Adsorption of Dyes

Abstract

Adsorption is one of the most commonly used, traditional separation technologies utilized for separation. Since it is an equilibrium-governed process, the process efficiency is excellent, but the throughput is relatively low. Nevertheless, because of its simplicity, this is one of the normally used technologies for dye removal from aqueous stream. Therefore, it is imperative to understand the modeling aspects of such adsorbent-based systems which is necessary for design and implementation of the technology. Additionally, the chapter describes the characteristics of the different commonly used adsorbents and its applicability.

The word “adsorption” was formulated in 1881 by German physicist Heinrich Kayser to differentiate between the surface phenomena and intermolecular penetration. Adsorption can be divided into physical and chemical adsorption. Physical adsorption is controlled by the physical forces such as van der Waals forces, hydrophobicity, hydrogen bond, polarity, static interaction, dipole-dipole interaction, π − π interaction, etc. When the species are adsorbed to the surface of the adsorbent by means of strong chemical interactions or bonding, it is referred to as chemisorption. The extent of adsorption depends on the nature of adsorbate such as molecular weight, molecular structure, molecular size, polarity, and solution concentration. It is also dependent on the surface properties of adsorbent such as particle size, porosity, surface area, surface charge, etc. The primary advantages of adsorption processes are:

• 1. Simple in design

• 2. Relatively safe and easy to operate

• 3. Inexpensive (compared to other separation processes)

• 4. Provides sludge-free cleaning operations (Gupta et al. 2000).

Selection of a suitable adsorbent is the primary concern for adopting adsorption in any process industries. The performance of the process is often limited by the equilibrium capacity of the adsorbent.

2.1 Application of Adsorption in the Treatment of Process Wastewater

Various low-cost adsorbents for treatment of effluent containing heavy metals have been studied by various researchers in the past. Most of these adsorbents are prepared from the waste or by-products of other process plants or naturally occurring materials. A list of most commonly used low-cost adsorbents for heavy metal removal that is prepared from naturally occurring materials and processes are presented in Table 2.1.

Table 2.1 Low-cost high-capacity metal ion adsorbents

2.2 Experimental Studies of Dye Adsorption

In the following sections, adsorption of chrysoidine, eosin, and Congo red by commercial activated carbon (CAC) has been presented.

2.2.1 Batch Adsorption

The batch adsorption is typically carried out in the solution phase containing dyes. The effects of agitation time and initial dye concentration on the percentage adsorption of dye by activated carbon at room temperature are shown in Figs. 2.1a, 2.1b, and 2.1c for chrysoidine, eosin, and Congo red. For all the cases, the percentage adsorption increases with agitation time for different initial dye concentration and attains equilibrium after some time.

Fig. 2.1a

Effects of agitation time and concentration of chrysoidine on percentage of adsorption (Reproduced from Purkait et al. (2004). With permission from Taylor & Francis Ltd)

Fig. 2.1b

Effects of agitation time and concentration of eosin on percentage of adsorption (Reproduced from Purkait et al. (2005). With permission from Elsevier)

Fig. 2.1c

Effects of agitation time and concentration of Congo red on percentage of adsorption (Reproduced from Purkait et al. (2007). With permission from Elsevier)

From Fig. 2.1a, it may be observed that for all initial chrysoidine concentration, the percentage adsorption is found to be constant beyond 80 min. This indicates that equilibrium is attained at about 80 min for initial dye concentration in the range of 100–400 mg/L. It is also clear that the extent of adsorption depends on the initial dye concentration. For dye solution of lower initial concentration (up to 100 mg/L), the adsorption is very fast and almost 100% adsorption is achieved quickly. The dye adsorption at equilibrium decreases from 100% to about 94% as the dye concentration increases from 100 to 400 mg/L.

It is clear from Fig. 2.1b that up to an initial eosin concentration of 100 mg/L, more than 99% adsorption is achieved within 5 min. For an initial concentration of 200 mg/L, the percentage adsorption increases until 90 min and becomes constant thereafter. For a feed concentration of 400 mg/L, the percentage adsorption increases rapidly for about 90 min, and the increase becomes gradual thereafter. For 210 min of operation, the dye adsorption is 99.6% for an initial dye concentration of 100 mg/L but only 72.3% for 400 mg/L.

Figure 2.1c describes the variation of Congo red adsorption with time for different initial dye concentration. The percentage adsorption of Congo red is found to be constant beyond 50 min. This indicates that the equilibrium is attained within 50 min for the range of initial dye concentrations. It is also clear that the extent of adsorption depends on the initial dye concentration. For dye solution of lower initial concentration, the adsorption is very fast and 90% of adsorption is achieved quickly. The percentage dye adsorption at equilibrium decreases from 90% to 28% as the dye concentration increases from 50 to 545 mg/L.

The effects of adsorbent dose on the extent of chrysoidine adsorption are shown in Fig. 2.2 for initial dye concentrations of 700 mg/L. It is clear from the figure that percentage adsorption increases with time up to 80 min and also with adsorbent dose. Percentage adsorption increases from about 77 to 99% when the adsorbent dose increases from 0.75 to 1.40 g/L. This increase in percentage adsorption may be due to the fact that the number of available sites for adsorption increases with adsorbent dose.

Fig. 2.2

Effects of agitation time and adsorbent dose on percentage adsorption (Reproduced from Purkait et al. (2004). With permission from Taylor & Francis Ltd)

The pH of the solution has significant influence in the rate of adsorption. The percentage dye adsorption at different pH is shown in Figs. 2.3a, 2.3b, and 2.3c for chrysoidine, eosin, and Congo red, respectively. Figure 2.3a describes the variation of chrysoidine adsorption at different pH for an initial dye concentration of 400 mg/L. The color of chrysoidine dye in aqueous medium is red (λ _max: 457 nm) in acidic pH but changes its color from red to yellow (λ_max: 442 nm) in basic pH. This is due to the presence of chromophore in the structure of chrysoidine. A chromophore is any structural feature (in this case, −N = N–) which produces light absorption in the ultraviolet region or color in the visible region. An auxochrome is any group (in this case –NH₂) which, although not a chromophore, leads to a red shift when attached to a chromophore. Thus, the combination of chromophore and auxochrome behaves as a new chromophore. Bathochromic effect (red shift) and hypsochromic effect (blue shift) are the shifting of the absorption band to the longer and shorter wavelengths (Finar 1973). Therefore, due to blue shift, chrysoidine changes its color in basic pH.

Fig. 2.3a

Effect of initial pH on percentage of adsorption for 400 mg/L of feed chrysoidine (Reproduced from Purkait et al. (2004). With permission from Taylor & Francis Ltd)

Fig. 2.3b

Effect of pH on the percentage adsorption of feed eosin (Reproduced from Purkait et al. (2005). With permission from Elsevier)

Fig. 2.3c

Effect of initial pH on percentage of adsorption for 200 mg/L of feed Congo red (Reproduced from Purkait et al. (2007). With permission from Elsevier)

Adsorption followed by desorption technique is generally used to get the more concentrated form of the dye solution. One of the most common desorption technique is the pH treatment. But problem arises for the dyes which are highly pH sensitive, like chrysoidine as discussed in the previous paragraph.

Most of the activated carbon contains some oxygen complexes on the surface, e.g., (a) strongly carboxylic groups, (b) carbonyl groups, and (c) phenolic groups (Motoyuki 1990). These groups are nucleophilic in nature and potential adsorbing sites. In acidic pH, these active sites get blocked by hydrogen ion leading to reduction in adsorption. Hence, adsorption of chrysoidine on activated carbon is less in acidic pH. It is found from Fig. 2.3a that at pH 2.6, adsorption is nearly 62% for the feed dye concentrations of 400 mg/L at the end of experiment. The percentage adsorption decreases from about 98 to 40%, when pH decreases from 11.1 to 2.6. From Fig. 2.3a, it may be observed that the adsorption of this dye is more at the basic pH.

The percentage of eosin adsorption at different pH levels are shown in Fig. 2.3b for an initial dye concentration of 100 mg/L. pH plays an important role on adsorption capacity by influencing the chemistry of both the dye molecule and the activated carbon in aqueous solution. Eosin is a dipolar molecule at low pH. Activated carbon contains oxygen complexes on its surface, e.g., strongly carboxylic groups, carbonyl groups, and phenolic groups (Motoyuki 1990). These groups are nucleophilic in nature. With decrease in pH of the dye solution, more dye molecules are protonated and get adsorbed on the surface of the activated carbon. It can be observed from Fig. 2.3b that at pH 2, adsorption is about 100% for an initial dye concentration of 100 mg/L. Percentage adsorption decreases with increase in pH. For the initial dye concentration of 100 mg/L, the removal is 91% for a pH of 12.

The percentage of Congo red adsorption at different pH has been shown in Fig. 2.3c for the initial dye concentrations of 200 mg/L. The initial pH of dye solution plays an important role particularly on the adsorption capacity by influencing the chemistry of both dye molecule and activated carbon in aqueous solution. Congo red is a dipolar molecular at lower pH and exists as anionic form at higher pH as shown in Fig. 2.3c. The sodium and potassium salt of anionic Congo red in aqueous medium is red in color in basic pH up to 10. Above the pH value of 10, the degree of red color changes from the original one. It has also been found that as the pH decreases, the color of Congo red solution changes from red to dark blue. Therefore, the pH of the medium needs to be maintained between 5 and 10 to treat Congo red. These variations of color with pH suggest that the extent and nature of ionic character of Congo red molecule depend on the pH of the medium. The variations in the extent of adsorption of Congo red on activated carbon with pH are due to the difference in ionic character of the dye molecule. With decrease in pH of dye solution, more dye molecules are protonated and chemisorbed on the nucleophilic sites of the surface of CAC. It is found from the figure that at pH 2, adsorption is about 100%. On the other hand, the percentage adsorption decreases with increase in pH of the dye solution. This is because at higher pH, dye molecules exist in anionic form, and due to interionic repulsion, less adsorption takes place. For the feed dye concentration of 200 mg/L, the percentage adsorption decreases to 25% at the end of the experiment when the pH is 12. From Fig. 2.3c, it may be observed that the adsorption of Congo red is maximum at the acidic pH. Therefore, when Congo red is present in the solution as red color, the operating pH for maximum adsorption should be kept at 5.

Effects of temperature on the extent of adsorption are shown in Figs. 2.4a, 2.4b, and 2.4c for chrysoidine, eosin, and Congo red, respectively. Adsorption experiments are carried out for aqueous solution of chrysoidine for two different concentrations (400 and 700 mg/L) at three different temperatures (30, 50, and 70 °C) and at a pH of 4.4. It has been observed that the adsorption capacity increases significantly with temperature as shown in Fig. 2.4a for the initial chrysoidine concentration of 400 mg/L. The percentage adsorption increases from about 94 to 99% for the feed dye concentration of 400 mg/L and about 80 to 87% for the feed dye concentration of 700 mg/L, at the end of experiment, when temperature is raised from 30 to 70 °C. This endothermic nature of adsorption is due to the positive ΔH ⁰ value as shown in Table 2.4a.

Table 2.4a Comparison of the first- and second-order adsorption rate constants, calculated and experimental q _e value for chrysoidine on activated charcoal

Fig. 2.4a

Effect of temperature on adsorption capacity for 400 mg/L of feed chrysoidine (Reproduced from Purkait et al. (2004). With permission from Taylor & Francis Ltd)

Fig. 2.4b

Effect of temperature on adsorption capacity for 100 mg/L of feed eosin (Reproduced from Purkait et al. (2005). With permission from Elsevier)

Fig. 2.4c

Effect of temperature on adsorption capacity for 100 mg/L of feed Congo red (Reproduced from Purkait et al. (2007). With permission from Elsevier)

In order to observe the effect of temperature on the adsorption capacity, experiments are carried out for 100 mg/L eosin at three different temperatures (30, 40, and 50 °C) using 1.0 g of activated carbon per liter of the solution. It has been observed that with increase in temperature, adsorption capacity decreases as shown in Fig. 2.4b. This is due to the negative value of ΔH ⁰ value (refer to Table 2.4b).

Table 2.4b Comparison of the first- and second-order adsorption rate constants, calculated and experimental q _e value for eosin on activated charcoal

Experiments are carried out to observe the effect of temperature on the extent of adsorption for Congo red of different initial concentration (50, 100, and 200 mg/L) in aqueous solution at three different temperatures (e.g., 30, 40, and 50 °C) and at neutral pH. It has been observed that the adsorption capacity increases significantly with temperature as shown in Fig. 2.4c for the initial Congo red concentration of 100 mg/L. This is because of positive ΔH ⁰ value as shown in Table 2.4c.

Table 2.4c Comparison of the first- and second-order adsorption rate constants, calculated and experimental q _e value for Congo red on activated charcoal

The thermodynamic parameters ΔG ⁰, ΔS,⁰ and ΔH ⁰ for the adsorption of chrysoidine, eosin, and Congo red have been determined by using the following equations (Khan et al. 1995):

$$ \Delta {G}^0=\Delta {H}^0-T\Delta {S}^0 $$

(2.1)

$$ \log \left({q}_{\mathrm{e}}/{C}_{\mathrm{e}}\right)=\frac{\Delta {S}^0}{2.303R}+\frac{-\Delta {H}^0}{2.303 RT} $$

(2.2)

where q _e is the amount of dye adsorbed per unit mass of activated carbon (mg/g), C _e is equilibrium concentration (mg/L), and T is temperature in Kelvin. q _e/C _e is called the adsorption affinity. It may be noted here that the experimental data considered here for the calculation of the thermodynamic parameters, namely, ΔG ⁰,ΔH ⁰ _, and ΔS ⁰, are in the linear range of the equilibrium adsorption isotherm (e.g., for chrysoidine, q _evaries from 2.5 to 3.0 mmol/g and C _evaries from 0.012 to 0.2 mmol/L as shown in Fig. 2.5a). The values of Gibbs free energy (ΔG ⁰) have been calculated by knowing the enthalpy of adsorption (ΔH ⁰) and the entropy of adsorption (ΔS ⁰). ΔS ⁰ and ΔH ⁰ are obtained from a plot of log(q _e/C _e) versus $\frac{1}{T}$, from Eq. (2.2). Once these two parameters are obtained, ΔG ⁰ is determined from Eq. (2.1). The values ofΔG ⁰, ΔH ⁰, and ΔS ⁰are listed in Tables 2.2a, 2.2b and 2.2c for chrysoidine, eosin, and Congo red, respectively.

Fig. 2.5a

Adsorption isotherms of chrysoidine on activated carbon (Reproduced from Purkait et al. (2004). With permission from Taylor & Francis Ltd)

Table 2.2a Thermodynamic parameters for adsorption of chrysoidine in activated charcoal at different temperature and dye concentrations

Table 2.2b Thermodynamic parameters for adsorption of eosin in activated charcoal at different temperature and 100 mg/L of eosin

Table 2.2c Thermodynamic parameters for adsorption of Congo red in activated charcoal at different temperature and dye concentrations

Gibbs free energy (ΔG ⁰) for all the three dyes is negative (as shown in Tables 2.2a, 2.2b, 2.2c). This indicates that the adsorption process is spontaneous for all the three dyes. Adsorption of chrysoidine and Congo red is endothermic in nature (since ΔH ⁰ value is positive; refer to Tables 2.2a and 2.2c). On the other hand, eosin adsorption is exothermic in nature (as ΔH ⁰ value is negative; refer to Table 2.2b). The positive value of ΔS ⁰ for chrysoidine and Congo red (refer to Tables 2.2a and 2.2c) dictates that the adsorbed dye molecules on the activated carbon surface are organized in a more random fashion compared to those in the aqueous phase. Similar observations have been reported in the literature (Bhattacharyya and Sharma 2003). For eosin, the negative value of ΔS ⁰ (refer to Table 2.2b) suggests decreased randomness at the solid solution interface during adsorption (Manju et al. 1998).

2.2.1.1 Langmuir Adsorption Isotherm

Langmuir adsorption isotherm is applicable to explain the equilibrium data for many adsorption processes. The basic assumption of this process is the formation of monolayer of adsorbate on the outer surface of adsorbent, and after that no further adsorption takes place. The expression of the Langmuir model is given as follows (Ozacar and Sengil 2003):

$$ {q}_{\mathrm{e}}=\frac{QbC_{\mathrm{e}}}{1+{bC}_{\mathrm{e}}} $$

(2.3)

A linear form of this expression is

$$ \frac{1}{q_{\mathrm{e}}}=\frac{1}{Q}+\frac{1}{Qb}.\frac{1}{C_{\mathrm{e}}} $$

(2.4)

where q _eis the amount of adsorbate adsorbed per unit weight of adsorbent (mg/g) and C _e is the equilibrium concentration of adsorbate (mg/L). The constant Q and b are the Langmuir constants and are the significance of adsorption capacity (mg/g) and energy of adsorption (l/mg), respectively. Values of Q and b are calculated from the intercept and slope of the plot 1/q _e versus 1/C _e.

2.2.1.2 Freundlich Adsorption Isotherm

This model is an indicative of the extent of heterogeneity of the surface of adsorbent and is given as follows:

$$ {q}_{\mathrm{e}}={K}_{\mathrm{F}}{C}_{\mathrm{e}}^{1/n} $$

(2.5)

where K _F and n are Freundlich constants. A linear form of the Freundlich expression is as follows:

$$ \log {q}_{\mathrm{e}}=\log {K}_{\mathrm{F}}+\frac{1}{n}\log {C}_{\mathrm{e}} $$

(2.6)

The constants K _F and n are the Freundlich constants and are the significance of adsorption capacity and intensity of adsorption, respectively. Values of K _F and n are calculated from the intercept and slope of the plot logq _e versus logC _e.

Adsorption isotherms of chrysoidine, eosin, and Congo red on activated carbon at 30 °C are shown in Figs. 2.5a, 2.5b, and 2.5c, respectively. The coefficients of these two isotherm models for the three dyes are given in Table 2.3. These data provide information on the amount of activated carbon required to adsorb a particular mass of dye under specified system conditions. Correlation coefficients are evaluated by fitting the experimental adsorption equilibrium data for three dyes separately using both Langmuir and Freundlich adsorption isotherms and are also shown in Table 2.3. It is found from the correlation coefficients (r ²) that adsorption isotherm for the present three dye-activated charcoal systems is best explained by Freundlich equation.

Fig. 2.5b

Adsorption isotherms of eosin on activated carbon

Fig. 2.5c

Adsorption isotherms of Congo red on activated carbon (Reproduced from Purkait et al. (2007). With permission from Elsevier)

Table 2.3 Langmuir and Freundlich isotherm constants for adsorption of dyes on activated charcoal

2.2.1.3 Adsorption Kinetics

The kinetics of adsorption of chrysoidine, eosin, and Congo red on activated carbon have been described using both first- and pseudo-second-order model. The Lagergren’s equation for first-order kinetics is as follows:

$$ \log \left({q}_{\mathrm{e}}-{q}_{\mathrm{t}}\right)=\log {q}_{\mathrm{e}}-\frac{k_1t}{2.303} $$

(2.7)

The expression for pseudo-second-order rate equation is given as (Ho et al. 1996)

$$ \frac{t}{q_{\mathrm{t}}}=\frac{1}{k_2{q}_{\mathrm{e}}^2}+\frac{t}{q_{\mathrm{e}}} $$

(2.8)

where q _e and q _t are the amounts of dye adsorbed (mg/g) at equilibrium and at any time t and k ₁ is the rate constant (min⁻¹). Figures 2.6a, 2.6b, and 2.6c show (t/q _t) versus t plot for pseudo-second-order kinetics for chrysoidine, eosin, and Congo red, respectively. In Eq. (2.8), k ₂ (g/mg min) is the rate constant for the pseudo-second-order adsorption kinetics. The slope of the plot (t/q _t) versus t gives the value of q _e , and from the intercept, k₂ can be calculated. The values of k₁, k₂, and correlation coefficients (r ²), both in the first and pseudo-second-order kinetics, are presented in Tables 2.4a, 2.4b, and 2.4c for chrysoidine, eosin, and Congo red, respectively. It may be observed from Tables 2.4a, 2.4b, and 2.4c that the adsorption of chrysoidine, eosin, and Congo red on activated carbon follows pseudo-second-order kinetics more closely.

Fig. 2.6a

Plot of the pseudo-second-order kinetic model for adsorption of chrysoidine on activated carbon (0.5 g/L). Feed chrysoidine: 200 and 400 mg/L (Reproduced from Purkait et al. (2004). With permission from Taylor & Francis Ltd)

Fig. 2.6b

Plot of the pseudo-second-order kinetic model for adsorption of eosin on activated carbon (1.0 g/L). Feed eosin: 200 and 400 mg/L (Reproduced from Purkait et al. (2005). With permission from Elsevier)

Fig. 2.6c

Plot of the pseudo-second-order kinetic model for adsorption of Congo red on activated carbon (1.0 g/L). Feed Congo red: 50 and 545 mg/L

2.2.2 Column Adsorption

Column adsorptions studies are essential for design of industrial scale fixed-bed adsorber system. Figure 2.7 shows the breakthrough curves for different bed depths. It may be observed from Fig. 2.8 that the breakthrough time (duration for zero column outlet concentration) increases from 18 to 39 h, when the bed depth is increased from 4.5 × 10⁻² m to 7.0 × 10⁻² m, for the same flow rate of 0.18 L/hr. The shape and the gradient of the breakthrough curves for the two bed depths are almost identical.

Fig. 2.7

Variation of the breakthrough curve with bed depth (Reproduced from Purkait et al. (2005). With permission from Elsevier)

2.3 Generalized Shrinking Core Model for Batch Adsorption Data

To develop a mathematical model that describes the adsorption dynamics, the following information are generally required:

• 1. A complete description of equilibrium behavior, i.e., the maximum level of adsorption attained in a sorbent/sorbate system as a function of the sorbate liquid-phase concentration

• 2. Mathematical representation of associated rate of adsorption, which is controlled by the resistances within the sorbent particles

In adsorption, mainly two resistances prevail – the external liquid film resistance and the resistance in the adsorbent particle. The intraparticle diffusion resistance may be neglected for solutes that exhibit strong solid to liquid-phase equilibrium solute distribution, in the initial period of operation. However, even for such systems, the above assumption leads to errors that are substantial beyond the first few minutes if the agitation is high (Mathews and Weber 1976). So, both the resistances are important for kinetic study (Chatzopoulos et al. 1993; McKay 1984; Costa et al. 1987; Komiyama and Smith 1974; Liapis and Rippin 1977).

The external liquid film resistance is characterized by the external liquid film mass transfer coefficient (k _f). The mass transport within the adsorbent particles is assumed to be a pore diffusion (Dedrick and Beckman 1967; Weber and Rumer 1965; Furusawa and Smith 1973; McKay 1982) or homogeneous solid diffusion process (McKay 1982; Hand et al. 1983; Kapoor et al. 1989).

The pore diffusion model outlined in this paper is based on the unreacted shrinking core model (Yen 1968; Levenspiel 1972) with pseudo-steady-state approximation. This model has mostly been applied to gas-solid non-catalytic reactions, but a number of liquid-solid reactions also have been analyzed using this model (Neretnieks 1976; Spahn and Schlunder 1975). In the pore diffusion model, there is adsorption of the adsorbate into the pores with a cocurrent solute distributed all along the pore wall.

The assumptions made in this model are as follows:

• (a) Pore diffusivity is independent of concentration.

• (b) Adsorption isotherm is irreversible.

• (c) Pseudo-steady-state approximation is valid.

• (d) The driving force in both film and particle mass transfer is directly proportional.

• (e) Adsorbent particles are spherical.

The major limitation of this model is that it is specific to the nature of isotherm. This means that the model available in literature is most suitable for Langmuir-type isotherm, i.e., formation of a monolayer of adsorbate on the adsorbent. Besides, this model is only applicable for higher initial adsorbate concentration in solution so that the batch process operating line intercepts the invariant zone of isotherm. For example, for Astrazone blue-silica system, the literature model is applicable for C ₀ >> 200 mg/lit (McKay 1984). The present model, which is more generalized, overcomes the above limitations. The model proposed, here in, can be applied to wide ranges of initial adsorbate concentrations for all possible nature of isotherms. The system reported here is adsorption of Astrazone blue dye on Sorbsil Silica.

The equations considered for the kinetics of the adsorption process for spherical adsorbent particles for the present model are as follows:

The mass transfer from external liquid phase can be written as

$$ N(t)=4\pi {R}^2{K}_{\mathrm{f}}\left({C}_{\mathrm{t}}-{C}_{\mathrm{et}}\right) $$

(2.9)

The diffusion of solute through the pores as per Fick’s law can be written as

$$ N(t)=\frac{4\pi {D}_{\mathrm{p}}{C}_{\mathrm{et}}}{\left[\frac{1}{R_{\mathrm{f}}}-\frac{1}{R}\right]} $$

(2.10)

where D _p is the effective diffusivity in the porous adsorbent (Fogler 1997).

The mass balance on a spherical element of adsorbate particle can be written as

$$ N(t)=-4\pi {R_{\mathrm{f}}}^2{Y}_{\mathrm{et}}\rho \left[\frac{dR_{\mathrm{f}}}{dt}\right] $$

(2.11)

The average concentration on adsorbent particle can be written as

$$ \overline{Y_{\mathrm{t}}}={Y}_{\mathrm{et}}\left[1-{\left(\frac{R_{\mathrm{f}}}{R}\right)}^3\right] $$

(2.12)

The differential mass balance over the system by equating the decrease in adsorbate concentration in the solution with the accumulation of the adsorbate in the adsorbent can be written as

$$ N(t)=-V\left(\frac{dC_{\mathrm{t}}}{dt}\right)=W\left(\frac{d\overline{Y_{\mathrm{t}}}}{dt}\right) $$

(2.13)

The dimensionless terms used for simplification are as follows:

$$ {C}_{\mathrm{t}}^{\ast }=\frac{C_{\mathrm{t}}}{C_0},r=\frac{R_{\mathrm{f}}}{R}, Bi=\frac{k_{\mathrm{f}}R}{D_{\mathrm{p}}},\kern0.5em Ch=\frac{W}{VC_0},{C_{\mathrm{et}}}^{\ast }=\frac{C_{\mathrm{et}}}{C_0}\ \mathrm{and}\ \tau =\frac{D_{\mathrm{p}}t}{R^2} $$

Simplifying Eqs. (2.9) and (2.10)

$$ {C_{\mathrm{et}}}^{\ast }=\frac{Bi\left(1-r\right)\kern0.1em {C}_{\mathrm{t}}^{\ast }}{\left[r+ Bi\left(1-r\right)\right]}={g}_1\left({C}_{\mathrm{t}}^{\ast },r\right) $$

(2.14)

Now differentiating the above equation with respect to τ

$$ \frac{{d C}_{\mathrm{et}}^{\ast }}{d\tau}=\frac{Bi\left(1-r\right)}{r+ Bi\left(1-r\right)}\frac{{d C}_{\mathrm{t}}^{\ast }}{d\tau}-\frac{{ Bi C}_{\mathrm{t}}^{\ast }}{{\left[r+ Bi\left(1-r\right)\right]}^2}\frac{d r}{d\tau} $$

(2.15)

From the equilibrium relationship

$$ {Y}_{\mathrm{e}}(t)={g}_2\left({C}_{\mathrm{e}\mathrm{t}}^{\ast}\right) $$

(2.16)

where g₂ is any equilibrium isotherm relationship. Simplifying Eqs. (2.9) and (2.11)

$$ \left(\frac{d r}{d\tau}\right)=\frac{- Bi\left(\frac{C_0}{\rho {Y}_{\mathrm{e}}}\right)\left({C}_{\mathrm{t}}^{\ast }-{C_{\mathrm{e}\mathrm{t}}}^{\ast}\right)}{r^2} $$

(2.17)

Simplifying Eqs. (2.12) and (2.13)

$$ \left(\frac{{d C}_{\mathrm{t}}^{\ast }}{d\tau}\right)+ Ch\left(1-{r}^3\right)\left(\frac{{d Y}_{\mathrm{et}}}{d\tau}\right)=3 Ch\cdot {Y}_{\mathrm{et}}{r}^2\left(\frac{d r}{d\tau}\right) $$

(2.18)

For Langmuir isotherm

$$ {Y}_{\mathrm{et}}=\frac{Y_{\mathrm{s}}{C}_{\mathrm{et}}}{1+{k}_0{C}_{\mathrm{et}}}=\frac{Y_{\mathrm{s}}{C}_0{C}_{\mathrm{et}}^{\ast }}{1+{k}_0{C}_0{C}_{\mathrm{et}}^{\ast }}=\frac{Y_{\mathrm{es}}{C}_{\mathrm{et}}^{\ast }}{1+{k}_0^{\ast }{C}_{\mathrm{et}}^{\ast }} $$

(2.19)

where Y _es = Y _s C ₀ and k ₀ ^* = k ₀ C ₀.

The time derivative of Eq. (2.19) becomes

$$ \frac{{d Y}_{\mathrm{et}}}{d\tau}=\frac{Y_{\mathrm{es}}}{{\left(1+{k}_0^{\ast }{C}_{\mathrm{et}}^{\ast}\right)}^2}\frac{{d C}_{\mathrm{et}}^{\ast }}{d\tau} $$

(2.20)

Combining Eqs. (2.14), (2.18), and (2.20) and after algebraic manipulation, the following expression is obtained (for Langmuir-type isotherm):

$$ \left(\frac{{d C}_{\mathrm{t}}^{\ast }}{d\tau}\right)=\left(N/M\right)\left(\frac{d r}{d\tau}\right) $$

(2.21)

where $M=1+ Ch\left(1-{r}^3\right)\frac{Y_{\mathrm{es}} Bi\left(1-r\right)}{{\left(1+{k}_0^{\ast }{C}_{\mathrm{et}}^{\ast}\right)}^2\left[r+\left(1-r\right) Bi\right]}$ and, $N=3{ChY}_{\mathrm{et}}{r}^2+\frac{ChY_{\mathrm{es}} Bi\left(1-{r}^3\right){C}_{\mathrm{t}}^{\ast }}{{\left(1+{k}_0^{\ast }{C}_{\mathrm{et}}^{\ast}\right)}^2{\left[r+\left(1-r\right) Bi\right]}^2}$. Using Eq. (2.14), Eq. (2.17) may be written as

$$ \frac{d r}{d\tau}=\frac{- Bi\left(\frac{C_0}{\rho {Y}_{\mathrm{e}}}\right)\left({C}_{\mathrm{t}}^{\ast }-{C_{\mathrm{e}\mathrm{t}}}^{\ast}\right)}{r^2}={f}_1\left({C}_{\mathrm{t}}^{\ast },r\right) $$

(2.22)

Using Eqs. (2.14) and (2.22), Eq. (2.21) may be expressed as

$$ \left(\frac{{d C}_{\mathrm{t}}^{\ast }}{d\tau}\right)=\frac{N\left({C}_{\mathrm{t}}^{\ast },r\right){f}_1\left({C}_{\mathrm{t}}^{\ast },r\right)}{M\left({C}_{\mathrm{t}}^{\ast },r\right)}={f}_2\left({C}_{\mathrm{t}}^{\ast },r\right) $$

(2.23)

The initial conditions for Eqs. (2.22) and (2.23), C ₀ = 1.0 and r = 1.0 at time, τ = 0.0. Equations (2.22) and (2.23) can be solved to find the bulk concentration at any time “t” if we know all the process parameters. The two process parameters – the external mass transfer coefficient (k _f) and internal effective diffusivity (D _p) – are unknown to us. These two parameters are estimated by optimizing the experimental concentration profile as outlined in the next section.

2.3.1 Numerical Analysis

The above set of equations are numerically solved using fourth-order Runge-Kutta of step size (dτ) of the order 10⁻⁵ along with a nonlinear optimization technique (Levenberg-Marquardt) to estimate the two process parameters described above, so that the experiment kinetic profile (i.e., bulk concentration versus time) is matched. For this purpose, optimization subroutine UNLSF/DUNLSF from IMSL math library has been used.

The adsorption systems studied here encompass Radke-Prausnitz isotherm (Tables 2.5 and 2.6). The systems considered here are (1) Astrazone blue dye on silica, (2) para-nitrophenol on granular activated carbon from Lurgi, and (3) toluene on F300 activated carbon. The experimental data on kinetics and the isotherm constants have been reported in literature (McKay 1984; Costa et al. 1987; Chatzopoulos et al. 1993).

Table 2.5 Radke-Prausnitz isotherm constants $\left[\left(\frac{1}{Y_{\mathrm{e}}}\right)=\left(\frac{1}{AC_{\mathrm{e}}}\right)+\left(\frac{1}{{BC_{\mathrm{e}}}^{\delta }}\right)\right]$

Table 2.6 Model parameters using Radke-Prausnitz isotherm at various temperatures

The adsorption of Astrazone blue on silica follows Langmuir isotherm (McKay 1984). The isotherm constants are Y _s = 0.5 lit/g and K ₀ = 0.016 lit/mg, where Y _e in mg/g and C _e in mg/l. For W = 17 g, V = 1.7 l, R = 0.3025 mm, and ρ = 2.2 g/cc, the concentration decay data for C ₀ = 520 mg/l has been used to determine the unknown process parameters using the above numerical procedure as shown in Fig. 2.8a. The estimated values of the parameters are as follows: k _f = 130.0 × 10⁻⁶ cm/s and D _p = 16.16 × 10⁻⁹ cm²/s. These values of k _f and D _p are used to simulate the adsorption kinetics for different operating conditions. It is interesting to note that the estimated values of k _f and D _p are close to the values reported by Mckay (1984), i.e., k _f = 80 × 10⁻⁶ cm/s and D _p = 18 × 10⁻⁹ cm²/s. The experimental observations and the model-simulated concentration profiles for different initial dye concentrations, masses of silica, and particle sizes of silica have been shown in Figs. 2.8b, 2.8c and 2.8d, respectively. From the above figures, it may be observed that beyond 120 min (2 h) of the process, the model underpredicts the bulk concentration profile. This may be due to the increase of the resistance inside the micropores which inhibits the process of adsorption. The present model can be used for multicomponent adsorption processes and also with concentration-dependent diffusivity. The model is useful to estimate k _f and D _p values, which are required for the design of fixed-bed adsorber.

Fig. 2.8a

Adsorption of Astrazone blue dye on silica

Fig. 2.8b

Effect of initial adsorbate concentration. Solid lines are the model predictions and symbols are the experimental data

Fig. 2.8c

Effect of the mass of adsorbent on concentration decay. Solid lines are the model predictions and symbols are the experimental data

Fig. 2.8d

Effect of silica particle size on concentration decay. Solid lines are the model predictions and symbols are the experimental data

2.4 Discussion of Mathematical Model Analysis

The adsorption experiments in the fixed-bed column are carried out to study the adsorption dynamics and quantify the breakthrough curve. One of the crucial aspects of design of adsorption columns for any separation process is the prediction of the breakthrough time. This is necessary to estimate the lifetime of the adsorption bed and its process efficiency. There have been several mathematical models developed in the past based on different assumption justifying the simplicity in the calculations.

2.4.1 Thomas Model (Thomas 1944)

Thomas solution is the most general and widely used equation for modeling performance of fixed-bed adsorption. The Thomas model assumes second-order reversible Langmuir kinetics of the adsorption-desorption process. Ideally the model is suitable for situations where the external and internal diffusion resistances are small. This is particularly true for adsorption scenarios in most liquid systems and therefore is most relevant for adsorption in aqueous environment. The expression describing the output concentration C _t /C ₀ is given by

$$ \frac{c_{\mathrm{t}}}{c_0}=\frac{1}{1+\exp \left({k}_{\mathrm{Th}}{q}_{\mathrm{e}}\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{$Q$}\right.-{k}_{\mathrm{Th}}{c}_0t\right)} $$

(2.24)

where k _Th is the model parameter obtained from nonlinear regression of the output concentration with time. The parameter q _e is the maximum adsorption capacity of the adsorbent.

2.4.2 Adams-Bohart Model (Bohart and Adams 1920)

The Adams-Bohart model considers that the adsorption rate is proportional to both the adsorbent leftover capacity and the concentration of the adsorbate species in the solution. The Adams-Bohart model is originally applied for prediction of adsorption behavior in gas-solid systems, but later on extended to liquid streams. It assumes that the adsorption rate is proportional to the residual capacity of the adsorbent and adsorbate concentration. Since the external mass transfer is not taken into account, it is particularly not suitable for describing the system at high flow rate and concentration. Theoretically, the model is applicable for predictions at early times, when C/C₀ << 1. The mathematical equation describing the output concentration is represented by Eq. (2.25):

$$ \frac{c_{\mathrm{t}}}{c_0}=\exp \left({k}_{\mathrm{AB}}{c}_0t-{k}_{\mathrm{AB}}{q}_{\mathrm{e}}\frac{Z}{Q}\right) $$

(2.25)

where k _AB is the model parameter obtained from nonlinear regression of the experimental data.

2.4.3 Yoon-Nelson Model (Yoon and Nelson 1984)

The Yoon-Nelson model is based on the assumption that the probability of adsorption for each molecule decreases proportionately on the probabilities of the adsorbate adsorption and breakthrough. One of the features of the model is that the product of the parameters K _yn τ _ynis constant for a particular adsorbent-adsorbate combination and independent on the operating conditions. This is a fairly simple model which does not require any knowledge of the adsorption capacity or type of the adsorbent:

$$ \frac{c_t}{c_0-{c}_{\mathrm{t}}}=\exp \left({k}_{\mathrm{YN}}t-\tau {k}_{\mathrm{YN}}\right) $$

(2.26)

where τ and k _YN are the model parameters obtained from nonlinear regression of the experimental data.

2.4.4 Clark Model (Clark 1987)

This model is based on the application of the mass transfer concept in combination with Freundlich equilibrium isotherm. The adsorption equilibrium isotherm satisfying Freundlich relationship can only be used for predicting the breakthrough profile of the adsorption column. The semiempirical relationship is presented in Eq. (2.27):

$$ \frac{c_{\mathrm{t}}}{c_0}={\left(\frac{1}{1+A\exp \left(- rt\right)}\right)}^{1/\left(n-1\right)} $$

(2.27)

where A and r are the model constants obtained from nonlinear regression analysis. The constant 1/n is obtained from the Freundlich isotherm equation.

2.4.5 Bed Depth/Service Time (BDST) Model (Goel et al. 2005)

The BDST model is the linearized form of the Adams-Bohart model. The main consideration here is the assumption that the intraparticle diffusion and external mass transfer resistance is negligible and the adsorption kinetics is controlled by the surface chemical reaction between the adsorbate and adsorbent, which is generally uncommon in real systems. The popularity of the BDST is due to its simplicity in predicting breakthrough behavior owing to its rapid analysis. The expression predicting the breakthrough profile (C _t/C ₀) is given by

$$ \frac{c_0}{c_{\mathrm{t}}}=1+{K}_{\mathrm{BDST}}\exp \left(\frac{q_eZ}{Q}-{c}_0t\right) $$

(2.28)

where K _BDST is the model parameter determined by the nonlinear regression analysis of the experimental data. Although the BDST model provides a simple and comprehensive approach for evaluating sorption column test, its validity is limited and does not involve any sound understanding of the implicit transport mechanism (Bohart and Adams 1920; Poots et al. 1976a; Faust and Aly 1987). One of the major limitations of this model is the symmetry of the logistic function (S-shaped curve) around its midpoint t = N ₀ Z/C ₀ U ₀ and C = C ₀/2, which is not true for most breakthrough profiles. Therefore, a more detailed adsorption bed modeling based on the physical transport laws of pore diffusion is necessary for accuracy of the model prediction and scaling up of the process.

2.4.6 Pore Diffusion-Adsorption Model

The 1D single species convective-diffusive equation (Kunii and Levenspiel 1991) is described by Eq. (2.29):

$$ \frac{\partial C}{\partial t}={D}_L\frac{\partial^2C}{\partial {z}^2}-v\frac{\partial C}{\partial z}-\left(\frac{3{k}_{\mathrm{f}}}{a_{\mathrm{p}}}\right)\left(\frac{1-\varepsilon }{\varepsilon}\right){\rho}_{\mathrm{s}}\left(C-{C}_{\mathrm{e}}\right) $$

(2.29)

where the generation term accounted is dependent on the solid-fluid mass transfer rate and is linearly proportional to the concentration difference and C _e is the adsorbate concentration at the adsorbent-bulk interface. The solution of Eq. (2.29) provides information of the transient solute concentration at various bed depths. In deriving Eq. (2.29), by the material balance analysis, it is inherently assumed that all the interparticle void space in the bed is saturated and the fluid velocity is uniform and unhindered throughout. The initial and boundary conditions of Eq. (2.29) are

$$ \mathrm{at}\ \mathrm{t}=0,C={C}_0\kern0.5em \mathrm{for}\ z=0\ \mathrm{and}\ C=0\ \mathrm{for}\kern1em 0<z\le L $$

(2.30a)

$$ \mathrm{at}\ z=0,\kern1em {D}_L\frac{\partial C}{\partial z}+V\left({C}_0-C\right)=0 $$

(2.30b)

$$ \mathrm{and}\ \mathrm{at}\ z=\mathrm{L},\kern1em \frac{\partial C}{\partial z}=0 $$

(2.30c)

The intra-pellet adsorption is described by the pore diffusion transport model. Intraparticle mass transport is characterized by the pore diffusion coefficient D _p. The mass balance equation for the liquid phase (pore) in a spherical particle can be written as

$$ {\varepsilon}_{\mathrm{p}}\frac{\partial {C}_{\mathrm{p}}}{\partial t}+\left(1-{\varepsilon}_{\mathrm{p}}\right){\rho}_{\mathrm{s}}\frac{\partial q}{\partial t}={D}_{\mathrm{p}}\left(\frac{\partial^2{C}_{\mathrm{p}}}{\partial {r}^2}+\frac{2}{r}\frac{\partial {C}_{\mathrm{p}}}{\partial r}\right) $$

(2.31)

where C _p is the contaminant concentration inside the particle and ε _pis particle porosity. Assuming instantaneous equilibrium $\frac{\partial q}{\partial t}=\frac{\partial {C}_{\mathrm{p}}}{\partial t}\frac{\partial q}{\partial {C}_{\mathrm{p}}}$. Modifying Eq. (2.31), we get (Singha et al. 2012),

$$ \frac{\partial {C}_{\mathrm{p}}}{\partial t}=\frac{1}{\left[1+\left(1-{\varepsilon}_{\mathrm{p}}\right){\rho}_s\frac{\partial q}{\partial {C}_{\mathrm{p}}}\right]}\left(\frac{D_{\mathrm{p}}}{\varepsilon_{\mathrm{p}}}\right)\left(\frac{\partial^2{C}_{\mathrm{p}}}{\partial {r}^2}+\frac{2}{r}\frac{\partial {C}_{\mathrm{p}}}{\partial r}\right) $$

(2.32)

The initial condition (t = 0) is given by C _p = 0 for 0 < r < a_p.

The symmetry condition at the particle center (r = 0) and continuity of the concentration on the external surface of the adsorbent bed are simultaneously expressed as

$$ \mathrm{at}\kern0.5em r=0,\frac{\partial {C}_{\mathrm{p}}}{\partial r}=0 $$

(2.33a)

$$ \mathrm{and}\ \mathrm{at}\kern0.5em r={a}_{\mathrm{p}},{k}_{\mathrm{f}}\left({C}_{\mathrm{p}}-{C}_{\mathrm{e}}\right)={D}_{\mathrm{p}}{\varepsilon}_{\mathrm{p}}\frac{\partial {C}_{\mathrm{p}}}{\partial r} $$

(2.33b)

2.5 Various Types of Adsorbents Used for Dye Adsorption

A summary of the various low-cost adsorbents for dye removal as studied by several researchers in the past is presented in Tables 2.7, 2.8 and 2.9. Natural materials or the wastes/by-products of industries or synthetically prepared materials, which cost less and can be used as such or after some minor treatment as adsorbents, are generally called low-cost adsorbents. Generally, the low-cost adsorbents are usually branded as substitutes for activated carbons because of their similar wide usage; however, in a clear sense, they are essentially substitutes for all available expensive adsorbents. These alternative low-cost adsorbents (Gupta et al. 2009) may be categorized in two ways (1) based on their availability, for, e.g., natural materials such as coal, wood, lignite, peat, etc., or agricultural/industrial/domestic wastes; or by-products such as sludge, slag, red mud, fly ash, etc., or synthesized products; and (2) depending on their nature, for, e.g., organic or inorganic. The adsorbents listed in Table 2.7, 2.8, and 2.9 provide useful information about the type and capacity of alternative adsorbents without going into too much detail of the preparation process.

Table 2.7 Adsorption capacities of commercial activated carbon and other alternative adsorbents for removal of acid dyes

Table 2.8 Adsorption capacities of commercial activated carbon and other alternative adsorbents for removal of basic dyes

Table 2.9 Adsorption capacities of commercial activated carbon and other alternative adsorbents for removal of dyes (apart from acid or basic dyes)