The Activated Sludge Model In Biological Wastewater Treatment And Its Numerical Simulation

Mathematical model in the field of polluted water microbial method often need us to solve the problems of quasi-linear partial differential equations. Today, there have been many kinds of numerical solution methods, such as volume finite element method, eigendifference method, the methods with moving mesh, alternating-direction method and so on.The basic numerical solution method of partial differential equations is difference method. The first question is how to make a proper difference scheme. By replacing differential coefficient with difference coefficient, we can get the difference schemes. We can get different difference schemes by choose the differential scheme. As for the difference schemes, we need to discuss the consistency, convergence and stability problem.In the thesis, we study the initial and boundary value problem of quasi-linear parabolic partial differential equations in activated sludge model in the field of polluted water microbial method: of which u ( x , t ) = (u₁ ( x , t ), , u_m( x , t)), f ( x , t , u , u_x)is a vector function of m dimensions, A( x , t , u )is m×m positive definite matrix, andψ₀ (t),ψ₁(t ),φ( x) are vector function of m dimensions.Supposes the condition:(I) A( x , t , u )is continuous at( x , t ),( x , t )in Q_T , u∈R^m, u∈R^m is continuous and differential.And there existσ₀ > 0,ξ∈R^m,for all ( x , t )∈Q_T and u∈R^m, has(ξ, A( x , t , u)ξ)≥σ₀|ξ|².(II) f ( x , t , u , p ) is continuous at( x , t ),( x , t )in Q_T ,( x , t )∈Q_T , u , p∈R^m are continuous,and is Lipschitz continuous where u , p∈Rm. And there exist C > 0, has for ( x , t )∈Q_T , u , p∈R^m,wheref ( x , t ) = f ( x , t,0,0).(III)φ( x )∈H¹[0, l]is a vector function,ψ₀(t )andψ₁(t ) is continuous and differential when t∈[0, T],ψ₀ (0) =φ(0),ψ₁(0) =φ(l)is satisfied.First, we make its difference scheme as follow: Where and we discuss the consistency, convergence and stability problem of the difference equations's solutions.Regarding multi-dimensional space in finite difference scheme, there exist similar produces.Then for the classical Newton's methods has an low order of convergence of solving systems of nonlinear equations. In chapter 3 we give an inexact modified Newton's methods to solve the nonlinear equations after the discretization process, that is to solve s_k,such that where is satisfied, and x_{k +1} = x_k + s_k, x₀ is initial value, 0≤η_k< 1.And we discuss the linear convergence. When x₀ is next to x * and 0≤η_k< 1 is satisfied,the inexact modified Newton's methods is convergent .Moreover, there are other methods such as the Newton method having parameter, enormous entropy algorithm and so on. In recent years, along with the development of the mathematics software Maple , Mathematica, and the computer mark computation ability,many scholars devote to use the mark to calculate the solution of the nonlinear equations.The active sludge method is one of most widespread methods in the biological treatment of waste water. The mathematical model has the important significance regarding the sewage biological treatment design and the movement management.We use the numerical methods to solve the initial and boundary value problem of differential equations in activated sludge model in the field of polluted water microbial method. In the equation, the hypothesis current capacity is at the steady state and the cross section is invariable along the regulation, then has the following equation regarding any water quality variable Ci , its item is with the ASM3 representative.Thus obtains the coupling quasi-linear partial differential equations:Through the analysis, we can draws up that using this method to solve the quasi-linear partial differential equations ,its procedure is more succinct, and is more accurate to the actual result. Compared with the real observed data, we conclude that it's a good simulated result.