Introduction

The presence of heavy metals in water sources poses a serious environmental and public health threat due to their toxicity, persistence, and bio accumulative nature1,2. Primarily originating from industrial and agricultural sources, these metals are often discharged above permissible limits, leading to contamination of water resources and the food chain3,4,5,6,7. Industries such as steel production, petrochemicals, agriculture, and mining require sustainable strategies, advanced treatment technologies, and strict compliance with environmental regulations to mitigate this pollution8. Heavy metals also possess significant economic value due to their hardness, conductivity, and high corrosion resistance9. Increasing demand across various industries9, coupled with the depletion of natural reserves8, highlights the need to recover these metals from secondary sources such as industrial wastewater, thereby supporting sustainable development and enhancing economic returns. Given the company’s focus on advanced materials and chemical technologies, strict environmental monitoring of heavy metal emissions is essential. Initial monitoring indicated that concentrations of these metals exceeded the limits set by environmental regulations. Therefore, the primary objective of this research is to assess the efficiency of the adsorption process in reducing Cu, Ni, Cd, and Zn concentrations to acceptable levels and to propose a practical treatment solution for this industrial effluent. Cu is widely used in various industries such as electroplating and plastic production6,10. Although it is biologically essential, excessive exposure has toxic effects4,11, including Wilson’s disease, liver and kidney dysfunction, insomnia, gastrointestinal disorders, lung cancer6,10,12, disruption of food chains, mutagenic effects13,14, Alzheimer’s disease12, and ecological damage to marine environments15. Zn is used in coatings, alloys, cosmetics, the food industry, and etc16,17. While it is essential in trace amounts8, high levels can cause stomach pain, vomiting, skin disorders, anemia, nausea, depression, neurological issues, and appetite loss10,18. Major sources include metal industries, coal combustion, waste incineration, and tire wear10,19. Ni is widely used in stainless steel, batteries, and electronics due to its corrosion resistance and conductivity9,20. If released into soil or water, it threatens ecosystems and enters the human body via inhalation, ingestion, or dermal contact21. Toxic effects include lung inflammation, allergic reactions, immune system disorders, and lung or nasal sinus cancers22. Cd originates from natural sources, but its major environmental release stems from mining and smelting activities23. Its high mobility enables contamination of water and soil23, entering the food chain and causing renal damage, bone disorders, and cancer24. Therefore, effective removal techniques are necessary25. Several techniques are currently employed to remove heavy metals from industrial effluent, including ion exchange, chemical precipitation, filtration, membrane processes26, adsorption27, electrochemical techniques, and advanced oxidation processes28, all of which have been developed for heavy metal removal2,8,29. Within these, adsorption is validated as the most efficient and cost-effective approach due to its simple design, affordable, high performance, and ability to eliminate trace contaminants (particularly when affordable adsorbents obtained from agricultural practices and industrial by-products are used)4,8,10,16,18,19,30. Ion exchange, considered a subtype of adsorption, is widely used in water treatment. It involves the exchange of dissolved ions with oppositely charged functional groups on a resin placed in a column31. Weakly acidic resins, such as CH030, are especially effective for recovering valuable metal ions such as Ni, Cu, Cd, and Zn. The performance of CH030 is attributed to its amino phosphonic functional groups9. Key parameters affecting adsorption efficiency (such as solution chemistry, adsorbent dosage, feed flow rate, column height, temperature, pH, and initial metal ion concentration) have been extensively studied1,8,16.

Previous studies by Singh et al. demonstrated that increasing column height significantly enhanced arsenic adsorption capacity, whereas higher inlet flow rates and initial concentrations led to reduced performance due to shorter contact time and rapid saturation of active sites32. Hameed et al. demonstrated that increasing the initial lead (II) concentration and feed flow rate accelerated column saturation and decreased adsorption capacity. However, increasing the column height extended contact time and improved lead removal efficiency33. Bayuo et al. investigated the optimization of key parameters affecting the adsorption of lead (II) from aqueous solutions using peanut shell biomass. The results indicated that a rise in the initial concentration resulted in a reduction in adsorption capacity and, consequently, a decrease in adsorption efficiency34. Sheibani et al. investigated iron (III) removal from aqueous solutions using hazelnut shell biomass. The column showed higher adsorption efficiency at lower initial concentrations, while saturation of adsorption sites at increased concentrations led to decreased performance35. Mehrmand et al. studied the adsorption of Pb(II), Cu(II), and Ni(II) using modified henna powder and found that higher initial metal concentrations led to saturation of adsorption sites, significantly reducing removal efficiency36. Dissanayake et al. demonstrated through simulation that increasing column height enhances contact time and adsorption efficiency, while increasing feed flow rate reduces contact time and removal efficiency37. Numerous studies have used RSM to optimize heavy metal adsorption by evaluating the interactive effects of process parameters. Zhou et al.38 utilized central composite design to optimize green adsorbents. Sathish et al.39 applied the Taguchi method to assess Cd removal using bacterial adsorbents. RSM-BBD was employed by Hosseinzadeh et al.40, Etemadi et al.41, and Niknam et al.42 to model and optimize Cd and arsenic removal using plant-based adsorbents. Jafari et al.43 also used RSM-BBD to optimize lead removal using bacterial biomass, demonstrating its strength in multivariable optimization.

The novelty of the present study lies in the integrated application of RSM and Aspen adsorption simulation to optimize and model the simultaneous removal of four heavy metals (Cu, Ni, Cd, Zn) from industrial wastewater using the weak-acid resin CH030. Unlike previous studies that typically focus on single-metal adsorption or rely solely on experimental trials, the present study uniquely combines statistical design of experiments, simulation-based modeling, and multivariable optimization within a single framework. In the first stage, RSM was used to systematically design the experiments, determine the optimal number of tests, and identify key operational variables, including column height, feed flow rate, and ion concentration. Subsequently, the adsorption process was modeled using Aspen adsorption software, and breakthrough curves were generated to assess system performance. A comprehensive RSM analysis was then conducted with three input variables (column height, feed flow rate, and ion concentration) and five output responses (the outlet-to-inlet concentration ratios for Ni, Cd, Cu, Zn, and total metals). This integrated approach allowed for a more detailed investigation of the simultaneous effects of operational parameters and led to the identification of optimal conditions for maximum adsorption efficiency. The methodology not only enables the prediction of breakthrough behavior under various operational conditions but also facilitates the identification of the optimal column height, feed flow rate, and ion concentration to maximize removal performance. This comprehensive framework offers a scalable and reproducible strategy for dynamic modeling and optimization of multi-metal wastewater treatment, contributing significantly to advancements in adsorption process modeling and environmental remediation.

Materials and methods

Adsorption

In this process, wastewater originating from an industrial facility containing heavy metals such as Cu, Ni, Cd, and Zn is introduced into an adsorption column, to achieve effluent concentrations within the regulatory limits established by the united states environmental protection agency (USEPA). To meet this goal, a packed bed column was designed, and key operational parameters, including column height, feed flow rate, and initial metal concentrations were investigated. The process uses adsorption, with the CH030 resin serving as the adsorbent. CH030 is a chelating resin that contains weakly acidic amino phosphonic groups (− CH2NCHH2PO3−) within a styrene and-divinylbenzene copolymer structure characterized by a specific microporous formation. This resin exhibits the ability to adsorb and chelate divalent cations across a wide range of solutions, including those with high ion concentrations. The relationship of the CH030 functional groups with metal cations is given in Eq. (1)9.

$${\left[ { - C{H_2}NCH{H_2}P{O_3}^ - } \right]^{2 - }} + {\text{ }}{M^{2 + }} \to {\text{ }}\left[ { - C{H_2}NCH{H_2}P{O_3}M} \right]$$
(1)

Optimization of process performance requires accurate determination of the optimal values for the aforementioned parameters. Achieving such conditions can significantly enhance the efficiency of ion separation, improve chelating resin performance, and increase both adsorption capacity and selectivity. To facilitate this, Aspen Adsorption software was utilized for process simulation, and RSM was employed to assess the effects of the variables and identify the optimal operating conditions.

Adsorption simulation

Aspen adsorption is a comprehensive software tool for modeling and simulating gas and liquid adsorption processes. It provides the capability to accurately evaluate the dynamic mass and energy balances throughout the complete adsorption and desorption cycle, owing to its robust framework of numerical and physical methods44. The software is specifically designed for simulating adsorption and ion exchange processes and has become widely adopted by engineers and researchers due to its intuitive graphical interface, which facilitates the design of process flow diagrams and interpretation of simulation results31,45. Furthermore, support for the IonX platform enhances its ability to accurately model ion exchange mechanisms and analyze multi-component isotherms31. In this study, Aspen adsorption was used to simulate the elimination of Ni, Cu, Zn, and Cd ions from industrial wastewater through the ion exchange method using a resin bed composed of CH030 resin provided by Canftech. The IonX bed feature within the software was utilized to carry out the simulation in Fig. 1.

Fig. 1
Fig. 1The alternative text for this image may have been generated using AI.
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Packed column simulation of the adsorption process.

The desired parameters, including bed height, feed flow rate, and input concentration, were configured based on the optimal conditions derived from the RSM method. In the feed and product blocks, specifications such as initial metal ion concentration, initial counterion concentration, and flow rate were defined. The metal ions and the counterion were the only components specified in the list, as water is included as the default component in this software. A counterion component was added to monitor its behavior during the adsorption of a corresponding metal ion, providing insight into adsorption rate and resin utilization before exhaustion. In the column specifications section, the governing parameters, solution algorithms, and isotherm models were selected to ensure optimal simulation conditions. The general settings included configuring the number of nodes and selecting the upwind differencing scheme (UDS1) to simulate the ion-exchange process46. This scheme was selected due to its high numerical stability and acceptable accuracy, particularly under varying flow conditions. However, increasing the number of nodes improves numerical accuracy by reducing the step size47. Thus, the appropriate selection of node count and discretization methods has a significant impact on both accuracy and computational efficiency31,46. To simulate the ion-exchange process, the material and momentum balance assumption was based on convection with estimated dispersion33,46. In this approach, the dispersion coefficient is allowed to vary along the bed length31. The software consolidates all resistances into a single overall resistance to model mass transfer46. enhancing simulation stability and realism across different scenarios47. The mass action equilibrium model was selected as the isotherm section. This isotherm model was selected due to its simplicity and low parameter requirement, making it suitable as a first approximation. However, it is important to note that this model is a simplification and may not accurately represent all adsorption systems47. The aggregated linear resistance kinetic model was chosen for simulation, with the mass transfer driving force defined by the concentration gradient in either the liquid or solid phase. For the film diffusion model, the driving force was defined based on concentration differences in the solid phase47. Additionally, internal diffusion resistances within the resin particles were modeled as a single lumped resistance term31. The mass transfer coefficient for the ions of Cu, Ni, Cd, and Zn in a fixed-bed column was calculated using the following Sherwood correlation33:

$$\:Sh=2+1.1{Re}^{0.6}{Sc}^{\frac{1}{3}}$$
(2)

Sherwood correlations vary depending on flow regime and geometry. Laminar flow typically exhibits different characteristics compared to turbulent flow, and this affects the correlations. Each Sherwood correlation may be designed for specific geometries, such as flat plates, tubes, or packed beds. For example, the correlation used for a flat plate is different from the one designed for a spherical particle. In this case, because the flow is laminar and the adsorbent particles are spherical, the above equation for Sherwood is applied at low Reynolds numbers. The number 2 represents mass transfer due to molecular diffusion, while the continuation of the equation accounts for mass transfer due to forced convection48. In this equation, \(\:Sh={k}_{c}{D}_{p}/{D}_{AB}\), \(\:Re={D}_{p}u\rho\:/\mu\:\), and \(\:Sc=\mu\:/\rho\:{D}_{AB}\). By substituting the definitions of Re, Sc, and Sh into the given equation, a new expression is obtained33:

$$\:\frac{{k}_{c}{D}_{p}}{{D}_{AB}}=2+1.1{\left(\frac{{D}_{p}u\rho\:}{\mu\:}\right)}^{0.6}{\left(\frac{\mu\:}{\rho\:{D}_{AB}}\right)}^{\frac{1}{3}}$$
(3)

In this formula, \(\:\text{D}\text{p}\) shows the particle diameter, \(\:{\text{D}}_{\text{A}\text{B}}\) represents the diffusion coefficient of the solute, \(\:\text{u}\) donates the fluid velocity, \(\:{\uprho\:}\) shows the solution density, \(\:{\upmu\:}\) is the solution viscosity, and kc is the mass transfer coefficient33. It is assumed that the solution is highly diluted, so the presence of metal ions does not influence its density or viscosity. Therefore, the density and viscosity of water at ambient temperature were considered. At this temperature, water has a density of 1003 \(\:kg.{m}^{-3}\)and a viscosity of 8.94 × 10⁻⁴ \(\:kg.{m}^{-1}.{s}^{-1}\)33,49. The diffusion coefficient DAB represents the mass diffusion of solute A in solvent B. However, in aqueous solutions, electrolytes dissociate into cations and anions. As a result, the entire diffusion of the electrolyte is set by the combined diffusion coefficients of both the cation and anion. Using the equation above, the total mass diffusion of the electrolyte in water was calculated as follows49.

$$\:{D}_{AB}=\frac{{n}_{+}+{n}_{-}}{\frac{{n}_{-}}{{D}_{+}}+\frac{{n}_{+}}{{D}_{-}}}$$
(4)

Here, \(\:{\text{n}}_{+}\) and \(\:{\text{n}}_{-}\) represent the charge of the cation and anion, respectively, while \(\:{\text{D}}_{\text{i}}\) denotes the diffusion coefficient of each individual ion. The diffusion coefficients of the dissociated ions for Cu²⁺, Ni²⁺, Cd²⁺, Zn²⁺, and NO₃⁻ were determined as 1.30 × 10⁻⁹ m²/s, 0.71 × 10⁻⁹ m²/s, 0.70 × 10⁻⁹ m²/s, 0.70 × 10⁻⁹ m²/s, and 1.902 × 10⁻⁹ m²/s, respectively. Consequently, the resulting mass diffusivity (\(\:{\text{D}}_{\text{A}\text{B}}\)) was calculated as 1.65 × 10⁻⁹ m²/s for Cu, 1.21 × 10⁻⁹ m²/s for Cd, 1.21 × 10⁻⁹ m²/s for Zn, and 1.21 × 10⁻⁹ m²/s for Ni. Similarly, the mass transfer coefficients for other metal ions were determined. The parameters related to the CH030 resin were obtained from Canftech company data. According to their specifications, a particle size range of 0.45 to 0.85 mm and a capacity of 1.2 eq/L. After conducting simulations, breakthrough curves were plotted. The results from Aspen Adsorption were analyzed using the RSM method to assess the simultaneous impacts of the autonomous variables, and the optimal conditions were identified through this software.

RSM modeling

RSM is a widely adopted statistical approach used for optimizing and modeling complex systems involving multiple interacting factors. It enables the analysis of the effects of independent variables on outcomes while requiring a reduced number of experimental runs. The methodology was significantly advanced by Box and Behnken, making it a standard tool across engineering, chemistry, and environmental studies50,51,52. The application of RSM typically follows a structured four-step framework: designing statistically reliable experiments with an appropriate sample size, constructing a predictive mathematical model with strong correlation accuracy, identifying optimal conditions to either maximize or minimize responses, and interpreting the individual and interactive effects of input variables. These stages collectively allow for precise modeling of nonlinear and interactive systems50. RSM serves as a powerful regression analysis technique that defines relationships between dependent and independent variables. Model selection depends on the nature of the system: linear regression is applied when relationships are direct, while multiple regression is employed for more complex interactions involving several independent variables. One of the most widely used experimental designs within RSM is the CCD, which facilitates empirical modeling and optimization, particularly useful for cases like Ni, Cu, Cd, and Zn adsorption, where minimizing the number of experimental runs is beneficial53,54. The quadratic equation commonly used to estimate the combined and interactive effects of input variables is shown in Eq. (5).

$$\:y={\beta\:}_{0}+\:\sum\:_{i=1}{\beta\:}_{i}{X}_{i}+\:\sum\:_{i=1}{\beta\:}_{ii}{X}_{i}^{2}+\:\sum\:_{i=1}\sum\:_{j=i+1}{\beta\:}_{ij}{X}_{i}{X}_{j}+\:\epsilon\:\:$$
(5)

In this equation, y represents the estimated response, β0 is the intercept term, Xi​ and Xj are the autonomous variables, and βii​ and βij​ are the interaction coefficients, in the same order. Additionally, ε is an unexpected variable that is established empirically9. The model’s predictive performance is typically validated using the coefficient of determination (R²), which measures the agreement between predicted and observed values, as given in Eq. (6)9,50,53,55,56:

$$\:{R}^{2}=\:\sum\:_{i=1}^{n}\frac{{{(X}_{predicted}-{X}_{actual})}^{2}}{{{(X}_{predicted}-{X}_{mean})}^{2}}$$
(6)

This metric evaluates the predictive reliability of the RSM model9,50,53,55,56. where Xactual and Xpredicted are the empirical and the estimated parameters by RSM, in the same order. Xmean is the average of the data and indicates the quantity of data values9. Table 1 provides the range settings for the independent variable, including column height, initial ion concentration and feed rate. Meanwhile, Table 2 outlines the corresponding output variables ranges, allowing for a comprehensive and systematic investigation of the design space. Determining appropriate factor levels in RSM involves multiple considerations, including feasibility, range, relevance, prior research insights, and statistical techniques. Feasibility ensures practical implementation within experimental constraints, while an adequately broad range captures potential nonlinear effects. Additionally, selecting levels based on prior studies prevents redundancy and enhances research validity. Statistical methods, such as design of experiments, optimize level selection to ensure sufficient precision in detecting significant relationships50. The selection of independent factors in RSM is guided by several criteria: relevance to the study, controllability within the experimental setup, classification as fixed or random effects, resource constraints, statistical significance, and potential interactions. Relevant factors directly impact the response variable, while controllable factors allow systematic manipulation for experimental accuracy. Statistical techniques like analysis of variance (ANOVA) assist in recognizing key factors and excluding insignificant ones. Furthermore, interrelations among variables are critical as they provide deeper insights into system behavior55. By considering these principles, RSM enables researchers to design experiments that are both efficient and applicable to real-world scenarios, ensuring valid conclusions and practical recommendations. Figure 2 illustrates the stepwise methodology employed for simulation and optimization using RSM and Aspen adsorption.

Table 1 The adoption process input factor.
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Table 2 The adoption process responses.
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Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
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Flowchart of the methodological framework.

Results and discussion

RSM results

In this study, the simultaneous elimination of Ni, Cd, Cu, and Zn ions from industrial wastewater at the same time was simulated using a fixed-bed adsorption column packed with CH030 resin. Initially, the number of experiments and the values of key process parameters, including bed height, feed flow rate, and preliminary ion concentration, were determined using RSM, based on relevant literature and empirical data. The software generated a total of 20 experimental runs, which are summarized in Table 3.

Table 3 The experimental runs determined using RSM.
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Subsequently, the 20 experimental runs were simulated using the Aspen Adsorption software. The bed height, flow rate and the concentration of each metal ion were recorded at multiple time intervals, and breakthrough curves were plotted. The simulation results were then analyzed using RSM with three input parameters A (bed height), B (initial concentration) and C (feed flow rate). Five output responses were evaluated, corresponding to the ratios of outlet to inlet concentrations for Ni, Cd, Cu, Zn, and total metal ions. The combined effects of these variables were investigated, and the optimal operating conditions were determined. In this process, outlier data points that did not conform to the overall pattern of the results were identified and removed from the model to prevent their adverse impact on model fitting. This procedure reduced the error and enhanced both the predictive accuracy and statistical validity of the analysis.

The RSM-based optimization identified the best set of operating conditions according to the experimental results55. In this study, modified quadratic models were used to describe the adsorption behavior of the target metals, and the results demonstrated that these models exhibited the best agreement with the simulated data57. The performance and reliability of the quadratic regression models were validated using ANOVA, incorporating statistical indicators such as the F-value, p-value9,58. For all five responses (C/C₀ of Cd, Cu, Ni, Zn, and total) the models demonstrated very high F-values and p-values below 0.0001, confirming their strong statistical significance59. The coefficients of determination (R²) values were also high for all responses: 0.9951 (Cu), 0.9939 (Ni), 0.9941 (Cd), 0.9941 (Zn), and 0.9943 (total), further supporting the robustness of the models. The regression equations, presented in coded form in Eq. 7 to 11, describe the fitted models but should not be directly used to interpret the relative influence of the parameters, as the coefficients are scaled around the center point of the design space9. However, an examination of Eq. 7 to 11 enables the identification of the most influential variables for each response. According to the equations, for Cu, Cd, Zn, and total metals, the dominant parameter was identified as feed flow rate but for Ni was identified as initial concentration. The equations provide linear coefficients separately from interaction and quadratic coefficients. If we only consider the linear coefficients, we may misinterpret the results, as these coefficients represent only a small portion of the overall effect.

$$\begin{gathered} \frac{{C\left( {Cd} \right)}}{{{C_0}\left( {Cd} \right)}} = 1.06 - 0.0651\;A + 0.0906\;B + 0.1354\;C + \;0.4901\;AB - 0.3816\;AC \hfill \\ + 0.0038\;BC - 0.0944\;{A^2} - 0.4906\;{B^2} - 0.1973\;{C^2} \hfill \\ \end{gathered}$$
(7)
$$\begin{gathered} \frac{{C\left( {Cu} \right)}}{{{C_0}\left( {Cu} \right)}} = 1.07 - 0.0543\;A + 0.0804\;B + 0.1448\;C + 0.5046\;AB - 0.4098\;AC \hfill \\ - 0.0015\;BC - 0.0820\;{A^2} - 0.5189\;{B^2} - 0.2084\;{C^2} \hfill \\ \end{gathered}$$
(8)
$$\begin{gathered} \frac{{C\left( {Ni} \right)}}{{{C_0}\left( {Ni} \right)}} = 1.05 - 0.1035\;A + 0.1222\;B + 0.1078\;C + 0.4905\;AB - 0.3035\;AC \hfill \\ - 0.0037\;BC - \;0.1529\;{A^2} - 0.4285\;{B^2} - 0.1602\;{C^2} \hfill \\ \end{gathered}$$
(9)
$$\begin{gathered} \frac{{C\left( {Zn} \right)}}{{{C_0}\left( {Zn} \right)}} = 1.06 - 0.0650\;A + 0.0905\;B + 0.1354\;C + 0.4902\;AB - 0.3816\;AC \hfill \\ - 0.0038\;BC - 0.0944\;{A^2} - 0.4906\;{B^2} - 0.1973\;{C^2} \hfill \\ \end{gathered}$$
(10)
$$\begin{gathered} \frac{{C\left( {Total} \right)}}{{{C_0}\left( {Total} \right)}} = 1.06 - 0.0703\;A + 0.0945\;B + 0.1321\;C + 0.4943\;AB \hfill \\ - 0.3726\;AC - 0.0031\;BC - 0.1035\;{A^2} - 0.4851\;{B^2} - 0.1924\;{C^2} \hfill \\ \end{gathered}$$
(11)

As shown in Table 4, A thorough statistical evaluation of the ANOVA results for all four responses (Cd, Cu, Ni, and Zn) confirms the robustness and adequacy of the developed quadratic models. The models demonstrate remarkably high Model F-values (4350.68 for Cd, 5219.22 for Cu, 4239.75 for Ni, and 4350.56 for Zn), with all p-values less than 0.0001, indicating that the combination of linear, interaction, and quadratic terms effectively captures the variation in the response variable (C/Co). The quadratic model was chosen as the most suitable model compared to the linear, 2FI, and cubic models, as the higher-order cubic model was aliased and statistically insignificant. All main effects (A: column height, B: initial ion concentration, and C: flow rate) were statistically significant (p < 0.0001). Among them, flow rate (C) exhibited the most substantial impact on all responses, as reflected by the highest F-values (1563.29 for Cd, 2050.91 for Cu, 991.66 for Ni, and 1562.82 for Zn). Furthermore, the interaction terms AB and AC showed pronounced synergistic effects across all metals, with F-values of 2242.16 (Cd), 2727.91 (Cu), 2249.03 (Ni), and 2242.28 (Zn) for AB, and 1917.02 (Cd), 2537.65 (Cu), 1214.58 (Ni), and 1916.80 (Zn) for AC. In contrast, the BC interaction was statistically insignificant for Cd, Cu, Ni and Zn (p > 0.5), indicating minimal combined influence of initial concentration and flow rate. The quadratic terms (A², B², and C²) were all highly significant (p < 0.0001), affirming the nonlinear behavior of the system. Among these, B² consistently exerted the strongest curvature effect, with F-values of 1442.68 (Cd), 1852.90 (Cu), 1102.47 (Ni), and 1442.81 (Zn), clearly identifying optimal concentration ranges for effective metal removal. These findings align well with multivariate adsorption optimization studies which underscore the critical role of second-order terms60. In the ANOVA table, the Pure Error values for all models were found to be extremely low (ranging from approximately 0.0007 to 0.0008), indicating high experimental accuracy and good reproducibility of the data. Moreover, the standard deviations of the predicted responses remained consistently minimal, suggesting that the quadratic models exhibited strong agreement with the experimental data and reliably described the overall trends. In addition, the coefficient of variation (CV) values were all below 10%, which is within the acceptable range, indicating that the models are reliable and the results can be considered precise and reproducible61. According to Table 5, the standard deviations of predicted responses remained consistently minimal across all models (ranging from 0.0259 to 0.0278), highlighting the high precision and reproducibility of the data. Adequacy of precision values (172.3322 for Cd, 186.5503 for Cu, 172.5037 for Ni, and 172.3247 for Zn) greatly surpassed the acceptable threshold of 4, confirming strong signal-to-noise ratios and the models’ reliability in navigating the design space. These exceptionally high values highlight the presence of a clear and powerful signal, ensuring that the models can be confidently applied for prediction and optimization without being influenced by random variations in the data61. Furthermore, the predicted R² values (e.g., 0.9937 for Cd, 0.9947 for Cu, 0.9935 for Ni, and 0.9937 for Zn) were in close agreement with their corresponding adjusted R² values, with differences well below 0.2, reinforcing the models’ predictive validity.

Table 4 ANOVA for quadratic model.
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Table 5 Statistical summary of RSM model fit indicators for C/C₀ responses of Cd, Cu, Ni, and Zn.
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Table 6 presents the Pearson correlation matrix, which illustrates the linear relationships among the input variables used in this study. The Pearson correlation matrix is a square matrix that quantifies the strength and direction of associations between pairs of variables and is commonly applied in fields such as social sciences, machine learning, and environmental engineering62,63. The Pearson correlation coefficient (r) measures the degree of a linear relationship between two continuous variables, ranging from − 1 to + 164. A value of + 1 shows a complete positive correlation, −1 expresses a complete negative correlation, and 0 means there is no linear relationship65. For instance, a correlation coefficient close to + 1 signifies that an increase in one variable is strongly associated with an increase in the other. In contrast, a value near − 1 implies that an increase in one variable is associated with a decrease in the other. A value near 0 suggests that there is no significant linear relationship between the two variables63. By definition, the diagonal elements of the correlation matrix are always + 1, as each variable is perfectly correlated with itself53. A comprehensive analysis of the Pearson correlation matrix for the variables column height (A), initial ion concentration (B), flow rate (C), and their interaction and quadratic terms (AB, AC, BC, A², B², C²) reveals weak to moderate relationships among the variables. This lack of strong correlation (multicollinearity) supports the robustness of the RSM model. The primary correlations between A, B, and C are notably weak, with a maximum of 0.393 between A and C, and near-zero between B and C (− 0.024), indicating a relatively independent experimental design. Quadratic terms, such as A² (0.675 with A) and B² (0.612 with B), exhibit moderate positive correlations, aligning with the nonlinear behavior observed in the RSM equations and underscoring the need for precise optimization of these variables. The AC interaction, with a correlation of 0.564 with B, highlights a significant interplay between concentration and flow rate, whereas other interactions (e.g., AB and BC) demonstrate weaker effects. These findings suggest that prioritizing the fine-tuning of B and further investigating the AC interaction could enhance the accuracy of process predictions and optimization strategies.

Table 6 Pearson correlation matrix where H, C, R represents the height, concentration and rate respectively.
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The normality and distribution of the residuals for Cu, Ni, Cd, and Zn were evaluated to identify potential outliers and detect regions with higher error rates for possible model improvement. For model validation, the residuals (defined as the differences between the actual and predicted values) were calculated. A comprehensive analysis of the perturbation plot involving column height (A), initial ion concentration (B), and flow rate (C) provides valuable insights into the relative influence of these variables on the response (C/Co), which represents the ratio of outlet to inlet concentration. Centered around a reference point (A = 193.51, B = 437.62, C = 5.54), the plot illustrates the sensitivity of C/Co to deviations from this baseline in coded units (−1 to + 1). Notably, the initial ion concentration (B), depicted by the blue curve, exhibits the most pronounced effect, with a clear increase in C/Co (about 0.25–0.3 units) as B shifts from − 1 to 0, followed by a further increase up to + 1. This behavior, driven by both the positive linear coefficient of B and the strong negative quadratic term (B²), reflects a nonlinear influence. It indicates that increasing the initial concentration leads to higher C/Co values, reducing removal efficiency, while lower concentrations (toward − 1) result in lower C/Co and higher efficiency. In contrast, column height (A), shown by the green curve, demonstrates a moderate effect with smoother variations (about 0.35–0.45 units), indicating a more linear and less dominant role; increasing the height gradually decreases C/Co, improving removal, while reducing it increases the output and decreases efficiency. Flow rate (C), represented by the gray curve, shows the weakest impact, with changes limited to about 0.2–0.25 units, reflecting its secondary influence; an increase in flow rate mildly elevates C/Co, reducing removal efficiency, whereas a decrease lowers C/Co, enhancing removal. This visual representation underscores that B is the critical factor, with its steep curvature highlighting the need for careful optimization, while A and C play supportive roles. These findings suggest that minimizing C/Co (enhancing removal efficiency) requires adjusting B to an optimal lower range, maximizing A, and minimizing C, providing a strategic foundation for further validation and process optimization.

The difference between equations and plots stems from the fact that each represents a different aspect of the data. Equations (7–11) present the coefficients of linear, interaction, and quadratic effects separately; thus, focusing solely on linear coefficients may lead to the incorrect assumption that variable C is more significant, whereas the nonlinear and interaction effects of variable B (such as B² and AB) actually play a more substantial role. In contrast, the perturbation plot provides a visual and integrated representation of the overall effect of each variable, encompassing its linear, interaction, and quadratic contributions. For instance, the negative coefficient of B², along with B’s interactions with other variables (such as AB), results in sharper curves in the plot, making B appear more influential. It is also worth noting that both approaches use coded units (ranging from − 1 to + 1), but the plots are more sensitive to nonlinear effects and reflect them more clearly.

Finally, as shown in Fig. 3, the initial concentration (B) variable exerts the most significant influence on the adsorption of all four metals, consistent with the findings of the ANOVA analysis. The residuals for all responses exhibited normal distribution, indicating both the statistical soundness and predictive reliability of the fitted models. Additional diagnostic plots are provided in the supplementary (Figure S1) to support these observations.

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
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Effect of operating variables on (a) Cd, (b) Cu, (c) Ni, (d) Zn and (e) total adsorption.

According to the RSM results, the optimal adsorption performance was achieved at a bed height of 288.27 cm, concentration of 301.06 ppm, and feed flow rate of 9.28 L/s. Given the significance of simultaneous adsorption of Ni, Cu, Cd, and Zn from industrial wastewater and the evaluation of the quadratic relationships among the key variables, the interactions and individual effects are illustrated in Figs. 4, 5 and 6.

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
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Response surfaces for (a) Cu, (b) Ni, (c) Cd, (d) Zn, and (e) total versus concentration and height.

Fig. 5
Fig. 5The alternative text for this image may have been generated using AI.
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Response surfaces for (a) Cu, (b) Ni, (c) Cd, (d) Zn, and (e) total versus rate and height.

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
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Response surfaces for (a) Cu, (b) Ni, (c) Cd, (d) Zn, and (e) total versus concentration and rate.

Based on the analysis of the response surface graphs, an increase in bed height generally leads to enhanced metal adsorption due to extended residence time, which improves contact between the adsorbent and metal ions. A higher bed height offers more binding sites and a larger surface area, resulting in improved metal removal efficiency. In contrast, at lower bed heights, the adsorption capacity declines due to reduced surface area and quicker saturation of the adsorbent, leading to shorter contact times. Furthermore, axial dispersion becomes more dominant at lower bed heights, limiting the effective diffusion of metal ions within the column. With regard to flow rate, higher velocities lead to more rapid filling of the adsorbent pores, as the fluid enters more aggressively. Although increased flow enhances external mass transfer, it simultaneously reduces contact time, resulting in earlier breakthrough and lower adsorption efficiency. At lower flow rates, the external film resistance increases, but the longer residence time allows for deeper penetration and more effective adsorption, thereby extending the saturation time. As for initial metal concentration, adsorption efficiency generally decreases as concentration increases. This is due to the limited capacity of the adsorbent, which becomes saturated more rapidly in the presence of a greater number of ions. According to Table 7, the operating parameter range was determined from experimental data and literature to minimize the ratio of the outlet to inlet concentration of the target metals. The optimum point was obtained by balancing these factors to maximize the overall adsorption efficiency.

Table 7 The parameter ranges for obtaining the optimal point.
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As previously noted, the optimal operating point was determined using the RSM, with the best conditions identified at a bed height of 288.27 cm, concentration of 301.06 ppm, and a feed flow rate of 9.28 L/s. Given the influence of flow rate on mass transfer coefficients, the values were calculated using Eq. (4). At the optimal flow rate of 9.28 L/s, the mass transfer coefficients for Cu, Ni, Cd, and Zn were computed as 0.012279, 0.010044, 0.009988, and 0.009988 per second, respectively. These optimal conditions were subsequently simulated using Aspen Adsorption software. The corresponding breakthrough curve was generated to evaluate the system’s performance under optimized parameters (Fig. 7).

Fig. 7
Fig. 7The alternative text for this image may have been generated using AI.
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The breakthrough curve at the optimal point.

The results of this study are in good agreement with the findings of previous researchers. For instance, Hameed et al.33 and Dissanayake et al.37 also reported that increasing column height significantly enhances removal efficiency, consistent with the results obtained in this work. Similarly, the negative impact of high inlet flow rates and elevated initial concentrations on adsorption efficiency observed in our study aligns with the trends reported by Singh et al.32 and Mehrmand et al.36. However, unlike some previous works that focus on single-metal systems or lack integrated modeling approaches, this study combines RSM optimization with Aspen Adsorption simulation for the simultaneous removal of Cu, Ni, Cd, and Zn, which provides a more comprehensive understanding of dynamic adsorption behavior and operational optimization.

Conclusion

This study investigated the simulation and analysis of the parameters influencing the removal of Cu, Ni, Cd, and Zn from industrial wastewater using the adsorption method with weakly acidic resin CH030. The process was simulated in Aspen Adsorption software, and the experimental design and optimization were carried out using the RSM with a quadratic model. The model yielded high R² values of 0.9951, 0.9939, 0.9941, 0.9941, and 0.9943 for Cu, Ni, Cd, Zn, and total metals, respectively, indicating strong predictive performance. Analysis of operational variables revealed that increasing bed height led to longer breakthrough times, thereby improving separation efficiency due to enhanced contact between metal ions and the adsorbent. In contrast, increasing the feed flow rate reduced residence time, leading to earlier saturation and decreased adsorption efficiency. Increasing the initial concentration of metal ions resulted in lower adsorption efficiency, primarily due to the limited capacity of the adsorbent. Overall, the RSM results identified initial concentration as the most influential parameter affecting adsorption efficiency for all four metals.