## Abstract

This work describes the successful application of the pore volume and surface diffusion (PVSD) model characterizing the batch adsorption of Cu(II) on a chemically modified *Cucurbita moschata* biosorbent. The PVSD model captures the convective transport of Cu(II) from the bulk solution to the biosorbent surface, followed by its surface and pore diffusion inside the biosorbent. The adsorption of Cu(II) is mimicked using the Langmuir isotherm. The algebraic, ordinary, and partial differential equations, involved in the PVSD model, are solved using the general process modeling system (gPROMS). The model simulation results, depicted by the Cu(II) concentration decay curve, show an excellent match with experimental data. The external mass transfer coefficient (≈10^{−3} m/s) indicated no restriction on approaching Cu(II) toward the biosorbent surface. Within the biosorbent, surface diffusion was dominant over pore volume diffusion. The statistical analysis of the PVSD model results has been done by calculating *R*^{2}, Chi-square value, normalized standard deviation, *p*-value, and root-mean-square error. The PVSD model approach presented in this work could be beneficial to other heavy metal–biosorbent systems.

## HIGHLIGHTS

Cu(II) adsorption on

*Cucurbita moschata*is characterized using the PVSD model.The adsorption mechanism is described using the Langmuir isotherm.

The simulation of the PVSD model was done using gPROMS.

The mass transfer coefficient value indicates no constraint on Cu(II) bulk transport.

Surface diffusion takes precedence over pore volume diffusion.

### Graphical Abstract

## INTRODUCTION

Industrialization and urbanization have significantly contributed to environmental pollution, which has become an important issue to be addressed (Sulyman *et al.* 2017; Aftab *et al.* 2021). Heavy metals are the most prevalent of the numerous industrial pollutants because of their high level of toxicity and non-degradability (Meitei & Prasad 2013). According to the WHO and IPCS, copper remains toxic like other heavy metals (Sulaymon *et al.* 2009). Despite being nutritious for the human body, an excess amount of copper can result in major health hazards such as nausea, diarrhea, epigastric pain, and vomiting (Bashir *et al.* 2020). Acute exposure to copper causes hepatic and renal failure, vascular injury, shock, coma, and sometimes death (Ng *et al.* 2002). A disease known as Wilson's illness is caused by the accumulation of copper in the brain, liver, and other tissues (Walshe 2006). Copper is also known to cause a disease called Vineyard sprayer's lungs, which is caused by workers getting exposed to copper sulfate (Shrivastava 2009). The drinking water threshold of WHO guidelines for copper is 1.00 mg/L (Khan & Rao 2017). Reverse osmosis (Johnston 1975), ion exchange (Tripp & Clifford 2006), chemical precipitation (Xanthopoulos *et al.* 2017), and adsorption (Verma *et al.* 2016) are possible routes for removing copper from polluted streams. However, these methods (apart from adsorption) have drawbacks, such as high costs and lower effectiveness at a low metal concentration of 1–100 mg/L. On the other hand, adsorption is the most preferred technique because of its ease of use, excellent selectivity, and regeneration (Abu Al-Rub *et al.* 2003; Verma *et al.* 2016).

The adsorption characteristics and transport processes for the adsorption of Cu(II) over diverse biosorbents can be studied by choosing an appropriate mathematical model (Qiu *et al.* 2009). Numerous mathematical models exist to define the adsorption process of heavy metals, which are categorized as the diffusional models and reaction models (Souza *et al.* 2017). The reaction models involve only the determination of reaction kinetics, whereas the diffusional models predict the realistic kinetics by considering external mass transfer followed by intraparticle diffusion, which involves pore volume diffusion and surface diffusion (Ocampo-Pérez *et al.* 2013; Danish *et al.* 2022a).

The various mathematical models (Elovich, Bangham's model, intraparticle diffusion model, and modified Freundlich model) for the batch mode including the pore volume and surface diffusion (PVSD) model have been reported in the literature (Ocampo-Pérez *et al.* 2013; Verma *et al.* 2016, 2017). These models were solved numerically in MATLAB through several computational steps, which involve converting the partial differential equations (PDEs) into ordinary differential equations (ODEs) using discretization methods and solving those ODEs again using ode15s (Souza *et al.* 2017, 2019). On the other hand, gPROMS (general process modeling system) (Process Builder, Academic 1.5.1) is capable of solving multiple PDEs and ODEs directly, thus avoiding additional discretization steps and minimizing the computational effects.

In this work, a PVSD model characterizing the dynamics of batch adsorption of Cu(II) on a chemically modified *Cucurbita moschata* biosorbent (Khan & Rao 2017) has been employed. The model includes a system of coupled algebraic, ordinary, and partial differential equations; the Langmuir isotherm is used to define Cu(II) adsorption (Khan & Rao 2017). These model equations have been solved in the gPROMS computation tool. The decay rates of Cu(II) on the biosorbent have been predicted concerning initial Cu(II) concentration and validated with the experimental data (Khan & Rao 2017). The mass transfer parameters such as mass transfer coefficient (), surface diffusion coefficient (), and effective pore volume diffusion coefficient () have been calculated at different initial Cu(II) concentrations. Also, the relative contributions of PVSD have been calculated for Cu(II) adsorption over the biosorbent. The consistency of predicted parameters was checked through *R ^{2}*,

*p*-value, root-mean-square error (RMSE), Chi-squared, and normalized standard deviation.

## MATHEMATICAL MODEL OF PVSD AND EXPERIMENTAL VALIDATION

As reported by Khan & Rao (2017), *C. moschata* biomass obtained locally from Aligarh, India was chemically modified (sodium carbonate) and tested for Cu(II) adsorption in batch mode. The stock solution of Cu(II) between 10 and 100 ppm were prepared for adsorption studies. The adsorbent amount was varied between 0.1 and 1 g. The batch adsorption experiments were conducted till 24 h; however, the equilibrium adsorption achieved within 30 min.

- 1.
Intraparticle diffusion of Cu(II) on the biosorbent occurs due to PVSD governed by Fick's law of diffusion.

- 2.
On active sites, the rate of adsorption is instantaneous.

- 3.
The biosorbent is considered to be homogeneous and spherical.

- 4.
As reported in the literature, Cu(II) coverage on adsorption surfaces is considered to be a monolayer (Khan & Rao 2017).

*et al.*2022b):

*V*is the volume of solution (mL), is the bulk liquid-phase concentration of Cu(II) (mg/L), is the effective mass transfer coefficient (m/s),

*m*is the mass of biosorbent (g), is Cu(II) concentration inside the biosorbent at a distance

*r*(mg/L), denotes the initial bulk liquid-phase concentration of Cu(II) (mg/L),

*t*is time (s), and

*S*is the external surface area of biosorbent per unit mass.

*et al.*2022b):where , , and

*r*are void fraction, density (kg/m

^{3}), and radius (m) of the biosorbent, respectively. and are the pore volume diffusion coefficient and surface diffusion coefficient (m

^{2}/s), respectively.

The left side of Equation (3) consists of an accumulation of Cu(II) in the pore volume of the biosorbent and the amount of Cu(II) adsorbed on the pore surface. The pore volume diffusion and surface diffusion are represented on the right side of Equation (3), respectively.

*R*is the radius of biosorbent (m).

*q*is the adsorption capacity at equilibrium (mg/g),

*b*is the Langmuir isotherm constant (L/mg), and is the monolayer capacity of Langmuir isotherm (mg/g).

*et al.*2013):where tortuosity factor () and molecular diffusion coefficient are calculated using the following expressions (Wilke & Chang 1955; Souza

*et al.*2017):where = association parameter for water = 2.6, is the molecular weight of water (g/mol), is the viscosity of solution (kg/m·s), is the molal volume of heavy metals at normal boiling point (cm

^{3}/g·mol), and

*T*is the temperature (K).

*g*is the gravitational acceleration (9.8 m/s

^{2}), is the density of the liquid, is the difference in the density between the wet particle and liquid density:

The model Equations (1)–(7) were solved in gPROMS by employing a backward finite difference scheme (Danish *et al.* 2022a); although not shown here, other finite difference schemes, i.e. forward and central, also yielded the same outputs. Table 1 lists the experimental conditions/parameters utilized in the simulation.

Operating parameters . | Value . |
---|---|

(kg/m^{3}) | 350 |

(μm) | 112.5 |

0.4 | |

(mL) | 20 |

(g) | 0.2 |

Langmuir parameters | |

(mg/g) | 20.41 |

(L/mg) | 0.11 |

Operating parameters . | Value . |
---|---|

(kg/m^{3}) | 350 |

(μm) | 112.5 |

0.4 | |

(mL) | 20 |

(g) | 0.2 |

Langmuir parameters | |

(mg/g) | 20.41 |

(L/mg) | 0.11 |

## RESULTS AND DISCUSSION

*C. moschata*biosorbent (Khan & Rao 2017) has been understood with a PVSD model. The mass transfer coefficient () is determined using Equation (11) at different initial Cu(II) concentrations. The value of the pore volume diffusion coefficient is calculated using Equation (8), and the value of the surface diffusion coefficient () is determined by dividing by the density of the adsorbent . And to get more precise values, the given values of and were specified as the initial value in the gPROMS program, and after several iterations, the numerical solution is fitted well with the experimental data providing new values of and . The values estimated from the curve fitting are of the same order as those calculated from the equation, thus strengthening the present simulation work. This exercise has been repeated for different initial metal concentrations and the concentration decay curve are shown in Figure 2 (Khan & Rao 2017). Table 2 shows the estimated transport parameters for Cu(II) adsorption at various initial concentrations. The mass transfer coefficient does not vary considerably while changing the initial Cu(II) concentration, and its lower value signifies that the intraparticle diffusion is predominant over the film resistance. It is worthwhile to note that the mass transfer coefficient () is also determined using Equation (14), and the value (2

**×**10

^{−3}m/s) is found to be of the same order of magnitude.

(mg/L) . | × 10^{3} (m/s)
. | × 10^{12} (m^{2}/s)
. | × 10^{10} (m^{2}/s)
. |
---|---|---|---|

100 | 4.12 | 4.0 | 4 |

70 | 4.08 | 1.0 | 4 |

50 | 4.06 | 0.9 | 4 |

30 | 4.06 | 0.4 | 1 |

10 | 3.53 | 0.25 | 1 |

(mg/L) . | × 10^{3} (m/s)
. | × 10^{12} (m^{2}/s)
. | × 10^{10} (m^{2}/s)
. |
---|---|---|---|

100 | 4.12 | 4.0 | 4 |

70 | 4.08 | 1.0 | 4 |

50 | 4.06 | 0.9 | 4 |

30 | 4.06 | 0.4 | 1 |

10 | 3.53 | 0.25 | 1 |

^{2}·s), respectively.

*et al.*2013; Souza

*et al.*2019).

The current PVSD model has also been compared with other kinetics and the diffusional model represented in Table 3, and the parameters estimated using linear regression are shown in Table 4.

Model . | Equation . | Plot . | Advantage . | Disadvantage . |
---|---|---|---|---|

Elovich | vs. | This model aids in predicting a system's activation and deactivation energy, mass, and surface diffusion. | The estimation of transport parameters like and was avoided. To calculate the model parameters, only curve fitting has been used. | |

Bangham | vs. | The dominance of pore diffusion in the adsorption process is assessed using this model. | The surface diffusion study is missing in the model. | |

Intraparticle diffusion | vs. | In intraparticle diffusion, the transport of heavy metal ions from the liquid phase to the solid phase is described. | The study of PVSD is devoided. | |

Modified Freundlich | vs. | The impact of heavy metal surface loading and ionic strength on the adsorption process is evaluated using the model. | The study of pore volume and surface diffusion is not present. |

Model . | Equation . | Plot . | Advantage . | Disadvantage . |
---|---|---|---|---|

Elovich | vs. | This model aids in predicting a system's activation and deactivation energy, mass, and surface diffusion. | The estimation of transport parameters like and was avoided. To calculate the model parameters, only curve fitting has been used. | |

Bangham | vs. | The dominance of pore diffusion in the adsorption process is assessed using this model. | The surface diffusion study is missing in the model. | |

Intraparticle diffusion | vs. | In intraparticle diffusion, the transport of heavy metal ions from the liquid phase to the solid phase is described. | The study of PVSD is devoided. | |

Modified Freundlich | vs. | The impact of heavy metal surface loading and ionic strength on the adsorption process is evaluated using the model. | The study of pore volume and surface diffusion is not present. |

is at any given moment the quantity of heavy metals absorbed per unit mass of biosorbent (mg/g). *α*, , *C*, , and *β* are constants. is the intraparticle diffusion rate constant (mg/g min^{0.5}). is the uptake rate constant (L/g min). is the Kuo–Lotse constant.

Model . | Parameters . | Initial concentration (mg/L) . | ||||
---|---|---|---|---|---|---|

100 . | 70 . | 50 . | 20 . | 10 . | ||

Elovich | α | 5.5615 | 3.8202 | 2.7337 | 1.0556 | 0.4695 |

β | 0.3623 | 0.5123 | 0.9423 | 1.8594 | 3.7565 | |

R^{2} | 0.761 | 0.775 | 0.868 | 0.657 | 0.896 | |

Bangham | α_{o} | 0.0087 | 0.0203 | 0.0168 | 0.0198 | 0.0069 |

k _{o} | 1.8764 | 1.8189 | 1.8290 | 1.7592 | 1.4804 | |

R^{2} | 0.870 | 0.963 | 0.8684 | 0.657 | 0.892 | |

Intraparticle diffusion | K _{id} | 1.6667 | 1.1802 | 0.8362 | 0.3249 | 0.1630 |

C | 2.5823 | 1.7600 | 1.2647 | 0.4886 | 0.2096 | |

R^{2} | 0.679 | 0.696 | 0.692 | 0.692 | 0.751 | |

Modified Freundlich | k_{3} | 0.0691 | 0.0530 | 0.0513 | 0.0718 | 0.1163 |

m_{1} | 0.3623 | 0.5129 | 0.7229 | 1.8594 | 3.7565 | |

R^{2} | 0.761 | 0.775 | 0.868 | 0.657 | 0.896 |

Model . | Parameters . | Initial concentration (mg/L) . | ||||
---|---|---|---|---|---|---|

100 . | 70 . | 50 . | 20 . | 10 . | ||

Elovich | α | 5.5615 | 3.8202 | 2.7337 | 1.0556 | 0.4695 |

β | 0.3623 | 0.5123 | 0.9423 | 1.8594 | 3.7565 | |

R^{2} | 0.761 | 0.775 | 0.868 | 0.657 | 0.896 | |

Bangham | α_{o} | 0.0087 | 0.0203 | 0.0168 | 0.0198 | 0.0069 |

k _{o} | 1.8764 | 1.8189 | 1.8290 | 1.7592 | 1.4804 | |

R^{2} | 0.870 | 0.963 | 0.8684 | 0.657 | 0.892 | |

Intraparticle diffusion | K _{id} | 1.6667 | 1.1802 | 0.8362 | 0.3249 | 0.1630 |

C | 2.5823 | 1.7600 | 1.2647 | 0.4886 | 0.2096 | |

R^{2} | 0.679 | 0.696 | 0.692 | 0.692 | 0.751 | |

Modified Freundlich | k_{3} | 0.0691 | 0.0530 | 0.0513 | 0.0718 | 0.1163 |

m_{1} | 0.3623 | 0.5129 | 0.7229 | 1.8594 | 3.7565 | |

R^{2} | 0.761 | 0.775 | 0.868 | 0.657 | 0.896 |

The statistical analysis of kinetics and diffusional model in this table shows that these model are not fitted well experimental data for Cu(II) adsorption over the *C. moschata* biosorbent.

*R*, normalized standard deviation (NSD),

^{2}*p*-value, and root-mean-square error (RMSE), are determined to ensure the accuracy of the simulated results of the PVSD model. The statistical parameters are given as follows (Verma

*et al.*2017):where denotes experimental , represents simulated bulk concentration, and

*N*is the number of experiments.

Table 5 shows the high *R ^{2}* and low Chi-square, RMSE values, and NSD for different initial Cu(II) concentrations, demonstrating that the proposed model precisely predicts batch adsorber behavior. Also, the

*p*-value is significantly lower than 0.001, which implies that the obtained simulated model is highly significant.

C (mg/L)
. _{0} | R^{2}
. | RMSE . | NSD . | Chi-square . | p-value
. |
---|---|---|---|---|---|

100 | 0.999 | 0.0015 | 6.36 | 0.0011 | 1.44 ×10^{−9} |

70 | 0.999 | 0.0017 | 9.86 | 0.0026 | 9.43 ×10^{−9} |

50 | 0.999 | 0.0030 | 8.85 | 0.0022 | 3.52 ×10^{−10} |

30 | 0.999 | 0.0010 | 11.51 | 0.0060 | 1.42 ×10^{−7} |

10 | 0.999 | 0.0224 | 7.65 | 0.0046 | 2.47 ×10^{−7} |

C (mg/L)
. _{0} | R^{2}
. | RMSE . | NSD . | Chi-square . | p-value
. |
---|---|---|---|---|---|

100 | 0.999 | 0.0015 | 6.36 | 0.0011 | 1.44 ×10^{−9} |

70 | 0.999 | 0.0017 | 9.86 | 0.0026 | 9.43 ×10^{−9} |

50 | 0.999 | 0.0030 | 8.85 | 0.0022 | 3.52 ×10^{−10} |

30 | 0.999 | 0.0010 | 11.51 | 0.0060 | 1.42 ×10^{−7} |

10 | 0.999 | 0.0224 | 7.65 | 0.0046 | 2.47 ×10^{−7} |

## CONCLUSIONS

The present work suggests monolayer Cu(II) adsorption over a chemically modified *C. moschata* biosorbent (Khan & Rao 2017) in batch mode. A mathematical model capturing the pore volume diffusion and surface diffusion coupled with the Langmuir isotherm has been developed and solved successfully using gPROMS. The concentration decay curves have been predicted at various initial concentrations, and the results fit the experimental data excellently (Khan & Rao 2017). The predicted values of the external mass transfer coefficient () suggest that there is no limitation for the external transport of Cu(II) on the biosorbent surface. Surface diffusion and pore volume diffusion were found to control the overall batch adsorption process. The variation of adsorption capacity as a function of radius shows that it remains maximum at the surface of the biosorbent and becomes constant as the adsorption process reaches equilibrium. The PVSD model shows that the influence of the surface diffusion is always greater than the pore volume diffusion. The statistical analysis of the PVSD model shows high *R ^{2}* (0.999) and low Chi-square value (0.006), RMSE (0.0015) value, and NSD (6.36–11.51%). A low

*p*-value (<0.001) indicates that the PVSD model precisely mimicked the Cu(II)–biosorbent system. The present PVSD model is advised for the characterization of different heavy metal–biosorbent systems.

## ACKNOWLEDGEMENTS

The first authors gratefully acknowledge the support provided by University Grant Commission (UGC) Non-NET.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.