Mathematical model of environmental waters purification

This article describes the use of the stochastic approach, in particular, mass service theory and the development of its methods, adapted directly to the coagulation process as a mathematical tool. The coagulation process will be concerned as a) supplying water to the mixer, b) processing it with reagents (coagulants), c) settling for the mathematical analysis of water clarification effectiveness.


Introduction
One of the most effective physico-chemical methods for removing suspended colloidal particles from natural and wastewaters is coagulation, which allows achieving a high degree of water clarification 1. The coagulation process will be concerned as a) supplying water to the mixer, b) processing it with reagents (coagulants), c) settling for the mathematical analysis of water clarification effectiveness. We use the stochastic approach [2], in particular, mass service theory and the development of its methods, adapted directly to the coagulation process as a mathematical tool. Containing dispersed particles, the reagent enters the mixer simultaneously with water supply.

Methods of experiment
As the average time of a particle service, we mean the average time t sev , determined by the gap length, from the moment the reagent enters the mixer until flakes precipitation. As incoming requirements we mean either particles that are already in the water and their average value (or concentration) which is already known, or entering into the mixer [3].
Formulation model. System S, consisting of storage and a service device, at random times receives requirements in accordance with the Poisson law with intensity a: The storage capacity is limited and equal to n requirements. If the incoming request finds the full storage, then it is lost. As soon as the service device is released (it finishes servicing the previous batch of requirements), it becomes the storage and accepts all the requirements there for servicing. [4] The service time of any requirement is random and distributed exponentially, with the parameter It is required to find   t Р k probability, at the moment of time t  k is in the storage, requirements, k = 0, 1, 2, … , n.
The system of equations corresponding to the model is: Initial conditions are defined as In order to solve system (1), we note that Given the initial conditions, the solution of this linear equation is Continuing the process of solving the equations of system (1) recurrently, we obtain for If k = n, we have a linear equation As we see, the formulas for the transitional regime are hard to predict and unsuitable for practical use.
However, they are greatly simplified for the stationary mode that takes place in practice,.
Considering formulas (6), we introduce and calculate the following quality indicators: Кdrive payload ratio, n N K  . (8) М -the reliability of the required storage capacity is equal to , where Р d. = 1 -,  -storage performance,  -incoming requirements.
We estimate the error from hypothesis that the process is stationary. The absolute error is determined from the formula The relative error will be equal to Suppose, given in m 3 , the volume of wash water contains, on average, =0,02% of colloid-dispersed particles, and the coagulants are able to effectively process per hour, on average,  = 0,025% of these particles contained in the mixer with volume n = 2 units of volume (per unit volume take 50m 3 ). The reliability of the required volume of the mixer  = 0,95. [5] Calculating the quality indicators, we have 1. probability that the drive is full: 2. the average volume of wash water contained in the drive    

Conclusion
It can be seen from the figure that the system quickly enters the stationary mode of operation and already at t> 30 minutes, you can use formulas (b) [9]. Thus, queuing theory models allow not only analyzing the coagulation process as a whole, but also effectively planning the water purification process [10].