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PhD Nemtsova À.À.*,
PhD
Ponomarenko
E.G.**
National
University of Pharmacy, Ukraine*
O.M. Beketov National University of
Urban Economy in Kharkiv, Ukraine**
Mathematical modelling of process of water quality
formation in the Kremenchug reservoir
In terms of active use of water resources for a variety of purposes,
including fisheries, drinking, household, agricultural etc., is important to
ensure the standard quality of these resources. An important role in the forecasting
of surface water quality in a changing environment water management issues is
the use of mathematical modelling methods in the solution of this problem.
Here proposes a model of water quality formation in reservoirs taking
into account chemical and biological factors. The model is developed on the
example of the Kremenchug reservoir. Complex physical, chemical and biological
processes in the aquatic environment influences the formation of water quality
reservoirs. The income of impurities from external sources, the characteristics
of flow and temperature of water influence this process too. Tool that allows
us to explore qualitative and quantitative characteristics of the processes of
water quality formation is a mathematical model representing the complex
kinetics of the transformation of substances involved in the cycle. The most
important substances from the point of view of the objectives of water
protection are substances for which approved water quality standards. For
example, nitrogen compounds included in the composition of bacteria, phytoplankton
and zooplankton involved in the nitrogen cycle.
Analysis of the literature showed that a decisive influence on the
content of nitrogen-containing substances in surface waters, provide the
following factors:
ü
the
nitrification process, i.e. a process of consecutive transformation of nitrogen
of dissolved organic matter to ammonium nitrogen, nitrite nitrogen and nitrate
nitrogen;
ü
the
transition nitrogen of suspended organic matter (detritus) in dissolved organic
matter;
ü
the
presence of feedback between nitrite nitrogen and dissolved organic matter;
ü
the
presence of feedback between the content of standardized forms of nitrogen and
consumption of intermediate forms;
ü
influence
of bacteria on the transformation of nitrogenous substances and on the balance
of organic substances;
ü
the
role of phytoplankton and zooplankton in nutrient cycle and balance of
dissolved oxygen in the aquatic environment;
As input data for the model used data from several
years of field observations conducted by the Institute of Hydrobiology Academy
of Sciences of Ukraine at the Kremenchug reservoir. In particular, the data on
the medium to seasonal concentrations such components of water quality:
ammonium nitrogen, nitrate nitrogen and nitrite nitrogen, bacteria Heterotrophs, Nitrosomonas, Nitrobacter, phytoplankton and zooplankton, suspended and
dissolved organic matter.
In addition, the
model uses data on temperature regime, the magnitude of water runoff,
hydrological characteristics of the reservoir and the sources of income-listed
component in the Kremenchug reservoir: by river runoff, from bottom sediments,
with surface runoff, with subsurface runoff of fertilizers, with industrial and
domestic waste waters, with precipitation.
Part of the
field data was used to verify the model; the remaining data was used to verify
its adequacy.
The basis of the
mathematical model is the model of the closed cycle of circulation of nitrogen-containing
substances in the water flow, previously presented by the authors in [1]. The
proposed model is a modification of the previously developed model for the conditions
of reservoirs.
The annual cycle
of nitrogen in a reservoir is a complex series of reactions of transformation
that has a great influence on the contents of its different forms in water.
Mineral nitrogen is assimilated by phytoplankton and bacteria and goes into protein
nitrogen. Albumen after dying of organisms is exposed to a mineralization, after
that nitrogen of protein substances is allocated in the form of ammonia. Ammonia
nitrogen is exposed to decomposition by nitrifying bacteria and turns at first
to nitrite nitrogen, and then in nitrate nitrogen. However, during the growing period, algae only at high
concentrations consume nitrate nitrogen, while ammonium nitrogen is consumed by
phytoplankton of 5-10 times more nitrate nitrogen.
The maximum
content of mineral forms of nitrogen is observed in the reservoir in the
spring, before the flood. This is due to the mineralization of organic matter
in the water column and in sediments and their accumulation during the cold
period of the year. In the period of spring flood, concentrations of mineral
forms of nitrogen begin to decline. Most intensively, this process occurs
later, when the activity of phytoplankton and bacteria increases. In addition,
the heterogeneity of the plankton development in the reservoir, the circulation
of water masses, discharge of sewage leads to uneven distribution of concentrations
of mineral nitrogen. With the end of the growing season, algae and the
beginning of death of organisms at all trophic levels there is an increased
concentration of ammonium nitrogen, nitrite nitrogen and nitrate nitrogen.
Dynamics of
organic nitrogen is closely related to the dynamics of aquatic organisms. The
concentration of organic nitrogen grows from early spring, reaching maximum
values in late summer – early autumn, and then observed for primary reduction.
The distribution
of dissolved oxygen in the water column in the spring is almost uniformly.
However, there may be a slight stratification, is more pronounced in the
shallows where the water warms up sooner and phytoplankton grows faster. During
the summer, two opposite processes significantly affect oxygen regime of the
reservoir: photosynthesis, in which water is enriched with oxygen, and
oxidation of organic substances, leading to decreased oxygen content in water.
Summer is especially strongly expressed stratification of the oxygen regime in
the reservoirs. The upper layers of water in which there is active
photosynthesis, is usually supersaturated with oxygen. In the lower layers,
especially for deep reservoirs, there may be a deficiency of oxygen.
In the autumn, the
"bloom" of algae leads to a predominance of processes of degradation,
which reduces the oxygen content in the water on certain sections and bays. In the
late autumn, the stratification is destroyed; the feeding of oxygen over the
entire thickness reaches almost 100%.
In the summer
the BOD reaches a maximum in the tail of the reservoir, the minimum value of
BOD is achieved in the spring. In the autumn, the values of the BOD are
reduced, but not all portions of the reservoir.
Seston is an
important functional component of aquatic ecosystems. The organic portion of
the seston, comprised of detritus and plankton, has a particular
effect on the status of aquatic ecosystems.
The part of
seston incoming in the Kremenchug reservoir is small compared to the seston
produced in the reservoir. The part of seston that enter to the Kremenchug
reservoir from without is small compared to the seston generated in the
reservoir. Values of seston are very different for the separate areas of the
reservoir. Especially sudden fluctuations of values of seston and its
components are observed at the bottom of the reservoir, where the most pronounced
bloom of freshwater algae. The proportion of detritus in the seston is of the
order of 60%, 72% and 67% in spring, summer and autumn respectively. Diatoms
and blue green algae, accounting for 65% of the total biomass of plankton in
the summer-autumn period, dominate a large part of the vegetation period in the
composition of the plankton. In the spring, bacteria dominate the reservoir.
According to the
Institute of Hydrobiology Academy of Sciences of Ukraine Kremenchug, water
storage refers to the mesotrophic type, so significant parts of the diet of
plankton are bacteria. The high content of bacteria Heterotrophs that transform organic matter into mineral forms determines their
interaction with dissolved organic matter, but also suspended organic matter.
The above
characteristics of the Kremenchug reservoir have been considered in the development
of a mathematical model. The model of water quality formation in the reservoir
was based on the following scheme of circulation of substances.
Nitrogen
dissolved organic matters and suspended organic substances by using bacteria Heterotrophs are transformed into ammonia nitrogen, next with the help of bacteria Nitrosomonas and Nitrobacter in nitrite nitrogen and nitrate nitrogen, respectively.
Dying in the life process bacteria enter in suspended organic matter. Ammonia nitrogen
and nitrite nitrogen consumed by phytoplankton, which is food for zooplankton.
The waste products of zooplankton replenish ammonia nitrogen, dying of phytoplankton
and zooplankton are transported into suspended organic matter, part of which
goes into dissolved organic matter and the rest using bacteria Heterotrophs go again in ammonia nitrogen. The cycle is completed. Bacteria,
phytoplankton and zooplankton consume oxygen for respiration. Dissolved oxygen
is consumed in the process of nitrification, which requires aerobic conditions.
The process of photosynthesis by phytoplankton increases the dissolved oxygen
concentration.
The model
accounts dynamics of plankton in the process of the growth of its population in
accordance with the principles of kinetics Michaelis - Menten- Mono according
to the formula
where m -
growth rate; mmax – maximum growth rate; N is the concentration of nutrients
in terms of nitrogen; KSN – constant of half-saturation, Y –
economic coefficient, taking into account the recycling of nutrients on the
growth of the biomass of the population (in this case bacteria).
The proposed
model of formation of water quality in the Kremenchug reservoir is based on the
law of conservation of matter in relation to the process of transformation of
various forms of nitrogen. It was developed for conditions of steady two-dimensional
flows, and taking into account the convective and diffusive transport of
substances in the aquatic environment and the influence of water temperature
and irradiance on these processes.
A linear
operator given bellow describes convective and diffusive transport of
substances in the aquatic environment
where uj
is averaged by depth flow velocity, D
is the longitudinal dispersion coefficient, x1
and x2 – coordinate is
measured along the length and width of flow respectively.
The effect of water temperature on the magnitude of
the main hydro chemical and hydro biological indices is estimated using the
temperature coefficients.
where T is water temperature in Celsius, q - experimental
constant.
The impact of solar radiation on the growth rate of
phytoplankton is estimated using the coefficient of irradiance Kf. The value of this
coefficient is equal to the relation of coefficient of solar radiation per
season to the average intensity of solar radiation during the completely
growing period.
Thus, the model
describes the following nitrogen cycle:
The cycle starts with of the transition of nitrogen,
which is part of suspended organic matter (SOM), into dissolved organic matter (DOM).
Part of the suspended matter is deposited in sediments, the other part is
dissolved. Both processes are evaluated according to the laws of reaction of
the first order. Dissolved nitrogen is a substrate for Heterotrophs bacteria, which, as a product of metabolism, are isolated in an aqueous
environment, ammonia nitrogen.
In the next stage of cycle (beginning of the
nitrification process) bacteria Nitrosomonas participate in the process. They use ammonia nitrogen
as a substrate, and excrete nitrite nitrogen as a product metabolism. Nitrobacter
bacteria play the main role on the second stage of nitrification. They consume
nitrite nitrogen and produce, as a product of the metabolism, nitrate nitrogen.
Plankton closes the cycle of transformation and cycling of nitrogen in aquatic
ecosystems. Both plankton (phytoplankton and zooplankton) play a dual role. On
the one hand, they are living organisms, and the laws of Michaelis - Menten –
Mono govern their growth. On the other hand, the phytoplankton is itself a
substrate for zooplankton. Phytoplankton uses for its development two types of
substrate – ammonium nitrogen and nitrate nitrogen. At the same time, it is
food for zooplankton and it replenishes the suspended organic matter after
death.
Zooplankton feeds on phytoplankton, in the process of
excretion excretes ammonia nitrogen, when death and defecation restock
suspended organic matter. The processes of decay of both types of plankton,
defecation and excretion of zooplankton obey the law of the first order. The
flow of the products of metabolism of plankton in the form of dissolved organic
matter is insignificant in comparison with other elements of the nitrogen
cycle, so they can be neglected.
Diagram of the nitrogen cycle given below.
DOM SOM NH4 NO2 NO3 Ht Ns Nb zooplankton phytoplankton
Checking the sensitivity
of the proposed model was presented in article [1].
The model uses
the following notation:
N1, N2, N3, N4,
N5 – concentrations (in terms of pure nitrogen) of ammonium nitrogen,
of nitrite nitrogen, of nitrate nitrogen, nitrogen of suspended organic matters
and nitrogen of dissolved organic matters, respectively.
X1, X2, X3, X4,
X5 – densities of the biomass of the bacteria Nitrosomonas, Nitrobacter, Heterotroph, phytoplankton and zooplankton, respectively.
mmax11
is maximum growth rate of Nitrosomonas when they consume ammonium nitrogen, mmax22
is maximum growth rate of Nitrobacter when they consume nitrite nitrogen, mmax34
is maximum growth rate of Heterotrophs when they consume nitrogen of suspended organic matters,
mmax35
is maximum growth rate of Heterotrophs when they consume nitrogen of dissolved organic matters,
mmax41 is maximum growth rate of phytoplankton
when he consumes ammonium nitrogen, mmax43 - maximum growth rate of phytoplankton when he consumes nitrate
nitrogen.
Y11 is economic coefficient Nitrosomonas when they consume ammonium nitrogen, Y22 is economic coefficient Nitrobacter
when they consume nitrogen
nitrite, Y35 is economic
coefficient Heterotroph when
they consume nitrogen of dissolved organic matters.
kS11 is constant of half-saturation for Nitrosomonas
when they consume ammonium
nitrogen, kS22 is
constant of half-saturation for Nitrobacter when
they consume nitrogen nitrite, kS35
is constant of half-saturation for Heterotroph when
they consume nitrogen of dissolved organic matters, kS41 is constant of half-saturation for phytoplankton
when he consumes ammonium nitrogen, kS43
is constant of half-saturation for phytoplankton when he consumes nitrate nitrogen,
kS53 is constant of
half-saturation for zooplankton when he consumes nitrogen of suspended organic
matters, kS54 is constant of half-saturation for zooplankton when
he consumes phytoplankton.
kd1,
kd2, kd3, kd4, kd5 are mortality rates of Nitrosomonas, Nitrobacter, Heterotroph, phytoplankton and è zooplankton, respectively.
k51 is a coefficient of excretion of zooplankton.
k45 is the coefficient of solubility of the suspended
solids.
Kf is the coefficient of irradiance.
f1, f2, f3, f4,
f5 are the external income of ammonium nitrogen, of nitrite nitrogen, of nitrate
nitrogen, of nitrogen of suspended organic matters and of nitrogen of dissolved
organic matters, respectively.
In the
mathematical model, equations for the substrates are written first, and then
the equations for the biomass of bacteria, phytoplankton and zooplankton.
The model
equations given below
The solution of
the equations of the model is obtained by numerical methods using MathCAD.
Literature
Nemtsova A.
A., Ponomarenko E.G. A
structured approach to the construction of models of transformation of
substances in the aquatic environment. Materials of XII
International research and practice conference, “Advanced science”, - 2016. 30
April – 07 May 2016. Volume 12. Sheffield, UK Science and Education Ltd., 2016.
pp. 80 – 94.