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Adsorption Thermodynamics and Kinetics of Light Hydrocarbons on Microporous Activated Carbon at Low Temperatures Florian Birkmann, Christoph Pasel, Michael Luckas, and Dieter Bathen Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00678 • Publication Date (Web): 23 May 2018 Downloaded from http://pubs.acs.org on May 23, 2018

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Industrial & Engineering Chemistry Research

Adsorption Thermodynamics and Kinetics of Light Hydrocarbons on Microporous Activated Carbon at Low Temperatures Florian Birkmann1*, Christoph Pasel1, Michael Luckas1, Dieter Bathen1,2 1

University of Duisburg-Essen, Lotharstraße 1, 47057 Duisburg, Germany 2

IUTA e.V., Bliersheimer Str. 58-60, 47229 Duisburg, Germany

Abstract The removal of light hydrocarbons from exhaust air and process gas is important for a variety of applications e.g. in natural gas treatment. However, particularly at lower concentrations the removal of C1- and C2-hydrocarbons is either very expensive or unfeasible with standard technology. Adsorption processes at temperatures below 0 °C may provide a technical solution, but until today no systematic study on the dynamics of trace adsorption at low temperatures is available. Therefore, in this work we present breakthrough curve experiments of ethane, propane, and n-butane over a temperature range from +20 to -80 °C and a concentration range from 5 to 1,000 Pa on a microporous activated carbon. Equilibrium loadings are calculated and modeled with the temperature dependent Toth equation. From dynamic simulation of the experimental breakthrough curves kinetic parameters are determined. Lowering of temperature results in slowdown of kinetics which however is overcompensated by a simultaneous capacity gain.

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1. Introduction The removal of light hydrocarbons such as C1- to C4-hydrocarbons in trace-level concentrations from process gas or exhaust air becomes an increasingly important issue in multiple technical applications. With regard to future environmental regulations, there is a demand for efficient separation techniques, e.g. the reprocessing of exhaust air from stationary and mobile tank systems. Other growth fields are the increasing demand for particularly high gas purities, e.g., for synthesis in the chemical industry, and the upswing of LNG technology1,2. For that purpose, adsorption in fixed beds would be generally suitable. However, adsorption capacities of industrial adsorbents for C1- and C2-hydrocarbons at ambient conditions are often poor compared to higher hydrocarbons. By lowering the adsorption temperature significantly below ambient temperature, as would be possible by energy integration at LNG terminals or low temperature rectification columns3,4, the adsorbent´s capacity could be increased as was shown in earlier work of the authors5. Even so, lowering the adsorption temperature also leads to slower kinetics, which may result in a shorter cycle time of the fixed bed. At present, an assessment of the opposing effects on process dynamics is not possible due to a lack of knowledge and data on the influence of temperature on thermodynamics and kinetics of adsorption in the abovementioned temperature range. In order to design technical adsorption processes, this knowledge is however critical. While adsorption capacity is accessible through experimental equilibrium measurements, kinetics of adsorption can be determined by dynamic simulation of experimental breakthrough curves based on mass transfer models. Modeling of adsorption kinetics of light hydrocarbons at ambient and higher temperatures was already investigated in several publications6–17. In many cases the load difference between equilibrium and present state is used as driving force (linear-driving-force model), while the mass transfer resistance is described formally by an effective mass transfer coefficient18.


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However, while such models can be easily implemented into simulations for designing of technical adsorption processes, predicting of the effective mass transfer coefficient is difficult and elaborate experiments are necessary to quantify it. In order to reduce the experimental expenditure in the future, a deeper knowledge of the mass transfer processes during adsorption is required. However, more complex mass transfer models require higher computational effort which leads to higher costs for designing such processes. Therefore, a common approach is to reduce the number of mass transport mechanisms to the limiting mass transfer mechanism. In adsorption on microporous materials, the limiting mass transfer mechanism is often identified to be intraparticle diffusion6,19,20. Moreover, by negligence of the diffusion due to viscous flow and assuming the adsorption step itself to be instantaneous, the intraparticle diffusion can be assumed to be the sum of diffusion in the gas phase and in the adsorbed phase, where diffusion in the adsorbed phase can account for up to 80% of the total diffusion flux6,21,22. For calculation of the diffusional flux in the gas phase several diffusion models are available21. In contrast, diffusion flux in the adsorbed phase depends on the concentration of the adsorbed phase11,16,23 and on temperature24,25 in a complex manner which complicates pre-calculation. Sophisticated theoretic models describing these dependencies were summarized in an overview article by Medved’ et al.26. However, the applicability of these models suffers in many cases from the high number of unknown energy parameters. For process design, the application-oriented models of Darken27 and Valiullin28 are more suitable due to their small number of accessible parameters. Although adsorption kinetics6–17 and thermodynamics6,29–37 of light hydrocarbons at ambient and higher temperatures are already discussed in several publications, the influence of concentration and temperature on the adsorption dynamics at trace concentrations and at low temperatures, has not been systematically investigated yet. Therefore, this work provides a


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systematic study on adsorption dynamics, including equilibrium and kinetic data, of the light hydrocarbons ethane, propane, and n-butane from nitrogen on a microporous activated carbon at partial pressures from 5 to 1,000 Pa and over a temperature range from +20 to -80 °C. Ethane adsorbs weakly at ambient conditions so it is expected to profit significantly from lower temperatures. Propane and n-butane adsorb more strongly, which is why adsorption may be used in industrial separation of these molecules even at ambient conditions. In this work, we have included propane and n-butane as reference adsorptives which allow to study adsorption behavior in a homologous series at low temperatures. 2. Experimental Section 2.1 Materials. The adsorptives ethane, propane, and n-butane were purchased in pressurized gas cylinders from Air Liquide Deutschland GmbH with purities of >99.5%. Table 1 shows the critical molecular diameters and polarizabilities of the adsorptives, which are important for discussing adsorption thermodynamics. As carrier gas, nitrogen of a purity >99.999% supplied by the university´s infrastructure was used. Table 1. Critical molecular diameters and polarizabilities of the adsorptives38 ethane

propane

n-butane

critical molecular diameter/Å

4.499

5.230

5.230

polarizability / C·m²·V

4.47

6.29

8.2

As adsorbent, the steam-activated carbon RX 1.5 Extra (Cabot Corporation, USA) was selected. The activated carbon is a fractured, cylindrical extrudate with an average diameter of 1.5 mm. The adsorbent´s main properties were determined using nitrogen adsorption at 77 K. Inner surface area was calculated by the BET method (DIN ISO 9277), the pore volume according to the Gurvich rule at a relative pressure of p/ps=0.98 and the micropore volume according to Dubinin-Radushkevich (DIN 66135-3). The adsorbent´s pore size distribution was


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evaluated by nonlocal density-functional theory (NLDFT) assuming slit pores and is given in the supporting information Table S1 and Figure S1. Results as well as basic physical properties of the adsorbent are presented in Table 2. Table 2. Basic physical properties of RX 1.5 Extra and results of nitrogen physisorption isotherms at 77 K

,

̅

,

,

/kg·m-3

/kg·m-3

/m

/-

/-

/-

/-

/-

/m

/cm3·g-1

/cm3·g-1

313

522

2.38E-3

0.3

0.75

1.12

2.8

2.8

7E-10

0.756

0.605

RX 1.5 Extra is a mainly microporous adsorbent with a multimodal pore size distribution having maxima at pore widths of 0.45-0.85 nm and 1.1-2.4 nm. As the critical diameter of the adsorptives is between 0.45 and 0.52 nm, steric hindrance is mainly excluded. Presumably, the activated carbon´s surface mainly consists of disordered, non-polar graphite-like areas forming slit pores. Dispersion interactions between the surface and the non-polar alkanes due to temporary polarization of the binding partner will dominate adsorption. In addition, few oxygen containing functional polar groups may be found on the surface leading to permanent induction forces. In micropores with pore widths in the range of the critical diameter of the adsorptive, adsorption is favored due to interactions of the adsorptive with more than one surface. So, owing to its multimodal pore size distribution, RX 1.5 Extra is energetically heterogeneous with adsorption sites of different energetic value.


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Preliminary to each set of experiments the activated carbon was outgassed at 175 °C in contact with atmosphere for 2h. Depending on the geometry of the adsorption column used, 30 to 95 g of adsorbent were filled into the adsorption column. 2.2 Apparatus and Experimental Procedure. Figure 1 shows a schematic flow sheet of the experimental set-up used for equilibrium and kinetic measurements, which was described in detail in an earlier paper5. In a tempered gas-mixing chamber (1) a defined mixture of adsorptive and nitrogen is provided by mass flow controllers. With a cooling thermostat (2) process

Figure 1. Flow sheet of the experimental set-up. 1, gas-mixing chamber; 2, cooling thermostat; 3, adsorption column; 4, flame ionization detector (FID). temperature is adjusted. Adsorption columns (3) with a height of 200 mm and inner diameter of 30 mm and 40 mm are available and are double walled (with cooling fluid) to minimize axial


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temperature gradients39,40. The hydrocarbon concentration in the carrier gas is continuously analyzed by a flame ionization detector (4). Initially, by bypassing the adsorption column, the hydrocarbon concentration in the raw gas was set. Subsequently, the experiment was started by leading the raw gas into the adsorption column. Thermodynamic equilibrium was assumed when the outlet concentration equaled the inlet concentration. Subsequently, additional equilibrium steps were measured by increasing the inlet adsorptive concentration. By plotting the outlet concentration over time, a characteristic breakthrough curve (BTC) is obtained. From the measured BTCs the excess adsorbed amount was determined using a mass balance around the adsorption column41. Here, the amount of adsorptive molecules remaining in the interparticle and intraparticle gas phase volume was neglected as its contribution is much smaller than the experimental error. By normalization to the mass of adsorbent [kg] the equilibrium load [mol·kg-1] was calculated. Equation 1 shows the mass balance, where

[mol·min-1] is the total mole flow, [mol ppm] is the inlet concentration, !" [mol ppm] is the measured outlet concentration at measuring point #, and ∆% [s] is the time span between two measuring points – usually 4 s.

= '( +

/,23 − ! / ∗, 1 ∗ ∆%/ * 1 − ! / /,4

(1)

2.3 Experimental Error. The measured quantities are afflicted with systematic and random errors leading to uncertainties of the equilibrium partial pressure and the equilibrium load. The systematic uncertainty arises from the uncertainties of the equipment used, namely the calibration gas (2%), the drift of the concentration signal of the FID (1%/24h), the pressure transducers (0.1%), the MFCs (1%), and the balance (0,025-0,15%), while the random error was estimated by Gaussian error propagation (3-5%). Additionally to the theoretical estimation of the


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experimental uncertainty, the reproducibility of the experimental set-up was investigated by replicate measurements. Uncertainties of 1% for the equilibrium partial pressure and up to 6% for the equilibrium load were found. 3. Physical Model 3.1 Adsorption Thermodynamics. For calculation of the equilibrium load as a function of partial pressure and temperature the Toth isotherm equation (equation 2), which has proved to be successful to model adsorption thermodynamics of alkanes on activated carbon21,37,42, was used.

(6, 7, ) = (6) ∗

9(6) ∗ 7,:

>

;1 + (9(6) ∗ 7,: ) (

0.00143 ∗ 6 >.uv ∗ w(t> )S> − (t_ )S> x_ >

> _

10Sy ∗ 7 ∗ √2 ∗ {(∑ ∆Z> )} + (∑ ∆Z_ )} ~

(12)

Aside from an exponential dependency on temperature, the surface diffusion coefficient is found to be dependent on the concentration of the adsorbed phase26 and on the bonding energy of the adsorptive23. Equations to directly calculate the surface diffusion coefficient ] (6, ) are not available yet. Therefore, the surface diffusion coefficient was fitted using the simplified models of Darken et al.27 and Valiullin et al.28, which describe a correlation between the surface diffusion coefficient and surface coverage. In the Darken model the surface diffusion coefficient can be expressed as the product of a ∗ (6), which represents diffusivity at zero loading and a limiting surface diffusion coefficient ],

thermodynamic correction factor Darken (Toth)

a a K`

(equation 13)27.

∗ (6) ∗ ], (6, ) = ],

X Y* 1 ∗ (6) ∗ = ], 1 − J X `

(13)

By substituting the thermodynamic correction factor with the Toth equation (equation 2) an expression for the load dependency of the surface diffusion coefficient is obtained (equation 13, RHS). From equation 13, it can be seen that an increase in loading leads to an increase of surface diffusivity. A limit value consideration shows: Surface diffusivity tends towards the limiting


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Industrial & Engineering Chemistry Research ∗ (6) for zero loading and approaches infinity for high surface surface diffusion coefficient ],

coverages. However, infinite surface diffusivity at high surface loadings is physically unreasonable. For the adsorption of ethane and propane on activated carbon, Do et al.11 found a much stronger dependency on surface coverage than proposed by Darken. Indeed, in preliminary investigations a stronger dependency on surface coverage for the stronger adsorbing components propane and n-butane was observed by the authors. In empirical investigations a better description of the experimental BTCs was obtained using an approach with stronger dependency on surface coverage, which is presented in equation 14 and referred to as the Empirical model. Empirical (Toth)

∗ (6) ], (6, ) = ], ∗

1 (1 − J )_

(14)

However, a stronger dependency on surface coverage further limits the range of validity of the surface diffusivity model as surface diffusivity reaches infinity at lower surface coverages compared to the Darken model. In PFG NMR measurements of tracer diffusivity Valiullin et al.28 observed increasing surface diffusivity with increasing surface coverage on heterogeneous surfaces. As more weak adsorption sites are occupied, the interactions between the adsorptive molecules and the surface get weaker and the molecules’ mobility gets higher:

Valiullin et al.

], (6, ) =

∗ (6) ],

(1 − J)_ Y* ∗ ∗ J Y*. (6)

μ: − * ∗ (6) ∗∗ (6) ], = ], ∗ AB7 F G E6

(15)

∗ (6) In equation 15, ], represents the reference diffusion coefficient including the limiting

∗∗ (6) diffusion coefficient ], at zero coverage and the dependence on the difference between the

standard chemical potential μ: and a constant average site energy * . Since μ: and * are ACS Paragon Plus Environment

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∗ (6) unknown, ], is also denoted as limiting diffusion coefficient for reasons of simplicity. By

substituting Y* in equation 16 using the Toth equation (equation 2), an expression only dependent on surface coverage is obtained: >

∗ (6) (1 − J)_ ], ], (6, ) = ∗1 9 ∗ Y*, 1 − J

(16)

Assuming all parameters but the surface coverage to be constant and < 0.5 as it was observed for all components in this work (see Section 4.1), a limit value consideration yields: • •

lim→: ], (6, ) =

∗ ( ], (6, ) = ∞

Therefore, a behavior similar to the Darken model and to the Empirical model is to be expected with small values of the surface diffusivity at low surface coverages and large values at high surface coverages. However, because of the different functional correlations of surface diffusivity and surface coverage, different BTC profiles are to be expected from the three model approaches. Furthermore, the tortuosity factors μl , μ and μ must be estimated. For diffusion in activated carbon Akani et al.49 proposed the following expressions, which were used in this work:

μl = 1 + 0.5 ∗ (1 − [ )

μ =

[ 0.4 ∗ [ − 0.0328

(17) (18)

Methods for estimation of the tortuosity factor of surface diffusion are not available yet, so that the tortuosity factor of surface diffusion was equated with the tortuosity factor of Knudsen diffusion.


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As unknown parameters in the heterogeneous model the limiting diffusion coefficients ∗ ∗ (6) ∗ (6) (6), ], ], und ], remain, which were determined by fitting the simulated BTCs to

the experimental BTCs. The following initial and boundary conditions were implemented into the model:

Y* (\, % = 0) = 0; Y* (\ = 0, %) = Y*, ; (\, %) = = Y%.

a a

;\ = , %= = 0; `(\, % = 0) = 0;

4. Results and Discussion 4.1 Adsorption Thermodynamics. Adsorption isotherms of ethane, propane, and n-butane on the activated carbon RX 1.5 Extra were measured between -80 °C and +20 °C in a concentration range from 5 to 1,000 Pa in nitrogen at 1 bar. The results are displayed in Figure 2. Equilibrium load in [mol·kg-1] is plotted versus equilibrium partial pressure in [Pa] at different temperatures. The symbols represent experimental data and the lines show the fit of the Toth isotherm equation. The isotherm parameters are given in Table 3. Since the partial pressure of the carrier gas nitrogen is much higher compared to the adsorptive’s, all isotherms show mixture adsorption equilibria of the alkane and nitrogen. However, the great similarity of adsorption enthalpies measured in pure alkane and alkane-nitrogen-mixture adsorption5,50 proves that the presence of nitrogen does not measurably affect alkane adsorption on activated carbon. Therefore, we regard the influence of adsorbed nitrogen to be insignificant.


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All isotherms exhibit a concave shape according to IUPAC type I. Over the experimental concentration range only n-butane at -80 °C reaches a marked plateau indicating saturation capacity. The slopes at low partial pressures increase with increasing chain length from ethane to n-butane. At 20 °C the ethane isotherm is still nearly linear while the n-butane isotherm already features a significant curvature.

Figure 2. Adsorption isotherms of ethane, propane, and n-butane from nitrogen on activated carbon RX 1.5 Extra at various temperatures. Lines show the fit of the Toth equation. We observe a marked increase of the adsorptive load for all components with decreasing temperatures. Again, the effect is most pronounced in case of ethane because for propane and nbutane the adsorbent already approaches saturation in the lower temperature range.


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In the homologues series of alkanes, the number of binding atoms as well as molecular polarizability increase with the chain length. This results in a growing ability to form dispersion interactions with the carbon surface leading to increasing equilibrium loads from ethane over propane to n-butane. The experimental data can be fitted to the temperature-dependent Toth isotherm equation, with the correlation factor R2 not worse than 99%. The affinity constant b0 is in the range of 102-10-3 Pa-1 and increases significantly with the chain length. The parameter Q is a measure for the heat of adsorption at zero coverage. Q increases incrementally with each carbon atom by about 7 kJ·mol-1. The heterogeneity parameter n0 is < 1 for all adsorptives and decreases with the chain length. These parameter characteristics are typical for the adsorption of hydrocarbons on microporous activated carbon21. Table 3. Temperature-dependent isotherm parameters of ethane, propane, and n-butane

adsorptive

6?@

(6?@ )

9: (6?@ ) /Pa-1

D

/kJ·mol-1

: (6?@ ) /-

/-

I

/K

/mol·kg-1

ethane

193

12.84

9.82E-3

22.16

0.43

0

propane

193

10.23

8.99E-1

28.87

0.39

0

n-butane

193

8.48

64.38

35.63

0.36

0

4.2 Adsorption Dynamics and Kinetics. In order to investigate the influence of concentration and temperature on kinetics in fixed bed adsorption at low temperatures, BTCs of ethane, propane, and n-butane were measured at different inlet concentrations and temperatures. For these experiments the adsorbent material was the same as for the equilibrium measurements. The experimental curves were then simulated using Aspen Custom Modeler. First, the numerically simpler homogeneous model was used to determine effective diffusion coefficients. Second, the


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more sophisticated heterogeneous model was applied to simulate some selected BTCs and to estimate surface diffusion coefficients according to the model approaches described in Section 3. A least squares method was used for the parameter fitting. The influence of flow rate and fixed bed dimensions on the kinetic parameter were investigated in preliminary studies. In all cases the kinetic parameters varied only slightly within the error margin. Therefore, an influence of different flow rates or column geometry were assumed to be insignificant. 4.2.1 Experimental Breakthrough Curves and Homogeneous Model. In Figure 3, experimental breakthrough curves (symbols) at different inlet concentrations and temperatures are presented. The lines depict the simulation. The y-axis represents normalized partial pressure and the x-axis experimental time. The experimental parameters as well as the fitted effective diffusion coefficients are tabulated in the supporting information Table S2.


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Figure 3. Experimental and simulated BTCs (homogeneous model) of ethane [a, b], propane [c, d], and n-butane [e, f] from nitrogen on activated carbon RX 1.5 Extra at various concentrations and a constant temperature of -20 °C [a,c,e] and various temperatures and a constant initial concentration of 500 Pa [b,d,f].


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All breakthrough curves exhibit an S-shaped pattern typical for concave isotherms. The curves can be modeled using the homogeneous model with a high overall accuracy (R2 >0.98). The accuracy is slightly better in the temperature range from +20 to -40 °C compared to the range from -60 to -80 °C. There, the model underestimates kinetics and does not describe accurately the initial breakthrough. The time required to reach equilibrium decreases as the inlet concentration increases. This is attributed to the higher driving force of adsorption because of higher equilibrium loads and a higher inlet mass flux of the adsorptive component. Additionally, broadening of the breakthrough curves is observed from higher to lower inlet concentrations. At low partial pressures with low equilibrium loads and slow equilibration the driving force of adsorption is smaller and also more time is available to broaden the mass transfer zone by axial dispersion. Moreover, if we interpret the effective diffusion coefficient as a fixed quantity, as is done in the homogeneous model, the isotherm slope appears in the denominator of equation (4). Consequently, if adsorption takes place at higher partial pressures, the flatter isotherm results in faster kinetics. Compared to ethane, the isotherms of propane and n-butane have a steeper initial slope and flatten more quickly, so kinetics of propane and n-butane are faster and the breakthrough curves are steeper, particularly at lower partial pressures. In contrast to the steeper breakthrough curves the effective diffusion coefficients decrease with increasing inlet concentrations (see Figure 4a). Contrary to expectations, in the range of low concentrations propane has the largest diffusion coefficients, followed by ethane and n-butane. The expected sequence of rising diffusion coefficients with declining molecular weight is only found in the range from 500 Pa onwards. This behavior cannot be explained within the framework of the homogeneous model where the effective diffusion coefficient has a constant


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value in a breakthrough experiment. Costa et al.6 expand the idea of diffusional mass flux in adsorption assuming that also the isotherm slope influences the effective diffusion coefficient. Following this proposal, the effective diffusion coefficient would not only be governed by molecular kinetic properties but also by adsorption thermodynamics. This idea corresponds to a transition from the homogeneous into the heterogeneous model according to equations 9 to 15. Therefore, we consider the influence of thermodynamics on mass flux using the heterogeneous model. While the adsorbent capacity increases with lower temperatures adsorption kinetics are expected to slow down resulting in a broader mass transfer zone51. The influence of temperature on adsorption kinetics and thermodynamics is discussed by the example of the breakthrough curves of ethane over a temperature range from +20 to -80 °C represented in Figure 3b. Lowering of temperature significantly increases ethane capacity. For example, at 500 Pa capacity rises by a factor of 20 from 0.25 mol·kg-1 to 5 mol·kg-1 while the width of the mass transfer zone (m0.985). The accuracy is lower for temperatures ≤ -60 °C, although compared to the homogeneous model the rise of the breakthrough curves can be much better modeled. While the effective diffusion coefficient of the homogeneous model has a constant numerical value in each experiment the effective diffusion coefficient in the heterogeneous model is expressed as the sum of diffusion in the gas phase and in the adsorbed phase, the latter being a function of surface coverage and isotherm slope (see equations 9-15). This way, also thermodynamics of adsorption have an effect on the diffusional mass flux in adsorption6.


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Figure 5. Experimental and simulated BTCs (heterogeneous darken model) of ethane [a, b], propane [c, d], and n-butane [e, f] from nitrogen on activated carbon RX 1.5 Extra at various concentrations and a constant temperature of -20 °C [a,c,e] and various temperatures and a constant inlet concentration of 500 Pa [b,d,f] . ACS Paragon Plus Environment

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Although the surface diffusion coefficients are smaller up to one order of magnitude than the gas phase diffusion coefficients, in this work the contribution of surface diffusion is clearly prevailing. This is attributable to the large proportion of small micropores dominating the mass flux in the adsorbent. The main mechanism of diffusion in macropores and mesopores (pore width > 2 nm) is molecular diffusion. In micropores (pore width < 2 nm), Knudsen diffusion dominates the transport in the gas phase. At the same time, in these small pores surface diffusion adds to the total mass flux. Despite small diffusion coefficients, the contribution of surface diffusion is substantial because of the high molar density in the adsorbed phase compared to the gas phase. Finally, in very small pores (pore width close to the critical molecular diameter) gas phase diffusion becomes more and more restricted and surface diffusion is the prevailing mechanism. According to the pore size analysis, 80% of the pore volume of the activated carbon RX 1.5 Extra used in this work is formed by micropores with an average pore width of 0.7 nm (see Table 2). The critical molecular diameters of the adsorptives range from about 0.45 to 0.55 nm (see Table 1) which is only slightly less than the average pore width of the adsorbent. Selected breakthrough curves of ethane, propane, and n-butane were simulated with the heterogeneous model using the approaches of Darken (het. Da model), Valiullin (het. V model) and an empirical approach (het. E model). Figure 6 compares the results by the example of the breakthrough curve of n-butane at -20 °C and 500 Pa. The experimental data are modeled by all approaches with good accuracy (R² >0.98). At the point of breakthrough, the curve predicted by the Darken model is slightly too steep and the breakthrough is too late. That means the Darken kinetics are too fast at low loadings. At high loadings however, close to equilibrium, the modeled curve is flatter than the experimental curve which means that in this range kinetics are too slow. A better description is provided by the Empirical model. This is due to the stronger load dependence of the surface diffusion coefficient


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in the Empirical model (see Section 3.2). Starting from a lower value at very low coverage, the surface diffusion coefficient of the Empirical model rises more quickly with coverage. The difference between the Darken and the Empirical model is particularly important at equilibrium

Figure 6. BTC of n-butane from nitrogen on RX 1.5 Extra at 20 °C and 500 Pa. Lines are BTCs simulated with the heterogenous model. surface coverages >0.3 which are reached at low temperatures and high concentrations. As discussed in Section 3.2 the surface diffusion coefficients of the Darken model and the Empirical model assume unphysically high values at very high surface coverage. For that reason, modeling of the systems in this work is limited to a coverage of 0.9 in case of the Darken model and 0.8 in case of the Empirical model.


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The surface diffusion coefficient according to Valiullin is small at very low loadings and also reaches infinity at very high loadings. However, at very high loadings the slope of the surface diffusion coefficient calculated with the Valiullin model is less than that calculated with the Darken model, so that very high surface diffusivities are only reached at higher surface coverages compared to the Darken model. Therefore, the Valiullin model allows satisfying descriptions over the entire range of surface coverages found in this work. Figure 6 illustrates that predicted kinetics are slightly too fast at the point of breakthrough. Close to equilibrium the modeled curve is significantly flatter than the experimental curve as the model’s kinetics are too slow. In case of the experiment with n-butane shown in Figure 6., this pattern is quite similar to the Darken prediction but deviations from the experiment are slightly more pronounced. In Figure 7, the fitted limiting surface diffusion coefficients ]∗ (6) according to the approaches of Darken and Valiullin and the Empirical approach are displayed as a function of concentration and temperature, respectively. The limiting surface diffusion coefficients obtained from the Darken model range from 10-9 to 10-11 m2·s-1. The Empirical model delivers coefficients which are one order of magnitude smaller. The coefficients of both models are in the range of values found in literature6,12,17. For the Valiullin approach, the fitted coefficients have much larger values from 10-6 to 10-7 m2·s-1.


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Figure 7. Limiting surface diffusion coefficients of ethane [a, b], propane [c, d], and n-butane [e, f] from nitrogen on activated carbon Norit RX 1.5 Extra at various concentrations and a constant temperature of -20 °C [a,c,e] and various temperatures and a constant inlet concentration of 500 Pa [b,d,f] .


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For ethane, the largest surface diffusion coefficients were determined, followed by propane and n-butane. This is in agreement with the assumption that diffusion is faster as the molecular weight decreases. The coefficients of all adsorptives rise with increasing concentration for all model approaches. The biggest rise is found for n-butane, the smallest for ethane. Weak surface bonding leads to higher molecular mobility on the surface. With increasing coverage weaker bonding sites are occupied so that the surface diffusion coefficients increase. The energetic surface heterogeneity is different for each adsorptive. It increases with the chain length since higher coverage is obtained and also weaker sites are occupied. This is reflected by the values of the Toth heterogeneity constant which is smallest for n-butane (see Table 3). The diffusion coefficients consistently rise with increasing temperature in case of all model approaches. Surface diffusion is regarded as an activated process describable by an Arrhenius approach26. The lines in the temperature-dependent plots of Figure 7 represent the respective fits. As the physical behavior of the fitted limiting surface diffusion coefficients shows the expected trends with respect to the concentration and temperature dependence, the discussion gives no clue to reject any of the model approaches due to a lack of physical meaning. Although the accuracies of simulation (R²) with the homogeneous and heterogeneous model are nearly identical, the effective diffusion coefficients from the homogeneous model do not seem to follow a physically consistent trend. Thus, for the design of technical adsorption processes, prediction of effective diffusion coefficients based on these trends would most probably lead to high uncertainties. In contrast, intraparticle diffusion in the heterogeneous model given as the sum of pore diffusion and surface diffusion is represented in a physical correct manner, as the surface diffusion coefficients show physically meaningful dependencies


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on concentration and temperature. Therefore, predicting of diffusion coefficients for unknown systems is assumed to be much more accurate. 5. Conclusion. The influence of thermodynamics and kinetics on the adsorption behavior of ethane, propane, and n-butane in trace level concentrations at low temperatures on a microporous activated carbon was examined by experiments and dynamic simulations. Adsorption isotherms were measured in a concentration range from 5 to 1,000 Pa and a temperature range from +20 to -80 °C and approximated by the temperature-dependent Toth model. All adsorptives have IUPAC type I isotherms. Capacity consistently increases with increasing chain length and with decreasing temperature. The mass transfer zone broadens as adsorption kinetics slow down when temperature is lowered. However, the kinetic effect is overcompensated by the capacity gain so that bed usage efficiency is consistently better. The strongest improvement is found for ethane with the highest capacity gain. All breakthrough curves can be well described using the homogeneous model. Even so, the fitted effective diffusion coefficients cannot be interpreted in terms of their physical meaning because the transport processes during adsorption are governed by molecular kinetic properties as well as by adsorption thermodynamics which is not reflected in the model. By using a more sophisticated heterogeneous model, intraparticle diffusion was described as the sum of pore diffusion and surface diffusion. While for determination of pore diffusion coefficients well known correlations are available, surface diffusion coefficients were modeled with three model approaches (Darken model, Empirical model, and Valiullin model). At low and medium surface coverages, all approaches provide good results. At very high coverages, the


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Valiullin model is most suitable, because the surface diffusion coefficient of the Darken model and the Empirical model take on unphysically high values. The fitted limiting surface diffusion coefficients rise with concentration and temperature, as expected. The temperature dependence can be accurately described by an Arrhenius approach. Although each of the three approaches may have a limited physical significance we have found no clue to generally reject any of them. Supporting Information. Supporting Information Available: Table and Figure of pore size distribution analysis of activated carbon and table of experimental parameters, effective diffusion coefficients, and surface diffusion coefficients. Corresponding Author. *E-mail: [email protected] Funding. The authors wish to express their thanks to DFG Deutsche Forschungsgemeinschaft e.V. for funding the research project BA 2012/8-1. Notes. The authors declare no competing financial interest. Nomenclature.

9

Affinity constant, Pa-1

Y*,

Adsorptive concentration corresponding to saturation pressure of adsorptive, kg·m-3

Y*

Adsorptive concentration, kg·m-3

̅

Average pore width, m

, ]^ ]*

Particle diameter of equivalent sphere, m Axial dispersion coefficient, m2·s-1 Inner diameter of adsorption column, mm


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Industrial & Engineering Chemistry Research ∗ ]@@

Effective diffusion coefficient, m2·s-1

]m

Molecular diffusion coefficient, m2·s-1

]

Overall gas phase diffusivity, m2·s-1

]∗

Limiting surface diffusion coefficient, m2·s-1

*

Average activation energy, J·mol-1

]

Knudsen diffusion coefficient, m2·s-1

]

Surface diffusion coefficient, m2·s-1

]∗∗

Limiting surface diffusion coefficient at infinite temperature, m2·s-1

b@@

* t

Effective mass transfer coefficient, s-1 Length of the fixed bed, m Mass of adsorbent, kg Molar Mass, g·mol-1

Heterogeneity constant of Toth isotherm

7

Total pressure, bar

Inlet mole flow, mol·s-1

7,:

Equilibrium partial pressure, Pa

E

Gas constant, J·mol-1·K-1

D

E² %

%T

% 6

6?@

Adsorption enthalpy at zero coverage, J·mol-1

Correlation coefficient time, s Time of breakthrough, s Time of equilibrium, s Temperature, K Reference Temperature, K


31


Z

, ,

∆Z

m

Adsorption Thermodynamics and Kinetics of Light ... - ACS Publications

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