flow) and RUTINGD (data on the stream tracing). · PDF filecases and using the existing data...

Application of developed methodology for the

study of non-stationary pollution of the river

Vrbas (B&H)

H. Bajraktarevic-DobranHydro-engineering Institute, Faculty of Civil Engineering Sarajevo,Bosnia and Herzegovina

Abstract

For the purposes of planning the protection against and control of water pollutionin streams, on the basis of theoretic and numeric analyses of the problem, amethodology has been worked out for defining incidental pollution spread inopen streams, which is generally applicable to non-stationary releases andspread of pollution in a stream. It covers the problem of longitudinal dispersionin natural streams in its totality. Attention has been paid to dissoluble orsuspended non-depositable polluting substances. Presented are the required field,laboratory and numeric investigations.

By the basic program DISPER a numeric model has been developed and theprogram INCH) made. The programs for calculation of dispersion coefficienthave been made and successfully applied: D (commonly measured values of theflow) and RUTINGD (data on the stream tracing).

The programs had been previously checked, assuming simple calculationcases and using the existing data on tracing in the river Miljacka. For furtherchecking and applicability of the methodology, measuring data have been used,obtained for a selected stretch of the river Vrbas for a length of 31,897m. Forprediction of possible incidental pollution spreading, assumed was anunpredicted release of a constant quantity of untreated waste cyanide waterduring 1600s.

On the basis of the data from one tracing reach of the river Vrbas, it has beenshown that by numeric calculation it is possible to obtain excellent concordanceof the measured and numeric curve. Therefore, it is necessary to usecorresponding hydraulic-hydrologic data, and importantly, to define reliablevalues of dispersion coefficients.

Development & Application of Computer Techniques to Environmental Studies VII, C.A. Brebbia, P. Zannetti & G.Ibarra-Berastegi (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-819-8

110 Computer Techniques in Environmental Studies

1 Introduction

An often-neglected fact is that water quality models require, amongst otherthings, the knowledge of corresponding parameters of the models (coefficients ofdiffusion and dispersion) which characterize the observed water recipient andwhich define accuracy and applicability of solutions.

The process of mixing of waste water and water of our streams, mostly ofmountainous type, shows the prevalence of a far field spreading (completemixing of waters in transverse direction). The far field is characterized bynormal flow which comes into a category of shear turbulent flows of boundarylayer type, primary in one direction. Suitable for such streams is a process that isboth very useful and economic: longitudinal turbulent dispersion.

A methodology [1] has been worked out for defining the spreading ofpollution wave in open streams. It covers the entire problem of longitudinal -one-dimensional dispersion in a natural stream, i.e. every case of non-stationaryrelease and spreading of pollution in a stream. Observed is the stationarycondition of water flow in a watercourse.

The theory of longitudinal turbulent dispersion is based on well-knowntheories of diffusion. In these models the assumption is that solutions ofmovement equations are known: hydrological data; simpler calculations andmeasurements both in the laboratory and in the field.

The stable transferable value has been observed - stable tracer and Reynoldsanalogy used. The aforementioned refers to release dissolved or suspendedsubstances that behave like stable tracers, while released waste waters behavelike parts of the basic fluid that transfers them.

2 Theoretic analysis of the given problem

The equation of one-dimensional longitudinal turbulent dispersion, made foruniform streams (a simple analytical solution) can be used in approximatelyuniform and non-uniform natural streams [2,3].

dc _dc _c^c (1)— + u — = D — -• ^ '

In eqn_(l) c represents concentration of the mass averaged over a cross-section, u and D represent mean values for the given reach of velocity andcoefficient of longitudinal dispersion (values averaged over a cross section andalong the given reach). Time and longitudinal - spatial co-ordinates arerepresented by t and x.

Eqn (1) can be used after the initial (diffusion) period, when deviation of thetime averaged concentration from its averaged value over a the cross section (c -c = c') is very small in relation to the averaged concentration over the crosssection, c « c.


Computer Techniques in Environmental Studies 1 1 1

Since most natural streams have a relation width/depth = 10 or more, aconclusion is that the primary cause of dispersion in them is the change ofvelocity in lateral-transverse direction.

For diffusion period a criterion can be given in the following form [2]:

(2)

where L is the distance from one point source or line source down the stream.The characteristic length i represents the distance from the point of maximumvelocity on surface to the farthest bank [2,3]. R is hydraulic radius and U*represents shear velocity given as: u * = -JdSg S is the slope of energy

gradient, d is the depth of the stream and g is gravity acceleration.A practical equation for defining of coefficient D, in natural streams, on the

basis of available parameters of the stream is [2]:

(3)^4 k=2

For a typical cross section it is necessary to define geometry of the cross-section,shear velocity U* and distribution of velocities along the cross section.

In eqn (3) q>' = 1/2 (d^ + di+0 Ui'; u, = mean velocity in the i^ vertical slice; u^= Ui - ii, u = mean velocity of the stream for the whole cross-section; d% = depthof the stream at the beginning of vertical slice; Az = the width of vertical slice;Da = 0.23 di U* = coefficient of turbulent diffusion in transverse directionbetween i -1 and i* slice and n = a number of vertical slices (not fewer than 20).For k = 1 the value between brackets is 0 as the initial integration point.According to this mostly analytical method, it is evident that: D = f (u, 1/Dz).

With this method the coefficient D can be predicted for laboratory conditionswith an error of up to 20 %, for uniform natural streams up to 30 %, and for verynon-uniform streams within a factor of 4 [2,3]. A drawback of the method is thatit requires that the coefficient of turbulent diffusion Dz be known, which is noteasy to obtain for natural streams.

To check the obtained coefficient D (eqn (3) or some other), a suggestion is touse an integration procedure [2] based on dispersion theory and tracer study.

To that purpose the one-dimensional dispersion model is used (eqn (1) and itscorresponding solution).The upstream curve is taken as the initial tracerdistribution, while the downstream curve is predicted. Adjustment of coefficientD is made several times by a computer, until the best adjustment of the measuredand predicted downstream curve is obtained. The equation made on the basis ofthe time-concentration curves is:



(4)

where, to is the mean time of the cloud passage at the measuring longitudinalstation Xo, C (Xi, t) is a predicted curve for station X% on the basis of themeasured curve at the station Xo, ti is the mean time of the cloud passage at X%and u dx element of integration.

In this way the best possible D can be obtained with accuracy of ± 2.5%.The expression for transverse mixing (Dzu) covers the total impact of

transverse turbulent diffusion (Dz) and transverse turbulent dispersion (Dzd),i.e. Dzu = Dz + Dzd.

On the basis of the existing and latest investigations [4] is has been found outthat non-dimensional coefficient of transverse mixing Dz/dU* with value of 0.23(eqn (3)) corresponds to: 1. straight and uniform natural watercourses, and 2.shallow, narrow, winding and non-uniform natural streams. Predominating in theprocess is transverse turbulent diffusion (e.g. the river Miljacka, width 8.6 - 13.5m, average depth 0.2-0.3 m).

For relatively uniform straight, almost straight and mildly winding naturalstreams, the average value of 0.5 of non-dimensional coefficient of transversemixing can be recommended. In such streams there is a joint impact of transversediffusion and dispersion (Dzu = Dz + Dzd).

In case of large natural non-uniform and/or winding streams (in width > 70m), actual - experimentally obtained values of transverse mixing coefficientshould be used. In them, apart from transverse turbulent diffusion, there is also aconsiderable turbulent dispersion (example: the river Bosna, width 37-76 m,average depth 0.53 - 1.14 m).

3 Numeric analysis of the given problem

3.1 Adopting the method of calculation and working out of the basicprogram DISPER

Adopted is a form of one-dimensional dispersion equation which can cover thechanges of velocity, depth and dispersion coefficient along a stream [1]. Thechosen equation is:

9c _8c 1 ,_ ,<.— + u — = -- \dD — L (5)

where c, u, d and D represent concentration, velocity, depth and dispersioncoefficient (averaged over a cross-section). The values of u, d and D arechangeable along the stream, x, dependant on characteristics of the observedreach of the stream, while c is changeable both in time and along the stream. The


Computer Techniques in Environmental Studies 113

eqn (5) can be well used for straight and winding streams, since being one-dimensional it is valid for both Descartes' and natural co-ordinate system. Eqn(4) is also concordant with the natural co-ordinate system.

Eqn (5) represents a linear partial differential equation of the second order, ofa parabolic type. The initial conditions refer to the initial moment of time, andthe limiting ones to margins of the observed space area (upstream limit - knownis the input, time dependent, concentration; downstream limit - passage ofconcentration through the limit).

For approximation of eqn (5) by finite-differences method, the implicitnumeric scheme of Harden and Shen [5] for longitudinal direction was usedbased on Stone's and Brian's numeric scheme [6], which is very well investigatedand adjusted to dispersion problems.

The size of weight coefficients (parameters of amortization of numericoscillations and numeric dispersion) for presentation of approximation of partialderivatives of the first order, was adopted according to the reliable Stones adBrians [6] method. At that, assumed is a cyclic use of a series of three finite-differences equations (the same values of coefficients for presentation 3c/dx, andthree successively different values of coefficients for presentation dc/di) for eachstep of the time (internal cycle- 3 At/3). As to the presentation of approximationd*c/dx the selected weights were not used, since the approximations of implicittype for the second order derivatives of are of high order of accuracy.

For solution of final approximation equations (system of linear algebraicequations) of the eqn (5), adopted is the standard Gauss elimination method(direct method). The process of elimination is represented by a rather simplealgorithm [7], the so called Thomas's algorithm or algorithm of three-diagonalmatrix (TDMA).

The numeric scheme is programed in FORTRAN IV for a stream reach with30 calculation points. For stability of the numeric scheme and for providing of agood convection presentation, which is a common practice [5,6], the criterionUo- At/(Ax) < 1 was used in selection of the time step ( UQ = referent velocity).The implicit scheme of dispersion member is unconditionally stable, so that noadditional criteria are required.

The program by which calculation of dispersion in one reach of a stream iscarried out is called DISPER.

An analysis [4] showed the quality of Stone and Brians [6] numeric scheme(accuracy and simplicity of use), which is confirmed by its continuous use[5,8,1].

3.2 Checking of the adopted method

The accuracy of calculation according to the program DISPER had beenpreviously checked by a simple computer example (D=0). Using numericcalculation of longitudinal dispersion for two reaches of the river Miljacka, andvarying the input data quality, an additional check-up and presentation ofDISPER program has been made. Available experimental data from previous



works have been used [3,4]. The calculation for one reach lasts for about aminute (180 steps of the time). Used were the best possible coefficients ofdispersion, defined by integration procedure (eqn (4)).

It has been shown that for natural streams, with input data of correspondingaccuracy, it is possible to very precisely estimate the dispersion of released traceralong the observed reach [1].

3.3 Making INCH) program and its checking

Using the basic program DISPER, a numeric model was worked out andprogram INCID made for calculation of incidental (or generally unstationary)pollution in the catchment area, along the main watercourse. It can cover astream consisting of 1 to 10 reaches with generally different flow Q. At theborders of the reaches, concentrated inflows from tributaries or groups of smallstreams are foreseen which contribute to the increase of the flow Q atcorresponding downstream reaches. Incidental pollution occurs at the entranceboundary of the first reach.

The developed numeric model has been checked in detail for characteristiccases: especially for D = 0 (a simple computer case), but also for D O (theexperimental data for the river Miljacka). Satisfactory results have been obtained

[1].

3.4 Making programs for defining of dispersion coefficient, Dand RUTINGD, and providing of required data

3.4.1 Making program and providing required dataProgram D has been made on the basis of eqn (3). The results of such acalculation for the observed stretch of the river Vrbas (Figure 1) at low waters(1st series) and medium waters (2nd series) are given in Table 1. For a betterinsight into the observed stretch, some basic hydrometric data are represented inTable 2 (stretch I-V: 49,222m).

Table 1: Coefficients of longitudinal dispersion, defined on the basis of theusually measured flow parameters

Measuringsection

IIIIIIIVV

1st SERIES

D(m /s)

2581.779.9324.338.66

Relationshipwidth/depth

b/daver5031252433

2nd SERIES

D(m'/s)

7.801.82-

43.0539.90

Relationshipwidth/depth

b/daver3927231628


Computer Techniques in Environmental Studies 1 15

Table 2: Hydrometric measurements

Ser

I

II

Measu-ring

section

IThemouth

IIIIIThemouthIVVI

Themouth

IIIIIThemouthIV

V

River

Vrbas

Bistrica

VrbasVrbas

Veseocica

VrbasVrbasVrbas

Bistrica

VrbasVrbas

Veseocica

Vrbas

Vrbas

Date(1985)

17. IX

18. IX

nn

19. IX20. IX10. XII

11. XII

n

12. XII

13. XII

Qnf/s

0.646

0.883

1.86145

1.09

5.088.23202

1.98

5.919.57

284

15.80

18.40

bm

9.50

6.50

13.5012.50

10.5

18.50210010.50

6.50

15.0014.00

10.00

18.00

2100

U-averm

0.19

0.24

0.430.49

0.16

0.760.690.27

0.32

0.550.61

0.29

1.09

0.82

Uaverm/s

036

0.57

0320.56

0.65

0360.520.71

0.96

0.721.12

0.99

0.80

0.97

S

0.00440

0.003520.00189

0.000440.001600.00440

0.004440.00156

0.00099

0.00244

Field and laboratory investigations, related to study of tracer spreading,precede the use of eqn (4) [1,3]. The results of tracing the river Vrbas withuranin refer to two tracing reaches (Figure 1), P1-P2 and P3-P4, for which time-concentration curves have been measured. Corresponding statistical values havebeen defined for the time-concentration curves on the basis of known statisticalmethods.As to the application of integration method according to eqn (4) the programRUTINGD [1,3] has been used. Average values of coefficients for sections IIand III (5.85 nf/s) and III and IV (17.13 nf/s) for the 1st series of measurement(Table 1) have been taken as initial dispersion coefficients for the reaches P1-P2and P3-P4 of the river Vrbas (Figure 1). The results of integration procedure forthe observed reaches, obtained rather quickly (after 25 and 41 iterations) arepresented in Table 3.



L e g e n d:

Catchment borderI H/drometnc measuring section

Waste water dischargeO Tracer injection,* Defining o1 marked cloud concentrationScale

70000 20000 m

Figure 1 Presentation of the river Vrbas (G. Vakuf - Torlakovac),selected hydrometric sections and other measuring points

Table 3: Coefficient of longitudinal dispersion obtained by integrationprocedure, 1st series of measurement

Hydro-metricand

tracingsections

II

PI

P2

III

P3

P4

IV

Charac.lengtht(m)

8.0

9.75

LengthscaleL(m)

969.0

1686.4

Averagetime of

passage ofthe down-stream

theoreticcurve^n(s)

-

22,224.4

-

-

15,968.0

-

Variance ofdownstreamtheoreticcurve

(A (s")n

-

4,035,725.7

-

-

3,086,537.7

-

Coefficient ofdispersionobtained byintegrationprocedureD(nf/s)

-

8.15

-

-

13.21



3.4.2 Mass balancePrior to the use of the measured time-concentration curves (Figure 2) it isnecessary to check their reliability. Defining of the mass balance is used to checkwhether the total quantity of the released tracer mass corresponds to the tracerquantity that passed through each observed tracing section, taking into accountpermissible errors (desirable up to 5 %, sometimes even up to 15%, withadditional explanations). For sections PI and P2 the differences between thepassed and released quantities amount to - 2.6% and - 5.62%. For the sectionsP3 and P4 the differences are + 7.0% and + 4.6%.

On the basis of Tables 1 and 3 and Chapter 2, it is evident that averagecoefficients D obtained by flow parameters (for the reaches II - III and III - IV:5.85 nf/s and 17.13 nf/s; 1st series of measurements) in comparison with thoseobtained by integration procedure (for the reaches P1-P2 and P3-P4: 8.15 nf/sand 13.21 nf/s) are predicted with an error smaller than 30%. This representsvery favourable results, taking into account the fact that this is a natural stream.The field investigation of tracer spreading have been carried out in diffusionperiod, in accordance with the theory of dispersion (eqns (2) and (4), Table 3 andFigure 1).

The assumption is that the coefficients D predicted via the flow parametersfor the 2nd series of measurements are defined within the limits of expectedaccuracy (Table 1). Due to certain problems, it has been concluded on the basisof an analysis that it is possible for the whole reach II - III to use, for practicalpurposes, the coefficient D = 1.82 nf/s that is defined for the section II rathersuccessfully.

4 Computer programs

Names of programs and corresponding sub-programs are given in Table 4.During the work on methodology, the structures of developed programs were

Numeric and measured curve at sectionPi Dalj an

Nurrwidly obtained curvesMeasured curve

Figure 2: Spreading of tracer in the river Vrbas,r* series of measurements - low water level



described and detailed description of their key elements presented, enabling thusa connection between numeric methods and computer laws. In this way abeneficiary of the presented models will be able to understand, and, if necessary,make additions or changes.

The programs written in FORTRAN IV language were made on a computerof the system DELTA - M VI.2 BL 26, but can easily be adjusted for IBM orsome other personal computer.

The basic program DISPER is made for 30 (29 real and 1 fictitious) pointsdistributed along the observed reach. The main program INCID, based on thebasic program DISPER is written for 30 points along each reach for themaximum of 10 reaches along the observed stretch.

Table 4: Review of programs and sub-programs

PROGRAM

Name

1DISPER

INCID

RUTINGD

D

Purpose

2Calculation oflongitudinaldispersion at onereach of the streamCalculation oflongitudinaldispersion at onestretch of the streamthat consists ofseveral reaches,based on the basicprogram DISPERDefining ofdispersion coefficienton the basis of thetracer study, byintegration procedure

Defining ofdispersion coefficientby flow parameters

SUB-PROGRAM

Name

3

OTVOR

OTVOR

UNOSF

OTVOR

STAT1

STAT2

OTVOR

ASSIGN

Purpose

L 4Choice of data bank fordataand reports

Choice of data bank fordataand reports

Creating data bank forTable F. i.e. for input datafor concentration at thebeginning of the stretchChoice of data bank fordataand reports

Calculation of theoreticdownstream curve

Calculation of the sum ofsquares of deviationsChoice of data bank fordataand reportsSystematic sub-program



5 Detailed check on the selected methodology

and its application to a watercourse

5.1 Numeric calculation of tracer spreading and comparison with measureddata - check No. 1.

The calculation of tracer spreading along the reach P3-P4 has been made byINCID program. The tracer time-concentration curve measured at the section P3has been used as input concentration, changeable in time. Other data were: QPS_P4= 5.08 mVs, u = 0.47 m/s, d = 0.76 m and D = 13.21 nf/s (Table 2, Table 3 andFigure 1). The reach was 3,375 m and the total of 90 steps of the time scale wasadopted, the time step being At = 210".

A graphic presentation of the carried out calculation is given in Figure 2.Presented are only some characteristic distance - concentration curves, forcharacteristic steps.

Figure 2 shows excellent concordance of the measured and numeric curvewhich proves the validity of adopted numeric scheme, carried out calculation andof dispersion coefficient defining. The peak concentration of the numeric curveis predicted with an error of « - 2.7% (time step No. 46 at 9660 s). Hereby isnumerically checked and described the whole process of injected tracer (uranin)at Curcica Lug, on its way from section P3 to P4 (Figures 1 & 2).

5.2 Defining of incidental pollution spreading along a selected stretch -checks 2 and 3

The stretch of the river Vrbas in the length of 31,897 m was used for calculationof spreading of simulated incidental pollution (i.e. for application of presentedmethodology) in open streams (Figures 1 and 3). The stretch is divided into fourcharacteristic reaches for calculation, Pi-P/i, in lengths of 2,832 m, 8,400 m,8,625 m and 12,000 m respectively.

Layout of observed stretchP3 P2 P1

Dotjani i s

'""ixoff

Bugojno GrataI

™.JY/>W.~t7

nica Pajtd Polj*i

1375 L 4055V " Q&O31697

1 *n ,/

¥*,

listn

m"7#

GCO/

'P

VakufKruSOcoI Vrt>as4

Waste wa,107 m

Jt

Calculation sc

fteO»r ,(l\> I

,0*.

^ "Qj Qj 0, I_J d625 J, W4.0 j 2832 j.

i. e 3 e n d.Section ~ I...,1VReach - P,, , P^Flow along fhe reach Q,- Q,Aggregate flow o1 smallertrtbutanes • Q^>Tracw mjeciton t—oDefining of marked cloud

wj10- concentration PI - Pi. \—*

Figure3: Layout of the observed stretch and calculation scheme



Table 5: Input concentration at the upstream point of the observed stretch

Seriesof

measurement

I

II

Flow of r.Vrbas atdischargepoint

Q(nf/s)

0.646

2.020

Dischargeof wastewater

q(1/s)

1.53

Concentration of cyanide

In wastewater

Co(mg/1)

100

In water of the r. Vrbas

At wastewaterreleasepointc

(mg/1)

0.237

0.076

Maximumpermissiblefor class IIof waterquality [9](mg/1)

0.01

Assumed was an unpredicted discharge of untreated waste cyanide water ofconstant quantity and concentration from the Factory of springs at Gornji Vakuf,directly into the stream during 26.6 min = 1600 s (Figure 3 and Table 5).

For the 1st series of hydrometric measurements, the calculation was made byINCID program for four reaches. As the length of the stretch was too big, thenumeric results were only discussed. Required data were taken from Tables1,2,3, and 5. Selected was the time step At = 160 s which guarantees the stabilityof numeric scheme. Calculation was made for the total of 490 steps.

For the first reach, PI, the measuring section I is characteristic, so its valueswere taken as constant for the whole reach (Table 1, D = 2.58 nf/s). The reaches?2 and PS are characterized by measuring sections II and III and have the samedispersion coefficient D = 8.15 nf/s obtained by integration procedure. SectionIV refers to ?4 as well as coefficient D = 13.21 nf/s obtained by integrationprocedure. The input concentration 0.237 mg/1 gets into 10 time steps.

From input concentration which is 23.7 times as big as the permissible one,already at the entrance of pollution cloud into the reach ?4, obtained is themaximum concentration lower than the permissible one (step no. 310).

In the 2nd series of measurements, presented were the results of numericcalculation for two reaches of the stretch, PI and ?2 (Table 5 and Figure 4).The input data were taken from tables 1,2, and 5. The sections I and II arecharacteristic for reaches PI and ?2. In this way also adopted were dispersioncoefficients D = 7.8 nf/s for PI and D = 1.82 nf/s for ?2 predicted on the basis offlow parameters (Table 1). The input concentration 0.076 mg/1 gets into 20 timesteps, the step being At = 80s. The calculation covered 470 time steps.

The curve for step no.20 represents the final moment of incidental pollutionentrance. It is evident that already at the end of ?2 (time step no. 204; 16,320 s =4.5 h after entrance of pollution) the concentration is 0.0217 mg/1 which is twiceas big as the permissible one (0.01 mg/1). Getting into the reach PS as a result ofdilution, it immediately becomes lower than the permissible maximum.



I — >N.»LSa 0. f?09 0. 100 -2t 0.080-| 0,060 ••| 0,040 -g. 0,020 -

Chamage

Reaches

Layoi/t

C. Vakvf

; 51 ep 20 Step 60i 7600 s, 3200 s

§k Cj| NK i\ @>1 6|& i| *

Reach • P\ 2832 mtWasie water discharge* ^ KruStica ^Bn

Pojit Polje GroCanica

Step GO '•/ MOO s :

i Step 1SOUtOO sStep 100

3000 s / / Step 20t- — I x' — '** ^ , / J 6 J ^ O s

i #| *£* <J>[ ^Reach - P^ 18UO m

fi TF v*// J|L Jisilica

Figure 4 .Spreading of incidental pollution at medium waters -2nd series of measurements

Comparison of these obtained results with the data which could be obtainedunder condition that there is no dispersion or that it is not taken intoconsideration (a very short cloud and far higher concentrations), clearly show theimportance of dispersion calculation and defining of as accurate values ofdispersion coefficients as possible.

6 Conclusions

On the basis of theoretic and numeric analyses and with a choice of a basicnumeric model and its further development, given is a methodology ofspreading of pollution cloud in open streams. The given methodology has beenchecked by presentation of the required field, laboratory and numeric researchesand very successfully applied to one concrete pollutant and a correspondingstretch for a case of simulated incidental pollution. In this way a possibility wascreated for a wide use of the proposed methodology in water management.

It has been shown that the given eqn (5) agrees very well with naturalstreams. Also, the selected, namely, developed numeric models and programsDISPER and INCID give results accurate enough, whether the dispersioncoefficient is zero, close to zero or much higher. It is necessary to mention herethat reliability and accuracy of previously defined dispersion coefficient(programs D and RUTINGD) play an important role in accuracy of calculation.



References

[1] Bajraktarevic-Dobran, H. & Sarajlic, M. Spreading of Incidental Pollutionin Streams, Published by "Svjetlost", Institute for Water ResourcesEngineering, Civil Engineering Faculty: Sarajevo, 1995.

[2] Ficher, H.B. Dispersion Predictions in Natural Streams. Journal of the/?7 ryE/?gmggnV?gD7Vz o/?, ASCE, 94 (SA5),pp.927-943, 1968.

[3] Bajraktarevic-Dobran, H. Dispersion in Mountainous Natural Streams.Journal of the Environmental Engineering Division, ASCE, 108 (EE3), pp.502-514, 1982.

[4] Bajraktarevic-Dobran, H. Prediction of Waste Water Spreading inMountainous Open Streams, Doctoral dissertation, University of Sarajevo -Faculty of Civil Engineering, 1991.

[5] Harden, T.O. & Shen, H.T. Numerical Simulation of Mixing in NaturalRivers. JowrW q/Y/ze /fK /z D/v/ o/?, ASCE, 105 (HY4), pp.393-408,1979.

[6] Stone, H.L. & Brian, P.L.T. Numerical Solution of Convective TransportProblems. A.I.Ch.E.Journal, 9 (5), pp.681-688, 1963.

[7] Patankar, S.V. Numerical Heat Transfer and Fluid Flow, McGraw-HillBook Company: New York, pp. 50-54, 1980.

[8] Lau, Y.L. & Krishnappan, B.G. Modeling Transverse Mixing in NaturalStreams. JowrW qff/ze T/ydrW/cs" Dzwsvo/?, ASCE, 107 (HY2), pp. 209-226, 1981.

[9] Sparavalo, Z. A Collection of Federal and Republic Regulations on Waterswith Comments and Typical Examples of Implemented Acts, WaterManagement of BiH Company - Institute for Water ResourcesDevelopment: Sarajevo, pp. 327-331, 1982.


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