Modelling Adsorption and Transport of Chrome VI onto Iron Oxide-Coated Sand filter

Natural decay of pollutants is rarely consider in mass transport equation. This paper develops a mathematical model of a filter based on iron oxide coated sand for the removal of chromium in groundwater and surface water. The natural decay of chromium was analyzed using a zero order decay reaction. The advection dispersion equation for this purpose is solved analytically using the Laplace transform method and numerically by the Fourth order Runge kutta method to determine the spatiotemporal distribution of the pollutant through the filter. The control parameters are the adsorption coefficient, the initial concentration and the chromium degradation coefficient. It has been proven that the chromium degradation coefficient strongly affect the concentration values of pollutants inside the filter. The results of this work should be used to manufacture low-cost filters based on iron oxide coated sand. Keywords— Filter, Sand, Iron oxide, Chromium, Adsorption, advection-dispersion, Laplace.


INTRODUCTION
Water is an essential component in the universe and plays an important role in the proper functioning of ecosystems [1].Despite this, access to drinking water is an increasingly recurrent challenge worldwide and it is estimated that nearly eight hundred and eighty (880) million people lack safe drinking water at home [2,3].Water pollution results mainly from poorly planned urban and rural development, industrial discharges, agricultural discharges, geological and environmental activities, domestic activities etc., it is a serious problem because it affects our lives [4].In addition, the World Health Organization (WHO) estimates that 80% of the diseases that affect the population are directly carried by water: 400 million people are permanently affected by gastroenteritis, 160 million from malaria and 30 million onchocerciasis [5].The industry uses heavy metals in various fields because they have a high electrical conductivity; however these heavy metals have some toxicity to humans, leading in particular toxic effects on the nervous system, carcinogenic effects on the blood and the bone marrow and kidney disorders.Chromium is one of the most widely used metals in industries, since it has enough advantages for tanneries, textile, wood processing, agribus iness... Chromium VI is the most problematic form of chromium since in this form chromium is very toxic and very soluble in water.This solubility gives it great mobility in ecosystems.Today, following a lack of compliance with laws or accidents, a large number of industrial sites are polluted by chromium.Therefore, the development of low-cost, easy-to-maintain technologies to reduce this pollutant in drinking water sources to an acceptable level for consumption is an interesting research topic.The methods implemented today to treat discharges polluted by chromium (VI) are generally aimed at recovering this metal, in order to reuse it and thus reduce its ecotoxicological impact.In parallel, physicochemical methods attempt to eliminate Cr (VI) and electrochemical methods attempt to reduce Cr (VI) to Cr (III).Iron oxide coated sand has proven its effectiveness for the elimination of various heavy metals (As, Cd, Cr, Pb, ...) at an elimination rates close to 100%.In this study, we develop a mathematical model to understand and predict the adsorption of chromium VI in a fixed column bed made witchiron oxide coated sand.As Chromium undergoing a natural decay, the discussion will be focus effect of degradation coefficient λ on the spatio-temporal variation of the pollutants inside the filter.

II. MATERIAL AND METHODS 2.1 Preparation of iron oxide coated sand
The iron oxide coated sand was prepared using a procedure similar to that of Bailey et al. (1992) [6].The author used washed and dried river sand with a weight of 200 g and a geometric size of 0.49 mm [7].It is obtained by mixing for 2 min, 80 ml of a solution of ferric nitrate 2M (Fe (NO3) 39H2O) to washed river sand and dried.He then placed the mixture in a drying oven at 110 ° C for 14 hours.He finally places the mixture in distilled water until the runoff is cleaned at 105 ° C and stored in closed bottles [7].

Mathematical description
The linear dispersion advection-equation is used to describe the transport of pollutants in this fixed column bed (Fig. 1).The aquation was derived to predict aqueous concentrations of contaminants over time and space in the porous medium.It is written as follows: [8] Where ρ is the density, θ the water content, D the hydrodynamic dispersion coefficient, V: average Darcy velocity and q the equilibrium adsorbed concentration which is described as follows [8]:  =    (2) Where k is the adsorption capacity and n the adsorption intensity.
In the absence of the iron oxide coated sand which is the adsorbent,Cr (VI) ion disappeared naturally from the solution.Given the fact that chromium is a naturally decreasing pollutant, this disappearance rate was assumed to be at zero order as follows: −   = ⁄ ;  =  0   = 0 (3) Where λ is the zero order rate constant or chromium degradation coefficient, c₀is the initial Cr(VI) concentration and c the Cr(VI) concentration at the time t.By adding a decay term λcto the equation above according to Runkel and by considering a linear adsorption isotherm (n = 1), after a few transformations, we obtain: The appropriate boundary conditions for the model are given by:  ( ,0) = 0 , t ≥ 0;  (0, t) = c0,  ≥ 0; (5)  (∞, 0) = 0 ,  ≥ 0

2.3Analytical solution of the mathematical model
The analytical solution is obtained using Laplace transform.It is use to transform one or more partial derivatives of the differential equation into algebraic expressions.We call Laplace transform of the function G denoted L (G), the function of complex variable P defined by: Where L denotes the Laplace transform, G (t) is a function of time,  ̅ ()is a function of the transformed space, it corresponds to the Laplace value.The operational property of the transformation that eliminates a partial derivative of the equation is as follows: Where G (0) is the value of G at time t = 0 To reduce equation ( 4) to a more familiar form, we take: By substituting equation (8) in equation ( 4), we obtain: The initial and boundary conditions (6) become: The Laplace transform of equation ( 9) is: Therefore, it's reduced to an ordinary differential equation The solution of the problem is: Substituting (13) in equation ( 8), the solution becomes: )]} (15)

Numerical solution of the mathematical model
The numerical solution allow to take into account more complex heterogeneous systems, and they can be thus more realistic and flexible to simulate the real conditions.It is not very obvious to obtain an analytical solution of linear PDE such as equation (4).However, there is an approximate solution for linear PDEs obtained from different numerical methods.Therefore, to solve the EDP of pollutant transp ort in porous media, we use numerical methods.The partial discretization method, where the finite difference approximation is used to obtain the discrete form of the spatial derivative.Then, the finite space difference diagram in space is used to have discrete shape spatial derivation equations of equation ( 4).Finally, the fourth order Runge Kutta system (RK4) is used to solve equation ( 4).This numerical technique is used to solve the ordinary differential equation expressed as follows: The RK4 algorithm is written as follows: ₁=Δt g(t, C(t)) Δt, C(t) + Where, Δt is the time step, C (t) is the concentration of the pollutant at time t, C (t + Δt) is the concentration at time (t + Δt).The partial discretization method of the finite difference approximation of first-order spatial derivatives is: The discrete form of derivative of the second order is: The indices (i) and (j) represent the discretization nodes along (x) and (t) respectively.Δx is the step of space.Thus, equation ( 4 The discrete version of the initial condition associated with equations ( 4) and ( 5) is expressed as follows:

Spatial and temporal variation of pollutants
Figure 1 represents the spatial evolution of the concentration of the pollutant (chromium) as a function of the depth x for the different times t = 1 day, t₁ = 5 days and t₂ = 30 days.c/c₀ is the ratio of the simulated value of the pollutant concentration to a depth x.The analysis of this figure shows that at a time t the concentration of the pollutant decreases exponentially inside the filter.In general, this concentration increases with the time.Considering 40% of the initial chromium concentration as a guideline, we found that the depths of the column x should be x1 = 10 cm for t1 = 1day, x2= 20 cm for t2= 5 days and x3 = 25, 5 cm for t3 = 30 days.Figure 2 shows the time evolution of the concentration of the pollutant at the filter outlet for different depth values of the filter: x1= 20cm, x2= 25 cm and x3= 30 cm.It is found that the concentration of the pollutant increases with time and decreases as the depth of the filter increases.Figure 7 shows that the concentration of the pollutant increases at the exit of the filter.The variation of theConcentration of Pollutants into Filter of Filter is due to the concentration at the filter entrance and to the characteristics of the material (ironoxide coated sand) used.Finaly, the results obtained shows that these two methods (analytical and numerical) are in good agreement.

I.4. Spatio-temporal evolution of the concentration of the pollutant
Fig. 8: Spatio-temporal evolution of the pollutant concentration in the filter for x = 30 cm and t = 20 days Figure 8 shows the spatio-temporal evolution of the concentration of the pollutant in the filter.The x-axis is associated to the time taken by the pollutant in the filter, and the y-axis represents the depth of the column.This figure is subdivided into three phases.The initial phase which starts from 0 to 30% of the initial concentration characterized by the blue color, the unsaturated phase from 30 to 70%, and the saturated phase which start from 70 to 100% characterized by the red color.Figure 8 showsthe concentration of pollutant increasewith time and decrease with the columndepth.This results could be used to design equipment for chromium removal in aqueous solution .
IV. CONCLUSION The degradation coefficient isan important factor which control the design of membrane filter to remove chromium VI in water.This coefficient must be known to optimize chromium VI adsorption in aniron oxide coated sand based filter on a technical scale.For adsorption of chromium VI on iron oxide coated sand, analytical and numerical data obtained from simulating a fixed bed adsorber showed that iron oxide coated sand was very effective in removing chromium VIfrom an aqueous solution.It is shows that the degradation coefficient of chromium λ has a significant effect on the spatio-temporal evolution of the concentration of the pollutant.This coefficient should be taken into account in models in order to avoid to overestimate the concentration of the pollutant in reactive media.A clear improvement is observed for a material having a high adsorption capacity and a high first order degradation coefficient.Affordable water treatment filters could be constructed using iron oxide coated sand which is an effective, inexpensive material for the removal of water pollutants or for the treatment of effluents from industries.

Fig. 1 :
Fig. 1: Evolution of chromium concentration in space for x = 30 cm

Fig. 4 :
Fig. 4: the spatial evolution of the chromium concentration for the values of λ = 0.0 and λ = 0.1 at t = 10 min and x = 30 cm.

Figure 3
Figure3represents the temporal evolution of the pollutant concentration in the filter with different values of the decay rate λ = 0.0 and λ = 0.1.We find that in the absence of this degradation coefficient i.e. when λ = 0, the concentration of the pollutant evolves rapidly inside the filter, in the presence of the latter, the concentration evolves slowly.

Figure 4
Figure4represents the spatial evolution of the concentration in the filter with different values of the decay rate λ = 0.0, 0.1 and λ = 2.The analysis of this figure reveals an exponential decay of the concentration inside the filter.The larger the lambda, the higher the concentration of the pollutant decreases rapidly inside the filter until reaching the asymptote c = 0. Therefore, the natural decay of pollutants plays an important role in wastewater treatment since it allows to avoid material waste.It is necessary for the design of industrial sorption columns