Environ. Sci. Technol. 2001, 35, 613-619
Modeling Effective Diffusivity of Volatile Organic Compounds in Activated Carbon Fiber M E H R D A D L O R D G O O E I , * ,† MARK J. ROOD,‡ AND M A S S O U D R O S T A M - A B A D I ‡,§ School of Environmental Science, Engineering & Policy, Drexel University, Philadelphia, Pennsylvania 19104, Department of Civil & Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, and Illinois State Geological Survey, Champaign, Illinois 61820
Volatile organic compounds (VOCs) comprise 67% of total hazardous air pollutants (HAPs) that are emitted by major industrial point sources into the U.S. atmosphere (1). Adsorption by activated carbon fiber (ACF) has been recognized as one of the feasible regenerative control processes to separate and recover VOCs for reuse. Characteristics of VOCs transport in ACFs are required to efficiently design ACF sorption systems. However, extensive resources are spent experimentally obtaining transient sorption data to design adsorption systems. As an alternative, this work develops a new model that predicts effective diffusivities of VOCs into ACFs. The diffusion process is modeled as Knudsen transport into the ACF open pore spaces coupled with activated surface diffusion on the ACF’s internal surface area. Temperature and Darken’s factors are included in the surface diffusion model to provide corrections for thermodynamic state and deviation from Fick’s Law, respectively. Depth of the adsorption potential well is considered as the product of the heat of adsorption of a reference VOC, an adsorption similarity factor, and a surface diffusion energy factor. Introduction of the adsorption similarity factor in the effective diffusivity model is a new concept providing a means to predict effective diffusivity of similar adsorption systems from a reference system. Experimental data from a short length column are used to determine effective diffusivity of acetone in ACF. Results from this diffusivity model are compared to experimental values for the acetone/ACF system to describe the degree of closure between modeled and experimental results.
Model Development The complex heterogeneous structure of ACF is replaced with an effective homogeneous structure having an equivalent rate of mass transfer (Figure 1). Coefficient of mass diffusion through the effective homogeneous ACF is defined as the effective diffusivity (Deff). Deff is then used to describe the mass transfer of VOC molecules through the original ACF complex structure. Mass transfer resistances inside the ACF need to be formulated to obtain Deff. The mass transfer resistances consist of (1) Knudsen diffusion in the ACF pore volume and (2) surface diffusion on the internal surface area. Molecular diffusion, another possible resistance to mass transfer, does not occur inside the ACFs because pore widths (e10 nm) are smaller than the mean free paths of VOC molecules. It is assumed that any point inside the ACF is at a pseudo adsorption equilibrium state (PAES). This enables one to correlate the local VOC concentrations in the Knudsen gas volume and on the pore surface by the VOC/ACF adsorption isotherms. PAES assumption is valid because the mean duration time (MDT) of VOC adsorption on the ACF surface is very short. Most of the VOCs targeted for air quality emission control or gas storage applications have molecular forms with sufficiently high relative pressure that they physically adsorb into ACFs. MTDs of such physical adsorption are < 10-6 s for typical VOC/ACF adsorption systems of interest at T > 298 K. The MDT can be estimated from MDT ) t0 exp(Ea/RuT) (3). The preexponential factor (t0) is equal to the vibrational period of the adsorbed molecule and can be calculated by statistical mechanics from the partition functions of the gaseous and adsorbed VOC molecules (4).
Model Components
Introduction Large amounts of volatile organic compounds (VOCs) are emitted into the atmosphere by industrial sources each year (1). Many of these VOCs are hazardous to human health and environment. Adsorption has been recognized as one of the most practical regenerative methods for separating and recovering VOCs from industrial flue gas streams. In recent * Corresponding author phone: (215)895-2280; fax: (215)895-2267; e-mail:
[email protected]. † Drexel University. ‡ University of Illinois at Urbana-Champaign. § Illinois State Geological Survey. 10.1021/es0012568 CCC: $20.00 Published on Web 12/21/2000
years, new adsorbent materials and sorption methods have been actively studied for efficient separation of VOCs from polluted air streams. Activated carbon fiber (ACF) is a promising adsorbent that can be used for the efficient separation, purification, recovery and storage of VOCs. Efficient utilization and design of an ACF/VOC sorption system requires prediction of transient mass transfer during sorption of the VOC. Effective diffusivity of the VOC in the ACF should be known to reliably predict such mass transfer. Typically, effective diffusivity data either are not available or are limited to a single thermodynamic phase condition. Effective diffusivity needs to be known as a function of VOC concentration and temperature to predict realistic sorption processes that occur concomitantly with heat transfer. Therefore, a new formulation is developed here to predict the effective diffusivity as a function of adsorbate concentration and temperature. Acetone and American Kynol ACF-20 (ACC-5092-20) (2) are used as the VOC and ACF to provide measured values for the effective diffusivity model and to evaluate closure between modeled and experimental results.
2001 American Chemical Society
Formulation of the effective Knudsen diffusion coefficient (DK,eff) requires knowledge about the ACF’s pore size distribution (PSD) (5). DK,eff for the nonadsorbed adsorbate located in the ACF’s pores is described by eq 1 when assuming a random distribution of PSD over the ACF cross-section (6)
( )
2 2RuT DK,eff ) DK ) 2DK ) τ 3 πMw
1/2
∫d [f(d )] p
p
2
d(dp) (1)
where f(dp) is the pore size density function and ∫ f(dp) d(dp) is the total porosity () of ACF. Note that the random pore VOL. 35, NO. 3, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 1. Electrical Polarization, Magnetic Susceptibility, and Similarity Factor for Adsorption of Some Organic Vapors in Activated Carbona
FIGURE 1. A circular plane model for diffusion of a VOC in an ACF. A typical gas phase and adsorbed phase concentration profiles of a VOC in the ACF are demonstrated. model describes the medium tortuosity factor (τ) as 1/ (6). However, 1/ is an estimate for τ and a more accurate value could be obtained from nonadsorbed gas-diffusion experiments. The exact value of τ cannot be independently predicted from first principles, even when details of the adsorbent structure are known (7). Surface diffusion can take place in a mobile or immobile adsorbed state on the surface of the pores parallel to the Knudsen diffusion. Mobile surface adsorption describes twodimensional adsorbate flow close to the pore surface when the binding energy of the adsorbate is less than its thermal energy of translation (RuT). Mobile surface diffusion is independent of temperature. In contrast to mobile surface diffusion, immobile surface diffusion occurs when the binding energy of the adsorbate is greater than its thermal energy of translation. In immobile surface diffusion, the adsorbed molecules hop between adjacent adsorption sites in the surface lattice. The adsorption sites arise from local variations in the binding energy along the surface (8). Adsorbate molecules must pass from the apex of the adsorption dispersion well to transfer from one adsorption site to another (9). The probability that VOC molecules overcome this hurdle increases with increasing temperature. Therefore, surface diffusivity (Ds,i) of immobile adsorption is strongly temperature dependent. Ds,i can be modeled as random jumping of adsorbed molecules between adjacent sites of different adsorption strength (8) referred to as the activated random walk model (eq 2).
( )
Ed 1 1 Ds,i ) νλs2 ) ν0λs2 exp 4 4 RuT
(2)
The jump frequency, ν0, and the jump distance, λs, are assumed to be independent of surface concentration. The value of ν0 is considered to be equal to the vibrational frequency of the adsorption following the Transition State Theory (TST), and it can be calculated from the statistical mechanics (4). Sladek et al. (10) compared several experimental results for immobile surface diffusivities of different adsorption systems resulting in the following empirical correlation
(
Ds,i ) 1.6 × 10-6 exp -0.45
Ea m RuT
)
(3)
where Ds,i is in m2/s, and m is a fitting factor, which can be different for different adsorption systems. Similar to Sladek et al.’s formulation, an Arrhenius form with no temperature dependency of the preexponential factor and a fixed activation energy of diffusion are often used for surface and overall effective diffusivities of microporous adsorbents (11). In this work, eq 2 is modified to provide an improved model describing the effective surface diffusivity, including 614
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compound
r × 1024, cm3
χ × 106
β
n-hexane TCA toluene cyclohexane pentane benzene cyclopentane n-butane MEK 1-butene propane acetone propene ethanol ethane ethylene methanol methane carbon
11.9 10.7 12.3 10.9 9.99 10.3 9.19 8.2 8.13 7.41 6.33 6.37 6.26 5.45 4.45 4.25 3.28 2.59 1.76
-92.5 -114.0 -65.9 -58.0 -63.3 -54.8 -59.2 -50.3 -45.6 -41.0 -38.6 -33.8 -30.7 -33.7 -27.09 -15.3 -21.4 -14.0 -6.00
2.13 2.09 1.94 1.72 1.67 1.62 1.55 1.36 1.30 1.18 1.05 1.00 0.952 0.905 0.735 0.563 0.555 0.409
a
Reference: acetone.
mobile and immobile diffusion, as a function of temperature and concentration (eq 4).
( )
(
)
Ea(T,C) 1 ∂lnC 1/2 Ds,eff (T,C) ) ν0λs2 T exp -ψβ 4 ∂lnq RuT
(4)
The dependence of surface diffusivity on temperature is now predicted with exponential and preexponential factors. The temperature functionality of the preexponential factor is consistent with TST (4) and observations of surface diffusivities of hydrocarbons in zeolites that could be correlated with T1/2 (12, 13). The temperature functionality of the exponential factor is related to the ratio of the isosteric heat of adsorption of the VOC/ACF over the VOC thermal energy of translation. In this model, the interaction between a VOC molecule and a surface is not necessarily limited to specific sites with a constant heat of adsorption since Ea is considered to be a function of VOC concentration and adsorption temperature. Ea can be extracted from the temperature dependency of isotherms and the Clausius-Clapeyron as given in eq 5:
Ea ) -RuT2[∂ ln CK/∂T]q
(5)
In this new surface diffusivity model, the surface diffusion energy factor ψ is related to the specific surface properties of the ACF including the surface tortuosity. The adsorption similarity factor β describes the relative depth of the adsorption potential wells of different adsorbates with respect to that of a reference adsorbate. The similarity factor β is predicted from the dispersion interaction theory (2, 14-15) as described in eq 6
β)
(
)
R Ro/χo + Rs/χs Ro R/χ + Rs/χs
(6)
with β ) 1 for the reference adsorbate. The R and χ are respectively the electrical polarization and the magnetic susceptibility of the VOC and carbon that control the strength of the physical adsorption. A compound with a higher value for β has a higher adsorption capacity and a smaller surface diffusivity. Calculated values of β for select diamagnetic VOCs are given in Table 1.
Increases in surface diffusion with increasing adsorbate concentration are predicted by an increase in Darken’s correction factor in the preexponential term and reduction in the heat of adsorption. Darken’s correction factor is related to the change in the activity of adsorbate on the pore surface of the ACF. Realizing that the driving force for diffusion is the gradient of chemical potential rather than that of concentration (16), Darken’s correction factor for the effective Fickian diffusivity can be derived as follows:
∂lnγ ≡ [1 + ( (∂lnC ∂lnq ) ∂lnq)] T
(7)
T
Often this correction factor has been applied to the overall effective diffusivity (e.g. refs 17 and 18). In this work, Darken’s correction factor is only applied to the surface diffusivity because the Knudsen diffusivity is independent of surface concentration. Reduction in the heat of adsorption with increasing surface concentration is due to progressive filling of pores with descending adsorption potential. Molecules that are more weakly bound to the surface will encounter a smaller energy barrier and consequently will be more mobile.
Overall ACF Effective Diffusivity The overall effective diffusivity model resolves heterogeneity of the ACF by considering a random distribution of pores over the cross-section of the ACF. It is assumed that the VOC molecules transfer through the homogenized and idealized slit pores by effective Knudsen diffusion in parallel with the effective surface diffusion (Figure 1). The longitudinal concentration gradient in the ACF is neglected due to the large aspect ratio of the fiber length to its diameter (. 100). As the adsorbate molecules diffuse through the pores by Knudsen diffusion, they develop a pseudoequilibrium adsorption state with the adsorbed molecules diffusing on the pore surface. Assuming Fickian transports for the effective Knudsen and surface diffusions, the mass transfer equation can be written as (2)
[
]
[
]
∂CK ∂q ∂2CK 1 ∂CK ∂2q 1 ∂q + ) DK,eff + + Ds,eff 2 + 2 ∂t ∂t r ∂r r ∂r ∂r ∂r (8)
The pseudoadsorption equilibrium state for the local gas and surface concentrations, discussed in the previous section, enables us to correlate the partial derivatives of gas phase and adsorbed phase concentrations by eq 9:
∂q ∂q ∂CK ∂q ∂T ) + ∂r ∂CK ∂r ∂T ∂r
(9)
∂T/∂r is practically zero through the ACF because the resistance to internal heat conduction is much less than the external heat convection resistance, as described by the ACF’s small Biot number (NBi ) hdf/k). NBi is the ratio of internal to external heat transfer resistances. For a typical ACF with diameter =10-5 m, NBi is =10-3 to 10-2. Such small NBi allows ∂q/∂CK to become an exact derivative. Then, dq/dCK can be calculated from slopes of isotherms. This allows one to separate the overall mass transfer equation (eq 8) into two equations using eq 9 (2), one for transport through an effective gas medium (eq 10) and one for transport through an effective adsorbent medium (eq 11):
∂CK ) ∂t
{(
)(
dq dq DK,eff + FfDS,eff / + Ff dCK dCK
)} [ ( )]
1 ∂ ∂CK r r ∂r ∂r (10)
{(
∂q ) ∂t
)(
dCK dCK + FfDS,eff / + Ff dq dq
DK,eff
)} [ ( )]
1 ∂ ∂q r r ∂r ∂r (11)
Substitutions of DK,eff and Ds,eff from eqs 1 and 4 into eqs 10 and 11 provide the final new model for predicting the effective diffusivities of VOCs in ACFs (2) as provided below:
{ ( )∫ ( )
1/2 2x2 RuT dp[f(dp)]2d(dp) + 3 πMw ∂lnCK dq ∂lnCK 1 F υ λ 2T1/2 exp -βψT 4 f o s ∂lnq T dCK ∂T
Deff,C )
[ ( ) ]}/ ( ) q
dq (12) dCK
+ Ff
Deff,q )
{ ( ) ∫ ( ) [ ( ) ]}/( 2x2 RuT 3 πM
1/2dC
∂lnCK 1 F υ λ 2T1/2 4 f o s ∂lnq
k
dq
T
dp[f(dp)]2d(dp) +
exp -βψT
∂lnCK ∂T
dCK + Ff dq
q
)
(13) Deff,C assumes that the fiber is an effective homogeneous gas medium, and Deff,q assumes that the fiber is an effective homogeneous solid medium. Deff,C is equivalent to Deff,q. This means that the computations of adsorption dynamics for an ACF column will provide the same results if the effective medium for the ACF is selected as a homogeneous solid with Deff ) Deff,q or a homogeneous gas with Deff ) Deff,C.
Model Parameters All of the thermodynamic parameters needed for the effective diffusivity model are obtained from the system thermal equation of equilibrium adsorption (TEEA) (2) except for ψ. The value of ψ is determined from nonequilibrium short length chromatography (SLC) experiments (2). The TEEA generates a thermodynamic surface in the 3-D space of adsorption capacity, adsorbate pressure and temperature. It is important to emphasize that the analytical form of an accurate TEEA enables one to reliably determine the required thermodynamic properties of VOC/ACF adsorption and pore size distribution of the adsorbent (5). The Dubinin-Astakhov isotherm was modified to provide a general TEEA (2) as given by the following equation:
{
()
(T - To) Fb qe ) F(T)Vooexp -2.30415 log Fc (Tc - Tb)
([
(Tc - T)
Ru A(Tc - T) + B
Tc0.5
(Tc - T)6
D
Tc5
1.5
(Tc - T)3
+C
Tc2
+
]/ ) }
Pc + Tln P
n(T)
βEo
(14)
Darken’s thermodynamic correction factor and the slope of isotherms (dq/dC) for adsorption of acetone in Kynol ACF20 (2) are computed using eq 14 and presented in Figure 2(parts a and b, respectively). Figure 2a shows how the surface diffusivity, which is proportional to Darken’s factor, could increase 300% due to the increase in this thermodynamic correction factor as the acetone concentration increases from zero to 10,000 ppmv. The slope of isotherms (dq/dC) has a dimension of specific volume and is a determining factor for controlling the magnitude of the overall effective diffusivity (Figure 2b). This point becomes clearer when comparing the orders of magnitudes of Knudsen diffusion, surface diffusion, dq/dC, and their contribution to the overall effective diffuVOL. 35, NO. 3, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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615
FIGURE 2. (a) Darken’s thermodynamic correction factor and (b) the slope of adsorption isotherms for diffusion of acetone into ACF-20.
FIGURE 3. Isosteric heat of acetone adsorption in ACF-20 as a function of adsorbed phase concentration and temperature. sivity models. Knudsen diffusivity is several orders of magnitude faster than the surface diffusivity. But dq/dC enhances the overall surface transport of VOC molecules in the ACF. The product of Ff dq/dC enhances the surface transport inside the ACF (i.e., the second term in the numerator of eq 12) 6 orders of magnitude for the acetone/ ACF-20 system at T ) 300-400 K and acetone partial pressure of 0-1000 Pa (Figure 2b). Figure 3 shows isosteric heat of adsorption of acetone in ACF-20 as a function of the amount of adsorbed acetone and temperature, computed from eqs 5 and 14. This figure shows the heat of adsorption reduces as the amount of adsorption increases. This is because ACF pores are filled in descending pore characteristic energy (2). The heat of adsorption for acetone in ACF-20 at 25 °C reduces from 62.5 kJ/mol to 51 kJ/mol when increasing the amount of adsorption from zero coverage to complete pore filling. Heat of adsorption for acetone on nonporous graphitized carbon black at zero coverage is 33.5 to 35 kJ/mol (19). The primary difference between the heat of adsorption on the carbon black and in the ACF is due to the overlapping and enhancement of the adsorption potentials inside the micropores. Figure 4 compares the heat of adsorption for select chemical compounds on graphitized carbon blacks and in activated carbons as provided by Stoeckli (20). Figure 4 also describes the heat of adsorption for acetone in ACF-20 reported here. Deviations from the observed correlation line are related to the variations of PSD in different activated carbons and concentration dependency of the heat of adsorption. Activated carbon samples with smaller pores have higher heat of adsorption values (5). The limiting heat of adsorption of acetone in ACF20 at zero coverage shows a larger deviation from the trend line observed. This deviation is presumably due to the presence of a higher partial pore volume of ultramicropores (dp < 7 Å) in ACF-20 than in the other tested activated carbons. 616
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FIGURE 4. Comparison of the heat of adsorption computed for acetone in the ACF-20 from this work and the limiting heats of adsorption of different chemical compounds on graphitized carbon blacks and activated carbons (20). Rectangle: acetone (this work), 1: Ne, 2: Ar, 3: N2, 4: O2, 5: CH4, 6: Kr, 7: Xe, 8: CO2, 9: C2H4, 10: H2O, 11: C3H8, 12: C4H10, 13: c-C6H12, 14: C5H12, 15: C6H6.
Determination of ψ from SLC Experiments Values for effective diffusivity (eq 12 and 13) for select values of ψ/RuT are provided in Figure 5. Mobile surface diffusion takes place inside the ACF when ψ is equal to zero. Effective diffusivity has a large value and increases with adsorbate partial pressure and decreases with temperature as a result of mobile surface diffusion. For higher ψ values, the strength of surface interaction energy increases and diffusivity decreases. At higher ψ values (ψ/RuT > 0.00032), the diffusivity increases with temperature and is weakly dependent on adsorbate partial pressure. Breakthrough curves (BTCs) from SLC experiments, containing a very small sample of the ACF, can be used to determine ψ and effective diffusivity for a VOC/ACF system. Careful experiments and data analysis are required to compute the effective diffusivity and then determine ψ. Measured effective diffusivities for similar systems by different methods or by other investigators have provided different results (11, 21). There are several factors that can contribute to sources of error in determining the effective diffusivity during the measurements and calculations. Some of these factors include, non-Fickian diffusivity, nonlinearity in the adsorption isotherm, nonisothermal adsorption due to the heat effect, film effect due to the flow velocity, change in the local concentration gradient, and longitudinal dispersion. An experiment was designed to measure the breakthrough concentration profiles of acetone through an SLC packed with the ACF-20. This method is similar to a regular gas chromatography except that a very small amount of adsorbent is used in the column (i.e. < 6 mg used in this work). The small sample size provided increased sensitivity of VOC measurement by reducing the secondary mass transfer resistances such as longitudinal dispersion and film diffusion. Experiments were performed for acetone mixed with N2 or He at 30, 40, 50, and 60 ppmv concentrations at 65 °C. High
FIGURE 5. Effective diffusivity as a function of gas-phase concentration and temperature for different ψ/RuT factors. The results are for a system having similar adsorption characteristics to the acetone/ACF-20 system. flow rates (i.e. ∼ average SLC pore velocity of 480 cm/s) were selected to reduce the film diffusion resistances to negligible values. The use of two inert carrier gases, N2 and He, while observing the same breakthrough concentration profiles, provided assurance for negligible film resistance. Duplicate blank tests (using the same amount of inert packing material and support screens) were also used to correct for adsorption that takes place on the inert packing, tubing and valves upstream of the VOC detector. A computational scheme was developed to simulate SLC breakthrough profiles based on variable effective diffusivities (2). The ψ factor was determined from fitting the effective diffusivities resulting from the computational and experimental breakthrough curves. By the method of separation of variables, the partial differential eq 11 that describes the mass transfer of VOC into the ACF was solved (2) to provide the following analytical formulation:
q qo
)
-2Deff R
∞
Jo (rλi)
∑ λ J (Rλ )exp(-D i
i)1
1
2
effλi
t
0
2
effλi
q j ) 4πDeff
∑ exp(-D i)1
effλi
2
t)
[
η) 1 -
∫ exp(D t
0
∫ exp(R
∑
]
qs(η) qo
effλi
2
An analytical form for the surface concentration qs(η) is not available; therefore, eq 16 is integrated semi numerically assuming a constant qs at each time step (2). The following
2 i
η)Qs(η) dη} +
∂C h (xj,τ)
∂C h (xj,τ) ∂xj
+
∂τ ) 0 (17)
where Pe′ (ζ) is the ratio of the SLC’s contact time (R/u) to the characteristic diffusion time of the VOC in the fiber (R2/ Deff). The TEEA is used to relate the overall local mass of adsorbed material q j to the local gas concentration C. A thirdorder accurate upwind discretization method is used to formulate the finite difference form of the transport equation as follows (2):
dη (15)
η)qs(η) dη (16)
τ
0
Pe′ζ
C h n+1j ) C h nj - ω
This equation has been developed for variable surface concentration of qs and an initial uniform preloading of qo. Often for simplicity, investigators assume a constant surface concentration when computing Deff from transient experimental results. However, this assumption serves as a source of error because adsorption can cause a transient change in the surface concentration through the column. The instantaneous amount of mass adsorbed in the ACF can be found by integrating eq 14 over the fiber cross sectional area. The temporal variability of adsorbed phase concentration for an initially pure ACF is described by ∞
∂ ∞ 4 exp(-Ri2τ) ω { ∂τ i)1
t)
i
∫ exp(D
normalized integro-differential equation is developed to define the SLC’s local adsorption kinetics assuming negligible overall axial diffusion due to the ACF’s small diameter and the SLC’s short length (2):
k
{
k
∑ exp(-R
i)1
[
dτ)Qni,j + 2(Qns,j +
i)1
1
∑R
Qn+1s,j)
2 1
2 i
-
1 Ri2
]
}
exp (-Ri2dτ) - Qnj,tot -
Pe′ζ∆τ n h nj-1 + 3C h nj + 2C h nj+1) (18) (C h j-2 - 6C 6∆xj where
Qn+1i,j )
2 n (Q s,j + Qn+1s,j)(1 - exp(-Ri2dt)) + Ri2 Qni,j exp(-Ri2dτ) (19)
The Qn+1s,j is related to the local normalized gas concentration C h n+1j by the TEEA. This numerical method is used to simulate the normalized BTCs of the SLC experiments. The blank BTCs are used as the SLC’s inlet boundary condition to obtain realistic simulation results for the BTCs with acetone. Several BTCs are generated with different diffusion time constants (Deff/ VOL. 35, NO. 3, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 6. Measured and modeled effective diffusional time constants of acetone in ACF-20 (from this research) compared with the measured diffusional time constants of nitrogen in a 5A carbon molecular sieve (CMS) (22) and Sladek’s model (eq 3).
adsorption systems with a dominant configurational diffusion (for adsorbents with primarily ultramicropores (5)). Figure 7 shows the modeled effective diffusivity of acetone as a function of its partial pressure and temperature using eq 13. The diffusivity shows 2 orders of magnitude variability for the given practical ranges of temperature and partial pressure. Prediction of effective diffusivities for other VOCs/ACF20 adsorption systems are now possible by using ψ/RuT ) 0.000195 from the acetone-ACF-20 experiments and the computed β similarity factors (Table 1). The β factor is used in the TEEA to predict the appropriate adsorption parameters as functions of temperature and VOC concentration (i.e., adsorption capacity, the isosteric heat of adsorption, isosteric entropy of adsorption, pore size distribution of the VOC/ ACF system, Darken’s thermodynamic correction factor, and slopes of isotherms) for the system of interest. Once these values are determined, the transient behavior of the VOC/ ACF sorption system can then be determined. Such novel approach allows for the prediction of VOC/ACF adsorption kinetics and dynamics without spending extensive amounts of experimental resources.
Nomenclature A
FIGURE 7. Modeled effective diffusivity of acetone through ACF-20 as a function of acetone partial pressure and temperature. R2). Note that the diffusion time constant is the property of the ACF/VOC system and is independent of the diameter of the ACF. The computed normalized BTCs from the simulated SLC are then used to determine the diffusion time constants that match the experimental BTCs (Figure 6). Least-square errors between the computed and measured BTCs are used to determine the best fits. The fits provided R2 coefficient of correlations greater than 0.996. Regression analysis between the measured effective diffusivities and predictions from eq 13 provides a surface energy factor ψ/RuT ) 0.000195 for the acetone/ACF-20 adsorption system. Results from the effective diffusivity model and SLC experiments obtained at 65 °C are provided in Figure 6. Increasing the effective diffusional time constant with increasing adsorption capacity is consistent with the modeled and experimental results. Figure 7 also compares the modeled and measured results with the diffusional time constants of N2 in a 5A carbon molecular sieve (CMS) at 60 °C (22). A consistent trend is observed for the functionality of the observed diffusional time constants with adsorption capacity and temperature. Adsorption capacity is shown as an important factor when describing the diffusional time constants of adsorbates in microporous adsorbents. Note that to achieve similar adsorption capacities for different adsorbates, the adsorption media should have either different physical (e.g. PSD) or chemical properties. For example ACF20 has an average pore size of 13 Å (2, 5). Therefore, in order for an activated carbon with equal total pore volume as ACF20 to have an adsorption capacity for nitrogen similar to that of the acetone/ACF-20, it must have smaller pores than 13 Å (5). This characteristic exists in the 5A CMS which has an average pore size of 5 Å. Regression analysis of the experimental data using Sladek et al.’s model (eq 3) is also provided in Figure 6. This surface diffusion model does not adequately describe the effective diffusional time constants of the acetone/ACF adsorption system at lower concentrations. This is because its slope is invariant of adsorption capacity and also it neglects Knudsen diffusion. Knudsen diffusion can only be neglected for 618
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a constant in the TEEA depends on the type of VOC
AC
cross sectional area of the SLC
B
a constant in the TEEA depends on the type of VOC
C
adsorbate concentration in the gas phase; a constant in the TEEA depends on the type of VOC
Ck
gas concentration in the Knudsen region
C h ) C/Ffqo
normalized gas-phase concentration
D
a constant in the TEEA depends on the type of VOC
Deff
overall fiber effective diffusivity
Dk
Knudsen diffusivity
Dk,eff
Knudsen effective diffusivity
Ds,eff
effective surface diffusivity
Deff,C
effective diffusivity for an effective homogenized gas medium replacing the fiber
Deff,q
effective diffusivity for an effective homogenized solid medium replacing the fiber
Ds,i
immobile surface diffusivity
df
fiber diameter
dp
pore diameter
E0
reference adsorbate pore characteristic energy of adsorption
f(dp)
pore size density function
Ea
heat of adsorption
Ed
activation energy of an immobile surface diffusion
i
subscript for Eigen values
j
subscript for the location of the computation grid node in the SLC
Jo
zero-order Bessel function of the first kind
J1
first-order Bessel function of the first kind
γ
adsorbate activity
m
a fitting constant in the Sladek model
ψ
surface energy factor
MACF
mass of ACF in the SLC
MW
VOC molecular weight
ACF porosity; SLC bulk porosity ) MACF/ FiACL
n
superscript for the computation time index
β
surface energy similarity factor
F
VOC density in liquid state
PC
critical pressure of the VOC
Fb
VOC density at boiling point
Pe′ ) uR/Deff ≡
local Peclet number
Fc
VOC density at critical point
Ff
fiber density
q
adsorbate concentration in the adsorbed state
λi
eigen value for the Bessel characteristic equation
qe
adsorption equilibrium capacity at a thermodynamic phase state
λs
mean free path of the surface diffusion
τ
tortuosity; normalized time of diffusion into ACF τ ) tDeff/R2
qo
a constant adsorbate concentration in the adsorbed state or the initial adsorbate preloading
χ
susceptibility of the test adsorbate
χo
susceptibility of the reference adsorbate
χs
susceptibility of the adsorbent (i.e. carbon)
qs
adsorbate concentration in the adsorbed state at the external surface of adsorbent
q j
average amount of adsorption in the fiber at time t
∆τ
computation normalized time interval
Q)q j /q jo )
∆x
computation distance interval
normalized local degree of saturation
∆xj
normalized computation distance interval
Qs
normalized local degree of saturation on the ACF external surface
r
radial distance in a cylindrical coordinate
Literature Cited
R
fiber radius
Ru
Universal Gas Constant
s
subscript for surface
T
temperature
T0
a reference temperature
Tb
boiling point temperature of the VOC
(1) USEPA. TRI Public Data Overview; www.epa.gov/opptintr/tri/ pdr95/drover01.htm, 1998. (2) Lordgooei, M. Ph.D. Dissertation, University of Illinois at Urbana-Champaign, Urbana, IL, 1999. (3) Chappuis, J. Multiphase Sci. Technol. 1982, 1, 387. (4) Laidler, K. J. Chemical Kinetics, 3rd ed.; Harper & Row: New York, 1987. (5) Lordgooei, M.; Rood, M. J.; Rostam-Abadi, M. submitted to the ASCE J. Environ. Eng., accepted for publication. (6) Froment, G. F.; Bischoff, K. B. Chemical reactor analysis and design, 2nd ed.; Wiley: New York, 1990. (7) Reyes, S. C.; Iglesia, E. Computer-Aided Design of Catalysts; M. Decker: New York, 1995. (8) Gilliland, E. R.; Baddour, R. F.; Perkinson, G. P.; Sladek, K. J. Ind. Eng. Chem. Fundam. 1974, 13, No. 2, 95. (9) Ehrlich, G. J. Chem. Phys. 1959, 31, 1111. (10) Sladek, K. J.; Gilliland, E. R.; Baddour, R. F. Ind. Eng. Chem. Fundam. 1974, 13, No. 2, 100-105. (11) Ruthven, D. M. Principles of adsorption and adsorption processes; Wiley: New York, 1984; p 138. (12) Xiao, J.; Wei, J. Chem. Eng. Sci. 1992, 47, No. 5, 1123. (13) Xiao, J.; Wei, J. Chem. Eng. Sci. 1992, 47, No. 5, 1143. (14) Muller, A. Proc. Royal Soc. (London) 1936, A154, 624-639. (15) Dubinin, M. M.; Polyakov, N. S.; Kataeva, L. I. Carbon 1991, 29 (4/5), 481-488. (16) Darken, L. Trans. AIME 1948, 175, 184. (17) Grenier, Ph.; Bourdin, V.; Sun, L. M.; Meunier, F. AIChE J. 1995, September, 41, No. 9, 2047-2057. (18) Pimenov, A. V.; Lieberman, A. I.; Scmidt, J. L.; Chen, H. Y. Separation Sci. Technol. 1995, 30(16), 3183-3194. (19) Avgul, N. N.; Kiselev, A. V. Chem. Phys. Carbon 1970, 6. (20) Stoeckli, F. Helv. Chem. Acta 1974, 57, Fasc., 7, Nr. 237, 21922195. (21) Hsu, L.-K. P.; Haynes, H. W., Jr. AIChE J. 1981, 27, No. 1, 81-91. (22) Kawazoe, K.; Suzuki, M.; Chihara, K. J. Chem. Eng. Jpn. 1974, 7, 151.
TC
critical temperature of the VOC
t
time
V0°
limiting volume of adsorption
u
velocity component in the direction of x
x
coordinate in the direction of bed length
xj ) x/L ≡
location ratio from the inlet of SLC
Greek Letters R
polarizability of the test adsorbate; thermal coefficient of limiting adsorption
Ro
polarizability of the reference adsorbate
Rs
polarizability of the adsorbent (i.e. carbon)
Ri ) Rλi ≡
eigen value of the normalized diffusion equation
ν
jump frequency for the surface diffusivity
νo
jump frequency constant
ζ ) R/L
ratio of the ACF’s radius to the SLC’s effective length
ω) η
1- ≡
representative of ACF’s bulk density in the SLC
Received for review May 10, 2000. Revised manuscript received October 4, 2000. Accepted November 8, 2000.
a dummy integration variable
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