Designing, Modeling and Developing Scale Models for the Treatment of Water Contaminated with Cr (VI) through Bacterial Cellulose Biomass

Abstract

The present research presents a method for scaling up a continuous treatment system with bacterial cellulose biomass for the removal of contaminants on a large industrial scale from effluents loaded with chromium (VI). This consisted of a laboratory-scale modeling process of the chromium (VI) adsorption processes, which would provide the necessary parameters to build a system on an industrial scale. The research also involved designing, modeling and developing scale models for the treatment of water contaminated with chromium (VI) through bacterial cellulose biomass. The results of the model indicated the specific route for the construction of a treatment system on an industrial scale, with the experimental data adjusted to achieve this objective. The pilot scale prototype was built using 450 g of biomass, including elution processes, with the data obtained from the aforementioned processes. In general, the excellent efficiency of the two models at different scales, together with the excellent elution results, suggests that this prototype could be presented to polluting industries for the treatment of water from different industrial effluents, being an advanced biotechnology for the treatment of industrial wastewater.

1. Introduction

Caring for water is a fundamental and intrinsic aspect of human existence. The conservation and reuse of water are crucial for the sustainability of our planet and the well-being of future generations. To achieve this, it is essential to implement effective treatment systems for industrial effluents to prevent irreversible damage to water systems, such as wetlands and rivers. Pollutants that cause these impacts are various heavy metals, which cause serious damage to the ecosystem in both animals and aquatic plants due to their bioaccumulation [1,2,3,4]. Therefore, it is of the utmost importance to be able to treat water contaminated with heavy metals in order to mitigate their increasingly harmful impacts. An unconventional form of treatment is through chemical adsorption processes with biomass, in which cationic exchanges occur between their multiple functional groups (OH) and the metals present in the water [5,6,7,8]. This is a biochemical technology [9]. The use of biomass capable of chemiadsorbing metals in an efficient, economical and easy way is a method in development [10,11].

These types of technologies are viable due to the ease of producing or obtaining them, as waste products are usually developed productively in the laboratory. This is exemplified by bacterial cellulose. This polysaccharide is produced from basic sources such as red tea, sugar and kombucha, where kombucha is a symbiotic colony formed by several species of bacteria (Acetobacter xylinum, Acetobacter aceti, Acetobacter pasteurianus, and Gluconobacter bluconicum) and yeasts (Brettanomyces, Brettanomyces bruxellensis, and Brettanomyces intermedius) [12,13,14,15]. Bacterial cellulose meets the three criteria previously described: it is easy to produce, economical, and, above all, it is effective in removing heavy metals from water due to its multiple hydroxyl groups (OH), which are responsible for chemisorbing these contaminants [16,17,18].

A number of investigations have been conducted worldwide that have employed bacterial cellulose in the removal of heavy metals, including Cu (II) [19], Pb (II) [20], Cd (III) [21], and Cr (VI) [22]. However, the implementation of these investigations on a larger scale has not yet commenced, with the majority remaining within the laboratory setting. One method for scaling this type of treatment is through the use of external-layer (K_f) and internal-layer (K_s) adsorption models, which are employed in mathematical modeling processes with the objective of parameterizing designs for the implementation of these treatment systems [23,24,25,26,27,28]. K_f is employed as a design and scaling parameter, as it relates the particle density, porosity and ideal diameter, in conjunction with the behavior of its isotherm. K_s is then identified as a parameter that relates the adsorption capacities, speeds and ideal treatment flow rates. This is to facilitate the design process for large-scale treatment systems [29,30,31,32,33,34,35]. By obtaining the design variables through laboratory-scale experimental processes, similarities in the contact time of the adsorbent and the hydrodynamic characteristics between the adsorption systems could be obtained, thus enabling the development of pilot-scale processes. It is feasible to use data at laboratory scale to calculate parameters and model the efficiency of larger column systems using vertical column systems [36,37]. Pilot-scale trials have been conducted using biomasses to treat this type of water, including waste cellulose biomass from the aquatic plant Eichhornia crassipes [38,39,40,41]. But currently, a treatment system based on extraparticle and intraparticle diffusion models with bacterial cellulose biomass has not been built; due to this, it is necessary to develop treatment systems on a real industrial scale with bacterial cellulose material through adsorption models in order to identify the optimal parameters for their implementation. In light of this, this project was initiated with the objective of designing, modeling and developing scale models for the treatment of water contaminated with Cr (VI) through bacterial cellulose biomass.

2. Materials and Methods

2.1. Production of Bacterial Cellulose (BC)

The production of bacterial cellulose BC was undertaken using a tea and sugar culture medium obtained from the bioprocess laboratory at the University Foundation Los Libertadores. A BC film was produced and subsequently cut into 20 mm × 20 mm × 5 mm pieces. These pieces were then placed in 8 L glass containers containing 6 L of sterilized water, 30 g of sugar, and 20 g of red tea. To produce wet bacterial cellulose films, 600 mL of kombucha sediment (approximately 200 mL per sheet of kombucha tea, with a total of 3 sheets) and 1.5 g of commercial yeast of (saccharomyces cerevisiae) were added. Samples of pH and temperature were taken in situ for internal control. After three weeks, the films were dried at 70 °C for 48 h to remove moisture (see Figure 1). The films were ground until they had a diameter of 0.216 mm. This process yielded 70 g of bacterial cellulose in two weeks.

2.2. The Research Parameters

The research parameters included initial chromium concentrations of 50, 100, 200, 300, and 600 mg (chromium/L). Samples were taken at each time interval, and the residual chromium concentration was analyzed. Samples of 20 µL were obtained and subsequently taken to the centrifuge (KASAI MIKRO 200, Hettich, Föhrenstr Germany). In the present investigation, the tests were conducted under neutral pH conditions. Given that the pH was neutral, the adsorption process was favored in this type of biomass.

Aliquots of the reaction mixture were analyzed for residual chromium concentration using a (UV84 Hettich, Föhrenstr Germany). The tests were conducted in accordance with the standard methods outlined by the American Public Health Association (APHA) for standard tests, specifically the Standard Methods for the Examination of Water and Wastewater. All procedures for determining chromium (VI) levels in water were carried out in accordance with the APHA procedure outlined in method 3500-Cr. The study’s measurement uncertainty indicates that heavy element measurements, specifically Cr (VI), can be performed with an uncertainty level of approximately 3.95%.

Batch adsorption experiments were conducted in a 100mL glass vessel with constant stirring (IKA Ks 4000 shaker, Hettich, Föhrenstr Germany) at 20 °C and 150 rpm. Data were recorded every 20 min until 180 min had elapsed. The sample size was 20 µm. All experiments were performed in triplicate, and the final values were averaged.

The amount of Cr (VI) residue was estimated by means of the diphenylcarbazide method. For this, a phosphate-buffered solution was prepared by adjusting it to a pH equal to 2 with (H₃PO₄) to 90%. In an eppendorf tube, 200 µL of 0.5% diphenylcarbazide (P/V acetone) was added, along with 900 µL of phosphate buffer and 100 µL of residual sample. The absorbance was measured at 540 nm, whereupon it was transferred to an adsorption cell.

The adsorption capacity was determined with a suspension of 0.3 g of biomass in 100 mL of Cr (VI) solution for 140 min at 200 rpm, taking samples every 20 min, before determining the residue and discarding the sediment. All procedures were performed in duplicate.

An Evolution 300 spectrophotometer was used to monitor changes in absorbance during the determination of chromium in water and substrates. The procedures followed the standard methods for the examination of water and wastewater as outlined by the American Public Health Association (APHA).

2.3. Desorption–Adsorption

Following the Cr (VI) adsorption process, the chromium-loaded biomass underwent an elution process. After each biomass was subjected to the adsorption process, it was washed with distilled water. The elution process was then carried out in an Erlenmeyer flask at 25 °C for 24 h with constant stirring, using 20 mL of 20 mL of EDTA. The biomass was subsequently separated using a filter [31].

2.4. Column Design and Experiments

Two treatment systems were developed using recyclable PET bottles of varying sizes. The smaller-scale treatment system (laboratory) had a biomass diameter of 2.3 cm and a biomass length between the two compartments of 25 cm. Through experimental tests on this, a larger-scale system was developed. The larger-scale treatment system (pilot) was 5 cm in diameter with 75 cm of biomass height. Both systems had meshes that separated the compartments. The laboratory-scale system comprised 45 g of bacterial cellulose biomass, which reached a height of 25 cm and had a volume of 50 mL (with an area of 2.33 cm²).

Subsequently, the column was scaled up to a larger laboratory scale, in accordance with the criteria of geometric, kinematic and dynamic similarity [33]. The design parameters of the two treatment systems were employed to ascertain the relationship between the heights and diameters of the biomasses, with a view to determining the optimal relationship criterion between the biomasses. The relationship between the densities of the biomasses was established as the primary similarity criterion through the identification of the most suitable particle diameter for the removal and elution processes. A pilot-scale prototype was constructed using 450 g of biomass, including the elution processes, with the data obtained from the aforementioned processes.

The biomasses of the bacterial cellulose samples were measured at the following diameters: 0.212 mm, 0.315 mm and 0.318 mm.

The three diameters resulted in three different biomass density ratios on a laboratory scale. Subsequent evaluations were conducted, and the diffusion constants K_f were established. The intra-particle diffusion constant K_s was established, allowing for the establishment of the relationship between height and contact area for the treatment of 50 liters of wastewater contaminated with 1000 g/L of chromium.

In accordance with the remaining criteria, the physical properties of the fluid and the linear velocity (v, cm min⁻¹) were maintained at a constant value in both columns, thus ensuring the same mass transfer and hydrodynamic conditions.

The initial experiments were conducted on a laboratory scale with a flow rate of 20 mL/min. The biomass particle diameter was 0.212 mm, 0.315 mm and 0.318 mm, with the objective of evaluating three different types of biomass densities. The initial concentration of Cr (VI) was 500, 300, 200 and 100 mg/L. The sampling period was set at 15 min, and the experiments were conducted in a downward flow configuration, with three repetitions.

2.5. Development of Adsorption Models

The dynamics of the adsorption process are reflected in Equation (1)

$${\mathrm{m}}_A{\displaystyle \frac{\partial {\mathrm{q}}_A}{\partial \mathrm{t}}}=-\mathrm{V}\mathrm{l}{\displaystyle \frac{\partial \mathrm{c}}{\partial \mathrm{t}}}$$

(1)

The development of adsorption models will be discussed. In Equation (1), m_A is the mass of the adsorbent, which in our case is bacterial cellulose, and Vl is the objective parameter, which is the volume of water to be treated. The equation links the change in adsorption capacity q_A in relation to the time spent between the biomass and the contaminant.

This model establishes that in external conditions, the contaminant adsorption process is carried out, where it has a concentration gradient (δ). Therefore, as long as equilibrium conditions are not established, the concentration in the adsorbent particle will be less than that in the fluid to be treated. This gradient extends around the outer layer. To establish this dynamism, the mass transfer equation for external diffusion or film is used, which can be derived from Fick’s law.

$${\mathrm{N}}_F=\mathrm{D}\mathrm{l}{\displaystyle \frac{\partial \mathrm{C}}{\mathsf{\delta }}}$$

(2)

The flux, N_f, is given in g/(m²* s), while the diffusion coefficient in the contaminant phase, Dl, is expressed in m²/s. Integrating the linear gradient δ within the boundary layer leads to the following equation:

$${\mathrm{N}}_F=K_f\left(\mathrm{c}\mathrm{o}-\mathrm{c}\right)$$

(3)

The mass transfer coefficient of the film, K_f, is the amount of contaminant chemisorbed in the fluid that has already been treated at this moment, and that is adsorbed in the biomass per unit of time.

This parameter is dependent on the density of the adsorbent and its porosity. It directly relates to the contaminant and the chemiadsorption attraction between heavy metals and the functional groups of the biomass [29,30,31]. The relationship between functional groups and the metal to be removed from the water is directly proportional to the density of the particle. Therefore, a model must be designed to comply with national or international discharge regulations, or even reach 0.

$${\mathrm{n}}_F={\displaystyle \frac{{\mathrm{N}}_F}{\mathrm{A}\mathrm{s}}}$$

(4)

The total external surface of all adsorbent particles within the designed bioreactor is also the external area of the volume of biomass occupied in the treatment. This is calculated using the following equation:

$$\mathrm{A}\mathrm{s}={\displaystyle \frac{3\mathrm{V}\mathrm{a}(1-\mathsf{\epsilon })}{\mathrm{r}\mathrm{p}}}$$

(5)

where Va is the volume occupied by the biomass in the reactor. The total volume of the reactor (including spaces between pores and air) is denoted by ε. This is the relationship between the densities of the particle and the density of the biomass occupied in the reactor [32,33,34]. The radius of the biomass occupied in the reactor is denoted by rp. The relationship between the densities is a design parameter that is fundamental to the assembly and commissioning processes of adsorbent systems. The relationship between the volume of the liquid to be treated and the total volume of the reactor is:

$$\mathsf{\epsilon }={\displaystyle \frac{\mathrm{V}\mathrm{l}}{\mathrm{V}\mathrm{a}}}$$

(6)

The relationship between the volume of the reactor (Vl) and the volume of the adsorbent (Va) is given by the following equation:

$${\displaystyle \frac{\mathrm{V}\mathrm{a}}{\mathrm{V}\mathrm{r}}}=1-\mathsf{\epsilon }$$

(7)

The volume of the liquid and the volume of the adsorbent is given by the following equation:

$${\displaystyle \frac{\mathrm{V}\mathrm{a}}{\mathrm{V}\mathrm{l}}}={\displaystyle \frac{1-\mathsf{\epsilon }}{\mathsf{\epsilon }}}$$

(8)

$${\displaystyle \frac{\mathrm{V}\mathrm{a}}{\mathrm{V}\mathrm{l}}}=pp{\displaystyle \frac{1-\mathsf{\epsilon }}{\mathsf{\epsilon }}}$$

(9)

$$\epsilon =1-{\displaystyle \frac{pb}{pp}}$$

(10)

This equation will be fundamental due to the relationship between the general biomass and its small particles. The simplest particle will have a direct relationship with the contaminant coupled with the biomass, whereby the more space there is between particles, the better it will be for the treatment. pp is determined by:

$$\rho p={\displaystyle \frac{mp}{Vp}}$$

(11)

The mass of the microparticle (mp) is its weight, and the volume of the microparticle vp is obtained by:

$$Vp={\displaystyle \frac{4\pi r^3}{3}}$$

(12)

The radius of the tiny particle (r) is a dependent variable in this research. Its diameter is obtained in meshes and classified by size. Returning to Equations (1) and (4), the relationship and differences that exist in the time of the concentrations is given by:

$${\displaystyle \frac{\partial \mathrm{C}}{\partial \mathrm{t}}}={\mathrm{k}}_F\ast {\mathrm{A}}_M{\displaystyle \frac{{\mathrm{M}}_A}{vL}}(c-cs)$$

(13)

where A_M is the total surface area related to the mass of adsorbent available in the reactor (A_M = As/mA); replacing Equations (5) and (10) in Equation (13) leaves the design equation unchanged:

$${\displaystyle \frac{\partial \mathrm{C}}{(c-cs)}}={\mathrm{k}}_F\ast {\mathrm{A}}_M{\displaystyle \frac{{\mathrm{M}}_A}{vL}}\partial \mathrm{t}$$

(14)

Integrating Equation (14),

$$Ln{\displaystyle \frac{\mathrm{C}\mathrm{o}}{\left(cs\right)}}=-{\mathrm{k}}_F\ast {\mathrm{A}}_M{\displaystyle \frac{{\mathrm{M}}_A}{vL}}\mathrm{t}$$

(15)

The natural logarithm of the initial concentration and final concentration of Cr (VI) (designing the final parameter of discharges), together with the surface area (As) of contact, the volume of treated water, and the objective parameter (Vl), the diffusion constant (K_f) of Cr (VI) in the biomass, are found by graphing this term. This parameter summarizes the design behavior of the following:

Surface area that the biomass will occupy in the reactor;

Mass necessary to meet the objective of treating the contaminant;

The target treatment volume;

The target concentration (mg/l) of the contaminant.

These are fundamental parameters when designing and implementing water treatment systems contaminated with heavy metals. The diffusion constant (K_f) summarizes these parameters. The other part of Equation (1), which takes into account intraparticle diffusion, is given by:

$${\displaystyle \frac{\partial \mathrm{q}}{\partial \mathrm{t}}}={\mathrm{K}}_{s}pp\left(\mathrm{q}\mathrm{s}-\mathrm{q}\right)$$

(16)

The density of the particle plays a fundamental role in the change of adsorption capacities over time, and the parameter K_s is obtained. This constant summarizes, like K_f, the design parameters. In order to establish the behavior of the adsorption capacity in saturation, the present investigation employs the isotherm equations, which are the Langmuir and Freundlich isotherms. Equations (15) and (16) will be grouped.

$$\mathrm{k}\mathrm{s}(\mathrm{q}\mathrm{s}-\mathrm{q})=-{\mathrm{k}}_F\ast {\mathrm{A}}_M{\displaystyle \frac{{\mathrm{M}}_A}{vL}}$$

(17)

In numerous studies of heavy metal adsorption, the principal focus has been the adsorption isotherm and its variables. Nonetheless, fixed bed or packed bed adsorbers’ design necessitates both the adsorption isotherm and the complete adsorption pace.

If it is the Freundlich Equation (18):

$$\mathrm{q}\mathrm{s}=\mathrm{K}{\left(\mathrm{C}\mathrm{s}\right)}^{\mathrm{n}}$$

(18)

The resulting expression with Freundlich modeling is Equation (19).

$$\mathrm{k}\mathrm{s}(\mathrm{K}{\left(\mathrm{C}\mathrm{s}\right)}^{\mathrm{n}}-\mathrm{q})=-{\mathrm{k}}_F\ast {\mathrm{A}}_M{\displaystyle \frac{{\mathrm{M}}_A}{vL}}\mathrm{t}$$

(19)

Equation (19) does not obtain the intraparticle diffusion constant when adjusted to this Freundlich isotherm

$${\mathrm{k}}_S={\displaystyle \frac{3{\mathrm{M}}_A{\mathrm{k}}_F{\mathrm{A}}_M}{\rho p\left(\mathrm{K}{Cs}^n-q\right)VL}}$$

(20)

But if it is the Langmuir Equation (21):

$$q_S={\displaystyle \frac{\mathrm{q}\mathrm{m}\mathrm{B}\mathrm{C}\mathrm{s}}{1+\mathrm{B}\mathrm{C}\mathrm{s}}}$$

(21)

where B is the Langmuir parameter, if there is a fit to the Langmuir isotherm, this equation must be used to determine the K_s. If it is the Langmuir Equation (22):

$${\displaystyle \frac{\mathrm{K}\mathrm{s}\mathrm{q}\mathrm{m}\mathrm{B}\mathrm{C}\mathrm{s}}{1+\mathrm{B}\mathrm{C}\mathrm{s}}}-\mathrm{K}\mathrm{s}\mathrm{q}={\mathrm{A}}_M{\displaystyle \frac{{\mathrm{M}}_A}{vL}}\mathrm{t}$$

(22)

The resulting expression with Langmuir modeling is Equation (23).

$${\mathrm{k}}_S={\displaystyle \frac{{3\mathrm{M}}_A{\mathrm{k}}_F\mathrm{B}{\mathrm{C}\mathrm{s}}^{2 }+{3\mathrm{M}}_A{\mathrm{k}}_F\mathrm{C}\mathrm{s}-{\mathrm{k}}_F\mathrm{C}\mathrm{s}+{\mathrm{k}}_F{\mathrm{C}\mathrm{s}}^{ }+{3\mathrm{M}}_A{\mathrm{k}}_F\mathrm{C}\mathrm{s}}{(\mathrm{p}\mathrm{p}\mathrm{q}\mathrm{m}\mathrm{B}\mathrm{C}\mathrm{s}-\mathrm{q}\mathrm{B}\mathrm{C}\mathrm{s}-\mathrm{q})vL}}$$

(23)

The intraparticle diffusion constant will be established through Equations (20) and (23), which will group design parameters such as:

• Biomass to use.
• Target volume.
• Target concentrations.
• Particle density.
• Maximum adsorption capacities.
• Constant K_f.

In order to ensure adequate staging and to obtain a full-scale treatment system, it is necessary to resolve various design parameters. These include determining the mass required to achieve a specific treatment objective or establishing the necessary treatment volume and adsorption capacity, taking into account all relevant variables. In the present investigation, Equation (24) was derived, which is ideal for establishing the biomass required for the effective treatment of wastewater contaminated with heavy metals.

$${\mathrm{M}}_A={\displaystyle \frac{(\mathrm{p}\mathrm{p}\mathrm{q}\mathrm{m}\mathrm{B}\mathrm{C}\mathrm{s}-\mathrm{q}\mathrm{B}\mathrm{C}\mathrm{s}-\mathrm{q})vL{\mathrm{k}}_S}{3({\mathrm{k}}_F\mathrm{B}{\mathrm{C}\mathrm{s}}^{2 }+{\mathrm{k}}_F\mathrm{C}\mathrm{s})}}$$

(24)

3. Results

3.1. Result of Removal of Cr (VI)

Figure 2 illustrates the removal processes by the bacterial cellulose biomass with three different density models. The diameters represented each of the biomasses, which were BC (1): 0.212 mm, BC (2): 0.315 mm and BC (3): 0.318 mm. All experiments were performed in triplicate, with the average product and its margin of error shown in Figure 2.

All biomasses demonstrated excellent removal of Cr (VI) ions, reaching initial equilibrium levels. All achieved removal times above 450 min of treatment. In treatment processes at this laboratory scale, similar results were recorded in [42,43,44,45,46,47].

The BC(1) biomass was ideal, achieving better treatment intensity due to a breaking point of around 550 min. It was able to treat 5 L of wastewater with Cr (VI) ions. The biomass with BC (2) was able to treat 4.7 L of water with a break point of 500 min, while BC (3) spent approximately 435 min treating 4.65 L of water. The 0.0212 particle demonstrated superior performance, likely due to its intimate interaction with the Cr (VI) contaminant. This particle diameter has been recorded in some investigations as 10 microns.

Table 1. Results of the analysis of the relationship between densities.

Diameters	M (g)	Volume Mass (Vb)	Density Mas (pb)	Ma Particle (mp)	Volume Particle (Vp)	Density Particle (pp)	$\mathsf{\epsilon }$
BC (1)	50	80	0.62	0.01	0.005	0.59	0.68
BC (2)	50	85	0.58	0.019	0.044	0.43	0.33
BC (3)	50	90	0.55	0.028	0.039	0.71	0.23

presents the final results of the analysis of the relationship between densities.

Equations (3)–(12) were used to obtain the relationships between densities (ε), between the biomass in general and the tiny particle. This relationship is linked to the removals recorded in Figure 1, where the BC (1) biomass obtained the best treatment records. This is due to the compactness of the biomass density in general and the particle diameter, which is ideal for the direct relationship with the contaminant Cr (VI) [48,49].

The next step was to implement dynamic modeling at the laboratory scale. The three models were implemented using Equation (15) to establish the degree of incidence and design criteria for the different variables and design parameters. The initial and final concentrations of Cr (VI) were graphed with their natural logarithms, relating the rupture time and volume of treated water. For the case of BC (1), this was 5000 mL of water.

Table 2. Summary of parameters given.

Biomass	As cm2	Kf (cm/min)	Volume Goal (L)	Time Break (min)
BC (1)	61.5	0.99	5	550
BC (2)	61.5	0.895	4.7	500
BC (3)	61.5	0.889	4.65	435

shows a summary of the variables in Equation (15). For illustrative purposes, let us consider the treatment with BC (1) as an example. In this case, Equation (15) becomes:

$$Ln{\displaystyle \frac{1000}{\left(1\right)}}=-k_F\ast {\displaystyle \frac{61.5 \ast 50}{50 \ast 5000}}550$$

where this equation can be solved to establish K_f or graph and through this the influences between the different variables can be observed, as shown in Figure 3.

The kinetic curve can be plotted according to Equation (3) in order to estimate K_f (Figure 3) in the which has the representative red color due to the breaking point. The two methods for establishing K_f were employed to complete the various processes, which are outlined in Table 2.

K_f is an adsorption rate that correlates with ideal diameters and densities in the removal processes of heavy metals in water. The K_f value for BC (1) was 0.99 cm/min, indicating that this biomass has an adsorption capacity for Cr (VI) ions. However, the K_f values for BC (2) and BC (3) were lower, and therefore, these biomass samples were not considered important in the design of treatment systems on a larger scale. In treatment systems employing plant biomass, the K_f for the Cr (VI) treatment was found to be less than 0.88 [50], indicating that the biomass of bacterial cellulose is greater under different circumstances of its particle sizes. K_f is the design and scaling parameter because it relates the particle density, porosity and ideal diameter, coupled with the behavior of its isotherm. Consequently, the K_s will be found to complement the scaling process in the design of a large-scale treatment system.

Subsequently, analysis of the removals must be carried out to establish how the isotherms in which the removals of Cr (VI) by the different biomasses occur behave. Figure 4 depicts the isotherms of each of the biomasses.

Figure 4 illustrates the relationship between bacterial cellulose and adsorption capacities (q_e) at different equilibrium concentrations (Ce) obtained in each of the initial concentrations of 400, 200, 100 and 50 mg/L of Cr (VI). It shows that adsorption capacities of more than 55 mg/g per biomass can be achieved, with the BC biomass (1) reporting the best results, at around 65 mg/g. The active sites (OH) in the bacterial cellulose biomass are filled by Cr (VI) ions, which is characteristic of this biomass. It is also homogenous in adsorption processes and adjusts to the Langmuir isotherm. It should be noted that this isotherm assumes that all active sites on the surface are energetically homogeneous. A representative fit of 0.99 R² can be seen in

Table 3. Isotherm representatives.

	Isotherm	Constant	R2
BC(1)	Langmuir	B = 0.03; qm; 65	0.99
	Freundlich	K = 0.17	0.91
	Isoterm	Constant	R2
BC (2)	Langmuir	B = 0.028; qm; 60	0.99
	Freundlich	K = 0.11	0.92
	Isoterm	Constant	R2
BC(3)	Langmuir	B = 0.027; qm; 58	0.98
	Freundlich	K = 0.18	0.96

.

For all biomasses, Langmuir isotherms were obtained due to the homogeneity of active sites in these; therefore, the models of Equation (23) were used to obtain the intra-particle design variable K_s, as shown in

Table 4. Isotherm equation representatives.

Biomass	Ks (1/s)	Isotherm Equation	Qm	pp
BC (1)	0.018	Langmuir	65	2
BC (2)	0.020	Langmuir	60	1.5
BC (3)	0.022	Langmuir	55	1.4

. Similarly to the variable K_f, the constant K_s plays a pivotal role in the design and scaling of the treatment system. This is because the intraparticle diffusion constant contains intertwined design parameters such as density variables and the influence of the contaminant.

To ascertain the impact of the EDTA eluent on recycling ratios and biomasses, a series of elution and recycling cycles were conducted, as illustrated in Figure 5. The design variables were established between densities after each elution process and subsequent reuse. Figure 5 illustrates the characteristics and elution processes of each biomass, along with their respective removal percentages, with its margin of error product of the arithmetic average.

It is evident that each elution process yielded a notable performance, with similar yields observed across the different cycles. All biomasses were tested with up to four elution and reuse cycles, from (4) cycles of Elutions/Recycling. The biomass with the best performance was BC (1), although the other biomasses also demonstrated promising results in the elution and reuse processes. The results of the investigations by [51,52] indicate that bacterial cellulose performs optimally with the EDTA eluent. In each adsorption and desorption process of each biomass, the same analyses were carried out to determine the influence of the EDTA eluent on the biomass, along with all the parameters previously detailed.

Table 5. The elutions and behavior of all biomasses.

Biomass	Elutions	Kf (cm/min)	Volume Goal (L)	Time Break (min)	Ks (1/s)	Isotherm Equation	Qm
BC (1)	1	0.98	4.5	500	0.017	Langmuir	58
	2	0.98	4.4	450	0.018	Langmuir	55
	3	0.96	4.4	440	0.019	Langmuir	52
	4	0.88	4.0	410	0.020	Langmuir	50
BC (2)	1	0.90	4.1	458	0.020	Langmuir	56
	2	0.90	4.0	450	0.021	Langmuir	54
	3	0.89	3.5	410	0.022	Langmuir	50
	4	0.80	3.0	390	0.022	Freundlich	44
BC (3)	1	0.90	4.1	450	0.022	Langmuir	55
	2	0.88	3.8	440	0.023	Langmuir	50
	3	0.88	3.5	380	0.023	Langmuir	45
	4	0.82	3.1	360	0.024	Freundlich	40

shows the elutions and the behavior of all biomasses.

The elutions did not compromise the behavior of their isotherms, but as the elutions were carried out, they approached the Freundlich isotherms and moved away from the Langmuir isotherm. This was due to the loss of adsorption capacities, in turn due to the gradual wear on the bacterial cellulose biomass caused by the same eluent. The K_s value was established using the equation developed by Freundlich.

3.2. Redesign of Process Treatment

The redesign conditions were established by considering the similarity relationship between the contact area, biomass and volume of water transferred, as well as the elution processes. The best conditions of K_f and K_s were then established; Figure 6 shows the scaling of both process.

In the Figure 6, in the which observed the characteristics of the biomass of bacterial cellulose, with chemical bonds in the adsorption of Cr (VI) [16]. The biomass to be treated, which was around 50 L of water, was calculated using Equation (15), which solved the mass of the equation. This was undertaken because we had K_f and K_s.

The values of K_f and K_s in the pilot-scale prototype were found to be of a similar magnitude to those found at the laboratory scale, due to similar conditions. The treatment was found to be very similar, with an inlet concentration of 500 mg/L and a flow rate that was the same.

The extension was carried out using the representative models of K_f and K_s, due to similar conditions in the treatment and relationship between contaminant and bacterial cellulose biomass. This proved to be an appropriate procedure in the treatment of 500 mg/L of Cr (VI) and around 50 l of water. Equation (23) was used, where all the biomass information was available and the

Table 6. Parameters of scaling.

Diameters	M (g)	Volume Mass (Vb)	Density Mass (pb)	Caudal mL/min	Volume Treat (L)	$\mathbf{E}\mathbf{l}\mathbf{u}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{s}$
BC (1)	50	80	0.62	20	5	4
Scaling	450	500	0.58	200	550	3

, shows the parameters of scaling.

$${\mathrm{M}}_A={\displaystyle \frac{(\mathrm{p}\mathrm{p}\mathrm{q}\mathrm{m}\mathrm{B}\mathrm{C}\mathrm{s}-\mathrm{q}\mathrm{B}\mathrm{C}\mathrm{s}-\mathrm{q})vL{\mathrm{k}}_S}{3({\mathrm{k}}_F\mathrm{B}{\mathrm{C}\mathrm{s}}^{2 }+{\mathrm{k}}_F\mathrm{C}\mathrm{s})}}$$

In the Table 6, is observed that, a pilot-scale prototype was constructed using 450 g of biomass, including the elution processes, with the data obtained from the aforementioned processes. The flow rate used was 200. Figure 7 shows the process of treatment.

The device created in this research carried out an effective treatment of 450 L of water contaminated with 1000 mg/L of Cr (VI) with four elution cycles under the same conditions as in the laboratory-scale treatment. The pilot-scale prototype was constructed using 450 g of biomass, including the elution processes, with the data obtained from the aforementioned processes.

4. Conclusions

The production of bacterial cellulose in a laboratory setting represents an economical and straightforward technique for the development of treatment systems for industrial wastewater contaminated with heavy metals. The high level of adsorption of this material, with a particle diameter of 0.212 mm, was confirmed due to the high level of contact between the contaminant and this biomass. Based on this information and mathematical modeling of extra-particle and intra-particle diffusion, together with representative isotherms, a treatment system was developed at a scale appropriate for the treatment of an effluent from a company that contaminates with Cr (VI). The results of the model indicated the specific route for the construction of a treatment system at an industrial scale, with the experimental data adjusted to achieve this objective. A pilot-scale prototype was constructed utilizing 450 g of biomass, incorporating elution processes and incorporating the data obtained from the aforementioned processes. The excellent efficiency of the two models at different scales, together with the excellent elution results where the design variables were implemented in the EDTA elution process, providing more complete adsorption capabilities, suggests that this prototype could be presented to polluting industries for the treatment of waters coming from different industrial effluents. This represents an advanced biotechnology for the treatment of industrial wastewater. It is anticipated that bacterial cellulose biomass will be applicable for the removal of diverse effluents contaminated with heavy metals and implemented in wastewater-polluting industries, representing a transformative model of advanced biotechnology in the treatment of industrial wastewater.