Prediction of the breakthrough curves of VOC isothermal adsorption on hypercrosslinked polymeric adsorbents in a fixed bed

Abstract

A mathematical model, which was based on the Linear Driving Force (LDF) model and Dubinin–Radushkevich (D–R) equation, was proposed to describe the removal of volatile organic compounds (VOCs) through a fixed bed packed with a hypercrosslinked polymeric resin (HPR) under isothermal conditions. The limiting volume adsorbed capacity (q₀) and the adsorption characteristic energy (E) of the D–R equation were correlated well with the properties of the VOCs and HPR using multi-linear regression (MLR). Therefore, no experimental effort was required to predict equilibrium adsorption capacities of VOCs over a wide range of relative pressures. The model was validated for the breakthrough adsorption of hexane, heptane, and butanone on HPR. The experimental and predicted breakthrough time had a deviation within 10%. This result indicates that the model can be used to determine the breakthrough curves and time as long as the known properties of VOCs and HPR were found in advance, without requiring any adsorption experiments.

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1. Introduction

The emission of volatile organic compound (VOC) vapors has caused not only severe air pollution but also a great loss of valuable chemicals. It is well known that adsorption is one of the most effective methods for separating and recovering VOCs from industrial waste gas. As a new kind adsorbent, hypercrosslinked polymeric resins (HPRs) represent a class of predominantly microporous organic materials with an extremely high specific surface area (>1000 m² g⁻¹), stable physical and chemical properties, as well as regeneration on site.^1–5 Therefore, hypercrosslinked polymeric resins have received increasing attention in recent years as adsorbents for purifying air. Simpson et al.,^6,7 Podlesnyuk et al.,⁸ Baya et al.⁹ and our research group^10–14 have investigated the adsorption characteristics of hypercrosslinked polymeric resins for VOCs. These studies indicated that hypercrosslinked polymeric resin is an efficient and competitive adsorbent for VOCs recovery from polluted vapor streams.

In practical applications, the fixed bed adsorption system has been widely used to control VOC emissions. To optimize design and operating conditions, the development of a mathematical model that can describe the dynamics of adsorption in a fixed bed with selected adsorbent is required. In general, a comprehensive mathematical model consists of coupled partial differential equations distributed over time and space representing material balances together with transport rates and equilibrium equations. It is worth noting that the adsorption equilibrium in a wide range of temperatures and concentrations is the most important process that controls the dynamics behavior of a packed column so that the general nature of a mass transfer zone is determined entirely by the equilibrium isotherm.^15–20 To obtain the adsorption equilibrium equation, gas–solid equilibrium experiments must be performed, which may tend to be expensive and time consuming for different combinations of adsorbate–adsorbent systems. To the best of our knowledge, it has not been reported to develop an adsorption isotherm equation from commonly available adsorbent and adsorbate properties for predicting breakthrough curves of VOCs in a fixed bed.

Although Langmuir or Freundlich equation is commonly used to describe adsorption data, these equations cannot predict the adsorption equilibrium capacity of adsorbent from fundamental adsorbate properties. Dubinin–Radushkevich (D–R) equation, deduced from the potential theory, has been proved to be an effective pool to describe the adsorption capacity of VOCs on microporous adsorbents.^21,22 According to D–R equation, the volumetric adsorption amount (q_v, mL g⁻¹) is related to the adsorption potential (ε) as given by eqn (1) and (2).

q_v = q₀ exp[−(ε/E)²] (1)

ε = RT ln(P_s/P) (2)

where q₀ corresponds to the limiting volume adsorbed capacity (mL g⁻¹); R stands for the gas constant (8.314 J mol⁻¹ K⁻¹); T is the absolute temperature (K); P_s represents the saturation vapor pressure (kPa); P is the equilibrium vapor pressure (kPa); and E corresponds to the adsorption characteristic energy of a specific adsorbate on adsorbent (J mol⁻¹).

To apply D–R equation to the prediction of adsorption capacity, the parameters q₀ and E must be obtained in advance. Previous researchers considered that E for different substances was related to affinity coefficient (β):¹⁵

E₁/β₁ = E₂/β₂ = E_i/β_i = E_reference/β_reference (3)

Benzene was usually taken as the reference compound with assumption β_benzene = 1, and E_i = β_iE_reference could be substituted into D–R equation. Affinity coefficients (β) can be approximated by ratio of molecular parachor, molar polarizability or molar volume of the specific adsorbate and reference compound (benzene).¹⁵ However, the selection of an appropriate reference adsorbate could be an arbitrary choice.^23–25 The choice of a reference adsorbate has an important effect on the accuracy of the D–R equation for predicting the adsorption capacity of organic compounds on adsorbents. In addition, according to the theory of volume filling of micropores,²⁶ there is a constants q₀ for any chemical at a given value of E_i/β_i. This method is suitable for carbonous adsorbent, and q₀ is generally assumed to be equal to the volume of micropore. However, for VOCs–HPR adsorption system, our previous studies^5,10–14,27 found that q₀ was not equal to but greater than microporous volume of adsorbent. Possible reason is that hypercrosslinked polymeric resin can swell strongly after adsorbing organic compounds, which results in larger actual microporous volume compared with the measured value based on N₂ adsorption experimental data. Therefore, in order to apply D–R equation to the prediction of the adsorption capacities of VOCs onto hypercrosslinked polymeric resin, it is important to obtain the parameters q₀ and E accurately.

The aim of the present study is to establish a model to predict the breakthrough curves of VOCs adsorption on hypercrosslinked polymeric adsorbent in a fixed bed. Mass transfer was described by the Linear Driving Force (LDF) model. Some studies have given good results for modeling dynamic adsorption with LDF model.^28–32 The adsorption equilibrium of VOCs on HPR was described by D–R equation; q₀ and E were estimated using multi-linear regression (MLR) based on the physical-chemistry properties of adsorbents and adsorbates. The accuracy of the predictions was assessed by comparing them with experimental breakthrough curves.

2. Theoretical approach

The mathematical model of the dynamic adsorption breakthrough process in a fixed bed is based on transient material balance, gas phase and intrapellet mass transfer, adsorption equilibrium relationship, boundary conditions, and initial conditions. In this study, both adsorption equilibrium and kinetic aspect are taken into account. Adsorption equilibrium and mass-transfer rate are described by D–R equation and linear driving force (LDF) model respectively. The assumptions listed below are made to develop this model.

(1) The pressure drop through bed is negligible.

(2) The plug flow model is used to describe the flow pattern. Adsorbate concentration, gas flow, porosity and temperature are uniform at any cross section of the bed.

(3) The physical properties of the adsorbent are constant, and accumulation of energy in the gas phase is negligible under constant temperature.

(4) The mass transfer rate described by the LDF model consists in a general expression that lumps together all diffusion mechanisms into one single parameter, k.³³

(5) The flow rate is constant and the effect of axial dispersion is neglected.

2.1 Mass balance

The mass balance on a portion of the fixed bed is given by

{mass flux} + {mass accumulation in the gas phase} + {mass adsorbed in the solid phase} = 0

where U is the linear gas velocity (m s⁻¹); C stands for the gas concentration (mg L⁻¹); z corresponds to axial position (m); ν stands for the porosity; t is the adsorption time (s); ρ_b represents the bed density (kg m⁻³); q is the adsorbed phase concentration (mg g⁻¹).

2.2 Mass transfer rate

LDF model was used to describe the mass transfer:where q is the adsorbed phase concentration (mg g⁻¹); t stands for the adsorption time (s); k corresponds to the mass transfer coefficient (s⁻¹); q_e represents equilibrium adsorbate concentration (mg g⁻¹).

The k parameter is an adjustable parameter.^14,26,32 It has been proven to be essential and can be described by following expression for the mass transfer rate:^31,32

where U is the linear gas velocity (m s⁻¹); d_p stands for the particles diameter of HPR (m); −ΔH_int corresponds to integral heat of adsorption (J mol⁻¹); R stands for the gas constant (8.314 J mol⁻¹ K⁻¹); T is the absolute temperature (K).

2.3 Adsorption isotherm

D–R equation can be defined as the following equations by combining eqn (1) and (2):where q is the equilibrium adsorption amount (mg g⁻¹); ρ stands for the adsorbate density in the adsorbed phase which is assumed to be the same as in the liquid phase (kg m⁻³).

The parameters of q₀ and E were estimated directly by using the multi-linear regression (MLR) based on the experiment data. MLR uses optimal combination of multiple independent variables to estimate the dependent variable, which is more effective than a single independent variable. MLR is a standard procedure that enables us to find the most significant variables (from the properties of VOCs and the hypercrosslinked polymeric adsorbent) on the measured response. Removing less significant variables leads to a more statistically reliable MLR model, which can be applied to a wide range of VOCs–resin systems. In this study, MLR models were established by using the statistical package SPSS19.

2.4 Initial and boundary conditions

The initial conditions are:

C = C₀ for z = 0 and t = 0

C = 0 for z > 0 and t = 0

q = 0 for z ≥ 0 and t = 0

The boundary condition is:

C = C₀ for z = 0 and t > 0 (8)

where C is the gas concentration (mg L⁻¹); C₀ stands for the inlet VOC concentration (mg L⁻¹); z corresponds to axial position (m); t is the adsorption time (s).

2.5 Numerical simulations

The set of eqn (4)–(7) enabled us to compute the amount of VOCs adsorbed onto the particles of adsorbent (q) and the concentration of VOCs in the gas-phase (C) along the column. The gas concentration could be obtained by using the finite-difference method. Each discrete point represents a certain region, whose concentration is a measurement of the average concentration of this region at a certain time. Therefore, the numerical accuracy of calculations strongly depends on the number of designated discrete points. The calculation process was quite complicated, and MATLAB R2010a was applied to solve the differential equation set.

3. Data set

3.1 Adsorbents and adsorbates

The adsorbents used in the experimental evaluation were the hypercrosslinked polymeric resins with different pore structure, which were prepared via a post-crosslinking step of low-crosslinked macroporous poly(styrene-divinylbenzene).¹⁴ The salient properties of hypercrosslinked polymeric resins are listed in Table 1.

Table 1 Salient properties of hypercrosslinked polymeric resins

Parameters	HY-1	HPsorbent	ND-100	Resin-1	Resin-2
a Synthesized by my colleagues in State Key Laboratory of Pollution Control and Resource Reuse, Nanjing University (NJU).
SBET (m^2 g^−1)	1244.42	1020.7	1325	811.9	944.35
Micropore volume (Vmicro, cm^3 g^−1)	0.541	0.394	0.423	0.400	0.430
Average pore diameter (nm)	2.35	2.52	2.54	2.92	2.73
Bulk density (kg m^−3)	294.9	335.4	294.8	376.0	306.8
Diameter of particles (mm)	0.4–0.6	0.4–0.6	0.4–0.6	0.4–0.6	0.4–0.6
Citation	14	11	32	NJU lab^a	NJU lab^a

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Fourteen common organic chemicals including hydrocarbons, halogenated hydrocarbons, ketones and esters were selected as adsorbates. Their physico-chemical property parameters used in modeling of adsorption capacities, shown in Table 2, were assembled from Lange's Handbook of Chemistry (Fifteenth Edition).

Table 2 Physico-chemical properties of adsorbates^a

Adsorbate	Density (kg m^−3)	Molar volume (Vm, mL mol^−1)	Molar polarizability (MP, cm^3 mol^−1)	Parachor (Pa, cm^3 g^1/4·s^−1/2 mol^−1)	Ionization potential (IP, eV)
a From Lange's Handbook of Chemistry (Fifteenth Edition).
Pentane	626	111.0	25.28	231.0	10.35
Hexane	659	127.5	29.90	270.8	10.13
Heptane	684	144.0	34.55	310.6	9.92
Benzene	879	89.4	26.27	207.2	9.25
Chlorobenzene	1106	101.3	31.15	243.1	9.06
Dichloromethane	1327	67.8	16.34	148.8	11.32
Trichloromethane	1500	79.5	21.46	184.6	11.37
Dichloroethane	1235	84.3	21.32	188.6	11.04
Trichloroethene	1460	89.1	25.76	210.4	9.47
Acetone	791	75.1	15.97	156.5	9.71
Butanone	805	91.6	20.60	196.3	9.51
Methyl acetate	934	81.5	17.72	176.2	10.27
Ethyl acetate	901	98.0	22.35	216.0	10.01
Propyl acetate	888	114.5	26.98	255.8	10.04

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3.2 Adsorption equilibrium data

Here, adsorption equilibrium data of pentane, hexane, heptane, benzene and butanone on HY-1, benzene and chlorobenzene on HPsorbent, methyl acetate, ethyl acetate and propyl acetate on ND-100 were collected from our previous publications.^11,13,27,34 Extra data measured by the authors for dichloromethane on HY-1, trichloromethane, dichloroethane and trichloroethene on Resin-1, and acetone and butanone on ND-100 were also used in this study. The measurements of the adsorption equilibrium of VOCs onto polymeric resins were conducted using a column adsorption method. The detailed experimental apparatus and adsorption procedure have been described previously.¹⁰ These data were used as the training set in building D–R model. Nonlinear regressions based on a least-squares criterion were performed on all of the data to obtain best fits for the parameters of E and q₀ appearing in the D–R equation. The obtained values of q₀ and E, shown in Table 3, will be correlated with properties parameters of adsorbates and adsorbents by using multi-linear regression.

Table 3 Parameters of q₀ and E of D–R model for different adsorbate/adsorbent systems

Adsorbent	Adsorbate	Temperature range (K)	q0 (mL g^−1)	E (kJ mol^−1)	References
HY-1	Pentane	293–323	0.706	12.80	13
	Hexane	293–323	0.685	12.89	13
	Heptane	293–323	0.678	15.00	13
	Benzene	303–328	0.544	11.98	29
	Dichloromethane	308–323	0.529	9.69	This study
	Butanone	303–328	0.558	12.17	29
HPsorbent	Benzene	303–333	0.465	13.73	11
	Chlorobenzene	303–333	0.505	13.95	11
ND-100	Methyl acetate	303–333	0.503	9.22	34
	Ethyl acetate	303–333	0.476	10.26	34
	Propyl acetate	303–333	0.476	10.77	34
Resin-1	Trichloromethane	303–333	0.500	10.68	This study
	Dichloroethane	303–333	0.486	10.53	This study
	Trichloroethene	303–333	0.407	10.47	This study
Resin-2	Acetone	308–328	0.509	8.58	This study
	Butanone	308–328	0.485	9.43	This study

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3.3 Breakthrough curve

In order to validate the mathematical model of the dynamic adsorption breakthrough process, the breakthrough curves of hexane, heptane on HY-1, and butanone on Resin-2, which are plots of the concentration measured at the outlet in the column versus time, were measured by the column adsorption method, in which the detailed experimental apparatus and adsorption procedure have been described previously.¹⁰ Fig. 1 shows the schematic diagram of the experimental apparatus. Briefly, it consisted of a VOCs vapor generator, an adsorption column, and a gas analysis system. The N₂ gas containing scheduled concentration of VOCs vapor passed through the adsorption column. The VOCs concentration at the outlet of the adsorption column was measured by using gas chromatography with a FID detector. The breakthrough curves were obtained by recording the concentration of VOCs consecutively at the outlet of adsorption column.

Fig. 1 Schematic diagram of the experimental apparatus.

4. Results and discussion

4.1 Multi-linear regression for q₀ and E of DR model

The MLR models for the q₀ and E of DR model are established as two functions of the properties of VOCs and hypercrosslinked polymeric adsorbents respectively. Two kinds of variables for the MLR model were considered in this case: porous parameters of adsorbents, such as micropore volume, specific surface area, average pore diameter, and physic-chemical properties of adsorbates.^35,36 Dubinin and co-workers^37,38 and others (see Wood¹⁵) suggested that the molar polarizability, molar volume, parachor, and ionization potential of adsorbates could be related to adsorption force. All these properties were chosen as preliminary parameters for MLR. Then practical model was established by removing less significant variables to achieve q₀ and E.

These observed q₀ and E data in Table 3 were regressed against the properties of VOCs and hypercrosslinked polymeric adsorbents by SPSS 19 software. Equations are given as the following eqn (8) and (9):

q₀ = 0.361V_micro + 0.0306IP + 0.0277MP + 0.0177V_m − 0.0103Pa − 0.109 (8)

R² = 0.927, adjusted R² = 0.891, s.d. = 0.028, F = 25.546, sig. = 0.000

E = 0.275MP + 0.00318S_BET + 1.509 (9)

R² = 0.819, adjusted R² = 0.791, s.d. = 0.862, F = 29.454, sig. = 0.000

V_micro – micropore volume, IP – ionization potential, MP – molar polarizability, V_m – molar volume, Pa – parachor, S_BET – specific surface area, where R² is square of correlation coefficient, s.d. is the standard deviation of the regression, F is the Fisher F-statistic, sig. is the P value of significance. The equations are dimensionless, as they are obtained by statistical approach and have no specific physical meaning. The analysis of variance (ANOVA) list for the MLR is showed in Table 4.

Table 4 ANOVA list for the MLR

		Sum of squares	df^a	Mean square	F^b	sig.^c
a Degrees of freedom.b Fisher F-statistic.c P value of significance.
q0	Regression	0.102	5	0.020	25.546	0.000
	Residual	0.008	10	0.001
	Total	0.110	15
E	Regression	44.935	2	21.870	29.454	0.000
	Residual	8.458	13	0.743
	Total	53.393	15

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The analysis of variance was performed for the regression significance test by F-test. If the significance level α is set as 0.05, F(q₀) = 25.546 > F_α(k, n − k − 1) = F_0.05(5, 10) = 3.33, F(E) = 29.454 > F_α(k, n − k − 1) = F_0.05(2, 13) = 3.80. What is more, P value of significance (sig.) equal to zero indicates that regression equation is highly significant. The relatively higher R² means satisfactory goodness of fit, demonstrating that the most difference of q₀ and E (dependent variables) can be credibly ascribed to the distinction of the properties of VOCs and hypercrosslinked polymeric adsorbents (explicative variables).

4.2 Verifying the accuracy of MLR model

By eqn (8) and (9), q₀ and E can be estimated for the adsorption of VOCs onto hypercrosslinked polymeric adsorbents without requiring any adsorption experiments conducted in advance. Fig. 2 shows the observed and calculated q₀ and E of various VOCs onto hypercrosslinked polymeric adsorbent. The accuracy of the predictions is assessed by comparing them with experimental adsorption isotherms. The errors between observed and calculated q₀ values range from −8.43% to 10.24%, and that of E values range from −12.75% to 10.78%. Relatively good consistency is shown between the calculated values and the experimental values of q₀ and E.

Fig. 2 Predictive ability of MLR model found from variables.

The calculated equilibrium adsorption amounts in the conditions of experiment are obtained by D–R equation with predicted q₀ and E. The comparison between calculated and experimental equilibrium adsorption amounts reflects the accuracy of the MLR equation more clearly. Fig. 3 shows the comparison between experimental and numerical results. It seems that the experimental and predicted equilibrium adsorption amounts in different conditions have the deviation mostly within 10%, while better agreement can be observed at high pressure.

Fig. 3 Observed and calculated adsorption quantity of VOCs onto hypercrosslinked polymeric adsorbents. HY-1 (black): – benzene, – butanone, – dichloromethane, – pentane, – hexane, – heptane; HPsorbent (red): – benzene, – chlorobenzene; ND-100 (blue): – methyl acetate, – ethyl acetate, – propyl acetate; Resin-1 (green): – trichloroethene, – dichloroethane, – trichloromethane; Resin-2 (turquoise): – acetone, – butanone.

4.3 Prediction of breakthrough curves

In order to predict the breakthrough curves, the column was divided into N equal increments (Δz = L/N), and the greater value of N makes the prediction closer to the actual situation. Predicted breakthrough curves almost do not change when N is greater than 100. In order to save computing time, we used N = 100 to ensure that the prediction was accurate enough.

For D–R equation, 0 is not acceptable for P, and complexed computation will occurred by minor increase of P value when P is very close to 0. Therefore, linear relation was used instead of D–R equation when P is between 0 and 0.01 kPa to make the equation could be solved efficiently.

During prediction process, we chose hexane and heptane adsorption on HY-1 (at 293 K, 308 K and 323 K), and butanone on Resin-2 (at 308 K, 318 K and 328 K) to produce results comparison. The prediction results of D–R equation estimated by the method in Section 4.1 were substituted into the prediction model of breakthrough curves.

The comparisons between experimental and numerical results of hexane and heptane adsorption on HY-1 and butanone on Resin-2 are shown in Fig. 4, respectively. Examples of the reasonably good agreement between the model and the experimental breakthrough curves can be observed. The variation of isotherm prediction can enlarge the deviation of breakthrough curve prediction when D–R equation being substituted into equation group. Nevertheless Fig. 5 shows that the experimental and predicted breakthrough time (t_b), which is defined as the time when C/C₀ = 0.01, has the deviation within 10%.

Fig. 4 Experimental and predicted breakthrough curves for hexane and heptane adsorption on HY-1 and butanone on Resin-2, respectively.

Fig. 5 Observed and calculated breakthrough time of VOCs onto hypercrosslinked polymeric adsorbents.

Because of the good agreement between the experimental and predicted data, it is possible to utilize the proposed model to analyze the relationship of breakthrough curve with time and bed thickness. During model application, these prediction models play an important role when there is no experimental data. As long as the related parameters of adsorbate and adsorbent are known, dimension of device and dosage of adsorbent can be calculated. In addition, enough hypercrosslinked polymeric resins may be filled into adsorption column to make the actual breakthrough time 10–20% longer than predicted in order to obtain the reliable treatment effect.

4.4 Parameter sensitivity

The sensitivity test is performed from the measurements of the breakthrough time t_b shown in Fig. 6, which is significant to determine the parameter which may signally influence the adsorption process. The breakthrough time t_b is predicted with the parameter values identical to those used in the computation except for the parameter to be tested for sensitivity. U, C₀, L (the length of column), q₀ and E appear to be the most sensitive parameters, and other factors are not very sensitive. The values of U, C₀ and L are definite parameters and can be calculated accurately from the operating conditions. Therefore, the unknown parameters q₀ and E in D–R equation become more important for predicting the breakthrough time accurately. In this study, the q₀ and E were related well with the properties of polymeric resins and VOCs using multi-linear regression method, and then the reasonable D–R equation was obtained for predicting the adsorption capacities of VOCs on polymeric resins. Therefore, the good accurate prediction for the adsorption capacities makes the breakthrough times prediction also have good accuracy.

Fig. 6 Significance of parameters on the breakthrough time t_b.

5. Conclusions

A mathematical model was built to simulate the adsorption breakthrough curves of VOCs onto hypercrosslinked polymeric adsorbents in a fixed bed. First of all, the q₀ and E value of D–R equation were related well with the structure properties of resin (micropore volume, specific surface area) and properties of VOCs (molar volume, molar polarizability, parachor, ionization potential) using multi-linear regression method. A reasonably good agreement was obtained between the predicted and experimental equilibrium data with the deviation lower than 10%. Secondly, based on predicted D–R equation and LDF model, the breakthrough curves of hexane, heptane, and butanone were successfully predicted. The breakthrough time had the deviation within 10% between experimental and theoretical results. This may indicate the potential applicability of the model. The main advantage of the approach developed in this work is that the model could be used to determine the breakthrough curves with time as long as the known properties of VOCs and hypercrosslinked polymeric resins without requiring any adsorption experiments conducted in advance.

Acknowledgements

This research was financially supported by National Natural Science Foundation of China (Grant No. 51578281).

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