NUMERICAL DETERMINATION OF HORIZONTAL SETTLERS PERFORMANCE

Dep. «Hydraulics and Water Supply», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 373 15 09, e-mail gidravlika2013@mail.ru, ORCID 0000-0002-1531-7882 Dep. «Hydraulics and Water Supply», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 373 15 09, e-mail kozachynav@yandex.ua, ORCID 0000-0002-6894-5532


Introduction
It's known, that sedimentation by gravity is one of the most common approaches for the removal of suspended solid particles from water in water treatment plants.This physical process is used in settlers.The engineers know that the performance of settlers would directly or indirectly affect the rest of water treatment process.The design of settlers with the high deposition rate is critical and has been the subject of many theoretical, experimental and numerical investigations [1,6,10,11,12,13,14].
Nowadays to receive more effective work of the horizontal settlers with comprehensive geomet-rical forms, different kinds of baffles and plates are proposed by designers.According to this the real lack of methods to calculate the efficiency of these settlers is an obvious problem.
To calculate the efficiency of horizontal settlers the empirical models are widely used in Ukraine [3,15].These models do not take into account the geometrical form of the horizontal settlers and the peculiarities of the sedimentation process.Therefore, it is important to develop CFD models having more capabilities to simulate the process of the waste waters treatment in settlers and which do not need much computational time for running and allow taking into account the geometrical form of settlers [1,10,9,16].

Purpose
The objective of this paper is the development of the effective computer models (CFD models) which are more effective than the employed in Ukraine models and which can be used for prediction of the horizontal settler efficiency.

Methodology
Mass transfer model.To simulate the process of water purification in the horizontal settler the transport equation ( 1) is used [1,7,10]: where C is the concentration; u, v are the velocity components in x, y direction respectively; w -is the settling velocity; σ -is the parameter taking into account the process of flocculation and decay; µ х , µ y are the coefficients of turbulent diffusion in x, y direction respectively; x i , y i are the Cartesian coordinates; The transport equation is used with the following boundary conditions [1,5,8,10]: -inlet boundary: , where C E is the known concentration (in the case study of this paper it is dimensionless and equal to 100 -outlet boundary: in numerical model the condition Here, C(i+1,j) is the concentration at the outlet boundary cell (this boundary condition means that we neglect the process of diffusion at this plane).

( )
, C i j is the concentration in the previous cell.
Fluid Dynamic Models.To simulate the flow in the horizontal settler three fluid dynamic models were used.
The governing equations of the first model are 1) Poisson equation for flow function [4,8]: x y 2) Equation of the vorticity transfer [4,8]: The components of the water flow velocity inside the settler are determined as follows: The initial and boundary conditions for these equations are shown in [4].
The governing equation of the second model is [1,4,6,8]: where P is the potential of flow.The boundary conditions for this equation are discussed in [4,8].The components of flow velocity inside the settler are calculated as follows [5] , The governing equations of the third model (Navier -Stokes equations) are equation ( 5) and equation (6).
Equation ( 5) is Poisson equation for flow function [11,12]: x y 1 Re where is Reynolds number.
Boundary and initial conditions of this fluid dynamic model are discussed in [8].
To model the process of flocculation the following equation is used [12,13] where C is concentration of primary (large) particles; C 1 -initial concentration; K A -experimentally determined coefficient for floc aggregation; K Bexperimentally determined coefficient for floc break up; G -mean velocity gradient.
Computation of settling velocity.
To compute the settling velocity the following model is used [12,14]  where K 1 , K 2 are experimental constants [5,12].Numerical solver.Numerical integration of governing equations is carried out using rectangular grid.The geometrical form of the horizontal settler in the numerical model is created using porosity technique (markers method) [1,8].
To solve the equation of potential flow (4) Samarskii A. A. implicit difference scheme is used [8].At the first step Laplace equation ( 4) is written in the following form where η is the ghost time.At the second step the time splitting procedure for equation (7) n n n n i j i j i j i j P P P P x y n n i j i j P P y Components of the velocity are calculated using expressions (10): , , To start the numerical integration of equation ( 7) it is necessary to set the initial condition in the form P=P 0 (for example P 0 =0).
To solve equation of the vorticity transfer (3) the two steps difference scheme is used [8]: − at the first step of splitting the difference equation is − at the second step of splitting the difference equation is The unknown meaning of vorticity is obtained from these expressions using the explicit formulae of «running calculation».
To solve Poisson equation (2) the following difference scheme of splitting is used [8]: − at the first step of splitting the difference equation is − at the second step of splitting the difference equation is − at the third step of splitting the difference equation is n n n n n n i j i j i j i j i j i j t x y − at the last step the difference equation is Velocity components are calculated using the following expressions , 1 , ; To solve equation ( 6) the change triangle difference scheme is used [8].First of all velocity components are written in the following form 2 2 After that the convective derivatives are approximated using the following expressions: The second order derivatives are written as following: The difference approximation of the equation ( 6) can be written as follows 1 , , Re we have the difference scheme which has the second order of accuracy in time.
The change triangle difference scheme for equation of vorticity transfer is written as follows ( ) ( ) Using these expressions the unknown meaning of vorticity is computed using «running calculation» [8].
To solve the mass conservation equation (1) the implicit difference scheme of splitting is used [1,8].At first step the physical splitting of equation (1) is carried out: At the second step the following approximation of the first order derivatives are used [5]: The second order derivatives are approximated as following: Here we use notation v=v-w.In these formu- las , , , , , , , . are the notations of the difference operators [8].
After the approximation the solution of the difference equation is splitted in 4 steps [1,8]: -at the first step 1 4 k = the difference equation is: ( ) -at the second step 1 2; 1 4 the difference equation is: ( ) -at the fourth step The developed numerical models where coded using FORTRAN.

Findings
The developed computer models were used to compute water purification in the horizontal settlers with comprehensive geometrical forms, different kinds of baffles and plates: The computational time was 10 sec -5 min to solve the fluid dynamics problems and masstransfer using the developed numerical models.
Results of numerical integration models equations described in Table

Originality and practical value
A new approach to investigate the mass transfer process in horizontal settler was proposed.This approach is based on the developed CFD models of different level.Three fluid dynamics models were used for the numerical investigation of flows in the settler.These models use the rectangular grid and porosity technique to create the form of the settler in the numerical model.The developed models have more capacity than the existing models in Ukraine.The developed models allow computing quickly the efficiency of water purification in settlers.The models are not computationally expensive.

Conclusions
Three CFD models were developed to compute the flow field in horizontal settler.These models are based on the equations of inviscid fluid dynamics models and Navier-Stokes equations.The process of mass transfer in the horizontal settlers is simulated using convection-diffusing equation.Numerical study based on the developed models was carried out.Results illustrate that the developed models can be used to simulate the process of water purification for settlers having comprehensive geometrical form.

Fig. 1 .Fig. 1 .
Fig. 1.Concentration field in the horizontal settler with baffle and vertical plate (inviscid vortex flow model) In Fig. 1-3 the concentration field in the settlers is shown.The concentration is presented using «Integer» form of number.Every number shows the percentage of the concentration in the computational cell.The maximum concentration is at the inlet cell (it's equal to «99») and the smallest concentration is in the outlet cell.This concentration shows the efficiency of the each settler.The computational time was 10 sec -5 min to solve the fluid dynamics problems and masstransfer using the developed numerical models.Results of numerical integration models equations described in Table1