Modeling of biological wastewater treatment on the basis of quick-computing numerical model

Purpose. The scientific paper involves the development of quick computing numerical model for prediction of output parameters of aeration tank. The numerical model may be used in predicting the effectiveness of aeration tank under different regimes of work. Methodology. To simulate the process of biological wastewater treatment in aeration tank numerical models were developed. The flow field in the aeration tank is simulated on the basis of potential flow model. 2-D transport equations are used to simulate substrate and sludge dispersion in the aeration tank. To simulate the process of biological treatment simplified model. For the numerical integration of transport equations implicit difference scheme was used. The difference scheme is built for splitting transport equations. Splitting of transport equation into two equations is carried out at differential level. The first equation of splitting takes into account the sludge or substrate movement along trajectories. The second splitting equation takes into account the diffusive process of substrate or sludge. To solve the splitting equations implicit difference scheme was used. For the numerical integration of potential flow equation the implicit scheme of conditional approximation was used. On the basis of constructed numerical model computer experiment was performed to investigate the process of biological treatment in aeration tank. Findings. Quick computing numerical model to simulate the process of biological treatment in the aeration tank was developed. The model can be used to obtain aeration tank parameters under different regimes of work. The developed model takes into account the geometrical form of the aeration tank. Originality. The numerical model which takes into account the geometrical form of aeration tank and fluid dynamics process was developed; the model takes into account substrate and sludge transport in aeration tank and process of biological treatment. Practical value. Efficient numerical model, so called «diagnostic models» was proposed for quick calculation of biological treatment process in aeration tank.


Introduction
Aeration tanks (AT) are widely used in practice for biological wastewater treatment at treatment plants. AT are used for industrial or municipal wastewaters treatment and may work under different regimes. There are different types of AT but in practice so called «vitesnitel» AT (AT of displacement type) is often used: In this AT influent (waste waters) and sludge, which is used for biological treatment, are supplied at one side of the AT (inlet boundary) and are discharged at the opposite side (outlet boundary) (Fig.1).

Literature review
Mathematical models which are used for aeration tank calculation can be separated in some classes. First of all we have to mention empirical models which were built on the basis of physical experiments [5,7]. These models have the form of simple algebraic formulae with some empirical coefficients. These models are widely used in Ukraine but for calculation of typical AT and for the typical regimes of work for which the empirical constants were obtained. We can't use these models for scientific research, for example, to predict the output parameters after treatment in the case which is out of the normal work of aeration tank.
The second class of the models includes mass balance models. These models can be named «zero-dimentional» models. The balance models are very popular [3,4,8,17] and take into account some important parameters of aeration tank work. These models are based on the ordinary differential equations which represent mass balance of sludge, admixture or oxygen in aeration tank. These differential equations can be solved analytically or numerically (for example using Runge -Kutte method). Some commercial codes can be used to perform calculations on the basis of these models. These models are very convenient for prediction of aeration tank output parameters but the models do not take into account the fluid dynamics process in AT.
The models of the third class are based on «one-dimentional» equations of mass transport to simulate, for example, substrate dispersion in AT [6,9]. The modeling equations are solved analytically. Fluid dynamics is taken into account in these models but for the case of constant velocity in AT.
CFD models are the most «powerful» models at present time to solve the problems of wastewater treatment [1,2,10,[12][13][14][15]. These models can reproduce the flow field in the AT and admixture transfer for different regimes of work with account of AT geometrical form. As a rule the CFD experiments are performed using commercial codes (for example, ANSYS, Fluent) [11,12,16,17]. CFD experiments comprise of two steps. The first step is computation of flow field. Very often this flow field is computed using of Navier -Stokes equations. The second step is simulation of admixture transfer on the basis of computed flow field. Application of Navier-Stokes equations needs much computing time (to solve some problems it may take from 90h to some weeks to perform CFD experiment). It is not convenient in case of many calculations during AT design or at stage of AT reengineering.

Purpose
The purpose of this work is the development of quick-computing numerical model to simulate the process of biological wastewater treatment in «vitesnitel» aeration tank (aeration tank of displacement type).

Mathematical model
To simulate the process of biological treatment in AT, at each time step of mathematical simulation, we separate the process in two stages. At first stage we consider the process of substrate and sludge movement in the aeration tank. It is so called «mass transfer» process. To simulate this process we use the following 2-D transport equations (plan model) [7,12]: where     cients of turbulent diffusion in x, y direction respectively; tis time. The boundary conditions for these equations are as following: 1. at the inlet opening the boundary condition is ,, where , in in CS are known concentrations of substrate and sludge respectively.
2. at the outlet opening the boundary condition in the numerical model ( Fig.2) is written as follows concentrations at the previous computational cell.
Boundary condition (4) means that we neglect the diffusion process at the outlet boundary.

  
where n is normal vector to the boundary.
The initial condition, for 0 t  , is 00 ,, CS are known concentrations of substrate and sludge respectively in computational domain.
At the second stage of mathematical simulation we consider the biological process in the aeration tank. To simulate this process in each computational cell inside the aeration tank we use the following simplified model where  is biomass growth rate; Y is biomass yield factor. To calculate biomass growth rate Monod law is used.
As the initial condition for each equation (5), (6), at each time step, we use the meaning of C, S obtained after computing Eq. 1, 2.
To solve Eq.1, 2 it is necessary to know the flow field in aeration tank. To simulate this flow field we use model of potential flow. In this case the governing equation is where Р is the potential of velocity.
The velocity components are calculated as follows: Boundary conditions for equation (7) are [5]: At the outlet boundary P=const.

Numerical model
To perform numerical integration of governing equations rectangular grid was used. Concentration of substrate, sludge and P were determined in the centers of computational cells. Velocity components u, v were determined at the sides of computational cells.
To solve equation (7) we used the difference scheme of «conditional approximation». To use this scheme we wrote Eq. 5 in «unsteady» form 22 22 , where t is «fictitious» time.
It's known that for t solution of Eq.9 tends to the solution of Eq. 7.
We split the process of Eq. 9 in two steps and difference equations at each step are as follows [4]: The calculation on the basis of these formulas is complete if the following condition is fulfilled: where ε is a small number; n is iteration number. Difference scheme of splitting (10), (11) is implicit but unknown value of P is calculated, at each step of splitting, using explicit formula of «running calculation». That is very convenient for programming the difference formulae.
To solve Eq. 9 it is necessary to set initial condition for fictitious time If we know field of P in computational domain we can compute velocity components at the side of computational cells , Main features of the implicit difference scheme to solve numerically Eq. 1, 2 we consider only for equation of substrate transport because Eq. 1 and 2 are similar from mathematical point of view. Before numerical integration we split transport equation in two equations. The scheme of splitting is as follows From the physical point of view, equation (14) takes into account substrate movement along trajectories, equation (15) takes into account the process of substrate diffusion in aeration tank. After that splitting the approximation of equation (12) is carried out. Time dependent derivative is approximated as follows: The convective derivatives are represented as: , At the next step we write the finite difference scheme of splitting: at the first step k=1/2: at the second step k=1, c=n+1/2: This difference scheme is implicit and absolutely steady but unknown concentration C is calculated using the explicit formulae at each step («method of running calculation»).
Further, Eq. (15) is numerically integrated using implicit difference scheme (10), (11). To solve Eq. 3, 4 we used Eurler method. On the basis of developed numerical model code «BIOTreat» was developed. FORTRAN language was used to code the solution of difference equations.

Description of computational procedure
Numerical solution of the whole problem is as follows: -Step 1: we compute potential P in aeration tank (Eq. 8,9) -Step 2: we compute velocity components (Eq. 10, 11) -Step 3: we compute biological process in aeration tank (Eq. transport in aeration tank (governing equations 1, 3; numerical equations 14, 15 (for C and S) and Eq. 8, 9 written for C and S) -Step 5: for the next time level t, the computational procedure repeats from step 3.
Case Study. Developed code «BIOTrea» was used to solve the following model problem. The aeration tank is filled with sludge (concentration S 0 =2) and substrate (concentration C 0 =100) at time t=0. All parameters of the problem are dimensionless. During time period from t=0 till t=2 the inlet and outlet openings are closed and no flow in the aeration tank. It means that for this time period only biological treatment takes place and we solve only Eq. 3, 4 of the model. At time t=2 the inlet and outlet openings are open and the transport process starts. At the inlet opening the substrate concentration is equal to C 0 =100 and sludge concentration is equal to S 0 =2. Also at this time five sources of sludge supply inside the aeration tank starts to work with intensity Q i . Position of these sources can be seen in Fig. 4 where the influence of these sources results in local 'deformation' of concentration field. This field has practically small concentration gradient in aeration tank everywhere except points where sources of sludge supply are situated.
In Fig. 3 we present sludge and substrate concentration change near the outlet opening of the aeration tank (point A in Fig.4). From Fig. 3 we can see that the process of biological treatment accelerates from t=2 and concentration of sludge at the outlet opening increases with time. In Fig. 4 the concentration field of sludge for time step t=4 is shown. It is well seen the zones of sludge sources influences. These zones have form of circles.

Findings
Quick computing numerical model was developed to simulate the wastewater treatment in aeration tank. The model does not take much time because the fluid dynamics process is simulated on the basis of potential flow model.

Originality and practical value
A new numerical model to predict the output parameters of aeration tank was developed. The model is based on the 2-D transport equations of substrate and sludge and simplified equations of biological treatment. The developed model takes into account geometrical form of aeration tank. The model can be useful in aeration tanks design.