Mathematical Model Describes Treatment of Waste Water Using Modified Activated Carbon

The proposed mathematical model covered in this paper includes the most important parameters associated with the rates of adsorption and desorption. Also, partial pressure is included since it is an important factor that affects rates of adsorption and desorption. The study focuses on the effects of the constant rates on adsorption of pollutant concentrations for benzene, nickel, cadmium, and copper using modified active carbon. When the rate constant of adsorption decreases, the pollutant concentration will also decrease, yielding high acceptable evidence of the logic of the proposed mathematical model. Also, the proposed model is compared with experimental data and other models to give good outcomes with high accuracy.

Amount of adsorbate adsorbed at equilibrium (mg/g).q max Maximum monolayer adsorption cap modified activate carbonity of the adsorbent (mg/g).

C e
Equilibrium concentration of adsorbate (mg/l).

K Ln
Langmuir's adsorption constant related to free energy adsorption (l/mg).

C TM
Temkin's constant related to the heat of sorption (J/mol).

R
Gas constant (8.314J/mol K).T Absolute temperature (K).k 1 First-order rate constant (min −1 ).q t Amount of adsorbate adsorbed at any time (mg/g).2005).Modified activated carbontivated carbon has been represented as one of the most effective modified activated carbontivated adsorption materials to be used to absorb carbon dioxide (Bog et al. 2006).Many researchers have used modified activated carbontivated carbon for waste water treatment (Afsaneh et al. 2008;Mohamed et al. 2008;Muhammad et al. 2008).A hybrid technique was used to produce a T-shirt model for waste water treatment.

Introduction
Other exploratory works have explored waste water treatment options through a combination of experimental and theoretical work.The first adsorption mathematical model which was proposed in 1906 by Freundlich has been used in hetrogenous surf modified activated carbon adsorbent systems where the binding sites are not equivalent.A form of the Freundlich's model can be represented as follows: The constants k fr and n can be evaluated from the intercept and the slope of the linear plot of experimental data of ln q e versus ln .e C A second important mathematical model for adsorption is Langmuir's isotherm model (Okieimen and Ogbeide 2009), which depends on an isothermal state when all the sites are homogenous compare to Freundlish's model, and all these sites are filled by molecules to be adsorbed.The linear form of the Langmuir isotherm can be represented by the following equation: (3) The values of the constants K L and q max can be evaluated from the intercept and the slope of the linear plot of experimental data of C e /q e versus C e.
Temkin and Pyzhev (Lalhruaitluanga et al. 2010)  ln ln The constants C TMI and C TM can be determined from the intercept and the slope of the linear plot of the experimental data of q e versus ln C e .The values of the constants C TMI and C TM are listed in Table 1.
The Lagergren mathematical model is represented as proportional to the first power of sorption cap modified activated carbonity of the adsorbent and can be expressed as follows (Khaled et al. 2009).(5) Integrating Eq. ( 5) for the initial and end conditions t = 0 to t = t and q t = 0 to q t = q t , and, after some rearrangement, a linear plot is obtained: . (6) values of k 1 and q e were obtained from the slope and intercept, respectively.Table 2 lists Lagergren's mathematical model constants.
. (6) Plots of log (q e − q t ) versus t for the Lagergren mathematical model where the values of k 1 and q e were obtained from the slope and intercept, respectively.Table 2 lists Lagergren's mathematical model constants.

Methodology
Deriving a mathematical model requires the provision of assumptions and a comparison with other models to create new ideas for a proposed model (Tables 3 and 4).Adsorption and desorption states represent active mechanisms for the system (Fig. 1) and can be derived as follows:

Adsorption State
This state depends on the properties of surface of adsorbent, partial pressure of fluid and rates constant of adsorption as seen in equations below: , , .

Desorption State
This state depends on the same properties of adsorption but the molecules of adsorbent left to the bulk flow in opposite direction of adsorption's flowrate as seen in equations below: , ., , , The mechanism of the rate of adsorption can be controlled in all states of the system.Thus, this is the most important assumption.0. From equation ( 16) Substitute ( 17) in ( 11) Representing the system as a continuous stirred-tank reactor (CSTR) process: The volumetric flow rate compared to the volume of the system is very limited, so 0 ( 2 3 ) Equation ( 22) will be ( 2 4 ) The rate of remodified activate carbonation is represented as (25) Substitute equation ( 21) in ( 25) Solve equation ( 26) *

Results and Discussion
The discussion that follows focuses on the experimental results associated with adsorbent benzene.The proposed mathematical model yielded good behavior for the experimental data from adsorbent benzene against the first order Lagergren model (Fig. 2).
Figure 3 shows results for adsorbent nickel using the proposed model and Lagergren model.
Also, the proposed mathematical model was highly accurate, with results close to experimental data as compared to other models for copper and cadmium (Figs. 4 and 5,respectively).
Rates of adsorption and desorption have a big effect on pollutant concentrations.The rate of adsorption decreased the change of pollutant concentrations and also decreased the benzene, nickel, cadmium and copper respectively).

Conclusion
This proposed mathematical model has enough ability to evaluate dynamic adsorption and desorption for modified activate carbon to give clear view about mechanism of the system and very acceptable results due to inclusion of all the rates types for adsorption and desorption compare to the other models.

Figure 2 .
Figure 2. Comparison between proposed mathematical model and lagergren model for benzene.

Figure 3 .
Figure 3.Comparison between proposed mathematical model and lagergren model for nickel. 0

Figure 4 .
Figure 4. Comparison between proposed mathematical model and lagergren model for copper.

Figure 5 .
Figure 5.Comparison between proposed mathematical model and lagergren model for cadmium.

Figure 6 .
Figure 6.Effect of different rates of adsorption and desorption on pollutant concentration of benzene.

Figure 7 .Figure 8 .
Figure 7. Effect of different rates of adsorption and desorption on pollutant concentration of nickel.

Figure 9 .
Figure 9.Effect of rates of adsorption and desorption on pollutant concentration of copper.

Table 1 .
Langmuir, Freundlich and Temkin models' constants and correlation coefficients for sorption of methylene blue (MB) into modified activated carbon.

Table 2 .
Correlation coefficients for adsorption of benzene on modified activated carbontivated carbon.

Table 3 .
The list of proposed mathematical model assumptions.The dynamics study is represented by the rates of adsorption and desorption at the surf modified activated carbon of modified activated carbon.8. Heat transfer is considered in the new mathematical model.

Table 4 .
Differences between the proposed mathematical model and the other models.