, S. 116 Cordery, I. 70 Crabtree, P.R. 167

Dein inger , R.A. 36, 39, 51 Dantz ig , G.B. 82, 163

F r ied , J.J. 55 Fuchs, L. 190

G i lbe r t , R.G. 143 Goodier, J.M. 63 Green , I. R.A. 64 Gr izzard, T.J. 70 Grosman, D.D. 86

Hadley, G. 162 Ha l l , G.C. 160 Helsel, D.R. 70 Henderson-Sel lers , B. 24 H i l t on , E. 27, 119 Hine, A.E. 197 Hinton, E. 149 Ho, G.E. 143 Hoehn, R.C. 70 Holton, M.C. 75 Hunter, J.V.I. 66

IBM 162 Ide lov i tch, E. 143

Joyce, J.M. 210

Kemp, P.H. 64 Kim, J.I. 70 Kleinecke, D. 41

Lance, J.C. 143 Larnbert, J.L. 66 Larnbourne, J.J. 66 Lange l i e r , W.F. 3, 5, 6 Lanyon, R. 75 Larson, T.J. 104 L a u r i a , D.T. 165 Leighton, J.P. 146, 149 Lettenmaier, D.P. 212 Lewis, R.W. 119, 149 L loyd , P.J. 1 Lo f t i s , J.C. 209, 212 Loucks, D.P. 116 Ludw ig , L. 9 Lynn , W.R. 116

Madisha, J.L. 75 Mathew, K. 143 McDonell, D.M. 56 McPherson, D.R. 41, 45 M icha i l , M. 143 Mika lsen, K.T. 75 Mrost, M. 1 MOller, D. 190

Neilsen, K.5. 210 Neurnann, A. 190 Newrnan, P.W.G. 143

O'Conner, B.A. 56 Orlob, G.T. 212

P a l i n g , W.A.J. 141, 143, 145 Pe l l e t i e r , R.A. 1 P e r r y , R. 66 Peters, C.J. 66 Petersen, 5.0. 190

Pomeroy, R.D. 212 Porges, J. 2 P ra t i sh thananda , S. 165

Quimpo, R.G. 211

Randa l l , C.W. 70 Rand Water Board 155 Revelle, C.S. 116 Rice, R.C. 143 Rinaldi, S. 116 Ryzner, J.W. 36

Sanders, T.G. 24, 209, 210, 212 Sar tor , J.D. 66 S c h i l l i n g , W. 190 Shar land, P.J. 41, 45 Sharp, W.E. 210

Pol ls , I. 75

21 8

S h a w , V.A. 167 Shoemaker, C.A. 146, 149 Simpson, D.E. 64 Smeers, Y. 116 Smith, A.A. 119, 149 Soncini-Sessa, R. 127 South Afr ican Bureau of Standards 72 Springer, N.K. 66 Steel, T.D. 209, 212 Stehfest, H. 127 Stephenson, D. 27, 66, 80, 81, 82,

115, 116, 117, 163, 175, 197, 200

Terstr iep, 66 Thomann, R.V. 39 Timoshenko, 5 . 55 Tyteca, D. 116

Uh l ig , H.H. 13 Van Staden, C.M.V.H. 2 Velz, C.J. 41

Waniel ista, M.P. 64, 146, 149 Ward, R.C. 209, 210, 212 Whipple, W. 66 Wang, L.K., 167

Yen, B.C., 190 Yevjevich, V . 209 Yu, S.L. 66

Zukovs, G. 190

21 9

SUBJECT INDEX

Acid 1 Addi t ives 6 Advection 21, 52 Aerobic 9 Agr icu l tu re 17 Antecedent moisture 66 A i r 1 A l k a l i n i t y 3 Al locat ion 79 Al loy 13 Ammonia 9 Anaerobic 9 Analyses 195 Ana ly t i ca l 39 Apartments 167 Aqui fer 141 Arsenic 17 A r t i f i c i a l recharge 141

Backwater 193 Bacter ia 9, 206, 207 Bar ium 16 Basin 209 Benefi ts 126 Bicarbonate 67 Biocide 9 Blend 89 Blowdown 2 BOD (biochemical oxygen demand) 37 Booster 152 Bottleneck 173 Boundaries 62 Bremen 193 Br ine 104, 122

Calcium carbonate 4 Ca l ib ra t ion 40 Cap i ta l 107, 157 Carbonaceous 38 Cathode 10 Catchment 64 Cellulose acetate 104 Character is t ic 39 Chelant 7 Chemical 67 Chlor ide 2, 67 Chlor ine 9 C i v i l engineer ing 107 Commerci a I 1 70 Cleaning 115 Computer 20, 115, 128 Concentration 2, 71, 159, 212 Conduct iv i ty 18 Conduit 175 Confidence 214 Constra in ts 41, 86 Conveyance 141 Cooling 20

Corre la t ion 66 Corrosion 3, 13 Cost 79, 107, 146 C r i t e r i a 211 Crop 17 Crump wei r 65 Crys ta l 7 Cyanide 16 Cycle 214

Data 177, 204 Dead water 24 Decomposition p r i n c i p l e 163 Desal inat ion 99, 115 Deter iorat ion 116 D i f fus ion 36 Disc 128 Dispersants 7 Dispersion 21, 166 D i s t i l l a t i o n 101 DO (d isso lved oxygen) 37, 206 Dissolved so l ids 206 Downstream 193 D r y d a y s 66 D r y weather 77

Economics 99 E lec t r i ca l corrosion 14 E I ect r o d i a I v s i s 105 Emulsion 10 Env i ronmenta I 193

Equipment 107 E r r o r 91 Estuar ies 37 Eu ler 57, 59 Evaporat ion 2 E x p l i c i t 39, 51

Fa l lou t 66 Faradays law 14 Feedback 205 F i e l d 45 F i n i t e d i f ference 55 F i n i t e elements 62 F i r s t f l u s h 70 F looding 193 Flow 166 Foam 8 Formulat ion 88 Fou l ing 9 Four po in t 51 Four ie r series 54, 169 Freezing 103 Frequency 21 1

Gain 23 Galvanic corrosion 13 Geochemical 1 Geohydrology 41

220

Graphics 118, 177 Groundwater 98, 112, 143 Gypsum 6

H ierarchy 209 H i l lb row 68 H y d r a u l i c 51, 167 Hydrodynamic 56 Hydrograph 166

IBM 150 I m p l i c i t 55 I n d u s t r i a l 1, 104, 112, 172 I n f i l t r a t i o n 143, 176 In f low 177 Inaccuracy 52 I n s t a b i l i t y 55 In teger Programming 141, 149 In terest r a t e 107 Ion exchange 105 I r o n 3, 16 I r r i g a t i o n 17 I te ra t ion 165

Johannesburg 167

K l i p r i v e r 40

Labora tory 205 Labour 108 Lange l ie r index 5 L a x a t i v e 16 Leach 1 , 26, 75 Lead 17 Leak 166, 176 Leap f r o g 51 Least squares 42 Leg is la t ion 204 L i n e a r programming 43, 85 Load fac to r 107 Loops 119

Maintenance 116 Make-up 26, 33 Manhole 167 Mass balance 20, 35, 64, 72, 161 Master programme 163 Mathematical models 20, 149, 158 Measurement 167 Membranes 105, 108 Meta product ion 191 Mine water 26, 117, 123 Min imize 41 Mixed f low 21 Mon i to r ing 204 Mul t i -s tage f l a s h d i s t i l l a t i o n 103 M u l t i step 61

Network 146, 205 N i t r a t e 17, 72

Nodes 119, 128, 160 Non conservat ive 35 Numerical 23, 51

d i f f u s i o n 35

Object ive 41 O i l 10 Opera t ing 157 Optimum 79 Opt imizat ion 116, 152, 162 Ore 30 Oxygen 10, 37, 40

Peak 146, 174 PH 3 Phenol 16 Phosphate 7 Photosynthesis 46 P i p i n g 2, 146 P l a n n i n g 149 P l a n t 122

Po l lu t ion 1 , 64 Pol l u t o g r a p h 23 Polymer 7 Polyphosphate 7 Populat ion 166 Potable 15 Pourba ix d iagram 12 P r o b a b i l i t y 167 Product ion system 190 Program 122, 128, 136, 174, 179 P u r i f i c a t i o n 143

P l u g f low 21

Rand Water Board 157 Random 212 Raw water 122 Reaction 14 Recharge 144 Recovery r a t i o 1 1 1 Reed beds 40 Regional 155 Regression 67 R e l i a b i l i t y 211 Reservoir 23 Resident ia l 167 Re-use 99 Reverse osmosis 81, 104 R ivers 37, 214 Rout ing 166, 175 Rule base 191 Runge Cutte 61 Runn ing 108 Runoff 67 Ryzner index 3

Sa l ts 102 San i ta t ion 195 S a n i t a r y eng ineer ing 206 Sample 68, 204, 205

221

Sampling frequency 206 Scale 102 Scal ing 3 Sea water 101 Sediment 8 Sens i t i v i t y 195 Separable programming 81, 95 Sens i t i v i t y 95, 165 Sewage 144, 176 Sewer 72, 166, 190, 196, 198 Shadow value 165 Shops 169 Simulation 31, 51, 166 Simplex method 89 Sink 42 Slack 82, 160 Software 195 Solut ion 82, 160 Source 43 Standards 15, 141 Station 210 Sta t is t i ca l 205 Stat is t ics 215 Steady s tate 20 Stormwater 64, 77, 166, 176 Stream 159, 204 Stream gauge 66 Streeter Phelps equat ion 37 Sub-programme 165 Sub-division 173 Sulphate 5, 16, 30, 67 Surcharge 168 Suspended 206 System 80 Systems a n a l y s i s 24, 118

Tape 128 Taste 16 Tay lo r series 53 TDS ( to ta l d isso lved so l ids) 2, 95 Temperature 3, 206, 107 Terminal concentrat ion 24 Time l a g 166 Topography 1% Toxic 16 T r a f f i c 69 Transpor tat ion programming 80 Treatment 141, 155, 157 T u r b i d i t y 206

Turbulence 8 Two step 39, 52

Unpredic tab le 64 Upstream 193

Vaal r i v e r 155 Vapour compression 102 Vegetables 18 Vent i la t ion 2

Washoff 67 Waste t i p 65 Waste water 99, 155 Water resources 79 Water supp ly 116 Waterways 190 Water vapour 2 Welding 13 Wi twatersrand 155

Zeoli tes 107 Z inc 16 Zooming 56

This Page Intentionally Left Blank

45197995 Book of Design Water System

Documents

Transcript of 45197995 Book of Design Water System