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P.h.d. Ponomarenko E. G.*,
p.h.d. Nemtsova A. A. **
* Department
of Urban Environmental Management & Engineering, State Academy of Municipal
Economy
* Department
of Information Technologies, National Pharmaceutical University
Mathematical Model of the Process of
Wastewater Treatment in the Biological Ponds for Automatic Control Purpose
Mathematical modelling of objects of
control is, as a rule, necessary stage of development of control systems. A
model of operational control of water quality is bound to describe the dynamics
of transport and transformations of substances in streams and has to permit using of control theory methods. It
has been known that in process units (excluding setting tanks) flow is turbulent.
In order to describe processes of water quality forming in turbulent streams
the dispersion equation is widely used:
(1)
t = time, s; x =
distance along the axes of stream, m; C
= cross-section average concentration of substance, g/m3; n = cross-section average of stream
velocity, m/s; k = decay rate
coefficient, s-1; f =
function of effluent input rate, g/(m3c);
D = dispersion coefficient, m2/s.
This is a distributed parameter equation. Its direct
use for solving the problem of operational control presents considerable
challenge of methodological and computational nature. Using of the models of
idealistic reactors instead of Equation (1) leads to somewhat simplified view
of the actual process of water quality forming. In this paper a method of
reduction of the one-dimensional dispersion equation to a lumped parameter
model is proposed.
The dispersion equation (1) is taken as the basis for
development of the lumped parameter model. Let us define effluent input of
substance as control influence, and change in background concentration of
substance as disturbance one. Water quality demands are usually made in fixed
points of an area. It provides a way to write transfer functions along the
channels of disturbance and control:
(2)
(3)
= cross section area, m2; p = parameter of Laplace transform; index
u
applies to the channel of control, and index d applies to the channel
of disturbance.
The processes
described by the transfer functions (2) and (3) are the distributed parameter
objects. The passage to the lumped parameter model can be realized through
approximation of the transfer function (2) and (3) by transfer functions with
lumped delay of the form:
(4)
= rational transfer function; t = delay time, s.
The method that is
given bellow allows achieving the required range of accuracy for approximating
the main problem. This method is based on a successive improvement of
approximation accuracy in raising order of the function Wa(p) and
using algorithm of choice of values of parameters of this function, which
ensure the best accuracy of this approximation. The foundation of this method
is the representation of the transfer function (4) as the generalized Fourier
series using the orthonormalized system of functions with deviating argument.
Use of the classical orthogonal fimctions does not permit to solve the problem
of choice of value of parameters of the transfer function. Therefore the necessity
is arisen in development of a special system of functions with deviating
argument which is orthonormalized on interval (t,¥) and including pre-indeterminated parameters of
the approximating transfer function as follows:
(5)
a and t are parameters to
be determined; 
, elements yij of vector Gi =(yi2,yi3,
... ,yii) are determined by the following expression:
(6)
In expression (6) vector H0 and matrix H are
determined by the following relationships:
Representing the
approximating transfer function as a result of the generalised Fourier series
expansion according to the system of functions (5) after series of
transformations we can obtain
(7)
where
From relationship
(7) follows that the approximating transfer function describes a parallel junction
of N aperiodical chains of the first order with delay. Values of unknown
parameters a and t can
be determined on the basis of moment method as a solution of the system of two
non-linear equations:
For the channel of control:
For the channel of
disturbance:
For both channels 
.
In such manner the
input distributed parameter equation can be reduced to the system of N lumped
parameter equations with delay.
The mathematical
model was used for control of oxygen
regime of a biological pond for waste-water treatment
The investigated
water body is the terminal block of the combined treatment units of one of the
large chemical plants in the city of Ivano-Frankovsk. After treatment in the
biological pond wastewater flows into the Dnister River which is a main source
of water supply for some western and southern regions of Ukraine and Moldova.
This is imposes heavy demand on quality of wastewater.
The volume of
biological pond is 260,000 m3 and depth is 5 m. Oxygen supply is
carried out with five mechanical aerators. Use of mechanical aerators derives
from the wish to achieve more intensive mixing of water. This allows
intensifying processes of biochemical oxidation and reducing secondary
pollution. Input of wastewater is 20,000 m3/day. The average content
of organic matter is 40 g/m3 as BOD20- The demands of
oxygen regime in biological ponds are determined by two following
circumstances:
1)
aerobic conditions in biological ponds necessary to maintain an
efficient course of biochemical oxidation processes;
2)
dissolved oxygen concentration in purified effluent before discharge in
the Dnister Rivermust satisfy the current standards for water quality.
In the both cases the limiting factors are minimal concentrations of
dissolved oxygen. Thus, in the majority of instances continuous aeration can be
unnecessary, and therefore, cost-ineffective. But it is impossible to determine
the effective mode of aeration on the design stage under conditions of
sufficient instability in chemical composition of wastewater resulted from
characteristics of chemical production. This dictates the utility of
construction of a system for automatic control of running of mechanical
aerators.
In order to develop
a model of the biological pond as the object of control an
experimental-analytical approach was used. According to this approach the
initial equations are derived on the base of mass balance and numerical values
of the equation parameters are determined in the course of experimental study
of the object.
A biological pond
can be considered as biochemical reactor. Processes into it depend upon hydraulic
processes and the kinetics of the processes of biochemical transformations. In
the general case mass balance in chemical reactors is described by the
dispersion equation adequately. The kinetics of biochemical oxidation on the
final stages of purification is described by the first-order reaction
satisfactorily. It makes possible to use Equation (1) as the basis for the
model of the object of control. This model can be written as modification of
the system of Phelps & Stritter equations for turbulent streams as follows:
(8)
L = value of BOD20,
g/m3; S = value of dissolved oxygen deficiency, g/m3; kR
= atmospheric re-aeration rate, s-1.
Numerical valuess of the
coefficients appearing in these equations were determined experimentally on
the hydraulic model. (See Fig. 1).
The aim of the experiments was to determine values of dispersion coefficient
D underworking and detached aerators. Results of experiments
confirmed the presence of the hydraulic regime in the
biological pond corresponding to turbulent mixing. Numerical values of
dispersion coefficient were obtained as 0.13 m2/s
under working aerators and as 0.025 m2/s under detached aerators.
Figure 1. Experimental hygraulic model of the biological pond
Since processes of
aeration do not lend itself to experimental study on physical models a simulation
of biochemical oxidation in the biological pond on the basis of the set of
equations (8) was carried out. Results of this simulation allowed determining
areas where dissolved oxygen concentrations were minimal. These areas and the
output of the biological pond were determined as points for setting dissolved
oxygen sensing devices.
The equation for
dissolved oxygen was represented as the model with lumped parameter in the form
(7) on the basis of above mentioned method. This model was used for design of
automatic control system of running of mechanical aerators. Using of this
control system allows reducing to 30% the running time of aerators. The
treatment efficiency does not deteriorate under these conditions.