Environ. Sci. Technol. 2009, 43, 1173–1179
Modeling the Heat and Mass Transfers in Temperature-Swing Adsorption of Volatile Organic Compounds onto Activated Carbons ´ ,* SYLVAIN GIRAUDET, PASCALINE PRE AND PIERRE LE CLOIREC Ecole des Mines de Nantes, GEPEA UMR CNRS 6144, 4 rue Alfred Kastler, BP 2072 44307 Nantes Cedex 3, France, Ecole de Chimie de Rennes (ENSCR), UMR CNRS 6226 Sciences Chimiques de Rennes, Campus de Beaulieu, Avenue du ´ eral ´ Gen Leclerc, 35700 Rennes, France
Received July 31, 2008. Revised manuscript received October 7, 2008. Accepted October 21, 2008.
A theoretical model was built to simulate the adsorption of volatile organic compounds (VOCs) onto activated carbons in a fixed bed. This model was validated on a set of experimental data obtained for the adsorption of acetone, ethyl formate, and dichloromethane onto five commercial activated carbons. The influence of operating conditions was modeled with various VOC contents at the inlet of the adsorber and superficial velocities of the gas-phase from 0.14 to 0.28 m.s-1. Breakthrough times and maximum temperature rises were computed with a coefficient of determination of 0.988 and 0.901, respectively. The simulation was then extended to the adsorption of mixtures of VOCs. From the comparison of simulation and experimental results, the advantage of accounting for dispersions of heat and mass is shown and the importance in taking into account the temperature effect on the equilibrium data is demonstrated.
Introduction Volatile organic compounds (VOCs) are air pollutants whose emissions are more and more strictly controlled because of their toxicity effect on health and their indirect contribution to radical reactions in the atmosphere. In 1997, the European rule directive fixed the target of a reduction of these emissions for France to 1050 t/year by 2010. Although it seems now reasonable to think that this goal will be effectively reached in Europe, it is likely that in the future these targets will be expanded to include every developed and emergent country. VOCs produced by industrial activities represent today 30% of the global emissions. They are commonly removed from industrial air streams by different processes and especially by adsorption onto activated carbons (1-4). This process is able to achieve high yields for a wide range of flow rates. Thanks to their low selectivity and relative low cost, activated carbons are used. Unfortunately, the efficiency of the separation strongly decreases with humidity and/or temperature (5, 6). When dealing with temperature swing adsorption (TSA), the influence of temperature is critical. Indeed, since desorption occurs by increasing temperature, adsorbers are thermally insulated in order to reduce the heat losses with surroundings. This insulation combined with the * Corresponding author phone: (+33) 251858252; fax: (+33) 251858299; e-mail:
[email protected]. 10.1021/es801494a CCC: $40.75
Published on Web 01/20/2009
2009 American Chemical Society
high (exothermic) heats of adsorption of VOCs onto activated carbons leads to significant temperature rises during the adsorption step. For instance, Delage et al. (5) measured temperature increases above 100 °C inside the adsorber for high VOC content. Such warming has a negative impact on the performance of the unit. Dynamic adsorption capacities were shown to be reduced about 10-15% because of excessive bed heating. Moreover, high temperature rises are responsible for the induction of oxidation reaction that may start in some cases in a rather low temperature range and result in bed ignition. In practice, numerous combustion accidents have been reported during the past decades. These accidents occur especially often in the case of treatment of ketones as these kinds of compounds drastically lower the oxidation temperature of activated carbons. For instance, points of initial oxidation were shown to be reduced from 200 to 100 °C in the case of coconut activated carbons saturated with methyl ethyl ketone (7-10). Good management and control of thermal conditions in the adsorber during the entire operation is thus a key point. For a TSA process to be handled properly and safely, fast and efficient simulations are needed to predict the best operating conditions. Numerous adsorption process models have been proposed in the literature which intended to describe the separation performance of the activated carbon filter. Starting with restrictive assumptions, models with analytical solutions were proposed (11, 12), that became progressively more complex and required numerical solutions (3). Former models focused on processes under isothermal or adiabatic conditions. However, for real TSA systems, conditions are not ideal and were regarded as near-adiabatic. Figure 1 illustrates the main differences between isothermal, adiabatic and nearadiabatic adsorbers (13, 14). The upper part of Figure 1 shows the evolution of the gas-phase VOC content (C) with time at one location in the column of adsorption. The second part of the diagram represents the temperature (T) profiles that are associated with the mass transfer zone for adiabatic and near-adiabatic conditions. A broader mass transfer zone is observed for near-adiabatic than for the isothermal conditions. In earlier modeling work (5, 6, 10), attention was paid to evaluating the influence of operating conditions on the efficiency and the safety of the adsorption process. A nonisothermal model was established and validated with tests conducted under a wide operating range, involving numerous VOCs adsorbed on a single type of adsorbent. The impact of initial moisture of the bed was pointed out as well as the initial concentration of VOC at the inlet of the column. This study is aimed at improving the simulation tool previously established by extending its applicability to various kinds of carbon adsorbents and solvent mixtures. Improvement in the predictive ability was also obtained thanks to the choice of a better numerical method to solve the mass and energy balance equations as well as the introduction of the axial dispersion coefficients of mass and heat.
Theoretical Model Mass Transfer from the Fluid to the Solid Phase. On the nanoscopic scale, the adsorption of organic molecules is led by diffusion mechanisms, which bring the molecules from the surrounding phase to the surface of the internal porosity of the adsorbent particle. Organic molecules reach adsorption sites in the most favorable state. The diffusion mechanisms are commonly distinguished as extra-particle, pore, surface, or Knudsen molecular diffusion (3, 15). These distinctions were developed with respect to the size of the pores. For instance, in the narrowest pores, surface VOL. 43, NO. 4, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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Mass Balance. Concerning mass transfer, the transport of the gas-phase through the adsorber mainly occurs by convection, which is sufficiently high to consider an axial dispersed-plug flow: -εDM
∂q ∂C ∂2C ∂C +u +ε + Fbed ) 0 ∂z ∂t ∂t ∂z2
(4)
Heat Balance. Likewise, a thermal balance is applied to the adsorber and results in -DH
FIGURE 1. Comparison of adiabatic, isothermal and nearadiabatic TSA processes (adapted from refs 13, 14). diffusion consists in the migration of the adsorbed molecule from one adsorption site to another without escaping from the attractive electrostatic force of the surface. On the contrary, Knudsen diffusion is due to the displacement of the adsorbate molecules along the mesopores resulting from their collisions with the walls. Numerous theoretical or empirical expressions were obtained for the rate of mass transfer while considering one or several of the adsorption mechanisms mentioned above. The combination of these partial resistances to the resulting global adsorption rate was compared to experimental results. Depending on the nature of the adsorbent and/or adsorbate as well as operating conditions, Knudsen diffusion (16, 17) or surface diffusion (18-22) were shown to be dominant for the adsorption of VOCs onto activated carbons. The linear driving force (LDF) model is the most common model used to describe adsorption rates. It consists in a general expression that lumps together all diffusion mechanisms into one single parameter, k (23): ∂q ) k(qe - q) ∂t
(1)
For the adsorption of VOCs onto the commercial activated carbon Pica NC60, surface diffusion was proven to be dominant and could be represented by the following expression for the coefficient of mass transfer (5): k)
0.45 60 × 1.23.10-9 uexp - 1.694.10-4 ∆Hads RT dp2
((
)
)
(2)
The LDF model is based on the amount of VOC adsorbed at equilibrium, qe, which represents the maximum loading of the adsorbent at a fixed temperature and VOC concentration in the gas-phase. In this study, the Langmuir isotherm as a function of temperature is used (24, 25):
( ) ( )
E1 C RT e qe ) E1 1 + k1exp C RT e qmk1exp
9
Dealing with a column at room temperature and free of VOC before experiment, the initial conditions are
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C(z ) 0, t ) 0) ) C0
(6)
C(z * 0, t ) 0) ) 0
(7)
T(z ) 0, t ) 0) ) T0
(8)
q(z ) 0, t ) 0) ) 0
(9)
At the inlet and the outlet of the column (length, L), the boundary conditions are defined by: C(z ) 0, t) ) C0
(10)
∂C(z ) L, t) )0 ∂z
(11)
T(z ) 0, t) ) T0
(12)
∂T(z ) L, t) )0 ∂z
(13)
Modeling Thermal Swing Adsorption of Binary Mixtures of VOCs. The adsorption of mixtures of VOCs is more complex than the adsorption of single VOCs. For instance, in the case of binary mixtures of VOCs, breakthrough curves are usually divided in five zones: (i) none of the components has broken through and their concentrations at the outlet of the column are zero; (ii) the mass transfer zone of the less adsordable component reaches the outlet of the column and its content increases; (iii) the concentration of this latter component increases above its level at the inlet of the column due to a partial desorption of this compound that is replaced by the most adsordable component of the mixture; (iv) the mass transfer zone of the most adsordable component reaches the outlet of the column and its concentration increases; (v) the adsorber is entirely saturated and the concentrations of each component at the outlet of the adsorption column equal that of the inlet. The model for the adsorption of single components was extended to describe the adsorption of mixtures (subscript i for each VOC of the mixture):
(3)
In addition to the diffusion of organic molecules into the particles of activated carbons, modeling the adsorption process in fixed bed requires establishing mass and heat balances over the entire adsorber. For that purpose, the following assumptions were taken into account: negligible pressure drops along the column (experimentally measured less than 2 kPa); no radial transfer of heat and mass; lumped coefficients of dispersion of heat and mass; constant heat of adsorption independent of loading and temperature; local thermal equilibrium between the gas and solid phases. 1174
∂T ∂2T ∂T + uFgCpg + (εFgCpg + FbedCps + FbedCpaq) ) ∂z ∂t ∂z2 h ∂q ex Fbed(-∆Hads) -4 (T - Tex) (5) ∂t d
-εDMi
∂2Ci ∂z
2
+u
∂Ci ∂qi ∂Ci +ε + Fbed )0 ∂z ∂t ∂t
(14)
∂2T ∂T + uFgCpg + (εFgCpg + FbedCps + ∂z ∂z2 ∂qi hex ∂T Cpaiqi) )Fbed (-∆Hadsi) Fbed -4 (T - Tex) (15) ∂t ∂t d i i
-DH
∑
∑
The rate of mass transfer was described by the LDF model applied independently to each component of the mixture. Numerical Simulations. Finally, this set of equations (1, 5, and 14) enabled to compute the amount of VOC adsorbed onto the particles of adsorbent (q), the concentration of the
TABLE 1. Main Characteristics of the GACs Tested, (after Giraudet et al., 2006) manufacturer
product
raw material
activation method
surface area (m2.g-1)
mesoporous volume (cm3.g)
mean radius of micropores (nm)
mass ratio % O/% C
Pica Pica Pica Norit Chemviron
NC60 NC100 BC120 GF40 BPL
coconut shell coconut shell wood olive stone coal
physicala physicala chemicalb chemicalb physicala
1220 1803 1975 1718 1106
0.354 0.470 1.503 0.796 0.404
0.97 1.11 1.12 1.15 0.93
3.6% 3.3% 35.4% 34.6% 4.1%
a Physical activation is a two-step process consisting in a pyrolysis followed by a steam activation around 800-1000 °C. Chemical activation consists in both cases in the addition of phosphoric acid to the raw materials, which are dried and pyrolyzed between 400-700 °C. b
TABLE 2. Coefficients of Heat Transfer at Wall and Coefficients of Heat Dispersion
average particle diameter (mm) bed voidage hex (W · m-2 · K-1) DH (W.m-1 · K-1)
FIGURE 2. Schematic of the adsorption column. VOC in the gas-phase (C) and the temperature (T) along the column. The singularity of the set of equations defined previously lies in the initial conditions, eqs 6 and 9. The repercussion of the discontinuity depends on the numerical method considered. Three numerical methods were tested (26, 27): finite volumes, numerical method of lines with a fixed space meshing, numerical method of lines with an adaptive space remeshing, which refines in regions where the solution gradient is changing most rapidly. Among these numerical methods, the latter was the most effective. Indeed, finite volumes method was suffering from false diffusion, which widespread the shape of the simulated profile. The numerical method of lines with a fixed space meshing was not able to overcome the discontinuity and large oscillations were computed. To conclude, the numerical method of lines with adaptative space remeshing was the most accurate with a negligible numerical diffusion and small oscillations. Furthermore, this numerical method was shown to be the fastest in terms of computation time.
Experimental Section Activated Carbons and Volatile Organic Compounds. Three granular activated carbons (GACs) were obtained from Pica Co. (namely, NC60, NC100, and BC120). Two other samples were supplied by Chemviron (BPL) and Norit (GF40). Their physical characteristics were reported elsewhere (28) and are partially described in Table 1. The five GACs selected were representative of a large panel of properties, resulting from the carbonization of various natural organic materials, followed by either physical or chemical activation. The porosity of GACs was analyzed using nitrogen adsorption isotherms at 77 K (Micromeritics ASAP 2010). The specific surface area was thus measured using the BET model, while the average radius of the micropores was evaluated from the density functional theory. The mesopore volume was measured for each sample by mercury porosimetry (Micromeritics Autopore IV 9500). The porosimeter enables the penetration of mercury up to a
Pica NC60
Pica NC100
Pica BC120
Chemviron BPL
Norit GF40
0.98
1.01
0.69
1.16
2.24
0.26 2.5 7.1
0.27 2.4 8.7
0.48 1.6 27.5
0.27 2.3 7.7
0.42 1.7 11.7
pressure of 2050 × 105 Pa, which corresponds to an approximate pore width of 8 nm, according to the Washburn equation. Three VOCs were chosen from different chemical species. Acetone, ethyl formate, and dichloromethane (from Aldrich Co.) were selected since these molecules have different adsorption energies (28). Therefore, the transfers of mass and heat, when adsorbed in the column, were expected to vary significantly from one VOC to the other. Heat Capacity of Activated Carbons. Heat capacities of the activated carbons were measured using a differential scanning calorimeter (Setaram DSC-111). Two crucibles were used, one containing the sample and the other used as a reference. A ramp of temperature was applied to the system between 32 and 151 °C. A constant flow rate of helium went through at 5 L · h-1. For each measurement, around 5 mg of dried activated carbons were used. Heat Losses at Wall. The thermal behavior of the fixed bed (without adsorption) was studied for each activated carbon with the objective to determine the heat losses at wall. These heat transfers depend on the size and shape of the particles, on the bed porosity and on the heat conductivity of the adsorbent. For that purpose, a column of adsorption, 46 mm inner diameter, was filled with activated carbons up to 200 mm high. The column was thermally insulated with glass wool of 10 mm width. Dry air (relative humidity less than 3%) was passing through the column at a controlled flow rate of 20 or 30 L · min-1 (flowmeter Brooks 5851E). Before entering the adsorber, air was heated through a heat exchanger immersed in a water bath, whose temperature was fixed at 80 °C. Finally, the thermal response of the fixed bed is recorded at seven locations in the bed, with thermocouples placed in the center of the cylindrical column every 40 mm. The thermal response along the column was modeled by the following heat balance and the parameter hex was then adjusted to fit the experimental data: -DH
∂ 2T ∂z2
+ uFgCpg
∂T ∂T + (εFgCpg + FbedCps) ) ∂z ∂t hex -4 (T - Tex) (16) d
Temperature Swing Adsorption. The adsorption of VOCs onto activated carbons was studied using the same column. VOL. 43, NO. 4, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 3. Modeling breakthrough curves and temperature profiles along the adsorption column: ethyl formate adsorbed onto Pica NC60.
FIGURE 4. Results of the simulationsbreakthrough times. The cylindrical adsorber was still filled with 200 mm of adsorbent. Air was loaded with VOC by bubbling into the liquid VOC solution, whose temperature was kept at 20 °C. The content of VOC in the air flow at the inlet of column is controlled by diluting the saturated flow with fresh air. As shown in Figure 2, each air flux is controlled by a flowmeter (Brooks 5851E and Brooks 5850S). The total air flow rate at the inlet of the column was ranging from 13 to 56 L · min-1. The thermal behavior of the adsorber during adsorption is recorded by seven thermocouples along the column, each 40 mm. At the same locations, the gas-phase was sampled and its VOC content was determined by an flame ionization detector FID (Cosma Graphite 355). For the adsorption of binary mixtures of VOCs, the same experimental setup was used (Figure 2). But, another line was added to load the second VOC. The gas-phase concentration was sampled and analyzed in a gas chromatograph (Perkin-Elmer Autosystem), so that the components of the mixture could be separated. Subsequently, the concentration of each component of the mixture was determined by an ionization flammable detector (FID).
Results and Discussion Heat Capacities of Activated Carbons. Heat capacities (Cps) were measured between 32 and 151 °C. Since only 5 mg of activated carbon were used, the heterogeneity of the materials resulted in experimental errors up to 100 J.kg-1 · K-1. Two groups of adsorbents were distinguished. On one hand, physically activated carbons (Pica NC60, NC100 and Chemviron BPL) were shown to have similar heat capacities, 1176
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FIGURE 5. Results of the simulationsmaximum temperature rises. around 1000-1100 J · kg-1.K-1, that were only slightly influenced by the temperature. On the other hand, the chemically activated carbons (Pica BC120 and Norit GF40) had higher heat capacities in the range 1300-1600 J.kg-1 · K-1. In these cases, a maximum was observed around 60-80 °C. At higher temperatures, the heat capacities decreased to reach a value close to those measured at 40 °C. The chemical composition of activated carbons is pointed out to explain these observations. In fact, for chemically activated carbons, the temperature of activation is lower and the volatilization of the chemical surface groups is incomplete. The resulting activated carbons possess more surface groups, as evidenced in our previous work (28). The chemical composition of Pica BC120 and Norit GF40 thus influences their heat capacities. On the other hand, physically activated carbons are activated at high temperatures and only few surface groups remain on the surface of these adsorbents. Heat Losses at Wall. The thermal response of the adsorber was modeled by equation (14). Two parameters were adjusted to fit experimental results hex and DH Table 2 gathers the coefficients of external heat transfer. Again, a distinction could be done between physically and chemically activated carbons; the latter having smaller coefficients. This observation can be explained by the differences in the bed voidage between the two types of granular adsorbents, as measured from mercury porosimetry, Table 2. Modeling the Adsorption Process of Single VOCs. The set of equations (1, 3, and 5) and the associated boundary conditions (6-13) were used to simulate the adsorption process. In order to fit the model to the experimental results,
FIGURE 6. Modeling breakthrough curves and temperature profiles: binary mixture of acetone and dichloromethane adsorbed onto Pica NC60. three adjustable parameters were chosen: the adsorption rate and the coefficients of axial dispersion of heat and mass. The main adjustable parameter is the adsorption rate coefficient, which was expressed as k ) γexp(
0.45∆Habs RT
(17)
During the calculations, γ was adjusted to reflect the influence of mass transfer kinetics. In order to fit the model to experimental results, the least-squares method was applied, based on the maximum coefficient of determination, whose expression is r2 )
Σ(yˆi - yj)2 Σ(yi - yj)2
(18)
An example of the adjusted breakthrough curves and temperature profiles is shown in Figure 3. To evaluate the efficiency of the model, breakthrough times at different locations along the column were gathered. Furthermore, two concentrations of VOCs at the inlet of column (50 and 100 g.m-3) and two superficial velocities of the gas-phase (0.14 and 0.28 m · s-1) were studied. As shown in Figure 4, the estimation of breakthrough times was accurate since all points are closely distributed along the line of perfect agreement and evenly dispersed. The coefficient of determination was computed between the experimental and simulated data sets and equals 0.988. Likewise, the efficiency of the model was determined toward the estimation of the maximum temperature achieved in the bed at each location along the column. This arbitrary criterion characterized the description of temperature profiles. Figure 5 compares the experimental and modeled maximum temperature rises. The corresponding coefficient of determination is 0.901. Indeed, the estimation of temperature profiles is less accurate than the description of breakthrough curves. Generally, modeled temperature rises are below the experimental values. However, a group of five data was pointed out for the higher modeled temperature rises between 60 and 70 °C. This phenomenon will be explained by a numerical artifact in the next paragraph. As stated above, the numerical simulation was fitted to experimental results by adjusting three parameters, namely, the rate of adsorption and the coefficients of dispersion of heat and mass. The rate of adsorption, γ, was in the range 32.4-1125 s-1, but no clear relationship with one operating conditions (type of adsorbate/adsorbent, superficial velocity, gas concentration) could be established.
This rate probably resulted from the combination of all these parameters. Furthermore, the coefficient, DM, for the axial dispersion of mass was between 0 and 0.0021 m2 · s-1 and its analog for the dispersion of heat, between 0 and 0.86 W · m-1 · K-1. For high dispersion of heat, the simulation gave a thermal peak in the initial stages of the process, which is rapidly dissipated and therefore did not affect the complete description of the whole temperature profiles. However, this phenomenon was a numerical artifact which was not observed experimentally. It ended up with an overvaluation of the maximum temperature rises for the simulation. It corresponds to the five points in Figure 5 that were discussed above. Modeling Thermal Swing Adsorption of Binary Mixtures of VOCs. The modeling of the adsorption of acetone and dichloromethane (38.2 and 44.1 g.cm-3 at the inlet of the column, respectively) onto Pica NC60 is presented in Figure 6. From this example, critical observations, which were verified in every case, could be pointed out: the typical behavior of adsorption of mixtures, as described above, was obtained experimentally and can be predicted by the model; the breakthrough curve of the most adsorbed compound was accurately described (coefficient of determination of 0.97 for acetone in Figure 6), whereas the prediction of the breakthrough curves of dichloromethane was unsatisfactory (dichloromethane exhibited a coefficient of determination around 0.72). Especially, the peak of desorption of the less adsorbed compound was overestimated; the temperature profiles along the adsorption column were described with an averaged coefficient of determination of 0.81. The modeling of the global thermal behavior of the bed agreed with the experimental values. However, the simulated profiles were linked to the poor description of the adsorption of dichloromethane. The maximum rises of temperature in the adsorber, were in good agreement with measurements (less than 5% of relative errors). Model Improvement and Limits: Axial Coefficients of Dispersion of Mass and Heat. Accounting for the dispersions of heat and mass gave two main advantages. First, when fitting experimental breakthrough curves and temperature profiles, significant enhancements were observed after the introduction of these coefficients of dispersion. Moreover, as reported in Table 2, it was observed that differences between adsorbents were explained with different coefficients of axial dispersion of heat. Second, the dispersion of mass and heat improved the stability of the numerical and reduced computing times. However a lack of published correlations able to predict accurately the dispersion coefficients in a system containing VOL. 43, NO. 4, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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microporous materials means that these parameters must be determined by fitting the model with the experimental data. ImprovementssModeling of Isotherms of Adsorption. For the simulation, the Langmuir model (eq 3) was used to describe adsorption equilibriums as a function of temperature. However, in another study, not described here, we reported that other isotherms such as [qe ) (a1T + a2T2)Ce1/n] were more efficient in describing the adsorption equilibrium data of VOCs onto activated carbons. Unfortunately, when these better expressions were introduced into the model, the resolution became divergent during the early stages of the adsorption process. Nevertheless, to illustrate the importance of the isotherm model, the results of simulation for the Langmuir and O’Brien-Myers isotherms were compared. The model, established by O’Brien and Myers ((29), is close to the Langmuir expression:
( ) ( )
(
( ) ( )
El C σ2(1 - klexp RT e 1+ qe ) El 1 + klexp Ce 2(1 + klexp RT qmklexp
El C) RT e El C) RT e
)
(19)
The efficiency of the O’Brien and Myers equation was applied to the adsorption of ethyl formate onto Pica NC60. In that particular case, the Langmuir model showed a coefficient of determination of 0.85, which was increased up to 0.93 for the O’Brien and Myers model. The accurate description of the adsorption equilibrium has a significant influence on the simulation of the adsorption process. To conclude, special attention must be paid to choose an appropriate temperature dependent equilibrium model, which must be compatible with the requirements of the numerical techniques for solution.
Acknowledgments We gratefully thank the French National Agency for Environment and Energy (ADEME) for financial support to this work.
Appendix A Nomenclature Ce
gas-phase concentration of VOC at equilibrium (mol · m-3) adsorbed-phase heat capacity (J · mol-1 · K-1) gas-phase heat capacity (J · kg-1 · K-1) solid-phase heat capacity (J · kg-1 · K-1) column diameter (m) diameter of the adsorbent particle (m) coefficient of axial dispersion of heat (W · m-1 · K-1) coefficient of axial dispersion of mass (m2 · s-1) Langmuir parameter (J · mol-1) coefficient of heat transfer at wall (W · m-2 · K-1) coefficient of mass transfer (s-1) Langmuir parameter (m3 · mol-1) amount of molecules adsorbed (mol · kg-1) amount of molecules adsorbed at equilibrium (mol · kg-1) maximum amount of molecules adsorbed (mol · kg-1) gas constant (J · mol-1 · K-1) coefficient of determination time (s) temperature (K) superficial gas-flow velocity (m · s-1) average data of measured values predicted values
Cpa Cpg Cps d dp DH DM El hex k kl q qe qm R r2 t T u yj yˆi 1178
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yj z ε Fbed Fg σ ∆Hads
measured values axial coordinate (m) bed porosity (-) bed density (kg · m-3) gas-phase density (kg · m-3) constant heat of adsorption (J · mol-1)
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