Nonisothermal Pore Diffusion Model for a Kinetically Controlled

Article pubs.acs.org/IECR

Nonisothermal Pore Diffusion Model for a Kinetically Controlled Pressure Swing Adsorption Process Mona Khalighi, Shamsuzzaman Farooq,* and Iftekhar A. Karimi Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576 ABSTRACT: A nonisothermal micropore diffusion model has been developed to simulate kinetically controlled pressure swing adsorption (PSA) processes. In this model, a dual-site Langmuir isotherm represents adsorption equilibrium and micropore diffusivity depends on adsorbate concentration in the solid phase according to the chemical potential gradient as the driving force for diffusion. The model has been validated with published experimental data for the kinetically controlled separation of propylene/propane on 4A zeolite. Its performance has also been extensively compared with that of a bilinear driving force (biLDF) model for the same system. The results clearly show that a nonisothermal micropore diffusion model with concentrationdependent diffusivity is comprehensive and complete for kinetically selective systems. The conditions under which the bi-LDF model predictions may significantly deviate from those of the pore diffusion model have also been discussed.

1. INTRODUCTION The separation of olefin/paraffin mixtures resulting from the thermal or catalytic cracking of hydrocarbons is a crucial operation in the petrochemical industry. A practically relevant example is the separation of the propylene/propane mixture, which is of immense economic significance owing to the wide use of separated propylene and propane. A major application of propylene is its use as the monomer feedstock for polypropylene elastomer. On the other hand, propane can be either recycled to the cracking step or used separately for various purposes such as fuel for engines, oxy-gas torches, barbecues, portable stoves, and residential central heating. The conventional method for separating the propylene/ propane mixture is cryogenic distillation.1 However, as the relative volatility of the mixture is close to unity (1.09−1.15), the process requires many (>100) contacting stages and large energy input for maintaining high reflux ratios.2 Cryogenic distillation consumes over 20 GJ of energy per tonne of propylene produced.3 It uses nonrenewable energy resources and emits significant greenhouse gases (GHGs) and criteria air contaminants (CACs). In fact, the U.S. Department of Energy has reported that propylene/propane separation is the most energy-intensive single distillation process practiced commercially.4 Therefore, it is desirable to have economical alternatives for this separation. Some alternative technologies with potential for commercialization are as follows: • absorption/stripping using aqueous silver nitrate solution5,6 • membrane separation7 • pressure swing adsorption (PSA) using zeolite molecular sieves Of the above three, the last (PSA) exhibits high selectivity (separation factor >10),8 and hence has the potential to offer a low energy option. Separation of gaseous mixtures by a PSA process is usually based on three possible separation mechanisms:2,9 equilibrium, kinetic, and steric. Equilibrium separation occurs due to the differing adsorption strengths of component gases. For © 2012 American Chemical Society

example, alumina-rich zeolites (e.g., 5A and 13X) show high equilibrium selectivity for propylene over propane.2,4 Propylene is adsorbed much more strongly than propane due to the electrostatic forces exerted by the exchangeable cations. Kinetic separation relies on the differing adsorption rates of component gases. For example, 4A zeolite is well-known for the separation of the propylene/propane mixture,10,11 where propylene diffuses much faster than propane due to its smaller size. In fact, 4A zeolite has one of the highest kinetic selectivities12 among the available adsorbents in the literature so far. Lastly, steric separation depends on the molecular sieving properties of crystalline microstructures. For example, AlPO4-14,9 an aluminophosphate with variations in crystalline structures, excludes propane due to its molecular shape and size, but allows propylene to adsorb due to its linear shape. Among the above three separation options, the equilibrium selective processes are the easiest to operate; hence, most large-scale adsorption-based separation processes exploit equilibrium selectivity. However, for the propylene/propane mixture, it has been shown in published studies13,14 that kinetic separation based on 4A zeolite is better than equilibrium separation based on 5A or 13X zeolites. Adequate representation of the mass transfer phenomena is essential for modeling a kinetically selective process. Adsorption-based processes involve three mass transfer resistances: external film, macropore, and micropore. Of these, the last (micropore) is usually the most dominant in kinetically selective processes.15 Assumptions of linear driving force (LDF)16 and constant micropore diffusivity17 have been used in the literature to model the intraparticle mass transfer resistance in kinetically controlled PSA separations. Kapoor and Yang16 advocated adjusting a cycle time dependent parameter, Ω, in their LDF rate constant Received: Revised: Accepted: Published: 10659

February 20, 2012 July 2, 2012 July 12, 2012 July 27, 2012 dx.doi.org/10.1021/ie3004539 | Ind. Eng. Chem. Res. 2012, 51, 10659−10670

Industrial & Engineering Chemistry Research

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2. PORE DIFFUSION MODEL A nonisothermal pore diffusion model is developed for a kinetically controlled PSA separation process. It allows for the concentration dependence of adsorbate diffusivity along the micropore radius and makes the following assumptions. 1. The ideal gas law applies. 2. The system is isobaric. 3. An axially dispersed plug flow model describes the flow pattern. 4. The adsorbent consists of uniform microporous crystals. 5. The chemical potential gradient is the driving force for diffusion along the micropore radius. 6. The macropore gas is in equilibrium with the bulk gas in the bed voids. 7. Temperature gradients along the radii of the column and microparticle are negligible. Farooq and Ruthven22 conducted breakthrough experiments in stainless steel columns with and without internal Teflon lining and confirmed that the major heat transfer resistance in the radial direction was at the inner side of the column wall. The radial temperature profiles were measured. Although a radial temperature gradient existed in the column, the inside wall film resistance to heat transfer was more important; hence a simple one-dimensional heat transfer model with a lumped heat transfer coefficient confined at the wall was sufficient to capture the experimentally measured temperature breakthrough behavior. 8. A finite heat transfer rate is introduced between the bulk gas and adsorbent particles. 9. Lumped coefficients account for the heat transfer between the bed and column wall and that between the column wall and external surroundings. Based on the above, the equations describing the PSA process are as follows. The terms with plus-or-minus signs (±) depend on the flow direction. The plus sign (+) applies for flow from z = 0 to L, and the minus sign (−) applies for flow from z = L to 0. mass balance for component i:

expressions to match their experimental results for methane/ carbon dioxide separation on a carbon molecular sieve (CMS). However, they found that their experimental estimates of Ω differed considerably from the predictions of a priori correlations developed by Nakao and Suzuki18 and Raghavan et al.19 These correlations were developed by forcing the LDF model solution to match the solution from the pore diffusion model based on constant diffusivity under different boundary conditions. Shin and Knaebel17 assumed constant diffusivity in their pore diffusion model for producing nitrogen via air separation on the molecular sieve RS-10, a modified form of 4A zeolite. However, the effective constant diffusivity values that gave overall best fits of their experimental PSA performance data over a wide range were different from the actual diffusivity values measured from low-concentration uptake experiments. To overcome the limitations arising from the assumptions of LDF and constant micropore diffusivity, Farooq and Ruthven20 developed a pore diffusion model in which micropore diffusivity varied with adsorbed concentration according to the chemical potential gradient as the driving force for diffusion. They applied their model with concentration-dependent pore diffusivity to simulate high-purity nitrogen production from air on a CMS. While the models of Kapoor and Yang16 and Shin and Knaebel17 applied some degree of data fitting to improve the agreement between experimental and simulation results, the experimental results were predicted reasonably well by the approach of Farooq and Ruthven20 that involved no parameter fitting. It merely used the parameters established from independent unary equilibrium and uptake experiments. Farooq et al.21 further demonstrated the predictive ability of this detailed micropore diffusion model by applying it to the air separation data of Shin and Knaebel17 using independently measured unary equilibrium and kinetic parameters from a different laboratory. The above discussion suggests that a complete and reliable prediction of a kinetically controlled PSA process, using independently measured equilibrium and kinetic data, is possible when the concentration dependence of micropore diffusivity is taken into account in a pore diffusion model. Although Farooq et al.20,21 successfully applied an isothermal model to air separation for nitrogen production on a CMS and modified 4A zeolite, this was possible only because oxygen and nitrogen have modest heats of adsorption in these adsorbents, and the net changes in their loadings during the cyclic operations were small. In order to account for larger heat effects in a kinetically controlled PSA separation, a nonisothermal micropore diffusion model including the concentration dependence of micropore diffusivity according to the chemical potential gradient as the driving force for diffusion is developed in this study, where the equilibrium isotherm is represented by an extended dual-site Langmuir model. The model has been applied to the experimental PSA separation data of the propylene/propane mixture on 4A zeolite reported by Grande and Rodrigues.12 Its performance has also been extensively compared with that of a nonisothermal bi-LDF model that treats the macropore and micropore resistances as separate. The importance of macropore and solid−fluid heat transfer resistances in a kinetically controlled separation is also investigated.

∂(ci) ∂(civ) ∂ ⎛ ∂yi ⎞ 1 − ε ∂qp̅ i =− − DL ρ ⎜C ⎟± ∂t ∂z ⎝ ∂z ⎠ ∂z ε p ∂t

(1)

overall mass balance: ∂qp̅ i ∂(C) ∂(vC) 1−ε ± =− ρp ∑ ∂t ∂z ε ∂t i

(2)

ρp qp̅ i = εpc pi + (1 − εp)ρc qc̅ i

(3)

where v is the interstitial velocity, DL is the axial dispersion coefficient, C is total concentration of the bulk phase, yi is the mole fraction of component i in the bulk gas phase, ci = (pyi)/ (RgTg) is its concentration in the bulk gas phase, p is the gas pressure, Tg is the gas temperature, Rg is the universal gas constant, cpi is the concentration in the macropore gas phase, qp̅ i is the average adsorbed concentration of component i per unit adsorbent particle mass calculated with eq 3,23 qc̅ i is the average adsorbed concentration of component i per unit crystal mass, ε is the bed porosity, and εp is the adsorbent particle porosity. Since the bulk gas is assumed to be in equilibrium with the macropore gas, we set cpi = ci. Note that v is computed from eq 2. The boundary conditions for eqs 1 and 2 depend on the PSA cycle and vary with each step of the cycle. Therefore, they are 10660

dx.doi.org/10.1021/ie3004539 | Ind. Eng. Chem. Res. 2012, 51, 10659−10670


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discussed later for a five-step PSA process used by Grande and Rodrigues.12

wall, Rw is the column (inside) radius, and λ is the axial heat dispersion calculated from the correlation by Wakao.25

crystal balance for component i:

solid phase energy balance:

∂qci

=

∂t

1 ∂ 2 (r Ji ) r 2 ∂r

(∑ cpgiρp qp̅ i + ρs cps)

(4)

i

where r is the radial distance along the crystal, qci is the adsorbed concentration of component i at r, and Ji is the diffusive flux. Using the chemical potential gradient as the driving force for diffusion and defining an imaginary partial pressure of component i, pim i , which is in equilibrium with the adsorbed concentration in the micropore, qci, the following equation is obtained:24 Ji = −Dc0i

∂ ln piim ∂qci ∂ ln qci

∂r

= −Dc0i

= −∑ cpgiTsρp

(5)

=0 (7a) (7b)

where rc is the micropore radius. The temperature dependence of micropore diffusivity follows the Eyring-type form: Dc0i = Dc∞i e−Ei / R gT

(8)

D∞ ci

where Ei is the activation energy of diffusion and is the temperature-independent pre-exponential constant. The average adsorbate accumulation in particles is equal to its accumulation in the macropores and flux into the microparticles: ρp

∂qp̅ i ∂t

= εp

∂c pi ∂t

+ (1 − εp)ρc

im 3Dc0i qci ∂pi rc piim ∂r

r = rc

(9)

gas phase energy balance: cpgC

∂Tg ∂t

=

2 ∂qp̅ i ∂Tg 1−ε λ ∂ Tg + cpgTg − cpgvC ρp ∑ 2 ∂z ∂t ε ∂z ε i

−

∂qc̅ i ∂t (11)

∂Tw ∂ 2Tw = Kw + αwihw (Tg − Tw ) ∂t ∂z 2 (12a)

αwi =

2R w e(2R w + e)

(12b)

αwo =

2(R w + e) e(2R w + e)

(12c)

where cpw and ρw are the specific heat and density of the column wall, respectively. αwi is the ratio of the internal surface area to the volume of the column wall, e is the wall thickness, αwo is the ratio of the external surface area to the volume of the column wall, h0 is the convection heat transfer coefficient between the wall and the surroundings, Kw is the wall conduction heat transfer coefficient, and T∞ is the constant ambient temperature. The wall heat balance given by eq 12a accounts for axial heat conduction along the column wall, heat exchange between the adsorbent bed and the column wall, and that between the column wall and the ambient air. The resistance of the metal wall to radial heat transfer has been neglected, which is a reasonable approximation in comparison to the film resistances on its two sides. Some studies in the literature12 have neglected the contribution of the axial conduction along the column wall, but have used an overall heat transfer coefficient in place of h0, presumably to include the metal wall resistance to radial heat transfer. However, for the propylene/propane system studied later, we find that the overall heat transfer coefficient fitted by Grande and Rodrigues12 to match their experimental observations best was 30−40% higher than the h0 that we fit to match the same observations. If the axial conduction along the column wall is indeed negligible and the purpose of introducing the overall heat transfer coefficient is to account for the metal wall resistance to radial heat transfer, then its value should be lower than that of h0. Thus, the higher value of the overall heat transfer coefficient is a clear indication that it is compensating for the neglected heat loss due to axial conduction. We, therefore, argue that eq 12a is a more appropriate description for the role of the column wall in heat transfer. 2.1. Boundary Conditions for a Five-Step PSA Process. As mentioned earlier, the boundary conditions for eqs 1, 2, 10, and 12a−12c will depend on the type of PSA cycle and its steps. Since, in this study, we will extensively use the data from the experimental study of Grande and Rodrigues12 on a fivestep PSA cycle shown in Figure 1, we present here the boundary conditions applicable for that process only. As shown

with boundary conditions

pim |r = rc = p

i

− αwoh0(Tw − T∞)

(6)

r=0

∑ (−ΔHi)(1 − εp)ρc

where −ΔHi is the isosteric heat of adsorption for component i. Finally, the wall heat balance is given by

In eq 5, Dc0i is the temperature-dependent limiting micropore diffusivity at zero adsorbate concentration. The imaginary gas phase pressure, pim i , can be calculated from an appropriate isotherm model. After substituting eq 5 in eq 4, the microparticle balance becomes

∂pim ∂r

∂t

+

+ a′hf (Tg − Ts)

ρw cp w

qci ∂piim ⎞ ∂qci 1 ∂ ⎛⎜ 2 ⎟ = 2 r D c0i im ∂t ∂r ⎟⎠ pi r ∂r ⎜⎝

∂qp̅ i

i

qci ∂piim piim ∂r

∂Ts ∂t

2hw 1−ε a′hf (Tg − Ts) − (Tg − Tw) ε εR w (10)

where cpg is the molar specific heat capacity of the gas mixture, a′ is the specific surface area of the pellet that is the area to volume ratio, and hf is the film heat transfer coefficient between the gas and the solid phase. Ts is the adsorbent (solid) temperature, Tw is the wall temperature, hw is the film heat transfer coefficient between the adsorption bed and the column 10661



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Kw

∂Tw ∂z

= −K w z=0

∂Tw ∂z

= βh0(Tw − T∞) z=L

(18a,b)

The boundary conditions for velocity in steps 1, 2, 3, 4, and 5, respectively (see Figure 1), are

= −v|z = 0 (yi |z = 0− − yi |z = 0 )

∂z

z=0

∂yi ∂z

(13a)

(15a)

= βh0(Tw − T∞) z=L

(e + R w )2 e(e + 2R w )

z=0

∂Tg ∂z

= z=0

(20)

for pressurization, evacuation, and (21a)

for high pressure adsorption and rinse

Assuming that appropriate isotherm models and adsorption and kinetic data are available for the adsorbed species, this completes our nonisothermal pore diffusion model for a kinetically selective PSA process. Since we will compare its performance with that of a bi-LDF model, we briefly describe the latter for the sake of completeness and highlight its differences.

3. BI-LDF MODEL Grande and Rodrigues12 proposed a bi-LDF model for studying propylene/propane separation using 4A zeolite in a five-step PSA process. While they took into account the bidispersity of a zeolite adsorbent by distinctly treating macropore and micropore resistances, they approximated the diffusive processes in these two resistances using linear models. Since macropore diffusion includes both Knudsen diffusion and molecular diffusion,15 Grande and Rodrigues12 used an effective LDF rate constant (kpi) that combines the effects of transport across the external fluid film and molecular and Knudsen diffusions in the macropores as follows:

(15b)

(15c)

where β is the ratio of the convection area to the conduction area at the column end and we have assumed the convection area as the total cross-section area of the column end. blowdown and evacuation steps: ∂z

(19e)

∑ ci = C ≠ f (t )

(14a)

Tw|z = 0 = Tfeed

=

v|z = L = 0

(21b)

(14b)

∂yi

(19d)

i

=0

β=

v|z = 0 = 0

blowdown

z=L

∂Tw ∂z

(19c)

∑ ci = C = f (t )

(13b)

λ ∂Tg = −Ccpgv|z = 0 (Tg|z = 0− − Tg|z = 0 ) ε ∂z

−K w

v|z = 0 = Gv0

where pI and pII are the initial and final pressures in the blowdown and evacuation steps, and a is computed by fitting eq 20 to the experimental pressure profiles of blowdown and evacuation steps. In eq 2

=0

∂z

(19b)

i

z=L

∂Tg

v|z = 0 = v0

p(t ) = pII + (pI − pII ) exp( −at )

in Figure 1, their PSA cycle consists of pressurization, feed, rinse with pure propylene, blowdown to intermediate pressure, and countercurrent evacuation for bed regeneration, where propylene product is withdrawn. For this particular PSA process, the following boundary conditions apply:26 pressurization, feed, and rinse steps: ∂yi

(19a)

It is assumed that bed pressure remains constant during the adsorption and rinse steps and linearly changes during pressurization. The following exponential form is used to compute the pressure profiles during the blowdown and evacuation steps:20

Figure 1. Schematic of the five-step PSA process including pressure− time history. PR = feed pressurization, HPA = high pressure adsorption, RI = rinse, BD = blowdown, and EV = evacuation.

Dl

v|z = L = 0

∂yi ∂z

(16a,b)

z=L

∂Tg ∂z

∂c pi

=0

∂t

=0 z=L

= k pi(ci − c pi)

k pi = εp (17a,b) 10662

(22)

15Dpi

Bii R p2 Bii + 1

(23a)



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where subscripts “1” and “2” represent the first and second sets of sites in the adsorbent, qsi is the temperature-independent saturation capacity of adsorbate i, and b1i = b01ie−ΔH1i/RgT and b2i = b02ie−ΔH2i/RgT are the isotherm constants with Arrhenius-type temperature dependence. The implicit MSL model requires a nonlinear equation solver to obtain the equilibrium loadings in the solid phase. This computational burden is reduced significantly by using the explicit dual-site Langmuir (DSL) model. The parameters (qs1i, qs2i, b01i, b02i, −ΔH1i, −ΔH2i) for the dual-site models are obtained from independent fits of the single-component equilibrium data to the unary form of the dual-site isotherm model. The fits of the DSL model to the experimental equilibrium data of propylene and propane on 4A zeolite are shown in Figure 2 with fitted parameters in Table 1. As shown in Figure 2, the DSL model provides a good fit. A perfect positive correlation is assumed for the binary prediction.29

R pk f

Bi =

5εpDpi

(23b)

where Bi is the Biot number, which represents the ratio of internal macropore to external film resistances, Rp is the adsorbent particle radius, Dpi is the effective macropore diffusivity corrected for tortuosity for component i, and kf is the mass transfer coefficient across the external film. For the micropores, they used the Darken equation to describe the concentration dependence of micropore diffusivity: ∂qc̅ i ∂t

=

15Dci rc 2

Dci = Dc0i

(qc*i − qc̅ i)

(24a)

∂ ln pi ∂ ln qc̅ i

(24b)

In contrast to the above, our pore diffusion model captures the strong influence of the concentration profiles on the diffusion in the microparticles.20,27 For the dual-site Langmuir isotherm, (∂ ln pi)/(∂ ln qc̅ i) in eq 24b is given by eq A10 in the Appendix. By combining the macropore and micropore resistances, Grande and Rodrigues12 obtained the following bi-LDF rate equation: ρp

∂qp̅ i ∂t

= k pi(ci − c pi) +

15Dci rc 2

(1 − εp)ρc (qc*i − qc̅ i) (25)

where qci* is the equilibrium adsorbate concentration of component i in the micropores corresponding to its concentration in the macropore gas, cpi. The boundary conditions and energy balance equations for bi-LDF model are the same as those for the pore diffusion model described previously. This completes our discussion of the two models (pore diffusion and bi-LDF) for describing kinetically selective PSA processes. In the following, we first validate our pore diffusion model with the experimental data of Grande and Rodrigues,12 and then compare its performance with that of the bi-LDF model. As mentioned before, Grande and Rodrigues12 studied the separation of propylene/propane using 4A zeolite. Therefore, we now identify the appropriate adsorption models and data for simulating this system.

4. PROPYLENE/PROPANE SYSTEM For this study, we assume nitrogen is inert on 4A zeolite. Adsorption equilibrium data, the controlling intraparticle transport mechanism, and the calculation of process performance indicators are discussed here. The external fluid film mass transfer coefficient, axial dispersion coefficient, Knudsen diffusivity, and axial heat dispersion coefficient calculations are detailed in the Appendix. 4.1. Adsorption Data. The experimental equilibrium data for propylene and propane on 4A zeolite have been reported by Grande et al.11 Grande and Rodrigues12 fitted the multisite Langmuir (MSL) isotherm model to these data. In this study, the following dual-site mixture equilibrium isotherm28 is used for both models (pore diffusion and bi-LDF). qs1ib1ipi qs2ib2ipi qc*i = + 1 + ∑j b1jpj 1 + ∑j b2jpj (26)

Figure 2. Experimental data for the adsorption equilibria of (a) propylene and (b) propane are well fitted by the dual-site Langmuir isotherm.

Grande and Rodrigues30 measured the individual transport parameters of propylene and propane on 4A zeolite by three different methods, namely zero length column (ZLC), column breakthrough, and gravimetry. The kinetic parameters obtained from these three techniques were in good agreement. 4.2. Controlling Transport Mechanism. With the above data, we can now confirm that micropore diffusion is indeed the controlling mass transfer mechanism for propylene/propane adsorption on 4A zeolite. Equations 27 and 28 represent the 10663



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Table 1. Parameters of Dual-Site Langmuir Isotherms for Propylene and Propane on 4A Zeolite qs1 (mol/kg)

gas propylene propane

qs2 (mol/kg)

0.7656 1.7527

b01 (kPa−1)

b02 (kPa−1)

−ΔH1 (kJ/mol)

−ΔH2 (kJ/mol)

−9

4.49 × 10−5 0

58.01 16.23

20.39 0

4.55 × 10 4.20 × 10−5

1.1866 0

tional intensity. The partial differential equations that describe the bulk fluid phase profiles are set up in the line geometry, whereas those for the microparticle profiles are set up in the square geometry. Five COMSOL files, representing the pressurization, adsorption, rinse, blowdown, and evacuation steps, are solved and exported as modules to MATLAB to execute the cycling of PSA steps. The cycle simulation sequence is started with the pressurization step. The initial gas phase concentration is assumed to be that of feed gas. Initial solid loading is assumed to be saturated with feed at low pressure. The bed profiles at the end of each step are transferred as the initial conditions for the subsequent step. The cycling is continued until the change in purity is less than 0.01% for five consecutive cycles. In the biLDF model, one-dimensional geometry is sufficient, since it does not allow any radial concentration variations. The number of cycles needed to reach the cyclic steady state (CSS) varied between 30 and 50 cycles. It required 3−4 h of CPU time for the pore model and 1−2 h for the bi-LDF model. The specifications of the computer used have been given at the beginning of this section. 5.1. Accuracy of Mass and Energy Balances. Mass balance and energy balance errors for the two models are computed first before conducting the simulation study. The mean resistance time, t,̅ of the adsorbate is calculated from the mass balance of an initially clean adsorption column where exit concentration and flow rate are monitored after introducing a concentration step change at the inlet.

ratios of diffusional time constants and capacities in the micropores and macropores, respectively:31 γ=

α=

Dc /rc 2 Dp /R p2 1 − εp εp

(27)

K (28)

where K is the dimensionless Henry’s law constant (limP→0[(ρpqp*)/(P/RgT)]). Micropore diffusion is the controlling mechanism for γ(1 + α) < 0.1, macropore diffusivity is controlling for γ(1 + α) > 10, and if 0.1 < γ(1 + α) < 10, both macropore and micropore should be taken into account. For propane and propylene on 4A Zeolite, we find γ(1 + α) < 0.1, which confirms that micropore resistance is the dominant mass transport mechanism in 4A zeolite particles. Thus, our pore diffusion model should be the more accurate model for describing the propylene/propane system. Furthermore, our assumption of macropore gas being in equilibrium with the bulk gas in the interparticle void spaces of the bed is valid. 4.3. Process Performance. For evaluating the performance of the PSA process, we use the purity, recovery, and productivity for propylene as defined by Grande and Rodrigues:12 propylene purity (%) t

=

∫0 evac CC3H6v|z = 0 dt t

t

∫0 evac CC3H6v|z = 0 dt + ∫0 evac CC3H8v|z = 0 dt

·100

propylene recovery (%) t

=

t

t

∫0 presszn CC3H6v|z = 0 dt + ∫0 feed CC3H6v|z = 0 dt

·100 (30)

propylene productivity (mol · h−1· kg −1) t

= 3600

t

∫0 evac CC3H6v|z = 0 dt − ∫0 rinse CC3H6v|z = 0 dt t totalVadsρads

τ

⎛ yv ⎞ L⎡ 1− ⎜⎜1 − e e ⎟⎟ dt = ⎢1 + yf vf ⎠ vf ⎢⎣ ε ⎝

ε ⎛ ρp qp̅ ⎞⎤ ⎟⎥ ⎜ ⎝ c f ⎠⎥⎦

(32)

In eq 32, L is the column length, vf and ve are inlet and outlet velocities, respectively, yf and cf are the mole fraction and feed concentration, ye is the adsorbate mole fraction in the exit stream, and qp̅ is the concentration in the adsorbed phase in equilibrium with cf. The left-hand side of eq 32 is obtained from the simulated breakthrough response, and the right-hand side is calculated from the operating isotherm data. The pore diffusion model has an error of 0.79% compared to 0.82% for the bi-LDF model. Breakthrough in an adiabatic column must also satisfy the following energy balance equation:

t

∫0 evac CC3H6v|z = 0 dt − ∫0 rinse CC3H6v|z = 0 dt

∫0

t̅ =

(29)

(31)

5. MODEL SOLUTIONS A Dell Optiplex 780 with an Intel Core 2 Quad CPU Q9400 @ 2.66 GHz Processor, 8 GB of RAM, is used for numerical simulation. The model equations detailed in section 3 are written in dimensionless form and solved in COMSOL Multiphysics, which uses the finite element method. Two dimensions (axial and radial) are used to implement the pore diffusion model. A square of unit length and width represents the axial direction (z) of the bed and the radial direction (r) of the particles (at each axial position within the bed), respectively. Using a square to represent the concentration profiles within the particle along the bed is helpful to overcoming the need to simulate a full sphere at each position in the bed, simplifying the problem, and reducing computa-

∫0

τ

⎛ T v ⎞ 1−ε L ⎜1 − ∞ ∞ ⎟ d t + ε v0T0 T0v0 ⎠ ⎝ ρp ∑i ( −ΔHi)(q∞ i − qi0) Ccpg

=

⎤ ⎡ L ⎢ 1 − ε ρp ∑i q∞ icpgi 1 − ε ρs cps ⎥ 1+ + v0T0 ⎢⎣ Ccpg ε ε Ccpg ⎥⎦ (T∞ − T0)

(33)

where subscript “∞” represents the final condition of the column (equilibrium condition) and “0” shows the initial 10664



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conditions. The heat balance errors for the bi-LDF and pore models are 1.5% and 0.01%, respectively.

6. BREAKTHROUGH RESULTS To validate our nonisothermal pore diffusion model with experimental data, we now simulate the binary breakthrough experiments reported by Grande and Rodrigues.12 The set of equations describing high pressure adsorption from z = 0 to z = L provides the necessary model for simulating the binary breakthrough experiments. The experiment chosen for this work has the feed composition of 25 mol % propylene, 24 mol % propane, and 51 mol % nitrogen. Feed rate, temperature, and pressure are 1.1 slpm, 423 K, and 250 kPa, respectively. The breakthrough results in Figure 3 clearly illustrate that the binary adsorption and diffusion of propylene and propane on Figure 4. Temperature profiles for the breakthrough experiments at the top (68 cm), middle (43 cm), and bottom (18 cm) of the column. The distances are measured from the feed end. The MSEs for model predictions are 0.059 (top), 0.156 (middle), and 0.207 (bottom) for the pore model and 0.250 (top), 0.829 (middle), and 0.717 (bottom) for the bi-LDF model.

summarized in Table 3. The equilibrium, kinetic, and heat transfer parameters in Tables 1 and 2 are used in the Table 2. Mass and Heat Transport Parameters Used in Simulating the Breakthrough Experiment with Propylene/ Propane Feed at 423 K, 250 kPa, and 7.5 cm/sa parameter Dc0 Dm Dk Dp kf Kg hf λ cp

Figure 3. Experimental measurements and simulated breakthrough responses for propylene and propane at 423 K and 250 kPa. The MSEs for model predictions are 1.05 × 10−4 (C3H6) and 4.81 × 10−5 (C3H8) for the pore model and 2.10 × 10−4 (C3H6) and 3.45 × 10−5 (C3H8) for the bi-LDF model.

4A zeolite are well represented by the pore diffusion model with the parameters estimated from the single component measurements. Moreover, the pore diffusion model has a better match with the experimental data for propylene compared to the bi-LDF model, which is quantitatively supported by a lower mean square error (MSE = 1.05 × 10−4 versus 2.10 × 10−4). It is evident from the almost instantaneous breakthrough of propane that its uptake in the adsorbent micropores in the observed time scale is practically negligible. The roll-up in its experimental breakthrough profile is a typical underdamped response of the flow meter at the column exit. Hence, comparing the mean square errors (MSEs) of the two models for propane breakthrough is not very meaningful. The temperature profiles measured at three different locations along the column length in the same breakthrough experiment are compared with the model predictions in Figure 4. The superior agreements with pore model predictions are even visually evident in addition to the pore model’s significantly lower MSE values (0.059 (top), 0.156 (middle), and 0.207 (bottom) vs 0.250 (top), 0.829 (middle), and 0.717 (bottom) for the bi-LDF model).

C3H6

C3H8 −12

5.50 × 10 0.073 3.078 0.035 3.32 3.9 × 10−4 1.7 × 10−2 1.11 × 10−2 84

unit −14

2.70 × 10 0.069 3.007 0.033 3.16 3.1 × 10−4 1.4 × 10−2 1.04 × 10−2 99

cm2/s cm2/s cm2/s cm2/s cm/s W/cm·K W/cm2·K W/cm·K J/mol·K

Kw = 0.21 W/cm·K; h0 = 2.4 × 10−3 W/cm2·K; hw = 6.0 × 10−3 W/ cm2·K.

a

simulations. The purity, recovery, and productivity of propylene are calculated using eqs 29−31. The measured pressure profiles are appropriately fitted to linear or exponential equations and used as inputs in the simulations for the pressure-changing steps. Representative pressure profiles in a cycle are shown in Figure 5. The experimentally observed effects of the nitrogen mole fraction and feed temperature on the purity and recovery of propylene are compared with two model predictions in Figures 6 and 7, respectively. Representative propylene and propane flow rates measured over a cycle after the cyclic steady state is reached in one experiment from each of the two sets are similarly compared with the model predictions in Figure 8. The total flow rate measured is converted to component flow rates using the measured compositions of these streams. Representative temperature profiles measured over a cycle at three different locations in the column after the cyclic steady state is reached are shown in Figure 9, where the model predications are also included. The overall observation from Figures 6−9 is that both models capture the experimental trends in the range of

7. PSA RESULTS The PSA experiments for propylene/propane separation reported by Grande and Rodrigues12 are simulated using the two models described earlier. The experimental conditions are 10665



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Figure 5. Experimentally measured pressure profiles and their linear or exponential fits used in the simulation. The a values in blowdown and the evacuation step are 6 and 0.15 s−1, respectively. For experimental details, see run 4 in Table 3.

Figure 6. Prediction of effect of nitrogen in the feed on the purity and recovery of propylene compared with experimental results. For experimental conditions, see runs 1−3 in Table 3.

Figure 8. Comparison of experimentally measured molar flow rates with model predictions over a cycle after reaching cyclic steady state. The results are from two different experimental runs: (a) run 6 and (b and c) run 4. For experimental details, see Table 3.

are quantitatively closer to the experimental results than those from the bi-LDF model. A perfect positive correlation has been assumed in this study for binary prediction using the DSL model. DSL constants of propylene (b11, b21) have higher values than DSL constants of propane (b12, b22): b22 = 0 and b11 > b21. As a result, with a perfect negative correlation the propane equilibrium is somewhat higher under binary conditions compared with a perfect positive correlation. For propylene, the effect is negligible on its binary equilibrium for the composition and pressure range covered in this study. The effect of using perfect negative correlations on PSA simulation is also shown in Figure 7. Although the qualitative trends are similar, the predictions

Figure 7. Prediction of effect of feed temperature on the purity and recovery of propylene compared with experimental results. For experimental conditions, see runs 3−5 in Table 3. PN is perfect positive correlation and PP is perfect negative correlation.

operating conditions investigated. In the purity−recovery plots shown in Figures 6 and 7, clearly the pore model predictions 10666



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Figure 9. Temperatures measured at three different locations in the column over a cycle after reaching cyclic steady state in run 4. See Table 3 for experimental details.

Figure 10. Concentration profiles of propylene and propane inside the crystal at z/L = 0.1 at the end of the high pressure adsorption (step 2) and the end of the evacuation (step 5) after reaching cyclic steady state in run 4 as detailed in Table 3.

with perfect negative correlations are quantitatively far removed from the experimental results. The component flow rates over a complete cycle compared in Figure 8 also suggest the marginal quantitative superiority of the pore diffusion model. In the case of the temperature profiles in Figure 9, the pore model also seems closer to the experimental data for the most part, except for the adsorption step where the temperature rise is rapid. It is important to note that, during a rapid change in temperature, the thermocouple readings are affected by their response times and the thermal conduction along the probe wall. In order to investigate the importance of fluid−solid heat transfer resistance, the fluid−solid heat transfer coefficient is varied over 3 orders of magnitude above the value given in Table 3. These perturbations do not affect the purity and recovery results in Figures 7 and 8 or the temperature profiles in Figure 9, which confirms that the adsorbent is, in fact, in thermal equilibrium with the fluid phase. With the exception of high flow rates in adiabatic operations, fluid−solid thermal equilibrium is a widely accepted assumption in adsorption process modeling.32 The small difference between the pore and bi-LDF model predictions for the present system may mislead to the conclusion that the latter model with the concentration dependence of micropore diffusivity accounted by eqs 24a and 24b will always be a good approximation of the more detailed pore diffusion model. A closer look at the representative concentration profiles of propylene and propane along the crystal radius shown in Figure 10 reveals that propane hardly enters the micropores during the cyclic operation. This means that the diffusion of propylene in the micropores is practically like a single-component diffusion.

As pointed out earlier, the pore diffusion model used here captures the strong influence of the concentration profiles of the two components in the microparticle during binary diffusion, which is not captured by eqs 24a and 24b used in the bi-LDF model. In the absence of propane in the micropore, it is therefore not surprising that the two models give such close results. To prove this point further, PSA simulations are carried out for the conditions of run 4 in Table 3 by gradually increasing the diffusivity of propane. The results are shown in Figure 11. Increasing propane diffusivity increases its diffusion into the micropores developing its concentration profile, which was previously absent. Hence, the difference between the two models grow larger, as expected. Figure 12 shows the solid concentration profiles of propylene and propane for all five PSA steps at the cyclic steady state of run 4 in Table 3. During pressurization (Figure 12a,b), propane does not have sufficient time to diffuse into the interior of the crystals. However, propylene, being faster, begins to diffuse into the crystals at the column inlet. Even during high pressure adsorption (Figure 12c,d), propane adsorption is limited to the crystal surface only and it breaks through almost immediately. Propylene does move into the crystal interior, but it does not have enough time to saturate the entire column. Thus, column capacity is not utilized fully, and a better performance may be obtained by increasing the time for adsorption. During rinse (Figure 12e,f), the propane concentration on the crystal surface near the column inlet is almost zero. This is because the pure propylene feed from the column inlet pushes propane out from the other end. Here, propylene gets sufficient time to diffuse into the crystals and it saturates most of the crystals in this step. During blowdown (Figure 12g,h), most of the propane desorbs and comes out from the column outlet. Decrease in the

Table 3. Operating Conditions of the PSA Experiments Taken from Grande and Rodrigues12 a

a

run no.

feed component C3H6/C3H8/N2

Phigh (kPa)

Pinter (kPa)

Plow (kPa)

tpr (s)

tad (s)

tri (s)

tbd (s)

tev (s)

T (K)

1 2 3 4 5 6

0.27/0.23/0.51 0.37/0.31/0.32 0.45/0.41/0.14 0.45/0.41/0.14 0.45/0.41/0.14 0.25/0.24/0.51

500 500 500 500 500 500

50 50 50 50 50 50

10 10 10 10 10 10

54 54 54 54 54 60

100 100 100 100 100 60

25 25 25 25 25 25

40 40 40 40 40 40

180 180 180 180 180 220

433 433 433 408 463 433

L = 87 cm; i.d. = 1.05 cm; rc = 1.9 × 10−4 cm; Rp = 0.08 cm; εp = 0.34; ε = 0.43. 10667



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center in the previous step. Therefore, it is clear that more recoverable propylene remains in the micropores at the end of the evacuation step. The above discussion clearly suggests that the column operating parameters are far from optimal. The process performance can be improved significantly by proper optimization. Figure 12 also gives us some idea of the conditions at the cyclic steady state (CSS). In this study, for the pressurization step in the first cycle of simulations, we assumed a bed saturated with the feed at the low pressure and gas phase at the feed conditions. These seem to be far from what we observe at CSS. It appears that a better initial condition for the pressurization step would be to assume a clean bed for the solid phase and feed gas for the gas phase. Such an initial condition may require fewer simulations to reach CSS. Comparing simulations from the two initial conditions, we observed that the former takes 35 cycles to reach CSS, while the latter takes 25 cycles.

Figure 11. Effect of propylene/propane diffusivity ratio on the purity and recovery predicted by the pore and bi-LDF models. The propane diffusivity was gradually increased, while the propylene diffusivity was held constant. The experimental conditions are same as in run 4 in Table 3.

8. CONCLUSION A nonisothermal micropore diffusion model has been developed for a kinetically controlled PSA separation process that allows for concentration dependence of micropore diffusivity according to the chemical potential gradient as the driving force for diffusion. The model equations have been solved in COMSOL, and the model has been experimentally verified with published separation data for propylene/propane separation on 4A zeolite. In comparison to the bi-LDF model advocated in the literature for this system, the pore diffusion model is quantitatively superior although the difference is not very large for the present system. Further analysis has revealed that the small difference is indeed unique to the propylene/ propane system on 4A zeolite, where propane practically does

pressure also makes propylene move from the interior to the surface of the crystals. Furthermore, propylene is also lost from the column outlet during this step. By the time of countercurrent evacuation (Figure 12i,j), little propane is left in the column. Propylene is withdrawn as the product, but most of it comes out from the region closer to the column inlet only. In other words, the duration is not enough to recover all the propylene. Furthermore, the propylene concentration is nearly zero at the crystal surface, and most of it still exists in the interior. The dimensionless concentration of propylene at the crystal center is near 0.1 and is the same as that at the crystal

Figure 12. Dimensionless adsorbate phase concentrations of propylene and propane in five steps of PSA run 4. 10668



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d ln pi

not enter the micropore due to its very low diffusivity and should not be mistaken as a general rule. Hence, the pore diffusion model developed and validated in this study is more appropriate for screening other potential adsorbents, such as certain eight-ring silica zeolites, ITQ3, ZSM58, SiCHA, and DD3R, reported in the literature for the industrially important propylene/propane separation.

d ln qc̅ i

Dm

= 2.0 + 1.1Re 0.6Sc1/3

B = (1 + bi2Pi + bj2Pj)(1 + bi1Pi + bj1Pj)

C = qsi1bi1(1 + bj1Pj)(1 + bi2Pi + bj2Pj)2 D = qsi2bi2(1 + bj2Pj)(1 + bi1Pi + bj1Pj)2 Nomenclature

(A1)

a′ = specific surface area of the pellet, cm−1 bij = isotherm constant of DSL isotherm for site i and component j, kPa−1 b0ij = pre-exponential constant of DSL isotherm for site i and component j, kPa−1 ci = concentration in the bulk gas phase of component i, mol/cm3 cpi = concentration in the macropore gas phase of component i, mol/cm3 cpg = molar specific heat capacity of the gas mixture, J/g·K cpw = specific heat of the column wall, J/g·K C = total concentration in the bulk gas phase, mol/cm3 dp = particle diameter, cm DL = axial dispersion coefficient, cm2/s Dc0i = temperature-dependent limiting micropore diffusivity, cm2/s D∞ ci = temperature-independent pre-exponential constant, cm2/s Dm = molecular diffusivity, cm2/s Dp = macropore diffusivity, cm2/s e = wall thickness, cm Ei = activation energy of diffusion, J/mol hf = film heat transfer coefficient between the gas and the solid phase, W/cm2·K hw = wall heat transfer coefficient, W/cm2·K h0 = convection heat transfer coefficient between wall and surrounding, W/cm2·K ΔHi = isosteric heat of adsorption for component i, J/mol Ji = diffusive flux, mol/cm2-s kg = gas thermal conductivity, W/g·K kf = external film mass transfer coefficient, cm/s Kw = wall conduction heat transfer coefficient, W/g·K K = dimensionless Henry’s law constant pim i = imaginary partial pressure of component i q*ci = equilibrium adsorbate concentration of component i, mol/g qp* = equilibrium adsorbate concentration, mol/g qsi = temperature-independent saturation capacity of adsorbate i, mol/g qp̅ i = average adsorbed concentration of component i per unit adsorbent particle volume, mol/g qc̅ i = average adsorbed concentration of component i per unit crystal volume, mol/g qci = adsorbed concentration of component i, mol/g rc = micropore radius, cm rp = macropore radius, cm Rg = universal gas constant, J/K·mol Rw = column (inside) radius, cm Rp = adsorbent particle radius, cm

where Dm is the molecular diffusivity, which can be calculated by using the Chapman−Enskog equation.33 Reynolds number: Re =

ρg vd p μg

(A2)

where dp is the particle diameter, ρg is the gas density, μg is the gas viscosity, and v is the velocity. Schmidt number: μg Sc = Dmρg

(A3)

hf is calculated from the Nusselt number: Nu =

hf d p kg

25

= 2.0 + 1.1Re 0.6Pr1/3 (A4)

Prandtl number: μg cpg Pr = kg The Knudsen diffusivity

(A5) 34

D k = 9700rp T /M

is calculated from the following: (cm 2·s−1)

(A6)

where rp is the macropore radius, which we take as 1 × 10 in this work. The macropore diffusivity equation is ⎛ 1 1 1 ⎞ = τ⎜ + ⎟ Dp Dm ⎠ ⎝ Dk

−4

cm

(A7)

where Dp is the macropore diffusivity that combines the contributions from molecular and Knudsen diffusivities and τ is the tortuosity factor. In this study, DL was calculated from the following equation:25 DL =

Dm (20 + 0.5Sc Re) ε

λ is the axial heat dispersion coefficient calculated by λ = kg(7.0 + 0.5Pr Re)

(A10)

A = qsi1bi1(1 + bi2Pi + bj2Pj) + qsi2bi2(1 + bi1Pi + bj1Pj)

■

Sh =

AB C+D

where

APPENDIX The external film mass transfer coefficient, kf, is calculated from the Sherwood number:25 kf dp

=

(A8) 25

(A9)

The micropore concentration dependent expression for DSL isotherm is 10669



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(13) Da Silva, F. A.; Rodrigues, A. E. Adsorption Equilibria and Kinetics for Propylene and Propane over 13X and 4A Zeolite Pellets. Ind. Eng. Chem. Res. 1999, 38 (5), 2051−2057. (14) Grande, C. A.; Poplow, F.; Rodrigues, A. E. Vacuum Pressure Swing Adsorption to Produce Polymer-Grade Propylene. Sep. Sci. Technol. 2010, 45 (9), 1252−1259. (15) Lamia, N.; Wolff, L.; Leflaive, P.; Leinekugel-Le-Cocq, D.; Gomes, P. S.; Grande, C. A.; Rodrigues, A. E. Equilibrium and Fixed Bed Adsorption of 1-Butene, Propylene and Propane Over 13X Zeolite Pellets. Sep. Sci. Technol. 2008, 43 (5), 1124−1156. (16) Kapoor, A.; Yang, R. T. Kinetic separation of methane-carbon dioxide mixture by adsorption on molecular sieve carbon. Chem. Eng. Sci. 1989, 44 (8), 1723−1733. (17) Shin, H. S.; Knaebel, K. S. Pressure Swing Adsorption: An Experimental Study of Diffusion-Induced Separation. AIChE J. 1988, 34 (9), 1409−1416. (18) Nakao, S.; Suzuki, M. Mass transfer coefficient in cyclic adsorption and desorption. J. Chem. Eng. Jpn. 1983, 16, 2. (19) Raghavan, N. S.; Hassan, M. M.; Ruthven, D. M. Numerical simulation of a PSA system using a pore diffusion model. Chem. Eng. Sci. 1986, 41 (11), 2787−2793. (20) Farooq, S.; Ruthven, D. M. Numerical simulation of a kinetically controlled pressure swing adsorption bulk separation process based on a diffusion model. Chem. Eng. Sci. 1991, 46 (9), 2213−2224. (21) Farooq, S.; Rathor, M. N.; Hidajat, K. A predictive model for a kinetically controlled pressure swing adsorption separation process. Chem. Eng. Sci. 1993, 48 (24), 4129−4141. (22) Farooq, S.; Ruthven, D. M. Heat effects in adsorption column dynamics. 2. Experimental validation of the one-dimensional model. Ind. Eng. Chem. Res. 1990, 29 (6), 1084−1090. (23) Qinglin, H.; Farooq, S.; Karimi, I. A. Prediction of binary gas diffusion in carbon molecular sieves at high pressure. AIChE J. 2004, 50 (2), 351−367. (24) Hu, X.; Do, D. D. Multicomponent adsorption kinetics of hydrocarbons onto activated carbon: effect of adsorption equilibrium equations. Chem. Eng. Sci. 1992, 47 (7), 1715−1725. (25) Wakao, N.; Funazkri, T. Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds: Correlation of Sherwood numbers. Chem. Eng. Sci. 1978, 33 (10), 1375−1384. (26) Wehner, J. F.; Wilhelm, R. H. Boundary conditions of flow reactor. Chem. Eng. Sci. 1956, 6 (2), 89−93. (27) Do, D. D. Adsorption Analysis: Equilibria and Kinetics; Imperial College Press: London, 1998. (28) Mathias, P. M.; Kumar, R.; Moyer, J. D.; Schork, J. M.; Srinivasan, S. R.; Auvil, S. R.; Talu, O. Correlation of Multicomponent Gas Adsorption by the Dual-Site Langmuir Model. Application to Nitrogen/Oxygen Adsorption on 5A-Zeolite. Ind. Eng. Chem. Res. 1996, 35 (7), 2477−2483. (29) Ritter, J. A.; Bhadra, S. J.; Ebner, A. D. On the Use of the DualProcess Langmuir Model for Correlating Unary Equilibria and Predicting Mixed-Gas Adsorption Equilibria. Langmuir 2011, 27 (8), 4700−4712. (30) Grande, C. A.; Rodrigues, A. E. Adsorption Kinetics of Propane and Propylene in Zeolite 4A. Chem. Eng. Res. Des. 2004, 82 (12), 1604−1612. (31) Silva, J. C.; Rodrigues, A. E. Analysis of ZLC technique for diffusivity measurements in bidisperse porous adsorbent pellets. Gas Sep. Purif. 1996, 10 (4), 207−224. (32) Suzuki, M. Adsorption Engineering; Chemical Engineering Monographs 25; Elsevier: Tokyo, 1990. (33) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. (34) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley & Sons: New York, 1984.

T∞ = constant ambient temperature, K Tg = temperature of the gas phase, K Ts = adsorbent (solid) temperature, K Tw = wall temperature, K v = velocity, cm/s yi = mole fraction of component i Greek Symbols

αwi = ratio of the internal surface area to the volume of the column wall, cm−1 αwo = ratio of the external surface area to the volume of the column wall, cm−1 β = ratio of the convection area to the conduction area ρg = gas density, g/cm3 ρp = particle adsorbent density, g/cm3 ρc = crystal adsorbent density, g/cm3 ρs = solid density, g/cm3 ρw = density of the column wall, g/cm3 μg = gas viscosity, g/cm·s τ = tortuosity factor λ = axial heat dispersion coefficient, W/cm·K ε = bed porosity εp = adsorbent particle porosity

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +65 6516-6545. Fax: +65 6779-1936. E-mail: chesf@nus. edu.sg. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Eldrige, R. B. Olefin/paraffin separation technology: a review. Ind. Eng. Chem. Res. 1993, 32 (10), 2208−2212. (2) Ruthven, D. M.; Reyes, S. C. Adsorptive separation of light olefins from paraffins. Microporous Mesoporous Mater. 2007, 104 (1− 3), 59−66. (3) Imtex. Olefin/Parafin Separation. In http://www. imtexmembranes.com/greenhouse_gas.php; 2009. (4) Jarvelin, H.; Fair, J. R. Adsorptive separation of propylenepropane mixtures. Ind. Eng. Chem. Res. 1993, 32 (10), 2201−2207. (5) Safarik, D. J.; Eldridge, R. B. Olefin/Paraffin Separations by Reactive Absorption: A Review. Ind. Eng. Chem. Res. 1998, 37 (7), 2571−2581. (6) Bryan, P. F. Removal of Propylene from Fuel-Grade Propane. Sep. Purif. Rev. 2004, 33 (2), 157−182. (7) Stoitsas, K. A.; Gotzias, A.; Kikkinides, E. S.; Steriotis, T. A.; Kanellopoulos, N. K.; Stoukides, M.; Zaspalis, V. T. Porous ceramic membranes for propane−propylene separation via the p-complexation mechanism: unsupported systems. Microporous Mesoporous Mater. 2005, 78, 235−243. (8) Knaebel, S. P.; Ko, D.; Biegler, L. T. Simulation and Optimization of a Pressure Swing Adsorption System: Recovering Hydrogen from Methane. Adsorption 2005, 11, 615−620. (9) Rege, S. U.; Yang, R. T. Propane/propylene separation by pressure swing adsorption: sorbent comparison and multiplicity of cyclic steady states. Chem. Eng. Sci. 2002, 57 (7), 1139−1149. (10) Padin, J.; Rege, S. U.; Yang, R. T.; Cheng, L. S. Molecular sieve sorbents for kinetic separation of propane/propylene. Chem. Eng. Sci. 2000, 55 (20), 4525−4535. (11) Grande, C. A.; Gigola, C.; Rodrigues, A. E. Propane−Propylene Binary Adsorption on Zeolite 4A. Adsorption 2003, 9 (4), 321−329. (12) Grande, C. A.; Rodrigues, A. E. Propane/Propylene Separation by Pressure Swing Adsorption Using Zeolite 4A. Ind. Eng. Chem. Res. 2005, 44 (23), 8815−8829. 10670


Nonisothermal Pore Diffusion Model for a Kinetically Controlled

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