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The Journal of Engineering Research (TJER), Vol. 14, No. 1 (2017) 94-104 

Mathematical Model Describes Treatment of Waste Water 
Using Modified Activated Carbon 

 
A.S. Ibrahim*, A.S. AL Buloshi, S.S. AL Zaabi and L.A. AL Yafai 

 
Chemical Engineering Department, Dhofar University, Salalah, Sultanate of Oman 

 
Received 7 March 2016; Accepted 2 November 2016 

 
Abstract: 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.  
 
Keywords: Mathematical model, Waste water, Adsorption, Desorption, Activate carbon. 
 
 

אא אאאא 
KאK*אKKאKK،אKK، 

 
 Wא אאא    א   א א אא  א

אאKאאאאKאא
א א ،א ،א   א א    א א

Kאאאאא،אא
     א א    Kא א   א

Kא 

אאW،אא،א،אאאא،א. 
 
 
 
 
 
 
* Corresponding author’s e-mail: ahmadsaadi47@yahoo.com 
 
 
 
 
 
 
 
 



A.S. Ibrahim, A.S. AL Buloshi, S.S. AL Zaabi and L.A. AL Yafai 
 

95 

Nomenclature 
 
Symbol  Definition (unit) 

 
A Waste water. 

,  Rate constant for adsorption of forward 
direction. 

CA.S Concentration of pollution of adsorption for 
surf modified activated carbon. 

CA Concentration of adsorption. 
CAi Initial concentration of adsorption. 
CV V modified activated carbon concentration. 

,  Rate for constant adsorption of a reverse 
direction. 

,  Rate for constant desorption of a reverse 
direction. 

,  Rate for constant desorption of a forward 
direction. 

PA Partial pressure. 
r  Rate	of adsorption. 
rd Rate of desorption. 
rA Rate of adsorption. 
S Modified activated carbonative carbon. 
Vi Volumetric flow rate. 
V Volume. 
t  Time (s). 
kfr Adsorption cap modified activated 

carbonity of the sorbent mg/g (l/mg)1/n. 
n Freundlich’s constants. 
qe Amount of adsorbate adsorbed at 

equilibrium (mg/g). 
qmax Maximum monolayer adsorption cap 

modified activate carbonity of the adsorbent 
(mg/g). 

Ce Equilibrium concentration of adsorbate 
(mg/l). 

KLn Langmuir’s adsorption constant related to 
free energy adsorption (l/mg). 

CTM Temkin’s constant related to the heat of 
sorption (J/mol). 

CTMI Temkin’s isotherm constant (l/g). 
R Gas constant (8.314 J/mol K). 
T Absolute temperature (K). 
k1 First-order rate constant (min−1). 
qt Amount of adsorbate adsorbed at any time 

(mg/g). 
 
 
 



Mathematical Model Describes Treatment of Waste Water Using Modified Activated Carbon 
 

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1. Introduction 
 
Waste water is one of the biggest pollution 
problems in the world, and suggestions for its 
treatment have been drawn from experimental 
and theoretical works. In experimental work, a 
simplified method for ion exchange has been 
used to evaluate kinetic data. Many techniques 
have been explored using modified activated 
carbontivated carbon to remove carbon 
dioxide. One of the most important techniques 
that has been used is semi-batch remodified 
activated carbonator fixed inside modified 
activated carbontivated carbon to adsorb 
methane and carbon dioxide (Prasetyo and 
Dod 1998). Another technique used biomass of 
sargassum fluitant to make bio sorption for 
heavy metals and tannin gel has been used to 
adsorb chromium (Nakano et al. 2001).  Some 
special materials have been used to remove 
cadmium and mercury ions from aqueous 
solutions using sorption on treated Pinus 
pinaster bark (Vazquez et al. 2002). Numerous 
experimental studies have used rice husk as an 
adsorbent for waste water treatment (Ajmal et 
al. 2003), and chitosan-cellulose hydrogel beads 
have been used to adsorb copper (Li and Bai 
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. 
     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: 
 

/                            (1) 
 

                                 (2) 

 

     The constants kfr and n can be evaluated 
from the intercept and the slope of the linear 
plot of experimental data of ln qe versus ln .eC  

  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 KL and qmax can 
be evaluated from the intercept and the slope 
of the linear plot of experimental data of Ce/qe 
versus Ce.  
     Temkin and Pyzhev (Lalhruaitluanga et al. 
2010) studied extensively the heat of 
adsorption and the adsorbate-adsorbent 
intermodified activated carbonation on 
adsorption isotherms. Temkin and Pyzhev’s 
mathematical model can be represented as 
follows: 
 

ln ln                   (4) 
 
     The constants CTMI and CTM can be 
determined from the intercept and the slope of 
the linear plot of the experimental data of qe 
versus ln Ce. The values of the constants CTMI 
and CTM 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).  
 
Table 1.  Langmuir, Freundlich and Temkin 
models’ constants and correlation coefficients 
for sorption of methylene blue (MB) into 
modified activated carbon. 
 

Isotherm Parameters Values 
Langmuir Qo (mg/g) 

KLn (l/mg) 
8.75 
0.23 

Fruendlich KFr 
n 

4.21 
4.13 

Temkin 
 

CTMI (l/g) 
CTM  

11.3 
1.5 

 



A.S. Ibrahim, A.S. AL Buloshi, S.S. AL Zaabi and L.A. AL Yafai 
 

97 

          (5) 

 
     Integrating Eq. (5) for the initial and end 
conditions t = 0 to t = t and qt = 0 to qt = qt, and, 
after some rearrangement, a linear plot is 
obtained: 

.
                  (6) 

 
 values of k1 and qe were obtained from the 
slope and intercept, respectively. Table 2 lists 
Lagergren’s mathematical model constants.  
 

.
                  (6) 

 
     Plots of log (qe − qt) versus t for the 
Lagergren mathematical model where the 
values of k1 and qe were obtained from the 
slope and intercept, respectively. Table 2 lists 
Lagergren’s mathematical model constants.  
 
2. 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: 
 
2.1. 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: 
 

	 ⇌
,

, 	. 	                                      (7) 
 

	 , 	 	 	 	 , 	 .         (8) 
 

	 , 	 	 	 	
.

,

,

		                       (9) 

    substitute (9) in (8) 
 

∗ 	 ,
,

         (10) 

     substitute (10) in (9) 
 

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

Initial 
concentration 

10 g/l 

qe exp. 
(mg/g) 
 

Pseudo-first-order kinetic 
model 

qe cal. 
(mg/g) 

k1 
(1/min) 

R2 

Benzene 4.14 2.3 0.003 0.915 
Nickel 6.31 3.8 0.003 0.898 
Copper 6.53 4.21 0.0037 0.927 

Cadmium 7.31 5.47 0.0028 0.934 
 

Table 3.  The list of proposed mathematical model assumptions. 
 

1. The fixed bed comprises liquid and solid phases. 
2. No chemical remodified activate carbonation occurs inside a fixed bed. 
3. Negligible radial temperature and concentration gradients exist in the fixed bed. 
4. Adsorption occurs to adsorbent solid particles inside holes of modified activated 

carbon. 
5. Desorption occurs to transfer solid particles from inside holes to the surf modified 

activated carbon of modified activated carbon. 
6. Mass transfer occurs from surf modified activated carbon of modified activated carbon 

to the bulk flow. 
7. 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. 

 



Mathematical Model Describes Treatment of Waste Water Using Modified Activated Carbon 
 

98 
 

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

No Functions 
New Mathematical 

model 
Langmuir 

model 

Freundlich 
isotherm 

model 

Pseudo-first-
order model 

1 Phases 
Liquid, gas and 
solid phase 

Liquid phase Liquid phase    Liquid phase 

2 Mass transfer  
Calculated without 
chemical remodified 
activate carbonation 

Not calculated 
Not 
calculated 

Calculated 
without 
chemical 
remodified 
activate 
carbonation 

3 Adsorption 

Mass transfer from 
surf modified 
activate carbon to 
the entrails holes of 
modified activate 
carbon  without  
chemical  

Considered Considered Considered 

4 Desorption 

Mass transfer from 
entrails holes of 
MODIFIED 
ACTIVATE 
CARBON to the 
surface activate 
carbon without 
chemical  

Not considered 
Not 
considered 

Not 
considered 

5 Energy transfer Considered  Not considered  Not 
considered 

Not 
considered 

R ate  of de sorption to
the bulk flow

R ate  of adsorption

 

Figure 1.  Steps of adsorption process. 
 
2.2 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: 

	 , 	 	 	 	
.

∗ 		       (11) 



A.S. Ibrahim, A.S. AL Buloshi, S.S. AL Zaabi and L.A. AL Yafai 
 

10 

 

. 	 ⇌
,

,                         (12) 

 

	 , 	 . 	 , 	        (13) 
 

	 , 	 .
,

,

	         (14) 

∗ 	 ,
,

             (15) 

 
Let 

	 	 , 	 . 	 ∗ 	                      (16) 

 
     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) 

C .
	
∗             (17) 

      Substitute (17) in (11) 

	 , 	 	 	 	 ∗	 ∗ 		       (18) 

	 .
	
∗         (19) 

	 	
∗

         (20) 

     Substitute (20) in (18) 

	 , 	 	 	
∗ 	 ∗   

	 , 	 	
∗ 	        (21) 

 
     Representing the system as a continuous 
stirred-tank reactor (CSTR) process: 
 

	 	 	 	 	 	        (22) 
 
     The volumetric flow rate compared to the 
volume of the system is very limited, so   
 

  0          (23) 

     Equation (22) will be 
 

          (24) 

 
     The rate of remodified activate carbonation 
is represented as 
 

	 	         (25) 

 
     Substitute equation (21) in (25) 
 

	 , 	 	
∗ 	 	 		      (26) 

 
     Solve equation (26) 

	 	 , 	 	
∗ 	 		 t constant		(27) 

 
     Boundary conduction  

At  t=0  ,  C C 	,       (28) 
 
     Substitute in (21) 

	 	 , 	 	
∗ 	 		 t C   

                                                                             (29) 

	 mt						 

	 	 , 	
∗ 		 					 

3. 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 (Figs. 6–9, 
respectively). 
 

99 



Mathematical Model Describes Treatment of Waste Water Using Modified Activated Carbon 
 

98 
 

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

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

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.8

1

1.2

1.4

1.6

1.8

2

Time (h)

C
o
n
c
e
n
tr

a
ti
o
n
 g

/l

 

 

Actual data

Propose model

Lagergren model

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (h)

C
o
n
c
e
n
tr

a
ti
o
n
 g

/l

 

 

Actual data

Propose model

Lagergren model

100 



A.S. Ibrahim, A.S. AL Buloshi, S.S. AL Zaabi and L.A. AL Yafai 
 

101 

 

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

 

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

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.8

1

1.2

1.4

1.6

1.8

2

Time (h)

C
o
n
c
e
n
tr

a
ti
o
n
 g

/l

 

 

Actual data

Propose model
Lagergren model

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.8

1

1.2

1.4

1.6

1.8

2

Time (h)

C
o
n
c
e
n
tr

a
ti
o
n
 g

/l

 

 

Actual data

Propose model

Lagergren model



Mathematical Model Describes Treatment of Waste Water Using Modified Activated Carbon 
 

102 
 

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

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

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
8.5

9

9.5

10

Time (h)

C
o
n
c
e
n
tr

a
ti
o
n
 g

/l

 

 

k(+ad)k(s)-k(-ad)=0.24

k(+ad)k(s)-k(-ad)=0.15

k(+ad)k(s)-k(-ad)=0.1

k(+ad)k(s)-k(-ad)=0.05

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (h)

C
o
n
c
e
n
tr

a
ti
o
n
 g

/l

 

 

k(+ad)k(s)-k(-ad)=0.18

k(+ad)k(s)-k(-ad)=0.15
k(+ad)k(s)-k(-ad)=0.1

k(+ad)k(s)-k(-ad)=0.05



A.S. Ibrahim, A.S. AL Buloshi, S.S. AL Zaabi and L.A. AL Yafai 
 

101 

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

 
Figure 9.  Effect of rates of adsorption and desorption on pollutant concentration of copper.
 
4.  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 

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Time (h)

C
o
n
c
e
n
tr

a
ti
o
n
 g

/l

 

 

k(+ad)k(s)-k(-ad)=0.18

k(+ad)k(s)-k(-ad)=0.15

k(+ad)k(s)-k(-ad)=0.1

k(+ad)k(s)-k(-ad)=0.05

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.8

1

1.2

1.4

1.6

1.8

2

Time (h)

C
o
n
c
e
n
tr

a
ti
o
n
 g

/l

 

 

k(+ad)k(s)-k(-ad)=0.18

k(+ad)k(s)-k(-ad)=0.15

k(+ad)k(s)-k(-ad)=0.1

k(+ad)k(s)-k(-ad)=0.05

103 



Mathematical Model Describes Treatment of Waste Water Using Modified Activated Carbon 
 

101 
 

the rates types for adsorption and desorption 
compare to the other models.  
 
References 
 
Afsaneh S, Mohammad R, Hosein D, Amin G, 

Mohammad H (2008), Adsorption of 
carbon dioxide using impregnated 
modified activate carbontivated  carbon 
promoted by Zinc. International Journal of 
Greenhouse Gas Control 3: 249-254. 

Ajmal M, Rak R, Jamal A, Ahmad R (2003), 
Adsorption studies on rice husk: removal 
and recovery of Cd (II) from wastewater. 
Bioresource Technology 86: 147–149. 

Bog G, Liping C, Kechang X (2006), Adsorption 
of carbon dioxide on modified activate 
carbontivated Carbon. Journal of Natural 
Gas Chemistry 15: 223-229. 

Lin N. Bai R (2005), Copper absorption on 
chitosan-cellulose  hydrogel  beads:  hydro- 
gel- beads: behaviors and mechanisms. 
Separation And Purification Technology 42: 
237–247. 

Lalhruaitluanga H, Jayaram K, Prasad M, 
Kumar K (2010), Lead (II) adsorption from 
aqueous solutions by raw and modified 
activate carbontivated charcoals of 
melocanna modified activate carboncifera 
roxburgh (bamboo)-A comparative study. 
Journal of Hazardous Materials 175: 311-318. 

Khaled A, Nemr A, Sikaily A, Abdelwahab O 
(2009), Treatment of artificial textile dye 
effluent containing Direct Yellow 12 by 
orange peel carbon. Desalination 238: 210-
232. 

Mohamed K, Wan M, Chun Y, Donni A (2008),  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 

Adsorption cap modified activate 
carbonites of carbon dioxide, oxygen, 
nitrogen and methane on carbon molecular 
basket derived from polyethyleneimine 
impregnation on microporous palm shell 
modified activate carbontivated carbon. 
Separation And Purification Technology 62: 
609–613. 

Muhammad R, Raziya N, Muhammad A, Tariq 
M, Khalil R (2008), Pb (II) biosorption from 
hazardous aqueous streams using 
gossypium hirsutum (Cotton) waste 
biomass. Journal of Hazardous Materials 161: 
88-94. 

Nakano Y, Takeshita K, Tsutsumi  T (2001), 
Adsorption mechanim of hevalent 
chromium by Redox within Condensed-
Tannin Gel, Water Research 35 :496-500. 

Okieimen C, Ogbeide S (2009), The 
Dependence On Temperature of 
Carbonization and chemical modified 
activate carbontivation on charmodified 
activate carbonteristics of granular 
modified activate carbontivated carbon 
charmodified activate carbonteristics. 
Advanced Materials Research 62-64 : 398-403. 

Prasetyo I,  Dod D (1998), Adsorption rate of 
methane and carbon dioxide on modified 
activate carbontivated carbon by the 
semibatch constant molar flow rate 
method. Chemical Engineering Science 53(19): 
3459-3467. 

Vazquez   G,  Gonzalez  A,  Freire  S,   Lopez,  
Antorrena G (2002), Removal of cadmium 
and mercury ions from aqueous solution 
by sorption on treated pinus pinaster bark: 
kinetics and isotherms. Bioresource 
Technology 822: 47-51.  

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