1 Introduction

Expanded bed adsorption (EBA) is a chromatographic technique used for the purification and separation of biomolecules such as proteins, enzymes, and antibodies [1]. EBA operates on the principle of fluidized bed technology, where the column is filled with porous beads that are fluidized by the upward flow of the feedstock. The advantages of EBA include higher productivity, reduced processing time, and lower operating costs compared to traditional chromatography techniques [2]. Nano adsorbents improve EBA performance by offering higher binding capacity, faster processing, better fluidization, and reduced fouling, making them ideal for large-scale bioseparations. Their tunable properties further enhance selectivity and efficiency in purifying biomolecules from complex mixtures [3]. Overall, EBA is a valuable tool in the biopharmaceutical industry for the purification of biomolecules, offering a more streamlined and efficient approach to chromatography [4].

Nano adsorbents are composed of nanoparticles, which have a high surface area and can effectively capture and remove contaminants from the environment [5]. Their small size and high reactivity make them ideal for use in advanced technologies for environmental remediation and resource recovery [6]. Nano adsorbents enhance EBA by improving bed stability (reducing channelling), accelerating binding kinetics (short diffusion paths), and increasing capacity (high surface area). They also prevent clogging (no interstitial blockage) and boost selectivity (tunable surface chemistry) while cutting costs (higher efficiency, reusability). Overall, they solve EBA’s key limitations—slow processing, fouling, and poor scalability—making them ideal for purifying biomolecules from crude feeds [7].

Most scientists today focus on developing high-density adsorbents and the outer shell. The inner core of these adsorbents is very high-density porous granules that are usually of good shape and excellent chemical resistance [8]. The outer layer of these adsorbents is hydrophilic polymeric materials with a Nano-porous structure such as polysaccharides. The remarkable advantage of this type of adsorbent is that the mass transfer path is shortened due to the addition of large nuclei, and therefore mass transfer occurs only in the outer layer [9]. Core-shell adsorbents enhance EBA by combining a dense core (e.g., silica, magnetic particles) for stable fluidization and a porous shell (e.g., polymer, hydrogel) for high binding capacity, improving both bed stability and adsorption efficiency. The shell’s tailored surface chemistry (e.g., ligands, ion-exchange groups) boosts selectivity while minimizing non-specific binding, crucial for purifying targets from complex feeds like cell lysates. Additionally, the core-shell design reduces fouling and enables reusability, lowering costs in large-scale bioprocessing [10].

Agarose is a type of polysaccharide that is commonly used in the production of adsorbent materials. The agarose coating can provide a protective layer for the adsorbent material, preventing it from degradation and increasing its stability and longevity. This can be particularly useful in harsh or demanding environmental conditions where the adsorbent material may be exposed to high temperatures, or corrosive substances [11]. Additionally, the agarose coating can also enhance the adsorption properties of the material, by providing additional binding sites for the target substances and improving the overall efficiency of the adsorption process. This can result in higher adsorption capacity and faster kinetics, making the two-layer adsorbent more effective in capturing and removing pollutants from liquids or gases [12].

Agarose-based resins are widely used in chromatography for the separation and protein purification [13]. The two-layer adsorbent with agarose coating can enhance the binding capacity and selectivity for proteins, leading to higher purity and yield in the purification process [14]. Additionally, the stability and longevity of the agarose coating can ensure consistent performance over multiple purification cycles, making it a reliable choice for protein purification applications [15].

Therefore, it is essential to design new methods to achieve higher efficiency, speed up, and reduce protein purification costs. Nanoparticle-based methods for separating cells and proteins are one of the most widely used purification methods that are also widely published. Temporarily, this study uses a simple and inexpensive way to prepare a new structure of Nano-adsorbents to purify a protein [16]. To achieve higher purity for the protein concerned, the initial and final purification steps of the protein are usually performed by liquid chromatography. These techniques have been developed to separate protein products from each other due to their differences in molecular properties [17].

Chromatography is a widely used technique in protein purification due to its ability to separate and purify proteins based on their specific interactions with the solid support material [18]. By adjusting the conditions such as pH, salt concentration, and temperature, the bound proteins can be selectively eluted from the column, allowing for the isolation and purification of the target protein [19].

Most recently, k-carrageenan-nickel adsorbent obtained by oil-in-oil emulsion method with a particle size of 60–320 nm and examined the morphology, structure, and porosity of the prepared matrix on the EBA [20]. In addition, the zirconia-agarose adsorbent was prepared for the synthesis of adsorption by albumin. The adsorbent synthesized by the RO107 ligand was immobilized and demonstrated excellent stability at pH 4.5. Investigation of the kinetics between albumin and ligand particles revealed that equilibrium was reached before one hour. The adsorption isotherm of the Langmuir model showed an adsorption capacity of about 222 mg/ml adsorbent [21]. In addition, they synthesized Fe3O4/SiO2 magnetic nanoparticles with agarose coating modified with Sodium dodecyl Sulphur for adsorption of cationic material [22].

The final geometric shape of the crust particles is sporadically dependent on an aqueous agarose solution containing nickel coated in the continuous phase. For the dispersion of an immiscible aqueous phase in a continuous turbulent period, the resulting drop size is reasonably well correlated with the maximum local specific energy dissipation rate (εT)max, based on Kolmogorov’s theory of local, isotropic turbulence. Kolmogorov’s method defines how energy transfers from larger to smaller eddies, how much power is controlled by eddies of a given size, and how much energy is degenerated by each measure.

In this study, a new composite of nickel coated with agarose (NAC) adsorbent was synthesized, and Lactoferrin (Lf) was used as a nanoparticle to investigate the adsorption capacity. Lf is a multifunctional protein found in milk and other secretions in the body. It is known for its ability to bind and transport iron, as well as its antimicrobial and immunomodulatory properties. In the field of biotechnology and pharmaceuticals, Lf has gained attention for its potential therapeutic applications. Lf present in plasma is predominantly derived from secondary granules of circulating neutrophils. Human milk is particularly rich in Lf, the concentration ranging from < 1 to 16 g/l in colostrum and about 1 g/l in mature milk. In bovine colostrum, the Lf content is, on average, 1.5 g/l, ranging from 0.2 to 5 g/l and decreases to about 0.1 g/l in mature milk. Very high concentrations (up to 50 g/l) of Lf are found in secretions of non-lactating human and bovine mammary glands.

The purification of Lf from its natural sources, such as milk or whey, often involves chromatography techniques, including affinity chromatography using specific ligands that can selectively bind to Lf. To stabilize the chemical structure, the NAC adsorbent was immobilized by the Cibacron Blue (CB) dye ligand, and 2@NAC-CB particles were prepared.

1.1 Modelling approach

Kolmogorov’s theory of isotropic turbulence postulated that in a fully turbulent flow, the kinetic energy cascades from large to small eddies in two flow regimes. The isotropic flow regime spans inertial and viscous ranges [23]. According to the Kolmogorov theory, the turbulent deformation stress is proportional to the root mean square (rms) of velocity fluctuation across distanced of the fluid eddy size equal to drop size:

$$\:{\uptau\:}\propto\:{{\uprho\:}}_{\text{c}}\overline{{{\text{u}}^{2}\left(\text{d}\right)}}$$
(1)

and since

$$\overline{{{\text{u}}^{2}\left(\text{d}\right)}}={\text{C}}_{1}{{{\epsilon}}_{\text{T}}}^{-2/3}{\text{d}}^{2/3}$$
(2)

where C1 is a constant, and for stirred tanks is reported to be 1.88.

Hinze predicted the maximum surviving droplet size dmax in the inertial flow regime as in Eq. (3):

$$\:{\text{d}}_{\text{m}\text{a}\text{x}}={\text{C}}_{1}{({\upsigma\:}/{{\uprho\:}}_{\text{c}})}^{0.6}{{{\epsilon}}_{\text{T}}}^{-0.4}$$
(3)

where \(\:{\upsigma\:}\) is the equilibrium interfacial tension, \(\:{{\uprho\:}}_{\text{c}}\) is the density of the continuous phase, and \(\:{{\epsilon}}_{\text{T}}\) is the average energy dissipation per unit mass of fluid (the same as the change in kinetic energy per unit time) in the stirred tank. The dmax was estimated to result from a balance of turbulent, fluctuating disruptive forces and cohesive surface forces. Assumptions for this theory were that the volume fraction of the dispersed phase was meager (< 0.01), the viscosity of the dispersed phase was close to that of the continuous period, and droplet breakage was the dominant process. The surface force was considered the significant counteractive force for turbulent deformation stresses of the fluid fluctuating velocities about the diameter of the droplet.

Shinnar and Church, using similar assumptions and Kolmogorov’s theory, rearranged Eq. (3) to include the Weber number and the Sauter mean diameter d32 for stirred tank geometry as in Eqs. (4) and (5). Both models were only valid with the assumption of local isotropy in fully turbulent flow at high Reynold’s number, Re > 10000:

$$\:\frac{{\text{d}}_{\text{m}\text{a}\text{x}}}{\text{D}}=\text{C}{\left(\text{W}\text{e}\right)}^{-0.6}$$
(4)
$$\:\frac{{\text{d}}_{32}}{\text{D}}={\text{C}}_{2}{\left(\frac{{{\uprho\:}}_{\text{c}}{\text{N}}^{2}{\text{D}}^{3}}{{\upsigma\:}}\right)}^{-0.6}={\text{C}}_{2}{\left(\text{W}\text{e}\right)}^{-0.6}$$
(5)

where C is a constant with values between 0.126 and 0.15 or 0.125, Eq. (5) is referred to as the Kolmogorov’s model, where C2 is a constant equal to 0.5, We is the Weber number, N is the impeller speed in revolutions per second, and D is the impeller diameter.

Generally, the calculation of the power of the stirrers was expressed in horsepower, and, in general, for the majority of reservoirs that fluctuated in the flow of fluid, the power can be calculated from the following equation:

$$\:{{\epsilon}}=\text{N}\text{p}\:{\uprho\:}\:{\text{N}}^{3}\:{\text{D}}^{5}$$
(6)

where\(\:\:\text{N}\text{p}\) is the power number that relates imposed forces to inertial forces and is a function of the Reynolds number and the Froude number. Established on data composed using a bladed, pitched blade turbo stirrer, Mak obtained the following results:

For

$$\:\frac{\text{D}}{\text{T}}=3\:\:\:\text{N}\text{p}=1.62{(\text{C}/\text{D})}^{-0.22}$$
$$2=1{.24(\text{C}/\text{D})}^{-0.4}$$
(7)
$$1.7=1{(\text{C}/\text{D})}^{-0.44}$$

where D is the stirrer diameter (m), where T is the tank diameter (m), and C is the distance between the stirrer and the bottom of the vessel (m). By comparing and summing up the above equations, we can show the following:

$$\:{\text{d}}_{\text{m}\text{a}\text{x}}\propto\:{{\upsigma\:}}^{0.6}{{{\epsilon}}_{\text{T}}}^{-0.4}\propto\:{{\upsigma\:}}^{0.6}{\text{N}}^{-1.2}$$
(8)

Which controls the interfacial tension. If the viscosity of the dispersed phase, µd, is so high that it is the dominant parameter resisting breakage, then:

$$\:{\text{d}}_{\text{m}\text{a}\text{x}}\propto\:{{\upmu\:}}^{0.75}{{{\epsilon}}_{\text{T}}}^{-0.25}\propto\:{{\upmu\:}}^{0.75}{\text{N}}^{-0.75}$$
(9)

With increasing concentration, coalescence generally occurs, and the equilibrium drop size increases. Because the measurement of dmax is difficult, it is usual to assume that dmax is proportional to some mean drop size, typically d32 (the Sauter mean diameter) and d43 (the De Brouckere mean diameter).

The experimental data were fitted to the power-law module:

$$\:{\upmu\:}=\text{K}\:{\text{y}{\prime\:}}^{\text{n}-1}$$
(10)

where µ is the apparent dynamic viscosity (Pas), y’ is the shear rate (s−1), K is the consistency index (Pasn), and n is the flow behavior index. The K values for 2% and 4% (w/v) agarose solutions were 0.035 and 0.29 Pasn, respectively, and both are weakly non-Newtonian (n ~ 0.94). It can, therefore, be assumed that, within the shear rate range associated with the agitation used during slurry preparation and emulsification procedures, the viscosities remained constant, i.e., the solutions could be considered Newtonian.

The Lagrangian pseudo-first-order equation is a commonly used kinetic model for data analysis of absorption systems [24]. This dynamic model holds for the physical adsorption of soluble material onto the adsorbent and indicates a weak interaction between the adsorbent and the adsorbent (van der Waals forces). The following equation defines this model.

$$\:\frac{dq}{\text{d}\text{t}}={k}_{1}({q}_{e}-{q}_{t})\:$$
(11)

where t is time (min), and k1 is the apparent rate constant (in min) of the pseudo-first-order equation. Integrating this equation gives:

$$\:{q}_{t}={q}_{e}(1-{e}^{-{k}_{1}t})\:$$
(12)

Many adsorption systems follow the pseudo-quadratic kinetics equation. This model is based on the assumption of adsorption coupled with the chemical reaction. In other words, it involves the incorporation of the Valencian forces. The sharing or exchange of electrons creates these forces. The following relation describes the pseudo-quadratic kinetics equation:

$$\:\frac{dq}{\text{d}\text{t}}={k}_{2}{({q}_{e}-{q}_{t})}^{2}\:$$
(13)

Integrating the above equation gives the following equation:

$$\:{q}_{t}=\frac{{k}_{2}{{q}_{e}}^{2}}{1+{k}_{2}{q}_{e}t}t$$
(14)

where k2 is the apparent rate constant (in min) of the quasi-quadratic equation, a complex function of concentration.

In addition to the coefficient of correlation, statistical parameters of mean relative error and root mean square error were also determined to determine the fitting quality of pseudo-first-order and pseudo-second-order kinetic models. These parameters are defined in terms of relationships as follows:

$$\:MRE=\frac{1}{N}\sum_{i=1}^{N}\frac{\left|{{q}_{i}}^{exp}-{{q}_{i}}^{cal}\right|}{{{q}_{i}}^{exp}}$$
(15)
$$\:RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}({{q}_{i}}^{exp}-{{q}_{i}}^{cal})}$$
(16)

In this equation, MRE is the relative mean error, RMSE is the root mean square error, N is the number of kinetic data, qiexp, is the first experimental adsorption capacity, and qical, is the first adsorption capacity calculated by the kinetic model. It should be noted that the lower these errors, the better the model fits the experimental data.

Accordingly, the correlation coefficient parameter (R2) can be calculated from the following relation:

$$\:{R}^{2}=\frac{\sum_{j=1}^{N}{({y}_{cal}-{\stackrel{-}{y}}_{exp})}^{2}}{\sum_{j=1}^{N}{({y}_{exp}-{\stackrel{-}{y}}_{exp})}^{2}}$$
(17)

In the above relation, yexp is the calculated value of the experiment, ̅yexp is the average value calculated, and ycal is the computed value in the model. It has been proved that the closer the correlation coefficient parameter to the amount of one, the model has more exceptional ability to fit the equation in question.

In this study, when the output concentration reaches 10% of the input concentration, it is considered the time of the break, and when the output concentration comes to 90% of the input concentration, it is regarded as the time of fatigue. Since the area beneath the crack curve represents the amount of unabsorbed protein coming out of the column, the dynamic adsorption capacity at the crack is obtained by 10% of the following relation. This number corresponds to a point in equilibrium symmetries as follows below:

$$\:\left(DBC\right){q}_{10\%}=\frac{{C}_{0}{\int\:}_{0}^{{V}_{10\%}}(1-C{/C}_{0})dV}{{V}_{d}}$$
(18)

Finally, the percentage of protein adsorption is obtained from the following relation:

$$\:\text{R}\%=\frac{{m}_{\text{a}\text{d}\text{s}\text{o}\text{r}\text{b}}}{{\text{m}}_{tot}}$$
(19)

The faster the slip curve, the slower the slope, and the earlier the fatigue time, the more efficient the adsorption process is. In addition, shorter mass transfer area, higher adsorption capacity, and adsorption percentage indicate better column efficiency. There is not much information about the effect of particle size on hydrodynamic conditions such as Reynolds number and flow type. Additionally, there is no significant effect on the size of the particles in this research with hydrodynamic conditions. Given this assumption, the analysis of the current results concerning the turmoil discussed in the introduction section is considered. Solid-phase particles are also assumed to prevent the breakdown of water-phase particles. The average specific energy dissipation rate \(\:{\stackrel{-}{{\epsilon}}}_{\text{T}}\), can be expressed as:

$$\:{\stackrel{-}{{\epsilon}}}_{\text{T}}\:=\:{\text{P}}_{0}\cdot\:{{\uprho\:}\:}_{\text{c}}\cdot\:{\text{N}}^{3}\:\cdot\frac{{\text{D}}^{5}}{\text{V}}$$
(20)

where P0 is the power number, and V is the fluid volume. For these Reynolds numbers at this scale, Po can be estimated from the work of Dyster et al. Assuming that d43 of the pellicular particles (as measured by the Malvern Mastersizer) is also proportional to dmax and representative of the droplet sizes, Eq. (8) takes the form:

$$\:{\text{d}}_{43}=\:\text{C}\:\:{\left({\stackrel{-}{{\epsilon}}}_{\text{T}}\right)\:}^{-0.4}$$
(21)

Dax is the axial dispersion coefficient, and the term Dax/UiH is called the vessel dispersion number. Error for this model is less than 5% if Dax/UiH < 0.01. However, for the higher values of dispersion, when there are larger deviations from plug flow, the σθ2 is correlated to the vessel dispersion number by (Levenspiel, 1999):

$$\:{{{\upsigma\:}}_{{\uptheta\:}}}^{2}=2\left(\frac{{\text{D}}_{\text{a}\text{x}}}{{\text{U}}_{\text{i}}\text{H}}\right)-2{\left(\frac{{\text{D}}_{\text{a}\text{x}}}{{\text{U}}_{\text{i}}\text{H}}\right)}^{2}\left[1-{\text{e}}^{-\left(\frac{{\text{U}}_{\text{i}}\text{H}}{{\text{D}}_{\text{a}\text{x}}}\right)}\right]$$
(22)

The reciprocal of vessel dispersion number (UiH/Dax) is called the Peclet number (Pe). Therefore, the higher Pe indicates lower mixing and less axial dispersion for the whole column. In the tanks in the series model, the σθ2 is correlated to the so-called number of theoretical plates (N) representing the number of equilibrium stages within a vessel or column by:

$$\:{{{\upsigma\:}}_{{\uptheta\:}}}^{2}=\frac{1}{\text{N}}$$
(23)

The number of theoretical plates (N) is calculated from the below formula:

$$\:\text{N}=\frac{{\text{t}}^{2}}{{{{\upsigma\:}}_{{\uptheta\:}}}^{2}}$$
(24)

The relation between N and the axial dispersion coefficient (Dax) is:

$$\:\text{N}=\frac{\text{u}\text{H}}{2{\epsilon}{\text{D}}_{\text{a}\text{x}}}$$
(25)

Better chromatography performance is expected when higher N is attained. The height equivalent to a theoretical plate (HETP) can be calculated by:

$$\:HETP=\frac{H}{N}$$
(26)

The hydrodynamic condition within the EBA column affects film mass transfer due to different factors such as flow velocity and viscosity, and their effects on film mass transfer (Kf) can be described by an equation developed by:

$$\:{K}_{f}=\frac{{D}_{m}}{{d}_{b}}\left\{2+\left[1.5{\left(\left(1-\epsilon\:\right){Re}_{b}\right)}^{1/2}{Sc}^{1/3}\right]\right\}$$
(27)

Reb is calculated by using Eq. (11). The Schmidt number (Sc) in Eq. (9) can be calculated from Eq. (10) and quantifies the ratio between viscous and buoyancy forces in a fluidized bed. Dm (cm2/s) is the molecular diffusion coefficient and can be obtained from Eq. (11). A different descriptor of hydrodynamic behavior in the column is the Bodenstein number (Bo), which correlates axial dispersion to linear velocity u0 and settles bed height. Bo was calculated according to Eq. (6), and the axial dispersion coefficient Daxl was numerically calculated according to Eq. (7), as described by Bruce & Chase. As axial dispersion decreases, chromatographic conditions become more favorable. A Bodenstein number > 40 indicates plug-like flow and negligible dispersion:

$$\:\text{B}\text{o}=\frac{\text{u}{\text{H}}_{0}}{{\text{D}}_{\text{a}\text{x}}}$$
(28)
$$\:\frac{{{{\upsigma\:}}_{{\uptheta\:}}}^{2}}{{\text{t}}^{2}}=\frac{2\left(\frac{{\text{D}}_{\text{a}\text{x}}}{{\text{u}}_{0}{\text{H}}_{0}}\right)+3{\left(\frac{{\text{D}}_{\text{a}\text{x}}}{{\text{u}}_{0}{\text{H}}_{0}}\right)}^{2}}{1+2\left(\frac{{\text{D}}_{\text{a}\text{x}}}{{\text{u}}_{0}{\text{H}}_{0}}\right)+{\left(\frac{{\text{D}}_{\text{a}\text{x}}}{{\text{u}}_{0}{\text{H}}_{0}}\right)}^{2}}$$
(29)

Thus:

$$\:{\text{D}}_{\text{a}\text{x}}=\frac{U\:H{{\:{\upsigma\:}}_{{\uptheta\:}}}^{2}\:}{2\:\:{\epsilon}\:{\text{t}}^{2}\:}$$
(30)

where\(\:{\:{\upsigma\:}}_{{\uptheta\:}}\), the variance and t, the time for A = 0.5Ao are identified in Fig. 8. ε and H represent bed voidage and the expanded bed height, respectively. Reynolds number can be assumed by the equation below:

$$\:Re=\frac{\rho\:Ud}{{\upmu\:}}$$
(31)

1.2 Taguchi array design and analysis of variance

A Taguchi orthogonal array design was used to identify the optimal conditions and to select the parameters having the principal influence on the particle size of Lf nanoparticles. Table 2 shows the structure of Taguchi’s orthogonal array design and the results of measurement. The variance of the particle size in Table 2 (analysis of variance) was calculated, and the results are shown in Table 3. The study of variance (ANOVA) is to investigate the factors that significantly affect the quality characteristic [25].

1.3  Bed expansion behavior

The first step to investigate the NAC in the developed bed is to draw a curve. Samples are taken at specific time intervals from the output of the column. Curves by plotting adsorbed concentration (the difference between input concentration and output concentration) or normalized concentration (defined as the ratio of output concentration to input concentration), as a function of time or volume of output current, for a given bed height obtained (Ct/C0). The shape of this diagram may change dramatically in different situations.

As expected, the hydrodynamic behavior of the EBA is the same as the bed behavior. In principle, mixing in the liquid phase is not a limiting factor as long as some of its requirements are met. One is a static height that should have a low altitude to adsorb adequate protein. Otherwise, the dispersion is limited, and a large percentage of the target protein will fail in the new developmental sectors. The fluidization characteristics of EBA columns generally start at different flow rates, and linear relationships are expected [26]. NACs with larger sizes and higher densities require more flow velocity to achieve specific development. However, there is usually a considerable deviation from the predicted fluidity rate of the Stokes equation. Richardson-Zaki formulated an empirical correlation to describe the effects of driver obstruction for identical beads of large glass size. This is often used to describe the expansion of the EBA bed. Thus:

$$\:{\text{U}}_{\text{s}}={\text{U}}_{\text{t}}{{\epsilon}}^{\text{n}}$$
(32)

While \(\:\text{n}\) is the constant of Richardson and Zaki’s equation. This equation is valid for the tiny beads to column diameters (dc), namely ratios (db/dc) > 0.01, where the wall effect can be ignored. The average voidage for an expanded bed can be calculated by using Eq. 12:

$$\:{\epsilon}=1-\frac{{\text{H}}^{\circ\:}}{\text{H}}(1-{{\epsilon}}^{\circ\:})$$
(33)

Here, \(\:{\text{H}}_{0}\) is settled bed height, H is expanded bed height and \(\:{{\epsilon}}_{^\circ\:}\) Is the voidage of the settled bed, which is generally assumed to be 0.4.

2 Experimental

2.1 Materials

Neutral Agarose with low gelling temperature was purchased from Invitrogen (USA). Sorbitan monooleate (Span 80), Silicone oil with a kinematic viscosity of 5 * 10−4 m/s at 25 °C, and Nickle particles with a bulk density of 1500–2600 Kg/m3 and an average particle size of 10 μm were commercially supplied by Sigma Chemical Co. (st. Louis, MO, USA). All solutions were prepared with DI water. Lf was purchased from Merck. The MCR rheometer series from Anton Paar was set to control the rheological properties of 2% and 4% (w/v) solutions in the water at 85 °C. Acetone obtained from Merck company. The sodium hydroxide (NaOH) and sulfuric acid (H2SO4) were purchased from Mitsui Chemicals Inc.

2.2 Methods

2.2.1 Fabrication of two-layer NAC

One of the ideal adsorbent properties is that by decreasing their size, the reactivity of the adsorbent particles is greatly enhanced and rapidly decomposes when exposed directly to the biological process. Therefore, proper coverage is necessary to avoid such restrictions. The formation of an inert agarose coating on the adsorbent surfaces has been shown to help prevent their accumulation in the liquid and improve their chemical stability. Another advantage of agarose coatings is that silane-terminated groups on the surface can be modified with various binding agents to attach specific ligands to the surfaces of these nanoparticles covalently. The fabrication of two layer adsorbent described earlier [27]. Briefly, we kept the inactive agarose-nickel beads in the refrigerator for 24 h at 4 °C. Simultaneously, in a double walled reactor connected to a water bath, 300 ml of silicone oil with 1 ml of span 80® was heated up to 70 °C and stirred at 500 rpm for 5 min. Next, 25 ml of distilled water was added to the heater at 70 °C and mixed with 1 g of agarose for 5 min and stirred at 500 rpm. The agarose nickel particles were added and stirred slowly for another 5 min. Then, the slurry solution was transferred to the oil-containing reactor and stirred for 15 min once more. Afterward, the reactor temperature was lowered to 15 °C, and stirring was continued for another 5 min. When a stably fluidized bed was achieved, the solution was transferred to the centrifuge, and it took 15 min at 11,000 rpm to separate the oil from the adsorbent beads. As a final point, the adsorbents were washed with acetone and rinsed by a sieve and cold water. Finally, the two layer adsorbent named 2@NAC was assessed as follows in Fig. 1.

Fig. 1
figure 1

Process design and total steps for fabrication of the structure

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2.2.2 Immobilizing of CB dye-ligand

Chemical meshing using epichlorohydrin was used to preserve the 2@NAC and mechanical strength. 2@NAC was mixed with an equal volume of 1 M sodium hydroxide containing 5 g/l sodium borohydride and placed in a shaker for 30 min at 150 rpm. Then epichlorohydrin (up to 2% by volume) was added to the mixture. The mixture was agitated at 25 ° C for 6 h in a shaker at a speed of 150 rpm. Afterward, the particles were washed with distilled water to separate the excess reactants. After washing with distilled water, to prepare the affinity adsorbents for use in the adsorption processes on the basic agarose-nickel bilayer structure, the pseudo-affinity ligands of CB protein products were separately fixed. For this purpose, the method proposed by Dean and Watson was used. First, 10 ml of distilled water was added to 0.3 g of ligand, and 10 ml of the agarose-nickel chemical structure was stirred entirely. Then 3 g of sodium chloride were added to the mixture, and after 30 min mixing at 150 rpm in the shaker, 12 ml of 1 mM sodium carbonate was added to the mixture. After 5 min of mixing, the final blend was stirred for 24 h in a brine bath at 65 °C. After this step; the agarose-nickel particles were washed several times with warm and cold water to obliterate the unsubstituted ligands. To prevent contamination, the resulting adsorbents on which the ligand was fixed were stored in 0.005% sodium azide solution at 4 °C. We named the agarose-nickel bilayer adsorbents functionalized with the CB affinity ligand 2@NAC-CB. The Streamline commercial structure was also functionalized in the same manner as the CB ligand to compare with 2@NAC-CB in the adsorption process, and we briefly named St-CB.

3 Results and discussions

3.1  Calibration curve

A spectrophotometric apparatus was used to measure protein concentration in the solution. For this purpose, the Lf adsorption calibration curve using a 450 nm spectrophotometer is shown in Fig. 2. Solutions with different concentrations were prepared and absorbed in a spectrophotometer analyzer at 450 nm to draw the calibration curve. The calibration curve for Lf was obtained by plotting the absorbance numbers.

Fig. 2
figure 2

Calibration curve of Lf at 450 nm wavelength

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A series of known Lf concentrations are prepared via serial dilution. A blank buffer without Lf is used to zero the spectrophotometer. Absorbance is measured at 450 nm. Lf itself did not absorb strongly at 450 nm, so a dye-binding assay used to generate a detectable signal. A good calibration curve have R2 = 1, indicating a strong linear relationship. As can be seen, R2 is close to one which is similar to earlier work [28].

3.2 Data analysis

In our previous work, four factors at a time were applied. It has been concluded that four essential elements, including impeller speed, the percentage of Agarose, the dosage of Nickel powder, and the concentration of span 80, affect the particle size. Therefore, to minimize the number of experiments Taguchi model was employed through Qualitek-4 software. Taguchi’s orthogonal array table was used by choosing four parameters that could affect the laminated adsorbents. Table 1 displays the settings and levels used in this experiment. The orthogonal array of the L16 type was used. L and subscript 16 mean Latin square and an experiment number, respectively.

Table 1 Parameters and levels used in this experiment
Full size table

3.3 Analysis of taguchi’s model

The effect of the size of the adsorbent particles was determined by changing the four main characteristics. In the first stage of the experiment, four different agitators were used for the emulsification and cooling process. Further, the rotational speed increased in attempts to reduce any accumulation and expanded to examine the relative importance of the stimulation conditions in two steps on the product. After particle screening by the Malvern Mastersizer apparatus, the particle size distribution was calculated. The results are summarized in Table 2. The factor, Re85, is the Reynolds number at the end of the emulsification stage at 85 °C (ʋ~), and Re15 refers to its value at 15 °C after cooling (ʋ~).

As seen in Table 3, the flow in the reservoir is transient in all conditions, and this can be justified by the Reynolds number obtained (10˂Re˂104). Turbulent flow is crucial for efficient adsorption because it enhances mass transfer, ensures uniform adsorbent utilization, and prevents flow channeling. While it demands more energy, the benefits in adsorption kinetics and capacity often justify its use in industrial and laboratory-scale processes.

Table 2 Introducing the number of experiments and the results of numerical calculations
Full size table
Table 3 The effect of stirring speed on particle size
Full size table
Fig. 3
figure 3

The effect of mean specific energy dissipation rate (εT) on particle mean diameter (d43)

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Using the experimental data in Fig. 3, non-linear Regression using Table Curve 2D from Sigma Plot gave an exponent of − 0.373 (R2 = 0.9278), a value very close to the theoretical one. Drop size distribution and liquid-liquid contacting conditions, such as those encountered in mechanically agitated tanks, were matched with experimental results measured by photographic skills. Good agreement was obtained at different flow velocities and screen geometries.

In this study, we compared the stirring speed with specific parameters to increase the cooling speed, reduce the particle size before the gelation process, and prevent particle adhesion. By comparing the absorbent particles under the light microscope, it can be concluded that with the increase in stirring speed, particle adhesion, and particle size begin to decrease, which is very important in the analysis of nano dimensions.

Energy consumption is the energy of each part that is transferred from the impeller to the fluid. This is a suitable quantity for mixing processes because the fluid needs mixing energy to move. It can be explained that the power factor in the mixer depends on variables such as tank size, impeller, and fluid properties such as stirring speed and viscosity. The relationship between the power number and several of these variables has been determined experimentally for different propellers. The work results in a Rushton turbine and baffled tank are shown in Fig. 4.

Fig. 4
figure 4

The Reynolds number column against the power number

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As shown above, the graph has a proper regression, and empirical data corresponds to what is obtained from the theoretical equations (Due to R2 being close to 1). The consequence of aqueous-phase surfactant concentration on the typical size of the particles is shown in Fig. 5. The particle size of colloidal systems, such as oil dispersions, plays an essential role in determining their specifications. The stability of this parameter over a long period indicates the security of a system. This study used various concentrations of span 80 as a surfactant to produce 2@NAC-CB. The average size of particles is in the range of 300–1200 nm. Besides, the used surfactant in oily nanocarriers plays a more critical role in controlling the crystallization process. The choice of stabilizers is a pivotal subject to be considered in the preparation of any nanoparticle origination to control the particle size and stabilization of the dispersions. If particle-particle osculates happen faster than surfactant molecules’ adsorption rate to cover the exposed hydrophobic patches, then holistic particle accumulation will occur due to hydrophobic attraction between the particles.

In this study, taking into account concentrations of 0.5, 1, 1.5, and 2 mg/ml, the effect of span percentage on particle size and hence the formation of the spherical shape of adsorbent particles were analyzed. As shown in Table 1, in some experiments, the spherical shape of the spherical particles was found in some others without form; this factor was obtained by observing the optic with an optical microscope and indicating to what extent our adsorbent is in the way of adsorbents is close to the commercial ones. Figure 5 shows the influence of surfactant percentage against d43.

Fig. 5
figure 5

Influence of surfactant concentration on the particle size

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The density of particles was determined using a density flask procedure. At the same time, pellicle depth was estimated by direct measurements taken from images of particles recorded on video and gravimetric analyses conducted following drying and combustion procedures. In many cases, with increasing surfactant concentration, the density of adsorbent particles also significantly increases. The distribution of particle size and relative quality of adsorbents by keeping the surfactant level constant with the lowest concentration of 0.5% w/v is shown in Fig. 6.

Fig. 6
figure 6

Investigate particle size in different experiments

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3.4 Particle morphology

Generally, the tentative experiments here suggest that the final particle size is predominantly determined by the hydrodynamic conditions impacting before the gelation of the Agarose. As seen in Fig. 7, a small amount of nickel as a metal core significantly affects the size of the adsorbent particles, which can be seen in part b. In part b, the low levels of Agarose and surfactant versus high doses of nickel powder cause the opaque layer to become organized, and the adsorbent is well visible in the sample. Low agarose concentration forms a looser, more porous polymer network with larger interstitial spaces, and light scatters less because the refractive index differences between agarose and water are minimized in a less dense matrix. In addition, agarose has a refractive index close to water. At low agarose concentration, the polymer-water mixture better approximates a homogeneous medium, reducing refractive index discontinuities.

In section c, which is our best test case, the effect of the proportion of nickel and Agarose can be understood. The agarose layer in this section is very transparent, and the metal core is distributed with an appropriate particle size that gives us an excellent spherical shape. Another successful test of agarose adsorption shows that the ratio of Agarose to nickel is well controlled.

Fig. 7
figure 7

Comparison of the microscopic shape of agarose adsorbents according to the samples a = Exp. No.1, b = 6, c = 11, d = 16

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As was indicated above, to achieve a high-efficiency separation, axial mixing within the expanded bed must be minimized to allow an approach to plug flow. The mixing time, tm, is the time needed until different fluids in a vessel are sufficiently mixed. The factors affecting that include design parameters, e.g., the flow distribution system and column diameter, and operational parameters, such as flow velocity, bed height, and fluid viscosity. The expansion of the tracer band within the column is evaluated by controlling the concentration of the tracer at the outlet (Fig. 8).

Fig. 8
figure 8

Concentration curve against the time at a fixed point

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3.5 SEM analysis

The 2@NAC-CB synthesized in this study was characterized using scanning electron microscopy (SEM) to analyze its surface roughness, softness, and shape at the nanoscale. The sample was deposited on a silicon water, sputter-coated with 5 nm Au/Pd layer to mitigate charging, and imaged using a field-emission SEM operated at 10 kV to enhance surface detail while minimizing beam damage.

The adsorbent particles exhibit a spherical morphology with diameters ranging from 50 to 180 nm, consistent with dynamic light scattering measurements. High-resolution imaging reveals a mesoporous surface with pore sizes of 15–25 nm, confirmed by BET analysis.

The surface topography displays nanoscale roughness, charachterized by granular protusions and interparticle voids, indicative of a high surface–area-to-volume ratio.

The SEM analysis depicted in Fig. 9 also provided insights into the mechanical properties of the 2@NAC-CB, allowing for the observation of its softness and the characterization of its material behavior. This information is crucial for understanding how the adsorbent interacts with target molecules and how it withstands environmental conditions during the adsorption process.

Furthermore, the shape and morphology of the 2@NAC-CB particles were accurately determined, and their size distribution was less than 200 nm. This information is essential for optimizing the design and performance of the adsorbent for specific applications and understanding its effectiveness in capturing and removing contaminants from different substances.

Fig. 9
figure 9

SEM images of 2@NAC-CB particles; (a) High-resolution image (200 nm scale), (b) Low-magnification view showing monodisperse particles (500 nm scale), (c) Aggregates and rough surface texture

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3.6 AFM analysis

AFM (Atomic Force Microscopy) image for 2@NAC-CB refers to a microscopic image obtained using the AFM technique that shows the surface morphology and topography of the adsorbent. The 2@NAC-CB was dispersed on a freshly cleaved mica substrate, air-dried, and imaged under ambient conditions using a multi-mode AFM (Bruker Dimension Icon) in PeakForce Tapping mode (scan rate: 0.5 Hz, resolution: 512 × 512 pixels). A silicon nitride cantilever (k ≈ 0.7 N/m, tip radius < 10 nm) was used to minimize tip convolution artifacts. The adsorbent particles exhibit quasi-spherical morphology with an average height of 51 nm (Fig. 10A), corroborating DLS measurements. Isolated particles dominate, with limited aggregation (Fig. 10C), suggesting effective stabilization. Heterogeneous phase contrast suggests material heterogeneity (e.g., core-shell structure or uneven ligand distribution).

The probe interacts with the surface, and the resulting forces are measured to create an image. This technique allows for the visualization of surface features at the nanoscale level [30].

Figure 10, which is an AFM image can provide valuable information about the surface characteristics. The image shows the topography of the 2@NAC-CB, revealing any roughness or irregularities on the surface. The AFM image also provides information about the size and distribution of pores or channels present in the adsorbent material. These features are crucial for understanding the adsorption capacity and efficiency of the material [31]. Additionally, the AFM image identifies changes in the surface morphology of the 2@NAC-CB after adsorption has taken place. It reveals the formation of aggregates of adsorbed molecules on the surface, indicating the effectiveness of the adsorption process. Overall, Fig. 11 shows the size range of 2@NAC-CB adsorbent, which is less than 120 nm, which is an appropriate achievement.

Fig. 10
figure 10

AFM characterization of 2@NAC-CB adsorbent; (A) Height image (scale bar: 500 nm), (B) Topography range (−32.9 to 26.9 nm), (C) 3D rendering highlighting surface roughness

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Fig. 11
figure 11

Nanoparticle ranges of 2@NAC-CB adsorbent

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3.7 XRD analysis

X-ray diffraction (XRD) analysis of NAC provides valuable information about its crystal structure, phase composition, and crystallite size. This technique involves shining X-rays onto the sample and measuring the angles at which the X-rays are diffracted, which can then be used to determine the arrangement of atoms within the material. The XRD analysis can help to identify the presence of specific crystalline phases, such as metal oxides or carbon-based materials, which are commonly used as adsorbents. It can also provide information about the size of the crystallites present in the material, which is important for understanding its surface area and reactivity. Figure 12 shows XRD analysis of NAC particles around 10 to 80 degrees from the angle of 2 theta. X-ray diffraction (XRD) analysis of nickel-agarose adsorbent in the angle 2 theta between 10 and 80 degrees can provide detailed information about its crystal structure and phase composition. At lower angles (10–30 degrees), the XRD pattern can reveal information about the long-range ordering and spacing of atoms within the crystal lattice of NAC. Agarose itself is predominantly amorphous, so its coating typically introduces a broad hump in the XRD pattern (centered around ~ 20° 2θ) characteristic of amorphous materials. This can help in determining the crystal structure and identifying any crystalline phases present in the material. In the mid-range angles (30–50 degrees), the XRD pattern can provide data on the phase composition of NAC, indicating the presence of any impurities or secondary phases that may affect its adsorption capacity. This range can also provide information about the orientation and stacking of crystal planes within the material. At higher angles (50–80 degrees), the XRD pattern can be used to determine the crystallite size of NACA. This is important for understanding the material’s surface area and its potential for adsorption of biomolecules. The peak broadening or narrowing in this range can provide insights into the size and distribution of crystallites within the material. If the substrate is crystalline (e.g., nanoparticles, metals, or inorganic crystals), the agarose coating may suppress sharp diffraction peaks due to dilution effect (the amorphous agarose dominates the signal, reducing the relative contribution of crystalline peaks.), and surface masking (agarose may partially obscure the crystalline substrate, reducing peak intensity without altering crystallinity.).

Fig. 12
figure 12

XRD pattern of 2@NAC-CB

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3.8 FTIR analysis

As shown in Fig. 13, the FTIR spectra of the 2@NAC-CB adsorbent were examined, to release the binding mechanism. The considerable peak around 3000 to 3200 cm−1 observed relates to the immobilization by CB. The peak at 1710 cm−1 relates to the two layer coating, and the peak at 1470 cm−1 relates to the agarose coating. In addition, the decreasing peak around 1000, 750, and 500 relates to water, as depicted in Fig. 13. Zhang et al. reported similar results [32]. The short downward peak around 3000 − 2500 is related to pH adjustment regulation, which is confirmed by the results of Rezvani. et al. [33]. The peak around 2000–2200 is related to N = H group. The lower peak close to 1500 is related to C = H group. The increased peaks between 1200 and 500, are related to C-C and C = O relatively.

Fig. 13
figure 13

FTIR spectra of 2@NAC-CB

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3.9 CFD modeling of double-layer reactor

Computational fluid dynamic (CFD) simulation of the double-layer reactor (T2) is done to understand better the physical and chemical conditions of what happens in it. The simulation of this reactor was carried out in a three-dimensional and stable state. The fluid of volume model was used to detect the interface between three phases of air, agarose, and silicone oil. The k-ε RNG model was used for high flow accuracy to model the current turbulence. The geometry consists of 432,500 structured grids, and the boundary conditions for the reservoir walls are fixed at constant temperature and for the stirrer in the rotating wall. A coupled algorithm was used to overlap between pressure and velocity. The discontinuities of continuity, momentum, and energy equations were selected as order upwind. The convergence criterion was considered for continuity and perturbation semantics and turbulence 1e-3 and energy 1e-6.

Fig. 14
figure 14

CFD simulations of the double-layer reactor; (a) temperature counter (b) velocity counter (c) fluid flow streamline

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As can be seen, Fig. 14a shows that to achieve the optimal temperature of the reactor, we need to increase the water bath temperature. As with the temperature distribution inside the reactor, it is determined that the heat around the stirrer is less than the temperature considered for testing. What is apparent in Fig. 14b is that the amount of mixing and mixing speed inside the reactor is well-tuned, and the desired result can be achieved, considering the rate. The contour shows the uniformity of the swirling speed throughout the reactor, which means that the agitator speed factor is a suitable and particle size-dependent index. The regular flow streamlines around the stirrer in Fig. 14c confirm this fact. The circular velocity lines with a proper distance from the agitator center and at a suitable velocity indicate the correct mixing of the material by the agitator. The similar works represented similar results [34].

4 Conclusion

Optimum production conditions of adsorbent coated with pure agarose were investigated in two separate stages (emulsification and cooling), and the design of 16 distinct experiments. The minimum particle size (optimum) was 50 rpm stirring speed, 6% agarose percentage, 0.5 mg/ml surfactant concentration, and 2 mg nickel dosage. Various chemical methods can fix the recycled adsorbent and proceed to the commercial industry. Agarose’s optimal porosity combined with compatibiblity makes it the gold standard for high-purity protein isolation in sensitive applications (e.g., pharmaceutical and food industries). Suitable porosity, biodegradability, and ease of process are the advantages of this recombinant adsorbent. Laboratory conditions such as density, particle size, size distribution, and graphic forms suggest that this study determines the best agarose adsorption conditions. A careful examination of the adsorbent particles and the measurement of all the qualitative parameters allow the use of the same test, depending on the application type and the adsorbent material, to be used. For example, when we use adsorbents to purify the small size of the adsorbent protein, we can use smaller-sized adsorbents suitable for the same protein. When using protein with an oil source, we can use an oil agarose adsorbent to reduce costs and use less raw material. Adsorbents with larger particle sizes or greater depths are more suitable for macromolecular protein absorption, and mutually smaller proteins are used in micro-sized proteins.