Research On Mass Transfer Theory For 2-D Stochastic Water Quality Model

Stochastic differential equation is an effective tool to solve complex problems about nonlinearity and uncertainty, and its research and application to water pollution control planning are still at the initial stage at home and abroad. After analyzing and reviewing the present research condition, mass-transfer theory, stochastic processes and diffusion theory of water quality model, a kind of two-dimensional stochastic water quality model based on mass-transfer theory is presented and established in this paper. It is mainly used in the simulation and predication of river pollution mixing zone (PMZ) and the real-time warning of emergency water pollution accidents. In this paper, the method and theory of modeling on water quality, as well as simulation and calculation of initial pollutant concentration ( c0 ) in the near field of the river's outlet, are researched of exploratory. All these not only provide a new approach to the simulation of water quality, but also help to improve the accuracy of present pre-warning water quality models in our country.Supposing that the pollutant concentration ( ct ) in the river vary as the geometric Brownian motion, a non-stationary stochastic process can be introduced. In this process, c0 and the synthetic attenuation coefficient (α) of the pollutant are taken as two indefinite constants. Analyzing the pollutant's diffusion process in the river in detail and supposing that the pollutant is only part depth homogeneous mixing before reaching whole depth homogeneous mixing, a theoretical formula to calculate c0 is proposed, and the part homogeneous mixing depth h p is a key parameter in this formula. In addition, in order to fully use the measure data of ct in the river, the first order model for unequal interval sequence is established and the time response function-expression for ct is proposed in this paper, basing on analysis of grey data treatment as well as its modeling way and the establishment, and a kind of the first accumulative generated sequence is constructed according to the geometrical meaning of integral in this model. Experimental computations show that this model's error on simulating the river's experimental pollutant concentration is generally less than 15%, and the concrete value of both c0 andαcan be obtained, which indicate that the modeling method is practical and effective.Since using transfer coefficient to approximately describe the diffusion process will be simpler, the transfer coefficient, which describe substances'speed at the transfer interface, and its theoretical research methods are introduced into the field of water pollution control planning in this paper, and the pollutant isoconcentration line in the river is taken as the transfer interface. The formation and development of the pollutant isoconcentration line in the river is analyzed in detail, the stochastic function which describes the pollutant's transfer speed is established, which shows that the pollutant's transfer speed is a non-stationary stochastic processes, and its value depends on the concentration difference between the two sides of interface. Also, in this paper, transfer coefficient and transfer interface are analyzed and compared in detail. And it is proposed that the transfer coefficient used in this paper is different from that classical one of mass-transfer science in the magnitude order because they are of different transfer interfaces. After comparison, the instantaneous transfer coefficient is used and taken as a constant in this paper, and the error which due to the fluctuation of the transfer coefficient is contained in the stochastic disturbance item. In addition, the synthetic attenuation coefficient of the pollutant in the river is analyzed in detail, and data series show that the dilution and diffusion actions of the water flow to the pollutant are more important than the biodegradation action of the pollutant itself.The formation and development process of the pollutant mixing zone (PMZ) in the river is divided into two stages, which are the stage of initial momentum transfer and the stage of pollutant's diffusion process, while the velocity of flow is supposed to be a stationary stochastic process to reflect the turbulence characteristic of rivers. So it is supposed that the absolute speed of the pollutant's diffusion equaled to the pollutant's diffusion and transfer speed superposed to the moving speed of the pollutant according to the water flow, and the development process of the PMZ is described by transfer ellipse and envelope line generated during moving. The PMZ is divided into two parts, which are the envelope line part and the transfer ellipse part. The long and short half axle of the transfer ellipse respectively show the pollutant's horizontal and vertical moving distances along the river because of the transfer process. So, comprehensively applying the method of system analysis and the theory of mass-transfer principle, stochastic functions which respectively describe the length, width and area of the pollutant zone in the river are established, correspondingly, their expected value and variance function-expression are deduced.In this paper, the measure data of the Rhodamine-B tracer in which pollutants are discharged continuously and stably in the center of the Jialing River during Chongqing (1989) , as well as the measure data in which pollutants are discharged in the riverside of Huangshaxi in the Yangtze River during Chongqing (1997), are used to determinate the parameters of the 2-D stochastic water quality model, and the concrete value of the transfer coefficient is that the horizontal one k x was( 0.5~1) m /s and the vertical one k y was ( 0.05~0.5 )m /s , and they are both proportional to u , the average velocity of the river flow. In addition, it is got that h p is ( 0.5 ~ 0.75 )h when the pollutants are discharged in the river center and respectively ( 0.5 ~ 0.75 )h in the riverside, the initial shape of the pollution cloud is a holosymmetric square or circle. The result of the model validation shows that the simulation error of the length x? ( t ) and the width y? ( t ) of the PMZ's isoconcentration line are less than 9% and 16% respectively, and the simulation error of the transfer ellipse's long and short half axle are both less than 21% at the end of the transfer process. This result indicates that the model can be used to simulate and calculate the river center PMZ and the riverside PMZ accurately.In this paper, the PMZs near the pollution discharge orifice at Taohuaxi and Lijiatuo in the Yangtze River during Chongqing, in low flow years before and after the impoundment of Three Gorges Reservoir, are respectively predicted and analyzed, and the areas of the two PMZs are calculated. The result indicates that, after the impoundment of Three Gorges Reservoir, the water flow's velocity in the reservoir region will decrease when the incoming flow and pollutant discharge conditions are constant, and the range of the riverside PMZs will be much larger than before, the added value depends on the concrete water flow condition of the outlet. Since the location of the outlet has important influence on the range of the PMZs, it should be avoided to locate the outlet in non-fluent regions in the river, especially back water regions and shallows.This model gives the stochastic function-expression to the range of PMZ, its expected value function is an analytical equation, it can predict the limits of PMZ conveniently and quickly, and it is especially adequate to the real-time warning of water quality. If the concrete value of the river's velocity and the pollutant concentration's stochastic disturbance strength can be got, the predication interval of the pollutant zone's length, width and area can be got as well, this will make the risk assessment and pre-warning of water quality more convenient.