A diffusion an mass transport model in stream with velocity and turbulent diffusion coefficient gradients
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 = ʆp vi hi, where ʆp is a proportionality coefficient, and its value is ʆp = (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.