A mini-review on the modeling of volatile organic compound adsorption in activated carbons:Equilibrium,dynamics,and heat effects

Shanshan Wang *,Liangliang Huang *,Yumeng Zhang ,Licheng Li ,Xiaohua Lu

1 State Key Laboratory of Material-Oriented Chemical Engineering,College of Chemical Engineering,Nanjing Tech University,Nanjing 211816,China

2 School of Chemical,Biological and Materials Engineering,University of Oklahoma,Norman,OK 73019,United States

3 College of Chemical Engineering,Nanjing Forestry University,Nanjing 210037,China

Keywords:Volatile organic compounds Activated carbon Adsorption equilibria Adsorption kinetics Adsorption exotherm

ABSTRACT The research on the adsorption equilibria,kinetics,and increase in process temperature of the volatile organic compound(VOC) adsorption in porous materials ensures safe production,thereby reducing production costs and improving separation efficiency.Therefore,it is critical in predicting the entire adsorption process based on minimal or no experimental input of the adsorbate and adsorbent.We discuss,in this review,the factors that affect the adsorption performance of VOCs in activated carbons,including the adsorption equilibrium,adsorption kinetics,and exotherm during adsorption.Subsequently,the existing prediction models are summarized and compared concerning the adsorption equilibrium,adsorption kinetics,and exothermic process of adsorption.We then propose a new prediction model based on intermolecular interaction and provide an outlook toward the design and manipulation of efficient adsorbents for the VOC system.

1.Introduction

With the ongoing concern of environmental safety,the control of volatile organic compound (VOC) pollution has become a key topic[1].VOCs are generally flammable and explosive,and are primarily emitted by industries such as organic chemicals,printing,shoemaking,and paint to cause considerable harm to the environment[2–4].VOC also poses a substantial threat to humans because it causes several health issues [5,6].Table 1 lists the allowable environmental concentration of common VOCs and their respective harm to humans.A large number of exhaust emissions,a wide range of pollution,and the difficulty to degrade require interdisciplinary collaboration.Countries worldwide have invested considerable workforce and material resources in this regard,devoting themselves to the research and development of VOC processing technology.

The leading VOC treatment/removal technologies are divided into two categories;one is the destruction methods,including biological methods and oxidation methods,and the other is the recovery techniques,in which the commonly used methods include adsorption,absorption,condensation,and membrane separation[10].Generally,the biological process can be used to remove water-soluble pollutants,while most organic solvents are either insoluble or have a low solubility in water.The oxidation method has high investment costs and does not have economic benefits,and the absorption method can easily cause secondary pollution.Compared with the methods mentioned above,the adsorption technology,which uses adsorption materials to interact with VOCs physically and chemically,is widely used in practical engineering because of its advantages of high efficiency,long-lasting effect,and low energy consumption [11,12].The adsorption technology realizes the enrichment and separation of VOCs through physical and chemical interactions between the adsorbent and exhaust gas.After the treatment,the adsorbent materials could be reactivated through thermal or vacuum desorption [13].

As a commonly used adsorbent for VOC adsorption,activated carbon (AC) has the following characteristics:strong selectivity,large specific surface area,wide pore size range,surface rich in multiple functional groups,stable performance,and regenerable[14–16].The adsorption performance is dominated by intermolecular interactions,including the van der Waals force and electrostatic interaction [17].Two aspects can significantly affect intermolecular interactions,that is,the complex structure of adsorbents and complex interaction between adsorbents and adsor-bates.For example,the increase in the number of surface functional groups on ACs can change many properties,including the polarity,hydrophilicity and hydrophobicity,surface charge distribution,and skeleton electron cloud density,which eventually affect the adsorption behavior[18].The physical properties differences between VOCs also alter their binding with the adsorbents and,in return,affect the macroscopic adsorption performance[19].The situation is complicated when water vapor is present.Water molecules are preferentially adsorbed on the AC surface,and VOCs are adsorbed on the adsorbents with preadsorbed water to replace water molecules partially [20].

Table 1 Maximum allowable VOC concentrations and VOCs’ harm to human health [7–9]

Considering the factors above,the understanding of the VOC removal mechanism via experiments can be relatively challenging.Predicting the adsorption behavior with minimal or no experimental input has been studied extensively in molecular simulation investigations.With the continuous improvement of supercomputing hardwares and molecular simulation algorithms [21,22],molecular simulation can provide necessary thermodynamic data and achieve a semiquantitative description of complex systems.For example,using Henry’s coefficient of CO2as a quantitative parameter,Lin et al.[23]screened a large number of zeolite imidazolate framework for CO2storage and separation.Wu et al.[24]proposed a concept of adsorbility to define the ratio of adsorption heat at infinite dilution(which characterizes the adsorbate–adsorbent interactions)and the porosity of the adsorbents(which represents structural characteristics of the adsorbents) and screened highly efficient metal–organic frameworks for CO2/CH4separation.Despite these achievements,obtaining satisfactory results when the molecular simulation is adopted to describe the kinetic properties,such as the adsorption and temperature-increasing rates quantitatively,is still challenging.This phenomenon is partly due to the unclear connection between structural/kinetic properties of the molecules and mechanisms of the adsorption processes.

We discuss,in this review,the factors that affect the adsorption performance of VOCs,such as adsorption equilibrium,kinetics,and adsorption exotherm.Subsequently,the existing prediction models are summarized and compared concerning adsorption equilibrium,adsorption kinetics,and the exothermic process of adsorption.We then propose a new prediction model based on intermolecular interaction and provide an outlook toward the design and manipulation of efficient adsorbents for the VOC system.

2.Factors Affecting Adsorption Performance of VOCs

The factors that affect the adsorption behavior of VOCs can be divided into the following three aspects(Fig.1):(1) the characteristics of the adsorbents,including specific surface area,pore volume,pore size distribution,and surface chemical heterogeneity;(2) the characteristics of the VOCs,such as molecular weight,molecular diameter,boiling point,polarity,and saturated vapor pressure of different VOCs;and(3)the influence of external conditions,including temperature,pressure,VOC concentration,and the impact of humidity.In this section,we briefly summarize these factors.It should be noted that we mainly investigated the modeling of the adsorption process in this work,and the presence of water vapor poses a challenge to the applicability of the model.Therefore,when discussing the influence of external conditions,we only examine the influence of humidity in detail in this work.

2.1.Characteristics of adsorbents

Pore structure(i.e.,specific surface area,pore volume,and pore size distribution)is an essential factor that determines the adsorption performance of ACs.A large specific surface area can generally provide additional adsorption sites,thereby resulting in excellent adsorption performance.Das et al.[14]found that the breakthrough time of toluene in AC fiber with a larger surface area (1 700 m2·g-1) is threefold higher than the adsorbents with a small surface area (1 000 m2·g-1).Lillo-Rodenas et al.[13]reported that the adsorption capacity of toluene and benzene increases with the pore volume of carbon materials,especially the microporous pore volume.Similarly,Chiang et al.[27]also found that the adsorption performance is closely related to the surface micropore structure,especially pore shape and size distribution.

Surface chemical heterogeneity refers to the chemical functional groups on the AC surface.Considering the existence of atomic unsaturation on the AC surface,they will chemically form various surface functional groups,thereby giving AC different adsorption characteristics.The chemical groups that have an important influence on the adsorption performance of AC are mainly oxygen-containing functional groups,which are divided into acidic and basic oxygen-containing functional groups.Acidic oxides give AC polar properties.In general,AC with acidic compounds in the oxygen-containing functional groups should have high efficiency when adsorbing polar compounds,while alkaline compounds tend to adsorb weak or nonpolar substances [28].The works of Liu et al.[29]and Azimi et al.[30]also confirmed this result.

2.2.Characteristics of VOCs

In addition to the characteristics of ACs,the intrinsic properties of VOCs,such as size,shape,polarity,boiling point,etc.,also play significant roles in the adsorption processes [14,31,32].Table 2 lists the different physical properties of common VOCs.As it is difficult to have a quantitative and unified description for the size property,thus we did not list the size information in Table 2.Qian et al.[26]quantitatively discussed the relationship between the molecular size of four VOCs and their adsorption capacity on activated carbons.Herein,the molecular size refers to a molecule cross-sectional area other than minimum dimensions such as molecule width,thickness,or length.It was found that the adsorption capacities of VOCs onto AC are negatively correlated to their molecular size [26](see Fig.2).

Fig.1.Factors that influence the adsorption performance of VOCs on carbon-based materials [25].Copyright (2017) Elsevier.

Table 2 Different physical properties of various VOCs

Fig.2.Relationship between the molecular cross-sectional area of VOCs and their adsorption capacity on activated carbons [26].Copyright (2015) Springer.

The polarity of adsorbents also has a noticeable influence on adsorption behavior [41].Bansode et al.[42]found that the phosphoric acid-AC exhibits higher adsorption capacity for nonpolar benzene and carbon tetrachloride than the polar bromodichloromethane,1,1,1-trichloroethane,chloroform,and 1,1-dichloroethane.The adsorption of methyl cyanide,trichloroform,and acetone has also been studied with different ACs,where various adsorption capacities on the same AC could vary significantly because AC generally has poor adsorption performance for polar molecules [43].Song et al.[44]measured the adsorption capacity of acetone and methyl hexanone on AC.They investigated the relationship between adsorption capacity and physical property parameters of the VOCs,such as molecular weight and polarity.Other studies found that various properties of VOCs,including molecular diameter,molecular weight,surface tension,electrostatic force,dipole distance,ionization potential,and saturated vapor pressure,could significantly influence adsorption performance [33,45].

2.3.Effects of humidity

Water vapor is ubiquitous in the environment and higher than the VOC concentration several orders of magnitude [46],in which case they could compete with VOCs on ACs.However,ACs are hydrophobic[47].The effect of water vapor on the adsorption performance of VOCs also depends on the characteristics of the adsorbents and VOCs.Significantly,when VOCs interact with polar chemicals,it largely depends on the relative hydrophobicity of the adsorbents.Yamamoto et al.[48]concluded that the effect of water vapor on carbonaceous adsorbents with larger pores is less than those with smaller pores.A dynamic study of zeolite adsorbent[49]and AC[50]found that the presence of hydrophilic bonding sites promotes the adsorption of water and increases the degree of competition for VOCs[51].Hunter-Sellars et al.[52]evaluated the adsorption of several VOCs by materials with various pore size distributions and chemical properties in dry and wet carrier gas conditions,as shown in Fig.3.The performance of sorbents with hydrophilic surface chemistry,such as silica gel and molecular sieve 13X,decreases significantly with a slight increase in preexposed humidity.Activated charcoal and high-silica faujasite Y retain their capacities for toluene at the relative humidity up to 50% and 70%,respectively.The performance of adsorbents with hydrophilic surface chemistry,such as silica gel and molecular sieve 13X,decreases significantly with even a small increase in humidity,while adsorbents with hydrophobic surface chemistry,such as AC high-silica faujasite Y,maintain high toluene capacity at relative humidity up to 50% and 70%.

Fig.3.Adsorption capacities for toluene (a) and ethanol (b) at different humidities.Samples studied:activated charcoal,AC;amorphous silica,AS;molecular sieve 13X,MS13X;zeolite Y,A88Y;ZSM-5 zeolite,A14Z.Reproduction from ref [52].Copyright (2020) Elsevier.

3.Modeling of Equilibria,Dynamics,and Exotherm of VOC Adsorption

To devise a cost-effective adsorption system,we first need to obtain the adsorption capacity in advance.Besides,the mathematical simulation of fixed-bed adsorption kinetics also plays a vital role in the prediction,design,and optimization of the adsorption process.According to mathematical simulation,industrial process simulation can be performed on the premise of obtaining necessary experimental data.The parameters obtained through model fitting can then be adopted to analyze the mass transport mechanism of the fixed bed.Moreover,AC is flammable;potential warming of the adsorption bed and a fire hazard can exist due to the exothermal nature of the adsorption process [53].Thus,the problem of predicting temperature change is an essential designing part of the safety of the adsorption process.

The efforts required to obtain adsorption capacity,dynamics,and increase in temperature through experiments are significantly high.This process is also costly and time-consuming.Reliable prediction models based on minimal or even no experimental input becomes crucial due to cost consideration and the variety of the characteristics of VOCs and ACs [30].In this section,we briefly summarize the modeling methods of the adsorption equilibria,dynamics,and exotherm of the VOC adsorptions.

3.1.Modeling of adsorption equilibria

Adsorption isotherms are the curves of the adsorption capacity changing with the equilibrium concentration[54].According to the isotherms,the surface area,pore volume,pore size distribution of the adsorbent,and adsorption performance can be determined.Typically,the mathematical correlation of the isotherms plays an important role in the operational design and applicable practice of the adsorption systems.Over the years,a wide variety of equilibrium isotherm models have been formulated [55],including the two parameter isotherm models:Langmuir [56],Freundlich [57],Dubinin-Radushkevich [58],Temkin [59],Flory-Huggins [60],Hill[61]isotherms,etc.;and three parameter isotherms models,such as Redlich-Peterson [62],Sips [63],Koble-Corrigan [64],Khan[65],and Radke-Prausnitz isotherms [66];and the multilayer physisorption isotherms,such as the Brunaue-Emmett-Teller isotherms [67].Table 3 listed the mathematical expressions of the commonly used four isotherm models.

The Langmuir equation assumes that monolayer adsorption occurs uniformly on the adsorbent surface,and no lateral interaction or steric hindrance exists between adsorbate molecules [68].However,the actual adsorption process is mostly multilayer adsorption.The Langmuir model follows Henry’s law at low concentrations.The two points are the main limitations of this model[69].Nevertheless,due to its simplicity,the Langmuir model is still popular for describing adsorption equilibrium [70].The Langmuir model has been extensively used in previous studies and has goodaccuracy [55].The Freundlich adsorption model,which is another well-known empirical model,can describe nonideal adsorptions without the restriction of the monolayer.However,this model does not comply with Henry’s law at low concentrations,which results in the equilibrium adsorption overestimation or underestimation at low pressure [71].The Brunauer-Emmett-Teller (BET)adsorption isotherm model is a multilayer molecular adsorption isotherm equation [72].The assumptions of the BET equation are as follows:the surface of the adsorbent is uniform,interaction force does not exist between the adsorbate molecules,the van der Waals force can form multilayer adsorption,and the total adsorption is the sum of the adsorption of each layer.The relative pressure above 0.35 may cause capillary condensation and is inconsistent with multilayer adsorption.Thus,the BET model is valid only for the relative pressures range of 0.05–0.35.In general,these popular isotherm equations are useful for fitting experimental isotherm data but provide almost no prediction function for adsorption capacities.

Table 3 List of adsorption isotherm models

Therefore,the Dubinin-Radushkevich (D-R) model is recommended for predicting adsorption capacities.The D-R model is based on well-known Polanyi’s potential and has been widely used to estimate the adsorption capacity for VOCs with ACs.The D-R model requires two parameters,namely,qs,which is the limiting pore volume,and kD,which is the D-R model parameter for the target adsorbate.To apply the D-R model to the prediction of the adsorption capacity,one shall obtain the two parameters in advance.The qsvalue can be obtained using one reference adsorbate,such as benzene,and it can be applied to the same adsorbent[73].The k value is only a function of the adsorbate[74]and can be obtained through the methods of the molar volume,molecular paracord,and electronic polarization [75].The D-R model is thermodynamically consistent at medium and high relative pressures,except for low loadings[76,77].Hung et al.[45]combined the D-R Equation with the Langmuir model and proposed that the D-RLangmuir(D-R-L)model successfully predict the adsorption capacity of aromatic and chlorinated hydrocarbons with BPL carbon and Sorbonorit B carbon over a broad range of relative pressures from 7.4 × 10-5to 0.03,as shown in Fig.4.

As discussed in Section 2,water has competitive adsorption with VOCs under humidity,while most mathematical models cannot explain the effect of moisture.The available models for predicting the competitive adsorption of VOCs and water vapor are generally empirical or semiempirical.When these isotherm models are integrated into the dynamic adsorption model,the process is tedious and difficult.Laskar et al.[79]used the Manes method,which is based on potential theory,to describe the competitive multicomponent adsorption of VOCs and water vapor with AC.The required inputs of the Manes method are as follows:(a) the single-component adsorption isotherms of VOCs by the modified D-R model,and (b) the adsorption isotherm of water vapor obtained by the Qi-Hay-Rood model.Fig.5 shows the prediction of the VOC adsorption capacity in the presence of water vapor and at different water relative humidity(RH),where the total average relative absolute errors are 1.9 % and 5.2 % for nonpolar and polar VOCs,respectively.

3.2.Modeling of adsorption kinetics

Studying the breakthrough curve can provide insight into mass transfer and the operation status of the adsorption bed.In general,the longer the breakthrough time is,the better the adsorption performance will be.The saturation capacity of the adsorbent can be calculated through the adsorption saturation time.Meanwhile,the size of the slope reflects the speed of the mass transfer rate.Satokawa et al.[80]studied the adsorption kinetics of dimethyl sulfide (DMS) and tert-butyl mercaptan (TBM) on the fixed bed of AgNa-Y molecular sieve.The results showed that the increase in silver ion loading of the molecular sieve extends the permeation time of DMS and TBM and eventually improves the adsorption capacity.As shown in Table 4,several empirical models have been proposed for fixed-bed adsorption breakthrough curves.

Generally,these empirical models have good fitting results only for specific systems and specific conditions.For example,Chowdhury compared the error function from the fitting results with the experimental equilibrium data using the first-order and pseudo-second-order kinetic models and found that the pseudofirst-order equation cannot provide accurate fitting of the experimental data on the malachite green chemically modified rice husk.Meanwhile,the pseudo-second-order kinetic model provides a good correlation [85].Yang et al.[86]presented that the quasifirst-order model is more suitable for describing the toluene adsorption process with ACs than the quasi-second-order model.The Boltzmann model has a better fitting for the experimental data than the Yoon-Nelson model [84].This result also indicates that VOCs with a weak resistance of mass transfer have an excellent utilization efficiency of the fixed bed [83].

Fig.4.(a)Predictions of D–R–L model for trichloroethene,o-xylene,and toluene adsorbed with activated carbons at 25°C.The solid lines denote Langmuir prediction and the dashed lines denote the D-R prediction.(b) Comparison of D–R–L predictions and experimental data by Yun et al.[78]for adsorption capacity of benzene,toluene,and pxylene with activated carbons.Reproduction from Ref.[45].Copyright (2007) Taylor &Francis.

Fig.5.Polar vs.nonpolar VOC:comparison of experimental and modeled equilibrium adsorption capacities of VOC at 0%,55 %,and 95 % relative humidity (RH)during their competitive adsorption on beaded activated carbons at 25 °C.Red labels denote the polar VOCs,and the blue labels denote the non-polar VOCs.Reproduction from Ref.[79].Copyright (2019) Elsevier.

Meanwhile,for VOC removal at a low potential flammable risk,carbon-silicon composite adsorbents are generally used.These cases have been described by the Yoon-Nelson model[87].In general,these empirical models provide satisfactory fitting results and are mathematically easy to solve.The main limitation is that these empirical models cannot effectively describe mass transfer and diffusion mechanisms of the fixed bed.Thus,these models cannot guide the design and optimization of the adsorption process nor the choice of the adsorbent.

For nonempirical models,the prediction of column dynamics requires the simultaneous solution of a set of coupled nonlinear partial differential equations (PDEs),to account for heat and mass balances [88],as follows:

Eq.(1)is the mass conservation equation,Eq.(2)is the heat conservation equation,and Eq.(3) is the mass transfer equation between the solid phase and the gas phase[the linear driving force(LDF) model],where C is the adsorbate inlet concentration,mol·m-3;Cais the adsorption heat capacity,J·mol-1·K-1;Cgis the relative heat capacity of gas,J·kg-1·K-1;Csis the relative heat capacity of solid,J·kg-1·K-1;H is the diameter of the adsorption column,m;h0is the heat dissipation coefficient of the column wall,W·m-2·K-1;k is the mass transfer coefficient,s-1;q is the adsorption capacity,mol·kg-1;qeis the equilibrium adsorption capacity,mol·kg-1;U is the apparent gas flow rate,m·s-1;ε is the bed porosity;ρbis the bed density,kg·m-3;ρgis the gas density,kg·m-3.

The model deals with 1D modeling of pollutant breakthrough curves and neglects the axial dispersion [88].As an extension of the model proposed by Tefera et al.[89]a 2D mathematical model,which consists of a competitive adsorption isotherm to predict the adsorption capacity,macroscopic mass,momentum,and energy conservation equations,was developed by Laskar et al.[19]to study the effects of relative humidity on the kinetics of VOCs.As illustrated in Fig.6,this model provides good predictions of breakthrough curves of several VOCs,including the nonpolar VOCs of toluene,n-butanol,and 1,2,4-trimethyl benzene and polar VOCs of acetone and 2-propanol.The mean relative absolute error under dry and wet conditions is 11.8% and 17.2%,respectively.Satisfactory agreements are observed between experimental and simulation results.

3.3.Modeling of the increase in temperature

Predicting the temperature change of the adsorption process is of importance for the safety issue.Over the years,most theoretical studies have targeted systems that have no significant thermal effects(＜10°C)[90,91].Although several studies have investigated the process with high thermal effects (up to 50 °C),the treatment methods used are generally hot purge regeneration [92].Information about the critical operating conditions,in which the adsorption process may cause a fire,is limited.Having a deep understanding of the various factors that cause the increase in temperature of the adsorption process is urgently needed [93].

Delage et al.[94]established a set of temperature change prediction models by using the differential heat of adsorption instead of integrated heat and investigated the heating rate of seven VOCs with different loadings(up to 100 g·m-3).As shown in Fig.7(a),the slope of the regression is slightly different from the perfect agreement.The theoretical parameter sensitivity test further illustrates that the increase in temperature depends mostly on the molar concentration of the VOCs,heat of adsorption,and volumetric heat capacity of the carrier gas.The rise in temperature of the AC bed can be predicted by combining these variables.The conclusions drawn from this work provide an adequate theoretical basis for designing a safe adsorber and preventing carbon bed ignition.

The study above was restricted to one compound adsorption with the dry AC bed.Experimental studies have shown that when the initial water content of the adsorbent is 10%,the increase in temperature rise to the exothermic nature of adsorption will besignificantly reduced.Delage et al.[95]then improved the original models and extended the application scope to a practical condition containing humidity.Fig.7 compares the experimental and theoretical temperature values of different types of VOCs in wet AC.Compared with Fig.7(a),the increase in temperature in wet AC does not exceed 25 °C,while that with dry AC is two-to threefold higher under the same conditions.Fig.7(b) also reveals that the theoretical prediction agrees well with the experiments;that is,the temperature difference is within 10 °C.Although the increase in the temperature of VOCs in wet AC does not precisely follow the experimental curve,the model still predicts the overall change in thermal behavior,which is observed under operating conditions.

Table 4 List of adsorption dynamics models

Fig.6.Comparison of experimental and modeled breakthrough curves of VOCs during competitive adsorption of water vapor,(a) 2-propanol,(b) acetone,(c)n-butanol,(d)toluene,and (e) 1,2,4-trimethyl benzene on one kind of activated carbon BAC at 25 °C and at different relative humidity (RH) [19].Copyright (2019) American Chemical Society.

Fig.7.Comparison of maximum temperature rise obtained from the prediction model (calculated) and experiments (observed) on (a) dry and (b) wet activated carbon.Reproduction from Refs.[94,95].Copyright (2000 and 2002) American Chemical Society.

To investigate the increase in temperature further,Le Cloirec[93]proposed that the thermal wave of the adsorption front can be visualized by an infrared camera,as illustrated in Fig.8.Initially,the exothermic nature of adsorption leads to an increase in temperature.Meanwhile,the temperature from the inlet to the top decreases with time because the heat generated due to adsorption is continuously transmitted along the adsorption bed through the air stream.From the visualization of the temperature profile,the temperature trajectory could be quickly recorded and interpreted.We shall obtain a good understanding of the adsorption kinetics with the penetration (～46 min) and saturation times (～70 min).

4.A Prediction Model Based on Intermolecular Interaction

Fig.8.Infrared camera visualization of temperature evolution in activated carbon fixed-bed with acetone adsorption.Reproduction from Ref.[93].Copyright(2017)Taylor&Francis.

In the development of the adsorption prediction model,the D-R model is not introduced into the PED sets.Instead,the Langmuir equation is adopted to facilitate the solution of these PDEs.The mass and heat conservation equations are solved simultaneously.Meanwhile,the adsorption isotherm is obtained by fitting to available experimental data.Given that the existing adsorption kinetics prediction model does not include the result of adsorption isotherm equations,the two predictions are unrelated,each of which requiring an experimental input in advance.

Meanwhile,when solving the PEDs,the mass transfer between gas and solid phases is generally described by the LDF model[96].As shown in Table 5,the mass transfer coefficient k1of the LDF model includes parameters,including the integrated adsorption heat ΔHintand particle size of the adsorbent dp.For the VOC/adsorbent systems with complex structures and interactions,the acquisition of these parameters is semiempirical with low accuracy.For each system,it needs to be measured in advance through experiments,which limits the predictability of the models.Therefore,a universal mass transfer model that can quantitatively describe heat and mass transfer,guide the preparation of adsorbents,and predict the adsorption process should be constructed.

4.1.Nonequilibrium thermodynamic mass transfer model

According to the study of interface transfer properties,Lu et al.[97,98]proposed a thermodynamic mass transfer model through the combination of the reciprocal relationship presented by Onsager and the principle of minimum entropy generation proposed by Prigogine,as shown in Table 5.Different from the LDF model,the driving force of thermodynamic mass transfer is the chemical potential gradient other than the concentration gradient,which can be extrapolated to predict other systems to complete the prediction function of the quantitative model.We will discuss below how the chemical potential can decouple the control factors more accurately than the concentration gradient as the driving force.The model parameters are grouped in the unified variable of the interface mass transfer coefficient k,and the parameters describing the interface structure are unified in the mass transfer distance δ.

The kinetics of the absorption/desorption process of CO2in the supported ionic liquid absorbent was studied based on the nonequilibrium thermodynamics model,and the effect of the ionic liquid film thickness on CO2mass transfer rate was quantitatively discussed [99].Fig.9 shows the mass transfer coefficient kμ based on the nonequilibrium thermodynamic mass transfer model and the mass transfer coefficient k based on the traditional dissolution-diffusion model during the CO2absorption process of the ionic liquid absorbent,namely,P25-[APMIm]Br.With different loadings (film thickness),three mechanisms exist for the absorption of CO2in ionic liquids,and the rate constant has changed by orders of magnitude.However,according to the traditional solvent diffusion model,the three evident types of mechanisms cannot be explained.The mechanism prevents further analysis of this process,as shown in Fig.9(b).The nonequilibrium thermodynamic model can reveal a complete mechanism behind experimental data than the traditional mass transfer model.The nonequilibrium thermodynamics model was also used to conduct the resistance analysis and obtain the mechanism of CO2permeation in choline proline/PEG200,which is an ionic liquid support membrane[100].These examples demonstrate the practical application of the nonequilibrium thermodynamic transfer model.However,the physical meaning of the parameters and generalization of the model has not been explained well at the microscopic scale.Thequantitative analysis and conception of the model shall be achieved only by associating the parameter of the nonequilibrium thermodynamic transfer model with the molecular interaction parameters.There is still a long way to go.And the most important part is the generalized description of the diffusion coefficient D.

Table 5 Comparison of mass transfer prediction models

Fig.9.Kinetics coefficients of CO2 absorption in P25-[APMIm]Br sorbents.(a) kμ from the mass transfer model based on nonequilibrium thermodynamics;(b) k from the traditional dissolution-diffusion model [99].Copyright (2015) American Institute of Chemical Engineers.

4.2.Generalized description of self-diffusion coefficient

Different from the bulk phase,the self-diffusion coefficient of species under confinement is relatively difficult to obtain.For example,Wang et al.[101]reported that water confined in a carbon nanotube with a diameter smaller than 1 nm would exhibit a discontinuous pseudo-gas state,which is similar to the state of bulk water at room temperature and under 0.1 kPa.Meanwhile,water in a carbon nanotube with a diameter of approximately 1.4 nm exhibited a discontinuous pseudo-liquid state,which is similar to that of the bulk water at room temperature and under 0.1 MPa.In the work of Long et al.[102,103],both the experiments and simulations suggest that the confinement experienced by an adsorbed phase confined within a carbon nanoporous material can be equivalent to the bulk phase pressure in equilibrium with the system.The statistical average of density in its microscopic state should be consistent with the value obtained by macroscopic adsorption experiments.Thus,the microstructure (density distribution) of the fluid under confinement can be equivalent to the state that is induced by pressure.A corresponding bulk system with the same density can be constructed by obtaining the fluid density under confinement.Therefore,the self-diffusion coefficient calculated from the bulk shall be equivalent to that of the confined system.With such an analogy,the issue is solved by calculating bulk self-diffusion coefficients with corresponding intermolecular parameters.Using this concept,Zhu et al.[104]investigated the diffusion coefficient from the Lennard–Jones model fluid to a binary system.They proposed a new equation for calculating the diffusion coefficient,as shown in Eqn 4 and Eqn 5.This approach can be adapted to obtain the self-diffusion coefficients of gas,liquid,and supercritical fluid under different temperature and density conditions.Once self-diffusion coefficients have been calculated for pure substances,one can use the mixing rule to obtain the self-diffusion coefficient for systems with multiple components.

where D is the diffusion coefficient,T*and ρ*are the dimensionless temperature and dimensionless density,respectively,N is the total number of particles in the simulated system,V is the volume of the simulated system,and σ and ε are Lennard-Jones parameters.

5.Conclusions

The understanding of adsorption balance,kinetics,and increase in process temperature of VOCs in porous materials are of theoretical and practical importance to ensure safe production,reduce production costs,and improve separation efficiency.This factor is also critical to predicting the adsorption process by using mainly physical and chemical characteristics of adsorbate and adsorbent.In this work,we discuss the factors that affect the adsorption performance of VOCs,including adsorption equilibrium,adsorption kinetics,and exotherm during adsorption.Existing prediction models are summarized and compared concerning adsorption equilibrium,adsorption kinetics,and exothermic process of adsorption.We then propose a new prediction model based on intermolecular interaction and provide an outlook toward the design and manipulation of efficient adsorbents for the VOC system.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We acknowledge the financial support from the National Natural Science Foundation of China (22008107,21838004) and DTRA through the grant HDTRA11910008 of the USA.