Mathematical model for estimation the concentration of heavy metals in soil

MJ Journal on Applied Mathematics 1 (1) (2016) 16-19 Website: www.math-journals.com/index.php/JAM doi: 10.14419/jam.v1i1.39 Research paper

Mathematical model for estimation the concentration of heavy metals in soil Luma. N. M. Tawfiq *, Farah. F. Ghazi College of Education for Pure Science Ibn Al-Haitham, Baghdad University *Corresponding author E-mail: dr.lumanaji@yahoo.com

Abstract The aim of this paper is to develop a model equation that can estimate the concentration of heavy metal in the soil. The model equation was developed can be considered as a good representation of the concentration of heavy metals in soils for different depth and times depending on the practical results. That the study which applied in Qanat Aljaeesh zone in Baghdad city in Iraq.

Keywords: Mathematical Model, Differential Equation, Soil Layers, Heavy Metals.

AMS Subject Classification: 34K05, 34K40, 37C65, 47N50, 32W99.

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].

In this paper we study and develop a model equation that can estimate the concentration of heavy metal in the soil. 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 [2].

Giordano and Mortvedt [3] 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.

This research work, therefore aimed at developing a mathematical model equation that estimate the concentration of heavy metals in the 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.  Development of mathematical model equations for the concentration of heavy metals in the soil.  Simulation of the model equation using computer software programmed, MathLab 2014 professional.

Compare the simulated result with the experimental data.

2. Soil layers Can three successive layers in soil discrimination, namely: 1) Surface layer Surface Soil: the layer that envelops the earth and depth does not exceed the kit centimeters, as they contain organic materials, and most live micro-organisms, worms and insects, and this layer is prone to erosion than others and vandalism.

2) Layer beneath the soil Subsoil: located directly under the surface layer and a small amount of humus and microorganisms compared to the surface layer.

Copyright © 2016 Luma. N. M. Tawfiq, Farah. F. Ghazi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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3) Mother rock layer Solid: Originally a class which consisted of soil that is less vulnerable to factors such as the soil is temperature, humidity and wind.

3. Heavy metals Means all heavy metals minerals that increase density 5 g / cm3, and at least it is called light metals. Some of these metals play an important role in the lives of the living and the effectiveness of different biological, The iron his well-known enzymes in the blood and the installation of importance, are all of the elements manganese, zinc, copper enzymatic catalysts, but these metals are toxic and dangerous to be in certain concentrations. Adding to the seriousness of these metals in the environment is not possible analyzed by bacteria and other natural processes as well as the authenticity of which enable them to spread over long distances for ton or sources sites, and perhaps the most dangerous thing which is due to susceptibility to each bio-accumulate in the tissues and organs of living organisms in the environment water or land. But some of the properties of radioactive heavy metal, that is, they serve as a radioactive isotopes Radioactive Isotopes, therefore, these metals will be charged double the risk to the environment in terms of being toxic and radioactive at the same time, as is the case in 65 of radioactive zinc, uranium 235 [4], [5].

Heavy metal contamination in soil may pose risks and hazards to human beings. Excessive concentrations of some heavy metals in biological systems, especially animals (human beings in particular) are highly dangerous to human health, and may even cause death [6]. For example, heavy metals such as: Cadmium, Nickel and Arsenic are carcinogenic. Table 1 gives a summary of some dangerous heavy metals that are commonly present in farm soils and their health impacts to human beings.

Table 1: Dangerous Heavy Metals and Their Health Impacts to Human Beings Heavy Metal Pb Cr As Zn Cd Cu Ni

Health Impact/s Mental lapse or even death Allergic dermatitis Skin damage, cancer, affects kidney and central nervous system Zinc shortages can cause birth defects Affects kidney, liver and GI tract Anaemia, liver and kidney damage, and stomach/intestinal irritation Various kinds of cancer

4. Mathematical model In this section we develop mathematical model of heavy metals material through soils, by using the classical convectiondispersion equation which defined as [7]: đ?œ•đ??ś đ?œ•đ?‘Ą

= đ??ˇđ??ż

đ?œ•2 đ??ś đ?œ•đ?‘Ľ 2

− đ?‘‰đ?‘Ľ

đ?œ•đ??ś đ?œ•đ?‘Ľ

−

đ?œŒđ?‘? đ?œ•đ?‘†

(1)

đ?œƒ đ?œ•đ?‘Ą

Where; đ?‘† = đ?‘˜đ?‘‘ . đ??ś

(2)

C: solute concentration (mg/L) đ??žđ?‘‘ is the distribution coefficient (đ?‘?đ?‘š2 /đ??žđ?‘”) θ: soil water content Vx: Darcy's flux (cm/hr) DL: hydrodynamic dispersion coefficient (đ?‘?đ?‘š2 /â„Žđ?‘&#x;) x: soil depth (cm) t: t ime (day ( p: Solid density Equation (1) can then be written as: đ?œ•đ??ś đ?œ•đ?‘Ą

= đ??ˇđ??ż

đ?œ•2 đ??ś đ?œ•đ?‘Ľ 2

− đ?‘‰đ?‘Ľ

đ?œ•đ??ś đ?œ•đ?‘Ľ

−

đ?œŒđ?‘? đ?œ• đ?œƒ đ?œ•đ?‘Ą

(đ??žđ?‘‘ đ??ś)

(3)

Since đ?‘˜đ?‘‘ is a constant, so rewrite equation (3) as following: đ?œ•đ??ś đ?œ•đ?‘Ą

= đ??ˇđ??ż

đ?œ•2 đ??ś đ?œ•đ?‘Ľ 2

− đ?‘‰đ?‘Ľ

And rearranging we get:

đ?œ•đ??ś đ?œ•đ?‘Ľ

−

đ??žđ?‘‘ đ?œŒđ?‘? đ?œ•đ??ś đ?œƒ

đ?œ•đ?‘Ą

(4)

18

MJ Journal on Applied Mathematics đ?œ•đ??ś đ?œ•đ?‘Ą

+

đ??žđ?‘‘ đ?œŒđ?‘? đ?œ•đ??ś đ?œƒ

đ?œ•đ?‘Ą

= đ??ˇđ??ż

đ?œ•2 đ??ś đ?œ•đ?‘Ľ 2

− đ?‘‰đ?‘Ľ

đ?œ•đ??ś

(5)

đ?œ•đ?‘Ľ

Then we get: đ?œ• đ?œ•đ?‘Ą

đ??ś(1 +

đ??žđ?‘‘ đ?œŒđ?‘? đ?œƒ

) = đ??ˇđ??ż

đ?œ•2 đ?œ•đ?‘Ľ 2

đ??ś − đ?‘‰đ?‘Ľ

đ?œ• đ?œ•đ?‘Ľ

đ??ś

(6)

That is: (1 +

đ??žđ?‘‘ đ?œŒđ?‘? đ?œ•đ??ś đ?œƒ

)

đ?œ•đ?‘Ą

= đ??ˇđ??ż

đ?œ•2 đ?œ•đ?‘Ľ 2

đ??ś − đ?‘‰x

∂ ∂x

C

(7)

Let Ď = (1 +

Kd Ď b θ

)

(8)

So, equation (7) can be written as: Ď

∂C ∂t

= DL

∂2 C ∂x2

− Vx

∂C ∂x

,0 < x < â&#x2C6;&#x17E; t > 0

(9)

Which is a second order linear PDE, with initial - boundary conditions: C(0, t) = C° and

â&#x2C6;&#x201A;C â&#x2C6;&#x201A;x

(â&#x2C6;&#x17E;, t) = 0.

C(x, 0) = Cx Where; C° : Initial concentration.

Cx: concentration for depth x (mg/L).

Vx : The average pore- water velocity, (cm/hr) x: soil depth (distance) (cm). t : time (day-1 (.

K Ď Ď : Retardation factor, with Ď = (1 + d b). θ The amount of each element retained by each soil s (mg / kg) was calculated from the initial concentration in solution (mg/ L) and the final concentration c in solution (mg/L).

5. The Impotency of the mathematical model Mathematical model was very important to describe different problems in life, and one of this problems is environmental contamination specially, contaminate of soils by heavy metals.

Equation (3.9), which can be represent a mathematical model to describe the concentration of heavy metals in the soil.

6. Results and discussion We applied the above study in Qanat Aljaeesh zone in Baghdad city in Iraq (see Figure 1), the experimental results of this project are as illustrated graphically in Figure 2. The experimental results show the presence of heavy metals in the soil as time increased from 0 to 8 months for different concentrations in the soil. For instance, the concentrations of heavy metals for ordinary soil (100% soil) at the initial time were found to be 0.173 mg/m 3. As time increased to 5 hours, the concentration of heavy metals had decreased to 0.131 mg/m3.

The comparisons between the experimental and simulated results are shown in the graph. The observations from the results showed there are good agreements between the experimental and simulated concentrations of heavy metals in the soil. For instance, when the experimental concentration of heavy metals was 0.13 mg/m 3 at the end of 5 hours, the simulated concentration was 0.1199 mg/m3. As time increased, the concentration of heavy metals had decreased because rain shedding.

The agreement between the experimental and simulated concentrations of heavy metals in the soil can also be confirmed by calculating the value of correlation coefficient on the two results (experimental and simulated). From the calculations, it was obtained that the correlation coefficient was 0.9983.

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Fig. 1: A Real Photo Showing the Sites of Study Areas in Qanat Aljaeesh Zone - Baghdad City.

Fig. 1: The Concentrations of Heavy Metals in the Soil for Different Times in Qanat Aljaeesh Zone.

7. Conclusion The model equation developed for the concentrations of heavy metals in the soil.

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 method also, by the correlation coefficient found to be 0.9983, for 100% soil.

In conclusion, the model developed can be considered to be a good representation of the estimating the concentrations of heavy metals in the soil.

References [1] Bokil, V. A., (2009), "Introduction to Mathematical Modeling", Spring.

[2] 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.

[3] 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. http://dx.doi.org/10.2134/jeq1976.00472425000500020011x.

[4] Wuana, R. A., and Okieimen, F. E., (2011), Heavy metals in contaminated soils: A review of sources, chemistry, risks and best available strategies for remediation, ISRN Ecology, vol. 2011, Article ID 402647, 20 pages.

[5] Kukkola, E., Raution, P., and Huttunen, S., 2000, Stress indications in copper and nickel exposed Scots pine seedlings, Environ.

Exp. Bot., Vol. 43, pp. 197-210. http://dx.doi.org/10.1016/S0098-8472(99)00057-X. [6] Ayeni, O. O., Ndakidemi, P. A., Snyman, and R. G., Odendaal, J. P., (2010), Chemical, biological and physiological indicators of metal pollution in wetlands. Scientific Research and Essays, Vol. 5, No.15, pp.1938-1949 [7] 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.