Modèle de transfert et de diffusion de masse dans un écoulement, en présence de gradients de vitesse et de gradients du coefficient de diffusion turbulente A diffusion an mass transport model in stream with velocity and turbulent diffusion coefficient gradients

Résumé

Le travail présente un modèle mathématique conceptuel de transfert et de diffusion de masse destiné à l'étude des migrations d'effluents en rivière. Ce modèle prend en compte l'existence d'écoulements cisaillés ainsi que la présence de gradients de diffusion turbulente. Il permet de calculer les champs de concentrations et les valeurs moyennes de concentration à travers toute section transversale de l'écoulement. La localisation et la taille relative du rejet sont respectées. L'influence des rives sur les processus de dispersion est prise en considération.Pour quantifier l'influence des berges, une relation est établie entre les concentrations calculées en écoulement de largeur infinie et les concentrations en écoulement d'extension limitée. La méthode utilisée est fondée sur l'emploi d'un champ de vitesse et d'un champ de coefficient de diffusion, symétriques par rapport à des lignes riveraines séparant le courant nul d'un courant fictif situé de part et d'autre de ces limites.Les résultats des tests du modèle mathématique, réalisés à l'aide du programme moniteur « Gradient 2 », sont présentés. Dans le cas d'écoulements cisaillés, on a constaté que la valeur moyenne de concentration d'effluent calculée au travers de sections transversales à l'écoulement n'était pas une quantité invariante tout au long de l'écoulement. Un gradient de vitesse négatif induit une augmentation de cette moyenne à mesure que l'on s'éloigne du rejet alors qu'un gradient positif produit l'effet inverse. Un gradient du coefficient de diffusion turbulente détermine un changement du profil de concentration à l'intérieur d'une section transversale donnée, sans en changer cependant la valeur moyenne. Un gradient négatif augmente la valeur maximale de la distribution des concentrations. Un gradient positif fait diminuer la valeur maximale en aplatissant l'allure du profil.Le modèle mathématique a ensuite été vérifié à l'aide d'un modèle physique. Un modèle réduit respectant les similitudes d'écoulement a été bâti. Les gradients de vitesse du fluide et les gradients du coefficient de diffusion étaient provoqués par l'introduction de tirants d'eau non uniformes dans chaque section transversale. Les mesures réalisées ont permis d'estimer les coefficients de diffusion turbulente.

Mots-clés:

• Modèle mathématique,
• convection,
• diffusion turbulente,
• gradient de vitesse d'écoulement,
• gradient du coefficient de diffusion

Abstract

The report shows mathematical model of diffusion and pollutants mass transport in rivers with transversal to main movement direction velocity and turbulent diffusion coefficient gradients. The occurence of the mentioned gradients in plane cross sections is a result of variable depth channel and stream interaction. The results of sample calculations using microcomputer program and the results of mathematical model verification on laboratory model were shown. The presented model serves to qualification of pollutants concentration distribution in rivers, below the outfall.

A mathematical model is based on differential equations of advection-diffusion. Fundamental equation includes real dimensions of outfall and riversides reaction. To take into account the influence of flow velocity and turbulent diffusion gradients, an intensity of transversal to the main flow direction diffusion flux was determined. An intensity of diffusion flux which flows in any stream filament « I » depends on its coordinate, flow velocity and turbulent diffusion coefficient in this stream filament. Basing on diffusion flux intensity equations with variable parameters vi and Di, a pollutant concentrations field in a plane cross section was determined. To determine the influence of stream boundaries, the method of ratio of concentration field in unlimited and limited by riversides stream was used. Using this method it is necessary to take into account the symmetry of turbulent diffusion coefficient and velocity distributions related to boundaries which separate the real stream from the imaginary streams situated beyond the boundaries. A condition of mirror reflection of turbulent diffusion coefficient and velocity distributions is accounted for transversal diffusion flux reflection off impermeable riverside and the change movement direction to reversal one.

Sample calculations were based on the example of river channel 30 m wide and the boundary outflow with real dimensions. To express explicitly the results of sample calculations and their analysis, a linear positive and negative diffusion coefficient (range 1.5 10^-3 to 80 10^-3 [ms^-2]) and velocity (range 0.2 to 0.8 [ms^-1]) gradients were assumed. Negative gradients mean decreasing flow velocity and diffusion coefficient with increasing y distance from the outflow. Positive gradient means increasing magnitude of D (y) and v (y) with increasing y coordinate.

Quantities of turbulent diffusion coefficient were estimated on the basis of mathematical model verification at hydraulic model described below. Sample calculations were dons using computer program « Gradient 2 ».

Basing on the analysis of testing results it was found out that the condition of concentration field continuity in the plane cross-section, which was valid with nongradient flow, was not satisfied with the occurence of flow velocity gradient. Negative velocity gradient causes a considerable increase of concentration field. The quantity of concentration field increases with distance from outfall to cross-section. Comparison results of sample calculations with carried out earlier for slot outfall shows lower concentrations field increments for mal dimension outfall. It is accounted for the presence of wedge with concentrations equal to the initial concentration in outfall. Positive flow velocity gradient causes concentration field decrease with the increase of distance between the stream plane cross-section and outfall.

The increase of concentrations field with negative flow velocity gradient is accounted for pollutants mass accumulation in stream filaments, which flow with lower velocity. With positive gradient an increased pollutant mass advection process occurs in stream filaments with higher velocity. Turbulent diffusion coefficient gradients do not change the quantity of concentration field, but change distribution of pollutant concentrations. Negative gradient increases the slope of concentrations distribution curve, increases maximum concentrations and decreases range of diffusion flux action. Positive diffusion coefficient gradients decrease the slope of concentrations distribution curve by decreasing maximum concentrations and increasing range of diffusion flux action.

Verification of mathematical model was clone using laboratory model of open channel. The conditions of dynamic similarity of pollutant mass propagation processes on hydraulic model and river channel were determined. Conditions of similarity were determined on the basis of identity of differential equations of advection-diffusion written for model end the real object. Provided there is a similarity condition of processes of advection and diffusion which is the same Peclet number (Pe = idem), the condition of flow similarity must be satisfied, which is the same Froude number (Fr = idem). Using the mentioned similarity conditions between the model and the object a similarity scale of concentrations is determined. Concentrations of pollutant mass in any stream point on hydraulic model are ʆ₁ times larger than concentrations in corresponding river point, where ʆ₁ is a scale of geometric similarity of model to object.

Verification measurements were carried out in laboratory channel 0.3 m in width and 5 m long. Flow velocity gradients were generated by various depth of trapesium channel. Reynolds number was ranging from 2.6 10^-3 to 2 10^-3. Pollutants were simulated by rhodamine B dye solution. Concentrations of rhodamine solution were measured using the spectral colorimeter, equipped with instrument for measuring fluorescence with accuracy of 10^-9 [N m^-3]. The measurements of flow velocity were done with using current flow meter, and the measurements of depth with the use of point limnimeter. Turbulent diffusion coefficient was estimated on the basic of measured parameters by empirical relation Di = ʆ_p vi hi, where ʆ_p is a proportionality coefficient, and its value is ʆ_p = (0.7 -1) 10^-2. The results of measurements of tracer concentrations, and results of mathematical model calculations show sufficient convergence for practical tanks. Maximum relative deviation of concentrations reached 18 %. For the detailed investigation of mathematical model suitability, the next stage of verification is predicted on the base of measurements in the natural channel, with gradient of flow velocity.

Keywords:

• Mathematical model,
• convection,
• turbulent diffusion,
• flow velocity gradient,
• diffusion coefficient gradient