Proceedings of the 23th specialization scientific Conference
in Al-Mustansirya University-Baghdad-Iraq
Mathematical Model for Estimation the Effect of Rainwaters in Contaminated Soil and its Application in Baghdad Luma Naji Mohammed Tawfiq* & Mohammed A. Hasan** College of Education for Pure Science Ibn Al-Haitham, Baghdad University, Baghdad, Iraq. *Author to whom correspondence should be addressed; Email: dr.lumanaji@yahoo.com ** alamery85@yahoo.com
Abstract The aim of this paper is to design a model equation that can estimate the effect of rainwaters in the contaminated soil by heavy metal. The model equation can be considered as a good representation to estimate concentration of heavy metals in soil depending on the practical results. The model equation was solved analytically by using Adomian Decomposition Method. That the study which applied in Bab Al-Muadham in Baghdad city and compared with the laboratory results to illustrate the accuracy, efficiency and easy implemented of suggested model. Keywords: Mathematical model, Adomian Decomposition Method, Soil layers, Heavy metals.
1. Introduction Mathematical Models are simplified representations of some real world entity can be in equations or computer code are intended to mimic essential features while leaving out inessentials, that is, models describe our beliefs about how the world functions. Mathematical modeling aims to describe the different aspects of the real world, their interaction, and their dynamics through mathematics [1]. Generally, nutritional metals do occur naturally in fruits and vegetables as essential trace elements needed for good health, but they could be toxic when their concentrations exceed limits of safe exposure; sixteen chemical elements are known to be important to a plant's growth and survival [2]. The sixteen chemical elements are divided into two main groups: non mineral and mineral. The non mineral nutrients are hydrogen (H), oxygen (O), and carbon (C), these nutrients are found in the air and water. The 13 mineral nutrients, which come from the soil, are dissolved in water and absorbed through a plant's roots. Heavy metals are that elements having specific gravity that is at least five times the specific gravity of water which is expressed as 1 at 4°C and refers to metallic elements with an atomic weight greater than iron (55.8 g/mol) [3]. In this paper we design a model equation that can estimate the effect of rainwaters in the contaminated soil by heavy metal such: lead (Pb), nickel (Ni), cadmium (Cd). Several worker have already investigated the mobility of heavy metal in the soil amended with sewage sludge and concluded that only relatively small amount of metal were available for transport in the soil water immediately after sludge application [4]. Giordano and Mortvedt [5] show that under excessive leaching condition, movement of heavy metal in soil is somewhat greater from inorganic than from complexes sources found in sewage sludge. Other works study the effect of rainwaters in soil depending on laboratory testing for many samples of soil before and after the rain such [6-8]. This research work, therefore aimed at design a mathematical model equation that estimate the concentration of heavy metals in the soil, then we compare the results before and after the rain to
Proceedings of the 23th specialization scientific Conference
in Al-Mustansirya University-Baghdad-Iraq
estimate the effect of rain in soil. This aim can be achieved through the realization of the following objectives:   
Collection of data showing the concentration of heavy metals at different percentage of the soil with respect to distance and time. Design mathematical model equations for the concentration of heavy metals in the soil. Simulation of the model equation using computer software programmed, Math Lab 2014 professional. Compare the simulated result with the experimental data to determine the accuracy of suggested technique.
2. Mathematical Modelling The suggested model descriptions of the contamination of soils by heavy metals which are distributed throughout the soil system remain in the soil solution as iron, organic and inorganic complexes. Some of these heavy metals are mobile for uptake by plants. This mobility and availability depends on several factors including soil texture and PH [9]. Also, in the industrial regions, where some types of factories are active, several chemical and petrochemical processes would be also active. As a result, industrial water becomes contaminated with various substances which are harmful; these are sources of environmental contamination, which added in the description of the model equation. Then, the model equation, can be written as the relation between the change of concentration C (
of heavy metals proportionally with the change of
time t ( day-1 ) and the change of concentration of heavy metals proportionally with the change of space x ( cm ) multiplicand with the average pore-water velocity V (
). Then adding the fluid
velocity in the reactor multiplicand with Dispersion parameter D (
. So, the model equation
can be written as: (1) Which is a second order linear PDE, with initial - boundary conditions. Now, how to choose initial and boundary conditions. A variety of conditions may be specified depending on the type of soil. There are different types of soil depending on properties, structure, characteristic or layers. In this thesis we classify the soil depending on characteristic and properties of soil as follows: Sand (fine and coarse), Silt, Loam, Crags and Clay [10]. If the soil is a loam land then the conditions are: C(x, 0) = Cx,0 =
Where; : Initial concentration (
).
Proceedings of the 23th specialization scientific Conference
in Al-Mustansirya University-Baghdad-Iraq
The amount of each element retained by each soil ( concentration in solution (
was calculated from the initial
and the final concentration C in solution (
Equation (1),
which can be represented as a mathematical model for spread of contamination through soil which can be used to determine the rate of contamination. The solution of model equation gives the concentration of the heavy metals in soil for any space and time. This model can be considered as an important model to give concentrations of heavy metals without spending in traditional laboratory inspecting. Thus, the comparison between the result of model equation (1) before and after the rain gives the effect of the rain of contaminated soil. 3. Solving the Model Equation We will use the Adomian Decomposition Method (ADM) to solve the equation (1). First we consider the linear differential equation in an operator form by Lt C = D Lxx C ˗ V Lx C
(2)
Where L, is the derivative symbol which assumed to be invertible. Take the inverse operator L−1 to both sides of equation (2), and using the initial condition: C(x, 0) = Cx,0 = We get:
+ D Lt˗1 Lxx C ˗ V Lt˗1 Lx C
C=
(3)
Related to Adomian method, we define the solution C by an infinite series of components given by: C= Then the equation (3) can be written as: (
+ D Lt˗1 Lxx
)=
˗ V Lt˗1 Lx
(4)
Where cx,0, cx,1, cx,2 , … can be determined as far as we like. As given in our model, the zeros component = Lt˗1 (D Lxx = Lt˗1 (
=
, then:
˗ V Lx
) )
= 2t = Lt˗1 (D Lxx (2t
) ˗ V Lx (2t
))
Proceedings of the 23th specialization scientific Conference
= Lt˗1 (2t
in Al-Mustansirya University-Baghdad-Iraq
+ 2t
= = Lt˗1 (D Lxx (
˗V Lx (
= Lt˗1 (
)) )
=8 And so on for other components. Consequently, the solution in a series form is given by: C=
+
=
+
……
+
+ 2t
+
+8
+ ….
That is: C=
…}
{ 1+(
i.e., C = Thus
C=
exp
=
exp
(5)
Thus, equation (5) represents the closed form solution of the model equation. Now, we choose D = 0.5
and V = 5.14 × 10˗6 ms˗1
, depending on results
of [11, 12]. Therefore, C = c0 exp So, C = c0 exp
(6) (7)
Thus, equation (7) represents the concentration of heavy metals in soil for any depth x and time t.
4. Application of the Mathematical Model The study area, Bab Al- Muadham, is located in the center of Baghdad. It has 3 central districts and near Mohammed Al- Kasim highway: residential land, commercial land and industrial land. Six sites were selected for study within Bab Al-Muadham area as illustrated in Figure 1. The distances of sampling sites from Mohammed Al- Qasim highway were (300, 600,750, 900, 900, and 950m). At each site, Soil samples were collected at a range of depths (0 -5), (5-10), (10-15), (15-20), & (20-25) cm, the soil was generally taken from (0-25) cm of the topsoil because much of
Proceedings of the 23th specialization scientific Conference
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the nutrient uptake by plants is from this depth. Samples were preserved in cleaned polyethylene bags and finally transported directly to the laboratory.
Figure 1: Locations of samples Sites in Bab Al- Muadham city, Baghdad
Now, in the equation (7) if, substituting the initial concentrations (C0) of heavy metals given in Table (1) before and after the rain, then we get the concentrations of this heavy metals for time t(d) and depth x(m). The practical results illustrated in Figure (2 – 4), when the figures represent the concentrations of heavy metals such: Cadmium (Cd), Nickel (Ni) and Lead (Pb) respectively in soil of Bab Al- Muadham city.
Table 1: Concentrations of heavy metals with soil depths for different sites in Bab Al- Muadham city.
sites
Depth(cm)
1 2 3 4 5 6
0-5 5-10 10-15 15-20 20-25 25-30
Pb
Ni
Cd
Before Rain
After Rain
Before Rain
After Rain(1)
After Rain(2)
Before Rain
After Rain
14 2.5 2.7 2 0.5 3
5 2 1.5 1 0.4 0.9
3 2.5 2.3 2.4 1.8 2.8
0.4 0.38 0.33 0.25 0.15 0.24
0.4 0.35 0.2 0.19 0.05 0.20
0.0065 0.0063 0.0060 0.0058 0.0020 0.0055
0.049 0.025 0.027 0.023 0.015 0.030
Figure2: Concentrations of Cd before & after the rain in Bab Al- Muadham city.
Proceedings of the 23th specialization scientific Conference
in Al-Mustansirya University-Baghdad-Iraq
Figure3: Concentrations of Ni before & after the rain in Bab Al- Muadham city.
Figure4: Concentrations of Pb before & after the rain in Bab Al- Muadham city.
5. Conclusions The analysis of the result shows that there is a very good level of agreement between the experimental and simulated results obtained. This can also be confirmed by the numerical analysis of the result by using interpolation. In conclusion, the model can be considered to be a good representation of the estimating the concentrations of heavy metals in the soil. The practical results show the effect of the rains on the soil. The results of the analysis of lead in soil in Bab Al- Muadham city, show the concentrations of this element has exceeded the permissible limits in some regions is due to the impact of divorced and industrial plants such as brick factories and smelters as well as the impact of divorced vehicles due to combustion. Also, observed by measuring the concentrations of Ni and Cd it had exceeded the permissible limits, which indicates the existence of these elements in the soil of this city, and the reason for the increase in the concentrations of these elements is due to several reasons including the impact of industrial activities, but by less than before rain due to suspension of many of these actors to work due to war conditions experienced by the country as well as the impact of adsorption by clay
Proceedings of the 23th specialization scientific Conference
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minerals and the presence of organic matter in the soil that play an important role in increasing concentrations of these elements as well as the weathering and irrigation and drainage. References [1] Bokil, V. A., (2009), "Introduction to Mathematical Modeling", Spring. [2] Ajmalkhan, M. U., and Showalter, A. M., (1999), Effect of salinity on growth, ion contact, and osmotic relation, Stapf. Journal of Plant Nutration, Vol. 22, No. 1, pp: 191-204. [3] Kabata, A., and Pendias, H., (2001), Trace Elements in Soils and Plants, 3rd Edition, CRC press, Washington, D.C., 550 p. [4] Sidle R. C., and Kardos L. T., (1977), Aqueous release of heavy metals from two sewage sludges, Water Air Pollution, Vol. 8, pp. 453 459. [5] Giordano, P. M., and Morvedt, J. J., (1979), Nitrogen effects on mobility and plant uptake of heavy metals in sewage sludge applied to soil columns, J. Env. Qual., Vol. 5, pp. 165-168. [6] Janette Worm, Tim van Hattum, (2006), Rainwater harvesting for domestic use, Agromisa Foundation and CTA, Wageningen, First edition. [7] Sivanappan, R. K., (2006), Rain Water Harvesting, Conservation and Management Strategies for Urban and Rural Sectors, National Seminar on Rainwater Harvesting and Water Management, Nagpur, Nagpur. [8] Agarwal, V., Agarwal, J. H., (2006), Water Issues and Related Concerns, National Seminar on Rainwater Harvesting and Water Management, Nagpur, Nagpur. [9] Selim, H. M., Amacher, M. C., and Iskandar, L. K., (1990), Modeling the Transport of Heavy Metals in Soils, Monograph 90-2,U.S. army corps of engineering. [10] Gzar, H. A., and Gatea, I. M., (2015), "Extraction of heavy metals from contaminated soils using EDTA and HCl", Journal of Engineering, Vol. 21, No. 1, pp: 45-61. [11] Meserecordias, W., L., Jasper, N., I., Karoli, N. N., and Patrick, A. N., (2014), Environmental Contamination by Radionuclides and Heavy Metals Through the Application of Phosphate Rocks During Farming and Mathematical Modeling of Their Impacts to the Ecosystem, International Journal of Engineering Research and General Science, Vol. 2, Issue 4, pp: 852 – 863. [12] Bohun, C. S., (2010), Mathematical models for an undisturbed soil-column, Mathematics-inIndustry Case Studies Journal, Vol. 2, pp: 1-15. [13] Maitham, A. S., (26-28 April 2010), Evaluation of Soil pollution by heavy metals in Baghdad city using GIS, The 1st International Applied Geological Congress, Department of Geology, Islamic Azad University, Mashad Branch, Iran, pp: 852-863. [14] Hiti, M., (1985), The quality of groundwater within the city of Baghdad, MSc. thesis, College of Science, Baghdad University, Iraq.