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Algorithmes · logiciel DESCAR · photos mer

                            

Applications 1: pollution marine et pollution de l'eau · ocean pollution · photos mer · pollution images · pollution et mer Méditerranée · pollution des eaux · pollution image · pollution photo · définition pollution · pollution carte · pollution maritime

 

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Il est fondé sur le modèle numérique Buoyant jet model
de l'Environmental Protection Agency des États-Unis (EPA) et stratifié modèle.
 

1. Buoyant jet model

A type of mathematical model that has been developed for sumerged round buoyant jets is the length-scale model.  Discharges flows can be divided into different regimes each dominated by particular flow properties. Within each regime, the flow may be approximated with simple mathematical relations describing the simplified problem. A model that uses asymptotic solutions is refered to as length-scale model because of length scales to delineate the extent of the regimes for which the mathematical expressions are valid. The pollutant concentration, in a certain instant, and at a distance x (meters) in the X-Axis and at a distance y(meters) in the Y-Axis will be given by: 

 

c =cc exp[-(r/b)2]  (1)

 

where c is the pollutant concentration, r is the distance from the point (that we are calculating) to the center of the line that forms the polluting plume, cc is the pollutant concentration in the center of the plume line and b is the plume half-width. We attempt to link the momentum dominated and buoyanvy dominated regimes into one relationship by using proposed relations for the transition where: 

 

z/Lb =24/3[(1/2)(x/Lb)2+(Lm/Lb)(x/Lb)]1/3  (2)

b/Lb =cb[(1/2)(x/Lb)2+(Lm/Lb)(x/Lb)]1/3    (3)

S=cs(uo/ua)[(1/2)(Lb/Lm)(x/Lm)2+(x/Lm)]1/3 (4)

Lb=plume-to-crossflow length scale

Lm=jet-to-crossflow length scale

x=horizontal downstream coordinate in global coordinate system

y=horizontal coordinate in coordinate system perpendicular to ambient crossflow

z=vertical coordinate

ua=ambient velocity

uo=discharge velocity

S=dilution along the plume centerline C/C0 being C the centerline pollutant concentration and the C0 initial pollutant concentration at the discharge.

cb=constant of proportionality that can be modified by the user (can be determined experimentally)

cs=constant of proportionality that can be modified by the user (can be determined experimentally)

cxy=constant of proportionality that can be modified by the user (can be determined experimentally)

We obtain solutions for a vertical buoyant jet in a crossflow. And buoyant jets discharged horizontally perpendiculat to crossflow.

z/Lb =cxy(x/Lm)1/3    (5) 

 

This model performs satisfactorily for simple flows with no shoreline interaction or attachment. Strong crosscurrents or limited depths causing attachment with the downstrean bank or strong initial buoyancy render this model invalid. In addition, they are incapable of simulating any far-field processes that occur after a certain distance.

 

 

2. Stratified model

 

This model is official in Spain and it follows Orden del 13 de Julio de 1993 del Ministerio de Obras Públicas y Transportes del Reino de ESPAÑA, B.O.E. Martes 27 de Julio de 1993, página 22861, I. Disposiciones generales. Proyecto de conducciones de vertidos desde tierra al mar.

 

The stratification phenomena is the existence of two homogeneous water layers and separated by a thin thermocline layer. In such a case, we can say that the water is stratified. There is no exchange of pollutants through this picnocline layer. In case we suppose that a picnocline layer exists, we will be able to check the stability by means of the application of the following equation:

 

[u02 B+ Ua2 H]/[(u0 B g’)2/3 H] <0.54

u0=effluent velocity (m/s)

H=water depth at discharge position (m)

g’=reduced gravity acceleration (m/s2), g’=g(ρa- ρ0)/ ρ0

g=9,81m/s^2 (gravity acceleration)

ρa=water density (Kg/m3)

ρ0=effluent density (Kg/m3)

Typical values (pollutants):

Organic matter as DBO5 - 350g/m3

Suspended matter - 600g/m3

E. Coli - 1012 /m3

N2 (total) - 30 gN/m3

Effluent velocity - entre 0.6 y 0.8 m/s

Port diameter - 6cm

 

Dispersion coefficients:

Horizontal dispersion: Ky(m2/s)=3x10-5 B4/3.

B=initial plume width(m)

 

Vertical dispersion: Kz(m2/s)=4x10-3 Ua e

e= thickness of the mixing layer

Ua=horizontal ambient velocity (m/s)

 

2.1 Water is stratified

 

2.1.1 Multiport diffuser.- We have three different cases: 

Case I: 

θ >=65º   F<=0.1 ó

θ <65º   F<=0.36

θ=angle between Ua vector and diffuser.

F=Fraude number F=Ua3 (g’q)-1

q=Unitary flor rate in the diffuser q=QLt-1 (m2/s)

Q=Discharge flow rate (m3/s)

Lt=Diffuser length (m)

S=initial dilution

In such a case, we have the next relationships

 

S=0.27 Ua H q-1 F-1/3

e=0.29H

B=SQ/eUa

 

Case II: 

 

25o=<θ <65º  F>0,36 (*)

 

In such a case, we have the next relationships

 

S=0.38 Ua H q-1

B=max[Ltsin θ; 0.93Lt F-1/3]

e=SQ/BUa

 

Case III: 

 

θ <25º  0,36<F=<20

 

In such a case, we have the next relationships

 

S=0.294 Ua H q-1 F-1/4

B=max[Ltsin θ; 0.93Lt F-1/3]

e=SQ/BUa

 

Case IV: 

 

θ <25º  F>20

 

In such a case, we have the next relationships

 

S=0.139 Ua H q-1

B=max[Ltsin θ; 0.93Lt F-1/3]

e=SQ/BUa

 

Case V: 

 

θ >65º  F>0,1

 

In such a case, we have the next relationships

 

S=0.58 Ua H q-1

B=max[Ltsin θ; 0,93Lt F-1/3]

e=SQ/BUa

 

From Case II to Case V, and if e>H, we take e=H and S= UaBH/Q.

 

2.1.2 Separated ports.- We will solve this case by means of a iterative mathematical method

 

B=max[Ltsin θ; 0,93Lt F-1/3]

S=0.089 g’1/3 (H-e)5/3 Qb-2/3 (***)

e=SQ/BUa

 

Qb= flow rate at each single port(m3/s).

 

The number of iterations can modify by means of the parameter N_it of the function Calculation parameters of the program.

Increasing N_it value, we increase the numeric convergence but we will need more time of calculation. We should look for an optimized value of N_it.

 

Single port.- In such a case, we have the next relationships

 

e=0.15H

S=0.089 g’1/3 (H-e)5/3 Q-2/3

B=SQ/eUa (*)

 

However, at high velocity values and if B<=0.3H, the approximation is not correct .

 

2.2 Water is not stratified

 

In this case, the picnocline or thermocline has been formed. We will distinguish the following cases:

 

2.2.1 Multiport diffuser.- In such a case, we have the next equations

 

ymax=2,84 (g’q)1/3 Г -1/2

S=0,31 g’1/3 ymax q-2/3

B=max[Ltsin θ; 0,93Lt F-1/3]

e=SQ/BUa

 

where Г=-(g/ρ)dρa/dy is the stratification coefficient (s-2) and ymax is the thickness of the mixing layer (m).

 

2.2.2 Separated ports.- In such a case, we have the next equations

 

ymax=3,98 (g’Qb)1/4 Г -3/8 (***)

S=0,071 g’1/3 y5/3max Qb-2/3

B=max[Ltsin θ; 0,93Lt F-1/3]

e=SQ/BUa

 

2.2.3 Single port.-

 

ymax=3,98 (g’Q)1/4 Г -3/8

S=0,071 g’1/3 y5/3max Q-2/3

e=0,13 ymax

B=SQ/eUa

 

For a profile of velocities different from the previous ones, it will be required a more complex method of numeric integration to solve the problem.

 

2.3 Near mixing zone and distant mixing zone

 

We need to know the place where the plume centerline crosses the water surface or picnocline layer. To calculate this point we will use Ua and the vertical velocity 

Multiport diffuser.- W=1,66(g’q)1/3 being W the vertical velocity of the effluent (m/s).

 

Separated ports.- W=6,3(g’Qb/H)1/3 .

 

Single port.- W=6,3(g’Q/H)1/3.

 

In the last two cases, H will be replaced by ymax when the water is stratified. The point localization with regard to the place where the plume centerline crosses the surface, gives us the near and distant mixing zone definitions.

 

2.4 Concentration calculation

 

The concentration value in a plume point is determined by the X,Y,Z coordinates and is given by the equation:

 

C(X,Y,Z)=(C0/S) F0(t)F1(t)F2(Y,t)F3(Z,t)

C0=pollutant concentration in the effluent

S=initial dilution

being t=X/Ua. F0(t) takes into account non-conservative pollutants and is equal to:

 

F0(t)=10-t/T90

    

The F0, F1, F2 and F3 functions depend on being in near or distant mixing zone.

 

(a) Near mixing zone:

In such a case, the equations are

 

F1(t)=1

F2(Y,t)=(1/2)[erf[(B/2+Y)/(σy21/2)]+ erf[(B/2-Y)/(σy21/2)]]

F3(Z,t)=(1/2)[erf[(e+Z)/(σz21/2)]+ erf[(e-Z)/(σz21/2)]]

 

being σy=(2Kyt)1/2 and σz=(2Kzt)1/2. The program calculats erf function by numerical integration. The the precision of integration method depends on the parameter N_int. Increasing N_int value, we increase the numeric convergence but we will need more time of calculation. We should look for an optimized value of N_int.

 

(b) Distant mixing zone:

 

In such a case, we approach

 

F1(t)=(2π)-1/2B σy-1/2

F2(Y,t)=exp[(-Y2/2σy2)]

F3(Z,t)=e/Hh

 

being σy=(B2/16+2Kyt)1/2. Here, we suppose that the plume was homogenized vertically when the water depth was Hh, that is the depth in the point where the thickness of the plume begins to occupy the whole layer of water. The program calculates considering the bottom of the sea like a flat surface. Then, Hh is the water depth at the location of the deepest outfall. If you want to consider a higher water thickness than outfall depth, you can draw a deeper outfall whose pollutant concentration is null. In asuch a case, the water depth is the depth of the deepest outfall. The calculation will not be affected by the null concentration of the deepest outfall.

 

 Errors and comments in the model: 

 

(*) We have found, in our opinion, typographic errors in Orden del 13 de Julio de 1993 del Ministerio de Obras Públicas y Transportes del Reino de ESPAÑA, B.O.E. Martes 27 de Julio de 1993, página 22861 that we have corrected considering mathematical consistency. The software assumes the present corrections in the calculation.

(**) Important note for DESCAR 3.0 (or lower versions):

In the approved model, T90 is un hours (this is used by the program). However, and in equation F0(t)=10-t/T90  of the approved model(1), time must be in seconds . Following criteria of mathematical coherence and results T90 must be expressed in seconds (multiply by 3600 seconds in one hour). At this point, the user can work following two different ways: using the approved model as is or rectify in the T90 input data. For example, for a T90=2 hours value, the user can introduce as input data:

(a) Following the approved model:

1/T90=0.5 hours-1 as input data. Then, write 0.5 in the window textbox for a T90=2 hours value .

(b) Following criteria of mathematical coherence and results:

1/T90=1/(2 x 3600)=0,000278 as input data. Then, write 0,000278 in the window textbox for a T90=2 hours value.

(1) Orden del 13 de Julio de 1993 del Ministerio de Obras Públicas y Transportes del Reino de ESPAÑA, B.O.E. Martes 27 de Julio de 1993, página 22861.

(***) In the model, it is not found a relationship between Qb y Q. This relation must be n (number of ports). The calculations assumes that Qb y Q are the same (that is always true for a single port). You can introduce a number of ports in the model parameters window.

 

References:

Orden del 13 de Julio de 1993 del Ministerio de Obras Públicas y Transportes del Reino de ESPAÑA, B.O.E. Martes 27 de Julio de 1993, página 22861, I. Disposiciones generales. Proyecto de conducciones de vertidos desde tierra al mar.

Gerard Kiely, 1999. Ingeniería Ambiental. Fundamentos, entornos, tecnologías y sistemas de gestión. Ed. McGraw-Hill.

E.N. Ramsden, 1996. Chemistry of the Environment. Ed.Stanley Thornes Ltd.

Geoff Hayward, 1992. Applied Ecology. Ed. Thomas Nelson and Sons Ltd.

Secretaría Provisional del Convenio de Estocolmo y la Unidad de Información para convenios del PNUMA, 2003. Eliminando los COP del Mundo: Guía del convenio de Estocolmo sobre contaminantes orgánicos persistentes.  Publicado por PNUMA.

IKSR 2000 : M. Braun, “The Pathways for the most important hazardous substances in the rhine basin (during floods)”, International Commission for the Protection of the Rhine, Koblenz, Germany, in Int. Symposium on River Flood Defence, Kassel, Kassel Reports of Hydraulic Engineering No. 9/2000

 

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