A fractional model for dye removal

Ji-Huan He; Zheng-Biao Li

doi:10.1016/j.jksus.2015.07.001

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Original article

28 (

); 14-16

doi:

10.1016/j.jksus.2015.07.001

A fractional model for dye removal

Ji-Huan He^{^a,^b,⁎}, Zheng-Biao Li^{^c}

a National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, China

b Nantong Textile Institute, Soochow University, Nantong, China

c College of Mathematics and Information Science, Qujing Normal University, Qujing, China

⁎Corresponding author at: National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, China. Tel.: +86 139 1315 2427. hejihuan@suda.edu.cn (Ji-Huan He), ijnsns@aliyun.com (Ji-Huan He),

Received: 2015-4-11, Accepted: 2015-7-5, Published: 2015-01-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

The adsorption process has a fractional property, and a fractional model is suggested to study a transport model of direct textile industry wastewater. An approximate solution of the concentration is obtained by the variational iteration method.

Keywords

Fractional calculus

Variational iteration method

Langmuir isotherm

Analytical solution

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1 Introduction

Textile industries produce huge amounts of polluted effluents that are normally discharged into surface water bodies and groundwater aquifers (Ardejani et al., 2007; Khaled et al., 2009; Gupta and Suhas, 2009; Sinha et al., 2013; Garaje et al., 2013 ). These wastewaters cause much damage to the ecological system and quality of the surface water obtained and create a lot of disturbance to groundwater resources (Ardejani et al., 2007).

To model the adsorption process of the direct textile industry wastewater, the following transport equation is normally adopted (Ardejani et al., 2007):

(1)

R \frac{\partial C}{\partial t} + KS ρ_{d} = 0

where

C = equilibrium concentration of the solution.
S = quantity of mass sorbed on the solid surface (mg/g).
R = the retardation factor.
K = delay constant.
$ρ_{d}$ = bulk density of the medium (1/1000 mg/mm³).
Langmuir isotherm reveals the relationship between C and S, which reads (Ardejani et al., 2007).

(2)

S = \frac{Q_{0} K_{L} C}{1 + K_{L} C}

where

Q₀ = maximum adsorption capacity.
K_L = Langmuir constant.
K_F = partition coefficient indicating adsorption capacity.

Combining Eqs. (1) and (2) together, we obtain the following nonlinear equation

(3)

R \frac{\partial C}{\partial t} + \frac{K ρ_{d} Q_{0} K_{L} C}{1 + K_{L} C} = 0, C (0) = C_{0}

It is pointed out, however, that the adsorption process is of fractional property, and can be modeled by a fractional differential equation (Quiroga et al., 2013a,b).

2 Fractional model for dye removal

According to the fractional statistical theory of adsorption (Quiroga et al., 2013a,b), we can modify Eq. (3) in the form:

(4)

R \frac{\partial^{α} C}{\partial t^{α}} + \frac{K ρ_{d} Q_{0} K_{L} C}{1 + K_{L} C} = 0

where the fractional derivative is defined as (He, 2014)

(5)

\frac{\partial^{α} C}{\partial t^{α}} = \frac{1}{Γ (n - α)} \frac{d^{n}}{{dt}^{n}} \int_{t_{0}}^{t} {(s - t)}^{n - α - 1} [C_{0} (s) - C (s)] ds

where

C_{0} (t)

is a known function.

By the fractional complex transform (Li and He, 2010; He and Li, 2012; Li et al., 2012 )

(6)

T = \frac{t^{α}}{Γ (1 + α)}

we can convert Eq. (4) into its differential partner, which reads

(7)

R \frac{\partial C}{\partial T} + \frac{K ρ_{d} Q_{0} K_{L} C}{1 + K_{L} C} = 0

We re-write Eq. (7) in the form

(8)

\frac{\partial C}{\partial T} + aC + bC \frac{\partial C}{\partial T} = 0, C (0) = C_{0}

where

a = \frac{K ρ_{d} Q_{0} K_{L}}{R}

b = K_{L}

Using the variational iteration method (He and Wu, 2007; He, 2006, 2012 ), we can construct the following iteration algorithm:

(9)

C_{n + 1} (T) = C_{n} (T) - \int_{0}^{T} e^{a (s - T)} (\frac{{dC}_{n} (s)}{ds} + {aC}_{n} (s) + {bC}_{n} (s) \frac{{dC}_{n} (s)}{ds}) ds

(10)

C_{n + 1} (T) = C_{0} (T) - b \int_{0}^{T} e^{a (s - T)} C_{n} (s) \frac{{dC}_{n} (s)}{ds} ds

Eq. (9) is called the variational iteration algorithm-I, and Eq. (10) the variational iteration algorithm-II. We begin with

(11)

C = C_{0} e^{- β T}

where

$β$ = an unknown constant to be determined later.

By the variational iteration algorithm-II, we have

(12)

C_{1} (T) = C_{0} e^{- β T} + b β C_{0}^{2} \int_{0}^{t} e^{a (s - T)} e^{- 2 β s} ds = C_{0} e^{- β T} + \frac{b β C_{0}^{2}}{a - 2 β} (e^{- 2 β T} - e^{- aT})

From Eq. (8), we have an additional initial condition, that is

(13)

\frac{dC}{dT} (0) = \frac{{aC}_{0}}{1 + {bC}_{0}}

Using this relationship, we can identify $β$ in Eq. (12):

(14)

C_{1}^{'} (0) = - β C_{0} + \frac{b β C_{0}^{2}}{a - 2 β} (- 2 β + a) = \frac{{aC}_{0}}{1 + {bC}_{0}}

From Eq. (14) $β$ can be solved, which is

(15)

β = \frac{- ({abC}_{0}^{2} - {aC}_{0} + 2 C_{0}^{'}) + \sqrt{{({abC}_{0}^{2} - {aC}_{0} + 2 {aC}_{0}^{'})}^{2} + 8 a^{2} C_{0} C_{0}^{'} (1 - {bC}_{0})}}{4 C_{0} (1 - {bC}_{0})}

We, therefore, obtain the following analytical solution:

(16)

C (T) = C_{0} e^{- β T} + \frac{b β C_{0}^{2}}{a - 2 β} (e^{- 2 β T} - e^{- aT})

Finally we have

(17)

C (t) = C_{0} \exp (- \frac{β t^{α}}{Γ (1 + α)}) + \frac{b β C_{0}^{2}}{a - 2 β} [\exp (- \frac{2 β t^{α}}{Γ (1 + α)}) - \exp (- \frac{{at}^{α}}{Γ (1 + α)})]

where

a = K ρ_{d} Q_{0} K_{L} / R

and

b = K_{L}

3 Conclusion

We give a fractional model for dye removal, when the fractional order $α = 1$ , the fractional model becomes the classic one. The approximate solution reveals that C changes with $t^{α}$ , and decreases approximately exponentially from $C_{0}$ at t = 0 to a final value C = 0 when t tends to infinity.

Acknowledgments

The work is supported by Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), National Natural Science Foundation of China under Grant Nos. 11372205 & 51463021 and Project for Six Kinds of Top Talents in Jiangsu Province under Grant No. ZBZZ-035, Science & Technology Pillar Program of Jiangsu Province under Grant No. BE2013072 and Yunnan province NSF under Grant No. 2011FB090.

References

Ardejani F.D., Badii Kh., Limaee N.Y., Mahmoodi N.M., Arami M., Shafaei S.Z., Mirhabibi A.R., . Numerical modelling and laboratory studies on the removal of Direct Red 23 and Direct Red 80 dyes from textile effluents using orange peel, a low-cost adsorbent. Dyes Pigments. 2007;73:178-185.
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Sinha A.K., Pradhan M., Sarkar S., Pal T., . Large-scale solid-state synthesis of Sn–SnO₂ nanoparticles from layered SnO by sunlight: a material for dye degradation in water by photocatalytic reaction. Environ. Sci. Technol.. 2013;47(5):2339-2345.
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Introduction

Fractional model for dye removal

Conclusion

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[1] Ardejani F.D., Badii Kh., Limaee N.Y., Mahmoodi N.M., Arami M., Shafaei S.Z., Mirhabibi A.R., . Numerical modelling and laboratory studies on the removal of Direct Red 23 and Direct Red 80 dyes from textile effluents using orange peel, a low-cost adsorbent. Dyes Pigments. 2007;73:178-185.
[Google Scholar]

[2] Garaje S.N., Apte S.K., Naik S.D., Ambekar J.D., Sonawane R.S., Kulkarni M.V., Vinu A., Kale B.B., . Template-free synthesis of nanostructured CdxZn1-xS with tunable band structure for H-2 production and organic dye degradation using solar light. Environ. Sci. Technol.. 2013;47(12):6664-6672.
[Google Scholar]

[3] Gupta V.K., Suhas, . Application of low-cost adsorbents for dye removal – A review. J. Environ. Manage.. 2009;90:2313-2342.
[Google Scholar]

[4] He J.H., . Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B. 2006;20:1141-1199.
[Google Scholar]

[5] He J.H., . Asymptotic methods for solitary solutions and compactons. Abstr. Appl. Anal. 2012 (916793)
[Google Scholar]

[6] He J.H., . A tutorial review on fractal spacetime and fractional calculus. Int. J. Theor. Phys.. 2014;53(11):3698-3718.
[Google Scholar]

[7] He J.H., Li Z.B., . Converting fractional differential equations into partial differential equations. Therm. Sci.. 2012;16(2):331-334.
[Google Scholar]

[8] He J.H., Wu X.H., . Variational iteration method: new development and applications. Comput. Math. Appl.. 2007;54:881-894.
[Google Scholar]

[9] Khaled A., El Nemr A., El-Sikaily A., Abdelwahab O., . Removal of Direct N Blue-106 from artificial textile dye effluent using activated carbon from orange peel: adsorption isotherm and kinetic studies. J. Hazard. Mater.. 2009;165:100-110.
[Google Scholar]

[10] Li Z.B., He J.H., . Fractional complex transform for fractional differential equations. Math. Comput. Appl.. 2010;15:970-973.
[Google Scholar]

[11] Li Z.B., Zhu W.H., He J.H., . Exact solutions of time-fractional heat conduction equation by the fractional complex transform. Therm. Sci.. 2012;16(2):335-338.
[Google Scholar]

[12] Quiroga E., Riccardo J.L., Ramirez-Pastor A.J., . Fractional statistics description applied to protein adsorption: effects of excluded surface area on adsorption equilibria. Chem. Phys. Lett.. 2013;585:189-192.
[Google Scholar]

[13] Quiroga E., Centres P.M., Ochoa N.A., Ramirez-Pastor A.J., . Fractional statistical theory of adsorption applied to protein adsorption. J. Colloid Interface Sci.. 2013;390(1):183-188.
[Google Scholar]

[14] Sinha A.K., Pradhan M., Sarkar S., Pal T., . Large-scale solid-state synthesis of Sn–SnO₂ nanoparticles from layered SnO by sunlight: a material for dye degradation in water by photocatalytic reaction. Environ. Sci. Technol.. 2013;47(5):2339-2345.
[Google Scholar]