#### 3.2.1. Mass Transfer Zone

During the adsorption process, a large number of metal ions are concentrated on the surface of the adsorbent material, respectively at the interface between the two phases, with an accumulation process taking place. The adsorption of the metal ions on the adsorbent is realized with maximum efficiency if: (i) the transport of the metal ions to the surface of the immobile layer is adequate, transport that can be realized by diffusion or dispersion; (ii) the metal ion transfer occurs at the level of the adsorbent interface; (iii) intraparticle diffusion takes place by moving the solution ions into the adsorbent pores [

27].

The performances of the adsorption process in the fixed-bed column are highlighted by the study of the breakthrough curves [

28,

29], which means following the evolution of the ratio between the residual concentration of Au (III) and its initial concentration (C

_{rez}/C

_{0}) according to the volume of effluent passed through the column, for three distinct quantities of material, respectively 3, 5 and 10 g (

Figure 3).

Figure 4 shows the case of the column in which 3000 mL of wastewater, with an Au (III) concentration of 60 (mg L

^{−1}), were passed over 10 g of XAD7-AcG material. The mass transfer zone (MTZ) is the active surface of the XAD7-AcG material where the Au (III) adsorption takes place. The waste solution Au (III) passes over a new and unused material. At the top of the column, XAD7-AcG material adsorbs Au (III) ions as soon as they come into contact; the area is called the primary sorption zone (PSZ) being delimited between the residual concentration of Au (III) at time 1 (one), C

_{1}, and the residual concentration of Au (III) at time t, C

_{t}. Thus, the first part of the liquid that is collected is without Au (III) ions, which means in this area the residual concentration of Au (III) tends towards zero. As the volume of Au (III) solution passing through the column increases, an adsorption zone begins to be defined in which mass transfer (MTZ) occurs. In this area, the adsorption process is complete, the concentration of Au (III) ions varies from the initial concentration (60 mg L

^{−1}) to zero, the saturation of the adsorbent material being total. This adsorption area extends over the entire height of the column depending on the contact time. The residual concentration at a given time, C

_{rez}, is zero and therefore the ratio C

_{rez}/C

_{0} is zero. When the residual solution passes through the whole layer of the adsorbent material, reaching its lower part, the Au (III) ions can no longer be completely adsorbed due to the saturation of the material. This moment is called the breakpoint moment and the surface obtained corresponds to the breakpoint curve. After a while, the column is completely saturated or exhausted and the adsorption of Au (III) is no longer performed. In this case the C

_{rez}/C

_{0} ratio is 1 (one) [

30,

31,

32].

#### 3.2.2. Adsorption Models for Column Study

The important parameters for evaluating the efficiency of an adsorbent material used in a dynamic regime are: the flow of the effluent in the column, the height of the fixed layer, and the contact time [

1,

33].

In the adsorption column, phenomena of axial dispersion, external resistance of the film and resistance to intraparticle diffusion can appear. Thus, the mathematical correlation of axial dispersion, mass transfer and intraparticle diffusion is rendered by mathematical models. In order to determine the adsorption mechanism of Au (III) and to design the adsorption process in a dynamic regime it is necessary to know the evolution of the residual concentration of the effluent in time. In this case, four models can be used, namely the Adams–Bohart, Yoon–Nelson, Thomas and Clark models to analyze the breakthrough curves and for the prediction of dynamic nature of the column [

34].

The Bohart–Adams Model [

35] is used to describe the first part of the column breakpoint curve. The Bohart–Adams equation is linearly expressed as:

where: C

_{0}—is the influent concentration, (mg L

^{−1}); C

_{t}—is the effluent concentration, (mg L

^{−1}); t—is time, (min); k

_{BA}—is the kinetic constant of the Bohart-Adam model, (L mg

^{−1} min

^{−1}); F—is the linear velocity calculated by dividing the flow rate by the column section area, (cm min

^{−1}); Z—is the bed height of column, (cm); N

_{0}—is the saturation concentration, (mg L

^{−1}).

In

Figure 5, the dependence ln (C

_{t}/C

_{0}) = f(time) was plotted. The results show that with the increase of the material layer height, respectively with the increase of the amount of material, there is a decrease of the N

_{0} value, but also an increase of the k

_{BA} value. The obtained regression coefficients R

^{2} show that the model is not the most suitable to describe the mechanism of the dynamic adsorption process of Au (III) on the XAD7-AcG material [

36].

The Yoon–Nelson Model [

37] is generally adopted to describe the breakpoint curve. It is a model used especially for the single component system and does not require information about the adsorbent, such as type, physical properties or other characteristics.

The Yoon–Nelson equation is linearly expressed as:

where: C

_{t}—is the solution concentration at time t, (mg L

^{−1}); C

_{0}—is initial solution concentration, (mg L

^{−1}); k

_{YN}—is the rate constant, (min

^{−1}); τ—is the time required for 50% adsorbate breakthrough, (min).

The parameters τ and k

_{YN} can be obtained from the plot of the function ln [C

_{t}/(C

_{0} − C

_{t})] = f(time), (

Figure 6). It is observed that with the increase of the height of adsorbent layer the time required to breakthrough increases, but not in direct proportion with the adsorbent layer height. The experimental data show that the most efficient layer is the one in which 5 g of adsorbent material is used, because when the adsorbent mass doubles, τ does not become double. This fact may be due to the higher probability of preferential drainage channels occurring with the increasing amount of adsorbent material. The regression coefficient R

^{2} is closer to 1, but we cannot say that the adsorption process mechanism is described in the best way by this model [

34].

The Thomas model [

38] is the most commonly used model to describe the adsorption column performance and to establish breakthrough curves. It is frequently used to determine the adsorption capacity of the material. The Thomas equation is linearly expressed as:

where: C

_{0}—is the solution concentration in the influent, (mg L

^{−1}); C

_{t}—is the solution concentration at time t in the effluent, (mg L

^{−1}); k

_{Th}—is the Thomas rate constant, (L min

^{−1} mg

^{−1}); q

_{Th}—is the equilibrium compounds uptake per g of the resin, (mg g

^{−1}); m—is the mass of adsorbent resin, (g); Q—is he flow rate, (mL min

^{−1}).

From the plot of ln [C

_{0}/(C

_{t} − 1)] = f(t) are determined k

_{Th} and q

_{Th} (

Figure 7). From the results obtained, it can be seen that the constant k

_{Th} decreases as the adsorbent layer height in the column increases, due to the adsorption driving force given by the difference between the concentration of Au (III) adsorbed on the material and the Au (III) concentration in the solution [

39,

40,

41].

At the same time, it is observed that the regression coefficient R^{2} decreases, but the values are close to 1 and the adsorption capacity has about the same value, about 13 (mg g^{−1}).

Another model reported in the literature for the adsorption study in the column is the Clark model. The main assumption of this model is the use of a mass-transfer concept in combination with the Freundlich isotherm [

1,

42,

43].

The linearized expression of the Clark model is:

where: C

_{0}—is the solution concentration in the influent, (mg L

^{−1}); C

_{t}—is the solution concentration at time t in the effluent, (mg L

^{−1}); n—is the Freundlich constant determined experimentally in batch; r—is the Clark model constant, (min

^{−1}); A—is the Clark model constant.

A previous batch adsorption study showed that the Freundlich constant was n = 2.5 [

26]. This value was used in the Clark model to estimate the model parameters for the Au (III) adsorption. The value of r and A parameters were evaluated by the slope and intercept of the linearized equation of the Clark model (

Figure 8). The obtained correlation coefficients have good values (R

^{2} > 0.97) for all the amounts of material studied. The increase of amounts of material leads to decrease of parameter r value and to increase of value of A parameter.

The difference in the models was based on the set parameters, not on matching the experimental data. Column parameters of all tested models for the adsorption process in dynamic regime are presented in

Table 1. It can be observed that all the applied four models fitted to a satisfactory extent the variation in the amount of adsorbent, respectively, the variation of the material layer height. All models have shown good values of the correlation coefficient which suggests their validity in this investigation. Additionally, in the case of the Thomas model, the adsorption capacity has about the same value for all the amounts of adsorbent used, about 13 (mg g

^{−1}). Therefore, it can be assumed that this model best describes the mechanism of the adsorption process in a dynamic regime [

41].

Maximum adsorption capacities of previously studied adsorbents for Au (III) recovery are presented in

Table 2. The value of this parameter is comparable to those previously reported in the literature, even higher than other similar adsorbents.

The mechanism of Au (III) recovery by adsorption is that in most cases ionic species come into contact with the solid surface of the material with adsorbent properties. Adsorption is determined by Van der Waals forces, which are manifested between the material with adsorbent properties and the Au (III) ion. The predominant species at pH = 2 is AuCl_{4}^{−}. This species does not adsorb to the XAD7-AcG surface in a certain position, but moves freely, by translation, at the interface. In the aqueous solution there are three types of interaction considered competitive, namely: (i) the interaction between Au (III) ions and water; (ii) the interaction between Au (III) ions and the surface of the material with adsorbent properties and (iii) the interaction between water and the surface of the XAD7-AcG material. The efficiency of the physical adsorption process is determined by the strength of the metal ion–adsorbent surface interactions compared to the strength of the surface adsorbent–water interactions.