1 Introduction

The quality of the water is decreasing as environmental contamination rises. Water pollution is caused by heavy metals, stormwater,  sewage water, radioactive materials, and treated wastewater discharges from industries  into rivers and the oceans (DhanaRamalakshmi et al. 2020; Janani et al. 2022; Say et al. 2003). Insecticides, toxic chemicals, heavy metal ions, radioactive nuclides, and other substances that are toxic to living beings are also frequent water contaminants (Puri et al. 2021). According to Lakherwal (2019), hazardous chemicals in water can harm the skin, brain, stomach, nervous system, heart, blood flow, bones, and kidney. Additionally, they may cause respiratory illness, mental retardation, nausea, exhaustion, thyroid dysfunction, elevated blood pressure, joint discomfort, tooth discoloration, anemia, blindness, and hearing. Water pollution also impacts the immunological and reproductive systems (Foo and Hameed 2010).

Many technologies are used for water treatment, including reverse osmosis, electrodialysis, membrane separation, filtration, coagulation, precipitation, and adsorption (He et al. 2020). Adsorption is the most feasible and commonly used method due to its ease of use. The major water purification systems are based on adsorption phenomena. Since German physicist Heinrich Kayser first used the word in 1881, adsorption has been thoroughly studied. The adsorption technique has since been widely used because of its simplicity and low cost (Rathore et al. 2016).

The adsorption process occurs through physical or chemical linkages between the adsorbent and the adsorbate. Adsorption is now highly relevant and valuable in the contemporary environment. It is utilized in several industrial processes, including wastewater treatment, flue gas stream purification, air pollution masks, heterogeneous catalysis, various chromatography methods, and many more (Bibi et al. 2015). Although single-component adsorption has been the subject of extensive research, most adsorption applications involve systems with multiple components. For example, arsenic and fluoride are found together in many places in India, China, Mexico, Japan and the USA. Lead and zinc, vanadium and chromium, zinc and nickel, cobalt and nickel and many other contaminants occur simultaneously in the sediments and unconfined aquifers leading to contamination of groundwater. Industrial wastewater usually consists of several pollutants such as lithium, phosphorus, chromium, nickel, cadmium, cobalt, copper, mercury along with other organic pollutants. These pollutants also interact with each other and may show synergistic or antagonistic behavior during adsorption. However, there are only few conclusive multi-component adsorption modeling studies for adsorption of such mixtures.  

The adsorption of contaminants may be done using several adsorbents, including activated carbon, zeolites, iron oxides, alumina, silica gel, and nanoparticles (Dhanasekaran and Sahu 2021). Materials such as biochar, agricultural wastes, modified sawdust, and activated alumina are commonly used for column studies. Also, activated carbon derived from various readily available materials, industrial-based materials such as iron oxide, manganese oxide, iron-coated sand (Jeppu et al. 2010), and ferric hydroxide with iron oxide are widely used for water treatment (Bibi et al. 2017).

Adsorption modeling studies are essential for research of the adsorption process (Abdoul Magid et al. 2021; Wang et al. 2017). The strength, maximum adsorption capacity, and adsorption affinity of the adsorbent system are quantified and provided through isotherm modeling. Adsorption isotherms give information on the adsorbent material's nature, including its surface area, pore size distribution, and the potency of adsorbent-adsorbate interactions. The performance of the materials in various applications, including catalysts, water filtration, and gas storage, may be predicted using this knowledge. Adsorption isotherms may be used to identify the ideal operating conditions for an adsorption process, including the temperature, pressure, and adsorbate concentration which aids in maximizing process effectiveness and lowering costs. Adsorption isotherm models may create adsorption systems tailored to a particular purpose. For instance, the models can predict the quantity of adsorbent needed to treat a specific volume of gas or liquid. Adsorption isotherms may be used to check the quality of adsorbent materials. The performance of the material under various situations may be predicted using the models, and the performances of multiple adsorbents can be compared. Adsorption isotherm modeling, in general, is a crucial tool for comprehending and improving, and assessing adsorption systems for several applications.

Modern research in multi-component or competitive adsorption still needs to advance, especially in comparison to the work done in single-component adsorption. A thorough interpretation of the isotherm models would significantly improve the competitive adsorption isotherm modeling (Ashour et al. 2022). Research on the simultaneous removal of dangerous contaminants by several adsorbents has recently been conducted (Adamu et al. 2018). In order to improve the design of the adsorbents and the adsorption system pattern, it is imperative and trustworthy to study the multi-component adsorption process (Chen et al. 2022b; Van Assche et al. 2018).

It should be easier to assess competing adsorption isotherm models thoroughly. There are limitations to the compilation of all multi-component isotherm models. However, their use is beneficial and practical (Qiao and Hu 2000). Adsorption considerations include temperature, pH, concentration, and how the adsorbent and adsorbate interact (Musso et al. 2019). Single-component systems are easier to model and implement than multi-component systems. Adsorption equilibrium conditions, the interaction coefficient between adsorbates and adsorbents, and the adsorbate distribution coefficient are a few of the numerous variables in multi-component adsorption systems. Affinity parameter of adsorption, non-interaction between adsorbents and adsorbates (Weerasundara et al. 2021; Wu et al. 2012), optimum adsorption capacity, selection of adsorbent for specific adsorbate, and antagonistic and synergetic behavior of adsorption are also included (Girish 2018). Adsorption models are nonlinear and multivariable, mainly when analyzing multi-component systems (groundwater, industrial effluents, and wastewater). As a result, adsorption data modeling is a complex mathematical issue requiring a nonlinear data regression process for determining model parameters with a correct error function and a thorough statistical analysis to determine model performance (Sellaoui et al. 2018).

The Scopus search engine was utilized in the literature search to retrieve published articles in the many subject areas. The search terms for the survey were adsorption, mono-component adsorption, and single-component adsorption isotherm models. A total of 20,894 scientific articles on adsorption were produced between 1986 and 2023. The disciplines with the highest levels of research include environmental science, chemistry, chemical engineering, engineering, and material science. A good number of papers have been written regarding single-component adsorption and multi-component adsorption (Figs. 1, 2 and 3).

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Trend of publication of mono-component adsorption in various research areas as per Scopus

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Trend of publication of multi-component adsorption in various research areas as per Scopus

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Trend of publication of multi-component adsorption modeling in various research areas as per Scopus

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Competitive adsorption or multi-component adsorption isotherm models are the search terms utilized in the survey. Between 1986 and 2023, only 401 research publications were published on multi-component isotherm models, showing fewer studies on multi-component adsorption. Most relevant publications are used in chemistry, environmental science, chemical engineering, engineering, and material science, demonstrating the need for this review. Therefore, there is a strong and increasing need for research on competitive adsorption. Due to a lack of literature on review of competitive adsorption models, a collection of isotherm models of competitive adsorption has become essential.

Between 1986 and 2023, nearly 357 research articles were published on multi-component adsorption. The search keywords used in this study are taken from the Scopus tool: competitive adsorption modeling, multi-component adsorption isotherm models, or competitive adsorption isotherm models. The acceptable subjects of study are chemistry, environmental science, chemical engineering, biochemistry, and engineering. It is essential to have adequate literature and simulation studies of competitive adsorption. Therefore, the need for research on competitive adsorption is substantial and expanding.

This review paper covers the modeling and simulation of multi-component adsorption systems. An extensive list of multi-component isotherm models is compiled. The most common single-component adsorption models are the Henry model, Langmuir isotherm model, Brunauer–Emmett–Teller (Lee and  Stenstrom 1996), Freundlich isotherm model, Langmuir Freundlich hybrid isotherm model (Al-Duri 1995), Modified Competitive Langmuir (Sohn and Kim 2005), Langmuir Extension First Approximation model (Lee and  Stenstrom 1996), Freundlich-Langmuir-Jovanovic (Field 2017), Modified Langmuir Freundlich model (Jeppu and Clement 2012), and LeVan–Vermeulen model (Jadhav and Srivastava 2019) which are stated with their significance and equations. Currently, 26 multi-component and 10 single-component isotherm models are discussed. Also, four multi-component isotherm models were used to simulate three binary adsorption systems.

2 Adsorption Isotherms Models

The adsorption behavior relies on solute-to-adsorbent molecular interactions in an adsorption system (Bonilla-Petriciolet et al. 2017). At a specific temperature, the equilibrium adsorbate concentrations in the aqueous phase and the solid phase are said to be isothermal. Isotherm equations are utilized to simulate the equilibrium adsorption data and examine adsorption information, adsorption mechanisms, and the maximum adsorption capacity of a system (Wang and Guo 2020). Isotherm equations provide a mathematical relationship between the adsorbate concentrations in the liquid phase and the adsorbent in the solid phase. The isotherm model seeks to establish the maximum adsorption capacity and suitability of particular adsorbates for the specific adsorbent. There are two phases of interaction, the first is solvent and solution, and the second is solid. It relies on the adsorption system's equilibrium concentrations and various component concentrations (Beyan et al. 2022). The complexity of all the components in the adsorption system needs to be analyzed. The equilibrium isotherms plot the component equilibrium concentrations and adsorption measured at various experimental conditions (Manning and Goldberg 1996). Adsorption isotherm models correlate adsorbate and adsorbent performance at certain experimental conditions, including pH, temperature, the concentration range of interest for adsorbent and adsorbates, and pressure. It also gives the flow rate in the case of column bed adsorption.

The isotherm models are classified as single and multi-component adsorption (Bonilla-Petriciolet et al. 2017). Both single-component and multi-component adsorption systems with substantial isotherm models are provided. It is proposed that isotherm models and their coefficients can be used to build adsorption systems (Al-Ghouti et al. 2010). It may contain linearized and non-linearized forms of equations. The equilibrium parameters, heterogeneity factor, interaction coefficient, model constants, distribution coefficient, and affinity parameter are all estimated. Non-interaction between adsorbents and adsorbates, antagonistic and synergetic behavior, optimum adsorption capacity, and selection of adsorbent for particular adsorbate are assessed (Baig et al. 2009; Weerasundara et al. 2021; Wu et al. 2007).

There are some criteria for selecting the optimum isotherm model. The isotherm function and the experimental data must correlate effectively is the first criterion for choosing the best isotherm model. Second, the isotherm's function ought to be thermodynamically rational. It must satisfy three requirements: first, the isotherm must be linear when the concentration approaches zero; second, there must be a finite capacity at the maximum concentration; and third, the slope must be positive at all concentrations (Al-Ghouti and Da’ana 2020). Usability is the third factor to consider when selecting the optimal isotherm model. Ideally, the isotherm should apply to all computations, such as converting concentration to find capacity and vice versa for only some cases.

Fitting of isotherm models to experimental data is traditionally done by linear regression. Consequently, linear regression has been recognized as one of the most practical methods for establishing the best-fitting relationship, quantifying the distribution of adsorbed materials, validating the accuracy of isotherm model predictions, and quantitatively analyzing the adsorption system. However, using parameter estimation tools and least square minimization of errors is a better approach (Jeppu and Clement 2012). The fit of isotherm models to experimental data is enhanced for a given concentration range by increasing the number of isotherm parameters and minimizing the error function (Chen et al. 2022b). The error is more if the experimental data and the data from the isotherm model are distinct; the error will be minor if they are close. The optimal model can be determined by calculating several mathematical error functions.

The literature cites many works and research on the simultaneous removal of pollutants. This work highlights the various conditions and factors that contribute to adsorption. Adsorbent type, adsorbate count, temperature, adsorbate and adsorbent concentration, agitation, pH, contact duration, adsorption capacity, and surface area are the components. Furthermore, this study also provides several examples of the simultaneous removal of contaminants from literature. The lower the initial concentration of the component, the better the adsorbent sorption affinity to the adsorbate molecule is (Girish 2018). The adsorbents' activity and regeneration are influenced by temperature. While activated carbon typically requires a lower temperature, zeolites demand a high temperature (Yang 2003). The higher the surface area of the adsorbent, the more the adsorption. The pH influences the metal solubility and the dissociation degree of functional groups on the surface of sorbents. Due to the electrostatic attraction between metal ions and protonated ones, the adsorption capacity rises with increasing pH but can also fall at a specific maximum pH (Qiao et al. 2014). As adsorbent dosage boosts, the surface area increases and the adsorption rate improves, providing more sites for heavy metal adsorption (Sun et al. 2019). Table 1 summarises the influencing factors of adsorption equilibrium.

Table 1 Influencing factors of adsorption equilibrium
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2.1 Single Component Adsorption Isotherm Models

Single-component adsorption isotherm models are mathematical models used to describe the relationship between the concentration of a solute in a solution and the amount of solute adsorbed onto a solid surface. The isotherm models are called "single components" because they consider the adsorption of only one solute at a time without assuming any interactions with other solutes. The adsorption of a single component on a uniform solid surface exhibits single adsorption energy and uniform adsorption sites (Babatunde et al. 2022). Understanding single-component equilibrium studies and the basic principles of retention processes of specific pollutant concentrations present in an adsorption system are essential (Piplai et al. 2018). The formation of electrostatic attraction and repulsion, hydrogen bond, and metal ion exchange between adsorbents and adsorbates at favorable conditions are shown in Fig. 4 (Cobbina et al. 2019). For the selection of isotherm models, theoretical and experimental guidance from the literature is available. Table 1 presents single-component removal models for one, two, and three parameters. Surface heterogeneity and homogeneity are basic assumptions in single-component models. In addition, it is also assumed that adsorption may be monolayer or multilayer (Van Assche et al. 2018). Single-component models are the foundation for many multi-component models. Table 1 represents the principal equilibrium isotherm models for single components and the prior significance, equations, and experimental factors influencing the equilibrium adsorption phenomena.

Fig. 4
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Schematic diagram of (a) Electrostatic attraction, (b) Electrostatic repulsion, (c) Hydrogen bond, and (d) Metal ion exchange between adsorbents and adsorbates

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2.2 Multi-Component Adsorption Isotherm Models

In the study of adsorption, multi-component adsorption is of crucial relevance. In addition to the solid–liquid adsorption phase, competition phenomena might occur in a solid–solid adsorption stage. When there is more than one contaminant, additional experimental work is necessary. Because there are more contaminants in the adsorption system, multi-component equilibria are more complicated (Bonilla-Petriciolet et al. 2017). The interaction of the various pollutants makes it impossible to measure them experimentally using current techniques since they depend on the concentration of the other component in the same adsorption system (Yang 2003). In a multi-component system, it is exceedingly difficult to predict the individual and competitive interaction during migration, the separation process, and the optimal experimental circumstances (Girish 2018). Despite this, several factories and industries release many pollutants simultaneously, such as organic and inorganic salts, heavy metals, proteins, and various behavioral characteristics. Therefore, they must simultaneously be removed from the polluted water (Girish and Murty 2016).

Competitive adsorption is influenced by the physical characteristics of the adsorbents, including surface charge density, surface area, particle size, and porosity (Bansal and Goyal 2005). Additionally, factors like adsorbent size, agitation rate, pollutant characteristics or ions that need to be removed, contaminant concentration, ion to ion interactions, affinity of an ion species for the adsorbent, and interest in the active sites of adsorbents are also considered (Bibi et al. 2015). Multi-component adsorption equilibria define the quantitative link between the adsorbates and adsorbents in the solution and the number of solutes adsorbed by the adsorbent (González-López et al. 2022). As shown in Table 2, many multi-component isotherms have a variety of assumptions and relevance.

Table 2 Single-component parameter-based nonlinear equations
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The most vital scenario of competitive adsorption model determinations depends on the parameters of the corresponding single-component isotherms. Comparatively single isotherm models are uncomplicated, accurate and precise. A schematic representation of multi-component adsorption is given in Fig. 5. Multi-component isotherm models are effectively used for the combination of pollutants. Some are the Extended Langmuir isotherm, Modified Competitive Langmuir isotherm, and Langmuir Extension First Approximation Equilibrium isotherm models. Twenty-six multi-component isotherms are presented in Tables 2 and 3, along with the nonlinear equations and their significance.  

Table 3 Multi-component adsorption isotherms models with nonlinear equations and their significance
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Fig. 5
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Schematic representation of multi-component adsorption

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3 Competitive Removal of Pollutants

The removal of multiple pollutants in a multi-component adsorption system is crucial. Multi-component adsorption is significant because the waste streams and groundwater contain a mixture of contaminants such as arsenic and fluoride, cadmium and nickel, and chromium and vanadium, and removing these pollutants requires effective adsorbents. Some case studies on the simultaneous removal of commonly co-occuring heavy metals are presented in Table 4.

Table 4 Summary of simultaneous adsorption of common heavy metals in the binary system, Isotherm models, experimental conditions, and adsorption capacity of heavy metals
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In case (i) Dhanasekaran and Sahu (2021) studied arsenic and fluoride adsorption on activated alumina-impregnated sawdust. The adsorption capacity of 0.10 mg/g of arsenic and 2.42 mg/g of fluoride was obtained in the binary system at pH 6–7. The Freundlich isotherm model was used to predict and simulate the equilibrium data. In case (ii) bagasse active carbon was successfully employed to extract zinc and cadmium at pH 4.5. The adsorption followed both Langmuir and Freundlich isotherm models. Comparatively, the Freundlich model provided a better fit (Mohan and Singh 2002). Further, Padilla-Ortega et al. (2013) demonstrated that vermiculite has stronger selectivity of Pb(II) in single-component adsorption than copper in case (iii). Even in competitive adsorption, Pb(II) has more affinity towards vermiculite (0.579 mg/g) than Cu(II) (0.513 mg/g). But both metals can be desorbed with presence of other metals in the adsorption system. Vermiculite removed about 88% of lead and 58% of copper at pH 4.

Similarly, Padilla-Ortega et al. (2013) also investigated the simultaneous removal of Cd and Ni onto bentonite in case (iv). Their research findings indicated that Cd(II) has a better affinity to bentonite than Ni(II) in a single-component system. In the competitive adsorption system, both metals displayed antagonistic behavior against the presence of other metals. Both metals have a relatively comparable affinity for bentonite. In case (v) Padilla-Ortega et al. (2013) gave the equilibrium adsorption studies of cadmium and zinc onto sepiolite in a single-component system at pH 7. It was demonstrated that cadmium was adsorbed 1.3 times more than zinc. However, in a competitive adsorption system, the behavior was different. Zn(II) presented assertive antagonistic behavior over Cd(II) adsorption. Cd(II) did not affect the adsorption of Zn(II). Nevertheless, in case (v), the presence of Zn(II) reduced the uptake of Cd(II) up to 16.5 times due to the same cationic sites on the adsorbent.

Roy and Bhattacharya (2013) used maghemite nanotubes to remove Cd(II) and Ni(II) from wastewater of chemical, metallurgical, and textile industries as in case (vi). Binary adsorption investigations showed Cd(II) affinity (88.14 mg/g) was better than that of Ni(II) (70.12 mg/g). The adsorption of a single ion system got better when the pH increased. Cd(II) gets strongly influenced by Ni(II). Correspondingly, Nickel and cobalt adsorption on γ-Fe2O3 nanotubes was also investigated in case (vii) by Roy and Bhattacharya (2013). Ni(II) absorbs way more than Co(II) in a single adsorbent system. The adsorption efficiency from a single to a binary system has almost no change for cobalt. The adsorption capacity for Ni(II) was 65.62 mg/g and for Co(II) 58.83 mg/g in a binary system. In this study, the higher the pH value of a single ion, the better the adsorption. The potential of maghemite nanotubes was also verified for cadmium and copper by Roy and Bhattacharya (2013). Cadmium adsorption was successfully higher than that of cobalt in a single-component system. The adsorption capacity for Cd(II) was 55.87 mg/g and for Co(II) 85.27 mg/g in a binary system. Cd(II) adsorption decreased firmly in the presence of Co(II) in case (viii). The adsorption of a single ion system got better when the pH increased. Table 4 summarizes the above case studies with different experimental conditions, isotherm models, and significance.

4 Interaction Mechanism of Competitive Adsorption

The mechanisms of pollutants are the most significant phenomena in multi-component adsorption equilibria. Gaining a deeper understanding of these mechanisms is essential in achieving better results and analyzing any adsorption system. One of the primary considerations for sorbent design is the van der Waals interactions since adsorption depends on the charge of the sorbate molecules and the atoms on the sorbent surface. The van der Waals radius and particle surface charges determine the interactions. The polarizability of the surface atom, which rises with the atomic weight of elements, enhances the dispersion interaction for a sorbate molecule (Yang 2003). Figure 6 shows the interaction between two adsorbents affecting adsorption in a binary system.

Fig. 6
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Schematic representation of interaction mechanisms of pollutants in a competitive adsorption system

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Synergism is one of the three primary methods through which adsorbate molecules interact. It describes how combining two components in a solution improves the ability of the pollutant to adsorb when compared to using only one (Bohli et al. 2017). Additionally, the process is referred to as no interaction when the presence of another pollutant does not impact how well the other contaminant is adsorbed in a binary solution. Every element of a single system is equally essential to a binary system. The last primary interaction is the opposite of synergism, called antagonistic. When a component is present in a solution with another element, its adsorption ability becomes less effective.

There is synergism when the ratio of the adsorption capacity of a component in a binary system to that in a single system is higher than one. If it is less than one, then there is antagonism. If this ratio is equal to one, then there is no interaction. This is called the additive rule. The relative percent adsorption RQ explains the interaction mechanism used (Saǧ et al. 1998). There is no molecular interaction if the value of RQ is 100. If the value of RQ is below 100, this indicates antagonistic interactions, whereas a value above 100 describes synergistic interactions (Girish 2018). Selectivity is a crucial component in the explanation of the adsorption mechanism. It characterizes a component's discrimination when it competes against two or more other elements in the same solution (Zhang et al. 2019).

The selectivity ratio determines the selective preference of an adsorbent for a component. This ratio decides between binary and single systems based on morphology, surface structure, and pore distribution (Girish 2018). Furthermore, the interactions of molecules are described by the P-factor, a streamlined model that uses a ratio of monolayer capacity (Qmax) for the correlation approach. The P-factor contrasts data from equilibrium with many components and is dimensionless (Choy et al. 2004). Similarly, the sorption affinity helps describe reactions. The adsorption capabilities of adsorbents in a binary system can be compared, and it may be considered for correlation metric with hydrophobicity, van der Waals contacts, and molecular connectivity. Adsorption affinity is expressed by the equilibrium ratio of the solid-phase concentration to the liquid-phase concentration (Saikia and Dutta 2008). Table 5 lists the interaction mechanism for multi-component systems.

Table 5 Expressions of interaction mechanism of a binary system
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5 Modeling of Competitive Adsorption for Various Data

Modeling and simulation consist of performing a sensitivity analysis to ensure that the model parameter values are optimum. Initializing the model parameters with the smallest possible objective function value, such as the sum of square errors (SSE), is the first step in a sensitivity analysis. In the perturbation concept, one parameter is changed over time while the other parameters are held constant. Nonlinear regression analysis finds the best solution for minimizing errors and perturbation concepts. For each perturbation, the objective function is re-estimated, and later, a graph between a model parameter's percentage of perturbation and the accurate function value is drawn. The global minimum is reached if all model parameters are perturbed at 0%; otherwise, it represents a poor model (Mukhtar et al. 2020). In this investigation, the "solver add-in" in Microsoft Excel is employed for fitting and predicting the isotherm models. The adsorption constant ‘K’ values were found using the Excel Solver by maintaining the sum of squared errors between the actual experimental values ‘Qe’ and the predicted values ‘Qe’ model. The errors are minimized using the Excel Solver, and correlation coefficients R2 are calculated. The comparison and the applicability of models for competitive adsorption are also learned.

In this study, the binary experimental data are taken from (Roy and Bhattacharya 2013). The single, binary, and ternary adsorption studies are carried out for cadmium, nickel, and cobalt using maghemite γ-Fe2O3 nanotubes as adsorbents. The extrapolated experimental data set is modeled using a few isotherm models, shown in Table 3. Roy and Bhattacharya (2013) presented the fit of the Langmuir and Freundlich isotherm model for the mono-component adsorption system and under the experimental conditions V = 100 mL, Co = 20 to 400 mg/L, pH = 6 ± 0.1, adsorbent dosage = 0.05 g, T = 30 °C, shaking time = 4 h. A binary combination of cadmium-nickel, nickel–cobalt, and cadmium-cobalt was considered with an equal concentration ratio of 1:1 (Roy and Bhattacharya 2013). The analysis focuses on interpreting the behavior of the interaction between the adsorbent and adsorbate, the adsorption scenario of isotherm models, and the prediction of isotherm models (Qu et al. 2020).

This study uses extrapolated data from Roy and Bhattacharya (2013) for modeling and simulation by four multi-component adsorption isotherm models. The isotherms such as the Modified Competitive Langmuir (MCL) isotherm model (Sohn and Kim 2005), Extended Langmuir Isotherm model (Srivastava and Tyagi 1995), Langmuir–Freundlich isotherm model (Costa et al. 2019), and Extended Freundlich isotherm model (Girish 2019) are used to fit experimental data obtained for cadmium-nickel, nickel–cobalt, and cadmium-cobalt. These models can determine competitive equilibrium, maximum adsorption capacity, and affinity parameters. They are based on nonlinear equation forms instead of linear forms, and the prediction of multi-component adsorption is initialized.

6 Modeling Results

6.1 Competitive Adsorption Isotherms of Cadmium-Nickel

Experimental data from Roy and Bhattacharya (2013) were used in multi-component adsorption modeling. Multi-component models were fitted using  MS Excel Solver by minimizing least square errors. Figure 7a shows the Modified Competitive Langmuir (MCL) isotherm model fit of adsorption of Cd(II) and Ni(II). The actual experimental values Qe and the predicted values Qe model (Fig. 7a) are plotted against the equilibrium concentration Ce, and the graph presents the fitted line and R2. The Modified Competitive Langmuir (MCL) isotherm model equation exhibited fit with a correlation coefficient of 0.99 and 0.99 for Cd(II) and Ni(II), respectively, with minimized errors. The maximum adsorption capacities of the adsorption system were 207.77 mg/g and 196.09 mg/g for Cd(II) and Ni(II), respectively. The high R2 values suggest a good fit in the adsorption system for Cd(II) and Ni(II). The fit is good if the R2 value is greater than 0.95. The K values reflect the affinity of maghemite γ-Fe2O3 nanotubes towards Cd(II) and Ni(II) in the binary system and are 0.022 L/mg and 0.033 L/mg, respectively.

Fig. 7
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a Modified Competitive Langmuir Isotherm, b Extended Langmuir Isotherm, c Langmuir–Freundlich Isotherm, and d Extended Freundlich Isotherm model fit of competitive adsorption of Cd(II) and Ni(II)

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Figure 7b shows the Extended Langmuir Isotherm/Non-modified Langmuir (EL) isotherm model fit of adsorption of Cd(II) and Ni(II). The Qe and Qe model (Fig. 7b) was plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2. The Extended Langmuir Isotherm/ Non-modified Langmuir (EL) isotherm model equation exhibited good fit with R2 values of 0.98 and 0.99 for Cd(II) and Ni(II), respectively, with minimized errors. The maximum uptake of the system was 203.63 mg/g, and 199.06 mg/g for Cd(II) and Ni(II), respectively. In the binary system, K values reflect the affinity of maghemite γ-Fe2O3 nanotubes for cadmium (0.0021 L/mg) and nickel (0.0022 L/mg).

Figure 7c shows the Langmuir–Freundlich isotherm model fit of Cd(II) and Ni(II) adsorption. The graphs present Qe and Qe model plotted against the equilibrium concentration Ce, and show the fitted line equation and R2 (Fig. 7c). The Langmuir–Freundlich isotherm model exhibited fit with R2 values of 0.99 and 0.99 for Cd(II) and Ni(II), respectively, with minimized errors. The maximum adsorption capacities of the adsorption system were 195.54 mg/g and 191.49 mg/g for Cd(II) and Ni(II), with K values of 0.0014 and 0.0027 (mg/g)/(mg/L)1/n, respectively.

Figure 7d shows the Extended Freundlich isotherm model fit of adsorption of Cd(II) and Ni(II). The Qe and Qe model (Fig. 7d) was plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2. The Extended Freundlich Isotherm model equation exhibited an excellent fit with R2 values of 0.99 and 0.99 for cadmium and nickel, respectively, with minimized errors.

6.2 Competitive Adsorption Isotherms of Cadmium-Cobalt

Experimental data from Roy and Bhattacharya (2013) were taken for multi-component adsorption modeling. Multi-component models were fitted using Excel Solver by minimizing least square errors. Figure 8a shows the Modified Competitive Langmuir (MCL) isotherm model fit of adsorption of cadmium and cobalt. The Qe and Qe model (Fig. 8a) was plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2. The Modified Competitive Langmuir (MCL) isotherm model equation exhibited fit with R2 values of 0.95 and 0.94 for cadmium and cobalt, respectively, with minimized errors. The maximum adsorption capacities of the adsorption system were 200.20 mg/g and 205.87 mg/g for Cd(II) and Co(II), respectively. In the binary system, the K values of Cd(II) (0.028 L/mg) and Co(II) (0.027 L/mg) reflect the affinity towards maghemite γ-Fe2O3 nanotubes.

Fig. 8
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a Modified Competitive Langmuir Isotherm, b Extended Langmuir Isotherm, c Langmuir–Freundlich Isotherm model, and d Extended Freundlich Isotherm model fit of competitive adsorption of Cd(II) and Co(II)

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Figure 8b shows the Extended Langmuir/Non-modified Langmuir (EL) isotherm model fit of adsorption of Cd(II) and Co(II). The actual experimental values Qe and the predicted values Qe model (Fig. 8b) were plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2. The Extended Langmuir Isotherm/ Non-modified Langmuir (EL) isotherm model equation exhibited fit with correlation coefficient values of 0.95 and 0.94 for Cd(II) and Co(II), respectively, with minimized errors. The maximum adsorption capacities of the adsorption system were 199.64 and 206.42 mg/g for Cd(II) and Co(II), respectively. Cadmium and cobalt affinities to maghemite γ-Fe2O3 nanotubes were 0.0046 and 0.0047 L/mg, respectively.

Figure 8c shows the Langmuir–Freundlich isotherm model fit of adsorption of Cd(II) and Co(II). The graph consists of the Qe and Qe model plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2 (Fig. 8c). The Langmuir–Freundlich isotherm model exhibited satisfactory fit with an R2 values of 0.97 and 0.96 for Cd(II) and Co(II), respectively, with minimized errors. The maximum uptake of this system was 189.20 mg/g and 209.74 mg/g, and the K values were 0.0028 and 0.0060 (mg/g)/(mg/L)1/n for cadmium and cobalt, respectively.

Figure 8d shows the Extended Freundlich isotherm model fit of adsorption of Cd(II) and Co(II). The actual experimental values Qe and the predicted values Qe model (Fig. 8d) were plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2. The Extended Freundlich Isotherm model equation exhibited good fit with R2 values of 0.96 and 0.98 for cadmium and cobalt, respectively, with minimized errors.

6.3 Competitive Adsorption Isotherms of Nickel–Cobalt

Experimental data from Roy and Bhattacharya (2013) was taken for multi-component adsorption modeling. Multi-component models were fitted using the Excel Solver by minimizing the least square errors. Figure 9a shows the Modified Competitive Langmuir (MCL) isotherm model fit of adsorption of nickel and cobalt. The Qe and Qe model (Fig. 9a) was plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2. The Modified Competitive Langmuir (MCL) isotherm model equation exhibited fit with R2 values of 0.98 and 0.99 for nickel and cobalt, respectively, with minimized errors. The maximum adsorption capacities of the adsorption system were 176.01 and 124.93 mg/g for Ni(II) and Co(II), respectively. In the binary system, the K values of Ni(II) (0.018 L/mg) and Co(II) (0.032 L/mg) reflect the affinity towards maghemite γ-Fe2O3 nanotubes.

Fig. 9
figure 9

a Modified Competitive Langmuir Isotherm, b Extended Langmuir Isotherm, c Langmuir–Freundlich Isotherm, and d Extended Freundlich Isotherm model fit of competitive adsorption of Ni(II) and Co(II)

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Figure 9b shows the Extended Langmuir Isotherm/Non-modified Langmuir (EL) isotherm model fit of adsorption of nickel and cobalt. The Qe and Qe model (Fig. 9b) was plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2. The Extended Langmuir Isotherm/ Non-modified Langmuir (EL) isotherm model equation exhibited fit with R2 values of 0.98 and 0.99 for nickel and cobalt, respectively, with minimized errors. The maximum adsorption capacities of the system were 147.21 and 144.61 mg/g for Ni(II) and Co(II), respectively. The high R2 value indicates a good fit in the adsorption system for nickel and cobalt. In this binary system, the K values reflect the affinities of maghemite γ-Fe2O3 nanotubes towards Ni(II) and Co(II), which were 0.0034 L/mg and 0.0031 L/mg, respectively.

Figure 9c shows the Langmuir–Freundlich isotherm model fit of adsorption of Ni(II) and Co(II). The graph presents Qe and Qe model plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2 (Fig. 9c). The Langmuir–Freundlich isotherm model exhibited fit with R2 values of 0.98 and 0.99 for nickel and cobalt, respectively, with minimized errors. The maximum adsorption capacities of the adsorption system were 165.13 mg/g and 134.10 mg/g, and K values were 0.0034 and 0.0032 (mg/g)/(mg/L)1/n for Ni(II) and Co(II), respectively.

Figure 9d shows the Extended Freundlich Isotherm model fit of adsorption of Ni(II) and Co(II). The Qe and Qe model (Fig. 9d) is plotted against the equilibrium concentration Ce, and the graph presents the fitted line equation and R2. The Extended Freundlich Isotherm model equation exhibited fit with R2 values of 0.98 and 0.98 for Ni(II) and Co(II), respectively, with minimized errors.

The four multi-component isotherm models modeled the binary adsorption system for cadmium-nickel, nickel–cobalt, and cadmium-cobalt. The isotherms such as the Modified Competitive Langmuir (MCL) isotherm model (Sohn and Kim 2005), Extended Langmuir Isotherm model (Srivastava and Tyagi 1995), Langmuir–Freundlich isotherm model (Costa et al. 2019), and Extended Freundlich isotherm model (Girish 2019) are modeled with the equations given in Table 3. As indicated in Table 6, the parameters are predicted using non-linearized equations by reducing errors. In this binary system of Cd(II)-Ni(II), the maximum uptake is by Cd(II), and the maghemite γ-Fe2O3 nanotubes have more affinity towards the Cd(II) than Ni(II), as in Table 4. The inferences of modeling studies of pollutants are given in Table 7. Similarly, in the competitive system of Ni(II)-Co(II) and Cd(II)-Co(II), Ni(II) adsorbed more than Co(II), and Co(II) uptakes more than Cd(II), respectively. The findings show that adsorption behavior in binary component systems is complicated, as it is expected. Metal ion uptake and characteristics are thought to be related by many parameters (Manning and Goldberg 1996). Competitive adsorption is caused by factors such as electronegativity of the Ni(II), Co(II), and Cd(II) which is 1.91, 1.88, and 1.69, respectively. Similarly, electrostatic attrition is caused by the charge-to-radius ratio, capacity to form metal hydroxide complex, and appropriate active site for adsorption on the adsorbent (Kong et al. 2020). The hydrated radii of Ni(II), Co(II), and Cd(II) are 4.04 Å, 4.23 Å, and 4.26 Å, respectively (Roy and Bhattacharya 2013). As a result, Ni(II) has a better probability than Co(II) and Cd(II) of accessing active sites and pores more quickly. Ionic radii of Ni(II), Co(II), and Cd(II) are 0.69 Å, 0.745 Å, and 0.97 Å, respectively. The adsorption rate decreases as the ionic radius increases (Mustafa et al. 2013). All four competitive isotherm models gave a good fit, as shown by the R2 values. This indicates that several isotherms may provide a good fit for multi-component isotherms. However, parameters from single-component systems cannot be directly used to model multi-component systems. Cd(II) exhibits a higher affinity in a single-component solution than Ni(II) for maghemite nanotubes. In a single-component solution, the success of Cd(II) adsorption on maghemite nanotubes is higher than that of Co(II). When Co(II) is present, Cd(II) adsorptions are strongly reduced. On γ-Fe2O3 nanotubes, Ni(II) absorbs more than Co(II) in a single adsorbent system. In a binary system, Co(II) adsorption effectivity changes.

Table 6 Modeling results of adsorption in the binary system of pollutants
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Table 7 Inferences of modeling studies of pollutants
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7 Discussion

Many of the contaminants are found to occur simultaneously in nature. For example, arsenic and fluoride, lead and zinc, vanadium and chromium, zinc and nickel, cadmium and zinc are co-occurring in groundwater. In the case study of Cd(II)-Ni(II) that we considered for modeling, Cd(II) showed a higher affinity than Ni(II). Similarly in the competitive adsorption systems of Ni(II)-Co(II) and Cd(II)-Co(II), Ni(II) adsorption was more than Co(II) adsorption, and Co(II) adsorption was more than Cd(II), respectively. Overall, in any multi-component adsorption system, presence of one contaminant in the solution can impact the adsorption of other pollutants. For instance, arsenic shows antagonistic behavior with fluoride during simultaneous adsorption. Zinc is also known to show intense antagonism in the presence of nickel. Identifying the behaviors of synergistic, antagonistic, and no- interaction of contaminants is possible by experimental studies and from inferences of previous research papers. The intricacy of competitive adsorption depends on several variables, like the competition to reach the active sites of the adsorbent surface, the component ionic radii and their bonding. Surface charge of the adsorbent will vary with the pH of the solution. The concentration of a particular hydrated specie will also change with the pH and with the concentration of other ions. Hence, pH will influence the adsorption significantly. When one component's concentration rises, another component's adsorption reduces, and interaction between the species in the solution determines how one component influences another (Roy and Bhattacharya 2013). The displacement effect, a changed adsorbent surface, and the addition of extra components to inhibit the active sites can change how adsorbent and adsorbate interact (Girish 2018). Multi-component adsorption equilibrium systems are dependent on  the functional group, structure, shape, and size of adsorbents, the electrostatic force of attraction for adsorption, the capacity to generate hydroxides, the ionic characteristics of pollutants, the hydrated radii, molecular weight, and the size of adsorbates.

The multi-component isotherm models are primarily based on assumptions and experimental conditions for interference and competition. The behavior of the Langmuir and Freundlich isotherms are combined in the Non-Modified Competitive Redlich Peterson Isotherm model. In some cases, this model behaves as both models. An extra-term interaction component is introduced into the Modified Competitive Redlich Peterson Isotherm model to better describe the competitiveness between the adsorbents and adsorbates. An enhanced version of the Jain-Snoeyink model, the modified Jain-Snoeyink Isotherm model includes the interaction term in the equation (Sheng et al. 2007). Under these conditions, a component's maximal adsorption capacity must be more significant than its counterpart. The Steric Mass Action Isotherm Model aims to adsorb proteins (Russell and Carta 2005). It was first applied to single-component adsorption, and then, it was further employed for multi-component adsorption, which proved ineffective. Other isotherm models have since been explicitly used in appropriate adsorption systems.

The specific equilibrium condition, the maximum adsorption (qmax), the concentration of molecules (C), the interaction factor (η) which depicts the competitive impact of adsorbates in the solution, and K which denotes the Langmuir constant for components are evaluated from the experimental data in the modified competitive Langmuir isotherm model. In the extended Langmuir isotherm model, the adsorbates have a non-interacting effect. The last assumption is that the pollutant molecules adsorb on the active sites. The drawback of this model is that the values of the isotherm parameters for various adsorbents might be drastically different. The formula has the same variables as the modified competitive Langmuir isotherm model without the interaction factor. A disadvantage of the extended Freundlich isotherm model is that additional fitting parameters are required for a binary system. The parameters K and n describe the isotherm constants for components, C is the concentration of components in an aqueous solution at equilibrium, and x, y, and z express the isotherm constants for components. The Langmuir–Freundlich isotherm model is a combination of Freundlich and Langmuir isotherm models. It could be possible to get better outcomes by merging the two models. The Langmuir–Freundlich model is generated with all parameters based on multi-component equilibrium data. The parameter KLF defines the intensity of adsorbent and adsorbate interactions, and n represents the heterogeneity and favorability.

Though widely used, adsorption isotherm models are black box type models that do not consider the mechanisms of adsorption. Adsorption isotherm models give good predictions for single-component adsorption systems, but they are unable to capture the complexities of multi-component systems. The limitations of a multi-component adsorption models are that they are unable to predict the interactions of different components (such as synergistic effects) and competitive effects. Most isotherm models cannot explain the competitive characteristics in multi-component adsorption systems, such as preferential adsorption, antagonism, and synergistic adsorption. Model parameters from single-component isotherms systems such as Langmuir and Freundlich cannot be used in multi-component systems, even for similar systems. Additionally, intrinsic adsorption heterogeneity becomes more complicated if the adsorbent is microporous (Jadhav and Srivastava 2019). Factors such as surface charge, structure, size, functional groups of the component, porosity and active sites on the adsorbent surface affect the components to each other in a system. Surface complexation modeling is a complicated method that can be used to determine multi-component adsorption surface reaction mechanisms (Jeppu and Clement 2012). However, the Log K values have to be determined for each system. It is essential to understand the effects of aqueous complexation between metals and ligands and the impact of surface charge change on adsorption owing to differences in solution ionic strength (Gun’ko 2007). Multi-component equilibrium adsorption takes account of a constant separation factor when one or several components are poorly adsorbed (Girish 2018). The pH of each isotherm is intended to remain constant, since pH is crucial to the adsorption system. No pH-dependent multi-component isotherm model systems exist (Pereira et al. 2023). Finding a suitable model for the competitive adsorption behavior of a component mixture is challenging since most existing non-linear models rely heavily on single-component data (Zhu et al. 1991).

8 Conclusions

Though there are several drinking water contaminants such as arsenic  and fluoride, which are known to occur simultaneously in nature, there is inadequate research on multi-component isotherm models for predicting simultaneous adsorption systems. In this review paper, models of multi-component systems were compiled, and case studies were considered. Several multi-component adsorption isotherms and competitive mechanisms and constraints were analyzed. Numerous adsorption models, including the Henry isotherm model, Langmuir isotherm model, Freundlich isotherm model, Langmuir Freundlich hybrid, Brunauer-Emmett-Teller, Modified Competitive Langmuir, Langmuir Extension First Approximation Model, Freundlich-Langmuir-Jovanovic, Modified Langmuir Freundlich, LeVan-Vermeulen model, were reviewed along with their equations and significance. The equations, parameters, and relevance of each isotherm model are summarized in this paper in Table 2 and Table 3.

In this study, the competitive adsorption of commonly found multi-component contaminant adsorption systems of cadmium-nickel, nickel–cobalt, and cadmium-cobalt is  considered and predicted using the various multi-component adsorption models such as Modified Competitive Langmuir (MCL), Extended Langmuir Isotherm (EL) model, Langmuir–Freundlich isotherm model, and Extended Freundlich Isotherm model (Table 6). The adsorbent in the multi-component system of Cd(II)-Ni(II) showed a higher affinity for Cd(II) than Ni(II), with a maximum absorption of 207.77 mg/g by Cd(II). Similarly, in the competitive adsorption of Ni(II)-Co(II) and Cd(II)-Co(II), Ni(II) adsorption was 176.01 mg/g, which is more than Co(II) adsorption, and Co(II) absorbed 205.87 mg/g, which is more than Cd(II), respectively. This suggests that some components are preferably adsorbed over others. The reasons for preferable adsorption and antagonism have to be further researched. The prediction of multi-component adsorption was done using isotherm models. Microsoft Excel Solver was used to determine the equilibrium systems' maximum adsorption capacity (Qm) and parameters such as K, n, and ƞ using the MS Excel Solver addin. The multi-component isotherm models provided a good fit as suggested by the coefficient of determination (R2) values. This suggests that several multi-component isotherms can provide a good fit for simultaneous adsorption experimental data though they do not consider the mechanism of adsorption, if they are fit for each dataset seperately.

Theoretical adsorption isotherm models are important for designing treatment plants, filtering units, and packed bed column studies. The knowledge of competitive isotherm models is needed in the design of multi-component adsorption and treatment systems. Current multi-component isotherm models are unable to  represent the interactions of various components such as synergistic, antagonistic and competitive effects.  Further advanced research is also required to elucidate the multi-component interaction mechanisms and competitive adsorption mechanisms at the surface, ionic and molecular level. Moreover, currently none of the multi-component isotherms are able to model adsorption under different pH conditions. So, further investigation has to be done to develop new equations for multi-component adsorption of co-occurring contaminants to overcome the above limitations.