ARTICLE pubs.acs.org/IECR
Separation of MethaneNitrogen Mixture by Pressure Swing Adsorption for Natural Gas Upgrading S. J. Bhadra and S. Farooq* Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576 ABSTRACT: A dynamic pressure swing adsorption simulation model has been developed that caters for a detailed transport mechanism of nitrogen and methane in the micropores of both ETS-4 and CMS adsorbents. Five adsorbents, namely, bariumexchanged ETS-4 dehydrated at 400 °C (Ba400), strontium-exchanged ETS-4 dehydrated at 190 °C (Sr190) and 270 °C (Sr270), BergbauForchung carbon molecular sieve (BF CMS), and Takeda carbon molecular sieve (Takeda CMS), have been selected for the simulation study in order to compare their performances for upgrading natural gas. The transport mechanisms of gases in CMS and ETS-4 adsorbents are different. In pressure swing adsorption (PSA) simulation, the binary equilibrium and kinetics are represented by the models that have recently been experimentally verified for methanenitrogen mixture in Ba400, Sr190, and Sr270 adsorbents [Majumdar, B.; Bhadra, S. J.; Marathe, R. P.; Farooq, S. Ind. Eng. Chem. Res. 2011, 50, 3021]. The multisite Langmuir isotherm is used for adsorption equilibrium. The kinetic model takes into account diffusion in both macropores and micropores and the concentration dependence of micropore diffusivity according to the chemical potential gradient as the driving force with constant limiting micropore diffusivity. It also has the provision to allow for the dual transport resistance and strong concentration dependence of the thermodynamically corrected micropore transport coefficients in CMS according to the published results [Huang, Q.; Farooq, S.; Karimi, I. A. Langmuir 2003, 19, 5722]. Operating conditions have been identified that favor high recovery while simultaneously meeting the required pipeline specification for methane purity. The performance of the best sample for methanenitrogen separation by PSA found from the simulation study, Ba400, is compared with published performances of ETS-4 and clinoptilolite. It has been found that, in addition to meeting pipeline specification, Ba400 also gives higher recovery, thus making this adsorbent a promising candidate for further exploration.
’ INTRODUCTION Natural gas, a vital source of the world’s supply of energy, is the cleanest among the fossil fuels. It consists primarily of methane, which, when combusted, produces carbon dioxide and water vapor. In contrast, for other fossil fuels such as coal and oil, being complex molecules containing higher carbon to hydrogen ratio, nitrogen, and sulfur, flue gas from their combustion contains a higher percentage of carbon dioxide and toxic gases such as sulfur dioxide, nitrogen oxides, carbon monoxide, etc. According to the 1998 report by the Energy Information Administration (EIA), the harmful emission levels from the combustion of oil and coal are higher than that of natural gas. The global market for natural gas is much smaller than for oil because gas transport is difficult and costly. The proven global reserve of natural gas is 6183 trillion cubic feet 1 or the equivalent of 6 368 490 trillion BTUs (British thermal units). The location of these reserves is distributed. World natural gas consumption, 100 trillion cubic feet in 2004, is increasing faster than that of any other fossil fuel. Natural gas consumption increased by an average of 1.8%/year from 2007 (EIA report, 2010) and is expected to grow even more in the near future as a result of new exploration and expansion of projects. The main sources of natural gas include oil fields, natural gas fields, and coal mines. Landfill gas is a potential source of methane mixed with nitrogen and other contaminants. The contamination of nitrogen above a certain level makes many natural gas reservoirs/sources unusable simply because they do not meet the pipeline specification ( Mg/Na (50/50) clinoptilolite > purified clinoptilolite > ETS-4. Butwell et al.13 in their patent have reported the selective removal of nitrogen from natural gas by two separate pressure swing adsorption stages, one containing a nitrogen-selective crystalline zeolite and a second containing a methane-selective adsorbent. CH4CO2N2 separation using a layered pressure swing adsorption process containing 13X zeolite and CMS 3K adsorbents was also studied using the simple Skarstrom type PSA cycle.14 In this work, a detailed PSA model has been developed to simulate a kinetically controlled Skarstrom PSA cycle consisting of pressurization, high-pressure adsorption, countercurrent blowdown, and purge steps for methanenitrogen separation. The PSA simulation model is used to carry out a comparative evaluation of the performances of five different CMS and ETS-4 adsorbents for methanenitrogen separation from a feed mixture that is representative of nitrogen-contaminated natural gas reserves. A parametric study has been also conducted in order to investigate the sensitivity of various independent process variables such as velocity, bed length, feed gas pressure, and purge gas pressure, etc., on the process performance indicators such as purity, recovery, and productivity. The performance of the best sample found from the simulation study is also compared with published performances of ETS-4 and clinoptilolite. 14031
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’ SELECTION OF ADSORBENTS AND PSA CYCLE Selection of Adsorbents. Five adsorbents, namely, Bergbau Forchung carbon molecular sieve (BF CMS), Takeda carbon molecular sieve (Takeda CMS, also known as Shirasagi MSC 3A), barium-exchanged ETS-4 dehydrated at 400 °C (Ba400), and strontium-exchanged ETS-4 dehydrated at 190 (Sr190) and 270 °C (Sr270) were selected for simulation study. Takeda and BF carbon molecular sieves are extensively used in industrial PSA air separation process for nitrogen production in which kinetic selectivity of oxygen over nitrogen is exploited. These adsorbents have also been recommended as potential candidates for methanenitrogen separation in the linear driving force (LDF) mass-transfer model based PSA simulation study by Ackley and Yang.4 LDF approximation is not the best mass-transfer model for a kinetically controlled separation process governed by a difference in micropore diffusion. A detailed evaluation of pure component and mixture equilibrium and kinetics in these CMS adsorbents by Huang et al.5,15 revealed new features that were previously not recognized. A reassessment of these adsorbents for methanenitrogen separation by incorporating these new features in the PSA simulation was, therefore, felt to be necessary. Ba400 8 and Sr270 16 were chosen because these were the best candidates among the dehydrated BaETS-4 and SrETS-4 samples in terms of ideal kinetic selectivity. In the case of Sr190,16 although the nitrogen to methane diffusivity ratio was very high, the adverse equilibrium selectivity favoring methane reduced the kinetic selectivity. Nevertheless, this sample was chosen to examine how much of the high-diffusivity ratio can be effectively exploited in a kinetically controlled PSA process despite the adverse equilibrium. In fact, on the basis of binary uptake of a methanenitrogen mixture in Sr190, Marathe et al.16 suggested that this sample could be suitable for a PSA process with a short cycle time. Selection of PSA Cycle. A typical PSA process involves a cyclic operation where a number of interconnected vessels containing one or more adsorbents undergo successive pressurization and depressurization steps in order to produce a continuous stream of purified product. The performance of any chosen adsorbent can be enhanced to some extent by choosing different types of cyclic configurations. But a preliminary simulation study using the basic Skarstrom cycle is generally useful for a comparative evaluation of several adsorbents for their potential to separate a particular gas mixture. Therefore, a two-bed, four-step Skarstrom cycle was chosen for this study to investigate the performances of the aforementioned adsorbents for natural gas upgrading. The four steps of the Skarstrom PSA cycle consist of feed pressurization (FP), high-pressure adsorption (HPA), blowdown (BD), and purge (Pu) at low pressure. The roles of these four steps are discussed in the context of a kinetically controlled separation. As schematically shown in Figure 1, in step 1, bed 2 is pressurized to a high pressure (PH) with feed from the feed end and at the same time, bed 1 is countercurrently blown down to a low operating pressure (PL). During pressurization to the high pressure, enrichment of a slower diffusing component in the gas phase at product end is observed. The countercurrent blowdown to the low pressure prevents contamination of the product end with more strongly adsorbed species. In step 2, high-pressure feed flows through the bed where the faster diffusing component is retained and a product stream enriched with less strongly adsorbed component is collected as a high-pressure raffinate product. Unlike in an equilibrium-controlled process, purging bed 1 (countercurrently to the feed flow) with a fraction of the purified effluent from bed 2 is not always beneficial in a kinetically
Figure 1. Schematic diagram of two-bed, four-step Skarstrom PSA process including the pressuretime history of bed 2: FP = feed pressurization, HPA = high-pressure adsorption, BD = blowdown, and Pu = purge.
controlled process. Instead, if bed 1 is left open at the low pressure for a period of time, it will allow the slower component to diffuse out of the adsorbed phase to give the effect of a self-purging (SP) step by sweeping along the faster diffusing component out of the voids. A simultaneous external purge will further improve the raffinate product purity, but at the expense of a significant drop in recovery. The operation of step 3 is the same as in step 1, the difference being that bed 2 is now subjected to countercurrent blowdown, while bed 1 undergoes pressurization. Similarly, the roles of the two beds in step 2 are interchanged in step 4.
’ MODELING AND SIMULATION The pressure swing adsorption process is transient in nature, and the equations describing the system dynamics are complex, particularly for a kinetically controlled separation. The performance of this process depends on several process variables as well as on the detailed transport mechanism of the adsorbates in the adsorbent micropores, which reduce the possibility of simulating a kinetically controlled PSA cycle using a simple approach. The dynamic PSA simulation models can be differentiated on the basis of the form of the mass-transfer rate equations chosen to describe adsorbate uptake in the adsorbent from the fluid phase. The choice of five adsorbents, namely, Ba400, Sr190, Sr270, BF CMS, and Takeda CMS, for natural gas upgrading have been detailed in an earlier section. All of the adsorbents had a bidispersed pore structure, and molecular diffusion was the transport mechanism in the macropores. The transport mechanisms of gases in ETS-4 and CMS adsorbents were, however, different. In ETS-4 adsorbents, the gas transport was controlled 14032
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Table 1. Equilibrium and Kinetic Parameters Used in the Simulationsa for given adsorbent Ba400
Sr190
Sr270
Takeda CMS
BF CMS
Equilibrium Isotherm Parameters qs for CH4 (mmol/cm3)
2.651
7.94
8.56
7.32
6.00
qs for N2 (mmol/cm3
4.4143
9.88
9.06
7.40
6.20
ΔU for CH4 (kcal/mol)
2.3051
3.50
0.91
5.98
6.31
ΔU for N2 (kcal/mol)
4.1627
2.40
2.33
4.72
4.61
a for CH4
5.540
3.18
2.90
3.55
3.47
a for N2
3.327
2.56
2.76
3.51
3.36
b0 for CH4 (cm3/mmol)
0.122820
0.012
0.215
5.02 104
3.37 104
b0 for N2 (cm3/mmol)
0.007589
0.025
0.025
7.40 104
8.04 104
26.8
0.0081
0.0341
2.77
63.12
Dc0/rc2 for N2 (1/s)
132109
81.16
41027
706.31
83.94
Ed for CH4 (kcal/mol)
9.0
3.47
4.5
8.43
10.24
Ed for N2 (kcal/mol)
9.1
5.26
10.43
8.42
7.32
βp for CH4
5.52
5.52
βp for N2
Transport Parameters 0
Dc0/rc2 for CH4 (1/s) 0
2.28
2.28
kb0 for CH4 (1/s) 0 kb0 for N2 (1/s)
468.74 819.83
7310.0 121.61
Eb for CH4 (kcal/mol)
9.85
11.10
Eb for N2 (kcal/mol)
6.88
5.62
βb for CH4
6.06
6.06
βb for N2
7.93
7.93
0
a
Equilibrium and kinetic parameters, and adsorbent properties of Sr190 and Sr270 from Marathe et al.,16 BF CMS and Takeda CMS from Huang et al.,5,15 and Ba400 from Majumdar et al.8
Table 2. Selected Common Parameters Used in the Simulations Bed Characteristics bed length (cm)
50
bed radius (cm)
1.9
bed voidage
0.4 Particle Characteristics
particle radius (cm)
0.16
particle voidage
0.4 (Sr190, Sr270, Ba400); 0.33 (CMS) Feed Gas Conditions
CH4 in feed gas (mole fraction)
0.9
N2 in feed gas (mole fraction) feed gas temperature (K)
0.1 300
Other Parameters adsorption pressure (atm)
59
desorption pressure (atm)
0.22.0
pressurization/blowdown time (s) 75150 adsorption/purge time (s)
75150
L/V0 ratio (s)
2545
purge to feed ratio (G)
00.6
by pore diffusional resistance distributed in the micropore interior,8,16 and a chemical potential gradient was the driving force for diffusion. As expected in inorganic crystalline systems with uniform micropores, the corrected diffusivities were constant.
In contrast, in the CMS samples the gas transport was controlled by a combination of barrier resistance at the entrance of the micropore mouth followed by a distributed pore interior resistance acting in series.5,15 The chemical potential gradient was the driving force for diffusion, but the thermodynamically corrected transport coefficients were also strongly concentration-dependent due to the micropore size distribution. The model equations for the cyclic process including those used to describe the transport of adsorbates (methane and nitrogen in the present study) in the micropores of both ETS-4 and CMS adsorbents and their solution procedure are detailed in the Appendix. The binary multisite Langmuir model17 is used as the equilibrium isotherm model based on the parameters obtained from unary equilibrium data published from this laboratory.5,8,15,16 The mixture equilibrium and kinetic models, which used parameters established from independent single-component measurements, were also validated with mixture experiments in these earlier studies. The single-component equilibrium and kinetic parameters used in the PSA simulation for the five adsorbents chosen in this study are compiled in Table 1 for convenience. Other common parameters including the range of process conditions that were varied in the parametric study are summarized in Table 2. The axial dispersion coefficients were calculated from eq A.1b, and binary molecular diffusivity of the methanenitrogen pair was estimated from the ChapmanEnskog equation.18,19 Raw natural gas emerges from wells at high pressure. It is then taken to the gas processing plant through a pipeline for producing a clean natural gas by separating the impurities and various 14033
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Figure 2. Effect of the length to velocity (L/V0) ratio on methane (a) purity, (b) recovery, and (c) productivity. The legends used in the bottom panel apply to all of the panels. PH = 9.0 atm, PL = 0.5 atm, T = 300 K, pressurization/blowdown time = 75 s, high-pressure adsorption/ purge time = 150 s, and purge to feed ratio (G) = 0.0.
non-methane hydrocarbons and fluids. The temperature of the raw natural gas in the transmission line is roughly the same as the ambient temperature. Depending on the location of the well, the gas temperature varies. An ambient temperature of 300 K was chosen as the operating temperature in this study. Natural gas is a gaseous natural resource, consisting mainly of methane and small amounts of higher hydrocarbons. The composition of natural gas varies from region to region. Some natural gas reserves contain a high percentage of nitrogen as well as carbon dioxide and hydrogen sulfide. At the natural gas well head, a nitrogen content of 520 mol % has been reported.20 Therefore, an intermediate nitrogen concentration of 10 mol % was chosen for this comparative evaluation study. Purity, recovery, and productivity were used as measures of PSA process performance for comparative evaluation of the five adsorbents. In this study, the purity of CH4 is defined as the average mole fraction of CH4 leaving the bed during the high-pressure adsorption step. On the other hand, recovery of CH4 is defined as the net number of moles of CH4 withdrawn as product during the highpressure adsorption step divided by the number of moles of CH4 fed to the PSA process during the pressurization and high-pressure adsorption steps. Finally, productivity is defined as the net volume of CH4 produced per unit volume of adsorbent per unit time. The integral equations corresponding to the above qualitative definitions
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Figure 3. Effect of pressurization time on (a) purity, (b) recovery, and (c) productivity. The legends used in the bottom panel apply to all of the panels. PH = 9.0 atm, PL = 0.5 atm, T = 300 K, high-pressure adsorption/ purge time = 150 s, L/V0 ratio = 35 s, and G = 0.0.
of product purity, recovery, and productivity are presented in the Appendix together with the detailed model equations.
’ RESULTS AND DISCUSSION Effect of Various Operating Parameters on PSA Performance. The effects of various independent process variables
such as velocity, bed length, feed gas pressure, purge gas pressure, and configuration of the process steps, etc., on the process performance indicators such as purity, recovery, and productivity have been evaluated. Several different combinations of the PSA process variables can be used to produce high-purity methane as well as high recovery and productivity. The sensitivity of the PSA process performance to the aforementioned process variables is analyzed for different adsorbents in the following sections. Effect of Length to Velocity Ratio. Figure 2 summarizes the net results for CH4 purity, recovery, and productivity as a function of length to velocity ratio (L/V0) for different adsorbents. The general trend is recovery and productivity increase with decreasing L/V0, but the trend is the opposite for purity. For a fixed column length, a decrease in L/V0 ratio means an increase in inlet velocity (or flow rate) to the bed. A closer observation revealed that a decrease in L/V0 from 45 to 25 s results in a change of exit flow rate from 40 to 75 cm3/s calculated at 1 atm and 300 K. With increasing feed velocity, there is less residence time in the bed for adsorption and more methane is collected 14034
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Figure 4. Mole fraction of methane in the gas phase as a function of dimensionless bed length at the end of pressurization (PR) and blowdown (BD) steps. The results are for Ba400. See the caption of Figure 3 for other operating conditions.
Figure 5. Effect of adsorption time on (a) purity, (b) recovery, and (c) productivity. The legends used in the bottom panel apply to all of the panels. PH = 9.0 atm, PL = 0.5 atm, T = 300 K, pressurization/blowdown time = 75 s, L/V0 ratio = 35 s, and G = 0.0.
during the given period of the high-pressure adsorption step. As a result, recovery is improved. However, an increase in velocity (or decrease in L/V0 ratio) also causes more nitrogen to travel to the product end which contaminates the product. Therefore, a reduced concentration of methane in the product during the high-pressure adsorption step is observed, which in turn diminishes the product purity. Effect of Pressurization/Blowdown Step Duration. In the present study, the traditional Skarstorm cycle is used where
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Figure 6. Mole fraction of methane as a function of dimensionless bed length at the end of the self-purge (SP) step. The results are for Ba400. See the caption of Figure 5 for other operating conditions.
pressurization and blowdown times are equal. Hence, the effect of a change in pressurization time should be analyzed together with a simultaneous change in blowdown time. From the results shown in Figure 3, it appears that the pressurization/blowdown time has very little effect on methane purity and recovery. Since the direction of mass transfer is from gas to solid during pressurization and vice versa during blowdown and the two steps have equal duration, a likely explanation for the observed trends in purity and recovery is the additional mass transfer due to a longer duration of one step gets canceled by an approximately equal effect in the opposite direction during the other step. Support for this explanation may be obtained from the respective gas-phase methane concentration profiles in the bed at the end of pressurization and blowdown steps of different duration shown in Figure 4. Longer blowdown time allows more of the slower diffusing methane to desorb from the adsorbent and hence at the end the gas phase has more methane, which is evident from Figure 4. The subsequent pressurization step, which has equal duration as the blowdown step, provides the necessary extra time for readsorption of the additional amount of methane desorbed during the longer blowdown. The gas-phase methane concentration profile along the bed, therefore, remains practically the same as what is attained with a shorter pressurization/blowdown time, as may be seen from Figure 4. Increasing pressurization/depressurization time increases the total cycle time without significantly affecting the methane product purity and flow rate, which decreases the productivity seen from Figure 3c. Effect of Duration of High-Pressure Adsorption/Purge Step. In a traditional Skarstorm cycle, both high-pressure adsorption and purge steps have equal duration, like the pressurization and blowdown steps. Therefore, it is not possible to independently vary the high-pressure adsorption and purge durations. The combined effect of equally changing the adsorption and purge steps on the performance of a PSA process is presented in Figure 5. For all five adsorbents, purity decreases, while recovery and productivity increase with increasing adsorption/purge step duration. This process variable appears to be an effective way of increasing methane recovery when there is some room to sacrifice methane purity in the product. Longer adsorption time means that the feed concentration front penetrates deeper into the bed. Progress of the concentration front in the gas phase gives a measure of how much of the bed has been equilibrated with respect to the feed composition. Since we are dealing with a kinetically controlled separation where the masstransfer resistance is high, there is a decrease in the adsorption rate as the bed length available for further adsorption decreases. Hence, the product flow rate increases with increasing highpressure adsorption time, and the productivity increases. Since 14035
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Figure 8. Mole fraction of methane in the gas phase as a function of dimensionless bed length at the end of (a) blowdown and (b) self-purge (G = 0.0) steps, showing the inadequacy of self-purge in most cases. See the caption of Figure 7 for other operating conditions.
Figure 7. Effect of purge to feed ratio on (a) purity, (b) recovery, and (c) productivity. The legends used in the bottom panel apply to all of the panels. PH = 9.0 atm, PL = 0.5 atm, T = 300 K, pressurization/blowdown time = 75 s, high-pressure adsorption/purge time = 150 s, and L/V0 ratio = 35 s.
other operating parameters such as L/V0 ratio and high and low operating pressures, etc., do not change, the total input of methane to the system during pressurization and high-pressure adsorption remains practically the same and, therefore, methane recovery also increases. A decreasing adsorption rate with increasing penetration of the feed concentration front causes more nitrogen to flow to the product end. As discussed earlier in relation to the effect of the L/V0 ratio, this increased flow of nitrogen to the product end can contaminate the methane product. However, to fully understand the reason for a drop in methane purity, one also needs to take into consideration the profiles in Figure 6, where it is shown that the bed contains more nitrogen at the end of a longer self-purge step compared to a shorter one. The area above each line in the figure gives the average mole fraction of nitrogen remaining in the bed after the self-purge step of corresponding duration. This residual nitrogen is pushed to the product end during the next pressurization step and is released with the high-pressure product. The drop in methane product purity with increasing adsorption time is, therefore, a combined effect of a decreasing adsorption rate and insufficiency of self-purge for methanenitrogen PSA separation on the adsorbents under investigation. Effect of Purge to Feed Ratio. In an equilibrium-controlled PSA separation operated on a conventional Skarstrom cycle,
Figure 9. Mole fraction of methane in the gas phase as a function of dimensionless bed length at the end of (a) blowdown and (b) purge (G = 0.6) steps, showing the improvements after introducing an external purge. See the caption of Figure 7 for other operating conditions.
a part of the high-pressure raffinate product is used to execute a countercurrent purge step following blowdown to the low operating pressure. The step is necessary to sufficiently remove the stronger adsorbate from the bed and ensure high raffinate product purity in the subsequent high-pressure adsorption step. The purge step is analogous to the light-component reflux in 14036
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Figure 10. Effect of adsorption pressure on (a) purity, (b) recovery, and (c) productivity. The legends used in the bottom panel apply to all of the panels. PL = 0.5 atm, T = 300 K, pressurization/blowdown time = 75 s, high-pressure adsorption/purge time = 150 s, L/V0 ratio = 35 s, and G = 0.0.
distillation or a similar countercurrent mass-transfer operation. In a kinetically controlled PSA cycle, the idea of self-purge (i.e., purge to feed velocity ratio, G, equal to zero in eq A.5c) is effective when the slower desorbing component (for example, nitrogen in the case of air separation using CMS) is in sufficient amount to effectively push out the faster desorbing component from the bed voids, which could otherwise contaminate the highpressure raffinate product in the next cycle. The effect of purge to feed ratio on methanenitrogen separation is compared for different adsorbents in Figure 7. The observed trends may be explained by closely looking at the results presented in Figures 8 and 9. The gas-phase concentration profiles along the bed length at the end of blowdown and self-purge (G = 0) steps shown in Figure 8 clearly show the inadequacy of self-purge for most adsorbents considered in this study. The shift in corresponding profiles in Figure 9a,b is a measure of how much nitrogen is pushed out of the voids in the bed by the slower diffusing methane. Except for Sr190, the residual amount of faster diffusing nitrogen in the gas phase at the end of the self-purge step is quite high in the case of all of the other four adsorbents, namely, Ba400, Sr270, BF CMS, and Takeda CMS. For these adsorbents, introduction of an external purge (G = 0.6) brings about significant changes to the methane concentration profiles at the end of the
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Figure 11. Effect of desorption pressure on (a) purity, (b) recovery, and (c) productivity. The legends used in the bottom panel apply to all of the panels. PH = 9.0 atm, T = 300 K, pressurization/blowdown time = 75 s, high-pressure adsorption/purge time = 150 s, L/V0 ratio = 35 s, and G = 0.0.
purge step, which is best appreciated by comparing the profiles in Figure 9b with those in Figure 8b. Hence, the external purge leads to better regeneration and, therefore, increased methane purity when the self-purge is inadequate. External purge, however, means that less high-pressure raffinate gas is available for withdrawal as product, which lowers the recovery and productivity. In the case of Sr190, self-purge is adequate and further improvement by introducing an external purge is negligible. Effect of Adsorption Pressure. Figure 10 shows the effect of adsorption pressure on the performance of methanenitrogen separation by PSA for the five adsorbents. It is clear from the figure that productivity gradually increases as the adsorption pressure is increased. On the other hand, except in the case of Sr190, recovery remains practically constant and there is a modest increase in purity with increasing adsorption pressure for the other adsorbents. In the case of Sr190, purity drops and recovery increases as adsorption pressure is increased. When the adsorption pressure is increased in a self-purged Skarstrom cycle without changing the L/V0 ratio, the low operating pressure, and the duration of various steps, it implies two things: (i) an increase in the bed capacity for adsorption and (ii) an increase in the input molar flow rate to the bed. Favorable equilibrium isotherm and dominance of mass-transfer resistance both contribute to a deeper penetration of the concentration front with increasing 14037
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Figure 12. Effect of the diffusivity of methane on the purity and recovery in the Ba400 sample. PH = 9.0 atm, PL = 0.5 atm, pressurization/blowdown time = 75 s, high-pressure adsorption/purge time = 150 s, L/V0 ratio = 35 s, and G = 0.0.
Figure 14. Plot of methane purity vs. recovery showing the effects of different parameters on the performance of a PSA system on (a) Sr270 and (b) Sr190 samples. The arrows indicate the increasing directions of the operating parameters. The legends used in the top panel apply to all of the panels. PH = 59 atm, L/V0 = 2545 s, PL = 0.32 atm, PR = 75150 s, high-pressure adsorption step = 75150 s, and G = 00.6.
Figure 13. Plot of methane purity vs. recovery showing the effects of different parameters on the performance of a PSA system on (a) BF CMS and (b) Takeda CMS samples. The arrows indicate the increasing directions of the operating parameters. The legends used in the top panel apply to all of the panels. PH = 59 atm, L/V0 = 2545 s, PL = 0.21 atm, PR = 75150 s, high-pressure adsorption step = 75150 s, and G = 00.6.
adsorption pressure, and there is an increase in the product flow rate for reasons that have already been discussed in relation to the effect of the high-pressure step duration. This explains the monotonic increase in methane productivity seen in Figure 10c. In this case, however, moles of feed gas (and therefore, methane) entering the adsorber during pressurization and high-pressure adsorption also increase with increasing adsorption pressure. Since recovery is defined as the ratio of methane output in product gas to that entering during pressurization and high-pressure adsorption, constant recovery implies that these quantities are linearly related to the high operating pressure. Closer examination of the simulation results revealed that this was indeed the case. Increase in purity with high operating pressure is a consequence of increased bed capacity for adsorption, which allows more adsorption of faster diffusing nitrogen compared to the slower diffusing methane. This results in purer methane at the product end.
Figure 15. Plot of methane purity vs. recovery showing the effects of different parameters on the performance of a PSA system on the Ba400 sample. The arrows indicate the increasing directions of the operating parameters. PH = 59 atm, L/V0 = 2545 s, PL = 0.32 atm, PR = 75150 s, high-pressure adsorption step = 75150 s, and G = 00.6.
Effect of Desorption Pressure. An adsorption bed is better regenerated by lowering the purge pressure. The capacity of the bed is increased for both adsorbates, but nitrogen, being the faster component, is adsorbed more relative to the slower methane. Hence, an increase in product purity at the expense of some drop in product recovery is expected with decreasing purge step pressure. The results in Figure 11 are consistent with these expectations. It is obvious that subatmospheric purge step pressure can be very effective for attaining high-purity methane product, but the energy penalty of running an industrial column at a very high vacuum should also be taken into consideration. Effect of Methane Diffusivity in Ba400 on a Self-Purge Cycle. It is clear from the discussion on the effect of the purge to feed ratio that the effectiveness of a self-purge step in a kinetically 14038
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Industrial & Engineering Chemistry Research controlled PSA process depends on the amount of slower component desorbed during this step. The amount desorbed depends first on the amount adsorbed in the first place during the high-pressure step and second on how fast it can come out over the duration of the self-purge step. The effect of the diffusivity of methane in Ba400 on the performance of a self-purge PSA cycle was, therefore, investigated, and the results are shown in Figure 12. The maximum diffusional time constant value on the x-axis was 0 calculated from eq A.14 using Dc0/rc2 and Ed values for Ba400 given in Table 1. There seems to be an optimum methane diffusivity value at which the self-purge cycle will be most effective and will give the maximum methane purity in the high-pressure product. Increased methane diffusivity will, however, result in a drop in recovery, which makes the overall effect comparable with introduction of an external purge. Hence, a modified BaETS-4 with somewhat higher methane diffusivity may not be advantageous over Ba400. Comparative Study of BaETS-4, SrETS-4, and CMS Adsorbents. The effects of several operating parameters on the performance of methanenitrogen separation by PSA have been investigated for all five adsorbents. The following general observations can be summarized in the light of preceding discussions: (i) It is very easy to attain >96% methane purity in Ba400 and Sr190. (ii) The performances in terms of purity, recovery, and productivity attained with two CMS samples are consistently lower than those of the other samples. (iii) Given the low enrichment attained using the CMS samples, it is not surprising that these samples give high methane recovery in all cases. What is noteworthy is the comparable high recovery achieved with Ba400 simultaneously with methane purity consistently above 96%. Sr190 gives the lowest recovery in all cases. (iv) The samples do not seem to differ significantly in terms of productivity. In addition to the above observations, the trends of the heat of adsorption and activation energy values seem to suggest that lower temperature will adversely affect the performance of the ETS samples but will favor the CMS samples. Having studied one by one the sensitivity of the process performance to various operating parameters, the next step is to find how these operating parameters can be optimally chosen to maximize both purity and recovery. To achieve this goal, the purity of methane is plotted as a function of the recovery of methane for six different parameters in Figures 1315, each plot representing one of the five adsorbents. The arrows in the figures indicate the increasing directions of the respective operating parameters. Here the target is set to produce, starting from a 90/10 CH4/N2 mixture, a product of at least pipeline-quality natural gas, at highest possible recovery. It is clear that, for the four-step cycle operation with no purge, the most efficient way to increase purity is to reduce desorption pressure since corresponding loss of recovery is relatively low as compared to the other cases. But reducing desorption pressure will increase energy consumption and hence the operating cost. The product purity may also be significantly improved by regenerating the adsorbent bed with product purge. However, the recovery is reduced by introducing purge. It is also evident from the figures that the best parameters to increase the recovery are high-pressure adsorption time and length to velocity ratio, although these parameters adversely affect the purity. Hence, a longer high-pressure step, high L/V0
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Figure 16. Plot of purity vs. recovery of methane for Ba400, clinoptilolite, and ETS-4 adsorbents. The arrows indicate the increasing directions of the operating parameters. For Ba400, L/V0 = 2545 s, PL = 0.32 atm, high-pressure adsorption step = 75150 s, G = 00.6, and total pressurization time = 75 s. For clinoptilolite and ETS-4, L/V0 = 1040 s, desorption pressure = 0.4 atm, adsorption pressure = 7 atm, pressurization time = 30 s, high-pressure adsorption time = 60 s, cocurrent blowdown time = 10 s, countercurrent blowdown time = 30 s, and desorption time = 60 s. Data for clinoptilolite and ETS-4 were taken from Jayaraman et al.21
ratio, and subatmospheric desorption pressure together with product purge are the desirable conditions for attaining the pipeline specification of methane concentration without a severe drop in its recovery. Figures 1315 show that the overall performances of the adsorbents decrease in the order Ba400 > Sr190 > Sr270 > Takeda CMS > BF CMS. The BF CMS sample cannot meet the pipeline specification at least when the Skarstrom cycle is used. It is possible to reach 96% methane purity with the Takeda CMS sample by resorting to a desorption pressure below 0.2 atm or introducing a small product purge at a desorption pressure of 0.2 atm. Sr190 and Sr270 adsorbents can still be used for methane nitrogen separation by PSA, but Ba400 adsorbent appears to be the best choice. In addition to giving high purity, it also gives a high recovery and obviously this improved performance is achieved by virtue of both equilibrium and kinetic selectivities favoring nitrogen over methane.8 The high sensitivity (indicated by sharp changes in purity or recovery) of some parameters seen in Figure 15 also confirms that these can be manipulated to further increase purity as well as recovery. In this study, a higher productivity of Ba400 is observed (although the difference with the other adsorbents is not large), indicating that this sample will require a relatively smaller bed for the same separation. Therefore, a considerable savings in both capital and operating cost may be expected for a PSA process designed for separating nitrogen from its mixture with methane using Ba400. Comparison with Published Performance. Having established Ba400 as the best adsorbent for natural gas cleaning among the five adsorbents studied here, the performance of this adsorbent is compared with that of purified clinoptilolite and ETS-4 studied by Jayaraman et al.21 In this simulation study, the authors evaluated the possibility of kinetic separation of the CH4/N2 mixture in the aforementioned microporous adsorbents. The model assumptions related to the representation of the external fluid phase flow were similar to those in the present study. However, they allowed heat effect in their model and, in contrast to the detailed pore diffusional model used in the present study, represented transport in the adsorbent micropores with the LDF model but corrected the diffusional time constants for concentration dependence using average loading values of the 14039
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Industrial & Engineering Chemistry Research adsorbates. Despite some differences, it was felt that a comparison of results from the two studies may still be useful. The simulation results for clinoptilolite and ETS-4 summarized in Figure 16 were reported for an 85/15 methane/nitrogen mixture. To ensure the same basis, new results were computed for Ba400 using the same feed composition. It is evident from the figure that, at the highest recoveries attained for clinoptilolite and ETS-4 samples, the product purity was below the pipeline specification of g96% methane (or e4% nitrogen). In contrast, the Ba400 sample, in addition to meeting the pipeline specification, is also able to attain up to 75% recovery. For the Ba400 sample, there is enough room to further increase purity by tuning some parameters such as G and PL, as shown in Figure 16. By increasing the high-pressure adsorption (HPA) time, it is also possible to increase the recovery beyond 75%, but the purity falls below the minimum limit. Another way to increase recovery without significant loss of purity is to use a fivestep PSA cycle as used by Jayaraman et al.21 to produce the results shown in Figure 16. In this five-step cycle, a cocurrent blowdown step was introduced between the high-pressure adsorption step and the countercurrent blowdown steps of the Skarstrom cycle. The inclusion of cocurrent blowdown step in the five-step cycle contributes to higher methane recovery. Therefore, by using this cycle and the Ba400 adsorbent, it is also possible to significantly increase recovery of methane. It is important to note that part of the methane-rich product from the five-step cycle is released at low pressure and needs to be repressurized for combination with the high-pressure product. Hence, the additional recovery comes at the expense of additional compression energy.
’ CONCLUSION A detailed PSA simulation model was developed on the basis of axially dispersed plug flow in the fluid phase, multi-site Langmuir (MSL) isotherm to represent binary equilibrium and bidispersed pore diffusion to represent adsorption kinetics including the features that correctly capture the binary transport of gases in the micropore of carbon molecular sieve and ion-exchanged ETS-4 adsorbents. Using this simulation model, an extensive study was conducted to compare the performances of five adsorbents, namely, BF CMS, Takeda CMS, Sr190, Sr270, and Ba400, for methanenitrogen separation by PSA. Among the adsorbents investigated, Ba400 and Sr190 were found to easily attain pipelinequality natural gas (g96% methane). The overall performance of Ba400 was, however, better than that of Sr190. The performance of the PSA system was very sensitive to process variables such as high-pressure adsorption time and length to velocity ratio, which were identified as the significant parameters to tune recovery. The purge to feed ratio as well as desorption pressure were found most sensitive for tuning purity. The performance of the best sample for methanenitrogen separation by PSA found from the simulation study, Ba400, was compared with published performances of ETS-4 and clinoptilolite. It was found that, in addition to meeting pipeline specification, Ba400 also provided higher recovery, thus making this adsorbent a promising candidate for further exploration. ’ APPENDIX: DETAILED DESCRIPTION OF THE MODEL EQUATIONS Assumptions. To develop a detail mathematical model for the PSA process, the following simplified assumptions were made. (1) The system was isothermal. The isothermal assumption is valid for kinetic separation due to the short cycle times and the low to moderate heats of adsorption.
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(2) The ideal gas law was applicable. (3) Frictional pressure drop along the bed was negligible. (4) The flow pattern was described by an axial dispersed plug flow model. (5) Velocity along the bed was assumed to vary due to adsorption/desorption. (6) Mass transfer between gas and adsorbed phases was accounted for in all steps, i.e., pressurization, high-pressure adsorption, blowdown, and purge steps. (7) The total column pressure remained constant during highpressure adsorption and purge steps. During pressurization and blowdown, the pressure profiles were assumed to change exponentially with time. (8) The feed gas contained two components only. In the present study, these were N2 and CH4. (9) Adsorption equilibrium was represented by the multisite Langmuir model. (10) Adsorbent particles were spherical. In addition to the preceding approximations, any additional approximations required for specific models are discussed separately. Bidispersed Adsorbent. The following assumptions were introduced in addition to the aforementioned assumptions. (1) Molecular diffusion dominated in the macropores. (2) The micropore surface was in equilibrium with the macropore gas. (3) The gradient of chemical potential was taken as the driving force for micropore diffusion. Generally, the PSA model is represented by a series of material balance equations. The equations are written in general terms for component i (=A for slower diffusing component and B for faster diffusing component) and step j (=1 for pressurization, 2 for high-pressure adsorption, 3 for blowdown, and 4 for purge). The ( sign is used to indicate the flow direction. The + sign represents the flow from the feed end (0) to the product end (L). Flow from L to 0 is indicated by the sign. Gas-Phase Component Mass Balance. For component i, mass balance over a differential volume element, subjected to axially dispersed plug flow assumption: ¼¼
DL
∂2 cij ∂cij Vj ∂cij 1 ε ∂qij ¼ 0 þ þ ( 2 ε ∂t ∂z ∂z ∂t
ðA.1aÞ
where the second term is positive for j = 1 and 2 and negative for j = 3 and 4. c i is the fluid-phase concentration of component i, V is the interstitial gas velocity, ¼q is the average loading of the adsorbent particle, ε is the bed voidage, z is the axial distance, and t is time. In the equation shown above, the effects of all mechanisms that contribute to axial mixing are lumped together into a single axial dispersion coefficient, DL . The axial dispersion term can be neglected for the case where mass-transfer resistance is significantly greater than the axial dispersion. The four terms on the left-hand side in the preceding equation represent contributions from axial dispersion, convection, accumulation in the gas phase, and accumulation in the adsorption particles, respectively, for component i. In this study, DL was calculated from the following equation:22 DL ¼ 0:7Dm þ 0:5Vdp 14040
ðA:1bÞ
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where dp is the diameter of the adsorbent particle and Dm is the molecular diffusivity, which can be calculated by using the 18,19 ChapmanEnskog equation: 1 1 1=2 ð1:86 103 ÞT 3=2 þ MA MB Dm ¼ ðA:1cÞ Pσ AB 2 ΩAB where T is the temperature, M A and M B are the molecular weights of gases A and B, P is the total pressure, Ω AB is the collision integral which is dependent on temperature, and σ AB is a constant in the Lenard-Jones potential energy function for pair AB and is calculated from the following expression: 1 ðσA þ σ B Þ 2
σAB ¼
ðA:1dÞ
where σ A and σ B are the collision diameters of gases A and B calculated from the Lenard-Jones potential. The standard (Danckwerts) inlet and exit boundary conditions were used. For j = 1 and 2 ∂cij DL ∂z
¼ Vj jz ¼ 0 ðcij jz ¼ 0 cij jz ¼ 0þ Þ and z¼0
∂cij ∂z
¼0 z¼L
the bed. Therefore, when the pressure is constant, ∂Vj 1ε þ ( Cj ε ∂z
z¼0
and
∂cij ∂z
For j = 4 DL
∂cij ∂z
¼0
and DL
z¼0
¼0
ðA:2bÞ
z¼L
∂cij ∂z
¼ Vj jz ¼ L ðcij jz ¼ Lþ cij jz ¼ L Þ
z¼L
ðA:2cÞ where for j = 1 and 2, cij|z=0 is the inlet concentration of the ith component that is known, and for j = 4, cij|z=L+ is the inlet concentration of the bed undergoing purge which is calculated from the following expression: ci4 jz ¼ Lþ ¼
PL ðci2 jz ¼ L Þ PH
ðA:2dÞ
Here j = 2 indicates that part of the product from the high-pressure step is expanded to the low pressure and used to purge the other bed. This equation is not applicable for a self-purge cycle since no external gas stream is involved. Continuity Condition. For negligible frictional pressure drop through the bed the continuity equation is as follows:
∑i cij ¼ cAj þ cBj ¼ Cj 6¼ f ðzÞ
ðA:3Þ
where C is the total concentration in the gas phase. It is also independent of time during high-pressure adsorption (j = 2) and purge (j = 4) steps but changes during pressurization (j = 1) and blowdown (j = 3) steps. Overall Material Balance. The overall material balance equation is required to capture this variation of velocity through
ðA:4aÞ
[j = 2 and 4; + when j = 2 and when j = 4]. When the pressure is time-dependent, ( Cj
∂Vj ∂Cj 1ε þ þ ε ∂z ∂t
∑i
¼¼
∂qij ¼0 ∂t
ðA:4bÞ
[j = 1 and 3; + when j = 1 and when j = 3]. Velocity boundary conditions are written on the basis of flow features of different steps. The product ends remain closed during pressurization and blowdown steps. In the case of the high-pressure adsorption step, the feed velocity is considered as an operating parameter. The boundary conditions are given as follows: For j = 1 and 3 V j jz ¼ L ¼ 0
ðA:5aÞ
For j = 2, V j jz ¼ 0 ¼ V 0
ðA:5bÞ
For j = 4, Vj jz ¼ 0 ¼ GV0
ðA:2aÞ For j = 3 ∂cij ¼0 ∂z
∑i
¼¼
∂qij ¼0 ∂t
ðA:5cÞ
For a self-purging cycle, G = 0. For a binary system, eqs A.1a, A.3, and A.4a or A.4b were used to solve V, cA, and cB as a function of time, t, and space, z. During constant-pressure operations, namely, high-pressure adsorption and purge steps, the total concentration, C, is a known constant. In the case of variable-pressure steps, two methods may be used to solve C as a function of time. The commonly used method is to directly provide the experimental pressuretime history, normally expressed in exponential form as shown in the following equations:23 P ¼ f ðtÞ ¼ PH ðPH PL Þea1 t a2 t
P ¼ f ðtÞ ¼ PL þ ðPH PL Þe
for j ¼ 1
ðA.6Þ
for j ¼ 3
ðA.7Þ
where PH and PL are the high and low pressures of the PSA operation, respectively, and a1 and a2 are the constants to be empirically determined by calibrating against experimental data. In the light of the assumptions of ideal gas and isothermal system, C is directly proportional to P and is no longer an unknown once the pressuretime history is given. The other method is to use pressurization and blowdown flow rates as inputs to solve the column pressures during these steps as unknowns. Mass-Transfer Rate across the External Film. The adsorbate gas crosses the external film and penetrates into the porous structure during adsorption and travels the same paths in the reverse direction during desorption. Therefore, accumulation in the particle is given by the amount transferred across the external film, which can also be expressed as the flux at the surface of the macropores. ¼¼ ∂ðcp Þij 3 ∂qij 3 ¼ kf ðcij ðcp Þij jR ¼ Rp Þ ¼ εp Dp Rp ∂t ∂R Rp
ðA:8Þ R ¼ Rp
where kf is the fluid-phase mass-transfer coefficient, Dp is the macropore diffusivity, (cp)i is the macropore concentration, and εp is the particle voidage. The mass-transfer coefficient, 14041
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kf, is calculated from the correlation given by Wakao and Funazkri:24 Sh ¼ 2:0 þ 1:1Sc1=3 Re0:6
The limiting diffusivity Dc0 is dependent on temperature according to the following equation: 0
ðA:8aÞ
Dc0 D ¼ c02 eEd =Rg T 2 rc rc
where
where Ed is the activation energy for diffusion in the micropore 0 interior and Dc0 is the preexponential constant. Boundary conditions for the micropore mass balance equation: ∂cim ij ¼ 0 and cim ðA:15Þ ij jr ¼ rc ¼ f ððcp Þij Þ ∂t
kf dp Dm μ Sc ¼ Schmidt no: ¼ Dm Fg dp V0 εFg Re ¼ Reynolds no: ¼ μ Sh ¼ Sherwood no: ¼
r¼0
Macropore Mass Balance. The macropore mass balance equation has the following form: " 2 # ∂ðcp Þij ∂ ðcp Þij 1 εp ∂qij 2 ∂ðcp Þij þ ¼ Dp ðA:9Þ þ R ∂R ∂t εp ∂t ∂R 2
Equilibrium Isotherm. The equilibrium relationship for both components is represented by the binary multisite Langmuir model. This model is an extension of the Langmuir model for singlecomponent and multicomponent equilibrium on microporous adsorbents that has created a provision for taking the variation of adsorbate sizes into account.17 The model has the following form:
where q ij is the average adsorbed concentration of component i in the micropore which is calculated from the following relation: ∂qij 3 ¼ Ji jr ¼ rc rc ∂t
ðA:10Þ
where J i is the diffusion flux of component i in the microparticles derived from the chemical potential theory by introducing an imaginary gas-phase concentration: 25 Ji
¼ ¼ ¼
∂qij ∂r ∂ ln cim ij ∂qij ðDc0 Þi ∂ ln qij ∂r
ðDc Þi
ðDc0 Þi
ðA:14Þ
ðA:11Þ
qij ∂cim ij cim ∂r ij
where c im ij is the imaginary gas-phase concentration in equilibrium with the adsorbed phase concentration and D c0 is the limiting diffusivity. The imaginary gasphase concentration is calculated from the multisite Langmuir model. Boundary conditions for the macropore mass balance equation: ∂ðcp Þij ¼ 0 and ∂R R¼0 ∂ðcp Þij εp ðDp Þi ¼ kf ðcij ðcp Þij jR ¼ Rp Þ ðA:12Þ ∂R R ¼ Rp
Micropore Mass Balance. Assuming spherical geometry and chemical potential gradient as the driving force for diffusion, the micropore mass balance equation has the following form: " !# ∂qij qij ∂cim 1 ∂ 2 ij r ðDc0 Þi im ¼ 2 ðA:13Þ r ∂r ∂t cij ∂r
bi cij ¼
qij =qsi n
ð1 ∑ ðqij =qsi ÞÞai
ðA:16Þ
i¼1
where bi is the affinity constant, qsi is the saturation capacity of component i, and qi is the distributed concentration in the micropore. Dual Resistance Model. The dual resistance model is an extension of gas transport in the micropores just described for a bidispersed adsorbent. In some adsorbents such as carbon molecular sieves, gas diffusion in the micropores is controlled by a combination of barrier resistance confined at the micropore mouth and a pore diffusional resistance distributed in the micropore interior.5,15 Assumptions 2 and 3 made for a bidispersed adsorbent were replaced with the following assumptions to accommodate dual resistance. (1) The transport mechanism in the micropores was viewed as a series combination of barrier resistance confined at the micropore mouth followed by the pore diffusional resistance in the interior of the micropores. (2) The micropore mouth was in equilibrium with the macropore gas. (3) The chemical potential gradient was the driving force for diffusion across the micropore mouth and in the micropore interior. The limiting micropore transport parameters were assumed to be increasing functions of adsorbent loading. The equations discussed for the bidispersed model are also applicable for the dual resistance model, but the boundary condition at the micropore mouth (at r = rc) changes to the following equation: im ∂cim 3ðDc0 Þi ∂cij ij ¼ ðkb0 Þi ðq qij Þjr ¼ rc ðA:17aÞ rc ∂r ∂qij i r ¼ rc
According to the study of Huang et al.15 on the diffusion of gases in CMS, the thermodynamically corrected transport coefficients, Dc0 and kb0, were also concentration-dependent according to the following correlations: ! θ Dc0 ¼ Dc0 1 þ βp ðA.17bÞ 1θ 14042
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!
kb0 ¼
θ 1 þ βb 1θ
kb0
ðA.17cÞ
Dc and kb are related to Dc0 and kb0 by the Darken equation and its equivalent for barrier coefficient, which combined with the preceding two equations take the following form: ! Dc θ d ln cim ðA.17dÞ ¼ 1 þ β p 1 θ d ln q Dc0 kb ¼ kb0
! θ d ln cim 1 þ βb 1 θ d ln q
ðA.17eÞ
where the values of βp and βb were obtained by fitting the experimental Dc/D*c0 vs θ and kb/k*b0 vs θ data, respectively. (d ln cim)/ (d ln q) is, therefore, related to the curvature of the equilibrium isotherm. βp and βb were introduced to take into account the effect of the pore size distribution experienced by different adsorbates. The form of concentration dependence mentioned in eqs A.17b and A.17c also ensured Dc0 (or kb0) = Dc0* (or k*b0) as θ f 0. In a subsequent communication, Huang et al.5 proposed the following multicomponent extensions of eqs A.17b and A.17c: ! n θ i ðβp Þi ðA.17f Þ ðDc0 Þi ¼ ðDc0 Þi 1 þ n i¼1 1 θj
∑
∑
j¼1
ðkb0 Þi ¼
ðkb0 Þi
1 þ
θi
n
∑ ðβb Þi
i¼1
1
n
∑
j¼1
θj
! ðA.17gÞ
where θi = qi/qsi and i = 1, 2, 3, ..., n. These correlations were validated with binary integral uptake experiments for oxygennitrogen mixture in BF CMS, methane carbon dioxide mixture in both BF and Takeda CMS, and also ternary uptake of methanecarbon dioxidenitrogen mixture in BF and Takeda CMS. These extensions were based on the assumption that the contributions of components in a multicomponent system are linearly additive. The input parameters were all obtained from unary differential uptake measurements, and no additional fitting parameters were involved when the proposed empirical model was applied to predict binary and ternary integral uptake results. The temperature dependence of the limiting diffusivity can be calculated from the Eyring equation (see eq A.14). Similarly, the temperature dependence of the limiting barrier coefficient can be calculated from the following equation: 0
Eb =Rg T
kb0 ¼ kb0 e
diffusion model when a large value is assigned to the barrier coefficient. Solution Method. All of the equations shown for the model were written in dimensionless form, and the set of partial differential equations was then converted to a set of coupled algebraic (linear and nonlinear) and ordinary differential equations by discretizing all the special variables (dimensionless forms of z, R, and r) using the orthogonal collocation scheme with 15, 5, and 15 internal collocation points along the bed, macropore radius, and micropore radius, respectively. For a detailed description of fixing these collocation points, the readers may consult the dissertation by Bhadra.26 The initial bed condition is known which is normally in equilibrium with the feed mixture at either high or low operating pressure. The algebraic and ordinary differential equations were then solved by using Gear’s variable step integration routine in the FORSIM (1976) integration package to obtain the gas-phase concentration as a function of dimensionless bed length (z/L) and adsorbed phase concentration as a function of both the dimensionless bed length and dimensionless macropore (R/Rp) and micropore (r/rc) particle radius for various values of time. A personal computer with Intel Core 2 CPU 6600 @ 2.40 GHz and 2 GB of RAM was used to solve the bidispersed PSA model which took 410615 CPU minutes to reach cyclic steady state for a set of operating conditions. Cycle steady state was typically reached in about 35 cycles. All of the simulation runs were conducted up to 40 cycles to ensure that cyclic steady state was reached. For each set of operating conditions, the concentration profiles of methane in the gas phase and that in the micropores at a representative location were plotted as functions of dimensionless bed length and dimensionless micropore radius in order to further confirm that cyclic steady state was reached. Numerical simulation of this rather complex model executed using single precision and the mass balance error at cyclic steady state was ∼0.5%. The performance indicators evaluated using the equations given below were calculated using results from the 41st cycle.
¼ f ððcp Þij Þ
volume averaged mole fraction of CH4 in the
ðA:20Þ
0
product recovery ¼ ½ðmoles CH4 from the high-pressure readsorption stepÞ ðmoles of CH4 used in the purge stepÞ =ðmoles of CH4 fed during the pressurization and high-pressure readsorption stepsÞ ! Z tadsorption
¼ ½ Ax εPH
ðA:18Þ
0
Z
diffusion across the barrier where Eb is the activation energy for 0 resistance at pore the mouth and kb0 is preexponential constant. q*i in eq A.17a is the equilibrium adsorbed phase concentration based on micropore volume corresponding to the macropore gas concentration. qi
¼
product purity
product from the high-pressure readsorption step ! Z tadsorption cCH4 Ax εPH V j dt 2 z¼L C 0 z¼L ! ¼ Z tadsorption V2 jz ¼ L dt Ax εPH
tadsorption
Ax εPL GV0 0
Z = Ax ε 0
ðA:19Þ
tpressurization
cCH4 jz ¼ L V2 jz ¼ L dt ! cCH4 jz ¼ L dt
PðtÞcCH4 jz ¼ 0 V1 jz ¼ 0 dt !
þ Ax εPH V0 cCH4 jz ¼ 0 t adsorption
The pore diffusion model is actually an extreme case of dual model. The dual model solution reduces to that of the pore 14043
ðA:21Þ
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¼
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ðvolume of CH4 producedÞ
=½ðvolume of adsorbent used in the bedÞ ðcycle timeÞ 20 ! Z tadsorption cCH4 4 @ ¼ Ax εPH V2 jz ¼ L dt C 0 z¼L 0 !3 Z tadsorption c CH 4 @Ax εPL GV0 dt 5=½Ax Lð1 εÞ ðcycle timeÞ C 0 z¼L
ðA:22Þ where Ax is the cross-sectional area of the bed and L is the length of the bed.
’ AUTHOR INFORMATION Corresponding Author
*Tel.: 65-65166545. Fax: 65-67791936. E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was carried out under Research Project R279000174112 funded by the Ministry of Education, Singapore. ’ NOMENCLATURE a = number of adsorption sites occupied by each molecule in the multisite Langmuir isotherm b = multisite Langmuir constant, cm3/mol b0 = preexponential constant for temperature dependence of b, cm3/mmol c = gas-phase concentration, mol/cm3 cim = imaginary gas-phase concentration, mol/cm3 cp = gas-phase concentration in macropores, mol/cm3 C = total concentration in the gas phase, mol/cm3 dp = particle diameter, cm Dc = micropore diffusivity, cm2/s Dc0 = limiting micropore diffusivity, cm2/s D*c0 = limiting micropore diffusivity in the smallest accessible pore (eq A.17b), cm2/s 0 Dc0 = preexponential constant for temperature dependence of diffusivity, cm2/s DL = axial dispersion, cm2/s Dm = molecular diffusivity, cm2/s Dp = macropore diffusivity, cm2/s Eb = activation energy for diffusion across the barrier resistance at the pore mouth, kcal/mol Ed = activation energy for diffusion in the micropore interior, kcal/mol J = diffusion flux, mol cm2 s1 kb = barrier coefficient, s1 k*b0 = limiting barrier coefficient at the smallest accessible pore (eq A.17c), s1 0 kb0 = preexponential constant for the temperature dependence of the barrier coefficient, s1 kf = fluid-phase mass-transfer coefficient, s1 L = column length, cm M = molecular weight, g/mol P = pressure, atm PH = highest pressure in PSA system, atm Pi = partial pressure of component i, atm PL = lowest pressure in the PSA system, atm
q = adsorbed phase concentration, mol/cm3 qs = monolayer saturation capacity according to multisite Langmuir model, mmol/cm3 qsi = saturation capacity of each adsorbate component according to the multisite Langmuir model, mol/cm3 q* = equilibrium adsorbed phase concentration based on micropore volume corresponding to the macropore gas concentration (eq A.17a), mol/cm3 q = average adsorbate concentration in the micropore, mol/cm3 ¼ q = adsorbed phase concentration averaged over the adsorbent particle, mol/cm3 r = radial distance coordinate of microparticle, cm rc = microparticle radius, cm R = radial distance coordinate in the macropores, cm Rg = universal gas constant, 82.05 cm3 atm mol1 K1; 1.987 cal mol1 K1 Rp = radius of adsorbent particle, cm t = time, s T = temperature, K ΔU = change of internal energy due to adsorption, kcal/mol V = interstitial gas velocity, cm/s z = space dimension, cm 0+, 0 = just outside of column inlet, just inside of column inlet L+, L = just outside of column exit, just inside of column exit Greek Letters
βb, βp = fitting parameters in empirical models defined by eqs A.17b and A.17c ε = bed voidage εp = particle voidage Subscripts and Superscripts
A = component A B = component B b = barrier c = micropore or microparticle f = fluid g = gas constant H = high i = component i (= 1, 2, 3, . . . n for a general n component system; = A and B for a binary system) im = imaginary j = step j (=1 for pressurization, =2 for high-pressure adsorption, =3 for blowdown, and =4 for purge) L = low P = macropore or particle s = saturation t = time x = cross-sectional area of the bed +, = just outlet or just inlet or vice versa
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