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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 R2, 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.

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  • 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

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adsorption, gPROMS, heavy metals, mass transfer parameters, PVSD model, wastewater treatment
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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 R2, p-value, root-mean-square error (RMSE), Chi-squared, and normalized standard deviation.

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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.

The schematic of the batch adsorption process and mass transfer mechanisms (external mass transfer, surface diffusion, and pore volume diffusion) are shown in Figure 1. Cu(II) gets transported from the bulk phase to the biosorbent surface through external mass transport, followed by its surface and pore diffusions in a typical adsorption process. As the Cu(II) reaches the surface of the biosorbent, competitive diffusion is likely to occur between the surface and the pore of the biosorbent. The model involves the following assumption:
  • 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).

Figure 1
Schematic of batch adsorption of Cu(II) over the biosorbent with the illustration of intraparticle diffusion.
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Schematic of batch adsorption of Cu(II) over the biosorbent with the illustration of intraparticle diffusion.

Figure 1
Schematic of batch adsorption of Cu(II) over the biosorbent with the illustration of intraparticle diffusion.
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Schematic of batch adsorption of Cu(II) over the biosorbent with the illustration of intraparticle diffusion.

Close modal
The transport of Cu(II) from the bulk phase to the adsorption surface is given as follows (Leyva-Ramos & Geankoplis 1985; Danish et al. 2022b):
(1)
The initial condition for Equation (1) is given as follows:
(2)
where 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.
Mass balance of Cu(II) over the porous biosorbent results in the following PDE (Leyva-Ramos & Geankoplis 1985; Danish et al. 2022b):
(3)
where ⁠, ⁠, and r are void fraction, density (kg/m3), and radius (m) of the biosorbent, respectively. and are the pore volume diffusion coefficient and surface diffusion coefficient (m2/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.

The initial and boundary conditions for Equation (3) are as follows:
(4)
(5)
(6)
where R is the radius of biosorbent (m).
The Langmuir isotherm for Cu(II) adsorption on the biosorbent is given as follows (Khan & Rao 2017):
(7)
where 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).
The pore volume diffusion coefficient (⁠⁠) can be determined using the following equation (Wilke & Chang 1955; Ocampo-Pérez et al. 2013):
(8)
where tortuosity factor (⁠⁠) and molecular diffusion coefficient are calculated using the following expressions (Wilke & Chang 1955; Souza et al. 2017):
(9)
(10)
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 (cm3/g·mol), and T is the temperature (K).
The external mass transfer coefficient (⁠⁠) is calculated by utilizing the approach proposed by Furusawa & Smith (1973):
(11)
The last term of Equation (11) can be calculated with the help of the initial points of the Cu(II) decay curve and can be expressed as follows:
(12)
The external surface area of the biosorbent per unit mass (⁠⁠) is expressed as follows:
(13)
The external mass transfer coefficient (⁠⁠) is also determined by using the following correlation (Tien 1994):
(14)
where is the biosorbent mass transfer coefficient while traveling at terminal velocity ⁠. To calculate ⁠, the following equation has been used (Harriott 1962; Tien 1994):
(15)
where is the particle diameter, is bulk-phase diffusivity, is fluid viscosity, and is the kinematic viscosity of the fluid.
The terminal velocity (⁠⁠) is given as follows (Nienow 1969; Tien 1994):
(16)
where g is the gravitational acceleration (9.8 m/s2), is the density of the liquid, is the difference in the density between the wet particle and liquid density:
(17)

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.

Table 1

Operating parameters for the biosorbent used for gPROMS simulation at 30°C (Khan & Rao 2017).

Operating parametersValue
(kg/m3) 350 
(μm) 112.5 
 0.4 
(mL) 20 
(g) 0.2 
Langmuir parameters  
(mg/g) 20.41 
(L/mg) 0.11 
Operating parametersValue
(kg/m3) 350 
(μm) 112.5 
 0.4 
(mL) 20 
(g) 0.2 
Langmuir parameters  
(mg/g) 20.41 
(L/mg) 0.11 
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The batch adsorption study of Cu(II) on a chemically modified 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.
Table 2

Estimated transport parameters for Cu(II) adsorption at different initial concentrations

(mg/L) × 103 (m/s) × 1012 (m2/s) × 1010 (m2/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) × 103 (m/s) × 1012 (m2/s) × 1010 (m2/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 
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Figure 2
Concentration decay curve for Cu(II) adsorption on a chemically modified C. moschata biosorbent at (a) 100 mg/L; (b) 70 mg/L; (c) 50 mg/L; (d) 20 mg/L; and (e) 10 mg/L.
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Concentration decay curve for Cu(II) adsorption on a chemically modified C. moschata biosorbent at (a) 100 mg/L; (b) 70 mg/L; (c) 50 mg/L; (d) 20 mg/L; and (e) 10 mg/L.

Figure 2
Concentration decay curve for Cu(II) adsorption on a chemically modified C. moschata biosorbent at (a) 100 mg/L; (b) 70 mg/L; (c) 50 mg/L; (d) 20 mg/L; and (e) 10 mg/L.
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Concentration decay curve for Cu(II) adsorption on a chemically modified C. moschata biosorbent at (a) 100 mg/L; (b) 70 mg/L; (c) 50 mg/L; (d) 20 mg/L; and (e) 10 mg/L.

Close modal
Figure 3(a) shows the adsorption of Cu(II) over the biosorbent as a function of radius at various time intervals. Figure 3(b) indicates that Cu(II) takes approximately 2 min to reach the center of the biosorbent. The adsorption capacity decreases as the time of adsorption increases since Cu(II) gets accumulated inside the biosorbent, leaving fewer sites for the Cu(II) adsorption. The adsorption capacity remains maximum at the surface of the biosorbent; then, it becomes constant until the adsorption process reaches equilibrium.
Figure 3
Cu(II) adsorption as a function of biosorbent radius at different time intervals taking initial Cu(II) of 100 mg/L.
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Cu(II) adsorption as a function of biosorbent radius at different time intervals taking initial Cu(II) of 100 mg/L.

Figure 3
Cu(II) adsorption as a function of biosorbent radius at different time intervals taking initial Cu(II) of 100 mg/L.
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Cu(II) adsorption as a function of biosorbent radius at different time intervals taking initial Cu(II) of 100 mg/L.

Close modal
The contribution of PVSD during Cu(II) adsorption is given by the following equations (Leyva-Ramos & Geankoplis 1985):
(18)
(19)
where and represent the mass flux of Cu(II) due to PVSDs (mg/m2·s), respectively.
The relative contribution of PVSD has been calculated with the following equations:
(20)
(21)
The intraparticle Cu(II) adsorption is apparently governed by both PVSD mechanisms. Initially, the pore volume diffusion plays an important role near the surface of the biosorbent; however, as time passes, the surface diffusion remains dominant throughout the process. This is because initially the more intraparticle surface is available for Cu(II) adsorption than pore volume, which decreases with time, as shown in Figure 4(a).
Figure 4
Contribution of (a) pore volume and (b) surface diffusion in the intraparticle diffusion at different radial positions for adsorption of Cu(II) on a chemically modified C. moschata biosorbent at 70 mg/L.
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Contribution of (a) pore volume and (b) surface diffusion in the intraparticle diffusion at different radial positions for adsorption of Cu(II) on a chemically modified C. moschata biosorbent at 70 mg/L.

Figure 4
Contribution of (a) pore volume and (b) surface diffusion in the intraparticle diffusion at different radial positions for adsorption of Cu(II) on a chemically modified C. moschata biosorbent at 70 mg/L.
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Contribution of (a) pore volume and (b) surface diffusion in the intraparticle diffusion at different radial positions for adsorption of Cu(II) on a chemically modified C. moschata biosorbent at 70 mg/L.

Close modal
The relative contribution of PVSD s for Cu(II) adsorption on biosorbent concerning time and the biosorbent radius is shown in Figure 4(b) and Supplementary Material, Figure S1, in which the contribution of surface diffusion is 57% at 70 mg/L. Figures 5 and 6 show PVSDs inside the biosorbent at different times, respectively. The time interval of 2–10 min is chosen because most of the adsorption process occurs in these time intervals. For 100 mg/L, the contributions of surface diffusion at the center and surface of the particle are 97 and 54%, respectively. As time passes and equilibrium is achieved, it decreases up to 76% at 10 min, which appeared to be uniform throughout the biosorbent. Supplementary Material, Figure S2 illustrates the ratio of surface diffusion to pore volume diffusion. As seen from the above results, surface diffusion takes precedence over pore volume diffusion throughout the process. Similar findings have also been observed elsewhere (Ocampo-Pérez et al. 2013; Souza et al. 2019).
Figure 5
Contribution of surface diffusion inside the biosorbent concerning radius and time.
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Contribution of surface diffusion inside the biosorbent concerning radius and time.

Figure 5
Contribution of surface diffusion inside the biosorbent concerning radius and time.
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Contribution of surface diffusion inside the biosorbent concerning radius and time.

Close modal
Figure 6
Contribution of pore volume diffusion inside the biosorbent concerning radius and time.
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Contribution of pore volume diffusion inside the biosorbent concerning radius and time.

Figure 6
Contribution of pore volume diffusion inside the biosorbent concerning radius and time.
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Contribution of pore volume diffusion inside the biosorbent concerning radius and time.

Close modal

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.

Table 3

Kinetics and diffusional model (Verma et al. 2016)

ModelEquationPlotAdvantageDisadvantage
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. 
ModelEquationPlotAdvantageDisadvantage
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 min0.5). is the uptake rate constant (L/g min). is the Kuo–Lotse constant.

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Table 4

Kinetics and diffusional model parameters for Cu(II) adsorption over the C. moschata biosorbent

ModelParametersInitial concentration (mg/L)
10070502010
Elovich α 5.5615 3.8202 2.7337 1.0556 0.4695 
β 0.3623 0.5123 0.9423 1.8594 3.7565 
R2 0.761 0.775 0.868 0.657 0.896 
Bangham αo 0.0087 0.0203 0.0168 0.0198 0.0069 
ko 1.8764 1.8189 1.8290 1.7592 1.4804 
R2 0.870 0.963 0.8684 0.657 0.892 
Intraparticle diffusion Kid 1.6667 1.1802 0.8362 0.3249 0.1630 
C 2.5823 1.7600 1.2647 0.4886 0.2096 
R2 0.679 0.696 0.692 0.692 0.751 
Modified Freundlich k3 0.0691 0.0530 0.0513 0.0718 0.1163 
m1 0.3623 0.5129 0.7229 1.8594 3.7565 
R2 0.761 0.775 0.868 0.657 0.896 
ModelParametersInitial concentration (mg/L)
10070502010
Elovich α 5.5615 3.8202 2.7337 1.0556 0.4695 
β 0.3623 0.5123 0.9423 1.8594 3.7565 
R2 0.761 0.775 0.868 0.657 0.896 
Bangham αo 0.0087 0.0203 0.0168 0.0198 0.0069 
ko 1.8764 1.8189 1.8290 1.7592 1.4804 
R2 0.870 0.963 0.8684 0.657 0.892 
Intraparticle diffusion Kid 1.6667 1.1802 0.8362 0.3249 0.1630 
C 2.5823 1.7600 1.2647 0.4886 0.2096 
R2 0.679 0.696 0.692 0.692 0.751 
Modified Freundlich k3 0.0691 0.0530 0.0513 0.0718 0.1163 
m1 0.3623 0.5129 0.7229 1.8594 3.7565 
R2 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.

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The current study's simulation results have been validated by the literature (Khan & Rao 2017) for the concentration decay curve, and the statistical parameters, such as Chi-square value (⁠⁠), R2, normalized standard deviation (NSD), 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):
(22)
(23)
where denotes experimental ⁠, represents simulated bulk concentration, and N is the number of experiments.

Table 5 shows the high R2 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.

Table 5

Statistical parameters of the PVSD model used for Cu(II) adsorption on the biosorbent

C0 (mg/L)R2RMSENSDChi-squarep-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 
C0 (mg/L)R2RMSENSDChi-squarep-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 
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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 R2 (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.

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

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

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The authors declare there is no conflict.

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This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY-NC-ND 4.0), which permits copying and redistribution for non-commercial purposes with no derivatives, provided the original work is properly cited (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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