CANARINA:
DESCAR:
Wikipedia:

Algorithmen: Es kann auf zwei verschiedene Rechenmodelle arbeiten: das klassische Modell Buoyant ( ist das Modell, dass die EPA, USA verwendet) und das Modell der Schichtung (Stratified).
1. Buoyant jet model A type of mathematical model that has been developed for sumerged round buoyant jets is the lengthscale 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 lengthscale 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 XAxis and at a distance y(meters) in the YAxis will be given by:
c =c_{c} 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, c_{c} is the pollutant concentration in the center of the plume line and b is the plume halfwidth. We attempt to link the momentum dominated and buoyanvy dominated regimes into one relationship by using proposed relations for the transition where:
z/L_{b} =2^{4/3}[(1/2)(x/L_{b})^{2}+(L_{m}/L_{b})(x/L_{b})]^{1/3} (2) b/L_{b} =c_{b}[(1/2)(x/L_{b})^{2}+(L_{m}/L_{b})(x/L_{b})]^{1/3} (3) S=c_{s}(u_{o}/u_{a})[(1/2)(L_{b}/L_{m})(x/L_{m})^{2}+(x/L_{m})]^{1/3} (4)
We obtain solutions for a vertical buoyant jet in a crossflow. And buoyant jets discharged horizontally perpendiculat to crossflow. z/L_{b} =c_{xy}(x/L_{m})^{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 farfield 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:
[u_{0}^{2} B+ U_{a}^{2} H]/[(u_{0} B g’)^{2/3} H] <0.54
Typical values (pollutants): Organic matter as DBO5  350g/m3 Suspended matter  600g/m3 E. Coli  10^{12} /m3 N2 (total)  30 gN/m3 Effluent velocity  entre 0.6 y 0.8 m/s Port diameter  6cm
Dispersion coefficients: Horizontal dispersion: K_{y}(m^{2}/s)=3x10^{5} B^{4/3}. B=initial plume width(m)
Vertical dispersion: K_{z}(m^{2}/s)=4x10^{3} U_{a} e e= thickness of the mixing layer U_{a}=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
In such a case, we have the next relationships
S=0.27 U_{a} H q^{1} F^{1/3} e=0.29H B=SQ/eU_{a}
Case II:
25^{o}=<θ <65º F>0,36 (*)
In such a case, we have the next relationships
S=0.38 U_{a} H q^{1} B=max[L_{t}sin θ; 0.93L_{t} F^{1/3}] e=SQ/BU_{a}
Case III:
θ <25º 0,36<F=<20
In such a case, we have the next relationships
S=0.294 U_{a} H q^{1} F^{1/4} B=max[L_{t}sin θ; 0.93L_{t} F^{1/3}] e=SQ/BU_{a}
Case IV:
θ <25º F>20
In such a case, we have the next relationships
S=0.139 U_{a} H q^{1} B=max[L_{t}sin θ; 0.93L_{t} F^{1/3}] e=SQ/BU_{a}
Case V:
θ >65º F>0,1
In such a case, we have the next relationships
S=0.58 U_{a} H q^{1} B=max[L_{t}sin θ; 0,93L_{t} F^{1/3}] e=SQ/BU_{a}
From Case II to Case V, and if e>H, we take e=H and S= U_{a}BH/Q.
2.1.2 Separated ports. We will solve this case by means of a iterative mathematical method
B=max[L_{t}sin θ; 0,93L_{t} F^{1/3}] S=0.089 g’^{1/3} (He)^{5/3} Q_{b}^{2/3}^{ (***)} e=SQ/BU_{a}
Q_{b}= flow rate at each single port(m^{3}/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} (He)^{5/3} Q^{2/3} B=SQ/eU_{a} (*)
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
y_{max}=2,84 (g’q)^{1/3 }Г^{ 1/2} S=0,31 g’^{1/3} y_{max} q^{2/3} B=max[L_{t}sin θ; 0,93L_{t} F^{1/3}] e=SQ/BU_{a}
where Г=(g/ρ)dρ_{a}/dy is the stratification coefficient (s^{2}) and y_{max} is the thickness of the mixing layer (m).
2.2.2 Separated ports. In such a case, we have the next equations
y_{max}=3,98 (g’Q_{b})^{1/4 }Г^{ 3/8}^{ (***)} S=0,071 g’^{1/3} y^{5/3}_{max} Q_{b}^{2/3} B=max[L_{t}sin θ; 0,93L_{t} F^{1/3}] e=SQ/BU_{a} _{ } 2.2.3 Single port.
y_{max}=3,98 (g’Q)^{1/4 }Г^{ 3/8} S=0,071 g’^{1/3} y^{5/3}_{max} Q^{2/3} e=0,13 y_{max} B=SQ/eU_{a}
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 U_{a} 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’Q_{b}/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)=(C_{0}/S) F_{0}(t)F_{1}(t)F_{2}(Y,t)F_{3}(Z,t)
being t=X/U_{a}. F_{0}(t) takes into account nonconservative pollutants and is equal to:
F_{0}(t)=10^{t/T90}
The F_{0}, F_{1}, F_{2} and F_{3} functions depend on being in near or distant mixing zone.
(a) Near mixing zone: In such a case, the equations are
F_{1}(t)=1 F_{2}(Y,t)=(1/2)[erf[(B/2+Y)/(σ_{y}2^{1/2})]+ erf[(B/2Y)/(σ_{y}2^{1/2})]] F_{3}(Z,t)=(1/2)[erf[(e+Z)/(σ_{z}2^{1/2})]+ erf[(eZ)/(σ_{z}2^{1/2})]]
being σ_{y}=(2K_{y}t)^{1/2} and σ_{z}=(2K_{z}t)^{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
F_{1}(t)=(2π)^{1/2}B σ_{y}^{1/2} F_{2}(Y,t)=exp[(Y^{2}/2σ_{y}^{2})] F_{3}(Z,t)=e/H_{h}
being σ_{y}=(B^{2}/16+2K_{y}t)^{1/}2. Here, we suppose that the plume was homogenized vertically when the water depth was H_{h}, 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, H_{h} 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 F_{0}(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 Q_{b } y Q. This relation must be n (number of ports). The calculations assumes that Q_{b } y Q are the same (that is always true for a single port). It is possible to use the program introducing different point sources with a single port each one, if the distance between the point sources is higher than 20% of the depth. In that way, it is possible to use the model. If it is neccessary the scale can be increased to introduce the point sources ne by one (as different sources) in a proper distance. We hope to introduce in a near future a parameter n in the program.
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. McGrawHill. 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
Canarina Algoritmos Numéricos, S.L. Software für Umweltsimulationen Kanarische Inseln, Spanien email: Kontakt
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